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Contents 1 Introduction to Carbon Materials 1 1.1 TheCarbonAtomanditsHybridisations . . . . . . . . . . . . 3 1.1.1 sp1hybridisation ..................... 4 1.1.2 sp2 hybridisation – graphitic allotopes . . . . . . . . . 6 1.1.3 sp3hybridisation–diamonds .............. 9 1.2 Crystal Structure of Graphene and Graphite . . . . . . . . . . 10 1.2.1 Graphene’shoneycomblattice.............. 10 1.2.2 Graphene stacking – the different forms of graphite . . 13 1.3 FabricationofGraphene ..................... 16 1.3.1 Exfoliatedgraphene.................... 16 1.3.2 Epitaxialgraphene .................... 18 2 Electronic Band Structure of Graphene 21 2.1 Tight-Binding Model for Electrons on the Honeycomb Lattice 22 2.1.1 Bloch’stheorem...................... 23 2.1.2 Latticewithseveralatomsperunitcell . . . . . . . . . 24 2.1.3 Solution for graphene with nearest-neighbour and next- nearest-neighourhopping................. 27 2.2 ContinuumLimit ......................... 33 2.3 Experimental Characterisation of the Electronic Band Structure 41 3 The Dirac Equation for Relativistic Fermions 45 3.1 RelativisticWaveEquations ................... 46 3.1.1 Relativistic Schr ̈odinger/Klein-Gordon equation . . . . 47 3.1.2 Diracequation ...................... 49 3.2 The2DDiracEquation...................... 53 3.2.1 Eigenstates of the 2D Dirac Hamiltonian . . . . . . . . 54 3.2.2 Symmetries and Lorentz transformations . . . . . . . . 55 iiiPDF Image | Physical Properties of Graphene
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