Physics of 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 (gt700
publications)
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2 view of Graphene
Graphite-graphene
Nanotube-graphene
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Outline I) Graphene Why Graphene is
interesting Theoretical background History Elab
oration Experimental Methods Graphene in
magnetic field (Dirac Fermions, Quantum Hall
effect) Applications 2) Graphite (vs
Graphene) Theory Experiment Dirac
Fermions Quantum Hall Effect
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2D
3D
1D
0D
(Nobel prize)
(Nobel prize)

• Why graphene is interesting ?
• Fundamental physics
• Applications (carbon-based microelectronics )

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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
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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 gt15 dB of
gain and lt1dB of noise figure Wafer yield of
the low noise amplifiers is more than 90
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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)
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November 2005
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Theoretical background
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Graphene Semimetal / Gapless Semiconductor
Special points of Brillouin zone
Linear Dirac spectrum
4-component (Dirac ????) wave function
DOS
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Free Relativistic Electrons
Dirac fermions"
"Normal electrons"
Dirac spinor
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Schroedinger cond-mat physics
Dirac cond-mat physics !!!
Gap formation, excitonic insulator, weak
ferromagnetism, ???
In magnetic field 2 component equations
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Minimal conductivity
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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)
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Problems
If 2D Graphene is stable?
STM
Graphene on a 6H-SiC(0001) substrate
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Experimental Methods
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ARPES angle resolved photo emission
spectroscopy
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Raman spectra of graphite
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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
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Graphene in Magnetic Field
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Landau quantization Normal vs Dirac
Normal electrons
gap
Dirac electrons
no gap !!!
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QHE effect Normal vs Dirac
Normal electrons,
sxy
1 / H
Dirac- like electrons (expected for graphene)
sxy
1 / H
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Graphene Half-Integer Quantum Hall Effect
Quantisation at ?N1/2
Novoselov et al, Nature 2005 Zhang et al, Nature
2005
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Possible applications
Nanoscopic device Ballistic regime, ultra-fast
electron dynamics etc
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Photonics???
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Igor Lukyanchuk, 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)
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GRAPHITE 3D semimetal or 2D multi graphene
stack ???
- Yes
Relation between QHE, Dirac fermions, Berry
phase. In graphite and graphene.
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Theoretical background 1950 - 60s Mc.Clure,
Slonczewski, Weiss, Nozieres, Dresselhaus,
Dresselhaus,
New Wave since 2004 (graphene synthesis)
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Graphite
Band structure Slonczewski-McClure Model
Fitting parameters
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holes
electrons
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EXPERIMENTAL BACKGROUND old Y. Kopelevich
2001-2005
Statement stack of graphene monolayers
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?(T), HOPG
In best samples ?c/ ?a gt 5x104 ?a 3 µO cm
(300K)
n3D3x1018 cm-3 n2D1011 cm-2 (1012-1013 in
Graphene) Mobility µ106cm2/Vs (104 in
Graphene)
Metals 300µO cm, Ioffe-Regel 1000 µO cm
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Field Induced Metal-Insulator Transition
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Magneto-resistance R(H)
SdH oscillations
Linear !!!
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Quantum Hall Effect, different samples (2003)
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Quantum oscillations and QHE in Graphite
Graphite vs Graphene

• Lukyanchuk and Y. Kopelevich
• - Phys. Rev. Lett. 93, 166402 (2004)

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Quantum oscillations What is usually studied ?
Profile Information about e-e interaction (in
2D)
Damping Information about e-scattering (Dingle
factor G )
Period Information about Fermi surface cross
section S(e)
and Phase ??? difficult to extract We propose
the method.!!!
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Generalized formula 2D, 3D, arbitrary spectrum
Lifshitz-Kosevich, Shoenberg, Mineev, Gusynin,
Sharapov, Lukyanchuk, Kopelevich
where
Fermi Surface cross section
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Falkovsky (65) Maslov- Berry phase
? for Normal electrons
? for Dirac electrons
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SdH Oscillations of ?xx (H) (1st harmonic)
Cyclotron mass (detection of e and h)
dHvA Oscillations of ? (H) (1st harmonic)
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Electrons or Holes ? Normal or Dirac ?
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Comparison of dHvA and SdH
SdH
dHvA
Pass-band filtering
In-phase
SdH
dHvA
spectrum
electrons
Out-phase
holes
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Fan Diagram for SdH oscillations in Graphite
Novoselov, 2005
Multilayer 5nm graphite
graphene
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Determination of phase f
Spectrum
No information about phase
Phase-shift function
Simultaneous determination of phase and frequency
!!!
Phase-frequency diagram
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Result spectrum of quantum oscillations in HOPG
Normal electrons
Dirac holes
e
h
Rxx, Kish
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Band interpretation
Normal electrons
Dirac holes
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2006 Confirmation Angle Resolved Photoemission
Spectroscopy
(ARPES)
Dirac holes
Normal electrons
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Sh gt Se
Problems with band interpretation
Se gt 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 ???
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Another confirmation of Dirac fermions
DiracNormal fermions in HOPG TEM results
E. Andrei et al. 2007, Nature Phys.
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2006
Graphite, interpretation, ??? gt
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QHE in graphite and in graphene

• Lukyanchuk and Y. Kopelevich
• - Phys. Rev. Lett. 97, 256801 (2006)

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QHE in graphite
Rxy
Rxx
Y. Kopelevich et al. Phys. Rev. Lett. 90, 156402
(2003)
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QHE Graphite vs multi graphene
HOPG, Y. Kopelevich et al. PRL2003
B0 4.68 T
Vs.
Few Layer Graphite (FLG) K.S.Novoselov et al.,
Science2004
B0 20 T, gt n 2x1012 cm-2
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Normal (Integer QHE)
GRAPHITE Normal vs Dirac carriers separation
Rxy
Dirac (Semi-integer QHE)
Rxx
Filtering
B (T)
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Normal QHE in graphite
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Dirac QHE in graphite
Graphene Y. Zhang, et al., Nature 438, 201
(2005)
Graphene Novoselov, et al. Nature 438, 197
(2005)
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Conclusion
? Both types of carriers (Normal and Dirac-like)
exist in Graphite. ? They have the same nature
as carriers recently identified in mono- and
bi-layer ? Graphene. Precursors of both types
of QHE exist in Graphite.
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Advantage of thin slabs of HOPG graphite

• Easy to fabricate
• Much better quality and purity
• Easier dopping control
• better mechanical stability