Theory of graphene: CERN on the desk

Graphene, a recently (2004) discovered two-dimensional allotrope of carbon (this discovery was awarded by Nobel Prize in physics 2010), has initiated a huge activity in physics, chemistry and materials science, mainly, for three reasons. First, a peculiar character of charge carriers in this material makes it a “CERN on the desk” allowing us to simulate subtle and hardly achievable effects of high energy physics. Second, it is the simplest possible membrane, an ideal testbed for statistical physics in two dimensions. Last not least, being the first truly two-dimensional material (just one atom thick) it promises brilliant perspectives for the next generation of electronics which uses mainly only surface of materials.
I will tell about the first aspect of graphene physics, some unexpected relations between materials science and quantum field theory and high-energy physics.
Electrons and holes in this material have properties similar to ultrarelativistic particles (two-dimensional analog of massless Dirac fermions). This leads to some unusual and even counterintuitive phenomena, such as finite conductivity in the limit of zero charge carrier concentration (quantum transport by evanescent waves) or transmission of electrons through high and broad potential barriers with a high probability (Klein tunneling). This allows us to study subtle effects of relativistic quantum mechanics and quantum field theory in condensed-matter experiments, without accelerators and colliders. Some of these effects were considered as practically unreachable. Apart from the Klein tunneling, this is, for example, a vacuum reconstruction near supercritical charges predicted many years ago for collisions of ultra-heavy ions and recently experimentally discovered for graphene.
It is demonstrated now both experimentally and theoretically that graphene is usually not flat but covered by ripples resulting from both intrinsic flexural instability of two-dimensional membranes and roughness of substrate. Thus charge carriers are not just Dirac fermions but Dirac fermions moving in a curved space. The effect of the corrugations on the electron spectrum can be described in terms of gauge (pseudo-magnetic) fields which result, in particular, in formation of pseudo-Landau levels recently predicted theoretically and already found experimentally. These gauge fields can be used for “strain engineering”, including tunable gap opening, quantum pumping and creation of valley-polarized current. Ripples can induce puddles, that is, charge inhomogeneities. The scattering by the ripples is also one of the limiting factors restricting the charge carrier mobility in graphene.