Band structure calculations Graphene layers AB stacked on

Band structure calculations Graphene layers AB stacked on Si. C (bulk terminated Si-face) Density functional theory - VASP code EG 0 Non conducting Buffer layer EG 1 Linear E(k) Graphene Electron doped EG 2 bilayer Similar results on the Si. C C-face F. Varchon, L. Magaud cond-mat/0702311

Graphene layers grow over the Si. C surface steps T. Seyller et al. , Surface Science 600, 3906 (2006).

STM image of the first graphene layer N doped (1018 cm-3) 6 H-Si. C(0001) substrate from Cree Research Graphitization in ultra-high vacuum (LEED + Auger) STM experiments at room temperature and 45 K 1 ML graphene P. Mallet and J. Y. Veuillen, cond-mat/0702406

Well ordered layers: Graphene on Si. C C-face Surface X-ray scattering - reflectivity J. Hass, E. Conrad et al. cond-mat/0702540 0 th layer = buffer graphene-substrate bond << the van der Waals distance not conducting (STM, ab initio calculation, photoemission) Smooth layers, atomically flat RMS roughness (over 2µm) s. G <± 0. 005 nm Long structural coherence length Lc>300 nm Layers are not AB stacked graphite graphene layer spacing d is not graphitic (d=0. 337 nm nearly turbostratic). Orientational disorder of the layers preferential orientations equal areas of rotated and non-rotated domains. mixture of stacking. Graphene growths over Si. C-steps (carpet-like) (from STM)

Relative transmission Landau level spectroscopy EF dependence of Landau levels c =1. 03 106 m/s ns≤ 4 1010 cm-2 EF <15 me. V - sharp Dirac cone Not graphite (B) line 1. 0 5 -7 layers B=1. 5 T Transmission Transition energy (me. V) 0. 8 1. 0 9 -10 layers 1. 4 T 50 layers 1. 5 T 0. 8 1. 0 0. 8 1. 5 T 1. 0 0. 8 HOPG ~ mm 100 200 300 400 500 600 Wavenumber (cm)-1 M. Sadowski et al. , PRL 97, 266405 (2006); cond-mat /0704. 0585 700

Pseudospin, chirality 2 equivalent sublattices A and B 2 inequivalent cones at K and K’ E E K' K k k K K’ Intravalley scattering: no back-scattering --> Weak anti-localization (note: long-range scattering preserves AB symmetry) Intervalley scattering: back-scattering --> Weak localization DR (note: warping, point defects break AB symmetry locally ) E. Mc. Cann et al. PRL 97, 146805 (2006) B (T) Phase coherence time : Intervalley scattering time : Warping-induced relaxation time

Weak antilocalization Graphene on C-face 1. 4 K 50 K 100 µmx 1000 µm R=137 ns=4. 6 1012 cm-2 µ=11600 cm 2/Vs 50 K Weak antilocalization 1. 4 K Weak localization tiv=1 ps ; tw=0. 28 ps ; t=0. 26 ps Weak anti-localization observed, in agreement with Dirac particle theory Long-range scatterers dominate (remote ions in substrate) Dephasing : e- e-scattering X. Wu et al. PRL 98, 136810 (2007) tee~C/T C=20 ps. K

Shubnikov de Haas oscillations wide Hall bar Small Sd. H amplitude in wide samples DR/R (%) 0. 1 100 µmx 1000 µm R = 141 /sq µ = 12000 cm 2/Vs 0 -0. 1 Field (T) 3. 8 1012 cm-2 1/B(T -1) Anomalous Berry’s phase Landau level spacing Resistance (Ω) Landau index (n) Landau plot B(T)

Shubnikov de Haas oscillations patterned Hall bar Grenoble High Magnetic Field Lab - D. Maud C. Berger et al. Phys. Stat Sol (a) in press Rxx (Ω/sq) Field (T) 1/B (T-1) 100 m. K DR/R=4% DR (Ω) 1µm x 6. 5µm R= 502 /sq ns= 3. 7 1012 cm-2 µ= 9500 cm/Vs 1/B (T-1)

Shubnikov de Haas oscillations patterned Hall bar 1µm x 5µm R=502 /sq

Magneto-transport of a narrow patterned Hall bar 15 Landau index (n) R(Ω/sq) 200 100 Width=500 nm 0 2 4 6 Field (T) 8 DR/R=10% 0 mobility µ*=27000 cm 2/Vs T(K) 4 6 9 15 35 58 10 5 0 0 0. 2 1/Bn (T-1) 0. 4 Anomalous Berry phase ns= 4 1012 cm-2 EF= 2500 K v. F= 106 m/s C. Berger et al. , Science 312, 1191 (2006)

Landau level spacing Level thermally populated Lifshitz-Kosevich DE Dirac Landau levels dispersion Width used = 270 nm Patterned width = 500 nm Field Confinement : theory experiment D. Mayou (2005) unpublished N. Peres et al. , Phys. Rev. B 73, 241403 (2006) C. Berger et al. , Science 312, 1191 (2006)

Long phase coherence length Quasi 1 d ribbon 0. 5µm x 5µm T(K) 4 6 9 15 35 58 Quantum Interference effects Phase coherence length determined from weak localization and UCF : l =1. 2 µm (4 K) Elastic mean free path ; boundary limited At higher temperatures l (T)~ T-2/3: e-e interactions cause dephasing.

Conductance fluctuations Fluctuations reproducible invariant by reversing field and inverting I-V contacts Width of CF ≈ width of weak localization peak Amplitude ≈ e 2/h Long coherence length 1080 0. 2µm x 1µm R=208 /sq 2 e 2/h R( ) 1060 4 K 1040 1020 90 K 1000 0 2 4 B(T) 6 8

Conductance fluctuations Fluctuations reproducible invariant by reversing field and inverting I-V contacts, Width of UCF ≈ width of weak localization peak, Amplitude ≈ 0. 8 e 2/h 0. 5µmx 5µm R=106 /sq 4 K

High mobility Mobility (m 2/ Vs) mobility as a function of width 5 T=4 K 3 Mobility (m 2/ Vs) µ=10000 -20000 cm 2/Vs at room temperature T=250 K 2 1 1 0. 1 1 10 Width (µm) 10 100 1500 100 R(Ω) 0. 1 1 Width (µm) Reduced width : - Enhanced back-scattering at ribbon edges - reduced back-scattering in quasi-1 D no back-scattering due to anomalous Berry’s phase; (Note that nanotubes are ballistic conductors). 1400 0 T. Ando J. Phys. Soc. Jpn, 67, 2857 (1998) W. de Heer et al. , cond-mat /0704. 0285 T(K) 300

Epitaxial graphene grown on Si. C Highly ordered and well-defined material (structural order and smooth layers on C-face) Transport layer protected (insulating buffer layer beneath - non charged layers above) Layers above are not graphite on C-face (orientational disorder / stacking faults) Graphene properties : Dirac - chiral electrons Sd. H : 1 frequency only, same carrier density as photoemission Anomalous Berry’s phase Weak anti-localization (long-range scattering) Landau level spectrum Long electronic phase coherence length Ballistic properties, high mobility Weak T-dependence Anomalous transport : no quantum Hall effect Small Shubnikov-de Haas oscillations, size dependent periodic and fractal-like spectrum for high mobility samples Electrostatic potentials cannot confine Dirac electrons.

Walt de Heer, Phillip First, Edward Conrad, Alexei Marchenkov, Mei-Yin Chou Xiaosong Wu, Zhimin Song, Xuebin Li, Michael Sprinkle, Nate Brown, Rui Feng, Joanna Haas, Tianbo Li, Greg Rutter, Nikkhil Sarma School of Physics - GATECH, Atlanta Thomas Orlando, Lan Sun, Kristin Thomson School of Chemistry - GATECH, Atlanta Jim Meindl, Raghuna Murali, Farhana Zaman Electrical Engineering - GATECH, Atlanta Gérard Martinez, Marcin Sadowski, Marek Potemski, Duncan Maud, Clément Faugeras CNRS - LCMI, Grenoble Didier Mayou, Laurence Magaud, François Varchon, Cécile Naud, Laurent Lévy, Pierre Mallet, Jean-Yves Veuillen, Vincent Bouchiat CNRS - Institut Néel, Grenoble Patrick Soukiassian, CEA - Saclay Jakub Kiedzerski, MIT-Lincoln Lab Joe Stroscio, Jason Crain, NIST Ted Norris, Michigan University Alessandra Lanzara, University Berkeley