1. INTRODUCTION

Nanostructured semiconductor surfaces are essential parts of modern technological products such as advanced integrated electronic circuits. Certain nanoscale structure sizes are smaller than 10 nm with production tolerances in the sub-nanometer scale [1]. Planar line gratings often serve as test structures for prototyping novel chip technologies and as test fields for in-line monitoring. This application case verifies the industrial production process of nanoscale structures and the metrological method used to characterize the dimensions of these structures. Suitable metrology methods can determine the dimensions of the structure’s form factor with corresponding accuracy for quality assessment (QA) [2]. Optical critical dimension (OCD) metrology enables fast and nondestructive inspections of structured surfaces for high volume manufacturing [3]. The limits of sensitivity of OCD metrology are in the nanometer range because it uses wavelengths in the visible range. Hard ${\rm{x}}$-rays with photon energies above about $1\,\,{\rm{keV}}$ are suitable for sub-nanometer resolution due to wavelengths below $0.1\,\,{\rm{nm}}$ [4]. Grazing-incidence small-angle ${\rm{x}}$-ray scattering (GISAXS) uses hard ${\rm{x}}$-rays to characterize surface nanostructures with sub-nanometer resolution [5]. X-rays of this energy range are scattered under grazing angles of incidences close to the critical angle of total reflection. This reduces the photons’ penetration depth and increases the surface sensitivity of the surface structure measured. However, the footprint of the photon beam may exceed the sizes of the structured areas or the region of interest due to the footprint elongation at small angles of incidence. Furthermore, the signal strength of the detected reflectance depends on the energy and sample material used and may not be sufficiently high. Alternative photon energies in the extreme ultraviolet (EUV) range are between about $30\,\,{\rm{eV}}$ and $250\,\,{\rm{eV}}$; soft ${\rm{x}}$-rays (SXRs) between about $250\,\,{\rm{eV}}$ and $1\,\,{\rm{keV}}$ may also be used [4]. Using photon energies of the EUV and SXR spectral range allows steeper angles of incidence. These large angles of incidence reduce the photon beam footprint and are suitable to be trimmed down or focused to the typical test field of $50\,\,{\rm{\unicode{x00B5}{\rm m}}} \times 50\,\,{\rm{\unicode{x00B5}{\rm m}}}$. EUV scatterometry and SXR scattering is a compromise between the photon beam footprint and the surface and structure sensitivity. This metrology method is suitable for determining the geometrical features of nanoscale gratings with accuracy comparable to that of GISAXS [5]. These kinds of measurement setups are typically located at synchrotron radiation facilities that can provide photon beams of high and defined quality. However, these facilities are limited in access and are primarily used for fundamental research. The synchrotron is suitable for reference measurements as a stable and calibrated source and allows a monochromatic photon beam to be generated. The goal of integrating EUV radiation-based nanometrology in industrial production processes is to create a scalable QA setup that does not depend on synchrotron facilities. Compact sources of radiation in the EUV spectral range have been available for several years and are used in lithography as well as metrology setups worldwide [6–8]. Furthermore, EUV scatterometers using such sources are being developed [9,10].

This paper compares such an EUV scatterometer that uses the EUV broadband emissions of a discharge-produced plasma (DPP). This source is a DPP (AIXUV GmbH), which is operated with pure xenon gas [11]. Compact, discharge-based EUV sources are currently not suitable for a stable operation, and the broadband emission varies in terms of intensity and spectral distribution, which in the used model can vary by more than $20\%$ from pulse to pulse. The EUV scatterometer that uses this source was recently presented by the chair for Technology of Optical Systems (TOS) at RWTH Aachen University [10]. The first challenge for measuring with this compact source setup is the fact that the source is unstable. This requires a complex monitoring system that uses the source emission before and after the sample interaction and an additional measurement of a reference sample to take the influence of optical components for the absolute reflectance determination into account. The detectors must be able to resolve the broadband spectrum of the plasma. Scatterometry normally uses the signal from several diffraction orders. To resolve the spectrum with the highest possible resolution, the detectors of the setup presented take only its specular reflection that is the zeroth order of diffraction. Thus, the detector of the setup measures the reflectance of samples under several angular orientations. The maximum angular range possible in the plane of incidence is from 5 to ${30^ \circ}$ with respect to the sample surface. But the low reflectance at angles above ${15^ \circ}$ limits the high-quality data range due to measuring noise effects. The sample can also be rotated in relation to the plane of incidence in the range of ${-}{90^ \circ}$ to ${90^ \circ}$. The number of angles used depends on the complexity of the structure investigated—the more complex the structure to be measured, the more angular orientations should be included in the measurement. In this paper, the setup’s detector measures reflectance of six different angular orientations. To qualify the measurements performed using the setup, they are compared to measurements performed under similar conditions using a scatterometer at the EUV/SXR radiometry beamline of the Physikalisch-Technische Bundesanstalt (PTB) at the BESSY II synchrotron facility. A nanoscale grating sample serves as the measurement object of both setups. The same method measures the sample and analyzes it by means of identical physical models. These models differ only in terms of the different properties of the source concerning the degree of polarization.

The measurement object of the comparison presented here is a fused-silica (${\rm{Si}}{{\rm{O}}_2}$) grating with a high aspect ratio and small spacing [Fig. 1]. This grating allows dimensional parameter reconstruction methods to be tested that are based on the spectral and angular resolved diffraction efficiency measured in the EUV spectral range. A central aspect of this study is to determine and compare the probability distributions for the grating’s geometrical feature sizes based on measurements performed with both setups. Bayesian statistics examine postulated uncertainty budgets using Markov chain Monte Carlo (MCMC) sampling. This method enables the comparison of the uncertainties of the dimensional reconstructions. One established approach to obtain grating characteristics used in this paper is to combine MCMC sampling with a suitable forward model. The forward model solves Maxwell’s equations with a finite-element approach for an infinitely extended representation of the simulated grating [5,12].

 

Fig. 1. Lamellar nanoscale grating of fused silica. (a) Top view of the grating structure via SEM imaging (RWTH Aachen, SEM Jülich). (b) Grating profile, image obtained via SEM imaging (RWTH Aachen, SEM Jülich). (c) Scheme showing values determined by SXR scattering with synchrotron radiation (see Table 1) for pitch $p$, line height $h$, critical dimension at half height $w$, sidewall angle $\beta$, and top as well as bottom corner rounding with radii ${r_{{\rm{top}}}}$ and ${r_{{\rm{btm}}}}$.

Download Full Size | PDF

In Section 2, a comparison of the compact source and synchrotron setups [Fig. 2] shows the similarity of the experimental details and the reflectance measured. Section 3 shows a comparison of the dimensional reconstruction based on measurement data from both facilities. The solutions of the dimensional reconstruction based on both setups with respect to the reflectance measured will be discussed in Section 4.

The software used for MCMC sampling and presentation in this work is open source [13–15].

2. MEASUREMENTS

The sample measured [Fig. 1(a)] is a pure, fused-silica (${\rm{Si}}{{\rm{O}}_2}$) nanoscale grating with a nominal grating period (pitch) of $p = 125\,\,{\rm{nm}}$. The sample is taken from a structured fused-silica wafer. A commercial setup (Eulitha) was used to structure this wafer using photolithography [10,16]. This setup makes use of displacement Talbot lithography [17] to create homogeneous samples several centimeters in size. The scanning electron microscope (SEM) image of a cross section of a sample from the same batch shows the grating profile [Fig. 1(b)].

The simplicity of the grating shape and material composition enables a comparison of different setups used for the shape characterization. Two different setups carry out scatterometry measurements in the EUV spectral range on the sample using near-identical reconstruction approaches including the choice of measurement points for optimization for optimal comparability.

The incident photon beam hits the sample at a grazing angle of incidence ${\alpha _i}$ measured from the surface [Figs. 3(a) and 4(a)]. The sample has an azimuthal orientation with respect to the plane of incidence described by the angle $\varphi$. At an initial orientation of ${\alpha _i}={ 0^ \circ}$ and $\varphi ={ 0^ \circ}$, the grating lines are parallel to the incident photon beam. In this scattering geometry, the final angles of the diffraction maxima, ${\alpha _f}$ and ${\vartheta _f}$ [Fig. 4(a)], of the scattered EUV radiation and SXRs can be derived from the reciprocal-space representation of the grating lines with the scattering vector ${\textbf{q}} = {{\textbf{k}}_f} - {{\textbf{k}}_i}$, where ${{\textbf{k}}_i}$ and ${{\textbf{k}}_f}$ are the initial and final wave vectors, respectively [18]. For $\varphi = 0$, the diffraction equation for the geometry follows from the equation for ${\vartheta _f}$. The diffraction equation and the relevant component of the scattering vector read as [19]

For the EUV scatterometry measurements, the measurand is the intensity of the zeroth diffraction order because the compact source setup is limited to the measurement of the specular reflection. The detectors of the compact source setup measure the reflectance of the sample for the broadband spectrum of the plasma [10]. The detectors of the scatterometer in PTB’s EUV/SXR radiometry beamline at the BESSY II synchrotron facility measure the reflectance of the sample for monochromatic synchrotron radiation. The beamline scatterometer provides the reference measurement because it measures the reflectance for each photon energy with a given bandwidth, achieves a higher signal-to-noise ratio, and is performed with a more precise adjustment process. PTB’s beamline setup is also able to carry out scattering measurements with SXRs at grazing angles of incidence. At the grazing angle of incidence ${\alpha _i}={ 5^ \circ}$ and the azimuth angle $\varphi ={ 0^ \circ}$, the goniometer detector takes reflectance data from the grating sample presented including signals from all diffraction orders measurable. For this purpose, the goniometer moves the detector along the final angles ${\vartheta _f}$ and ${\alpha _f}$. Figure 4(a) shows the positions of the diffraction maxima.

Figure 2 shows both measurement setups with their different length scales. The setups differ in terms of source, vacuum, beam path, beam footprint, and detector type.

 

Fig. 2. Setups for measuring reflectance of a sample. (a) Compact source setup using EUV radiation from a DPP. (b) PTB’s soft EUV/SXR radiometry beamline at the BESSY II synchrotron facility and the scatterometer.

Download Full Size | PDF

A. Experimental Setup: Compact Source EUV Scatterometer

Figure 2(a) shows the compact source setup that achieves a vacuum of $5 \times {10^{- 5}}\,\,{\rm{mbar}}$ at the source (head) and $3 \times {10^{- 5}}\,\,{\rm{mbar}}$ at the sample chamber (body) during operation. The EUV source is a DPP [6]. The magnetic self-compression of xenon ignites the plasma in a pseudospark-like electrode geometry. The broadband emission of the plasma is unpolarized. The radiation curve of the emission is the result of the combination of bremsstrahlung, recombination radiation, and line radiation [20–22]. The radiation curve described by Planck’s law for blackbody radiation at the temperature of about $20\,\,{\rm{eV}}$ envelops the spectrum with a difference of radiation intensity up to 1 order of magnitude [23–25]. The characteristic spectral lines of xenon are in the upper energy limit in the EUV spectral range. During the measurement, a background noise correction based on dark images is performed to reduce the noise level. The source spectrum is monitored with the first detector and considered in the reflectance calculation with Eq. (3).

The photon beam from the source passes the first aperture and a zirconium filter $200\,\,{{\unicode{x00B5}{\rm m}}}$ thick to filter radiation with photon energies lower than $62\,\,{\rm{eV}}$ and higher than $248\,\,{\rm{eV}}$. A first diffraction grating mounted in the direction of the dispersion splits off a part of the photon beam to collect the incoming spectrum at a first charge-coupled device (CCD) detector. The grating is a gold-coated holographic grating with alternating lines and spaces [26]. The grating surface is concave to focus the zeroth diffraction order of the photon beam in the plane of incidence onto the sample. The beam divergence at the sample is $1.7\,\,{\rm{mrad}}$ [10]. For the grazing angle of incidence of ${\alpha _i}={ 7^ \circ}$, the beam spot on the sample is $3\,\,{\rm{mm}} \times 410\,\,{{\unicode{x00B5}{\rm m}}}$ (width by height). If necessary, the spot size can be reduced to $50\,\, {\rm{\unicode{x00B5}{\rm m}}} \times 410\,\,{\rm{\unicode{x00B5}{\rm m}}}$ (width by height) by using an additional aperture. In this paper, this was not necessary because the whole sample was structured. Furthermore, a bigger spot size allows for more radiation throughput and, thus, increased accuracy due to a higher signal-to-noise ratio.

After the sample interaction, a palladium-coated silicon mirror (M1) directs the split-off part of the incoming photon beam to the first detector (reference spectrum). This detector is completely enclosed in the vacuum of the body. A second palladium-coated silicon mirror (M2) directs the reflected photon beam to a second diffraction grating, which splits off a part of the photon beam for the second detector (reflected spectrum). Both cameras have an E2V chip with a pixel size of $13.5\,\,{\rm{\unicode{x00B5}{\rm m}}}$. The second detector is the monitor that measures the reflected spectrum. Both detectors monitor the source, which does not have an output spectrum temporally stable enough for scatterometry without source monitoring. A measurement with a well-known reference sample is performed to calibrate the setup by taking into account the influence of the other optical components. PTB calibrated this reference sample with unpolarized synchrotron radiation [27]. Source monitoring and calibration allow the sample’s absolute reflectance spectrum to be detected.

Although the compact setup monitors the source spectrum ${S_{{\rm{source}}}}$ and the reflected spectrum ${S_{{\rm{reflected}}}}$ as well, none of the components were absolutely calibrated recently, and instead another approach is used to determine the absolute reflectance. Each measurement is performed two times, once with the sample measured and once with a reference sample of known reflectance. Afterward both measurements are used to determine the absolute reflectance of the sample measured by equation Eq. (3) without the need to know the performance of the other optical components (like diffraction grating or mirrors) [28],

This procedure is robust to degradation of all the optical components besides the reference sample, which is frequently characterized by the PTB.

The contributions to the relative intensity uncertainty are mainly the detector noise and the thermal instability of the first in-vacuum CCD [29]. This work assumes that contributions to the relative uncertainty are made by the detector inhomogeneity and that the spatial and spectral inhomogeneous stray light from the source and the sample degradation are negligible.

B. Reference Setup: Scatterometer at the EUV/SXR Radiometry Beamline

The reference setup is a scatterometer located at PTB’s EUV/SXR radiometry beamline at the BESSY II synchrotron facility [Fig. 2(b)]. The values in the following brief overview correspond to [30].

The degree of linear polarization of the synchrotron radiation at the beamline is about $98.7\%$ with respect to the plane of incidence. The monochromator optics select monochromatic photon beams with photon energies between 50 and $1771\,\,{\rm{eV}}$. A given photon energy has a deviation that varies between 0.05 and $0.25\%$. The monochromatic photon beam reaches the sample in a vacuum chamber with a lubricant-free goniometer. At the sample, the vacuum is about ${10^{- 7}}\,\,{\rm{mbar}}$. The photon beam divergence onto the sample is horizontally $1.6\,\,{\rm{mrad}}$ and vertically $0.4\,\,{\rm{mrad}}$. For a grazing angle of incidence of ${\alpha _i}={ 7^ \circ}$, the beam spot on the sample is $1.2 \,\,{\rm{mm}} \times 8 {\rm{mm}}$ (width by height) in size. The mechanical sample stage system in this chamber can orient the sample to the incoming photon beam with an angular accuracy of ${0.01^ \circ}$. A calibrated photodiode detector allows the scatterometer to measure the sample’s absolute reflectance. This detector measures the intensity of the direct photon beam before and after each measurement step to normalize the signal from the reflected photon beam. Moreover, the photodiode detector can be placed at defined positions relative to the sample to detect the intensity of scattered and diffracted radiation.

Contributions to the relative uncertainty of the measured intensity that are not negligibly small come from the monochromator optics and the photodiode detectors. At the monochromator optics, these contributions are photon losses caused by diffuse scattering $(u = 0.12\%)$. The photodiodes contribute inhomogeneity $(u = 0.06\%)$ to the relative uncertainty [31].

C. Experimental Results

Both measurement setups (Fig. 2) take reflectance data from EUV radiation with photon energies from 73 to $146\,\,{\rm{eV}}$. Figure 3 shows a comparison of the reflectance curves from both setups. The filled area around the curves indicates the measurement uncertainty $(\pm 2\sigma)$ as taken from the uncertainty analysis. For the scattering measurements from the compact source setup, the given step width corresponds to the energy resolution at the detector. The beamline setup, which has a monochromatic measurement principle, covers the reflectance curves with a step width of $0.5\,\,{\rm{eV}}$. The setups perform variations of the angular orientations of the sample with respect to the photon beam to obtain reflectance curves for different grazing angles of incidence and azimuth angles. To achieve a sufficiently high reflectance, the photon beam hits the sample under grazing angles of incidence up to a maximum of ${15^ \circ}$.

 

Fig. 3. Scheme of EUV scatterometry used in the different setups presented (Fig. 2) and comparison of the measured reflectance of the specular reflection over the photon energy from the fused-silica nanoscale grating shown in Fig. 1. (a) The incident photon beam (red arrow) hits the grating surface under the grazing angle of incidence ${\alpha _i}$ and the azimuth angle $\varphi$. At $\alpha ={ 0^ \circ}$ and $\varphi ={ 0^ \circ}$, the photon beam is parallel to the direction of the grating lines. (b) Reflectance curves from the compact source (red) and beamline (blue) setup in the EUV spectral range for different angular orientations of the sample to the incident photon beam. The filled areas indicate the measurement uncertainty with ${\pm}2\sigma$. The gray marked areas indicate the energy range where optimization data are taken from.

Download Full Size | PDF

With respect to the independence of and difference between the two setups, the reflectance curves in the energy range of 80 to $120\,\,{\rm{eV}}$ show an almost identical trend for all orientations (Fig. 3). For some angular orientations of the sample, the reflectance curves from the two setups, shown in Fig. 3(b), differ by more than the measurement uncertainty. This fact can be explained by deviations between the nominal $({\alpha _i},\varphi)$ and actual $({\alpha ^\prime _i},\varphi ^\prime)$ values of the grazing angle of incidence and the azimuth angle that are due to alignment uncertainties. The differences between actual and nominal values, $\Delta {\alpha _i} = {\alpha ^\prime _i} - {\alpha _i}$ and $\Delta \varphi = \varphi ^\prime - \varphi$, are setup-related parameters, which can be included into the model that describes the measurement (see Section 3). Thus, alignment uncertainties should not have a significant impact to the reconstruction uncertainty of the grating dimensions.

Although the curves shown in Fig. 3(b) have the same features near the silicon absorption edge around $105\,\,{\rm{eV}}$, the reflectance curve from the compact source EUV scatterometer does not resolve sharp features over the photon energy, which are visible at the reflectance curve of the beamline scatterometer. Furthermore, within the full energy range of 73 to $146\,\,{\rm{eV}}$, the reflectance curve from the beamline-scatterometer has a higher signal-to-noise ratio than the curve from the compact source EUV scatterometer. For photon energies smaller than $80\,\,{\rm{eV}}$ and larger than $120\,\,{\rm{eV}}$, this ratio further decreases for the reflectance measured for the compact source setup. Therefore, only data between 84 and $119\,\,{\rm{eV}}$ were used for the reconstruction of the nanoscale grating geometry in the remainder of this paper. The same restriction applied for the data of the beamline setup for the sake of comparability (see Section 3), although the signal-to-noise ratio of this setup did not require it.

 

Fig. 4. Scheme of SXR scattering used in the beamline setup [Fig. 2(b)] and measured diffraction efficiency over the photon energy from the fused-silica nanoscale grating shown in Fig. 1. (a) At $\alpha ={ 0^ \circ}$ and $\varphi ={ 0^ \circ}$, the photon beam is parallel to the direction of the grating lines. The grating diffracts the photon beam into spots of diffraction maxima (blue points) at the final angles ${\alpha _f}$ and ${\vartheta _f}$. (b) SXR scattering under ${\alpha _i}={ 5^ \circ}$ and $\varphi ={ 0^ \circ}$. The bars indicate the measurement and model uncertainties with ${\pm}\sigma$.

Download Full Size | PDF

Within the energy range of 400 to $600\,\,{\rm{eV}}$, PTB’s beamline setup takes reflectance data from the grating sample presented at the fixed grazing angle of incidence ${\alpha _i}={ 5^ \circ}$ up to the fifth order of diffraction. The purple graphs in Fig. 4(b) are the reflectance curves with a step width of $20\,\,{\rm{eV}}$. The number of diffraction signals changes depending on the photon energy.

3. MODELING AND RECONSTRUCTION OF GEOMETRICAL GRATING FEATURES

A. Maxwell Solver

The incoming radiation is described by plane electromagnetic waves of amplitudes with specified polarization and incident photon energy ${E_i}$. This model of the radiation defines the polarization with respect to the plane of incidence as percentages of the perpendicular $S$ and the parallel $P$ part. The direction of the plane waves is given by the grazing angle of incidence ${\alpha _i}$ and the azimuth angle $\varphi$. From the model view, radiation from the sources of the compact source and the beamline setup differ from each other only in terms of their polarization state. For the compact source setup, $S$ and $P$ equal $50\%$. For the beamline setup, $S = 100\%$ and $P = 0\%$ are assumed to reduce the computational time, and the effect of about $2\%$ $P$ polarization is negligible. To account for the angular sample alignment uncertainty (see Section 2.C), the resulting systematic offsets of the grazing angle of incidence and the azimuth angle, $\Delta {\alpha _i}$ and $\Delta \varphi$, are taken as model parameters.

 

Fig. 5. Calculated reflectance (zeroth diffraction order) of a representation of the fused-silica grating (Fig. 1) with angular orientation $({\alpha _i}={ 5^ \circ},\varphi ={ 0^ \circ})$ for incident photon energies in the EUV and SXR spectral range. For energies over the silicon L-edge and oxygen K-edge, the intensity distributions of the standing wave fields [(a)–(f)] of a computational domain around the line profile are assigned to points on this reflectance curve. The mesh in (a) indicates the finite elements on which the finite-element Maxwell solver calculates polynomials with fixed order, here 3, to describe the electric field amplitude.

Download Full Size | PDF

The incoming and reflected radiation interfere in the plane that intersects the grating lines perpendicularly. The grating profile influences the resulting standing wave fields depending on the sample’s angular orientation and the incident photon energy [Figs. 5(a)–5(f)]. The model uses a Maxwell solver (JCMsuite) based on the finite-element method to carry out simulations of these fields [32]. The Maxwell solver calculates the electric field strength of the standing wave field by solving the time-harmonic Maxwell’s equations with a finite-element approach. The solver used is suitable for the dimensional reconstruction as its numerical accuracy can be adjusted. Thus, this solver is an absolute method with predictable calculation errors [5]. Figure 5 shows the intensity of the standing wave fields of two of the periodic computational domains for three different photon energies around the silicon L-edge (EUV) and three around the oxygen K-edge (SXR) at a fixed angular orientation of the sample. Around both edges, the intensity modulations of the standing wave field expand within the material with increasing photon energy due to increasing absorption. Behind the edge, the intensity changes are the inverse of those described above. Furthermore, modulations of the electric field strength in the vacuum undergo a slight shift in the plane of the field.

Far from the standing wave field, the scattering signal spreads over diffraction orders via a Fourier transform according to those shown in Fig. 4(a). A dimensional reconstruction based on a scattering measurement compares the signals calculated for all diffraction orders to the signals measured.

B. Grating Model

The model of the grating assumes there are equal and symmetric lines over the entire sample area. Thus, the grating profile can be represented in the Maxwell solver as shown in Fig. 5(a). The schematic [Fig. 1(c)] shows the grating model. The pitch $p = 125\,\,{\rm{nm}}$, which is well-defined by the lithography process, determines the grating structure. Therefore, its uncertainty is negligibly small, and its value can be fixed in the grating model [33]. The geometrical parameters of line height $h$, linewidth $w$, and sidewall angle $\beta$ determine the shape of the grating lines. The model defines the linewidth as the critical dimension at half the height of the line. The top and bottom corner radii, ${r_{{\rm{top}}}}$ and ${r_{{\rm{btm}}}}$, complete the description of the line shape. As the SEM images of the sample [Figs. 1(a) and 1(b)] are taken from samples of the same batch, initial guesses for the values of the parameters can be derived from these images. Line edge roughness, charging effects, and break-off edge skewness of the sample make the grating SEM profile inaccurate. Thus, initial guesses for the parameters intervals are $h \approx ({38 \ldots 42}) \,\,{\rm{nm}}$, $w \approx ({50 \ldots 54})\,\,{\rm{nm}}$, $\beta \approx {({9 \ldots 15})^ \circ}$, ${r_{{\rm{btm}}}} \approx ({5 \ldots 15})\,\,{\rm{nm}}$, and ${r_{{\rm{top}}}} \approx ({3 \ldots 9})\,\,{\rm{nm}}$.

The line edge roughness can be seen in Fig. 1(a) as displacements of the line edges. An initial guess for the standard deviation of these displacements can be derived from an image analysis of Fig. 1(a), yielding $\xi \approx ({0.6 \ldots 2.6})\,\,{\rm{nm}}$. The roughness parameter $\xi$ corresponds to the standard deviation of the line displacements. The random line displacements in turn correspond to thermal displacements of atoms in a crystal lattice. The Debye–Waller factor was first introduced to describe the temperature dependence of scattering intensities of Bragg reflection [34]. A previous study showed the applicability of the Debye–Waller factor for describing the influence of line edge roughness on the scattering signal of hard $\rm x$-rays [35]. The roughness parameter $\xi$ is the damping coefficient in the Debye–Waller factor; for small values, it determines the damping of the diffraction intensities caused by the varying pitch. For an accurate model of the damping of the diffraction intensities without knowledge about the roughness distribution, the first component of the scattering vector $({q_x})$ [see Eq. (2)] should fulfill the condition $(q_x^2{\xi ^2} \lt 1)$ [35]. In this paper, the model assumes that this condition also applies for SXRs because ${q_x}$ can still fulfill this condition. For example, $\xi = 3\,\,{\rm{nm}}$, $\lambda = 2.2 \,\,{\rm{nm}}$, and $m = 4$ yield $q_x^2{\xi ^2} \approx 0.36$. Thus, SXR scattering fulfills this condition.

The intensity of the zeroth diffraction order $(m = 0)$ is not affected by this factor because the intensity of the specular reflection only depends on the structure and the shape in the EUV and SXR spectral range. In this study, the Debye–Waller factor does not apply to the EUV scatterometry measurements because these measurements only detect the intensity of the specular reflection. Thus, the impact of line edge roughness is only relevant for the SXR scattering measurement.

The grating model assumes a homogeneous sample material with a mass density ${\rho _{{\rm{Si}}{{\rm{O}}_2}}}$. Although this model could potentially describe the material’s optical properties via constants taken from a chunk of fused silica [36], the model used describes the material’s optical properties via optical constants of a measured chunk of crystalline silica (${\rm{Si}}{{\rm{O}}_2}$) for the crystal’s (100) and (001) planes [37]. The model takes these optical constants because they were determined around the silicon L-edge with the highest possible resolution via reflectometry. For the optimization (see Section 3.C), the model uses the average of the optical constants from the ${\rm{Si}}{{\rm{O}}_2}$ crystal’s planes as approximation for fused silica. For the comparison between calculated and measured reflectance, the calculation is based on the weighted average for a representation of a uniform crystal-orientation distribution reading:

The quartz optical constants (including their resolution) are necessary to reduce the interpolation uncertainty at the silicon L-edge. The averaged optical constants of quartz are divided by their mass density, ${\rho ^\prime _{{\rm{Si}}{{\rm{O}}_2}}} = 2.65\,\,{\rm{g}}\,{\rm{c}}{{\rm{m}}^{- 3}}$. The model factors the normalized optical constants with the mass density parameter ${\rho _{{\rm{Si}}{{\rm{O}}_2}}}$. The expected value of the bulk density of fused silica $({\rho _{{\rm{Si}}{{\rm{O}}_2}}} = 2.2 \,\,{\rm{g}} {\rm{c}}{{\rm{m}}^{- 3}})$ will probably not apply to the measured part of the sample because the measurements performed are surface sensitive. Thus, the density is a free-floating parameter that gives greater flexibility to the dimensional reconstruction.

C. Optimization

The residuals of the measured intensity ${I_{i,{\rm{e}}}}$ and the calculated intensity ${I_{i,{\rm{c}}}}$ of the reflectance signal is part of the objective function given as

The calculated intensities depend on 10 free parameters: six sample-related ($h$, $w$, $\beta$, ${r_{{\rm{btm}}}}$, ${r_{{\rm{top}}}}$, ${\rho _{{\rm{Si}}{{\rm{O}}_2}}}$) and four setup-related parameters ($\Delta \alpha$, $\Delta \varphi$, $a$, $b$). For a dimensional reconstruction based on SXR scattering, the optimization problem also includes the sample-related roughness parameter $\xi$.

Table 1. Listed Sample-Related Parameters and Their Values Determined from the Optimization Based on the EUV Scatterometry Measurements with the Compact Source and the Beamline Setups as well as the SXR Scattering Measurement with the Beamline Setup on the Fused-Silica Grating^a$^{^,}$^b$^{^,}$^c$^{^,}$^d

View Table

A previous investigation using the sample presented here applied the Levenberg–Marquardt algorithm to ${\chi ^2}$ to find a solution in the form of a single value for each sample-related parameter ($h$, $w$, $\beta$, ${r_{{\rm{btm}}}}$, ${r_{{\rm{top}}}}$) [10]. This work uses MCMC sampling to find the estimated probability distribution for each parameter. This distribution is proportional to the actual probability density function [40]. The correlated distributions of the parameters serve as solutions as well as estimations of the reconstruction uncertainties of these solutions. Via MCMC sampling, the likelihood distribution ${\cal L}$ of the parameters is obtained by maximizing the logarithmic likelihood function,

 

Fig. 6. Fits from the dimensional reconstruction based on compact source EUV scatterometry at the fused-silica grating with a geometry according to Fig. 1(c) in the EUV spectral range. The filled areas indicate the measurement and model uncertainties with ${\pm}2\sigma$.

Download Full Size | PDF

Figures 6 and 7 present the fit results for reflectance data sets from spectroscopic and monochromatic reflectance. The term ${({a{I_{i,{\rm{c}}}}})^2} + {b^2}$ from Eq. (6) calculates the uncertainties $(\sigma _{i,c}^2)$ of the values. These uncertainties depend on the solutions of the parameters $a$ and $b$. The filled areas around the fit curves represent the $2{\sigma _{i,c}}$ uncertainties. The fits show the ability of the model described to be optimized for three data sets. Exceptions exist for minor differences between the model and the data measured behind the silicon L-edge. Minor differences between the measured and fitted reflectances also appear for SXR scattering behind the oxygen K-edge [Fig. 4(b)]. These differences are visible for the signal of higher diffraction orders as well.

 

Fig. 7. Fits from the dimensional reconstructions based on synchrotron EUV scatterometry at the fused-silica grating with a geometry according to Fig. 1(c) in the EUV spectral range. The filled areas indicate the measurement and model uncertainties with ${\pm}2\sigma$.

Download Full Size | PDF

A likely reason for these differences is the uncertainty of the optical constants used. The model used does not take these uncertainties into account in detail. These uncertainties may contribute to the uncertainties of the calculated intensity. The differences show the importance of well-determined optical constants with known uncertainty.

4. RECONSTRUCION RESULTS AND DISCUSSION

For the grating’s dimensional reconstruction, the optimization method used takes equal-sized data sets from each measurement presented in Section 2. For the EUV scatterometry measurements performed at the compact source and the beamline setup, data are taken from the gray marked area of the energy range in Fig. 3(b). For the SXR scattering measurement, the optimization method takes data from 400 to $560\,\,{\rm{eV}}$ for the geometrical reconstruction. Projections of probability distributions from MCMC sampling with weak correlations are like Gaussian distributions and give well-defined solutions for the reconstruction parameters. The violin plots in Fig. 8 show normalized density distributions from the sample-related parameters based on the different measurements for a visual comparison of the reconstruction uncertainties [41]. Estimations of the kernel density using a Gaussian kernel with a varying bandwidth yield the violin plots presented [42]. The medians of the projected posterior distributions are the parameter solutions. Here, the confidence interval of $68.27\%$ is the projected uncertainty. For Gaussian distributions, this would result in a ${\pm}1\sigma$ range. This applies for the line height $h$, the width $w$, and the mass density of the material ${\rho _{{\rm{Si}}{{\rm{O}}_2}}}$ in all cases within the limited statistics. Other parameters show a slight asymmetry where ${+}\sigma$ and ${-}\sigma$ are different. Table 1 lists the reconstruction results and the uncertainties determined for the sample-related parameters based on the different measurements. Solutions for ${r_{{\rm{btm}}}}$ and ${r_{{\rm{top}}}}$ from the reflectivity measurements in the EUV spectral range cannot be determined from the distributions.

 

Fig. 8. Violin plots of the normalized probability density from MCMC sampling for the relevant fused-silica grating parameters (Fig. 1). The plots show the reconstruction results based on three different measurements (color code). In cases where the distributions touch the boundary limits, these limits (gray lines) are visible. The quantiles for the uncertainty estimation are 15.87, 50, and $84.14\%$ $(\pm \sigma)$. The white-edged black circles show the medians of the distributions.

Download Full Size | PDF

Figure 8 shows that the parameter’s solutions based on the compact source (red) and the synchrotron (blue) EUV scatterometry for $h$, $w$, $\beta$, ${r_{{\rm{btm}}}}$, and ${r_{{\rm{top}}}}$ match, as each confidence interval $(\pm \sigma)$ from one solution contains the other solution (Fig. 8). The most striking result is that the height $h$ and the critical dimension $w$ of the grating can be determined accurately for both setups. The uncertainties determined are practically the same. The accuracy of the parameter $\beta$ determined is about $50\%$ higher for the synchrotron than for the compact source EUV scatterometry. This may follow from the noisier signal in the latter case. Neither of the two measurements resolves the corner radii, as they do not give Gaussian-like distributions for the solutions. For small feature sizes, the sensitivity is still limited by the wavelength in the EUV spectral range. The size of the intensity modulations (Fig. 5) that scales with the wavelength is the resolution criterion for feature sizes. Thus, ever shorter wavelengths of the incident photons would enable a better sampling of the structure, which further increases the sensitivity for small feature sizes [compare Figs. 5(a)–5(c) with Figs. 5(d)–5(f)]. Still, the reconstructions based on both measurements yield large distributions that show trends for more squared top corners and rounder bottom corners, as expected from the lithography process. In sum, the two setups show similarity with respect to the geometrical parameters determined and their uncertainties. Based on a locally minimized ${\chi ^2}$, the Levenberg–Marquardt algorithm used in [10] finds solutions for ${r_{{\rm{btm}}}}$ and ${r_{{\rm{top}}}}$ within the given parameter range. But MCMC sampling shows that these solutions are not certain.

The solutions of the sample-related parameters based on the SXR scattering measurement (purple) lie within the distributions of the solutions based on the EUV scatterometry measurements, except for the density parameter. Here, the most striking feature is the uncertainties estimated by the solutions, which are about 1 order of magnitude smaller than the uncertainties from the first two reconstructions. A trend to higher linewidths $w$ and sidewall angles $\beta$ is visible from the reconstruction results based on the EUV scatterometry measurement at the compact source setup (red) to the soft SXR scattering measurement at the beamline setup (purple). But the solution for $w$ based on the SXR scattering measurement still lies within the $95\%$ confidence interval based on the EUV scatterometry measurement with the compact source setup. The solutions of ${r_{{\rm{btm}}}}$ and ${r_{{\rm{top}}}}$ are well-defined and, as expected, ${r_{{\rm{btm}}}} \gt {r_{{\rm{top}}}}$. The solutions of the corner radii have uncertainties as small as those of the other geometrical parameters.

Figure 9 shows the line profiles from parameter values taken from the MCMC samples compared to the profile from the SEM image [Fig. 1(b)]. The profiles based on the compact source EUV scatterometry and synchrotron SXR scattering measurements fit to the image within the uncertainties. However, the lower corner rounding based on the EUV scatterometry measurement is undefined. The profiles based on the SXR scattering measurement show comparably sharp edges that lie within the edges based on the EUV scatterometry measurement.

 

Fig. 9. Comparison of the line profile of part of a SEM image representing the measured fused-silica grating (Fig. 1) and plotted profiles from MCMC sampling based on the compact source EUV scatterometry (red) and the synchrotron SXR scattering measurements (purple).

Download Full Size | PDF

The results of the MCMC approach suggest that using higher photon energies and taking the signal from higher diffraction orders increases the sensitivity for determining the geometrical parameters. The sensitivity analysis in Appendix A shows the sensitivities for EUV scatterometry and SXR scattering. The comparison between the two energy ranges shows that the relative changes of the zeroth-order diffraction signals increase by about $100\%$, while the relative changes of the higher order diffraction signals increase by 1 order of magnitude.

For the compact source EUV scatterometry measurements, the solutions for the mass density include the expected mass density of ${\rho _{{\rm{Si}}{{\rm{O}}_2}}} = 2.2\,\,{\rm{g}} {\rm{c}}{{\rm{m}}^{- 3}}$ for fused silica. For the synchrotron EUV scatterometry measurements, the solution includes the value ${\rho _{{\rm{Si}}{{\rm{O}}_2}}} = 2.3\,\,{\rm{g}}\,{\rm{c}}{{\rm{m}}^{- 3}}$, which is given by (Filatova et al. [36]). The value of ${\rho _{{\rm{Si}}{{\rm{O}}_2}}}$ from the solution based on SXR scattering is low compared to this value. This relatively large deviation of the reconstructed value from the expected value could be explained by the determination problem of the optical constants (see Section 3.C) and the assumption of similarity between fused and crystalline silica (see Section 4).

Finally, the solutions from the compared reconstructions shown in Table 1 were used to calculate the reflectance showing the fits [Figs. 4(b), 6, and 7]. Figure 6 shows that the fit curves for the reflectance of the compact source EUV scatterometry agree with the curves from the measurements apart from the region around the silicon L-edge. As the detectors of the compact source setup smooth out the measured reflectance in this region, the curves do not agree with the shape of the fit curves based on the used optical constants. Figure 7 shows that measured reflectance of the synchrotron EUV scatterometry fits better at the silicon L-edge with respect to following the shape compared to the spectroscopic reflectance. Exceptions exist for some angular orientations, especially ${\alpha _i}={ 5^ \circ}$ and $\varphi ={ 0^ \circ}$ (the highest reflectance). Between 110 and $119\,\,{\rm{eV}}$, the calculated reflectance has a different shape than the measured reflectance. This occurs for the reflectance from compact source EUV scatterometry as well but at slightly higher photon energies. Possible sample contamination between measurements may influence the parameter solutions in such a way that the reconstruction should be based on a model that takes a contamination layer into account.

 

Fig. 10. Sensitivity study for the compact source EUV scatterometry reconstruction result.

Download Full Size | PDF

5. CONCLUSION

The fused-silica nanoscale grating characterized in this work represents one of the most important metrological challenges for modern semiconductor nanostructures. The sample is used to benchmark a compact source EUV scatterometer. The reference setup used is a synchrotron EUV scatterometer that is one part of PTB’s EUV/SXR radiometry beamline at the BESSY II synchrotron facility. The comparison shows that EUV scatterometry with a compact source setup can characterize nanoscale gratings with reconstruction uncertainties comparable with those from the beamline scatterometer for the same applied angular and energy limitations caused by the compact source setup. The EUV scatterometry data for the relevant spectral region around the silicon L-edge have comparable accuracy. The optimization method used to reconstruct the dimensions of the grating line profile and the considered measurement points during optimization is kept as close as possible for optimal comparability. Differences between the measurements with respect to fluctuations and angular offsets are visible in the data. Despite these differences, possible problems with the optical constants in edge regions, and possible sample contamination in between the measurements, the dimensional reconstructions yield coincident solutions for the sample-related parameters.

The comparison shows that the compact source setup is suitable for characterizing a nanoscale grating that complies with current QA requirements for semiconductor fabrication. The compact source setup presented in this paper can determine grating features with uncertainties as low as the sub-nanometer range. Exploring portable sources for SXRs and compact source setups for scatterometry could lead to future setups for feature determination with even lower uncertainties.

APPENDIX A

To compare the capabilities of different measurement techniques, which use the measured reflectance (or diffraction efficiency) $R$, as a basis for the reconstruction of entirety of the grating parameters $p$, introducing the so-called relative sensitivity ${S_{{\rm{rel}}}}$ can be useful [43]. The sensitivity of a single grating parameter follows from the variation of the parameter value ${\rm{d}}{p_i}$ according to

(A1)$${S_{{\rm{rel}}}} = \frac{{\frac{{R(p) - R(p + {\rm{d}}{p_i})}}{{\left[{R(p) + R(p + {\rm{d}}{p_i})} \right]/2}}}}{{{\rm{d}}{p_i}}} = \frac{{({{\rm{d}}R/R} )}}{{{\rm{d}}{p_i}}}.$$

This equation also applies for $| {R(p) - R(p + {\rm{d}}{p_i})} |$ and $| {{\rm{d}}R} |$. The relative sensitivity is a technique-independent measure for the expected response of the measured reflectance signal to an absolute change of a relevant grating parameter. This measure can be transformed to an estimation of the uncorrelated uncertainty [44]. In the scope of this paper, a more advanced and reliable approach to determine the reconstruction uncertainty for each parameter was performed that does not require more than the assumption of a model. Furthermore, the simple investigations in terms of a relative sensitivity can provide valuable information as well. On the one hand, the simple equation is applicable for different setups by replacing the reflectance $R$ by the main measurement of the corresponding technique. On the other hand, sensitivity investigations can provide insights on wavelength and/or angular regions of the measurement technique, which exhibit a stronger change for a parameter variation. Therefore, they are more suitable for optimization tasks.

 

Fig. 11. Sensitivity of SXR scattering with the beamline setup.

Download Full Size | PDF

 

Fig. 12. Sensitivity of scatterometry over long energy range.

Download Full Size | PDF

For this purpose, different sensitivity studies were performed in this paper. The first sensitivity study shown in Fig. 10 is performed with the compact source EUV scatterometry setup with nominal parameters reconstructed from the laboratory measurements (parameters in Table 1).

The first plot (top left) shows the diffraction efficiency for the zeroth and the minus first diffraction order for the nominal parameters (although only the zeroth diffraction order signal was used during optimization). The other plots provide information on the relative change of diffraction efficiency due to a parameter change of $1\,\,{\rm{nm}}$, $1\,\,{\rm{g}} {\rm{c}}{{\rm{m}}^{- 3}}$, respectively, ${1^ \circ}$. All simulations are conducted for a parameter change of ${\pm}0.1\,\,{\rm{nm}}$, ${\pm}0.1\,\,{\rm{g}} {\rm{c}}{{\rm{m}}^{- 3}}$, respectively, ${\pm}{0.1^ \circ}$ to stay close to the nominal parameters, but the relative reflectance change is scaled to a change of $1\,\,{\rm{nm}}$, $1\,\,{\rm{gc}}{{\rm{m}}^{- 3}}$, respectively, ${1^ \circ}$. Depending on the parameter under variation, the maximal relative sensitivity for the zeroth diffraction order within this photon energy range varies from about $2\,\,{\rm{g}}\,{\rm{c}}{{\rm{m}}^{- 3}}$ $({\rho _{{\rm{Si}}{{\rm{O}}_2}}})$ to about $0.05\,\,{\rm{nm}}$ $({r_{{\rm{btm}}}})$ for the angle of the highest sensitivity. The parameters exhibit sensitivities with differences of about 1 order of magnitude for the zeroth diffraction order, which corresponds nicely with the reconstructed uncertainties based on the MCMC approach. In addition, the possible benefit to include higher diffraction orders is visible for the height, corner radius, illumination angles, and sidewall angle parameters due to their higher sensitivity. The actual sensitivity is strongly related to the assumed grating parameters, and even minor changes can influence the quantitative results of a sensitivity study. In any case, including higher diffraction orders in the optimization process can lead to a more accurate reconstruction by increasing the sensitivity of those parameters and by a proper choice of photon energy and angle.

In contrast to the sensitivity study of the compact source EUV scatterometry setup, Fig. 11 shows the sensitivity based on the SXR scattering measurements. Again, the simulations are performed for the reconstructed parameters provided in Table 1 as the nominal parameters, and all parameter variations are kept the same as before.

The plots show a higher maximal sensitivity for almost each parameter (besides top corner radius) compared to the compact source EUV scatterometry reconstruction (considering only the zeroth order) and a higher averaged sensitivity for each parameter. This observation is in good agreement with the reconstructed uncertainties from MCMC sampling. Only a single illumination angle was measured during the SXR scattering measurement to provide the same amount of data points for the optimization process and to reach a higher averaged sensitivity due to the additional diffraction orders present at those high energies.

The last sensitivity study in Fig. 12 is carried out to investigate the influence of the probing photon energy on the expected sensitivity and, therefore, on the reconstruction uncertainty. Here, the nominal parameters as well as the illumination conditions reconstructed from the SXR scattering measurements are chosen (see Table 1).

 

Fig. 13. Optical constants of fused silica from $1\,\,{\rm{eV}}$ to $600\,\,{\rm{eV}}$ [37,45–47].

Download Full Size | PDF

Figure 12 shows that the highest sensitivities are reached within the EUV and SXR spectral range when at least one diffraction order is present. In particular, the plus-minus first diffraction order provides the highest sensitivity around $200\,\,{\rm{eV}}$ $(6\,\,{\rm{nm}})$ due to a very small intensity of this diffraction order in this spectral range, while the sensitivity within the vacuum ultraviolet (VUV) to visible (VIS) spectral range remains almost zero at this grazing incidence angle. Typical OCD techniques use larger grazing angles of incidence to achieve sensitivities suitable to characterize nanoscale gratings with dimensions down to a few tens of nanometers, but this study should only consider the influence of the probing radiation.

To enable the simulations over this long photon energy range, three different sets of optical constants were used [37,45–47]. These data sets are corrected for the reconstructed density (SXR scattering), stitched together, and interpolated afterward to achieve a smooth transition of optical constants shown in Fig. 13.

Funding

H2020 Leadership in Enabling and Industrial Technologies (875999); European Metrology Programme for Innovation and Research (20IND04).

Acknowledgment

The authors acknowledge that this project has received funding from the Electronic Component Systems for European Leadership Joint Undertaking under grant agreement No. 875999–IT2 (IC Technology for the 2 nm Node). The Joint Undertaking receives support from the European Union’s Horizon 2020 Research and Innovation Programme alongside the Netherlands, Belgium, Germany, France, Austria, Hungary, the United Kingdom, Romania, and Israel. The authors also acknowledge that this project has received funding from the European Metrology Programme for Innovation and Research under grant agreement No. 20IND04—ATMOC (Traceable metrology of soft $\rm x$-ray to IR optical constants and nanofilms for advanced manufacturing). This program is co-financed by the European Union’s Horizon 2020 Research and Innovation Programme and the participating states. The authors thank their colleagues for their support for the measurements on the EUV/SXR radiometry beamline and Lukas Bahrenberg, who developed the software and hardware for the operation of the compact source EUV scatterometer. The authors also thank Analía Fernández Herrero and Qais Saadeh for their fruitful discussion.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

REFERENCES AND NOTES

1. “International roadmap for devices and systems: more Moore,” (IEEE, 2021), https://irds.ieee.org/editions/2021/more-moore.

2. Y. Shimizu, L. C. Chen, D. W. Kim, X. Chen, X. Li, and H. Matsukuma, “An insight on optical metrology in manufacturing,” Meas. Sci. Technol. 32, 042003 (2020). [CrossRef]  

3. “International roadmap for devices and systems: metrology,” (IEEE, 2021), https://irds.ieee.org/editions/2021/metrology.

4. D. T. Attwood, X-Rays and Extreme Ultraviolet Radiation: Principles and Applications, 1st ed. (Cambridge University, 1999), Chap. 1, pp. 2.

5. V. Soltwisch, A. Fernández Herrero, M. Pflüger, J. Probst, C. Laubis, M. Krumrey, and F. Scholze, “Reconstructing detailed line profiles of lamellar gratings from GISAXS patterns with a Maxwell solver,” J. Appl. Crystallogr. 50, 1524–1532 (2017). [CrossRef]  

6. K. Bergmann, G. Schriever, O. Rosier, M. Müller, W. Neff, and R. Lebert, “Highly repetitive, extreme-ultraviolet radiation source based on a gas-discharge plasma,” Appl. Opt. 38, 5413–5417 (1999). [CrossRef]  

7. H. Legall, H. Stiel, U. Vogt, H. Schönnagel, P.-V. Nickles, J. Tümmler, F. Scholz, and F. Scholze, “Spatial and spectral characterization of a laser produced plasma source for extreme ultraviolet metrology,” Rev. Sci. Instrum. 75, 4981–4988 (2004). [CrossRef]  

8. S. Hädrich, M. Krebs, A. Hoffmann, A. Klenke, J. Rothhardt, J. Limpert, and A. Tünnermann, “Exploring new avenues in high repetition rate table-top coherent extreme ultraviolet sources,” Light Sci. Appl. 4, e320 (2015). [CrossRef]  

9. Y.-S. Ku, C.-L. Yeh, Y.-C. Chen, C.-W. Lo, W.-T. Wang, and M.-C. Chen, “EUV scatterometer with a high-harmonic-generation EUV source,” Opt. Express 24, 28014–28025 (2016). [CrossRef]  

10. L. Bahrenberg, S. Danylyuk, S. Glabisch, M. Ghafoori, S. Schröder, S. Brose, J. Stollenwerk, and P. Loosen, “Characterization of nanoscale gratings by spectroscopic reflectometry in the extreme ultraviolet with a stand-alone setup,” Opt. Express 28, 20489–20502 (2020). [CrossRef]  

11. R. Lebert, C. Wies, B. Jaegle, L. Juschkin, U. Bieberle, M. Meisen, W. Neff, K. Bergmann, K. Walter, O. Rosier, M. C. Schuermann, and T. Missalla, “Status of EUV-lamp development and demonstration of applications,” Proc. SPIE 5374, 943–953 (2004). [CrossRef]  

12. J. Pomplun, S. Burger, L. Zschiedrich, and F. Schmidt, “Adaptive finite element method for simulation of optical nano structures,” Phys. Status Solidi B 244, 3419–3434 (2007). [CrossRef]  

13. D. Foreman-Mackey, D. Hogg, D. Lang, and J. Goodman, “EMCEE: the MCMC hammer,” Publ. Astron. Soc. Pac. 125, 306–312 (2013). [CrossRef]  

14. J. D. Hunter, “Matplotlib: a 2D graphics environment,” Comput. Sci. Eng. 9, 90–95 (2007). [CrossRef]  

15. “Inkscape [Open-source computer software],” (2022) https://inkscape.org.

16. H. Solak, “Photonics enabler technology for fabrication of photonic nanostructures,” Adv. Manuf. Technol. 31, 5–6 (2010).

17. P.-M. Coulon, B. Damilano, B. Alloing, P. Chausse, S. Walde, J. Enslin, R. Armstrong, S. Vézian, S. Hagedorn, T. Wernicke, J. Massies, J. Zúñiga-Pérez, M. Weyers, M. Kneissl, and P. A. Shields, “Displacement Talbot lithography for nano-engineering of iii-nitride materials,” Microsyst. Nanoeng. 5, 52 (2019). [CrossRef]  

18. M. Pflüger, V. Soltwisch, J. Probst, F. Scholze, and M. Krumrey, “Grazing-incidence small-angle x-ray scattering (GISAXS) on small periodic targets using large beams.,” IUCrJ 4, 431–438 (2017). [CrossRef]  

19. M. H. Pflüger, “Using grazing incidence small-angle x-ray scattering (GISAXS) for semiconductor nanometrology and defect quantification,” Ph.D. thesis (Humboldt Universitaet zu Berlin, 2020).

20. O. Rosier, “Pinchplasma mit hohlkathodenzündung als strahlungsquelle im extremen ultraviolett,” Ph.D. thesis (RWTH Aachen University, 2004).

21. G. Schriever, “Laserproduzierte plasmen als euvstrahlungsquellen,” Ph.D. thesis (RWTH Aachen University, 1998).

22. K. Walter, “Auslegung und charakterisierung eines extremultraviolet-transmissionsmikroskops,” Ph.D. thesis (RWTH Aachen University, 2005).

23. C. Kittel and H. Kroemer, Thermal Physics, 2nd ed. (Wiley, 1970), pp. 94–95.

24. U. Stamm, H. Schwoerer, and R. Lebert, “Strahlungsquellen für die EUV-lithographie,” Phys. J. 1, 33–40 (2002).

25. T. Krücken, K. Bergmann, L. Juschkin, and R. Lebert, “Fundamentals and limits for the EUV emission of pinch plasma sources for EUV lithography,” J. Phys. D 37, 3213–3224 (2004). [CrossRef]  

26. M. Banyay, “Surface and thin film analysis by spectroscopic reflectometry with extreme ultraviolet emitting laboratory sources,” Ph.D. thesis, (RWTH Aachen University, 2011).

27. M. Tryus, “Extreme ultraviolet reflectometry for structural and optical characterization of thin films and layer systems,” Ph.D. thesis (RWTH Aachen University/Università degli Studi di Padova (2018).

28. S. Danylyuk, S. Herbert, P. Loosen, R. Lebert, A. Schäfer, J. Schubert, M. Tryus, and L. Juschkin, “Multi-angle spectroscopic extreme ultraviolet reflectometry for analysis of thin films and interfaces,” Phys. Status Solidi C 12, 318–322 (2015). [CrossRef]  

29. S. Schröder, L. Bahrenberg, N. K. Eryilmaz, S. Glabisch, S. Danylyuk, S. Brose, J. Stollenwerk, and P. Loosen, “Accuracy analysis of a stand-alone EUV spectrometer for the characterization of ultrathin films and nanoscale gratings,” Proc. SPIE 11517, 115170S (2020). [CrossRef]  

30. A. Haase, “Multimethod metrology of multilayer mirrors using EUV and x-ray radiation,” Ph.D. thesis (Technische Universität Berlin, Inst. Optik und Atomare Physik, 2017).

31. F. Scholze, J. Tümmler, and G. Ulm, “High-accuracy radiometry in the EUV range at the PTB soft x-ray beamline,” Metrologia 40, S224–S228 (2003). [CrossRef]  

32. “JCMsuite [Commercial computer software],” (2021) https://jcmwave.com/.

33. E. Buhr, W. Michaelis, A. Diener, and W. Mirandé, “Multi-wavelength VIS/UV optical diffractometer for high-accuracy calibration of nano-scale pitch standards,” Meas. Sci. Technol. 18, 667–674 (2007). [CrossRef]  

34. C. Kittel, Introduction to Solid State Physics, 8th ed. (Wiley, 2005), pp. 696–697.

35. A. F. Herrero, M. Pflüger, J. Probst, F. Scholze, and V. Soltwisch, “Applicability of the Debye-Waller damping factor for the determination of the line-edge roughness of lamellar gratings,” Opt. Express 27, 32490–32507 (2019). [CrossRef]  

36. E. Filatova, V. Lukyanov, R. Barchewitz, J.-M. André, M. Idir, and P. Stemmler, “Optical constants of amorphous SiO₂ for photons in the range of 60–3000 eV,” J. Phys. Condens. Matter 11, 3355–3370 (1999). [CrossRef]  

37. A. Andrle, P. Hönicke, J. Vinson, R. Quintanilha, Q. Saadeh, S. Heidenreich, F. Scholze, and V. Soltwisch, “The anisotropy in the optical constants of quartz crystals for soft x-rays,” J. Appl. Crystallogr. 54, 402–408 (2021). [CrossRef]  

38. M.-A. Henn, H. Gross, F. Scholze, M. Wurm, C. Elster, and M. Bär, “A maximum likelihood approach to the inverse problem of scatterometry,” Opt. Express 20, 12771–12786 (2012). [CrossRef]  

39. A. Fernández Herrero, M. Pflüger, J. Puls, F. Scholze, and V. Soltwisch, “Uncertainties in the reconstruction of nanostructures in EUV scatterometry and grazing incidence small-angle x-ray scattering,” Opt. Express 29, 35580–35591 (2021). [CrossRef]  

40. J. S. Speagle, “A conceptual introduction to Markov chain Monte Carlo methods,” arXivarXiv:1909.12313 (2019). [CrossRef]  

41. M. C. Thrun, T. Gehlert, and A. Ultsch, “Analyzing the fine structure of distributions,” PLOS ONE 15, e0238835 (2020). [CrossRef]  

42. F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay, “Scikit-learn: machine learning in Python,” J. Mach. Learn. Res. 12, 2825–2830 (2011).

43. L. Bahrenberg, S. Glabisch, S. Danylyuk, M. Ghafoori, S. Schröder, S. Brose, J. Stollenwerk, and P. Loosen, “Nanoscale grating characterization through EUV spectroscopy aided by machine learning techniques,” Proc. SPIE 11325, 113250X (2020). [CrossRef]  

44. R. Silver, T. Germer, R. Attota, B. M. Barnes, B. Bunday, J. Allgair, E. Marx, and J. Jun, “Fundamental limits of optical critical dimension metrology: a simulation study,” Proc. SPIE 6518, 65180U (2007). [CrossRef]  

45. B. Henke, E. Gullikson, and J. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at e = 50-30,000 ev, z = 1-92,” At. Data Nucl. Data Tables 54, 181–342 (1993). [CrossRef]  

46. L. V. R. de Marcos, J. I. Larruquert, J. A. Méndez, and J. A. Aznárez, “Self-consistent optical constants of SiO₂ and Ta₂O₅ films,” Opt. Mater. Express 6, 3622–3637 (2016), [CrossRef]  

47. Numerical data in [46] kindly provided by Juan Larruquert.