Introduction to Graphene Bjarni Hannesson FacultyFaculty of of Physics Physics UniversityUniversity of of Iceland Iceland 20162016 INTRODUCTION TO GRAPHENE Bjarni Hannesson 16 ECTS thesis submitted in partial fulfillment of a B.Sc. degree in physics Advisors Lárus Thorlacius Valentina Giangreco M. Puletti Faculty of Physics School of Engineering and Natural Sciences University of Iceland Reykjavík, June 2016 Abstract In this report some properties of graphene, a single atomic layer of carbon, are outlined. The structure of graphene is reviewed and its dispersion relation within the tight-binding approximation and the effective mass approximation. The ob- served half-integer quantum Hall effect in graphene is illustrated and finally bilayer graphene is discussed briefly. Útdráttur Í þessari ritgerð verður fjallað um grafín sem er einnar frumeindar þykkt lag af kolefni. Frumeindirnar í grafíni mynda tvívíða sexhyrningagrind og nota má einfalt tvívítt líkan með víxlverkunum milli næstu og þarnæstu nágranna til að lýsa hegðun rafeinda í grafíni. Einnig verður fjallað um skömmtuð Hallhrif í grafíni og að lokum stuttlega um tveggja laga grafín. v Contents 1. Introduction 1 2. Lattice structure 3 3. Energy spectrum 5 3.1. The massless Dirac fermions . .6 4. The Quantum Hall effect in Graphene 9 4.1. Classical Hall effect . .9 4.2. Landau levels . 10 4.3. Quantum Hall Effect . 11 4.4. Landau levels in graphene . 12 4.5. The anomalous quantum Hall effect in graphene . 15 4.6. Symmetry breaking . 16 4.7. Shubnikov-de Haas oscillations . 18 5. Bilayer Graphene 19 5.1. Fabrication . 19 5.2. Identification of Layers . 20 5.3. Tight-binding model . 22 6. Summary 25 A. 2D fermion in an external magnetic field 27 B. 2D Dirac fermion in an external magnetic field 29 Bibliography 31 vii 1. Introduction Graphene is a sheet of carbon atoms, ordered in a hexagonal pattern and only one atom thick. We have all come in contact with graphene in our daily lives in its bulk form, named graphite. Graphite is simply a collection of graphene stacks or multiple layers of graphene weakly bound together. Graphite is in your pencil and these weak bonds are the reason the pencil leaves a trace. Although the pencil was invented in the 16-th century, graphene was only first observed in 2004 by Novoselov et.al [1]. Novoselov and Geim received the Nobel prize in Physics in 2010 for their groundbreaking studies regarding graphene. This report will review the structure of graphene in section 2 and its dispersion relation and charge carriers, which behave as massless Dirac fermions, in section 3. We will also discuss the anomalous quantum Hall effect in graphene in section 4 and briefly review bilayer graphene in section 5. This report is mainly based on the review papers by Abergel et.al, Properties of graphene: a theoretical perspective, published in Advances in Physics in 2010 [2] and by Castro Neto et.al, The electronic properties of graphene, published in Reviews of Modern Physics in 2009 [3]. 1 2. Lattice structure Graphene is a single atomic layer of carbon. The carbon atoms form a hexagonal or honeycomb lattice making the whole sheet resemble chickenwire. The graphene sheet (2D) can be rolled up to form a carbon nanotube (1D) or to a complete sphere to form fullerene (zero-D). To form a sphere some pentagons must replace the hexagons for the pentagons introduce a necessary curvature to the sheet, just like a soccer ball. The sheet of graphene has many qualities such as high electric and thermal conductivity as well as being transparent and robust. Graphene is even more robust than diamond and is in fact carbons most stable form. That is if we excite a diamond with a little energy above a threshold it would break down and form graphene or graphite. Although fullerenes and nanotubes have a lot of interesting features we will only consider the two dimensional sheet of graphene. Each carbon atom has three nearest neighbours arranged with a 120◦ angle between them, as seen in figure 2.1. This 2 trigonal plane is caused by sp hybridization of the 2s; 2px and 2py orbitals. These orbitals form strong σ bonds with a separation of 1:42 Å. The σ band forms a full and deep valence band which is responsible for the robustness of graphene. The last valence electron is in the 2pz orbital. A weak π band forms when half filled 2pz orbitals overlap other 2pz orbitals. The hexagonal lattice of graphene is bipartite and can be divided into two triangular sublattices, A and B, as shown in figure 2.1. Each sublattice is a Bravais lattice and the unit cell consist of two atoms. Since the unit cell has two atoms each cell has two 2pz electrons and the π band is completely filled. The basis vectors of sublattice A with lattice constant a = 2:46Å, are a p a p ~a1 = 3; 1 ; ~a2 = 3; −1 : (2.1) 2 2 The location of an A site atom can be expressed as a linear combination of these basis vectors. The location of a B site atom can be expressed as a linear combination of the basis vectors plus an offset, ~ a 1 ~ a 1 ~ 1 δ1 = p ; 1 ; δ2 = p ; −1 ; δ3 = a −p ; 0 : (2.2) 2 3 2 3 3 3 2. Lattice structure Figure 2.1: The honeycomb lattice of graphene in real space and reciprocal space. a) The left figure shows the hexagonal lattice, the A and B sublattices, the basis vectors and the unit cell (dotted lines). b) The corresponding reciprocal lattice and its basis vectors. The shaded area is the first Brillouin zone and its high-symmetry points Γ, M and K are illustrated. Figure from [2]. The respective reciprocal vectors are ~ 2π 1 ~ 2π 1 b1 = p ; 1 ; b2 = p ; −1 ; (2.3) a 3 a 3 p with a reciprocal lattice constant j~bj = 4π= 3a. The first Brillouin zone is a hexagon with degeneracy points appearing at the corners. These are called K points or Dirac points and, as we shall see below, they are the positions of the apex of so-called Dirac cones in the electron dispersion relation. Only two of the corners are inequivalent of each other and their position in momentum space are 2π 1 2π 1 K~ = p 1; p ; K~ 0 = p 1; −p : (2.4) 3a 3 3a 3 4 3. Energy spectrum To study graphene we use the tight-binding approximation, where an electron is only allowed to travel between its nearest or next-nearest neighbours. The tight-binding Hamiltonian for the π electrons in graphene was first studied by Wallace [4] in 1947 and gave the energy dispersion of the π band. The Hamiltonian is 1 X y 0 X y y Htight-binding = −t (aσ;ibσ;j + H:c:) − t (aσ;iaσ;j + bσ;ibσ;j + H:c:): (3.1) hi;ji,σ hhi;jii,σ y In the Hamiltonian aσ;i and aσ;i are the annihilation and creation operators for an ~ y electron with spin σ ="; # at site Ri on sublattice A. Likewise, the bσ;i and bσ;i operators refer to sublattice B. The hopping integral, t, is the energy needed to hop to the nearest neighbour and is t ≈ 2:8eV . The next-nearest neighbour hopping 0 integral, t’, is not well known but calculations in [5] found that 0:02 t . t . 0:2 t. For the tight-binding Hamiltonian the corresponding wavefunction for graphene is a linear combination of Bloch functions for each sublattice. In the nearest neighbour approximation the energy eigenvalues can be obtained in closed form [6,7] " p #1=2 3kxa kya 2 kya E(kx; ky) = ±t 1 + 4 cos cos + 4 cos : (3.2) 2 2 2 If we take into account the next-nearest neighbour hopping an electron-hole asym- metry in the K points appears. For a complete picture of the band structure the σ bands should be added since they are the lowest energy bands near the center of the Brillouin zone. The interesting features of graphene however occur in the low energy limit at the K points so we ignore this in this report. Expanding the energy dispersion (3.2) around the Dirac points such that the wave vector is ~q = K~ +~k and ka 1, where k = j~kj, gives the linear dispersion relation, p ~ 3 E±(k) ≈ ± tak = ±vF k: (3.3) 2 ~ 1The brackets and double brackets under the sum denote nearest and next-nearest neighbour approximation, respectively, indicating to only sum over neighbouring sites. 5 3. Energy spectrum p 6 m The Fermi velocity is vF = 3ta=2~ ≈ 10 =s and is independent of charge density and momentum. Note that the energy varies linearly with respect to momentum which indicates that we have relativistic massless particles. Figure 3.1: a) The energy dispersion of graphene plotted. The π∗ band refers to holes and the filled π band to electrons. The bands meet at the K and K’ points forming Dirac cones. b) The dispersion along the high-symmetry points ΓMK. Figure from [2]. The π and π∗ bands are shown in figure 3.1 and they meet at the Dirac points forming the Dirac cones. These cones are a direct result of a linear dispersion relation around the Dirac points. In undoped graphene the Fermi energy level lands exactly at the apex of the cones. Toggling the Fermi level slightly up or down the cones one finds the Fermi surface to be a circle. 3.1. The massless Dirac fermions The linear dispersion relation indicates that the charge carriers are massless Dirac fermions.