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Klein Tunneling in Graphene

Explore Klein tunneling theories and observations in graphene for massless particles, including behavior at sharp and smooth barriers. Discover applications, resistance studies, and conductance oscillations in ballistic n-p junctions.

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Klein Tunneling in Graphene

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  1. KleinTunnelinginGraphene 物理学院 谷平凡 1500011413

  2. Contents • TheoriesofKleintunneling • Kleintunnelingformasslessparticles • KleintunnelinginGraphene • Sharpbarrier • Smoothbarrier • EvidencesfortheobservationofKleintunneling • Resistance of a smooth ballistic n-p junction • Conductance oscillations and magneto-resistance across a ballistic n-p-n junction

  3. QuantumTunneling:WKBApproximation I II III

  4. KleinTunneling • 1929,proposedbyOscarKlein. • If the potential is of the order of the electronmass, V∼mc2, the barrier is nearly transparent. Moreover, as the potential approaches infinity, the reflection diminishes and the electron is always transmitted. • Application:Rutherford‘sproton-electron model for neutral particles within the nucleus.

  5. KleinTunnelingforMasslessParticles T=1？ R=0？ Matching between electron and positron wavefunctions across the barrier.

  6. ObservationofKleinTunneling • Directobservation:V/cm. • Positronproductionaroundsuper-heavynucleiwithchargeZ ≥ 170. • Evaporationofblackholesthroughgenerationofparticle–antiparticlepairsneartheeventhorizon.

  7. KleinTunnelinginGraphene • Masslessrelativisticelectrons. • Holesbehaveasa condensed- matter equivalent of positrons. • Thespinindexisusuallyreferredtopseudospin. • Thebarriercanbeeasilyproducedbygatevoltage.

  8. KleinTunnelinginSingleLayerGraphene

  9. KleinTunnelinginSingleLayerGraphene The matching between directions of pseudospin σ for quasiparticles inside and outside the barrier results in perfect tunneling. Pseudospin:S=1/2

  10. KleinTunnelinginBilayerGraphene • Massive quasi-particles with a finite density of states at zero energy. • Holesbehaveasa condensed- matter equivalent of positrons. • Thespinindexisusuallyreferredtopseudospin.

  11. KleinTunnelinginSingleLayerGraphene

  12. KleinTunnelinginSingleLayerGraphene The charge conjugation requires a propagating electron with wavevector k to transform into a hole with wavevectorik (rather than −k), which is an evanescent wave inside a barrier. Pseudospin:S=1

  13. TunnelinginNonchiralParticles The drastic difference between the three cases is essentially due to different chiralities or pseudospins of the quasiparticles involved!

  14. Application:HighOn-offRatioFET • Local gates and collimators used in electron optics in 2D gases areneeded. • Singlelayergraphene: • 90°:low resistance and no significant changes in it with changing gate voltage • 45°:higher resistance anda number of tunneling resonances as a function of gate voltage. • Bilayergraphene: • 90°:highresistance • 45°:resonances as a function of gate voltage.

  15. TransmissionProbabilitythroughaPNJunction

  16. LandauerFormulaforBallisticP-NJunction

  17. ConductanceofDiffusiveP-NJunction

  18. FabricationoftheDevice • Two layers of PMMA with different molecular weights are spun on the flake. • Two different exposure doses were used in the areas of the span and pillars of the bridge. • The ‘lift off’ removes PMMA leaving the bridge with a span up to 2 μm sup- ported by two pillars.

  19. TransportMeasurementacrossthePotentialBarrier

  20. TransportMeasurementacrossthePotentialBarrier Ballisticmodelcalculationresults. Experimentresults. Diffusivemodelcalculationresults.

  21. TransportMeasurementacrossthePotentialBarrier • Characteristic length of the junction:2t=40nm. • Meanfreelengthofthreesamples:100nm,75nm,45nm. • (b)、(c)、(d)representtransfercurveofS1,S2,S3,respectively. • Dottedlinesareresultscalculatedbasedonballisticmodel,whilepointsbasedondiffusivemodel.

  22. Fabry-Perot Interference in Graphene Heterojunctions • TheconductionoscillateswithincreasingVtg. • The net backreflection phase exihibitesaπ-shiftwithincreasingmagneticfield.

  23. QuantumInterferencein Graphene Heterojunctions

  24. PhaseShiftofOscillation • They-momentumisn’ttoolarge. • Thecurvedtrajectoriesdominatetheoscillation.

  25. PhaseShiftofOscillation Thephaseshiftsat0.3T,inagreementwiththetheoreticalestimation!

  26. Conclusion • Thebarrierisnearlytransparentforaparticleifitissufficientlylarge,whichiscalledKleintunneling. • Kleintunnelingiscausedbythematching between electron and positron wavefunctions across the barrier,whileingraphene,holesplaytheroleofpositrons. • Theresistance of a smooth ballistic n-p junction,theconductance oscillations and magneto-resistance across a ballistic n-p-n junction confirmtheKleintunneling. Thanksforyourlistening.

  27. References • Chiral tunneling and the Klein paradox in graphene. Nature Physics 2, 620 • Quantum interference and Klein tunnelling in graphene heterojunctions.Nature Physics 5, 222 • Evidence for Klein Tunneling in Graphene p-n Junctions. PRL 102, 026807 (2009) • Klein tunneling in driven-dissipative photonic graphene. Phys. Rev. A 96, 013813 (2017) • Klein tunneling in graphene: optics with massless electrons.PhysicsofCondensedMatter 83(3) • Klein Backscattering and Fabry-Perot Interference in Graphene Heterojunctions. PRL 101, 156804 (2008) • Colloquium: Andreev reflection and Klein tunneling in graphene. Rev. Mod. Phys. 80, 1337

  28. References • Transport Measurements Across a Tunable Potential Barrier in Graphene. PRL 98, 236803 (2007) • Size, Shape, and Low Energy Electronic Structure of Carbon Nanotubes. Physical Review Letters, 03/1997 • Conductance of p-n-p graphene structures with ‘air-bridge’ top gates. Nano Lett., 2008, 8 (7) • Selective transmission of Dirac electrons and ballistic magnetoresistance of n-p junctions in graphene. PHYSICAL REVIEW B 74, 041403(R)(2006) • Electronic Transport and Quantum Hall Effect in Bipolar Graphene p-n-p Junctions. PRL 99, 166804 (2007) • Unconventional quantum Hall effect and Berry’s phase of 2π in bilayer graphene.Nature Physics volume 2, pages 177–180 (2006)

  29. References • Landau level degeneracy and quantum Hall effect in a graphite bilayer. Phys. Rev. Lett. 96, 086805 • Unconventional quantum Hall effect and Berry’s phase of 2 in bilayer graphene. Nature Physics volume 2, pages 177–180 (2006) • Gate-tunable topological valley transport in bilayer graphene. DOI: 10.1038/NPHYS3485 • First Direct Observation of a Nearly Ideal Graphene Band Structure. DOI: 10.1103/PhysRevLett.103.226803 • Electrical Properties of Graphene for Interconnect Applications. Appl. Sci. 2014, 4, 305-317; doi:10.3390/app4020305 • Disorder, pseudospins, and backscattering in carbon nanotubes. Physical Review Letters, 12/1999

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