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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 view of Graphene. Graphite-graphene. Nanotube-graphene. Outline I) Graphene
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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 view of Graphene Graphite-graphene Nanotube-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
2D 3D 1D 0D (Nobel prize) (Nobel prize) • Why graphene is interesting ? • Fundamental physics • Applications(carbon-based microelectronics )
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…”
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%
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)
Brillouin zone Graphene: Semimetal / Gapless Semiconductor Special points of Brillouin zone Linear Dirac spectrum 4-component (Dirac ????) wave function DOS
Dirac equation Schrödinger equation Free Relativistic Electrons “Dirac fermions" "Normal electrons" Dirac spinor
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
Klein effect: Metal (semiconductor) Semimetal: No electron localization !!! Minimal conductivity
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)
Problems… If 2D Graphene is stable? STM Graphene on a 6H-SiC(0001) substrate
ARPES – angle resolved photo emission spectroscopy
double-resonant graphite 2.33 eV G D‘ G‘ D Raman spectra of graphite
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
Landau quantization: Normal vs Dirac Normal electrons ‘’gap’’ Dirac electrons no ‘’gap’’ !!!
QHE effect : Normal vs Dirac Normal electrons, sxy 1 / H Dirac- like electrons (expected for graphene) sxy 1 / H
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
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
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)
GRAPHITE: 3D semimetal or 2D multi graphene stack ??? - Yes Relation between QHE, Dirac fermions, Berry phase…. In graphite and graphene….
Theoretical background 1950 - 60s Mc.Clure, Slonczewski, Weiss, Nozieres, Dresselhaus, Dresselhaus, + « New Wave » since 2004 (graphene synthesis)
Graphite: Band structure: Slonczewski-McClure Model Fitting parameters
holes electrons
EXPERIMENTAL BACKGROUND: old + Y. Kopelevich 2001-2005 Statement: = stack of graphene monolayers
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
Magneto-resistance R(H) SdH oscillations Linear !!!
Quantum oscillations and QHE in Graphite: Graphite vs Graphene • Luk’yanchuk and Y. Kopelevich • - Phys. Rev. Lett. 93, 166402 (2004)
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.!!!
Generalized formula: 2D, 3D, arbitrary spectrum Lifshitz-Kosevich, Shoenberg, Mineev, Gusynin, Sharapov, Lukyanchuk, Kopelevich where Fermi Surface cross section
Falkovsky (65) – Maslov- Berry phase ► for Normal electrons ► for Dirac electrons
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)
Experiment: SdH dHvA Electrons or Holes ? Normal or Dirac ?
Comparison of dHvA and SdH SdH dHvA Pass-band filtering In-phase SdH dHvA spectrum electrons Out-phase holes
Normal Dirac Fan Diagram for SdH oscillations in Graphite Novoselov, 2005 Multilayer 5nm graphite graphene
f n Determination of phase f Spectrum No information about phase Phase-shift function Simultaneous determination of phase and frequency !!! Phase-frequency diagram
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 (ARPES) Dirac holes Normal electrons
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 ???