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

2 view of Graphene Nanotube-graphene Graphite-graphene

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

Why graphene is interesting ? - Fundamental physics - Applications (carbon-based microelectronics ) 3D 2D 1D0D (Nobel prize)

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

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% $ HP, Intel, IBM… Wanted:

Graphene, history of discovery From ancient time … Graphite in pencils, nuclear reactors, lubrification etc Theory of 2D and 3D graphite (Mc. Clure, Slonczwski, Weiss, Nozieres, Dresselhaus 2 ) 1962 HOPG, synthesis of graphite monocristal (Ubbelohde] 1985 Fullerens [Kroto, Curl, Smalley] 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)

November 2005

Theoretical background

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

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

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

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

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

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

Experimental Methods

ARPES – angle resolved photo emission spectroscopy

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

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

Graphene in Magnetic Field

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

QHE effect : Normal vs Dirac Normal electrons, Dirac- like electrons (expected for graphene) 1 / H  xy 1 / H  xy

Graphene: Half-Integer Quantum Hall Effect Quantisation at =N+1/2 Novoselov et al, Nature 2005 Zhang et al, Nature 2005  xy (4e 2 /h)  xx (k  ) n (10 12 cm -2 ) T

Possible applications: Nanoscopic device: Ballistic regime, ultra-fast electron dynamics etc Graphene: Mobility: μ~10 4 cm 2 /Vs Concentration: n 2D ~10 13 cm -2 -Nanoimprint lithography -Naoribons etc…

Photonics???

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

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

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

Band structure: Slonczewski-McClure Model Graphite: Fitting parameters

holes electrons

EXPERIMENTAL BACKGROUND: old + Y. Kopelevich Statement: = stack of graphene monolayers

ρ(T), HOPG In best samples ρ c / ρ a > 5x10 4 ρ a ~ 3 μΩ cm (300K) n 3D ~3x10 18 cm -3 n 2D ~10 11 cm -2 ( in Graphene) Mobility: μ~10 6 cm 2 /Vs (10 4 in Graphene) Metals: 300μΩ cm, Ioffe-Regel 1000 μΩ cm

Field Induced Metal-Insulator Transition

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

Quantum Hall Effect, different samples (2003)

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

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

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

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

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

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

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

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

Determination of phase  Phase-frequency diagram Spectrum Phase-shift function No information about phase Simultaneous determination of phase and frequency !!! 

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

Band interpretation Normal electrons Dirac holes

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

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

E. Andrei et al. 2007, Nature Phys. Dirac+Normal fermions in HOPG TEM results: Another confirmation of Dirac fermions:

2006 Graphite, interpretation, ??? =>

QHE in graphite and in graphene I.Luk’yanchuk and Y. Kopelevich - Phys. Rev. Lett. 97, (2006)

QHE in graphite R xx R xy Y. Kopelevich et al. Phys. Rev. Lett. 90, (2003)

HOPG, Y. Kopelevich et al. PRL´2003 B0 = 4.68 T Few Layer Graphite (FLG) K.S.Novoselov et al., Science´2004 B 0 = 20 T, = > n ~ 2x10 12 cm -2 Vs. QHE: Graphite vs multi graphene

GRAPHITE: Normal vs Dirac carriers separation Filtering Rxy Rxx B (T) Normal (Integer QHE) Dirac (Semi-integer QHE)

Normal QHE in graphite Bi-layer graphene Novoselov, et al. Nature Physics 2, 177 (2006)

Dirac QHE in graphite Graphene: Y. Zhang, et al., Nature 438, 201 (2005) Graphene: Novoselov, et al. Nature 438, 197 (2005)

► Both types of carriers (Normal and Dirac-like) exist in Graphite. ► They have the same nature as carriers recently identified in mono- and bi-layer ► Graphene. Precursors of both types of QHE exist in Graphite. Conclusion:

Advantage of thin slabs of HOPG graphite: - Easy to fabricate - Much better quality and purity - Easier dopping control - better mechanical stability