1. Electronic properties and the quantum Hall effect in bilayer graphene Vladimir Falko

2. Geim’s group at Manchester Novoselov et al - Nature 438, 197 (2005) Novoselov et al - Nature Physics 2, 177 (2006) Kim-Stormer group at Columbia University NY Zhang et al - PRL 94, 176803 (2005) Zhang et al - Nature 438, 201 (2005) Morpurgo’s group at TU-Delft S-Graphene-S Josephson effect transistor Conference ‘Graphene Week’, MPI-PKS Dresden (2006) Ultra-thin graphitic films: from flakes to micro-devices Novoselov et al - Science 306, 666 (2004)

3. Monolayer and bilayer graphene Berry phase, degeneracy of the zero-energy Landau level, and the QHE McCann, VF - PRL 96, 086805 (2006); Abergel, VF - PR. B 75, 155430 (2007) Novoselov, McCann, Morozov, VF, Katsnelson, Zeitler, Jiang, Schedin, Geim - Nature Physics 2, 177 (2006) Relevance of Fermi surface warping and symmetry-breaking defects for weak localisation and WL magnetoresistance in graphene: monolayer McCann, Kechedzhi, VF, Suzuura, Ando, Altshuler - PRL 97, 146805 (2006) bilayer Kechedzhi, McCann, VF, Altshuler – PRL 98, 176806 (2007) NP junctions: focusing, caustics and Veselago lens for electrons Cheianov, VF - PR B 74, 041403 (2006) Cheianov, VF, Altshuler - Science 315, 1252 (2007) Random resistor network model for the minimal conductivity of graphene with inhomogeneous charge density Cheianov, VF, Altshuler, Aleiner (2007) Specifics of Friedel oscillations in monolayer graphene Cheianov, VF – PRL 97, 226801 (2006)

4. Berry phase, degeneracy of the zero-energy Landau level, and the QHE in bilayer graphene McCann, VF - PRL 96, 086805 (2006); Abergel, VF - PR. B 75, 155430 (2007) Novoselov, McCann, Morozov, VF, Katsnelson, Zeitler, Jiang, Schedin, Geim - Nature Physics 2, 177 (2006) Tight-binding-model analysis leading to ‘chiral’ electrons characterised by the Berry phase Jπ. Landau levels and quantum Hall effect in bilayer graphene. Trigonal warping in bilayer graphene. FIR magneto-optical properties of bilayer graphene.

5. - bonds hybridisation forms strong directed bonds which determine a honeycomb lattice structure. Carbon has 4 electrons in the outer s-p shell orbitals determine conduction properties of graphite

6. Wallace, Phys. Rev. 71, 622 (1947) Slonczewski, Weiss, Phys. Rev. 109, 272 (1958)

8. ARPES: heavily doped graphene synthesized on silicon carbide A. Bostwick et al – Nature Physics, 3, 36 (2007)

9. Bilayer [ Bernal (AB) stacking ]

10. Closest neighbour intra-layer hops

11. Bilayer [ Bernal (AB) stacking ] Closest neighbour approximation (questions about the effect of the next-neighbour hops are welcome!)

12. ARPES: heavily doped bilayer graphene synthesized on silicon carbide T. Ohta et al – Science 313, 951 (2006) (Rotenberg’s group at Berkeley NL) McCann, VF PRL 96, 086805 (2006) Fermi level in undoped bilayer graphene

13. McCann, VF PRL 96, 086805 (2006)

14. Berry phase Jπ (for a monolayer π for a bilayer 2π ) Degree of chirality J

15. Monolayer graphene Bilayer graphene Tight-binding-model analysis: ‘chiral’ electrons and the Berry phase Jπ. Landau levels and quantum Hall effect in bilayer and monolayer graphene. Effect of trigonal warping Infra-red and FIR magneto-optics in graphene.

16. 2D Landau levels semiconductor QW / heterostructure (GaAs/AlGaAs)  xy ( ) -2-3 231 -2 3 1 2 integer QHE in semiconductors a energies / wave functions

17. Landau levels and QHE

18. 2D Landau levels of chiral electrons J=1 monolayer J=2 bilayer valley index also, two-fold real spin degeneracy 4J-degenerate zero-energy Landau level

19. McClure, Phys. Rev. 104, 666 (1956) Haldane, Phys.Rev.Lett. 61, 2015 (1988) Zheng and Ando Phys. Rev. B 65, 245420 (2002) McCann and VF Phys. Rev. Lett. 96, 086805 (2006) 4J-degenerate zero-energy Landau level for electrons with degree of chirality J

21. -2-440 n (10 12 cm -2 ) 2 2 4 6 0  xx (k  ) -2-440 n (10 12 cm -2 ) 2 2 4 6 0  xx (k  ) 1L graphene 2L graphene db EE pp  xy (4e 2 /h) 1 2 -2 -4 0 -3 4 c 3  xy (4e 2 /h) 1 2 -2 -4 0 -3 4 a 3 Unconventional quantum Hall effect and Berry’s phase of 2π in bilayer graphene K.Novoselov, E.McCann, S.Morozov, V.Fal’ko, M.Katsnelson, U.Zeitler, D.Jiang, F.Schedin, A.Geim Nature Physics 2, 177 (2006)

22. How robust is the degeneracy of Landau level in bilayer graphene? Direct inter-layer A hops (warping term, Lifshitz trans.) McCann, VF - PRL 96, 086805 (2006)

23. Monolayer graphene Bilayer graphene Tight-binding-model analysis: ‘chiral’ electrons and the Berry phase Jπ. Landau levels and quantum Hall effect in bilayer and monolayer graphene. Effect of trigonal warping Infra-red and FIR magneto-optics in graphene.

24. Hops between A and via Direct inter-layer hops between A and

25. Inter-layer asymmetry (electric field across the structure, effect of a substrate/overlayer) ‘trigonal warping’ term strong magnetic field weak magnetic field Lifshitz transition

26. 8-fold degenerate zero-energy Landau level

28. How robust is the degeneracy of Landau level in bilayer graphene? Direct inter-layer A hops (warping term, Lifshitz trans.) Distant intra-layer AA,BB hops McCann, VF - PRL 96, 086805 (2006) Inter-layer asymmetry (substrate, gate) Spontaneous symmetry breaking due to e-e interactions

29. T. Ohta et al – Science 313, 951 (‘06) (Rotenberg’s group at Berkeley NL) Interlayer asymmetry gap in bilayer graphene McCann, VF - PRL 96, 086805 (2006) inter-layer asymmetry gap (controlled using electrostatic gate)

30. Lifting degeneracy of Landau levels in bilayer graphene McCann, VF - PRL 96, 086805 (2006) McCann - cond-mat/0608221 longer than next neighbour in- plane AA and BB hops (weak) inter-layer asymmetry (controlled using gate voltage)

31. Monolayer graphene Bilayer graphene Tight-binding-model analysis: ‘chiral’ electrons and the Berry phase Jπ. Landau levels and quantum Hall effect in bilayer and monolayer graphene. Effect of trigonal warping Infra-red and FIR magneto-optics in graphene.

32. Abergel, VF - PR. B 75, 155430 (2007) Infrared absorptions due to inter-LL transitions σ +, M z =+1 σ -, M z =-1

33. Electronic Properties of Graphene-Based Nanostructures ICTP Trieste Italy, 25-29 August 2008 ESF Conference Graphene Week ‘08 Obergurgl, Austria, XX March or YY April 2008 (if we and ESF agree on dates)

37. Berry phase π ‘trigonal warping’ valley symmetry of wave vector K is lower than the hexagonal crystalline symmetry Berry phase 2π

38. Weak localisation correction may be suppressed by the intervalley scattering due to atomically sharp scatterers or edges can be suppressed only by decoherence Berry phase π killed by trigonal warping reflecting the asymmetry in each valley Berry phase 2π

39. Berry phase π ‘slow’ inter-valley scattering: neither WL nor WAL magnetoresistance ‘fast’ inter-valley scattering: usual WL magnetoresistance cut at Weak localisation magnetoresistance E. McCann, K.Kechedzhi, V.Fal'ko, B.Altshuler, in preparation E. McCann, K.Kechedzhi, V.Fal'ko, H.Suzuura, T.Ando, B.Altshuler, cond- mat/0604015

40. S.V. Morozov et al, cond-mat/0603826 (Manchester group) Weak localisation magnetoresistance