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Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x9" Mack and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. Bell & Howell Information and Learning 300 North Zeeb Road. Ann Arbor, Ml 48106-1346 USA UIVQ800-521-0600 Auto-oscillation’s behavior in YIG films from the stabihty point of view. DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Nikolay Piskun, M.S. in Physics ***** The Ohio State University 1999 Dissertation Committee: Approved by Dr. P.E. Wigen , Adviser Dr. A.J.Epstein A d ^ Dr. J.Shigemitsu Department of Physics Dr. B.R.Patton UMi Number 9951713 Copyright 2000 by Piskun, Nikolay All rights reserved. UMI* UMI Microform9951713 Copyright 2000 by Bell & Howell Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. Bell & Howell Information and teaming Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 ABSTRACT In recent years the interest on chaotic phenomena has sparked a new wave of theoretical as well as experimental work. Despite the wide diversi^ of nonlinear systems they usually undergo a common dynamic evolution. Starting from the steady state the system evolves to a linear periodic motion and then experiences a series of “phase transitions” known as bifurcation which lead to a final chaotic state. On the other hand it is well known that thin magnetic films display nonlinear behavior ultimately leading to chaos. The whole range o f experimental results has been reported as for synchronization of chaotic behavior as well as stabilization of unstable periodic orbits. This work investigates the dynamics of such a complex magnetic system as yttrium iron garnet films and a numerical study of bifurcation to chaos in auto-oscillations, based on the experimental work is presented. Experiments and numerical analysis have been performed with thin circular films using perpendicular high-power ferromagnetic resonance. Starting with the low power FMR spectrum and analyzing the behavior as a function of power and magnetic film the creation and evolution of auto-oscillations is demonstrated. The numerical analysis is based on six coupled equations of motion for three complex magneto-static modes. Taking the magnetic field as a variable parameter the bifurcation route to chaos has been 11 investigated. Along with the modeled signal, the time dependence of complex Lyapunov exponents has been traced. It has been found that during one trajectory cycle the complex Lyapunov exponents have two different sets of solutions. The first set is a complex conjugate pair with the same large negative real part. The second is a pair of two real distinct solutions one of which quickly goes above zero. The corresponding imaginary parts are equal to zero over this time range. It is during this period when period the real exponents of the system are positive and modifying its periodic orbit the system dramatically changes its behavior, thus spanning through the whole attractor of the nonlinear system. It has been found that auto-oscillations in YIG films follow the Feigenbaum (logistic map) route to chaos. It can be concluded that even though this system is represented by multi-dimensional equations of motion, the underlying nonlinear dynamic can be described by a one-dimensional Poincare map and can be controlled as a one-dimensional system. The possibilities for controlling chaos with periodic perturbation have been studied and the reason for desynchronization bursts is discussed. Ill Dedicated to my family. IV ACKNOWLEDGMENTS I first want to thank my lovely wife for her support, understanding, and patience. Secondly, I must thank my parents for their steadfast support and encouragement for all my endeavors. My advisor. Dr. Phil Wigen provided countless hours of discussion, assistance, support and direction over the past years, and I am thankful for that. Dr. Derrick Peterman have provided a great deal of assistance and motivation during my first years while we worked together in the lab. Dr. Ursula Ebels and Dr. Rich Bomfireund also participated in many precious discussions and made valuable suggestions. Tanya Mishurova, Brad Turpin, Nalan Ozkan and Donglei Li added to a great atmosphere to work in. Special thanks go to Brian Keller and Brian Dunlap in the Computer Facility, who helped me a lot in my computer education. I would also like to acknowledge the R.J. Yeh fund for financial support of this research. VITA Januar 1 1969...........................Bom, Azov, USSR May 1994.................................. M.S. B.S., with Honors in Microelectronics, Moscow Institute of Physics And Technology (MIPT) 1992 - 1994..............................Research Associate, Physics and Technology Institute of the Academy of Sciences o f Russia. 1994 — 1996 Teaching Assistant, Physics Department of Physics, Ohio State University, Columbus, Ohio 1996 - present .........................Research Assistant, Physics Department of Physics, Ohio State University, Columbus, Ohio PUBLICATIONS Piskun N. Y and Wigen PÆ. "Bifurcation to chaos in auto-oscillations in circular YIG films" J. of Appl, Phys. pp. 4521-4523 v. 85 No. 6,1999 Piskun N.Y and Wigen P.E. "Frequency versus Lyapunov exponent map: new approach to investigate dynamics of nonlinear magnetic systems." J. of Appl,Phys. pp.6590-6593, v 83; No 11,1998; Piskun N.Y., Peterman D.W. and Wigen P.E. "Thickness dependence of auto oscillations in circular yttrium-iron-gamet films” J. of Applied Physics pp. 1087- 3848, V 81; No 8//28, 1997; VI Piskun N. Y., Baryshev Y.P., Kiivospitskij AJ).,Orlikovskij A.A. "60-nm trenches in Si02 made with die plasma-chemical etching" Mikroelektronika. p 27-28, v 23 n 5 Sept-Oct 1994. FIELD OF STUDY Major Field: Physics Condensed Matter Physics V» TABLE OF CONTENTS ABSTRACT................................................................................................................ ii DEDICATION............................................................................................................... iv ACKNOWLEDGEMENTS...................................................................................... v VITA............................................................................................................................. vi LIST OF TABLES.............................................................................................................x LIST OF FIGURES.................................................................................................... xü CHAPTERS: 1 Introduction ................................................................................................................. I 2 Magnetic resonance ............................................................................................... 3 2.1 Ferromagnetic resonance in thin film s ..................................................... 3 2.2 Spinwaves ................................................................................................. 5 2.3 High power resonance phenomena ....................................................... 10 2.4 Hamiltonian of the system ........................................................................ 13 2.5 Equation of motion ..................................................................................... 19 3 Chaos and control ......................................................................... 23 3.1 Definition of chaos ................................................................................... 23 3.2 Poincare m aps ............................................................................................. 25 3.3 Strange attractors, Lyapunov exponents and divergence of nearby trajectories ............................................................................................... 26 3.4 Control of chaos ..................................................................................... 30 4 Origin of auto-oscillations ....................................................................................... 37 4.1 Complex Lyapunov exponents ............................................................. 37 4.2 Lyapunov exponent versus frequency m ap .............................................43 VIII 5 Bifurcation to chaos ............................................................................................