Preface ...................................................... xiii
Chapter 1. Basic Theoretical Description of
Flexoelectricity in Solids ...................................... 1
P.V. Yudin and A.K. Tagantsev
1 Introduction ................................................. 1
2 Static Bulk Flexoelectric Effect ............................. 2
2.1 Thermodynamic Phenomenological Description .............. 4
2.2 Microscopies of Static Bulk Flexoelectric Effect ........ 6
3 Flexoelectric Effect in an Acoustic Wave ..................... 9
3.1 Dynamic Flexoelectric Effect ........................... 10
3.2 Phenomenological Description of the Flexoelectric
Effect in an Acoustic Wave ............................. 11
3.3 Microscopic Description of the Flexoelectric Effect
in an Acoustic Wave .................................... 12
4 Flexoelectric Effect in a Finite Sample ..................... 16
4.1 Direct Flexoelectric Effect in a Finite Sample ......... 16
4.2 Converse Flexoelectric Effect in a Finite Sample ....... 20
5 Recent Advances in the Field ................................ 24
5.1 Electronic Contribution to the Flexoelectric Effect .... 25
5.2 Effect of Field Inhomogeneity in a Short-Circuited
Sample ................................................. 26
5.3 Details of Converse Flexoelectric Effect in a Finite
Sample ................................................. 26
6. Conclusions ................................................. 27
Acknowledgments ............................................. 28
References .................................................. 28
Chapter 2. First-Principles Theory of Flexoelectricity ......... 31
Massimiliano Stengel and David Vanderbilt
1 Introduction ................................................ 32
2 Theory and Methods .......................................... 33
2.1 Strain, Strain Gradients, and Responses ................ 33
2.2 Long-Wave Approach ..................................... 35
2.3 Macroscopic Electric Fields ............................ 40
2.4 Lattice Response ....................................... 43
2.5 Electronic Response .................................... 47
2.5.1 Spherical term, pseudopotential dependence,
and the non-interacting spherical-atom paradox .. 51
2.6 Surface Effects ........................................ 57
2.6.1 Electronic surface response ..................... 58
2.6.2 Lattice surface response ........................ 62
2.7 Electronic and Lattice Response Revisited:
Curvilinear Coordinates ................................ 64
2.7.1 A simple 1D example ............................. 65
2.7.2 General formalism in 3D ......................... 69
2.7.3 Microscopic polarization response ............... 72
2.7.4 Atomic relaxations .............................. 75
2.7.5 Electrostatics in a curved space ................ 76
2.7.6 Treatment of the macroscopic electric fields .... 78
2.8 Surface Effects in Curvilinear Coordinates ............. 79
2.8.1 Surface polarization and metric ................. 79
2.8.2 Atomic relaxations .............................. 86
2.9 Summary ................................................ 89
3 Application to SrТіО3 ....................................... 90
3.1 General Methodology .................................... 90
3.1.1 Bulk calculations ............................... 91
3.1.2 Truncated-bulk slab calculations ................ 92
3.1.3 Relaxed-ion slab calculations ................... 95
3.2 Computational Parameters ............................... 96
3.3 Results ................................................ 97
3.3.1 Bulk calculations ............................... 97
3.3.2 Truncated-bulk slab calculations ................ 99
3.3.3 Relaxed-ion slab calculations .................. 102
4 Conclusions and Outlook .................................... 105
Acknowledgments ............................................ 108
References ................................................. 108
Chapters. A Continuum Theory of Flexoelectricity .............. 111
Q. Deng, L. Liu, and P. Sharma
1 Introduction ............................................... 111
2 Continuum Theory of Flexoelectricity ....................... 116
2.1 Continuum Kinematics .................................. 116
2.2 Maxwell's Equations ................................... 117
2.3 Free Energy of an Electromechanical System ............ 119
2.4 Governing Equations ................................... 120
2.5 Internal Energy Density ............................... 128
3 Piezoelectric Thin Film Super Lattices Without Using
Piezoelectric Materials .................................... 129
4 Compression of a Truncated Cone, Effective Piezoelectric
Response, and Determination of Material Properties ......... 135
5 Flexoelectric Beam Bending, Vibration, and Energy
Harvesting ................................................. 142
6 Flexoelectric Membranes .................................... 151
6.1 Е3 = Е3(х1) ............................................ 154
6.2 Е3 = Е3(r) ............................................ 154
7 Flexoelectricity in Soft Materials ......................... 155
7.1 One-dimensional Formulation for Soft Materials ........ 155
7.2 Flexoelectric Effects in a Soft Bilayer Structure ..... 160
8 Concluding Remarks ......................................... 163
References .................................................... 164
Chapter 4. Mechanical Boundary Conditions for a Case When
Thermodynamic Potential Depends on Strain Gradients ........... 169
A.S. Yurkov
1 Introduction ............................................... 169
2 General Description of the Mechanical Equilibrium
of the Elastic Body ........................................ 174
3 Additional Surface Coordinate System and Boundary
Conditions in Cartesian Tensor Components .................. 177
4 Boundary Conditions within Covariant Formalism ............. 181
5 Boundary Conditions in Local Coordinate System ............. 185
6 Conclusion ................................................. 186
Acknowledgments ............................................... 187
Appendix A. Relation between Tensor γij and Gauss Second
Fundamental Form ........................................... 187
Appendix B. Curvilinear Coordinates and Covariant
Formalism .................................................. 190
Appendix C. Equivalence of Boundary Conditions Derived
by Two Different Approaches ................................ 194
Appendix D. Some Specific Coordinate System ................... 196
References .................................................... 199
Chapter 5. Flexoelectric Deformations of Finite-Size Bodies
in Framework of the Theory of Continuum ....................... 201
A.S. Yurkov
1 Introduction ............................................... 201
2 Thermodynamic Potential, Differential Equations of
Elastic Equilibrium, and Boundary Condition ................ 204
3 Exact Solution for Homogeneously Polarized Ball of
Isotropic Dielectric ....................................... 207
4 Approximate Method to Calculate Flexoelectric
Deformations of Finite Size Bodies ......................... 217
5 Comparison of Exact and Approximate Solutions for
Homogeneously Polarized Ball ............................... 219
6 Flexoelectric Bending of Homogeneously Polarized Circular
Rod ........................................................ 223
7 Flexoelectric Bending of Homogeneously Polarized Circular
Plate ...................................................... 226
8 Conclusion ................................................. 229
Acknowledgment ................................................ 231
Appendix A. A Simplified Description of the Converse
Flexoelectric Eff'ect in Finite-size Bodies ................... 231
References .................................................... 239
Chapter 6. Flexoelectricity and Phonon Spectra ............... 241
P.V. Yudin, A. Kvasov, and A.K. Tagantsev
1 Introduction ............................................... 242
2 Landau Theory of Bulk Flexoelectric Coupling in Crystals ... 243
2.1 Static Flexoelectric Effect ........................... 244
2.2 Dynamic Flexoelectric Effect .......................... 246
3 Flexoelectric Coupling and Phonon Spectra .................. 248
4 Upper Bounds for the Static Bulk Flexocoupling
Coefficients ............................................... 252
5 Calculations of Flexoelectric Coefficients using Phonon
Dispersion ................................................. 255
6 Calculations of Flexoelectric Coefficients using
Dynamical Matrix ........................................... 259
7 Conclusions ................................................ 262
Acknowledgments ............................................... 263
References .................................................... 263
Chapter 7. Impact of Flexoelectric Effect on Electro-
mechanics of Moderate Conductors .............................. 265
A.N. Morozovska, O.V. Varenyk, and S.V. Kalinin
1 Introduction ............................................... 265
2 Thermodynamic Approach for the Flexoelectrochemical
Effect Description ......................................... 268
3 Flexo-electrochemical Effect Manifestation in
Electrochemical Strain Microscopy .......................... 271
4 Summary Remarks ............................................ 277
Acknowledgments ............................................... 278
Appendix A .................................................... 278
References .................................................... 281
Chapter 8. Role of Flexoelectricity in Multidomain
Ferroelectrics ................................................ 285
R. Ahhiwalia, A.K. Tagantsev, P. Yudin, N. Setter, N. Ng,
and D.J. Srolovitz
1 Introduction ............................................... 286
2 Model ...................................................... 289
3 Simulations of Bulk Domain Patterns ........................ 293
3.1 Coupling of Polarization with Shear Strain
Gradients: f44 ........................................ 295
3.2 Coupling of Polarization with Uniaxial Strain
Gradients: ƒ11 ......................................... 297
4 Single Mode Analysis ....................................... 300
5 Influence of Flexoelectricity on Domain Switching .......... 305
6 Summary and Conclusions .................................... 307
References .................................................... 308
Chapter 9. Flexoelectricity Impact on the Domain Wall
Structure and Polar Properties ................................ 311
A.N. Morozovska, S.V. Kalinin, and E.A. Eliseev
1 Overview ................................................... 312
1.1 Polarization Structure and Electronic Properties at
the Domain Wall in Ferroics ........................... 312
1.2 Flexoelectricity Impact on the DW Structure and
Electronic Properties ................................. 314
2 Free Energy and Coupled Equations .......................... 317
3 Neel Component Impact on the DW Structure and Energy in
ВаТіО3 ..................................................... 321
4 DW Static Conductivity in ВаТіО3 ........................... 328
5 Flexoelectric Coupling Reveals the New Symmetry of DW ...... 330
6 Summary .................................................... 333
References .................................................... 333
Chapter 10. Bending-Induced Giant Polarization
in Ferroelectric MEMS Diaphragm ............................... 337
Zhihong Wang and Weiguang Zhu
1 Introduction ............................................... 338
2 Diaphragm Fabrication and Characterization ................. 339
3 Microstructure of PZT Film and Multilayer Structure
of the PZT Diaphragm ....................................... 341
4 Curvature of the Diaphragm and Polarization ................ 342
5 Evolution of Polarization .................................. 344
5.1 Bending Induced Electromechanical Couphng in a Flat
Diaphragm Subjected to High Tensile Stress ............ 344
5.2 Reversed Remnant Polarization in a Bent Diaphragm
Subjected to High Compressive Stress .................. 349
6 Estimation of Flexoelectric Coefficient .................... 354
6.1 Central Deflection and Radius of the Curvature in
Relation to the External Bias ......................... 355
6.2 Asymmetric C-V Curves ................................. 357
6.3 Negative Shift of P-E Hysteresis at Low Maximum
Voltage ............................................... 358
6.4 Calculation of Flexoelectric Coefficient .............. 359
7 Conclusions ................................................ 364
References .................................................... 364
Chapter 11. Quasi-Amorphous Materials ........................ 367
David Ehre, Ellen Wachtel, and Igor Lubomirsky
1 Introduction ............................................... 367
2 Preparation of Quasi-Amorphous Thin Films .................. 368
3 Physical Properties of Quasi-Amorphous Thin Films .......... 370
4 The Role of Stress and Strain in the Formation of
Quasi-amorphous Films ...................................... 371
4.1 Pre-nucleation Expansion .............................. 371
4.2 Suppression of Crystallization in Amorphous
Titanate Films ........................................ 372
4.3 The Essential Role of the Temperature Gradient
During Heat Treatment: Flexoelectricity ............... 375
4.3.1 Suppressing crystallization in the presence
of as-deposited voids .......................... 375
4.3.2 Formation of a permanent electric
polarization ................................... 376
5 The Structure of Quasi-Amorphous Materials: Theory of
the Random Network of Local Bonding Units .................. 380
5.1 Local Bonding Units in High Density Amorphous (HDA)
and Quasi-Amorphous Films ............................. 380
5.2 The Role of Ba and Sr as Random Network Stabilizers
in the HDA ............................................ 382
5.3 Charge Redistribution in the HDA to LDA
Transformation ........................................ 385
6 Compositional Criteria for the Quasi-Amorphous Phase ....... 385
7 Conclusion ................................................. 388
References .................................................... 388
Index ......................................................... 391
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