Carbon Hybridization to Tight-Binding to Dirac Solid

Abstract

We make a pedagogical survey on why the charge carriers (electrons) in graphene are called massless Dirac fermions. Our analysis begins at the beginning, namely, we start from the quantum chemistry of two nearby carbon (C) atoms and show how their hybridized orbitals ‘valence-bond’ with each other to form an energy-band in the solid-state. This yields a two-dimensional honeycomb lattice of graphene, which can be viewed as two inter-penetrating triangular sublattices. That recognition provides a perfect setting for describing the dynamics of the last weakly-localized valence electron of C in a tight-binding model, which captures all the unusual electronic phenomena of graphene. The latter emerges from a resemblance to the relativistic Dirac theory of electrons because, in the long-wavelength limit, the energy dispersion is linear in the wave vector. We build up - step by step - this remarkable transition of a carbon-based material to an exotic two-dimensional Dirac solid, in which much of the quantum aspects of modern condensed matte physics can be tested in the laboratory.