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

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Chiral Tunneling and the Klein Paradox in
GrapheneM.I. Katsnelson, K.S. Novoselov, and
A.K. GeimNature 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

• 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
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

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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

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

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 0808.0855v3. 2008.
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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 40,313-321 (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. 2015 (1988).
• Novselov, K.S. et al. Unconventional quantum
  Hall effect and Berrys phase of 2p 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, 086805 (2006)
• Sakurai, J.J. Advanced Quantum Mechanics.
  Addison-Wesley Publishing Company, Inc. Redwood
  City, CA. 1984.