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Resistor Network Approach to Electrical Conduction and. Breakdown Phenomena in Disordered Materials. C. Pennetta, E. Alfinito and L. Reggiani Dip. di Ingegneria dell’Innovazione,Universita’ di Lecce , Italy INFM – National Nanotechnology Laboratory, Lecce, Italy. Motivations:.

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Resistor Network Approach to Electrical Conduction and

Breakdown Phenomena in Disordered Materials

C. Pennetta, E.Alfinito and L. Reggiani

Dip. di Ingegneria dell’Innovazione,Universita’ di Lecce, Italy

INFM – National Nanotechnology Laboratory, Lecce, Italy



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 inconductor - insulator composites,ingranular 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 ofthe system.


Resistor Network Approach:




R = network resistance

rn = resistance of the n-th resistor

I = stress current (d.c.), kept constant

T0 = thermal bath temperature


two-species of resistors:

rreg (Tn) = r0[1 +  (Tn -Tref) ]


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

rregrOP defect generation probabilityWD=exp[-ED/kBTn]

rOPrreg defect recovery probabilityWR =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:

  • on the external conditions (I, T0)
  • on the material parameters (r0,,A,ED,ER)


<p> , <R>



p fraction of broken resistor, pC percolation threshold

sets the level of intrinsic disorder (<p>0)

here max=6.67

flow chart of computations
Flow Chart of Computations

I 0

change T

Initial network

t=0, R(T0)


rreg rOP


t = t +1

Change T



Save R,p


Solve Network




rOP rreg


Solve Network



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

Experiments and Simulations

Simulated Failure

Experimental failure

Lognormal Distribution

Tests under accelerated conditions

Qualitative and quantitative agreement


Resistance evolutionat increasing bias

Average resistance <R>:



Steady state

Distribution of resistance

fluctuations, R = R-<R>

at increasing bias

 probability density function (PDF)


Effect of the recovery energy:

Effect of the initial film resistance:

=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 






We have found that is:

independent on the initial resistance of the film

independent on the bias conditions

dependent on the temperature coef. of the


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)


Effect on the resistance noise

of the bias conditions and of

the temperature coefficient of

the resistance 






Non-Gaussianity of resistance fluctuations

Bramwell, Holdsworth and Pinton

(Nature, 396, 552, 1998):

universalNG fluctuationdistribution

in systems near criticality


Denoting by:


a=/2, b=0.936, s=0.374, K=2.15

BHP distribution: generalization of Gumbel

a, b, s, K :

fitting parameters

Bramwell et al. PRL, 84, 3744, 2000


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:

At increasing levels of disorder

(decreasing  values) the PDF

at the breakdown threshold

approaches the BHP

Pennetta et al., Physica A, in print


Power spectral density of resistance fluctuations


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 statedetermined by the competition of two biased stochastic

processes, accordingto the BSRN model.

We have considered systems ofdifferent sizes and under different stress

conditions, from the linear response regime up to the threshold for electrical


A remarkablenon-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 non-Gaussianity 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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