École polytechnique de Louvain (EPL) Development of a tight-binding model to study Hofstadter’s butterfly in graphene on h-BN exhibiting a moiré pattern Dissertation presented by Nicolas MOREAU for obtaining the Master’s degree in Physical Engineering Supervisor(s) Jean-Christophe CHARLIER Reader(s) Simon DUBOIS, Benoît HACKENS Academic year 2016-2017 Acknowledgements First, I want to thank Simon for all the time he spent for me during this year. I am deeply grateful to him for his availability to answer my questions, to modify quickly his code for solving the problems I faced, and finally for helping me construct my model. This work would not have been possible without him. I also thank Professor Jean-Christophe Charlier for all his remarks and advice since the beginning of my master. HIs inputs have helped me greatly during these two last years, especially for this thesis. I am also grateful to him for having proposed to me this highly interesting subject which accords well with my interests. I take this occasion to thank Professor Benoît Hackens for the time he spent constructing with me my doctorate project, as well as for the time he spent reading my work. More than ever, I thank my parents for having supported me throughout my studies and in all my choices. My fulfillment in my studies and in my life, including the opening of fresh horizons, are also their success. And deeply I am grateful to my family for their encouragement. Also, I thank all my friends, particularly Thérèse, for having made these five years of study the best of my life. Finally, I thank Peter for his precious corrections that restore dignity to Shakespeare’s language in my thesis. i Contents Abstract v Introduction 1 1 The tight-binding model for graphene 3 1.1 Generalities about the tight-binding method . 3 1.1.1 The tight-binding Hamiltonian . 3 1.1.2 Slater-Koster parameters . 5 1.1.3 The effect of a magnetic field . 5 1.1.4 The second quantization . 7 1.2 Derivation of the band structure of graphene . 7 1.2.1 The structure of graphene . 7 1.2.2 Tight-binding model for graphene . 8 1.2.3 Emergence of Dirac fermions . 10 1.3 Implementation of the tight-binding model . 11 1.3.1 The case of graphene . 11 1.3.2 The case of hBN . 12 2 Energy quantizations in solid state physics 15 2.1 The quantization in a periodic structure . 15 2.1.1 The Bloch bands of the tight-binding model . 15 2.1.2 The effect of a weak potential . 15 2.2 The effect of a strong magnetic field . 17 2.2.1 Electrons and magnetic field in the second quantization formalism . 17 2.2.2 Quantization for classical electrons . 18 2.2.3 Quantization for Dirac fermions . 19 2.2.4 Quantization of the conductance . 20 3 Hofstadter’s butterfly 23 3.1 Hofstadter butterfly in a simple square lattice . 23 3.1.1 Harper’s equation . 23 3.1.2 Plotting the butterfly . 25 3.1.3 Wannier diagram . 26 3.1.4 Study of the conductance . 28 3.2 Hofstadter’s butterfly in graphene . 30 3.2.1 Wannier diagram for graphene . 31 3.2.2 Hofstadter’s butterfly for relativistic Landau levels . 31 3.3 Observation of Hofstadter’s butterfly . 32 3.3.1 The moiré pattern . 33 3.3.2 Bilayer graphene on hBN . 34 3.3.3 Single graphene on hBN . 36 4 The models 39 4.1 Initial hypotheses . 39 4.1.1 Hypotheses on the geometrical structure . 39 4.1.2 Hypotheses on the tight-binding model . 40 iii Contents 4.2 Construction of the commensurate supercell . 40 4.2.1 Commensurate and incommensurate lattices . 40 4.2.2 Derivation of the parameters of a commensurate lattice . 41 4.2.3 The different stacking configurations . 43 4.2.4 Interlayer distance . 44 4.3 Implementation of the magnetic field in the tight-binding model . 44 4.3.1 Construction of a periodic potential vector . 45 4.3.2 Precautions to follow for the implementation . 46 4.3.3 Construction of a commensurable flux lines lattice . 47 4.4 The Ewald potential model . 47 4.4.1 Construction of an electrostatic potential . 48 4.4.2 The potential in the tight-binding model . 49 4.5 The complete tight-binding model . 49 4.5.1 Development of the general interaction Hamiltonian . 50 4.5.2 Definition of the tight-binding Hamiltonian . 50 4.5.3 Decimation of the interaction terms . 52 4.5.4 Bilayer graphene on hBN . 53 5 Results and discussions 55 5.1 Electronic properties for the Ewald potential model . 55 5.2 Electronic properties for the complete tight-binding model . 57 5.2.1 Band structure and density of states . 57 5.2.2 Effect of different parameters on the electronic propreties of graphene . 58 5.2.3 Bilayer graphene on hBN . 62 5.3 Hofstadter’s butterfly . 63 Conclusion and future perspectives 67 Appendices 71 A Brief presentation of the DFT calculation 71 A.1 Resolution of the Schrödinger equation . 71 A.2 Density Functional Theory . 72 A.3 The pseudopotentials . 73 B A few words about the Dirac equation 75 B.1 From Schrödinger to Dirac . 75 B.2 The Weyl equation in two dimensions . 77 C Implementation of the Tight-binding model 79 C.1 The code SuperTB .......................................... 79 C.2 Results of the fits on DFT band structures . 80 C.3 Calculation of the density of state . 81 C.4 Contributions of this work to the code . 82 D The second quantization 85 D.1 From the first to the second quantization . 85 D.2 For the bosons . 86 D.3 For the fermions . 86 D.4 The operators . 87 E Presentation of the codes 89 E.1 Graphene band structure . 89 E.2 Construction of a graphene on hBN commensurate supercell . 89 E.3 Ewald potential . 92 Bibliography 95 iv Abstract When submitted both to a magnetic field and a periodic potential, the energy spectrum of electrons exhibits a fractal-like pattern known as Hofstadter’s butterfly. Proofs of the existence of this butterfly were reported in a moiré lattice of graphene on hexagonal-boron-nitride (hBN) in 2013. In this master thesis, two main tight-binding (TB) models are developed to study the butterfly in similar moiré systems. The first model considers the effect of hBN as a periodic electrostatic potential that modifies the on-site terms of graphene TB Hamiltonian. The second model is based on a complete TB approach. Interactions between graphene and hBN are performed by interlayer bond energies. The values of those interlayer parameters have not yet been reported in literature. In this thesis, they are determined with DFT calculations. Based on those two models, the electronic properties of graphene on hBN are discussed first. Character- istic features, such as the gap opening at the Dirac point, the appearance of secondary Dirac points, and the geometrical influence of the moiré pattern are properly described by the complete TB model. In the latter part of the thesis, the signature of the butterfly is highlighted and compared with corresponding experimental results. v Introduction In a periodic lattice, energy of electrons is quantized in Bloch bands due to the presence of the periodic atomic potential. A magnetic field also leads to a quantization of energy into discrete energy levels, called Landau levels. Together, the effect of lattice quantization and magnetic quantization leads to a fractal-like energy spectrum. This recursive behaviour of energy was highlighted theoretically for the first time in 1976 by D.R. Hofstadter [1]. Due to the characteristic shape of the fractal-like graph, it is called Hofstadter’s butterfly. The appearance of a fractal-like spectrum depends on the ratio between two characteristic dimensions. These are the magnetic length and the unit cell area [2, 3]. By taking the size of a classical unit cell area, the magnetic field required to observe the recursive behaviour of energy is far too large. Nevertheless, by breaking the lattice symmetry resulting in the appearance of large supercells, the required magnetic field reaches values available in laboratory. As already proposed by Hofstadter in his paper [1], breaking the symmetry can be achieved by applying an external potential having the desired periodicity. Using this technique, experiments performed on two-dimensional electron gas (2DEG) in GaAs/AlGaAs heterostructures have highlighted the presence of Hofstadter’s butterfly [4-6]. Since then, a new way to observe Hofstadter’s butterfly has been discovered with the arrival of a material with great potential: graphene. Graphene is a two-dimensional material composed of carbon atoms that form a honeycomb lattice. It was isolated from graphite for the first time in 2004 [7]. Graphene has exceptional electrical properties and is forseen as a predominant actor in electronics [8, 9]. However, when graphene is placed on a classical silicon-based substrate, its electronic properties are significantly lowered [10-12]. This problem has been overcome by using hexagonal-boron-nitride (hBN) as substrate [12-14]. hBN is composed of 2D atomic layers that are isomorphic to graphene, with a lattice mismatch of about 1.8% between the two lattices. When placed on hBN, graphene is extremely flat and its electronic conductivity is close to that in suspended graphene [15]. The lattice mismatch between the two lattices are of major interest for observation of Hofstadter’s butterfly. Indeed, it breaks the graphene lattice symmetry, leading to a moiré pattern when graphene is placed on top of hBN [16-18].