Electrons in bilayer graphene - Physics at Lancaster University

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E. McCann et al. / Solid State Communications 143 (2007) 110–115 113Fig. 3. (a) Schematic of the Fermi line at low energy in the valley K, ξ = 1, fordifferent values of the Fermi energy. Note that the asymmetry of the Fermi lineat the other valley, ξ = −1, is inverted. (b) The low energy bands plotted alongthe line p y = 0. They are obtained by taking into account intralayer hoppingwith velocity v, B1A2 interlayer coupling γ 1 , A1B2 interlayer coupling γ 3[with v 3 /v = 0.1] and zero layer asymmetry . Dashed lines show the bandsobtained by neglecting γ 3 [i.e. with v 3 /v = 0].densities n < n L ∼ (v 3 /v) 2 n ∗ ∼ 1 × 10 11 cm −2 . In thisestimation of n L , the constant of proportionality is of the order 1as determined by the strongly warped shape of the Fermi line atthe Lifshitz transition. For n < n L , the central part of the Fermisurface is approximately circular with area A c ≈ πɛ 2 /(¯hv 3 ) 2 ,and each leg part is elliptical with area A l ≈3 1 A c. The lowenergy part of the band structure is plotted in Fig. 3(b) alongthe line p y = 0. Taking the line p y = 0, φ = 0, at the firstvalley ξ = 1 gives ɛ ± (1) ≈ ±|v 3 p − p 2 /(2m)|. It shows that,at zero energy, the leg pocket of the Fermi surface developsat p = 2mv 3 = γ 1 v 3 /v 2 , Fig. 3(a), and that the overlapbetween the conduction and valence bands, Fig. 3(b), is givenby 2ɛ L ≈ (γ 1 /2)(v 3 /v) 2 ≈ 2 meV [15] using γ 1 ≈ 0.4 eV andv 3 /v ≈ 0.1.5. Stability of the bilayer against ferroelectric ordering anda voltage-controlled gap in the spectrumThe parameter in the Hamiltonian equation (1) takesinto account a possibly-externally-controlled asymmetry =ɛ 2 − ɛ 1 between on-site energies in the two layers, ɛ 2 =2 1 ,ɛ 1 = −2 1 . The electronic bands near the K point, Eq. (2), areshown in Fig. 4(a) for a large value of the layer asymmetry .For simplicity, we neglect A1B2 interlayer coupling γ 3 :√ɛ ± (α)2 ≈ γ 12 2 + 24 + v2 p 2 + (−1) α γ14 4 + v2 p 2 ( γ1 2 + 2) .The energies of the bands exactly at the K point are |ɛ ± (2)√(p =0)| = γ1 2 + 2 /4 and |ɛ ± (1) (p = 0)| = ||/2: the lowenergy bands, ɛ ± (1) , are split by the layer asymmetry at theK point [30].In an asymmetrical bilayer, the electronic densities on theindividual layers, n 1 and n 2 , are given by an integral withrespect to momentum p = ¯h|k| over the circularly symmetricFermi surface, taking into account the relative weight of theFig. 4. (a) Schematic of the electronic bands near the K point in the presenceof finite layer asymmetry (for illustrative purposes a very large asymmetry = γ 1 is used) obtained by taking into account intralayer hopping withvelocity v and B1A2 interlayer coupling γ 1 , but neglecting A1B2 interlayercoupling γ 3 . Dotted lines show the bands for zero asymmetry = 0. (b)Dependence of the function f Λ (x), Eq. (14), describing the density dependenceof the layer asymmetry on the argument x for different values of the screeningparameter Λ.wave functions:n 1(2) = 2 ∫π ¯h 2(p dp |ψ A1(2) (p) | 2 + |ψ B1(2) (p) | 2) , (8)where we have included a factor of four to take into accountspin and valley degeneracy. By determining the wavefunctionamplitudes on the four separate atomic sites we find∫n 1(2) = dp p g ∓ (ɛ, p), (9)g ∓ (ɛ, p) = ɛ ∓ /2π ¯h 2 ɛ[(ɛ 2 − 2 /4 ) 2]∓ 2v 2 p 2 ɛ − v 4 p 4(ɛ 2 − 2 /4 ) ,2 + v 2 p 2 2 − v 4 p 4where the minus (plus) sign is for the first (second) layer.We establish the stability of an undoped, gapless bilayersystem with respect to ferroelectric ordering (redistributionof charge density between the two layers) by comparing thegain in the total energy due to the opening of a gap in thelow energy spectrum with the energetic cost due to Coulombenergy E c equal to that of a capacitor with oppositely chargeplates. We estimate the charging energy by approximatingthe excess electronic densities on the individual layers, n 1and n 2 , as uniformly distributed within infinitesimally thin 2dlayers. Then, the charging energy is E c = Q 2 /2C b whereQ = en 1 L 2 = −en 2 L 2 is the excess charge on one of thelayers in the presence of finite asymmetry = ɛ 2 − ɛ 1 , andC b = ε r ε 0 L 2 /c 0 is the capacitance of a bilayer with interlayerseparation c 0 and area L 2 . For an undoped system, with Fermienergy ɛ F = 0 and zero excess total density n 1 = −n 2 , we onlyneed to consider the valence bands ɛ −(1) and ɛ(2) − . On integratingEq. (9) from zero momentum to a large momentum p ∞ , andusing an expansion in /γ 1 , we find that the change in thedensity of the valence bands for finite , as compared to = 0,isn 1(2) ≈ ±γ (14π ¯h 2 v 2 ln 4γ1||), (10)
114 E. McCann et al. / Solid State Communications 143 (2007) 110–115andE c ≈( ) (e 2 L 2 γ12 L 2 2 ) ( )32π 2 C b ¯h 2 v 2 ¯h 2 v 2 ln 2 4γ1. (11)||The change of the total single-particle energy may beestimated by integrating the valence bands energies, ɛ −(1) andɛ − (2) , with respect to momentum over the circularly symmetricalFermi surface:∫E ≈ 2L2 []π ¯h 2 p dp ɛ −(1) () + ɛ(2) − () − ɛ(1) − (0) − ɛ(2) − (0) .On integrating from zero momentum to a large momentum p ∞ ,and expanding in /γ 1 , the change in the single particle energyfor finite , as compared to = 0, isE ≈ − γ (1 L 2 2 ) [ ( )]18π ¯h 2 v 2 2 + ln 4γ1. (12)||The logarithmic dependence of the density equation (10)appears as the square of the infrared logarithm in the chargingenergy equation (11). Since the single particle energy equation(12) contains only a single infrared logarithm, the largeenergetic cost of charging an undoped bilayer ensures stabilitywith respect to the opening of a gap in the absence of an appliedexternal electric field.The gap between the conduction and valence bandsarises from layer asymmetry so that, in contrast to monolayergraphene, there is a possibility of tuning the magnitude of thegap using external gates. Indeed, a gate is routinely used inexperiments to control the density of electrons n on the bilayersystem [1,2,8] and, in general, this will produce a simultaneouschange in . In the following we use a self-consistent Hartreeapproximation to determine the electronic distribution on thebilayer and the resulting band structure in the presence of anexternal gate.As shown in Fig. 5, we consider the graphene bilayer,with interlayer separation c 0 , to be located a distance d froma parallel metallic gate. The application of an external gatevoltage V g = end/ε r ′ ε 0 induces a total excess electronic densityn = n 1 + n 2 on the bilayer system where n 1 (n 2 ) is theexcess density on the layer closest to (furthest from) the gate(we use SI units). Here ε 0 is the permittivity of free space,ε r ′ is the dielectric constant of the material between the gateand the bilayer, and e is the electronic charge. We assume thatthe screening of the effective charge density ρ + = en fromthe metallic gate is not perfect, leading to an excess electronicdensity n 2 on the layer furthest from the gate. Considering aGaussian surface, such as that shown with the dashed lines inFig. 5, shows that the excess density n 2 gives rise to an electricfield with magnitude E = en 2 /ε r ε 0 between the layers whereε r is the nominal dielectric constant of the bilayer. There is acorresponding change in potential energy U = e 2 n 2 c 0 /ε r ε 0that determines the layer asymmetry [16]: (n) = ɛ 2 − ɛ 1 ≡ e2 n 2 c 0ε r ε 0. (13)Fig. 5. Schematic of the graphene bilayer, with interlayer separation c 0 , locateda distance d from a parallel metallic gate. The gate voltage V g induces a totalexcess electronic density n = n 1 + n 2 on the bilayer system where n 1 (n 2 ) isthe excess density on the layer closest to (furthest from) the gate. The dashedline shows a Gaussian surface, from which it can be deduced that the magnitudeof the electric field between the layers of the bilayer is E = en 2 /ε r ε 0 .In terms of the capacitance of a bilayer of area L 2 , C b =ε r ε 0 L 2 /c 0 , this may be written (n) = e 2 n 2 L 2 /C b .We self consistently calculate the excess densities n 1 , n 2 ,n = n 1 + n 2 , Eq. (9), and the gap , Eq. (13). This has beendone numerically in Ref. [16]. Analytically, for moderately lowdensity, 4π ¯h 2 v 2 |n| < γ1 2 , we find ≈ e2 L 2 n2C bf Λ(¯h 2 v 2 π|n|γ121)f Λ (x) ≈).1 + Λ(x − 1 2 ln x, (14)The function f Λ (x) is plotted in Fig. 4(b) for different valuesof the dimensionless parameter Λ = e 2 L 2 γ 1 /(2π ¯h 2 v 2 C b )which describes the effectiveness of the interlayer screening ofthe bilayer. The limit Λ → 0 describes poor screening whenthe density on each layer is equal to n/2 whereas for Λ → ∞there is excellent screening, the density lies solely on the layerclosest to the external gate and = 0. Note that f Λ (x) → 0 asx → 0 because of the logarithm, meaning that the effectivenessof interlayer screening increases upon lowering density. Usingtypical experimental parameters [22] we find Λ ≈ 1.3. Weestimate that the addition of density n ∼ 10 12 cm −2 yields agap ∼ 10 meV [31].6. ConclusionsWe reviewed the results of tight-binding model studies ofbilayer graphene and its low-energy electronic band structure.The optical absorption coefficient of a bilayer displays featuresrelated to the band structure on the energy scale of the orderof the interlayer coupling γ 1 ≈ 0.4 eV [18,19], in contrast tothe featureless absorption coefficient of monolayer graphene.At much lower energies, ɛ L ∼ 1 meV, trigonal warping of theband structure might produce a Lifshitz transition in which theisoenergetic line about each valley is broken into four pockets.Inter-layer asymmetry creates a gap between the conductionand valence bands: bilayer graphene is a semiconductor with atuneable gap of up to about 0.4 eV. However, by comparingthe charging energy with the single particle energy, it is
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