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**Chiral Tunneling and the Klein Paradox in Graphene M. I. Katsnelson, K**

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 2006

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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 Dirac’s relativistic quantum mechanics Consider potential step on right Relativistic QM gives V x Pauli pointed out that for x>0, p^2 = (V-E)^2 – m^2, so group velocity vg = dE/dp = p/(E-V). For positive vg and V > E, must take negative root for momentum. This implies the k in the slide. Don’t 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 10^16 V/cm needed in conventional QED, ~10^5 V/cm needed for graphene

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**Brief review of Dirac physics**

R – spin parallel to momentum L – spin anti-parallel to momentum

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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 Charge conjugation symmetry: q -> -q, so right handed electron -> right handed hole Normally electrons and holes would have different wave functions and effective masses, so changing sign of charge would change the system Flipping spin would require short-range potential which would act differently on each sublattice

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**Solution to Dirac Equation**

n = .5 x 10^12 cm^-2, p = 1 x 10^12 (red) and p = 3 x 10^12 cm^-2 (blue) Fermi energy of incident electrons then ~ 80 meV Right: Transmission probability through 100 nm wide barrier as a function of incident angle for electrons with E ~ 80 meV. V0 = 200 meV V0 = 285 meV

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**Bilayer Graphene No longer massless fermions Still chiral**

Four solutions Propagating and evanescent Electrons still described by spinor wavefunctions Normal (i.e. perpendicular) scattering for V < E is the same as for electrons described by Schrodinger equation

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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 n = .5 x 10^12 cm^-2, p = 1 x 10^12 (red) and p = 3 x 10^12 cm^-2 (blue) Fermi energy of incident electrons then ~ 17 meV Spin 1 reference: bilayer graphene has double degenerate landau level at zero energy – double degenerate implies spin 1 bosons? Charge conjugation takes k_electron -> ik_hole (exponentially decaying in barrier) V0 = 50 meV V0 = 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**

Tunneling amplitude as function of barrier thickness Different pseudospins key Single layer graphene: chiral, behave like spin ½ Bilayer graphene: chiral, behave like spin 1 Conventional: no chirality Resonances possible for conventional case, but they oscillate with barrier thickness as opposed to single layer graphene which always has perfect transmission for normally incident electrons and bilayer graphene which always exponential decay for normal incidence. Lack of chirality implies electrons can have their momentum flipped because there is no longer a short range potential required to flip their pseudospin “Spin 1” implies electrons scattered to evanescent hole states in barrier “Spin ½” implies electrons scattered to hole states travelling in opposite direction in barrier Red: single layer graphene Blue: bilayer graphene Green: Non-chiral, zero-gap semiconductor

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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 Still expect collimation Oscillating component of conductance due to FP resonance (no FP resonances for normally incident particles because no reflection) G_osc max for not too big R or T (i.e. marker 1) Add B field - Adds Aharonov-Bohm phase - Reflectionless tunneling manifested as phase shift in transmission resonances

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Conductance θrf = klein back-reflection phase, θwkb = semi-classical phase from pathlength difference, T+- is transmission at + and – interface, lLGR = mean free path in locally gated region, L = width of LGR Perfect transmission at zero incidence angle -> reflection amplitude changes sign with incident angle, pick up π phase shift in reflection amplitude, if R changes sign, it had to be zero, so T=1. Phase shift is seen in phase shift of Gosc Berry phase explanation: pick up pi phase shift when path in momentum space encloses origin – singularity at origin gives Berry phase of pi Marker 1: at low B field, most of Gosc comes from particles with not too big or small ky

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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 Berry’s phase of 2π in bilayer graphene. Nature Physics 2, 177 (2006) McCann, E. and Fal’ko, 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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