# Non-destructive depth reconstruction of Al-Al2Cu layer structure with
nanometer resolution using extreme ultraviolet coherence tomography

Johann J. Abel  Institute of Optics and Quantum Electronics, Friedrich
Schiller University Jena, Max-Wien-Platz 1, 07743 Jena, Germany  Jonathan
Apell  Otto Schott Institute of Materials Research, Friedrich Schiller
University Jena, Löbdergraben 32, 07743 Jena, Germany  Institute of Materials
Science and Engineering, Chemnitz University of Technology, Erfenschlager Str.
73, 09125 Chemnitz, Germany  Felix Wiesner  Institute of Optics and Quantum
Electronics, Friedrich Schiller University Jena, Max-Wien-Platz 1, 07743 Jena,
Germany  Julius Reinhard  Institute of Optics and Quantum Electronics,
Friedrich Schiller University Jena, Max-Wien-Platz 1, 07743 Jena, Germany
Helmholtz Institute Jena, GSI Helmholtzzentrum für Schwerionenforschung GmbH,
Fraunhofer Str. 8, 07743 Jena, Germany  Martin Wünsche  Institute of Optics
and Quantum Electronics, Friedrich Schiller University Jena, Max-Wien-Platz 1,
07743 Jena, Germany  Helmholtz Institute Jena, GSI Helmholtzzentrum für
Schwerionenforschung GmbH, Fraunhofer Str. 8, 07743 Jena, Germany  Indigo
Optical Systems GmbH, Moritz-von-Rohr-Str. 1a, 07745 Jena, Germany  Nadja
Felde  Fraunhofer Institute for Applied Optics and Precision Engineering IOF,
Albert-Einstein-Straße 7, 07745 Jena, Germany  Gabriele Schmidl  Leibniz
Institute of Photonic Technology e.V., Albert-Einstein Straße 9, 07745 Jena,
Germany  Jonathan Plentz  Leibniz Institute of Photonic Technology e.V.,
Albert-Einstein Straße 9, 07745 Jena, Germany  Gerhard G. Paulus  Institute of
Optics and Quantum Electronics, Friedrich Schiller University Jena, Max-Wien-
Platz 1, 07743 Jena, Germany  Helmholtz Institute Jena, GSI Helmholtzzentrum
für Schwerionenforschung GmbH, Fraunhofer Str. 8, 07743 Jena, Germany
Stephanie Lippmann  Otto Schott Institute of Materials Research, Friedrich
Schiller University Jena, Löbdergraben 32, 07743 Jena, Germany  Silvio Fuchs
Institute of Optics and Quantum Electronics, Friedrich Schiller University
Jena, Max-Wien-Platz 1, 07743 Jena, Germany  Helmholtz Institute Jena, GSI
Helmholtzzentrum für Schwerionenforschung GmbH, Fraunhofer Str. 8, 07743 Jena,
Germany  Indigo Optical Systems GmbH, Moritz-von-Rohr-Str. 1a, 07745 Jena,
Germany  Laserinstitut Hochschule Mittweida, University of Applied Science
Mittweida, Technikumplatz 17, 09648 Mittweida, Germany

# Non-destructive depth reconstruction of Al-Al2Cu layer structure with
nanometer resolution using extreme ultraviolet coherence tomography

Johann J. Abel  Institute of Optics and Quantum Electronics, Friedrich
Schiller University Jena, Max-Wien-Platz 1, 07743 Jena, Germany  Jonathan
Apell  Otto Schott Institute of Materials Research, Friedrich Schiller
University Jena, Löbdergraben 32, 07743 Jena, Germany  Institute of Materials
Science and Engineering, Chemnitz University of Technology, Erfenschlager Str.
73, 09125 Chemnitz, Germany  Felix Wiesner  Institute of Optics and Quantum
Electronics, Friedrich Schiller University Jena, Max-Wien-Platz 1, 07743 Jena,
Germany  Julius Reinhard  Institute of Optics and Quantum Electronics,
Friedrich Schiller University Jena, Max-Wien-Platz 1, 07743 Jena, Germany
Helmholtz Institute Jena, GSI Helmholtzzentrum für Schwerionenforschung GmbH,
Fraunhofer Str. 8, 07743 Jena, Germany  Martin Wünsche  Institute of Optics
and Quantum Electronics, Friedrich Schiller University Jena, Max-Wien-Platz 1,
07743 Jena, Germany  Helmholtz Institute Jena, GSI Helmholtzzentrum für
Schwerionenforschung GmbH, Fraunhofer Str. 8, 07743 Jena, Germany  Indigo
Optical Systems GmbH, Moritz-von-Rohr-Str. 1a, 07745 Jena, Germany  Nadja
Felde  Fraunhofer Institute for Applied Optics and Precision Engineering IOF,
Albert-Einstein-Straße 7, 07745 Jena, Germany  Gabriele Schmidl  Leibniz
Institute of Photonic Technology e.V., Albert-Einstein Straße 9, 07745 Jena,
Germany  Jonathan Plentz  Leibniz Institute of Photonic Technology e.V.,
Albert-Einstein Straße 9, 07745 Jena, Germany  Gerhard G. Paulus  Institute of
Optics and Quantum Electronics, Friedrich Schiller University Jena, Max-Wien-
Platz 1, 07743 Jena, Germany  Helmholtz Institute Jena, GSI Helmholtzzentrum
für Schwerionenforschung GmbH, Fraunhofer Str. 8, 07743 Jena, Germany
Stephanie Lippmann  Otto Schott Institute of Materials Research, Friedrich
Schiller University Jena, Löbdergraben 32, 07743 Jena, Germany  Silvio Fuchs
Institute of Optics and Quantum Electronics, Friedrich Schiller University
Jena, Max-Wien-Platz 1, 07743 Jena, Germany  Helmholtz Institute Jena, GSI
Helmholtzzentrum für Schwerionenforschung GmbH, Fraunhofer Str. 8, 07743 Jena,
Germany  Indigo Optical Systems GmbH, Moritz-von-Rohr-Str. 1a, 07745 Jena,
Germany  Laserinstitut Hochschule Mittweida, University of Applied Science
Mittweida, Technikumplatz 17, 09648 Mittweida, Germany

###### Abstract

Non-destructive cross-sectional characterization of materials systems with a
resolution in the nanometer range and the ability to allow for time-resolved
in-situ studies is of great importance in material science. Here, we present
such a measurements method, extreme ultraviolet coherence tomography (XCT).
The method is non-destructive during sample preparation as well as during the
measurement, which is distinguished by a negligible thermal load as compared
to electron microscopy methods. Laser-generated radiation in the extreme
ultraviolet (XUV) and soft x-ray range is used for characterization. The
measurement principle is interferometric and the signal evaluation is
performed via an iterative Fourier analysis. The method is demonstrated on the
metallic material system Al-Al2Cu and compared to electron and atomic force
microscopy measurements. We also present advanced reconstruction methods for
XCT which even allow for the determination of the roughness of outer and inner
interfaces.

_Keywords_ Non-destructive characterization $\cdot$ microstructure $\cdot$ XUV
$\cdot$ extreme ultraviolet coherence tomography $\cdot$ transmission electron
microscopy $\cdot$ atomic force microscopy

##  1 Introduction

Transmission electron microscopy is undeniably one of the most important tools
for microstructural analysis in materials science. It provides information on
key aspects such as local composition, crystal structure (including phase
identification) and crystal orientation Williams and Carter (2009). A number
of imaging and diffraction modes are available for this purpose, providing a
large number of contrasts and structural information. In addition,
spectroscopy methods such as energy dispersive X-ray spectroscopy or electron
energy loss spectroscopy provide detailed information about the composition
Williams and Carter (2009). These techniques are indispensable for pursuing
scientific and technological questions in materials science. However, there
are also limitations due to specimen damage during sample preparation and
measurement, including heat deposition and amorphization during ion milling
Mayer et al. (2007); Viguier and Mortensen (2001) as well as knock-on damage,
radiolysis, and specimen heating during the measurement Williams and Carter
(2009); Egerton (2019); Neelisetty et al. (2019); Jiang (2015).

For in-situ measurements, high spatial and high temporal resolution are in
conflict with each other. For example, it is not possible to measure the
velocity of interfaces during rapid phase transformations and to conclude on
the thermodynamic state of the interface, as relevant time and length scales
are on the order of $1\,\mathrm{\upmu s}$ and $2-10\,\mathrm{nm}$,
respectively Gamsjäger et al. (2007); Rettenmayr and Buchmann (2008);
Mathiesen et al. (2012); Thi et al. (2003). The limitation concerns both
solid/liquid phase transformations with pronounced jumps in the chemical
potential, triggered by high temperature or concentration gradients, and solid
state transformations with partial or complete interfacial control
(martensitic and massive transformation).

Materials characterization with extreme ultraviolet (XUV) radiation is an
excellent way to overcome some of these limitations. It provides information
on local parameters, e.g., chemical and stoichiometric compositions Yakunin et
al. (2014), depth structure Le Guen et al. (2010), surface and interface
roughness Sertsu et al. (2015) and thus allows conclusions on phases present
and their spatial extension. Usually, such measurements are performed using
XUV radiation at synchrotron facilities. However, laser-based XUV sources
increasingly enable characterization methods established at synchrotrons to be
transferred to university-scale laboratories. The process of high harmonic
generation (HHG) is a key technology in this respect. Our method of XUV
coherence tomography (XCT) is based on white light interferometry and utilizes
the broad bandwidth of HHG for non-destructive nanoscale subsurface imaging.

With the development of XUV microscopy in general and laser based XCT in
particular, entirely new possibilities become available. The methodology
combines a depth resolution in the few nanometer range Fuchs et al. (2017);
Wachulak et al. (2018); Skruszewicz et al. (2021) with the option of extremely
high temporally resolved measurements in the pico- to femtosecond range. At
the same time, heat insertion is minimal, as a power of only up to
$50\,\mathrm{W/m^{2}}$ is used.

We describe and apply the method of XCT for non-destructive characterization
of a metallic layer system composed of an intermetallic phase (Al2Cu) and a
solid solution ($\alpha$-Al). In the current state of development, the focus
is on the achievable material contrast and the quantification of sample depth
profiles with nanometer resolution. We further report on the reconstruction of
layer thickness and roughness parameters of present native surface and
interface oxides, using a novel reconstruction algorithm based on a step wise
optimization routine. Retrieved sample parameters from non-destructive XCT
measurements are validated and compared to atomic force microscopy (AFM) as a
surface sensitive method and transmission electron microscopy (TEM) as an
invasive characterization method.

##  2 Materials and Methods

###  2.1 Extreme ultraviolet coherence tomography (XCT)

####  2.1.1 Principle of XCT

Extreme ultraviolet coherence tomography (XCT) is a non-destructive
tomographic imaging technique, well suited for depth profile reconstruction of
nanoscale layered samples with high resolution Fuchs et al. (2016); Wiesner et
al. (2022). It is based on the principles of white light interferometry and
uses a Fourier-domain common path interferometric setup Fuchs et al. (2012),
which is depicted in Figure 1. Scattered light from different interfaces
interferes in the detected reflection beam and causes spectral modulations,
which carry the structural depth information.

The complex field reflectivity response $r(\omega)$ of a layer structured
sample can be expressed by the sum of $n$ independent interface reflectivities
$r_{j}$ located at depths $z_{j}$ and reads

$r(\omega)=\sum_{j=1}^{n}r_{j}(\omega)\cdot\exp\left[2ik_{z}(\omega)z_{j}\right],$
(1)

where the real part of the wave vector component $k_{z}(\omega)$ approximates
the dispersion of the dominant component within the sample. The axial sample
structure $r(z)$ can be reconstructed by the Fourier transformation of (1).
When using a common-path interferometer, the phase information of the response
is lost and thus only the intensity reflectivity $R(\omega)=|r(\omega)|^{2}$
is measured. However, reconstruction of the field response from $R(\omega)$ is
possible using a three-step, one-dimensional, iterative phase retrieval
algorithm to recover the lost phase information and subsequently determine the
unique sample response $r(z)$ Fuchs et al. (2017).

Figure 1: Principle of depth profile reconstruction with XCT based on common-
path interferometry. Scattered radiation from different interfaces interferes
with each other, which leads to modulations in the recorded reflectivity
spectrum. The frequencies of spectral modulations carry the structural depth
information of the sample, which can be extracted by Fourier-based algorithms.

The maximum achievable depth resolution $\Delta z$ depends mainly on the
bandwidth of the radiation source used and the XUV transmission bandwidth of
the sample’s dominant material. It is given by

$\Delta z\,\sim\,\frac{\lambda_{0}^{2}}{\Delta\lambda}\,\sim\,\Delta E.$ (2)

Thus, broadband short wavelength radiation in the spectral region of XUV and
soft x-rays enables imaging with nanometer resolution Skruszewicz et al.
(2021). Additionally, the refractive index in this spectral region is
dominated by characteristic atomic levels of inner K-, L- and M-shells.
Therefore, the individual reflectivities of different materials provide a
unique elemental material contrast. However, since this spectral region is
dominated by strong absorption of most elements, XCT is restricted to sample
inspection under vacuum conditions, a limited penetration depth of a few
micrometers, and spectral transmission windows depending on the absorption of
the materials in the sample.

####  2.1.2 Experimental XCT setup

The experimental method rests on recording reflection spectra, which have been
measured using a laser-based broadband XUV beamline Nathanael et al. (2019).
An overview of the used setup is shown in Figure 2. The high-harmonic
radiation between 30 and 100 eV is generated by focusing an intense ultrashort
laser pulse ($E_{P}$ = 2 mJ, $\tau$ = 35 fs, $\lambda$ = 1300 nm) into an
argon gas jet (G) by means of a lens with $f$ = 30 cm focal length (L).
Residual light of the driving laser is removed by a 200 nm thin Al filter foil
(F), while the transmitted XUV radiation is focused by a 1-m focal length
toroidal mirror (TM), thus illuminating a spot size of around 50 µm on the
sample under investigation. The use of Al for filtering limits the useful
transmission window to $<$ 72.6 eV, which is connected to the fundamental
L-absorption. Consequently, an Al dominated sample itself is restricted to be
measured within this window. The used bandwidth in the experimental data
supports an axial (i.e. depth) resolution of 35 nm.

Figure 2: Basic XCT setup consisting of a high harmonic XUV source, the
toroidal focusing mirror TM, the sample and an imaging XUV spectrometer.

Finally, the radiation is reflected at the sample at an angle of incidence of
15∘ and is subsequently detected using a high-resolution imaging XUV
spectrometer Wünsche et al. (2019). It includes a cylindrical focusing mirror
(CM), a variable line space grating (VLS-G) and a CCD camera (C). In order to
cover the full spectral window, quasi-continuous XUV radiation is generated by
fast wavelength shifting of the driving laser field during the sample exposure
Wünsche et al. (2017).

An additional 20 nm thin Si3N4-membrane, which is placed directly in front of
the sample is used to split the incoming beam in order to enable a
simultaneous reference measurement of the source spectrum Abel et al. (2022).
This procedure provides precise data on absolute broadband reflectivity
values, which is essential for the subsequent phase retrieval and sample
reconstruction.

####  2.1.3 Signal analysis and sample reconstruction

An overview of the further data processing after the measurement is shown in
Figure 3. The absolute reflectance of the sample $R(\omega)$ is determined
from the simultaneously recorded and calibrated spectra of the reference
membrane and the sample and serves as a starting point for further sample
reconstruction.

First, the spectral field response $r(\omega)$ of the sample is recovered
using one-dimensional phase retrieval algorithms Fuchs et al. (2017). The
iteratively determined solution with a phase that minimizes the error to the
measured spectrum allows the unambiguous assignment of the sample’s depth
structure using a Fourier transform. At this point, the depth profile of the
sample under investigation is unambiguously revealed for the first time.

Figure 3: Flowchart of structural sequential sample reconstruction based on
XCT measurements and simulations from a sample model. The left edge of the
figure merges into Figure 1.

Additionally, one can extract material specific spectral information of the
individual interfaces. This is realized by truncated Fourier transform of
$r(z)$. The according interfaces are selected by sequential spatial filtering
of $r(z)$ before the Fourier transform, which reveals $r_{j}(\omega)$ of the
addressed interface Wiesner et al. (2021).

The depth structure together with the additional spectral information for each
interface can further be used for model-based fitting of structural parameters
such as thickness and interface roughness for each layer. The respective
procedure starts with the measured depth profile. Information on material
composition of the sample is included based on prior knowledge or from the
just determined characteristic spectral reflectivities $r_{j}(\omega)$. From
the sample model, which includes the materials, thicknesses $d_{i}$ as well as
the surface and the interface roughnesses $S_{q,i}$, the field reflectivity is
calculated using the transfer matrix formalism Saleh and Teich (2019) together
with Nevot-Croce factors Nevot and Croce (1980) and tabulated XUV dispersion
data from Henke et al. (1993). This model-based data is then processed in
exactly the same way as the measurement data and allows an optimization based
on minimizing an error function between model-based data and the measurement.

Structural properties such as layer thicknesses $d_{i}$ and roughnesses
$S_{q,i}$ of the sample are determined sequentially from top to bottom in the
optimization presented here. This means that initially only the properties of
the first layer are determined. Subsequently, the reconstructed parameters
($d_{1},S_{q,1}$) of the first layer are used to improve the sample model. On
this basis, the next layer beneath is optimized. Since dispersion, absorption,
and scattering losses due to the thickness and roughness of the layer above
are already known from the first optimization step, only the parameters of the
lower layer need to be further optimized. The procedure is repeated until the
lowest layer has been addressed. In this way the sample is reconstructed step
by step from top to bottom.

Employing an optimization of the sample model to fit our measurements
additionally allows for the detection of very subtle changes in layer
thickness far below the nominal (i.e. model-free) depth resolution of XCT. The
two interfaces of a layer with a thickness below the depth resolution of XCT
can not be imaged, only a superposition $\tilde{r}_{\text{j}}(\omega)$ of both
interface reflectivities is reconstructed by the phase-retrieval algorithm.
However, using the knowledge about the existence of a thin layer, it is
possible to also reconstruct its thickness. A variation in the range of single
nanometers has a significant effect on $\tilde{r}_{\text{j}}(\omega)$, since
the thickness has a fairly strong influence on the phase difference as shown
in Figure 4. Ultimately, the algorithm achieves sub-nanometer accuracy for the
layer thickness.

Figure 4: The complex-valued field reflectivity $\tilde{r}_{\text{j}}(\omega)$
of a thin layer. The phase strongly depends on the layer thickness $d$, which
allows for the characterization of ultra thin layers even below the limit of
depth resolution.

The sequential optimization routine presented here is fundamentally different
from a direct multidimensional fit of the reflected spectrum. The fitting
procedure based on distinct spectral functions $r_{j}(\omega)$ is very
advantageous since the spatial restrictions during the filtering process
strongly reduce numerical instability. In the picture of frequency space, the
truncated Fourier-transform acts as a sharp frequency filter and only the
modulation that corresponds to the related interface under investigation for
the single optimization step is further evaluated. Thus, the presented
reconstruction offers the advantage to optimize physically measured quantities
by single sequential optimization steps.

###  2.2 Al-Al2Cu sample preparation

An Al-Al2Cu layer system was prepared for investigation with XCT. Al2Cu is an
incongruently melting phase whose preparation as a pure bulk material is only
possible by directional solidification in a temperature gradient Löffler et
al. (2014) or by solidification in the metastable Prince et al. (2020); Fang
et al. (2022). In this work, an induction levitation furnace with a cold-wall
crucible that allows the suppression of the primary phase was used. The Al2Cu
substrate was prepared from raw materials of $\geq$99.99% purity. A 20 g ingot
with the composition Al-33.3Cu (in at.%) was inductively melted in a cold-wall
crucible in Ar atmosphere. The ingot was cut into samples with dimensions of
$10\,\mathrm{mm}\times 20\,\mathrm{mm}\times 2\,\mathrm{mm}$ with a high-speed
saw. Sample surfaces were ground with SiC abrasive paper successively up to
1200 grit and polished to a mirror finish with $9\,\mathrm{\upmu m}$,
$3\,\mathrm{\upmu m}$, $1\,\mathrm{\upmu m}$, and $0.25\,\mathrm{\upmu m}$
polycrystalline diamond suspensions. The polished Al2Cu sample surfaces were
coated with a $\sim 290\,\mathrm{nm}$ thick Al layer using DC magnetron
sputtering in a redesigned Perkin-Elmer 4480 Sputter Deposition Systems
equipped with three DeltaTM cathode positions. The argon gas pressure in the
deposition chamber was $2.5\cdot 10^{-2}$ mbar and a background pressure of
$2\cdot 10^{-7}$ mbar was used. The sputtering power was kept at $1.5$ kW,
resulting in a deposition rate of $0.34$ nm/s. After sputter coating, a final
polishing step was carried out with $50\,\mathrm{nm}$ alkaline colloidal
silica suspension (Cloeren Technology, Germany) in order to achieve a surface
root mean square roughness below $5\,\mathrm{nm}$ while preserving most of the
Al layer.

###  2.3 Sample characterization with atomic force microscopy and transmission
electron microscopy

In order to compare the results obtained from the XCT reconstruction with
results obtained from established characterization methods, additional
characterization of layer thickness and roughness was carried out. The root
mean square roughnesses $S_{q}$ of the substrate before and after coating and
polishing were determined in an area of $1\,\mathrm{\upmu m}\times
1\,\mathrm{\upmu m}$ using atomic force microscopy (AFM, Bruker Dimension
Icon). The AFM was operated in tapping mode utilizing single crystalline
silicon probes with a nominal tip radius of $7\,\mathrm{nm}$.
Cross sections of the Al-Al2Cu samples were prepared for transmission electron
microscopy (TEM) using a dual beam focused ion beam system (FEI Helios NanoLab
600i). The sample surface was covered with protective Pt before the cross
section preparation to minimize damage during sample preparation. The average
thicknesses of the Al layer and the observed oxide layers were determined from
10 measurements along the TEM lamella. High resolution transmission electron
microscopy (HRTEM) and scanning transmission electron microscopy (STEM)
imaging was carried out on the Al-Al2Cu sample (JEOL NEOARM 200 F). Nano beam
electron diffraction (NBED) was used to record diffraction patterns of the
substrate and the sputter-coated Al layer. Crystallographic phases were
identified from NBED patterns and overlaid with corresponding calculated
diffraction patterns using SingleCrystal 4.1.2 (CrystalMaker Software Ltd.,
Begbroke, UK) to verify the identified phases. STEM equipped with energy-
dispersive X-ray spectroscopy (EDXS) was used to map the concentration
distributions in the sample. EDXS concentration profiles were extracted from
the STEM EDXS maps.

##  3 Results

###  3.1 Structural sample reconstruction with XCT

XCT measurement, data processing, and final structural sample reconstruction
were performed as described in section 2.3. The structural model of the sample
is shown in Figure 5a. In addition to the already specified Al2Cu and Al
layer, the existence of a surface oxide layer and a buried oxide layer between
Al2Cu and Al was considered. Al2O3 is the equilibrium oxide for both phases,
i.e. native Al2O3 layers (typically less than 10 nm thick) are expected to
form almost instantaneously when in contact with air Seyring et al. (2021).
Prior to the measurement and sample reconstruction, the thicknesses of the
oxide layers, just like that of the metallic layers, is unknown, only their
possible existence was postulated.

The measured reflection spectrum between 40 and 72 eV is shown in blue in
Figure 5b. The spectral width of 32 eV provides an axial resolution of 35 nm.
Thus, the depth of the interface (including the possible oxide layer) can be
reconstructed by a Fourier transform. For reconstruction of the thicknesses of
the oxide layers, we used the optimization procedure described above based on
the sample model, in which the thicknesses as well as layer roughnesses are
included as free parameters.

Figure 5: (a) Structural model of the sample’s depth profile. (b) Measured
(blue) and calculated reflectivity curve from the reconstructed sample model
(red). The modulations encode the buried interfaces. (c) The sample’s depth
profile $r(z)$ is reconstructed by a phase retrieval algorithm applied to
measured reflectivity data (blue). The calculated response (red) is obtained
from reflection data using the final reconstructed sample model. (d) Truncated
Fourier transform retrieves the effective spectral reflectivity of the surface
(green) and the buried oxide layers (orange). Dashed lines are measured data.
The solid lines represent calculated data from the sample model and dotted
lines display their corresponding error intervals.

For the present sample, this works as follows: A phase retrieval is used to
determine the absolute values of the samples field response $|r(z)|$, which is
shown in blue in Figure 5c. This forms the starting point for the
determination of material-specific spectral reflectivity curves of the
individual interfaces within the sample. For this purpose, the respective peak
is first selected and filtered between the first minima on the left and on the
right in spatial domain. While the surface peak can be Fourier transformed
directly and results in the effective surface reflectivity
$\tilde{r}_{\text{1}}(\omega)$ (dashed green curve in Figure 5d), the depth
signal of the buried oxide needs to be shifted to zero first. As a result, the
solid orange curve for the effective interface reflectivity
$\tilde{r}_{\text{2}}(\omega)$ is obtained.

Theoretical intensity and field reflectivities from a specific sample model
structure are calculated using the transfer matrix formalism Saleh and Teich
(2019) and XUV dispersion data from Henke et al. (1993). Subsequently, the
calculated field reflectivity obtained from the model is processed in the same
way as the measured data including spatial filtering and Fourier transform.
This allows a direct comparison between the XCT measurement and calculations
from the sample model.

Starting from the surface, the sample structure is reconstructed and the
sample model is updated step by step. In the first step, only the surface
roughness $S_{q,1}$ and surface oxide thickness $d_{1}$ are optimized, by
minimizing the deviation between measured and simulated
$\tilde{r}_{\text{1}}(\omega)$ of the surface. The optimization was performed
in a spectral range from 47 to 65 eV, since the edges of the spectral signal
feature larger deviations due to strong spatial filtering. The optimization
yields a surface roughness of $S_{q,1}=2.4\pm 0.2$ nm and an oxide thickness
of $d_{1}=2.1\pm 0.3$ nm. These values are now fixed in the model during
further optimization. The resulting calculated effective surface reflectivity
$\tilde{r}_{\text{1}}(\omega)$ of the optimized surface oxide in the sample
model is shown as a solid green line in Figure 5d.

Subsequently, the thickness $d_{2}$ and roughness $S_{q,2}$ of the buried
oxide layer is optimized. Therefore, the structural sample model is refined by
including the previously reconstructed values for the surface oxide thickness
and roughness. Again the deviation between model based calculations and the
measurement is minimized to recover $d_{2}$ and $S_{q,2}$. The sum of oxide
thickness and Al layer thickness is kept constant, since the absolute depth is
already known from $|r(z)|$ (see Figure 5c). A thickness of
$d_{2}=3.7\begin{smallmatrix}+1.4\\\ -1.0\end{smallmatrix}$ nm and roughness
of $S_{q,2}=3.5\begin{smallmatrix}+0.1\\\ -0.2\end{smallmatrix}$ nm is
obtained for the buried interface oxide and results in the solid orange
simulated curve in Figure 5d. The thickness of the Al layer is determined to
be $230.5\begin{smallmatrix}+0.7\\\ -0.9\end{smallmatrix}$ nm.

###  3.2 Cross section analysis with TEM

Characterization using TEM allows for independently determining layer
thicknesses and compositions with high resolution to verify the XCT
reconstruction results. STEM bright field images show the Al2Cu substrate
below the sputter coated Al layer (Figure 6a and b). Dark regions in the Al2Cu
substrate are a Cu-rich phase, $\eta$-AlCu, that formed in the last steps of
the solidification process due to a local solute enrichment. The Al layer has
a homogeneous thickness across the width of the TEM lamella. The Al layer
thickness excluding oxide layers is measured to be $233\pm 2\,\mathrm{nm}$.
Phase identification using NBED of the Al2Cu substrate confirms the tetragonal
Al2Cu phase with space group I4/mcm (Figure 6c). The Al layer is confirmed to
be in face centered cubic structure in space group Fm-3m (Figure 6d).
Calculated diffraction patterns fit well with the experimentally recorded NBED
patterns. STEM EDXS analysis of the region marked in Figure 6a and given in
Figure 6b reveals oxide layers at the Al2Cu/Al interface and on the Al surface
(Figure 6e). The extracted EDXS profile along the orange shaded area in Figure
6b confirms a layer of pure Al on an Al2Cu substrate. The oxide layers at the
surface and the Al2Cu/Al interface are Al-rich and contain no Cu. The low O
content observed in the Al layer and Al2Cu substrate likely stems from native
oxide layers forming on the top and bottom of the TEM lamella during
thepreparation process.

Figure 6: TEM results of the Al2Cu-Al sample. (a,b) STEM bright field images
showing the Al-Al2Cu-Al layer system. NBED patterns recorded from the Al2Cu
substrate (c) and the Al layer (d). The experimental and calculated
diffraction patterns according to the phase identification are overlaid. (e)
STEM EDXS maps of the region marked in (a) and shown in (b). (f) Extracted
EDXS profile along the orange shaded region in (b).

In order to characterize the thin oxide layers at the surface and Al2Cu/Al
interface in detail, HRTEM was carried out. Both Al-rich oxide layers are
found to be amorphous (Figure 7). The thickness of the oxide that formed on
the sample surface (Figure 7a) is measured to be $d_{1}^{\text{TEM}}=4.1\pm
0.3\,\mathrm{nm}$. The oxide at the Al2Cu/Al interface (Fig. 7b) is slightly
thicker at $d_{2}^{\text{TEM}}=6.0\pm 0.7\,\mathrm{nm}$.

Figure 7: HRTEM images of the oxide layer on the Al surface (a) and at the
Al2Cu/Al interface (b).

###  3.3 Roughness analysis of substrate, coated and polished samples with AFM

Characterization using AFM allows for independent determination of surface
roughness of the sample before and after sputter coating. The AFM scan of the
Al2Cu substrate before coating shows a surface with visible polishing grooves
and a low roughness of $0.9\,\mathrm{nm}$, corresponding to the roughness of
the inner oxide layer $S_{q,2}$ (Figure 8a). After sputter coating of the
290-nm Al layer, the Al surface is made up of individual grains with a
significantly larger surface roughness of $S_{q,\mathrm{sc}}=8.1\,\mathrm{nm}$
(Figure 8b). In order to enable an XCT measurement on the Al2Cu-Al sample, a
further polishing step was carried out as described above. After polishing,
the sample surface is much smoother with an achieved final roughness of
$2.5\,\mathrm{nm}$, corresponding to the surface roughness $S_{q,1}$ (Figure
8c).

Figure 8: AFM scans and measured RMS roughnesses $S_{q}$ for the Al2Cu
substrate (a), the sputter-coated Al surface (b), and the polished Al surface
(c).

##  4 Discussion

The XCT method presented in this paper enables precise, non-destructive
characterization of nanometer-scale layered systems. A complete reconstruction
of the Al-Al2Cu layer system including thickness and roughness values has been
demonstrated. First, the layer thicknesses and interfacial roughnesses
determined by XCT are compared with results obtained by TEM and AFM
measurements (section 4.1). Additionally, the error propagation and the
confidence intervals of thickness and roughness values are discussed (section
4.2). Specific requirements for the surface quality of the sample under
investigation and their influence on the sample reconstruction are discussed
in section 4.3. Finally, the presented XCT method is compared to established
characterization methods and possible applications are highlighted (section
4.4).

###  4.1 Comparison of layer thickness and roughness values obtained by XCT
reconstruction, TEM, and AFM

A comparison of the layer thickness and roughness values attained by the XCT
measurement and subsequent reconstruction with the values measured using TEM
and AFM is shown in Table 1. The Al layer thicknesses as determined separately
by XCT and TEM show very good agreement. Slight deviations in the measured
thickness may result from slight lateral variations in Al layer thickness on
the sample as both XCT ($\sim 50\,\mathrm{\upmu m}$ spot size) and TEM ($\sim
10\,\mathrm{\upmu m}$ lamella width) measure locally. However, lateral
scanning with XCT to investigate variations in thickness and roughness over
the sample surface is straight forward.

Both XCT and TEM confirm the buried oxide thickness $d_{2}$ to be larger than
the surface oxide thickness $d_{1}$. The buried oxide thickness $d_{2}$
measured by TEM is close to the upper error limits of $d_{2}$ obtained from
the XCT reconstruction. Differences in the measured surface oxide thickness
$d_{1}$ can arise from effects during preparation of the TEM lamella, e.g.,
due to the Pt layer deposition on the sample surface during FIB preparation.
Similar to the Al layer thickness, slight lateral variations in oxide layer
thicknesses may also contribute to differences in the values measured with
different techniques.

Surface roughness values $S_{q,1}$ determined by XCT and AFM are in excellent
agreement. The substrate roughness $S_{q,2}$ as measured with AFM is
significantly lower than the roughness of the buried oxide layer $S_{q,2}$ as
reconstructed with XCT. This difference can result from the formation of a
transition layer between the substrate (Al2Cu) and the thin film material (Al)
during the sputter coating. This is supported by the TEM results, as a thin
transition layer is visible as a dark contrast in Figure 6b and below the Al-
rich oxide in Figure 7b. This may lead to an increased buried interface
roughness in XCT measurements when compared to the substrate roughness before
sputter coating. In addition, lateral variations in the substrate roughness
due to polishing grooves may influence the roughness results obtained by AFM,
as the $1\,\mathrm{\upmu m}\times 1\,\mathrm{\upmu m}$ measurement area is
small compared to the XCT spot size.

measured parameter (nm) | XCT | TEM | AFM
---|---|---|---
Al layer thickness | $230.5\begin{smallmatrix}+0.7\\\ -0.9\end{smallmatrix}$ | $233\pm 2.0$ | -
surface oxide thickness $d_{1}$ | $2.1\pm 0.3$ | $\,4.1\pm 0.3$ | -
buried oxide thickness $d_{2}$ | $3.7\begin{smallmatrix}+1.4\\\ -1.0\end{smallmatrix}$ | $\,6.0\pm 0.7$ | -
surface roughness $S_{q,1}$ | $2.4\pm 0.2$ | - | $2.5$
substrate/buried oxide roughness $S_{q,2}$ | $3.5\begin{smallmatrix}+0.1\\\ -0.2\end{smallmatrix}$ | - | $0.9$
Table 1: Layer thickness and roughness values attained by XCT, TEM, and AFM.

###  4.2 Error estimation

To estimate the error of the measured effective layer reflectivities
$\tilde{r}_{\text{1}}$ and $\tilde{r}_{\text{2}}$, the measured reflection
spectrum $R(\omega)$ is modified using different randomized spectral noise
terms that have been characterized for the used setup in Abel et al. (2022).
For each spectrum from the obtained ensemble of spectra, phase reconstruction
and truncated Fourier transform are performed and thus mean and standard
deviation for $\tilde{r}_{\text{1}}$ and $\tilde{r}_{\text{2}}$ are determined
(Fig. 5d). Confidence intervals for the reconstructed layer parameters $d_{j}$
and $S_{q,j}$ are then obtained by the model-based optimization. However,
since the optimization is done sequentially, the error propagation for each
subsequent step needs to be included as well. It is mainly determined by the
transmission of the already determined sample structure above the respective
interface. In each optimization step $j$ the sets of parameters $d_{k}$ and
$S_{q,k}$ with $k<j$ that cause maximal and minimal transmission are
considered for further optimization. In this way the confidence interval for
the optimization step $j$ also includes the error propagation from earlier
steps. Larger confidence intervals are typically caused by a low depth signal
strength of the respective buried layer. Therefore, besides the roughness of
the interface itself, the detection limit of an extremely thin layer is
limited by the transmission of the overlying layer system. This also implies a
strong dependence on roughnesses of the overlying interfaces, as illustrated
for example in section 4.3 using the surface roughness of the Al-Al2Cu layer
system.

Whereas the 35 nm depth resolution in the XCT measurement is limited by the
spectral width of the measured reflection spectrum and describes the minimum
detectable distance between two interfaces, the lower limit for a reliable
layer thickness determination using the model-based reconstruction algorithm
is basically given by the confidence interval of retrieved sample parameters.
In the case of the surface oxide, the limit of a reliable thickness
determination is roughly 0.5 nm, wheras the limit for the interface oxide is
already 1.0 nm.

###  4.3 Sample requirements

In order to achieve successful XCT measurements and a valid sample
reconstruction, a certain surface quality is an important prerequisite, since
broadband XUV reflectivity measurements were performed in near-normal
incidence geometry. Since the XUV illumination is focused, inhomogeneities on
larger scales can be detected in advance and excluded. Homogeneous surface
conditions need to be fulfilled over the illumination spot size of around 50
µm. Smaller variations such as pores will influence the measured reflectivity
and could possibly be detected. However, they will not be spatially resolved.

Figure 9: Influence of the surface roughness $S_{q,1}$ on the XCT
reconstruction based on simulations of the investigated sample structure. (a)
An increasing surface roughness leads to a reduced modulation depth in the
spectral reflectivity. (b) This is transferred to the sample response. (c) The
detection limit due to observed noise level in the reconstructed response of
experimental data specifies the detection limit due to the surface roughness,
which needs to be smaller than $\sim$ 5 nm RMS. (d) The amplitude of the
surface response $r(0)$ as a function of the surface roughness is shown. The
XCT reconstruction fits very well to simulated data including the measured
surface roughness with AFM. Measurements with the unpolished substrate are not
possible, since detected signal is below the noise level.

The surface roughness of a layer-structured sample strongly influences its XUV
reflectivity. Figure 9 illustrates the influence of the surface roughness on
the course of the XUV field reflectivity (a) as well as the reconstructed
depth structure (b). The reflectivity data in Figure 9a are obtained from
calculations based on the structure of the Al2Cu-Al system investigated in the
present work. The surface reflectivity was weighted by a Nevot-Croce factor to
include roughness into the model Nevot and Croce (1980).

The Al2Cu substrate had a low roughness of $S_{q,2}=0.9\,\mathrm{nm}$,
enabling an XCT measurement with sufficient reflectivity. However, after
sputter coating of the Al layer, the increased surface roughness of
$S_{q,\mathrm{sc}}=8.1\,\mathrm{nm}$ leads to a strongly decreased
reflectivity, preventing a successful XCT measurement. This is consistent with
the extracted detection limit due to the surface roughness of approximately
$S_{q}=5\,\mathrm{nm}$, which fits the $\lambda/4$ value of 20 nm radiation
(62 eV). It is determined by the amplitude of residual noise in the depth
response $|r(z)|$ obtained from the measurement as depicted in Figure 9c and
d. Only after the subsequent polishing step, which reduced the roughness to
$S_{q,1}=2.5\,\mathrm{nm}$, a strong reflectivity was observed and an XCT
measurement was possible.

This illustrates a central point regarding sample design and preparation for
XCT measurements for non-destructive depth reconstruction. Thicker sputtered
layers are generally associated with a higher surface roughness Lita and
Sanchez (1999); Elsholz et al. (2005), which has to be taken into account when
preparing a layer system. When the initial surface roughness is too high, the
sample can still be made suitable for XCT measurements with a subsequent
polishing procedure. However, care has to be taken to choose a polishing
procedure that decreases the surface roughness while preserving most of the
sputter coated layer.

###  4.4 Advantages of layer system characterization using XCT and comparison
with established methods

In this study, we successfully applied the XCT method to characterize a layer
system, evaluating both layer thicknesses and roughness down to a nanometer
scale. In order to contextualize the novel method, a comparison with
established characterization methods used for thin layers is presented,
focusing on TEM, atom probe tomography (APT), and AFM.

The XCT method is non-destructive, allowing for further analysis of the same
sample location using different techniques after an XCT measurement, thus
facilitating a variety of analytical approaches. This non-invasive technique
differs from traditional methods that often require destructive sample
preparation. Whereas methods such as TEM or APT require ion milling with a
focused ion beam Mayer et al. (2007); Gault et al. (2021), a process that can
potentially inflict significant damage to the specimen — including ion
implantation or amorphization near the surface Mayer et al. (2007) — XCT
avoids these issues entirely. It maintains the surface of the sample without
any artifacts that are generally associated with ion milling.

In addition to artifacts from the sample preparation, beam damage during TEM
analysis can result in knock-on damage, radiolysis, and localized specimen
heating Williams and Carter (2009); Egerton (2019); Neelisetty et al. (2019);
Jiang (2015). The extent of specimen heating during TEM analysis depends on
numerous variables, particularly the thermal conductivity of the sample under
investigation and the beam conditions utilized Williams and Carter (2009).
Specimen heating is typically negligible for metallic materials with high
thermal conductivity. However, it can be significant for materials with lower
thermal conductivity like ceramics or polymers Williams and Carter (2009).
Using XCT eliminates the risk of unwanted local specimen heating.

TEM and APT are known for their high spatial resolution, which facilitates the
characterization of thin layers at the nanoscale. Although the depth
resolution of XCT used in the present work is $35\,\mathrm{nm}$, the
subsequent reconstruction algorithm allows for very accurate depth profile
reconstruction comparable with TEM, as discussed in section 4.1. The small
illumination spot of XCT ($\sim 50\,\mathrm{\upmu m}$) also enables the
acquisition of localized data from small sample volumes, notably making it
easier to laterally scan the sample surface. This approach simplifies the
comprehensive 3D analysis of layered systems by scanning the sample over a
2-dimensional lateral grid Fuchs et al. (2017).

Moreover, the XCT technique has several distinct advantages. It can evaluate
layer thicknesses and discern roughness values, while also probing buried
interface roughness, as highlighted in this study. Unlike AFM
characterization, which is limited to the sample surface, XCT explores below,
revealing unprecedented details.

The distinct material contrast provided by the XUV spectrum allows for precise
material identification despite variations in composition. This enables the
method to be applied to other material systems, including semiconductors and
technologically relevant alloy systems such as Fe-, Cu-, Ni-, Mg- and Ti-base
alloys. In particular, XCT is highly effective in characterizing thin oxide
layers on metal or alloy substrates, indicating its potential as a key tool in
the study of oxidation-resistant alloys. Such studies are increasingly
significant, particularly in the investigation of the early stages of
oxidation and thus thin oxide layers Apell et al. (2023); Yu et al. (2020);
Apell et al. (2021). Furthermore, a far-reaching characterization of different
coating processes of thin layers becomes possible. It includes the
simultaneous non-destructive determination of layer parameters such as
thickness and roughness, which also allows conclusions to be drawn on the
buried interface quality of processed coated sample substrates.

Finally, the use of an HHG source provides the additional potential for
ultrafast time-resolved XCT measurements, since the pulse duration of the
produced XUV radiation can be on the femto- or even attosecond time scale.

##  5 Conclusions

With XCT, a new non-destructive method for materials characterization is
presented that is complementary to TEM and AFM. TEM investigation requires a
destructive sample preparation and can be limited by specimen damage during
preparation and measurement, including heat insertion, amorphization, knock-on
damage, and radiolysis. Unwanted specimen damage and heating is avoided using
XCT, as heat insertion with the method is negligible. XCT offers information
on roughnesses of internal and external interfaces, where AFM provides
information on external interfaces only. The element specific reflectance
contrast in the XUV range also allows a conclusion on the composition without
any prior knowledge. By combining a sub-microsecond time scale and a nanometer
length scale with XUV contrast based on material composition, the new
methodology offers the possibility to study fundamental aspects in materials
science, e.g., an in-situ measurement of the velocity of off-equilibrium
interfaces in the field of rapid phase transformations that at present is by
far out of reach for any other approach.

## Acknowledgments

This work was supported by the German Research Foundation (grant numbers
390918228 and PA 730/13-1); European Social Fund (ESF) with Thüringer
Aufbaubank (2018FGR008, 2015FGR0094), and Bundesministerium für Bildung und
Forschung (VIP “X-CoherenT”). J. Apell gratefully acknowledges the State of
Thuringia for providing a state scholarship. F. Wiesner is part of the Max
Planck School of Photonics supported by BMBF, the Max Planck Society, and the
Fraunhofer Society.

## Disclosures

The authors declare that there are no conflicts of interest related to this
article. M. Wünsche and S. Fuchs are shareholders of the Indigo Optical
Systems GmbH.

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