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The latter scattering event would require the pseudospin to be? In the 1. Indeed, in the absence of pseudospin-? This is shown in Fig. This perfect tunnelling can be understood in terms of the conservation of pseudospin. In the limit of high barriers V0 E, the expression for T can be simpli? More signi? The latter is the feature unique to massless Dirac fermions and is directly related to the Klein paradox in QED. Requiring the continuity of the wavefunction by matching up coe? The barrier heights V0 are a and b 50 meV red curves and a and b meV blue curves.įor simplicity, we assume in 2 in? The sharp-edge assumption is justi? Such a barrier can be created by the electric? Importantly, Dirac fermions in graphene are massless and, therefore, there is no formal theoretical requirement for the minimal electric? To create a well-de? It is straightforward to solve the tunnelling problem shown in Fig. The electron concentration n outside the barrier is chosen as 0. Neglecting many-body e? The general scheme of such an experiment is shown in Fig. This allows the introduction of chirality12, that is formally a projection of pseudospin on the direction of motion, which is positive and negative for electrons and holes, respectively. The conical spectrum of graphene is the result of intersection of the energy bands originating from sublattices A and B see Fig. In contrast, electron and hole states in graphene are interconnected, exhibiting properties analogous to the chargeconjugation symmetry in QED10- There are further analogies with QED. At negative energies, if the valence band is not full, its unoccupied electronic states behave as positively charged quasiparticles holes, which are often viewed as a condensed-matter equivalent of positrons. Above zero energy, the current carrying states in graphene are, as usual, electron-like and negatively charged. The blue? The spectrum is isotropic and, despite its parabolicity, also originates from the intersection of energy bands formed by equivalent sublattices, which ensures charge conjugation, similar to the case of single-layer graphene.Īlthough the linear spectrum is important, it is not the only essential feature that underpins the description of quantum transport in graphene by the Dirac equation. The Fermi level dotted lines lies in the conduction band outside the barrier and the valence band inside it. The three diagrams in a schematically show the positions of the Fermi energy E across such a barrier. The purpose of this paper is to show that graphene-a recently found allotrope of carbon9-provides an e? The red and green curves emphasize the origin of the linear spectrum, which is the crossing between the energy bands associated with crystal sublattices A and B. The essential feature of quantum electrodynamics QED responsible for the e? This fundamental property of the Dirac equation is often referred to as the charge-conjugation symmetry. This relativistic e? Matching between electron and positron wavefunctions across the barrier leads to the high-probability tunnelling described by the Klein paradox7. In this case, the transmission probability, T, depends only weakly on the barrier height, approaching the perfect transparency for very high barriers, in stark contrast to the conventional, non-relativistic tunnelling where T exponentially decays with increasing V0. Owing to the chiral nature of their quasiparticles, quantum tunnelling in these materials becomes highly anisotropic, qualitatively different from the case of normal, nonrelativistic electrons. Here we show that the effect can be tested in a conceptually simple condensed-matter experiment using electrostatic barriers in single- and bi-layer graphene. 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 continuum outside4-6. This relativistic effect can be attributed nuclear and astro-physics but direct tests of the Klein to the fact that a sufficiently strong potential, being repulsive paradox using elementary particles have so far proved for electrons, is attractive for positrons and results in positron states inside the barrier, which align in energy with the electron impossible. The T, depends only weakly on the barrier height, approaching the perfect transparency for very high barriers, in stark contrast to phenomenon is discussed in many contexts in particle, the conventional, non-relativistic tunnelling where T exponentially decays with increasing V0. In this case, the transmission probability, consequences of quantum electrodynamics.