Novel Topological Valleytronics in Moiré Heterostructures

Novel topological valleytronics in moiré heterostructures

Chen Hu Centre for the Physics of Materials Department of Physics McGill University Montréal, Québec Canada

A Thesis submitted to the Faculty of Graduate Studies and Research in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy

c Chen Hu, 2020

Je me souviens

Contents

Abstract ix

Résumé xi

Statement of Originality xiii

Acknowledgments xv

1 Introduction 1

1.1 Carbon-based electronics ...... 1

1.2 Topological insulators ...... 7

1.3 Topological valleytronics in 2D ...... 11

1.4 Topological Zak phase in 1D ...... 15

1.5 Basic theorems of density functional theory ...... 19

1.6 Outline of the thesis ...... 22

2 Moiré valleytronics: realizing dense topological arrays 25

2.1 Introduction ...... 25

2.2 Gr/hBN moiré and topological electronic structure ...... 27

2.3 Topological analysis of moiré valleytronics ...... 30

2.3.1 Berry curvature and interlayer interaction ...... 30

2.3.2 Valley Chern number and bulk-edge correspondence ...... 33

2.3.3 Topological phase transition in periodic moiré patterns ...... 34

2.4 Generality of moiré valleytronics ...... 36

2.4.1 Moiré system of strained bilayer graphene ...... 36

v vi Contents

2.4.2 Moiré system of silicene/hBN ...... 38

2.5 Structural robustness of moiré valleytronics ...... 39

2.6 Summary ...... 41

3 Theoretical design of topological heteronanotubes 43

3.1 Introduction ...... 43

3.2 Geometry and electronic properties of THTs ...... 44

3.3 Valley-dependent topological analysis ...... 47

3.4 Spiral-oriented THTs and topological solenoid ...... 50

3.5 Generality and robustness of THTs ...... 53

3.5.1 Other diameters ...... 53

3.5.2 Multi-period THTs ...... 55

3.5.3 Reverse-ordered THTs ...... 56

3.5.4 Commensurate double-wall heteronanotubes ...... 56

3.5.5 Zigzag and chiral heteronanotubes ...... 57

3.5.6 Topological properties of double-wall CNTs ...... 57

3.6 Summary ...... 59

4 Topological Zak phase of zigzag carbon nanotubes 61

4.1 Introduction ...... 61

4.2 Z2, Wannier center and Zak phase ...... 63

4.3 Topological Zak phase of zigzag CNT ...... 64

4.3.1 Two-band model of zigzag CNT ...... 65

4.3.2 Parity analysis of eigenstates ...... 67

4.3.3 The 2N -rule ...... 70

4.3.4 Wannier center analysis ...... 71 Contents vii

4.4 Edge states in ﬁnite zigzag CNTs ...... 72

4.4.1 FES and TES under gating electric ﬁeld ...... 74

4.4.2 FES and TES under random edge-potential environment ...... 76

4.5 Further discussions on diﬀerent terminations ...... 78

4.6 Summary ...... 80

5 Dirac electrons in 2D moiré superlattice 81

5.1 Introduction ...... 81

5.2 Dirac electronic states in ﬂat-sheet moiré patterns ...... 82

5.3 Dirac electronic states in wavelike moiré patterns ...... 89

5.4 Summary ...... 92

6 Conclusion 93

A calculations of berry curvature 97

B calculations of zak phase 100

Bibliography 104

Abstract

Two-dimensional (2D) van der Waals (vdW) heterostructures have attracted great attentions in recent years. By stacking diﬀerent 2D materials to weakly bond via the vdW force, the resulting artiﬁcial bilayers and/or multi-layers create novel material platforms for fundamental as well as technological exploration. A nearly ubiquitous feature of the vdW heterostructures is the moiré pattern caused by lattice mismatch or relative rotation of the two stacking lattices which creates a periodic lateral modulation on the electronic potential in the heterostructure. Such moiré modulation can form large-scale 2D and 1D superlattice leading to interesting new physics.

In this thesis, we propose and theoretically investigate the notion of topological valleytronics in 2D vdW heterostructures where moiré patterns are predicted to induce arrays of topologically protected pathways of valley electronic states. Systematic and first-principles calculations on the graphene/hBN systems are carried out to establish a basic understanding of the formation of topological moiré edges, and the novel moiré topological physics is also found in a broad range of other 2D vdW materials. Importantly, due to the structural robustness, the moiré topological physics can be extended to 1D: where the 2D graphene/hBN materials are rolled up into tubular structures to form topological heteronanotubes (THT). Due to essentially infinite possibilities of the tube index, interesting valley topological states are found and modulated by various 1D moiré patterns with different moiré periods and chiral orientations. We also show that the novel valley-Dirac physics resulted from moiré patterns show up in other aspects, including the appearance of secondary Dirac cones induced by the large-scale honeycomb graphene/hBN moiré superlattice. The topological order of the 2D moiré systems are analyzed by calculating the valley Chern number of the electronic bands. For 1D carbon nanotube (CNT) systems, the topological classification can be built by calculating the topological Zak phase. Interestingly, we discover that the Zak phase of zigzag CNTs can be quantized to take 0 or π, suggesting topological differences in the insulating phases of the zigzag CNT, and a 2N -rule is established both analytically and numerically that topologically classifies the insulating phases.

We conclude that moiré patterns which appear naturally on 2D vdW double-layer materials and/or 1D tubular carbon structures lead to very rich and novel valley topological

ix x Abstract states which are protected by certain topological orders and thus can sustain large external perturbations. Our researches in this thesis provide a general, robust and experimentally feasible platform for both investigating fundamental science of low-dimensional topological physics and practical applications in reliable low-power carbon-based nanodevices and nanoelectronics. Résumé

Les hétérostructures bidimensionnelles (2D) de van der Waals (vdW) ont attiré l’attention dans les dernières années. En empilant diﬀérents matériaux 2D liés faiblement par la force de vdW, on obtient des bicouches ou des multicouches artiﬁcielles qui créent autant de su- jets recherche fondamentale et d’avenues de développement technologique. Une caractéris- tique presque commune à toutes les hétérostructures de vdW est le moiré provoqué par le mésappariement des structures cristallines ou la rotation relative des couches, ce qui crée une modulation latérale périodique du potentiel électronique dans l’hétérostructure. Un tel moiré peut ainsi former une super-structure cristalline 1D ou 2D produisant d’intéressants phénomènes physiques.

Dans cette thèse, nous proposons et étudions théoriquement la notion de vallétronique topologique dans les hétérostructures 2D de vdW, selon laquelle les moirés induisent des mosaïques de canaux de conductions topologiques pour les états électroniques des vallées. Des calculs systématiques à partir de principes premiers sont effectuées sur des systèmes graphène/hBN pour établir une compréhension de base de la formation de canaux topologiques de moiré, et les nouveaux phénomènes topologiques des moirés se retrouvent également dans une vaste gamme de matériaux 2D de vdW. En effet, en raison de la robustesse struc- turelle, la physique topologique des moirés peut aussi être observée en 1D: où les matériaux graphène/hBN 2D sont enroulés de sorte à former des structures tubulaires qu’on nomme des hétéronanotubes topologiques (THT). à cause des possibilités infinies de l’indice des tubes, des états de vallée topologiques intéressants sont présents en 1D et modulés par les moirés avec diverses périodes et orientations chirales. Nous montrons également que la nou- velle physique des points de Dirac provenant des moirés se manifeste sous d’autres formes, y compris en induisant des cônes de Dirac secondaires dans les super-structures cristallines graphène/hBN en nid d’abeille. L’ordre topologique des moirés 2D est analysé en calculant le nombre Chern de vallée des bandes électroniques. Pour les nanotubes de carbone 1D (CNT), la classification topologique peut être établie en calculant la phase topologique de Zak. Fait intéressant, nous découvrons que la phase de Zak des CNT orientés zigzag est quantifiée et prend une valeur de 0 ou π, ce qui suggère la présence de différentes topologies parmi les phases isolantes des CNT orientés zigzag, et une règle 2N qui classe topologiquement les phases isolantes est établie à la fois analytiquement et numériquement.

xi xii Résumé

Nous concluons que les moirés qui apparaissent naturellement dans les matériaux bicouches 2D de vdW et les structures de carbone tubulaires 1D mènent à des états topologiques de vallée nouveaux et variés qui sont protégés par certains ordres topologiques, et qui peuvent donc resister à de grandes perturbations. Les recherches exposées dans cette thèse fournissent un aperçu général, robuste et expérimentalement faisable permettant d’étudier la science fondamentale de la topologie de dimension réduite et les applications pratiques que sont les nanodispositifs à base de carbone et la nanoélectronique ﬁables de faible puissance. Statement of Originality

In this thesis, by analytical as well as numerical ﬁrst-principles techniques, I have theoretically investigated novel valleytronics, topological physics and Dirac electronics in emerging low- dimensional carbon-based materials. My original contributions include:

• I proposed and investigated the idea of moiré valleytronics - a rather general and robust material platform to realize high-density arrays of 1D topological helical channels. I developed methods to numerically calculate the Berry curvature and the topological Chern invariants based on the k · p perturbation theory within our state-of-the-art large-scale DFT codes. Results are summarized in Chapter 2.

• I proposed and investigated the idea of topological heteronanotubes (THT) formed by a carbon nanotube (CNT) inside a boron nitride nanotube (BNNT). I discovered that such THTs are low-dissipation 1D conductors and the spiral-type THTs could serve as nanoscale solenoids generating remarkable magnetic ﬁelds. Results are summarized in Chapter 3.

• I discovered a 2N -rule that topologically classiﬁes the insulating phases of zigzag CNTs, using the topological Zak phase. I have demonstrated that topological zigzag CNTs generate very robust topological end states that survive strong external perturbations. I have also analytically derived an eﬀective low-energy model of zigzag CNTs leading to a closed formula of the 2N -rule. Results are summarized in Chapter 4.

• I investigated the Dirac electronic properties modulated by 2D moiré patterns in graphene/hBN double-layer materials. Both ﬂat-sheet moiré and wavelike moiré structures were calculated by a state-of-the-art density functional theory (DFT) method for systems as large as over twelve thousand atoms. Results are summarized in Chapter 5.

My researches in this thesis are summarized in the following four articles [1, 2, 3, 4]:

1. Chen Hu and Hong Guo, 2N-rule: Topological classiﬁcation of zigzag carbon nanotubes (manuscript under journal review, 2020);

xiii xiv Statement of Originality

2. Chen Hu, Vincent Michaud-Rioux, Wang Yao, and Hong Guo, Theoretical design of topological heteronanotubes, Nano Letters 19, 4146 (2019);

3. Chen Hu, Vincent Michaud-Rioux, Wang Yao, and Hong Guo, Moiré Valleytronics: Realizing Dense Arrays of Topological Helical Channels, Physical Review Letters 121, 186403 (2018);

4. Chen Hu, Vincent Michaud-Rioux, Xianghua Kong, and Hong Guo, Dirac electrons in Moiré superlattice: From two to three dimensions, Physical Review Materials 1, 061003 (R) (2017) (Rapid communication).

In addition to the above theoretical work, during my Ph.D studies I also carried out several collaborative projects with experimental groups to investigate quantum transport in single- molecule devices. In these collaborations, I was responsible for all the theoretical formulations and calculations. While these works are not discussed in this thesis, some results have been published in the following three papers:

1. Hongliang Chen*, Haining Zheng*, Chen Hu*, Kang Cai, Yang Jiao, Long Zhang, Feng Jiang, Indranil Roy, Yunyan Qiu, Dengke Shen, Yuanning Feng, Fehaid M. Alsubaie, Hong Guo, Wenjing Hong, J. Fraser Stoddart, Giant conductance Enhancement of intramolecular circuits through interchannel gating, Matter 2, 1(2020) (Cover page);

2. Linan Meng*, Na Xin*, Chen Hu*, Jinying Wang, Bo Gui, Junjie Shi, Cheng Wang, Cheng Shen, Guangyu Zhang, Hong Guo, Sheng Meng, Xuefeng Guo, Side-Group chemical gating via reversible optical and electric control in a single-molecule transistor, Nature Communications 10, 1450 (2019);

3. Chunhui Gu*, Chen Hu*, Ying Wei, Dongqing Lin, Chuancheng Jia, Mingzhi Li, D- ingkai Su, Jianxin Guan, Andong Xia, Linghai Xie, Abraham Nitzan, Hong Guo, X- uefeng Guo, Label-free dynamic detection of single-molecule nucleophilic substitution reaction, Nano Letters 18, 4156 (2018). Acknowledgments

First and foremost, I would like to thank my supervisor, Professor Hong Guo for giving me the opportunity to study in this distinguished group; for helping me build solid knowledge, skills and background in a broad range of research areas: theoretical and computational condensed matter physics, material and chemical physics and quantum transport theory and modeling; for sharing his deep physical insights, rich academic experience and novel research ideas with me; for guiding me on all of my PhD projects; for passing me strong passion and interest about science; for proving me wide collaborations and relationships with excellent theoretical and experimental groups all over the world; for enlightening me on a future academic career path. All in all, Prof. Guo teaches me how to think as a physicist.

Next, I would also like to thank my collaborators and coauthors. Theory: Prof. Wang Yao for greatly inspiring me by his interesting ideas and strong knowledge of topological physics; Dr. Vincent Michaud-Rioux for providing me the state-of-art RESCU package, and giving me long-time help on my DFT coding, parallel computing and research projects; Dr. Xianghua Kong for her much help on DFT applications and useful discussions about material chemistry. Experiment: Prof. Xuefeng Guo, Prof. Fraser Stoddart, Dr. Hongliang Chen, Dr. Na Xin and Dr. Chunhui Gu for their excellent experiments on single-molecular devices. Collaborating with them largely broadens my horizons and lets me learn a lot of multi-discipline knowledge on chemistry, engineering and nanoscience.

Then, I would like to thank all my colleagues and professors with whom I have learned and discussed during my years at McGill. To: Prof. Kirk H. Bevan for giving me useful DFT courses and good training about DFT coding; Dr. Lei Liu for his help on NEGF- DFT theory and Nanodcal package; Chenyi Zhou for a lot of inspiring discussions on many- body theory, DFT theory and material sciences, and for his long-time "peer pressure" which absolutely makes me be better; Dr. Qing Shi for his help on LAMMPS package and interesting discussions on device physics; Dr. Ying-Chih Chen for providing optimized ONCV atomic basis; Dr. Peng Kang for help on building moiré patterns; Maoyuan Wang for the help on topological number calculations; Prof. Manuel Smeu, Stuart Shepard, Ulises Torres Herrera, Dr. Saeed Bohloul, Dr. Mohammed Harb, Raphaël Prentki, Daniel Abarbanel, Xiaodong Xu, Dr. Eric Zhu, Prof. Haoran Chang Prof. Huichao Li, and Prof. Dawei Kang for lots of

xv xvi Acknowledgments help and useful discussions on various research ﬁelds.

Also, I want to thank the members of "McGill Meal Group": Chenyi Zhou, Qing Shi, Zezhou Liu, Yuning Zhang and Peng Kang. PhD life in Physics is not easy, and the time with you helps me survive better abroad. I also want to thank our "Guo’s Daily Walking Group", from which I beneﬁt a lot.

Finally, many thanks to my family for their huge support and suggestions on my career life. Specially to dear Yueming, thank you for your warm company, great support, continuous encouragement, and kind understanding all the way. Physical Constants and Units

1 Å = 10−10 m

a0 (Bohr radius) = 0.5292 Å −31 me (electron mass) = 9.1096 × 10 kg e (electron charge) = 1.6 ×10−19 C h (Planck’s constant) = 6.626 × 10−34 J · s −23 kB (Boltzmann’s constant) = 1.38 × 10 K

kBT (at 300 K ) = 0.026 eV c (speed of light) = 2.9979 × 108 m/s

Atomic units are used throughout this thesis unless otherwise indicated. In this system of units, e = me =h ¯ = 1.

1 unit of Length = a0 = 0.5292 Å −31 1 unit of Mass = me = 9.1096 ×10 kg 1 unit of Charge = e = 1.6 ×10−19 C 1 unit of Angular momentum = h¯ = 1.0546 ×10−34 J · s 1 unit of Energy = 1 Hartree = 27.2 eV

xvii List of Abbreviations

(1,2,3)D (One-, Two-, Three-) Dimensional 1DMP One-Dimensional Moiré Pattern a.u. Atomic Units BNNT Boron Nitride Nanotube BZ Brillouin Zone CNT Carbon Nanotube DFT Density Functional Theory D(S)ZP Double- (Single-) Zeta Polarized FET Field-Eﬀect Transistors GGA Generalized Gradient Approximation hBN hexagonal Boron Nitride h.c. Hermitian Conjugate KS Kohn-Sham LAMMPS Large-scale Atomic/Molecular Massively Parallel Simulator LCAO Linear Combination of Atomic Orbitals LDA Local Density Approximation N(T)I Normal (Topological) Insulator O(S)DC Original (Secondary) Dirac Cone ONCV Optimized Norm Conserving Vanderbilt QH Quantum Hall QSH Quantum Spin Hall PBC Periodic Boundary Condition SCF Self-Consistent Field SOI Spin-Orbital Interaction SSH Su, Schrieﬀer, Heeger TKNN Thouless, Kohmoto, Nightingale, Nijs THT Topological Heteronanotube vdW van der Waals WC Wannier Center XC Exchange-Correlation

xviii List of Figures

1.1 The graphene hexagonal real-space lattice (a) and k-space lattice (b). In (a), two groups of carbon atom sublattices, A and B, are denoted by light and dark gray balls respectively.√ a1 and a2 are two lattice vectors, with the magnitude of |a1| = |a2| = a = 3aCC , where aCC is the C-C bond length (1.42 Å). The red-line zone is the origin unit cell and the orange-dashed-line zones are the nearest-neighbor cells. In (b), b1 and b2 represent the k-space lattice vectors. The high-symmetry points in the BZ, Γ, M, K and K0 are shown in the ﬁgure. The Dirac cones are located at two groups of hexagonal corners (K and K0)...... 2

1.2 Band structure of graphene. Dashed hexagonal zone denotes the location of the BZ. At each valley (K or K0), the conduction and valence bands touch and the dispersion of the bands is linear...... 4

1.3 (a) The structure of graphene and CNTs. A CNT can be formed by rolling a ribbon of graphene along the Ch. In the plot, an example case is shown: Ch(n1, n2) = (4, 2). k⊥ and kk are the wavevectors along the circumference and transnational direction (T ), respectively. (b) The band structures from graphene to CNTs. The quantization of the circumferential momentum, k⊥, leads to the formation of a set of discrete energy subbands for each nanotube (red parallel lines). If those lines pass through the Dirac points (K or K0), the nanotube is metallic; otherwise, the nanotube is a semiconductor, as the case of (b). Adapted from Ref. [6]...... 7

1.4 (a) The interface between a quantum Hall (QH) insulator C = 1 and a normal insulator C = 0 holds a topological conducting state (black arrow). (b) Band structure of (a), where a single topological edge mode connecting the valence band to the conduction band. (c) The interface between a quantum spin Hall (QSH) insulator Z2 = 1 and a normal insulator Z2 = 0 holds a pair of counterpropagating topological conducting states (blue arrow for spin up and the green arrow for spin down). (d) Band structure of (c), where two topological modes crossing the Fermi level: spin-up mode with positive group velocity and spin-down mode with negative group velocity. Note that both in (a) and (c), vacuum can serve as the normal insulator and then the topological edge states occur in isolated ﬁnite topological systems. Rearranged from Ref. [20]...... 10

1.5 The k-space distribution of the Berry curvature Ω...... 13

xix xx List of Figures

1.6 (a) Schematic of the prototypical bilayer graphene devices with dual-split gating, where opposite polarities of gating voltage (V ) are added on each side of the device. (b) Theoretical energy-dispersion calculation of the configuration in (a), with the gapless topological helical states crossing the Fermi level. (c) Experimental realization of the dual-split-gating bilayer graphene device. Lower panels are scanning electron micrograph images of the junction. (d) Measured conductance in different gating configurations: "(+−)", i.e. the red curve, denotes the case of opposite polarities of gating V (helical channels appears); while (−−)", i.e. the blue curve, is the control group with same polarity of gating (the absence of helical channels). Rearranged from Refs. [31, 37]...... 14

1.7 (a) The chemical structure of the polyacetylene. (b) The sketch conﬁguration of the SSH model, where the purple box denotes one unit cell with intracell coupling v and intercell coupling w...... 16

1.8 (a-c) Band structures of the SSH model: (a) v = 1.5, w = 1; (b) v = 1, w = 1; (c) v = 1, w = 1.5. (d-f) Corresponding dispersion diagrams in parameter space (dx − dy plane). In each case, the path of the endpoints of the vector d(k) represents the SSH Hamiltonian (Eq.1.23 and Eq.1.24) as the wavevector k is swept across the BZ, i.e. k = 0 → 2π (the path of a closed loop). The eigenenergy E equals to the amplitude of d(k). The quantities, k, v and w are also shown in the plot...... 18

1.9 (a-c) Energy spectra of ﬁnite SSH chains for (a): v = 1.5 and w = 1, i.e. a normal insulator; (b) v = 1 and w = 1.5, i.e. a topological insulator. The length of both chains is taken as 50 unit cells in Fig.1.7(b). The gray lines are the bulk energy levels with a same gap Eg = 2|v − w| = 1. In (b), a group of topological edge states (red lines) occur at the Fermi level, protected by the bulk-edge correspondence...... 19

1.10 Flowchart of self-consistent solution procedure in DFT calculation...... 21

2.1 (a) The graphene and hBN lattices, strained along the zigzag direction to match while left free along the armchair direction (insets) to form the 1D moiré pattern. a and b are the lattice constants of primitive graphene, with the value of 2.4595 and 4.26 Å, respectively. (b) The 1D moiré pattern periodic along the armchair direction formed by three diﬀerent high-symmetry local stacking conﬁgurations indicated as N-h, B-h and B-N stacking...... 27 List of Figures xxi

2.2 (a) Band structure of the 1DMP. The linear bands near the Fermi level (energy zero), red and blue lines, are the two valley helical channels, and the gray lines are bulk 0 bands. The valley pointsK and K are at kx = 1/3 and 2/3, respectively. (b) Wave 2 functions |ψy| along the armchair (y) direction of the two helical channels with the same color scheme; vertical dashed lines indicate central positions of the three high- symmetry local stacking conﬁgurations. The left (right) panels are for k1 = 0.3329 (k2 = 0.3338), located at the left (right) of the K valley. (c) Illustration of valley helical channels and valley current. Upper: Solid (empty) circles denote carriers with positive (negative) group velocity, i.e. right movers (left movers). Solid (dashed) lines are helical channels with valley index K or K0, respectively. Lower: The red (blue) color is for a channel located at edge 1 (2); purple and green arrows indicate valley currents. (d) Topological valley currents ﬂow at the moiré edges...... 29

2.3 Topology analysis of the lattice-matched Gr/hBN bilayer. (a) Three high-symmetry stacking conﬁgurations in the moiré structure, where α and β are two subsites of the graphene lattice. (b) Corresponding k-resolved Berry curvature Ω(k) of the linear conducting band...... 31

2.4 Bulk-edge correspondence in the valley-dependent topological phase diagram. (a-b) Lattice-matched stackings of (a) opposite topology and (b) same topology. (c-d) Band structures corresponding to (a) and (b), respectively. (e) Spatial distribution 2 of |ψy| of the two helical channels in (c), red (blue) curves are for the red (blue) channel respectively, and the vertical dashed lines denote edges between the two lattice-matched conﬁgurations. The left (right) two panels are taken at k = 0.3325 (0.3375), located at left (right) of K valley...... 34

2.5 (a) Valley-dependent topological phase diagram parametrized by r0, which is the lattice displacement between Gr and hBN layers, and CV is the calculated valley- dependent Chern number of K valley. (b)Schematic demonstration of topological valley currents ﬂowing at the moiré edges [referring to Fig.2.2(d)]...... 36

2.6 (a) Bilayer graphene lattice. The black arrows denote the uniaxial tensile strain added along the armchair direction on the layer 1. (b) The bilayer graphene moiré structure. AB and BA are two local stackings. In this ﬁgure purple and grey balls represent carbon atoms in Layer 1 and 2, respectively...... 37

2.7 (a) Graphene lattice. (b) Silicene lattice. d is the buckled parameter...... 38

2.8 (a) Relaxed freestanding Gr/hBN bilayer with out-of-plane (z-direction) corrugation of about 13 Å. (b) Left: Band structure; right: spatial distribution of modular squared wave functions of the two helical channels, where the plotted isosurface (pink and light blue areas) is 4 × 10−9 a.u. (c) Topological valley current in the freestanding moiré structure...... 40 xxii List of Figures

3.1 (a-c) Geometries of the armchair single-wall CNT, single-wall BNNT and THT respectively, where THT is a double-wall CNT(inner)@BNNT(outer) tube. (d-f) Calculated band structures of the three tubes with n = 96 and m = 1. The Fermi level is shifted to energy zero, and the valley points K and K’ are at kz = 1/3 and 2/3, respectively. Bulk bands are the gray lines. In (f), the linear red and green bands near the Fermi level are the two valley-dependent topological helical states. a = 2.4595 Å is the lattice constant of the transnational vector T. For armchair CNT and BNNT, the small lattice diﬀerence of T (1.8%) is neglected. (g-i) Spatial distribution of modular squared wave functions |ψ|2 on the tube circumference, which are the Bloch states indicated in the inset at chemical potential µ = E. The solid (empty) circles denote carriers with positive (negative) group velocity, i.e. right movers (left movers), and solid (dashed) lines are helical states with valley index K or K’, respectively. In (i), since |ψ|2 is from carbon atoms, only CNT circumference is plotted for clearer vision. (j-l) Schematics of charge transport in these nanotubes. Arrows on the wires denotes electron ﬂow, curved white arrows on the nanotubes indicate backscattering from one conducting channel to another...... 45

3.2 (a) Schematics of rolling and unrolling processes, building a mapping between 1D THT and its corresponding 2D flat double-layer moiré structure. (b) 2D moiré mapping with n = 96. Ch is the circumference direction (chiral vector), and T is the tube- axis direction (translational vector). Some representative high-symmetry local atomic registries are shown in green circles, where gray, pink and blue balls denote carbon, boron and nitrogen atoms respectively. (c) Valley-dependent Chern numbers (CV) for valley K of lattice-matched configurations at the corresponding real-space locations. (d) Circular distribution of CV for valley K rolled up from the flat-sheet in (c). In (c) and (d), purple and light blue regions denote CV = -1/2 and 1/2 respectively. (e) Spatial distribution of modular squared wave functions |ψ|2 along the circumference of the topological helical states in Fig.3.1(f)...... 48

3.3 (a) Illustration for creating spiral THTs with topological moiré edge vector e (indicated by yellow arrow) not parallel to the transitional vector T . The orange rectangle denotes the supercell of the 2D mapping of the spiral THT, containing 8448 atoms. (b) Band structure of the spiral THT. (c) Spatial distribution of modular squared wave functions |ψ|2 of the topological helical states where the plotted isosurfaces (red and green areas) are 1.8 × 10−11 a.u. (d) Schematic demonstration of the spiral THT device in a circuit to serve as a topological solenoid or nanomagnet. Arrows denote the direction of electron ﬂow...... 52

3.4 (a) Geometric structure of CNT(22,22)@BNNT(23,23), with the inner diameter of 2.98 nm as shown in the cross-section view of (b). (c) Dependence of total energy on the inter-wall distance. (d)Calculated band structure. Gray lines are bulk bands and red (green) lines are helical channels - same as those of Fig.3.1. (e) Spatial distribution of modular squared wave functions |ψ|2 along the circumference of the topological helical channels in (d). The plotted isosurfaces are 7 × 10−8 a.u. . . . . 54 List of Figures xxiii

3.5 (a-c) 2D mapping of THTs with diﬀerent values of m. n = 96 for all cases. (d-e) Calculated spatial distribution of modular squared wave functions |ψ|2 of multiple topological helical states along the circumference, corresponding to the cases of (a-c) respectively. The plotted isosurfaces are 2.2 × 10−9 a.u...... 55

3.6 Band structure of the reverse-ordered THT: CNT(n, n)@BNNT(n+m,n+m) with n = 96 and m = −1. Gray lines are bulk bands and red (green) lines are helical channels - same as those of Fig.3.1...... 56

3.7 Calculated band structure of the commensurate double-wall heteronanotubes: CN- T(n, n)@BNNT(n+m,n+m) with n = 96 and m = 0...... 57

3.8 (a) Band structure of a zigzag-type THT: CNT(n,0)@BNNT(n+m,0) with n=96 and m=1. b is the lattice constant of the transnational vector of 4.26 Å. (b) Band structure of a chiral-type THT: CNT(100,25)@BNNT(104,26). a is the lattice constant of the transnational vector with a value of 6.50 Å. Gray lines are bulk bands and red (green) lines are helical channels - same as those of Fig.3.1...... 58

3.9 (a) Armchair double-wall CNT of CNT(n,n)@CNT(n+m,n+m). Here we take m=1 for example. The carbon atoms of outer CNT are plotted by purple balls to distinguish from those of the inner CNT wall (gray color). (b) The 2D mapping of (a). AB and BA are two local stackings. (c) A schematic plot of a radial electric ﬁeld (orange arrows) that induces the topological states...... 59

3.10 Schematic summary of the topological heteronanotube (THT) scheme...... 60

4.1 Atomic conﬁgurations of the graphene (a) and the zigzag CNT (b). The graphene sheet in (a) is rolled up along the y direction to form the zigzag CNT in (b). The red- dashed zone of (a) and (b) denotes one unit cell. In (a), "1", "2", "3" and "4" label four atomic sites in one hexagonal benzene-like ring. Two groups of carbon atoms form two sublattices, A and B, as denoted by light and dark gray balls respectively.√ a1 and a2 are two lattice vectors, with the magnitude of |a1| = |a2| = a = 3aCC where aCC is the C-C bond length 1.42 Å...... 66

4.2 Complete ﬂow chart of phase determination of zigzag CNTs...... 71

4.3 Diagram between the sum of occupied Wannier centers x/d¯ and index n of zigzag CNTs(n,0). d is the 1D lattice constant, with the value of 4.26 Å...... 72

4.4 Energy spectra of ﬁnite zigzag CNTs: (a) CNT(17,0) and (b) CNT(16,0). As shown in the Table 4.2, CNT(17,0) is a normal insulator while CNT(16,0) is a topological insulator. The length of both tubes are taken as the same: L = 16.8 nm (40 unit cells). Gray lines are the bulk states with a bulk band gap: 0.609 eV for the CNT(17,0) and 0.622 eV for CNT(16,0). The color lines near the Fermi level denote the in-gap edge states: Fujita edge states (FES) and/or topological edge states (TES)...... 73 xxiv List of Figures

4.5 Energy spectra of finite zigzag CNTs under different gating electric fields E. The unit of E is Volt/Angstrom (V/Å). (a-c) are the cases of normally-insulating CNT(17,0), and (d-f) are the cases of topologically-insulating CNT(16,0). The length of both tubes are taken as the same: L = 16.8 nm (40 unit cells). Gray lines are the bulk states. The red and green lines near the Fermi level denote the topological edge states (TES) and the Fujita edge states (FES) respectively, and the orange lines are the mixing of them...... 75

4.6 Energy splitting gap of ﬁnite zigzag CNTs under random edge potential with diﬀer- ent maximal amplitudes Vmax. For each Vmax, we perform 30 sets of calculations including totally random number arrays as the edge potential. Red and blue lines denote the cases of topological insulator CNT(16,0) and normal insulator CNT(17,0), respectively. The length of both tubes are taken as the same: L = 16.8 nm (40 unit cells)...... 76

4.7 DFT total energies of finite CNT(17,0) systems versus tube lengths. The red and blue lines respectively represent different edges: Type A and Type B, corresponding to the configurations shown in Table.4.3. Large supercells of finite CNT systems (including up to 2040 atoms) are calculated by the real-space Kohn-Sham density functional theory (KS-DFT) as implemented in the RESCU package [52]...... 79

5.1 The flat-sheet Gr/hBN vertical heterostructure and the moiré patterns on top of the lattice, where the hBN and graphene have a natural lattice mismatch of 1.8%. Different patterns represent different high-symmetry stacking configurations. The red, green and gray balls denote boron, nitride and carbon atoms, respectively. The orange line zone is the calculated supercell with dimensions 137.73 Å × 137.73 Å that contains 12,322 atoms...... 83

5.2 The calculated band structures. (a) For the ﬂat moiré pattern. (b) For the strained, lattice-matched structure. Γ, M, K are high-symmetry k-points, located at the center, the edge center, and the corner points of the hexagonal Brillouin zone, respectively. In (a), S1, S2 and S3 are the SDC bands, O1 and O2 are the ODC bands, and there is a gap of 7.5 meV between O1 and O2 that is too small to be clearly resolved in the ﬁgure...... 85

2 5.3 Left column: spatial distribution of modular squared wave functions |ψn(r)| of the flat moiré system, where n is the label of the SDC and ODC bands (S1-S3, O1, O2). The isosurfaces used are 7.6 × 10−6 a.u. for SDCs and 2.7 × 10−6 a.u. for 2 ODCs. Middle column: amplified view of |ψn(r)| at selected regions as indicated. The representation of atoms is the same as those in Fig.5.1. Right column: indicating the stacking configurations. B-site, N-site and h-site represent carbon locations over boron atoms, nitrogen atoms and hollow sites, respectively...... 87

2 5.4 Spatial distribution of wave functions of the ODC band states |ψCBM,V BM (r)| in the lattice-matched system (B-h stacking). The isosurface value of wave functions and the representation of atoms are the same as those in Fig.5.3(d) or (e)...... 89 List of Figures xxv

5.5 (a) The relaxed freestanding Gr/hBN heterostructure. (b) The out-of-plane corrugation amplitude of hBN and Gr layers, unit in Å. (c) The calculated band structure of the freestanding structure. (d-e) Spatial distribution of wave functions of ODC band 2 states |ψCBM,V BM (r)| . The isosurface value of wave functions and the representation of atoms are the same as those in Fig.5.3(d-e)...... 90 List of Tables

4.1 Phase table of zigzag CNTs. M and I are the abbreviations of metal and insulator respectively...... 63

4.2 Complete phase table of zigzag CNTs. M, TI and NI are the abbreviations of metal, topological insulator and normal insulator respectively...... 70

4.3 Geometries and topological properties of two types of terminations (A or B) whose unit cells hold the spatial symmetries. In second row, the red boxes denote one bulk unit cell. From A to B, the unit cell is shifted by 1/4. In third row, the types of spatial symmetries for each bulk cell are presented. In the forth row, we show the edge configurations corresponding to each bulk unit cell (the second row), where red balls denote the edge carbon atoms. In Type A edge atoms are bonded with each other while in Type B edge atoms are dangling. The edge stability of finite CNTs and topological property of each type are demonstrated in the fifth and sixth row respectively...... 78

xxvi 1

Introduction

In this chapter, we will give introductions on (1) novel Dirac electronics in carbon-based materials; (2) topological insulators and topological bulk-edge correspondence; (3) topological valleytronics in 2D hexagonal lattices; (4) topological Zak phase in 1D systems; (5) basic theorems of Kohn-Sham density functional theory (KS-DFT). In the last section, an outline of the thesis is summarized.

1.1 Carbon-based electronics

In the last few decades, dramatic advances in microelectronics have fundamentally changed almost every aspect of our lives. By continuous miniaturization of the size of electronic devices such as the field-effect transistors (FETs) - following the Moore’s law, the integrated density, computing speed and performance have been exponentially improved. However, the technology scaling of silicon-based devices has been slowing down in recent years, due to a number of limitations of fundamental science as well as technological nature. Among them, the major hindrances to the Moore’s law scaling is the power consumption when billions of device units are integrated into a circuit [5]. Today, a very important and focused discussion in the scientific community is concerning a possible future device technology that is beyond the silicon-based Moore’s law.

In this discussion and its related researches, carbon electronics are interesting and may provide a viable route toward an emerging nanotechnology beyond silicon [5, 6, 7]. Carbon

1 2 1 Introduction nano-materials, namely the one-dimensional (1D) carbon nanotubes (CNTs) and/or two- dimensional (2D) graphene, have very unique and outstanding electronic properties for device application and can serve as conducting channels of FETs [8, 9, 10, 11], and interconnect wires or contacts between micro- and nano-scale units [12, 13].

Fabricated in 2004 by A. Geim and K. Novoselov using a mechanical exfoliation method [14], graphene has already attracted great attentions. Graphene not only provides an excellent 2D carbon-based material for high-performance nanodevices [13], but also opens a new world of exploring and searching emerging and novel 2D or quasi-2D physics, given the almost inﬁnite possibilities of synthesizing artiﬁcial 2D van der Waals (vdW) heterostructures [15]. Graphene has exceptional properties such as ultra-high room-temperature mobility, remarkable long mean-free path of the charge carriers, and massless Dirac fermion behaviors [13]. These features can be well understood by studying the electronic structures of graphene.

(a) (b) 푘푦 푏1 A B 푲 풂ퟐ Г 푀 푘푥 풂ퟏ 퐾’

푏2

Figure 1.1: The graphene hexagonal real-space lattice (a) and k-space lattice (b). In (a), two groups of carbon atom sublattices, A and B, are denoted by light and√ dark gray balls respectively. a1 and a2 are two lattice vectors, with the magnitude of |a1| = |a2| = a = 3aCC , where aCC is the C-C bond length (1.42 Å). The red-line zone is the origin unit cell and the orange-dashed-line zones are the nearest-neighbor cells. In (b), b1 and b2 represent the k-space lattice vectors. The high-symmetry points in the BZ, Γ, M, K and K0 are shown in the ﬁgure. The Dirac cones are located at two groups of hexagonal corners (K and K0).

The real-space lattice of graphene sheet and its corresponding Brillouin zone (BZ) are plotted in Fig.1.1. Graphene has a hexagonal structure with two carbons, A and B, in one unit cell. Only considering the nearest-neighbor hopping, a tight-binding Hamiltonian of the 1.1 Carbon-based electronics 3 graphene can be written as:

X † X ˆ † ˆ X † ˆ HGr = µA aî aî + µB bj bj − t (a î bj + h.c.), (1.1) i j i,j

† ˆ ˆ † where aˆi (aˆi ) and bj (bj ) are the annihilation (creation) operator of sublattice A and B respectively. h.c. is the abbreviation of hermitian conjugate. t is the C-C hopping energy

(bond energy) with the value of about 2.8 eV [13]. µA ans µB are the chemical potentials of sublattice A and B respectively. In pristine graphene, site A and B are equivalent (spatial- inversion symmetry preserved), thus µA = µB. For a clear energy reference, they are set to zero in the following discussions. Therefore, the Hamiltonian becomes:

X † ˆ HGr = −t (a ˆi bj + h.c.). (1.2) i,j

By Fourier transform, Eq.1.2 is transformed from real-space representation to the k-space representation (momentum space):

 √ √  ia(− 3 k + 1 k ) ia(− 3 k − 1 k ) 0 1 + e 2 x 2 y + e 2 x 2 y HGr(k) = −t ·  √ √  , ia( 3 k + 1 k ) ia( 3 k − 1 k ) 1 + e 2 x 2 y + e 2 x 2 y 0 (1.3) where kx and ky are the components of Bloch wave vectors in the x and y directions respectively. By using the pseudospin notation of the two-band model, Hamiltonian Eq.1.3 can be rewritten as:

HGr(k) = −td(k) · σ, (1.4) where σ (σx, σy and σz) are Pauli matrices, and the components of the parameter vector d(k) are:  √ d (k) = 1 + 2 cos 1 ak cos 3 ak ,  x 2 y 2 x  √ 1 3 dy(k) = 2 cos aky sin akx , (1.5)  2 2   dz(k) = 0. 4 1 Introduction

It is noted that dz(k) = 0 is guaranteed by the combination of spatial-inversion symmetry and time-reversal symmetry [13]. Solving this Hamiltonian, the eigenenergies of graphene are obtained:

s √ q 1 1 3 E (k) = ±t· d2(k) + d2(k) = ±t· 1 + 4 cos2 ak + 4 cos ak cos ak . (1.6) ± x y 2 y 2 y 2 x

The corresponding eigenstates of graphene are:

  1 e−iφ(k) u±(k) = √   . (1.7) 2 ∓1

The notations + and − denote the conduction band and valence band, respectively. The function of e−iφ(k) is a phase factor having the form:

d (k) − id (k) e−iφ(k) = x y . (1.8) q 2 2 dx(k) + dy(k)

Figure 1.2: Band structure of graphene. Dashed hexagonal zone denotes the location of the BZ. At each valley (K or K0), the conduction and valence bands touch and the dispersion of the bands is linear. Using Eq.1.6 we calculated the electronic band dispersion as shown in Fig.1.2. The most 1.1 Carbon-based electronics 5 prominent feature in the band structure is the Dirac cones (linear dispersion) near two groups of BZ corners: K = ( √2π , 2π ),K0 = ( √2π , − 2π ). Close to the Dirac points (such as K), where 3a 3a 3a 3a k = K + q with |q| |K|, a linear expansion of the full dispersion Eq.1.6 can be obtained:

E±(q) ≈ ±vF |q|, (1.9)

where q is the momentum measured relatively to the Dirac points and vF is the Fermi velocity, √ 6 given by vF = 3at/2 with a value of about 10 m/s [13]. Such high vF and massless Dirac- type dispersion lead to the ultrahigh carrier mobility in graphene. The linear dispersion also makes quasi-particles in graphene hold diﬀerent electronic properties as compared to those of conventional three-dimensional materials which possess parabolic dispersions near the band minimum.

By rolling graphene into a tubular structure produces a carbon nanotube (CNT) which is another emerging carbon-based electronic material. A CNT typically has a nanometer diameter but µm length [16, 17]. In fact, S. Iijima experimentally discovered CNTs in 1991 [18], many years before the single-layer graphene was exfoliated. The research ﬁeld of CNT has strongly beneﬁted from not only a broad fundamental interest of 1D physics but also the rich breakthroughs in CNT-based short-channel FETs [8, 9, 10, 11] and ideal interconnects [12].

As shown in Fig.1.3(a), from the viewpoint of atomic geometries, a 1D CNT is formed by rolling up a 2D graphene lattice along a particular direction, called “chiral vector": Ch = n1a1 + n2a2 ≡ (n1, n2), where a1 and a2 are the hexagonal lattice vectors and n1 and n2 are integers (tube index) [16]. Distinct from the semi-metallic pristine graphene which is gapless, CNTs can demonstrate different electronic structures depending on configurations characterized by Ch. Along the tube circumference (the Ch direction), periodic boundary condition (PBC) gives a spatial confinement leading to quantization of the allowed quantum states:

ik⊥·Ch ψk⊥ (r + Ch) = e ψk⊥ (r) = ψk⊥ (r), (1.10) 6 1 Introduction where the r and k⊥ are the components of the position and momentum along the circumference. A direct conclusion yields:

k⊥ · Ch = 2πv, (1.11) where v is an integer. Due to the restriction of Eq.1.11, for each particular Ch, only a set of discrete k⊥ are allowed, giving rise to the subbands shown in Fig.1.3(b). As for a metallic CNT, the Dirac points (K or K0) are included in the subband lines which means

K · Ch = 2πv. Then a very useful relationship can be obtained:

n1 − n2 = 3N, (1.12) where N is an integer. This relation is called the "3N "-rule which was ﬁrst proposed in

1990s [19], it provides a useful scheme to classify diﬀerent phases of CNT(n1,n2): if n1 − n2 is multiple of 3, it is metallic; otherwise it is an insulator with a ﬁnite gap Eg ∝ 1/D where

D is the diameter of the CNT [6, 16]. By the 3N -rule, armchair-type CNTs with n1 = n2, are always metallic. In experiments and practical applications, semiconducting-CNT-based FETs have been demonstrated [8, 9, 10, 11], and metallic nanotubes can be used as high- performance interconnects [12]. 1.2 Topological insulators 7

(a) (b)

푛1a1

푛2a2

Figure 1.3: (a) The structure of graphene and CNTs. A CNT can be formed by rolling a ribbon of graphene along the Ch. In the plot, an example case is shown: Ch(n1, n2) = (4, 2). k⊥ and kk are the wavevectors along the circumference and transnational direction (T ), respectively. (b) The band structures from graphene to CNTs. The quantization of the circumferential momentum, k⊥, leads to the formation of a set of discrete energy subbands for each nanotube (red parallel lines). If those lines pass through the Dirac points (K or K0), the nanotube is metallic; otherwise, the nanotube is a semiconductor, as the case of (b). Adapted from Ref. [6].

1.2 Topological insulators

Having brieﬂy discussed graphene and CNT, in this section we review the physics of topological insulators. The idea of classifying phases of matter using some topological invariances has in fact a long history [20] and the 2016 Nobel Prize of physics was awarded to three pioneers, D. J. Thouless, F. D. M. Haldane and J. M. Kosterlitz.

Early topological systems are built on the broken time-reversal symmetry. The quantum Hall (QH) state is a ground breaking example, which occurs when electrons confined to two dimensions are placed in a strong magnetic field [21]. The quantized Landau levels lead to insulating phases of the material which are essentially different from conventional (called topologically trivial or simply "trivial") insulators such as atomic insulators or vacuum. The fundamental theory to explain such difference was firstly given by D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs (TKNN) in 1982 [22], namely gapped band structures can be classified topologically by considering the equivalence classes of Hamiltonian 8 1 Introduction that can be continuously deformed from one to another without closing the energy gap [20]. In other word, if two insulating Hamiltonians belong to the same topological class, they are adiabatically equivalent and there must be an adiabatic deformation connecting them. For example, the topological classes of quantum Hall states can be characterized by non-zero integers, called Chern invariant. In Bloch band theory, the Chern number Cn for the band index n is calculated by the surface integral of the Berry curvature Ωn(k) [20, 23]:

1 Z C = Ω (k)d2k, (1.13) n 2π n

where Ωn(k) can be obtained by:

Ωn(k) = ∇k × ihunk|∇k|unki, (1.14)

where |unki is the periodic part of the Bloch states of the band with index n. The total Chern number C is by summing Cn over all occupied states, which has an important relation with 2 the quantum Hall conductivity [22] ρxy = Ce /h, where e and h are the electron charge and Plank’s constant, respectively. This relationship provides an accurate approach to measure the total Chern number C by experiments.

In a later paper, F. D. M. Haldane proposed the famous Haldane’s model for realizing quantum Hall states in graphene-based lattices [24]. By adding a staggered periodic magnetic flux to the 2D crystal (the average flux is thus zero), non-zero Chern number arises and the graphene becomes a topological insulator. Phases of states in both TKNN’s model and Haldane’s model are topologically classified by the Chern numbers. The key to realizing such Chern insulator is the broken time-reversal symmetry, which can be understood by the properties of the Berry curvature under symmetry operations [23]:

  PΩˆ n(k) = Ωn(−k), (1.15)  TΩˆ n(k) = −Ωn(−k). 1.2 Topological insulators 9

Pˆ and Tˆ are the operators of spatial-inversion symmetry and time-reversal symmetry, respectively. According to Eq.1.15, if a system has the time-reversal symmetry, Ωn(k) is an odd function and the integral of it, i.e. the Chern number (see Eq.1.13), must be zero. However, it doesn’t mean that no topological physics can happen in this situation. A famous counterexample is the quantum spin Hall (QSH) phase which can be characterized by a Z2 topological invariant [20]: Z2 = 1 or 0 corresponds to topologically nontrivial or trivial respectively. In 2005, C. L. Kane and E. J. Mele proposed a prototypical QSH system: graphene with spin-orbital iteration (SOI), which is also called Kane-Mele model [25]. In this model, the SOI leads to a new mass term to the graphene Hamiltonian (Eq.1.4) which opens a nontrivial band gap proportional to the strength of SOI. In experiments, materials with heavy elements having strong SOI are competitive candidates for the QSH phase, such as the Hg/Cd/Te quantum well structures [26].

A fundamental consequence of the topological classiﬁcation of gapped band structures is the existence of in-gap topologically-protected edge states at interfaces where the topological order changes [20]. Such physical phenomenon is called bulk-edge correspondence, which is a general and ubiquitous principle in all types of topological matters. Fig.1.4 demonstrates the appearance of the topological edge states in QH and QSH systems. In 2D systems, the edge states are in fact 1D conducting channels (or called helical channels) propagating along the interface. For a QH system, a single conducting channel appears while for a QSH system, a pair of channels with opposite spins are counter-propagating at the edges [20, 25]. The existence of such topological channels is deeply related to the topological phase transition at the interface. This relation can be understood by an intuitive physical picture. Namely, from the topological insulators to normal insulators, the Hamiltonian or gapped band structures cannot be adiabatically connected, which means that somewhere along the way the energy gap has to vanish because otherwise it is impossible for the topological invariant to change. There will therefore be low-energy electronic states bound to the region where the energy gap passes through zero. Fundamental and detailed theories of the bulk-edge correspondence can be found in Ref. [27]. 10 1 Introduction

(a) (b)

퐶 = 0

Normal insulator

QH insulator 퐶 = 1

푍2 = 0

Normal insulator

QSH insulator 푍2 = 1

Figure 1.4: (a) The interface between a quantum Hall (QH) insulator C = 1 and a normal insulator C = 0 holds a topological conducting state (black arrow). (b) Band structure of (a), where a single topological edge mode connecting the valence band to the conduction band. (c) The interface between a quantum spin Hall (QSH) insulator Z2 = 1 and a normal insulator Z2 = 0 holds a pair of counterpropagating topological conducting states (blue arrow for spin up and the green arrow for spin down). (d) Band structure of (c), where two topological modes crossing the Fermi level: spin-up mode with positive group velocity and spin- down mode with negative group velocity. Note that both in (a) and (c), vacuum can serve as the normal insulator and then the topological edge states occur in isolated ﬁnite topological systems. Rearranged from Ref. [20]. 1.3 Topological valleytronics in 2D 11

One of the most attractive features of the topological edge states is their topological robustness. The appearance of such states are protected by topological orders, therefore insensitive to most disorders, impurities, defects or other perturbations. From the viewpoint of charge transport, the topologically protected conducting channels are thus said to be "dissipationless". To briefly explain, we take the QSH system for example. As shown in Fig.1.4(c) and (d), two helical channels (blue for spin up and green for spin down) are counter propagating. In this case, the backscattering from one channel to another needs a simultaneous spin flipping, which is difficult in most situations, i.e. the backscattering is largely suppressed [20, 27]. Therefore, a ballistic transport character will occur in the conducting process of the topological helical channels. Practically, this should be quite useful for producing devices with low power-dissipation.

1.3 Topological valleytronics in 2D

Despite of the novel dissipationless topological channels, experimental realization of topological phases in carbon-based systems is not trivial. Due to the natural presence of time-reversal symmetry at zero magnetic ﬁeld, the QH state vanishes. Because carbon is a light element with extremely weak SOI, the graphene in the Kane-Mele model is in fact not suitable for experimental observations and practical applications of the QSH phase. A recent experimental study reported that the intrinsic spin-orbit bulk gap of graphene is 42.2 µ eV, which means that QSH cannot happen in graphene unless the temperature gets as low as 0.5 K [28].

In 2D hexagonal systems, the valley-related topological physics provides a promising route for realizing and studying novel topological phenomena [3, 29, 30, 31]. The concept of "valley" indicates the local minimum in the conduction band and/or local maximum in the valence band. When carriers (electrons or holes) occupy diﬀerent valleys, one may use the valley index to label these carriers. Analogous to to spin in spintronics, it is thus possible to use the valley degree of freedom to encode, process and store information, leading to a new ﬁeld to manipulate and investigate the valley-related electronics, called valleytronics [32]. Recently, 12 1 Introduction rapid progresses on valleytronics have been achieved in low-dimensional condensed matters [33, 34, 35, 36, 37].

As shown in Fig.1.2, in graphene, two inequivalent valleys labelled by K and K0, occur at the corners of BZ. Because of their large separation in the momentum space, intervalley scattering is strongly suppressed [4,5], implying that the valley index K or K0 can be treated as good quantum numbers. Close to the Dirac points, the Hamiltonian of graphene Eq.1.4 can be linearly expanded: √ 3 H(q) = at(q τ σ + q σ ), (1.16) 2 x z x y y where a is the lattice constant, t is the hopping parameter, q is the momentum measured from one valley point, and σ is the Pauli matrices. τz is the valley index (+1 for K, -1 for K0). By introducing a stagger potential term, the spatial symmetry can be broken in this system: √ 3 ∆ H(q) = at(q τ σ + q σ ) + σ , (1.17) 2 x z x y y 2 z where ∆ is the staggered potential describing the on-site potential diﬀerence between two sublattice sites (A and B in Fig.1.1). Solving this Hamiltonian and performing the topological analysis according to Eq.1.14, the Berry curvature of the conduction band is derived [29, 30]:

3a2∆t2 Ω(q) = τ . (1.18) z 2(∆2 + 3q2a2t2)3/2

This formula shows that diﬀerent valleys hold the Berry curvatures with the same magnitude but opposite signs (opposite directions +ˆz or −zˆ), as guaranteed by the time-reversal symmetry. Fig.1.5 plots the calculated Berry curvature in k-space, from which it is clear that the total integral of the Berry curvature (total Chern number C) vanishes. Nevertheless, for each valley, one can still deﬁne a valley-dependent Chern number CV by integrating the Berry curvature around that valley[29, 30].

Z 1 2 CV = Ω(q)d q. (1.19) 2π valley 1.3 Topological valleytronics in 2D 13

푲 퐾’

Figure 1.5: The k-space distribution of the Berry curvature Ω.

˜ Such valley-dependent Chern number CV (or called valley-resolved topological charge N3 in some literature, e.g. Ref. [30]) is nonzero, which can provide an eﬀective way to classify diﬀerent topological phases in hexagonal 2D lattices. By Eq.1.18, we can also conclude that the sign of stagger potential ∆ determines the valley-dependent topological order. This ∆ term naturally exists in monolayer transition metal dichalcogenides (TMDs) [38], and can be realized by a hBN substrate in monolayer graphene [3] or a gating voltage in bilayer graphene [31, 36, 37, 39, 40], as well as in buckled honeycomb systems [41] such as silicene, germanene and stanene.

The bulk-edge correspondence guarantees the appearance of valley-dependent topological states at the interface where the sign of the stagger potential ∆ changes. In 2008, I. Martin et al. theoretically proposed a dual-split-gate scheme in bilayer graphene [31] to realize such valley-dependent helical channels, as shown in Fig.1.6. Opposite polarities of a gating voltage (V ) on each side give rise to opposite signs of the stagger potential ∆ and different valley-dependent Chern numbers, leading to a topological phase transition at the interface of the junction. Therefore, gapless helical states occur inside the band gap [see Fig.1.6(b)]. Despite of the clear and interesting physics, the experimental realization of such scheme is very challenging, because precise alignments of four nanoscale splitting gates are required. The first experimental observation was done by J. Li et al. in 2016 [37]. The authors found that the topological helical channels have much higher transport efficiency than the normal 14 1 Introduction

(a) (b)

Figure 1.6: (a) Schematic of the prototypical bilayer graphene devices with dual-split gating, where opposite polarities of gating voltage (V ) are added on each side of the device. (b) Theoretical energy-dispersion calculation of the configuration in (a), with the gapless topological helical states crossing the Fermi level. (c) Experimental realization of the dual-split-gating bilayer graphene device. Lower panels are scanning electron micrograph images of the junction. (d) Measured conductance in different gating configurations: "(+−)", i.e. the red curve, denotes the case of opposite polarities of gating V (helical channels appears); while (−−)", i.e. the blue curve, is the control group with same polarity of gating (the absence of helical channels). Rearranged from Refs. [31, 37]. 1.4 Topological Zak phase in 1D 15 states, as shown in Fig.1.6(d). This is precisely because the topological states are robust and have ballistic character with largely suppressed backscattering as discussed above. This experiment also shows that the high-efficiency conducting regime can be up to 400 nm length and survive noncryogenic temperatures.

In Chapters 2 and 3, we shall respectively demonstrate that by utilizing the topological valleytronics we can realize dense topological arrays in moiré patterns and controllable topological pathways in heteronanotubes.

1.4 Topological Zak phase in 1D

So far we discussed the topological invariance in 2D systems based on the Berry curvature and Chern number. On the other hand, in 1D or quasi-1D systems, the intrinsic topological orders P are characterized by the total Zak phase γ, summed by all the occupied states: γ = n γn, where n is the band index. The Zak phase of each band γn is obtained by the integral of the

Berry connection, i umk|(∂umk/∂k) , across the 1D Brillouin zone (BZ) [42, 43]:

Z D ∂umk E γm = i umk| dk. (1.20) BZ ∂k

In general, the Zak phase takes any value. However, when a system has spatial symmetries (such as inversion and/or mirror), the total Zak phase γ can be quantized to 0 or π (modular 2π), corresponding to topologically trivial or nontrivial cases, respectively.

One of the most extensively studied topological 1D system is the polyacetylene, a conduc- tive polymer (see Fig.1.7). In 2000, A. G. MacDiarmid, H. Shirakawa and A. J. Heeger won the Nobel Prize in Chemistry for the discovery and development of this promising material. A famous model to describe the properties of polyacetylene was proposed by W. P. Su, J. R. Schrieﬀer and A. J. Heeger in 1979, called SSH model [44], which is written as: 16 1 Introduction

(a)

(b) 풗 풘

Figure 1.7: (a) The chemical structure of the polyacetylene. (b) The sketch conﬁguration of the SSH model, where the purple box denotes one unit cell with intracell coupling v and intercell coupling w.

N N N−1 X † ˆ † ˆ X † ˆ X ˆ † H = µ (a î aî + bi bi) + v (a î bi + h.c.) + w (bi aiˆ+1 + h.c.), (1.21) i=1 i=1 i=1

† ˆ ˆ † where aî (aî ) and bi (bi ) are the annihilation (creation) operators of the left and right site in the unit cell in Fig. 1.7(b), respectively. N is the total number of unit cell. µ, v and w is the chemical potential, intracell coupling and intercell coupling respectively. For a clear energy reference, µ is set to zero in the following. As for an infinite-length SSH chain, the periodic boundary condition can be used. Transforming Eq.1.21 to k-space, we get:

  0 v + w e−ik H(k) =   . (1.22) v + w eik 0

By using the pseudospin notation of the two-band model, Eq.1.22 can be rewritten as:

H(k) = d(k) · σ, (1.23)

where σ (σx, σy and σz) are Pauli matrices, and the components of the parameter vector d(k) are:   dx(k) = v + w cos k,  dy(k) = w sin k, (1.24)    dz(k) = 0. 1.4 Topological Zak phase in 1D 17

It is noted that dz(k) = 0 is guaranteed by the spatial symmetry (chiral symmetry) [45]. Solving this Hamiltonian matrix, we can obtain the eigenenergies of the SSH model:

√ 2 2 E±(k) = ±|d(k)| = ± v + w + 2vw cos k. (1.25)

The notation + and − denote the conduction and valence band respectively. The energy dispersion can be represented in a parameter space diagram, as shown in Fig.1.8. When v = w, the system is metallic with no gap, and in this case the dispersion path crosses the origin point (E = 0) in the dx − dy plane [see Fig.1.8(e)]. As long as the hopping amplitudes are staggered, i.e. v 6= w, there is an energy gap of Eg with:

Eg = 2|v − w|. (1.26)

The staggered hopping amplitudes occur naturally in many 1D solid state systems such as the polyacetylene, by what is known as the Peierls instability [45].

Very interestingly, although for both cases of v < w and v > w the band structures show the same insulating character, they are in fact topologically different in the Hamiltonian representation in d-space: the circular path loop of v < w winds around the origin but the loop of v > w does not, shown in Fig.1.8(f) and (d) respectively. The winding number is used to characterize different topological classes: the path circles of v < w and v > w cannot be smoothly connected to each other unless they cross the origin (close the gap), i.e. they are adiabatically non-equivalent and belong to different topological classes [27, 45].

We then use Eq.1.20 to compute the intrinsic 1D topological Zak phase of the SSH model and obtain:   π v < w, γ = (1.27)  0 v > w.

We therefore conclude that the cases of v < w and v > w correspond to topological insulator and normal insulator, respectively. According to the topological bulk-edge correspondence, a 18 1 Introduction

(a) 푣 > 푤 (b) 푣 = 푤 (c) 푣 < 푤

푘 푘 푘

(d) (e) 푑 (f) 푦 푑푦 푑푦 푤 푤 푘 푤 푑 푑푥 푥 푑푥 푣 푣 푣

Figure 1.8: (a-c) Band structures of the SSH model: (a) v = 1.5, w = 1; (b) v = 1, w = 1; (c) v = 1, w = 1.5. (d-f) Corresponding dispersion diagrams in parameter space (dx − dy plane). In each case, the path of the endpoints of the vector d(k) represents the SSH Hamiltonian (Eq.1.23 and Eq.1.24) as the wavevector k is swept across the BZ, i.e. k = 0 → 2π (the path of a closed loop). The eigenenergy E equals to the amplitude of d(k). The quantities, k, v and w are also shown in the plot. 1.5 Basic theorems of density functional theory 19

1D ﬁnite-length topological insulator has topologically protected edge states inside the bulk gap, as shown in Fig.1.9. For any ﬁnite length 1D material, its "edge" is in fact the "end" of the 1D material, thus the topological edge state is also called topological end-state.

In Chapter 4, by calculating the topological Zak phase we can classify diﬀerent insulating phases and investigate robust topological edge states in 1D CNT materials.

(a) 푣 = 1.5 , 푤 = 1 (b) 푣 = 1, 푤 = 1.5

Figure 1.9: (a-c) Energy spectra of ﬁnite SSH chains for (a): v = 1.5 and w = 1, i.e. a normal insulator; (b) v = 1 and w = 1.5, i.e. a topological insulator. The length of both chains is taken as 50 unit cells in Fig.1.7(b). The gray lines are the bulk energy levels with a same gap Eg = 2|v − w| = 1. In (b), a group of topological edge states (red lines) occur at the Fermi level, protected by the bulk-edge correspondence.

1.5 Basic theorems of density functional theory

In this thesis, we shall also calculate electronic structures and topological invariants of materials by the Kohn-Sham density functional theory (KS-DFT) methods and a brief review is in order. Since it was ﬁrstly proposed in 1965 [46], KS-DFT has become an extremely suc- cessful scheme for predicting mechanical, electronic and optical properties of materials and been widely applied in condensed matter physics, material science and quantum chemistry. Extensive literatures can be found for comprehensive and detailed introductions, derivations and implementations of KS-DFT methods [47, 48, 49]. In this section, we will only review 20 1 Introduction some basic theorems and formulas.

The core of KS-DFT is the well-known Kohn-Sham (KS) equation [46]:

1 − ∇2 + V (r) + V (r) + V (r) ψ (r) = ψ (r) . (1.28) 2 H ext XC n n n

Here, atomic units are applied. The ﬁrst, second, third and fourth terms in the left describes the kinetic energy, Hartree potential, external potential (which includes electron-ion interaction) and exchange-correlation potential, respectively. ψn (r) and n are the KS eigenstate and KS eigenenergy of the state labelled by n. The Hartree potential VH (r) gives the classical Coulomb interaction between electrons:

Z ρ (r0) V (r) = dr0 , (1.29) H |r − r0| which is the solution to the Poisson equation:

2 −∇ VH (r) = 4πρ(r), (1.30) where ρ(r) is the electron density:

X 2 ρ (r) = fFD(n)|ψn (r)| . (1.31) n fFD is the Fermi-Dirac distribution. The exchange-correlation potential VXC (r) is deﬁned as: δE [ρ] V (r) = XC (r) , (1.32) XC δρ where EXC is the exchange-correlation energy including all the non-classical and many-body eﬀects. EXC is a functional of density ρ(r), but its exact form is unknown. Nevertheless, in practical calculations, two most common approximations are applied to EXC : the local LDA LDA density approximation (LDA) [50] having EXC = EXC [ρ(r)] and the generalized gradient GGA GGA approximation (GGA) [51] having EXC = EXC [ρ(r), ∇ρ(r)]. 1.5 Basic theorems of density functional theory 21

Initial density 휌0(풓)

KS equation

New density 휌(풓)

휌(풓) NO converged?

YES

Save data

Figure 1.10: Flowchart of self-consistent solution procedure in DFT calculation.

Above discussions show that the eﬀective potential Veff (r) = VH (r) + Vext (r) + VXC (r) depends on the density ρ(r) or the solutions of the KS equation. The KS equation is con- sequently nonlinear and must be solved self-consistently by some iterative approach. This process is called self-consistent ﬁeld (SCF) calculation which is summarized in Fig.1.10.

When performing self-consistent KS-DFT calculation of a large material system, i.e. with more than a few hundred atoms, the huge bottleneck is diagonalizing the large Hamiltonian matrix when solving the KS equation Eq.1.28. In this thesis, however, the moiré pattern materials contain up to ten thousand atoms which was beyond most of the KS-DFT solvers - unless further approximations are made. Recently, our group developed a powerful real- space KS-DFT solver, RESCU [49, 52], that overcomes the bottleneck by using the Cheby- shev ﬁltering technique [53] and other advanced computational mathematics to drastically increase the computational eﬃciency, making it possible for calculating very large systems self-consistently. For more technical details we refer the interested readers to the original 22 1 Introduction reference [52].

In crystals, the transnational symmetry is preserved. The Bloch’s theorem dictates that under an external periodic potential (such as the crystal potential), the eigenstate of the system can be written as a periodic function multiplied by a plane wave:

ψnk (r) = exp(ik · r)unk (r) , (1.33)

where k is the crystal wavevector. unk (r) is the periodic part of the wave function. Combin- ing Eq.1.28 and Eq.1.33, we obtain the k-dependent KS equation under periodic boundary conditions [49]:

1 − (∇ + ik)2 + V (r) + V (r) + V (r) u (r) = u (r) . (1.34) 2 H ext XC nk nk nk

For each k point, the unk (r) is individually obtained by solving Eq.1.34. The k-dependent

Hamiltonian H(k) and wave function unk (r) play an important role when analyzing electronic structures and topological properties of materials, which will be demonstrated in detail in the following chapters.

1.6 Outline of the thesis

In this thesis, by both analytical and ﬁrst-principles techniques, we systematically investigate novel valleytronics, topological physics and Dirac electronics in emerging low-dimensional carbon-based systems, including 2D graphene, 1D carbon nanotubes, as well as a broad range of their van der Waals (vdW) heterostructures and moiré superlattices. This thesis is organized as follows:

• In Chapter 2, we propose a general and robust platform, the moiré valleytronics, to realize high-density arrays of 1D topological helical channels in real materials at room temperature. We demonstrate the idea using a long-period 1D moiré pattern of graphene 1.6 Outline of the thesis 23

on hBN by ﬁrst-principles calculation. Through the Berry curvature and valley topological charge of the electronic structure associated with various local graphene/hBN stackings in the moiré pattern, it is revealed that helical channel arrays originate in- trinsically from the periodic modulation of the local topological orders by the moiré pattern. For a freestanding wavelike moiré pattern, two groups of helical channel arrays are spatially separated out of plane, validating the structural robustness of the moiré topology. The generality and experimental feasibility of moiré valleytronics are demonstrated by investigating a broad range of moiré systems.

• In Chapter 3, we propose and investigate the idea of topological heteronanotube (THT) for realizing a one-dimensional (1D) topological material platform that can pave the way to low-power carbon nanoelectronics at room temperature. We predict that the coaxial double-wall heteronanotube - a carbon nanotube (CNT) inside a boron nitride nanotube (BNNT), can act as a THT. Dissipationless topological conducting pathways on the THT are protected by a valley-dependent topological Chern invariance that originates from local topological phase transitions of the CNT modulated by the CNT- BNNT interaction. Spiral THTs, where topological current flows spirally around the tube, function as nanoscale solenoids to induce remarkable magnetic fields due to the dense moiré nanopatterning. The generality and robustness of the THT materials are demonstrated by investigating different tube diameters, tube indexes, tube types as well as topological-pathway orientations through first-principles calculations.

• In Chapter 4, we discover and report an interesting 2N -rule that topologically classifies the insulating phases of the zigzag CNTs(n,0). For even n, n = 2N where N is an integer, the zigzag CNT is a topological insulator; and for odd n it is a normal insulator. The 2N -rule is established by calculating 1D topological Zak phase through both analytical and first-principles techniques. Cutting CNTs produces edge states at end of the tubes. In topological zigzag CNTs, topological edge states occur which are remarkably robust against strong gating electric fields and complicated edge environment, in comparison with the traditional Fujita edge states. 24 1 Introduction

• In Chapter 5, we investigate the Dirac electronic properties modulated by the 2D moiré pattern of graphene/hBN. We find that the local high-symmetry stacking configurations of the moiré pattern play the key role on the behaviors of the secondary Dirac cone (SDC) and the original Dirac cone (ODC). In the flat sheet, the SDC dispersion emerges due to the stacking-selected localization of SDC wave functions, while the original Dirac cone (ODC) gap is suppressed due to an overall effect of ODC wave functions. In the freestanding wavelike moiré structure, we predict that a specific local stacking will dominate, leading to opposite behaviors of moiré-modulated Dirac electrons.

• In Chapter 6, we summarize the thesis. 2

Moiré valleytronics: realizing dense topological arrays

In this chapter, we propose a general and robust platform, the moiré valleytronics, to realize high-density arrays of 1D topological helical channels in real materials at room temperature. Valley-dependent topological analysis reveals the local topological phase transition modulated by the periodic moiré potential in moiré superlattices. The generality, structural robustness and experimental feasibility of moiré valleytronics are demonstrated by investigating a broad range of moiré systems [3].

2.1 Introduction

Two-dimensional (2D) van der Waals (vdW) heterostructures have attracted great attention in the past few years [15]. By stacking different 2D materials to weakly bond via the vdW force, the resulting artificial bilayers and/or multilayers create novel material platforms for fundamental as well as technological exploration. A nearly ubiquitous feature of the vdW heterostructures is the moiré patterns caused by lattice mismatch or the relative rotation of the two stacking lattices [15, 54, 55, 56]. Moiré patterns can create a periodic lateral modulation on the electronic property in the heterostructure leading to interesting new physics. These include the appearance of secondary Dirac cones [57] and the self-similar Hofstadter butterfly states [58] in a graphene(Gr)/hBN bilayer, the nanoscale patterned optical properties and spin-orbit coupled exciton superlattices [59] and moiré-defined superstructures of quantum spin Hall insulators [60] in transition metal dichalcogenides, helical networks under

25 26 2 Moiré valleytronics: realizing dense topological arrays an interlayer bias [61] and the newly found correlated insulator behavior [62] and superconductivity [63] in a twisted graphene bilayer. Since a moiré superlattice can be experimentally created, they provide extraordinary opportunities for generating novel electronic properties that are hard to obtain otherwise.

In hexagonal 2D crystals, the valley degree of freedom, i.e. the degenerate Bloch band extremum at K and K0 corners of the hexagonal BZ as discussed in Chapter 1, is of particular interest [29, 38, 64, 41]. In graphene, the nontrivial topological properties of the Dirac cones at the two valleys have led to intriguing phenomena, such as the bulk topological valley current when a gap arises through inversion symmetry breaking [33, 34, 35]. In the gapped graphene system, the bulk topological properties of the valley also give rise to in-gap valley helical channels at topological line defects [31, 39, 40, 30] analogous to the quantum spin Hall edge states, and were experimentally observed in bilayer graphene [36, 37]. Such valley helical modes are protected from backscattering by the large momentum space separation, and can be exploited as conducting channels for quantum electronics. However, so far in the literature, realizations of the topological physics rely on random local stacking faults [36] or on using a dual-split-gate structure (see Fig.1.6) that is extremely challenging to fabricate [37]. Moreover, these schemes can host a single pair of helical channels, limiting the applicability because high-density valley channels and currents are required in circuit integration.

In this chapter, the scheme of moiré valleytronics is proposed and investigated, which can realize high-density arrays of 1D topological helical channels at room temperature and in realistic easily fabricatable materials. Speciﬁcally, an 1D graphene(Gr)/hBN moiré pattern and several other moiré systems are shown to establish the topological moiré valleytronics. For the 1D graphene(Gr)/hBN moiré pattern, we show by ﬁrst-principles calculations that a long-period moiré pattern realizes dense arrays of helical channels inside the noncryogenic bulk band gap that is opened by local inversion symmetry breaking due to the hBN. These helical channels arise from the periodical modulation in the topological order of local electronic structure, due to the variation of atomic stacking in the moiré pattern. The generality and structural robustness of the topological moiré valleytronics are demonstrated by investigating 2.2 Gr/hBN moiré and topological electronic structure 27 a broad range of moiré systems and freestanding Gr/hBN moiré structures.

2.2 Gr/hBN moiré and topological electronic structure

(a) Graphene hBN Armchair

B Zigzag N 풙 a a 풚 b 1.018b (b)

B-N B-h N-h B-N B-h N-h B-N

Figure 2.1: (a) The graphene and hBN lattices, strained along the zigzag direction to match while left free along the armchair direction (insets) to form the 1D moiré pattern. a and b are the lattice constants of primitive graphene, with the value of 2.4595 and 4.26 Å, respectively. (b) The 1D moiré pattern periodic along the armchair direction formed by three diﬀerent high-symmetry local stacking conﬁgurations indicated as N-h, B-h and B-N stacking.

Fig.2.1(a) shows the orientation of the graphene and the hBN lattices, and the latter has a lattice constant mismatch of 1.8% larger than that of the graphene. We will focus on the 1D moiré pattern (1DMP) formed by stacking graphene on hBN along the armchair direction [see Fig.2.1(a)]. To do so, hBN is compressed along the zigzag direction (x direction) to match the graphene, while its armchair direction (y) is left to freely and vertically stack on graphene, resulting in a periodic 1DMP shown in Fig.2.1(b). Along the armchair direction, the atomic registry of the two lattices varies gradually and continuously, giving rise to the periodic appearance of three high-symmetry local stacking configurations: the carbon atoms on top of boron or nitrogen atoms (B-N), the carbon atoms on top of boron or hollow sites (B-h), and carbon on nitrogen and hollow sites (N-h), see Fig.2.1(b). Each moiré period has a real space width of 238.56 Å, containing 56 graphene primitive cells stacked on 55 hBN primitive 28 2 Moiré valleytronics: realizing dense topological arrays cells, and Fig.2.1(b) shows two such moiré periods. The moiré pattern extends periodically and infinitely in the 2D x-y plane, but shall be referred to as 1DMP due to its quasistripe shape along the y direction. Finally, the distance between graphene and hBN is 3.22 Å, which corresponds to the equilibrium distance of the most stable local stacking configuration (B-h) [65]. Experimentally, to obtain a small uniaxial strain, using a polyethylene terephthalate (PET) substrate is effective [66].

In this chapter, electronic structures and topological properties are calculated using the real-space Kohn-Sham density functional theory (KS-DFT) as implemented in the RESCU package [52], which was recently developed by our group. RESCU uses the Chebyshev ﬁltering technique [53] and other advanced computational mathematics that drastically increases the computational eﬃciency, making it possible for calculating large moiré systems self- consistently. For more technical details we refer the interested readers to the original reference [52]. In our calculations, double-zeta polarized (DZP) atomic orbital basis set and optimized norm conserving Vanderbilt (ONCV) pseudopotentials [67] are employed. The exchange- correlation is treated at the generalized gradient approximation level (GGA-PBE) [51]. The real-space resolution is set to 0.3 Bohr and 15 × 1 × 1 Monkhorst-Pack k-point sampling is adopted.

The electronic band structure of the 1DMP is plotted in Fig.2.2(a), showing characteristic Dirac cones at the K and K0 valleys. Since the two valleys are far apart in the BZ, they can serve as a new degree of freedom when describing the low-energy electronic states [29, 30]. Most prominently, besides the noncryogenic bulk band gap (261.6 meV) opening due to local spatial symmetry breaking, there are two linear metallic bands near the Fermi level (red and blue lines) at each Dirac point which turn out to be helical channels (see below), one with positive and the other with negative group velocity. The spatial distribution of the

2 2 wave function |ψy| is shown in Fig.2.2(b). By tracking the peaks of |ψy| near the K valley (and similarly at the K0 valley), one can ﬁnd that the two linear band states are periodically located on the 1DMP: One (red) is between the B-N and B-h stackings, while the other (blue)

2 between N-h and B-N stackings. In the rest of this chapter, the peak regions of |ψy| shall be 2.2 Gr/hBN moiré and topological electronic structure 29

(a) (c)

K K’ K K’ Edge 1

Edge 2 K’ K

(b)

BN Bh Nh BN Bh Nh BN BN Bh Nh BN Bh Nh BN (d)

B-N B-h N-h B-N B-h N-h B-N

Figure 2.2: (a) Band structure of the 1DMP. The linear bands near the Fermi level (energy zero), red and blue lines, are the two valley helical channels, and the gray lines are bulk bands. The valley pointsK and 0 2 K are at kx = 1/3 and 2/3, respectively. (b) Wave functions |ψy| along the armchair (y) direction of the two helical channels with the same color scheme; vertical dashed lines indicate central positions of the three high-symmetry local stacking configurations. The left (right) panels are for k1 = 0.3329 (k2 = 0.3338), located at the left (right) of the K valley. (c) Illustration of valley helical channels and valley current. Upper: Solid (empty) circles denote carriers with positive (negative) group velocity, i.e. right movers (left movers). Solid (dashed) lines are helical channels with valley index K or K0, respectively. Lower: The red (blue) color is for a channel located at edge 1 (2); purple and green arrows indicate valley currents. (d) Topological valley currents flow at the moiré edges. 30 2 Moiré valleytronics: realizing dense topological arrays named topological moiré edges. The property of the K0 valley is easily envisioned, since wave functions at K and K0 are connected by time-reversal symmetry. Fig.2.2(c) sketches that the helical channels can generate a pure topological valley current. At each topological moiré edge, carriers with different valley index (K or K0) have opposite group velocities (moving directions), leading to a pure topological valley-current. Namely, at different edges the valley current propagates oppositely. Fig.2.2(d) illustrates two groups of topological valley current flowing oppositely at the moiré edges. Analogous to the spin-dependent helical states in a quantum spin Hall system [60, 20], the valley current is quantized and topologically robust against small chemical potential variations. Interestingly and importantly, in contrast with most previous schemes where only a single topological edge can be found or created at the boundary of the crystals, the periodic 1DMP can, in principle, support dense arrays of such topological moiré edges, with an integrated density of two per moiré period (238.56 Å). Thus, multiple helical channels can be implemented periodically and simultaneously, which is greatly beneficial for applications in quantum electronics and high-density integrated circuits.

2.3 Topological analysis of moiré valleytronics

Having presented the calculated results which strongly suggest topological moiré valleytronics, in this section we present detailed analysis based on the topological valley-dependent Chern number, using the theoretical formulation of Chapter 1.

2.3.1 Berry curvature and interlayer interaction

To establish the topological nature and understand the origin of the helical channels of 1DMP in Fig.2.2(a), we analyze the topological properties of the electronic structure in the 1DMP. We begin by considering perfectly lattice matched Gr/hBN systems: we build them by using the high-symmetry staking conﬁgurations of B-N staking or B-h stacking or N-h stacking, as shown in Fig.2.3(a). The topological nature can be quantiﬁed by the Berry curvature Ω(k) 2.3 Topological analysis of moiré valleytronics 31 using the perturbation theory and Kubo formula [23, 68, 69]:

X hψnk|vx|ψmkihψmk|vy|ψnki Ωn,xy(k) = −2Im 2 , (2.1) [mk − nk] m6=n

where m and n are the band indices, k is the eigenvalue of the eigenstate |ψki, and vx and vy are the components of velocity operators, the elements of which are deﬁned as:

∂H(k) hψnk|vα|ψmki = unk| |umk , (2.2) ∂kα

−ik·r ik·r where α can be taken as x or y, H(k) = e He , and |uki is the periodic part of the Bloch states. As shown in Fig.2.3(b), for every lattice-matched conﬁguration, the Berry curvature Ω(k) of the conduction band is strongly concentrated at the valley point with the equal amplitude but opposite sign for the K and K0 which is protected by the time-reversal symmetry. Comparing the three panels, one can ﬁnd two types of Ω polarity: Ω of the K valley is negative for the B-N stacking (middle panel) but positive for the other two.

(a) N-h B-N B-h

(b) 휶 휷 휶 휷 휶 휷 3 1 8

1.5 0.5 4 a.u.)

4 4 0 0 0

(10 -1.5 -0.5 -4 Ω -3 -1 -8 0 K 0.5K’ 1 0 K 0.5K’ 1 0 K 0.5 K’ 1 풌풙 (2π/a) 풌풙 (2π/a) 풌풙 (2π/a)

Figure 2.3: Topology analysis of the lattice-matched Gr/hBN bilayer. (a) Three high-symmetry stacking con- ﬁgurations in the moiré structure, where α and β are two subsites of the graphene lattice. (b) Corresponding k-resolved Berry curvature Ω(k) of the linear conducting band.

This polarity diﬀerence originates from the interlayer interaction between the graphene and hBN. To explain, let’s consider the model of massive Dirac Hamiltonian of graphene [29], 32 2 Moiré valleytronics: realizing dense topological arrays i.e. Eq.1.17 which is reproduced below: √ 3 ∆ H = at(q τ σ + q σ ) + σ , (2.3) 2 x z x y y 2 z where a is the lattice constant, t is the hopping parameter, q is the momentum measured from one valley point, ∆ is the staggered potential, and σ is the Pauli matrices. τz is the valley index (+1 for K, -1 for K0). Analytically, the Berry curvature can be obtained by the following expression:

Ωn(k) = ∇k × ihunk|∇k|unki. (2.4)

Then, the Berry curvature of the conduction band of Hamiltonian Eq.(2.3) can be analytically derived [29]: 3a2∆t2 Ω(q) = τ . (2.5) z 2(∆2 + 3q2a2t2)3/2

The quantity ∆ is the staggered potential between the graphene sublattices, given by Eα−Eβ, where Eα,Eβ are the on-site energies of the α and β sites in graphene. As shown in Fig.2.3(a), α and β sites are N-sites (C atoms on the top of N atoms) and h-sites (C atoms on the hollow site of hBN) respectively, for the N-h stacking; B-sites (C atoms on top of B atoms) and N-sites for B-N stacking; h-sites and B-sites for B-h stacking. In the hBN layer, B (N) atoms are positively (negatively) polarized, due to a large diﬀerence in their electronegativity. The π orbital of the graphene layer has attractive and repulsive interactions with B cations and N anions respectively [4, 70]. Therefore, electrons in the graphene preferentially locate at the B-sites and avoid N-sites to reduce electrostatic energy.

As a result, an on-site energy ordering is set: EN−site > Eh−site > EB−site. Thus for the B-N stacking, ∆ = Eα − Eβ = EB−site − EN−site < 0; but for other two stackings ∆ > 0. In one words, due to the interlayer interactions with the hBN layer, the graphene sublattices obtains a finite staggered on-site potential ∆. Different local stackings give rise to different signs of ∆, which lead to different Ω polarities in Fig.2.3(b).

On the technical side, note that we have given two methods for calculating the Berry curvature: Eq.2.1 and Eq.2.4. Eq.2.4 is a simpler form and usually used to calculate the 2.3 Topological analysis of moiré valleytronics 33

Berry curvature analytically. The other formula, Eq.2.1 is usually applied in numerical calculations of the Berry curvature to avoid the gauge problem (random phase of numerical wave functions) [68]. The equivalence of them can be proved by the k·p perturbation theory (details can be seen in Appendix A).

2.3.2 Valley Chern number and bulk-edge correspondence

The Chern number C - the physical indicator used to classify topological phases, can be obtained from the ﬂux of the 2D Berry curvature:

1 Z C = Ω (k)dk dk . (2.6) 2π xy x y

In the Gr/hBN bilayer, the total Chern number is zero since Ω(k) = −Ω(−k) as protected by the time-reversal symmetry. In valleytronics, as discussed in the Chapter 1, one can define a valley-depedent Chern number CV for each valley [29, 30], which is computed by integrating the Berry curvature flux around one particular valley. In the rest of this chapter, we specifically refer CV to the K valley. In our KS-DFT calculations, to obtain CV by integrating the Berry curvature flux, we adopt a fine k-mesh of 666 × 400 in the BZ. Our calculation results show that for the B-N stacking, CV = −1/2; while for N-h stacking and

B-h stacking, CV = +1/2. This means that they have diﬀerent valley-dependent topological orders and belong to diﬀerent topological phases.

To investigate the bulk-edge correspondence of the topological valleytronics, we compare two cases: opposite topology and same topology. In the opposite topology, we choose the N-h and B-N stacking: they belong to diﬀerent topological phases. As shown in Fig.2.4(a), we construct a hybridized system by combining these two lattice matched stacking bilayers together, and the helical channels are visible clearly in the calculated band structure in Fig.2.4(c). The wave function analysis in Fig.2.4(e) shows that the helical channels are exactly located at the edges where the topological order changes. In contrast, Fig.2.4(b) is a same topology structure by combining the B-h and N-h stacking, which belong to the same 34 2 Moiré valleytronics: realizing dense topological arrays topological class. Therefore, as shown in Fig.2.4(d), no helical channels are found in the calculated band structure.

(a) N-h B-N N-h B-N (b) B-h N-h B-h N-h (c) (d)

(e)

Nh BN Nh BN Nh BN Nh BN 풚 풚

Figure 2.4: Bulk-edge correspondence in the valley-dependent topological phase diagram. (a-b) Lattice- matched stackings of (a) opposite topology and (b) same topology. (c-d) Band structures corresponding 2 to (a) and (b), respectively. (e) Spatial distribution of |ψy| of the two helical channels in (c), red (blue) curves are for the red (blue) channel respectively, and the vertical dashed lines denote edges between the two lattice-matched conﬁgurations. The left (right) two panels are taken at k = 0.3325 (0.3375), located at left (right) of K valley.

2.3.3 Topological phase transition in periodic moiré patterns

In the previous subsection we investigated valley-dependent topological properties of lattice- matched Gr/hBN. Here we study the moiré systems. In a long period moiré pattern with a period much larger than the lattice constant of graphene, the electronic structure in a local region, with a length scale small compared to the moiré period but larger than the lattice constant, can be approximated by that of the lattice-matched Gr/hBN of the corresponding 2.3 Topological analysis of moiré valleytronics 35 atomic registry. Fig.2.5(a) plots the calculated valley topological phase diagram, showing how the valley-dependent Chern number of K valley varies with the atomic registry. Clearly, there are two topological phase transitions indicated by the jumping points in the plot when the atomic registry varies by one moiré period: one is between the B-N and B-h stacking while the other is between the N-h and B-N stacking.

The helical channels and topological valley current anticipated in Fig.2.2(d) are direct manifestation of the topological phase transition due to variations of the atomic registry. Comparing the phase transition points in Fig.2.5(a) and the locations of the topological moiré edges in Fig.2.5(b), we conclude that the valley-dependent helical channels found in the 1DMP originate from the change of the topological order across the moiré edges. For 1DMP, the moiré period (238.56 Å) is much greater than the lattice constant (4.26 Å), therefore the local atomic conﬁguration varies so slowly that it has a negligible diﬀerence compared to the lattice-matched structures. In moiré superlattices, the local electronic property can be described quite well by using corresponding lattice-matched stackings [60]. From this point of view, one can track the local topological properties of the moiré super-lattice according to the topological phase diagram of the lattice-matched stacking in Fig.2.5(a). Along the direction of the moiré super-lattice (y in our case), two local topological phases appear alternatively and periodically, giving rise to two groups of “boundaries" where topological phase transitions happen. This leads to arrays of valley-dependent helical channels (or the topological moiré edges) in Fig.2.2. 36 2 Moiré valleytronics: realizing dense topological arrays

(a) B-N B-h N-h B-N B-h N-h B-N 1 0.5 0

-0.5 -1 0 1/3 2/3 1 4/3 5/3 2 풓ퟎ/b (b)

B-N B-h N-h B-N B-h N-h B-N

Figure 2.5: (a) Valley-dependent topological phase diagram parametrized by r0, which is the lattice displacement between Gr and hBN layers, and CV is the calculated valley-dependent Chern number of K valley. (b)Schematic demonstration of topological valley currents ﬂowing at the moiré edges [referring to Fig.2.2(d)].

2.4 Generality of moiré valleytronics

The key for realizing moiré valleytronics and dense arrays of 1D helical channels is a moiré structure (1DMP) that has different local stacking configurations generating different topological orders. That is, different local topological phases coexist in one long-period moiré super-lattice simultaneously. This situation is in fact available in a broad range of 2D materials besides the Gr/hBN studied above. In this section, we demonstrate two different real material systems for moiré valleytronics: the strained bilayer-graphene moiré pattern and the silicence/hBN moiré pattern, where multiple moiré edges supporting dense arrays of helical channels exist between different local stacking.

2.4.1 Moiré system of strained bilayer graphene

Consider a pristine bilayer graphene system, by adding an uniaxial tensile strain along the armchair direction on one layer [Layer 1 in Fig.2.6(a)] while no strain on the other layer 2.4 Generality of moiré valleytronics 37

(a)

Layer 2

Layer 1

(b) AB BA AB BA

Figure 2.6: (a) Bilayer graphene lattice. The black arrows denote the uniaxial tensile strain added along the armchair direction on the layer 1. (b) The bilayer graphene moiré structure. AB and BA are two local stackings. In this ﬁgure purple and grey balls represent carbon atoms in Layer 1 and 2, respectively.

(Layer 2), an 1D moiré superlattice is formed as shown in Fig.2.6(b). In this moiré pattern, two opposite local stackings - AB and BA, appear periodically as shown in Fig.2.6(b). This moiré system has already been fabricated experimentally [71].

Next, it is obvious that if an external uniform electric ﬁeld is applied vertically to the bilayer, the AB and BA local stacking conﬁgurations obtain opposite staggered potentials [36]:

∆AB = −∆BA. (2.7)

According to Eq.2.7, the polarities of the Berry curvature are opposite:

ΩAB = −ΩBA, (2.8) which gives rise to opposite Berry curvature ﬂux and/or opposite valley-dependent Chern number [via Eq.2.6]. We conclude that between these two local stacking conﬁgurations, multiple moiré edges emerge to support helical channels and topological valley currents. 38 2 Moiré valleytronics: realizing dense topological arrays

Note that the topological eﬀect in bilayer graphene has attracted a lot of attention recently [36, 37, 72, 61], and the strained bilayer graphene paves a new way to theoretical and experimental researchers in this material ﬁeld.

2.4.2 Moiré system of silicene/hBN

(a) Top view Side view

(b)

Figure 2.7: (a) Graphene lattice. (b) Silicene lattice. d is the buckled parameter.

We may easily envision another moiré system which has the same topological physics as the above. As shown in Fig.2.7, silicene has a buckled honeycomb structure with a buckled parameter d about 0.45 Å [73]. By adding a vertical electric ﬁeld through a gate bias, the pristine silicene obtains a staggered potential ∆E = Ed, where E is the electric ﬁeld. Analogous to Gr/hBN, bilayer silicene/hBN has a natural moiré structure due to lattice mismatch. For every location r on the moiré pattern, the total staggered potential is:

∆(r) = ∆E + ∆Moire(r) = Ed + ∆Moire(r), (2.9)

where ∆Moire(r) is the intrinsic moiré staggered potential at real-space location r. Since polarity of ∆ determines the topological order, the sign-changing location of ∆(r) is where 2.5 Structural robustness of moiré valleytronics 39 moiré edges appear and topological phase transition occurs. Therefore we expect that in this system the gate-dependent ∆ generates a gate tunable moiré valleytronics.

In particular, two cases are predicted to exist. (a) Switching OFF state: when the electric

field is such that E > |max{∆Moire(r)}/d|, ∆(r) > 0 for any r, i.e. no topological phase transition and moiré edges as well as helical channels disappear. The moiré valleytronics is thus switched off. (b) Switching ON state: when the above condition is not satisfied, the moiré edges may be found. Furthermore, by manipulating the electric field, the sign-changing locations of ∆(r) can be modulated. Therefore, an electric field can be applied to control the real-space distribution of the moiré edges and the helical channels.

Monolayer materials of other group IV elements such as germanene and stanene, also have buckled honeycomb lattices with bucked parameter larger than silicene (0.69 Å for germanene and 0.85 Å for stanene) [73]. The gate manipulation of moiré valleytronics is expected to be even more eﬀective for these materials which calls for further studies.

2.5 Structural robustness of moiré valleytronics

In experiments, different fabrication techniques can lead to different structural geometries. In this section, we turn to another situation which could happen experimentally: the freestanding Gr/hBN moiré super-lattice. By freestanding, the bilayer can fluctuate into the third dimension (z) thus changing the local stacking distribution. We determine the atomic structure of free-standing bilayers using a force-field method as implemented in the large-scale atomic/molecular massively parallel simulator (LAMMPS) [74]. In the structural relaxation, the in-plane C-C and B-N interactions are treated using many-body Tersoff potentials [75, 76]. The interlayer vdW interaction is well described by the optimized Morse potential whose parameters are taken from Ref. [4]. We set equilibrium distances of the C-B and C-N bonds to 3.56 Å and 3.54 Å, respectively. For a free-standing film, this is to keep the interlayer distance near 3.2 Å, consistent with the flat-sheet situation. This way we can focus on the 40 2 Moiré valleytronics: realizing dense topological arrays

(a) N-h B-N B-h

(b)

(c)

Figure 2.8: (a) Relaxed freestanding Gr/hBN bilayer with out-of-plane (z-direction) corrugation of about 13 Å. (b) Left: Band structure; right: spatial distribution of modular squared wave functions of the two helical channels, where the plotted isosurface (pink and light blue areas) is 4 × 10−9 a.u. (c) Topological valley current in the freestanding moiré structure. 2.6 Summary 41 eﬀects of large corrugations in freestanding ﬁlms. The relaxation is deemed completed when residual force on each atom is less than 1.0 × 10−6 eV/Å.

As shown in Fig.2.8(a), the freestanding bilayer is 3D wavelike with a large out-of-plane corrugation (> 13 Å). This corrugation originates from an energetic competition between maximizing the favorable stacking configuration (B-h) and minimizing the elastic energy of the lattice. By out-of-plane deformation, the area of B-h stacking is enlarged and pushed to the bottom of the corrugated structure, while the N-h and B-N stacking are reduced and pushed to the top [4, 77]. Experimentally, this 3D wavelike moiré structure has been realized on suspended bilayers [77]. Fig.2.8(b) shows that in the 3D wavelike bilayer the valley-dependent helical channels still appear in the band structure, and the wave functions of the helical channels [right panel in Fig.2.8(b)] show that the two groups of helical channel arrays are vertically separated. As a consequence, the two groups of counter-propagating topological valley currents can flow at different heights in the corrugated bilayer as shown in Fig.2.8(c), leading to an extra dimension to manipulate and detect the helical states. Most importantly, this result shows the robustness of the topological moiré edges against a huge structural corrugation, making possible practical applications of the proposed moiré valleytronics in fluctuated 2D systems (such as Refs.[71, 77]) besides the flat structures.

2.6 Summary

In this chapter, we report a general and robust scheme, the moiré valleytronics, to realize dense arrays of 1D topological helical channels. For the 1DMP of a zigzag-oriented Gr/hBN bilayer, ﬁrst-principles calculation shows that the two groups of helical states are located at two groups of topological moiré edges, one located between the B-N and B-h stackings while the other between the N-h and B-N stackings. A topological phase diagram based on Berry curvature and valley-dependent Chern number reveals the intrinsic dependence between the topological order and the atomic stacking conﬁguration. Moreover, the generality and experimental feasibility have been discussed and proved by investigating several moiré 42 2 Moiré valleytronics: realizing dense topological arrays systems. Finally, we predict that in the freestanding wavelike moiré structure, two groups of moiré edge arrays carrying counter propagating helical channels are spatially separated in the out-of-plane direction, indicating the structural robustness of moiré valleytronics. This study paves a new way for modulating valley electronic states and realizing multiple topological helical channels which are relatively easily accessible experimentally. The results suggest that topological valleytronics based on moiré patterns to be a rich research direction in material science. 3

Theoretical design of topological heteronanotubes

In the previous chapter, we presented moiré-induced topological arrays in 2D heterostructures. In this chapter, we will show that such moiré topological physics can be extended to 1D tubular structures to form topological heteronanotubes (THTs) - a dissipationless 1D conductor. A wide range of THT systems are investigated to demonstrate various topological eﬀects in 1D nanotube systems [2].

3.1 Introduction

The discovery of carbon nanotubes (CNTs) by Sumio Iijima [18] in 1991 opened an exciting field of carbon nanoelectronics where transistors and interconnect wires are entirely made of CNTs [7, 78, 79]. From the fundamental physics point of view, CNT is of particular interest due to its one-dimensional (1D) nature, very long carrier mean free path, and massless Dirac electrons at low energy. Recently, high-performance single-wall CNT field-effect transistor (FET) with 5 nm gate length [10] and low sub-60 millivolts per decade switching [11], were experimentally realized. With these and other [8, 9, 80, 81, 82, 83, 84] impressive progresses, CNT may well provide a viable route toward an emerging electronics beyond silicon [5, 6].

More recently, double-wall heteronanotubes made by a single-wall CNT conﬁned inside a boron nitride nanotube (BNNT), were experimentally synthesized using in situ electron irradiation [85] and chemical vapor deposition [86]. The original motivation of producing such composite CNT@BNNT tubes was to use the outer BNNT wall as a protective insu-

43 44 3 Theoretical design of topological heteronanotubes lating shell against environmental perturbations to the inner CNT. In this chapter we shall theoretically show that CNT@BNNT heteronanotubes can provide a new material platform for realizing topological low-dissipation CNT nanoelectronics. The outer BNNT wall, which was considered to have little interaction with the inner CNT, can in fact strongly modulate the electronic structure and carrier transport of the CNT, i.e. make it become an intrinsic topological material for a wide range of tube indexes. 1D topological materials are fundamentally important for investigating the Majorana zero modes [81, 87], topological quantum computing [88] and they are also practically signiﬁcant for fabricating low-dissipation 1D nanoelectronic devices due to substantially reduced carrier scattering [12]. Therefore, 1D topological materials have attracted a lot of attentions in the past years, including quantum wires [89] and heavy-element nanotubes [90].

In this chapter, we propose and investigate the idea of topological heteronanotube (THT) for realizing a one-dimensional (1D) topological material platform that can pave the way to low-power carbon nanoelectronics at room temperature [2]. We predict that the experimentally feasible systems, coaxial double-wall heteronanotube - a carbon nanotube (CNT) inside a boron nitride nanotube (BNNT), can act as THT. Dissipationless topological conducting pathways on the THT are protected by a valley-dependent topological Chern invariance that originates from local topological phase transitions of the CNT modulated by the CNT- BNNT interaction. Spiral THTs, where topological current flows spirally around the tube, function as nanoscale solenoids to induce remarkable magnetic fields due to the dense moiré nanopatterning. The generality and robustness of the THT materials are demonstrated by investigating different tube diameters, tube indexes, tube types as well as topological-pathway orientations through first-principles calculations.

3.2 Geometry and electronic properties of THTs

As discussed in Chapter 1, the geometry and electronic property of nanotubes are unique- ly characterized by the chiral vector Ch = n1a1 + n2a2 ≡ (n1, n2), where a1 and a2 are 3.2 Geometry and electronic properties of THTs 45

(a) (b) (c) CNT(풏, 풏) BNNT(풏+풎, 풏+풎) CNT(풏, 풏)@BNNT(풏+풎, 풏+풎)

(d) (e) (f)

// // // //

(g) (h) (i) ퟐ 흍 ퟐ 흍 ퟐ 흍

푬 푬 푬 K K’ K K’ K K’

K × (j) K K’ K (k) (l) K’ × K’ K’ × K

Trivial conductor Insulator Topological conductor

Figure 3.1: (a-c) Geometries of the armchair single-wall CNT, single-wall BNNT and THT respectively, where THT is a double-wall CNT(inner)@BNNT(outer) tube. (d-f) Calculated band structures of the three tubes with n = 96 and m = 1. The Fermi level is shifted to energy zero, and the valley points K and K’ are at kz = 1/3 and 2/3, respectively. Bulk bands are the gray lines. In (f), the linear red and green bands near the Fermi level are the two valley-dependent topological helical states. a = 2.4595 Å is the lattice constant of the transnational vector T. For armchair CNT and BNNT, the small lattice diﬀerence of T (1.8%) is neglected. (g-i) Spatial distribution of modular squared wave functions |ψ|2 on the tube circumference, which are the Bloch states indicated in the inset at chemical potential µ = E. The solid (empty) circles denote carriers with positive (negative) group velocity, i.e. right movers (left movers), and solid (dashed) lines are helical states with valley index K or K’, respectively. In (i), since |ψ|2 is from carbon atoms, only CNT circumference is plotted for clearer vision. (j-l) Schematics of charge transport in these nanotubes. Arrows on the wires denotes electron ﬂow, curved white arrows on the nanotubes indicate backscattering from one conducting channel to another. 46 3 Theoretical design of topological heteronanotubes

the hexagonal lattice vectors and n1 and n2 are integers (tube index) [16]. Let’s begin by considering an armchair CNT(n,n) in Fig.3.1(a) and an armchair BNNT(n+m,n+m) in Fig.3.1(b); together they form a THT in Fig.3.1(c). Here, m is a non-zero integer denoting the difference in circumferential unit cell number between the two tubes. To make a specific calculation as one example without losing generality, we start from the case of n = 96 and m = 1, thus the THT is CNT(96,96)@BNNT(97,97). We emphasize that the generality and robustness of THT shall be demonstrated (see later sections) by investigating different tube diameters, tube indexes, tube types as well as topological-pathway orientations. For the CNT(96,96)@BNNT(97,97), the optimized inter-wall distance of this THT is 3.2 Å by structural relaxation. The CNT(n,n) is metallic, the BNNT(n+m,n+m) is insulating, as shown by the band structures in Fig.3.1(d,e) obtained by DFT calculations [52]. Compared with band structures of individual CNT or BNNT, the band structure of THT in Fig.3.1(f) is significantly different. First, there opens a noncryogenic bulk band gap of 150.6 meV due to local sublattice symmetry breaking in the THT. Second, there are two linear bands (red and green lines) near the Fermi level at each Dirac point K and K0, which turn out to be valley-dependent helical states (see below). Third, wave functions of these linear bands of THT are concentrated at two locations on the circumference as shown by red and green isosurfaces coded in Fig.3.1(i). This is to be compared to that of the individual CNT where the wave functions spread over the entire circumference, as shown by the yellow isosurface coded in Fig.3.1(g).

In this chapter, the electronic structures and topological properties are calculated using the real-space KS-DFT solver as implemented in the RESCU package [52]. In our calculations, single-zeta polarized (SZP) atomic orbital basis set and optimized norm conserving Vanderbilt (ONCV) pseudopotentials [67] were employed. The exchange-correlation was treated at the generalized gradient approximation level (GGA-PBE) [51]. The real-space resolution was set to 0.3 Bohr and we adopted 15 × 1 × 1 Monkhorst-Pack k-point sampling.

Based on the calculated electronic structure, quantum transport properties of the three nanotubes in Fig.3.1 can be established. The BNNT is an insulator with a large band gap 3.3 Valley-dependent topological analysis 47 shown in Fig.3.1(h), thus no current flows in the circuit of Fig.3.1(k). Metallic CNT(n,n) is a conductor, but intravalley backscattering at K with positive velocity to the same K with negative velocity can occur due to interaction (the same is true for K’), which decreases conductivity as illustrated in Fig.3.1(j). For THT of Fig.3.1(l), on other hand, each topological conducting pathway (red or green regions on the tube) supports one pair of counterpropagating helical states: Carriers with different valley index (K or K’) have opposite group velocities, the intravalley scattering is significantly suppressed due to the spatial separation of states in the same valley but with opposite velocities [Fig.3.1(i)]. Meanwhile, since the two valleys are far apart in the Brillouin zone (BZ), the intervalley scattering from K to K’ or vise versa, are largely suppressed on each topological edge [29, 30]. Therefore, THT behaves as a low-dissipation CNT conductor for ballistic transport with little or no back-scattering for

4e2 low energy excitation, giving a quantized conductance of G = 2G0 = h (spin degenerate), where h is the Planck’s constant and e the electron charge. Analogous to the spin-dependent helical states in a quantum spin Hall system [20] and the topological valleytronics discussed in Chapter 2, the 1D valley-dependent topological current is robust and holds a high transport eﬃciency against small disorders and chemical potential variations [36, 37]. Finally, carriers with K and K’ index travel in diﬀerent pathways as shown in Fig.3.1(l), thus THT can also be exploited as a perfect valley splitter.

3.3 Valley-dependent topological analysis

In this section, we discuss the physical origin of the topological channels in THT. As for the low-energy electronic structure of CNT, because of the hexagonal rolled-up conﬁgurations, the valley topological invariant presented in Chapter 1 and Chapter 2 can be employed for this purpose [29, 30]. We build a mapping between the THT and its corresponding 2D double-layer ﬂat sheet, shown in Fig.3.2(a), to obtain an intuitive perspective into the local atomic registries of the circular tube [82]. As shown in Fig.3.2(b), the 2D mapping of a THT CNT(n,n)@BNNT(n+1,n+1) is a moiré pattern with one moiré period. Along the 48 3 Theoretical design of topological heteronanotubes

(a)

(b) CNT(풏, 풏)@BNNT(풏+1, 풏+ퟏ) T

C풉

Figure 3.2: (a) Schematics of rolling and unrolling processes, building a mapping between 1D THT and its corresponding 2D flat double-layer moiré structure. (b) 2D moiré mapping with n = 96. Ch is the circumference direction (chiral vector), and T is the tube-axis direction (translational vector). Some representative high-symmetry local atomic registries are shown in green circles, where gray, pink and blue balls denote carbon, boron and nitrogen atoms respectively. (c) Valley-dependent Chern numbers (CV) for valley K of lattice-matched configurations at the corresponding real-space locations. (d) Circular distribution of CV for valley K rolled up from the flat-sheet in (c). In (c) and (d), purple and light blue regions denote CV = -1/2 and 1/2 respectively. (e) Spatial distribution of modular squared wave functions |ψ|2 along the circumference of the topological helical states in Fig.3.1(f). 3.3 Valley-dependent topological analysis 49

circumference direction Ch (armchair direction), the atomic registry varies gradually and continuously, giving rise to various local stacking conﬁgurations. In a long-period moiré structure with moiré period much larger than each unit cell of the double-wall tube, the electronic structure and topological nature in a local region can be well determined by that of the lattice-matched stacking conﬁguration of the corresponding atomic registry, as depicted in the green circles in Fig.3.2(b) [3, 60, 91]. The topological properties of the latter is described by the valley-dependent Chern number CV is again calculated by:

1 Z CV = Ωxy(k)dkxdky. (3.1) 2π valley

The above integral is around one particular valley (valley K in this chapter). Ω(k) is the Berry curvature calculated by Eq.2.1 which is reproduced here:

X hψνk|vx|ψµkihψµk|vy|ψνki Ων,xy(k) = −2Im 2 . (3.2) [µk − νk] µ6=ν

Here µ and ν are band indexes, k is the eigenvalue of eigenstate |ψki, and vx, vy are the components of velocity operators, the elements of which are deﬁned as:

∂H(k) hψνk|vα|ψµki = uνk| |uµk , (3.3) ∂kα

−ik·r ik·r where α can be taken as x or y, H(k) = e He , |uki is the periodic part of the Bloch states. For calculating the Chern number by integrating the Berry curvature ﬂux in the DFT frame, we adopt a ﬁne 666 × 400 k-mesh in the Brillouin zone. These details are the same as those in Chapter 2.

As shown in Fig.3.2(c), our calculation results show that various local stackings give rise to diﬀerent signs of the valley Chern number CV : purple and light blue regions denote CV =

-1/2 and 1/2 respectively. Such dependence between the valley Chern number CV and local atomic registry can be understood by investigating the interwall interaction between CNT and BNNT (or equivalently graphene and hBN lattices in the 2D mapping). Using the same 50 3 Theoretical design of topological heteronanotubes analysis as in Section 2.3.1 of Chapter 2, we conclude that the topological phase transition

- the change of sign of CV, originates from the sign of the stagger potential changing along the smooth variation of local atomic registries between the CNT and BNNT walls. From the

2D sheet to the 1D tubular system, the circular spatial distribution of CV can be obtained in Fig.3.2(d). Comparing Fig.3.2(d) and Fig.3.2(e), it can be clearly seen that the calculated locations of topological conducting pathways of the THT match well with the topological phase transition points, revealing the moiré-related topological bulk-edge correspondence in 1D tubular systems.

3.4 Spiral-oriented THTs and topological solenoid

The great richness of THT is further seen by controllable orientations of topological transport pathways. As a demonstration, we roll up the 2D mapping sheet of the THT with the index of CNT(16,16)@BNNT(17,17) along the particular chiral vector direction (Ch) shown in Fig.3.3(a). It is clear that in this structure the variation of atomic-registry is tilted from T so that the topological moiré edge vector e is not parallel to the transitional vector T . Fig.3.3(b) shows that in this spiral THT, topological helical states still appear (albeit with some band-folding). The calculated wave functions of these helical states, shown in Fig.3.3(c), are located spirally wrapping the tube. Such spiral THTs act as nanoscale solenoids where the "coil" is the topological conducting pathways. When a current I passes through, the magnitude of the magnetic ﬁeld B per unit length is obtained by Ampere’s law:

B = µ0NI, (3.4)

where µ0 is the permeability of free space, N is the number density of the coil turns depending on the transitional vector T which can be manipulated by various chiral angle and THT index. In the case of Fig.3.3, T = 7.87 nm; Thus N is as large as 1.27 × 108 per meter which is extremely large (as a comparison, the traditional macroscopic solenoids usually have the coil density N of about 103). Such a dense nanocoil would generate a magnetic ﬁeld 3.4 Spiral-oriented THTs and topological solenoid 51 of 1.6 Gauss for a 1 µA current which should be remarkable to experimental observations. Most importantly, the current ﬂowing in the transport pathways are topologically protected against backscattering and such spiral THT realizes a dense moiré nanopatterning technique to produce "topological solenoids".

It is noted that a spiral THT has an extremely large supercell. For example, the supercell of the CNT(16, 16)@BNNT(17, 17) THT in Fig.3.3, shown by the orange rectangle in Fig.3.3(a), contains 8448 atoms. Such a large system presents a serious computational challenge for DFT calculations. The RESCU package [52] which was recently developed by our group uses the Chebyshev ﬁltering technique [53] and other advanced computational mathematics to drastically increase computational eﬃciency, making it possible for calculating such large moiré systems self-consistently by KS-DFT. Interested readers can refer to Ref. [52]. 52 3 Theoretical design of topological heteronanotubes

(a) T

C풉

(b) (c) 흍 ퟐ (c)

(d) (d)

N S

Topological solenoid

Figure 3.3: (a) Illustration for creating spiral THTs with topological moiré edge vector e (indicated by yellow arrow) not parallel to the transitional vector T. The orange rectangle denotes the supercell of the 2D mapping of the spiral THT, containing 8448 atoms. (b) Band structure of the spiral THT. (c) Spatial distribution of modular squared wave functions |ψ|2 of the topological helical states where the plotted isosurfaces (red and green areas) are 1.8 × 10−11 a.u. (d) Schematic demonstration of the spiral THT device in a circuit to serve as a topological solenoid or nanomagnet. Arrows denote the direction of electron ﬂow. 3.5 Generality and robustness of THTs 53

3.5 Generality and robustness of THTs

In this section, we discuss the generality and robustness of the THT, by investigating several diﬀerent nanotube systems: various diameters, multi-period THTs, reverse-ordered THTs, commensurate double-wall heteronanotubes, zigzag-type heteronanotubes, and double-wall CNTs.

3.5.1 Other diameters

The general form of armchair THT is CNT(n,n)@BNNT(n+m,n+m), where nanotube indexes (m and n) provide essentially infinite possibilities fof THTs. Different values of n correspond to different tube diameters. In Fig.3.1, to be specific but without losing generality, we studied a particular case of n = 96 as an example to demonstrate the topological physics of THT.

Experimentally, smaller diameter CNTs, typically with a diameter below 4-5 nm, are most common and easily fabricated due to their stable circular geometry [92, 93]. Here we investigate a small diameter case: CNT(22,22)@BNNT(23,23), whose diameter is 2.98 nm as shown in Fig.3.4(b). Structural relaxation of Fig.3.4(c) shows that the optimized inter-wall distance is around 3 Å. In Fig.3.4(d) and (e), it is shown that the topological helical channels still appear in the band structure and their wave functions are located at two different regions (topological conducting pathways), similar to the large diameter THT in Fig.3.1. In the small diameter THT, the geometric curvature effect is much larger than that of large diameter case. Our result reveals that THT is general to a broad range of tube diameters, with structural robustness against the curvature effect of small diameter tubes. In Chapter 2, a similar property of topological structural robustness was found in 2D freestanding wavelike moiré patterns where the moiré induced topological valleytronics is protected from the huge structural corrugation [3]. 54 3 Theoretical design of topological heteronanotubes

(a) (b) CNT(22, 22)@BNNT (23, 23)

2.98 nm

(e) 흍 ퟐ

Figure 3.4: (a) Geometric structure of CNT(22,22)@BNNT(23,23), with the inner diameter of 2.98 n- m as shown in the cross-section view of (b). (c) Dependence of total energy on the inter-wall distance. (d)Calculated band structure. Gray lines are bulk bands and red (green) lines are helical channels - same as those of Fig.3.1. (e) Spatial distribution of modular squared wave functions |ψ|2 along the circumference of the topological helical channels in (d). The plotted isosurfaces are 7 × 10−8 a.u. 3.5 Generality and robustness of THTs 55

3.5.2 Multi-period THTs

(a) (b) (c)

CNT(풏, 풏)@BNNT(풏+2, 풏+2) CNT(풏, 풏)@BNNT(풏+3, 풏+3) CNT(풏, 풏)@BNNT(풏+4, 풏+4) T

C풉

(d) (e) (f) 흍 ퟐ 흍 ퟐ 흍 ퟐ

Figure 3.5: (a-c) 2D mapping of THTs with diﬀerent values of m. n = 96 for all cases. (d-e) Calculated spatial distribution of modular squared wave functions |ψ|2 of multiple topological helical states along the circumference, corresponding to the cases of (a-c) respectively. The plotted isosurfaces are 2.2 × 10−9 a.u.

In the general form of CNT(n,n)@BNNT(n+m,n+m), the value of m equals to the number of the moiré periods on the circumference [as shown in Fig.3.5(a-c)]. When m = 1 as discussed in the Fig.3.1 and Fig.3.2, one moiré period locates along one circumference, which we shall name as 1-THT (single-period THT). In Fig.3.5, we investigate several multi-period THTs with m = 2, 3, 4. Each moiré period is n/m unit-cell length which includes two topological phase transition points (two topological moiré edges). Thus, the number of helical states is also determined by m. Fig.3.5(d-f) show that multiple pairs of topological conducting pathways occur in the multi-period THTs. This suggests that quantized conductance takes

4me2 the value of G = 2mG0 = h (spin degenerate).

Note that the condition of establishing topological physics in THT is that two stable and robust topological phases coexist on the tube circumference. Therefore, each moiré period (n/m unit-cell length) or each topological phase region should be large enough to support a stable topological phase [60]. 56 3 Theoretical design of topological heteronanotubes

// //

Figure 3.6: Band structure of the reverse-ordered THT: CNT(n, n)@BNNT(n+m,n+m) with n = 96 and m = −1. Gray lines are bulk bands and red (green) lines are helical channels - same as those of Fig.3.1.

3.5.3 Reverse-ordered THTs

The THTs discussed so far are formed by inner CNT and outer BNNT (m > 0). On the experimental side, another system with a reversed order, outer CNT and inner BN- NT, is also feasible [94]. In this case, m is usually negative in the general form CN- T(n,n)@BNNT(n+m,n+m). In such systems, the key condition of the moiré topological physics, i.e. the moiré atomic registries varying along the circumference, still exists. As a result, the helical channels arise in the same manner as shown in Fig.3.6.

3.5.4 Commensurate double-wall heteronanotubes

When m = 0 in the general form CNT(n,n)@BNNT(n+m,n+m), along the circumference the atomic registry is commensurate, i.e. not a moiré structure. A bulk band gap still appears at the Dirac points as shown in Fig.3.7, due to the sublattice symmetry breaking. However, no topological helical state appears since no moiré structure is formed in such a tube. 3.5 Generality and robustness of THTs 57

// //

Figure 3.7: Calculated band structure of the commensurate double-wall heteronanotubes: CNT(n, n)@BNNT(n+m,n+m) with n = 96 and m = 0.

3.5.5 Zigzag and chiral heteronanotubes

The necessary condition for the appearance of 1D helical channels is the moiré induced topological phase transition located at topological moiré edges of the THT. In the above we mainly discussed the armchair heteronanotubes. For zigzag heteronanotubes, i.e. CN- T(n,0)@BNNT(n+m,0), as shown in Fig.3.8(a), no helical state appears inside the band gap. This is because indices K and K’ are mixed together due to folding of reciprocal space [3, 95]. For chiral heteronanotubes, the topological moiré edges are along the chiral direction. The topological helical states appear as shown in Fig.3.8(b), similarly as the armchair THT in Fig.3.1.

3.5.6 Topological properties of double-wall CNTs

In fact, the same topological physics is expected in other tubular materials. In this section we investigate a double wall CNT system where two opposite local stackings (AB and BA) locate on the circumference of the double-wall tube, according to the 2D mapping in Fig.3.9(b). Due to its excellent mechanical strength and thermal/chemical stability, double wall CNT is 58 3 Theoretical design of topological heteronanotubes

(a) (b)

Figure 3.8: (a) Band structure of a zigzag-type THT: CNT(n,0)@BNNT(n+m,0) with n=96 and m=1. b is the lattice constant of the transnational vector of 4.26 Å. (b) Band structure of a chiral-type THT: CNT(100,25)@BNNT(104,26). a is the lattice constant of the transnational vector with a value of 6.50 Å. Gray lines are bulk bands and red (green) lines are helical channels - same as those of Fig.3.1. considered a promising material for nanodevices [96].

When an electric ﬁeld is pointing along the radial direction, the AB and BA local stacking conﬁgurations obtain opposite staggered potentials: ∆AB = −∆BA. Similarly to Eq.2.7 of Chapter 2, one can obtain

CV (AB) = −CV (BA). (3.5)

We therefore conclude that between these two local stacking configurations, the topological moiré edges emerge to support helical channels. All the topological properties in the THT of CNT@BNNT can be also established in double-wall CNT. The double-wall CNT is not a heteronanotube since both walls are carbon, therefore a finite stagger potential and relevant topological physics require a radial electric field to establish. Experimentally, we expect two techenical schemes may practically achive such field: an atomic chain enclosed inside a CNT [97] or the gate-all-around (GAA) gating method [98]. 3.6 Summary 59

(a) (c) E

(b) AB BA

Figure 3.9: (a) Armchair double-wall CNT of CNT(n,n)@CNT(n+m,n+m). Here we take m=1 for example. The carbon atoms of outer CNT are plotted by purple balls to distinguish from those of the inner CNT wall (gray color). (b) The 2D mapping of (a). AB and BA are two local stackings. (c) A schematic plot of a radial electric ﬁeld (orange arrows) that induces the topological states.

3.6 Summary

In this chapter, we propose a general, robust and experimentally feasible material platform - the topological heteronanotube (THT), for naturally realizing 1D topological helical states at room temperature. By first-principles calculations on the prototypical CNT@BNNT system, two groups of valley-dependent helical states are discovered to form inside a noncryogenic bulk band gap. By analyzing wave functions and inferring valley topological invariance, these helical states are found to locate at two groups of topological moiré edges, due to the local topological phase transition associated with the local atomic registry that is varying along the circumference of the tubes. In principle, low-energy carriers in the helical states supported by the topological moiré edges do not suffer intravalley and intervalley scattering, thus the THT is an ideal 1D topologically protected ballistic conductor where transport can be close to dissipationless. The spiral THTs have topological current flowing spirally around the tube to function as topological nanosolenoids with ultrahigh coil density, giving rise to a 60 3 Theoretical design of topological heteronanotubes

CNT@BNNT

Straight Spiral

K K’ ×

K’ × K N S

Topological conductor Topological solenoid

Figure 3.10: Schematic summary of the topological heteronanotube (THT) scheme. large induced magnetic ﬁeld. Our results suggest that 1D topological heteronanotubes should provide very rich opportunities for both fundamental science of 1D topological physics and practical applications in low-power nanodevices. 4

Topological Zak phase of zigzag carbon nanotubes

In the previous chapter, using an outer BNNT shell we discovered that the armchair and chiral CNTs become topological materials [2]. We established that the valley-related topological physics is deﬁned in the 2D hexagonal graphene lattice, thus it is possible to use the 2D topological physics to discuss the 1D THT, due to the structural robustness against the curvature eﬀect of the cylinder geometry. More generally in fact, for 1D or quasi-1D systems, the intrinsic topological order can be characterized by the so-called Zak phase [42]. In this chapter, we investigate the topological Zak phase and related robust topological edge states in zigzag CNTs [1].

4.1 Introduction

As discussed in Section 1.4 of Chapter 1, for 1D or quasi-1D systems, the total Zak phase P γ is obtained by summing all the occupied states: γ = m γm where m is the band index. The Zak phase of each band, γm, is obtained by the integral of the Berry connection, i umk|(∂umk/∂k) , across the 1D BZ [42, 43]:

Z D ∂umk E γm = i umk| dk, (4.1) BZ ∂k where umk is the periodic part of the Bloch states of band index m. In general, the Zak phase can take any value. However, when a system has some spatial symmetries such as inversion and/or mirror symmetry, the total Zak phase γ can be quantized to 0 or π (modular 2π),

61 62 4 Topological Zak phase of zigzag carbon nanotubes corresponding to topological trivial or nontrivial cases respectively. Therefore, in spatial symmetry preserving systems, total Zak phase is used to classify diﬀerent 1D topological classes, i.e. 1D topological insulators (γ = π) or 1D normal insulators (γ = 0).

As discussed in Chapter 1 and 3, because of the periodic boundary condition along the tube circumference, CNTs show different electronic properties depending on their unique configurations characterized by the chiral vector Ch = n1a1 + n2a2 ≡ (n1, n2), where a1 and a2 are the hexagonal lattice vectors and n1 and n2 are integers called tube index [19, 16]. In particular, the 3N -rule governs the metal-insulator classification [19] of CNTs: when n1 − n2 = 3N where N is an integer, the CNT is metallic; otherwise it is insulating. As a result, all armchair CNTs(n,n) are metallic. For the zigzag CNTs(n,0), one-third of them are metallic - for example with n = 6, 9, 12 ··· , and two-thirds of them are insulating with a gap - i.e. with n = 7, 8, 10, 11 ··· . The metal-insulating phase diagram of zigzag CNTs is summarized in Table 4.1. The 3N-rule is quite robust - it works as along as the tube radius is not extremely small to cause cross-diameter interaction between the carbon atoms. The 3N-rule is also remarkably useful because it immediately tells if a CNT is metallic or not without any theoretical calculation or experimental measurement.

Note that there are two-thirds of zigzag CNTs having n 6= 3N: all of them have an intrinsic band gap. Apart from the value of the gap which depends on the diameter of the tube, there appears to have no further classification of these insulating tubes. An extremely interesting question is if there are any qualitative difference between the insulating zigzag CNTs beyond the quantitative value of the gap? It is the purpose of this chapter to understand this issue. By topological analysis based on both analytical and first-principles numerical approaches, we discover that the insulating zigzag CNTs (n, 0) are, in fact, not topologically equivalent. In particular, we discover a general "2N "-rule that governs the topological phase determination [1], namely, of all insulating zigzag CNTs, those with n = 2N (even) where N is integer, are topologically nontrivial; while those with n odd are topologically trivial. Cutting the topological CNTs generates topologically protected edge states. These edge states are extremely robust: enduring strong gating electric field and highly random edge environments, showing 4.2 Z2, Wannier center and Zak phase 63 significant consistency and robustness against external perturbations. In comparison, cutting the topologically trivial zigzag CNTs produces the well-known Fujita edge states [99] which are easily destroyed by small edge perturbations.

CNT(푛, 0) ⋯ 7 8 9 10 11 12 13 14 15 16 17 ⋯

Phase ⋯ I I M I I M I I M I I ⋯

Table 4.1: Phase table of zigzag CNTs. M and I are the abbreviations of metal and insulator respectively.

4.2 Z2, Wannier center and Zak phase

We have used two diﬀerent methods to calculate the total Zak phase γ of CNT. The ﬁrst method is by computing another equivalent topological invariant, Z2 [100]. The relationship between them is: eiγ = (−1)Z2 . (4.2)

The Z2 number can only take values of 0 or 1, corresponds to the total Zak phase of 0

(normal) or π (topological). In systems with time reversal symmetry, the Z2 number is a powerful tool to determine the topological phase, as seen in theories of 2D and 3D quantum spin Hall systems [20, 100]. If the system holds a spatial symmetry, the evaluation of the

Z2 number is greatly simpliﬁed as found by Fu and Kane in 2007 [100]. The "Fu-Kane method" demonstrates that the Z2 number can be determined from knowledge of the parity of the occupied Bloch wave functions at the time-reversal invariant points in the BZ. In a 1D system, it is written as:

Z2 Y (−1) = [pm(Γ)pm(X)], (4.3) m where Γ(k = 0) and X(k = 0.5) are two time-reversal invariant points in the 1D BZ, and pm(Γ) and pm(X) are the parity eigenvalues of the Bloch state in the m-th band at Γ and X, respectively. The product is over all occupied bands. As mentioned above, spatial symmetry is a necessary precondition for getting well-deﬁned (quantized) Zak phase in 1D systems [42]. 64 4 Topological Zak phase of zigzag carbon nanotubes

All our studied systems, i.e. zigzag CNTs, have spatial symmetry, thus the Fu-Kane method can be applied to simplify the topological analysis. In the following we shall employ the

Fu-Kane method to analytically investigate the topological Z2 number and the Zak phase of the insulating zigzag CNTs.

The second scheme to compute the total Zak phase γ is by using numerical Wannier functions within the KS-DFT framework [101]. For an 1D system, the center of charge of a Wannier function, or called Wannier center (WC), can be written as:

Z d D ∂umk E x¯m = i umk| dk, (4.4) 2π BZ ∂k where x¯m is the WC of the band with index m, d is the 1D unit cell size. Compare Eq.4.1 and Eq.4.4, we obtain a relationship between the Zak phase γ and the WC x¯ [42, 101]:

2π γ = x¯ . (4.5) m d m

Eq.4.5 is the basic formula of computing the Zak phase by Wannier center method. In this chapter, we apply the maximally localized Wannier functions as implemented in the Wannier90 package [102] within the DFT calculation performed by the Vienna Ab initio simulation package (VASP) [103]. In Appendix B, we give full details of theoretical derivations and technical parameters of the Wannier function method. This Wannier method will verify the analytical rule of topological phase classiﬁcation in CNTs, which will be introduced in Section 4.3.4.

4.3 Topological Zak phase of zigzag CNT

In this section we analyze the Zak phase of zigzag CNTs by two approaches. First, based on a two-band Hamiltonian of graphene, the Zak phase is derived analytically. Second, using the Wannier function method and KS-DFT, we numerically calculate the Zak phase which compares perfectly with the analytical results. 4.3 Topological Zak phase of zigzag CNT 65

4.3.1 Two-band model of zigzag CNT

We start from the electronic structure of graphene. The low-energy eﬀective two-band Hamil- tonian of graphene is given in Eq.1.4, which is reproduced here for clarity of discussions:

HGr(k) = −td(k) · σ, (4.6)

where σ = (σx, σy, σz) are Pauli matrices, and the components of the parameter vector d(k) are:  √ d (k) = 1 + 2 cos 1 ak cos 3 ak ,  x 2 y 2 x  √ 1 3 dy(k) = 2 cos aky sin akx , (4.7)  2 2   dz(k) = 0.

Note that dz(k) = 0 is guaranteed by the combination of spatial inversion symmetry and time reversal symmetry. Solving this Hamiltonian, the eigenenergy of the valance band of graphene is obtained:

s √ q 1 1 3 E (k) = −t· d2(k) + d2(k) = −t· 1 + 4 cos2 ak + 4 cos ak cos ak . (4.8) Gr x y 2 y 2 y 2 x

The eigenstates of the valance band is:

  1 e−iφ(k) uGr(k) = √   , (4.9) 2 1 where the function of e−iφ(k) is a phase factor having the form:

d (k) − id (k) e−iφ(k) = x y . (4.10) q 2 2 dx(k) + dy(k)

We now build a model of zigzag CNT according to the above formulation of graphene. From the view point of atomic structures, a zigzag CNT(n,0) is a rolled up ﬁnite graphene 66 4 Topological Zak phase of zigzag carbon nanotubes

(a) (b)

A B 풂ퟐ

풂ퟏ 3 4 1 2 3 4 1 2 …

Figure 4.1: Atomic conﬁgurations of the graphene (a) and the zigzag CNT (b). The graphene sheet in (a) is rolled up along the y direction to form the zigzag CNT in (b). The red-dashed zone of (a) and (b) denotes one unit cell. In (a), "1", "2", "3" and "4" label four atomic sites in one hexagonal benzene-like ring. Two groups of carbon atoms form two sublattices, A and B, as denoted by light and√ dark gray balls respectively. a1 and a2 are two lattice vectors, with the magnitude of |a1| = |a2| = a = 3aCC where aCC is the C-C bond length 1.42 Å.

strip along the zigzag direction [y direction in Fig.4.1(a)]. Along the circumference of CNT, the number of hexagonal benzene-like rings is n, and each of them consists of four atomic sites labeled 1-4 in Fig.4.1. For CNTs, the wave function is the same as Eq.4.9, but due to the periodic boundary condition around the circumference (rolled-up direction, y in Fig.4.1), ky can only take discrete values: 2π m k (m) = · , (4.11) y a n where m is the subband index: m = 1, 2, 3...2n for all the occupied states. Note that only occupied states contribute to the calculation of the topological Z2.

According to Eq.4.9, we set the amplitudes of the eigenstates on site 1 and site 2 are:

−iφ(k) φ#1 = e , φ#2 = 1. By adding suitable Bloch phases, we can deduct that φ#3 = 4.3 Topological Zak phase of zigzag CNT 67

−ika1 ika2 φ#2 · e , φ#4 = φ#1 · e . Then the eigenstates of zigzag CNTs can be expressed as:

    e−iφ(k) e−iφ(k)          1   1  √ 1   1  3 1   −ika1   −ia( kx− ky)  u(k) = √  e  = √  e 2 2  . (4.12) 2 n 2 n √    3 1   −iφ(k) ika2   −iφ(k) ia( kx+ ky) e · e  e · e 2 2      ··· ···

The ﬁrst, second, third and fourth components of above are the amplitudes of site 1, 2, 3 and 4, respectively.

4.3.2 Parity analysis of eigenstates

To obtain the topological Z2 and the Zak phase of zigzag CNT, the Fu-Kane method (Eq.4.3) will be used. First, we consider the Γ(0, 0) point of the 1D BZ where kx = 0. According to Eq.4.7, we get:  mπ  dx(Γ) = 1 + 2 cos( ), n (4.13)  dy(Γ) = 0.

Deﬁne f(k) ≡ e−iφ(k), therefore: d (Γ) f(Γ) = x . (4.14) |dx(Γ)| According to Eq.4.12, the wave functions (with band index m) of the zigzag CNT(n,0) at Γ can be written as:   f(Γ)      1  1    i mπ  um(Γ) = √  e n  . (4.15) 2 n    i mπ  f(Γ) · e n    ···

Due to the spatial mirror symmetry of zigzag CNTs, the wave functions of Eq.4.15 should be eigenstates of some parity operator Pˆ, namely Pˆ is the mirror reﬂection operator. When 68 4 Topological Zak phase of zigzag carbon nanotubes operating on one quantum state, Pˆ switches the amplitudes of wave functions on the atomic sites located at mirror reﬂecting positions, such as sites 1 and 2, or sites 3 and 4. That is:

  1      f(Γ)  1   ˆ  i mπ  P um(Γ) = √ f(Γ) · e n  = f(Γ)um(Γ). (4.16) 2 n    i mπ   e n    ···

It can be easily veriﬁed that the function f(Γ) equals the eigenvalues of Pˆ:

pm(Γ) = fm(Γ). (4.17)

When d (Γ) = 1 + 2 cos( mπ ) < 0 (see Eq.4.13), f (Γ) = dx(Γ) = −1. We may further x n m |dx(Γ)| 2n 4n deduce m ∈ ( 3 , 3 ). For an insulate zigzag CNT(n,0), there are two possible cases: n = 3s+1 2 4 and n = 3s + 2, where s an integer. For n = 3s + 1, m ∈ (2s + 3 , 4s + 3 ); for n = 3s + 2, 4 8 m ∈ (2s + 3 , 4s + 3 ). In both cases, the number of allowed m is 2s + 1. We therefore obtain, at the Γ(0, 0) point:

Y 2s+1 pm(Γ) = (−1) = −1. (4.18) m

Next, let’s consider the X = X(0.5, 0) point where k = √π , by Eq.4.7 we obtain: x 3a

  dx(X) = 1, (4.19) mπ  dy(X) = 2 cos( n ).

The function fm(X) and the eigenenergy Em(X) of band m are:

mπ 1 − 2i cos( n ) fm(X) = , (4.20) p 2 mπ 1 + 4 cos ( n ) 4.3 Topological Zak phase of zigzag CNT 69

r mπ E (X) = −t 1 + 4 cos2( ). (4.21) m n The wave function of band m at X is:

  f(X)      1  1    i mπ  um(X) = √  −ie n  . (4.22) 2 n    i mπ  if(X) · e n    ···

Obviously, um(X) in this form is not an eigenstate of the parity (mirror) operator. How- ever, if we consider the band of index m and the band of index m + n, it is found that they are degenerate (according to Eq.4.21) and orthogonal (according to Eq.4.22) at the X point:

Em(X) = Em+n(X), (4.23)

um(X)|um+n(X) = 0. (4.24)

As a result, through a unitary transformation, these two states um(X) and um+n(X) can form two states with opposite parity (one pair), giving rise to:

pm(X)pm+n(X) = −1. (4.25)

Since m = 1, 2, 3...2n, the total number of such pairs is n. Thus we obtain:

Y n pm(X) = (−1) . (4.26) m

Combining Eq.4.18 and Eq.4.26, we obtain:

Y n+1 [pm(Γ)pm(X)] = (−1) . (4.27) m 70 4 Topological Zak phase of zigzag carbon nanotubes

Recalling Eq.4.2 and Eq.4.3, we deduce a relationship between the tube index n of zigzag CNT(n,0) and the total Zak phase γ:

π γ = [1 − (−1)n+1]. (4.28) 2

This is a key prediction for zigzag CNTs.

4.3.3 The 2N-rule

CNT(푛, 0) ⋯ 7 8 9 10 11 12 13 14 15 16 17 ⋯

Phase ⋯ NI TI M TI NI M NI TI M TI NI ⋯

Table 4.2: Complete phase table of zigzag CNTs. M, TI and NI are the abbreviations of metal, topological insulator and normal insulator respectively.

Using Eq.4.28 to deduce γ for zigzag CNTs(n,0), a very surprising "2N -rule" is obtained where N is integer. Namely, if the tube index n is multiple of 2 (i.e. n is even), we obtain γ = π, indicating such a zigzag tube is in the topologically insulating (TI) phase; on the other hand if n is odd, we obtain γ = 0, the zigzag tube is in the normal insulating (NI) phase. Combined with the well-known 3N -rule discussed above, our analysis gives a complete phase behavior of zigzag CNTs shown in Table 4.2. Comparing to the old table (Table 4.1), in addition to the metallic phase giving by n = 3N, the insulating phases of the zigzag CNTs turn out to be further classified into two categories by the topological Zak phase: TI and NI. In later sections, we will show that the electronic properties of finite TI systems have remarkable difference from the NI ones. Therefore, distinguishing TI and NI is crucial for predicting and studying the behaviors of CNTs in actual experiments and practical applications.

As introduced in Section 4.1, the 3N -rule has been used to determine the phases (metal or insulator) of zigzag CNTs(n,0) for nearly three decades [19]. Here we have discovered that the 3N -rule is in fact not yet complete and the process of topological phase classification should be added. In Fig.4.2, we propose a complete flow chart of phase determination of 4.3 Topological Zak phase of zigzag CNT 71 zigzag CNTs, combining previous 3N -rule that classifies metal and insulator, with a new 2N -rule that further classifies the insulator phase into TI and NI.

퐶푁푇(푛, 0)

푌푒푠 “3푁” 푟푢푙푒 푛 = 3푁 ? 푀푒푡푎푙

푁표

퐼푛푠푢푙푎푡표푟

푌푒푠 “2푁” 푟푢푙푒 푛 = 2푁 ? 푇퐼

푁표

푁퐼

Figure 4.2: Complete ﬂow chart of phase determination of zigzag CNTs.

4.3.4 Wannier center analysis

Before showing important physical consequences of the TI/NI classification, we carry out first- principles DFT calculations to verify the analytical result of the 2N -rule. These calculations capture the microscopic details left out of the analytical model such as the curvature effects of CNT which can play a non-negligible role [17]. To this end, we apply the numerical Wannier function method within the KS-DFT, as discussed in Eq.4.5 of Section 4.2.

The first-principles calculated γ/2π is shown in Fig.4.3, fully consistent with the analytical 2N -rule. Interestingly, the curvature effects of small diameter metallic CNTs give rise to a small but finite band gap due to the cross-diameter interaction between the carbon atoms, yielding the so-called quasi-metallic phase [17]. From Fig.4.3 and due to the quasi-metallic gap, we observe that the 2N -rule also holds for the quasi-metallic tubes (n = 9, 12, 15...), validating the complete generality of the 2N -rule in all zigzag CNTs. In Appendix B, we 72 4 Topological Zak phase of zigzag carbon nanotubes give full details of theoretical derivations and technical parameters of the Wannier function method.

// //

Figure 4.3: Diagram between the sum of occupied Wannier centers x/d¯ and index n of zigzag CNTs(n,0). d is the 1D lattice constant, with the value of 4.26 Å.

4.4 Edge states in ﬁnite zigzag CNTs

Having established the 2N -rule, in the rest of this chapter we investigate physical differences of the TI/NI classes of zigzag CNT. To this end, we note that in a finite sized 1D graphene system (e.g. graphene nanoribbon), zigzag edges induce some in-gap edge states, which were theoretically predicted by Fujita et al. in 1996 [99]. These edge states are called Fujita edge states (FES), they are flat bands (without dispersion) inside the bulk band gap and locate around the Fermi level, and their density of states (DOS) and charge density are strongly located at the zigzag-edge atomic sites. Eexperimentally, FES was detected in graphene nanoribbons by scanning tunneling microscopy (STM) and spectroscopy (STS) [104]. Due to the nature of half-occupation and the location of FES being around the Fermi level, the in-gap zigzag edge states (FES) were found to play an important role in physical properties, including the optical properties [105, 106], the magnetic effects [107], and near-field phenomenon [108] etc. Most importantly, in nanoscience and nanotechnology, the size of systems is typically very small,thus the role of edge states becomes more significant. Despite of great efforts, experimental works reported that FES are strongly affected and/or destroyed by external 4.4 Edge states in finite zigzag CNTs 73 perturbations such as edge roughness and interaction with supporting substrate [109, 110]. Moreover, the experimentally observed spectroscopic features of FES (e.g. splitting energy gap) vary greatly and show clear behaviors of huge inconsistency [109, 110, 111, 112].

푪푵푻(ퟏퟕ, ퟎ) 푪푵푻(ퟏퟔ, ퟎ)

FES FES TES

Figure 4.4: Energy spectra of ﬁnite zigzag CNTs: (a) CNT(17,0) and (b) CNT(16,0). As shown in the Table 4.2, CNT(17,0) is a normal insulator while CNT(16,0) is a topological insulator. The length of both tubes are taken as the same: L = 16.8 nm (40 unit cells). Gray lines are the bulk states with a bulk band gap: 0.609 eV for the CNT(17,0) and 0.622 eV for CNT(16,0). The color lines near the Fermi level denote the in-gap edge states: Fujita edge states (FES) and/or topological edge states (TES).

All finite length zigzag CNTs exhibit FES due to the zigzag atomic configurations. On the other hand, the class of topological zigzag CNTs (the TI phase) which are found in this work can host additional topological edge states (TES), which are protected by the topological bulk-edge correspondence [114, 20]. To demonstrate TES, we cut a long zigzag CNT through the cross-section to expose its circular edge. To compare the conventional FES and the new TES, we choose two zigzag CNTs having close chiral vectors and similar geometric parameters: CNT(17,0) and CNT(16,0). The diameter of them are 1.33 nm and 1.25 nm, respectively. According to the 2N -rule, their insulating phases differ qualitatively: CNT(17,0) is a normal insulator while CNT(16,0) a topological insulator, see Table 4.2. Using the effective Hamiltonian Eq.(4.6), we determine the energy spectra of a long but finite segment of these two tubes. Due to similar diameter, their bulk gaps are quite close. As shown in 74 4 Topological Zak phase of zigzag carbon nanotubes

Fig.4.4(a) and (d), the edge states (colorful lines) of ﬁnite CNT(17,0) and CNT(16,0) appear in very similar manner but are in fact qualitatively diﬀerent. For CNT(17,0) - a normal insulator, only FES appear [green lines in Fig.4.4(a)]; but for CNT(16,0) - a topological insulator, additional TES exist, accompanying the conventional FES to form some mixed states [indicated by orange lines in Fig.4.4(b)].

In the following, we subject FES and TES to external perturbations including a gating electric ﬁeld and some random edge-potential environments. Since TES are protected by the topological Zak phase but FES are not, we will observe that these perturbations have essentially little eﬀect to TES but easily destroy FES.

4.4.1 FES and TES under gating electric ﬁeld

Let’s consider an external gating electric field to the FES and TES. Applying a uniform electric field E along the y direction in Fig.4.1(b), the effective Hamiltonian becomes:

X † H = HfCNT + µi cˆi cˆi, (4.29) i

† where HfCNT is the Hamiltonian of the finite zigzag CNT, and cî (cî ) is the annihilation

(creation) operator at site Ri ( i belongs to all atomic sites). µi is the chemical potential variation induced by the electric ﬁeld at site Ri, and takes the value:

µi = E yi, (4.30)

where yi is the coordinate component of Ri.

As shown in Fig.4.5(a-c), when E is applied, the FES (green lines) of the normal CNT(17,0) split and open an energy gap. As E increases, the spitting gap becomes larger and when it reaches 0.2 V/Å, the FES completely merge into bulk states of the CNT and no measurable edge states exist inside the bulk gap, as shown in Fig.4.5(c). We conclude that FES are 4.4 Edge states in ﬁnite zigzag CNTs 75

(a) (b) (c) 푬 = ퟎ 푬 = ퟎ. ퟏ 푬 = ퟎ. ퟐ

(d) (e) (f) 푬 = ퟎ 푬 = ퟎ. ퟏ 푬 = ퟎ. ퟐ

Figure 4.5: Energy spectra of finite zigzag CNTs under different gating electric fields E. The unit of E is Volt/Angstrom (V/Å). (a-c) are the cases of normally-insulating CNT(17,0), and (d-f) are the cases of topologically-insulating CNT(16,0). The length of both tubes are taken as the same: L = 16.8 nm (40 unit cells). Gray lines are the bulk states. The red and green lines near the Fermi level denote the topological edge states (TES) and the Fujita edge states (FES) respectively, and the orange lines are the mixing of them. 76 4 Topological Zak phase of zigzag carbon nanotubes sensitive to the gating field E for the case of the normal insulator CNT(17,0).

In comparison, TES of the topological CNT(16,0) show entirely different features under external field. Without E, the FES and TES all sit together at the Fermi level [orange line in Fig.4.5(d)]. At any finite E, i.e. 0.1 V/Å in Fig.4.5(e), the FES (green lines) split into two just as those of the normal insulator CNT(17,0); while the TES (red lines) remain at zero energy without being influenced by the external electric field. Increasing E = 0.2 V/Å in Fig.4.5(f), the FES are shifted into the bulk states and disappear from the bulk gap, but the TES remain unperturbed. We conclude that the TES are robust against (very strong) external electric fields while FES are not. Experimentally, the gating field can be used as an effective way to separate FES and TES (filter out FES).

4.4.2 FES and TES under random edge-potential environment

(a) (b)

Figure 4.6: Energy splitting gap of ﬁnite zigzag CNTs under random edge potential with diﬀerent maximal amplitudes Vmax. For each Vmax, we perform 30 sets of calculations including totally random number arrays as the edge potential. Red and blue lines denote the cases of topological insulator CNT(16,0) and normal insulator CNT(17,0), respectively. The length of both tubes are taken as the same: L = 16.8 nm (40 unit cells).

The second perturbation we examine is random edge-potential environment. In experi- 4.4 Edge states in finite zigzag CNTs 77 mental situations, the CNT edge may be subjected to complicated and inevitable non-ideal environments such as impurities, dangling bonds, other defects, end-cap structures, etc., due to fabrication processes. We may simulate some of these uncertainties by a random potential environment on the edge atoms. To this end, we build an effective model including external random edge potential: X † H = HfCNT + µi cî cî, (4.31) i∈ES

† where HfCNT is the Hamiltonian of the finite zigzag CNT, and cî (cî ) is the annihilation

(creation) operator at site Ri [ i belongs to edge sites (ES)]. µi is the chemical potential variation induced by the complicated edge environment at site Ri, and take the random value:

µi = Random(0,Vmax), (4.32) where Vmax is the maximal amplitude of the random potential.

As shown in Fig.4.6(a) and (b), under diﬀerent sets of random edge potentials, the energy splitting gap of FES (blue lines) for the normal insulator CNT(17,0) varies very strongly and randomly from sample to sample. We conclude that the FES are fragile and sensitive to the edge environment, totally consistent with existing experiments where the FES are strongly susceptible to external environments [109].

In comparison, for the topological insulator CNT(16,0), the TES is remarkably robust against the random edge potential as shown by the red line in Fig.4.6(a), and the robustness is almost perfect even at very large random perturbations as shown in Fig.4.6(b). We conclude that the topological zigzag CNTs, protected by the topological Zak phase, have significantly better reliability and repeatability for edge engineering which are important for practical applications of CNT in quantum dots and nanotube devices. We also noticed that the topological insulator CNT(16,0) can be selectively and efficiently fabricated by using structure-defined catalysts in experiments [113]. 78 4 Topological Zak phase of zigzag carbon nanotubes

4.5 Further discussions on diﬀerent terminations

Termination type A B

Unit cell shape … … … …

Bulk symmetry Inversion/mirror Inversion

Edge configuration

Edge stability High Low

Topological property “2N” rule All TI

Table 4.3: Geometries and topological properties of two types of terminations (A or B) whose unit cells hold the spatial symmetries. In second row, the red boxes denote one bulk unit cell. From A to B, the unit cell is shifted by 1/4. In third row, the types of spatial symmetries for each bulk cell are presented. In the forth row, we show the edge configurations corresponding to each bulk unit cell (the second row), where red balls denote the edge carbon atoms. In Type A edge atoms are bonded with each other while in Type B edge atoms are dangling. The edge stability of finite CNTs and topological property of each type are demonstrated in the fifth and sixth row respectively.

Before ending this chapter, we discuss an interesting but subtle point. In zigzag CNTs, there are in fact two types of unit cells with different terminations holding the spatial symmetries: named "Type-A" and "Type-B" in Table.4.3. The CNTs with unit cell of Type-A are what we have investigated so far in previous sections, they are governed by the 2N -rule for the topological phase classification. It turns out that the 1D Zak phase can depend on the termination type of the bulk unit cell, i.e. the choice of the unit cell shape [43, 114], which 4.5 Further discussions on different terminations 79 calls for a further investigation of zigzag CNTs with Type-B termination. We found that all CNT(n,0) with Type-B termination are topological, independent with the tube index n (See Appendix B for details). As for the 1D topological bulk-edge correspondence, the edge type of finite system should be commensurate with the terminal type of the bulk unit cell [43, 114]. Accordingly, we can plot the edge configurations in finite systems of Type-A and -B, as shown in the fourth row of Table.4.3.

In Fig.4.7, we compare the energetic stability of finite zigzag CNTs with Type-A and Type-B edges. It is clearly that the system with Type-A edge is more energetically stable than those with Type-B edge. This is due to the different edge atomic configurations (red balls) shown in the fourth row of Table.4.3. Atoms in Type-A edge are bonded with each other while atoms in Type-B edge are dangling which hold double unpaired electrons. In fact, in most previous studies of finite zigzag CNTs [108, 116], only Type-A edges (energetically favorable) were observed. Therefore, from a practical point of view, topological properties of the Type-A edge are relevant.

Figure 4.7: DFT total energies of finite CNT(17,0) systems versus tube lengths. The red and blue lines respectively represent different edges: Type A and Type B, corresponding to the configurations shown in Table.4.3. Large supercells of finite CNT systems (including up to 2040 atoms) are calculated by the real- space Kohn-Sham density functional theory (KS-DFT) as implemented in the RESCU package [52]. 80 4 Topological Zak phase of zigzag carbon nanotubes

4.6 Summary

In this chapter, by calculating the Zak phase of zigzag CNT(n,0) using analytical and first- principles techniques, a very interesting and surprising 2N -rule was discovered that classifies the insulating phases of the zigzag CNTs. For even n, n = 2N where N is an integer, the zigzag CNT is a topological insulator; and it is a normal insulator for odd n. In finite length topological zigzag CNTs, topological edge states occur which are remarkably robust against strong gating electric fields and random edge environment, distinctly different from the traditional Fujita zigzag edge states. Therefore, for CNTs, the complete phase behavior is classified by the 3N -rule which classifies the tubes into metallic and insulating phases, followed by the 2N -rule applicable to the zigzag tubes that further classifies the insulating tubes into topological and normal phases. The robust TES should be very useful for potential applications of CNTs in electronics and optoelectronics devices. 5

Dirac electrons in 2D moiré superlattice

In Chapter 2, we have analyzed topological physics induced by stripe-like moiré patterns of graphene/hBN bilayer with a small uniaxial strain. In this chapter, we investigate another interesting moiré-induced phenomenon - modulated behaviors of Dirac electrons in 2D hexagonal graphene/hBN moiré superlattice without strain [4].

5.1 Introduction

As mentioned before, 2D van der Waals (vdW) heterostructure has attracted great attention [15, 58, 117, 118, 119]. By stacking diﬀerent 2D materials to bond via the vdW force, these heterostructures provide very interesting material phase space where new and high performance nano-materials are expected to emerge[15]. The moiré pattern, caused by lattice mismatching or relative rotation of two stacking layers, is often observed in 2D heterostructures [15, 54, 55, 56]. In visual arts, the moiré pattern is an optical perception of a new pattern formed on top of two similar stacking patterns. In 2D vdW materials, the moiré pattern can be a physical superlattice at the nanometer moiré wavelength scale, which brings about novel electronic properties such as self-similar Hofstadter butterﬂy states [58] and topological states as presented in Chapter 2.

A prototypical vdW heterostructure is a ﬂat sheet of graphene (Gr) stacking on a ﬂat- sheet hexagonal boron nitride (hBN) [120, 121]. hBN has a large band gap and relatively weak interaction with Gr, thus the Gr/hBN heterostructure can provide a clean platform

81 82 5 Dirac electrons in 2D moiré superlattice for investigating the behavior of Dirac-like electrons in the graphene[57]. Most interestingly, while Gr and hBN both have honeycomb lattice, the moiré pattern on 2D Gr/hBN can form a third one, a honeycomb superlattice - albeit at a much larger extension of the moiré wavelength as shown in Fig.5.1. In such a material, two generations of Dirac electron states in the Brillouin zone (BZ) are thus expected - the original Dirac cone (ODC) of the graphene and the secondary Dirac cone (SDC) of the larger scale hexagonal moiré supperlattice [57, 122, 123, 124, 125, 126]. This very distinct property has attracted great attention both experimentally by measurements [54, 57, 120, 121, 122, 127], and theoretically by model analysis [123, 124, 125, 128, 129, 130, 131, 132, 133].

From a theoretical point of view, first-principles methods are necessary for providing unambiguous information about quantum states in the moiré pattern. However, due to a small lattice mismatch of 1.8% between Gr and hBN, the moiré pattern in Gr/hBN has a superlattice unit cell with extremely large number of atoms, for instance, there are 12,322 atoms in one moiré cell, as shown in Fig.5.1. For this reason, the behavior of the Dirac electron states of the Gr/hBN moiré system has not been predicted from atomic first principles. It is the purpose of this chapter to fill this void [4].

5.2 Dirac electronic states in ﬂat-sheet moiré patterns

We begin by investigating the flat-sheet moiré pattern. The heterostructure without relative rotation is shown in Fig.5.1. As discussed in Chapter 2, there are three high-symmetry local stacking configurations coexisting on the moiré pattern plus the transitional regions between them. The three high-symmetry local stacking configurations are shown in the right panel of Fig.5.1 and are designated as the N-h stacking - meaning carbon atoms over nitrogen atoms and hollow sites; B-h stacking - meaning carbon atoms over boron atoms and hollow sites; B-N stacking - meaning carbon atoms over boron atoms and also over nitrogen atoms. The orange zone in Fig.5.1 is the calculated supercell: since the lattice constants of graphene and BN are 2.4595 Å and 2.5042 Å respectively and differing by 1.8%, the supercell contains 56 5.2 Dirac electronic states in flat-sheet moiré patterns 83

B-N Pattern

B-h Pattern

N-h Pattern

Carbon Boron Nitrogen

Figure 5.1: The flat-sheet Gr/hBN vertical heterostructure and the moiré patterns on top of the lattice, where the hBN and graphene have a natural lattice mismatch of 1.8%. Different patterns represent different high-symmetry stacking configurations. The red, green and gray balls denote boron, nitride and carbon atoms, respectively. The orange line zone is the calculated supercell with dimensions 137.73 Å × 137.73 Å that contains 12,322 atoms.

× 56 repeated primitive cells of graphene and 55 × 55 of hBN. For the flat-sheet Gr/hBN, the distance between the two layers is 3.22 Å, which is the equilibrium distance of the most stable stacking (B-h) [65]. These structural parameters are verified by relaxation using the force field method of large-scale atomic/molecular massively parallel simulator (LAMMPS) [74], in particular the in-plane deformation of atomic positions is found to be negligible due to the strong intralayer sp2 bonds. For this moiré structure, the supercell (orange zone) is the smallest unit to correctly construct the moiré pattern periodically residing on the infinitely large bilayer. Thus all the band structures to be presented below are the supercell band structures. We also calculate the lattice-matched Gr/hBN for comparison purposes where the lattice constant of hBN is compressed to that of the Gr: in this case there is only one stacking for which we choose the most stable one - the B-h stacking. Since the main electronic properties - SDC and ODC of the Gr/hBN bilayer, come from graphene states, to build the lattice-matched structure we compress hBN without changing the graphene lattice.

The calculation of the moiré superlattice that involves over ten thousand atoms is made possible by the real-space KS-DFT method RESCU [52], which was recently developed by our group. Real-space implementation avoids the global communication required by Fourier transforms in plane-wave codes so that parallelization eﬃciency is drastically enhanced. The 84 5 Dirac electrons in 2D moiré superlattice

Chebyshev ﬁltering technique [53] and other advanced computational mathematics are employed to avoid solving large eigenvalue problems in the full Hilbert space. These advances allow KS-DFT to be solvable for material systems having many thousands of atoms at the same accuracy as other KS-DFT methods. For more technical details we refer the interested readers to the original reference [52].

In our calculation, the effect of vdW interaction is included in the exchange-correlation potential by using the Grimme’s method, which accurately describes the structural and energetic properties of graphene and many other 2D materials[118, 134]. For self-consistent field (SCF) calculation, we adopt a single Γ-point reciprocal space sampling since the BZ is extremely small for such large systems. To further reduce the computational burden but without compromising accuracy (which we confirmed), a single-zeta polarized (SZP) atomic orbital basis set is generated to describe the carbon atoms. For the real-space part, the grid resolution is set at 0.51 Bohr. We use periodic boundary conditions along the in-plane dimensions (x,y) while using the von Neumann boundary conditions in the stacking direction which allows us to decrease the vacuum thickness to 5.4 Å. Self-consistent KS-DFT calculation is considered converged when all elements of the Hamiltonian matrix and the density matrix are converged to less than 10−5 a.u.

The calculated band structure of the flat-sheet moiré pattern in Fig.5.1 is shown in Fig.5.2(a). The SDC is located at the Γ point and formed by the bands labeled as S1 and S2, which are separated by a gap of 21.0 meV, and they are curved to give SDC dispersion. For comparison, Fig.5.2(b) plots the band structure of the strained, lattice-matched bilayer, for which the S1, S2 and S3 bands are degenerate at Γ and there is no SDC dispersion. The ODC located at the K point in Fig.5.2, is formed by the O1 and O2 bands in Fig.5.2(a). As for the flat-sheet moiré pattern, there is a very small ODC gap of 7.5 meV. On the contrary, the lattice-matched bilayer has a much larger ODC gap at 38.3 meV as seen in Fig.5.2(b). It is noted that the locations of ODC and SDC in BZ rely on the band folding of the supercell. Comparing the lattice-matched sheet to the 2D flat-sheet moiré pattern, their local stacking configurations are dramatically different, even though only 1.8% lattice mismatching exists 5.2 Dirac electronic states in flat-sheet moiré patterns 85

(a) (b) 0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

O1 0 0

S1 O2

Energy (eV) Energy Energy (eV) Energy -0.2 S2 -0.2 S3 -0.4 -0.4

-0.6 -0.6

-0.8 -0.8 M K M M K M

Figure 5.2: The calculated band structures. (a) For the flat moiré pattern. (b) For the strained, lattice- matched structure. Γ, M, K are high-symmetry k-points, located at the center, the edge center, and the corner points of the hexagonal Brillouin zone, respectively. In (a), S1, S2 and S3 are the SDC bands, O1 and O2 are the ODC bands, and there is a gap of 7.5 meV between O1 and O2 that is too small to be clearly resolved in the figure. 86 5 Dirac electrons in 2D moiré superlattice in the latter. Two important outcomes arise in the moiré pattern: the emergence of SDC band dispersion and the suppression of the ODC gap. The dispersion indicates SDC to be more free-electron-like rather than Dirac-electron-like, while suppression of the ODC gap makes ODC states more Dirac-electron-like. For comparison, exactly opposite conclusions are reached for the lattice-matched bilayer, as clearly seen in Fig.5.2(b). The calculated band gaps in this work are generalized gradient approximation (GGA) gaps which present some differences compared to previous DFT calculations based on the local density approximation (LDA) [65]. Experimentally, diverse gaps were reported in the literature, ranging from 30 meV [58] to 160 meV [57], depending on the measurement method, sample quality, and other material factors. Nevertheless, the qualitative features of the moiré pattern on the gap modulation at SDC and ODC reported in this chapter should not be significantly affected by the precise gap values.

The origin of the SDC dispersion and the suppression of the ODC gap in the moiré pattern can be traced to the nature of quantum wave functions. Fig.5.3 plots the spatial

2 distribution of the modular squared wave functions |ψn(r)| where n is the label of SDC and ODC bands (S1 - S3, O1, O2). Most importantly, we ﬁnd that in real space the SDC states are rather localized on some, but not all, local high-symmetry stacking conﬁgurations in the

2 moiré pattern. For instance, |ψS1(r)| of the S1 band shown in Fig.5.3(a) is clearly located at carbon atoms in the B-N and B-h stacking, but is essentially invisible in the N-h stacking

2 [see middle column of Fig.5.3(a)]. Similarly, the S2 and S3 states |ψS2,S3(r)| in Fig.5.3(b,c) are invisible in the B-N and B-h stacking respectively but present in the N-h stacking. In other words, the wave functions of SDC bands show spatial-stacking-selected localization to depict hexagonal inversion asymmetry of the moiré superlattice. The interlayer interactions at these stackings are quite diﬀerent and nonuniform over the moiré pattern, resulting in the lifting of degeneracy at SDC and opening up the SDC gap as well as introducing SDC dispersion. For the lattice-matched structure, there is only one stacking and thus degeneracy at the SDC is preserved as shown in Fig.5.2(b). The wave functions of ODC bands are

2 shown in Fig.5.3(d,e). Notably diﬀerent from SDC, |ψO1,O2(r)| are moiré - delocalized. For 5.2 Dirac electronic states in ﬂat-sheet moiré patterns 87

(a) B-N (N-site)

B-h (h-site) S1

N-h (none)

(b) B-N (none)

B-h (B-site) S2

N-h (N-site)

B-h (none) S3

N-h (h-site)

(d) B-N (N-site)

B-h (B-site) O1 N-h (h-site)

(e) B-N (B-site)

B-h (h-site) O2

N-h (N-site)

2 Figure 5.3: Left column: spatial distribution of modular squared wave functions |ψn(r)| of the flat moiré system, where n is the label of the SDC and ODC bands (S1-S3, O1, O2). The isosurfaces used are 7.6×10−6 −6 2 a.u. for SDCs and 2.7×10 a.u. for ODCs. Middle column: amplified view of |ψn(r)| at selected regions as indicated. The representation of atoms is the same as those in Fig.5.1. Right column: indicating the stacking configurations. B-site, N-site and h-site represent carbon locations over boron atoms, nitrogen atoms and hollow sites, respectively. 88 5 Dirac electrons in 2D moiré superlattice

O1 which gives the conduction band minimum (CBM) of the ODC and O2 which gives

2 the valence-band maximum (VBM), |ψO1,O2(r)| spread over all three high-symmetry local stacking configurations to depict inversion asymmetry of the pristine sub-lattice. The moiré delocalization of wave function is because both the CBM and VBM bands are contributed by atomic orbitals of all the three stackings, giving rise to an overall effect that suppresses the ODC gap. Due to the complicated moiré structures, the ODC wave functions also have small inhomogeneities, leading to the gap of a finite nonzero value [7.5 meV in Fig.5.2(a)] Experimentally, the ODC gap suppression was indeed observed [120, 121].

2 The ordering of SDC bands can also be well understood by analyzing |ψn(r)| . The S1 state is located at N-sites of the B-N stacking and h-sites of the B-h stacking, and the S2 (S3) state located at B-sites of the B-h (B-N) stacking and N- (h-) sites of the N-h stacking [Fig.5.3(a-c)]. In the hBN layer, B atoms are positively polarized and N atoms are negatively polarized, due to large differences in their electronegativity. The π orbital of the graphene layer has attractive and repulsive interactions with B cations and N anions, respectively[70]. Therefore electrons in graphene preferentially locate at the B-sites and avoid N-sites to reduce electrostatic energy. An on-site energy ordering is therefore set: N-site > h-site > B-site. This leads SDC bands to naturally order in such a way that the S1 band (N-sites and h-sites) has the highest band energy while the S3 band (B-sites and h-sites) has the lowest. For the ODC bands, the on-site energy can further substantiate the overall effect in the moiré structure: both the two ODC bands include all the three sites as shown in Fig.5.3(d) and (e), which suppresses the band-energy difference.

As a comparison, shown in Fig.5.4, in the lattice-matched system the wave functions of the ODC bands (VBM and CBM) are located at B-sites and h-sites respectively. This is called as individual eﬀect, leading to a much larger band gap in Fig.5.2(b). 5.3 Dirac electronic states in wavelike moiré patterns 89

(a)

h-site CBM

(b)

B-site VBM

2 Figure 5.4: Spatial distribution of wave functions of the ODC band states |ψCBM,V BM (r)| in the lattice- matched system (B-h stacking). The isosurface value of wave functions and the representation of atoms are the same as those in Fig.5.3(d) or (e).

5.3 Dirac electronic states in wavelike moiré patterns

More recently, Argentero et al. observed experimentally [77] that a freestanding Gr/hBN is in fact a wavelike sheet rather than a flat sheet. Namely, the freestanding Gr/hBN heterostructure forms undulations in the out-of-plane direction, having an effective corrugation as large as 8 Å [77]. Such a large corrugation greatly changes the local high symmetry stacking configurations of Gr on hBN, and compared with the 2D moiré patterns in a flat sheet, the wavelike moiré superlattice and its associated Dirac electrons on the wavelike sheet are expected to be rather different. Understanding Dirac electrons in the moiré superlattices of flat sheets and wavelike sheets is of fundamental importance for their applications in flexible electronics, wearable electronics and 2D nanoelectronics.

In the following we turn our attention to the freestanding wavelike moiré structure. Be- fore calculating electronic properties, we determine its atomic structure using the LAMMPS package [74]. The in-plane C-C and B-N interactions are treated using many-body Tersoﬀ potentials [75, 76]. The interlayer vdW interaction is described by a Morse potential whose parameters are taken from the Ref.[77] with some exceptions. We set equilibrium distances 90 5 Dirac electrons in 2D moiré superlattice

(a) (c) ͲǤͺ B-N B-h N-h

ͲǤ͸

ͲǤͶ

ͲǤʹ

Ͳ (b)

BN Gr (eV) Energy ͺǤͲ͸Ͳ ͳͳǤ͵ͺͲ ǦͲǤʹ

͸ǤͲͶͷ ͻǤ͵ͶͲ ǦͲǤͶ

ͶǤͲ͵Ͳ ͹ǤʹͻͲ ǦͲǤ͸ ʹǤͲͳͷ ͷǤʹͶͷ

ͲǤͲͲͲ ͵ǤʹͲͲ ǦͲǤͺ ડ (d)

B-N (B-site)

B-h (h-site) CBM

N-h (N-site)

(e) B-N (N-site)

B-h (B-site) VBM

N-h (h-site)

Figure 5.5: (a) The relaxed freestanding Gr/hBN heterostructure. (b) The out-of-plane corrugation amplitude of hBN and Gr layers, unit in Å. (c) The calculated band structure of the freestanding structure. (d-e) 2 Spatial distribution of wave functions of ODC band states |ψCBM,V BM (r)| . The isosurface value of wave functions and the representation of atoms are the same as those in Fig.5.3(d-e). 5.3 Dirac electronic states in wavelike moiré patterns 91 of the C-B and C-N bonds at 3.56 Å and 3.54 Å, respectively. For the freestanding film situation, this is to keep the interlayer distance near 3.2 Å to be consistent with the flat-sheet situation. By this way we can focus on the effects of the large corrugation for the freestanding films. The relaxation is completed when the forces are less than 1.0 × 10−6 eV/Å.

As shown in Fig.5.5(a,b), relaxation of the freestanding moiré structure leads to a large out-of-plane corrugation - up to ∼8.1 Å, along the periodicity of the moiré superlattice. The hBN and graphene layers corrugate with essentially the same shape and very similar amplitude. This corrugation originates from an energetic competition between maximizing the favorable stacking configuration (B-h) and minimizing the elastic energy of the lattice. By the out-of-plane deformation, the area of B-h stacking relative to other stacking configurations is enlarged, consistent with the experimental observation [77]. We therefore take this structure for further electronic analysis. This new wavelike moiré pattern influences the band structure very significantly, as shown in Fig.5.5(c). Now, the ODC gap becomes 31.1 meV, which is much larger than the 7.5 meV of the flat-sheet moiré pattern reported above. Fig.5.5(d,e) show that the wave function of the CBM (VBM) is located at the h- (B-) site of the B-h stacking, the B- (N-) site of the B-N stacking, and the N- (h-) site of the N-h stacking, different from the flat-sheet moiré pattern in Fig.5.3(d,e). At the same time, we note from Fig.5.5(c) that the SDC band dispersion and SDC gap all vanish in the freestanding wavelike moiré structure.

Why does the freestanding wavelike moiré pattern behave so differently from the flat-sheet counterpart? This is, again, due to the role played by local high-symmetry stacking configurations. In the freestanding wavelike film, the enlargement of B-h stacking and shrinkage of other stackings, significantly promotes the interlayer interaction contributed by the B-h stacking. This dominance of the B-h stacking can be verified by spatial distribution of the wave function in Fig.5.5(d,e). In the lattice-matched system (B-h stacking), the wave functions of VBM and CBM are located at B-sites and h-sites respectively (see Fig.5.4). This is exactly the same as the result of the B-h stacking configuration in the wavelike moiré case. For the wavelike moiré structure, its SDC dispersion is significantly weakened as seen 92 5 Dirac electrons in 2D moiré superlattice in Fig.5.5(c) - i.e. becoming similar to that of the lattice-matched flat sheet; and its ODC band gap is found to be 31.1 meV - i.e. approaching that of the lattice-matched case (38.3 meV). These are not accidental because the lattice-matched flat sheet consists of a pure B-h stacking in the entire film. Therefore, by corrugating into the third dimension, B-h local stacking becomes dominant on the wavelike moiré pattern of the freestanding film, which leads to an electronic structure that is more similar to the lattice-matched flat sheet than to the flat moiré pattern. This interesting prediction should be experimentally verifiable.

5.4 Summary

In this chapter, the Dirac electrons modulated by the 2D moiré patterns of the Gr/hBN are investigated from atomic first principles, for both the flat sheet and the freestanding wavelike films. We find that the local high-symmetry stacking configurations of the moiré pattern play the key role. In a flat sheet, SDC dispersion of the moiré pattern emerges due to the stacking-selected localization of SDC wave functions; at the same time, the ODC gap is suppressed due to an overall effect of the moiré - delocalized ODC wave functions over all local high-symmetry stacking configurations. For a freestanding wavelike moiré pattern, the B-h local stacking configuration dominates the moiré superlattice at the expense of other local stackings, leading to an electronic structure more similar to that of the lattice-matched flat sheet than to the flat moiré pattern. This is significant for applications of the film in flexible and wearable electronics, since the low-energy electronic structure is sensitive to the local high-symmetry stacking configurations, which in turn depends on whether the film is constrained to a 2D plane or freestanding with possible corrugation into the third spatial dimension. 6

Conclusion

In this thesis, by both analytical and ﬁrst principles techniques, we systematically investigate novel valleytronics, topological physics and Dirac electronics in emerging low-dimensional carbon-based materials, including 2D graphene, 1D carbon nanotubes, as well as a broad range of their van der Waals (vdW) heterostructures and moiré superlattices.

In Chapter 2, we propose a general and robust scheme, the moiré valleytronics, to realize high-density arrays of 1D topological helical channels at room temperature and in realistic easily fabricatable materials. Specifically, an 1D graphene(Gr)/hBN moiré pattern and several other moiré systems are shown to establish moiré valleytronics. For the 1D graphene(Gr)/hBN moiré pattern, we show by first-principles calculations that a long- period moiré pattern realizes dense arrays of helical channels inside the noncryogenic bulk gap opened by local inversion symmetry breaking due to hBN. These helical channels arise from a periodical modulation on the topological order of local electronic structure, due to variations of atomic stacking in the moiré pattern. A topological phase diagram based on the calculations of Berry curvature and valley-dependent Chern number reveals the intrinsic dependence between the topological order and the atomic stacking configuration. Moreover, the generality and experimental feasibility are discussed and theoretically substantiated by investigating a broad range of moiré systems including strained bilayer-graphene moiré pattern and the silicence/hBN moiré pattern. Finally, we predict that in the freestanding wavelike moiré structure, two groups of moiré edge arrays carrying counter-propagating helical channels are spatially separated in the out-of-plane direction, which demonstrates the structural

93 94 6 Conclusion robustness of moiré valleytronics. This study paves a new way for modulating valley electronic states and realizing multiple topological helical channels using moiré patterns which are relatively easily accessible in experiments. The results suggest that valleytronics based on moiré patterns, i.e. moiré valleytronics, to be a rich research direction in material physics.

In Chapter 3, we propose and investigate the idea of topological heteronanotube (THT) for realizing a one-dimensional topological material platform that can pave the way to low- power carbon nanoelectronics at room temperature. We predict that the experimentally feasible system, coaxial double-wall heteronanotube - a CNT inside a boron nitride nanotube (BNNT), can act as a THT. By first-principles calculations on the prototypical CNT@BNNT system, two groups of valley-dependent helical states are discovered to form inside a noncryogenic bulk band gap. By analyzing wave functions and inferring valley topological invariance, these helical states are found to locate at two groups of topological moiré edges, due to the local topological phase transition associated with the local atomic registry that is varying along the circumference of the tubes. The local topological order of the CNT is modulated by the CNT-BNNT interaction. In principle, low-energy carriers in the helical states supported by the topological moiré edges do not suffer intravalley and intervalley scattering, thus THT is an ideal 1D topologically protected ballistic conductor where transport suffers less dissipation. The spiral-oriented THTs have topological current flowing spirally around the tube to function as topological nanosolenoids with a ultrahigh coil density, giving rise to a large induced magnetic field. By investigating small-diameter THT systems, we verify the topological robustness of THT against curvature and tube diameter. By studying multi- period THTs, reversed-ordered THTs, various-orientation THTs and different tube materials, the generality and experimental feasibility are demonstrated through first-principles calculations. Our results suggest that the 1D THT should provide very rich opportunities for both fundamental science of 1D topological physics and practical applications in low-power nanodevices.

In Chapter 4, we discover a very interesting and surprising 2N -rule that topologically classifies the insulating phase of zigzag CNTs(n,0). For even n, n = 2N where N is an 95 integer, the zigzag CNT is a topological insulator; and it is a normal insulator for odd n. The 2N -rule is established by calculating the 1D topological Zak phase through both analytical and first-principles techniques. As for the analytical scheme, an effective low-energy model of zigzag CNTs is applied and parity analysis of occupied eigenstates is performed; for first-principles calculations, we employ numerical Wannier functions within the Kohn-Sham density functional theory. They both consistently reveal the 2N -rule for the topological insulator (TI) and normal insulator (NI) classification. Furthermore, we show the important physical consequences of the TI/NI classification. Cutting the topological CNTs generates topologically protected edge states. These edge states are extremely robust: enduring strong gating electric field and highly complicated edge environments, showing significant consistency and robustness against external perturbations. In comparison, the non-topological Fujita edge states are easily destroyed by external perturbations. The robust topological edge of the zigzag CNTs should be very useful for potential applications in electronics and optoelectronics.

In Chapter 5, we investigate the Dirac electronic properties modulated by the 2D moiré pattern of graphene(Gr)/hBN. To capture the overall picture of the 2D moiré super-lattice, supercells containing 12,322 atoms are simulated by first-principles, based on the large-scale KS-DFT method RESCU, which was recently developed by our group. We find that the local high-symmetry stacking configurations of the moiré pattern play the key role on the behaviors of the secondary Dirac cone (SDC) and the original Dirac cone (ODC). In a flat-sheet moiré pattern, the SDC dispersion emerges due to the stacking-selected localization of SDC wave functions. In comparison, the ODC wave functions spread over the entire structure, i.e. moiré - delocalized, leading to the suppression of the ODC gap. In the freestanding wavelike moiré pattern, we predict that a specific local stacking in the moiré superlattice is enlarged at the expenses of other local stackings, leading to an electronic structure more similar to that of the lattice-matched flat Gr/hBN than that of the flat-sheet moiré pattern: SDC gap vanishes and ODC gap is promoted. This is significant for applications of the film in flexible and wearable electronics, since the low-energy electronic structure is sensitive to the 96 6 Conclusion local high-symmetry stacking configurations, which in turn depends on whether the film is constrained to a 2D plane or freestanding with possible corrugation into the third spatial dimension. Our results demonstrate that the moiré pattern can provide a new platform to generate and modulate novel Dirac electronics in nanodevices.

Overall, this thesis suggests that there exists very rich topological physics and other novel states in carbon-based materials, especially due to moiré patterns. We envision many further investigations to be in order. Several of them are listed below:

1. Moiré-modulated spin-valley physics: For monolayers formed by heavier Group-IV elements, such as silicene, germanene or stanene, larger spin-orbital interaction (SOI) makes the spin-related topological physics possible [41]. By building suitable moiré systems, such as silicene/hBN in Section 2.4.2 of Chapter 2, we expect the spin-dependent and valley-dependent topological states to coexist and be modulated simultaneously by moiré potentials, leading to new and interesting physical eﬀects which call for further studies.

2. Moiré-modulated Zak phase: In Chapter 4, we discovered the topological 2N -rule in single-wall zigzag CNTs. In the future, we will investigate the topological Zak phase in double-wall or multi-wall CNTs with diﬀerent stacking conﬁgurations. Furthermore, analogous to the moiré valleytronics investigated in our thesis, moiré-induced Zak phase transition and relevant phenomena should also be an exciting area.

3. Moiré optoelectronics in MoS2 heteronatoubes: The ﬁeld of heteronanotubes opens a new door for exploring hybrid 1D systems. A broad range of tube materials, such as

MoS2-nanotubes, can be fabricated eﬃciently in experiments [86]. Moiré heteronan-

otubes formed by MoS2 should hold novel optical and excitation-related properties, similarly to their corresponding 2D moiré heterostructures [59]. This may provide a future platform for investigating 1D optoelectronics. A

calculations of berry curvature

This Appendix is to provide details of the numerical calculations of Berry curvature in Chap- ter 2 and 3, based on the k · p perturbation theory.

The Berry curvature Ωn(k) of band n is deﬁned by Eq.2.4, which is reproduced in the following:

Ωn(k) = ∇k × ihunk|∇k|unki, (A.1) where unk is the periodic part of the Bloch states. In 2D systems (x-y plane), the equation can be rewritten as: ∂unk ∂unk Ωn(k) = −2Im , (A.2) ∂kx ∂ky where kx and ky are the x and y components of the wavevector k. The core part of Eq.A.2 is the derivative ∂unk/∂kα (α = x, y), which is valid for direct calculation only when unk has a smooth gauge (usually in analytical calculations), e.g. the computation of Eq.2.5 in Chapter 2. In numerical calculations such as DFT, for each k point, the Hamiltonian is diagonalized individually, making the gauge of unk random [68]. Thus the derivative ∂unk/∂k in numerical approaches needs more treatments.

According to the k · p perturbation theory, very near a particular k point, i.e. ∆k is small, the Hamiltonian can be written as:

H(k + ∆k) = H(k) + ∆k · p. (A.3)

97 98 A calculations of berry curvature

Note that atomic units are applied here. p is the momentum operator:

p = −i∇. (A.4)

We take ∆k · p as the perturbation potential, by computing the ﬁrst-order perturbation term of the wave function, we can obtain:

X hum(k)|∆k · p|un(k)i un(k + ∆k) = un(k) + um(k) , (A.5) mk − nk m6=n where m and n are the band indices, k is the eigenvalue. From Eq.A.5, the derivative of the wave function can be calculated:

un(k + ∆k) − un(k) X hum(k)|p|un(k)i ∇kun(k) ≈ = um(k) . (A.6) ∆k mk − nk m6=n

The x and y components of ∇kun(k) are:

 ∂un(k) P hum(k)|px|un(k)i  = um(k) ,  ∂kx m6=n mk − nk ∂u (k) hu (k)|p |u (k)i (A.7)  n P m y n  = um(k) , ∂ky m6=n mk − nk

where px = −i∂/∂x and py = −i∂/∂y. Combining Eq.A.2 and Eq.A.7, we obtain another formula for calculating the Berry curvature:

X hunk|px|umkihumk|py|unki Ωn,xy(k) = −2Im 2 . (A.8) [mk − nk] m6=n

According to the k · p model, ∂H(k)/∂kα = pα + kα (α = x, y), an equivalent form of Eq.A.8 yields [23, 68, 69]:

X hunk|∂H(k)/∂kx|umkihumk|∂H(k)/∂ky|unki Ωn,xy(k) = −2Im 2 . (A.9) [mk − nk] m6=n 99

As for Eq.A.9, when we calculate the Berry curvature Ωn,xy(k) at one k point, all we need is the information of this k point: Hamiltonian H(k), eigenstate uk and eigenenergy k, thus avoiding the gauge problem. As a result, Eq.A.9 provides a practical scheme for numerically calculating the Berry curvature, which was unitized in Chapter 2 (Eq.2.1) and Chapter 3 (Eq.3.2). B

calculations of zak phase

This Appendix is to provide details of the numerical calculation of the Zak phase in Chapter 4, based on the Wannier function method.

The Zak phase of a 1D system is calculated by Eq.4.1, which is reproduced in the following:

Z γn = Ankdk, (B.1) BZ where γn is the Zak phase of band n. Ank is the Berry connection deﬁned as:

D ∂u E A = i u | nk , (B.2) nk nk ∂k

where unk is the periodic part of the Bloch states of band index n. Similarly to the Berry curvature calculation shown in Appendix A, direct calculation of the derivative part ∂unk/∂k is valid only when unk has a smooth gauge, e.g. the analytical computation on Zak phase of the SSH model in Section 1.4 of Chapter 1. In numerical calculations such as DFT, for each k point, the Hamiltonian is diagonalized individually, making the gauge of unk random and direct derivative ∂unk/∂k impossible.

Consider a Bloch state ψnk in a periodic 1D system. By Fourier transformation, we obtain the corresponding Wannier function:

Z d −ik·R wnR = e ψnk dk, (B.3) 2π BZ

100 101 where d is the lattice constant of the 1D system. wnR is the Wannier function of band index n and lattice vector index R. R = 0 denotes the home unit cell.

The eigenstate of the position operator ˆr in Wannier representation is:

D E r¯n = wn0|ˆr|wn0 , (B.4)

where r¯n is also called Wannier center (WC). In 1D Bloch representation (k-space), the position operator is: ˆr = i ∂/∂k. (B.5)

Combing Eq.B.3, Eq.B.4 and Eq.B.5, we get:

Z d D ∂unk E r¯n = i unk| dk. (B.6) 2π BZ ∂k

Inserting Eq.B.2 to Eq.B.6, the formula is rewritten as:

d Z r¯n = Ankdk. (B.7) 2π BZ

Comparing Eq.B.1 and Eq.B.7, a fundamental relationship between the Zak phase γn and the WC r¯n can be established [42, 101]:

2π γ = r¯ . (B.8) n d n

Eq.B.8 is the basic formula of computing the Zak phase by Wannier center method. In 1D system, r¯n is in fact in one direction (such as x−direction in Fig.4.3 of Chapter 4).

The issue of gauge freedom when numerically calculating the Bloch eigenstate unk discussed above will also aﬀect Eq.B.3 for computing the Wannier function. This makes the Wannier function not unique. Practically, the scheme of maximally localized Wannier func- 102 B calculations of zak phase tions (MLWF) is mostly utilized. We explain it in the following.

The second moment of the Wannier function can be calculated as:

D E ¯2 ˆ2 r n = wn0|r |wn0 , (B.9)

Combing Eq.B.4 and Eq.B.9, we can deﬁne the total spread (delocalization) of all the Wannier functions δ:

X ¯2 2 δ = [r n − ¯rn]. (B.10) n When the value of spread δ approaches minimum, the MLWF are found. This process has been implemented in many DFT codes, such as the most famous Wannier90 package [102].

In Chapter 4, we apply the MLWF as implemented in the Wannier90 package within the DFT calculation performed by the Vienna Ab initio simulation package (VASP) [103]. In the DFT calculation, projector augmented wave (PAW) method and Perdew-Burke-Ernzerhof generalized gradient approximation (PBE-GGA) [51] are taken. The plane wave basis is set to a kinetic energy cutoﬀ of 350 eV and we we adopt 16 × 1 × 1 Monkhorst-Pack k-point sampling.

In Table.4.3 of Chapter 4, for a zigzag CNT(n,0), two types of bulk unit cells with diﬀerent terminations are demonstrated. As for the Type A, a 2N -rule occurs by the Wannier functions calculation as shown in Fig.4.3:

  0 n = odd, x/d¯ = (B.11)  0.5 n = even, where d is the lattice constant of the 1D system, and x¯ is the total WC in x−direction P summed by all the occupied states: x¯ = x¯n. As shown in Table.4.3, from the Type-A to n∈occ Type-B, the unit cell is shifted by 1/4, which makes the coordinates of all WCs also shift by 1/4 at the same time. As for a zigzag CNT(n,0), the number of WCs is 2n. Therefore, the 103 shift of total WC ∆¯x = 2nd/4 = nd/2. The total WC of Type-B is:

  0.5 n = odd, x/d¯ = (B.12)  0.5 n = even.

Note that the value of x/d¯ is modulo d. Therefore, we conclude that all CNT(n,0) with Type-B termination are topological (γ = π), independent with the tube index n. Bibliography

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