Under Supervision of:

University of Qom

In the name of allah

Excitons in Single Wall Carbon Nanotubes

Dr. Fazeli and Dr. Mozaffari

Nasim Moradi

OutlineIntroduction to carbon nanotubes and excitons

Theoretical backgrounds

ABINIT code

Results and Our Model

Excitons In Single – Walled Carbon Nanotubes

Carbon nanotubes were discovered in 1991 by Iijima

GraphiteA single layer of graphite, graphene

introduction to nanotubes ■□□□□□□□□□□

Nasim Moradi

1

Excitons In Single – Walled Carbon Nanotubes

a carbon nanotube made of a single graphite

layer rolled up into a hollow cylinder

Animation from S. Maruyama’s carbon nanotube site

with diameter as small as

nm

Length : few nm to

microns

introduction to nanotubes ■□□□□□□□□□ Nasim Moradi

2

Excitons In Single – Walled Carbon Nanotubes

J. Charlier et al. , “electronic properties of nanotubes”, Rev. Mod. Phys. 79, ( 2007 )

multi-walled nanotube (mwnt)Diameter ~ 10 – 50 nm

Single-walled nanotube (swnt)

Diameter ~ 0.5 - 2nm

Images of nanotubes

3

Chiral vector

SWNT’s geometry

specified by a pair of

integers (n , m) :

a = lattice constant of the honeycomb networka = (, the C-C bond length)

diameter

𝑑𝑡=¿ 𝑐h∨¿/𝜋=𝑎𝜋√𝑛2+𝑛𝑚+𝑛2 ¿

1 2hC na ma

4

Excitons In Single – Walled Carbon Nanotubes

SW

NT

’s g

eom

etry

5

Excitons In Single – Walled Carbon Nanotubes

ethan minot , “Tuning the band structure of cnt”, PhD Thesis, cornell univ (2004 ), page : 28

𝑘⊥=2𝜋 𝑙

¿𝑐h∨¿¿

(7,7)

(7,7) (7,0)

Metallic: Semiconducting:

3n m l

3 1n m l

Electronic properties □■□□□□□□□□ Nasim Moradi

6

Excitons In Single – Walled Carbon Nanotubes

optical properties □□■□□□□□□□ Nasim Moradi

Ratio problem :22

11

2E

E

Kataura plot

H. Kataura et al., “Optical Properties of Single-Wall Carbon Nanotubes”, Synth. Met. 103, (1999)

C. L. Kane et al., “ Ratio Problem in Single Carbon Nanotube Fluorescence Spectroscopy”, Phys. Rev. Lett. 90, (2003)

7

Excitons In Single – Walled Carbon Nanotubes

optical properties □□■□□□□□□□ Nasim Moradi

An exciton is a bound state of an electron and hole which are attracted

to each other by the electrostatic coulomb force.

ex gap bondingE E E

exciton

Many-body effects modify the physical picture :

Grosso, Solid state physics, page: 2338

Excitons In Single – Walled Carbon Nanotubes

Bethe – Salpeter Equation

The equation was actually first

published in 1951.

describes the bound states of a two-body

(particles) system in a formalism.

[ ] s eh s s sck vk vck v c k vck

v c k

E E A vck K v c k A A

Hans Bethe Edwin Ernest Salpeter 1906 - 2005

1924 - 2008

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Excitons In Single – Walled Carbon Nanotubes

How can we obtain optical spectra by solving

the Bethe-Salpeter equation?

Problem definition □□□■□□□□□□ Nasim Moradi

10

Excitons In Single – Walled Carbon Nanotubes

Nasim MoradiABINIT code □□□□■□□□□□

acell bdgw bs_algorithm bs_freq_mesh ecut ecuteps enunit ixc iscf nsppol nstep nsym soenergy symsigma xred zcut znucl F. . gwcalctyp gwmem kptopt ngfft ngkpt npwkss

ABINIT is an ab initio computational package, using the pseudopotentials and

plane-wave basis set. It allows one to find the total energy , charge density and

electronic structure of systems within Density Functional Theory.

11

X. Gonze et al. “ABINIT: First-principles approach to material and nanosystem properties”, Comput. Phys. Comm. 180 (2009)

The Carbon atom has six

electrons .1 s  𝟐   2s  𝟐   2 p  𝟐×192 128

Unit cell contains 32 carbon atoms

.

(8,0)

Structures producted by jmol software 12

y

x

Excitons In Single – Walled Carbon Nanotubes

Nasim MoradiDensity Functional Theory (DFT) □□□□□■□□□□

( )n r

2

1

( )occ

ii

n r

Self – Consistent?

( )effV n

KS EquationSolve

 Output Quantities

No

Yes

13

2

( [ ]) ( ) ( )2 eff i i iV n r r

[ ] ( ) [ ] [ ]eff ext Hartree xcV n V r V n V n

[ ( )]( )

( )xc

xcE n r

V rn r

LDA: Teter Pade parametrization

Local density approximation

• Kohn-Sham equations

exchange-correlation energy

R. M. Martin, Electronic Structure, page 173

Excitons In Single – Walled Carbon Nanotubes

Nasim MoradiABINIT code □□□□□■□□□□

Band structure obtained with ABINIT

0.57 eVgapE

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Excitons In Single – Walled Carbon Nanotubes

Nasim MoradiMany-Body Perturbation Theory (MBPT) □□□□□□■□□□

1( )

2) : 1 | ( )

3) :

) :

)

1 ( )

(

KS

KS KS KS KSi i i xc iE E

KS KS KS KSi i

QP KS KS QP KSi i i i xc

i i xc i

i

EE E V

E

E Z E V

E V

2

[ ( ) ( ) ( )] ( ) ( )2

QP QP QP QPext H i i i iV r V r r r

• Dyson Equation or Quasiparticle (QP) Equation :equation:

• First - Order Perturbation Theory :

15W. G. Aulbur, Quasiparticle calculations in solids

Excitons In Single – Walled Carbon Nanotubes

GW approximation □□□□□□■□□□ Nasim Moradi

L. Hedin, “New Method for Calculating the One-Particle Green's Function

with Application to the Electron-Gas Problem”, Phys. Rev. 139 , (1965)16

0 0 ModelG W

Excitons In Single – Walled Carbon Nanotubes

Nasim MoradiABINIT code □□□□□□■□□□

KSS fileSC

R fil

e

GW file

WG, G (q,)G, G 1 (q,)v(q G )

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Excitons In Single – Walled Carbon Nanotubes

Nasim MoradiABINIT code □□□□□□■□□□

GW Band structure obtained with ABINIT18

1.76 eVgapE

Excitons In Single – Walled Carbon Nanotubes

□□□□□□□■□□Bethe-Salpeter Equation

19

[𝑬¿¿𝒄𝒌−𝑬 𝒗𝒌]𝑨𝒗𝒄𝒌𝒔 +∑

𝒗 ′𝒄 ′𝒌 ′

¿¿ ¿

𝑯 𝑩𝑺¿𝝋𝒔 ⟩=𝜴𝒔 ¿𝝋𝒔 ⟩

𝑯 𝑩𝑺=𝑯𝟎+𝑲𝒆𝒉

¿𝝋𝒔 ⟩=∑𝒗𝒄𝒌

𝑨𝒗𝒄𝒌𝒔 ¿𝒗𝒄𝒌 ⟩

𝑲 𝒆𝒉=𝟐𝐕−𝐖

Nasim Moradi

Excitons In Single – Walled Carbon Nanotubes

□□□□□□□■□□Bethe-Salpeter Equation

20

Tamm-Dancoff approximation

Nasim Moradi

Excitons In Single – Walled Carbon Nanotubes

Optical spectra by solving the BSE □□□□□□□□■□ □□□□□□□□□

21

Nasim Moradi

11 1.33 E eV

22 1.57 E eV

1.18Ratio

C. Spataru, S. Ismail-Beigi, L. X. Benedict, S. G. Louie

“ Excitonic Effects and Optical Spectra of SWCNTs ” , Phys. Rev. Lett. 92 ( 2004 )22

ab initio E11=1.55(eV)

E22=1.80(eV)

Ratio=1.16

exp E11=1.60(eV)

E22=1.88(eV)

Ratio=1.17

S. M. Bachilo et al., Science. 298 (2002)

Deformation

1b

a

Structures producted by v_sim software

0.8b

a 0.7

b

a

0.9b

a

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□□□□□□□□□■Results and our model

24

Nasim Moradi

Semiconductor-Metal Transition

This result reported in :Yaroslav V. Shtogun, Lilia M. Woods, “Electronic Structure Modulations of Radially Deformed Single

Wall Carbon Nanotubes under Transverse External Electric Fields”, J. Phys. Chem. C. 113 (2009)

0.26 eVgapE 0.49 eVgapE

□□□□□□□□□■

25

Results and our model Nasim Moradi

□□□□□□□□□■

25

Results and our model Nasim Moradi

Red Shift

DFT• Time and memory

GW• Time and memory

BSE• Time and memory

Cputime: 5.5h

Mem: 1.5 GB

Cputime: 1.5-15h

Mem: 4-12 GB

Cputime: 438h

Mem > 20 GB

26

Thanks foryour

attention

Excitons In Single – Walled Carbon Nanotubes

Nasim MoradiABINIT code □□□□□□□□□

Soenergy

GW LDAgap gapSoenergy E E

KSS fileSC

R fil

e

MDF file