1. Introduction

In science and technology, the characterization of very thin layers is a recurrent topic. To list just a few fields: water purification, anti-reflective coatings, biosensors, or questions connected to immunology. For instance, the height of a layer consisting of flatly adsorbed antibodies like immunoglobulin G is about 3 nm in air (IgG has a Y-shape, dimension in water are height = 14.5 nm, width = 8.5 nm, thickness = 4.0 nm) [1]. However, if the antibodies bind to antigens and reorient, then they form a 11 nm high monolayer. Another example concerns polyelectrolytes which adsorb from aqueous solution onto solid substrates and form either patches or homogeneous films. If they form patches the average index of refraction of the film is not the one of a homogeneous film, but between the one of a homogeneous film and the surrounding medium. In general, the properties of films can be tuned by an accurate control of the film thickness, including its optical properties and its mechanical strength.

Among the techniques which are most suited to analyse thin films on substrates are quartz crystal microbalance (QCM), surface plasmon resonance (SPR) and ellipsometry. QCM measures the mass increase of a rigid film using the Sauerbrey equation [2]. However, in case of films which are soft compared to their substrate or immersed in solution both the mass of the adsorbed molecules and the mass of immobilised solution molecules contribute to the mass increase. Thus, the surface coverage (mg/m²) cannot be determined directly. For data interpretation, QCM with dissipation (QCM-D) is used. The film thickness is determined using viscoelastic models, like the Kevin-Voigt model or the Voinova model [3], SPR measures the optical thickness, i.e. the product of film thickness and local index of refraction. Data interpretation is based on the Fresnel formulas combined with those of the plasmons [4].

Ellipsometry is an optical technique. Measurements and data analysis are based on classical optics. This scientific field started about 200 years ago, with the work of Augustin Jean Fresnel (1788–1827). His experiments led him to use the wave theory of light and abandon the particle theory. He derived his now famous formulas for the amplitude of the reflected and transmitted light in the early nineteenth century [5]. The empirical knowledge on light and electromagnetism obtained by Fresnel and many others was summarized and extended a generation later by James Clark Maxwell (1839–1879) in a single set of equations. He could show, purely theoretically, that light could propagate as a transverse wave. The existence of electromagnetic waves with long wavelengths was verified by Heinrich R. Hertz (1857–1894) by generating and detecting them in a series of experiments. Sir George Biddell Airy (1801–1892) considered layered systems [6]. He used Fresnel’s reflection and transmission coefficients to calculate the superposition of light waves, which are reflected and transmitted at the boundaries of a thin film.

“The Theory of Optics” [7] by Paul K. L. Drude (1863–1906) was the first book which “attempts a complete development of the electromagnetic theory with all its bearings” (Robert A. Millikan in the preface of the English edition). Drude observed that the measured reflection coefficients of bare substrates “can better be represented if a transition layer is taken into account”. He based his calculations on the assumption [7] that “the thickness of this transition layer is so small that all terms of higher order than the first in thickness may be neglected.”

Today, Drude’s transition layer is the thin film under investigation. Ellipsometry delivers direct experimental access to both amplitude and phase information of the reflected light. Thus, the investigated films may be significantly smaller than the wavelength of the light used. The variables which are directly measured by ellipsometry are the two ellipsometric angles Δ and Ψ. These angles are related to the ellipsometric ratio $\rho =r_p/r_s$ via

Although Δ and Ψ, and thus ρ, are experimentally immediately accessible, they are also abstract. They cannot be converted directly into film properties, such as thickness and refractive indices of layered media on a surface. The physically relevant parameters are typically determined numerically. For this purpose, it is common practice to (i) predefine a slab model of the refractive index profile perpendicular to the sample surface, (ii) calculate the associated ellipsometric angles using Fresnel's and Airy's formulas [6,8] and (iii) vary the parameters of the slab model until the calculated angles match the measured ones. Naively, the conversion of two independently measured variables (Δ and Ψ) into two unknown parameters (thickness d_F and refractive index n_F of the surface film) should be simple. However, the simultaneous determination of both film parameters is only possible in a straightforward way, if the film thickness is sufficiently large [9–11]. A threshold value of 15 nm is reasonable. Below this threshold the film thickness and refractive index are coupled parameters and cannot be determined independently.

The dilemma of coupled film parameters is illustrated in Fig. 1(a). Shown is the squared deviation χ² between measured and calculated ellipsometric angles for a silicon wafer with a native oxide film. The actual film parameters are n_F = 1.457 and d_F = 1.6 nm. Yet all pairs (n_F, d_F) with low χ² (χ² < 0.01, i.e. blue region) are equally suited to match the measured ellipsometric angles. Additionally, in this case small experimental errors (which are inevitable) have a particularly strong effect on the measured ellipsometric angles [12]. As a consequence, the resulting values are subject to a considerable systematic error for both film thickness and refractive index. This problem is solved, if one of the two film parameters (refractive index or thickness) is independently determined by another method or is known a priori. Under these conditions, the other unknown parameter can be determined by ellipsometry with high accuracy. However, this is only a special case, not the general case.

 

Fig. 1 Color-coded representation of the squared deviation χ² between measured and calculated ellipsometric angles. The investigated sample is a silicon wafer with native oxide film. The actual film properties are d_F = 1.6 nm and n_F = 1.457. (a) Model parameter for the calculation of ellipsometric angles are d_F and n_F. The solid white line corresponds to all pairs (d_F, n_F), for which the first ellipsometric moment is constant E₁ = 11 nm. (b) Model parameters are the first and second ellipsometric moment, E₁ and E₂ defined in Eq. (4). The position of the minimum is E₁ = 11 nm and E₂ = 575 nm, respectively (white point). The white ellipse corresponds to all pairs (E₁, E₂), which lead to the constant deviation χ² = 0.008.

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To overcome this limitation, we exploit Drude’s approach to calculate the complex reflection coefficients from electromagnetic theory. This approach allows determining those two film quantities E₁ and E₂ on which ellipsometry delivers independent information. Due to this decisive property we like to denote these quantities as first and second ellipsometric moments. A thin film approximation reveals a new technique to quantify thickness and index of refraction of a non-absorbing nm-thick layer via ellipsometry. This technique amounts to the plotting of the real part of the ellipsometric ratio against the square of the imaginary part for a series of samples that can be as small as the uncoated substrate and one value of film thickness. The slope yields the refractive index while the interval between the points gives the thickness.

To demonstrate the wide applicability of our approach, we characterize nm-thick silicon oxide films, observe the growth of a polyelectrolyte multilayer, and quantify a parameter map obtained by imaging ellipsometry.

2. Equations

First, for the sake of simplicity, substrate and ambient media are separated by only one single homogenous film with negligible surface roughness. Thus, the interfacial film is completely characterized by its thickness d_F and its refractive index n_F. Second, in order to consider the influence of this interfacial film on the ellipsometric ratio $\rho =r_p/r_s$, we use Drude's reflection coefficients [7] (listed in Eq. (36) on page 290 in Drude's 1902 book [7])

In his 1902 textbook Drude showed that Eq. (2) offers an alternative to Fresnel's and Airy's formulas, as long as the parallel component of the electric field (E-field) is approximately constant (i) at the interface ambient/film, (ii) within the film, and (iii) at the interface film/substrate. Also the normal component of the electric displacement field (D-field) at these positions across the film needs to be approximately constant [7]. These continuity conditions are met exactly in the absence of the interfacial film. In fact, in the limiting case of a pristine ambient/substrate interface (d_F → 0) the imaginary terms of Eq. (2) vanish leaving the Fresnel reflection coefficients of the interface ambient/substrate. However, with increasing film thickness these components of E- and D-field at the ambient/film and the film/substrate interface start to differ due to a slight phase shift of the light wave traversing the film. Yet, the continuity conditions are still approximately satisfied as long as the film thickness is small compared to the wavelength ($2\pi d_F/\lambda <<1$). Hence, Eq. (2) is a thin film approximation.

For use in ellipsometry, not the reflection coefficients themselves are relevant, but only their ratio. Therefore, it is obvious to use Eq. (2) to express ρ = r_p / r_s in a power series expansion of the relative film thickness 2πd_F/λ. Already Drude did this to the first order of $2\pi d_F/\lambda$, resulting in a good approximation for films thinner than 5 nm (listed as Eq. (48) on page 292 in Drude’s 1902 book [7]). Drude's 1st-order approximation had since found numerous scientific applications in the fields of native oxide layers and lipid monolayers [13–17].

However, for films with a thickness greater than 5 nm a series expansion of higher order is required to obtain a meaningful thin film approximation. Using only elementary trigonometric relationships, Snell's law n₁sinα_inc = n₂sinα_tra as well as Eq. (2) one can show the validity of the approximation in 2nd order of $2\pi d_F/\lambda$

The constant term ρ₀ describes the ellipsometric ratio of the ambient/substrate interface and can be calculated with Fresnel's reflection coefficients. The linear coefficient ${\rho }^{\prime }$ and the two quadratic coefficients ${{\rho }^{″}}_A$ and ${{\rho }^{″}}_B$ are given by

Figure 2(a) shows the comparison between experimental ellipsometric data, measured on oxidized silicon wafers with variable film thickness, and the various theoretical models: Airy's formula serves as reference and describes the measured trajectory of ρ as a function of d_F in the entire range studied. The 2nd-order approximation (red solid line) is in good agreement with measured values up to d_F = 30 nm. In contrast, Drude's 1st-oder approximation (blue solid line) deviates significantly from the experiment for d_F > 5 nm.

 

Fig. 2 Influence of film thickness d_F on ellipsometric angles in case of (a) SiO₂ and (b) polymer films. Ellipsometric angles Δ and Ψ are converted to Reρ and (Imρ)², respectively. Measured data (black points) are compared to calculated trajectories using Airy's formula, Drude approximation and Eq. (5). ρ₀ is calculated (not measured) using Fresnel's reflection coefficients and literature values (red open circle). The nominal incidence angle is α_inc = 60° and wavelength λ = 633 nm.(c) Apparent refractive index n_F of a polymer film as function of the film thickness. The actual value is n_F = 1.57 (solid line). A conventional data analysis (black squares) gives systematically wrong values for films thinner than 15 nm. By means of a reference measurement on the uncoated substrate and the use of Eq. (7), reliable values are obtained for films thicker than d_F = 5 nm.

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According to Snell's law the transmission angle α_tra is a function of n₁, n₂ and α_inc. In particular, α_tra is independent of the film. If one assumes λ, α_inc as well as the optical properties of ambient and substrate (n₁ and n₂, respectively) to be constant values, the individual terms of the series expansion depend only on the thin film (characterized by parameters n_F and d_F). Interestingly, in 2nd-order approximation all contributions of the film can be found in two factors, namely

Supposing ambient, film and substrate media are transparent, n₁, n_F and n₂ are real numbers. Then the constant and quadratic terms of Eq. (3) are purely real, while the linear summand is purely imaginary. Hence, Eq. (3) can be simplified and the complex number ρ can be directly separated in its real and imaginary part, i.e. $\mathrm{Re}\rho ={\rho }_0+({{\rho }^{″}}_A+{{\rho }^{″}}_B)\cdot {(2\pi d_F/\lambda )}^2$and $\mathrm{Im}\rho ={\rho }^{\prime }\cdot 2\pi d_F/\lambda$. In the next step, the terms of Re ρ which are proportional to (2πd_F/λ)² can be substituted by the square (Imρ)².

3. Observations

On the other hand, Reρ and Imρ are experimentally directly accessible by applying Euler's famous formula on Eq. (1), i.e. exp(iΔ) = cosΔ + i sinΔ. This idea is demonstrated in Fig. 2 (a,b) on two series of thin film samples with variable film thickness: (a) oxidized silicon wafers and (b) polymer covered silicon wafers, respectively. Note that Eq. (5) remains valid even if silicon wafers are used as substrates. Silicon wafers are not transparent, which can be seen with the naked eye (n₂ = 3.882 – i 0.02 for λ = 633 nm) [18]. However, the imaginary part is two orders of magnitude smaller than the real part and thus negligible for all practical purposes. The polymer films are polyelectrolyte multilayers. For the chosen polyelectrolytes and preparation conditions, the film thickness increases linearly with the number of deposition steps [19,20], while the composition of the film remains the same [17]. The measurements are performed with either Multiskop (OptrelGbR, Kleinmachnow, Germany) in case of homogeneous films or imaging ellipsometry using EP3 (Accurion, Göttingen, Germany) in case of Fig. 3. All measurements are carried out in PCSA configuration (polarizer, compensator, sample, analyzer) with a He-Ne laser (power = 4 mW; wavelength λ = 632.8 nm). Measurements are performed (for a fixed compensator) for two different pairs of polarizer and analyzer positions, i.e. two different ellipsometric zones [10]. In both cases, converting the measured ellipsometric angles Δ and Ψ into Reρ and (Imρ)² results in a straight line whose slope S is a measure for n_F. Furthermore, in this plot the distance between a particular point and ρ₀ is a measure for the corresponding film thickness d_F (note that Imρ is proportional to $2\pi d_F/\lambda$).

 

Fig. 3 Position-resolved ellipsometric angles (a) Δ and (b) Ψ of a polymer film partially covering a Si wafer. The boundary line between uncoated substrate (region B) and polymer film (region A) is located at x = 500 µm. (c) Average ellipsometric angles Δ and Ψ of region A and B converted to Reρ and (Imρ)², respectively. Thus, n_F = 1.57 and d_F = 5.2 nm is obtained.

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To conclude: by plotting Reρ against (Imρ)² and extrapolating the data to (Imρ)² = 0, the value of ρ₀ can be obtained from the intercept at (Imρ)² and n_F can be obtained from the slope S. Ideally, ρ₀ and the intercept at (Imρ)² = 0 are identical. But in experiments, there are uncertainties: In the classical data evaluation via Fresnel's and Airy's formulas, the value ρ₀ is calculated using predetermined or tabulated values α_inc, n₁, n₂ and λ. Of these, especially α_inc and n₂ are, despite careful adjustment, subject to slight, yet finite errors. Provided the uncertainty in α_inc and n₂ is roughly ± 0.05° and ± 0.01, respectively, the resulting systematic error in ρ₀ is of the order of ± 0.001. Even though this error is small, it dominates the data evaluation in the case of thin films. In the experiment depicted in Fig. 2(b) the actual value ρ₀ was underestimated by 0.0009. Based on the estimated (and error-prone) value of ρ₀ the trajectory ρ(d_F) calculated by means of Airy's formula runs too steep (cf. Figure 2(b), black lines). This is equivalent to a systematic error in n_F, which in return leads to a systematic error in d_F. In short, there are two different sources of error: (i) a slight uncertainty in the experimental determination of ρ₀ and (ii) the thin film correlation between d_F and n_F.

The first of these error sources can be eliminated with the aid of one additional reference measurement of the bare substrate and the use of Eq. (5). For this purpose it is necessary to perform two different measurements (A and B) in order to gain two different pairs of ellipsometric angles (Δ_A, Ψ_A as well as Δ_B, Ψ_B). Both measurements, A and B, are carried out on the same sample, which is not moved between the two measurements. This ensures that both measurements are performed at exactly the same angle of incidence. The two measurements differ solely with respect to the thickness of the thin film: while measurement A is the thin film of interest, the reference measurement B takes place on the uncoated substrate. Converting the pair (Δ_A, Ψ_A) into the corresponding values Reρ_Α and (Imρ_Α)² (and analogous for B) allows for a direct experimental quantification of the slope $S=\left(\mathrm{Re}{\rho }_A-\mathrm{Re}{\rho }_{B\text{}}\right)/\left({(\mathrm{Im}{\rho }_A)}^2-{(\mathrm{Im}{\rho }_B)}^2\right)$without predefining the optical parameters α_inc, n₁, n₂ and λ. The additional information obtained by the reference measurement B allows to determineρ₀ from the intercept and is equivalent to a direct measurement of the actual value ρ₀.

Determining the film refractive index n_F from the measured value S is carried out analytically in two steps: (i) According to Eq. (5) the slope S is a function of E₂/E₁, the ratio of second to first ellipsometric moment. Solving the slope S in Eq. (5) for E₂/E₁ yields

Figure 2(c) shows the results of this data evaluation using the example of polymer covered silicon wafers. When using Airy's formula d_F needs to exceed 15 nm in order to obtain meaningful results. In contrast, Eq. (7) delivers the reliable value n_F = 1.57 for films down to 5 nm in thickness. Once n_F of a particular sample is quantified via Eq. (7), the corresponding film thickness can be determined numerically using Fresnel's and Airy's formulas.

However, a series of samples with increasing film thickness, as shown in Fig. 2(a,b), is often not of interest. Instead, frequently the task is to quantify the film parameters of a specific sample. Figure 3 shows that it is possible to meet the requirements of Eq. (7), by examining an inhomogeneously covered sample by imaging ellipsometry. Only a part of the surface is coated while the complementary part of the surface remains native. Investigating the boundary line between coated and uncoated surfaces with an imaging ellipsometer delivers two different pairs of ellipsometric angles (Δ_A, Ψ_A and Δ_B, Ψ_B). For the sample depicted in Fig. 3 the algorithm described above gives n_F = 1.57 and d_F = 5.2 nm, which is in good agreement with the expected film thickness for this polymer film as well as the refractive indices obtained at higher film thickness [19,21,22].

4. Conclusion

For transparent homogeneous thin films, we used an alternative way to calculate the thickness and the index of refraction from the ellipsometric ratio ρ. We abandoned Fresnel’s definition of the reflection and transmission coefficients. Instead, we used Drude’s definition, and did a series expansion with respect to the relative film thickness $2\pi d_F/\lambda$ up to the second order. A linear relationship between Reρ and (Imρ)² is found. The slope of the straight line depends on n_F, but not on d_F. Knowing n_F, one can calculate d_F which depends on the distance between points. To minimize experimental errors, one additional reference measurement is necessary. Our approach allows to determine the parameters n_F and d_F independently, even for very thin films which could not be described unambiguously up to now (thickness ≥ 5 nm).