8420
J. Phys. Chem. B 2000, 104, 8420-8429
Microstructure Evolution in Diamond CVD: Computer Simulations of 111 Surface Site Formation on a Growing Diamond-100 Surface Ronald C. Brown† and Jeffrey T. Roberts* Department of Chemistry, UniVersity of Minnesota, Minneapolis, Minnesota 55455-0431 ReceiVed: March 15, 2000; In Final Form: June 6, 2000
A Monte Carlo simulation strategy was used to explore the kinetics and mechanisms of 111-surface site formation on a growing diamond-100 surface. The starting point of the simulations was a 30 × 30 carbon atom array with the structure of an ideally reconstructed, diamond-(100)-(2×1) surface. The growth mechanism included 69 reactions relevant to diamond growth from CH3, H, and H2. Several of the growth steps involved migration of C1-containing fragments across a diamond surface. The simulations were run for growth conditions similar to those found in hot filament and microwave plasma diamond chemical vapor deposition (CVD) reactors. The temperature and gas-phase composition were varied to investigate the dependence of 111surface site formation on growth conditions. The extent of 111-surface site formation was inferred from the height distributions of films that had been grown to mean heights of 10 carbon monolayers. Results of the simulations imply that, for typical hot filament and microwave plasma conditions, 111-surface site formation increases with growth temperature, decreases with H-atom concentration, and is relatively independent of methyl concentration. Much of the temperature and composition dependence vanishes when reactions allowing for carbon surface migration are prohibited. It is concluded that surface migration plays an important role in microstructure development in diamond CVD.
Introduction Chemical vapor deposition (CVD) of diamond has been the subject of intense study for over a decade, and many aspects of the diamond growth mechanism are now well understood. In particular, relatively simple kinetic models have been developed that predict linear diamond growth rates over an astonishingly broad range of growth conditions.1-5 To be sure, most of these kinetic models are in some respects unsatisfactory. For instance, with few exceptions, they generally allow for diamond growth only on an idealized 100-diamond surface, and they are usually unable to account for growth species other than CH3. Nevertheless, the fact that simple models can be used to make reasonably accurate predictions of linear growth rate over a broad range of conditions suggests that they successfully capture the important bond-breaking and bond-making events that lead to the accumulation of carbon in a diamond CVD film. The success of the above-mentioned kinetic models notwithstanding, there are important issues related to diamond CVD that continue to be poorly understood. One of these is microstructure development. By microstructure, we mean the shape and size distributions of the crystallites in a polycrystalline film. Scanning electron microscopy (SEM) images of diamond CVD films deposited on non-diamond substrates establish that such films are polycrystalline, with crystallite sizes that are generally in the nanometer-to-micrometer range. The crystallite shapes are almost always those of cubes or truncated cubooctahedra. Scanning tunneling microscopy (STM)6-9 and atomic force microscopy (AFM)10-12 have demonstrated that the surfaces of * To whom correspondence should be addressed. Telephone: (612) 6252363. Fax: (612) 626-7541. E-mail:
[email protected]. † Current address: Department of Chemistry, Mercyhurst College, 501 E. 38th, St. Erie, PA 16546. Tel: (814) 824-2389. E-mail: rbrown@ mercyhurst.edu.
crystallite facets having 3- and 4-fold symmetry are mostly diamond-111 and diamond-100, respectively. The surfaces may, however, be highly defective, and there is some evidence that the defect density on diamond-111 generally exceeds that on diamond-100. Koidl defined a useful index of crystallite shape, the R-parameter:13
V(100) V(111)
R ) x3
(1)
where V(100) and V(111) are the perpendicular growth rates of the 100- and 111-surfaces. Koidl and others have produced maps of R as functions of various process parameters (e.g., temperature and gas-phase composition).13-16 It is clear from these maps that R varies in a systematic way with growth conditions, from which one can conclude that microstructure is at least partly a function of the steady-state deposition chemistry. In this paper, we describe a strategy for simulating one aspect of microstructure development, namely the appearance of 111 sites on a growing diamond-100 surface. The strategy makes use of a Monte Carlo simulation scheme involving 69 reactions relevant to diamond growth from a gas-phase mixture of CH3, H, and H2. The growth mechanism is limited in that it allows for diamond growth only in the 100 direction; once a 111 site appears, all subsequent growth at that site stops. This limitation makes the growth mechanism unphysical and therefore inappropriate for calculating such quantities as R and the linear growth rate. However, the simulations are useful because of what they reveal about microstructure development in diamond CVD. Specifically, the simulations provide an explanation for why certain growth conditions favor the formation of 111 surface sites on a growing diamond-100 surface. Such information is crucial if we are to gain a clear understanding of how
10.1021/jp000978i CCC: $19.00 © 2000 American Chemical Society Published on Web 08/12/2000
Microstructure Evolution in Diamond CVD
J. Phys. Chem. B, Vol. 104, No. 35, 2000 8421
TABLE 1: Gas-Phase Compositions for the Simulations Reported in This Work composition set
[H]/mol‚cm-3
[H2]/mol‚cm-3
[CH3]/mol‚cm-3
i ii iii iv
2 × 10-9 2 × 10-9 2 × 10-9 2 × 10-10
2 × 10-7 2 × 10-7 2 × 10-7 2 × 10-7
5 × 10-11 1 × 10-10 2 × 10-10 1 × 10-10
microstructure develops in diamond CVD, and also of how microstructure development relates to Koidl’s R-parameter. Simulation Strategy Overview. The diamond growth simulations were run on a cluster of IBM RS6000 computers at the University of Minnesota, using software written by the authors. The simulation strategy is described in detail in the paragraphs below, but we begin with a brief overview of how the simulations worked. The starting point was a hypothetical, flat slab of 900 C-H groups arranged to have the structure of the hydrogenterminated, 100-(2×1)-diamond surface. The gas phase, whose composition was unchanging over the course of a simulation, consisted of a mixture of CH3, H, and H2. The diamond growth mechanism consisted of 69 reactions, some of which were between the surface and the gas phase and others of which were confined to the surface. The mechanism attempted to account for structural effects in surface reactivity. Thus, for instance, the rate parameters for H-atom desorption from a C-H group on the original slab differed from those for H-atom desorption from absorbed methyl. Growth was simulated in a series of discrete time steps. Within any time step, appropriate statistical methods were used to identify which surface carbon atoms underwent one of the growth reactions. Inputs to the simulations included temperature, the concentrations of gas-phase CH3, H, and H2, and the Arrhenius rate parameters of the growth reactions. The gas-phase composition and growth temperatures were chosen to resemble those in a typical hot-filament or microwave diamond CVD reactor. Table 1 summarizes the four sets of gas-phase composition explored in this work; simulations were carried out for each composition set at temperatures of 1050, 1200, and 1300 K. The simulation outputs included indices describing the local structure and Cartesian coordinates of all accumulated carbon atoms. The simulations were terminated after the addition of 9000 carbon atoms, or 10 monolayer equivalents of carbon. Growth Surface. The initial growth surface was a flat slab that contained 900 carbon atoms in a 30 × 30 atom array. The arrays were configured so that all of the dimer bonds were oriented in the same direction. Simulated diamond growth led to the formation of other types of surface carbon, and for this reason the surface carbon atoms were indexed according to three properties: structure, orientation, and degree of saturation. Six types of site structure were defined (Figure 1): (a) the 100 site, in which carbon is part of a 100-dimer pair, (b) the 100-addition site, which results from the addition of carbon to a 100 site, (c) the incorporated bridge site, which can be thought of as having been formed by the insertion of carbon into a 100-dimer bond, (d) the bridge addition site, which results from the addition of carbon to an incorporated bridge site, (e) the 111 site, which is formed at the positions adjacent to an incorporated bridge site, and (f) the 111-addition site, which results from the addition of carbon to a 111 site. The orientation indices accounted for the fact that the vectors shown in Figure 1a-f may be parallel or perpendicular to the dimer bonds of the original template and pointed toward or away from the template origin. (Note
Figure 1. The six types of surface site investigated in this work: (a) the 100 site, (b) the 100-addition site, (c) the incorporated bridge site, (d) the bridge addition site, (e) the 111 site, and (f) the 111-addition site. The value of x establishes whether carbon is saturated, and the vector establishes the orientation of carbon with respect to the initial surface, as described in the text.
that the incorporated bridge and bridge addition sites have only two possible orientations, since they are not pointed toward or away from the template origin.) Finally, the surface carbon atoms were defined to be closed shell (x ) 1, y ) 2, or z ) 3 in Figure 1) or radical (x ) 0, y )1, or z ) 2), depending on whether they were saturated with hydrogen. Growth Mechanism. The growth mechanism includes 69 reactions relevant to diamond growth from gas-phase CH3, H, and H2; 30 of the reactions involve gas-surface interactions, and 39 of them are exclusively surface reactions. The reactions account for the following critical steps in diamond growth: (1) activation of the otherwise inactive diamond surface via the formation of surface radical sites, (2) addition of carbon or carbon-containing fragments to the activated surface, (3) incorporation of carbon addition fragments into the diamond lattice, (4) migration of surface hydrogen via transfer from a surface C-H bond to an adjacent radical site, and (5) migration of adsorbed CH2, also via a radical-mediated process. The rate constants of the 69 reactions were assumed to obey the Arrhenius expression. The rate parameters of the 30 gas-surface reactions are given in Table 2; those of the 28 H-transfer and 11 CH2 migration and carbon incorporation reactions are given in Figures 3 and 4, respectively. Two sources of surface activation were considered, abstraction of surface hydrogen via reaction with gas-phase atomic hydrogen, and desorption of hydrogen atoms from the diamond surface. The abstraction and desorption reactions are formulated generically below as eqs 2 and 3, respectively:
C-H(surf) + H(g) a C•(surf) + H2(g)
(2)
C-H(surf) a C•(surf) + H(g)
(3)
where C-H(surf) represents a C-H bond in a hydrogen-saturated site at the diamond surface, and C•(surf) is a surface radical site. As is implied by eqs 2 and 3, both reaction classes are reversible under diamond CVD conditions. For this reason, the surface radical concentration, and thus the carbon accumulation rate, is partly controlled by the concentrations of the gas-phase species H and H2. Because the kinetics and thermodynamics of H-atom desorption and abstraction are in principle highly structure-sensitive, it is necessary to account for the different kinds of C-H bond that are present on a growing diamond surface. Rate parameters were included for H-atom abstraction and desorption from the six types of surface site described above. The sources of the H-abstraction and H-desorption reaction rate parameters were as follows. For abstraction (and its reverse) at the 100, 111, 100-addition, and 111-addition sites, and for
8422 J. Phys. Chem. B, Vol. 104, No. 35, 2000
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TABLE 2: Descriptions and Reaction Parameters for Gas-Surface Reactionsa reaction no.
description
prefactor
prefactor source
barrier
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
hydrogen abstraction from 100 site reverse of 1 hydrogen abstraction from 111 site reverse of 3 hydrogen abstraction from an incorporated bridge site reverse of 5 hydrogen abstraction from an absorbed methyl on a 100 site reverse of 7 hydrogen abstraction from an absorbed methyl on a 111 site reverse of 9 hydrogen abstraction from a methyl absorbed on a bridge site reverse of 12 hydrogen association with radical 100 site hydrogen desorption from 100 site hydrogen association with radical 111 site hydrogen desorption from 111 site hydrogen association with an incorporated bridge site radical hydrogen association from an incorporated bridge site hydrogen association with an absorbed CH2 on a 100 site hydrogen desorption from an absorbed methyl on a 100 site hydrogen association with an absorbed CH2 on a 111 site hydrogen desorption from an adsorbed methyl on a 111 site hydrogen association with an adsorbed CH2 bridge site hydrogen desorption from a methyl adsorbed on a bridge site methyl adsorption on radical 100 site methyl desorption from a 100 site methyl adsorption on radical 111 site methyl desorption from a 111 site methyl adsorption to an incorporated bridge site radical methyl desorption from an incorporated bridge site
1.3 × 1014 9.2 × 107 1.3 × 1014 9.2 × 107 1.3 × 1014 9.2 × 107 2.8 × 107T2 1.0 × 101T2 2.8 × 107T2 1.0 × 101T2 2.8 × 107T2 1.0 × 101T2 1.0 × 1013 1.5 × 1015 1.0 × 1013 1.5 × 1015 1.0 × 1013 1.5 × 1015 1.0 × 1013 1.5 × 1015 1.0 × 1013 1.5 × 1015 1.0 × 1013 1.5 × 1015 5.0 × 1012 7.7 × 1016 5.0 × 1012 7.7 × 1016 5.0 × 1012 7.7 × 1016
ref 3 ref 3 ref 3 ref 3 reaction 3 reaction 4 ref 3 ref 3 ref 3 ref 3 reaction 9 reaction 10 ref 3 ref 3 ref 3 ref 3 reaction 15 reaction 16 reaction 13 reaction 14 reaction 15 reaction 16 reaction 21 reaction 22 ref 3 ref 3 ref 3 ref 3 ref 27 reaction 28
9.0 6 6.3 7.5 6.3 7.5 9.7 17.8 12.1 16.2 12.1 16.2 0.0 101.1 0.0 97.2 0.0 97.2 0.0 91.9 0.0 92.7 0.0 92.7 0.0 75.7 0.0 68.9 0.0 68.9
a
cm3‚mol-1‚s-1
s-1
barrier source b b b b reaction 3 reaction 4 c c c c reaction 9 reaction 10 c c reaction 16 c c reaction 22 c c reaction 28
kcal‚mol-1. b
Prefactor units are (for odd-numbered reactions) or (for even-numbered reactions); barrier units are From ref 17 with 2.0 kcal‚mol-1 subtracted from transition states. c From ref 18 with transition states corrected by -2.0 kcal‚mol-1 and methyl desorption barriers corrected by -10.0 kcal‚mol-1.
Figure 3. A false-color topographic map of a diamond film simulated under conditions that favor layered 100-surface growth. The growth temperature was 1050 K. The gas-phase composition was [CH3] ) 5 × 1011 mol‚cm3, [H] ) 2 × 109 mol‚cm3, and [H2] ) 2 × 107 mol‚ cm3. Figure 2. Some of the reactions that lead to CH2 migration across a diamond surface or carbon incorporation into a diamond lattice: (a) dimer opening via a β-scission reaction, (b) the SR-SOR reaction (see text), (c) reaction of adsorbed CH2 with an across-trough, unsaturated 111 site, (d) reaction of adsorbed CH2 with an across-trough, unsaturated 100 site, and (e) CH2 migration up or down a row, with migration occurring between two 100 sites.
desorption (and its reverse) at the 100 and 111 sites, the reaction prefactors were taken from the work of Srolovitz and coworkers. The activation energies of the same reactions, as well as those for H-desorption (and its reverse) from the 100- and 111-addition sites, were derived from the results of an ab initio electronic structure study,17,18 the goal of which was to obtain an accurate accounting of the differences in energetics between reactions occurring on the 111- and 100-diamond surfaces. There
are no good theoretical or experimental estimates of the other rate parameters. The prefactors and activation energies of reactions at the incorporated bridge site were therefore assumed to be equal to those of the analogous reactions at the 111 site. Similarly, the rate parameters of reactions at the bridge addition site were assumed to be equal to those of the analogous reactions at the 111-addition site. Finally, the prefactors for H-atom desorption (and its reverse) from the 100- and 111-addition sites were taken as equal to those of the other H-atom desorption and addition reactions. The H-abstraction barriers calculated by ab initio theory were too high by several kcal‚mol-1, because of well-known limitations associated with the theoretical treatment of reaction transition states. For this reason, a uniform -2.0 kcal‚mol-1 correction was applied to the calculated barriers
Microstructure Evolution in Diamond CVD
J. Phys. Chem. B, Vol. 104, No. 35, 2000 8423
TABLE 3: Descriptions and Reaction Parameters for Hydrogen Transfer Surface Reactionsa reaction no.
description
prefactor
prefactor source
barrier
barrier source
31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58
hydrogen transfer from 100 adsorbed CH3 to 100 site across trough reverse of 31 hydrogen transfer from 100 adsorbed CH3 to 111 site across trough reverse of 33 hydrogen transfer from 100 adsorbed CH3 to radical bridge site across trough reverse of 35 hydrogen transfer from 111 adsorbed CH3 to 100 site across trough reverse of 37 hydrogen transfer from 111 adsorbed CH3 to 111 site across trough reverse of 39 hydrogen transfer from 111 adsorbed CH3 to bridge site across trough reverse of 41 hydrogen transfer from bridge site to 111 site across trough reverse of 43 hydrogen transfer from bridge site to 100 site across trough reverse of 45 hydrogen transfer from 111 site to 100 site across trough reverse of 47 hydrogen transfer from 100 adsorbed CH3 to 100 on same dimer reverse 49 hydrogen transfer from 100 adsorbed CH3 to 100 site down row reverse 51 hydrogen transfer from 111 adsorbed CH3 to 100 site down row reverse of 53 hydrogen transfer from 111 adsorbed CH3 to 111 site down row reverse of 55 hydrogen transfer from 100 adsorbed CH3 to 111 site down row reverse of 57
1.7 × 1012 4.8 × 1012 1.7 × 1012 4.8 × 1012 1.7 × 1012 4.8 × 1012 1.7 × 1012 4.8 × 1012 1.7 × 1012 4.8 × 1012 1.7 × 1012 4.8 × 1012 1.2 × 1012 1.2 × 1012 1.2 × 1012 1.2 × 1012 1.2 × 1012 1.2 × 1012 2.1 × 1012 1.2 × 1012 2.1 × 1012 1.2 × 1012 2.1 × 1012 1.2 × 1012 2.1 × 1012 1.2 × 1012 2.1 × 1012 1.2 × 1012
ref 20 ref 20 ref 20 ref 20 ref 33 ref 34 ref 31 ref 32 ref 33 ref 34 ref 33 ref 34 ref 20 ref 20 ref 20 ref 20 ref 20 ref 20 ref 19 ref 19 reaction 49 reaction 50 reaction 49 reaction 50 reaction 49 reaction 50 reaction 55 reaction 56
16.3 25.1 12.9 9.0 12.9 9.0 16.3 25.1 12.9 9.0 12.9 9.0 7.0 11.5 23.6 38.2 30.1 39.4 37.5 50.5 37.5 50.5 37.5 50.5 52.5 65.5 52.5 65.5
ref 20 ref 20 ref 20 ref 20 ref 33 ref 34 ref 31 ref 32 ref 33 ref 34 ref 33 ref 34 ref 20 ref 20 ref 20 ref 20 ref 20 ref 20 ref 19 ref 19 reaction 49 reaction 50 reaction 49 reaction 50 b b reaction 55 reaction 56
a
Prefactor units are s-1, and barrier units are kcal‚mol-1. b From ref 20 with 15 kcal‚mol-1 added to barrier, for reasons described in the text.
Figure 4. A false-color topographic map of a diamond film simulated under conditions that favor the appearance of 111-surface sites. The growth temperature was 1300 K. The gas-phase composition was [CH3] ) 5 × 1011 mol‚cm3, [H] ) 2 × 109 mol‚cm3, and [H2] ) 2 × 107 mol‚cm3.
for abstraction and its reverse. In estimating the H-atom desorption activation energies, we relied on the fact that there is no barrier for the reverse reaction, i.e., H-atom addition. The activation energies for H-atom desorption from the diamond surface were therefore obtained directly from the ab initio C-H bond dissociation energies. In order to keep the computer simulations as simple as possible, diamond growth was assumed to occur exclusively via a growth-by-methyl mechanism. Other carbon-containing species, most notably acetylene (C2H2), play an important role in diamond growth as well, but their role in microstructure development was not considered in this work. Methyl addition, which at high enough temperatures is reversible, may be described by the generic reaction:
C•(surf) + CH3(g) a C-CH3 (surf)
(4)
In this work, methyl addition was allowed to occur at the 100,
111, and incorporated bridge sites only; the 100- and 111addition sites were inactive for methyl addition. The rate parameters of the three adsorption and three desorption reactions considered in this work are summarized in reactions 25-30 of Table 2. The prefactors of the methyl addition and desorption reactions are taken or derived from the work of Srolovitz and co-workers. The activation energies for methyl addition and desorption from the 100- and 111-addition sites were derived from the ab initio work cited earlier,17,18 in which methyl addition to a radical surface site was shown to be a barrierless process. A -10.0 kcal‚mol-1 correction was applied to the CH3 desorption barriers to account for a systematic overestimation of the C-C bond dissociation energies. The prefactors and activation energies for methyl adsorption and desorption from incorporated bridge sites were assumed to be equal to those of the analogous reactions on the 111 sites. The simulations took into account various types of H-atom transfer reactions on the diamond surface; the rate parameters of these reactions are summarized in Table 3. Many types of H-atom transfer reaction are possible. Some are too slow to affect the equilibrium established by the abstraction and addition reactions,19 and they were therefore not included in the present model. Others, for instance H-atom transfer across a dimer trough, have rate constants large enough to make them potentially important in diamond growth and were included in the model. Activation energies and reaction prefactors are taken from the work of Frenklach and co-workers.19,20 For reactions that do not explicitly appear in the literature, the rate parameters were estimated using as a guide those of similar hydrogen transfer reactions. Reactions 59-65 (Table 4) define a mechanism for carbon incorporation into a growing diamond lattice. They also facilitate CH2 migration across the diamond surface. The rate parameters for reactions 59, 60, and 62-65 were taken from the work of Frenklach and co-workers,19,20 and those for reaction 61 were
8424 J. Phys. Chem. B, Vol. 104, No. 35, 2000
Brown and Roberts
TABLE 4: Descriptions and Reaction Parameters for CH2 Migration and Incorporation Reactionsa reaction no.
description
prefactor
prefactor source
barrier
barrier source
59 60 61 62 63 64 65 66 67 68 69
β-scission reaction on dimer adsorbed CH2 (Figure 2a) reverse of 59 bridge formation of 111 adsorbed CH2 with 100 site across trough bridge formation of C)CH2 with 100 site across trough (Figure 2d) bridge formation of C)CH2 with radical bridge site across trough (Figure 2b) reverse 63 bridge formation of C)CH2 with 111 site across trough (Figure 2c) CH2 migration from 100 site to 100 site down row (Figure 2e) CH2 migration from 111 site to 111 site down row migration of CH2 form C)CH2 down row to 100 site migration of CH2 form C)CH2 down row to 111 site
9.8 × 1012 2.7 × 1011 1.0 × 1013 1.1 × 1012 1.1 × 1012 6.1 × 1013 1.1 × 1012 5.5 × 1011 5.5 × 1011 5.5 × 1011 5.5 × 1011
ref 19 ref 19 ref 4 b ref 19 ref 19 ref 20 b reaction 66 reaction 66 reaction 67
15.3 2.9 46.0 26.6 12.3 36.3 3.2 30.0 45.0 30.0 45.0
19 19 4 20 19 19 20 19 c reaction 66 reaction 67
a Prefactor units are s-1, and barrier units are kcal‚mol-1. b Prefactor estimated from similar reactions. c From ref 20 with 15 kcal‚mol-1 added to barrier, for reasons described in the text.
taken from the work of Garrison and co-workers.21 Taken in sequence, reactions 59 and 63 comprise the mechanism for methyl addition to dimer sites proposed by Garrison and Brenner.22 Reaction 59 (shown schematically in Figure 2a) can be classified as a β-scission reaction, and reaction 63 (Figure 2b) is a radical addition to a double bond (Figure 2b). Reaction 63 was termed the surface-radical with surface-olefin (SRSOR) reaction by Harris and co-workers.23 The reverse reactions of this mechanism (reactions 60 and 64) have been included in the present model. The intermediate in the Garrison-Brenner mechanism is a CH2 fragment that is double-bonded to a surface carbon. This intermediate can react with other types of sites across a 100-(2×1) trough. For example, reactions 65 and 62 (Figure 2, c and d, respectively) result in the formation of new incorporated bridge sites via the cross-trough reactions of CH2 with unsaturated 111 and 100 sites. The other CH2 migration steps, which result in CH2 migration up and down the dimer rows of the diamond-100-(2×1) surface, are defined by reactions 66-69 (e.g., Figure 2e). Reactions 6669 are not believed to have a profound effect on microstructure development, but they were included for completeness. Their rate parameters were taken from Frenklach1,20 or assumed to be equal to those of similar reactions, as indicated in Table 4. One of the principal goals of this work was to explore the importance of surface migration to the development of microstructure in diamond CVD. For this reason, some simulations were conducted in which no H or CH2 migration was allowed. For these simulations, the following rate constants were set to zero: those associated with hydrogen transfer (i.e., reactions 31-58), those leading to CH2 transport up and down the rows of the dimer troughs (reactions 66-69), and the bridge site opening reaction (reaction 64). Setting the rate constant of reaction 64 equal to zero makes carbon incorporation irreversible: once adsorbed CH2 forms two bonds to the diamond surface, it is forever bound in that position. A secondary goal of this work was to determine the importance of accounting for different types of surface structural motifs in calculating the rates of important gas-surface reactions. For this reason, a series of simulations was carried out in which the rate constants of all reactions occurring on the 111 and incorporated bridge sites were set equal to those of analogous reactions occurring on the 100 site. Simulation Algorithm. The diamond growth simulations treated the gas-surface and surface-only reactions separately, within two loops of the simulation algorithm. The simulation program alternated between the two loops until a predetermined number of carbon atoms, usually 9000, had been added to the original 900-atom slab. The gas-surface and surface-only reactions were treated separated because, over extremely broad
ranges of CH3, H, and H2 concentration (10-11-10-6 mol‚cm-3), the gas-surface reactions were several orders of magnitude slower than the surface-only reactions. The “slow loop” thus took into account the critical steps of surface activation and carbon accumulation, and the “fast loop” described the various migration and carbon incorporation reactions. The simulation algorithm divided diamond growth into a series of discrete timesteps, with each complete algorithm loop accounting for one time-step. The time-step, ∆t, was chosen to have a value no greater than 50% of the reciprocal of the largest first-order or pseudo-first-order Arrhenius rate constant in the slow reaction manifold. (Reactions involving the addition or activation of gasphase CH3, H, or H2 are second order overall, because their absolute rates are dependent on surface coverage and gas-phase composition. Pseudo-first-order rate constants were obtained for these reactions by multiplying the second-order Arrhenius rate constants by the CH3, H, or H2 concentration.) For the whole range of conditions studied, the gas-surface reactions all proceeded with rate constants of less than 5 × 105 s-1. The time interval was thus set to 1 µs for all of the simulations. The slow loop of the simulation algorithm worked as follows. First, the structure and saturation indices of every surface carbon atom were retrieved. The indices were used in conjunction with a look-up table to determine which of the 30 slow reactions were available to each surface carbon. The next step was to determine the probabilities of the different reactions available to each surface atom. Suppose that surface atom d could react via n of the slow reactions. Then Pdl, the probability of reaction l occurring at atom d over time interval ∆t, was
Pld )
()
kld (1 - e-kd∆t) kd
(5)
where kd is the total first-order rate constant at atom d, obtained by summing over the n first-order rate constants. The simulation algorithm then assigned randomly chosen integers, Id, to each surface carbon atom. The integers, which fell between 0 and 105 inclusive, were compared to the reaction probabilities so as to establish where and how the surface reacted during ∆t. Specifically, the algorithm identified reaction l as having occurred at d when both of the following inequalities were satisfied: l-1
Pid e ∑ i)1
Id 105
l
