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**C. Pennetta, E. Alfinito and L. Reggiani**

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

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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.

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The model

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**Resistor Network Approach:**

THIN FILM OF RESISTANCE R 2D SQUARE LATTICE RESISTOR NETWORK R = network resistance rn = resistance of the n-th resistor I = stress current (d.c.), kept constant T0 = thermal bath temperature

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**two-species of resistors:**

rreg (Tn) = r0 [1 + (Tn -Tref) ] rn rOP = 109 rreg (broken resistor) Tn = local temperature = temperature coeff. of the resistance

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**Resistor Network (BSRN) Model:**

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

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**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

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**Flow Chart of Computations**

change T Initial network t=0, R(T0) no rreg rOP rreg(T) Change T t = t +1 t>tmax? yes Save R,p end Solve Network no yes Solve Network rOP rreg rreg(T) R>Rmax? end

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Results

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**Network evolution for the irreversible breakdown case**

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**SEM image of electromigration damage in Al-Cu interconnects**

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

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**Experiments and Simulations**

Evolution and TTFs Simulated Failure Experimental failure Lognormal Distribution Tests under accelerated conditions Qualitative and quantitative agreement

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Steady State Regime

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**Resistance evolution at increasing bias Average resistance <R>:**

Ib Steady state Distribution of resistance fluctuations, R = R-<R> at increasing bias probability density function (PDF)

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**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

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**=0 =0 0 0 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

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**All these features are in good agreements with electrical **

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 = 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)

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**Relative variance of resistance fluctuations**

<R2>/<R>2

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**=0 =0 0 0 Effect on the resistance noise**

of the bias conditions and of the temperature coefficient of the resistance =0 =0 0 0

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**Non-Gaussianity of resistance fluctuations**

Bramwell, Holdsworth and Pinton (Nature, 396, 552, 1998): universal NG fluctuation distribution 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

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**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

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**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

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**Power spectral density of resistance fluctuations**

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

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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 non-Gaussianity is found to persist in the large size limit and the PDF is well described by the universal Bramwell-Holdsworth-Pinton distribution.

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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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**References: 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, , 2001. 5) B. Portelli, P. C. W. Holdsworth, M. Sellitto, S.T. Bramwell, Phys. Rev. E, 64, (2001). 6) T. Antal, M. Droz, G. Györgyi, Z. Rácz, Phys. Rev. Lett., 87, (2001) 7) T. Antal, M. Droz, G. Györgyi, Z. Rácz, Phys. Rev. E, 65, (2002). 8) V. Eisler, Z. Rácz, F. Wijland, Phys. Rev. E, 67, (2003). 9) K. Dahlstedt, H Jensen, J. Phys. A 34, (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, , 2003. 17) C. Pennetta, Fluctuation and Noise Lett., 2, R29, 2002. 18) C. Pennetta, L. Reggiani, G. Trefan, E. Alfinito, Phys. Rev. E, 65, , 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).

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