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C. Pennetta, E. Alfinito and L. Reggiani

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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.

THIN FILMOFRESISTANCER

2D SQUARE LATTICE

RESISTORNETWORK

R = network resistance

rn = resistance of the n-th resistor

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

T0 = thermal bath temperature

rreg (Tn) = r0[1 + (Tn -Tref) ]

rn

rOP = 109 rreg (broken resistor)

Tn = local temperature

= temperature coeff. of the resistance

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)

STEADY STATE

<p> , <R>

IRREVERSIBLE

BREAKDOWN, pC

p fraction of broken resistor, pC percolation threshold

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

here max=6.67

Flow Chart of Computations

I 0

change T

Initial network

t=0, R(T0)

no

rreg rOP

rreg(T)

t = t +1

Change T

t>tmax?

yes

Save R,p

end

Solve Network

no

yes

R>Rmax?

rOP rreg

rreg(T)

Solve Network

end

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

Simulated Failure

Experimental failure

Lognormal Distribution

Tests under accelerated conditions

Qualitative and quantitative agreement

Resistance evolutionat increasing bias

Average resistance <R>:

I0

Ib

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

=0

=0

0

0

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):

universalNG fluctuationdistribution

in systems near criticality

BHP

Denoting by:

Gaussian

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

networks NxN with: N=50, 75, 100, 125

Gaussian in the linear regime

NG at the electrical breakdown:

vanishes in the large size limit

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

Lorentzian:

the corner frequency

moves to lower values at increasing

levels of disorder

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

breakdown.

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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