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Invariant Tori in a Network of Two Spin-Torque Nano Oscillators

  • James TurtleEmail author
  • Antonio Palacios
  • Patrick Longhini
  • Visarath In
Conference paper
Part of the Lecture Notes in Networks and Systems book series (LNNS, volume 6)

Abstract

Over the past few years it has been shown, through theory and experiments, that the AC current produced by spin torque nano-oscillators (STNO), coupled in an array, can lead to feedback between the STNOs causing them to synchronize and that, collectively, the microwave power output of the array is significantly larger than that of an individual oscillator. Other works have pointed, however, to the difficulty in achieving synchronization. In particular, Persson et al. [17] shows that the region of parameter space where the synchronization state exists for even a small array with two STNOs is rather small. In this work we explore in more detail the nature of the bifurcations that lead into and out of the synchronization state for the two-array case. The bifurcation analysis shows bistability between in-phase and out-of-phase oscillations. A more detailed analysis of the out-of-phase solutions reveals both limit-cycles and invariant tori that are responsible for anti-phase and quasi-periodic oscillations respectively. A continuation of unstable tori demonstrates a portion of the separatrix bounding the basins of attraction for the in- and out-of-phase limit-cycles.

Keywords

Hopf Bifurcation Spin Valve Synchronization State Free Layer Saddle Node 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    K. Beauvais, A. Palacios, R. Shaffer, J. Turtle, V. In, P. Longhini, Coupled spin torque nano-oscillators: stability of synchronization, in Interdisciplinary Topics in Applied Mathematics, Modeling and Computational Science (Springer, Seattle, WA, 2015), pp. 43–48Google Scholar
  2. 2.
    L. Berger, Emission of spin waves by a magnetic multilayer traversed by a current. Phys. Rev. B 54, 9353 (1996)CrossRefGoogle Scholar
  3. 3.
    G. Bertotti, I. Mayergoyz, C. Serpico, Analytical solutions of landau-lifshitz equation for precessional dynamics. Phys. B 343, 325–330 (2004)CrossRefGoogle Scholar
  4. 4.
    E. Doedel, Auto: a program for the automatic bifurcation analysis of autonomous systems. Congr. Numer. 30, 265–284 (1981)MathSciNetzbMATHGoogle Scholar
  5. 5.
    B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students (Siam, 2002)Google Scholar
  6. 6.
    J. Grollier, V. Cros, A. Fert, Synchronization of spin-transfer oscillators driven by stimulated microwave currents. Phys. Rev. B 73 (2006)Google Scholar
  7. 7.
    S. Kaka, M.R. Pufall, W.H. Rippard, T.J. Silva, S.E. Russek, J.A. Katine, Mutual phase-locking of microwave spin torque nano-oscillators. Nature 437, 389–392 (2005)CrossRefGoogle Scholar
  8. 8.
    B. Krauskopf, H.M. Osinga, Computing Invariant Manifolds Via the Continuation of Orbit Segments (Springer, 2007)Google Scholar
  9. 9.
    B. Krauskopf, H.M. Osinga, E.J. Doedel, M.E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz, O. Junge, A survey of methods for computing (un)stable manifolds of vector fields. Int. J. Bifurcat. Chaos 15, 763–791 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, vol. 112 (Springer, 2013)Google Scholar
  11. 11.
    M. Lakshmanan, The fascinating world of the landau-lifshitz-gilbert equation: an overview. Philos. Trans. R. Soc. A 369, 1280–1300 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    M. Lakshmanan, K. Nakamura, Landau-lifshitz equation of ferromagnetism: exact treatment of the gilbert damping. Phys. Rev. Lett. 53, 2497 (1984)CrossRefGoogle Scholar
  13. 13.
    C.-S. Liu, K.-C. Chen, C.-S. Yeh, A mathematical revision of the landau-lifshitz equation. J. Mar. Sci. Technol. 17, 228–237 (2009)Google Scholar
  14. 14.
    S. Murugesh, M. Lakshmanan, Bifurcation and chaos in spin-valve pillars in a periodic applied magnetic field. Chaos 19, 043111 (2009)CrossRefGoogle Scholar
  15. 15.
    S. Murugesh, M. Lakshmanan, Spin-transfer torque induced reversal in magnetic domains. Chaos, Solitons Fractals 41, 2773–2781 (2009)CrossRefGoogle Scholar
  16. 16.
    J. Neimark, On some cases of periodic motions depending on parameters, in Dokl. Akad. Nauk SSSR 129, 736–739 (1959)Google Scholar
  17. 17.
    J. Persson, Y. Zhou, J. Akerman, Phase-locked spin torque oscillators: Impact of device variability and time delay. J. Appl. Phys. 101, 09A503 (2007)CrossRefGoogle Scholar
  18. 18.
    W. Rippard, M. Pufall, S. Kaka, T. Silva, S. Russek, J. Katine, Injection locking and phase control of spin transfer nano-oscillators. Phys. Rev. Lett. 95, 067203 (2005)CrossRefGoogle Scholar
  19. 19.
    R.J. Sacker, On invariant surfaces and bifurcation of periodic solutions of ordinary differential equations. Technical report, DTIC document (1964)Google Scholar
  20. 20.
    F. Schilder, H.M. Osinga, W. Vogt, Continuation of quasi-periodic invariant tori. SIAM J. Appl. Dyn. Syst. 4, 459–488 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    F. Schilder, B.B. Peckham, Computing arnold tongue scenarios. J. Comput. Phys. 220, 932–951 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    C. Serpico, R. Bonin, G. Bertotti, M. Aquino, I. Mayergoyz, Theory of injection locking for large magnetization motion in spin-transfer nano-oscillators. IEEE Trans. Magn. 45, 3441–3444 (2009)CrossRefGoogle Scholar
  23. 23.
    J.Z. Sun, Spin-current interaction with a monodomain magnetic body: a model study. Phys. Rev. B 62, 570–578 (2000)CrossRefGoogle Scholar
  24. 24.
    V. Tiberkevich, A. Slavin, E. Bankowski, G. Gerhart, Phase-locking and frustration in an array of nonlinear spin-torque nano-oscillators. Appl. Phys. Lett. 95, 2505 (2009)CrossRefGoogle Scholar
  25. 25.
    J. Turtle, K. Beauvais, R. Shaffer, A. Palacios, V. In, T. Emery, P. Longhini, Gluing bifurcations in coupled spin torque nano-oscillators. J. Appl. Phys. 113, 114901 (2013)CrossRefGoogle Scholar
  26. 26.
    A.E. Wickenden, C. Fazi, B. Huebschman, R. Kaul, A.C. Perrella, W.H. Rippard, M.R. Pufall, Spin torque nano oscillators as potential terahertz (thz) communications devices. Technical report, DTIC document (2009)Google Scholar
  27. 27.
    Z. Zeng, P.K. Amiri, I.N. Krivorotov, H. Zhao, G. Finocchio, J.-P. Wang, J.A. Katine, Y. Huai, J. Langer, K. Galatsis et al., High-power coherent microwave emission from magnetic tunnel junction nano-oscillators with perpendicular anisotropy. Acs Nano 6, 6115–6121 (2012)CrossRefGoogle Scholar
  28. 28.
    I. Žutić, J. Fabian, S.D. Sarma, Spintronics: fundamentals and applications. Rev. Mod. Phys. 76, 323 (2004)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • James Turtle
    • 1
    Email author
  • Antonio Palacios
    • 2
  • Patrick Longhini
    • 3
  • Visarath In
    • 3
  1. 1.Predictive Science Inc.San DiegoUSA
  2. 2.San Diego State UniversitySan DiegoUSA
  3. 3.Space and Naval Warfare Systems Center PacificSan DiegoUSA

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