# Shape- and element-sensitive reconstruction of periodic nanostructures with
grazing incidence X-ray fluorescence analysis and machine learning

A. Andrle<EMAIL_ADDRESS>Physikalisch-Technische Bundesanstalt (PTB),
Abbestr. 2-12, 10587 Berlin, Germany  P. Hönicke  Physikalisch-Technische
Bundesanstalt (PTB), Abbestr. 2-12, 10587 Berlin, Germany  G. Gwalt  Helmholtz
Zentrum Berlin für Materialien und Energie (HZB), Department Optics and
Beamlines, Albert-Einstein-Str. 15, 12489 Berlin, Germany  P.-I. Schneider
JCMwave GmbH, Bolivarallee 22, 14050 Berlin, Germany  Zuse Institute Berlin,
Takustrasse 7, 14195 Berlin, Germany  Y. Kayser  Physikalisch-Technische
Bundesanstalt (PTB), Abbestr. 2-12, 10587 Berlin, Germany  F. Siewert
Helmholtz Zentrum Berlin für Materialien und Energie (HZB), Department Optics
and Beamlines, Albert-Einstein-Str. 15, 12489 Berlin, Germany  V. Soltwisch
Physikalisch-Technische Bundesanstalt (PTB), Abbestr. 2-12, 10587 Berlin,
Germany

###### Abstract

The characterization of nanostructured surfaces with sensitivity in the sub-nm
range is of high importance for the development of current and next generation
integrated electronic circuits. Modern transistor architectures for e.g.
FinFETs are realized by lithographic fabrication of complex, well ordered
nanostructures. Recently, a novel characterization technique based on X-ray
fluorescence measurements in grazing incidence geometry has been proposed for
such applications. This technique uses the X-ray standing wave field, arising
from an interference between incident and the reflected radiation, as a
nanoscale sensor for the dimensional and compositional parameters of the
nanostructure. The element sensitivity of the X-ray fluorescence technique
allows for a reconstruction of the spatial element distribution using a
finite-element method. Due to a high computational time, intelligent
optimization methods employing machine learning algorithms are essential for a
timely provision of results. Here, a sampling of the probability distributions
by Bayesian optimization is not only fast, it also provides an initial
estimate of the parameter uncertainties and sensitivities. The high
sensitivity of the method requires a precise knowledge of the material
parameters in the modeling of the dimensional shape provided that some
physical properties of the material are known or determined beforehand. The
unknown optical constants were extracted from an unstructured but otherwise
identical layer system by means of soft X-ray reflectometry. The spatial
distribution profiles of the different elements contained in the grating
structure were compared to scanning electron and atomic force microscopy and
the influence of carbon surface contamination on the modeling results were
discussed.

##  I Introduction

Since nanotechnology and thus nanostructures of different kind are relevant in
many areas of science and technology, metrology techniques that can support
design, research and fabrication of such nanostructures are of high
importance. Especially in the semiconductor industry, which is probably the
most popular field of application for nanotechnology as well as a very strong
driver for research in this field, complex 2D and 3D nanostructures with
feature sizes in the single-digit nanometer regime Natarajan et al. (2014);
Markov (2014) are employed in order to keep Moore’s law Moore (2006) alive.
The performance of these nanostructures crucially depends on a well-controlled
fabrication, both in terms of targeted dimensional parameters and 3D element
compositions (e.g. dopant distributions). Thus, there is a strong need for
metrology techniques which allow to characterize these parameters with
sufficient sensitivity Orji et al. (2018).

Typical analytical or dimensional techniques used in this context are scanning
and transmission electron microscopy (SEM, TEM) Takamasu et al. (2011); Erni
et al. (2009), atomic force microscopy (AFM) Ukraintsev et al. (2005) and
techniques that also address elemental distributions such as secondary ion
mass spectroscopy (SIMS) Franquet et al. (2016), atom probe tomography (APT)
Vandervorst et al. (2014) and energy-dispersive X-ray spectroscopy (EDX)
combined with scanning electron microscopy (STEM) Mertens et al. (2017). All
these techniques have different advantages and disadvantages regarding sample
preparation and consumption, achievable spatial resolution, required duration
and other experimental parameters (e.g. tip sizes). Optical metrology based on
light-matter interaction has a significant advantage in terms of measurement
speed (with respect to slow techniques, e.g. APT and STEM) and the ability to
statistically measure large areas in contrast to scanning techniques. Optical
reflectometry also known as optical critical dimension (OCD) Silver et al.
(2007) metrology is still used and continuously improved despite the
resolution limits that have been reached. In order to keep pace with shrinking
structures, intensive research is being carried out to reduce the wavelength
of the employed radiation and to increase the sensitivity via the dispersion
of the periodic nanostructured surface. This is called deep ultraviolet (DUV)
or extreme ultraviolet (EUV) scatterometry and can be extended into the X-ray
spectral range. In X-ray scattering techniques, a distinction is made between
measurements in transmission, also known as critical dimension small angle
X-ray scattering (CDSAXS) Sunday et al. (2015), and reflection mode known as
grazing incidence small angle X-ray scattering (GISAXS) Levine et al. (1989);
Renaud et al. (2009). Both techniques have already shown that they allow for
the dimensional reconstruction of nanostructures with an uncertainty in the
sub-nm range Soltwisch et al. (2017); Herrero et al. (2019), but both require
a special sample thinning or sample design, which limits the application
possibilities Pflüger et al. (2020). Usually, these techniques also employ
rather high photon energy X-rays, which limits the optical contrast between
different materials within the investigated nanostructures.

In this paper, we employ the grazing incidence X-ray fluorescence analysis
(GIXRF) de Boer et al. (1995) method and soft X-rays ($<1$keV) instead to
analyze the nanostructured surface. This non-destructive technique allows
reconstructing the composition of the sample in terms of both their
dimensional properties as well as the distribution of different elements,
since the emitted X-ray fluorescence (XRF) is characteristic for each element.
GIXRF uses the X-ray standing wave (XSW) field Bedzyk et al. (1989);
Golovchenko et al. (1982), which results from the interference between the
incident and the reflected X-ray beam as a nanoscale sensor. The intensity
modulation inside the XSW field significantly influences the X-ray
fluorescence intensity of an element depending on its spatial position within
the electromagnetic field distribution. Recent studies have shown the
potential of the GIXRF technique for the dimensional and compositional
nanometrology of periodic 2D Soltwisch et al. (2018) and 3D Nikolaev et al.
(2020); Hönicke et al. (2020) nanostructures.

In this work, we further develop this approach towards being a reliable
metrology technique by employing a combined element sensitive reconstruction
for the fluorescence signals of oxygen, nitrogen and in parts also carbon from
within a silicon nitride grating structure. The integration of machine
learning (ML) techniques, such as the Bayesian optimization Shahriari et al.
(2016) algorithm based on Gaussian processes in combination with a finite-
element Maxwell solver Schneider et al. (2019) allows to control the
computational modeling effort and to derive first estimates of the
uncertainties of the parameterized nanostructure. For an even increased
reliability, we are experimentally determining the optical constants of the
employed materials instead of using tabulated data as these are often not
realistic for nanolayers and nanostructures Zhang et al. (2012). Finally, we
compare and validate the results against AFM and SEM cross sections.

##  II Experimental

As an example of two-dimensional nanostructures, a lithographically patterned
silicon nitride grating on a silicon substrate was investigated. It was
manufactured by means of electron beam lithography (EBL) at the Helmholtz-
Zentrum Berlin. The nominal pitch of the grating is $p=100$
$\mathrm{nm}\text{/}$, the nominal height is $h=90$ $\mathrm{nm}\text{/}$ and
the nominal line width of the sample is $w=50$ $\mathrm{nm}\text{/}$. For the
manufacturing of the gratings, a silicon substrate with a 90 nm-thick Si3N4
layer was used. ZEP520A, a positive resist (organic polymer), was spin coated
on the substrate and developed with a Vistec EBPG5000+ e-beam writer, operated
with an electron acceleration voltage of 100 kV. The grating was etched via
reactive ion etching using CHF3 and to remove the remaining resist an oxygen
plasma treatment was applied. A sketch of the cross-section is shown in Fig. 1
a). The total structured area of the grating was 1 mm x 15 mm and the sample
area outside the patterned region consists of the originally deposited Si3N4
layer. Directly after fabrication, cross-section SEM images have been recorded
on a sister sample.

We performed GIXRF and X-ray reflectometry (XRR) measurements in PTB’s
laboratory Beckhoff et al. (2009) at the BESSY II electron storage ring using
the plane-grating monochromator (PGM) beamline Senf et al. (1998) for
undulator radiation. The sample was mounted in an ultrahigh-vacuum (UHV)
measurement chamber Lubeck et al. (2013), where a 9-axis manipulator allows
for an accurate sample alignment with respect to the direction of incident
X-ray beam. The incidence angle $\theta$ is defined as the angle between the
X-ray beam and the sample surface. The azimuthal angle $\varphi$ is defined as
the angle between the incident beam and a plane, which is normal to the sample
surface and parallel to the direction of the grating lines such that
$\varphi=0^{\circ}$ is defined as the orientation parallel to the plane of
incidence. Both sample rotation axes can be aligned with an uncertainty below
0.01∘.

As we have employed radiometrically calibrated X-ray fluorescence (XRF)
instrumentation, we can perform reference-free GIXRF Hönicke et al. (2019) and
gain a quantitative access to the elemental mass depositions present on the
sample Beckhoff (2008). At each angular position for $\theta$ and $\varphi$, a
fluorescence spectrum is recorded with a calibrated silicon drift detector
(SDD) Scholze and Procop (2009) and the incident photon flux is monitored by
means of a calibrated photodiode. The GIXRF-measurements were performed at an
incident photon energy of $E_{i}=680$ $\mathrm{eV}\text{/}$, allowing for the
excitation of N-K${\alpha}$ as well O-K${\alpha}$ fluorescence radiation,
which mainly originates from the surface oxide layer on the grating structure.
In Fig. 4 the obtained and normalized N-K${\alpha}$ a) and O-K${\alpha}$ b)
fluorescence intensities for different angles $\theta$ and $\varphi$ are
shown.

In addition, we have performed XRR experiments on the non-structured Si3N4
layer next to the grating at the same photon energy ($E_{i}=680$
$\mathrm{eV}\text{/}$). From this we can determine the optical constants of
the SiO2 layers and the Si3N4 layer, which are expected to be more reliable in
the soft X-ray spectral range than using only tabulated data as in Andrle et
al. (2020).

For an independent validation of the dimensional GIXRF reconstruction results,
additional AFM measurements were performed with an Nanosurf Nanite 25x25. The
sample was measured under tapping mode condition and a standard pyramidal
shaped silicon probe with a tip radius $<10$ $\mathrm{nm}\text{/}$ was used,
as it is commonly applied for the inspection of diffraction gratings Siewert
et al. (2018). The AFM-probe is characterized by a resonance frequency of
$190$ $\mathrm{Hz}\text{/}$ and force constant of $48$
$$\mathrm{N}\text{/}$/$\mathrm{m}\text{/}$$. The inspected sample area was
$500$x$500$ $$\mathrm{n}\mathrm{m}$^{2}$ in size covering about 5 grating
lines. The height profile, obtained by averaging two in juxtaposition located
AFM line profiles, is shown in Fig. 6.

Figure 1: a) Cross-section with the finite-element mesh-grid showing the
layout used for the simulation. The height $h$, the width $w$, the sidewall
angle $swa$, the oxide layer thicknesses in the groove $t_{g}$ and the on the
grating line $t_{t}$ were optimized as independent parameters. The oxide layer
on the substrate $t_{s}$ was kept constant during the optimization. b) The
calculated electric field strength inside and outside the structure is shown
for $\theta=1.2^{\circ}$ and $\varphi=1.6^{\circ}$. For the nitrogen and
oxygen fluorescence the electric field strength is integrated and the
reabsorption is calculated with the distance ydis the photon has to travel to
leave the structure.

##  III Simulation and optimization of fluorescence intensities

The GIXRF signals are directly related to the XSW field intensity distribution
which, besides the incident photon energy and the incidence and azimuthal
angles, depends on the shape and material composition of the illuminated
nanostructured surface. To reconstruct these sample features from the
experimental data, we applied a finite-element method (FEM) based forward
calculation of the XSW and optimized the structural parameters to reproduce
the experimental data.

In Fig. 1, the basic principle of the FEM procedure is shown. The mesh grid
and the model parameters are displayed in Fig. 1 a). The FEM solver calculates
the electric near-field for a given structure (Fig. 1 b)). The amount of
fluorescence photons generated at a given coordinate depends on the local
electric near-field intensity ($E(x,y)$) and the compositional and fundamental
parameters of the respective material (mass fraction of the fluorescent
element in the material $W_{k}$, photo-ionization cross section for the
incident photon energy $\tau(E_{i})$ and the fluorescence yield $\omega_{k}$,
taken from databases Schoonjans et al. (2011) or dedicated experiments Hönicke
et al. (2016)). These photons can be reabsorbed on the path through the sample
towards the detector (Fig. 1 b)) with a probability depending on the materials
mass attenuation coefficient $\mu(E_{f})$ for the fluorescence photon energy
$E_{f}$, the density $\rho$ of the material and the distance to the surface of
the nanostructure $y_{dis}(x,y)$. Eventually, the fluorescence photon will be
detected with a given detection efficiency $\epsilon(E_{f})$ if it is arriving
within the effective solid angle of detection $\Omega$. The overall emitted
fluorescence intensity of course also depends on the incident photon flux
$N_{0}$.

Thus, the measured emitted fluorescence intensity $\Phi(\theta,\varphi,E_{i})$
(derived from the detected count rate $F(\theta,\varphi,E_{i})$) can be
modeled using the integration over the full area of the fluorescent material
of the nanostructure as described by the modified Sherman equation Soltwisch
et al. (2018); Hönicke et al. (2020):

$\Phi(\theta,\varphi,E_{i})=\underbrace{\frac{4\pi\sin\theta}{\Omega}\frac{F(\theta,\varphi,E_{i})}{N_{0}\epsilon(E_{f})}}_{\begin{subarray}{c}I_{exp}\end{subarray}}=\underbrace{\frac{W_{k}\rho\tau(E_{i})\omega_{k}}{\sum
dx}\cdot\sum_{x}\sum_{y}|E(x,y)|^{2}\cdot\exp\left[-\rho\mu(E_{f})y_{dis}(x,y)\right]dxdy}_{\begin{subarray}{c}I_{model}\end{subarray}}\textrm{.}$
(1)

The equation 1 is applied for nitrogen and oxygen fluorescence and has been
implemented directly in the Maxwell solver to eliminate errors due to
conversion to a regular cartesian grid. The calculated emitted fluorescence
intensities $I_{model}$ are then compared against the experimental value
$I_{exp}$. By using a global optimization algorithm such as Bayesian
optimization (BO) Shahriari et al. (2016); Schneider et al. (2019), an optimal
set of the model parameters can then be determined. BO uses a stochastic
model, a Gaussian process, of an unknown objective function to be minimized in
order to determine promising parameter values Shahriari et al. (2016); Garcia-
Santiago et al. (2018). In a previous work Schneider et al. (2019); Andrle et
al. (2019), it was shown that the BO performs much better than other meta-
heuristic optimization approaches with respect to the computing time needed to
find the global minimum. Since BO considers all previous function evaluations,
it can be more efficient than other meta-heuristic global optimization
strategies Schneider et al. (2019) and local optimization strategies Schneider
et al. (2019). This is a crucial benefit here, as one model calculation takes
several minutes on a standard desktop computer. Here, we use an implementation
of BO which is part of the JCMsuite software package JCMwave GmbH (2020).

The error function $\chi^{2}$

$\chi^{2}(\vec{gp})=\sum_{\theta,\varphi}{\frac{(I_{exp}(\theta,\varphi)-I_{model}(\vec{gp},\theta,\varphi))^{2}}{\sigma_{N}^{2}(\theta,\varphi)}}$
(2)

was minimized with respect to the different model parameters of $\vec{gp}$
described earlier (see Fig. 1 a)) and model errors for the nitrogen
$\epsilon_{N}$ and the oxygen fluorescence signals $\epsilon_{O}$). These
model error parameters are introduced to consider potential errors e.g. the
uncertainty of the employed atomic fundamental parameters or the influence of
a thin surface contamination layer into account. Even though material
dependent parameters like optical constants or material densities deviate most
likely from the tabulated bulk data for the grating materials (SiO2 and
Si3N4), we do not include these as free parameters in the model. Indeed, as
this would drastically increase the number of model parameters and thus
prolong the necessary calculation times, we rather determine these parameters
separately by XRR (described in the next section). $\sigma_{N}$ is the
calculated experimental error consisting of an error estimation for the
effective solid angle of detection $\sigma_{\Omega}(\theta)$, the error
contributions originating from counting statistics $\frac{\sqrt{F}}{F}$ for
the respective fluorescence line as well as for the spectra deconvolution
$\sigma_{num}$.

$\sigma_{N}(\theta)^{2}=\left(\frac{\sqrt{F(\theta)}}{F(\theta)}\right)^{2}+\sigma_{\Omega}(\theta)^{2}+\sigma_{num}(\theta)^{2}$
(3)

By inverting the Hessian matrix ${\bf H}_{kj}$ Henn et al. (2012) of the error
function,

${\bf H}_{kj}=\frac{\partial^{2}\chi^{2}(\vec{gp})}{\partial gp_{j}\partial
gp_{k}}$ (4)

at the minimum of $\chi^{2}$ it is possible to determine the confidence
intervals $\left(\bf H\right)^{-1}$ of the model parameters, if they are
gaussian distributed as assumed.

The standard deviation or noise level at the global minimum parameter set is
defined as ${\rm STD}=\sqrt{\frac{\chi^{2}}{\rm DOF}}$, where ${\rm DOF}=N-M$
is the difference between the number of measurement points $N$ and the number
of free model parameters $M$ and thus the degrees of freedom.

The model parameter confidence interval can then be calculated by

$\sigma_{\vec{gp}}={\rm STD}\sqrt{{\rm diag}(\bf\left(\bf H\right)^{-1})}$ (5)

Based on the Gaussian process model of $\chi^{2}(\vec{gp})$ we determine an
estimate of the Hessian matrix by computing all second derivatives of the
Gaussian process model of $\chi^{2}$ at the minimum and can then estimate the
error covariance matrix $\left(\bf H\right)^{-1}$ and the parameter confidence
interval.

##  IV Results and Discussion

###  IV.1 Validation of the optical material parameters

From the angle-dependent measurement of reflection intensities on a layered
system, information about layer thicknesses, densities or even their optical
constants can be obtained Chason and Mayer (1997). Fig. 2 b) shows the
experimental data in comparison with the best simulation. The high frequency
oscillation visible in the XRR curve, is a clear indication for a multi-layer
system. Due to the native oxide layer of the Si substrate and the removal of
the photo resist, resulting in an oxidized surface on the Si3N4 layer Kennedy
et al. (1999), we apply a three layer model for the XRR simulation (as shown
in the inset of Fig. 2 a)). Since the calculation of the reflectivity for a 1D
layer system is several orders of magnitude faster than the FEM based 2D GIXRF
modeling, statistical analysis methods of the posterior distributions can be
used for a large number of parameters such as layer thickness, roughness and
optical constants. We applied the Markow Chain Monte Carlo method (MCMC)
Foreman-Mackey et al. (2013) to determine the individual parameter
uncertainties and to resolve possible inter parameter correlation effects
Haase et al. (2016). In Fig. 2 a) the posterior distribution determined in
this procedure is shown as projections of the refractive index
$n(Si_{3}N_{4})=1-\delta+i\beta$ and thickness $h$ of the $Si_{3}N_{4}$ layer.
The almost perfect gaussian-like shape of the distributions, which is also
present in all other parameters, allows the determination of uncertainties
directly from the measurements. The relative uncertainties of the experimental
data reconstructed with a linear error model ($ax+b$) Henn et al. (2012) are
with $(0.6\pm 0.2)\%$ exactly in the expected range.

From this modeling we derived optical constants for the top $SiO_{2}$
($\delta$ = $(8.73\pm 0.03)10^{-4}$, $\beta$ = $(2.58\pm 0.04)10^{-4}$) and
the $Si_{3}N_{4}$ ($\delta$ = $(12.59\pm 0.05)10^{-4}$, $\beta$ = $(2.58\pm
0.02)10^{-4}$) layers. By comparing the experimentally determined optical
constants with tabulated Henke data Henke et al. (1993), one can estimate the
densities of the respective materials and their deviation from the respective
bulk densities. For $Si_{3}N_{4}$, a relative density of $(0.89\pm 0.01)$ and
for $SiO_{2}$ a relative density of $(0.79\pm 0.01)$ was found. This is in
line with the already observed material density reduction discussed in Ref.
Soltwisch et al. (2018). The reduced densities as well as the experimental
optical constants are used for the GIXRF reconstruction.

Figure 2: a) The posterior distributions for relevant parameters
$\delta_{Si_{3}N_{4}}$, $\beta_{Si_{3}N_{4}}$ and the $Si_{3}N_{4}$ thickness
$h$ obtained from the MCMC sampling. The red line marks the mean of the
distribution and the dotted black lines in the histogram indicates a 3$\sigma$
interval. In the top right corner, a sketch of the used layer stack is shown.
b) Comparison of the experimental data (black stars) and the model calculation
(red line) as obtained by the MCMC.

###  IV.2 GIXRF reconstruction results

####  IV.2.1 Virtual Experiment

Before we apply the reconstruction model to real experimental data, we apply
it to an artificial dataset in order to test the reconstruction method and to
study, whether an increased incident photon energy, capable of exciting also
oxygen fluorescence signal, is beneficial. In our previous study Soltwisch et
al. (2018), measurement data at $E_{i}=520$ $\mathrm{eV}\text{/}$ was analyzed
and an indirect sensitivity to the surface oxide layer was found even though
no fluorescence signal originating from it was used for the reconstruction.
The indirect sensitivity was merely due to the attenuation behavior of the
oxide layer and thus, we expect an increased sensitivity if also a direct
signal originating from it is used. For this purpose, we generate artificial
experimental data by calculating model curves using the reconstruction model
for a given set of parameters and the two incident photon energies of
$E_{i}=680$ $\mathrm{eV}\text{/}$ and $E_{i}=520$ $\mathrm{eV}\text{/}$. To
mimic experimental noise, we apply a Gaussian disturbance with a width of 3%
(see Tab. LABEL:tab:results_GIXRF_sim). By reconstructing the artificial
datasets, we can now analyze the influence of the increased incident photon
energy on the reconstruction results and their confidence intervals without
the influence of any experimental error contributions. In Fig. 3 the
corresponding artificial GIXRF curves and reconstruction results are
displayed. They match the artificial data well for both photon energies and
characteristic features, e.g. the first local maxima are very well retrieved.
From the BO reconstruction, we are able to calculate the confidence intervals
of the reconstructed parameters and, as shown in Tab.
LABEL:tab:results_GIXRF_sim, they agree well with the initial parameters
within the derived confidence interval.

From a comparison of the determined confidence intervals for the two different
configurations (Configuration A, $E_{i}=520$ $\mathrm{eV}\text{/}$, only the
nitrogen signal is modeled and Configuration B, $E_{i}=680$
$\mathrm{eV}\text{/}$, both nitrogen and oxygen signals are modeled, see Tab.
LABEL:tab:results_GIXRF_sim), the positive influence of also considering the
oxygen fluorescence signal is obvious. Especially for the height and the
groove oxide thickness, the achieved confidence intervals are significantly
smaller for configuration B. Nevertheless, also for configuration A, the
angle-dependent nitrogen fluorescence contains all relevant information about
the dimensional properties of the nanostructure and even the surface oxide
layer, as already pointed out in our earlier work Soltwisch et al. (2018).

Table 1: The values of the geometrical parameters of the synthetic data from the GIXRF BO reconstruction with the model parameter confidence intervals for $E_{i}=520$ $\mathrm{eV}\text{/}$ (Configuration A, only the nitrogen signal is modeled) and $E_{i}=680$ $\mathrm{eV}\text{/}$ (Configuration B, both nitrogen and oxygen signals are modeled). The pitch was set to $p=50$ $\mathrm{nm}\text{/}$. (height $h$, width $w$, sidewall angle $swa$, oxide layer in the groove $t_{g}$ and oxide layer on the grating line $t_{t}$) Parameter | Intial | Config. | Conf. Interval
---|---|---|---
Name | Value | A | B | Ratio $\frac{B}{A}$
$h/$\mathrm{nm}\text{/}$$ | $90$ | $89.4(6)$ | $90.3(4)$ | 0.67
$w/$\mathrm{nm}\text{/}$$ | $25$ | $24.87(7)$ | $25.08(6)$ | 0.85
$swa/^{\circ}$ | $88$ | $88.1(1)$ | $87.9(1)$ | 1.0
$t_{t}/$\mathrm{nm}\text{/}$$ | $3$ | $3.13(6)$ | $3.02(5)$ | 0.83
$t_{g}/$\mathrm{nm}\text{/}$$ | $5$ | $5.3(6)$ | $5.0(1)$ | 0.17
$\epsilon_{N}$ | $1$ | $1.02(1)$ | $1.00(1)$ | 1
$\epsilon_{O}$ | $1$ | - | $0.99(1)$ |

####  IV.2.2 Real experimental data

Using the same model and methodologies, also the real experimental data,
measured according to configuration B, was modeled. A comparison of the two
experimental datasets and the resulting modeled data is shown in Fig. 4. The
corresponding optimal parameters and confidence intervals are summarized in
Tab. 2. Again, the confidence intervals of the structure parameters derived
from the BO posterior distribution are very similar to those determined in the
virtual experiment.

Similarly as already for the virtual experiment, a higher sensitivity for the
line width is observed as compared to the height. This is expected to be a
result of the relatively large line height with respect to the achieveable
information depth in the soft X-ray regime. Thus, the nitrogen fluorescence
radiation from the very bottom of the grating line does not contribute
significantly to the overall observed signal. Nevertheless all calculated
confidence intervals are in the sub-nm regime.

It should also be noted that the two modeling error parameters $\epsilon_{N}$
and $\epsilon_{O}$ deviate from unity within a range of 10 %. This is the same
magnitude one would expect the relative uncertainty of the employed
fluorescence production cross sections (product of fluorescence yield and
photo ionization cross section) to be in.

Systematic errors not taken into account may increase the final uncertainties.
This problem is often called model error and refers to the whole physical
model or virtual experiment that is applied and is not limited to the finite
element model. The next section shows that these modeling uncertainties can
have a significant impact on the reconstruction parameters.

Table 2: The values of the geometrical parameters of the nanostructures from the GIXRF BO reconstruction with the model parameter confidence intervals (one sigma). The parameters are height $h$, width $w$, sidewall angle $swa$, oxide layer in the groove $t_{g}$ and oxide layer on the grating line $t_{t}$.  Parameter | Reconstructed
---|---
Name | Value
$h/$\mathrm{nm}\text{/}$$ | $97.5(5)$
$w/$\mathrm{nm}\text{/}$$ | $49.77(7)$
$swa/^{\circ}$ | $83.54(9)$
$t_{t}/$\mathrm{nm}\text{/}$$ | $2.84(3)$
$t_{g}/$\mathrm{nm}\text{/}$$ | $5.82(9)$
$\epsilon_{N}$ | $0.918(6)$
$\epsilon_{O}$ | $1.059(7)$
Figure 3: Comparison of the expected artificial disturbed simulated
N-K$\alpha$ (green points) or O-K$\alpha$ (blue points) fluorescence
intensities for the different excitation energies and the corresponding BO
reconstruction. Clearly visible is how the first peak in the nitrogen signal
shifts with increasing energy to smaller incident angles. Figure 4: Comparison
of the measured and simulated fluorescence maps for N-K a) and O-K b) based on
the reconstructed parameter set. c) Comparison of the experimental N-K$\alpha$
(green) or O-K$\alpha$ (blue points) fluorescence intensities for
$\varphi=0^{\circ}$ to the reconstructions from the Bayesian optimization (red
lines).

####  IV.2.3 Contamination

In addition to oxygen and nitrogen signals in the fluorescence spectra, also
carbon fluorescence was observed. A presence of carbon on the sample surface
is likely as the sample is stored under normal ambient conditions. A lateral
scan of the sample (Fig. 5 a)) at a fixed incident angle $\theta=15^{\circ}$
reveals that this carbonaceous contamination is not homogeneously distributed
over the patterned area. In the center of the grating area ($x=-0.15$
$\mathrm{mm}\text{/}$) a strong increase of the carbon signal and a slight
increase of the oxygen signal can be observed, whereas the nitrogen signal is
practically constant. In an earlier benchmark study, the sample under
investigation was measured in various scatterometers and electron microscopes
around the world. A punctual contamination of the grating surface by these
techniques can therefore not be excluded. Here, we take advantage of this
artificially created contamination layer to demonstrate the sensitivity and
the influence of model errors.

At each lateral position shown in the figure, we performed a GIXRF angular
scan (at $\varphi$ = 0∘) and performed the reconstruction including the
confidence interval calculation without considering any contamination. Fig. 5
b) shows the obtained differences between the reconstructed model parameters
at each position to the reference position at $x=0$ $\mathrm{mm}\text{/}$
(position where the data shown in Fig. 4 was taken). The model parameters at
$-0.15$ $\mathrm{mm}\text{/}$, which is the position with maximal carbon and
oxygen contamination, differ clearly with respect to less contaminated areas.
The difference is larger than the calculated confidence intervals from the
reconstruction and they are also much larger than the expected lateral
inhomogeneities of the sample. In addition, the nearly constant nitrogen
fluorescence signal in X-direction clearly indicates a homogeneous overall
amount of $Si_{3}N_{4}$, which is in contradiction to the larger cross-
sectional area of the grating determined by the reconstruction from the GIXRF
model. The reconstructed heights and widths increase by more than $4$
$\mathrm{nm}\text{/}$.

This behavior can be explained considering the XSW near field distribution
inside the grooves: The slight increase of the oxygen due to the contamination
signal can only be incorporated by increasing the oxide layer thicknesses.
However, an increase of the oxide layer thickness weakens the penetrating
field inside the $Si_{3}N_{4}$ and thus the emitted nitrogen fluorescence. To
compensate for this, a larger grating cross section is reconstructed. One may
think, that the reconstruction algorithm could circumvent this by simply
increasing only the groove oxide layer thickness, which does not affect the
nitrogen signal so much. But as shown in part c) of the figure, where the
angular oxygen fluorescence signal contributions from the different parts in
the structure model are shown, the groove oxide has a significantly different
angular behavior as compared to the oxide on the grating line surface.
Especially the features at about $1^{\circ}$ and $2.7^{\circ}$ are very
distinct. For this reason, the reconstruction algorithm must increase both in
order to account for the higher oxygen signal in the contaminated area
throughout the angular range.

This shows how sensitive the GIXRF method is, but also how carefully the
models have to be developed for a realistic uncertainty estimation. However,
even if the model is not accurate enough, a spacial resolved reconstruction
can reveal flaws of the model.

Figure 5: a) Here the C-K$\alpha$, N-K$\alpha$ and O-K$\alpha$ fluorescence
intensity at $\theta=15^{\circ}$ where no XSW needs to be considered from
different positions on the sample are compared.
b) The reconstruction results for the width, the height, the thickness of the
oxide in the groove and the thickness of oxide around the line for the
different position are plotted.
c) The various contributions from the oxide from the line, the groove and the
oxide layer of the silicon wafer to the total oxygen fluorescence are
compared.

###  IV.3 Comparison with SEM and AFM

For a validation of the dimensional and compositional parameters as derived
from the GIXRF modeling without the thick carbon contamination, we have
performed AFM measurements on the sample and use also SEM (on a witness
sample). As AFM is an established method to map the surface topology of
nanostructures with sensitivities down to the $\mathrm{nm}\text{/}$ regime, we
can use an AFM to verify the GIXRF reconstructed total line heights
(difference between top surface and groove surface). From the AFM data, we
determined a total line height of $98.4$ $\mathrm{nm}\text{/}$. This is
consistent with the GIXRF reconstructions within the uncertainties. Here, it
should be noted that the AFM measurement is only representative for the small
area of the grating where it was performed whereas the GIXRF result is
averaged over a much larger area due to the elongated beam footprint. Due to
the interplay of the tip shape with the nanostructure, other dimensional
parameters as e.g. line width or sidewall angle are not straightforward to be
deduced Martin and Wickramasinghe (1994).

In Fig. 6 the GIXRF-reconstructed shape of the sample as well as the AFM
profile is overlayed onto the SEM cross section. Therefore the GIXRF line
profile and the AFM were scaled to the SEM image in order to match the line
pitch while keeping the aspect ratios constant. Also here, the agreement with
respect to line height, sidewall angle, line width and even the oxide layer
thicknesses (assuming the bright areas in the SEM to be the oxide) is very
good. The reconstructed thicker groove oxide can also be seen in the SEM
picture.

Figure 6: Comparison of the AFM data with the result of the GIXRF
reconstruction and a SEM picture from a witness sample.

##  V Conclusion

Here, we have demonstrated how the GIXRF based methodology for a dimensional
and compositional characterization of regular nanostructures can be enhanced
with respect to the achieveable sensitivities by incorporating fluorescence
signals of different elements from within the nanostructure. In addition, the
incorporation of supporting experiments as e.g. XRR for optical constant
verification and machine learning techniques such as Bayesian optimization
decrease the necessary computational effort of the FEM based reconstruction.
The BO allows for an intelligent and fast scanning of the parameter space as
compared to other optimizer approaches.

Initial steps towards determining a reliable uncertainty budget for the
reconstructed parameter set were taken by deriving confidence intervals for
the parameters from the Gaussian process model. We have shown, how the
incorporation of the oxygen signal shifts the achievable sensitivities well
into the sub-$\mathrm{nm}\text{/}$ regime. The obtained GIXRF reconstruction
results agree very well with results from SEM and AFM indicating the validity
of the methodology.

In addition, we have shown that the methodology is also somewhat sensitive
towards unexpected effects on the nanostructure using the example of the
carbon contamination. Especially the element sensitivity of X-ray fluorescence
but also the behavior of the reconstruction results indicate if unexpected
effects are present on the nanostructure.

By developing more sophisticated techniques to quantify the corresponding
model error influences to the final parameter uncertainties this can be a
promising technique for nanostructure characterization. In fact, by combining
it with techniques such as soft X-ray GISAXS one may even enhance the
obtainable sensitivities and even learn about important quality parameters
e.g. line roughnesses Herrero et al. (2019).

##  VI Acknowledgements

This project has received funding from the Electronic Component Systems for
European Leadership Joint Undertaking under grant agreement No 826589 —
MADEin4. This Joint Undertaking receives support from the European Union’s
Horizon 2020 research and innovation programme and Netherlands, France,
Belgium, Germany, Czech Republic, Austria, Hungary, Israel.

JCMwave acknowledges support by the Central Innovation Programm of the Federal
Ministry for Economic Affairs and Energy on a basis of a decision by the
German Bundestag (project QUPUS, grant no. ZF4450901RR7).

The authors would like to thank Jürgen Probst for producing the samples.

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