Chiral Tunneling and the Klein Paradox in Graphene M.I. Katsnelson, K.S. Novoselov, and A.K. Geim...
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Transcript of Chiral Tunneling and the Klein Paradox in Graphene M.I. Katsnelson, K.S. Novoselov, and A.K. Geim...
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.