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On-Off Intermittency and Criticality in Early Stage Electromigration

On-Off Intermittency and Criticality in Early Stage Electromigration. Wouter Arts Peter Koelman. Index. General introduction Introduction to intermittency Theoretical statistical behavior Numerical simulations Real life situation Comparison Conclusion. General introduction.

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On-Off Intermittency and Criticality in Early Stage Electromigration

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  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 Chaos - 3T220

  3. General introduction • Article: On-Off Intermittency and Criticality in Early Stage Electromigration • By: Eric Dalton, Ian Clancy, David Corcoran, ArousianArshak and George Gooberman • Main focus on on-off intermittency Chaos - 3T220

  4. Introduction to intermittency • Logistic map: • Three times iterated • Characterized by laminar/turbulent region Chaos - 3T220

  5. Chaos - 3T220

  6. Noise driven logistic map • Use of multiplicative noise: • Additive noise • Used in article Chaos - 3T220

  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: Chaos - 3T220

  8. Criticality • Noise-driven logistic map: • Taking the natural logarithm, this can be written as: • Critical value if • Substituted • Thus Chaos - 3T220

  9. Power law • PDF of the length of the laminar phase • Laminar phase: Chaos - 3T220

  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: Chaos - 3T220

  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: Chaos - 3T220

  12. Numerical simulations • PDF of return time • Theory reasonably well of Chaos - 3T220

  13. Numerical simulations • Probability density of change (Δx) • Close to theoretical prediction of Chaos - 3T220

  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 Chaos - 3T220

  15. Comparison Article Our simulation Chaos - 3T220

  16. Comparison Article Our simulation Chaos - 3T220

  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) Chaos - 3T220

  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) Chaos - 3T220

  19. The End • Questions? Chaos - 3T220

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