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4144 with vF = √ Graphene – Synthesis, Characterization, Properties anWdill-bAe-psept-lbiyc-IaN-tTiEoCnHs 3γ0r/2h ̄ the energy independent Fermi velocity (Jiang et al. (2007); Novoselov & Geim (2005); Zhang et al. (2005); Zhang & Brar et al. (2008)). Note that in the K point the pseudospin direction is parallel to the momentum for electrons (states with Ek > 0) and antiparallel for holes (states with Ek < 0). This property is the result of the fact that in each valley the particles wavefunction is an eigen function of the helicity operator. Consequently, at the K’ point the pseudospin direction is antiparallel to the momentum for electrons and parallel for holes. The absence of a band-gap in the energy dispersion of graphene implies that the conduction in this material cannot be simply switched on or off by means of a gate voltage which acts on the position of the Fermi level, limiting the use of graphene in conventional transistor applications. Indeed, even when the Fermi level in graphene devices is at E = 0, the current in graphene is far from being completely pinched-off. However, the gapless energy dispersion of graphene is a consequence of the assumption that the electron onsite energy between the A and B sublattice carbon atoms are equal. Whenever HAA ̸= HBB a band-gap opens in the energy spectrum of graphene, see Fig. 1c. A viable way to experimentally engineer such a band-gap consists in growing and/or depositing graphene on a commensurate honeycomb lattice formed by chemically inequivalent atoms which ultimately will originate a difference in the onsite energy between the A and B sublattices. So far, hexagonal BN is considered to be one of the most promising candidates for graphene band-gap engineering since it has an almost commensurate crystal structure to the one of graphene and it has two different elements in each sublattice (Giovannetti et al. (2007)). However, despite the growing interest in graphene on h-BN, no experimental evidence has been reported yet of a band-gap opening in graphene on h-BN (Xue et al. (2011)). On the other hand, bilayer graphene offers a unique alternative to the problem of band-gap engineering for reasons which will become clear when considering the bilayer non-interacting tight binding description (McCann (2006)). The most common stacking of graphite planes found in nature is of Bernal type, where the A atoms in one layer are aligned on top of B atoms of an adjacent layer. The unit cell of bilayer graphene consists of a basis of four atoms labelled A1, B1, A2 and B2 belonging to different atomic planes as indicated by the numerical index, see Fig. 2a and b. Similarly to the case of a single layer we adopt the LCAO method with Bloch wavefunctions 1 ik·d corresponding to the sublattice carbon atoms of the form: φi = √N ∑n e in φi(r − din) with i = A1, B1, A2, B2 and din the translation vector of the i sublattice. For simplicity, we start by considering non-zero only the nearest neighbour coupling -i.e. γ0 hopping from A1 to B1 and from A2 to B2 atomic sites- and γ1 ≈ 0.39eV the interlayer coupling between A2 and B1 atoms (Castro Neto et al. (2009)). We also assume that all the carbon atoms lattice sites are energetically equivalent -i.e. HAiAi = HBjBj with i, j = 1, 2. In this case we consider the linear combination of Bloch wavefunctions of the form φ = aφA1 + bφB1 + cφA2 + dφB2, with a, b, c and d coefficients. The solution of the Schrödinger equation is readily obtained projecting the solutions of Eq. 3 onto the states < φi | with i = A1, B1, A2, and B2. The equivalent system to Eq. 5 for the case of bilayer contains 4 equations, and imposing the condition that the secular determinant is zero leads to the set of the bilayer four bands: √ 3γ0r/2h ̄ . The energy dispersion in Eq. 9 shows that the lowest energy conduction and valence band of bilayer graphene touch each other in each of the two K and K’ valleys. γ1 E ( k ) = ± 2 ± γ12 4 + ( h ̄ v F k ) 2 ( 9 ) with vF =PDF Image | GRAPHENE SYNTHESIS CHARACTERIZATION PROPERTIES
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