Acta mater. Vol. 45, No. 11, pp. 4441-4452,1997
0 1997Acta MetallurgicaInc.
Published by ElsevierScience Ltd. All rights reserved
Printed in Great Britain
1359-6454/97
$17.00+ 0.00
PII: s1359-6454(97)00156-0
Pergamon
VACANCY
FORMATION DURING
DEPOSITION
X. W. ZHOU, R. A. JOHNSON
VAPOR
and H. N. G. WADLEY
Department of Materials Science, School of Engineering and Applied Science, University of Virginia,
Charlottesville, VA 22903, U.S.A.
(Received 30 August 1996: accepted 14 April 1997)
hybrid modeling approach combining two-dimensional atomistic molecular dynamics
simulations of vacancy formation with a continuum analysis of vacancy diffusion has been used to predict
the vacancy content of vapor deposited nickel as a function of deposition rate/temperature and incident
flux energy/angle. The hybrid approach uses a previously developed molecular dynamics technique to
obtain the vacancy concentration in the surface of a film formed during a brief period of very high rate
deposition. The structure is then annealed and the decrease in surface vacancy concentration calculated
by solving continuum diffusion equations. By varying the annealing period, a good approximation to the
surface vacancy concentration as a function of deposition rate is obtained. The vacancy profile through
the thickness of a thick film is then obtained by solving continuum diffusion equations using the surface
vacancy concentration as a moving boundary condition. In contrast to molecular dynamics alone, the
hybrid approach enables calculation of the vacancy content for arbitrary low deposition rates. It reveals
the existence of a deposition rate dependent temperature where the vacancy content exhibits a minimum
value. The 1 ppm iso-vacancy contour in the process variable space is found to be a very steep function
of deposition rate and temperature, and depends only weakly on incident flux energy or angle. 0 1997
Abstract-A
Acta Metallurgica Inc.
1. INTRODUCTION
Physical vapor deposition
(PVD) processing
has
grown to become an important
technology
for the
synthesis
of many
of the materials
used in
microelectronic
components,
giant magnetoresistive
devices, photovoltaic
cells, and high temperature
superconducting
films. The defects/microstructures
in
these
PVD
materials
significantly
affects
their
performance in these applications. The defects/microstructures
populations
are dependent
upon the
conditions
used for processing,
i.e. the deposition
rate, the substrate temperature,
and the incident flux
energy/angle
(to the substrate normal). Thus, highly
controllable
vapor
deposition
technologies
are
required for these and the many other materials that
are now synthesized by PVD methods. Sputtering [l],
electron beam evaporation [2] and new directed [3]/jet
[1] vapor deposition
processes are being developed
for the deposition
of a wide range of materials,
including
metals,
ceramics,
and semiconductors.
Directed/jet
deposition
processes
are of interest
because they use inexpensive, high rate evaporation
methods in combination
with transonic helium jets to
create concentrated
molecular beams of an evaporant. If the jet velocity is high (say Mach l-3) and the
evaporant
has an atomic mass significantly greater
than that of the helium, then the evaporant impinges
directly on to the substrate with an incident angle
defined by the jet axis and the substrate normal, and
with an incident (kinetic) energy directly related to
the jet velocity. The process conditions of directed/jet
vapor deposition processes can be widely varied. For
example, the substrate
temperature
can be varied
from below 0.2-0.6 T,,, (where T, is the melting
temperature)
or more, the deposition
rate can be
changed from less than 0.1-100 p/min, the incident
energy can be controlled between 0.1 and 2.0 eV, and
the incident flux angle can be adjusted to lie anywhere
between 0 and 90” [3].
Vacancies
can be easily incorporated
in films
synthesized
by any of the PVD methods.
These
vacancies increase the electrical resistivity of conducting films, and modify many of the other mechanical/
physical properties of deposited structures. They may
also aggregate to form voids or dislocation loops and
they can play an important role in the development
of film stress [4, 51. The concentration
of these defects
appears to be a sensitive function
of processing
conditions.
Experiments
[6, 71 reveal that defect
concentrations
increase when the substrate temperature/incident
adatom energy decreases, or when the
deposition rate/incident
angle increases. Thus, efforts
to more rapidly deposit thin films often result in a loss
of performance
due to the incorporation
of defects.
A detailed understanding
of vacancy concentration
as
a function of processing
variables, as well as the
underlying
mechanisms
controlling
such relations,
are needed to identify optimal processing conditions
for all PVD processes.
4441
4442
ZHOU et al.:
VACANCY FORMATION
Several modeling strategies could be used to explore
processing-defect relationships for PVD film synthesis. Simulations based on continuum diffusion
approaches [8] are computationally
fast and well
suited to the analysis of long duration (hours)
processes. However, these approaches are not able to
provide any insight about vacancy formation. The
population of point defects in a vapor deposited film
results from kinetically limited atomic scale assembly
events on a film surface that is being continually
buried beneath new adatoms. Simple atomistic
concepts have revealed that the surface assemblies
formed during deposition
can vary drastically
depending on the deposition conditions and on the
form of the interaction between the depositing and
the substrate atoms. For example, a film can grow by
either the Frank-van der Merwe (FM) layer-by-layer
mode [9], the Volmer-Weber (VW) three-dimensional
nucleation mode [lo], or by the Stranski-Krastanov
(SK) (a mixture of FM and VW) mode [l 11.
While these approaches help understand surface
morphology, they unfortunately provide little quantitative information about point defect formation rates.
To understand how vacancies are trapped in thin
films during vapor deposition requires a detailed
examination of the mechanisms of adatom surface
diffusion on the deposition surface. An initially
energetic adatom that arrives at a surface may
redistribute by kinetic energy induced jumps, by
thermally activated diffusion, or by the collapse of
metastable configurations of atoms [12, 131. During
thermally activated diffusion, the adatom’s immediate environment governs the diffusion rate [1420].
For instance, the binding energies are a function of
the atomic configuration surrounding the atom and
the activation energies for diffusion are different for
an atom that is moving along a flat surface, a ledge,
or from/to a kink, etc. [14, 151. The activation
energies and the pre-exponential diffusion coefficients
are also functions of the crystallographic type of
surface plane, the jump direction, and the jump
character (i.e. in channel, cross channel, direct hop,
exchange, etc.) [14-191. Atoms can also move by the
mechanism of dislocation motion or as clusters on
surfaces [20]. High vacancy formation results when
these transport processes are insufficient to move
atoms into all the vacant lattice sites before the
surface is buried. Unfortunately,
none of these
atomistic transport processes can be directly addressed in a continuum or simple atomistic model.
However, with the emergence of faster computers, it
is now possible to conduct more complex simulations
at an atomistic level that capture the complexity of
the vacancy formation
process during vapor
deposition [12, 13,21-48].
Several atomistic modeling approaches can be used
to simulate vacancy formation during vapor deposition. The Monte Carlo (MC) method has been
widely used for the study of various surface problems
[21-281. A Monte
Carlo
procedure
can be used to
DURING
VAPOR
determine the vacancy diffusion parameters in alloys
[49]. Recent Kinetic Monte Carlo (KMC) methods
[21,22] are able to incorporate many of the kinetic
features of vapor deposition, and are very successful
in revealing the development of surface roughness
and defect structures as a function of substrate
temperature, deposition rate and flux incidence angle
[22]. Voter’s [23, 241 multistate Transition-State
Theory (TST) can even account for the dynamics on
a short-time scale and so provides a more accurate
version of the method. However, the underlying
drawback of all MC methods is that they are based
on an Ising lattice, and the MC moves are usually
predetermined based on approximations to a set of
selected thermally activated jump events [22]. Since
the MC methods do not explicitly account for the
interaction of an energetic adatom as it impacts the
surface, they are also unable to provide insight about
the important
role of adatom energy on the
formation of vacancies [13].
The Molecular Dynamics (MD) method traces the
time dependent
trajectory of atoms during a
deposition process based on user defined interatomic
potentials and Newton’s force law. Hence, the
approach rather precisely describes the history of all
the atom locations in a microstructure [12, 13,2946]. Recent studies have used the MD method for
detailed
two-dimensional
(2D) simulations
of
vacancy formation during Ni on Ni vapor deposition
over a range of deposition temperatures,
flux
energies and incident angles [13]. However, the MD
approach suffers from a serious time scale problem.
In order to deposit a sufficiently thick film suitable
for analysis within a practical computational period
(say about a day on a typical workstation), a very
high deposition rate (- lo9 pm/min) has to be
employed [13]. This deposition rate is nine orders of
magnitude or higher than those used in many
deposition processes and therefore is of questionable
value for real processes.
Here, we develop a MD-continuum hybrid model
to predict the dependence of vacancy content on
processing conditions. The approach uses atomistic
calculations to obtain the surface vacancy concentration in a few monolayers
formed at high
deposition rates. Continuum methods are used to
evolve the vacancy concentration to that of a low
deposition rate process. This result is then used as a
moving boundary condition in a numerical analysis
of bulk vacancy diffusion during subsequent film
growth. The approach enables the prediction of a
vacancy concentration
profile through a film’s
thickness as well as study of the effects of substrate
temperature, incident flux energy and incident angle
upon vacancy content for the entire range of
deposition rates accessible with modern vapor
deposition techniques. We illustrate the method with
a systematic study of the 2D deposition of nickel
since detailed MD results have recently been
presented for this system [13].
ZHOU et al.:
VACANCY FORMATION
2. THE HYBRID MODEL CONCEPT
Since diffusion in the bulk is much slower than that
at a free surface, the vacancy content of thin films
synthesized by physical vapor deposition processes is
governed by a competition
between kinetically
limited adatom surface diffusion (to incipient vacancy
sites) and the burial of these sites by depositing
atoms. Since the rate of diffusion is a sensitive
function of temperature, the vacancy content will
depend on deposition temperature and deposition
rate. Earlier work [13] has shown that the incident
energy and incident angle are also important. The
adatom incident kinetic energy provides additional
energy to facilitate local short time scale atomic
reconstruction
of the surface. The incident angle
controls the surface geometry (through a shadowing
effect) which influences the density of incipient
vacancy sites. Thus, deposition can be thought of as
a process in which an instantaneous
(non-equilibrium) vacancy concentration (determined primarily by the incident energy and angle of the adatom)
is created in the surface of a film. Subsequent thermal
migration permits some annealing of the high energy
atomic configurations leading to a reduction of the
instantaneous vacancy concentration.
One example of these two processes is illustrated in
Fig. 1. In Fig. l(a), a vapor atom “a” ballistically
arrives at point A on a surface. If the adatom has
sufficient energy (roughly a sum of the incident
kinetic energy, E,, and a fraction of the potential
energy or latent heat transferred to the adatom upon
(a) Vacancy formation dunng impact
(b) Vacuay
elimrsation during annealing
Fig. 1. Schematic of surface vacancy configuration upon
deposition and its subsequent evolution during annealing.
DURING
VAPOR
4443
adsorption), the adatom at point A may overcome
the energy barrier impeding its diffusion to the lattice
site B, leaving a vacancy at V. Figure l(a) suggests
that the chance of creating a vacancy will be a
function of both E, and the incident angle, t?. For
instance, if E, is high and 0 corresponds to a low
energy path, then atom “a” may penetrate the lattice
to directly achieve lattice site “V” without creating
any buried vacancies. Unlike a thermally activated
event which is “vibrational” and may take significant
time to succeed, such a kinetic energy assisted jump
is an “immediate” jump and is associated with a very
short time scale (- 10-12s) [30,4&48]. Thus, the
vacancy at V as described above can be viewed as
formed instantly after atom “a” is added, and the
corresponding vacancy concentration is termed the
instantaneous concentration, C,. Figure l(b) indicates
that annealing of the configuration
at a given
temperature might lead to the elimination of the
vacancy at V due to the consecutive thermally
activated jumps of two surface atoms. Even though
activation barriers to surface diffusion are low
compared to those for bulk diffusion (see [22] for
details), a successful jump may still require many
jump attempts and so the vacancy at V could persist
for a significant time.
If deposition is rapid, surface atoms near the
incipient vacancy become buried requiring higher
activation energy bulk diffusion to then remove
vacancies. The net vacancy concentration is then
determined by an interplay between high atom
mobility near the surface (which increases with
temperature) and the time available before the
surface is covered by additional atoms (which
decreases with deposition rate). The result is that thin
films deposited at higher rates normally contain more
defects [6], although continued bulk diffusion could
continue within the film during subsequent annealing,
albeit with slower kinetics.
From the example above, a model for calculating
the net vacancy concentration during physical vapor
deposition must address three issues:
1. The “instantaneous” formation of vacancies. The
complicated atomic assembly sequences must be
obtained using methods that allow the role of the
incident atom kinetic energy to be involved. The
results of earlier MD calculations [13] provide a
convenient empirical formula for the instantaneous
vacancy concentration, C,, of a high rate deposition
process as a function of the incident energy, the
incident angle, and the substrate temperature.
2. The low deposition rate of realistic processes.The
MD calculated surface vacancy concentrations
correspond to unrealistically high deposition rates.
However, the instantaneous concentrations in a thin
layer of “pulse” deposited material can be annealed
by simulating diffusion. Thus, if thin layers are
deposited at the MD time scale and are then
interspersed with periods of simulated diffusion with
4444
ZHOU et al.:
VACANCY FORMATION
no deposition, the vacancy concentration can be
obtained at the surface for any deposition rate (by
adjusting the annealing time). The result would be a
vacancy concentration in the near surface region, C,,
which can be much less than Ci for low rate/high
temperature processes. The approximation mimics
the real case of constant low rate deposition which
can essentially be viewed as the deposition of single
atoms (at high rates) with annealing in between.
3. The vacancy concentration profile through the
film thickness. The contribution of bulk diffusion
varies through the thickness of a film. However, this
can be analyzed by solving the diffusion equation
using a moving boundary condition that the surface
vacancy concentration remains at C, throughout the
period of deposition.
3. MODEL
DURING
VAPOR
Fig. 2. Geometry of “pulsed” deposition layer.
STRUCTURE
3.1. Molecular dynamics deposition
A method for calculating the vacancy concentration of a thin film under deposition rates amenable
to MD calculations (10 m/s) has been developed
based on a full Embedded-Atom Method (EAM)
model for nickel [13]. To incorporate the effects of
surface roughening due to shadowing, thin films 16 or
more monolayers thick were deposited to enable an
accurate estimate of the vacancy concentration.
Two-dimensional calculations were carried out as a
function of the incident energy E, (in the range from
0.1 to 2.0 eV), incident angle 0 (in the range from 0
to 60”), and substrate temperature T (in the range
from 100 to 1100 K). Since almost no annealing
accompanied these simulations, an analysis of these
results led to a convenient empirical formula relating
the instantaneous
vacancy concentration
Ci (in
expression as an atomic fraction, i.e. number of
vacancies/number of lattice sites) to Ei, 0 and T:
CULBJ’) = K(E,,B)C$
[ 1 - exp(
-ff,)] (1)
where
K(Ei,O) =
(2)
k is Boltzmann constant, Co = 0.0086, EO= 1.0 eV,
Ep = 0.0273 eV, and & = 30”. The angular dependence factor, K, is 1 for Q = 0” (normal incidence),
and Ep is sufficiently small to yield only a minor
temperature dependence of C,.
3.2. Pulsed atomic d$fiision
A surface layer of thickness, h, with a vacancy
content, C,, obtained from the above MD simulations
of deposition (at a deposition rate of 10 m/s), can be
annealed
for a time t,, = h/R to obtain an
approximation of the layer vacancy content at a
lower rate, R, of deposition. An important issue here
is the appropriate selection for h. MD simulations
indicate that a high incident energy of 2.0eV may
cause the reconstruction
of atomic configuration
within about five monolayers near surface. At a low
incident energy of 0.1 eV, surface roughness of five or
more monolayers frequently develops. To account for
the effects of both incident energy induced atomic
reconstruction and surface roughness, an h of 20 8,
(about 10 monolayers) thick is used. Detailed
simulations have revealed that the results are not
highly sensitive to h in the range of 5-15 monolayers.
Annealing between “pulsed” deposition can be
simulated by solving the continuum equations of
diffusion. The geometry of the “pulsed” deposition
layer is illustrated in Fig. 2. The one-dimensional
diffusion problem involves starting with a constant
concentration of vacancies, Ci, in an infinite plate
(thickness in the y direction) and calculating the
decay in this concentration as a function of position
and time. The boundary conditions for the diffusion
are that the flux from the newly deposited surface
layer into the preexisting surface is zero and that the
vacancy concentration at the surface is the thermal
equilibrium
value C, = exp(Ef/kT),
where Ef is
vacancy formation energy. The zero flux boundary
condition does not mean that there will be no vacancy
diffusion into the bulk in the long term, but that such
diffusion is negligible in the short time period of the
transient effect being studied here. The surface
boundary condition has a negligible effect on the
results of this calculation under the processing
conditions of interest because the vacancy concentration in the interior of this thin surface layer is
much greater than the thermal equilibrium value. The
constant C, surface boundary condition is used
because this ensures that an equilibrium vacancy
concentration
is achieved in the limit of low
deposition rate or high temperature. The initial
condition of Ci implies that at a very high deposition
ZHOU
et al.:
VACANCY
FORMATION
rate, little diffusion occurs and the average vacancy
concentration in the layer will be high (close to C,).
Thus, the continuum diffusion problem is specified
by:
ac(vt) = $[D(*)Yj
-+
C(y:O) =
(3)
c,
(4)
C(h,t > 0) = c,
aC(Y,t)_
at
(5)
=
o
ay
4445
D
aw
)
t)
R
I
acbd
(9)
aY2
C(y < 0,O) =
c,
(10)
C(O,t > 0) =
c,
(11)
v _ 0.1
where D(y) is a position dependent diffusion
coefficient, and C = C(y,t) is the vacancy concentration as a function of position and time.
The diffusion equation can be solved using finite
difference methods and yields a vacancy concentration profile within the layer of thickness h. An
average surface vacancy concentration,
C,, in the
layer after time th is thus given by
cs= ;
VAPOR
value C,) can be treated by solving the continuum
diffusion equation with the position of the surface
continuously
growing at the rate R and with
boundary
conditions
that the surface vacancy
concentration
remain constant at C, while the
concentration deep within the substrate remains at
the equilibrium value, C,. The problem of diffusion
in the substrate and in a film growing in the y
direction can then be written as
and
acw
DURING
s
hC(y,th)d,
0
with th = h/R.
The diffusion coefficients, D, as a function of
temperature is obtained through Arrhenius equation
and
C(-m,t)
= c,
(12)
where the growing surface is defined by y = 0 and D
is taken to be the position independent bulk diffusion
coefficient. Again finite difference methods can be
used to solve for the vacancy concentration profile in
the preexisting substrate and in the growing film. The
use of the boundary condition (11) results in a small
transient anomalous enhancement of the vacancy
content. This arises because the vacancy profile
across the substate interface starts out as a step
function which introduces a vacancy flux greater than
that of the physical deposition process. However, this
transient very rapidly decays with deposit thickness
and makes a negligible contribution to the results.
3.4. Temperature scaling
Molecular Dynamics and Molecular Statics can be
employed to determine the vacancy formation
energy, Ef, the pre-exponential coefficient, DO, and
activation energy for vacancy migration, E,,,. Three
sets of pre-exponential coefficients and activation
energies were considered; those for diffusion within a
distance 6 of the surface, those between 6 and 26 of
the surface, and those for bulk diffusion at distances
greater than 26 of the surface. The {1 11} interplanar
distance in nickel is used for 6 (S = (3~*/8)“~, where
the lattice constant, a, is reduced from 3.52 to 3.42 8,
due to the 2D approximation [13]). Details of the
vacancy formation energy and the diffusion coefficients calculations are given in the Appendix. A
vacancy formation energy of 1.182 eV is obtained.
The activation energy for diffusion in the two atomic
layers closest to the surface is essentially for vacancy
diffusion in the direction towards the surface. This is
a reasonable approximation
since it will be the
dominant flux direction in the transient period under
study. The results obtained
for the diffusion
parameters are summarized in Table 1.
3.3. The vacancy concentration projle
Bulk diffusion within the growing film and
substrate (with a starting vacancy concentration
The intent of the present work is to obtain
guidelines for optimizing physical vapor deposition
of many materials rather than to carry out
calculations specific only to nickel. It is therefore
desirable to give the results in terms of homologous
temperature T/T,,,, where T, is the absolute melting
temperature. Although the 2D EAM model is based
on parameters fitted to physical quantities for
three-dimensional
(3D) nickel, the calculated 2D
vacancy properties are significantly lower than the 3D
values. Therefore, the melting temperature
for
determining
homologous temperatures
has been
scaled by multiplying the nickel melting temperature
by the ratio of the self-diffusion activation energy for
computational 2D nickel (the sum of the vacancy
formation and migration energies) to the experimental value for 3D nickel. A calculated vacancy
formation energy of 1.182 eV and an activation
energy for bulk diffusion of 0.908 eV yields a 2D
Table
1. Diffusion
parameters
for vacancy
different positions
diffusion
Vacancy jump paths
TO
Surface plane
2nd plane
> 2nd plane
FtW??
2nd plane
3rd plane
>2nd plane
6.325
5.925
6.525
21.142
12.994
51.239
0.583
0.642
0.908
at
4446
ZHOU et al.:
DURING
1
10-l
1O-2
103
10-4
10s
104
0”
5
‘E
2
0
lo-7
'g
10-a
q
VACANCY FORMATION
VAPOR
Empiricalcalculation
10-9
It-l-to
T = 600K
10-1*Li
0
’
I
20
t
I
40
(a) Temperature Effect
I
I
I
I
100
60
80
I
Empiricalcalculation
1
0”
5
“0
E
.o
g
I
I
10-l
10-2
103
10-4
lo-5
IO+
I
??
)._.._*_L.A--JI
1
I
I
Empiricalcalculation
lo-7
Numericalcalculation
1P
T= 500K
E;=O.leV
ti
(c) Incident
Angle Effect
-
Deposition rate R (pm / min)
Fig. 3. Surface vacancy concentration
self-diffusion activation energy of 2.09 eV, while the
3D experimental value is 2.88 eV [50]. The ratio of
these values is used to scale the melting temperature
from the physical value of T,,, = 1726 K to
T,,, = 1250 K for 2D nickel.
4. RESULTS
AND DISCUSSION
4.1. Surface vacancy content
The surface vacancy content, C, calculated from
equation (7) as a function of deposition rate is shown
as a function of deposition rate.
in Fig. 3. The five curves in Fig. 3(a) correspond to
five different deposition temperatures but the same
incident energy of 0.1 eV and incident angle of 0”.
The three curves in Fig. 3(b) show the effect of
incident energy for a fixed deposition temperature of
500 K and an incident angle of O”, while the three
curves in Fig. 3(c) show the effect of incident angle
at a fixed deposition temperature of 500 K and an
incident energy of 0.1 eV. It can be seen from Fig.
3(a) that at a low temperature of 300 K (T/
T,,, = 0.24), the vacancy concentration remains high
ZHOU et al.: VACANCY
4441
FORMATION DURING VAPOR
and is more or less independent of rate for the entire
deposition rate range considered. This arises because
the diffusion coefficient is very small at low
temperatures, and thus vacancies cannot migrate out
of the film, even when more annealing time is
provided by a low deposition rate. Figure 3(a) shows
that as the temperature is increased, first to 450 K
(T/T,,, = 0.36) and then to 500 K (T/T, = 0.40), the
vacancy concentration decreases and also becomes a
decreasing function of decreasing deposition rate.
This is consistent with more vacancy diffusion out of
the film as the temperature and annealing time
increase. Interestingly, when the temperature is raised
to 550 K (T/T,,, = 0.44) vacancy concentration
becomes almost independent of rate in the low rate
regime, and when the temperature is raised to 600 K
is almost
(T/T, = 0.48) vacancy concentration
independent of rate for the entire deposition rate
range studied. This is because the equilibrium
vacancy concentration is very nearly achieved at the
low rate at 550 K and even at the highest rate at
600 K. It is intriguing to note that since the
equilibrium vacancy concentration
increases with
temperature, the vacancy concentration at the lowest
deposition rate is higher for 600 K than for 500 or
550 K. We also note that in the intermediate
temperature range (450-550 K), the vacancy concentration drops quickly when the deposition rate is less
than a temperature dependent critical value. The
calculations predict that in order to produce films
with a low point defect concentration (say less than
1 ppm) at temperatures of 500 K, the deposition rates
should be less than 5 pm/min, which is commonly
used in experiments.
Figure 3(b) and (c) shows that at high deposition
rates both the adatom kinetic energy and the angle of
incidence also influence the vacancy concentration
(through their effect on C,). The vacancy concentration is seen to decrease with increase in incident
energy and decrease in incident angle. As the
deposition rate is decreased, diffusion during the
period of annealing at 500 K becomes sufficient to
merge the curves and the surface vacancy content
becomes a function of the deposition rate and is
independent of incident energy/angle.
The behavior of C, as a function of temperature
calculated from equation (7) is shown in Fig. 4. The
six curves in Fig. 4(a) correspond to six different
deposition rates at the same incident energy of 0.1 eV
and incident angle of 0”. In Fig. 4(b), the effect of
incident energy for an incident angle of 0” and a
deposition rate of 1 pm/min is shown, while in Fig.
4(c), the effect of incident angle at an incident energy
of 0.1 eV and a deposition rate of 1 pm/min is
revealed. Figure 4 indicates that there exists a critical
temperature above which the vacancy concentration
drops rapidly with temperature until it intersects a
line corresponding to the equilibrium concentration
of vacancies. This critical temperature is also a strong
function of deposition rate. Lower deposition rates
have significantly lower critical temperatures because
of the greater available diffusion time. For a fixed
incident energy and an incident angle, the vacancy
concentrations
for the different deposition rates
converge to a single value (CJ at low temperatures
because of the absence of apprciable diffusion. In this
low temperature regime, E, and 0 both contribute to
C, and the surface vacancy concentration
thus
becomes a function of both but is independent of R
and T. The data in the figures show that at high
temperatures, all of the vacancy concentration curves
again converge on to a single line determined by the
equilibrium vacancy concentration only. In this high
temperature
regime, the vacancy concentration
becomes independent of all process variables except
deposition temperature.
The surface vacancy concentration is an important
parameter to control in practical aplications, and an
approximate formula for its calculation is desirable.
Solutions for equations (3)-(6) can be analytically
found for the case where the constant bulk diffusion
coefficient, D, is used throughout the plate:
If the decay of the n = 0 term is taken as emkf, the
decay of the n = 1 and n = 2 terms goes as e-9i’ and
e - 2Sk’respectively,
,
and higher order terms decay even
faster. Thus, after a brief transient, the decay in
vacancy concentration is controlled by the n = 0 term
in equation (13) and
C(y,t) =
c, + 4(C, n-
G)
cosz
exp[ -D (G)il]
(14)
The average vacancy concentration
time th is then
in the layer at
G=C~+$(C--C.)exp[-D($--$I
(15)
This term alone yields reasonable agreement with the
detailed numerical calculations but neglects both the
transient and the region within 26 of the surface in
which the diffusion is significantly faster. However,
with this solution as a guide, the functional form
cs= c, +
a = D
&e-a
(16)
(17)
has been compared to the predictions of the full
model in Figs 3 and 4. Overall, it has been found to
give an excellent fit to the numerical results over all
parameter ranges tested.
4448
4.2.
ZHOU
et al.: VACANCY
FORMATION
‘The vacancy concentration profile
I
VAPOR
constant temperature of 550 K, an incident energy of
0.1 eV, and an incident flux angle of 0”. The thickness
coordinate, y, shown in Fig. 5 is chosen so that y < 0
represents the substrate. It can be seen that a fraction
of vacancies diffuses into the substrate, causing a
smooth transition near the substrate/deposit interface. Although a longer time is available for bulk
diffusion at a lower deposition rate, the total
vacancies transported to the substrate (i.e. the shaded
area in Fig. 5) decrease with decreasing deposition
rate. Since vacancies more easily diffuse to and
investigate the role of bulk diffusion upon the
vacancy concentration in subsurface material, numerical calculations were conducted to calculate the
vacancy concentration profile through the thickness
of a growing film. For these calculations, 10 pm thick
films were deposited under various conditions and the
system of equations (9H12) w&e solved using a
constant surface concentration
of vacancies C,.
Figure 5 shows the effect of decreasing the deposition
rate on the vacancy profile in a film deposited at a
To
I
DURING
I
I
!
I
(a) Deposition Rate
tI=O” f,=O.leV
300
350
400
450
550
500
600
-
650
(b) Incident Energy Effect
Empiricalcalculation
Numericalcalculation
8=0”
300
350
400
450
500
R=l~m/min
550
600
-
650
(c) Incident Angle Effect
Empiricalcalculation
Numerical calculation
R= l~m/min
300
350
400
450
Temperature
Fig. 4. Surface
vacancy
concentration
550
500
Ei=O.leV
600
T (K)
as a function
of substrate
temperature,
650
ZHOU
et al.:
VACANCY
FORMATION
4x10-7
*
Deposit
Thickness y (Fm)
Fig. 5. Vacancy concentration
profile along 0.01 mm thick
films deposited at different deposition rates.
disappear
at the surface over the available diffusion
time, decreasing the deposition rate causes the surface
vacancy concentration,
C,, and hence the vacancy
concentration
gradient
at the substrate/deposit
interface to be reduced as shown in Fig. 5. This
gradient reduction accounts for the lower vacancy
flux into the substrate.
Simulation of the vacancy concentration
profile at
various substrate temperatures
but fixed deposition
rate and incident energy/angle
indicates that since
increasing
temperature
and decreasing
deposition
rate both enhance
diffusion,
increasing
substrate
temperature
has a similar
effect as decreasing
deposition rate on the vacancy concentration
profile
in the film.
An important finding in Fig. 5 is that the overall
variation of vacancy concentration
through the entire
film thickness is very little. This conclusion is general
for other process conditions.
In conditions
where
significant
diffusion does not occur, the vacancy
concentration
is almost uniform through the film
thickness. In conditions
where significant diffusion
occurs, the surface vacancy concentration,
C,, is so
low that
the vacancy
concentration
difference
between surface and substrate is small.
These
calculations
do not
account
for all
mechanisms of vacancy transport. A small fraction of
the vacancies at ‘vacancy concentrations
well above
the equilibrium
concentration
will tend to cluster,
create
vacancy
dislocation
loops,
annihilate
at
existing
dislocations
or grain
boundaries
and
segregate to regions of compressive
stress. None of
the contributions
of these transport mechanisms are
included in the analysis and so the results slightly
overestimate
the vacancy content.
4.3.
Vacancy content-processing
DURING
4449
VAPOR
incident angle 0 = O’, results in a 3D plot relating the
log of the surface vacancy concentration,
C,, to the
log of R and T/Tm, as shown in Fig. 6. It can be seen
that the C, surface is sharply divided into regions of
high and low vacancy content. Within the ranges of
the parameters
studied here, the factors other than
deposition
rate and homologous
temperature
only
have a minor effect because, at the 1 ppm level, the
concentration
changes by an order of magnitude in
4 K at the lower deposition
rates and 7 K at the
higher deposition
rates. Thus, even a variation in
other parameters which yields an order of magnitude
change in vacancy concentration
only shifts the
dividing curve by several degrees. It is interesting to
note that the curves similar to the ones shown in Fig.
6 are found in radiation
damage
studies with
vacancies frozen into the lattice at low homologous
temperatures,
annealing
occurring
with increasing
temperature,
and thermal equilibrium dominating at
high temperatures.
To a good approximation
the vacancy content will
be high (i.e. C, > 10m6) if
0
R
log R,
1
‘A-“TiT,
where R,,, a scaling factor, is 1 pmjmin,
the factor B is approximated
by
B
=
EmlkTm
ln(lO)
(18)
A = 5.2, and
(19)
Since the vacancy migration energy, E,,,, scales with
melting temperature
with E,,,/kT,,, x 8.5 for most
metals (8.4 for the present parameters), B has a value
relations
The results reported
above indicate
that the
variation of vacancy concentration
along the film
thickness
is small compared
with
C,. Hence,
dependence of vacancy content on process variables
is well represented
by C,. Combining
calculations
under various substrate temperatures
and deposition
rates for a fixed incident energy E, = 1 eV and an
R, = (pm / min.)
Fig. 6. Surface vacancy concentration
substrate temperature
and deposition
and 0 = 0”.
as a function
of
rate at E, = 1.0 eV
4450
ZHOU et al.:
VACANCY FORMATION
of 3.7. Thus, only the factor A is material dependent.
While many approximations
used in this study,
including the 2D lattice dynamics, will have an effect
on its value, they should have little effect on the form
of equation (18). Even here, the magnitude of A can
be estimated by
A =
log(s)- log~~(4~8)~]}
(20)
where Ck is the critical concentration chosen to divide
high and low concentrations (e.g. 1 ppm or 10-6),
which yields physically realistic results. These
approximate forms for A and B are found by
simplifying equations (l), (16) and (17) to
Administration
u=26R
through
REFERENCES
Engineering, 1997,.B28, 57.
4. Johnson, R. A., Surf. Sci., 1996, 355, 241.
5. Smith, R. W. and Srolovitz, D. J., J. Appl. Phys., 1996,
79, 1448.
6. Ratnaparkhi, P. and Wadley, H. N. G., Acta mater. (in
9.
(22)
10.
11.
D
(D. Brewer, Program Monitor)
In Handbook of Deposition Technologies for Films and
Coatinns, ed. R. F. Bunshah. Noves Publications. Park
Ridge,-1994, p. 1.
2. In Electron Beam Technology, ed. S. Schiller, U. Heisig
and S. Panzer. Wiley, New York, 1982, p. 1.
3. Groves, J. F. and Wadley, H. N. G., Composites
(21)
and
VAPOR
NASA grant NAGW1692.
8.
C, = C,e-a!,
DURING
preparation).
Thornton, J. A., J. Vat. Sci. Technol. 1975, 12, 830.
Wang, Y., Wang, H. Y., Chen, L. Q. and Khachaturyan, A. G., J. Am. Ceram. Sot., 1995, 78, 657.
Frank, F. C. and van der Merwe, J. H., Proc. R. Sot.,
1949, A198, 205.
Volmer, M. and Weber, A., Z. Phys. Chem., 1926,119,
277.
Stranski, J. N. and Krastanov, L., Ber. Akad. Wiss.
Wien, 1938, 146, 797.
(23)
with D specified by equation (8). Although model
dependent, equation (18) indicates the functional
dependence
on other processing and material
parameters.
5. CONCLUSIONS
A hybrid modeling approach has been developed
to predict the effect of process conditions upon the
defect concentration of vapor deposited metals. The
approach uses continuum
analyses of vacancy
diffusion combined with atomistic calculations of
vacancy formation to predict the vacancy content of
vapor deposited nickel as a function of deposition
rate/temperature and flux incident energy/angle. The
model leads to a universal relationship for the
vacancy content of metal films and predicts:
1. The existence of a low temperature regime
(T < 0.25 T,) where the vacancy content is independent of deposition rate and temperature and depends
weakly on incident adatom energy and angle.
2. The presence of minimum vacancy concentration determined by the deposition rate and
substrate temperature only. In general, reducing the
deposition rate decreases the temperature at which
the minimum vacancy content occurs.
3. For most practical applications, a vacancy
content below 1 ppm is sufficient. The 1 ppm
iso-vacancy contour in process space is a very
sensitive function of deposition temperature and rate,
but depends only weakly on flux energy and angle.
4. Only small variations of vacancy concentration
are observed through the thickness of deposited films.
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
12. Gilmore, C. M. and Sprague, J. A., Phys. Rev., 1991,
B44, 8950.
13. Zhou, X. W., Johnson, R. A. and Wadley, H. N. G.,
Acta mater. 1997, 45, 1513.
14. Liu, C. L. and Adams, J. B., Surf. Sci., 1993, 294, 197.
15. Liu, C. L. and Adams, J. B., Surf. Sci., 1992, 265, 262.
16. Liu, C. L.. Cohen. J. M.. Adams. J. B. and Voter. A.
F., surf. sci., 199i, 253,‘334.
17. Xu, W. and Adams, J. B., Surf. Sci., 1994, 319, 58.
18. Shiang, K. D., Wei, C. M. and Tsong, T. T., Surf. Sci.,
1994, 301, 136.
19. Perkins, L. S. and DePristo, A. E., Surf. Sci., 1994,317,
L1152.
20. Hamilton, J. C., Daw, M. S. and Foiles, S. M., Phys.
Rev. Lett., 1995, 74, 2760.
21. Chason, E. and Dodson, B. W., J. Vat. Sci. Technol.,
1991, A9, 1545.
22. Yang, Y., Johnson, R. A. and Wadley, H. N. G., Acta
mater. 1997, 45, 1455.
23. Voter, A. F., Phys. Rev., 1986, B34, 6819.
24. Voter, A. F., Modeling of Optical Thin Films, 1987,821,
214.
25. Rittner, J. D., Foiles, S. M. and Seidman, D. N., Phys.
Rev., 1994, B50, 12,004.
26. Faux, D. A., Gaynor, G., Carson, C. L., Hall, C. K. and
Bernholc, J., Phys. Rev., 1990, B42, 2914.
27. Dodson, B. W. and Taylor, P. A., Phys. Rev., 1986,
B34, 2112.
28. Bolding, B. C. and Carter, E. A., Surf. Sci., 1992, 268,
142.
29. Rafii-Tabar,
H., TambyRajah, A. L., Kamiyama, H.
and Kawazoe, Y., Modeling Simul. Mater. Sci. Engr.,
1996, 4, 101.
30. Gilmer, G. H. and Roland, C., Radiation Effects and
Defects in Solids, 1994, 130/131, 321.
31. Gilmore, C. M. and Sprague, J. A., J. Vat. Sci.
Technol., 1995, A13, 1160.
32. Rongwu, L., Zhengying, P. and Yukun, H., Phys. Rev.,
1996, B53, 4156.
33. Gilmer, G. H., Roland, C., Stock, D., Jaraiz, M. and de
la Rubia, T. D., Mater. Sci. Engr., 1996, B37, 1.
34. Averback, R. S., Ghaly, M. and Zhu, H., Radiation
Eficts and Defects in Solids, 1994, 130/131, 211.
35. Katagiri, M., Kubo, M., Yamauchi, R., Miyamoto, A.,
Nozue, Y., Terasaki, O., Coley, T. R., Li, Y. S. and
Newsam, J. M., Jpn. J. Appl. Phys., 1995, 34, 6866.
36 Muller, K. H., Surf. Sci., 1987, 184, L375.
ZHOU
et al.:
VACANCY
FORMATION
37. Muller, K. H., Phys. Rev., 1987, B35, 7906.
38. Taylor, P. A. and Dodson, B. W., Phys. Rev., 1987,
B36, 1355.
39. Raeker, T. J. and DePristo, A. E., J. Var. Sci. Technol..
1992, AlO, 2396.
40. Raeker, T. J., Sanders, D. E. and DePristo, A. E.. J.
Vat. Sci. Technol., 1990, AS, 3531.
41. Raeker, T. J. and DePristo, A. E., Surf: Sri., 1991, 248,
134.
42. Blandin, P., Massobrio, C. and Ballone, P., Phys. Rev.,
1994, B49, 16,637.
43. Haftel, M. I. and Rosen, M., Phys. Rec., 1995, B51,
4426.
44. Schneider, M., Rahman, A. and Schuller, I. K.. Phys.
Rev. Left., 1985, 55, 604.
45. Hara,
K., Ikeda, M., Ohtsuki,
O., Terakura,
K.,
Mikami, M., Tago, Y. and Oguchi, T., Phys. Rev., 1989,
B39, 9476.
46. Gilmer, G. H. and Roland, C., Appl. Phys. Left., 1994,
65, 824.
47. Dodson, B. W., Phys. Rev., 1987, B36, 1068.
1987. B5,
48. Dodson,
B. W., J. Vat. Ski. Technol.,
1393.
49. Zhao, L., Najafabadi,
R. and Srolovitz, D. J., Acta
mater., 1996, 44, 2737.
50. Wycisk,
W. and Feller-Kniepmeier,
M., J. Nucl.
Mater., 1978, 69/70, 616.
51. Johnson. R. A., Phys. Rev., 1989, B39, 12,554.
52. Fletcher, R. and Reeves, C. M., Comput. J., 1964, 7,
149.
DURING
4451
VAPOR
[ii21
t
I
[oii]
APPENDIX
The vacancy formation energy and diffusion parameters can
be determined
using Molecular
Dynamics/Statics
simulations, with the atomic potential
defined by Johnson’s
EAM model [13, 511. The net effect of decreasing
the
deposition rate is to promote the surface equilibrium which
is inversely proportional
to vacancy
concentration.
By
extrapolating
the vacancy type vs vacancy concentration
data obtained
from MD simulation
[13] to low vacancy
concentration,
the vacancies at low rate are found to be
dominated
by mono-vacancies.
Corresponding
to different
positions with respect to the surface, three different vacancy
types are considered in the present work as shown in Fig.
Al(b)(d).
The 2D arrays seen in Fig. Al are defined by a
(111) plane containing
20 [i lo] atomic rows stacked along
[iI2] direction, each with 20 atoms. Figure Al(a)(d)
shows
a perfect crystal, a bulk vacancy, a vacancy at the first plane
from surface, and a vacancy
at the second plane from
surface, respectively. The initial nearest atomic spacing was
set to be 2.42 A [ 131, and the total potential energy and the
atomic
spacings
of the equilibrium
2D lattice were
determined by Molecular Statics (MS) energy minimization
procedure
using a Conjugate
Gradients
Method [52] with
periodic boundary conditions along both x and y directions.
The vacancy
formation
energy was calculated
as the
potential energy difference between the equilibrated
crystal
with a vacancy far from the surface and the equilibrated
perfect crystal composed
of the same number of atoms,
Er = 1.182 eV. To calculate the activation energy for jumps
in different vacancy configurations,
the atoms marked by 1
in Fig. Al(b) and (d) were stepped and constrained
along
the [Oli] direction toward the vacant site. To prevent the
shift of the whole crystal, a fixed boundary
condition was
used for the bottom surface and the two side surfaces. To
allow the relaxation
of atoms near surface,
the free
boundary
condition
was used for the top surface. The
relaxed crystal configurations
and the associated potential
energies were determined by MS. The activation energies of
the jumps were determined as the energy peaks during the
stepping.
Similar procedure
was used to determine
the
Fig. Al. Crystals for calculation
of vacancy
formation
energy and diffusion parameters:
(a) a perfect crystal; (b) a
bulk vacancy; (c) a vacancy at the 1st plane from surface;
(d) a vacancy at the 2nd plane from surface.
activation energy of jumps for the vacancy shown in Fig.
Al(c). Since the diffusion path of atom 1 is not clear in this
case, it was shifted along the [ilO] direction,
and its
movement along this direction was constrained
in the MS
calculation.
This ensures that this atom migrates naturally
4452
ZHOU et al.: VACANCY FORMATION
along the low energy path. The calculated activation
energies E, are shown in Table 1.
If only the vacancy diffusion along the vertical direction
is considered, the diffusion coefficient can be expressed as
D = hFG*
(Al)
where F is the jump frequency of the nearby atom to the
vacant site, 6 is the distance a vacancy can move along the
vertical direction per jump, and h is a constant. h is taken
to be 2 for the vacancy in Fig. Al(b) since both the atoms
marked by 1 and 2 jumping to the vacant site can cause the
vacancy to move forward. h is taken to be 1 for the surface
vacancy in Fig. Al(c) since the atom marked by 1 jumping
to the vacant site is themost probable event for the motion
of the vacancy. h is taken to be 0.5 for the subsurface
vacancy in Fig. Al(d) since the atom marked by 1 can jump
to two lattice sites, only one of which causes the forward
motion of the single vacancy.
DURING
VAPOR
The jump frequency of atoms
temperature can be expressed by
F=vexp
as a function
of
( >
-n
E,
where v is the vibrational (jump attempt) frequency. The
vibration times of atom 1 were measured directly using MD
calculation at 100 K for 40 ps for the crystals shown in Fig.
Al(b)-(d) under periodic boundary conditions along the
horizontal direction and a free surface boundary condition
in the vertical direction. The vibration frequencies of this
atom were then determined as the vibration times divided by
40. Combining equations (Al) and (A2), the pre-exponential
coefficients for the three diffusions were determined. The
vibration frequencies v and pre-exponential coefficients Do
are also included in Table 1.