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Nanoscale Ferroelectrics and Multiferroics Key Processing and Characterization Issues, and Nanoscale Effects Edited by Miguel Algueró J. Marty Gregg Liliana Mitoseriu VOLUME 1

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Nanoscale Ferroelectrics

and MultiferroicsKey Processing and Characterization Issues,

and Nanoscale Effects

Edited byMiguel AlgueróJ. Marty Gregg

Liliana Mitoseriu

VOLUME 1

Nanoscale Ferroelectrics

and MultiferroicsKey Processing and Characterization Issues,

and Nanoscale Effects

Edited byMiguel AlgueróJ. Marty Gregg

Liliana Mitoseriu

VOLUME 2

Nanoscale Ferroelectricsand Multiferroics

Nanoscale Ferroelectricsand Multiferroics

Key Processing and Characterization Issues,and Nanoscale Effects

VOLUME I

Edited by

MIGUEL ALGUEROJ. MARTY GREGG

LILIANA MITOSERIU

This edition first published 2016© 2016 John Wiley & Sons Ltd

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Library of Congress Cataloging-in-Publication Data

Names: Alguero, Miguel, 1969- editor. | Gregg, J. Marty (John Marty), 1969- editor. | Mitoseriu, Liliana, editor.Title: Nanoscale ferroelectrics and multiferroics : key processing and characterization issues, and nanoscale effects / [compiled

by] Miguel Alguero, J. Marty Gregg, Liliana Mitoseriu.Description: Hoboken : John Wiley & Sons, Inc., 2016. | Includes bibliographical references and index.Identifiers: LCCN 2015042479 (print) | LCCN 2015046191 (ebook) | ISBN 9781118935750 (cloth) | ISBN 9781118935675

(ePub) | ISBN 9781118935705 (Adobe PDF)Subjects: LCSH: Ferroelectric devices. | Nanoelectromechanical systems.Classification: LCC TK7872.F44 N36 2016 (print) | LCC TK7872.F44 (ebook) | DDC 621.381–dc23LC record available at http://lccn.loc.gov/2015042479

A catalogue record for this book is available from the British Library.

ISBN: 9781118935750

Set in 10/12pt Times by Aptara Inc., New Delhi, India.

1 2016

Contents

VOLUME I

List of Contributors xviiPreface xxiii

Introduction: Why Nanoscale Ferroelectrics and Multiferroics? xxvMiguel Alguero, J. Marty Gregg, and Liliana Mitoseriu

PART A 1

Nanostructuring: Bulk

1 Incorporation Mechanism and Functional Properties of Ce-DopedBaTiO3 Ceramics Derived from Nanopowders Prepared by the ModifiedPechini Method 13Adelina-Carmen Ianculescu, Daniela C. Berger, Catalina A. Vasilescu, MariusOlariu, Bogdan S. Vasile, Lavinia P. Curecheriu, Andreja Gajovic, andRoxana Trusca

1.1 Why Cerium-Doped BaTiO3? 131.2 Sample Preparation, Phase and Nano/Microstructural Characterization 18

1.2.1 Powders 181.2.2 Ceramics 22

1.3 Dielectric Properties 301.4 Raman Investigation 371.5 Conclusions 39Acknowledgments 40References 40

2 Synthesis and Ceramic Nanostructuring of Ferroic and MultiferroicLow-Tolerance-Factor Perovskite Oxides 44Teresa Hungria, Covadonga Correas, and Alicia Castro

2.1 Introduction 442.2 Synthesis of Perovskites Oxides 47

vi Contents

2.2.1 General Information on the Preparation of Low-tolerance-FactorPerovskite Oxides 47

2.2.2 Mechanochemical Activation and Mechanosynthesis Applied toLow-Tolerance-Factor Perovskite Oxides 48

2.3 Processing of Ferroic and Multiferroic Materials: From the CeramicMethod to the Current Assisted Methods 542.3.1 SPS Applied to the Preparation of Ferroic Ceramic Materials 60

2.4 Combination of Mechanosynthesis and Spark Plasma Synthesis: TheRight Track to the Nanoscale in Ferroic Materials 62

2.5 Conclusions 63Acknowledgments 64References 64

3 Core–Shell Heterostructures: From Particle Synthesis to Bulk Dielectric,Ferroelectric, and Multiferroic Composite Materials 72Vincenzo Buscaglia and Maria Teresa Buscaglia

3.1 Introduction 723.2 Liquid-Phase Synthesis of Core–Shell Particles 74

3.2.1 Controlled Assembly of Preformed Nanoparticles 743.2.2 Precipitation 75

3.3 BaTiO3@polymer Particles and Composites 773.4 Inorganic Core–Shell Particles with a Ferroelectric Core 783.5 Multiferroic Core–Shell Particles and Composites 833.6 Conclusions and Outlook 90References 92

4 Modeling of Colloidal Suspensions for the Synthesis of the FerroelectricOxides with Complex Chemical Composition 100Gregor Trefalt, Bosiljka Tadic, and Barbara Malic

4.1 Introduction 1004.2 Solid-State Synthesis 1024.3 Colloidal Interactions and Aggregation 103

4.3.1 DLVO Theory 1034.3.2 Modeling of Aggregate Formation and Their Structure 105

4.4 Aggregation in the Three-Component System 1064.5 Applying the Modeling Results to Enhance Properties of Ferroelectric

Complex Oxides 1094.5.1 PMN Synthesis by Controlled Aggregation of Reagent

Particles 1094.5.2 PMN–PT Synthesis by Controlled Aggregation of Reagent

Particles 1104.6 Conclusions and Outlook 112Acknowledgments 114References 114

Contents vii

Nanostructuring: Thin Films

5 Self-Assemblage and Patterning of Thin-Film Ferroic Nanostructures 118Theodor Schneller

5.1 Introduction 1185.2 Short Survey on Classical Top-Down Approaches 1195.3 Non-invasive Procedures 121

5.3.1 Nanomask Techniques 1245.3.2 Direct Self-Assembly of Ferroelectric Nanostructures 1265.3.3 Nanoseeds Assisted Growth 131

5.4 Embedded Ferroelectric Nanograins 1355.4.1 Embedded Lead Titanate-Based Nanograins 1365.4.2 Further Embedding Concepts 137

5.5 Conclusions 139References 140

6 Thin-Film Porous Ferroic Nanostructures: Strategies andCharacterization 147Alichandra Castro, Paula Ferreira, Stella Skiadopoulou, Liliana P. Ferreira,Margarida Godinho, Brian J. Rodriguez, and Paula M. Vilarinho

6.1 (Nano)Microelectronics Considerations 1476.2 Ferroics and Nanoferroic Structures 1486.3 Meso- and Nanoporosity in Advanced Functional Materials 1516.4 Nanoporosity in Ferroelectric Thin Films 153

6.4.1 Lead Titanate (PbTiO3) – Nanoporous and Nanopatterned ThinFilms 153

6.4.2 Lead-Free Ferroelectric and Single-Phase Multiferroic – BismuthFerrite 156

6.5 Looking Ahead 158Acknowledgments 160References 160

7 Low-Temperature Photochemical Solution Deposition of Ferroelectricand Multiferroic Thin Films 163Christopher De Dobbelaere, An Hardy, Marlies K. Van Bael, Inigo Bretos,Ricardo Jimenez, and M. Lourdes Calzada

7.1 Introduction 1637.2 Low-Temperature Processing of Ferroelectrics and Multiferroics 1657.3 Low Environmental Impact Precursor Chemistry 1657.4 Aquous Solution–Gel Precursors 166

7.4.1 Metal Ions in Water 1667.4.2 Gel Formation and Decomposition 1697.4.3 UV Activity 170

viii Contents

7.5 Photosensitive Sol–Gel Precursors 1717.5.1 Addition of UV-Absorbing Molecules to the Precursor Solution 1727.5.2 Synthesis of Highly Photosensitive Metal Complexes 175

7.6 Photochemical Solution Deposition (PCSD) 1787.6.1 Working Principle 1787.6.2 Low-Temperature Ferroelectric Oxide Films Prepared from

Photosensitive Sol–Gel Precursors 1827.6.3 Low-Temperature Ferroelectric and Multiferroic Oxide Films

Prepared from Aqueous Photochemical Solution Deposition(Aqueous PCSD) 185

7.7 Final Remarks 190References 191

Nanostructuring: Fibers and Wires

8 Synthesis and Properties of Ferroelectric Nanotubes and Nanowires:A Review 200Vincenzo Buscaglia and Maria Teresa Buscaglia

8.1 Introduction 2008.2 Synthesis of Ferroelectric Nanowires and Nanotubes 202

8.2.1 Direct Growth from the Liquid Phase 2028.2.2 Templated Growth 2088.2.3 Topochemical Solid-State Reactions 2118.2.4 Electrospinning 2138.2.5 Focused Ion Beam Lithography 213

8.3 Crystal Structure, Phase Transitions, and Ferroelectric Properties 2138.4 Applications 219

8.4.1 Nanogenerators for Energy Harvesting and Other Devices 2198.4.2 Dielectric Composites and Other Applications 221

8.5 Summary and Conclusions 222References 223

9 Fabrication of One-Dimensional Ferroelectric Nano- and Microstructuresby Different Spinning Techniques and Their Characterization 232Tony Lusiola and Frank Clemens

9.1 Introduction 2329.2 Fiber Synthesis 233

9.2.1 One-Dimensional Nanostructures 2339.2.2 One-Dimensional Microstructures 236

9.3 Investigation of Nanostructure, Microstructure, and Phase Composition 2409.3.1 Microscopy 2419.3.2 Phase Composition 243

9.4 Investigation of Mechanical Properties 2459.4.1 Characterization of Nanowires/Fibers 2469.4.2 Characterization of Microfibers 248

Contents ix

9.5 Ferroelectric Characterization 2499.5.1 Characterization of Nanowires/fibers 2509.5.2 Characterization of Microfibers 252

9.6 Applications 2559.6.1 Potential Applications for Nanowires 2559.6.2 Applications for Microfibers 258

9.7 Summary and Outlook 259References 260

PART B 269

Characterization (of the Nanostructured Materials): Crystal Structure

10 Structural Characterization of Ferroelectric and MultiferroicNanostructures by Advanced TEM Techniques 275Etienne Snoeck, Axel Lubk, and Cesar Magen

10.1 Introduction: Advanced TEM Techniques for the Analysis ofFerroelectric and Multiferroic Materials 275

10.2 Transmission Electron Microscopy 27610.2.1 TEM and HRTEM Basics 27610.2.2 TEM and HRTEM Imaging of Ferroelectrics 280

10.3 HAADF–STEM Structure Determination 28810.3.1 Basics of HAADF–STEM Imaging 28910.3.2 Solid-State Properties 29410.3.3 Case Studies 297

10.4 Electron Energy Loss Spectroscopy 30710.4.1 EELS Basics 30710.4.2 EELS Applications in the Characterization of Multiferroics 310

10.5 Future and Challenges 317References 319

11 Raman Spectroscopy of Nanostructured Ferroelectric Materials 325Marco Deluca and Andreja Gajovic

11.1 Introduction 32511.2 Raman Spectroscopy Fundamentals 326

11.2.1 The Raman Effect 32611.2.2 Selection Rules 32811.2.3 Phase Transitions 329

11.3 Raman Analysis of Relaxors 33111.3.1 Lead-Based Relaxors 33311.3.2 Lead-Free Relaxors 34011.3.3 Wrap-Up: Raman Analysis of Relaxors 351

11.4 Raman Analysis of Nanostructured Ferroelectrics 35211.4.1 Ferroelectric Nanostructured Ceramics 353

x Contents

11.4.2 Ferroelectric 2D Nanostructures 35711.4.3 Wrap-Up: Raman Analysis of Nanostructured Ferroelectrics 358

11.5 Tip-Enhanced Raman Spectroscopy (TERS) of NanoscaleFerroelectrics 359

11.6 Summary 362Acknowledgments 363References 363

12 Neutron and Synchrotron X-Ray Scattering Studies of Bulk andNanostructured Multiferroic and Ferroelectric Materials 375Evagelia G. Moshopoulou, Pascale Foury-Leylekian, Katharine Page,Celine Doubrovsky, Martha Greenblatt, and Alan J. Hurd

12.1 Introduction 37512.2 Synchrotron X-Ray and Neutron Facilities for Structural

Characterization of Ferroelectrics and Multiferroics 37812.3 Crystal Structure of NdMn2O5 in Powder and Single Crystalline Forms 381

12.3.1 Multiferroelectricity in RMn2O5 Compounds 38112.3.2 Synthesis of NdMn2O5 38312.3.3 Synchrotron X-Ray Powder Diffraction Data Collection 38412.3.4 Synchrotron X-Ray Powder Diffraction Data Analysis 38412.3.5 Results 38512.3.6 Discussion 385

12.4 Neutron Total Scattering Investigations of the BaTiO3@SiO2Nanocomposites 39012.4.1 Neutron and Synchrotron X-Ray Probes for Nanostructure

Determination 39012.4.2 Ferroelectricity in BaTiO3 39212.4.3 Total Neutron Scattering Data Collection 39312.4.4 Total Neutron Scattering Data Analysis 39412.4.5 Results and Discussion 394

12.5 Concluding Comments 395Acknowledgments 395References 396

Characterization (of the Nanostructured Materials): Domains

13 Advanced Characterization of Multiferroic Materials by Scanning ProbeMethods and Scanning Electron Microscopy 400Michael R. Koblischka and Anjela Koblischka-Veneva

13.1 SPM-Related Methods in Advanced Materials Research 40013.2 Magnetic Force Microscopy (MFM) and Related Methods 403

13.2.1 Magnetic Imaging 40313.2.2 Magnetic Force Microscopy (MFM) 40613.2.3 Further Developments (HF-MFM, MRFM) 406

Contents xi

13.3 Electrostatic Force Microscopy (EFM) and Piezo Force Microscopy(PFM) 40813.3.1 Electrostatic Force Microscopy (EFM) 40813.3.2 Piezo Force Microscopy (PFM) 408

13.4 Scanning Tunneling Microscopy (STM) and Related Methods 40913.5 Imaging of Crystallographic Orientations (SEM/EBSD) 412

13.5.1 EBSD Technique 41213.5.2 Examples of EBSD Measurements on Multiferroics 41613.5.3 EBSD Analysis of a Composite Multiferroic 41813.5.4 Future Directions 425

13.6 New Developments in the Field of EBSD 42613.6.1 3D EBSD 42613.6.2 Transmission EBSD (t-EBSD) 427

References 428

14 Electrostatic and Kelvin Probe Force Microscopy for Domain Imagingof Ferroic Systems 435Brian J. Rodriguez

14.1 Introduction 43514.2 Electrostatic Force Microscopy and Kelvin Probe Force Microscopy 438

14.2.1 Electrostatic Force Microscopy 43814.2.2 Kelvin Probe Force Microscopy 438

14.3 EFM of Ferroelectric Materials 43914.4 KPFM of Ferroelectric Materials 44114.5 Recent Advances 445

14.5.1 Open-Loop KPFM 44514.5.2 Time-Resolved KPFM 447

14.6 Summary and Outlook 447Acknowledgments 448References 448

VOLUME II

List of Contributors xviiPreface xxiii

Introduction: Why Nanoscale Ferroelectrics and Multiferroics? xxvMiguel Alguero, J. Marty Gregg, and Liliana Mitoseriu

PART C 461

Nanoscale Effects: Bulk Properties

xii Contents

15 Nanostructured Barium Titanate Ceramics: Intrinsic versus ExtrinsicSize Effects 473Liliana Mitoseriu and Lavinia P. Curecheriu

15.1 Introduction 47315.2 Applications of BaTiO3 Ceramics; Actual Demands for Passive

Components 47615.3 Size-Dependent Phenomena in Ferroelectrics 478

15.3.1 General Properties 47815.3.2 Theoretical Description of Size-Dependent Phenomena in

Ferroelectrics 47915.4 Size-Dependent Properties in BaTiO3 Ceramics: Intrinsic versus

Extrinsic Size Effects 48615.4.1 Size-Dependent Properties of BaTiO3 Ceramics with Grain Size

in the Range of Micrometers 48615.4.2 Size-Dependent Properties of BaTiO3 Nanoceramics 487

15.5 Conclusions 501Acknowledgments 501References 501

16 The Effects of Ceramic Nanostructuring in High-SensitivityPiezoelectrics 512Harvey Amorın, Ricardo Jimenez, Jesus Ricote, Alicia Castro, andMiguel Alguero

16.1 Technological Drive for Ceramic Nanostructuring of High-SensitivityPiezoelectrics 512

16.2 State of the Art Ferroelectric Materials for Piezoelectric Technologies 51416.2.1 Commercial, PZT-Based Hard and Soft Piezoceramics 51416.2.2 Novel High-Sensitivity Piezoelectrics 516

16.3 Nanostructuring Effects in Perovskite Systems with PZT-like MPBs 51916.3.1 Basic Phenomenology 51916.3.2 Grain Size Dependence of Room Temperature Properties 52516.3.3 Grain Size Against Boundary Effects 533

16.4 Nanostructuring Effects in Perovskite Systems with Intrinsic ChemicalInhomogeneity 540

16.5 Summary and Future Perspectives 545Acknowledgments 546References 547

17 Correlation between Microstructure and Electrical Properties ofFerroelectric Relaxors 554Jelena D. Bobic, Jan Macutkevic, Robertas Grigalaitis, Maksim Ivanov,Mirjana M. Vijatovic Petrovic, Juras Banys, and Biljana D. Stojanovic

17.1 Introduction 55417.2 Perovskite Relaxors 558

Contents xiii

17.2.1 Nanosized PMN Powders and Nanostructured Ceramics 55817.2.2 Nanostructured PbSc 1

2Nb 1

2O3 Ceramics 566

17.3 Aurivillius Lead-Free Ferroelectric Relaxors 57017.3.1 Relaxor BaBi4Ti4O15 Ceramic System 57217.3.2 Grain Size-Dependent Electrical Properties of BaBi4Ti4O15

Ceramics 57417.3.3 BaBi4Ti4O15 Doped Ceramics 577

17.4 Conclusions 580References 581

18 Local Field Engineering Approach for Tuning Dielectric andFerroelectric Properties in Nanostructured Ferroelectrics andComposites 588Leontin Padurariu and Liliana Mitoseriu

18.1 Introduction 58818.2 Finite Element Method 59018.3 The Role of Local Electric Field Inhomogeneity on Tunability

Properties 59318.3.1 Nanostructured Ferroelectric Ceramics 59618.3.2 Porous Ferroelectric Ceramics 59918.3.3 Composites with Non-linear Matrix and Conductive

Inclusions 60018.4 Switching Properties in Inhomogeneous Ferroelectrics 603

18.4.1 Classical Preisach Model 60318.4.2 Switching in Nanostructured Ferroelectric Ceramics 60818.4.3 Switching in Porous Ferroelectrics 609

18.5 Conclusions 610Acknowledgments 611References 611

Nanoscale Effects: Thin-Film Properties

19 Ferroelectric Phase Transitions in Epitaxial Perovskite Films 617Marina Tyunina

19.1 Introduction 61719.2 Experimental Study of Phase Transition 618

19.2.1 Bulk Ferroelectrics and Relaxors 61819.2.2 Epitaxial Thin Films 622

19.3 Dielectric Permittivity in Thin Films 62319.3.1 Thin-Film Capacitor 62319.3.2 Derivative of Inverse Permittivity 62719.3.3 Thermo-optical Behaviour 634

19.4 Phase Transitions in Epitaxial Films 63519.4.1 Pb0.5Sr0.5TiO3 63519.4.2 BaTiO3 637

xiv Contents

19.4.3 KTaO3 63819.4.4 PbSc0.5Nb0.5O3 638

19.5 Concluding Remarks 639Acknowledgments 640References 640

20 Interfaces in Epitaxial Ferroelectric Layers/Multilayers and Their Effecton the Macroscopic Electrical Properties 645Lucian Pintilie, Andra G. Boni, Cristina Chirila, Luminita M. Hrib, AlinIuga, Lucian Trupina, Ioana Pintilie, Iuliana Pasuk, Raluca Negrea, CorneliuGhica, Mihaela Botea, Nicoleta Apostol, and Cristian M. Teodorescu

20.1 Introduction 64520.2 Electrode Interfaces in Ferroelectric Capacitors 647

20.2.1 Interface Effects on the Macroscopic Properties 64720.2.2 Analysis of Free Surfaces and of Electrode Interfaces by

AFM/PFM, XPS, and TEM 66020.3 Interfaces in Complex Structures 667

20.3.1 PZT/ZnO 66720.3.2 Ferroelectric/Multiferroic Multilayers 669

20.4 Conclusions 671Acknowledgments 671References 671

21 Electric Field Control of Magnetism Based on Elastically CoupledFerromagnetic and Ferroelectric Domains 677Kevin J.A. Franke, Tuomas H.E. Lahtinen, Arianna Casiraghi, Diego LopezGonzalez, Sampo J. Hamalainen, and Sebastiaan van Dijken

21.1 Introduction 67721.2 Domain Pattern Transfer 67821.3 Elastic Coupling Between Magnetic and Ferroelectric Domain Walls 68121.4 Domain Size Scaling of Pattern Transfer 68621.5 Electric Field Control of Ferromagnetic Domains 68821.6 Electric Field Driven Magnetic Domain Wall Motion 69021.7 Summary and Outlook 693Acknowledgments 695References 695

Nanoscale Effects: Novel Phenomena and Applications

22 Ferroelectric Vortices and Related Configurations 700Sergei Prosandeev, Ivan I. Naumov, Huaxiang Fu, Laurent Bellaiche,Michael P.D. Campbell, Raymond G.P. McQuaid, Li-Wu Chang,Alina Schilling, Leo J. McGilly, Amit Kumar, and J. Marty Gregg

22.1 Insights from Simulations and Theory 70022.1.1 Introduction 70022.1.2 Methods 701

Contents xv

22.1.3 Discovery of Unusual Patterns in Ferroelectric Nanostructuresand Their Associated Order Parameters 704

22.1.4 Controlling and Switching of Single, Double, and MultipleVortex States 708

22.1.5 Other Phenomena Related to Vortices and Complex DipolarPatterns 712

22.1.6 Summary from the Simulations and Theory Perspective 71422.2 Insights from Experiments 714

22.2.1 Introduction 71422.2.2 Vortices and Topological Winding Numbers 71522.2.3 Continuous versus Sudden Dipole rotation 71822.2.4 Unique Core Singularity 71922.2.5 Summary and Conclusions from an Experimental Perspective 723

Acknowledgments 723References 723

23 Reentrant Phenomena in Relaxors 729Alexei A. Bokov and Zuo-Guang Ye

23.1 Introduction 72923.2 Reentrant Phases in Condensed Matter 73023.3 Relaxor Ferroelectrics 73623.4 Reentrant Relaxor Behavior in Ferroelectrics 741

23.4.1 Properties of Materials with Paraelectric–Ferroelectric–RelaxorSequence of States 742

23.4.2 Models and Theories 75223.5 Transverse Glassy Freezing and Canonical Relaxors 75323.6 Reentrant Dipolar Glass State in Quantum Paraelectrics Doped with

Dipolar Impurities 75723.7 Summary and Concluding Remarks 758Acknowledgments 759References 759

24 Functional Twin Boundaries: Multiferroicity in Confined Spaces 765Ekhard. K.H. Salje and Xiangdong Ding

24.1 Introduction 76524.2 Ferroelectric Twin Boundaries 766

24.2.1 Polar Twin Walls in CaTiO3 76824.2.2 Polar Twin Walls in SrTiO3 771

24.3 Conducting Twin Boundaries 77324.4 Thermal Conductivity 77524.5 Landau Theory of Coupled Order Parameters in Domain Walls 77624.6 Chiral Twin Walls 77824.7 High Wall Densities 78024.8 Summary 783Acknowledgment 784References 784

xvi Contents

25 Novel Approaches for Genuine Single-Phase Room TemperatureMagnetoelectric Multiferroics 789Lynette Keeney, Michael Schmidt, Andreas Amann, Tuhin Maity, NitinDeepak, Ahmad Faraz, Nikolay Petkov, Saibal Roy, Martyn E. Pemble, andRoger W. Whatmore

25.1 Introduction to Single-Phase Multiferroic Materials 78925.2 Aurivillius Phase Materials – Candidate Single-Phase Multiferroics? 79625.3 Magnetoelectric Coupling in Multiferroic Bi6TixFeyMnzO18 Systems at

Room Temperature 80125.3.1 Fabrication and Structural Analysis of

Bim+1Ti3(Mn/Fe)m-3O3m+1 Thin Films 80125.3.2 Ferroelectric Investigations of Bi6TixFeyMnzO18 thin films 80425.3.3 Assessment of Ferromagnetism in Bi6TixFeyMnzO18 Thin Films 80425.3.4 Room Temperature Magnetoelectric Coupling in

Bi6Ti2.8Fe1.52Mn0.68O18 Thin Films 81025.4 Confidence Level Assessment of Genuine Single-Phase Multiferroicity 810

25.4.1 Statistics of a Single Measurement 81225.4.2 The Inclusion Size Distribution Function 81325.4.3 Series of Measurements at Varying Resolutions 81325.4.4 Upper Limit on the Contribution to Remanent Magnetization 814

25.5 Potential Devices/Applications Based on Single-Phase MagnetoelectricMultiferroics 816

25.6 Summary and Conclusions 820References 821

26 Semiconducting and Photovoltaic Ferroelectrics 830Andrew R. Akbashev, Vladimir M. Fridkin, and Jonathan E. Spanier

26.1 Introduction 83026.2 Basic Principles of Photovoltaic Effect in Ferroelectrics 83126.3 Experimental Evidence of the Bulk Photovoltaic Effect in Ferroelectrics 835

26.3.1 BiFeO3 83526.3.2 Niobates 838

26.4 The Role of the Schottky Barrier and Depletion Layer 84026.5 Photovoltaic Effect Mediated by Ferroelectric Domain Walls 84026.6 Ferroelectric Solid Solutions: Chemical Tuning of Band-Gap 84326.7 Conclusions 847Acknowledgment 847References 847

Index 851

List of Contributors

Andrew R. Akbashev Drexel University, USA

Miguel Alguero Instituto de Ciencia de Materiales de Madrid (ICMM), Consejo Superiorde Investigaciones Cientıficas (CSIC), Spain

Andreas Amann Tyndall National Institute, University College Cork, Ireland; and Schoolof Mathematical Sciences, University College Cork, Ireland

Harvey Amorın Instituto de Ciencia de Materiales de Madrid (ICMM), Consejo Superiorde Investigaciones Cientıficas (CSIC), Spain

Nicoleta Apostol National Institute of Materials Physics, Romania

Juras Banys Faculty of Physics, Vilnius University, Lithuania

Laurent Bellaiche Physics Department, University of Arkansas, USA

Daniela C. Berger Departement of Inorganic Chemistry, Physical-Chemistry and Electro-chemistry, “Politehnica” University of Bucharest, Romania

J.D. Bobic Institute for Multidisciplinary Research, University of Belgrade, Serbia

Alexei A. Bokov Department of Chemistry and 4D LABS, Simon Fraser University,Canada

Andra G. Boni National Institute of Materials Physics, Romania

Mihaela Botea National Institute of Materials Physics, Romania

Inigo Bretos Instituto de Ciencia de Materiales de Madrid (ICMM), Consejo Superior deInvestigaciones Cientıficas (CSIC), Spain

Maria Teresa Buscaglia Institute of Energetics and Interphases, National ResearchCouncil, Italy

Vincenzo Buscaglia Institute of Energetics and Interphases, National Research Council,Italy

Michael P.D. Campbell School of Maths and Physics, Queen’s University Belfast, North-ern Ireland, United Kingdom

Arianna Casiraghi NanoSpin, Department of Applied Physics, Aalto University Schoolof Science, Finland

xviii List of Contributors

Alichandra Castro CICECO – Aveiro Institute of Materials, Department of Materials andCeramic Engineering, University of Aveiro, Portugal

Alicia Castro Instituto de Ciencia de Materiales de Madrid (ICMM), Consejo Superior deInvestigaciones Cientıficas (CSIC), Madrid, Spain

Li-Wu Chang School of Maths and Physics, Queen’s University Belfast, Northern Ireland,United Kingdom

Cristina Chirila National Institute of Materials Physics, Romania

Frank Clemens Laboratory for High Performance Ceramics, Empa, Swiss Federal Labo-ratories for Materials Science and Technology, Switzerland

Covadonga Correas Instituto de Ciencia de Materiales de Madrid (ICMM), Consejo Supe-rior de Investigaciones Cientıficas (CSIC), Spain; and College of Engineering, SwanseaUniversity, United Kingdom

Lavinia P. Curecheriu Department of Physics, “Alexandru Ioan Cuza” University,Romania

Nitin Deepak Tyndall National Institute, University College Cork, Ireland

Marco Deluca Materials Center Leoben Forschung GmbH, and Institut fur Struktur- undFunktionskeramik, Montanuniversitaet Leoben, Austria

Xiangdong Ding State Key Laboratory for Mechanical Behavior of Materials, Xi’anJiaotong University, China

Christopher De Dobbelaere Inorganic and Physical Chemistry Group, Institute for Mate-rials Research, Universiteit Hasselt and imec vzw, Division imomec, Belgium

C. Doubrovsky Laboratoire de Physique des Solides, Universite de Paris Sud, Campusd’Orsay, France

Ahmad Faraz Tyndall National Institute, University College Cork, Ireland

Liliana P. Ferreira Biosystems and Integrative Sciences Institute, Department of Physics,University of Coimbra, Portugal

Paula Ferreira CICECO – Aveiro Institute of Materials, Department of Materials andCeramic Engineering, University of Aveiro, Portugal

Pascale Foury-Leylekian Laboratoire de Physique des Solides, Universite de Paris Sud,Campus d’Orsay, France

Kevin J. A. Franke NanoSpin, Department of Applied Physics, Aalto University Schoolof Science, Finland

Vladimir M. Fridkin Drexel University, USA; and Russian Academy of Sciences,Moscow, Russian Federation

Huaxiang Fu Physics Department, University of Arkansas, USA

Andreja Gajovic Molecular Physics Laboratory, Institute Rudjer Boskovic, Croatia

List of Contributors xix

Corneliu Ghica National Institute of Materials Physics, Romania

Margarida Godinho Biosystems and Integrative Sciences Institute, Department ofPhysics, Faculty of Sciences, University of Lisbon, Portugal

Martha Greenblatt Department of Chemistry and Chemical Biology, Rutgers, The StateUniversity of New Jersey, USA

J. Marty Gregg School of Maths and Physics, Queen’s University Belfast, Northern Ire-land, United Kingdom

R. Grigalaitis Faculty of Physics, Vilnius University, Lithuania

Sampo J. Hamalainen NanoSpin, Department of Applied Physics, Aalto UniversitySchool of Science, Finland

An Hardy Inorganic and Physical Chemistry Group, Institute for Materials Research,Universiteit Hasselt and imec vzw, Division imomec, Belgium

Luminita M. Hrib National Institute of Materials Physics, Romania

Teresa Hungria Instituto de Ciencia de Materiales de Madrid (ICMM), Consejo Superiorde Investigaciones Cientıficas (CSIC), Spain; and Centre de Microcaracterisation RaimondCastaing, UMS 3623, France

Alan J. Hurd Los Alamos National Laboratory, USA

Adelina-Carmen Ianculescu Department of Oxide Materials Science and Engineering,“Politehnica” University of Bucharest, Romania

Alin Iuga National Institute of Materials Physics, Romania

Maksim Ivanov Faculty of Physics, Vilnius University, Lithuania

Ricardo Jimenez Instituto de Ciencia de Materiales de Madrid (ICMM), Consejo Superiorde Investigaciones Cientıficas (CSIC), Spain

Lynette Keeney Tyndall National Institute, University College Cork, Ireland

Michael R. Koblischka Institute of Experimental Physics, Saarland University, Germany

Anjela Koblischka-Veneva Institute of Experimental Physics, Saarland University,Germany

Amit Kumar School of Maths and Physics, Queen’s University Belfast, Northern Ireland,United Kingdom

Tuomas H.E. Lahtinen NanoSpin, Department of Applied Physics, Aalto UniversitySchool of Science, Finland

Diego Lopez Gonzalez NanoSpin, Department of Applied Physics, Aalto UniversitySchool of Science, Finland

M. Lourdes Calzada Instituto de Ciencia de Materiales de Madrid (ICMM), Consejo Supe-rior de Investigaciones Cientıficas (CSIC), Spain

xx List of Contributors

Axel Lubk Triebenberg Laboratory, Technische Universitat Dresden, Germany

Tony Lusiola Laboratory for High Performance Ceramics, Empa, Swiss Federal Labora-tories for Materials Science and Technology, Switzerland

Leo J. McGilly Ceramics Laboratory, Ecole Polytechnique Federale de Lausanne (EPFL),Switzerland

Raymond G.P. McQuaid School of Maths and Physics, Queen’s University Belfast, North-ern Ireland, United Kingdom

J. Macutkevic Faculty of Physics, Vilnius University, Lithuania

Cesar Magen Instituto de Nanociencia de Aragon-ARAID, Universidad de Zaragoza,Spain

Tuhin Maity Tyndall National Institute, University College Cork, Ireland

Barbara Malic Electronic Ceramics Department, Jozef Stefan Institute, Slovenia

Liliana Mitoseriu Faculty of Physics, University “Alexandru Ioan Cuza”, Romania

Evagelia G. Moshopoulou Institute of Nanoscience and Nanotechnology, National Centerfor Scientific Research “Demokritos”, Greece; and Laboratoire de Physique des Solides,Universite de Paris Sud, Campus d’Orsay, France

Ivan I. Naumov Carnegie Institution of Washington, USA

Raluca Negrea National Institute of Materials Physics, Romania

Marius Olariu Department of Electrical Measurements & Materials, Technical UniversityGh. Asachi Iasi, Romania

Leontin Padurariu University “Alexandru Ioan Cuza”, Romania

Katharine Page Spallation Neutron Source, Oak Ridge National Laboratory, USA

Iuliana Pasuk National Institute of Materials Physics, Romania

Martyn E. Pemble Tyndall National Institute, University College Cork, Ireland; andDepartment of Chemistry, University College Cork, Ireland

Nikolay Petkov Tyndall National Institute, University College Cork, Ireland

M.M. Vijatovic Petrovic Institute for Multidisciplinary Research, University of Belgrade,Serbia

Ioana Pintilie National Institute of Materials Physics, Romania

Lucian Pintilie National Institute of Materials Physics, Romania

Sergei Prosandeev Physics Department, University of Arkansas, USA and Physics Depart-ment and Research Institute of Physics, Southern Federal University, Russia

Jesus Ricote Instituto de Ciencia de Materiales de Madrid (ICMM), Consejo Superior deInvestigaciones Cientıficas (CSIC), Spain

List of Contributors xxi

Brian J. Rodriguez Conway Institute of Biomolecular and Biomedical Research, Univer-sity College Dublin, Ireland

Saibal Roy Tyndall National Institute, University College Cork, Ireland

Ekhard. K.H. Salje Department of Earth Sciences, University of Cambridge, UnitedKingdom

Alina Schilling School of Maths and Physics, Queen’s University Belfast, Northern Ireland,United Kingdom

Michael Schmidt Tyndall National Institute, University College Cork, Ireland

Theodor Schneller Institut fur Werkstoffe der Elektrotechnik II, RWTH Aachen Univer-sity, Germany

Stella Skiadopoulou CICECO – Aveiro Institute of Materials, Department of Materialsand Ceramic Engineering, University of Aveiro, Portugal

Etienne Snoeck Centre d’Elaboration de Materiaux et d’Etudes Structurales, CNRS,France

Jonathan E. Spanier Drexel University, USA

Biljana D. Stojanovic Institute for Multidisciplinary Research, University of Belgrade,Serbia

Bosiljka Tadic Department of Theoretical Physics, Jozef Stefan Institute, Slovenia

Cristian M. Teodorescu National Institute of Materials Physics, Romania

Gregor Trefalt Electronic Ceramics Department, Jozef Stefan Institute and Center ofExcellence NAMASTE, Slovenia; and University of Geneva, Switzerland

Lucian Trupina National Institute of Materials Physics, Romania

Roxana Trusca S.C. METAV – Research and Development, Romania

Marina Tyunina Microelectronics and Materials Physics Laboratories, University of Oulu,Finland

Marlies K. Van Bael Inorganic and Physical Chemistry Group, Institute for MaterialsResearch, Universiteit Hasselt and imec vzw, Division imomec, Belgium

Sebastiaan van Dijken NanoSpin, Department of Applied Physics, Aalto UniversitySchool of Science, Finland

Bogdan S. Vasile Department of Oxide Materials Science and Engineering, “Politehnica”University of Bucharest, Romania

Catalina A. Vasilescu Department of Oxide Materials Science and Engineering,“Politehnica” University of Bucharest, Romania

Paula M. Vilarinho CICECO – Aveiro Institute of Materials, Department of Materials andCeramic Engineering, University of Aveiro, Portugal

xxii List of Contributors

Roger W. Whatmore Tyndall National Institute, University College Cork, Ireland; Depart-ment of Chemistry, University College Cork, Ireland and Department of Materials, RoyalSchool of Mines, Imperial College London, United Kingdom

Zuo-Guang Ye Department of Chemistry and 4D LABS, Simon Fraser University, Canada

Preface

Multiferroics have been at the cutting edge of research and development in materials forICTs for over a decade. During this period steady improvements in fundamental knowl-edge have been made. At the same time, nanoscale phenomena have assumed an increasingimportance. Progress has benefited from the strong synergies with activities in nanoscaleferroelectrics, which are at a more mature stage. Multifunctionality and nanoscaling arewidely acknowledged at present as the keys to the miniaturization of solid-state electronics,and specifically nanoferronics, which is emerging as a new area with large technologicalpotential. The topic has now reached a maturity level that allows, and actually requires,books that provide a comprehensive revision of the topic, and an in-depth analysis of futuretrends. These are the objectives of Nanoscale Ferroelectrics and Multiferroics: Key Pro-cessing and Characterization Issues, and Nanoscale Effects.

It is intended to provide the increasing number of scientists and engineers, who areapproaching the topic from a range of backgrounds, with a reference/guide text that shouldhelp them to roadmap their R&D activities. The volume reviews the key issues in processingand characterization of nanoscale ferroelectrics and multiferroics, and provides a compre-hensive description of their properties. An emphasis is put on differentiating size effectsof extrinsic ones like boundary or interface effects. Nanoscale novel, recently described,phenomena that are bound to be behind major advancement in the field during the comingyears are also addressed.

The book is devised to stress, and take full advantage of, the synergies between nanoscaleferroelectrics and multiferroics. It covers materials nanostructured at all levels, fromceramic technologies like ferroelectric nanopowders, bulk nanostructured ceramics andthick films, and magnetoelectric nanocomposites, to thin films, either polycrystalline layerheterostructures or epitaxial systems, and to nanoscale free-standing objects with specificgeometries, such as nanowires and tubes at different levels of development, but all tech-nologically relevant. Nanostructuring is a requirement of the current tendency to minia-turization of ceramic technologies for microelectronics that imposes stringent conditionson processing, and has a deep impact on functional properties. Also, nanostructuring ulti-mately results in the ever-decreasing processing temperatures of thin films, a key issue tothe integration of these multifunctional oxides with silicon devices and flexoelectronics.Last but not least, a range of novel physical phenomena has been described in nanoscaleferroelectrics and multiferroics that have the potential to enable a range of disruptive tech-nologies, like magnetoelectric memory. Overall, the book reviews the current state of theart of these materials, stressing a range of specific topics at the cutting edge of research.

This project springs from the high-level European scientific knowledge platform builtwithin the COST Action Single and Multiphase Ferroics and Multiferroics with Restricted

xxiv Preface

Geometries (SIMUFER, ref. MP0904), active between March 2010 and May 2014. COST(European Cooperation in Science and Technology) is a pan-European networking instru-ment that allows researchers from COST member countries and cooperating states to jointlydevelop their ideas and initiatives in a field or topic of common interest. SIMUFER estab-lished a multidisciplinary scientific network of groups from 24 European countries and7 non-COST countries, experienced in synthesis, advanced characterization, and model-ing of all nanoscale ferroics, single-phase multiferroics, and ferroic-based combinations ofdissimilar materials. This book project arises primarily from their expertise, though it hasbeen open to world renowned experts when necessary. Chapter contributors have been care-fully selected and have all made major contributions to knowledge of the respective topics;overall, they are among the most respected scientists in the field.

IntroductionWhy Nanoscale Ferroelectrics

and Multiferroics?

Miguel Alguero1, J. Marty Gregg2, and Liliana Mitoseriu3

1Instituto de Ciencia de Materiales de Madrid (ICMM), Consejo Superior de InvestigacionesCientıficas (CSIC), Spain

2School of Maths and Physics, Queen’s University Belfast, Northern Ireland, United Kingdom3Faculty of Physics, University “Alexandru Ioan Cuza”, Romania

I.1 Ferroics and Multiferroics

Single-phase ferroics are compounds that present one of the three (currently expandedto four) ferroic properties: ferroelectricity, ferromagnetism, or ferroelasticity, to whichferrotoroidicity has recently been added. The common feature of the four types of ferroicsis the appearance of the ferroic order; either it is a spontaneous electrical polarization,magnetization, strain, or toroidal moment, in a phase transition from a high-temperatureprototype phase to the low-temperature ferroic phase, related by a group/subgrouprelationship. This transition is always accompanied by a decrease in symmetry and thesplitting of the ferroic phase into domains (regions with a different orientation of theorder parameter). A second feature, the direct consequence of the switchable nature ofthe order parameter and of the domain dynamics, is the characteristic ferroic hysteresisloop; that is, a distinctive hysteretic dependence of the order parameter on its conjugatedfield (electric or magnetic field, mechanical stress or toroidal source vector, respectively),with two remnant states of opposite sign [1]. The four phenomena are schematicallyshown in Figure I.1. Ferroics are highly topical, advanced functional materials that havenot only enabled a range of mature and ubiquitous related technologies (like magnetic

Nanoscale Ferroelectrics and Multiferroics: Key Processing and Characterization Issues, and Nanoscale Effects,First Edition. Edited by Miguel Alguero, J. Marty Gregg, and Liliana Mitoseriu.© 2016 John Wiley & Sons, Ltd. Published 2016 by John Wiley & Sons, Ltd.

xxvi Introduction

Figure I.1 Schematics of domains and hysteretic switching for the four ferroic phenomena.Adapted by permission of IOP Publishing from [1]. © IOP Publishing. All rights reserved.

or ferroelectric information recording, ceramic ultrasound transducers, or shape memoryalloys, to name only a few examples) but are also under extensive research for a numberof novel, potentially disruptive, applications [2].

There are also compounds that simultaneously present two (or more) ferroic phenomena,known as multiferroics, among which those showing coexistence of ferroelectricity andferromagnetism (initially termed ferroelectromagnets) are receiving increasing attention[3,4]. This is not only because of their inherent multifunctionality but also for the fact thatthey are liable to show magnetoelectric coupling, and have thus the potential to enable theelectrical control of magnetism (and the magnetic control of polarization) [5].

This book specifically deals with ferroelectrics and ferroelectromagnets (either robustferromagnetic materials or canted antiferromagnets showing weak ferromagnetism), thoughthe general term multiferroics will be used following the current tendency to name

Introduction xxvii

ferroelectric–ferromagnetic materials, and in general any type of magnetic ordering com-pounds, in this way.

The choice of addressing them together only acknowledges the deep-rooted relationshipbetween the two sets of materials; multiferroism requires ferroelectricity and thus multi-ferroics have to be electrically insulating to be functional (an issue not always acknowl-edged). This feature is not easily found in magnetic materials, most of them being metallicor narrow-band gap semiconductors. Indeed, chemical bonding requirements suggest thetwo ferroic phenomena to be incompatible [6]; transition metal or rare earth atomic specieswith partially filled outer d or f electronic shells (and unpaired electrons) are necessary formagnetism, while model ferroelectric perovskite oxides are characterized by covalentlybonded transition metals (to oxygen) with empty d orbitals [7]. Nevertheless, an ever-increasing number of multiferroic single-phase materials have been reported over the lastdecade, exploring alternative mechanisms of ferroelectricity.

I.2 Ferroelectric Materials and Related Technologies

Ferroelectrics are thus materials that present a spontaneous electrical polarization, whosedirection can be reversed with an electric field (by nucleation and growth of inversiondomains, resulting in the distinctive ferroelectric hysteresis). The ferroic phase appears ata ferroelectric transition, driven by electrical polarization [8], which can be either of a dis-placive type, the most common one, associated with a crystal structure instability inducedby condensation of a transverse optical phonon (the soft mode) [9], or of an order–disordertype. Its macroscopic phenomenological description according to Landau’s theory of phasetransitions can be found in Chapter 19.

All ferroelectrics are also pyroelectric and piezoelectric, as well as electrooptic, whichturns them into a prototype of multifunctionality (even before magnetic order is added).Moreover, they are the only materials that can present these properties, intrinsically linkedto the crystal structure, in polycrystalline form (thanks to the ability to reorient the polar-ization under an electric field).

Though ferroelectricity was first described for hydrogen-bonded compounds (Rochellesalt being the first one in 1921), and there are also examples among tellurides, fluorides,and iodides [10], a number of electroactive polymers like poly(vinylidene fluoride) [11],and recent reports of ferroelectric metal organic frameworks [12], clearly oxides stand outas the ones that have enabled a range of successful ferroelectric technologies.

I.2.1 Ferroelectric Bulk Technologies

Perhaps the best-known ferroelectric, and also the first oxide shown to be so in 1944, isBaTiO3 with a perovskite structure. This model compound presents the ferroelectric transi-tion at ∼393 K and a succession of low-temperature polymorphic phase transitions betweenferroelectric phases with decreasing symmetry, from tetragonal to orthorhombic and torhombohedral, for which the polar axis (and thus the direction of the spontaneous polariza-tion defined by the displacement of the Ti4+ cation from the centre of the oxygen octahedra)changes. Polymorphism is a quite common phenomenon in ferroelectric perovskite oxidesand plays a very important role in their functionality. BaTiO3 is also the base composition

xxviii Introduction

of multilayer ceramic capacitors (after the chemical tailoring of the ferroelectric transitiondown to room temperature), one of the two large-scale, mature bulk ceramic ferroelectrictechnologies. This material and its modifications are extensively addressed in this book (seeChapters 1, 11, 12, 15, and 18), for the miniaturization of these capacitors is a case study ofthe current trends in microelectronics that require the nanostructuring of the ceramic lay-ers. In the last few years, nanostructured BaTiO3 and its solid solutions have become themain candidates for active materials used in capacitive building blocks for energy storageapplications.

The second successful technology is piezoelectric ceramics for electromechanical trans-duction. The state of the art material for these applications, which range from sensorsand actuators (like accelerometers or positioning systems for scanning probe microscopy,respectively) and their combination in smart systems (to implement active vibration damp-ing), to ultrasound generation and sensing (for medical imaging or non-destructive test-ing), and to submarine acoustics, is Pb(Zr,Ti)O3, which also has a perovskite structure.This is an oxide solid solution, for which the best properties are found at a morphotropicphase boundary (MPB) between rhombohedral and tetragonal ferroelectric polymorphs, forwhich a monoclinic phase has been recently described [13]. This material can be regarded asa modification of PbTiO3, a second model ferroelectric oxide that also shows a successionof polymorphic phase transitions, yet induced by hydrostatic pressure instead of temper-ature [14], which has been placed at one of these ferroelectric instabilities (the MPB) bybuilding up chemical pressure (achieved by substitution of Zr4+ for Ti4+). As a matter offact, the very good electromechanical response of this material is a combination of twoeffects, a crystal contribution, associated with the existence of a transverse lattice insta-bility at the monoclinic tetragonal boundary [15], and an extrinsic contribution, associatedwith the fact that ferroelectric perovskite oxides are also ferroelastic (and therefore multi-ferroic, but not ferroelectromagnets). The spontaneous strain develops at the ferroelectrictransition along with the polarization (the two parameters are intrinsically coupled), and asa consequence ferrroelectric–ferroelastic domains appear (in addition to polarization inver-sion domains).The non-180◦ (90◦ in the tetragonal case) domain walls are mobile understress and electric field (unlike 180◦ walls that only move under an electric field), givingway to a wall contribution to the piezoelectric effect [16]. Moreover, the domain dynamics isenhanced at the morphotropic phase boundary. In addition, chemical (or doping) engineer-ing of Pb(Zr,Ti)O3 has been developed that enables a range of soft and hard piezoelectricceramics with tailored properties for specific applications. Piezoelectric ceramics are alsobeing considered for novel applications, such as energy harvesting [17] and magnetoelec-tric composites (see later). Further explanations of the mechanisms, along with a review ofalternative materials, can be found in Chapter 16. This technology is not oblivious to thegeneral miniaturization trend and nanostructuring can also be anticipated.

Other examples of ferroelectric ceramic bulk technologies are infrared (IR) cameras fornight vision (and, in general, IR detectors for a range of applications exploiting the pyro-electric effect) and electrooptic devices. Modifications of PbTiO3 like (Pb,La)TiO3 andtransparent (Pb,La)(Zr,Ti)O3 are usually the material choices, respectively. An excellentreview of ferroelectric ceramics and related technologies can be found in [18].

Also successful ferroelectric single-crystal technologies are presently available. The bestexamples are surface acoustic wave (SAW) devices for radio frequency and microwavesignal conditioning, based on ferroelectric LiNbO3 substrates. At the very end of the