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Chiral Tunneling and the Klein Paradox in Graphene
M.I. Katsnelson, K.S. Novoselov, and A.K. GeimNature Physics Volume 2 September 2006
John Watson
Outline
• Background, main result
• Details of paper
• Authors’ proposed future work
• Reported experimental observations
• Summary
Background
• Klein paradox implied by Dirac’s relativistic quantum mechanics
• Consider potential step on right– Relativistic QM gives
V
x
• Don’t get non-relativistic exponential decay
Calogeracos, A.; Dombey, N.. Contemporary Physics, Sep/Oct99, Vol. 40 Issue 5
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
Brief review of Dirac physics
LR
LR
k
k
tiH
I
ImcicH
,
0
0
0
0,2
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
fivH
Solution to Dirac Equation
22220 / yfx kvVEq
V0 = 200 meV
V0 = 285 meV
Right: Transmission probability through 100 nm wide barrier as a function of incident angle for electrons with E ~ 80 meV.
22
2
sincos1
cos
DqT
x
Bilayer Graphene
• No longer massless fermions
• Still chiral
• Four solutions – Propagating and
evanescent
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.
EVV
ET 0
2
0
,2sin
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 grapheneGreen: Non-chiral, zero-gap semiconductor
Tunneling amplitude as function of barrier thickness
Predicted experimental implications
• Localization suppression– Possibly responsible for
observed minimal conductivity
• Reduced impurity scattering
Diffusive conductor thought experiment with arbitrary impurity distribution
Proposed experiment
• Use field effect to modulate carrier concentration
• Measure voltage drop to observe transmission
Dark purple: gated regionsOrange: voltage probesLight purple: graphene
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.
Fabry-Perot Etalon• Collimation still expected
• “Oscillating” component of conductance expected
• Add B field
Conductance
Observed and theoretical phase shifts
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
Summary
• Young et al.– Resistance measurements no good – need
phase information– Observe phase shift in conductance to find T
= 1
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 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, 086805 (2006)
• Sakurai, J.J. Advanced Quantum Mechanics. Addison-Wesley Publishing Company, Inc. Redwood City, CA. 1984.