Elementary Reaction Mechanism for Growth of Diamond (100

Jul 1, 1994 - CCl Radicals As a Carbon Source for Diamond Thin Film Deposition ... Boron Incorporation at a Diamond Surface: A QM/MM Study of Insertio...
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J. Phys. Chem. 1994,98, 7073-7082

7073

Elementary Reaction Mechanism for Growth of Diamond (100) Surfaces from Methyl Radicals Sergei Skokov, Brian Weiner,t and Michael Frenklach’ Department of Materials Science and Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802 Received: March 21, 19940

Chemical reactions of methyl radicals on (100) diamond surfaces have been investigated theoretically. Quantummechanical calculations a t the PM3 semiempirical level were performed on a series of small- and large-size clusters to explore possible reaction steps responsible for diamond growth a t conditions typical of chemical vapor deposition. Among a variety of possible chemisorption sites considered, surface dimer radicals not only were the most favorable on kinetic grounds but appeared to be the only type capable of sustaining the subsequent incorporation of adsorbed methyl groups into the diamond lattice. Surface migration of H atoms, radical sites, and chemisorbed CHz groups proved to be important for diamond growth. A new reaction mechanism of diamond (100) growth from methyl radicals is proposed which offers a plausible explanation for the experimentally observed fast and smooth growth of diamond surfaces. The mechanism consists of two principal features, conversion of dimer sites into bridge sites and surface migration of bridge sites toward continuous bridge chains; it does not require any particular order of dimer formation but establishes the governing role of surface diffusion.

Introduction In spite of substantial progress in technological aspects of chemical vapor deposition (CVD) of diamond films,’ the underlying chemical reaction mechanism of diamond growth remains poorly understood. Generally, Cl-hydrocarbons have been assumed to be the sole growth species, basically following a one-carbon-at-a-timepicture. Recent isotope labeling experiments,23 reporting that about 90% of deposited diamond originates with methane and its radicals, support this point of view. A series of less direct experiments4echo this conclusion. Proposals have been made by invoking CH5+ ions5 and CH4 molecules6 but without suggesting possible reactions. The mechanism suggested by Tsuda et al. that implicated either gaseous methyl cations7or the presence of a positive charge on the surfaces faces two major difficulties. First, the critical role of ions or electrostatic charges has not been corroborated by experiment, and second, a prohibitively strong repulsion exists among methyl groups required to be adsorbed at three neighboring sites of the (1 11) surface. This repulsion has been computed at different levels of theory: empirical, using Brenner p~tential;~ quantum semiempirical, using AM1 10 and ASED-MOII methods; and local density approximation12 (LDA). It was determined in these studies that the chemisorption energy per methyl group is decreased by 11-23, 29, and 28 kcal/mol, respectively, when CHI groups are placed at three adjacent (1 11) surface sites. No stable states for a (1 1 1) surface totally covered by CH3 groups were found in the calculations of Mehandru and Anderson.” While the adsorptionof three neighboring methyl groups seems to be essentially forbidden, the adsorption of two neighboring methyl groups meets only a moderate r e p u l ~ i o n . I ~ -The ~~ formation of a C-C bond between these two methyls encounters a substantial potential energy barrier, 50 kcal/mol, if only one of the two is activated by H abstraction.13 Such relatively large barriers are typical for the addition of a primary hydrocarbon radical to a hydrogen-saturated carbon at0m.1~9~5When both methyl groups undergo H abstraction, the C-C bond should be formed as quickly and irreversibly as is typical of radical ~ombinati0n.I~ However, the produced - C H & H r adspecies

-

~~

is kinetically unstable under the conditions of diamond CVD.16 Abstraction of an H atom from one of these CH2 sites will be followed by a @-scission and ultimate desorption of the -CH=CH2 group. This sequence of events limits the rest of the proposed CH3-basedmechanisms of diamond growth on (1 1 1) and (1 10) surfaces.17-19 The most promising in terms of mechanistic understanding appears to be the growth on (100) surfaces. The growth on (100) surfaces by incorporationof CIH, hydrocarbonradicals has been advocated from considerations of surface morphology.20.21 HarrisZ2proposed a specific reaction mechanism in which the growth is initiated by the addition of CH3 to a radical created on a dihydride surface. After an H atom is abstracted from the adsorbed CH3 and another H from the neighboring surface site, the two radicals combine, forminga next-layer diamond site. The principal difficulty with this mechanism is the strong repulsion exerted on the incoming CH3.23 In fact, the dihydride surface itself, due to the repulsion between closely spaced hydrogen atoms, was suggested to be thermodynamicallyunstable at typical CVD conditions and to be reconstructed to a monohydride phase.24 Such reconstruction was observed e~perimentallyz”~and supported by theoretical c a l c u l a t i o n s . ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ On the basis of these considerations,it has been proposed23J4 that the growth takes place on a reconstructed, (100)-(2 X 1) surface, initiated by the addition of CH3 to a radical site of the surface dimer unit. Following an H abstractionfrom the adsorbed CH3, the produced CH2 adradical is incorporated into the diamond lattice by opening the dimer and forming a bridge between the dimer carbons. We will refer to such a site as a CH2-bridge site or simply, in the context of the present study, a bridge site. In one of the proposed mechanism^,^^ the CHI adradical forms a C-C bond with the second carbon site of the dimer simultaneously with the cleavage of the dimer bond. This one-step reaction was found to possess a substantial potential energy barrier. In the other me~hanism,’~ the reaction proceeds in two steps: dimer opening through @-scission,

~~

t Department of Physics, The PennsylvaniaState University, DuBois, PA 15801. *Abstract published in Advance ACS Abstracts, June 15, 1994.

and methylene bridging the former dimer carbon atoms,

0022-3654/94/2098-7073$04.50/0 0 1994 American Chemical Society

7074 The Journal of Physical Chemistry, Vol. 98, No. 28, 1994

To avoid lengthy descriptions, we will refer to reaction l a as dimer @-scission,reaction l b as methylene bridging, and the sequenceof reactions 1a and 1b as “insertion”into dimer (we will keep insertion in quotation marks to distinguish it from a carbenetype insertion). This two-step reaction mechanism appears to be energetically and kinetically feasible, as determined by using empirical,34 quantum ab initio,35 and quantum semiempirical16potentials; for instance, the respectivepotential energy barriers obtained in these calculations for the second step are 3.9, 8.8, and 12.3 kcal/mol. The feasibility of this reaction is further supported by the thermodynamic stability of the dimer monoradica1,16*32defying in this case the @-scissionrule. However, reaction 1 alone cannot explain the propagation of diamond growth, as it only “fills in” the dimer sites and does not address the growth between the nascent bridge sites. Two proposals were made regarding this issue. In the mechanism of Harris and Goodwin,35 after one CH3 group is “inserted” into a dimer, another CH3 is chemisorbed to a nearby radical of the next-row dimer and following an H abstraction forms a bridge site next to the first one. However, as will be shown later, once CH3 is chemisorbed to a dimer radical, it is more likely to “insert” into that dimer than to bridge to the adjacent bridge site. To resolve this difficulty, Zhu et al.36suggested that after the dimer @-scissionstep and before the bridging, surface radical migration H H

c

/ \

H d‘‘

/H Cd

c

/ \

H, cd

SH2 cd

/H Cd

H \ ~ cd

tH2 H\

/H

H\.

Cd

cd +d

cd

(2)

takes place, followed by the eventual addition of the lone chemisorbed CH2 group at the initial bridge site. That reaction 2 is facile was based on a relatively small endothermicity, 6 kcal/ mol, computed36 using MM3 molecular mechanics.3’ However, the analysis indicates that reaction 2 is not effective in competing with reaction 1b. Furthermore, neither the mechanism of Harris and Goodwin35 nor that of Zhu et al.36can resolve the difficulty arising from CH3 being adsorbed not at the nearby site but at the remote site of the dimer, as shown below

HH

H.

H

The configuration formed in such a case of two bridge sites separated by a void site has the geometry of a dihydride (100) site and, as mentioned above, exerts a large steric repulsion on the adsorption of CH3. Thus, the occurrence of reaction 3 should result in a highly defective film or cease the growth altogether, contrary to the experimental evidence reviewed by Harris and Goodwin35and Zhu et A remedy to such disruptive growth, suggested by the latter authors,36is the removal of undesirable bridge sites by the reverse of reaction 1. This requirement, however, should substantially slow down the growth, again contrary to experiment. Another possibility would be to “fill in” thevoid sites by speciesother than methyl radicals. For instance, Imai et al.38performed PM3-levelquantum chemical calculations

Skokov et al.

Figure 1. Schematic diagram of a diamond (100) surface model used in the MD simulations. Gray-shaded circles represent quantum mechanically described carbon atoms, white circles empirical carbon atoms, hatchedcirclesempirical carbon atoms of the thermostat, and black circles empirical carbon atoms held in fixed positions throughout the simulations.

and found no potential energy barriers for the addition of CH or CHI to a CllHl8 cluster simulating such a dehydrogenated void site. Yet, the concentrations of these species are expected to be orders of magnitude lower than that of CH3 in the deposition zone of hot-filamentreactors,3w2Le., under the conditionscapable of producing high-quality diamond films. In any event, we will leave these possibilities to future studies and focus here on methyl radicals. The brief analysisgiven aboveshows that the growth of diamond (100) surfaces by reactions of methyl radicals remains an open question. The objective of the present study was to systematically investigate the underlying chemistry and identify the most plausible reactions responsible for the growth of diamond. Our analysis employed for this purpose was based on semiempirical quantum-chemical calculations, the level of theory well suited for establishingreaction mechani~ms.~3 Specifically,we analyzed the kinetics of the following gas-surface and surface reactions: adsorption of CH3 at a series of possible surface sites; migration of H atoms, surface radicals, and chemisorbed CH2 groups; and the growth of adjacent chains of bridge sites. The results indicate that it is imperative to consider migration of H atoms and CH2 groups for the growth of diamond. The latter process, in particular, appeared to be responsible for the smoothness of the growth. Method The reaction rate coefficients were calculated using nonvariational transition-state theory (TST).& Similarly to our previousstudy of acetylenereactions on diamond (100) surfaces,16 the input data were obtained by quantum-chemical calculations performed on two kinds of model clusters. The surface was modeled by a composite cluster illustrated by the schematic diagram in Figure 1. The central gray-shaded atoms in the top three layers were described by a PM3 semiempirical quantumchemical H a m i l t ~ n i a nand ~ ~ the surrounding atoms by an empirical potential fitted to PM3. The form and parameters of the latter were the same as reported previously.32 The basic quantum-region clusters employed in the present study, CmH52 and C45H60, are shown in Figure 2. These quantum clusters were embedded into 320- and 545-atom empiricalclusters,respectively. The geometries of the CmH52 and C45Ha clusters were chosen to accommodate different types of surface-active sites used in analysis. A variety of such sites were created from the two surfaces shown in Figure 2 by incorporation of appropriate atomic steps. Minimum-energy configurations of the model surfaces were obtained by unrestricted Hartree-Fock (UHF) calculationsusing a combined-force molecular dynamics (MD) method, described and tested by us previously in studies of diamond (100) surface reconstr~ctions,3~ structures of oxygenated ( 100) surfaces,46and reactions of acetylenewith (100) surfaces.16 In thesecalculations, the bottom-layer atoms, shown in black in Figure 1, were held at fixed bulk positions. Periodic boundary conditions were

The Journal of Physical Chemistry, Vol. 98, No. 28, 1994 7075

Growth of Diamond (100) Surfaces

chemical similaritySoof gas-phase and gas-surface reactions and accurate first-principal calculations of gas-surface reaction rate coefficients, the latter still being unrealistic for large-sizeclusters. Results and Discussion I

I

I

I

Before we begin the analysis, there are several issues that need to be addressed. First, although the present method is not limited to any specific set of conditions, for the sake of clarity, the present study will focus on the conditions typical of diamond CVD:35~51*52 temperature 1200 K, [HI = le9mol/cm3, and [CH3] = mol/cm3. Second, the fraction of surface radical sites, r, can be expressed41953 as the ratio of kb: to k&, where the latter are rate coefficients of H abstraction,

and H addition,

reactions, respectively. Here c d represents a carbon atom of a diamond surface. At 1200 K,the value of kFb is on the order of54-573 X 10l2cm3 mol-' s-l and that of kzd is about 10-fold larger.58-61 Thus, under the control of reactions 4 and 5, the fraction of surface radical sites or, equivalently, the probability of a given surface site to be a radical is r 0.1. Third, for the concentrations of gaseous H atoms and CH3 radicals adopted above, the per-site rates of H-abstraction, H-addition, and CH3 combination with a surface radical are 3 x 103, R& 3 x 104, and ~22 1 x 103 s-1, respectively, assuming a rate coefficient 0F4 1 X 10'3 cm3 mol-' s-I for the radical combination. These per-site rates provide critical time scales against which other surface reactionswill becompared. Thus, for instance, surface reactions much faster than 3 X lo4 s-1 can be assumed to maintain the state of partial equilibrium. Under such conditions,the knowledgeofexact rateconstant values becomes less critical. Finally, as mentioned in the Introduction, the two carbon atoms of the (100)-(2 X 1) dimer unit behave rather inde~endently3~ of each other, and hence the probability of two sites chosen at random both being radicals, or the probability of having a doubleradical site, is62 r2 0.01, and the per-site formation rate of such a site is rR,Hb, 3 x 102 s-1. Adsorption of C H 3 on (100) Surfaces. As a result of a continuous evolution of the growing surface, an incoming methyl radical can encounter numerous possible combinations of individual surface sites. A series of such circumstances, those of interest to the present study, were examined. Figure 3 depicts them-combinations of dimer, bridge, and lone dihydride sites-arranged into three groups: A-C. In group A, a dimer radical to which CH3 is being added is paired with a dimer, bridge, or dihydride site, hydrogenated or dehydrogenated. Similarly structured is group B, where the adsorption of CH3 takes place at a bridge radical site. Group C is designed to study the addition of CH3 to a lone dihydride radical site. Chemisorption energies, potential energy barriers, reaction equilibrium constants, and reaction rate constants obtained for the cases listed in Figure 3 are reported in Table 1. As explained in the preceding section, the reactions enthalpies and potential energy barriers were computed by employing large clusters with the combined-force molecular dynamics. The vibrational frequencies used in the determination of the reaction-rate and equilibrium constants were obtained in static calculations performed on small-sizeclusters,those depicted in Figure 4: cluster I was used for the addition of CH3 to sites A, cluster I1 to sites

-

Figure 2. Quantum-mechanically described C40H52 (top) and C45Hm (bottom) clusters representing diamond (100) surfaces. White circles represent top-layercarbon atoms, gray circlesbottom-layercarbon atoms, and black circles hydrogen atoms. Surrounding empirical atoms are not shown for clarity.

implementedby the minimum imageconvention in two dimensions across planes perpendicularto the ( 110) directions. Temperature control was accomplished by the method of Berendsen et al.?' maintaining a prescribed temperature, 1200 K in the present simulations, of the four near-bottom layers shown hatched in Figure 1. Transition-state geometries were found by constrained optimization, where only certain atoms were allowed to relax. The choice of relaxed atoms was dictated by the specific surface site and reaction of interest. The other surface atoms were fixed at the positions of the minimum-energy configuration obtained in the MD simulations for the initial model surface. The use of large-size clusters is beneficial in the determination of potential energy minima; however, due to the constraints imposed by boundary conditions of the MD simulations, vibrational frequencies cannot be evaluated from these clusters reliably. To obtain normal frequencies, the calculations were repeated using small clusters,modeled after C9H12 examined in our previous studies.16f2 In most cases, all atoms were allowed to relax in these calculations. Transition-state geometries for both large- and small-cluster models were found to be close to each other, thus justifying the use of the MOPAC48vibrational frequencies from the smaller cluster. Low frequencies that originated from internal rotations were substituted by appropriate free rotations with rotational constants evaluated from computed moments of inertia. The ability of PM3 to reproduce vibrational frequencies was tested by Coolidge et al.?9 who found that PM3 predictions for hydrocarbons are within a reasonable accuracy, 9.3%. Thus, in the TST calculations, the potential energies of ground and transition states were those obtained in large-cluster calculations and the corresponding vibrational frequencies were taken from small-cluster calculations. Such an approach is a compromise between the usually adopted, ad-hoc hypothesis of

-

-

--

-

Skokov et al.

7076 The Journal of Physical Chemistry, Vol. 98, No. 28, 1994

TABLE 1: Enthalpies of Adsorption (AH), Potential Energy and Desorption (4)Rate Barriers (E),Adsorption (&a) Constants,and Equilibrium C~~tants ( ) for the Reactions of CHSwith Different Sites on a Diamon (100) Surface at

%

1200 K

surface site'

HH

cluster I A1 A2 A3 A4

A5 A6

cluster I1 B1 B2 B3 B4 B5 B6

cluster 111

c1

B1

c2

c3

c4

HH

c5

B2

AH, E, kadm kcal/mol kcal/mol cm3 mol-' s-l -74.7 -73.6 -75.4 -65.7 -73.7 -66.2 -74.1 -25.8 -24.2 46.8 10.9 -30.9 9.5 -27.8 -66.7 -34.5 8.1 -25.8 -39.3 -16.9

0.0 0.0 0.0 0.0 0.0 0.0 0.0 15.9 14.8 2.4 24.2 8.6 22.6 13.9 1.8 8.3 29.3 10.0 7.2 12.6

x

2.8 x 5.3 x 1.0 x 1013b 2.2 x 1.0 x 1013b 1.5 x 1.0 x 1013b 5.1 x 1.0 x 101136 1.2 x 1.0 x 1013b 4.3 x 4.0 X lo6 4.3 X 6.4 X lo6 3.8 X 1.2 x 109 5.4 x 1.2 X los 1.8 X 8.6 X lo7 3.2 X 2 . 4 ~ 105 2.1 x 9.4 X 106 1.2 X 1.5 X lo9 1.7 X 9.8 X 10' 7.8 X 1.5 X 104 6.7 X 3.9 X lo7 1.5 X 1.6 X 10' 1.7 X 1.5 X lo7 1.9 X 1.0

KW

kd, s-l

1013b

1.0 x 1013b

cm3 mol-'

103 103 103 105 103 105 103

lo8 10' io6

10'3 10' 1013 10'

lo3

lo7 10"

lo9 lo7 10'0

3.6 x 1.9 x 4.5 x 6.7 x 2.0 x 8.3 x 2.3 x 9.3 X 1.7 X 2.2 x 6.7 X 2.7 X 1.1 x 7.8 X 8.8 X 1.3 2.2 X 2.6 X 9.4 7.9 X

109 109 109 107 109 107 109 10-3 10-' 102 10-9 10-I

10-8 10-2 105 10-8 10-2

lo"

a The site notations A-C are those of Figure 3; AH and E listed for these sites are computed using large-size surface models while the vibrational frequencies are obtained with small-size clusters 1-111, respectively,depicted in Figure 4. The value is assigned to that typical of gas-phase radical recombination reactions.54

*

B3

B4

HH

I11

x

B5

HH B6

HH

HH

HH

Figure 3. Combination of surface sites considered for methyl adsorption: (a, top) CH3 is added to a radical of the dimer positioned on the left; (b, middle) CH3 is added to a radical of the bridge site positioned on the left; (c, bottom) CHI is added to a radical of the lone dihydride carbon unit positioned in the middle.

B, and cluster I11 to sites C. For comparison, the results obtained with clusters 1-111 are also listed in Table 1. The computed energy of CH3 chemisorption to a dimer radical varies from 65.7 to 75.4 kcal/mol, depending on the nature of the

Figure 4. Small-size clusters used in the calculations of vibrational frequencies. White circles represent carbon atoms and black circles hydrogen atoms on reacting carbon atoms. Hydrogen atoms saturating dangling bonds of the lower-level carbon atoms are not shown for clarity.

dimer's neighbor. The value of 75.4 kcal/mol, calculated for case A2 representing adsorption of CH3 on the ideal (100)-(2 X 1) surface, is in good agreement with corresponding values obtained in ASED-MO calculations of Mehandru and Anderson,63 77.9 kcal/mol, and MM3 calculations of Zhu et al.,36 76.1 kcal/ mol, and is somewhat lower than the MM2 result of Harris and Goodwin,35 80.9 kcal/mol. Similar to the preceding comparison, our chemisorption energy of 65.7 kcal/mol for site A3 is lower than the 70.9 kcal/mol reported by Harris and Goodwin35 for the same type of site. All the chemisorption energies for dimer sites are markedly smaller than that of a typical C-C bond in hydrocarbons, largely due to the repulsion among H atoms of the CH3 group and H atoms of the surface. Inspection of Table 1 reveals one of the most significant findings: the absence of potential energy barriers for the addition of CH3 to all of the considered dimer radical sites, irrespective of the neighboring site. Such a situation has been implied in a kinetic analysis35 but without proof. On the MNDO level this reaction was found23 to possess a sizable barrier, 33 kcal/mol, and a low chemisorption energy, 30 kcal/mol, results that are

Growth of Diamond (100) Surfaces due to the overestimation by the MNDO potential of steric repulsion in the vicinity of van der Waals distances.64 The postMNDO methods, AM1 and PM3, correct for this deficiency.64 As a further test, a molecular dynamics simulation was carried out for theadsorption of CH3 at the A2 dimer site. A freegaseous CH3radical was placed 3 A above the dimer radical of the (100) surface, which was preliminarily equilibrated at 1200 K for a duration of 500 fs. During the MD simulation, the temperature of the four bottom layers was maintained at 1200K by the method of Berendsen et al.,47 while the top-layer surface atoms and the atoms of the incoming methyl radicals were allowed to move adiabatically. It was found that adsorption takes place even with the initial kinetic energy of CH3 set to zero. The kinetic energy of the incoming CH3 and the order of the forming C-C bond computed during the MD simulation are shown in Figure 5 . Inspection of Figure 5 shows that the kinetic energy peak of about 50 kcal/mol corresponds to the creation of the new bond between the surface radical and CH3. After that, almost all the excess energy dissipates during the first several tens of femtoseconds, though full thermal equilibration occurs on a picosecond time scale. A similar picture was observed for the adsorption of a superheated gas-phase CH3 radical onto a room-temperature (100) diamond surface.65 These results show that the rate of energy transfer for the adsorption of CH3 even at temperatures as high as 1200 K is much larger than the expected rate of CH3 thermal desorption (101-102 s-l; see, e.g., ref 54), thus indicating that under the conditionsof diamond CVD the methyl adsorption f desorption reaction should be in thermal equilibrium with bulk diamond. The latter consideration underlies the use of transition-state theory for evaluation of the corresponding rate coefficients assuming the high-pressure limit. On the basis of the above results, the rate constant for the adsorption of CH3 at a dimer radical site, regardless of its neighbor, was assumed to be 1013cm3 mol-' s-l, typical for barrierless radical recombinations.54 The rate constants for the reverse direction, i.e. for CH3 desorption, were calculated from the principle of detailed balancing. The reaction equilibrium constants were computed, as described earlier, using vibrational frequencies obtained for a prototype reaction of CH3 with a small cluster, C9H13(cluster I in Figure 4), and enthalpy changes from corresponding large-cluster calculations. Although the environment of the dimer radical affects the desorption rate, as demonstrated by the results reported in Table 1, the computed residence times for the adsorbed CH3 are usually on the order of the characteristic time of H abstraction, meaning that CH3 adsorbed at dimer sites have sufficient time for "insertion" into dimers. Unlike dimer sites, adsorption of CH3 at the bridge (B-type sites) and dihydride(C-type sites) radical sites encounterspotential energy barriers (see Table 1). The vibrational frequencies for the calculation of the rate and equilibrium constants listed in Table 1 for these sites were obtained using analogous reactions of CH3 with two small clusters, C10H17 and C ~ H Ishown ~ , as clusters I1 and 111,respectively, in Figure 4. In these calculations, the distance between the top-layer atoms was kept at 2.52 A to mimic the diamond (100) surface structure. The local environment of the bridge and dihydride sites was found to be critical. The chemisorption energy varies from -46.8 to 10.9 kcal/mol for the bridge sites and from -39.3 to 8.1 kcal/ mol for the dihydride sites and the potential energy barrier from 2.4 to 24.2 kcal/mol for the bridge sites and from 7.2 to 29.3 kcal/mol for the dihydride sites (Table 1). Of particular interest in the present study are the void sites (B3 and B4 in Figure 3). Adsorption of CH3 at these sites must be ineffective for diamond growth due to the much reduced adsorption rates (and in the case of site B3 also due to a small equilibrium constant and hence a short residence time). The reason for this is again the strong

The Journal of Physical Chemistry, Vol. 98, No. 28, 1994 7077 repulsion among H atoms of the CH3 group and the surrounding surface H atoms. To explore the repulsion in greater detail, we examined the barrier to rotation of CH3 adsorbed at site B3 around the C-C axis. The potential energy as a function of the rotational angle is shown in Figure 6. As can be seen in this figure, the rotational barrier was found to be 6.9 kcal/mol, which is significantly higher than the barrier of 1.8 kcal/mol calculated at the same level of theory for the relative rotation of methyl groups in a free gaseous ethane molecule (for comparison, the experimental value of the latter is 2.9 kcal/mol"). The general mechanistic conclusions we draw from the above results are as follows. The adsorption of methyl groups at dimer surface radicals, A-type sites, should be essentially uninhibited and independent of the nature of the neighboring sites. The adsorbed CH3 groups are relatively stable, yet their lifetimes are comparable to and sometimes lower than the time scale of H abstraction, the latter being the necessary step toward incorporation of the CH3 into the diamond lattice. In light of this, the surface reactions discussed below may play a significant role in determining the fate of the chemisorbed CH3. In contrast to that at the A sites, CH3 adsorption at the B and C sites is both kinetically too slow and thermodynamicallyunstable to contribute significantly to diamond growth. Migrationof H. Migration of a hydrogen atom between carbon sites appeared to play an important role in surface rearrangements during acetylene adsorption onto a diamond (100)-(2 X 1) surface.16 It is reasonable to anticipate that such H migrations are rather common in surface chemistry of diamond growth. In the context of the present investigation, we examined two types of H migrations. First are the following migrations of H atoms between carbon sites of the surface itself

H H

H H

Heats of formation, potential energy barriers, and rate and equilibrium constants computed for these reactions are reported in Table 2. As a point of comparison, the barrier of 7.0 kcal/mol calculated for reaction 9 is somewhat lower than the 11.2 kcal/ mol obtained for a similar reaction on the MNDO level.23 We also note that the sum of reactions 7 and 9 is reaction 8, expecting then A H 7 + A H 9 = A H 0 and K7K9 = Ks. However, the enthalpy balance is off by 0.8 kcal/mol and the corresponding equilibrium constants product by about 20%. These deviations are the result of slight differences in the constraints of surface models used in the calculations of the individual reactions. Inspection of these results indicates that the forward and reverse rates of reactions 7-9 are much larger than the respective persite rates of H and CH3 addition reactions, R.Hddand RFZ. This implies that under the conditions of diamond CVD, these H migration reactions should be in a state of partial equilibrium, and which particular configuration prevails is determined by the corresponding equilibrium constant. On the contrary, due to a large potential energy barrier, 66.3 kcal/mol, reaction 6 appears

Skokov et al.

7078 The Journal of Physical Chemistry, Vol. 98, No. 28, 1994 to be very slow and thus insignificant on the time scale of diamond CVD. The large barrier of this reaction is due to the large separation between the neighboring dimer carbon atoms that the H needs to traverse. The immediate implication of the fast H migrations, reactions 7-9, concerns the discussed above expression r = KtJkfIdd. For equivalent surface sites, it defines the fraction of surface radical sites as well as the probability of an individual site to be a radical. However, the fast H migrations distort the local balance of reactions 4 and 5. As a result, although still representing the average fraction of radical sites, r no longer determines the individual site probability. Based on thevalues of the equilibrium constants reported in Table 2 for reactions 7-9, the probability of being a radical decreases from dihydride to bridge to dimer sites. That is, once an A3 or A5 radical site is formed, it immediately converts to a less reactive radical site B1 or C1, respectively; or stated in numerical terms, the rate of CH3 adsorption to an A3 or A5 site is reduced by a factor of 5.5 X 101or 4.4 X 102due to reaction 7 or 8, respectively. This implies that the adsorptionof CH3occurs with a higher relative frequency at biradical sites A4 and A6. A similar conclusion was reached by Harris and Goodwin.35 The second type of H migration reactions considered in the present study is transfer of H from adsorbed CH3 to the neighboring surface radical site: fH3

H

/‘tlH2

The computed reaction heats of formation, potential energy barriers, and rate and equilibrium constants are listed in Table 2. The principal result is that the rates of reactions 10-13 are much faster than the per-site rates of H abstraction, R:b, and H addition, RZd,indicating that these reactions will be in a state of partial equilibrium under the conditions typical of diamond CVD. Reaction 12 is of particular interest to the present study. CHJ adsorbed near a bridge site provides a situation conducive to filling the void. In fact, the reactant of reaction 12 is the key intermediate of the “trough” mechanism proposed by Harris and Goodwin.35 According to their mechanism, abstraction of an H from the adsorbed CH3 forms a CH2 adradical which then combineswith the neighboringsurface bridge radical; this is shown as the left-hand-side reaction sequence in Figure 7. However, the present results indicate that the rate of the H migration from the adsorbed CH3 to a neighboring surface radical, reaction 12, is about 106times faster than the rate of H abstraction. The CH2 adradical formed in this case “inserts” into the dimer and not into thevoid site, as illustrated by the right-hand-side reaction sequence in Figure 7. The CH2 “insertion” steps, reactions l a and lb, are as fast16J4.35as reaction 12, and hence the entire sequence of reactions between the chemisorbed CH3 site (reactant in Figure 7) and the product (B3 site, the final product of the right-hand-

:I: ; g

*

: 0.4

-

0.2

0.0

;

0

2

”

6

0

0

8

0

0

1

m

Tim (fs)

Rtsults of an adiabatic molecular dynamic simulation of CHJ approaching an A2 site: (top panel) kinetic energy of CH3 in the centerof-mass coordinate system; (bottom panel) bond order between a chemisorbedradical and a surfacecarbon, defined as a sum of the squares of density matrix elements connecting the two carbon atoms. Fipure 5.

side reaction sequence in Figure 7) is in a state of partial equilibrium. At the present level of theory, theoverall equilibrium constant is about 6 (or 11 using the thermodynamic data of ref 3 3 , showing that only a small part of the chemisorbed CH3 groups will follow the ‘trough” mechanism of Harris and Goodwin35 and most will lead to the creation of void sites. The relatively high rates predicted for reactions 10 and 11 mean that the H migration will dominate over the H abstraction in converting the adsorbed CH3 into a CH2 adradical. In other words, the incorporation of CH3 adsorbed at a dimer site does not necessarily require the abstraction of an H atom from CH3 itself, which is much slower than H abstraction from a tertiary dimer carbon.54 Finally, reaction 13 provides an interesting possibility of filling a void site, by following a &scission that forms a doubly-bonded =CH2 group which then adds to a nearby surface radical, as will bediscussed later in the text. However,theinitial step, adsorption of CH3 to a C-type radical that forms the reactant of reaction 13, is about 6 orders of magnitude slower than the addition of CH3 to a dimer radical, at an A-type site, as discussed in the preceding subsection. Therefore, reaction 13 is also unable to explain the rate of diamond growth observed experimentally. Migration and Combination of Surface Radicals. As already mentioned in the Introduction, in order to overcome the difficulty of filling the void sites, Zhu et a126 suggested migration of surface dimers, reaction 2. The key element of their proposal is the reaction

which is presumed to compete with and thus to prevent the “insertion” of the chemisorbed CH2 into the dimer, reaction 1, and thereby to impel the bridging over the void by the CH2. Our PM3 calculations resulted in a considerable potential energy barrier, 37 kcal/mol, and consequently the rate of this radical migration, reaction 2, is several orders of magnitude lower than therateoftheCH2 “insertion”intodimer, reaction 1. Nonetheless, the reaction rates of both forward and reverse directions are sufficiently fast to uphold the state of partial equilibrium. What

The Journal of Physical Chemistry, Vol. 98, No. 28, 1994 7079

Growth of Diamond (100) Surfaces

TABLE 2 Enthalpy Changes (AH), Potential Energy Barriers ( E ) , Forward (kf)and Reverse (kr)Rate Constants, and Equilibrium Constants (I&) for Surface Reaction on a Diamond (100) Surface at 1200 K small-cluster model reaction no. 1a' 2 6 7 8 9 10 11 12 13 15 16b 17 18c 19 20

large-cluster model AH, kcal/mol E, kcal/mol 12.4 7.7 0. -9.3 -14.6 -4.5 -10.4 -8.8 3.9 12.1 -43.0 34.1 -13.6 24.0 -28.3 -13.4

15.3 37.0 66.3 30.1 23.6 7.0 29.8 16.3 12.9 14.1

0. 62.0 0.7 36.3 3.2 14.9

kf,s-l 1.6 X 1Olo 8.8 X lo5 1.1 4.1 x i o 6 6 . 2 ~107 6.6 X 1Olo 4.6 X lo6 1.8 X lo9 7.4 x 109 4.5 x 109 8.9 X 10" 8.O 7.5 X 10" 1.5 x 107 2.9 X 10" 2.2 X lo9

k,, s-l 8.0 X 2.2 X 1.1 7.5 x 1.4 1.O X 6.0 X 1.3 X 1.1 x 2.0 x 2.2 X 8.8 X 1.2 X 6.5 x 1.1 X 4.2 X

1Olo

lo7

104 ~ 105 1O'O

lo4 lo8

1011 1012

lo6 lo7 10" 109

lo8 lo8

K,

AH, kcal/mol

E, kcal/mol

2.0 X 10-l 4.0 X 1. 5.5 x 101 4 . 4 ~io2 6.6 7.7 X 10' 1.4 X 10' 6.7 x 10-2 2.3 x 10-3 4.0 X lo5 9.1 X 1od 6.3 2.3 x 10-3 2.6 X lo6 5.2

6.0 0. 0. 0. 0. 0. -6.5 9.9 9.9 9.9 -36.0 46.3 -29.4 19.8 -19.8 -19.8

16.0 45.2 18.7 18.7 18.7 14.9 32.8 12.3 12.3 12.3 0.8 59.2 4.0 29.1 9.3 9.3

base model in Figure 4 I VI I11 I11 111

IIIC I IV IV IV IIId

I11 VI V V V

From ref 16. The units of k, are cm3 mol-' s-l and of K mol cm-3. C The distance between the two top carbon atoms was fixed at 2.7 A. The distance between the two top carbon atoms was fixed at 2.521. The parameters of this reaction are those calculated for the reverse of reaction 1b.

Figure 7. Comparison of the trough mechanism35 (left-hand-side reactions) with the one initiated by H migration (right-hand-side reactions).

8,

t

TABLE 3 Enthalpy Changes (AH), Potential Energy Barriers ( E ) , Forward (kf)and Reverse (kr)Rate Constants, and Equilibrium Constants (Kq)for Reaction l b at 1200 KI adjacent sites

AH, E, kcal/ kcal/ mol mol

kf,s-l

monohydridedimersb -24.0 12.3 6.5 X bridgingCH2groups 4.5 31.7 1.9 X bridging CH2 and CH' -6.6 20.5 2.1 X "0

30

60

90

120

150

180

Angle (degree) Figure 6. (Top panel) CH3 adsorbed at a B3 surface site. The white circle represents a carbon atom of the chemisorbed CH3 group, gray circles represent surfacecarbon atoms, and black circlestop-layer hydrogen atoms. Hydrogen atoms saturating dangling bonds on the cluster edges are not shown for clarity. (Bottom panel) Cluster relativepotential energy versus the angle of rotation of CH3 around the C-C axis; the maximum in energy corresponds to the H of CH3 passing through the surface H.

makes this reaction essentially ineffective is its slight endothermicity, 7.7 (PM3, see Table 2) or 6 (MM3, ref 36) kcal/mol, compared to the moderate exothermicity of reaction lb, -24.0 (PM3, see Table 3) or -28.3 (MM3, ref 35). Thus, with the reactant configuration of reaction 2 as a starting point, a rough estimate for the ratio of the fluxes toward the formation of the

lo9 lo6 lo8

k,,s-1 1.5 X 6.8 X 7.0 X

lo7 lo8 lo8

K, 4.3 X lo2 2.8 X lP3 3.0 X 10-l

The calculations of the vibrational frequencies were performed on cluster V in Figure 4. Reaction 1b in the present study. The data listed are from ref 16.

bridge site and toward the product configuration of reaction 2 is Klb/K2, which is equal to 1 X lo4, based on the present level of theory, or 1 X 105, using the MM3 results.35J6 Considering also that the lifetime of the dimer radical next to a bridge site is lowered due to the H migration, as established earlier in this paper, we must conclude that reaction 2 does not resolve the problem of filling the void sites during diamond CVD. We note that theendothemicity of reaction 2 is consistent with (and actually caused by) the row-zigzag rearrangement of the dimer pattern studied by us previously.32 To complete the analysis, we also considered the following combinations of surface radicals,

7080 The Journal of Physical Chemistry, Vol. 98, No. 28, 199‘4

H

H\ Cd

cd

‘d&

(16)

H H

c

aCd / \

+H

cd

H H

H

Skokov et al.

cd‘H e H Cd +d \

’c

\c,/”

(17)

Heats of formation, potential energy barriers, and rate and equilibrium constants computed for reactions 15-17 are reported in Table 2. Reaction 16 is a possible contributor to the reconstruction of dihydride surface sites to a monohydride (100)(2 X 1) surface dimer structure;23 however, the rate of this reaction was calculated to be very low, 8 s-l. The present predictions of 62.0 and 59.2 kcal/mol for the potential energy barrier of reaction 16 are somewhat higher than the previous estimate of 49.7 kcal/ mol at the MNDO level.23 Reaction 15 on the other hand, as expected, was calculated to be very fast, 10I2s-l. Limited by dehydrogenation of the dihydride surface sites, the formation rate of a biradical by two consecutive H abstractions is, as estimated earlier, -3 X 102 s-1. This indicates, by comparison to the much slower rate of reaction 16, that reaction 15 is a more likely candidate for participation in the reconstruction. Reaction 17 is another viable possibility for the combination of surface radicals into dimers: this reaction is kinetically fast and thermodynamically shifted to the right. Its existence was also observed in our previous MD simulation^.^^ Reaction 17 may play an important role in surface migration of CH2 groups, as discussed next. Migration of Surface CH,. Invoking additional reactions introduced above-chemisorption of CHS on different surface sites, migration of H atom, and migrationof surface radicals-does not resolve the problem of filling the void sites. Indeed, with a perfect (100)-(2 X 1) dimer surface as a starting point, CH3 will be ”inserted” into dimers by reactions 10-12, la, and lb. After some time, when the surface is covered by dimer and bridge sites, the rate of CH3 adsorption at dimer sites will start to fall due to the influence of the H migration, reactions 7 and 8. Finally, the surface will be covered by bridge sites separated by the voids. Adsorption of CH3 at the void sites is ineffective due to steric repulsion, as discussed previously in the text. A possible recovery from such a situation was suggested by Zhu et to be the etching (e.g., by the reverse of reaction 1) of undesirable bridge sites. However, with reaction 2 being ineffectivetocompete with the formation of bridge sites separated by voids, as discussed in the preceding subsection, the removal of bridge sites does not create viable alternatives for filling the voids. Searching for a possible solution to this problem, we discovered that migration of CH2, as shown below,

not only is feasible on kinetic and thermodynamic grounds (see Table 2) but also provides the most likely explanation for the formation of smooth surfaces observed experimentally. Each of these reactions is much faster, in both forward and reverse directions,than the H addition to a surface radical. This indicates that reactions 18 and 19 should be in a state of partial equilibrium at theconditions typical of diamond CVD. The overallequilibrium

... Figure 8. Illustration for the growth of a continuous chain of bridge sites.

constant, KlsK19,is equal to about 6, showing that the sequence of these two reactions is shifted toward a bridge-next-to-bridge configuration, i.e., toward the formation of a smooth surface. We note that even if the overall equilibrium constant happens to be smaller than unity, for example K18K19 0.1 (and we do not expect the accuracy of the present method to produce deviations larger than 1-2 orders of magnitude), the sequence of reactions 18 and 19 is still capable of filling the void sites, although with a somewhat reduced efficiency. The only requirement for the CHI migration is the creation of a biradical at the void site, which is not formidable in light of the probability for such an Occurrence discussed earlier in the text. Another reaction that may contribute to the surface migration is the addition of chemisorbed CH2 to a dimer radical as shown below

-

H H

This reaction is fast in both directions, and its equilibrium at 1200 K is shifted to the right (see Table 2). Let us now demonstrate that the migration of CH2 is capable of sustaining the growth of a continuous chain of bridge sites. We begin with a chain of dimer sites, shown as the initial structure in Figure 8. Formation of surface radicals, adsorption of gaseous CH3, migration of H atoms, and “insertion” of chemisorbed CH3 into dimers create bridge sites separated by the voids. A possible fragment of such an evolving chain is shown as the second structure in Figure 8. Adsorption of CH3 at one of the carbon atoms of the void site between the bridges is extremely slow, as established earlier in this study. After a period of time required for both of the void H atoms to be abstracted, the CH2 bridge in the center migrates to the left, following reactions 18 and 19, and the two surface radicals-one of the newly formed bridge and the other of the left-behind lone dihydride unit-combine according to reaction 17. The dimer formed in this manner undergoes the H abstraction and addition of a gaseous CH3 with the eventual conversion of this dimer into a bridge. This elementary-reaction sequence that creates a group of three consecutive bridge sites from the initial configuration of two bridges separated by a void can be viewed as a filling-in-the-void overall step. When the dimer unit located to the right of the chain fragment shown in Figure 8 converts to a bridge, occurring either simultaneously with or consequentlyto the previously described sequenceof events, its CH2 group undergoes a similar migration toward the already established bridge chain, and so on. The above discussedprocess can beviewed as surface “diffusion” of bridge sites, whose rate is limited by the H additions and H abstractions. The precise interaction among these processes and the resulting pattern formation shall be the subject of a future investigation. An immediate conclusion though can be made regarding the effect of temperature. The surface migration of CH2 hasa higher activationenergy than H abstraction, and hence a decrease in reaction temperature should slow down the mobility of CH2 faster than the activation of surface sites. This predicts that a decrease in temperature should lead to a stiffer yet reactive

Growth of Diamond (100) Surfaces

Figure 9. Schematic diagram for a configuration undergoing reaction

1bbetween twodihydrideunitsona (100)surface. Whitecirclesrepresent carbon atoms and black circles hydrogen atoms.

surface, forming an increasing number of lattice defects and eventually non-diamond carbon, in agreement with experiment.’ In summary, the general implication of the CH2 migration suggested by the present study is that the reaction mechanism of diamond (100) growth consists of two principal features: conversion of dimer sites into bridge sites and surface migration of the bridge sites toward continuous bridge chains. This mechanism does not require any particular order of dimer formation (which may include reactions of gaseous species other than methyl, like, e.g., acetylene16)but establishes the governing role of surface diffusion. This conclusion is in harmony with those drawn from experimental 0bservations.6~In fact, considering that the fast reactions of H atoms kinetically limit the migration of CH2 groups to rather small distances, our results are in remarkable accord with van Enckevort et al.67that “surface diffusion over a distance far less than ...2 4 nm ...is rate limiting in { 100) diamond growth”. Growth of Adjacent Chains of Bridge Sites. The analysis of surface reactions presented above is applied to the conditions of a single chain of dimers and bridges on a diamond (100)-(2 X 1) surface. However, during the growth process, as the dimers are being converted into bridges over the entire surface, one can no longer assume that reactions take place in isolation-along a single chain-and the influence of the neighboring-chain bridge sites must be taken into account. Of particular interest in the present study is the effect of hydrogen repulsion exerted on the CH2 group forming a bridge site, reaction lb, by the CH2 groups of the adjacent bridge sites as shown in Figure 9. Table 3 presents the results of our calculations for a case of completely hydrogenated sites, as shown in Figure 9, and for a case when a hydrogen atom from one of the adjacent bridging CH2 groups is removed. To complete the comparison, also listed in Table 3 are the results for an undisturbed case, computed in our previous studyl6 with adjacent sites being monohydride dimers. Inspection of the data presented in Table 3 indicates that with the adjacent dimers replaced by CH2 bridges the potential energy barrier of reaction 1b is increased by about 20 kcal/mol and the reaction rate drops by more than 3 X lo3. Still, the absolute values of the reaction rates in both forward and reverse directions are relatively high, larger than the per-site rates of H addition. What does make the difference for the growth of diamond is the switch from being a highly exothermic to endothermic step, manifested in the dramatic decrease, by more than 5 orders of magnitude, in the equilibrium constant. With such a large change, the reaction in essenceruns in reverse and thus becomes ineffective for diamond CVD. With one of the H atoms removed from the adjacent bridging CH2 groups, the energy barrier is lowered and, what is more important, the reaction step becomes exothermicagain. The value of the equilibrium constant, 0.3, is still lower than unity but not necessarily formidable: the rate of growth is roughly given as rR:!KlaKlb, which is equal in this case to 0.1 X 1 X lo3 s-l X 0.2 X 0.3 = 6 s-1. We also found that it does not matter which H atom of the three CH2 groups involved is removed, as the decrease in repulsion in all such cases is about the same. The repulsion is removed essentially completely’when another H is abstracted from one of the remaining CH2 groups.

The Journal of Physical Chemistry, Vol. 98, No. 28, 1994 7081 While the full impact of the repulsion discussed above on the growth of diamond must await a detailed stochastic analysis, the present results are generally consistent with experimental observations of the decreasing diamond film quality with the increase in the growth rate. As long as the surface migration of CH2 is much faster than the adsorption of gaseous CH3, the aggregation of bridge sites into continuous chains should assure smooth-surface growth. With the increase in the CH3 addition rate (e.g., by increasing the gaseous CH3 concentration), adsorbed CH3 begin to interfere with CHI migration and, as a result, the growth of the next layer will proceed before all the voids are completely filled. Conclusions The local environment of reactive sites significantly affects chemisorption of CH3on diamond (100) surfaces. Chemisorption of CH3 onto bridge and lone-dihydride-unit radicals was found to be ineffective for diamond growth due to excessive steric repulsion. Only dimer radical sites are capable of adsorbing gaseous CH3 radicals with sufficient frequency and keeping them adsorbed for a sufficient time to become incorporated into the diamond lattice. Migrations of H atoms from one surface site to another were found to be much faster than the H addition and H abstraction reactions and therefore to provide redistribution of radical sites on the reacting surface. Thus, due to fast H migrations, the probability of having a dimer radical near a bridge site or near a lone dihydride surface unit is decreased by more than 1 or 2 orders of magnitude, respectively. The decreased lifetime of a dimer radical when it neighbors a bridge site decreases the adsorption rate of gaseous CH3 at such sites. Migration of an H atom from chemisorbed CH3 groups to neighboring surface radicals was also found to be faster than the H addition and H abstraction reactions. Such H migrations further decrease the rate of the CH3 incorporation into void sites as compared to the CH3 “insertion” into dimers. In other words, the present results do not support the “trough” mechanism suggested by Harris and G ~ o d w i nto~ explain ~ the growth of diamond. A radical migration, reaction 2, proposed by Zhu et as an alternative mechanism for diamond growth was found to be ineffective to compete with the CH3 “insertion” into dimers accompanied by the formation of void sites and thus unlikely to play a significant role in diamond CVD. Probably the most important finding of the present study is the migration of CH2 groups on diamond (100) surfaces. The migration was found to be feasible on both kinetic and thermodynamic grounds and offers the most likely explanation for the formation of smooth surfaces observed experimentally. Being an essentialpart of the diamond growth mechanism, the migration of CH2 can be viewed as surface “diffusion”of bridge sites, whose rate is limited by the H addition and H abstraction reactions. Hydrogen repulsion which is responsible for the reconstruction of a dihydride (1 X 1) surface was also shown to impede the “insertion” of CH2 into dimers surrounded by bridge sites. With all of the involved carbon atoms fully hydrogenated, the CH2 “insertion” reaction is essentially prohibited. It becomes feasible when at least one of these hydrogen atoms is removed. Thegeneral implication of the present study is that the reaction mechanism of diamond (100) growth consists of two principal features: conversion of dimer sites into bridge sites and surface migration of the bridge sites toward continuous bridge chains. The mechanism does not require any particular order of dimer formation but establishes the governing role of surface diffusion. The two principal elements identified in the present study, the conversion of dimers to bridges and the CH2 migration, are not unique tothegrowth frommethyl radicals andarelikely to govern the growth of diamond (100) surfaces irrespective of the specific

7082 The Journal of Physical Chemistry, Vol. 98, No. 28, 1994 nature of the gaseous growth species. For instance, the addition of acetylene to (100) biradicals'6 not only converts dimer sites into bridge sites but also creates chemisorbed CH2 groups ready to migrate, according to the mechanism presented here. In this sense, the CHI migration provides the unifying element for the growth from different gaseous precursors. Acknowledgment. The work was supported by the Innovative Scienceand Technology Program of the Ballistic Missile Defense Organization via the US. Office of Naval Research, under Contract No. NO0014-92-5- 1420. References and Notes (1) For recent reviews and articles see: Spear, K. E., Dismukes, J. P., Eds.; Synthetic Diamond: Emerging CVD Science and Technology;Wiley: New York, 1994. Dismukcs,J. P., Ravi, K. V., Eds.; Proceedings of the Third

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