Self-consistent optical constants of SiO2 and Ta2O5 films

Luis V. Rodríguez-de Marcos; Juan I. Larruquert; José A. Méndez; José A. Aznárez

doi:10.1364/OME.6.003622

1. Introduction

Oxides are among the most commonly used materials in optical coatings, and SiO₂ and Ta₂O₅ are among the most common pair of oxides to be combined in a multilayer coating. This is so due to their wide spectral transparency range, to their contrasting refractive indices in the transparency region, and to the stability of multilayer coatings made with these materials. This explains the number of papers providing optical constants of these materials in the visible and adjacent ranges, although over these papers there is a strong dispersion on the reported extinction coefficient k values in the transparency range; k values smaller than ~0.01 are difficult to determine because of the small dependence on them of reflectance, transmittance, and ellipsometry parameters.

There are spectral ranges, including the far and the extreme ultraviolet (FUV-EUV, λ in the 115-200- and 10-115-nm range, respectively), where little or no data are available. The extended transparency of SiO₂ down to the FUV makes it a suitable material for multilayer coatings in applications such as free electron lasers [1] and space instrumentation [2,3]. In some of these applications, dichroic optics may require the availability of materials switching from transparent to reflective or absorptive, so that optical constants in extended ranges are needed. Other than for applications in the FUV or EUV, the latter ranges have to be covered because in order to obtain self-consistent optical constants through the Kramers-Kronig (KK) analysis, one function such as k must be known in the whole spectrum. Experimental data in the FUV-EUV are particularly important because this is a range of strong absorption for most materials in nature, and hence this range has a heavy weight in KK analysis.

Thin films of the present materials can be deposited by different methods, and their optical constants somewhat depend on the specific deposition conditions. Two usual deposition techniques for these materials are electron-beam (e-beam) evaporation and sputtering, and usually the deposition is performed under an oxygen atmosphere in order to get stoichiometric films; in both cases the growing film is provided with extra energy through heating the substrate and/or with ion assistance in order to get compact films. E-beam deposition on hot substrates was selected here. Other techniques like chemical vapor deposition (CVD), atomic layer deposition, pulsed laser deposition, sol gel methods, etc., have been also successfully used to deposit films of these materials.

Literature reports data on optical constants in various spectral ranges for SiO₂ films deposited by different techniques. Reactive e-beam evaporation was used by Zukic et al. [4] (O₂), Thielsch et al. [5] (O₂) and Scherer et al. [6] (H₂O or O₂). E-beam deposition was typically performed on heated substrates. Unspecified e-beam was used by Kolbe et al. [7], Günster et al. [1], Philipp [8], and Cai et al. [9]. Optical constants have been also reported for e-beam films deposited with some ion assistance [1, 5, 10–14], as well as for films deposited by sputtering [1, 7, 11, 12, 15–19], CVD [20, 21], and by heating a Si film in oxidizing atmosphere [22–24]. The above papers report optical constants of SiO₂ films in specific parts of the spectrum and only the works of Phillipp et al. [8], Tarrio and Schnatterly [20], and Filtatova et al. [23, 24] provided self-consistent data that intended to cover the whole spectrum, although details on film deposition are either unknown or far away from standard techniques for multilayer coatings. For films deposited by e-beam on substrates at temperatures close to 573 K, the technique used in this research, we found no FUV or EUV data.

Many investigations were also conducted on the optical constants of Ta₂O₅ films prepared with various techniques, focussed on the visible and adjacent ranges. Reactive e-beam was used by Gürtler et al. [25], Malacara and Baumeister [26], Kruschwitz and Pawlewicz [27], Porqueras et al. [28], Woo et al. [29], Asghar et al. [30], Chandrasekharan et al. [31], and Cheng et al. [32]; deposition was performed on substrates that were either heated and/or post-annealed. Zhang et al. [33] used unspecified e-beam deposition. Optical constants were also reported for e-beam films deposited with some ion assistance [10, 12, 13, 25, 29, 34], as well as for films deposited by sputtering [12, 15, 19, 35–42], atomic layer deposition [43], sol-gel [44], and by other techniques [45–47]. No self-consistent set of optical constant data was found for Ta₂O₅ films. Regarding short wavelengths, we found no optical-constant data for Ta₂O₅ films shortwards of its cut-off at around 260 nm. For this material, optical constants are reported in the literature in its transparency range, i.e., from the near UV to the near infrared (NIR).

After reviewing the above literature, we did not find self-consistent optical–constant data for Ta₂O₅, and we found few sources reporting it for SiO₂, and only for films deposited at room temperature or at a very high temperature in Refs [22, 23], unsuitable for multilayer deposition. The lack of self-consistent optical constants may result in an inaccurate computation of the optical properties of multilayer coatings. This research is devoted to obtain self-consistent optical constants of SiO₂ and Ta₂O₅ films deposited by reactive e-beam onto substrates at 573 K in a broad spectral range. The films were deposited in a reactive O₂ atmosphere. The optical constants were obtained from measurements between the EUV and the NIR ranges, and they were extended to a larger spectrum with literature data and extrapolations, in order to construct wide sets of self-consistent optical constants. Section 2 presents the preparation conditions of the samples and the experimental techniques used to measure them. Section 3 presents the procedure followed to obtain self-consistent optical constants. Section 4 successively presents the experimental measurements and the calculated optical constants of SiO₂ and Ta₂O₅, along with the sum-rule tests for the evaluation of the global and local consistency of the optical constants.

2. Experimental techniques

Thin films of SiO₂ and Ta₂O₅ were prepared using a 3-kW electron beam. Crucibles of alumina and glassy carbon were used for the evaporation of SiO₂ and Ta₂O₅, respectively. Substrates were heated to 573 K during deposition. Various samples were simultaneously prepared on different substrates that were coated in the same run: a c-cut MgF₂ crystal was used for transmittance measurements, a piece cut from a Si wafer was used for ellipsometry measurements; reflectance measurements were made on films deposited on a piece of Si wafer (SiO₂) or on a piece of glass (Ta₂O₅).

For SiO₂, a three-sample set and an additional single sample (a silicon wafer substrate was used for the latter) were prepared with film thicknesses of 56 nm and 125 nm, respectively; deposition rates were 0.43 and 1.71 nm/s, respectively. For Ta₂O₅, a three-sample set was prepared with a common film thickness of 53 nm and a deposition rate of 0.44 nm/s. The crucible-substrate distance was 150 mm; sample centers were within 27 mm over the three samples, so that the thickness difference over them is estimated to be within 1%.

Depositions were performed in a UHV chamber pumped with a cryopump. The base pressure was ~8x10⁻⁷ Pa for the deposition of SiO₂ and 1.2x10⁻⁵ Pa for the deposition of Ta₂O₅; the difference arises in that the chamber had been baked to ~420 K before the deposition of SiO₂, but not before the deposition of Ta₂O₅, since in fact the chamber bake-out was not considered necessary for the deposition of the present materials. During deposition, pure O₂ was let to flow to reach a pressure of 10⁻² Pa, as measured with a cold-cathode gauge. All samples were let to cool down at a rate of ~1 K/min before removing them from the vacuum chamber. Film thicknesses were obtained through stylus profilometry, and from the fit to both grazing-incidence x-ray reflectometry and ellipsometry measurements. The available measurements were averaged to obtain final thicknesses. Thickness uncertainty is estimated to be within ± 2%.

A reflectometer described elsewhere [48] was used to measure transmittance in the FUV and reflectance in the FUV and EUV ranges. It has a grazing-incidence, toroidal-grating monochromator, in which the entrance and exit arms are 146° apart. The monochromator covers the 12.5-200 nm spectral range with two Pt-coated diffraction gratings that operate in the long- (250 l/mm) or in the short-wavelength (950 l/mm) spectral sub-ranges. The reflectometer operates with spectral lines that are generated in a windowless discharge lamp. The lamp is fed with various pure gases or gas mixtures with which the lamp can generate many spectral lines to cover the spectral range of interest. The beam divergence was ~1.7 mrad. The sample holder can fit samples up to 50.8x50.8 mm². A channel electron multiplier with a CsI-coated photocathode was used as the detector. Reflectance (transmittance) was obtained by alternately measuring the incident intensity and the intensity reflected (transmitted) by the sample. FUV-EUV reflectance measurements were performed at 5° from the normal, whereas transmittance was measured at normal incidence. A Lambda-9 and a Lambda-900 Perkin-Elmer double-beam spectrophotometers were used to measure regular transmittance and specular reflectance with the universal reflectance accessory, respectively, in the 190-1500-nm range. For reflectance and transmittance measurements, samples were situated at 8° and 1.5° from normal incidence, respectively. FUV-EUV and long-wavelength reflectance and transmittance uncertainty are estimated to be ~ ± 2%. Ellipsometry measurements in the 190-950 nm range were performed with a SOPRALAB GES5E spectroscopic ellipsometer. The incidence angles at which measurements were performed, 76° and 77° from the normal incidence for SiO₂ and 63° from the normal incidence for Ta₂O₅, were optimized by confining symmetrically around zero the spectral distribution of ellipsometry parameter cos Δ in order to maximize accuracy.

Samples were mostly stored in vacuum until completing all sets of measurements. For ellipsometry and EUV-FUV measurements, samples were exposed to the atmosphere not more than ~1 h. For reflectance measurements in the near UV to the infrared, samples were exposed to the atmosphere for about one day.

Grazing incidence x-ray reflectometry measurements were performed at Centro de Asistencia a la Investigación, Universidad Complutense de Madrid. The diffractometer was a PANalytical X`pert PRO MRD. The source was a Cu anode under 45 kV discharge. The Cu Kα (λ = 0.154 nm) line was selected by means of a graphite monochromator. Measurements were performed at the grazing incidence angles from 0.15° to 2.5°, with a step of 0.005°.

3. Optical constants calculation procedure

The optical constants of the two oxides were calculated following an iterative double KK analysis. This iterative process was repeated twice for each material and the result was a self-consistent {n,k} set that fitted all experimental photometry and ellipsometry measurements. KK analysis was performed both in reflectance and in the optical constants:

φ (E) = - \frac{E}{π} P \int_{0}^{\infty} \frac{\ln [R (E')]}{E'^{2} - E^{2}} d E'

n (E) - 1 = \frac{2}{π} P \int_{0}^{\infty} \frac{E' k (E')}{E'^{2} - E^{2}} d E'

where P stands for the Cauchy principal value. Equations (1) and (2) relate reflectance with its phase, and the extinction coefficient k with the real part of the refractive index n, respectively. R^1/2 and φ stand for the modulus and phase of complex reflectance r = R^1/2exp(iφ). Both integrals are presented in terms of photon energy E instead of wavelength to show them in the most common way. Each KK analysis requires the availability of reflectance or k data over the whole spectrum.

The iterative, double KK process was aimed at solving the problems that can be anticipated: initial iterations with the available measurements were affected by either the contribution of substrate to reflectance in the FUV range (iteration 1) or by the lack of k data in the EUV range (iteration 2), the latter arising from the lack of transparent massive substrates. We note here that to obtain n and k from R and φ, the contribution from the inner interfaces to reflectance must be negligible. With the present procedure, each iteration provided the necessary data to be used in the next one. At the end, a number of four iterative KK processes were enough to obtain a self-consistent data set which sufficiently fitted the experimental photometric and ellipsometry measurements.

The next paragraphs summarize the motivation of the choice of data sources over the various KK analyses; the spectrum is divided into several spectral ranges for KK integration:

i) Between 0.041 nm (3·10⁴ eV) and 30 nm, k and reflectance for an opaque film were obtained from the semiempirical Henke database (downloaded from the Web site of Center for X-Ray Optics (CXRO) at Lawrence Berkeley National Laboratory [49]). k was extended even more up to ~4·10⁵ eV with data obtained from Chantler et al. [50]. To use these databases, densities are required; oxide densities for the present oxide films were determined by grazing incidence x-ray reflectance measurements and are given in sub-section 4.3.
ii) Between 30 nm and 115 nm, k could not be obtained from transmittance measurements, due to the lack of transparent massive substrates in most of the range, so that reflectance measurements were used instead. For this reason, we started our iteration with an R-φ KK analysis. With R-φ data at near-normal incidence we can obtain n and k from the well-known reflectance equation: $r = \frac{(n + i k) - 1}{(n + i k) + 1}$
where r = R^1/2exp(iφ); in Eq. (3) it is assumed that the oxide layer is opaque. By inverting Eq. (3), we immediately get the optical constants:
$n + i k = \frac{1 + r}{1 - r}$
Then we obtain a first set of k (along with n) data, that is introduced in later iterations and is subsequently refined. Regarding reflectance, experimental values for both materials were used in the two R-φ KK analyses.
iii) Between 115 and 190 nm k was calculated from transmittance measurements. In a first iteration, k was obtained using the well-known Beer-Lambert law: $\frac{T_{f s}}{T_{s}} = \exp (- \frac{4 π k z}{λ})$
where T_s and T_fs represent the transmittance of the uncoated MgF₂ substrate and of the substrate coated with the film, respectively; λ is the radiation wavelength in vacuum; z stands for the film thickness. Equation (5) neglects reflectance effects; these effects are somewhat reduced when working with normalized transmittance compared to the bare transmittance. After a preliminary set of optical constants was available, exact transmittance, including internal interferences and reflectance, could be calculated, so that reflectance was not neglected in the next iteration and the simplified view of Eq. (5) was not needed any more. For Ta₂O₅, k obtained from transmittance measurements was strongly dependent on small variations of the refractive index to be determined, even when internal interferences and reflectance were taken into account in iteration 3. To reduce this dependence, in this spectral range we averaged k data obtained either from Eq. (5) [iteration 2] or from exact transmittance calculations including interferences and reflectance [iteration 4] with k data obtained from R-φ KK in the previous iteration; the use of averaged k data resulted in a more satisfactory combined fitting of FUV reflectance and transmittance measurements.
Regarding reflectance, experimental values were used in the first R-φ KK analysis for Ta₂O₅ and SiO₂; in the second R-φ KK analysis, calculated reflectance values using Eq. (3) (with optical constants obtained from iteration 2) were used for SiO₂ and calculated reflectance values using Eq. (3) averaged with experimental reflectance were used for Ta₂O₅. Calculated reflectance of an opaque film was used instead of experimental reflectance since the latter was affected by the substrate contribution, particularly for SiO₂ which turns transparent in this spectral range. As a reminder, to obtain n and k from reflectance using Eq. (4), the contribution from the inner interfaces must be negligible.
iv) Between 190 and 950 nm, ellipsometry measurements were fitted with one (SiO₂) or three (Ta₂O₅) Lorentz oscillators. From the fit, k or reflectance was calculated for both materials, and used in all iterations.
v) Above 950 nm we extrapolated the fitted Lorentz oscillator(s) in the 190-950-nm range, which were used to calculate k or reflectance for both materials.

The sequence of R-φ and n-k KK integrals through successive iterations is explained in the following.

An initial {R₁} set was used for a first R-φ KK integration [Eq. (1)], from which an initial {n₁,k₁} set was obtained using Eq. (4). Then for the first n-k KK integration, a {k₂} data set was gathered, from which a second {n₂,k₂} set was obtained using Eq. (2). Next, a new {R₂} set was gathered for a second R-φ KK integration using Eq. (1), from which a third {n₃,k₃} set was obtained using Eq. (4). Finally, {k₄} data set was gathered and used in a second n-k KK integration, from which a fourth and final {n_final, k_final} set was obtained using Eq. (2). {k₂} and {k_final} data sets included parts of {k₁} and {k₃} data obtained in the previous iteration, respectively, as displayed in Table 1. {R₂} included parts of calculated reflectance with {n₂,k₂}. The above procedures and data sources are summarized in Table 1.

Table 1. Scheme of the procedure and data sources for the iterative KK analysis.

View Table

4. Experimental results

4.1 Optical constants of SiO₂

Figure 1(a) displays the transmittance of the 56-nm thick film normalized to the transmittance of the bare MgF₂ substrate; half of the MgF₂ substrate was left uncoated for this purpose. Figure 1(b) displays the reflectance of the 125-nm thick film deposited on the Si wafer. The 125-nm thick sample was selected to measure reflectance to ensure the opacity of the material in the EUV range. Figure 2 displays both the ellipsometry measurements at 77° on the 56-nm thick film and at 76° on the 125-nm thick film deposited on Si wafers, and the fit performed with a single Lorentz oscillator; the latter provided a precise fit to the experimental data for both films. The oscillator parameters were used in the iterative process described in section 3 to calculate the optical constants in the 190-950-nm spectral range and also at longer wavelengths. Silicon wafer substrates were previously characterized: measurements were performed on wafer spots where no coating was deposited, from which the thickness of the native SiO₂ was obtained; optical constants of Si [51] and SiO₂ [8] were used for modeling the Si wafer and its native oxide.

Fig. 1 Comparison between the experimental measurements and calculations with SiO₂ optical constants obtained in this subsection. a: transmittance (normalized to the transmittance of the bare MgF₂ substrate) of a 56-nm thick film on a MgF₂ substrate. b: reflectance of a 125-nm thick film on a Si substrate. Wavelength is in log scale.

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Fig. 2 Ellipsometry measurements at 77° on a 56-nm thick SiO₂ film (a) and at 76° on a 125-nm thick SiO₂ film (b) deposited on Si wafers, along with the fits performed with a single Lorentz oscillator and calculations with optical constants obtained in this subsection.

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The final self-consistent set of optical constants, referred to as (n_final, k_final), obtained after iteration 4, is displayed in Fig. 3.

Fig. 3 SiO₂ optical constants (a: linear axis; b: log-axis) versus wavelength in log axis.

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With the final optical constants, normalized transmittance, reflectance, and ellipsometry parameters were calculated and are also plotted in the above figures. There is an excellent match between calculation and experimental values.

4.2. Optical constants of Ta₂O₅

Figure 4(a) displays the transmittance of the film normalized to the transmittance of the bare MgF₂ substrate. Figure 4(b) displays the reflectance of the film deposited on a glass substrate. Figure 5 displays both the ellipsometry measurements at 63° on the film deposited on the Si wafer and the fit performed with three Lorentz oscillators. The Si wafer was previously characterized as described above. For this material, a single oscillator did not provide a good fitting of the experimental data, so that the number of oscillators in the model was increased to three.

Fig. 4 Comparison between the experimental measurements and calculations with Ta₂O₅ optical constants obtained in this subsection. a: transmittance (normalized to the transmittance of the bare MgF₂ substrate) of a 53-nm thick film on a MgF₂ substrate. b: reflectance of a 53-nm thick film on a glass substrate. Wavelength is in log scale.

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Fig. 5 Ellipsometry measurements at 63° on a 53-nm thick Ta₂O₅ film deposited on a Si wafer, along with the fit performed with 3 Lorentz oscillators and calculations with the optical constants obtained in this subsection.

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The final self-consistent set of optical constants, referred to as {n_final, k_final}, obtained after iteration 4, is displayed in Fig. 6.

Fig. 6 Ta₂O₅ optical constants (a: linear axis; b: log-axis) versus wavelength in log axis.

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With the final optical constants, normalized transmittance, reflectance, and ellipsometry parameters were calculated and are also plotted in Figs. 4 and 5. We find somewhat larger discrepancies between calculations and experimental data than for SiO₂ films; normalized transmittance presents some deviation above ~270 nm, and calculated ellipsometry parameters are close to the experimental ones but discrepancies could be observed too. Experimental ellipsometry and transmittance measurements seem to be somewhat inconsistent for Ta₂O₅ in a range around 270 nm.

When we compare Ta₂O₅ k in the transparency range with literature data available for films prepared in close conditions [27, 30, 32], it seems that the deposition parameters used might not have been optimal. Unfortunately, a large uncertainty is often obtained when measuring k values that are smaller than ~0.01, as mentioned in the introduction, which complicates this comparison. What is distinct in the present research is that the optical constants are self-consistent and have been obtained in a broad spectrum. Reviewing the data of Cheng et al. [32], k (calculated from the absorption coefficient α) displays a minimum at ~450 nm and it grows towards longer wavelengths, which can be considered unexpected and might be the result of measurement uncertainty; in Ref [32], we interpreted α data in their Fig. 2 by correcting units from cm⁻¹ to nm⁻¹ and we avoided the oscillations originated in interferences by taking middle values through them. In the paper of Asghar et al. [30], they used the flow of O₂ during deposition as a variable and obtain k, along with n, as a function of this parameter; however, for most flows the dependence of k on wavelength is flat, which looks inconsistent. At 550 nm, a usual wavelength to compare optical constants, Asghar data oscillate between 0.0018 (for an O₂ flow of 20 sccm) and 0.022 (for an O₂ flow of 5 sccm), which is over an order of magnitude difference. At the same wavelength, Cheng data amounts to 0.018 and the present data stays at 0.011. In spite of the above wide data differences, the data of Asghar et al. brought them to the conclusion that there is an inverse correlation between oxygen flow during deposition and k. Such an optimization was not performed here, and hence the obtained k data might not be the minimum possible values at the present deposition temperature.

4.3. Optical constant consistency

The consistency of the optical constants was evaluated with sum rules. First we start applying the standard f- and inertial sum rules to the final sets of optical constants in order to obtain a global evaluation of the optical-constant consistency. In a second step, we will apply novel sum rules that can be tuned in the desired spectral range to evaluate the local consistency of optical constants [52].

f-sum rule is expressed by:

n_{e f f} = \frac{4 ε_{0} m}{π N e^{2} ℏ^{2}} \int_{0}^{\infty} E' k (E') d E'

where e and m are the electron charge and mass, respectively, ε₀ is the permittivity of vacuum,

ℏ

is the reduced Planck’s constant, and N is the electron density. Grazing incidence x-ray reflectance measurements at Cu Kα line (λ = 0.154 nm) were performed to obtain density for the Ta₂O₅ and for the 56-nm-thick SiO₂ film samples. Measurements were fitted with IMD software [53], and the density of the samples was obtained from total reflection critical angle. A density value of 2.28 g/cm³ was obtained for SiO₂ which is 85.4% of its bulk density, and a value of 7.72 g/cm³ was obtained for Ta₂O₅, which is 93.2% of its bulk density. f-sum rule indicates that n_eff should be 30, and 186 for SiO₂ and Ta₂O₅, respectively, i.e., the total number of electrons in the molecule. When the relativistic correction on scattering factors is taken into account, n_eff is somewhat modified: the theoretical effective number of electrons is reduced to 29.96 (SiO₂) and 184.32 (Ta₂O₅) [54]. The integral of Eq. (4) with the present k data results in 29.65 for SiO₂ and 187.19 for Ta₂O₅. The deviations are ∼1.0% and ~1.6% for SiO₂ and Ta₂O₅, respectively, which can be considered acceptable numbers.

Another proof of global consistency is obtained through the inertial sum rule:

\int_{0}^{\infty} [n (E) - 1] d E = 0

The following normalization parameter was used to evaluate the fulfillment of Eq. (7) [55]:

ζ = \frac{\int_{0}^{\infty} [n (E) - 1] d E}{\int_{0}^{\infty} | n (E) - 1 | d E}

Shiles et al. [56], proposed that a satisfactory value for ζ should stand within ± 0.005. ζ values of 6.5 × 10⁻⁴ and 1.4 × 10⁻⁵ for SiO₂ and Ta₂O₅, respectively, were obtained with the present n data set, well within Shiles's limit.

Other than the previous global sum rules, a procedure to spectrally tune sum rules to evaluate the local consistency of optical constants has been recently developed [52]. It enables enhancing the weight of a desired spectral range within the sum-rule integral. The procedure consists in multiplying the complex refractive index with an adapted function, which is named window function. In the following we use the window functions proposed in Ref [52]:

H_{1} (E) = \frac{1}{π} [L (E; E_{2}, c) - L (E; E_{1}, c)]

and

H_{2} (E) = \frac{1}{π} [L (E; E_{2}, c_{2}) + L (E; E_{1}, c_{1}) - 2 L (E; \sqrt{E_{1} E_{2}}, \frac{c_{1} + c_{2}}{2})]

with:

L (E; E_{j}, c) \equiv \ln (E_{j}^{2} - E^{2} - i c E) j = 1, 2

where ln stands for natural logarithm, which is understood as the logarithm principal value, i.e., its imaginary part lies in the interval (−π,π]. [E₁,E₂] is the range where the window function has larger values, in order to give more weight to the optical constants in that range in the sum-rule calculation. The shape of H₁ and H₂ can be seen in a plot in Ref [52].

With H₁ and H₂ window functions, the following new sum rules have been used here [52]:

\int_{0}^{\infty} E'^{- 2} Re {H_{2} (E') [N (E') - 1]} d E' = 0

\int_{0}^{\infty} E'^{- 1} Im {H_{2} (E') [N (E') - 1]} d E' = 0

\int_{0}^{\infty} Re {H_{2} (E') [N (E') - 1]} d E' = 0

\int_{0}^{\infty} E' Im {H_{2} (E') [N (E') - 1]} d E' = 0

\int_{0}^{\infty} E'^{2} Re {H_{2} (E') [N (E') - 1]} d E' = 0

\int_{0}^{\infty} E'^{3} Im {H_{1} (E') [N (E') - 1]} d E' = \frac{ℏ^{2} ω_{p}^{2}}{4} (E_{1}^{2} - E_{2}^{2})

Among these sum rules, the last one is a modified version of f-sum rule and all others are akin to the inertial sum rule. ω_p stands for the plasma frequency, with ω_p² = Ne²/mε_o, which depends on electron density N, electron mass and charge, and vacuum permittivity. What is distinct with the new sum rules is that the range of larger contribution to the integral is selected at will through the election of E₁ and E₂.

The new sum rules are applied here by making the window functions to continuously scan the spectrum. Hence we center the window function at a variable energy E_w and we scan E_w over a wide spectral range. In this way, for each E_w we obtain a measurement of the consistency of the optical constants mostly in the spectral range surrounding E_w and we extend this information by scanning E_w over the spectrum. Figure 7 plots ζ vs E_w for the above sets of optical constants of SiO₂ and Ta₂O₅ using the five sorts of inertial-like sum-rules (Eqs. (12) to 16) with window function H₂ and window parameters given by E_w = (E₁E₂)^0.5, E₂/E₁ = 3, and c₁₍₂₎ = E₁₍₂₎/10. A straightforward generalization of Shiles’s evaluation parameter given by Eq. (6) has been used [52]. The figure highlights the limits suggested by Shiles et al. at ± 0.005 [56]. Parameter ζ comfortably stands within these limits in the whole spectral range investigated for all five sum rules, which indicates that the optical constants are locally consistent in this wide spectrum. The exception is sum rule given by Eq. (16) in the limit of high-energy windows, where it diverges.

Fig. 7 ζ versus the central energy E_w for sum rules represented through Eqs. (12) to (16) calculated with H₂ window function and with the optical constants of SiO₂ (a) and of Ta₂O₅ (b). The five inertial-like sum rules are identified in the legend by the power of photon energy in the integral. Window function parameters at each E_w are given by: E_w = (E₁E₂)^0.5, E₂/E₁ = 3, c₁₍₂₎ = E₁₍₂₎/10.

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For the new f-like sum rule, Eq. (17) is rewritten to display it in a manner similar to Eq. (6):

n_{e f f} = \frac{4 ε_{0} m}{N e^{2} ℏ^{2} (E_{1}^{2} - E_{2}^{2})} \int_{0}^{\infty} E'^{3} Im {H_{1} (E') [N (E') - 1]} d E'

Figure 8 plots the new f-like sum rule given by Eq. (18) with H₁ window function applied to the optical constants of both SiO₂ and Ta₂O₅. n_eff is plotted as a function of the central energy of the window function E_w, with the edges satisfying E_w = (E₁E₂)^0.5, E₂/E₁ = 3, and c = E₁/5. The sum of electrons in the molecules, once corrected for relativistic effects, was mentioned to be 29.96 and 184.32 for SiO₂ and Ta₂O₅, respectively. Looking at data plotted in Fig. 8, the deviation from these numbers reaches a maximum of ~1.1% and 1.9% for SiO₂ and Ta₂O₅, respectively, which can be considered satisfactory numbers and are not much larger at any energy than the global ones reported at the beginning of this sub-section. This again indicates that the optical constants are locally consistent in the wide spectrum plotted in the figure. The curves tend to diverge at high energies, which, along with the similar behavior observed in Fig. 7 for sum rule (16), suggests that the extrapolation using data of Chantler et al. [50] and/or the power extrapolation at even larger energies are not fully consistent. This hypothesis is supported by the fact that sum rules corresponding to Eqs. (16) and (18) are strongly biased towards high energies due to the terms E² and E³ in the integrand, respectively. Nevertheless, since this lack of self-consistency is found far away from the spectral range covered in this research, a negligible effect is expected on the latter.

Fig. 8 n_eff versus the central energy E_w for sum rule represented through Eq. (18) calculated with H₁ window function and with the optical constants of SiO₂ (left axis) and of Ta₂O₅ (right axis). Window function parameters at each E_w are given by: E_w = (E₁E₂)^0.5, E₂/E₁ = 3, c = E₁/5.

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In conclusion, both classical f- and inertial-sum rules, along with new sum rules involving window functions, provided a satisfactory evaluation, which suggests a good consistency of the present optical constants of SiO₂ and Ta₂O₅. This good consistency is found both globally and also locally in a broad spectral range.

The optical constants are available upon request at the following e-mail address: j.larruquert@csic.es.

4. Conclusions

Self-consistent optical constants of thin films of SiO₂ and Ta₂O₅ deposited by reactive electron-beam evaporation onto substrates heated at 573 K were obtained in a wide spectral range. The optical constants were obtained from a combination of transmittance (FUV range), reflectance (EUV and FUV ranges), and ellipsometry (from NUV to NIR ranges) measurements, along with extrapolations, from which data sets of k and reflectance were gathered over the whole spectrum. A consistent set of optical constants for each oxide was obtained by means of an iterative double Kramers-Kronig analysis, one relating reflectance with its phase and the other relating k with n. The obtained sets of optical constants enabled us to satisfactorily reproduce experimental reflectance, transmittance, and ellipsometry measurements. The so-obtained sets of optical constants of SiO₂ and Ta₂O₅ thin films can be widely applied in multilayer design for the everyday use of these materials in multilayer coatings. Furthermore, the wide range of optical constants so produced can be useful in applications for which, in addition to a desired performance in the NUV-to NIR ranges, a certain performance in a secondary range, such as in the EUV or FUV ranges, is required.

Satisfactory self-consistency evaluation parameters were obtained for the optical constants of SiO₂ and Ta₂O₅ with the use of sum rules. Successful global consistency of optical-constant data over the spectrum was found through standard f- and inertial sum rules; additionally, successful local consistency at each photon energy range was also found through the use of sum rules with window functions. To the best of our knowledge, these are the first self-consistent sets of optical constants of SiO₂ and Ta₂O₅ thin films deposited onto substrates heated at a temperature close to 573K that involve experimental data from the EUV to the NIR.

Funding

Spanish Programa Estatal de Investigación Científica y Técnica de Excelencia, Secretaría de Estado de Investigación, Desarrollo e Innovación (AYA2013-42590-P).

Acknowledgments

We gratefully acknowledge J. Campos and A. Pons for measurements with the spectrophotometers, and I. Carabias and Centro de Asistencia a la Investigación, Universidad Complutense de Madrid, for performing the grazing incidence x-ray reflectometry measurements.

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Iteration	Spectral range					Produced optical constants
Iteration	λ>950 nm	950-190 nm	190-115 nm	115-30 nm	λ<30 nm	Produced optical constants
1. R-φ KK. {R₁}:	R from Lorentz oscillator(s) extrapolation	R from Lorentz oscillator(s)	R experimental		R CXRO	{n₁, k₁}
*2. k-n* KK.** {k₂}:	k from Lorentz oscillator(s) extrapolation	k from Lorentz oscillator(s)	k from transmittance, neglecting R) (SiO₂)	k from iteration 1 (k₁)	k CXRO	{n₂, k₂}
*2. k-n* KK.** {k₂}:	k from Lorentz oscillator(s) extrapolation	k from Lorentz oscillator(s)	k from iteration 1 (k₁) averaged with k from transmittance, neglecting R (Ta₂O₅)	k from iteration 1 (k₁)	k CXRO	{n₂, k₂}
3. R-φ KK. {R₂}:	R from Lorentz oscillator(s) extrapolation	R from Lorentz oscillators(s)	R calculated from iteration 2 with n₂, k₂ (SiO₂)	R experim-ental	R CXRO	{n₃, k₃}
3. R-φ KK. {R₂}:	R from Lorentz oscillator(s) extrapolation	R from Lorentz oscillators(s)	R experimental averaged with R calculated from iteration 2 with n₂, k₂ (Ta₂O₅)	R experim-ental	R CXRO	{n₃, k₃}
4. k-n KK. {k_final}:	k from Lorentz oscillator(s) extrapolation	k from Lorentz oscillator(s)	k from transmittance. Exact calculations including reflectance and interferences (SiO₂)	k from iteration 3 (k₃)	k CXRO + Chantler et al. [50]	{n_final, k_final}
4. k-n KK. {k_final}:	k from Lorentz oscillator(s) extrapolation	k from Lorentz oscillator(s)	k from iteration 3 (k₃), averaged with k from transmittance. Exact calculations including reflectance and interferences (Ta₂O₅)	k from iteration 3 (k₃)	k CXRO + Chantler et al. [50]	{n_final, k_final}

Self-consistent optical constants of SiO₂ and Ta₂O₅ films

Abstract

1. Introduction

2. Experimental techniques

3. Optical constants calculation procedure