Pergamon
PII: S1359-6454(96)00256-X
A MONTE
CARLO SIMULATION
OF THE PHYSICAL
VAPOR DEPOSITION
OF NICKEL
Y. G. YANG, R. A. JOHNSON
Department
Acta mafer. Vol. 45, No. 4, pp. 1455-1468, 1997
0 1997 Acta Metallurgica
Inc.
Published by Elsevier Science Ltd
Printed m Great Britain. All rights reserved
1359-6454197 $17.00 + 0.00
of Materials
Science
and H. N. G. WADLEY
and Engineering,
University
U.S.A.
(Received
4 March
1996; accepted
of Virginia,
Charlottesville,
VA 22903.
15 July 1996)
Abstract-A
two-step Monte Carlo method for atomistically
simulating
low energy physical vapor
deposition processes is developed and used to model the two-dimensional
physical vapor deposition of
nickel. The method consists of an impact approximation
for the initial adatom adsorption
on a surface
and a multipath
diffusion analysis to simulate subsequent
surface morphology
and interior atomic
structure evolution. An embedded atom method is used to determine the activation energies for each of
the many available diffusional paths. The method has been used to predict the morphology/structure
evolution of nickel films over the length and time scales encountered
in practical deposition processes.
The modeling approach has enabled determination of the effect of vapor processing variables such as flux
orientation,
deposition rate and substrate temperature
on deposit morphology/microstructure
as defined
by packing density, surface roughness and growth column width (which appears closely related to grain
size). Several aspects of the empirical MovchanDemchishin
structure zone model are well predicted by
this approach.
fi 1997 Acta Metallurgica Inc.
1. INTRODUCTION
vapor
deposition
(PVD) via either electron
beam
evaporation
[l], resistive
heating
[2] or
sputtering [3] is widely used for depositing metal films
at low to intermediate
rates (< 1 pm/min). Interest is
growing in the development
of potentially very high
rate (l-100 pm/min)
physical
vapor
deposition
processes such as jet vapor deposition (JVDTM) [4-S]
and directed
vapor deposition
(DVD) [9, lo] for
synthesizing
metal multilayers
with minimal interlayer diffusion. In all these processes, the atomic flux
(or equivalently
the deposition
rate),
the flux
incidence angle, the incident atom kinetic energy and
the substrate temperature
can all be independently
varied. Since these parameters
govern the kinetic
phenomena
involved in the atomic assembly and
reconstruction
of surfaces during film growth, many
options are available for controlling the morphology/
microstructure
of a deposit
during vapor phase
manufacturing.
The many variables of the processes
make it difficult to identify the conditions that result
in acceptable morphologies/microstructures.
The prediction of these morphologies/microstructures, and their dependence
on material
system
properties and process parameters
has proven to be
very difficult. The structures predicted by classical
arguments [l l] (e.g. layer-by-layer
or Frank-van
der
Merwe [12], island or Volmer-Weber
[13], and the
layer-plus-island
or StranskiLKrastanov
[ 141 growth
modes) are normally
applicable
only to the early
stages of the nucleation
and growth of epitaxy.
However, the empirical Structure Zone Model (SZM)
Physical
of Movchan,
Demchishin
and Thornton
[15. 161
indicates
that,
depending
upon
the deposition
temperature,
the actual microstructures
consist of
either tapered crystals with domed tops separated by
voided growth boundaries (zone I), columnar grains
separated by metallurgical grain boundaries (zone II),
or equiaxed grains corresponding
to a fully annealed
structure (zone III). These microstructures
may also
be twinned, contain stacking-fault
structures [17, 181
or other types of defected structures [19, 201.
This research contributes
to the development
of
an atomistic-based
modeling approach
that might
eventually
be used to identify optimal deposition
conditions
for any material
system of interest.
Thus, we seek a model that, when given a substrate
temperature,
a deposition
rate, an incidence angle,
the adatom
kinetic energy, a substrate
geometry
and a material
system as inputs, is capable
of
predicting
the morphology/microstructure
of the
resulting deposit. As a start, this work concentrates
on low incident energy processes such as evaporation
in which incident atoms have only thermal energy.
Both Molecular Dynamics (MD) [21] and Monte
Carlo (MC) [22] methods are able to simulate some
aspects of the microstructure
of a vapor deposition
process,
Each method
has its advantages
and
disadvantages
[21-231. In MD calculations,
the
atomic configuration
of the deposit is represented by
the coordinates
and velocities of the atoms and the
dynamics
are completely
determined
by these
coordinates
and the interatomic
potential
function
that is used to represent
the interaction
between
simulations
of
the atoms [21]. The model permits
1455
1456
YANG et ~1.: MONTE CARLO SIMULATION
granular structure [24], epitaxial growth [25], twin
formation
[24], and stress development
[26, 271
and their sensitivity to the diffusion processes [28],
energetic bombardment
modification
[27729]. However, since the lattice atoms vibrate in this approach,
the forces on atoms must be calculated several times
per lattice vibration period (i.e. every femtosecond
or so). Because of this, computations
in reasonable
times (less than
several
hours
of workstation
computation)
can only be conducted
for problems
with a limited physical elapsed time and a small
system size. Generally, only systems with less than
several thousand atoms deposited in a period of less
than a nanosecond
or two can be simulated. This
results in the simulation of a vapor deposition process
with unrealistically
high deposition rates of l-10 m/s
[24, 281.
While MD methods are deterministic,
the many
variants of the Monte Carlo method are based on
probabilities.
Among those addressing
low energy
processes, all use a form of Henderson’s
model [30]
to deduce the initial adatom configuration.
Some
strictly follow the Metropolis procedure in which the
atoms are moved in accordance
with Boltzmann
statistics
but the kinetic
path of evolution
is
physically
meaningless
[31, 321. Others
follow a
possible path taken by the system and generate a
more
correct,
usually
nonequilibrium
structure
[20, 3040].
Regardless
of approach,
the common
feature is that the detailed interatomic forces need not
be evaluated
and the movements
of atoms are
determined
by a set of statistical rules established
beforehand.
The benefits of this method are that
simulations
can be carried out rapidly and a very
large number of atoms can be simulated in relatively
short periods of time depending
upon the type of
algorithm used. MC methods have been applied to
studies of condensation
phenomena
[37], the occurrence of anisotropy
[30], the scaling properties
of
vapor deposition [35], step coverage in metallization
processes
[40,41] and the reproduction
of the
columnar
structure
in vapor deposition
processes
the MC method
also
suffers
[20]. However,
limitations;
in particular,
it imposes
severe constraints on the crystal structure. In most simulations,
an Ising model [42] is used in which a crystal lattice
is selected in advance,
and the allowed atomic
configurations
are described
by specifying
the
occupancy
of each crystal
lattice site. In this
approximation
it is assumed
that crystal growth
only involves a very small subset of all the possible
atomic configurations.
Because of this, it cannot
then model amorphous
systems, dislocations,
twins,
grain orientations
and other problems (e.g. stress)
involving
atoms
that are not on bulk lattice
sites [22, 231. Another limitation of the model as it
used
arises
from
the sometimes
is normally
extreme approximations
made to simplify diffusional
processes.
This has resulted in the absence of a
connection
between the simulation and the physical
OF PHYSICAL VAPOR DEPOSITION
time scale and real temperature
of the growth
processes [21].
Although MD and MC applications
can be used
to explore
many important
features
of vapor
deposition
processes, their use to identify the best
practical process parameters
is still difficult. A start
toward this end was made by Mi_iller [34], who
sought to introduce meaningful temperature and time
scales into a MC approach. Although his approach
was capable
of identifying
a transition
between
zone I and zone II structures defined in the SZM
model, it suffered from a number of deficiencies.
For example, the bond counting method used to
assess the jump’s activation energy was too simple
and could not take the Schwoebel barrier [43] into
account. Also, atomic jump decisions were made by
comparing
the thermal
energy
with the bond
counting
activation
barrier and always making a
possible
jump
into the site with the highest
coordination.
In addition, a proper link between the
real deposition
time and the diffusive process time
was not made. Another
effort toward realistically
modeling
higher energy sputtering
processes
has
been made by Fang and Prasad [41] in which a
hybrid numerical scheme was used to account for the
entire deposition
process. In the first step, a MC
algorithm
was used to calculate the transport
of
sputtered
atoms
in a deposition
chamber.
A
simplified
MD approach
along with a nuclear
scattering theory was adopted to approximate
the
collision
between
the incident
atom
and the
substrate/deposited-film.
Finally,
a mobility
parameter was calculated
in order to determine
the
extent of hopping before an atom reached the lowest
energy neighbor
site. The method has now been
successfully
used to predict
step coverage
of
sub-micron
contact
holes. Although
the kinetic
energy was appropriately
treated, an explicit treatment of the substrate temperature and the deposition
rate was not developed.
Two phenomena
must be incorporated
in a MC
simulation of physical vapor deposition.
First, it is
necessary
to characterize
the outcome
of the
adatom-substrate
collision process that results in
an initial
adatom
configuration.
Secondly,
the
subsequent diffusion of atoms both on the growing
surface and within the previously deposited bulk of
the film must be modeled. For low incident energy
processes, the Henderson,
Brodsky and Chaudhari
model [30], demonstrated
in Fig. 1, has been widely
used in the past for the first step. Atoms (represented
by solid discs) are randomly dropped from above the
substrate. They are assumed to travel in a straight line
(at a defined angle to the substrate normal) until
coming in contact with an already deposited disc on
the substrate. The incident disc is assumed to remain
in contact with the disc with which it first makes
contact, but is allowed to relax to the nearest cradle
formed by the first contacted
disc and any other
previously deposited disc in the surface. After this
YANG
et al.:
MONTE
CARLO
SIMULATION
relaxation
process is completed,
the next incident
hard disc is introduced
and so on.
to use their
Henderson
et al. initially intended
model to simulate the entire atomic arrangement
process in random networks or dense random packed
arrays to demonstrate
the natural occurrence
of
anisotropy
and voids in film growth. However, they
and others subsequently
found that the packing
densities of the simulated films were much smaller
than those of real films [30, 33, 44,451. Furthermore,
the angle of inclination of the column structures they
predicted was only slightly less than the flux incidence
angle [30]. This disagreed with experimental
observations [46]. In an effort to improve this model, Kim
et al. [33] allowed the incident particle to “bounce”
according
to a pre-assigned
probability
after first
impact. In the bouncing process, atoms then had a
chance to move to sites with a higher coordination.
They found that the film density increased as the
probability of bouncing was increased. Gau et al. [44]
introduced
a migration parameter
i (defined as the
ratio of the average distance traveled by the atom
after impingement
to the diameter D of an atom) to
allow incident atoms to migrate across the surface
until a stable location was reached. By choosing
different values for d, they crudely simulated different
adatom mobility conditions.
In a similar approach,
Dew et al. [40] approximated
long range surface
diffusion by introducing
an average diffusion length
and using a calibration factor to make the simulated
column size comparable to that in a real growth case.
All these models had the common feature that after
an adatom eventually came to rest, it never again
moved. Thus the structure was frozen after the initial
adsorption/relaxation
event, and further diffusional
relaxation
as deposition
progressed
was precluded.
OF PHYSICAL
VAPOR
Model
Momentum
Fig. I. Schematic
illustration
and comparison
of the
Henderson model and the momentum
scheme for the initial
relaxation
in the two-step Monte Carlo simulation.
The
shadowed disk represents an atom in flight and the dashed
disk an atom hitting a previously deposited atom. Path B
stands for the Henderson model and path A the momentum
scheme. a is the incidence angle.
1457
These schemes can therefore be considered one-step
models. In such models, it is impossible to precisely
connect
a migration
parameter
or an average
diffusion length to the deposition
temperature
and
rate. Thus, the microstructures
they predict are only
valid when diffusion is relatively insignificant
(i.e.
very high rate or low homologous
temperature
deposition).
Two-step models that allow continued
atomic movements
after the initial deposition event
are needed
to more
realistically
simulate
the
phenomena
that occur under
conditions
where
diffusion is likely to be active.
A significant start in the development
of two-step
MC models in low energy processes began with work
by Miiller [34], who permitted unlimited adjustment
of an “average diffusion length” and was therefore
able to approximately
incorporate
the consequences
of high homologous
temperature
or low rate
deposition.
However, when the homologous
deposition temperature
was very low and/or the adatoms
possessed very limited mobilities, this scheme again
simply reduced to the Henderson model which may
be an inaccurate approximation
of the initial impact
process.
A two-step MC simulation model for low energy
(< 1.0 eV) deposition
is proposed
in the present
report in which the actual process parameters can be
used as inputs. It involves the use of a momentum
scheme to identify the initial surface configuration
of an adatom
and then a multipath
analysis of
all subsequent
diffusion.
The results of a twodimensional
(2-D) model are presented
first for
simplicity; subsequent work will extend the approach
to the full three-dimensional
case and sputtering
process. The results of this work will be shown to
provide a deeper insight into many of the different
physical
processes
involved
in physical
vapor
deposition
and enables exploration
of the effects
of process conditions upon morphology
and microstructure.
2. ADATOM
Henderson
DEPOSITION
INCORPORATION
In the present scheme to simulate low energy
processes, the interaction
at the moment of adatom
incidence has been modified from the Henderson
model, and the relaxation step is treated in a more
general way than that used by Miiller. As shown in
Fig. 1, an incident atom 0 travelling along the cd
direction inclined at an angle CI from the normal to
the substrate
will first touch atom F located at
positi0n.f. With Henderson’s nearest cradle criterion,
the atom was assumed to go to the nearest stable site
which is B. This method results in a very low packing
density which is not observed in experimental
work.
We note that the adatoms deposited
in even low
energy deposition process can still carry considerable
energy when they arrive at the substrate.
For
example, Zhou, Johnson and Wadley [47] using MD
simulations
found that a nickel atom with thermal
1458
YANG
et al.:
MONTE
CARLO
SIMULATION
energy of 0.1 eV can be accelerated to a kinetic energy
as high as 1.OeV in the attractive force field of the
surface atoms before being incorporated
in the
lattice.
This same phenomenon
has also been
observed for silicon deposition by Gilmer [48], again
through
MD simulation.
Thus, conservation
of
momentum
is an important
consideration
and
suggests that the atom 0 will more likely move
towards the cradle site A when the landing point is
to the left side of line ef(for a head-on collision), even
though the distance of path oA is larger than that of
oB. This is clearly
an approximation
to the
complicated
binary collision processes with a rough
surface at low energy situations [38].
This simple modification,
which we term the
momentum
scheme, generates configurations
having
significant differences with those of the Henderson
model. Two sample configurations
are shown in
Fig. 2, both with an incidence angle a of 45”. In the
momentum
scheme, the column orientation
angle
with respect to the normal to the substrate,
fl, is
significantly less than that with the Henderson model.
This trend continues for a range of M as shown by
Fig. 3(a). From a plot of packing density (defined as
the fraction of atoms occupying the lattice sites in the
deposit region, with sufficient number of surface
layers and the substrate excluded from the measurement; the packing density = 1 if fully packed) as a
function of incidence angle, Fig. 3(b), it is seen that
structures generated by the momentum scheme have
considerable
higher packing densities
than those
generated by the Henderson model when a is less than
about 30’. When c( approaches 50’, the density tends
to be similar for both models. Thus the momentum
scheme provides a better initial relaxation model for
prediction of density [30, 33, 44-461.
The decrease of the column orientation
angle can
be indirectly confirmed by experiments.
Nakhodkin
and Shaldervan
[46] using evaporation
process
conducted
an extensive
study on the effect of
condensation
condition on the profile of condensed
films. One of their results was that the column
orientation
angle decreased with the increase of the
substrate temperature,
which is in essence related to
the increase of the surface mobility of adatoms. Since
adatoms can relax to a distant cradle site rather than
to a near one, the momentum
scheme essentially
increases the surface mobility that leads to the result
of a smaller angle. Therefore,
in simulating vapor
deposition, atomic kinetic energy must be taken into
account in an appropriate
way even in very low
energy processes.
3. DIFFUSION
3.1.
Activated
MODEL
difSusion process
As shown in Section 2, the structures generated by
both the Henderson
model [30] and the momentum
scheme do not account for atomic diffusion and have
low densities. They can only represent the situations
OF PHYSICAL
VAPOR
DEPOSITION
4
a
P
a = 45”
(a) Henderson Model
(b) Momentum Scheme
5nm
,
Fig. 2. Representative
configurations
simulated using (a) the
Henderson
model and (b) the momentum
scheme. An
incidence angle IX of 45” is used for both cases and B is
indicated for column orientation
angle.
where diffusion
is unimportant,
i.e. at very low
homologous
temperatures
and/or very high deposition rates. In practice, most deposition processes are
conducted under conditions
where atomic diffusion
occurs simultaneously
with deposition and diffusion
must therefore
be taken
into account
in the
simulation.
The methodology
should address the
many diffusional
pathways
available (e.g. due to
the atomic coordination
number dependence
of the
YANG
et al.:
MONTE
CARLO
SIMULATION
OF PHYSICAL
VAPOR
1459
DEPOSITION
jumping activation energy). It should also connect
with the temperature
and deposition rate since these
control the available time for rapid surface diffusional processes before the surface becomes covered
by a new adatom layer.
If Boltzmann statistics are assumed to govern the
diffusional processes, the probability per unit time for
a possible jump, i, to take place is given by
from among all the possible jumps weighted by their
relative probability
of occurrence.
By following the
ensuing discrete jump path for the system, accumulating the residence time of the system along the path,
and linking the duration of this history to the adatom
arrival interval (or the deposition rate), the diffusion
process can be realistically simulated.
The calculation
begins by determining
a time
interval
between
adatom
arrivals
based on the
deposition
rate. The average time interval between
the arrival of two atoms in a 2-D lattice model is
where 1’”is the effective vibration frequency (taken to
be 5 x lO”/s for all the cases in this work), E, is the
activation
energy for the ith type of jump, k is
Boltzmann’s
constant and T the absolute temperature. If the different diffusional pathways each have
different E, values, they must also have different p,
values.
The reciprocal of an atomic jump probability
per
unit time (equation (1)) is a residence time for an
atom that moves by that specific type of jump. Since
the jump probabilities
of all the different types of
jumps are independent,
the overall probability
per
unit time for the system to change its state by any
type of jump step is just the sum of all the possible
specific jump type probabilities,
and so the residence
time for the system in a specific configuration
is the
reciprocal of this overall jump probability.
The next
diffusional step is determined by randomly choosing
At=&
80
I
/
I
I
/
(a) Column orientation angle
1’
60 -
&
I
Henderson model 7
I
A”
0.9
/
1
,A’
p
I
/d
I
I
(b) Packing density
0.80 I3 c
-
n
I
20
0
/
I
40
Momentum scheme
/
60
b
80
Incidence angle CL (deg)
Fig. 3. Comparison
of the difference between the Henderson
model and the momentum
scheme in terms of (a) the
relationship
between
column
orientation
angle /II and
incidence angle r and (b) the relationship
between packing
density and incidence angle, a.
where R is the deposition
rate (deposited thickness
per unit time), a is the nearest neighbor distance and
n is the number of atoms comprising a monolayer in
a close-packed simulation system. Clearly, the higher
the deposition
rate, the smaller the time interval
between two consecutive
deposition
events in the
model. Time periods are then related to the diffusion
process through a net residence time, t,,, given by
where N is the number of different types ofjumps (i.e.
different diffusion pathways). In this model, a single
jump is allowed only to vacant nearest-neighbor
sites
or over a ledge at the surface, i.e. a Schwoebel jump
[43]. The two time definitions (equations (2) and (3))
are then linked for the simulation.
An atom is next dropped from a random position
above the surface. It travels to the surface in a
straight line and impacts the substrate at an angle r*
to the normal. It is then initially relaxed using the
momentum
scheme.
The set of atoms
in the
simulation system are then monitored for diffusional
modification prior to the arrival of the next atom, i.e.
in the time At. If the system has a net jump
probability
greater than 1 in the At time period, a
jump is made and a time equal to the net residence
time as calculated using equation (3) is subtracted
from At so that there is less time remaining for further
jumps in the allotted time period. This process is
iterated until the probability of making any jump in
the remaining time is less than one. Whether or not
any jump is made in this time period is then
determined by random choice based on the remaining
time and the net time for that particular state of the
system. Whenever a jump is to be made, the specific
one is determined
by random choice based on the
relative probabilities of all potential jumps. When the
remaining
time reaches zero, the clock is turned
ahead by At and another atom is then deposited.
3.2. Calculation
migration
of the
activation
energies
Each of the probabilities,
p,, for the
migration
steps are required
as input
,for
possible
for the
1460
YANG
et al.:
MONTE
CARLO
SIMULATION
calculation
of diffusion by this MC method.
To
obtain physically
reasonable
values for the 2-D
problem being studied, atomistic calculations for the
activation energy for migration for different diffusion
paths (defined by the initial and final configurations)
have been carried out.
A 3-D EAM model [49] with parameters for nickel
has been modified for use with a close packed 2-D
array constrained
to remain in a plane. The energy
of the 2-D array was minimized
to find a 2-D
equilibrium
lattice constant
and cohesive energy.
No adjustments
were made to the 3-D EAM
parameters,
and static calculations were then carried
out to track the energy barrier for a number of
possible jump configurations.
Since at this stage
the calculations
are designed
to indicate overall
behavior and not provide exact results for each jump
type, the static calculations
used an approximate
relaxation scheme that resulted in activation energies
that were accurate within several hundredths
of an
eV.
The basic equations
of the EAM [50] in the
notation used by Johnson (511 are
E=&%)f~KG
I ,ir
P6= cf(y,)
i”
OF PHYSICAL
Table
I.
Calculated
Configurational
transition
VAPOR
DEPOSITION
activation energies for possible jumps
2-D nickel EAM
Bonds
(fromto)
usmg a
Energy
(eV)
I
2-2
0.44
2
Z-3
3-2
0.38
0.91
3
3-3
0.85
4
24
4-2
0.3 I
1.34
444
0.96
6
3-4
4-3
0.71
1.21
I
4-s
5-4
0.48
0.93
8
3-5
5-3
0.20
I .02
9
5-5
0.70
10
2-s
s-2
Spontaneous
Unstable, > I .30
11
bulk 3-3 bulk
0.80
12
bulk 5-5 bulk
0.83
(4)
where E is the total internal energy, 4(r,,) is the
two-body
potential
between atoms i and j, P,, is
the distance between atoms i and j, F(p,) is the
embedding energy of atom i, p, is the electron density
at atom i from all other atoms, and f(~,,) is the
contribution
to the electron density at atom i from
atom ,j. Although
the physical interpretations
are
different,
these equations
have the same form as
those developed
by Finnis and Sinclair [52]. The
model used above has been “normalized”,
so that
independent
equilibrium
is attained
from
the
two-body
interaction
and from the embedding
function [53].
Analytic forms for the two-body
potential,
the
electron density, the embedding
function and the
parameters
for a nearest-neighbor
EAM model for
nickel are used in the present calculations [50]. When
applied to an infinite 2-D close-packed
plane of
atoms, the equilibrium
spacing decreases from 2.49
to 2.42 8, and the cohesive energy decreases from
4.45 to 3.515 eV. The 2-D bulk vacancy formation
and migration
energies
are 1.14 and 0.83 eV,
respectively,
and the 2-D surface energy, assuming
the 2-D sheets are stacked as 3-D {ll l}planes, is
1.4 J/m’.
With a normalized nearest-neighbor
EAM model,
the 2-body potential
yields equilibrium
in a 2-D
model at the 3-D lattice spacing and will produce no
forces for relaxation
at a surface. The embedding
energy favors the bulk 3-D electron density at each
site, so the decrease in the lattice constant for 2-D
equilibrium
is caused by the system adjusting
to
increase the electron density at each atom, and is
13
14
2-3 via 24
and 4-3
2-3 via 2-l-3
0.57,
Reverse I .06
0.66,
Reverse 1. I5
counterbalanced
by the consequent increase in energy
in the two-body bonds as the system contracts. This
same effect leads to an inward displacement of atoms
at a surface.
The basic jump configurations
were identified by
the number of nearest neighbors
to the jumping
atom before and after the jump. The arrangement
closest to the surface was chosen with two bulk cases
included for comparison.
One of the bulk cases was
just vacancy migration;
the other was the most
bulk-like 3 to 3 atom coordination
configuration.
The
numerical results are summarized
in Table 1 where
each run number corresponds
to a different diffusion
path. Run 9, which in bond number is like bulk
vacancy migration, involved a vacancy in the second
row moving parallel to the surface. Runs 13 and 14
have the same initial and final configuration:
the
configuration
changes from there being one atom on
top but at the edge of a one-atom high terrace (i.e.
at the terrace ledge) to the terrace being extended by
one atom. In run 13, the ledge “crumbles” with two
YANG
et al.:
MONTE
CARLO
SIMULATION
atoms moving concurrently
so that the atom which
had been on top of the terrace replaces the atom
below it at the edge of the terrace, with atom moves
to extend the terrace. In run 14, the single atom goes
over the top of the ledge. The activation energy for
crumbling is less than that for over the top, but for
purposes of the diffusion calculation, this is taken as
a two-bond
to one-bond
case followed by spontaneous decay from the unstable one-bond configuration. Both jumps represented by runs 13 and 14 are
considered as Schwoebel jumps [43]. The Schwoebel
barrier
is defined [54] as the activation
energy
difference between a Schwoebel jump (either run 13
or run 14) and a two-bond
to two-bond jump on
smooth surface (run 1) and so that the barrier energy
is either 0.13 or 0.22 eV depending upon mechanism.
4. RESULTS
In this two-step
MC model,
the momentum
scheme, an approximation
for low energy processes,
is used for the initial adsorption
of the impinging
atom and the diffusion model is used for simultaneous annealing during the deposition. Desorption
is ignored because simulations
are carried out at
relatively
low temperatures.
All calculations
are
based on the parameters for the 2-D nickel model and
lateral periodic boundary conditions are employed to
account for the limited system size. The substrate
consisted
of a perfect array of 200 close-packed
atoms. A representative
incidence angle, c( = 38’, was
used for all the calculations,
and 8000 atoms were
deposited during each run. Five runs were carried out
for each data point to determine the statistical spread
in the data. Since this is a 2-D model, the actual
temperatures
used for deposition
are not directly
related to those of a 3-D system. To compensate,
we
note that the 2-D vacancy formation energy is about
2/3 of the corresponding
3-D value [55]. Thus, a
homologous
temperature
is also given, based on a
T,,,, of 1150 K, which is 2/3 of
melting temperature,
the 3-D nickel melting temperature
of 1726 K.
Two series of computer experiments
are reported.
In one, the deposition
rate was held fixed and the
substrate
temperature
was varied. In the other,
the substrate
temperature
was held fixed and the
deposition
rate was varied.
Since this model
incorporates
a fixed (Ising) lattice of sites which are
either
occupied
or not, dislocations
and grain
boundaries cannot occur. However, mounding of the
surface profile yields a characteristic
width that is
used as a measure of the column size. The column
width was approximated
by dividing the width of the
system by the number
of mounds.
The surface
W is defined as the standard deviation of
roughness
the surface height [35, 541, UR = CN-‘(h, - @, with
k the average height of the surface layer, h, the height
of ith site and N the total number of surface sites.
Since a hexagonal
rather than the usual square
grid is chosen in the simulation, when a part of the
OF PHYSICAL
VAPOR
DEPOSITION
surface is perfectly smooth, the heights
ing sites are adjusted to be equal to
higher one for proper averaging. Finally,
density is measured in the same way as
Section 2.
4.1. Eflect
of temperature
1461
of neighborthat of the
the packing
that used in
on microstructure/mor-
phology
The effect of the substrate temperature
on the thin
film structure has been simulated by changing the
temperature
systematically
at three considerably
different deposition rates. The basic physical properties reported are packing density, surface roughness
and column width. For some simulations,
tracer
atoms were used to reveal the evolution
of the
internal
structure
of the deposit and to indicate
possible mechanisms
by which the evolution occurs.
The tracer atoms are a set of marked adatoms
deposited during a specific time period. Their position
is observed
throughout
the remainder
of the
simulation run. They are shown as filled circles while
the remainder of the atoms are open circles in the
configuration
displays. When an adatom arrives at
the growing surface, it is first adsorbed
and then
diffuses to some extent depending
upon its local
configuration
and the computer experimental
conditions. The pattern of development can be visualized
by motion of the tracer atoms.
4.1.1. Structural conjiguration, packing density and
surface roughness. The final configurations
at four
deposition
temperatures
are shown in Fig. 4 for a
deposition
rate of 10 pm/min.
At a fairly low
temperature
of 250 K (T/T,,, = 0.22) Fig. 4(a), a low
density
structure
(packing
density = 0.79) with
voided growth boundaries,
typical of zone I in the
SZM model, is found. This structure forms because
of self-shadowing
[16]. With an increase of temperature to 300 K (T/T,,, = 0.26) Fig. 4(b), the voided
growth
boundaries
are found
to have almost
disappeared (packing density = 0.97) and the internal
columnar boundaries have become better defined, as
demonstrated
by the surface contour in relation to
the tracer atom contour, although the surface is still
quite rough. With a further increase of temperature
to 400 K (T/T, = 0.35) a fully dense columnar
structure
(packing density = 1.0) and a facet-like
surface typical of zone II in the SZM model was
observed, Fig. 4(c). The column boundaries
within
the bulk can be seen with the aid of the tracer atoms:
the columns nucleate at the beginning of deposition,
are tilted
toward
the flux, and
retain
their
approximate
width throughout
the run. At a still
higher temperature of 550 K (T/T, = 0.48), Fig. 4(d),
the surface becomes fairly flat and the tracer atoms
show a rapid increase of the column width. Tracer
atoms also show that only limited bulk diffusion
occurs at this temperature.
The packing density as a function of temperature
for this deposition rate (10 pm/min) is represented by
the filled circles in Fig. 5. The fitted curve exhibits
1462
YANG
et al.:
MONTE
CARLO
SIMULATION
OF PHYSICAL
VAPOR
L
DEPOSITION
a
R= lOpm/min
I(a) T = 250 K (T/T, = 0.22)
(b) T = 300 K (T/T, = 0.26)
(d) T = 550 K (TIT, = 0.48)
12.5 nm
Fig. 4. Representative
2-D configurations
of Ni growth at various substrate temperatures
at a deposition
rate of 10 pm/min and an incidence angle c( of 38”. The momentum
scheme is used to treat the effect of
low kinetic energy (< 1.O eV) in the deposition process. Atoms, tagged as tracers to show sample atom
movements,
are shown as solid circles.
YANG
et al.:
MONTE
CARLO
Homologous temperature
0.00
0.10
I Dewxition
0.20
rate
R=
i
f
i4
lb
1
2;O
I
OF PHYSICAL
VAPOR
0.40
1
0.50
0.10
0.20
0.30
I
I
I
0
I
100
0.40
0.t 50
I
a=38”
pm / min
I
0.6 /
1463
DEPOSITION
Homologous temperature T/T,,,
T/T,
0.30
O.k5
a=38'
SIMULATION
I
I
I
I
200
300
400
500
Deposition temperature
1
I
100
300
I
I
400
500
6C IO
Substrate temperature T (K)
60( 1
T (K)
Fig. 6. Surface roughness
vs substrate
temperature
at
various
deposition
rates for s( = 38’. The momentum
scheme is used to treat the effect of low kinetic energy
(< 1.0 eV) in the deposition process.
Fig. 5. Packing density vs substrate temperature
at various
deposition rates for d( = 38-. The momentum scheme is used
to treat the effect of low kinetic energy (< 1.O eV) in the
deposition process.
three regions. When the temperature
is below about
150 K, the porous structure has a fairly constant
density of about 0.68, whereas a fully dense structure
is obtained
when the temperature
is above about
350 K. A transition
occurs in the region between
these two temperatures
and marks the change from
a porous
columnar
structure
to a fully dense
columnar structure. The transition temperature,
T,, is
defined as the onset of full structural densification,
in
conformity
with the characteristic
columnar boundary difference of zone I and zone II [16]. Using this
definition,
r, = 350 K (T/T, = 0.3) for a deposition
rate of 10 pm/min. This temperature
is defined by
both the mobility of the adatoms and the deposition
rate. It delineates
a temperature
where adatom
mobility is sufficient to fill shadowed
regions. The
shadowed
regions grow at a rate proportional
to
R and so T, corresponds
to the temperature
at which
a balance between mobility and deposition
rate is
achieved,
and so it must
be deposition
rate
dependent.
The results of simulations
conducted
for both a
much higher
(250 pm/min)
and a much lower
deposition
rate (0.05 pm/min)
are also plotted in
Fig. 5. The faster deposition
is seen to shift the
transition
region
to a higher
temperature
(Tr/
T,, = 0.35) while the slower deposition
shifts it to a
lower temperature (7’,/T, = 0.24). The transition also
occurs over a greater temperature
spread as the
deposition
rate increases. This zone I-II transition
was seen by Miiller [34] as well from an analysis of
packing density in his 2-D calculations,
although at
a significantly higher temperature
partly because of
his use of the Henderson
model and the bond
counting activation barrier energy.
I
200
The surface morphology (or roughness) also varied
with the processing parameters and is well known to
exhibit transition
phenomena
[16]. The effect of
temperature on surface roughness predicted using the
MC model is shown in Fig. 6. It can be seen that in
each case the roughness is approximately
constant at
low temperature,
decreases
with an increase
of
temperature
and approaches a final saturation value
of about 6 A. The roughness transition is coincident
with the transition
in the density
plot: for a
deposition
rate of 10 pm/min,
the roughness
is
Homologous temperature T/T,,,
0.20
0.30
0.25
I
200
/
I
Deposition
rate
R = 0.06
0.35
I
Km I min
a=38’
160
0.40
T
z
E
‘ci
120
E’
3
8
g
i.
80
-
S
P
a
40
0
b
I
I
100
200
I
300
/
400
1
I
500
600
Substrate temperature T (K)
Fig. 7. Average column size vs substrate temperature
at the
same deposition
rate of 0.06 pm/min as that used in the
experiments [55]. The momentum scheme is used to treat the
effect of low kinetic energy (< 1.0 eV) in the deposition
process.
1464
YANG
et al.:
MONTE
CARLO
SlMULATION
OF PHYSICAL
VAPOR
DEPOSITION
Grovenor et al. [55] suggested that the universal
nature of their graph of grain size vs homologous
temperature
is due to the scaling of the activation
energy for bulk diffusion with T,. Since there is little
bulk diffusion
in the present
calculations,
the
agreement of these results with their data indicates
that surface diffusion activation energies, which also
scale with T,, [46], are controlling the microstructural
development
at lower temperatures.
4.2. Eff;ct
Minimum
\
observed
grain size
01
I
,
I
I
I
I
I
I
0
2
4
6
8
10
12
14
16
1100
18
T, / T
Fig. 8. (a) Plot of measured
grain size variation
with
substrate temperature
for Ni films deposited at a deposition
rate of 0.06 pm/min using an e-beam evaporation process
(ref. [55] with permission)
compared
with the simulated
column size plotted in Fig. 7. (b) Comparisons
with other
metals.
greatest
below about
150 K (where densification
starts) and becomes lowest at and above 350 K when
densification
is complete.
Surface roughness
thus
provides an alternative way to measure the zone I-II
transition.
4.1.2. Comparison of simulated column size and
experimentally
observed grain size. Since a comprehensive
experimental
study
of the relationship
between grain size of Ni thin films and processing
parameters
has been carried out [55], simulations
were run at the same deposition rate of 0.06 pm/min
used in the experiments.
The relationship
between
average column size and substrate
temperature
is
plotted in Fig. 7. The column size does not experience
any significant change when the temperature
is at or
below about 200 K, but increases
rapidly above
about 250 K.
These results are compared with the experimental
results from the literature [55], Fig. 8. Figure 8(a)
shows that at all substrate temperatures
a range of
grain sizes was observed and the difference between
the maximum and the minimum observed grain sizes
increased
with the temperature.
A reasonable
agreement between the calculation and the minimum
observed grain sizes can be seen. This suggests that
a simulation
approach
that uses the activated
processes of atomic diffusion and a relatively large
system size can predict the trends in microstructure/
morphology
surprisingly
well. Figure 8(b) suggests
that when the deposition temperature is scaled by the
melting temperature,
the grain sizes of many metals
collapse on to a master curve and all can be fitted by
a nickel model prediction.
of deposition rate on microstructure
Some insight into the effect of the deposition rate
on thin film structures
has already been given. A
more detailed view has been obtained
by using a
range of deposition
rates and fixing the substrate
temperature
and the system
size. Examples
of
representative
structural
configurations
for deposition at 350 K are given in Fig. 9. As shown in
Fig. 9(a), a fairly porous columnar structure (packing
density = 0.96) results when a rapid deposition rate
of 250 pm/min is used. The rate of 10 pm/min, typical
of the high rate JVD”“‘/DVD processes, results in a
fully dense columnar
structure
with a facet-like
surface morphology,
Fig. 9(b). As indicated
in
Fig. 9(c), further decreasing the rate to 0.5 pm/min
yields a structure with an increased column size and a
facet-like surface. At a still lower rate of 0.05 pm/min
(typical deposition
rate for a sputtering process), a
structure
with a further enlarged column size is
generated, Fig. 9(d).
The corresponding
variations
of the packing
density, the surface roughness and the column size
with deposition
rate are plotted
in Fig. 10. A
transition from a porous columnar structure to a fully
dense columnar structure occurs at a deposition rate
of about 50 pm/min, Fig. 10(a). This is similar to the
transition due to the change of substrate temperature
observed earlier.
The roughness
values show a plateau over a
considerable
range of deposition rates, as illustrated
in Fig, 10(b). This plateau can be ascribed to the
presence of a facet-like surface morphology,
Fig. 9.
That is, although the column size changes with the
rate, the standard
deviation of the surface height,
which is used as the measure of surface roughness in
this work, remains fairly constant. The variation of
the column size with deposition rate is illustrated in
10(c). It increases
approximately
as the
Fig.
logarithm
of the deposition
rate at rates less than
about 200 pm/min, and is fairly constant at higher
rates.
5. DISCUSSION
The computer
experiments
above describe
the
atomistic
evolution
processes
during low energy
deposition
over a wide range of substrate temperatures and deposition
rates. A dependence
of the
density and roughness with deposition rate, flux angle
or substrate temperature,
a coarsening phenomenon
YANG
et al.:
MONTE
CARLO
SIMULATION
at relatively high temperatures
and low deposition
rates as well as structural transitions are all exhibited
by this model. These observations
can be explained
physically as the effect of the initial impact followed
by thermal activation
leading to atomistic surface
diffusion.
Bulk diffusional
contributions
to the
OF PHYSICAL
VAPOR
DEPOSITION
1465
microstructural
evolution are relatively insignificant
for the range of deposition
rates and temperatures
investigated
here.
Although the activation energy parameters used in
the present work are for nickel and the model results
apply strictly only to this metal, the study indicates
T = 350 K
(a) R = 250 pnhnin
(b) R = 10 pm/min
(c) R = 0.5 pmhnin
(d)R= 0.05 pn/min
12.5nm
Fig. 9. Representative
2-D configurations
of Ni film growth at various deposition rates at a substrate
temperature
of 350 K and an incidence angle a of 38 The momentum
scheme is used to treat the effect
of low kinetic energy (< 1.O eV) in the deposition process.
1466
YANG
et a/.:
Substrate temperature
MONTE
CARLO
T = 350K,
c( = 38”
SIMULATION
OF PHYSICAL
VAPOR
DEPOSITION
process. When the deposition
temperature
is sufficiently low, relative to the deposition
rate, the
residence times for the adatoms are relatively large
compared to the adatom arrival interval so that there
is limited relaxation/diffusion
on the surface before it
is covered with additional atoms. Since every adatom
essentially sticks close to where it arrives, the growth
is dominated by the transport of the depositing atoms
to the substrate
and in particular
by the set of
directions
from which these atoms arrive at the
substrate. As a result, the self-shadowing
effect is very
important
and void networks
develop. Thus, the
structure in zone I is generally characterized
by low
density poorly aligned crystals, a very rough domed
top surface, and small column sizes. In each column,
there are essentially
several smaller sub-columns
aligned in a similar direction, as reflected both in the
(cj Average column size
experimental
observations
[55-591 and the computer
A
150
‘B
simulations.
We find that this type of porosity (due
to the limited rearrangement
of adatoms) is difficult
to eliminate
[45, 601. Even when the films are
subjected to post-deposition
annealing, these voids
tend
only
to
change
shape
since
the overall structural
h
01
I
I
.,.,,,
I
,.,,,
.-I
10'
IO'
102
103
104 framework has been previously established.
102
100
Deposition rate R (pm / min)
The residence times of adatoms at a surface site
decrease with an increase in temperature.
When the
Fig. 10. Deposition rate vs (a) packing density, (b) surface
temperature
is
increased
to
the
point
where
the
roughness and (c) column size. A substrate temperature
of
residence
is comparable
to the adatom
arrival
350 K and an incidence angle ix of 38- are used for all cases.
The momentum
scheme is used to treat the effect of low
interval, At, the porous columnar structure begins to
kinetic energy (< 1.0 eV) in the deposition process.
change to a fully dense columnar structure separated
by distinct, dense and intercrystalline
boundaries.
This is the transition reflected in the density plot of
Fig. 5 and in the roughness plot of Fig. 6.
the observed patterns of behavior for physical vapor
The zone II structure
is controlled
by surface
deposition are applicable to a broad range of metals.
diffusion, as indicated by the smooth contour of the
Experimental
results for vapor deposition
demontracer atoms in Fig. 4(c). Activation
energies for
strate such a general pattern
regardless
of the
surface diffusion are significantly less than those for
material and the crystal structure used [55]. As shown
bulk diffusion
[56]. Accordingly,
at a specific
in Fig. 8(b), the variation of grain size as a function
deposition rate, a temperature
range normally exists
of T,,,/T,
(where T,,,
is melting point and T,substrate
which is high enough
so that surface diffusion
temperature)
is very similar for 10 elements including
dominates over arrival rates, and the coating atoms
a variety of b.c.c., hexagonal and f.c.c. metal films
lose memory
of their arrival
directions.
This
and motivated the development
of the SZM model.
temperature
is still low enough (TJTm < 0.5)that
The results of the present calculation are also plotted
bulk diffusion rates remain orders of magnitude less
on this graph as the solid squares and show the same
than surface diffusion rates and are consequently
pattern.
negligible. The direct effect is that the columns grow
Thin film growth is generally considered to occur
bigger with increasing
temperature,
Figs 4(b)-(d),
in a series of steps [56]: transport of coating atoms to
and facet-like faces are usually present in this zone.
the substrate,
adsorption
of these atoms on the
Alternatively,
a zone II structure can be obtained by
growing surface, their diffusion over the surface,
fixing the temperature
and increasing
the arrival
eventual incorporation
into the coating or release
interval, as shown in Fig. 9. The roughness is nearly
from the surface by thermal desorption or sputtering,
constant in zone II, Fig. 10(b), and can therefore be
and finally movement
of atoms to lower energy
used as a delimiter of this range.
positions within the lattice by bulk diffusion and solid
It is found that the formation of facet-like mounds
state reactions. In an actual process, however, the
in zone II structures
is quite universal and was
simulations
show that the surface morphology
of a
present in every simulation carried out in this work.
film and its internal structure may be dominated by
Experimentally,
this also appears
to be a fairly
just one of these steps.
general phenomenon,
occurring
in a number
of
The simulation confirms that in the SZM model,
different systems [57, 58, 61-661. We attribute
this
the zone I structure represents the result of limited
phenomenon
to both self-shadowing
and the presence
atomic
rearrangement
in the vapor
deposition
c
,,,(,
.I,.,,
(.,,I,
t
YANG
et al.:
MONTE
CARLO
SIMULATION
of Schwoebel
[43] or step edge barriers
in the
material. They make it more difficult for adatoms to
move from one terrace to a lower one. For the two
possible Schwoebel jumps considered
in this work,
Table 1, the Schwoebel barriers are 0.13 and 0.22 eV,
respectively as indicated earlier.
Zone III conditions
occur at higher temperatures
where bulk diffusion
dominates
over all other
processes so that atoms lose all memory of the initial
events associated with their condensation.
Zone III
conditions
are not modeled in this work, in part
because the activated character of the diffusion rate
intrinsically
demands
a high computational
cost.
Zone III conditions are less likely in many advanced
applications due to concerns about thermal stress and
the stability of the substrate [57, 58, 671.
OF PHYSICAL
surface diffusion
can
microstructure/morphology
1467
DEPOSITION
predict well the
evolution.
trends
in
Acknowledgements-We
are grateful for the support of this
work by the Advanced Research Projects Agency (A. Tsao,
Program Manager) and the National Aeronautics
and Space
Administration
(D. Brewer, Program
Monitor)
through
grant NAGW1692.
REFERENCES
1.
2.
3.
4.
5.
6.
6. CONCLUSIONS
A two-step
Monte
Carlo
method
has been
proposed and shown to generate reasonably realistic
low energy deposition simulations over a wide range
of deposition
conditions.
The model incorporates
a
momentum
scheme in which the effect of low atomic
kinetic energy (< 1.0 eV) at the instant of adatom
impact with the substrate
is approximated.
This
yields initial packing densities and column orientation
angles that are closer to experimentally
reported
values
than
those
yielded
from
the
Henderson
model used in prior calculations.
Using
basic kinetic considerations
of solid-state diffusion, a
multipath
diffusion model has been developed
to
provide a fundamental
link between the deposition
rate and atomic diffusive process on, and within, the
sample. An embedded atom method is employed to
calculate the activation
energies for diffusion of a
variety of atomic configurations.
This two-step
approach
makes
possible
the
simulation
of low energy deposition
over a broad
range of physically realistic deposition
parameters.
The approach
provides
a practical
method
to
simulate
several aspects of the vapor deposition
processes. In particular, it enables determination
of
the effect of vapor processing
variables
such as
deposition
flux density (deposition
rate), flux angle
and substrate temperature
upon deposit microstructure/morphology
parameters such as packing density,
surface roughness and column size.
Although a two dimensional
model is used in this
work,
the simulation
results
demonstrate
the
transition from a porous columnar structure to a fully
dense columnar structure in a way that agrees well
with the SZM model. The transition is found to occur
at a higher temperature
as the deposition
rate
increases. The width of the columns in the simulated
microstructures
appears to correlate well with grain
size, and the temperature
dependence
of the width
correlates closely with experimental
grain size data
for many metals. This suggests that a Monte Carlo
simulation that uses the activated processes of atomic
VAPOR
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
R. F. Bunshah, J. Var. Sci. Technol. 11, 633 (1974).
J. F. Butler. J. Vat. Sci. Technol. 7, S52 (1970).
J. A. Thornton,
SAE Truns. 82, 1787 (1974).
J. J. Schmitt, U.S. Patent No. 4, 788, 082, Nov. 29
(1988).
B. L. Halpern, J. J. Schmitt, Y. Di, J. W. Golz, D. L.
Johnson,
D. T. McAvoy, D. Wang and J. Z. Zhang,
Metal Finishing 12, 37 (1992).
L. M. Hsiung, J. Z. Zhang, D. C. McIntyre, J. W. Golz,
B. L. Halpern,
J. J. Schmitt and H. N. G. Wadley,
Scripta Metall. et Mater. 29, 293 (1993).
B. L. Halpern and J. J. Schmitt, in Handbook qf
Deposition Technologies for Films and Coatings (edited
by R. F. Bunshah), p. 822. Noyes Publications,
Park
Ridge, New Jersey (1994).
H. N. G. Wadley, L. M. Hsiung and R. L. Lankey,
composites. Engr. 5, 935 (1995).
J. F. Groves and H. N. G. Wadley, Composites. Engr.
to be published.
J. F. Groves, S. H. Jones, T. Globus, L. M. Hsiung and
H. Wadley, J. Electrochem. Sot. 142, L173 (1995).
E. Bauer,.2.
Krist. 110, 372 (1958).
F. C. Frank and J. H. Van der Merwe. Proc. Rev. Sot.
A198, 205 (1949); A200, 125 (1949).
M. Volmer and A. Weber, Z. Phl;s. Chem. 119, 277
(1926).
J. N. Stranski and L. Krastanov,
Brr. Akad. Wits. Wien
146, 797 (1938).
B. A. Movchan
and A. V. Demchishin,
Phys. Met.
Metallogr. 28, 83 (1969).
J. A. Thornton,
Ann. Rec. Mater. Sci. 7, 239 (1977).
R. W. Vook and F. Witt, J. Vat. Sci. Technol. 2, 49
(1965).
S. D. Dahlgren, J. Vat. Sci. Technol. 11, 832 (1974).
K. L. Chopra,
Thin Film Phenomenu. McGraw-Hill
Book Company,
New York (1969).
H. J. Leamy, G. H. Gilmer and A. G. Dirks, in Current
Topics in Materials Science 6 (edited by E. Kaldis),
p. 309. North-Holland,
Amsterdam
(1980).
G. H. Gilmer, in Handbook of’Crystal Growth 1 (edited
by D. T. J. Hurle), p. 583. (1993).
B. W. Dodson,
Critical Reviews in Solid State and
Materials Science 16, 115 (1990).
J. B. Adams,
A. Rockett,
J. Kieffer, W. Xu, M.
Nomura,
K. A. Kilian,
D. F. Richards
and R.
Ramprasad,
J. Nuclear Materials 216, 265 (1994).
X. W. Zhou, R. A. Johnson and H. N. G. Wadley,
accepted for publication
in Acta Metall. et Mater.
M. Schneider, Ivan K. Schuller and A. Rahman. Phys.
Rev. B36, 1340 (1987).
K. H. Miiller, J. Appl. Phys. 62, 1796 (1987).
C. C. Fang, F. Jones and V. Prasad, J. Appl. Phys. 74,
4472 (1993).
G. H. Gilmer, M. H. Grabow and A. G. Bakker. Mater.
Sci. and Engr. B6, 101 (1990).
K. H. Miiller, Phys. Rec. B35, 7906 (1987).
D. Henderson, M. H. Brodsky and P. Chaudhari,
Appl.
Phys. Lett. 25, 641 (1974).
G. S. Crest. M. P. Anderson
and D. J. Srolovitr,
in
Computer Simulation of Microstructural
Evolution
(edited by David J. Srolovitz) (1986).
1468
YANG
et al.:
MONTE
CARLO
SIMULATION
32. Y. Sasajima, T. Tsukida, S. Ozawa and R. Yamamoto,
Applied Su@zce Science 60161, 653 (1992).
Thin Solid
33. S. Kim, D. J. Henderson and P. Chaudhari,
Films 47, 155 (1977).
34. K. H. Mtiller, J. Appl. Phys. 58, 2573 (1985).
L. M. Sander and R. C. Ball,
35. P. Meakin, P. Ramanlal,
Phys. Rev. A34, 5091 (1986).
H. van Kranenburg
and J. C.
36. S. Miiller-Pfeiffer,
Lodder, Thin Solid Films 213, 143 (1992).
37. R. D. Young and D. C. Schubert, J. Chem. Phys. 42,
3943 (1965).
Computer
Simulution
of Ion-Solid
38. W. Eckstein,
Interactions. Springer-Verlag,
Berlin (1991).
39. E. Chason and B. W. Dodson, J. Vat. Sci. Technol.
A9(3), 1545 (1991).
40. S. K. Dew, T. Smy and M. J. Brett, IEEE Transactions
on Electron Devices 39, 1599 (1992).
41. C. C. Fang and V. Prasad, in Modeling und Simulation
of Thin-Film Processing 389 (edited by D. J. Srolovitz,
C. A. Volkert, M. J. Fluss and R. J. Kee), D. 131. Mat.
Res. Sot. Symp. Proc., San Francisco (i995).
42. K. Huang,
Statistical
Mechanics.
John Wiley, New
York (1963).
43. R. L. Schwoebel and E. J. Shipsey, J. Appl. Phys. 37,
3682 (1966).
44. J. S. Gau and B. Liao, Thin Solid Films 176, 309 (1989).
45. Y. G. Yang, R. A. Johnson and H. N. G. Wadley,
unpublished
work.
and A. 1. Shaldervan,
Thin Solid
46. N. G. Nakhodkin
Films 10, 109 (1972).
47. X. W. Zhou, R. A. Johnson and H. N. G. Wadley, to
be published.
48. G. H. Gilmer and C. Roland, Radiation Effects and
D#cts
in Solicis 13&131, 321 (1994).
49. R. A. Johnson, Phys. Rev. B39, 12,554 (1989).
50. M. S. Daw and M. I. Baskes, Phys. Rev. B29, 6443
(1984).
OF PHYSICAL
VAPOR
DEPOSITION
51. R. A. Johnson, Phys. Rev. B37, 3924 (1988).
52. M. W. Finnis and J. E. Sinclair, Philos. Mag. A-50, 45
(1984); 53, 161 (1986).
53. R. A. Johnson and D. J. Oh, J. Mater. Res. 4, 1195
(1989).
54. A.-L. Barabasi and H. E. Stanley, Fractal Concepts in
Surface Growth. Cambridge
University
Press, Cambridge (1995).
55. C. R. M. Grovenor, H. T. G. Hentzell and D. A. Smith,
Acta Metall. 32, 773 (1984).
56. J. A. Thornton, SPIE Vol. 821 Modeling of Optical Thin
Films, 95 (1987).
57. A. G. Dirks, R. A. M. Wolters and A. E. M. De
Veirman, Thin Solid Films 208, 181 (1992).
58. Y. Shigesato, I. Yasui and D. C. Paine, J. Metals. 47,
47 (1995).
59. R. Messier and J. E. Yehada, J. Appl. Phys. 58, 3739
(1985).
60. K. H. Miiller,
J. Vu.
Sci. Technol.
A3, 2089
(1985).
61. C. Orme, D. Johnson, J. L. Sudijono, K. T. Leung and
B. G. Orr, Appl. Phys. Lett. 64, 860 (1994).
62. M. D. Johnson,
C. Orme, A. W. Hunt, D. Graff, J.
Sudijono, L. M. Sander and B. G. Orr, Phys. Rev. Lett.
72, 116 (1994).
63. G. W. Smith, A. J. Pidduck, C. R. Whitehouse,
J. L.
Glasper and J. Spowart,
J. Cryst. Growth 127, 966
(1993).
J. L.
64. G. W. Smith, A. J. Pidduck, C. R. Whitehouse,
Glasper and A. M. Keir, Appl. Phys. Lett. 59, 3282
(1991).
65. M. A. Cotta, R. A. Hamm, T. W. Staley, S. N. Chu,
L. R. Harriott, M. B. Panish and H. Temkin, Phys. Rev.
Lett. 70, 4106 (1993).
66. H. J. Ernst, F. Fabre, R. Folkerts and J. Lapujoulade,
Phys. Rev. Lett. 72, 112 (1994).
67. S. M. Rossnagel,
Thin Solid Films 263, 1 (1995).