Dirac fermions in Graphite and Graphene

Igor Lukyanchuk Amiens University

I. Lukyanchuk, Y. Kopelevich et al.

- Phys. Rev. Lett. 93, 166402 (2004)- Phys. Rev. Lett. 97, 256801 (2006)

Graphene2005

Novoselov, et al. Nature 438, 197 (2005

Y. Zhang, et al., Nature 438, 201 (2005

Why graphene is interesting ?

- Fundamental physics

- Applications (carbon-based microelectronics )

3D 2D 1D 0D

(Nobel prize) (Nobel prize)

2 view of Graphene

Nanotube-grapheneGraphite-graphene

“ “:

November 2005

• 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:

Linear Dirac spectrum

Graphene: (2D graphite monolayer, Semimetal)

Special points of Brillouin zone

Brillouin zone 4-component (Dirac ????) wave function

"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

Band structure: Slonczewski-McClure Model

Graphite:F

ittin

g pa

ram

eter

s

holes

electrons

ρ(T), HOPG

In best samples

ρc/ ρa > 50000 (instead of 300 in Kish)

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

Novoselov, K. S. et al. Nature 438, 197 (2005); Zhang, Y. et al. Nature 438, 201 (2005).

2005: Discovery of Quantum Hall Effect in 2D Graphene Due to Dirac fermions …

From: - phase analysis - semi-integerr QHE

Quantum Hall Effect, different samples (2003)

0 1 2 3 4 50

1

2

3

4

5

Filling Factor

-

Gx

y /

G0

xy

Normal QHE

-8

-4

0

4

8

-

Rx

x

( m

)

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

B0 = 4.68 T

Few Layer Graphite (FLG)K.S.Novoselov et al., Science´2004

B0= 20 T, = > n ~ 2x1012 cm-2

.

QHE: Graphite vs multi graphene

HOPG, Y. Kopelevich et al. PRL´2003

Do Dirac Fermions Exist in Graphite ?

Normal electrons

Dirac electrons

Landau quantization: Normal vs Dirac

‘’gap’’

no ‘’gap’’ !!!

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

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:

2006 Confirmation: Angle Resolved Photoemission Spectroscopy

Dirac holes

Normalelectrons

(ARPES)

E. Andrei et al. 2007, Nature Phys.

Dirac+Normal fermions in HOPGTEM results:

Another confirmation of Dirac fermions:

Interlayer tunneling spectroscopy of Landau levels in graphiteYu. I. Latyshev1, A. P. Orlov1, V. A. Volkov1, A. V. Irzhak2, D. Vignolles3, J. Marcus4 and T. Fournier4

0 1 2 3 4 5 6-400

-300

-200

-100

0

100

200

300

400

Mag.trans 2x(01) STM 2x(01) STM 2x(02) Inter.tunn -11 Inter.tunn -22 Inter.tunn -33

V (

mV

)

B (T)

Graphite #1

OPTICAL PROPERTIES

- Visible- Infrared- Raman

Graphite

Graphene

C=

Reflectance and transmitance coefficients

Optical properties are defined by HF conductivity

πα ≈ 2.3%

INFRARED SPECTROSCOPY

nE)n(sign

nBe2v)n(signE

10

Fn

2006

Graphite, interpretation, ??? =>

RAMAN SPECTROSCOPY

RAMAN SPECTROSCOPY

« Graphene Fingerprint »

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

HOPG, Raman

model