Exciton effect of the resonance

window of SWNTs

1Department of Physics, Tohoku University, Sendai, Japan2Department of Mechanical Engineering, University of Tokyo, Tokyo, Japan

J. S. Park1, K. Sato2, R. Saito1

The 2nd GCOE International Symposium

• No destructive, No contact Measurement

– Room Temperature at Ambient Pressure

– Quick (1min)

• Standard tool for Nano-Technology

Raman Spectroscopy of CNTM. S. Dresselhaus, et.al. Physics Reports, 409, 47-99, (2005)

Background

One dimensional structure

Chiral vector (n,m) determines physical properties of CNT.

Metallic and semiconducting electronic structure

Carbon nanotube

Raman Spectroscopy

1nm

1m

(5,5)

Metallic Semiconducting

EFEF

(9,0) (10,0)

Ch

T

Ch=na1+ma2= (n,m)

na1ma2

As-grown purified

CoMoCAT SWNTsSDS + SWNTs

Population (P) =IEXP(RBM)

ICAL(RBM)

S:M=1.2:1 S:M=2.4:1

If consider the calculated RBM intensity,

S:M=11:1

A. Jorio et al., PRB (2005)

Motivation: Population of SWNTs with RRS

Population by summation of experimental RBM intensity

(n,m) dependent P(IEXP/ICAL)

M

S

M

S

Precise nanotubes population

Calculation of precise Raman cross-section

Exciton-phonon matrix, excition-optical matrix, (Resonance width)Jiang, et.al. PRB (2007) Park, et.al. PRB (2006)

M

G+G-

G-

S-SWNT

M-SWNT

G+

[cm-1] G+ G-

S-SWNTs 1590 1570

M-SWNTs 1590 1550

Kohn anomaly effect in

Metallic SWNTsS.D.M. Brown et al., Phys. Rev. B63, 155414 (2001)

A. Jorio et al., Phys. Rev. B65, 155412 (2002)

Chirality dependence of

the experimental G band

Z. Yu, et al. J. Phys. Chem. B,

105, 6831 (2001)

Outline of this presentation

1. Raman spectra of single wall carbon nanotubes

- Radial Breathing Mode and G band

- Kohn anomaly effect of metallic-SWNTs

2. Exciton effect of Raman resonance window- Introduction of previous calculation

- Exciton-phonon scattering process

Intra-valley scattering

Inter-valley scattering (in future work)

3. Summary

Calculation of Raman spectra (RBM, G)

Raman Intensity I(, EL)

phonon

energy K ELk

j

a

b

resonance

window

Laser

energy

ex-ph

matrix

optical

matrixoptical

matrix

K. Sasaki, et al. Phys. Rev. B 77, 245441 (2008)

ex

Original

frequencyFrequency shift

2

Kohn anomaly

effect for metallic tube

Raman spectra = Raman intensity + Spectral width (const.)

G band Raman spectra

ELaser = E11Low

TOLO

LO

J. S. Park, et al. Phys. Rev. B 80, 081402 (2009)

TO

LO

M-SWNT ELaser = E22S-SWNT

k11

10 Mel-ph

(15,0)TO

k11

10 Mel-ph

(15,0)LO

Mel-phk

Chiral angle dependence of KA for G band

Zigzag Armchair

Why TO becomes hard for zigzag NT?

Q dependent el-ph interaction

LO:

TO:

Chiral

K. Ishikawa and T. Ando, J. Phys. Soc. Jpn. 75, 84713 (2006).

T. Ando, J. Phys. Soc. Jpn. 77, 14707 (2008).

K. Sasaki, R. Saito, et al. Phys. Rev. B, 77, 245441 (2008)

TOLO

TO

LO LO

EF>0

EF<0

(11,8)

Gate voltage dependence of G band Raman spectraJ. S. Park, et al. Phys. Rev. B 80, 081402 (2009)

Metallic RBM Raman spectra

ELaser = E11Low

Ex-ph matrix elements &

Resonance windows

Zigzag

Armchair

(15,0)

(10,10)

(13,4)

(11,8)

(12,6)

(14,2)

q=0

RBM

A

Z

k22

10 Mel-ph

(10,8)

S1

k22

10 Mel-ph

(10,9)

S2

S1 Type

S2 Type

Chiral angle dependence of ex-ph interaction

semiconducting

S1 Type S2 Typemod(2n+m,3)=1 mod(2n+m,3)=2

S1

S2

Experimental 2D RBM Raman plot

ii

Resonance

window

EL=Eii EL Eph

ab

i

valence

band

conduction

band

Resonance

point

ħ

ħ

ħ

el-ph AD1 AD2

additional effect

(ex. el-op scattering)

Resonance window

- Inverse lifetime of a

photo-excited electron

Resonance Raman window

MIT

Experimental resonance window

Brazil

Calculated values

(S1) > (S2) in the diameter (>0.9nm) range

Resonance window for S-SWNTs

For 0.6 < dt < 1.5nm

S1 type

S2 type

semiconducting

(8,0), (7,2), (9,1), and (11,0) SWNTs

: small value compared to other S1 tubes

Diameter and chirality

dependence

0.9 nm

S1 Type

S2 Type

mod(2n+m,3)=1

mod(2n+m,3)=2

Comparison with the experiment

Semiconducting SWNTs

Correlation

Ex

pe

rim

en

t

Calculation

Experiment(Brazil)

EX 15 meV

S1 Type

S2 Type

Experiment (MIT)

F=38(S)

F=30(M)

E33S

E11ML

E22S

Calculation

Exciton-phonon scattering processes

Intra-valley scattering

k

kq

kq1

k

kq

kq2

Bright

exciton

Bright

exciton Bright

excitonDark

exciton

Intervalley Scattering

A exciton

ehK

KK

E exciton

e

h

K

KK

not vertical transition

A- : bright exciton

A+, E and E*: dark excitons

RBM and LO phonon

LO

RBM

TO

S1

S2

Open circle : el-ph

Filled circle : ex-ph

TO phonon

Ex-ph Intra-valley scattering

Exciton-phonon

interactionElectron-phonon

interaction

Mel-phk

k

Mex-ph

Why ex-ph is smaller than el-ph?

Mel-ph

Constant for k

k

Mel-ph

Mex-ph

Not constant for k

Exciton-phonon interaction

Summary

1. Chirality dependence of RBM and G band Raman intensity

RBM : Zigzag > Armchair

G : only one peak appears in Zigzag CNT.

2. Resonance window

exciton-phonon Intravalley scattering

: Bright exciton (A-)

exciton-phonon Intervalley scattering

: Bright(A-) and Dark (E or E*) exciton

3. Ex-ph interaction < el-ph interaction

- Localization of exciton

- El-ph interaction dependence on k.

Gate voltage dependence of metallic RBM Raman spectra

(12,0) (9,6) (8,8)

EF=0.6eV

EF=0.6eV

Ram

an inte

nsity, a. u.

200180160140

Raman shift, cm -1

0 V

-0.5 V

0.5 V

-1.5 V

1.5 V

-1.0 V

1.0 V

EXP

Sasaki-san

Jiang-san

RBM for (10,10)

Goff = 6.4 eV

Gon = 3.0 eV

Resonance window- Inverse lifetime of a photo-excited electron (uncertainty principle)

EL=Eii EL Eph

ab

i

valence

band

conduction

band

Resonance

window

ii

Raman excitation profile

Resonance

point

ħ : relaxation time of transition

from an initial k state to all possible

final k states

ħ

ħ

ex-ph AD1 AD2

additional effect

(ex. ex-op scattering)

Fermi Golden rule

ex

Symmetry of Excitons

A

Bright and dark exciton

A- : bright exciton

A+, E and E*: dark excitons

J. Jiang et al. Phys. Rev. B75 035405 and 035407(2007)

E exciton

e

h

K

KK

C2

ehK

KK eh

K

KK

ehK

KK

00exciton dipole transition matrix

Eigen states are irreducible representation

for C2 rotation (odd or even).

not vertical transition

A+ even A- odd

oddodd

A exciton

ehK

KK

Resonance window for intravalley scattering

Type I

Type II

O Ec2-c1

Eph < Ec2-c1

c2

c1

Resonance

Exciton-phonon interaction

Exciton-phonon scattering processes

Intravalley scattering (2) Intervalley scattering(2)phonon(6)=24

1. Intravalley scattering

O Ec2-c1

Eph < Ec2-c1

c2

c1

Resonance

2. Intervalley scattering

kk+q

Ram

an inte

nsity, a. u.

200180160140

Raman shift, cm -1

0 V

-0.5 V

0.5 V

-1.5 V

1.5 V

-1.0 V

1.0 V

Gate dependent resonance Raman excitation profile

M. Kalbac et al, unpublished.

Ra

ma

n in

ten

sity,

a.u

.

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Electrode potential, V vs. Ag/Ag+

2.067 eV

2.084 eV

2.120 eV

2.138 eV

2.102 eV

M

Ram

an in

tensi

ty,

a.

u.

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Electrode potential, V vs. Ag/Ag+

1.653 eV

1.687 eV

1.699 eV

1.710 eV

1.722 eV

1.734 eV

1.746 eV

1.676 eV

Ram

an in

tensi

ty, a. u.

200150

Raman shift, cm -1

16001500

x10

S

A- : bright exciton

A+, E, E*: dark excitons

E exciton

e

h

K

KKA++A- exciton

ehK

KKeh

K

KK

A+-A- exciton

E* exciton

e

h

K

KK

inter-valley

scattering

(fast!)

D-band, G’-band

exciton-phonon interaction

- breaking symmetry -

EF dependence of the G band

Raman spectra for metallic tubes

LO

softeningTO

noshiftLO

softening

TO

hardening

LO

softening

Armchair – TO mode not shift

Chiral tube – both TO and LO shift

Zigzag – LO softening

EL

j

a

b

G band Raman scattering process

a point

: resonance point

ab process

: el-ph scattering

Original

frequencyFrequency shift

2

Kohn anomaly

effect for metallic tube

What is the Kohn Anomaly

Hard

Soft

Low energy el-hole pair – TO hardening

High energy el-hole pair – LO softening

El-ph matrix for the el-hole pair creation

F : Fermi velocity

: Pauli matrix

g: el-ph coupling constant

A: Deformation-induced gage field

u: Relative displacement vector

El-hole pair

energy

K. Sasaki, et al. Phys. Rev. B 77, 245441 (2008)

El-ph matrix elements for Raman scattering processK. Sasaki, et al. Phys. Rev. B 77, 245441 (2008)

Armchair (=30)

QR(k)

QR(k)

Calculation of Raman spectra (RBM, G)

G band Raman spectra = Intensity Spectral width

1. Raman Intensity I(, EL)

phonon

energy

K ELk

j

a

b

resonance

window

Laser

energy

ex-ph

matrix

optical

matrixoptical

matrix

2. Spectral width

Phonon energy

Original

frequency

Correction frequency

Including el-ph coupling

Spectral

width

Electron-hole pair creation matrix

by el-ph interaction

electron

energy

hole

energy

Fermi distribution function

Spectral width is given by the decay length .

K. Sasaki, et al. Phys. Rev. B 77, 245441 (2008)

Kohn anomaly effect

ex