Strain-phase relations in lead-free ferroelectric
KxNa1−xNbO3epitaxial films for domain engineering
vorgelegt von
Master of Science
Dorothee Braun
geboren in Berlin
von der Fakult¨at II - Mathematik und Naturwissenschaften
der Technischen Universit¨at Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
Dr. rer. nat.
genehmigte Dissertation
Promotionsausschuss:
Vorsitzende: Prof. Dr. Marga Cornelia Lensen
Gutachter: Prof. Dr.-Ing. Matthias Bickermann
Gutachter: PD Dr. Martin Schmidbauer
Gutachter: Prof. Dr. Roger W¨ordenweber
Tag der wissenschaftlichen Aussprache: 17. Oktober 2017
Berlin 2017
D. Braun
Abstract
The aim of this thesis is to demonstrate the potential of thin films for technical ap-
plications by tuning the ferroelectric film properties on the basis of the strain-phase
diagram. For this purpose, a fundamental understanding of the relation between incor-
porated lattice strain by epitaxial growth of thin films and the evolution of ferroelectric
phases has been pointed out. As an exemplary lead-free material, potassium sodium
niobate (KxNa1−xNbO3) is considered. This material is of high technological interest
due to its large coupling constant [1] and Curie temperature [2]. Furthermore, calcu-
lations predict the appearance of monoclinic phases under anisotropic epitaxial lattice
strain [3]. They are attractive due to their inherent flexibility for the arrangement
of the electrical polarization vector yielding e.g. huge piezoelectric responses [4] and
flexible domain wall formation [5]. However so far, thin films have rarely been investi-
gated and the influence of lattice strain on ferroelectric properties has not been studied
systematically. In this work, thin films were grown by metal-organic chemical vapor
deposition technique (MOCVD) on different oxide single-crystalline substrates.
For a targeted choice of appropriate film-substrate combinations, theoretical consider-
ations with regard to film orientation and phase symmetry are essential. To predict
the energetically most favorable film unit cell orientation on a substrates, linear elas-
ticity theory was used in this thesis. For engineering the domain structure, a misfit
strain-misfit strain phase diagram was calculated for potassium niobate by means of
the Landau-Ginzburg-Devonshire (LDG) theory with major accuracy compared to ex-
isting predictions [3]. The result manifests a diversity of different, mainly monoclinic
phases for KNbO3.
By means of the calculated misfit strain-phase diagram, particular KxNa1−xNbO3com-
positions were epitaxially grown by MOCVD with nearly perfect structural ordering
and stoichiometry. Use of different 0.1◦off-oriented (001) SrTiO3, (110) DyScO3, (110)
TbScO3, (110) GdScO3and (110) NdScO3single-crystalline substrates provides an ex-
perimental verification of the calculated strain-phase diagram. This was realized for
the first time for the promising KxNa1−xNbO3material system.
As a proof of concept, two different film-substrate combinations have been inves-
tigated in detail mainly with the piezoresponse force microscope (PFM). Together
with elaborated x-ray diffraction measurements a detailed analysis of the ferroelec-
tric domain structure inherently coupled to the crystal symmetry was possible. First,
K0.75Na0.25NbO3on TbScO3with nearly uniaxial, medium compressive lattice strain
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D. Braun
leads to periodically ordered, monoclinic MAstripe domains. A second 90◦rotated
variant occurs only to a minor fraction which is attributed to the small strain energy
density difference of both (001)pc orientations. Second, K0.90Na0.10NbO3deposited on
NdScO3displays a combination with degenerated strain energy densities for the (100)pc
and (001)pc orientation. The result is a nested ferroelectric herringbone pattern with
alternating, monoclinic a1a2/Mcdomains. The longitudinal piezoelectric coefficient d0
33
was locally determined in the system K0.90Na0.10NbO3/SrRuO3/NdScO3to d0
33 = 29pm
V
which is promising in comparison to established lead-based ferroelectrics [[6], [7], [8],
[9], [10]]. Moreover, the domain wall inclination angle differs significantly from those
of other symmetries and was discussed as a function of potassium concentration x in
the framework of a model established by Bokov and Ye [5] yielding a first experimental
proof. Furthermore, the composition x = 0.90 on NdScO3enabled the first investiga-
tion of the hierarchy and scaling behavior of complex, monoclinic multi-rank pattern.
ii
D. Braun
Zusammenfassung
Das Ziel dieser Arbeit ist das Potential d¨unner Filme f¨ur technische Anwendungen zu
zeigen, wenn deren ferroelektrische Eigenschaften auf der Grundlage eines Verspannung-
Phasen Diagrammes gezielt eingestellt wurden. Zu diesem Zweck muss ein grundle-
gendes Verst¨andnis f¨ur die Beziehung von eingebrachter Gitterverspannung mittels
epitaktischem Wachstum d¨unner Filme und der Entstehung ferroelektrischer Phasen
gewonnen werden. Als beispielhaftes bleifreies Material wurden Kaliumnatriumniobat
(KxNa1−xNbO3) betrachtet. Dieses ist von großer technologischer Relevanz auf Grund
seiner hohen Kopplungskonstante [1] und Curietemperatur [2].
Zudem sagen theoretische Betrachtungen unter anisotroper Gitterverspannung das
Auftreten monokliner Phasen f¨ur epitaktische Filme vorher [3]. Diese sind begehrt,
da sie eine inh¨arente Flexibilit¨at f¨ur die Ausrichtung des Polarisationsvektors besitzen,
was zu sehr hohen piezoelektrischen Auslenkungen [4] und flexibler Ausrichtung von
Dom¨anenw¨anden [5] f¨uhren kann. Jedoch wurden d¨unne Kaliumnatriumniobatfilme
bisher wenig untersucht und ebenso wenig der Einfluss von Gitterverspannung auf die
ferroelektrischen Eigenschaften. In der vorliegenden Arbeit wurden D¨unnfilme mit-
tels metallorganischer Gasphasendeposition (MOCVD) auf verschiedenen einkristalli-
nen Oxidsubstraten abgeschieden.
F¨ur eine zielgerichtete Wahl interessanter Substrat-Filmzusammensetzungen bedarf es
theoretischer Betrachtungen zur Filmorientierung und Phasensymmetrie. Zur Vorher-
sage der energetisch g¨unstigsten Orientierung der Filmeinheitszelle auf einem Substrat
wurde die lineare Elastizit¨atstheorie in dieser Arbeit angewandt. Zum Zweck des ”do-
main engineering”, wurde ein Verspannung-Phasen Diagramm f¨ur Kaliumniobat auf
der Grundlage der Landau-Ginzburg-Devonshire-Theorie mit h¨oherer Genauigkeit als
bereits existierende Daten [3] berechnet. Das Ergebnis best¨atigt eine Vielzahl ver-
schiedener, haupts¨achlich monokliner Dom¨anenarten f¨ur KNbO3.
Auf der Basis dieser Berechnungen wurden zielgerichtet KxNa1−xNbO3Zusammenset-
zungen epitaktisch mittels MOCVD gewachsen. Die Verwendung verschiedener 0.1◦
fehlorientierter (001) SrTiO3, (110) DyScO3, (110) TbScO3, (110) GdScO3und (110)
NdScO3Substrate erm¨oglicht die experimentelle ¨
Uberpr¨ufung der theoretischen Ergeb-
nisse. Diese Untersuchung wurde erstmals f¨ur das vielversprechende KxNa1−xNbO3-
System durchgef¨uhrt.
Um die G¨ultigkeit des Phasendiagramms zu testen, wurden im Detail zwei Film-
Substratkombinationen mit dem Piezoresponse Force Mikroskop untersucht. Zusam-
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D. Braun
men mit komplexen R¨ontgenbeugungsexperimenten war so eine detaillierte Analyse
der ferroelektrischen Dom¨anenstruktur als Folge der Kristallstruktur m¨oglich. Im er-
sten Beispiel werden K0.75Na0.25NbO3Filme auf TbScO3mit nahezu uniaxialer, mit-
tlerer kompressiver Gitterverspannung gezeigt, die periodisch angeordnete, monok-
line MAStreifendom¨anen aufweisen. Eine zweite, um 90◦rotierte Variante tritt mit
geringerer H¨aufigkeit auf, was auf den eher geringen Verspannungsenergiedichteunter-
schied beider (001)pc Orientierungen zur¨uckgef¨uhrt werden kann. Das zweiten Beispiel,
K0.90Na0.10NbO3auf NdScO3, zeigt eine Kombination entarteter Energiedichten beider
pseudokubischen Orientierungen. Das Ergebnis ist ein verschachteltes, ferroelektrisches
Fischgr¨atenmuster aus alternierend angeordneten, monoklinen a1a2/McDom¨anen. Der
longitudinale piezoelektrische Koeffizient d0
33 wurde lokal im Schichtsystem
K0.90Na0.10NbO3/SrRuO3/NdScO3zu d0
33 = 29pm
Vbestimmt. Dieses Ergebnis ist vielver-
sprechend im Vergleich zu existierenden bleibasierten Ferroelektrika [[6], [7], [8], [9],
[10]]. Dar¨uber hinaus unterscheidet sich der Neigungswinkel der Dom¨anenw¨ande sig-
nifikant vom dem anderer Symmetrien und wurde erstmals experimentell als Funktion
des Kaliumgehaltes x im Rahmen des Modelles von Bokov und Ye [5] diskutiert. Ab-
schließend wurden zum ersten Mal die Dom¨anenhierarchie sowie das Skalierungsver-
halten dieser komplexen, monoklinen Fischgr¨atenmuster untersucht.
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D. Braun
Publications in international refereed journals
1. D. Braun, M. Schmidbauer, M. Hanke and J. Schwarzkopf, ”Hierarchy and scaling
behaviour of ferroelectric multi-rank domain patterns in strained K0.9Na0.1NbO3
epitaxial films”, submitted to Nanotechnology, 2017
2. D. Braun, M. Schmidbauer, M. Hanke, A. Kwasniewski and J. Schwarzkopf,
”Tunable ferroelectric domain wall alignment in strained monoclinic KxNa1−xNbO3
epitaxial films”, Appl. Phys. Lett. 110, 232903 (2017)
3. J. Schwarzkopf, D. Braun, M. Hanke, R. Uecker and M. Schmidbauer, ”Strain
engineering of ferroelectric domains in KxNa1−xNbO3epitaxial layers”, Frontiers
4, 26 (2017)
4. M. Schmidbauer, D. Braun, T. Markurt, M. Hanke, and J. Schwarzkopf, ”Strain
Engineering of Monoclinic Domains in KxNa1−xNbO3Epitaxial Layers: A Path-
way to Enhanced Piezoelectric Properties”, Nanotechnology 28, 24LT02 (2017)
5. M. Schmidbauer, M. Hanke, A. Kwasniewski, D. Braun, L. von Helden, Ch.
Feldt, S. J. Leake and J. Schwarzkopf, ”Scanning X-Ray Nanodiffraction from
Ferroelectric Domains in Strained K0.75Na0.25NbO3Epitaxial Films Grown on
(110) TbScO3”, J. Appl. Cryst. 50, 519 −524 (2017).
6. B. Cai, J. Schwarzkopf, E. Hollmann, D. Braun, M. Schmidbauer, T. Grell-
mann, and R. W¨ordenweber, ”Electronic characterization of polar nanoregions
in relaxor-type ferroelectric NaNbO3films”, Phys. Rev. B 93, 224107 (2016)
7. J. Schwarzkopf, D. Braun, M. Hanke, A. Kwasniewski, J. Sellmann and M.
Schmidbauer, ”Monoclinic MAdomains in anisotropically strained ferroelectric
K0.75Na0.25NbO3films on (110) TbScO3grown by MOCVD”, J. Appl. Cryst. 49,
375–384 (2016).
8. J. Schwarzkopf, D. Braun, M. Schmidbauer, A. Duk, and R. W¨ordenweber, ”Fer-
roelectric domain structure of anisotropically strained NaNbO3epitaxial thin
films”, J. Appl. Phys. 115, 204105 (2014)
9. M. Schmidbauer, J. Sellmann, D. Braun, A. Kwasniewski, A. Duk, and J. Schwarzkopf,
”Ferroelectric domain structure of NaNbO3epitaxial thin films grown on (110)
DyScO3substrates”, Phys. Stat. Sol. RRL 8, 522 −526 (2014)
v
D. Braun
10. J. Sellmann, J. Schwarzkopf, A. Kwasniewski, M. Schmidbauer, D. Braun and A.
Duk, ”Strained ferroelectric NaNbO3thin films: Impact of pulsed laser deposi-
tion growth conditions on structural properties”, Thin Solid Films 570, 107-113
(2014)
Conference presentations
1. D. Braun, M. Schmidbauer, M. Hanke, C. Feldt and J. Schwarzkopf, ”Coexistence
of monoclinic domains in KxNa1−xNbO3on NdScO3grown by MOCVD”, (oral),
Joint IEEE International Symposium on the Applications of Ferroelectrics, Eu-
ropean Conference on Applications of Polar Dielectrics & Workshop on Piezore-
sponse Force Microscopy (ISAF/ECAPD/PFM), Darmstadt, Germany, August
22-25, 2016
2. D. Braun, M. Schmidbauer, M. Hanke, C. Feldt and J. Schwarzkopf, ”Alter-
nating monoclinic domains in K0.9Na0.1NbO3thin films grown on NdScO3by
MOCVD”, (poster), Joint IEEE International Symposium on the Applications
of Ferroelectrics, European Conference on Applications of Polar Dielectrics &
Workshop on Piezoresponse Force Microscopy (ISAF/ECAPD/PFM), Darm-
stadt, Germany, August 22-25, 2016
3. D. Braun, M. Schmidbauer, A. Kwasniewski, P. M¨uller, J. Sellmann, M. Hanke,
H. Renevier, and J. Schwarzkopf, ”Formation and switching behavior of mon-
oclinic domains in strained K0.90Na0.10NbO3epitaxial thin films on NdScO3”,
(oral), 13th European Meeting on Ferroelectricity (EMF), Porto, Portugal, June
28 - July 3, 2015
4. D. Braun, M. Schmidbauer, P. M¨uller and J. Schwarzkopf, ”Domain structure in
anisotropically strained K0.75Na0.25NbO3thin films on TbScO3”, (oral), Deutsche
Physikalische Gesellschaft (DPG), Fr¨uhjahrstagung, Berlin, Germany, March 15-
20, 2015
vi
D. Braun Contents
Contents
Abstract i
List of Abbreviations ix
Introduction 1
I. Ferroelectricity & Materials 5
1. Fundamentals of Piezo- and Ferroelectrics 5
1.1. Crystalsymmetry.............................. 5
1.2. Phasetransitions .............................. 6
1.3. Ferroelectricdomains............................ 8
1.4. Ferroelectric switching . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5. Piezoelectric coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2. Ferroelectric thin films 14
2.1. Perovskitestructure............................. 14
2.2. Introduction to Linear elasticity theory . . . . . . . . . . . . . . . . . . 15
2.3. Strain engineering in perovskite materials . . . . . . . . . . . . . . . . . 16
2.4. Potassium sodium niobate KxNa1−xNbO3................. 20
2.5. Enhancement of piezoelectric responses . . . . . . . . . . . . . . . . . . 22
II. Growth & Characterization methods 27
3. Thin film deposition 27
4. Atomic Force Microscopy 28
5. Piezoresponse Force Microscopy 30
5.1. Contributions of a PFM signal . . . . . . . . . . . . . . . . . . . . . . . 32
5.2. Experimentally influencing the PFM signal . . . . . . . . . . . . . . . . 34
5.3. Resonance enhancement of PFM signal . . . . . . . . . . . . . . . . . . 36
5.4. Evaluation of the local piezoelectric coefficient d0
33 ............ 37
vii
D. Braun Contents
6. X-ray diffraction 39
6.1. High-resolution x-ray diffraction (HRXRD) . . . . . . . . . . . . . . . . 39
6.2. Grazing incidence x-ray diffraction (GIXD) . . . . . . . . . . . . . . . . 42
III. Results & Discussion 44
7. Theoretical Considerations 44
7.1. Calculating lattice parameter from Vegard’s law . . . . . . . . . . . . . 44
7.2. Calculation of the strain energy density F(ε)(x) ............. 45
7.3. Calculation of the strained vertical lattice parameter dstrained
⊥...... 46
7.4. Calculation of the misfit strain-misfit strain phase diagram for KNbO347
8. Uniaxial strain: K0.75Na0.25NbO3films on (110) TbScO351
9. Energy density degeneration for (100)pc and (001)pc orientation:
K0.90Na0.10NbO3films on (110) NdScO364
10.Tunable ferroelectric domain wall alignment in monoclinic systems 80
11.Hierarchy and scaling behavior of herringbone domain patterns in strained
K0.90Na0.10NbO3ferroelectric epitaxial films 86
IV. Summary 96
V. Outlook 98
References 99
Acknowledgement 113
Selbstst¨andigkeitserkl¨arung 115
viii
D. Braun Contents
List of Abbreviations
1D One Dimensional
2D-FFT Two Dimensional Fast Fourier Transformation
ac Alternating Current
AFM Atomic Force Microscopy
cCubic
DART Dual AC Resonance Tracking
dc Direct Current
DHO Damped Harmonic Oscillator
DRAM Dynamic Random Access Memory
ESRF European Synchrotron Radiation Facility
FeRAM Ferroelectric Random Access Memory
GIXD Grazing Incidence X-Ray Diffraction
(hkl) Miller Indices
HRXRD High Resolution X-Ray Diffraction
InvOLS Inverse Optical Lever Sensibility
LPFM Lateral Piezoresponse Force Microscopy
MOCVD Metal Organic Chemical Vapor Deposition
MPB Morphotropic Phase Boundary
NSO Neodymium Scandate
oOrthorhombic
pc Pseudocubic
PFM Piezoresponse Force Microscopy
PLD Pulsed Laser Deposition
PR Piezoresponse
PSD Position Sensitive (Photo-)Detector
PZT Lead Zirconium Titanate
RSM Reciprocal Space Map
SEM Scanning Electron Microscopy
STEM Scanning Transmission Electron Microscopy
TEM Transmission Electron Microscopy
TSO Terbium Scandate
VPFM Vertical Piezoresponse Force Microscopy
XRD X-ray Diffraction
ix
D. Braun Contents
Introduction
In times of a technological revolution where applications can be tailor-made for every
single part in daily life, functional materials are needed in huge quantity and perfect
quality. Consequently, in the past two decades major effort was put into the research
on materials with piezo - and ferroelectric properties and their defect-free synthesis
[11]. Oxides with perovskite-like structure represents a class of materials with huge
variety of possible compositions yielding a multitude of functional properties ranging
from di-, piezo- and ferroelectric to optical or even multiferroics. As a result, they can
apply for e.g. in non-volatile memory applications (FeRAM), as high-k materials in
random-access memory devices (DRAM) or as optical waveguides to name only a few
[[12],[13]]. Yet, most commonly used for such applications are lead-based materials due
to their outstanding electric properties combined with excellent coupling coefficients
[14]. However, they suffer from degradation and fatigue effects limiting the life time
that seem not to be overcome [[15],[16]]. An additional problem is the toxic effect of
lead itself [17]. Since it poses a serious environmental hazard, the European Union has
restricted the use of lead (ratified 2003) in electronic devices. However, the search for
alternative, lead-free materials that could compete with lead-based ferroics turns out
to be difficult.
One promising lead-free alternative in this context is KxNa1−xNbO3. In common to
other alkaline-based oxides, potassium sodium niobate has been investigated as single
crystal or as ceramic. Already in bulk form, the versatile and outstanding properties
could be proved. However, the ongoing trend of device miniaturization requires the
growth of high crystal quality thin films.
Additionally, the functional properties of thin layers can be tailored due to the incor-
porated lattice strain by epitaxial growth. Consequently, novel, artificial and enhanced
features can be generated that differ significantly compared to their bulk counterparts.
One focus point is thereby the introduction of a monoclinic symmetry because these
phases have revealed protruding properties like giant piezoelectric responses when they
where discovered at the morphotropic phase boundary (MPB) in lead zirconium ti-
tanate [[18],[19],[20]]. Theoretical predictions revealed the possibility to overcome the
absence of such a MPB in many lead-free materials by using anisotropic lattice strain
for the purpose of monoclinicity. This procedure to modify material’s properties by
applying epitaxial lattice strain is denoted as strain or domain engineering [21].
Mandatory for engineering the ferroelectric properties is a detailed understanding of
1
D. Braun Contents
strain provoked phase changes and the evolving ferroelectric ordering. Because do-
mains are formed in order to neutralize charge or to reduce lattice strain, they dictate
the macroscopic and microscopic properties of a material. In particular, the domain
wall periodicity scales with the film thickness yielding a much higher domain wall
density in thin films compared to bulk crystals. Therefore, calculations are essen-
tial that reveal the strain-phase relationship for a distinct material system. Different
approaches exist to describe a ferroelectric material as density-functional theory [22],
time-dependent Ginzburg-Landau theory [23] or most commonly the modified Landau-
Ginzburg-Devonshire theory [24]. In the framework of the latter, Bai and Ma calculated
a strain-phase diagram for KNbO3[3].
However, detailed investigations of the promising material system KxNa1−xNbO3in
thin film form are still missing. Especially, the approach to reveal structure-phase rela-
tionships in nearly perfect films grown by the industrial compatible metal-organic chem-
ical vapor deposition technique (MOCVD) method is novel, since KxNa1−xNbO3films
are rarely studied yet. This method provides low defect concentrations and smooth and
sharp interfaces, due to the growth close to the thermodynamic equilibrium, as well as
uniform and homogeneous depositions on large scale. However, oxide film deposition
by MOCVD is still challenging yet due to the stability and availability of suitable metal
organic precursors.
For material research not only a sophisticated deposition techniques is necessary, rather
also adequate characterization methods for thin film properties on a nanoscale are indis-
pensable. In terms of thin film characterization, new methods for detecting local piezo
- and ferroelectric properties on sub-micron scale are needed. Hereby, several prop-
erties are of special interest. The local piezoelectric responses have to be determined
precisely. Likewise the ferroelectric domain structure as well as the local polarization
switching behavior are mandatory to be investigated in terms of prospective applica-
tion in high-density non-volatile memories. Several techniques at the nanoscale exist
such as scanning electron microscopy (SEM), transmission electron microscopy (TEM)
or various etching methods [[25],[26],[27]]. But most of these methods are either de-
structive, need sophisticated sample preparation or suffer from low lateral resolution.
Indeed, non of them yield information about the local piezoelectric response or ferro-
electric switching. Therefore, the upgrade of an atomic force microscope (AFM) with
a piezoresponse force module (PFM) was the breakthrough in thin ferroelectric film
research [28]. It has become the primary tool for imaging of domains and spectroscopy
of piezo- and ferroelectric materials with high lateral resolution and without any spe-
2
D. Braun Contents
cific sample preparation needed beforehand [[29],[30],[31],[32]].
Within the framework of this thesis, the liquid-delivery MOCVD technique was applied
to grow high quality, ferroelectric KxNa1−xNbO3epitaxial thin films on 0.1◦off-oriented
(001) SrTiO3, (110) DyScO3, (110) TbScO3, (110) GdScO3and (110) NdScO3single-
crystalline substrates. The aim was to get a fundamental understanding of the impact
of anisotropic strain on ferroelectric domain formation. For that purpose, detailed PFM
measurements were performed and correlated to structural data gained from elaborated
XRD measurements. The experimental data agree with the predictions drawn from
theoretical considerations of strain-phase relations and linear elasticity theory.
This thesis is divided into five parts.
Part I takes a look into the world of crystal symmetries and introduces their crucial
impact on ferroelectric properties. Furthermore, the concept of strain engineer-
ing in perovskite thin films is described in general and how it is performed in
particular for the case of potassium sodium niobate. Moreover, a brief discussion
of how to enhance the piezoelectric response is given.
Part II explains the growth and characterization methods. The liquid delivery metal
organic chemical vapor deposition technique is briefly illustrated. Since PFM rep-
resents the main characterization method, challenges and advantages of piezore-
sponse force microscopy are discussed in more detail. Moreover a basic introduc-
tion into x-ray diffraction is given.
Part III begins with theoretical considerations. As starting point, the linear elasticity
theory is introduced to illustrate a theoretical concept for strain-dependent pre-
dictions of the energy density and thus the preferred film unit cell orientation.
Moreover, a misfit strain - misfit strain phase diagram was calculated on the
basis of Landau-Ginzburg-Devonshire theory for potassium niobate because the
existing literature data [3] were calculated with insufficient increment. On the
basis of this diagram, the impact of substrate stress on the structural properties
of a lattice mismatched, thin film and thus on the ferroelectric phase is ana-
lyzed in detail for two selected examples. The experimental results for strained
KxNa1−xNbO3thin films grown by liquid-delivery MOCVD on (110) TbScO3and
(110) NdScO3substrates are discussed and compared to the theoretical predic-
tions. Furthermore, the dependence of the domain wall inclination angle from the
3
D. Braun Contents
potassium concentration x was evaluated for the herringbone arrangement evolv-
ing for KxNa1−xNbO3on NdScO3. Eventually, the thickness dependent evolution
of such a monoclinic herringbone pattern is described in the exemplary system
K0.90Na0.10NbO3on (110) NdScO3revealing novel stable domain configurations
and a multistage relaxation process.
Part IV summarizes the general conclusions of the experimental results.
Part V presents some ideas in form of an outlook for further work.
4
D. Braun I. Ferroelectricity & Materials
Part I.
Ferroelectricity & Materials
1. Fundamentals of Piezo- and Ferroelectrics
The basis of this thesis is an effect that has been observed by Pierre and Jacques Curie
nearly 200 years ago in tourmaline crystals [33]. They showed that this material re-
sponds to a mechanical deformation with an accumulation of electric charges on the
surface, whereby the amount is directly proportional to the applied stress. A few years
later, also the reverse effect was discovered and was then named direct and indirect
piezoelectric effect, respectively.
In 1920 J. Valasek demonstrated that Rochelle salt shows the same hysteresis behavior
as it was known from ferromagnetics - but now under an electrical voltage applied [34].
As a result, this group of materials got the name ferroelectrics although they have
nonessentially something to do with iron.
What should be shown later, these piezoelectric and ferroelectric properties are di-
rectly related to the crystal structure. Therefore, at this point a brief insight into the
crystallography shall be given.
1.1. Crystal symmetry
Based on symmetry considerations 32 crystallographic point groups can be distin-
guished that can not be further converted into each other by translation, mirroring
or rotation [35]. The first classification can be made using the symmetry element of
inversion. It turns out that eleven of 32 crystal groups have an inversion center and
may be characterized as non-polar. Whereas the remaining 21 crystal classes have
no centrosymmetry and are therefore referred to as polar. From the last group 20
lattices show a linear behavior between the applied electric voltage and mechanical
deformation (or vice versa) and are therefore considered to be piezoelectric accord-
ing to the experiments of the Curie brothers from 1880. Within the latter, one half
exhibits a single, unique polar axis. This circumstance leads to the emergence of a
temperature-dependent (in the absence of a mechanical stress) polarization in these
crystals, referred as pyroelectric effect. In some of these materials, the polarization can
moreover be switched reversibly by an applied external electric voltage between - at
5
D. Braun I. Ferroelectricity & Materials
least - two equilibrium states. This group is named ferroelectrics and is the lynchpin
of this thesis.
1.2. Phase transitions
Behind the classification of materials into a symmetry class stands the dependence on
intensive variables such as temperature and pressure, since the formation of a crystal
lattice always obeys energetic boundary conditions.
The transformation of crystal lattices under the mentioned variables is known as phase
transition according to Ehrenfest’s theorem. Generally, the system is described by a
thermodynamic potential G. Changing the environment parameters, the system reacts
- either continuously or discontinuously - with a revised potential landscape and thus
novel energy minima. The order of the phase transition is named after the level of the
lowest derivative of G for a chosen intensive parameters for whom this transition expires
discontinuously. Physically relevant are usually only phase transitions of 1st and 2nd
order. The Landau theory is used here to differentiate. The key of this theory is to
introduce an order parameter whose non-zero value characterizes the low temperature
phase while it is zero in the high temperature phase [36]. Mathematically, it is based
on a Taylor expansion of the Gibbs free energy G as a function of the order parameter.
In the case of the paraelectric-to-ferroelectric phase transition, this order parameter is
the polarization P. Due to the fact that the ferroelectric phase is non-polar, the energy
of a polarization and its opposite must be equal, so that the odd parameters must equal
zero and drop out the equation [37].
∆G(P) = G0+a
2
∂2G
∂P2P2+b
4
∂4G
∂P4P4+.... (1)
Landau-Ginzburg-Devonshire relate the coefficients to ferroelectric properties trough
an expansion of the Taylor series to 6th order [38]:
∆G = α1(P2
1+ P2
2+ P2
3) + α11(P4
1+ P4
2+ P4
3) + α12(P2
1P2
2+ P2
2P2
3+ P2
3P2
1)
+α111(P6
1+ P6
2+ P6
3) + α112[P4
1(P2
2+ P2
3)+P4
2(P2
1+ P2
3)+P4
3(P2
2+ P2
1)]
+α123(P2
1P2
2P2
3)−1
2sD
11(σ2
1+σ2
2+σ2
3)−sD
12(σ1σ2+σ2σ3+σ1σ3)
−1
2sD
44(σ2
4+σ2
5+σ2
6)−E1P1−E2P2−E3P3
−Q11(σ1P2
1+σ2P2
2+σ3P2
3)−Q12[σ1(P2
2+ P2
3) + σ2(P2
1+ P2
3) + σ3(P2
2+ P2
1)]
−Q44(σ4P2P3+σ5P1P3+σ6P1P2)
(2)
6
D. Braun I. Ferroelectricity & Materials
where α’s are the dielectric stiffnesses, Qij the electrostrictive constants, sD
ij the elastic
compliances, σithe stress components, Pmis the polarization and Emis the electric
field.
In order to facilitate the understanding of eq. 2, the physical meaning of these coeffi-
cients shall be explained in more detail.
First, the dielectric stiffness displays the maximum electric field that a material can
withstand without breaking down. Within the latter, bound electrons become free due
to the incorporated energy of the electric field yielding conductive paths through the
material.
The electrostrictive constant Qij describes the deformation of a dielectric material in
dependence of the applied electric field. In contrast to the piezoelectric effect that
reflects the linear mechanic response under an electric field, Qij considers only the part
independent of the direction of the electric field and thus the part proportional to E2.
The third term, the elastic compliances sD
ij , classify the resistance a material opposes its
mechanical deformation. Hereby, only a reversible deformation is taken into account.
Last, the stress tensor σis a tensor of second rank, describing the mechanic deforma-
tion of material in reference to the initial ground state.
It should be noted that the tensors in eq. 2 and in the following are written in Voigt
notation as exemplary shown in eq. 3.
[cαβ]V=
c11 c12 c13 c14 c15 c16
c21 c22 c23 c24 c25 c26
c31 c32 c33 c34 c35 c36
c41 c42 c43 c44 c45 c46
c51 c52 c53 c54 c55 c56
c61 c62 c63 c64 c65 c66
V
:=
c1111 c1122 c1133 c1123 c1113 c1112
c2211 c2222 c2233 c3311 c2213 c2212
c3311 c3322 c3333 c3311 c3313 c3312
c2311 c2322 c2333 c3311 c2313 c2312
c1311 c1322 c1333 c3311 c1313 c1312
c1211 c1222 c1233 c3311 c1213 c1212
(3)
When ∆G is plotted as a function of the polarization, the temperature dependent min-
ima can be extracted. As an example, I calculated the energy landscape of Pb(Zr0.40Ti0.60)O3
at room temperature revealing only one stable phase which can be related to the tetrag-
onal phases as illustrated in fig. 1.
To link these theoretical predictions to the experiment it can be described as fol-
lowed: Phenomenologically, at a 1st order phase transition the polarization changes
discontinuously or abruptly and the permittivity of the ferroelectric has a finite max-
imum in the intersection between the phases. In contrast, the phase transition of 2nd
7
D. Braun I. Ferroelectricity & Materials
Figure 1: Gibbs free energy ∆G plotted as a function of the polarization Pifor the tetragonal
phase of Pb(Zr0.40Ti0.60)O3at room temperature. No strain or electric field is applied.
order is characterized by a continuous change in the polarization and a break of per-
mittivity. Examples include BaTiO3and LiTiO3showing a 1st and 2nd order phase
transition, respectively.
1.3. Ferroelectric domains
These transitions are usually accompanied by a structural transformation. Especially
evident is this phenomenon for perovskites ABO3, as they are studied in this work
(for more details see chapter 2.1). An example to be discussed at this point is barium
titanate. At high temperatures, the perovskite unit cell is arranged as a face-centered
cubic one. The titanium ion forms the inversion center and is coordinated by eight
barium ions in the cube corners and six oxygen ions in the cube faces. This phase is
non-polar and is classified as paraelectric. If the temperature decreases, the system un-
dergoes from a temperature T on, which is equal to the so-called Curie temperature Tc
(T = Tc), a structural transition with breaking the symmetry. The unit cell is slightly
stretched in one direction and loses the former cubic symmetry under manifestation
of a tetragonal structure. Consequently, the titanium ion is shifted from the central
position and a local electric dipole moment develops. The entirety of these dipoles is
referred to as crystal spontaneous polarization Ps.
With the formation of a spontaneous polarization the process has not been completed
in terms of energy minimization. A uniform orientation of all local dipole moments
would result in charged surfaces (and interfaces). Therefore, as a counter-force a de-
polarization field is formed. A balance of both forces is only able if the spontaneous
8
D. Braun I. Ferroelectricity & Materials
polarization is equally distributed. This process is called formation of ferroelectric do-
mains.
In the case of the discussed barium titanate with its tetragonal structure below the
Curie temperature two opposite polarization direction along the long axis may occur.
This configuration is called 180◦- or c-domains.
In other crystal classes, for example, in orthorhombic or rhombohedral lattices, the
spontaneous polarization can be aligned along the two perpendicular in-plane lattice
vectors, which is then described as 90◦- or a1(or a2) domains. Likewise a lateral do-
main is formed, when the polarization vector is along the in-plane face diagonal referred
as a1a2formation. Moreover, ferroelectric phases appear that have both a vertical and
a lateral polarization component. These inclined arrangements are known as a1c, a2c
and r domains.
Due to the fact, that the nomenclature arose from tetragonal systems, theses domain
types got sometimes different notations in other symmetries. For example, the latter
ones are usually depicted as Mcand MAphases in monoclinic crystals.
Basically, there are also cases in which different structures so-called multidomain ar-
rangement as e.g. a/c pattern.
Additionally, in all these cases the urge for electrical neutrality has to be considered.
This means that the cost of a separating domain wall is only efficient if two positive
(or negative) charges are isolated. This boundary condition can be fulfilled when the
electric polarization vectors are arranged in a ”head-to-tail” configuration. This for-
mation is illustrated for different domain structures in fig. 2.
Figure 2: Schematic illustration of the domain wall formation by means of (a) 180◦domain walls
in e.g. c+/c−domain structures and (b) 90◦walls in pure lateral a1/a2and alternating
a/c domain structures in ”head-to-tail” configuration (d: crystal thickness, D: domain
width). Taken from [39].
9
D. Braun I. Ferroelectricity & Materials
The question which domain width screens effectively the charge and is economically
to be build up can be answered only theoretically from energetic considerations. In
order to solve the equations analytically, the configuration has to be restricted to two
opposing domains.
On one side of the energy balance is the expenditure for the construction of the domain
wall Ewall. On the other side stands the energy gain by minimizing the depolarization
field Efield. The two opponents can be written as followed [40]:
Ewall =σV
D(4)
Efield =εDV P2
0
t(5)
whereby σis the energy of the domain wall per unit cell, D the domain width, P0the
polarization, εis the dielectric constant of the material, V denote the crystal volume
and t expresses the film thickness. Under equilibrium conditions, Ewall = Efield applies
and in the following a description for the domain width D can be calculated [40]:
D=stσ
εP2
0(6)
This root dependency D ∝√t was former predicted by Kittel [41] for ferromagnets
and could be confirmed experimentally.
In a real crystal structure, additional factors such as defects, tension or free charge
carriers occur that will affect the domain formation (and thus the periodicity) and will
moreover complicate the structure. As one example, herringbone pattern are discussed
in detail in chapter 11.
1.4. Ferroelectric switching
The switching behavior of the electric polarization Pas a function of the applied electric
field Eis shown schematically in fig. 3. First, the application of an electrical voltage
leads to an alignment of the local dipoles along the electric field vector. Finally, all
the dipoles are oriented and a saturation is reached. This condition is referred to as
ferroelectric poling. The fascinating thing about ferroelectrics is that with decreasing
of voltage, the initial state can never be achieved because some dipoles remain in their
switched position. Thus, even when the applied electric field is zero a polarization is
still measurable. This effect is called remnant polarization PR. When the material-
specific coercivity is reached, the distribution of dipoles is identical. The experimental
10
D. Braun I. Ferroelectricity & Materials
Figure 3: Polarization in dependence of the electric field for a ferroelectric material. A typical
hysteresis shape develops for ideal ferroelectrics.
course in counter-rotating applied field looks the same and closes the curve in the P-
E-diagram to a hysteresis.
What happens microscopically during this process in the crystal is shown in a model-
like sequence in fig. 4. Based on an idealized monodomain state between a top and
Figure 4: Ideal ferroelectric domain switching process in a sandwich structure shaped as elec-
trode/ferroelectric/electrode structure. (a) First step is the nucleation of small domains
with polarizations opposite to the surrounding material at the ferroelectric interface.
(b) In the following, the domains grow needle-like towards the reverse electrode. (c)
During the last step, the lateral domain expansion affiliates yielding complete reversion
of the initial macroscopic polarization. Taken from [39].
a bottom electrode, applying an electric voltage leads to nucleation of domains with
opposite polarization. Thereby, nucleation is preferred especially on defects due to the
presence of kink positions. In the next step, the domains grow needle-like through the
film to the mutual electrode. Due to the narrow lateral extension, the macroscopic
polarization change is not significant. After reaching the backside of the crystal, the
horizontal movement of the domain walls starts with rapid growth in lateral domain
size. Finally, adjacent domains unite to a complete reversal of the crystal. This process
11
D. Braun I. Ferroelectricity & Materials
can take nanoseconds to microseconds for ferroelectric oxides.
1.5. Piezoelectric coefficient
A piezoelectric material can theoretically be distorted mechanically in all three di-
mensions xiwith i = 1,2,3 by applying an electrical voltage. Since this is a multi-
dimensional problem, the mathematical description of this process needs the use of
tensor geometries. The charge density Dias a result of stress σjk can be written as
follows:
Di=dijk ·σjk (7)
where the indices i, j, k correspond to the three spatial directions. dijk is a third-rank
tensor and reflects the system’s symmetry reducing the unknown parameter immensely.
Furthermore, the tensor elements in Voigt notation are quite descriptive:
if a voltage is applied perpendicular to the film in the z=3-direction and the mechanical
stress in the same direction is of interest, the tensor entry d33 has to be calculated.
That is exactly the situation later in the PFM experiment.
Since the coordinate system in the film does not have to match the laboratory system,
in addition a coordinate transformation is necessary to evaluate the correct d0
33.
d0
ijk =A·dijk ·NT(8)
The matrix A rotates the system along all three coordinate axes [6]:
A =
a11 a12 a13
a21 a22 a23
a31 a32 a33
(9)
A =
cos ϕcos φ−cos θsin ϕsin φcos ϕsin φ+ cos θsin ϕcos φsin θsin ϕ
−sin ϕcos φ+ cos θcos ϕsin φ−sin ϕsin φ+ cos θcos ϕcos φcos ϕsin θ
sin θsin φ−sin θcos φcos θ
(10)
12
D. Braun I. Ferroelectricity & Materials
and NTis calculated from the matrix entries of A according to [42]:
NT=
a2
11 a2
21 a2
31 2a21a31 2a31a11 2a11a21
a2
12 a2
22 a2
32 2a22a32 2a32a12 2a12a22
a2
13 a2
23 a2
33 2a23a33 2a33a13 2a13a23
a12a13 a22a23 a32a33 a22a33 + a32a23 a12a33 + a32a13 a22a13 + a12a23
a13a11 a23a21 a33a31 a21a33 + a31a23 a31a13 + a11a33 a11a23 + a21a13
a11a12 a21a22 a31a32 a21a32 + a31a22 a31a12 + a11a32 a11a22 + a21a12
In case of an orthorhombic symmetry, the coefficient d0
33 can be estimated as follows:
d0
33 = cos θsin θ2sin φ2(d24 +d32) + sin θ2cos φ2(d15 +d31) + d33 cos θ2(12)
Figure 5: Angular dependency of the longitudinal piezoelectric coefficient d0
33 for the orthorhom-
bic KNbO3crystal without any strain or electric field applied in a temperature range
between T = 270 −490 K. Calculated on the basis of eq. 12 and the material constants
taken from [38].
As a result from these calculations, the measured vertical piezoelectric coefficient d0
33
is significantly influenced by shear components.
I plotted the angular and temperature-dependent d0
33 for KNbO3as depicted in fig.
5. Herefrom, the direction of the maximal response can directly be read. As a result,
information can be gained if the maximal longitudinal response can be expected along
the polar axis or tilted away in a non-polar direction.
The piezoelectric coefficients have the unit m
Vand are therefore directly suitable for
classifying the piezoelectric response in dependence of the applied field in the respec-
tive spatial direction.
13
D. Braun I. Ferroelectricity & Materials
2. Ferroelectric thin films
2.1. Perovskite structure
Perosvkite is the name for CaTiO3, denoted after the Russian mineralogist Lew Alex-
ejewitsch Perowski. However, the structure is inherent in a wide range of oxides (e.g.
all used compositions in this work). Consequently, perovskite became a generic term
for oxides crystallizing in the general composition formula ABO3structure with A and
B being mono- to trivalent and tri- to pentavalent cations, respectively [43]. The ion
formation within the unit cell can be schematically illustrated as shown in fig. 6.
Figure 6: The perovskite unit cell for the general composition ABO3in the case of a cubic sym-
metry.
Depending on chemical composition and external parameter as pressure and temper-
ature, the crystal structure exhibits cubic, tetragonal, orthorhombic or rhombohedral
symmetry. In reality, the distortion is limited and can be estimated by Goldschmidt’s
factor κwhich constitutes a dimensionless quantity that describes the ratio of cation
radii - rAand rB- to the anion radius rO[44]:
κ=rA+rO
√2(rB+rO)(13)
Hereby, the parameter κcan only take values between 0.71 ≤κ≤1 for a perovskite
material [45]. For the crystal lattice, the changing Goldschmidt factor ultimately leads
to a variation of the crystal symmetry from cubic to orthorhombic to rhombohedral.
14
D. Braun I. Ferroelectricity & Materials
The end member κ= 1 means cubic structure as built in strontium titanate.
However, fascinating regarding eq. 13 is that different physical properties like fer-
roelectricity or magnetism can arise depending on the material composition, but all
perovskites have similar lattice parameters. This circumstance allows the sequential
deposition of different perovskite materials without large lattice mismatch and even-
tually the combination of different functional layers.
2.2. Introduction to Linear elasticity theory
Under an external force, the atoms within the crystal lattice are displaced from their
equilibrium position by the vector u(r). In order to determine the relationship between
u(r) and the elasticity tensor εlm, the displacement between two points within an
infinitely small volume has to be considered. Mathematically, the position of two
points shifted relatively to one another can be solved for each direction individually by
means of a Taylor expansion [[46],[47]]. Due to the small volume under investigation,
the calculation is truncated after the linear part. Consequently, the theory is named
as ”linear” elasticity theory.
Summarizing the distortion for the three dimensional, atomistic arrangement in tensor
notation yields eq. 14:
u(r) =
∂ux
∂x
1
2∂ux
∂y+∂uy
∂x1
2∂ux
∂z+∂uz
∂x
1
2∂uy
∂x+∂ux
∂y∂uy
∂y
1
2∂uy
∂z+∂uz
∂y
1
2∂uz
∂x+∂ux
∂z1
2∂uz
∂y+∂uy
∂z∂uz
∂z
x
y
z
(14)
u(r) =
εxx εxy εxz
εyx εyy εyz
εzx εzy εyy
r(15)
Eqs. 14 and 15 describe the relationship between displacement uiand strain εlm. As a
result, the respective strain components can be written as follows:
εlm =1
2∂ul
∂xm
+∂um
∂xll, m =x, y, z = 1,2,3 (16)
Experimentally, if the strain should be calculated for an epitaxial film along the sub-
strate directions, thus for identical l = m, eq. 16 can be simplified. For the in-plane
lattice parameter of the film afilm and substrate asubstrate, eq. 16 can be written as:
εll =asubstrate −afilm
afilm
(17)
15
D. Braun I. Ferroelectricity & Materials
In conclusion, the application of an external force as e.g. epitaxial stress on a mate-
rial leads to the displacement of the unit cell atoms from their equilibrium position. In
consequence, the inherent symmetry of the material can be artificially modified. Es-
sential for this symmetry transformation is the direction of the executed force. Hence,
in the example of epitaxial growth the difference in lattice parameter and if relevant
the discrepancy between substrate and film symmetry.
A special but important aspect is the application of anisotropic stress. This way, a low-
ering of the symmetry can be achieved which may enhance the piezoelectric properties
as it should be presented in more detail in the following chapters.
2.3. Strain engineering in perovskite materials
During epitaxial growth, the film adapts first to the in-plane lattice parameter of the
substrate. This process takes as long as the critical film thickness is not exceeded.
The adoption of the ”foreign” lattice parameter of the substrate stresses the lattice
artificially. This process is schematically illustrated in fig. 7a for a perovskite unit cell
without the oxygen ions for a better visibility of the strain effect.
As can be seen in fig. 7a, the incorporated strain can be either compressive or tensile
nature. In the former one, the film lattice has do adopt onto smaller in-plane substrate
lattice parameter compared to the initial film in-plane lattice values. In consequence,
the film is elongated perpendicularly to the film-substrate interface.
In the case of tensile strain, the film lattice has to be stretched to fit onto the larger
in-plane lattice parameter of the substrate yielding a parallel elongation of the unit
cells in respect to the film-substrate interface.
In any case, the result is a distorted lattice structure that can undergo hereby a struc-
tural phase transition. In the case of perovskites, especially the BO6octahedra are
likely to be tilted which may lead to enormous changes in film properties such as Curie
temperature or piezoelectric properties. In addition, properties which are not apparent
for the bulk crystal can be introduced into thin films via lattice-mismatched growth.
This process is called ”strain engineering”.
A popular example is the artificially introduced ferroelectricity in strained strontium
titanate. The respective misfit strain-temperature phase diagram is illustrated in fig.
7b. In the unstressed case, SrTiO3has a cubic symmetry at room temperature and thus
a paraelectric behavior. But, if the material is grown heteroepitaxially on a substrate,
16
D. Braun I. Ferroelectricity & Materials
SrTiO3can be stressed this way that a paraelectric-to-ferroelectric phase transition is
achieved as it is shown in fig. 7b.
Figure 7: (a) Influence of lattice mismatched growth on a perovskite unit cell. For a better
visibility of the strain effect, the perovskite unit cell is drawn without the oxygen ions.
(b) Misfit strain-temperature phase diagram for SrTiO3. Taken from [48].
In the Leibniz Institute for Crystal Growth the unique possibility exists to have access
to a huge variety of different substrate materials and thus in-plane lattice parameter.
Most of them have orthorhombic symmetry. Due to the oftentimes huge orthorhombic
unit cells, the calculation of appropriate substrate-film combinations is impeded. For
that reason, usually a transfer into the pseudocubic notation is made to describe only
one perovskite unit. In order to simplify the calculations, octahedral tilts are neglected.
The conversion between orthorhombic ”o” and pseudocubic ”pc” lattice parameters is
17
D. Braun I. Ferroelectricity & Materials
executed in accordance to the equations of Vailionis [49]:
apc =co
2(18)
bpc =pa2
o+b2
o−2aobocos γ0
2(19)
cpc =ra2
o+b2
o−2b2
pc
2(20)
αpc = arccos b2
pc +c2
pc −a2
o
2bpccpc (21)
If γ0= 90◦is chosen, eqs. 18-21 can be simplified to:
apc =co
2(22)
bpc =pa2
o+b2
o
2=cpc (23)
αpc = arccos 1−a2
o
2b2
pc (24)
Hereby, fig. 8 serves for a better illustration of the relationship between orthorhombic
and pseudocubic notation.
Figure 8: The relationship between orthorhombic and pseudocubic notation for the case of γ0=
90◦.
Due to the fact, that the orthorhombic lattice parameter are not equal (ao6= bo), the
pseudocubic unit cell is actually monoclinically distorted. Therefore, the angle αpc
corresponds to the monoclinic distortion angle βvia:
β= 90◦−αpc (25)
18
D. Braun I. Ferroelectricity & Materials
Using eqs. 22-24, the pseudocubic lattice parameters for the (110)ogrowth plane of
different, available substrates can be calculated and are depicted in tab. 1. The values
in tab. 1 clarify that most substrates have anisotropic in-plane lattice parameter, po-
tentially yielding biaxial anisotropic strain.
Table 1: Orthorhombic and pseudocubic in-plane lattice parameter of different substrates for the
(110)ogrowth plane.
Substrate aoboco[Ref.] apc bpc
NdGaO35.428 5.498 7.708 [50] 3.854 3.863
SrTiO3(cubic) 3.905 - - [50] 3.905 3.905
DyScO35.442 5.719 7.904 [51] 3.952 3.947
TbScO35.466 5.731 7.917 [51] 3.959 3.960
GdScO35.480 5.746 7.932 [51] 3.966 3.970
SmScO35.527 5.758 7.965 [51] 3.991 3.983
NdScO35.575 5.776 8.003 [51] 4.014 4.002
The availability of these substrates leads to the central question:
Which material is interesting both from a technological point of view as well as a
fundamental study of ferroelectricity?
Hereby, two aspects have to be satisfied. The material should (i) be lead-free and (ii)
should yield high piezoelectric coefficients.
One candidate will be presented in chapter 2.4.
19
D. Braun I. Ferroelectricity & Materials
2.4. Potassium sodium niobate KxNa1−xNbO3
Lead-free potassium sodium niobate was grown as bulk crystal first in 1954 [52].
The most promising piezoelectric properties have been found for the composition
x=0.5. Here, a sort of morphotropic phase boundary exists whereby the symmetry
changes from orthorhombic (Pm) to orthorhombic (Amm2) via an oxygen octahedral
tilt [[53],[54]].
This concentration is characterized by a relatively high Curie temperature of Tc=
415◦C [2] and a high coupling coefficient kp= 0.51 [1]. Moreover, the longitudinal
piezoelectric coefficient amounts d33 = 80pC/N [1].
Quite similar to other known perovskites, KxNa1−xNbO3with x ≥0.5 passes the poly-
morphic phase sequence cubic (T >690 K), tetragonal (T between 690 −465 K),
orthorhombic (T between 465 −123 K) and rhombohedral (T <123 K) [55]. Thus,
KxNa1−xNbO3(x ≥0.5) crystallizes in the orthorhombic symmetry at room temper-
ature. Below x = 0.5, the crystal symmetries and phase transitions are more compli-
cated, especially for x = 0 (NaNbO3). Therefore, they are not considered here.
Potassium sodium niobate is studied as bulk single crystal [[56],[57]] or as ceramic
[[58],[59]], but rather less is known about the fabrication and electrical properties in
thin film form [[60],[61]]. Typically, KxNa1−xNbO3thin films have been grown by phys-
ical methods like sputtering [[62],[63]] or pulsed laser deposition [[64],[65]]. Few papers
report growth by chemical sol-gel techniques [[66],[67]]. All of these films exhibit low
crystalline quality. Only MOCVD films grown by Schwarzkopf et al. [68] have been
shown to be single crystalline and almost stoichiometric.
Comparing the in-plane lattice parameter in the pseudocubic notation of KxNa1−xNbO3
to those of the substrates depicted in tab. 1 yields fig. 9. It can be seen that the strain
can be tailored from compressive to tensile in small steps by changing either the potas-
sium concentration x or the substrate, especially the rare-earth scandates.
The evolving ferroelectric phase as a consequence of changing epitaxial strain can be
read from a misfit strain-misfit strain phase diagram as it was calculated by Bai and Ma
[3] for KNbO3(see fig. 10). As can be seen in fig. 10, under biaxial anisotropic strain
monoclinic phases (like r0, ac, a1a2) can be provoked in KNbO3. This result is novel
and only possible by extending the Gibbs free energy function to 8th order. Indeed,
such an anisotropy can be incorporated with the use of orthorhombic substrates as they
are depicted in tab. 1. A similar strain-phase relation is expected for KxNa1−xNbO3
with x ≥0.5.
20
D. Braun I. Ferroelectricity & Materials
Figure 9: Number line of the average in-plane lattice parameter for: on the top, the end members
of the solid solution KxNa1−xNbO3, on the bottom, the substrates depicted in tab. 1.
In addition, the position of in-plane lattice parameters of PbZr0.50Ti0.50O3according
to [69].
The material mostly used for electronic application is lead zirconium
titanate (PbZrxTi1−xO3). At about 50% zirconium concentration, a MPB separates
the rhombohedral and tetragonal phase symmetry. In between, a small composition
range exists where a monoclinic symmetry evolves that is accompanied by giant piezo-
electric responses. In detail, excellent piezoelectric properties as a high longitudinal
piezoelectric coefficient d33 = 413 pm/V and a coupling constant kp= 0.68 [70] have
been measured. However, it is difficult to obtain exclusively this monoclinic phase [71].
The phase transition point between tetragonal-to-monoclinic-to-rhombohedral exists
at room temperature, whereby the tetragonal phase seems to be the most stable one.
Here, the lattice parameters are at= 4.037 ˚
A and ct= 4.138 ˚
A [69]. As it can be seen
in fig. 9, these values are extremely large in comparison to the depicted rare-earth
scandate substrates. Thus, epitaxial growth of fully strained thin films on commonly
used substrates is only possible for very thin films. In addition, strain engineering of a
tetragonal system is predicted to be rather inefficient in regard of the enhancement of
the piezoelectric response as it will be denoted in chapter 2.5 [[38],[72],[73]].
In conclusion, KxNa1−xNbO3is lead-free, non-toxic, has a Curie temperature far
above room temperature and a comparable coupling coefficient to lead zirconium ti-
tanate. Moreover, monoclinic phases can be introduced via anisotropic lattice strain.
This factor enables to enhance the piezoelectric coefficient as it will be discussed in
chapter 2.5 in order to be an alternative to lead zirconium titanate.
21
D. Braun I. Ferroelectricity & Materials
Figure 10: Misfit strain - misfit strain phase diagram calculated by Bai and Ma [3] for KNbO3at
room temperature. Taken from [3].
How the piezoelectric response can be enhanced, should be discussed in a general
manner in chapter 2.5.
2.5. Enhancement of piezoelectric responses
Giant piezoelectric responses were observed first in monoclinic systems at the mor-
photropic phase boundary in ferroelectric lead zirconium titanate (PZT)-based solid
solutions [[69],[74],[75],[76]]. There, the crystal symmetry changes from rhombohedral
to tetragonal via a monoclinic bridging phase.
The question why this monoclinicity is related to enhanced piezoelectric amplitudes
has been debated for a long time and should be discussed at this point.
The most popular picture of piezoelectricity is a field applied along the polar axis
and the subsequent expansion or shrinking of the material in the same direction. This
behavior can be described mathematically.
The spontaneous strain Sij = QijklPkPl, or maybe more schematically the piezoelectric
22
D. Braun I. Ferroelectricity & Materials
response, changes as a result of an electric field by ∆Sij:
Sij + ∆Sij = Qijkl(PkPl+ ∆Pk∆Pl) (26)
∆Sij ≈Qijkl(∆PkPl+ Pk∆Pl)=Epdpij (27)
For a polarization distribution P= (0,0,P3), an electric field E= (0,0,E3) leads to:
∆S33 = 2Q3333∆P3P3= E3d333 (28)
⇔∆P3=E3d333
2Q3333P3
(29)
⇔∆P3
P3
=E3d333
2Q3333P2
3
=E3ε0χ33
P3
(30)
(31)
Consequently, the response depends on the piezoelectric coefficient d333 = d33 and
thus on the dielectric susceptibility χ33. Such a behavior is called an ”Extender”.
In the next step, the case of a lateral applied electric field E= (E1,0,0) should be
considered:
∆S13 = 2Q1313∆P1P3= E1d113 (32)
⇔∆P1=E1d113
2Q1313P3
(33)
⇔∆P1
P3
=E1d113
2Q1313P2
3
=E1ε0χ11
P3
= tan θ(34)
As a result, the polarization vector rotates away from the former polar axis and the
response depends on the piezoelectric coefficient d113 = d15 and thereby from χ11. This
means, the response can be higher along a non-polar axis compared to the polar one
for a material whose χ11 > χ33. This behavior was something completely unexpected.
These materials are named ”Rotators”.
Within the aforesaid model, enhanced piezoelectric responses known from phase tran-
sition points can be understood: the lattice softens when approaching the phase tran-
sition which is accompanied or expressed in extremely large susceptibilities χij yielding
enhanced piezoelectric properties.
It can be divided immediately between these two species on the basis of polfigures
like fig. 5 or energy landscapes like fig. 1. For the former one, barium titanate is an
impressive example. It is tetragonal between T = 279−365 K, so the polar axis is along
23
D. Braun I. Ferroelectricity & Materials
[001], but the maximal d0
33 response can be far away from the polar axis namely when
the phase transition is near (see fig. 11). The reason is a large dielectric susceptibility
χ11. Then the maximal piezoelectric response can be expected 51.6◦(at T = 279 K)
Figure 11: Angular dependency of the longitudinal piezoelectric coefficient d0
33 for the tetragonal
phase of BaTiO3without any strain or electric field applied in a temperature range
between T = 279 −365 K. Taken from [38].
or 43.1◦(at T = 315 K) tilted away from the polar axis.
Rotators can be identified likewise when the Gibbs free energy is plotted as a function
of polarization. As an example, I depicted Pb(Zr1−xTix)O3at room temperature for
different compositions x in fig. 12. In the rhombohedral phase in fig. 12a, the four
h111idirections are energetically preferred, whereas in the tetragonal phase in fig. 12c,
the both c-domain states have the lowest energy. In between, when the composition
approaches the morphotropic phase boundary at x = 0.5 a clear propensity for a polar-
ization rotation can be seen where the transition pathways along the monoclinic mirror
planes are extremely flat.
So, indeed, the Gibbs free energy instability is the thermodynamic origin of the
enhanced piezoelectric response. This flat energy landscape can be achieved only in
monoclinic symmetries. In contrast to high symmetric point groups, here, the prior
symmetry element is no axis but the monoclinic mirror plane. Consequently, the po-
larization vector is able to rotate continuously within this plane. That discussion
explains the observed giant piezoelectric responses at the morphotropic phase bound-
ary in Pb(Zr,Ti)O3where the material crystallizes in the monoclinic orientation.
24
D. Braun I. Ferroelectricity & Materials
Figure 12: Gibbs free energy in dependence of the polarization Pifor (a) x = 0.2, (b) x = 0.4
and (c) x = 0.6. Calculated according to the material constants [77].
It should be noted, to calculate the same energy surface as in fig. 12 for monoclinic
phases would need (i) an expansion of eq. 2 to the 8th order and (ii) the required
material specific constants. Due to the lack of the latter, it can not be presented here.
If a material will behave like an extender or rotator can not only be estimated from
the aforementioned polfigures or energy landscapes but can be calculated. Mathemat-
ically, the turning point has to be evaluated:
∂d0
33
∂θ = 0 (35)
The solution is approximately the same for all crystal symmetries. A material behave
25
D. Braun I. Ferroelectricity & Materials
like a rotator when: d15
d33 ≥3
2−Q1313
Q3333 ≈3
2(36)
To have an impression which material is suited, the ratio 36 is calculated for some
examples in tab. 2 according to the piezoelectric coefficients depicted in [73]:
Table 2: Ratio of the shear and longitudinal piezoelectric coefficient d15 and d33, respectively, for
different materials [73].
PbTiO3BaTiO3KNbO3PMN −33PT
d15
d33 0.7 6.0 7.0 22.0
It has to be noted, that PMN−33PT is a relaxor material and is therefore not suitable
for classical domain engineering. In consequence, potassium niobate is a perfect rota-
tor candidate and hence a good combination of the two aspects lead-free and expected
high piezoelectric responses.
26
D. Braun II. Growth & Characterization methods
Part II.
Growth & Characterization methods
3. Thin film deposition
The metal organic chemical vapor deposition technique (MOCVD) is a coating method
related to the group of chemical deposition processes. The central point is the growth
of a single layer from the vapor phase. The latter consists of a chemical compound of
a metal-organic precursor and the material to be deposited. The film stoichiometry
can be controlled by the composition in the gas phase. Thus, it is possible to produce
films with high crystalline quality and precise stoichiometry.
Until now, this method has not been applied for potassium sodium niobate due to
both the low availability of suitable MO precursor as well as to the high volatility of
the alkali components. Challenging is that the source material exists oftentimes only
in the solid state and has high vapor pressures. So a slightly modified variant of the
typical MOCVD is applied as illustrated in fig. 13.
Figure 13: Schematic sketch of the MOCVD system. For details see text. Taken from [39].
In the liquid-delivery MOCVD, the metal-organic precursor is previously dissolved in
an organic solution. For the deposition of potassium sodium niobate, K(thd) and
Na(thd) ((thd) = 2,2,6,6-tetramethyl-3,5-heptanedione) and Nb(EtO)5were dissolved
27
D. Braun II. Growth & Characterization methods
in dry toluene and the flash evaporators were heated up to 180◦or 190◦, respectively.
Via two peristaltic pumps the fluid is conveyed into independent flash evaporators and
is dropped onto a hot plate for vaporizing. In the next step, the gaseous components
were directed by means of an inert carrier gas, in this case argon, into the reaction
chamber. Simultaneously, pure oxygen was introduced in the same chamber. The gas
mixture now passed a shower head whose core is a combination of a porous - and a hole
plate to ensure uniform distribution of oxygen (see fig. 13). On an underlying carrier
made of silicon nitride the substrates are fixed with silver glue and heated up by a rear
filament. This carrier is put in continuous rotation and swirled the gas mixture of MO
precursor, argon and oxygen and thereby ensured a homogeneous composition.
Common parameters for film deposition were a gas pressure in the reaction chamber
of 2.6 kPa and a ratio of gases O2/Ar = 0.63. Furthermore, the carrier was operated
with 560 revolutions per minute and the depositions were performed at 700◦C.
All substrates used have an intentional off-orientation of 0.1◦towards the [1¯
10]o
direction and are first cleaned with acetone and 2-propanol. The subsequent annealing
in pure oxygen for one hour at T = (1050−1150)◦C leads to the formation of a regular
step-terrace surface structure with atomic steps (≈4˚
A) and terraces of about 200 nm
in width. This structure favors the nucleation and results in an increased growth rate.
Another advantage of annealing is that a single surface termination remains. For more
details, see [68].
4. Atomic Force Microscopy
The atomic force microscope (AFM) is a scanning probe method, which was developed
in 1985 by Gerd Binnig, Calvin Quate and Christoph Gerber [78]. It becomes one of
the most important tools in surface science for mechanical scanning of surfaces. By
measuring atomic forces, characteristic properties of materials can be investigated on
the nanometer scale. So, in regard of the achievable resolution it represents a bridging
technique between the light - and electron microscopy.
A schematic sketch of an atomic force microscope is shown in fig. 14a.
The main components are either a white light-emitting diode or a laser (depending
on the design), the photodetector and the probing tip. The latter forms the core of the
machine and consists mostly of a rectangular cantilever beam with dimensions (length
x width) of (100 −200)µm×(10 −40)µm. An ultra-sharp tip is mounted at one end
28
D. Braun II. Growth & Characterization methods
Figure 14: (a) Scheme of a typical AFM setup. For a detailed description see text. (b) left:
deflection and right: torsion of the cantilever caused by forces acting on the tip from
different directions. The resulting reflected cantilever beam into the detector is marked
by red dots. Adapted from [39].
of the beam with a diameter of typically (10 −20)nm. To manufacture these probes
usually silicon or silicon nitride are used. Depending on the force to be detected, the
cantilever can be additionally coated, eg with conductive metals. This entire beam is
characterized by the spring constant k with a typical range of k = (0.01 −40)Nm−1.
The sample is placed on a scanning stage which can be moved in x-y direction man-
ually or by piezo stack actuators. Next, a probing tip is brought into contact with the
surface.
During the measurement, the nanoscopic needle is scanned line by line in a defined
grid over the sample surface. As a result of the changing morphology, the beam is dif-
ferently deflected. Via the reflection of the laser spot on the backside of the cantilever,
this deflection is recorded on the detector as depicted in fig. 14b. The latter consists
of four segments A-D in combination with a differential amplifier and is basically a
diode. The task is to convert the incident light signal into a voltage pulse in order to
scale the topography in meters.
The changing light intensity between the upper two sections A + B and the lower ones
C + D yields information on the vertical deflection of the cantilever whereas the differ-
ence between the areas of A + C and B + D reflects the torsional forces. In sum, the
measurement method plots a three-dimensional map of the surface.
In order to ensure a resolution on the nanometer scale, the measuring system must
be shielded against external influences. For this purpose, the AFM is positioned on
an active, vibration damping table (in our lab an Accurion Halcyonics micro 40) for
29
D. Braun II. Growth & Characterization methods
absorption of mechanical vibrations, e.g. subsonic noise. On the other hand, the entire
setup is enclosed in an acoustic hood made of steel for shielding of acoustic vibrations.
5. Piezoresponse Force Microscopy
Basically, the piezoresponse force microscopy (PFM) is a special mode of the AFM as
it is shown in fig. 15. The imaging in PFM mode is based on the inverse piezoelectric
Figure 15: Model of the PFM mode as complement to the basic AFM setup shown in fig. 14a.
Taken from [39].
effect. As discussed in chapter 1, the application of an electrical voltage leads to a
mechanical deformation in piezoelectric materials. In the PFM, the bias is applied via
a conductive tip which is in contact with the surface. For that purpose, an alternating
electric field is used stimulating an oscillation in the material underneath, which in
turn enables the cantilever to vibrate. When the applied potential difference has the
form of:
Vtip =Vdc +Vac ·cos(ωt) (37)
the 1st Harmonic d of the cantilever can be calculated as follows [30]:
d=d0+A·cos(ωt +φ) (38)
In eq. 38, d0represents the static component due to the dc component, A is the
amplitude and φis the phase of the signal. This means that the material gives a piezo-
electric response to the applied ac voltage with the characteristic values: amplitude
A and phase shift φ. To measure and especially separate these signals from both the
30
D. Braun II. Growth & Characterization methods
input signal and the topography a lock-amplifier is used. As shown in fig. 16 important
conclusions can be gathered from these two parameters on the domain structure. If the
Figure 16: Piezoresponse signal (black) (a) in-phase (φ= 0◦) and (b) out-of-phase (φ= 180◦)
with the applied alternating voltage marked in red color. Taken from [39].
electric field vector Eof the applied voltage is parallel to the polarization vector Pthe
sample will extend in the vertical direction by means of a lateral reduction as can be
seen in fig. 17a. In the case of an antiparallel arrangement, the solid will shrink in the
Figure 17: The converse piezoelectric effect as a result of an applied voltage between the conduc-
tive tip and a bottom electrode in a piezoelectric material. The induced tip deflection
as well as the corresponding position of the reflected beam on the photodiode are
displayed by red dots. The effect is shown for (a) vertical and (b) lateral polarization
alignments. Taken from [79].
z direction and expand in the x direction. In the detector, this response is recorded as
amplitude, namely ∆z, and phase shift ∆φ. This allows a distinct characterization of
vertical domains in view of the orientation of the polarization vector and is referred to
vertical PFM (VPFM). If the electric field vector is aligned parallel to the polarization
31
D. Braun II. Growth & Characterization methods
vector, there will be no phase shift ∆φ= 0◦between the signals, whereas in antiparallel
position a phase shift of ∆φ= 180◦occurs. The amplitude, however, reveals some-
thing about the strength of polarization. This longitudinal displacement is related to
the tensor element d33.
However, if the polarization of the sample is aligned in the film plane, and consequently
perpendicular to the electric field, this will lead to a shearing of the sample surface
and therefore to the torsion of the cantilever. As already stated in the AFM chapter
4, this rotation of the tip can be measured as a light intensity difference in the areas
between A + C and B + D. This signal thus reveals information about a right- or
leftward orientation of polarization. The strength of the twist, hence the amplitude,
is associated with the shear tensor element d15. This measurement method is called
lateral PFM or LPFM.
In the case of an orientation of the polarization vector neither parallel nor perpendic-
ular to the electric field, the measurement signal will have both a lateral and vertical
component.
Oftentimes, the piezoresponse PR is calculated from both the amplitude and the
phase according to eq. 39:
PR =amplitude ·cos(phase) (39)
Simultaneously to the piezoelectric response, the surface topography is recorded by
feeding a low-pass filter with the low-frequency cantilever motion. As a result, both
the surface topography and the polarization are measured in a single measuring step
and give rise to spatially resolved polarization maps.
An important note for interpreting the domain structure is that all subsequent mea-
surements (if not stated otherwise) have been carried out at room temperature.
5.1. Contributions of a PFM signal
Contrary to the theoretical idea presented in chapter 5 that pure electromechanical
forces occur in the PFM, electrostatic forces also act in the experiment, which can
falsify the measuring signal. These can be divided into different categories and should
be discussed at this point on the basis of [80].
The total amplitude A of a conductive coated cantilever in contact with a sample
32
D. Braun II. Growth & Characterization methods
surface can be described as follows:
A=Apiezo +Ael +Anl (40)
Hereby, Apiezo stands for the purely electromechanical deflection of the cantilever
due to the piezoelectric effect. Ael describes the electrostatic interaction of the tip with
the surface and Anl is the non-local part of the electrostatic interaction due to the
capacitive coupling between tip and surface.
Eq. 40 consists of the following summands: According to the piezoelectric effect,
the longitudinal deflection Apiezo of a sample is linked linearly to the dielectric tensor
element d0
33 at a perpendicularly applied voltage Vac [[81],[82]]:
Apiezo =d0
33 ·Vac (41)
The symbol ’refers to the different coordinate systems of sample and laboratory.
The two further terms in eq. 40 describe the electrostatic interactions as follows:
Ael =Floc ·Vac ·(Vtip −Vloc) (42)
Anl =Fnl ·Vac ·(Vtip −Vav) (43)
In the case of local influences, the local potential gradient Vloc and the local capacitive
forces between tip and sample Floc play the decisive role. For non-local interactions
(=influences that exceed the cantilever dimension), the average surface potential Vav
and the capacitive forces between tip and sample Fnl dominate the signal.
Which changes can appear in the measured PFM signal? If the vertical PFM signal
is measured over a surface with c+, c−and a domains, the mentioned interactions can
lead to falsifications of the measurement signal as depicted in fig. 18. In the case of
pure electromechanical coupling, the forces Floc = Fnl = 0. This means that the ampli-
tude signal is equal for c+and c−domains, and zero for a domains. The corresponding
phase signal shows a phase shift of 180◦between the vertical c domains and is zero (or
in the range of noise) for lateral a domains.
If there is a weak non-local electrostatic interaction Fnl 6= 0 (fig. 18b), it will result
in a slightly different amplitude signal for vertical c+and c−domains, while the phase
shift is still correctly displayed with 180◦. In addition, a non-zero signal for lateral
a-domains evolves in the vertical PFM, which falsifies additionally the phase contrast.
Similarly behavior is expected for weak local electrostatic forces Floc 6= 0 (fig. 18c).
33
D. Braun II. Growth & Characterization methods
Figure 18: Representation of the piezoresponse divided in amplitude and phase signals over a
c+/c−/c+/a domain pattern for three different electrostatic contributions. Taken from
[39].
However, if this component dominates, there won’t be any phase shift and a strongly
distorted amplitude signal.
This discussion clearly shows that a trustworthy measurement can only be obtained
with a dominant electromechanical response. It is therefore unavoidable to minimize
the electrostatic interactions.
How this can be achieved experimentally will be discussed below.
5.2. Experimentally influencing the PFM signal
First of all, a good electrical contact between the sample surface and the tip must be
ensured. An intermediate layer between the two components can affect the measure-
ment signal. However, they naturally occur in the form of, for example, thin water
films on the surface or oxidation of the tip, since the measurement is not taken under
vacuum conditions. Thus further experimental parameters have to be optimized.
One approach is to make use of the different contrast mechanisms between electrome-
chanical and electrostatic interaction. In the first case, this is a purely stress-induced
effect which leads to a mechanical deflection. In the second case, forces between the
tip and the sample play the dominant role and are therefore influenced by the probe
geometry or the measuring frequency. At this point it can be hooked up.
Solving the differential equation of a clamped beam [80] yield information how the
34
D. Braun II. Growth & Characterization methods
electromechanical and electrostatic contributions scale with the modulation frequency
ω[80]:
ivertical components of the electromechanical force ∝k
ω
ii local electrostatic force ∝1
ω
iii non-local electrostatic force ∝1
ω3
2
All contributions decrease with increasing measuring frequency. Furthermore, the fre-
quency dependency is the same for purely mechanical and local electrostatic contribu-
tions, but they differ for uneven spring constants. Thus, this can not be separated here
by choosing a suitable frequency of the alternating voltage, but by choosing proper
cantilever properties. For dominant electromechanical response the spring constant k
should be [80]:
k >> LwUε0
48H2d33
(44)
For typical measuring conditions for weak piezoelectric materials, such as thin films,
where the cantilever dimensions are in the range of length L = 225µm, width w = 30µm
and height H = 15µm as well as a voltage of U = 1V and a resulting response of
d = 5pm
V, this means a stiffness larger than 1.1N
mmust be chosen. This can be well
done experimentally and has been realized in the present experiment by the choice of
cantilever with k1= 2.8N
mor k2= 5.0N
m.
A further possibility to reduce unwanted electrostatic influences is the use of top
electrodes. Since these could not be deposited routinely in our institute, this method
was rejected.
Basically this is a rather static view on the problem, neglecting dynamic and spatial
effects like buckling. This would, however, make an analytical solution impossible and
would have to be treated numerically by differential equations adapted to the system,
which should not be discussed here.
In summary, the experimental signal can be shifted to high electromechanical pro-
portions when the cantilever has a sufficient stiffness and the measuring frequency
is chosen to be correspondingly high. The latter can be driven up to the resonant
frequency, which will be explained in more detail in the following section.
35
D. Braun II. Growth & Characterization methods
5.3. Resonance enhancement of PFM signal
The measuring mode in the PFM is based on inducing vibrations into a piezoelectric
sample by an electrical field and evaluating the strength as well as the direction of
the response with regard to the piezoelectric effect. It is precisely this vibration that
can be driven through a suitable choice of voltage up to the resonance case where the
electromechanical interaction reaching its natural maximum. As a result, the signal-
to-noise ratio is minimized, which is almost unavoidable for an evaluable signal in the
case of weak piezoelectric samples, such as thin films.
However, the main problem is that the contact resonance frequency f0depends on the
sample-tip contact and is therefore, for example, topography-dependent.
To avoid this obstacle, Rodriguez et al. [83] developed a method in which the contact
resonance frequency is tracked for each measurement point. For this purpose, the tip-
sample contact is described by two frequencies f1and f2, where f1is the same measure
smaller than f0as f2is larger than f0. The corresponding amplitudes A1and A2are
selected in the manner that A1= A2is always valid. The latter are used to track the
resonant frequency f0. For a better understanding, all sizes are shown in fig. 19. When
Figure 19: Schematic sketch of the piezoelectric resonance peak. Central point of the DART
mode is tracking the contact-resonance amplitude peak by dual-frequency excitation.
For the tracking, the amplitudes on both sides of the flank and the corresponding
frequencies are essential. Taken from [83].
the resonance peak shifts due to changing characteristics of the tip-surface contact, the
amplitude equality is no longer valid (A0
16= A0
2). This result is fed into a feedback loop
and results in a changed signal frequency until A0
1= A0
2holds.
36
D. Braun II. Growth & Characterization methods
This resonance tracking was technically implemented in the Asylum MFP-3D classic as
a dual AC resonance tracking (DART) method and was used for each of the subsequent
measurements.
5.4. Evaluation of the local piezoelectric coefficient d0
33
In this work, the local piezoelectric coefficient d0
33 of thin films was measured via
resonance-amplified PFM. The amplitude A measured in DART mode can be described
as follows [84]:
A=d0
33 ·Vac ·Q(45)
In eq. 45, Q corresponds to the amplification factor of the measurement signal. De-
pending on the cantilever used, typical values for Q are in the range of 10-100. This
value must be known for each single measurement point and be calculated off. Since
this value is different at each measurement point as a result of tracking the system’s
resonance, the operation is not trivial or can be made manually.
If a surface is measured in the DART-PFM mode by tracking the first resonant fre-
quency, four quantities A1,2and f1,2can be obtained from a single measurement. If the
tip-sample-contact during the oscillation is described as a damped harmonic oscillator
(DHO), the individual measured variables Aiand fiare composed as follows [85]:
A1,2(f1,2) = ADrivef2
0
q(f2
0+f2
1,2)2+ (f0f1,2/Q)2
(46)
φ1,2(f1,2) = tan−1f0f1,2
Q(f2
0−f2
1,2)+φDrive (47)
Where f0, ADrive,φDrive and Q are the four unknown parameters of the resonance fre-
quency, amplitude, and phase and the quality factor.
Since four quantities are measured and four unknown ones remain, the equation system
is uniquely solvable and Q can be determined for each point [86]. This conversion is
carried out numerically and provides a picture with quasi subtracted resonance.
In order to compare several samples, however, the sensitivity of the detector on the
deflection of the cantilever has to be calibrated. The most frequent method for this
purpose is the inverse optical lever sensibility, InvOLS, as it should be explained on
the basis of [[87],[88],[89]].
Experimentally, the laser or white light diode spot is reflected from the rear side of the
37
D. Braun II. Growth & Characterization methods
cantilever into the position - sensitive detector (PSD), that detects changes in the laser
position as a voltage change ∆V. The measured electric signal has to be converted
into the cantilever deflection x, which is called the InvOLS calibration.
In scanning force microscopes this is realized by the force τ, which can be described
with Hooke’s law in dependence of the deflection x and the spring constant k:
τ=kx (48)
This means that the two quantities x and k must be known in order to measure the
force at the same time for each measurement.
First, a force-distance curve τ(x) is measured. For this purpose, the cantilever is
brought into contact with the surface and then retracted by a defined distance. From
this measurement, the voltage ∆V in the PSD is gained as a function of the distance
x. It follows from Hooke’s law that the contact regime between sample and peak is
linear in the force-distance curve. InvOLS can be obtained from the slope of exactly
this linear part of the curve:
τ=kx =k·InvOLS ·∆V(49)
In a second step, the spring constant k will be determined. For this task, a thermal
spectrum is recorded. Hereby, the resonance frequency of the cantilever is measured
by applying an iterative series of frequencies and finally averaging the result. The
outcome is a real-time measurement of the amplitude of the cantilever as a function of
the applied frequency (power spectrum). In the following, the 1st Harmonic is fitted
with a Gaussian function and the resonance frequency and finally the spring constant
k is determined via the software.
In the end, the detector is calibrated in the sense that a defined voltage in the
PSD can be recalculated into a real deflection. Thus, the measured amplitude can be
depicted in meter and after division by the drive voltage, the longitudinal piezoelectric
coefficient d0
33 can be gained with spatial resolution.
38
D. Braun II. Growth & Characterization methods
6. X-ray diffraction
6.1. High-resolution x-ray diffraction (HRXRD)
In the following chapter, a brief and basic introduction should be given to the struc-
tural investigation of the crystal lattice by means of high-resolution x-ray diffraction
(HRXRD).
X-rays where discovered rather accidentally by Conrad Wilhelm R¨ontgen in 1895 dur-
ing a study on cathode rays. He discovered luminous effects that can not be related
to the actual experiment itself and called them therefore x-rays. Their wave length
λis in the range of 1 pm to 10 nm and thus on the same scale as atomic distances
in a crystal. As a result, the three dimensional crystal lattice can serve as diffraction
grating for the x-rays to examine the lattice structure itself in detail. In contrast to
measurements with electrons or neutrons, the access to x-rays are comparable easy and
the penetration depth in the range of microns is rather large.
The scattering process can be described as follows. The electrons of the atoms
are excited to forced vibrations as Hertzian’s dipoles within the electromagnetic field
of the incident x-rays. As a result, they reemit x-rays of the same frequency that
can interfere. Depending on the distance of the atoms in the lattice, unequal path
differences result for the diffracted waves. Since crystals consist of three-dimensional,
periodically arranged structural units, constructive interference is limited to particular
angles θ. In a schematic model, crystal planes can be spanned between these atoms
denoted as net planes. The distance d of these planes is the decisive parameter for the
path difference. Mathematically, the constructive interference during the diffraction
process can be described by the Bragg equation 50:
nλ = 2dsin θ n ∈Z(50)
When the path difference d of two net planes is an integer multiple of the x-ray wave
length λ, constructive interference occur. This scenario is restricted to selected angles
towards the net plane for the incident beam that are called Bragg angles θB.
Transferring this model into reciprocal space requires the usage of vector algebra [90]
as depicted in fig. 20. A monochromatic, collimated incident x-ray beam with a wave
vector kand length 2π
λhits the sample under an inclination angle ω, defined between
the x-ray source and the sample surface. For an elastic scattering process as the one
discussed in this chapter, the reemitted wave has the wave vector k0with the same
39
D. Braun II. Growth & Characterization methods
Figure 20: Schematic illustration of the HRXRD scattering geometry in 2D section of the Ewald
sphere. Adapted from [90].
length 2π
λ.
Constructive interference between waves reemitted from two points in the reciprocal
space with the spatial difference vector K=k0−kcan be expected exclusively when
Kis equal to a reciprocal lattice vector q. This restriction is denoted as Laue equation
51:
K=q(51)
Consequently, the scattering vector q=k0-kdepends on ωand 2θas well. Splitting q
into the components parallel (=in-plane) and normal (=out-of-plane) to the sample
surface yields [90]:
qk=2π
λ[cos(2θ−ω)−cos ω] (52)
q⊥=2π
λ[sin(2θ−ω) + sin ω] (53)
Two different measurement procedures are typically used. First, when the sample and
the detector are both rotated with an angle-relation of 1:2, the direction of qis kept
constant while its length changes. Thus, the Bragg reflection is scanned alonsg radial
direction which is named ω−2θ- or θ−2θscan. This case is depicted in fig. 20 by a
blue double arrow.
Second, if the detector position 2θis fixed but the sample is rotated and thus ωis
changing (ω-scan), the angle between kand k0remains unmodified. As a result, the
length of qis fixed while the scanning direction is angular along the Ewald sphere [91].
40
D. Braun II. Growth & Characterization methods
This measurement principle is called ω-scan and is illustrated by a double orange arrow
in fig. 20.
Due to the fact, that the ω−2θ- and ω-scan are locally perpendicular, a combined
measurement yields a surface in the reciprocal space. This technique is called recipro-
cal space mapping.
In epitaxial growth, the strain state of the film is of crucial interest. This information
can be derived from the relative distance between film and substrate peak (whose
angular position is known). From eq. 50 it follows:
∆d
d=∆2θ
tan θ⊥
(54)
with ∆2θ= 2θ⊥−2θsand ∆d = d⊥−ds. Hereby, θ⊥and θsare the Bragg positions of
the film and the substrate, respectively. Similar, d⊥and dsdenote the vertical lattice
parameter of the film and the substrate, respectively. Using eq. 54, the vertical lattice
parameter of the film can be calculated via:
d⊥= ds·1−2θ⊥−2θs
2·(tan θ⊥)−1(55)
=nλ
2 sin θs·1−2θ⊥−2θs
2·(tan θ⊥)−1(56)
If the grown films are of good crystalline quality (<1 nm rms roughness) and thus
having smooth interfaces and surfaces, thickness oscillations emerge on each side of the
film reflection. These Kiessig fringes results from the fact, that the experiment is equal
to the diffraction on a slit yielding a modulation by sin θ
θ2. Evaluating the distance
of the minima of the fringes ∆θmin in a symmetric reflex yields the layer thickness t:
t=λsin θ⊥
∆θmin ·sin(2θ⊥)(57)
The high resolved x-ray diffraction measurements were performed on a D8 Discover
diffractometer from Bruker Corporation. X-ray CuKαradiation provided by a copper
tube was employed as source. If relative changes in the vertical lattice parameter of
∆d
d= 3 ·10−4˚
A should be measured, the angular resolution of the diffractometer has
to be ω= 0.003◦[91]. Hence, the x-ray beam has to be collimated. The primary beam
is precollimated by a Goebel mirror and than transmits a 2-bounce Si channel cut
monochromator for minimizing the divergence of the x-ray beam. In the end, CuKα1
radiation (λCuKα1= 1.54056 ˚
A) is collimated to <20 arcsec. The diffracted beam
enters a scintillation counter trough a collimating slit system.
41
D. Braun II. Growth & Characterization methods
In order to evaluate the in- and out-of-plane lattice strain state of the epitaxial film,
reciprocal space maps have to be recorded. In contrast to the high resolution x-ray
diffraction scans described above, a one-dimensional Dectris Mythen 1K detector was
use for that purpose. The advantage of this detector is the measurement of the respec-
tive Bragg reflection within an angular range of 8.7◦[91]. In consequence, to map a
surface in reciprocal space do not require both - an ω−2θ- and ω-scan - but rather a
single movement of the ωaxis. This procedure decreases the measuring time immensely.
6.2. Grazing incidence x-ray diffraction (GIXD)
In the chapter before, the large penetration depth was depicted as one among many
advantages of x-rays. Unfortunately, in case of thin films, that property is not always
desirable because of the huge background signal of the substrate. To avoid the latter,
the measurement geometry has to be slightly modified in this way that the x-ray hits
the sample surface at a very grazing angle of incidence ϑiand an in-plane angle θi(see
fig. 21a). Indeed, ϑihas to be close to the angle of total external reflection. As a
Figure 21: Schematic illustration of the GIXD scattering geometry. Adapted from [90].
result, diffraction occurs solely from net planes that are almost perpendicular to the
sample surface.
As illustrated for the reciprocal space in fig. 21b, the scattering vector qthus depends
on the in-plane entrance and exit angle θiand θf, respectively. Experimentally, a linear
position-sensitive detector records the exit angle ϑfwith respect to the surface. Simul-
taneously, a crystal analyzer selects the in-plane angle θfof the diffracted beam. This
geometry allows to picture a three-dimensional map of the reciprocal space [90].
Eventually, GIXD offers the possibility to investigate the in-plane lattice parameter
and in-plane lattice distortions as they can occur in monoclinic phase.
Generally, these x-ray measurements - and in particular the latter - require good col-
limation of the incidence beam both in vertical and lateral direction. Moreover, chal-
42
D. Braun II. Growth & Characterization methods
lenging is the rather weak scattered x-ray signal. Therefore, it is oftentimes useful to
measure with bright and high collimated x-ray beams as it is inherent in synchrotron
radiation [90]. In consequence, most of the GIXD mappings presented in this thesis
were recorded at synchrotron radiation facilities as the European Synchrotron (ESRF,
Grenoble, France) and Bessy II (Helmholtz Center, Berlin, Germany).
The presented x-ray measurements have been performed by Dr. Martin Schmid-
bauer, Albert Kwasniewski, Dr. Michael Hanke and Christoph Feldt.
43
D. Braun III. Results & Discussion
Part III.
Results & Discussion
The greater goal of the presented study is the investigation of changing ferroelectric
properties in dependence on the incorporated lattice strain to learn how to perform
”strain engineering”. Therefore, different paths can be taken: lattice strain can be
modified by varying the lattice parameter (i) of the substrate (e.g. by different rare-
earth scandates) and/or (ii) of the film by changing the composition.
7. Theoretical Considerations
7.1. Calculating lattice parameter from Vegard’s law
The KxNa1−xNbO3thin films investigated in this work exhibit all an orthorhombic
symmetry. Using eqs. 22-24, the pseudocubic parameters can be calculated from
the former orthorhombic ones. According to Vegard’s law, the pseudocubic lattice
parameters of potassium sodium niobate films aKxNa1−xNbO3
pc , bKxNa1−xNbO3
pc , cKxNa1−xNbO3
pc
and βKxNa1−xNbO3
pc with 0.5≤x≤1 can be estimated from a linear interpolation:
aKxNa1−xNbO3
pc =x·aKNbO3
pc + (1 −x)·aK0.5Na0.5NbO3
pc (58)
bKxNa1−xNbO3
pc =cKxNa1−xNbO3
pc =x·bKNbO3
pc + (1 −x)·bK0.5Na0.5NbO3
pc (59)
βKxNa1−xNbO3
pc = 90◦−arccos
1−aKxNa1−xNbO3
o2
2bKxNa1−xNbO3
pc 2
(60)
with the boundary lattice parameter [[92],[93]]:
Table 3: Orthorhombic and pseudocubic lattice parameter for KNbO3and K0.5Na0.5NbO3
[[92],[93]] serving as boundary parameter for Vegard’s law.
aKNbO3bKNbO3cKNbO3aK0.5Na0.5NbO3bK0.5Na0.5NbO3cK0.5Na0.5NbO3
orthorhombic 3.971 ˚
A 5.697 ˚
A 5.722 ˚
A 3.940 ˚
A 5.640 ˚
A 5.567 ˚
A
pseudocubic 3.974 ˚
A 4.036 ˚
A 4.036 ˚
A 3.940 ˚
A 4.000 ˚
A 4.000 ˚
A
44
D. Braun III. Results & Discussion
7.2. Calculation of the strain energy density F(ε)(x)
Which energy cost is needed for the epitaxy and thus which crystallographic orientation
will be favorable can be determined by the linear elasticity theory. The resulting
strain energy density F(ε) indicates the effort expended to elastically distort the crystal
structure. This process depends on the material characterizing stiffness tensor cijkl and
the strain in both in-plane directions εmn.
F(ε) = 1
2X
ijkl
cijklεijεkl (61)
By using the Voigt notation depicted in eq. 3 and applying symmetry considerations,
the problem can be simplified. The elasticity tensor can be written as followed in case
of an orthorhombic lattice:
cV oigt =
c11 c12 c13 c14 c15 c16
c21 c22 c23 c24 c25 c26
c31 c32 c33 c34 c35 c36
c41 c42 c43 c44 c45 c46
c51 c52 c53 c54 c55 c56
c61 c62 c63 c64 c65 c66
→corth
V oigt =
c11 c12 c13 000
c12 c22 c23 000
c13 c23 c33 000
0 0 0 c44 0 0
0000c55 0
00000c66
(62)
Moreover, boundary conditions for the applied stress can be defined.
If the stress from the substrate onto the film lattice shall be calculated, the problem
can be regarded as first approximation as pure interfacial effect. Hence, the following
limitations should be valid [[94],[95]]: for in-plane biaxial strain the contributions εxz
and εyz should be neglected. Likewise, for in-plane stress σzz =σxz =σyz = 0 (see eq.
64) should be used. Moreover, the layer should grow epitaxially without any in-plane
shearing (εxy = 0).
The latter is an approximation for the potassium sodium niobate films under investiga-
tion. In reality, they possess monoclinic symmetry and hence a monoclinic distortion
angle β. For (100)pc oriented films, βlies in-plane. However, the substrates illus-
trated in tab. 1 have a rectangular surface. In consequence, the rectangular growth
of KxNa1−xNbO3requires an additional energy contribution to elastically distort the
lattice which is not considered in the following calculations. Indeed, it was shown for
NaNbO3films of only a 1-3 nm that the monoclinic angle is forced to β∼
=0 [96] on a
DyScO3substrate.
45
D. Braun III. Results & Discussion
As a consequence, eq. 61 can be reduced to:
F(ε)orth =1
2ε2
xxc11 +ε2
yyc22 + 2εxxεyyc12(63)
for orthorhombic films. Hereby, the strain εii is calculated between the film lattice
parameter and those of the substrate illustrated in tab. 1 according to eq. 17.
In the orthorhombic symmetry, three different orientations can exist according to the
unit cell axes a,b,c differing in the out-of-plane axis. Indeed, either of these can occur
in two 90◦in-plane rotated variants. In consequence of the likewise changing stress
applied from the substrate, the energy density F(ε)orth will vary. Hence, eq. 63 allows
to predict the most favorable orientation on the basis of the minimal energy density of
the six possible variants.
It has to be noted, that the elastic constants are reported for KNbO3only [92], while
they are not available for KxNa1−xNbO3. But since we assume similar coefficients for
the KxNa1−xNbO3solid-solutions, we also refer for prediction of preferred film unit cell
orientations for these materials to the coefficients of KNbO3[92].
In consequence of the discussed assumption to solve eq. 61, eq. 63 is strictly spoken
only an approximation for the discussed KxNa1−xNbO3in this thesis.
The elasticity coefficients used for the calculations are depicted in tab. 4:
Table 4: Directional dependent elastic coefficients cij (GPa) of KNbO3taken from [92].
c11 c22 c33 c12 c13 c23
224 GPa 273 GPa 245 GPa 102 GPa 182 GPa 130 GPa
From tab. 4 the conclusion can be drawn, that KNbO3possesses different stiffnesses
along the unit cell axes denoted in tab. 4 as 1,2,3 because c11 6= c22 6= c33 is valid.
Consequently, the material has an inherent elastic anisotropy which has to be consid-
ered additionally to the anisotropic in-plane lattice parameter of a rare-earth scandate
substrate (see tab. 1).
Furthermore, eq. 63 enables to calculated F(ε)orth for every potassium concentration
x yielding the functionality F(ε)(x). The result is an x-dependent information on the
preferred growth mode.
7.3. Calculation of the strained vertical lattice parameter dstrained
⊥
Within Hooke’s theory, the strained vertical lattice spacing can be calculated. Taking
the eq. 61 in notation 64:
σij =
3
X
k=1
3
X
l=1
cijklεkl (64)
46
D. Braun III. Results & Discussion
and the elasticity tensor for orthorhombic media illustrated in eq. 62, yield the
transformation from stress to strain:
σxx
σyy
σzz
σyz
σxz
σxy
=
c11 c12 c13 000
c12 c22 c23 000
c13 c23 c33 000
0 0 0 c44 0 0
0000c55 0
00000c66
·
εxx
εyy
εzz
2εyz
2εxz
2εxy
(65)
If the stress results exclusively from the underlying substrate and thus by an in-
plane force on the unit cell, it can be concluded that σzz = 0. Thus, the strain in
vertical direction as a consequence of in-plane stress in an orthorhombic lattice can be
calculated according to eq. 67:
σzz = 0 = c13 ·εxx + c23 ·εyy + c33 ·εzz (66)
εzz =−1
c33
(c13 ·εxx + c23 ·εyy) (67)
The strained vertical lattice parameter dstrained
⊥can be calculated from the unstrained,
bulk lattice parameter d⊥with eq. 68:
dstrained
⊥= (1 + εzz)·d⊥(68)
As discussed in chapter 7.2, the elastic constants cijkl are listed in the literature
only for KNbO3[92], but we assume similar coefficients for the KxNa1−xNbO3solid-
solutions. As a result, the parameter listed in tab. 4 were taken for the calculation of
the strained vertical lattice value.
7.4. Calculation of the misfit strain-misfit strain phase diagram for
KNbO3
As already described in chapter 1.3, several different domains can appear in a ferro-
electric material. For pure potassium niobate, a prediction from Bai and Ma [3] exists,
that illustrates the most favorable domain state in dependence of the biaxial misfit
strain (depicted in fig. 10). However, comparing the experimental results to this the-
oretical predictions yielded oftentimes no agreement. Probably, the transitions points
47
D. Braun III. Results & Discussion
were plotted with major increment. Hence, the calculation is to rough. Therefore, I
calculated and plotted the diagram with minor step size.
The basis of the former calculations is Gibbs free energy in the framework of the
Landau-Ginzburg-Devonshire theory as discussed in chapter 1.2. Due to the fact that
the film is clamped to a bulky substrate, parts of the energy functions do not longer
describe a free stranding material. In consequence, a Legendre transformation of G
has to be calculated [[22],[97]]. In contrast to the initial function of G, the modified
thermodynamic potential ˜
G depends on different variables compared to eq. 2. These
changes are denoted by the index * whereby some of them correlate directly with the
biaxial strain εxx and εyy [98] as denoted by the index ** on the respective summand
[98].
∆˜
G = α∗∗
1P2
1+α∗∗
2P2
2+α∗∗
3P2
3+α∗
11(P4
1+ P4
2) + α∗
33P4
3+α∗
13(P2
1+ P2
2)P2
3+α∗
12P2
1P2
2
+α111(P6
1+ P6
2+ P6
3) + α112[P4
1(P2
2+ P2
3)+P4
2(P2
1+ P2
3)+P4
3(P2
2+ P2
1)]
+α123(P2
1P2
2P2
3) + α1111(P8
1+ P8
2+ P8
3)
+α1112[P6
1(P2
2+ P2
3)+P6
2(P2
1+ P2
3)+P6
3(P2
2+ P2
1)]
+α1122(P4
1P4
2+ P4
1P4
3+ P4
2P4
3) + α1123(P4
1P2
2P2
3+ P4
2P1
2P2
3+ P4
3P2
1P2
2)
+s11(ε2
xx +ε2
yy)−2s12εxxεyy
2(s2
11 −s2
12)
(69)
To obtain a misfit strain-misfit strain phase diagram, ˜
G has to be minimized in respect
to the respective polarization component to determine the equilibrium thermodynamic
state of the thin film. In detail, that means that for every strain configuration (εxx, εyy)
the minimum minimorum has do be calculated with regard to P1, P2and P3. This
global minimum can then be compared to the equilibrium states that may occur in
thin films [[3],[24],[99]]:
(i) p phase (P1= P2= P3= 0); (ii) a1phase (P16= 0 and P2= P3= 0); (iii) a2phase
(P26= 0 and P1= P3= 0); (iv) c phase (P36= 0 and P1= P2= 0); (v) a1a2phase
(P16= 0, P26= 0 and P3= 0); (vi) a1c or Mcphase (P16= 0, P36= 0 and P2= 0); (vii)
a2c or Mcphase (P26= 0, P36= 0 and P1= 0) and (viii) r or MAphase (P16= 0, P26= 0
and P36= 0).
One possibility is to plot the transitions points between the equilibrium phases to
separate the favored ferroelectric states as depicted in fig. 22a for the material constants
48
D. Braun III. Results & Discussion
of KNbO3[3] and a tgemperature of T = 25◦C.
Figure 22: (a) Misfit strain - misfit strain phase diagram of single-domain KNbO3thin films
calculated for T = 25◦C. (b) Multilayer plot consisting of the elastic strain energy
density F(ε) for KNbO3in dependence if the biaxial strain εij together with the phase
transition points calculated according to eq. 69.
From consideration 63 further predictions can be inferred. It immediately pro-
vides information about the energy density surface in dependence of the biaxial strain
F(εxx, εyy). If moreover the stability regions for individual ferroelectric domain are
calculated numerically according to eq. 69, a multi-dimensional plot can be derived as
shown in fig. 22b. Herefrom, it can be concluded that the strain as discrete variable
alone may not be the only decisive parameter whether a domain configuration will be
stable and occur. Rather the elastic energy needed to distort the material under an
applied stress has to be considered likewise.
Meanwhile, it shows impressively, that monoclinic phases as they would be highly de-
sirable because of the high piezoresponses (see chapter 2.5) are energetically favorable
in KNbO3.
To summarize chapter 7, the elastic deformations enforced during epitaxial growth
dictate the crystal film symmetry and thereby the functional properties. Consequently,
the substrate stress induced distortions in the film unit cell have to be understood for a
precise prediction of the ferroelectric ordering. This way, strain engineering can become
a sophisticated tool for the targeted fabrication of functional thin films.
In the following, KxNa1−xNbO3films were grown on (001) SrTiO3, (110) DyScO3,
(110) TbScO3, (110) GdScO3and (110) NdScO3single-crystalline substrates. As a
result of varying misfit strain, the ferroelectric domain structure changes drastically as
can be seen in fig. 23. For example, although the change of the average strain is rather
49
D. Braun III. Results & Discussion
Figure 23: Lateral PFM amplitude measurements of KxNa1−xNbO3on (a) (001) SrTiO3, (b)
(110) DyScO3, (c) (110) TbScO3, (d) (110) GdScO3and (e) (110) NdScO3single-
crystalline substrates. εis the average in-plane strain.
minor between films on TbScO3and NdScO3, the domain pattern reveals enormous
differences. The reasons has to be associated to the energy density situation as they
represent different strain states. Therefore, to verify the theory and to eventually prove
the possibility to systematically adjust the ferroelectric properties, these both examples
on TbScO3and NdScO3should be discussed in detail in chapter 8 and 9, respectively.
50
D. Braun III. Results & Discussion
8. Uniaxial strain: K0.75Na0.25NbO3films on (110)
TbScO3
In the ε(x) diagram 24a the film composition with x = 0.75 on (110) TbScO3(TSO)
determines an unique point. The in-plane lattice parameter of the film calculated
according to eq. 58 to apc = 3.957 ˚
A and cpc = 4.018 ˚
A can be arranged on the (110)
TbScO3surface with its apc = 2×3.958 ˚
A and bpc = 2×3.959 ˚
A lattice parameters this
way that the strain is almost zero along the [001]TSO direction and −1.47% compressive
for the orthogonal [1¯
10]TSO direction as it is illustrated in fig. 24a. In the end, it marks
the unique possibility to investigate uniaxial strain and was therefore discussed in detail
[100].
From the elastic strain energy density plotted in fig. 24b for films on TbScO3in
dependence of the potassium content x, it becomes obvious that for x = 0.75 a (001)pc
orientation is the preferred growth mode.
Figure 24: (a) Strain in dependence of the potassium concentration x in the solid solution
KxNa1−xNbO3for 0.5≤x≤1 on TbScO3substrate. The composition with x = 0.75
is marked with a dotted line. (b) Estimation of the elastic strain energy density F(ε)
for four possible orientations of KxNa1−xNbO3on TbScO3substrate versus potassium
content x for 0.5≤x≤1. As inset a magnification of the elastic energy density for
both 90◦rotated (001)pc variants in the vicinity of x = 0.75 is shown.
The atomic force microscopy (AFM) was used to study the surface topography in
ambient air shown in fig. 25a. The AFM image of the film illustrates very smooth
surfaces with an average root mean square surface roughness of 0.3 nm. To determine
the growth mode, a detailed thickness series should have been performed. However,
from the AFM image (fig. 25a) it can be suggested that no step flow growth occurred
51
D. Braun III. Results & Discussion
Figure 25: (a) Surface morphology of a respective film on a (1 ×1) µm2scale. (b) θ−2θHRXRD
pattern of a K0.75Na0.25NbO3films on (110) TbScO3in the range from 2θ= 20◦−
23.5◦. The red and blue dashed lines indicate the nominal angular position of the bulk
vertical lattice parameter in case of an (100)pc- and (001)pc orientation, respectively,
for K0.75Na0.25NbO3. The blue dotted line indicates the nominal angular position
of the strained vertical lattice parameter dstrained
⊥for a (001)pc orientation in case of
K0.75Na0.25NbO3/TbScO3.
but rather a transition from 2D growth on each step to island growth.
In fig. 25b the θ−2θHRXRD pattern of the K0.75Na0.25NbO3film around the
(110)TSO Bragg reflection of the orthorhombic TbScO3substrate is shown in the range
from 2θ= 20◦−23.5◦. The diffraction measurement reveals a sharp substrate peak
around 2θ= 22.44◦and a single film peak appearing at about 2θ= 21.99◦. A quanti-
tative analysis according to the procedure presented in eq. 56 and 57 yields a vertical
lattice parameter of d⊥= 4.040 ˚
A and film thickness t = 29 nm. To clarify the strain
state, the strained vertical lattice spacing of the (001)pc orientated K0.75Na0.25NbO3
phase was estimated from eq. 67 to dstrained
⊥= 4.047 ˚
A and transferred into the θ−2θ
HRXRD pattern in fig. 25b as dotted line. In regard of the incertitude arising from
the calculation e.g. by taking the elastic constants of KNbO3instead of those from
K0.75Na0.25NbO3, it can be concluded that the film contribution matches rather ex-
actly the theoretical value. This demonstrates that the film has been grown under
compressive strain with indeed (001)pc-orientation, confirming the energetic consider-
ations presented in fig. 24b.
Evaluating the strain state along both substrate directions from fig. 24a and intro-
52
D. Braun III. Results & Discussion
ducing the values as black circles in the strain-phase diagram in fig. 26a yields the
prediction of monoclinic Mcdomains for a fully strained film. In the monoclinic phase,
Figure 26: (a) 3D plot of the elastic strain energy density F(ε) for KxNa1−xNbO3in dependence
of the biaxial strain εxx and εyy. On the energy landscape the strain-phase diagram
presented in fig. 22a for KNbO3was plotted. The filled black circles indicate the
strain state in K0.75Na0.25NbO3on a (110) TbScO3substrate. PFM measurements
of a K0.75Na0.25NbO3/TbScO3film, picture size (1 ×1) µm2: vertical amplitude (b)
and piezoresponse PR (c). (d)-(h) are lateral measurements whereby (d) and (g) show
amplitude images and (e) and (h) the piezoresponse images. The cantilever orientation
is illustrated in the upper left of part of the amplitude images. (f) 2D-FFT of the
lateral amplitude image (d).
the electric polarization vector is not constrained to highly symmetric crystallographic
directions as it is the case for rhombohedral, orthorhombic or tetragonal phases. In
contrast, the polarization can rotate unhamperedly within the mirror plane of the mon-
oclinic unit cell [101]. In the special case of Mcdomains this means the {100}pc planes
[5].
The VPFM and LPFM measurements are illustrated in fig. 26b-h separated in
amplitude and piezoresponse signal both for a sample area size of (1 ×1) µm2. Stripe
domains become visible in the vertical amplitude in fig. 26b but no resulting phase
contrast can be detected (see fig. 26c). Eventually, such a missing 180◦phase shift is
not unusual and can be caused by several phenomena:
53
D. Braun III. Results & Discussion
iOne explanation could be, that all polarization vectors are aligned equally and hence
a vertical monodomain has formed. But, in this case, a complete up or down
alignment should result in a charged surface and interface. Consequently, it is
energetically not favorable.
ii During the PFM mode, a drive voltage in the range of Vdrive = 0.2−1 V is applied
to a tip with an effective tip radius of r ≈30 nm. Finally, enormous electric
fields result that can lead to a strong electrostatic contribution to the signal
misrepresenting the vertical response as discussed in chapter 5.1.
iii A third effect that can not be neglected in PFM measurements is that the pressure
of the tip in contact mode leads immediately to a mechanically induced switching
process and hence to a poling of the film.
In principle, all three explanations could be responsible for the missing phase contrast
in fig. 26. However, a buckling effect [102] can be neglected, since the cantilever axis
is not parallel to the stripes.
In the end, from the vertical images 26b,c it can be concluded that the polarization
exhibits a vertical component but its orientation can not be determined.
Contrariwise, in the lateral mode presented in fig. 26d,e well pronounced, almost
perfectly ordered stripe domains arise. The phase signal shown in fig. 26e depicts
an alternating left - and rightwards torsion of the cantilever manifesting a periodical
change of the in-plane polarization component. The same lateral behavior can be de-
tected on a 90◦rotated sample illustrated in fig. 26g,h.The periodicity of the stripe
domains can be evaluated to (50 ±1) nm.
In both modes, the domain walls are aligned along [1 ¯
12]TSO. Measurements on a larger
scale reveal the perfectness as the straight walls are running along microns. The proof
is given in the periodic arrangement of spots in the 2D-FFT image of the lateral am-
plitude signal in fig. 26f. This effect can be attributed to both the defect-poor growth
mode and the high oxygen partial pressure in the MOCVD process. As a result, the
films have a low defect density leading to ferroelectric pattern ranging from the nano
- to the micron scale without any domain wall bending around defect cluster as it was
theoretical predicted [[103],[104]] and experimentally confirmed [[96],[105]] for other
systems. This perfectness is unique and was rarly achieved before.
54
D. Braun III. Results & Discussion
From the PFM measurements we conclude that the K0.75Na0.25NbO3films on (110)
TbScO3result in a ferroelectric pattern with both a vertical and a lateral polarization
component. Consequently, four in-plane domain arrangements are possible as can be
seen in fig. 27a-d: two 90◦(figs. 27a,b) and two 180◦configurations (figs. 27c,d).
However, only two of them (figs. 27a and c) fulfill the requirement of uncharged do-
Figure 27: Possible in-plane arrangement of the electric polarization vector for stripe domains:
(a), (b) 90◦arrangement, (c), (d) 180◦pattern. (e) Schematic presentation of the
diffraction pattern in reciprocal space in case of a cubic and monoclinic unit cell in
real space.
main walls. In order to distinguish between the remaining two opportunities, in-plane
GIXD and out-of-plane HRXRD experiments were performed to get detailed informa-
tion about the film lattice structure. Hereby, the questions have to be answered in
which orientation the film is grown and if indeed monoclinic domains occur as pre-
dicted in fig. 24c. As a proof of monoclinic film symmetry a respective lattice splitting
has to be detected as it is schematically illustrated for real and reciprocal space for
cubic and monoclinic symmetry in fig. 27e. Depending on the film orientation, thus
either (100)pc or (001)pc orientation, the according splitting appears in the in-plane or
out-of-plane reciprocal space maps, respectively.
For the following discussion, it should be claimed that in reference to the substrate
lattice parameter, the x-ray data are denoted in the orthorhombic notation. For the
in-plane characterization, reciprocal space maps were recorded in the vicinity of the
(008)TSO (fig. 28a), (4¯
40)TSO (fig. 28b) and (2¯
24)TSO (fig. 28c) Bragg reflections.
In order to guard against misunderstandings, it should be mentioned that the experi-
mental resolution element indicated in fig. 28a-c in every left bottom corner leads to
a slight blurring of the diffraction pattern although the film is almost perfectly ordered.
55
D. Braun III. Results & Discussion
Figure 28: GIXD intensity distribution patterns for the (a) (008)TSO, (b) (4¯
40)TSO and (c)
(2¯
24)TSO in-plane reciprocal lattice points of the substrate. (d) to (f) Plots of the
corresponding line scans along the green dashed lines in part (a), (b) and (c), re-
spectively. Black arrows mark the positions of the main film peaks, while red arrows
indicate the satellite peaks. Adapted from [100].
In all measured Bragg reflections, a distinct central peak emerges that can be identi-
fied as the film peak. This film peak reveals no in-plane peak splitting as it is schemat-
ically drawn in fig. 27b. Consequently, no monoclinic in-plane distortion exists.
Furthermore, periodic satellite peaks along [¯
110]pc and [110]pc occur in the proximity
of the central peaks for all presented in-plane reciprocal lattice points. Evaluation of
the distance of these satellites by means of a line scans along the dashed green lines
in figs. 28a,b,c and plotting them in fig. 28d,e,f, respectively, reveals an equidistant
appearance. In addition, the spacing is identical for all Bragg reflections. Herefrom,
it can be concluded that the satellites are cause by a positional correlation. For the
real space, this can be translated into identical film in-plane lattices in neighboring
domains independent of the domain state. Thus, they are due to the periodical for-
mation of the ferroelectric domain pattern along the [110]pc = [1 ¯
12]TSO directions in
the film plane. Transferring the distance between the satellite peaks in the reciprocal
space of ∆q = 0.0129˚
A−1into the real space yields a periodicity of the domain pat-
tern of (49±1) nm, which is in excellent agreement with the PFM value of (50±1) nm.
Due to the fact, that the monoclinicity could not be proven in the in-plane GIXD
56
D. Braun III. Results & Discussion
data, further out-of-plane measurements are needed. The reciprocal space maps in the
vicinity of the (44¯
4)TSO, (42¯
2)TSO, (422)TSO, (220)TSO and (620)TSO Bragg reflection
points are plotted in figs. 29a-e, respectively. For a better illustration of their arrange-
ment and the relationship to the pseudocubic film notation, the corresponding (hkl)
of the in-plane reciprocal lattice points are depicted schematically in fig. 29i by red
circles. The first result is that the in-plane component of the reciprocal lattice vectors
Figure 29: X-ray reciprocal space maps of the (a) (44¯
4)TSO, (b) (42¯
2)TSO, (c) (422)TSO, (d)
(220)TSO and (e) (620)TSO Bragg peaks. In the lower image the line scans along
Q001 in the (f) (44¯
4)TSO, (g) (42¯
2)TSO and (h) (422)TSO mappings are plotted. (i)
A sketch of the in-plane arrangement of reciprocal lattice points around the (220)TSO
diffraction point. The recorded reciprocal space maps are indicated by red circles and
additionally referred to the pseudocubic notation. Adapted from [100].
of the film equals that of the substrate in all measured Bragg reflection. Herefrom, it
can be denoted that the film lattice is grown fully matched to the in-plane dimensions
of the substrate unit cell.
Moreover, equidistant satellite peaks of the film contribution appear in the asymmetric
(44¯
4)TSO and (620)TSO mappings. The examination of the satellite spacings and con-
verting them into real space yield a domain periodicity of (48 ±1) nm. This result is
in full accordance to the values obtained from GIXD and PFM measurements.
Furthermore, the film peak substructure was investigated in detail by line scans along
Q001 as they are presented in fig. 29f-h. The Bragg reflections around the (42¯
2)TSO,
57
D. Braun III. Results & Discussion
(422)TSO and (220)TSO do not show any film peak splitting. Hence, here only one sin-
gle film peak exists. In contrast, the (44¯
4)TSO, (42¯
2)TSO and (620)TSO reciprocal space
maps reveal a vertical splitting of the film peak as it should be indicated exemplary by
arrows in fig. 29a. In consequence, the film lattice is indeed monoclinically distorted
in out-of-plane direction.
How these appearances of splitting and single film peaks can be summarized into a
structural model to obtain the domain type, should be examined in the following.
A structural model is presented in fig. 30a. The black cube displays the undistorted,
Figure 30: (a) View on an unstrained (black) and a monoclinically distorted (red) pseudocubic
unit cell. For a better understanding, the monoclinic shearing angle βis illustrated in
an exaggerated manner. (b) Schematic illustration of the domain pattern in top view
with white arrows denoting the in-plane component of the polarization vector. (c),
(d) Cross-sectional views of the monoclinically distorted pseudocubic unit cells along
the [¯
110]pc and [110]pc directions, respectively. For (b) −(d) the shearing direction of
unit cells in the respective domains is indicated with bold red and blue arrows. (e),
(f) Transformation of the cross-sectional views from (c) and (d) into reciprocal space.
Black and blue/red circles indicate the reciprocal lattice points of the substrate and
the film, respectively, in accordance to the color code in fig. 27e. Adapted from [100].
pseudocubic unit cell. The observed peaks splittings can be understood this way that
pseudocubic unit cells of the films are sheared along the ±[110]pc direction, while the
rectangular in-plane symmetry of the unit cell is preserved as illustrated by red dashed
58
D. Braun III. Results & Discussion
lines.
Furthermore, the occurrence of satellite peaks only in the case of the (44¯
4)TSO and
(620)TSO Bragg reflection points indicate that the shearing of the film unit cell has
to be in [1¯
10]pc direction. Simultaneously, unit cells with the opposite shearing along
±[1¯
10]pc have to exist. For illustration of the effect, a cubic unit cell is sheared parallel
to the film plane. The involving distortion is transferred from real space - depicted
in figs. 30c,d - into reciprocal space in figs. 30e,f. Hereby, figs. 30c,d enable a
view on the cross-section in the [110]pc and [¯
110]pc directions, respectively. For a
better discrimination, the red and blue color code imply a shearing of the unit cell
by an angle +βand −β, respectively. A monoclinic distortion in the real lattice
as shown in fig. 30d leads to a splitting of the film contribution into red and blue
circles in the reciprocal space exactly for the above mentioned Bragg points as can be
seen in fig. 30f. Simultaneously, for the chosen geometry in fig. 30d the scattering
plane is perpendicular to the direction of domain periodicity. Consequently, the lateral
domain periodicity can not be detected preventing satellite reflections in the (42¯
2)TSO
diffraction pattern. As a result of a distortion along ±[¯
110]pc, the unit cell meshes
along the (001)pc planes are still rectangular (fig. 30c). In contrast, for the opposite
direction the unit cell meshes in the (110)pc planes form parallelograms with angles
90◦±β(fig. 30d).
The monoclinic angle βcan be evaluated directly from the vertical peak splitting
2∆Q001 using the equation:
tan β=∆Q001
Qk
(70)
whereby Qkis the horizontal component of the scattering vector Qhkl in the [¯
110]pc
direction. The results are presented in tab. 5.
Table 5: Measured Q001 and Qkvalues and calculated monoclinic distortion angle βfor the
(44¯
4)TSO, (620)TSO and (42¯
2)TSO Bragg reflection.
(44¯
4)TSO (620)TSO (42¯
2)TSO
2∆Q001 (˚
A−1) 0.0210 0.0193 0.0190
Qk(˚
A−1)2.19 2.20 2.26
β(0.27 ±0.02)◦(0.25 ±0.02)◦(0.24 ±0.02)◦
As a result of tab. 5, the averaged monoclinic angle can be calculated to βfilm =
(0.26 ±0.02)◦which is slightly below βbulk = 0.30◦for bulk K0.75Na0.25NbO3. The
discrepancy can be understood in terms of elastic relaxation of the film lattice: the
monoclinic βfilm angle increases with increasing film thickness as it was former discussed
59
D. Braun III. Results & Discussion
for NaNbO3films on DyScO3and TbScO3single crystalline substrates [[96],[106]].
Combining the structural and the electrical measurements yield the following domain
model. The lateral PFM shows a distinct phase contrast between adjacent domains
(see fig. 26e). According to the x-ray data, these neighboring domains are formed by
unit cells with the opposite shearing along ±[1¯
10]pc. Because the unit cell distortion is
different by 180◦the electrical polarization has to change likewise alternatingly by 180◦.
This model is shown in fig. 30b in plane view. Regarding the domain arrangements
in fig. 27, the model confirms variant c in fig. 27. In combination with the vertical
response, the conclusion is drawn that the electrical polarization is located within the
{110}pc planes. In regard to the literature [5], the domain pattern is described by
monoclinic MAor MBdomains, with periodically arranged domain walls along the
±[110]pc direction and a periodicity of averaged 50 nm.
In order to distinguish between the two monoclinic phases, the ratio between the out-
of-plane and in-plane (pseudocubic) lattice parameters is decisive [107]:
iapc >cpc and bpc >cpc means tensile stress and result in MBdomains.
ii apc <cpc and bpc <cpc refers to compressive strain and and MAdomains.
In the case of K0.75Na0.25NbO3, films are grown epitaxially under compressive in-
plane strain on the TbScO3substrate with in-plane lattice parameters equal to the
those of the substrate of apc = 3.959 ˚
A and bpc = 3.960 ˚
A. As a result of strain, the
out-of-plane lattice parameter amounts to cpc = 4.040 ˚
A. In consequence, apc <cpc
and bpc <cpc and we conclude that the observed monoclinic phase is a MAphase.
Interestingly, the x-ray results predict a coexistence of both (001)pc-orientations be-
cause the evaluation of the RSM in the vicinity of the (4¯
40)TSO in-plane reciprocal
lattice point reveals equidistant satellite peaks in both ±[¯
110]pc directions.
Prospective, the following notation should be used for further discussion: the in-plane
film orientation with [100]pc||[001]TSO and [010]pc||[1¯
10]TSO will be referred as 0◦variant
and the 90◦rotated variant with [010]pc||[001]TSO and [100]pc||[1¯
10]TSO will be noted
as 90◦variant.
But, the experimental intensity of the satellite peaks of the 90◦variant is about one
order of magnitude lower than that of the 0◦variant. Surprisingly, they exhibit the
same periodicity as those in ±[110]pc direction.
From the discussed model in fig. 27c the deduction can be drawn that the domain wall
60
D. Braun III. Results & Discussion
alignment has to be collinear with the shearing direction of the pseudocubic unit cell.
This implies that a 90◦in-plane rotation of the (001)pc film orientation should lead to
a likewise 90◦in-plane rotation of the stripe domains.
Indeed, the x-ray diffraction results could be confirmed by recording several LPFM
images at different film positions. A 90◦rotated domain variant with domain walls
running along the ±[1¯
12]TSO direction was found in rather few cases. One example is
shown in fig. 31, where the 90◦rotated variant is visible as a bright bundle.
The reason for the uneven appearance of both variants can be found by using the
Figure 31: (a) A lateral piezoresponse image with a scan size of (2×2) µm2revealing both (001)pc
orientations further labeled as 0◦and 90◦variant; (b), (c) Schematic presentations
of the in-plane component of the electric polarization vector (white arrows) in the
domains for both variants with highlighted 180◦domain walls within one variant. (d)
Higher magnification of the lateral piezoresponse image shown in (a) on a scale of
600×600 nm on the transition between the 0◦and 90◦variant. (e) The lateral change
of the electric polarization vector at the boundary between the two variants forming
a 90◦domain wall. Adapted from [100].
linear elasticity theory. Zooming in fig. 24b as illustrated by the inset displays that the
energy density difference between the two 90◦rotated (001)pc variants is rather small.
Hence, both variants may be stable but obviously different energy densities lead to a
preference of the 0◦variant.
Regarding fig. 31, it seems that the domains in both variants have a different domain
width. While in the 0◦area the sheared MAdomains have equal size, the 90◦variant
61
D. Braun III. Results & Discussion
seems to be built up with broad and narrow stripes. This result is in contradiction to
the x-ray data where the same domain periodicity is predicted for both types.
Why the domains of both variants can not be imaged in equal resolution with the PFM
technique in fig. 31 can only be estimated. Regarding the domain wall model in fig.
31b,c, no reason can be found why there should be a different detection sensitivity
for both variants as long as the cantilever is aligned as depicted in fig. 31a. However,
already the lateral amplitude image in fig. 26b shows a clear evidence of strong electro-
static contribution leading to a misrepresentation of equal domains in the amplitude
image [108]. As discussed in chapter 5.1, the amplitude signal measured over equal
domains should not differ. But as can be seen in fig. 26b already for the 0◦variant
a bright-dark contrast appears that is not real. So far it can be assumed, that the
measurement background is large and shifts the zero level of the amplitude signal in
such a manner that the domains and the respective walls are measured in a false color
code.
Returning to the starting point for K0.75Na0.25NbO3thin films on (110) TbScO3, the
question arises why the experiment yields MAdomains and the theory predicts a Mc
phase. Several explanations can be discussed:
iThe strain-phase diagram from Bai et al. [3] was calculated for pure KNbO3and could
therefore deviate for films with 25% sodium in the solid solution. Regarding the
average strain - phase diagram from Di´eguez et al. [22] calculated for pure KNbO3
and NaNbO3supports the assumption. There, changes between both alkaline
niobates become apparent that will lead to a shift of the transition points in fig.
26a for x = 0.75.
ii It can not be ruled out that the composition delivered into the reaction chamber dur-
ing the MOCVD process is not originally transferred into the film stoichiometry.
A slightly different ratio between sodium and potassium can originate leading to
changed lattice parameter of the unstrained film unit cell. Thus, the stress from
the TbScO3substrate can likewise differ. Considering the strain in dependence
of the potassium content illustrated in fig. 24a shows clearly that with decreasing
potassium content, the compressive strain decreases likewise. Hence, transferring
a strain state with less potassium into the strain-phase diagram e.g. in fig. 24c
can yield indeed a monoclinic MAphase in contrast to the theoretically predicted
Mcstructure in case of a potassium content of x = 0.75.
62
D. Braun III. Results & Discussion
However, in contradiction, the fact that the experimental vertical lattice param-
eter for the fully strained film d⊥= 4.040 ˚
A equals the theoretically predicted
strained vertical lattice spacing of dstrained
⊥= 4.047 ˚
A for x = 0.75 yield the result
of rather a the targeted stoichiometry.
iii MAdomains occur oftentimes as metastable ferroelectric phase. In the literature,
reasons like excessive mechanic pressure, exaggerated voltage or post-growth cool-
ing in electric fields are claimed [[5],[107],[109]]. Likewise, the growth conditions
could lead to metastable ferroelectric states. The deposition temperature was
adjusted to TGrowth = 650◦C in the case discussed beforehand. If TGrowth is cho-
sen to low, the mobility of the molecules on the substrate surface is to small to
arrange in the best manner. Such, the growth does not take place at the thermo-
dynamic equilibrium. As a proof, K0.75Na0.25NbO3films on TbScO3substrates
have to be deposited at higher temperatures.
63
D. Braun III. Results & Discussion
9. Energy density degeneration for (100)pc and (001)pc
orientation:
K0.90Na0.10NbO3films on (110) NdScO3
A second unique strain state occurrence is the film composition with x = 0.90 on (110)
NdScO3(NSO). Calculating the apc and cpc bulk lattice parameters of K0.90Na0.10NbO3
according to eq. 58 yields apc = 3.967 ˚
A and cpc = 4.029 ˚
A. In contrast to TbScO3
substrates, neodymium scandate offers a rather high anisotropic in-plane parameter
on the (110) growth plane with apc = 2 ×4.0015 ˚
A and bpc = 2 ×4.0138 ˚
A (see tab.
1). Evaluating the elastic strain energy density when depositing a KxNa1−xNbO3film
with (100)pc or (001)pc orientation on the (110) NdScO3surface reveals an energetic
degeneration (fig. 32b) at x = 0.87. Since the elastic strain energy densities at this
point have the same value for (100)pc and (001)pc oriented films, no surface orientation
is preferred; this may lead to a coexistence of both phases. Calculating the strain for
Figure 32: (a) Strain in dependence of the potassium concentration x in the solid solution
KxNa1−xNbO3for 0.5≤x≤1 on NdScO3substrate. (b) Estimation of the elastic
strain energy density F(ε) for KxNa1−xNbO3on NdScO3substrate versus potassium
content x for 0.5≤x≤1. For (a) and (b) the position with x = 0.90 is marked.
both orientations leads to the strain-composition diagram illustrated in fig. 32a for
x≥0.5. As a result of the large anisotropic substrate lattice parameters, the film
unit cell is likewise highly anisotropically strained. For x = 0.90, the (100)pc phase
is compressively strained along both substrate directions counting ε[001]NSO
xx =−0.67%
and ε[1¯
10]NSO
yy =−0.36%. However, the (001)pc orientation expires tensile strain of about
ε[001]NSO
xx = +0.87% and compressive strain of ε[1¯
10]NSO
yy =−0.36%. As a result, the film
64
D. Braun III. Results & Discussion
composition with x = 0.90 on (110) NdScO3facilitates the coexistence of a (100)pc
phase under average compressive strain and a (001)pc orientation with mean tensile
strain.
In order to study the surface morphology a (1×1)µm2area was measured in ambient
air with the atomic force microscopy and the scan is represented in fig. 33a.
Figure 33: (a) Surface morphology of a K0.90Na0.10NbO3film on a (110) NdScO3substrate on a
(1 ×1) µm2scale. (b) θ−2θHRXRD pattern of a K0.90Na0.10NbO3films on (110)
NdScO3in the range from 2θ= 20◦−23.5◦. The red and blue dotted lines indicate the
nominal angular position of the strained vertical lattice parameter dstrained
⊥for both a
(100)pc and (001)pc orientation, respectively, in case of K0.90Na0.10NbO3/NdScO3.
The AFM image of the film reveals very smooth surfaces with an average root mean
square surface roughness of 0.5 nm. As explained in chapter 8, the determination of
the growth mode for K0.90Na0.10NbO3films on (110) NdScO3would require a detailed
thickness series. However, from the AFM image (fig. 33a) it can be reasoned that a
transition occurs from 2D growth on each single step to island growth.
For the structural investigation, a θ−2θHRXRD scan of the K0.90Na0.10NbO3film
around the (110)NSO Bragg reflection of the orthorhombic NdScO3substrate was per-
formed. The measurement is shown in the range from 2θ= 20◦−23.5◦in fig. 33b. The
diffraction pattern is characterized by a sharp substrate peak around 2θ= 22.13◦and
a single film peak shaped as a shoulder appearing at about 2θ= 22.22◦. Measuring the
diffraction pattern until 2θ= 70◦reveals no secondary phases proving a pure perovskite
structure of the film. Extracting the angular values needed for eq. 56 and 57 from fig.
33b yield a vertical lattice parameter of d⊥= 4.013 ˚
A and film thickness t = 29 nm
65
D. Braun III. Results & Discussion
for the K0.90Na0.10NbO3film. For comparison, the theoretical strained vertical lattice
spacing for both, a (100)pc and (001)pc orientation, were estimated in accordance to eq.
67 to d(100)pcstrained
⊥= 3.987 ˚
A and d(001)pcstrained
⊥= 4.018 ˚
A. These values are transferred
into the θ−2θ-scan in fig. 33b as red and blue dotted lines, respectively.
As expected from the linear elastic theory, the experimental parameter is between both
theoretical ones. In consequence, indeed, both orientation could have been occurred.
Transferring the strain values obtained for x = 0.90 from fig. 32a into the multi-
dimensional misfit strain - misfit strain phase and energy density diagram in fig. 34a
yields the prediction of a monoclinic Mcphase for both orientations. Indeed, a Mc
Figure 34: (a) 3D plot of the elastic strain energy density F(ε) for KxNa1−xNbO3in dependence
of the biaxial strain εxx and εyy. On the energy landscape the strain-phase diagram for
KNbO3from fig. 22a was plotted. The filled red circles indicate the (001)pc strain state
in K0.90Na0.10NbO3/NdScO3. PFM measurements of a K0.90Na0.10NbO3/NdScO3
film, picture size (1 ×1) µm2: (b), (c) vertical, (d)-(h) lateral amplitude and phase
image, respectively. The cantilever orientation is illustrated in the upper left of part
of the image. (f) 2D-FFT of the lateral amplitude image (d).
domain structure is characterized by a polarization vector that can freely rotate within
the {100}pc monoclinic mirror planes and exhibits a Pm symmetry. Problematically,
the misfit strain - misfit strain phase diagram was calculated exclusively for a (001)pc
orientation because the relevant coefficients for eq. 69 are only available for KNbO3
66
D. Braun III. Results & Discussion
in the c-orientation. In conclusion, the prediction of Mcdomains for the (100)pc phase
may not be correct.
However, to get an idea about the expectable domain type in the a-orientation, the
unit cell distortion should be inspected. Both regarded orientations differ by at least
one 90◦rotation of the film unit cell. As a result, the monoclinic angle is in-plane
for the (100)pc oriented cells. This distortion would fit to the monoclinic a1a2forma-
tion. Meanwhile, the latter exhibits Pm symmetry in agreement to the Mcdomains
for (001)pc orientation. Indeed, for a (100)pc oriented film a a1a2domain structure
was confirmed experimentally [[96],[106]]. Therefore, it can be assumed that rather a
monoclinic a1a2pattern is going to develop for the (100)pc phase.
The vertical and lateral PFM images are displayed in fig. 34b-h distinguished into
amplitude and phase signal on a scale of (1 ×1) µm2. Already on a first sight, these
electric images show apparently a domain pattern which is rather complex and differs
significantly from those of K0.75Na0.25NbO3/TbScO3.
In the lateral phase image in fig. 34e comparatively large domain bundles colored in
yellow and violet can be resolved. They are tentatively aligned along the [001]NSO
direction with typical lateral sizes in [1¯
10]NSO direction between 50 and 200 nm. The
color code implies a distinct left - and rightwards torsion of the cantilever caused by a
changing direction of the lateral piezo active component from left to right. Already in
the lateral phase image 34e it can be assumed, that the domain pattern is not contin-
uously one domain type along the [001]NSO direction, but interrupted by tiny stripes
running rather along the [1¯
10]NSO direction. Indeed, a particular substructure arises
in the corresponding vertical and lateral amplitude figs. 34b and d. The large domain
bundles consist of a highly periodic array with alternating narrow and broad domains
that are shaped like honeycombs. The periodicity in [001]NSO direction amounts to
about LPFM = (30 ±1) nm.
Of major interest is if these narrow stripes are domains or represent only domain walls.
Analyzing both amplitude images 34b and d in detail on the basis of the color code
enables a distinct classification. The lateral amplitude image 34d reveals that the
bundles along [001]NSO are separated by pure domain walls because the amplitude de-
creases to zero. In contrast, the narrow domains within the bundles including an angle
to the [1¯
10]NSO direction are dyed in white for both the vertical and lateral amplitude
manifesting a high response. Therefore, it can be concluded that they are domains oth-
erwise the amplitude has to drop to rougly zero as in the case discussed beforehand. It
67
D. Braun III. Results & Discussion
seems, that the domain walls there are too small to be solved within the resolution of a
piezoresponse force microscope. That circumstance is not unusual because the width of
a ferroelectric domain wall can be about a few unit cells only [[110],[111],[112]]. Indeed,
the same situation arose in the K0.75Na0.25NbO3films on TbScO3hence it seems to be
common in theses films.
In addition to the white colored narrow domains in the vertical and lateral amplitude
images 34b and d, the broad stripe domains are dyed differently in both modes. They
have a low response in the vertical and a high signal in the lateral image. Herefrom
it can be concluded that two different domain types occur which are alternatingly
arranged. On one hand, the narrow domains with high response in the vertical and
lateral mode manifesting a phase with a distinct in- and out-of-plane signal. On the
other hand, comparably broad stripes that exhibit mainly in-plane polarization.
It should also be noted that the interpretation with only one domain variant and
domain walls would be incomprehensible from the theoretical background of domain
wall formation energies and - algorithms. Then, the walls would separate domains of
the same type with the same in-plane polarization alignment as can be concluded from
the lateral phase figure 34e. Such a structure would be accompanied by remarkably
energetic costs caused by the formation of a huge density of domains walls.
Unusually, these embedded stripes are tilted off against the [001]NSO direction by
α=±75◦as it is schematically illustrated in green in fig. 34d. This circumstance
can be likewise evaluated from the two branches in the 2D-FFT image in fig. 34f.
Such a tilting is completely unknown from classical ferroelectric systems in tetragonal,
orthorhombic or rhombohedral materials where the domain walls are fixed to the high
symmetric directions and therefore only tilting angles of α= 45◦and 90◦can occur.
In order to clarify the orientation(s) of the film, grazing incidence in-plane x-ray
diffraction (GIXD) has been performed at the European Synchrotron Radiation Fa-
cility (ESRF) using an x-ray wavelength of λ= 1.239 ˚
A. In-plane x-ray reciprocal
space maps have been measured in the vicinity of the (008)NSO and (2¯
24)NSO in-plane
Bragg reflections and are shown in fig. 35a,b. In contrast to the film on TbScO3, a
characteristic peak splitting P1and P2can be observed for both measured reciprocal
lattice points. As discussed in chapter 8, the lattice splits as a result of monoclinic
symmetry as illustrated schematically in fig. 27e.
68
D. Braun III. Results & Discussion
Figure 35: GIXD in-plane x-ray reciprocal space maps in the vicinity of the (a) (008)NSO and (b)
(224)NSO substrate Bragg reflections. (c) Simulation of reciprocal space map around
(008)NSO, for details see text. (d) Out-of-plane reciprocal space map in the vicinity of
(440)NSO substrate Bragg reflection. Taken from [113].
A detailed analysis of the observation of peak splitting near the (008)NSO and (2¯
24)NSO
Bragg reflection points together with the absence of a film peak splitting in the vicinity
of the (4¯
40)NSO reflex (not shown here) reveals that P1and P2are caused by domains
with alternate in-plane shearing. The monoclinic cell exhibits a distortion angle of
β=±0.12◦and a shearing along the [001]NSO direction whereas the film crystal lattice
remains coherent along the [1¯
10]NSO direction. In conclusion, the observed in-plane
shearing of the film lattice is attributed to the presence of a monoclinic phase with
(100)pc orientation and thus a1a2domains. This is in agreement with the strong lateral
piezoelectric response of the bundles in the PFM measurements.
As an evidence of the periodic ordering of the domains well pronounced satellite
reflections emerge in the vicinity of P1and P2for both the (008)NSO and (2¯
24)NSO
in-plane reciprocal lattice points. From the satellite peak spacing a periodicity of
LGIXD = (30 ±1) nm along the [001]NSO direction can be deduced. This result is in
accordance with the PFM data LPFM = (30 ±1) nm shown in fig. 34b-h.
69
D. Braun III. Results & Discussion
Interestingly, for each peak only one satellite branch emerges. For P1it is tilted off by
α= +75◦while for P2it is tilted off by α=−75◦against the [001]NSO direction. As
a result, the sign of the monoclinic shearing can be directly related to the observation
in the piezoresponse force micrographs. The ”yellow” domains in fig. 34e exhibit an
in-plane lattice shearing of β=−0.12◦whereas the ”violet” domains can be associated
with an in-plane lattice shearing of β= +0.12◦.
In order to prove the model, the domain arrangement as depicted schematically in fig.
35 was simulated by means of the Hosemann function [114] for the (008)NSO Bragg
reflection by Dr. Martin Schmidbauer and Dr. Michael Hanke. The result can be seen
in fig. 35c and is in good agreement to the respective experimental result in fig. 35a.
To unveil the structure of the narrow subdomains, plan-view transmission electron
microscopy (TEM) investigations using a FEI Titan 80-300 operating at 300 kV were
carried out by Dr. Toni Markurt (for further information see [113]). Hereby, the sam-
ple has to be ”destroyed” for preparation and therefore a similar sample with the same
deposition parameters were used as for the sample discussed before. In order to avoid
elastic relaxation of the strained K0.90Na0.10NbO3film as a result of thinning down the
substrate during specimen preparation, TEM specimens with a comparatively thick
(t >100 nm) substrate layer were used. A dark field contrast pattern obtained by
using the excitation vector g= (002)NSO is illustrated in fig. 36. In excellent agree-
ment with the PFM measurement shown in fig. 34c and e the same characteristic
features appear. A substructure of narrow stripe domains arise within the matrix of
large domain bundles as can be seen in the dark field image 36a, in the in-plane shear
component εxy map as well as in the in-plane strain εxx image in fig. 36c,d, respectively.
This confirms that the TEM specimen preparation has not changed significantly the
local strain state of the film and thus the domain structure is still comparable to the
as-grown sample state. The analysis in fig. 36c shows that the large domain bundles
colored in yellow and violet in the PFM (fig. 34e) - and red and green in the TEM
results - differ in their in-plane shear angle εxy. This result is in agreement with GIXD
observations discussed beforehand: There, the (100)pc orientation of the pseudocubic
unit cell was determined with an alternate shearing along ±[001]NSO. From the STEM
images, the corresponding monoclinic shear angle difference between adjacent domain
bundles can be evaluated to ∆εxy = 2β= (0.4±0.1)◦. In comparison to the value
obtained by GIXD 2β= (0.2±0.1)◦, these data are slightly larger but is still in satisfac-
tory agreement. The deviation could result from the fact that for both measurements
70
D. Braun III. Results & Discussion
Figure 36: Plan-view (S)TEM micrographs of a K0.90Na0.10NbO3film on (110) NdScO3substrate:
(a) Dark field TEM image and (b) high-resolution STEM ADF measurement. The
inset in (b) shows a magnification whereby white intensity maxima correspond to the
Nb atomic columns of the film. The in-plane shear strain maps divided in εxx (d) and
εxy (c). In addition the in-plane axial strain maps εyx (e) εyy (f) obtained from a
geometric phase analysis of (b). Taken from [113].
different samples were used that slightly differ.
A major information that can be gained from these STEM measurements is that
the broad and narrow subdomains exhibit each a different in-plane lattice parameter
along the [001]NSO direction. The evaluation shows that the narrow areas exhibit a
reduced lattice parameter of ∆εTEM
xx = (−1.0±0.3)% in comparison to the broader
ones. In contrast, complementary maps along the orthogonal [1¯
10]NSO direction reveal
no significant contrast within the accuracy as depicted in fig. 36e,f. This behavior
would be likewise in agreement to the prediction of an additional (001)pc orientation
because the ε(x) characteristic in fig. 32a shows no strain difference for both orienta-
tions along the [1¯
10]NSO direction. These results strongly suggest a (001)pc orientation
of the pseudocubic unit cell in the narrow subdomains.
71
D. Braun III. Results & Discussion
To identify a (001)pc orientation, x-ray diffraction in normal geometry was per-
formed. The c-orientation should have a different vertical lattice parameter com-
pared to the (100)pc symmetry as it was discussed in the beginning of this chapter
(d(100)pcstrained
⊥= 3.987 ˚
A and d(001)pcstrained
⊥= 4.018 ˚
A). Moreover, the (001)pc orienta-
tion is characterized by an out-of-plane monoclinic distortion in contrast to the (100)pc
phase. As a result, four Bragg reflection points should occur: one each for the substrate
and the (100)pc phase and two for the monoclinically distorted unit cells with (001)pc
orientation. A reciprocal space map around the symmetrical (440)NSO out-of-plane
substrate Bragg reflection was measured and illustrated in fig. 35d. However, since the
Bragg reflection peaks of the (001)pc phase are interfered by the substrate peak and the
Bragg peak of the (100)pc phase, the monoclinic distortion of the (001)pc orientation
can not be resolved.
But, a complex pattern with satellite peaks arises. The latter can only be explained,
if a period arrangement of two orientations with different out-of-plane parameter ex-
ists. Moreover, the spacing between these two orientations can be evaluated from the
separation of the satellite peaks along Q001 to a periodicity of 30 nm. Together with
the information from GIXD, where a distinct (100)pc orientation was detected and the
alternating domain pattern determined in PFM as well as in STEM mode, the only
possibility is a (001)pc phase in the narrow subdomains.
Summarizing the electrical and structural data into a domain model yields fig. 37.
The analysis shows that the large bundles in fig. 34e, marked likewise by yellow and vi-
olet colors in fig. 37, exhibit (100)pc orientation. As displayed in fig. 37 at the bottom,
for this type the film unit cell features an in-plane monoclinic distortion accompanied
with an in-plane electric polarization. Between the bundles, the in-plane polarization
changes by 90◦as can be extracted from the color code in the lateral phase image
34f. Such a ferroelectric pattern is assigned as a1a2domain. In the adjacent narrow
domain, schematically sketched in light blue and green, the film lattice is rotated by
90◦into a (001)pc orientation. Simultaneously, the polarization rotates and exhibits
then a pronounced vertical component. Due to the fact, that the vector is aligned
within the {100}pc planes, the corresponding domain pattern is indeed a so-called Mc
pattern. This result is moreover in agreement with the calculated misfit strain-misfit
strain phase diagram shown in fig. 34a. From the crystallographic view, both orienta-
tions have Pm symmetry and can be transformed into each other easily by a rotation.
This point may facilitate the growth of both types by means of elastic deformation and
72
D. Braun III. Results & Discussion
Figure 37: Schematic plan view of the domain pattern arising in the corresponding
K0.90Na0.10NbO3depicted in cross sectional view. Moreover, sheared pseudocubic
unit cells for both (100)pc and (001)pc orientations. The a1a2domains with (100)pc
orientation are dyed in yellow and violet colors whereas the Mcdomains with (001)pc
orientation are colored in blue and green. The red arrows indicate the orientation of
the electrical polarization vector.
thus without defects.
In contrast to ferroelectric materials with tetragonal or rhombohedral symmetry, in
K0.90Na0.10NbO3films on NdScO3distinctive differences can be observed:
imonoclinic symmetry resulting in a1a2/Mcdomains instead of a a/c domain pattern
ii due to the monoclinic symmetry domain walls do not necessarily run along main
crystallographic axes, e.g. parallel to [001]NSO, [1¯
10]NSO or [112]NSO. As a conse-
quence, the film in fig. 34 exhibits an inclination angle of α= 75◦in respect to
the [001]NSO direction. This phenomenon will be discussed in detail in chapter
10.
One important factor is the switching behavior of a ferroelectric film under an exter-
nal bias. Experimentally, a direct voltage is applied to the tip and a pattern is written
on the film surface. For vertical c-domains, the result is the well-known switching of
the polarization vector by 180◦[[115],[116]].
73
D. Braun III. Results & Discussion
If and how a mainly lateral domain pattern reacts on a vertically applied voltage is
rather less studied [[117],[118],[119]]. Therefore, the discussed domain pattern formed
in K0.90Na0.10NbO3on (110) NdScO3is ideally suited for a fundamental investigation
of lateral poling.
The lateral PFM measurements before and after a writing pulse of Vdc = +2 V are
shown in fig. 38. Within the red square in fig. 38a the voltage was applied exclusively.
For a better correlation of the PFM signal to the lithography area, the red square was
illustrated in all LPFM images in fig. 38.
The poling field was situated this way on the ferroelectric herringbone pattern that
it has overlap with ”ending” bundles aligned along [001]NSO. The result can be seen
Figure 38: Lateral PFM images of a K0.90Na0.10NbO3film deposited on a (110) NdScO3substrate
presented (a)-(b) before poling and (c)-(d) after poling each distinguish into amplitude
and phase signal. The red square marks the lithography area. The bias applied
amounts Vdc = 2 V.
in fig. 38c,d. The former ”ending” bundles grow in [001]NSO direction far behind the
red poling square (see red arrows). Interestingly, the positive voltage does not lead to
a single polarization alignment, a so-called monodomain, within the red square. As
can be seen in the lateral phase image in fig. 38d, the in-plane polarization within the
written square leads to a torsion of the cantilever both left - and rightwards. Even-
tually, the highly periodic and nested herringbone structure leads immediately to a
compensation of the polarization after switching to remain stable.
74
D. Braun III. Results & Discussion
How can it be explained that a vertically applied bias lead to a lateral poling? It has to
be noted that in contrast to the poling model in chapter 1.4, the K0.90Na0.10NbO3film
was grown epitaxially and fully strained on the bare substrate. Hence, the deposition
of a bottom electrode in-between film and substrate was rejected. Consequently, the
introduced field may be distributed completely different in comparison to the domain
switching model in chapter 1.4. This effect is schematically drawn in fig. 39. While the
Figure 39: Electric field (a) between a top and bottom electrode and (b) without any electrode.
electric field is homogeneously distributed between two electrodes (fig. 39a), it could
propagate not directly to the substrate in the case without any electrode (fig. 39b). In
the latter, defects and film distortions can influence the energetically favored path. In
consequence, the lateral expansion of the electric field in the film can be much higher.
From fig. 38c,d the conclusion can be drawn that the electric field propagates well in
lateral dimension yielding a switching far behind the poling area. Hence, the missing
bottom electrode could be beneficial for the lateral switching.
Generally, lateral domains reduce mainly the strain. How it works to pole an a1a2
domains from ”yellow” to ”violet” is not completely understood so far. Regarding fig.
37, the (100)pc unit cell has to rotate by 90◦although it should be pinned to the sub-
strate. Perhaps, this rotation is not needed because the monoclinic unit cell distortion
is rather small with β= 0.12◦. Hence, the distortion difference between ”yellow” to
”violet” is about β=±0.12◦only and can be switched by means of an elastic progress.
Meanwhile the bundle width in [1¯
10]NSO remains constant as well as the periodicity
of the stripe domains within the bundles. This effect is maybe due to the nested multi-
domain structure itself. Considering the theoretical calculations for these so-called
”multi-rank” or ”herringbone” structures [[120],[121],[122],[123]], domain walls behave
75
D. Braun III. Results & Discussion
differently. To explain this characteristic, the following notation should be used. As
can be seen in fig. 40, the domain walls separating the bundles shall be denoted as d2
whereas the small herringbones should be confined by d1walls. The theoretical cal-
culations on domain switching in such a confinement as the herringbone arrangement
come to the conclusion that d1and d2have different freedom to move [[120],[121]]. The
reason is due to the pursuit of energetic neutralization. In consequence, the shape and
domain volume have to be kept constant during domain growth/switching. This aim
can be easily fulfilled when the d1domain walls move, but is difficult for a movement
of d2domain walls as it should be illustrated schematically in fig. 40. Indeed, the same
Figure 40: Schematic illustration of the movability of different domain walls in ferroelectric her-
ringbone pattern according to [[120],[121]].
behavior can be observed during poling of the K0.90Na0.10NbO3film grown on a (110)
NdScO3substrate. Hence, the development exclusively along [001]NSO seems to be in
order of energetic neutralization.
Finally, the lithography area was scanned for hours to see if a back switching occurs.
But the domain pattern remains stable. Hence, a vertically applied bias can yield to
an irreversible lateral switching.
It shall be mentioned, that no experimental difference can be observed if the applied
voltage has a positive or negative sign.
To sum up, the commonly known switching from vertical domains into one polar-
ization state does not appear. This circumstance may be explained by the missing
bottom electrode and the nested herringbone pattern.
Because of the problem how the lateral switching could progress, it was considered to
be impossible. But, the example K0.90Na0.10NbO3on NdScO3prove impressively the
possibility to switch also in lateral direction.
76
D. Braun III. Results & Discussion
Such an alternating monoclinic domain pattern of a1a2/Mcdomains is potentially of
huge technological relevance. As already described in chapter 2.5 and 8, monoclinic
phases are accompanied by giant piezoelectric responses attributed to the free rota-
tion of the polarization vector. A similar rotation can be artificially mimicked by a
domain structure with alternating exclusive in-plane and out-of-plane polarization. At
the domain walls discontinuities exist that provoke a polarization rotation enhancing
the piezoresponse in the same manner. Such a behavior was observed in tetragonal a/c
domain pattern [[124],[125]]. Meanwhile, a piezo-enhanced signal was also detected in
vortex structures [126].
However, an even larger longitudinal piezoelectric coefficient is expected for the com-
bination of both - a monoclinic alternating domain structure with polarization discon-
tinuities at the domain walls due to the exclusive in-plane and inclined polarization
components in adjacent domains as it was theoretically predicted by Koukhar et al.
[97]. Indeed, the presented pattern is the first experimental realization of such a struc-
ture.
Therefore, after proving the concept of coexisting monoclinic phases predicted by
linear elasticity theory and experimentally verified by XRD, TEM and PFM, a similar
film with a thickness of t = 25 nm was deposited on a conducting SrRuO3bottom
electrode deposited with pulsed laser deposition for comparison. The resulting vertical
and lateral PFM images are depicted in fig. 41a-d. The same herringbone pattern as
described before for the deposition of K0.90Na0.10NbO3on the bare NdScO3substrate
arises on the multi-layer structure K0.90Na0.10NbO3/SrRuO3/NdScO3(fig. 41a-d). It
becomes obviously that the ordering is worse on the sample coated with SrRuO3com-
pared to the 29 nm sample thick sample discussed in fig. 34b-h. This behavior could be
explained by several effects. Defects could be introduced from the intersection SrRuO3
layer into the K0.90Na0.10NbO3film partially hindering the perfect ferroelectric or-
dering. Indeed, the bottom electrode was grown vial pulsed laser deposition (PLD)
yielding oftentimes off-stoichiometric films with reduced crystal quality. Moreover, the
films have to be transported from the PLD chamber to the MOCVD reactor yielding
the possibility of further environmental dust on the surface. Additionally, the electri-
cal boundary conditions are different with a conducting electrode underneath the film
which may influence the domain formation. Indeed, a plastic lattice relaxation of the
SrRuO3film reducing the final stress on the K0.90Na0.10NbO3film can be neglected as
77
D. Braun III. Results & Discussion
the SrRuO3is only 10 nm thick and reciprocal space maps performed in the University
of Groningen showed no onset of relaxation processes.
Figure 41: PFM measurements of a K0.90Na0.10NbO3film on a SrRuO3bottom electrode on a
NdScO3substrate, picture size (1 ×1) µm2: (a), (b) vertical, (c)-(f) lateral amplitude
and phase image, respectively. The measurements (a)-(d) are taken before writing the
red square with Vdc = 2 V on the sample surface. The lateral PFM images (e) and
(f) are measured afterward. The orange, dotted square shows a magnification of the
lateral amplitude image. (g) Phase-Voltage diagram extracted from a local hysteresis
measurement. The phase shift of ∆φ= 180◦is illustrated in red.
In order to reveal the influence of the bottom electrode on the lateral switching pro-
cess, the K0.90Na0.10NbO3/SrRuO3/NdScO3stack was written with a square of Vdc = 2
V. The resulting lateral PFM measurement is depicted in fig. 41e,f indicating the lithog-
raphy area with a red square.
Obviously, the influence of a bias is different with a conducting bottom electrode un-
derneath. First, the bias induced changes are limited to the poling area. Second, the
bundles do not grow neither in [001]NSO nor in [1¯
10]NSO direction. But, the small stripe
domains within the bundles are affected. They are partially rotated as it is highlighted
in an orange doted square for a specific position yielding a cross of stripe domains as
it can be seen in the magnified orange doted square. A similar rotation was observed
before by Xu et al. [127] but as an intermediate step for a complete reversal. Therefore,
78
D. Braun III. Results & Discussion
it can be assumed that the lateral PFM measurement in fig. 41e,f shows a metastable
switching state. Why it was ”frozen” in this state can only be supposed. Probably, it
was pinned by defects, maybe arising from the SrRuO3electrode. This disadvantage
from first sight, can provoke conducting domain walls. This could not be proven on
this sample so far but is part of further research.
The bottom electrode enables further measurements. First, a ferroelectric hysteresis
could be measured as it is shown in fig. 41g for the phase part. The hysteresis reveals
a phase difference of ∆φ= 180◦which reflects at least two stable phases of the polar-
ization that can be achieved via an external bias proving the ferroelectric nature of the
film on a different scale.
Moreover the longitudinal piezoelectric coefficient d0
33 can be recorded. According
to the theoretical calculations of Koukhar et al. [97], the highest d0
33 values can be
achieved indeed in such an alternating, monoclinic a1a2/Mcstructure. Due to a lack
of top electrodes, the vertical piezoelectric coefficient was determined locally with the
PFM according to the procedure explained in chapter 5.4 yielding a value for the piezo-
electric response in z-direction of d0
33 = 29pm
V. This value is promising because it is
in the range of bulk KNbO3crystals [[6],[7],[8],[9]] and can compete with piezoelectric
coefficients recorded for Pb(Zr,Ti)O3thin films with similar thickness [10]. Therefore,
further effort will be spent in optimizing the growth conditions for achieving a com-
parable ordering of the domains on the bottom electrode to those on the bare substrate.
Regarding the unusual domain wall inclination angle, I performed an intense study
of αas a function of potassium content x [128] presented in chapter 10.
79
D. Braun III. Results & Discussion
10. Tunable ferroelectric domain wall alignment in
monoclinic systems
In regard of charge neutrality, domains have to be formed as twins with equal shape and
volume. Hereby, the alignment of domain walls is restricted to the symmetry elements
of the crystallographic point group of the material under investigation. Hence, in com-
mon ferroelectric materials crystallizing in high symmetric tetragonal or orthorhombic
phases, the electrical polarization vector is linked to a crystallographic axis. As a re-
sult of this restriction the corresponding domain walls can be tilted against the main
crystallographic axes exclusively by angles of 45◦or 90◦[[111],[129]] although the angle
between both adjacent polarization vectors could be different as apparent in rhom-
bohedral systems [[130],[131]]. In contrast, in the low symmetric monoclinic phase,
the mirror plane is the primary symmetry element. As a result, for ferroelectric sys-
tems with monoclinic symmetry, the domain walls are solely fixed to this mirror plane
permitting arbitrary directions to the main crystallographic axes. Consequently, the
included angle between both can be almost continuously modified.
Indeed, in a theoretical work, Bokov and Ye [5] showed that domain wall angles in
monoclinic lead zirconium titanate may vary in the range of several tens of degree.
They basically depend only on the lattice parameters and the monoclinic distortion
angle β.
Lattice parameter in KxNa1−xNbO3can easily be varied by changing the composition
x or the underlying substrate. Here, the spatial dimension as well as the shearing angle
of the monoclinic unit cell were modified through a variation of the potassium content
x between x = 0.80 to 0.95 in the KxNa1−xNbO3thin films on NdScO3substrates. The
comparatively narrow composition range is based on the requirement of coexisting a1a2
and Mcphases with Pm symmetry to induce the pronounced herringbone domain pat-
tern. However, despite the small variation in the potassium content x, the monoclinic
distortion angle βand the vertical lattice parameter do change significantly.
The surface morphology of the films was analyzed via atomic force microscopy. The
surface topology of all films exhibit rather smooth surfaces with a root mean square
roughness well below 0.5 nm. The ferroelectric domain structure was investigated with
the piezoresponse force microscopy. The lateral piezoresponse images of all samples
are shown in fig. 42a-d.
In accordance to the measurements presented before for x = 0.90, a coexistence of a1a2
80
D. Braun III. Results & Discussion
and Mcdomains is observed which is interlaced to a herringbone pattern. Obviously,
the stripe domains do not include an angle of 45◦or 90◦to the [001]NSO directions. In
order to quantitatively evaluate the inclination angles, αexp
PFM, of the stripe domains,
two-dimensional Fourier analysis of the PFM images shown in fig. 42a-d were per-
formed. As one example, the 2D-FFT calculation for the sample with 90% (fig. 42b)
is shown in fig. 42f. According to the model from Bokov and Ye [5], αdepends on the
Figure 42: (a)-(d) Lateral piezoresponse force amplitude images ((1 ×1) µm2) of KxNa1−xNbO3
thin films grown on NdScO3orthorhombic substrate with varying potassium concen-
tration x. Insets in (b) and (c) show the definition of in-plane domain wall inclination
angle αand the orientation of the NdScO3substrate, respectively. (e) Grazing Inci-
dence in-plane x-ray diffraction intensity distribution recorded in the vicinity of the
(2¯
24)NSO reciprocal lattice point for x = 0.90. (f) 2D-FFT calculation of the PFM
image shown in fig. 42b. Taken from [128].
monoclinic lattice parameters and the distortion angle β. For a better imagination,
the monoclinic shearing angle β(exaggerated plotted) and the domain wall inclination
81
D. Braun III. Results & Discussion
angle αare sketched in fig. 43. To validate their consideration, the monoclinic lattice
Figure 43: Schematic view of the domain pattern with embedded domain wall inclination angle
αand the respective KxNa1−xNbO3(sheared) pseudocubic unit cells (cross sectional
view). The monoclinic a1a2domains with (100)pc orientation are colored in blue, the
Mcdomains with (001)pc orientation in are depicted with yellow color. The monoclinic
distortion angle βis illustrated for both orientations and shearing directions. Taken
from [128].
parameters were experimentally determined. As in-plane lattice parameter, the surface
lattice parameter of the neodymium scandate substrate are used because the films are
grown fully strained.
In order to evaluate the monoclinic in-plane distortion angle β, grazing incidence in-
plane x-ray diffraction has been performed. In order to distinguish between strain
and morphology related features in reciprocal space a variety of Bragg reflections, e.g.
(004)NSO, (008)NSO, and (2¯
24)NSO, were investigated. Exemplary, the in-plane inten-
sity distributions recorded in the vicinity of the (2¯
24)NSO reciprocal lattice point is
presented in fig. 42e for the potassium content of x = 0.90. A complicated diffraction
pattern evolves. The periodic arrangement of the domain walls leads to two inclined
intensity branches (marked by white dashed lines) which are intersected by broad cor-
relation peaks. From the inclination of the two branches the domain wall angle αexp
GIXD
is obtained (indicated in fig. 42e-f). Moreover, the separation ∆Q1¯
10 of the central
peaks P1and P2of the two branches is related to the monoclinic distortion angle β
82
D. Braun III. Results & Discussion
via:
tan β=∆Q1¯
10
2·Q001
(71)
In this way, βhas been measured for all four samples depicted in fig. 42a-d. The
corresponding βangles are summarized in table 6.
Table 6: Structural parameters of the samples under investigation. The in-plane lattice parame-
ters of the film are equal to the in-plane lattice parameters of the (110) NdScO3surface
unit cell (d1¯
10 = 4.0124 ˚
A and d001 = 4.0025 ˚
A).
sample 1 sample 2 sample 3 sample 4
K atomic concentration x 0.80 0.90 0.92 0.95
Vertical lattice spacing d (˚
A) 3.969 3.986 3.989 3.992
Monoclinic angle β(degrees) 0.145 0.11 0.08 0.077
The experimental domain wall angle obtained from PFM (αexp
PFM) and GIXD (αexp
GIXD)
analysis are illustrated in fig. 44 as blue circles and red triangles, respectively, as a
function of monoclinic distortion β. The error bars result from the reading accuracy
Figure 44: In-plane domain wall inclination angle αderived from PFM and GIXD along with
calculated values as a function of the monoclinic distortion angle β(for details see
text). Taken from [128].
in figs. 42e,f. It is striking, that (i) the experimental data evaluated from PFM and
GIXD are in excellent agreement and (ii) the domain wall angle is quite sensitive to
83
D. Braun III. Results & Discussion
the monoclinic distortion of the film unit cell.
The correlation between monoclinic shearing angle βand in-plane domain wall angle
αhas been theoretically considered by Fousek et al. [132] for the arrangement of
compatible ferroelectric domain walls depending on their respective symmetry and by
Sapriel [133] for ferroelastics. The special case of monoclinic symmetry was discussed
later by Bokov and Ye [5]. On the basis of these geometrical considerations, compatible
monoclinic domain walls can be identified and the resulting domain wall angles αcan
be derived for Pm symmetry.
Thereby, two different domain walls S1and S2can appear whose plane equations are
denoted as follows:
S1→(a−b)(x±y) = ±2dz (72)
S2→(c−a)(x±y) = ±2dz (73)
The difference is a quantitative change of the tilting angle of the domain wall around
the h001ipc directions. In eq. 72 and 73, the parameter set a, b, c and d denotes the
components of the so-called spontaneous strain tensor [134]. These quantities can be
expressed by the pseudocubic [49] lattice parameters apc, bpc, cpc and the monoclinic
distortion angle βvia:
a=bpc −p
p(74)
b=apc −p
p(75)
c=cpc −p
p(76)
2d=π
2−β(77)
p=apc +bpc +cpc
3(78)
Furthermore, x, y and z describe likewise the main crystallographic axes as well as
the indices of the planes in the pseudocubic notation. The domain wall angle αwith
respect to the h001ipc directions can be evaluated as traces of these walls on e.g. the
x = 0 planes as described in eqs. 79-80:
S1→tan α=a−b
±2d(79)
S2→tan α=c−a
±2d(80)
84
D. Braun III. Results & Discussion
As mentioned before for fully strained films, the dimensions of the surface unit cell
of the NdScO3substrate can be used as in-plane lattice parameters of the film, while
the vertical lattice parameters have been determined from θ/2θhigh resolution x-ray
diffraction scans (see table 6). Introducing these values in eqs. 74-80, the domain wall
angles αtheo were calculated and are displayed in fig. 44 as green squares with error bars
given by the experimental uncertainties of βof uβ=±0.010◦. Very good agreement
between experimental and calculated values are achieved.
The remaining difference should be briefly discussed. In general, the theoretical descrip-
tions [[5],[132]] are based on a single-domain and free-standing crystal. In consequence,
no epitaxial, elastic stresses are considered as they apply for the epitaxial, fully strained
film discussed in this thesis. Beyond that, Mokry and Fousek [135] calculated the case
of elastic stresses resulting from domain wall alignment in the case of more than two
ferroelectric/ferroelastic domains. They conclude that in the presence of these ”do-
main quadruplets” the ferroelastic domains prohibit a stress-free coexistence. Indeed,
the a1a2and Mcdomains have different monoclinic distortion requiring an elastic re-
laxation process at the domain walls. As a result, the pure geometric aspects of the
models [[5],[132]] do not reflects the experimental conditions and a deviation from the
prediction seems reasonable.
In conclusion, the domain wall angle in a herringbone pattern in monoclinic KxNa1−xNbO3
thin films can significantly deviate from 45◦or 90◦. Indeed, it could even be verified that
the domain wall alignment is flexible and can be adjusted according to the theoretical
model of Bokov and Ye [5]. This emphasizes the outstanding role of monoclinic phases
in ferroelectric materials. The symmetry inherent property to tilt the domain walls
nearly arbitrarily against main crystallographic axes results in an additional degree of
freedom with regard to ”domain engineering” and allows for targeted manipulation of
the ferroelectric, periodic domain pattern on a nanometer scale.
In order to tune the monoclinic lattice parameter, a second pathway would be to vary
the monoclinic distortion βwith film thickness [[96],[106]]. Above the critical thickness,
the lattice parameters of a fully strained films relaxes to those of an unstrained film
crucially influencing the domain pattern. Rather less is known about the thickness
dependent evolution of such hierarchical ferroelectric pattern. Therefore a thickness
series of K0.90Na0.10NbO3on (110) NdScO3was deposited and studied in detail for the
first time presented in chapter 11 and [136].
85
D. Braun III. Results & Discussion
11. Hierarchy and scaling behavior of herringbone
domain patterns in strained K0.90Na0.10NbO3
ferroelectric epitaxial films
For epitaxial layers with thicknesses in the range of a few tens of nanometers, classi-
cal scaling theories calculated for laminar domain formations [[137],[138]] predict that
the lateral domain widths and film thickness are of the same order of magnitude. In
consequence, domain walls represent a significant volume fraction of the film and their
contribution to the total piezoelectric response is of particular relevance [[139],[140]].
By contrast, very little is known about the evolution and scaling behavior of multi-
rank [[122],[123]] structures with competing, different electric polarization alignments -
especially if domains exhibit a monoclinic symmetry. Indeed, the formation of ferroelec-
tric, monoclinic herringbone domain structures is highly interesting for possible tech-
nological applications as structuring devices [141] or building energy harvester [142].
However, the impact of film thickness on complex ferroelectric, monoclinic multi-rank
arrangements has not been investigated yet in detail. In this context, open questions
are, if there exists a hierarchy for the inset of each single ferroelectric domain. Which
film thickness provokes what kind of domain formation? These questions are manda-
tory to answer for a targeted film deposition in nanoscale applications.
With regard to the questions raised above, K0.90Na0.10NbO3thin films on a (110)
NdScO3substrate as discussed in chapter 9 represent an interesting model system. In
order to study the thickness dependence of ferroelectric herringbone pattern for the
first time, thin films with x = 0.90 were deposited on NdScO3in the range of 7 to 52
nm.
In order to directly correlate the electrical and structural properties, PFM and GIXD
measurements have been performed.
Generally, the surface topology of all films is comparable to fig. 33a and exhibits rather
smooth surfaces with a root mean square roughness well below 0.5 nm.
The piezoresponse force microscopy measurements are shown in fig. 45.
For the 7 nm thin film, the lateral PFM amplitude image (fig. 45a) indicates the
onset of ferroelectric domain formation. The pattern reveals randomly distributed do-
mains, which are elongated along [001]NSO and exhibit a typical width of about 65 nm
along [1¯
10]NSO. However, the vertical PFM image (fig. 45b) does not show any sig-
nificant contrast variation. This is attributed either to a marginal amplitude contrast
86
D. Braun III. Results & Discussion
Figure 45: Piezoresponse force amplitude micrographs (PFM) of (a-b) 7 nm, (c-d) 14 nm, (e-f)
29 nm K0.90Na0.10NbO3thin films grown on orthorhombic (110) NdScO3substrate
in lateral (top) and vertical (bottom) mode. For the (g) 38 nm and (h) 52 nm thin
films lateral PFM amplitude measurements are depicted. The measurement area was
chosen to (1 ×1)µm2. Taken from [136].
due to lower sensitivity of the vertical mode [143] or even the absence of a vertical
polarization component. LPFM phase images (not shown here) indicates that in the
dark and bright areas the in-plane component of the electric polarization vector is ro-
tated by 90◦. In accordance to the former results on a 29 nm thick film in chapter
9, the conclusion is drawn of exclusively in-plane oriented a1a2domains at very thin
film thicknesses. Grazing incidence x-ray diffraction on this sample results in a sharp
peak S (fig. 46a), which is caused by the coherently grown K0.90Na0.10NbO3film on
the underlying NdScO3substrate. It is surrounded by pronounced diffuse scattering,
which is presumably caused by fluctuations of the irregular domains sizes and of the
(still very small) in-plane monoclinic angles of the a1a2domains.
With increasing thickness (14 nm), the domains form periodic bundles which are aligned
along [001]NSO, clearly observed in the lateral PFM image (fig. 45c). A detailed anal-
ysis of the PFM images indicates, however, that the bundles are regularly disrupted
along [001]NSO by slanted lines. This can be also detected in the vertical PFM image
(fig. 45d). Since the slanted lines show a low amplitude signal in the lateral PFM
image but appear in bright color (= large amplitude signal) in the vertical PFM (see
magnified inset in fig. 45c) measurement, they are unlikely be caused by a crosstalk
effect. Consequently, an experimental artifact can be excluded. Hence, at 14 nm film
87
D. Braun III. Results & Discussion
thickness, the onset of a second ferroelectric phase can be observed with a strong ver-
tical electrical polarization component. According to the intense study in chapter 9,
the bright and dark domains in VPFM can be identified as monoclinic Mcand a1a2
domains, respectively. The occurrence of (001)pc oriented Mcdomains in 14 nm thick
films substantially changes the corresponding GIXD intensity distribution (fig. 46b).
The a1a2domains provoke a distinct peak splitting (P1and P2) along Q1¯
10. The corre-
sponding monoclinic distortion angle, which describes the alternate in-plane shearing
of the unit cells along the [001]NSO direction, is determined to β= 0.075◦. However,
the slanted lines, which is assigned to the onset of Mcdomain formation, do not lead
to a detectable signal in the x-ray diffraction data. This can be explained by the very
small scattering volume accompanied with the Mcphase.
When the film thickness is further increased to 29 nm, a pronounced herringbone
domain pattern arises in both the lateral (fig. 45e) and vertical (fig. 45f) PFM im-
ages. It consists of a highly periodic array of alternating a1a2/Mcdomains, where the
corresponding domain walls are tilted by an angle ±αwith respect to the [001]NSO
direction (see inset in fig. 45f). Moreover, the periodic a1a2/Mcdomain arrangement
is embedded in large bundles which are aligned along [001]NSO forming a herringbone
pattern. These bundles exhibit a uniform width along [1¯
10]NSO, which has already
been indicated in fig. 45c and d for the 14 nm thick film. Grazing incidence x-ray
measurements on the same film exhibit an increased peak splitting (P1and P2) (fig.
46c) pointing to a larger monoclinic shearing angle. Furthermore, pronounced satellite
peaks occur which are aligned in branches inclined by ±αwith respect to the [001]NSO
direction (see dashed white lines in fig. 46c). These are attributed to the periodic ar-
rangement of the a1a2/Mcdomains with corresponding inclined domains walls. Again,
a direct scattering signal from the Mcdomains cannot be detected in the x-ray data.
The overall amount of the Mcphase is still too small as to lead to a detectable scat-
tering signal. This behavior is consistent with the sample investigated in chapter 9.
Compared to the 29 nm thick film, the sample with a film thickness of 38 nm does
not reveal remarkable modifications of the domain pattern (see the lateral PFM image
in fig. 45g). The widths of both Mcand a1a2domains along [001]NSO have further
increased, however, the relative volume fraction of Mcdomains has been grown. The
corresponding characteristic x-ray diffraction pattern with two distinct peaks P1and
P2and the associated satellite branches are maintained (fig. 46d). The intensity of the
satellite peaks is increased indicating an increased periodicity of the domain pattern
as compared to the 29 nm sample. This is in agreement with the lateral PFM micro-
88
D. Braun III. Results & Discussion
Figure 46: Grazing-incidence in-plane X-ray diffraction (GIXD) intensity distribution in the vicin-
ity of the (2¯
24)NSO substrate Bragg reflection for the (a) 7 nm, (b) 14 nm, (c) 29 nm,
(d) 38 nm and (e) 52 nm K0.90Na0.10NbO3thin films grown on (110) NdScO3(NSO)
substrate. (f) Out-of-plane intensity distribution in the vicinity of the (444)NSO sub-
strate Bragg reflection for the 52 nm film. Taken from [136].
graphs displayed in fig. 45g (38 nm) and fig. 45e (29 nm). However, importantly, an
additional peak (denoted as P3in fig. 46d) can be observed.
Increasing the film thickness to 52 nm leads to a further growth of the relative frac-
tion of Mcdomains, hence the widths of a1a2and Mcdomains are approximately of
about the same size. However, the overall arrangement of the a1a2/Mcdomains has
now transformed from a herringbone (α < 90◦) to a more checkerboard-like pattern
(fig. 45h), where the domain walls are tilted by angles αof about ±90◦with respect
to the [001]NSO direction. This is reflected in the in-plane X-ray pattern, which has
changed significantly. The satellite branches initially observed for the 29 nm and 38
nm films have disappeared. Instead, P3is now surrounded by satellite peaks, which
89
D. Braun III. Results & Discussion
form a rectangular grid in reciprocal space in agreement with the transformation from
a herringbone to a checkerboard-like pattern with almost rectangular domain walls.
Regarding the diffraction pattern of the 52 nm sample, more details can be evaluated:
(i) the peak P3is even more pronounced (fig. 46e) compared to the intensity measured
for the 38 nm film, while its position has not changed. From the experimental position
(Q001, Q1¯
10) of P3peak, the horizontal dimensions of the film unit cell can be evaluated
to d001 = (3.9675 ±0.0005) ˚
A and d1¯
10 = (4.0135 ±0.0005) ˚
A. By comparing these
film lattice values to (i) the bulk lattice parameter for x = 0.90 calculated in chapter
9 to apc = 3.967 ˚
A and cpc = 4.029 ˚
A and (ii) taking into account that the P3peak
only occurs for larger Mcdomain widths, we conclude, that the P3peak is caused by
Mcdomains. They consist of (001)pc oriented unit cells with the epitaxial relationship
[100]pc||[001]NSO and [010]pc||[1¯
10]NSO. However, the tensile lattice strain in [001]NSO
of the Mcunit cells has been relaxed, while they are fully compressively strained in
[1¯
10]NSO direction:
(ii) The peaks P1and P2provoked by the a1a2phase have slightly shifted to smaller
Q001 values while their position with respect to the Q1¯
10 direction remains unchanged
(compared to Q001 and Q1¯
10 determined for the thinner films). This can also be ob-
served in the out-of-plane intensity distribution in the vicinity of (444)NSO (fig. 46f)
where the film reflection (P) is shifted to smaller Q001 values as compared to the sub-
strate reflection (S). This proves an increased size of the pseudocubic unit cell along
[001]NSO while it stays fully (compressively) strained along the orthogonal [1¯
10]NSO di-
rection.
Based on the presented PFM and XRD results several aspects with regard to domain
evolution will be discussed now in more detail.
First, I will consider the different domain sizes and hierarchy of domain evolution. The
domain widths along the [001]NSO direction have been evaluated from PFM images
as a function of the film thickness (fig. 47a). It is obvious that the widths of a1a2
(closed circles) and Mc(open circles) domains continuously increase with film thick-
ness. A closer look, however, reveals that for comparatively small film thicknesses, a
strong asymmetry in the domain widths is observed. The Mcdomains are significantly
smaller than the a1a2domains. For the 7 nm film, Mcdomains could not be even
observed anymore. However, the Mcdomain widths increase remarkably faster in the
studied thickness range than those of the a1a2domains, eventually leading to almost
equal sized domains in the 52 nm film. If only elastic strain energy density arguments
are considered (for K0.90Na0.10NbO3the elastic strain energy density for both (001)pc
90
D. Braun III. Results & Discussion
Figure 47: (a) Domain width w001 of the a1a2(black filled circles) and Mc(black empty circles)
domains along the [001]NSO direction as a function of film thickness t. (b) In-plane
monoclinic distortion angle βof the a1a2domains. (c) Domain periodicity P001 along
the [001]NSO direction versus film thickness t evaluated from PFM (black triangle)
and GIXD (black square) measurements. The corresponding solid lines represent best
data fits using a power law P001 ∝tνwith ν= 0.83 ±0.08 (PFM, red line) and
ν= 0.80 ±0.10 (GIXD, green line). Taken from [136].
91
D. Braun III. Results & Discussion
and (100)pc oriented phases is equal [see chapter 9]), the (100)pc oriented a1a2and the
(001)pc oriented Mcdomains should appear with identical probability independent on
the film thickness or at least as long as both domains types are fully strained. This is
in clear contradiction to the experimental observations, especially for the thinner films.
However, additional energy terms may lead to a distinct shift of the energy balance.
For example, ferroelastic domains are energetically more favorable than ferroelectric
domains owing to the lack of any depolarization field. Indeed, Mcdomains exhibit a
strong vertical component of electrical polarization. They would require an increased
amount of electric field energy (depolarization field). Therefore, at very small film
thicknesses the a1a2domains should appear first, whereas the Mcdomains should be
strongly suppressed. This is in agreement with the experimental results obtained for
the 7 nm thin film (fig. 45a, fig. 47a).
With increasing film thickness, the epitaxial strain can be effectively reduced by the
combined formation of a1a2and Mcdomains. This can be easily understood by the dif-
ferent strain states of both domain types: For (100)pc orientation the K0.90Na0.10NbO3
pseudocubic unit cell is compressively strained, while for (001)pc orientation the pseu-
docubic unit cell is tensely strained on average. Strictly speaking, it is weakly com-
pressively strained along [1¯
10]NSO and strongly tensely strained along [001]NSO. The
coexistence of phases with compressive and tensile strain, however, would enable an
efficient reduction of elastic strain energy. This can be achieved via the formation of a
domain pattern consisting of both types of surface orientation leading to a herringbone
pattern. Furthermore, it has to be noted, that in the thickness range between 14 nm
and 38 nm the a1a2domains do not release strain, while the Mcdomains are already
fully relaxed at 38 nm in [001]NSO. This is the reason, why in this thickness range
the overall amount of Mcdomains increases more rapidly with film thickness than the
amount of a1a2domains. When both domain sizes have equalized the herringbone
pattern transfers to a checkerboard-like pattern in this monoclinic system, which is
predicted to be unstable for tetragonal materials [142].
Secondly, the domain period P001 in the direction along [001]NSO, i.e., the sum of the
individual a1a2and Mcdomain widths, is evaluated from PFM (red triangles) and
GIXD (green circles) measurements and presented as function of film thickness in fig.
47a in a double logarithmic scale. The experimental data can be fitted by a power law
P001 ∝tνwith ν= 0.83 ±0.08 (using PFM data) and ν= 0.80 ±0.10 (using GIXD
data). These values agree well but are slightly larger than ν= 0.5 as predicted in clas-
sical scaling theories [[137],[138]]. However, these calculations treat simplified cases,
92
D. Braun III. Results & Discussion
only. The model by Roitburd [137] describes the scaling of periodic domain structures
for limiting cases of very large and very small domains compared to the film thickness.
On the other hand, Pertsev and Zembilgotov [138] have treated laminar, tetragonal
a/c or a1/a2domains. These boundary conditions are certainly not applicable to our
case of thin films and multi-rank domain arrangements, as e.g. herringbone patterns.
A second important point is that these considerations were made for tetragonal and
orthorhombic systems. In monoclinic materials, the arrangement of domain walls is
much more flexible and is not restricted to highly symmetric crystallographic directions
[128].
A third aspect concerns the lattice relaxation. From XRD measurements, it was de-
rived that for the Mcdomains (grown under tensile and compressive lattice in [001]NSO
and [1¯
10]NSO direction, respectively) the strain relaxation process exclusively proceeds
along the [001]NSO direction while the lattice remains, at least up to a thickness of 52
nm, fully coherent along the [1¯
10]NSO direction. These observations are in full qualita-
tive and quantitative agreement with the investigations in chapter 9 where a reduction
of the horizontal lattice spacing of the film along [001]NSO of about 1% was found.
Likewise, in a1a2domains, compressively strained in both in-plane directions, the film
lattice has also relaxed in [001]NSO, but not in [1¯
10]NSO direction valid even for the
52 nm film. This means, for the a1a2and Mcdomains the strain relaxation along the
[001]NSO direction proceeds in opposite directions: For the compressively strained a1a2
domains an increased in-plane lattice parameter (along [001]NSO) is achieved, while
for the tensely strained Mcdomains the strain relaxation leads to a reduced in-plane
lattice parameter. This, eventually, leads to an effective reduction of strain energy.
On first sight, it is surprising, that the a1a2do not release strain in the thickness regime
between 14 nm and 38 nm. However, for monoclinic a1a2domains with (100)pc surface
orientation an additional elastic strain relaxation mechanism has to be taken into ac-
count: For (100)pc surface orientation the monoclinic distortion of the pseudocubic unit
cell is located in-plane. The corresponding in-plane shearing angle βmay continuously
increase with increasing film thickness as has been observed for NaNbO3thin films
grown on (110) TbScO3[106] and (110) DyScO3[96] substrates. This strain relaxation
effect is also observed for our samples. The experimental peak splitting ∆Q1¯
10 of the
thin film reflection is related to the in-plane shear angle βvia:
tan(2β) = ∆Q1¯
10
Q001
(81)
93
D. Braun III. Results & Discussion
In fig. 47b, the in-plane shear angle βis plotted as a function of the film thickness. It
is obvious, that the monoclinic distortion increases monotonically with increasing film
thickness indicating elastic strain relaxation. The increase of βleads to an effective
reduction of the epitaxial stress induced by the underlying substrate, which exhibits
a rectangular surface unit cell. However, this energy gain is exclusively attainable
for a1a2domains with (100)pc surface orientation. By contrast, for Mcdomains with
(001)pc surface orientation the monoclinic shearing arises in a vertical lattice plane.
Consequently, for Mcdomains, the epitaxial stress cannot elastically be reduced by
a variation of the shearing angle. This suggests that the a1a2domains preferentially
release epitaxial strain via enlargement of the in-plane shear angle β, while the Mc
domains have to release epitaxial strain via reduction of the in-plane lattice parameter
along [001]NSO. However, with increasing thickness the in-plane monoclinic shearing
of the a1a2domains monotonically increases and, eventually, at a thickness of 52 nm
the monoclinic angle of the a1a2domains has reached the bulk value for unstrained
K0.90Na0.10NbO3of βbulk = 0.27◦(fig. 47b). At this point a further reduction of strain
energy can be only achieved by an increase of the in-plane lattice parameter (along
[001]NSO). This is exactly what we observe for the 52 nm film, where the in-plane
lattice parameters of the a1a2domains start to relax (fig. 46e,f). A very similar 2-step
relaxation process has been observed for a a1a2domains in NaNbO3films grown on
TbScO3[106].
In conclusion, ferroelectric, monoclinic K0.90Na0.10NbO3thin films grown epitaxially
on (110) NdScO3substrates show a thickness dependence of the domain pattern as
well as of the lateral size of the individual domains. At comparatively low thicknesses
around 14 nm, the domain pattern shows a one-dimensional stripe like symmetry with
predominate evolution of a1a2domains. For film thicknesses between 29 and 38 nm
a complex, highly regular herringbone pattern consisting of alternating a1a2/Mcmon-
oclinic domains is observed while the relative fraction of Mcdomains is increasing
with film thickness. At a thickness of 52 nm, the domain pattern transforms into
a checkerboard-like arrangement with equal fractions of a1a2and Mcdomains. The
latter is not stable in tetragonal structures [142] proving the unique properties of mon-
oclinic ferroelectric materials. The evolution of the domain pattern with film thickness
strongly correlates with strain relaxation inside the film. For the a1a2domains, strain
relaxation is achieved by a continuous increase of the in-plane shear angle with film
thickness. Plastic strain relaxation starts at about 52 nm film thickness when the in-
plane shear angle has reached the bulk value of K0.90Na0.10NbO3. For the Mcdomains,
94
D. Braun III. Results & Discussion
this shear-relaxation mechanism is not accessible and epitaxial strain can be released
in this plane exclusively through plastic lattice relaxation. Hence, for Mcdomains,
plastic strain relaxation occurs already at lower thicknesses well below 38 nm.
The presented systematic investigation enables to targeted adjust the film thickness in
order to provoke one of the stable monoclinic multi-rank arrangements for a specified
application.
95
D. Braun IV. Summary
Part IV.
Summary
Within this work a fundamental understanding of the relationship between incorpo-
rated lattice strain and ferroelectric phase has been gained for the aim of domain for-
mation in a targeted way known as ”domain engineering”. The technologically relevant
potassium sodium niobate (KxNa1−xNbO3) was chosen as film material. Although, it
has promising piezo - and ferroelectric properties as bulk crystal, this thesis is the first
detailed investigation of the thin film form both from a theoretical and experimental
point of view.
As a starting point for a fundamental comprehension of strain-phase relations, the for-
malism to determine the preferred film unit cell orientation of KxNa1−xNbO3under
an applied mechanic stress was developed within the framework of the linear elastic-
ity theory. In addition, this calculation was used to determine the resulting vertical
strained lattice parameter which can be expected for fully strained films. The value
was compared to the experimentally measured lattice parameter to obtain quantitative
information about the strain state of the film. For the purpose of domain engineering, I
calculated a close meshed misfit strain-misfit strain phase diagram in the framework of
the Landau-Ginzburg-Devonshire theory. The result differs from the work of Bai and
Ma [3] by significantly more detailed determination of the phase boundaries. Strictly
spoken, the strain-phase diagram is valid for pure KNbO3. However, due to the similar
properties with KxNa1−xNbO3for x ≥0.5, it was successfully applied for this material
system as well. On the basis of the misfit strain-phase diagram, targeted KxNa1−xNbO3
film compositions were grown epitaxially by liquid-delivery spin metal organic chemical
vapor deposition.
As a proof of concept, two examples have been investigated in detail:
(i) Uniaxial, compressive strain can be achieved when a K0.75Na0.25NbO3film is de-
posited on a (110) TbScO3substrate. The PFM measurements revealed periodic
stripe domains running along the ±[1¯
12]TSO direction for several microns. In
agreement with the structural data from x-ray measurements, the domain struc-
ture was explained as MAformation. Moreover, a second 90◦rotated variant of
such MAdomains was observed via PFM, however with lower fraction. Indeed,
the linear elastic strain energy density reveals only a slight difference for the two
96
D. Braun IV. Summary
90◦rotated (001)pc orientations, which explains the occurrence of a preferred do-
main orientation. Reasons for the deviation from the theoretical prediction of
Mcdomains was discussed: (i) too low growth temperature and (ii) deviations of
the strain-phase diagram calculated for KNbO3to the case of 25% sodium in the
reported film.
(ii) A point of degenerated strain energy densities for the (100)pc - and (001)pc orien-
tations can be found according to the linear elasticity theory in KxNa1−xNbO3
on (110) NdScO3for x ≈0.90. The PFM and x-ray measurements depict a
herringbone arrangement consisting of monoclinic a1a2/Mcdomains. The incli-
nation angle αbetween the herringbones and the [001]NSO direction amounts
to α= 75◦which differs significantly from commonly observed angles in mate-
rials with tetragonal, orthorhombic or rhombohedral symmetry. In a detailed
study, I showed that αcan be systematically adjusted between α= 49◦−76◦
with changing the potassium concentration from x = 0.80 to 0.95. This effect
is attributed to the monoclinic symmetry and was explained within the model
of Bokov and Ye [5]. This flexibility in domain arrangement has not been ex-
perimentally observed in thin films yet. Furthermore, the thickness dependent
domain evolution evolving for x = 0.90 on (110) NdScO3in the range of t = 7−52
nm was studied for the first time. Although the elastic strain energy density is
degenerated for both pseudocubic orientations, a hierarchy for the onset of each
phase was found. The exclusively in-plane oriented a1a2domains form first, fol-
lowed by Mcdomains with an inclined electric polarization vector. The reason
is the lack of depolarization field energy in the case of pure in-plane domains
favoring their prior appearance. In this context, a thickness dependent evolution
from stripe domains to a herringbone pattern to a checkerboard-like structure
could be observed. This sequence is in contrast to tetragonal systems where the
latter is energetically forbidden [142]. Moreover, the domain periodicity P001 in
dependence of the film thickness t was fitted with P001 = tνand ν= 0.8 for
both the PFM and GIXD data. This exponent deviates from classical models
[[137],[138]]. This is attributed to the more complex multi-domain pattern as
well as the monoclinicity of the film enabling elastic and plastic lattice relaxation
processes. Eventually, PFM measurements on a film with a SrRuO3bottom elec-
trode yield a longitudinal piezoelectric coefficient of d0
33 = 29pm
V. This value is
rather high compared to established ferroelectric materials [[6],[10]].
97
D. Braun V. Outlook
Part V.
Outlook
The described results clearly show a relationship between lattice strain and ferroelec-
tric domain formation. This intense study is novel for KxNa1−xNbO3epitaxial films
and represents an important step regarding domain engineering in lead-free materials.
However, this is just a first but essential step to grow thin films with adjusted functional
properties. On the basis of these fundamental results and theoretical considerations,
different approaches for a deeper understanding and/or practical application have been
identified.
1. Depositions on SrRuO3covered substrates is needed to have a defined potential
and determination of macroscopic ferro-/piezoelectric properties.
2. The same strain energy density degeneration of the (100)pc - and (001)pc orienta-
tion observed in K0.90Na0.10NbO3on (110) NdScO3can be obtained in K0.57Na0.43NbO3
on SmScO3. But film composition and strain state are different. With x nearly
x≈0.5, a larger longitudinal piezoelectric coefficient d0
33 is expected [93]. This
may give the chance to enhance d0
33.
3. Rather less is known if the piezoelectric response can be increased with increasing
strain. Therefore, a fixed composition could be deposited on various SrRuO3
covered substrates to study the strain dependent evolution of d0
33.
4. First, preliminary experiments have shown that the stripe domains as discussed
for K0.75Na0.25NbO3/TbScO3(εxx = 0.0003, εyy =−0.0147) yield a large signal-
to-noise ratio and promising Raleigh velocities in surface acoustic waves devices.
However, open questions remain as: Do both 90◦rotated domain variants lead
to different Curie temperatures? Moreover, no fundamental investigation of the
Raleigh velocity as a function of strain exists so far. Therefore, can this effect be
enhanced in the system K0.66Na0.34NbO3/GdScO3with the same coexistence of
both (001)pc orientation but different strain state (εxx = 0.003, εyy =−0.0103)?
5. For a more reliable ”domain engineering” it would be highly desirable to calcu-
late the misfit strain-misfit strain phase diagram for different sodium concentra-
tions for an unfailing forecast. However, for that purpose, elastic constants of
KxNa1−xNbO3have to be determined as function of x.
98
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Acknowledgement
Ich m¨ochte Herrn Prof. Dr. Roberto Fornari daf¨ur danken, dass er mich als Dok-
torandin am Leibniz-Institut f¨ur Kristallz¨uchtung angenommen hat. Im weiteren Ver-
lauf der Arbeit ¨ubernahm Herr Prof. G¨unther Tr¨ankle die Leitung des Institutes und
ich danke ihm f¨ur diese T¨atigkeit und die aufbauenden Worte in der Instituts - und
Doktorandenversammlung.
Des Weiteren m¨ochte ich Herrn Prof. Dr.-Ing. Matthias Bickermann meinen Dank
aussprechen, dass er diese Doktorarbeit als Hauptgutachter ¨ubernommen hat und mir
in organisatorischen und rechtlichen Fragen jederzeit zur Seite stand.
Ganz besonders m¨ochte ich mich bei der Gruppenleiterin ”Ferroelektrische Oxidschichten”
Dr. Jutta Schwarzkopf bedanken. Sie hat mich als meine direkte Betreuerin angeleitet
und mich herausgefordert, weiter und anders zu denken. Sie hat unerm¨udlich an der
Interpretation der Ergebnis mitgewirkt und zahlreiche Texte mit viel Geduld konstruk-
tiv verbessert. Ich danke ihr f¨ur alles, was ich in den Jahren lernen durfte.
Ebenso m¨ochte ich PD Dr. Martin Schmidbauer danken f¨ur sein Interesse und Durch-
halteverm¨ogen, die Kristallstruktur unserer Filme zu verstehen und in Korrelation
mit der ferroelektrischen Anordnung zu bringen. Ich danke ihm f¨ur die vielen Mod-
elle, die zahlreichen R¨ontgenmessungen und sein offenes Ohr f¨ur Unverstandenes. In
diesem Rahmen m¨ochte ich auch PD Dr. Michael Hanke meinen Dank aussprechen
f¨ur seine beharrliche Unterst¨utzung bei R¨ontgenmessungen und Perfektionierung von
Ver¨offentlichungen.
Dar¨uber hinaus w¨are diese Arbeit nicht m¨oglich gewesen ohne den unerm¨udlichen Ein-
satz von Michaela Klann und Sebastian Markschies an der MOCVD-Anlage. Vielen
Dank hierbei ebenso an Raimund Gr¨uneberg, der als helfende Hand fortw¨ahrend zur
Verf¨ugung stand.
Den Start in diese Arbeit hat mir Dr. Andreas Duk sehr erleichtert, indem er mir einen
fundierten Einblick ins PFM gegeben hat. Vielen Dank f¨ur die exzellente Einf¨uhrung!
Ich m¨ochte mich ebenso bei Prof. Roger W¨ordenweber bedanken, der sich nicht nur als
Gutachter dieser Arbeit bereiterkl¨art hat, sondern mich mit seinem Optimismus und
zahlreichen Ideen f¨ur Anwendungen fasziniert hat.
Die gewonnen Erkenntnisse fußen auf zahlreichen Diskussionen und weiterf¨uhrenden
Messungen. Ich danke Albert Kwasniewski, Christoph Feldt, Jan Sellmann, Philipp
Kehne und Leonard von Helden f¨ur R¨ontgenbeugungsmessungen. Zudem m¨ochte ich
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D. Braun
meine Dankbarkeit an Philipp M¨uller, Dr. Toni Markurt und Dr. Martin Albrecht
f¨ur die Pr¨aparation, Durchf¨uhrung und Interpretation von TEM-Messungen zum Aus-
druck bringen.
Die Grundlage dieser Arbeit waren im wahrsten Sinne des Wortes Oxidsubstrate aus
der AG ”Oxide und Fluoride”. F¨ur diese herausragende Arbeit m¨ochte ich mich bei
allen Mitgliedern dieser Gruppe bedanken. Besonders seien hier Dr. Reinhard Uecker
und Isabelle Schulze-Jonack erw¨ahnt.
F¨ur die elektrischen Messungen unverzichtbar waren leitf¨ahige SrRuO3-Schichten von
Dr. Arnoud Everhardt von der Universit¨at Groningen. Vielen Dank f¨ur die Abschei-
dung.
Als wichtige Messpl¨atze m¨ochte ich Bessy II in Berlin und die ESRF in Grenoble dank-
end erw¨ahnen. Vielen Dank f¨ur die M¨oglichkeit, dort zu experimentieren und die vielen
helfenden Mitarbeiter, die zu jeder Uhrzeit zur Verf¨ugung standen.
Abschließend m¨ochte ich mich beim Abteilungsleiter Dr. G¨unter Wagner f¨ur seine
fr¨ohliche Art, unsere Abteilung und so auch meine Arbeit zu leiten, bedanken.
Grunds¨atzlich m¨ochte ich allen f¨ur die sch¨one Zeit danken und freue mich, dass es so
viel zu lachen gab.
Letztlich w¨are ich ohne meine Familie nie so weit gekommen. Ich danke meinem Vater
f¨ur die Liebe zur Wissenschaft und meiner Mutter f¨ur das unersch¨utterliche Vertrauen,
dass ich es so weit schaffen kann. Mein gr¨oßter Dank gilt meinem Mann Johannes,
der jedes Hoch und Tief dieser Arbeit miterleben musste und dennoch nie die Ruhe
verloren hat.
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D. Braun
Selbstst¨andigkeitserkl¨arung
Hiermit erkl¨are ich, dass ich die vorliegende Arbeit selbstst¨andig verfasst und nur die
angegebene Literatur und Hilfsmittel verwendet habe. Ich habe mich an keiner an-
deren Universit¨at um einen Doktorgrad beworben und besitze keinen entsprechenden
Doktorgrad. Ich habe Kenntnis von der dem Verfahren zugrunde liegenden Promo-
tionsordnung der Fakult¨at II - Mathematik und Naturwissenschaften der Technischen
Universit¨at Berlin.
Berlin, 26. Juni 2017
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