  The so-called Klein paradox - unimpeded penetration of relativistic particles
through high and wide potential barriers - is one of the most exotic and
counterintuitive consequences of quantum electrodynamics (QED). The phenomenon
is discussed in many contexts in particle, nuclear and astro- physics but
direct tests of the Klein paradox using elementary particles have so far proved
impossible. Here we show that the effect can be tested in a conceptually simple
condensed-matter experiment by using electrostatic barriers in single- and
bi-layer graphene. Due to the chiral nature of their quasiparticles, quantum
tunneling in these materials becomes highly anisotropic, qualitatively
different from the case of normal, nonrelativistic electrons. Massless Dirac
fermions in graphene allow a close realization of Klein's gedanken experiment
whereas massive chiral fermions in bilayer graphene offer an interesting
complementary system that elucidates the basic physics involved.
