Contents

Cover

Title page

Copyright page

Preface

Chapter 1: Characterization of Phase Transition Points for Topological Gapped Systems

1.1 Introduction
1.2 General Definition of Topological Invariant of Phase Transition Points
1.3 Phase Transition Points of One-Dimensional Systems
1.4 Phase Transition Points of Two-Dimensional Systems
1.5 An Example of 3D Topological Insulators
References

Chapter 2: Topological Insulator Materials for Advanced Optoelectronic Devices

2.1 Excellent Electronic Properties
2.2 Excellent Optical Properties
2.3 Advanced Optoelectronic Devices
2.4 Conclusion and Outlook
References

Chapter 3: Topological Insulator Thin Films and Artificial Topological Superconductors

3.1 Theoretical Background
3.2 Introduction of the Experimental Methods
3.3 Topological Insulator Thin Films
3.4 Artificial Two-Dimensional Topological Superconductor
3.5 Discovery of Majorana Zero Mode
6 Summary
Acknowledgements
References

Chapter 4: Topological Matter in the Absence of Translational Invariance

4.1 Introduction
4.2 Topological Insulator and Real-Space Topology
4.3 Layer Construction: Dimensional Crossovers of Topological Properties
4.4 Effects of Disorder
4.5 Critical Properties of Topological Quantum Phase Transitions
4.6 Phase Diagrams Obtained from Machine Learning
4.7 Summary and Concluding Remarks
References

Chapter 5: Changing the Topology of Electronic Systems Through Interactions or Disorder

5.1 Introduction
5.2 Change of an Insulator’s Topological Properties by a Hubbard Interaction
5.3 Effects of Disorder on Chern Insulators
5.4 Topological Superconductors
5.5 Conclusions
5.6 Acknowledgements
References

Chapter 6: Q-Switching Pulses Generation Using Topology Insulators as Saturable Absorber

6.1 Introduction
6.2 Fiber Laser Technology
6.3 Topology Insulator (TI)
6.4 Pulsed Laser Parameters
6.5 Bi₂Se₃ Material as Saturable Absorber in Passively Q-Switched Fiber Laser
6.6 Q-Switched EDFL with Bi₂Te₃ Material as Saturable Absorber
6.7 Conclusion
References

Chapter 7: Topological Phase Transitions: Criticality, Universality, and Renormalization Group Approach

7.1 Generic Features Near Topological Phase Transitions
7.2 Topological Invariant in 1D Calculated from Berry Connection
7.3 Topological Invariant in 2D Calculated From Berry Curvature
7.4 Universality Class of Higher Order Dirac Model
7.5 Topological Invariant in D-Dimension Calculated From Pfaffian
7.6 Summary
References

Chapter 8: Behaviour of Dielectric Materials Under Electron Irradiation in a SEM

8.1 Introduction
8.2 Fundamental Aspects of Electron Irradiation of Solids
8.3 Electron Emission of Solid Materials
8.4 Electron Emission of Solid Materials
8.5 Trapping and Charge Transport in Insulators
8.6 Application: Dynamic Trapping Properties of Dielectric Materials Under Electron Irradiation
8.7 Conclusion
References

Chapter 9: Photonic Crystal Fiber (PCF) Is a New Paradigm for Realization of Topological Insulator

9.1 Introduction
9.2 Structure of Photonic Crystal Fiber
9.3 Result and Discussion
9.4 Conclusion
References

Chapter 10: Patterned 2D Thin Films Topological Insulators for Potential Plasmonic Applications

10.1 Introduction
10.2 Fundamentals of Plasmons
10.3 Plasmons at Structured Surfaces
10.4 Nanostructured Thin Films and Its Applications
10.5 Summary
References

Index

Also of Interest

End User License Agreement

List of Illustrations

Chapter 1

Figure 1.1 (a) The spectrum of the SSH model versus δ under OBC. Insects show...
Figure 1.2 Spectrum of Eq. (1.4) with an extra term H₀(k) = cos...
Figure 1.3 Sketch of the integral path of the defined topological invariant....
Figure 1.4 The winding of h(k) with different t′ and...
Figure 1.5 The phase diagram of the extended SSH model with ν_d on...
Figure 1.6 The phase diagram of the Creutz ladder model with υ_d on...
Figure 1.7 The paths of the Hamiltonian for 1D 2-band models. (a) and (b) show the cases of...
Figure 1.8 Energy spectrum of the extended Kitaev model with µ = 1,...
Figure 1.9 Phase diagram of the extended Kitaev model, with Δ₁ = t₁...
Figure 1.10 Spectrum of the DIII model with Δ_p = 1 and µ = 0.5....
Figure 1.11 Phase diagram of the DIII model, with Δ_p = 1 and...
Figure 1.12 Spectrum E and the Berry phase γ of the lower (occupied) band of...
Figure 1.13 A sketch of the lattice of Haldane model. The lines with arrow indicate the...
Figure 1.14 Spectra of the Haldane model with PBC along x direction and OBC along y direction....
Figure 1.15 The energy spectrum of the Haldane model with t₁ = 1 and...
Figure 1.16 (a) and (b) show the phase diagram of the Haldane model with...
Figure 1.17 The energy spectrum of the extended QWZ model with µ = 1....
Figure 1.18 The phase diagram of the extended QWZ model. The solid and dashed lines are trajectories...
Figure 1.19 The torus in Eq. (1.49) in the parameter space (k, Δη...
Figure 1.20 A sketch of the winding number of L_NL given by k, the...
Figure 1.21 Geometric relations between L_NL (a single closed loop) and...
Figure 1.22 Phase diagrams of model (22) in respect of ν_NL and...

Chapter 2

Figure 2.1 Optical parameters of Bi_1.5Sb_0.5Te_1.8Se_1.2...
Figure 2.2 Physical mechanism of the Sb₂Te₃ thin film cavity. Diagram...
Figure 2.3 Spectroscopy characterization of the Bi₂Se₃ nanosheets on...
Figure 2.4 Extinction coefficients of the microribbon arrays of topological insulator...
Figure 2.5 Simulation of electromagnetic field distribution in Bi_1.5Sb_0.5Th_1.8Se_1.2...
Figure 2.6 Plasmon resonances enhanced light absorption in ultrathin a-Si solar cells simulated...
Figure 2.7 Nanometric Sb₂Te₃ thin film holograms and reconstructed images....
Figure 2.8 Ultra-short pulse generation by topological insulator material Bi₂Te₃...

Chapter 3

Figure 3.1 Molecular beam epitaxy of topological insulator thin films (a), the crystal structure...
Figure 3.2 The experimental ascertainment of thickness limit of a topological insulator thin film....
Figure 3.3 Artificial two-dimensional topological superconductor composed by...
Figure 3.4 Bi₂Te₃ / NbSe₂ based artificial topological superconductor...
Figure 3.5 Detection of a Majorana zero mode in vortex core state. (a) a zero bias dI/dV map...
Figure 3.6 The proof of the spin-polarized nature of a Majorana zero mode in the center of...

Chapter 4

Figure 4.1 Energy spectrum of a cylinder topological insulator. (a) Energy gap associated with...
Figure 4.2 Non-trivial degeneracy of the electronic spectrum on the surface of a spherical...
Figure 4.3 Penetration of the surface wave function. The surface state of a topological insulator...
Figure 4.4 Phase diagrams of Z₂ topological insulator (STI/WTI) (a) and Weyl semimetal/Chern...
Figure 4.5 Topological index map for stacked (a) QSH and (b) QAH model [33]. The anisotropy...
Figure 4.6 The lowest-energy wave function in the terrace of WTI(001). 1D helical channel appears...
Figure 4.7 Phase diagram of disordered 3D Z₂ topological insulator. The horizontal...
Figure 4.8 Averaged conductance as a function of mass in disordered topological insulators....
Figure 4.9 (a) Phase diagram of disordered topological insulator with anisotropic hopping parameter...
Figure 4.10 Density state scaling. (a) Density of states calculated on the Dirac semimetal line...
Figure 4.11 (a) Phase diagram obtained by CNN, corresponding to the case...
Figure 4.12 Phase diagram for Weyl semimetal. The left panel is for the case α = 1/2....

Chapter 5

Figure 5.1 Spectrum for the model given by Eq. (5.3) in the ribbon geometry: (a) for...
Figure 5.2 Hall conductance, in units of e²/h, as a function of interaction strength,...
Figure 5.3 The topological phases for the model given in Eq. (5.25) with a Hubbard interaction....
Figure 5.4 Phase diagram of the Haldane model as a function of the parameters M and φ...
Figure 5.5 Level statistics analysis for the Haldane model with Anderson disorder W = 3 t equally...
Figure 5.6 Localization transition studied through level statistics for the Haldane model with...
Figure 5.7 Evolution of the Chern number for Anderson disorder. (a) Chern number as a function...
Figure 5.8 Level spacing statistics analysis for the Haldane model with Anderson disorder selectively...
Figure 5.9 (a) Level spacing variance for increasing disorder strengths W/t in the selectively...
Figure 5.10 Hall conductivity of the resulting metallic state emerging from the Chern insulator...
Figure 5.11 The 40 lowest energy states as a function of model parameters: exchange coupling...
Figure 5.12 Evolution of the MZEM from YSR state: both panels show |ψ|² on...
Figure 5.13 Top row: Chern number as a function of concentration and magnetic field for magnetic...

Chapter 6

Figure 6.1 The growth of fiber laser over the past 25 years....
Figure 6.2 Loss characteristic of silica fiber and emission bands of some rare-earth ions....
Figure 6.3 Illustration of energy level diagram of Er³⁺ ions in silica fibers....
Figure 6.4 Basic structure of passive Q-switch laser cavity....
Figure 6.5 The evolution of gain and losses in passive Q-switching technique....
Figure 6.6 The important pulsed laser parameters....
Figure 6.7 FESEM image of the Bi₂Se₃ composite film. Insert shows the...
Figure 6.8 Linear transmission of the free standing Bi₂Se₃-PVA film....
Figure 6.9 Raman spectrum for the free standing Bi₂Se₃-PVA film....
Figure 6.10 Nonlinear absorption characteristic for the Bi₂Se₃-PVA film....
Figure 6.11 Configuration of the Bi₂Se₃ PVA film based Q-switched EDFL....
Figure 6.12 Output spectrum of the EDFL configured with and without Bi₂Se₃...
Figure 6.13 Oscilloscope trace for the EDFL with Bi₂Se₃ SA at 116.6 mW...
Figure 6.14 A single pulse envelop for the EDFL with Bi₂Se₃ SA at...
Figure 6.15 Pulse Width and Repetition Rate as a function of pump power....
Figure 6.16 Pulse energy and peak output power as a function of pump power....
Figure 6.17 RF spectrum for the EDFL with Bi₂Se₃ SA at 116.6 mW pump power....
Figure 6.18 FESEM image of Bi₂Te₃ composite film. Insert shows a high...
Figure 6.19 Linear transmission of the free standing Bi₂Te₃-PVA film....
Figure 6.20 Raman spectrum of the free standing Bi₂Te₃-PVA film....
Figure 6.21 Nonlinear absorption profile for the Bi₂Te₃ PVA film....
Figure 6.22 Configuration of the Q-switched EDFL with Bi₂Te₃ PVA film....
Figure 6.23 Emission spectrum of the EDFL with Bi₂Te₃ film-based SA at...
Figure 6.24 Typical pulses train for the EDFL with Bi₂Te₃ film-based SA...
Figure 6.25 A single envelop of the pulse for the EDFL with Bi₂Te₃ film-based...
Figure 6.26 RF spectrum for the EDFL with Bi₂Te₃ film-based SA at...
Figure 6.27 The pulse repetition rate and pulse width against pump power....
Figure 6.28 The average output power and the corresponding single-pulse energy against pump power....

Chapter 7

Figure 7.1 Schematic graph of the evolution of the topological invariant C and the...
Figure 7.2 Two different ways to tell the number of knots in a messy closed string: (a) Going...
Figure 7.3 An intuitive picture of the topologically trivial phase (left) of the SSH model,...
Figure 7.4 (a) The proposed scaling process, Eq. (7.16), applied to the topologically nontrivial...
Figure 7.5 (a) The integrand r| W (r) |² in Eq. (7.22) for...
Figure 7.6 (a) The RG flow dµ/dl (orange and light blue curves) of Kitaev chain...
Figure 7.7 (top) d vector of 2D 2 × 2 Dirac models with higher orbital...
Figure 7.8 (a) The Pfaffian Pf [m (k, M)] of the BHZ model described...

Chapter 8

Figure 8.1 (a) Representation of some trajectories of the primary electrons in the irradiated...
Figure 8.2 Bombardment of a thin target and resulting signals....
Figure 8.3 Energetic distribution, N(E), of electrons emitted by a beam of primary electrons...
Figure 8.4 Energy spectra of backscattered electrons for different target elements....
Figure 8.5 Variation of the backscatter coefficient η as a function of the atomic number...
Figure 8.6 Variation of the backscatter coefficient with the incident energy for various...
Figure 8.7 Detectable secondary electron emission zones....
Figure 8.8 Depth distribution of energy losses for a silver sample irradiated at 20 kV [14]....
Figure 8.9 Simplified schema of the band structure of the vacuum-insulator interface. An electron...
Figure 8.10 Illustration of the variation in the energy loss functions for different penetration...
Figure 8.11 Inelastic interaction of the primary electron with an electron of an atomic orbital:...
Figure 8.12 Schematic course of the secondary, backscattered and total electron emission yields...
Figure 8.13 Illustration of the energy loss function variation for different penetration depths...
Figure 8.14 Schematic representation of the variation of the possible state density, n (E),...
Figure 8.15 Schematic representation of the density of possible states in a crystalline insulator...
Figure 8.16 Schematic representation of the density of possible states for an insulator containing...
Figure 8.17 Measurement of a trapped charge by the electrostatic influence method (EIM) in...
Figure 8.18 (a): Cutaway view of the experimental arrangement for measuring displacement and...
Figure 8.19 (a): Time variation of displacement and leakage currents during and after electron...
Figure 8.20 Time variation of the displacement (a) and leakage (b) currents recorded on PMMA...
Figure 8.21 (a): Time variation of the trapped charge for glass under and after electron irradiation...
Figure 8.22 Time variation of total electron emission yield σ (open triangle) and of...
Figure 8.23 Enlarged view of the change of the displacement and the leakage currents recorded...
Figure 8.24 A Comparison (Dashed arrow) of the enlarged view of image charge, Q_im,...
Figure 8.25 Time decay of the displacement and the leakage current recorded on PET polymer...
Figure 8.26 Inverse of the experimental resistivity values as a function of the primary beam...
Figure 8.27 Stored charge versus time during and after electron irradiation for glass at different...
Figure 8.28 Variation of σas a function of irradiation time for glass at 13.7 keV primary...
Figure 8.29 X-ray intensity decay curves acquired using a SiLi x-ray detector for glass at...
Figure 8.30 Sketch illustrating the effect of current density on sodium migration using Bethe...

Chapter 9

Figure 9.1 (a) Hall voltage in magnetic field (b) Bending of electron in presence of magnetic field....
Figure 9.2 parallel line is the normal resistance independent of B, (ii) Red colour linear...
Figure 9.3 Realization of integer quantum Hall effect....
Figure 9.4 localization and non-localization of electrons in structure....
Figure 9.5 Edge state at the boundary when defect is present....
Figure 9.6 Band structure and spin behavior of electron....
Figure 9.7 Two edge stage effectively making no magnetic field....
Figure 9.8 E versus K curve....
Figure 9.9 (a) Basic structure to understand the geometry of insulator (b) Basic structure...
Figure 9.10 Cross-sectional view of conventional and photonic crystal fiber....
Figure 9.11 Photonic crystal structure to realize photonic topological insulator....
Figure 9.12 Electricfield distribution in photonic crystal fiber at wavelength 13(a) 400 nm,...

Chapter 10

Figure 10.1 A typical setup for near-field optical imaging of SPP fields at a metal/air interface....
Figure 10.2 Experimental setup for diffuse scattering....
Figure 10.3 Classifications of Materials....

List of Tables

Chapter 1

Table 1.1 Ten-fold way symmetry classes, classified in terms of the presence or absence of...

Chapter 4

Table 4.1 Wigner-dyson classification [60, 77] According to the presence/absence of the time...
Table 4.2 Scaling relations in the semimetal-metal transition at...
Table 4.3 Critical behaviors of the Anderson transition....
Table 4.4 Comparison of the Anderson metal-insulator (M-I) transition and the semimetal-metal...

Chapter 6

Table 6.1 Rare-earth ions with common host glasses and emission wavelength ranges....

Chapter 7

Table 7.1 Summary of the critical behavior of various quantities in a 2d 2 × 2 dirac...

Chapter 8

Table 8.1 Calculated penetration depth Z_m using Kanaya-Okayam expression, the...
Table 8.2 Trapped charge Q_S and time constant T recorded on PET polymer at the...
Table 8.3 The parameters used for the calculation of the trapping cross section....
Table 8.4 Trapped charge, surface potential, leakage current...

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Library of Congress Cataloging-in-Publication Data

ISBN 978-1-119-40729-4

Preface

Topological insulators are one of the most exciting areas of research in condensed matter physics. Topological insulators are materials with nontrivial symmetry-protected topological order that behaves as insulators in their interior but whose surface contains conducting states, meaning that electrons can only move along the surface of the material. The underlying cause is time-reversal symmetry: their physics is independent of whether time is flowing backward or forward. These surface states are robust, maintained even in the presence of surface defects. In the bulk of a non-interacting topological insulator, the electronic band structure resembles an ordinary band insulator, with the Fermi level falling between the conduction and valence bands. On the surface of a topological insulator there are special states that fall within the bulk energy gap and allow surface metallic conduction. Carriers in these surface states have their spin locked at a right-angle to their momentum (spin-momentum locking).

During the past decade, myriad reliable theoretical and experimental data have been accumulated on topological insulators. The time is now right to gather together this information into a handbook to make it readily available for researchers and students preparing to work in this area of condensed matter physics, quantum information and materials science. Presenting the latest developments, this book covers most introductory experiments and applications in topological insulators and provides a foundation for understanding the field.

The book begins with the characterization of phase transition points for topological insulating systems. In chapter 1 reviews the studies of topological properties of phase transition points of topological quantum phase transitions by assigning a topological invariant defined for these points. It moves on to show how to use topological insulator materials for advanced optoelectronic devices in chapter 2. The focus is on the excellent electronic and optical properties of topological insulator materials and their wide applications in advanced optoelectronic devices. Chapter 3 explains what topological insulator thin films and artificial topological superconductors are. It discuss the experimental results on topological insulator thin films and two-dimensional topological superconductor based on two prototypical materials, Bi₂Se₃ and Bi₂Te₃. Atomically-precisely controlled growth was realized in Bi₂Se₃ and Bi₂Te₃ single crystalline thin films by molecular beam epitaxy, furthermore the minimum thicknesses which maintain the topology of these materials were experimentally determined. Chapter 4 introduces the topological matter in the absence of translational invariance. Dimensional crossover of topological properties in thin films of topological insulators (TI) and Weyl semimetals, electronic properties on the surface of TI nanoparticles and TI nanowires as a constrained electronic system are discussed. The effects of disorder are also highlighted.

Chapter 5 shows that a purely local interaction can cause topological transitions by renormalizing kinetic energy terms alone, without phase transitions associated with order parameters. Disorder is also a mean of changing the topology of Chern insulators, as it localizes every state except for those carrying the topological invariant. With increasing disorder, states with opposite topological invariant meet and annihilate. But considering the sub-lattice degree of freedom, Chern insulators may evade localization: an anomalous Hall metal may be stabilized with strong disorder in one sublattice, while the disorder in the other sublattice remains below some critical value. Chapter 6 presents two Q-switched Erbium-doped fiber lasers utilizing topology insulators as a saturable absorber. Two different passively Q-switched Erbium-doped fiber lasers are demonstrated using a few-layers Bi₂Se₃ and Bi₂Te₃ based saturable absorbers to exploit the wideband saturable-absorption characteristic of the topology insulators. Chapter 7 introduces several statistical aspects related to the critical phenomena of topological phase transitions. The concept is based on the observation that a curvature function used to calculate topological invariants diverges at where the band gap closes as the system approaches a topological phase transition. Introducing a renormalization group procedure for the curvature function, scale invariance allows us to characterize the topological phases and to define in a natural way a correlation function based on the Wannier functions. Similar to the standard critical behavior we can define critical exponents and universality classes. We will demonstrate the generality of these aspects by applying it to a number of systems in different dimensions and symmetry classes. The volume ends with 3 applications chapters on “Behavior of Dielectric Materials Under Electron Irradiation in a SEM”; “Photonic Crystal Fiber (PCF) Is a New Paradigm for Realization of Topological Insulator”; and “Patterned 2D Thin Films Topological Insulators for Potential Plasmonic Applications”.

It is hoped that the data tabulations and other information gathered together in this book will have a significant influence on expediting the progress of future research.

I would like to express my gratitude to all the contributors for their collective and fruitful work. It is their efforts and expertise that have made this book comprehensive, valuable and unique. I am also grateful to the managing editors, the International Association of Advanced Materials and publisher for their help and useful suggestions in preparing the “Advanced Topological Insulators.”

Huixia Luo
December 2018