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