Band structure calculations
Graphene layers AB stacked on SiC (bulk terminated Si-face)
Density functional theory - VASP code
EG0
Non conducting
Buffer layer
EG1
Linear E(k)
Graphene
Electron doped
Similar results on the
SiC C-face
EG2
bilayer layer
F.Varchon, L. Magaud cond-mat/0702311
Graphene layers grow over the SiC surface steps
T. Seyller et al. , Surface Science 600, 3906 (2006).
STM image of the first graphene layer
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
Well ordered layers: Graphene on SiC C-face
Surface X-ray scattering - reflectivity
J. Hass, E. Conrad et al. cond-mat/0702540
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) sG <±0.005nm
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 SiC-steps (carpet-like)
(from STM)
Landau level spectroscopy
B dependence of Landau levels
EF

c =1.03 106 m/s
ns≤4 1010 cm-2
EF <15 meV - sharp Dirac cone
Not graphite
(B) line
1.0
5-7 layers
B=1.5T
Transition energy (meV)
Transmission
0.8
1.0
9-10 layers
1.4T
50 layers
1.5T
0.8
1.0
0.8
1.5T
1.0
0.8
HOPG
~ mm
100
B(T1/ 2 )
200
300
400
500
600
Wavenumber (cm)-1
M.Sadowski et al., PRL 97, 266405 (2006);cond-mat /0704.0585
700
Pseudospin, chirality
2 inequivalent cones at K and K’
2 equivalent sublattices A and B
E
E
K'
K
k
k

K’
K
s .p
Intravalley scattering: no back-scattering
--> Weak anti-localization
p
s .p
1
p
 1
(note: long-range scattering preserves AB symmetry)
Intervalley scattering: back-scattering
--> Weak localization


DR
(note: warping, point defects break AB symmetry locally )
E. McCann et al. PRL 97, 146805 (2006)
B
(T) Phase coherence time
: Intervalley scattering time
: Warping-induced relaxation time
Weak antilocalization
Graphene on C-face
1.4K
50K
100 µmx1000 µm
R=137 W
ns=4.6 1012cm-2
µ=11600 cm2/Vs
50K
Weak antilocalization
1.4K

Weak localization
tiv=1ps ; tw=0.28ps ; t0.26ps
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)
tee~C/T
C=20ps.K
Shubnikov de Haas oscillations wide Hall bar
Small SdH amplitude in wide samples
DR/R (%)
0.1
100 µmx1000 µm
R = 141 W/sq
µ = 12000 cm2/Vs
0
-0.1
Field (T)
Landau level spacing
Landau index (n)
Landau plot
3.8 1012 cm-2
1/B(T -1)
Anomalous Berry’s phase
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
Field (T)
Rxx (Ω/sq)
1µm x 6.5µm
R= 502 W/sq
ns= 3.7 1012cm-2
µ= 9500 cm/Vs
1/B (T-1)
100 mK
1/B (T-1)
Shubnikov de Haas oscillations patterned Hall bar
1µm x 5µm
R=502 W/sq
Magneto-transport of a narrow patterned Hall bar
R(Ω/sq)
15
200
10µm
100
Width=500 nm
0
2
4
6
Field (T)
5
8
0
DR/R=10%
0
10
mobility µ*=27000 cm2/Vs
T(K)
4
6
9
15
35
58
0
0.2
1/Bn (T-1)
0.4
Anomalous Berry phase
ns= 4 1012cm-2
EF= 2500 K
vF= 106 m/s
C.Berger et al. , Science 312, 1191 (2006)
Landau level spacing
u
2 2 k B T
;u 
Level thermally populated Lifshitz-Kosevich A(T)  A0
sinh( u)
DE(B)
7T n=5
5T n=7
1T n=23-24
0
Dirac Landau levels dispersion
E n (B)   0 2enB

DE
1
0
20
40

60
Temperat ure ( K)
300
Width used = 270 nm
Patterned width = 500 nm
Field
200
Confinement :
E n (W )   0 k   0
100
n
W
theory
experiment
0
0
2
4
Field ( T)
6
8
 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 1d 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 ≈ e2/h
Long coherence length
1080
2e2/h
0.2µm x 1µm
R=208 W/sq
R(W)
1060
4K
1040
1020
90K
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 e2/h
H
1120
0.5µmx5µm
R=106 W/sq
4K
Resist ance ( W)
H
1100
1080
1060
-4
-3
-2
-1
0
1
Field(Tesla)
2
3
4
High mobility
Mobility (m2/ Vs)
mobility as a function of width
5
T=4 K
3
Mobility (m2/ Vs)
µ=10000-20000 cm2/Vs at room temperature
T=250K
2
1
1
0.1
1
10
100
Width (µm)
1500
0.1
1
10
Width (µm)
100
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).
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 SiC
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
SdH : 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