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LOCAL STRUCTURE OF PEROVSKITE FERROELECTRIC CERAMICSAS REVEALED BY 3D ELECTRON DIFFUSE SCATTERING

Alexandra Neagu

Local structure of perovskiteferroelectric ceramics as revealed by3D electron diffuse scattering

A walk in between the Bragg peaks

Alexandra Neagu

Doctoral Thesis 2018 Department of Materials and Environmental Chemistry Arrhenius Laboratory, Stockholm University SE-106 91 Stockholm, Sweden

Cover: Electron diffuse scattering from Pb-based relaxor perovskite ceram-ic, recorded by the author.

Faculty opponent: Prof. Kenji Kaneko Department of Materials Science and Engineering Kyushu University, Japan

Evaluation committee Docent Martina Lattemann Principal R&D Engineer, Metal Cutting Modeling Sandvik Coromant, Sweden

Prof. Johanne Mouzon Department of Civil, Environmental and Natural Resources Engineering Luleå University of Technology, Sweden

Prof. Muhammet Toprak Department of Applied Physics KTH Royal Institute of Technology, Sweden

Substitute Prof. Gunnar Westin Department of Chemistry-Ångström Laboratory Uppsala University, Sweden

© Alexandra Neagu, Stockholm University 2018 ISBN print 978-91-7797-302-7 ISBN PDF 978-91-7797-303-4 Printed in Sweden by Universitetsservice US-AB, Stockholm 2018 Distributor: Department of Materials and Environmental Chemistry, Stockholm University

Nobody ever figures out whatlife is all about, and it doesn'tmatter. Explore the world.Nearly everything is reallyinteresting if you go into itdeeply enough. Richard Feynman

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Abstract

Local structural disorder in crystalline materials plays a crucial role in un-derstanding their properties. The means to study structural disorder in recip-rocal space is through diffuse scattering (DS). Even though DS had been observed since the early days of X-ray diffraction the weak intensity and the sheer number of different kinds of disorder hindered the development of a unique solution strategy. However, the advent of synchrotron X-rays and neutron sources together with recent advances in automated electron diffrac-tion techniques have unraveled a new world in between Bragg peaks where a wealth of information is available.

In this thesis electron diffraction is used to explore the different kinds of DSin three-dimensions, for several perovskite ferroelectric ceramics. Based on the information presented in the reconstructed 3D reciprocal space volumes, disordered atomic structures were proposed and verified by the calculated electron diffraction patterns. A complex structural model for the local disor-der in 85Na0.5Bi0.5TiO3-10K0.5Bi0.5TiO3-5BaTiO3 piezoceramic was devel-oped by analyzing the morphology and intensity of electron DS in 3D. Next, the influence of potassium-content on the octahedral-tilt disorder for three different piezoceramics was studied by a combination of dark-field imaging and electron diffraction. Further on, the temperature-dependence of electron DS for 95Na0.5Bi0.5TiO3-5BaTiO3 piezoceramic revealed a local structural phase transition that was correlated with the depolarization mechanism. Lastly, strong electron DS was recorded from a Pb-based relaxor and simula-tions of disordered atomic structures showed that the local structure resem-bles a dipolar glass state. These demonstrate that electron diffraction is a powerful tool for the study of local structural disorder in crystalline materi-als, especially for ceramics. The major advantage is that we are able to rec-ord single-crystal electron diffraction data from individual grains. Moreover, since we are analyzing a 3D reciprocal volume several orientations can be studied simultaneously and we are not limited to zero-Laue zones. Finally, the models for local structural disorder provided valuable insight into how macroscopic properties are influenced by local structural disorder, in addi-tion to the average structure.

Keywords: Electron diffraction, Diffuse scattering, Local structural disor-der, Perovskite, Ferroelectric ceramics, Transmission electron microscopy.

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List of Papers

This thesis is based on papers I to IV:

I. Local disorder in Na0.5Bi0.5TiO3-piezoceramic determined by 3D elec-tron diffuse scattering Alexandra Neagu and Cheuk-Wai Tai Scientific reports 7 (2017): 12519 My contribution: I designed and conducted all experiments, data analysis and wrote the manuscript.

II. The influence of potassium content on octahedral-tilt disorder in Na0.5Bi0.5TiO3-solid solutions near morphotropic phase boundary Alexandra Neagu and Cheuk-Wai Tai Scripta Materialia 152 (2018): 49–54My contribution: I conceived the idea of the study, designed and con-ducted all experiments, data analysis and wrote the manuscript.

III. Investigation of local structural phase transitions in 95Na0.5Bi0.5TiO3-5BaTiO3 piezoceramics by mean of in-situ transmission electron mi-croscopy Alexandra Neagu and Cheuk-Wai Tai SubmittedMy contribution: I conceived the idea of the study, designed and con-ducted all experiments, data analysis and wrote the manuscript.

IV. Study of local structural disorder in 0.6PbIn½Nb½O3-0.4 PbMg⅓Nb⅔O3 ceramic as revealed by 3D electron diffuse scatteringAlexandra Neagu and Cheuk-Wai Tai Submitted My contribution: I designed and conducted all experiments, data analysis and wrote the manuscript.

Reprints were made with permission from the publishers.

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Papers not included in this thesis:

V. Two-step photochemical inorganic approach to the synthesis of Ag-CeO2 nanoheterostructures and their photocatalytic activity on tribu-tyltin degradation Eva Raudonyte-Svirbutaviciene, Alexandra Neagu, Vida Vickackaite, Vi-talija Jasulaitiene, Aleksej Zarkov, Cheuk-Wai Tai and Arturas Katelni-kovas Journal of Photochemistry and Photobiology A 351 (2018): 29-41

VI. High-temperature structural and electrical characterization of re-duced oxygen-deficient Ruddlesden-Popper nickelates Ekaterina Kravchenko, Alexandra Neagu, Kiryl Zakharchuk, Jekabs Grins, Gunnar Svensson, Vladimir Pankov and Aleksey Yaremchenko Accepted

VII. Effect of seed layers on dynamic and static magnetic properties of Fe65Co35 thin filmsSerkan Akansel, Vijayaharan A. Venugopal, Ankit Kumar, Rahul Gupta, Rimantas Brucas, Sebastian George, Alexandra Neagu, Cheuk-Wai Tai, Mark Gubbins, Gabriella Andersson and Peter Svedlindh Accepted

VIII. Highly porous amorphous calcium carbonate and its ability to en-hance the dissolution of poorly soluble drugsRui Sun, Peng Zhang, Éva Bajnóczi, Alexandra Neagu, Cheuk-Wai Tai, Ingmar Persson, Ocean Cheung and Maria Strømme Submitted

IX. Hydride reduction of BaTiO3–oxyhydride vs O-vacancy formationReji Nedumkandathil, Aleksander Jaworski, Jekabs Grins, Diana Bernin, Maths Karlsson, Carin Österberg, Alexandra Neagu, Cheuk-Wai Tai, Andrew J. Pell and Ulrich Häussermann In manuscript

X. Hollow spheres of silicon carbide prepared from spherical particles of hydrothermally carbonized glucoseWenming Hao, Yongsheng Liu, Alexandra Neagu, Zoltan Bacsik, Cheuk-Wai Tai, Zhijian Shen and Niklas Hedin In manuscript

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Contents

Abstract ............................................................................................................ i

List of Papers .................................................................................................. ii

Abbreviations ................................................................................................. vi

1. Introduction ................................................................................................. 1 1.1 Aim of this thesis ....................................................................................................... 3

2. Perovskite ceramics .................................................................................... 4 2.1 The perovskite crystal structure ................................................................................ 4 2.2 Distortions of the perovskite structure ....................................................................... 5 2.2.1 Octahedral tilting .................................................................................................... 5 2.2.2 Cation site ordering ................................................................................................ 7 2.2.3 Off-center displacements of atoms ........................................................................ 8 2.3 Properties of perovskite ceramics ............................................................................. 9 2.3.1 Ferroelectricity ........................................................................................................ 9 2.3.2 Piezoelectricity .....................................................................................................10 2.3.3 Relaxors ...............................................................................................................10

3. Electron diffuse scattering .........................................................................13 3.1 Basic crystallography ..............................................................................................13 3.2 Diffuse scattering .....................................................................................................16 3.2.1 General intensity expressions ..............................................................................17 3.3 Modeling structural disorder ....................................................................................18

4. Experimental .............................................................................................19 4.1 The transmission electron microscope ....................................................................19 4.2 TEM sample preparation .........................................................................................20 4.3 Bright-field and centered dark-field imaging ...........................................................20 4.4 Rotation electron diffraction (RED) .........................................................................21 4.4.1 RED data collection ..............................................................................................21 4.4.2 RED data processing ...........................................................................................22 4.5 In-situ TEM measurements .....................................................................................24 4.6 Simulations ..............................................................................................................24 4.6.1 Creating the initial supercell .................................................................................24 4.6.2 Monte Carlo simulations .......................................................................................24 4.6.2.1 Chemical short-range order (CSRO) ................................................................. 26

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4.6.2.2 Local atomic displacements ..............................................................................26 4.6.3 Domain subroutine ...............................................................................................27 4.6.4 Fourier transform ..................................................................................................28

5.(95-x)Na0.5Bi0.5TiO3-xK0.5Bi0.5TiO3-5BaTiO3 (papers I, II & III) ...................29 5.1 Literature overview ..................................................................................................29 5.2 Local structure of 85NBT-10KBT-5BT ceramic (paper I) ........................................31 5.2.1 Analysis of SSRs and 3D electron diffuse scattering ...........................................32 5.2.2 CDF-imaging of antiphase and in-phase tilted nanodomains ..............................35 5.2.3 Highlights of experimental results ........................................................................36 5.2.4 Modeling of local structural disorder ....................................................................37 5.2.5 Summary ..............................................................................................................43 5.3 Influence of K-content on octahedral-tilt disorder (paper II) ....................................44 5.3.1 SAED patterns ......................................................................................................44 5.3.2 CDF-imaging of antiphase and in-phase tilted nanodomains ..............................45 5.3.3 Electron diffuse scattering ....................................................................................47 5.3.4 Highlights of experimental results ........................................................................48 5.3.5 Modeling of local structural disorder ....................................................................49 5.3.6 Summary ..............................................................................................................51 5.4 Local phase transitions in 95NBT-5BT ceramic (paper III) .....................................51 5.4.1 Temperature-dependence of SSRs .....................................................................51 5.4.2 Temperature-dependence of FE domains ...........................................................53 5.4.3 Temperature-dependence of electron diffuse scattering .....................................54 5.4.4 Highlights of experimental results ........................................................................55 5.4.5 Modeling of local structural disorder ....................................................................56 5.4.6 Summary ..............................................................................................................58

6. xPbIn½Nb½O3-(100-x)PbMg⅓Nb⅔O3 (paper IV) ........................................59 6.1 Literature overview ..................................................................................................59 6.2 Local structure of 60PIN-40PMN ceramic ...............................................................60 6.2.1 Electron diffuse scattering ....................................................................................60 6.2.2 Modeling of local structural disorder ....................................................................64 6.2.3 Summary ..............................................................................................................66

7. Conclusions ...............................................................................................67

8. Future perspectives ...................................................................................68

Populärvetenskaplig sammanfattning ...........................................................69

Acknowledgments .........................................................................................71

References ....................................................................................................73

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Abbreviations

3D Three-dimensionalADT Automated diffraction tomographyBF Bright-fieldBT BaTiO3C CubicCCD Charge-coupled deviceCDF Centered dark-fieldCSRO Chemical short-range orderDS Diffuse scatteringDFT Density functional theoryEXAFS Extended X-ray absorption fine structureKBT K0.5Bi0.5TiO3M MonoclinicMC Monte CarloMPB Morphotropic phase boundaryNBT Na0.5Bi0.5TiO3NMR Nuclear magnetic resonance spectroscopyNNI Nearest-neighbor IsingPT PbTiO3PDF Pair distribution functionPIN PbIn0.5Nb0.5O3PMN Pb(Mg1/3Nb2/3)O3PNRs Polar nanoregionsPZN Pb(Zn1/3Nb2/3)O3PZT PbZr1-xTixO3R RhombohedralRT Room temperatureRED Rotation electron diffractionRMC Reverse Monte CarloSAED Selected-area electron diffractionSSRs Superstructure reflectionsT TetragonalTEM Transmission electron microscopyZA Zone axis

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1. Introduction

Local structure and disorder in crystalline materials have been increasingly recognized to play a crucial role in understanding their complex functional properties1-3. Typically, in order to obtain a comprehensive model of the crystal structure the short-range order/disorder needs to be reconciled with the average long-range structure. On the one hand, the term average structure refers to long-range atomic order with three-dimensional (3D) translational periodicity which is commonly described by space group symmetry. Diffrac-tion experiments using X-rays, neutrons or electrons, aimed at unravelling the long-range average structure in crystalline materials, have become rela-tively routine techniques. On the other hand, the local structure is associated with short-range order/disorder and dynamics that lead to deviations from the average structure at scales ranging from less than 1 Å to a few nm. Gain-ing information about the local structure is not a routine task. The conven-tional crystallographic approach, which relies upon the analysis of Bragg peaks and assumes 3D translational periodicity, cannot be applied for deter-mining the local structure which is aperiodic by definition. The diffraction signal for the local structure is revealed by the diffuse scattering (DS) inten-sity. DS can be defined as the coherently scattered intensity that is not local-ized on the average reciprocal lattice and is the result of two-body correla-tions in a crystalline material4,5.

Despite the fact that DS was observed since the early days of X-ray diffrac-tion, as shown by the extensive investigations of Lonsdale6,7, the develop-ment of methods for recording and analyzing DS has lagged well behind when compared to Bragg diffraction. One reason is that DS intensities are typically two to four orders of magnitudes weaker than Bragg reflections, so long exposure experiments were needed in order to reveal the weak diffuse features. Another reason is the sheer number of different kinds of disorder that appear in nature which rendered impossible the development of a unique solution strategy that would work for all kinds of disorder. Nowadays, with the advent of X-ray synchrotron sources, pulsed neutron sources and large-area detectors, 3D volumes of reciprocal space can be recorded very quickly in a relatively routine manner on single crystals8,9. However, in this case large single crystals (~5 μm3 for X-rays and ~0.5 mm3 for neutrons) are needed in order to record high quality DS intensities. For polycrystalline materials the total-scattering approach is used, where both Bragg reflections

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and DS intensity are recorded into one-dimensional diffraction patterns and analyzed simultaneously10. The Fourier transform of this function is the pair distribution function (PDF) which describes the probability of finding two atoms separated by a certain distance and provides the distribution of intera-tomic distances. Even though PDF data is able to provide insights into the local structure of materials, when compared to single crystal DS it is less sensitive11. An alternative is to use electron diffraction in a transmission electron microscope to record DS intensity12. This method can be used for both single crystals and polycrystalline materials. Moreover, new electron crystallographic techniques like automated diffraction tomography (ADT)13,14 and rotation electron diffraction (RED)15,16 have made the fast collection of high quality 3D volumes of reciprocal space possible using electron diffraction. The use of electrons instead of X-rays or neutrons for DS studies has several advantages: (i) electrons interact stronger with matter and this is of significant importance since DS intensity is very weak, 2-4orders of magnitude weaker than Bragg reflections; (ii) crystals as small as ~5 nm3 can be analyzed in a transmission electron microscope; and (iii) elec-tron diffraction can be correlated with various imaging and spectroscopy techniques available in the microscope.

An important class of materials where structural disorder has a significant impact on the macroscopic properties is perovskite ferroelectrics. These ma-terials are used in a variety of applications such as energy harvesting, non-volatile memories, transducers and medical imaging, to name a few17-22.Indeed, structural inhomogeneity was reported to be responsible for the en-hanced piezoelectricity observed in PbZr1-xTixO3 (PZT) close to the mor-photropic phase boundary (MPB)23. A morphotropic transition is an abrupt change in the structure of a solid solution with variation in composition. The steep line boundaries separating adjacent ferroelectric phases have been re-ferred to as MPB. Furthermore, although BaTiO3 (BT) is one of the most studied ferroelectric ceramics only recently it was confirmed that even in the cubic phase Ti-Ti and Ti-O correlated displacements are present along <100> directions24. This is an important detail since the long-range ferroe-lectric order in BT arises from concerted off-centering of Ti ions within the oxygen octahedron. Relaxors like Pb(Mg1/3Nb2/3)O3 (PMN), Pb(Zn1/3Nb2/3)O3 (PZN) and their solid solutions with PbTiO3 (PT) are a subclass of ferroelectric materials that have recently attracted tremendous attention due to their enhanced dielectric and electromechanical properties25,

26. A large number of studies have focused on unravelling their local short-range structure which was proven to be closely related to the relaxor proper-ties. A popular description of the local structure in these materials makes use of the concept of polar nanoregions (PNRs) with different shapes and sizes27-

31. However, other studies32,33 have questioned the idea of PNRs and were able to explain the observed DS with a local structure more akin to spin

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glass, so the discussion is still ongoing. The use of Pb-based ferroelectrics in electronic devices has several environmental drawbacks related to the toxici-ty of Pb. Hence, there is an intensive search for Pb-free ferroelectric materi-als that can replace PMN, PZN, PZT and their derivatives in electronic de-vices34,35. One of the most promising candidates is Na0.5Bi0.5TiO3 (NBT) and its solid solutions with BT and K0.5Bi0.5TiO3 (KBT). Similar to Pb-based ferroelectrics enhanced dielectric and piezoelectric properties were observed close to the MPB36-38. These properties were related to the complex short-range structure that comprises of oxygen octahedral tilt disorder39-41, corre-lated displacements of A- and B-cations and short-range chemical order on the A-site42-44.

1.1 Aim of this thesis This thesis aims at investigating the local short-range structure in several perovskite ferroelectric ceramics as revealed by 3D electron DS. The power of automated 3D electron diffraction techniques for DS studies combined with computer simulations of disordered atomic structures is demonstrated. Also, an attempt is made to reconcile the local short-range structure with the average long-range structure of perovskite ferroelectrics and relate it to the functional properties reported in the literature. Chapter 2 covers the back-ground on perovskite ceramics. Further on in Chapter 3 the differences be-tween Bragg diffraction and DS are highlighted and the available methods for interpretation of DS data are discussed. The experimental methods used for this thesis are presented in Chapter 4. In Chapter 5 a detailed study of local structural disorder in (95-x)Na0.5Bi0.5TiO3-xK0.5Bi0.5TiO3-5BaTiO3 pie-zoelectric ceramics is presented including comprehensive models of local structural disorder at room temperature (paper I), the effect of K-addition on the local octahedral-tilt disorder (paper II) and local structural phase transi-tions investigated by in-situ transmission electron microscopy (TEM) (paper III). Chapter 6 extends the methodology used in the study of NBT-compounds to another ferroelectric system namely, xPb(Mg1/3Nb2/3)O3-(1-x)Pb(In1/2Nb1/2)O3 solid solutions, in order to assess the local structural dis-order at room temperature (paper IV). Finally, the general conclusions of this thesis are highlighted in Chapter 7 and future perspectives are discussed in Chapter 8.

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2. Perovskite ceramics

The perovskite family of ceramics has interested researchers for over 70 years, both because their seemingly simple average structure45 hides rich structural complexity46, and because they exhibit a plethora of important physical properties like ferroelectricity, piezoelectricity, nonlinear optical characteristics, tunability, high dielectric permittivity, pyroelectricity and photovoltaic behavior. Increasingly, the local structure of perovskite oxides has been the main focus of many studies that showed a direct correlation between local structural disorder and physical properties47,48. The following sections describe some general structural characteristics and properties of perovskite ceramics relevant for this thesis.

2.1 The perovskite crystal structure Perovskite have the general formula ABO3 and can be regarded as frame-work structures composed of corner-sharing BO6 octahedra with A-cations occupying 12-coordinated interstices, where A is the larger cation and B is the smaller one. Figure 2.1 illustrates a schematic representation of the basic perovskite unit cell with Pm-3m cubic symmetry.

Figure 2.1. The basic perovskite unit cell with symmetry Pm-3m. The B-cation (gray) occupies the body-centered position (0.5, 0.5, 0.5), A-site cations (green) are placed in the cube corner positions (0, 0, 0) and the oxygen anions (red) are placed at the face centers of the cubic unit cell (0.5, 0.5, 0) and form an octahedron around the B-site, as shown in gray.

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2.2 Distortions of the perovskite structure The unique ability of the perovskite structure to accommodate most elements from the periodic table gives rise to a rich variety of structural disorder and related properties. This high compositional and structural flexibility makes the perovskite structure ideal for systematic substitutions of different ions, either on the A- or the B-sites. The basic perovskite structure can then be tuned relatively easily by substituting cations with different sizes, charge or electronic structure. The most common mechanisms of structural distortion which modify the basic perovskite structure are octahedral tilting, cation site ordering and off-center displacements of atoms. Since in the ideal perovskite structure the atoms occupy special positions the symmetry is lowered and tetragonal, orthorhombic or rhombohedral distortions of the perovskite struc-ture are known. In most perovskites however, a combination of these mech-anisms contributes to the structural departure from the ideal perovskite struc-ture which leads to complex local structures.

2.2.1 Octahedral tiltingOctahedral tilting distortions occur when there is a size mismatch between A- and B-cations. By treating A and B ions as hard spheres it is possible to estimate the distortion from the cubic perovskite structure. This was done by Goldschmidt49 who introduced the tolerance factor, τ, that is widely used to assess the geometric stability and distortion of the perovskite crystal struc-ture. The tolerance factor is defined as:

� �OB

OA

rrrr�

��

2� (2.1)

Where rA, rB and rO are the ionic radii of A, B and O ions. In the perovskite family, the value of τ lies between approximately 0.78 and 1.05 with τ = 1for the ideal cubic perovskite structure50. As the tolerance factor departs from 1 the cubic structure becomes less stable and octahedral tilting occurs to min-imize the electrostatic ion-ion interactions. Glazer46 introduced a standard notation, which was later refined by Woodward50,51, to describe octahedral tilting in perovskite oxides. This notation characterizes a tilt system by rota-tions of the BO6 octahedron about the [100]pc, [010]pc and [001]pc axes of the pseudocubic perovskite unit cell. The magnitudes of the tilts about the three pseudocubic axes are indicated by a set of three letters. In the general case when unequal tilts occur about all three axes the notation is abc. In Glazer notation the letters are accompanied by superscripts of 0, + or – to discern the phase of the octahedral tilting in adjacent layers. A positive superscript indicates in-phase tilting in adjacent layers, which means that all layers are

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tilted the same way about the respective axis. A negative superscript indi-cates antiphase tilting in adjacent layers, which means that consecutive lay-ers are tilted with the same tilt angle but in opposite directions. The super-script 0 is an indication that no tilting occurs along that axis. Schematic rep-resentations of the non-tilted BO6 octahedra framework together with the in-phase tilt system a0a0c+ and antiphase tilt system a–a–a– are shown in Figure 2.2.

Figure 2.2. Schematic representation of (a) non-tilted BO6 octahedra framework, (b)a0a0c+ in-phase tilt system and (c) a–a–a– antiphase tilt system.

Tilting of the oxygen octahedra causes doubling of the unit cell axes52-54 and as a result, superstructure reflections (SSRs) are produced which lie at half-integral reciprocal lattice planes. Depending on the phase, two distinct clas-ses of SSRs can be established. Hence, in-phase tilting gives rise to ½(ooe)SSRs while antiphase tilting produces ½(ooo) SSRs (where o stands for odd and e stands for even). These SSRs are indexed with respect to the funda-mental perovskite reflections hkl. In the case of kinematical diffraction, the following rules apply for in-phase and antiphase tilting:

� a+ produces superstructure reflections of type ½(eoo) with k ≠ ±l� b+ produces superstructure reflections of type ½(oeo) with h ≠ ±l� c+ produces superstructure reflections of type ½(ooe) with h ≠ ±k� a– produces superstructure reflections of type ½(ooo) with k ≠ ±l� b– produces superstructure reflections of type ½(ooo) with h ≠ ±l� c– produces superstructure reflections of type ½(ooo) with h ≠ ±k

When in-phase and antiphase tilting occur about more than one axis or in the case of mixed tilt systems (i.e. a+b–b–) these simple rules change as de-scribed in detail by Woodward et al.55 For example in the case of mixed tilt systems additional reflections appear due to the combination of in-phase and antiphase tilting at ½(oee) positions. When analyzing SSRs one should keep in mind that there are two other possible sources for SSRs: antiparallel dis-placements of cations and chemical ordering of A- or B-site cations.

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2.2.2 Cation site ordering Besides the aforementioned structural flexibility, perovskites also exhibit rich compositional flexibility. So, chemical substitutions have been widely used to expand the perovskite family and improve physical properties. Cati-on substitutions can be made on both A- and B-sites which can result in ei-ther ordered or random arrangements53,54. Whereas A-site cation ordering is quite rare and usually involves vacancy ordering, B-site cation ordering in perovskite oxides is very common56-58. An ordered arrangement is obtained when there is a significant difference in the coordination preference, valence or size of the two cations that share the same site. Three simple patterns of ordering can be envisioned for the A- and B-site cations: (i) layered or (001) ordering, (ii) columnar or (110) ordering and (iii) rock salt or (111) ordering. Figure 2.3 illustrates these three ordering patterns for both A- and B-site cations.

Figure 2.3. Cation ordering patterns in perovskite oxides. Layered (001) ordering is shown in (a) and (b), columnar (110) ordering is shown in (c) and (d) and rock salt (111) ordering is shown in (e) and (f).

The most common type of ordering for complex perovskites oxides is (111) ordering, which is the most favorable one from the point of view of electro-static considerations. An example of technologically important perovskite oxides, which present B-site ordering, are 1:2 mixtures of divalent and pen-tavalent cations A2+(B2+

1/3B5+2/3)O3 where A = Sr, Ba, Pb; B2+ = Mg, Zn, Ni

and B5+ = Nb, Ta. The importance of the 1:2 perovskites arises from their relaxor ferroelectric and piezoelectric properties (e.g. PbMg1/3Nb2/3O3- PMN) and their application as low-loss dielectrics in wireless microwave communications devices (e.g. BaZn1/3Ta2/3O3- BZT)56. It is interesting to note that even though they present the same site stoichiometry, in Ba-based

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compounds layered (001) ordering is established59 while in Pb-based com-pounds the rock salt (111) ordering prevails60. Moreover, it was suggested that Pb-based compounds present rather short-range B-site ordering in con-trast to Ba-based compounds which present long-range B-site ordering. The reason for this is thought to be the interaction of the 6s2 lone pair of Pb2+

with the oxygen atoms bonded to the lower charged B2+ cations60.

2.2.3 Off-center displacements of atoms The magnitude of off-center displacements of cations from the high sym-metry positions in the cubic perovskite structure is strongly correlated with ferroelectric properties like spontaneous polarization and the Curie tempera-ture (Tc)61,62. As a first estimation the tolerance factor shown in equation 2.1 can be used to assess the atomic displacements. If the tolerance factor is larger than 1 the B-site cation is too small for the space inside the oxygen octahedron and has the possibility to displace. On the other hand, a tolerance factor smaller than 1 indicates that the A-site cation has too much space in its coordination polyhedron and hence can displace. The atomic off-center displacements are usually accompanied by a change in lattice parameters and tetragonal, orthorhombic or rhombohedral distortions of the perovskite struc-ture are known63. A classic example is BaTiO3, which at room temperature presents a distorted perovskite structure with tetragonal symmetry (space group P4mm)45 as shown in Fig. 2.4. The B-site cation (Ti4+) is displaced along the [001]pc direction which gives rise to ferroelectricity. Other com-plex perovskite oxides have been engineered to incorporate ions on the A-site with a tendency to distort, such as lone pair species like Pb2+ and Bi3+.Studies of both Pb-based perovskites like PMN28 and PZN32 and Bi-based perovskites like NBT42 and KBT64 revealed local off-center displacements of A-site cations.

Figure 2.4. The tetragonal structure of BaTiO3. The purple arrows indicate that Ti4+

(gray) is displaced within the oxygen octahedra along the [001]pc direction.

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2.3 Properties of perovskite ceramics

2.3.1 Ferroelectricity Ferroelectrics are polar materials that possess a non-zero spontaneous elec-tric polarization in the absence of an external electric field. By applying a suitable electric field, the polarization can be reoriented. The reversibility of the polarization is represented by the hysteresis loop (Fig. 2.6a) and gives rise to a non-linear dielectric behaviour65. This reversibility is a consequence of the fact that the polar structure of a ferroelectric is a slightly distorted nonpolar structure. In general, a ferroelectric material consists of domains and domain walls, in order to minimize the total energy. These are regions of homogenous polarization that differ only in the direction of the polarization. By applying an electric field a macroscopic polarization is generated66. Fer-roelectricity is limited to ten crystal classes (1, 2, m, mm2, 3, 3m, 4, 4mm, 6, and 6mm) which possess a polar axis. Thus, the existence of a dipole mo-ment is symmetrically allowed67. Most ferroelectric materials undergo a phase transition from a paraelectric (i.e. showing no spontaneous polariza-tion) state to a ferroelectric state at a critical temperature, denoted as Curie temperature (Tc). The phase transition is most often accompanied by changes in crystal structure and may occur in several steps through intermediate dis-torted phases. A classic example is BaTiO3 which undergoes several phase transitions with decreasing temperature45,68,69.

Figure 2.5. Schematic representation of phase transitions in BaTiO3.

The different distortions of the perovskite structure during phase transitions in BaTiO3 are shown in Fig. 2.5. Above Curie temperature (120°C) the struc-ture is cubic and paraelectric. At Tc a phase transition occurs from the cubic paraelectric phase to a tetragonal ferroelectric phase. At around 5°C another phase transition takes place from a tetragonal ferroelectric phase to an ortho-rhombic ferroelectric phase. A final phase transition follows at -90°C when the structure becomes a rhombohedral ferroelectric. The cubic structure isparaelectric with no spontaneous polarization. The tetragonal, orthorhombic

10

and rhombohedral phases are ferroelectric with the spontaneous polarization direction along <001>pc, <110>pc and <111>pc, respectively.

2.3.2 Piezoelectricity Piezoelectric materials exhibit an electromechanical coupling and as a con-sequence the electric polarization varies in response to an applied strain70.The direct piezoelectric effect entails applying a mechanical strain to a pie-zoelectric material and as a result a proportional alteration in the electric polarization is created. The converse piezoelectric effect arises when an ap-plied external electric field generates mechanical deformation in a piezoelec-tric material. Piezoelectricity is observed in many materials and all of the 20 non-centrosymmetric crystallographic point groups are capable of exhibiting piezoelectricity. Hence, all ferroelectric materials also present piezoelectric properties. A distinction needs to be made between piezoelectricity and elec-trostriction. In the case of piezoelectricity the mechanical deformation is linear with respect to the applied electric field and changes sign if the elec-tric field is reversed. In contrast, in the case of electrostriction the sign of the deformation is independent of the polarity of the electric field and is propor-tional to the square of the electric field. Electrostriction is a phenomenon that is very weakly present in all materials, whether amorphous or crystalline, centrosymmetric or non-centrosymmetric63. The piezoelectric properties of ferroelectric ceramics are used for a wide variety of applications especially in transducers for converting electrical energy into mechanical response and vice versa. Other remarkable applications are found in the field of medical imaging, micro-motors and sensing, ultrasonic devices and telecommunica-tions19,25,34,71.

2.3.3 Relaxors Relaxors represent a particular class of ferroelectric materials that have been intensively studied due to their outstanding dielectric and electromechanical properties26,72-74. In contrast to normal ferroelectrics they present a slim hys-teresis loop (Fig. 2.6b) with low remanent polarization (Pr) after removing the applied electric field. In the case of ferroelectrics, the saturation and remanent polarizations decrease with increasing temperature and vanish at the Curie point (Tc), as shown in Fig. 2.6c. No ferroelectric domains exist above Tc. In the case of relaxors though the field-induced polarization de-creases smoothly with increasing temperature and it retains finite values well above Tm, which represents the temperature at which the dielectric permittiv-ity is maximum (Fig. 2.6d). Another significant difference between ferroe-lectrics and relaxors concerns the temperature dependence of the dielectric

11

permittivity (ε') as illustrated in Fig. 2.6(e, f). The dielectric permittivity of a ferroelectric crystal is frequency (f) independent and exhibits a sharp maxi-mum at Tc accompanied by a long-range phase transition. Relaxors, however present a very broad maximum in the dielectric permittivity at Tm and strong frequency (f) dispersion with no long-range phase transition.

Figure 2.6. Comparison between physical properties of normal ferroelectrics and relaxors. The P(E) hysteresis loop for (a) ferroelectric and (b) relaxor. Temperature dependence of polarization for (c) ferroelectric and (d) relaxor. Temperature de-pendence of dielectric permittivity at selected frequencies for (e) ferroelectric and (f)relaxor. Adapted from [Ref. 72] with permission from IOP Publishing.

In order to explain the peculiar physical properties of relaxors several mod-els have been proposed. One of the first models75-77 suggests that chemically different nanoregions give rise to different local Curie temperatures, which lead to a broad maximum in ε'(T) at a mean value around Tm. However, chemical inhomogeneity alone cannot explain the strong frequency disper-sion in the vicinity of Tm. Another model reported by Burns et al.78,79 intro-duces the concept of polar nanoregions (PNRs) which appear at a tempera-ture well above Tm, denoted as the Burns temperature TB. Moreover, in anal-ogy to the superparamagnetic state Cross80 and later Bokov81 proposed a

12

model for relaxors known as the dipolar glass model. They suggest that above Tm dynamic PNRs or nanodomains are embedded in a paraelectric matrix. The random interactions between electric dipole clusters give rise to the electric relaxation. Theoretical work on relaxors has put forward two approaches to explain their peculiar behavior. The first one is the quenched-random-field approach82,83 which suggests that the existence of polar nanodomains is due to the interplay between the surface energy of the nanodomain walls and the bulk energy of the nanodomains in the presence of weak random fields created by chemical inhomogeneity. The second ap-proach is the random field theory84 which is based on electrostatic interac-tions. At high temperatures the crystal is composed of random reorientable dipoles embedded in a highly polarizable matrix, similar to the dipolar glass model. The main difference is that the dipole-dipole interactions are not di-rect but indirect via the matrix leading to uniformly directed local fields.

13

3. Electron diffuse scattering

3.1 Basic crystallography A conventional ideal crystal can be thought of as a 3D periodic arrangement of atoms that can be divided into a set of identical entities, denoted as unit cells. The unit cell of a crystal is characterized by 3 non-coplanar vectors (a,b, c) and the angles between them (α, β, γ). Based on unit cell parameters and symmetry, crystals can be classified into 7 different crystal systems: cubic, hexagonal, rhombohedral, tetragonal, orthorhombic, monoclinic and triclinic. The way the unit cells are packed next to each other defines the lattice. In 3D there are 14 lattice types, called Bravais lattices. However, the lattice type is not enough to completely describe the symmetry of a crystal. By taking into consideration both translational and non-translational symme-tries, 230 space groups are generated for 3D crystals. An important concept in crystallography is the reciprocal space. Just as in real space a unit cell can be used to describe the crystal in reciprocal space. A lattice point in recipro-cal space corresponds to a set of parallel atomic planes in real space. In order to describe how atomic planes cut the lattice, Miller indices are used. These indices are three integers usually denoted h, k, and l. Three different atomic planes with their Miller indices are schematically illustrated for a cubic crys-tal system, as shown Fig. 3.1.

Figure 3.1. Different crystallographic planes and their Miller indices for a cubic crystal system.

Diffraction occurs when electrons of a given wavelength (λ) are scattered by a set of atomic planes with the interplanar distance dhkl, leading to interfer-ence. When the difference in the ray path is equal to an integer multiple of

14

the wavelength, the interference will be constructive and the Bragg condition is fulfilled, see Fig. 3.2. Bragg’s law describes the relationship between wavelength (λ), interplanar distance (dhkl) and scattering angle (2θ), Equation (3.1).

� sin2 hkldn � (3.1)

Figure 3.2. Two parallel electron waves scattered by equivalent atomic planes will interfere constructively when the path difference, 2dhklsinθ, is equal to an integer multiple of the wavelength.

The Ewald sphere is a geometrical construction which demonstrates the rela-tionship between the wave vector of the incident (k0) and diffracted (k) elec-tron beams, the diffraction angle for a given reflection (θ) and the reciprocal lattice of the crystal, as illustrated in Fig. 3.3. The scattering vector ghkl rep-resents the difference between the diffracted wave vector and the incident wave vector. Diffraction occurs when the Ewald sphere passes through a reciprocal space “point” and as a result a reflection is observed in the diffrac-tion pattern. In case of electron diffraction it is more accurate to consider reciprocal space relrods. This implies that reflections may appear in an elec-tron diffraction pattern even when the Bragg condition is not exactly fulfilled (Ewald sphere does not crosses the middle of the relrod). An electron dif-fraction pattern is a 2D-section of the reciprocal lattice with its origin in-dexed 000, i.e. the transmitted beam. Each reciprocal lattice point is de-scribed by its scattering vector:

*** cbag lkhhkl �� (3.2)

where a*, b* and c* are the reciprocal lattice vectors and h, k, l are the Miller indices. The relationship between real space and reciprocal space is as fol-lows:

�sin21

��hkl

hkl dg (3.3)

15

Most electron diffraction experiments imply tilting the crystal along a par-ticular zone axis. A zone axis is a lattice row parallel to the intersection of two (or more) families of lattices planes. A zone axis [uvw] is parallel to a family of lattice planes with Miller indices (hkl) if:

n = wl+ vk + uh (3.4)

If n = 0 all the reflections in the zero-order Laue zone (ZOLZ) are lying on a reciprocal lattice plane through the reciprocal lattice origin, i.e. the 000 re-flection. In addition to the ZOLZ the Ewald sphere might also cut other par-allel planes giving rise to more diffraction spots (n = 1 is called the first-order Laue zone FOLZ).

Figure 3.3. Schematic representation of the Ewald sphere in 2D cutting through the reciprocal lattice. The resultant electron diffraction (ED) pattern is shown below.

The scattering vectors attributed to reflections in an electron diffraction pat-tern give information about equivalent sets of lattice planes in real space, while the reflections’ intensities depend on the type of atoms present in the structure and their position in the unit cell. Electron diffraction patterns carry information only about the amplitude of the resultant wave interference. The phase information is lost. In electron diffraction we measure the amplitudes of the structure factors of the electrostatic potential. The structure factor of a crystal is defined as follows:

� � ��

��N

jjjjj lzkyhxihklfhklF

1

2exp)()( (3.5)

16

where fj is the atomic scattering factor of the j atom, hkl are Miller indices and xj yj and zj fractional atomic coordinates of the j atom. The kinematical relationship between the intensity of reflections and the structure factor is:

2)()( hklFhklI � (3.6)

Other factors like crystal size, thickness, absorption and dynamical scattering can also affect the intensities of reflections in an electron diffraction pattern.

3.2 Diffuse scattering The premise of Bragg diffraction is that crystals consist of a 3D-network of identical repeating unit cells, resulting in diffraction patterns with well-defined peak intensities (Bragg reflections). However, perfect crystals are rare and the occurrence of disorder in crystals needs to be taken into consid-eration for a realistic understanding of their atomic structure. Disorder refers to any kind of local deviation from the average long-range structure.

Figure 3.4. Comparison between Bragg reflections and DS intensity. Structures of a cubic binary compound with 70% red atoms and 30% black atoms occupying the same site (0, 0, 0) for (a) random configuration and (c) structure with a small degree of CSRO along <100> and <110> directions and their respective electron diffraction patterns in (b) and (d). Schematic representation of atomic displacements for a cubic structure with one atom at (0, 0, 0) in the case of (e) random displacements and (g)correlated displacements along <100> directions and their corresponding electron diffraction patterns in (f) and (h).

The local disorder in real space cannot be described in terms of a repeating unit cell that represents the average structure but is only locally correlated over a finite number of unit cells. Information about structural disorder is

17

revealed by DS intensity. Local structural deviations produce diffracted in-tensities that are not localized to a single, discrete peak but rather diffuse in reciprocal space with intensities that are usually orders of magnitude weaker than those of Bragg reflections. A comparison between diffraction from structures with random occupational disorder or random displacements and structures with chemical short-range (CSRO) or correlated displacements is shown in Fig. 3.4. A broad diffuse background is observed in the case of random occupational disorder/displacements while the structures that present CSRO and correlated atomic displacements show structured DS besides the sharp Bragg reflections.

3.2.1 General intensity expressions Similar to Bragg diffraction the theoretical background of DS has been well developed only in the case of X-rays. Nevertheless, within the limits of the kinematical approximation the same equations can be employed in the case of electron DS. The general expression for the total scattering intensity ac-counts for both chemical short-range order and local atomic displacements85,86, as shown in Equation (3.7).

� � � � ��

��N

n

N

mnnmmhklnmhkl iffI

1 1

exp ururgg (3.7)

I is the total scattering intensity, ghkl is the scattering vector, N is the total number of atoms in the crystal, fm and fn are the complex atomic scattering factors of the atoms m and n, rm and rn are the real space vectors for atoms mand n, um and un represent small amounts by which atoms m and n are dis-placed from their average positions. Equation (3.7) includes both Bragg and diffuse intensity. Bragg peaks correspond to scattering from the average long-range structure while DS is a result of local deviations from this aver-age structure. By subtracting the intensity contribution from Bragg diffrac-tion an expression for the DS intensity can be written as follows (for details see ref 80):

� � � � �� lpknhmilZkYhXi

lZkYhXiffccNI

ijijij

ij mnp

ijmnp

ijmnp

ijmnp

ijmnpjijiPDiffuse

��

��

��

��

�

2exp2exp

2exp1*

(3.8)

By definition the term in curly brackets, vanishes at large interatomic vectors so the summation in Equation (3.8) is only over relatively short lengths and describes a smoothly varying diffuse intensity. The first summation is over the different types of pairs consisting of atoms i and j while the second one is

18

over all unique interatomic vectors in real space, mnp. The vector mnp re-fers to the average interatomic vector between two atomic sites. The averag-es indicated by angle brackets are over all NP atom pairs that have the same site separation. ci and cj are the overall concentrations of atoms i and j and fiand fj are their corresponding complex atomic scattering factors. αij

mnp are CSRO parameters defined by Cowley87 as:

j

ijmnpij

mnp cP

��1� (3.9)

The term Pijmnp represents the probability of finding an atom of type j at the

end of an interatomic vector mnp from the origin, where an atom of type isits. Xij

mnp, Yijmnp and Zij

mnp are components of the relative displacement of atoms i and j, and they are defined as:

xixjmnp

ijmnp uuX 0�� ; yiyj

mnpij

mnp uuY 0�� ; zizjmnp

ijmnp uuZ 0�� (3.10)

Here uxjmnp is the displacement in the x direction of atom j situated at the end

of the mnp vector, while uxi0 is the displacement of atom i also in the x direc-

tion situated at the origin of the mnp vector.

3.3 Modeling structural disorder The study of average structures from conventional Bragg diffraction is now-adays carried out routinely with the use of powerful algorithms implemented in user friendly programs. In the case of disordered crystals however there are no such programs for the analysis of DS intensity. The most common approach to deal with DS from disordered crystals is to build a model of the disordered structure, using chemical and any other a priori knowledge avail-able, and compare the simulated diffuse intensity with the experimental data. This initial model can be further refined against the DS data. A number of computational methods like Monte Carlo (MC) simulations, reverse Monte Carlo (RMC) simulations and molecular dynamics have been used to build disordered atomic structures for a variety of materials. In this work the MC method was used to develop all disordered atomic structures and will be discussed in details in Chapter 4.

19

4. Experimental

4.1 The transmission electron microscope A typical transmission electron microscope (TEM) can be thought of as con-sisting of three main components: the illumination system, the objective lens/stage and the imaging system (Fig. 4.1a).

Figure 4.1. (a) Schematic representation of constituting components for a typical TEM. (b) Schematic drawings of ray paths for the formation of diffraction patterns (left) and images (right).

The function of the illumination system is to produce a suitable beam. First the electrons are extracted from an electron source and further accelerated by the applied high voltage. Subsequently the beam is focused using the con-denser lenses. Since electrons are charged particles electromagnetic lenses are used to focus the beam. By adjusting the strength of the condenser lenses a parallel or convergent illumination can be produced. Furthermore, the con-denser aperture is used to control the fraction of the beam which is allowed to hit the specimen. After interaction with the specimen the electrons are focused by the objective lens. This is the most important lens in the micro-scope since the performance of the objective lens largely determines the overall performance of the microscope. Two other apertures are placed bel-low the objective lens in order to select the desired part of the beam. The objective aperture is situated in the back focal plane (bfp) while the selected

20

area electron diffraction (SAED) aperture is situated in the first image plane.By adjusting the focus of the intermediate lens either a diffraction pattern or image can be formed (Fig. 4.1b). The projector lenses are used to magnify the image or diffraction pattern onto the fluorescent screen or selected detec-tor. All TEM experiments have been performed using either a JEOL JEM-2100F microscope equipped with a Schottky-type field-emission gun or a JEOL JEM-2100 microscope equipped with a thermionic source (LaB6 filament). When compared to LaB6 thermionic sources, field-emission guns provide higher current density and brightness with a highly spatially coherent beam and lower energy spread.

4.2 TEM sample preparation All materials investigated in this thesis were polycrystalline in ceramic form. The (95-x)Na0.5Bi0.5TiO3-xK0.5Bi0.5TiO3-5BaTiO3 ceramics with x = 0, 5, 10 were prepared via conventional solid state method as described in details in Ref [36]. In the case of the xPbIn½Nb½O3-(100-x)PbMg⅓Nb⅔O3 system with x = 60, the ceramics were synthesized by a two-step solid state reaction as reported in Ref [93]. TEM specimens were prepared via mechanical polish-ing to a thickness of ~30 μm, followed by Ar ion milling (Fischione Model 1050) until electron transparency was achieved. A beam voltage between 5-6kV was used for ~4 hours followed by gentle ion-milling at 100 eV for ~1 minute. The milling angles for top and bottom Ar guns were +10° and -10° respectively.

4.3 Bright-field and centered dark-field imaging A combination of bright-field (BF) and centered dark-field (CDF) imaging was used to investigate the general morphology of the ceramics, ferroelectric domain morphology and the size and distribution of octahedral-tilted nanodomains. BF-images are formed by electrons from the direct beam while CDF-images are formed by electrons from a specific diffracted beam,see Fig. 4.2. In the second case the incident beam is tilted so that the scat-tered beam is on the optical axis. The objective aperture is used to select the beam of interest. To record BF- and CDF-images a JEOL JEM-2100F microscope equipped with a Gatan Orius 200D CCD camera was used. Weak SSRs of the type ½(ooe) and ½(ooo) were used to record CDF-images of octahedral-tilted domains with long exposure times of up to 30 sec.

21

Figure 4.2. Ray diagrams showing how the objective lens and objective aperture are used to produce (a) BF-images formed from the direct beam and (b) CDF images formed from a specific scattered beam. The diffraction patterns are shown below illustrating the area selected by the objective aperture.

4.4 Rotation electron diffraction (RED)

4.4.1 RED data collection RED data collection combines the goniometer’s coarse mechanical tilt (1-2°) with a fine beam tilt (0.1-0.4°) to record 3D electron diffraction data, as schematically shown in Fig. 4.3a. The small beam tilt increment ensures a fine sampling in reciprocal space. At each beam tilt a 2D SAED pattern is recorded. Data collection is semi-automated within the REDacquisition software16. During acquisition the z-height is corrected manually so that the specimen is always at eucentric height. Also, the specimen position is ad-justed to ensure that SAED patterns are collected from the same area during the entire data collection. Bragg reflections and DS intensity are recorded simultaneously. Moreover, since the RED method does not require zone axis (ZA) alignment, DS intensity is recorded in the presence of fewer Bragg reflections excited simultaneously. This also reduces the dynamical effects. The acquisition time per SAED pattern should be optimized so that the DS intensity has a high enough intensity. High exposure times might be needed leading to saturated Bragg reflections. The thickness of the specimen has a significant impact on the observed DS intensity and only very thin samples

22

should be used for data collection. Complex structural disorder might pro-duce DS intensity that is not visible/present at low order ZAs. Hence, 3D diffraction data like the one recorded with the RED method offers a substan-tial advantage when it comes to the interpretation of DS intensity.

Figure 4.3. (a) Schematic representation of working concept for RED method. (b)General view of reconstructed 3D reciprocal space volume from RED data contain-ing both DS intensity and Bragg reflections. Adapted from [Ref. 16] with permission of International Union of Crystallography.

A Gatan Orius 200D CCD camera was used to record the SAED patterns during RED data collection. A summary of the acquisition parameters for RED data collection is shown in Table 4.1.

Table 4.1. Acquisition parameters for RED data collection.

4.4.2 RED data processing The 3D reconstruction of reciprocal space volumes from RED data was done using the REDprocessing software16. The reconstruction simply involves combining the individual 2D SAED patterns after shift correction. Hence, the reconstructed result does not suffer from distortions produced by the missing wedge as is the case for tomography where back-projection algo-rithm is used. The SAED patterns were usually scaled to half of their initial

23

size (i.e. 1024×1024 pixels) so that the final 3D volume has a reasonable dat size that can be handled by regular 3D visualization software. Before export-ing the 3D reconstructed reciprocal space volume the contrast of all SAED patterns was adjusted by linearly scaling the pixel values between upper and lower limits. Pixel values that are above or below this range are saturated to the upper or lower limit value, respectively. The typical lower limit was 0 while the upper limit varied from 800-2000 depending on the used exposure time. The goal is to obtain a good contrast for both intense Bragg reflections and weak DS features. The final 3D reciprocal space volume contains both Bragg reflections and DS intensity and can be read by most 3D visualization software. In our case the Kitware VolView software was used to visualize and analyze the 3D reconstructed reciprocal space volume. A big advantage of 3D diffraction data is the option to cut slices/volume-sections that are not at zero Laue-zone. This simplifies significantly the interpretation of DS. Figures 4.4(a–c) illustrate a schematic representation of the 3D reciprocal space volume as to differentiate between a slice/volume-section through the 3D reciprocal space volume and the whole reciprocal space volume. A slice along a certain plane is equivalent to a typical SAED pattern. A thin volume-section, centered along a particular slice/ZA, was generally used to highlight certain DS features that were not clearly visible in the single slice. On the other hand, viewing the whole reciprocal-space volume oriented along dif-ferent directions enabled a 3D analysis of DS features. To further enhance the visualization of weak DS all RED data presented in this thesis are shown on a logarithmic scale.

Figure 4.4. Schematic representation of (a) slice, (b) volume-section and (c) whole reciprocal space volume. Adapted from [Paper I] with permission from Springer Nature.

24

4.5 In-situ TEM measurements The temperature study presented in paper III was performed using a double tilt heating holder (model 652, Gatan Pleasanton CA). During the experi-ments 2D SAED patterns, BF-images of ferroelectric domains and RED data were recorded at constant temperature using a Gatan Orius 200D CCD cam-era. The temperature was increased from room temperature to 500°C with a heating rate of 10°C/min. Due to the specific geometry of the sample holder the measured temperatures have a high uncertainty of ~ ±10°C. For a semi-quantitative analysis of the SSRs’ relative intensity with increasing tempera-ture, SAED patterns were also recorded using Timepix camera (Amsterdam Scientific Instruments B.V.) which allows very fast acquisition with a high signal-to-noise ratio. The SSRs intensities were integrated using the Profile tool in DigitalMicrograph software from a sum of 100 SAED patterns rec-orded with 0.1 sec acquisition time.

4.6 Simulations All simulated disordered structures were developed using DISCUS software which is a versatile tool aimed to simulate defect crystal structures and their corresponding intensity distribution in reciprocal space94, 95. The software is able to simulate a wide range of defect structures such as chemical short-range order, local displacements of atoms away from their average positions, modulated structures, stacking faults, domain structures and nanoparticles. The following sections will focus only on DISCUS tools that have been used for this work.

4.6.1 Creating the initial supercell The initial supercell was usually based on the ideal perovskite structure with cubic symmetry (space group Pm-3m). The unit cell parameters were those of the average cubic structure reported in the literature for the investigated materials. The initial crystal was 50×50×50 unit cells large with A-site, B-site and O atoms at their average positions as for the ideal perovskite struc-ture, namely (0, 0, 0), (0.5, 0.5, 0.5) and (0, 0.5, 0.5) respectively. The supercell was generated from the contents of the asymmetric unit cell.

4.6.2 Monte Carlo simulations The MC algorithm88 is used to minimize the energy of a model system (i.e. initial supercell) in order to generate disordered structures. A schematic rep-

25

resentation of the MC algorithm within DISCUS software is shown in Fig. 4.5. The supercell’s total energy E is expressed as a function of random vari-ables such as site occupancies or local atomic displacements away from the average positions. Next, a site within the model crystal is chosen at random, and the associated variables are altered by a random amount. The energy difference ΔE between the initial configuration and the configuration after the change is computed. The new configuration is accepted if the probability P, given by equation (4.1), is less than a random number chosen uniformly in the range [0, 1].

� �� kTE

kTEP/exp1

/exp��

�� (4.1)

Here T is the system’s temperature and k is Boltzmann's constant. If kT is zero, only changes that decrease the energy of the system will be accepted. A typical value of 1 was used for the temperature during all MC simulations. Within DISCUS an MC move represents the modification of a single varia-ble such as occupancy or atom position. One MC cycle corresponds to the number of moves necessary to visit every site within the supercell once on average. The number of MC cycles was usually 1000, which was often enough as no changes in the system’s energy were observed in the final 200 cycles. The different MC energies used to create disordered structures in this work are presented in the following sections.

Figure 4.5. Schematic diagram illustrating the concept of MC algorithm.

26

4.6.2.1 Chemical short-range order (CSRO) CSRO is created within DISCUS94 using the Ising model which was origi-nally used to model magnetism. During MC simulations two different atoms are selected at random and their positions are switched. The MC energy for CSRO takes the following form:

jii i ijj

ijiocc JHE ��

��,

(4.2)

The sums are over all sites i and all nearest-neighbors j in the supercell. σiand σj represent Ising spin variables (+1/–1) of site i and its neighboring site j. Jij are pair interaction energies corresponding to the neighboring vector defined by i and j while H represents the energy of a single site and controls the overall concentration. If two neighboring sites are occupied by the same atom type the product σi·σj=1. A positive correlation means that two neigh-boring sites are occupied by the same atom type while a negative correlation will cause the two sites to be preferably occupied by different atom types. A certain degree of ordering is obtained by adjusting the value of Jij/kT. For a quantitative analysis of the modeled CSRO one can calculate the occupa-tional correlation coefficient cCSRO

ij between site pairs i and j defined by the following equation:

� ��

��

�1

2ijCSRO

ij

Pc (4.3)

Where Pij represents the joint probability that both atom sites i and j are oc-cupied by the same atom type and θ is the overall occupancy. Negative val-ues of cCSRO

ij indicate that the two sites i and j tend to be occupied by differ-ent atom types while positive values indicate that sites i and j tend to be oc-cupied by the same atom type. A correlation value of zero describes a ran-dom distribution.

4.6.2.2 Local atomic displacements To produce local atomic displacements away from average positions DIS-CUS offers different energies or potentials that can be used in MC simula-tions. For this work a simple harmonic potential (Eqn. 4.4) or a LennardJones potential (Eqn. 4.5) was used to create atomic displacements. In the case of displacive disorder a randomly selected atom is shifted by a random amount along user defined directions. The harmonic potential is defined as follows:

� �20�� i j

ijijjh ddkE � (4.4)

27

The sums are over all sites i in the supercell and all neighbors j around site i.dij represents the distance between neighboring atoms whiled0 is the average distance. The term kj is a force constant for each individual neighbor type j.The desired displacements are defined by the factor τij. The Lennard-Jones potential is defined as follows:

��

�

�

�

��

�

�

��

!""#

$��

�

!""#

$�

i ijj ij

ij

ij

ijij dd

DE,

612

2��

(4.5)

The sums are over all atom sites i in the supercell and all neighboring sites j.D represents the potential depth which must be negative, τij is the target dis-tance where the Lennard-Jones potential has its minimum and dij represents the user defined distance between two neighboring atoms. Similar to the case of CSRO one can calculate displacive correlation coeffi-cients relating the atomic displacements away from the average positions for two sites, as shown in the following equation:

22ji

jidisplaciveij

xx

xxc � (4.6)

In this case xi represents the displacements of the atom at site i from its aver-age position in a certain direction and the average is calculated over the en-tire supercell. As for CSRO, a value of zero describes the case of random displacements. Positive values indicate that atoms are displaced in the same direction while negative values indicate that atoms are displaced in opposite directions.

4.6.3 Domain subroutine Within DISCUS a domain is defined as any part of the structure that has been replaced by another structure. This very general definition allows creat-ing a wide range of different domains ranging from small clusters of atoms to intergrowth of different phases. By systematically replacing a part of the crystal one can create complex disordered crystal structures. The domain subroutine allows to easily define the domain position within the host struc-ture, the shape and size of the domain and the transformation matrix between the host and the domain. In the simulation process each domain’s atom is placed into the host structure and all host atoms within a user defined dis-tance are removed.

28

4.6.4 Fourier transform Electron diffraction patterns were computed using DISCUS software which basically calculates the Fourier transform according to the standard formula for kinematic scattering:

4

1

2

2

)()(h

rhhhBN

i

ii eefF i

�

�

� �� (4.7)

The sum goes over all atoms N in the supercell and is calculated over all points h in reciprocal space within the plane of interest. if represents the atomic form factor for atom i and ri are the fractional coordinates of the at-om. Optionally the Debye-Waller factor can be calculated, where B is the isotropic thermal coefficient. In the case of disordered structures at high temperatures, before calculating the Fourier transform, all atoms were ran-domly displaced as expected from pure random thermal disorder. This is a rather artificial way to account for the increase in temperature which leads to an increase in the random thermal displacements of atoms. The directions of atomic displacements were distributed with a uniform random spherical dis-tribution while the amplitude was distributed by a Gaussian distribution. The mean of the Gaussian distribution was zero and its sigma was calculated from the isotropic thermal coefficient B, of each atom as follows:

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8 Bu � (4.8)

The form factors are tabulated96 and calculated at each h for all atoms pre-sent in the supercell. The scattering intensity is simply computed as I(h) = F(h)·F*(h). The finite-size effect was avoided by applying periodic boundary conditions: Δh = 1/Δx. Where Δh is the grid size of the calculated reciprocal space plane and Δx is the dimension of the supercell. The KUPLOT routine94

was used for graphical representation of simulated electron diffraction pat-terns. In order to reduce the anisotropic contribution to the diffraction inten-sities final electron diffraction patterns were computed by averaging the results from 20 different simulations carried out with the exact same parame-ters.

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5.(95-x)Na0.5Bi0.5TiO3-xK0.5Bi0.5TiO3-5BaTiO3(papers I, II & III)

5.1 Literature overview Since its discovery in the 1960s Na0.5Bi0.5TiO3 (NBT)97 and its solid solu-tions with BaTiO3 (BT) and K0.5Bi0.5TiO3 (KBT) have been studied as poten-tial replacements of Pb-based piezoelectric ceramics18,35. This research has intensified in the last decades owing to environmental regulations against the use of Pb-based ceramics in electronic applications due to high toxicity of Pb and its derivatives34,98. As a result, restrictions of the use of certain hazardous substances (including Pb, Cd, Hg, Cr etc.) in electrical and electronic equip-ment have been legislated for the near future in many countries around the world, including the European Union and Japan99,100. Pure NBT is ferroelec-tric at room temperature with a high remnant polarization (Pr~38 μC/cm2)and a large piezoelectric coefficient (d33~73 pC/N). However, the use of NBT in practical applications is hindered by the low depolarization tempera-ture (Td~200°C) which is lower than the Curie temperature (Tc~320°C) and high coercive field (above 50 kV/cm)101,102. To overcome these shortcomings binary and ternary NBT-based solid solutions have been synthesized. In particular, the (95-x)Na0.5Bi0.5TiO3-xK0.5Bi0.5TiO3-5BaTiO3 system shows promising dielectric and piezoelectric properties which are comparable with those of Pb-based ceramics. Enhanced properties are observed close to the MPB where two ferroelectric phases coexist: one with rhombohedral sym-metry (R3c) and one with tetragonal symmetry (P4bm)36-38,103-105.Even though the average and local structures of NBT and NBT-based solid solutions have been the topic of many studies, there is still an ongoing de-bate over their exact nature. The initial reported room temperature average structure for pure NBT presents a rhombohedral symmetry (R3c) with polar cation displacements along <111>pc (where pc stands for pseudo-cubic) di-rections and antiphase oxygen octahedral rotations (a–a–a– tilt system)106-108.Recent high-resolution X-ray studies suggest on the contrary a monoclinic symmetry (Cc) with a–a–c– octahedral tilting109,110. Another proposed average structure indicates that the equilibrium state of NBT at room temperature is best described by a coexistence of R3c and Cc phases111. Different TEM studies confirmed both rhombohedral112 and monoclinic113 symmetries con-tributing to the pronounced discrepancy within reported results. Moreover,

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TEM also revealed a tetragonal phase (P4bm) at room temperature with a a0a0c+ tilt system40,114. The discrepancy amongst reported average structures of NBT and NBT-based solid solutions stems from a complex local structure that significantly deviates from the average one. Many studies have tried to tackle this problem by means of EXAFS115,116, NMR41, X-ray DS42,43,117-119,TEM39,40,114,120, PDF121,122 or DFT calculations123-126. A highly distorted local coordination for Bi atoms was indicated by EXAFS measurements115 and further confirmed by RMC refinement of PDF data122. Additionally, broad DS intensity in the vicinity of fundamental perovskite reflections was rec-orded using synchrotron X-ray radiation42,43,117,118,127 and continuous DS rods along <00l>* directions together with two types of SSRs were observed in electron diffraction patterns120,128,129, indicating that NBT-based solid solu-tions present a complex local structure. Several models have been proposed to explain the local structural disorder present in these types of materials. Kreisel et al.42 invoked planar defects of a few unit cells thick with mono-clinic symmetry (Cc), resembling Guinier-Preston zones (GPZ), to explain the observed DS. In their model there is a competition between the rhombo-hedral distortion of the Ti-O framework with displacements along <111>pcand the tetragonal Na/Bi displacements along <100>pc directions. Further-more, Dorcet et al.40 have carried out a detailed TEM study for pure NBT and suggested that locally the structure comprises of platelets of tetragonal symmetry imbedded in a rhombohedral matrix. Another model proposed by Levin et al.39 describes the local disorder as nanoscale domains with a-a-c+

tilting where the coherence length of in-phase tilting is just a few unit cells large and is much shorter than for antiphase tilting. Moreover, there is also a mismatch between the macroscopic properties of NBT-based ceramics and the temperature-dependent long-range structural phase transitions. The commonly accepted phase transition sequence for pure NBT was proposed by Jones et al.108. Two long-range structural phase transitions occur: one at ~296°C from a ferroelectric rhombohedral structure (R3c) to a ferroelectric tetragonal structure (P4bm) and the second one at ~566 °C from a ferroelectric tetragonal P4bm structure to a paraelectric cu-bic Pm-3m structure. Rhombohedral/tetragonal and tetragonal/cubic phase coexistence was observed over a large temperature range (i.e. ~150°)108, 130.On the other hand, a completely different phase transition sequence was reported by Dorcet et al. from in-situ TEM investigations120, 131. They pro-pose that at ~200°C the ferroelectric R3c structure transforms into an antifer-roelectric modulated phase and that at ~280°C an orthorhombic structure develops with space group Pnma. The assignment of this space group was based on observed ½(oee) SSRs. At 320°C another transition to a paraelec-tric phase (P4/mbm + P42/mnm) occurs and above 520°C the structure is cubic (Pm-3m). However, the validity of this model still needs further clari-fication since the ½(oee) SSRs have not yet been reported elsewhere. Anoth-er study by Aksel et al.132 suggests that a subtle change in the local structure

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of NBT occurs at ~195°C while the long-range structural transition from rhombohedral to tetragonal symmetry is observed at 280°C and the transition to cubic symmetry occurs above ~510°C. Furthermore, temperature-dependent studies of PDF data for pure NBT revealed that the local envi-ronment of Bi3+ and Na+ significantly changes with temperature and deviates from the long-range average structure121. This indicates that temperature-dependent studies of phase transitions should not neglect the changes in the local structure. Even though no long-range structural phase transition was observed below 280°C, the piezoelectricity in poled NBT ceramics is significantly dimin-ished or completely disappears upon heating to ~187°C, the so called depo-larization temperature132,133. The same phenomenon is observed in other NBT-based ceramics, for example the 95NBT-5BT composition has a depo-larization temperature, (Td) of ~120°C134,135. Earlier studies have suggested a ferroelectric-antiferroelectric transition as an explanation for the depolariza-tion mechanism based on the double hysteresis-loop observed at tempera-tures close to Td

36,103,136,137. However, no satellite reflections characteristic for a commensurate or incommensurate antiferroelectric phase have been rec-orded either by X-ray, neutron or electron diffraction. Moreover, a double hysteresis-loop does not necessarily imply the existence of an AFE phase and can be explained by several other mechanisms. For example, a double hysteresis-loop can occur for solid solutions where the ferroelectric and paraelectric phases coexist like in (PbxCa1-x)TiO3

138. Hence, due to the lack of structural evidence more recent studies have questioned the existence of the ferroelectric-antiferroelectric transition. Tai et al.139 suggested that the depolarization is associated with the reduction of oxygen octahedral tilting which leads to a weakening of the macroscopic ferroelectric domains. On the other hand, Jo et al.140 argue that the depolarization temperature is in fact the temperature where a long-range ferroelectric state, induced by an electric field, reverts back to the initial relaxor state upon heating. Another study, focused on the Young’s modulus temperature-dependence, proposes a ferro-electric-relaxor transition as an explanation for the depolarization mecha-nism141.

5.2 Local structure of 85NBT-10KBT-5BT ceramic (paper I) This paper focused on developing a detailed model for the local structural disorder present in 85NBT-10KBT-5BT ceramic as revealed by 3D electron DS. First, the insights provided by analyzing the 3D reciprocal space volume and how DS features are correlated to the real space local structure will be presented. Next, DF-imaging was used to further confirm the presence of

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antiphase and in-phase tilted nanodomains. Finally, a comprehensive model for the local structure of 85NBT-10KBT-5BT ceramic was developed using DISCUS software. This model explains all observed diffraction features and sheds light onto the ongoing debate over the nature of the local structure for NBT-based solid solutions.

5.2.1 Analysis of SSRs and 3D electron diffuse scattering The 85NBT-10KBT-5BT ceramic has a composition near the MPB36,38

where rhombohedral (R3c) and tetragonal (P4bm) symmetries coexists. The rhombohedral symmetry exhibits antiphase tilting (a–a–a–) while the tetrago-nal symmetry exhibits in-phase tilting (a0a0c+). Since a–a–a– tilt system pro-duces ½(ooo) SSRs and the a0a0c+ produces ½(ooe) SSRs, the presence of different types of SSRs can help us identify what kind of oxygen octahedral tilting exists in the structure. A detailed analysis of SSRs was initially done using conventional 2D SAED patterns oriented along different high sym-metry ZAs, as shown in Fig. 5.1. Two types of SSRs were identified namely, ½(ooe) as seen in [001]pc and [111]pc ZAs and ½(ooo) as seen in [110]pc ZA. This indicates a coexistence of in-phase and antiphase octahedral tilting. Moreover, all three variants of ½(ooe) SSRs are simultaneously observed in the [111]pc ZA which is proof that in-phase tilting occurs along all three main pseudo-cubic directions according to a0a0c+, a0c+a0 and c+a0a0 tilt sys-tems.

Figure 5.1. SAED patterns for 85NBT-10KBT-5BT ceramic recorded along (a)[001]pc, (b)[110] pc and (c) [111]pc ZAs. The different types of SSRs are marked with circles. Indexing throughout the thesis was done with respect to the pseudo-cubic (pc) unit cell (Pm-3m). [Paper I]-Reproduced with permission of Springer Na-ture.

The whole 3D reciprocal space volume reconstructed from RED data rec-orded for a grain of 85NBT-10KBT-5BT ceramic is shown in Fig. 5.2. The

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reciprocal space volume is projected along three equivalent ZAs namely, [001]pc, [010]pc and [100]pc. Besides Bragg reflections which are divided into fundamental perovskite reflections and SSRs, two extra diffraction features can be observed: (i) broad DS intensity near the fundamental perovskite reflections and (ii) continuous DS rods passing through the SSRs along [00l]*

pc, [0k0]*pc and [h00]*

pc directions. The broad DS intensity can be clear-ly observed only for reflections close to the direct beam i.e. 110, -110, 1-10, -1-10 (Fig. 5.2a) and tends to be concentrated at positions in between the fundamental perovskite reflections.

Figure 5.2. Whole 3D reciprocal space volume projected along the three main pseu-do-cubic ZAs.: (a) [001]pc, (b) [010]pc and (c) [100]pc. [Paper I]-Reproduced with permission of Springer Nature.

Figure 5.3 shows the whole reciprocal space volume projected along [100]pcZA together with a volume-section centered on the hk0 reciprocal plane and the hk0.5 slice. Here, the power of 3D diffraction data is revealed as we are able to separate the different DS features. Broad DS intensity is observed in the hk0 volume-section together with sharp ½(ooe) SSRs while in the hk0.5slice we observe only the continuous DS rods. Moreover, one can clearly see that along the DS rods increased intensity is found at positions of ½(ooe)SSRs as indicated by white arrows in Fig. 5.3c. The use of volume-sections was necessary in order to better visualize the broad DS intensity observed near the fundamental perovskite reflections. This feature is easily observed for reflections close to the direct beam i.e. the region close to the direct beam depicted by the white square in Fig. 5.3b. The lack of visibility for the broad DS intensity in single slices can be explained by taking a closer look at the individual 2D SAED patterns recorded during RED data collection.

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Figure 5.3. Analysis of 3D electron DS intensity by means of volume-sections and slices cut from the reciprocal space volume. (a) Whole 3D reciprocal space volume projected along [100]pc ZA. (b) Volume-section centered on the hk0 slice, outlined by the dashed orange rectangle in (a) and (c) hk0.5 slice outlined by the magenta dotted line in (a). [Paper I]-Reproduced with permission of Springer Nature.

The individual SAED patterns recorded during RED data collection reveal that for patterns close to exact ZA condition (Fig. 5.4a) almost no DS inten-sity is visible while for patterns a few degrees away from exact ZA condition (Fig. 5.4b) broad DS intensity can be easily observed for fundamental perov-skite reflections close to the direct beam. The same phenomenon was report-ed before for a number of different materials12,142,143. One explanation pro-posed for the lack of visibility of DS intensity at low order ZAs suggests that compositional and/or displacive disorder is averaged out along the atomic columns leading to no net modulation12, 142. Another explanation is that for thick specimens (≥200 nm) inelastic scattering leads to a smearing out of DS intensity143. In our case either of these two explanations or a combination of both could account for the lack of visibility for DS intensity at low order ZAs.

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Figure 5.4. Individual SAED patterns recorded (a) close to [001]pc ZA showing no DS intensity and (b) approximately 3° away from [001]pc ZA, showing broad DS intensity near the fundamental perovskite reflections, as highlighted for 110 reflec-tion. [Paper I]-Reproduced with permission of Springer Nature.

5.2.2 CDF-imaging of antiphase and in-phase tilted nanodomains CDF-images of antiphase and in-phase tilted nanodomains were recorded in order to estimate their size and concentration. Figure 5.5b is a representative CDF image of antiphase nanodomains, which was recorded using the -3/23/2 1/2 reflection along [110]pc ZA as highlighted by the black circle in Fig. 5.5a. Figure 5.5d is a representative CDF image of in-phase nanodomains which was recorded using the -3/2 1/2 1 reflection along [111]pc ZA, as high-lighted by the blue circle in Fig. 5.5c. The bright speckles in Fig. 5.5(b, d) represent the antiphase and in-phase tilted nanodomains. The dark contrast in both CDF images corresponds to regions with octahedral tilting about a dif-ferent axis than the antiphase/in-phase tilting or regions without octahedral tilting. A non-uniform distribution can be observed in the case of antiphase nanodomains with regions that have higher concentrations of bright speck-les, like the region depicted by a black circle in Fig. 5.5b. In contrast the in-phase nanodomains present a uniform distribution. The size of the antiphase nanodomains was less than ~8 nm across while the size of in-phase nanodomains was less than ~12 nm across. All three variants of ½(ooe)SSRs are observed in ZA [111]pc indicating that in-phase tilting occurs along three orthogonal directions rather than a single nanodomain tilted along a unique orientation. Moreover, these SSRs are slightly elongated as can be seen in Fig. 5.5c which is an indication that the in-phase nanodomains have a plate-like shape. No direct correlation between the antiphase/in-phase nao-domains and the macroscopic ferroelectric domains was found which sug-gest that the octahedral tilted nanodomains are confined to the nanoscale and no long-range tilting order exists. Even though CDF images in Fig. 5.5(b, d)

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were taken from different grains we can still estimate that the in-phase nanodomains have a higher concentration than the antiphase nanodomains.

Figure 5.5. Centered dark-field images of antiphase and in-phase nanodomains. SAED patterns recorded along (a) [110]pc and (c) [111]pc ZAs. (b) DF image of antiphase nanodomains recorded using the SSR depicted by a black circle in (a). (d)DF image of in-phase nanodomains recorded using the SSR depicted by a blue circle in (c). A region with high concentration of antiphase nanodomains is highlighted by the black circle in (b). The inset in (c) shows an enlarged view of the slightly elon-gated ½(ooe) SSR. [Paper I]-Reproduced with permission of Springer Nature.

5.2.3 Highlights of experimental results The experimental data revealed two type of SSRs, namely ½(ooe) and ½(ooo). This is an indication that both antiphase (a–a–a–) and in-phase (a0a0c+) tilting are present in the structure. Mixed octahedral tilting has been excluded as no ½(oee) SSRs were observed by SAED or RED. Moreover, all three variants ½(ooe), ½(oeo) and ½(eoo) were simultaneously observed in the [111]pc ZAs suggesting that in-phase tilting occurs along three orthogo-nal axes rather than a unique orientation. Hence, in-phase tilting is produced by three equivalent tilt systems: a0a0c+, a0c+a0 and c+a0a0. The RED data further confirmed the presence of three variants for ½(ooe) SSRs as this type

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of SSRs could be observed along all three [001]pc, [010]pc and [100]pc ZAs. A peculiar feature is that ½(ooe) variants are slightly elongated suggesting that planar defects are present in the structure. This guess is confirmed by the continuous DS rods observed in the hk0.5 slice. These rods pass through the SSRs which is an indication that octahedral tilting is the main contributor to their DS intensity. Based on the SAED patterns and RED data we can conclude that plate-like in-phase nanodomains tilted about the three pseudo-cubic axes are present in the structure. Another feature observed in the hk0volume-section was broad DS intensity near the fundamental perovskite reflections which suggests that local atomic displacements and/or CSRO are also present in the structure. Finally, DF images of antiphase and in-phase nanodomains were recorded as means to estimate their size and concentra-tion. The size of the antiphase nanodomains was less than ~8 nm across while the size of in-phase nanodomains was less than ~12 nm across. A higher concentration of in-phase domains was observed when compared to the antiphase nanodomains.

5.2.4 Modeling of local structural disorder The model of local structural disorder in 85NBT-10KBT-5BT ceramic was developed using DISCUS software, according to the following methodology. First an antiphase tilted matrix was created. Then plate-like in-phase tilted nanodomains were introduced into the structure. Lastly, the MC algorithm was employed to create CSRO and local atomic displacements of Bi3+ and Na+ atoms. The initial supercell (50×50×50 unit cells) was created by ex-panding the cubic perovskite asymmetric unit cell (Pm-3m, a = 3.8894 Å) with Bi3+, Na+, Ba2+ and K+ sharing the (0.5, 0.5, 0.5) position, Ti4+ at (0.0, 0.0, 0.0) and O2- at (0.5, 0.0, 0.0). Further, the antiphase tilted matrix was created by tilting all oxygen octahedra according to a–a–a– tilt system, i.e. a single operation about [111]pc axis with alternating layers tilted by 8° and -8°. To ensure an anisotropic antiphase tilting the matrix was created from three kinds of antiphase tilted nanodomains (6×6×6 unit cells) where alter-nating layers tilted by 8° and -8° were stacked along [001]pc, [010]pc or [100]pc directions. These antiphase nanodomains are randomly distributed and cover the entire crystal. The next step was to introduce in-phase tilted nanodomains into the structure. Three different variants were created accord-ing to the tilt requirements of a0a0c+, a0c+a0 and c+a0a0 tilt systems. The same tilting angle as for antiphase tilting was chosen. The in-phase nanodomains were randomly distributed into the matrix using an intrinsic function of the software which returns Gaussian distributed pseudo-random numbers with mean zero.

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The first parameter tested was the antiphase/in-phase tilting ratio as shown in Fig. 5.6. For these simulations in-phase nanodomains 10×10×2, 10×2×10 and 2×10×10 unit cells large were used. The short dimension represents the direction along which the oxygen octahedra were tilted, namely [001]pc,[010]pc or [100]pc. Comparing the morphology and intensity of DS rods in the simulated and experimental hk0.5 reciprocal plane one can conclude that the concentration of in-phase nanodomains needs to be higher than 50% for a good agreement.

Figure 5.6. Simulated electron diffraction patterns for hk0.5 reciprocal plane with different antiphase/in-phase tilting ratios. All patterns have been calculated from a 50×50×50 supercell with in-phase nanodomains of 10×10×2 unit cells large but with different antiphase/in-phase ratios: (a) 100/0, (b) 90/10, (c) 70/30, (d) 50/50, (e)30/70 and (f) 10/90. [Paper I]-Reproduced with permission of Springer Nature.

The second tested parameter was the size of in-phase nanodomains as can be seen in Fig. 5.7. For these simulations the antiphase/in-phase tilting ratio was kept at 30/70 but the initial size of in-phase nanodomains varied from 4×4×2, 6×6×2, 8×8×2 to 10×10×2 unit cells large. Initial size refers to the minimum size of an isolated nanodomain. Since overlapping was allowed, larger naodomains can be obtained when compared with the initial minimum size. A good match between experimental and simulated data was obtained for sizes larger than 4×4×2 unit cells.

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Figure 5.7. Simulated electron diffraction patterns for hk0.5 the reciprocal plane with different sizes of in-phase nanodomains. All patterns have been calculated from a 50×50×50 supercell with an antiphase/in-phase ratio of 30/70 but using different initial sizes for the in-phase nanodomains: (a) 4×4×2 unit cells, (b) 6×6×2 unit cells, (c) 8×8×2 unit cells and (d) 10×10×2 unit cells. [Paper I]-Reproduced with per-mission of Springer Nature.

The last tested parameter was the shape of in-phase nanodomains. By shape is meant that the short dimension of in-phase nanodomains was varied from 2 to 10 unit cells large (Fig. 5.8). The short coherence length along the tilting direction (~1 nm) for in-phase nanodomains was crucial to obtain a good agreement between simulated and experimental patterns. By increasing the short dimension of in-phase nanodomains from 2 up to 10 unit cells the elongated DS feature gradually transforms into sharp ½(ooe) SSRs.

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Figure 5.8. Simulated electron diffraction for hk0.5 reciprocal plane with different shapes for in-phase nanodomains. All patterns have been calculated from a 50×50×50 supercell with an antiphase/in-phase ratio of 30/70 but using different shapes for in-phase nanodomains: (a) 10×10×2 unit cells (b) 10×10×4 unit cells (c)10×10×6 unit cells, (d) 10×10×8 and (e) 10×10×10 unit cells. [Paper I]-Reproduced with permission of Springer Nature.

Based on these simulations we have chosen an antiphase/in-phase tilting ratio of 30/70 and a size/shape of 10×10×2, 10×2×10 and 2×10×10 unit cells for the in-phase variants. The distribution of in-phase nanodomains within the antiphase tilted matrix is illustrated in Fig. 5.10b. The size of the anti-phase nanodomains was less than ~10 nm across while for in-phase nanodomains the size was less than ~14 nm across. To further check that these parameters are reasonable electron diffraction patterns were calculated along [001]pc, [110]pc and [111]pc ZAs (Fig 5.9). The experimental SAED patterns (Fig. 5.1) suffer from dynamical effects so kinematically forbidden reflections can be observed, like 1/2 1/2 1 SSR in [001]pc ZA and 1/2 1/2 1/2SSR in [110]pc ZA. Nevertheless, within the limits of the kinematical ap-proximation all SSRs are accounted for by a coexistence of antiphase and in-phase tilted nanodomains. Moreover, all three variants of ½(ooe) SSRs are obtained in the calculated pattern along [111]pc ZAs together with the slight elongation which was also observed in the experimental pattern (Fig. 5.5c).

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Figure 5.9. Simulated electron diffraction patterns along (a) [001]pc, (b) [110]pc and (c) [111]pc ZAs. The dotted lines are a visual guide with fundamental perovskite reflections at their intersection. [Paper I]-Reproduced with permission of Springer Nature.

The last step was to use MC simulations to create CSRO on the A-site and to induce local atomic displacements. CSRO of Na/Bi atoms was created by using a positive correlation of 0.1 for <001>pc directions and a negative cor-relation of -0.1 for <111>pc directions. The degree of CSRO was in fact very small but significantly different from a random distribution. Ba and K atoms were randomly distributed leading to a slight suppression of Na/Bi A-site CSRO. A 2D layer illustrating the distribution of A-site atoms is shown in Fig. 5.10c. A small tendency of alternating rows with Na- and Bi-containing unit cells can be observed along <001>pc directions. Furthermore, Bi-Bi dis-tances along <001>pc and <111>pc directions were increased by 1-2% while Na-Na distances were decreased by the same amount. Ba, K and Ti atoms were assumed to maintain their average positions.

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Figure 5.10. Local structure of 85NBT-10KBT-5BT ceramic. (a) Schematic repre-sentation of different octahedral tilt systems used during simulations. (b) 2D layer containing only TiO6 octahedra illustrating the distribution of in-phase nanodomains within the antiphase tilted matrix. (c) Distribution of A-site cations. Adapted from [Paper I] with permission of Springer Nature.

Figures 5.11(a, b) illustrate the electron diffraction patterns along hk0 and hk0.5 reciprocal planes calculated from the final structure. A good agree-ment was obtained between the simulated and experimental data as can be seen in Fig. 5.11(c-h). Both the morphology and intensity distribution along the DS rods (i.e. higher elongated intensity at positions of ½(ooe) SSRs) present in hk0.5 reciprocal plane are accounted for by the model (Fig. 5.11(g, h)). In the case of the broad DS intensity near the fundamental per-ovskite reflections the calculated DS intensity (Fig. 5.11(e,f)) deviates slight-ly from the experimental result (Fig. 5.11(c,d)). One reason might be that this very weak DS feature is more drastically affected by dynamical effects and absorption which were not included in our calculations. Also, a single wavelength (0.025 Å) was used to calculate the electron diffraction patterns while in reality the electron beam presents an energy spread of ~0.9 eV.

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Figure 5.11. Simulated electron diffraction patterns. (a) hk0 reciprocal plane. (b)hk0.5 reciprocal plane. (c), (d) Enlarged images of the regions depicted by white squares in the experimental hk0 pattern (Fig. 5.3b) compared with the corresponding calculated region (e), (f). (g) Enlarged image of the region depicted by the white square in the experimental hk0.5 pattern (Fig. 5.3c) compared with the correspond-ing calculated region (h). [Paper I]-Reproduced with permission of Springer Nature.

5.2.5 Summary A model of local structural disorder for 85NBT-10KBT-5BT ceramic was developed by analyzing the morphology and intensity of experimental DS features and SSRs. Electron diffraction data were recorded using the RED method which provides a 3D reciprocal space volume. The option to study electron diffraction data in 3D combined with cutting slices and volume-sections enables us to separate different DS features. A better understanding of the morphology of DS was obtained which helped translating the result into a real space disordered structure. The final model comprises of three variants of in-phase nanodomains tilted about the three pseudo-cubic axes

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and randomly distributed into an antiphase tilted matrix. Furthermore, a small degree of Na/Bi CSRO was introduced on the A-site. Lastly, correlated Na- and Bi-displacements along <001>pc and <111>pc directions were creat-ed. This model explains all observed diffraction features within the limits of kinematical approximation. This work shows that electron diffraction can be used to record high-quality DS intensity in 3D for polycrystalline materials. This data can then be used as input for simulating disordered atomic struc-tures using dedicated software, like DISCUS. The methodology developed here for creating antiphase and in-phase tilted nanodomains was also used in papers II&III.

5.3 Influence of K-content on octahedral-tilt disorder (paper II) In this work SAED, RED, CDF-imaging and modeling of disordered atomic structures were combined to investigate the influence of K-content on the local structure of (95-x)Na0.5Bi0.5TiO3-xK0.5Bi0.5TiO3-5BaTiO3 solid solu-tions, with x = 0, 5, 10. The concentration and size of antiphase and in-phase tilted nanodomains were estimated from SAED patterns and CDF-images. Furthermore, the intensity and morphology of DS rods observed in hk0.5reciprocal plane were analyzed for the different compositions. Finally, mod-els of octahedral-tilt disorder showed that with increasing K-content a trans-formation occurs from a structure with predominant antiphase tilting to a structure with predominant in-phase tilting.

5.3.1 SAED patterns SAED patterns were recorded along [112]pc ZA for 95NBT-5BT, 90NBT-5KBT-5BT and 85NBT-10KBT-5BT ceramics, in order to study the differ-ent types of octahedral tilting present in the structure. As can be seen in Fig. 5.12, two kinds of SSRs are observed, namely ½(ooo) and ½(ooe). This in-dicates that antiphase (a–a–a–) and in-phase (a0a0c+) tilting coexist for all compositions. Moreover, kinematically forbidden reflections can be ob-served as a result of double diffraction (i.e. -1/2 1/2 0 and 1/2 1/2 -1/2). The insets in Fig. 5.12(a-c) show an enlarged view of the two kinds of SSRs. For 95NBT-5BT ceramic ½(ooo) SSRs present higher intensity when comparedto ½(ooe) SSRs (Fig. 5.12a). Increasing the K-content the relative intensity is reversed and 85NBT-10KBT-5BT ceramic presents higher intensity for ½(ooe) SSRs (Fig. 5.12c). For the composition at the MPB the two kinds of SSRs present comparable intensities (Fig. 5.12b). It should be mentionedthat only SSRs’ intensities relative to each other are compared in order to

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estimate the relative phase fraction of antiphase and in-phase nanodomains. Although SAED patterns suffer from dynamical effects and thickness varia-tions, which makes a quantitative analysis very difficult, the contribution to the scattering intensities is not dominant. Hence, the trend of intensity change for the two kinds of SSRs with increasing K-content could still be followed.

Figure 5.12. SAED patterns recorded along [112]pc ZA for (a) 95NBT-5BT, (b)90NBT-5KBT-5BT and (c) 85NBT-10KBT-5BT ceramics. The insets in the upper-right corner show an enlarged view of ½(ooo) SSRs while the insets in the lower-right corner show an enlarged view of ½(ooe) SSRs. [Paper II]-Reproduced with permission of Elsevier Inc.

5.3.2 CDF-imaging of antiphase and in-phase tilted nanodomains The power of the TEM technique resides in the possibility of combining diffraction and imaging within the same instrument. So, in order to visualize the antiphase and in-phase nanodomains indicated by the SSRs observed in [112]pc ZA, CDF-images were recorded for all ceramics. The antiphase nanodomains (Fig. 5.13(a-c)) were recorded close to [110]pc ZA and the in-phase nanodomains (Fig. 5.13(d-f)) were recorded close to either [001]pc or [111]pc ZAs. The bright speckles indicate the tilted regions which scatter the electrons into selected ½(ooo) or ½(ooe) SSRs. The dark regions present either no octahedral tilting or different tilting than the selected antiphase/in-phase tilt system. The distribution of antiphase and in-phase nanodomains is rather uniform and no direct correlation to the macroscopic FE domains was found. The octahedral tilted nanodomains intimately coexist at the nanoscale which produces a disruption in the long-range octahedral tilting. On average the size of antiphase nanodomains was found to be ~2-5 nm for all composi-tions. In the case of in-phase nanodomains we observed slight differences for the three compositions: ~2-3 nm for 95NBT-5BT, ~2-6 nm for 90NBT-5KBT-5BT and ~2-7 nm for 85NBT-10KBT-5BT. This information was

46

used as a starting point for the simulation of local octahedral-tilt disorder for the different ceramics.

Figure 5.13. CDF images of ½(ooo) SSRs for (a) 95NBT-5BT, (b) 90NBT-5KBT-5BT and (c) 85NBT-10KBT-5BT and of ½(ooe) SSRs for (d) 95NBT-5BT, (e)90NBT-5KBT-5BT and (f) 85NBT-10KBT-5BT. The insets in (a-c) show an en-larged view of antiphase tilted clusters while the insets in (d-f) show an enlarged view of in-phase tilted clusters. [Paper II]-Reproduced with permission of Else-vier Inc.

BF-images of macroscopic FE domain walls are shown in Fig. 5.14. A com-plex domain configuration can be observed with herringbone-like domains and lamellar domains. The domain boundaries reside on the {100}pc or {110}pc planes similar to the case of rhombohedral NBT40.

Figure 5.14. BF TEM images of FE domain walls close to [110]pc ZA for (a)95NBT-5BT, (b) 90NBT-5KBT-5BT and (c) 85NBT-10KBT-5BT ceramics. The main crystallographic directions are indicated according to the corresponding SAED pattern from the same area. [Paper II]-Reproduced with permission of Elsevier Inc.

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5.3.3 Electron diffuse scattering The influence of K-addition on the nanoscale octahedral-tilt disorder was further investigated using the RED method. Electron diffraction data were recorded in 3D and the whole reciprocal space volume projected along [100]pc ZA, in the case of 90NBT-5KBT-5BT ceramic, is shown in Fig. 5.15a. The hk0.5 reciprocal plane is depicted by the yellow dashed line. The experimental hk0.5 reciprocal planes cut from the 3D volumes for the differ-ent ceramics are shown in the left-hand side of Fig. 3(b-d). The initial orien-tation for each grain is different and due to the limited tilting-range (±21°) only parts of the hk0.5-slices were recorded.

Figure 5.15. (a) Whole 3D reciprocal space volume projected along [100]pc ZA, depicting the perovskite unit cell and the hk0.5 reciprocal plane (yellow dashed line). Comparison of experimental DS data (left hand side) and simulated (right hand side) electron diffraction patterns for (b) 95NBT-5BT, (c) 90NBT-5KBT-5BT and (d) 85NBT-10KBT-5BT along the hk0.5 reciprocal plane. [Paper II]-Reproduced with permission of Elsevier Inc.

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Continuous DS rods along [00l]*pc, [0k0]*

pc and [h00]*pc directions appear in

the hk0.5 reciprocal planes for all compositions. They pass through the SSRs so they are associated with TiO6 octahedra tilting. The intensity distribution along the DS rods is not uniform and it changes with K addition. Sharp spot-like intensity (white circles in Fig. 5.15) is observed at positions of ½(ooo)SSRs. At positions of ½(ooe) SSRs an elongated DS feature (white arrows in Fig. 5.15) can be observed. With K-addition a decrease in the spot-like in-tensity occurs together with an increase in intensity for the elongated DS feature. For 95NBT-5BT ceramic the spot-like intensity is higher than the intensity of the elongated DS feature. A reversal in intensity is observed for the other two compositions. An elongation in reciprocal space is usually associated with a planar defect in real space. Hence, the continuous DS rods along [00l]*

pc, [0k0]*pc and [h00]*

pc directions indicate that three variants of plate-like in-phase tilted nanodomains are present in the real space structure.

5.3.4 Highlights of experimental results SAED and RED data indicate that a–a–a– and a0a0c+ tilting coexist for all compositions. Moreover, the DS rods observed in hk0.5 reciprocal planes are evidence of planar defects in the real space structure. Since, these DS rods pass through the SSRs and were observed in hk0.5 planes, oxygen octahedral tilting is the main contribution to their intensity. Hence, plate-like in-phase nanodomains tilted along all three main pseudo-cubic axes are developed within an antiphase tilted matrix. Both the relative intensity of SSRs and the intensity of DS rods changes with K-addition. This is an indication that a structural transformation occurs with K-addition from a structure dominated by antiphase tilting to a structure dominated by in-phase tilting. Therefore, the fraction of antiphase nanodomains decreases with K addition while the fraction of in-phase nanodomains increases. CDF-images suggest that the size of the antiphase nanodomains does not change significantly. By con-trast, slightly larger clusters of in-phase nanodomains could be observed for 85NBT-10KBT-5BT ceramic. These findings are in good agreement with the long-range average structures reported for these ceramics. 95NBT-5BT ceramic presents a rhombohedral symmetry (R3c), 90NBT-5KBT-5BT has a composition at the MPB where the average structure was described as a phase coexistence of rhombohedral (R3c) and tetragonal (P4bm) symmetries and 85NBT-10KBT-5BT present long-range tetragonal (P4bm) symmetry36,

144,145. Based on these observations a model of oxygen octahedral-tilt disor-der was developed for the different ceramics.

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5.3.5 Modeling of local structural disorder Since the main contribution to the observed SSRs and DS intensity is oxygen octahedral tilting, during the simulations only this type of disorder was con-sidered. The methodology developed in paper I, was used for creating the antiphase tilted matrix and for distributing the plate-like in-phase nanodomains. The size of the antiphase nanodomains was 6×6×6 unit cells and was kept the same for all simulations. In the case of in-phase nanodomains a good agreement between simulated and experimental data was obtained for an initial size of 6×6×2 unit cells for 95NBT-5BT, 8×8×2 unit cells for 90NBT-5KBT-5BT and 10×10×2 unit cells for 85NBT-10KBT-5BT. Three variants of in-phase nanodomains were introduced in the structure, i.e. 6×6×2, 6×2×6 and 2×6×6. As already mentioned, tilting wasapplied with respect to the short dimension (~1 nm) which was chosen based on the detailed simulations presented in paper I. The fraction of antiphase nanodomains decreases with K-addition while the fraction of in-phase nanodomains increases, as revealed by the experimental data. Hence, the final structures had an antiphase/in-phase tilt ratio of 70/30 for 95NBT-5BT, 50/50 for 90NBT-5KBT-5BT and 30/70 for 85NBT-10KBT-5BT. However, from our observations a quantitative estimation of the fractions for antiphase and in-phase nanodomains is not possible. The antiphase and in-phase nanodomains distribution is illustrated in Fig. 5.16(a-b). For 95NBT-5BT ceramic a small number of in-phase nanodomains are embedded in an anti-phase-tilted matrix (Fig. 5.16a). The situation is reversed for 85NBT-10KBT-5BT where antiphase tilted nanodomains are surrounded by an in-phase tilted “matrix” (Fig. 5.16c). In the case of 90NBT-5KBT-5BT ceramic the model suggest an intimate coexistence of antiphase and in-phase tilting (Fig. 5.16b).

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Figure 5.16. Distribution of in-phase nanodomains within the antiphase tilted matrix for (a) 95NBT-5BT, (b) 90NBT-5KBT-5BT and (c) 85NBT-10KBT-5BT. Only theTiO6 octahedra are shown along [001]pc. The numbers in the upper-right corners denote the antiphase/in-phase tilt ratio. (d) Schematic representation of a–a–a– tiltsystem (black), a0a0c+ tilt system (blue), a0c+a0 tilt system (green) and c+a0a0 tiltsystem (red). [Paper II]-Reproduced with permission of Elsevier Inc.

Electron diffraction patterns were calculated using the kinematical approxi-mation. A good agreement between simulated and experimental patterns was obtained, as can be seen in Fig. 5.15(b-d). The morphology and intensity distribution along the DS rods could be explained by a model comprising of in-phase tilted nanodomains with a thickness of ~ 0.8 nm distributed into an antiphase tilted matrix. Our model suggests that the concentration and size of in-phase nanodomains increases with K-addition. This leads to a transfor-mation from a structure with predominant antiphase tilting (95NBT-5BT) to a structure with predominant in-phase tilting (85NBT-10KBT-5BT). The long-range antiphase octahedral order is disrupted by the substitution with K on the A-site which favors the creation of plate-like in-phase nanodomains.

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5.3.6 Summary Oxygen octahedral-tilt disorder present as plate-like in-phase nanodomains, develops in (95-x)Na0.5Bi0.5TiO3-xK0.5Bi0.5TiO3-5BaTiO3 solid solutions with increasing K-content. The analysis of SSRs and DS intensity provided the necessary input for creating models of disordered structures. CDF-images further confirmed the existence of antiphase and in-phase nanodomains and were used for estimating their size. A coexistence of antiphase and in-phase tilting at the nanoscale was obtained for all compositions suggesting that the local structure deviates from the average one. The RED data provided the vital clue that DS rods are present in hk0.5 reciprocal planes and their inten-sity changes with K-content. A structural transformation takes place with increasing K-content from a structure with predominant antiphase tilting (95NBT-5BT) to a structure with predominant in-phase tilting (85NBT-10KBT-5BT). This study shows that electron diffraction is well suited for analyzing weak SSRs and DS intensity which provided information about oxygen octahedral-tilt disorder on the nanoscale.

5.4 Local phase transitions in 95NBT-5BT ceramic (paper III)In this study in-situ TEM was used to investigate how the local structure in 95NBT-5BT ceramic changes with temperature. The temperature-dependence of SSRs, DS intensity and macroscopic FE domains was ana-lyzed and used as a base for simulations of disordered structures at different temperatures. A subtle local structural phase transition was observed at~110°C, characterized by a change in the dominant octahedral tilt system from antiphase to in-phase tilting. We propose that a small fraction of plate-like in-phase nanodomains are present in the structure, even at room temper-ature, and that with increasing temperature this fraction increases until acritical value is reached. Moreover, the depolarization mechanism character-istic for NBT-based solid solutions was reported around the same tempera-ture suggesting that it is mainly influenced by octahedral-tilt disorder.

5.4.1 Temperature-dependence of SSRs Figures 5.17(a-c) show the [112]pc ZA for 95NBT-5BT ceramic recorded at different temperatures. This particular ZA was chosen as it presents both ½(ooo) and ½(ooe) SSRs which are characteristic for the rhombohedral (R) phase (R3c) and the tetragonal (T) phase (P4bm), respectively. Hence, the temperature-dependence of antiphase (a–a–a–) and in-phase (a0a0c+) tilting can be studied simultaneously. At room temperature strong ½(ooo) and weak

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½(ooe) SSRs are observed (Fig. 5.17a). With increasing temperature, the SSRs’ relative intensity is reversed as can be seen at 170°C (Fig. 5.17b). At 350°C only very weak ½(ooe) SSRs are observed (Fig. 5.17c). A semi-quantitative analysis of the temperature-dependence of SSRs’ relative inten-sity is shown in Fig. 5.17d. In the investigated temperature-range three re-gions can be distinguished:

I. From RT to ~100°C the intensity of ½(131) SSR is higher when compared with ½(132) SSR. Also a gradual decrease in intensity is observed for ½(131) SSR together with a slight increase for ½(132) SSR. This indicates that in this region the R- and T-phases coexist but with a predominant R-phase.

II. Between ~110-250°C the SSRs’ relative intensity is reversed and the ½(132) SSR has a higher intensity than ½(131) SSR. The intensity of ½(131) SSR continues to gradually decrease. In the case of ½(132) SSR the slight increase in intensity is ob-served until ~180°C where its intensity begins to also decrease. In this region the predominant phase is the T-phase with a small fraction of R-phase. In order to account for the decrease in in-tensity of both types of SSRs we propose that an additional cu-bic (C) phase develops in this region.

III. For temperatures higher than ~270°C only weak ½(ooe) SSRs are observed and their intensity gradually decreases with in-creasing the temperature further. This is an indication that the fraction of the C-phase gradually increases on the expense of the T-phase.

The intensity of ½(ooo) and ½(ooe) SSRs is compared only relatively to each other in order to estimate the relative fractions of the different phases. The transition from region I to region II is correlated with the depolarization mechanism which was reported around the same temperature for 95NBT-5BT ceramics134,135. Furthermore, the transition from region II to region III is associated with the onset of the long-range phase transition from T- to C-phase which should finalize at ~330°C135. However, the presence of weak ½(ooe) SSRs at temperatures as high as 500°C indicates that the T-phase has not completely vanished.

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Figure 5.17. SAED patterns for 95NBT-5BT ceramic recorded along [112]pc ZA at (a) RT, (b) 170°C and (c) 350°C. The blue circle depicts the ½(131) SSR while the red square depicts the ½(132) SSR. (d) Temperature-dependence of ½(131) and ½(132) SSRs’ relative intensity.

5.4.2 Temperature-dependence of FE domains BF-images of FE domains were recorded at different temperatures at the same time as the SAED patterns shown in Fig. 5.17. At room temperature 95NBT-5BT ceramic presents a complex domain configuration with her-ringbone-like domains and small lamellar domains, as can be seen in Fig. 5.18a. The domain boundaries lie on the {100}pc or {110}pc planes which is known to reduce strain in R-phases146. A high concentration of small do-mains with curved walls which do not follow any crystallographic direction can also be observed indicating the presence of 180° domains or other phas-es. At around 130°C some of the small lamellar domains merge and the her-ringbone-like domains are more visible. Also some the FE domains begin to disappear (Fig. 5.18b). At 150°C most of the FE domains have disappeared (Fig. 5.18c) and at 350°C no traces of FE domains can be seen (Fig. 5.18d). The temperature at which the FE domains start to vanish corresponds to the temperature at which the reversal in intensity for SSRs was observed. This

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suggests that the depolarization mechanism is correlated with the local phase transition characterized by a transformation in the dominant octahedral-tiltsystem from antiphase to in-phase tilting.

Figure 5.18. BF-images of FE domains for 95NBT-5BT ceramic recorded close to [112]pc ZA at (a) RT, (b) 130°C, (c) 150°C and (d) 350°C. The main crystallograph-ic directions are given according to the SAED pattern recorded from the same area.

5.4.3 Temperature-dependence of electron diffuse scattering In order to further investigate the local structural changes in 95NBT-5BT ceramic indicated by the temperature-dependence of SSRs, RED data were recorded at three selected temperatures. Figures 5.19(a-c) display the hk0.5reciprocal plane at RT, 180°C and 340°C. These are representative tempera-tures for the three regions depicted in Fig. 5.17d. The hk0.5 reciprocal planes present DS rods along [00l]*

pc, [0k0]*pc and [h00]*

pc directions that pass through the SSRs. Along the DS rods two features can be distinguished: (i) spot-like intensity (white circle) at positions of ½(ooo) SSRs and (ii) elon-gated features (white arrows) at positions of ½(ooe) SSRs. At room tempera-ture the spot-like intensity is significantly higher when compared to the elongated features that barely can be distinguished. At 180°C a reversal in intensity is observed with high intensity for the elongated features and a reduced intensity for the spot-like intensity (Fig. 5.19b). By increasing the

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temperature further to 340°C the spot-like intensity vanishes and a slight decrease in intensity for the elongated features is observed (Fig. 5.19c). The temperature-dependence of DS rods in hk0.5 reciprocal planes indicates that with increasing the temperature a local structural transition occurs. This tran-sition is characterized by the development of plate-like in-phase nanodomains normal to all three pseudo-cubic axes of the perovskite cell. Since strong DS scattering rods are observed only for temperatures in region II and III we can conclude that at room temperature the structure is predomi-nant R with antiphase tilting. At 180°C a high concentration of in-phase nanodomains is present with a small fraction of antiphase nanodomains. Moreover, at 340°C the spot-like intensity has vanished indicating that the R-phase has disappeared and a C-phase develops on the expense of the T-phase.

Figure 5.19. Experimental and simulated electron diffraction patterns at selected temperatures. The experimental hk0.5 reciprocal plane recorded at (a) RT, (b) 180°C and (c) 340°C. Simulated electron diffraction patterns along hk0.5 reciprocal plane for the structures at (d) RT, (e) 180°C and (f) 340°C.

5.4.4 Highlights of experimental results The in-situ TEM experiments of SSRs, FE domains and DS intensity have established three distinct regions.

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I. Region I from RT to ~100-150°C characterized by strong ½(ooo) SSRs and weak ½(ooe) SSRs. In this region a gradual decrease for ½(ooo) SSRs’s intensity is observed together with a slight increase in intensity for ½(ooe) SSRs. Furthermore, BF-images of FE domains reveal a complex domain configuration with herringbone-like domains and small lamellar domains showing that long-range polar order is present. The RED data showed that weak DS rods are present in the hk0.5 reciprocal plane confirming that the predominant phase in this region is the R-one characterized by antiphase (a–a–a–) tilting. Small traces of a T-phase are present as plate-like in-phase nanodomains (a0a0c+, a0c+a0 and c+a0a0) tilted about the three pseudo-cubic axes.

II. Region II expands from ~110-150°C to ~230-270°C and is char-acterized by strong ½(ooe) SSRs and weak ½(ooo) SSRs. The intensity of ½(ooe) SSRs slightly increases until ~180°C fol-lowed by a gradual decrease. On the other hand the intensity of ½(ooo) SSRs gradually decreases in the entire temperature range of region II. This reversed relative intensity indicates that a sub-tle local phase transition occurs and the dominant octahedral-tiltsystem changes from a–a–a– to a0a0c+. In this region we observe the disappearing of FE domains which indicates that the long-range polar order has been disrupted. Furthermore, the RED data confirms that the fraction of in-phase nanodomains has signifi-cantly increased as shown by the strong intensity of DS rods. In order to account for the gradual decrease of both types of SSRs with increasing temperature we propose that an additional C-phase is developed in this region. The predominant phase in re-gion II is the T-phase.

III. For temperatures higher than ~230-270°C region III is defined. In this region only weak ½(ooe) SSRs are observed that are gradually decreasing in intensity. The FE domains have com-pletely disappeared so no long-range polar order is present. The intensity of DS rods is also gradually decreasing suggesting that the fraction of the C-phase is increasing on the expense of the T-one. At high enough temperature we assume that a predominant C-phase will be reached.

5.4.5 Modeling of local structural disorder Models of local structural disorder were developed at selected temperatures based on the analysis of the temperature-dependence of SSRs, FE domains and electron DS. Since the main contribution to the experimental SSRs and

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DS rods is oxygen octahedral tilting, only this type of disorder was consid-ered during simulations. The same methodology as described for paper I was employed to develop the antiphase matrix and the in-phase nanodomains. The antiphase tilted matrix was created from nanodomains 6×6×6 unit cells large. The initial size of the of in-phase nanodomains variants was kept the same for all simulations and was: 6×6×2, 6×2×6 and 2×6×6 unit cells large. The parameter that was varied for simulated structures at different tempera-ture was the fraction of the antiphase, in-phase and non-tilted octahedra. At room temperature the structure is predominantly R so the final model struc-ture contained 70% antiphase tilted nanodomains and 30% in-phase tilted nanodomains. These values were estimated from the relative intensities of SSRs and DS features. Nevertheless, from our observations a quantitative estimation for the fraction of different phases is not possible. At 180°C the predominant phase is the T-phase so the fraction of in-phase nanodomains was increased to 70% and the fraction of antiphase nanodomains was de-creased to 20%. An additional C-phase was also included with a small frac-tion of 10%. At 340°C we assume that no traces of the R-phase are present so the in-phase nanodomains are embedded in a cubic matrix. The cubic matrix is basically the initial perovskite supercell. The ratio was 50% in-phase nanodomains and 50% cubic matrix. The distribution of antiphase, in-phase and non-tilted octahedra for different temperatures is shown in Fig. 5.20(b-d).

Figure 5.20. (a) Schematic representation of octahedral-tilt systems used during simulations. The non-tilted octahedra are designated yellow, antiphase tilted octahe-dra are black and the three variants of in-phase tilted octahedra are blue, green and red. Distribution of antiphase, in-phase and non-tilted octahedra for 95NBT-5BT ceramic at (b) RT, (c) 180°C and (d) 340°C. For clarity only the TiO6 octahedra are shown projected along the [001]pc direction.

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Figures 5.19(e-g) illustrate the simulated electron diffraction patterns along the hk0.5 reciprocal plane calculated from the model structures for RT, 180°C and 340°C. The correct morphology and intensity distribution along the DS rods was obtained for each temperature. However, since a quantita-tive analysis of DS intensity is not possible with the current data, slight dif-ferences between the simulated and experimental patterns can be observed. One reason is the limited size of the model crystal when compared with the grain size from which RED data were collected. Another factor is dynamical effects which strongly affect the intensity for thick samples. Nevertheless, the proposed structure models are able to account for the temperature-dependence of DS intensity. At room temperature we obtain weak elongated features and strong spot-like intensity along the DS rods similar to the exper-imental result. With increasing temperature, the intensity is reversed and the elongated features have stronger intensity than the spot-like intensity. At 340°C the spot-like intensity has disappeared and the intensity of the elon-gated features was reduced. In summary, our simulations suggest that in region I a predominantly R-phase coexists with a small fraction of T-phase present as plate-like in-phase nanodomains with a thickness of ~1 nm. The transition from region I to region II occurs through a subtle local phase tran-sition characterized by a change in the dominant octahedral-tilt system from antiphase to in-phase tilting. In region II the structure is predominantly T with small traces of R- and C-phases. In region III the fraction of the C-phase gradually increases at the expense of the T-phase. We propose that at high enough temperatures a predominant C-phase will be attained.

5.4.6 Summary In-situ TEM experiments for 95NBT-5BT ceramic have revealed a subtle local structural phase transition around the temperature at which FE domains disappear. This indicates that the depolarization mechanism is associated with this local phase transition. Simulations of disordered atomic structures showed that this transition is characterized by a change in the dominant oc-tahedral-tilt system from antiphase tilting to in-phase tilting. The transition occurs at the nanoscale where plate-like in-phase nanodomains tilted about the three pseudo-cubic axes are gradually developed until a critical volume fraction is reached. The in-phase nanodomains disrupt the long-range polar order and as a result the polarization is significantly decreased or even re-duced to zero. The RED data were able to unequivocally confirm the subtle local phase transition indicated by the temperature-dependence of SSRs. This work shows that structural studies of NBT-based ceramics need to go beyond the average structure in order to explain macroscopic physical prop-erties like the depolarization mechanism.

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6. xPbIn½Nb½O3-(100-x)PbMg⅓Nb⅔O3 (paper IV)

6.1 Literature overview Lead-based relaxors exhibit superior electromechanical and dielectric prop-erties which make them appealing for electronic applications19,25,26. In con-trast to normal ferroelectrics, they present a slim hysteresis loop with low remnant polarization and a high dielectric response which is strongly fre-quency dependent with a broad maximum in temperature72. Many studies have tried to tackle the nature of structure-property relationship in these ma-terials but up to date no general consensus has been reached. Nevertheless, most studies agree that local structural disorder is an inherent characteristic for Pb-based relaxors and is a key feature in determining their bulk response. B-site cation ordering is of paramount importance and there are many exam-ples, like PbSc½Ta½O3

77,147 and PbIn½Nb½O3148,149

, where the degree of chem-ical ordering can be tailored by thermal treatment leading to changes in the macroscopic ferroelectric properties. Another important factor is local atom-ic displacements. One popular description of local atomic displacements in Pb-based relaxors makes use of the concept of PNRs which are thought to be a few nm in size with the polarization parallel to high symmetry directions27,

28,150. Another physical model attributes the relaxor behavior to a dipolar glass state with randomly interacting polar nanodomains in the presence of random electric fields32,33.Due its technological importance, the average and local structures of the prototype relaxor PbMg⅓Nb⅔O3 (PMN) have been the topic of many studies. At room temperature PMN was reported to present a cubic (Pm-3m) long-range average structure with a ≈ 4.05 Å151,152. Another study proposed on the contrary a rhombohedral symmetry (R3m)153 for the long-range average structure of PMN. Moreover, recent investigations using synchrotron X-ray diffraction suggested that the average structure of PMN is best described by a phase coexistence of a local rhombohedral (R3m) phase dispersed in a paraelectric cubic (Pm-3m) matrix154. The lack of consensus over the long-range average structure of PMN is due to the complex local structure which significantly deviates from the average structure. The B-site in PMN is oc-cupied by Mg2+/Nb5+ which take a 1:2 stoichiometric ratio in order to achieve an average valence of 4+ for charge conservation. Differences in

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valence and ionic radii produce 1:1 CSRO which leads to local charge im-balances and strong random fields31,155-159. Another reason for the highly disordered local structure in PMN is the tendency of the A-site cation (Pb2+)to shift away from its average position160-162. Unlike other relaxors the degree of chemical ordering in PMN cannot be tailored through thermal treatment but only by cation substitution163,164. Hence, it is common to mix PMN with other complex perovskites in order to tailor its physical properties. For ex-ample, the degree of chemical order in PbIn½Nb½O3 (PIN) can be enhanced or reduced by thermal treatment. In its disordered state PIN behaves as a relaxor whereas in its ordered state it transforms into an antiferroelectric148,165. The room temperature structure of disordered PIN was reported to be rhombohedral (R3m)166 while the ordered PIN presents rock-salt cation ordering and is characterized by an orthorhombic symmetry (Pbam)149. The xPbIn½Nb½O3-(1-x)PbMg⅓Nb⅔O3 (xPIN-(1-x)PMN) system was first introduced by Lee and Kim167 and reported to have negligible hys-teresis demonstrating promising electrostrictive properties. Moreover, die-lectric and ferroelectric measurements revealed typical relaxor behav-ior168,169. Also several studies have shown that the degree of chemical order-ing in xPIN-(1-x)PMN solid solutions can be modified by thermal treatment leading to changes in the dielectric and electrostrictive properties93,170,171.

6.2 Local structure of 60PIN-40PMN ceramic This study is focused on the local structure of 60PIN-40PMN ceramic as revealed by 3D electron DS scattering. This particular composition was cho-sen as it was reported that chemically ordered nanosized domains are present in the structure resulting in strong random fields93. Hence, it is expected that the local structure in this composition deviates significantly from the average structure, which makes it a good candidate for electron DS experiments. Sharp ½(ooo) SSRs and the typical relaxor features like butterfly-shaped DS in the vicinity of h00 type reflections and ellipsoidal-shaped DS in the vi-cinity of hk0 type reflections were recorded. Here, we present a structural model comprising of CSRO on the B-site and local Pb-displacements to explain the observed SSRs and DS intensity.

6.2.1 Electron diffuse scattering The RED method was used to record 3D electron diffraction data from indi-vidual grains of 60PIN-40PMN ceramic. First, a careful analysis of individ-ual SAED patterns recorded during RED data collection was made. Reduced visibility for DS intensity was observed for patterns close to exact ZA condi-tion as can be seen in Fig. 6.1. For example close to [001]pc and [110]pc ZAs

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only traces of DS intensity can be distinguished (Fig 6.1(a, c)). In contrast, a few degrees away from exact ZA condition strong DS intensity can be ob-served in the vicinity of Bragg reflections (Fig 6.1(b, d)). We observed a similar trend in the case of 85NBT-10KBT-5BT ceramic. In order to over-come this problem we analyzed thin volume-sections centered with respect to a ZA of interest. The thickness of the volume-sections was around ~0.05 Å-1 in reciprocal space. This way the visibility of DS intensity is drastically improved making the interpretation of DS more straightforward.

Figure 6.1. SAED patterns recorded during RED data collection for 60PIN-40PMN ceramic. (a) SAED pattern close to [001]pc ZA showing very weak DS intensity. (b)SAED pattern recorded away from [001]pc ZA showing strong DS intensity, for example in the vicinity of 210 Bragg reflection. (c) SAED pattern recorded close to [110]pc ZA with almost no visible DS intensity and (d) SAED pattern recorded away from [110]pc ZA showing strong DS intensity in the vicinity of for example the 001 Bragg reflection.

Thin-volume sections centered along [001]pc [110]pc and [110]pc ZAs are shown in Fig. 6.1(a-c). The diffuse ring close to the direct beam is due to a thin amorphous layer produced as an artefact during TEM sample prepara-tion. Sharp SSRs of the type ½(ooo) are observed in the volume-sections centered along [110]pc and [112]pc ZAs, as depicted by white circles in Fig. 6.1(b, c). This indicates that 1:1 CSRO is present on the B-site which is shared by 3 different cations Nb5+/Mg2+/In3+. Furthermore, strong DS streaks

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are observed along <110>pc* directions passing through the fundamental

perovskite reflections. Analogous to other Pb-based relaxors27, 172 we observe in the volume-section centered along [001]pc ZA (Fig. 6.2a), the butterfly-shaped DS in the vicinity of h00 reflections and the ellipsoidal-shaped DS in the vicinity of hk0 reflections. A detailed view of these two DS features can be seen in Fig. 6.2d, for 100 and 210 reflections. Similar DS features are observed in the volume-section centered along [110]pc ZA (Fig. 6.2b) with butterfly-shaped DS for 00l type reflections and ellipsoidal-shaped DS for hkl type reflections. A stronger asymmetry can be seen for the DS in this volume-section as shown in Fig. 6.2d for 001 and 222 reflections.

Figure 6.2. Experimental thin volume-sections for 60PIN-40PMN ceramic cut from the 3D reciprocal space volume, centered along (a) [001]pc, (b) [110]pc and (c)[112]pc ZAs. (d) Details of diffuse scattering around selected Bragg reflections, as recorded in the raw SAED patterns, close to [001]pc, [110]pc and [111]pc ZAs.

Moreover, forbidden ½(ooe) SSRs seem to be present in the volume-sections shown in Fig. 6.2(a, b) as depicted by white arrows. However, a slice-by-slice analysis of the 3D reciprocal space volume reveals that the stronger intensity observed at position of ½(ooe) SSRs is actually due to the intersec-

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tion of <110>pc* DS streaks. This can be clearly seen in the hk0.1 and hh0.3

slices, Fig. 6.3 (b, d). Here, the advantage of analyzing 3D diffraction data is demonstrated.

Figure 6.3. Slice-by-slice analysis of the 3D reciprocal space volume for 60PIN-40PMN ceramic. Slices are centered along the (a) hk0, (b) hk0.1, (c) hh0, (d) hh0.3,(e) hhl and (f) hhl.2 reciprocal planes. The white squares in (e) and (f) highlight the “triangular”-shaped DS which is barely visible for hhl slice but can be clearly ob-served in the hhl.2 slice.

The DS intensity observed in the volume-section centered along [112]pc ZA seems at first quite different from the DS observed in the other two volume-sections. A strongly asymmetric “triangular”-shaped DS intensity is present in between Bragg peaks as indicated by white arrows (Fig. 6.2c). However, a slice-by-slice analysis revealed that these are actually traces of <110>pc

* DS streaks near Bragg reflections as viewed along [111]pc ZA (Fig. 6.3(e, f)). A detailed view of this DS feature is shown in Fig. 6.2d for 121 and 011 reflec-tions. The DS streaks along <110>pc

* directions are continuous in reciprocal space and they extend the entire way in between the Bragg peaks with stronger intensity in the close proximity of Bragg reflections and at positions of ½(ooe) SSRs. Although, the intensity of <110>* DS streaks can vary for different Bragg reflections and certain streaks may even be absent. The pres-ence of these DS streaks is an indication of correlated local atomic dis-placements in real space.

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6.2.2 Modeling of local structural disorder The initial supercell was based on the ideal perovskite structure (space group Pm-3m and a = 4.09 Å) with a total size of 50×50×50 unit cells large. Initial-ly, all atoms occupy the ideal positions as for the cubic perovskite structure with Pb2+ at (0, 0, 0), Nb5+/Mg2+/In3+ at (0.5, 0.5, 0.5) and O2- at (0, 0.5, 0.5). MC simulations were used to create CSRO on the B-site and correlated local atomic displacements. The short-range B-site cation ordering was developed using MC simulations with a simple nearest-neighbor-Ising (NNI) model. The degree of chemical order that can be induced with this model is limited by the systems’ stoichiometry. Moreover, since there is no energy gain from the next-nearest-neighbors being the same, large 1:1 chemical domains are avoided and only short-range chemical order is obtained. The obtained B-site cation distribution is shown if Fig. 6.4a. Small 1:1 chemical domains are present with a maximum size of ~35 Å. Occupational correlation coefficients were calculated for Nb-In, Nb-Mg and Mg-In atom pairs. Negative values were obtained for all atoms pairs namely -0.65 for Nb-In, -0.45 for Nb-Mg and -0.5 for Mg-In. This shows that the nearest-neighbor sites tend to be occupied by different atom types.

Figure 6.4. Local structure of 60PIN-40PMN ceramic. (a) Distribution of B-site cations in the xy plane with black color designating Nb atoms, blue color In atoms and red color Mg atoms. (b) Black arrows illustrate local displacements of Pb atoms away from their average Pm-3m positions.

The main contribution to the experimental DS streaks comes from local Pb-displacements. So, during the MC simulations only Pb atoms were consid-ered. The B-site cations and O atoms were kept at their average positions. Pb-displacements were created using a simple harmonic potential with dis-placements limited to <110>pc and <111>pc directions. Figure 6.4.b illus-trates the local atomic displacements of Pb atoms (black arrows) away from their average positions. No separate domains or PNRs are observed but ra-ther a system that is more akin to a dipolar glass state. This is in good agreement with previous work reported for pure PMN31,173. The average atomic shifts of Pb atoms along <110>pc and <111>pc directions were 0.15 Å

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and 0.08 Å, respectively. Correlation coefficients were calculated for <001>pc (nearest-neighbors) and <110>pc (next- nearest-neighbors) direc-tions. Positive values were obtained of 0.289 for nearest-neighbors and 0.266 for next- nearest-neighbors. This suggests that Pb atoms have a small ten-dency to shift in the same direction but overall no preferred directions for the atomic shifts were observed.

Figure 6.5. Simulated electron diffraction volume-sections projected along (a)[001]pc, (b) [110]pc and (c) [112]pc reciprocal planes. (d) Comparison between exper-imental and simulated DS intensity around selected Bragg reflections from [001]pc

and [110]pc ZAs.

Electron diffraction patterns were calculated from a structure containing both CSRO and local Pb-displacements. To account for the fact that we are ana-lyzing thin volume-sections cut from the experimental reciprocal space vol-ume the patterns shown in Fig. 6.5 were calculated by integrating 19 electron diffraction patterns slightly above and below [001]pc, [110]pc and [112]pcZAs. A discrete step of 0.01 reciprocal space units (~0.0024 Å-1) was used. For example, the diffraction pattern shown in Fig. 6.5a represents the sum of

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an electron diffraction pattern along [001]/hk0 ZA, 8 diffraction patterns slightly above (hk0.01, hk0.02 etc.) and 8 diffraction patterns slightly below (hk-0.01, hk-0.02 etc.). A visual inspection reveals that a good agreement was obtained between the experimental and simulated patterns, especially for the low order volume-sections. All ½(ooo) SSRs are accounted for and continuous DS streaks along <110>pc

* directions are observed for the vol-ume-sections projected along [001]pc and [110]pc ZAs (Fig. 6.5(a, b)). Stronger DS intensity is obtained in the vicinity of Bragg reflections and at the intersection of DS streaks, as depicted by white arrows in Fig. 6.5(a, b). Both butterfly-shaped and the ellipsoidal-shaped DS features are obtained with our model as can be seen in Fig. 6.5d, where a comparison between the experimental and simulated DS intensity in the vicinity of selected Bragg reflections is shown. Slight differences between the experimental and simu-lated data can be observed for the volume-section centered along [112]pc ZA (Fig. 6.5c) especially for the region close to the origin. These differences diminish further away from the direct beam. Nevertheless, the most signifi-cant diffraction features are explained by the current simple model compris-ing of 1:1 B-site cation ordering and local Pb-displacements.

6.2.3 Summary By means of the RED technique high-quality 3D electron DS data were rec-orded from individual grains of 60PIN-40PMN ceramic. We were able to study the butterfly-shaped and the ellipsoidal-shaped DS features character-istic for Pb-based relaxors using electron diffraction rather than synchrotron radiation. The electron diffraction data were used as input for simulations of disordered structure and we were able to explain all significant diffraction features with a structural model comprising of 1:1 B-site cation ordering and local Pb-displacements. Electron diffraction patterns were calculated along several reciprocal planes and compared with the experimental data. In agreement with other recent studies of PMN, PMN-PT and PZN, we showed that the local structure in 60PIN-40PMN is more akin to a dipolar glass state rather than separate PNRs.

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7. Conclusions

Electron diffraction is a powerful tool to study local structural disorder. It was proved that high-quality 3D electron DS can be recorded using the RED method in the case of perovskite ferroelectric ceramics. The strong electron-matter interaction facilitates recording of weak DS intensity that can be col-lected from individual grains in a transmission electron microscope. A de-tailed study of the morphology and intensity of DS features was carried out by analyzing the reconstructed reciprocal space volume in 3D or by cutting slices/volume-sections centered along several orientations.

This thesis is concerned with studying the local structure of two perovskite ferroelectric systems by means of 3D electron DS. It is important to not limit the collection of electron diffuse scattering intensity to zone axis studies as it was shown that certain DS features are not visible at zone axis condition. A 3D reciprocal space volume enabled us to separate certain features which facilitated the interpretation of DS intensity. The information from reciprocal space was translated into real space by means of modeling of disordered atomic structures. The calculated electron diffraction patterns along several orientations and the comparison with experimental data verify the proposed structural disorders. The most common mechanisms of structural distortion which modify the basic perovskite structure are octahedral tilting, cation site ordering and off-center displacements of atoms. For the (95-x)Na0.5Bi0.5TiO3-xK0.5Bi0.5TiO3-5BaTiO3 system all three mechanisms lead to local structural changes. Moreover, octahedral tilting is dependent on both A-site cation substitution and temperature. Based on electron DS data we were able to prove that octa-hedral tilting is present in the structure as plate-like in-phase nanodomains tilted about the three pseudo-cubic axes embedded in an antiphase tilted matrix. On the other hand for the xPbIn½Nb½O3-(100-x)PbMg⅓Nb⅔O3 sys-tem only cation site ordering and off-center displacements of atoms contrib-ute to the observed DS intensity. In this case we demonstrated that Pb-displacements lead to a local structure which resembles more a dipolar glass state rather than separated polar nanoregions. Even though the long-range average structure of these two systems can be described as a distorted cubic perovskite structure, the local structures are fundamentally different. This proves that structural studies need to reconcile both the average and local structures for a realistic description of the 3D structure.

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8. Future perspectives

The methodology used in this thesis can be employed for a wide variety of crystalline materials that present disordered structures. The only requirement is that they can withstand TEM sample preparation without significant struc-tural changes. This way, high-quality three-dimensional electron diffraction volumes containing both Bragg reflections and diffuse scattering intensity can be recorded even for polycrystalline materials, such as ceramics. A three-dimensional reciprocal space volume contains a wealth of information that is just not available in conventional selected-area electron diffraction patterns. Studies of local structural disorder play an important role in under-standing the structure-property relationship in the case of ferroelectric perov-skite ceramics. The analysis of electron diffuse scattering in 3D provided the necessary information for building a model of the local structural disorder present in these materials. A better understanding of the local structure in the case of perovskite ferroelectrics can lead to a better understanding of their macroscopic properties. These properties could then be deliberately tailored by changing the local structure or by synthesizing new materials that contain a desired type of disorder.

Even though a lot of information can be retrieved from electron diffuse scat-tering intensity the current analysis is only semi-quantitative due to dynam-ical scattering. Both data collection and processing strategies need to be im-proved to obtain more reliable electron diffuse scattering intensities that can then be used as input for refinement strategies. However, the structural mod-el obtained from analyzing electron diffuse scattering in 3D can be used for example, as an additional input in refining PDF data. Another issue is that there is no unique solution strategy that works for all kinds of disorder. This led to the fact that the modelling of single-crystal diffuse scattering relies quite heavily on an investigator with wide experience of diffuse scattering and disorder. Further work needs to be carried out in order to reach a more user-friendly approach for studying disordered atomic structures.

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Populärvetenskaplig sammanfattning

Evolutionen av den mänskliga civilisationen är tätt förknippat med förståel-sen av sambandet mellan materials egenskaper och struktur och användandet av denna kunskap för att ta fram nya material med bättre egenskaper. På stenåldern användes mestadels naturmaterial som sten, lera och trä följt av brons när bronsåldern började ca 3000 f.Kr. i och med att koppar upptäcktes. Nästa steg var upptäckten av järn och stål början på järnåldern omkring 1200 f.Kr. Idag lever vi i informationsåldern. En utveckling som inte skulle ha varit möjligt utan betydande framsteg inom materialvetenskapen. Ett steg var upptäckten att kisel och germanium kan användas som halvledarmaterial vilket spelade en avgörande roll vid den miniatyrisering av elektroniska komponenter som möjliggjort informationssamhället.

För att förstå materialens unika egenskaper måste man först förstå deras struktur på mikroskopisk och atomär nivå. Rent generellt kan strukturen hos ett ideal kristallint material beskrivas med ett perfekt 3D-nätverk av atomer. I verkligheten förekommer alltid avvikelser från denna genomsnittliga struk-tur och denna lokala struktur är ofta väldigt viktig för materialets egenskap-er. I denna avhandling undersöktes en viss materialklass, nämligen ferroe-lektriska keramer med perovskitstruktur. Deras unika egenskaper som ferroelektricitet, piezoelektricitet och elektrostriktion gör att dessa material används i tillämpningar som anordningar för energiskörd (omvandling av lågvärdig energi till elektricitet), icke-flyktiga minnen, transduktorer och medicinsk bildbehandling. Transmissionselektronmikroskopi användes för att studera atomstrukturen hos flera Bi- och Pb-baserade ferroelektriska perovskit-keramer. Ett särskilt fokus har legat på att studera hur den lokala strukturen hos dessa material avviker från den genomsnittliga. Detta har gjorts genom att jämföra den diffusa spridning i elektrondiffraktionsmönster insamlade med 3-dimensionell rotationselektrondiffraktiontomografi med simuleringar av oordnade atomstrukturer. Modellering av lokal strukturell oordning kan hjälpa oss att förstå hur den lokala strukturen är korrelerad med materialets fysikaliska egenskaper.

De vanligaste förekommande mekanismerna för strukturell distorsion, frånden ideala kubiska perovskitstrukturen, är lutning av oktaedrarna, kationord-ning och off-center förskjutningar av atomer. I avhandlingen har modeller tagits fram och utvecklats för att beskriva lokal strukturell oordning hos

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olika perovskiter, baserade på de tre mekanismerna, med hjälp av simule-ringar. En komplex strukturell modell för att beskriva den lokala oordningenhos keramen 85Na0.5Bi0.5TiO3-10K0.5Bi0.5TiO3-5BaTiO3 (piezoelektrisk) togs fram genom att analysera form och intensiteten hos den diffusa spridningen i 3-dimensionella elektrondiffraktionsmönster. Därefter studerades hur kaliu-minnehållet inverkar på den oordnade oktaedriska lutningen för tre olika piezokeramer genom en kombination av mörkfältsavbilning och elektrondiffraktion. Vidare avslöjades ett temperaturberoende hos den dif-fusa elektronspridningen, en lokal strukturfasövergång för piezokeramen 95Na0.5Bi0.5TiO3-5BaTiO3, som kunde korreleras med depolarisationsmekan-ismen. Depolarisationsmekanismen innebär för Na0.5Bi0.5TiO3-baserade ke-ramer att piezoelektriciteten minskas signifikant eller helt försvinner vid uppvärmning över ~180°C, (depolarisationstemperaturen), trots att ingen tydlig strukturell fasövergång observeras nära depolarisationstemperaturen. Slutligen studerades den starka diffusa elektronspridning från en Pb-baserad relaxor med perovskittypstruktur, där simuleringar av oordnade atomstruk-turer visade att den lokala strukturen liknar ett di-polärt glas tillstånd. En relaxor tillhör till en viss klass av ferroelektriska material som visar enastå-ende dielektriska och elektromekaniska egenskaper. I motsats till vanliga ferroelektriska material uppvisar de låg restpolarisation och en hög dielekt-riskt respons som är starkt frekvensberoende med ett brett temperaturmaxi-mum. Eftersom inga makroskopiska ferroelektriska domäner har observerats har idén om polära nanodomäner introducerats för att förklara relaxorbete-endet. Men den här studien har visat att den lokala strukturen liknar mer ett di-polärt glas tillstånd och inga polära nanodomäner observerades.

Studierna som presenterats i denna avhandling visar att elektrondiffraktion är ett mycket kraftfullt verktyg för studier av lokal strukturell oordning ikristallina material, speciellt för keramer. Detta har varit möjligt eftersom det nu går att samla 3-dimensionella elektrondiffraktionsdata från enskilda kristaller (korn) i keramen. Eftersom 3-dimensionell volym analyseras kan flera orienteringar studeras samtidigt och man är inte begränsad till nollte ordningens Laue-zoner av typen hkl där l, k eller l = 0. Modellerna för lokal strukturell oordning har gett en värdefull inblick i hur makroskopiska egen-skaper påverkas av lokal strukturell oordning, vid sidan av den genomsnitt-liga strukturen.

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Acknowledgments

First of all I would like to thank my supervisor Dr. Cheuk-Wai Tai who in-troduced me to a fascinating new world viewed through the transmission electron microscope. You helped me become an independent researcher and always encouraged me to pursue my own ideas regardless they were bad or good. I will forever be grateful for your never ending patience and availabil-ity throughout my five years as a PhD student. Without your tremendous knowledge and guidance this thesis would not have been possible.

Special thanks go to my assistant supervisor Prof. Gunnar Svensson for fruit-ful discussions and constant encouragement to study Swedish. Thank you for helping me with the Swedish summary for this thesis.

I would also like to thank my former supervisor in Romania Prof. Liliana Mitoseriu and her group. You were my very first role model as a scientist and I always found your love for research inspiring. You believed in me and encouraged me to go and study abroad and for this I will forever be grateful.

My appreciation goes to all professors and senior researchers (past and pre-sent) at MMK for all the interesting discussions and courses that contributed to my development as a scientist and as a person: Wei Wan, Xiaodong Zou, German Salazar Alvarez, Niklas Hedin, Ulrich Häussermann, Jekabs Grins, Lars Eriksson, Sven Hovmöller, Mats Johnsson, Osamu Terasaki, Junliang Sun, Kjell Jansson, Tom Willhammar, Trung-Dung Tran, Andrew Kentaro Inge, Thomas Thersleff, Hongyi Xu, Lennart Bergström, Mattias Eden, Ar-nold Maliniak, Josef Kowaleski, Anja-Verena Mudring, Aji Mathew and James Zhinjian Shen.

Special thanks to Ken and Thomas for proof reading my thesis and for their helpful feedback.

I offer my appreciation to the entire administrative staff at MMK for all theirhelp and their very effective way of solving any problem: Tatiana Bulavina,Helmi Frejman, Ann Loftsjö, Daniel Emanuelsson, Camila Berg, Rolf Eriks-son, Anne Ertan, Hans-Erik Ekström and Per Jansson.

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To all my friends at MMK (past and present) I cannot thank you enough for making my time as a PhD student colorful and exciting: Neda, Dickson, Yuan, Eva, Elina, Ning, Yifeng, Magda, Hani, Viktor, Mylad, Eleni, Stef, Ken, Yulia, Arnaud, Yunxiang, Tom, Thomas, Fei, Junzhong, Ahmed, Reji, Laura, Yang, Henrik, Erik, Bin, Taimin, Jingjing, Jonas, Varvara, Fredrik, Mirva, Korneliya, Valentina, Przemyslaw, Amber, Chris, Inna, Istvan, Zhehao, Yunchen, Sumit, Kristina, Ekaterina, Leifeng, Peng, Alfredo, Arto,Fabian, Anne, Dariusz, Jon, Jie. Listing all of these people I am humbled and overwhelmed of how many wonderful people I met during my doctoral stud-ies and hope I did not forget anyone.

Last but not least I am grateful to my family and friends for their constant support and love. I am who I am thanks to my family and without them I would not be here. Special thanks, to my life partner Mateusz who has al-ways been by my side throughout this entire journey both for the happy and difficult moments. I cannot thank you enough for all the encouragement and love that helped me reach this moment.

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