Igor Lukyanchuk Amiens University Dirac fermions in Graphite

Igor Lukyanchuk Amiens University Dirac fermions in Graphite and Graphene I. Lukyanchuk, Y. Kopelevich et al. - Phys. Rev. Lett. 93, 166402 (2004) - Phys. Rev. Lett. 97, 256801 (2006) Graphene 2005 Novoselov, et al. Nature 438, 197 (2005 Y. Zhang, et al. , Nature 438, 201 (2005

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

2 view of Graphene Nanotube-graphene Graphite-graphene

November 2005

HP, Intel, IBM… 30 000 $ Wanted: • Graphene active area covering an entire 8 -inch wafer • Carrier mobility of the FET exceeding 15, 000 cm 2/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 d. B of gain and <1 d. B of noise figure • Wafer yield of the low noise amplifiers is more than 90%

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

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

Schroedinger cond-mat physics Dirac cond-mat physics !!! Gap formation, excitonic insulator, weak ferromagnetism, … ? ? ? In magnetic field: 2 component equations Abrikosov Phys. Rev. B 60, 4231 (1999) B 61, 5928 (2000) González, Guinea, Vozmediano, Phys. Rev. Lett. 77, 3589 (1996) Khveshchenko, Phys. Rev. Lett. 87, 206401 (2001); 87, 246802 (2001)

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

Graphite: Fitting parameters Band structure: Slonczewski-Mc. Clure Model

holes electrons

ρ(T), HOPG In best samples ρc/ ρa > 50000 (instead of 300 in Kish) ρa ~ 3 μΩ cm (300 K) n 3 D~3 x 1018 cm-3 n 2 D~1011 cm-2 (1012 -1013 in Graphene) Mobility: μ~106 cm 2/Vs (104 in Graphene)

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

Quantum Hall Effect, different samples (2003)

HOPG, Y. Kopelevich et al. PRL´ 2003 B 0 = 4. 68 T QHE: Graphite vs multi graphene. Few Layer Graphite (FLG) K. S. Novoselov et al. , Science´ 2004 B 0= 20 T, = > n ~ 2 x 1012 cm-2

Do Dirac Fermions Exist in Graphite ?

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

Sd. H: Oscillations of xx (H) (1 st harmonic) Phase depends on : ► Spectrum : Cyclotron mass (detection of e and h) { Normal: Dirac: = 1/2 = 0 ► Dimensionality : d. Hv. A: Oscillations of (H) (1 st harmonic) { 2 D: = 0 3 D: = ± 1/8

Electrons or Holes ? Normal or Dirac ? Experiment: Sd. Hv. A

Comparison of d. Hv. A and Sd. Hv. A Sd. H Pass-band filtering In-phase spectrum electrons Out-phase holes d. Hv. A

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

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

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

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

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 Colors Graphene

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

INFRARED SPECTROSCOPY

2006 Graphite, interpretation, ? ? ? =>

RAMAN SPECTROSCOPY

RAMAN SPECTROSCOPY « Graphene Fingerprint »

Raman spectra of graphite double-resonant graphite D 2. 33 e. V G D‘ G‘

HOPG, Raman

model