1. 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

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

3. 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

4. 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

5. Brief review of Dirac physics
R – spin parallel to momentum
L – spin anti-parallel to momentum

6. 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

7. 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

8. 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

9. 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.

10. 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

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

12. 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

13. 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

14. 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

15. 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

16. Observed and theoretical phase shifts

17. 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

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

19. 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