Electrons in bilayer graphene - Physics at Lancaster University

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E. McCann et al. / Solid State Communications 143 (2007) 110–115 111Fig. 1. (a) Schematic of the bilayer lattice containing four sites in the unit cell:A1 (white circles) and B1 (grey) in the bottom layer, and A2 (grey) and B2(black) in the top layer. (b) Schematic of the low energy bands near the Kpoint obtained by taking into account intralayer hopping with velocity v, B1A2interlayer coupling γ 1 , A1B2 interlayer coupling γ 3 [with v 3 /v = 0.1] andzero layer asymmetry .employ the tight-binding model of graphite [20] by adaptingthe Slonczewski–Weiss–McClure parameterization [21,22] ofrelevant couplings in order to model bilayer graphene. In-planehopping is parameterized by coupling γ A1B1 = γ A2B2 ≡ γ 0and it leads to the in-plane velocity v = ( √ 3/2)aγ 0 /¯h wherea is the lattice constant. In addition, we take into account thestrongest inter-layer coupling, γ A2B1 ≡ γ 1 , between pairs ofA2–B1 orbitals that lie directly below and above each other.Such strong coupling produces dimers from these pairs ofA2–B1 orbitals, leading to the formation of high energy bands[7]. We also include weaker A1–B2 coupling γ A1B2 ≡ γ 3that leads to an effective velocity v 3 = ( √ 3/2)aγ 3 /¯h wherev 3 ≪ v. Here, we write the Hamiltonian [7] near the centresof the valleys in a basis corresponding to wave functionsΨ = (ψ A1 , ψ B2 , ψ A2 , ψ B1 ) in the valley K [23] and of Ψ =(ψ B2 , ψ A1 , ψ B1 , ψ A2 ) in the valley ˜K :⎛− 1 ⎞2 v 3π 0 vπ Ďv 3 π Ď 1H = ξ2 vπ 0⎜ 0 vπ Ď 1, (1)⎝2 ξγ 1 ⎟vπ 0 ξγ 1 − 1 ⎠2 where π = p x + ip y , π Ď = p x − ip y , p = (p x , p y ) ≡p(cos φ, sin φ) is the momentum measured with respect to theK point, ξ = +1(−1) labels valley K ( ˜K ). The Hamiltoniantakes into account asymmetry = ɛ 2 − ɛ 1 between on-siteenergies in the two layers, ɛ 2 = 1 2 , ɛ 1 = − 1 2 .At zero magnetic field, the Hamiltonian H has four valleydegeneratebands [7], ɛ ± (α) (p), α = 1, 2, withɛ ± (α)2 = γ ( )12 2 + 24 + v 2 + v2 3p 2 + (−1) α √ Γ2Γ = 1 (γ 1 2 4− v2 3 p2) 2 [ + v 2 p 2 γ1 2 + 2 + v3 2 p2]+ 2ξγ 1 v 3 v 2 p 3 cos 3φ. (2)They are plotted in Fig. 1(b) for = 0 and v 3 /v = 0.1. Thedispersion ɛ ± (2) describes two bands with energies ɛ(2) + ≥ γ 1 and≤ γ 1: they do not touch at the K point. These bands are theɛ (2)−result of strong interlayer coupling γ A2B1 ≡ γ 1 which forms‘dimers’ from pairs of A2–B1 orbitals that lie directly belowand above each other [7].The dispersion ɛ ± (1) (p) describes low energy bands thattouch at the K point in the absence of layer asymmetry = 0.In the intermediate energy range, 1 4 γ 1(v 3 /v) 2 , || < |ɛ 1 | < γ 1 ,it can be approximated [7] withɛ (1)± ≈ ±1 2 γ 1[√1 + 4v 2 p 2 /γ 2 1 − 1 ]. (3)This interpolates between a linear spectrum |ɛ ± (1) | ≈ vp at highmomenta and a quadratic spectrum |ɛ ± (1) | ≈ p2 /2m, wherem = γ 1 /2v 2 . Such a crossover happens at p ≈ γ 1 /2v, whichcorresponds to the carrier density n ∗ ≈ γ1 2/(4π ¯h 2 v 2 ). Thisis lower than the density at which the higher energy bandɛ (2) becomes occupied n (2) ≈ 2γ1 2/(π ¯h 2 v 2 ) ≈ 8n ∗ . Usingexperimental graphite values [22] gives n ∗ ≈ 4.36×10 12 cm −2and n (2) ≈ 3.49 × 10 13 cm −2 . The estimated effective mass mis light: m = γ 1 /2v 2 ≈ 0.054m e .3. Optical absorption of bilayer grapheneThe electromagnetic field absorption in graphene has beenstudied in a number of publications [24,18,19,25,26]. We showthat the bilayer absorption coefficient can be described asg 2 = (2πe 2 /¯hc) f 2 (ω), (4)and it is qualitatively different from the featureless monolayerabsorption coefficient g 1 = πe 2 /¯hc, since g 2 reflects thepresence and dispersion of two pairs of bands in bilayergraphene [18,19] as described in Section 2 and Fig. 1(b).We characterize the polarization of the EM field E ω =lEe −iωt by denoting the vector l ⊕ = (l x − il y )/ √ 2 tocorrespond to right-hand circular polarization and the vectorl ⊖ = (l x + il y )/ √ 2 to be the left-handed circularly polarizedlight. In a 2D electron gas with conductivity σ (ω) ≪c/2π, absorption of an EM field arriving along the directionantiparallel to a magnetic field can be characterized by the ratioof the Joule heating and the energy flux S = cE×H/4π = −Sl ztransported by the EM field so that g ≡ E i E ∗ j σ i j(ω)/S. Usingthe Keldysh technique, we expressg = 8e2cω R ∫FdɛN̂Tr{}ˆυ i l i Ĝ R (ɛ) ˆυ j l ∗ j Ĝ A (ɛ + ω) ,where ˆv = ∂ p H is the velocity operator, ̂Tr includes thesummation both over the sublattice indices “tr” and over singleparticleorbital states, N is the normalization area of the sampleand F = n F (ɛ) − n F (ɛ + ω) takes into account the occupancyof the initial and final states and includes spin and valleydegeneracies.We write the retarded and advanced Green’s functions ofelectrons in the bilayer as Ĝ R/A (p, ɛ) = [ɛ±i¯h/(2τ)−H(p)] −1and the trace operation as ̂Tr = ∫ d 2 pN tr. Neglecting(2π ¯h) 2the momentum transfer from light (since ɛ/c ∼ 3 × 10 −3 ),reproduces [24,25] the constant absorption coefficient g ‖ 1 =
112 E. McCann et al. / Solid State Communications 143 (2007) 110–115between the A1–B2 sites:(Ĥ 2 = − 1 0(π Ď) 2)+ ĥ w + ĥ a ; (6)2m π 2 0( ) 0 πĥ w = ξv 3π Ď , where π = p x + ip y ;0[ ( ) (1 1 0ĥ a = −ξ− v2 π Ď ) ] π 02 0 −1 0 −ππ Ď .γ 2 1Fig. 2. Absorption coefficient of bilayer and monolayer graphene in the opticalrange of frequencies. The insets illustrate the quasiparticle dispersion branchesin the vicinity of ɛ F and possible optical transitions.πe 2 ¯hc ( f 1 = 1 2) in monolayer graphene. Taking into accountall four bands in the bilayer we arrive at the expression forthe absorption coefficient for light polarized in the plane of thegraphene sheet:g ‖ 2 = 2πe2¯hc f 2(Ω), Ω ≡ ¯hωγ 1> 2|ɛ F|γ 1, (5)f 2 = Ω + 2 θ(Ω − 1)+2(Ω + 1) Ω 2 +(Ω − 2)θ(Ω − 2),2(Ω − 1)where θ(x < 0) = 0 and θ(x > 0) = 1 which agrees withthe calculation by Nilsson et al. [18] in the clean limit andT = 0. The frequency dependence [27] of the bilayer opticalabsorption is illustrated in Fig. 2 in comparison to that in amonolayer. It is the electron–hole excitation between the lowenergy band ɛ ± (1) and the split band ɛ(2) ± which provides thestructure in the vicinity of ¯hω = γ 1 (γ 1 ≈ 0.4 eV [22]). Athigh photon energies, ¯hω ≫ γ 1 , the frequency dependencesaturates at f = 1. The absorption coefficient for the left- andright-handed light are the same over the whole spectral interval,so Eq. (5) is also applicable to light linearly polarized in thegraphene plane.4. Effective low energy HamiltonianTo describe the transport properties of bilayer graphene, itis convenient to use a low energy Hamiltonian that describeseffective hopping between the non-dimer sites, A1–B2, i.e.those that do not lie directly below or above each other and arenot strongly coupled by γ 1 . This two component Hamiltonianwas derived in [7] using Green’s functions. Alternatively(and equivalently), one can view the eigenvalue equation ofthe four component Hamiltonian equation (1) as producingfour simultaneous equations for components ψ A1 , ψ B2 , ψ A2 ,ψ B1 . Eliminating the dimer state components ψ A2 , ψ B1 bysubstitution, and treating γ 1 as a large energy, gives thetwo component Hamiltonian [7] describing effective hoppingThe effective Hamiltonian Ĥ 2 is applicable within the energyrange |ɛ| < 1 4 γ 1. In the valley K , ξ = +1, we determineΨ ξ=+1 = (ψ A1 , ψ B2 ), whereas in the valley ˜K , ξ = −1,the order of components is reversed, Ψ ξ=−1 = (ψ B2 , ψ A1 ).The Hamiltonian Ĥ 2 describes two possible ways of A1 ⇋ B2hopping. The first term takes into account A1 ⇋ B2 hoppingvia the A2B1 dimer state. Consider A1 to B2 hopping asillustrated with the thick solid line in Fig. 1(a). It includes threehops between sites: an intralayer hop from A1 to B1, followedby an interlayer transition via the dimer state B1A2, followedby an intralayer hop from A2 to B2. Since the two intralayerhops are both A to B, the first term in the Hamiltonian containsπ 2 or (π Ď ) 2 on the off-diagonal with the mass m = γ 1 /2v 2reflecting the energetic cost γ 1 of a transition via the dimerstate. This term in Ĥ 2 yields a parabolic spectrum ɛ = ±p 2 /2mwith m = γ 1 /2v 2 . It has been noticed [7] that quasiparticlesdescribed by it are chiral: their plane wave states are eigenstatesof an operator σ n 2 with σ n 2 = 1 for electrons in the conductionband and σ n 2 = −1 for the valence band, where n 2 (p) =(cos(2φ), sin(2φ)) for p = (p cos φ, p sin φ). Quasiparticlesdescribed by this term acquire a Berry phase 2π upon anadiabatic propagation along a closed orbit, thus charge carriersin a bilayer are Berry phase 2π quasiparticles, in contrast toBerry phase π particles in a monolayer [5].The second term ĥ w in the Hamiltonian equation (6)describes weak direct A1B2 coupling, γ A1B2 ≡ γ 3 ≪ γ 1 .This coupling γ A1B2 ≡ γ 3 leads to the effective velocityv 3 = ( √ 3/2)aγ 3 /¯h where v 3 ≪ v, Eq. (2). In a similar wayto bulk graphite [21,28], the effect of coupling γ 3 is to producetrigonal warping, which deforms the isoenergetic lines alongthe directions φ = φ 0 , as shown in Fig. 3(a). For the valley K ,φ 0 = 0, 2 3 π and 4 3 π, whereas for ˜K , φ 0 = π, 1 3 π and 5 3 π.The effective low energy Hamiltonian equation (6) yields thefollowing energy for = 0,ɛ ± (1)2 ≈ (v 3 p) 2 − ξv 3 p 3 ( p2) 2m cos (3φ) + , (7)2mwhich agrees with Eq. (2) in the low energy limit |ɛ| ≪ γ 1 . Atvery low energies |ɛ| < ɛ L = 1 4 γ 1(v 3 /v) 2 ≈ 1 meV, the effectof trigonal warping is dramatic. It leads to a Lifshitz transition:the isoenergetic line is broken into four pockets, which canbe referred to as one “central” and three “leg” parts [28,7,29].The central part and leg parts have minimum |ɛ| = 1 2|| atp = 0 and at |p| = γ 1 v 3 /v 2 , angle φ 0 , respectively. Forv 3 ≪ v, we find [7,22] that the separation of the 2D Fermiline into four pockets would take place for very small carrier
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