corbett c. battailestaff.ustc.edu.cn/~fuzp/course/paper/kinetic monte carlo...corbett c. battaile...

7 Jun 2002 8:22 AR AR162-12.tex AR162-12.SGM LaTeX2e(2002/01/18)P1: GJB10.1146/annurev.matsci.32.012102.110247

Annu. Rev. Mater. Res. 2002. 32:297–319doi: 10.1146/annurev.matsci.32.012102.110247

Copyright c© 2002 by Annual Reviews. All rights reserved

KINETIC MONTE CARLO SIMULATION OF

CHEMICAL VAPOR DEPOSITION

Corbett C. BattaileMaterials and Process Modeling Department, Sandia National Laboratories,Albuquerque, New Mexico 87185-1411; e-mail: [email protected]

David J. SrolovitzPrinceton Materials Institute and Department of Mechanical and Aerospace Engineering,Princeton University, New Jersey 08544-2006; e-mail: [email protected]

Key Words film, growth, model, review

■ Abstract The kinetic Monte Carlo method is a powerful tool for exploring theevolution and properties of a wide range of problems and systems. Kinetic MonteCarlo is ideally suited for modeling the process of chemical vapor deposition, whichinvolves the adsorption, desorption, evolution, and incorporation of vapor species atthe surface of a growing film. Deposition occurs on a time scale that is generallynot accessible to fully atomistic approaches such as molecular dynamics, whereasan atomically resolved Monte Carlo method parameterized by accurate chemical ki-netic data is capable of exploring deposition over long times (min) on large surfaces(mm2). There are many kinetic Monte Carlo approaches that can simulate chemi-cal vapor deposition, ranging from coarse-grained model systems with hypotheticalinput parameters to physically realistic atomic simulations with accurate chemical ki-netic input. This article introduces the kinetic Monte Carlo technique, reviews someof the major approaches, details the construction and implementation of the method,and provides an example of its application to a technologically relevant depositionsystem.

INTRODUCTION

Monte Carlo (MC) methods have been used in a wide variety of science and non-science disciplines, including materials science, nuclear physics, economics, andtraffic flow. Just as gaming in the casinos of Monte Carlo rests upon the statisticalproperties of random events, MC-based methods are stochastic techniques. Sim-ulations based on the MC method provide means of approximating solutions tomathematical problems through statistical sampling experiments on a computer.The most widespread application of the MC method in materials science is deter-mining the equilibrium structure or thermodynamic properties of materials. Theseapplications are based on the idea that the probability of different configurations

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298 BATTAILE ¥ SROLOVITZ

occurring in a system depends both on the relative energies of those configurationsand on their relative abundances in the phase space.

Monte Carlo methods can also be used to study non-equilibrium and kineticphenomena. For kinetic phenomena, the exploration of phase space must be per-formed along a Markov chain of states such that each state is accessible fromthe preceding one along the chain, given the available degrees of freedom. Asan example of the use of MC methods to describe kinetic phenomena, consi-der the diffusive motion of a single interstitial atom in a close-packed metal crys-tal. The motion of the interstitial atom is usually limited to two types: vibration ofthe atom around the center of the interstitial hole in which it resides and hops tonearest-neighbor interstitial sites. If the interstitial atom undergoes a large numberof vibrations between hops, the individual hops are thermalized and, hence, uncor-related. Because the atom is equally likely to hop into any of the nearest-neighborinterstitial sites, it executes a random walk. In an MC simulation of this diffusionprocess, the new position of the interstitial atom is chosen at random from a listof the adjacent interstitial sites. In practice, this is commonly done by choosing arandom number,ξ , on the interval from 0 to 1 [ξ ∈ (0, 1)] and then moving theatom to site numberi, for which (i − 1)/n≤ ξ < i/n, wheren is the total numberof accessible interstitial sites. Following this procedure for many steps yields onepossible trajectory, whereas a different set of random numbers usually leads toanother trajectory. Averaging over many trajectories reveals that the mean squareddisplacement of the atom is proportional to the number of hops that it makes.Although this is a characteristic of random walk diffusion phenomena, it providesno information about how much time the kinetic process takes (or, ultimately, thediffusivity of the interstitial atom). To uncover this information, we must providean estimate of how much time passes between hops or, equivalently, the hoppingrate. Thus the two main ingredients in a kinetic Monte Carlo (kMC) simulation arethe identification of all of the possible events that can occur and the determinationof the rates at which these events can occur. In the present review, we focus onthe application of MC methods to describe the temporal evolution of atomic-scalesurface structures. While the approach discussed herein is general, we further limitour discussion to a single application that has received a great deal of attentionin recent years: modeling the chemical vapor deposition (CVD) of thin films. Be-cause CVD typically involves a large number of competing chemical reactionsoccurring on vastly different time scales, kMC is an ideal simulation methodologyfor describing the molecular-scale surface evolution during CVD.

Chemical vapor deposition is widely used as a means of depositing films forapplications ranging from superconductors to ferroelectric films to hard coatings.In a CVD reactor, gaseous reactants are brought into contact with a solid surface,which catalyzes one or more key reactions. In some cases, substantial vapor-phasereactions occur before the reactants reach the surface. These are often catalyzed byinteractions with a plasma, flame, hot filament, or other heat source that leads to theproduction of free radicals. In other cases, the only reactions of importance occuron the solid surface, which is often allowed to reach a high temperature in order to

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KMC SIMULATION OF CVD 299

achieve a reasonable deposition rate. A complete model of the CVD process mustinclude a model of the reactive flow of the vapor-phase species on the scale of theentire reactor, a model for the boundary layer near the growing surface, a modelfor the evolution of the atomic structure at the growth surface, and a model for thereaction kinetics in the vapor and at the surface. The reacting flow in the reactor andthe boundary layer at the surface have been successfully modeled for several CVDprocesses via continuum methods (1–5). Prior to the application of kMC methodsto CVD, the true atomic structure of the surface was replaced with a small numberof atomic configurations (usually only one) and a limited set of reactions. Thiscommonly led to a one-dimensional model of the true three-dimensional growingmaterial (6, 7). Kinetic Monte Carlo methods can be applied to the CVD processin order to remove as many of these ad hoc assumptions as possible.

Two of the most challenging problems in modeling CVD are predicting therate of growth of the film and capturing the morphology of the surface as a func-tion of reactor conditions. These are both governed primarily at the atomic scale.Because the atomic structure of the surface can evolve during growth, the sur-face structure and morphology are established by kinetic, rather than equilibrium,considerations. It is, in principle, possible to describe the evolution of the surfaceusing molecular dynamics (MD) simulations. However, since typical growth ratesin CVD are on the order of microns per hour or monolayers per second, even today’smost heroic MD simulations are capable of simulating the deposition of only avery small fraction of a monolayer within reasonable computation times. On theother hand, kMC simulations can easily model the deposition of thousands oflayers, as we show below. Even so, the switch from MD to kMC comes at a cost:The true dynamical information (associated with atomic momenta) is lost, and thefull (continuous) atomic trajectories are replaced with a set of discrete positionschosen from a set of pre-determined possible sites. This cost can be sustained pro-vided that each event is thermalized before another event occurs nearby. Nonethe-less, MD simulations can help determine the list of possible/probable events forinclusion in kMC simulations. Just as the validity of MD simulations depends onthe adequacy of the description of the atomic/molecular interactions, the fidelityof kMC simulations depends on the accuracy of relevant reaction rate data. For-tunately, it is much easier to determine reaction rates from experiments or first-principles calculations than it is to obtain the full atom configuration–dependentenergy required for MD simulations. Below, we describe how the requisite reactionrate data are obtained for kMC modeling of CVD.

The goal of this article is twofold: to provide a brief review of the application ofthe kMC method to CVD processes and to provide a pedagogical description of thefoundations of kMC for CVD applications. The next section provides a review ofthe application of kMC methods to CVD processes, focusing on what we considerto be the main modeling issues, the approaches that have been used, and theadvantages and disadvantages of each type of model. (This review is, however, byno means exhaustive.) In the third section, we provide a generic description of theimplementation of kMC CVD simulations, and outline the theoretical foundations

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of this approach. Next, we summarize some of the options for obtaining the kineticdata that are the necessary input to kMC models. Finally, we discuss the applicationof the kMC model to the CVD of diamond, relying heavily upon our own results.

BACKGROUND

A full description of the CVD process cannot be provided by simulations that fo-cus on a single length or time scale. Rather, it requires an integrated set of modelsthat address the chemistry and transport of species in the vapor within the reactor(macro-scale), the molecular and atomic processes at the growing surface (nano-scale), and the microstructural and morphological evolution of the film (micro-scale) (8–13). Reactor-scale models of CVD focus on reactive hydrodynamics,vapor-phase chemistry, and vapor-surface chemistry to predict the temperatureand composition profiles within the vapor at and near the growth surface. This in-formation provides the link between reactor design and the structure of the growingfilm and serves as necessary input to molecular-scale simulations of film growth.The surface orientation–dependent growth rates predicted by the molecular-scalemodels can serve as input to models that track the evolution of the growth fronton the microstructural scale. If the growth temperature is sufficiently low, the filmmicrostructure can be viewed as a historical record of the state of the surface atthe time of growth.

Although our focus herein is on the molecular-scale kMC simulation approach,it is appropriate to briefly consider the other components of a full multi-scalestrategy for modeling the CVD of thin films. The earliest CVD reactor-scale mod-els were one-dimensional and included detailed vapor-phase and vapor-surfacechemistry. These provide insight into the chemical kinetics governing the deposi-tion process and the relationships between mass transfer and chemistry (1). Reactormodels in two and three dimensions can, in principle, provide more accuracy andcan be used as reactor design tools but often balance the greater computationaldemands of the higher dimensionality with simplified vapor-phase chemistry anddeposition models (4). There are CVD reactor-scale models for simulating arc jet,plasma, and hot filament systems (2, 3, 5), to name but a few. In addition, severalapproaches have been proposed for tracking the evolving surface morphology andfilm microstructure. These include two-dimensional ballistic deposition modelsof polycrystalline films (14) and front tracking methods based upon growth ratesof different surfaces (12, 15–17). Front tracking is much more difficult in threedimensions than in two because of the much greater topological complexity as-sociated with tracking individual grains. As a result, new methods based uponlevel sets (18) have been developed to avoid the difficulties associated with theevolving topology (19, 20). Other simulation methods have been developed totrack the evolution of more macroscopic surface features, which is important intechnological problems such as the filling of vias in microelectronic applications(21).

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Both MD and MC models have been used to provide a molecular view of thegrowing surface. The earliest MD studies of molecular reactions on surfaces wereperformed in the 1970s (22, 23). Garrison et al., among others, investigated theadsorption of such species as H, H2, and CH3 on diamond surfaces via MD todetermine the rates of some of the important reactions in diamond CVD (24–28).Although MD provides one approach to determining reaction mechanisms forCVD, off-lattice MC methods yield another. Following the methods of Fichthorn& Weinberg (29), Scott et al. (30, 31) employed off-lattice MC on a continuouspotential surface, combined with local relaxations, to determine binding sites, com-petitive reactions, and reaction rates for diamond CVD. Molecular dynamics andoff-lattice MC methods can be combined (32) to provide the necessary individualreaction rates needed to make the kMC predictions quantitative. Both MD andoff-lattice MC simulations have provided important information about reactionpathways that were previously unknown and were later included in kMC simula-tions. Although not directly relevant to our present focus on atomic-scale growthcalculations, it is interesting to note that MC simulations have also been used topredict other aspects of CVD. For example, Dong & Zhang (33) used MC to predictthe distributions of electron energies and velocities in electron-assisted CVD.

All molecular-scale models of the CVD process must include descriptions ofadsorption, desorption, and reaction. In some cases, the transport of molecularspecies along the surface (i.e., surface diffusion) may also be important. Con-ceptually, kMC models can be divided according to the level of realism used todescribe the chemical reactions and the structure of the growing film. Computa-tionally, kMC models fall into two main categories: fixed and variable time stepmethods. We now discuss these different classes of kMC models.

Kinetic Monte Carlo simulations of CVD have been performed using descrip-tions of reaction kinetics ranging from those that are essentially schematic to thosein which the parameters were taken from first-principles calculations. Greatly sim-plified kinetic models were used for simulations that focus on generic aspects ofCVD (34, 35) and those for which little is known about the reaction mechanismand chemistry, e.g., MOCVD of ferroelectrics (36). In some CVD kMC simula-tions, important kinetic features were neglected in order to determine the effectsof particular variables (37). In many cases, the real microscopic reaction pathwaysare extremely complex, such that there is no alternative to using lumped kineticparameters that may actually represent a large number of basic reaction steps.One extreme approach is to simply replace the true reactions with effective onesthat are meant solely to mimic experimentally measured growth rates. Althoughthis approach can be used to predict how substrate topography is covered by thegrowing film (38), it can usually predict little else. A related approach is to replacea microscopic description of the CVD kinetics with effective sticking coefficientsor reaction probabilities. Applications of this method range from those that define asingle sticking coefficient for all sites on a surface (39) to those in which the effec-tive sticking coefficients include substantial insight into the stereochemistry (40).These approaches all represent methods for dealing with the dearth of knowledge

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about true reaction mechanisms and microscopic reaction rates associated withmost CVD processes and systems.

Fortunately, a few cases do exist for which there is a substantial body of micro-scopic measurements and predictions for the individual reactions that control filmgrowth. A prime example of this is diamond CVD because an extensive set of hy-drocarbon reaction kinetics parameters are available for the vapor phase from thecombustion literature, and because there have been periods of substantial researchinterest (and funding) in diamond CVD for a range of technological applications.The vapor-phase kinetic data can be manipulated for application to reactions on thesurface of growing diamond by adjusting the reaction cross-sections to account forthe fact that some of the reactants are fixed on the surface (41, 42). These estimatescan be improved by correcting for surface relaxations using molecular mechan-ics methods (42). As described above, MD simulations can also provide usefulkinetic data. Even with this wealth of kinetic input, the predicted growth ratesdo not always match experiment, and additional mechanisms must therefore besought and the corresponding kinetic parameters determined from first-principlescalculations (43, 44). Kinetic Monte Carlo models based on these more completedescriptions of the microscopic kinetics have had substantial success in modelingsurface orientation–dependent growth rates, surface morphologies, point defectincorporation rates, etc. (11, 13, 26, 43, 45–51).

Many of the early CVD kMC simulations focused on general features of filmgrowth such as the evolution of roughness and the role of surface diffusivity ver-sus deposition rate, rather than describing particular materials [especially in thephysical vapor deposition (PVD) arena]. Because these models were generic, mostwere based on simple descriptions of the crystal lattice. Of these, perhaps the mostwidespread was the so-called solid-on-solid (SOS) model (52). In this model, thesurface profile is represented in three dimensions by a height function,h(x, y),whereh is the height of the growing surface above the flat substrate (thez =0 plane) as a function of its in-plane coordinates, andh, x, andy represent posi-tions on a three-dimensional (usually simple cubic) lattice. No overhangs, voids,or vacancies can be represented in this framework. Although this is a reasonableapproach for an idealized theoretical model, it has several shortcomings for mod-eling the growth of real films. Nonetheless, SOS models have found widespreaduse in kMC simulations of CVD (38, 53–56). Alternatively, kMC models that arebased on a complete representation of the appropriate crystal lattice (unlike mostSOS models) can be used to determine the correct bonding configurations that arekey to properly accounting for the stereochemistry. This is especially importantfor non-flat surfaces where, for example, steps on{111} can have the same bond-ing configuration as ledges on{011}. Such effects are particularly important forpredicting phenomena such as faceting (46). It is also possible to represent surfacereconstructions within the framework of the perfect crystal lattice (26, 45). Clearly,the surface orientation dependence of growth rates cannot be predicted withoutan appropriate description of the crystal symmetry. As an aside, we note that careis required in predicting film growth rates in high-symmetry orientations because

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truly singular surfaces (e.g., perfect{111}) of any appreciable area are practicallynever found in reality, which always presents at least some degree of vicinality.[Even slight miscuts can lead to large changes in growth rate (48).]

Increasing the realism of a kMC model for CVD generally comes at the costof increasing computational burden. There are no universal rules that tell us whenenough is enough. The degree of realism required clearly depends on the ques-tions being asked. For example, consider the determination of local environmentsof topologically equivalent structures, such as may exist between atomic structuresof terraces on one surface, steps on another, and kinks on a third. While nearest-neighbor topology might be sufficient to identify the environment in stiff, covalentmaterials, long-range interactions between defects (such as steps) may break thisapparent degeneracy in other materials. The importance of such effects may dependon what type of material is being examined and whether the dominant kinetic phe-nomena are chemical reactions or surface diffusion. Because potentially limitlesssets of kinetic events could conceivably be treated in a kMC simulation, it is impor-tant to determine the sensitivity of the simulation results to the chosen set (57).

We now turn from considering basic kMC model issues to more computationaltopics. There are two distinct classes of on-lattice kMC models (i.e., the typewe focus on below): fixed and variable time step approaches. Fixed time stepmethods are based upon the calculation of the probability that a particular eventwill occur within a constant time period,1t, and have been widely used in kMC ofCVD (26, 35, 40, 58). However, the convenience of these methods incurs a penaltyin efficiency. The fixed time step must be small enough to avoid missing anyimportant events. The growth surface can evolve at very different rates within thespan of a single simulation, and a fixed time step must be chosen to accommodatethe fastest possible evolution. Furthermore, because the time increment is fixed andthe evolution of the surface is inherently stochastic, there are some time steps inwhich the surface changes and others during which nothing happens whatsoever.These issues point to the appeal of variable time step methods. In the variable timestep approaches, the rate at which each event occurs is calculated, and an event israndomly chosen and executed with a probability that is given by its rate relative tothose of all other possible events. The time is then advanced by an increment thatis inversely proportional to the total rate at which any event occurs. This methodhas two main advantages: An event occurs with unit probability during each timestep, and the time step gets longer or shorter as the evolving system demands.Thus variable time step approaches are usually more efficient than fixed time stepmethods. The variable time step method is now widely used in kMC simulationsof CVD (36, 45, 50, 59) and is an extension of the approach suggested (apparentlyindependently) by Bortz et al. (60) and Gillespie (61). The implementation of sucha method is described in the next section.

Another important issue for the computational efficiency of a kMC simulationis associated with the potentially large range of rates with which the allowed eventscan occur. If a small set of events is much faster than all of the others, then mostof the computational time will be spent executing those fast events, and relatively

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few of the more rare (and often crucial) events will ever be seen. This is a potentialproblem if the fast events do not lead to growth. For example, consider the growthof diamond in a hot filament CVD reactor. In this case, the adsorption, desorption,and abstraction of atomic hydrogen from the surface occurs about 10,000 timesmore frequently than anything else (32). Thus the events that lead to growth arevery rare (1 in 10,000 at best), and this is clearly undesirable because diamondgrowth, and not hydrogen evolution, is the phenomenon of interest. Nonetheless,the hydrogen events are important because they stabilize the formation of diamondrather than graphite (62, 63). If the fast events are very much faster than the slowevents that determine the process of interest, the kMC algorithm can be acceleratedby assuming that these fast events are in equilibrium. In the diamond CVD case,this involves assigning a hydrogen occupancy probability to each of the sites wherea hydrogen atom can possibly reside, randomly assigning an appropriate concen-tration of hydrogen to those sites, and removing all the hydrogen reactions from thelist of possible events that determine the kinetics. This simplification can acceleratethe kMC of diamond growth simulations by roughly a factor of 104. If the disparityin rates between the fast and slow events is not great, a similar procedure based onintegrating part of the Markov chain can be used, as outlined by Gillespie (64).

The next section contains a heuristic introduction to the kMC method, especiallyas it applies to CVD. The focus is on what we believe to be the best practice ofthis method for most molecular-scale CVD applications, as reviewed above. Theapplication of the method to diamond CVD is then outlined as an example, andsome results are presented.

IMPLEMENTATION

A thermodynamic MC approach samples random system configurations in anattempt to lower the energy, whereas a kMC scheme tracks the temporal evolutionof a system by stochastically choosing from the state-dependent sets of transitionsavailable to the system. While the former generally requires information aboutthe energy of the system in all possible configurations, the kMC scheme needsthe rates of all the possible state transitions. A spatially discrete kMC simulationof CVD is a specific incarnation, and the state transitions are represented by thechemical reactions occurring on the deposition surface. Thus a kMC simulation ofCVD generally requires descriptions of the atomic structure, vapor environment,and surface chemistry of the depositing material. In this section, we describe insome detail the implementation and combination of each of these components, andthen present a real-world application of a model of this sort.

Methodology

The ideal (i.e., perfect) atomic structure of most crystalline deposits is relativelystraightforward to represent in the computer. The details of this endeavor are notaddressed here, and in the discussion that follows we assume an ideal crystal lattice

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for both the deposited film and the positions of the adsorbed species. A more real-istic treatment of extended defects, including the free surface, grain boundaries,and dislocations, is possible to an extent (especially in the former case) but is notconsidered here for simplicity. (A common approach to simulating polycrystallinedeposition is to “tag” individual nuclei or grains on an otherwise uniform lattice,and while this approach can capture some of the features of grain formation andevolution, it can address only the most basic features of the grain boundaries them-selves.) Likewise, an explicit treatment of off-lattice configurations for adsorbedspecies is tractable but generally requires coupling the kinetic scheme to a fullyatomistic approach (31, 32) to relax adsorbate positions and is not addressed inthis discussion. Furthermore, the atomic-level details of nucleation in thin filmgrowth are complex and remain somewhat mysterious, and thus we consider onlyhomoepitaxy.

The foundation of any kinetic CVD scheme is the set of chemical reactions thatare relevant to the growth process, i.e., the reaction mechanism. The mechanismis comprised of a set of coupled chemical reactions that contain (or lead to) somereaction(s) whose product(s) represent deposited solid material, i.e., growth. Ina purely chemical kinetic approach (without any explicit treatment of the atomicstructure), the set of rate equations (i.e., partial differential equations) presented bythe reaction mechanism is solved numerically, and the rate of the growth reaction isused to calculate the growth rate. However, in the spatially discrete kinetic approachpresented here, the reaction mechanism is used to define rates for the fundamentalsteps in the evolution of individually resolved surface species during deposition.Because a discrete simulation can include a description of the underlying structureof the material, its behavior depends not only on chemical but also spatial (i.e.,steric) concerns. In this way, for example, the growth behavior on different crystalfaces (e.g.,{001} and{111}) and surface features (e.g., steps and kinks) of thesame material can be (and often is) very different.

A kinetic model of deposition requires quantitative information about the rates,in units of reactions per second (i.e., s−1), of all the viable chemical processespertinent to the deposition system of interest. These rates are generally obtainedfrom a transition state theory, wherein a reaction is viewed as a transition of thesystem between adjacent local energy minima. Specifically, consider the energycurve in Figure 1. A system (or local subsystem) in stateS1 will make thermallyinduced energy excursions about the minimum energy reaction coordinate atS1,and these excursions can be viewed as attempts by the system to escape the energywell. These attempts occur at an average frequency,A, and the rate at which thesystem successfully escapes the energy well atS1 and falls intoS2 is expressed as

kf = A exp

(− E

RT

), 1.

wherekf is the rate constant (the subscript,f, denoting the forward transition; seebelow),E is the energy barrier between statesS1 andS2, R is the universal gas con-stant equaling 1.987 cal mol−1 K−1, andT is the temperature. The pre-exponential,

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Figure 1 The energy curve for a transition between statesS1

andS2.

A, and thus the rate constant,k, are conventionally expressed in units comprisedof mol, cm3, and s. The specific combination thereof depends on the nature ofthe reaction; a discussion on this point is best left to treatises on rate theory. Forthe present purposes the parameters,A and k, are in s−1 for reactions with novapor-phase reagents, and mol cm−3 s−1 for reactions with a vapor-phase reagent.

A discrete simulation needs information about the absolute per-site rates of eachreaction in s−1. Thus a rate constant in units of s−1 likely needs no further conversionbefore incorporation into a discrete kinetic model. However, a rate constant in anyother units, e.g., mol cm−3 s−1, must be adjusted to yield an absolute per-site rate.In the case of vapor-phase deposition, this is accomplished using the density ofthe vapor-phase reagent. Specifically, if a reaction involves a vapor-phase reagentwhose concentration in the vapor isC, then the rate of the reaction in units of s−1

is approximately

r = CP

RTk, 2.

whereP is the vapor pressure in units of cal cm−3 (obtained, for example, bymultiplying torr by 3.18757× 10−5 cal torr−1 cm−3).

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Most reactions are reversible and should be considered in practice as two sepa-rate reactions: the forward and the reverse. The procedure for determining therate of a reverse reaction from that of the forward is perhaps more complicatedthan one might guess. Consider again the reversible state transitionS1 ↔ S2 inFigure 1. The energy barrier to the forward transitionS1→ S2 is, as defined above,E. Assuming that1G is the change in free energy upon the forward transitionS1 → S2, such that1G is negative in Figure 1, then the energy barrier to thereverse transitionS1← S2 is E − 1G. The free energy change is a function ofthe changes in enthalpy and entropy,1H and1S respectively, such that1G =1H − T1S, and thus the rate of the reverse transition is

kr = αA exp

(−E −1G

RT

)= αAexp

(−1S

R

)exp

(−E −1H

RT

), 3.

where the constant,α, is a conversion factor to account for the fact that the thermo-dynamic parameters,1H and1S, are conventionally computed for standard statesof 1 atm. The value ofα is 1.016× 10−5 mol cm−3 atm−1 for reactions where thenumbers of reagents and products differ, and unity otherwise (42).

Consider the simulation cell depicted in Figure 2. The simulation space isdiscretized into a simple square lattice, and the sites are tagged to represent asubstrate covered by an adsorbate in contact with a vapor. To approximate thebehavior of a much larger surface than can be represented in a computer simulation,periodic boundary conditions are often applied in the directions parallel to thesubstrate. Imagine that the substrate in Figure 2 is composed of some solid materialA(s) covered by B(s) and in contact with a dilute vapor containing B(g) atoms,B2(g) molecules, AB2(g) radicals, and an AB3 precursor, in the concentrationslisted in Table 1 and at a pressure of 6.8 torr. The designation (s) denotes an atomon or below the surface, and (g) denotes an atom in the vapor.

Now imagine that the only appreciable reactions that can occur at an A(s) surfacein contact with the aforementioned vapor are listed in Table 2. Notice that neitherB2 nor AB3 plays a role in the deposition process, i.e., they are inert. The species∗(s) denotes a radical, or dangling bond on the surface, and is listed in the reactionmechanism for clarity. Reaction 1 involves adsorption (forward) and desorption(reverse) of B to and from the vapor. Reaction 2 is the adsorption and desorptionof AB2. Reaction 3 entails the desorption of a B atom from an adsorbed AB2

TABLE 1 Composition of a hypothetical vapor

Species Concentration

B 0.010

B2 0.900

AB2 0.001

AB3 0.089

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TABLE 2 Chemistry for the deposition of A from a vapor of B and AB2a

Reaction A E 1H 1S kf rf kr rr

1. ∗(s)+ B(g) ↔ 1× 1012 5 −20 −1.34 1× 1011 1× 102 2× 102 2× 102

B(s)

2. ∗(s)+ AB2(g) ↔ 1× 1012 0 −29 −6.40 1× 1012 1× 102 4× 102 4× 102

AB2(s)

3. AB2(s)→ 4× 103 5 4× 102 4× 102

A(s)+ B(s)+ B(g)

aA andk are in units of mol, cm3, and s (see text);E and1H in kcal mol−1;1Sin cal mol−1 K−1; andr in s−1. Thermodynamicquantities,1H and1S, are calculated at the standard state ofT = 1200 K andP = 1 atm. Reaction rates,r, calculated atT = 1093 K,P = 6.8 torr, and the vapor composition in Table 1.

molecule, and the subsequent incorporation of the A atom into the film. The latter isthe reaction that results in the incorporation of adsorbed species into the solid film,i.e., it is the growth reaction. The implication of Reaction 3 is that any AB(s) (partof the implied intermediate state for the reaction) will instantaneously transforminto a solid A(s) terminated by a B(s) adsorbate. Physically, this representationof the growth reaction implies some bond rearrangement between the A atomand the underlying substrate, but for the purposes of this example, we forgo anyelaboration or justification of these details (important though they might be to anaccurate mechanistic picture). Reaction 3 is irreversible in this example, but wereit not, then the reverse would represent a pathway that attacks or etches the solid byB(g). Note that there is no provision in Table 2 for surface diffusion, and althoughthis is certainly a critical component in many deposition mechanisms, it is omittedhere for simplicity.

The chemical kinetic parameters for each reaction are provided in Table 2, andthese are used, along with the concentrations in Table 1 and the vapor pressure of6.8 torr, as described above, to calculate the rate constants,k, and reaction rates,r,listed in the table. The physical system depicted in Figure 2, the reaction mechanismand associated rates in Table 2, and the aforementioned growth rule that providesfor the instantaneous conversion of AB(s) to A(s)+ B(s) provide all the informa-tion needed to construct a simple simulation of chemical vapor deposition. In thisexample, we use a kMC scheme based on the N-Fold Way approach (60), i.e., avariable time step approach. The general recipe for the simulation is to (a) checkthe system configuration against the chemical reactions in the mechanism to iden-tify all the possible events (i.e., all possible reactions at all reactive sites), (b) chooseone event (i.e., one specific reaction at one specific site) at random, (c) update thesystem to reflect the event, (d) update the simulation clock, and (e) repeat. Toclarify this procedure, we examine a few simulation steps in more detail.

The lattice in Figure 2 is reproduced in Figure 3a with some of the siteslabeled a–e. A cursory examination of the lattice in Figure 3a and the

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information in Table 2 reveals that the only reactive species in Figure 3a are thefive B(s) adsorbates. In order to take a simulation step, we must choose one ofthese B(s) adsorbates to desorb via the reverse of Reaction 1 in Table 2. To do this,we must weight the probability of choosing an event by its rate, so that events withhigher rates are chosen with higher probabilities. (In the configuration depicted inFigure 3a, the weighting is trivial because all the possible events occur at the samerate.) This concept is easily grasped by a pictorial representation of the selectionprocess. In Figure 4a, all five possible choices are placed along a line according

Figure 4 Lists of possible events for the configurations in Figure 3, plotted bythe events’ rates normalized by the total rate,0. t is the simulation time. Negativereaction numbers denote reverse reactions.

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to their cumulative rates (normalized by the sum of the rates). To select one of thefive possible events in Figure 4a, we generate a random numberξ1 ∈ [0, 1) andselect one event accordingly. For example, if the random number were generatedsuch thatξ1 = 0.52, which is between 2/5 and 3/5, then the reverse of Reaction1 would be executed at Sitec. Formally, the rate of the chosen event,i, obeys

i−1∑j=1

r j

0< ξ1 ≤

i∑j=1

r j

0, 4.

whererj is the rate of eventj. 0 is the sum of the rates of all possible events at thestart of the simulation step of interest,

0 =n∑

j=1

r j , 5.

wheren is the total number of possible events. After the chosen event, i.e., thereverse of Reaction 1 is executed at Sitec, the system changes to that in Figure 3band the event list to that in Figure 4b. The only remaining task for this simulationstep is to update the simulation clock.

A more conventional MC scheme, e.g., using the Metropolis algorithm (65),would step uniformly through simulation time and attempt a random event at eachstep with conditional success. The first simulation step in the present example,however, implies a different approach: One of the known possible events is ex-ecuted at every single simulation step (unconditionally), which suggests that thetime increment in the present approach should not be constant, but rather shoulddepend on the state of the system at each simulation step. One can demonstraterigorously (60) that this approach is nothing more than a procedural reorderingof the Metropolis algorithm and that the appropriate time increment for a givensimulation step is

1t = − ln (ξ2)

0, 6.

where ξ2 is another random number such thatξ2 ∈ (0, 1). In the Metropolisalgorithm, the acceptance of each transition is stochastic and the time incrementis constant. In the present N-Fold Way approach, the acceptance probability isconstant (unity) and the time increment is stochastic. Zero is excluded from thepossible values ofξ2 to avoid the obvious singularity. Note that the expectationvalue of the numerator in Equation 6 is〈ln (ξ2)〉 = −1. For the present example,assumeξ2 = 0.27, and note from the value ofrr for Reaction 1 in Table 2 that thesum of the rates of all five possible events at the start of the first simulation step is0 = 1 ms−1. Given these values, the first time increment is1t = 1.3 ms, and atthe end of the first simulation step, the simulation time is simplyt = 1.3 ms.

This procedure can now be repeated using the updated information in Figures 3band 4band continued as long as is possible or desirable. For example, the sequenceof random numbers (including the ones enumerated above) of (ξ1, ξ2) = (0.52,

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0.27), (0.58, 0.13), (0.74, 0.85) results in the progression of configurations inFigures 3 and 4. Notice that the value of the random number,ξ2, greatly affectsthe time increment: Small values ofξ2 yield large time increments and vice versa.Phenomenologically, this randomization of the time increment corresponds to thefact that even though we know how fast each reaction will occur on average, eachindividual realization of a reaction might occur slower or faster than the average dueto the inherently stochastic nature of the underlying thermal state transitions. Alsonotice that in this simplified model of deposition, we have made an implicit “unitedatom” approximation and in doing so have taken certain liberties in implementingthe growth reaction. For example, in Figure 3d, where AB2(s) has converted toA(s)+ B(s)+ B(g) to yield deposition, the B atom that was previously incorpo-rated into the reagent AB2(s) site has “magically” appeared in the site above thedeposited A(s) product. Although this sort of united atom approach is useful andconvenient, it is certainly possible to construct a more elaborate model whereineach atom occupies its own site and is thus accounted for separately. One can evengo so far as to consider deposition off-lattice and to incorporate realistic interatomicinteractions (31, 32), but these modifications are beyond the scope of the presentexample.

For the purposes of this example, we have chosen convenient (though not phys-ically unrealistic) values for the vapor-phase composition and the chemical pa-rameters,A, E,1H, and1S. In practice, these inputs must be derived from somesource that is meaningful to the problem at hand. The composition of the vaporis commonly obtained either from experimental mass spectrometry data (66) orfrom computer simulations of the vapor in the CVD reactor (1). In some cases,the rate parameters,A andE, can be gleaned from experimental techniques suchas temperature programmed desorption. Otherwise, if the surface reaction of in-terest can be closely matched to an analogous vapor-phase process for which rateinformation is known, the rate parameters for the vapor-phase analogue can beadjusted for application to the surface reaction by accounting for differences insymmetry and collision frequency (41, 42), as mentioned above. Specifically, therate constant,ks, for the surface reaction is approximately

ks = βkgns

√µ

mg, 7.

wherekg is the rate constant of the vapor-phase reaction,ns is the number ofequivalent reaction sites on the vapor-phase molecule,µ is the reduced mass ofthe vapor-phase collision partners,mg is the mass of the vapor-phase molecule,andβ is a constant equal to one half for radical recombination reactions and unityotherwise. Yet another option for determiningAandE is computer simulation, e.g.,using quantum mechanics. In this case, the activation barrier,E, can be determinedby a carefully executed probe for the saddle point along the reaction pathway, whichcorresponds to the maximum in the curve in Figure 1. (The reaction coordinateis, by definition, a mapping of the trajectory of the transitioning species onto theaxis.) In general, this procedure demands some knowledge of the reaction path,

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and sophisticated techniques such as the nudged elastic band method (67) or anaccelerated dynamics approach (68, 69) can be used to aid in the automation androbustness of this approach. The reaction attempt frequency,A, can be estimatedfrom the phonon properties of the system in atomistic calculations that accountfor atomic vibrations. The reaction thermodynamic quantities,1H and1S, can becomputed similarly using the atomic configurations before and after the transition(at statesS1 and S2 in Figure 1). The best method for obtaining the chemicalparametersA, E,1H, and1Sdepends on the availability of pertinent experimentaldata and on the CVD system of interest, and different choices are often appropriatefor each individual reaction in a given problem.

APPLICATION: DIAMOND CHEMICAL VAPOR DEPOSITION

The preceding discussion describes the basic steps involved in constructing and im-plementing a very simple kMC deposition model. In general, real systems are muchmore complex than this, and to illustrate the application of kMC CVD modelingwe summarize the approach and associated results for the deposition of diamondin a hot-filament reactor. This technique for growing diamond is commonly ac-complished by injecting a vapor containing H2 and CH4 into the reactor, whereappreciable concentrations of H, CH3, C2H2, and various inert species (such as H2

and CH4, and often impurity species such as O and N which we do not considerhere) are generated by interaction with the hot filament. A simplified depositionreaction mechanism for this system is presented in Table 3. The rates in Equations1 and 3 are modified slightly to include a power law dependence on temperaturefor certain reactions, such that

kf = ATn exp

(− E

RT

)8.

and

kr = αATn exp

(−1S

R

)exp

(−E −1H

RT

). 9.

This mechanism was used previously (43, 53) to simulate diamond depositionand is based on information from several sources (42, 70, 71). Note that with-out a thorough understanding of diamond growth chemistry, the notation and themechanism in Table 3 might appear cryptic, and the reader is referred elsewhere(24, 42, 43, 45, 53) for more detail on the subject.

The notation in Table 3 is as follows. As in the previous example, (s) denotesan atom or molecule adsorbed to the deposition surface, and (g) denotes a speciesin the vapor. The notation (d) represents a species adsorbed to a dimer pair on the{001} surface,∗2(s) is the corresponding biradical on{001}, and C(s) is a carbonin the diamond film (but exposed to the surface). Reactions 1 and 2 representthe abstraction and adsorption, respectively, of H at the surface. Reactions 6–9represent these same processes at adsorbed hydrocarbons, as do Reactions 12 and13 at sites on dimer pairs. Reactions 4 and 5 are the adsorption of methyl (CH3)

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TABLE 3 Chemistry for the deposition of diamond from a precursor of H2 and CH4a

Reaction A n E 1H 1S

1. H(s)+ H(g) ↔ ∗(s)+ H2(g) 1.3× 1014 0 7.3 −9.9 5.3

2. ∗(s)+ H(g) ↔ H(s) 1.0× 1013 0 0 −96.9 −32.8

3. CH2(s)+ H(g) ↔ ∗(s)+ CH3(g) 3.0× 1013 0 0 −24.6 7.9

4. ∗(s)+ CH3(g) ↔ CH3(s) 5.0× 1012 0 0 −70.9 −42.0

5. ∗(s)+ C2H2(g) ↔ C2H2(s) 4.5× 1011 0 6.9 −28.5 −1.9

6. CHy(s)+ H(g) ↔ CHy−1(s)+ H2(g) 2.8× 1014 2 7.7 −11.3 6.6

7. CHy(s)+ H(g) ↔ CHy+1(s) 1.0× 1013 0 0 −83.0 −34.1

8. C2Hy(s)+ H(g) ↔ C2Hy−1(s)+ H2(g) 9.0× 106 2 5.0 −8.9 8.7

9. C2Hy(s)+ H(g) ↔ C2Hy+1(s) 2.0× 1013 0 0 −47.7 −36.2

10. C2Hy(s)+ H(g) ↔ CHy−2(s)+ CH3(g) 3.0× 1013 0 0 −24.6 7.9

11. CHy(s)+ CH3(g) ↔ C2Hy+3(s) 5.0× 1012 0 0 −70.9 −42.0

12. H(d)+ H(g) ↔ ∗(d)+ H2(g) 2.5× 1014 0 7.3 −6.2 6.7

13. ∗(d)+ H(g) ↔ H(d) 1.0× 1013 0 0 −100.6 −34.2

14. CH2(d)+ H(g) ↔ ∗(s)+ CH3(g) 3.0× 1013 0 0 −17.8 8.0

15. ∗(d)+ CH3(g) ↔ CH3(d) 5.0× 1012 0 0 −81.0 −42.2

16. CxHy(d) ↔ ∗2(s)+ CxHy(s) 1.0× 1013 0 0 4.9 0.4

17. ∗2(s)+ CxHy(s)→ C(s)+ Cx−1Hy(s) 2.0× 1013 0 8.8

18. C(s)H2(s)+ H(g)→ ∗2(s)+ CH3(g) 2.5× 109 0 28.3

aA is in units of mol, cm3, and s (see text);E and1H in kcal mol−1; 1S in cal mol−1 K−1. Thermodynamic quantities,1Hand1S, are calculated at the standard state ofT = 1200 K andP = 1 atm.

and acetylene (C2H2), and Reaction 3 is the abstraction of CH2. Reactions 14 and15 are analogous to Reactions 3 and 4 but apply to dimer sites. Reactions 10 and11 address the abstraction and adsorption of hydrocarbons at other hydrocarbons.Reaction 16 is aβ scission reaction that serves to break a dimer bond on the{001}surface (24), and Reaction 17 allows for the subsequent rotation of the adjacenthydrocarbon into the{001} biradical site [∗2(s)]. Reaction 18 represents surfaceetching, i.e., the abstraction of an incorporated (solid) surface carbon. Note thatReactions 5–11 apply to either type of surface site [(s) or (d)].

To simulate the deposition of diamond by the mechanism in Table 3, a diamondcubic lattice in the desired orientation is constructed in the computer. A few layersat one face of the orthorhombic simulation cell are labeled as diamond [C(s)] tocreate a substrate, and periodic boundary conditions are imposed in the directionsparallel to the substrate. All the unsatisfied bonds at the diamond surface are termi-nated with hydrogen [H(s)]. The remaining lattice sites are labeled as vapor. Thegrowth surface is assumed to be in contact with a vapor of constant composition,and, as in the previous example, there is no feedback between the surface reactions

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and the vapor composition. (Coupling the deposition simulation to a calculation ofthe vapor chemistry is certainly possible but is not addressed here.) The simulationproceeds as in the previous example, and when a molecule from the vapor adsorbsto the surface, all of its atoms are explicitly represented in separate lattice sitesabove the surface. Some of the individual atoms (e.g., H) and subcomponents (e.g.,CH2 in C2Hy) in an adsorbed hydrocarbon can react individually with the vapor(see Table 3), and a very limited set of structural transitions (e.g., Reaction 17; seeabove) are allowed for adsorbed molecules. As hydrocarbons adsorb and evolve onthe growth surface, and whenever a situation arises such that any two C(s) atomsare connected to one another by a chain of adsorbed C, all the C atoms in theadsorbed chain are instantaneously converted to C(s). (Because of the diamondcrystal structure and the kinetics of the hydrocarbon reactions in Table 2, theseconverted C chains are always very short, usually containing one or two, and oc-casionally three, C atoms.) Thus Reaction 17 will always lead to the incorporationof C into the film, as will any reaction that deposits a hydrocarbon directly intoa biradical site. (The latter case is not explicitly represented in Table 3 but canarise naturally owing to the local atomic configuration at the adsorption site.) Notethat reactions such as 12–17 in Table 3, which are specific to{001} orientations,are not restricted to{001} substrates because any surface can potentially developlocal configurations (at steps and kinks) that are nearly identical to those on a flat{001} surface.

One of the many advantages of a discrete atomic-scale deposition simulationis its ability to predict atomic surface morphologies. Figure 5 contains such im-ages for simulated diamond deposition on substrates with different orientations.These simulations provide much more information than just surface morpholo-gies, including more common data such as film growth rates. Figure 6 shows thedramatic effect of C2H2 concentration on the growth rates of the different ori-entations. The range of simulated vapor compositions (see Figure 6 caption) isunusual in that it contains less C2H2 than is commonly measured in hot-filamentdiamond CVD systems, but these results serve to illustrate the rich variety ofgrowth behaviors that are produced in a single deposition environment by sim-ply changing the surface orientation and vapor composition. Figure 7 containssimulations performed using a more typical vapor composition and shows a plotof the rates of diamond growth on the three principal surface orientations, alongwith various experimental measurements (72–74) for comparison. The agreementbetween the simulated and experimental data is very good, considering the sim-ulation used no adjustable parameters to fit the magnitudes, or the trends in theorientation dependence, of the measured growth rate. However, it is clear thatthe fine details of the experiments are not faithfully reproduced, and the normalcaveats about uncertainty in the experimental data and the assumptions used inthe reactor and atomic-scale simulations apply. Given these facts, the agreementin Figure 7 is noteworthy. Clearly, results like those in Figures 5–7 would bemuch more difficult to capture using any simulation approach that is not per-formed at the atomic scale, in three dimensions, and on the true diamond crystallattice.

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Figure 5 Surface morphologies from simulations of diamond deposition in a vaporat T = 1100 K andP = 20 torr containing 0.5% H, 0.05% CH3, and 0.005% C2H2.The substrate orientations are (a) {001}, (b) {0 1 10}, (c) {011}, (d ) {0 9 10}, (e) {1 1010}, ( f ) {111}, (g) {9 9 10}, (h) {9 10 11}, and (i) {10 11 11}. Dark gray shading isC and light gray shading is H.

DISCUSSION

In its simplest form, a discrete kMC model of deposition is relatively straightfor-ward to construct and implement. As mentioned above, tackling real-world prob-lems is considerably more difficult but not intractable. However, it is important

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Figure 6 Simulated diamond growth rates on different substrateorientations as functions of C2H2 concentration. The vapor is atT =1100 K andP = 20 torr, and contains 0.5% H and 0.05% CH3.

to bear in mind the many drawbacks and limitations of this approach, and some ofthe more important of these are discussed below.

First, it is very difficult, even using the most sophisticated quantum mechanicalapproaches, to accurately estimate chemical reaction rate parameters. Thus anydeposition mechanism is, by necessity, approximate at best. It is also virtuallyimpossible to know a priori all the reactions (even just the important ones) thatcan occur at the surface in a given deposition system. Therefore, any reactionmechanism represents a limited and simplified version of the real chemistry.

Second, performing a simulation of this type on a fixed lattice makes the prob-lem tractable, but it also neglects a whole host of factors that might be critical to thedeposition process. For example, adsorbates are confined to the small number ofconfigurations offered by the simulation lattice near the adsorption site. [However,there are hybrid approaches (31, 32) that alleviate this shortcoming.] Although thereaction mechanism can be constructed to include as much detail as is available,steric and other local effects on the reaction kinetics are difficult to robustly incorpo-rate into a model of this type. Furthermore, the use of a lattice is not conducive to thetreatment of defects, e.g., surface relaxations, surface reconstructions, grain bound-aries, twins, and dislocations, which are sometimes crucial to the growth behavior.

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Figure 7 Simulated and measured diamond growth rates as func-tions of temperature. The vapor is atP = 20 torr and contains0.25% H, 0.01% CH3, and 0.25% C2H2.

Third, the diamond CVD example presented here is perhaps one of the moststraightforward CVD systems to simulate, primarily because the molecules in-volved are relatively simple and because a considerable database of hydrocarbonchemistry is already available from combustion experiments. (A similar argumentcan be made for simulating the deposition of Si from SiH4.) However, the vastmajority of technologically relevant CVD systems are much more complex, and inmost cases the basics of the reaction mechanisms are unknown. The situation is ingeneral somewhat better for PVD systems; however, the challenge in simulatingCVD at the atomic scale is often not in developing the deposition algorithm butrather in constructing a meaningful chemical reaction mechanism.

ACKNOWLEDGMENTS

CCB acknowledges the support of the United States Department of Energy un-der Contract DE-AC04-94AL85000. DJS would like to thank the United StatesDepartment of Energy, DE-FG02-01ER54628 for continuing support.

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The Annual Review of Materials Researchis online athttp://matsci.annualreviews.org

LITERATURE CITED

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17 Jun 2002 8:8 AR AR162-12-COLOR.tex AR162-12-COLOR.SGM LaTeX2e(2002/01/18)P1: GDL

Figure 2 A two-dimensional depositionsimulation cell.Graysites are substrate,redare adsorbates, andwhiteare vapor.

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Figure 3 A progression of reactions that leads to growth on the substrate depicted inFigure 1 in the vapor described by Table 1, subject to the chemistry in Table 2.t is thesimulation time.Gray sites are substrate,redsites are B(s) adsorbates,greensites areradical *(s),purplesites are AB2(s) adsorbates, andwhiteareas represent vapor.

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corbett c. battailestaff.ustc.edu.cn/~fuzp/course/paper/kinetic monte carlo...corbett c. battaile...

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