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C. Pennetta, E. Alfinito and L. Reggiani Dip. di Ingegneria dellInnovazione,Universita di Lecce, Italy INFM – National Nanotechnology Laboratory, Lecce,

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Presentation on theme: "C. Pennetta, E. Alfinito and L. Reggiani Dip. di Ingegneria dellInnovazione,Universita di Lecce, Italy INFM – National Nanotechnology Laboratory, Lecce,"— Presentation transcript:

1 C. Pennetta, E. Alfinito and L. Reggiani Dip. di Ingegneria dellInnovazione,Universita di Lecce, Italy INFM – National Nanotechnology Laboratory, Lecce, Italy

2 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.

3 The model

4 2D SQUARE LATTICE 2D SQUARE LATTICE RESISTOR NETWORK RESISTOR NETWORK R = network resistance r n = resistance of the n-th resistor I = stress current (d.c.), kept constant T 0 = thermal bath temperature THIN FILM OF RESISTANCE R Resistor Network Approach:

5 = temperature coeff. of the resistance two-species of resistors: rnrn r OP = 10 9 r reg (broken resistor) r reg (T n ) = r 0 [1 + (T n -T ref ) ] T n = local temperature

6 defect generation probability r reg r OP defect generation probability W D =exp[-E D /k B T n ] defect recovery probability r OP r reg defect recovery probability W R =exp[-E R /k B T n ] defect generation probability r reg r OP defect generation probability W D =exp[-E D /k B T n ] defect recovery probability r OP r reg defect recovery probability W R =exp[-E R /k B T n ] T n =T 0 + A[ r n i n 2 +(B/N neig ) m (r m,n i m,n 2 - r n i n 2 )] Gingl et al, Semic. Sc. & Tech. 1996; Pennetta et al, PRL, 1999 Gingl et al, Semic. Sc. & Tech. 1996; Pennetta et al, PRL, 1999 Biased and Stationary Biased and Stationary Resistor Network (BSRN) Model: Resistor Network (BSRN) Model: Biased and Stationary Biased and Stationary Resistor Network (BSRN) Model: Resistor Network (BSRN) Model: biased percolation: Pennetta et al, UPON, Ed. D. Abbott & L. B. Kish, 1999 Pennetta et al, UPON, Ed. D. Abbott & L. B. Kish, 1999 Pennetta et al. PRE, 2002 and Pennetta, FNL, 2002 Pennetta et al. PRE, 2002 and Pennetta, FNL, 2002

7 STEADY STATE,, IRREVERSIBLE BREAKDOWN, p C The network evolution depends: a)on the external conditions (I, T 0 ) b)on the material parameters (r 0,,A,E D,E R ) p fraction of broken resistor, p C percolation threshold sets the level of intrinsic disorder ( 0 ) here max =6.67

8 Initial network t=0, R(T 0 ) Save R,p R>Rmax ? end r reg r OP r reg (T) Solve Network Change T yes no I 0 change T r OP r reg r reg (T) t = t +1 t>tmax? no end yes Flow Chart of Computations

9 Results

10 Network evolution for the irreversible breakdown case

11 SEM image of electromigration damage in Al-Cu interconnects Granular structure of the material Atomic transport through grain boundaries dominates Transport within the grain bulk is negligeable Film: network of interconnected grain boundaries Observed electromigration damage pattern

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

13 Steady State Regime

14 Average resistance : Resistance evolution at increasing bias I0I0 IbIb probability density function (PDF) Distribution of resistance fluctuations, R = R- at increasing bias Steady state

15 Effect of the recovery energy : Effect of the initial film resistance: In the pre-breakdown region: I = = =

16 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

17 = 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 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)

18 Relative variance of resistance fluctuations / 2 / 2

19 Effect on the resistance noise of the bias conditions and of the temperature coefficient of the resistance =0 0 0

20 a= /2, b=0.936, s=0.374, K=2.15 Non-Gaussianity of resistance fluctuations Denoting by: 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

21 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

22 Role of the disorder: Pennetta et al., Physica A, in print At increasing levels of disorder (decreasing values) the PDF at the breakdown threshold approaches the BHP

23 Power spectral density of resistance fluctuations Lorentzian: the corner frequency moves to lower values at increasing levels of disorder

24 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.

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

26 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, ) 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, ) S. T. Bramwell, J. Y. Fortin, P. C. W. Holdsworth, S. Peysson, J. F. Pinton, B. Portelli and M. Sellitto, Phys. Rev E, 63, , ) 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, ) C. D. Mukherijee, K.K.Bardhan, Phys. Rev. Lett., 91, , ) C. Pennetta, Fluctuation and Noise Lett., 2, R29, ) C. Pennetta, L. Reggiani, G. Trefan, E. Alfinito, Phys. Rev. E, 65, , ) Z. Gingl, C. Pennetta, L. B. Kish, L. Reggiani, Semicond. Sci.Technol. 11, 1770, ) C. Pennetta, L. Reggiani, G. Trefan, Phys. Rev. Lett. 84, 5006, ) C. Pennetta, L. Reggiani, G. Trefan, Phys. Rev. Lett. 85, 5238, ) 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), ) 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, ) 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, ) S. T. Bramwell, J. Y. Fortin, P. C. W. Holdsworth, S. Peysson, J. F. Pinton, B. Portelli and M. Sellitto, Phys. Rev E, 63, , ) 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, ) C. D. Mukherijee, K.K.Bardhan, Phys. Rev. Lett., 91, , ) C. Pennetta, Fluctuation and Noise Lett., 2, R29, ) C. Pennetta, L. Reggiani, G. Trefan, E. Alfinito, Phys. Rev. E, 65, , ) Z. Gingl, C. Pennetta, L. B. Kish, L. Reggiani, Semicond. Sci.Technol. 11, 1770, ) C. Pennetta, L. Reggiani, G. Trefan, Phys. Rev. Lett. 84, 5006, ) C. Pennetta, L. Reggiani, G. Trefan, Phys. Rev. Lett. 85, 5238, ) 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), ) 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). References:


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