f.varchon, l. magaud cond-mat/0702311 band structure calculations eg0 non conducting buffer layer...
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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