Multimethod Metrology of Multilayer Mirrors
Using EUV and X-Ray Radiation
vorgelegt von
Diplom-Physiker
Anton Haase
geboren in Berlin
Von der Fakultät II - Mathematik und Naturwissenschaften
der Technischen Universität Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
Dr. rer. nat.
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr. Norbert Esser
Gutachter: Prof. Dr. Stefan Eisebitt
Gutachter: Prof. Dr. Mathias Richter
Gutachterin: Dr. Saša Bajt
Tag der wissenschaftlichen Aussprache: 30. Oktober 2017
Berlin 2017
Abstract
Multilayer mirrors for the extreme ultraviolet (EUV) spectral range are essential optical
elements of next-generation lithography systems and in scientific applications, e.g. water
window microscopes. Their failure so far to reach theoretically predicted peak reflectivity
values significantly hinders their applicability and raises the question of the reasons
behind that limited performance. This thesis introduces a combination of indirect metro-
logical characterization techniques using EUV and X-ray radiation to enable unambiguous
judgments on the structural properties and interface morphologies of those multilayer
systems, providing possible answers.
The approach was used to study two sets of unpolished and interface-polished
Mo/Si/C multilayer systems designed to reflect EUV radiation with
13.5nm
wave-
length. These were fabricated with increasing molybdenum thickness from sample to
sample. By examining the combination of EUV reflectivity and X-ray reflectivity (XRR),
and considering experimental uncertainties, structural parameters were reconstructed
and validated through the deduction of confidence intervals. By establishing a method
for the analysis of EUV diffuse scattering, an observed minimum in the peak reflectance
for some samples could be related to variations in layer thickness and interface rough-
ness associated with crystallization in the molybdenum layers. Increased roughness for
samples at the crystallization threshold and intermixing were identified as impeding the
measured reflectance.
Furthermore, the new methodology was applied to Cr/Sc multilayer mirrors for the
water window spectral range having individual layer thicknesses in the sub-nanometer
regime. The combination of the analysis of EUV reflectivity and of XRR based on a
binary layer model was shown to be insufficient to describe this system. The model was
extended to explicitly take into account gradual interface profiles and strong intermixing.
It was demonstrated by structural characterization and systematic validation of the
extended model parameters, based on the analysis of EUV reflectivity, resonant extreme
ultraviolet reflectivity (REUV), XRR and X-ray fluorescence (XRF) experiments, that only
the combination of those analytic methods yields a consistent result. Augmenting the
characterization through the EUV diffuse scattering analysis explains the low reflectivity
as resulting from a theoretical model that is too simplistic.
Zusammenfassung
Mehrschichtspiegel für den EUV Wellenlängenbereich sind wichtige optische Komponen-
ten für die nächste Halbleiterlithografiegeneration und kommen auch im wissenschaftli-
chen Bereich, beispielsweise in Mikroskopen für das Wasserfenster, zum Einsatz. Deren
verminderte Reflektivität im Vergleich zu den theoretisch möglichen Werten schränkt ihre
Einsatzfähigkeit ein und wirft die Frage nach den Ursachen dafür auf. In der vorliegenden
Dissertation wurde eine Kombination von metrologischen indirekten Charakterisierungs-
techniken unter Anwendung von EUV und Röntgenstrahlung eingeführt. So wurden
Rückschlüsse auf die Struktur und Grenzflächenmorphologie der Mehrschichtsysteme
eindeutig möglich.
Die Methodik wurde zur Untersuchung von Mo/Si/C-Mehrschichtsystemen mit po-
lierten und unpolierten Grenzflächen eingesetzt, welche als Spiegel für EUV-Strahlung
mit
13.5nm
Wellenlänge dienen. Die Mehrschichtsysteme wurden mit wachsender Mo-
lybdänschichtdicke von Probe zu Probe hergestellt. Die kombinierte Analyse von EUV-
Reflektivität und Röntgenreflektivität unter Berücksichtigung der experimentellen Unsi-
cherheiten ermöglichte eine Bestimmung der strukturellen Modellparameter und deren
Konfidenzintervalle. Die Einführung einer Methode zur Analyse diffuser EUV Streuung
erlaubt ferner die Korrelation beobachteter Reflektivitätseinbrüche in bestimmten Proben
mit Variationen der Schichtdicken und der Grenzflächenrauigkeit durch Kristallisation in
den Molybdänschichten. Erhöhte Rauigkeit an der Kristallisationsschwelle und Durch-
mischung an den Grenzflächen konnten als Ursache der beeinträchtigten Reflektivität
eindeutig identifiziert werden.
Die hier etablierte Methodologie wurde desweiteren auf Cr/Sc-Mehrschichtspiegel
für das Wasserfenster angewandt. Die Kombination von EUV- und Röntgenreflekti-
vität basierend auf einem binären Schichtmodell stellte sich bei diesem System als
unzureichende Beschreibung heraus. Daher wurde das Modell erweitert, um graduelle
Grenzflächenprofile und starke Vermischung explizit zu berücksichtigen. Auf Grundlage
der Strukturanalyse mittels EUV-Reflektivität, resonanter EUV-Reflektivität, Röntgen-
reflektivität und Röntgenfluoreszenz und anschließender Validierung konnte gezeigt
werden, dass nur die Kombination all dieser analytischen Methoden ein konsistentes
Ergebnis liefert. Die Erweiterung dieser Charakterisierung durch diffuse EUV-Streuung
erklärt eindeutig die Ursachen für die geringe Reflektivität.
Contents
1 Introduction 1
2 Theoretical Description of EUV and X-ray Scattering 7
2.1EUV and X-ray Radiation 7
2.2Interaction of EUV and X-ray Radiation With Matter 8
2.2.1Elastic Scattering 11
2.2.2Absorption and Fluorescence 13
2.3
Specular Reflection from Surfaces and Interfaces in Layered Systems
14
2.4Diffuse Scattering in Layered Systems 19
2.5Grazing-incidence X-ray Fluorescence 28
3 Experimental Details and Analytical Toolset 33
3.1Synchrotron Radiation 34
3.2The Instrumentation for the EUV Spectral Range 37
3.2.1The EUV Beamlines at BESSY II and MLS 37
3.2.2
The Experimental Endstations at the EUVR and SX700 Beam-
lines 41
3.3Grazing-incidence X-ray Fluorescence at the FCM Beamline 43
3.4Sample systems 44
3.4.1Choice of the Chemical Species and Multilayer Design 45
3.4.2Multilayer Deposition by Magnetron Sputtering 47
3.5Analytical Tools 48
4 Characterization of the Multilayer Structure for Different Systems 51
4.1Reconstruction Based on Specular EUV Reflectance 52
4.1.1Multilayer Model and Particle Swarm Optimization 54
4.1.2Model Uniqueness and Maximum Likelihood Estimation 58
4.2Molybdenum Thickness Variation in Mo/Si/C Multilayers 62
4.2.1Sample Systems and Experimental Procedure 63
4.2.2Combined Analysis of X-ray and EUV reflectance 64
4.2.3Optimization Results 69
4.3
Analysis of Cr/Sc Multilayers with Sub-nanometer Layer Thickness
72
vii
Contents
4.3.1Reconstruction with a Discrete Layer Model Approach 73
4.3.2
Extending the Model to Graded Interfaces and Interdiffusion
76
4.3.3Addition of Complementary Experimental Methods 81
4.3.4Reconstruction and Maximum Likelihood Evaluation 83
5 Analysis of Interface Roughness Based on Diffuse Scattering 93
5.1Near-normal Incidence Diffuse Scattering 94
5.1.1Mapping Reciprocal Space for the Mo/B4C/Si/C Sample 97
5.1.2Kiessig-like Peaks and Resonant Effects 99
5.1.3
Reconstruction of the PSD and the Multilayer Enhancement Fac-
tor 104
5.2
Differently Polished Mo/Si/C Multilayers with Molybdenum Thickness
Variation 109
5.2.1Reconstruction of the Interface Morphology 112
5.2.2Discussion of the Results 115
5.3Roughness and Intermixing in Cr/Sc Multilayers 117
5.3.1
Estimation of the Vertical Roughness Correlation and the PSD
119
5.3.2Results and Conclusions 121
6 Summary 123
References 127
viii
List of Figures
2.1Illustration of X-ray fluorescence for an atom. . . . . . . . . . . . . . . . . . 13
2.2Illustration of Snell’s law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3Field amplitudes in the exact solution for a multilayer system. . . . . . . . 16
2.4Schematic layout of periodic multilayer systems . . . . . . . . . . . . . . . 18
2.5Specular reflectivity from periodic multilayer systems . . . . . . . . . . . . 19
2.6Scattering geometry and definition of the scattering vector. . . . . . . . . . 19
2.7Illustration of the four scattering processes of the DWBA. . . . . . . . . . 23
2.8
Illustration of the perturbation potential
Vi
r(
~
r)
at the
i
th interface of a
multilayer system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.9Qualitative illustration of the Hurst factor. . . . . . . . . . . . . . . . . . . 25
2.10 Illustration of correlated roughness in a binary periodic multilayer stack. 26
2.11 Illustration of orthogonal and non-orthogonal correlated roughness. . . . 27
2.12 Calculation scheme for the X-ray fluorescence . . . . . . . . . . . . . . . . 29
2.13 Principle of X-ray standing wave fluorescence analysis. . . . . . . . . . . . 31
3.1Theoretical synchrotron radiation radiant power spectra . . . . . . . . . . 34
3.2Schematic overview of BESSY II. . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3Schematic principle of insertion devices. . . . . . . . . . . . . . . . . . . . . 37
3.4Schematic overview of the MLS . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.5Schematic setup of the SX700 beamline. . . . . . . . . . . . . . . . . . . . . 39
3.6Radiant power of the SX700 beamline. . . . . . . . . . . . . . . . . . . . . . 40
3.7Schematic optics of the SX700 and EUVR beamlines. . . . . . . . . . . . . 41
3.8The EUV reflectometer end station of the EUVR beamline. . . . . . . . . . 42
3.9The EUV ellipso-scatterometer end station at the SX700 beamline. . . . . 43
3.10 FCM beamline scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.11 The GIXRF chamber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.12 Refractive indices of Cr and Sc in the water window. . . . . . . . . . . . . 46
3.13 Refractive indices of Mo and Si for wavelengths from 12.4nm to 14.0nm. 47
3.14 Schematic setup of a magnetron sputtering deposition system. . . . . . . 48
3.15 Photograph of a Mo/Si multilayer sample. . . . . . . . . . . . . . . . . . . 48
4.1Spectrally resolved reflectance of the Mo/B4C/Si/C multilayer sample. . 53
4.2Model of the Mo/B4C/Si/C multilayer stack. . . . . . . . . . . . . . . . . . 54
4.3Theoretical EUV reflectance curve for the Mo/B4C/Si/C sample. . . . . . 57
4.4Influence of the model parameters on the simulated EUV reflectivity curve.58
ix
List of Figures
4.5
Results of the maximum likelihood estimation for Mo and Si thicknesses
of the Mo/B4C/Si/C sample. . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.6
Results of the maximum likelihood estimation for the remaining model
parameters of the Mo/B4C/Si/C sample. . . . . . . . . . . . . . . . . . . . 61
4.7
Measured EUV reflectivity data for the polished and unpolished Mo/Si/C
samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.8XRR data for all unpolished and polished Mo/Si/C samples. . . . . . . . 65
4.9Model of the Mo/Si/C multilayer stack. . . . . . . . . . . . . . . . . . . . . 66
4.10 Combined analysis of XRR and EUV reflectivity for the Mo/Si/C samples. 68
4.11 Correlation of silicon and carbon layer thickness parameters in the model. 69
4.12
Experimental EUV reflectivity data in comparison with the theoretical
curves for an unpolished Mo/Si/C sample. . . . . . . . . . . . . . . . . . . 69
4.13 Fitted dMo and Dvalues for both Mo/Si/C sample sets. . . . . . . . . . . 70
4.14
Peak reflectance values for each Mo/Si/C sample in comparison with
theoretical expectation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.15 Model of the Cr/Sc multilayer stack. . . . . . . . . . . . . . . . . . . . . . . 72
4.16 EUV and XRR data recorded for the Cr/Sc sample system. . . . . . . . . . 74
4.17 Fitted EUV reflectance curves for the Cr/Sc sample. . . . . . . . . . . . . . 75
4.18 Comparison of EUV and XRR fitting results for the binary Cr/Sc model. 76
4.19 Binary and gradual Cr/Sc multilayer models. . . . . . . . . . . . . . . . . 77
4.20
Comparison of the numerical uncertainty with the experimental uncer-
tainty for the graded Cr/Sc model. . . . . . . . . . . . . . . . . . . . . . . . 79
4.21 Reconstruction for the gradual model based on EUV reflectivity and XRR. 79
4.22 EUV peak deformation for a constant drift in the Cr/Sc period thickness. 80
4.23 Measured resonant EUV reflectivity curves across the Sc L2and L3-edge. 81
4.24
Measured relative XRF curves for the Cr and Sc K-lines across the first
Bragg peak of the Cr/Sc sample. . . . . . . . . . . . . . . . . . . . . . . . . 83
4.25 Full data set from the Cr/Sc sample used in the combined analysis. . . . 84
4.26
Measured data and optimized theoretical curves the combined analysis of
the Cr/Sc system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.27
Matrix representation of the maximum likelihood analysis for the Cr/Sc
sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.28
Correlation of the roughness and intermixing parameter in the Cr/Sc sample.
89
4.29 Illustration of the confidence intervals for the Cr/Sc model parameters. . 89
4.30 Multilayer structure for best binary and gradual model results. . . . . . . 90
5.1Co-planar measurement geometries for the diffuse scattering. . . . . . . . 95
5.2Schematic measurement paths in reciprocal space. . . . . . . . . . . . . . . 95
5.3Schematic illustration of the appearance of Bragg-sheets. . . . . . . . . . . 96
5.4Measured intensity map of a detector scan of the Mo/B4C/Si/C sample. 98
5.5Illustration of dynamic scattering processes. . . . . . . . . . . . . . . . . . 99
5.6Measured reflectivity curve of the Mo/B4C/Si/C multilayer mirror. . . . 100
5.7Calculated positions of the Kiessig-like lines in the reciprocal space maps. 101
5.8
Calculated diffuse scattering intensity distribution at
qz=0.93
nm
−1
for
the Mo/B4C/Si/C mirror. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.9Calculated diffuse scattering intensity along a vertical cut in qz. . . . . . . 104
5.10 Averaged diffuse scattering intensity along the Bragg sheet resonance. . . 105
5.11 Multilayer enhancement factor for three different measurement geometries.106
x
List of Figures
5.12
Diffuse scattering intensity corrected for the multilayer enhancement factor.
106
5.13
Measured reciprocal space maps for the detector scan geometry and the
rocking scan geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.14 Measured diffuse scattering distributions of the Mo/Si/C samples. . . . . 110
5.15
Diffuse scattering maps of the Mo/B
4
C/Si/C sample for two rotational
orientations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.16 Direct comparison of the measured and calculated reciprocal space maps. 113
5.17
Root mean square roughness and Névot-Croce factor results from the
analysis of the diffuse scattering for the two sample sets. . . . . . . . . . . 115
5.18 Diffuse scattering measurement for the Cr/Sc sample. . . . . . . . . . . . 118
5.19
Diffuse scattering measurement and DWBA calculation for the Cr/Sc mirror.
119
5.20 Measured data and calculations at the vertical cut. . . . . . . . . . . . . . . 119
5.21 Comparison of the extracted effective PSDs. . . . . . . . . . . . . . . . . . 120
xi
List of Tables
3.1Beamline parameters of the two EUV beamlines in comparison. . . . . . . 41
4.1Multilayer parametrization and parameter limits . . . . . . . . . . . . . . . 55
4.2Results for the optimized parameters for the Mo/B4C/Si/C sample. . . . 57
4.3
MCMC results obtained by the analysis of the EUV reflectivity for the
Mo/B4C/Si/C sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4Parametrization of the Mo/Si/C multilayer samples. . . . . . . . . . . . . 66
4.5Nominal molybdenum layer thicknesses in the two Mo/Si/C sample sets. 70
4.6Parametrization of the Cr/Sc binary multilayer model. . . . . . . . . . . . 73
4.7PSO fit results for the discrete layer Cr/Sc multilayer model. . . . . . . . . 75
4.8Multilayer parametrization and parameter limits . . . . . . . . . . . . . . . 78
4.9Optimized model parameters of the capping layers in the Cr/Sc system. . 84
4.10
Optimized model parameters with confidence intervals for the Cr/Sc system.
85
5.1Parameters and limits of the DWBA analysis. . . . . . . . . . . . . . . . . . 108
5.2
List of the reconstructed molybdenum layer thicknesses in the selected
samples in both Mo/Si/C sets. . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.3
Results for the DWBA model parameters with the respective confidence
intervals for both sample sets. . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.4
Best model parameters and confidence intervals of the PSD for the gradual
Cr/Sc system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
xiii
1
Introduction
In 1959, Jack S. Kilby made an invention at the root of the technological revolution in
the years that would follow. His development of the first integrated circuit was the
realization of a logical element known as flip-flop, capable of storing a single bit, by
implementing a layout that could host all required circuits on a single semiconductor
wafer piece [75]. His achievement paved the way for the miniaturization of electronic
circuits that enabled the technological advancements we have experienced over the
past 57 years, and was recognized as part of the Nobel prize in physics in 2000 [135].
Only two years after the original invention, Robert N. Noyce submitted a patent on the
fabrication of integrated circuits in monolithic single crystals, using photo lithography to
create the necessary artificial structures [103]. This technique of using light to transfer
a pattern from a photomask onto a semiconductor wafer has prevailed over the course
of the technological development and is still the primary method for the fabrication of
computer chips today [91]. As the technology has improved over time, progress has
roughly followed Moore’s law of doubling the transistor count on a unit area of the wafer
every two years [99]. Consequently, the structure size on the wafers has shrunken, to
accommodate the large number of circuits on a single chip. Today, structure sizes in the
lower nanometer regime have been reached [69], but only through the implementation of
additional methods augmenting the optical lithography. With the extreme decrease in
size, Moore’s law now threatens to break down [90,114]. The technological requirements
on the lithography systems used to fabricate those chips in mass production have thus
increased significantly.
A basic principle of optical resolution known as the Rayleigh criterion states that the
minimum structure size achievable with a purely optical system is proportional to
the wavelength used [86]. Consequently, while the first lithography systems used in
the semiconductor industry operated in the visible spectrum, wavelengths have been
reduced to the deep ultraviolet (DUV) regime, the current standard at
193nm
, in order
to keep pace with Moore’s law. However, with required feature sizes of only a few
tenths of nanometer, a significant further reduction of the wavelength is unavoidable,
as lithography at optical wavelengths has reached its physical limits. Next-generation
lithography uses wavelengths in the extreme ultraviolet (EUV) spectral range of
13.5nm
.
1
INTRODUCTION
This radiation is strongly absorbed by all materials, including air, challenging the design
of the optical lithography systems by effectively ruling out any optical design based on
transmission lenses for focusing and imaging. With the semiconductor industry at the
verge of a major technological change, the topic of reflective optical elements for EUV
radiation has received significant attention and extensive research efforts [10].
In 1972, Eberhart Spiller proposed a new design for efficient mirror systems working
at incidence angles near the surface normal for strongly absorbed radiation such as
EUV. The idea was based on fabricating artificial layer systems reflecting portions of the
incoming radiation at each interface that would interfere constructively at acceptable
absorption levels, overcoming the extremely low reflection otherwise seen from single
surfaces [128]. The result are multilayer Bragg reflectors, which fulfill the Bragg condition
for constructive interference for specific pairs of wavelength and angle of incidence,
and thus require specific design. At angles close to the surface normal, layers with a
thickness on the order of half the wavelength are necessary, which requires fabrication
methods capable of precisely depositing layers of only several nanometers thick. Since
the original proposal, multilayer systems have been realized using evaporation and
sputtering techniques, and have been demonstrated to increase reflection [129,139]. As
the technology developed and more advanced sputtering techniques became available
to fabricate at the necessary precision [133], the first important applications of focusing
multilayer mirrors were space probes used for the observation of the sun in the EUV
spectrum [32,33,130].
Theoretical models and calculations of candidate systems for large reflectivity close to
normal incidence at a wavelength of
13.5nm
show peak values of approximately
72%
,
by using multilayer systems based on molybdenum (Mo) and silicon (Si) [12,13,50].
State of the art systems reach values slightly above
70%
[8,29,30,49,95], which is
still a few percentage points below the theoretical limit. This is of particular concern
for the usage in EUV lithography systems, where
11
near-normal incidence reflections
from the source to the wafer are required to image a structure [71,142]. Even at the
theoretical threshold, with
11
reflections only
3%
of the radiation reaches the wafer. Thus,
even a small difference to the theoretical reflection limit has a large impact on the total
radiant power at the wafer level. This is a very crucial point in the development of the
next-generation lithography using EUV radiation.
While the semiconductor industry without doubt is a strong driving force in the
development of EUV multilayer optics for
13.5nm
wavelength, mirrors for other spectral
ranges suffer the same problem. A relevant system to this work is a mirror designed
to reflect radiation in the range of the so-called water window which is found between
2.3
nm and
4.4
nm. The water window is of special interest, because radiation in this
spectral range shows low absorption in water, while it is absorbed by many elements,
most importantly carbon and nitrogen, naturally occurring in organic molecules such
as proteins [76]. This allows the study of biological systems in their native environment
(water), where many proteins are biologically active. With the ability to produce radiation
at those wavelengths at free-electron laser (FEL) sources [2,117], more applications with
strong and coherent pulses are coming into reach. High resolution imaging of protein
samples, in addition to the required short wavelength, needs sufficient reflected radiation
intensity, and more generally, optical elements capable of focusing and magnification.
This can be achieved with high reflectance multilayer mirrors [63,80]. A candidate
system, relevant to this spectral range, is made from chromium (Cr) and scandium (Sc),
applying the very same principle as introduced above, albeit with a much thinner layer
2
thicknesses, corresponding to the shorter wavelength. While at
3.1nm
wavelength, the
theoretical reflectance limit is calculated to reach values above
50%
[113], state of the
art mirrors only show reflectivities below
20 %
[48,145], that is less than half of the
theoretically possible values.
The main reasons for radiation loss, beyond unavoidable absorption inside the materials
of both the Mo/Si and Cr/Sc multilayer systems, are imperfections at the interfaces,
such as compound formation, intermixing, and roughness. As a result, the perfect
multilayer system is distorted, since the interfaces are not chemically abrupt anymore.
Thus, intermixing and compound formation lead to a diminished optical contrast and
consequently to lower reflectance at the respective interface [101]. This is a known
problem for multilayer optics, and measures taken to counteract this effect are the
introduction of barrier layers hindering the formation of intermixing layers in some of
the systems [29,30]. In the case of roughness, the result of reduced optical contrast at the
interfaces is the same on average for the impinging wavefield, with additional scattering
outside the specular beam direction [124]. This scattering is not present in the case of
pure intermixing.
To minimize interface distortions and to ultimately increase the reflectivity of the
respective systems, research and industry groups concerned with fabricating multilayer
mirrors require detailed information on the structural properties and interface mor-
phology of their samples. The characterization of those multilayer systems is thus a
cornerstone in the effort for improve mirror reflectivity, and for the fundamental under-
standing of the effects involved. There are several characterization techniques that have
been applied to assess and quantify the structure of the layer system, roughness and
intermixing of materials at the interfaces of multilayer mirrors in the past. They can be
roughly categorized as direct scanning methods and indirect ensemble methods.
Some widely used example in the first category is transmission electron microscopy
(TEM), which establishes a microscopic approach to the problem of assessing the interface
morphology with a resolution at the nanoscale [9,134]. By imaging the layer stack,
interface imperfections can be made visible directly. In combination with high-resolution
electron energy loss spectroscopy (HREELS), element-specific interface profiles can be
deducted, giving insight into the intermixing behavior of two (or more) materials at the
interfaces [43,108]. A large downside of both methods, however, is the intrinsically local
area of the image and thus the characterization of only very small local portions of the
entire sample. Apart from that, the stack needs to be cut open to apply these techniques
and thus leads to a destruction of the sample.
Another popular method, before and after deposition of a multilayer stack, is atomic
force microscopy (AFM) [19]. It is a scanning technique with nanometer resolution,
allowing to determine the morphology of a surface and thus to investigate its roughness.
However, it faces the same locality obstacle as TEM or HREELS, and can only operate
on exposed areas. Thus, the morphology of buried structures remains hidden to this
method. Nevertheless, it is applied to determine the initial substrate roughness and the
condition of the final top surface as an important prerequisite for high-quality multilayer
mirror fabrication [9,88].
Apart from the direct and local scanning techniques, indirect ensemble methods based
on the elastic scattering of radiation are accurate and extensively used in multilayer
characterization. Examples include X-ray reflectivity (XRR) and EUV reflectivity with
resonant extreme ultraviolet reflectivity (REUV) as a variation of the latter. They are
employed as standard methods in multilayer mirror fabrication and the subsequent
3
INTRODUCTION
device characterization [9,30,84,115]. Other techniques, sensitive to structural proper-
ties, are spectroscopic ellipsometry and X-ray fluorescence (XRF). Ellipsometry delivers
information on the optical constants and layer thicknesses by measuring the altering of
the polarization state of the impinging radiation after reflection from the sample [7,85].
With XRF, fluorescence radiation of the materials inside the multilayer stack is excited
with X-rays energetically slightly above the materials’ respective absorption edges, and
subsequently detected to analyze the structure [72,77]. The major advantage of all
these techniques is that they are non-destructive and contactless, and quickly obtain
information on the buried structure, as well as on the top-surface condition. Furthermore,
statistical information across a large area depending on the beam footprint of the im-
pinging radiation is obtained, in contrast to the aforementioned local methods. However,
it is no longer possible to directly gain information on the multilayer stack, as in all of
the above examples, theoretical models are required to calculate the expected results
from a certain model, and to compare that to the measurement outcome. This is known
as the inverse problem. Reconstruction of the model parameters by fitting calculations
to the experimental data raises the question of uniqueness and accuracy of the found
solution. The fundamental applicability of the model itself and its limitations are of
great importance to these considerations. Studies have shown that the combination of
EUV reflectivity and XRR can lead to significant improvements in the accuracy com-
pared to standalone measurements with each technique individually [144]. Similarly, by
XRF further complementary information can be added to assist in the solution of this
problem [54].
While these experiments allow to obtain structural information on the layer stack
through reconstruction of a theoretical model, only limited information is gained on the
roughness of the interfaces, which cannot be distinguished from intermixing. However,
as only roughness causes diffuse scattering, the analysis of the off-specular intensity
upon irradiation of a multilayer stack is a natural tool for the characterization of the
interface morphology. Significant theoretical and experimental work has been conducted
in towards the study of diffuse scattering from multilayer samples in grazing incidence
geometries using X-rays, e.g. by grazing-incidence small-angle X-ray scattering (GISAXS),
at small incidence angles [22,24,83,96,112,123,124], but also in the optical and
EUV regime [4,5,45,46,118,119], to deduce the desired information on the interface
roughness.
This work was performed at the Physikalisch-Technische Bundesanstalt (PTB). As the
German national metrology institute (NMI), the PTB is dedicated to precise measurements
related to all fields of physics and technology providing metrology as its core mission.
In fact, the international metrology organization, the Bureau International des Poids et
Mesures, defines
*
metrology as “the science of measurement, embracing both experimental
and theoretical determinations at any level of uncertainty in any field of science and technology.”
In the PTB, over a quarter-century of experience and expertise in the field of metrology
with synchrotron radiation exists [17], with a particular focus on industry applications
such as next-generation lithography.
In this spirit, the scope of this thesis is to provide methods of metrology for multilayer
optics. It is dedicated to the accurate and complete characterization of the structural
properties and the interface morphology of multilayer mirrors to gain insight into the
origin of their limited performance. The uniqueness-problem associated with any model-
based indirect characterization approach has remained largely unanswered and requires
*Source: http://www.bipm.org/en/worldwide-metrology/
4
a response. In this thesis, the data from different indirect experiments was analyzed
with the goal of answering the question for multilayer mirror systems. Experimental
uncertainties, inevitably associated with any measurement, and model uncertainties had
to be investigated with respect to the effect on the results obtained from each method.
Based on theoretical optimization algorithms, confidence intervals for each reconstructed
parameter of the underlying models can be deducted, which allows to validate the
results of established characterization techniques. Improvement of the models and the
exploitation of several experimental techniques such that unequivocal judgments on the
causes of the reduced multilayer reflectance can be made are thus the major focus of this
work.
The thesis is structured in the following way. Chapter 2introduces the fundamental
theoretical concepts underlying the interaction of multilayer systems with EUV and X-ray
radiation. The theoretical basis of the analytic experiments (EUV reflectivity, REUV, XRR,
XRF and EUV diffuse scattering) conducted in this thesis to characterize the various
samples is given. In chapter 3, the different experimental setups in the PTB laboratories
at the two storage rings metrology light source (MLS) and electron storage ring for
synchrotron radiation (BESSY II) used in obtaining the analyzed data are presented.
Samples fitting in two major categories of multilayer mirrors for two different spectral
ranges were investigated. They were fabricated using a sputtering technique, which is
briefly reviewed. Furthermore, the extensive software that was developed over the course
of this thesis is summarized. The first relevant sample systems designed to operate as
mirrors at
13.5nm
wavelength are two sets of Mo/Si/C multilayers with an increase
of the molybdenum layer thickness from sample to sample from nominally
1.7nm
to
3.05nm
, crossing the threshold for crystallites forming in these layers. The second set
was treated using an ion polishing technique during deposition, with the goal to reduce
roughness at the interfaces. The methods employed for this system were compared to the
reconstruction of a state-of-the-art Mo/B
4
C/Si/C multilayer mirror. The second major
investigated sample system are Cr/Sc multilayer mirrors for the water window with
nominal layer thicknesses in the sub-nanometer regime. The structural reconstruction
of the Mo/Si and Cr/Sc multilayer mirrors based on the combination of the different
experiments is presented in chapter 4. Here, the validity of the models and the accuracy
of the reconstructed parameters with their confidence intervals is discussed in depth.
Chapter 5addresses the evaluation of the interface morphology of these samples based on
the EUV diffuse scattering measurements and the models reconstructed in the previous
chapter. The summary and conclusions of this thesis are in chapter 6.
Large parts of this thesis have been published in peer-reviewed journals and conference
contributions [58–61]. A reference to the relevant publications is given at the end of each
chapter.
5
2
Theoretical Description of EUV and
X-ray Scattering
This chapter summarizes the aspects of the interactions of electromagnetic radiation with
matter, relevant for this investigation. Since this thesis specifically covers the interaction
of EUV and X-ray radiation with multilayer systems, in particular the basic principles
of specular reflection and transmission through a stack of layers are given. Then, the
diffuse scattering theory for multilayer systems is derived based on the well established
distorted-wave Born approximation. Finally, the generation of fluorescence radiation and
its exploitation for the analysis of multilayer compositions is described.
2.1 EUV and X-ray Radiation
EUV and X-ray radiation is electromagnetic radiation, which only differs by its wave-
length. The different names for these parts of the electromagnetic spectrum are mostly of
historic origin. However, differences in energy and, thus, reflectance, transmission and
absorption properties in matter still justify this differentiation today from a technical per-
spective. For the sake of consistency within this thesis and the lack of a unique definition
of the terms used in literature, we shall define EUV radiation as electromagnetic radiation
within the spectral range from
1nm
to
100nm
vacuum wavelength (corresponding to
photon energies of approximately
12.4eV
to
1240eV
). Consequently, the radiation with
the wavelengths below
1.0nm
(photon energies above
1.24keV
) shall be called X-rays. In
both cases the theoretical description is identical and is thus presented here independent
of this naming convention.
The entirety of electrostatic fields and electromagnetic radiation is described by
Maxwell’s equations. In vacuum they are defined as
∇·~
E=0, ∇·~
B=0,
∇×~
E=−∂~
B
∂t,∇×~
B=µ0e0∂E
∂t,
7
Chapter 2 THEORETICAL DESCRIPTION OF EUV AND X-RAY SCATTERING
with the electric constant
e0
and the magnetic constant
µ0
and the electric field
~
E
and
the magnetic field
~
B
. By taking the curl of these equations and using the identity
∇×(∇×~
X) = ∇(∇·~
X)−∆~
X
and the Laplacian
∆=∇2
for an arbitrary vector field
~
Xthe Maxwell equations yield the wave equations
∆~
E−1
c2
∂2~
E
∂t2=0, ∆~
B−1
c2
∂2~
B
∂t2=0, (2.1)
with c=1/õ0e0, the speed of light in vacuum.
All scattering processes and charge densities in this thesis are considered to be time-
independent. The wave equations Eq.
(2.1)
can thus be further simplified by separating
the explicit time dependence of the fields as
~
E(
~
r,t) = ~
E(
~
r)eiωt,~
B(
~
r,t) = ~
B(
~
r)eiωt, (2.2)
where~
ris a vector to a point in space. The time-independent wave equations then read
(∆+k02)~
E=0, (∆+k02)~
B=0, (2.3)
where
k0=ω/c=2π/λ
, i.e. the absolute value of the vacuum wave vector. A very
important and often applied solution to this wave equation is the monochromatic plane
wave. Hence, for Eq. (2.3) we obtain
E(
~
r,t) = E0eiωt−i
~
k·~
r,B(
~
r,t) = B0eiωt−i
~
k·~
r, (2.4)
where
E0
and
B0
are the initial electric and magnetic field amplitudes, respectively, and
|~
k|=k0[27].
2.2 Interaction of EUV and X-ray Radiation With Matter
The wave equations Eq.
(2.3)
still hold for the propagation of radiation inside an isotropic
*
,
homogeneous medium in slightly modified form. The Maxwell equations contain the
electric permittivity and magnetic permeability, which are different for electric and
magnetic fields inside a medium compared to the respective quantities in vacuum
(electric and magnetic constants). The equations inside a medium are therefore obtained
by replacing e0→e=ere0and µ0→µ=µrµ0,
∇×~
E=−∂~
B
∂t,∇×~
B=µrµ0ere0∂E
∂t, (2.5)
where
er
is the relative electric permittivity and
µr
is the relative magnetic permeability.
These quantities are defined through the electric displacement field
~
D=e~
E
, which
remains unchanged at the interface of vacuum and matter, and the magnetic field relation
~
B=µ~
H
(for para- and diamagnetic materials with a magnetization parallel to the field
lines). In case of electromagnetic waves in the EUV and X-ray spectral range, the latter
does not differ significantly from one and is often approximated by
µr≈1
[18]. The
electric permittivity, however, can take significantly different values inside matter than in
vacuum. An electric field entering a medium causes a polarization field
~
P
of that matter
*
The general case including anisotropic materials can also be described with the wave equation. In that
case the scalar coefficients for isotropic materials become tensors.
8
Interaction of EUV and X-ray Radiation With Matter 2.2
depending on the respective polarizability. The displacement field is given as
~
D=~
E+~
P
and remains constant at the interface as mentioned above. Hence, the relative electric
permittivity is directly related to the susceptibility
χ=er−1
, which is defined as the
proportionality in the relation of the dielectric polarization density and the electric field
~
P=e0χ~
E, (2.6)
and thus a measure for the polarizability of a material with respect to an electric field.
In terms of the derivation of the wave equations, the electric permittivity and magnetic
permeability enter in the speed of light
c=1/√ere0µrµ0
(for
er
and
µr
being real numbers),
which is different inside a medium than in vacuum. Also, the changes in polarization of
matter under a changing electric field will not be instantaneous but occur with a delay
depending on the material. Thus, the electric permittivity will in general be a function of
the frequency
ω
(or equivalently a function of the photon energy) , i.e.
e=e(ω)
, also
known as dielectric function. In turn, while being a constant in vacuum with respect
to the energy, the speed of light becomes energy dependent once the wave enters the
medium [18]. This dispersion has consequently also an effect on the value of the wave
number
k
inside the medium in comparison to the vacuum equations in Eq.
(2.3)
, which
yields
k=1
õrer
k0=nk0, (2.7)
where
n
is the index of refraction taking into account the changes of the wave vector
~
k
of
an electromagnetic field at the interface of vacuum and matter. The delay in polarization
response of the material due to electromagnetic waves can be described by a complex
valued dielectric function
e(ω) = e1(ω) + ie2(ω)
, which accounts for the phase difference
in the polarization density with respect to the electric field and dissipative effects in
matter. In consequence the wave number
k
and the index of refraction
n
become complex
quantities, with the imaginary part describing the absorption of the electromagnetic
radiation during the propagation.
The index of refraction can then be written as,
n=1−δ−iβ, (2.8)
where its real part
δ
accounts for the deviation from the vacuum index of refraction
and its imaginary part
β
for the absorption. The origin of the values of these two parts
is strongly dependent on the material and the spectral range of the electromagnetic
radiation. Later, we will quickly summarize this dependence for the interaction of matter
with EUV and X-ray radiation due to the atomic electronic structure in condensed matter.
9
Chapter 2 THEORETICAL DESCRIPTION OF EUV AND X-RAY SCATTERING
Interaction processes
The continuum approach above describes the propagation of X-rays and EUV radiation
through vacuum and matter in a macroscopic picture. Based on the aforementioned
refractive index, the reflective, refractive and dissipative processes at interfaces and
in homogeneous materials will be treated for the special case of multilayer systems.
However, it is necessary to also give a more general description on the interaction of a
photon with the atoms, and more importantly the electrons, of a medium to describe the
origin of the fluorescence processes, which are not covered by the continuum description
above.
When a photon hits an atom or molecule with its electrons three
*
very important
processes can occur, that need to be distinguished.
Elastic Scattering
The photon interacts with the matter in an energy conserving way.
Two limiting cases of a free and a bound electron are distinguished as scatterers. In
the first case, the photon may be scattered out of its original direction by interaction
with a single free electron retaining its wavelength (and equivalently its energy).
This process is also known as Thomson scattering. More generally however, instead
of interacting with free electrons, it might encounter a bound electron of an atom
forming a dipole with the positive charge of the atom core. In the latter example,
the interactions due to the bound nature of the electron have to be considered and
affect the scattering process. This scattering by a bound electron is called Rayleigh
scattering or dipole scattering, which is highly photon energy dependent in its
scattering cross section. Both scattering processes can be described within the wave
description of the impinging radiation.
Inelastic Scattering Inelastic scattering refers to the case where the photon exchanges
a portion of its energy with the system it interacts with resulting in a loss of photon
energy and, thus, increased wavelength for the scattered photon. Considering the
case of high-energy X-ray photons colliding with free electrons, the total momentum
of the system (photon and electron) needs to be taken in to account. A portion of
the momentum of the photon (depending on the scattering direction) is transferred
to the electron making it recoil. This process is known as Compton scattering and it is
the result of the particle-wave-duality of electromagnetic radiation. The momentum
transfer and thus the change in wavelength depend on the rest mass of the electron.
In the low-energy limit, this process becomes negligibly small resulting in simple
elastic Thomson scattering.
Absorption
The third possibility is that the photon is absorbed by ejecting a bound
core shell electron from the atom leaving a vacancy. This is known as photoelectric
effect. It requires a photon energy exceeding the binding energy of the electron
for allowing it to be ejected from the atom. The vacancy on the inner shells is
filled by relaxation of electrons from energetically higher core shell states leading
to the emission of radiation of lower energy than the initial photon energy. This
is called X-ray fluorescence, where the emitted photons energy is specific for the
element of the atom due to the specific binding energies in the core shell for each
element. Another process competing with the emission of fluorescence radiation is
the Auger effect. Here, instead of emitting the energy of the core shell relaxation as
*
Other processes, e.g. magnetic scattering, can occur as well. However, the description here is limited to the
relevant aspects for this work.
10
Interaction of EUV and X-ray Radiation With Matter 2.2
fluorescence radiation, it is transmitted to second electron, which is in turn ejected
with reduced energy compared to the photon of the competing X-ray fluorescence
process.
2.2.1 Elastic Scattering
Angular resolved scattering of an incoming plane wave is described by the differential
scattering cross section, defined as
dσ
dΩ(θ,ϕ) = Is(θ,ϕ)
Φ0∆Ω , (2.9)
where
Is
is the scattered intensity into the solid angle
∆Ω
and
Φ0
is the total flux of
incoming photons of the primary wave per unit area. The differential cross section gener-
ally has an angular dependence with respect to the position of the observer (detector), the
distribution of the scattering matter and the direction of the incoming beam. Here,
θ
and
ϕ
are angular coordinates in an coordinate system with its origin at the scattering center.
Due to this proportionality, the goal of calculating the scattering intensity is achieved
by determining the differential cross section for the scattering problem at hand. As an
example the differential cross section of scattering from a single free electron is briefly
demonstrated and that description is extended to scattering from an arbitrary electron
density ρe(
~
r)of free and bound electrons.
Thomson scattering from single free electrons
The scattering cross section in case of a single free electron is given by
dσ
dΩ(θ,ϕ) = e2
4πe0mc22|~
ei·~
es|2=re2|~
ei·~
es|2, (2.10)
where
e
is the electron charge and the unit vectors
~
ei
and
~
es
describe the direction of the
electric field vector before and after the scattering process, respectively. The differential
cross section in the case of Thomson scattering is proportional to the square of the classical
electron radius
re=e2/4πe0mc2
. Depending on the polarization properties of the impinging
radiation, the scalar product of the two unit vectors yields
|~
ei·~
es|2=
1 electric field perpendicular to scattering plane
cos2(∆Ψ)electric field parallel to scattering plane
1
21+cos2(∆Ψ)unpolarized radiation
, (2.11)
where
∆Ψ(θ,ϕ)
is the total angle between the incoming beam and the scatter direction [3]
and lies in the scattering plane spanned by the propagation direction of the incoming
and scattered waves.
11
Chapter 2 THEORETICAL DESCRIPTION OF EUV AND X-RAY SCATTERING
Rayleigh scattering from bound electrons and Born approximation
In general, the scattering from a single free electron will not be an accurate description
for most scattering problems of EUV and X-ray radiation impinging on matter. Instead
electrons are bound in an atom or molecule (or in the band structure of a solid) and the
radiation is scattered by an electron density associated with the distribution of electrons
bound in an atom. The bound nature of the electrons also influences the scattering
cross section as it shall be summarized here. The result is that the differential cross
section obtained for Thomson scattering has to be modified by the form factor
f(~
q)
defined
through,
dσ
dΩ(θ,ϕ) = re2|f(~
q)|2|~
ei·~
es|2, (2.12)
where
~
q=~
kf−~
ki
the wavevector transfer or scattering vector. Let us first consider the case
of a free electron cloud. A plane wave impinging on a distributed charge distribution will
be scattered from all positions of that distribution. The observer located far away from
the scatterer detects a superposition of this radiation scattered at each position within the
charge density. The individual scattered waves have a path difference from the scatter
center to the detector resulting in a phase difference. The form factor, which we shall
denote f0(~
q), is then given by
f0(~
q) = Zρe(
~
r)e−i~
q·~
rd
~
r. (2.13)
The exponential function in Eq.
(2.13)
accounts for the aforementioned phase difference
between different scattering centers in the spatial electron distribution [38]. The scattering
from a free electron cloud is thus characterized by the Fourier transform of the electron
density spatial distribution. In the limiting case of a singular isolated electron (described
by a delta function for the electron density), the scattering cross section will just yield
the Thomson scattering formula in Eq.
(2.10)
. It is important to note here, that the
form factor found in Eq.
(2.13)
is only valid if the scattering is weak compared to the
primary incident wave. For solving the corresponding wave equation one approximates
the incoming field at all positions
~
r
of the electron density with the initial primary wave
neglecting any scattered contributions from other positions
~
r0
. This is called the Born
approximation. It implicitly corresponds to considering only one single scattering event
per incident photon. Multiple scattering processes are not included in this description
(kinematic scattering). Later, we will generalize this approximation to more complex,
exactly solvable scattering problems instead of considering only the kinematic processes.
The differential cross section in Eq.
(2.12)
with the form factor
f0(~
q)
is only valid for free
electrons. In case of bound electrons in a atom, molecule or solid, electronic resonances
exist which affect the scattering. For EUV and X-ray radiation dipole scattering on light
elements, the core shell energy levels are close to the energy of the impinging radiation.
In that case the electron response will no longer be that of a free or quasi free electron
but influenced due to the fact that it is tightly bound. This effect is called dispersion and
results in two additional wavelength dependent dispersion factors in the atomic form
factor [3,38], which is now a complex quantity including absorption effects described as
f(~
q,λ) = f0(~
q) + f0(λ) + i f 00(λ). (2.14)
12
Interaction of EUV and X-ray Radiation With Matter 2.2
The atomic scattering factors
f0(λ)
and
f00(λ)
are strongly dependent on the element of
the atoms involved in the scattering process. The first factor
f0(λ)
accounts for the
modified response of an electron close to an electronic resonance, often described in
analogy to a driven harmonic oscillator close to its eigenfrequency. The second factor
f00(λ)
describes dissipative processes into the atomic system. It is associated with the
absorption of radiation in matter. In fact, both factors, while being related through the so
called Kramers-Kronig relation, define the complex index of refraction (expressed here for
a single element) of the continuum theory introduced above at the beginning of Sec. 2.2
through
n=1−δ−iβ=1−re
2πλ2naf(0,λ), (2.15)
where nais the number of atoms per unit volume [136].
2.2.2 Absorption and Fluorescence
Absorption of electromagnetic radiation, more specifically X-ray radiation, in matter
is the third main interaction process mentioned here apart from elastic and inelastic
scattering. In that case, the incoming photon transfers all its energy to an electron leaving
it in a energetically excited state. If the energy of the incoming photon is sufficient to
excite the electron into the continuum above the binding energy, that electron is ejected
from the atom leaving a vacancy at one of the core shells and, thus, leaving the ion in an
exited state. The relaxation of electrons in energetically higher shells into the vacancy
causes the release of energy. This can happen through two competing processes known
as X-ray fluorescence and the Auger effect. The general principle of X-ray fluorescence is
illustrated in Fig. 2.1.
E
K-shell
L-shell
vacany
electron
Kα1fluorescence radiation
(2s)
(2p1/2)
(2p3/2)
(1s)
Figure 2.1 |
Illustration
of X-ray fluorescence
for an atom. As an ex-
ample, the relaxation of
an
L
-shell electron into
the
K
-shell vacancy is
shown. This leads to the
emission of characteris-
tic
Kα1
fluorescence ra-
diation at three differ-
ent energies according
to the dipole transition
selection rules. The elec-
tron configuration of
the two shells is given in
brackets of the respec-
tive energy level (figure
not to scale).
Each material exhibits a steady decrease of the interaction cross section when irradiated
with radiation of increasing photon energy known as normal dispersion. However, at
certain material dependent energies, sharp increases can be observed, also referred to as
resonances or ranges of anomalous dispersion. Those jumps correspond to absorption
edges like the
K
,
L
and
M
excitations of the core shell electrons leading to photoionization
of that particular atom creating the above mentioned vacancy. Since the electronic
structure of the core shell is specific to a particular element, the emitted fluorescence
13
Chapter 2 THEORETICAL DESCRIPTION OF EUV AND X-RAY SCATTERING
radiation is characteristic for the material in the sample. That fact is exploited in the
XRF analysis, where the amount of a specific chemical element inside of matter can be
determined by measuring the spectral distribution of the fluorescence radiation.
Finally, instead of emitting fluorescence radiation the energy of the relaxation process
into the vacancy can be transferred radiation less to a secondary electron with lower
binding energy than the primary, excited electron. In that case, given sufficient energy, the
secondary electron can also be ejected with a overall reduced kinetic energy compared to
the primary electron. This is the Auger process. In principle, since the binding energy of
the secondary electron is specific for the chemical element, Auger electron spectroscopy
also offers the possibility for material analysis. However, a limitation is the small median
travel distance of electrons in matter making this technique highly surface sensitive and
thus unpractical for the analysis of buried material.
The two processes of fluorescence and Auger emission compete. For elements with low
atomic number
Z
, the Auger process dominates while almost no fluorescence is present.
With increasing atomic number the ratio reverses resulting in a higher fluorescence yield
than Auger electron yield for high Zelements and inner shells.
2.3 Specular Reflection from Surfaces and Interfaces in
Layered Systems
As mentioned above in the beginning of Sec. 2.2the reflection and transmission of EUV
and X-ray radiation will be treated here with a continuum approach based on the index
of refraction. Before we treat specular reflectance and transmittance in multilayer systems,
lets recapitulate reflection and transmission through a single surface. Fig. 2.2gives the
necessary definitions for radiation passing through an abrupt interface. The coordinate
system was chosen such that the surface is perpendicular to the
z
-direction and
z=0
is at
the surface. The refraction process in that case is entirely governed by Snell’s law known
Figure 2.2 |
Illustra-
tion of Snell’s law. The
parallel component of
the wave vector
k(0)
x=
k(1)
x=kx
remains un-
changed when the radia-
tion enters the medium.
The perpendicular com-
ponent changes accord-
ing to the index of re-
fraction (see main text).
matter j=1,n(1)=1−δ(1)−iβ(1)
~
k(1)
t
~
k(0)
i
~
k(0)
r
zvacuum j=0,n(0)=1
k(0)
z
k(0)
x
k(1)
x
k(1)
z
from classical optics [27]. Since all measurements in this thesis were conducted with
highly linearly polarized light, the description of the refraction processes is given only
for the specific conditions found in our experiments. In our case, the electric field vector
oscillates perpendicular to the scattering plane defined by the incoming wave vector
~
ki
and
the surface normal. This geometry is referred to as s-polarization. For the opposite case of
an electric field vector oscillating parallel to the aforementioned scattering plane, known
as p-polarization, modified forms of the corresponding equations apply not mentioned
14
Specular Reflection from Surfaces and Interfaces in Layered Systems 2.3
here.
Considering the interface of vacuum and material, the condition of continuity of both
the electric field amplitude and its derivative need to be fulfilled [27,55]. From that
follows that the parallel component of the wave vector
k(j)
x≡kx∀j
does not change at
the interface. With the solutions of the wave equation for propagation in homogeneous
media in the beginning of Sec. 2.2, Snell’s law can be expressed in terms of the wave
vector by
k(j)
z=rn(j)k02−k2
x, with kx=sin(αi)k0, (2.16)
and the angle of incidence
αi
defined from the surface normal (cf. Fig. 2.6) and
n(j)
is the
complex index of refraction of layer j.
Together with Eq. 2.7this yields a relation for the perpendicular component of the
wave vector and of the electric field amplitudes in vacuum (layer
j=0
) and the medium
(layer j=1) through the Fresnel coefficients of reflection r(0)and transmission t(0)via
E(1)
t
E(0)
r!=t(0)E0
r(0)E0, (2.17)
where
E0
is the field amplitude of the incident field with wave vector
~
k(0)
i
,
E(1)
t
is the
transmitted field amplitude in layer
j=1
with wave vector
~
k(1)
and
E(0)
r
is the reflected
field amplitude with wave vector
~
k(0)
r
. For the transmission and reflection at any two
interfaces jand j+1 the Fresnel coefficients in s-polarization read
r(j)=k(j)
z−k(j+1)
z
k(j)
z+k(j+1)
z
, (2.18)
t(j)=2k(j)
z
k(j)
z+k(j+1)
z
. (2.19)
For the sake of completeness, we shall also give the corresponding Fresnel coefficients
in case of p-polarized light impinging on the surface [27],
r(j)
p=k(j+1)
z−(n(j+1)/n(j))2k(j)
z
k(j+1)
z+ (n(j+1)/n(j))2k(j)
z
, (2.20)
t(j)
p=2k(j)
z
(n(j+1)/n(j))k(j)
z+ (n(j)/n(j+1))k(j+1)
z
. (2.21)
15
Chapter 2 THEORETICAL DESCRIPTION OF EUV AND X-RAY SCATTERING
Matrix algorithm for multilayer systems
In this part the calculation above is extended to a system of multiple layers on top of a
substrate which is assumed to be infinite. This provides the exact fully dynamic solution
of the wave equation for an ideal multilayer system with abrupt interfaces. Thus, all
reflections and transmissions at all interfaces are considered, including multiple events.
The EUV and X-ray fields were calculated based on the well-established matrix algorithm
which is an extension of the above Fresnel coefficient method [27,96]. The field inside
each layer
j
is described similarly to Eq.
(2.17)
by their reflected and transmitted field
components as
E(j)(
~
r) = ei
~
kk·~
rk(E(j)
t(z) + E(j)
r(z)), (2.22)
where
~
kk
is the wave vector component parallel to the interfaces (in the two-dimensional
geometry of Fig. 2.2above was
~
kk=~
kx
) and
~
rk
is the position perpendicular to the
z
-direction. Here, the exponential function in Eq.
(2.22)
takes into account the changes in
phase and the absorption inside the material for the wave components traveling parallel
to the surface. The two field components are further described by the transmitted and
reflected field amplitudes Tjand Rjas
E(j)
t(z) = Tjeik(j)
zz, (2.23)
E(j)
r(z) = Rje−ik(j)
zz, (2.24)
where
E(j)
t(z)
describes the field component propagating towards the substrate and
E(j)
r(z)
is the reflected field component in each layer propagating towards the vacuum.
The field amplitudes and layer thicknesses are illustrated in Fig. 2.3. The components of
Figure 2.3 |
Illustration
of the field amplitudes
in the exact analytical
solution of field propa-
gation through a multi-
layer stack. The verti-
cal coordinate
z
is de-
fined to be zero at the
substrate interface. The
field amplitude of the in-
cident field in the vac-
uum
T0
is known. Inside
the infinite substrate
no reflected field ampli-
tude exists, i.e.
RN+1=
0
. The layer thicknesses
are denoted
dj
for the
jth layer.
layer j−1
layer j
Tj−1Rj−1
Tj
Rj
layer j+1
Tj+1
Rj+1
TjRj
surface layer j=1
TN+1
TNRN
T1
R1
T0R0
substrate j=N+1
d1
dj−1
dj
dj+1
z
z=0
vacuum j=0
16
Specular Reflection from Surfaces and Interfaces in Layered Systems 2.3
two adjacent layers are connected by the propagation matrix Mj
Mj=1
t(j)1r(j)
r(j)1 e−ik(j+1)
zdj+11
1eik(j+1)
zdj+1!, (2.25)
through the relation
E(j)
t
E(j)
r!=Mj E(j+1)
t
E(j+1)
r!. (2.26)
The field propagation matrix in Eq.
(2.25)
includes the Fresnel coefficients from Eq.
(2.18)
and Eq.
(2.19)
accounting for the reflection and transmission process at the interface.
In between two interfaces a homogeneous layer was assumed so that the field is only
propagated by the phase factor
e±ik(j)
zdj
along the
z
-direction and the layer thickness
dj
.
The system of equations in Eq.
(2.26)
becomes solvable by replicated application of the
field propagation matrix to relate the known incident field amplitude
E0
, the total reflected
field amplitude in the vacuum
ER
and the transmitted field in the substrate
ET
. Since
there can not be a reflected field inside the substrate the system of equations Eq.
(2.26)
reads
E0
ER=∏
j
MjET
0, (2.27)
with two unknowns
ER
and
ET
which can be calculated based on this relation. Thereby
all field amplitudes at each interface can be obtained. The total reflectance
R
and
transmittance
T
can then be calculated as the quotient of the (known) incoming field
E0
with the reflected ERand transmitted field ET, respectively, as
R=|ER/E0|2,
T=|ET/E0|2. (2.28)
Accounting for roughness and interdiffusion
The calculation above yields an exact solution of the problem of reflecting and trans-
mitting EUV or X-ray radiation from and through a generic multilayer. However, in
a realistic sample the interfaces will not be perfectly flat and abrupt. Instead the two
materials could mix or the interfaces could be rough. Both effects lead to a diminished
reflectance of each interface and thus reduce the reflected field amplitudes which changes
their interference behavior. These two processes of roughness and interdiffusion can be
treated within the framework of the matrix algorithm presented above by using modified
Fresnel coefficients. A detailed calculation for arbitrarily rough interface profiles along
the z-direction can be found in [141], for example.
For our calculations a Gaussian distribution function of the roughness and interdif-
fusion is assumed. The general expression found in [141] for the modified Fresnel
coefficients then yields the result of Névot and Croce [36,102]. The Gaussian distribution
function corresponds to the assumption of the interdiffusion and roughness profile to be
17
Chapter 2 THEORETICAL DESCRIPTION OF EUV AND X-RAY SCATTERING
of error-function like shape, which leads to the modified Fresnel coefficients
˜
r(j)=r(j)exp(−2k(j)
zk(j+1)
zσ2
j),
˜
t(j)=t(j)exp((k(j)
z−k(j+1)
z)2σ2
j/2), (2.29)
where
r(j)
and
t(j)
are the unmodified Fresnel coefficients for an ideal multilayer system
at each interface
j
from Eq.
(2.18)
and Eq.
(2.19)
. The parameter
σj
is the mean square
roughness or mean square intermixing, respectively at the
j
th interface. It should be
mentioned, that this parameter describes both, the roughness and the interdiffusion
as they have the same average effect in the impinging radiation beam footprint on the
specular reflectivity. It is thus not possible to distinguish those two based on σj.
Specular reflectivity from periodically layered systems
Based on the formalism described within this section, the specular reflectivity from
periodically layered systems can be calculated. In the course of this thesis, two systems
are of relevance for the studies presented. In Fig. 2.4those two multilayer systems are
defined, which have periodic alternating layers of the materials chromium and scandium,
as well as, molybdenum and silicon. The specular reflectivity calculated using the matrix
Figure 2.4 |
Schematic
layout of a periodic
layer structure. a)
Shows an example for
periodically layered
structures involving the
materials chromium
(Cr) and scandium (Sc).
The periodic part of
the stack is replicated
N=400
times. b) Sim-
ilar layout with thicker
layers of molybdenum
(Mo) and silicon (Si) with
a number of periods of
N=65.
substrate substrate
Cr, dCr =0.787 nm
Si
Sc, dSc =0.787 nm
Sc
Sc
Cr
Cr
Cr
Si, dSi =3.9 nm
Mo, dMo =3.0 nm
Si
a) Cr/Sc, N=400 b) Cr/Sc, N=65
Cr
periodic replication periodic replication
periodic replicationperiodic replication
formalism at different angles of incidence and within different wavelengths ranges for the
two examples are shown in Fig. 2.5. The calculations assume perfectly shaped interfaces
and thus do not include a description of roughness or interdiffusion.
Clearly, due to the periodic layout of the layered systems, constructive interference
leads to a high reflectivity at certain wavelengths, depending on the thickness of the layers
and the angle of incidence. Based on this principle it is thus possible to construct mirrors
for those EUV wavelengths, where otherwise only very low reflectivity is observed from
single surfaces. These systems can therefore serve as reflective optical elements for the
respective spectral range and are known as multilayer mirrors.
18
Diffuse Scattering in Layered Systems 2.4
3.06 3.10 3.14 3.18 3.22
wavelength λ/ nm
0.0
0.2
0.4
0.6
0.8
1.0
reflectivity
a) αi=1.5◦
Cr/Sc
12.0 12.5 13.0 13.5 14.0
wavelength λ/ nm
b) αi=15◦
Mo/Si
Figure 2.5 |
Calculated
specular reflectivity
curves considering the
periodically layered
multilayer systems
shown in Fig. 2.4 acting
as mirrors in a certain
bandwidth. a) Shows
the resulting reflectivity
off the Cr/Sc multilayer
system irradiated at
an angle of incidence
αi=1.5°
from the
surface normal. b)
Calculated theoretical
reflectivity by irradiat-
ing the periodic Mo/Si
system at αi=15°.
2.4 Diffuse Scattering in Layered Systems
For the characterization of a scattering process in general, but here in particular from
surfaces or interfaces, it is necessary to define the coordinate system of the momentum
transfer. The scattering process from a single surface in reflection geometry is depicted
in Fig. 2.6. The incoming beam irradiating the sample under the angle of incidence
αi
is
described by the wave vector
~
ki
. The direction of this vector is the propagation direction
z
x
αi
~
kiαf
sample
~
kf
~
ki
θf
~
kf
y
x
a) side view b) top view
sample
Figure 2.6 |
Scattering
geometry for the defi-
nition of the scattering
vector ~
q.
of the incident radiation, where its absolute value is the wavenumber
k=|~
ki|=2π
λ
. A
detector positioned at a different angle, typically called scattering angle
αf
, detects the
scattered radiation. The outgoing or scattered beam is described by the wavevector
~
kf
with direction towards the detector, again in accordance with the propagation direction of
the radiation. In case of an elastic, i.e. energy conserving, scattering process its absolute
value is the wavenumber of the incoming beam
|~
kf|=|~
ki|=k0
. This general scattering
process is characterized by its momentum transfer vector
~
q=~
kf−~
ki, (2.30)
19
Chapter 2 THEORETICAL DESCRIPTION OF EUV AND X-RAY SCATTERING
also known as scattering vector. From this definition the components of this three
dimensional vector can be expressed by the involved angles and wavelengths as
qx=kcosθfsin αf−sinαi,
qy=ksinθfsin αf,
qz=kcosαf+cos αi.
(2.31)
The momentum transfer vector is a characteristic quantity for scattering processes. Its
three components in Eq. (2.31) span the so called reciprocal space.
Modified wave equation and the distorted-wave Born approximation
Diffuse scattering in the special case of layered systems is the result of imperfections
of surfaces or interfaces, which otherwise show only specular (coherent) reflectance. In
Sec. 2.2.1the elastic scattering of EUV and X-ray radiation on an electron density was
elaborated. An important assumption for the results obtained, the Born approximation,
is that the scattering is weak with respect to the incoming primary wave. The scattering
process thus only considers the primary wave, typically a plane wave, and not the
total wave field including the scattered radiation in the theoretical description of the
process. This is equivalent to the assumption of a single scattering event ignoring
multiple scattering, also known as kinematic scattering. In the context of layered systems,
diffuse scattering is described within the framework of perturbation theory with a similar
approach.
The existence of a multilayer structure is different from scattering on a simpler system,
e.g. an isolated electron cloud. The wave field at the interfaces significantly differs from
that of a plane wave due to multiple reflection and transmission processes occurring
in a multilayer system. This alternation of the wave field can no longer be considered
weak and the Born approximation fails. Instead, the theoretical description of the diffuse
EUV scattering from multilayers is based on the distorted-wave Born approximation
(DWBA) [65,67], widely used in the analysis of hard X-ray scattering. The DWBA is an
extension of the above mentioned Born approximation in which the interfacial roughness
is considered to be a small deviation from the ideal multilayer system. In general, the
wave equation for a multilayer system is
(∆+k02)E(
~
r) = V(
~
r)E(
~
r), (2.32)
with the potential
V(
~
r) = k01−n2(
~
r), (2.33)
describing the different materials inside the layer system through their index of refraction
n
[106]. The DWBA is based on the principle that part of this potential leads to a wave
equation which can be solved analytically, while a small disturbance to that potential
remains to be treated as perturbation. In case of a multilayer the exact solution of a
system with ideal interfaces can indeed be found and is given in Sec. 2.3. The potential
can be separated into a strong part
Vid(
~
r)
for which an analytical solution exists and a
small perturbation
Vr(
~
r)
describing the interfacial roughness as deviation from the ideal
20
Diffuse Scattering in Layered Systems 2.4
layer system, i.e.
V(
~
r) = Vid(
~
r) + Vr(
~
r). (2.34)
In analogy to the Born approximation, the scattering process is then evaluated consid-
ering the wave fields obtained from the solution with the ideal potential
Vid(
~
r)
only
and calculating a first iteration. Thus, the analytic solution of the multilayer wave equa-
tion (“distorted wave”) in the DWBA takes the place of the plane wave in the Born
approximation. In that way, the scattering from the perturbations are still considered
kinematically (single scattering approximation), however, the incoming distorted waves
are exact solutions of the transmittance and reflectance at all layers of the multilayer
system.
The distorted-wave Born approximation scattering cross section
The detailed derivation of the diffuse (incoherent) differential scattering cross section for
rough multilayer systems can be found in Pietsch, Holý and Baumbach [106] and the
corresponding publications [65,125], as well as in de Boer [24] and Mikulík [96]. Here, a
summarized version illustrating the application to near-normal incidence scattering is
given and the corresponding approximations leading to the determination of a roughness
power spectral density (PSD) for the interfaces in a multilayer system are described.
The derivation of the diffuse scattering cross section is done by applying the mathe-
matical tools from the quantum mechanical formalism for perturbation theory. There,
the transition probability from one state into another is described as the expectance
value of the transition matrix. In case of the scattering problem at a multilayer this
translates to considering the incoming wave field, given by the exact solution of the wave
equation for a multilayer system and calculating the expectance value for scattering into
a scattered state arriving at the detector. The latter is generally unknown. However, the
reciprocity theorem [79,87] of classical electrodynamics states that an unknown field at
an detector generated by a known dipole source, i.e. the incident field induced dipole at
a perturbation of an interface causing the emission of scattered radiation, can be replaced
by the time-inverted known field caused by a single dipole source at the detector position
(“detector beam”) [37,65,125]. The latter is just the time-inverted solution of the same
wave equation of the ideal multilayer as for the regular solution. Thus, two independent
solutions of the wave equation
(2.32)
with
V(
~
r) = Vid(
~
r)
are considered and they are
expressed in Dirac notation [42] as
|E(1)
id i
and
|E(2)
id i
, where the superscript
(1)
denotes
the regular solution obtained via the matrix algorithm in Sec. 2.3and the index
(2)
indicates the time-inverted solution for the scattering angle
αf
of the detector position
with respect to the surface.
*
According to Eq.
(2.22)
, Eq.
(2.23)
and Eq.
(2.24)
the two
solutions can be expressed in terms of the reflected and transmitted field amplitudes as
|E(1)
id i=ei
~
kk,(1)·~
rkT(1)
jeik(j)
zz+R(1)
je−ik(j)
zz, (2.35)
|E(2)
id i=hE(2)
id |∗=e−i
~
kk,(2)·~
rkT(2)∗
je−ik(∗j)
zz+R(2)∗
jeik(∗j)
zz. (2.36)
These solutions are the basis for the calculation of the differential scattering cross section,
*
In regard to the matrix algorithm in Sec. 2.3the solution for the time-inverted “detector beam” is obtained
by replacing the vacuum wave vector component
kx
in Eq.
(2.16)
with the corresponding component for the
scattering angle αfinstead of the angle of incidence αi.
21
Chapter 2 THEORETICAL DESCRIPTION OF EUV AND X-RAY SCATTERING
which is given by the covariance of the matrix element of the perturbation potential [106]
as
dσ
dΩ
DWBA
=Cov(hE(2)
id |Vr|E(1)
id i). (2.37)
The explicit expression for the covariance can be calculated based on Eq.
(2.35)
and
Eq.
(2.36)
and yields the full DWBA differential scattering cross section for the diffuse
(incoherent) scattering considering all transmitted and reflected fields, i.e. all first order
dynamic effects, as
dσ
dΩ
DWBA
=Aπ2
λ4
N
∑
j=1
N
∑
i=1
(n2
j−n2
j+1)∗(n2
i−n2
i+1)(T(1)
j+R(1)
j)∗(T(2)
j+R(2)
j)∗
×(T(1)
i+R(1)
i)(T(2)
i+R(2)
i)Sij(~
qk;q(j)
z,q(i)
z), (2.38)
where
A
is the illuminated sample area and
Sij(~
qk;q(j)
z,q(i)
z)
is the result of the averaging
over the perturbation potential
Vr(
~
r)
in evaluation of the covariance in Eq.
(2.37)
, as
outlined below in the following paragraph. For the multilayer system this perturbation is
roughness at the interfaces, which can be correlated vertically throughout the stack, as
well as in-plane of a single interface. A detailed derivation of the explicit form of that
form factor is given in the following paragraph.
In the case of small reflectivity amplitudes, dynamic multiple reflections are often
neglected and the dominant term in the decomposition is diffuse scattering of the
transmitted fields at the roughness of each interface. The so-called semi-kinematic
approximation [125] yields an explicit expression for Eq. (2.37) with
semi-kinematic
dσ
dΩ
DWBA
=Aπ2
λ4
N
∑
j=1
N
∑
i=1(n2
j−n2
j+1)∗(n2
i−n2
i+1)
×T(1)∗
jT(2)∗
jT(1)
iT(2)
iSij(~
qk;q(j)
z,q(i)
z). (2.39)
The semi-kinematic approximation is similar to the conventional Born approximation,
except that it considers the exact transmitted field amplitudes at a certain interface instead
of a plane wave. The comparison of this expression with the full first-order DWBA term
in Eq.
(2.38)
is useful to evaluate the contribution of dynamic effects to the scattering
cross section and consequently the measured diffuse scattering distribution.
An illustration of the four scattering processes included in the full first-order DWBA
is shown in Fig. 2.7at the example of the interface of layer
j
and
j+1
in the multilayer
system.
22
Diffuse Scattering in Layered Systems 2.4
TT∗RT∗TR∗RR∗
semi-kinematic dynamic
z
layer j+2
layer j−1
layer j
layer j+1
T(1)
jT∗(2)
jT∗(2)
jT(1)
j
R(1)
j+1R∗(2)
j+1R∗(2)
j+1
R(1)
j+1
Figure 2.7 |
Illustration of the four scattering processes of the DWBA
a
. The
TT∗
process on the left
is purely kinematic in nature and equivalent to the Born approximation. The three other processes
RT∗
,
TR∗
and
RR∗
are purely dynamic and not described by kinematic theory. It should be noted
here, that the illustration shows a simplified picture. The reflection and transmission amplitudes in
the respective layers contain all reflections and transmission of all preceding and following interfaces.
They represent the full field in the respective interface with all components propagating towards the
vacuum (R,T∗) and the substrate (T,R∗).
aFigure similar to Pietsch, Holý and Baumbach [106].
Calculation of the roughness power spectral density
The perturbation potential describes the derivation of the actual interface profile in the
multilayer from the perfectly flat case of an ideal system. Thus, this potential is only
non-vanishing if roughness is present between the layers
i
and
j
at only in the vicinity
of an interface. Let us consider
hi(
~
r)
as the interface height profile (in
z
direction) of
the interface between the
i
th and
j
th layer with
hi(x,y,zi) = 0
at the position of the
ideal interface
zi
as illustrated in Fig. 2.8. Then the perturbation potential, considering
z
zi
layer j,Vj
id
layer i,Vi
id
hi(
~
r)>0
hi(
~
r)<0
Figure 2.8 |
Illustra-
tion of the perturbation
potential
Vi
r(
~
r)
at the
i
th interface of a multi-
layer system. The ideal
(mean) interface posi-
tion is at
z=zi
with
a height profile
hi(
~
r)
in-
dicating the deviation
from that ideal interface
due to roughness.
Eq. (2.34) and Eq. (2.33), at that interface can be calculated to be
Vi
r(
~
r) =
Vj
id −Vi
id for hi(
~
r)>z>zi
Vi
id −Vj
id for hi(
~
r)<z<zi
0 for z<hi(
~
r)<ziand zi<hi(
~
r)<z
,
=
k0(n2
i−n2
j)for hi(
~
r)>z>zi
k0(n2
j−n2
i)for hi(
~
r)<z<zi
0 for z<hi(
~
r)<ziand zi<hi(
~
r)<z
. (2.40)
With the explicit form of
Vi
r(
~
r)
in Eq.
(2.40)
, the averaging in the covariance of Eq.
(2.37)
23
Chapter 2 THEORETICAL DESCRIPTION OF EUV AND X-RAY SCATTERING
can be calculated. However,
hi(
~
r)
is generally unknown. For the multilayer systems
under investigation in this thesis, this perturbation is interfacial roughness and thus
hi(
~
r)
a random quantity. Sinha et al. [125], D. K. G. d. Boer [22] and Mikulík [96] have shown,
that by assuming a Gaussian probability distribution of the height values in
hi(
~
r)
around
z=zi
at each interface
i
, the covariance can be calculated explicitly as given in Eq.
(2.38)
with
Sij(~
qk;q(j)
z,q(i)
z) =
exph−((q(j)∗
z)2σ2
j+ (q(i)
z)2σ2
i)/2i
q(j)∗
zq(i)
z
×Zd2~
Xexp[q(j)∗
zq(i)
zCij(~
X)] −1exp(i~
qk·~
X), (2.41)
where
q(i)
z
is the
z
-component of the scattering vector
~
q
at the
i
th interface,
~
X=~
r−~
r0
is
the lateral distance vector and
Cij(
~
r−~
r0) = hhi(
~
r)hj(
~
r0)i
is the height correlation function
of the interface profiles
h(
~
r)
of the interfaces
i
and
j
, respectively. The factor
σj
is the root
mean square (r.m.s.) roughness of the jth interface.
We consider the situation, where the roughness is small in relation to the scattering
vector. This assumption is valid especially for high-quality multilayer systems as the
mirrors considered in the framework of this thesis. This is the so-called small roughness
approximation. In that case, the product of roughness and the
z
-component of the
scattering vector is small, i.e.
q(j)
zσj1
. We therefore can approximate the first part of
Eq. (2.41) by
exph−((q(j)∗
z)2σ2
j+ (q(i)
z)2σ2
i)/2i
q(j)∗
zq(i)
z≈1
q(j)∗
zq(i)
z
(2.42)
and Taylor expand the integrand as
exp[q(j)∗
zq(i)
zCij(~
X)] −1≈q(j)∗
zq(i)
zCij(~
X)
. With these
approximations Eq. (2.41) reduces to
Sij(~
qk)≈Zd2~
XCij(~
X)exp(i~
qk·~
X). (2.43)
Sij(~
qk)
is, thus, the Fourier transform of the correlation function
Cij(~
X)
. Assuming
identical growth for the individual layers, i.e. a material independent propagation of
roughness along the
z
-direction,
Sij(~
qk)
can be expressed in terms of the lateral PSD
Ci(~
qk)and a vertical replication factor c⊥
ij (~
qk)[131],
Sij(~
qk) = c⊥
ij (~
qk)Cmax(i,j)(~
qk). (2.44)
PSD functions based on different models of lateral interface roughness correlation
have been proposed, e.g. by Sinha et al. [125]. We follow the approach by de Boehr et
al. [22,23] for fractal interface roughness, where the lateral correlation function of the
i
th
interface is given by
˜
Ci(~
X) = PiξHi
k|~
X|HiKHi|~
X|/ξk. (2.45)
Hi
is the Hurst factor providing a measure for the jaggedness of the interface [125] as
illustrated in Fig. 2.9,
KHi
are the modified Bessel functions of the order
Hi
,
ξk
is a lateral
24
Diffuse Scattering in Layered Systems 2.4
z
high H≤1
low H≥0
Figure 2.9 |
Qualitative
illustration of the Hurst
factor
H
describing the
jaggedness of the inter-
face. The Hurst factor
H
is defined between
0≤H≤1
, with
small values describing
strongly jagged rough-
ness profiles as shown
for the bottom inter-
face and large values
approaching unity for
smooth (Gaussian type)
interfaces.
correlation length and
Pi=σ2
i
ξHi−1
k2Hi−1Γ(1+Hi)/Hi
. (2.46)
The multilayer mirror samples investigated within this thesis are highly-reflective
samples fabricated with state of the art deposition processes. Therefore, a highly periodic
and highly-stable vertical replication of roughness is expected. While a distinction of the
roughness at the individual interfaces is theoretically possible, the experimental method
always irradiates all interfaces simultaneously. Thus, reconstructing the parameters
describing roughness of all interfaces individually is not possible due to the indistin-
guishability of the contribution of separate interfaces. Such a model would be ill-defined
based on the scattering data recorded. Instead, our goal is to determine a single average
power spectral density. Thus, identical roughness properties for all interfaces in our
model are assumed. Hence
σj=σ
,
Hj=H
and
Cmax(i,j)(~
qk) = C(~
qk)
. The PSD is given
by the Fourier transform of Eq.
(2.45)
with respect to
qx
, which yields the closed analytic
form
C(~
qk) = 4πHσ2ξ2
k
(1+|~
qk|2ξ2
k)1+H. (2.47)
Vertical correlation of roughness
The high degree of thickness stability for well-defined multilayers as is necessary for high-
performance mirrors implies a high degree of vertical correlation of individual interfaces
roughness throughout the stack. To better illustrate the correlation of roughness, Fig. 2.10
shows two situations where a weak and a strong correlation exist. In order to derive the
replication factor in Eq.
(2.44)
, we follow Stearns et al. [132]. In this model, the evolution
of the surface roughness
w(x,y)
during the growth of a single layer is described by the
Langevin equation. In its Fourier transformed form,
∂w(f)
∂t=−4π2v f 2w(f) + ∂η(f)
∂t, (2.48)
where
v
is a diffusion-like parameter,
η(f)
is random noise normalized to the layer
thickness
t
and
w(f)
describes the roughness evolution in dependence of the spatial
25
Chapter 2 THEORETICAL DESCRIPTION OF EUV AND X-RAY SCATTERING
z z
high vertical rougness correlation
low vertical rougness correlation
Figure 2.10 |
Illustration of correlated roughness in a binary periodic multilayer stack. If the roughness
is fully correlated, each interface replicates the morphology of the previous. A mathematical expression
introducing a vertical correlation length parameter is used to characterize this property.
frequency
f
. The roughness evolution during the growth of a single layer of a specific
material can then be evaluated by discretizing Eq.
(2.48)
for the successive deposition of
material of thickness δd
wi(f) = c⊥(f;δd)wi−1(f) + η(f), (2.49)
where
c⊥(f;δd)
is the replication factor of roughness for a single deposition. In the limit
of repeated infinitesimal depositions until the full
n
th layer of thickness
dn
is grown,
c⊥(f,dn)can be evaluated to be [131]
c⊥(f,dn) = exp(−4π2f2v dn)
=exp(−|~
qk|2v dn), (2.50)
with
|~
qk|2=4π2f2
. Assuming identical diffusion-like behavior
v
for all materials of a
multilayer and defining
ξ⊥(~
qk) = 1/(v|~
qk|2)
, the replication factor in Eq.
(2.44)
is given
by
c⊥
ij (~
qk) = exp −
max(i,j)−1
∑
n=min(i,j)
dn/ξ⊥(~
qk)!. (2.51)
Here,
ξ⊥(~
qk)
can be interpreted as a spatial frequency dependent vertical correlation
length, describing the distance perpendicular to the stack until the replication factor
decreased to 1/e.
Off-axis vertical roughness correlation
Gullikson et al. [57] observed that the direction of the vertical replication of roughness
can be tilted with respect to the surface normal as shown in schematically in Fig. 2.11.
Including this effect in the differential cross section, requires a coordinate transformation
in reciprocal space to account for the tilt angle βaccording to
qz=qz−ˆ
e·~
qktan β, (2.52)
where
ˆ
e
is a unit vector in direction of the roughness replication. Since the vertical
scattering vector components enter the calculations through the Fresnel coefficients in
Eq.
(2.18)
and Eq.
(2.19)
, an additional factor appears in the calculation of Eq.
(2.44)
26
Diffuse Scattering in Layered Systems 2.4
z
orthogonal rougness correlation
z
non-orthogonal rougness correlation
β6=0
β=0
Figure 2.11 | Illustration of orthogonal and non-orthogonal correlated roughness.
through substitution by
Sij(qx) = exp −iˆ
e·~
qktan β(zi−zj)Sij(qx), (2.53)
where ziis the z-position of the ith interface.
Full DWBA expression for near-normal incidence scattering
Taking together all the above findings and inserting them into Eq.
(2.38)
, the full explicit
expression for the DWBA scattering cross section on high-quality multilayer systems is
given by
dσ
dΩ
DWBA
="Aπ2
λ4
N
∑
j=1
N
∑
i=1
(n2
j−n2
j+1)∗(n2
i−n2
i+1)(T(1)
j+R(1)
j)∗(T(2)
j+R(2)
j)∗
×(T(1)
i+R(1)
i)(T(2)
i+R(2)
i)exp−iqxtan β(zi−zj)cij
⊥#C(qx). (2.54)
Since all experiments in this thesis have been conducted in in co-planar geometry, i.e. for
in-plane scattering measurements with a vanishing azimuthal angle
θf
in Fig. 2.6, the
parallel component of the scattering vector
~
q
is given by its
qx
component only, i.e.
~
qk≡qx
,
by choice of the coordinate system for the reciprocal space. We define the
x
,
y
and
z
components of the reciprocal space vector in Eq.
(2.31)
to be parallel to the respectively
labeled real space vectors in Fig. 2.6. The angle
β
is thus determined based on that
scattering direction only and dependent on the direction from which the sample is
irradiated. The replication factor cij
⊥and the PSD then read
cij
⊥(qx) = exp −
max(i,j)−1
∑
n=min(i,j)
dnq2
x
ξ⊥!, (2.55)
where the definition ξ⊥=1/vholds and
C(qx) = 4πHσr2ξk2
(1+q2
xξk2)1+H, (2.56)
in the explicit expression of Eq. (2.54).
In addition it should be noted here, that Eq.
(2.54)
separates the contribution to the
27
Chapter 2 THEORETICAL DESCRIPTION OF EUV AND X-RAY SCATTERING
scattering distribution of the multilayer and vertical correlation (in square brackets) on
the one hand and the in-planar roughness represented through the PSD
C(qx)
on the
other hand.
2.5 Grazing-incidence X-ray Fluorescence
X-ray fluorescence analysis is an established method to characterize the chemical com-
position of materials through the irradiation of samples with X-rays. In this thesis,
the focus is on the treatment of fluorescence emission by periodic multilayer systems,
which posses a Bragg resonance, i.e. a pair of angle of incidence and photon energy
which cause constructive interference at the interfaces. The requirement to excite the
Bragg resonance thus intrinsically connects the angle of irradiation with the wavelength
of the radiation, in our case this requires grazing angles of incidence. Based on the
measured signal, the structure of the multilayer sample can be inferred. The technique
is based on the emission of characteristic fluorescence radiation, as elaborated on in
Sec. 2.2.2. By irradiating an unknown sample with photons of sufficiently high energy,
those photons are absorbed leaving the characteristic vacancies in the
K
,
L
and
M
core
shells. The following recombination processes causes the emission of fluorescence radia-
tion, which can be detected outside the sample. A quantitative analysis was developed
by Sherman [121] and refined by others [35,110,122]. The Sherman equation links the
emitted and measured characteristic fluorescence radiation to the material concentration
of a specific element via fundamental parameters and the measurement characteristics
(experimental parameters) [111]. The quantitative analysis requires a detailed knowledge
of the fundamental parameters as well as all experimental parameters and only considers
absorption according to the Beer-Lambert law [3]. This is a very elaborate procedure.
However, for the purpose of analyzing special distribution of different materials in
periodic multilayer structures irradiated under grazing incidence, a relative analysis of
the measured fluorescence yield already delivers valuable spatial information on the
distribution of chemical species.
Before entering the details, let us review the aspects of generation of fluorescence
radiation. The appearance of fluorescence radiation is linked with the electromagnetic
field intensity of the impinging radiation for each infinitesimal volume element inside
the sample through a proportionality
IXRF =CZ|E(
~
r)|2ρ(
~
r)d3r, (2.57)
where
C
is a constant,
|E(
~
r)|2
is the field intensity and, here,
ρ(
~
r)
is the relative density
at the position
~
r
of the chemical species of which the characteristic fluorescence radiation
intensity is measured. This expression is an approximation, since it ignores any self-
absorption effects that may occur during the propagation of the fluorescence radiation
through the material before reaching the detector. However, for strongly periodic systems
as we shall discuss in this chapter, and a relative comparison of the intensities the effect
of self-absorption does not change if the angle is varied and the excitation wavelengths is
kept fixed and can be omitted.
In case of laterally infinitely extended and invariant multilayer systems, i.e. for
samples which are larger than any impinging beam and have the same layer stacking for
all these points, the field intensity
|E(
~
r)|2
only varies with the vertical coordinate
z
and
28
Grazing-incidence X-ray Fluorescence 2.5
thus reduces to
|E(
~
r)|2=|E(z)|2
for a given angle of incidence and photon energy [21].
If the layer stacking is known, that intensity can be calculated with the matrix formalism
elaborated on in Sec. 2.3. The intensity of the fluorescence radiation from those systems
simplifies Eq. (2.57) to
IXRF =˜
C
D
Z
0|E(z)|2ρ(z)dz, (2.58)
where
D
is the total thickness of all layers of the stack and
E(z) = E(i)
t(z) + E(i)
r(z)
is
given by Eq.
(2.23)
and Eq.
(2.24)
for the respective layer
i
depending on the coordinate
z
.
We define
di
as the thickness of each layer
i
, and thus
D=∑idi
. This formula is only
valid if the fluorescence radiation is not emitted by the substrate on which the multilayer
stack was deposited, since in that case the wave intensity inside the substrate has to be
considered as well.
The integral in Eq.
(2.58)
has to be evaluated for all points
z
inside each layer
i
. The
matrix algorithm as outlined in Sec. 2.3yields only the field amplitudes at the interfaces
of two materials of the stack. To numerically evaluate the integral we discretize the
multilayer stack by subdividing the whole stack and thereby each layer into equidistant
sublayers with a sufficient number of samples
*
. Fig. 2.12(a) shows an exemplary Cr/Sc
multilayer system with the relative Sc density
ρSc
in each layer. In order to numerically
calculate the integral in Eq.
(2.58)
, the system was divided into sublayers of thickness
di
and bottom interface positions
zi
as described above and illustrated in Fig. 2.12(b). The
(a) multilayer system
z z z
(b) sublayers (c) graded ρ(z)
ρSc =1.0
ρSc =0.0
ρSc =1.0
ρSc =a
ρSc =b
ρSc =a
a≥ρ(zi)≥b
b≤ρ(zi)≤a
Sc
Cr
Sc
0.0 ≤b<a≤1.0
zidi
Figure 2.12 |
Multilayer scheme to illustrate the method of calculating the X-ray fluorescence yield
by the example of Sc in a Cr/Sc multilayer. The multilayer system (a) is split into multiple equidistant
sub-layers (b) to obtain the field intensity at discrete points inside the Sc and Cr layers by applying the
matrix algorithm in Sec. 2.3. The relative density of Sc
ρSc
is multiplied by the respective intensity inside
each sublayer. In case of a more a realistic intermixed system, the intensity is calculated similarly to
(b) for discrete equidistant sub-layers. However, they differ by their relative Sc density, which is now
generally different for each layer (c).
integral thus turns into a discrete sum as
IXRF =˜
C
M
∑
i|E(zi)|2ρ(zi)di, (2.59)
*
The necessary number of samples for a given system can be determined heuristically by evaluating the
calculated fluorescence signal for an increasing number of sublayers until the numerical change of the result
saturates.
29
Chapter 2 THEORETICAL DESCRIPTION OF EUV AND X-RAY SCATTERING
where Mis the total number of sublayers of the whole stack.
In case of a theoretical multilayer system with perfectly sharp interfaces and no
interdiffusion, the relative density of
ρSc
in the given example will be binary, i.e. either
ρSc =1.0
in the Sc (sub-)layers or
ρSc =0.0
in the Cr layers. That, however, does not
reflect a realistic situation, where interdiffusion of interface imperfections could lead to a
mixture of the two materials in the sublayers. The third example in part (c) of Fig. 2.12
refers to a more realistic case, where the two materials interdiffuse with asymmetric
interface regions. There, the relative Sc density
ρSc
varies gradually from the highest
value
a
to its lowers value
b
. In general,
a
and
b
will not attain the values
1.0
and
0.0
,
respectively, as in the cases (a) and (b). That is, because the possibility exists that the two
materials interdiffuse so strong, that no region with pure Sc or pure Cr remains.
The X-ray standing wave analysis of periodic multilayer systems
The section above covers the case of general multilayer systems. Here, we now shall
consider the special case of strongly periodic layers [41,54], such as multilayer mirror
systems for the examples given in Fig. 2.12(a,b) and 2.12(c). For the exemplary calculation,
first we choose the layer thicknesses to be
dSc =0.6
nm and
dCr =0.7
nm periodically
replicated
N=400
times with perfectly sharp interfaces, i.e. the case (a) and (b) of
Fig. 2.12, we obtain a one-dimensional artificial Bragg crystal. Radiation of
6.25
keV is
well above the
K
-absorption edges of both materials and thus causes the emission of the
K
-line fluorescence radiation. For grazing angles of incidence between
αGI
i=3.7◦
and
αGI
i=3.9◦
and for this photon energy the reflected field amplitudes at each interface
interfere constructively causing the appearance of the first order Bragg peak of the
periodic layer structure. The corresponding calculation employing the matrix method
from Sec. 2.3is shown in Fig. 2.13(a).
The constructive interference in the Bragg condition, in addition to resulting in a
high reflectance, causes the formation of a X-ray standing wave inside the layers. The
corresponding intensity distribution in the top first few layer pairs for the example given
here is shown in Fig. 2.13(c). The standing wave intensity shifts through the individual
layers thereby selectively exciting fluorescence radiation in the respective elements while
changing the angle of incidence across the Bragg peak. Thus, this yields a method for
material selective composition analysis with spatial resolution in the sub-nanometer
regime called X-ray standing wave (XSW) analysis. The response curves of the respective
relative fluorescence radiation intensity across the Bragg peak for both materials is shown
in Fig. 2.13(b). In this example, the fluorescence radiation was calculated according to
the discrete sum in Eq.
(2.59)
with a sublayer setup as indicated in Fig. 2.12(b) and
30
sublayers per layer pair in each period.
The example above was extended to the case of imperfect layer stacks with interdiffu-
sion and strongly asymmetric interface region thickness. The corresponding calculation is
then calculated according to the scheme given in Fig. 2.12(c) and is added for comparison
to the reflectance and fluorescence yield curves in Fig. 2.13(a) and 2.13(b) as dotted lines.
The diminished contrast causes a decrease in the peak reflectance of that multilayer
system as well as changes in the fluorescence yield. Based on this analysis technique, it is
possible to gain information on the distribution of a specific element inside the multilayer
stack averaged over the irradiated sample area. In the particular, strong intermixing
with potentially asymmetric interface regions, as described in the latter case, can be
distinguished from sharp interfaces through the decrease in amplitude and the symmetry
30
Grazing-incidence X-ray Fluorescence 2.5
3.77 3.78 3.79 3.80 3.81 3.82 3.83 3.84 3.85
grazing angle of incidence αGI
i/◦
0
1
2
3
4
5
depth z/ nm
Cr
Sc
Cr
Sc
Cr
Sc
(c)
0.4
0.6
0.8
1.0
1.2
1.4
1.6
relative F.Y.
(b) Cr
Sc
0.00
0.05
0.10
0.15
0.20
0.25
0.30
reflectivity
(a)
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
relative Intensity
Figure 2.13 |
Illustra-
tion of the grazing in-
cidence X-ray standing
wave fluorescence anal-
ysis. The exemplary sys-
tem shown is a bilayer
multilayer mirror of Cr
and Sc irradiated with a
6.25
keV photon beam at
grazing angles. Chang-
ing the grazing angle
of incidence
αGI
i
across
the first Bragg peak (a)
causes a standing wave
inside the multilayer (to-
tal intensity in first top
layers shown in (c)) and
cause a relative fluores-
cence yield for the two
different materials as
shown in (b). The dot-
ted lines in (a) and (b) in-
dicate the case of inter-
diffusion and strongly
asymmetric interface re-
gions for comparison.
of the measured fluorescence signal when changing the incidence angle. For periodic
multilayer samples with very thin individual layers, this is of special relevance as the
intermixing region can be of the same order of thickness as the layers itself. The XRF
analysis, thus, is a suitable tool to analyze and reconstruct the structural properties of
such periodic multilayer samples.
31
3
Experimental Details and Analytical
Toolset
Within this chapter, an overview of the various experimental setups used for the char-
acterization of the samples of this thesis is given. The data evaluated in the course
of this thesis resulted from experiments that were performed at the BESSY II and the
MLS, which are third generation synchrotron radiation sources
*
. Depending on the
spectral range of the radiation, different beamlines with specialized endstations and
different monochromatization methods were used to perform the experiments. The
existing experimental setups at the various beamlines and their endstations are reviewed
in the following paragraphs. First, an introduction on the basic principles governing the
generation of synchrotron radiation and their specific application to metrology tasks is
given. Second, the instrumentation at the laboratories of the PTB used for the experiments
in this thesis is presented.
Additionally, the most important details of the multilayer fabrication principle are
described as they determine the quality of the sample. The theoretical description
from chapter 2already indicated the requirements of optical contrast to achieve high
reflectivities of multilayer systems within a certain bandwidth. The details of the sample
fabrication and their composition is therefore described in the second part of this chapter.
Finally, a software package was developed to evaluate the data extracted from the
measurements and to reconstruct the model parameters describing the layer systems.
This software implements all theoretical methods introduced in the previous chapter
and allows to quantify structural parameters of the samples based on the experimental
data. The last part of this chapter gives a brief description of the individual modules
developed in the framework of this thesis.
*
In addition to the experiments conducted as part of this thesis, there was XRR data taken into account
during the analysis. This data was the result of precharacterization experiments done using lab instruments
operated by the sample fabricators.
33
Chapter 3 EXPERIMENTAL DETAILS AND ANALYTICAL TOOLSET
Figure 3.1 |
Theoret-
ical synchrotron radia-
tion radiant power spec-
tra for the MLS and
BESSY II in comparison
to black body radiation
a
.
The curves show the
radiant power of emis-
sion from bending mag-
nets at both electron
storage ring facilities for
different electron ener-
gies. The curve marked
WLS shows the radiant
power from the
7
Tesla
wavelength shifter inser-
tion device installed at
BESSY II.
a
Image taken from
Beckhoff et al. [17]
3.1 Synchrotron Radiation
The radiation emitted by a relativistic charged particle, usually electrons, accelerated
on an orbit through an external magnetic field is called synchrotron radiation. This
radiation is polarized and emitted tangentially to the orbital movement of the charged
particle in forward direction. In the history of synchrotron radiation, sources have
evolved from parasitic use of particle accelerators to the extend of building electron
storage rings dedicated for the sole purpose of generating this radiation [100]. Its most
prominent features are the high brilliance, that is the number of photons per second per
unit particle beam cross section and per unit solid angle within
0.1%
bandwidth at a
specific wavelength, and its huge spectral range of emission. Depending on the energy of
the relativistic particles forced on an orbit, in modern electron storage rings typically in
the order of one to several GeV, the emission covers the range from the terahertz into the
hard X-ray regime. The PTB operates two laboratories at the dedicated sources BESSY II
and MLS [28]. The two third-generation synchrotron radiation sources provide maximum
electron energies of
1.7
GeV (BESSY II) and
0.6
GeV (MLS), respectively. Theoretical
emission spectra for a single dipole magnet (bending magnet) are shown in Fig. 3.1in
comparison to black body radiation.
A very important theoretical aspect of synchrotron radiation, apart from the high
brilliance and broad spectrum, is the fact that the emission can be calculated exactly
from first principles of classical electrodynamics and special relativity. The theory for
synchrotron radiation was developed by Schwinger [120] and we shall review its most
important aspects here. Given all the fundamental and experimental parameters are
known, the total emitted radiant power per relativistic particle can be calculated exactly
as
P=1
4πe0
2
3
e2c
R2E
m0c24, (3.1)
where
e
is the elementary charge,
c
is the speed of light in vacuum,
E
is the particles
34
Synchrotron Radiation 3.1
energy,
m0
is the rest mass of the particle and
R
is the radius of the circular trajectory
imposed by the magnetic field. The radiant power is thus inversely proportional to the
fourth power of the particles rest mass, which explains the usage of light electrons in
comparison with significantly heavier protons in synchrotron radiation sources. Apart
from the total emitted radiant power, an additional characteristic quantity of synchrotron
radiation is the critical energy or critical wavelength [120], respectively,
EC=3hc
4πRE
m0c23. (3.2)
It marks the point in the spectrum, where the integrated radiant power for all values
above and below the critical energy are equal [11]. This formula quantifies the shift
towards higher energies in Fig. 3.1due to the increase of the electron energy comparing
the MLS and BESSY II emission spectra. Apart from the spectral distribution, the emitted
radiation is linearly polarized with an electric field vector oscillating parallel to the
orbital plane. This property, however, is only strictly valid for the emission inside this
plane. For radiation above or below, a vertical polarization component (parallel to the
surface normal of the orbital plane) exists and the radiation becomes elliptically polarized.
The intensity
I(λ,Ψ)
emitted by a single electron on a circular orbit in direction of the
azimuthal angle Ψat the wavelength λis described by
I(λ,Ψ) = 27e2γ8
36π3R3λc
λ41+ (γΨ)2K2
2/3(ζ) + (γΨ)2
1+ (γΨ)2K2
1/3(ζ), (3.3)
where
γ=E/m0c2
and
Ψ
is the angle between the orbital plane and the observation
direction outside of that plane [120]. The characteristic wavelength
λc=hc/Ec
is given
by the critical photon energy defined in Eq.
(3.2)
. The argument of the modified Bessel
functions of second kind Kx(ζ)is defined as
ζ=λ
λc1+ (γΨ)23
2. (3.4)
The ability to calculate the emission and polarization properties of synchrotron radia-
tion based on Eq.
(3.3)
with a given electron current and acceptance angle have another
very valuable side effect for the field of metrology. It enables the use of synchrotron
radiation as a primary standard for electromagnetic radiation within the available spectral
range, which is in fact exploited by the PTB [137] to provide absolute radiometry.
The dedicated synchrotron radiation facilities, such as BESSY II and the MLS provide
additional possibilities of generating synchrotron radiation beyond a simple bending
magnet through different insertion devices. Fig. 3.2gives a schematic overview of the
storage ring BESSY II. At each of the marked dipole magnets, synchrotron radiation is
produced according to the theory presented above. The radiation is transmitted through
outlet systems towards a large number of beamlines, which monochromatize and focus
the radiation for experimental applications. Undulators or wigglers are inserted in
the straight sections of the BESSY II storage ring with a large number of periodically
arranged magnets with alternating polarization forcing the electrons on a beam path
alternating in direction, e.g. on a sinusoidal path. The goal of these insertion devices
is to shift the critical energy of the storage ring towards higher energies or increase
the radiated power (wigglers). An undulator, is the limiting case of a wiggler, where
the emitted radiation can interfere constructively dramatically increasing the brilliance
35
Chapter 3 EXPERIMENTAL DETAILS AND ANALYTICAL TOOLSET
electromagnetic lens
insertion device
(undulator or wiggler)
accelerator synchrotron
electron storage ring
LINAC
example beamline
dipole (bending) magnet
quadrupole magnet
sextupole magnet
cavity
Figure 3.2 |
Schematic overview of the electron storage ring facility BESSY II
a
. The synchrotron
accelerates the electrons coming from the linear accelerator (LINAC), which are then injected in the
electron storage ring with their full desired energy. Electromagnetic lenses focus and stabilize the
beam, as well as deflecting it onto the circular orbit while emitting synchrotron radiation at each dipole
(bending) magnet. Cavities reaccelerate the electrons in the storage ring to compensate the energy
loss due to the radiation emission.
a
Original image by Helmholtz-Zentrum Berlin (HZB), Ela Strickert, source:
https://www.
helmholtz-berlin.de/mediathek/bildarchiv/
within a significantly smaller spectral range compared to bending magnets. The different
effect of the undulators and wigglers on the generated spectrum is determined by the
magnetic field strength
B0
and the distance between two identical periodic arrangements
of the magnets of alternating polarization
λ0
. The deflection parameter quantifies this
relation through
K∝B0λ0
. Undulators typically have deflection parameters with a small
value
K
, while in case of wigglers
K
is very large [100]. Technically, the magnetic field
strength can be varied by changing the distance (“gap”) between the magnets vertically.
By changing the vertical alignment of the magnetic field direction with respect to the
beam path, it is even possible to affect the polarization properties of the emitted radiation
to obtain circularly or elliptically polarized radiation. The effect of these insertion devices
is illustrated in Fig. 3.3.
The most advanced light source available today, also known as fourth generation
source, is following the concept of a FEL as first invented by Madey [93]. In that case,
radiation is produced by a typically single very long undulator after a linear accelerator
instead of a comparatively short straight section of a storage ring. The concept was first
demonstrated by Deacon et al. [39]. FEL sources produce highly coherent radiation in
the X-ray regime. A possible operation scheme is through the principle of self-amplified
spontaneous emission (SASE) [26,40]. In short, the emitted radiation inside the long
undulator has a feedback effect on the electron bunch traveling along the beam path
36
The Instrumentation for the EUV Spectral Range 3.2
bending magnet
wiggler
undulator
free electron laser
Figure 3.3 |
Schematic illustration
a
of the generation of synchrotron radiation in bending magnets,
insertion devices and in free electron lasers. In a bending magnet, synchrotron radiation is produced
through the acceleration in the magnetic field. The wiggler has alternating magnetic fields, thus causing
an alternating trajectory of the electron increasing the radiated power. The undulator is the limiting
case of the wiggler, where the generated radiation interferes constructively increasing its brilliance. By
increasing the lengths of the undulator, a feedback effect of the generated radiation is exploited to
produce extremely strong radiation peaks. The latter case is known as free electron laser (see main
text).
aImage taken from https://www.helmholtz-berlin.de
(cf. bottom part of Fig. 3.3). The result is an exponential amplification of the emitted
radiation connected with a (random) wavelength within a certain spectral range defined
by the undulator properties until a saturation level is reached [98]. The resulting emission
spectrum shows several extremely strong spikes of amplified wavelengths with a low
noisy background.
3.2 The Instrumentation for the EUV Spectral Range
3.2.1 The EUV Beamlines at BESSY II and MLS
The application of radiation generated in bending magnets or in insertion devices
of synchrotron radiation sources typically requires monochromatization and focusing
trough a series of optical elements depending on the experimental requirements or
designated use cases. In the specific case of radiation in the EUV spectral range, quick
absorption during propagation under atmospheric conditions is an additional problem
to be considered in the technical setup. It is thus necessary to maintain a high vacuum
from the source point to the experiment and the detector. The two PTB beamlines for
the EUV spectral range at the two storage rings BESSY II and MLS operate on the broad
spectrum emitted by bending magnets at each facility. The experiments conducted in the
37
Chapter 3 EXPERIMENTAL DETAILS AND ANALYTICAL TOOLSET
Figure 3.4 |
Schematic
overview of the electron
storage ring facility MLS
microtron
straight for insertion device
RF cavity
bending magnet
beamline
framework of this thesis were performed at both beamlines, as they are optimized for
different spectral ranges within the EUV window as defined in the beginning of chapter 2.
Depending on the required spectral range of the respective experiments, those had to be
conducted on the respective instrument. Nevertheless, the two beamlines share many
technical and design aspects. Thus, the description here will introduce most of these
aspects with respect to the SX700 beamline at BESSY II. The differences of the extreme
ultraviolet beamline (EUVR) at the MLS will be given below.
The Soft X-ray Beamline SX700
The soft x-ray beamline (SX700) at BESSY II provides a monochromatic beam in the
spectral range from
0.7nm
to
25nm
wavelength (corresponding to photon energy range
from
50eV
to
1800eV
) [17]. The beam size at the entrance aperture to the reflectometer
(experimental end station) is variable only in vertical direction through the setting of the
exit slit. In the standard setting, the beam spot is approximately
1mm
by
1mm
[116] and
can be reduced vertically (grating dispersion direction) to
0.25mm
, which is the lower
limit of the standard settings.
The monochromatization of the radiation is achieved by a plane grating monochromator
with a blazed line grating with 1200 lines per millimeter mounted with its rotational axis
parallel to the plane of the storage ring and illuminated perpendicular to the grating
lines, yielding the dispersive direction being perpendicular to the storage ring plane. The
schematic layout of the beamline is illustrated in Fig. 3.5including the plane grating
position of the monochromator, the focusing mirrors and slit positions. The selection
of the desired wavelength is done by the exit slit of the monochromator, which limits
vertically and thus allows only a portion of the dispersed radiation to pass through. The
achievable relative bandwidth depends on the size of this slit as well as on the selected
wavelength. It varies between values of
0.5 ×10−3
and
2.5 ×10−3
relative bandwidth.
38
The Instrumentation for the EUV Spectral Range 3.2
178◦
toroidal mirror plane mirror
plane grating
focussing mirror
filter
entrance aperture
cooled
bending
magnet apertures
exit slit
Figure 3.5 |
Schematic setup of the SX700 beamline at BESSY II in top view (upper part) and side view
(lower part) a.
aOriginal image taken from Scholze et al. [116]
As mentioned above, the monochromator grating disperses the incoming broad band
radiation into the vertical direction with respect to the storage ring plane. The blaze of
the grating ensures high grating efficiency in the first diffraction order. However, higher
diffraction orders are still part of the selected vertical angular range selected by the exit
slit leading to a diminished spectral purity. For the purpose of suppression of these
higher grating orders, thin metal films in transmission geometry acting as filters are
installed close before the exit slit suppressing radiation energetically above the respective
absorption edges of the material. In consequence, several different metal thin films
have to be used to ensure the spectral purity across the spectral range of the beamline
depending on the monochromator setting.
The SX700 beamline has only one focusing mirror per horizontal and vertical direction,
which differs in position in the beamline and produces different focal points for the
two directions. The focusing in horizontal direction is done trough the toroidal mirror
(cf. Fig. 3.5), which also serves as a collector mirror for both axis and parallelizes the
beam in vertical direction. The focal point is located in the entrance aperture (about
2m
behind the exit slit in propagation direction), which allows to cut off any unwanted stray
light at this position. The vertical focusing is done by an additional focusing mirror after
the monochromator grating. The vertical focal point is located in the exit slits, which
ensures high energy resolution through the selection process explained above. Due to
the large distance of the two focusing elements to the experimental station and the low
acceptance of the toroidal mirror, a low divergence of the beam of about
1.6
mrad
×0.4
mrad is achieved. The total radiation power of this beamline is shown in Fig. 3.6.
39
Chapter 3 EXPERIMENTAL DETAILS AND ANALYTICAL TOOLSET
Figure 3.6 |
Radiant
power of the SX700
beamline at BESSY II in
W per mA storage ring
ring current. The two
curves differ by the coat-
ing on the premirror
in the beamline. The
Pt coating ensures high
radiant power in the
high energy range up to
1800 eV
, while the TiO
coating absorbs a large
amount of high-energy
radiation and reduces
the radiation power den-
sity on the monochro-
mator grating at lower
photon energy settings.
101102103104
photon energy / eV
10−11
10−10
10−9
10−8
10−7
radiant power / W mA−1
SX700 Pt premirror
SX700 TiO premirror
The Extreme Ultraviolet Beamline EUVR
The general layout and operation principle of the EUVR beamline is identical to that
of the SX700 beamline described in the previous paragraph with some differences in
the focusing and radiant power, which are described in the following. Due to the lower
electron energy in the MLS storage ring, the spectrum of the bending magnets for
both beamlines differs with the critical energy shifted to smaller values (cf. Fig. 3.1).
Consequently, the wavelength range covered by the EUVR beamline is between
5nm
to
50nm
(corresponding to photon energies from approximately
25eV
to
248eV
), to
make use of the higher radiant power available in that range compared to the BESSY
II spectrum. The toroidal mirror (collector mirror) of the EUVR beamline does have a
larger aperture and is positioned significantly closer to the source point. Through this
modification, an increase of the total acceptance angle by two orders of magnitude is
achieved. This, however, increases the divergence of the beam to approximately
4
mrad
in both directions. In contrast to the SX700 beamline, the foci for horizontal and vertical
direction are both at the position of the exit slit with an additional refocusing mirror
behind that slit to counteract the strong divergence. Together with the shifted bending
magnet spectrum of the MLS, this different setup allows higher photon flux. In addition,
the refocusing allows the spot size of the EUVR beamline to be adjusted (by closing the
cooled apertures shown in Fig. 3.5) at acceptable reduction of the overall photon flux.
Furthermore, through an off-center positioning of the cooled aperture opening outside
of the orbital plane of the ring, different polarization degrees may be chosen. The key
properties of both beamlines and their differences are given in Table 3.1. The optics of
both beamlines in comparison are shown in Fig. 3.7.
40
The Instrumentation for the EUV Spectral Range 3.2
Table 3.1 | Beamline parameters of the two EUV beamlines EUVR and SX700 in comparison.
Parameter SX700 EUVR
Wavelength range 0.7nm to 24.8nm 5nm to 50nm
Spot size (standard settings) 1mm ×1mm 0.1mm ×0.1mm to
2mm ×2mm
Beam divergence 1.6mrad ×0.4mrad 4mrad ×4mrad
Linear polarization (horizontal) 98% 40% to 98%
a) SX700 beamline
b) EUVR beamline
Figure 3.7 | Schematic optics of the SX700 and EUVR beamlines in direct comparison.
3.2.2 The Experimental Endstations at the EUVR and SX700 Beam-
lines
All experiments in the EUV spectral range within the framework of this thesis were
conducted at the beamlines EUVR and SX700. Each of the beamlines is equipped with
an experimental end station containing the detectors, mounts for charge coupled device
(CCD) cameras and a goniometer to adjust the angle of the sample holder with respect to
the beam. Due to the high absorption of the EUV radiation in air, both chambers need to
be kept under high vacuum conditions, typically below the limit of 3 ×10−6mbar.
The end stations differ in the size and weight of samples, which can be mounted
on the sample holder. The large reflectometer at the EUVR beamline was designed
with heavy and large samples in mind, whereas the ellipso-scatterometer at the SX700
beamline covers a larger angular range for both the detector and the sample holder,
due to additional axis allowing measurements anywhere in between and including
perpendicular to the orbital plane (s-polarization direction) and parallel to the orbital
plane (p-polarization direction). In the following the two different setups with their
41
Chapter 3 EXPERIMENTAL DETAILS AND ANALYTICAL TOOLSET
primary features are summarized.
The large reflectometer at the EUVR beamline
The large reflectometer serving as the end station at the EUVR beamline was designed
for reflectometry and scatterometry measurements for samples with a weight of up to
50 kg
and a maximum diameter of
550mm
in mind [138]. The available axis of movement
and rotation are shown in Fig. 3.8. The sample holder plate allows for linear movement
(b) Axis labels and movement directions
of the sample holder and the detector
(a) Photograph of the goniometer with sample
holder and the detector arm
Figure 3.8 |
The EUV reflectometer end station of the EUVR beamline at the MLS. A photograph of the
internal mechanics and the vacuum chamber is shown in (a). The schematic layout of the goniometer
axes and the detector arm are shown in (b).
in all three orthogonal directions as well as angular rotations in three axis. The rotation
around the
Θ
-axis covers the range from
−30°
to
95°
relative to the incoming beam. Thus,
enabling reflectometry and scatterometry from normal incidence to grazing incidence
angles together with the detector arm rotation around the
2Θ
-axis from
−5°
to
190°
. The
rotation of the sample holder around the
ϕ
-axis (cf. Fig. 3.8) with an angular range from
0°
to
360°
, allows to measure a sample mounted in the center of the sample holder with
radiation impinging from all directions. The distance of the detector to the sample is
variable through the Det-R axis from a minimum value of 150mm to 550mm.
The detector mount is equipped with up to
4
diodes, which can be rotated to face
either the sample or the incoming beam. The diodes used within the framework of
this thesis are
4.5mm ×4.5mm
and, optionally
10mm ×10mm
, GaAsP photodiodes.
The detector holder can be moved along the
Θ
and
2Θ
rotational axes, labeled as Det-X
direction in Fig. 3.8, which allows to take measurements in the out-of-plane
*
direction in
s-polarization.
*
The out-of-plane scattering direction refers to radiation scattered outside of the scattering plane spanned
by the surface normal of the sample and the impinging beam direction.
42
Grazing-incidence X-ray Fluorescence at the FCM Beamline 3.3
The Ellipso-scatterometer at the SX700 beamline
The ellipso-scatterometer is a reflectometer similar to the large reflectometer providing
the end station for the SX700 beamline but it operates with hydrocarbon-free mechanics
reducing a source of contamination for the measured samples. Its capabilities differ
from the large reflectometer by a wider reachable angular range for both the detector
movement as well as the sample movement. The angular and linear movements and
axes are shown in Fig. 3.9together with a photograph of the goniometer and detector
arm. In contrast to the large end station at the EUVR beamline, it can hold samples
(b) Axis labels and movement directions
for the sample holder and the detector
(a) Photograph of the goniometer with sample
holder and the detector arm
Figure 3.9 |
The EUV ellipso-scatterometer end station at the SX700 beamline at BESSY II. The internal
mechanics of the goniometer and the detector arm are shown in (a). The schematic layout of the axes
an the movable detector arm are given in (b).
with a a maximum of
5kg
in weight. However, the rotational movement of both the
detector and the sample holder allow for a larger angular range. In particular, the
detector holder may be moved on large parts of the upper hemisphere above the sample
holder. In consequence, measurements in s-polarization and well as p-polarization can
be conducted on the same sample. With the capability to mount a polarization analyzer
at the detector holder, polarization resolved measurements are thus possible [127].
3.3 Grazing-incidence X-ray Fluorescence at the FCM Beam-
line
The grazing incidence X-ray fluorescence (GIXRF) measurements of the Cr/Sc sample
systems were performed at the four crystal monochromator (FCM) bending magnet
beamline [78] in the BESSY II laboratory. The necessary photon energies to excite the K-
edge X-ray fluorescence of chromium and scandium, are well above the spectral range of
the EUVR and SX700 beamlines in the order of several
keV
. The general setup and design
of the FCM beamline is very similar to that of the SX700 beamline, with the exception of
the four crystal monochromator, which replaces the plane grating monochromator in the
X-ray spectral range. It offers tunable photon energies from
1.75keV
to
10.0keV
. A high
43
Chapter 3 EXPERIMENTAL DETAILS AND ANALYTICAL TOOLSET
energy resolution of
E/∆E=104
is attained by the combination of four exchangeable
crystal Bragg reflections. The monochromator can be equipped with two monochromator
crystal types. For the high energy range above approximately
3.5keV
to
10.0keV
silicon
is used. In the lower energy range between
1.75keV
and
3.5keV
higher radiant power is
available through the usage of a InSb crystal and more specifically, the silicon K-edge at
1.84keV
becomes accessible. A schematic overview of the FCM beamline can be found in
Fig. 3.10.
Figure 3.10 |
Schematic
layout
a
and optical path
of the FCM beamline
at BESSY II. The setup
is comparable to the
SX700 beamline, but
uses a four-crystal
monochromator setup
instead.
a
Original image taken
from Krumrey [78].
The end station used for the GIXRF experiments is a specialized chamber for GIXRF,
total reflection x-ray fluorescence (TXRF) and XRR [89] depicted in Fig. 3.11. It is
equipped with a detector arm and a sample goniometer allowing to measure grazing
incidence angles of
0°
to
60°
. The detector arm holds a diode allowing XRR measurements.
Perpendicular to the beam direction, an energy dispersive silicon drift detector (SSD) is
mounted close to the sample surface. It allows to detect fluorescence radiation emitted
from the sample energetically resolved. The samples can be rotated with respect to the
storage ring plane in order to allow a variable polarization impinging on the surface,
very similar to the axis movements possible with the ellipso-scatterometer end station at
the SX700 beamline.
3.4 Sample systems
The samples studied in the framework of this thesis are designed to work as near-normal
incidence mirrors for the EUV spectral range. The underlying principle of an artificial
one dimensional Bragg crystal requires the deposition of thin layered systems with high
periodicity and stability. The experiments presented here were conducted on two sets of
sample types as prototypes of mirrors for two different spectral ranges. In the theoretical
description of the principle of multilayer mirrors in Sec. 2.3of Ch. 2is was outlined, that
optical contrast, i.e. a large as possible difference in the real part of the refractive index
n
,
is required to achieve high reflectivities while maintaining a low absorption. The latter
is of special importance, as stacking of several layers is only beneficial if the radiation
can reach deep into the layer stack. Nevertheless, a compromise between low absorption
and optical contrast has to be found specific for the application and the desired spectral
range. While high peak reflectivities in a relatively small spectrum require many layers
to contribute, broadband mirrors with smaller peak reflectance may work better with a
44
Sample systems 3.4
(b) GIXRF sample holder and manipulator
(a) Exterior of the GIXRF chamber
Figure 3.11 |
The schematic layout of the dedicated GIXRF chamber
a
. This end station can be mounted
to the FCM or plane grating monochromator (PGM) beamlines to conduct grazing-incidence XRF
experiments. In (a), the schematic exterior layout and, in (b), the interior layouts are shown.
aImages taken from Lubeck et al. [89].
material with higher absorption and higher optical contrast with fewer layers contributing
to the reflectivity.
3.4.1 Choice of the Chemical Species and Multilayer Design
This thesis investigates systems designed to reflect radiation in two spectral ranges, the
water window with wavelengths from
2.2nm
to
4.4nm
and the range from
12.4nm
to
14.0nm
with a wide range of applications, e.g. for the next-generation lithography. The
choice of the chemical species for the multilayer systems, apart from trivial properties
such as non-toxicity and solidity, is largely influenced by the electronic structure of the
respective materials, since large changes in the refractive index, i.e. large optical contrast
with respect to a second material, can be expected close to resonances in the electronic
structure. The demand for low absorption also requires species, where the absorption
edges are energetically higher or far lower than the desired spectral range of operation.
For a well defined interface it is also necessary that the two materials are mostly inert and
do not react with one another or alternatively, if reactive materials are the only reasonable
choice, that mechanisms for avoiding strong intermixing exist.
45
Chapter 3 EXPERIMENTAL DETAILS AND ANALYTICAL TOOLSET
Cr/Sc multilayer system
In case of the water window spectral range, samples designed for a peak reflectance
at a wavelength closely above
3.14nm
, where the L3edge of scandium (Sc) is found,
were investigated. Fig. 3.12 shows the refractive index of Sc and the second material
chromium (Cr) in the water window spectral range. The periodic multilayers of the
Figure 3.12 |
Refractive
indices of Cr and Sc with
the water window spec-
tral range. The marked
absorption edges are
the L2 and L3 edge
of Sc. The imaginary
part of the refractive in-
dex accounts for the ab-
sorption and is shown
for Sc. At wavelengths
only slightly larger than
that of the L3 edge is
the highest contrast for
the two materials pro-
viding the highest poten-
tial reflectivity in a peri-
odic multilayer arrange-
ment.
3.0 3.1 3.2 3.3 3.4 3.5
wavelength / nm
0.990
0.995
1.000
1.005
1.010
1.015
real part of n
L2
L3
Re(nSc)
Re(nCr)
0.000
0.002
0.004
0.006
0.008
0.010
0.012
imaginary part of n
Im(nSc)
systems investigated here, were therefore binary alternating layers of Cr and Sc. The
required nominal period thickness
D
, i.e. the thickness of each periodically repeated
layer stack, for the design goal of a peak reflectivity at
λ=3.14nm
is
D=1.573nm
with a layer thickness ratio of
Γ=0.5
of both materials. To protect the Sc layers from
oxidation, an additional Cr capping layer of approximately
dcap =3nm
was added as
the surface layer. The multilayer is composed of alternating layers of Cr and Sc with
periodic replication of the bilayer stack by
N=400
times. The substrate is a Si wafer
piece. The sample dimensions measure approximately
(20 ×20)
mm
2
. More details can
be found elsewhere [108]. The multilayer mirror was designed to reflect radiation in the
water window, at wavelengths just above the Sc L edge, close to a
3.1
nm at an angle of
incidence (AOI) of αi=1.5°.
Mo/Si multilayer systems
The second set of systems under investigation in this thesis is composed out of
50
to
65
bilayers molybdenum (Mo) and silicon (Si). Si shows a very low absorption in the range
from
12.4nm
to
14.0nm
, with the Si L2edge forming the lower wavelength limit for the
usage as a mirror system in this combination. The Mo layers absorb stronger than the Si
layers but provide the optical contrast required for high peak reflectance, as outlined at
the beginning of this section. The respective refractive indices are given in Fig. 3.13.
Finally, specifically Mo and Si are a choice of materials, which indeed do react and
form MoSi
x
compounds at the interfaces. This reduces the optical contrast and has to be
avoided. For that purpose the sample systems can contain additional materials, which
serve as barrier layers preventing this effect. The two species used for our samples are
Boroncarbite (B
4
C) and Carbon (C). Those two materials do not have any absorption
edges in the given relevant spectral range and additionally show low contrast to the
46
Sample systems 3.4
12.5 13.0 13.5 14.0
wavelength / nm
0.90
0.95
1.00
1.05
1.10
real part of n
L3
L2
Re(nMo)
Re(nSi)
0.000
0.005
0.010
0.015
0.020
imaginary part of n
Im(nSi)
Figure 3.13 |
Refrac-
tive indices of Mo and
Si in the wavelength
range from
12.4 nm
to
14.0 nm
. The L3 absorp-
tion edge of Si marks the
lower wavelength limit
for the applicability of
this material combina-
tion in multilayer mirror
systems.
spacer material Si. The details of the respective sample layouts are discussed in the
corresponding sections of the following chapters.
3.4.2 Multilayer Deposition by Magnetron Sputtering
The multilayer samples investigated here were fabricated by the DC magnetron sputtering
technique [133] by two different multilayer and optics groups. The Mo/Si multilayer
samples were fabricated by Stefan Braun at the Fraunhofer IWS, Dresden, Germany and
the Cr/Sc samples are by Saša Bajt from the Optics Group at CFEL, DESY, Hamburg in
Germany.
Magnetron sputtering is a physical vapor deposition technique. A vacuum chamber
is equipped with a substrate to be coated, in our case silicon, and one or more sputter
targets. Depending on the intended design of the multilayer to be deposited, those targets
are the respective materials, which later form the individual layers. In the DC magnetron
sputtering system, a strong electric field is applied between the substrate and the sputter
targets. The vacuum chamber containing those parts is then filled with a sputter gas,
typically ultrapure Ar gas (
99.999 %
), with partial pressures in the range from
10−3mbar
to
10−2mbar
[133]. The strong electric field ionizes the sputter gas causing the ions to
be accelerated towards the sputter targets (cathode) and form a charged plasma. Upon
impact in the target, atoms and electrons of the condensed matter phase of the respective
material are released and travel towards the substrate. The released atoms condense
there, forming bonds and creating a slowly growing layer. The thickness of the layer can
be fine tuned through the deposition time. The additionally released electrons, while
being accelerated towards the substrate (anode), collide with the sputter gas atoms and
cause further ionization. In order to avoid damage of the forming layer at the substrate,
strong magnetic fields are applied to the sputter targets. This confines the movement of
the charged particles (the plasma) sputter gas ions and electrons to the region close to the
target surface. Thereby, increasing the collision (ionization) probability of electrons and
the gas atoms through the helical movement in the magnetic field while keeping those
particles away from the substrate. To ensure homogeneous layer deposition, the substrate
is kept under permanent rotation. A schematic DC magnetron sputtering system is
depicted in Fig. 3.14.
For both samples, silicon serves as the substrates for the deposition process. The sample
size and shape differ for the systems investigated. In case of the Mo/Si mirror samples,
47
Chapter 3 EXPERIMENTAL DETAILS AND ANALYTICAL TOOLSET
Figure 3.14 |
Schematic
setup of a magnetron
sputtering deposition
systema.
a
Original image taken
from Stearns et al. [133]
vacuum chamber
spinner motor assembly
substrate platter
target
magnetron source
crystal monitor
substrate (Si)
wafer pieces of approximately
20mm×20mm
(photograph shown in Fig. 3.15) and round
substrates, so-called GO optical flats, of
1/4
inch in thickness and with approximately
1
inch (
≈25mm
) in diameter were used. In case of the Cr/Sc systems, wafer pieces of
Figure 3.15 |
Photo-
graph of a Mo/Si multi-
layer mirror sample on
20 mm ×20 mm
wafer
substrate.
varying size but approximately 10mm ×20mm served as the substrate.
3.5 Analytical Tools
In this thesis, several experiments are conducted on different sample systems requiring
a dedicated analytical toolset to analyze the sets of data and implement the theoretical
calculations based on the models introduced in chapter 2. For that purpose, dedicated
software was developed to enable the quantitative analysis conduced in the following
chapters. Here, an overview of the software packages and their relation to those already
existing and integrated into the framework is given.
All software was written in the Python programming language using the Numpy
and Scipy frameworks [143] for data analysis and scientific computing. The graphical
representation of the data and calculations was done using the Matplotlib [68] framework.
The packages developed may be coarsely categorized in the calculation of the electro-
magnetic field inside and outside a multilayer system following the matrix algorithm
explained in Sec. 2.3, the implementation of the DWBA as described in Sec. 2.4and
the optimization algorithms partly using existing software packages. All modules were
combined using the framework provided by iPython Notebooks [104], which allow to
integrate the modules necessary to analyze a sample system including the results of the
calculations and their graphical representation within a single code file. The individual
modules and descriptions of the functions provided by them is listed below.
48
Analytical Tools 3.5
matrixmethod
Implementation of the matrix algorithm for calculating electromagnetic
fields inside a multilayer system. The theoretical fundamentals of this module are
described in detail in Sec. 2.3. The functions provided here require a predefined
layer system with the respective optical constants. At each of the interfaces, a
roughness/interdiffusion parameter (Névot-Croce parameter) may be considered.
reflectivity
This module serves as an interface to the matrixmethod module. It provides
functions to construct a periodic layer system based on the specification of the
layer materials, periodicity, densities as well as substrate material. Based on
this, the models described in the following chapters can be implemented and
the electromagnetic fields outside and inside the systems may be calculated. In
addition to the periodic part of the system, capping layers can be considered
explicitly. Furthermore, the module provides functions to consider graded interfaces
of different thickness by introducing a given amount of sublayers. Those provide an
automatic gradual sinusoidal transition from the optical constants of one material
to the next in the stack. Based on the resulting model, the reflectivity depending on
angle of incidence and wavelength as well as all field components at each interface
are returned. This allows to calculate the reflectivity at any specified photon energy
and angle of incidence, but due to the availability of the full field components also
the X-ray fluorescence to be expected according to the method described in Sec. 2.5.
helper
Several often used functions are bundled in this module. This includes the
unit conversion from electron volt to wavelength for the impinging radiation, the
calculation of the wave vectors and the implementation of Snell’s law. In addition,
this module contains an interface to the periodictable
*
module to obtain the optical
constants for the materials specified for the reflectivity module from the Henke
database [62].
dwba
Implementation of the DWBA as introduced in Sec. 2.4. This module executes
the dynamic and semi-kinematic calculations described in the theory part. For
that purpose it requires the full set of field amplitudes that are calculated within
the reflectivity module. In addition, a PSD function needs to be specified which is
calculated within the integrals module described below and a vertical correlation
length value as well as the off-normal roughness correlation angle
β
. The result
of those calculations are absolute intensities of diffusely scattered radiation de-
pending on the specified detector distance and solid angle, as well as the incidence
and exit angles and wavelengths. The result may thus be directly compared to
correspondingly measured data.
integrals
Due to the separation of the roughness contributions and the contribution
due to the multilayer nature of the sample to the diffusely scattered radiation, a
separate calculation of the PSD is possible as explained in Sec. 2.4and performed
by this module. The input parameters of this calculation are the values for the r.m.s.
roughness, the Hurst factor and a lateral correlation length. The result enters the
calculations done in the dwba module.
pso
Implementation of the particle swarm optimization (PSO) algorithm following the
detailed description in the publication by Carlisle and Dozier [31]. The details of
*
The periodictable module was developed by the DANSE/Reflectometry team,
http://www.reflectometry.org/danse/elements.html
49
Chapter 3 EXPERIMENTAL DETAILS AND ANALYTICAL TOOLSET
the application of this optimization algorithm are described in Sec. 4.1. Due to the
implementation of that algorithm within the Python programming language, above
modules can be directly incorporated and used during an optimization of theoretical
curves based on experimental data. This allows to perform all calculations highly
parallelized and achieve reasonable calculation times. The implementation of the
parallel computing applied here is provided through the iPython toolset.
fitting
The most used residual functions for fitting data from EUV and XRR mea-
surements are contained in this model for convenience. This module requires
the reflectivity module (including the model specifications) and input data with
specified angle of incidence and wavelength range.
Based on the toolset of modules given here, all calculations within this thesis were
conducted. As mentioned above, for any given system an iPython notebook was created
bundling all measured data. The modules above provide the required access to simulate
and calculate any reflectivity, fluorescence or diffuse scattering experiment conducted
in this work and were optimized for highest possible performance. Within each of
the notebooks, residual functions were defined constituting an optimization functional
for the individual analysis of a single experiment or any combination of experiments.
Here, any parameters defining the respective model are specified and can be varied. All
specific systems analyzed within the thesis are described in the respective following
chapters in detail. Apart from the PSO algorithm implemented in the pso module, the
Python-based implementation emcee by Foreman-Mackey et al. [51] of a Markov-chain
Monte Carlo (MCMC) algorithm was used. Again, for any details of the application of
this method I refer the reader to the following chapters. With this purely Python-based
architecture, it was possible to accelerate any calculation of reflectivity, diffuse scattering
and fluorescence necessary within the optimization algorithms using the parallelization
framework provided by iPython. For that purpose, several available Linux machines
distributed across the PTB network were used in parallel to combine their computing
power for solving the optimization problems within this work in a reasonable time.
50
4
Characterization of the Multilayer
Structure for Different Systems
In this chapter, the structural properties of different multilayer systems are analyzed. The
samples investigated here are highly periodic multilayer systems designed as mirrors to
reflect radiation in different spectral ranges. The basic theory behind the principle of a
one-dimensional Bragg crystal exploited to achieve high reflectance values, is described
in chapter 2. All systems were fabricated using the magnetron sputtering technique
discussed in chapter 3with nominal layer thicknesses and chemical species, depending
on the desired reflection angle and spectral range. The different samples serve as mirrors
for two different wavelength ranges within the EUV spectrum, the range from
12.4nm
to
14.0nm
and the so-called water window range from
2.2nm
to
4.4nm
. As discussed
in Sec. 2.3, the individual layer thicknesses and the required number of periods are
intrinsically connected to the spectral range and angles, where maximum reflectance
shall be achieved. In case of the systems analyzed within this chapter, the individual
layer thicknesses are ranging from approximately
4nm
down to
0.5nm
. In addition, each
periodic part of the multilayer system is composed out of two to four of these individual
layers repeated with a number of periods of
50
to
400
. For the performance of a multilayer
mirrors system, the surface and interface morphology and the actual layer thicknesses
and densities of all these layers play a crucial role and affect the reflectivity behavior.
Small deviations of the perfect layer layout such as intermixing of the materials or
roughness at the interfaces are therefore a significant reason for a diminished reflectivity.
While the deposition through the magnetron sputtering process is a well established
technique for mirror fabrication, the actual layer thicknesses in the sample may differ
from the nominal values and furthermore have imperfections at the interfaces. For an
improvement of the deposition process, it is thus essential to assess the morphology and
potential intermixing of these highly complex samples.
Based on the matrix algorithm introduced in Sec. 2.3of Ch. 2, the electromagnetic
fields inside and outside an arbitrary layer system upon irradiation with EUV or X-ray
radiation can be calculated. Most importantly, this allows to calculate the expected
specular reflectance curves across angular or spectral ranges for a given layer model and
51
Chapter 4 CHARACTERIZATION OF THE MULTILAYER STRUCTURE FOR DIFFERENT SYSTEMS
even fluorescence expected from certain materials inside the stack. The comparison of
these calculated curves to measured data thus allows to obtain information about the
actual layer properties in a given sample with a destruction free approach. However, the
detected reflectance values in a specular reflection experiment, for example, are typically
very simple curves with only a very limited amount of information contained about the
rather complex samples. It is thus not possible to directly reconstruct the layout of the
sample with the measured reflection curve. This is known as the inverse problem of
scatterometry. Reconstructing the layer properties is therefore an attempt of solving this
inverse problem by accumulating prior knowledge about the sample, such as the nominal
design goals during the fabrication process, into a model of that system. Starting from
this model, the theoretically calculated curve is compared to the measured reflectance
and optimized iteratively.
This chapter is structured as follows. First, the information content within a simple
reflectivity curve for the design wavelength of the mirrors systems is discussed in
Sec. 4.1at the example of a mirror for the EUV range between
12.5nm
and
14.0nm
. A
reconstruction of the model for that particular system is presented and discussed in
conjunction with methods to assess the uniqueness and parameter accuracy. Second,
in Sec. 4.2, the investigation of a more complex set of samples designed for the same
spectral range is conducted. Here, the individual layer thicknesses inside the samples
were varied and different polishing methods affecting the interface morphology were
applied during fabrication. Based on the analytical experiments conducted here, an
improved reconstruction could be obtained by incorporating data from additional XRR
experiments. Finally, in Sec. 4.3, multilayer mirrors with sub-nanometer layer thicknesses
for the water window spectral range are investigated as limiting case of very thin layer
systems. There, the combination of multiple analytical experiments is required to deduct
a consistent reconstruction of the model.
4.1 Reconstruction Based on Specular EUV Reflectance
In this section, the reconstruction of a multilayer system designed as near-normal in-
cidence mirror for the wavelength range between
12.4nm
and
14.0nm
based solely on
experimental data of EUV reflectivity is demonstrated. The mirror was designed to
achieve a peak in the reflectance at a wavelength of
λ=13.5nm
for an angle of incidence
of
αi=6°
with respect to the surface normal. That combination is of relevance for
optical setups in the next generation lithography for the semiconductor industry, for
which this sample served as a prototype. The multilayer coating was deposited with
magnetron sputtering on a polished silicon substrate. The sample contains a periodic
layer stack of molybdenum (Mo) and silicon (Si). Due to the problem of intermixing and
resulting loss of interface definition, additional barrier layers of boroncarbite (B
4
C) and
carbon (C) were included at the Mo to Si and Si to Mo interfaces, respectively. We shall
therefore refer to this sample with the layer sequence within one period from bottom
to top as Mo/B
4
C/Si/C. The number of periods for that system is
N=65
, while the
65th
(capping) layer period does not posses a carbon layer but terminates at the vacuum
interface with the silicon layer and a natural SiO
2
oxide layer. A detailed schematic figure
of the layer layout can be found in the description of the corresponding theoretical model
in Fig. 4.2.
The sample was measured with respect to its reflectivity across the spectral range
52
Reconstruction Based on Specular EUV Reflectance 4.1
mentioned above at an angle of incidence of
αi=15°
from the surface normal. The
measurement was conducted at the EUVR beamline at the MLS. The reflectivity was
evaluated by first measuring the intensity of the direct beam in the reflectometer with
the photo diode detector. Then, the reflected radiation at an detector angle of
30°
was measured in reference to the direct beam signal. To ensure the stability of the
result, the direct beam was measured again afterwards and compared to the data of
the first measurement. The normalized results are shown in Fig. 4.1. The measurement
uncertainty with this experimental method is within
0.15%
(one standard deviation, i.e. a
coverage factor of
k=1
) of the peak reflectance value [115] and an angular uncertainty
of
0.01°
, which leads to an upper limit for the uncertainty of the reflectivity curve of
0.4%
in the peak flanks. Consequently, the total uncertainty margin is within the line
thickness of the data presentation in Fig. 4.1.
12.5 13.0 13.5 14.0
wavelength λ/ nm
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Reflectance
measured data
Figure 4.1 |
Spectrally
resolved reflectance of
the Mo/B
4
C/Si/C multi-
layer sample. The mea-
surement was conduced
under a fixed angle of in-
cidence αi=15.0◦.
The reflectivity curve shows a broad peak attaining its maximum value at a wavelength
of approximately 13.1nm, which is lower than the design peak reflectance of 13.5nm at
αi=6°
. That is due to the different angle of incidence,
αi=15°
, used in the experiment.
Apart from the main peak, side fringes are visible. They originate from the superposition
of waves being reflected at the top surface and the substrate interface. They are thus
directly related to the total thickness of the multilayer coating and well known as Kiessig
fringes [74]. Based on the data obtained through this spectrally resolved reflectivity
experiment, we shall attempt to reconstruct the unknown layer layout in the following
sections. The nominal fabrication parameters serve as starting values for the analysis to
build a reasonable model for the reconstruction.
The reconstruction of a given model based on the evaluation of EUV (or XRR) reflec-
tivity data is a well established method for the characterization of multilayer systems
[9,30,84]. In most cases a model is optimized applying gradient methods such as the
Levenberg-Marquardt method [82,94]. Those optimization algorithms typically operate
with a set of start parameters within the parameter space and iteratively improve an
optimization functional, usually termed
χ2
, describing the sum of the squared absolute
value of the difference between the theoretical calculation and the experimental data. This
is done by calculating the gradient of a that functional in all directions in the parameter
space and changing the parameters accordingly in direction of smaller
χ2
values. This
approach has the major disadvantage that the end result is strongly dependent on the
choice of starting values and may not represent a global minimum of
χ2
but only a local
optimum. While estimations of the quality of the fit results within the (local) optimum
are possible, no estimation can be given globally for the given model. For those reasons,
53
Chapter 4 CHARACTERIZATION OF THE MULTILAYER STRUCTURE FOR DIFFERENT SYSTEMS
this characterization strategy has only limited applicability and alternative approaches
are required.
In contrast to those gradient methods, heuristic optimization algorithms exist. Instead
of operating with predefined starting values, from which a gradient approach minimizes
the
χ2
functional, they operate distributed on the whole parameter space with often
randomly initialized parameters within given boundaries, instead. In the following
we shall apply those heuristic optimization routines to obtain the reconstruction of the
Mo/B
4
C/Si/C sample and elaborate their application to the characterization of multilayer
systems in detail.
4.1.1 Multilayer Model and Particle Swarm Optimization
For the purpose of reconstructing the layer layout of the Mo/B
4
C/Si/C sample, a
parametrized model is needed entering the theoretical calculations to obtain the reflec-
tivity curve according to the matrix algorithm. The model is largely based on prior
knowledge available from the fabrication process. For the multilayer sample investigated
here, the nominal layer design is known and a schematic representation is shown in
Fig. 4.2. As introduced above, the multilayer coating consists of a periodic arrangement
Figure 4.2 |
Model of
the multilayer stack in-
cluding the substrate
and the capping lay-
ers. The periodic part
is enclosed between the
dashed lines with four
layers in each period re-
peated
N=64
times.
The capping period does
not include an interdif-
fusion layer but does re-
flect the natural oxida-
tion through the addi-
tion of a SiO2layer.
Si (substrate)
Mo
Si
C (buffer layer)
SiO2
D
dC
dSi
dMo
capping layer
periodic replication
B4C (buffer layer)
dB4C
z
of four layers replicated 64 times. With the top period being different from the others
through the missing carbon interdiffusion layer on the top surface. Since the sample
was exposed to ambient conditions, a passivization of the top silicon surface through
oxidation has to be taken into account through a silicondioxide layer. The parametrization
of that model is given by the thicknesses of each layer within one period as well as for
the capping silicondioxide layer. Each of the deposited layers may vary in density with
respect to the bulk density of that material [30], which also needs to be reflected in the
model. Finally, the Névot-Croce factor
σ
accounting for roughness and intermixing at the
interfaces as introduced in Eq.
(2.29)
of chapter 2is also included. The required optical
constants, i.e. the indices of refraction, of the respective materials in the relevant spectral
range are taken from tabulated values by Henke et al. [62] and are used for the theoretical
calculations based on the matrix algorithm. At this point, it should be noted that the
tabulated optical constants itself come with an uncertainty, which is generally unknown
here. In order to account for this, all models within this thesis contain a variable density
54
Reconstruction Based on Specular EUV Reflectance 4.1
Table 4.1 | Multilayer parametrization and parameter limits
Parameter Definition Lower bound Upper bound
dMo / nm Mo layer thickness 0.0 7.0
dSi / nm Si layer thickness 0.0 7.0
dC/ nm C buffer layer thickness 0.0 5.0
dB4C/ nm B4C buffer layer thickness 0.0 5.0
σ/ nm Névot-Croce parameter 0.0 2.0
(identical for all interfaces)
ρMo Mo density w.r.t. bulk density 0.5 2.0
ρSi Si density w.r.t. bulk density 0.5 2.0
ρCC density w.r.t. bulk density 0.5 2.0
ρB4CB4C density w.r.t. bulk density 0.5 2.0
Capping layer
dSiO2(cap) / nm SiO2capping layer thickness 0.0 5.0
ρSiO2(cap) =ρSi (identical to Si density)
parameter for each material taken from the Henke database. This acts as a factor on the
optical constants and thus takes the uncertainties into account. This parameter is used
across all wavelengths in this thesis. As shall be demonstrated later, this does not pose a
limitation on the structural reconstruction, as the sensitivity with respect to the optical
constants of the different experiments conducted here is important in the EUV spectral
range, but negligible for high photon energies.
A full list of the model parameters for the multilayer sample can be found in table 4.1
together with physically plausible limits for each of the parameters. Due to the fact
that the EUV reflectivity curve shown in Fig. 4.1shows the first order Bragg peak of
the layer system, none of the layers can be thicker than
7nm
, i.e. in the order of half of
the wavelength. The barrier layers were designed to attain thicknesses below
1nm
. The
density of the various materials within this model was constrained to values between
50%
and
200%
with respect to their bulk density. Due to the high peak reflectance
close to the theoretical limit, i.e. the reflectance calculated for a given model without any
roughness or intermixing present, of the multilayer sample in the EUV measurement, the
maximum value of the Névot-Croce factor was limited to be below
σ≤2nm
. With its
upper limit, the measured peak reflectance can not be attained within this model thus
not limiting the generality.
The minimization functional and particle swarm optimization
As introduced above, the reconstruction of the model for the multilayer is primarily an
optimization problem. Based on the measured reflectivity data an optimization functional
defines the goodness of the model with respect to the measured data. The quality is
asserted based on the method of least squares [20,52,81] and the functional is defined as
the reduced ˜
χ2
˜
χ2=1
M−P∑
m
(Imodel
m−Imeas
m)2
˜
σ2
m, (4.1)
55
Chapter 4 CHARACTERIZATION OF THE MULTILAYER STRUCTURE FOR DIFFERENT SYSTEMS
where
M
is the number of measurement points,
P
is the number of parameters used in
the model,
Imodel
m
is the calculated intensity for the corresponding measurement point
with index
m
having the measured intensity
Imeas
m
. The calculated intensity fur the EUV
reflectivity curve above
Imodel
m
follows directly from the matrix algorithm and the quantity
R
in Eq.
(2.28)
in chapter 2. Each point is calculated based on the angle of incidence and
wavelength associated with measurement point
m
. The experimental uncertainty for each
measurement point is described by ˜
σm.
For the minimization of the functional in Eq.
(4.1)
a global optimization algorithm
known as PSO [73] is applied. In contrast to the aforementioned gradient based methods,
the PSO operates on the whole parameter space as defined by the upper and lower
parameter limits, which are given in table 4.1for the particular example here, without
specific starting parameters influencing the convergence result. The PSO algorithm was
implemented based on the draft by Carlisle and Dozier [31]. The basic mechanism of
the algorithm is the definition of a swarm of individual particles, i.e. positions in the
parameter space associated with a directional vector, which are initialized randomly
distributed between the defined space limits. Initially, each of those particles calculates the
minimization functional at its random position retaining that result including a random
start velocity. In an iterative process, the global best solution (“social component”) found
as well as the individual best solution (“cognitive component”) of each particle are used
to calculate an updated and weighted velocity vector within the parameter space for each
particle. Within that iteration each of the particle thus moves to a new position, where the
minimization functional is again evaluated and compared the the individual and global
best solutions. If a better value is found, the respective retained results are updated with
the new value and the next iteration is performed. While following that process the
particles eventually converge to the global best solution, which may or may not be the
global best optimum of the whole optimization problem. Due to the combination of social
and cognitive component, fast convergence into a local optimum can be avoided. The
state of full convergence is reached, when either all particles occupy the same place in the
parameter space or if stagnation is reached. Due to the heuristic nature of the algorithm,
it may happen that the global best optimum found is not necessarily the global minimum
of the optimization problem. The result may be verified, however, by repeated application
of the algorithm or simply by reaching a satisfactory solution through comparison of the
measured and calculated curves and thus small ˜
χ2values.
Model reconstruction based on the EUV reflectivity data
This optimization procedure was applied to the Mo/B
4
C/Si/C sample and the measured
EUV reflectivity curve. The fit result is shown together with the measured data in Fig. 4.3.
The parameter results are listed in table 4.2. The solution does indeed provide a very
good agreement with the measured data. However, by repeated evaluation of the PSO
procedure, significantly different results for the optimal parameter set with comparable
agreement and very similar
˜
χ2
values were found. Three examples are listed in table 4.2
with their respective
˜
χ2
values. Clearly, this is no desirable situation, since no definite
answer of the actual thicknesses found in the sample can be made. To complete the
characterization additional methods of model verification are thus required. We shall
therefore discuss an additional approach to the optimization problem in the following
section on how the model validity and the information content of the measured data can
be asserted based on the example of the PSO results obtained here.
56
Reconstruction Based on Specular EUV Reflectance 4.1
12.5 13.0 13.5 14.0
wavelength λ/ nm
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Reflectance
measured data
PSO fit
Figure 4.3 |
Theoreti-
cal reflectance curve for
the Mo/B
4
C/Si/C sample
based on the optimal
model parameters ob-
tained from the particle
swarm optimization.
Table 4.2 |
Results for the optimized parameters based on the PSO of the EUV reflectivity for the
Mo/B4C/Si/C sample.
Parameter Definition PSO results
dSiO2(cap) / nm SiO2capping layer thickness 3.194 3.418 3.558
dMo / nm Mo layer thickness 2.460 2.748 3.082
dSi / nm Si layer thickness 2.421 2.617 1.997
dC/ nm C buffer layer thickness 0.811 0.709 0.818
dB4C/ nm B4C buffer layer thickness 1.308 0.923 1.129
σ/ nm Névot-Croce parameter 0.322 0.249 0.177
ρMo Mo density w.r.t. bulk density 0.989 0.919 0.944
ρSi Si density w.r.t. bulk density 0.883 0.974 0.749
ρCC density w.r.t. bulk density 0.833 0.971 0.608
ρB4CB4C density w.r.t. bulk density 0.909 0.973 0.936
˜
χ2reduced χ2value 17.87 17.89 18.27
57
Chapter 4 CHARACTERIZATION OF THE MULTILAYER STRUCTURE FOR DIFFERENT SYSTEMS
4.1.2 Model Uniqueness and Maximum Likelihood Estimation
With the ambiguous reconstruction result of the previous section, the demand for a
verification of the model with respect to the measured data becomes apparent. To clarify
the problem of uniqueness of the solution, it is instructive to investigate the influence
of the individual model parameters on the theoretical reflectivity curve. In Fig. 4.4a
subset of the parameters is varied starting from the best PSO solution from Sec. 4.1.1.
In each of the subfigures, one parameter or a quotient of parameters is varied while all
others are kept fixed. By comparison of Fig. 4.4a, 4.4b, 4.4c and 4.4e it becomes clear
0.0
0.2
0.4
0.6
0.8
Reflectance
a) σ/ nm
0.0
0.5
1.0
b) dcap / nm
0.0
4.0
8.0
12.0 12.5 13.0 13.5 14.0
wavelength λ/ nm
0.0
0.2
0.4
0.6
0.8
Reflectance
e) ρMo / rel. dens.
1.0
0.8
0.6
12.0 12.5 13.0 13.5 14.0
wavelength λ/ nm
f) dMo / nm
2.5
2.4
2.3
0.0
0.2
0.4
0.6
0.8
Reflectance
c) Γ=dMo/dSi
0.7
1.0
1.4
d) N/ layers
65
50
35
Figure 4.4 |
Influence of the change of model parameters on the simulated EUV reflectivity curve. In
each of the figures, all parameters were kept constant at the values listed in table 4.2 varying only the
respective shown parameter.
that a reduction of the peak reflectivity can originate in either a large roughness and
intermixing parameter
σ
or similarly from the thickness of the capping layer, the silicon
to molybdenum layer thickness ratio of the molybdenum density. A reconstruction based
on a single EUV reflectivity therefore intrinsically produces a highly ambiguous result
with strong parameter correlations. The available data, a single EUV reflectivity curve
in this case, does not allow for a unique set of parameters of the model minimizing
the
χ2
functional. In reality multiple solutions with very similar values for
˜
χ2
exist as
shown above. Clearly, this raises the question of how accurately a reconstruction may be
achieved here.
58
Reconstruction Based on Specular EUV Reflectance 4.1
Maximum likelihood
A solution of the aforementioned problem requires to determine the value of
˜
χ2
in
vicinity of the PSO solution or possibly the whole parameter space. This is approached
by numerically sampling the functional based on a MCMC method [56]. An application
of this technique to the design process of multilayer mirrors has been demonstrated by
Hobson and Baldwin [64]. In our case, the match of model and experimental result is
evaluated based on a non-centered
χ2
distribution assuming independent measurements.
It was further assumed that any measured point is distributed around the actual reflectiv-
ity curve following a Gaussian distribution, i.e. Gaussian uncertainties for the experiment
are assumed. The corresponding probability density function for a measurement result
matching with the actual reflectivity curve, which is assumed to be obtainable exactly
through the theoretical calculation, is then of Gaussian form [1]. Thus, the likelihood
that the measured values match with the theoretical curve under the assumption that the
model is correct is proportional to
L(E|M(~
x)) ∝exp −˜
χ2(~
x)/2, (4.2)
where
E
denotes the experiment, i.e. the measured data and
M(~
x)
represents the model
given through parameter set
~
x
, e.g. the parameters of the model in table 4.1. In our case
however, we seek to evaluate the likelihood
L(M(~
x)|E)
that the model
M(~
x)
with a given
set of parameters
~
x
is valid assuming the experiment
E
yields the correct curve (the so
called “posterior distribution”). Those two quantities are linked through the Bayesian
theorem [16,97] stating
L(M(~
x)|E)∝L(E|M(~
x))L(M(~
x)), (4.3)
where
L(M(~
x))
denotes the likelihood for the model to be valid for a specific set of
parameters
~
x
(the so called “prior distribution”). The prior distribution does contain any
prior knowledge about the model and allowed parameters. For the example of the model
parameters in table 4.1, the prior distribution is
L(M(~
x)) → −∞
for any parameter
set outside the listed boundaries and
L(M(~
x)) = 1
everywhere else. In addition, the
maximum total period thickness is limited, i.e. the sum of all layers in one period to
only allow the appearance of the first Bragg peak within the measured spectral range
through the same condition. Combining Eq.
(4.2)
and Eq.
(4.3)
then yields the likelihood
functional
L(~
x) = L(M(~
x)|E)∝exp−˜
χ2(~
x)/2L(M(~
x)). (4.4)
Solving the optimization problem posed in the previous section within this context
is then, equivalently to the minimization of
˜
χ2
, the maximization of the likelihood
L(~
x)
.
The MCMC method poses a statistical approach on evaluating (mapping) the likelihood
across the parameter space within the previously defined limits as in the PSO approach.
It was proven that after a theoretical number of infinite iterations, the distribution of the
individual samples within the MCMC algorithm, corresponds to the likelihood functional
in Eq.
(4.4)
[34,92]. With a limited number of iterations, a numerical approximation of
that distribution is obtained after reaching an equilibrium state in the algorithm [51]. It
thus yields an alternative method on solving the optimization problem by extracting the
maximum likelihood from the final result. However, in addition to the maximum value,
59
Chapter 4 CHARACTERIZATION OF THE MULTILAYER STRUCTURE FOR DIFFERENT SYSTEMS
the likelihood distribution in parameter space is obtained allowing to extract confidence
intervals for each of the parameters. Thereby, the aforementioned ambiguity of solutions
can be quantified within the defined model and the available experimental data. The
confidence intervals are defined as the one- or two-sigma standard deviations of the
respective distributions for each parameter.
Confidence intervals for the Mo/B4C/Si/C sample
An existing implementation of the MCMC algorithm by Foreman-Mackey et al. [51] was
applied to the EUV measurement of the Mo/B
4
C/Si/C sample in Fig. 4.1with the model
in Fig. 4.2. The likelihood, as defined in Eq.
(4.4)
with the
˜
χ2
functional from Eq.
(4.1)
,
is sampled in a high-dimensional space depending on the number of parameters in the
model. We therefore need to project the distribution for each parameter by marginalizing
over all other parameters. Alternatively, two-parameter correlations can be visualized by
projecting on a two-dimensional area, again marginalizing across all other parameters.
The projection for the Si and Mo layer thicknesses are shown in Fig. 4.5b and 4.5c. In
both cases, a well defined distribution is obtained. In the two-dimensional projection in
Fig. 4.5a, no correlations are apparent and a two-dimensional Gaussian-like shape results.
In all cases, the one-sigma standard deviations for Gaussian distributions are shown
12345
dSi / nm
1
2
3
4
5
dMo / nm
1σ
2σ
a) PSO fit
12345
dSi / nm
b)
1 2 3 4 5
dMo / nm
0.0
0.2
0.4
0.6
0.8
1.0
likelihood / arb. units
c) ±1σ
50%
Figure 4.5 |
Results of the maximum likelihood estimation obtained via the MCMC procedure. a) Two
dimensional projection of the likelihood distribution for the parameter pair
dSi
and
dMo
. The projection
was obtained by marginalizing over all other parameters of the model. The black contours indicate
the areas for one and two standard deviations (one and two sigma contours). The blue lines in all
three sub-figures indicate the best parameter set found with the PSO method. b) One dimensional
projection of the likelihood distribution for the silicon layer thickness
dSi
. The solid black line marks
the center position (
50%
percentile) of the distribution. The dotted lines are the limits of one standard
deviation. c) The one dimensional distribution similarly to b) for the molybdenum layer thickness.
together with the weighted center, i.e. the
50
th percentile. The PSO result is also indicated,
which is compatible with the one sigma standard deviation, but does not match the
center of the likelihood result. The reason for that lies in higher order correlations of the
parameters. In Fig. 4.6, all one-dimensional projections of the likelihood distribution are
shown for all remaining parameters. Clearly, while a reasonably small confidence interval
(again, one standard deviation for all distributions) can be found for the thickness of
the carbon and boroncarbite layers, the off-center value for the silicon thickness of the
PSO result in Fig. 4.5c is compensated by a larger than center value for the boroncarbite
layer in Fig. 4.6. Thus, the thicknesses are correlated and are no independent model
parameters. Nevertheless, confidence intervals can be obtained within the given model
60
Reconstruction Based on Specular EUV Reflectance 4.1
0 1 2 3 4 5 6 7
dSiO2(cap)/ nm
0 1 2 3 4 5 6 7
dC/ nm
0 1 2 3 4 5 6 7
dB4C/ nm
0.0 0.5 1.0 1.5 2.0
σ/ nm
likelihood / arb. units
0.5 0.8 1.1 1.4 1.7 2.0
ρSi
0.5 0.8 1.1 1.4 1.7 2.0
ρMo
0.5 0.8 1.1 1.4 1.7 2.0
ρC
0.5 0.8 1.1 1.4 1.7 2.0
ρB4C
likelihood / arb. units
Figure 4.6 |
In analogy to Fig. 4.5b and 4.5c the one dimensional projections of the likelihood
distribution estimation for the Mo/B
4
C/Si/C sample are shown for the remaining parameters of the
model with the PSO result, the center value and one standard deviation.
and the given prior (the boundaries listed in table 4.1) and are listed accordingly in
table 4.3for one and two standard deviations. Within the allowed boundaries, some
parameters remain entirely undefined with similar likelihood for any parameter value,
such as the SiO
2
capping layer thickness, the silicon, carbon and boroncarbite relative
densities. Their corresponding total confidence intervals thus cover almost exactly
68.2%
(one standard deviation) and
95.4%
(two standard deviations) of the allowed respective
parameter range. Hence, with respect to the model defined and the measured EUV
reflectivity curve, no reliable value for those sample properties can be determined.
Table 4.3 |
MCMC results obtained by the analysis of the EUV reflectivity for the Mo/B
4
C/Si/C sample.
The center values (
50%
percentile) together with confidence intervals (c.i.) of one and two standard
deviations are shown.
Parameter PSO result center value 1σc.i. 2σc.i.
dSiO2(cap) / nm 3.194 3.139 (−1.077/+1.108) (−2.704/+3.378)
dMo / nm 2.460 1.998 (−0.422/+0.429) (−0.789/+0.945)
dSi / nm 2.421 2.910 (−0.529/+0.473) (−1.162/+0.862)
dC/ nm 0.811 1.190 (−0.516/+0.459) (−0.947/+0.968)
dB4C/ nm 1.308 0.894 (−0.531/+0.560) (−0.825/+1.060)
σ/ nm 0.322 0.456 (−0.211/+0.206) (−0.376/+0.399)
ρMo 0.989 1.086 (−0.098/+0.147) (−0.183/+0.340)
ρSi 0.883 0.851 (−0.219/+0.253) (−0.330/+0.491)
ρC0.833 0.941 (−0.297/+0.418) (−0.421/+0.846)
ρB4C0.909 1.115 (−0.435/+0.572) (−0.588/+0.845)
It should be noted, that the given center values here are not a good solution to the
optimization problem. The reason for that is, that the parameters are highly correlated.
61
Chapter 4 CHARACTERIZATION OF THE MULTILAYER STRUCTURE FOR DIFFERENT SYSTEMS
The center values of the one-dimensional projections may therefore not be suitable pa-
rameters for the model based on the low amount of data available. A valid optimization
result can therefore only be obtained by either applying the PSO routine or by iterative
application of the MCMC procedure. The latter may be achieved by fixing single parame-
ters according to their maximum likelihood value found in the previous iteration and
obtaining the resulting likelihood distributions for the remaining parameters according
to that restricted prior distribution.
The results listed in table 4.3serve as the model parameters for the analysis of diffuse
scattering from the Mo/B4C/Si/C sample in chapter 5.
4.2 Molybdenum Thickness Variation in Mo/Si/C Multi-
layers
For the engineering of a near-normal incidence mirror, the ratio of molybdenum layer
thickness to total period thickness has a clear impact on the reflectivity curve as seen
from the theoretical simulations in Fig. 4.4c. Studies have shown, that an optimal value
for high reflectivity is achieved by depositing
40%
molybdenum layer thickness
dMo
with
respect to the total period thickness
D
[9,30]. During the deposition process, the layer
of molybdenum grows in thickness and at a certain threshold, crystallites may begin to
form [9,140] inside the layer. Those may affect the interface morphology of the layer
system at the boundaries to the molybdenum layer and possibly at further interfaces
through correlation effects. This potentially increases the roughness and thus the loss of
specularly reflected radiation to diffuse scatter.
In the following, the reconstruction procedure discussed in the above section shall
be applied and extended to the problem of multilayer sample systems deposited with
varying molybdenum layer thicknesses from sample to sample. The samples discussed
in this section were designed to investigate the impact of the crystallization on the
performance on Mo/Si multilayer mirrors systems. Two sets of samples were fabricated.
One with the standard magnetron deposition technique, which we shall refer to as
unpolished samples and another set with an additional ion polishing step applied within the
deposition of each period to counteract the roughening expected from the crystallization
process. Consequently, the latter samples are referred to as polished samples. Both sets
were deposited with linearly increasing molybdenum thickness across all periods from
sample to sample. The details of the sample layout and the reflectivity measured from
each sample are described in detail in Sec. 4.2.1.
The goal of this investigation is to analyze the interface morphology in each sample
and asses the effect of the crystallization process and the polishing treatment. For
that, the nominally deposited layer thicknesses are verified and the model including
the determination of the densities and the intermixing and roughness parameters is
reconstructed in Sec. 4.2.2based on specular analytic experiments and the application
of the PSO and the MCMC methods introduced above. The findings are shown and
discussed in Sec. 4.2.3. The results presented here are part of the published work in A.
Haase, V. Soltwisch, S. Braun, C. Laubis, and F. Scholze: ‘Interface morphology of Mo/Si
multilayer systems with varying Mo layer thickness studied by EUV diffuse scattering’.
EN. in: Optics Express 25.13 (June 2017), pp. 15441–15455.doi:10.1364/OE.25.015441.
62
Molybdenum Thickness Variation in Mo/Si/C Multilayers 4.2
4.2.1 Sample Systems and Experimental Procedure
Two sets of several samples of Mo/Si/C multilayer mirrors with C interdiffusion barriers
with thicknesses of nominally below 0.5nm at the Mo on Si interfaces (a detailed figure
of the model for those samples is given below in Fig. 4.9of the following sections) were
prepared. As mentioned above, the samples under investigation here were fabricated with
increasing relative Mo thickness from sample to sample while keeping the nominal period
thickness
D≈7
nm constant by correspondingly reducing the silicon layer thickness. In
this study, two sets of samples are investigated. In the first set, the magnetron sputtered
layers were deposited one after another for each sample. In the second set, during
deposition, an additional polishing process was used once during sputtering each period
to counteract the possible roughening due to the crystallization. The nominal thickness
values of the molybdenum layers in the two sample sets were varied in equidistant steps,
from
1.7nm
to
2.9nm
across nine unpolished samples. The polished set includes an
additional sample with
3.05nm
molybdenum thickness with identical values as in the
unpolished case, otherwise. The nominal layer thicknesses refer to the value aimed at
during production of the multilayer systems, realized by different deposition times for
the respective layer material.
Spectrally resolved EUV reflectivity curves at an angle of incidence from the surface
normal of
αi=15°
and in the wavelength range from
12.4nm
to
14.0nm
have been
measured for all samples at the EUVR beamline at the MLS. The data obtained is shown
in Fig. 4.7sorted by the nominal molybdenum layer thickness. The reflectivity curves
1.70 1.85 2.00 2.15 2.30 2.45 2.60 2.75 2.90
nom. Mo thickness / nm
12.4
12.6
12.8
13.0
13.2
13.4
13.6
13.8
Wavelength / nm
a)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Reflectivity
1.70 1.85 2.00 2.15 2.30 2.45 2.60 2.75 2.90 3.05
nom. Mo thickness / nm
12.4
12.6
12.8
13.0
13.2
13.4
13.6
13.8
Wavelength / nm
b)
Figure 4.7 |
a) Mea-
sured reflectivity curves
for the unpolished sam-
ples across the wave-
length at a fixed angle
of incidence of
αi=15◦
from the surface nor-
mal. The nine samples
differ by the nominal
Mo layer thickness in-
dicated at the bottom
axis. b) Measured reflec-
tivity curves of the ten
polished samples mea-
sured under the same
conditions as for the
first sample set.
in Fig. 4.7a and Fig. 4.7b have the characteristic curve shape of periodic EUV multilayer
mirrors with a main broad maximum and side fringes, very similar to the mirror sample
discussed in Sec. 4.1above. In direct comparison of the measured reflectivity data,
shifts of the peak center position are clearly visible. As illustrated in Fig. 4.4above,
several properties of a multilayer stack, e.g. molybdenum content and period thickness,
contribute to such a difference. Clear differences in the maximum reflectivity value
can also be observed in the two subfigures, with strong increases at
dnom
Mo =2.45nm
for
the unpolished set and at
dnom
Mo =2.00nm
for the polished set. In all samples the only
nominal difference, i.e. the only parameter changed during the deposition process, is the
relative molybdenum thickness. The increase in reflectance, peak broadening and the
jump of the peaks center position are therefore indicators for an abrupt change in the
63
Chapter 4 CHARACTERIZATION OF THE MULTILAYER STRUCTURE FOR DIFFERENT SYSTEMS
multilayer properties.
For the purpose of obtaining additional information about the samples, in addition
to the EUV reflectivity curves above, all samples were measured by Stefan Braun after
deposition using a lab-based Cu-K
α
X-ray diffractometer at the Fraunhofer IWS Dresden,
Germany. The XRR data is shown in Fig. 4.8for the set of unpolished and polished
samples in direct comparison. The position of the Bragg peaks and their respective
intensity contain additional information on the layer stack thicknesses and its interface
properties. In both cases, shifts of the peak positions and intensities similar to those
observed in the EUV curves become apparent. Especially the higher orders towards larger
grazing incidence angles show distinct differences. In addition, in direct comparison of
the XRR curves for the respective sample with
dnom
Mo =3.05nm
from the unpolished and
polished sets (on the top of Fig. 4.8a and Fig. 4.8b), a higher intensity for higher-order
Bragg peaks above grazing angles of incidence of
αGI
i>7°
can be observed for the
polished sample. This hints towards an improved interface definition and sharpness due
to the polishing process and consequently a lower roughness or intermixing.
In the following section we shall analyze the EUV and XRR data discussed here to
reconstruct a model of the samples using the MCMC approach introduced in Sec. 4.1.1
in order to verify the observations made here.
4.2.2 Combined Analysis of X-ray and EUV reflectance
To obtain the actual layer thicknesses in the samples, the data of the EUV reflectivity and
XRR experiments was analyzed and these parameters were reconstructed by combined
analysis of the measured data. The reflectivity curves for the different measurements
are calculated by introducing a model for the multilayer system and applying the matrix
formalism described in detail in the theory part of this thesis, Sec. 2.3.
The thicknesses of the Mo layers inside the stack were varied nominally from
1.7
nm to
3.05
nm from sample to sample, where the unpolished sample set lacks the last
nominal thickness. The stacking of the different layers in the multilayer consists of the
Mo and Si layers, as well as an additional C buffer layer at the Mo on Si interface to
prevent interdiffusion. For the Si on Mo interfaces, no buffer layers were included since
interdiffusion is usually less in this case [105]. However, for the theoretical description
of the sample stack an additional MoSi
2
layer is considered in the model, which is
well known to form during the deposition process [9]. The full model used in the
reconstruction is illustrated in Fig. 4.9with the thickness parameters for each layer. To
account for any contamination on the top sample surface, an additional carbon-like layer
as the upper most layer was considered. In addition to the thicknesses of each layer
a variation of the layer density between
80%
and
100%
of the bulk density was also
allowed for, in agreement with findings in literature [30]. The model parameters and
their boundaries entering in the optimization procedure are listed in table 4.4. Similar to
the Mo/B
4
C/Si/C in Sec. 4.1.1, a Névot-Croce damping factor was assumed to account
for specular reflectivity loss due to interface imperfections.
64
Molybdenum Thickness Variation in Mo/Si/C Multilayers 4.2
10−8
10−6
10−4
10−2
100
reflectivity
a) unpolished samples
1.70
1.85
2.00
2.15
2.30
2.45
2.60
2.75
2.90
dnom
Mo / nm
reflectivity
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
0 1 2 3 4 5 6 7 8 9
grazing angle of incidence / ◦
10−8
10−6
10−4
10−2
100
reflectivity
10−8
10−6
10−4
10−2
100
reflectivity
b) polished samples
grazing angle of incidence / ◦
1.70
1.85
2.00
2.15
2.30
2.45
2.60
2.75
2.90
3.05
dnom
Mo / nm
reflectivity
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
0 1 2 3 4 5 6 7 8 9
grazing angle of incidence / ◦
10−8
10−6
10−4
10−2
100
reflectivity
Figure 4.8 |
XRR data for all unpolished and polished samples shown in dependence on the nominal
molybdenum layer thickness
dnom
Mo
and the grazing angle of incidence
αGI
i
at the Cu-K
α
photon energy of
Eph =8048 eV
. In each of the subfigures a) and b) the XRR measurements for the sample with smallest
and largest
dnom
Mo
are shown on the bottom and the top of the subfigure, respectively. In between, the
XRR curves are shown in as a color map plot.
65
Chapter 4 CHARACTERIZATION OF THE MULTILAYER STRUCTURE FOR DIFFERENT SYSTEMS
Figure 4.9 |
Model of
the multilayer stack in-
cluding the substrate
and the capping lay-
ers. The periodic part
is enclosed between the
dashed lines with four
layers in each period re-
peated 49 times. The
capping period does not
include an interdiffu-
sion layer but has a
natural SiO
2
layer and
a carbon-like layer ac-
counting for contamina-
tion on the top surface.
Si (substrate)
Mo
Si
C (buffer layer)
SiO2
D
dC
dSi
dMo
capping layer
periodic replication
MoSi2(interdiffusion layer)
dMoSi2
C (contamination)
z
Table 4.4 |
Parametrization of the Mo/Si/C multilayer samples with varying molybdenum layer thick-
nesses.
Parameter Definition Lower bound Upper bound
dMo / nm Mo layer thickness 0.0 4.5
dSi / nm Si layer thickness 0.0 7.0
dC/ nm C buffer layer thickness 0.0 0.6
dMoSi2/ nm MoSi2interdiffusion layer thickness 0.0 0.6
σ/ nm Névot-Croce parameter 0.0 0.5
(identical for all interfaces)
ρMo Mo density w.r.t. bulk density 0.8 1.0
ρSi Si density w.r.t. bulk density 0.8 1.0
ρCC density w.r.t. bulk density 0.8 1.0
ρMoSi2MoSi2density w.r.t. bulk density 0.8 1.0
Capping layer
dC(cap) / nm C capping layer thickness 0.0 3.0
dSiO2(cap) / nm SiO2capping layer thickness 0.0 1.5
ρC(cap) C density w.r.t. bulk density 0.0 1.0
ρSiO2(cap) =ρSi (identical to Si density)
66
Molybdenum Thickness Variation in Mo/Si/C Multilayers 4.2
Optimization functional and procedure
The data analysis was conduced similarly to the procedure described in Sec. 4.1.1.
However, for the samples studied here, two separate experiments and data sets were
measured with the goal to improve the reconstruction of the model. Due to the increased
amount of data through the additional XRR measurements, a definition for a combined
χ2
functional is required to allow an analysis based on both data sets. The two data
sets, i.e. the EUV and XRR reflectivity curves have significantly different number of
data points, which are not entirely independent of each other. In case of the XRR curve
increasing the number of data points, e.g. by reducing the angular step size by half does
not lead to better statistics due to systematic errors. Defining a
χ2
functional as the total
sum of all measured data point residuals, i.e. both the EUV data and the XRR data would
therefore create an unwanted weighting due to the large amount of XRR data points in
comparison to far fewer EUV data points. To avoid this effect, the combined
χ2
functional
is defined as the sum of the reduced
˜
χ2
functionals. The
˜
χ2
is equivalently defined to
Eq.
(4.1)
for each of the datasets separately. The reduced
˜
χ2
can be interpreted as the
average of the squared residuals of model prediction and experiment. Thereby, each
experiment is reduced to a single comparable quantity. By the definition of
χ2=˜
χ2
EUV +˜
χ2
XRR, (4.5)
we are therefore enabled to obtain confidence intervals for the parameters of the model,
which represent a conservative (upper limit) estimation for the combined analysis of both
experiments, similarly to the procedure for a single EUV curve as described in Sec. 4.1.1
above. The combined χ2functional enters the likelihood through Eq. 4.4.
The solution to the inverse problem of reconstructing the optimal model parameters is
conducted by minimizing the
χ2
functional (or equivalently maximizing the likelihood).
To minimize the functional with respect to the best choice of parameters, the MCMC
method as described above is applied for the Mo/B
4
C/Si/C sample system. The analysis
is not started with a PSO optimization, since the sample system is numerically simpler
due to the decreased amount of layers and interfaces. The MCMC method itself yields an
optimization result, although slower in convergence, as mentioned in the discussion of
the procedure above in Sec. 4.1.2. As a starting point, again a random set of parameters
is generated with respect to predefined boundaries listed in table 4.4. The limits are
chosen in reference to prior knowledge and physical plausibility. Confidence intervals for
each value within the underlying model are estimated from the likelihood distribution
resulting from the MCMC as one standard deviation of the sample distribution in each
parameter.
We shall discuss the results of the optimization procedure at the example of the
unpolished sample with nominal molybdenum layer thickness of
dnom
Mo =3.05nm
. The
results of the MCMC maximum likelihood estimation for the other samples were found
to show the same properties and the same findings discussed in the following with the
only distinction of broader or even improved distributions in some cases. The latter
causes the confidence intervals to be different for the respective parameters.
As a first step, the MCMC procedure was performed within the defined boundaries
for all parameters. An unambiguous result was only found with respect to the thickness
parameters of Mo, with the smallest confidence intervals in comparison to all other
parameters, and Si, as well as for the Névot-Croce parameter
σ
, whereas all other param-
eters show broad likelihood distributions within the predefined boundaries not allowing
67
Chapter 4 CHARACTERIZATION OF THE MULTILAYER STRUCTURE FOR DIFFERENT SYSTEMS
a unequivocal parameter determination. Therefore, the best model was obtained in a two-
step process. First the MCMC optimization was performed including all parameters as
mentioned above. Proceeding from this, the value of the Mo thickness with its confidence
interval was obtained by marginalizing over all other parameters, yielding the most pre-
cise parameter estimation from the procedure, i.e. the smallest confidence interval. The
results for the molybdenum and silicon layer thickness parameters are shown in Fig. 4.10.
In comparison to the analysis based on only EUV data for the Mo/B
4
C/Si/C in Fig. 4.5,
12345
dSi / nm
1
2
3
4
5
dMo / nm
a)
12345
dSi / nm
b)
1 2 3 4 5
dMo / nm
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
likelihood / arb. units
c)
fit
±1σ
50%
Figure 4.10 |
Results of the maximum likelihood estimation obtained via the MCMC procedure similar
to Fig. 4.5 but for the combination of EUV and XRR data. a) Two dimensional projection of the likelihood
distribution for the parameter pair
dSi
and
dMo
. The projection was obtained by marginalizing over
all other parameters of the model. The black contours indicate the areas for one and two standard
deviations (corresponding to a coverage factor
k=1
and
k=2
, respectively). The blue lines in all
three sub-figures indicate the best parameter set. b) One dimensional projection of the likelihood
distribution for the silicon layer thickness
dSi
. The solid black line marks the center position (
50%
percentile) of the distribution. The dotted lines are the limits of one standard deviation. c) The one
dimensional distribution similarly to b) for the molybdenum layer thickness.
the inclusion of additional XRR measurements lead to significantly smaller confidence
intervals and thus higher accuracy of the reconstruction. The method of combining the
analysis of two datasets of EUV and XRR measurements has been previously applied
by others [144], which have come to the same result of a significantly improved model
reconstruction. Each of the methods does provide different sensitivity for the different
model parameters. As an example, EUV measurements are sensitive to the Mo and Si
layer thicknesses due to the large optical contrast in that spectral range. On the other
hand, high accuracy can be expected from the XRR measurements with respect to the
period thickness parameter D.
In a second step, another MCMC optimization was performed on a reduced parameter
set, fixing the determined molybdenum layer thickness to its optimal value, i.e. the
50%
percentile of its distribution. Finally, the layer thicknesses of the C barrier layer and the
MoSi
2
interdiffusion layer were fixed to their nominal values of
dC=dMoSi2=0.5
nm.
Due to the broad distribution result for the likelihoods of those parameters, this comes
without a limitation of the generality for this analysis, since any value is valid within
the predefined boundaries. Additionally, this ensures comparability of the models for
all samples without constraining the applicability of the model with respect to the data
available.
The results of the second MCMC procedure of the restricted model yield the remaining
values for the model parameters by obtaining the globally best solution found. The final
result is indicated by the blue solid lines in Fig. 4.10. Due to the choice to restrict the
68
Molybdenum Thickness Variation in Mo/Si/C Multilayers 4.2
2.0 2.5 3.0 3.5 4.0 4.5
dSi / nm
0.0
0.1
0.2
0.3
0.4
0.5
dC/ nm
fit
Figure 4.11 |
Two-
dimensional likelihood
distribution indicating
the correlation of
silicon and carbon layer
thickness. The distribu-
tion was obtained by
marginalizing over all
remaining parameters
of the model. The
blue lines indicate the
fit obtained through
the two-step MCMC
optimization procedure
(see main text).
model to a buffer layer thickness of
dC=0.5nm
, the optimal solution for the silicon layer
thickness is found at the limit of one standard deviation in Fig. 4.10b. The distributions
shown represent the MCMC results of the unrestricted model, where the silicon and
carbon layer thicknesses are strongly correlated as shown in Fig. 4.11. By fixing the carbon
layer thickness to its nominal value, this correlation is resolved and the corresponding
silicon layer thickness is well within the interval of one standard deviation as indicated
through the solid black contours in Fig. 4.11.
4.2.3 Optimization Results
The theoretical reflectivity curves calculated from the optimal model parameters for
the unpolished sample with
dnom
Mo =3.05nm
are shown in Fig. 4.12. Overall, a very
good agreement of the two experiments with the theoretical curves is obtained. The full
12.5 13.0 13.5 14.0
wavelength / nm
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
reflectivity
αi=15◦
0123456789
grazing angle of incidence / ◦
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
reflectivity
Eph =8.04 keV
Figure 4.12 |
Experi-
mental data in compari-
son with the theoretical
curves calculated with
the model parameters
obtained from the com-
bined analysis of EUV
and XRR data. The
data shown here was
measured on the unpol-
ished sample with nom-
inal molybdenum thick-
ness of dnom
Mo =3.05 nm.
list for all nominal molybdenum layer thicknesses for all samples with the respective
69
Chapter 4 CHARACTERIZATION OF THE MULTILAYER STRUCTURE FOR DIFFERENT SYSTEMS
experimental values and their confidence intervals is given in table 4.5.
Table 4.5 |
List of nom-
inal molybdenum layer
thicknesses in the two
sample sets. Both sets
were fabricated with a
equidistant increase in
thickness from
1.70 nm
to
3.05 nm
with 9 unpol-
ished and 10 polished
samples.
nom. dMo / nm EUV & XRR EUV & XRR
(unpolished) (polished)
1.70 1.81(−0.12/+0.24)1.77(−0.22/+0.19)
1.85 1.98(−0.15/+0.14)1.91(−0.12/+0.17)
2.00 2.08(−0.11/+0.22)2.29(−0.28/+0.13)
2.15 2.31(−0.22/+0.21)2.45(−0.43/+0.06)
2.30 2.43(−0.09/+0.16)2.60(−0.12/+0.14)
2.45 2.68(−0.13/+0.16)2.58(−0.21/+0.15)
2.60 2.91(−0.17/+0.12)2.87(−0.22/+0.12)
2.75 3.02(−0.15/+0.15)3.03(−0.16/+0.14)
2.90 3.22(−0.13/+0.11)3.15(−0.13/+0.13)
3.05 - 3.47(−0.19/+0.13)
The optimal parameters for the molybdenum layer thickness
dMo
and the period
thickness
D
found for both sample sets in the two-step MCMC analysis are shown in
Fig. 4.13. The confidence intervals shown in Fig. 4.13a are one standard deviation of
1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.20
nom. Mo thickness
1.5
2.0
2.5
3.0
3.5
dMo / nm
a)
nom. Mo thickness
unpolished
polished
2.00 2.50 3.00 3.50
fitted Mo thickness
6.80
6.85
6.90
6.95
7.00
7.05
7.10
D/ nm
b)
samples with diffuse
scattering analysis
Figure 4.13 |
a) Fitted Mo thickness values for both sample sets resulting from the MCMC analysis (see
text). The nominal Mo layer thickness is shown in comparison in good agreement with the obtained
thicknesses. b) Fitted total period thickness
D
for both sample sets. For both sample sets, clear jumps
can be observed at approx.
dnom
Mo =2.00
nm and
dnom
Mo =2.38
nm, respectively, which is attributed to
the crystallization threshold (see text). The marked (circle) samples were measured and analyzed with
respect to the diffuse scattering.
the likelihood determined for the Mo layer thickness by the first-step MCMC procedure,
i.e. for the unrestricted model with the parameter limits as listed in table 4.4. The
results show the desired linear increase in molybdenum layer thickness, however at
a systematically higher thickness than the nominal values. A possible cause for that
observation, consistent with the model reconstruction results, is the possible interdiffusion
of the molybdenum layer with the silicon and carbon during deposition and a lower
molybdenum density. The reduced relative density of the molybdenum layer is indeed
found in the reconstruction results for all samples showing systematically reduced
density values of
ρMo ≈90%
w.r.t. the Mo bulk density. It is reasonable to assume, that
the magnetron sputtered Mo layer, which is in mostly amorphous or polycrystalline
70
Molybdenum Thickness Variation in Mo/Si/C Multilayers 4.2
state, leads to density reduced layers compared to fully crystalline bulk molybdenum.
Thus, the nominal amount of deposited molybdenum leads to higher thicknesses than
desired. In Fig. 4.13b the fitted period thicknesses
D
are shown in dependency of the
fitted molybdenum thicknesses.
For both sets, distinct jumps can be observed between
dMo ≈1.9nm
and
dMo ≈
2.3nm
for the polished samples and between
dMo ≈2.3nm
and
dMo ≈2.7nm
for the
unpolished set. To better understand this observation, Fig. 4.14 shows the maximum
peak reflectance of all EUV measurements as a function of the reconstructed Mo layer
thickness. The identical blue solid line in both subfigures indicates the maximum
0.64
0.66
0.68
0.70
0.72
max. Reflectivity
a)
1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75
fitted Mo thickness / nm
0.64
0.66
0.68
0.70
0.72
max. Reflectivity
b)
Figure 4.14 |
Peak
reflectance values
for each sample ob-
tained from the EUV
measurements for the
unpolished sample set
(a) and the polished
sample set (b). The
maximum theoretical
reflectance is shown in
both subfigures for a
perfect (no roughness
or interdiffusion) layer
system with the same
specifications as the
samples.
peak reflectance attainable for a perfect multilayer system with the respective Mo layer
thickness without any interdiffusion or roughness. For the calculation, a carbon capping
layer of
dC(cap) =2.0
nm and a relative density of
ρC(cap) =0.5
and a silicon dioxide layer
of
dSiO2=2.0
was considered. The dashed curves in both figures show the expected
maximum peak reflectance values for the two sample systems calculated by adding the
respective roughness/interdiffusion to the model and varying the molybdenum thickness
accordingly. In both cases, a significant dip with respect to the expected value can be
observed starting at thicknesses of
dMo =2.31(−0.22/+0.21)
nm for the unpolished
samples in Fig. 4.14a and at
dMo =1.77(−0.22/+0.19)
nm for the polished samples in
Fig. 4.14b. This significantly diminished peak reflectance is attributed to the process of
crystallization as the most likely cause. These values are consistent with the increase
observed in the period thickness for both cases. Possibly, the deposition is affected
by the crystallization threshold causing the increase in period thickness. The values
measured here for the dip in peak reflectance are in agreement with earlier observation of
molybdenum crystallization in literature [9] for the unpolished sample set. The polishing
process shifts that threshold to lower thicknesses by approximately 0.2nm to 0.3nm.
For a deeper investigation of the interface morphology at the presumed crystallization
threshold, EUV diffuse scattering experiments have been conducted for selected samples
of the respective set. To gain a deeper understanding of the reflectivity dip and the
71
Chapter 4 CHARACTERIZATION OF THE MULTILAYER STRUCTURE FOR DIFFERENT SYSTEMS
period increase, the samples in vicinity of this feature in the Fig. 4.14 and Fig. 4.13 were
investigated in comparison to reference samples above and below the threshold. The
selection is marked with open circles in Fig. 4.13. This analysis is the topic of chapter 5of
this thesis and is described and discussed in detail there based on the model parameters
obtained here.
4.3 Analysis of Cr/Sc Multilayers with Sub-nanometer
Layer Thickness
In the previous sections, multilayer systems designed to reflect radiation in the EUV
spectral range from
12.5nm
to
14.0nm
wavelength were characterized. There, three to
four layer systems per period with period thicknesses of
D≈7nm
were used to achieve
constructive interference at the desired reflection angles. We shall now apply the analysis
to a different system, multilayer mirrors designed to reflect radiation in the spectral
range between
2.2nm
and
4.4nm
wavelength, the so called water window. Those systems
share the basic principle of a one-dimensional Bragg crystal with the Mo/Si multilayer
stacks from the previous sections, but differ in the selection of materials and their layer
thicknesses. The intrinsic relationship between spectral range and period thickness to
achieve constructive interference, requires period thicknesses of
D≈1.5nm
for this case
and higher number of period replications.
The system investigated here is a bilayer stack of chromium (Cr) and scandium (Sc).
A detailed description of the sample preparation process and the choice of the layer
materials can be found in Ch. 3, Sec. 3.4.1. The sample is optimized to reflect radiation
of
λ=3.14nm
at an angle of incidence of
αi=1.5°
. It has
N=400
bilayer periods,
where the last period has a larger Cr capping layer thickness. The model of the sample is
shown in Fig. 4.15. The small period thickness of only
D≈1.5nm
for this type of sample
Figure 4.15 |
Model
of the Cr/Sc multilayer
stack including the sub-
strate and the capping
layers. The periodic part
is enclosed between the
dashed lines with two
layers in each period
repeated
N=400
times. The capping pe-
riod does include the
thicker chromium layer
deposited to avoid ox-
idation of the periodic
stack. Furthermore, ox-
ide and carbon contami-
nation layers are consid-
ered at the top surface.
Si (substrate)
Sc
Cr
CrO (oxide layer)
D
dCr
dSc
capping layer
periodic replication
C (contamination)
z
yields individual layer thicknesses in the sub-nanometer regime, for a bilayer period
with approximately equal individual layer thicknesses. This is a significant difference
to the Mo/Si systems treated in the beginning of this chapter, where the molybdenum
72
Analysis of Cr/Sc Multilayers with Sub-nanometer Layer Thickness 4.3
and silicon layers were well above
>1.7nm
. The exception in the previous case were the
buffer and interdiffusion layers, which nominally have thicknesses below one nanometer
and could not be characterized based on the methods employed above. In the Cr/Sc
system investigated here, all nominal layer thicknesses are in that order of magnitude
and are thus challenging to characterize. We shall therefore first compare the results
obtained with an approach similar to the methods in the previous sections to establish a
limit to the applicability of discrete layer models.
4.3.1 Reconstruction with a Discrete Layer Model Approach
In analogy to Sec. 4.2, we seek to reconstruct the individual layer thicknesses based on
experimental data. For this we construct a discrete layer model as illustrated in Fig. 4.15
in analogy to the procedure applied for the Mo/Si multilayer systems. The parameters of
this discrete layer model are listed in table 4.6together with the upper and lower bound
for the particle swarm optimization procedure.
Table 4.6 | Parametrization of the Cr/Sc binary multilayer model.
Parameter Definition Lower bound Upper bound
dCr / nm Cr layer thickness 0.0 1.5
dSc / nm Sc layer thickness 0.0 1.5
σ/ nm Névot-Croce parameter 0.0 0.5
(identical for all interfaces)
ρCr Cr density w.r.t. bulk density 0.5 2.0
ρSC Sc density w.r.t. bulk density 0.5 2.0
Capping layer
dC (cap) / nm C capping layer thickness 0.0 1.0
dCrO (cap) / nm SiO2capping layer thickness 0.0 1.5
dCr (cap) / nm SiO2capping layer thickness 0.0 3.0
ρC (cap) C density w.r.t. bulk density 0.0 2.0
ρCrO (cap) CrO density w.r.t. bulk density 0.0 2.0
ρCr (cap) Cr (cap) density w.r.t. bulk density 0.5 2.0
The reflectivity of the sample in the water window spectral range from
3.12nm
to
3.16nm
was measured at the SX700 beamline at BESSY II. The angle of incidence was
αi=
1.5°
(corresponding to a grazing angle of incidence of
αGI
i=88.5°
), which corresponds
to the design goal for this mirror prototype. In addition, similar to the Mo/Si samples,
a XRR measurement was conducted in the group of Saša Bajt at the DESY laboratory
using a laboratory-based X-ray diffractometer (X’Pert PRO MRD, Panalytical). The
diffractometer is equipped with a high-resolution goniometer and uses Cu-K
α
radiation
as a source. The XRR intensities were recorded using a PIXcel counting detector. The
dynamic range achieved in the measurements extended down to a reflectance of
10−6
for
grazing angles of incidence of αi=0◦to αi=3◦.
Both measurement curves are shown together in Fig. 4.16. Due to the short period of the
multilayer sample, only two Bragg peaks could be observed in this angular range in the
XRR curve. All expected higher order peaks were below the detection threshold of
10−6
in reflected intensity. The dominating experimental uncertainty was the inhomogeneity
of the sample stack across the sample area. The given uncertainty values for each of
73
Chapter 4 CHARACTERIZATION OF THE MULTILAYER STRUCTURE FOR DIFFERENT SYSTEMS
Figure 4.16 |
EUV and
XRR data recorded for
the Cr/Sc sample sys-
tem. a) The EUV curve
was obtained at an an-
gle of incidence
αi=
1.5°
. b) The XRR curve
was recorded using a Cu-
K
α
source with a pho-
ton energy of
Eph =
8048 eV.
3.12 3.13 3.14 3.15 3.16
λ/ nm
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
Reflectivity
a) EUV, αi=1.5◦
Data
0123456
αGI
i/◦
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Log. Reflectivity
b) XRR, Eph =8048 eV
the measurement points were estimated, by measuring the peak reflectance of the EUV
reflectivity curve on positions marking a cross of
2mm
by
2mm
in the sample center.
This data was compared to theoretical expectance value based on a PSO fit of the discrete
layer model above (for details of the optimization results see below). From this a drift
of the period thickness of
D=2pm
was obtained and uncertainties were calculated as
the difference of two theoretical curves attaining the maximum and minimum
D
values.
Similarly, uncertainties for the XRR curve were calculated by simulating theoretical curves
based on the same period drifts.
In comparison, the most remarkable difference with respect to the Mo/Si mirrors is the
significantly reduced measured peak reflectance of the EUV curve in Fig. 4.16a compared
to the curves in Fig. 4.1and Fig. 4.14. The maximum experimental value attained is only
approximately Rmax ≈15% while it is up to Rmax ≈70% for the Mo/Si systems.
To better illustrate the differences to the Mo/Si systems, an analysis based on the
discrete layer model of a Cr/Sc multilayer was conducted as described above. The
particle swarm optimization was done based on the EUV data shown in Fig. 4.16a and
the parameters and limits listed in table 4.6. The resulting parameters are listed in
table 4.7. The capping layer results were obtained in a combined PSO analysis based on
the EUV and XRR data excluding the areas of the Bragg peaks. This grazing incidence
reflectivity data has a very high sensitivity for the top surface layers, which can not be
deducted from an EUV curve alone as demonstrated in Sec. 4.1.1.
The theoretical curve obtained from the PSO procedure is shown in Fig. 4.17 in direct
comparison with the theoretically achievable maximum reflectivity curve. The latter
was obtained by calculating the resulting reflectivity based on the parameter results in
table 4.7, but without any roughness or interdiffusion, i.e. by requiring
σ≡0.0
. The Sc to
Cr ratio was found to be
ΓSc =dSc/dCr =0.48
with a r.m.s. value of
σ=0.385
nm for
the Névot-Croce factor. While the EUV reflectance curve shows excellent agreement with
the measured data, there is a significant offset to the theoretically achievable maximum
reflectance. For the particular model derived above, theoretical reflectance values of
Rmax >50%
are possible. This large difference, especially compared to Mo/Si systems
74
Analysis of Cr/Sc Multilayers with Sub-nanometer Layer Thickness 4.3
Parameter PSO result
dCr / nm 0.8224
dSc / nm 0.7510
σ/ nm 0.375
ρCr 0.876
ρSc 0.957
Capping layer
dC (cap) / nm 0.462
dCrO (cap) / nm 1.143
dCr (cap) / nm 2.322
ρC (cap) 0.502
ρCrO (cap) 0.618
ρCr (cap) 0.851
Table 4.7 |
PSO fit re-
sults for the discrete
layer Cr/Sc multilayer
model.
which are very close to the theoretically achievable maximum reflectance (cf. Fig. 4.14),
hints at strong roughness or intermixing of the two materials. To verify the applicability
of the discrete (binary) layer model used here, the calculated curves for both experiments,
the EUV and XRR curve, are shown together in Fig. 4.18.
3.12 3.13 3.14 3.15 3.16
λ/ nm
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Reflectivity
Fit (binary model)
Max. theo. reflectivity
Data
Figure 4.17 |
Fitted
experimental EUV re-
flectance curves across
the wavelength of the
radiation impinging at
αi=1.5◦
from normal,
based on the binary
model. The green curve
shows the maximum
theoretical reflectance
assuming a perfect mul-
tilayer system without
roughness or interdiffu-
sion.
Again, the EUV data is matched rather good, while in the case of the XRR measurement
only the first Bragg peak is found to be matched by the model also in the X-ray regime.
However, the second Bragg resonance, clearly visible with a peak reflectance value of
approximately
10−3
is not represented by the model at all. A fully combined analysis
similarly to the approach in Sec. 4.2did not yield a consistent result. The r.m.s. value for
σ
required to reduce the theoretical EUV reflectance down to the measured level could
not be brought into agreement with the existence of the second Bragg peak in the XRR
curve. In a strictly binary model like this one with a layer thickness ratio of
ΓSc ≈0.5
, the
second Bragg peak is additionally suppressed due to symmetry reasons. Thus, there is a
clear mismatch of the model reconstruction and the experimental observations, mostly
due to the complementary data delivered through the measurement of the second Bragg
75
Chapter 4 CHARACTERIZATION OF THE MULTILAYER STRUCTURE FOR DIFFERENT SYSTEMS
0123456
AOI / ◦
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Log. Reflectivity
b) XRR Fit (binary model)
Data
3.12 3.13 3.14 3.15 3.16
λ/ nm
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
Reflectivity
a) EUV
Figure 4.18 |
a) Measured EUV reflectivity curve for the near-normal angle of incidence of
αi=1.5°
together with the theoretical curve based on the PSO optimized binary multilayer model. b) Measured
and calculated XRR curves for the same sample and model parameters at grazing angles of incidence
using radiation at the Cu-K
α
wavelength. A clear mismatch of the theoretical curve and the measured
data can be observed for the second Bragg peak between αGI
i=5.0° and αGI
i=6.0°.
peak of the XRR curve. This is a strong indicator, that the simple model as defined above
does not suffice to describe the sample. Therefore, a more elaborate model is required
introducing additional parameters to account for the increased complexity of the samples
layer properties compared to the Mo/Si sample systems above.
4.3.2 Extending the Model to Graded Interfaces and Interdiffusion
The physical structure of Cr/Sc multilayer systems with individual layer thicknesses in
the sub-nanometer regime is significantly different than in case of the comparably large
thicknesses of several nanometers in the Mo/Si case of the two preceding sections. It is
well known [108], that magnetron sputtered Cr and Sc multilayer systems, similarly to the
Mo/Si systems, suffer from imperfect interfaces. Phase diagrams of Cr/Sc systems show,
that the two materials do not like to mix or form composites at the interfaces [25]. That
makes them an ideal candidate for chemically abrupt multilayer structures as needed for
multilayer mirrors. However, due to the very thin layer structure, both materials are in an
amorphous state and intermixing was in fact observed for multilayer structures similar
to the one discussed here [53]. Another possible reason is the magnetron sputtering
deposition, which has shown to cause intermixing upon deposition [47]. In addition,
roughness at the interfaces exists and further diminishes an ideal chemically abrupt
transition from one material to the next. Due to the small layer thicknesses required
to achieve the first Bragg resonance upon near-normal incidence with radiation of
λ=3.14nm
, roughness and interdiffusion may occur over a zone as large as the total
layer thickness itself. The results from the specular EUV and XRR measurements shown
above, clearly demonstrate that a binary model with only a Névot-Croce damping
parameter
σ
does not provide an accurate model for the physical structure. Instead,
a more complex model is required. Here, a periodic model is defined to account for
possible interdiffusion gradients and intermixing between the two materials in the stack.
The symmetry of two identically thick layers within one period in the simple model
above leads to a suppression of the second order Bragg peak. Nevertheless, physically
76
Analysis of Cr/Sc Multilayers with Sub-nanometer Layer Thickness 4.3
this symmetry effect can be broken by interdiffusion zones with different thicknesses,
depending on whether Cr was deposited on Sc or vice versa. Thereby, the second Bragg
peak is no longer suppressed even though both layers have the same thickness if the
interdiffusion zones are asymmetric. The model used to reconstruct the Cr/Sc multilayer
sample measured above is illustrated in Fig. 4.19 in direct comparison to the simple
model used before. The interdiffusion zones are modeled following a sinusoidal profile,
dCr
dSc
sSc
sCr
D
z
nSc nCr
˜
nCr
˜
nSc
n
dCr
dSc
D
z
nSc nCr
n
b)
gradual interface
profile
ideal interface
profile
a)
Figure 4.19 |
a) Binary Cr/Sc multilayer model with total period thickness
D
and the individual layer
thicknesses
dSc
and
dCr
. b) Model with explicit gradual interfaces following a sinusoidal profile. The
ideal interface profile is approximated through discrete sublayers as indicated in red, forming the actual
gradual interface profile entering the electric field calculations. The thickness of the interdiffusion
zones can differ for the top and bottom interface in each period. Their total thicknesses are given by
sSc and sCr. The effective index of refraction for both layers is given by ˜
nSc and ˜
nCr, respectively.
which represents a smooth transition from the refractive index of the Cr layer to the Sc
layer and vice versa. The thickness of those zones is given by the parameters
sSc
and
sCr
.
For the calculation of the electromagnetic fields inside the stack, the interface region is
sampled with a fixed number of equally spaced points in
z
-direction, effectively creating
a region of thin sublayers with a gradually changing index of refraction (illustrated by the
red stepped function in Fig. 4.19). To take into account intermixing extending across the
full period, an intermixing parameter
η
is introduced. The effective indices of refraction
of the individual Cr and Sc layers are then given through
˜
nCr = (η/2)nSc + (1−η/2)nCr,
˜
nSc = (1−η/2)nSc + (η/2)nCr, (4.6)
for η∈[0,1],
where
nCr
and
nSc
are the tabulated values [62] with densities
ρCr
and
ρSc
. Similarly as
discussed in the case of the Mo/Si multilayer systems, the densities serve to consider
unknown uncertainties in the tabulated values of the optical constants with respect to
the actual case in the samples.
With the definition of the model as outlined above, natural restrictions arise for the
77
Chapter 4 CHARACTERIZATION OF THE MULTILAYER STRUCTURE FOR DIFFERENT SYSTEMS
parameters. As an example, the interdiffusion zone region can not extend across half of
the thickness of the original layers total thickness described by the parameter
dCr
or
dSc
,
respectively. Instead, the intermixing parameter would have to be increased to account
for that situation. The model is therefore parametrized according to the list of effective
parameters given in table 4.8together with their allowed ranges for the optimization
procedure in analogy to the analysis conducted in the previous sections. The range
limits arise either from physical plausibility or are intrinsic properties of the parameter
definition. Here,
D
is the full period thickness,
dSc
and
dCr
are the nominal layer
Table 4.8 | Multilayer parametrization and parameter limits
Parameter Definition Lower bound Upper bound
D/ nm =dSc +dCr 1.5 1.6
ΓSc =dSc/D0.0 1.0
sd/ nm =sSc +sCr 0.0 1.6
Γσ=sSc/sd0.0 1.0
ηlayer intermixing 0.0 1.0
σr/ nm r.m.s. roughness 0.0 0.5
ρSc Sc density w.r.t. bulk density 0.5 2.0
ρCr Cr density w.r.t. bulk density 0.5 2.0
thicknesses of the Cr and Sc layers as indicated in Fig. 4.19, and
ρSc
and
ρCr
their respective
densities with respect to their bulk densities
˜
ρSc =2.989
g/cm
3
and
˜
ρCr =7.19
g/cm
3
[62].
The loss of specular reflectance due to roughness-induced scattering is considered through
the Névot-Croce factor using
σr
identical at each interface. This is necessary to account for
diffusely scattered light, which is missing in the measured specularly reflected radiation
but can not be attributed to contrast loss due to interdiffusion. The parameter
ΓSc
indicates the portion of the Sc layer thickness with respect to the full period thickness
D
, which together uniquely define the thickness
dCr
;
Γσ
describes the asymmetry of
the widths of the interdiffusion zones at the Cr on Sc and Sc on Cr interfaces and is
intrinsically limited to the interval
Γσ∈[0,1]
. Note that
sSc
and
sCr
are half periods
of the sinus functions used to describe the interface profiles. Therefore the condition
sSc +sCr ≤Dholds.
The discretization of the smooth interface profile in the interdiffusion zones introduces
an additional numerical uncertainty through the number of discretization points
n
required to reflect the physical situation of a smooth transition. To assert a lower limit
for this number, the mean error introduced was evaluated by coarse sampling. The most
accurate experiment of the analysis within this chapter is given by the EUV reflectivity
curve, which serves as a reference for this assertion through the sum of the squared
uncertainty of each data point in Fig. 4.16a, ∑m˜
σm.
The numerical error of the model depending on the interface sampling through gradual
sublayers was evaluated by comparing the sum of squares
χn=∑
m
(In=100
m−In
m)2(4.7)
of the difference of the theoretical EUV curves with increasing numbers of gradual
interfaces and an “ideal” smooth transition represented by
100
sublayers. The model
parameters used for this analysis were obtained through a PSO optimization of the model
with respect to the EUV reflectivity curve. As illustrated in Fig. 4.20, the experimental
78
Analysis of Cr/Sc Multilayers with Sub-nanometer Layer Thickness 4.3
0 5 10 15 20 25 30
number of sublayers
0.000
0.001
0.002
0.003
0.004
0.005
numerical uncertainty
χn
∑m˜
σ2
m
Figure 4.20 |
Compari-
son of the numerical un-
certainty with the exper-
imental uncertainty for
the graded Cr/Sc model.
uncertainty dominates at the lower limit of
n=10
sublayers for the interface zone. For
the analysis is this chapter, and due to reasons of numerical effort required to calculate
the electromagnetic field for all measurements discussed here,
n=15
sublayers are used
for all calculations. At that value, the experimental uncertainty is clearly dominant and
only a marginal additional numerical error is acquired due to insufficient sampling.
As a verification of the applicability of the model to the problem of accurately represent-
ing the physical structure that could describe the EUV and XRR data shown in Fig. 4.16
above, the combined analysis technique has been applied to the two data sets described
in Sec. 4.2.2based on the improved gradual model. The particle swarm optimization
approach is used to obtain a global solution for the model parameters by minimizing
the functional defined in Eq.
(4.5)
. The results found for the gradual model are shown
in Fig. 4.21. The EUV reflectivity curves show visually indistinguishable fits for both,
0123456
αGI
i/◦
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Log. Reflectivity
b) XRR data
PSO fit
3.12 3.13 3.14 3.15 3.16
λ/ nm
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
Reflectivity
a) EUV
Figure 4.21 |
Measured and calculated curves based on the reconstruction results for the gradual
model and the EUV and XRR data. a) Measured EUV reflectivity curve for and near-normal angle of
incidence of
αi=1.5°
together with calculated curve of the PSO-based gradual model reconstruction.
b) Measured and calculated XRR curves for the same sample and model parameters at grazing angles
of incidence using radiation at the Cu-Kαwavelength.
the binary model shown in Fig. 4.18 and the gradual model in Fig. 4.21a. For the binary
model, we have seen the distinct mismatch with the second order Bragg peak. For the
gradual interface model, we see a significant improvement of the optimized result with a
perfect match in both Bragg peaks of the XRR curve in Fig. 4.21d while also maintaining
an excellent agreement with the EUV curve.
79
Chapter 4 CHARACTERIZATION OF THE MULTILAYER STRUCTURE FOR DIFFERENT SYSTEMS
Based on the example of a combined analysis of EUV and XRR data in this section, the
gradual interface model clearly provides a more accurate representation of the sample
than the binary approach by offering a reconstruction satisfying both data sets. At the
same time, the results show that a verification of the model only becomes possible by
adding complementary information. In case of the example above, that information is
provided through the appearance of a second Bragg peak in the XRR curve. Thereby,
the limiting case of the binary model, which is still possible for the new gradual model,
can be excluded with certainty through the comparison shown in Fig. 4.21. The main
difference of both models is the local gradual change of the index of refraction, which
attributes for the fact that both materials may intermix. More importantly, both materials
may intermix differently with respect to the specific interface, i.e. the situation where
Cr is deposited on top of Sc or vice versa. A key element of obtaining a reconstruction
of that particular model is thus the application of experimental techniques, which can
deliver information on the spacial distribution of the materials within one period.
At that point, it should be noted that other distortions of a perfect layer system can
be imagined, which are not covered by a strictly periodic model as the one introduced
above. Those include drifts of the period thickness
D
across the stack or other systematic
aperiodicities. In that case, however, a broadening of the peak or a distortion of the
peaks symmetry, most prominently in the EUV curve, will be observed, which is not
the case (for example, cf. Fig. 4.22 below). Although situations may occur, where the
aperiodicities could lead to effects compensated by tuning the parameters of the gradual
interface model, this assumption would assume a more complex situation than the simple
assumption of periodicity and thus lead to a more complex, ill-defined model which
could not be reconstructed. To further strengthen that argument, we shall calculate
the distortion occurring through a drift in the deposition process. This is a plausible
systematic error, which could be caused by instabilities in the deposition process. Fig. 4.22
shows the peak distortion for a drift of the total period thickness
D
across the whole stack
of
N=400
periods by
ddrift =0.005nm
based on the model parameters for the curves in
Fig. 4.21c keeping the mean period thickness constant. Clearly, already this drift would
cause a distortion of the peak symmetry with a significant minimum at
λ≈3.136nm
and an additional shoulder at λ≈3.153nm, which is not observed in the data.
Figure 4.22 |
EUV peak
deformation assuming
a constant drift of
ddrift =0.005 nm
across
the total multilayer
stack keeping the mean
period thickness
D
constant.
3.12 3.13 3.14 3.15 3.16
λ/ nm
0.00
0.05
0.10
0.15
0.20
Reflectivity
data
no ddrift
ddrift =0.005 nm
80
Analysis of Cr/Sc Multilayers with Sub-nanometer Layer Thickness 4.3
4.3.3 Addition of Complementary Experimental Methods
Due to the increased complexity of the model, the question arises how accurately any
parameter of the model can be determined and whether correlations exist and can
be resolved (cf. Fig. 4.11 as an example for correlated model parameters in case of
Mo/Si multilayer systems) based on the available data and whether further analytical
measurements can improve the result as this was clearly the result for the combination
of EUV and XRR experiments shown above. For the particular case of the gradual
interface model for periodic multilayer systems with sub-nanometer layer thicknesses, in
total four experiments were conducted to study the applicability of each method with
respect to finding a unequivocal reconstruction including confidence intervals. Only by
systematically analyzing the strength and weaknesses of the employed analytic methods,
a reconstruction of the model resembling the reality inside the sample becomes possible.
Resonant EUV Reflectivity
As seen for the four layer system discussed in Sec. 4.1.1, confidence intervals for the
individual layer thicknesses in the range below
1nm
could not be obtained by exclusively
analyzing the EUV curve. Similarly, the combined analysis of EUV and XRR experiments
in Sec. 4.2.2did improve the result but still shows fairly large confidence intervals
concerning the small total layer thickness in the Cr/Sc systems. For the particular system
discussed here with possibly strong interdiffusion, a technique is required that yields the
total amount of Sc and equivalently Cr within a single period. For that purpose, resonant
reflectivity experiments in the EUV spectral range are promising. The knowledge of the
optical constants are a necessary requirement for deducting quantitative information
from that kind of experiment. In case of Sc, those were measured precisely for the Sc L3
and L2absorption edges at approximately
λSc-L ≈3.1nm
and below by Aquila et al. [6].
The real and imaginary parts obtained from that experiments are shown together with
the respective optical constants of Cr in Fig. 3.12 of Sec. 3.4.1. To exploit the information
contained in the optical constants of Sc, angular resolved reflectivity curves across the
first Bragg peak were recorded at several wavelengths across the Sc L-edge. As the
Cr dispersion is changing only marginally and smoothly across that wavelength range,
any change of contrast and absorption can be attributed to the Sc in the multilayer.
The corresponding measurements are shown in Fig. 4.23. Each reflectivity curve was
αi/◦
0
5
10
15
20
λ/ nm
2.95 3.00 3.05 3.10
log. reflectivity
10−4
10−3
10−2
10−1
100
Figure 4.23 |
Measured
resonant EUV reflectiv-
ity curves across the Sc
L2 and L3-edge in log-
arithmic representation.
At each equidistant pho-
ton energy point, an an-
gular resolved reflectiv-
ity curve was recorded
across the Bragg peak.
recorded within the interval from
αi=2.5°
to
αi=19.0°
, with varying upper and lower
81
Chapter 4 CHARACTERIZATION OF THE MULTILAYER STRUCTURE FOR DIFFERENT SYSTEMS
boundary depending on the selected wavelength to incorporate only the range of the
Bragg peak. The wavelength range was chosen between
λ=2.986nm
and
λ=3.128nm
including the Sc L2and L3edges. The resulting data is analyzed in analogy to the EUV
reflectivity curves in Sec. 4.3.2by applying the matrix algorithm on basis of the gradual
layer model and the optical constants by Aquila et al. [6]. The experimental uncertainties
taken into account for the REUV experiment were estimated on basis of the multilayer
inhomogeneity deducted as described for the EUV experiment in Sec. 4.3.1. It should
be noted, that uncertainties for the measured optical constants were not given by the
authors of the respective publication. Nevertheless, again by allowing a variation of the
densities of the respective materials in the model, those are accounted for in the analysis.
The variation, however, uses the same value for this density parameter for all analyzed
curves. A mismatch of the individual reflectivity curves at the different energies with the
theoretical calculation based on the results by Aquila et al. is thus possible. This leads to
a broadening of the likelihood distribution, and thereby an increase of the confidence
intervals reflecting the uncertainty in the optical constants determination. The details of
the reconstruction based on this dataset are shown below in this section.
Grazing Incidence X-ray Fluorescence
In addition to the reconstruction of the Sc content via the REUV experiment, spacial
resolved measurements are necessary to deduct the interface profile in the gradual layer
model. As discussed in Sec. 4.3.2, asymmetric interface regions provide a possibility to
observe a second Bragg peak in the XRR measurement, even though both layers in the
period have equal nominal thickness. To obtain information on that spacial distribution of
both materials within a period, XRF experiments exploiting the formation of a standing
wave when scanning across the first Bragg peak were performed. The details of the
method and how spacial sensitivity can be obtained are described in detail in Ch. 2,
Sec. 2.5.
The sample was measured exciting fluorescence of the Sc and Cr K-lines, which
show the highest fluorescence yield for the core shell transitions. The K-edges for both
materials are at energies of
ESc-K =4492eV
and
ECr-K =5989eV
[44]. The experiment
was therefore conducted at the FCM beamline at BESSY II in grazing incidence geometry
at photon energies of
Eph =5500eV
and
Eph =6250eV
, well above the respective edges
as described in Ch. 3, Sec. 3.3. Depending on which energy was used, the Bragg peak
is found at grazing angles of incidence of
αGI
i≈4.12°
and
αGI
i≈3.62°
, respectively. The
measured relative fluorescence yield in the vicinity of the first Bragg peak is shown in
Fig. 4.24 for both photon energies and materials. Here, due to the grazing angles of
incidence, the method is referred to as GIXRF. Since the photon energy of
Eph =5500eV
is below the K-edge of Cr, only data for the Sc K-fluorescence exists. In the second case,
fluorescence from both materials was detected. The measurement uncertainties were
estimated from the scattering of the data for regions away from the Bragg resonance,
where a flat curve is theoretically expected.
The fluorescence curves for Cr and Sc show distinctly different behavior and the
expected curve shape (cf. Fig. 2.13). For the analysis, the result at photon energies of
Eph =5500eV
(Fig. 4.24a) was not taken into account, as the information is redundant to
the result at
Eph =6250eV
(Fig. 4.24b). As mentioned above, the theoretical description
on how the relative fluorescence is calculated based on the gradual model is elaborated
on in detail in Ch. 3, Sec. 3.3.
82
Analysis of Cr/Sc Multilayers with Sub-nanometer Layer Thickness 4.3
4.00 4.05 4.10 4.15 4.20 4.25 4.30
αGI
i/◦
0.94
0.96
0.98
1.00
1.02
1.04
FY of Sc / a.u.
a) Eph =5500 eV
3.50 3.55 3.60 3.65 3.70 3.75
αGI
i/◦
0.96
0.98
1.00
1.02
1.04
FY of Sc / a.u.
b) Eph =6250 eV
3.50 3.55 3.60 3.65 3.70 3.75
αGI
i/◦
0.96
0.98
1.00
1.02
1.04
FY of Cr / a.u.
c) Eph =6250 eV
measured relative fluorescence yield
Figure 4.24 |
Measured relative X-ray fluorescence curves for the Cr and Sc K-lines across the first
Bragg peak. a) Relative fluorescence yield of the Sc K-line at a primary photon energy of
Eph =5500 eV
.
b), c) Relative fluorescence yield for both materials at an primary photon energy of Eph =6250 eV.
4.3.4 Reconstruction and Maximum Likelihood Evaluation
With the two additional measurements described above, five data sets (EUV, XRR,
REUV, GIXRF (Sc) and GIXRF (Cr)) are available for the Cr/Sc multilayer sample to
reconstruct the parameters of the gradual interface model. The full dataset, repeating and
summarizing the relevant experimental results of this section in one figure, is compiled
in Fig. 4.25.
As in the combined analysis conducted for the Mo/Si/C systems in Sec. 4.2.2, we
define the minimization functional for the combined analysis of all the datasets as
χ2=˜
χ2
EUV +˜
χ2
XRR +˜
χ2
REUV +˜
χ2
GIXRF(Sc) +˜
χ2
GIXRF(Cr), (4.8)
where each of the reduced functionals is defined as given in Eq.
(4.1)
. This functional
corresponds to the combined
χ2
functional defined in
(4.5)
, augmented by the additional
measurements conducted here.
Firstly, similar as for the other two sample systems treated in this chapter, the parame-
ters of the model, here the gradual interface model with the parameters and their limits
listed in table 4.8, were obtained using the PSO method to find a solution reproducing
the experimental results. Secondly, following the maximum likelihood approach em-
ploying the MCMC method as detailed in Sec. 4.1.2, starting in the vicinity of this result
the uniqueness and confidence intervals for each parameter were obtained. The final
parameter results were obtained by taking the
50%
percentile of the resulting likelihood
distribution for each parameter.
Through the minimization of the combined
χ2
functional in Eq.
(4.8)
via the PSO
method, the best model parameters were obtained. It should be noted here, that in case
of the XRR curve, the analyzed data set was restricted to the two visible Bragg peaks
which contain the information on the periodic part of the layer system. The data in
83
Chapter 4 CHARACTERIZATION OF THE MULTILAYER STRUCTURE FOR DIFFERENT SYSTEMS
αi/◦
0
5
10
15
20
λ/ nm
2.95 3.00 3.05 3.10
log. reflectivity
10−4
10−3
10−2
10−1
100
c) REUV
0.96
0.98
1.00
1.02
1.04
FY Sc / a. u.
d) GIXRF, Sc Eph =6.25 keV
3.50 3.55 3.60 3.65 3.70 3.75
αGI
i/◦
0.96
0.98
1.00
1.02
1.04
FY Cr / a. u.
e) GIXRF, Cr Eph =6.25 keV
3.12 3.13 3.14 3.15 3.16
λ/ nm
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
Reflectivity
a) EUV, αi=1.5◦
0123456
αGI
i/◦
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Log. Reflectivity
b) XRR, Eph =8048 eV
Figure 4.25 |
Full data set used in the combined analysis. Due to redundancy, the XRF data for the Sc
at a photon energy of Eph =5500 eV was omitted.
between those does reflect the top surface layer thicknesses and was therefore analyzed
separately to obtain the capping layer thicknesses after the optimization of the periodic
part. The results for the capping layer thicknesses listed in table 4.9was consequently
used throughout the theoretical analysis for all experiments described here.
Table 4.9 |
Optimized
model parameters ob-
tained by PSO analysis
of the XRR data to de-
termine the structure of
the capping layers.
Parameter XRR (areas in between the peaks)
dC (cap) / nm 0.709
dCrO (cap) / nm 0.913
dCr (cap) / nm 2.495
ρC (cap) 0.527
ρCrO (cap) 0.548
ρCr (cap) 0.791
Confidence Intervals and Evaluation of the Experimental Methods
As discussed numerously throughout this chapter, the PSO ideally delivers the global
minimum of the respective optimization functional. However, no information is obtained
about the uniqueness and accuracy of the solution or correlation of parameters causing
ambiguity of the results. Consequently, in addition to fitting the data with a particle
swarm optimizer, the result was verified based on the MCMC method described above
to evaluate the confidence intervals for each parameter. To assess the performance
of each of the experimental methods individually, the two step process, i.e. the PSO
fitting procedure followed by the MCMC sampling, was conducted for each standalone
experiment as well as for the combined optimization problem stated in Eq. (4.8).
The results are compiled in Table 4.10. The confidence intervals were calculated by
evaluating the probability distribution as a result of the MCMC procedure for each
84
Analysis of Cr/Sc Multilayers with Sub-nanometer Layer Thickness 4.3
Table 4.10 |
Optimized model parameters with confidence intervals derived from MCMC validation
for each individual experiment and the combined analysis
Parameter Combined EUV XRR REUV GIXRF
D/ pm 1.5742+0.0007
−0.0007 1.5742+0.0022
−0.0033 1.5742+0.0056
−0.0044 1.5740+0.0021
−0.0022 1.5793+0.0046
−0.0049
ΓSc 0.47+0.04
−0.04 0.46+0.16
−0.17 0.54+0.27
−0.34 0.46+0.09
−0.08 0.49+0.09
−0.09
sd/ nm 1.31+0.19
−0.25 0.89+0.55
−0.85 0.60+0.78
−0.56 0.92+0.56
−0.84 1.27+0.24
−0.37
Γσ0.13+0.30
−0.12 0.28+0.63
−0.27 0.46+0.51
−0.44 0.45+0.53
−0.42 0.52+0.47
−0.49
η0.58+0.06
−0.15 0.55+0.15
−0.35 0.47+0.36
−0.44 0.55+0.14
−0.31 0.50+0.20
−0.41
σr/ nm 0.09+0.13
−0.09 0.19+0.15
−0.18 0.13+0.16
−0.12 0.18+0.14
−0.16 0.23+0.23
−0.22
ρSc 0.95+0.18
−0.13 1.12+0.85
−0.59 1.20+0.76
−0.67 1.08+0.31
−0.22 1.40+0.57
−0.82
ρCr 1.07+0.17
−0.12 1.16+0.53
−0.34 1.10+0.75
−0.45 1.07+0.33
−0.28 1.38+0.52
−0.55
parameter. The confidence intervals given here represent percentiles of the number of
samples found in the interval defined by the upper and lower bounds used for the PSO
procedure for each parameter. In the case of a centered Gaussian distribution, percentiles
of
2.3%
and
97.8%
of the integrated number of samples forming the distribution, mark
the interval of four times the standard deviation, i.e.
±2σ
in statistical terms. Due to
potential asymmetries in the actual distributions found by the MCMC method, explicit
upper and lower bounds of the confidence intervals are given in table 4.10 based on these
percentiles. The best model value is calculated by the MCMC sampling by taking the
50% percentile, of the distribution of the numerical parameter samples.
Before discussing the achieved reconstruction and the corresponding confidence inter-
vals of each of the methods in detail, we shall view the theoretical curves calculated from
the best model of the combined analysis. The curves are shown in direct comparison with
the data from Fig. 4.25 including the respective experimental uncertainties in Fig. 4.26.
Clearly, the data and the solution found in the optimization procedure show excellent
agreement indicating that the gradual interface model indeed provides a very good
representation of the multilayer structure with respect to the experiments conducted here.
Nevertheless, differences can be observed. The reason lies in the fact that the model is
potentially still rather ideal. Small variations during the deposition process, for example,
could lead to imperfections, which are not described in a strictly periodic model. How-
ever, including these by explicitly breaking the periodicity would lead to an ill-defined
model with a vastly increased number of parameters and is thus not practical. Another
reason is the deviation in the homogeneity of the sample, e.g. a varying period across the
sample, which causes mismatches if the measurement position varies slightly between
the different experimental setups. The latter effects were considered in the uncertainties
of the individual measurements by measuring the EUV reflectivity at positions
±2
mm
from the center position and fitting the model. The result was a
∆D=2
pm shift in the
period over 4 mm across the sample.
85
Chapter 4 CHARACTERIZATION OF THE MULTILAYER STRUCTURE FOR DIFFERENT SYSTEMS
3.12 3.13 3.14 3.15 3.16
λ/ nm
0.00
0.05
0.10
0.15
0.20
Reflectivity
a) EUV
0123456
AOI / ◦
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Log. Reflectivity
b) XRR
3.50 3.55 3.60 3.65 3.70 3.75
AOI / ◦
0.92
0.94
0.96
0.98
1.00
1.02
1.04
1.06
1.08
FY of Sc / a.u.
c) GIXRF, Sc
3.50 3.55 3.60 3.65 3.70 3.75
AOI / ◦
0.92
0.94
0.96
0.98
1.00
1.02
1.04
1.06
1.08
FY of Cr / a.u.
d) GIXRF, Cr
70 72 74 76 78 80 82 84 86 88
AOI / ◦
10−4
10−3
10−2
10−1
100
Log. Reflectivity
e) REUV Combined fit
Data
Figure 4.26 |
Measured reflectance and fluorescence yield curves in direct comparison with the calcu-
lated reflectance and intensity curves for the optimized parameters obtained through the combined
analysis of all experiments as listed in table 4.10.
86
Analysis of Cr/Sc Multilayers with Sub-nanometer Layer Thickness 4.3
Parameter correlations in the combined analysis
With the optimized model parameters listed in table 4.10 and shown in Fig. 4.26 for
the combined analysis, a model reconstruction could be obtained explaining the data
for each of the experiments. The MCMC sampling of the likelihood functional based
on the
χ2
definition in Eq.
(4.8)
yields the corresponding confidence intervals for all
parameters given through the upper and lower bound as described above. Here, we shall
illustrate and discuss in detail the resulting likelihood distributions obtained from the
combined analysis, as they show that correlations of the parameters could be resolved
and only persist for a single important case. For that, Fig. 4.27 shows the full matrix of
two- and one-dimensional likelihood distribution projections marginalizing over all other
parameters. The details of how this figure is to be interpreted are described in detail
above in Sec. 4.1.2for the example of Mo and Si layer thicknesses obtained through fitting
EUV reflectivity data. Here, all possible gradual interface model parameter combinations
are shown as two dimensional histograms together with the one-dimensional projection
at the diagonal of the plot matrix. The solid blue line represents the values of the
optimized model as listed in table 4.10 for the combined analysis column.
Generally, most of the parameter combinations do not show distinct correlations but
approach the shape of a two-dimensional Gaussian distribution, which would be expected
for a unique solution with corresponding uncertainty. It should be noted that in some
cases, the distribution is truncated by parameter limits, which follow from physical or
mathematical restrictions on the parameters as discussed in Sec. 4.3.2, such as for the
densities
ρSc
and
ρCr
as well as for the interface region ratio
Γσ
. In addition, the latter
parameter shows a bimodal distribution for all two-dimensional histograms with clear
emphasis on the lower value. That is a particularly interesting result of the combined
analysis as it clearly demonstrates that only strongly asymmetric interface regions are
minimizing the χ2functional and it may thus be concluded that this corresponds to the
actual structure present in the sample.
Finally, the parameter set of the r.m.s. roughness
σr
and the interdiffusion parameter
η
show a “banana shaped” correlation significantly broadening the confidence intervals for
both parameters in table 4.10. Fig. 4.28 shows a magnification of that particular histogram
to better illustrate this property. The broad spectrum of values covered by the distribution
in both parameters hints at a indistinguishability of those two model parameters, and
consequently physical properties of the sample, based on the analyzed data. In fact, this
conclusion can easily be understood as none of the applied experimental methods can
separate the effect of roughness and interdiffusion. For better understanding this, we
shall consider the relatively large beam footprint, with the smallest one of all experiments
covering and area of approximately
1mm
by
1mm
, in comparison to the roughness
dimensions and frequencies expected in the order of nanometers. Thereby, any reflected
radiation or fluorescence radiation excited within the multilayer always represents an
average of the rough interface morphology. That, however, can not be distinguished from
a homogeneous layer with gradual interdiffusion along the surface normal of the sample.
The solution to this problem of distinction is the analysis of diffuse scattering from the
sample in addition to the combined analysis, which is the topic of the Ch. 5of this thesis.
87
Chapter 4 CHARACTERIZATION OF THE MULTILAYER STRUCTURE FOR DIFFERENT SYSTEMS
D=1.574+0.001
−0.001
0.3
0.4
0.5
0.6
η
η= 0.58+0.06
−0.15
0.40
0.44
0.48
0.52
ΓSc
ΓSc = 0.47+0.04
−0.04
0.75
1.00
1.25
1.50
sd
sd= 1.32+0.19
−0.25
0.2
0.4
0.6
0.8
Γσ
Γσ= 0.14+0.50
−0.13
0.06
0.12
0.18
0.24
σr
σr= 0.09+0.13
−0.09
0.75
0.90
1.05
1.20
ρSc
ρSc = 0.94+0.17
−0.13
1.570
1.572
1.574
1.576
D
0.8
1.0
1.2
1.4
ρCr
0.3
0.4
0.5
0.6
η
0.40
0.44
0.48
0.52
ΓSc
0.75
1.00
1.25
1.50
sd
0.2
0.4
0.6
0.8
Γσ
0.06
0.12
0.18
0.24
σr
0.75
0.90
1.05
1.20
ρSc
0.8
1.0
1.2
1.4
ρCr
ρCr = 1.07+0.16
−0.13
Figure 4.27 |
Matrix representation of the result of the maximum likelihood analysis based on the
MCMC method for all parameter combinations. At the top of each column, the one-dimensional
projection of the likelihood distribution for the respective parameter is shown in analogy to the figures
4.6 or 4.10. The dotted red lines indicate the
±2σ
interval, i.e. two standard deviations from the center
value (
50 %
percentile). The latter is indicated through the solid blue lines. In the two dimensional
projections, the solid black contours mark the areas for one and two standard deviations, respectively.
For a discussion of the observed features see main text
88
Analysis of Cr/Sc Multilayers with Sub-nanometer Layer Thickness 4.3
0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65
η
0.00
0.05
0.10
0.15
0.20
0.25
σ/ nm
1σ
2σ
Figure 4.28 |
Magnified
two dimensional projec-
tion histogram for the
correlation of the inter-
diffusion parameter
η
and the r.m.s. rough-
ness parameter
σr
from
Fig. 4.27. Again, the ar-
eas of one and two stan-
dard deviations are indi-
cated together with the
50 %
percentile as solid
blue lines.
Confidence intervals depending on the employed method
The confidence intervals of each experimental method differ significantly as table 4.10
shows. The reason behind this is the different sensitivity of the methods to the specific
physical properties described by the respective model parameter. To better illustrate the
information compiled in the table above, for each method and each parameter the total
confidence interval is shown in Fig. 4.29 in a radial plot. The total confidence interval
is defined as the difference of the upper and lower values as listed in table 4.10 for
each experiment and parameter. The plot shows the four relevant experiments and the
D/ pm
η
ΓSc
sd/ nm
Γσ
σr/ nm
ρSc
ρCr
2.2 4.4 6.6 8.8 11.0
Combined
EUV
XRR
REUV
XRF
0.17
0.35
0.52
0.7
0.87
0.13
0.27
0.4
0.54
0.67
0.31
0.62
0.93
1.23
1.54
0.210.420.630.841.05
0.1
0.2
0.3
0.4
0.5
0.32
0.63
0.95
1.27
1.59
0.26
0.53
0.79
1.05
1.32
Figure 4.29 |
Visual rep-
resentation of the to-
tal confidence intervals
for each of the parame-
ters with respect to each
of the individual exper-
iments as well as the
combined analysis.
combined analysis results. Any value closer to the origin of the radial plot indicates
a smaller confidence interval and thus a better accuracy of the solution found for the
respective parameter.
89
Chapter 4 CHARACTERIZATION OF THE MULTILAYER STRUCTURE FOR DIFFERENT SYSTEMS
It is worth noting that the confidence interval for the combined analysis is significantly
smaller compared to the individual experiments for all parameters and therefore yields
the best result. This is especially true for the parameter
Γσ
describing the asymmetry of
the interdiffusion layers. Within each of the individual experiments this parameter has a
large uncertainty and can not be determined, whereas the combined analysis delivers a
significant result of a clearly asymmetric interdiffusion layer thickness. In combination
with the observations made above for the respective histograms in Fig. 4.27, we can
conclude that this asymmetry is indeed a significant result and that the remaining fairly
large confidence interval mainly results from the fact of having a bimodal distribution as
the dotted lines in the respective histogram Fig. 4.27 prove. A possible explanation for
this asymmetry is the deposition process through magnetron sputtering. The elements
Cr and Sc have different mass and thus different momentum when deposited onto each
other. A similar effect is known from the deposition of Mo/Si multilayer systems, where
the heavier Mo shows higher penetration into the Si layer than vice versa [105]. In the
case of Cr/Sc multilayers, the Cr is heavier and thus has higher momentum leading to a
broader interdiffusion layer, which is indeed also the interface region found to be the
broadest by the analysis conducted here.
The comparison of the sensitivity, i.e. the size of the confidence intervals, of each
method further reveals, that the density parameters
ρSc
and
ρCr
can not be determined
based on methods using X-ray radiation, such as XRR and XRF. This proves the claim
made at the beginning of this chapter that the uncertainties of the optical constants, while
relevant and considered through these density parameters for EUV experiments, do not
impede the structural reconstruction using X-ray radiation.
The final result of this structural analysis of Cr/Sc multilayer systems with sub-
nanometer layer thicknesses is shown in Fig. 4.30 by the depth dependence of the
index of refraction in direct comparison with the initial binary model. As mentioned
Figure 4.30 |
Real part
of the index of refrac-
tion
n
based on the re-
sults of the optimized
parameters listed in Ta-
ble 4.10 for the com-
bined analysis for a se-
lected wavelength. The
gradual interface model
is shown in direct com-
parison to the binary
model optimized for the
EUV reflectance curve
over three full peri-
ods. The resulting
strong asymmetry in the
width of the interface re-
gions is clearly visible
(see text). The gray and
white shaded areas indi-
cate the Cr and Sc lay-
ers, respectively, for the
binary model.
01234
relative depth z/ nm
0.996
0.998
1.000
1.002
1.004
Re (n)at λ=3.142 nm
Cr Sc graded Cr/Sc interface model
binary Cr/Sc model
before, the most remarkable result of the combined analysis is the strong asymmetry of
the interdiffusion layers. This can only be shown by the combination of all analytical
experiments conducted here. In addition, the comparison shows that at no point within
90
Analysis of Cr/Sc Multilayers with Sub-nanometer Layer Thickness 4.3
the periodic multilayer stack pure Sc or pure Cr layers are observed, but always a mixture
of both. In the context of answering the question of poor reflectivity with respect to the
theoretical possible maximum, this shows that intermixing is the main reason. The loss
of contrast with respect to the binary model, causes the diminished reflectivity. However,
it should noted that due to the correlation between roughness and interdiffusion this
result is still to be verified by the aforementioned analysis of diffuse scattering. This is
the topic of chapter 5and analyzed there for the Cr/Sc system.
The experiments, methods and findings of this section are part of the publication
A. Haase, S. Bajt, P. Hönicke, V. Soltwisch, and F. Scholze: ‘Multiparameter char-
acterization of subnanometre Cr/Sc multilayers based on complementary measure-
ments’. en. In: Journal of Applied Crystallography
49
.6(Dec. 2016), pp. 2161–2171.doi:
10.1107/S1600576716015776
91
5
Analysis of Interface Roughness
Based on Diffuse Scattering
So far, no distinction could be made between intermixing and roughness at the surface
or interfaces. As discussed in detail in Sec. 4.3.4, this distinction based on the employed
structural characterization methods from chapter 4, such as EUV reflectivity, REUV, XRR
and XRF, is in fact not possible. This is due to the lack of sensitivity of the experiments
conducted there, which is not even existent for the combination of all methods. Due to the
comparatively large beam footprint on the sample in comparison to interfacial roughness
on the nanoscale, any specular reflection measurement, and even the measurement of
fluorescence radiation generated by a standing wave field, is only sensitive to the average
of the interfacial profile and can thus not distinguish from horizontally homogeneous
intermixing. Effectively, both cases can be described with a gradual profile in the
optical constants at the interfaces. Consequently, all methods applied so far rely on a
horizontally homogeneous medium model, which was reconstructed. The correlation
of the roughness parameter
σr
and the intermixing parameter
η
in Fig. 4.28 nicely
demonstrate that assessment.
Within this chapter, the diffuse scattering contribution measured from all samples
studied in chapter 4is investigated. While none of the experiments conducted there could
yield a distinction criterion, diffuse scattering can only be observed from rough surfaces
or interfaces, while intermixing does not cause any off-specular intensity contribution.
The analysis of the diffuse scattering, here in particular scattering in the EUV spectral
range, therefore serves as a natural tool to implement the distinction of intermixing on
the one hand and roughness on the other. In addition, the distribution of the scattered
intensity contains information on the morphology of roughness which is of particular
interest to understand the effect on the reflectivity as observed in the previous chapter.
First, the analysis of the Mo/B
4
C/Si/C sample is continued based on the layer model
derived in Sec. 4.1.1and the effects observed in case of diffuse EUV scattering from
multilayer systems are demonstrated in detail with an analysis based on the theory
introduced in Sec. 2.4. In the second part, the two sample sets with systematically varied
molybdenum thickness from Sec. 4.2are investigated. The focus is on the role of the
93
Chapter 5 ANALYSIS OF INTERFACE ROUGHNESS BASED ON DIFFUSE SCATTERING
interface morphology in the diminished reflectivity observed for some of the samples
in the two sets of Mo/Si/C systems. Furthermore, the effect of the polishing process in
one of the sample sets is addressed. Finally, the parameter correlation of intermixing
and roughness for the Cr/Sc sample is investigated and the characterization made in
Sec. 4.3.4is finalized based on the diffuse scattering from that sample.
5.1 Near-normal Incidence Diffuse Scattering
The goal of the investigation of the diffuse scattering intensity is to gain information on
the interface morphology in the sample. In the theoretical description of diffuse scattering
in chapter 2, the characterization of the scatter intensity from a multilayer sample was
elaborated. In Sec. 2.2.1, the measured scattering intensity
Is
is described in terms of
the differential scattering cross section
(dσ
dΩ)
, which is given explicitly for the problem
of interfacial and surface roughness in multilayer samples in Eq.
(2.38)
. As indicated
there, the full theoretical description is based on the introduction of the reciprocal space
as an adapted set of coordinates for the scattering problem. This space is spanned by
the coordinates qx,qyand qz. Those are the components of the momentum transfer due
to the scattering process (cf. Sec. 2.4) and are related to the experimental parameters
wavelength λ, as well as the angle of incidence αiand the exit angle αfof the scattering
experiment. Based on the theory developed in Sec. 2.4, a mapping of reciprocal space
along the two coordinates
qx
and
qz
is required to obtain information on the samples
interface morphology.
In order to discuss the diffuse scattering experiments and enable a theoretical analysis,
we shall therefore first give some definitions of measurement geometry and how it is
related to the reciprocal space coordinates. So far, any scattering measurement (excluding
the XRF experiment) of chapter 4was conducted in the specular reflection geometry,
where incidence and exit angle are equal, i.e. at
qx=qy≡0
. Diffusely scattered radiation
caused by roughness, however, is scattered to off-specular angles. The experiments
conducted here are exclusively done in a co-planar geometry since the roughness in the
samples under investigation is assumed to be isotropic in the directions parallel to the
surface (cf. Sec. 2.4). Thus, any scattered radiation is only measured in the scattering
plane defined by the incidence wave vector and the surface normal of the multilayer
sample. Two different types of measurements need to be distinguished as they relate
to different paths through reciprocal space, the detector scan geometry and the rocking
scan geometry both indicated in Fig. 5.1. The detector scan describes a movement of the
detector inside the scattering plane recording radiation scattered to the exit angle
αf
,
while keeping the incidence angle
αi
constant and is indicated by the red shaded area in
Fig. 5.1. The rocking scan refers to a rotation of the sample around the axis perpendicular
to the scattering plane while keeping the detector position fixed with respect to the
incident beam (indicated by the blue shaded are in Fig. 5.1). The angle between detector
and the incident beam is referred to as
∆Θ =αi+αf
, while the tilt angle of the sample is
ω
. By changing
ω
, the incidence angle
αi
and the exit angle
αf
are changed accordingly.
In both cases this leads to incidence and exit angles, which are no longer equal and, thus,
non-vanishing values for the
qx
vector component. The out-of-plane angle
θf
(cf. Fig. 2.6
in Ch. 2) remains zero in those experiments and consequently qy≡0.
The corresponding paths through reciprocal space, calculated according to Eq.
(2.31)
,
are different for these two cases. They are shown schematically in Fig. 5.2for two
94
Near-normal Incidence Diffuse Scattering 5.1
ω
z
y
x
αi
∆Θ
incoming beam
detector
αf
in-plane scattering
sample
reflected beam
Figure 5.1 |
Co-planar
measurement ge-
ometries. By keeping
the opening angle
∆Θ =αi+αf
between
incident and exit beam
and the detector fixed,
respectively, a rocking
scan can be performed
by changing the sample
angle
ω
. In a detector
scan the sample angle
ω
is kept fixed and
defines the angle of
incidence while the
detector is moved along
Θ.
exemplary experimental parameter sets of incidence angle
αi
and opening angle
∆Θ
,
respectively, as well as wavelength for the two scan types. Clearly, for a mapping of the
0−0.2
0.6
0.4
αf(αi=6.75◦,λ=13.5 nm)
ω(∆Θ =30◦,λ=13.5 nm)
λ
λ
0.8
1.0
0.2 qx/ nm−1
qz/ nm−1Figure 5.2 |
Schematic
positions in reciprocal
space (cf. Eq.
(2.31)
)
in dependence on the
measurement geometry.
The dashed path rep-
resents a rocking scan
with the angle
ω
. The
solid line shows the
movement in
q
-space
when changing the de-
tector angle
αf
at a fixed
angle of incidence. By
tuning the wavelength
at each angular posi-
tion, the
qz
-direction be-
comes accessible as in-
dicated by the dotted ar-
rows.
two-dimensional space spanned by
qx
and
qz
it does not suffice to perform only angular
scans. In addition, wavelength scans (
λ
-scan) have to be performed at each angular
position. By changing the wavelength and the angle in the same measurement, both
degrees of freedom (qxand qz) in reciprocal space become accessible.
Based on the theory in Sec. 2.4, interface roughness contributes to the scattering inten-
sity by introducing momentum transfer parallel to the surface. The PSD, describing this
statistical roughness, is thus in our co-planar geometry only dependent on
qx*
, i.e. the
momentum transfer within the interface planes. In the theory chapter, an expression
for the PSD was derived, which describes an average value across all interfaces of the
multilayer. While individual PSDs can be described within the theoretical framework,
this poses an ill-defined model for the experiments conducted here. In all measurements
taken, many interfaces contribute to the diffuse scattering intensity simultaneously. The
periodic character of the multilayer systems, does not allow to distinguish the individual
contributions as the wave field inside the multilayer exhibits the same periodicity. The
*The PSD is generally dependent on qk=qq2
x+q2
y, which reduces to qk≡qxin co-planar geometry
95
Chapter 5 ANALYSIS OF INTERFACE ROUGHNESS BASED ON DIFFUSE SCATTERING
experiment thus delivers a contribution across all interfaces, which makes a distinction of
individual interfaces impossible. In addition to that assessment, the model of a single av-
erage PSD is found to be justified in case of high degree of vertical correlation throughout
the stack, which we shall confirm through the appearance of the corresponding resonant
features in our experiments.
Based on the PSD as derived in Eq.
(2.47)
with the dependence only on
qx
, we should
expect to be able to extract its values from the measured data as cuts along the
qx
axis
anywhere in a measured reciprocal space map. However, it was observed in grazing
incidence diffuse X-ray scattering experiments, that vertical correlation of roughness
causes an additional intensity modulation of the scattering in reciprocal space along the
qz
direction, the so-called Bragg sheets [65,66,70,112]. As the interfaces have periodic
0−0.2
0.6
0.4
0.8
1.0
0.2 qk/ nm−1
qz/ nm−1
z
0−0.2
0.6
0.4
0.8
1.0
0.2 qk/ nm−1
qz/ nm−1
2π/˜
D
2π/˜
D
z
high vertical rougness correlation
low vertical rougness correlation
δqz
δqz
D=di+dj
D=di+dj
Figure 5.3 |
Schematic illustration of the appearance of Bragg-sheets in the off specular scattering
intensity at the
qz
value fulfilling the Bragg condition of the multilayer stack. The width
δqz
of the sheet
is dependent on the degree of vertical correlation of roughness.
distances along the surface normal of the sample, roughness correlation poses a Bragg
condition with respect to the
qz
component of the momentum transfer vector enhancing
the diffuse scattering where fulfilled. The expected diffuse scattering distribution in
reciprocal space is schematically depicted in Fig. 5.3. Since the periodicity of the interfaces
is the multilayer periodicity, those Bragg sheets are expected to appear, where the first and
higher order Bragg condition of the multilayer is fulfilled, i.e. where inside the multilayer
qz=m2π/˜
D
. Here,
m
is the integer number of the Bragg order and
˜
D=˜
nidi+˜
njdj
is the optical multilayer period thickness in real space, where
˜
ni
and
˜
nj
are the real
part of the index of refraction of the respective layer
i
and
j
. Those sheets of increased
intensity appearing in reciprocal space vary in width along
qz
, depending on the strength
of the correlation of roughness along the vertical direction in the sample. The higher the
correlation length
ξ⊥(~
qk)
, the thinner is the Bragg sheet in
qz
direction (cf. upper and
lower part of Fig. 5.3), i.e. the smaller is
δqz
. In the theoretical treatment of the diffuse
scattering in Sec. 2.4, this vertical roughness correlation length
ξ⊥(~
qk)
enters through
the replication factor
c⊥
ij (~
qk)
in Eq.
(2.44)
and was explicitly given in Eq.
(2.51)
. Due to
96
Near-normal Incidence Diffuse Scattering 5.1
the strong enhanced intensity in those Bragg sheets, the PSD is preferably extracted as
a vertical cut along
qx
at the
qz
position of the sheet [112,123]. Consequently, in the
following we shall focus on the mapping of reciprocal space in the vicinity of the first
Bragg resonance to observe the expected Bragg-sheet intensity distribution and analyze
the interface morphology.
In the studies cited above, the reciprocal space maps of multilayer diffuse scattering
were obtained in a grazing incidence geometry using X-rays. The major disadvantage
of this technique is that curved samples are not accessible in that way, since no grazing
incidence measurement can be conducted if the sample is convexly curved. Here, the
diffuse scattering is studied using EUV radiation impinging at near-normal incidence.
Thereby, this disadvantage is overcome. However, as explained above, using near-normal
incidence radiation reduced the accessible
qz
range for constant wavelength, which
can be compensated by tuning the wavelength accordingly, whereas grazing incidence
studies reveal the Bragg sheets in the out-of-plane direction at fixed photon energies,
e.g. Siffalovic et al. [123].
5.1.1 Mapping Reciprocal Space for the Mo/B4C/Si/C Sample
In this section, the EUV diffuse scattering from the Mo/B
4
C/Si/C sample, structurally
characterized using EUV reflectivity in Sec. 4.1, is investigated as an example for the
analysis of near-normal scatter intensity from multilayer samples. Diffuse scattering
measurements in three different geometries were conducted at the SX700 beamline at
BESSY II. From the experimental data, the respective reciprocal space coordinates were
calculated. The corresponding maps and the experimental details are given in Fig. 5.4.
The reciprocal space maps for the rocking scan (b) at an opening angle of
∆Θ =13.5◦
and the rocking scan (c) at an opening angle of
∆Θ =30◦
and for the detector scan (a)
with the angle of incidence
αi=6.75◦
clearly show different symmetries rather than
the expected Bragg sheet from Fig. 5.3. Thus, the measurements conducted here stand
in contrast to the expectation derived from grazing incidence experiments. Instead, a
strong enhancement in the off-specular scattering is observed around
qx≈ ±0.1
nm
−1
(cf. Fig. 5.4(a) and (c)), which is not replicated on the negative
qx
-axis in case of (a).
The rocking scans (b) and (c) are symmetric with respect to the specular axis at
qx=0
,
however, no enhanced scattering appears in (b). The latter map shows a triangular-shaped
intensity distribution for both the positive and negative
qx
range. A minimum in the
width, i.e. in the quantity
δqz
defined in Fig. 5.3, with respect to the
qz
direction can
be observed in (b) around
qx≈ ±0.2
nm
−1
. The triangular shape also appears for the
positive
qx
range of the detector scan in (a), where the minimum in width coincides
with the intensity maximum. Clearly, the measurement of diffuse scattering at EUV
wavelengths and near-normal incidence differs from the observations made for grazing
incidence experiments using X-rays (cf. Salditt et al. [112] or Jiang et al. [70]), which are
independent of the measurement geometry.
The measurement geometry used influences the measured reciprocal space maps, even
though all maps were recorded for the same sample with the same spot position. This
indicates that the intensity distributions seen here, cannot be the result of multilayer
roughness properties alone, i.e. the PSD, which do not change due to changes in the
illumination geometry. Furthermore, the results from Fig. 5.4are clearly deviating from
the expectation sketched in Fig. 5.3. This indicates, that an additional effect causing
a modulation of the diffuse scattering intensity is observed. In fact, the additional
97
Chapter 5 ANALYSIS OF INTERFACE ROUGHNESS BASED ON DIFFUSE SCATTERING
0.90
0.91
0.92
0.93
0.94
0.95
0.96
0.97
qz/ nm−1
a)
0.90
0.91
0.92
0.93
0.94
0.95
0.96
0.97
qz/ nm−1
b)
−0.3 −0.2 −0.1 0.0 0.1 0.2 0.3
qx/ nm−1
0.90
0.91
0.92
0.93
0.94
0.95
0.96
0.97
qz/ nm−1
c)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Reflectivity / sr−1
×10−5
Figure 5.4 |
Measured intensity map of a detector scan of the Mo/B
4
C/Si/C multilayer mirror at an
angle of incidence
αi=6.75◦
(a) and obtained through rocking scans at an opening angle between
detector and incident beam of
∆Θ =13.5◦
(b) and
∆Θ =30◦
(c). The area close to the specular axis
was excluded from the datasets due to its strong intensity compared to the diffuse scattering shown
here. The detector scan (a) was performed at an angle of incidence of
αi=6.75°
moving the detector
from
αf=−3.75°
to
αf=46.75°
(corresponding to detector angles from
∆Θ =3.0°
to
∆Θ =40.0°
)
in steps of
0.5°
. The access to the negative
qx
-axis in (a) was limited due clipping of the incoming
beam with the detector. The angular ranges for the rocking scan (b) with opening angle
∆Θ =13.5°
correspond to angles of incidence from
αi=−18.0°
to
αi=31.5°
in steps of
∆αi=0.5°
. In terms of
the rocking angle
ω
this range corresponds to values from
ω=−24.75°
to
ω=24.75°
, where
ω=0.0°
corresponds to the specular reflection geometry (
αi=αf
). For the second rocking scan geometry (c)
with
∆Θ =30.0°
, the angle of incidence was varied from
αi=−3.0°
to
αi=27.5°
(corresponding to
ω=−18.0°
to
ω=12.5°
) in steps of
0.5°
. At each angular position of the aforementioned angular
scan geometries, a wavelength scan between
12.4 nm
and
14.0 nm
was conducted using a step size of
∆λ=0.01 nm
. A GaAsP photo diode with an active area of
4.5 mm ×4.5 mm
at a distance to the sample
of 250 mm was used as a detector for the diffusely scattered radiation.
98
Near-normal Incidence Diffuse Scattering 5.1
modulations of the scatter intensity are caused by the direction from which the radiation
impinges on the multilayer structure itself, rather than the roughness properties. We shall
therefore investigate this observation to give an indication for the results found here.
5.1.2 Kiessig-like Peaks and Resonant Effects
To explain the observed off-specular intensity distribution for the multilayer sample,
additional effects exceeding the description of Bragg sheets need to be taken into account.
So far, the description of diffuse scattering and enhancement due to correlated roughness
was under the assumption of kinematic scattering, i.e. a single diffuse scattering event.
However, multiple reflections at the interfaces may not be ignored. To clarify that, we
shall consider two additional processes, which may happen before and after a diffuse
scattering event at the interface. Fig. 5.5illustrates two situations, where the impinging
or exiting (diffusely scattered) radiation is in resonance with the multilayers Bragg
condition, i.e. a situation of strongly enhanced in intensity. In the first case (a), the
z
layer j+2
layer j−1
layer j
layer j+1
resonant reflection
diffuse scattering
resonant reflection
impinging radiation
impinging radiation
qzqz
diffuse scattering
a) b)
Figure 5.5 |
Illustration of dynamic scattering processes. In (a), the impinging radiation is resonantly
reflected from the multilayer structure by fulfilling the Bragg condition. In (b), certain parts of the
diffusely scattered radiation from the interface roughness again fulfills the Bragg condition and is
enhanced in intensity.
impinging radiation fulfills the Bragg condition with respect to angle of incidence and is
consequently resonantly reflected from the multilayer mirror. Through this, any diffusely
scattered radiation measured at any exit angle would be significantly stronger compared
to the situation, where the incidence angle or wavelength does not fulfill the Bragg
condition, despite the fact that the roughness itself did not change. In the second case (b),
depending on the wavelength some of the diffusely scattered radiation fulfills the Bragg
condition of the multilayer and is again reflected resonantly from it causing a major
intensity increase. These two processes are a special case of the two situations considered
more generally within the DWBA theory illustrated in Fig. 2.7, termed
RT∗
and
TR∗
.
It should be noted, however, that the processes described there take into account any
reflection and transmission at the respective interface. Here, the focus is on the case,
where either reflection fulfills the Bragg condition and, thus, is resonantly enhanced.
The effects seen here are the result of multiple (dynamic) reflections inside the multi-
layer system. They were observed as resonantly enhanced streaks, so-called Bragg-like
lines, and intense Bragg-like peaks. The latter case occurs, where both the conditions
illustrated in Fig. 5.5are fulfilled simultaneously, i.e. where the Bragg-like lines cross
each other. These two phenomena were often observed in diffuse scattering maps from
multilayer samples recorded in grazing incidence geometry with X-rays [65]. The
theoretical principle leading to these off-specular enhancements is also known as the
99
Chapter 5 ANALYSIS OF INTERFACE ROUGHNESS BASED ON DIFFUSE SCATTERING
process of Umweganregung [14,15]. As the fulfillment of the Bragg condition for each
Bragg-like line is only dependent on two of the three experimental parameters, i.e. either
the incidence angle
αi
or the exit angle
αf
, in both cases together with the wavelength.
The position of those enhancements is different in the reciprocal space map depending
on the measurement geometry. In literature [14,15,65,96], such enhancements were so
far only observed from the main Bragg resonance of the multilayer, i.e. the fulfillment of
the Bragg condition of the periodic stack. In our case, no higher-order Bragg resonances
can be observed, as they would appear as Bragg-like peaks in the off-specular scattering
far away from the accessible
qk
range of our experiment. However, the two Bragg-like
lines corresponding to the first order Bragg peak cross at the position of the specular
reflex and otherwise amount to broad bands in the diffuse map as elaborated in the
following paragraphs.
Figure 5.6 |
Measured
reflectivity curve of the
Mo/B
4
C/Si/C multilayer
mirror at an angle of in-
cidence
αi=6.75◦
. The
solid black lines mark
the positions of the first
two Kiessig-fringes at
each side of the main
maximum. The dashed
lines indicate the full
width at half maximum
(FWHM) position of the
main Bragg peak.
12.5 13.0 13.5 14.0 14.5
λ/ nm
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
reflectivity
measured data
Apart from the main Bragg peak, additional resonances are observed in the EUV
reflectivity curve as shown in Fig. 5.6(marked with solid vertical lines). Those side
peaks, known as Kiessig fringes [74], correspond to the interference of radiation reflected
from the top surface and the substrate interface, as previously discussed in Sec. 4.1.
The dynamic enhancement, equivalent to the Bragg-like lines and Bragg-like peaks for
the main maximum, expected for those side fringes is very well within the measured
reciprocal space ranges of our measurements geometries and wavelengths. In analogy to
the names given to those effects originating from the main Bragg resonance, they shall be
called Kiessig-like lines and Kiessig-like peaks here. In Fig. 5.7, the positions where those
enhancements are to be expected in the maps (shown originally in 5.4) are indicated as
white solid lines for the first two fringes on either side of the reflectivity curve maximum.
In addition, the FWHM of the main Bragg maximum was marked with dashed lines,
both in Fig. 5.6and in Fig. 5.7to indicate the limits of the two aforementioned Bragg-like
lines observable in this scattering map.
Clearly, the off-specular enhancement observed in Fig. 5.7a and 5.7c fits to some of
the theoretically predicted appearances of the Kiessig-like peaks, i.e. at the crossing
points of the Kiessig-like lines (white solid lines). However, at other crossing points or in
Fig. 5.7b no strong visible enhancement appears. The reason for that is, that the diffuse
scattering map is the result of several overlapping effects. A strong enhancement is only
observed where, in addition to the Kiessig-like peaks, also a Bragg sheet, due to correlated
roughness in the sample, appears. The intensity distribution along
qx
for the Bragg sheet,
as outlined above, is governed by the PSD and decays with increasing absolute values
100
Near-normal Incidence Diffuse Scattering 5.1
−0.3 −0.2 −0.1 0.0 0.1 0.2 0.3
qx/ nm−1
0.90
0.91
0.92
0.93
0.94
0.95
0.96
0.97
qz/ nm−1
c)
0.90
0.91
0.92
0.93
0.94
0.95
0.96
0.97
qz/ nm−1
b)
0.90
0.91
0.92
0.93
0.94
0.95
0.96
0.97
qz/ nm−1
a)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Reflectivity / sr−1
×10−5
Figure 5.7 |
Measured intensity maps of Fig. 5.4 with the calculated positions of the Kiessig-like lines
(solid lines) for the Kiessig fringes marked in Fig. 5.6 and the Bragg-like lines (bands between the
dashed lines). The positions, where the solid lines cross show the Kiessig-like peaks positions. The
area contained within the dashed lines in the center of each plot correspond to the Bragg-like peak of
the first Bragg order of the multilayer and explain the triangular or diamond shaped area of increased
intensity (see main text).
101
Chapter 5 ANALYSIS OF INTERFACE ROUGHNESS BASED ON DIFFUSE SCATTERING
of
qx
in positive and negative direction. Consequently, while an enhancement due to
Kiessig-like peaks also exists in Fig. 5.7b, their positions are at larger positive and negative
qx
values, where the intensity of the Bragg sheet has already decayed. A similar case can
be made for Kiessig-like peaks far away from the vertical, i.e.
qz
, position of the Bragg
sheet. As discussed above, highly correlated roughness limits the width
δqz
of the sheet.
Thus, Kiessig-like peaks above or below the
qz
position of the sheet, where its intensity
has dropped, may cause enhancement, but it is below the detection threshold.
The aforementioned broad bands corresponding to the Bragg-like lines of the main
Bragg resonance appear in between the dashed lines. Indeed, most prominently visible
in Fig. 5.7b, the triangular shaped intensity distribution in the center of the map is in
fact the result of resonant enhancement due to the first order Bragg-like peak, which
extents across a large area of the map in this case. The diffuse scattering distribution
in the reciprocal space maps is thus a combination of dynamic effects (the first-order
Bragg-like peak and the Kiessig-like peaks) and kinematic effects (Bragg sheets).
As indicated above, the processes described here are contained in the theoretical
description given in Eq.
(2.54)
in Sec. 2.4. They correspond to the contributions of
the DWBA differential cross section through the processes shown in Fig. 2.7, labeled
RT∗
and
TR∗
(Kiessig-like lines, Bragg-like lines) and
RR∗
(Kiessig-like peaks, Bragg-
like peaks). The Bragg-sheets, however, are described as a simple fulfillment of the
Bragg condition due to the momentum transfer at the interfaces according to the semi-
kinematic description labeled
TT∗
. In order to assess the contribution of dynamic
multiple reflections within the stack, the semi-kinematic approximation in Eq.
(2.39)
was compared with the dynamic calculations in Eq.
(2.54)
. In the semi-kinematic case,
all multiple reflection effects are ignored in the differential cross section. The result is
the intensity distribution as expected from the kinematic case, however including the
accurate transmitted field amplitudes at each interface instead of only a plane wave field
amplitude as in the simple Born approximation.
To evaluate and illustrate the contribution of multiple (dynamic) reflections due to
the subsidiary maxima in comparison to the semi-kinematic case, which ignores those
effects. Fig. 5.8b shows a calculated intensity distribution along
qx
at
qz=0.93
nm
−1
for the sample investigated here, employing the theoretical framework of the DWBA, as
introduced in Sec. 2.4. This calculation corresponds to a horizontal cut at
qz=0.93
of the
measured reciprocal space map shown in Fig. 5.7c, i.e. the rocking scan geometry with
an opening angle of
∆Θ =30°
. The structural parameters used in this calculation were
determined in Sec. 4.1for this sample. At this point, no explanation was given yet on how
the parameters of the PSD, required to perform this calculation, were obtained. Instead, to
first emphasize the origin and impact of the dynamic effects, this will be postponed here
and discussed in detail in the following Sec. 5.1.3of this chapter. The EUV reflectivity
curve with the marked positions of the Kiessig fringes and the FWHM of the main
Bragg peak are repeated in Fig. 5.8a for reference. The solid blue line corresponds to the
dynamic theory, while the dotted blue line is the result of the semi-kinematic calculation.
The dashed vertical lines indicate the limits of the main Bragg peaks FWHM. The vertical
black lines show the position of the Kiessig-like lines intersecting the cut position. Each
of the marked fringes in Fig. 5.8a appears on the negative and positive
qx
-axis in Fig. 5.8b.
This is caused by the incidence and exit angle, respectively, being at the resonance angle
of the various Kiessig maxima in the reflectivity curve as illustrated in Fig. 5.5. A strong
increase with respect to the semi-kinematic approximation is observed. The position
of the dynamic contribution from the first Kiessig fringes on either side of the main
102
Near-normal Incidence Diffuse Scattering 5.1
−0.2 −0.1 0 0.1 0.2
qx/ nm−1
0
1
2
3
4
5
6
reflectivity ×10−5/ sr
b) dynamic
semi-kinematic
12.6 13.0 13.4 13.8
λ/ nm
0
0.2
0.4
0.6
0.8
reflectivity
a) measured data
Figure 5.8 |
a) EUV reflectivity curve with the positions of the FWHM of the Bragg peak (dashed black
lines) and the positions of the first two Kiessig fringes on each side of the main maximum (solid black
lines) similar to Fig. 5.6. b) Calculated scattering intensity distribution at
qz=0.93
nm
−1
. The solid blue
line shows the result of the dynamic calculation for a rocking scan with an opening angle of
∆Θ =30◦
.
The dashed blue line represents the calculation applying the semi-kinematic approximation, ignoring
any multiple reflections within the multilayer. The dashed vertical lines are the position of the main
Bragg peaks FWHM, while the solid vertical lines show the position of dynamic contributions of the
Kiessig fringes close to the main maximum. Each Kiessig fringe marked in the inset appears for the
corresponding positive and negative
qx
value. The strong intensity at
|qx| ≈ 0.1
nm
−1
results from the
overlap of the dynamic maxima of two different Kiessig fringes (see text).
resonance exhibits a pronounced maximum in the diffuse scattering. These fringes
contribute most due to their high overall relative intensity compared with the fringes
further away from the reflectivity maximum. In addition, the position in reciprocal space
coincides with the first two Kiessig fringes marked on either side of the main maximum.
The contribution by the main Bragg resonance, i.e. the Bragg-like peak amounts to
approximately
100%
intensity increase at
qx=0
. The comparison to the semi-kinematic
case reveals another reason for the strong intensity of the Kiessig-like peaks compared
to the Bragg-like peak. In between the dashed lines on the positive and negative
qx
axis in Fig. 5.8b, a significant decrease of kinematically scattered radiation is observed.
The reason for that is a strongly diminished penetration depth of the radiation into the
multilayer at the Bragg resonance, which causes less rough interfaces to contribute to
the diffuse scattering. This directly counteracts the resonant enhancement due to the
Bragg-like peak and leads to an overall lower scattering contribution at these positions in
reciprocal space.
Similarly to the calculated intensity distribution for a horizontal cut, a vertical cut
at fixed
qx
position further emphasizes the importance of taking dynamic effects into
account. Fig. 5.9shows that line cut, perpendicular to the one shown in Fig. 5.8, along the
qz
at
qx=0.05
nm
−1
assuming a measurement geometry corresponding to the rocking
scan with opening angle
∆Θ =30°
(cf. Fig. 5.7c). Again, the structural data was taken
from the analysis in Sec. 4.1. The results of the calculation including the dynamic effects
show distinct differences with an increase up to
100%
of the calculated scattered intensity
close to the multilayer resonance at
qz=0.93
nm
−1
compared to the semi-kinematic
calculation. Hence the dominance of the dynamic contributions in the vicinity of the
103
Chapter 5 ANALYSIS OF INTERFACE ROUGHNESS BASED ON DIFFUSE SCATTERING
Figure 5.9 |
Calculated
scattering intensity
along a vertical cut in
qz
with fixed
qx=0.05
nm
−1
for the dynamic
and semi-kinematic cal-
culations for a rocking
scan of the Mo/B
4
C/Si/C
sample at ∆Θ =30◦.
0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98
qz/ nm−1
0
0.5
1.0
1.5
2.0
2.5
3.0
Intensity (I/I0)×10−5
dynamic DWBA,
ξ⊥(0.05) = 3004 nm
dynamic DWBA,
ξ⊥(0.05) = 100 nm
semi-kinematic DWBA,
ξ⊥(0.05) = 3004 nm
Bragg resonance is also observed here. In addition to comparing the dynamic and semi-
kinematic calculations, a dynamic calculation assuming a reduced vertical correlation of
roughness was added as dashed blue curve. As discussed in the beginning of Sec. 5.1, the
Bragg sheet width is strongly dependent on the amount of correlated interfaces. Clearly,
a broadening and reduction of scatter intensity is seen for this case here (dashed line
in Fig. 5.9). This shows, that the Bragg sheet is in fact still visible but obscured by the
dominant structure in the diffuse scattering caused by the dynamic effects explained
above.
5.1.3 Reconstruction of the PSD and the Multilayer Enhancement
Factor
Within the framework of the DWBA, considering the dynamic effects, the full expression
for the differential cross section of the diffuse scatter is given in Eq.
(2.54)
. As discussed
above, the power spectral density only becomes accessible through the diffuse scattering
measurements, if the structural properties of the multilayer are known. Those were
determined for all samples in this thesis with the methods described in chapter 4. Based
on those results, the differential cross section allows to calculate the scattering intensity
maps, which were measured here and therefore enable the reconstruction of the PSD.
The large impact of resonant effects on the off-specular scattering intensity, measured
in the three geometries shown above, prove that multiple reflections have to be taken into
account to extract the contribution of the interface morphology and determine a PSD.
To better understand the effects involved here, we shall analyze the intensity curves for
all three measurement geometries based on a horizontal cut along
qx
at the position of
qz=0.93nm−1
, which corresponds to the momentum transfer at the multilayer resonance.
As illustrated in the schematic explanation in Fig. 5.3, this coincides with the horizontal
position of the the Bragg sheet and its maximum intensity. In the case shown there,
the PSD could just be extracted from a horizontal cut of the scatter intensity at that
position. This has been done by Siffalovic et al. [123], for example, in the case of GISAXS
measurements. Similarly to this approach, the calculated intensity curves corresponding
to such a horizontal cut in the three geometries measured here including the dynamic
104
Near-normal Incidence Diffuse Scattering 5.1
effects, are shown in direct comparison in Fig. 5.10. The strong off-specular enhancement
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4
qx/ nm−1
0
1
2
3
4
reflectivity / sr−1
×10−5
∆Θ =30◦
∆Θ =13.5◦
αi=6.75◦
Figure 5.10 |
Averaged
diffuse scattering inten-
sity along
qx
in the inter-
val
qz= (0.930 ±0.003
)
nm
−1
corresponding to
the resonance of the
multilayer. The data
shown are two rock-
ing scan and one detec-
tor scan geometries (see
text for details).
of scattering intensity obstructing the underlying PSD is clearly visible here for the
detector scan geometry and the rocking scan with opening angle of
∆Θ =30.0°
. In case
of the second rocking scan with
∆Θ =13.5°
, only a small shoulder can be observed at
qx≈ ±0.2nm−1.
In the theoretical treatment of the diffuse scattering in Sec. 2.4, an expression for the
differential cross section based on the DWBA in Eq.
(2.54)
was derived. It separates the
dynamic enhancements and penetration depth considerations from the power spectral
density contribution. It can be divided in two parts of interest. The factor contained in
rectangular brackets is the dynamic and kinematic part due to the scattering properties
from a multilayer and only dependent on the multilayer layout and vertical roughness
correlation, we shall therefore refer to it as multilayer enhancement factor. The remaining
term,
C(qx)
, is the average power spectral density and describes the average interface
morphology.
To illustrate the impact due to the presence of the multilayer and the geometry
dependence, the result of calculations of the multilayer enhancement factor alone, based
on the layer model of our multilayer sample, is shown in Fig. 5.11 for the detector
scan and the two rocking scan configurations. The multilayer enhancement factor was
normalized with respect to
qx=0
, i.e. the calculated diffuse scattering contribution
on the specular axis. It should be noted here, that the abrupt decrease observed for
each of the curves towards higher
qx
values is not the result of a breakdown of vertical
correlation. Instead, it marks the point in reciprocal space for each geometry, respectively,
where the photon energy is in resonance with the Si-L edge causing a strong increase of
absorption and thus a sharp decrease of the penetration depth into the multilayer. As a
result, diffuse scattering intensity is decreased significantly.
The results of the calculation above show that the diffuse scattering from these multi-
layer mirror systems at near-normal incidence exhibits strong enhancement due to the
intrinsically limited bandpass of reflectivity and high reflectance. If both the incidence
and exit angle is out of the Bragg resonance, the higher penetration depth of the multi-
layer causes an increase in the number of interfaces contributing to the diffuse scattering
intensity. Thus higher total scattering is observed. The Kiessig fringes and the main Bragg
105
Chapter 5 ANALYSIS OF INTERFACE ROUGHNESS BASED ON DIFFUSE SCATTERING
Figure 5.11 |
Enhance-
ment factor due to the
specific properties of
multilayer reflectivity
for three different mea-
surement geometries.
The simulations shown
here were normalized
with respect to the
diffuse contribution to
the specular reflectivity
at qx=0.
−0.4 −0.2 0 0.2 0.4
qx/ nm−1
0
1
2
3
4
5
6
7
rel. multilayer enhancement / a. u.
rocking scan, ∆Θ =30◦
rocking scan, ∆Θ =13.5◦
detector scan, αi=6.75◦
peak cause modulations in the enhancement factor resonantly increased by the purely
dynamic processes described in the previous section. Based on these calculations, the
cuts along
qx
in the measured maps shown in Fig. 5.10 could be normalized by diving the
measurement through the calculated multilayer enhancement factor to extract the PSD of
the sample. The result is shown in Fig. 5.12 for the positive
qx
range. Clearly, this result
Figure 5.12 |
Diffuse
scattering intensity cor-
rected for the multi-
layer enhancement fac-
tor. The blue solid
line corresponds to a
power spectral density
with
ξk=5.6
nm,
H=1.0
,
σ=0.2
nm
and a vertical correla-
tion length of
ξ⊥(qx) =
7.5/q2
xnm−1.
10−210−1100
qx/ nm−1
10−1
100
101
102
PSD / nm4
PSD best model
rocking scan, ∆Θ =30◦
rocking scan, ∆Θ =13.5◦
detector scan, αi=6.75◦
shows a consistent determination for the PSD independent of the measurement geometry
applied. The individual cuts are in agreement within the measurement uncertainty.
Based on the calculation of the multilayer enhancement factor, experimental curves for
the PSD can be extracted as shown in Fig. 5.12 without applying a specific model for
the interface morphology. However, the measurements conducted here only deliver
data in a limited range in the reciprocal space, depending on the selected geometry and
wavelengths. To characterize the interface morphology, it is therefore necessary to model
the measured data and deduct parameters that relate to the roughness properties. To
obtain the PSD best model reconstruction, the PSO method was employed similarly to
the reconstructions shown in chapter 4.
106
Near-normal Incidence Diffuse Scattering 5.1
Reconstruction of the Power Spectral Density
It is the goal of this analysis to deduct key properties of the interface roughness, such
as vertical and lateral correlation lengths and the r.m.s. roughness value
σr
. The latter
is directly related to the Névot-Croce parameter
σ
, which was introduced in Sec. 2.3
and determined in the structural reconstruction of chapter 4. There, the roughness
is described using this factor. However, intermixing at the interfaces is additionally
contained as it can not be distinguished from the roughness. On average of the beam
footprint of the specular and fluorescence methods described there, both effects lead
to the same decrease in sharpness of the interfaces. Based on the analysis of the PSD
through the diffuse scattering analysis, this distinction can be made. To reconstruct the
PSD, a suitable model has to be introduced for the interface morphology. Here, a fractal
interface model is applied, which was found to adequately describe the roughness in case
of sputter deposited multilayer systems [22,24,125]. It should be noted, that the PSD
for a two dimensional surface should be two-dimensional itself and consider possibly
different roughness properties in
x
(
qx
) and
y
(
qy
) direction. The samples investigated
here, however, are fabricated using magnetron sputtering and on rotating sample holders
as shown in Sec. 3.4.2. This is important to achieve a homogeneous deposition. It is
therefore concluded, that roughness on the surfaces and interfaces does not have any
predominant direction and may be assumed to be isotropic, i.e. only dependent on the
absolute value of the lateral momentum transfer vector
qk=qq2
x+q2
y
. The PSD can
then be expressed in the closed analytical one-dimensional form as and was introduced
in Sec. 2.4and explicitly given in Eq.
(2.47)
. The three parameters describing the fractal
nature of the roughness are the lateral correlation length
ξk
, the r.m.s. roughness
σr
and
the Hurst factor
H
. The vertical correlation of the roughness parameter
ξ⊥
and the off-
axis roughness correlation angle
β
, defined through Eq.
(2.51)
, Eq.
(2.55)
and Eq.
(2.52)
,
however, are not included in the PSD as they are part of the multilayer enhancement
factor. Illustrations and explanations of the meaning and effect of these parameters can
be found in Sec. 2.4. In order to fully characterize the system, the full data set comprising
all data points measured for the reciprocal space maps is analyzed. As explained above,
the maps were measured by performing wavelength scans at each angular position of
the rocking or detector scans. The result are intensity curves
I(αi,αf)(λ)
, for each set of
angular positions in dependence on the wavelength. The minimization functional
˜
χ2
for
each of the three experiments (three diffuse scattering maps), is thus given by
˜
χ2=1
M−P∑
(αi,αf)
∑
mImodel
m(αi,αf,λ)−Imeas
m(αi,αf,λ)2
˜
σ2
m
, (5.1)
where
M
is the total number of measurement points,
P
is the number of optimization
parameters,
(αi,αf)
indicates a specific position in the angular detector or rocking
scans and
˜
σ2
m
denotes the experimental uncertainty of measurement point
m
. The
reconstruction was achieved by applying the structural reconstruction from Sec. 4.1and
the PSO technique on the combined set of measurements from all three experiments,
i.e. minimizing the functional
χ2=˜
χ2
a+˜
χ2
b+˜
χ2
c
. The letter indices a, b and c refer to the
reciprocal space maps shown in Fig. 5.4. The optimization model parameters are listed in
table 5.1together with the converged results found. In Fig. 5.13, the measured reciprocal
space maps in the detector scan geometry and the rocking scan geometry are shown in
direct comparison with the theoretically calculated maps based on the best model results.
107
Chapter 5 ANALYSIS OF INTERFACE ROUGHNESS BASED ON DIFFUSE SCATTERING
Table 5.1 |
Parameters of the DWBA analysis. The lower bound (LB) and upper bound (UB) specify the
PSO parameter space limits.
Parameter Definition LB UB PSO result
σr/ nm root mean square roughness 0.0 1.0 0.201
ξk/ nm lateral correlation length 0.0 20.0 5.579
ξ⊥/ nm−1
vertical correlation parameter yielding ver-
tical correlation length trough
˜
ξ⊥(qk) =
ξ⊥/q2
k
0.0 20.0 7.512
HHurst factor 0.0 1.0 1.000
β/◦
angle for off-axis vertical roughness corre-
lation −10.0 10.0 −0.152
−0.2 −0.1 0.0 0.1 0.2 0.3
qx/ nm−1
d) DWBA calculation
−0.1 0.0 0.1 0.2 0.3
qx/ nm−1
0.90
0.91
0.92
0.93
0.94
0.95
0.96
0.97
qz/ nm−1
c) DWBA calculation
b) measured data
0.90
0.91
0.92
0.93
0.94
0.95
0.96
0.97
qz/ nm−1
a) measured data
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Reflectivity / sr
×10−5
Figure 5.13 |
Measured reciprocal space maps for the detector scan geometry (a) and the rocking scan
at an opening angle of
∆Θ =30◦
(b). The corresponding calculated maps based on the PSO results are
shown in direct comparison in (c) and (d) for the respective scan geometries.
The calculated reciprocal space maps are in good agreement with the measured data.
The results reveal a strong vertical correlation of the roughness throughout the multilayer
stack. Indeed, the correlation length parameter
ξ⊥=7.512nm−1
suggests, that the
roughness correlation extends across the whole multilayer stack up to spacial frequencies
of
qk≈0.13nm−1
. The total stack thickness based on the structural reconstruction of the
individual layers and the periodicity with a multiplication by
n=65
is
Dtot =455nm
.
Using the relation
˜
ξ⊥(qk) = ξ⊥/q2
k
, the perpendicular correlation length of roughness
can be calculated to be
˜
ξ⊥(0.128nm−1)≈458nm
. For higher values of that spacial
frequency
qk>0.128
the correlation length reduces to values lower than the total stack
thickness. This is physically plausible, as higher spacial frequency roughness replicates
worse throughout the stack upon deposition of the layers than low spacial frequency
roughness as indicated in the calculation in Sec. 2.4for vertical roughness correlation.
Apart from the vertical correlation observed, the average PSD parameters obtained
108
Differently Polished Mo/Si/C Multilayers with Molybdenum Thickness Variation 5.2
show a r.m.s. roughness of
σr=0.201nm
, which is in agreement with the value
σ=
0.214nm(−0.143nm/+0.201nm)
obtained in the MCMC analysis conducted in Sec. 4.1.1
for the Névot-Croce parameter. In thus may be concluded that roughness is the dominant
disturbance relevant for diminished reflectivity for that sample and the interdiffusion
barriers provide effective means to hinder intermixing.
In conclusion, the analysis of diffuse scatter presented here provides a powerful
method for the reconstruction of the average PSD of the interfaces inside the multilayer.
In comparison to techniques such as AFM, which solely measure at the top surface,
it can deliver data on the interface properties inside the multilayer. In addition it
provides information on a large area of the surface and the interfaces. The near-normal
incidence angles used in the measurement allow to study potentially strongly curved
multilayer mirrors, which are often implemented in optical setups, and thus provides
an advantage to established grazing-incidence methods of measuring diffuse scattering.
Due to the experimental access to the interface morphology based on this technique,
the assessment of which interface disturbances cause a loss of reflectivity compared to
simulations based on perfect chemically abrupt interfaces provides interesting insights
on the sample properties and extends the capabilities of characterization established in
chapter 4. An analysis of the confidence intervals for the respective PSD and correlation
parameters will be additionally given for the Mo/Si/C and Cr/Sc sample systems in
the following sections. With the determination of the roughness properties introduced
here, the improvement of the fabrication of such optics may become possible, knowing
which effects need to be counteracted to reach higher reflectivities. Parts of the results
of the analysis in Sec. 4.1and the findings of this section were published in A. Haase,
V. Soltwisch, C. Laubis, and F. Scholze: ‘Role of dynamic effects in the characterization of
multilayers by means of power spectral density’. In: Appl. Opt.
53
.14 (2014), pp. 3019–
3027.doi:10.1364/AO.53.003019.
5.2 Differently Polished Mo/Si/C Multilayers with Molyb-
denum Thickness Variation
In Sec. 4.2, the multilayer model of two sample sets of polished and unpolished Mo/Si/C
multilayer mirrors with a varying relative thickness of the molybdenum layer from sample
to sample was reconstructed. The findings there show the appearance of significant
drops in the peak reflectivity at certain thickness values correlated with jumps in the
total period thickness, different depending on to which set, polished or unpolished, the
samples belong to. Here, we shall apply the method to analyze the diffuse scattering
detailed above to the two sample sets investigated in the previous chapter. The goal
of this is to assess the effect of the presumed crystallization at a certain molybdenum
thickness threshold on the interface morphology and, thus, investigate the origins of the
reflectivity drops that are shown in Fig. 4.14.
For that purpose, only for selected samples in the vicinity of the presumed crystal-
lization threshold in both sets, as well as far away from that molybdenum thickness
range, the diffuse scattering maps analogous to the previous section were measured. The
respective samples are marked with open circles in Fig. 4.13b. In both cases, scattering
maps were taken from the samples with lowest and highest Mo layer thickness, respec-
tively, in addition to maps taken from the samples with Mo thicknesses right before, at
and right after the presumed crystallization threshold. Table 5.2lists the reconstructed
109
Chapter 5 ANALYSIS OF INTERFACE ROUGHNESS BASED ON DIFFUSE SCATTERING
molybdenum thicknesses corresponding to these samples derived in Sec. 4.2.3.
Table 5.2 |
List of the
reconstructed molybde-
num layer thicknesses
in the selected samples
in both sets investigated
with the diffuse scatter-
ing analysis in relation
to the nominal thick-
ness.
nominal reconstructed dMo / nm reconstructed dMo / nm
dMo / nm (unpolished) (polished)
1.70 1.81(−0.12/+0.24)1.77(−0.22/+0.19)
1.85 - 1.91(−0.12/+0.17)
2.00 - 2.29(−0.28/+0.13)
2.15 2.31(−0.22/+0.21)-
2.30 2.43(−0.09/+0.16)2.60(−0.12/+0.14)
2.45 2.68(−0.13/+0.16)-
2.60 - -
2.75 - -
2.90 3.22(−0.13/+0.11)-
3.05 - 3.47(−0.19/+0.13)
All selected samples were measured in the rocking scan geometry with an opening
angle of
∆Θ =30°
. This is analogous to the measurement of the Mo/B
4
C/Si/C sample
shown in Fig. 5.4c. In that geometry, a large off-specular increase due to the Kiessig-like
peaks was observed. Due to that enhancement, the measured intensity is stronger and
further away from the detection threshold of the photodiode. However, as shown above,
any other geometry would be equivalently applicable. As discussed in the previous
section, it is sufficient to measure only one half space of the maps shown there as the PSD
only depends on the absolute value of
qx
by the assumption of isotropic roughness in all
directions lateral to the interfaces. Thus, the interface morphology may be reconstructed
based on this smaller data set reducing the experimental effort. The resulting maps are
shown in the reciprocal space representation for both sets in comparison in Fig. 5.14. The
0.0
0.4
0.8
1.2
1.6
2.0
2.4
2.8
3.2
3.6
4.0
reflectivity
×10−5
×10−5
−0.4−0.3−0.2−0.1 0.0
qx/ nm−1
0.88
0.90
0.92
0.94
0.96
qz/ nm−1
dMo =1.81 nm
a) unpolished samples
−0.3−0.2−0.1 0.0
qx/ nm−1
dMo =2.31 nm
−0.3−0.2−0.1 0.0
qx/ nm−1
dMo =2.43 nm
−0.3−0.2−0.1 0.0
qx/ nm−1
dMo =2.68 nm
−0.3−0.2−0.1 0.0
qx/ nm−1
dMo =3.22 nm
−0.4−0.3−0.2−0.1 0.0
qx/ nm−1
0.88
0.90
0.92
0.94
0.96
qz/ nm−1
dMo =1.77 nm
b) polished samples
−0.3−0.2−0.1 0.0
qx/ nm−1
dMo =1.91 nm
−0.3−0.2−0.1 0.0
qx/ nm−1
dMo =2.30 nm
−0.3−0.2−0.1 0.0
qx/ nm−1
dMo =2.59 nm
−0.3−0.2−0.1 0.0
qx/ nm−1
dMo =3.47 nm
Figure 5.14 |
Measured diffuse scattering distributions in reciprocal space representation shown on
linear false-color scale. The selected unpolished samples are shown in a) with increasing Mo layer
thickness
dMo
. The selected samples for the polished set are shown in b) also in order of increasing
Mo thickness
dMo
. The samples with strongest scattering are shown in larger detail in Fig. 5.16. The
diffuse scattering was measured by keeping the detector angle with respect to the incoming beam
fixed at
∆Θ =30◦
, while the sample was tilted from an AOI of
αi=15◦
to
αi=38◦
with a step size
∆αi=0.5◦
. At each angular position, a wavelength scan from
λ=12.35
nm to
λ=14.0
nm in steps of
∆λ=0.01 nm was performed to map the diffuse scattering distribution.
110
Differently Polished Mo/Si/C Multilayers with Molybdenum Thickness Variation 5.2
maps in Fig. 5.14a show the scattering distribution from the unpolished samples marked
with the fitted Mo layer thickness as listed in table 5.2. The polished samples are shown
in Fig. 5.14b.
A very prominent observation in both sets, is that one sample in each series shows
significantly stronger overall scattering than the others. In addition, both sets show
distinctly different scattering distributions clearly differentiating the polished from the
unpolished samples. In the case of the polished samples, significantly less scattering
than for the unpolished ones can be observed for higher spatial frequencies
qx
, whereas
more intensity is measured for smaller frequencies. A recognizable characteristic of the
off-specular scattering intensity is the observation of a downward tilted Bragg sheet
in case of the unpolished samples, which is in contrast to the rocking scan map of
the unpolished Mo/B
4
C/Si/C sample from Sec. 5.1. This is due to a non-orthogonal
roughness correlation throughout the stack with respect to the surface and interfaces
first observed by Gullikson and Stearns [57]. The theoretical aspects of this effect were
discussed in Sec. 2.4, but we shall investigate this behavior for the specific set of samples
studied here.
The downward tilt of the Bragg sheet is clearly observed for all samples in the
unpolished series with a similar direction. All samples were measured along the same
nominal
x
axis, i.e. along the same direction with respect to their mounting orientation
during the deposition process. Due to in-plane measurement of the diffuse scattering, the
non-orthogonal roughness correlation angle
β
can only be evaluated along the projection
of its directional vector onto the
x
-
z
-plane. However, the vertical correlation direction
vector may not necessarily lie in that plane. To verify this property, we shall investigate
the corresponding diffuse scattering distribution from the strongest scattering sample
with
dMo =2.43nm
by rotating it by
90°
around its surface normal onto the sample
holder and repeat the mapping of reciprocal space. Fig. 5.15 shows the comparison of
the map obtained earlier with the map from the rotated sample. The tilt direction is
−0.3 −0.2 −0.1 0.0
qk/ nm−1
0.88
0.90
0.92
0.94
0.96
qz/ nm−1
a) measured data
dMo =2.43 nm
−0.3 −0.2 −0.1 0.0
qk/ nm−1
b) measured data
dMo =2.43 nm
rot. 90◦
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Reflectivity
×10−5
Figure 5.15 |
a) Diffuse scattering map for the unpolished sample (
d=2.43 nm
) with strongest total
scattering intensity measured at the same orientation as in Fig. 5.14. b) Corresponding reciprocal
space map for the same sample but irradiated from a different angle by rotating the sample by
90°
around the surface normal. Clear differences in the tilt angle of the Bragg sheet can be observed
associated with the vertical roughness correlation direction.
clearly different for the map of the rotated sample, where a similarly horizontal Bragg
sheet as for the Mo/B
4
C/Si/C sample in Sec. 5.1is obtained. Based on the evaluation
of the tilt angle in both maps, it is possible to deduce the direction and total angle
β
of the roughness correlation direction with respect to the surface normal and the
111
Chapter 5 ANALYSIS OF INTERFACE ROUGHNESS BASED ON DIFFUSE SCATTERING
orientation directions of the sample during the measurement. This angle is given by the
two orthogonally measured Bragg sheet tilt angles β0° and β90° through
tan2(β) = tan2(β0°) + tan2(β90°). (5.2)
These two independent measurements can be additionally used to verify the results of the
reconstruction. We shall thus perform the analysis described in Sec. 5.1and deduce the
PSD parameters including the vertical correlation length as well as the non-orthogonal
correlation direction for this sample in particular. For all other measured samples it
was proceeded in the same way, where here only the in-planar Bragg sheet tilt angle is
determined.
5.2.1 Reconstruction of the Interface Morphology
The theoretical analysis was performed based on the method described in the first part
of this chapter. Instead to applying the PSO method to reconstruct the parameters
characterizing the interface morphology, the MCMC procedure was applied to obtain
the optimized parameter values and their confidence intervals. The basic principle is
identical to that used in chapter 4and relies on the minimization functional stated in
Eq.
(5.1)
, which enters the likelihood according to the definition in Eq.
(4.4)
. This method
is computationally more challenging than applying only the PSO procedure, but becomes
possible with the smaller number of periods (
N=50
) for the samples investigated here,
since the effort scales with the order of
O(N2)
. As starting values, the walkers for the
MCMC algorithm were distributed randomly across the parameter space given by the
limits listed in table 5.1with the exception of the Hurst parameter. The latter is limited
between
0.8
and
1.0
, where the upper limit is the intrinsic theoretical limit representing
Gaussian type roughness. The measurements conducted here only allow a limited access
to the Hurst parameter, as it is determined by the asymptotic behavior of the PSD towards
higher lateral roughness frequencies. For that spacial frequency range, however, no data
exists as the vertical correlation of roughness is reduced and the detector threshold is
reached so no asymptotic data can be recorded. Others [109] have observed Hurst values
in that range for similar samples with values close to the case of Gaussian roughness.
The results for the Mo/B
4
C/Si/C sample shown in Fig. 5.12 are in good agreement
with these findings resulting in a Hurst factor of
H=1.0
. There, an overall higher
in-plane correlation length
ξk
compared to the unpolished samples here, allows a better
determination of the asymptotic behavior of the PSD. Due to these results, the samples
from both sets are analyzed by fixing the Hurst parameter to
H=1.0
, i.e. by applying
a roughness model for Gaussian roughness only. However, in the determination of the
confidence intervals, the range from
H=0.8
to
H=1.0
was considered to reflect this
uncertainty in the determination of the parameters.
The results of the ideal model for each sample system entering the DWBA calculation
were obtained from the analysis in Sec. 4.2.3. The optimization was conducted by
applying the MCMC method with respect to the vertical correlation length
ξ⊥
in the
vertical correlation function
c⊥(qk)
, the tilt angle
β
and all PSD parameters in
C(qk)
.
For the two samples, the maps with the strongest scattering from each set are shown in
comparison to the best model DWBA calculation found this way. The resulting maps in
Fig. 5.16 from the unpolished (a) and polished (c) samples show very good agreement
with the theoretical calculations in (b) and (d), respectively, including the tilted Bragg
sheet observed for the unpolished sample. All parameter values obtained from the
112
Differently Polished Mo/Si/C Multilayers with Molybdenum Thickness Variation 5.2
−0.3 −0.2 −0.1 0.0
qk/ nm−1
0.88
0.90
0.92
0.94
0.96
qz/ nm−1
a) measured data
dMo =2.43 nm
−0.3 −0.2 −0.1 0.0
qk/ nm−1
b) best model DWBA calculation
−0.3 −0.2 −0.1 0.0
qk/ nm−1
0.88
0.90
0.92
0.94
0.96
qz/ nm−1
c) measured data
dMo =1.91 nm
−0.3 −0.2 −0.1 0.0
qk/ nm−1
d) best model DWBA calculation
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Reflectivity
×10−5
Figure 5.16 |
Direct comparison of the measured reciprocal space maps with the DWBA calculation
resulting from the parameters obtained with the MCMC optimization procedure (see text). a) shows
the maps of the unpolished sample with strongest diffuse scattering. Similarly, b) shows the maps of
the polished sample at the respective presumed crystallization threshold with strongest scattering.
113
Chapter 5 ANALYSIS OF INTERFACE ROUGHNESS BASED ON DIFFUSE SCATTERING
MCMC optimization procedure are compiled in table 5.3.
Table 5.3 |
Results for the DWBA model parameters with the respective confidence intervals for both
sample sets.
nom. Mo thickness / nm σr/ nm ξk/ nm ξ⊥/ nm−1β/◦
(fitted Mo thickness / nm)
Unpolished samples
1.70 (1.81[−0.12/+0.24]) 0.227+0.010
−0.003 3.14+0.45
−0.06 3.69+0.15
−0.16 −4.62+0.05
−0.06
2.15 (2.31[−0.22/+0.21]) 0.232+0.009
−0.002 3.72+0.44
−0.05 4.88+0.17
−0.18 −5.02+0.04
−0.04
2.30 (2.43[−0.09/+0.16]) 0.329+0.009
−0.003 4.51+0.45
−0.06 4.44+0.17
−0.17 −5.67+0.05
−0.06
verification 90◦0.317+0.011
−0.004 4.56+0.48
−0.10 3.62+0.18
−0.19 +0.55+0.07
−0.07
2.45 (2.68[−0.13/+0.16]) 0.211+0.009
−0.003 3.61+0.46
−0.06 3.80+0.15
−0.16 −5.06+0.06
−0.06
2.90 (3.22[−0.13/+0.11]) 0.243+0.009
−0.002 2.89+0.43
−0.03 5.72+0.14
−0.17 −5.06+0.03
−0.03
Polished samples
1.70 (1.77[−0.22/+0.19]) 0.129+0.009
−0.002 7.05+0.55
−0.23 0.53+0.03
−0.02 −1.19+0.28
−0.28
1.85 (1.91[−0.12/+0.17]) 0.195+0.008
−0.002 10.66+0.56
−0.19 0.76+0.04
−0.04 −1.50+0.25
−0.26
2.00 (2.29[−0.28/+0.13]) 0.105+0.005
−0.001 8.95+0.52
−0.13 0.76+0.03
−0.03 −2.28+0.16
−0.14
2.30 (2.60[−0.12/+0.14]) 0.106+0.006
−0.001 8.22+0.52
−0.17 0.86+0.04
−0.04 −2.90+0.16
−0.07
3.05 (3.47[−0.19/+0.13]) 0.088+0.005
−0.001 10.29+0.58
−0.19 1.47+0.13
−0.11 −1.62+0.16
−0.16
The verification measurement of the rotated sample appears in the row below the
respective sample and shows very good agreement with the original measurement. The
only exception is the vertical correlation parameter
ξ⊥
, which is lower in case of the
rotated sample. This is due to a truncation of the scattering intensity to higher values
of
|qx|
, because of the absorption due to the Si L
2
-edge. From the two measurements, a
total non-orthogonal tilt angle of the vertical roughness correlation of
β= (5.70 ±0.06)°
is obtained by applying Eq.
(5.2)
. This clearly indicates an anisotropy of the deposition
process, which is likely due to non-central mounting of the sample on the sample
holder during fabrication. For both sample sets, table 5.3shows a significant increase
of roughness
σr
at the crystallization threshold at nominal molybdenum thicknesses
of
dnom =2.30nm
for the unpolished samples and
dnom =1.85nm
for the polished
samples. This coincides with the lowest reflectance for that sample in each set shown
in Fig. 4.14. Interestingly, the roughness returns to the previous value for further
increasing Mo layer thicknesses. This indicates, that the roughening due to the formation
of nanocrystallites at the threshold is compensated for even larger thicknesses. A
restored peak reflectance was also observed in Fig. 4.14 in that case. For the polished
samples, the formation of crystallites can be observed with similar effects, but at lower
Mo layer thickness with overall significantly lower root mean square roughness
σr
. It
should be noted, that the strong roughness increase is only observed from the diffuse
scattering measurement in Fig. 5.14 for one of the samples in each set in table 5.3.
This is despite the fact, that a reduced peak reflectance deviating from the expected
theoretical values is seen for two samples out of each set in Fig. 4.14, that were associated
with the crystallization threshold. In both cases, only the sample with the thicker
114
Differently Polished Mo/Si/C Multilayers with Molybdenum Thickness Variation 5.2
molybdenum layer of the two shows stronger roughness, which we shall discuss in the
following subsection. Another clear difference between the polished and unpolished
sets prominently shown in table 5.3, is the large gap between the vertical correlation
factors
ξ⊥
. In the unpolished case, values between
ξ⊥=3.62(+0.18/ −0.19)
nm
−1
and
ξ⊥=5.72(+0.14/ −0.17)
nm
−1
are found, whereas for the polished samples these values
range between
ξ⊥=0.53(+0.03/ −0.02)
nm
−1
and
ξ⊥=1.47(+0.13/ −0.11)
nm
−1
. As
is to be expected, the polishing process largely reduces the roughness correlation between
different interfaces as it alters the morphology of the interface. This situation corresponds
to the low vertical roughness correlation illustrated in the left part of Fig. 2.10. In
the case of unpolished growth, almost the entire stack is correlated (see the right part
of Fig. 2.10) for the observable spatial frequencies. The large values for the in-planar
correlation length
ξk
for the polished samples (between
ξk=7.05(+0.55/ −0.23)
nm and
ξk=10.66(+0.56/ −0.19)
nm) are also a direct result of the polishing process as high
spacial frequencies
qx
are successfully reduced due to the smoothing of the polishing
process.
5.2.2 Discussion of the Results
Finally, we shall interpret the results and relate them to the findings made in Sec. 4.2. To
illustrate the relation between the peak reflectance (originally shown in Fig. 4.14) of each
sample and the r.m.s. roughness reconstructed here, Fig. 5.17 shows these parameters in
comparison with each other. In addition, the Névot-Croce parameter from the specular
reflectance analysis performed in Sec. 4.2.3is included through its confidence interval to
allow the distinction of intermixing and roughness.
1.50 2.00 2.50 3.00 3.50
fitted Mo thickness / nm
0.65
0.67
0.69
0.71
max. reflectivity
b)
0.0
0.1
0.2
0.3
0.4
0.5
σr/ nm
a) unpolished
polished
Figure 5.17 |
a) Root mean square roughness results from the analysis of the diffuse scattering for
the two sample sets together with the full confidence intervals of the Névot-Croce damping factor as
obtained from the structural analysis in Sec. 4.2. In each set, an increase of roughness is observed at
the crystallization threshold. For comparison, the max peak reflectance for each sample set is shown
in b). The increase in roughness clearly correlates with a significant dip in the peak reflectance as
indicated by the dashed vertical lines.
The reconstructed roughness values shown in the figure have the expected increase
at the presumed crystallization threshold for each set. In Fig. 4.13(b), a simultaneous
jump in the total period thickness
D
at this threshold for both sample sets was observed
115
Chapter 5 ANALYSIS OF INTERFACE ROUGHNESS BASED ON DIFFUSE SCATTERING
at the molybdenum layer thickness around
dMo =2.5
nm for the unpolished samples
and
dMo =2.2
nm for the polished samples. The evaluation of the diffuse scatter
revealed increased roughness throughout the multilayer stack for the samples just at the
thickness jump. In comparison to the suspected trend of the peak reflectance with
dMo
in
Fig. 4.14 two samples with lower reflectance in both sets were observed, one exactly at
the position of this increased scatter at
dMo =2.43(+0.16/ −0.09)
nm for the unpolished
samples and
dMo =1.91(+0.17/ −0.12)
nm for the polished samples, and the other
sample with nominally
0.15
nm lower thickness at
dMo =2.31(+0.21/ −0.22)
nm for
the unpolished samples and
dMo =1.77(+0.19/ −0.22)
nm for the polished samples,
respectively. At least for the unpolished samples, this higher roughness
σr
at
dMo =
2.43(+0.16/ −0.09)
nm is not observed from evaluating the specular reflectance alone,
where the reflectance is diminished by the combined effects of roughness, intermixing
and compound formation, which is represented by an effective
σ
-value in the Névot-
Croce factor. In contrast, for the polished samples the enhanced scatter is also observed in
the total Névot-Croce damping factor as indicated by the confidence interval in Fig. 5.17
at
dMo =1.91(+0.17/ −0.12)
nm. In the polished system, the magnitude of the peak
reflectance decrease visible as the difference between theoretical expectation and actually
measured value in Fig. 4.14, however, is also significantly higher for the two respective
samples, compared to the rest of the set, than for the unpolished samples. This may
explain that an increase in the damping factor is less pronounced. The roughness
amplitudes
σr
, as derived from the diffuse scatter, however, have much smaller values
than the Névot-Croce factor
σ
. The comparison of the Névot-Croce parameters and
the roughness values reveals, that the polishing successfully reduced the roughness
contribution to the overall damping factor. It should be noted, that the Névot-Croce
parameters confidence intervals have reduced less than the roughness
σr
, compared to
the series of unpolished samples. This shows, that the remaining interface distortions
through compound formation and intermixing are largely responsible for the gap between
theoretically possible maximum reflectance and actual measured values for the polished
set.
The interpretation of these findings is in line with the observation of the formation
of crystallites in the molybdenum layer at around
2
nm thickness reported by Bajt et al.
[9]. Particularly, the threshold is assigned to the lower thickness where the reflectance
first decreases (at
dMo =2.31(+0.21/ −0.22)
nm for the unpolished set and at
dMo =
1.77(+0.19/ −0.22)
for the polished set) without an observation of increased roughness
by diffuse scatter. This is explained by the crystallization process starting with increased
intermixing and small seeds corresponding to a short correlation length. This yields a
high spacial frequency roughness, which is not correlated throughout the stack. The
corresponding scatter is, thus, not resonantly enhanced. Without the enhancement, it
is below the detection threshold of the diffuse scattering experiment conducted here.
With increasing crystallites, the diffuse scatter becomes observable at slightly higher
molybdenum thickness. Note that for the unpolished sample, the threshold coincides
with the point where the ideal Mo-to-Si ratio should yield the highest reflectance in
agreement with the findings in [9] and the theoretical calculations shown in Fig. 4.14.
For the polished samples, this threshold is shifted to thinner molybdenum layers around
dMo =1.77(−0.22/ +0.19)
nm. This is beneficial for the peak reflectance, which is higher
at the optimum ratio, than for the unpolished set. In both cases, a smoothing occurs
for even larger molybdenum thickness, restoring the roughness to its value below the
threshold. The evaluation of the diffuse scatter shows an overall lower roughness for
116
Roughness and Intermixing in Cr/Sc Multilayers 5.3
the polished samples than for the unpolished ones and, particularly, a destruction of
vertical roughness correlation
ξ⊥
throughout the stack shown in table 5.3and an increase
of the in-planar correlation length
ξk
(describing a reduction of high-spacial frequency
roughness), as intended by the polishing.
Finally, it should be noted that based on the analysis methods introduced in Sec. 4.2
and the diffuse scatter analysis explained in beginning of this chapter, it is possible to
consistently determine the molybdenum layer thickness and the average power spectral
density roughness for the interfaces throughout the full multilayer stack. The application
of these methods to Mo/Si multilayer samples with varying molybdenum thickness
with/without polishing confirmed previous findings on the onset of molybdenum
crystallization in the literature. The results presented in Sec. 4.2and the analysis of the
diffuse scatter discussed here are published together in A. Haase, V. Soltwisch, S. Braun,
C. Laubis, and F. Scholze: ‘Interface morphology of Mo/Si multilayer systems with
varying Mo layer thickness studied by EUV diffuse scattering’. EN. in: Optics Express
25.13 (June 2017), pp. 15441–15455.doi:10.1364/OE.25.015441.
5.3 Roughness and Intermixing in Cr/Sc Multilayers
In Sec. 4.3, a robust method to characterize the ultra-thin multilayer systems structure with
subnanometer layer thicknesses unambiguously was demonstrated. Layer thicknesses
in the subnanometer region are necessary for near-normal incidence reflective mirrors
in the water window spectral range. However, they come with the cost of increasing
susceptibility to disturbances in the interfaces at the layer boundaries. This limits
the achievable reflectance to values well below the theoretical threshold. The main
mechanisms for diminished reflectance are intermixing and roughness. With these effects
ranging on the order of the layer thickness, models based on binary layer stacks become
inadequate to describe the physical situation. In order to find a proper representation
of the multilayer sample, more sophisticated models with an explicit description of the
gradual interdiffusion layers are necessary. This inevitably increases the number of
parameters, as shown in table 4.8, to be determined in analytical experiments. Finding an
unambiguous solution is challenging and can only be achieved with a combined analysis
of several non-destructive techniques. The results obtained in Sec. 4.3.4are listed in
table 4.10 with their confidence intervals. The latter are illustrated in Fig. 4.29, to indicate
the accuracy of the structural reconstruction. The large confidence interval found for the
r.m.s. roughness parameter
σr=0.09(+0.13/ −0.09)
nm and the intermixing parameter
η=0.58(+0.06/ −0.15)
in the combined analysis (Tbl. 4.10) show that, as for the Mo/Si
systems investigated in this thesis, the methods applied in Ch. 4do not yield a possibility
to distinguish roughness from intermixing. Even though, the spatially resolved methods
such as XRF did in combination yield the interface profile asymmetry, a correlation
between the intermixing parameter
η
and the r.m.s. roughness parameter
σr
remains, as
seen in Fig. 4.28.
Based on the analysis of the diffuse scatter, as it was shown above, that distinction
becomes possible. This chapter is finalized with the analysis of the Cr/Sc sample system
studied in Sec. 4.3and completes the characterization with respect to that parameter
correlation. For that purpose, the diffuse scattering intensity from the Cr/Sc sample was
measured similarly to the Mo/Si sample systems above in a rocking scan geometry. As
the theoretical model for the DWBA calculation, the gradual interface model was applied
117
Chapter 5 ANALYSIS OF INTERFACE ROUGHNESS BASED ON DIFFUSE SCATTERING
as defined in the previous chapter with the optimal parameters listed in table 4.10 for the
combination of all analytic experiments conducted there.
The reciprocal space map was taken at an opening angle of
∆Θ =3°
, where the
specular reflectance condition corresponds to the situation where the EUV reflectivity
was evaluated in the previous chapter. This is necessary for this particular sample system
to fulfill the Bragg condition without decreasing the wavelength to values below the Sc
L-edge, where absorption would eliminate the possibility to measure diffuse scattering
from the multilayer. The measurement results are shown in Fig. 5.18. A clear difference
Figure 5.18 |
Diffuse
scattering measure-
ment for the Cr/Sc
sample.
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45
|qk|/ nm−1
3.96
3.98
4.00
4.02
4.04
4.06
qz/ nm−1
measured data, ∆Θ =3◦
10−5
10−4
10−3
reflectivity / sr−1
to the maps in case of Mo/Si samples is the lack of Kiessig-like peaks and a similar
triangular shaped intensity distribution due to the Bragg-like peak. Here, rather the
expectation issued at the beginning of the chapter of the observation of a well formed
Bragg sheet is met. The reasons behind this different behavior are the fundamental
differences with respect to the quality as a mirror of the Cr/Sc system compared to
the Mo/Si system and the different measurement geometry. With a EUV reflectance
in the peak maximum of only about
15%
, only approximately
27%
of the maximum
theoretical reflectivity is attained (cf. Fig. 4.17). In the previous chapter, it was found
that the gradually shaped interfaces regions and intermixing of the sub-nanometer thick
layers play a fundamental role in diminishing the reflectivity. This, however, does also
crucially reduce the impact of multiple dynamic reflections, which were found to have a
strong impact on the measured diffuse scatter intensity for the Mo/Si multilayer systems
investigated above. In addition, this also causes significantly higher penetration depth
allowing more layers to contribute to the diffuse scatter, even if the Bragg condition is
fulfilled for both the incidence and exit angles. Apart from the general lack of dynamic
effects due to bad reflectivity at the interfaces, the non-appearance of Kiessig-like peaks
is also related to the rocking scan geometry with a significantly smaller opening angle.
In the comparison of geometries done at the beginning of this chapter in Fig. 5.7, it was
shown that the resonance conditions move to higher absolute values of
qk
, if the opening
angle is reduced. Thus, no peaks are to be expected in the accessible range for the scan
geometry chosen here.
118
Roughness and Intermixing in Cr/Sc Multilayers 5.3
5.3.1 Estimation of the Vertical Roughness Correlation and the PSD
The sample investigated here is represented by the gradual interface model introduced
in Sec. 4.2above. With total number of
N=400
bilayers and subsequent subdivision
in sublayers, a substantial increase of interfaces has to be considered for the DWBA
analysis as compared to the Mo/Si systems. As pointed out above, the computation cost
growth quadratically with the order
O(N2)
and thus renders the MCMC method very
unpractical for this particular system. However, in order to deduct an estimate of the
PSD and the vertical roughness correlation, we shall apply the approach introduced in
Sec. 5.1at the beginning of the chapter by analyzing only selected cuts of the map to
obtain the relevant parameters. The best PSD model parameters are then obtained by
analyzing a horizontal cut of the Bragg sheet divided by the multilayer enhancement
factor. The two cut positions are shown in Fig. 5.19 as dashed lines in both, the measured
maps and the best model DWBA calculation that was obtained with this approach. The
3.96
3.98
4.00
4.02
4.04
4.06
qz/ nm−1
a) measured data
10−5
10−4
10−3
reflectivity / sr−1
0.0 0.1 0.2 0.3 0.4
|qk|/ nm−1
3.96
3.98
4.00
4.02
4.04
4.06
qz/ nm−1
b) DWBA calculation
Figure 5.19 |
a) Dif-
fuse scattering measure-
ment in
q
-space repre-
sentation and log scale.
b) DWBA calculation of
the optimal PSD model
based on the gradual in-
terface model with the
multilayer parameters
for the combined anal-
ysis listed in table 4.10.
comparison shows good agreement of the model with the measured data.
The data and the simulation results at the vertical cut position are shown in detail in
Fig. 5.20. The solid red line represents the measured data extracted at the aforementioned
3.94 3.96 3.98 4.00 4.02 4.04 4.06 4.08 4.10
qz
0
1
2
3
4
5
Reflectivity ×10−5
measured data
model uncertainty
(vert. corr. length)
DWBA calculation
(best model)
measurement uncertainty
Figure 5.20 |
Measured
data and calculations
at the vertical cut in-
dicated by the vertical
white dashed line in
Fig. 5.19. The blue
dashed lines show two
limiting cases for the
value of the vertical cor-
relation length. The re-
sult leads to the model
uncertainty in the PSD.
vertical cut position. The best model result was obtained using the PSO algorithm and
119
Chapter 5 ANALYSIS OF INTERFACE ROUGHNESS BASED ON DIFFUSE SCATTERING
is shown as solid blue line providing a good match with the data. The measurement
uncertainty is indicated through the red shaded area. Due to the very high computational
cost of the MCMC procedure mentioned above, instead two limiting cases of the vertical
correlation were calculated to assess the confidence interval for that parameter. The
results of that calculation are shown as dashed blue lines framing the measurement
uncertainty at the position of the peak. This parameter enters the calculation of the
multilayer enhancement factor, which is the term in the square brackets of Eq.
(2.54)
,
and thus affects the absolute values of the PSD extracted from the horizontal cut by
dividing through that term. It thus introduces a numerical or model uncertainty to
the deduction of the PSD. Proceeding from here, the measured PSD was evaluated. To
deduct the effective power spectral density, the cut along the Bragg sheet was taken as
indicated by the horizontal white dashed lines in the reciprocal space maps in Fig. 5.19.
The extracted scattering intensity was divided by the multilayer enhancement factor,
leaving the contribution of the effective PSD
C(qk)
to the diffuse scattering. Again, the
two limiting cases are shown as red dashed curves in Fig. 5.21 including the PSD deduced
from the best model value for
ξ⊥
as a solid red curve. Here, the red dashed curves
in Fig. 5.21 correspond to the experimental PSD curves resulting from evaluating the
limiting cases for the vertical correlation lengths evaluated in Fig. 5.20. They are thus
considered a model uncertainty affecting the extraction process for the measured curves.
The actual measurement uncertainty is only shown as red shaded area for the extraction
based on the best model value for ξ⊥.
Figure 5.21 |
Compari-
son of the extracted ef-
fective PSDs from the
diffuse scattering mea-
surement. The uncer-
tainty interval for the ex-
tracted power spectral
density is shown by the
two dashed PSD profiles
(see main text).
10−210−1100
|qk|/ nm−1
10−1
100
101
PSD / nm4
measured Data
model uncertainty
DWBA calculation
measurement uncertainty
The r.m.s. roughness
σr
, which we seek to determine here to solve the correlation
problem illustrated in Fig. 4.28 above, is given by the two-dimensional integral of the
PSD as
σr=1
2πrZ∞
0qkC(qk)dqk. (5.3)
The uncertainty of the PSD due to the vertical correlation leads to an uncertainty in
the r.m.s. roughness when evaluating the integral. Due to the limited
qk
range where
measurements can be taken, the PSD model of Eq.
(2.47)
was fitted to the resulting data
by applying the PSO method based on the parameter limits shown in table 5.1. The addi-
tional uncertainty introduced through the model estimate causes a systematic deviation
120
Roughness and Intermixing in Cr/Sc Multilayers 5.3
Parameter Best model values Confidence interval
σr/ nm 0.17 (−0.01/ +0.02)
ξk/ nm 3.93 (−0.42/ +0.33)
ξ⊥/ nm−110.5 (−3.5/ +3.5)
H1.0 (−0.03/ +0.0)
β/◦0.0 -
Table 5.4 |
Best model
parameters and confi-
dence intervals of the
PSD as a result of the
diffuse scattering analy-
sis for the gradual Cr/Sc
system.
of the PSD extraction and confidence intervals for the parameters were determined by
separately fitting the resulting alternative PSDs. The tilt angle beta was fixed to
β=0°
in this analysis, since no non-orthogonal roughness correlation (tilted Bragg sheet) was
determined by comparison of vertical cuts at different
qk
positions in the map in Fig. 5.18
at this sample orientation. After that, the integration over the full
qk
range was performed
for the best model. The deviation of the integration for the PSD model fit and the data in
the available range were negligible. The best model results for the vertical replication
factor and the power spectral density are given in table 5.4, together with their estimated
uncertainties.
5.3.2 Results and Conclusions
A rigorous analysis of several experimental methods to determine the model parameters
representing one Cr/Sc sample was performed. The optimal set of parameters was
determined by applying a particle swarm optimizer in conjunction with a Markov-chain
Monte Carlo method to verify the uniqueness of the solution and derive confidence
intervals for all parameters in all experiments. Within the verified confidence intervals
the MCMC method reveals a remaining correlation between the intermixing parameter
η
and the roughness factor
σr
, which could not be resolved with the experiments in
specular geometry and the fluorescence measurements. Here, therefore, a measure-
ment of the off-specular diffuse scattering was performed to distinguish between the
roughness and the intermixing similarly to the approach used for the Mo/Si systems.
The r.m.s. roughness value found with the analysis of the diffuse scattering is identical
within its confidence interval to the value obtained from the combined analysis and
thus confirms the intermixing and roughness parameters listed in table 4.10. The results
of this analysis further reveal a high degree of roughness correlation throughout the
multilayer, which is in agreement with observations made for the unpolished Mo/Si
systems and hints at a strong roughness replication during deposition of each layer. It
should also be noted here that the intermixing width
sd
is much larger than the roughness
values
σr
. Also none of the layers was found to have the index of refraction of pure Cr
or Sc, respectively. This is reflected through the non-vanishing intermixing parameter
η>0
. Thus, it can be concluded that while roughness still exists, intermixing and
interdiffusion of the two materials in these sub-nanometer layer systems are the main
cause of diminished reflectance for the Cr/Sc multilayer system studied here.
The findings made in Sec. 4.3together with the diffuse scattering analysis presented
here have been published in A. Haase, S. Bajt, P. Hönicke, V. Soltwisch, and F. Scholze:
‘Multiparameter characterization of subnanometre Cr/Sc multilayers based on com-
plementary measurements’. en. In: Journal of Applied Crystallography
49
.6(Dec. 2016),
pp. 2161–2171.doi:10.1107/S1600576716015776.
121
6
Summary
This thesis has treated the characterization of Mo/Si and Cr/Sc multilayer mirror systems
by combining several indirect methods based on reflection, fluorescence and scattering
of extreme ultraviolet (EUV) and X-ray radiation. Its focus was to validate and improve
the applied theoretical models and determine the experimental techniques required to
achieve an unambiguous solution to the inverse problem. For the reconstruction of the
layer systems structure, a particle swarm optimization (PSO) was applied to fit the model
parameters to the measured data from EUV reflectivity, X-ray reflectivity (XRR), resonant
extreme ultraviolet reflectivity (REUV) and X-ray fluorescence (XRF) experiments. A
Markov-chain Monte Carlo (MCMC) algorithm was further employed to deduct the
maximum likelihood distribution and thereby to obtain confidence intervals based on
the measurement and model uncertainties. It was found that different methods and
models had to be applied depending on the system under investigation. The values and
confidence intervals determined for each parameter of the respective model allowed to
draw conclusions on the structural layout of the samples.
The structural characterization methods were able to yield layer thicknesses, densities
and even the distortion of the interfaces. However, they lack in the ability to identify
these distortions as either roughness or intermixing. This distinction could only be
achieved by combining the results of the structural characterization with a method
sensitive to roughness and re-validating the accuracy of the result. This issue was
approached through the analysis of EUV diffuse scattering with radiation impinging with
near-normal incidence, as a suitable technique to deliver this distinction method. The
method was introduced by analyzing the state-of-the-art Mo/B
4
C/Si/C mirror reaching
(68.5±0.7)%
peak reflectance at its operation wavelengths of
13.5nm
. It was revealed
that the high quality, and thus reflectivity, of the sample causes resonant enhancement
of diffusely scattered radiation within the stack, which significantly contributes to the
diffuse scattering intensities. These dynamic effects must be considered in the analysis by
employing the theoretical framework of the distorted-wave Born approximation (DWBA),
including multiple reflections at the interfaces of the multilayer. With this approach, the
roughness properties for the samples could be extracted consistently. By comparing and
combining the results of the structural characterization and the roughness analysis a
123
SUMMARY
consistent characterization of the multilayer mirrors could be achieved. Thus, the analysis
in this thesis was able to explain the lack of peak reflectivity compared to the theoretical
expectation for an ideal system for both sample systems.
In the unpolished and polished set of the Mo/Si/C multilayer mirrors, it was revealed
that the combination of EUV reflectivity and XRR yields an unambiguous result for the
molybdenum layer thickness confirming the nominal trend in both sets. The confidence
intervals for the molybdenum thickness could be determined ranging from
0.43nm
to
0.24nm
, depending on the sample. In comparison, the analysis of EUV reflectivity for the
Mo/B
4
C/Si/C sample only yielded a confidence interval of approximately
1nm
. This
demonstrated the need for combining multiple datasets, despite an excellent agreement
of the calculated and measured curves, since multiple solutions exist. The sum of
the thicknesses of all layers in a period shows a distinct increase for both sets at a
certain molybdenum layer thickness, associated with a minimum in peak reflectance with
respect to the theoretical expectation. This effect, while observed in both sets, happens
at significantly different molybdenum thicknesses, comparing the unpolished with the
polished samples.
The analysis of the diffuse scattering intensity allowed for an assessment of the interface
morphology for these samples. The comparison with the structural analysis revealed an
increase of roughness, associated with the sudden increase in the period thickness and
the minimum in peak reflectance, which is compensated again at larger thicknesses in
both sets. At this point, it may be concluded that these effects are caused by the onset
of crystallization in the molybdenum layer, causing increased interface disturbances
through roughness. In the analysis of the ion polished set, this threshold was shown to
have moved towards lower molybdenum thicknesses. This is beneficial to the reflectance
at the optimum molybdenum ratio with respect to the rest of the layers in a period,
which in the polished set is now unaffected trough roughening due to crystallization.
Nevertheless, comparing the roughness values found in the diffuse scattering analysis
with the Névot-Croce factor, i.e., with the single root mean square (r.m.s.) value
σ
for the
amount of intermixing and roughness at the interfaces, from the optimized layer structure
model, it became clear that while overall roughness was reduced significantly and led to
a significant increase of the reflectivity in the polished set, the Nevót-Croce parameter
was only reduced slightly, indicating that intermixing is still largely responsible for the
remaining gap to the theoretically achievable reflectivity.
In the case of the Cr/Sc multilayers for the water window spectral range, nominal layer
thicknesses within a bilayer period are between
0.7nm
and
0.8nm
and thus noticeably
thinner than for the Mo/Si systems. It was shown that an approach to the structural
characterization based on a discrete layer model for the chromium and scandium layers
does not yield a solution valid for both the EUV reflectivity and XRR experiments, with
the same set of parameters. That is, a solution fitting the EUV reflectivity experiment
fails to describe the XRR curve and vice versa. Thus, the discrete layer model is not
suitable to describe the physical structure of the sample. Any solution found for either
one of the experiments can therefore not be related to the physical properties of the
sample. Instead, a model describing a gradual interface profile and layers composed of a
mixture of both materials was introduced. Based on this gradual model, the intermixing
and roughness were parametrized separately and asymmetric interface profiles could be
described explicitly.
It was found through the uniqueness and accuracy analysis that the increased variability
of the improved model requires more complementary information than the analysis
124
of the Mo/Si samples. The goal of unambiguity of the solutions was achieved by
performing EUV reflectivity, REUV, XRR and XRF experiments. Confidence intervals
were determined, by evaluating each dataset individually and by combining all in a single
analysis. The found solutions and confidence intervals prove that only the combination
of all datasets can yield a consistent result. It was found that none of the regions within
the Cr/Sc stack are pure chromium or scandium. Furthermore, the interface regions
show a strong asymmetry, which could not be determined with the required significance
by any of the aforementioned standalone analytic experiments. Not even the combined
analysis of these methods could distinguish between roughness and intermixing. Those
two parameters were shown to have a strong correlation. To determine roughness and
intermixing, the EUV diffuse scattering was measured and analyzed similarly as for
the Mo/Si samples. The result shows a roughness value of
σr=0.17(−0.01/ +0.02)
nm. Consequently, the intermixing could be determined to be
47(−4/ +3)%
, leaving
any of the nominal chromium or scandium layers of the stack to contain large amounts
of the other material on average. In conclusion, the roughness determined here is
comparable to the values found for the polished Mo/Si/C samples. There, this roughness
amplitude evidently allowed reflectivities much closer to the theoretical maximum value.
Consequently, intermixing could be identified as the main cause for the small reflectivity
achieved with Cr/Sc multilayer mirrors for the water window.
In summary, the work presented in this thesis proves the importance of assessing the
uniqueness and accuracy of indirect metrological characterization methods to deduct a
meaningful result. As shown on several occasions in the analysis of the multilayer mirrors,
even reconstructions in very good agreement with the data curves show ambiguities
and inconsistencies. This was revealed by adding complementary information from
other experiments, or even by analyzing the data of a single experiment with global
optimization algorithms. With the approach of combining multiple analytic techniques
and determining confidence intervals of the reconstructed parameters, conclusions on
the physical properties of the samples could be drawn reliably. This thesis augments the
existing characterization methods for multilayer mirrors in that respect.
Finally, with the inclusion of EUV diffuse scattering, a technique to assess the interface
morphology was established. It is suitable for characterization near-normal incidence,
offering an alternative to grazing-incidence methods such as grazing-incidence small-
angle X-ray scattering (GISAXS). This has some unique advantages, as any measurement
using small incidence angles is inherently limited to flat or convex surfaces. Focusing
mirrors, however, usually are concavely curved and thus characterization techniques
with grazing angles of incidence are not applicable. Instead, with EUV diffuse scattering
with radiation impinging near normal incidence, it is possible to extract the roughness
information for those samples as well. In addition, radiation at the wavelengths of
operation for these mirrors is suitable to conduct this experiment.
As an outlook extending the scope of this work, it would be interesting to evaluate
the gain in accuracy and uniqueness of the solutions by applying the compilation of
techniques used for the Cr/Sc system, also to the two Mo/Si/C sample sets. This may
prove to be beneficial to further reduce the confidence intervals on the results, most
importantly on the thickness of the barrier layers. In particular, as a straightforward
approach, the improved model for the Cr/Sc mirrors could be carried over to these
systems. Thereby, the role of the barrier and compound layers in the crystallization
could be investigated based on validated reconstruction parameters. This could augment
the analysis conducted on similar systems elsewhere [9]. In general, including further
125
SUMMARY
methods would deliver additional complementary information. Ellipsometry, for example,
could yield results on the optical constants of the various materials in the layer stack.
126
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Acknowledgement
At this point, I would like to express my gratitude to all of those who directly or indirectly
contributed to the successful completion of this thesis.
First and foremost, I would like to thank Dr. Frank Scholze, head of the EUV radio-
metry group at the Physikalisch-Technische Bundesanstalt, for giving me the chance
to conduct the work leading to this PhD thesis under his supervision. Our numerous
scientific discussions, his valuable ideas and his constructive criticism bundled with the
opportunity to conduct experiments even on a short notice, contributed significantly to
the success of this thesis.
Furthermore, I would like to thank Prof. Dr. Mathias Richter for his support and the
examination of this thesis. He always had an open ear and valuable advise for the course
of my scientific work and the near future.
I am very grateful to Prof. Dr. Stefan Eisebitt for supporting and evaluating my disser-
tation and to Dr. Saša Bajt for the fruitful discussions and collaboration, for providing
me with the Cr/Sc samples for my experiments and her willingness to serve as evaluator
of this thesis. In addition, I thank Dr. Stefan Braun for contributing the Mo/Si multilayer
mirror samples.
I also like to acknowledge all of my current and former colleagues and fellow graduate
students, first of all my mentor, Dr. Victor Soltwisch, who supported me during the
past years. I would also like to thank Analía Fernández Herrero, Raül García Diez,
Dr. Christian Gollwitzer, Dr. Philipp Hönicke, Mika Pflüger and Dr. Jan Wernecke. Our
many intense discussions and the collaborative atmosphere they helped to establish
improved my research significantly.
I am sincerely grateful to all members of the working group 7.12, Christian Buchholz,
Ayhan Babalik, Anja Babuschkin, Martin Biel, Benjamin Dubrau, Andreas Fischer, Anne
Hesse, Sina Jaroslawzew, Florian Knorr, Dr. Christian Laubis, Jana Lehnert, Heiko Mentzel,
Jana Puls, Anja Schönstedt, Christian Stadelhoff. Without their support and patience in
many late-night beamtimes in the laboratory, this work would not have been possible.
My honest thanks also go to all other colleagues of the PTB in Berlin-Adlershof.
Finally, I am in dept to all of my friends and family for their endless support and their
distractions during my studies and over the course of my PhD thesis. Most importantly I
would like to name Michl, Michael, Paul, Laura, Tim, Anna, Leo and my parents Detlev
and Martina. Last but not least, I am deeply grateful to my grandfather Dr. Walther
Neudert for inspiring me and his early encouragement of my scientific career.
Eidesstattliche Versicherung
Hiermit versichere ich an Eides statt, dass ich die vorliegende Arbeit selbstständig verfasst
und keine anderen als die in der Dissertation angegebenen Quellen und Hilfsmittel
benutzt habe. Alle Ausführungen, die anderen veröffentlichten oder nicht veröffentlichten
Schriften wörtlich oder sinngemäß entnommen wurden, habe ich kenntlich gemacht.
Die Darstellung des Eigenanteils an bereits publizierten Inhalten in meiner beigefügten
Erklärung ist zutreffend.
Berlin, den 4. November 2017 Anton Haase
Declaration
Parts of this dissertation were previously published in peer-reviewed journals and
conference contributions. I attach information on previous publications according to §2
(4) of the Promotionsordnung of TU Berlin and regulations of Faculty II.
List of publications containing parts of the dissertation and the detailed contributions
of the co-authors to each publication:
1)
A. Haase, V. Soltwisch, C. Laubis und F. Scholze: „Role of dynamic effects in the
characterization of multilayers by means of power spectral density“. In: Appl. Opt.
53.14 (2014), S. 3019–3027.doi:10.1364/AO.53.003019
VS and FS concieved the study. AH conducted the EUV reflectivity and diffuse
scattering measurements, developed the software, performed the data analysis and
wrote the manuscript. VS, CL and FS contributed trough discussion of the results.
All authors read, approved and contributed to the final manuscript.
2)
A. Haase, V. Soltwisch, F. Scholze und S. Braun: „Characterization of Mo/Si mirror
interface roughness for different Mo layer thickness using resonant diffuse EUV
scattering“. In: Proc. SPIE. Bd. 9628.2015.doi:10.1117/12.2191265
AH, FS and SB designed the study. FS organized the samples. SB fabricated the
samples and conducted the XRR measurements. AH conducted the EUV reflectivity
and diffuse scattering measurements, analyzed the data and drafted the manuscript.
VS and FS contributed trough discussion of the results. All authors read, approved
and contributed to the final manuscript.
3)
A. Haase, S. Bajt, P. Hönicke, V. Soltwisch und F. Scholze: „Multiparameter charac-
terization of subnanometre Cr/Sc multilayers based on complementary measure-
ments“. en. In: Journal of Applied Crystallography
49
.6(Dez. 2016), S. 2161–2171.doi:
10.1107/S1600576716015776
AH, SB and FS developed the study. AH organized the samples, conducted the
EUV reflectivity, resonant EUV reflectivity and diffuse scattering measurements.
SB fabricated the samples and conducted the XRR measurement. PH and AH
performed the XRF experiment. AH developed the model and the corresponding
software, analyzed the data of all experiments and wrote the manuscript. VS and
FS contributed through discussion of the results. All authors read, approved and
contributed to the final manuscript.
4)
A. Haase, V. Soltwisch, S. Braun, C. Laubis und F. Scholze: „Interface morphology
of Mo/Si multilayer systems with varying Mo layer thickness studied by EUV
diffuse scattering“. EN. In: Optics Express
25
.13 (Juni 2017), S. 15441–15455.doi:
10.1364/OE.25.015441
AH, FS and SB designed the study. FS organized the samples. SB fabricated the
samples and conducted the XRR measurements. AH conducted the EUV reflectivity
and diffuse scattering measurements, analyzed the data, developed the software
and drafted the manuscript. VS, CL and FS contributed trough discussion of the
results. All authors read, approved and contributed to the final manuscript.
List of peer-reviewed publications not part of the dissertation:
1)
M. Prasciolu, A. Haase, F. Scholze, H. N. Chapman und S. Bajt: „Extended asymmetric-
cut multilayer X-ray gratings“. EN. In: Optics Express
23
.12 (Juni 2015), S. 15195–
15204.doi:10.1364/OE.23.015195
2)
V. Soltwisch, A. Haase, J. Wernecke, J. Probst, M. Schoengen, S. Burger, M. Krumrey
und F. Scholze: „Correlated diffuse X-ray scattering from periodically nanostruc-
tured surfaces“. In: Physical Review B
94
.3(Juli 2016), S. 035419.doi:
10.1103/
PhysRevB.94.035419
Berlin, den 4. November 2017 Anton Haase
