Monitoring the thickness of white Sn thin island films by spectroscopic ellipsometry Open Access

Metallic nanoparticles have been widely studied for their striking electromagnetic properties,¹ as well as applications such as in catalysis² and surface enhanced Raman spectroscopy.³ These applications require well-defined particle features such as size and shape because their physical properties vary strongly with these features. In other situations, such as contamination control in extreme ultraviolet (EUV) optics,^4–12 the quantity of metal on the surface is the main parameter of interest. Accurate control during processing can be obtained by in situ monitoring. This article studies the specific case of metallic nanoparticles on a dielectric substrate, so-called thin island films (TIFs).

In situ spectroscopic ellipsometry (SE) enables noninvasive, fine control over the growth or etching of TIFs by monitoring the sample in real time.¹³ The interpretation of SE data is well-developed for stacks of closed layers and for rough dielectric layers,¹³ but the analysis of SE data for metal island films is more challenging. The reason is that the Bruggeman and Maxwell Garnett models, which provide an effective dielectric function¹³ for layers consisting of mixed materials, do not account for plasmonic interactions^14,15 between islands in a two-dimensional array.

The Bedeaux–Vlieger (BV) model¹⁵ avoids the need for an effective dielectric function altogether by not considering the TIF as a layer in a stack but instead representing it as an excess polarizability at an interface. While this produces highly accurate model spectra and allows extraction of the size, shape, and distribution of individual islands, the downside is incompatibility with real-time monitoring software based on layer-stack formalism. Furthermore, the model is not turn-key but requires a priori input of material parameters and expert decisions on which phenomena and corrections to include and with what assumptions. If many parameters are fitted, the time to search for the global minimum may be longer than desired for real-time control. In some situations, the trade-off to extract fewer parameters with a simpler and more broadly usable model is preferred.

The question arises whether analysis of SE data with a simple effective dielectric function is possible for metal TIFs. This problem has been studied^16,17 for metals such as Au and Ag, but to our knowledge, it is not yet for white tin (Sn) TIFs. These TIFs have attracted interest for plasmonic applications such as the enhancement of infrared absorption,^18,19 as templates for the growth of nanowires,^20–22 and as contaminants in extreme ultraviolet (EUV) lithography tools.^4–12 

Therefore, this article addresses the question of SE data interpretation for Sn TIFs. Specifically, a model of low complexity is desired that can be used to monitor the quantity of Sn on Si, in real time, and using the layer-stack formalism that is commonly applied in commercial SE analysis software. The cited works by Pachecka and co-workers^9–11 directly show the need for such a model to study contaminants on EUV optics. They found a clear correlation between the raw ellipsometric parameter Δ and the quantity of Sn in a TIF but have not yet taken the next step to develop a fit model, instead calibrating Δ directly against the quantity of Sn determined by other methods.

The present work starts from the observation that a Lorentzian dielectric function successfully fits a series of SE spectra acquired during in situ monitoring of the deposition of Sn on Si. This Lorentzian function finds a physical basis, as explained in Sec. II. The fitted thickness of the effective layer increases linearly, in agreement with an observed linear mass increase on a reference microbalance. These observations lead to the question of whether the effective Lorentzian (EL) model can be generally applied to Sn on Si or only under specific conditions. Second, to what extent can the fitted “optical thickness” d_EL be interpreted in order to track the progress of growth, since this is the thickness of an imagined layer? These questions are studied using both experimental and simulated data of TIFs at different stages of growth, all with a known mass of Sn for comparison to the optical thickness.

After discussing the theoretical basis and the experimental approach in Secs. II and III, this article considers the following questions in Secs. IV A to IV D : (1) how does the optical thickness d_EL relate to the mass of Sn in experimental TIFs; (2) how well do simulations match the experimental spectra and d_EL; (3) can the observed linear increase of d_EL during deposition be considered a general feature of Sn TIFs, at all coverages and shapes accessible to the simulation; (4) to what extent can the fit model and, in particular, the optical thickness, be used to monitor the growth of the TIF?

A priori, a metallic TIF can be expected to reflect light very differently compared to a closed metallic film as a result of particle plasmon resonance (PPR). In short, (1) the external field displaces free electrons in each island, polarizing it and (2) a counteracting field (depolarization field) arises in the island due to Coulomb forces. The resulting oscillation^23,24 enhances the polarizability of the island and, thus, affects its optical properties near the resonance frequency.²⁵ 

For spheres of a Drude metal, the polarizability as a function of photon energy shows a Lorentzian peak.¹⁷ If the restoring force on the free electrons is reduced by the growth of the island,²⁵ island-island interactions,^17,26 and/or island-substrate interactions,²⁷ the PPR shifts to lower energy. In the stratified layer limit, the Lorentzian shifts to 0 eV, which means that a regular Drude response is obtained instead of a PPR. Mathematically, the Drude dielectric function equals a Lorentzian function at 0 eV; equations for both functions are given in the supplementary material.

A distinction should be made between the polarizability parallel (γ) and perpendicular (β)¹⁵ to the TIF-substrate interface, if one desires to accurately reproduce the details of the optical spectra. The present work has a different focus, and Sec. II B only describes γ. The PPR in β tends to shift to higher energy as a TIF grows,^17,26,28 which an isotropic EL model (as used here) cannot capture. The interested reader is referred to the cited literature.

By assuming that d_EL/d_m must be real and inserting a Drude function for ε in Eq. (1), oscillator parameters for $ε E L$ are obtained in agreement with the values given by Wormeester and Oates¹⁷ for spheres. The resulting d_EL > d_m. However, Eq. (1) does not in general provide a fixed value for d_EL/d_m and does not hold exactly for real TIFs that interact with neighboring islands and the substrate.

The first goal of this work is, therefore, to test to what extent the proportionality between d_EL and d_m holds for real TIFs of Sn, given the following deviations from the ideal TIF. Actual Sn islands on Si resemble truncated spheres rather than spheres; they may have significant interactions with one another and the substrate; and ε contains interband transitions in addition to the Drude term. A consequence of these deviations is that the equality in Eq. (1) will no longer be exact. The intention of the EL model is to describe the dominant PPR in the ellipsometric spectra in order to monitor a deposition. The reproduction of fine effects in the spectra is out of the scope of this work.

A second goal of this work is to study the ratio d_EL/d_m as a function of the island shape and coverage. When a TIF transitions toward the stratified layer limit, as explained above, $ε E L$ transitions toward $ε$⁠. Furthermore, L decreases because the depolarization field in the islands decreases^15,17 when the growing islands approach one another. Inserting L = 0 (stratified layer limit) and $ε E L=ε$ into Eq. (1) yields d_EL = d_m. Therefore, a gradual transition from d_EL/d_m > 1 toward d_EL/d_m = 1 might be expected during growth.

This study applies the theory of Bedeaux and Vlieger¹⁵ (“BV theory”) to simulate SE data of Sn TIFs with known d_m. The simpler effective Lorentzian (EL) model is fitted to the SE data (experimental and simulated from BV) to determine d_EL. The BV model is more sophisticated than the EL model and reproduces effects such as the splitting of the PPR energy but because its sophistication is not always chosen for data analysis, particularly, when the purpose is process control as opposed to detailed study of optical properties. Details of the two models can be found in the supplementary material.

White Sn (Advent Research Materials, 99.95%) was deposited on Si(100) by evaporation from an effusion cell (Prevac EF 40C1, source at 980 °C, 0.52 nm/min growth as measured by quartz balance) in the setup described by Pachecka and co-workers.^9,11 The base pressure of the chamber was 5 × 10⁻⁶ Pa. In total, 25 samples were coated, varying the duration of deposition (and thus d_m) and the surface of the substrate (three types: 1.5 nm native oxide on Si, 17 nm thermally grown oxide, or hydrogen-terminated Si after treatment with 1% HF prior to deposition). Ellipsometric spectra (J. A. Woollam Co. M2000) were acquired in situ during deposition (single location on the sample, 75° or 80° angle of incidence, 13 spectra/min) and ex situ after deposition (mapping of 25 locations on a wafer). Ex situ measurements and simulations were made at 50°, 75°, and 80° to match the angles for the in situ ports in our etching and deposition chambers. All angles were included in a single fit per sample or simulation. Samples were further studied by ex situ x-ray photoelectron spectroscopy (XPS, Physical Electronics Quantera SXM, Al Kα source), x-ray fluorescence [XRF, Bruker S8 Tiger, Sn Lα1 with Rh anode and LiF(200) crystal], and high-resolution scanning electron microscopy (HR-SEM, Zeiss Merlin, detection of secondary and backscattered electrons).

Figure 2 shows an experimental and a simulated spectrum, with matching R, Θ, and C, as determined by scanning electron microscopy (SEM) for the experimental samples. Despite the observed polydispersity in these parameters for the experimental samples, the SE spectra show appreciable agreement with the simulations. Furthermore, the EL model fits the spectra and determines optical thicknesses 12.4 ± 0.2 nm (experimental), whereas the same model fit on the simulated spectrum yields 16.3 ± 0.5 nm. The simulations are further discussed and compared to the experimental results in Secs. IV B–IV D.

To further investigate the relation between d_EL and d_m, 35 samples were produced and characterized ex situ by XRF and SE. The duration of the deposition of Sn, and the surface condition of the Si, were varied. Six samples were exposed to H atoms in a different vacuum chamber after deposition, to test whether the etching and potential change in morphology²² of the Sn affect the SE analysis.

The results in Fig. 3 support a direct proportionality between d_EL and d_m. Therefore, the progress of a deposition can be followed for these samples by SE. The fitted values for d_EL/d_m depend on the surface of the silicon substrate: 2.33 ± 0.11 (thermal SiO₂), 1.89 ± 0.12 (native oxide), and 1.6 ± 0.2 (hydrogen termination). The results for H-exposed TIFs fall within the trends, suggesting that in situ SE could also be used to study the etching of the Sn by H atoms.

To conclude, a linear relationship between d_EL and d_m is found for the studied experimental samples. The slope d_EL/d_m varies with the condition of the Si surface. The next questions of interest are whether the proportionality is more generally valid for Sn TIFs, and what determines the slope. These questions will be addressed using simulated spectra, which are introduced in Sec. IV B.

In order to conduct a more generalized investigation of the relationship between d_EL and d_m by simulations, it is desired to validate the simulations by comparison to experimental data. GranFilm software³¹ is used to compute SE spectra for monodisperse Sn TIFs of predefined geometries (R, Θ, and C) and thus known d_m. Subsequently, the simpler and more approximate EL model is fitted to the simulated data to determine d_EL.

As illustrated by Fig. 2(a), the simulated spectra have a comparable shape but a slightly lower Δ than the experimental spectra with the same geometrical parameters. An interpretation of a lower Δ is that a larger quantity of Sn is present on the surface.¹¹ Indeed, fits of the simulated data find a higher d_EL for a given d_m. In Fig. 2(a), d_EL/d_m amounts to 2.1 for the experiment and 2.7 for the simulation. A possible explanation is that oxidation of a Sn island reduces the total number of free electrons and may thus be expected to reduce d_EL, whereas d_m from XRF also counts Sn atoms in the oxide shell. The simulations do not account for oxidation. Based on XPS (see supplementary material for further details), we estimate that 17% of all Sn atoms on the sample surface reside in an oxide shell, which is the right order of magnitude to explain the difference between the simulated and experimental d_EL/d_m for Sn on Si:H [e.g., Fig. 2(a)]. However, the difference is also observed in the vacuum chamber (in situ SE) when oxidation is expected to be limited. An alternative explanation is that the experimental sample has shorter island-island distances as a result of polydispersity, which should weaken the depolarization field in the islands and decrease d_EL/d_m.

In conclusion, a lower Δ in the simulated spectra leads to a higher d_EL/d_m compared to experimental data for which oxidation and polydispersity provide plausible explanations. Despite this quantitative difference, the simulations allow further investigation of the trend in d_EL/d_m with varying TIF configurations.

This section describes a more generalized investigation of d_EL/d_m using simulations. Figure 4 shows the results of six series of spectra simulated with the BV model and fitted with the EL model. Within each set, Θ and C are kept constant. An increase in d_m [due to increasing radius R; see Eq. (2)] leads to a proportional increase in d_EL and a linear fit through the origin determines d_EL/d_m. The key observation is that the optical thickness rises with the mass thickness within each series but at a different rate d_EL/d_m depending on Θ and C.

The simulations support the expectation from the theory that d_EL/d_m > 1 and that this ratio declines with increasing coverage but does not provide a general rule to quantitatively determine d_m at any Θ and C. Furthermore, minor deviations from linearity are observed within the series. These may be attributed to the increasing difficulty of the EL model to describe the spectra, in which higher order effects may appear as the island-island distance decreases. Simulations for C > 0.5 and Θ < 0 tended to converge poorly and are not included in this work.

Figure 5 further generalizes the results in Fig. 4 to the full range of combinations of (Θ, C) that could be simulated using the approach described in the supplementary material. The key trends are that the ratio d_EL/d_m consistently increases with truncation Θ and decreases with increasing coverage C. The first trend matches the experimental finding in Fig. 3 that d_EL/d_m is larger for spherical Sn islands on silicon dioxide (Θ = 0.8 ± 0.3 for native oxide and Θ = 0.7 ± 0.2 for thermal oxide) than for hemispherical islands on hydrogen-terminated silicon (Θ = 0.3 ± 0.4). The second trend is consistent with the expectation that d_EL/d_m should decrease with increasing island-island interactions. This effect is not identified in Fig. 3 because it is relatively small compared to the error bars in the experimental data.

To conclude, the experimentally observed linear increase in d_EL during deposition of Sn on hydrogenated Si cannot be generally expected to hold. Island truncation most strongly affects d_EL/d_m but may be assumed constant during processes that do not affect the wetting angle of the islands on the substrate. The island coverage has a minor effect on d_EL/d_m so that some degree of deviation from linearity can occur during a deposition or etch.

The next step is to evaluate to what extent the SE growth curve deviates from linearity during growth. For this, a growth scenario must be assumed that describes the evolution of C and Θ with time. The scenario shown in Fig. 6 assumes that a given number of islands (2.5 × 10¹⁰ per cm²) grows with a constant increase in island mass per unit time t. Then, the island radius grows with t^1/3 and the coverage with t^2/3. The truncation Θ is assumed constant. The growth rate is matched to the experimental rate.

The results in Fig. 6 confirm that island-island interactions diminish d_EL/d_m as the growth proceeds. This effect is consistent with the expectation that d_EL/d_m should eventually converge to 1 (the value for a closed layer) due to the decline of the depolarization field in the islands. For hemispherical islands, the ratio plateaus between 4 and 16 min (1.8–7.2 nm mass thickness). Consequently, d_EL can be used to monitor the film growth in this region with a minor systematic error of up to 10% if d_EL/d_m is not known.

For TIFs at higher Θ (i.e., nearly full spheres), the absolute value of d_EL/d_m is higher and the downward trend during growth is more pronounced. This should translate to a noticeably nonlinear increase in d_EL during growth on poorly wetted substrates. Indeed, in Fig. 7, the experimental evolution of d_EL during growth on a silicon substrate with its native oxide (Θ = 0.8, from SEM) is markedly nonlinear compared to growth on a hydrogenated silicon surface. Hence, the study of d_EL can provide in situ information on the wetting of the substrate by Sn.

If d_m can be independently determined, for instance with a quartz microbalance, the combined knowledge of d_EL and d_m can provide further insight into the growth process. For instance, a comparison with Fig. 6 may indicate whether the assumed scenario appropriately describes the growth process; and if not, an alternative scenario may be proposed by reference to Fig. 5.

In conclusion, the scenarios in Fig. 6 clearly show that a linear dependence between d_EL and d_m cannot be expected. However, the deviations from linearity can be acceptably small in a production environment with well-controlled process conditions so that Θ does not vary. In such conditions, d_EL can be used as a direct indicator of the amount of Sn deposited.

In a research environment, with more widely varying process conditions, Θ cannot be assumed constant. As a consequence, d_EL/d_m varies strongly with Θ and may be used as an indicator for this parameter. This requires independent knowledge of the deposition rate (e.g., using a microbalance or a known deposition rate from ex situ measurements). Although outside the scope of this work, offline analysis may yield additional information from the SE data directly, using more complex fit models in combination with a priori known material parameters.

The conclusions from this study are likely applicable to TIFs of other metals for which the optical constants are dominated by the Drude component, such as Pt and Fe. Here, the method applied to Sn provides a basis for further studies into the interpretation of SE spectra for TIFs of other conductors or on different substrates.

The spectroscopic ellipsometry data on thin island films of white Sn fit well when modeled as an effective continuous tin layer with a Lorentzian dielectric function. Both experiments and simulations show that the optical thickness for Sn on hydrogenated Si is linearly proportional to the mass thickness of Sn to good approximation. For example, the ratio between the two thicknesses does not deviate by more than 10% between 4–16 min in Fig. 6. For Sn on oxide, the optical thickness is a more qualitative indicator of growth, although with relatively large error bars (Fig. 3) may appear proportional to mass thickness as well.

This work takes a novel approach to study the interpretation of the SE data, and, in particular, the fitted optical thickness during a deposition, by validating the EL fit model on simulated data. The use of GranFilm software allows the simulation of a wide variety of particle shapes and coverages with known mass thickness. Although the simulated spectra do not account for polydispersity and do not quantitatively reproduce Δ, the fitted simulations do reproduce the trend with which d_EL evolves in an appreciable experimental window. The ratio d_EL/d_m > 1 and increases strongly with increasing truncation parameters. The ratio declines weakly with increasing island coverage, which the simulations show can be attributed to island-island interactions, as expected since d_EL/d_m for a fully closed layer should be unity.

The simulations support the experimental findings that the optical thickness linearly increases during the growth of hemispherical island films (Sn on Si:H) but not for spherical islands (Sn on oxide). These results also show that the degree of linearity can provide an indication of the particle shape. For spherical islands the ratio d_EL/d_m is high and decays significantly during growth, whereas for hemispherical islands, the ratio starts lower and decays less strongly, leading to a nearly linear evolution of d_EL during deposition.

In conclusion, while earlier studies monitoring the deposition and etching of Sn used the raw ellipsometric Δ value compared to a calibration, our model-based approach enables real-time fitting of the data to extract a measure for the quantity of Sn on the surface.

The online supplementary material gives details for the simulations in GranFilm, method and accuracy of the EL fits, reproducibility of the ellipsometry data, the effect of angle of incidence on the EL fits, the characterization of TIFs by SEM, and the determination of the oxide shell thickness by XPS as well as simulations of these XPS spectra.

The authors thank Małgorzata Pachecka, Herbert Wormeester, Stefan Kooij, and Dirk Gravesteijn for discussions, and the staff of the XUV Optics group at the University of Twente for the use of the Sn deposition setup. Mark Smithers, Gerard Kip, and Tom Velthuizen are acknowledged for SEM, XPS, and XRF measurements. Tom Aarnink is thanked for exposing samples to atomic hydrogen. This work was supported by the Netherlands Organization for Scientific Research, Domain Applied and Engineering Sciences (NWO TTW Project No. 13900) and by ASML.

The authors have no conflict of interest to disclose.

A. J. Onnink: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). J. Schmitz: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). B. Terbonssen: Data curation (equal); Investigation (equal). A. Y. Kovalgin: Conceptualization (equal); Funding acquisition (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.