Lecture 5 : Graphene : Electronic band structure and Dirac fermions

A suspended sheet of pure graphene – a plane layer of C atoms bonded together in a honeycomb lattice – is the “most two-dimensional” system imaginable. Such sheets have long been known to exist in disguised forms – in graphite (many graphene sheets stacked on top of one another), C nanotubes (a graphene sheet rolled into a cylinder) and fullerenes (buckyballs), which are small areas of a graphene sheet sewn together to form an approximately spherical surface. Moreover, it was appreciated that every time one writes on a sheet of paper with a “lead” (i.e. graphite) pencil one probably produces the odd flake of graphene, along with a whole lot of other by-products; however, until 2004, it was generally believed (a) that an extended graphene sheet would not be stable against the effects of thermal and other fluctuations, and (b) that even if they were stable, it would be impossible to isolate them so that their properties could be systematically studied. In that year, André Geim and his colleagues at the University of Manchester in the UK demonstrated that both these beliefs were false: they created single graphene sheets1 by peeling them off a graphite substrate using scotch tape, and characterized them as indeed single-sheet by simple optical microscopy on top of a SiO2 substrate. (This procedure, which turns out to be exquisitely sensitive to the details of the substrate, was an essential ingredient in the success of the whole operation. A more recent and faster selection technique is Raman spectroscopy). Subsequently it was found that small graphene sheets do not need to rest on substrates but can be freely suspended from a scaffolding; furthermore, bilayer and multilayer sheets can be prepared and characterized. As a result of these developments, the number of papers on graphene published in the last few years exceeds 3000.