In this chapter we start with a presentation of the so-called Klein tunneling mechanism, which is one of the most striking properties of graphene. Later we give an overview of ballistic transport both in graphene and related materials (carbon nanotubes and graphene nanoribbons). After presenting a simple real-space mode-decomposition scheme, which can be exploited to obtain analytical results or to boost numerical calculations, we discuss Fabry-Pérot interference, contact effects, and the minimum conductivity in the 2D limit.

The Klein tunneling mechanism

The Klein tunneling mechanism was first reported in the context of quantum electrodynamics. In 1929, physicist Oskar Klein (Klein, 1929) found a surprising result when solving the propagation of Dirac electrons through a single potential barrier. In non-relativistic quantum mechanics, incident electrons tunnel a short distance through the barrier as evanescent waves, with exponential damping with the barrier depth. In sharp contrast, if the potential barrier is of the order of the electron mass, eV ~ mc2, electrons propagate as antiparticles whose inverted energy–momentum dispersion relation allows them to move freely through the barrier. This unimpeded penetration of relativistic particles through high and wide potential barriers has been one of the most counterintuitive consequences of quantum electrodynamics, but despite its interest for particle, nuclear, and astro-physics, a direct test of the Klein tunnel effect using relativistic particles still remains out of reach for high-energy physics experiments.