Physics of Graphene*

1. Igor Lukyanchuk L.D.Landau Inst. for Theor. Phys. & Amiens University Physics of Graphene* *Monolayer of Graphite, synthesized in 2005, " new wave "in cond-mat physics (>700 publications)

2. 2 view of Graphene Graphite-graphene Nanotube-graphene

3. Outline I) Graphene Why Graphene is interesting Theoretical background History Elaboration Experimental Methods Graphene in magnetic field (Dirac Fermions, Quantum Hall effect) Applications 2) Graphite (vs Graphene) Theory Experiment Dirac Fermions Quantum Hall Effect

4. 2D 3D 1D 0D (Nobel prize) (Nobel prize) • Why graphene is interesting ? • Fundamental physics • Applications(carbon-based microelectronics )

5. QED in a Pencil Trace Nature: “… Erasing electron mass…” “…Electrons in Carbon sheets behave like Massless Particles….” La Recherche: “…La relativité dans une mine de crayon ….” Google: (Dirac Fermions, graphite…) “… Einstein's relativity theory proven with the 'lead' of a pencil…”

6. HP, Intel, IBM… Wanted: 30 000 000 $ • Graphene active area covering an entire 8-inch wafer • Carrier mobility of the FET exceeding 15,000 cm2/V-s • Drain voltage of the FET smaller than 0.25 V • ft and fmax both larger than 500 GHz • W-band low noise amplifier with >15 dB of gain and <1dB of noise figure • Wafer yield of the low noise amplifiers is more than 90%

8. Graphene, history of discovery From ancient time … Graphite in pencils, nuclear reactors, lubrification etc. 50-60 Theory of 2D and 3D graphite (Mc. Clure, Slonczwski, Weiss, Nozieres, Dresselhaus2) 1962 HOPG, synthesis of graphite monocristal (Ubbelohde] 1985 Fullerens [Kroto, Curl, Smalley] 91-93 Nanotubs [Iijima] 2003 Quantum Hall Effect (QHE) in Graphite (!) 2004 Dirac Fermions in Graphite (!) 2005 Prediction of Semi-integer QHE in 2D graphite (Gusynin, Sharapov)

9. November 2005

10. Theoretical background

11. Brillouin zone Graphene: Semimetal / Gapless Semiconductor Special points of Brillouin zone Linear Dirac spectrum 4-component (Dirac ????) wave function DOS

12. Dirac equation Schrödinger equation Free Relativistic Electrons “Dirac fermions" "Normal electrons" Dirac spinor

13. AbrikosovPhys. Rev. B60, 4231 (1999)B61, 5928 (2000) González, Guinea, Vozmediano, Phys. Rev. Lett. 77, 3589 (1996) Khveshchenko, Phys. Rev. Lett. 87, 206401 (2001); 87, 246802 (2001) Schroedinger cond-mat physics Dirac cond-mat physics !!! Gap formation, excitonic insulator, weak ferromagnetism, … ??? In magnetic field: 2 component equations

14. Klein effect: Metal (semiconductor) Semimetal: No electron localization !!! Minimal conductivity

16. Graphene elaboration, 2 methods - Exfoliation Technique K.S. Novoselov et al;, Science 306, 666 , (2004). EPITAXIAL GRAPHENE ON SIC D.Mayou, V. Olevano, L. Levy, P. Darancet (IN), B. Ngoc Nguyen, N. Wipf, C. Berger, E. Conrad W. de Heer (Gatech, Atlanta, USA)

17. Problems… If 2D Graphene is stable? STM Graphene on a 6H-SiC(0001) substrate

18. Experimental Methods

19. ARPES – angle resolved photo emission spectroscopy

20. double-resonant  graphite 2.33 eV G D‘ G‘ D Raman spectra of graphite

21. Scanning force microscope 2 1 1 m Spatially resolved Raman spectroscopy of single- and few-layer graphene D G D‘ 2 double-layer graphene single-layer graphene Experiment: Davy Graf, Françoise Molitor, and Klaus EnsslinSolid State Physics, ETH Zürich, Switzerland Christoph Stampfer, Alain Jungen, and Christofer HieroldMicro and Nanosystems, ETH Zürich Theory: Ludger Wirtz Institute for Electronics, Microelectronics, and Nanotechnology, Lille 1

22. Graphene in Magnetic Field

23. Landau quantization: Normal vs Dirac Normal electrons ‘’gap’’ Dirac electrons no ‘’gap’’ !!!

24. QHE effect : Normal vs Dirac Normal electrons, sxy 1 / H Dirac- like electrons (expected for graphene) sxy 1 / H

25. xx (k) xy (4e2/h) 3.5 12T 2.5 10 1.5 0.5 -0.5 5 -1.5 -2.5 -3.5 0 -4 -2 0 2 4 n (1012 cm-2) Graphene: Half-Integer Quantum Hall Effect Quantisation at =N+1/2 Novoselov et al, Nature 2005 Zhang et al, Nature 2005

26. Graphene: Mobility: μ~104cm2/Vs • Nanoimprint lithography • Naoribons etc… Concentration: n2D~1013 cm-2 Possible applications: Nanoscopic device: Ballistic regime, ultra-fast electron dynamics etc

27. Photonics???

28. Igor Luk’yanchuk, Yakov Kopelevich Dirac Fermions in Graphite and Graphene: Implications to QHE Experiment: Kopelevich et al. - Phys. Rev. Lett. 90, 156402 (2003) Interpretation and analysis - Phys. Rev. Lett. 93, 166402 (2004) - Phys. Rev. Lett. 97, 256801 (2006) Graphite (2004)

29. GRAPHITE: 3D semimetal or 2D multi graphene stack ??? - Yes Relation between QHE, Dirac fermions, Berry phase…. In graphite and graphene….

30. Theoretical background 1950 - 60s Mc.Clure, Slonczewski, Weiss, Nozieres, Dresselhaus, Dresselhaus, + « New Wave » since 2004 (graphene synthesis)

31. Graphite: Band structure: Slonczewski-McClure Model Fitting parameters

32. holes electrons

33. EXPERIMENTAL BACKGROUND: old + Y. Kopelevich 2001-2005 Statement: = stack of graphene monolayers

34. ρ(T), HOPG In best samples ρc/ ρa > 5x104 ρa ~ 3 μΩ cm (300K) n3D~3x1018 cm-3 n2D~1011 cm-2 (1012-1013 in Graphene) Mobility: μ~106cm2/Vs (104 in Graphene) Metals: 300μΩ cm, Ioffe-Regel 1000 μΩ cm

35. Field Induced Metal-Insulator Transition

36. Magneto-resistance R(H) SdH oscillations Linear !!!

37. Quantum Hall Effect, different samples (2003)

38. Quantum oscillations and QHE in Graphite: Graphite vs Graphene • Luk’yanchuk and Y. Kopelevich • - Phys. Rev. Lett. 93, 166402 (2004)

39. Quantum oscillations: What is usually studied ? Profile: Information about e-e interaction (in 2D) Damping: Information about e-scattering (Dingle factorG ) Period: Information about Fermi surface cross section S(e) and Phase ??? … difficult to extract We propose the method.!!!

40. Generalized formula: 2D, 3D, arbitrary spectrum Lifshitz-Kosevich, Shoenberg, Mineev, Gusynin, Sharapov, Lukyanchuk, Kopelevich where Fermi Surface cross section

41. Falkovsky (65) – Maslov- Berry phase ► for Normal electrons ► for Dirac electrons

42. Phase depends on : { ► Spectrum : Normal:  = 1/2 Dirac: = 0 { ► Dimensionality : 2D: = 0 3D:  = ± 1/8 SdH: Oscillations of xx (H) (1st harmonic) Cyclotron mass (detection of e and h) dHvA: Oscillations of  (H) (1st harmonic)

43. Experiment: SdH dHvA Electrons or Holes ? Normal or Dirac ?

44. Comparison of dHvA and SdH SdH dHvA Pass-band filtering In-phase SdH dHvA spectrum electrons Out-phase holes

45. Normal Dirac Fan Diagram for SdH oscillations in Graphite Novoselov, 2005 Multilayer 5nm graphite graphene

46. f n Determination of phase f Spectrum No information about phase Phase-shift function Simultaneous determination of phase and frequency !!! Phase-frequency diagram

47. Result: spectrum of quantum oscillations in HOPG Normal electrons Dirac holes e h Rxx, Kish

48. Band interpretation Normal electrons Dirac holes

49. 2006 Confirmation: Angle Resolved Photoemission Spectroscopy (ARPES) Dirac holes Normal electrons

50. Sh > Se Problems with band interpretation Se > Sh 1) 2) H: point Dirac Spectrum Phase volume ~0 no Dirac Fermions should be seen in experiment holes Normal Spectrum electrons Another possibility: Independent layers ???