Presentation on theme: "C. Pennetta, E. Alfinito and L. Reggiani"— Presentation transcript:
1 C. Pennetta, E. Alfinito and L. Reggiani Resistor Network Approach to Electrical Conduction andBreakdown Phenomena in Disordered MaterialsC. Pennetta, E. Alfinito and L. ReggianiDip. di Ingegneria dell’Innovazione,Universita’ di Lecce, ItalyINFM – 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.
4 Resistor Network Approach: THIN FILM OF RESISTANCE R2D SQUARE LATTICERESISTOR NETWORKR = network resistancern = resistance of the n-th resistorI = stress current (d.c.), kept constantT0 = thermal bath temperature
5 two-species of resistors: rreg (Tn) = r0 [1 + (Tn -Tref) ]rnrOP = 109 rreg (broken resistor)Tn = local temperature = temperature coeff. of the resistance
6 Resistor Network (BSRN) Model: Biased and StationaryResistor Network (BSRN) Model:Pennetta et al, UPON, Ed. D. Abbott & L. B. Kish, 1999Pennetta et al. PRE, 2002 and Pennetta, FNL, 2002rreg 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
7 The network evolution depends: on the external conditions (I, T0) on the material parameters (r0,,A,ED,ER)STEADY STATE<p> , <R>IRREVERSIBLEBREAKDOWN, pCp fraction of broken resistor, pC percolation thresholdsets the level of intrinsic disorder (<p>0 )here max=6.67
8 Flow Chart of Computations change TInitial networkt=0, R(T0)norreg rOPrreg(T)Change Tt = t +1t>tmax?yesSave R,pendSolve NetworknoyesSolve NetworkrOP rregrreg(T)R>Rmax?end
10 Network evolution for the irreversible breakdown case
11 SEM image of electromigration damage in Al-Cu interconnects Observed electromigration damage patternGranular structure of the materialAtomic transport through grain boundaries dominatesTransport within the grain bulkis negligeableFilm: network of interconnectedgrain boundariesSEM image of electromigrationdamage in Al-Cu interconnects
12 Experiments and Simulations Evolution and TTFsSimulated FailureExperimental failureLognormal DistributionTests under accelerated conditionsQualitative and quantitative agreement
14 Resistance evolution at increasing bias Average resistance <R>: IbSteady stateDistribution of resistancefluctuations, R = R-<R>at increasing bias probability density function (PDF)
15 Effect of the recovery energy: Effect of the initial film resistance: =2.0 0.1In the pre-breakdown region: I=3.7 0.3
16 =0 =0 0 0 Effect on the average resistance of the bias conditions (constantvoltage or constant current) and of the temperature coefficient of theresistance =0=000
17 All these features are in good agreements with electrical We have found that is:independent on the initial resistance of the filmindependent on the bias conditionsdependent on the temperature coef. of theresistancedependent on the recovery activation energy= 1.85 ± 0.08All these features are in good agreements with electricalmeasurements up to breakdown in carbon high-density polyethylene composites(K.K. Bardhan, PRL, 1999 and 2003)
18 Relative variance of resistance fluctuations <R2>/<R>2
19 =0 =0 0 0 Effect on the resistance noise of the bias conditions and ofthe temperature coefficient ofthe resistance =0=000
20 Non-Gaussianity of resistance fluctuations Bramwell, Holdsworth and Pinton(Nature, 396, 552, 1998):universal NG fluctuation distributionin systems near criticalityBHPDenoting by:Gaussiana=/2, b=0.936, s=0.374, K=2.15BHP distribution: generalization of Gumbela, b, s, K :fitting parametersBramwell et al. PRL, 84, 3744, 2000
21 Effects of the network size: networks NxN with: N=50, 75, 100, 125Gaussian in the linear regimeNG at the electrical breakdown:vanishes in the large size limit
22 Role of the disorder: At increasing levels of disorder (decreasing values) the PDFat the breakdown thresholdapproaches the BHPPennetta et al., Physica A, in print
23 Power spectral density of resistance fluctuations Lorentzian:the corner frequencymoves to lower values at increasinglevels of disorder
24 Conclusions :We have studied the distribution of the resistance fluctuations of conductingthin films with different levels of internal disorder.The study has been performed by describing the film as a resistor networkin a steady state determined by the competition of two biased stochasticprocesses, according to the BSRN model.We have considered systems of different sizes and under different stressconditions, from the linear response regime up to the threshold for electricalbreakdown.A remarkable non-Gaussianity of the fluctuation distribution is found nearbreakdown. This non-Gaussianity becomes more evident at increasing levelsof disorder.As a general trend, these deviations from Gaussianity are related to thefinite 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 welldescribed by the universal Bramwell-Holdsworth-Pinton distribution.
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