Abstract
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. 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 using electrostatic barriers in single- and bi-layer graphene. Owing to the chiral nature of their quasiparticles, quantum tunnelling in these materials becomes highly anisotropic, qualitatively different from the case of normal, non-relativistic 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.
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References
Klein, O. Die reflexion von elektronen an einem potentialsprung nach der relativistischen dynamik von Dirac. Z. Phys. 53, 157–165 (1929).
Greiner, W., Mueller, B. & Rafelski, J. Quantum Electrodynamics of Strong Fields (Springer, Berlin, 1985).
Grib, A. A., Mamayev, S. G. & Mostepanenko, V. M. Vacuum Effects in Strong Fields (Friedmann, St-Petersburg, 1994).
Su, R. K., Siu, G. C. & Chou, X. Barrier penetration and Klein paradox. J. Phys. A 26, 1001–1005 (1993).
Dombey, N. & Calogeracos, A. Seventy years of the Klein paradox. Phys. Rep. 315, 41–58 (1999).
Calogeracos, A. & Dombey, N. History and physics of the Klein paradox. Contemp. Phys. 40, 313–321 (1999).
Krekora, P., Su, Q. & Grobe, R. Klein paradox in spatial and temporal resolution. Phys. Rev. Lett. 92, 040406 (2004).
Page, D. N. Hawking radiation and black hole thermodynamics. New J. Phys. 7, 203 (2005).
Novoselov, K. S. et al. Electric field effect in atomically thin carbon films. Science 306, 666–669 (2004).
Slonczewski, J. C. & Weiss, P. R. Band structure of graphite. Phys. Rev. 109, 272 (1958).
Semenoff, G. W. Condensed-matter simulation of a three-dimensional anomaly. Phys. Rev. Lett. 53, 2449–2452 (1984).
Haldane, F. D. M. Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the ‘parity anomaly’. Phys. Rev. Lett. 61, 2015–2018 (1988).
Novoselov, K. S. et al. Two-dimensional gas of massless Dirac fermions in graphene. Nature 438, 197–200 (2005).
Zhang, Y., Tan, Y. W., Stormer, H. L. & Kim, P. Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature 438, 201–204 (2005).
Vonsovsky, S. V. & Katsnelson, M. I. Quatum Solid State Physics (Springer, Berlin, 1989) Sect. 4.6.6.
Boyanovsky, D., Blankenbecler, R. & Yahalom, R. Physical origin of topological mass in 2+1 dimensions. Nucl. Phys. B 270, 483–505 (1986).
Ando, T., Nakanishi, T. & Saito, R. Berry’s phase and absence of back scattering in carbon nanotubes. J. Phys. Soc. Japan 67, 2857–2862 (1998).
McEuen, P. L., Bockrath, M., Cobden, D. H., Yoon, Y. G. & Louie, S. G. Disorder, pseudospins, and backscattering in carbon nanotubes. Phys. Rev. Lett. 83, 5098–5101 (1999).
Tworzydlo, J., Trauzettel, B., Titov, M., Rycerz, A. & Beenakker, C. W. J. Quantum-limited shot noise in graphene. Phys. Rev. Lett. 96, 246802 (2006).
Novoselov, K. S. et al. Unconventional quantum Hall effect and Berry’s phase of 2π in bilayer graphene. Nature Phys. 2, 177–180 (2006).
McCann, E. & Falko, V. I. Landau-level degeneracy and quantum Hall effect in a graphite bilayer. Phys. Rev. Lett. 96, 086805 (2006).
Esaki, L. New phenomenon in narrow germanium para-normal-junctions. Phys. Rev. 109, 603–604 (1958).
Meyer, J. R., Hoffman, C. A., Bartoli, F. J. & Rammohan, L. R. Type-II quantum-well lasers for the midwavelength infrared. Appl. Phys. Lett. 67, 757–759 (1995).
Teissier, R. et al. Experimental determination of gamma-X intervalley transfer mechanisms in GaAs/AlAs heterostructures. Phys. Rev. B 54, 8329–8332 (1996).
Ziman, J. M. Models of Disorder (Cambridge Univ. Press, Cambridge, 1979).
Lifshitz, I. M., Gredeskul, S. A. & Pastur, L. A. Introduction to the Theory of Disordered Systems (Wiley, New York, 1988).
Lee, P. A., Altshuler, B. L. & Webb, R. A. (eds) Mesoscopic Phenomena in Solids (North-Holland, Amsterdam, 1991).
Berry, M. V. & Mondragon, R. J. Neutrino billiards—time reversal symmetry-breaking without magnetic fields. Proc. R. Soc. London A 412, 53–74 (1987).
Spector, J., Stormer, H. L., Baldwin, K. W., Pfeiffer, L. N. & West, K. W. Electron focusing in 2-dimensional systems by means of an electrostatic lens. Appl. Phys. Lett. 56, 1290–1292 (1990).
Dragoman, D. & Dragoman, M. Optical analogue structures to mesoscopic devices. Prog. Quantum Electron. 23, 131–188 (1999).
Acknowledgements
We are grateful to A. C. Neto, V. Fal’ko, P. Guinea and D. Khveshchenko for illuminating discussions. This work was supported by EPSRC (UK) and FOM (Netherlands).
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Katsnelson, M., Novoselov, K. & Geim, A. Chiral tunnelling and the Klein paradox in graphene. Nature Phys 2, 620–625 (2006). https://doi.org/10.1038/nphys384
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DOI: https://doi.org/10.1038/nphys384
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