1. On-Off Intermittency and Criticality in Early Stage Electromigration Wouter Arts Peter Koelman

2. Index General introduction Introduction to intermittency Theoretical statistical behavior Numerical simulations Real life situation Comparison Conclusion 16/11/2010Chaos - 3T2202/19

3. General introduction Article: On-Off Intermittency and Criticality in Early Stage Electromigration – By: Eric Dalton, Ian Clancy, David Corcoran, Arousian Arshak and George Gooberman Main focus on on-off intermittency 16/11/2010Chaos - 3T2203/19

4. Introduction to intermittency Logistic map: Three times iterated Characterized by laminar/turbulent region 16/11/2010Chaos - 3T2204/19

5. 16/11/2010Chaos - 3T2205/19

6. Noise driven logistic map Use of multiplicative noise: Additive noise Used in article 16/11/2010Chaos - 3T2206/19

7. Statistical behavior Three characteristic statistical properties: – Theoretically derived: Critical value of stability is e PDF of return time to turbulent phase is a power law of: – Simulated: PDF of return time Fractional resistance change has a power law of: 16/11/2010Chaos - 3T2207/19

8. Criticality Noise-driven logistic map: Taking the natural logarithm, this can be written as: Critical value if Substituted Thus 16/11/2010Chaos - 3T2208/19

9. Power law PDF of the length of the laminar phase Laminar phase: 16/11/2010Chaos - 3T2209/19

10. Power law This set can be rewritten using the following notation: The set of the laminar phase becomes: Shifting the origin: The chance of finding a certain length: 16/11/2010Chaos - 3T22010/19

11. Power law Around criticality the noise is logarithmic additive: This leads to the simplification of: Direct substitution: This function is solved in an article by Heagy et al. The solution is proportional to: 16/11/2010Chaos - 3T22011/19

12. Numerical simulations PDF of return time Theory reasonably well of 16/11/2010Chaos - 3T22012/19

13. Numerical simulations Probability density of change (Δx) Close to theoretical prediction of 16/11/2010Chaos - 3T22013/19

14. Real life situation Article: Early stage electromigration Short bursts of fractional resistance change Signal is intermittent Statistics experimentally derived Statistics compared to simulations and theory 16/11/2010Chaos - 3T22014/19

15. Comparison ArticleOur simulation 16/11/2010Chaos - 3T22015/19

16. Comparison ArticleOur simulation 16/11/2010Chaos - 3T22016/19

17. Conclusion Numerical simulations made of intermitted signal Theoretical derivation made of return time Statistics derived with simulations Numerical simulation closely match theoretical predictions and experimental data No theoretical prediction could be made of probability density of change (Δx) 16/11/2010Chaos - 3T22017/19

18. Reference List C. Toniolo, A. Provenzale, and E.A. Spiegel, Phys. Rev. E 66, 066209 (2002). J.F. Heagy, N. Platt, and S.M. Hammel, Phys. Rev. E 49, 1140 (1994) 16/11/2010Chaos - 3T22018/19

19. The End Questions? 16/11/2010Chaos - 3T22019/19