Graphene, part I

Graphene is one of the hottest materials out there right now in condensed matter physics, and I'm trying to figure out what tactic to take in making some blog postings about it.  One good place to start is the remarkably fast rise in the popularity of graphene.  Why did it catch on so quickly?  As far as I can tell, there are several reasons.

1. Graphene has a comparatively simple electronic structure.  It's a single sheet of hexagonally arranged carbon atoms.  The well-defined geometry makes it extremely amenable to simple calculational techniques, and the basic single-particle band structure (where we ignore the fact that electrons repel each other) was calculated decades ago.
2. That electronic structure is actually pretty interesting, for three reasons.  Remember that a spatially periodic arrangement of atoms "picks out" special values of the electron (crystal) momentum.  In some sense, electrons with just the right (effective) wavelength (corresponding to particular momenta) diffract off the lattice.  You can think of the hexagonal graphene lattice as a superposition of two identical sublattices off-set by one carbon-carbon bond length.  So, the first interesting feature is that there are two sets of momenta ("sets of points in reciprocal space") that are special - picked out by the lattice, inequivalent (since the two sublattices really are distinct) but otherwise identical (since it's semantics to say which sublattice is primary and which is secondary).  This is called "valley degeneracy", and while it crops up in other materials, the lattice symmetry of graphene ends up giving it added significance.  Second, when you count electrons and try filling up the allowed electronic states starting at the lowest energy, you find that there are exactly two highest energy filled spatial states, one at each of the two lowest-momentum inequivalent momentum points.  All lower energy states are filled; all higher energy states are empty.  That means that graphene is exactly at the border between being a metal (many many states forming the "Fermi surface" between filled and empty states) and a semiconductor (filled states and empty states separated by a "gap" of energies for which there are no allowed electronic states).  Third and most importantly, the energy of the allowed states near those Fermi points varies linearly with (crystal) momentum, much like the case of an ultrarelativistic classical particle, rather than quadratically as usual.  So, graphene is in some ways a playground for thinking about two-dimensional relativistic Fermi gases.
   
3. The material is comparatively easy to get and make.  That means its accessible, while other high quality two-dimensional electron systems (e.g., at a GaAs/AlGaAs interface) require sophisticated crystal growth techniques.
4. There is a whole literature of 2d electron physics in Si and GaAs/AlGaAs, which means there is a laundry list of techniques and experiments just waiting to be applied, in a system that theorists can actually calculate.
5. Moreover, graphene band structure and materials issues are close to that of nanotubes, meaning that there's another whole community of people ready to apply what they've learned.
6. Graphene may actually be useful for technologies!

Next time I post about this, I'll try to talk more about the special aspect of the valley degeneracy.