Multiscale modeling of chemical vapor deposition
Seth T. Rodgers and Klavs F. Jensena)
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139
~Received 2 July 1997; accepted for publication 26 September 1997!
An integrated method for modeling chemical vapor deposition on length scales ranging from
microns to meters has been developed. The macroscale problem of flow and transport in a single
wafer, low pressure chemical vapor deposition reactor is solved using the finite element method.
However, on the feature scale, continuum models of flow and transport are not valid and discrete
particle transport models must be employed. The two transport regimes, continuum and discrete
particle, are linked by an effective reactivity function e which includes both effects of multiscale
surface heterogeneity and microscale transport resistance. A hybrid ballistic transport, Monte Carlo
method, is developed permitting calculation of e for any set of reaction pathways occurring over
microelectronic device features of any geometry. Surface topography, that is, feature scale
calculations, are combined to yield an effective reactivity map over the surface of the substrate. This
map is subsequently used to formulate a flux boundary condition on the macroscopic model.
Iteration between macroscopic and microscopic models is used to assure a consistent set of
conditions at the macro–micro interface. The resulting linked models are used for investigation of
local and reactor wide loading effects. Matched macroscopic solutions provide inputs for profile
evolution calculations during fill of trenches and vias. © 1998 American Institute of Physics.
@S0021-8979~98!04401-6#
I. INTRODUCTION
Chemical vapor deposition ~CVD! is a complex operation with key processes occurring on widely separated length
and time scales. At each of the length scale regimes important in CVD, a particular set of modeling techniques and
assumptions is appropriate ~see Fig. 1!.1 Frequently though,
processes of different scales interact, and thus many problems of interest cannot be cast within the context of a single
scale.2,3 This work focuses on interactions between macroscopic flow and transport, where the gas phase may be regarded as a continuum, and phenomena at the micron length
scale, where the discrete nature of the gas phase becomes
apparent.
For most CVD processes, flow and transport in the reactor at large can be described by macroscopic conservation
equations for momentum, mass, and energy. The resulting
coupled nonlinear partial differential equations are then
solved with the finite element method ~FEM!. However,
these continuum models are not valid as length scales approach and shrink below the gas phase mean-free path l.
This is an important consideration since microelectronic device features are frequently submicron, while l under low
pressure CVD conditions can be several hundred mm. This
limitation on the use of continuum models is widely
recognized4,5 and has led to the development of discreteparticle transport models using both ballistic integral equations and Monte Carlo ~MC! techniques.
Assuming sufficient computational resources are available, MC models may be formulated as direct or near-direct
analogs to the physical processes of interest. While this generally results in models that are expensive relative to other
a!
Electronic mail: kfjensen@mit.edu
524
J. Appl. Phys. 83 (1), 1 January 1998
strategies, considerable physical detail may be easily included. MC methods are thus well suited to systems that are
difficult to describe with deterministic sets of equations, but
whose physical nature is well known. Monte Carlo models
have been used for some time to address discrete particle
transport problems, especially rarefied gas flow and neutron
transport.6,7 Within the context of CVD models, MC methods have been employed to examine precursor transport inside microelectronic device features.8,9 More recent efforts
have employed MC models to simultaneously examine feature scale transport and the microstructure of deposited thin
films.10,11 While MC methods have seen widespread acceptance, many problems do not require all the detail available
in a MC calculation. In these cases, MC models are usually a
poor choice if the desired calculation can be preformed by
other means.
The slow convergence of MC methods has led to formulation of ballistic integral based methods. These methods ignore detail at scales below the micron scale, and neglect
collisions in the gas phase, modeling molecular transport as a
line-of-sight process. Thus, these models focus on processes
at the feature scale, and calculations are much faster than
MC simulations. These methods have been successfully applied by several investigators4,12 to predict CVD film profiles. Although both MC and ballistic integral based models
have enjoyed success at the feature scale, they are ~a! not
appropriate ~ballistic integral! or ~b! too expensive ~MC! for
use at the reactor scale.
Because use of both MC and ballistic integral techniques
is confined to scale, both types of models require descriptions of the macroscopic environment, such as temperature
and concentration profiles as inputs. Hasper et al.13 addressed this problem for tungsten deposition from both
WF6 /H2 and WF6 /SiH4. After solving the macroscopic con-
0021-8979/98/83(1)/524/7/$15.00
© 1998 American Institute of Physics
Downloaded 08 Aug 2001 to 18.63.1.162. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp
FIG. 2. Schematic of the control surfaces for the micro- and macroscale
interface boundary condition @Eqs. ~2!–~4!#.
FIG. 1. Schematic illustrating the coupling of micro- and macroscale phenomena in chemical vapor deposition processes.
servation equations to find conditions at the substrate surface, they performed feature scale calculations to determine
step coverage. However, no information was fed back to the
reactor scale simulation, resulting in a ‘‘sequential model’’
approach that does not treat the time-dependent effects of
microscale heterogeneity on the reactor at large. In cases
where reactor scale and feature scale processes interact, obtaining the correct inputs for feature scale simulation
amounts to finding a consistent set of conditions at the border between macroscopic and microscopic models.
Multiscale resolution is difficult since in many CVD operations, deposition occurs over integrated circuit device layouts that are heterogeneous patchworks on length scales
ranging from centimeters to microns. For example, areas
where device features are dense may border areas with no
features, or masked regions may abut active deposition areas.
Each region may present a somewhat different reactive surface area to film precursors, giving rise to concentration gradients, i.e., loading effects, at the edges of pattern fields.2,3
Efforts to simultaneously resolve both reactor and feature scale environments using a mesoscale model as an intermediate, have been undertaken by Cale et al.2 In their
work, the macroscale problem is solved with FEM, and mesoscale models are created by local refinement of the finite
element mesh in the vicinity of clusters of identical features.
The effects of feature scale heterogeneity are treated with an
averaging scheme based on asymptotic analysis. Microscale
transport resistance is treated with a generalized Damkoehler
number. Some difficulty with this approach may be encountered, e.g., on an actual deposition surface, the reaction area
will vary on several length scales with the type of surface, as
well as the geometry of features present. Averaging schemes
would have to be redone at each scale of detail present, and
J. Appl. Phys., Vol. 83, No. 1, 1 January 1998
at each level decisions must be made as to what information
to retain. Finally, arguments based on reactive area are not
sufficient to completely describe the effects of feature scale
heterogeneity, as microscale effects that restrict access of
precursors to the growth interface, such as ‘‘shadowing,’’
must be considered.
The linking method presented here is based on determining the number and nature of encounters that molecules entering from the macroscopic environment have, on average,
with the substrate surface. This concept of feature scale precursor transport and reaction leads to the introduction of an
effective reactivity function e. It is calculated with a hybrid
MC approach, and includes the effects of feature scale heterogeneity and transport resistance. Feature scale simulations
are used to construct a macroscopic map of e over the substrate surface. After accounting for feature scale topography
in this fashion, the reactor scale problem is solved with
FEM. Macro- and microscale results are then matched iteratively to assure a consistent set of conditions across the
macro–micro interface. CVD of tungsten from WF6 reduced
by H2 serves as a case study. A blanket W film is deposited
over fields of trenches and vias. The presence of patterned
areas is seen to affect both local and reactor-wide concentration profiles.
II. MULTISCALE MODEL
A. Problem structure and model formulation
At the macroscopic level, continuum descriptions of
fluid flow, heat transfer, and mass transfer apply. For the
dilute gas systems common in CVD, only the transport of
chemical species is affected by microscale heterogeneity, and
is thus the only problem to require multiscale resolution. The
macroscopic species balance is14
]
~ r Y k ! 1“• ~ r ûY k ! 52¹•jk1G k .
]t
~1!
Here, Y k represents the mass fraction of species k in the gas
phase, jk is the diffusion flux, and G k is the generation term
in the gas phase. The flow velocity û and the temperature are
determined by FEM solutions of the momentum and energy
conservation equations.1,14 The density r is found from the
ideal gas law.
Microscale heterogeneity affects the macroscopic solution through the boundary conditions imposed at reactive
surfaces. A mass balance may be written for each species k
at the surface of the heterogeneous substrate having surface
area à ~Fig. 2!.
S. T. Rodgers and K. F. Jensen
525
Downloaded 08 Aug 2001 to 18.63.1.162. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp
E
E
~2!
r s dÃ5 ˜ j k dÃ,
˜A k
A
where r sk represents the local surface reaction rate and j k the
local reagent flux out of the macroscopic environment. However, when heterogeneity is present at scale below l, the
expression in Eq. ~2! cannot be combined with a continuum
model of chemical species transport. Introducing a control
surface having surface area A just above the substrate results
in another mass balance:
E
r s dÃ5
˜A k
E
A
j k dA5
Ee
A
k P rh~ x ! f k dA,
~3!
where f k represents the total flux of molecules crossing the
control surface. The product e k P rh(x) represents the probability that a molecule crossing surface A will be incorporated into the growing film before escaping back to the bulk
gas phase. P rh is a reference probability for incorporation on
a homogeneous surface. Thus, e k represents the ratio between the actual and reference incorporation probabilities.
Although gas phase reactions have been neglected in Eq. ~3!,
they may easily be included. Equating the fluxes at the control surfaces, one obtains
j k 5 e k P rh~ x ! f k .
~4!
Note e k is thus viewed from the macroscopic environment as
the ratio between the local fluxes to heterogeneous and homogeneous surfaces. This flux term now becomes the boundary condition on the macroscopic finite element calculation.
In order for the boundary condition represented by Eq.
~4! to be useful, variations in e must be resolvable on a finite
element mesh. That is, e must include information from all
scales where continuum transport models do not apply, but
must itself be compatible with a continuum approach. The
distinction between microscale and macroscale transport is
made based on the local Knudsen number, Kn•Kn represents
the ratio of l to L, the distance over which flow conditions
change. When Kn is !1, the surrounding gas phase may be
regarded as only slightly perturbed from collisional equilibrium. In this situation species transport through the gas phase
is well described as a combination of thermal and ordinary
diffusion.14 Where surface heterogeneity causes the local environment to vary abruptly, transport and reaction are summarized with effective reactivity functions, which, in turn,
are fed to the macroscopic model.
The calculation of the effective reactivity function at a
point S above the substrate is described in Fig. 3. Statistics
are recorded for some number b of test trajectories initiated
at S. The history of each test trajectory yields an estimate of
e (S,t), the effective reactivity of the surface with respect to
a molecule at position S. Each simulated trajectory a will
consist of k steps ~collisions!. The locations of these collisions are stored as the x i and define the test particle’s history,
Eq. ~5!.
a 5 ~ x 1 ,x 2 ,...,x k ! .
~5!
Trajectories are assumed to terminate, either by surface reaction or escape to the macroscopic environment, within
526
J. Appl. Phys., Vol. 83, No. 1, 1 January 1998
FIG. 3. Schematic displaying the simulation procedure for surface encounters of given a particle with initial location, S. Statistics from many test
particles determine the effective reactivity of the substrate surface when
viewed from point S.
some finite k. Note e can be expressed as a random variable
on the space of all possible trajectories a. The expected
value of e E( e ) is given in Eq. ~6!.
E~ e !5
(a P ~ a ! e ~ a ! .
~6!
This expectation value of e represents the average environment seen by particles entering from the macroscopic gas
phase.
Each collision may contribute differently to a history and
thus, e. For example, gas phase collisions with inert molecules may change a trajectory’s direction, while surface interactions might influence both direction and reactive history. For a given trajectory a,e~a! is given by the following
expression:
e~ a !5
( ki51 P r ~ x i !
P rh
~7!
.
Throughout the course of the trajectory, if a collision occurs,
the local probability of reaction, P r (x i ) is added to the history. At each collision a random number f is compared to
P r (x i ), if it is less than P r (x i ), the particle is said to have
reacted and a new trajectory begins. A reference reaction
probability P rh is obtained by comparison with a homogeneous surface under the same macroscopic conditions.
To simplify subsequent arguments, free molecular flow
will be assumed at the feature scale, reducing the space of
possible trajectories by excluding gas phase collisions.
Deposition kinetics are taken to be first order decomposition
of a single precursor, resulting in constant reaction probability over the substrate surface. Thus, all surface encounters
are equally productive. Under these conditions, e can be
given a geometric interpretation, it is the effective area of the
growth interface exposed to reactive precursors, after accounting for microscale transport resistance. If the system of
interest does not exhibit simple chemistry, due to fluxdependent reaction probabilities, or the presence of multiple
reactions, reaction probability may be expressed in the style
of Cooke and Harris.15
S. T. Rodgers and K. F. Jensen
Downloaded 08 Aug 2001 to 18.63.1.162. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp
For a collisionless gas phase, examination of the initial
velocity of each test particle will decide if the test particle
will impinge inside a feature. Since only particles that enter
surface features may experience multiple collisions, only
these histories need be simulated. The estimator for e above
can now be rewritten:
E ~ e ! 511
b*
b
HF(
a
G J
P * ~ a ! g ~ a ! 21 ,
~8!
where P has been changed to P * to denote simulation only
of the fraction of trajectories from a Maxwellian source,
( b * / b ), that intersect the mouth of a feature, and thus might
undergo multiple surface collisions. The number of surface
collisions on a trajectory, a is g~a!. The term in square
brackets is then the average number of collisions a particle
entering the feature is expected to suffer. The advantages of
this formulation are twofold. For collisionless flows, the
view factor term is easily computed analytically for most
geometries or can be done as a MC integral, if analytic methods prove inconvenient. Because the fraction of molecules
entering the feature depends only on feature geometry and
source position, the feature scale transport problem is uncoupled from the surface chemistry. Secondly, if re-emission
is assumed to be independent of incident trajectory, e.g., cosine re-emission, the re-emission term depends only on feature geometry and reaction kinetics. In cases where angular
distributions for the re-emitted molecules are known a priori,
and gas–gas collisions are rare, as in thermal CVD, low
discrepancy Sobol sequences can be employed to considerable advantage for variance reduction.16
B. Mapping microscale to the macroscale
The effective reactivity of a surface with several features
may be computed with the following approach, exemplified
for a via of aspect ratio 3 ~see Fig. 4!. Beginning with a
single feature, an effective reactivity map is found by generating 102 – 103 test particles from several hundred map
points. If the reference value of e, e 0 is subtracted from this
map, the difference represents the marginal contribution of
the feature to the effective reactivity. Once individual feature
scale e functions have been generated for each feature geometry present, they can be superimposed, serving as ‘‘building
blocks’’ to map the effective reactivity of heterogeneous
fields of features to the reactor scale.
The resulting reactivity maps must meet two criteria.
First, the MC results must exhibit variance, or ‘‘scatter,’’
small enough to allow superposition. Equation ~9! was taken
as an estimate of the error d in e.17
d5
A
E ~ e 2 ! 2 ~ E ~ e !! 2
.
b
~9!
Sufficiently small errors were achieved at most map
points using less than 23103 test particles. Second, as mentioned previously, gradients in e must be resolvable on a
finite element mesh to allow formulation of a flux boundary
condition on the continuum model, Eq. ~4!. For collisionless
flows, these gradients in e are dictated by the starting height
h of the test trajectories shown previously in Fig. 3. The
source plane should be located at the smallest h, where graJ. Appl. Phys., Vol. 83, No. 1, 1 January 1998
FIG. 4. Example of the superposition of feature scale calculations for construction of macroscale effective reactivity maps.
dients in e become resolvable upon superposition ~any closer
and the continuum equations would be invalid! while a
source farther away would result in simulation of a large
number of gas phase collisions, and thus excessive computational effort. In this work, the source plane height was
taken to be at 0.33l. Use of a ballistic transport integral to
determine ( b * / b ) shows that gradients in e will have a scaling length of about 20h or just less than 7l. While this is
very near the limit of a continuum formulation,17 it should be
noted that collisionless flows present the steepest gradients
possible, and as such represent a ‘‘worst case’’ picture of
loading effects. If gas–gas collisions are allowed, gradients
in e will be smoothed, facilitating representation of e on a
finite element mesh.
Given e to complete the boundary condition in Eq. ~4!,
the macroscale model can now be solved with FEM to provide concentration and temperature profiles to serve as inputs
for the microscale simulation. Iteration between the two domains will produce mutually consistent micro- and macroscopic environments, allowing tracking of feature scale morphology and its effects on macroscopic quantities of interest
throughout the course of deposition. The resulting algorithm
for consistent multiscale simulations is shown in Fig. 5. In
the example below, the majority of simulation time was
spent in calculations at the feature scale. Approximately 30
min was required for profile evolution at each time step, for
each trench and 150 min per via monitored. Computing e
functions required about 1 h for each feature geometry
present. Computations were executed on an IBM workstation
capable of 22 Mflops, and could be parallelized to run considerably faster than the figures reported here.
S. T. Rodgers and K. F. Jensen
527
Downloaded 08 Aug 2001 to 18.63.1.162. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp
FIG. 5. Flowchart of procedure for matching feature scale and macroscale
results in an iterative fashion.
III. RESULTS AND DISCUSSION
Although deposition of tungsten from hydrogen-reduced
tungsten hexaflouride has been chosen as a case study, in
general, the methods presented here are suitable for any
CVD problem requiring linked treatment of transport at the
continuum and discrete particle levels. We have chosen CVD
of tungsten because it is relatively familiar, in particular because information on typical deposition conditions is available in the literature.15
The operating conditions for this example were chosen
to be typical for CVD tungsten with a few modifications.
Temperature was taken to be 750 K, this is at the high end of
the range of temperatures commonly used, resulting in
poorer than usual film conformality. This choice was made
to illustrate the importance of microscale transport resistance. Total pressure was 1 Torr, the mole fraction of WF6 at
the inlet was taken as 0.005, with the remaining carrier gas
90% H2 and 10% Ar. As the goal of this work was to develop a linking methodology, and not to explore tungsten
chemistry, we chose a simplified kinetic model for WF6 /H2
deposition from the considerable body of work available.
Deposition was modeled with a single heterogeneous decomposition reaction, and the rate of reaction was calculated with
an expression of the form proposed by Charra and Cale:18
WF613H2→6HF1W,
S
D
0.5
PH
P
8300
2 WF6
R s 52.0 exp 2
.
T
111000P WF6
~10!
~11!
The chosen partial pressure of WF6 is somewhat lower than
that normally used, as we wished to minimize the error in
treating the kinetics as first order at the feature scale.
A single wafer axisymmetric reactor, consistent with
current CVD reactor designs, is chosen as the macroscale
system. The substrate is a 200 mm diameter wafer, with two
annular fields of features chosen to provide microscale heterogeneity typical of microelectronic device features. Field 1
528
J. Appl. Phys., Vol. 83, No. 1, 1 January 1998
FIG. 6. Computed WF6 profiles at the start of deposition. ~a! Over flat
wafer, ~b! over patterned wafer as predicted with effective reactivity calculations, ~c! over patterned wafer with microscale transport resistance neglected, i.e., employing a simple ‘‘exposed area’’ concept.
is comprised of 0.33 mm wide trenches spaced 0.33 mm
apart. This field covers the substrate from r517 to 33 mm.
The first half of field 1 consists of aspect ratio 3 trenches, the
second half consists of aspect ratio 2 trenches. This geometry
highlights the effects of microscale transport resistance, i.e.,
when reaction probabilities are high, there is little difference
between deep and shallow trenches. Both appear as ‘‘reagent
sinks’’ when viewed from the macroscopic environment.
Field 2, ranging from r550 mm to r570 mm, is composed
of aspect ratio 3 trenches interspersed with lines of aspect
ratio 3 vias. The layout of this field illustrates calculation of
e over features of varying geometry; in particular, the combination of fully three-dimensional effective reactivity maps
over circular vias and with the two-dimensional maps obtained over long trenches.
The results of linked model calculations for WF6 profiles
at the start of deposition are shown in Fig. 6. Curve A is the
solution obtained for an unpatterned, ~homogeneous! substrate, and is shown for reference. Curve B represents the
WF6 concentration profile at the start of deposition. Not only
do patterned regions deplete reagents significantly faster than
flat areas, the growth rate over the entire wafer surface is
depressed, as observed in the work of Kleijn et al.19 for
silane-reduced tungsten.
Film uniformity suffers considerably with the introduction of microscale heterogeneity. Maximum growth rate differences are approximately 25% for the patterned substrate
B, compared to less that 1% for the flat substrate represented
by curve A. Furthermore, considerable growth rate variation
~about 15%! is seen from the center to edge of feature fields.
The importance of microscale transport resistance is apparent
when comparing curve B and curve C. Curve C derives from
a ‘‘simple perimeter’’ concept to account for the additional
surface area offered by device features. By neglecting microscale transport resistance, precursor consumption and
loading effects are overestimated. Another example of microscale transport resistance is shown in the shape of curve
B. Note that the first field is nearly symmetric about its midS. T. Rodgers and K. F. Jensen
Downloaded 08 Aug 2001 to 18.63.1.162. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp
FIG. 7. Time evolution of computed WF6 profiles. ~a! Over flat wafer, ~b!
over patterned surface at the beginning of deposition, ~c! over patterned
surface at the end of deposition.
point, r525 mm, in spite of the inner half-trenches being
composed of aspect ratio 3, while the outer trenches are aspect ratio 2.
As deposition progresses, features are filled with deposited films, and the effective area presented to the macroscopic environment changes. The time evolution of WF6
concentration profiles appears in Fig. 7. The circular vias in
the outer field begin to close rapidly, causing the outer field
to quickly resemble a flat substrate.
FIG. 9. Calculated feature profiles at the end of deposition ~outer field!.
Film thickness at field edges is roughly 110% of field center. Clockwise
from upper left: ~a! aspect ratio 3 via at r550 mm; ~b! aspect ratio 3 via at
r560 mm; ~c! aspect ratio 3 trench at r550 mm; ~d! aspect ratio 3 trench at
r560 mm.
Examination of feature profiles from field 1 at the end of
deposition shows that very little flux actually penetrates to
the bottoms of the aspect ratio 2 or 3 features. The features
behave as ‘‘sinks,’’ and thus the extra area offered by the
deeper features is not ‘‘seen’’ by the macroscopic environment ~see Fig. 8!.
Postdeposition profiles from field 2 also show a significant difference—roughly 10% from center (r560 mm) to
the edge (r550 mm). Shadowing is particularly pronounced
in the vias, which received almost no bottom or sidewall
deposition ~see Fig. 9!.
IV. CONCLUSIONS
FIG. 8. Calculated feature profiles at the end of deposition ~inner field!.
Film thickness at field edges is roughly 115% of that at the field center.
Clockwise from upper left: ~a! aspect ratio 3 trench at r517 mm; ~b! aspect
ratio 3 trench at r525 mm; ~c! aspect ratio 2 trench at r525 mm; ~d! aspect
ratio 2 trench at r533 mm.
J. Appl. Phys., Vol. 83, No. 1, 1 January 1998
A method for linking continuum and discrete particle
transport models has been developed. The resulting linked
model can be used to investigate interactions between reactor
scale and feature scale processes in low pressure CVD. Such
interactions affect film conformality and growth rate on both
the microscale and over the wafer as a whole, thus exerting a
large influence on the quality of finished devices. The linking
algorithm presented employs an effective reactivity function
e, which is computed through a hybrid MC method. While
submicron trenches and vias were taken as typical structures
for this work, a map of e can be calculated for any feature
suitable for MC simulation. Calculations at the feature scale
include microscopic transport resistance, which cannot be
treated by continuum models because of the particulate nature of the gas phase at the micron length scale. Under conditions of free molecular flow at feature scale, e calculations
S. T. Rodgers and K. F. Jensen
529
Downloaded 08 Aug 2001 to 18.63.1.162. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp
over individual features can be superimposed to yield maps
over macroscopic fields of features. In this fashion, the effective reactivity of an entire substrate can be mapped. The
effective reactivity concept presented here can be applied to
other CVD processes where feature scale heterogeneity is
important at the macroscale, such as selective growth. Treatment of collisional flows is also possible within the framework presented, but collisionless calculations are valuable as
a worst case picture of loading effects, in the absence of
smoothing of the e functions resulting from gas phase collisions.
ACKNOWLEDGMENTS
The authors thank Ajit Balakrishna ~MIT!, Dan Coronell
~Motorola!,and Sadasivan Shankar ~Intel! for technical discussions, and the Semiconductor Research Corporation for
financial support.
1
D. G. Coronell and K. F. Jensen, J. Computer-Aided Materials Design 1,
3 ~1993!.
2
M. K. Gobbert, C. A. Ringhofer, and T. S. Cale, J. Electrochem. Soc. 143,
2524 ~1996!.
3
L. Buydens, P. Demeester, M. Van Ackere, A. Ackaert, and P. Van Daele,
J. Electron. Mater. 19, 317 ~1990!.
530
J. Appl. Phys., Vol. 83, No. 1, 1 January 1998
4
T. S. Cale, G. B. Raupp, and T. H. Gandy, J. Appl. Phys. 68, 3645
~1990!.
5
J. C. Rey, L. Y. Cheng, J. P. McVittie, and K. C. Saraswat, J. Vac. Sci.
Technol. A 9, 1083 ~1991!.
6
J. Spanier and E. M. Gelbard, Monte Carlo Principles and Neutron Transport Problems ~Addison-Wesley, Reading, Massachusetts, 1964!.
7
G. A. Bird, Astronaut. Aeronaut. 74, 239 ~1981!.
8
M. Ikegawa and J. Kobayashi, J. Electrochem. Soc. 136, 2982 ~1989!.
9
H. C. Wulu, K. C. Saraswat, and J. P. McVittie, J. Electrochem. Soc. 138,
1831 ~1991!.
10
T. Smy, S. K. Dew, and M. J. Brett, MRS Bull. XX, 65 ~1995!.
11
P. J. Stout and M. J. Kushner, J. Vac. Sci. Technol. A 11, 2562 ~1993!.
12
M. Mazhar Islam Raja, M. A. Capelli, J. P. McVittie, and K. C. Saraswat,
J. Appl. Phys. 70, 7137 ~1991!.
13
A. Hasper, J. Holleman, J. Middelhoek, C. R. Kleijn, and C. J. Hoogendoorn, J. Electrochem. Soc. 138, 1728 ~1991!.
14
C. Kleijn, Computational Modeling in Semiconductor Processing, edited
by M. Meyappan ~Artech House, Boston, 1995!.
15
M. J. Cooke and G. Harris, J. Vac. Sci. Technol. A 7, 3217 ~1989!.
16
W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. H. Flannery,
Numerical Recipes ~Cambridge University Press, Cambridge, 1992!.
17
G. A. Bird, Molecular Gas Dynamics ~Oxford University Press, Oxford,
1995!.
18
T. S. Cale and M. B. Chaara, Thin Solid Films 220, 19 ~1992!.
19
J. Holleman, A. Hasper, and C. R. Kleijn, Advanced Metallization for
ULSI Applications, edited by V. V. S. Rana, R. V. Joshi, and I. Ohodomari
~Mater. Res. Soc. Proc. ULSI-VII, Pittsburgh, PA, 1992!, p. 241.
S. T. Rodgers and K. F. Jensen
Downloaded 08 Aug 2001 to 18.63.1.162. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp