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Chiral Tunneling and the Klein Paradox in Graphene M.I. Katsnelson, K.S. Novoselov, and A.K. Geim Nature Physics Volume 2 September 2006 John Watson

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Outline Background, main result Details of paper Authors proposed future work Reported experimental observations Summary

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Background Klein paradox implied by Diracs relativistic quantum mechanics Consider potential step on right –Relativistic QM gives V x Dont get non-relativistic exponential decay Calogeracos, A.; Dombey, N.. Contemporary Physics, Sep/Oct99, Vol. 40 Issue 5

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Main result Graphene can be used to study relativistic QM with physically realizable experiments Differences between single- and bi-layer graphene reveal underlying mechanism behind Klein tunneling: chirality

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Brief review of Dirac physics

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Graphene and Dirac Linear dispersion simplifies Hamiltonian Electrons in graphene like photons in Dirac QM Pseudospin refers to crystal sublattice Electrons/holes exhibit charge-conjugation symmetry

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Solution to Dirac Equation V 0 = 200 meV V 0 = 285 meV Right: Transmission probability through 100 nm wide barrier as a function of incident angle for electrons with E ~ 80 meV.

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Bilayer Graphene No longer massless fermions Still chiral Four solutions –Propagating and evanescent

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Klein paradox in bilayer graphene Electrons still chiral, so why the different result? Electrons behave as if having spin 1 Scattered into evanescent wave V 0 = 50 meV V 0 = 100 meV Right: Transmission probability through 100 nm wide barrier as a function of incident angle for electrons with E ~ 17 meV.

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Conclusion on mechanism for Klein tunneling Different pseudospins key –Single layer graphene: chiral, behave like spin ½ –Bilayer graphene: chiral, behave like spin 1 –Conventional: no chirality Red: single layer graphene Blue: bilayer graphene Green: Non-chiral, zero-gap semiconductor Tunneling amplitude as function of barrier thickness

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Predicted experimental implications Localization suppression –Possibly responsible for observed minimal conductivity Reduced impurity scattering Diffusive conductor thought experiment with arbitrary impurity distribution

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Proposed experiment Use field effect to modulate carrier concentration Measure voltage drop to observe transmission Dark purple: gated regions Orange: voltage probes Light purple: graphene

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Graphene Heterojunctions Used interference to determine magnitude and phase of T and R –Resistance measurements not as useful Used narrow gates to limit diffusive transport Young, A.F. and Kim, P. Quantum interference and Klein tunneling in graphene heterojunctions. arXiv: v

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Fabry-Perot Etalon Collimation still expected Oscillating component of conductance expected Add B field

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Conductance

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Observed and theoretical phase shifts

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Summary Katsnelson et al. –Klein tunneling possible in graphene due to required conservation of pseudospin –Single layer graphene has T = 1 at normal incidence by electron wave coupling to hole wave –Bilayer graphene has T = 0 at normal incidence by electron coupling to evanescent hole wave –Suggests resistance measurements to observe

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Summary Young et al. –Resistance measurements no good – need phase information –Observe phase shift in conductance to find T = 1

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Additional References Calogeracos, A. and Dombey, N. History and Physics of the Klein paradox. Contemporary Physics 40, (1999) Slonczewski, J.C. and Weiss, P.R. Band Structure of Graphite. Phys. Rev. Lett. 109, 272 (1958). Semenoff, Gordon. Condensed-Matter Simulation of a Three- Dimensional Anomaly. Phys. Rev. Lett. 53, 2449 (1984). Haldane, F.D.M. Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of a Parity Anomaly. Phys. Rev. Lett (1988). Novselov, K.S. et al. Unconventional quantum Hall effect and Berrys phase of 2π in bilayer graphene. Nature Physics 2, 177 (2006) McCann, E. and Falko, V. Landau Level Degeneracy and Quantum Hall Effect in a Graphite Bilayer. Phys. Rev. Lett. 96, (2006) Sakurai, J.J. Advanced Quantum Mechanics. Addison-Wesley Publishing Company, Inc. Redwood City, CA

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