Transcript Les materiaux ferroelectriques: du bulk vers nano

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 (>700 publications)
2 view of Graphene
Nanotube-graphene
Graphite-graphene
Outline
I) Graphene
Why Graphene is interesting
Theoretical background
History
Elaboration
Experimental Methods
Graphene in magnetic field (Dirac Fermions, Quantum Hall effect)
Applications
2) Graphite (vs Graphene)
Theory
Experiment
Dirac Fermions
Quantum Hall Effect
3D
2D
1D
(Nobel prize)
0D
(Nobel prize)
Why graphene is interesting ?
- Fundamental physics
- Applications (carbon-based microelectronics )
QED in a Pencil Trace
Nature:
“… Erasing electron mass…”
“…Electrons in Carbon sheets
behave like Massless Particles….”
La Recherche:
Google:
(Dirac Fermions, graphite…)
“…La relativité dans une mine de
crayon ….”
“… Einstein's relativity theory proven
with the 'lead' of a pencil … ”
HP, Intel, IBM…
30 000 000 $
Wanted:
• 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%
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
Graphene:
Semimetal / Gapless Semiconductor
Brillouin
zone
Special points
of Brillouin zone
Linear Dirac
spectrum
4-component (Dirac ????) wave function
DOS
Free Relativistic Electrons
“Dirac fermions"
"Normal electrons"
Dirac
spinor
Schrödinger equation
Dirac equation
Schroedinger cond-mat physics
Dirac cond-mat physics !!!
Gap formation, excitonic insulator, weak ferromagnetism, … ???
In magnetic field:
2 component equations
Abrikosov Phys. Rev. B60, 4231 (1999) B61, 5928 (2000)
González, Guinea, Vozmediano, Phys. Rev. Lett. 77, 3589 (1996)
Khveshchenko, Phys. Rev. Lett. 87, 206401 (2001); 87, 246802 (2001)
Klein effect:
Metal (semiconductor)
U(x)
electron
Ef
U(x)
electron
Semimetal:
Ef
hole
hole
No electron localization !!!
Minimal conductivity
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)
Problems…
If 2D Graphene is stable?
STM
Graphene
on a 6H-SiC(0001) substrate
Experimental Methods
ARPES – angle resolved
photo emission spectroscopy
Raman spectra of graphite
double-resonant
Intensity (a.u)

graphite
D
0
2.33 eV
G
D‘
1500
Raman shift (cm-1)
G‘
3000
Spatially resolved Raman spectroscopy of single- and
few-layer graphene
Scanning force microscope
2
1
1 m
Experiment:
Davy Graf, Françoise Molitor,
and Klaus Ensslin
Solid State Physics, ETH Zürich, Switzerland
Christoph Stampfer, Alain Jungen, and Christofer Hierold
Micro and Nanosystems, ETH Zürich
Theory:
Ludger Wirtz
Institute for Electronics, Microelectronics, and
Nanotechnology, Lille
Intensity (a.u) Intensity (a.u)
D
1200
D‘
G
2
double-layer graphene
single-layer graphene
1
1600
2000
2400
Raman shift (cm-1)
2800
Graphene in Magnetic Field
Landau quantization: Normal vs Dirac
Normal electrons
‘’gap’’
Dirac electrons
no ‘’gap’’ !!!
QHE effect : Normal vs Dirac
Normal electrons,
xy
5
4.5
4
3.5
 xy
3
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
2.5
1/H
3
3.5
4
4.5
5
1/H
Dirac- like electrons
(expected for graphene)
xy
1/H
Graphene: Half-Integer Quantum Hall Effect
Quantisation at =N+1/2
xx (k)
xy (4e2/h)
3.5
12T
2.5
10
1.5
0.5
-0.5
5
-1.5
-2.5
-3.5
0
-4
-2
0
n (1012 cm-2)
2
4
Novoselov et al, Nature 2005
Zhang et al, Nature 2005
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???
Igor Luk’yanchuk, Yakov Kopelevich
Dirac Fermions in Graphite and Graphene: Implications to QHE
Experiment: Kopelevich et al.
- Phys. Rev. Lett. 90, 156402 (2003)
Graphite
(2004)
Interpretation and analysis
- Phys. Rev. Lett. 93, 166402 (2004)
- Phys. Rev. Lett. 97, 256801 (2006)
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)
Graphite:
Fitting parameters
Band structure:
Slonczewski-McClure Model
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)
Field Induced Metal-Insulator Transition
Magneto-resistance R(H)
SdH oscillations
Linear !!!
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 ?
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.!!!
Generalized formula: 2D, 3D, arbitrary spectrum
Lifshitz-Kosevich, Shoenberg, Mineev, Gusynin, Sharapov, Lukyanchuk, Kopelevich
where
Fermi Surface cross section
Falkovsky (65) – Maslov- Berry phase
► for Normal electrons
► for Dirac electrons
SdH: Oscillations of xx (H) (1st harmonic)
Phase
depends on :
► Spectrum :
Cyclotron mass
(detection of e and h)
{
Normal:
Dirac:
 = 1/2
 = 0
► Dimensionality :
dHvA: Oscillations of  (H) (1st harmonic)
{
2D:  = 0
3D:  = ± 1/8
Electrons or Holes ?
Normal or Dirac ?
Experiment:
SdH
dHvA
Comparison of dHvA and SdH
SdH
dHvA
SdH
dHvA
In-phase
Pass-band
filtering
spectrum
electrons
Out-phase
holes
Fan Diagram for SdH oscillations in Graphite
Novoselov, 2005
Normal
Multilayer 5nm graphite
Dirac
graphene
Determination of phase f
Spectrum
No information about phase
Phase-shift function
f
Simultaneous determination
of phase and frequency !!!

Phase-frequency diagram
Result: spectrum of quantum oscillations in HOPG
Normal
electrons
Dirac
holes
e
h
2.5
Rxx, Kish
2
1.5
1
0.5
0
29.3722
58.7444
88.1166
117.4888
146.861
Band interpretation
Normal
electrons
Dirac
holes
2006 Confirmation: Angle Resolved Photoemission Spectroscopy
(ARPES)
Dirac
holes
Normal
electrons
Sh > Se
Problems with band interpretation
1)
Se > Sh
2)
H: point
Dirac Spectrum
Phase volume ~0
holes
no Dirac Fermions
should be seen in experiment
Normal Spectrum
electrons
Another possibility:
Independent layers ???
Another confirmation of Dirac fermions:
Dirac+Normal fermions in HOPG
TEM results:
E. Andrei et al. 2007, Nature Phys.
2006
E n  sign (n ) v F 2eB n
 sign (n )E10 n
Graphite, interpretation, ??? =>
QHE in graphite
and in graphene
I.
Luk’yanchuk and Y. Kopelevich
- Phys. Rev. Lett. 97, 256801 (2006)
QHE in graphite
Rxy
Rxx
Y. Kopelevich et al. Phys. Rev. Lett. 90, 156402 (2003)
QHE: Graphite vs multi graphene
HOPG, Y. Kopelevich et al. PRL´2003 B0 = 4.68 T
Vs.
Few Layer Graphite (FLG)
K.S.Novoselov et al., Science´2004
Fig. 1
B0= 20 T, = > n ~ 2x1012 cm-2
9
8
7
HOPG, Y. Kopelevich et al. PRL´2003
- Gxy/G0xy
6
B0 = 4.68 T
5
4
3
Few Layer Graphite (FLG)
K.S.Novoselov et al., Science´2004
2
2
1
12
B0= 20 T, = > n ~ 2x10 cm
1
0
0
1
2
3
4
B0/B
5
6
-2
7
8
Normal (Integer QHE)
5
GRAPHITE: Normal vs Dirac
carriers separation
8
4
3
0
2
-  Rxx ( m )
- Gxy / G0xy
4
-4
1
Normal QHE
0
0
1
2
3
4
-8
5
Filling Factor 
Rxy
Dirac (Semi-integer QHE)
5
8
4
xy
0
2
-4
1
Dirac QHE
0
0
1
2
3
Filling Factor 
4
-8
5
-  Rxx ( m )
3
/G
0xy
4
- G
B (T)
Filtering
Rxx
Normal QHE in graphite
Bi-layer graphene
Novoselov, 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)
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.
in Graphite.
Precursors of both types of QHE exist
Advantage of thin slabs of HOPG graphite:
- Easy to fabricate
- Much better quality and purity
- Easier dopping control
- better mechanical stability