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

22D SQUARE LATTICED SQUARE LATTICE RRESISTORESISTOR NETWORKNETWORK

R = network resistancern = resistance of the n-th resistorI = stress current (d.c.), kept constantT0 = thermal bath temperature

THIN THIN FILMFILM OFOF RESISTANCERESISTANCE RR

Resistor Network Approach:Resistor Network Approach:Resistor Network Approach:Resistor Network Approach:

= temperature coeff. of the resistance

two-species of two-species of resistors:resistors:

rn

rOP = 109 rreg (broken resistor)

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

Tn = local temperature

rreg rOP defect defect generation probabilitygeneration probability WD=exp[-ED/kBTn]

rOP rreg defect defect recovery probabilityrecovery probability WR

=exp[-ER/kBTn]

rreg rOP defect defect generation probabilitygeneration probability WD=exp[-ED/kBTn]

rOP rreg defect defect recovery probabilityrecovery probability WR

=exp[-ER/kBTn]

Tn =T0 + A[ rn in2 +(B/Nneig)m(rm,nim,n

2 - rnin

2)] Tn =T0 + A[ rn in

2 +(B/Nneig)m(rm,nim,n2 -

rnin2)] Gingl et al, Gingl et al, Semic. Sc. & TechSemic. Sc. & Tech. 1996; Pennetta et al, . 1996; Pennetta et al, PRLPRL, , 19991999

Biased and Stationary Biased and Stationary Resistor Network (BSRN) Resistor Network (BSRN) Model:Model:

Biased and Stationary Biased and Stationary Resistor Network (BSRN) Resistor Network (BSRN) Model:Model:

biased percolation:biased percolation:

Pennetta et al, UPON, Ed. D. Abbott & L. B. Kish, Pennetta et al, UPON, Ed. D. Abbott & L. B. Kish, 1999 1999 Pennetta et al. Pennetta et al. PRE,PRE, 2002 and Pennetta, 2002 and Pennetta, FNL,FNL, 20022002

STEADY STATESTEADY STATE <p> , <R><p> , <R>

IRREVERSIBLEIRREVERSIBLEBREAKDOWNBREAKDOWN, p, pCC

The network evolution depends:The network evolution depends:a)a) on the external conditions (I, Ton the external conditions (I, T00) ) b)b)on the material parameters on the material parameters

(r(r00,,,A,E,A,EDD,E,ERR))

p fraction of broken resistor, pC percolation threshold

0Tk

EE

B

RD

0Tk

EE

B

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

here max=6.67max 00 ( / )D BE k T max 00 ( / )D BE k T

Initial network t=0, R(T0)

Save R,p

R>Rmax?

end

rreg rOP

rreg(T)

Solve Network

Solve Network

Change T

yesno

I 0change T

rOP rreg rreg(T)

t = t +1 t>tmax?no

end

yes

Flow Chart of ComputationsFlow Chart of Computations

ResultsResults

Network evolution for the irreversible breakdown case

SEM image of electromigrationdamage in Al-Cu interconnects

Granular structure of the material

Atomic transport through grain boundaries dominates

Transport within the grain bulkis negligeable

Film: network of interconnectedgrain boundaries

Observed electromigration damage pattern

Experiments and Simulations

Tests under accelerated conditions

Tests under accelerated conditions

Experimental failure Experimental failure

Qualitative and quantitative agreement

Qualitative and quantitative agreement

Evolution and TTFs

Simulated Failure

Lognormal DistributionLognormal Distribution

Steady State Regime

Average resistance <R>:Resistance evolution at increasing bias

I0 Ib

probability density function (PDF)

Distribution of resistancefluctuations, R = R-<R>at increasing bias

Steady state

Effect of the recovery energy: Effect of the initial film resistance:

00 I

Ig

R

R )/(1)/( 00 IIaIIg

In the pre-breakdown region: In the pre-breakdown region: II=3.7 =3.7 0.3 0.3 In the pre-breakdown region: In the pre-breakdown region: II=3.7 =3.7 0.3 0.3

=2.0 =2.0 0.1 0.1 =2.0 =2.0 0.1 0.1

Effect on the average resistanceof the bias conditions (constantvoltage or constant current) and of the temperature coefficient of theresistance

=0 =0

0 0

= 1.85 ± 0.08= 1.85 ± 0.08

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 resistance dependent on the recovery activation energy

0

R

R b

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

<<RR22>/<R>>/<R>22 <<RR22>/<R>>/<R>22

Effect on the resistance noiseof the bias conditions and of the temperature coefficient of the resistance

=0 0=0

0

)()( yy )()( yy

mmy

mmy

a=/2, b=0.936, s=0.374, K=2.15 a=/2, b=0.936, s=0.374, K=2.15

Non-Gaussianity of resistance fluctuations

Denoting by:

)()()(sybesybaKey

)()()(sybesybaKey

BHP distribution: generalization of Gumbel

Bramwell, Holdsworth and Pinton (Nature, 396, 552, 1998):

Bramwell et al. PRL, 84, 3744, 2000 a, b, s, K : fitting parameters

a, b, s, K : fitting parameters

universal NG fluctuation distribution in systems near criticality

BHP

Gaussian

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:

Pennetta et al., Physica A, in print

2

00

0

I

I

p

pp2

00

0

I

I

p

ppAt increasing levels of disorder (decreasing values) the PDFat the breakdown threshold approaches the BHP

max0 max0 0Tk

EE

B

RD

0Tk

EE

B

RD

Power spectral density of resistance fluctuations

Lorentzian: the corner frequency moves to lower values at increasinglevels of disorder

Conclusions :Conclusions :

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.

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.

Laszlo Kish (A&T Texas), Zoltan Gingl (Szeged), Gyorgy Trefan Fausto Fantini (Modena), Andrea Scorzoni (Perugia), Ilaria De Munari (Parma) Stefano Ruffo (Firenze)

Acknowledgments : :

1) M. B. Weissman, Rev. Mod. Phys. 60, 537 (1988). 2) S. T. Bramwell, P. C. W. Holdsworth and J. F. Pinton, Nature, 396, 552, 1998.3) S. T. Bramwell, K. Christensen, J. Y. Fortin, P. C. W. Holdsworth, H. J. Jensen, S.Lise, J. M. Lopez, M. Nicodemi, J. F. Pinton, M. Sellitto, Phys. Rev. Lett. , 84, 3744, 2000.4) S. T. Bramwell, J. Y. Fortin, P. C. W. Holdsworth, S. Peysson, J. F. Pinton, B. Portelli and M. Sellitto, Phys. Rev E, 63, 041106, 2001.5) B. Portelli, P. C. W. Holdsworth, M. Sellitto, S.T. Bramwell, Phys. Rev. E, 64, 036111

(2001).6) T. Antal, M. Droz, G. Györgyi, Z. Rácz, Phys. Rev. Lett., 87, 240601 (2001) 7) T. Antal, M. Droz, G. Györgyi, Z. Rácz, Phys. Rev. E, 65, 046140 (2002).8) V. Eisler, Z. Rácz, F. Wijland, Phys. Rev. E, 67, 56129 (2003).9) K. Dahlstedt, H Jensen, J. Phys. A 34, 11193 (2001). 10) V. Aji, N. Goldenfeld, Phys. Rev. Lett. 86, 1107 (2001).11) N. Vandewalle, M. Ausloos, M. Houssa, P.W. Mertens, M.M. Heyns,Appl. Phys.Lett.

74,1579 (1999).12) L. Lamaignère, F. Carmona, D. Sornette, Phys. Rev. Lett. 77, 2738 (1996).13) J. V. Andersen, D. Sornette and K. Leung, Phys. Rev. Lett, 78, 2140 (1997).14) S. Zapperi, P. Ray, H. E. Stanley, A. Vespignani, Phys. Rev. Lett., 78, 1408 (1997)15) C. D. Mukherijee, K.K.Bardhan, M.B. Heaney, Phys. Rev. Lett.,83,1215,1999. 16) C. D. Mukherijee, K.K.Bardhan, Phys. Rev. Lett., 91, 025702-1, 2003.17) C. Pennetta, Fluctuation and Noise Lett., 2, R29, 2002.18) C. Pennetta, L. Reggiani, G. Trefan, E. Alfinito, Phys. Rev. E, 65, 066119, 2002.19) Z. Gingl, C. Pennetta, L. B. Kish, L. Reggiani, Semicond. Sci.Technol. 11, 1770,1996.20) C. Pennetta, L. Reggiani, G. Trefan, Phys. Rev. Lett. 84, 5006, 2000. 21) C. Pennetta, L. Reggiani, G. Trefan, Phys. Rev. Lett. 85, 5238, 2000.22) C. Pennetta, G. Trefan, L. Reggiani, in Unsolved Problems of Noise and Fluctuations, Ed. by D. Abbott, L. B. Kish, AIP Conf. Proc. 551, New York (1999), 447.23) C. Pennetta, E. Alfinito, L. Reggiani, S. Ruffo, Semic. Sci. Techn., 19, S164 (2004).24) C. Pennetta, E. Alfinito, L. Reggiani, S. Ruffo, Physica A, in print.25) C. Pennetta, E. Alfinito, L. Reggiani, Unsolved Problems of Noise and Fluctuations, AIP Conf. Proc. 665, Ed. by S. M. Bezrukov, 480, New York (2003).

1) M. B. Weissman, Rev. Mod. Phys. 60, 537 (1988). 2) S. T. Bramwell, P. C. W. Holdsworth and J. F. Pinton, Nature, 396, 552, 1998.3) S. T. Bramwell, K. Christensen, J. Y. Fortin, P. C. W. Holdsworth, H. J. Jensen, S.Lise, J. M. Lopez, M. Nicodemi, J. F. Pinton, M. Sellitto, Phys. Rev. Lett. , 84, 3744, 2000.4) S. T. Bramwell, J. Y. Fortin, P. C. W. Holdsworth, S. Peysson, J. F. Pinton, B. Portelli and M. Sellitto, Phys. Rev E, 63, 041106, 2001.5) B. Portelli, P. C. W. Holdsworth, M. Sellitto, S.T. Bramwell, Phys. Rev. E, 64, 036111

(2001).6) T. Antal, M. Droz, G. Györgyi, Z. Rácz, Phys. Rev. Lett., 87, 240601 (2001) 7) T. Antal, M. Droz, G. Györgyi, Z. Rácz, Phys. Rev. E, 65, 046140 (2002).8) V. Eisler, Z. Rácz, F. Wijland, Phys. Rev. E, 67, 56129 (2003).9) K. Dahlstedt, H Jensen, J. Phys. A 34, 11193 (2001). 10) V. Aji, N. Goldenfeld, Phys. Rev. Lett. 86, 1107 (2001).11) N. Vandewalle, M. Ausloos, M. Houssa, P.W. Mertens, M.M. Heyns,Appl. Phys.Lett.

74,1579 (1999).12) L. Lamaignère, F. Carmona, D. Sornette, Phys. Rev. Lett. 77, 2738 (1996).13) J. V. Andersen, D. Sornette and K. Leung, Phys. Rev. Lett, 78, 2140 (1997).14) S. Zapperi, P. Ray, H. E. Stanley, A. Vespignani, Phys. Rev. Lett., 78, 1408 (1997)15) C. D. Mukherijee, K.K.Bardhan, M.B. Heaney, Phys. Rev. Lett.,83,1215,1999. 16) C. D. Mukherijee, K.K.Bardhan, Phys. Rev. Lett., 91, 025702-1, 2003.17) C. Pennetta, Fluctuation and Noise Lett., 2, R29, 2002.18) C. Pennetta, L. Reggiani, G. Trefan, E. Alfinito, Phys. Rev. E, 65, 066119, 2002.19) Z. Gingl, C. Pennetta, L. B. Kish, L. Reggiani, Semicond. Sci.Technol. 11, 1770,1996.20) C. Pennetta, L. Reggiani, G. Trefan, Phys. Rev. Lett. 84, 5006, 2000. 21) C. Pennetta, L. Reggiani, G. Trefan, Phys. Rev. Lett. 85, 5238, 2000.22) C. Pennetta, G. Trefan, L. Reggiani, in Unsolved Problems of Noise and Fluctuations, Ed. by D. Abbott, L. B. Kish, AIP Conf. Proc. 551, New York (1999), 447.23) C. Pennetta, E. Alfinito, L. Reggiani, S. Ruffo, Semic. Sci. Techn., 19, S164 (2004).24) C. Pennetta, E. Alfinito, L. Reggiani, S. Ruffo, Physica A, in print.25) C. Pennetta, E. Alfinito, L. Reggiani, Unsolved Problems of Noise and Fluctuations, AIP Conf. Proc. 665, Ed. by S. M. Bezrukov, 480, New York (2003).

ReferencesReferences::