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Study of theSi fullerene cage isomers

Zacharias G. Fthenakis1,2,

Remco W. A. Havenith3, Madhu Menon4,5 and Patrick W. Fowler6

1Institute of Electronic Structure and Laser - FORTH, Greece2Department of Physics, University of Crete, Greece3Debye Institute, Utrecht University, The Netherlands

4Department of Physics and Astronomy, University of Kentucky, USA5Center for Computational Sciences, University of Kentucky, USA

6Department of Chemistry, University of Exeter, UK

Study of the Si fullerene cage isomers – p. 1

Fullerenes

Definition:

Study of the Si fullerene cage isomers – p. 2

Fullerenes

Definition:

Closed-cage structuresconstructed by only

pentagonal and hexagonal rings

Study of the Si fullerene cage isomers – p. 2

Fullerenes

Definition:

Closed-cage structuresconstructed by only

pentagonal and hexagonal rings

(All the atoms are three-fold co-ordinated=⇒ sp2-like bonding)

Study of the Si fullerene cage isomers – p. 2

C60 Buckminsterfullerene

H.W.Kroto, J.R.Heath,

S.C.O’Brien, R.F.Curl

and R.E.Smalley,

Nature,318, 162, (1985)

Study of the Si fullerene cage isomers – p. 3

. . . but not only C60

E.A.Rohlfing,

D.M.Cox and

A.Kaldor,

J.Chem.Phys.81,

3322, (1984)

Study of the Si fullerene cage isomers – p. 4

How many pentagons and hexagons ?

As a consequence of the Euler’s theoremfor the geometrical solids, an N-atom fullerene has:

⇓

Study of the Si fullerene cage isomers – p. 5

How many pentagons and hexagons ?

As a consequence of the Euler’s theoremfor the geometrical solids, an N-atom fullerene has:

⇓

• 12 pentagons

• N2−10 hexagons

N = 20, 24, 26, . . . , 2n, n ∈ N

Study of the Si fullerene cage isomers – p. 5

How many pentagons and hexagons ?

As a consequence of the Euler’s theoremfor the geometrical solids, an N-atom fullerene has:

⇓

• 12 pentagons

• N2−10 hexagons

N = 20, 24, 26, . . . , 2n, n ∈ N

Different arrangement of the pentagons and thehexagons gives different structures (isomers)

Study of the Si fullerene cage isomers – p. 5

How many isomers are they ?

N Isomers N Isomers N Isomers

20 1 42 45 62 2385

24 1 44 89 64 3465

26 1 46 116 66 4487

28 2 48 199 68 6332

30 3 50 271 70 8149

32 6 52 437 72 11190

34 6 54 580 74 14246

36 15 56 924 76 19151

38 17 58 1205 78 24109

40 40 60 1812 80 31924

P.W.Fowler and

D.E.Manolopoulos,

An atlas of fullerenes,

(Oxford: Clarendon Press),

1995

Study of the Si fullerene cage isomers – p. 6

What about Si fullerenes ?Bonding:C : sp1, sp2, sp3

Si : mainly sp3 + dangling bonds

Study of the Si fullerene cage isomers – p. 7

What about Si fullerenes ?Bonding:C : sp1, sp2, sp3

Si : mainly sp3 + dangling bonds

General Conclusions for Sifullerenes:

Study of the Si fullerene cage isomers – p. 7

What about Si fullerenes ?Bonding:C : sp1, sp2, sp3

Si : mainly sp3 + dangling bonds

General Conclusions for Sifullerenes:

• Thermodynamically unstable structures (localminima of the PES)

Study of the Si fullerene cage isomers – p. 7

What about Si fullerenes ?Bonding:C : sp1, sp2, sp3

Si : mainly sp3 + dangling bonds

General Conclusions for Sifullerenes:

• Thermodynamically unstable structures (localminima of the PES)

• They look like "puckered" balls (the Si atomsmove radially inwards and outwards)

Study of the Si fullerene cage isomers – p. 7

What about Si fullerenes ?Bonding:C : sp1, sp2, sp3

Si : mainly sp3 + dangling bonds

General Conclusions for Sifullerenes:

• Thermodynamically unstable structures (localminima of the PES)

• They look like "puckered" balls (the Si atomsmove radially inwards and outwards)

However ...Study of the Si fullerene cage isomers – p. 7

Three-fold co-ordinated and/orfullerene-like Si systems

• Si surfaces

Study of the Si fullerene cage isomers – p. 8

Three-fold co-ordinated and/orfullerene-like Si systems

• Si surfaces• Si-C heterofullerenes and nanotubes

Study of the Si fullerene cage isomers – p. 8

Three-fold co-ordinated and/orfullerene-like Si systems

• Si surfaces• Si-C heterofullerenes and nanotubes• Si nanowires and nanotubes

Study of the Si fullerene cage isomers – p. 8

Three-fold co-ordinated and/orfullerene-like Si systems

• Si surfaces• Si-C heterofullerenes and nanotubes• Si nanowires and nanotubes• Si fullerene-like endohedral clusters

Study of the Si fullerene cage isomers – p. 8

Three-fold co-ordinated and/orfullerene-like Si systems

• Si surfaces• Si-C heterofullerenes and nanotubes• Si nanowires and nanotubes• Si fullerene-like endohedral clusters• Si clathrates

Study of the Si fullerene cage isomers – p. 8

Three-fold co-ordinated and/orfullerene-like Si systems

• Si surfaces• Si-C heterofullerenes and nanotubes• Si nanowires and nanotubes• Si fullerene-like endohedral clusters• Si clathrates

Consequently: Si fullerenes are of interest

Study of the Si fullerene cage isomers – p. 8

To date studied Si fullerenes

• Not any systematic study of Si fullerenes andtheir isomers

Study of the Si fullerene cage isomers – p. 9

To date studied Si fullerenes

• Not any systematic study of Si fullerenes andtheir isomers

• Focused on highly symmetric starting structures(for exampleIh Si60)

Study of the Si fullerene cage isomers – p. 9

To date studied Si fullerenes

• Not any systematic study of Si fullerenes andtheir isomers

• Focused on highly symmetric starting structures(for exampleIh Si60)

• Studied structures:SiN , N = 20 − 32, 36, 44, 50, 60, 70

Study of the Si fullerene cage isomers – p. 9

The system under consideration

Si38

all the 17 isomers

and

Si20

for comparison

Study of the Si fullerene cage isomers – p. 10

The method for Global Optimization I• Molecular Dynamics At Constant Temperature

md2

ri

dt2= −∇iV − mγ

EK − ET

EK

dri

dt

EK =

N∑

i=1

1

2miv

2

i ET =f

2kBT

H.J.C.Berendsen, J.P.M.Postma, W.F. van Gunsteren, A.DiNola and J.R.Haak, J.Chem.Phys.81, 3684, (1984)

Study of the Si fullerene cage isomers – p. 11

The method for Global Optimization I• Molecular Dynamics At Constant Temperature

md2

ri

dt2= −∇iV − mγ

EK − ET

EK

dri

dt

EK =

N∑

i=1

1

2miv

2

i ET =f

2kBT

H.J.C.Berendsen, J.P.M.Postma, W.F. van Gunsteren, A.DiNola and J.R.Haak, J.Chem.Phys.81, 3684, (1984)

• If first neigbour’s distances > cut off distancethen freezing the motions (i.e.vi = 0)

Study of the Si fullerene cage isomers – p. 11

The method for Global Optimization I• Molecular Dynamics At Constant Temperature

md2

ri

dt2= −∇iV − mγ

EK − ET

EK

dri

dt

EK =

N∑

i=1

1

2miv

2

i ET =f

2kBT

H.J.C.Berendsen, J.P.M.Postma, W.F. van Gunsteren, A.DiNola and J.R.Haak, J.Chem.Phys.81, 3684, (1984)

• If first neigbour’s distances > cut off distancethen freezing the motions (i.e.vi = 0)

• Evolution ofV = V (t) in time under constanttemperature

Study of the Si fullerene cage isomers – p. 11

The method for Global Optimization II• Minima of V (t) configurations: Initials for

damping molecular dynamics

Study of the Si fullerene cage isomers – p. 12

The method for Global Optimization II• Minima of V (t) configurations: Initials for

damping molecular dynamics

• Damping Molecular Dynamics

md2

ri

dt2= −∇iV vi(t + δt) = 0.99vi(t)

Study of the Si fullerene cage isomers – p. 12

The method for Global Optimization II• Minima of V (t) configurations: Initials for

damping molecular dynamics

• Damping Molecular Dynamics

md2

ri

dt2= −∇iV vi(t + δt) = 0.99vi(t)

• Configuration of the global energy minimum

Study of the Si fullerene cage isomers – p. 12

The method for Global Optimization II• Minima of V (t) configurations: Initials for

damping molecular dynamics

• Damping Molecular Dynamics

md2

ri

dt2= −∇iV vi(t + δt) = 0.99vi(t)

• Configuration of the global energy minimum

In agreement with the Cambridge Cluster Database

for the first 100 Lenard-Jones clusters

(http://www-wales.ch.cam.ac.uk/CCD.html)

Study of the Si fullerene cage isomers – p. 12

38-atom fullerene isomers

Study of the Si fullerene cage isomers – p. 13

38-atom fullerene isomers

Study of the Si fullerene cage isomers – p. 14

Optimized Structures

OTBMD GTBMD DFT/B3LYP

Si38

Si20

Study of the Si fullerene cage isomers – p. 15

Cohesive Energy

0 5 10 15 20Isomer number

0

20

40

60C

ohes

ive

Ene

rgy

(meV

)

GTBMDDFT/B3LYP

Study of the Si fullerene cage isomers – p. 16

Comparison between Si38 and Si20

Cohesive energyMethod DifferenceOTBMD 0.1171 eVGTBMD 0.0385 eVDFT/B3LYP 0.0809 eV

Study of the Si fullerene cage isomers – p. 17

Comparison between Si38 and Si20

Cohesive energyMethod DifferenceOTBMD 0.1171 eVGTBMD 0.0385 eVDFT/B3LYP 0.0809 eV

Marsen and Sattler assumptions:• The smallest Si fullerenes are the most stable• For Si a fused pentagon rule can replace the IPR

of C fullerenes

B. Marsen and K. Sattler, Phys. Rev. B60, 11593, (1999)

Study of the Si fullerene cage isomers – p. 17

Bond Lengths

2.2 2.4 2.6 2.8Bond Length Threshold (A)

0

10

20

30

40

50

60

70N

umbe

r of

bon

ds

OTBMDGTBMDDFT/B3LYP

Si38

Si20

Study of the Si fullerene cage isomers – p. 18

Sum of the three bondassociated angles for ideal cases

Fornp pentagons (np = 0, 1, 2, 3)sum =360o − 12onp

3 hexagons 360o

2 hexagons + 1 pentagon 348o

1 hexagon + 2 pentagons 336o

3 pentagons 324o

tetrahedral arrangement 328.41o

not hybridized bonding 270o

Study of the Si fullerene cage isomers – p. 19

Distribution of the sum of thethree bond associated angles

200 240 280 320 360Sum of the three bond angles (

o)

0

50

100

150

200

Num

ber

of s

ums

200 240 280 320 3600

5

10

15

20

OTBMD

Study of the Si fullerene cage isomers – p. 20

Acknowledgments• Prof. A.N.Andriotis• This work was partially supported by EU TMR

Network "USEFULL"

Study of the Si fullerene cage isomers – p. 21

Study of the Si fullerene cage isomers – p. 22

Study of the Si fullerene cage isomers – p. 23

The Si(110) surface

M.Menon, N.N.Lathiotakis,

and A.N.Andriotis,

Phys. Rev. B56,

1412, (1997)

BackStudy of the Si fullerene cage isomers – p. 24

Si-C nanotubes

Back

M.Menon, E.Richter, A.Mavrandonakis, G.Froudakis, and A.N.Andriotis,

Phys. Rev. B69, 115322, (2004)

Study of the Si fullerene cage isomers – p. 25

Si nanowires and nanotubes

B.Marsen and

K.Sattler, Phys. Rev.

B 60, 11593, (1999)

Back

Study of the Si fullerene cage isomers – p. 26

Si fullerene-like endohedral clusters

Q.Sun, Q.Wang, P.Jena, B.K.Rao, and Y.Kawazoe,

Phys. Rev. Lett.90, 135503, (2003)

BackStudy of the Si fullerene cage isomers – p. 27

Si clathrates

Back

A.San-Miguel, P.Keghelian, X.Blase, P.Melinon, A.Perez, J.P.Itie, A.Polian,

E.Reny, C.Cros, and M.Pouchard, Phys. Rev. Lett.83 5290, (1999)

Study of the Si fullerene cage isomers – p. 28

Si20Active

X

Y

Z

Study of the Si fullerene cage isomers – p. 29