Graphene is a simple honeycomb lattice of carbon that displays a number of remarkable physical properties. The essential reason for all the fascinating properties of graphene is that the low energy excitations behave as if governed not be the Schrödinger equation, but instead by an effective Dirac-Weyl equation. This latter equation describes particles of zero rest mass which (therefore) travel at the speed of light.

Much of the strange physics of this equation transfers to the physics of graphene, producing several novel properties (Klien tunneling, absence of back scattering). On the other hand, the more humble origin of this this effective equation can itself still be seen at several places in the physics of graphene which, therefore, consists of an mixture of exotic Dirac-Weyl physics with standard solid state physics.

Note that by an effective equation we simply mean that while the electrons are, of course, governed by the Schrödinger equation they are conspiring in some way to produce low energy behavior that appears as if they were governed by a quite different equation. Since we are dealing with an effective equation the Dirac-Weyl equation that describes low energy graphene excitations contains not the speed of light c, but an effective constant (which turns out to be just the Fermi velocity).

The following link provides a derivation of the origin of the Dirac-Weyl in graphene physics, and allows one to see how various properties of the quasi-particles such as their chirality emerge.
Continuum approximation description of graphene

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Reduced density-matrix functional theory

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