Tight-binding description of patterned graphene

Semicond. Sci. Technol. 23 (2008) 075026B Gharekhanlou et alFigure 3. Showing a Dirac point in the band structure of graphene by taking first neighbors.Figure 4. Opening of an energy gap at Dirac points due to vacancies.structure of patterned graphene, E(κ)may be expanded aroundany of the Dirac points located at K asE c (κ) = + E c + ¯h2 [m∗ −12 c,x |κ x − K x | 2+ m ∗ c,y −1 |κ y − K y | 2] , (2a)E v (κ) = + E v + ¯h2 [m∗ −12 v,x |κ x − K x | 2+ m ∗ v,y −1 |κ y − K y | 2] , (2b)respectively, for conduction and valence bands. Here, E c,vcorresponds to the shifts in the peaks of conduction and valencebands due to patterning. Hence, the energy gap E g will besimply given byE g = E c + E v . (3)Please note the difference of (2a), (2b) to the bandstructure of unpatterned graphene where the energy dispersionaround the Dirac points is given by the linear expressionE(κ) =±¯hV |κ − K|, with V being the 2D Fermi velocity[22].The energy band structure (2a), (2b) of graphene wouldeventually determine not only the effective mass of electronsand holes, but also the effective velocity of Fermionic Diracparticles which normally propagate in unpatterned graphene ata speed of c/300, with c being the velocity of light in vacuum[1]. Also, as we later observe, there will be a noticeableanisotropy in the effective mass in the patterned graphene.In the tight-binding model applied here, it is assumed thatthe full Hamiltonian of the system may be approximated by theHamiltonian of an isolated atom centered at each lattice point,and only the effects of the first-nearest neighbors are involved.3