Dirac fermions in Graphite and Graphene Igor Lukyanchuk Amiens University I. Lukyanchuk, Y. Kopelevich et al. - Phys. Rev. Lett. 93, (2004) - Phys. Rev. Lett. 97, (2006) Graphene 2005 Novoselov, et al. Nature 438, 197 (2005 Y. Zhang, et al., Nature 438, 201 (2005

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

2 view of Graphene Nanotube-graphene Graphite-graphene

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November 2005

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:

Linear Dirac spectrum Graphene: (2D graphite monolayer, Semimetal) Special points of Brillouin zone Brillouin zone 4-component (Dirac ????) wave function

"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

Band structure: Slonczewski-McClure Model Graphite: Fitting parameters

holes electrons

ρ(T), HOPG In best samples ρ c / ρ a > (instead of 300 in Kish) 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

Novoselov, K. S. et al. Nature 438, 197 (2005); Zhang, Y. et al. Nature 438, 201 (2005). 2005: Discovery of Quantum Hall Effect in 2D Graphene Due to Dirac fermions … From: - phase analysis - semi-integerr QHE

Quantum Hall Effect, different samples (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. QHE: Graphite vs multi graphene HOPG, Y. Kopelevich et al. PRL´2003

Do Dirac Fermions Exist in Graphite ?

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

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

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:

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

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

Interlayer tunneling spectroscopy of Landau levels in graphite Yu. I. Latyshev 1, A. P. Orlov 1, V. A. Volkov 1, A. V. Irzhak 2, D. Vignolles 3, J. Marcus 4 and T. Fournier 4

OPTICAL PROPERTIES - Visible - Infrared - Raman

Graphite Graphene

C= Reflectance and transmitance coefficients Optical properties are defined by HF conductivity

πα ≈ 2.3%

INFRARED SPECTROSCOPY

2006 Graphite, interpretation, ??? =>

RAMAN SPECTROSCOPY

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

HOPG, Raman

model