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Massless fermions Klein tunneling

  1. Sep 23, 2015 #1
    "In 1929, physicist 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 tunneling into a barrier is observed, with exponential damping. However, Klein’s result showed that if the potential is of the order of the electron mass, [PLAIN]https://upload.wikimedia.org/math/5/e/a/5ead404e0410a13fa323d3e8de6b2ced.png, [Broken] the barrier is nearly transparent. Moreover, as the potential approaches infinity, the reflection diminishes and the electron is always transmitted."

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


    However, the Dirac electrons found in graphene can tunnel through energy barriers regardless of their width and energy height; a phenomenon called Klein tunneling, described theoretically for 3D massive Dirac electrons by the Swedish physicist Oskar Klein in 1929. Graphene was the first material in which Klein tunneling was observed experimentally, as massive Dirac electrons required energy barriers too large to be observed.

    If electrons in graphene can tunnel throug energy barriers regardless of their width and energy height, does it mean they could tunnel through any distance? So, we could make a graphene tunnel diode with 1 km distance between cathode and anode and an empty space (air) in between?
    Last edited by a moderator: May 7, 2017
  2. jcsd
  3. Sep 23, 2015 #2
    Your springer link has incorrect wording (for some reason). It's not electrons that are tunneling, but quasiparticles in graphene (of which, of course, electrons are the physical substratum, but electrons aren't actually tunneling). The potential barrier also is of an electrical nature, not just "any" barrier::http://arxiv.org/abs/cond-mat/0604323 "Such a barrier can be created by the electric field effect using a thin insulator or by local chemical doping".
  4. Sep 23, 2015 #3
    They claim they created graphene tunneling diode with very thing layer of dielectric between graphene sheets.

    http://www.researchgate.net/publication/260520206_High_performance_vertical_tunneling_diodes_using_graphenehexagonal_boron_nitridegraphene_hetero-structure [Broken]

    If electrons tunnel through dielectric, does it mean they do actually tunnel through, as you said? And if yes, then their unusual properties apply only to tunneling through electric barrier, but when they tunnel through dielectric, classical tunneling properties stay in the force?
    Last edited by a moderator: May 7, 2017
  5. Sep 23, 2015 #4
    Again, the electrons aren't tunneling, the graphene quasiparticles resulting from the collective behavior of the electrons are tunneling. If actual electrons were to tunnel, then the non-relativistic way of tunneling with exponential dampening of the amplitude applies, but it's most likely irrelevant to that article if the purpose is Klein tunneling.
  6. Sep 23, 2015 #5
    So, graphene quasiparticles could tunnel through any kind of electric barrier (electric field?), but through dielectric barrier they tunnel on very small distance only, like regular tunneling particles?
  7. Sep 23, 2015 #6
    I think the quasiparticles also tunnel Klein-like through the dielectric barrier (not sure, the abstract isn't very clear). It's just the electrons that tunnel normally.
  8. Sep 23, 2015 #7
    1) Could this quasiparticles carry energy or information while they tunnel?
    2) Could you use just an empty air as a dielectric (in principle)?
    3) If they Klain-tunneling through a dielectric barrier, what maximal thickness of such barrier could be acheived?
  9. Sep 24, 2015 #8
    The answer to 1) is yes. The Dirac theory predicts 2) - yes; 3) - infinite. But graphene is a realistic case, it's probably different. You should ask in the atomic and solid state subforum.
  10. Sep 24, 2015 #9


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    The (quasi-)electrons in graphene behave like 2D Dirac particles. However, the arrangement they are talking about in this article is a 3D structure. So that's certainly not Klein tunneling.

    Furthermore, you certainly can't use a dielectric barrier. Rather, the barrier has to be "vacuum like", i.e. graphene like in our case , up to some applied electric potential.

    In graphene, the electrons only behave Dirac like in a small energy band around the Dirac point. This limits the height of the barrier. So there will be no complete transmission.
    Last edited: Sep 24, 2015
  11. Sep 25, 2015 #10
    So, Klein tunneling (of electrons or charged quasiparticles) through the air is not possible under any practical conditions?
    This one article mentions Photonic Klein tunneling through a dielectric slab.

    And if it would be possible, what distance could we achieve?

    If common tunneling of electrons through dielectric barrier is possible, then why Klain tunneling of either electrons or quasi-particles is impossible?
    Last edited: Sep 25, 2015
  12. Sep 25, 2015 #11


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    Klein tunnelling is possible for electrons through air (or even better vacuum), but not for quasi-particles for the simple reason that the definition of a quasi particle requires a specific medium (like graphene). No specific medium, no quasi-particles, hence also no tunnelling of quasi particles through no medium (=vacuum).
  13. Sep 25, 2015 #12
    But they claim created graphene tunneling diode in which quasiparticles(?) tunnel through dielectric barrier (just few atoms thick). How do you explain that?

    Here we report resonant tunnelling of Dirac fermions through a boron nitride barrier, a few atomic layers thick, sandwiched between two graphene electrodes.
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