Electronic properties and topological phases in graphene based van der waals heterostructures

... that the Rashba effect in BiTeI is maintained, and a significant spin splitting in graphene s linear band is found, which can be exploited in designing graphene- based spintronic devices to fully... BiTeI /graphene, Bi2 Se3 /graphene, bilayer graphene, and BN /graphene We discover interesting properties, such as the topological phase transition, Rashba splitting, band gap induced by band inversion,... unique properties, is still standing in the center of the stage In this thesis, using first-principle calculations, we investigate electronic properties and possible topological phase transitions in

ELECTRONIC PROPERTIES AND TOPOLOGICAL PHASES IN GRAPHENE-BASED VAN DER WAALS HETEROSTRUCTURES MEINI ZHANG (B.Sc., Anhui University) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF PHYSICS NATIONAL UNIVERSITY OF SINGAPORE 2014 I extremely appreciate the kind and beneficial help from my supervisor, Prof. Feng Yuanping, for giving me the opportunity to explore my research interest and the guidance to avoid getting lost in my exploration. Prof. Feng shows his support in a number of ways to my study, research and life. From the deepest part of my heart, I always feel the encouragements from him. It is a great experience and a precious treasure for me to learn and to grow up under his guidance, and it would definitely exert positive influence on my future career and life. I would like to thank Dr. Zeng Minggang for teaching me the knowledge and mindset to carry out research, as well as the time he spent in discussing with me to provide quite plenty of helpful suggestions. The majority of this thesis is comleted with his cooperation. I also acknowledge Prof. Wang Xuesen for offering me the research assistant position and inspiring me to think in experimental viewpoint. It is a pleasure to thank my group members, Dr. Yang Ming, Dr. Shen Lei, Mr. Wu Qingyun, Miss Li Suchun, Miss Chintalapati Sandhya, Miss Linghu Jiajun, Miss Qin Xian, Mr. Zhou Jun, Mr. Le Quy Duong, Mr. Luo Yongzheng for their help and valuable discussions, as well as the happy time we spent together. Finally, I would like to express my sincerest gratitude to my family. It is their support and thoughtfulness that motivate me to keep improving and never give up. i Table of Contents Abstract iv List of Figures vii 1 Introduction 2 1 1.1 Two-dimensional (2D) system and van der Waals heterostructure . . . . 1 1.2 Topological insulator and topological phase transition in 2D materials . 2 1.3 Rashba effect, spin-orbit coupling and BiTeI . . . . . . . . . . . . . . . 5 1.4 Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Motivation of our work . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Methodology 12 2.1 First-principles calculations . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.1 Earlier approximation . . . . . . . . . . . . . . . . . . . . . . 13 2.1.2 Density functional theory (DFT) . . . . . . . . . . . . . . . . . 14 2.1.3 The exchange-correlation functional approximation . . . . . . . 17 2.1.4 Bloch’s theorem and supercell approximation . . . . . . . . . . 19 2.1.5 Brillouin zone sampling . . . . . . . . . . . . . . . . . . . . . 20 2.1.6 Plane-wave basis sets . . . . . . . . . . . . . . . . . . . . . . . 21 ii 2.1.7 3 4 5 The pseudopotential approximation . . . . . . . . . . . . . . . 22 2.2 VASP software packages . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3 Implementation of van der Waals correction in computation . . . . . . . 26 The heterostructure of graphene/BiTeI 27 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 The heterostructure of graphene/Bi2 Se3 51 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.3 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 The heterostructure of bilayer graphene and BN/graphene 61 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.2.1 Bilayer graphene . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.2.2 BN/graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.3 6 Conclusion remarks and future work 73 6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 References 76 iii Abstract Fruitful progresses on graphene since its successful fabrication in 2004 have stimulated great research interest in other two-dimensional (2D) materials, such as isolated monolayers and few-layer crystals of hexagonal boron nitride (hBN), molybdenum disulphide (MoS2 ), other dichalcogenides and layered oxides [1]. The 2D materials have a great range of intriguing properties since the charge and heat transportation both happen in a plane, such as reduced dimensionality and symmetry, sensitivity to adatoms and defects, high electron mobility, topologically protected states [2]. Layered structures with stacking 2D hexagonal lattices separated by the van der Waals (vdW) interaction can be fabricated in experiments by techniques, such as epitaxy growth. Thus, it motivates us to consider what will be lying in heterostructures based on these 2D materials. Heterostructures of BiTeI/graphene, Bi2 Se3 /graphene, bilayer graphene, and BN/graphene are investigated in this project. The parent compounds in these heterostructures possess exotic properties. BiTeI is a semiconductor exhibiting remarkable bulk Rashba splitting and changes to topological insulator under proper pressure. Bulk Bi2 Se3 is a typical topological insulator with a large energy gap in the bulk state and a Dirac point in the surface state, while the Dirac point vanishes and an energy gap is formed in ultrathin films of Bi2 Se3 . Bilayer graphene iv has a parabolic band dispersion without a gap at K. Hexagonal BN can act as a perfect substrate for graphene-based electronic devices due to its wide energy gap as large as ∼5.2 eV and excellent lattice match with graphene. The common compound in the studied four types of heterostructure, graphene, as an attractive research frontier for a whole decade due to its unique properties, is still standing in the center of the stage. In this thesis, using first-principle calculations, we investigate electronic properties and possible topological phase transitions in BiTeI/graphene, Bi2 Se3 /graphene, bilayer graphene, and BN/graphene. We discover interesting properties, such as the topological phase transition, Rashba splitting, band gap induced by band inversion, tunability of Fermi level, etc., by altering the distance between layers, changing the stacking configurations, interlayer sliding, and applying external electric field. For the system of BiTeI/graphene heterostructure, six stacking configurations, depending on whether Te or I of BiTeI facing to graphene and its relative lateral position of Te/I and C, are investigated. At equilibrium state, it inherits the properties of its parent materials, BiTeI and graphene, presenting a linear band with Dirac point at K and Rashba splitting around G. However, when the interlayer distance is reduced by compression, some novel properties emerge. The compression and release process provides a dynamic mechanism to control electronic properties in these heterostructures. Reducing the interlayer distance, the gap at K is enlarged while the gap around G is decreased. Moreover, the Fermi level can be tuned in the process. More attractively, for heterostructures in which Te faces to graphene, a band inversion can be observed, leading to a gap opening at the crossing point of the inverted bands. The band inversion is attributed to spin-orbit coupling (SOC) and enhanced interaction due to smaller v interlayer distance. In addition, charge transfers can be found in these heterostructures; and the charge transfer in systems in which Te faces to graphene is stronger than that in systems in which I faces to graphene, due to the fact that the electronegativity of Te (∼2.1 eV) is smaller than that of the elements I (∼2.66 eV) and C (∼2.55 eV). What’s more, the electronic properties of the heterostructures are not sensitive to an applied electric field; and only the Fermi level can be controlled by a positive electric field, pointing perpendicularly from BiTeI to graphene. For the system of Bi2 Se3 /graphene heterostructure, the heterostructure displays a new Dirac point and a gapped graphene-derived Dirac state even without considering the SOC effect. Turning on the SOC leads to Rashba splitting and band inversion, which gives rise to gap opening due to the hybridization effect of energy bands. For the systems of bilayer graphene and BN/graphene heterostructures, a new way to realize gap opening is discovered. In the process of interlayer sliding, the energy gap can be manipulated to become larger or smaller accordingly. The successful realization of band gap and its controllable feature are promising to make graphene more practical for use and are appropriate for switching applications and mechanical sensor devices. Our results show some attractive properties for designing future electronic devices. Although there may be some difficulties in realizing these systems in experiments at the moment, our theoretical prediction still adds some valuable insight to the development of 2D heterostructures. vi List of Figures 1.1 Schematic of the topological quantum phase transition in 2D HgTe/CdTe quantum well as a function of the thickness of HgTe layer. . . . . . . . 2.1 Schematic illustration of all electron and pseudoelectron potentials and their corresponding wavefunctions. . . . . . . . . . . . . . . . . . . . . 3.1 3.7 32 Band structures of GBTI-IH: (a) without SOC; (b) with SOC along the k-path of G-K and G-M. . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 31 The band structure of BiTeI/graphene at equilibrium state with the ˚ . . . . . . . . . . . . . . . . . . . . . . . interlayer distance d0 =3.4 A. 3.5 30 The energy curve as a function of the interlayer distance for all the six investigated configurations. . . . . . . . . . . . . . . . . . . . . . . . . 3.4 29 (a, b) Top and side views of graphene/BiTeI heterostructures. (c) The summary of the six investigated configurations. . . . . . . . . . . . . . 3.3 23 Schematic of the stacking structure of the BiTeI/graphene heterostructure and the Brillouin zone. . . . . . . . . . . . . . . . . . . . . . . . . 3.2 5 34 Schematic of the net charge transfer from BiTeI into graphene plane in terms of graphene plane. . . . . . . . . . . . . . . . . . . . . . . . . . 35 Band structures of GBTI-IH as a function of the interlayer distance (d). 36 vii 3.8 Band structures of GBTI-IA: (a) without SOC; (b) with SOC. . . . . . . 37 3.9 Band structures of GBTI-IB: (a) without SOC; (b) with SOC. . . . . . . 38 3.10 Band structures of GBTI-TH: (a) without SOC; (b) with SOC. . . . . . 40 3.11 The weight analysis on the orbital projections of the band structure of GBTI-TH without SOC. . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.12 The weight analysis on the orbital projections of the band structure of GBTI-TH with SOC. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.13 Band structures of GBTI-TA: (a) without SOC; (b) with SOC. . . . . . 42 3.14 The weight analysis on orbital projections of C-pz of GBTI-TA: (a) without SOC; (b) with SOC. . . . . . . . . . . . . . . . . . . . . . . . 42 3.15 The weight analysis on orbital projections of C-pz of GBTI-TA with ˚ and 2.6 A, ˚ respectively. . . . . . . . . . . . interlayer distance as 2.4 A 44 3.16 The band structures of GBTI-TA with a sequence of reduced interlayer distance d with SOC along G-K. . . . . . . . . . . . . . . . . . . . . . 45 3.17 The band structures of GBTI-TA with a sequence of reduced interlayer distance d without SOC along G-K. . . . . . . . . . . . . . . . . . . . 46 3.18 The influence of applying external gate voltage on GBTI-TA heterostructure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.19 Band structures of GBTI-TB: (a) without SOC; (b) with SOC. . . . . . 48 3.20 The band structures of GBTI-TB with a sequence of reduced interlayer distance d with more dense points along G-K. . . . . . . . . . . . . . . 49 4.1 A quintuple layer (QL) of Bi2 Se3 . . . . . . . . . . . . . . . . . . . . . 52 4.2 The energy curve as a function of the interlayer distance for all the three investigated configurations. . . . . . . . . . . . . . . . . . . . . . . . . 54 viii 4.3 The band structure of GBS-A: (a)without SOC; (b)with SOC. . . . . . . 4.4 The weight analysis on orbital projections of GBS-A without and with 56 SOC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.5 The band structure of GBS-B: (a)without SOC; (b)with SOC. . . . . . . 58 4.6 The weight analysis on orbital projections of GBS-B with SOC. . . . . 59 4.7 The band structure of GBS-H: (a)without SOC; (b)with SOC. . . . . . . 59 4.8 The weight analysis on orbital projections of GBS-H with SOC. . . . . 60 5.1 The band structure of Bernal-stacking bilayer graphene with interlayer ˚ (b)2.4 A. ˚ . . . . . . . . . . . . . . . . . . . . . . distance as (a)3.4 A; 5.2 The energy curve for all the three investigated configurations of bilayer graphene as a function of the interlayer distance. . . . . . . . . . . . . . 5.3 63 64 Change the configuration of bilayer graphene by interlayer sliding, ˚ . . . . . . . . based upon the system with interlayer distance as 2.4 A. 66 5.4 The schematic illustration of a device to realize the sliding operation. . . 67 5.5 The structure of the nanoribbon for calculating the edge state using bridge-stacking bilayer graphene. . . . . . . . . . . . . . . . . . . . . . 5.6 The band structure of the nanoribbon constructed by the bridge-stacking graphene. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 5.9 68 (a) The detailed picture of the crossing point in Figure 5.6; (b) The density of states of each orbitals. . . . . . . . . . . . . . . . . . . . . . 5.8 67 69 The energy curve for all the three investigated configurations of graphene/BN as a function of the interlayer distance. . . . . . . . . . . . . . . . . . . 71 The band structures of BN/graphene in three different configurations. . 72 ix Chapter 1 Introduction 1.1 Two-dimensional (2D) system and van der Waals heterostructure Dimensionality plays a significant role in deciding fundamental properties of materials, such as phase transition and electronic correlation [3]. Physics behind low-dimensional materials and in the crossover from 3D to 2D materials is worth more attention, especially in such an era that various characterizations and experimental techniques are advanced enough to reach atomic layer accuracy [4]. It is interesting to explore 2D materials since they host unique phenomena when the charge and heat transports are confined in a plane [2]. The extreme of 2D materials is single-layer materials which are completely surface area; and the influences from substrate, local electronic environment and mechanical deformations are crucial [2]. However, the growth of low-dimensional 1 Chapter 1. Introduction crystals is unfavorable in nature. In the process of crystal growth, thermal fluctuations in high temperature processing are detrimental for the stability of macroscopic 1D and 2D crystals. Fortunately, with the advancement of techniques, it is completely possible to realize 2D materials artificially. One route is to separate mechanically from layered materials such as graphite, a method known as scotch-tape technique by which graphene was initially successfully realized. Another way is to grow 2D materials epitaxially on top of other crystals and then to remove the substrate by chemical etching [5]. Other approaches include chemical exfoliation by exposing to a solvent which has appropriate surface tension and molecule/atom intercalation [2]. Recently, van der Waals (vdW) heterostructures have been realized in experiments, kicking a promising research field [1]. Two-dimensional heterostructures, such as MoS2 /graphene, WS2 /graphite and BN/Graphene have been reported [6–12], from which a number of novel phenomena emerge. For example, BN/graphene enhances graphene’s mobility from normally achieved µ ≈ 100,000 cm2 V−1 s−1 up to 500,000 cm2 V−1 s−1 at low temperature [12, 13], and its high spatial uniformity allows graphenebased capacitors with size over 100 µm2 to show quantum oscillations in a magnetic field as low as 0.2 T [14]. 1.2 Topological insulator and topological phase transition in 2D materials Topological insulator (TI) is an exotic quantum phase state of matter, with charge excitation gaps in the bulk and gapless Dirac edge (surface) states residing in the 2 Chapter 1. Introduction bulk gap of 2D (3D) materials [15–19]. The Dirac states are protected by the time reversal symmetry and hence robust against non-magnetic perturbation and disorder. Mathematically, in materials with an energy gap and inversion symmetry, the topological invariant can be characterized by the Z2 topological number [18]. An exotic property in TI is the suppression of backscattering of electrons by weak disorders [20–23]. Electrons in topological surface or edge states do not suffer localization by nonmagnetic impurities [24]. Therefore, TIs are very promising in applications of spintronics and quantum computation. In terms of the controllability in charge transport, 2D-TI is superior to 3D-TI due to the fact that the charge in 2D-TI is confined to move along its 1D metallic edges in two opposite and well-defined directions. However, the research progress of 2D TIs is not as prosperous as that of 3D TIs. HgTe/CdTe quantum well is the first experimentally demonstrated 2D-TI. Some other promising materials are put forward, such as ultrathin Bi (111) films [25]. On the other hand, Bi2 Se3 , Bi2 Te3 , Sb2 Te3 are typical examples of 3D-TI [19]. Gap opening of topological gapless states can give rise to some novel phenomena, such as anomalous quantum Hall effect and surface plasmon excitation. Introducing magnetic impurity can destroy the time-reversal symmetry and thus open an energy gap at the Dirac point. Such gap openings have been observed in TI materials doped with bulk or surface magnetic impurities [26, 27]. Reducing the thickness of 3D TIs to the 2D limit can also open a Dirac gap due to the interaction between top and bottom surfaces, and moreover, linear Dirac bands and massless fermions are replaced by conventional parabolic bands and massive fermions [4, 28, 29]. For example, when the thickness of Bi2 Se3 is decreased to 6 quintuple layers (QLs), the Dirac point vanishes and an energy gap appears. 3 Chapter 1. Introduction Another intriguing effect in TI is topological phase transition characterized by the parity exchange between conduction bands and valence bands. Band bending effect due to non-magnetic impurity adsorption can drive a phase transition from topologically trivial to nontrivial state [30, 31]. Adiabatic changes by strain or controlling the spinorbit coupling (SOC) strength can also lead to the topological phase transition. For example, a trivial-to-nontrivial phase transition was predicted theoretically and verified experimentally in Bi-Sb alloy, in which the SOC strength can be manipulated to tune the parity exchange [18, 32]. Reducing the dimensionality of crystals not only enhances the ratio of surface states to bulk states, but also plays a significant role in determining its topological properties [33]. It has been reported that decreasing the thickness of Bi2 Se3 or Bi2 Te3 thin-films leads to oscillation between topological trivial states with a normal energy gap and quantum spin Hall (QSH) states [34]. In the case of HgTe/CdTe quantum well, as shown in Figure 1.1, when the thickness of HgTe layer is smaller than the critical thickness, the band gap of the bulk states (∆ < 0) closes (∆ = 0), and then reopens (∆ > 0) with a parity inversion between the conduction band and the valence band [17, 24, 33]. Similar thickness-dependent topological phase transition is also found in Bismuth (Bi). Despite the topological triviality of bismuth crystal, bilayer bismuth thin films exhibit QSH states and have been realized in experiments [25, 35]. Angle-resolved photoemission spectroscopy (ARPES) is a powerful tool to observe surface states in 3D-TIs. Adding spin resolution to ARPES can give us additional spin texture information [36, 37]. The first experimentally reported 3D-TI is disordered Bi-Sb alloy. Recently, Bi2 Se3 and Bi2 Te3 have attracted more research interest due to their simple Dirac cone and large bulk gap observed by ARPES [19, 38, 39]. Edge 4 Chapter 1. Introduction ¨>0 ¨=0 ¨ dc Film thickness d Figure 1.1: Schematic of the topological quantum phase transition in 2D HgTe/CdTe quantum well as a function of the thickness of HgTe layer. At d = dc , the band gap closes (∆ = 0) and the parity of conduction bands and valence bands is exchanged [33]. states in 2D-TIs are difficult to be directly measured by ARPES and the observation of edge states may rely on other plausible approaches, such as scanning tunneling microscopy/scanning tunneling spectroscopy (STM/STS) and transport measurements [25]. 1.3 Rashba effect, spin-orbit coupling and BiTeI The Rashba effect is characterized by a momentum-dependent spin splitting as a consequence of atomic SOC and an effective electric field arisen from asymmetry of potential at surface or interface of semiconductors, or from bulk materials with broken inversion symmetry. An electron with momentum k and spin σ in a Rashba system 5 Chapter 1. Introduction experiences a magnetic field induced by internal electric field Ez , whose Hamiltonian can be represented as HR = λσ·(Ez × k), where λ is the coupling constant [40]. The resulting spin-polarized band dispersions are denoted as E(k)± = E0 (k)±α×|k|, where the first term E0 (k) = 2 |k|2 /2m∗ , with m∗ denoting the effective mass of electrons, represents the energy without Rashba effect; and the second correction term indicates a momentum-dependent energy shift. The coefficient α is the Rashba parameter, which is proportional to the strength of SOC and the potential gradient. The spin-splitting sub-bands depend both on the direction of spins and the electron momentum (k). Moreover, this spontaneous magnetization occurs without any external magnetic fields. Therefore, Rashba effect makes it possible to manipulate spin-polarized electrons via electric field rather than magnetic field. A typical example of utilizing Rashba effect is the Datta-Das-type spin field-effect-transistor (FET), in which spinpolarized currents can be turned on or off in terms of phase difference in spin precession motion, which can be controlled by the strength of Rashba SOC [41]. Increasing αR to create a large splitting between spin sub-bands allows to tune the chemical potential over a broad energy range [42], which is useful for gate-controlled devices, such as FETs. However, for the traditional narrow-gap semiconductors, αR is of the order about 10−1 ˚ which is too small for devices operating at room temperature. To enhance αR , eV/A, some methods have been proposed, like incorporating heavier elements and adjusting the interface or surface of 2D systems in order to enlarge the potential gradient. For example, diluted doping of Bi atoms into GaAs can strongly increase the spin splitting of electronic states [43]. A remarkable breakthrough in developing materials or systems with large αR is the ˚ This very recently reported BiTeI, a bulk material with αR as large as 3.8 eV/A. 6 Chapter 1. Introduction large αR makes BiTeI a prominent candidate for future advanced spintronic devices [40]. Currently, three techniques, i.e. standard vertical Bridgman, modified horizontal Bridgman, and vapour transport, can be employed to grow high-quality BiTeI crystals with electronic properties ranging from metallic to insulating state [44]. For an example, Kishizaka [40, 45] measured the in-plane electric resistivity of a crystalline BiTeI sample and found a metallic behavior down to 2 K. Moreover, the Hall coefficient is temperature-independent, and the electron concentration nH is estimated to be 4.5×1019 cm−3 , implying BiTeI behaves as a degenerate n-type semiconductor. The Rashba effect in BiTeI crystal has been observed by STM/STS and ARPES measurements. Based upon Ishizaka and his colleagues’ work, a giant Rashba splitting is found in bulk BiTeI with ˚ 2 and the spin splitting of ∼400 meV. Moreover, the Rashba energy ER kCBM = ±0.05 A is larger than 100 meV, which is among the highest found so far [40]. These results and techniques pave the way for further research in topological phases in BiTeI. The mechanism of Rashba effect in BiTeI comes from its non-centrosymmetric structure. BiTeI has space group of P 3m1 and consists of a sequence of layers of I, Bi and Te, each of which forms a trigonal planar lattice. As a consequence, BiTeI has a spontaneous built-in potential between the positively charged (BiTe)+ layer and the negatively charged (BiI)− layer. This considerably large internal potential gradient ∆V along the layer stacking direction, in addition to the large SOC of Bi atoms, leads to a large Rashba effect in BiTeI. Besides the built-in potential and large SOC, Bahramy et al. suggested another two contributing factors. One is the band gap, the other is the symmetry of top valence and bottom conduction bands. Materials with large Rashba spin splitting can be found in a narrow-gap semiconductor with the same symmetry between top valence and bottom conduction bands, such as BiTeI, in which the requirement of 7 Chapter 1. Introduction same band symmetry (pz type) can be realized by the negative crystal field splitting (CFS) arisen from the strong anisotropic ionicity of atomic bondings in BiTeI or from external influences, such as pressure-induced structure distortion [46]. Besides BiTeI, there are some research efforts on the series of BiTeX (X = Cl, Br, I) materials. Sakano et al. reported that the Rashba parameters (αR ) of BiTeCl and BiTeBr are significantly smaller than that of BiTeI. In BiTeX, the lower set of valence bands is mainly dominated by X atoms [47], while the upper bands above the Fermi level (EF ) mainly come from three Bi-6p orbitals [40]. Furthermore, the band gap of BiTeX depends on the X ions. BiTeX with heavier atom X has a smaller band gap. The combined effect of narrower band gap and larger atomic SOC in Bi-6p orbitals results in an enhanced spin splitting in BiTeI, compared with BiTeBr and BiTeCl, as demonstrated by Bahramy et al. using the second-order perturbative k · p calculations [46]. In addition to the large Rashba splitting effect, BiTeI also has other intriguing properties. Wang et al. pointed out that the Rashba parameter α can be tuned by adjusting the Bi-Te bond length, and the Fermi level can be adjusted by doping, applying an electric field or controlling the surface termination [48]. Electron accumulation (depletion) tends to occur at the Te- (X-) terminated surface, and thus, n-type (p-type) of surface states with Rashba spin splitting have been observed. Moreover, pressure-induced topological phase transition in BiTeI has been predicted. Bahramy et al. found topological nontrivial states in BiTeI with topological invariant of Z2 = 1;(001) [49]. It is of great interest to investigate the interplay of topological surface states with Rashba spin splitting, in both of which the SOC plays an essential role. 8 Chapter 1. Introduction 1.4 Graphene Graphene, an one-atom-thick allotrope of carbon, is an epoch-making material which opens up a completely new research field by virtue of its various unique and superior properties. Graphene is the thinnest and strongest material in the universe we have ever known and discovered. It maintains current densities up to six orders of magnitude higher than that of copper, and has outstanding thermal conductivity of ∼5000 WmK−1 [50], breaking strength of ∼40 N/m, which reaches the theoretical limit, Young’s modulus of ∼1.0 TPa [51], and room-temperature electron mobility of 2.5×105 cm2 V−1 S−1 [12]. It keeps a good balance between competing qualities such as brittleness and ductility as well. Graphene also shows the feature of being impermeable to gases. The electronic properties of graphene are very unique due to its linear band dispersion. Its charge carriers have giant intrinsic mobility near the Dirac point and can travel several micrometers without scattering at room temperature. The behaviors of lowenergy electrons in graphene can be described by Dirac equation instead of Schr¨odinger equation. The zero effective mass of electrons makes graphene an excellent platform to study relativistic quantum phenomena [52]. Currently, there are several methods to massively produce graphene, including liquid phase and thermal exfoliation, chemical vapor deposition, synthesis on SiC, mechanical exfoliation, molecular assembly and etc. [53]. However, for graphene-based electronic devices, such as transistors, one of the bottlenecks lies in the fact that graphene remains metallic even at the neutrality point. The absence of a band gap greatly limits graphene’s uses in electronics. Opening a large-enough energy gap without degrading the excellent 9 Chapter 1. Introduction properties of graphene is a challenge to be conquered. Several mechanisms, such as spatial confinements by patenting nanoribbon [52, 54], single electron transistor [55, 56], lateral-sublattice potential by bilayer control [57, 58] and chemical modification [59, 60], have been proposed to generate an energy gap. 1.5 Motivation of our work The first system to be considered in the project is BiTeI/graphene. Graphene provides an excellent transport channel for spintronic devices, while strong Rashba effect in BiTeI can be utilized to control spin polarized currents. Moreover, graphene is semimetallic, while BiTeI is semiconducting. The combination of graphene and BiTeI offers us some unique effects without degrading the intrinsic properties of graphene and BiTeI. Furthermore, graphene and BiTeI share similar structures. All of these motivate us to investigate possible interesting phenomena in BiTeI/graphene heterostructures by changing the distance between graphene and BiTeI, applying electric field and altering the stacking configurations. Our results show that the Rashba effect in BiTeI is maintained, and a significant spin splitting in graphene’s linear band is found, which can be exploited in designing graphene-based spintronic devices to fully utilize the excellent transport properties of graphene. More interestingly, a topological phase transition is discovered at critical interlayer distance or with SOC effect, indicating the system changes from topological trivial to nontrivial state. Furthermore, there is a band gap opening. The magnitude of the band gap and the Fermi level can be tuned by altering the distance between 10 Chapter 1. Introduction graphene and BiTeI. The gap is as large as ∼300 meV, which is quite close to the widest band gap achieved in graphene so far (∼360 meV). The controlling of the band structure via only adjusting the interlayer distance is superior to previous approaches, such as introducing defects or dopants, because the adsorbed molecules may have an extremely high mobility and adhesion in ambient atmosphere [61], which is detrimental for reliable device fabrication and may lower the carrier mobility of graphene by 3 orders of magnitude to ∼10 cm2 /V·s [2]. We are also interested in graphene/Bi2 Se3 heterostructure. Bulk Bi2 Se3 is a prototype of 3D-TI, while ultrathin film Bi2 Se3 is a conventional semiconductor with a trivial band gap, which motivates us to explore what will happen when the interaction is strong enough between graphene and Bi2 Se3 . Surprisingly, a new Dirac point emerges in the system. Furthermore, topological phase transition and significant Rashba splitting in Bi2 Se3 /graphene are found. Graphene is a prominent 2D material for electronics, however, the absence of band gap is an annoying snag. Bilayer graphene and appropriate substrates are possible solutions researchers seek for. From calculations of different configurations about bilayer graphene and graphene/BN with smaller interlayer distance, large band gap is predicted and the gap may result from band inversion. It provides a new way to control graphene’s energy band and the continuous transition of electronic properties by changing stacking configurations can be employed to design switching applications and mechanical sensors. 11 Chapter 2 Methodology Our first-principles calculations are implemented in VASP software package, which is based on the density functional theory (DFT). The first-principles calculation have been extensively applied to investigate electronic and magnetic properties of materials [62]. 2.1 First-principles calculations First-principles calculations have successfully improved computational science, which is the third pillar to explore the nature and science besides theoretical and experimental aspects. In physics, the first-principles, or ab initio approach, indicates to start directly from the most fundamental laws of physics without any assumptions such as empirical model and fitting parameters. 12 Chapter 2. Methodology 2.1.1 Earlier approximation In theory of quantum mechanics, the electron behaviors can be perfectly described by the Schr¨odinger equation for a many-body system: ˆ = EΨ HΨ (2.1) ˆ is where E is the energy eigenvalue, Ψ = Ψ(r1 , r2 , ..., rN ) is the wave function, and H the Hamiltonian operator, given by ˆ = H l 1 Pˆl2 + 2ml 2 l=l′ ql ql′ , |rl − rl′ | (2.2) where the summation is over all electrons and nuclei in the system, ml is the mass of an electron or nucleus, and ql is its charge. However, it is impossible to solve Schr¨odinger equation accurately, due to the complexity coming from the interactions between a huge number of particles. Thus, some important approximations were come up with. The first important approximation is the Born-Oppenheimer approximation, which is intended to ignore the nuclei in the system and treats them adiabatically due to the fact that the nuclei are much heavier so that their motion is quite slower than that of electrons. In this approximation, the kinetic energy of the nuclei is neglected and the interaction between the nuclei can be handled classically. In that case, the many-body problem described in Eq. (2.1) is reduced to one system in which the interacting electrons moving in an external potential field V (r), formed by a frozen-in ionic configuration. The Born-Oppenheimer approximation makes it possible to break the wavefunction of a complicated system into its electronic and nuclear components, which can be calculated 13 Chapter 2. Methodology in two more simplified consecutive steps. However, it is still not an easy job to solve the equations due to the electron-electron interaction. Hartree approximation reduces the many-body equation into independent equations regarding each individual electrons, in which Φ can be represented as a product of every individual electron wavefunction based upon the assumption that the electrons interact with each other only via the Coulomb force: Φ = Φ(r1 , r2 , ..., rN ) = Φ1 (r1 )Φ2 (r2 ) · · · ΦN (rN ) (2.3) However, Hartree potential does not take the exchange interaction into consideration. Adding Fermi statistics to Hartrees method develops the Hartree-Fock approximation, which is also called the self-consistent field method, since the procedure to derive the ground state electron wavefunction is to use a trial wavefunction to solve the equation until the convergence is realized, in which the final field computed from the charge distribution should be self-consistent with the assumed initial field. But comparing with the calculated energy from the many-body Schr¨odinger equation under the Born-Oppenheimer approximation, there is a deviation for Hartree-Fock equation. The difference in total energy is called correlation energy, which, together with the exchange energy (the energy difference for exchanging electrons in solving HartreeFock equations), is proved to be quite difficult to calculate in a complex system. 2.1.2 Density functional theory (DFT) In order to overcome the deficiencies of the above-mentioned approximation methods and develop a method which can describe the electron-electron interaction more 14 Chapter 2. Methodology precisely and also reduce the Schr¨odinger equation into a much easier format, the density functional theory comes to the center of the stage. Instead of dealing with the complicated multi-dimensional N-electrons wave function in Hartree-Fock method, DFT focuses on the electron density, a simple scalar field. In 1964, Hohenberg and Kohn summarized two theorems in their paper on DFT [63]. The first Hohenberg and Kohn theorem states that the total energy, including exchange and correlation energy of a system of electrons and nuclei, is a unique functional of the electron density n(r). E = E(n(r)) = T [n(r)] + Ene [n(r)] + Eee [n(r)] (2.4) = n(r)Vne (r)dr + FHK [n(r)] + Ecorr [n(r)] where FHK [n(r)] = T [n(r)] + Eexc [n(r)] + Ecoul [n(r)] (2.5) where T [n(r)] is the kinetic energy and Eee [n(r)] is the electron-electron interaction energy which contains the Coulomb interactions Ecoul [n(r)] given by: Ecoul [n(r)] = e2 2 n(r1 )n(r2 ) dr1 dr2 r12 (2.6) The second Hogenberg and Kohn theorem is a variational statement for the energy in terms of the density, that is, there exist a universal functional n(r) which can be minimized by the ground state density n 0(r). DFT demonstrates that the energy is given exactly by electron density and the groundstate energy can be obtained via a variational principle. However, accurate calculational 15 Chapter 2. Methodology implementations of the density functional theory are far from easy to achieve due to the fact that the functional FHK [n(r)] is hard to come by in explicit form. Kohn and Sham provided a useful computational scheme by replacing the many-body problem with an exact equivalent set of self-consistent one-electron equations [64, 65]. The Khon-Sham (KS) total-energy functional for a set of doubly occupied electronic states Φi can be written as 2 E[Φi ] = 2 (Φi [− i e2 + 2 2m ]∇2 Φi )d3 r + Vion (r)n(r)d3 r n(r)n(r′ ) 3 3 ′ d rd r + Exc [n(r)] + Eion (RI ) r12 (2.7) where Vion is the static total electron-ion potential, Exc [n(r)] is the exchange-correlation functional, and Eion is the Coulomb energy associated with interactions among the nuclei (or ions) at positions RI . The minimum of the KS total-energy functional is achieved by solving the KS equations self-consistently: 2 [(− 2m ∇2i ) + Vion (r) + VH (r) + Vxc (r)]Φi (r) = εi (r)Φi (r) (2.8) where, Φi (r) is the wave function of electronic state i, εi is the KS eigenvalue, and VH (r) is the Hartree potential of the electrons given by VH (r) = e2 n(r′ ) 3 ′ dr |r − r′ | (2.9) The exchange-correlation potential Vxc (r) is given by the functional derivative Vxc (r) = δExc (r) δn(r) (2.10) In the KS equations, the effective potential is the KS potential: VKS (r) = VH (r) + Vxc (r) = e2 n(r′ ) 3 ′ δExc (r) dr + |r − r′ | δn(r) (2.11) 16 Chapter 2. Methodology The KS equation maps an interacting many-electron system into a single electron system moving in the KS effective potential formed by the nuclei and the other electrons. Comparing with the many-body Schr¨odinger equation in Eq. (2.1), solving the KohnSham equation is much easier for a practical system. Given that the functional depends on the density, the Kohn-Sham equation forms a set of nonlinear coupled equations, and the standard procedure to solve it is iterating until self-consistency is achieved. Starting from an assumed density n(r), firstly VH (r) and Vxc (r) are calculated, then solve the KS Eq. (2.8) for the wavefunctions. With these wavefunctions, a new density can be constructed by n(r) = i |ψi (r)|2 , (2.12) where the index i goes over all occupied states. This procedure is repeated until selfconsistency, i. e. the consistency between the constructed and the initial densities, is achieved. 2.1.3 The exchange-correlation functional approximation There is an exact functional Exc [n(r)] in KS equation for electron exchange and correlation interaction in any situation. Despite we cannot derive the exact functional from DFT, there exist several reasonable approximations we can take advantage of and fortunately, the results based upon these approximations are in good agreement with experiments, indicating their effectiveness. Among the various approximations, local density approximation (LDA) is the simplest, which is based on the assumption that, for a system whose density varies slowly, the 17 Chapter 2. Methodology electron density in a small region near point r can be treated as if it is homogeneous [64]. Therefore, the Exc [n(r)] can be written as Exc [n(r)] = ǫxc (r)n(r)d3 r, (2.13) where ǫxc (r) is the exchange-correlation energy per electron. Even if the form of LDA is quite simple, it is amazingly successful in plotting the structural and electronic properties of materials. However, several problems also emerge in the application of LDA. For example, LDA fails to provide a good description for excited states and underestimates the band gap of semiconductor and insulator. LDA tends to overbind, resulting in the underestimated lattice parameter and the overestimated cohesive energy. In addition, the LDA does not show satisfied performance for the van der Waals interactions and it would also give wrong predictions for some strongly correlated magnetic systems. An improvement for the LDA is the generalized gradient approximation (GGA), in which the exchange-correlation functional is considered as the functional of the electron density and the gradient of the electron density: Exc [n(r)] = f (n, ∇n)d3 r, (2.14) A variety of choices for f (n, ∇n) can be made [66, 67]. The GGA has been turned out to be quite successful in improving some of the deficiencies of the LDA, such as the overbinding tendency and the wrong predictions for the non-magnetic ground state of Fe and etc. However, the GGA is far from being ideal. Its underbinding problem gives rise to the overestimation of the lattice parameter and the underestimation of the cohesive energy. What’s more, the GGA cannot solve situations concerning the strong correlated systems, the excited states and the van der Waals interactions properly, neither. Moreover, finding an accurate and universally-applicable Exc remains 18 Chapter 2. Methodology a significant challenge in DFT. 2.1.4 Bloch’s theorem and supercell approximation With the help of DFT, the many electron Schr¨odinger equation can be transformed into N one-electron KS equations, which is more practical for us to handle. However, the number of particles in solids in reality usually reaches the order of 1023 , which means we have to solve the KS equations for a huge number of times and thus seems incredible. Blochs theorem helps to get rid of that concern. Blochs theorem states that the wavefunction for an electron moving in a periodically repeated potential environment can be represented as: ψk (r + R) = eik·r uk (r), (2.15) where uk (r) has the same periodicity as the atomic structure of the solid. Since the electron energy is periodic in the k-space and thus each k-point outside the first Brillouin zone (1BZ) can be mapped to its counterpart in the 1BZ, all the wavevector k can be folded into the 1BZ when calculating the electronic properties. The number of kpoints in the 1BZ is equal to the number of unit cells in real space. As the number of unit cells is as large as ∼1022 , the k-points in the 1BZ can be considered as quasi-continuous. Many systems do not have periodic structure along certain directions of all three dimensions. For example, an interface system just has periodicity along the plane direction, and an isolated atom or molecule is totally aperiodic at all. For these systems, the supercell approximation, which intends to artificially model a periodic structure on 19 Chapter 2. Methodology the aperiodic dimension, is utilized. For the interface system mentioned above, the supercell approximation is implemented by modeling the interface with periodically arranged slabs which are separated by vacuum layers. To minimize the artificial Coulomb interactions between neighboring surfaces, the vacuum layer should be thicker ˚ than 10 A. 2.1.5 Brillouin zone sampling The Blochs theorem changes the problem of calculating an infinite number of electronic wave functions to one of calculating a finite number of electronic wave functions at an infinite number of k-points. In principle, each k-point in 1BZ have to be considered for the calculations, due to the fact that the occupied states at each k-point contribute to the electronic potential in the bulk solid, typically, through the following integrals, I(ε) = 1 ΩBZ BZ F (ε)δ(εnk − ε)dk, (2.16) where ΩBZ is the volume of the BZ. Fortunately, the electronic wavefunctions at k-points which are very close are almost identical, which makes it possible to represent the wavefunctions over a region of k space by the wavefunction at a single k-point and thus enables us to deal with only a special finite number of k-points in the 1BZ to calculate the electronic potential: 1 ΩBZ f (k)dk =⇒ BZ ωki f (ki ), (2.17) ki where ωki is the weight of ki . Any errors of the magnitude in the total energy due to the inadequacy of the k-points 20 Chapter 2. Methodology sampling can always be reduced by deploying a denser set of k-points. In other words, the k-points mesh should be increased until the calculated total energy is converged. Various methods have been proposed for the k-points sampling in the BZ [68, 69], among which Monkhorst-Pack scheme is one of the most widely used. The basic idea of Monkhorst-Pack scheme is to construct equally spaced k-points (N1 ×N2 ×N3 ) in 1BZ according to k= n2 + 1/2 n3 + 1/2 n1 + 1/2 b1 + b2 + b3 , N1 N2 N3 (2.18) where n1 , n1 , and n3 =0,···, Ni -1. Symmetry is used to map equivalent k-points with each other, which is helpful to reduce the total number of k-points significantly generated by Eq. (2.18). 2.1.6 Plane-wave basis sets Based upon Blochs theorem, the electronic wavefunctions at each k-point can be expanded in terms of a discrete plane-wave basis sets: ψn,k (r) = G cn,k+G exp[i(k + G) · r]. (2.19) After expanding the wavefunction as Eq. (2.19), the KS equation can be reduced to the secular equation: 2 [ G′ 2m |k + G|2 δGG′ +V (G − G′ )+VH (G − G′ )+Vxc (G − G′ )]cn,k+G′ = εn,k cn,k+G . (2.20) In principle, an infinite plane-wave basis set is required to expand the electronic wavefunctions. Fortunately, the coefficient cn,k+G for the plane-wave with small kinetic 21 Chapter 2. Methodology energy 2 2m |k + G|2 are typically more important than those with large kinetic energy. Thus, the plane-wave basis set can be truncated to include only plane-waves that have kinetic energies less than some particular cutoff energy Ecut = 2 2m |k + Gcut |2 . The truncation of the plane-wave basis set at a finite cutoff energy will lead to an error and the error can be reduced by increasing the cutoff energy. Normally, the cutoff energy should be increased until the total energy has converged. 2.1.7 The pseudopotential approximation Although the plane-wave basis sets allow us to expand the electronic wavefunction using a discrete set of plane-waves, a very large number of plane-waves is needed to expand the tightly-bound core orbitals and to follow the rapid oscillations of wavefunctions of the valence electrons in the core region due to the strong ionic potential in this region, which would make the all-electron calculations very expensive and even impractical for a large system. Fortunately, the pseudopotential approximation allows the expansion of electronic wavefunctions with much smaller number of plane-wave basis sets [70, 71]. Based upon the fact that most of the physical properties of solids are mainly determined by the valence electrons, the pseudopotential approximation ignores the core electrons and strong ionic potential, and replaces them by a weaker pseudopotential that acts on a set of pseudo wavefunctions rather than the true valence wavefunctions, as Figure 2.1 shows. The nonlocal and angular momentum dependent pseudopotentials are usually generated from isolated atoms or ions, but can be used in other chemical environment such as 22 Chapter 2. Methodology Figure 2.1: Schematic illustration of all electron (solid lines) and pseudoelectron (dash lines) potentials and their corresponding wavefunctions. 23 Chapter 2. Methodology solids (transferability of the pseudopotentials). The general form for a pseudopotential is VN L = lm |lm Vl lm|, (2.21) where |lm is the spherical harmonics and Vl is the pseudopotential for the angular momentum l. The construction of the pseudopotential within the core radius rc should preserve the scattering properties, which means that the scattering properties for the pseudo wavefunctions should be identical to that of the ion and core electrons for the valence wavefunctions. The scattering properties are satisfied automatically in the region outside the rc since the pseudopotential and the true potential there are the same. In order to improve the transferability, the norm-conserving condition: rc rc 2 0 |φn,k (r)| dr = 0 |ψn,k (r)|2 dr (2.22) should be fulfilled. Norm-conserving pseudopotentials are very successful for solids of s, p-bond main group elements, but they do not give ideal results for the first-row elements, transition metals and rare-earth elements due to the highly localized valence orbitals in these elements. Fortunately, this difficulty has been overcome by introducing the so-called ultrasoft pseudopotentias proposed by Vanderbilt [72]. More recently, a more accurate and efficient pseudopotential formalism, the projector augmented-wave (PAW) [73, 74], has been developed by Blochl. The augmentation procedure is generalized in that partial-wave expansions are not determined by the value and the derivative of the envelope function at some muffin-tin radius, but rather by the overlap with localized projector functions [75]. It combines the versatility of the linear augmented-plane-wave 24 Chapter 2. Methodology (LAPW) method with the formal simplicity of the pseudopotential approach. Compared with the ultrasoft pseudopotentials, PAW has smaller radial cutoffs (core radii), and also exactly reconstructs the valence wave function with all nodes in the core region, which makes PAW more accurate and efficient than ultrasoft pseudopotentials in many systems while the calculations using PAW are not more expensive. 2.2 VASP software packages VASP (Vienna ab initio simulation package) is a first-principle calculation code within density functional theory frame, which uses ultrasoft pseudopotentials or the PAW method and a plane wave basis set. The approach implemented in this code is based on the (finite-temperature) LDA with the free energy as variational quantity and an exact evaluation of the instantaneous electronic ground stat at each molecular dynamics (MD) time step. It also uses efficient matrix diagonalisation schemes and an efficient Pulay/Broyden charge density mixing. Thus, it can give information about total energies, forces and stresses on an atomic system, as well as calculating optimum geometries, band structures, optical spectra, etc. It can also perform first-principles molecular dynamics simulations. In this thesis, VASP has been utilized to calculate the electronic properties of various systems. The band structure calculations were performed within the density functional theory with the generalized gradient approximation (GGA) by utilizing the accurate projector augmented-wave (PAW) method in the VASP package. 25 Chapter 2. Methodology 2.3 Implementation of van der Waals correction in computation Since van der Waals (vdW) effect commonly exists between the investigated heterostructure layers, which is important and cannot be ignored, we take vdW correction into consideration when relaxing the structures by setting relative parameters in VASP. The inclusion of vdW effect makes the calculated electronic properties more reliable and closer to real situations. After relaxation with vdW effect, the equilibrium state can be achieved. For example, in the case of BiTeI/graphene, for all the investigated configurations, the vdW layer spacing distance of equilibrium state is relaxed to about ˚ (see details in next chapter). 3.4 A 26 Chapter 3 The heterostructure of graphene/BiTeI 3.1 Introduction Free-standing graphene has Dirac cones at two inequivalent high symmetry points (K and K′ ) of the first Brillouin zone. Intrinsic SOC in graphene can open an energy gap with opposite parity at K and K′ , and give rise to quantum spin Hall states. However, the extremely small energy gap opened by intrinsic SOC makes quantum spin Hall effect in graphene very difficult to achieve. In contrast, the strong SOC effect in BiTeI results in a giant Rashba splitting for energy bands around the G point. In the following illustration, G is used to represent Γ for easy editing purpose. Is there any interesting properties if we combine graphene with BiTeI to form a BiTeI/graphene heterostructure, as shown in Figure 3.1, i.e. a layer of BiTeI is above a layer of graphene? BiTeI has space group 156 of P 3m1 and consists of a sequence of 27 Chapter 3. The heterostructure of graphene/BiTeI layers of I, Bi and Te, each of which forms a trigonal planar lattice. As a consequence, BiTeI has a spontaneous built-in potential between the positively charged (BiTe)+ layer and the negatively charged (BiI)− layer. Graphene shares the same lattice structure with BiTeI. Figures 3.2(a) and (b) show the top and side views of the graphene/BiTeI heterostructure, respectively, which consists of a single layer of graphene and BiTeI. The interaction of both Bi/Te and Bi/I is the type of ionic bonding, while that of Te/I is mainly dominated by van der Waals (vdW) effect which is much weaker than ionic bonding, so that it is easier to separate BiTeI layers between Te and I. In that case, in the single layer BiTeI, Bi atoms lie in the middle position with two sides terminated by Te and I atoms, respectively. Therefore, the BiTeI layer is stacked along the perpendicular → (− z ) direction on top of graphene sheet, with either Te or I of BiTeI facing to graphene. Moreover, as shown in Figure 3.2(a), we have three positions: (i) the atom of BiTeI facing to graphene is on top of the carbon atom in A sublattice of graphene; (ii) the atom of BiTeI facing to graphene is on top of the carbon atom in B sublattice of graphene; (iii) the atom of BiTeI facing to graphene is on top of the hole center of graphene’s hexagonal lattice. Figure 3.2(c) summarizes the six investigated configurations, named as GBTI-XY, in which GBTI indicates the system of graphene/BiTeI, X indicates the atom of BiTeI facing to graphene, and Y indicates the location of the atom X. Thus, X can be Te or I, and Y can be hole (H), A-sublattice (A) or B-sublattice (B). 3.2 Results and discussion We first calculate the interlayer distance (d0 ), the vertical distance between graphene and BiTeI layer, at equilibrium state. After taking vdW correction into consideration, d0 is 28 Chapter 3. The heterostructure of graphene/BiTeI (a) (b) Graphene Figure 3.1: (a) Schematic of the stacking structure of the BiTeI/graphene heterostructure, i.e. a layer of BiTeI is above a layer of graphene. (b) Brillouin zone. Adapted from the work of Ishizaka et al [40, 46]. ˚ for all the investigated configurations (see details as shown in Figure relaxed to ∼3.4 A 3.3). The large value of d0 indicates the very weak bonding between graphene and BiTeI. Therefore, in the equilibrium state, graphene/BiTeI heterostructures inherit the electronic properties of their parent compounds. A linear band from graphene appears at K; and spin splitting states from BiTeI can be found at G, as shown in Figure 3.4. These two unique features make graphene/BiTeI an interesting system for electronic applications. By tuning the gate voltage, non-spin-polarized carriers with very-high mobility from the linear band of graphene, or spin polarized carriers with gate-tunable procession from spin splitting subbands of BiTeI, can be induced. Moreover, in the equilibrium state, we find electronic properties of graphene/BiTeI heterostructures are stable against horizontal displacement of graphene or BiTeI layer in the xy-plane. 29 Chapter 3. The heterostructure of graphene/BiTeI (a) (b) B I(Te) H Bi A Te (I) d (c) Configuration Nearest atom Locationof nearestatom GBTIͲIH I Hole GBTIͲIA I SublatticeͲA GBTIͲIB I SublatticeͲB GBTIͲTH Te Hole GBTIͲTB Te SublatticeͲB GBTIͲTA Te SublatticeͲA Figure 3.2: (a, b) Top and side views of graphene/BiTeI heterostructures. (c) The six investigated configurations, named as GBTI-XY, in which GBTI indicates the system of graphene/BiTeI, X indicates the atom of BiTeI facing to graphene, and Y indicates the location of the atom X. Thus, X can be Te or I, and Y can be hole (H), A-sublattice (A) or B-sublattice (B). 30 Chapter 3. The heterostructure of graphene/BiTeI -70.2 GBTI-IA GBTI-IB -70.4 GBTI-IH -71.6 GBTI-TA -70.6 GBTI-TB GBTI-TH -71.7 -71.0 -71.8 -71.2 E tot (eV) -70.8 3.0 3.3 3.6 -71.4 -71.6 -71.8 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 d (Angstrom) Figure 3.3: The energy curve as a function of the interlayer distance for all the six investigated configurations. The system GBTI-TH, in which the atoms Te of BiTeI face to graphene and are above the hole position of graphene, is the most stable configuration. 31 Chapter 3. The heterostructure of graphene/BiTeI Figure 3.4: The band structure of BiTeI/graphene at equilibrium state with the interlayer ˚ distance d0 =3.4 A. Enhancing the interaction between graphene and BiTeI layers might lead to some intriguing phenomena, such as charge doping and increasing the SOC strength. Decreasing the interlayer distance d, which may be realized by applying external uniaxial pressure, is an effective method for this purpose. We calculate the energy curve as a function of the interlayer distance shown in Figure 3.3 and find that the total energy increases with decreasing d. Among all the six investigated configurations, the system GBTI-TH, in which the atom Te of BiTeI faces to graphene and is above the hole center of graphene’s hexagonal lattice, is the most stable configuration. The approximately quadratic increase, rather than exponential increase, indicates the mechanical force required for a smaller d may be realized in experiments. Furthermore, upon decreasing d, the lack of a turning-point in which total energy sharply drops indicates the absence of strong chemical bonding between graphene and BiTeI. Therefore, once the pressing 32 Chapter 3. The heterostructure of graphene/BiTeI force is released, the vdW separation of the graphene/BiTeI heterostructure will return to its equilibrium value. The compression and release processes provide a dynamic mechanism to control electronic properties of graphene/BiTeI heterostructure. Without otherwise specified, the electronic properties of the system, whose vdW ˚ is given in this chapter as typical separation between graphene and BiTeI is 2.4 A, ˚ is found to make the interaction between examples. The interlayer distance of 2.4 A graphene and BiTeI strong to present interesting properties after the system is examined ˚ to 1.0 A. ˚ We firstly investigate electronic properties with a series of distances from 3.4 A of GBTI-IH, i.e., graphene/BiTeI heterostructure in which I atoms of BiTeI facing to graphene are above the hole center of graphene’s hexagonal lattice. Figures 3.5(a) and (b) show band structures of GBTI-IH without and with SOC, respectively. Without SOC, the Dirac point of graphene locates exactly at the Fermi level; and the Dirac state at K disperses linearly from K to G and merges with electronic states from BiTeI. The SOC effect strongly influences electronic properties of GBTI-IH. As expected, SOC results in significant spin splitting in bands around G. Near the Fermi level, these energy bands are mainly contributed by the BiTeI layer. Surprisingly, the SOC effect arisen from the heavy elements of Bi greatly modifies the linear Dirac band at K. An energy gap around 60 meV is open for the Dirac state. Moreover, the Fermi level crosses the lower Dirac band, indicating p-doping of graphene due to charge transfers between graphene and BiTeI. Furthermore, spin splitting of the Dirac state is found, even though the system is nonmagnetic. We attribute these unique features to the enhanced interaction and charge transfer between graphene and BiTeI. The net charge transfer from BiTeI into graphene plane is shown in Figure 3.6. As can be seen, the net charge at A (blue-filled circles) and B (black-filled circles) 33 Chapter 3. The heterostructure of graphene/BiTeI (b) With SOC K G M Figure 3.5: Band structures of GBTI-IH: (a) without SOC; (b) with SOC along the kpath of G-K and G-M. subatoms is very different. As a result, the difference of local potential between A and B sublattice breaks the A-B sublattice symmetry. The broken A-B sublattice symmetry and the enhanced SOC strength due to the hybridization between C-pz and Bi-p orbitals lead to a significant band gap for Dirac states. Furthermore, the enhanced SOC effect also results in Rashba spin splitting of the BiTeI states. To further illustrate the electronic properties in relation to the interlayer distance d, we calculate the band structures of graphene/BiTeI heterostructure with different layer separations (d). As shown in Figure 3.7, decreasing d can narrow the energy gap between the conduction band and valence band at G. In contrast, the energy gap of the Dirac state is enlarged with decreasing d. The Fermi level of the system can also be tuned. As ˚ and strong spin-splitting a result, carriers from both Dirac states (d = 2.4 and 2.5 A) 34 Chapter 3. The heterostructure of graphene/BiTeI 0.001 -0.0005 Figure 3.6: Schematic of the net charge transfer from BiTeI into graphene plane in terms of graphene plane. The blue (black) filled circles represent the A (B) sublattice, respectively. The warm color describes the electron accumulation, while the cold color describes the electron depletion. 35 Chapter 3. The heterostructure of graphene/BiTeI EͲEF (eV) (a) d=2.2Å d=2.3Å d=2.4Å 2 (b) 2 1.5 1.5 1.5 1.5 1 1 1 1 0.5 0.5 0.5 0.5 0 0 0 0 −0.5 −0.5 −0.5 −0.5 −1 −1 −1 −1 −1.5 −1.5 −1.5 −1.5 −2 (c) 2 −2 K G M K −2 G M d=2.5Å (d) 2 −2 K 14/6/2014 G M K G M 15 Figure 3.7: Band structures of GBTI-IH as a function of the interlayer distance (d). ˚ can be realized. This offers an alternative scheme to control Rashba states (d = 2.2 A) the type of carriers by a gate voltage traditionally. Next, we investigate electronic properties of GBTI-IA and GBTI-IB, in which I atoms of BiTeI facing to graphene are respectively above the carbon atoms in the A and B sublattices of graphene. These two configurations can obviously break the A-B sublattice symmetry of graphene. As shown in Figure 3.8(a), an energy gap around 300 meV is found at K in GBTI-IA even without SOC. Turning on SOC results in spin splitting in both graphene and BiTeI’s bands (see Figure 3.8(b)). In addition, the Fermi level crosses the conduction band and the valence band at G and K, respectively. These two bands are separately dominated by the BiTeI and graphene layers and have opposite ˚ is a semi-metal. Besides, as shown in Figure parities. Therefore, GBTI-IA with d = 2.4 A 3.9, GBTI-IB has very similar energy bands as GBTI-IA, which reflects the equivalence of A-B sublattices in graphene. 36    sublattice (a) 3 (b) 3 2 2 1 1 E−EF (eV)      E−EF (eV)  Chapter 3. The heterostructure of graphene/BiTeI 0 0 −1 −1 −2 −2 −3 −3 K G K M K K G K M K Figure 3.8: Band structures of GBTI-IA: (a) without SOC; (b) with SOC. 37     sublattice (a) 3 (b) 3 2 2 1 1 E−EF (eV)      E−EF (eV)  Chapter 3. The heterostructure of graphene/BiTeI 0 0 −1 −1 −2 −2 −3 −3 K G K M K K G K M K Figure 3.9: Band structures of GBTI-IB: (a) without SOC; (b) with SOC. 38 Chapter 3. The heterostructure of graphene/BiTeI The fourth configuration of graphene/BiTeI heterostructures is GBTI-TH, in which Te atoms of BiTeI faces to graphene and are above the hole center of the graphene’s hexagonal lattice. The calculated band structures of GBTI-TH is shown as Figure 3.10. Without SOC, the Dirac state is gapped and the lower Dirac band mixes with the energy bands of BiTeI. In order to gain insight of the bands, analysis on the weight of the orbital projections is carried out. Figure 3.11 shows the weight of orbital-projected band structures of GBTI-TH without SOC. The size and number of the points indicate the presence of the according orbitals. As can be seen, below the Fermi level and at the G point, the presence of both C-pz and (Bi,Te)-p orbitals indicates strong interaction between graphene and BiTeI. As a result, the Dirac point of graphene shifts below the Fermi level; and the distorted lower Dirac cone is anisotropic with smaller Fermi velocity compared to the upper Dirac cone. The SOC leads to spin splitting, especially for electronic states at G (see Figure 3.12). The spin states of the upper Dirac cone split slightly along M-K-G. In contrast, the lower Dirac cone displays highly anisotropic spin splitting. A significant spin splitting along K-G is found. More interestingly, topological phase transition is found in the last two configurations, i.e. GBTI-TA and GBTI-TB. The band structure of GBTI-TA shown in Figure 3.13(a) presents a very large energy gap for the Dirac state. Moreover, the lower Dirac band almost disappears, only leaving a small band tail at K. To understand the physics behind this unusual feature, we plot the orbital projection of C-pz in Figure 3.14(a). In the situation without SOC, the electronic state of C-pz mainly distributes at K and a part of C-pz states appears at the valence-band maximum at G. As shown in Figure 3.13(b) and Figure 3.14(b), SOC leads to spin splitting. The large SOC effect in GBTI-TA generates band crossing between the top valence band (from 39  Chapter 3. The heterostructure of graphene/BiTeI Withou SOC (b) 3 2 2 1 1 E−EF (eV) (a) 3 E−EF (eV)  WithSOC 0 0 −1 −1 −2 −2 −3 −3 K G K M K K G K M K  Figure 3.10: Band structures of GBTI-TH: (a) without SOC; (b) with SOC. 40 Chapter 3. The heterostructure of graphene/BiTeI EͲEF (eV) CͲPz TeͲP BiͲP 1 1 1 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 −0.2 −0.2 −0.2 −0.4 −0.4 −0.4 −0.6 −0.6 −0.6 −0.8 −0.8 −0.8 −1 M −1 G K M −1 M G K M M G K M Figure 3.11: The weight analysis on the orbital projections of the band structure of GBTI-TH without SOC. EͲEF (eV) CͲPz TeͲP BiͲP 1 1 1 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 −0.2 −0.2 −0.2 −0.4 −0.4 −0.4 −0.6 −0.6 −0.6 −0.8 −0.8 −0.8 −1 M −1 G K M M −1 G K M M G K M Figure 3.12: The weight analysis on the orbital projections of the band structure of GBTI-TH with SOC. 41 Chapter 3. The heterostructure of graphene/BiTeI  2 2 1 1 E−EF (eV) (b) 3 E−EF (eV)  (a) 3 0 0 −1 −1 −2 −2 −3 −3 K G K M K K G K M K  Figure 3.13: Band structures of GBTI-TA: (a) without SOC;   (b) with SOC.   (b) 3 3 2 2 1 1 0 0 Ͳ1 Ͳ1 Ͳ2 Ͳ2 EͲEF (eV) (a) K G K M K K G K M K Figure 3.14: The weight analysis on orbital projections of C-pz of GBTI-TA: (a) without SOC; (b) with SOC. 42 Chapter 3. The heterostructure of graphene/BiTeI C-pz orbitals) and the bottom conduction band (from (Bi,Te)-p orbitals). Due to the hybridization between these C-pz and (Bi,Te)-p orbitals, the crossing point is removed; and band inversion in a gapped system is realized, as reflected in Figure 3.14. The band inversion is related to not only the SOC effect, but also the hybridization effect between graphene and BiTeI, which is determined by the interlayer distance. As shown ˚ shows a normal in Figure 3.15, with SOC, the band structure of GBTI-TA with d = 2.6 A band gap with C-pz states occupying the top valence band at G. However, reducing the ˚ gives rise to a band inversion characterized by a significant vdW spacing (d) to 2.4 A amount of C-pz states occupying the bottom conduction band at G. The band inversion indicates a phase transition of GBTI-TA heterostructure from a topologically trivial state into a quantum spin Hall system. As shown in Figure 3.16, further reducing d can enlarge the inverted band gap at the price of rapidly increasing ˚ an inverted energy gap as large as ∼0.75 the compressing force. When d = 1.8 A, eV can be obtained. Besides, the band inversion can also be observed from band structures without SOC when reducing the distance as described in Figure 3.17. In the process of compressing the two layers, the energy gap gets smaller and smaller, ˚ is a critical distance. Usually, trending to close, and then reopens, in which d = 1.8 A a monotonous change of band gap size with decreased interlayer distance is expected; and consequently, the reversal of the size of the energy gap is an anomaly implying the emergence of band inversion. Our results show that the vdW spacing can act as a new degree of freedom to control the topological properties of materials. Alloying with heavy element is a conventional method to realize topological phases. However, it is difficult to control the exact chemical ratio and the structure ordering of the alloyed compounds. Compared with alloying, compression reduces the vdW spacing, but the 43 Chapter 3. The heterostructure of graphene/BiTeI 0.4 EͲEF (eV) 0.3 0.2 d=2.4Å 0.1 0 Ͳ0.1 Ͳ0.2 K G 0.6 EͲEF (eV) 0.4 0.2 d=2.6Å 0 Ͳ0.2 Ͳ0.4 K G Figure 3.15: The weight analysis on orbital projections of C-pz of GBTI-TA with ˚ and 2.6 A, ˚ respectively. interlayer distance as 2.4 A 44 Chapter 3. The heterostructure of graphene/BiTeI    2 2 d=1.4000Å EͲEF (eV)  2 d=1.8000Å 2 d=2.2000Å d=2.6000Å 1.5 1.5 1.5 1.5 1 1 1 1 0.5 0.5 0.5 0.5 0 0 0 0 −0.5 −0.5 −0.5 −0.5 −1 −1 −1 −1 −1.5 −1.5 −1.5 −1.5 −2 −2 K G −2 K G −2 K G K G Figure 3.16: The band structures of GBTI-TA with a sequence of reduced interlayer distance d with SOC along G-K. symmetry of the heterostructure is unaffected. What’s more, the method of compression provides a dynamic scheme to control the electronic properties and topological phases of heterostructures. This unique feature can be utilized for pumping of spin currents. BiTeI is reported to have a topological insulator phase under proper pressure [49], which is a similar and effective method to enhance the SOC and the interaction between atoms. In addition, we investigate the gate-voltage-controlled electronic properties in the GBTITA heterostructure. As shown in Figure 3.18, the dispersion of energy bands remains almost the same with an applied electric field along the stacking direction, which only tunes the Fermi level of the GBTI-TA heterostructure. The insensitivity to negative electric field, which points perpendicularly from graphene to BiTeI layer, stems from the screening effect of graphene. Comparatively, a positive electric field, pointing from BiTeI vertically to graphene layer, induces electrons accumulating at BiTeI and leads to 45 Chapter 3. The heterostructure of graphene/BiTeI d = 1.8000 Å d = 2.2000 Å d = 2.6000 Å E-EF (eV) d = 1.4000 Å Critical distance K G K G K G K G Figure 3.17: The band structures of GBTI-TA with a sequence of reduced interlayer distance d without SOC along G-K. 46 Chapter 3. The heterostructure of graphene/BiTeI EͲEF (eV) (a) EFIELD=Ͳ0.4V/Å (b) EͲfieldeffect EFIELD=0V/Å EFIELD=0.2V/Å 1 (c) 1 (d) 1 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0 0 0 0 −0.2 −0.2 −0.2 −0.2 −0.4 −0.4 −0.4 −0.4 −0.6 −0.6 −0.6 −0.6 −0.8 −0.8 −0.8 −0.8 1 −1 M Figure 3.18: −1 G K M M −1 G K M M EFIELD=0.4V/Å −1 G K M M G K M The influence of applying external gate voltage on GBTI-TA heterostructure. The negative electric field points from graphene to BiTeI layer, and vice versa. a n-doping. This bipolar effect may be used to design field-effect transistors, in which transport channels can be turned on by a positive electric field. Due to the sublattice symmetry of graphene, the electronic properties of GBTI-TB (see Figure 3.19) are very similar to GBTI-TA, implying that the Te atoms of BiTeI facing to graphene are the main factor in determining electronic properties of graphene/BiTeI heterostructure. Besides, without SOC, GBTI-TB has nearly the same energy gap (∼0.28 eV) as GBTI-TA. The difference between GBTI-TB and GBTI-TA is their energy gap when considering the SOC effect. As shown in Figure 3.19(b), the spinsplitting valence and conduction bands approach to each other by the SOC effect. The band calculations with more dense points along G-K show that two spin-splitting subbands cross each other between G and K, as plotted in Figure 3.20. These two 47   Chapter 3. The heterostructure of graphene/BiTeI (b) 3 2 2 1 1 E−EF (eV) (a) 3 E−EF (eV)    0 0 −1 −1 −2 −2 −3 K −3 G K M K K G K M K Figure 3.19: Band structures of GBTI-TB: (a) without SOC; (b) with SOC. ˚ to subbands come from Bi-p and (Te,C)-pz , respectively. Surprisingly, from d = 2.4 A ˚ the crossing point is robust against the electron-electron interaction, which is d = 2.2 A increased with reduced d due to the larger overlap of electron wavefunctions. However, ˚ can finally remove the strong-enough electron-electron interaction in the case of d =2 A band crossing point and result in a quantum spin Hall phase with an inverted band gap. 48 Chapter 3. The heterostructure of graphene/BiTeI EͲEF (eV) (a) 2 d=2.0Å (b) 2 d=2.2Å (c) 2 d=2.4Å (d)2 d=2.6Å (e) 2 1.5 1.5 1.5 1.5 1.5 1 1 1 1 1 0.5 0.5 0.5 0.5 0.5 0 0 d= 3Å BiͲP 0 0 0 −0.5 −0.5 −0.5 −0.5 −0.5 −1 −1 −1 −1 −1 −1.5 −1.5 −1.5 −1.5 −1.5 −2 −2 K G Te,CͲ pz −2 K G −2 K G −2 K G K G Figure 3.20: The band structures of GBTI-TB with a sequence of reduced interlayer distance d with more dense points along G-K. 3.3 Chapter summary In conclusion, six configurations of graphene/BiTeI heterostructure are explored. The heterostructures in their equilibrium states inherit the properties of their parent materials, presenting linear bands at K and Rashba splitting around G. Compressing the systems to reduce the distance between layers of the heterostructures gives rise to intriguing properties due to enhanced interactions between graphene and BiTeI layers. The compression and release process provides a dynamic mechanism to control the electronic properties. When the interlayer distance is reduced, the band gap at K becomes larger, which is ascribed to broken sublattice symmetry of graphene and enhanced SOC due to orbital hybridization, while the gap at G shrinks. Besides, the Fermi level can also be tuned by changing the distance. More interestingly, in systems in which the atoms Te of BiTeI 49 Chapter 3. The heterostructure of graphene/BiTeI faces to graphene, a band inversion occurs, resulting in a gap opened at the crossing point of the inverted bands. This provides a new way to realize topological phase comparing with traditional methods, like incorporating heavy elements. In addition, charge transfer coming from the strong interaction between the graphene and BiTeI layers can be observed. The transfer is more prominent in GBTI-TY (T: Te, Y represents the location) series of systems than that in GBTI-IY series, since the electronegativity of Te (∼2.1 eV) is far less than that of C (∼2.55 eV) and I (∼2.66 eV). The result from the net charge transfer shows obvious broken sublattice symmetry, leading to the gap opening at K. What’s more, the band structures of these systems are not sensitive to an applied electric field; and only the Fermi level can be tuned by an electric field along the positive direction point from BiTeI to graphene. 50 Chapter 4 The heterostructure of graphene/Bi2Se3 4.1 Introduction Bi2 Se3 crystal has a rhombohedral structure with the space group of D 53d (R¯3m) [19]. The ABC stacking of rhombohedral Bi2 Se3 unit cell gives the layered Bi2 Se3 structure with hexagonal lattice plane [19]. Along the stacking direction, the periodic unit is a quintuple layer (QL), which consists of two equivalent Se atoms (indicated as Se1 and Se1′ in Figure 4.1), two equivalent Bi atoms (indicated as Bi1 and Bi1′ ) and a third Se atom (indicated as Se2) which serves as the center of inversion symmetry. Given an inversion operation with respect to the symmetry center Se2, Se1 is mapped to Se1′ and Bi1 is mapped to Bi1′ . The coupling between each atomic layer within one QL is strong, while that between every QL is much weaker, mainly dominated by the van der Waals (vdW) interaction. 51 Chapter 4. The heterostructure of graphene/Bi2 Se3 C Se1’ A B Se1 Bi1 C Se2 A Bi1’ B Se1’ C Se1 Figure 4.1: A quintuple layer (QL) of Bi2 Se3 . Given an inversion operation with respect to the symmetry center Se2, Se1 is mapped to Se1′ and Bi1 is mapped to Bi1′ [19]. 52 Chapter 4. The heterostructure of graphene/Bi2 Se3 Theoretical prediction and subsequent experimental observation of the topological surface states in Bi2 Se3 demonstrate that Bi2 Se3 is a prototype of 3D topological insulator with a large bulk gap (∼0.3 eV) and the simplest Dirac cone. This Dirac cone can be described by a four-band effective model [19, 37, 39]. When the thickness of bulk Bi2 Se3 is reduced below 6 QLs, a gap was observed in the topologically protected metallic surface states due to the surface coupling [4]. Furthermore, the parity of ultrathin films of Bi2 Se3 exhibits an oscillatory behavior as a function of the thickness. ˚ below which a normal band is formed, The critical thickness for Bi2 Se3 is dc = 25 A, transited from topological inverted bands [76]. If the structural inversion symmetry is removed, such as by a substrate, a sizable Rashba splitting of surface states would be induced; and the Rashba strength can be tuned by the potential between two surfaces. This motivates us to investigate graphene/Bi2 Se3 heterostructure. In the system, one QL Bi2 Se3 is stacked on monolayer graphene. √ √ Considering the lattice match, a 3 × 3 graphene supercell is designed to match a 1 × 1 Bi2 Se3 . In this chapter, three configurations labeled as GBS-X are considered, in which GBS represents graphene/Bi2 Se3 and X indicates the position of the atom Se1′ of Bi2 Se3 facing to graphene. The position X can be A, representing the atom Se1′ of Bi2 Se3 facing to graphene is above the carbon atom in the A sublattice; it can be H, indicating the atom Se1′ of Bi2 Se3 facing to graphene is above the hole center of graphene’s hexagonal lattice; and it can be B, as bridge position, which means the atom Se1′ of Bi2 Se3 facing to graphene is above the middle position of two adjacent carbon atoms in the A and B sublattices in graphene. It is found from the energy analysis shown in Figure 4.2 that the system GBS-H, in which the atom Se1′ of Bi2 Se3 facing to graphene is above the hole center of graphene, is the most stable configuration comparing with the other two. 53 Chapter 4. The heterostructure of graphene/Bi2 Se3 -82.6 GBS-A GBS-B -82.8 -83.0 -83.2 E tot (eV) GBS-H -83.4 -83.6 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 d (Angstrom) Figure 4.2: The energy curve as a function of the interlayer distance for all the three investigated configurations. The system of GBS-H is the most stable configuration. Without otherwise specified, the systems with the vdW spacing between graphene and ˚ are shown as examples in this chapter, since the distance d=2.4 A ˚ Bi2 Se3 as d=2.4 A is found to be ideal to present interesting properties after the systems with a series of interlayer distances were examined. 4.2 Results and discussion Gao et al. have reported that the band structure of free-standing 1 QL Bi2 Se3 has a large indirect energy gap (∼0.3 eV) [25]. When it is stacked onto monolayer graphene and the 54 Chapter 4. The heterostructure of graphene/Bi2 Se3 distance between graphene and Bi2 Se3 layers is decreased to smaller value such as d = ˚ the resulted graphene/Bi2 Se3 system shows distinct features compared to their 2.4 A, parent compounds. We first consider the system of GBS-A and plot its bandstructures without and with spin-orbit coupling (SOC) in Figure 4.3. Surprisingly, even without SOC, a Dirac cone appears at the Fermi level. Moreover, besides this Dirac cone, we find gapped linear bands in the vicinity of the Fermi level. To understand the origin of these electronic states, we plot the weight of orbital projections in Figure 4.4. Figures 4.4(a-c) indicate the gapped Dirac states come from the pz electron orbitals of graphene. Due to the band folding of the supercell, the Dirac state is moved to the G point and gapped because of the broken sublattice symmetry. Interestingly, the new Dirac cone is contributed by both graphene and Bi2 Se3 , indicating the strong hybridization between them. The M-shaped lower Dirac band suggests the existence of quantum confinement effect acting on the interface states between graphene and Bi2 Se3 . The SOC effect shown in Figure 4.3(b) and Figures 4.4(d-f) leads to remarkable spin splitting on the hybridized bands of the Dirac cone. As a consequence, the spin subbands cross each other; their interaction removes the crossing point and results in parity inversion. Interestingly, in the vicinity of the Fermi level, we find negligible spin splitting on the linear bands of C-pz orbital. The SOC effect removes the old Dirac cone and shifts the lower branch of the gapped C-pz states close to the Fermi level. As a result, as shown in Figure 4.3(b), linear Dirac states with slightly splitting stay in a parity-inverted system, which may host a hidden topological phase. Next, we investigate the electronic properties of GBS-B, in which the Se atom of Bi2 Se3 facing to graphene is above the middle position of two adjacent carbon atoms in the A and B sublattices of graphene. Without SOC, the band structure of the system GBS-B 55 Chapter 4. The heterostructure of graphene/Bi2 Se3 Withou SOC EͲEF (eV) (a) 2 (b) 2 1.5 1.5 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −1.5 −1.5 −2 M WithSOC −2 G K M M M GG KK M M Figure 4.3: The band structure of GBS-A: (a)without SOC; (b)with SOC. shown in Figure 4.5(a) has a trivial energy gap of ∼0.18 eV. With SOC, the energy bands near the Fermi level change greatly. As described in Figure 4.5(b), strong anisotropic spin splitting between G-M and G-K is found. The spin splitting along G-K is stronger than that along G-M. The weight analysis on orbital projections in Figure 4.6 shows the hybridization effect between graphene C-pz and (Bi,Se)-p orbitals. The hybridization results in a large energy gap (∼0.2 eV) for the Dirac state of graphene. The spin-splitting conduction bands near the Fermi level disperse into this energy gap. The strong SOC and hybridization effects greatly redistribute the electronic states of C-pz orbitals, leading to significant mixing states with Se-p or Bi-p states. Moreover, the Bi-p state dips deeply into the valence bands, as shown in the middle panel of Figure 4.6. In contrast, the Se-p state rises to the conduction bands. In bulk Bi2 Se3 , the band inversion results in topological surface states in a bulk gap. Here, we find the lower Dirac cone of graphene and its mixing states with (Bi,Se)-p orbitals reside in the inverted band gap. Inside 56 Chapter 4. The heterostructure of graphene/Bi2 Se3 ͲWnj ŝͲW ;ďͿ ^ĞͲW ;ĐͿ Ͳ& ;ĞsͿ ;ĂͿ D ' D < ' < ;ĞͿ D ' < D ' < ;ĨͿ Ͳ& ;ĞsͿ ;ĚͿ D ' < D ' < Figure 4.4: The weight analysis on orbital projections of GBS-A without (a, b, c) and with (d, e, f) SOC. 57 Chapter 4. The heterostructure of graphene/Bi2 Se3 Fig33 Withou SOC EͲEF (eV) (a) 2 (b) 2 1.5 1.5 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −1.5 −1.5 −2 M WithSOC −2 G G KK M M M M G G KK M M Figure 4.5: The band structure of GBS-B: (a)without SOC; (b)with SOC. the inverted gap, the existence of strong anisotropic Rashba splitting states with Dirac feature in the vicinity of G is an interesting aspect of our results. The last studied graphene/Bi2 Se3 heterostructure in this chapter is GBS-H, in which the Se atom of Bi2 Se3 facing to graphene is above the hole center of graphene. As shown in Figure 4.7, the electronic properties of GBS-H are similar to those of GBS-B. Without SOC, there is a large energy gap for GBS-H. The SOC effect results in parity inversion between Bi-p and Se-p orbitals. Moreover, the interaction between Bi-p and Se-p orbitals opens an energy gap, which is filled by the lower branch of Dirac band of graphene. Compared with GBS-B, the lower branch of Dirac band in GBS-H displays a better linear dispersion and smaller spin splitting. A nearly perfect linear band inside a large inverted gap makes GBS-H an attractive system. 58 Chapter 4. The heterostructure of graphene/Bi2 Se3 M G K M G K M K G Figure 4.6: The weight analysis on orbital projections of GBS-B with SOC. Fig35 Withou SOC EͲEF (eV) (a) 2 (b) 2 1.5 1.5 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −1.5 −1.5 −2 M WithSOC −2 G G KK M M M M G G KK M M Figure 4.7: The band structure of GBS-H: (a)without SOC; (b)with SOC. 59 Chapter 4. The heterostructure of graphene/Bi2 Se3 M G K M G K M G K Figure 4.8: The weight analysis on orbital projections of GBS-H with SOC. 4.3 Chapter summary In this chapter, three configurations of graphene/Bi2 Se3 heterostructure are explored. Compared with 1 QL free-standing Bi2 Se3 , a new Dirac state is found due to the interaction between graphene and Bi2 Se3 . With SOC effect, remarkable Rashba splitting and topological phase transition can be observed. Besides, SOC leads to anisotropic splitting along the k-path in Brillouin zone. In addition, the hybridization of the inverted energy bands gives rise to a band gap opening at the crossing-avoided points. 60 Chapter 5 The heterostructure of bilayer graphene and BN/graphene 5.1 Introduction It is widely known that single-layer graphene has exotic properties, such as the Dirac point and linear bands, which indicates the electrons have very high mobility. However, there is an annoying snag greatly limiting the uses of graphene in electronics, i.e. the absence of a band gap. Without a band gap, the devices cannot be turned on/off. To search for an energy gap, previous researchers pay much attention to bilayer graphene and graphene-substrate systems. Hexagonal boron nitride (h-BN) is an excellent substrate for high quality graphene devices due to its high dielectric constant and very small lattice mismatch with graphene [77]. Doping potassium atoms into one of the two layers of graphene is useful to introduce a band gap [57], but it is not easy to control the 61 Chapter 5. The heterostructure of bilayer graphene and BN/graphene chemical ratio and the doped atoms may exert some unexpected influence on the original system. Besides, applying an electric field onto bilayer graphene and BN/graphene can also realize the gap opening [58, 78]. Furthermore, in bilayer graphene, Zheng et al found that the conductance fluctuates with the sliding motion [79] and Son et al also discovered an electronic topological transition in the process of sliding [80]. These motivate us to explore whether it is possible to control electronic properties by interlayer sliding and thus achieve a gap opening in systems of bilayer graphene and BN/graphene. 5.2 Results and discussion 5.2.1 Bilayer graphene Motivated by previous research on BiTeI/graphene and Bi2 Se3 /graphene in which smaller interlayer distance may lead to stronger interaction and thus give rise to interesting properties like topological phase transition, we firstly study electronic properties of Bernal (or AB-) stacked bilayer graphene when reducing the interlayer ˚ In the ˚ (the equilibrium state demonstrated by Figure 5.2) to 2.4 A. distance from 3.4 A configuration of Bernal stacking, carbon atoms in A sublattice of one layer are exactly on the top of carbon atoms in the B sublattice of the other layer. In the situation of d ˚ four energy bands near Fermi level can be observed clearly in Figure 5.1(a), = 3.4 A, with two bands touching at the Fermi level and the other two having an energy gap of ˚ ∼0.6 eV. When the two layers of graphene get closer to each other with d = 2.4 A, the energy spacing between two conduction or valence bands increases, which reflects the enhanced interaction in bilayer graphene. Interestingly, in the vicinity of the Fermi 62 Chapter 5. The heterostructure of bilayer graphene and BN/graphene d = 3.4 Å M G d = 2.4 Å K M M G K M Figure 5.1: The band structure of Bernal-stacking bilayer graphene with interlayer ˚ (b)2.4 A. ˚ distance as (a)3.4 A; level, an crossover between the conduction and valence bands occurs due to the band inversion. The increased electron-electron interaction of graphene π-electrons shifts the energy level of antibonding (bonding) π-states down (up). Such a band crossing is robust against the SOC effect due to the protection of three-fold rotational symmetry. ˚ hosts a unique 2D Dirac point which is a Therefore, bilayer graphene with d = 2.4 A topological phase due to band inversion and is protected by crystal symmetry. These features are very different from single layer graphene, in which Dirac cone sits at K momentum point without the requirement of crystal symmetry protection and can be gapped by the SOC effect. We expect different electronic phases can be induced by breaking the crystal symmetry ˚ To verify it, we change the configuration of bilayer of bilayer graphene with d = 2.4 A. 63 Chapter 5. The heterostructure of bilayer graphene and BN/graphene -39.4 AB-stacking bridge AA-stacking -39.8 -40.0 E tot (eV) -39.6 -40.2 -40.4 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 d (Angstrom) Figure 5.2: The energy curve for all the three investigated configurations of bilayer graphene as a function of the interlayer distance. The AB-stacking is the most stable configuration. graphene by sliding one layer graphene sheet along the direction indicated by the black arrow shown in Figure 5.3(a). It is found from the energy analysis displayed in Figure 5.2 that the AB-stacking is the most stable configuration. An energy barrier of 40 meV/cell is required for structure transition from AB-stacking to AA-stacking at the ˚ and the energy barrier for transition from AB- to AA-stacking separation of d = 2.4 A increases rapidly with decreasing the interlayer distance. The system has a transition from metal to semiconductor and then back to metal. Figure 5.3(b) shows the energy gap as a function of the sliding distance from the starting point 64 Chapter 5. The heterostructure of bilayer graphene and BN/graphene (AB stacking) to the ending point (AA stacking), in which the metal-semiconductormetal transition can be seen clearly. Zheng et al. have reported a similar behavior of oscillatory conductance due to the sliding effect in bilayer graphene [79]. This metalsemiconductor-metal transition can be used to design strain sensor or for switching applications. The schematic illustration of a device designed by Zheng et al. [79] shown in Figure 5.4 may provide a possible way to realize the sliding operation. Besides, Son et al. [80] also reported that the topology of energy bands undergoes a very sensitive transition when a slight sliding occurs in bilayer graphene. The sliding between two layers of graphene splits the Dirac cone into several cones and changes the topology of Dirac points around the corners of Brillouin zone. For AB-stacking and AA-stacking, the gaps tend to close. More interestingly, the gap opening in bridge-stacking configuration (the carbon atoms of one layer of graphene are above the middle position of two adjacent carbon atoms in the A and B sublattices of another layer of graphene) reflects a topological phase transition due to the broken threefold rotational symmetry. Without the symmetry protection, the interaction between the intersected valence and conduction bands is hybridized strongly to open a gap. An ˚ as inverted energy gap indicates the bridge-stacking bilayer graphene with d = 2.4 A a possible quantum spin Hall system. In order to confirm this topological phase, we calculate edge states of a nanoribbon (see Figure 5.5) using the bridge-stacking bilayer ˚ graphene. The width of the nanoribbon is 43.5 A. The result in Figure 5.6 shows that there is a band crossing at the Fermi level. The zoomin energy band near the Fermi level in Figure 5.7(a) shows that the Fermi level crosses the energy bands three times. This odd-number of crossing indicates that the edge states are 2D non-trivial topological states. Moreover, the crossing bands are distinct from the 65 Chapter 5. The heterostructure of bilayer graphene and BN/graphene end (a) (c) bridge AB-stacking AA-stacking start (b) Gap (eV) K-M G-K AB AA M G K M M G K M M G K M d = 2.4 Å without SOC Figure 5.3: Change the configuration of bilayer graphene by interlayer sliding, based ˚ (a) Schematic of the sliding operation; upon the system with interlayer distance as 2.4 A. (b) the magnitude of gaps opened along the k-path of K to G and K to M; (c) the band structures for three typical configurations, i.e. AB-stacking, bridge-stacking (the carbon atoms of one layer of graphene are above the middle position of two adjacent carbon atoms in the A and B sublattices of another layer), and AA-stacking. 66 Chapter 5. The heterostructure of bilayer graphene and BN/graphene - (a) + Substrate (b) Sliding direction Figure 5.4: (a) The schematic illustration of a device designed by Zheng et al. [79] may provide a possible way to realize the sliding operation. (b) Side view. Figure 5.5: The structure of the nanoribbon for calculating the edge state using bridgestacking bilayer graphene (the carbon atoms of one layer of graphene are above the middle position of two adjacent carbon atoms in the A and B sublattices of another layer ˚ of graphene). The width of the nanoribbon is 43.5 A. 67 Chapter 5. The heterostructure of bilayer graphene and BN/graphene G X Figure 5.6: The band structure of the nanoribbon constructed by the bridge-stacking graphene. The edge state disperses into the gap from the bulk. typical Dirac point, showing a very low Fermi velocity and strong anisotropy. Besides, the lack of high DOS peak at the fermi level in Figure 5.7(b) indicates the edge states may not be strictly localized at the edges. As a result, we find that the bilayer graphene nanoribbon has a nonmagnetic ground state. 5.2.2 BN/graphene Last, we present our preliminary results on graphene/BN heterostructure. Three ˚ are considered by changing configurations of BN/graphene heterostructure (d = 2.4 A) the horizontal positions of hexagonal BN layer so that the N atoms are above the top of carbon atoms (either in A or B sublattice), the bridge position (the middle position of two adjacent carbon atoms in the A and B sublattices in graphene) and the hole 68 Chapter 5. The heterostructure of bilayer graphene and BN/graphene (a) (b) 20 s px,py pz 10 G X 0 -12 -8 -4 0 4 8 E-EF (eV) Figure 5.7: (a) The detailed picture of the crossing point in Figure 5.6; (b) The density of states of each orbitals. position of graphene sheet, respectively. It is found from the energy analysis shown in Figure 5.8 that the system, in which the nitrogen atoms of BN are above the hole position of graphene, is the most stable configuration comparing with the other two. As shown in Figure 5.9, a considerable gap can be found in all these three configurations. The energy bands at the G momentum point are almost unaffected by changing the configurations. On the contrary, the energy bands at the K momentum point are much more sensitive to the different stacking positions. Among these three configurations, the graphene/BN heterostructure in which the atom N is above the top (hole) position shows the largest (smallest) energy gap. The energy gap can be ascribed to the broken A-B sublattice symmetry, and its size is related to the potential difference between the A and B sub-atoms of graphene. Besides, more attractively, when changing N atoms of hexagonal BN sheet to the bridge position (the middle position of two adjacent carbon atoms in the A and B sublattices in graphene), the energy bands at K point has a feature of band inversion, which is quite similar with our previous discussion in bilayer graphene. Due to the lack of inversion symmetry in the graphene/BN heterostructure, an adiabatic process by tuning the sliding parameters to close and reopen the energy gap, 69 Chapter 5. The heterostructure of bilayer graphene and BN/graphene accompanied by the band inversion, is required to search for the possible topological phase transition. Alternatively, we can calculate the edge state of nanoribbons based on this band inverted graphene/BN heterostructure. A topological edge state in a band inverted insulator can act as an indicant of 2D topological insulator. All of these will be considered in our future studies. 5.3 Chapter summary In summary, a new and straightforward way is discovered to open a gap in graphene and to control the electronic properties of graphene. The gap opening is resulted from the inverted bands so that the topological phase transition may shed light on the mechanism of the gap. Besides, by interlayer sliding, continuous transition of electronic properties in bilayer graphene can be realized, which is promising to be utilized in the design of switching applications and mechanical sensors. 70 Chapter 5. The heterostructure of bilayer graphene and BN/graphene -37.1 Top Bridge Hole E tot (eV) -37.2 -37.3 -37.4 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 d (Angstrom) Figure 5.8: The energy curve for all the three investigated configurations of graphene/BN as a function of the interlayer distance. The system, in which the nitrogen atoms of BN are above the hole position of graphene, is the most stable configuration comparing with the other two. Top: the N atoms are above the top of carbon atoms (either in A or B sublattice) of graphene. Bridge: the N atoms are above the bridge position (the middle position of two adjacent carbon atoms in the A and B sublattices in graphene). Hole: the N atoms are above the hole position of graphene sheet. 71 Chapter 5. The heterostructure of bilayer graphene and BN/graphene Top M G bridge K M M G Hole K M M G K M Figure 5.9: The band structures of BN/graphene in three different configurations by changing the nitrogen atoms of BN onto top, bridge (the middle position of two adjacent carbon atoms in the A and B sublattices in graphene) and hole positions of graphene. 72 Chapter 6 Conclusion remarks and future work 6.1 Conclusions Motivated by the research interest starting from graphene and the subsequent 2D materials, we carry out first principle calculations on the electronic properties and topological phases of heterostructures of graphene/BiTeI, graphene/Bi2 Se3 , bilayer graphene and BN/graphene, which are comprised of single layer graphene with single layer BiTeI, Bi2 Se3 , graphene and BN. In equilibrium state, the systems preserve their parent compounds’ properties well. However, compressing the layers, taking SOC into consideration and changing the stacking configurations can give rise to band inversion in all the investigated systems, opening a topological non-trivial gap. In the process of reducing the interlayer distance in BiTeI/graphene and bilayer graphene, it is found that smaller distance results in 73 Chapter 6. Conclusion remarks and future work stronger interaction between layers. With stronger interaction, the SOC splits the bands showing a prominent Rashba effect due to the broken inversion symmetry, which stems from the substrate effect of graphene in the systems of graphene/BiTeI and graphene/Bi2 Se3 . In graphene/BiTeI, the SOC enhances the hybridization between conduction and valence bands around G, shrinking the band gap, while enlarging both the band gap at K and the splitting of graphene’s Dirac state. The SOC induces giant Rashba splitting and anisotropic effect along G-M and G-K in both graphene/BiTeI and graphene/Bi2 Se3 heterostructures. The SOC makes the hybridization between C-pz and Bi-p in BiTeI or (Bi,Se)-p in Bi2 Se3 stronger and redistributes the orbitals, leading to band inversion and gap opening at the band-crossing points. Besides, the influence of interlayer distance on band inversion can be clearly observed in BiTeI/graphene and ˚ with SOC and d = 1.8 A ˚ without SOC bilayer graphene. The separation d = 2.4 A are critical distances for occurrence of band inversion in BiTeI/graphene, and in bilayer ˚ a band crossing can be observed, too. In addition, the applied graphene at d = 2.4 A electric field exerts little influence on electronic properties in graphene/BiTeI but tunes the Fermi level. To make the family of 2D graphene-based heterostructures more diversified, bilayer graphene and graphene/BN are studied with the focus on band structures of different stacking configurations. Similar band inversion and the induced band gap are spotted in both systems, which provides a new and straightforward way to open a gap and to control the electronic properties comparing with the previously reported methods, such as doping potassium atoms and applying electric field. To confirm the topological phase, an edge state is predicted to exist in bilayer graphene. The continuous transition of electronic properties in bilayer graphene in the process of sliding can be utilized in the 74 Chapter 6. Conclusion remarks and future work design of switching applications and mechanical sensors. 6.2 Future work Although we discover some interesting properties in our calculations, it has to be admitted that there are still some challenges waiting to be conquered. 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Furthermore, graphene and BiTeI share similar structures All of these motivate us to investigate possible interesting phenomena in BiTeI /graphene heterostructures by changing the distance between graphene and BiTeI, applying electric field and altering the stacking configurations Our results show that the Rashba effect in BiTeI is maintained, and a significant spin splitting in graphene s linear band is... bilayer graphene and graphene/ BN with smaller interlayer distance, large band gap is predicted and the gap may result from band inversion It provides a new way to control graphene s energy band and the continuous transition of electronic properties by changing stacking configurations can be employed to design switching applications and mechanical sensors 11 Chapter 2 Methodology Our first-principles... Implementation of van der Waals correction in computation Since van der Waals (vdW) effect commonly exists between the investigated heterostructure layers, which is important and cannot be ignored, we take vdW correction into consideration when relaxing the structures by setting relative parameters in VASP The inclusion of vdW effect makes the calculated electronic properties more reliable and closer to... explore what will happen when the interaction is strong enough between graphene and Bi2 Se3 Surprisingly, a new Dirac point emerges in the system Furthermore, topological phase transition and significant Rashba splitting in Bi2 Se3 /graphene are found Graphene is a prominent 2D material for electronics, however, the absence of band gap is an annoying snag Bilayer graphene and appropriate substrates are... be considered in the project is BiTeI /graphene Graphene provides an excellent transport channel for spintronic devices, while strong Rashba effect in BiTeI can be utilized to control spin polarized currents Moreover, graphene is semimetallic, while BiTeI is semiconducting The combination of graphene and BiTeI offers us some unique effects without degrading the intrinsic properties of graphene and BiTeI... amazingly successful in plotting the structural and electronic properties of materials However, several problems also emerge in the application of LDA For example, LDA fails to provide a good description for excited states and underestimates the band gap of semiconductor and insulator LDA tends to overbind, resulting in the underestimated lattice parameter and the overestimated cohesive energy In addition,... be exploited in designing graphene- based spintronic devices to fully utilize the excellent transport properties of graphene More interestingly, a topological phase transition is discovered at critical interlayer distance or with SOC effect, indicating the system changes from topological trivial to nontrivial state Furthermore, there is a band gap opening The magnitude of the band gap and the Fermi... calculating an infinite number of electronic wave functions to one of calculating a finite number of electronic wave functions at an infinite number of k-points In principle, each k-point in 1BZ have to be considered for the calculations, due to the fact that the occupied states at each k-point contribute to the electronic potential in the bulk solid, typically, through the following integrals, I(ε) = 1 ΩBZ... Rashba spin splitting have been observed Moreover, pressure-induced topological phase transition in BiTeI has been predicted Bahramy et al found topological nontrivial states in BiTeI with topological invariant of Z2 = 1;(001) [49] It is of great interest to investigate the interplay of topological surface states with Rashba spin splitting, in both of which the SOC plays an essential role 8 Chapter 1 Introduction... 1.4 Graphene Graphene, an one-atom-thick allotrope of carbon, is an epoch-making material which opens up a completely new research field by virtue of its various unique and superior properties Graphene is the thinnest and strongest material in the universe we have ever known and discovered It maintains current densities up to six orders of magnitude higher than that of copper, and has outstanding thermal