Download - Physics of Graphene*
Physics of Graphene*
Igor Lukyanchuk
* Monolayer of Graphite, synthesized in 2005, " new wave " in cond-mat physics (>700 publications)
L.D.Landau Inst. for Theor. Phys. & Amiens University
2 view of Graphene
Nanotube-grapheneGraphite-graphene
Outline
I) Graphene
Why Graphene is interestingTheoretical backgroundHistoryElaborationExperimental MethodsGraphene in magnetic field (Dirac Fermions, Quantum Hall effect)Applications
2) Graphite (vs Graphene)
TheoryExperimentDirac FermionsQuantum Hall Effect
Why graphene is interesting ?
- Fundamental physics
- Applications (carbon-based microelectronics )
3D 2D 1D 0D
(Nobel prize) (Nobel prize)
QED in a Pencil Trace
“… Einstein's relativity theory proven with the 'lead' of a pencil … ”
Google: (Dirac Fermions, graphite…)
“…La relativité dans une mine de crayon ….”
La Recherche:
“…Electrons in Carbon sheets behave like Massless Particles….”
“… Erasing electron mass…” Nature:
• 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 >15 dB of gain and <1dB of noise figure• Wafer yield of the low noise amplifiers is more than 90%
30 000 000 $
HP, Intel, IBM…
Wanted:
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)
November 2005
Theoretical background
Linear Dirac spectrum
Graphene: Semimetal / Gapless Semiconductor
Special points of Brillouin zone
Brillouin zone 4-component (Dirac ????) wave function
DOS
"Normal electrons" “Dirac fermions"
Schrödinger equation Dirac equation
Dirac spinor
Free Relativistic Electrons
Gap formation, excitonic insulator, weak ferromagnetism, … ???
Abrikosov Phys. Rev. B60, 4231 (1999) B61, 5928 (2000)
Khveshchenko, Phys. Rev. Lett. 87, 206401 (2001); 87, 246802 (2001)
González, Guinea, Vozmediano, Phys. Rev. Lett. 77, 3589 (1996)
In magnetic field: 2 component equations
Schroedinger cond-mat physics
Dirac cond-mat physics !!!
Klein effect:
U(x)
U(x)
Ef
Ef
electron
electron
holehole
Metal (semiconductor)
Semimetal:
No electron localization !!!
Minimal conductivity
- Exfoliation Technique
K.S. Novoselov et al;, Science 306, 666 , (2004).
EPITAXIAL GRAPHENE ON SIC
Graphene elaboration, 2 methods
D.Mayou, V. Olevano, L. Levy, P. Darancet (IN), B. Ngoc Nguyen, N. Wipf, C. Berger, E. Conrad W. de Heer (Gatech, Atlanta, USA)
Graphene on a 6H-SiC(0001) substrate
STM
Problems…
If 2D Graphene is stable?
Experimental Methods
ARPES – angle resolved photo emission spectroscopy
E =
2.3
3 e
V
0.4 0 0.20.400.2-2
-1
0
1
2
E (
eV
)
KK MM M
q’
q
q’’A
B C
double-resonant
0 1500 3000Raman shift (cm-1)
Inte
nsity
(a.
u) graphite 2.33 eV
D
G
D‘ G‘
Raman spectra of graphite
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
1
2
Scanning force microscope
1 mIn
tens
ity (
a.u)
D
single-layer graphene
double-layer graphene2
1
G D‘
1200 1600 2000 2400 2800Raman shift (cm-1)
Inte
nsity
(a.
u)
Spatially resolved Raman spectroscopy of single- and few-layer graphene
Graphene in Magnetic Field
Normal electrons
Dirac electrons
Landau quantization: Normal vs Dirac
‘’gap’’
no ‘’gap’’ !!!
QHE effect : Normal vs Dirac
Normal electrons,
Dirac- like electrons(expected for graphene)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
1/H
xy
1 / H
xy
1 / H
xy
Graphene: Half-Integer Quantum Hall Effect
Quantisation at =N+1/2
Novoselov et al, Nature 2005Zhang et al, Nature 2005
xy (4e2/h)xx (k)
n (1012 cm-2)
-2-4 40 2
5
10
0
1.5
-1.5
-2.5
-3.5
-0.5
2.5
3.5
0.5
12T
Possible applications:
Nanoscopic device: Ballistic regime, ultra-fast electron dynamics etc
Graphene: Mobility: μ~104cm2/Vs
Concentration: n2D~1013 cm-2
-Nanoimprint lithography-Naoribons etc…
Photonics???
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)
Igor Luk’yanchuk, Yakov Kopelevich
Graphite (2004)
GRAPHITE: 3D semimetal or 2D multi graphene stack ??? - Yes
Relation between QHE, Dirac fermions, Berry phase….In graphite and graphene….
Theoretical background
1950 - 60s
Mc.Clure, Slonczewski, Weiss,
Nozieres, Dresselhaus, Dresselhaus,
+ « New Wave » since 2004 (graphene synthesis)
Band structure: Slonczewski-McClure Model
Graphite:F
ittin
g pa
ram
eter
s
holes
electrons
EXPERIMENTAL BACKGROUND:
old + Y. Kopelevich 2001-2005
Statement: = stack of graphene monolayers
ρ(T), HOPG
In best samples
ρc/ ρa > 5x104
ρa ~ 3 μΩ cm (300K)
n3D~3x1018 cm-3
n2D~1011 cm-2 (1012-1013 in Graphene)
Mobility:
μ~106cm2/Vs (104 in Graphene)
Metals: 300μΩ cm, Ioffe-Regel 1000 μΩ cm
Field Induced Metal-Insulator Transition
Magneto-resistance R(H)
Linear !!!
SdH oscillations
Quantum Hall Effect, different samples (2003)
Quantum oscillations and QHE in Graphite:
Graphite vs Graphene
I. Luk’yanchuk and Y. Kopelevich
- Phys. Rev. Lett. 93, 166402 (2004)
Quantum oscillations: What is usually studied ?
Period: Information about Fermi surface cross section S()
Profile: Information about e-e interaction (in 2D)
Damping: Information about e-scattering (Dingle factor )
and Phase ??? … difficult to extractWe propose the method.!!!
Generalized formula: 2D, 3D, arbitrary spectrum
where
Lifshitz-Kosevich, Shoenberg, Mineev, Gusynin, Sharapov, Lukyanchuk, Kopelevich
Fermi Surface cross section
► for Normal electrons
► for Dirac electrons
Falkovsky (65) – Maslov- Berry phase
SdH: Oscillations of xx (H) (1st harmonic)
Normal: = 1/2Dirac: = 0► Spectrum : {
2D: = 03D: = ± 1/8► Dimensionality :{
Phase depends on :
dHvA: Oscillations of (H) (1st harmonic)
Cyclotron mass(detection of e and h)
SdHdHvA
Experiment:
Electrons or Holes ?
Normal or Dirac ?
SdH dHvA
dHvASdH
Pass-band filtering
spectrum
Comparison of dHvA and SdH
electrons
holes
In-phase
Out-phase
Fan Diagram for SdH oscillations in Graphite
Dirac
Normal
Novoselov, 2005
graphene
Multilayer 5nm graphite
Determination of phase
Phase-frequency diagram
Spectrum
Phase-shift function
No information about phase
Simultaneous determinationof phase and frequency !!!
Result: spectrum of quantum oscillations in HOPG
Normalelectrons
Dirac holes
29.3722 58.7444 88.1166 117.4888 146.8610
0.5
1
1.5
2
2.5
e
h
Rxx, Kish
Band interpretation
Normalelectrons
Dirac holes
2006 Confirmation: Angle Resolved Photoemission Spectroscopy
Dirac holes
Normalelectrons
(ARPES)
holes
electrons
Dirac Spectrum
Normal Spectrum
H: point
Phase volume ~0
no Dirac Fermionsshould be seen in experiment
Problems with band interpretation
Se > Sh1)
2)
Sh > Se
Independent layers ???
Another possibility:
E. Andrei et al. 2007, Nature Phys.
Dirac+Normal fermions in HOPGTEM results:
Another confirmation of Dirac fermions:
nE)n(sign
nBe2v)n(signE
10
Fn
2006
Graphite, interpretation, ??? =>
QHE in graphite and in graphene
I. Luk’yanchuk and Y. Kopelevich
- Phys. Rev. Lett. 97, 256801 (2006)
QHE in graphite
Rxx
Rxy
Y. Kopelevich et al. Phys. Rev. Lett. 90, 156402 (2003)
0 1 2 3 4 5 6 7 80
1
2
3
4
5
6
7
8
9
-
Gxy
/G0
xy
B0/B
HOPG, Y. Kopelevich et al. PRL´2003
Few Layer Graphite (FLG)K.S.Novoselov et al., Science´2004
B0= 20 T, = > n ~ 2x1012cm-2
B0 = 4.68 T
Fig. 1
1
2
HOPG, Y. Kopelevich et al. PRL´2003 B0 = 4.68 T
Few Layer Graphite (FLG)K.S.Novoselov et al., Science´2004
B0= 20 T, = > n ~ 2x1012 cm-2
Vs.
QHE: Graphite vs multi graphene
GRAPHITE: Normal vs Dirac carriers separation
Filterin
g
Rxy
Rxx
B (T)
Normal (Integer QHE)
Dirac (Semi-integer QHE)
0 1 2 3 4 50
1
2
3
4
5
Filling Factor
-
Gxy
/ G
0xy
Normal QHE
-8
-4
0
4
8
-
Rxx
(
m
)
0 1 2 3 4 50
1
2
3
4
5
Filling Factor
- G
xy /
G0x
y
Dirac QHE
-8
-4
0
4
8
-
Rxx
( m
)
Normal QHE in graphite
Bi-layer grapheneNovoselov, et al. Nature Physics 2, 177 (2006)
Dirac QHE in graphite
Graphene:Y. Zhang, et al., Nature 438, 201 (2005)
Graphene:Novoselov, et al. Nature 438, 197 (2005)
► 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.
Conclusion:
Advantage of thin slabs of HOPG graphite:
- Easy to fabricate- Much better quality and purity- Easier dopping control- better mechanical stability