F.Varchon, L. Magaud cond-mat/0702311

Band structure calculations

EG0Non conducting

Buffer layer

EG1

Linear E(k) Graphene

Electron doped

EG2

bilayer layer

Graphene layers AB stacked on SiC (bulk terminated Si-face)Density functional theory - VASP code

Similar results on the SiC C-face

Graphene layers grow over the SiC surface steps

T. Seyller et al. , Surface Science 600, 3906 (2006).

N doped (1018 cm-3) 6H-SiC(0001) substrate from Cree Research

Graphitization in ultra-high vacuum (LEED + Auger)STM experiments at room temperature and 45K

1ML graphene

P. Mallet and J.Y. Veuillen, cond-mat/0702406

STM image of the first graphene layer

0th 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) G <±0.005nm

Long structural coherence length Lc>300 nm

Layers are not AB stacked graphite

graphene layer spacing is not graphitic

(=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 SiC-steps (carpet-like) (from STM)

Well ordered layers: Graphene on SiC C-face

J. Hass, E. Conrad et al. cond-mat/0702540Surface X-ray scattering - reflectivity

M.Sadowski et al., PRL 97, 266405 (2006);cond-mat /0704.0585

€

B(T1/ 2)

Tra

nsiti

on e

nerg

y (m

eV)

Re l

a tiv

e tr

ansm

issi

onLandau level spectroscopy

Wavenumber (cm)-1

dependence of Landau levelsc =1.03 106 m/sns≤4 1010 cm-2

EF <15 meV - sharp Dirac cone Not graphite€

B

100 200 300 400 500 600 700

1.5T

1.5T

1.4T

0.8

1.0

1.0

1.0

0.8

0.8

0.8

HOPG~ m

50 layers

5-7 layers

9-10 layers

1.0B=1.5T

Tra

nsm

issi

on

(B) line

EF

2 equivalent sublattices A and B

Pseudospin, chirality

KK'

2 inequivalent cones at K and K’

(T) Phase coherence time : Intervalley scattering time : Warping-induced relaxation time

E. McCann et al. PRL 97, 146805 (2006)

Intravalley scattering: no back-scattering --> Weak anti-localization (note: long-range scattering preserves AB symmetry)Intervalley scattering: back-scattering --> Weak localization (note: warping, point defects break AB symmetry locally )

E

k

K

€

.p

p=1

€

.p

p= −1

R

B

E

k

K’

iv=1ps ; w=0.28ps ; ps

Weak antilocalization

€

Weak antilocalization

Weak localization

ee~C/TC=20ps.K

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. PRL98, 136810 (2007)

100 µmx1000 µm

R=137 ns=4.6 1012cm-2

µ=11600 cm2/Vs

1.4K

50K

Graphene on C-face

50K

1.4K

Shubnikov de Haas oscillations wide Hall bar

100 µmx1000 µm

Anomalous Berry’s phase

Landau plot

1/B(T -1)

La

nd

au

ind

ex

(n)

3.8 1012 cm-2

R/R

(%

)

0

-0.1

0.1

Field (T)

Small SdH amplitude in wide samples

R= 141 /sq µ = 12000 cm2/Vs

Res

ist a

nce

( Ω)

B(T)

Landau level spacing

R (

Ω)

Rxx

(Ω/s

q)

Field (T)

1/B (T-1)

R/ R

=4%

Shubnikov de Haas oscillations patterned Hall bar

1µm x 6.5µmR= 502sqns= 3.7 1012cm-2

µ= 9500 cm/Vs

1/B (T-1)

Grenoble High Magnetic Field Lab - D.MaudC.Berger et al. Phys.Stat Sol (a) in press

100 mK

1µm x 5µmR=502sq

Shubnikov de Haas oscillations patterned Hall bar

Magneto-transport of a narrow patterned Hall bar

C.Berger et al. , Science 312, 1191 (2006)

T(K)469

153558

Width=500 nm

10µm

0 2 4 6 80

100

200

Field (T)

R(Ω

/sq

)

R/R

=10

%

0

5

10

15

0 0.2 0.41/Bn (T-1)

La

nd

au

in

de

x (n

)

Anomalous Berry phasens= 4 1012cm-2

EF= 2500 KvF= 106 m/s

mobility µ*=27000 cm2/Vs

Landau level spacing

C.Berger et al. , Science 312, 1191 (2006)

0 2 4 6 80

100

200

300

Field (T)

0 20 40 600

1 7T n=5

1T n=23-245T n=7

Temperature (K)

€

A(T) = A0

u

sinh(u);u =

2π 2kBT

ΔE(B)Level thermally populated Lifshitz-Kosevich

D. Mayou (2005) unpublished N. Peres et al. , Phys. Rev. B 73, 241403 (2006)

Confinement :

€

En (W ) = hν 0k = hν 0

nπ

W

€

En (B) = ν 0 2enB

Dirac Landau levels dispersion

Field

E

Width used = 270 nmPatterned width = 500 nm

experimenttheory

Phase coherence length determined from weak localization and UCF : l=1.2 µm (4 K)Elastic mean free path ; boundary limitedAt higher temperatures l(T)~ T-2/3: e-e interactions cause dephasing.

T(K)469

153558

Long phase coherence length

Quantum Interference effects

0.5µm x 5µm

Quasi 1d ribbon

Conductance fluctuations

Fluctuations reproducible invariant by reversing field and inverting I-V contactsWidth of CF ≈ width of weak localization peakAmplitude ≈ e2/hLong coherence length

0.2µm x 1µmR=208 /sq

2e2/h

R

1080

1060

1040

1020

1000

0 2 4 6 8B(T)

4K

90K

Conductance fluctuations

-4 -3 -2 -1 0 1 2 3 4

1060

1080

1100

1120

Field(Tesla)

Resistance ()

H

H

Fluctuations reproducible invariant by reversing field and inverting I-V contacts,Width of UCF ≈ width of weak localization peak,Amplitude ≈ 0.8 e2/h

4K0.5µmx5µmR=106 /sq

mobility as a function of width

µ=10000-20000 cm2/Vs at room temperature

Reduced width :- Enhanced back-scattering at ribbon edges- reduced back-scattering in quasi-1D no back-scattering due to anomalous Berry’s phase; (Note that nanotubes are ballistic conductors).

High mobility

T=4 K

Mob

ility

(m

2/

Vs)

Width (µm)

1

3

5

10.1 10 100

T=250K

Width (µm)

Mob

ility

(m

2/

Vs)

1

2

10.1 10 100

T. Ando J. Phys. Soc. Jpn, 67, 2857 (1998)

1500

14000 300

R( Ω

)T(K)

W.de Heer et al., cond-mat /0704.0285

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 electronsSdH : 1 frequency only, same carrier density as photoemissionAnomalous Berry’s phaseWeak anti-localization (long-range scattering)Landau level spectrum

Long electronic phase coherence lengthBallistic properties, high mobilityWeak T-dependence

Anomalous transport : no quantum Hall effectSmall Shubnikov-de Haas oscillations, size dependentperiodic and fractal-like spectrum for high mobility samples

Electrostatic potentials cannot confine Dirac electrons.

Epitaxial graphene grown on SiC

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 ThomsonSchool of Chemistry - GATECH, Atlanta

Jim Meindl, Raghuna Murali, Farhana ZamanElectrical 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