Force Field Development for Silicon
Carbides, Bulk Silicon and Oxidized
Silicon surfaces with Graphite
Santiago Solares, Adri van Duin and
William A. Goddard III
California Institute of Technology
Objectives
• To study graphite-silicon systems (vdw
interactions and reactions)
• To optimize Reax FF for silicon carbide systems
(molecular and bulk systems)
• To optimize Reax FF for all-carbon systems
(including free radicals and resonant structures)
• To compile a bonded force field to be used in
mechanical systems under high stresses
AFM Microscopy
5.5 nm
40 nm
Full Width 3.1 nm, Height 1.9 nm
Resolution = 1.2 nm
AFM Microscopy
Interactions to be optimized in Reax
Bonds:
Angles:
• Si-C
•
•
•
•
•
•
•
– Regular bond in H3SiCH3
– Simultaneous breaking of 2
bonds in Si2H4-C2H4
• Si=C
– H2Si=CH2
C-Si-Si
C-C-Si
C-Si-C
Si-C-Si
Si-C-H
C-Si-H
Future work: angles
involved in double bonds
Parameter Optimization Procedure
Si-C dissociation curve in H4Si2-C2H4 (for 2 bonds)
200
Energy, kcal/mol
150
singlet
triplet
Reax fit
100
50
0
1.0
1.5
2.0
2.5
3.0
Radius, Ang
3.5
4.0
4.5
5.0
Reax Fit Results
Si-C Bond Dissociation Curve
in H3 Si-CH3
Energy, kcal/mol
200
150
Reax
100
QM
50
0
1.0
1.5
2.0
2.5
3.0
Bond Length, Angstrom
3.5
4.0
4.5
Reax Fit Results
Si=C Double Bond Dissociation Curve
in H2 Si=CH2
Energy, kcal/mol
200
150
Reax
100
QM
50
0
1.0
1.5
2.0
2.5
3.0
3.5
Bond Length, Angstroms
4.0
4.5
5.0
Reax Fit Results
C_C_Si Angle Bend Curve
in H3C-CH2-SiH3
30
Energy, kcal/mol
25
20
Reax
15
QM
10
5
0
80
90
100
110
120
Angle, degrees
130
140
150
Reax Fit Results
C_Si_C Angle Bend Curve
in H3C-SiH2-CH3
30
Energy, kcal/mol
25
20
Reax
15
QM
10
5
0
75
85
95
105
115
Angle, degrees
125
135
145
155
Reax Fit Results
C_Si_Si Angle Bend Curve
in H3CSiH2SiH2
30
Energy, kcal/mol
25
20
Reax
15
QM
10
5
0
75
85
95
105
115
Angle, degrees
125
135
145
155
Reax Fit Results
Si_C_Si Angle Bend Curve
in H3SiCH2SiH3
30
Energy, kcal/mol
25
20
Reax
15
QM
10
5
0
75
85
95
105
115
Angle, degrees
125
135
145
155
Reax Fit Results
Si_C_H Angle Bend Curve
in H3CSiH2CH3
30
Energy, kcal/mol
25
20
Reax
15
QM
10
5
0
75
85
95
105
115
Angle, degrees
125
135
145
155
Reax Fit Results
C_Si_H Angle Bend Curve
in H3SiCH2SiH3
30
Energy, kcal/mol
25
20
Reax
15
QM
10
5
0
75
85
95
105
115
Angle, degrees
125
135
145
155
Reax FF Crystal Fits (in progress)
Energy Vs. Lattice - Silicon Carbide Crystal
(periodic PBE)
Energy Vs. Lattice - Silicon Crystal (periodic PBE)
120
80
DESIRED RANGE
70
80
Energy, kcal/mol/atom
Energy, kcal/mol/atom
100
60
40
20
0
4.0
5.0
6.0
-20
7.0
8.0
60
USEFUL RANGE
50
40
30
20
10
0
-10
3.5
4.0
Lattice constant, Ang.
Future calculations: Crystal cohesive energy
Also available: Diamond crystal
4.5
5.0
5.5
Lattice constant, Ang.
6.0
6.5
Energy
(kcal/mol)
Bond formation between two C20-dodecahedrons
100
DFT
ReaxFF
50
0
Energy
(kcal/mol)
1.5
2
2.5
100
DFT
ReaxFF
50
0
1.5
2
2.5
C-C distance (Å)
- ReaxFF properly describes the coalescence reactions between C20-dodecahedrons
Diamond to graphite conversion
Calculated by expanding a 144 diamond supercell in the c-direction and relaxing
the a- and c axes
QC-data: barrier 0.165 eV/atom
(LDA-DFT, Fahy et al., PRB 1986, Vol. 34, 1191)
DE (eV/atom)
0.2
0.15
graphite
diamond
0.1
0.05
0
10
15
c-axis (Å)
-ReaxFF gives a good description of the diamond-to-graphite reaction path
20
Relative stabilities of graphite, diamond, buckyball and nanotubes
a:
Compound
ERef (kcal/atom)
EReaxFF
Graphite
0.00a
0.00
Diamond
0.8a
0.52
Graphene
1.3a
1.56
10_10 nanotube
2.8b
2.83
17_0 nanotube
2.84b
2.83
12_8 nanotube
2.78b
2.81
16_2 nanotube
2.82b
2.82
C60-buckyball
11.5a
11.3
Experimental data; b: data generated using graphite force field (Guo et al. Nature 1991)
- ReaxFF gives a good description of the relative stabilities of these structures
Bonded Force Field Remarks
• Silicon force field (Hessian-Biassed Method)
– LJ 6-12 (vdw), Morse (bond), cosine harmonic (angle), dihedral
(torsion), r-cosine (stretch-bend-stretch), r-r (stretch-stretch),
cosine2 (bend-bend), coulomb, 2-center Ang-Ang (not available in
Cerius2)
• Graphite force field (optimized for graphite and CNT’s)
– Morse (vdw and C-C bond), cosine harmonic (angle), dihedral
(torsion), no inversion, r-cosine (stretch-bend-stretch – not used for
CNT’s), r-r (stretch-stretch – not used for CNT’s), coulomb
• Vdw Cross Terms (C-O, C-Si, C-H) – Bonds not considered
– Bond length: arithmetic combination rule
– Well depth: geometric combination rule
– Used LJ_6-12 function (instead of Morse Potential)
Force Field Energy Terms
• LJ 6-12: E = Ar-12 – Br-6
• Morse:
E = Do { (1 – e-B(r-r ))2 – 1}
• Cosine harmonic:
E = 0.5 Kq ( cos q – cos q o )2
• Dihedral: E = Sj 0.5 Bj ( 1 – Dj cos (nj f) )
• Cosine-2: E = Kbb (q jil – qjilo) (q kil – qkilo)
• r-r:
E = Kss (Rij – Rijo) (Rjk – Rjko)
• r-cosine: E = (cos q – cos qo) [Cij (Rij – Rijo) + Cjk (Rjk - Rjko)]
• 2-center Ang-Ang:
E = Faa (cos ijk – cos ijko) ( cos ikl – iklo)(1 – 2 cosf)/3
• Coulomb: E = C q1 q2 / (r12)2
o
LJ6-12 Vs. Morse Potential
Comparison of LJ 6-12 and Morse Potentials
10
Energy, kcal/mol
LJ Energy = Ar-12-Br-6
Morse Energy = Do{ [1 – e-B(r-ro)]2 –1}
5
LJ 6-12
Morse
0
2.50
3.50
4.50
5.50
-5
Interatomic Distance, Ang.
6.50
LJ6-12 Vs. Morse Potential
Comparison of LJ 6-12 and Morse Potentials
(Behavior near r = 0)
1.E+06
Energy, kcal/mol
8.E+05
E,F  Infinity
6.E+05
LJ 6-12
E,F finite
Morse
4.E+05
2.E+05
-1.0
-5.E+00
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Interatomic Distance, Ang.
LJ Energy = Ar-12-Br-6
Morse Energy = Do{ [1 – e-B(r-ro)]2 –1}
3.5
AFM Tip Equation of Motion
m z” = -k z – (m wo / Q) z’ + Fts + Focos(w t)
m = mass
k = harmonic force constant
z = tip-sample separation
wo = cantilever resonance frequency
Q = cantilever quality factor
Fts = tip-sample interaction force
Focos(w t) = external force
30,30 CNT AFM Tip (vertical)
• 35,200 total atoms
• 30,30 CNT on Si(100)-OH
surface
• CNT diameter = 40.69 Ang
• Tip length = 40 nm
• ~145 hours of computer
time
CNT Tip on CNT (20,20)
Energy Vs. Position Curve
Energy Vs. Tip Position
30,30 CNT Tip on 30,30 CNT
6000
5000
Energy, kcal/mol
CNT Readjustments
4000
Down
3000
Up
2000
1000
0
-35
-30
-25
-20
-15
-10
Tip Position (above CNT), Ang.
-5
0
5
Force Vs. Position Curve
Force Vs. Position, 30,30 CNT Tip on 30,30 CNT
70
Strong Interaction with the Surface
60
50
CNT Readjustments
Force, nN
40
30
Down
20
Up
10
0
-50
-40
-30
-20
-10
-10
0
-20
Tip Position (above CNT), ang.
10
20
Interpretation and prediction of AFM Behavior
Selective Phase Angle Inversion
Initial conditions
Surface = CNT on Si
Tip = Ntb tip
DF = 59.45 KHz
ASP =1.440
Sensitivity = 21.82 nm / V
Q 148
Rp = Asp/DA = 0.6
DA= 653.2 mV
ASP=0.1V (small value implies
oscillation close to the surface)