PDF | The so-called Klein paradox-unimpeded penetration of relativistic particles through high and wide potential barriers-is one of the most. This plot shows the transmission coefficient for a barrier of height in graphene as a function of the angle of a plane wave incident on the barrier. Title: Chiral tunnelling and the Klein paradox in graphene. Author(s): Katsnelson, M.I. ; Novoselov, K.S. ; Geim, A.K.. Publication year: Source: Nature.

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Inphysicist Oskar Klein [1] obtained a surprising result by applying the Dirac equation to the familiar problem of electron scattering from a potential barrier.

In nonrelativistic quantum mechanics, electron chital into a barrier is observed, with exponential damping. Moreover, as the potential approaches infinity, the reflection diminishes and the electron is always transmitted.

Chiral Tunneling and the Klein Paradox in Graphene – Wolfram Demonstrations Project

The immediate application of the paradox was to Rutherford’s proton—electron model for neutral particles within the nucleus, before the discovery of the neutron. The paradox presented a quantum mechanical graphdne to the notion of an electron confined within a nucleus.


The meaning of this paradox was intensely debated at the time. Both the incoming and transmitted wave functions are associated with positive group velocity Blue lines in Fig. Green lines in Fig.

Condensed Matter > Mesoscale and Nanoscale Physics

We now want to calculate the transmission and reflection coefficients, TR. One interpretation of the paradox is that a potential step cannot reverse the direction of the group velocity of a massless relativistic particle.

This explanation best suits the single particle solution cited above. Other, more complex interpretations are suggested in klin, in the context of quantum field theory where the unrestrained tunnelling is shown to occur due to the existence of particle—antiparticle pairs at the potential.

For the massive case, the calculations are similar to the above.

The results are as surprising as in the massless case. The transmission coefficient is always larger than zero, and approaches 1 as the potential step goes to infinity.

This strategy was also applied to obtain analytic solutions to the Dirac equation for an infinite square well.

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These results were expanded to higher dimensions, and to other types of potentials, such as a linear step, a square barrier, a smooth potential, etc. Many experiments in electron transport in graphene rely on the Klein paradox for massless particles.


From Wikipedia, the free encyclopedia. This article needs attention from an expert in physics. The specific problem is: The diagrams and interpretation presented here need confirmation. WikiProject Physics may be able to help recruit an expert. This section needs expansion. You can help by adding to it.

Chiral tunnelling and the Klein paradox in graphene

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