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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
Full Width 3.1 nm, Height 1.9 nmResolution = 1.2 nm
5.5 nm
40 nm
AFM Microscopy
Interactions to be optimized in Reax
Bonds:
• Si-C– Regular bond in H3SiCH3
– Simultaneous breaking of 2 bonds in Si2H4-C2H4
• Si=C– H2Si=CH2
Angles:
• 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)
0
50
100
150
200
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Radius, Ang
En
erg
y, k
cal/m
ol
singlet
triplet
Reax fit
Reax Fit Results
Si-C Bond Dissociation Curve
in H3Si-CH3
0
50
100
150
200
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
Bond Length, Angstrom
En
erg
y, k
ca
l/mo
l
Reax
QM
Reax Fit ResultsSi=C Double Bond Dissociation Curve
in H2Si=CH2
0
50
100
150
200
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Bond Length, Angstroms
En
erg
y, k
cal/
mo
l
Reax
QM
Reax Fit Results
C_C_Si Angle Bend Curvein H3C-CH2-SiH3
0
5
10
15
20
25
30
80 90 100 110 120 130 140 150
Angle, degrees
En
erg
y, k
cal/m
ol
Reax
QM
Reax Fit Results
C_Si_C Angle Bend Curvein H3C-SiH2-CH3
0
5
10
15
20
25
30
75 85 95 105 115 125 135 145 155
Angle, degrees
En
erg
y, k
cal/m
ol
Reax
QM
Reax Fit Results
C_Si_Si Angle Bend Curvein H3CSiH2SiH2
0
5
10
15
20
25
30
75 85 95 105 115 125 135 145 155
Angle, degrees
En
erg
y, k
cal/m
ol
Reax
QM
Reax Fit Results
Si_C_Si Angle Bend Curvein H3SiCH2SiH3
0
5
10
15
20
25
30
75 85 95 105 115 125 135 145 155
Angle, degrees
En
erg
y, k
cal/m
ol
Reax
QM
Reax Fit ResultsSi_C_H Angle Bend Curve
in H3CSiH2CH3
0
5
10
15
20
25
30
75 85 95 105 115 125 135 145 155
Angle, degrees
En
erg
y, k
cal/m
ol
Reax
QM
Reax Fit ResultsC_Si_H Angle Bend Curve
in H3SiCH2SiH3
0
5
10
15
20
25
30
75 85 95 105 115 125 135 145 155
Angle, degrees
En
erg
y, k
cal/m
ol
Reax
QM
Reax FF Crystal Fits (in progress)
Energy Vs. Lattice - Silicon Crystal (periodic PBE)
-20
0
20
40
60
80
100
120
4.0 5.0 6.0 7.0 8.0
Lattice constant, Ang.
En
erg
y, k
cal/m
ol/a
tom
Energy Vs. Lattice - Silicon Carbide Crystal (periodic PBE)
-10
0
10
20
30
40
50
60
70
80
3.5 4.0 4.5 5.0 5.5 6.0 6.5
Lattice constant, Ang.E
ner
gy,
kca
l/mo
l/ato
m
Future calculations: Crystal cohesive energyAlso available: Diamond crystal
USEFUL RANGE
DESIRED RANGE
0
50
100
1.5 2 2.5
DFTReaxFF
0
50
100
1.5 2 2.5
DFTReaxFF
C-C distance (Å)
Ene
rgy
(kca
l/m
ol)
Ene
rgy
(kca
l/m
ol)
Bond formation between two C20-dodecahedrons
- ReaxFF properly describes the coalescence reactions between C20-dodecahedrons
0
0.05
0.1
0.15
0.2
10 15 20
c-axis (Å)
E (
eV/a
tom
)
diamond
graphite
Diamond to graphite conversionCalculated 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)
-ReaxFF gives a good description of the diamond-to-graphite reaction path
Relative stabilities of graphite, diamond, buckyball and nanotubes
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
a: 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-ro))2 – 1}• Cosine harmonic:
E = 0.5 K ( cos – cos o )2
• Dihedral: E = j 0.5 Bj ( 1 – Dj cos (nj ) )
• Cosine-2: E = Kbb ( jil – jilo) ( kil – kilo)
• r-r: E = Kss (Rij – Rijo) (Rjk – Rjko)
• r-cosine: E = (cos – cos o) [Cij (Rij – Rijo) + Cjk (Rjk - Rjko)]
• 2-center Ang-Ang:
E = Faa (cos ijk – cos ijko) ( cos ikl – iklo)(1 – 2 cos)/3
• Coulomb: E = C q1 q2 / (r12)2
LJ6-12 Vs. Morse Potential
Comparison of LJ 6-12 and Morse Potentials
-5
0
5
10
2.50 3.50 4.50 5.50 6.50
Interatomic Distance, Ang.
En
erg
y, k
cal/
mo
l
LJ 6-12
Morse
LJ Energy = Ar-12-Br-6
Morse Energy = Do{ [1 – e-B(r-ro)]2 –1}
LJ6-12 Vs. Morse PotentialComparison of LJ 6-12 and Morse Potentials
(Behavior near r = 0)
-5.E+00
2.E+05
4.E+05
6.E+05
8.E+05
1.E+06
-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
Interatomic Distance, Ang.
En
erg
y, k
cal/
mo
l
LJ 6-12
Morse
LJ Energy = Ar-12-Br-6
Morse Energy = Do{ [1 – e-B(r-ro)]2 –1}
E,F Infinity
E,F finite
AFM Tip Equation of Motion
m z” = -k z – (m wo / Q) z’ + Fts + Focos(w t)
m = massk = harmonic force constantz = tip-sample separation
wo = cantilever resonance frequencyQ = 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 CurveEnergy Vs. Tip Position
30,30 CNT Tip on 30,30 CNT
0
1000
2000
3000
4000
5000
6000
-35 -30 -25 -20 -15 -10 -5 0 5
Tip Position (above CNT), Ang.
En
erg
y,
kc
al/
mo
l
Down
Up
CNT Readjustments
Force Vs. Position CurveForce Vs. Position, 30,30 CNT Tip on 30,30 CNT
-20
-10
0
10
20
30
40
50
60
70
-50 -40 -30 -20 -10 0 10 20
Tip Position (above CNT), ang.
Fo
rce
, n
N
Down
Up
CNT Readjustments
Strong Interaction with the Surface
Interpretation and prediction of AFM BehaviorSelective Phase Angle Inversion
Initial conditionsSurface = CNT on SiTip = Ntb tipDF = 59.45 KHzASP =1.440Sensitivity = 21.82 nm / VQ 148Rp = Asp/DA = 0.6
DA= 653.2 mVASP=0.1V (small value impliesoscillation close to the surface)