The Role of Topology in Materials (eBook)

Prof. Sanju Gupta is Associate professor and has substantial experience and expertise in functional nanomaterials and characterization with an emphasis on energy, water and biophysics- related research. Additionally, she is working in the area of topological and geometrical aspects of materials and co-authored publications on this topic including an invited feature article in MRS Bulletin and the recent News Feature in MRS Bulletin with the co-editor (Dr. Avadh Saxena) about the underpinning role(s) of topology in the 2016 Physics and Chemistry Nobel Prizes. She has co-organized several conferences and symposia on this topic as well as on advanced nanomaterials and applications. She has won many accolades (e.g. Recipient of NSF Fellowship, NIH Fellowship, DOE Fellowship, JSPS Fellowship and Junior Investigator Award, WISE Award) since her graduate study in addition to being well-placed and known in the materials community for more than two decades including the topology community. As a result, she is co-organizing an Applied Topology conference with colleagues in the Physics and Mathematics departments at WKU in 2018. Dr. Avadh Saxena is Group Leader of the Condensed Matter and Complex Systems theory group at Los Alamos National Laboratory in New Mexico. He has extensive experience in working on topology related problems both in materials science and condensed matter physics with a large number of publications in this field. He is a Los Alamos National Laboratory Fellow and a Fellow of the American Physical Society. He serves on the advisory boards of several international conferences, has organized numerous workshops and symposia related to topology and functional materials (APS, MRS, SIAM, among others), has co-authored an MRS Bulletin article on this topic and a news Feature on 2016 Nobel Prizes, has co-edited four books (with Springer) and many special issues of research journals.  His recent work on topological defects, specifically the magnetic textures called skyrmions, has been featured in Discovery News. He holds Adjunct Professor positions at the University of Barcelona, University of Arizona and Virginia Tech.  Recently he was granted the prestigious Affiliate Professorship at the Royal Institute of Technology (KTH), Stockholm. He also serves as an advisor for the National Institute for Materials Science, Tsukuba, Japan. 

Foreword 6
Preface 7
Contents 10
Contributors 16
Introductory 18
1 Importance of Topology in Materials Science 19
1.1 Introduction 19
1.2 Essentials of Topology 20
1.2.1 Genus and Euler Characteristics 20
1.2.2 Network Topology 20
1.2.3 Geometry-Topology Interrelationship 21
1.3 Topological Taxonomy of Functional Materials 21
1.3.1 Nanocarbons 21
1.3.2 Soft and Polymeric Materials 26
1.3.3 Minimal Periodic Surfaces 28
1.4 Topological Phases in Condensed Matter 30
1.4.1 Real-Space Topological Materials 31
1.4.2 Dirac Materials 31
1.4.3 Topological Insulators and Topological Superconductors 33
1.4.4 Weyl Semimetals 34
1.4.5 Other Topological Materials 35
1.5 Metrology and Techniques 39
1.5.1 High-Resolution Electron Microscopy 39
1.5.2 Nonlinear Optical Imaging 39
1.5.3 X-Ray Tomography and Electron Holography 39
1.5.4 X-Ray and Neutron Scattering 40
1.5.5 Elasticity and Deformation Energy Characterization 40
1.5.6 Topological Correlators and Other Metrics 42
1.6 Computational Topology of Materials 42
1.6.1 Topological Databases and Visualizing Topology 43
1.6.2 Miscellaneous Topics 43
1.7 Conclusion 45
References 46
2 Topology and Geometry in Condensed Matter 50
2.1 Topology 50
2.1.1 Introduction 50
2.1.2 Classification of Vector Fields with Homogeneous Boundary Conditions 52
2.1.3 Classification of Defects in Vector Fields (Mainly Spin Fields) 53
2.1.4 Defects and Homogeneous Boundary Conditions 53
2.2 Geometry 54
2.2.1 Energy 54
2.2.2 Geometry with Intrinsic Length: The Cylinder 55
2.2.3 Geometry with Intrinsic Length: Plane with a Disc Missing 57
2.2.4 Interaction Between Geometry and Physical Field 59
2.2.5 Chirality of 1d Spin Configurations 60
2.3 Quantum Potential, Thin Tubes, Knots 63
2.4 Conclusions 65
References 65
Condensed Matter Materials Physics 66
3 Topology-Induced Geometry and Properties of Carbon Nanomaterials 67
3.1 Introduction 67
3.1.1 Carbon as a Building Block 67
3.1.2 Defect in sp2 Nanocarbon 68
3.2 Topology-Induced Geometry in sp2 Nanocarbon 69
3.2.1 Surface Curvature Generation in Graphene Sheets 69
3.2.2 Plastic Deformation of Carbon Nanotubes 71
3.3 Stone-Wales Defect 73
3.3.1 Symmetry Breaking by C–C Bond Rotation 73
3.3.2 Formation Energy 74
3.3.3 Out-of-Plane Displacement 75
3.3.4 Microscopic Observation 77
3.4 Defect of 5–7 Paired Type 78
3.4.1 Dissociation of a SW Defect 78
3.4.2 As a Seed of Surface Curvature 79
3.5 Peanut-Shaped C60 Polymers 80
3.5.1 Fusion of C60 Molecules 80
3.5.2 TLL State in C60 Polymers 81
3.5.3 Topology-Based Understanding 82
3.5.4 Curvature-Based Understanding 83
3.5.5 Electron-Phonon Coupling in C60 Polymers 85
3.6 Carbon Nanocoil 86
3.6.1 Benefit from Coiled Structure 86
3.6.2 Atomistic Modeling 86
3.6.3 Experimental Realization 88
3.6.4 Theoretical Prediction 88
3.7 State-of-the-Art Curved sp2 Nanocarbons 90
3.7.1 Nano-“Pringles” 90
3.7.2 Nano- “Tetrapod” 91
3.7.3 Nano-“Schwarzite” 92
3.8 Perspective 93
References 94
4 Topology by Design in Magnetic Nano-materials: Artificial Spin Ice 99
4.1 Introduction 99
4.2 Frustration, Topology, Ice, and Spin Ice 102
4.3 Simple Artificial Spin Ices 106
4.3.1 Kagome Spin Ice 106
4.3.2 Square Ice 110
4.4 Exotic States Through Vertex-Frustration 112
4.5 Emergent Ice Rule, Charge Screening, and Topological Protection: Shakti Ice 114
4.6 Dimensionality Reduction: Tetris Ice 118
4.7 Polymers of Topologically Protected Excitations: Santa Fe Ice 119
4.8 Conclusions 121
References 121
5 Topologically Non-trivial Magnetic Skyrmions in Confined Geometries 127
5.1 Introduction 127
5.2 Topological Effect in Magnetic Skyrmions 130
5.2.1 Topology in Magnetic Materials 130
5.2.2 Topological Stability of Magnetic Skyrmions and Emergent Magnetic Monopoles 131
5.2.3 Topological and Skyrmion Hall Effect 133
5.2.4 Skyrmion-Based Racetrack Memory (RM) 134
5.3 Origin of Magnetic Skyrmion 135
5.3.1 Magnetic Phase Diagram in Chiral Magnets 135
5.3.2 Mechanism of DM Interaction 138
5.4 Magnetic Skyrmions in Confined Geometries 140
5.4.1 Sample Fabrication Techniques 140
5.4.2 Lorentz TEM 142
5.4.3 Off-Axis Electron Holography for Imaging Magnetic Contrast 145
5.4.4 Edge-Mediated Skyrmion Phase and Field-Driven Cascade Phase Diagram 147
5.4.5 High Flexibility of Geometrically-Confined Skyrmions 149
5.5 Conclusions 153
References 153
6 Topological Phases of Quantum Matter 155
6.1 Introduction 155
6.2 Topology in Condensed Matter Physics 156
6.3 HgTe/CdTe Quantum Wells and Quantum Spin-Hall Insulators 159
6.4 Z2 Topological Insulators in Three Dimensions 160
6.5 Topological Crystalline Insulators 163
6.6 Topological Semi-metals 165
6.7 Topological Superconductivity 169
6.8 Strongly Correlated Topological Materials 171
6.9 Outlook and Conclusions 172
References 172
Biology and Mathematics 184
7 Theoretical Properties of Materials Formed as Wire Network Graphs from Triply Periodic CMC Surfaces, Especially the Gyroid 185
7.1 Introduction 185
7.1.1 Classical Geometry of the Gyroid and Graph Approximation for the Channels 187
7.1.2 P and D Surfaces 190
7.2 Theory 191
7.2.1 Overview 191
7.2.2 Summary of the Methods 192
7.2.3 The Gyroid Without Magnetic Field 193
7.2.4 Enhanced Symmetries from a Re-gauging Groupoid 197
7.2.5 Slicing, Chern Classes and Stability Under Perturbations 199
7.2.6 Possible Experimental Verification 200
7.2.7 The P Wire Network Without Magnetic Field 201
7.2.8 The D Wire Network and the Honeycomb Lattice Without Magnetic Field 201
7.3 Noncommutative Approach in the Presence of a Magnetic Field 205
7.3.1 Gyroid in the Presence of a Magnetic Field 206
7.3.2 P Wire Network in a Magnetic Field 207
7.3.3 D Wire Network in a Magnetic Field 207
7.3.4 Honeycomb in a Magentic Field 208
7.3.5 Possible 3d Quantum Hall Effect 209
7.4 General Theory and Possible Material Design 209
7.5 Discussion and Conclusion 210
References 211
8 Entangled Proteins: Knots, Slipknots, Links, and Lassos 213
8.1 Introduction 213
8.2 Entanglement in Proteins 214
8.3 Proteins with Knots and Slipknots 216
8.3.1 Classification and Description of Knots 216
8.3.2 Proteins with Knots and Slipknots – KnotProt Server and Database 220
8.3.3 Knotting Fingerprint for Knots and Slipknots 221
8.3.4 Folding of Knotted Proteins 224
8.3.5 Function of Knotted Proteins 227
8.4 Links in Proteins 228
8.5 Proteins with Lassos 231
8.6 Conclusions 234
References 235
Soft Matter and Biophotonics 239
9 Topology in Liquid Crystal Phases 240
9.1 Introduction 240
9.2 Schlieren Textures and Two-Dimensional Nematics 243
9.3 The Homotopy Theory of Defects 246
9.3.1 Point Defects: Hedgehogs 247
9.3.2 Disclination Loops 248
9.3.3 The Pontryagin–Thom Construction 250
9.4 Illustrations in Liquid Crystals 251
9.4.1 Skyrmions 251
9.4.2 Colloids 252
9.4.3 Torons and Hopf Textures 254
9.5 Smectics 255
9.6 Geometry of Line Fields 258
9.6.1 Umbilics 259
9.6.2 Chirality Pseudotensor 260
9.7 Cholesterics 261
9.7.1 ? Lines: Defects in the Pitch 262
9.8 Knotted Fields 262
9.8.1 Homotopy Classification 263
9.8.2 Construction of Knots in Nematics 264
References 266
10 Topologically Complex Morphologies in Block Copolymer Melts 269
10.1 Introduction 269
10.2 AB Block Copolymers 271
10.3 ABC Block Copolymers 273
10.4 Blending Molecular Architectures 279
10.5 Concluding Remarks 280
References 284
11 Topology of Minimal Surface Biophotonic Nanostructures in Arthropods 285
11.1 Introduction 286
11.2 Topology of Arthropod Biophotonic Nanostructures 287
11.3 Self-assembly of Minimal Surface Biophotonic Nanostructures 291
11.4 Biomimetic Potential of Minimal Surface Biophotonic Nanostructures 295
11.5 Conclusion 296
References 297
Index 301