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Dirac Cone Systems

Miguel Monteverde

LPS, Univ. Paris-Sud, CNRS, UMR 8502, F-91405 Orsay Cedex, France.

Dirac Cones on graphene

Outline

Introduction to graphene and applications

Dirac Cones on -(BEDT-TTF)2I3

Introduction 2/3What is graphene?

Graphene


Graphite

Graphene

Strong bond

Weak bonds(Van der Waals)


Graphite

Graphene

Novoselov & Geim (2005)Strong bond

Weak bonds(Van der Waals)

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

Introduction 2/3What makes graphene so interesting?

Year

Discovery

Number of graphene publications**@arXiv with “graphene” in their title

Introduction 2/3Graphene applications potentiality

Composites

Large size graphene production

Flexible and transparent electronics

Supercapacitors

Fast electronics

10 um

Bistables Memories

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?

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

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

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

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

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


Outline



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.

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 :

Conventional 2DEG band structure

massive fermions physics

Conventional2DEG

massive fermionselectron-hole asymmetry

Ee=ħ 2kF2 / 2 me

*

electrons

holes

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


Ee=ħ 2kF2 / 2 me

*

electrons

holes

Dirac cone band spectrum

Graphene

massless fermionsvF ~106 m/s

E=ħ vF kF (m*=0)

Conventional2DEG


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

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

Novoselov, et.al. (2005)

Is (VG) in graphene understood ?

holes

electrons



At Dirac point

Conductivity quantization = 4e2/h

no charge is present (ballistic)

perfect transmission across barriers via evanescent modes

→→

holes

electrons



At Dirac point

min = 4e2/h

Conductivity quantization = 4e2/h


perfect transmission across barriers via evanescent modes

→→

Theory

Experimentmin = 4e2/hThe mystery of the

missing pi ()

Geim, et.al. (2007)

holes

electrons



At Dirac pointelectrons

J.Martin, et.al. (2007)

holes electrons

holes


Vg ↔ ‹ n › → nWhen‹ n › ~ 0

Measurement of the local electrostatic potential

SET



At Dirac pointelectrons


holes electrons

Vg ↔ ‹ n › → nWhen‹ n › ~ 0

Morpurgo (2008)

holes


Measurement of the local electrostatic potential

SET



holes

electrons

~ VG Out of Dirac pointDiffusive ~ 103 cm2/Vs → what type of impurities?

Diffusive ~ 103 cm2/Vs → what type of impurities?



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

~ kF2 ~ 2Vg

= 2 vF kF tr

tr-1 ~ D(EF) ~ kF



Out of Dirac point

holes

electrons


→


~ 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


= 2 vF kF tr

tr-1 ~ D(EF) ~ kF



Out of Dirac point

holes

electrons


→


~ const

~ VG

= 2 vF kF tr

tr-1 ~ U2D(EF) ~ kF

-1


→


U ~ qTF-1 ~ kF

-1

(T)Ethanol 2555

Geim (2009)

Expected

MeasuredEthanol (T)


~ kF2 ~ 2Vg

= 2 vF kF tr

tr-1 ~ D(EF) ~ kF



Out of Dirac point

holes

electrons


→


~ const

~ VG

= 2 vF kF tr

tr-1 ~ U2D(EF) ~ kF

-1


→


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)


~ kF2 ~ 2Vg

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?


Outline



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.

Graphene Dirac Point

Theoretical Dirac‐Point

Experimental Dirac‐Point

EFEF•

• •TF

Graphene Dirac Point

Theoretical Dirac‐Point

Experimental Dirac‐Point

EFEF•

• •TF

EF

Graphene Dirac‐Point

TF ~ 100K


holes electrons

Dirac-cone systems

stacking conducting ET layers and insulating Iodine layers

Organic conductor a‐(ET)2I3

bulk material with strong 2D conductance

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)

Magneto-conductance of a-ET2I3

Only one type of carrier:

Magneto-conductance:J.S.Kim, et.al. (1993)

xx(0)



Magneto-conductance:J.S.Kim, et.al. (1993) Monteverde, et.al. (2012)



Two types of carriers:

Magneto-conductance:

a-ET2I3 is a multicarrier system

J.S.Kim, et.al. (1993) Monteverde, et.al. (2012)



Two types of carriers:

Magneto-conductance:

measurements of the carrier density

J.S.Kim, et.al. (1993) Monteverde, et.al. (2012)

T

T 2


the carrier density depend on temperature

Dirac-cone band structure

Conventional2DEG


EM=ħ 2kF2 / 2 m*

electrons

holesDM=gvgsm* / 2ħ 2 = constant

(T»TF)nM(T) =∫ f D dE T

Dirac-cone band structure

the carrier density depend on temperature

massless fermions

E=ħ vF kF (m*=0)m*~1/ (d2E/dk2)

Conventional2DEG


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

T

T 2

Massive fermions


Coexistence of Massive and Dirac fermions

(T»TF)nM(T) T

Dirac fermions(T»TF)

nD(T) T 2

T

T 2


Coexistence of Massive and Dirac fermions

(T»TF)nM(T) TMassive fermions

Dirac fermions(T»TF)

nD(T) T 2

Graphene vs a-ET2I3 under pressure

a‐ET2I3 @ pressure

EFTF ~ 1KEF

Graphene

TF ~ 100K

Dirac‐Point

rs ~ 0.5


vF ~ 105 m/s

rs Dirac ~ e2

ε ħ vFvF ~ 106 m/s

rs ~ ?

a‐ET2I3 @ pressure Graphene

Electron Correlations


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

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.

Thanks for your attention

Merry Christmas and

Happy New Year

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