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Unformatted text preview: 1 ECE 407 Spring 2009 Farhan Rana Cornell University Handout 11 Energy Bands in Graphene: Tight Binding and the Nearly Free Electron Approach In this lecture you will learn: The tight binding method (contd) The -bands in graphene FBZ Energy ECE 407 Spring 2009 Farhan Rana Cornell University Graphene and Carbon Nanotubes: Basics a 3 a a x y 1 a v 2 a v a y x a + = 2 1 2 3 1 v a y x a = 2 1 2 3 2 v A B Graphene is a two dimensional single atomic layer of carbon atoms arranged in a Honeycomb lattice (which is not a Bravais lattice) The underlying Bravais lattice is shown by the location of the black dots and is a hexagonal lattice There are two carbon atoms per primitive cell, A and B (shown in blue and red colors, respectively) Graphene can be rolled into tubes that are called carbon nanotubes (CNTs) CNT CNT a = 2.46 A 2 ECE 407 Spring 2009 Farhan Rana Cornell University Graphene: Sp2 Hybridization All carbon atoms are all sp2 hybridized (one 2s orbital together with the 2p x and the 2p y orbitals generate three sp2 orbitals) All sp2 orbitals form -bonds with the sp2 orbitals of the neigboring carbon atoms The bonding orbital associated with each -bond is occupied by two electrons (spin-up and spin-down) There is one electron per carbon atom left in the 2p z orbital x y Sp2 hybridization in carbon: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) r r r r r r r r r r r...
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- Spring '08