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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