C. Pennetta, E. Alfinito and L. Reggiani
Dip. di Ingegneria dell’Innovazione,Universita’ di Lecce, Italy
INFM – National Nanotechnology Laboratory, Lecce, Italy
Motivations:
To study the electrical conduction of disordered materials
over the full range of the applied stress, by focusing on
the role of the disorder.
To investigate the stability of the electrical properties and
electrical breakdown phenomena in
conductor - insulator
composites,in granular metals and in nanostructured materials.
To establish the conditions under which we expect failure
precursors and to identify these precursors.
To study the properties of the resistance fluctuations,including
their non-Gaussianity and to understand their link with other
basic features of the system.
The model
Resistor Network Approach:
THIN FILM OF
RESISTANCE R
R
rn
I
T0
2D SQUARE LATTICE
RESISTOR NETWORK
= network resistance
= resistance of the n-th resistor
= stress current (d.c.), kept constant
= thermal bath temperature
two-species of resistors:
rreg (Tn) = r0 [1 +  (Tn -Tref) ]
rn
rOP = 109 rreg (broken resistor)
Tn = local temperature
 = temperature coeff. of the resistance
Biased and Stationary
Resistor Network (BSRN) Model:
Pennetta et al, UPON, Ed. D. Abbott & L. B. Kish, 1999
Pennetta et al. PRE, 2002 and Pennetta, FNL, 2002
rreg
rOP defect generation probability WD=exp[-ED/kBTn]
rOP
rreg defect recovery probability
WR =exp[-ER/kBTn]
biased percolation:
Tn =T0 + A[ rn in2 +(B/Nneig)m(rm,nim,n2 - rnin2)]
Gingl et al, Semic. Sc. & Tech. 1996; Pennetta et al, PRL, 1999
The network evolution depends:
a) on the external conditions (I, T0)
b) on the material parameters (r0,,A,ED,ER)
STEADY STATE
<p> , <R>
IRREVERSIBLE
BREAKDOWN, pC
p fraction of broken resistor, pC percolation threshold

ED  ER
k BT0
sets the level of intrinsic disorder (<p>0 )
0    max  ( ED / kBT0 ) here  =6.67
max
Flow Chart of Computations

I 0
change T
Initial network
t=0, R(T0)
no
Change T
t = t +1
t>tmax?
rreg

rOP
rreg(T)
yes
end
Save R,p
Solve Network
rOP  rreg
rreg(T)
Solve Network
no
R>Rmax
?
yes
end
Results
Network evolution for the irreversible breakdown case
Observed electromigration damage pattern
Granular structure of the material
Atomic transport through grain
boundaries dominates
Transport within the grain bulk
is negligeable
Film: network of interconnected
grain boundaries
SEM image of electromigration
damage in Al-Cu interconnects
Experiments and Simulations
Evolution and TTFs
Experimental failure
Simulated Failure
Lognormal Distribution
Tests under
accelerated
conditions
Qualitative and
quantitative
agreement
Steady State Regime
Resistance evolution at increasing bias
Average resistance <R>:
I0
Steady state
Distribution of resistance
fluctuations, R = R-<R>
at increasing bias
 probability density function (PDF)
Ib
Effect of the recovery energy:
Effect of the initial film resistance:
 I 
R
 g  
 R 0
 I0 
g ( I / I 0 )  1  a( I / I 0 )
=2.0  0.1
In the pre-breakdown region: I=3.7  0.3
Effect on the average resistance
of the bias conditions (constant
voltage or constant current) and of
the temperature coefficient of the
resistance 
=0
=0
0
0
We have found that

 R b
 R 0
is:
independent on the initial resistance of the film
independent on the bias conditions
dependent on the temperature coef. of the
resistance
dependent on the recovery activation energy
= 1.85 ± 0.08
All these features are in good
agreements with electrical
measurements up to breakdown
in carbon high-density
polyethylene composites
(K.K. Bardhan, PRL, 1999 and 2003)
Relative variance of resistance fluctuations
  <R2>/<R>2
Effect on the resistance noise
of the bias conditions and of
the temperature coefficient of
the resistance 
=0
=0
0
0
Non-Gaussianity of resistance fluctuations
Bramwell, Holdsworth and Pinton
(Nature, 396, 552, 1998):
universal NG fluctuation distribution
in systems near criticality
BHP
Denoting by:
y 
m  m 

Gaussian
 ( y )   ( y )

( y)  Ke
a b ( y  s )  e b ( y s )

a=/2, b=0.936, s=0.374, K=2.15
BHP distribution: generalization of Gumbel
Bramwell et al. PRL, 84, 3744, 2000
a, b, s, K :
fitting parameters
Effects of the network size:
networks NxN with: N=50, 75, 100, 125
Gaussian in the linear regime
NG at the electrical breakdown:
vanishes in the large size limit
Role of the disorder:


ED  ER
k BT0
 p    p 0  I 
  
 p 0
 I0 
2
Pennetta et al., Physica A, in print
0    max
At increasing levels of disorder
(decreasing  values) the PDF
at the breakdown threshold
approaches the BHP
Power spectral density of resistance fluctuations
Lorentzian:
the corner frequency
moves to lower
values at increasing
levels of disorder
Conclusions :
We have studied the distribution of the resistance fluctuations of conducting
thin films with different levels of internal disorder.
The study has been performed by describing the film as a resistor network
in a steady state determined by the competition of two biased stochastic
processes, according to the BSRN model.
We have considered systems of different sizes and under different stress
conditions, from the linear response regime up to the threshold for electrical
breakdown.
A remarkable non-Gaussianity of the fluctuation distribution is found near
breakdown. This non-Gaussianity becomes more evident at increasing levels
of disorder.
As a general trend, these deviations from Gaussianity are related to the
finite size of the system and they vanish in the large size limit.
Near the critical point of the conductor-insulator transition, the nonGaussianity is found to persist in the large size limit and the PDF is well
described by the universal Bramwell-Holdsworth-Pinton distribution.
Acknowledgments :
Laszlo Kish (A&T Texas), Zoltan Gingl (Szeged), Gyorgy Trefan
Fausto Fantini (Modena), Andrea Scorzoni (Perugia), Ilaria De Munari (Parma)
Stefano Ruffo (Firenze)
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