Klein tunneling in graphene: optics with massless electrons
Institut d’Électronique Fondamentale, Univ. Paris-Sud, CNRS UMR
2 Laboratoire de Physique des Solides, Univ. Paris-Sud, CNRS, UMR 8502, 91405 Orsay, France
a e-mail: email@example.com
Received in final form: 17 August 2011
Published online: 10 October 2011
This article provides a pedagogical review on Klein tunneling in graphene, i.e. the peculiar tunneling properties of two-dimensional massless Dirac electrons. We consider two simple situations in detail: a massless Dirac electron incident either on a potential step or on a potential barrier and use elementary quantum wave mechanics to obtain the transmission probability. We emphasize the connection to related phenomena in optics, such as the Snell-Descartes law of refraction, total internal reflection, Fabry-Pérot resonances, negative refraction index materials (the so called meta-materials), etc. We also stress that Klein tunneling is not a genuine quantum tunneling effect as it does not necessarily involve passing through a classically forbidden region via evanescent waves. A crucial role in Klein tunneling is played by the conservation of (sublattice) pseudo-spin, which is discussed in detail. A major consequence is the absence of backscattering at normal incidence, of which we give a new shorten proof. The current experimental status is also thoroughly reviewed. The Appendix contains the discussion of a one-dimensional toy model that clearly illustrates the difference in Klein tunneling between mono- and bi-layer graphene.
© EDP Sciences, Società Italiana di Fisica and Springer-Verlag, 2011