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Force Field Development for Silicon Carbides_ Bulk Silicon and

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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

        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                            •   C-Si-Si
    – Regular bond in H3SiCH3      •   C-C-Si
    – Simultaneous breaking of 2
                                   •   C-Si-C
      bonds in Si2H4-C2H4
                                   •   Si-C-Si
                                   •   Si-C-H
                                   •   C-Si-H
 • Si=C                            •   Future work: angles
    – H2Si=CH2                         involved in double bonds
Parameter Optimization Procedure
                                  Si-C dissociation curve in H4Si2-C2H4 (for 2 bonds)

                      200



                      150
   Energy, kcal/mol




                                                                                             singlet
                      100                                                                    triplet
                                                                                             Reax fit


                       50



                        0
                            1.0    1.5    2.0    2.5       3.0       3.5   4.0   4.5   5.0
                                                       Radius, Ang
                                     Reax Fit Results
                                       Si-C Bond Dissociation Curve
                                               in H3 Si-CH3

                   200


                   150
Energy, kcal/mol




                                                                                      Reax
                   100
                                                                                      QM

                    50


                     0
                         1.0   1.5    2.0       2.5      3.0        3.5   4.0   4.5
                                            Bond Length, Angstrom
                                     Reax Fit Results
                                     Si=C Double Bond Dissociation Curve
                                                 in H2 Si=CH2

                   200
Energy, kcal/mol




                   150

                                                                                       Reax
                   100
                                                                                       QM

                   50


                    0
                         1.0   1.5    2.0     2.5    3.0    3.5      4.0   4.5   5.0
                                            Bond Length, Angstroms
                                  Reax Fit Results
                                         C_C_Si Angle Bend Curve
                                             in H3C-CH2-SiH3

                   30

                   25
Energy, kcal/mol




                   20
                                                                               Reax
                   15
                                                                               QM
                   10

                   5

                   0
                        80   90    100      110       120    130   140   150
                                            Angle, degrees
                                  Reax Fit Results
                                        C_Si_C Angle Bend Curve
                                             in H3C-SiH2-CH3

                   30

                   25
Energy, kcal/mol




                   20
                                                                                       Reax
                   15
                                                                                       QM
                   10

                   5

                   0
                        75   85    95   105        115         125   135   145   155
                                              Angle, degrees
                                  Reax Fit Results
                                       C_Si_Si Angle Bend Curve
                                            in H3CSiH2SiH2

                   30

                   25
Energy, kcal/mol




                   20
                                                                                       Reax
                   15
                                                                                       QM
                   10

                   5

                   0
                        75   85   95    105        115         125   135   145   155
                                              Angle, degrees
                                  Reax Fit Results
                                       Si_C_Si Angle Bend Curve
                                            in H3SiCH2SiH3

                   30

                   25
Energy, kcal/mol




                   20
                                                                                       Reax
                   15
                                                                                       QM
                   10

                   5

                   0
                        75   85   95    105        115         125   135   145   155
                                              Angle, degrees
                                  Reax Fit Results
                                       Si_C_H Angle Bend Curve
                                            in H3CSiH2CH3

                   30

                   25
Energy, kcal/mol




                   20
                                                                                      Reax
                   15
                                                                                      QM
                   10

                   5

                   0
                        75   85   95   105        115         125   135   145   155
                                             Angle, degrees
                                  Reax Fit Results
                                       C_Si_H Angle Bend Curve
                                            in H3SiCH2SiH3

                   30

                   25
Energy, kcal/mol




                   20
                                                                                      Reax
                   15
                                                                                      QM
                   10

                   5

                   0
                        75   85   95   105        115         125   135   145   155
                                             Angle, degrees
Reax FF Crystal Fits (in progress)
                              Energy Vs. Lattice - Silicon Crystal (periodic PBE)                                             Energy Vs. Lattice - Silicon Carbide Crystal
                                                                                                                                            (periodic PBE)

                        120
                                                                                                                  80
                        100                                                                                                        DESIRED RANGE
                                                                                                                  70
Energy, kcal/mol/atom




                                                                                                                  60




                                                                                          Energy, kcal/mol/atom
                        80
                                                                                                                                               USEFUL RANGE
                                                                                                                  50
                        60
                                                                                                                  40
                        40                                                                                        30

                        20                                                                                        20
                                                                                                                  10
                         0
                                                                                                                   0
                              4.0          5.0            6.0             7.0       8.0
                        -20                                                                                             3.5        4.0      4.5       5.0      5.5       6.0   6.5
                                                                                                                  -10
                                                 Lattice constant, Ang.                                                                     Lattice constant, Ang.




Future calculations: Crystal cohesive energy
Also available: Diamond crystal
          Bond formation between two C20-dodecahedrons




             (kcal/mol)
               Energy
                          100




                                                                  DFT
                           50
                                                                  ReaxFF




                            0
                                1.5          2              2.5




                          100
             (kcal/mol)
               Energy




                                                                  DFT
                          50
                                                                  ReaxFF




                           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)

                 0.2
 DE (eV/atom)




                0.15
                                                                         graphite
                 0.1        diamond

                0.05



                  0
                       10                     15                               20
                                          c-axis (Å)
-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-r ))2 – 1}
                                     o



• 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
                   LJ6-12 Vs. Morse Potential
                           Comparison of LJ 6-12 and Morse Potentials

                   10

                                LJ Energy = Ar-12-Br-6
                                Morse Energy = Do{ [1 – e-B(r-ro)]2 –1}
Energy, kcal/mol




                   5

                                                                                 LJ 6-12
                                                                                 Morse

                   0
                    2.50     3.50          4.50          5.50             6.50



                   -5
                                    Interatomic Distance, Ang.
                   LJ6-12 Vs. Morse Potential
                                  Comparison of LJ 6-12 and Morse Potentials
                                            (Behavior near r = 0)

                           1.E+06

                           8.E+05
                                                            E,F  Infinity
Energy, kcal/mol




                           6.E+05
                                                                                            LJ 6-12
                                         E,F finite
                                                                                            Morse
                           4.E+05

                           2.E+05

                           -5.E+00
                   -1.0   -0.5     0.0      0.5       1.0   1.5   2.0    2.5    3.0   3.5
                                            Interatomic Distance, Ang.


                                           LJ Energy = Ar-12-Br-6
                                           Morse Energy = Do{ [1 – e-B(r-ro)]2 –1}
 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
                                 CNT Readjustments
Energy, kcal/mol




                                                                            4000

                                                                                           Down
                                                                            3000
                                                                                           Up

                                                                            2000

                                                                            1000

                                                                               0
                   -35    -30   -25        -20     -15     -10         -5          0   5
                                      Tip Position (above CNT), Ang.
                  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
                                                         40
Force, nN




                                                         30                     Down
                                                         20                     Up

                                                         10

                                                          0
            -50     -40        -30        -20     -10          0   10      20
                                                         -10

                                                         -20
                                Tip Position (above CNT), ang.
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)

				
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