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.