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)