Band model of the graphene bilayer

18Chapter 3Band model of the graphene bilayerMany of the special properties of the graphene bilayer have their origins in its latticestructure that results in the peculiar band structure that we willdiscussindetailinthischapter. First we repeat the observation from Chapter 2 that the graphene bilayer in theA-B stacking 1 is just the unit cell of graphite that we depict in Fig. 3·1. Therefore, if theB2A2A1B1Figure 3·1: Lattice structure of the graphene bilayer. The A (B) sublatticesare indicated by the darker (lighter) spheres and the planes are labeledby 1 and 2.two planes are equivalent the symmetry analysis of graphite is also valid for the graphenebilayer. Thus we can directly use the spinor of Eq. (2.15) and the Hamiltonian in Eq. (2.17)1 Also known as Bernal stacking. Other stackings are also possible but the A-B stacking seems to be themost energetically favorable.

19with Γ = 1 and γ 2 = γ 5 =0,leadingto:⎛⎞∆ v F pe iφ t ⊥ −v 4 v F pe −iφv F pe −iφ 0 −v 4 v F pe −iφ v 3 v F pe iφH 0 (p) =. (3.1)⎜ t ⊥ −v 4 v F pe iφ ∆ v F pe −iφ⎟⎝⎠−v 4 v F pe iφ v 3 v F pe −iφ v F pe iφ 0Another way of arriving at Eq. (3.1) is to use the tight-binding model directly in the bilayer.Since the system is two-dimensional only the relative position of the atoms projected onto the x-y-plane enters into the model. The projected position of the different atoms areshown in Fig. 3·2. Since the A atoms are sitting right on top of each other in the lattice, theB2B1A1A2Figure 3·2: The real space lattice structure of the graphene bilayer projectedonto the x-y plane showing the relative positions of the differentsublattices.hopping term between the A1 andA2 atomsarelocalinrealspaceandhenceaconstantthat we denote by t ⊥ in momentum space. Referring back to Section 2.1 we note that thehopping B1 → A1 [A1 → B1] gives rise to the factor ζ(k) [ζ ∗ (k)], with ζ(k) definedinEq. (2.6). Since the geometrical role of the A and B atoms are interchanged between plane1andplane2weimmediatelyfindthatinFourierspacethehopping A2 → B2 [B2 → A2]gives rise to the factor ζ(k) [ζ ∗ (k)]. Furthermore, the direction in the hopping B1 → B2

20(projected on to the x-y plane) is opposite to that of hopping B1 → A1. Thus we associateafactorv 3 ζ ∗ (k) tothehoppingB1→ B2, where the factor v 3 = γ 3 /γ 0 is needed becausethe hopping energy is γ 3 instead of γ 0 = t. Similarly, the direction of hopping B1 → A2(projected on to the x-y plane) is the same as B1 → A1 and therefore the term −v 4 ζ(k)goes with the hopping B1 → A2. 2 Continuing to fill in all the entries of the matrix the fulltight-binding Hamiltonian in the graphene bilayer becomes:⎛⎞∆ ζ(k) t ⊥ −v 4 ζ ∗ (k)ζ ∗ (k) 0 −v 4 ζ ∗ (k) v 3 ζ(k)H t.b. (k) =, (3.2)⎜ t ⊥ −v 4 ζ(k) ∆ ζ ∗ (k)⎟⎝⎠−v 4 ζ(k) v 3 ζ ∗ (k) ζ(k) 0where ∆ parametrizes the difference in energy between the A and B atoms. We come backto Eq. (3.1) upon expanding this expression close to the K point. The typical behavior ofthe bands obtained from Eq. (3.1) is shown in Fig. 3·3. Two of the bands (labeled 3 and4inthefigure)aremovedawayfromtheDiracpointbyanenergythat is given by theinterplane hopping term t ⊥ .E31tkFigure 3·3: Band dispersions near the K-points in the bilayer. Bands arelabeled by the numbers 1 − 4asinthetext.242 The minus sign in front of v 4 follows from the conventional definition of γ 4 (Partoens and Peeters, 2006).

213.1 Simplified modelAsimplifiedmodelthatonlyconsiderstheinterplanehoppingtermbetweenA atoms employsa matrix of the form⎛⎞0 v F pe iφ(p) t ⊥ 0v F pe −iφ(p) 0 0 0H 0 (p) =. (3.3)⎜ t ⊥ 0 0 v F pe −iφ(p)⎟⎝⎠0 0 v F pe iφ(p) 0From now on in this section, we use units such that v F =1asdiscussedinSection2.1. ThisHamiltonian has the advantage that it allows for relatively simple calculations. Some of thefine details of the physics might not be accurate but it will work as a minimal model andcapture most of the important physics. It is important to know thequalitativenatureoftheterms that are neglected in this approximation, this will be discussed later in this Chapter.It is also an interesting toy model as it allows for (approximately) “chiral” particles withmass (i.e., a parabolic spectrum) at low energies as we will discuss in the next Section. Foralargepartofthisthesiswewillstudythepropertiesofthesystem with this simplifiedHamiltonian.3.2 Approximate effective two-band modelsThere are two main reasons for constructing approximate two-band models:First, onphysical grounds the high-energy bands (far away from the Dirac point) should not be veryimportant for the low-energy properties of the system. Second, it is often easier to workwith 2 × 2matricesinsteadof4× 4matrices. Inthissection,wederivethelow-energyeffective model by doing degenerate second order perturbation theory. The quality of theexpansion is good as long as v F p ≪ t ⊥ ≈ 0.35 eV. We first present the general expressionfor the second-order 2 × 2effectiveHamiltonian,thereaftervarioussimplifiedformswillbeintroduced. Analyses similar to the one presented here were presented in (McCann andFal’ko, 2006) and (Nilsson et al., 2006c).

22Derivation of effective modelsFirst we decompose Eq. (3.1) into a high-energy part K 0 and two low-energy parts K 1 andK 2 according to H 0 = K 0 + K 1 + K 2 ,with⎛⎞∆ 0 t ⊥ 00 0 0 0K 0 =, (3.4)⎜t ⊥ 0 ∆ 0⎟⎝⎠0 0 0 0⎛⎞0 pe iφ 0 −v 4 pe −iφpe −iφ 0 −v 4 pe −iφ 0K 1 = v F , (3.5)⎜ 0 −v 4 pe iφ 0 pe −iφ⎟⎝⎠−v 4 pe iφ 0 pe iφ 0⎛⎞0 0 0 00 0 0 v 3 pe iφK 2 = v F . (3.6)⎜0 0 0 0⎟⎝⎠0 v 3 pe −iφ 0 0The usual manipulations (Sakurai, 1994) then given the Hamiltonian matrix in the lowenergy subspace as K low = K 2 −K † 1 P 1(1/K 0 )P 1 K 1 ,whereP 1 is the projection out of thelow-energy subspace (explicitly P 1 =Diag[1, 0, 1, 0]). The result is:⎛⎞0 0 0 0K low =v2 F p20 2t ⊥ v 4 +∆(1+v4 2t 2 ⊥ − ) 0 −[ t ⊥ (1 + v4 2)+2v 4∆ ] e −2iφ+ K ∆2 ⎜0 0 0 02 ,⎟⎝0 − [ t ⊥ (1 + v4 2)+2v 4∆ ] ⎠e 2iφ 0 2t ⊥ v 4 +∆(1+v4 2) (3.7)

23and because this is really just a 2 × 2matrixinthelow-energysubspacewecanwriteitas:H eff =⎛ ⎞{ v2 F p2 [2t⊥t 2 ⊥ − v 4 +∆(1+v 2∆2 4 )] ⎝ 1 0 ⎠0 1⎛− [ t ⊥ (1 + v4)+2v 2 4 ∆ ] ⎝ 0The corresponding eigenvalues are:E eff,± ≈v2 F p2 [t 2 ⊥ − 2t⊥ v 4 +∆(1+v4) 2 ]∆2√{ v2± (v 3 v F p) 2 + Fp 2[ t ⊥ (1 + v4 2)+2v 4∆ ]e−i2φe i2φ 0⎞ ⎛ ⎞}⎠ + v 3 v F p ⎝ 0 eiφ⎠ . (3.8)e −iφ 0} 2 2v 3 v −F 3 p3[ t ⊥ (1 + v4 2)+2v 4∆ ]t 2 ⊥ − ∆2 t 2 ⊥ − ∆2cos(3φ).(3.9)This expression shows that v 4 and ∆ weakly breaks the particle-hole symmetry of thesystem and that v 3 is responsible for breaking the cylindrical symmetry and the so-called“trigonal warping” of the energy bands.A simplified model that takes only the termsinvolving t ⊥ and the trigonal warping v 3 into account isH eff = − v2 F p2t ⊥⎛⎝ 0e−i2φe i2φ 0⎞ ⎛ ⎞⎠ + v 3 v F p ⎝ 0 eiφ⎠ . (3.10)e −iφ 0An even simpler model which neglects both the electron-hole asymmetry and the trigonalwarping is:H eff = − v2 F p2t ⊥⎛⎝ 0e−i2φe i2φ 0⎞⎠ . (3.11)This form is interesting since it gives rise to massive “chiral” quasi-particles (McCann andFal’ko, 2006). Here “chirality” means that there exist an operator Ĉ defined by⎛Ĉ≡−⎝ 0e i2φ 0e−i2φ⎞⎠ , (3.12)

24that has the eigenvalues ±1 andcommuteswiththeHamiltonian. Thisimpliesthatthereis another quantum number (in addition to the energy) with which one can label the statesof the system.3.3 Band structure comparisonsAcomparisonofthebandsobtainedfromthesimpleHamiltonian in Eq. (3.3) and thoseobtained from the full Hamiltonian in Eq. (3.1) on a large scale is shown in Fig. 3·4. Thisfigure clearly shows that the gross features of the bands are correctly captured in the simpleminimal model.0.40.40.20.2EeV0-0.2EeV0-0.2-0.4-0.4-0.2 -0.1 0 0.1 0.2k eV(a)-0.2 -0.1 0 0.1 0.2k eV(b)Figure 3·4: Comparison between the bands obtained from the full Hamiltonianin Eq. (3.1) and those of the simple minimal model in Eq. (3.3)alongthe direction φ =0intheBZ.(a)FullHamiltonian. (b)MinimalmodelHamiltonian.That the low-energy effective theory in Eq. (3.8) and Eq. (3.9) is accurate for lowenergies is shown in Fig. 3·5. But as one moves away from the Dirac point deviations fromthe real spectrum is clearly visible.We show, in Figure 3·6 andinFigure3·7, a comparison of the bands obtained from thesimple minimal Hamiltonian in Eq. (3.3) and those of the full Hamiltonian in Eq. (3.1) forlow energies. This shows that especially γ 3 ,whichgivesrisetothe“trigonaldistortion”,significantly changes the behavior at the lowest energies. A normal Dirac cone is found atp =0atthelowestenergies,butnowtheFermi-Diracvelocityis v 3 v F . There is also the

25EeV0.040.030.020.010-0.1 -0.05 0 0.05 0.1k eV(a)EeV0.0080.0060.0040.0020-0.04 -0.02 0 0.02 0.04k eV(b)Figure 3·5: Comparison between the bands obtained from the full Hamiltonianin Eq. (3.1) and those of the effective model in Eq. (3.8) alongdifferentdirections in the BZ. Solid line – [φ =0,Eq.(3.1)],Dash-dottedline–[φ =0,Eq.(3.8)],Dashedline–[φ = π/6, Eq. (3.1)], Dotted line –[φ = π/6, Eq. (3.8)]. (a) Larger energy scale. (b) Zoom in at low energies.extra band crossings in the directions φ =0andφ = ±2π/3 whichgivesrisetoellipticalDirac cone 3 away from the point p =0. Thisstructureispresentatasmallenergyscaleofthe order of ∼ 1meV, therefore experimental probes that are sensitive to this energy scaleare necessary to be able to detect these features. Moreover, as we will see in Chapter 5,different forms of disorder can easily generate energies of this scale or larger in the realexperimental samples, thus this structure might be hard to detect experimentally.Astudyoffew-layergraphenesystems(includinggraphenebilayers) with plots similarto those in this chapter can be found in (Partoens and Peeters, 2006). Afirst-principlesstudy with both similar scope and results also exists (Latil and Henrard, 2006).3 This means that there are two inequivalent perpendicular axes (1 and 2) in the cone, with two differentvalues of the Fermi velocity v F1 ≠ v F2. Thespectrumisthen± p v 2 F1 p2 1 + v2 F2 p2 2 .

26EeV0.020.010-0.01-0.02-0.1 -0.05 0 0.05 0.1k eV(a)EeV0.0030.0020.0010-0.001-0.002-0.04 -0.02 0 0.02 0.04k eV(b)Figure 3·6: Comparison between the low-energy bands obtained from thesimple model in Eq. (3.3): solid lines; and those of the full Hamiltonian inEq. (3.1) along three different directions in the BZ: φ =0(dashed),φ = π/9(dashed-dotted), and φ =2π/9 (dotted). (a)Largerenergyscale. (b)Zoomin at low energies.k y meV6040200-20-40-60k y meV6040200-20-40-60-60 -40 -20 0 20 40 60k x meV(a)-60 -40 -20 0 20 40 60k x meV(b)Figure 3·7: Comparison between the bands obtained from the full Hamiltonianand those of the simple minimal model at low energies using a contourplot: (a) Full Hamiltonian (b) Simplified Hamiltonian. The contours are atthe energies: 0.1, 0.3, 0.5, 1, 2, 3, 4meV(black lines, orderedfrom thesolidline to the more dotted), 5, 10, 20 meV (gray lines, same ordering).