1. Introduction

Information about the thickness of nano-films and layers is very important for precise determining the structural features of composite material, especially in the frequent case of the materials non-transparent for visible/NIR light as Si/SiO_x interfaces [1], uniform metallic thin films [2], some of the polymers [3], including important thin-film applications and devices [4,5]. Typically, a sample cross-section is directly investigated after some sort of masking a part of the sample during the thin-film deposition [4], or by a precise cut with e.g. focused ion beam (FIB) [5]. Then the dimensions of the revealed step in the material can be directly extracted and visualized with either electron (SEM, TEM) or atomic force (AFM) [6] microscopy. The solutions avoiding sample destruction or even modification are frequently required and these are not offered by the aforementioned invasive masking or cutting techniques. As a consequence, the optical methods [7], as structured-illumination microscopy [8] or white-light phase-shift interferometry [9] are often used.

Ellipsometry [10], is a very well established optical method and frequently used as the method of choice for investigation of thin films and material characterization [11], dimensional metrology [12], process control in material science [13], microelectronics [14] or the solar industry [15]. Real-time monitoring and characterization of thin-film growth [16], as well as the process diagnostics, including etching and thermal oxidation [17,18], belong to the standard application fields of the ellipsometry. However, its indirect nature, due to the fact that the data analysis requires a predefined optical model characterized by the optical constants, constitutes the major drawback of the method. Additionally, the data analysis using such an optical model is usually complex. Finally, the spot size of a light beam used for spectroscopic ellipsometry is typically large, several millimeters in diameter [19]. This noticeably limits the spatial resolution of the measurement. For these reasons, methods that are not based on refraction or phase/polarization change gain importance. As an example, the thickness of the carbon coatings on silicon materials was determined by the hard X-ray photoelectron spectroscopy [20]. A fluorescence-yield depth-resolved soft X-ray absorption spectroscopy technique, based on the principle of probing depth of the sample by changing the detection angle of the fluorescence soft X-rays emission was used for the analysis of FeCo thin films [21]. The X-ray photoelectron spectroscopy was also demonstrated as the tool determining the effective attenuation length of the photoelectrons in SiO₂ on top of Si substrate with a synchrotron being the radiation source [22].

The thickness of thin films can also be obtained directly from absorption measurement by exploiting its very prominent character in the extreme ultraviolet (EUV) region and on the low-energy-side of the soft X-ray (SXR) spectrum. The absorption dynamics unveils even small changes in the thickness of the investigated layer of the order of single nanometers [23], [24]. Hence, obtaining precise information about the sample thickness is possible from the spectroscopic studies by utilizing the direct absorption at a particular wavelength of interest [25] or by analysis of a broadband absorption presented in a form of a spectral optical density (OD) [26,27].

In this work, we follow the absorption approach by analyzing the spectral OD curves, to obtain accurate information about the thickness distribution within a specific area of the TiO₂ layer on the top of a SiN membrane. We propose a new “average” approach to increase the accuracy of the measurement on the layer thickness. The output of the method compares favorably with more standard measurement methods. We revealed the basics of the data processing to explain the reduction in error by analysis of the signal bias present in the reference and transmitted signals/spectra of the laser-plasma X-ray source (LPXS). The absorption spectroscopy is represented in the method by the data acquisition with the near edge X-ray absorption fine structure (NEXAFS). Finally, the presented approach has been extended by the possibility to obtain 2-D maps of the sample thickness with micrometer lateral resolution.

2. Theoretical background

The optical density spectrum extracted from the NEXAFS data carries information about the spectral absorption in the form of the product of medium length and the absorption coefficient. We are looking for one of these parameters – sample thickness. The spectrum is typically obtained from the broadband SXR signal transmitted through the investigated sample (Fig. 1(a)). One can express this dependence by OD(E) = - ln[T(E)] = - ln[I_s (E) / I_r (E)], where OD(E) is the spectral optical density, T(E) is the transmission of the investigated sample, I_s(E) is the spectral intensity of the radiation transmitted through the sample, I_r (E) is the reference spectrum from the source illuminating the sample. Typically, the sample consists of a thin film and a substrate (membrane support) as can be seen in the boxed region in Fig. 1(a). For such sample the total transmission can be expressed in the form T(E) = T₁(E)·T₂(E), with T₁(E) denoting the spectral transmission of the supporting membrane, in principle of known thickness.

 

Fig. 1. Scheme of the experimental setup for NEXAFS-based thickness mapping a). A small inset shows the schematic structure of the investigated sample. Two spectra (the sample I_s (E) and reference I_r (E) spectrum, image b)) are recorded simultaneously to obtain an accurate OD spectrum. The typical raw data image, depicting both spectra is also presented. A 2-D mapping was obtained by shifting the sample in x-direction and partial-binning of the available sample spectrum in the y-direction.

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Following the Lambert-Beer absorption law, and assuming isotropic absorption of the film and the supporting membrane, the total absorption of any multi-layer can be represented by

Then, Eq. (2) can be converted to the system of two equations with d₂ denoting the layer thickness to be determined and T₁(E) being the substrate transmission at the given energy/wavelength:

3. Experimental setup

The experimental setup for the NEXAS-based thickness mapping is depicted in Fig. 1(a). The SXR radiation is produced using an LPXS based on a double-stream Kr/He gas puff target in an arrangement described in detail in [30]. An efficient SXR krypton emission in the energy range of ∼230–500 eV (Fig. 1(b)) allows for reaching the nitrogen K- and titanium L-edges, Fig. 2(a). The Nd:YAG laser (NL 303 HT laser system from EKSPLA, Lithuania) with a pulse energy of 600 mJ, pulse duration of 3 ns, and a wavelength of 1064 nm was used for plasma generation. The laser energy stability was < 1%, measured as a standard deviation from 20 consecutive laser pulses. The plasma was formed by focusing laser pulses with a 0.5 inch in diameter lens, f = 25 mm, onto a double stream gas puff target, 1.5 mm above the nozzle, formed by a set of two colinear nozzles (0.4 mm diameter for the inner nozzle and 0.7-1.5 mm for the outer, ring-shaped nozzle). The backing pressure for Kr gas was 3 bar and for He gas 5.5 bar. For the given laser parameters and a 100 µm in diameter focal spot, a 2.5·10¹² W/cm² power density was reached, sufficient to produce a plasma capable of emitting an SXR radiation in the photon energy range from ∼200 to 500 eV, as depicted in Fig. 1(b) – I_r(E) reference spectrum. Due to the plasma instabilities and mechanical imperfection of the valve, the SXR energy, and spectral stability were measured to be ∼5% for the optimized source. The description of the optimization procedure can be found in [30].

 

Fig. 2. A CCD image showing simultaneously acquired sample and reference spectra a). Photon energy information, the position of both OD edges - OD(E₁) and OD(E₂), as well as the spectral binning ranges for both reference (200 lines) and sample (10 bins, 20 pixels wide each), were overlaid for clarity. Typical OD spectrum b) obtained from the sample and reference spectra using theoretical [33] transmission of 100 nm thick SiN support and absorption length of 200 nm TiO₂ anatase were also depicted. Variables, used in Eqs. (4)–(7) were also indicated. For their explanation please see the text.

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The spectral measurements were performed by direct SXR irradiation of the sample located ∼52 cm from the source without any thin-film metal filters. The sample was a thin, approximately 200 nm in thickness titanium dioxide TiO₂ (anatase form) layer, spin-coated on the top of a 100 nm thick silicon nitride SiN membrane with dimensions 1.5×1.5 mm². The thickness value obtained from previously calibrated SEM (FEI Quanta 3D FEG), supported by the FIB-sectioning, was treated as the reference value. The thickness measurements in 10 different locations over the surface of the TiO₂ sample resulted in an average layer thickness of 196.9 ± 35.4 nm. For the NEXAFS measurements, the sample was mounted in the special holder facilitating the acquisition of the SXR light transmitted through the sample (sample beam), and the direct SXR plasma emission (reference beam) as it is seen in Fig. 2(a).

The spectra were recorded using a home-made SXR spectrometer equipped with a grazing-incidence flat-field diffraction grating with 2400 lines/mm (Hitachi High Technologies America, Inc., USA) in the standard configuration [31]. The grating size allowed for using a 15 mm long and 100 µm wide entrance slit, located ∼72 cm from the plasma, which, in turn, enabled simultaneous entering the spectrometer by the sample and reference beams to give two spectra recorded with a 16 bit GE 20482048 CCD camera (greateyes GmbH, Germany) (Fig. 1). The camera has a 2k x 2k pixels chip and each pixel is 13×13 µm² in size. The detector dynamic range was 3.3·10⁴. For spectral calibration the Ar:N₂:O₂ (1:1:1 by volume) gas mixture was used with the oxygen line λ = 2.1602 nm from O⁶⁺ ion, two well-defined and the most intense SXR nitrogen lines: λ = 2.489 nm and λ = 2.878 nm from N⁵⁺ ions and argon lines λ = 4.873 and λ = 4.918 nm from Ar⁸⁺ ions delivering data for the calibration curve. The resolving power of the SXR spectrometer near the nitrogen absorption edge (∼400 eV, λ=3.1 nm) was estimated by measuring the isolated line at λ = 2.878 nm from N⁵⁺ ions, at the photon energy of 431 eV. The measured line width (FWHM) of 7 pm resulted in the energy resolution of E/ΔE ∼410 in this particular spectral region [32]. The quantum efficiency of this camera in the investigated spectral range is equal to 75% at 400 eV and ∼80% at 460 eV, which corresponds to 110-126 photoelectrons, generated per absorbed photon. The chip was cooled during the experiment down to -20°C to reduce its internal noise and the background. The two-beam approach limits the influence of the source energy fluctuations and mechanical vibrations on the unpredicted spectral shifts.

2-D thickness mapping over the defined region of interest (ROI) requires registration of the OD(E) spectra in both x and y directions. The x-direction sample scan was realized by the motorized translation stage (Standa) with recording the spectral data in fourteen 100 µm-wide steps. This covered almost the entire width of the SiN window equal to 1.5 mm in this direction. The slit was de-magnified onto the sample having an equivalent width of ∼72 µm - smaller than the step size by ∼30%. Such a spatial sampling reduction prohibits any “cross-talk” between the data from neighboring sample positions in the x-direction. The y-direction was explored by taking advantage of the spectrometer's a very long slit. As can be seen in the spectral data (Fig. 2(a)), this enabled simultaneous access to the spectrum from the entire height of the sample (also 1.5 mm, indicated in the horizontal direction here) covering ∼200 pixels. Thus, in the y-direction scanning was not necessary and the OD(E) data in this direction were produced by arbitrary binning the 200-pixel wide signal area. Specifically, 10 equidistant bins were chosen for this demonstration.

4. Measurement and analysis

4.1 Validation tests

The test calculations utilizing Eqs. (4)–(7) were performed with the theoretical values of sample transmission, and the absorption coefficients taken from the CXRO database for the materials used in the experiment [33], i.e. for d₁=100 nm-thick SiN membrane and the thin layer of TiO₂ (anatase) with the reference thickness of 200 nm taken from the SEM measurement. The theoretical nitrogen edge of SiN is located 5 eV above that observed in the experiment (Fig. 2(b)) and hence, E₁= 410 eV, while E₂ for Ti-L edge was located at E₂= 454 eV, in accordance to the CXRO database.

The transmission of the 100 nm thick membrane (density of 2.33 g/cm³) just above the K edge (at the position of the OD(E₁) peak) was equal to T₁(E₁) = 0.613, while at energy E₂ it was equal to T₁(E₂) = 0.688. The theoretical transmission of the anatase layer at the same energies was equal to T₂(E₁) = 0.779 and T₂(E₂) = 0.25. The same procedure conducted for d_t = 200 nm (reference thickness) layer of anatase (density of ρ = 3.78 g/cm³) resulted in the total transmissions of double-layer (SiN + TiO₂) sample equal to T(E₁) = 0.477 and T(E₂) = 0.172. Calculated optical densities at E_1,2 are thus equal to OD(E₁) = 0.74 and OD(E₂) = 1.758. The absorption coefficients of the materials were taken from the database as the inverse absorption lengths (distance at which absorption falls to the 1/e level). Thus, we obtained µ₂(E₁) = 1.25 µm^-1 and µ₂(E₂) = 6.923 µm^-1. One has to keep in mind, that the theoretical values for a bulk material of a given density may differ slightly from the actual real deposited material of slightly different density and structure. However, the values used for calculating d were taken with the highest available precision. Substituting these values into Eqs. (4)–(5) resulted in the film thicknesses of d_2- = 199.993 nm and d₂₊ = 199.994 nm. Thus, the average value [Eq. (6)], as expected, is equal to ≅200 nm.

Extraction of the TiO₂ sample thickness by the absorption jump approach by using Eq. (7) for the energies E₀ = 453.68 eV and E’₀ = 453.92 eV and applying other data taken from the database l_abs(E₀) = 0.989 µm, l_abs(E’₀) = 0.14 µm (µ₂(E₀) = 1.011 µm^-1, and µ₂(E’₀) = 7.126 µm^-1, respectively), as well as the theoretical OD(E₀) = 0.588 and OD(E’₀) = 1.81, gave the expected value of d₀ = 200 nm as well. This proves numerically that Eqs. (4)–(7) are correct.

4.2 Experimental results

The experiment was conducted to deliver the necessary data for Eqs. (4)–(7) and estimate the thickness of the TiO₂ layer over the spatial region of interest (ROI). The values of OD(E₁) = 1.054 and OD(E₂) = 1.819, at the photon energies E₁ = 405 eV (µ₂(E₁) = 1.283 µm^-1), and E₂ = 465 eV, (µ₂(E₂) = 6.762 µm^-1) were obtained from the spatially constrained OD(E) curve (Fig. 2(b)). It has to be stressed that E₁ and E₂ energies are not the same as those selected for testing the theoretical data. Small shifts of 5 eV ad 11 eV, marked in Fig. 2(b), indicate the difference between the theoretical transmission data and the experimental values. The reason for that is that the CXRO calculations do not take NEXAFS features, emerging from a quantum mechanical scattering of the photoelectron wave, into account. The experimental OD curve was obtained by summing up nine individual traces to improve the signal to noise ratio (SNR) by a factor of 3. One such a typical image used in the spectrum extraction is presented in Fig. 2(a). Each image was acquired by an accumulation of 200 SXR pulses during 20 seconds of exposure. A [2 × 19] median filter was applied to the images to remove the “salt and pepper”-type noise (e.g. single bright and dark pixels). The filter is asymmetric for a purpose; the width of 2 pixels in the dispersion direction was chosen to limit spectral smoothing and to minimize the influence on the spectral information. For each point (x,y) reference and sample regions were defined. The reference region of 200 pixels in width was unchanged for all x-positions in the scan and all y-bins to provide larger SNR. The sampling region was always 20 pixels-wide. The signal bias (constant parasite background) was estimated for both the sample and the reference spectra assuming that there is no SXR emission above 630 eV, Fig. 1(b), and then the corresponding correction was introduced. The assumption was based on the observation of the source spectral emission in the range up to ∼1 keV. Such corrected spectra were used for calculation of OD(E, x, y), from which OD(E₁, x, y) and OD(E₂, x, y) were obtained. Thickness values for each spatial point (x,y) for “subtractive” and “additive” approach, denoted as d_2-(x, y) and d₂₊(x, y), respectively, were retrieved from the experimentally determined OD, given the theoretical values of SiN transmissions T(E₁) and T(E₂), as well as the attenuation coefficients µ₂(E₁) and µ₂(E₂) of the anatase. The thickness maps d_2-(x, y) and d₂₊(x, y) were then spatially re-scaled taking into account the x-scan step and the de-magnified widths of the y-bins. These maps, of the thickness for “subtractive” (Fig. 3(a)) and “additive” approach, Fig. 3(c), as well as the corresponding errors (2-D error maps for each spatial point (x,y)) for these two approaches, were depicted in Figs. 3(b) and 3(d), respectively. The error maps were obtained under the assumption of the sample thickness equal to 200 nm.

 

Fig. 3. TiO₂ thickness spatial mapping using the SXR NEXAFS data (top row) and the error maps (bottom row) using the “subtractive” approach a, b) – Eq. (4), “additive” approach (c, d) – Eq. (5) and average approach (e, f) – Eq. (6).

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In the ideal case, the changes in the OD data are related only to the sample thickness. The sample thickness varies from point to point, thus the OD related to a particular point carries information about the sample thickness at that corresponding spatial location. Using the scanning/binning hybrid technique (see Section 3) it is possible to retrieve the thickness information, generating 2-D thickness maps, presented in Fig. 3. The retrieved thickness for every spatial point is not exactly the same, see Fig. 3 (top row), and is characterized by a certain spread of the measured values. Thus, the average thicknesses of the TiO₂ layer within the entire investigated (x,y) domain was equal to 155.1 ± 28 nm and 245.9 ± 23.1 nm when estimated by applying the “subtractive” (d_2-(x, y)) and “additive” (d₂₊(x, y)) approach, respectively.

It is easily seen, that the average thicknesses for both approaches and corresponding 2-D thickness maps (d_2- and d₂₊), presented in Figs. 3(a) and 3(c), deviate significantly in opposite directions from the expected value. The “subtractive” approach produced the average thickness equal to 78.8% of the reference thickness measured using the SEM + FIB approach, while the “additive” approach - 124.9%. This is caused by difficulty in obtaining bias-free experimental spectral data (OD), especially using an LPXS-based system. Moreover, neither Eq. (3) nor Eq. (4), could be separately used to find the absolute value of the thickness, but only for the bias-free data.

The average approach (Eq. (6)) was used to generate the thickness d(x, y) and error map presented in Figs. 3(e)–3(f), respectively. The average thickness over the entire area was equal to 200.5 ± 24.9 nm and the error could be further reduced from ∼12% to ∼10% by reducing the region of interest (ROI), and rejecting the most noise-affected pixels close to the membrane borders (edges). Such an action limits in our case the error but increases slightly the expected value to 207 nm.

Comparison of the average approach with the absorption-jump ratio method using Eq. (7) and taking the required parameters at the photon energies of E_0’ = 455 eV and E’_0’ = 465 eV works in favor of the average approach. The average thickness of the TiO₂ layer in the (x, y) domain corresponding to ROI obtained using absorption-jump ratio method was equal to 224.5 ± 25.4 nm, hence on average ∼27 nm higher than the real thickness of the TiO₂ layer. This means that the simple/standard way is also prone to the errors caused by accidental and uncontrolled bias in the transmission and OD data. All the discussed methods are compared to the SEM measurements and shown in Table 1.

Table 1. Comparison between TiO₂ layer thickness measurements using the “typical”, “averaged” approaches based on the NEXAFS data and the SEM measurements.

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This can also be visualized by the histograms of the spatial maps presented in Fig. 4 for all approaches.

 

Fig. 4. The histograms of the 2-D spatial maps were presented for a) “subtractive” approach, b) “additive” approach, c) standard (absorption-jump ratio) approach, and d) average approach. Gray region indicated the statistically bounded range of thicknesses defined by the FIB+SEM measurements (see Table 1).

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The histograms were overlaid with the results of the FIB-assisted SEM measurements, indicated using a gray box. The “subtractive” and “additive” approaches yield the center of the mass of the histogram shifted towards smaller and larger thicknesses, respectively. The standard approach is also slightly shifted from the correct thickness range. The best match for the SEM data was obtained by the average approach.

Importantly, the presented average approach has also the advantage that the space-resolved OD data contains the information not only about the thickness, but, due to the NEXAFS spectrum, also the positions of the absorption edges. Moreover, the NEXAFS features above the edge (here, π* - 400-410 eV, and σ* band of N-K edge from 410 eV to 440 eV, or L₂ and L₃ spectral features of Ti-L edge in Fig. 2(b) can be useful in bond identification and chemical mapping. Such a possibility was demonstrated recently in [34] as a novel approach in contrast to a well-established energy-scanning SXR microscopy for chemical mapping [35].

4.3 Discussion

A comparison of the results obtained by applying the average approach with those extracted from the theoretical considerations (with the use of the database values) allows for the quantitative estimate of the accidental and uncontrolled signal bias. Such a signal bias is behind the fact that d_2- ≠ d₂₊ when the experimental data is used. At the same time difference between the parameters calculated with the experimental and theoretical data is the measure of the bias.

Thus, adjusting these values of the input parameters to match the “experimental” and “theoretical” values of d_2-, d₂₊, and d will result in a clear and direct indication of the bias scale, as seen in Table 2. It can be deduced, separately at E₁ and E₂, that during the measurement the total measured transmission (SiN + TiO₂ layer) deviated by |ΔT(E₁)| = 12.8% at E₁, but only by |ΔT(E₂)| ∼1% at E₂.

Table 2. Influence of the OD bias on the thickness measurements. OD(E) experimental values were found by matching the experimentally obtained d_2-, d₂₊, and d values. Afterward, ΔOD(E) and ΔT(E) were found.

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As the values of d_2-, d₂₊ are a linear combination of the OD(E₁) and OD(E₂) parameters, it is possible to find the corrected values of these parameters fulfilling Eq. (8) and giving the right value of the sample thickness d. The necessary procedure requires perturbation of the theoretical transmission values by introducing an error ɛ, i.e. T(E_i) → T(E_i)+ɛ_i giving

The influence of these a priori introduced error components is depicted in Fig. 5. The measure of the sample thickness is a deviation (Δd) from the reference thickness value d_t equal to 200 nm. Obviously, in case of no error, Δd should be equal to zero. Both d_2- (Fig. 5(a)) and d₂₊ (Fig. 5(b)) show monotonic behavior in opposite directions for such small transmission changes, indicating Δd = 0 for multiple combinations of ɛ₁ and ɛ₂. As a consequence, using the “subtractive” or “additive” approach will subtract or add the errors and the averaging process of both gives a more accurate estimate of the sample thickness. Moreover, the changes of Δd are much slower for the average approach (c) in a wide range of ɛ₁ and ɛ₂ values. The case for (ɛ₁, ɛ₂) = (0, 0) denotes the theoretical transmissions without any bias to the signal (the ideal case). Thus, the single (marked) point on these three maps is the unique solution to find a correct bias-free OD(E) and T(E) corresponding to the experimental parameters given by (d_2-, d₂₊, d) = (155.3, 245.9, 200.6) nm. Such solution was found as corresponding to bias (ɛ₁, ɛ₂) = (-0.128, -0.01), as can be seen in Table 2, justifying the aforementioned conclusion and |ΔT(E₁)| and |ΔT(E₂)| values.

 

Fig. 5. Visualization of the influence of small changes of the sample transmission ɛ₁ and ɛ₂ at two considered energies E₁ and E₂ on the sample thickness calculated using “subtractive” a), “additive” b) and average approaches c). A point indicated in c) denotes transmission changes ɛ₁ and ɛ₂, which correspond to the experimental data indicated in Table 2, for which d_2-, d₂₊, and d₀ are matched to the experimental data.

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5. Conclusions

In conclusion, we have shown the possibility to perform 2-D spatial thickness mapping with nanometer accuracy using the data from SXR NEXAFS arrangement utilizing LPXS compact system. The thickness measurements were performed based on an OD(E) data near the absorption edges, in contrast to the commonly applied methods of measuring the transmission. The well-defined absorption edges allow for reducing the error of transmission estimation near the edge comparing to such an attempt in the other spectral region, due to a possible spectral miscalibration. The thickness map was obtained in a hybrid way, where one spatial direction was assessed using typical scanning, while the other one was obtained directly from the spatially extended spectrum through spectral binning. The experiment showed also the importance of the bias signal, and its influence on the optical density data and the thickness calculations. We also proposed a way to mitigate the spectral bias problem by the average approach, which allowed us to measure the nanometer-range thickness of the investigated layer directly from NEXAFS data with an accuracy of ∼10%. Such an approach might be useful in the case of systems, in which absolute values of transmission in a single measurement are difficult to obtain.