Domains in Ferroic Crystals and Thin Films (eBook)

Preface 5
Acknowledgments 7
Contents 8
A Preview of Concepts and Phenomena 15
Fundamentals of Ferroic Domain Structures 25
2.1 Structural Phase Transitions and Domain States: Basic Concepts and Classifications 25
2.1.1 Structural Changes at Phase Transitions: Ferroics 25
2.1.2 Ferroelectric Phase Transitions 30
2.1.3 Ferroelastics and Ferrobielectrics 33
2.1.3.1 Ferroelastic Phase Transition 33
2.1.3.2 Natural Spontaneous Strain 34
2.1.3.3 Aizu’s Definition of Spontaneous Strain 39
2.1.3.4 Ferrobielectrics 42
2.1.4 Higher Order Ferroics 42
2.1.5 Relation Between the Symmetries G and F: Order Parameter 46
2.1.5.1 Transition Without Multiplication of the Unit Cell 47
2.1.5.2 Transition with Multiplication of the Unit Cell 48
2.1.6 Overview of Different Kinds of Phase Transitions: Species 49
2.1.7 Domain States 50
2.1.7.1 Basic Concepts 50
2.1.7.2 Left Coset Approach 55
2.1.8 Ferroic Species 57
2.1.9 Ferroelectric Species 61
2.2 Coexisting Domain States 65
2.2.1 Twinning Operations 66
2.2.2 Twin Laws for Nonferroelastic Domain Pairs 67
2.2.3 Domain Wall Orientation: Electrical Compatibility 72
2.2.4 Domain Wall Orientation: Mechanical Compatibility 76
2.2.5 Ferroelastic Domains in Physical Contact 80
2.2.6 Examples of Domain Wall Orientations: Nonferroelastic Walls 83
2.2.7 Examples of Domain Wall Orientations: Ferroelastic Walls 88
2.3 Thermodynamic Approach 95
2.3.1 Single-Component Order Parameter 95
2.3.2 Uniaxial Proper Ferroelectric (Nonferroelastic) 102
2.3.3 Uniaxial Proper Ferroelectric-Ferroelastic 105
2.3.4 Multiaxial Proper Ferroelectric-Improper Ferroelastic 106
2.3.5 Uniaxial Improper Ferroelastic-Ferroelectric 111
2.3.6 Limitation of Traditional Thermodynamic Approach: Pseudo-proper and Weak Ferroelectricity 114
Ferroic Materials 122
3.1 Sources of Information and Statistics 122
3.2 Table of Selected Ferroic Materials 123
Methods for Observation of Domains 134
4.1 Introductory Remarks 134
4.2 Surface Etching Techniques 135
4.3 Other Methods Based on Surface Relief 142
4.4 Surface Decoration Techniques 144
4.4.1 Colloidal Suspensions 145
4.4.2 Decoration by Sublimation and Vacuum Evaporation 147
4.4.3 Deposition in Liquids 149
4.4.4 Condensation of Vapor 149
4.4.5 Decoration by Liquid Crystal Layers 150
4.5 Scanning Force Microscopy-Based Techniques 155
4.5.1 Electrostatic Force Microscopy (EFM) 156
4.5.2 Scanning Surface Potential Microscopy (SSPM) 160
4.5.3 Contact Domain Imaging 162
4.5.4 Lateral Force Microscopy (LFM) 163
4.5.5 Domain Imaging via Surface Topography 164
4.5.6 Domain Imaging via Nonlinear Dielectric Response (SNDM) 166
4.5.7 Domain Imaging via Static Piezoresponse 168
4.5.8 Domain Imaging via Dynamic Piezoresponse (PFM) 171
4.6 Polarized Light Microscopy Based on Unperturbed Linear Optical Properties 174
4.6.1 Birefringence 174
4.6.2 Spatial Dispersion 182
4.6.3 Optical Activity 183
4.6.4 Optical Absorption and Observation in Reflected Light 185
4.7 Optical Methods Based on Higher Order Optical Properties 186
4.7.1 Perturbed Linear Optical Properties: Electro-optics and Elasto-optics 186
4.7.2 Nonlinear Optical Properties 189
4.7.3 Photorefractive Properties 192
4.8 Electron Microscopy 195
4.8.1 Scanning Electron Microscopy 195
4.8.2 Transmission Electron Microscopy 199
4.8.2.1 Bright Field Imaging, Dark Field Imaging, Weak Beam Dark Field Imaging, and Selected Area Electron Diffraction 200
4.8.2.2 High-Resolution Transmission Electron Microscopy 204
4.8.3 Electron Mirror Microscopy 206
4.9 Methods Based on Interactions with X-Rays 207
4.10 Pyroelectric Mapping 210
4.11 Scanning Optical Microscopy 213
4.12 Additional Methods and Concluding Remarks 216
Static Domain Patterns 220
5.1 Introductory Remarks and Scheme of the Chapter 220
5.2 Equilibrium 180? Domain Patterns in a Ferroelectric Plate: Theories 221
5.3 Domain Patterns Connected with Phase Boundaries 233
5.3.1 Perfect Matching 233
5.3.2 Matching on Average 235
5.4 Selected Observations of Domains in Crystalline Ferroic Samples 237
5.4.1 Uniaxial Ferroelectrics (Nonferroelastic) with the Second-Order Transition 238
5.4.1.1 TGS, Triglycine Sulfate 238
5.4.1.2 Other Representatives 246
5.4.2 Ferroelastics with a Small Number of Domain States 248
5.4.2.1 Ferroelastics with Two Ferroelastic Domain States 248
5.4.2.2 Ferroelastics with Three Domain States (Not Including Perovskite Material) 255
5.4.2.3 KH2PO4, A Ferroelectric Ferroelastic 258
5.4.3 Perovskite Ferroics 266
5.4.4 R Cases 275
5.4.5 Quartz 278
5.4.6 Tweed Patterns 280
Domain Walls at Rest 284
6.1 Thickness and Structure of Domain Walls: Methods and Data 285
6.1.1 Direct Optical Observations 286
6.1.2 X-Ray and Neutron Scattering 291
6.1.3 X-Ray Topography 294
6.1.4 Raman Scattering 295
6.1.5 Electron Holography 296
6.1.6 Transmission Electron Microscopy 298
6.1.7 Surface Methods 301
6.1.8 Comments on Available Data 304
6.2 Macroscopic Theories of Domain Walls 304
6.2.1 Order Parameter Profile in Domain Walls 305
6.2.1.1 Single-Component Order Parameter 305
6.2.1.2 Multi-component Order Parameter 309
6.2.2 Effects of Strain Induced by the Order Parameter 313
6.2.3 Domain Walls in Selected Ferroics 318
6.2.3.1 Domain Walls in Uniaxial Nonferroelastic-Ferroelectrics 318
6.2.3.2 Domain Wall in a Multiaxial Ferroelectric: Barium Titanate 320
6.2.3.3 Domain Wall in an Improper Uniaxial Ferroelastic Ferroelectric: Gadolinium Molybdate 323
6.2.3.4 Domain Wall in Nonferroelectrics 326
6.2.4 Concluding Remarks 327
6.3 Microscopic Theories of Domain Walls 328
6.4 How Flat Is the Wall? 332
6.4.1 Mathematical Problem 334
6.4.2 Nonferroelectric/Nonferroelastic Walls 335
6.4.3 Walls in Ferroelectrics and Ferroelastics 338
6.4.4 Experimental Data on Roughening of Ferroic Domain Walls and Experimental Observations 341
Switching Properties: Basic Methods and Characteristics 343
7.1 Introduction 343
7.2 Ferroelectric Hysteresis Loop 344
7.3 TANDEL Effect 351
7.4 Pulse Switching 352
7.5 Ferroelastic Hysteresis Loops 356
7.6 More Involved Methods 361
Switching Phenomena and Small-Signal Response 363
8.1 Introduction and Overview of Switching Mechanisms 363
8.2 Basics of Domain State Reorientation 366
8.2.1 Driving Force for Processes of Domain State Reorientation 366
8.2.2 Pressure Acting on a Domain Wall 369
8.3 Single Domain Wall in Motion 372
8.3.1 Experimental Techniques Used to Measure Domain Wall Velocity 372
8.3.2 Motion of Ferroelectric Nonferroelastic Walls 376
8.3.2.1 Planar Walls 376
8.3.2.2 Growing Domains 389
8.3.3 Motion of Ferroelastic Walls in Ferroelectrics 394
8.3.3.1 Planar Walls 394
8.3.3.2 Needle-Shaped and Wedge-Shaped Domains 400
8.4 Theories of Single Wall Motion 403
8.4.1 Two Regimes of Wall Motion 404
8.4.2 Wall Mobility in Activated Regime. Miller-Weinreich Theory 406
8.4.3 Wall Mobility in Activated Regime. Advanced Theories and Present Understanding of the Problem 409
8.4.3.1 The Role of Shape of Critical Nuclei 410
8.4.3.2 The Nucleation Rate vs. Wall Velocity Relation 411
8.4.3.3 Departure from the Exponential Law and Temperature Dependence of the Velocity 413
8.4.4 Domain Wall Motion in Non-activated Regime 416
8.4.4.1 Low-Field Wall Mobility in Non-activated Regime 416
8.4.4.2 Exact Solution for a Moving Wall Modification of the Profile of Moving Wall
8.4.4.3 Interpretation of the Theoretical Results for Wall Motion in Non-activated Regime. Additional Factors Influencing the Phenomenon 419
8.4.5 Domain Wall Motion Influenced by the Ferroelectric/Electrode Interface 422
8.4.6 Motion of Curved Domain Walls 427
8.5 Defect Pinning and Creep of Domain Walls 430
8.5.1 Non-thermally Assisted Regime, Weak and Strong Pinning 431
8.5.2 Finite Temperatures: Weak Pinning and Creep 434
8.5.3 Finite Temperatures: Strong-Pinning Regime 438
8.5.4 Weak and Strong Pinning with Flexible Defects 439
8.5.5 Experimental Evidence on Weak Pinning and Creep of Ferroelectric Domain Walls 440
8.6 Switching Process in Selected Materials 441
8.6.1 BaTiO3 442
8.6.2 TGS and TGFB 450
8.6.3 LiTaO3 and LiNbO3 455
8.6.4 KDP and Isomorphous Crystals 458
8.7 Theory and Modeling of Switching 460
8.7.1 Introduction 460
8.7.2 Domain Nucleation 463
8.7.3 Domain Coalescence 469
8.7.4 Pulse Switching 471
8.7.5 Classical Polarization Hysteresis Loops 478
8.7.6 Rayleigh Loops 482
8.7.7 Piezoelectric Hysteresis Loops 487
8.7.7.1 Thermodynamic Piezoelectric Loops and the Case of Abrupt Switching 488
8.7.7.2 Piezoelectric Loops Affected by Non-saturation, Backswitching, and Partial Switching 489
8.7.8 Ferroelectric Breakdown 493
8.8 Extrinsic Contribution to Small-Signal Dielectric Response in Bulk Ferroelectrics 495
8.8.1 Introduction 495
8.8.2 Fully Immobile Domain Pattern 497
8.8.2.1 Excessive Polarizability of Domain Walls 497
8.8.2.2 Electrocaloric Effect 498
8.8.2.3 Piezoelectric Clamping 500
8.8.3 Contributions from Moving Domain Walls in Ideal Crystals 502
8.8.3.1 Wall Near the Bottom of Peierls Potential 503
8.8.3.2 Dielectric Response of Moving Domain Walls and C-V Curves 503
8.8.3.3 Dispersion of the Dielectric Response Due to Free Domain Walls 508
8.8.4 Quasistatic Bending Contribution from ‘‘Firmly’’ Pinned Domain Walls 510
8.8.5 Limited Motion of Free Domain Wall 513
8.8.6 Wall Motion in Random Potential and Dispersion of the Dielectric Response (Experimental Findings and Interpretation) 516
8.8.7 Domain Freezing 523
8.8.8 Dielectric Response Associated with Mobile Ferroelastic Domain Walls in a Clamped Multidomain Ferroelectric 527
Ferroelectric Thin Films 532
9.1 Introduction 532
9.2 Experimental Studies on the Static Domain Pattern in Thin Films 534
9.2.1 Domain Structure in (001) Thin Films of Tetragonal Ferroelectric Perovskites 534
9.2.1.1 Configuration of Ferroelastic Domain Patterns 535
9.2.1.2 Composition of Ferroelastic Domain Patterns 541
9.2.1.3 Antiparallel Domain Patterns 544
9.2.2 Ferroelastic Domain Patterns in (001) Rhombohedral and (111) Tetragonal Thin Films of Ferroelectric Perovskites 547
9.2.3 Domain Structure in Other Systems 553
9.3 Domain Pattern and Elastic Effects 555
9.3.1 Strained State of Ferroelectric Film and Dislocation-Assisted Stress Release 555
9.3.2 Single-Domain State in a Strained Film 564
9.3.3 Domain Formation Driven by Elastic Effects: Basic Concepts 571
9.3.3.1 Factors Governing Domain Formation Driven by Elastic Effects 571
9.3.3.2 Domain Formation Driven by the Energy of Macroscopic Stress 573
9.3.3.3 Effect of Film Thickness on the Stress-Driven Formation of Domain Structure 578
9.3.4 Domain Formation Driven by Elastic Effects: Advanced Theoretical Results 584
9.3.4.1 Results of the Mean-Strain Approach 585
9.3.4.2 Theory of Domain Pattern of Arbitrary Density for Cubic-Tetragonal Transition in (001) Films 586
9.3.4.3 Theories Taking into Account the Stress Dependence of the Order Parameter 590
9.3.4.4 Theory of Domain Patterns in (001) Rhombohedral and (111) Tetragonal Films of Ferroelectric Perovskites 596
9.3.5 Domain Formation Driven by Elastic Effects: Theory vs. Experiment 600
9.3.5.1 Domain Fraction of a/c-Pattern in (001) Tetragonal Perovskite Films as a Function of Temperature 600
9.3.5.2 Thickness Dependence of the Domain Fraction of a/c-Pattern in (001) Tetragonal Perovskite Films 605
9.3.5.3 Periodicity of a/c-Patterns in (001) Tetragonal Perovskite Films 607
9.3.5.4 Periodicity of Domain Patterns in (111) Tetragonal and (001) Rhombohedral Perovskite Films 609
9.4 Domain Pattern and Electrostatic Effects 610
9.4.1 Equilibrium Domain Pattern in Ferroelectric/Dielectric Sandwich Structure 610
9.4.2 Equilibrium Domain Pattern in Ferroelectric Films on Insulating Substrates 614
9.4.3 Limitations of Hard-Ferroelectric Approximation and Results Obtained Beyond This Approximation 615
9.5 Switching and Polarization Hysteresis 619
9.5.1 Pulse Switching 619
9.5.2 Ferroelectric Hysteresis Loops Size Effects
9.5.2.1 Size Effect on Nucleation of the Reverse Domains 628
9.5.2.2 Surface Pinning Size Effect 629
9.5.2.3 Depletion-Assisted Nucleation of Reverse Domains 629
9.5.2.4 Surface Passive Layer 631
9.5.2.5 Insulating Passive Layer 632
9.5.2.6 Passive Layer with Threshold Conduction 635
9.5.3 Effects of Internal Bias and Imprint 638
9.5.3.1 Voltage Offset Caused by Nearby-Electrode Trapped Charge 639
9.5.3.2 Electrode-Adjacent Injection Model for Imprint 642
9.5.3.3 Poling Effect of Misfit Dislocations 648
9.5.3.4 Voltage Offset Due to Reorientation of Random Field Defects 653
9.5.3.5 Voltage Offset Due to Depletion Effect 654
9.5.3.6 Internal Bias Field and Imprint-Experimental Observations 656
9.6 Small-Signal Response 664
9.6.1 Intrinsic Contribution-Effect of Passive Layer 665
9.6.2 Intrinsic Contribution-Depletion Effect 673
9.6.3 Intrinsic Contribution-Strain Effect 675
9.6.4 Extrinsic Contribution-Mechanical Effects 678
9.6.5 Extrinsic Contribution-Electrostatic Effects 682
9.7 Polarization Fatigue in Thin Ferroelectric Films 688
9.7.1 How Can Imperfections Influence Polarization Switching in Ferroelectric Capacitor? 689
9.7.1.1 Reduction of the Effective Electrode Area 689
9.7.1.2 Reduction of the Electrical Field Seen by Ferroelectric 690
9.7.1.3 Mechanisms of Switching Modification 690
9.7.1.4 Wall Pinning Mechanism 691
9.7.1.5 Seed Inhibition Mechanism 692
9.7.1.6 Experimental Evidence for Wall Pinning and Seed Inhibition Mechanisms 693
9.7.1.7 Fatigue as ‘‘Local Imprint’’ 696
9.7.2 What Are These Imperfections and How Do They Affect the Switching Performance of Ferroelectric Capacitor? 697
9.7.2.1 Oxygen Vacancy Redistribution Mechanism 698
9.7.2.2 Injection Mechanism 699
9.7.3 Overall Picture of Polarization Fatigue in PZT Thin Films 699
9.8 Scanning Force Microscopy Study of Polarization Reversal 701
9.8.1 Top-Electrode-Free PFM 703
9.8.2 Through-Electrode PFM 711
9.9 Films of Proper Ferroelectric-Ferroelastics 715
Appendix A 32 Point Groups: Notations, Symmetry Elements, Crystalline Classes, Group Orders, Subgroups, and Supergroups 723
Appendix B Ferroic Species 725
Appendix C Phase Transitions into Ferroelectric Phases 731
Appendix D Spontaneous Polarization, Spontaneous Strain, and Orientation of Domain Walls in Ferroic Species 738
Appendix E Piezoelectric Coefficients in Ferroelectric Phases of BATiO3-Type Perovskites 784
Appendix F Tensors: Properties and Notations 788
Transformation Laws for Tensors 788
Voight Notations for Tensors 788
Notation for Symmetry of Tensors 789
References 791
Index 820

"Chapter 5 Static Domain Patterns (p. 207-208)

5.1 Introductory Remarks and Scheme of the Chapter

After discussing in some detail the theoretical aspects of properties of domain states and after describing a number of methods to observe domains, we now wish to deal with some real domain structures in single crystals. Several thousands of papers have been published on observations of domain patterns in different kinds of ferroics,1 offering a large amount of interesting data for materials listed in Chap. 3 and many others. Some of them are just observations as it stands, others were performed with the aim to create situations corresponding to theoretically defined conditions.

When treating properties of domain patterns in real ferroic samples, it is necessary to distinguish features of stable domain structures from those of dynamic domain phenomena. In the present chapter we have primarily in mind static and quasistatic domain patterns which can be observed in the absence of intentionally applied external forces that would tend to change their geometry or sizes. We define static or quasistatic domain patterns arbitrarily as those which do not appreciably change on the time scale of hours.

These are the patterns whichmay correspond to the thermodynamic equilibrium of the sample or which are metastable with long lifetimes because of large energy barriers that would have to be overcome to reach more stable configuration. Available data on domain patterns can be, in some approximation, classified into three categories. First, we can observe domains in a sample as it stands, meaning that its history (sample preparation, thermal record, applied forces) is not known. Second, and perhaps most often, the sample has been treated in a way which has been planned or which at least is known.

Third, the sample quality and the external conditions (e.g., thermal history) are well defined and carefully prepared so that we may expect the resulting domain structure to correspond to minimum energy harmonizing with its intrinsic properties and external conditions; this is often referred to as the ‘‘equilibrium domain pattern.’’ In the present chapter we first discuss, in Sect. 5.2, theoretical aspects of the last mentioned case, paying attention to the simplest example of equilibrium domain pattern in ferroelectric samples containing only domains with antiparallel orientation of PS vectors (‘‘1808 domains’’).

Such patterns have been studied extensively in ferroelectrics, both nonferroelastic and ferroelastic, with the aim to obtain regular patterns corresponding to thermodynamic equilibrium. This research was, in its early stages, inspired by successful treatments of equilibrium domain structures in ferromagnets. The role of the energy of demagnetizing field has its electrical counterpart treated in some detail in the following section. However, in ferroelectrics the situation is different because of the existence of free charge carriers that may contribute in a decisive way to the reduction of depolarization energy. This issue will be addressed in Sect. 5.2 as well."