5 Chapter 2 Introduction to carbon physics Carbon is in many ways a unique element. Most importantly, it is crucial for life on earth as we know it since the human body is to a large extent made up out ofcarbon.Scientifically, the whole huge field of organic chemistry deals entirely with carbon-based compounds; and in the field of physics carbon is one of the most intensively studied materials. There even exists a journal named “Carbon” devoted exclusively to carbon that only in the year of 2006 consisted of over 3000 pages. Pure carbon compounds comes in many different incarnations with different effective dimensionalities. There are molecules and structures made up entirely of carbon atoms, some of which have generated much excitement over the last twodecades.Oneexample is the Fullerenes that were discovered in 1985 (Kroto et al., 1985). Carbon nanotubes is another example. These are tubes made out of carbon, with a typical diameter of a few nanometers (from which the name stems) and their lengths can be as large as a few mm. Although they were discovered a long time ago,1 the really big interest came with their rediscovery in 1991 (Iijima, 1991). Carbon nanotubes are interesting in many ways. For example, depending on the diameter and how the tube is rolled up it can be both metallic [i.e., the electronic excitation spectrum is gapless and thenanotuberespondstoanelectric field like a 1-dimensional (1D) metal] or semiconducting (i.e., the electronic spectrum has agapthatcanbeusedforphotonicapplicationssincethebandgap it typically in the right range for optics). There has also been experimental demonstrations of transistors made out of carbon nanotubes [for a recent review, see e.g. (McEuen et al., 2002) or (Avouris et al., 2003)]. These have been shown to exhibit good characteristics, but unfortunately there is a 1For a thorough discussion of where the credit is due, see (Monthioux and Kuznetsov, 2006). 6 problem with producing nanotubes with specific properties. Anumberofreviewsandbooks about carbon nanotubes have been produced over the years, seeforexample(Dressehaus et al., 2000). Two other well-known pure carbon materials aregraphiteanddiamond. Graphite in particular is important enough to our topic to deserve a special section to which we soon will turn. 2.1 Graphene Graphite is a layered material made up of weakly coupled planar sheets of carbon atoms that are arranged in a hexagonal lattice structure. It was noted by Wallace already in 1947 that a good starting point for studying graphite is to study the single sheet of graphite, this is the material that we today call graphene. Graphene wasforalongtimeassumed not to exist (or being stable) by itself. But in 2004, in an experimental breakthrough it was shown that atomically thin, large ( 10 µm) two-dimensional crystals of carbon could be ∼ produced by a fairly simple technique (Novoselov et al., 2004). It is also worth to mention that the same method can also be used to obtain two-dimensional crystals of other layered materials (Novoselov et al., 2005b) . In this section, we willdiscusssomeoftheelectronic properties of graphene starting from the lattice structure and a simple tight-binding model for the electronic motion. The discussion here will only serve as a brief review to set the stage for the treatment of the graphene bilayer, for a more thorough treatment we refer the reader to e.g. (Peres et al., 2006b) and references therein. In ideal graphene the carbon atoms are arranged in a planar hexagonal lattice. Each carbon atom has four electrons in the outer shell. Out of thesethreehybridizetoform the directed orbitals that dictates the hexagonal lattice structure and are responsible for the formation of the lattice (Pauling, 1960). The remaining electrons occupy the π-orbitals that are sticking out of the plane. These electrons are relatively free to move around in the plane and are responsible for the low-energy electronic properties of graphene. The hexagonal lattice is not a Bravais lattice. To fit the system into the usual Bloch state picture it is therefore necessary to describe the system in terms of a triangular lattice with 7 two atoms in the unit cell, the two sublattices we call A1andB1.2 Asimplemodelforthe electronic properties of graphene consists of a nearest neighbor tight-binding Hamiltonian considering only the π-orbital on each atom. The real-space lattice structure is depicted in Fig. 2 1. A possible choice of the real-space lattice vectors are: · a a = (3, √3), (2.1a) 1 2 a a = (3, √3). (2.1b) 2 2 − Here a( 1.4 A)˚ denotes the distance between nearest carbon atoms, and thevectorscon- ≈ necting atoms that are nearest neighbors are: a δ = (1, √3), (2.2a) 1 2 a δ = (1, √3), (2.2b) 2 2 − δ = a( 1, 0), (2.2c) 3 − which we take (by convention) to connect the A1-atoms to the B1-atoms. To construct the tight-binding model for the motion of the electrons it is convenient to introduce the operator that creates (annihilates) an electron on the lattice site at position Ri and lattice site αj as c† (c ).3 Here α =(A, B)denotestheatomsublatticeandj (j =1) αj,Ri αj,Ri denotes the plane. The tight-binding Hamiltonian then reads: † t.b. = t c R c R δ +h.c. (2.3) H − A1, i B1, i+ j R j=1,2,3 !i ! " # Here t ( 3eV)istheenergyassociatedwiththehoppingofelectronsbetween neighboring ≈ π orbitals. We now define the Fourier-transformed operators 1 ik·R c R = e i c , (2.4) αj, i √ αj,k N k ! 2The subindex 1 is not really necessary at this point, but it is necessary later on when we need a layer index in the graphene bilayer. 3Because the spin is irrelevant for the independent electron problem that we are currently studying we only consider spin-less electrons in this chapter. 8 a1 a2 δ1 A1 δ3 δ2 B1 Figure 2 1: The real-space lattice structure of graphene is that of a two- · dimensional planar honeycomb lattice. The A1(B1) atoms are indicated by the dark (light) circles. where N is the number of unit cells in the system. Throughout this thesis we will use units such that ! = kB =1unlessspecifiedotherwise.Thetight-bindingHamiltonian in this basis reads = ζ(k)c† c + ζ∗(k)c† c , (2.5) Ht.b. A1,k B1,k B1,k A1,k k !$ % where δ k a√3 ζ(k)= t eik· i = teikxa/2 2cos( y )+e−i3kxa/2 . (2.6) − − 2 i ! $ % Because of the sublattice structure it is often convenient todescribethesystemintermsof † † † aspinor:Ψk = cA1,k,cB1,k ,inwhichcasetheHamiltoniancanbewrittenas " # † 0 ζ(k) t.b. = Ψk Ψk. (2.7) H ∗ k ζ (k)0 ! The reciprocal lattice is shown if Fig. 2 2alongwiththefirstBrillouinzone(BZ)whichis · ahexagon.Alternativelyonecanusethediamond-shapedreciprocal unit cell shown in the 9 figure to label the states. A choice of the reciprocal lattice vectors are 2π b = (1, √3), (2.8a) 1 3a 2π b = (1, √3). (2.8b) 2 3a − The encircled (K and K#)cornersoftheBZofFig.22havethecoordinates · 2π K = (√3, 1), (2.9a) 3√3a 2π K# = (√3, 1). (2.9b) 3√3a − We now expand close to these corners of the BZ according to k = K + q and k = K# + q# with the result that 3at 3at ζ(q) e−iπ/6+iqxa/2(q + iq )= e−iπ/6(q + iq )=v qei(φ−π/6), (2.10a) ≈ 2 x y 2 x y F 3at 3at ! ζ(q#) e−iπ/6+iqxa/2(q# iq# )= e−iπ/6(q# iq# )=v qe−i(φ +π/6),(2.10b) ≈ 2 x − y 2 x − y F −1 where we have defined φ = φ(q)=tan (qy/qx). The Fermi-Dirac velocity is given by vF =3ta/2intermsoftheparametersinthetight-bindingmodel.Theextra phase of π/6 can be absorbed into the phases of the B1wave-functions.Furthermore,becauseitisthe same for both the K point and the K# point, we will not have to worry about it even when we are constructing wave functions that have components in both valleys.4 Thus from now on we never write the π/6. The only time that the direction of φ is important in graphene is for large values of q where the linear approximation to ζ(q)breaksdown.Inbilayer | | graphene the direction of φ is important also for lower energies when one is considering the so-called “trigonal distortion” that we will discuss in Chapter 3. Alternative convention Since two Bloch states that are separated by a reciprocal lattice vector are equivalent it is possible to use other pairs of the corners of BZ to label the states. If we instead choose to 4The two inequivalent corners of the BZ are often referred to as“valleys”. 10 describe the system using the pair of corners in the BZ denotedbystarsinFig.22, namely · the K and K# points defined by 4π K = (0, 1), (2.11a) 3√3a − K# = K, (2.11b) − we find 3at ζ(q) ( q + iq )=iv qeiφ, (2.12a) ≈ 2 − y x F 3at ! ζ(q#) (q# + iq# )=iv qe−iφ . (2.12b) ≈ 2 y x F This convention has the advantage that time reversal symmetry is easier to implement with this particular choice. In particular time reversal takes k k,andthereisinaddition →− the complex conjugate operation of an anti-unitary operator(Sakurai,1994).Therefore time reversal just exchange the K and the K# points in this convention. K’ b1 K K K’ K’ b2 K Figure 2 2: The reciprocal lattice of graphene is a triangular lattice result- · ing in a hexagonal Brillouin zone. The two choices of the K and K# points discussed in the text are shown as the circles and the stars in the corners of the BZ.