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ACTA UNIVERSITATIS UPSALIENSIS Uppsala Dissertations from the Faculty of Science and Technology
112
Ferromagnetic Resonance as a Probe of Magnetization Dynamics
A Study of FeCo Thin Films and Trilayers
Yajun Wei
Dissertation presented at Uppsala University to be publicly examined in Häggsalen,Ångströmlaboratoriet, Uppsala, Wednesday, 13 May 2015 at 13:15 for the degree of Doctorof Philosophy. The examination will be conducted in English. Faculty examiner: ProfessorSujeet Chaudhary (Indian Institute of Technology Delhi).
AbstractWei, Y. 2015. Ferromagnetic Resonance as a Probe of Magnetization Dynamics. A Studyof FeCo Thin Films and Trilayers. Uppsala Dissertations from the Faculty of Science andTechnology 112. 125 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-554-9195-6.
The high frequency dynamic magnetic responses of FeCo thin films and structures have beeninvestigated mainly using ferromagnetic resonance (FMR) technique.
The FMR resonance condition and linewidth are first derived from the dynamic Landau-Lifshitz-Gilbert equation, followed by a study of the conversion between FMR field andfrequency linewidths. It is found that the linewidth conversion relation based on the derivativeof resonance condition is only valid for samples with negligible extrinsic linewidth contribution.The dynamic magnetic properties obtained by using FMR measurements of FeCo thin filmsgrown on Si/SiO2 substrates with varying deposition temperatures is then presented. Theeffective Landé g-factor, extrinsic linewidth, and Gilbert relaxation rate are all found to decreasein magnitude with increasing sample growth temperature from 20oC to about 400–500oC andthen on further increase of the growth temperature to increase in magnitude. Samples grownat about 400–450oC display the smallest coercivity, while the smallest value of the Gilbertrelaxation rate of about 0.1 GHz is obtained for samples grown at 450–500oC. An almostlinear relation between extrinsic linewidth and coercivity is observed, which suggests a positivecorrelation between magnetic inhomogeneity, coercivity and extrinsic linewidth. Another majordiscovery in this study is that the Gilbert relaxation decreases with increasing lattice constant,which is ascribed to the degree of structural order in the films.
A micromagnetic model is established for an asymmetric trilayer system consisting of twodifferent ferromagnetic (FM) layers separated by thin non-magnetic (NM) layer, treating themagnetization in each FM layer as a macrospin. Based on the model, numerical simulationsof magnetization curves and FMR dispersion relations, of both the acoustic mode wheremagentizations in the two FM layers precess in phase and the optic mode where they precessout-of-phase, have been carried out. The most significant implication from the results is thatthe coupling strength can be extracted by detecting only the acoustic mode resonances at manydifferent unsaturated magnetic states using broadband FMR technique.
Finally, trilayer films of FeCo(100 Å)/NM/FeNi(100 Å) with NM=Ru or Cu were preparedand studied. The thickness of the Ru and Cu spacer was varied from 0 to 50 Å. For the Ruspacer series, the film with 10 Å Ru spacer shows antiferromagnetic coupling while all otherfilms are ferromagnetically coupled. For the Cu spacer trilayers, it is found that all films areferromagnetically coupled and that films with thin Cu spacer are surprisingly strongly coupled(the coupling constant is 3 erg/cm2 for the sample with 5 Å Cu spacer). The strong couplingstrength is qualitatively understood within the framework of a combined effect of Ruderman-Kittel- Kasuya-Yosida interaction and pinhole coupling, which is evidenced by transmissionelectron microscopy analysis. The magnetic coupling constant decreases exponentially withincreasing Cu spacer thickness, without showing an oscillatory thickness dependence. Theresults have implications for the design of multilayers for spintronic applications.
Keywords: ferromagnetic resonance (FMR), Gilbert damping, magnetic coupling, FeCo
Yajun Wei, Department of Engineering Sciences, Box 534, Uppsala University, SE-75121Uppsala, Sweden.
© Yajun Wei 2015
ISSN 1104-2516ISBN 978-91-554-9195-6urn:nbn:se:uu:diva-247238 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-247238)
List of papers on which Chapters 5-8 are based
I Y. Wei, R. Brucas, K. Gunnarsson, I. Harward, Z. Celinski, and P. Svedlindh,Static and dynamic magnetic properties of bcc Fe49Co49V2 thin films onSi(100) substrates,Journal of Physics D: Applied Physics, 46 495002 (2013)
II Y. Wei, R. Brucas, K. Gunnarsson, Z. Celinski, and P. Svedlindh, Posi-tive correlation between coercivity and ferromagnetic resonance extrinsiclinewidth in FeCoV/SiO2 films,Applied Physics Letters, 104 072404 (2014)
III Y. Wei, S. Jana, R. Brucas, Y. Pogoryelov, M. Ranjbar, R. K. Dumas, P.Warnicke, J. Åkerman, D. A. Arena, O. Karis, and P. Svedlindh,Magnetic coupling in asymmetric FeCoV/Ru/FeNi trilayers,Journal of Applied Physics, 115 17D129 (2014)
IV Y. Wei, S. Akansel, T. Thersleff, I. Harward, R. Brucas, M. Ranjbar, S.Jana, P. Lansaker, Y. Pogoryelov, R. Dumas, K. Leifer, O. Karis, J. Åk-erman, Z. Celinski, and P. SvedlindhExponentially decaying magnetic coupling in sputtered thin film FeNi/Cu-/FeCo trilayers,Applied Physics Letters, 106 042405 (2015)
V Y. Wei, and P. Svedlindh,On the frequency and field linewidth conversion of ferromagnetic reso-nance spectra,Submitted (2015)
VI Y. Wei, and P. Svedlindh,Numerical studies of magnetization curves and ferromagnetic resonancedispersion relations of coupled asymmetrical trilayers,In manuscript (2015)
List of pedagogical papers that are not included
• Y. Wei, Z. Zhai, K. Gunnarsson, P. Svedlindh,A guided enquiry approach to introduce basic concepts concerning mag-netic hysteresis to minimize student misconceptions,European Journal of Physics, 35 065015 (2014)
• Y. Wei,A simple demonstration of terminal velocity: an experimental approachbased on Lenz's law,Physics Education, 47 265 (2012)
Notes on author contributions to the included papers
• Paper (I)Idea and Design (Svedlindh)Experiments and Data (Wei, Brucas, Gunnarsson, Harward, Celinski)Interpretation and Theory (Wei, Svedlindh, Brucas)Manuscript writing (Wei, Svedlindh, Brucas)Comments and revision (All authors)
• Paper (II)Idea and Design (Wei)Experiments and Data (Wei, Brucas)Interpretation and Theory (Svedlindh, Wei, Brucas, Gunnarsson, Celin-ski)Manuscript writing (Wei, Svedlindh)Comments and revision (All authors)
• Paper (III)Idea and Design (Svedlindh)Experiments and Data (Wei, Brucas, Svedlindh, Ranjbar, Jana, Pogo-ryelov)Interpretation and Theory (Wei, Svedlindh, Karis)Manuscript writing (Wei)Comments and Revision (All authors)
• Paper (IV)Idea and Design (Wei, Svedlindh)Experiments and Data (Wei, Akansel, Thersleff, Harward, Brucas, Ran-jbar)Interpretation and Theory (Wei, Svedlindh, Celinski, Thersleff, Leifer)Manuscript writing (Wei)Comments and Revision (All authors)
• Paper (V)Idea and Design (Wei)Experiments and Data (Wei)Interpretation and Theory (Wei, Svedlindh)
Manuscript writing (Wei)Comments and Revision (Wei, Svedlindh)
• Paper (VI)Idea and Design (Wei, Svedlindh)Modelling and Calculation (Wei)Interpretation and Theory (Wei, Svedlindh)Manuscript writing (Wei)Comments and Revision (Wei, Svedlindh)
PREFACE
The monograph includes 8 chapters. Chapters 1-4 give an introduction to thegeneral theory of magnetism, with a focus on ferromagnetic resonance (FMR).I transferred from an optical engineering (mainly optical sensing) backgroundto magnetism. Going through the theory of magnetism and deriving generalexpressions for FMRhave been quite helpful forme to catch upwith the subjectof magnetism. Such a detailed derivation of ferromagnetic resonance condi-tions is usually not presented in a textbook and thus it might also be helpful tothose who have little background in magnetism.Chapter 5 discusses the frequency and field linewidth conversion of FMR
spectra. Chapter 6 presents the dynamicmagnetic properties of FeCo thin filmsinvestigated mainly using FMR technique. Chapters 7-8 are numerical andexperimental studies of the magnetic coupling between the two ferromagneticlayers of an asymmetrical trilayer system. Chapter 9 includes a short summaryof the results and an outlook for future work.Chapters 5-8 are based on the author's own work and they are recent find-
ings. Therefore, each of these chapters are written independently and self-explanatory. This means that the readers can read each of the chapters 5-8without the need to refer to other chapters for definitions, equations, figures ortables. The main purpose of including the Preface is to point this out.
Contents
1 Basics of magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2 Magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.3 Magnetic moment and magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.3.1 Magnetic moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.3.2 Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4 Magnetic response of materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.4.1 Magnetic induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.4.2 Magnetic flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.4.3 Magnetic susceptibility and permeability . . . . . . . . . . . . . . . . . . . . . . 201.4.4 Types of magnetic materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5 Magnetic hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.6 Models of exchange interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.6.1 Exchange interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.6.2 Superexchange interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.6.3 Ruderman-Kittel-Kasuya-Yosida interaction . . . . . . . . . . . . . . . . . 241.6.4 Double exchange interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.6.5 Dipole-dipole interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.7 Experimental techniques for measuring magnetization andinduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.7.1 Superconducting Quantum Interference Device
(SQUID) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.7.2 Vibrating Sample Magnetometer (VSM) . . . . . . . . . . . . . . . . . . . . . . . 261.7.3 Alternating Gradient Magnetometer (AGM) . . . . . . . . . . . . . . . . . 281.7.4 Magneto-Optic Kerr Effect (MOKE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.7.5 X-ray Magnetic Circular Dichroism (XMCD) . . . . . . . . . . . . . . 29
2 Magnetization dynamics in thin films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2 Magnetic energies of thin films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.1 Exchange energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2.2 Zeeman energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2.3 Demagnetizing energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.2.4 Cubic anisotropy energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.2.5 Uniaxial anisotropy energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.2.6 Perpendicular anisotropy energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.2.7 Other energy terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Magnetization dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.3.1 Effective magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.3.2 Dynamic equation of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4 Experimental techniques in magnetization dynamiccharacterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.4.1 Ferromagnetic resonance (FMR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.4.2 Time resolved-XMCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.4.3 Other techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 Kittel approach to FMR resonance condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.1 Solving the LLG equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.1 Simplifying the equation for ideal thin films within-plane uniaxial anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.2 Solving for resonance condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.2 Resonance frequency and linewidth in realistic measurements . . 49
3.2.1 Inhomogeneities of excitation field: Nonuniform spinwave excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2.2 Inhomogeneities of effective field: Dispersion ofmagnetic anisotropy field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3 Field-swept FMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4 Smit-Suhl approach to FMR resonance condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.1 The general resonance condition and intrinsic linewidth . . . . . . . . . . . . 554.2 Energy density of a single layer thin film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.3 FMR with in-plane applied magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3.1 Film with both uniaxial and cubic anisotropy . . . . . . . . . . . . . . . 614.3.2 Film with only uniaxial anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.3.3 Film with only cubic anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4 FMR with out-of-plane applied magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.4.1 Film with uniaxial anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.4.2 Film with cubic anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5 Conversion between the field and frequency linewidth of FMR spectra 725.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.4 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6 Dynamic properties of FeCo films studied by ferromagnetic resonance 806.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.2 Film preparation, structural characterization and static magnetic
properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.2.1 Film preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.2.2 Structural characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.2.3 Static magnetic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.3 FMR study of dynamic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
7 Effects of coupling between magnetic layers in a trilayer structure . . . . . . 897.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897.2 Micromagnetic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897.3 Simulated results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.3.1 Ferromagnetic coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 927.3.2 Antiferromagnetic coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.4 Conclusion and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
8 Magnetic coupling of FeCo/Ru/FeNi and FeCo/Cu/FeNi trilayersystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 998.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 998.2 FeCo/Ru/FeNi trilayer films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
8.2.1 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 998.2.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
8.3 FeCoV/Cu/FeNi trilayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1028.3.1 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1028.3.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
8.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
9 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Svenska sammanfattning (Summary in Swedish) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114Dynamiska egenskaper hos tunna filmer av en järn-kobolt legering . . . 115Koppling mellan magnetiska lager i tunnfilmsstrukturer - inverkan påstrukturernas dynamiska egenskaper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
1. Basics of magnetism
1.1 IntroductionModern technology would be unthinkable without magnetic materials and mag-netic phenomena.--Rolf E. Hummel, Understanding Materials Science, Springer, Germany,
2006
Magnetism is a fascinating subject whose first application dates back to the11th century when the Chinese invented amagnetic device consisting of a lode-stone placed in a spoon and a smooth plate of bronze housing the spoon. Thedevice was used for geomancy, a technique for aligning buildings in order tobe in harmony with the surrounding environment and to bring good luck forthe residents of the building. This device was later used for orientation deter-mination and navigation.Today, magnetic materials and devices have found their applications in al-
most all technology areas and industrial sectors. Some examples of these ap-plications include medical imaging technology, magnetic storage technology,telecommunications industry and so on. We understand that magnetism orig-inates from spin and orbital magnetic moments, which are connected to themotion of the electrons. The magnetism research is driven by the need to de-velop smaller, faster, cheaper and more efficient materials and devices.
1.2 Magnetic fieldA magnet always has two poles, the north pole and the south pole. Magneticmonopoles does not exist according to the Maxwell equations. Still, one maythink of a single magnetic pole as one end of a long bar magnet. Michell in1750, and Coulomb in 1785, independently found that the interactive force be-tween two magnetic poles is proportional to the product of their pole strengthsand inversely proportional to the square of the distance r separating them. Inmathematical form, this empirical relation, in SI units, is written as
F =µ0
4π
p1p2r2
, (1.1)
where µ0 is called the permeability of free space, and has the value 4π× 10−7
Wb/Am or asF =
p1p2r2
(1.2)
17
in cgs units. In cgs units, a pole of unit strength is one that applies a force of 1dyne (1× 10−5 N) on another unit pole located 1 centimeter away. In SI unitsthe pole strength is given in A·m, while in cgs units it is given by emu/cm.This relation between magnetic force and pole strength has the same form
as the Coulomb’s law for electric charges. In fact, the electric and magneticinteractions are interlinked and are referred to as electromagnetic force. Elec-tromagnetic interaction, alongside with gravitation, strong and weak interac-tion make the four fundamental forces in nature. Similar to electric field, theconcept of magnetic field was invented to describe the non-contacting interac-tions. For the case of two magnetic poles, one may think of pole 1 generatinga magnetic field, which then exerts a force on pole 2. In cgs units a magneticfield of unit strength is one which exerts a force of 1 dyne on a unit pole. Theunit of magnetic field is oersted (Oe) in cgs units and A/m in SI units, with 1Oe = 103/4π A/m.
1.3 Magnetic moment and magnetization1.3.1 Magnetic momentMagnetic moment is measured in Am2 in SI units and in emu in cgs units.When a magnetic moment, such as a dipole, is placed in a magnetic field asshown in Fig. 1.1, the moment will have a potential energy relative to the par-allel position as the torque tends to align the moment along the field direction.The work done to turn the moment by a small angle of dθ against the field willbe
dEp = (pH sin θ)l
2dθ + (pH sin θ)
l
2dθ = mH sin θdθ (1.3)
in cgs units anddEp = µ0mH sin θdθ (1.4)
in SI units.Conventionally the potential energy zero was set at θ = 90o and the potential
energy of a magnetic moment in a magnetic field is
Ep =
∫ θ
90o
mH sin θdθ = −mH cos θ = −m · H (1.5)
in cgs units and
Ep = µ0
∫ θ
90o
mH sin θdθ = −µ0m · H (1.6)
in SI units.
18
Figure 1.1. Magnetic dipole in a uniform magnetic field. Plus and minus signs desig-nate north and south poles, respectively.
1.3.2 MagnetizationThe degree to which a body is magnetized is described by the magnetizationM . It is defined as the magnetic moment per unit volume, i.e.
M =m
V. (1.7)
The unit of magnetization in cgs units is gauss (G) and in SI units it is A/m,with 1 G = 103/4π. The magnetization of a material is also an indicator of howlarge a torque would be exerted on the magnet when it is placed in a uniformmagnetic field.
1.4 Magnetic response of materials1.4.1 Magnetic inductionWhen amagnetic fieldH is applied onto amaterial, the response of thematerialis called magnetic induction B. This response includes creating a magnetiza-tionM and a magnetic fieldHi inside the sample. Magnetic induction is givenby,
B = µ0(Hi +M) (1.8)
in SI units andB = Hi + 4πM (1.9)
in cgs units. Outside the material (M=0), B = H in cgs units and B = µ0Hin SI units. The SI unit of B is tesla (T) and the cgs unit of B is G, with 1T=104 G.
19
1.4.2 Magnetic fluxMagnetic flux, denoted Φ, is the surface integral of the normal component ofmagnetic induction B passing through the surface. The unit of magnetic fluxin cgs units is maxwell (Mx) and in SI units it is weber (Wb), with 1Mx = 10−8
Wb. Thus B is also called the magnetic flux density. The Faraday's law statesthat the electromotive force ε is equal to the rate of change of the magneticflux,
ε = −dΦ
dt. (1.10)
1.4.3 Magnetic susceptibility and permeabilityWhen placed in an varying external magnetic field, the response of a materialvaries, i.e. generating different M and Hi. This response is, to some extent,characterized by the susceptibility χ and permeability µ. Susceptibility is de-fined as the ratio of the change of magnetizationM over the change of intrinsicfield Hi,
χ =∂M
∂Hi. (1.11)
Permeability is the ratio of the change of induction B over the change of in-trinsic field Hi,
µ =∂B
∂Hi. (1.12)
1.4.4 Types of magnetic materialsMagnetic materials are classified according to their response to a magneticfield as diamagnetic materials, paramagnetic materials, ferromagnetic materi-als, ferrimagnetic materials and antiferromagnetic materials. In fact, all mate-rials display diamagnetism. In any atom or ion, the orbital motion of electronscreates atomic-scale current loops, which generate magnetic fields. When anexternal magnetic field is applied to the material, these current loops will tendto align in such a way as to create weak magnetic fields opposing the appliedfield. This may be viewed as an atomic version of Lenz's law: Induced mag-netic fields tend to oppose the external field change that created them. There-fore all materials are inherently diamagnetic, but only materials in which thisdiamagnetic effect is the only magnetic response are called diamagnetic ma-terials. Other materials have another stronger response (paramagnetism, an-tiferromagnetism, ferromagnetism or ferrimagnetism) besides diamagnetism.These stronger effects are then always dominant, with the exception of Pauliparamagnetism. Fig. 1.2 summarizes the features of each type of magneticmaterial.
20
• Diamagnetic materials: The magnetic moments of individual atoms orions are randomly oriented so there is no net magnetization. When anexternal field is applied, the moments will act in the way to counter theapplied field, resulting in a negative susceptibility. Except for supercon-ductors, the susceptibility is very small (typically in the order of -10−6
[1] in SI units).• Paramagnetic materials: The magnetic moments of individual atoms orions are randomly oriented so there is no net magnetization. When an ex-ternal field is applied, the moments will partially align with it, yielding apositive susceptibility. But the susceptibility is also very small (typicallyin the order of 10−5∼10−3 in SI units [1]) because the thermal energy thattends to align the moments randomly is much larger than the magneticenergy that tends to align them parallel to the external field. However, inconductors the electrons are itinerant rather than localized. When placedin an external magnetic field, the conduction band splits into a spin-upband and a spin-down band due to the difference in magnetic potentialenergy for spin-up and spin-down electrons. As the Fermi level mustbe identical for both spin-up and spin-down bands, there will be a smallsurplus of the type of spins in the band that moved downwards. Thisleads to a weak form of paramagnetism known as Pauli paramagnetism.Paramagnetism is temperature dependent, with the susceptibility χ ofparamagnetic materials inversely proportional to the temperature. Thisis known as Curie's law. But Pauli paramagnetism is an exception to thislaw, with its susceptibility almost independent of temperature.
• Ferromagnetic materials: The magnetic moments of individual atoms orions align parallel to each other below a magnetic ordering temperaturereferred to as the Curie temperature, resulting in a large net magnetiza-tion. The susceptibility is large and positive.
• Antiferromagneticmaterials: Themagneticmoments of individual atomsor ions are aligned in antiparallel below a magnetic ordering temperaturereferred to as the Néel temperature, implying that there is no net magneticmoment. When an external field is applied, the angle between neighbor-ing magnetic moments will decrease slightly from π, resulting a smallpositive susceptibility.
• Ferrimagnetic materials: The magnetic moments form many sublatticepairs between which the moments are aligned in antiparallel. However,there is discrepancy in the magnitudes of the magnetic moments in thetwo sublattices, resulting in a net magnetization. Ferrimagnetic materialsbehave macroscopically like ferromagnets with large positive suscepti-bility while its microscopical configuration is similar to that of antifer-romagnets.
21
Figure 1.2. The magnetic ordering, magnetization curves and example materials ofdifferent types of magnetism. Note that the magnetization curves are schematic illus-trations for each type of material [2]. They do not correspond to the magnetizationcurves of the example material(s).
22
Figure 1.3. A typicalM vs. H and B vs. H curve of a ferromagnetic material [3].
1.5 Magnetic hysteresisFor a ferromagnetic or ferrimagnetic material, the magnetic response to an ex-ternal field is not linear, as can be seen from Fig. 1.2. Even when the field isremoved, part of the spin alignment remains: The material has become mag-netized. The phenomenon that the magnetization M (and induction B) lagsbehind the change of external field H is referred to as hysteresis and the M(B) vs. H curves of ferro- and ferrimagnetic materials are called hysteresisloops [2]. A typical hysteresis curve of a ferromagnet is shown in Fig. 1.3.When the applied field is reduced to zero after saturation, the magnetizationdecreases from a saturated value Ms to the remanent magnetization Mr (in-duction decreases to remanent induction Br). The reversed field required toreduce the magnetic induction to zero is called the coercivity,Hc. The reversefield required to drive the magnetization to zero is called the intrinsic coerciv-ityHci [3]. Depending on the value of the coercivity, ferromagnetic materialsare classified as hard magnetic materials and soft magnetic materials. A hardmagnet needs a large field to reduce its induction to zero (or to saturate themagnetization). A soft magnet is characterized by small coercivity. It can beeasily saturated and also easily demagnetized.
1.6 Models of exchange interaction1.6.1 Exchange interactionThe quantum mechanical exchange energy between two electrons Jab is de-fined as twice the energy difference between their symmetric and antisymmet-ric two-body wavefunctions [2]. In mathematic form it is
Jab = ⟨ϕa (ri)ϕb (rj)|Hij |ϕb (ri)ϕa (rj)⟩ , (1.13)
where ri and rj are the position vectors of the two electrons, ϕa and ϕb areelectron wave functions and Hij is the Hamiltonian for the electron-electronrepulsive interaction. Since a system tends to minimize its energy to stablize,
23
when Jab is positive the exchange energy favors electrons with parallel spinsand causes ferromagnetic interaction. When Jab is negative the exchange en-ergy favors electrons with antiparallel spins and the interaction is antiferro-magnetic. Exchange interaction requires overlapping electron wave functions.
1.6.2 Superexchange interactionSuperexchange is an indirect exchange interaction between the nearest neigh-boring magnetic cations mediated by chemical bonding through an intermedi-ate anion (eg. oxygen). It results from partial covalent bond formation betweenthe electrons on the magnetic ions and those on the intermediate anions. Thisinteraction can lead to either strong antiferromagnetic coupling or weak ferro-magnetic coupling, depending on the occupancy of the interacting cation or-bitals and on the bond-angle between the interacting cations. Following the so-called Goodenough-Kanamori-Andersson rules [4], the 180o superexchange(the magnetic cation-anion-cation angle is 180o) of two magnetic ions withhalf-filled d orbitals is strongly antiferromagnetic, while a 90o superexchangeinteraction is ferromagnetic and much weaker.
1.6.3 Ruderman-Kittel-Kasuya-Yosida interactionRuderman-Kittel-Kasuya-Yosida (RKKY) interaction refers to the interactionbetween a localized magnetic moment (nuclear magnetic moments, or local-ized d or f shell electron spins) and an electron gas, through their mutual ex-change [5, 6, 7]. The interaction between the electron gas and other localizedmoments in the system can also lead to an effective interaction between the lo-calized moments. An important characteristic of this interaction is that the signand magnitude of the coupling is oscillatory, causing either ferro- or antiferro-magnetic coupling depending on the spacing between the local moments andthe carrier density in the electron gas. RKKY interaction is usually responsiblefor the interlayer magnetic coupling of trilayers (two ferromagnetic layers sep-arated by a nonmagnetic layer), which are basic building blocks of magneticstorage devices.
1.6.4 Double exchange interactionDouble exchange is an interaction that occurs in mixed valence materials. Insuch materials double exchange arises from the delocalization of electronsfrom the low valence to high valence ions as it lowers the kinetic energy ofthe system. The electrons are itinerant. Like for superexchange interaction,the interaction between magnetic cations is mediated by the anions. However,double exchange causes only ferromagnetism.
24
1.6.5 Dipole-dipole interactionDipole-dipole interaction is a direct anisotropic long-range interaction betweentwo magnetic dipoles. It is also called dipolar coupling. Two dipoles thatare close to each other in space experience the magnetic fields of each other,which leads to a slightly different effective magnetic fieldHeff experienced bya dipole. The field depends on the orientation of the two magnetic dipoles.The strength of the dipole-dipole coupling depends strongly on the distancebetween the two dipoles.
1.7 Experimental techniques for measuringmagnetization and induction
Measurements of magnetic quantities are usually based on the determinationof either the magnetic flux density (magnetic induction) or the magnetization.
1.7.1 Superconducting Quantum Interference Device (SQUID)The SQUID magnetometer [8] is currently one of the most widely used andmost sensitive instruments for magnetic flux measurement. This device usesthe quantization ofmagnetic flux in a superconducting loop together with Joseph-son junctions to measure magnetic flux density, achieving a resolution of about10−15 T/
√Hz or better. Fig. 1.4 is a sketch of a DC SQUID sensing element.
Figure 1.4. Sketch of a SQUID sensor.
A constant bias current is fed through the ring and divided between two equalbranches, each passing through one Josephson junction. Due to the wave na-ture of the superconducting current, the resistance of the ring appears to bea periodic function of the magnetic flux threading the superconducting ring.The voltage across the SQUID element oscillates with the changes in phase
25
Figure 1.5. Block diagram of a SQUID system. Through a variable flow valve, heliumis drawn to the vacuum insulated chamber for accurate control of the temperature insidethe sample chamber.
of the superconducting wave function at the two junctions, which is depen-dent on the change in the magnetic flux. By using a flux-locked operationit is possible to resolve minute fractions ( 10−5) of a single flux quantum ϕ0
(ϕ0 = 2.0678× 10−15 Wb) [9]. A block diagram of a SQUID magnetometeris shown in Fig. 1.5.
1.7.2 Vibrating Sample Magnetometer (VSM)The VSM is another commonly used instrument for measuring magnetic fluxdensity [10]. In a VSM measurement, a sample is placed inside an externalmagnetic field to be magnetized and an oscillatory motion of the sample is
26
created, typically through the use of a piezoelectric material. The vibratingsample therefore causes a time varying magnetic field in its surrounding, anda voltage will be induced in a pickup coil placed close to the sample. Fig. 1.6
Figure 1.6. Principle of a VSM.
Figure 1.7. Block diagram of a VSM system.
illustrates the working principle of a VSM and in Fig. 1.7 a block diagram ofa simple VSM system is presented. It is shown that for constant oscillationamplitude and frequency, the voltage induced in the coil will be directly pro-portional to the magnetic moment of the sample. By measuring the voltagefrom the pickup coil and the volume of the sample, one is able to obtain the
27
value of the magnetization. The VSM is sometimes called Foner magnetome-ter after its inventor Simon Foner [10].
1.7.3 Alternating Gradient Magnetometer (AGM)In contrast to SQUID and VSM that measure magnetic flux, the AGM mea-sures the sample magnetic moments directly [11]. Fig. 1.8 shows the workingprinciple of an AGM. In a measurement, a sample is mounted on an extendedsample holder, which is placed inside a fixed DC magnetic field. At the sametime, a pair of coils are used to generate an alternating field gradient along they-axis on the order of 1 Oe/mm. This field gradient then produces an alternat-ing force exerted on the sample. The force is given by
F = −∇E = −∇(m · H) (1.14)
orFy = −m
∂H
∂y, (1.15)
where m is the magnetic moment of the sample, H is the DC magnetic fieldand E is the magnetic energy of the sample. It is immediately clear that thealternating force is directly proportional to themagnetic moment of the sample.The force is converted to a voltage signal by a piezoelectric sensor and thenanalyzed by a lock-in amplifier, and thus the magnetic moment is measured.
Figure 1.8. Principle of an AGM.
1.7.4 Magneto-Optic Kerr Effect (MOKE)The MOKE technique is based on the Kerr effect first discovered by JohnKerr in 1877 [12]. When a linearly p-polarized light beam is reflected by a
28
ferromagnetic sample, it will consist of an s-component (electrical field Es
component perpendicular to the plane of incidence) in addition to the originalp-component (electrical field Ep component perpendicular to the plane of in-cidence), withEs/Ep being defined as the Kerr rotation. The magnitude of theKerr rotation is proportional to the magnetization of the sample. The MOKEmeasures the Kerr rotation and yields the magnetization value of the sample.A schematic diagram of a MOKE setup is shown in Fig. 1.9.
Figure 1.9. A MOKE experimental setup.
Since the light beam (usually generated by a red laser) has a limited pene-tration depth into the sample, only surface magnetic properties are detected.Therefore MOKE is also referred to as SMOKE (surface MOKE) [13]. Thesensitivity of aMOKE is not as good as for a SQUIDmagnetometer. However,a MOKE system is often constructed in such a way to allow one to rotate thesample with respect to the applied magnetic field, making it possible to studyanisotropy of magnetic materials.
1.7.5 X-ray Magnetic Circular Dichroism (XMCD)Circularly polarized light carries angular momentum that is either parallel orantiparallel to the direction of light propagation, denoted by left- and right-handed circularly polarized light, respectively. Usually a photon's angular mo-mentum can only be transferred to the electron orbital momentum and the spinof the electron is conserved. However, this is not necessarily true for elec-trons with a strong spin-orbit coupling such as the 2p core level electrons intransition metals. For right-handed circularly polarized light at the L3 edge62.5% of the excited electrons have spin up, while for left-polarized light spindown electrons account for the same percentage. In magnetic transitionmetals,there are more unoccupied 3d states for one spin direction than the other. Sincethe X-ray absorption cross-section is proportional to the number of empty 3d
29
Figure 1.10. Diagram of an XMCD process. Courtesy of R. Knut [16].
states, the absorption of circularly polarized X-ray is sensitive to the magne-tization of the sample. The difference in absorption between right- and left-handed circular polarized light is called X-ray magnetic circular dichroism.Fig. 1.10 illustrates the XMCD process. The cross section of X-ray absorptionalso depends on the the degree of the circular polarization of the synchrotronradiation.In anXMCDmeasurement, the experiment can be performed either by switch-
ing the polarization of light while keeping the magnetization constant or bychanging the magnetization direction of the sample while keeping the lightpolarization constant. The most important differences between this techniqueand other magnetization measurement techniques such as SQUID and VSMare that the XMCD technique is element specific and that it enables one toseparate spin and orbital magnetic moment contributions [14, 15].
30
2. Magnetization dynamics in thin films
2.1 IntroductionThe technological advancements in the field of high quality thin films as wellas in micro- and nanostructuring of films have been substantial during the pastdecades. For instance, optical lithography pushes down the miniaturizationinto the nanometer regime. This advancement undoubtedly brings promisingopportunities for miniaturization of novel magnetic memory devices and sen-sors [17, 18] that offer high speed, high storage density and low energy con-sumption.Since the switching current of a spintronic magnetic memory cell is pro-
portional to the damping of the spin dynamics of the magnetic material, forfast writing and reading speed as well as for low energy consumption, a smallmagnetization damping is preferred. As a consequence, a comprehensive un-derstanding of the dynamic response of small magnetic elements to high fre-quency magnetic fields will be essential to be able to push the device perfor-mance closer to the fundamental limits.This chapter first gives a brief introduction to the magnetization dynamics
equation, starting from the energies of a thin magnetic film. Then the phe-nomenon of ferromagnetic resonance is explained. Finally a couple of exper-imental techniques for characterizing magnetization dynamics of thin films inboth frequency and time domains are briefly described.
2.2 Magnetic energies of thin filmsThe total magnetic energy density of a thin film is given by
εtot = εex + εzee + εdem + εc + εu + εp, (2.1)
where εex is the exchange energy density, εzee is the Zeeman energy density,εdem is the demagnetizing energy density, εc is the cubic magnetocrystallineanisotropy energy density, εu is the uniaxial phanerocrystalline anisotropy en-ergy density and εp is the perpendicular anisotropy energy density. These termsare explained in the following subsections.
2.2.1 Exchange energyThe phenomenon whereby individual atomic magnetic moments will attemptto align with all other atomic magnetic moments within a material is referred
31
to as the exchange interaction. The energy associated with this interaction isknown as the exchange energy, which is given by
Eex = −N∑ij
Jij
(Si · Sj
), (2.2)
where Jij is the exchange integral, Si and Sj are vectors pointing along thedirections of the magnetic moments of atoms i and j, respectively. The sum-up is usually only to the nearest neighbors.A positive sign of Jij indicates that the material exhibits ferromagnetic be-
haviour and the exchange energy is at a minimum when two neighbouringmoments are in a parallel alignment. Antiferromagnetic materials, on the con-trary, exhibit a negative sign of the exchange integral, and therefore have aminimum exchange energy when two neighbouring moments are aligned inopposite directions.In classic form, the exchange energy density is expressed as
εex =A
V
∫dV
(∇ M
Ms
)2
, (2.3)
where V is the volume of the material and A is the exchange constant (alsocalled exchange stiffness) given by [19]
A =JijS
2
a(2.4)
for a simple cubic lattice with a being the lattice constant, or
A =2JijS
2
a(2.5)
for a body-centered cubic lattice, or
A =4JijS
2
a(2.6)
for a face-centered cubic lattice.
2.2.2 Zeeman energyThe Zeeman energy is due to the interaction of the sample magnetization Mwith the external field Hext. It is given by
εzee = −µ0
V
∫dV M · Hext. (2.7)
This energy is minimized by aligning the magnetization M along the directionof external field Hext.
32
2.2.3 Demagnetizing energyThe demagnetizing energy is due to dipolar interaction between the magneticmoments and is expressed as
εdem = − µ0
2V
∫sample volume
dV M · Hdem. (2.8)
For a homogeneous magnetization and demagnetization field, the demagnetiz-ing energy is then given by
εdem =µ0
2M ·N · M, (2.9)
where N is the so called demagnetization tensor that is dependent on the geo-metrical shape of the magnet.The demagnetizing field is written as
Hdem = −N · M. (2.10)
For ellipsoids and by choosing an appropriate coordinate system, the demag-netizing tensor N can be simplified as
N =
Nx 0 00 Ny 00 0 Nz
, (2.11)
where the non-zero tensor elements correspond to demagnetization factors alongthe principal axes.The demagnetizing energy density is also referred to as shape anisotropy en-
ergy. The demagnetizing energy density of an ellipsoid is of particular interest.If the coordinate system is chosen to be coincident with the ellipsoid axes a, band c, the demagnetizing energy density is given by
εdem_ellipsoid =µ0M
2s
2
(Nxcos2αx +Nycos2αy +Nzcos2αz
), (2.12)
where Nx, Ny, and Nz are the demagnetizing factors (Nx + Ny + Nz = 1)and αx, αy and αz correspond to the angles between the magnetization vectorand the ellipsoid axes. With oblate-like objects (x ≫ z and y ≫ z) such asthin films, we have Nx = Ny = 0. Thus the tensor reduces to a scalar. Thedemagnetizing energy for a uniformly magnetized thin film reads
εdem_film =µ0M
2s
2cos2αz. (2.13)
2.2.4 Cubic anisotropy energyMagnetocrystalline anisotropy refers to the tendency of the magnetization toalign along one (some) particular direction(s) due to crystal symmetry. Cubic
33
magnetocrystalline anisotropy energy, mostly called cubic anisotropy energyfor short, is the dependence of the energy on the magnetization direction withrespect to crystal axes for materials with cubic symmetry. Cubic anisotropyenergy is usually written as
εc = K1
(cos2αcos2β + cos2βcos2γ + cos2γcos2α
)+K2cos2αcos2βcos2γ, (2.14)
where K1 and K2 are the 1st and 2nd order cubic anisotropy constants of thematerial, dependent on temperature, and α, β and γ are the angles that themagnetization vector makes with the crystal axes. For many cases,K2 is neg-ligible in comparison to K1 and is set to be 0. When K1 dominates over K2,the easy (hard) axis is along ⟨100⟩ (⟨111⟩) if K1 > 0 and the easy (hard) axisis along ⟨111⟩ (⟨100⟩) ifK1 < 0.Anisotropy decreases with increasing temperature in all materials and ap-
proaches zero near the Curie temperature as there is no preferred orientationfor the magnetization in a paramagnetic state.
2.2.5 Uniaxial anisotropy energySimilar to cubic anisotropy energy, uniaxial anisotropy energy is associatedwith magnetocrystalline energies of materials with uniaxial symmetry,
εuni = Kusin2α, (2.15)
whereKu is the uniaxial anisotropy constant of thematerial, dependent on tem-perature, and α is the angle that M makes with the easy axis of magnetization(often this axis is defined as x axis). ForKu > 0, the x axis is the easy axis; forKu < 0, the x axis is the hard axis (and the y− z plane corresponds to an easyplane of magnetization). Besides intrinsic magnetocrystalline anisotropy, uni-axial anisotropy may also include contribution from strain-induced anisotropyand growth-induced anisotropy [20].
2.2.6 Perpendicular anisotropy energyThe perpendicular anisotropy energy εp is any energy contribution that tendsto align the magnetization normal to the film plane. The main origin of per-pendicular anisotropy is the surface and interface anisotropy. The broken sym-metry of the surface and the bottom of the film leads to surface anisotropy, aphenomenon appreciable only in thin enough films but not in bulk materials.Surface anisotropy energy usually leads to a perpendicular easy axis of magne-tization and a hard plane of magnetization (the film plane). This energy densityterm is given by
εsur =Ks
d, (2.16)
34
where Ks (or in some literature referred to as the phenomenological uniaxialperpendicular anisotropy parameterK⊥) is the surface anisotropy constant andd is the thickness of the film. For films thicker than 50 nm, εsur is usually smallenough to be neglected.
2.2.7 Other energy termsThe six energy terms listed above are common to many magnetic thin film sys-tems. The latter three are all anisotropy energies. In some special cases, someother not-so-common energy terms such as unidirectional anisotropy mightneed to be included. In principle, the lowest order symmetry possible for mag-netic anisotropy is uniaxial, which has 180o symmetry. However, by takingadvantage of exchange coupling between the magnetic material of interest andan antiferromagnet with substantial anisotropy, such as in a bilayer or mul-tilayer, a unidirectional anisotropy with a symmetry of 360o can be created[21, 20].
2.3 Magnetization dynamics2.3.1 Effective magnetic fieldThe total effective magnetic field Heff acting on the magnetic moment insidea material is the result of the combined effect of the above interactions (en-ergies), namely, exchange, demagnetizing, Zeeman and anisotropy energies.The equilibrium state of the magnetic moments will be such that the total en-ergy due to the above interactions is minimal, as only at the energy minimumthe system can be stable. Therefore, Heff ends up being the derivative of thetotal energy density εtot with respect to the magnetization M ,
Heff = − 1
µ0∇M εtot. (2.17)
2.3.2 Dynamic equation of motionThemagnetization vector M will tend to align with the effective magnetic fieldHeff, in order tominimize the energy of the system. The spin and orbital motionof the electron generate an angular momentum that points along the directionof the magnetic moment (magnetization). When a small time varying radiofrequency (RF) field h is applied perpendicular to the static magnetic field (andthus the magnetization), it will apply a torque on the magnetic moment andinduce a precessional motion of the magnetization around the external staticfield. The motion of magnetization is described by the Landau-Lifshitz (LL)
35
equationdM
dt= −γµ0M × Heff, (2.18)
where γ is the gyromagnetic ratio with γ = g |e|/2me. e is the electron charge,me is the electron mass and g is the Lande g-factor whose value depends onthe material. The value of g for free electrons is equal to 2.0023. For othermaterials, it is usually slightly larger. Then γ ≈ 1.76 × 1011 rad· s−1·T−1.The LL model results in a precessional motion of the magnetization.With damping taken into account, the dynamics is described by the famous
Landau-Lifshitz-Gilbert (LLG) equation
dM
dt= −γµ0M × Heff +
α
Ms
(M × dM
dt
), (2.19)
where α is the dimensionless Gilbert damping parameter. This equation indi-cates that the magnitude of the magnetization is conserved but the precessionalcone angle decreases with time due to damping.α indicates how fast the precessional magnetization M relaxes to equilib-
rium (approaches the direction of the effective field Heff). The term (Gilbert)relaxation rate G is introduced and is related to α by
G = γαMs, (2.20)
where G has the dimension of frequency and is measured in Hz in both cgsand SI units. The damping constant or relaxation rate is very important as itcharacterizes the dynamics of the magnetization and determines the switchingcurrent if the material is used as a magnetic electrode in a magnetic randomaccessory memory (MRAM).
2.4 Experimental techniques in magnetization dynamiccharacterizations
2.4.1 Ferromagnetic resonance (FMR)FMR is one of the most powerful experimental techniques that is widely usedto investigate the magnetic dynamics in frequency domain. The first reportof an experimental observation in English-language literature was made byGriffiths [22] in 1946 and he is usually credited for the discovery. It is believed[23, 24] that FMRwas unknowingly discovered by Arkad'yevin in 1911 [25] ashe was the first to detect resonant absorption of ultra-high frequency radiationby ferromagnetic materials. The initial formalism of FMR was achieved byKittel [26] and Suhl [27]. Since then, FMR has become a well establishedtechnique, both in theory and experiment.
36
Figure 2.1. Image of the home built VNA-FMR setup at Uppsala University. Theinset at the left bottom of the photo is a magnified view of the picoprobes and coplanarwaveguide without sample on it.
Ferromagnetic resonance phenomenonWhen a static magnetic field H is applied on a ferromagnet, the magnetizationM in the material will tend to align with H in order to minimize the energy ofthe system (in fact due to the other energy terms, M might not be completelyaligned with H , but instead with Heff).When a weak RF field h is applied perpendicular to the static magnetic field
(and thus the magnetization), it will exert a torque on the magnetization and in-duce a precessional motion as shown in Fig. 2.3(a). The induced precessionalmotion of the magnetization implies that microwave energy is absorbed. Thismotion is damped due to interaction with other degrees of freedom, eg. relax-ation can occur due to electron-electron interaction (interband and intrabandexcitations often explain the intrinsic Gilbert damping). When the frequencyof the RF field matches the natural frequency of the system, the absorption willbe maximum and it is called FMR. This is observed as a Lorentzian line shapein the absorption spectrum, from which one can extract important propertiesof the material, such as the effective magnetization, anisotropy fields, Landeg-factor and the dynamic damping constant.
Cavity FMRA schematic of a cavity-based FMR system is shown in Fig. 2.2. A microwaveis generated by a microwave source and directed by a waveguide to a metalliccavity resonator, whose dimensions are carefully chosen to achieve impedance
37
Figure 2.2. A schematic of a cavity based FMR measurement. A microwave is gener-ated by the microwave source and directed to the cavity with the help of a circulator.The circulator also makes sure that the microwaves reflected from the cavity is directedto the microwave detector.
match between the waveguide and the cavity. This makes sure that the mi-crowaves in the cavity add up constructively to make a strong RF field. Themicrowave frequency is usually tunable within a small range. Before a mea-surement, the source microwave is set at cavity resonance to ensure that themicrowave couples well inside the cavity. A DC magnetic field is swept us-ing an electromagnet. When the DC field satisfies the ferromagnetic resonancecondition, the absorption of microwave energy by the sample will reach a max-imum, leading to a dip in the reflected microwave power, which is measuredby the microwave detector diode.In such a system, one is able to rotate the sample by employing a goniome-
ter, keeping the DC field in the film plane. Also, one may place the sample onthe sample holder in such a way that the sample rotates from parallel to per-pendicular with respect to the magnetic field. This will enable one to preciselydetermine the anisotropy fields of the sample.
Broadband VNA-FMRAn alternative to the cavity FMR is the broadband vector network analyzerbased FMR (VNA-FMR). This technique utilizes a broadband VNA to gen-erate, detect and analyze the microwave signal. The signal is directed usingcoaxial cables to a coplanar waveguide (CPW), on which the sample is placedface-down, as illustrated in Fig. 2.3. The field-swept measurement is similarto that used in a cavity FMR system. The only difference is that the transmit-
38
Figure 2.3. Schematics of a vector network analyzer (VNA) based FMR setup.
ted, rather than the reflected, microwave signal is measured in this case. Sincethe VNA is a broadband microwave source, one can easily make field-sweptmeasurements at many different frequencies. The VNA-FMR also enables oneto make frequency-swept measurements in which one keeps the DC field con-stant and sweeps the frequency instead.In general, the VNA-FMR technique is not as sensitive as cavity-FMR, but
it offers better flexibility and efficiency due to its broadband nature.
2.4.2 Time resolved-XMCDTime resolved XMCD is based on a bunch clock signal from a synchrotronlight source and the XMCD signal of the magnetic elements in the sample.It is an element specific technique used to study magnetization dynamics inthe time domain. A block diagram of such a setup is presented in Fig. 2.4 toillustrate the working principles and instrumentations.A bunch clock is an electronic signal from the synchrotron. The signal has
a frequency f0 (say f0 = 100 MHz, as for beamline I1011 at the MAX-labsynchrotron facility, Lund University, Sweden) and can count time to a res-olution of 1/f0 (10 ns). The bunch clock signal is sent to a delay controllerthat can adjust the time delay of the signal. The signal in frequency domainis not altered after passing the delay controller. Then the signal is sent to acomb generator, which splits the power of the signal into harmonics with fre-quencies f0, 2f0, 3f0… (100 MHz, 200 MHz, 300 MHz…) and so on. Nextthe signal goes through a band pass filter. The band pass filter is set to selectone of the frequency components between the 10th and 20th harmonics as thefiltered signal (say we select 1 GHz). The filtered signal is then passed to anamplifier and an isolator. The purpose of the isolator is to prevent reflectedsignals from reaching the amplifier.The amplified signal is passed to a grounded CPW. The electronic signal
(time varying current passing through the CPW) will produce an RF magneticfield that is parallel to the sample plane. This magnetic field acts as a drivingfield to the precession of the sample magnetization. The period of magnetiza-
39
Synchrotron
Bu
nch
Clo
ck
Ph
oto
n
be
am
RF
sw
itch
Lo
ck-in
Am
plif
ier
De
lay
co
ntr
olle
r
Ba
nd
pa
ss
filte
rC
om
b
ge
ne
rato
rA
mp
lifie
r
Iso
lato
r
Co
mp
ute
rA
DC
Dire
ctio
na
l
co
up
ler
Ph
oto
dio
de
/
Ph
oto
co
un
ter
X-r
ay p
ho
ton
sS
am
ple
CP
W
Figure 2.4. Block diagram of the time resolved XMCD technique. An electromagnetused to magnetize the sample is not shown in the diagram to increase readability. Alock-in amplifier is usually added to improve signal to noise ratio of the fieldmodulatedFMR measurement.
40
tion precession is the same as the period of the driving signal (1 ns in case of a1 GHz signal). The precession amplitude is maximum in the vicinity of FMR(when the DC magnetic field is approximately equal to the resonant field). AnFMR scan is first performed and then the DC magnetic field is set to a valueclose to the resonance field to maximize the precession amplitude. The pho-ton energy of the X-ray is set at the XMCD absorption peak of the element ofinterest to maximize the photon absorption.The photon bunches from the same synchrotron beam is used to probe the
precessional motion of the magnetization. The bunch length (defined as thehalf value of the time a photon bunch lasts) is typically about 0.1 ns. Theshorter the bunch length is, the better the time resolution will be.The number of photons absorbed by the sample is dependent on the angle
between the precessing magnetization and the incident photons. Absorptionpeaks when the precessing magnetization vector is parallel with the incidentphoton direction and is at a minimum when it is antiparallel with the photondirection.At a certain delay time, there is a specific direction of the precessing mag-
netization vector with respect to the incident photons. Since the electrical sig-nal has a frequency that is an integer multiple of the frequency of the photonbunches, the time interval between two adjacent photon bunches is always aninteger of the precessing period of the magnetization. As a result, consecutivephoton bunches will experience the same orientation (angle) of the precess-ing magnetization vector. Counting the number of transmitted photons over atime period of a few seconds (and thus the number of photons absorbed by thesample can be obtained) reveals the direction of the precessing magnetizationvector. By scanning the delay time via the delay controller, the transmittedphotons for each bunch of photon vs. delay time (or direction of the precess-ing magnetization vector with respect to the direction of incident light) can beplotted. This gives information about the dynamics of the magnetization, i.e.,how the magnetization vector precesses with time.
2.4.3 Other techniquesApart from the frequency domain FMR and time domain TR-XMCD tech-niques, there exist many other experimental techniques that can address mag-netization dynamics both in the frequency domain, e.g. Brillouin Light Scat-tering (BLS), and in the time domain, e.g. Time Resolved Scanning Kerr Mi-croscopy (TR-SKEM) [28, 29].
41
3. Kittel approach to FMR resonance condition
3.1 Solving the LLG equationTo get the resonance condition, one needs to obtain the natural frequency ofthe system in a particular external field via solving the LLG equation. Findinga general analytical solution for the LLG equation is not really practical. Butwith appropriate assumptions, an expression for the resonance condition canbe derived. In this section we solve the LLG equation for a thin film with theexternal magnetic field applied along the in-plane uniaxial easy axis.
3.1.1 Simplifying the equation for ideal thin films with in-planeuniaxial anisotropy
The following assumptions are made for the derivation in this section:• For the thin film, Nx = Ny = 0 and Nz = 1;• the film exhibits an in-plane uniaxial magnetocrystalline anisotropy fieldHu with easy axis along x and Hu ≪ Ms;
• a surface anisotropy with an easy plane in the film plane exists charac-terized by the anisotropy constantKp;
• the external field Hext is applied along the x direction. Themagnetizationpoints along the x direction before the excitation field is applied;
• a weak uniform microwave field h is applied along the y direction.The geometry is illustrated in Fig. 3.1. As h is small, the precessionalmotion
of M will be characterized by a small cone angle and we will then haveMy ≪Mx andMz ≪ Mx. Therefore,
M = Mxx+Myy +Mz z
= Msx+Myy +Mz z
=1
Ms(mxx+myy +mz z)
≈ 1
Ms(x+myy +mz z) , (3.1)
where m is defined as
m =M
Ms. (3.2)
42
x
y
z
M
M = Ms = M
s x
Hext
Hu
(a) Without microwave magnetic field.
x
y
z
MH
ext
Hu
h
M = Mx x + M
y y + M
z z
(b) With microwave magnetic field.
y + My + Mx + Mx + Mx y z
= Ms x + M
y y + M
z zy + My + Mx + Mx + M
Mx
My
Mz
Figure 3.1. FMR setup geometry. The CPW stripline is along the x direction and thesample is placed in the x− y plane.
43
The total energy density is
εtot = −µ0M ·(Hext + h
)+
µ0
2Mz
2 +Kusin2α−Kp
dcos2α
= −µ0M ·(Hext + h
)+
µ0
2Mz
2 +Ku
[1− (m · x)2
]−
Kp
d
(Mz
Ms
)2
= −µ0M ·(Hext + h
)+
µ0
2
(M · z
)2+
Ku
Ms2
[1−
(M · x
)2]−
Kp
Ms2d
(M · z
)2. (3.3)
Substituting the above equation into the expression for the effective field, i.e.Heff = − 1
µ0∇M εtot, yields,
Heff = Hext + h−(M · z
)z +
2Ku
µ0Ms2
(M · x
)x+
2Kp
µ0Ms2
(M · z
)z
= Hextx+ hy −Mzz +2Ku
µ0Msx+
2Kp
µ0Ms2Mz z
=
(Hext +
2Ku
µ0Ms
)x+ hy −
(Ms −
2Kp
µ0Ms
)Mz
Msz. (3.4)
Introducing the in-plane uniaxial anisotropy field
Hu =2Ku
µ0Ms(3.5)
and the effective magnetization
Meff = Ms −2K⊥µ0Msd
, (3.6)
we can simplify Heff and write it as
Heff = (Hext +Hu) x+ hy −MeffMz
Msz. (3.7)
44
Inserting the above expressions for Heff (Eq. (3.7)) and M (Eq. (3.1)) into thefirst and second terms of the LLG equation gives
−γµ0M × Heff = −γµ0
∣∣∣∣∣∣x y zMx My Mz
Heff_x Heff_y Heff_z
∣∣∣∣∣∣= −γµ0
∣∣∣∣∣∣x y zMx My Mz
Hext +Hu h −MeffMz
Ms
∣∣∣∣∣∣= −γµ0
{[−MyMeff
Mz
Ms−Mzh
]x
+
[Mz (Hext +Hu) +MxMeff
Mz
Ms
]y
+ [Mxh−My (Hext +Hu)] z} (3.8)
and
α
Ms
(M × dM
dt
)=
α
Ms
∣∣∣∣∣∣x y zMx My MzdMx
dtdMy
dtdMz
dt
∣∣∣∣∣∣=
α
Ms
{(My
dMz
dt−Mz
dMy
dt
)x
+
(Mz
dMx
dt−Mx
dMz
dt
)y
+
(Mx
dMy
dt−My
dMx
dt
)z
}. (3.9)
The LLG equation can then be written as
dMx
dt= −γµ0
(−MyMeff
Mz
Ms−Mzh
)+
α
Ms
(My
dMz
dt−Mz
dMy
dt
)dMy
dt= −γµ0
[Mz (Hext +Hu) +MxMeff
Mz
Ms
]+
α
Ms
(Mz
dMx
dt−Mx
dMz
dt
)dMz
dt= −γµ0 [Mxh−My (Hext +Hu)] +
α
Ms
(Mx
dMy
dt−My
dMx
dt
).
(3.10)
As h ≪ Hext,Mz andMy are much smaller thanMx(≈ Ms), we assume thatthe precession is only in the y-z plane, i.e. dMx
dt = 0. Substituting dMx
dt = 0
45
andMx(≈ Ms) into Eq. (3.10) yields
0 = −γµ0
(−MyMeff
Mz
Ms−Mzh
)+
α
Ms
(My
dMz
dt−Mz
dMy
dt
)dMy
dt= −γµ0 [Mz (Hext +Hu) +MzMeff]− α
dMz
dtdMz
dt= −γµ0 [Msh−My (Hext +Hu)] + α
dMy
dt. (3.11)
Also, as h, My and Mz are very small compared to other quantities in theabove equations, we can neglect the second order terms (i.e. terms includinga product of any two or three of the three quantities h,My andMz) and obtain
dMy
dt= −γµ0 [Mz (Hext +Hu) +MzMeff]− α
dMz
dtdMz
dt= −γµ0 [Msh−My (Hext +Hu)] + α
dMy
dt. (3.12)
This basically leads to the disappearance of the 1st equation while leaving the2nd and 3rd equations unaltered.
3.1.2 Solving for resonance conditionThe microwave is usually a sinusoidal signal. It is then more convenient towrite the microwave magnetic field and magnetization in a complex form,
h → h = |h| eiωtMx → Mx = |Mx| eiωtMy → My = |My| eiωtMz → Mz = |Mz| eiωt.
(3.13)
Inserting these expressions into Eq. (3.12) givesiωMy = −γµ0
[Mz (Hext +Hu) + MzMeff
]− αiωMz
iωMz = −γµ0
[Msh− My (Hext +Hu)
]+ αiωMy.
(3.14)
Rearranging the terms yields iωMy + [γµ0 (Hext +Hu) + iαw + γµ0Meff] Mz = 0
[iωα+ γµ0 (Hext +Hu)] My − iωMz = γµ0Msh
. (3.15)
46
The equations can be written in a matrix form as[iω γµ0 (Hext +Hu) + iαw + γµ0Meff
iωα+ γµ0 (Hext +Hu) −iω
]×(
My
Mz
)= γµ0Ms
(0
h
), (3.16)
or1
γµ0Ms
[iωα+ γµ0 (Hext +Hu) −iω
iω γµ0 (Hext +Hu) + iαw + γµ0Meff
]×(
My
Mz
)=
(h0
).
.(3.17)
To make the matrix look a bit more tidy, let us introduce the following deno-tations
ϖH = γµ0 (Hext +Hu) (3.18)ϖeff = γµ0Meff (3.19)ϖM = γµ0Ms. (3.20)
Then the matrix equation is
1
ϖM
(iωα+ϖH −iω
iω ϖH +ϖeff + iωα
)(My
Mz
)=
(h0
). (3.21)
This also indicates that the precessional motion is only in y-z plane, consistentwith the assumption we made in the very beginning.If we write h = (h, 0) and Myz = (My, Mz), then
Myz = χ h, (3.22)
where χ is the susceptibility tensor, according to its definition. Rewriting Eq.(3.22) in matrix form gives(
My
Mz
)=
(χyy χyz
χzy χzz
)(h0
). (3.23)
The susceptibility tensor χ can be computed from the above matrix equationas
χ =
(χyy χyz
χzy χzz
)=
[1
ϖM
(iωα+ϖH −iω
iω ϖH +ϖeff + iωα
)]−1
= ϖMϖH(ϖH+ϖeff)−w2+iωα(2ϖH+ϖeff)
(ϖH +ϖeff + iωα iω
−iω iωα+ϖH
)(3.24)
47
4 5 6 7 8 9-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Rel
ativ
e su
scep
tibilit
y (a
.u.)
Frequency (GHz)
real part imag part
f
fres
Figure 3.2. Simulated real and imaginary parts of the susceptibility spectrum. Pa-rameters used in the simulation are µ0Ms = 0.95 T ≈ µ0Meff, µ0Hext = 0.05 T andα = 0.01.
and
χyy = χ′yy + i χ′′
yy
=ϖM (ϖH +ϖeff)
[ϖH (ϖH +ϖeff)− w2
][ϖH (ϖH +ϖeff)− w2]2 + α2w2(2ϖH +ϖeff)
2
+i2wϖM
[w2 + (ϖH +ϖeff)
2]
[ϖH (ϖH +ϖeff)− w2]2 + α2w2(2ϖH +ϖeff)2. (3.25)
To give an impression of the frequency dependence of the susceptibility, asimulated complex susceptibility curve is shown in Fig. 3.2. While the realpart of susceptibility χ′
yy represents the dispersion and is antisymmetric withrespect to ωres [30], the imaginary part of the susceptibility χ′′
yy represents theprecessional amplitude [30] and has a Lorentzian shape with its maximum atω2 = ϖH (ϖH +ϖeff).So,
ω2res = ϖH (ϖH +ϖeff) (3.26)
is called the resonance condition and ωres is referred to as the resonant angularfrequency. This is a special case (for oblate objects such as thin films whereNx = Ny = 0 and Nz = 1 ) of the famous Kittel formula, which reads
ω2res = [ϖH + (Nz −Nx)ϖeff] [ϖH + (Ny −Nx)ϖM] . (3.27)
48
Inserting Eq. (3.18) and (3.19) into Eq. (3.26) gives
ω2res = ϖH (ϖH +ϖeff)
= γµ0 (Hext +Hu) [γµ0 (Hext +Hu) + γµ0Meff]
= γ2µ02 (Hext +Hu) [(Hext +Hu) +Meff] .
(3.28)
The resonance frequency reads
f =γµ0
2π
√(Hext +Hu) [(Hext +Hu) +Meff]. (3.29)
The angular frequency linewidth (∆ω) of the susceptibility is defined as the fullwidth at half maximum (FWHM) of the imaginary part (χ′′
yy) and is directlyrelated to the damping constant α by
α =∆ω
2ϖH +ϖeff
=∆ω
2γµ0 (Hext +Hu) + γµ0Meff
=2π∆f
γµ0 (2Hext + 2Hu +Meff)(3.30)
(or in more general form α = ∆ω/[2ϖH + (Nz +Ny − 2Nx)ϖeff] for nonoblate shaped samples).Thus we can see from Eq. (3.30) that the frequency linewidth is proportional
to the external field,
∆f =γµ0
2πα [2Hext + 2Hu +Meff] . (3.31)
The damping constant α can be obtained by fitting a straight line to ∆f vs.Hext.In cgs units, the expressions for the resonance frequency and linewidth are
f =γ
2π
√(Hext +Hu) [(Hext +Hu) + 4πMeff] (3.32)
and∆f =
γ
2πα (2Hext + 2Hu + 4πMeff) , (3.33)
respectively. Figs. 3.3 and 3.4 show examples of the field dependence of theresonance frequency and frequency linewidth, respectively.
3.2 Resonance frequency and linewidth in realisticmeasurements
In the previous section, we discussed only uniform magnetization motion, i.e.all magnetic moments are aligned perfectly in parallel and they are precessing
49
0 2 4 6 8 10 12 14 160
10
20
30
40
50
60
70
External field (kOe)
Res
onan
ce fr
eque
ncy
(GH
z)
4πMeff
=22 kG, Hu=0 kOe
4πMeff
=9.5 kG, Hu=0 kOe
4πMeff
=22 kG, Hu=0.5 kOe
4πMeff
=9.5 kG, Hu=0.5 kOe
Figure 3.3. Simulated FMR resonance frequency as a function of external field forfour example samples with different effective magnetizations and uniaxial anisotropyfield values.
0 2 4 6 8 10 12 14 160
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
External field (kOe)
Fre
quen
cy li
new
idth
(G
Hz)
4πM
eff=22 kG, H
u=0 kOe, α=0.005
4πMeff
=9.5 kG, Hu=0 kOe, α=0.005
4πMeff
=9.5 kG, Hu=0.5 kOe, α=0.01
4πMeff
=9.5 kG, Hu=0.5 kOe, α=0.01
4πMeff
=22 kG, Hu=0 kOe, α=0.01
4πMeff
=9.5 kG, Hu=0 kOe, α=0.01
Figure 3.4. Simulated FMR frequency linewidth as a function of external field for fourexample samples with different effective magnetization and uniaxial anisotropy fieldvalues.
50
coherently and with the same phase. This uniform precessional motion can bedepicted using a wave of infinite wavelength. In experimental measurements,one in most cases observes deviations from the prediction of Eq. (3.31). Ina realistic system, frequency shift and linewidth broadening will be induceddue to the fact that the precession is not ideally uniform and coherent, which,in turn, is caused by factors such as sample defects and inhomogeneity of themicrowave magnetic field. Two major contributions for linewidth broadeningare briefly discussed below.
3.2.1 Inhomogeneities of excitation field: Nonuniform spin waveexcitation
For amagnetic thin film sample whosewidth exceeds that of the CPW stripline,the excitation field generated by the microwave current is not uniform acrossthe sample. The non-uniformity of the microwave field causes a nonuniformprecessional motion of magnetic moments in the sample located a bit awayfrom the stripline, which is referred to as a spin wave.Spin waves in a magnetically ordered system, in classical approach, are the
analog of lattice waves in solid systems; and just as a quantized lattice waveis called a phonon, a quantized spin wave is called a magnon. Spin wavescontribute to the modification of the magnetic microwave response. Counil et.al. [31] calculated the frequency shift and linewidth broadening due to spinwave excitation. The measured resonance frequency is shifted by an amountof δω, which reads,
δω =ϖ2
eff8
(kmaxωres
), (3.34)
where ϖeff is given by Eqs. (3.6) and (3.19), ωres is the intrinsic resonancefrequency (resonance frequency without spin waves) and kmax is the maximumspin wave vector that can be estimated by kmax = π/wsample, where wsample isthe width of the sample.The measured frequency linewidth, with the contribution from spin waves
taken into account, is given by
∆fmea = ∆f
√1 +
(fs (kmax, θ)− fres
∆f
)2
, (3.35)
where ∆f is the intrinsic linewidth (resonance linewidth without spin wavecontribution), θ is the angle between the bias external field Hext and the RF fieldh, fres is the intrinsic resonance frequency and fs (kmax, θ) is the maximumspin wave frequency calculated using
f2s (kmax, θ) = f2
res −γ2µ2
0Meff
8π2
[Hext − (Hext +Meff)sin2θ
]kmaxd. (3.36)
51
When the microwave magnetic field is perpendicular to the external DCmagnetic field (θ = 90o), and in the limit of small dispersion in wave vec-tor, Eq. (3.35) can be simplified to
∆fmea = ∆f + C
(kmaxd
fres
)2
, (3.37)
where C is a constant whose value depends on ϖeff and ∆f . This equationexplicitly shows that the linewidth broadening caused by spin wave excita-tion scales down proportionally with fres. This means that the data acquiredat lower frequencies are not that reliable. The frequency shift and linewidthbroadening caused by this effect are small and negligible in most VNA-FMRmeasurements at high frequencies.If the microwave magnetic field is not perfectly perpendicular to the exter-
nal field, the measured frequency linewidth will, according to Eqs. (3.35) and(3.36), become more broadened. In this case, the microwave field h can bedecomposed into two components; h∥, that is parallel to the external field andh⊥, that is perpendicular to the external field. h∥ is directly added to the ex-ternal field but since it is much smaller than the external field no significantdifference would be noticed. h⊥ will act as the effective excitation field. Witha smaller value compared with the perfect perpendicular geometry, the am-plitude of the resonance signal will decrease. When the microwave field isparallel with the external field, the resonance phenomenon will disappear. Insummary, a continuous decrease of amplitude and broadening of the resonancesignal, with the resonance condition unmodified, can be expected if the anglebetween h and the external field Hext decreases from 90o to 0o.
3.2.2 Inhomogeneities of effective field: Dispersion of magneticanisotropy field
If the magnetic sample is inhomogeneous, the local effective fields vary dueto the local variations of the energy contributions. This will modify both theresonance condition and linewidth by causing a process named two magnonscattering. Two magnon scattering is induced by one, or a combination of thefollowing mechanisms [32]:
• Local variations of exchange interaction caused by eg. spatial composi-tion variations, large grain size, etc.;
• Local variations of magnetic anisotropy fields caused by eg. oxidationinduced exchange bias (also called unidirectional anisotropy), strain in-duced anisotropy, etc.
Besides two magnon scattering, nonuniformity of the external DC magneticfield alters the microwave response of the magnetic sample as well.
52
3.3 Field-swept FMRIf the FMR setup makes use of the field-swept mode, the resonance field is stillgiven by Eq. (3.26) or Eq. (3.29),
f =γµ0
2π
√(Hres +Hu) [(Hres +Hu) +Meff]. (3.38)
The field linewidth caused by the Gilbert damping is directly proportional tothe damping constant,
∆H =2ωresα
γµ0=
4πfα
γµ0. (3.39)
However, similar to the discussion in Section 3.2, the field linewidth is alsoinfluenced by extrinsic contributions such as spin waves and two magnon scat-tering (sample inhomogeneities). Among these factors, sample inhomogeneityis usually the most substantial contributor [33]. The extrinsic contribution tofield linewidth is generally observed and modeled [32] by a frequency inde-pendent term ∆H0, which is very sensitive to the microstructure (influencedby thermal history) of the material. This∆H0 is directly added to the intrinsiclinewidth∆H yielding
∆Hmea = ∆H0 +4πfα
γµ0. (3.40)
Material g (dimensionless) γ′ (GHz/kOe) Sourcefree electron 2.002 2.80 Ref.[34]bulk Fe 2.10 2.94 Ref.[35]bulk Ni 2.21 3.09 Ref.[36]bulk Co 2.18 3.05 Ref.[36]film Fe 2.09 2.93 Ref.[35]film Ni 2.208 3.09 Ref.[37]film Co 2.146 3.004 Ref.[35]film FeNi 2.02-2.20 2.83-3.08 Ref.[38, 39]film FeCo 2.08-2.20 2.91-3.08 Ref.[35, 40]
Table 3.1. g-factor and gyromagnetic ratio of common ferromagnetic materials.
In cgs units, Eqs. (3.38) and (3.40) are then given by
f =γ
2π
√(Hres +Hu) [(Hres +Hu) + 4πMeff] (3.41)
and∆Hmea = ∆H0 +
4πfα
γ, (3.42)
53
where ∆Hmea and ∆H0 are measured in kOe, f is measured in GHz and γ isthe gyromagnetic ratio with a value in the vicinity of 17.6 GHz/kOe.For convenience, some authors prefer to write Eq. (3.42) in the following
form,∆Hmea = ∆H0 +
2α
γ′f, (3.43)
with γ′ = γ/2π ≈ 2.9 GHz/kOe. But the value of γ′ varies slightly withmaterial and Table 3.1 lists the values of γ′ for some common ferromagneticmaterials.Eq. (3.42) enables one to determine the value of the Gilbert damping con-
stant by fitting the∆Hmea vs. f data to a straight line. ∆H0 is mainly attributedto inhomogeneities and is referred to as the extrinsic contribution to the FMRlinewidth.
54
4. Smit-Suhl approach to FMR resonancecondition
Though a straightforward derivation, the Kittel approach to the resonance con-dition (Eq. (3.29)) is based on specific geometries and therefore the applicationof the formula is restricted in many ways. For instance, the external field mustbe applied along the in-plane easy axis direction. After Kittel's paper, manyscientists have gradually increased the complexity of the explicit solutions byweakening the restrictions for their validity. In 1955, Smit, Suhl and Beljers[41, 27] treated the general case for any ferromagnet and obtained a simpleform for the resonance condition in terms of second derivatives of the totalfree energy density εtot with respect to the polar angle θM and the azimuthalangle ϕM of the equilibrium magnetization M .
4.1 The general resonance condition and intrinsiclinewidth
When an external field Hext is applied, the magnetization vector M will startto approach the external field until it reaches equilibrium, where the system'stotal magnetic energy is at minimum. This equilibrium condition is thereforedetermined by
∂εtot∂θM
= 0 and∂εtot∂ϕM
= 0. (4.1)
A microwave signal leads to the generation of a small magnetic field h thatinteracts with a ferromagnet. As long as h has a component normal to M ,it will disturb the equilibrium and cause the magnetization vector to undergoprecessional motion. At the same time, microwave energy is absorbed by theferromagnet and the absorption exhibits a maximum (resonance occurs) whenthe microwave frequency matches the natural frequency of the ferromagneticsystem. The derivation of this resonance condition can also start from the LLequation. In principle, LLG equation shall be used here. But for simplicity weuse the LL equation, omitting the damping term, as damping will not influencethe resonance condition.In cgs units, the LL equation (Eq. (2.18)) is given by
dM
dt= −γM × Heff. (4.2)
55
x
y
z
M
M
M
r
Figure 4.1. The polar coordinate system used when deriving the resonance conditions.
Choosing a polar coordinate system as shown in Fig. 4.1, Eq. (4.2) can thenbe written as
dM
dt= −γ (Msr)×
(Hrr +Hθθ +Hϕϕ
), (4.3)
where Ms is the magnitude of the magnetization vector M . In matrix form,Eq. (4.2) is written as
dMsdt
MsdθMdt
Ms sin θM dϕdt
= −γ
∣∣∣∣∣∣r θ ϕMs 0 0Hr Hθ Hϕ
∣∣∣∣∣∣ . (4.4)
The total free energy density is
εtot = −M · Heff
= −γ (Msr) ·(Hrr +Hθθ +Hϕϕ
).
(4.5)
If the magnetization rotates by a small angle ∆θ, the free energy densitybecomes
εtot′ = −γ
(Msr
′) · (Hrr +Hθθ +Hϕϕ)
= −γ[Ms
(r +∆θMθ
)]·(Hrr +Hθθ +Hϕϕ
). (4.6)
The change of free energy density is then
∆εtot = εtot′ − εtot = −MsHθ∆θM. (4.7)
56
Thus, Hθ can be expressed using εtot as
Hθ = − 1
Ms
∂εtot∂θM
. (4.8)
Following a similar analysis yields
Hϕ = − 1
Ms sin θM∂εtot∂ϕM
. (4.9)
By substituting Eqs. (4.8) and (4.9) into Eq. (4.4), one obtains dMsdt
MsdθMdt
Ms sin θM dϕdt
= −γ
∣∣∣∣∣∣r θ ϕMs 0 0
Hr − 1Ms
∂εtot∂θM
− 1Ms sin θM
∂εtot∂ϕM
∣∣∣∣∣∣ . (4.10)
Eq. (4.10) can be also written in the form of a system of equations as
dMs
dt= 0
− Ms
γsin θM
dθMdt
=∂εtot∂ϕM
Ms
γsin θM
dϕ
dt=
∂εtot∂θM
.
(4.11)
The first step to solve the above system of equations [42] for θM and ϕMis to determine the equilibrium angle values of θM,equi and ϕM,equi using Eq.(4.1). As the deviation of θM and ϕM from their equilibrium positions θM,equiand ϕM,equi are small under the influence of the microwave magnetic field, wecan write {
θM = θM,equi + δθMϕM = ϕM,equi + δϕM.
(4.12)
Also, we can expand ∂εtot∂θM
and ∂εtot∂ϕM
into Taylor series and keep only the linearterms to get
∂εtot∂θM
=∂2εtot∂θ2M
δθM +∂2εtot
∂θM∂ϕMδϕM
∂εtot∂ϕM
=∂2εtot
∂ϕM∂θMδθM +
∂2εtot∂ϕ2
MδϕM. (4.13)
One should note that the above expansion is performed at θM,equi and ϕM,equi.Substituting Eqs. (4.12) and (4.13) into Eq. (4.11) yields
− Ms
γsin θM
dδθMdt
=∂2εtot
∂ϕM∂θMδθM +
∂2εtot∂ϕ2
MδϕM
Ms
γsin θM
dδϕMdt
=∂2εtot∂θ2M
δθM +∂2εtot
∂θM∂ϕMδϕM. (4.14)
57
The above system of homogeneous equations will have a periodic solutionδθM, δϕM ∼ eiωt, with ω being the frequency of the microwave magnetic field,if the determinant of the characteristic system of equations is equal to zero.That is
∂2εtot∂ϕM∂θM
− ∂2εtot∂θ2M
∂2εtot∂ϕ2
M+
ω2Ms2
γ2sin2θM = 0. (4.15)
This directly gives the characteristic, or the resonance, frequency of the oscil-lation as(ωres
γ
)2
=1
Ms2sin2θM
[∂2εtot∂θ2M
∂2εtot∂ϕ2
M−(
∂2εtot∂θM∂ϕM
)2]∣∣∣∣∣
θM=θM,equi, ϕM=ϕM,equi
,
(4.16)where θM,equi and ϕM,equi are calculated using Eq. (4.1). First derived by Smit,Suhl and Beljers, Eq. (4.16) is commonly known as the Suhl-Smit or Suhl-Beljers-Smit equation.The frequency linewidth was derived by Vonsovskii to be [42]
∆ω =αγ
Ms
[∂2εtot∂θ2M
+1
sin2θM∂2εtot∂ϕ2
M
]∣∣∣∣θM=θM,equi, ϕM=ϕM,equi
. (4.17)
The field linewidth is then given by [27]
∆H =α
|∂ωres/∂Hext|γ
Ms
[∂2εtot∂θ2M
+1
sin2θM∂2εtot∂ϕ2
M
]∣∣∣∣θM=θM,equi, ϕM=ϕM,equi
.
(4.18)Thus, for a ferromagnetic film with any experimental geometry, the Suhl-
Smit solution allows one to find out the resonance condition and linewidth bytaking the following procedures:(a) For a given film geometry and external field Hext (Hext, θH , ϕH ) the free
energy density εtot should be calculated.(b) The equilibrium orientation of the magnetization vector M (θM = θM,equi,
ϕM = ϕM,equi) can be determined from the equilibrium condition Eq (4.1).(c) If the microwave excitation field has a component perpendicular to the
magnetisation vector, precessional motion of the magnetization can be ex-cited. Then the resonance condition for the precessional motion and thefield linewidth can be obtained using the Suhl-Smit formula Eq. (4.16)(differentiate εtot with respect to the angles and then set their values toθM = θM,equi and ϕM = ϕM,equi) and Eqs. (4.17) and (4.18), respectively.
4.2 Energy density of a single layer thin filmNow let us consider a ferromagnetic thin film placed in an external field, wherethe angles are defined as in Fig. 4.2. The total energy density includes the Zee-man energy εzee, demagnetizing energy εdem, effective perpendicular uniaxial
58
Figure 4.2. The coordinate system of a single layer film with a mixture of uniaxialanisotropy and cubic anisotropy. The x-y-z axes are chosen to be along the cubic easyaxes.
anisotropy energy εp, uniaxial magnetocrystalline anisotropy energy εu andcubic magnetocrystalline anisotropy energy εc. These energy terms are givenby
εzee = −Hext · Ms= −HextMs [sin θH sin θM cos(ϕH − ϕM) + cos θH cos θM] ,
(4.19)
εdem = 2πM2s cos
2θM, (4.20)
εp = Kpsin2θM =1
2HpMssin2θM, (4.21)
where Hp = 2Kp/Ms is defined as the perpendicular anisotropy field, and
εc = Kc1
(cos2αcos2β + cos2βcos2γ + cos2γcos2α
)= Kc1
[(sin θM cosϕM sin θM sinϕM)2 + (sin θM sinϕM cos θM)2
+(cos θM sin θM cosϕM)2]
=1
4Kc1(sin4θMsin22ϕM + sin22θM)
=1
8HcMs(sin4θMsin22ϕM + sin22θM), (4.22)
where Hc = 2Kc1/Ms is defined as the cubic uniaxial anisotropy field, and
εu = Kusin2α′
= Ku
[1− sin2θMcos2
(ϕM − ϕ0
′)]=
1
2HuMs
[1− sin2θMcos2
(ϕM − ϕ0
′)] , (4.23)
59
where Hu = 2Ku/Ms is the uniaxial anisotropy field and ϕ0′ is the angle
between in-plane uniaxial easy axis and cubic easy axis (x axis).The total energy density reads (angular invariant terms are omitted),
εtot = εzee + εdem + εp + εu + εc
= −HextMs [sin θH sin θM cos(ϕH − ϕM) + cos θH cos θM]
+1
24πMs
2cos2θM − 1
2HpMssin2θM
−1
2HuMssin2θMcos2
(ϕM − ϕ0
′)+1
8HcMs(sin4θMsin22ϕM + sin22θM)
= −HextMs [sin θH sin θM cos(ϕH − ϕM) + cos θH cos θM]
+1
24πMeffMscos2θM
−1
2HuMssin2θMcos2
(ϕM − ϕ0
′)+1
8HcMs(sin4θMsin22ϕM + sin22θM), (4.24)
where the effective perpendicular uniaxial anisotropy fieldHp and the satura-tion magnetization 4πMs are incorporated into a single term effective magne-tization 4πMeff = 4πMs −Hp.The equilibrium condition of such a system becomes,
∂εtot∂θM
= −HextMs [sin θH cos θM cos(ϕH − ϕM)− cos θH sin θM]
− 1
24πMeffMs sin 2θM
− 1
2HuMs sin 2θMcos2
(ϕM − ϕ0
′)+
1
8HcMs(2sin2θM sin 2θMsin22ϕM + 2 sin 4θM)
= 0
∂εtot∂ϕM
= −HextMs sin θH sin θM sin(ϕH − ϕM)
+1
2HuMssin2θM sin 2
(ϕM − ϕ0
′)+
1
4HcMssin4θM sin 4ϕM
= 0. (4.25)
60
The second derivatives are given by
∂2εtot∂θ2M
= HextMs [sin θH sin θM cos(ϕH − ϕM) + cos θH cos θM]
− 4πMeffMs cos 2θM−HuMs cos 2θMcos2
(ϕM − ϕ0
′)+
1
2HcMs
[2 cos 4θM + sin2θM (1 + 2 cos 2θM) sin22ϕM
]∂2εtot∂ϕ2
M= HextMs sin θH sin θM cos(ϕH − ϕM)
+HuMssin2θM cos 2(ϕM − ϕ0
′)+HcMssin4θM cos 4ϕM
∂2εtot∂θM∂ϕM
= −HextMs sin θH cos θM sin(ϕH − ϕM)
+1
2HuMs sin 2θM sin 2
(ϕM − ϕ0
′)+
1
2HcMssin2θM sin 2θM sin 4ϕM. (4.26)
In principle, substituting Eq. (4.26) into Eq. (4.16) and then setting the val-ues of θM and ϕM as the equilibrium angles obtained from Eq. (4.25) enablesone to obtain the resonance condition for such a thin film with external fieldapplied along an arbitrary direction. But in reality an analytical expression forthe resonance condition is almost impossible to be obtained due to the com-plexity of the equations above. In practice, FMR measurements are carriedout with special geometries, which significantly simplify the equations. Thefollowing two sections discuss two cases with special geometries.
4.3 FMR with in-plane applied magnetic field4.3.1 Film with both uniaxial and cubic anisotropyConsider the case where the external field is applied in the film plane. In thiscase, the polar angle of the external field is fixed at θH = π/2. Even thoughthere is perpendicular anisotropy that tends to align the magnetization M out ofplane, for most cases this effect is much smaller than the large demagnetizingfield that tends to align M in-plane ( Hp ≪ 4πMs ). Therefore θM = π/2.Under these conditions, the equilibrium values of the angles, according to the
61
equilibrium condition Eq. (4.25), can be determined asθM =
π
2
−HextMs sin(ϕH − ϕM) +1
2Hu sin 2(ϕM − ϕ′
0)
+1
4Hc sin 4ϕM = 0. (4.27)
The second derivatives of the total energy density, according to Eq. (4.26), aregiven by
∂2εtot∂θ2M
= HextMs cos(ϕH − ϕM) + 4πMeffMs
+HuMscos2(ϕM − ϕ0
′)+HcMscos22ϕM∂2εtot∂ϕ2
M= HextMs cos(ϕH − ϕM)
+HuMs cos 2(ϕM − ϕ0
′)+HcMs cos 4ϕM∂2εtot
∂θM∂ϕM= 0. (4.28)
Substituting Eq. (4.28) into Eq. (4.16) yields the resonance condition,(ωres
γ
)2
= [Hext cos(ϕH − ϕM) + 4πMeff
+Hucos2(ϕM − ϕ0′) +
1
2Hc (1 + cos 4ϕM)
]×[Hext cos(ϕH − ϕM) +Hu cos 2(ϕM − ϕ0
′)
+Hc cos 4ϕM] , (4.29)
where the value of ϕM is calculated using Eq. (4.27). The effective magne-tization 4πMeff includes the contribution from the perpendicular anisotropy,4πMeff = 4πMs −Hp.Here, one can see that the resonance field (or frequency) is dependent on both
the uniaxial and cubic anisotropy fields. By fitting the experimental resonancefield vs. in-plane angle data to the resonance condition, one is able to determinethe uniaxial and cubic anisotropy fields precisely.
4.3.2 Film with only uniaxial anisotropyIn this case,Hc = 0 and we can set the uniaxial easy axis as the x-axis (ϕ′
0 =0). The equilibrium condition is then given byθM =
π
2Hu sin 2ϕM = 2Hext sin(ϕH − ϕM). (4.30)
62
Using θH = π/2 and θM = π/2, Eq. (4.26) can be simplified to
∂2εtot
∂θM2 = HextMs cos(ϕH − ϕM) + 4πMeffMs +HuMscos2ϕM
∂2εtot
∂ϕM2 = HextMs cos(ϕH − ϕM) +HuMs cos 2ϕM
∂2εtot∂θMϕM
= 0. (4.31)
Inserting Eq. (4.31) into the Suhl-Smit equation yields the resonance conditionfor in-plane geometry,(
ωres
γ
)2
= [Hext cos(ϕH − ϕM) +Hu cos 2ϕM]
×[Hext cos(ϕH − ϕM) + 4πMeff +Hucos2ϕM
], (4.32)
where ϕM is the equilibrium direction of the magnetization, which is given byEq. (4.30). Eq. (4.32) can also be directly obtained by setting Hc = 0 andϕ′
0 = 0 in Eq. (4.29).Combining Eq. (4.30) and (4.32) allows one to decide the resonance condi-
tion for any in-plane configuration. Appropriate information about the sample(eg. effective magnetization and anisotropy field) can be extracted by perform-ing field or frequency swept measurements, or by performing a ϕH scan andfitting the data to Eqs. (4.30) and (4.32).For films with comparably small anisotropy field (Hu ≪ 4πMeff), the reso-
nance condition for the in-plane geometry becomes much simpler,(ωres
γ
)2
= Hext × (Hext + 4πMeff) . (4.33)
The frequency linewidth is
∆ω = αγ[2Hext cos(ϕH − ϕM) + 4πMeff +Hu
(3cos2ϕM − 1
)]. (4.34)
For the case that the external field is along the easy axis (ϕH = 0 and thusϕM = 0), then one has
∆ω = αγ [2Hext + 4πMeff + 2Hu] (4.35)
or∆f = αγ′ [2Hext + 4πMeff + 2Hu] , (4.36)
which is identical to Eq. (5.5) derived through the Kittel approach.
63
From Eq. (4.32),
∂ωres
∂Hext=
γ2
2ωres
[2Hextcos2(ϕH − ϕM) +Hu cos 2ϕM cos(ϕH − ϕM)
+(4πMeff +Hucos2ϕM
)cos(ϕH − ϕM)
]=
γ2
2ωrescos(ϕH − ϕM) [2Hext cos(ϕH − ϕM) + 4πMeff
+Hu(3cos2ϕM − 1
)].(4.37)
Substituting Eq. (4.37) and (4.31) into Eq. (4.18) yields the field linewidth
∆H =2αωres
γ
1
cos(ϕH − ϕM), (4.38)
where ϕM is the equilibrium direction of the magnetisation.In summary, with the external field applied in-plane of a ferromagnetic film
with comparably small anisotropy field (Hu ≪ 4πMeff), the equilibrium di-rection ϕM can be obtained from Eq. (4.30). Then this value can be substitutedinto Eq. (4.33) and (4.38) to get the resonance condition and linewidth for thisgeometry.Fig. 4.3 shows an example of the dependence of resonance field and linewidth
on the orientation of the in-plane external field. It can be seen that since themisalignment angle (defined as the angle between the external field and themagnetization vector at equilibrium) is rather small, the intrinsic linewidth'sdependence on the in-plane azimuth angle is almost not appreciable. How-ever, for the resonance field, the variation is quite significant. For many cases,the external field is much smaller than the saturation magnetization (Hext ≪4πMeff), then it is not difficult to note from Eq. (4.32) thatHu can be estimatedas roughly a half of the difference between resonance fields applying the fieldalong the hard and easy axis.When the external field is applied along the easy axis of the film (θH = π/2,
ϕH = 0), then θM = π/2 and ϕM = 0, following Eq. (4.30). In this case, thein-plane resonance condition (Eq. (4.32)) reduces to(
ωres
γ
)2
= (Hext +Hu) (Hext +Hu + 4πMeff) , (4.39)
which is essentially identical to Eq. (3.29). The in-plane intrinsic linewidthexpression, Eq. (4.33), reduces to
∆H =2αωres
γ=
4πα
γf. (4.40)
It can be seen that both the expression for the resonance frequency and intrinsiclinewidth have the simplest form when the external field is applied along the
64
Figure 4.3. Simulated resonance field and intrinsic linewidth as a function of azimuthangle when the external field is rotated in-plane. The parameters used here come froma FeCo film grown on a SiO2 substrate. 4πMeff = 20.5 kG,Hu = 0.08 kOe, γ′ = 3.08GHz/kOe. The excitation microwave frequency is 24 GHz.
easy axis if the film has in-plane uniaxial anisotropy.
When the external field is applied along the hard axis (θH = π/2, ϕH = π/2),then θM = π/2 and ϕM = π/2, following Eq. (4.30). The resonance conditionEq. (4.32) then reduces to
(ωres
γ
)2
= (Hext −Hu) (Hext + 4πMeff) . (4.41)
This is different from the resonance condition for the case when the externalfield is applied along the easy axis, though the magnetization vector is alsoaligned with the external field. The intrinsic linewidth expression (Eq. (4.33))reduces to
∆H =2αωres
γ=
4πα
γf, (4.42)
which is identical to the case where the external field is applied along the easyaxis (Eq. (4.40)).
65
4.3.3 Film with only cubic anisotropyIn this case, Hu = 0 and ϕ′
0 = 0. The equilibrium condition is then given byθM =
π
2
−HextMs sin(ϕH − ϕM) +1
4Hc sin 4ϕM = 0. (4.43)
The second derivatives of the total energy density, according to Eq. (4.28), aregiven by
∂2εtot∂θ2M
= HextMs cos(ϕH − ϕM) + 4πMeffMs
+HcMscos22ϕM∂2εtot∂ϕ2
M= HextMs cos(ϕH − ϕM) +HcMs cos 4ϕM
∂2εtot∂θM∂ϕM
= 0. (4.44)
Substituting Eq. (4.44) into Eq. (4.16) yields the resonance condition,(ωres
γ
)2
=
[Hext cos(ϕH − ϕM) + 4πMeff +
1
2Hc (1 + cos 4ϕM)
]× [Hext cos(ϕH − ϕM) +Hc cos 4ϕM] , (4.45)
where the value of ϕM is calculated using Eq. (4.43). The effective magne-tization 4πMeff includes the contribution from the perpendicular anisotropy4πMeff = 4πMs −Hp.The frequency linewidth is
∆ω = αγ[2Hext cos(ϕH − ϕM) + 4πMeff +Hc
(3cos22ϕM − 1
)](4.46)
and the field linewidth is
∆H =2αωres
γ
1
cos(ϕH − ϕM). (4.47)
4.4 FMR with out-of-plane applied magnetic fieldIn this case, the external field is usually applied in the x-z plane (ϕH = 0).
4.4.1 Film with uniaxial anisotropyIn the case that Hc = 0, we can set the uniaxial easy axis as the x-axis (thusϕ′
0 = 0 ) and the equilibrium condition for this case becomes{2Hext sin (θM − θH)− (4πMeff +Hu) sin 2θM = 0ϕM = 0.
(4.48)
66
Substituting the equilibrium angle values obtained from Eq. (4.48) into the2nd derivatives of the total energy density yields
∂2εtot∂θ2M
= −HextMs cos(θH − θM)− (4πMeff +Hu)Ms cos 2θM
∂2εtot∂ϕ2
M= HextMs sin θH sin θM +HuMs sin θ2M
∂2εtot∂θM∂ϕM
= 0. (4.49)
The resonance frequency for the geometry where the external field is rotatingout-of-plane and always orthogonal to the hard axis for a film with in-planeuniaxial anisotropy then reads:(
ωres
γ
)2
= [Hext cos(θH − θM) − (4πMeff +Hu) cos 2θM]
×(Hext
sin θHsin θM
+Hu
). (4.50)
By using the equilibrium condition (Eq. (4.48)), it is not difficult to prove(see Appendix 1) that Eq. (4.50) can also be written as(
ωres
γ
)2
=[Hext cos(θH − θM)− (4πMeff +Hu) cos2 θM +Hu
]× [Hext cos(θH − θM)− (4πMeff +Hu) cos 2θM] . (4.51)
This form of the resonance condition (Eq. (4.51)) is actually more popular inliterature.Fig. 4.4 shows simulation results of how the resonance field depends on the
out-of-plane angle for different microwave frequencies.The frequency linewidth is
∆ω = αγMs
{Hext
[cos(θH − θM) +
sin θHsin θM
]−4πMeff cos 2θM +Hu (1− cos 2θM)
}. (4.52)
The field linewidth is
∆H =2ωα
γ
Hext [p+ q]− 4πMeff cos 2θM +Hu(1 + cos 2θM)2Hextpq − 4πMeffq cos 2θM +Hu (p− q cos 2θM)
, (4.53)
where p = cos(θH − θM) and q = sin θH/sin θM are introduced to make theequation fit into the page.
67
f (GHz)
f (GHz)
(a)
(b)
f (GHz)
f (GHz)
Figure 4.4. Simulated dependence of resonance field on out-of-plane angle for differ-ent microwave frequencies for an isotropic magnetic thin film with effective magneti-zation of (a) 4πMeff = 9.5 kG and (b) 4πMeff = 22 kG.
68
For the special geometry where the external field is normal to film plane(θH = 0, ϕH = 0), the equilibrium condition for the magnetization vector is(i) {
θM = 0ϕM = 0
(4.54)
when Hext ≥ 4πMeff +Hu, and(ii) cos θM =
Hext
4πMeff +Hu
ϕM = 0 (4.55)
when Hext < 4πMeff +Hu.For situation (i), Eq. (4.50) becomes(
ωres
γ
)2
= (Hext − 4πMeff −Hu) (Hext +Hu) . (4.56)
We can also see that the condition Hext ≥ 4πMeff +Hu must be fulfilled forthe above equation to be physically meaningful.For situation (ii), Eq. (4.50) becomes(
ωres
γ
)2
= (4πMeff +Hu)
[Hu −Hu
(Hext
4πMeff +Hu
)2]. (4.57)
Also in this case it is obvious that the condition Hext ≤ 4πMeff +Hu must befulfilled for the equation to be physically meaningful. Situation (ii) is morecommon for ferromagnetic films having higher saturation magnetisation (eg.FeCo).In summary, for an uniaxial film with external field applied normal to the
film plane, the resonance frequency depends on the magnitude of the externalfield with respect to the effective magnetization of the film as given by Eqs.(4.56) and (4.57).Fig. 4.5 shows a few examples of the resonance frequency vs. external
field curves when the external field is applied normal to the film plane. Ascan be seen, for samples without a large in-plane anisotropy, the resonancefor situation (ii) occurs only at rather low frequency (f < 5 GHz), which isusually unobservable in an FMR experiment. Also, it is worthwhile to pointout that for FMR scans performed in the perpendicular geometry, the extrinsiclinewidth contribution from two-magnon scattering is minimized [43].
4.4.2 Film with cubic anisotropyIn this case, the simplest configuration is that the external field rotates in thex-z plane (ϕH = 0). Due to the cubic anisotropy field, it is easy to see that
69
0 5 10 15 20 25 300
5
10
15
20
25
30
35
40
45
50
Hres
(kOe)
f res (
GH
z)
4πMeff
=22 kG, H
u=0.50 kOe.
4πMeff
=22 kG, H
u=0.05 kOe.
4πMeff
=9.5 kG,H
u=0.50 kOe.
4πMeff
=9.5 kG,H
u=0.05 kOe.
Figure 4.5. Simulated resonance frequency vs. external field curves with external fieldapplied normal to the film plane.
ϕM = 0 must be a solution to the equilibrium condition. The equilibriumcondition is, in this case, given by
0 = −HextMs (sin θH cos θM − cos θH sin θM)
− 1
24πMeffMs sin 2θM +
1
4HcMs sin 4θM
ϕM = 0. (4.58)
The second derivatives of the total energy density read,
∂2εtot∂θ2M
= HextMs cos(θH − θM)− 4πMeffMs cos 2θM
+HcMs cos 4θM∂2εtot∂ϕ2
M= HextMs sin θH sin θM +HcMssin4θM
∂2εtot∂θM∂ϕM
= 0. (4.59)
The resonance condition is then given by(ωres
γ
)2
= [Hext cos(θH − θM)− 4πMeff cos 2θM +Hc cos 4θM]
×(Hext
sin θHsin θM
+Hcsin2θM). (4.60)
70
The frequency linewidth reads
∆ω = αγMs
{Hext
[cos(θH − θM) +
sin θHsin θM
]− 4πMeff cos 2θM
+Hc(cos 4θM + sin2θM
)}(4.61)
and the field linewidth is
∆H =2ωα
γ
Hext [p+ q]− 4πMeff cos 2θM +Hc
(sin2θM + cos 4θM
)2Hextpq − 4πMeffq cos 2θM +Hc
[psin2θM + q cos 4θM
] ,(4.62)
where p = cos(θH − θM) and q = sin θH/sin θM are introduced to make theequation fit into the page.
71
5. Conversion between the field and frequencylinewidth of FMR spectra
5.1 IntroductionFerromagnetic resonance (FMR) is a popular and powerful experimental tech-nique for characterizing properties of ferromagnetic materials. Since its initialexperimental observation by Arkad'yevin [25] and Griffiiths [22] and formal-ism developed by Kittel [26] and Suhl [27], the technique has been widely usedto measure effective magnetization, anisotropy fields, gyromagnetic ratio (orLande g-factor), magnetic coupling, sample inhomegeniety and magnetic re-laxation [44, 45, 46, 47]. In an FMR measurement, either a field-swept scan,with magnetic field swept at a fixed microwave frequency, or a frequency-swept scan, with the frequency swept at a fixed magnetic field can be per-formed. In the former, one obtains the resonance field and field linewidth ofthe spectra while in the latter one obtains the resonance frequency and fre-quency linewidth.For an isotropic ferromagnetic film with in-plane magnetization, the FMR
resonance condition is given by the famous Kittel equation as
2πf
γ=√
H (H + 4πMeff), (5.1)
where γ is the gyromagnetic ratio, f is the resonance frequency, H is the ex-ternal magnetic field and 4πMeff is the effective magnetization including thecontributions from the saturation magnetization 4πMs and the perpendicularanisotropy field Hp (4πMeff = 4πMs − Hp). It is well established that bothfield and frequency linewidths are related to the Gilbert damping parameter,therefore a relation between the two linewidths should exist. It is widely be-lieved that the ratio between the frequency and field linewidths is simply thedifferentiation of the resonance frequency with respect to the resonance exter-nal field. In other words, frequency and field linewidths are related to eachother by
∆f =∂f (H)
∂H∆H
= γ
√1 +
(γMeff
f
)2
∆H (5.2)
72
and
∆H =∂H (f)
∂f∆f
=1
γ
√1 +
(γMefff
)2 ∆f. (5.3)
This type of conversion was firstly used by Patton [48] and thereafter by manyother researchers [49, 50, 51, 52, 53, 54, 55]. Though there are many articlesdedicated to comparing the results of different FMR measurements [56, 57,52], the validity of this linewidth conversion relation has not been carefullyverified.In this study, we prepared a set of FeCo thin films with different growth tem-
peratures and thicknesses and compare the FMR frequency and field linewidthsof these samples. We managed to obtain films with negligible, moderate andsignificant extrinsic linewidth contributions. We find that the widely usedconversion relation is only valid for samples exhibiting negligible extrinsiclinewidth contributions (frequency independent field linewidth ∆H0 ≈ 0).Furthermore, a close look at the literature reveals that previously reported re-sults support our findings.
5.2 ExperimentalFour different FeCo films were prepared under different experimental condi-tions by varying growth temperature, film thickness and choice of substrate.The growth conditions for the different samples are listed in Tab. 5.1.
Label Substrate Growth temperature ThicknessSample 1 Si/SiO2 400 oC 25 nmSample 2 Si/SiO2 400 oC 10 nmSample 3 Si 200 oC 100 nmSample 4 Si 450 oC 100 nm
Table 5.1. Growth parameters of the studied samples.
All FeCo filmswere deposited from a Fe49Co49V2 (purity 99.95%) target us-ing DC magnetron sputtering. The substrates were introduced into the growthchamber and out-gassed at about 700◦C for 1 h under ultrahigh vacuum con-ditions and then cooled down to the growth temperature.The samples were investigated by X-ray reflectivity (XRR) and diffraction
(XRD) measurements. All XRR scans were performed using CuKα=1.5404Å radiation on a Siemens D5000 diffractometer in Bragg-Brentano geometrywith a secondary monochromator. XRD scans with the small incidence angle
73
(grazing incidence) of 1.5◦ with respect to the sample surface were carriedout on a parallel-beam setup (CuKα radiation) equipped with a Soller slit with0.40◦ divergence.Room temperaturemagnetization versus fieldmeasurementswere performed
using a Quantum Design MPMS-XL SQUID magnetometer. FMR measure-ments were performed in a Bruker field-swept cavity system with a microwavefrequency of 9.8 GHz. This system is equipped with a goniometer, makingit possible to orient the sample with an angular resolution of 0.125o with re-spect to the external magnetic field. FMR spectra were also recorded using abroadband vector network analyzer (VNA) based FMR systems. Frequencyscans (2-20 GHz) at a number of different external fields up to 1.2 kOe wererecorded. Also, field-swept FMR scans were performed at a few fixed fre-quencies ranging between 5 and 45 GHz using a different VNA-based setup.
5.3 ResultsThe XRD results show that the films are polycrystalline. The magnetizationversus field measurements show that all films have high saturation magnetiza-tion and small coercivity, typical for FeCo films. The in-plane angular depen-dent FMR results indicate that all films exhibit negligible in-plane anisotropy.Figs. 5.1(a) and (b) show frequency-swept and field-swept FMR spectra, re-spectively, of Sample 2 as a typical example. The resonance frequency versusfield curve can be well fitted to the Kittel equation (Eq. (6.1)) as shown inFig. 5.1(c). The fitting yields an effective magnetization value of 19.3 kG,which is smaller than the saturation magnetization of FeCo films [40]. For a10 nm thick film, this discrepancy can be explained by an interface inducedperpendicular anisotropy field [31].Both frequency and field linewidth values for all four samples are shown
in Fig. 5.2. Also shown in the figure is the field linewidth converted fromthe measured frequency linewidth using Eq. (5.3). Comparing the convertedfield linewidth and the measured field linewidth clearly demonstrates that theconverted linewidth is consistent with the measured linewidth when∆H0 ≈ 0(Sample 1) and is larger than the measured linewidth when∆H0 = 0 (samples2, 3 and 4). It therefore seems that the derivative based linewidth conversionrelation holds only for samples with negligible extrinsic linewidth contribu-tions (∆H0 ≈ 0). This is further supported by an obvious qualitative conclu-sion one may draw from Fig. 5.2: The larger the ratio between the extrinsiclinewidth and intrinsic damping, the bigger the mismatch between the mea-sured and converted linewidths.To get a quantitative impression, we define the ratio between the frequency
independent linewidth (intercept at f = 0) and slope of the field linewidth vs.
74
f (GHz)
Figure 5.1. Field-swept (a) and frequency-swept (b) FMR spectra measured for Sam-ple 2. (c) shows the measured resonance frequency as a function of resonance field,with symbols representing experimentally measured values and the solid curve a fit ofthe experimental results to the Kittel formula.
75
6 8 10 12 140.7
0.8
0.9
1.0
6 8 10 12 140.7
0.8
0.9
1.0
0 5 10 15 20 25 30 35 40 450.0
0.1
0.2
0.3
0.4
6 8 10 12 141.6
1.8
2.0
2.2
0 5 10 15 20 25 30 35 40 450.0
0.1
0.2
0.3
0.4
0 5 10 15 20 25 30 35 40 450.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 5 10 15 20 25 30 35 40 450.0
0.1
0.2
0.3
0.4
6 8 10 12 14
2.83.23.64.04.4
f (GHz)
f (G
Hz)
f (GHz)
f (G
Hz)
H (k
Oe)
f (GHz)
f (G
Hz)
f (GHz)
H (k
Oe)
f (GHz)
f (GHz)
H (k
Oe)
H (k
Oe)
(d)(c)
(a)
f (GHz)
(b)
f (G
Hz)
f (GHz)
Figure 5.2. Directly measured field linewidth (blue triangles) and converted fieldlinewidth values (red circles) as a function of resonance frequency for Sample 1 (a),Sample 2 (b), Sample 3 (c) and Sample 4 (d). The insets show frequency-swept FMRmeasured frequency linewidth values from which the converted field linewidths arecalculated.
frequency curve askw = ∆H0/(
4πα
γ), (5.4)
henceforth referred to as the linewidth contribution ratio. Apparently, kw willhave the dimension of frequency. From Fig. 5.2 one extracts kw for Samples1-4 to be 1, 19, 42 and 87 GHz. Comparing these values with the results shownin Fig. 5.2 enables one to conclude that a linewidth contribution ratio of lessthan about 10 GHz will result in a reasonable agreement between measuredand converted field linewidth values.
5.4 DiscussionsThe frequency linewidth is related to the intrinsic Gilbert damping α as
∆f =γ
2πα (2H + 4πMeff) . (5.5)
The relationship is derived directly from the Landau-Lifshitz-Gilbert equation[30] and is well established.
76
The field linewidth can be related to intrinsic Gilbert damping using
∆H =α
|∂ω/∂H|γ
Ms
[∂2εt∂θ2M
+1
sin2θM∂2εt∂ϕ2
M
]∣∣∣∣ θM=θM,equiϕM=ϕM,equi
, (5.6)
where εt is the total magnetic energy density of the film and ω = 2πf is theangular frequency of the microwave field at resonance. θM and ϕM are the po-lar and azimuthal angles of the magnetization, respectively, with θM,equi andϕM,equi corresponding to their values at equilibrium. In the case when the ex-ternal field is applied in-plane of an isotropic film, the linewidth is simply
∆H =4πf
γα. (5.7)
A comparison between Eqs. (5.5) and (5.7) indeed leads to a relation betweenfrequency and field linewidths as indicated by Eqs. (5.2) and (5.3). The ex-perimental results for Sample 1 supports this relation. In both Eqs. (5.5) and(5.7), the linewidth contribution includes only the intrinsic part originatingfrom Gilbert damping. Clearly the underlying physics of the conversion rela-tion is connected to the Gilbert damping, which is the origin of both frequencyand field linewidth.However, extrinsic linewidth contributions are observed in most FMR mea-
surements, both in frequency- and field-swept ones [58, 24, 31, 59]. Extrin-sic linewidth contributions may originate from magnetic anisotropy disper-sion [60], surface roughness [61], non-uniform spin wave excitation [31], two-magnon scattering [32] and eddy currents [24]. The observed field linewidthis often modeled as a frequency independent extrinsic contribution ∆H0 anda frequency dependent intrinsic contribution 4πf
γ α originating from Gilbertdamping [33].To the best of our knowledge, theoretical work generalizing the linewidth
conversion relation based on the resonance derivative conditions to circum-stances with frequency independent linewidth contributions (∆H = ∆H0 +4πfγ α) has not been presented to date. We argue that existing physical modelsdoes not support the derivative based conversion in the presence of a frequencyindependent linewidth and the conversion relation has been incorrectly appliedto cases with substantial extrinsic linewidth contributions. Our experimentalresults clearly indicate that the linewidth conversion is only valid for sampleswith negligible frequency independent linewidth broadening contribution.There are many reports (see e.g. Refs. [50, 51, 52, 53, 54, 55]) that have used
the conversion to perform FMR data analysis of samples both with and withoutextrinsic frequency independent linewidth contributions. In Fig. 5.3 we showtwo typical linewidth comparisons as examples. In Ref. [52], the frequencylinewidths are converted to field linewidth values. The converted linewidthsare in good agreement with the measured ones and they can be well fitted to astraight line. An important feature that can be noticed from Fig. 5.3 (a) is that
77
Figure 5.3. Two example figures from work published by others demonstrating thefrequency and field linewidth conversion. (a) Measured and converted FWHM fieldlinewidth of a FeNi film with∆H0 ≈ 0. Figure extracted from Ref.[52]. (b) Compar-ison of measured and converted frequency linewidths. Triangles and circles representthe measured peak-to-peak frequency linewidth for thin films of Fe and FeV, respec-tively. The dashed and solid curves correspond to frequency linewidths convertedfrom the measured field linewidths (shown in the inset) for the Fe and FeV films, re-spectively. Data extracted from Fig. 1 in Ref. [53].
78
the zero frequency intercept is very small, with the linewidth contribution ratiobeing only 0.4 GHz. Other published works that find consistence between theconverted and measured linewidths also have the feature that the frequency in-dependent linewidth is either zero or much smaller than the intrinsic linewidth[50, 55].Another explicit example is connected to the attempt of Scheck et al. [53] to
convert the field linewidth of Fe and FeV ultrathin films to frequency linewidthusing Eq. (5.2). It was found that the converted frequency linewidth is largerthan the measured frequency linewidth, as shown in Fig. 5.3(b). The authorsfound that they needed to reduce the extrinsic linewidth by 50 % to make theconverted and measured frequency linewidth values consistent and they at-tributed this discrepancy to a smaller sampling area in the frequency-sweptmeasurement (the measured response originates from the vicinity of the copla-nar waveguide signal line) than in the field-swept measurement (all of the sam-ple contributes to the measured response as a cavity resonator was used). Wethink this is highly unlikely as the sampling area in the coplanar waveguidebased FMR measurement has recently been demonstrated to be in the orderof tens or hundreds of micrometers [24, 62], a few orders of magnitude largerthan the dimension of typical grains. Instead, we believe that the disagreementis due to the invalidity of the conversion equation used since the kw values forthese samples are 28 (Fe) and 34 (FeV) GHz.
5.5 ConclusionIn conclusion, our experimental results from frequency- and field-swept FMRmeasurements on a set of FeCo thin films grown under different conditionsindicate that the widely used conversion relation based on the derivative of theresonance condition is only valid for cases obeying the condition that the fre-quency independent linewidth is negligible (∆H0 ≈ 0). Furthermore, a closeexamination of the literature reveals that previously reported results supportour findings, with successful conversions related to samples with negligibleextrinsic contributions and unsuccessful conversions related to samples withsignificant extrinsic contributions to the linewidth. A linewidth contributionratio kw that is defined as the ratio between the zero-frequency intercept andthe slope of the field linewidth versus frequency curve is introduced as an indi-cator of the validity of linewidth conversion. It is suggested that a value of kwsmaller than about 10 GHz would yield a reasonable conversion. The findingsare of great significance in providing guidance for FMR linewidth measure-ments and data analysis.
79
6. Dynamic properties of FeCo films studiedby ferromagnetic resonance
6.1 IntroductionThe ever increasing demand for high performance magnetic memory deviceshas led to extensive research on the high frequency properties of magnetic thinfilms, which are closely related to the performance of magnetic memory de-vices. It is well established that the switching speed of a ferromagnetic mem-ory device relates to the damping of the precessional motion of the magneticmoments [63, 64, 65]. This precessional motion is described by the Landau-Lifshitz-Gilbert (LLG) equation [66, 67]. Ferromagnetic resonance (FMR) isa powerful and convenient technique for characterizing the spin dynamics offerromagnetic films and heterostructures.The Ni0.8Fe0.2 alloy known as Permalloy has attracted much attention due
to excellent soft magnetic properties. However, further miniaturization andincrease of memory storage density as well as improved performance of elec-tromagnetic MEMS devices require magnetic materials with higher saturationmagnetization [68]. The magnetic properties of the bulk-like Fe49Co49V2 al-loy (known as Permendur) are well known. Being a soft magnetic materialwith high permeability, high saturation magnetization (4πMs = 24.5 kG) andhigh Curie temperature [69], Permendur is very attractive for many technolog-ical applications and has been studied extensively as a bulk alloy [70, 71]. Itis well known that the magnetic properties of Permendur in thin film form canbe drastically affected by the microstructure (texture, grain size and strain).Numerous studies have been carried out to investigate Fe1−xCox alloy thinfilms of different concentrations [72]. It has been found that Fe50Co50 filmsexhibit the highest saturation magnetization but that as-deposited films usu-ally show relatively large coercivity and are magnetically isotropic in-plane[73, 74]. Many attempts have been made to improve the magnetic propertiesof the films by varying film thickness [73], substrate bias [75], seed layers[76] or even by using ion irradiation [77, 78] and ion beam mixing of Co/Femultilayers [79] to modify the magnetic properties. However, there is still alack of systematic studies regarding the relation between magnetic and struc-tural properties. Also, in contrast to the static magnetic properties much lessattention has been paid to the dynamic magnetic properties.We performed high frequency (20-50 GHz) FMR measurements of FeCo
films grown on Si/SiO2 substrate. In recent years, thin films of FeCo haveshown high potential for applications in spintronic memory devices [80, 81].
80
While previous studies on thin films of the FeCo system suggest that the dy-namic properties are largely dependent on composition [82], seed layer [76],film thickness [73] and heat treatment [83], this study reports the influence ofgrowth temperature on the dynamic properties of the films. Also, of particu-lar interest is that a nearly linear relation between the FMR extrinsic linewidthand coercivity, to the best of our knowledge the first of its kind, is reported anddiscussed.
6.2 Film preparation, structural characterization andstatic magnetic properties
6.2.1 Film preparationThe FeCo films were grown onto Si/SiO2 substrates using dc magnetron sput-tering from a Fe49Co49V2 (purity 99.95%) target. High-quality SiO2 layerswere prepared by high temperature dry oxidation of Si(100) wafers. The Sisubstrates with 100 nm of SiO2 were introduced into the growth chamber andout-gassed at about 700 ◦C for one hour under ultra-high vacuum and thencooled down to the deposition temperature (20, 100, 200, 300, 400, 450, 500,550 and 600 ◦C). The background pressure was typically 2× 10−9 Torr. Dur-ing deposition, the argon (of 99.99% purity) flow was controlled for obtaininga partial pressure of 2 × 10−3 Torr resulting in a deposition rate of 0.42 Å/s.The deposition rate was determined from X-ray reflectivity measurements ona FeCoV calibration sample. The thickness of all samples was 100± 1 nm asdetermined from the deposition rate. In order to prevent the films from oxida-tion, the films were cooled down to a temperature of about 50 ◦C and a thin (4nm) capping layer of Pd was deposited on the films.
6.2.2 Structural characterizationFig. 6.1 shows grazing incidence X-ray diffraction (XRD) spectra of FeCofilms grown at 20 ◦C and 600 ◦C. The XRD scans show only reflections fromthe bcc (110), (200) and (211) planes indicating that the films grow as a purebcc alloy in the whole temperature range. The relatively high intensity of the(110) diffraction peak indicates that the films exhibit some degree of (110)texture. As can be seen in the insert of Fig. 6.1, the peak position of the (110)reflection shifts to lower angle with increasing growth temperature in the rangefrom RT to 400 ◦C, implying that the lattice constant increases. The calculatedlattice constant for the film deposited at RT was a = 0.2846(9) nm, a valuethat is smaller than that of the bulk Fe50Co50 alloy, ab = 0.2855(1) nm.The lattice constants for all films deduced from the (110) reflection and nor-
malized to the value of bulk FeCo are shown in Fig. 6.2. The lattice constant in-creases with increasing growth temperature and at about 400-500 ◦C approach
81
44 45 46
30 40 50 60 70 80 90 100
20oC 100oC 200oC
2 (deg)
300oC 400oC 500oC 600oC
I (a.
u.)
I (a.
u.)
(deg)
20 oC 600 oC
(200) (211)(220)
(110)
Figure 6.1. X-ray diffraction spectra of the FeCo films grown on Si/SiO2 substrates at20◦C and 600 ◦C. The inset shows the dependence of the (110) peak on film growthtemperature for all films. Vertical shifts (offsets) are applied to both the main figureand the inset.
0 100 200 300 400 500 6000.997
0.998
0.999
1.000
a (d
imen
sion
less
)
t (oC)
Figure 6.2. Lattice constant a (normalized to the value for the bulk Fe50Co50 alloy,ab = 0.2855 nm) vs. film growth temperature t of the Fe49Co49V2 films.
82
0 200 400 6002021222324
-200 -100 0 100 200-1.0
-0.5
0.0
0.5
1.0
0 200 400 60020
40
60
80
4s (
kG)
t (oC)
20oC 200oC 400oC 600oC
m (d
imen
sion
less
)
H (Oe)
Hc (
Oe)
t (oC)
Figure 6.3. Normalized hysteresis loops for some of the FeCo films measured withthe magnetic field H applied along the easy magnetization axis. The top left (rightbottom) inset shows the easy axis coercivity (saturation magnetization) as a functionof sample growth temperature.
the bulk value. Above 500 ◦C the lattice constant decreases with growth tem-perature. The dependence of the lattice constant on growth temperature maybe explained by an increase (a decrease) in the proportion of the disordered(ordered) bcc A2 (B2) phase in the FeCo films (to be further discussed below)[84, 85, 86]. The linewidth of the diffraction peaks decreases with increasingtemperature, indicating an increase of the grain size in the films. The grain sizeestimated using Scherrer equation isD = 15± 2 nm for the sample depositedat 20 ◦C, which gradually increases with temperature to D = 23 ± 2 nm forthe film grown at 600 ◦C.
6.2.3 Static magnetic propertiesThe static magnetic properties of the films were characterized by supercon-ducting quantum interference device (SQUID) magnetometry, with a selectionof magnetization curves shown in Fig. 6.3. The coercive fields and saturationmagnetizations of the films were determined from hysteresis measurements,with the results presented in the inserts of Fig. 6.3. Initially the saturationmagnetization increases with growth temperature and saturates at 4πMs ≈ 23kG. The change of the coercivity vs. growth temperature is more complex. Thefilms grown at 400 and 450 ◦C exhibit the best soft magnetic properties (small-est coercivity). The in-plane magnetic anisotropy of the films was studiedusing an X-band cavity based FMR system with an RF frequency of 9.8 GHz.This system is equipped with a goniometer, making it possible to rotate the
83
0 60 120 180 240 300 3600.50
0.51
0.52
0.53
0.54
0.55
measured fit
H (o)
Hre
s(kO
e)
Figure 6.4. Resonance field Hres as a function of in-plane azimuth angle of the ex-ternal applied field ϕH for the sample grown at 450 ◦C. The solid line is a fit of themeasured data to Eqs. (4.27) and (4.29), yielding an uniaxial anisotropy field of 9 Oe.The inset shows the two dimensional raw data of FMR spectra, with the horizontalaxis representing ϕH , vertical axis representing external magnetic field and the colorrepresenting amplitude of the 1st derivative of microwave absorption.
sample with an angular resolution of 0.125 degree with respect to the appliedmagnetic field. The resonance field as a function of in-plane angle was fitted tothe in-plane angular dependent resonance condition given by Eq (4.29). The re-sults indicate that all samples exhibit a weak uniaxial and/or cubic anisotropy,with the largest anisotropy field at roughly 20 Oe. No systematic dependenceof the anisotropy fields on sample growth temperature could be revealed. Fig.6.4 shows the resonance field as a function of the orientation of the externalfield for the sample grown at 450 ◦C as an example.
6.3 FMR study of dynamic propertiesWe also used a broadband vector network analyzer based FMR system to in-vestigate the dynamic magnetic properties of the films. Field scans at variousmicrowave frequencies ranging from 20 to 50 GHz were performed. Since theuniaxial and cubic anisotropy fields are much smaller than the external fieldH and saturation magnetization 4πMs, the resonance condition can be simplygiven as (
2πf
γ
)2
= H (H + 4πMs) , (6.1)
where f is the frequency and γ is the gyromagnetic ratio.
84
2 3 4 5 6 7 8 9 10
0 10 20 30 40 50
0.20
0.25
0.30
2 4 6 8 10
1000
2000
S21
der
ivat
ives
(a.u
.)
H (kOe)
25GHz 30GHz 35GHz 40GHz 45GHz 50GHz
data fit
(kO
e)
f (GHz)
datafit
f 2
(GH
z 2 )
H (kOe)
Figure 6.5. FMR spectra at different frequencies for the sample grown at 450 ◦C.Bottom left inset: Frequency squared vs. resonance field. The solid line corresponds toa fit of the experimental results to Eq. (6.1). Bottom right inset: The total peak-to-peaklinewidth as a function of frequency, with the solid circles representing experimentaldata and the solid line representing a fit to Eq. (3.42).
From the FMR measurements at different frequencies, the gyromagnetic ra-tio γ (and thus the Landé g-factor) and resonance linewidth were extractedfor each sample, with the spectra for the 450 ◦C film shown as an example inFig. 6.5. The experimentally measured peak to peak FMR linewidth,∆H , isgenerally believed to consist of a frequency independent extrinsic contributiondue to magnetic inhomogeneity [32, 33] and a frequency dependent intrinsiccontribution described by Gilbert relaxation (G),
∆H = ∆H0 +4π√3
G
γ2Msf. (6.2)
The Gilbert relaxation G, measured in Hz, is related to the dimensionlessGilbert damping parameter α by α = G/γMs.By fitting the measured total linewidth vs. frequency to Eq. (6.2) (see inset
of Fig. 6.5), we obtained the extrinsic linewidth and Gilbert relaxation rate forall samples. The extrinsic linewidth ∆H0 vs. easy axis coercive field for thedifferent films is shown in Fig. 6.6. The extrinsic linewidth varies between0.15 and 0.35 kOe for the different films, comparable to the values obtainedin other studies on similar films [87]. A substantial decrease in the extrinsiclinewidth occurs when the growth temperature is above 200 ◦C. This behaviorindicates improved sample homogeneity. However, when the growth temper-ature is extended to 500◦C and above, the linewidth increases with tempera-ture. It is interesting to note that the extrinsic linewidth and coercivity exhibita similar dependence on growth temperature. We therefore plot the extrinsic
85
0 200 400 6000.1
0.2
0.3
0.4
10 20 30 40 50 60 70
0.2
0.3
0.4
H0 (
kOe)
t (oC)
H0 (
kOe)
Hc (Oe)
Figure 6.6. Peak to peak extrinsic linewidth ∆H0 vs. easy axis coercive field Hc.The dashed line is a guide to the eye. The top left inset shows the dependence of theextrinsic linewidth on growth temperature.
linewidth as a function of coercivity in Fig. 6.6, revealing an approximatelylinear dependence. Yuan and Bertram [88] numerically studied the coercivitymechanism of Permalloy thin films and found a positive correlation betweensample inhomogeneity and coercivity. Our experimental results clearly sup-port the correlation found in simulations and furthermore give evidence for apositive correlation between sample inhomogeneity and extrinsic linewidth.The extracted Landé g-factor vs. film growth temperature is shown in Fig.
6.7. Note that the g-factors are slightly larger than 2 for all samples, reachinga minimum between 300-400 ◦C. The values are larger than that of bulk FeCo,but comparable to values obtained for similar magnetic films, such as FeCoand FeCoB films [82].The Gilbert relaxation rate G estimated from the the FMR linewidth is also
presented in Fig. 6.7. The value varies between 0.1 and 0.25 GHz (corre-sponding to α=0.003∼0.008) depending on growth temperature. In the rangefrom RT to 500 ◦C the damping decreases with increasing growth temperature,while for even higher growth temperatures the trend is opposite. The valuesare in agreement with those determined by other researchers from studies ofthe FeCo film system [89, 82]. Comparing with results derived for FeCo filmsdirectly grown on Si substrates, there is a reduction of the Gilbert relaxationrate values for films grown on Si/SiO2 substrates [58].It is well established that in non-magnetic metals the damping mechanism
is attributed to spin-orbit coupling [90, 91, 92], with the Gilbert relaxationproportional to (g − 2)2. For ferromagnetic metals, this dependence is stillunder debate due to a lack of systematic theoretical and experimental work.Although a reliable quantitative evaluation of the relationship between G and
86
0.997 0.998 0.999 1.000
0.1
0.2
0.3
0 100 200 300 400 500 600
2.05
2.10
2.15
2.20
2.25
2.30
t (oC)
g (d
imen
sion
less
)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
G (G
Hz)
G (G
Hz)
a (dimensionless)
Figure 6.7. Gilbert relaxation rate G (green squares) and Landé g-factor (red trian-gles) as a function of growth temperature t. The inset presents the Gilbert relaxationrate as a function of the normalized lattice constant, with the blue solid circles beingexperimental data and the dashed line being a guide to the eye.
g for the present samples is not possible, one can easily see from Fig. 6.7 thatthe two quantities are positively correlated. This finding is consistent with theresults presented by Oogane et al. [89] for CoFeB/SiO2 films and by Pelzl etal. [35] for FeCo films. It is also interesting to note the similar dependenceon growth temperature for the Gilbert relaxation and lattice constant (see insetof Fig. 6.7). The fact that Gilbert relaxation decreases with increasing lat-tice constant suggests a correlation with the order-disorder transformation (A2→ B2 structural transformation) in the Fe-Co binary system. Studies of thistransformation performed on Fe50Co50 fine particles [85], applying the anoma-lous X-ray dispersion effect, showed that the as-synthesized particles have adisordered structure and that the disordered structure could be transformed toordered phase by annealing at temperatures above 400 ◦C. Moreover, studieson bulk FeCo annealed at different temperatures and then quenched into icedbath showed that the degree of order (the amount of B2 phase) varied with an-nealing temperature [86], the degree of order disappeared at 720 ◦C and hada maximum at about 550 ◦C. The final piece of information needed to under-stand the present FeCo results is that the lattice constant for the ordered B2phase in comparison to the disordered A2 phase is larger [84]. A plausible ex-planation for the results presented here is that the degree of order (the amountof B2 phase) varies with growth temperature and exhibits a maximum at about450 ◦C. Considering the difference in sample preparation, one can not expectan exact correspondence between sputtered FeCo films and bulk FeCo. Thecorrelation in results is nevertheless convincing, which leads us to conclude
87
that the variation in coercivity, Gilbert relaxation rate, extrinsic linewidth andLandé g-factor are all connected to the degree of structural order in the films.
6.4 SummaryIn summary, magnetic properties of 100 nm thick FeCo alloy films grown onSi/SiO2 substrates by dc magnetron sputtering have been investigated by FMRtechnique and SQUID magnetometry. The films exhibit a weak in-plane uni-axial and/or cubic anisotropy with anisotropy fields less than 20 Oe. Whilethe influence of growth temperature on the saturation magnetization is weak,it greatly affects the coercivity and damping. The extrinsic FMR linewidth de-creases with growth temperature from 20 ◦C to 450 ◦C, but starts to increasewith further increase of growth temperature. An almost linear relationshipbetween coercivity and extrinsic linewidth is observed, which suggests thatmagnetic inhomogeneity is a major cause for both FMR extrinsic linewidthand coercivity. The intrinsic Gilbert relaxation rate of the prepared films ex-hibits values in the range 0.1∼0.25 GHz (α=0.003∼0.008), with the smallestdamping obtained for samples grown at 450 and 500 ◦C. The intrinsic Gilbertrelaxation decreases with increasing lattice constant, which is a sign of theorder-disorder structural transformation.
88
7. Effects of coupling between magnetic layersin a trilayer structure
7.1 IntroductionA trilayer system with two ferromagnetic (FM) layers separated by a non-magnetic (NM) spacer layer has various technological applications such asmagnetic recording devices, magnetic sensors, non-volatile magnetic randommemories and spin-torque oscillators [17, 93, 94, 95]. Understanding the in-fluence of material parameters such as magnetic anisotropy, saturation mag-netization and interlayer coupling on the overall magnetic response of the tri-layer system to an external field is of critical importance for engineering ofFM/NM/FM trilayer based magnetic devices. Therefore, such systems havesparked immense research interest [96, 97, 98], especially after the discoveryof interlayer exchange coupling [99] and its oscillatory variation with respectto the thickness of the non-magnetic spacer [100]. Most of the investigationshave been carried out on symmetrical trilayers with the two FM layers madefrom the same material. Nevertheless, there are studies of trilayers with differ-ent FM layers, either by varying the thickness of the the two FM layers or thematerials from which they are made. In this chapter, we present results on theinfluence of interlayer coupling on the static and dynamic magnetic responsesof general asymmetric trilayer systems obtained by numerical simulation ofa simple micromagnetic model. Static magnetization curves and ferromag-netic resonance dispersion relations are calculated for different coupling typesand strengths. Many interesting features are found from the simulated results,which are valuable references for experimental work (see Chapter 8 for exam-ple).
7.2 Micromagnetic modelIn a trilayer system, besides the Zeeman energy, anisotropic energy and de-magnetizing energy, there is an additional energy term, stemming from theinterlayer exchange coupling, which is given by
εex = −J1MA · MB
MAMB− J2
(MA · MB
MAMB
)2
, (7.1)
89
Figure 7.1. The geometry and coordinate system of the film.
where MA and MB are the magnetization vectors of the two FM layers. J1 isthe bilinear coupling constant and J2 is the biquadratic coupling constant, bothgiven in units erg/cm2.We consider a trilayer system with two FM layers of the same thickness t
separated by a NM spacer, with thickness tNM. The geometry and coordinatesystem employed in the study are shown in Fig. 7.1. Taking into account theZeeman energy εzee, exchange energy εex and demagnetizing energy εdem, thetotal energy per unit area of the system is
εtot = εzee + εex + εdem
= −tHMA [sin θH sin θAcos (ϕH − ϕA) + cos θH cos θA]−tHMB [sin θH sin θBcos (ϕH − ϕB) + cos θH cos θB]−J1 [sin θA sin θBcos (ϕA − ϕB) + cos θA cos θB]−J2[sin θA sin θBcos (ϕA − ϕB) + cos θA cos θB]2
+2πt(M2
Acos2θA +M2Bcos2θB
). (7.2)
θA, θB , ϕA and ϕB correspond to the polar and azimuth angles of the magneti-zation vectors, as indicated in Fig. 7.1. H , θH and ϕH refer to the magnitude,polar angle and azimuth angle of the external magnetic field, respectively.If J1 dominates over J2, a positive value of J1 will lead to the system favour-
ing a parallel configuration of the two magnetization vectors in order to min-imize exchange coupling energy. Therefore a positive J1 corresponds to fer-romagnetic coupling. On the contrary, a negative J1 will make the two mag-netization vectors to align antiparallel with respect to each other to minimizetheir exchange energy, corresponding to antiferromagnetic coupling [101]. If,on the other hand, J2 dominates over J1 and is positive, then the minimum en-ergy occurs when the magnetizations are oriented perpendicular to each other(90 degree type coupling).For a given set of parameters characterizing the trilayer system (J1, J2,MA
and MB), the equilibrium orientations of the magnetizations at any external
90
field can be calculated numerically by finding the values of θA, θB , ϕA andϕB that minimize εtot. Using these values, the in-plane static magnetization ofthe film can be obtained as
m∥ =M∥(H)
Ms=
MA sin θA cosϕA (H) +MB sin θB cosϕB (H)
MA +MB. (7.3)
Similarly, the out-of-plane static magnetization of the film is given by
m⊥ =M⊥(H)
Ms=
MA cos θA (H) +MB cos θB (H)
MA +MB. (7.4)
Ferromagnetic resonance (FMR) experiments make use of a microwave sig-nal generating an RF magnetic field h that is perpendicular to the static field.Resonance absorption occurs when the microwave frequency matches the nat-ural frequency of the system, which is determined by the magnetic propertiesof the sample and the static external magnetic field. By substituting the to-tal energy density (Eq. (7.2)) and the equilibrium angles into the Suhl-Smitequation [27] the resonance condition is obtained as a fourth order polynomialequation of the microwave angular frequency ω, which reads
aω4 + cω2 + e = 0. (7.5)
The coefficients of this 4th order polynomial equation are
a =
(t2MAMB
γAγB
)2
,
c = −t2MAMB
[(hA1 h
A2
γ2B+
hB1 hB2
γ2A
)+
2C0C2
γAγB
+C1
(tMAh
B2
γ2A+
tMBhA2
γ2B
)+C2
(tMAh
B1
γ2A+
tMBhA1
γ2B
)(7.6)
+C1C2
(MA
MBγ2A+
MB
MAγ2B
)],
e =[t2MAMB + C2
(tMAh
A2 + tMBh
B2
)]×[t2MAMBh
A1 h
B1 + C1
(tMAh
A1 + tMBh
B1
)+(C21 − C2
0
)],
with γA and γB being the gyromagnetic ratio of the FM layer A and B, respec-tively. The parameters in Eq. (7.6) are defined as
C0 = J1 + 2J2 cos(ϕA − ϕB),
C1 = J1 cos(ϕA − ϕB) + 2J2cos2(ϕA − ϕB), (7.7)C2 = J1 cos(ϕA − ϕB) + 2J2 cos 2(ϕA − ϕB),
91
and
hA1 = H cosϕA + 4πMA,
hB1 = H cosϕB + 4πMB, (7.8)hA2 = H cosϕA,
hB2 = H cosϕB.
Among the four solutions to Eq. (7.5), only two are physically meaningful.These two solutions are referred to as the acoustic resonant mode, which cor-responds to the two magnetization vectors in the two FM layers precessingin-phase and the optic resonant mode, which corresponds to the two magne-tization vectors precessing out-of-phase. So one would expect two distinctresonance peaks on the FMR spectrum. In the case where the two FM layersare identical, the resonance field difference of the two modes is the value ofthe exchange field.
7.3 Simulated resultsFig. 7.2 and 7.3 present some simulated curves for FM and AFM coupled tri-layers, respectively. From the figures, many features expected to be observedin FMR studies on trilayer systems can be seen.
7.3.1 Ferromagnetic couplingFor ferromagnetic coupling, as one can see clearly from Fig. 7.2 (a), the twomagnetizations are always aligned in parallel with the applied external DCfield. The in-plane magnetization is always saturated, regardless of magneticfield and exchange coupling magnitudes (neglecting coercivity and magne-tocrystalline anisotropy), as demonstrated by Fig. 7.2 (b). In a de-coupledsystem (i.e. very thick spacer layer), the FMR dispersion relation is describedby a distinct Kittel formula for each layer, as shown by the blue curves in Fig.7.2 (c).In a coupled trilayer, the optical branch of the dispersion curves is signifi-
cantly pushed to higher frequencies. The two resonance conditions for a FMcoupled trilayer system are both dependent on the strength of the coupling insuch a way that a stronger coupling leads to a higher value of fres/γH . Inother words, the resonance frequencies of both the acoustic and optic modesincrease with increasing coupling in a frequency-swept FMR measurement ata fixed magnetic field. While in a field-swept FMR measurement with fixedfrequency, the resonance fields of both modes decrease with increasing cou-pling. This is in agreement with experimental findings for a variety of trilayerfilms [102, 101, 103, 104]. It is evident from Fig. 7.2 (c) that the resonance
92
0 500 1000 1500 20000
5
10
15
20
25
Magnetic field (Oe)
Re
son
an
ce f
req
ue
ncy
(G
Hz)
(c)
0 500 1000 1500 2000
−100
0
100
Magnetic field (Oe)An
gle
s b
etw
ee
n m
ag
ne
tiza
tio
ns
an
d a
pp
lied
ma
gn
eti
c fi
eld
(d
eg
)
(a)
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
Magnetic field (Oe)
Re
du
ced
ma
gn
eti
zati
on
(d
ime
nsi
on
less
)
(b)
Figure 7.2. Simulated results of magnetization orientation (a), magnetization (b) andFMR resonance frequency (c) as a function of applied magnetic field for a few FMcoupled trilayers. Solid lines correspond to the dispersion curves of the acoustic modeand dashed lines corresponds to that of the optic mode. The magnetization values ofthe two FM layers are 4πMA = 22 kG and 4πMB = 9.5 kG. The coupling constantsused are: J1 = 0, J2 = 0 erg/cm2 (blue); J1 = 0.2, J2 = 0 erg/cm2 (green); J1 = 0.6,J2 = 0 erg/cm2 (red); J1 = 0.2, J2 = 0.2 erg/cm2 (magenta). The red and magentacurves overlap with each other in panel (c). In panel (a) and (b), curves for all foursets of coupling constants overlap.
93
condition of the optic branch is more sensitive to coupling strength than thatof the acoustic branch. It is found that as the spacer thickness decreases, theoptic mode resonance peak becomes broader and its intensity decreases [105].As one can see from Eqs. (7.7) and (7.8), for ferromagnetically coupled
films where ϕA = ϕB = 0, the contributions from J1 and J2 can no longerbe separated. Therefore the coupling strength is described by the effectivecoupling constant Jeff = J1+2J2 [101]. This is demonstrated by the completeoverlap of the red and magenta curves in Fig. 7.2.Ultimately, in the case that the two FM layers are very strongly coupled,
the optic mode will disappear (or move to frequencies much higher than canbe experimentally measured), leaving only the sharp acoustic mode being ob-servable [96]. In this case, only the equivalent effective magnetization andthe equivalent gyromagnetic ratio of the whole system can be extracted byFMR technique, while the magnetic parameters of the individual layers cannotbe separated. Following Lavadi [96], the in-plane resonance condition for anisotropic system in this case reads,(
ω
γeqv
)2
= H[H − (4πMeff)eqv
], (7.9)
whereγeqv =
4πMA + 4πMB
4πMA/γA + 4πMB/γB(7.10)
and(4πMeff)eqv = (4πMs)eqv −Hp,eqv, (7.11)
with the equivalent saturation magnetization (4πMs)eqv and equivalent per-pendicular anisotropy field Hp,eqv being
(4πMs)eqv =(4πMA)
2 + (4πMB)2
4πMA + 4πMB(7.12)
andHp,eqv =
Hp,A (4πMA) +Hp,B (4πMB)
4πMA + 4πMB. (7.13)
7.3.2 Antiferromagnetic couplingIn the case of antiferromagnetic coupling, the orientations of the magnetiza-tions at equilibrium are more complicated than that for ferromagnetically cou-pled trilayers. The behavior of the magnetizations is dependent on the relativemagnitude of the bilinear and biquadratic coupling constants. We discuss onlytwo special cases for the sake of simplicity: J1 ≫ J2 and J2 ≫ J1.When J2 ≫ J1, the magnetizations of the two FM layers make a 90o angle
with each other at zero external field in order to minimize the energy of the
94
0 500 1000 1500 20000
5
10
15
20
Magnetic field (Oe)
Re
so
na
nce
fre
qu
en
cy (
GH
z)
(c)
0 500 1000 1500 20000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Magnetic field (Oe)
Re
du
ce
d m
ag
ne
tiza
tio
n (
dim
en
sio
nle
ss)
(b)
0 500 1000 1500 2000-200
-150
-100
-50
0
50
Magnetic field (Oe)
An
gle
s b
etw
ee
n m
ag
ne
tiza
tio
ns a
nd
fie
ld(d
eg
)
(a)
Angle between MA and H
Angle between MB and H
Figure 7.3. Simulated results of magnetization orientations (a), magnetization (b) andFMR resonance frequency (c) as a function of applied magnetic field for four AFMcoupled trilayers. Solid lines correspond to the dispersion curves of the acoustic modeand dashed lines corresponds to that of the optic mode. The magnetization values ofthe two FM layers are 4πMA = 22 kG and 4πMB = 9.5 kG. The coupling constantsused are: J1 = −0.2, J2 = 0 erg/cm2 (blue); J1 = −0.6, J2 = 0.04 erg/cm2 (green);J1 = 0, J2 = −0.2 erg/cm2 (red); J1 = −0.04, J2 = −0.3 erg/cm2 (magenta). 95
system. With increased external field, the two magnetization vectors begin torotate close to the external field, as shown in 7.3 (a). There are two equilib-rium phases of the magnetization vectors depending on the external field value,relative to the exchange field.
• When the external field is smaller than the saturation fieldHsat, the mag-netizations make non-zero angles with the external field.
• When the external field is larger thanHsat, the magnetizations are alignedwith the external field.
The magnetization curve and dispersion relation also exhibit two phases, asshown in Figs. 7.3 (b) and (c) with the red and magenta curves. After satu-ration the acoustic mode resonance position is also independent of couplingstrength, while at unsaturated condition both the acoustic and optic modes arequite sensitive to coupling strength. When the external field is larger thanHsat,a stronger coupling leads to a lower value of fres/γH for the optic mode. Inother words, the resonance frequency of the optic mode decreases with in-creasing coupling strength in a frequency-swept FMR measurement, while ina field-swept FMR measurement, the resonance fields of both modes increasewith increasing coupling strength.When J1 ≫ J2, the magnetizations of the two FM layers are aligned in an-
tiparallel at zero external field. In an external field, there are three equilibriumphases of the magnetization vectors depending on the external field value, rel-ative to the exchange field.
• When the external field is smaller than a certain critical value Hcrit, themagnetizations in the two FM layers are still aligned in antiparallel, dueto the effect of the negative coupling J1. The total magnetization remainsthe same as at remanence.
• When the external field is larger than Hcrit but smaller than Hsat, themagnetizations make non-zero angles with the external field.
• When the external field is larger thanHsat, the magnetizations are alignedwith the external field.
The magnetization curve and dispersion relation also exhibits three distinctphases, as shown in Fig. 7.3 (b) and (c) with the blue and green curves. Thisresult is in agreement with experimental and calculated results by other re-searchers [106, 107]. It is interesting that after saturation the acoustic moderesonance position is almost independent of coupling strength, while at un-saturated condition both the acoustic and optic modes are quite sensitive tocoupling strength. When the external field is larger than Hsat, the optic moderesonance condition for an AFM coupled trilayer is dependent on the strengthof the coupling in such a way that a stronger coupling leads to a lower value offres/γH . In other words, the resonance frequency of the optic mode decreaseswith increasing coupling strength in a frequency-swept FMR measurement,while in a field-swept FMR measurement, the resonance fields of both modesincrease with increasing coupling strength.
96
For the case when the two FM layers are very strongly AFM coupled, theoptic mode resonance peak will also disappear. Therefore, the trilayers systemcan also be treated as an equivalent single layer with its resonance conditionidentical to Eq. (7.9). But the definitions of equivalent effective magneti-zation (4πMeff)eqv, equivalent gyromagnetic ratio γeqv, equivalent saturationmagnetization (4πMs)eqv and equivalent perpendicular anisotropy fieldHp,eqvare different and given by [96]
γeqv =4πMA − 4πMB
4πMA/γA + 4πMB/γB, (7.14)
(4πMeff)eqv = (4πMs)eqv −Hp,eqv, (7.15)
(4πMs)eqv =(4πMA)
2 + (4πMB)2
4πMA − 4πMB(7.16)
and
Hp,eqv =Hp,A (4πMA) +Hp,B (4πMB)
4πMA − 4πMB. (7.17)
7.4 Conclusion and DiscussionIn the micromagnetic model, it is important to note the assumptions made.Therefore the results are valid only for isotropic trilayer films that are uni-formly magnetized. In the experimental studies of realistic films, slight devi-ations from the simulated results are expected due to domain walls, interfacecontributions and anisotropy energies. Nevertherless, the simplemodel revealsmany features of the magnetic response of trilayers. For the FM coupling case,the equilibrium orientations of the magnetizations are always aligned with theexternal field. For the 90 degree coupling (J2 dominated AFM coupling), theequilibrium orientations of the magnetizations undergo two phases with in-creasing external field, from a non-zero angle to being aligned with externalfield. In the case of AFM coupling (J1 dominated AFM coupling), the equi-librium orientations of the magnetizations undergo three distinct phases withincreasing external field, from antiparallel alignment to both making non-zeroangles with external field to both being aligned with external field. The totalmagnetization and ferromagnetic resonance follow the phases of the equilib-rium orientations of the magnetization vectors accordingly. The resonancecondition is related to the interlayer coupling strength in such a way that astronger coupling leads to a higher (lower) value of fres/γH in FM (AFM)coupling. The optic mode where the two magnetizations precess out-of-phaseis sensitive to coupling strength while the acoustic mode where the two mag-netizations precess in phase is insensitive to the coupling strength (except forthe unsaturated AFM coupling).
97
Another point that should be noted is that in an AFM coupled film, if thetwo ferromagnetic layers are exactly identical, the optic mode cannot be ob-served in the FMR spectra. In a real trilayer film, there will always be somedifferences between the two ferromagnetic layers so a weak optic resonancepeak is expected. However, many researchers have found it difficult to ob-serve the optic mode resonance. In order to overcome this difficulty, a com-monly used method is to make an asymmetrical structure (eg. different layerthickness or different choices of material for the two FM layers) so that the ef-fective magnetization of each ferromagnetic layer is different from one to theother [108, 109]. However, the optic mode is still difficult to observe even inthose cases, partly due to a broader linewidth and lower intensity for this mode[110, 108]. In the case of different materials of the two FM layers, the fact thatthe two FM layers might have different precessional cone angles [111] makesthe magnetization components along the transverse direction similar in sizeand opposite in direction in the optic mode resonance. Zhang et al. developeda technique called longitudinal pumping, which applies the RF field along thesame direction of the external bias field [112]. This technique makes the opticmode much more detectable.Nevertheless, an important implication from this study is that in some situa-
tions the observation of the optic mode resonance is not always necessary. Forinstance, if the purpose of the measurement is to determine the strength of thecoupling, the fact that the acoustic mode resonance before saturation is quitesensitive to coupling strength offers a possibility to extract the coupling con-stant by measuring only the acoustic mode dispersion relation. To do this, oneneeds to carry out measurements at many different external fields before sat-uration, which is not very practical with conventional FMR systems. But thebroadband FMR based on a vector network analyzer, intensively used duringthe recent years, makes this easily implementable.
98
8. Magnetic coupling of FeCo/Ru/FeNi andFeCo/Cu/FeNi trilayer systems
8.1 IntroductionA trilayer system with two ferromagnetic (FM) layers separated by a non-magnetic (NM) spacer layer is of great interest due to its use in applicationssuch as magnetic recording devices and non-volatile magnetic random mem-ories [113, 114]. The interlayer coupling between the two FM layers plays animportant role for engineering the magnetic properties of the trilayer systemsand there have been numerous studies on coupled trilayers. However, mostof the investigations have been carried on symmetrical trilayers with the twoFM layers made from the same material. Systematic investigations into theexchange coupling in trilayers with different FM layers are rare. In this chap-ter, we present our investigation of the interlayer coupling in trilayer films ofFeCo/Ru/FeNi and FeCo/Cu/FeNi with varying spacer thicknesses. The cou-pling across Ru is found to oscillate with spacer thickness, as expected. How-ever, the oscillatory behavior of the coupling is not observed for Cu spacedtrilayers. Instead, all FeCo/Cu/FeNi films are found to be ferromagneticallycoupled, with the coupling strength decreasing exponentially with increasingCu spacer thickness.
8.2 FeCo/Ru/FeNi trilayer films8.2.1 ExperimentalThe films were deposited at room temperature using dc magnetron sputtering.The base pressure of deposition was 5 × 10−8 Torr. The thin films consist ofSi/FeCo(100 Å)/Ru(tRu Å)/FeNi(100 Å), with a 10 Å Ru layer added as cap.The chemical composition of the FeCo and FeNi alloys are Fe49Co49V2 andFe19Ni81, respectively. The Ru layer thickness was varied between 0 and 200Å. The Ru spacer layer was deposited at low sputtering rate (0.4 Å/s) and lowAr gas pressure (3 mTorr) for optimal uniformity and interface smoothness.Room temperature magnetization vs. field curves were measured using a
SQUID magnetometer. We also performed XMCD studies at the Fe, Ni andCo L2,3 edges. This element specific technique allows us to determine themagnetic state of the separate magnetic layers as each layer contains a mag-netic element that is not present in the other layer; Ni in the FeNi layer and Co
99
in the FeCo layer. The samples were also studied in a field-sweep cavity fer-romagnetic resonance (FMR) system with an RF frequency of 9.8 GHz. Angledependent FMR measurements were performed using a goniometer, keepingthe DC magnetic field parallel to the film plane. FMR spectra using a vectornetwork analyzer based FMR system (VNA-FMR)were additionally recorded.Frequency scans at different magnetic fields ranging from 0.2 to 1.2 kOe wereperformed.
8.2.2 Results and discussionThe results from angle dependent field-sweep cavity-FMRmeasurements showthat all trilayer samples are isotropic in-plane. The FeCo/FeNi bilayer sampleexibits a small in-plane uniaxial anisotropy characterized by an anisotropy fieldof Hu = 27 Oe. The in-plane magnetization curves for some of the films arepresented in Fig. 8.1. The hysteresis loops of the films indicate that all films areferromagnetically coupled except the one with 10 Å Ru spacer, which showsantiferromagnetic coupling instead. In addition, the XMCD studies performedin remanence were found to agree with the hysteresis measurements. Analyz-ing the difference spectra between the left and right circularly polarized lightcollected at the Ni and Co L edges (not shown), we find that Ni and Co mo-ments are aligned antiparallel to each other for the sample with 10 Å thick Ruspacer, while for the other samples they are aligned in parallel.The antiferromagnetically coupled sample is characterized by a bilinear cou-
pling constant of −0.57 erg/cm2 and a small biquadratic coupling constant of−0.05 erg/cm2. The coupling constants were obtained by matching the calcu-lated magnetization curve, using Eqs. (7.2) and (7.3), with the experimentaldata; the right bottom inset in Fig. 8.1 shows ϕA and ϕB as function of externalfield for this sample. The saturation magnetization values used in the calcula-tions are 4πMA = 22 kG for FeCo [58] and 4πMB = 9.5 kG for FeNi [115].
Fig. 8.2 shows the dispersion relations (resonance frequency vs. magneticfield) for some of the trilayer samples. The coupling constants were obtainedby fitting the experimental cavity-FMR and VNA-FMR results to Eq. (7.5);the bilinear coupling constants are presented in Fig. 8.3. The simulated res-onance frequency vs. magnetic field curves fit the experimental data quitewell, without introduction of the biquadratic coupling constant for any of theferromagnetically coupled samples. However, for the antiferromagneticallycoupled sample, a small biquadratic coupling constant of -0.05 erg/cm2 wasnecessary to fit the experimental data. It is important to note that this valueis much smaller compared to the bilinear coupling constant, implying that thecoupling is dominated by the bilinear contribution. Interestingly, the best fitobtained from the FMR data yields J1 = -0.62 erg/cm2, slightly different fromthe one obtained from the magnetization curves. This can be explained by the
100
Figure 8.1. Some typical magnetization curves of the FeCo/Ru/FeNi trilayers. Thetop left inset shows the geometry and coordinate system of the film samples.The rightbottom inset shows ϕA and ϕB as function of external field for the tRu = 10Å sample.
0 500 1000 1500 20000
5
10
15
20
25
30
Magnetic field (Oe)
Res
onan
ce fr
eque
ncy
(GH
z)
0 1000 2000Magnetic field (Oe)
Abs
orpt
ion
(a.u
)
acoustic mode
opticmode
Figure 8.2. The dispersion relations of FeCo/Ru/FeNi trilayers with tRu = 0Å (green),tRu = 10 Å (red), tRu = 15 Å (blue), tRu = 20 Å (magenta). Open and solid markersare experimental data obtained from VNA-FMR and cavity-FMR, respectively. Linescorrespond to simulated curves using the micromagnetic model discussed in Section7.2, with solid lines representing the acoustic modes and dotted lines the optic modes.The inset shows the cavity FMR spectrum of the sample with tRu = 15 Å.
101
Figure 8.3. Bilinear coupling constant as a function of tRu. The solid line is a guide tothe eye.
fact that in the micromagnetic model, discussed in Section 7.2, we treated eachlayer of the film as a single magnetic domain (a microspin) and neglected ef-fects of a multidomain state. Another comment we would like to add is that theVNA-FMR setup was unable to resolve the optic mode resonance peaks dueto limitations in instrument sensitivity and the low intensity of the optic moderesonance [108, 106].As one can see from Fig. 8.3, the FM coupling for the FeCo/FeNi sample is
comparably strong with J1 =2 erg/cm2. The coupling constant changes signfor the sample with 10 Å Ru thickness but changes back to FM coupling forlarger Ru thickness. For Ru spacer thickness larger than 20 Å, the static RKKYcoupling strength vanishes. Despite this, the films with Ru spacer thickness inthe range 20 Å to 50 Å are still coupled as the FMR spectra do not reveal twodistinct resonance peaks for the FeCo and FeNi layers. This is tentatively ex-plained by so-called dynamic exchange coupling interaction [116]. The cou-pling constants of our asymmetric trilayers exhibit values similar to that ofsymmetric FeNi/Ru/FeNi trilayers, but its dependence on the Ru thickness isshifted towards to the thicker end [117, 101].
8.3 FeCoV/Cu/FeNi trilayers8.3.1 ExperimentalThe films were deposited at room temperature using DC magnetron sputter-ing. The base pressure of deposition was 5 × 10−8 Torr. The thin films con-sist of Si/SiO2/FeNi(100 Å)/Cu(tCu Å)/FeCo(100 Å), with a 30 Å Ta layeradded as cap. The chemical composition of the FeNi and FeCo alloys areFe19Ni81 and Fe49Co49V2, respectively. The Cu layer thickness was varied,with tCu = 5, 7, 8, 10, 15, 50 Å. The Cu spacer layer was deposited at low rate
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(0.4 Å/s) and low Ar gas pressure (3 mTorr) for optimal uniformity and inter-face smoothness.The samples were first studied in a field-sweep cavity FMR system using an
RF frequency of 9.8 GHz to extract information regarding anisotropy fields.This system is equipped with a goniometer, making it possible to rotate thesample with respect to the applied field. The results (not shown) from in-plane angular dependent field-sweep FMR measurements indicate that the in-plane uniaxial anisotropy fields for all trilayer samples are small; the observedanisotropy fields cover the range 5 − 25 Oe. Therefore the anisotropy is ne-glected in the analysis of the coupling strength.Both room temperature in-plane and out-of-planemagnetization curves were
measured using a superconducting quantum interference device (SQUID)mag-netometer. FMR spectra using a vector network analyzer based broadbandFMR system (VNA-FMR) [118] were additionally recorded with the externalfield applied in the film plane. Frequency scans (1-30 GHz) at different mag-netic fields ranging from 0.7 to 2 kOe were performed. Also, field scans (1-10kOe) at different frequencies ranging from 15 to 55 GHz were carried out.
8.3.2 Results and discussionSome typical room temperature magnetization curves are presented in Fig. 8.4.The hysteresis loops indicate that all films are ferromagnetically coupled, ex-cept the one with tCu = 50 Å. The two-step transition of the magnetizationcurve for the tCu = 50 Å shows that this sample is decoupled, with the twomagnetization vectors switching independently.Fig. 8.5 shows the dispersion relations (resonance frequency vs. magnetic
field) measured by FMR technique for some of the trilayer samples. Thecoupling constants were obtained by fitting the experimental FMR results toEq. (7.5). For ferromagnetically coupled films, neither the FMR nor M(H)fitting techniques allow one to separate the J1 and J2 exchange constants.Thus, the coupling strength is described by the effective coupling constantJeff = J1 + 2J2 [101]. The simulated resonance frequency vs. magneticfield curves fit the experimental data quite well (as shown in Fig. 8.5), withthe estimated effective coupling constants presented in Fig. 8.6.The saturation magnetization values used in the calculations were 4πMA =
21 kG for FeCo and 4πMB = 9.2 kG for FeNi. These values were taken froma single layer Si/SiO2/FeCo film [40] measured previously and a single layerSi/SiO2/FeNi/Cu/Ta control sample measured in this study. The choice of thesaturation magnetization values is also justified by the fact that the experimen-tal out-of-planeM vs. H data compare reasonably well to the simulated resultusing the above saturation magnetization values (see the inset in Fig. 8.4).As one can see from Fig. 8.5, all films are ferromagnetically coupled since
for a given field, the optic mode resonance frequency is higher than that of
103
-50 -40 -30 -20 -10 0 10 20 30 40 50
-1.0
-0.5
0.0
0.5
1.0
0 10 20 300.00.20.40.60.81.0
mII (
dim
ensi
onle
ss)
H (Oe)
tCu=5Å tCu=15Å tCu=50Å
m(d
imen
sion
less
)H (kOe)
exp sim
Figure 8.4. Some typical in-plane easy axis reduced magnetization m∥ vs. appliedfieldH curves of the FeNi/Cu/FeCo trilayers. The inset shows the measured and sim-ulated out-of-plane magnetization curves for the tCu = 50 Å sample.
0 1 2 3 4 5 6 7 8 90
10
20
30
40
50
8 12 16 200.00.30.6
f (G
Hz)
H (kOe)
r(d
imen
sion
less
)
f (GHz)
Figure 8.5. The dispersion relations of FeNi/Cu/FeCo trilayers with tCu = 5 Å (blue),tCu = 7 Å (purple), tCu = 15 Å (green) and tCu = 50 Å (red). Solid triangles andcircles are experimentally measured acoustic and optic resonance modes, respectively.Lines correspond to simulated curves using Eq. (7.5), with solid lines representingthe acoustic modes and dashed lines the optic modes. The inset shows the frequencyswept FMR spectrum of the sample with tCu = 15 Å at an external field of 1.1 kOe.
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0 10 20 30 40 50
0
1
2
3
20 30 40
0.0
0.1
0.2
0.3
0.4
J eff (
erg/
cm2 )
tCu (Å)J
(erg
/cm
2 )
tPt (Å)
data fit
Figure 8.6. Effective coupling constants Jeff as a function of spacer thickness tCu. Thesolid line is a fit of y = Ae−x/λ to the experimentally measured coupling constants.The inset shows the coupling constant J for a Co/Pt/Co sandwich system vs. Pt thick-ness tPt, as adapted from Ref. [119]. The dashed line is a fit of the exponential functionto the data (no such fitting was attempted in Ref. [119]).
the acoustic mode for all samples. The extracted effective coupling constantsare presented in Fig. 8.6. The figure clearly shows that the oscillatory cou-pling is absent for the here studied FeNi/Cu/FeCoV system. The ferromag-netic coupling for the sample with 5 Å Cu spacer is very strong with Jeff = 3erg/cm2. The coupling constant decreases monotonically with increasing Cuspacer thickness and approaches zero for a spacer thickness of tCu = 50 Å.It is widely believed that the magnetic coupling between two FM layers os-
cillates between ferromagnetic and antiferromagnetic (AFM) coupling withrespect to the NM spacer thickness [120, 100]. Also, experimental investiga-tions into various FM/NM/FM systems [121, 102] find an oscillatory behaviorof the coupling strength. However, Parkin claimed, without giving detailedexplanation or experimental results, that for the FeNi/Cu system prepared byDC magnetron sputtering the oscillatory behavior of the coupling would beabsent [122]. Our results constitute an experimental evidence to support thisclaim.In research literature, both the magnitude and sign of the coupling constant
scatter considerably for FeNi/Cu multilayers and trilayers. The coupling con-stant is found to be not only dependent on the Cu thickness, but also manyother factors such as film deposition method, FM layer thickness, interfacelayer, etc. For example, Nagamine et al. identified both FM and AFM cou-pling in RF sputtered NiFe/Cu (9 Å)/NiFe films depending on the thickness ofthe NiFe layer [123]. Gonzalez-Chavez et al. studied RF sputtered FeNi(200Å)/Cu(tCu Å)/FeNi(200 Å) IrMn (150Å) films with tCu=7.5, 10 and 25 Å and
105
found only AFM coupling [124]. Heinrich et al. found that Fe/Cu/Fe trilayersare ferromagnetically coupled for Cu thickness less than 9 monolayers (about16 Å) [125]. Parkin found that the interlayer coupling of the FeNi/Cu systemcan be dramatically enhanced by adding a Co interface layer [122] and Diaoet al. [126] observed that buffer layers may play a significant role in coupledNiFeCo/Cu multilayers. Also, although a rare observation, a monotonicallydecreasing ferromagnetic coupling with increasing spacer thickness has pre-viously been observed for other metals, such as Pd [33] and Pt [119]. ForFe/Pd/Fe trilayers the increase of the average terrace size, through the modi-fication of the growth conditions, resulted in the observation of an AFM cou-pling. This result indicated that for a specific thickness of the ferromagneticlayers, antiferomagnetic coupling can be observed depending on the roughnessat the ferromagnet/nonmagnetic metal interface [127, 128].Besides the fact that the coupling does not oscillate with spacer thickness, the
coupling observed in this study displays two other marked features, large mag-nitude in coupling strength and exponential spacer thickness dependence, asshown in Fig. 8.6. The strength of the coupling is much larger than the valuesexpected from Ruderman-Kittel-Kasuya-Yosida (RKKY) coupling. We sus-pect that the RKKY coupling for small Cu thickness is masked by the strongferromagnetic coupling due to pinholes formed in the Cu spacer layer. Thepresence of such pinholes in Co/Cumultilayers was confirmed by transmissionelectron microscopy (TEM) [129]. Atom probe microscopy analysis directlyobserved the ferromagnetic atoms segregating a few Ångstroms into the Culayer at the CoFe/Cu [130] interface, leading to the formation of pinholes inthe Cu layer. Kikuchi et al. suggested pinholes as the cause of diminishing an-tiferromagnetic coupling in Cu/Co multilayers and their model indicated thatthe pinhole coupling strength is roughly inversely proportional to the thicknessof the spacer as it arises from direct exchange coupling [131]. This is qualita-tively consistent with our results. We believe that pinholes play a significantrole in increasing the coupling strength.To verify the existence of pinholes experimentally, high resolution TEM
analysis was carried out including acquisition of Electron Energy Loss Spec-troscopy Spectral Images (EELS-SI) acquisition. Fig. 8.7 shows the elementalmaps of the samples with Cu spacer thickness of 7 and 15 Å extracted fromEELS-SI datacubes (three-dimensional array of values). To the left in the fig-ure, a High Angle Annular Dark Field (HAADF) image of the scanned regionis presented along with the scale bar for each sample. This data can be in-terpreted as mostly Z-contrast, and it already reveals differences between thetwo ferromagnetic layers. Elemental maps, presented to the right in the figure,were generated from a reduced dimensionality datacube reconstructed fromthe eight principle spectral components in the original hyperspectral dataset,as calculated by the Principle Component Analysis (PCA) technique [132].The Cu spacer layer is clearly visible after data treatment and appears to haveinterfacial roughness on the order of 1-2 nm in both films. The film with 7 Å
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Figure 8.7. Elemental maps generated from EELS-SI datacubes.
Cu spacer exhibits clearly a number of discontinuities. Within these disconti-nuities, Fe is observed, confirming the existence of ferromagnetic pinholes inthe spacer layer.The Bruno model suggests that the magnetic coupling across an insulating
or semiconductor spacer should exhibit an exponential dependence on spacerthickness instead of an oscillatory dependence [120]. The coupling strengthbetween two Fe layers across FeSi was observed to be strong and exponentiallydecaying with FeSi thickness [133, 134]. However, the coupling in these caseswas AFM and the exponential decay was only observed for large spacer thick-ness (> 13Å). Experimentally, the results of Bloemen et al. [119] on Co/Pt/Cosandwiches grown on glass substrate resemble ours. We present their results inthe inset of Fig. 8.6 and find that the results can be reasonably well describedby an exponentially decaying curve. The lack of oscillations in coupling as afunction of Cu thickness suggests that in the studied samples we have nano-sized regions with different thickness of Cu, in consistence with our EELSimages (Fig. 8.7). As a result oscillations are washed out. However, there isno theoretical model that provides an explanation for a ferromagnetic couplingstrength decaying exponentially with increasing metallic spacer thickness.
8.4 ConclusionIn conclusion, we have investigated the magnetic anisotropy and interlayercoupling in FeCo/NM/FeNi (NM=Ru,Cu) trilayers as a function of spacer thick-ness in this chapter. For the Ru spaced trilayer series, the film without Ruspacer shows a weak in-plane uniaxial anisotropy with an anisotropy field of27 Oe, while all of the trilayer samples were magnetically isotropic in-plane.The films are all ferromagnetically coupled except the onewith 10ÅRu spacer,which shows a relatively strong antiferromagnetic coupling characterized by abilinear coupling constant J1= -0.6 erg/cm2 and biquadratic coupling constantJ2= -0.05 erg/cm2. The bilayer FeCo/FeNi sample is strongly ferromagneti-
107
cally coupled with a coupling constant of about 2 erg/cm2. The static RKKYcoupling strength vanishes for a Ru spacer thickness larger than 20 Å.For the Cu spacer series, we found that all the films with Cu thickness rang-
ing from 5-50 Å show ferromagnetic coupling and the coupling strength of thistrilayer system does not exhibit the expected oscillation with spacer thickness.The effective coupling constants are large (3 erg/cm2 for the sample with 5Å Cu spacer) and decay exponentially with Cu spacer thickness. While thehigh strength of the coupling at small spacer thickness might be explained bypinhole coupling, we lack a theoretical model that could quantitatively explainhow fluctuations of Cu thickness across the spacer area (the roughness) leadto an exponential dependence of the coupling on the spacer thickness.
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9. Summary and Outlook
9.1 SummaryTo summarize, the magnetization dynamics of FeCo thin films and structureshave been investigated in this study, mainly using ferromagnetic resonance(FMR) technique.The FMR resonance condition and linewidth for films with uniaxial and cu-
bic anisotropy fields are derived from the dynamic Landau-Lifshitz-Gilbertequation for different orientations of the external field with respect to the sam-ple surface. An analysis of the validity of the conversion between FMR fieldand frequency linewidths is performed and it is found that the linewidth con-version relation based on the derivative of the resonance condition is only validfor samples with a negligible frequency-independent linewidth contribution.The dynamic magnetic properties of FeCo thin films grown on Si/SiO2 sub-
strates with varying deposition temperatures are characterized. The effectiveLandé g-factor, extrinsic linewidth, and Gilbert relaxation rate are all foundto decrease in magnitude with increasing sample growth temperature from 20oC to about 400–500 oC and then on further increase of the growth temper-ature to increase in magnitude. Samples grown at about 400–450 oC displaythe smallest coercivity, while the smallest value of the Gilbert relaxation rateof about 0.1 GHz is obtained for samples grown at 450–500 oC. An almostlinear relation between extrinsic linewidth and coercivity is observed, whichsuggests a positive correlation between magnetic inhomogeneity, coercivity,and extrinsic linewidth, which is ascribed to the degree of structural order inthe films.A micromagnetic model is established for an asymmetric trilayer system
consisting of two different ferromagnetic (FM) layers separated by a thin non-magnetic (NM) layer, treating each FM layer as a macrospin. Based on themodel, numerical simulations of magnetization curves and FMR dispersionrelations, of both the acoustic mode where magentizations in the two FM lay-ers precess in phase and the optic mode where they precess out-of-phase, havebeen carried out. The most significant implication from the results is that thecoupling strength can be extracted by detecting only the acoustic mode reso-nances at many different unsaturated states using broadband FMR technique.Trilayer films of FeCo(100 Å)/NM/FeNi(100 Å) with NM=Ru or Cu were
prepared and studied. The thickness of the Ru and Cu spacer was varied from0 to 50 Å. Magnetization curves and FMR dispersion relations are measured,with data being compared to the micromagnetic model. For the Ru spacer se-ries, the film with 10 Å Ru spacer shows antiferromagnetic coupling while
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all other films are ferromagnetically coupled. For the Cu spacer trilayers,it is found that all films are ferromagnetically coupled and that films withthin Cu spacer are surprisingly strongly coupled, with the coupling constantfor the sample with 5 Å Cu spacer being 3 erg/cm2 . The strong couplingstrength is qualitatively understood within the framework of a combined effectof Ruderman-Kittel- Kasuya-Yosida interaction and pinhole coupling, which isevidenced by transmission electron microscopy analysis. The magnetic cou-pling constant decreases exponentially with increasing Cu spacer thickness,without showing an oscillatory thickness dependence. The results have impli-cations for the design of multilayers for spintronic applications.
9.2 Future workIn this study, the effect of growth temperature on the dynamics magnetic prop-erties of FeCo has been systematically investigated. Previous studies indicatethat thermal treatment also plays an important role in modifying the structuraland magnetic properties of FeCo [135]. For future work, the effect of anneal-ing on the dynamic properties of the films can be an interesting topic to study.Specifically, samples grown at 450–500 oC are in this study found to exhibitthe best dynamic magnetic properties with a Gilbert relaxation rate of about0.1 GHz. It is natural to ask the question if annealing FeCo films grown atabout 450–500 oC would lead to a further decrease of Gilbert relaxation [136].Thus, samples grown in this temperature interval could be the focus of futurework.Unusually large extrinsic linewidth contributions are observed in the FeCo
films with a thickness of 100 nm. Though the thickness of the metal films ismuch thinner than the skin depth at typical microwave frequencies, it has beendemonstrated, both theoretically and experimentally, that eddy current shield-ing broadens the linewidth [24]. The effect of eddy current on the FMR spec-tra due to nonmagnetic metal capping layers have been investigated by others[137]. Our investigation of the influence of the thickness of FeCo magneticlayers on the FMR response and linewidth broadening due to eddy currents isin progress.In the study of FeCo/Cu(Ru)/FeNi trilayers, the simulation results from the
the micromagnetic model suggests that in antiferromagnetically coupled tri-layers, the acoustic mode resonance frequency is quite sensitive to the cou-pling strength before the trilayer is magnetically saturated. This finding pro-vides new possibility for extracting the coupling strength of AFM trilayerseven though the optic mode is absent in the experimental observations. Thisnew possibility is important since it is difficult to detect the optic mode in AFMcoupled trilayers. An experimental confirmation of the idea of extracting theAFM coupling strength by measuring only the acoustic mode resonance beforesaturation would be a meaningful future study.
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Appendix 1
Statement:If 2Hext sin (θM − θH)− (4πMeff +Hu) sin 2θM = 0,then
Hextsin θHsin θM
= Hext cos(θH − θM)− (4πMeff +Hu) cos2 θM. (9.1)
Proof:Eq. (9.1) can be written as
Hext
[cos(θH − θM)−
sin θHsin θM
]= (4πMeff +Hu) cos2 θM. (9.2)
Since4πMeff +Hu = Hext
sin(θM − θH)
sin 2θM(9.3)
it is equivalent to prove that
Hext
[cos(θH − θM)−
sin θHsin θM
]= Hext
2 sin(θM − θH) cos2 θMsin 2θM
(9.4)
orsin θM cos(θH − θM)− sin θM
sin θM=
cos θM sin(θM − θH)
sin θM. (9.5)
Thus, if one can prove that
sin θM cos(θH − θM)− sin θH = cos θM sin(θM − θH) (9.6)
the proof will be complete.Rearranging the terms, Eq. (9.6) can be written as
sin θM cos(θH − θM)− cos θM sin(θM − θH)− sin θM − sin θH = 0. (9.7)
Using trigometric relations, Eq. (9.7) is found to be equivalent to
sin [θM + (θH − θM)]− sin θH = 0, (9.8)
which simplifies tosin θH − sin θH = 0 (9.9)
and thus the proof is complete.
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Acknowledgement
First of all, I would like to thank Prof. Peter Svedlindh for offering me theopportunity to work on the spin dynamics project under his supervision. Prof.Peter Svedlindh has solid knowledge and profound experience in experimentalmagnetism. He gave me plenty of guidance in both theory and experimentsin this project. I highly appreciate his availability and patience during ourdiscussions as well as his invaluable comments and suggestions on my work.Also in particular for this dissertation, I must say that I (and also the readers, ifany, would) appreciate very much his careful and patient corrections for errors,which made it the way it is now. In addition, his attitude towards science willdefinitely impose positive influence on my future career.Also, I would like to say thanks to my co-supervisor, the enthusiastic scien-
tist Dr. Klas Gunnarsson, for his guidance, mentoring, inspiring discussions,as well as for helping me translate the lunch menus at the Rullan restaurant andfor translating the Swedish summary. I wish to thank another co-supervisor ofme, Prof. Per Nordblad, for his guidance and comments on my study plan.Thanks go also to my project co-workers Dr. Rimantas Brucas and soon-to-
be-Dr. Serkan Akansel. Our work together have always been pleasant and Ilearned many things from you. I also want to show my gratitude to our col-laborators Prof. Olof (Charlie) Karis, Dr. Ronny Knut, Dr. Somnath Jana andDr. Yvengen Pogoryelv at the Department of Physics and Astronomy, UppsalaUniversity, Dr. Peter Warnicke and Dr. Dario Arena at Brookhaven NationalLaboratory and Dr. Mojtaba Ranjbar, Dr. Randy Dumas and Prof. JohanÅkerman at the University of Gothenburg and Royal Institute of Technology(KTH). Among these people, special thanks to Ronny should be noted for be-ing my guide to X-ray magnetic circular dichroism technique and for the timespent at the Maxlab pingpong table.I also owe appreciations to Prof. Zbigniew Celinski for hosting me during
my time at the University of Colorado and for inspiring and fruitful discussionsand comments. Dr. Chen and Dr. Harward are also acknowledged for theirhelp during my stay. I owe my gratitude to Prof. Mikhail Kostylev for hostingme during my time at the University of West Australia, for generously supply-ing the calculation code, for helping me setup the experiments and for helpfuldiscussions, and of course also for finding me a nice apartment. I would liketo also thank Gunnar at Maxlab and David at Argonne National Laboratory forbeamline supports. I would like to thank the Maxlab restroom as some of mybest sleeping times were spent in it. The online distant help with my Matlabcode from Dr. Zhai Zhaohui at Nankai University and Chinese Academy ofEngineering Physics is appreciated.
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The Knut and Alice Wallenberg (KAW) Foundation and the Swedish Re-searchCouncil (VR) are the funders to be acknowledged for the research projects.The Ericsson Research Foundation, C F Liljevalch's stipendiefonder, AnnaMaria Lundins stipendienämnd (and Smålands Nation) and gradSAM21 areacknowledged for funding collaborative research projects and conference trav-els. Also, as a doctoral degree (usually) marks the end of one's formal educa-tion, I take the opportunity to thank the Nankai University's Scholarship Officefor partially financing my bachelor level education and the European Com-mission's Erasmus Mundus Scholarship Scheme for fully financing my masterlevel education.Thanks to all the colleagues in theDivision of Solid State Physics ofÅngström
lab, Bengt, Bozhidar, Mikael, Miguel, Anil, Malin, Matthias, Erik, Delphine,Roland, Zareh, Ruitao, Umut, David, JoakimM, JoakimW, Christopher, Tarja,Magnus, Bo, Junxin, Sergey, Mattias, Rebecca, Christopher, Carlos, Sofia,Jose, Annica, Ankit, Claes, Ewa, Gunnar, Per, Pia, Yuxia, Rasmus, Carl-Gustaf,Arne, Andreas and Shuxi, for creating such a friendly and pleasant working en-vironment. The small talks during and after working hours with my officemateJoakim were interesting. Special thanks to Delphine, for organising many ac-tivities and get-togethers. Many thanks toMaria Skoglund, Ingrid Ringård, Se-bastian Alonso, Inger Ekberg, Evika Bruciene and Ylva Johansson for keepingmany of the administrative stuffs moving smoothly and in good order. Thanksto Jonatan Bagge for technical support related to computer stuff.I enjoyed my time in Uppsala very much and wish to thank my friends
Shuangwei, Ruitao, Sabina, Esbjörn, Yue, Lisa, Allen, Yueming, Shummetand Miette for your hospitality, caring and friendship.The support from my family is important and a big thanks should go to all
my family members.Finally, the love and care from Jing is of utmost importance, without which
my contribution to the scientific knowledge base during the past 3 years wouldbe absolute zero rather than negligible.
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Svenska sammanfattning (Summary inSwedish)
Det finns tillämpningar för magnetiska material inom nästan alla teknik- ochindustri-områden. Några exempel på applikationer är medicinsk avbildning-steknik, magnetisk datalagringsteknik och tillämpningar inom telekommunika-tion. En egenskap som är viktig för funktionella komponenter är deras snabb-het. Helst ska en komponent kunna utföra sin funktion snabbt, ibland kandet vara fråga om att en speciell funktion ska kunna utföras på en miljondelssekund eller kanske ännu snabbare, en miljarddels sekund. Ett exempel ärhårddiskarna i våra hemdatorer. Informationen som vi sparar på en hårddiskfinns lagrad som ettor och nollor på en tunn magnetisk film och området somvisar en etta (eller en nolla) är idag ungefär en tiotusendels millimeter långt.Information lagras genom att området magnetiseras av en liten elektromag-net med ett luftgap som är en tiotusendels millimeter brett. Dessutom måsteden lagrade information kunna läsas och idag används en magnetisk sensorsom är lika liten som den lagrade informationen och där man utnyttjar att sen-sorns elektriska motstånd ändras då den påverkas av ett magnetfält. Ett an-nat exempel inom datorvärlden är magnetiska minnen som kallas för MRAM(förkortning av Magnetoresistive Random Access Memory) och som nyligenhar introducerats på marknaden. Minnet, vars byggstenar är två magnetiskaelektroder separerade av ett tunt elektriskt isolerande skikt, konkurrerar med dehalvledarminnen som idag finns som arbetsminnen i datorer eller som lagrings-media i digitala kameror. Idag används begreppet spinntronik som är baseratpå att elektroner förutom laddning även har magnetiska moment. Idén är attanvända elektronensmagnetiskamoment och laddning för att åstadkommamerän vad vanlig elektronik klarar av.Inom spinndynamikforskning studeras hur materialens magnetiska moment
svarar på snabba förändringar av yttre magnetfält. Magnetfälten som användsvarierar ofta periodiskt med tiden med karakteristiska frekvenser som över-stiger många miljarder svängningar per sekund, vilket innebär frekvenser i in-tervallet 1-50 GHz. Ferromagnetisk resonans (FMR) är en mycket kraftfulloch praktisk experimentell teknik som används för karakterisering av spinndy-namik hos ferromagnetiska filmer och heterostrukturer. En FMR mätning kanexempelvis innebära att man låter en mikrovågssignal med konstant frekvenspåverka en magnetisk film samtidigt som ett yttre magnetfält ökar i styrka.Mäter man transmission av mikrovågor kommer transmissionen uppvisa ettminimum för en viss styrka på magnetfältet, vilket innebär att materialet ab-sorberar mikrovågsenergi och att materialets magnetisering precesserar runt ett
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effektivt magnetfält (se figur). I denna avhandling presenteras och diskuterasgrundläggande dynamiska egenskaper och bakomliggande fysikaliskamekanis-mer för järn-kobolt (FeCo) baserade tunna filmer och heterostrukturer främstundersökta med hjälp av FMR.
Dynamiska egenskaper hos tunna filmer av enjärn-kobolt legeringDen ständigt ökande efterfrågan på högpresterande magnetiska minnesenheterhar lett till omfattande forskning om högfrekventa egenskaper hos tunna mag-netiska filmer som är nödvändiga för tillverkning av magnetiska minnen. Min-nets hastighet bestäms av hur snabbt man kan ändra materialets magnetisering,vilket i sin tur bestäms av de magnetiska momentens dynamiska egenskaper,exempelvis vet vi att momentens precessionsrörelse som beskrivs av rörelsensfrekvens och relaxation spelar en viktig roll för minneshastigheten. FeCo ärett material som har egenskaper som gör det intressant för att användas i mag-netiska minnen men de dynamiska egenskaperna hos FeCo-filmer har hittillsinte studerats systematiskt. I det här forskningsprojektet har jag undersökthur de magnetiska filmernas tillväxttemperatur påverkar de dynamiska mag-netiska egenskaperna hos FeCo-filmerna samt de underliggande fysikaliskamekanismerna som styr påverkan. Jag har funnit att den effektiva Landé g-faktorn, extrinsiska linjebredden och Gilbert-dämpningen alla minskar i stor-lek med ökande temperatur från 20oC till 400-500 oC men att ytterligare ökn-ing av tillväxttemperaturen gör att de ökar i storlek. Detta beroende liknarmycket tillväxttemperaturens inverkan på gitterkonstanterna, vilket indikeraratt tillväxttemperaturens inflytande på de dynamiskamagnetiska egenskapernaberor på förändringar av den strukturella ordningen i FeCo filmerna.
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Koppling mellan magnetiska lager i tunnfilmsstrukturer- inverkan på strukturernas dynamiska egenskaperTreskikts-system med två ferromagnetiska (FM) skikt separerade av ett icke-magnetiskt (NM) skikt är av stort intresse på grund av deras användning i ap-plikationer såsom magnetiska inspelningsenheter och MRAM minnen. Ommellanskiktet är tillräckligt tunt kan de två FM skikten interagera så att mag-netiseringarna i skikten kommer att vara parallella eller motriktade, vilket er-bjuder en möjlighet att kontrollera de magnetiska egenskaperna hos treskikts-system genom att variera NM skiktets tjocklek. Dock har de flesta under-sökningarna genomförts på symmetriska trelagerstrukturermed de två FM skik-ten tillverkade av samma material. I denna studie har vi undersökt mellanskik-tets inverkan i treskikts-filmer tillverkade av FeCo/Ru/FeNi och FeCo/Cu/FeNimed varierande tjocklekar på det NM skiktet. Resultat från magnetiseringskur-vor och FMR mätningar har applicerats på mikromagnetiska modeller och förden Ru-baserade serien observerades den förväntade oscillerande kopplingen.För deCu-åtskilda trelagerstrukturernaminskar emellertid denmagnetiska kop-plingskonstanten exponentiellt med ökande tjocklek på Cu-lagret, utan att visaett oscillerande tjockleksberoende. Delvis kan detta beroende förklaras av enbetydande ojämnhet mellan gränsskikten som suddar ut den förväntade os-cillerande kopplingen mellan de FM skikten. Interaktionens styrka kan förståssom en kombinerad effekt av Ruderman-Kittel-Kasuya-Yosida växelverkanoch ”pin-hole” koppling, där den senare kopplingen innebär att det finns nanome-terstora kanaler i det NM skiktet som innehåller FM material. Interaktionenmellan de FM skikten innebär också att spinndynamiken i trelagerstruktur-erna är kollektiv och två kopplade svängningar har studerats i min avhandling;akustiska och optiska svängningar som innebär att magnetiseringarna i de tvåFM skikten svänger i fas respektive i motfas.
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References
[1] J. Svoboda, Magnetic techniques for the treatment of materials. SpringerScience and Business Media, 2004.
[2] N. A. Spaldin, Magnetic materials: fundamentals and applications.Cambridge University Press, 2010.
[3] Y. Wei, Z. Zhai, K. Gunnarsson, and P. Svedlindh, ``A guided enquiry approachto introduce basic concepts concerning magnetic hysteresis to minimizestudent misconceptions,'' Eur. J. Phys., vol. 35, no. 6, p. 065015, 2014.
[4] J. B. Goodenough et al., Magnetism and the chemical bond, vol. 98.Interscience publishers New York, 1963.
[5] M. A. Ruderman and C. Kittel, ``Indirect exchange coupling of nuclearmagnetic moments by conduction electrons,'' Phys. Rev., vol. 96, no. 1, p. 99,1954.
[6] T. Kasuya, ``Electrical resistance of ferromagnetic metals,'' Progr. Theor.Phys., vol. 16, no. 1, pp. 58--63, 1956.
[7] K. Yosida, ``Magnetic properties of Cu-Mn alloys,'' Phys. Rev., vol. 106, no. 5,p. 893, 1957.
[8] J. Magnusson, C. Djurberg, P. Granberg, and P. Nordblad, ``A low fieldsuperconducting quantum interference device magnetometer for dynamicmeasurements,'' Rev. of Sci. Instrum., vol. 68, no. 10, pp. 3761--3765, 1997.
[9] T. Sheahen, Introduction to high-temperature superconductivity. SpringerScience and Business Media, 1994.
[10] S. Foner, ``Versatile and sensitive vibrating-sample magnetometer,'' Rew. Sci.Instrum., vol. 30, no. 7, pp. 548--557, 1959.
[11] H. Zijlstra, ``A vibrating reed magnetometer for microscopic particles,'' Rew.Sci. Instrum., vol. 41, no. 8, pp. 1241--1243, 1970.
[12] J. Kerr, ``On rotation of the plane of polarization by reflection from the pole ofa magnet,'' Philos. Mag., vol. 3, no. 5, p. 339, 1877.
[13] Z. Qiu and S. Bader, ``Surface magneto-optic kerr effect,'' Rev. Sci. Instrum.,vol. 71, no. 3, pp. 1243--1255, 2000.
[14] P. Carra, B. T. Thole, M. Altarelli, and X. Wang, ``X-ray circular dichroismand local magnetic fields,'' Phys. Rev. Lett., vol. 70, pp. 694--697, 1993.
[15] B. T. Thole, P. Carra, F. Sette, and G. van der Laan, ``X-ray circular dichroismas a probe of orbital magnetization,'' Phys. Rev. Lett., vol. 68, pp. 1943--1946,1992.
[16] R. Knut, New Materials for Spintronics: Electronic structure and magnetism.PhD thesis, Uppsala University, Sweden, 2012.
[17] J. Åkerman, ``Toward a universal memory,'' Science, vol. 308, no. 5721,pp. 508--510, 2005.
[18] J. G. Zhu, Y. Zheng, and G. A. Prinz, ``Ultrahigh density verticalmagnetoresistive random access memory,'' J. Appl. Phys., vol. 87, no. 9,pp. 6668--6673, 2000.
117
[19] D. J. Craik,Magnetism: principles and applications. Wiley-VCH, 2003.[20] K. H. Buschow, Concise encyclopedia of magnetic and superconducting
materials. Elsevier, 2005.[21] J. Nogués and I. K. Schuller, ``Exchange bias,'' J. Magn. Magn. Mater,,
vol. 192, no. 2, pp. 203--232, 1999.[22] J. H. Griffiths, ``Anomalous high-frequency resistance of ferromagnetic
metals,'' Nature, vol. 158, no. 4019, p. 670, 1946.[23] N. Mo and C. E. Patton, ``Perspective: Local ferromagnetic resonance
measurement techniques,'' Rev. Sci. Instrum., vol. 79, no. 4, p. 40901, 2008.[24] I. S. Maksymov and M. Kostylev, ``Broadband stripline ferromagnetic
resonance spectroscopy of ferromagnetic films, multilayers andnanostructures,'' Physica E, 2014.
[25] V. Arkad'yev, ``Absorption of electric waves in parallel wires,'' J. Russian Soc.Phys. Chem., vol. 44, p. 165, 1912.
[26] C. Kittel, ``On the theory of ferromagnetic resonance absorption,'' Phys. Rev.,vol. 73, no. 2, p. 155, 1948.
[27] H. Suhl, ``Ferromagnetic resonance in nickel ferrite between one and twokilomegacycles,'' Phys. Rev., vol. 97, pp. 555--557, 1955.
[28] M. R. Freeman, W. K. Hiebert, and A. Stankiewicz, ``Time-resolved scanningkerr microscopy of ferromagnetic structures,'' J. Appl. Phys., vol. 83, no. 11,pp. 6217--6222, 1998.
[29] Y. Acremann, C. H. Back, M. Buess, O. Portmann, A. Vaterlaus, D. Pescia,and H. Melchior, ``Imaging precessional motion of the magnetization vector,''Science, vol. 290, no. 5491, pp. 492--495, 2000.
[30] I. Neudecker,Magnetization dynamics of confined ferromagnetic systems.PhD thesis, University of Regensburg, Germany, 2006.
[31] G. Counil, J. V. Kim, T. Devolder, C. Chappert, K. Shigeto, and Y. Otani,``Spin wave contributions to the high-frequency magnetic response of thinfilms obtained with inductive methods,'' J. Appl. Phys., vol. 95, no. 10,pp. 5646--5652, 2004.
[32] B. Heinrich, J. F. Cochran, and R. Hasegawa, ``FMR linebroadening in metalsdue to two-magnon scattering,'' J. Appl. Phys., vol. 57, no. 8, pp. 3690--3692,1985.
[33] Z. Celinski and B. Heinrich, ``Ferromagnetic resonance linewidth of Feultrathin films grown on a bcc Cu substrate,'' J. Appl. Phys., vol. 70, no. 10,pp. 5935--5937, 1991.
[34] B. Odom, D. Hanneke, B. dUrso, and G. Gabrielse, ``New measurement of theelectron magnetic moment using a one-electron quantum cyclotron,'' Phys.Rev. Lett., vol. 97, no. 3, p. 030801, 2006.
[35] J. Pelzl, R. Meckenstock, D. Spoddig, F. Schreiber, J. Pflaum, and Z. Frait,``Spin--orbit-coupling effects on g-value and damping factor of theferromagnetic resonance in Co and Fe films,'' J. Phys.: Condens. Matter,vol. 15, no. 5, p. S451, 2003.
[36] R. Naik, C. Kota, J. S. Payson, and G. L. Dunifer, ``Ferromagnetic-resonancestudies of epitaxial Ni, Co, and Fe films grown on Cu (100)/Si (100),'' Phys.Rev. B, vol. 48, no. 2, p. 1008, 1993.
[37] J.-M. Beaujour, W. Chen, K. Krycka, C. C. Kao, J. Z. Sun, and A. D. Kent,
118
``Ferromagnetic resonance study of sputtered Co|Ni multilayers,'' Eur. Phys. J.B, vol. 59, no. 4, pp. 475--483, 2007.
[38] C. Kittel, ``On the gyromagnetic ratio and spectroscopic splitting factor offerromagnetic substances,'' Phys. Rev., vol. 76, no. 6, p. 743, 1949.
[39] T. Silva, C. Lee, T. Crawford, and C. Rogers, ``Inductive measurement ofultrafast magnetization dynamics in thin-film Permalloy,'' J. Appl. Phys.,vol. 85, no. 11, pp. 7849--7862, 1999.
[40] Y. Wei, R. Brucas, K. Gunnarsson, Z. Celinski, and P. Svedlindh, ``Positivecorrelation between coercivity and ferromagnetic resonance extrinsic linewidthin FeCoV/SiO2 films,'' Appl. Phys. Lett., vol. 104, no. 7, p. 072404, 2014.
[41] J. Smit and H. G. Beljers, ``Philips res.,'' Rep., vol. 10, p. 113, 1955.[42] S. V. Vonsovskii, Ferromagnetic resonance. Pergamon Press Oxford, 1966.[43] R. D. McMichael and P. Krivosik, ``Classical model of extrinsic ferromagnetic
resonance linewidth in ultrathin films,''Magn., IEEE Tran., vol. 40, no. 1,pp. 2--11, 2004.
[44] V. Siruguri and S. Kaul, ``Detailed ferromagnetic resonance study ofamorphous Fe-rich alloys. ii: critical behaviour and uniaxial anisotropy,'' J.Phys.: Condens. Matt., vol. 8, no. 25, p. 4567, 1996.
[45] S. Mizukami, Y. Ando, and T. Miyazaki, ``Ferromagnetic resonance linewidthfor NM/NiFe/NM films (NM= Cu, Ta, Pd and Pt),'' J. Magn. Magn. Mater.,vol. 226, pp. 1640--1642, 2001.
[46] G. Stenning, L. Shelford, S. Cavill, F. Hoffmann, M. Haertinger, T. Hesjedal,G. Woltersdorf, G. Bowden, S. Gregory, C. Back, et al., ``Magnetizationdynamics in an exchange-coupled NiFe/CoFe bilayer studied by x-ray detectedferromagnetic resonance,'' New J. Phys., vol. 17, no. 1, p. 013019, 2015.
[47] Y. Wei, S. Akansel, T. Thersleff, I. Harward, R. Brucas, M. Ranjbar, S. Jana,P. Lansaker, Y. Pogoryelov, R. K. Dumas, et al., ``Exponentially decayingmagnetic coupling in sputtered thin film FeNi/Cu/FeCo trilayers,'' Appl. Phys.Lett., vol. 106, no. 4, p. 042405, 2015.
[48] C. E. Patton, ``Linewidth and relaxation processes for the main resonance inthe spin-wave spectra of NiFe alloy films,'' J. Appl. Phys., vol. 39, no. 7,pp. 3060--3068, 1968.
[49] M. Van Kampen, C. Jozsa, J. Kohlhepp, P. LeClair, L. Lagae, W. De Jonge,and B. Koopmans, ``All-optical probe of coherent spin waves,'' Phys. Rev.Lett., vol. 88, no. 22, p. 227201, 2002.
[50] D. Twisselmann and R. McMichael, ``Intrinsic damping and intentionalferromagnetic resonance broadening in thin permalloy films,'' J. Appl. Phys.,vol. 93, no. 10, pp. 6903--6905, 2003.
[51] B. Kuanr, R. Camley, and Z. Celinski, ``Narrowing of the frequency-linewidthin structured magnetic strips: Experiment and theory,'' Appl. Phys. Lett.,vol. 87, no. 1, p. 012502, 2005.
[52] S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L. Schneider, P. Kabos,T. Silva, and J. P. Nibarger, ``Ferromagnetic resonance linewidth in metallicthin films: Comparison of measurement methods,'' J. Appl. Phys., vol. 99,no. 9, p. 093909, 2006.
[53] C. Scheck, L. Cheng, I. Barsukov, Z. Frait, and W. Bailey, ``Low relaxationrate in epitaxial vanadium-doped ultrathin iron films,'' Phys. Rev. Lett., vol. 98,
119
no. 11, p. 117601, 2007.[54] D. I. Mircea and T. Clinton, ``Near-field microwave probe for local
ferromagnetic resonance characterization,'' Appl. Phys. Lett., vol. 90, no. 14,p. 142504, 2007.
[55] D. Marko, K. Lenz, T. Strache, R. Kaltofen, and J. Fassbender, ``Measuringthe saturation magnetization in samples with unknown magnetic volume,''Magn., IEEE Trans., vol. 46, no. 6, pp. 1711--1714, 2010.
[56] C. Bilzer, T. Devolder, P. Crozat, C. Chappert, S. Cardoso, and P. Freitas,``Vector network analyzer ferromagnetic resonance of thin films on coplanarwaveguides: Comparison of different evaluation methods,'' J. Appl. Phys.,vol. 101, no. 7, p. 074505, 2007.
[57] I. Neudecker, G. Woltersdorf, B. Heinrich, T. Okuno, G. Gubbiotti, andC. Back, ``Comparison of frequency, field, and time domain ferromagneticresonance methods,'' J. Magn. Magn. Mater., vol. 307, no. 1, pp. 148--156,2006.
[58] Y. Wei, R. Brucas, K. Gunnarsson, I. Harward, Z. Celinski, and P. Svedlindh,``Static and dynamic magnetic properties of bcc FeCoV thin films on Si(100)substrates,'' J. Phys. D: Appl. Phys., vol. 46, no. 49, p. 495002, 2013.
[59] E. Mancini, F. Pressacco, M. Haertinger, E. Fullerton, T. Suzuki,G. Woltersdorf, and C. Back, ``Magnetic phase transition in iron--rhodium thinfilms probed by ferromagnetic resonance,'' J. Phys. D: Appl. Phys., vol. 46,no. 24, p. 245302, 2013.
[60] J. Fermin, A. Azevedo, F. De Aguiar, B. Li, and S. M. Rezende,``Ferromagnetic resonance linewidth and anisotropy dispersions in thin Fefilms,'' J. Appl. Phys., vol. 85, no. 10, pp. 7316--7320, 1999.
[61] A. Y. Dobin and R. Victora, ``Surface roughness induced extrinsic damping inthin magnetic films,'' Phys. Rev. Lett., vol. 92, no. 25, p. 257204, 2004.
[62] M. Bailleul, ``Shielding of the electromagnetic field of a coplanar waveguideby a metal film: Implications for broadband ferromagnetic resonancemeasurements,'' Appl. Phys. Lett., vol. 103, no. 19, p. 192405, 2013.
[63] D. C. Ralph and M. D. Stiles, ``Spin transfer torques,'' J. Magn. Magn. Mater.,vol. 320, no. 7, pp. 1190--1216, 2008.
[64] G. Bertotti, C. Serpico, I. D. Mayergoyz, A. Magni, M. dAquino, and R. Bonin,``Magnetization switching and microwave oscillations in nanomagnets drivenby spin-polarized currents,'' Phys. Rev. Lett., vol. 94, no. 12, p. 127206, 2005.
[65] A. D. Kent, B. Ozyilmaz, and E. Del Barco, ``Spin-transfer-inducedprecessional magnetization reversal,'' Appl. Phys. Lett., vol. 84, no. 19,pp. 3897--3899, 2004.
[66] T. L. Gilbert, ``A lagrangian formulation of the gyromagnetic equation of themagnetization field,'' Phys. Rev., vol. 100, p. 1243, 1955.
[67] T. L. Gilbert, ``A phenomenological theory of damping in ferromagneticmaterials,''Magn., IEEE Trans., vol. 40, no. 6, pp. 3443--3449, 2004.
[68] P. C. Andricacos and N. Robertson, ``Future directions in electroplatedmaterials for thin-film recording heads,'' IBM J. Res. Dev., vol. 42,pp. 671--680, Sept. 1998.
[69] B. D. Cullity and C. D. Graham, Introduction to magnetic materials.Wiley-IEEE Press, 2011.
120
[70] T. Sourmail, ``Near equiatomic feco alloys: Constitution, mechanical andmagnetic properties,'' Prog. Mater. Sci., vol. 50, no. 7, pp. 816--880, 2005.
[71] R. V. Major and C. M. Orrock, ``High saturation ternary cobalt-iron basaltalloys,'' Magn., IEEE Trans., vol. 24, no. 2, pp. 1856--1858, 1988.
[72] D. Kim, D. Y. Park, B. Y. Yoo, P. T. A. Sumodjo, and N. V. Myung, ``Magneticproperties of nanocrystalline iron group thin film alloys electrodeposited fromsulfate and chloride baths,'' Electrochim. Acta, vol. 48, no. 7, pp. 819--830,2003.
[73] C. L. Platt, A. E. Berkowitz, D. J. Smith, and M. R. McCartney, ``Correlationof coercivity and microstructure of thin CoFe films,'' J. Appl. Phys., vol. 88,no. 4, pp. 2058--2062, 2000.
[74] H. S. Jung, W. D. Doyle, and S. Matsunuma, ``Influence of underlayers on thesoft properties of high magnetization FeCo films,'' J. Appl. Phys., vol. 93,no. 10, pp. 6462--6464, 2003.
[75] H. Du, G. Pan, and R. Buckley, ``Effect of substrate bias and magneticannealing on soft magnetic properties of rf-sputtered Fe49Co49V2 films,'' J.Appl. Phys., vol. 91, no. 10, pp. 7827--7829, 2002.
[76] H. S. Jung, W. D. Doyle, J. E. Wittig, J. F. Al-Sharab, and J. Bentley, ``Softanisotropic high magnetization Cu/FeCo films,'' Appl. Phys. Lett., vol. 81,no. 13, pp. 2415--2417, 2002.
[77] R. Gupta, K. H. Han, K. P. Lieb, G. A. Muller, P. Schaaf, and K. Zhang,``Influence of ion implantation on the magnetic properties of thin FeCo films,''J. Appl. Phys., vol. 97, no. 7, pp. 073911--073911, 2005.
[78] R. Gupta, K. P. Lieb, G. A. Muller, P. Schaff, and K. Zhang, ``Xenon-ioninduced magnetic and structural modifications of ferromagnetic alloys,''Hyperfine interactions, vol. 160, no. 1-4, pp. 107--121, 2005.
[79] K. Zhang, K. P. Lieb, M. Marszalek, V. Milinovic, and V. Tokman, ``Ion beammixing of Co/Fe multilayers: Magnetic and structural properties,'' Thin SolidFilms, vol. 515, no. 2, pp. 700--704, 2006.
[80] G. V. Stepanov, A. V. Chertovich, and E. Y. Kramarenko,``Magnetorheological and deformation properties of magnetically controlledelastomers with hard magnetic filler,'' J. Magn. Magn. Mater., vol. 324, no. 21,pp. 3448--3451, 2012.
[81] H. C. Wang, T. Zhao, M. Li, and K. Zhang, ``Tmr device with lowmagnetorestriction free layer,'' Apr. 11 2012. the U.S. Patent App. 13/444,497.
[82] F. Schreiber, J. Pflaum, Z. Frait, T. H. Mühge, and J. Pelzl, ``Gilbert dampingand g-factor in FexCo1−x alloy films,'' Solid State Commun., vol. 93, no. 12,pp. 965--968, 1995.
[83] C. Bilzer, T. Devolder, J. V. Kim, G. Counil, C. Chappert, S. Cardoso, and P. P.Freitas, ``Study of the dynamic magnetic properties of soft CoFeB films,'' J.Appl. Phys., vol. 100, no. 5, pp. 053903--053903, 2006.
[84] I. Ohnuma, H. Enoki, O. Ikeda, R. Kainuma, H. Ohtani, B. Sundman, andK. Ishida, ``Phase equilibria in the Fe--Co binary system,'' Acta Mater., vol. 50,no. 2, pp. 379--393, 2002.
[85] G. B. Chon, K. Shinoda, S. Suzuki, and B. Jeyadevan, ``Order-disordertransformation in Fe50Co50 particles synthesized by polyol process,''Mater.Trans., vol. 51, no. 04, pp. 707--711, 2010.
121
[86] N. S. Stoloff and R. G. Davies, ``The plastic deformation of ordered feco andFe3Al alloys,'' Acta Metal., vol. 12, no. 5, pp. 473--485, 1964.
[87] J. Rantschler, Y. Ding, S. C. Byeon, and C. Alexander, ``Microstructure anddamping in FeTiN and CoFe films,'' J. Appl. Phys., vol. 93, no. 10,pp. 6671--6673, 2003.
[88] S. W. Yuan and H. N. Bertram, ``Inhomogeneities and coercivity of softpermalloy thin films,''Magn., IEEE Trans., vol. 28, no. 5, pp. 2916--2918,1992.
[89] M. Oogane, T. Wakitani, S. Yakata, R. Yilgin, Y. Ando, A. Sakuma, andT. Miyazaki, ``Magnetic damping in ferromagnetic thin films,'' Jpn. J. Appl.Phys., vol. 45, p. 3889, 2006.
[90] R. J. Elliott, ``Spin-orbit coupling in band theory-character tables for some,''Phys. Rev., vol. 96, pp. 280--287, 1954.
[91] V. Kambersky, ``On the Landau-Lifshitz relaxation in ferromagnetic metals,''Can. J. Phys., vol. 48, no. 24, pp. 2906--2911, 1970.
[92] F. Beuneu and P. Monod, ``The Elliott relation in pure metals,'' Phys. Rev. B,vol. 18, no. 6, p. 2422, 1978.
[93] T. Kimura and M. Hara, ``Nonvolatile multiple-valued memory device usinglateral spin valve,'' Appl. Phys. Lett., vol. 97, no. 18, p. 182501, 2010.
[94] R. L. Stamps, S. Breitkreutz, J. Åkerman, A. V. Chumak, Y. Otani, G. E.Bauer, J.-U. Thiele, M. Bowen, S. A. Majetich, M. Kläui, et al., ``The 2014magnetism roadmap,'' J. Phys. D: Appl. Phys., vol. 47, no. 33, p. 333001, 2014.
[95] S. M. Thompson, ``The discovery, development and future of gmr: The NobelPrize 2007,'' J. Phys. D: Appl. Phys., vol. 41, no. 9, p. 093001, 2008.
[96] A. Layadi, ``Ferromagnetic resonance modes in single and coupled layers withoblique anisotropy axis,'' Phys. Rev. B, vol. 63, no. 17, p. 174410, 2001.
[97] C. Bezerra, C. Chesman, E. Albuquerque, and A. Azevedo, ``Effects of themagneto-crystalline anisotropy on the magnetic properties of Fe/Cr/Fe (110)trilayer,'' Eur. Phys. J. B, vol. 39, no. 4, pp. 527--533, 2004.
[98] M. Matczak, B. Szymański, M. Urbaniak, M. Nowicki, H. Głowiński,P. Kuświk, M. Schmidt, J. Aleksiejew, J. Dubowik, and F. Stobiecki,``Antiferromagnetic magnetostatic coupling in Co/Au/Co films withperpendicular anisotropy,'' J. Appl. Phys., vol. 114, no. 9, p. 093911, 2013.
[99] P. Grünberg, R. Schreiber, Y. Pang, M. Brodsky, and H. Sowers, ``Layeredmagnetic structures: evidence for antiferromagnetic coupling of Fe layersacross Cr interlayers,'' Phys. Rev. Lett., vol. 57, no. 19, p. 2442, 1986.
[100] S. S. Parkin, N. More, and K. P. Roche, ``Oscillations in exchange couplingand magnetoresistance in metallic superlattice structures: Co/Ru, Co/Cr, andFe/Cr,'' Phys. Rev. Lett., vol. 64, no. 19, p. 2304, 1990.
[101] M. Belmeguenai, T. Martin, G. Woltersdorf, M. Maier, and G. Bayreuther,``Frequency-and time-domain investigation of the dynamic properties ofinterlayer-exchange-coupled Ni81Fe19/ Ru/Ni81Fe19 thin films,'' Phys. Rev. B,vol. 76, no. 10, p. 104414, 2007.
[102] J. Lindner, Z. Kollonitsch, E. Kosubek, M. Farle, and K. Baberschke, ``In situdetection of two ferromagnetic resonance modes in coupled Ni/Cu/Co/Cu(001) trilayer structures,'' Phys. Rev. B, vol. 63, no. 9, p. 094413, 2001.
[103] Z. Zhang, L. Zhou, P. E. Wigen, and K. Ounadjela, ``Angular dependence of
122
ferromagnetic resonance in exchange-coupled Co/Ru/Co trilayer structures,''Phys. Rev. B, vol. 50, no. 9, p. 6094, 1994.
[104] B. Heinrich and J. A. Bland, Ultrathin Magnetic Structures II, vol. 3. Springer,1994.
[105] J. B. Youssef and A. Layadi, ``Ferromagnetic resonance study ofPermalloy/Cu/Co/NiO spin valve system,'' J. Appl. Phys., vol. 108, no. 5,p. 053913, 2010.
[106] S. M. Rezende, C. Chesman, M. A. Lucena, A. Azevedo, F. M. De Aguiar, andS. S. Parkin, ``Studies of coupled metallic magnetic thin-film trilayers,'' J.Appl. Phys., vol. 84, no. 2, pp. 958--972, 1998.
[107] A. Layadi, ``Effect of biquadratic coupling and in-plane anisotropy on theresonance modes of a trilayer system,'' Phys. Rev. B, vol. 65, no. 10, p. 104422,2002.
[108] J. F. Cochran, J. Rudd, W. B. Muir, B. Heinrich, and Z. Celinski, ``Brillouinlight-scattering experiments on exchange-coupled ultrathin bilayers of ironseparated by epitaxial copper (001),'' Phys. Rev. B, vol. 42, no. 1, p. 508, 1990.
[109] Y. Wei, S. Jana, R. Brucas, Y. Pogoryelov, M. Ranjbar, R. K. Dumas,P. Warnicke, J. Åkerman, D. A. Arena, O. Karis, et al., ``Magnetic coupling inasymmetric FeCoV/Ru/FeNi trilayers,'' J. Appl. Phys., vol. 115, no. 17,p. 17D129, 2014.
[110] K. Lenz, T. Toliński, J. Lindner, E. Kosubek, and K. Baberschke, ``Evidenceof spin-pumping effect in the ferromagnetic resonance of coupled trilayers,''Phys. Rev. B, vol. 69, no. 14, p. 144422, 2004.
[111] D. A. Arena, E. Vescovo, C. C. Kao, Y. Guan, and W. E. Bailey, ``Weaklycoupled motion of individual layers in ferromagnetic resonance,'' Phys. Rev. B,vol. 74, no. 6, p. 064409, 2006.
[112] Z. Zhang, L. Zhou, P. E. Wigen, and K. Ounadjela, ``Using ferromagneticresonance as a sensitive method to study temperature dependence of interlayerexchange coupling,'' Phys. Rev. Lett., vol. 73, no. 2, p. 336, 1994.
[113] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton,S. Von Molnar, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger,``Spintronics: A spin-based electronics vision for the future,'' Science,vol. 294, no. 5546, pp. 1488--1495, 2001.
[114] B. N. Engel, J. Akerman, B. Butcher, R. W. Dave, M. DeHerrera, M. Durlam,G. Grynkewich, J. Janesky, S. V. Pietambaram, N. D. Rizzo, et al., ``A 4-Mbtoggle MRAM based on a novel bit and switching method,''Magn., IEEETrans., vol. 41, no. 1, pp. 132--136, 2005.
[115] E. Klokholm and J. Aboaf, ``The saturation magnetostriction of permalloyfilms,'' J. Appl. Phys., vol. 52, no. 3, pp. 2474--2476, 1981.
[116] B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas, R. Urban, and G. E.Bauer, ``Dynamic exchange coupling in magnetic bilayers,'' Phys. Rev. Lett.,vol. 90, no. 18, p. 187601, 2003.
[117] J. Fassbender, F. Nörtemann, R. L. Stamps, R. E. Camley, B. Hillebrands,G. Güntherodt, and S. S. Parkin, ``Oscillatory interlayer exchange coupling ofCo/Ru multilayers investigated by brillouin light scattering,'' Phys. Rev. B,vol. 46, no. 9, p. 5810, 1992.
[118] I. Harward, T. OKeevan, A. Hutchison, V. Zagorodnii, and Z. Celinski, ``A
123
broadband ferromagnetic resonance spectrometer to measure thin films up to70 GHz,'' Rev. of Sci. Instrum., vol. 82, no. 9, p. 095115, 2011.
[119] P. J. Bloemen, E. A. van Alphen, and W. J. de Jonge, ``Interlayer coupling inCo/Pt/Co trilayers investigated by ferromagnetic resonance,'' J. Magn. Magn.Mater., vol. 104, pp. 1775--1776, 1992.
[120] P. Bruno, ``Theory of interlayer magnetic coupling,'' Phys. Rev. B, vol. 52,no. 1, p. 411, 1995.
[121] W. Weber, R. Allenspach, and A. Bischof, ``Exchange coupling across Cu(100): A high-precision study,'' Europhys. Lett., vol. 31, no. 8, p. 491, 1995.
[122] S. S. Parkin, ``Dramatic enhancement of interlayer exchange coupling andgiant magnetoresistance in Ni81Fe19/Cu multilayers by addition of thin Cointerface layers,'' Appl. Phys. Lett., vol. 61, no. 11, pp. 1358--1360, 1992.
[123] L. C. Nagamine, J. Geshev, T. Menegotto, A. A. Fernandes, A. Biondo, andE. B. Saitovitch, ``Ferromagnetic resonance and magnetization studies inexchange-coupled NiFe/Cu/NiFe structures,'' J. Magn. Magn. Mater., vol. 288,pp. 205--213, 2005.
[124] D. E. Gonzalez-Chavez, R. Dutra, W. O. Rosa, T. L. Marcondes, A. Mello, andR. L. Sommer, ``Interlayer coupling in spin valves studied by broadbandferromagnetic resonance,'' Phys. Rev. B, vol. 88, no. 10, p. 104431, 2013.
[125] B. Heinrich, Z. Celinski, J. F. Cochran, W. B. Muir, J. Rudd, Q. M. Zhong,A. S. Arrott, K. Myrtle, and J. Kirschner, ``Ferromagnetic andantiferromagnetic exchange coupling in bcc epitaxial ultrathin Fe (001)/Cu(001) Fe (001) trilayers,'' Phys. Rev. Lett., vol. 64, no. 6, p. 673, 1990.
[126] Z. T. Diao, S. Goto, K. Meguro, S. Tsunashima, and M. Jimbo, ``Role of thebuffer layers in determining the antiferromagnetic coupling andmagnetoresistance of NiFeCo/Cu superlattices,'' J. Appl. Phys., vol. 81, no. 5,pp. 2327--2335, 1997.
[127] Z. Celinski and B. Heinrich, ``Exchange coupling in Fe/Cu, Pd, Ag, Au/Fetrilayers,'' J. Magn. Magn. Mater,, vol. 99, no. 1, pp. L25--L30, 1991.
[128] D. Lucic, N. Cramer, R. E. Camley, Z. Celinski, P. Kabos, and C. E. Patton,``Influence of the interface morphology on the exchange coupling in Fe/Pd/Fe(001) structures,'' J. Magn. Magn. Mater,, vol. 198, pp. 418--420, 1999.
[129] J. F. Bobo, H. Kikuchi, O. Redon, E. Snoeck, M. Piecuch, and R. L. White,``Pinholes in antiferromagnetically coupled multilayers: Effects on hysteresisloops and relation to biquadratic exchange,'' Phys. Rev. B, vol. 60, no. 6,p. 4131, 1999.
[130] D. J. Larson, P. H. Clifton, N. Tabat, A. Cerezo, A. K. Petford-Long, R. L.Martens, and T. F. Kelly, ``Atomic-scale analysis of CoFe/Cu and CoFe/NiFeinterfaces,'' Appl. Phys. Lett., vol. 77, no. 5, pp. 726--728, 2000.
[131] H. Kikuchi, J. F. Bobo, and R. L. White, ``The effect of pinholes on theproperties of Co/Cu multilayers,''Magn., IEEE Trans., vol. 33, no. 5,pp. 3583--3585, 1997.
[132] H. Abdi and L. J. Williams, ``Principal component analysis,''WileyInterdisciplinary Reviews: Computational Statistics, vol. 2, no. 4,pp. 433--459, 2010.
[133] J. J. De Vries, J. Kohlhepp, F. J. Den Broeder, R. Coehoorn, R. Jungblut,A. Reinders, and W. J. De Jonge, ``Exponential dependence of the interlayer
124
exchange coupling on the spacer thickness in MBE-grown Fe/SiFe/Fesandwiches,'' Phys. Rev. Lett., vol. 78, no. 15, p. 3023, 1997.
[134] R. R. Gareev, D. E. Burgler, M. Buchmeier, R. Schreiber, and P. Grünberg,``Very strong interlayer exchange coupling in epitaxial Fe/Fe1−xSix/Fetrilayers (x = 0.4− 1.0),'' J. Magn. Magn. Mater., vol. 240, no. 1,pp. 235--237, 2002.
[135] M. Cooke, L. Wang, R. Watts, R. Zuberek, G. Heydon, W. Rainforth, andG. Gehring, ``The effect of thermal treatment, composition and substrate onthe texture and magnetic properties of feco thin films,'' Journal of Physics D:Applied Physics, vol. 33, no. 12, p. 1450, 2000.
[136] M. Kostylev, ``In private communication,'' Private Communication, 2015.[137] I. S. Maksymov, Z. Zhang, C. Chang, and M. Kostylev, ``Strong eddy--current
shielding of ferromagnetic resonance response in sub--skin--depth--thickconducting magnetic multilayers,'' IEEE Magn. Lett., vol. 5, no. 1, p. 3500104,2014.
125
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