special symmetries: carbon nanotubes

Special symmetries: carbon nanotubes

Yves Noël

and Raffaella Demichelis

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Overview

I.What are carbon nanotubesa. Definitionsb. Vectors and the flat latticec. The helical symmetry

II.The exploitation of symmetry in CRYSTALa. Structure constructionb. Integrals calculationc. Diagonalisationd. Performance

III.Some results on carbon nanotubesa. Structureb. Band gapc. Vibrational frequencies

What are nanotubes?Nanotubes are " regular cylindrical nanometric structures "

Experimentally discovered by Iijima in 1991

Inorganic nanotubes

Single wall /multiple wall nanotubes

Periodicity and lattices

Tube:1D lattice

flat equivalent:

The graphene sheet~ 2D lattice

+ cyclicity

The graphene sheet as reference

R,L define the nanotube cellcell basis.

Tube axis

Rolling vector R

Longitudinal vector L

Nanotube cell

R and L are

orthogonal


Vectors are expressed in the graphene basis.

R

L

Grapheneunit cell

R

R = R/N= (2,1)n1=6 n2=3

R = (n1,n2) = (6,3)

L = (l1,l2) = (-4,5)

In general n1 and n2are not coprime:

N is the greatest common divisor

R passes N=3 times through nodes


The chiral angle is measured with respect to the zigzag rolling and ranges from 0 to 30°.

(n,n) : arm

chair

=30°

(n,0) : zig zag =0°

(6,3) : Chiral

=19.1°


Tube axis

a1

a2R

L

R , H : a new basis more appropriated to the nanotube

R

H


The nanotube cell and R,H basis

The cell contains 42 nodes

R

HL=1/14

HR=3/14

H


Order :

R/R = N = 3

The R translation can be applied N times before reaching the starting node

R

1 2 3

Correspondence betweenFlat 2D and cylindrical 1D lattice

Graphene Nanotube

Translation R/NTranslation R = R/3

Rotation 2/NRotation 2/3

The graphene sheet referenceThe H translation can be applied 14 times before reaching the starting node

HL=1/14

HR=3/14

H

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Translation R/pTranslation R = R/3

Translation HR = 3R/14

Translation L/KTranslation HL = L/14

General translationTranslation H

Correspondence between Flat 2D and cylindrical 1D lattice

Graphene Nanotube


Rotation 6/14


RototranslationRototranslation (6/14 , L/14)

The graphene sheet referenceThe H+R translation can be applied 42 times before reaching the starting node

HL=1/14

HR=3/14

HR

1

2

3

4

25

27

11

12

13

14

5

6

7

8

9

10

21

22

23

24

26

30

31

15

16

17

18

19

20

29

32

33

34

35

36

38

40

37

39

41

4228

R+H

Correspondence betweenFlat 2D and cylindrical 1D lattice

Graphene Nanotube

Translation R/NTranslation R = R/3

Translation HR = 3R/14


General translationTranslation H


Rotation 6/14


RototranslationRototranslation (6/14 , L/14)

Translation H+RRototranslation (3/14.2+1/3.2 , L/14)

Rototranslation (23/42.2 , L/14)

Group symmetry

The nanotube symmetry group is the direct product of 2 cyclic groups:

An helical cyclic group of order M = N x K

A pure rotation of order N(R is larger than R )

An helical group of order K(coming from H)

Large atom number but high symmetry

15.39

15.39

11.29

15.39

27.98

4.27

L(Å)

6.2213.92081042(12,4)

4.2513.9156782(9,3)

3.1119.184422(6,3)

2.8313.9104522(6,2)

2.577.6172862(6,1)

2.350.024122(6,0)

R(Å)

deg)

Nb atNbsym

Nb irr

Exploitation of symmetry in CRYSTAL

I. For the tube constructionDISTINCTIVE NANOTUBE VECTORS EXPRESSED IN THE SLAB UNIT CELL BASIS A AND B

A BROLL UP DIRECTION, R 9 3HELICAL DIRECTION, H 2 1CYLINDER AXIS, L 1 5

LENGTHS IN ANGSTROMSCYLINDER RADIUS 3.112PERIOD ALONG THE CYLINDER AXIS, |L| 11.289THICKNESS OF THE SLAB 0.000

NUMBER OF ATOMS 84CHIRAL ANGLE 19.107 DEG

NANOTUBE SYMMETRYNUMBER OF SYMMETRY OPERATORS 42NUMBER OF ATOMS IN THE ASYMMETRIC UNIT 2THE HELICAL OPERATOR:ORDER 42ANGLE ( 23/ 42) x

2PI RADTRANSLATION 1/ 14 OF

THE CYLINDER PERIOD

DISTINCTIVE NANOTUBE VECTORS EXPRESSED IN THE SLAB UNIT CELL BASIS A AND B

A BROLL UP DIRECTION, R 9 3HELICAL DIRECTION, H 2 1CYLINDER AXIS, L 1 5

LENGTHS IN ANGSTROMSCYLINDER RADIUS 3.112PERIOD ALONG THE CYLINDER AXIS, |L| 11.289THICKNESS OF THE SLAB 0.000

NUMBER OF ATOMS 84CHIRAL ANGLE 19.107 DEG

NANOTUBE SYMMETRYNUMBER OF SYMMETRY OPERATORS 42NUMBER OF ATOMS IN THE ASYMMETRIC UNIT 2THE HELICAL OPERATOR:ORDER 42ANGLE ( 23/ 42) x

2PI RADTRANSLATION 1/ 14 OF

THE CYLINDER PERIOD

CRYSTAL computes the helical symmetry of the rolling vector given in the input and apply all the symmetry operators of the group to fill the tube.


II. For integrals calculation

CRYSTAL uses the symmetry to compute the mono and bielectronic integrals of the irreducible Fock matrix only

Irr F

+ Symmetry

F


III. For the diagonalisation in the reciprocal space of the Fock matrix

If a Symmetry Adapted Crystalline Orbitals (SACO) basis is used instead of a Bloch Functions basis, F(k) becomes block diagonal ; each block corresponding to an irreducible representation.

Example of the (6,0) SWCNT :

61111G :22 AOs24 atoms

F(k) size = 528x528

SACO12 blocks

44x44BF

1 block 528x528

Several small matrices are more easily diagonalised than a unique big matrix.


The unit cell of the (24,0) SWCNT contains 96 atoms.

FREQUENCY CALCULATION: 96x3+1=289 SCF calculations

With the helical symmetry exploitation, the calculation is performed on 2 irreducible atoms:

FREQUENCY CALCULATION: 2x3+1=7 SCF calculations

IV. building of the dynamic matrix (vibration frequency calculation)þ

SCF cycle time scaling

Times of the different parts of an scf cycle(seconds as a function of the atoms number) þ

Results

Computation conditions:Basis : 6 1111 G*Hybrid hamiltonian B3LYP

Studied carbon nanotubes : Zigzag nanotubes (n,0) up to n=24

References:1) Y. Noel, P. D�arco, R. Demichelis, C.M. Zicovich-Wilson, R. Dovesi On the use of symmetry in the ab initio quantum mechanical simulation of

Nanotubes and related materials, J. Comp. Chem, 2009, in press

2) Demichelis, D�arco, Noel, Dovesi � in preparation

Results: Relaxation energy

E = Erelaxed - Eunrelaxed

Relaxation energy E (kJ/mol by cell), as a function of the radius (Å)

Results : Stability

E = Erelaxed - Egraphene

Relative stability E (kJ/mol by cell) of the tube with respect to graphene, as a function of the radius (Å)

Cyclicity and Brillouin zone

K

K

M

Graphene

b1

b2

K

SWCNT (n,0)

(6,0)

K

(7,0)

(9,0)

K

(8,0)

Cyclicity and Brillouin zone

(4,4)

SWCNT (n,n)

K

K

M

Graphene

b1

b2

(3,3)K

Cyclicity and Brillouin zone(6,0)

K

(3,3)

No moreon K Still on K

Semi conductor with small gap

conductor

But the Brillouin zone is deformed and is no more rigorously hexagonal because of the tube curvature

Metallic if 2n1+n2 or n1+2n2 is a multiple of 3

Results : Band gaps

Energy gap as a function of the radius (Å)

Unpublished material removed

* Demichelis, D�arco, Noel, Dovesi � in preparation

Results : Band gaps

(a) Weisman et al. Nano Lett. 2003, 3, 1235(b) Ouyang et al. Science 2001,292,702

Pure DFT methods systematically underestimate Eg; the inclusion of the exact HF exchange (hybrid methods) redresses this lack.

Hybrid functionals fail in describing small Eg; pure DFT provide more reasonable results.

Brothers et al. J. Phys. Chem. B

2006, 110, 12860

Unpublished material removed

* Demichelis, D�arco, Noel, Dovesi � in preparation

Results : vibrational frequenciesFrequency calculation of (n,0) zig-zag SWCNT

645 (inactive) þ

0

1634

GRAPHENE

� 3N- 4 vibrational frequencies (N=atoms in unit cell; N=4n for (n,0) SWCNT) þ

� 15 active modes for all (n,0) SWCNT

� Frequencies tend to the ones of graphene for increasing SWCNT radii

6 modes

3 modes

6 modes

Unpublished material removed* Demichelis, D�arco, Noel, Dovesi � in preparation

Practical: a nanotube input

Specific keywords SWCNT

construction

NANOTUBE

construction

NANORE

Use a previous geometry

NANOJMOL

Graphical output

Usefull keywords ROTCRY

Position of the atomswith respect to theundistorded surface

ATOMROT

Which side in/out

special symmetries: carbon nanotubes

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