Interpretation of the Raman spectra of graphene and carbon nanotubes: the effects of Kohn anomalies and non-adiabatic effects

S. Piscanec

Cambridge University Engineering Department: Centre for Advanced Photonics and Electronics, Cambridge, UK

G-band in graphite and nanotubes

Graphite:

one single sharp G peak corresponding to q==0, mode E2g

Nanotubes:

• Two main bands, G+ and G-.• Modes derived from graphite E2g

• Metallic semiconducting

Common interpretation: curvature

G- diameter dependence TO circumferential

G+: no diameter dependence LO axial

Jorio et al. PRB 65, 155412 (2002)

Common interpretation: Fano resonance

In metallic tubes the G- peak is:

• Downshifted

• Broader

• Depends on diameter

Interpretation

• Fano resonance

• Phonon-Plasmon interaction

Electron-phonon coupling and Kohn anomalies have to be considered

Kohn anomalies

• Atomic vibrations are screened by electrons

• In a metal this screening abruptly changes for vibrations associated to certain q points of the Brillouin zone.

• Kink in the phonon dispersions: Kohn anomaly.

• Graphite is a semi-metal

• Nanotubes are folded graphite

• Nanotubes can as well be metallic

Kohn anomalies: when?

k1

k2 = k1+ q

q

Fermi surface

q = phonon wavevector

k = electron wavevector

1. k1 & k2= k1+q on the Fermi surface

2. Tangents to the Fermi surface at k1 and k2= k1+ q are parallel

Everything depends on the geometry of the Fermi surface

•W. Kohn, Phys. Rev. Lett. 2, 393 (1959) bold

Kohn anomalies in graphite

EF

’

•Graphite is a semi metal:

•Fermi surface = 2 points: K and K’ = 2 K

Kohn Anomalies for: • q = - = 0 =

• q = ’- -

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

1200

1300

1400

1500

1600

1700

Calculations IXS data

Phonon Wave Vector (2/a0)

Fre

quen

cy (

cm-1)

Kohn anomalies in graphite

• 2 sharp kinks for modes E2g at and A1’ at

IXS data: J. Maultzsch et al. Phys. Rev. Lett. 92, 075501 (2004)

E2g

E2g

A’1

Kohn Anomaly EPC ≠ 0

Kohn anomalies in nanotubes

•Metallic tubes: two Giant Kohn anomalies predicted

•Semi-conducting tubes: NO Kohn anomalies predicted

Ef

Metallic tubes: same geometrical conditions as graphite

Metallic tubes: LO-TO splitting

0.0 0.1 0.2 0.3 0.4 0.5

1190

1260

1330

1400

1470

1540

1610

Pho

non

Fre

quen

cy (

cm-1)

Phonon Wavevector (2/a units)

10

LO:

• Axial• strong EPC• G-

TO:

• Circumferential• No KA• G+

Opposite Interpretation

Dynamic Effects

• Frozen phonons• Finite differences

• Density functional perturbation theory

Staticapproaches

Rely on Born-Oppenheimer approximation: electrons

see fixed ions

For 3D crystals this is 100% OK

This is no longer true for 1D systems

• The dynamic nature of phonons can be taken into account• Beyond Born-Oppenheimer…

Dynamic effects in nanotubes

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

1540

1550

1560

1570

1580

1590

1600

1610

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

1500

1530

1560

1590

1620

1650

LO

Ph

on

on

Fre

qu

en

cy (

cm-1)

Dynamic Static EZF (static)

(11,11) 315K

a)

Phonon wavevector (2/a0 units)

b) (11,11) 315K

TO

Dynamic Static EZF (static)

•KA@LO: smeared•New KA@TO

•LO: increased•TO: decreased

Dynamic effects

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

1540

1550

1560

1570

1580

1590

1600

1610

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

1500

1530

1560

1590

1620

1650

LO

Ph

on

on

Fre

qu

en

cy (

cm-1)

Dynamic Static EZF (static)

(11,11)

315K

a)

Phonon wavevector (2/a0 units)

b) (11,11)

315KTO

Dynamic Static EZF (static)

0.00 0.01 0.02 0.03

1550

1560

1570

1580

1590

a)

LO

Ph

on

on

Fre

qu

en

cy (

cm-1)

T=30K T=300K T=1000K

0.00 0.01 0.02 0.03

1500

1550

1600

1650

1700

1750

b)

Phonon Wavevector (2/a0 units)

TO

0.00 0.01 0.02 0.03 0.04 0.05

1530

1560

1590

1620

0.00 0.01 0.02 0.03 0.04 0.05

1440

1530

1620

1710

a)

TO

Ph

on

on

Fre

qu

en

cy (

cm-1)

d=0.8 nm d=1.6 nm d=2.4 nm

LO

b)

Phonon Wavevector (2/a0 units)

Phonons are not static deformations

•KA@LO: smeared•New KA@TO

•T increases:•KA@LO: weaker•KA@TO: no changes

•d increases:•KA@LO: weaker•KA@TO: weaker

LO and TO frequencies

Th Vs Exp: Room Temperature

1530

1540

1550

1560

1570

1580

1590

1600

0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.41520

1530

1540

1550

1560

1570

1580

1590

1600

G-

Diameter (nm)

Semiconducting

Ram

an S

hift

(cm

-1)

LO

TO

Metallic

G+

G-

G+

LO

TO

0.6 0.9 1.2 1.5 1.8 2.1 2.40

50

100

150

Brown [10] Jorio [11] Maultzsch [14] Oron-Carl [17] Doorn [18]

FW

HM

(G- )

(cm

-1)

Diameter (nm)

Metallic tubes

• Metallic tubes: G-LO & G+TO

• Semiconducting tubes: G- TO & G+ LO

• Fermi golden rule:•EPC FWHM(G-)

Interpretation of Raman spectra

1450 1500 1550 1600 1650 1700

MetallicSWNT

1550 1587G- G+

Raman Shift (cm-1)

1450 1500 1550 1600 1650 1700

Semiconducting SWNT

1570

1592

G-

G+

Raman Shift (cm-1)

TO – circumferential

TO – circumferential

LO – axial

LO – axial

Semiconducting:

• LO-TO splitting curvature• G+ axial• G- circumferential

Metallic:

• LO-TO splitting Kohn an.• G+ circumferential• G- axial (KA)• FWHM(G-) EPC

G- interpretation: EPC and notPhonon-plasmon resonance

Piscanec et al. PRB (2007)

G- band Vs T: experiments

• Metallic SWNTs

• Dielectrophoresis• HiPCo SWNTs (Houston), d~1.1nm• Vpp = 20 V and f=3MHz

• Raman Spectroscopy

• = 514 nm (resonant with semicon.)• = 633 nm (resonant with metallic)• Linkam stage: 80K < T < 630K

Krupke et al. Science 301, 344 (2003)

G- band Vs T: experiments

• Semiconducting tubes: G+ - G- constant Anharmonicity• Metallic tubes: G+ - G- increases with T ??? (EPC)

Th Vs Exp: Temperature Dependence

0 150 300 450 600 750 900

25

30

35

40

45

50

55

60

65

70

Semiconducting d=1.1nm

G+-G

- (cm

-1)

Temperature (K)

Metallicd=1.0nm

0 150 300 450 600 750 900

25

30

35

40

45

50

55

60

65

70

Semiconducting d=1.1nm

static

G+-G

- (cm

-1)

Temperature (K)

Metallicd=1.0nm

0 150 300 450 600 750 900

25

30

35

40

45

50

55

60

65

70

Semiconducting d=1.1nm

static dynamic

G+-G

- (cm

-1)

Temperature (K)

Metallicd=1.0nm

Metallic tubes from R. Krupke

Conclusions

• Measurement of the Raman G-band Vs T Metallic tubes from dielecrophoresis Semiconducting tubes G+ - G- = constant Metallic tubes G+ - G- changes with T

• Kohn anomalies and electron phonon coupling and dynamic effects

Interpretation of G-band in SWNTs Raman spectra Explanation of the T-dependence of the G- in metallic SWNTs

•