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. We also show the diamond as another choice of the reciprocal unit cell and the approximation of covering it with two circles.

In any convention, after the proper rotation of the coordinate system the form of Hamil- tonian close to the K point is that of a 2D Dirac Hamiltonian of massless Fermions:

0 qx + iqy = v Ψ† Ψ . (2.13) HDirac F q   q q q iq 0 ! x − y   11

The resulting spectrum is linear in the momentum E (q)= v q (as measured from the ± ± F K-point). A linear dispersion relation is unconventional in condensed matter systems and has therefore generated a lot of excitement in the community.Thebandsaresketchedin Fig. 2 3andgivesrisetotheso-called“Diracpoint”atzeroenergywhere the two bands · touch. Another interesting feature is that in neutral graphene the chemical potential is sitting exactly at the Dirac point, this results in a Fermi point instead of the more generic case of a Fermi surface.BecausetheFermisurfaceiswhatgivesrisetothestabilitytothe concept of the Fermi liquid (Shankar, 1994), it is expected that there will be deviations from the Fermi liquid paradigm in graphene.

0.1

0.05 #

eV 0 " E -0.05

-0.1

-0.1 -0.05 0 0.05 0.1 k ! "eV#

Figure 2 3: The “Dirac cone” dispersion of the quasiparticles in graphene. ·

Clearly the Hamiltonian in Eq. (2.13) is not valid for all points in the BZ. When the momentum is far away from the corners of the BZ, the lattice will appear in the dispersion and it will no longer be cylindrically symmetric. Nevertheless, to study the low-energy properties near the Dirac point the linear spectrum should beagoodapproximation.But since the band width is finite it is sometimes necessary to introduce a cutoffΛto regularize the theory at high energies. The linear spectrum is then assumed to be valid for momenta such that v q Λ. A simple estimate of Λis obtained by demanding that the number of F & states in the BZ is conserved within the linear approximation. Thus two circles of radius ΛshouldcovertheBZassketchedinFig.22. More explicitly 2πΛ2 =(2π)2/A ,where · u 2 Au = √27a /2istheareaofthereal-spaceunitcell.Inthenaturalunitswhere vF =1 this implies that Λ t π√3 7eV. ≈ ≈ * 12

Like in high-energy physics we often use “natural” units setting ! = vF =1.This imply that energies and frequencies have the same units as momenta and wave numbers. Therefore we often give the momenta in units of eV. For an electron (or hole) pocket with aradiusgivenbyQ in eV, the corresponding electron (hole) density is

Q2 Q 2 4 n = = Q2 7.8 1013 cm−2, (2.14) π t 9πa2 ≈ × + , which includes both the two valleys (K-points) and the two spin projections.

2.2 Graphite

Graphite is a material with a long scientific history that has been studied extensively in the past. Numerous reviews on graphite have been written over the years, see for example (Brandt et al., 1988; Chung, 2002). In this section, we briefly review the theory of the graphite band structure, which is very relevant to the physics of the graphene bilayer.

Band model of graphite

In graphite the graphene planes are stacked in the A-B-configuration with the interplane distance d 3.44 A˚5 (Brandt et al., 1988). The lattice structure is sketched in Figure 2 4. ≈ · Thus there are now two planes and hence four atoms in the unit cell. Actually the real space unit cell of graphite is the same as that of a graphene bilayer since the graphite lattice is made up by stacking graphene bilayers. The Brillouin zone is also shown in Figure 2 4 · along with the high-symmetry point K [H]atk⊥ =0[k⊥ = π/(2d)]. The atoms in the unit cell are now labeled by A1, B1, A2andB2; where 1 and 2 denotes the layer and A and B the sublattice in each layer. The convention that we use to label the atoms is shown in Figure 2 4[seealsoFigure31]. Therefore the A and B atoms in each plane are now · · inequivalent, in particular the A atoms have a nearest neighbor directly above and below it in the neighboring layer whereas the B atoms are sitting in the middle of the hexagons

5In graphite with imperfect stacking (turbostratic graphite) the interplane spacing can vary down to d ≈ 3.36A.˚ 13

H H’

B2 K A2 K’

H H’ A1 B1 (a) (b)

Figure 2 4: (a) The lattice structure of graphite. (b) The hexagonal Bril- · louin zone of graphite, including the positions of the K and the H points.

of the neighboring layers.6 Using now a four-component spinor defined by

† † † † † Ψk = cA1,k,cB1,k,cA2,k,cB2,k , (2.15) $ % the Hamiltonian can be written generally as

† kin = Ψq 0(q)Ψq. (2.16) H q H !

6The advantage of this notation is that one can talk collectively about the A (B)atomsthatareequivalent in their physical properties such as the weight of the wave functions and the distribution of the density of states etc. This notation was used in the early work of e.g., (McClure, 1957; Slonczewski and Weiss, 1958). Many authors use a notation similar to A1 → A, B1 → B, A2 → B,andB2 → A.Inthisnotationthe e e relative orientation within the planes of the A (A)andB (B)atomsarethesame,butfortheotherphysical e e properties the equivalent atoms are instead A (B)andB (A). Since the other physical properties are often e e more relevant for the physics than the relative orientation within the planes we choose to use the more “natural” convention. 14

Here (p)isa4 4-matrix that close to the K-point can be parametrized as H0 ×

∆+γ Γ2/2 v peiφ γ Γ v v pΓe−iφ 5 F 1 − 4 F  −iφ 2 −iφ iφ  vFpe γ2Γ /2 v4vFpΓe v3vFpΓe 0(p)= −  , (2.17) H  iφ 2 −iφ   γ1Γ v4vFpΓe ∆+γ5Γ /2 vFpe   −   iφ −iφ iφ 2   v4vFpΓe v3vFpΓe vFpe γ2Γ /2  −    where v = γ /γ and v = γ /γ .Microscopically,Γ 2cos(k d)canbeobtained 3 3 0 4 4 0 ≡ ⊥ from a tight-binding dispersion in the direction perpendicular to the layers. The angle φ φ(p)isobtainedasinEq.(2.10)orEq.(2.12).Whetheroneshoulduseeiφ or e−iφ ≡ is dictated by the relative orientation of the appropriate pair of atoms projected on to the

±iφ x-y-plane as will be discussed in Chapter 3. The entries goinglikevFpe comes from the in-plane graphene dispersion. Hopping terms correspondingtonearestneighboringplanes are different depending on the pair of atoms in question: γ (A A), γ (B B), γ 1 ↔ 3 ↔ 4 (A B). γ (γ )denotesahoppingbetweennextnearestneighboringplanesfor the B ↔ 2 5 (A)atoms.Finally∆denotesthedifferenceinon-siteenergiesof the A and B atoms due to their different crystal environments. The matrix

1000   0 e−iφ(p) 00 1(p)=  , (2.18) M   0010    iφ(p) 000e      can be used to perform the gauge transformation = † (φ) (φ)thatmovesallof H1 M1 H0M1 the e±iφ phase-factors (and hence the information of the orientationofthelattice)tothe

γ3-term:

∆+γ Γ2/2 v pγΓ v v pΓ 5 F 1 − 4 F  2 3iφ vFpγ2Γ /2 v4vFpΓ v3vFpΓe 1(p)= −  . (2.19) H  2   γ1Γ v4vFpΓ∆+γ5Γ /2 vFp   −   −3iφ 2   v4vFpΓ v3vFpΓe vFpγ2Γ /2   −    15

This form has the advantage that it makes it manifest that the γ3-term is responsible for the “trigonal distortion” of the bands and actually breaks the cylindrical symmetry of the bands. To make contact with the graphite literature one further performs two unitary transformations. First, symmetric/antisymmetric combinations of the A-atoms leading to bonding/antibonding bands are generated by the matrix:

1/√20 1/√20 −   01 0 0 2 =   . (2.20) M  √ √  1/ 20 1/ 20      00 0 1     We further permute the bands with the matrix

10 0 0   00 10 3 =  −  , (2.21) M   01 0 0     00 0 1  −    so that the transformed Hamiltonian = † † becomes H2 M3 M2 H1 M2 M3

∆+γ Γ2/2+γ Γ0(v Γ 1)v p/√2(v Γ 1)v p/√2 5 1 4 − F 4 − F  2  0∆+γ5Γ /2 γ1Γ(1+v4Γ)vFp/√2 (1 + v4Γ)vFp/√2 2 =  − −  . H  √ √ 2 3iφ   (v4Γ 1)vFp/ 2(1+v4Γ)vFp/ 2 γ2Γ /2 v3vFpΓe   −   −3iφ 2   (v4Γ 1)vFp/√2 (1 + v4Γ)vFp/√2 v3vFpΓe γ2Γ /2   − −   (2.22) Except for some minor notational differences and the overall gauge transformation by (2.18) this is the Slonczewski-Weiss-McClure model for graphite (McClure, 1957; Slonczewski and 16

Weiss, 1958), which is usually written as

∗ E1 0 H13 H13  ∗  0 E2 H23 H23 SWMC =  −  , (2.23) H  ∗ ∗  H13 H23 E3 H33     ∗  H13 H23 H33 E3   −    where

1 E =∆+γ Γ+ γ Γ2, (2.24a) 1 1 2 5 1 E =∆ γ Γ+ γ Γ2, (2.24b) 2 − 1 2 5 1 E = γ Γ2, (2.24c) 3 2 2 1 iα H13 = ( γ0 + γ4Γ)e ζ, (2.24d) √2 − 1 iα H23 = (γ0 + γ4Γ)e ζ, (2.24e) √2 iα H33 = γ3Γe ζ. (2.24f)

Typical values of the parameters from the graphite literature are shown in Table 2.2, see also (Partoens and Peeters, 2006) for a discussion on the connection between the tight- binding parameters and those of the Slonczewski-Weiss-McClure model. The accepted

γ0 γ1 γ2 γ3 γ4 γ5 γ6 =∆ &F 3.16 0.39 -0.02 0.315 0.044 0.038 0.008 -.024 3.12 0.377 -0.020 0.29 0.120 0.0125 0.004 -.0206

Table 2.1: Values of the Slonczewski-Weiss-McClure parameters for the band structure of graphite. Upper row from (Brandt et al., 1988) and lower row from (Chung, 2002).

parameters from the graphite literature results in electrons near the K-point [k⊥ =0]and holes near the H-point [k = π/(2d)] in the BZ as sketched in Figure 2 4. These electron ⊥ · and hole pockets are chiefly generated by the coupling γ2 that in the tight-binding model corresponds to a hopping between the B-atoms of next-nearest planes. Note that this process involves a hopping of a distance as large as 7 A.˚ ∼ 17

Finally, it is interesting to note that at the H-point in the BZ, Γ= 0, and therefore the two planes “decouple” at this point. Furthermore, if one neglects ∆the spectrum is that of massless Dirac fermions just like in the case of graphene. Note that in graphite A and B atoms are different however, and that the term parametrized by∆willgenerallyopenup agapinthespectrumleadingtomassiveDiracfermionsattheH-point. Since the value of ∆intheliteratureisquitesmall,thealmostlinearmasslessbehaviorshouldbeobserved by experimental probes that are not sensitive to these small energy scales. The values of the parameters used in the graphite literature are consistent with a large number of experiments. The most accurate ones are various magneto-transport measure- ments discussed in (Brandt et al., 1988). More recently, ARPES was used to directly visualize the dispersion of massless Dirac quasi-particlesneartheH-point and massive quasi-particles near the K-point in the BZ (Zhou et al., 2006a; Zhou et al., 2006b). The band structure of graphite has been calculated and recalculated many times over the years, a recent reference is (Charlier et al., 1991). It isalsoworthtomentionthat because of the (relatively) large contribution of the non-local van der Waals interaction to the interaction between the layers in graphite, the usual local density approximation or semilocal density approximation schemes are offby an order ofmagnitudewhenthebinding energy of the planes are calculated and compared with experiments. For a discussion of this topic and a possible remedy see (Rydberg et al., 2003).