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Dirac Cone Systems
Miguel Monteverde
LPS, Univ. Paris-Sud, CNRS, UMR 8502, F-91405 Orsay Cedex, France.
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Dirac Cones on graphene
Outline
Introduction to graphene and applications
Dirac Cones on -(BEDT-TTF)2I3
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Introduction 2/3What is graphene?
Graphene
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Introduction 2/3What is graphene?
Graphite
Graphene
Strong bond
Weak bonds(Van der Waals)
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Introduction 2/3What is graphene?
Graphite
Graphene
Novoselov & Geim (2005)Strong bond
Weak bonds(Van der Waals)
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Start with a graphite flake
Put in a scotch tape
Exfoliate many times
Apply to clean wafer(SiO2 ~ 300nm)Search for graphene with opticalmicroscope / Raman spectra
Geim & Kim (2008)
Fabrication Process :
Exfoliation techniqueHow we make graphene?
Graphene
Few LayersGraphene
20um
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Introduction 2/3What makes graphene so interesting?
Year
Discovery
Number of graphene publications**@arXiv with “graphene” in their title
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Introduction 2/3Graphene applications potentiality
Composites
Large size graphene production
Flexible and transparent electronics
Supercapacitors
Fast electronics
10 um
Bistables Memories
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Introduction 2/3
Very special electrical conductor :Dirac band structure Wallace (1947)
Best thermal conductor
Optically transparent (Absorption 2.3%)
k ~ 5000 W/mK A.A.Balandin, et.al. (2008)
Best electrical conductor at room Temperature ~105 cm2/Vs
j ~mA/m
Strongest materialChanggu Lee, et.al. (2008)
Y ~ 1 TPaU ~ 42 N/m
P.Blake, et.al. (2008)
Graphene properties :
What makes graphene so interesting?
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Composite materials with 0.1% graphene :
Composites
- large contact surface
- Is strong and stiff
- It avoids crack propagation
Luke S. Walker, Victoria R. Marotto, Mohammad A. Rafiee, Nikhil Koratkar, and Erica L. CorralACS Nano 5 (2011), 3182
Mohammad A. Rafiee, Javad Rafiee, Zhou Wang, Huaihe Song, Zhong-Zhen Yu, and Nikhil KoratkarACS Nano 3 (2009), 3884
Crack propagation rate is reduced by 1 to 2 order of magnitude
Fracture Toughness
Fracture Energy
+53%+40%+126%
Tensile strength
Epo
xy
0.1%
SW
NT
0.1%
MW
NT
0.1%
Gra
phen
e
Si3N4 + 1.5 vol% graphene
0.1%
Gra
phen
e
0.1%
Gra
phen
e
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1100 10 0.1 0.01 0.001Pulse width (kHz)
RH/RL ~106
Bistable Memories
Y. Li, A. Sinitskii and J.M. TourNature Materials 7 (2008), 966K.S.Vasu, S. Sampath and A.K. SoodSolid State Communications 151 (2011), 1084X. Wang, W. Xie, J. Du, C. Wang, N. Zhao and J.B. XuAdvance Materials 24 (2012), 2614
- High On/Off ratio- Mass production (repeatability)- Resilient:
- Large retention time (+months)- Temperature (200°C)- Radiation (+20 Mrad @ 8keV)
- Average writing and deleting speed- High Resistance
Graphene memories based on Reduced Graphene Oxide and CVD graphene
5 um
Reduced Graphene Oxide
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Flexible & transparent electronics
Sukang Bae, Hyeongkeun Kim, Youngbin Lee, Xiangfan Xu, Jae-Sung Park, Yi Zheng, Jayakumar Balakrishnan, Tian Lei, Hye Ri Kim, Young Il Song, Young-Jin Kim, Kwang S. Kim, Barbaros O¨ zyilmaz, Jong-Hyun Ahn, Byung Hee Hong and Sumio IijimaNature Nanotechnology 5 (2010), 574
GraphenePET
PET y electrodes
x electrodes
Touch screen
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Sukang Bae, Hyeongkeun Kim, Youngbin Lee, Xiangfan Xu, Jae-Sung Park, Yi Zheng, Jayakumar Balakrishnan, Tian Lei, Hye Ri Kim, Young Il Song, Young-Jin Kim, Kwang S. Kim, Barbaros O¨ zyilmaz, Jong-Hyun Ahn, Byung Hee Hong and Sumio IijimaNature Nanotechnology 5 (2010), 574
Graphene vs indium tin oxide (ITO):
- ITO is brittle- Indium becomes rare and expensive while CVD graphene is low cost- ITO is less transparent and more resistive- ITO is not flexible- Graphene has long cycle-life (+1000 bends)
Applications :flat-panel displaystouch screensorganic light-emitting diodes (OLEDs)solar cells
(for use as a transparent conductive coating)
Flexible & transparent electronics
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Industrial production of large size Graphene
1012
109
106
103
1002005 2006 2007 2008 2009 2010
US$
/ m
2
2010 CVD
1 m210 um2
2005 Exfoliated 2008 Epitaxial (on SiC)
1 cm2
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Dirac Cones on graphene
Outline
Introduction to graphene and applications
Dirac Cones on -(BEDT-TTF)2I3
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Quantum transport in graphene
Miguel Monteverde, C. Ojeda Aristizabal, R. Weil, M. Ferrier, S. Gueron, H. Bouchiat, J.N. Fuchs and D. Maslov
LPS, Univ. Paris-Sud, CNRS, UMR 8502, F-91405 Orsay Cedex, France.
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Introduction 2/3What makes graphene so interesting?
Very special electrical conductor :Dirac band structure Wallace (1947)
Best thermal conductor
Optically transparent (Absorption 2.3%)
k ~ 5000 W/mK A.A.Balandin, et.al. (2008)
Best electrical conductor at room Temperature ~105 cm2/Vs
j ~mA/m
Strongest materialChanggu Lee, et.al. (2008)
Y ~ 1 TPaU ~ 42 N/m
P.Blake, et.al. (2008)
Graphene properties :
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Conventional 2DEG band structure
massive fermions physics
Conventional2DEG
massive fermionselectron-hole asymmetry
Ee=ħ 2kF2 / 2 me
*
electrons
holes
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Graphene electronic band structure
massless fermions physics
Graphene
massless fermionsvF ~106 m/s
E=ħ vF kF
kF=0 → F=∞semiclasical physics not valid !
(m*=0)
m*~1/ (d2E/dk2)
Conventional2DEG
massive fermionselectron-hole asymmetry
Ee=ħ 2kF2 / 2 me
*
electrons
holes
Dirac cone band spectrum
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Graphene
massless fermionsvF ~106 m/s
E=ħ vF kF (m*=0)
Conventional2DEG
massive fermionselectron-hole asymmetry
Ee=ħ 2kF2 / 2 me
*
Bilayer Graphene
E=ħ 2kF2 / 2 m*
(Low energy)m*=0.03 me
electrons
holes
Dirac cone band spectrum
massive vs. massless fermions
Why few-layer graphene ?Graphene electronic band structure
massive fermions
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Graphene
massless fermions
Conventional2DEG
massive fermions
Bilayer Graphene
(Low energy)
electrons
holes
massive vs. massless fermions
Why few-layer graphene ?Graphene electronic band structure
massive fermions
VG
zero gap
electron-hole symmetry
tunable carrier density and type
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Novoselov, et.al. (2005)
Is (VG) in graphene understood ?
holes
electrons
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Novoselov, et.al. (2005)
Is (VG) in graphene understood ?
At Dirac point
Conductivity quantization = 4e2/h
no charge is present (ballistic)
perfect transmission across barriers via evanescent modes
→→
holes
electrons
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Novoselov, et.al. (2005)
Is (VG) in graphene understood ?
At Dirac point
min = 4e2/h
Conductivity quantization = 4e2/h
no charge is present (ballistic)
perfect transmission across barriers via evanescent modes
→→
Theory
Experimentmin = 4e2/hThe mystery of the
missing pi ()
Geim, et.al. (2007)
holes
electrons
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Novoselov, et.al. (2005)
Is (VG) in graphene understood ?
At Dirac pointelectrons
J.Martin, et.al. (2007)
holes electrons
holes
no charge is present (ballistic)
Vg ↔ ‹ n › → nWhen‹ n › ~ 0
Measurement of the local electrostatic potential
SET
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Novoselov, et.al. (2005)
Is (VG) in graphene understood ?
At Dirac pointelectrons
J.Martin, et.al. (2007)
holes electrons
Vg ↔ ‹ n › → nWhen‹ n › ~ 0
Morpurgo (2008)
holes
no charge is present (ballistic)
Measurement of the local electrostatic potential
SET
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Novoselov, et.al. (2005)
Is (VG) in graphene understood ?
holes
electrons
~ VG Out of Dirac pointDiffusive ~ 103 cm2/Vs → what type of impurities?
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Diffusive ~ 103 cm2/Vs → what type of impurities?
Novoselov, et.al. (2005)
Is (VG) in graphene understood ?
holes
electrons
~ VG
= 2 vF kF tr
tr-1 ~ D(EF) ~ kF
Out of Dirac point
Fermi golden rule & Drude
→
Neutral-short range and weak impurities:
~ const
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~ kF2 ~ 2Vg
= 2 vF kF tr
tr-1 ~ D(EF) ~ kF
Novoselov, et.al. (2005)
Is (VG) in graphene understood ?
Out of Dirac point
holes
electrons
Fermi golden rule & Drude
→
Neutral-short range and weak impurities:
~ const
~ VG
= 2 vF kF tr
tr-1 ~ U2D(EF) ~ kF
-1
Thomas-F. approximation, Fermi golden rule & Drude
→
Charged (screened) impurities:
U ~ qTF-1 ~ kF
-1
Diffusive ~ 103 cm2/Vs → what type of impurities?
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= 2 vF kF tr
tr-1 ~ D(EF) ~ kF
Novoselov, et.al. (2005)
Is (VG) in graphene understood ?
Out of Dirac point
holes
electrons
Fermi golden rule & Drude
→
Neutral-short range and weak impurities:
~ const
~ VG
= 2 vF kF tr
tr-1 ~ U2D(EF) ~ kF
-1
Thomas-F. approximation, Fermi golden rule & Drude
→
Charged (screened) impurities:
U ~ qTF-1 ~ kF
-1
(T)Ethanol 2555
Geim (2009)
Expected
MeasuredEthanol (T)
Diffusive ~ 103 cm2/Vs → what type of impurities?
~ kF2 ~ 2Vg
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= 2 vF kF tr
tr-1 ~ D(EF) ~ kF
Novoselov, et.al. (2005)
Is (VG) in graphene understood ?
Out of Dirac point
holes
electrons
Fermi golden rule & Drude
→
Neutral-short range and weak impurities:
~ const
~ VG
= 2 vF kF tr
tr-1 ~ U2D(EF) ~ kF
-1
Thomas-F. approximation, Fermi golden rule & Drude
→
Charged (screened) impurities:
U ~ qTF-1 ~ kF
-1
measurements of transport scattering times for both graphene and bilayer
Neutral-short range (R) and strong impurities:
= 2 vF kF tr
tr ~ kF ln2(kF R) →Most probable ad-atoms (binding affinity is improved by corrugation caused by the substrate).
Graphene
Bilayer
M.Monteverde, et.al. (2010)
~ Vg ln2(RVg0.5)
Diffusive ~ 103 cm2/Vs → what type of impurities?
~ kF2 ~ 2Vg
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Massless and Massive Fermions differences
n Dirac = 4 (n+1/2) B/0n Massive = 4 n B/0
Quantum Hall Effect Universal Conductance Fluctuations
Induced Superconductivity
BC Dirac Vg-1/4
BC Massive Vg-1/2
2
2/3
M.Monteverde, et.al. (2010) C.Ojeda-Aristizabal, et.al. (2010) C.Ojeda-Aristizabal, et.al. (2009)
Specular Andreev Reflexions?
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Dirac Cones on graphene
Outline
Introduction to graphene and applications
Dirac Cones on -(BEDT-TTF)2I3
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Dirac Cones on ‐(BEDT‐TTF)2I3
Miguel Monteverde, M.O. Goerbig, P. Auban‐Senzier, F.Navarin, H.Henck, C.R. Pasquier, C.Mézière, and P.Batail
LPS, Univ. Paris‐Sud, CNRS, UMR 8502, F‐91405 Orsay Cedex, France.
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Graphene Dirac Point
Theoretical Dirac‐Point
Experimental Dirac‐Point
EFEF•
• •TF
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Graphene Dirac Point
Theoretical Dirac‐Point
Experimental Dirac‐Point
EFEF•
• •TF
EF
Graphene Dirac‐Point
TF ~ 100K
J.Martin, et.al. (2007)
holes electrons
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Dirac-cone systems
stacking conducting ET layers and insulating Iodine layers
Organic conductor a‐(ET)2I3
bulk material with strong 2D conductance
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Dirac-cone systems
Organic conductor a‐(ET)2I3
Tilted Dirac‐cones under pressure.
Fermi level will be at Dirac point .
P.Alemany, et.al. (2012)
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Magneto-conductance of a-ET2I3
Only one type of carrier:
Magneto-conductance:J.S.Kim, et.al. (1993)
xx(0)
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Magneto-conductance of a-ET2I3
Only one type of carrier:
Magneto-conductance:J.S.Kim, et.al. (1993) Monteverde, et.al. (2012)
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Magneto-conductance of a-ET2I3
Only one type of carrier:
Two types of carriers:
Magneto-conductance:
a-ET2I3 is a multicarrier system
J.S.Kim, et.al. (1993) Monteverde, et.al. (2012)
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Magneto-conductance of a-ET2I3
Only one type of carrier:
Two types of carriers:
Magneto-conductance:
measurements of the carrier density
J.S.Kim, et.al. (1993) Monteverde, et.al. (2012)
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Magneto-conductance of a-ET2I3
Only one type of carrier:
Two types of carriers:
Magneto-conductance:
measurements of the carrier density
J.S.Kim, et.al. (1993) Monteverde, et.al. (2012)
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T
T 2
Magneto-conductance of a-ET2I3
the carrier density depend on temperature
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Dirac-cone band structure
Conventional2DEG
massive fermionselectron-hole asymmetry
EM=ħ 2kF2 / 2 m*
electrons
holesDM=gvgsm* / 2ħ 2 = constant
(T»TF)nM(T) =∫ f D dE T
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Dirac-cone band structure
the carrier density depend on temperature
massless fermions
E=ħ vF kF (m*=0)m*~1/ (d2E/dk2)
Conventional2DEG
massive fermionselectron-hole asymmetry
EM=ħ 2kF2 / 2 m*
electrons
holes
Dirac - Cone Band structure
Dirac-Point :
DM=gvgsm* / 2ħ 2 = constant
(T»TF)nM(T) =∫ f D dE T
DDirac=gvgsE / 2(ħvF)2 E
(T»TF)nDirac(T) =∫ f D dE T2
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T
T 2
Massive fermions
Magneto-conductance of a-ET2I3
Coexistence of Massive and Dirac fermions
(T»TF)nM(T) T
Dirac fermions(T»TF)
nD(T) T 2
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T
T 2
Magneto-conductance of a-ET2I3
Coexistence of Massive and Dirac fermions
(T»TF)nM(T) TMassive fermions
Dirac fermions(T»TF)
nD(T) T 2
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Graphene vs a-ET2I3 under pressure
a‐ET2I3 @ pressure
EFTF ~ 1KEF
Graphene
TF ~ 100K
Dirac‐Point
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rs ~ 0.5
Graphene vs a-ET2I3 under pressure
vF ~ 105 m/s
rs Dirac ~ e2
ε ħ vFvF ~ 106 m/s
rs ~ ?
a‐ET2I3 @ pressure Graphene
Electron Correlations
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Graphene vs a-ET2I3 under pressure
Electron Correlations
~ T -1
~ T -2
Dirac / Massive system:Calculating the scattering times using Fermi golden rule…
T10-3(T~TF ) (T>TF )
a‐ET2I3 @ pressure
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Conclusions
Electronic correlations are not only relevant but needs of both types of carriers to explain the physics.
TF Dirac ~1 K (2 order of magnitude lower than graphene)
vF Dirac ~105 m/s (1 order of magnitude lower than graphene)
Mobility ratioTemperature dependence of the Mobility ratio
Electronic correlations may be relevant !
Coexistence of Dirac and Massive carriers
Dirac Point Physics can be studied in others systems than graphene that can have an homogenous Fermi level.
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Thanks for your attention
Merry Christmas and
Happy New Year