- Email: [email protected]
Simulation of the deposition and aging of thin island films
COMPUTATIONAL MATERIALS SCIENCE
ELSEVIER
Computational Materials Science 10 (1998) 139-143
Simulation of the deposition and aging of thin island films P. Bruschi *, A. Nannini Dipartimento di Ingegneria dell’lnformazione: Elettronica, Informatica, Telecomunicazioni, Universitci degli Studi di Piss, via Diotisalvi 2, l-56126 Piss, Italy
Abstract Wo-dimensional atomistic simulations of the growth and post-deposition behaviour of island metal films are described. The program includes a module for the calculation of the film resistance on the basis of a charge limited tunnelling model. The dependence of the resistance on various deposition parameters is investigated. Examples of simulated post-deposition resistance drift are shown. Copyright 0 1998 Elsevier Science B.V.
1. Introduction Island metal films have been studied for their intriguing electrical and optical properties[ 11. Recently these films have been the object of extensive studies since they represent the first stage of growth of many continuous films of practical interest [2,3]. Discontinuous metals have also raised interest for practical applications
in the field of strain sensors for their high
gage factor [4]; so far, long resistance drift due to aging effects, high sensitivity to temperature and lack of reproducibility has prevented the fabrication of reliable devices based on these materials. In this paper we present a series of two-dimensional Monte Carlo simulation of the growth and postdeposition aging of island films. The program can be divided into two main modules: the first is aimed to simulate the microscopical film evolution, the second is used to calculate the resistance of the film. The program permits to relate important parameters such *Corresponding author. Tel.: +39 50 568538; fax: +39 50 568522; e-mail: [email protected].
as the deposition
effects of some deposition parameters size distribution and morphology.
not only to the
on the cluster
In our program, electrical conduction of the films is ascribed, as generally accepted, to a process of charge limited tunnelling
of electrons between the islands [6].
The problem of calculating
the resistance of a film por-
tion is solved by replacing the film itself with an electrical network where the charge exchange between the islands is modelled by equivalent conductances which depend on the size and spacing of the grains. The novelty of this work with respect to previous studies based on a similar approach [7,8] stands in the combination of a very efficient program for morphological simulations with a resistance extractor which operates directly on the simulated films. The simulations have been devoted to show the dependence of the resistance on the fractional coverage of films deposited at different substrate temperatures. In order to reduce statistical fluctuations the
0927-0256/98/$19.00 Copyright 0 1998 Elsevier Science B.V. All rights reserved PII SO927-0256(97)00126-Z
rate and temperature
morphology of the clusters but also to the overall resistance. The structure of the first module is described in details in [5] where it is used for predicting the
I? Bruschi,
140
resistances
A. Nunnini/Computational
Materials
Science
10 (1998)
139-143
has been averaged over a set of values ob-
tained by changing only the seed of the random number generator.
The standard deviation
of each set of
values has been reported to provide an indication
of
the reproducibility. Preliminary
data about the application
gram to the prediction of the post-deposition
0a
of the probehaviour
of the resistance have also been shown.
2. Model of film growth and electrical conduction The substrate is modelled as an N x N array of sites mapped on a rectangular portion of a triangular lattice. At each simulation step, three type of transitions are allowed: (i) arrival of an atom at the substrate, (ii) diffusion of an adatom to an empty nearest neighbour and (iii) re-evaporation
of an adatom from the sub-
strate. The rate r, of atom arrivals is fixed and given by r, = R x N2. where R is the number of impinging atoms per site and unit time. The rate yCd) for a diffusion transition r(d) = ,@) 0
is given by the equation
exp
(1)
’
where r;“’ IS an attempt frequency which is considered to be independent of the occupancy of the neighbours [9], kn is the Boltzmann constant and T the absolute temperature. Indicating with cl-c5 the occupancies the five nearest neighbours of hopping particles, shown in Fig. l(a), the diffusion calculated Ecd’ = Et’
by the following + (cj +
cc4
+ (Cl + C5)EL..
activation
of as
energy is
23kL grain 1 /‘\
grain 2 i 1L
~
c,,
Cl
1
(W
c2
I
Fig. I. (a) Symbolic representation of a particle hop in a triangular lattice showing the five neighbours affecting the activation energy. (b) Equivalent electrical circuit used to calculate the energy of a pair of charged grains.
of Amar an Family [2] and Breeman et al. [3] dealing with square lattices. The algorithm used to determine the rate for reevaporation transitions is described in [5] where it is shown that introducing re-evaporation leads to cluster size distributions which are not in agreement with experimental data. For this reason re-evaporation rate is set to zero in this work. A transition is randomly selected according to a
formula:
+ ccz)E~ (2)
where Ehd) IS a fixed contribution due to the interaction with the substrate, EB the binding energy between two adatoms and EL is the activation energy for diffusion of adatoms along cluster borders. The occupancy ci is defined as 0 if site i is empty and 1 if it is occupied and 6 = 1 - ci. The distinction between the two different contributions EB and EL of the neighbours to the activation energy is also present in the models
probability proportional to the transition rate. At each step the simulation time is incremented by an amount given by the inverse of the sum of all the transition rates. The equivalent electrical network utilized for calculating the resistance is obtained by associating a node to each cluster and introducing conductances between the nodes to model the electron exchange occurring by tunnel effect. The conductance between a pair of clusters is calculated by summing
I? Bruschi, A. Nannini/Computational
up the rates of all the possible volving
hopping
charge transition
of one electron
in-
from a grain to the
other. As shown in [lo], transitions
can be grouped
in “pair of antagonist
each pair gives
the following
transitions”;
contribution
Gi to the total low field
conductance:
Materials Science 10 (1998) 139-143
p. =
exP(Ei/kT)
’
Cj exp(Ej/kT)’
where
141
(4)
the sum includes
all the possible
l%o extra nodes are included
in order to repre-
sent two parallel contacts placed at two opposite sides of the sample: each contact is connected ductance
Gi = Go exp (-2~s) x p, (AEjkT - 1) exp (AE/kT) I (exp (AE/kT) - 1)2
+ 1 (3)
where Go is a constant, x the decay rate of the electron wave function in the insulator, s the minimum distance between the two clusters, probability
states of
charge.
of the initial
the electrostatic
Pi the occurrence
to all the clusters which are less than s,,n
away from the corresponding energies required to determine
side. The electrostatic these conductances are
calculated by modelling the contacts as infinite conductive planes. The network is then solved using the Fogelholm
method [ 1 l] to obtain the equivalent
resis-
tance between the contacts.
state of charge and AE is
energy variation
caused by the tran-
sition. The distance s is the length of the shortest line segment connecting the two grains without intersecting other clusters. Conductances are included only if the distance s is smaller than a cut-off distance %ff. The electrostatic energy of a given state of charge is calculated modelling each pair of clusters with the
3. Results The deposition
simulations
per are completely
mensionless
deposition
time &jr0(d) (where t,j is the
time) and the ratios EB/E~),
the capacitance
and R/r-f’. All the simulations on 400 x 400 samples.
of cluster 1 to cluster 2 while capac-
two clusters is calculated
ing them by two spheres of equivalent
by replac-
volume spaced
in this pa-
by a set of five
dimensionless parameters, namely the dimensionless temperature T/To (where TO = Ef)/kB), the di-
deposition
itors Ct and Cz represent the self-capacitance of the clusters which are estimated by summing up all the capacitance to the surrounding clusters. The capac-
described
characterized
circuit shown in Fig. l(b): the capacitor Ct2 represents
itance between
by a con-
The ratios EL/ Ef’ 0.5, respectively, made in [5].
EL,/E~),
have been performed
and EB/ Ef’ are fixed to 0.2 and
in conformity
As far as the conduction
with the assumptions
parameters
in Fq. (3) are
x is set to 0.3 x 10” m-t
and the lattice
by s. The volume of the clusters is estimated by assuming that each adatom gives a contribution equal to a3, where a is the lattice constant. An exact solution
concerned,
for the capacitance between two spheres is adopted [8]. Capacitance between clusters are included only
A first series of simulations was devoted to determine the effect of the deposition parameters on the resistance of the films. In Fig. 2 the calculated
if their distance s is smaller than the cut-off distance Soff.
A given state of charge is identified by the total charges Qt and Q2 in node 1 and 2, respectively; the set of possible states is restricted by allowing Qt and Q2 to assume only the values 0, +e, -e, where e is the electron charge. The occurrence probability Pi of a given state of energy Ei is calculated by the Boltzmann statistics:
constant to 0.5nm. at 300 K.
All the resistances
are calculated
resistance, normalized to Ro = Go’, is plotted as a function of the fractional coverage X, for four different settings of the deposition parameters, indicated in the figure. Since each curve refers to a fixed deposition time, the coverage has been varied by changing the deposition rate R. Each resistance value is averaged over a set of nine samples. For each curve the maximum coverage shown approaches the
142
p Bruschi,
1013
’
,
1012 C
A. Nannini/ComputationaI
-1_ T/T, = 0.043, tdrotd) = 10’
1o'O
+ a
109
a
T/T, = 0.046, tdrofd)= 10’
lo5
2.5 -
+
T/T, = 0.046,
tdroCd)= 1 014
_ +
T/T,, = 0.043,
t,r,(d) = 1013
T/T, = 0.043, tdroCdl = 10’
139-143
lo7 cT
10 (1998)
Eo
108 lo6
Science
I, 0 1 I, I, I, I -& T/T, = 0.036, t,r,(d) = 1 014
10"
g
Materials
z
1.5 -
g
%NC .g 1.0 us g 5 0.5 -
lo4 lo3 lo2
0.0 II
10' 100 -0.1
0.2
0.3 0.4 0.5 0.6 0.7
11
0.2
0.8
8
0.3
I
18
s
0.4
0.5
percolation
threshold for the corresponding
‘I
0.6
0.7
X
x Fig. 2. Plot of the resistance, normalized to RI) = Go’. as il function of the fractional coverage x for various dimensionless temperatures and deposition times.
1
Fig. 3. Standard deviation of the resistance normalized to the average resistance, calculated over a set of nine samples. The deposition parameters are indicated in the legend.
deposition
parameters. Comparison with the experimental dependence of the sheet resistance of island film on the equivalent
temperature,
the higher the resistance. This
is clearly the result of temperature activated island agglomeration [ 121 occurring during film growth and causing the inter-island distance to increase. Reduc-
x= 0.4 (annea*
s- 9 z tz7 5
ing the deposition time produces similar effects to reducing the deposition temperature since less time is available for agglomeration.
3 1
]
PI
II- 4L-----
The reproducibility of the deposition experiments is represented in Fig. 3, where the normalized standard deviation of the resistance, calculated over sets of nine samples is shown for the same deposition conditions as in Fig. 2. The high uncertainty is due to the small size of the samples compared with macroscopical films. The curves exhibit an abrupt increase at the percolation threshold: this can be a limiting factor for exploiting the small sensitivity to temperature which can be theoretically predicted for films near the transition to a metallic regime of conduction [ 11. For
f
x = 0.5
thickness [ 1,4] reveals a qualitative agreement. It can be observed that, for a given coverage, the higher the deposition
I
x = 0.4
0.0
2.0x10i7
-I 4.0x1017
tro(d) Fig. 4. Simulated post deposition resistance variations, normalized to the initial value, as a function of the dimensionless time t x rr’ for samples of different fractional coverage x. The deposition and aging temperature is T/To = 0.043 for all the curves. In one case, indicated in the legend, an annealing at T/To
for a dimensionless time f x rid’ = lOI before the aging simulation. I
= 0.052
performed
was
P Bruschi, A. Nannini/Computational
coverages far from the percolation threshold it can be roughly observed that the lower the temperature, the lower the uncertainty. This was experimentally observed for gold films [13] and suggested as a viable method to improve reproducibility. Simulation of the post-deposition aging are performed by the same program, setting the deposition rate R to zero. Fig. 4 shows the resistance variations of samples of different coverage maintained at T/To= 0.043for a dimensionless time f x P-?’ = 4 x 1017. All the samples were deposited at T/To= 0.043 for a deposition time fd x r.Cd)= 1014. One aging simulation, indicated in the figure, was performed after annealing the samples at T/To= 0.052 for a dimensionless time of 10t5: the annealing produced a large resistance increase (about 28 times the initial value) but improved the stability. All the curves have been averaged over a set of four independent experiments. These preliminary results provide an atomistic support to the hypothesis that the continuous resistance increase observed in ultra-thin metal films [ 141 is due to cluster aggregation caused by surface diffusion of adatoms and clusters.
Materials Science 10 (1998) 139-143
143
References [l] Z.H. Meiksin, in: Phys. of Thin Films, eds. G. Hass, M. Francombe and R.W. Hoffman, Vol. 8 (Academic Press, New York, 1975) p. 9. [2] J.G. Amar and F. Family, Thin Solid Films 272 (1996) 208. [3] M. Breeman, G.T. Barkema, M.H. Langelaar and D.O. Boerma, Thin Solid Films 272 (1996) 195. [4] G.R. Witt, Thin Solid Films 22 (1974) 133. [5] P. Bruschi, P. Cagnoni and A. Nannini, Phys. Rev. B 55 (1997) 7955. [6] C.J. Adkins, J. Phys. C 15 (1982) 7143. [7] P Sheng, Philosofical Magazine B 65 (1992) 357. [8] P Smilauer, Thin Solid Films 203 (1991) 1. [9] A.M. Bowler and E.S. Hood, J. Chem. Phys. 94 (1991) 5162. [lo] P Bruschi and A. Nannini, Thin Solid Films 201 (1991) 29. [ 111 R. Fogelholm, J. Phys. C 13 (1980) L57 1. [ 121 J.G. Skofronick and W.B. Phillips, J. Appl. Phys. 38 (1967) 4791. [13] Z.H. Meiksin, E.J. Stolinski, H.B. Kuo, R.A. Mirchandani and K.J. Shah, Thin Solid Films 12 (1972) 85. [14] M. Pattabi, MS. Murali Sastry and V. Sivaramakrishnan, J. Appl. Phys. 63 (1988) 983.
ELSEVIER
Computational Materials Science 10 (1998) 139-143
Simulation of the deposition and aging of thin island films P. Bruschi *, A. Nannini Dipartimento di Ingegneria dell’lnformazione: Elettronica, Informatica, Telecomunicazioni, Universitci degli Studi di Piss, via Diotisalvi 2, l-56126 Piss, Italy
Abstract Wo-dimensional atomistic simulations of the growth and post-deposition behaviour of island metal films are described. The program includes a module for the calculation of the film resistance on the basis of a charge limited tunnelling model. The dependence of the resistance on various deposition parameters is investigated. Examples of simulated post-deposition resistance drift are shown. Copyright 0 1998 Elsevier Science B.V.
1. Introduction Island metal films have been studied for their intriguing electrical and optical properties[ 11. Recently these films have been the object of extensive studies since they represent the first stage of growth of many continuous films of practical interest [2,3]. Discontinuous metals have also raised interest for practical applications
in the field of strain sensors for their high
gage factor [4]; so far, long resistance drift due to aging effects, high sensitivity to temperature and lack of reproducibility has prevented the fabrication of reliable devices based on these materials. In this paper we present a series of two-dimensional Monte Carlo simulation of the growth and postdeposition aging of island films. The program can be divided into two main modules: the first is aimed to simulate the microscopical film evolution, the second is used to calculate the resistance of the film. The program permits to relate important parameters such *Corresponding author. Tel.: +39 50 568538; fax: +39 50 568522; e-mail: [email protected].
as the deposition
effects of some deposition parameters size distribution and morphology.
not only to the
on the cluster
In our program, electrical conduction of the films is ascribed, as generally accepted, to a process of charge limited tunnelling
of electrons between the islands [6].
The problem of calculating
the resistance of a film por-
tion is solved by replacing the film itself with an electrical network where the charge exchange between the islands is modelled by equivalent conductances which depend on the size and spacing of the grains. The novelty of this work with respect to previous studies based on a similar approach [7,8] stands in the combination of a very efficient program for morphological simulations with a resistance extractor which operates directly on the simulated films. The simulations have been devoted to show the dependence of the resistance on the fractional coverage of films deposited at different substrate temperatures. In order to reduce statistical fluctuations the
0927-0256/98/$19.00 Copyright 0 1998 Elsevier Science B.V. All rights reserved PII SO927-0256(97)00126-Z
rate and temperature
morphology of the clusters but also to the overall resistance. The structure of the first module is described in details in [5] where it is used for predicting the
I? Bruschi,
140
resistances
A. Nunnini/Computational
Materials
Science
10 (1998)
139-143
has been averaged over a set of values ob-
tained by changing only the seed of the random number generator.
The standard deviation
of each set of
values has been reported to provide an indication
of
the reproducibility. Preliminary
data about the application
gram to the prediction of the post-deposition
0a
of the probehaviour
of the resistance have also been shown.
2. Model of film growth and electrical conduction The substrate is modelled as an N x N array of sites mapped on a rectangular portion of a triangular lattice. At each simulation step, three type of transitions are allowed: (i) arrival of an atom at the substrate, (ii) diffusion of an adatom to an empty nearest neighbour and (iii) re-evaporation
of an adatom from the sub-
strate. The rate r, of atom arrivals is fixed and given by r, = R x N2. where R is the number of impinging atoms per site and unit time. The rate yCd) for a diffusion transition r(d) = ,@) 0
is given by the equation
exp
(1)
’
where r;“’ IS an attempt frequency which is considered to be independent of the occupancy of the neighbours [9], kn is the Boltzmann constant and T the absolute temperature. Indicating with cl-c5 the occupancies the five nearest neighbours of hopping particles, shown in Fig. l(a), the diffusion calculated Ecd’ = Et’
by the following + (cj +
cc4
+ (Cl + C5)EL..
activation
of as
energy is
23kL grain 1 /‘\
grain 2 i 1L
~
c,,
Cl
1
(W
c2
I
Fig. I. (a) Symbolic representation of a particle hop in a triangular lattice showing the five neighbours affecting the activation energy. (b) Equivalent electrical circuit used to calculate the energy of a pair of charged grains.
of Amar an Family [2] and Breeman et al. [3] dealing with square lattices. The algorithm used to determine the rate for reevaporation transitions is described in [5] where it is shown that introducing re-evaporation leads to cluster size distributions which are not in agreement with experimental data. For this reason re-evaporation rate is set to zero in this work. A transition is randomly selected according to a
formula:
+ ccz)E~ (2)
where Ehd) IS a fixed contribution due to the interaction with the substrate, EB the binding energy between two adatoms and EL is the activation energy for diffusion of adatoms along cluster borders. The occupancy ci is defined as 0 if site i is empty and 1 if it is occupied and 6 = 1 - ci. The distinction between the two different contributions EB and EL of the neighbours to the activation energy is also present in the models
probability proportional to the transition rate. At each step the simulation time is incremented by an amount given by the inverse of the sum of all the transition rates. The equivalent electrical network utilized for calculating the resistance is obtained by associating a node to each cluster and introducing conductances between the nodes to model the electron exchange occurring by tunnel effect. The conductance between a pair of clusters is calculated by summing
I? Bruschi, A. Nannini/Computational
up the rates of all the possible volving
hopping
charge transition
of one electron
in-
from a grain to the
other. As shown in [lo], transitions
can be grouped
in “pair of antagonist
each pair gives
the following
transitions”;
contribution
Gi to the total low field
conductance:
Materials Science 10 (1998) 139-143
p. =
exP(Ei/kT)
’
Cj exp(Ej/kT)’
where
141
(4)
the sum includes
all the possible
l%o extra nodes are included
in order to repre-
sent two parallel contacts placed at two opposite sides of the sample: each contact is connected ductance
Gi = Go exp (-2~s) x p, (AEjkT - 1) exp (AE/kT) I (exp (AE/kT) - 1)2
+ 1 (3)
where Go is a constant, x the decay rate of the electron wave function in the insulator, s the minimum distance between the two clusters, probability
states of
charge.
of the initial
the electrostatic
Pi the occurrence
to all the clusters which are less than s,,n
away from the corresponding energies required to determine
side. The electrostatic these conductances are
calculated by modelling the contacts as infinite conductive planes. The network is then solved using the Fogelholm
method [ 1 l] to obtain the equivalent
resis-
tance between the contacts.
state of charge and AE is
energy variation
caused by the tran-
sition. The distance s is the length of the shortest line segment connecting the two grains without intersecting other clusters. Conductances are included only if the distance s is smaller than a cut-off distance %ff. The electrostatic energy of a given state of charge is calculated modelling each pair of clusters with the
3. Results The deposition
simulations
per are completely
mensionless
deposition
time &jr0(d) (where t,j is the
time) and the ratios EB/E~),
the capacitance
and R/r-f’. All the simulations on 400 x 400 samples.
of cluster 1 to cluster 2 while capac-
two clusters is calculated
ing them by two spheres of equivalent
by replac-
volume spaced
in this pa-
by a set of five
dimensionless parameters, namely the dimensionless temperature T/To (where TO = Ef)/kB), the di-
deposition
itors Ct and Cz represent the self-capacitance of the clusters which are estimated by summing up all the capacitance to the surrounding clusters. The capac-
described
characterized
circuit shown in Fig. l(b): the capacitor Ct2 represents
itance between
by a con-
The ratios EL/ Ef’ 0.5, respectively, made in [5].
EL,/E~),
have been performed
and EB/ Ef’ are fixed to 0.2 and
in conformity
As far as the conduction
with the assumptions
parameters
in Fq. (3) are
x is set to 0.3 x 10” m-t
and the lattice
by s. The volume of the clusters is estimated by assuming that each adatom gives a contribution equal to a3, where a is the lattice constant. An exact solution
concerned,
for the capacitance between two spheres is adopted [8]. Capacitance between clusters are included only
A first series of simulations was devoted to determine the effect of the deposition parameters on the resistance of the films. In Fig. 2 the calculated
if their distance s is smaller than the cut-off distance Soff.
A given state of charge is identified by the total charges Qt and Q2 in node 1 and 2, respectively; the set of possible states is restricted by allowing Qt and Q2 to assume only the values 0, +e, -e, where e is the electron charge. The occurrence probability Pi of a given state of energy Ei is calculated by the Boltzmann statistics:
constant to 0.5nm. at 300 K.
All the resistances
are calculated
resistance, normalized to Ro = Go’, is plotted as a function of the fractional coverage X, for four different settings of the deposition parameters, indicated in the figure. Since each curve refers to a fixed deposition time, the coverage has been varied by changing the deposition rate R. Each resistance value is averaged over a set of nine samples. For each curve the maximum coverage shown approaches the
142
p Bruschi,
1013
’
,
1012 C
A. Nannini/ComputationaI
-1_ T/T, = 0.043, tdrotd) = 10’
1o'O
+ a
109
a
T/T, = 0.046, tdrofd)= 10’
lo5
2.5 -
+
T/T, = 0.046,
tdroCd)= 1 014
_ +
T/T,, = 0.043,
t,r,(d) = 1013
T/T, = 0.043, tdroCdl = 10’
139-143
lo7 cT
10 (1998)
Eo
108 lo6
Science
I, 0 1 I, I, I, I -& T/T, = 0.036, t,r,(d) = 1 014
10"
g
Materials
z
1.5 -
g
%NC .g 1.0 us g 5 0.5 -
lo4 lo3 lo2
0.0 II
10' 100 -0.1
0.2
0.3 0.4 0.5 0.6 0.7
11
0.2
0.8
8
0.3
I
18
s
0.4
0.5
percolation
threshold for the corresponding
‘I
0.6
0.7
X
x Fig. 2. Plot of the resistance, normalized to RI) = Go’. as il function of the fractional coverage x for various dimensionless temperatures and deposition times.
1
Fig. 3. Standard deviation of the resistance normalized to the average resistance, calculated over a set of nine samples. The deposition parameters are indicated in the legend.
deposition
parameters. Comparison with the experimental dependence of the sheet resistance of island film on the equivalent
temperature,
the higher the resistance. This
is clearly the result of temperature activated island agglomeration [ 121 occurring during film growth and causing the inter-island distance to increase. Reduc-
x= 0.4 (annea*
s- 9 z tz7 5
ing the deposition time produces similar effects to reducing the deposition temperature since less time is available for agglomeration.
3 1
]
PI
II- 4L-----
The reproducibility of the deposition experiments is represented in Fig. 3, where the normalized standard deviation of the resistance, calculated over sets of nine samples is shown for the same deposition conditions as in Fig. 2. The high uncertainty is due to the small size of the samples compared with macroscopical films. The curves exhibit an abrupt increase at the percolation threshold: this can be a limiting factor for exploiting the small sensitivity to temperature which can be theoretically predicted for films near the transition to a metallic regime of conduction [ 11. For
f
x = 0.5
thickness [ 1,4] reveals a qualitative agreement. It can be observed that, for a given coverage, the higher the deposition
I
x = 0.4
0.0
2.0x10i7
-I 4.0x1017
tro(d) Fig. 4. Simulated post deposition resistance variations, normalized to the initial value, as a function of the dimensionless time t x rr’ for samples of different fractional coverage x. The deposition and aging temperature is T/To = 0.043 for all the curves. In one case, indicated in the legend, an annealing at T/To
for a dimensionless time f x rid’ = lOI before the aging simulation. I
= 0.052
performed
was
P Bruschi, A. Nannini/Computational
coverages far from the percolation threshold it can be roughly observed that the lower the temperature, the lower the uncertainty. This was experimentally observed for gold films [13] and suggested as a viable method to improve reproducibility. Simulation of the post-deposition aging are performed by the same program, setting the deposition rate R to zero. Fig. 4 shows the resistance variations of samples of different coverage maintained at T/To= 0.043for a dimensionless time f x P-?’ = 4 x 1017. All the samples were deposited at T/To= 0.043 for a deposition time fd x r.Cd)= 1014. One aging simulation, indicated in the figure, was performed after annealing the samples at T/To= 0.052 for a dimensionless time of 10t5: the annealing produced a large resistance increase (about 28 times the initial value) but improved the stability. All the curves have been averaged over a set of four independent experiments. These preliminary results provide an atomistic support to the hypothesis that the continuous resistance increase observed in ultra-thin metal films [ 141 is due to cluster aggregation caused by surface diffusion of adatoms and clusters.
Materials Science 10 (1998) 139-143
143
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