# Dynamic Magnetics:Microwave Interactions with Magnetic Structures on both a Macroscopic and Microscopic Scale

код для вставкиСкачатьDynamic Magnetics: Microwave Interactions with Magnetic Structures on both a Macroscopic and Microscopic Scale by Timothy J. Fal B.S., University of Colorado at Colorado Springs, May 17, 2004 M. S., University of Colorado at Colorado Springs, December 16, 2006 A dissertation submitted to the Graduate Faculty of the University of Colorado at Colorado Springs in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Applied Science Department of Physics 2011 UMI Number: 3450521 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI 3450521 Copyright 2011 by ProQuest LLC. All rights reserved. This edition of the work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106-1346 ii © Copyright By Timothy J. Fal and Robert E. Camley 2011 All Rights Reserved iii This dissertation for Doctor of Philosophy in Applied Science degree by Timothy J. Fal has been approved for the Department of Physics by ___________________________________________ Robert E. Camley, Chair ___________________________________________ Zbigniew Celinski ___________________________________________ Marek Grabowski ___________________________________________ Anatoliy Glushchenko ___________________________________________ Anatoliy Pinchuk ________________________ Date iv Fal, Timothy J. (Doctor of Philosophy in Applied Science) Dynamic Magnetics: Microwave Interactions with Magnetic Structures on both a Macroscopic and Microscopic Scale Dissertation directed by Professor Robert E. Camley Microwave interactions with magnetic materials are an important topic with research being done for signal processing devices and magnetic storage. There is an interest in how magnetic materials affect electromagnetic signals and in turn, how the signals affect the magnetic material. In this work, we present analytical and numerical calculations for electromagnetic waves interacting with thin-film magnetic materials as well as magnetic nano structures. The first part of this dissertation focuses on how a microwave signal is affected by a magnetic material. These results can be applied to signal processing devices such as notch filters, band pass filters, phase shifters, and isolators. The materials considered in the analysis are iron, M-type barium hexagonal ferrite, permalloy, and yttrium iron garnet. The second part of this dissertation focuses on how a magnetic material is affected by a microwave signal. Micromagnetic simulations are used to track the magnetization direction of a magnetic nano structures as it evolves with time in an oscillating magnetic field. The results demonstrate the possibility of microwave assisted magnetic reversal for magnetic nano squares and bilayer exchange spring structures. I dedicate this work to my parents, Liz and John Fal, and to my wife Holly. vi AKNOWLEDGEMENTS I would like to thank my graduate advisor Bob Camley for giving me so many opportunities, a lot of insight, and his tireless work for this graduate program. I would also like to thank Zbigniew Celinski for all of his support of the graduate program, fellow students, and myself. Thank you to Karen Livesy and Veera Venugopal for collaborating with me on projects and giving me sound, useful advice. Thank you to Sam Milazzo for a significant part of my undergraduate physics education. Thank you as well to Marek Grabowski who instructed me in a majority of my graduate level classes. Much of this work was supported by the DOA Grant No. W911NF-04-1-0247, the US Army Research Office Grant No. DAAD19-02-1-0174, and the NSF Grant No. DMR 0907063. Finally, I would like to thank Evan McHugh for introducing me to physics many years ago. vii TABLE OF CONTENTS CHAPTER I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 II. BACKGROUND: Dynamic Magnetic Permeability . . . . . . . . . . . . . . 8 The Landau-Lifshitz Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Ferromagnetic Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 The Dynamic Permeability Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Permeability with Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Demagnetizing Fields and Thin Film FMR . . . . . . . . . . . . . . . . . . . 20 III. Thin Films Structures and Boundary Condition Calculations . . . . . . 24 Calculation for a Small 3-Layer Ferromagnetic Notch filter . . . . . 24 Transmission Results for the small 3-Layer Notch Filter . . . . . . . . 30 Reflections and Insertion Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 IV. Devices with Hexagonal-ferrites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 The Calculation for the Hexagonal Ferrite waveguide . . . . . . . . . . 43 Transmission Results for the Hexagonal Ferrite Waveguide . . . . . 50 Using the waveguide as a Phaseshifter . . . . . . . . . . . . . . . . . . . . . . . 60 Experimental Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 viii V. Non-reciprocal Ferromagentic Devices using Attenuated Total Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Surface Polaritons and Three-layer ATR . . . . . . . . . . . . . . . . . . . . . 72 Four-layer ATR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 VI. BACKGROUND: Micromagnetic Simulations . . . . . . . . . . . . . . . . . . . 95 The Finite Difference Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 The Anisotropy Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 The Effective Exchange Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 The Dipole Field and the FFT Method . . . . . . . . . . . . . . . . . . . . . . .100 VII. Accessing Multiple States in Magnetic Squares with Microwaves . . .107 The Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .108 The Multi-State Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .110 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .115 VIII. Microwave Assisted Switching in an Exchange Spring Bilayer. . . . . . .116 Theoretical Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .118 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .120 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .129 ix IX. SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .130 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .132 APPENDICES Appendix- A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .139 x TABLES Table 5.1 Magnetic Layer Thickness and Return Loss Differences . . . . . . . . 92 xi FIGURES Figure 1.1 Basic microstrip geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Oersted fields around a microstrip . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Schloemann notch filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Simple bit patterened media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1 Simple macrospin precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Small angle precession of magnetization . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Small angle precession with an oscillating magnetic field . . . . . . . . 14 2.4 Macrospin precession with the direction of damping . . . . . . . . . . . 17 2.5 Real parts of the permeability tensor elements . . . . . . . . . . . . . . . . 19 2.6 Imaginary parts of the permeability tensor elements . . . . . . . . . . . . 20 2.7 Thin film geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.1 Three layer notch filter geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Transmission loss vs. frequency for different positions of an Fe layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3 Maximum transmission loss vs. magnetic film thickness . . . . . . . . 32 3.4 Rejection bandwidth vs. magnetic layer position . . . . . . . . . . . . . . . 34 3.5 Cross-section of a two region waveguide . . . . . . . . . . . . . . . . . . . . . . 36 3.6 Theoretical return loss due to insertion vs. frequency . . . . . . . . . . . 38 3.7 Experimental loss due to insertion vs. frequency . . . . . . . . . . . . . . . 39 4.1 Notch filter geometry with BaM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 xii 4.2 Transmission vs. frequency for 0.5 μm of BaM with different fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.3 Transmission vs. frequency for 4.0 μm of BaM with different fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.4 Resonant frequency vs. BaM filling factor . . . . . . . . . . . . . . . . . . . . 54 4.5 Resonant frequency vs. linewidth . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.6 Transmission vs. frequency for 0.5 μm of out-of-plane permalloy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.7 Transmission vs. frequency for 0.5 μm of in-plane BaM . . . . . . . . . 60 4.8 Phase difference vs. applied field for 0.8 μm of BaM for different frequencies 4.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Phase difference vs. applied field with highly attenuated results removed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.10 Phase difference vs. applied field for 1.6 μm of BaM with highly attenuated results removed . . . . . . . . . . . . . . . . . . . . . . 63 4.11 Phase difference vs. applied field for 0.8 μm of BaM for higher frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.12 Phase difference vs. applied field for 1.6 μm of BaM for higher frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.13 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.14 Theoretical results for experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.1 Three-layer ATR geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.2 Four-layer ATR geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 xiii 5.3 Polariton Dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.4 Return Loss vs. frequency for a three-layer ATR geometry using YIG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.5 Return Loss map of frequency vs. wavevector for a three-layer ATR geometry using YIG . . . . . . . . . . . . . . . . . . . 83 5.6 Four-layer ATR geometry comparison with the three-layer ATR geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.7 Return Loss map of frequency vs. wavevector for a four-layer ATR geometry using BaM. . . . . . . . . . . . . . . . . . . . . 86 5.8 Return Loss vs. frequency for a four-layer ATR geometry using 0.1 cm of BaM . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.9 Return Loss vs. frequency for a four-layer ATR geometry, comparing different applied fields . . . . . . . . . . . . . 88 5.10 Return Loss map of frequency vs. wavevector For a four-layer ATR geometry using 0.01 cm of BaM . . . . . . . . . . 89 5.11 Return Loss vs. frequency for a four-layer ATR geometry using 0.01 cm of BaM . . . . . . . . . . . . . . . . . . . . . . . . 90 5.12 Return Loss vs. frequency for a four-layer ATR geometry using 0.001 cm of BaM with bandpass behavior . . . 91 5.13 A typical ATR geometry and lateral ATR waveguide . . . . . . . . . . . 93 6.1 Magnetic structure divided into cells for the finite difference method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 xiv 6.2 A row of spins with an effective exchange length between two selected spins . . . . . . . . . . . . . . . . . . . 99 6.3 Average magnetic field in one volume created by another volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101 6.4 Average dipolar field in one cell due to all other cells . . . . . . . . . . .102 6.5 Time comparison between the double iteration method and the FFT method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .106 7.1 Nano-square geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .108 7.2 The three states of the nano-square . . . . . . . . . . . . . . . . . . . . . . . . .110 7.3 Time evolution of the z-component of the magnetization . . . . . . . .112 7.4 Square state based on driving field vs. applied field . . . . . . . . . . . .114 8.1 The geometry of the exchange spring bilayer . . . . . . . . . . . . . . . . . .118 8.2 Hysteresis loops for a structure made of high anisotropy material and an the exchange spring bilayer structure . . . . . . . . . .121 8.3 Frequency spectrum for standing spin wave modes across the bilyaer structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .122 8.4 Spatial profile for standing spin wave modes . . . . . . . . . . . . . . . . . .123 8.5 Time evolution of the z-component of magnetization in the exchange spring bilayer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .124 8.6 Final state of bilayer structure based on the strength of the driving field . . . . . . . . . . . . . . . . . . . . . . . . . . . . .125 8.7 Hysteresis loops for the bilayer structure without a driving field and with a driving field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .127 CHAPTER I INTRODUCTION Ferromagnetic materials have played a large role in developing solid-state technology in the last 50 years. There continues to be a great deal of interest in ferromagnetism for technology today, particularly in layered and patterned structures that can be grown with thicknesses on the order of microns. Communication devices based on these structures are being developed to work in the microwave range of 1-100 GHz and promise to be more efficient and compact than the large, bulky devices currently implemented.1-11 Growing layers of magnetic materials for potential devices can be time consuming and requires costly equipment.12 To reduce the time and cost it is beneficial to explore analytic and numerical results for the dynamics of these materials. In this dissertation we present analytical and numeric results for the interactions between ferromagnetic materials and electromagnetic waves in the microwave frequency range. We consider a variety of structures that may be implemented in devices. The focus of the first part of this dissertation is on how magnetic materials change microwaves. The focus for the second part is on how microwaves can change magnetic materials. We will consider different microwave signal processing devices, such as notch filters, band pass filters, phase shifters, and isolators, as well as how magnetic storage and bit reversals are affected by microwaves. Chapters III, IV, V, VII, and VIII present new results on these topics and have been submitted for review and accepted in scientific journals. The signal processing devices discussed in Chapters III and IV cam be constructed from a variety of geometries, including microstrips, striplines and co-planar 2 waveguides. The microstrip geometry is a thin strip of conductive material, typically mounted on a dielectric with a ground plane below that. The top strip has an oscillatory current running through it, leading to electromagnetic fields in the region near the strip. The oscillating current, possibly being brought into the strip from some kind of receiver like an antenna, creates the signal of interest. The signal comes in through one port and leaves through a second port, seen in Fig. 1.1. Figure 1.1: Basic microstrip geometry. One can obtain an intuitive understanding of the microstrip in the following way: the current in the strip (signal line) creates Oersted fields around it that oscillate at the same frequency as the current, as shown in Fig. 1.2. 3 Figure 1.2: Oersted fields oscillating around a conductive strip. The oscillating magnetic field, in turn, creates oscillating electric fields in an electromagnetic wave. A magnetic material in close proximity to the strip can potentially modify the signal. Chapters III and IV explore the interaction of a magnetic material with a microwave that could be generated by a microstrip. Schloeman et. al. created some of the first microstrip filters with a magnetic layer next to the signal line.2 The dimensions of that device were on the order of 3 cm in length and a few hundred microns in thickness. The results of those experiments showed that the incident signals were absorbed at frequencies matching the resonant frequencies of the magnetic material. Since then the microstrip geometry using a magnetic thin film has been improved and made more efficient.7-11 4 Figure 1.3: Notch filter device from Schloemann et at [2]. A microstrip of 2 μm of Al or Au with 0.1 μm of Fe directly below it. An applied magnetic field orients the magnetization of the magnetic material and sets the magnetic resonance frequency. Some of the types of signal processing devices that can be made from microstip structures are notch filters, bandpass filters, and phase shifters. A notch filter, or bandstop filter, is a device that blocks a small range of frequencies from passing through it. In other words, it makes a notch in the transmission spectrum. A bandpass filter works in the opposite way. The bandpass filter allows a small range, or band, of frequencies to pass through it, blocking off all other frequencies. A phase shifter is a device that will shift the phase of a signal. By modifying something about the device, for example an applied H-field, the phase of the signal is altered. The last type of signal processing device we will consider is the isolator. An isolator is a device that will allow a signal to pass in one direction through the device, but not in the opposite direction. 5 In Chapter III we attempt to find a way to improve the notch filter. We look for an optimal placement of the magnetic layer to get the greatest signal damping in the notch. Chapter IV examines notch filters and phase shifters using M-type barium hexagonal ferrite (BaM). Hexagonal ferrite allows for the processing of higher frequencies than the materials previously used due to its high crystalline anisotropy. Chapter V proposes a possible modified microstip device that would use attenuated total reflection to achieve notch filter, bandpass filter, and isolator behavior. Chapter V concludes the investigation of the topic on how a microwave is changed by a magnetic material and the remainder of the dissertation considers the inverse problem - whether a microwave signal can be used to alter the magnetic material. These remaining chapters deal with magnetic reversals and patterned magnetic nanostructures. Reversing magnetic orientations has been of great interest for many years13-15, particularly in data storage. Because of new film growth and lithography techniques, bit patterned media shows promise for being a method to create high density magnetic storage media.21-26A bit patterned medium is one where every single bit in the device has a well defined magnetic structure. Current techniques can make magnetic squares or dots with side lengths or diameters on the order of 20-50 nm in size. With further reductions in size, this could produce magnetic storage densities of around 2000 Giga-bits per square inch, which is four times the areal densities of most hard drives made today. This neglects the stacking of disks, which effectively increases areal density by a multiple of how many disks are stacked, one on top of the other. 6 Figure 1.4: Diagram of a bit patterned media. An array of small magnetic squares. There are thermal stability issues with a bit’s magnetization as it gets smaller, particularly for structures below 50 nm in scale. In order to make these structures more stable, materials with high crystalline magnetic anisotropy are being considered. Unfortunately, the large anisotropy field means a very large magnetic field is normally required to reverse the magnetization direction. Therefore, if such a material is used, there is a need to find new ways to temporarily reduce the magnetic stability. One possible method to do this is to use oscillating magnetic fields in order to destabilize a magnetic state to aid magnetization switching. This technique is called microwave assisted magnetic reversal (MAMR).16-20 In order to evaluate such a process, micromagnetic computer simulations can be used. Chapter VI outlines the principles used in these simulations and Chapters VII and VIII explore the use of microwaves to aid in the switching of magnetic orientations. 7 Chapter VII examines magnetic nano-sized squares with three possible stable magnetic states. The square is subjected to microwave oscillating magnetic fields at different strengths and frequencies, and this can determine the which of the states remain stable. . Chapter VIII focuses on bi-layer structures. Each layer has a different crystalline anisotropy, one with a strong anisotropy the other with a weak one. The simulations in Chapter VIII are concerned with reducing the strength of magnetic fields used to switch the magnetic structure by applying an additional oscillating magnetic field. Before looking at applications and results, Chapter II reviews the physics involved with in dynamic calculations of magnetic materials. The concepts in that chapter will be used in all the following chapters. 8 CHAPTER II BACKGROUND: Dynamic Magnetic Permeability In order to make the reader more familiar with microwave interactions with ferromagnetic materials, we present some of the theory involved.3,27-30 The magnetic properties of a material depends on the magnetic moments of each individual atom in the material. In general, there can be both spin and orbital angular momentum contributions to the magnetic moment. However, in the common elements of Fe, Ni and Co, the spin contribution is dominant. In ferromagnetic materials the magnetic moments of the individual atoms are nearly parallel to each other. This alignment is a quantum mechanical effect produced by the overlapping electron wave functions of the atoms and the Pauli Exclusion principle. This is called the exchange interaction and will become important in the following chapters, but for now we will simply say that it aligns the atomic magnetic moments in the material. The volumetric density of the magnetic moments is the material’s magnetization M. The Landau-Lifshitz Equation The magnetic field, B, inside a magnetic material is the result of two terms. In cgs units, the terms are: B H 4 M (2.1). 9 There is a contribution from a field H and the magnetization of the material M . The magnetization of the material is influenced by H and can, in some limits, be represented as a linear function of the H-field through the magnetic susceptibility M H In general, the susceptibility (2.2). is a tensor. When Eq. (2.2) is put into Eq. (2.1), we have the expression with the permeability of the material: B H (2.3) where μ is the permeability of the material 1 4 (2.4). We are particularly interested in how a magnetic material would respond to an oscillating field h . Such an interaction can be contained in Eq. (2.4). We can find a dynamic susceptibility if we have a microscopic model of how an individual magnetic moment moves in response to an H-field. We note that the torque on an individual moment m is given by m H (2.5) The torque, in turn, is given by the time derivative of the angular momentum. For a single electron with angular momentum based on spin only, the angular momentum is given by S . The magnetic moment can be related to the angular momentum through the Bohr magneton B 10 m g e S 2mc g( / ) S B (2.6) where the Bohr magneton μB, is given by e 2m e c B (2.7) and me is the mass of an electron, the speed of light is c, and the charge of an electron is e. The parameter g arises from relativistic effects and is typically about 2 for most transition metal ferromagnets. Using Eqs (2.5) – (2.7) we arrive at the Landau-Lifshitz equation (LL equation) for the motion of a magnetic moment in an applied field dm dt Here g B m H (2.8). / and the negative sign in the electron charge is made explicit in Eq. (2.8) by using the absolute value and the minus sign. If the magnetization is a region of space is uniform, then Eq. (2.8) can be written with M replacing m . Ferromagnetic Resonance The conditions that the magnetization behaves as one giant macrospin and that angular momentum must be conserved, requires that the magnitude of the total magnetization is conserved. Because torque is defined as the change of angular momentum with time, in order for the angular momentum to change it must do so by 11 changing its direction. This is what magnetic moments do in an applied H-field, instead of reducing their magnitude, they change their direction and precess around the field, as illustrated in Fig. 2.1. Figure 2.1: The precession of a macrospin, represented by the magnetization M of a magnetic structure, about the applied H-field H. The vector denoting the direction of the change of magnetization with time is given by the LL equation. For given |γ| and magnetization we can find the precession frequency in an infinitely extended material. In Fig. 2.2, we consider a small precession for the magnetization. The H-field is in the z-direction with the magnetization mostly saturated in that direction. The angle of precession is very small, much less than 1o, so we can say that Mz is approximately equal to the material’s saturation magnetization Ms. The time dependent components of the magnetization are the x and y components. We represent them here as lowercase mx and my to remind us that the values of these components are very small. The LL equation then becomes: t mx mx 0 my Mz my Mz 0 H (2.9). 12 Figure 2.2: Simplified model of small angle precession of magnetization about applied field H. Vector mr is the time dependant component of the magnetization. If the angle θ is small enough, we can approximate Mz as Ms. The components mx and my are assumed to have the same time dependence to allow precession at particular frequency ωo, which gives them the form: mi m i0e i ot (2.10). Performing the time derivative and cross product gives us two equations. i o mx my H (2.11) i o my mx H (2.12) Substituting one equation into the other we find the frequency is o H (2.13). This frequency is the ferromagnetic resonance (FMR) of an infinite ferromagnet. The only material parameter in this case is the gyromagnetic ratio |γ|, which is around 2.9 GHz/kOe for many ferromagnetic materials. So, a ferromagnetic material with a 2 kOe 13 magnetic H-field applied to it would precess at about 5.8 GHz, which falls into the microwave range. This is a simple result for a simple model. The FMR frequency for a finite magnetic structure could differ from this form. H is not just an applied external field, but the total effective field acting on the magnetization. The H-field could include interactions unique to the material or a particular geometry. The Dynamic Permeability Tensor We have discussed how the magnetization of a material responds to a static Hfield. The devices we are interested in also make use of the dynamic H-fields from electromagnetic waves,. So to account for a response to a dynamic field we add an oscillating magnetic field to our simple model. Fig. 2.3 is just like Fig. 2.2, except we have now added a driving magnetic field h. Just like the time dependant magnetization components, the magnitude of the driving h-field is small (hence the lower case representation). The LL equation becomes: t mx mx hx my Mz my Mz hy H hz (2.14) We give the driving h-field a typical time dependant form: hi h i0e i t (2.15) 14 Figure 2.3: Simplified model of small angle precession about an applied field H with a small oscillating magnetic field h. The precession will have the same frequency as that of the oscillating driving field. We assume that the time dependence of the magnetization and the driving field will be the same. mi mi0e i t (2.16) Performing the time derivative and cross product gives us: i mx m y (H h z ) M s h y my 0 M s h x m x (H h z ) mx h y myh x (2.17) We linearize this set of equations by setting any product of a small m component and small h component to zero. This simplifies Eq. (2.17) into two equations: mx my i m yH Msh y Msh x m x H (2.18) Substituting one equation into the other, we arrive at solutions for mx, and my. 15 mx Ms my 2 i hy 2 ( H2 2 2 i hx Ms 2 ( hxH H2 (2.19) ) h yH 2 (2.20) ) Separating out the h components and writing the equations in a matrix form, we get: 2 mx my Ms H 2 o i 2 2 o 2 2 i 2 o 2 2 o H hx hy (2.21) 2 where ωo is given by Eq (2.13). Eq. (2.21) is Eq. (2.19) and (2.20) rewritten in a matrix form. This is what we were looking for, an expression that gives us the dynamic magnetization as it relates to an oscillating h-field. The matrix that relates the two vectors (magnetization and driving field) is the magnetic material’s susceptibility. According to Eq. (2.4), this also gives us the permeability: i 1 i 0 t 0 t 0 1 1 0 (2.22) Where the matrix elements μ1 and μt are given by: 2 1 1 4 Ms 2 o H 2 (2.23) 16 t 4 Ms 2 o (2.24) 2 The most obvious problem with this expression is when the driving frequency ω equals the FMR frequency ωo and both Eqs. (2.22) and (2.23) blow up. These results are based on “perfect” materials where all the work done by the driving field is transferred to the magnetic material. According to the second law of thermodynamics, not all of the energy pumped into the system will be turned into work. There will be losses and they need to be accounted for in the equations of motion. Permeability with Damping The complete Landau-Lifshitz Equation typically has a term added to handle the damping dM dt M H M M [ M H] (2.25). The term α is the damping factor, a real unitless number. The damping of a material is often measured by the width of an observed absorption peak in a scan over either the frequency of the signal or of an external applied H-field. This provides either a frequency linewidth or field linewidth. The damping factor α can be related back to these linewidths. The double cross product term M [M H] supplies a new vector that moves M toward Ms, as seen in Fig. 2.4. The resulting motion is a spiral decay into smaller and 17 smaller precession angles until the magnetization comes to rest in the direction of the applied H-field. Figure 2.4: Macrospin precession with the direction of the damping vector. If we were to add a driving field, as we did in the previous section, the resulting equation of motion would be: mx my t Mz mx my Mz hx hy H hz Mz mx my Mz mx my Mz hx hy H hz (2.26). We follow the same procedure used to find the permeability of an undamped magnetic material, we must linearize the equations. The permeability with the damping terms in them can be found as30 i 1 i 0 t t 1 0 0 0 1 (2.27). 18 where μ1, and μt are: 2 1 1 4 Ms t 4 Ms 2 2 H 2 (1 H 2 (1 2 H(1 ) 2i H 2 (2.28) ) 2i H 2 (2.29). 2 2 ) i Using the permeability elements, we can gain information about the material's resonance, and we can express the material’s damping in terms of linewidth. If we were to keep ω fixed frequency and plot μ1 and μt as functions of the applied field H, there is a signature near an applied field of H / | | . How the signature spreads out near that field is determined by the damping constant α. This is the connection between the experimental parameter of a material’s linewidth and the damping factor α. Fig. 2.5 shows the real parts of both μ1 and μt as a function of applied field using Fe as the magnetic material with a Ms = 1.7 kG, |γ| = 2.9 GHz/kOe, the damping set to α = 0.03, and the frequency f = 8 GHz. For both the real parts, there is a maximum and minimum centered around about 2.76 GHz. For one of the elements we measure the difference between one minimum and maximum, we get a value of 0.165 kOe. 19 linewidth = 0.165 kOe 80 Realative Permeability 60 Real part of 40 20 0 -20 -40 Real part of -60 -80 2.0 2.5 3.0 3.5 Applied Field (kOe) t 4.0 Figure 2.5: Real parts of the permeability tensor elements μ1 and μt. The linewidth between the minimums and maximums is 0.165 kOe. Fig. 2.6 is the plot of the imaginary parts of μ1 and μt. The signature is now a peak or dip centered around 2.76 GHz. Measuring the width of the peak across half of its maximum value (full width at half max), we again come up with 0.165 kOe. Again, this spreading of the function is due to the damping constant α, which is related to the experimental linewidth. In this case, the damping constant of α = 0.03 is related to a linewidth of 0.165 kOe. The value of the of the field-linewidth is given by H 2 In the next chapters, we will use these forms for the permeability. (2.30). 20 Realative Permeability 150 linewidth = 0.165 kOe Imaginary part of 100 50 0 -50 -100 -150 2.0 Imaginary part of 2.5 t 3.0 3.5 4.0 Applied Field (kOe) Figure 2.6: Imaginary parts of the permeability tensor elements μ1 and μt. The linewidth between the minimums and maximums is 0.165 kOe, typical full width at half max measurement. Demagnetizing Fields and Thin Film FMR We now consider the ferromagnetic resonance for a thin film. The film is infinite in the y and z plane, but has a thickness in the x-plane. Figure 2.7: Geometry of a thin film. The y and z directions are infinite, and there is a finite thickness in the x-direction. 21 As we will see, the most significant change in the calculation comes from the static and dynamic demagnetizing fields introduced by effective magnetic surface charges at the boundaries. We start with H-field being represented as the divergence of a scalar magnetic potential H (2.31). M The potential’s value at a point in space (r) is determined by the surrounding magnetization (the magnetization at all points r'). The potential is then given by the integral M(r ) M (r ) 1 dV r r V (2.32). Making an approximation that the magnetization is uniform everywhere in the volume, we can write an expression for the average dipolar H-field in the volume, V, surrounding the observation point r: Hd 1 V dV M V V r 1 dV r N M (2.33). The tensor N is a strictly geometric relationship between volume V and V’. The elements of this tensor are given by these integrals N ij 1 V dV V V xi 1 dV xj r r (2.34). These volume integrals are generally changed to surface integrals using a version of Guass’s theorem 22 N 1 V dV V V 1 dV r r 1 dA dA VA r r A (2.35). This is a general demagnetization tensor. When the observation point lies inside the magnetic volume the elements of the tensor N are known as demagnetizing factors. The general tensor relationship between a small volume centered around r' and an observation point at r is known as the Newell tensor.29 Using the results obtained from the Newell tensor for an infinitely extended film in the yz plane but with a finite thickness in the x-direction and a uniform magnetization, the average H-field from the dipolar interactions is Hd 4 m x x̂ (2.36). This is the demagnetizing field in a thin film with uniform magnetization. Assuming that the static magnetization is in the plane of the film, along the z direction, one notes that the demagnetizing field is dependent on a component of magnetization that precesses. Hence, this is a dynamic demagnetizing field. It might be a little disturbing that the field has no apparent dependence on the thickness of the film, but this is because of the infinite components in the y and z directions. Essentially the result is the same as that for the electric field inside an infinitely extended capacitor. If the thin film or the capacitor were not infinite in the y and z directions, the results would depend on thickness. This method for finding the demagnetizing field can also be applied to dipolar fields from non-local volumes of magnetic materials and becomes critical in the micromagnetic simulations. 23 To observe the effects of the demagnetizing fields, we consider the usual geometry: a thin film with the z-component of magnetization Mz in-plane and an applied static magnetic field also along the z-axis. Assuming there is small angle precession around the applied field, the Mz component is taken to be the saturation magnetization, and the demagnetizing field depends on the component of magnetization normal to the film. With the damping set to zero for simplicity, we get the following LL equation: t mx mx 4 mx my Mz my Mz 0 H (2.37). Using the same procedure that gave us Eq. (2.14), we get an expression for the FMR of a thin film FMR H( H 4 M s ) (2.38). This is more complicated compared to the bulk FMR case given by Eq. (2.14). The most notable change is that it now includes a dependence on the magnetization of the material. So, if we used the 2.9 GHz/kOe gyromagnetic ratio |γ| and applied a 2 kOe field to thin film of Fe with a Ms of 1700 G, we would get a FMR of 19.8 GHz. The goal of this chapter was to give the reader an understanding of the magnetic permeability tensor which will be used extensively in the following work. In the following chapters, there may be differences in the explicit forms of the presented expressions, but this is due to different choices of coordinate systems and definitions of damping. These choices were each made on an individual basis and the work is presented in its original form. CHAPTER III Thin Films Structures and Boundary Condition Evaluation The first device presented in this chapter is the notch filter. We give results predicting the performance of a notch filter device with different magnetic components. We model an electromagnetic wave forced into a planar waveguide. The device is composed of two conducting plates with one or more dielectric spacer along with a magnetic thin film sandwiched between the conducting plates. The magnetic film interacts with the wave and changes it. We evaluate what happens to the wave in the guide by applying boundary conditions set by Maxwell’s electrodynamic equations. We investigate a different structure from the Schloemann geometry mentioned in Chapter I.2 The ferromagnetic film is placed in an arbitrary position between the outer conductive layers of the waveguide as is shown in Fig. 3.1 and is surrounded by dielectric material on both sides. This geometry produces a significant improvement in attenuation compared to the filter where the ferromagnet is directly next to the conductive layers. In addition, the width of the frequencies absorbed by the filter is reduced slightly, giving one better frequency resolution. Calculation for a Small 3-Layer Ferromagnetic Notch filter The geometry of the waveguide is shown in Fig. 3.1. The conductive layers are silver (Ag). The silver sandwiches two dielectric spacer layers, which we model as silicon dioxide (SiO2). The magnetic component is in-between the spacers, which we 25 have chosen to be iron (Fe). A transverse magnetic (TM) mode propagates down the z axis parallel to the external applied field Ho. For the TM mode the oscillating magnetic field is directed along the x axis only. The electric field has a longitudinal component, Ez, and a transverse component, Ex. Figure 3.1: Geometry of the three layer notch filter. The wave propagates along the z axis, parallel to the applied field. The outer metallic layers are assumed to be a highly conductive material such as Ag. The behavior of the magnetic material is described by a permeability tensor given by30-32 1 ( ) 4 M( H 0 i ( H0 i )2 4 M i ( H0 i )2 0 ) 2 2 4 M ( H0 i )2 4 M( H 0 i 1 ( H0 i )2 0 i 0 2 ) 2 i 1 0 1 i 0 t t 1 0 0 0 1 (3.1). 26 H0 is the applied field, M is the saturation magnetization, γ is the gyromagnetic ratio, ω is the frequency of the electromagnetic (EM) wave, and Γ is the damping factor. The damping factor for a set frequency can be related to the measured full width at half maximum ferromagnetic resonance (FMR) line width (ΔH) of a material. In the range of small damping, Γ can be taken to be the measured full width at half maximum linewidth. We will assume that Γ is linearly dependant on ω and has this form: ( H0 ) (3.2) Here ΔH0 is the FMR linewidth at zero frequency and α is a parameter that determines how much the FMR linewidth changes with frequency. The use of the complete tensor would make the problem complicated and has been shown to be unnecessary.3 One can instead use a scalar permeability given by ( xx yy 2 1 2 t v ) , (3.3) 1 (3.4) 1. zz Using the two curl equations from Maxwell's equations we find the following wave equation for the system: ( H) 2 c2 t 2 v Hx Hy Hz v (3.5) The equation has the same form for all regions but the permeability μ and permittivity ε change from region to region. Because we are dealing with a TM mode only Hx is nonzero.30,31 Furthermore we assume the width of the guide is sufficiently large that there is no variation in the x direction. These equations then reduce to one equation 27 2 2 Hx y2 2 Hx z2 v 2 c t2 (3.6) Hx . Because Hx is a wave propagating along the z axis it has the form Hx H x ei ( k z z t ) ik y y e (3.7) . When we substitute this equation into the wave equation we find two solutions for k y which are the negative of each other. 2 ky i k 2 z v (3.8) c2 We expect three independent values of ky, one for each region. These values are denoted by ky1, ky2, and ky3, each with respect to the region in which they are found. Eq. (3.10) then applies in all regions with the appropriate choices for μ and ε. In each of the regions, the x-component of the H field can be represented as the sum of two terms with different ky values, each with an unknown amplitude: Hx (A e ik y1y A e ik y1y )e i ( k z z t) . (3.9) Using the curl of the H-field, we can represent the z component of the E field by the same unknown amplitudes: Ez (A e ik y1y A e ik y1y )e i ( k z z t) . (3.10) Where η is a constant defined by the permittivity of the region, the k y of the region, the frequency of the electromagnetic wave (ω), and the speed of light (c). ck y (3.11) 28 The constraints on what waves can propagate in this geometry are determined by the boundaries where the layers meet. We can relate the values of Hx and Ez at the first boundary of a region (y = yo) to the second boundary of the same region (y = y1). We evaluate Eqs. (3.11) and (3.12) at the second boundary, y1, and solve for the constants A+ and A- in terms of Hx(y1) and Ez(y1). We then evaluate Eqs. (3.11) and (3.12) at the first boundary, y0, and using the results for A+ and A- we obtain a relationship that can be described with a matrix transferring the values Hx and Ez at one boundary into terms of Hx and Ez at the next boundary. H x (y y0 ) E z (y y0 ) i sin( k y y) cos(k y y) i sin( k y y) cos(k y y) H x (y y1 ) E z (y y1 ) (3.12) Eq (3.14) is a well known relationship.33,34 Where Δy is y1-yo. This supplies us with the form of the transfer matrix relating Hx and Ez at the boundaries on either side of a single region.35 mn cos(k yn y n ) i sin( k yn y n ) n i sin( k yn y n ) (3.13) cos(k yn y n ) Because the tangential components of E and H are continuous, the transfer matrix in Eq. (3.15) also relates Hx and Ez at the first boundary of region to the first boundary of region two. With appropriate transfer matrices, we can make a connection from the region one boundaries to the region two boundaries and then to the region three boundaries. This can be represented nicely in a progression of transfer matrices. To relate the fields at first boundary in region one to those at the last boundary in region three, we must include the transfer associated with each layer: 29 H x (y y0 ) E z (y y0 ) H (y y3 ) m1 m 2 m 3 x E z (y y3 ) (3.14) The resulting transfer matrix is a multiplication of all three transfer matrices. M m1 m 2 m 3 M11 M 21 M12 M 22 (3.15) At the outer boundaries, y0 and y3, the boundary conditions are that Ez must be zero because the Ag is considered as highly conductive. Thus H x (y y0 ) 0 H (y y3 ) M x 0 (3.16) This gives the relation that the element M21 of the transfer matrix must be zero. Expanding out M21, we get: iM 21 1 sin( k y1 y1 ) cos(k y 2 y 2 ) cos(k y 3 y 3 ) 2 cos(k y1 y1 ) sin( k y 2 y 2 ) cos(k y 3 y 3 ) 3 cos(k y1 y1 ) cos(k y 2 y 2 ) sin( k y 3 y 3 ) 1 3 (3.17) sin( k y1 y1 ) sin( k y 2 y 2 ) sin( k y 3 y 3 ) 2 0 One can make additional approximations, e.g. k y1 y1 and k y3 y 3 are both small and then expand out the sine terms in Eq. (3.19). However, this does not substantially simplify the resulting equation. We find a solution to Eq. (3.19) numerically for kz as a function of . One picks a real frequency and a complex wavevector kz. This generates all the ky values in each region. These values are then plugged into the matrix element M21. If the result is zero, our guess for kz was good and we can pick a different frequency and start over. If not, we 30 guess a new value for kz. The guessing is done efficiently by using a root finding method which searches the complex kz space. Transmission Results for the small 3-Layer Notch Filter In all the following graphs the same parameters were used, unless otherwise stated. The permittivity of SiO2 is 4.0 (ε1 = ε3 = 4.0) the total thickness of the two dielectric layers adds up to 4.5 microns, the applied field is 1000 Oe, the saturation magnetization for Fe is 1700 G, the gyromagnetic ratio is γ 1.803 x 107 rad/G-sec, and the permittivity for the Fe layer changes as a function of frequency and conductivity ζm = 107 /Ωm; ε2 = 1 + i ζm/ ε0ω. The damping factor Γ has been set by the parameters ΔH0 = 120 Oe and α=0.016. The numerical method, outlined in the previous section, provides a complex propagation wavevector kz for every frequency ω. The real part of kz gives information on the wavelength and the imaginary part of kz determines the attenuation of the wave as it propagates. In Fig. 3.2 we plot the transmission loss of the wave as a function of frequency for a set of filters where the magnetic element is placed in different positions. If the value is zero, then there are no transmission losses, and the EM wave gets through the waveguide at the same intensity it went in with. By examining the dips of the transmission, we notice that the largest attenuation occurs at the ferromagnetic resonance frequency (ωres). res H 0 (H 0 4 M) (3.18) 31 Rejection Band at -10 dB/cm Transmission Loss (dB/cm) 0 -15 Fe at the edge. -30 (.75 microns from the edge) -45 -60 -75 (1.5 microns from the edge) 9 12 15 Frequency (GHz) Fe in the middle. (2.25 microns from the edge) 18 Figure 3.2: Transmission as a function of frequency for different positions of the magnetic Fe layer. The Fe film is 0.1 microns thick. There are four positions graphed, each with the Fe film moved a certain distance away from one edge of the waveguide: 0 microns away, 0.75 microns, 1.5 microns, and 2.25 microns (right in the middle). The attenuation at resonance is greatest for when the Fe film is in the center of the waveguide, and the dip width is the narrowest at the same position. We also notice that the largest attenuation occurs when the film is positioned directly in the middle of the waveguide with equal amounts of dielectric on either side of the film. 32 Fig. 3.3 shows transmission loss at the resonant frequency for varying thicknesses of the magnetic material while keeping the amount of dielectric constant (4.5 microns total thickness). Transmission Loss (dB/cm) 0 -20 -40 Difference between the middle and the edge results Fe at the edge of the waveguide -60 -80 Fe in the middle of the waveguide 0.0 0.1 0.2 0.3 0.4 0.5 Thickness of Magnetic Film (microns) Figure 3.3: Transmission loss at resonance as a function of the thickness of the Fe film. The sum of the total thickness for all dielectric layers is 4.5 microns. For the magnetic material at the edge, the attenuation reaches a maximum at 0.10 microns. In the middle, the attenuation reaches a maximum at 0.20 microns. The results show that for extremely thin films (between 0.01-0.05 microns) the position of the magnetic film doesn't make much difference, both the middle and the edge positions have the same attenuation. For thicker films, the difference between attenuation in the two positions increases. After 0.2 microns we find that the attenuation for both the 33 middle and the edge positions have reached their maximum values, and there is about a constant 30 dB/cm difference between the two. The saturation value for attenuation when the Fe film is at the edge of the waveguide is about -55 dB/cm, and the saturation value for the Fe in the middle is about -86 dB/cm. This could be the result of the skin depth in the Fe.11,31 A skin depth calculation for Fe (which is frequency dependent), 2 (3.19) gives about 0.1 microns at the FMR frequency. The EM wave must exist inside the Fe film to be absorbed, but due to the skin depth it can only penetrate a limited distance. The amount of the Fe the EM wave "sees" determines how much power is lost to the film. If the film is on the edge of the waveguide, then the EM wave only sees one side of the film. If the Fe film is in the center of the waveguide, then the EM wave penetrates both sides of the film. From Fig. 3.3 we see that when the Fe film is at the edge of the waveguide the attenuation reaches a constant value at a thickness of about 0.1 microns. This distance is a measure of the skin depth in the Fe at resonance. In contrast, when the film is in the center of the waveguide the attenuation becomes constant at a thickness of about 0.2 microns. This further demonstrates the fact that the EM wave penetrates from both sides of the Fe film. Another important issue is how narrow or wide the transmission dips are. The width of the dip will characterize the rejection band, which is the range of frequencies that a notch filter effectively blocks out. This chapter uses the criterion that the rejection band spans the region where the transmission loss is below -10 dB/cm. 34 As seen in Fig. 3.2, the transmission dip for the Fe film in the middle of the waveguide is narrower than the dip for the film at the edge of the waveguide. Fig. 3.4 illustrates this characteristic by plotting the width of the rejection band as a function of the position of the magnetic film. The graph shows that the width reaches a minimum when the Fe film Rejection Band at -10 dB/cm (GHz) is at the center of the waveguide. 6.0 Fe at the edges of the waveguide 5.9 5.8 5.7 Fe in the Center of the Waveguide 5.6 5.5 5.4 5.3 0 1 2 3 4 Position of the Fe film from an edge of the waveguide (microns) Figure 3.4: Rejection bandwidth as a function of the magnetic film's distance from the edge of the waveguide. The bandwidth decreases as the magnetic film approaches the center of the waveguide. The total sum of the dielectric layers is 4.5 microns. The Fe film is 0.1 microns thick. We have seen that increasing the magnetic film thickness (up to the skin depth) increases attenuation. The increase in film thickness also increases the width of the rejection band. In Fig. 3.5 we plot transmission as a function of frequency with the 35 magnetic film in the middle of the waveguide for four different thicknesses of the Fe film: 0.075 microns, 0.100 microns, and 0.150 microns. Clearly the smallest thickness has the smallest rejection band. Fig. 3.6 summarizes this behavior and displays the rejection bandwidth as a function of the Fe film thickness. The rejection bandwidth increases almost linearly as a function of thickness 0.05 microns. Reflections and Insertion Loss Using the filter described above in an application, the wave would likely be guided to the filter using a standard microstrip waveguide. This raises the issue of reflection of the electromagnetic waves as they move from one structure to another. As seen in Fig. 3.5, we again apply boundary conditions to find an approximation of the reflection properties at the interface between the two waveguides (one with the magnetic material, one without). As a quick approximation, let us assume the wave-guide passage consists primarily of dielectric. In this case, there is only one allowed traveling wave of TM character. For thin waveguides, this mode is nearly TEM in nature, and we consider a case where there are no components of the H or E fields that are parallel with the direction of propagation. The wave number k1 in the fully dielectric waveguide is: k1 c (3.20) We consider the case where the wave starts in Region 1 of the waveguide (all dielectric) and then encounters the structure with the magnetic layer, Region 2. An incident electromagnetic wave in the first section traveling in the +z direction encounters the interface at z = 0 and is partially reflected. For the nearly TEM mode we assume that 36 there is only a y component of the electric field. This y component can be used to find the components of the H field. From the equation, 1 B c t E (3.21) we find that only an x component of the H field exists. Hx Ey ic v z (3.22) In this case we are not using the complete tensor for the permeability. Instead we use the scalar permeability from Eq (2). Figure 3.5: Cross-section of the waveguide divided into two regions. Region 1 does not have a magnetic layer. Region 2 has a magnetic layer. The wavevector in region 1 will be different than it is in region 2. The wave in region one is a superposition of both the incident and reflected parts. However, after the interface, the wave will continue on with a different wavevector. So in the two regions, denoted by subscripts 1 and 2, we can represent our E and H fields as: 37 E y1 H x1 E y2 A I eik1z ic v ARe (k1A I e ik1z ik1z (3.23) k 1A R e ik 1z ) A T eik 2z ic H x2 v (3.24) (3.25) (k 2 A T e ik 2z ) (3.26) Because the parallel components of E and H must be continuous over the interface we are able to solve for the amplitudes of the fields in terms of the wave numbers. The reflection coefficient is ratio of the intensity of the reflected portion of the E field and of the incident E field. We find R IR II k1 k 2 k1 k 2 2 (3.27) The wavevector k1 is found from Eq (15) and k2 comes from the results of our calculations in the first part of this paper. Note that k2 is complex and both the real and imaginary parts are necessary for an appropriate result. We compare the results of the approximation in Fig. 3.6 with an experiment in Fig. 3.7. The experiment was performed using a microstrip geometry where a 0.1 micron layer of Fe and 4.5 microns layer of SiO2 were used. The Fe film was at the top of the dielectric, just under the metallic signal line. 38 The experiment measured the return loss of the signal entering the microstrip as a function of the frequency for different applied fields. Using the boundary condition method we find slightly different results from the experiment. Return Loss (dB) -4 0.715 kOe -6 1.057 kOe -8 1.745 kOe -10 -12 -14 -16 0.385 kOe 0 Fe 0.1 microns SiO2 4.5 microns 10 20 30 Frequency GHz 40 Figure 3.6: Return Loss vs. Frequency for an EM wave in a SiO2 waveguide (providing a k1 wave number) interfacing with a waveguide geometry consisting of a 0.1 microns thick layer of Fe and a 4.5 microns thick layer of SiO2 (which provides the parameters for the calculation of a k2 wave number). The results for the boundary condition method in Fig 3.6 show a rise in the reflected signal and a peak right around the FMR frequency. The maximum reflection is between –8 dB and –5 dB. The maximum return loss again decreases as the applied field is increased. After the peak, the return loss drops by about 7-8 dB’s in a matter of 3-4 39 GHz, reaches a minimum and then begins to increase again. The return loss at higher frequencies converges for all the curves at a value of around -11 dB. -4 Return Loss (dB) -6 0.385 kOe -8 -10 0.715 kOe 1.057 kOe -12 1.745 kOe -14 -16 Fe 0.1 microns SiO2 4.5 microns -18 0 5 10 15 20 25 30 35 40 Frequency GHz Figure 3.7: Experimental Data of Return Loss vs. Frequency for a microstrip geometry using a 0.1 microns thick layer of Fe and a 4.5 microns thick layer of SiO 2. The experimental results in Fig. 3.7 display similar trends. The return loss peaks in the experiment occur slightly after the predicted FMR frequency. The maximum reflected signal decreases as the field is increased, this is in agreement with the boundary condition method. At frequencies beyond the peak, the return loss drops sharply by about 6-9 dB. After the drop, the return loss rises again, converging on a value of near –8 dB, also agreeing with the calculation. Qualitatively, the results agree on the most important points: 1) The maximum reflected signal occurs near the FMR frequency and 2) The maximum reflected signal becomes smaller as the external field is increased. The values 40 for the maximum reflected signal ranged from -5 to -7 dB for the boundary condition method, and from -7 to -11 dB in the experiment. Given the approximate nature of the calculations, this level of agreement seems appropriate. These results validate the assumption that the signal can be inserted into the waveguide and the results of our calculations within the structure are appropriate. Conclusions In conclusion, for a fixed thickness of the magnetic film, the attenuation at resonance is at a maximum when the magnetic layer is at the center of the waveguide. In addition, when the magnetic film is at the center of the structure, the rejection bandwidth is minimized. These features can be understood in terms of the skin depth of the magnetic film. The attenuation of the EM wave is determined, in part, by how far it can penetrate the magnetic material. If a film is thinner than the material’s skin depth, then it does not matter where the film is positioned in the waveguide geometry. However, if a thicker layer is used, it must be positioned away from the conductive edge in order for the additional material to effectively contribute to the absorption of the wave. We have also calculated the reflection of a wave entering from a non-magnetic waveguide into a waveguide with a magnetic layer. We find that the reflection is largest slightly above the FMR frequency. The experimental results show the reflection is always below -7 dB. The theoretical results also show a maximum reflected signal of about -5 dB, but this could be reduced if the damping in the calculations was increased. CHAPTER IV Devices with Hexagonal-ferrites We also investigated a notch filter using a hexagonal-ferrite as the ferromagnetic layer in the waveguide. M-type Barium hexagonal ferrites are of interest because of their high crystalline anisotropy and low losses. The strong anisotropy of the Barium hexagonal ferrite (BaM) allows for the orientation of its magnetization in a thin film to point out of the plane of the film. We investigated a waveguide with a film of a hexagonal ferrite and a film of a dielectric material (SiO2) put between two conductive layers in a microstrip-like geometry as shown in Fig. 4.1. The hexagonal ferrite is assumed to be grown with its c axis out-of-plane. In this case, the magnetization and a large anisotropy field will also point out-of-plane. Furthermore, in contrast to most of the previous work using ferromagnetic metals, a magnetic field is applied perpendicular to the surface of the waveguide. There are several advantages to using a hexagonal ferrite: 1) The conductivity is low, therefore, there are no issues with nonmagnetic losses. Because of this, the insertion loss in the device is generally smaller compared to a filter using ferromagnetic metals. 2) Hexagonal ferrite film thicknesses comparable to metallic films produce transmission loss dips with greater symmetry than their metallic counterparts. 3) The internal anisotropy fields produce an operational frequency close to 47 GHz at low fields (at least 1 kOe), compare this to the original Schloemann’s filter that required a large field (10 kOe) to reach the same operating frequency.2 42 For comparison, we also present results for a few different waveguides. In one addition case the hexagon ferrite film has an in-plane anisotropy and magnetization. The other addition case uses a metallic magnetic layer that has its magnetization forced out of plane by an external field. We also evaluate devices made from hexagonal ferrites as phase shifters for the magnetization oriented both in-plane and out-of-plane. The main results are the following: 1) The structure outlined here can be used as a tunable notch filter, with an attenuation of over 50 dB/cm at the notch. This is true even with fairly thin hexagonal ferrite films around one micron thin and relatively poor linewidths of 400 Oe. 2) There is a small shift in the frequency of the notch as the thickness of the hexagonal ferrite film is changed. The thicker the film, the greater the shift in frequency. 3) It is possible in the same device to get substantial phase shifts, on the order of 360o/cm with relatively small losses, less than 2dB/cm. 4) A geometry where the magnetization and anisotropy field lie in-plane requires a smaller external field to obtain the same operational frequency as the out-of-plane structure. We point out that the structure considered here is small, with thicknesses of the hexagonal ferrite on the order of one micron. There has been a lot of work designed to grow high quality hexagonal ferrite films with thicknesses of several microns or more using a variety of different methods.36-38 In addition there have been some phase shifters built with hexagonal ferrite films which are .2 mm thick. 39-40 This theoretical work shows that thicker films may not be necessary, at least for some devices. 43 The Calculation for the Hexagonal Ferrite waveguide The geometry of the waveguide is shown in Fig. 4.1. The magnetic field is directed along the z axis, as is the magnetization. To be specific, the propagation of the electromagnetic wave is along the y axis, although by symmetry all directions in the plane are equivalent. Figure 4.1: Geometry of the notch filter. The wave propagates along the y axis, perpendicular to the applied field, Ho. The outer metallic layers are assumed to be a highly conductive material such as Ag. The behavior of the magnetic material is described by a dynamic permeability tensor given by3 i 1 i 0 t t 1 0 0 0 1 (4.1). 44 2 1 1 4 M t 4 M H 2 2 i 2 i i 2 H 2 H H H H (4.2) (4.3) M is the saturation magnetization. H is the sum of the applied external field (H0), the internal anisotropy field (Ha), and the static demagnetizing field (-4πM). |γ| is the absolute value of the gyromagnetic ratio, ΔH is the measured full width half maximum ferromagnetic resonance (FMR) linewidth for the material, and ω is the frequency of the interacting electromagnetic wave. There is a resonance in the permeability elements when Δω2 = 0. Δω2 is given by 2 2 H2 2 (4.4). Here the factor δ is close to 1 if the linewidth is small and is given by 1 2 H2 4 2 (4.5), The FMR linewidth is generally frequency dependent. Based on experiments36,41, we can assume that the FMR linewidth for barium hexagonal ferrite is linearly dependent on frequency. We will assume it has the form: H H0 | | (4.6) where ΔH0 is the FMR linewidth at zero frequency and ψ is a unitless parameter that determines how the linewidth changes with frequency. 45 If there is no damping, then ΔH = 0 and δ = 1. If this is the case, the denominator for both μ1 and μt goes to zero when ω = |γ|H. This gives us a resonance frequency for our material, | |H res (4.7). The final results of these calculations will show significant absorption near this frequency. In order to find the behavior of an electromagnetic wave in our device, we must solve the wave equations which arise from Maxwell's equations using the boundary conditions at the interfaces between materials. The wave equations of our system are found from the two curl equations of Maxwell's equations. hy y z x x hz y hx z hx y hy z hz x z x y hx z hy x hz y hz x hx y hy 2 c2 t 2 1 hx i thy i thx hz 1 hy (4.8) z The equation has the same form for all regions but the permittivity ε, and the permeability components μ1 and μt change from region to region. The main difference between this calculation and the calculation in the previous chapter is that we use the full tensor. We assume that the h-field for our EM wave can be represented by this form: h i ( k r h 0e t) (4.9) where the vector h0 represents the amplitude components of the h-field. Because we are dealing with an infinite planar geometry and the permeability is oriented out-of-plane, the 46 direction of propagation in the plane is arbitrary. Therefore, we can choose the wave to propagate in the y-direction. This constraint makes the x-component of the wave vector zero (kx=0) and the wave equation from the double curl equations can be rewritten for each region 2 k 2 y k 2 z 2 i 1 c2 2 i 0 t c2 hx 2 k t c2 2 z k yk z c2 hy hz 2 0 k yk z k 2 y c2 0 (4.10). 1 This matrix containing the permittivity, the permeability, frequency, the k y’s and kz’s will be referred to as matrix Q. The kz’s are the wavevectors for the standing wave modes in the different layers and ky is the wavevector describing propagation. The relationship can be simply written as: Qh 0 (4.11) For a non-trivial solution to this equation the determinant of matrix Q is equal to zero. This gives us a dispersion relationship in each region: 2 kz 4 2 ky ( 1 1) 2 c 2 1 2 kz 2 ky 2 2 c ky 2 2 1 c 2 ( 2 1 2 t ) 0 (4.12) We define the following two quantities with wavevector units: 2 q1 ky 2 1 ky 2 2 (4.13) c2 2 q2 ky 2 c2 1/ 4 2 ky 2 1 c2 ( 2 1 2 t ) (4.14) 47 Eq. (4.12) becomes: kz 4 q12 k z 2 q 42 0 (4.15) This gives four solutions for kz in each region. q12 kz q14 2 4q 42 (4.16) We can find relationships between the components of the h-field. In other words, for a given kz we can find hy and hz in terms of hx. For example, we can write this as hy (k z )h x hz (k z )h x (4.17) (4.18) where α and β are coefficients with a dependence on permeability, permittivity, frequency and ky. These are found from the Q-matrix acting on the h-field in Eq. (4.10). We let hx be a superposition of the four waves associated with the four kz vectors 4 hx Ane ik z n z ik y y e (4.19) n 1 Then we can represent the solution of the components hy and hz as they relate to the solution for hx: 4 hy (k zn )A n e ik z n z ik y y e (4.20) n 1 and 4 hz (k z n )A n e n 1 ik z n z ik y y e (4.21). 48 The h-field can now be represented in each region of our waveguide in terms of the xcomponent of the field, hx, with four unknowns A1 to A4 One now must consider the transverse magnetic wave in the dielectric layer. Examining Eq. (4.12) with 1 1 and t 0 we find the usual dispersion relationship 2 kz 2 k 2 y c2 0 . This leads to two solutions for kz: 2 kz i k 2y (4.22) c2 In addition, there is partial diagonalization of the matrix in Eq. (4.10) which shows that hy and hz have no dependence on hx. This means that there are two independent modes that can exist. One mode has only an hx component, and the other mode has only hy and hz components. Because of these modes are possibile, the h-fields for the dielectric must be considered a superposition of these two modes with the weight of their importance determined by coefficients similar to those we have used in Eq. (4.19). The h-field components in a dielectric region can be written as: hx [A5eik z z hy [A7 eik zz hz [ ( k z )A 7eik zz A 6e A8e ik z z ik z z ]e ]e ik y y (4.23) ik y y ( k z ) A8e (4.24) ik z z ]e ik y y (4.25) where η is the coefficient relating hz to hy, found from Eq. (4.10). Because there are two regions (hexagonal ferrite, and the SiO2), there are four different kz values and four different A coefficients for each region. This gives a total of 49 eight A coefficients. We will see that there are eight boundary conditions which will allow us to obtain a complete set of equations for the unknown A values. The E and h fields in each region need to satisfy the boundary conditions at the interfaces between materials. These conditions are that the tangential components of E and h are continuous. We can find the E fields for each region from the h-field by using Maxwell's equations and substituting in Eqs. (19-21) or (23-25), based on the region. We assume the tangential E field is zero at the interfaces between the dielectric and the top, highly conductive, layer and between the hexagonal ferrite and the bottom, highly conductive, layer. This condition will give us four equations: two for the Ex and Ey components at the top of the waveguide and two for the Ex and Ey components at the bottom of the wave guide. Looking at the boundary of the dielectric layer and the hexagonal ferrite, we have four additional equations: two equations for the Ex and Ey components and two for the hx and hy components. Thus there are a total of 8 equations. We can write this set of equations in a matrix form, i.e. GA 0 where the elements of the vector A are defined in Eqs. (19-21) and Eqs. (23-25). The elements of G are given in Appendix A. Every element of G is a function of frequency, ω, and wavevector, ky, (all kz’s are functions of ky). For a non-trivial solution, the determinant of the matrix G must be zero: det G( , k y ) 0 (4.26) This condition provides us with the dispersion relation, the relationship between a wave’s frequency and wavevector. The solution is found through the following numerical method. We choose a particular frequency ω and guess a value for ky. From the wave equations in the different regions, we can find a solution for kz in terms of ω and ky. 50 These values are substituted into the matrix G, and the determinant of G is calculated. If it is zero, the guess for ky was good and we can pick a different frequency and start over. If it is not zero, we guess a new value for ky. The guessing is done efficiently by using a root finding method which searches the complex ky space. Transmission Results for the Hexagonal Ferrite Waveguide In all the graphs we have used the following parameters. The reported permittivity of barium hexagonal ferrite varies according to different sources.42,43 Based on this we chose a permittivity of 10 for M-type barium hexagonal ferrite. The saturation magnetization of the M-type barium hexagonal ferrite is 0.334 kG, and the internal anisotropy field is 16.4 kOe. We do not consider variations in the magnetization or anisotropy field due to changes in temperature. However, it has been shown in the cataloged properties of hexaferrites that the Curie temperatures for these materials are around 700-750 K. In a range of 100 K centered on room temperature, the anisotropy field increases by about 3% and the magnetization decreases by about 23% as the temperature increases.44 This would lead to a small shift of about 4-5 GHz in the resonant frequency over the whole range. The permittivity of SiO2 is 4.0. The SiO2 could be replaced by a variety of dielectrics with different permitivities. The point of this layer is to provide a portion of the waveguide that facilitates propagation of the electromagnetic wave. In some previously grown devices the dielectric layer doubled as the substrate on which the magnetic material was grown2 but this is not always the case, as seen in more recent thin film devices.8 We have chosen SiO2 for our results because it is commonly used as the filler layer in microstrip devices. 51 We explore the transmission of the filter as a function of frequency for changes in the thicknesses of the dielectric layer, the thickness of the hexagonal ferrite thin film, the external applied field (H0), and the damping in the hexagonal ferrite (determined by ΔH). The material’s FMR linewidth (ΔH) is set by the parameters ΔH0 and ψ from Eq. (4.6). The numerical method, outlined in the previous section, provides a complex propagation wavevector ky for every frequency ω. The real part of ky gives information on the EM wave’s phase at a given position y, and the imaginary part of k y determines the transmission loss of the wave as it propagates. For a notch filter device, the interest is the transmission as a function of frequency. Fig. 4.2 gives the results for a filter with 4.0 microns of dielectric and 0.5 microns of barium hexagonal ferrite and plots the transmission at different applied fields. The linewidth at 40 GHz is 400 Oe (ψ = 200 Oe |γ|/(40GHz), ΔH0 = 200 Oe). The results show a substantial symmetric dip or notch in the transmission, with the position of the notch tunable with an external field. For this geometry the maximum transmission loss is about 80 dB/cm. The largest attenuation occurs near the ferromagnetic resonance frequency (ωres ). A closer examination shows that the greatest transmission loss for each dip occurs about 0.8 GHz past the calculated FMR frequency from Eq.(4.4). 52 Transmission (dB/cm) 0 -20 4.0 microns SiO2 -40 0.5 microns BaM 200 Oe linewidth -60 1 kOe -80 5 kOe 3 kOe -100 30 35 40 45 50 Frequency (GHz) 55 60 Figure 4.2: Transmission as a function of frequency for three different values of the applied field. The thickness of the BaM is 0.5 μm. The thickness of the SiO2 film is 4.0 μm. The linewidth of the BaM is set at 400 Oe at 40 GHz by the parameters ΔH0 = 200 Oe and ψ/|γ| = 200 Oe /40 GHz. Fig. 4.3 shows results for a waveguide where the thicknesses of the dielectric layer and the hexagonal ferrite layer are reversed compared to those used in Fig. 4.2. The SiO2 layer is 0.5 microns thick and the hexagonal ferrite layer is 4.0 microns thick. The results show the same features as in Fig. 4.2, except the losses are all much greater, from about 80 dB/cm in Fig. 4.2 to about 600 dB/cm in Fig. 4.3, and the shift in frequency of the maximum transmission loss above the FMR frequency is also greater. In Fig. 4.3 the position of the notch is about 5 GHz higher than the calculated FMR frequency. Fig. 4.3 also shows that the notch is no longer symmetric in the case of the thicker hexagonal ferrite films. 53 Transmission (dB/cm) 0 -100 -200 -300 0.5 microns SiO2 -400 4.0 microns BaM 200 Oe linewidth -500 1 kOe -600 -700 35 3 kOe 40 5 kOe 45 50 55 60 Frequency (GHz) 65 Figure 4.3: Transmission as a function of frequency for three different values of the applied field. The thicknesses of the BaM and SiO2 are reversed from Fig. 4.2. The thickness of the BaM is 4.0 μm. This thickness of the SiO2 film is 0.5 μm. The linewidth of the BaM is the same as in Fig. 4.2. Note the attenuation at the notch is significantly larger than that seen in Fig. 4.2. The expected FMR behavior for ferromagnets is that the resonant frequency of the material shifts linearly with the applied field. However, in comparing the results from Figs. 4.2 and 4.3, a surprising feature emerges - the notch position depends not only on the applied field, but also on the thickness of the hexagonal ferrite. Fig. 4.4 plots the frequency of the maximum transmission loss versus the ratio of thickness of the BaM to the total thickness of the waveguide in a 3 kOe applied magnetic field. The graph shows a significant shift in the frequency of maximum attenuation away from the calculated ferromagnetic resonance frequency, Eq. (4.4). The shift is nearly linear as a function of the filling fraction. The maximum attenuation frequency of around 48.5 GHz occurs 54 when the hexagonal ferrite fills the entire waveguide. This is 6.0 GHz above the calculated 42.5 GHz FMR for a 3 kOe applied field. Frequency of Maximum Transmission Loss (GHz) 50 48 Ho = 3 kOe 46 44 42 0.0 0.2 0.4 0.6 0.8 1.0 BaM filling fraction Figure 4.4: Frequency of the notch as a function of the filling fraction ratio of the thickness of the BaM divided by the entire thickness between the metallic layers in the notch filter. The applied field is 3kOe, and the linewidth is the same as in Fig. 2 and Fig. 4.3. This shift in the notch position is a result obtained from the full boundary condition calculation. In an FMR experiment the effective wavevector is zero, i.e. one has a uniform oscillating magnetic field, and one expects the maximum absorption at the resonant frequency. However, in this experiment k ≠ 0. The propagation wavevector k y is dependent on all the kz’s (standing waves) in each region and on the thicknesses of the 55 various films through the boundary conditions. This relationship between the changes in thickness and the changes in standing waves influences the frequency of the transmission loss of the wave. We can obtain some idea of why the frequency of the maximum transmission loss is not simply at the resonance frequency given by Eq. (4.4). If one considers the propagation of magnetostatic waves in an infinite material, then Maxwell's equations may be reduced to an equation for the magnetic scalar potential φ which obeys an anisotropic Laplace equation: 1 2 2 2 x2 y2 z2 0 (4.27) A wave in this material will have the form Ae i(k y y k zz) e i t (4.28) and substitution of this form into Eq. (4.27) gives a connection between the propagation wavevectors and the frequency through the permeability. 1( ) k 2z (4.29) k 2y If we use Eq. (2) and let the damping go to zero, we obtain ( / )2 H2 4 MH k 2z 1 k 2y (4.30) One should note two a special limits: 1) if propagation is parallel to the magnetic field, then ω = |γ|H and 2) for propagation perpendicular to the magnetic field ω2 = |γ|2H ( H + 4πM ). The point is that the frequency of the wave is not necessarily just |γ|H , but 56 depends on the direction of propagation. A similar analysis can be found in Xu Zuo et al’s work.39,40 The effective direction of propagation in the dynamic problem depends on the full use of Maxwell's equations. For an out-of-plane hexagonal ferrite filter with the dimensions used in Fig. 2, our results show that the ratio of k z2/ky2 is typically in the 1-13 range. According to the magnetostatic case given by Eq. (4.30), this would produce resonance at a frequency shifted about 0.5-2.5 GHz above the FMR predicted by Eq. (4.7). According to Fig. 4.2 and Fig. 4.4 the frequency of maximum attenuation is shifted by 0.8 GHz above the predicted frequency, which falls into the 0.5-2.5 GHz range. For the parameters in Fig. 4.3, our results kz2/ky2 show magnitudes between 0.2 and 5.0. According to Eq. (4.30), this would make the shift range between 1-5 GHz. The shift of 6.0 GHz shown in Figs. 4.3 and 4.4 is just out of this range. Because we are not examining a magnetostatic case, the results are not expected to match perfectly. However, Eq. (4.30) does give results for wavevectors in the range of our calculation results that are shifted off the Eq. (4.7) results by the same order of magnitude. These results give an intuition that the frequency of maximum absorption is close to ω = |γ|H but increases slightly from that value depending on the values of kz and ky, which are needed to solve the boundary conditions. This implies that the resonant frequency is indeed related to the thickness of the waveguide. There is also a shift in the frequency of the maximum transmission loss from the damping. The solid line in Fig. 4.5 is a graph of the frequency of the notch versus the FMR linewidth. The linewidth is for a frequency of 40 GHz for a 3 kOe field. The dashed line in this figure is derived from Eq. (4.4) with the damping included. There is 57 approximately 0.8 GHz difference between them at a 1000 Oe linewidth. That difference increases to about 1.6 GHz difference at a linewidth of 5000 Oe. At low linewidths, the difference between these two values can be attributed to the shift in frequency as a function of thickness as discussed earlier. As the linewidth increases it also affects the Frquency of Maximum Transmission Loss (GHz) wavevector, making the thickness shift greater. 44.5 H0= 3 kOe 44.0 4.0 SiO2 43.5 0.5 BaM 43.0 42.5 42.0 Predicted Resonance from Equation 4 1000 2000 3000 4000 5000 FMR linewidth at 40.0GHz (Oe) Figure 4.5: Frequency of the maximum attenuation as a function of linewidth. The thickness of the BaM and SiO2 layers is the same as in Fig 4.2, as are the linewidth parameters. The solid line is the frequency of maximum attenuation obtained from the full boundary condition calculation. The dashed line is the calculated FMR frequency from Eq. (4.7). In addition to the results for the out-of-plane hexagonal ferrite, we also consider results for a magnetic layer that is metallic with an out-of-plane magnetization. Such an 58 orientation could be useful, even for the metallic ferromagnet, because this geometry allows one to use a large out-of-plane magnetic field over a short distance, i.e. the thickness of the film and substrate. In contrast, the in-plane geometry requires a magnetic field extending over several millimeters. These results are based on the same two-layer geometry shown in Fig. 4.1, except that the hexagonal ferrite is replaced by Permalloy. The parameters used for Permalloy are M = 0.850 kG, ζ = 1.67 x 105 C2/Nm2-sec for its conductivity, and |γ| = 1.80 x 107 rad/G-sec. Fig. 4.6 shows a similar behavior for transmission as that seen in Figs. 4.2 and 4.3. However, the operational frequencies for devices made out of Permalloy are considerably lower than for the BaM devices, even when much larger external fields are used. There are two reasons for this. Permalloy does not have the large anisotropy field of BaM and the larger M in Permalloy also means a larger demagnetizing field which must be overcome by the external field. Barium hexagonal ferrite can also be grown to have its anisotropy field in-plane. The theory for this geometry has been published earlier, for more discussion on this consider references.3,30,45 For the purpose of this paper, we only present the results of this theory. Fig. 4.7 shows the transmission loss results for the same film thicknesses and damping parameters as used in Fig. 4.2; the difference is that the permeability used for the calculation is for an in-plane magnetization with an in-plane anisotropy field. The most obvious feature is that the in-plane geometry results in a higher operating frequency for the same applied field. For the in-plane case, the static demagnetizing field does not reduce the effect of the external field, and the dynamic demagnetizing fields also increase the frequency. The magnitude of the maximum transmission loss is about 20 dB more for 59 the out-of-plane case (Fig. 4.2) compared to the in-plane case in Fig. 4.7. We note that in Fig. 4.7 the magnitude for maximum attenuation decreases as the external field is increased, this is in contrast with the results in Fig. 4.2. The behavior of the linewidth and the maximum attenuation can be complicated. It can either increase or decrease as a function of field or frequency. This effect is discussed by Kuanr et al.41 Transmission (dB/cm) 0 -15 -30 4.0 m SiO2 -45 0.5 m permalloy 400 Oe linewidth -60 13 kOe 14 kOe -75 8 10 12 15 kOe 14 16 18 20 Frequency (GHz) Figure 4.6: Transmission as a function of frequency for three different fields for a notch filter with Permalloy as the active element. The external field is directed out-of-plane as shown in Fig. 4.1. Note that the frequencies are all considerably lower for the Permalloy film compared to the BaM film in Fig. 4.2. 60 Transmission (dB/cm) 0 -10 -20 4.0 microns SiO2 -30 0.5 microns BaM 400 Oe linewidth -40 -50 1 kOe 3 kOe 2 kOe -60 45 50 55 60 65 Frequency (GHz) Figure 4.7: Transmission as a function of frequency for a BaM filter with three different values of the applied field. The anisotropy axis and the applied field are now directed in-plane, and parallel to the propagation direction. The thickness of the BaM is 0.5 μm. The thickness of the SiO2 film is 4.0 μm. Note that the frequencies are all higher than those found for the BaM film with the outof-plane geometry as shown in Fig. 4.2. Using the waveguide as a Phaseshifter The planar waveguides with metallic ferromagnetic films can also be used as phase shifters, however the performance in the typical microstrip filter shows significant damping. In this section will explore the phase shifting properties of a microstrip filter using BaM as the active material. 61 Fig. 4.8 shows, for fixed frequencies, the phase difference in the real part of the propagation wavevector, ky, as a function of a change in the external magnetic field. To be more specific, Fig. 4.8 plots the phase difference of ky(Ho = 2kOe) – ky(Ho). Phase difference (rad/cm) 20 15 10 5 0 -5 -10 -15 48 GHz 45 GHz 51 GHz 4.0 microns SiO2 0.8 microns BaM 200 Oe linewidth 0 2 4 6 Field Difference from 2 kOe Figure 4.8: Phase shift at different frequencies as a function of the change in the magnetic field from 2 kOe. The thickness of the BaM is 0.8 μm. The thickness of the SiO2 film is 4.0 μm. The linewidth of the BaM is set at 200 Oe at 40 GHz by the parameters ΔH0 = 100 Oe and ψ= 100 Oe |γ| /40 GHz. These results are for the out-of-plane geometry. The results shown are found with the boundary condition and root finding method outlined in the theory section. The BaM layer in Fig. 4.8 is thicker than the one used in Fig. 4.2 (0.8 μm compared to 0.5 μm) and the damping is lower (ΔH = 200 Oe, with ψ = 100 Oe |γ|/(40 Ghz), and ΔH0 = 100 Oe). One sees that substantial phase shifts are possible; however the largest values all occur near the ferromagnetic resonance 62 frequency. This means that there is a lot of attenuation, which makes most of the shifted signal unusable. For practical devices, a transmission loss of no more than -2 dB/cm is a reasonable cut off. Fig. 4.9 is Fig. 4.8 redrawn with all phase shifts that operate at a field with transmission losses below -2 dB/cm removed. The phase shifting range has been substantially limited. For the 45 GHz signal, the phase can be shifted from 0 rad/cm to 0.3 rad/cm at 0.16 kOe away from 2 kOe. For larger field differences the phase shift ranges from 5 rad/cm to 3 rad/cm. For higher frequencies, the general trends of the Phase difference (rad/cm) behavior are the same but, the phase shifts become smaller. 5 4 3 2 1 0 -1 -2 -2 dB/cm cut off 45 GHz 45 GHz 48 GHz 51 GHz 4.0 microns SiO2 0.8 microns BaM 200 Oe linewidth 48 GHz 51 GHz 0 2 4 6 Field Difference from 2 kOe Figure 4.9: Phase shift at different frequencies as a function of the change in the magnetic field from 2 kOe. This figure uses the same data as Fig. 4.8, except that it omits frequencies that produce transmissions less than -2 dB/cm. These results are for the out-of-plane geometry. 63 To achieve larger phase shifts, one approach might be to make the layers of the BaM thicker. In addition, to have a wider range of usable applied fields, the damping could be decreased, thereby decreasing the linewidths of the transmission loss dips. These two strategies are used in Fig. 4.10 which shows the same type of graph as Fig. 4.9 but, with a BaM layer that is twice as thick (BaM = 1.6 μm), and has half the damping (ΔH = 100 Oe, with ψ = 50 Oe |γ|/(40GHz), and ΔH0 = 50 Oe). Furthermore, the investigated frequencies are increased by +3 GHz each. The phase shifts are noticeably larger, and the width of the lower field ranges has also been increased. For example, the 48 GHz phase shift now ranges from 8.0 rad/cm to 4.4 rad/cm as the field difference changes from +3.5 kOe to +8 kOe. Again the phase shifts at higher frequencies are somewhat smaller. Phase difference (rad/cm) -2 dB/cm cut off 8 6 4 2 0 -2 -4 51 GHz 48 GHz 54 GHz 4.0 microns SiO2 54 GHz 48 GHz 51 GHz 1.6 microns BaM 100 Oe linewidth 0 2 4 6 8 Field Difference from 2 kOe Figure 4.10: Phase shift at different frequencies as a function of the change in the magnetic field from 2 kOe. Although this figure is the same type of graph as Fig. 9, the parameters in this figure differ from Fig. 9. The thickness of the BaM is 1.6 μm. This thickness of the SiO2 film is 4.0 μm. The linewidth of the BaM is set at 100 Oe at 40 GHz by the parameters ΔH 0 = 50 Oe and ψ = 50 Oe |γ| /40 GHz. These results are for the out-of-plane geometry. 64 We have also studied the phase shifting behavior for the geometry where the magnetization and anisotropy are in-plane. Because the demagnetizing field is no longer opposing the anisotropy field, results are readily obtained using lower applied fields. Fig. 4.11 compares differences in phase between an original zero-applied-field state and one at a higher field of Ho. This figure assumes that the magnetization has already been established and is being maintained by the anisotropy field. For 57 GHz, the high field phase shift varies from 5.5 rad/cm to 4.0 rad/cm as the field changes from 3.4 kOe to 7 Phase Difference (rad/cm) kOe. 7 6 5 4 3 2 1 0 -1 -2 -2 dB/cm cut off 57 GHz 60 GHz 63 GHz 57 GHz 4.0 m SiO2 0.8 m BaM 200 Oe linewidth 60 GHz 0 63 GHz 1 2 3 4 5 6 Applied H field (kOe) 7 Figure 4.11: Phase shift at different frequencies as a function of the change in the magnetic field from 0 kOe. The parameters are the same as in Fig. 4.9, but the geometry is for an in-plane magnetization. 65 Fig. 4.12 graphs the phase difference versus the field difference for a thicker, less damped BaM layer in the in-plane geometry. The new parameters are 1.6 μm of BaM with a damping of ΔH = 100 Oe (ψ = 50 Oe |γ|/(40GHz ), and ΔH0 = 50 Oe). As expected, one finds larger phase shifts and a larger field range compared to the results of Fig. 4.11. For example, for 57 GHz, the high field shifts now range from 10.5 rad/cm to Phase Difference (rad/cm) 7.0 rad/cm in a range of fields from 3.5 kOe to 7.0 kOe. 12 10 8 6 4 2 0 -2 -4 -2 dB/cm cut off 57 GHz 57 GHz 60 GHz 63 GHz 4.0 microns SiO2 1.6 microns BAM 100 Oe linewidth 60 GHz 63 GHz 0 1 2 3 4 5 6 Applied H field (kOe) 7 Figure 4.12: Phase shift at different frequencies as a function of the change in the magnetic field from 0 kOe. The parameters in this graph are the same as in Fig. 4.10, but the geometry is for an in-plane magnetization. 66 Experimental Comparison A notch filter experiment with BaM components was performed by Z. Wang et al 46 . The experiment transmitted a signal down a co-planar wave guide with a 5 μm thick film of BaM placed on top of it, followed by a thick layer of sapphire (>150 μm). There was also be a small air gap in between the co-planar waveguide and the BaM. Fig. 4.13 shows a) the transmission of that signal as a function of frequency, and b) the phase of the signal as a function of frequency. The experiment was performed with four different applied fields: 4.62 kOe, 5.17 kOe, 5.62 kOe, and 6.31 kOe. We compare these experimental results with Fig. 4.14, theoretical results we calculated for an infinite planar wave guide with three layers: a 5 μm thick layer of BaM , 150 μm thick layer of sapphire, and a 5 μm air gap. Figure 4.13: Experimental results for a coplanar waveguide with 5 μm of BaM on top of it. Figure 4.14: Theoretical results for a infinite planar wave guide with a 5 μm of BaM magnetic layer. 67 The theoretic results match quite well with the experimental results. The transmission dips have an error of about ± 1 GHz in frequency. The magnitude of the transmission dips have an error less than ± 1 dB. The magnitudes of the phases don’t match but, this was expected due to the experiment having sections of the signal line without BaM present and the theory only considered the regions with BaM. Even though the magnitudes of phase seems not to match, they are still qualitatively close in frequency behavior. Conclusions We have theoretically studied the performance of tunable notch filters and phase shifters which use BaM hexagonal ferrites as the active element in a small planar device. The devices operate at frequencies in the 40-60 GHz range with relatively low applied magnetic fields. We have considered anisotropy fields both out-of-plane and in-plane. As a notch filter, the device has significant attenuation at the notch, typically in the 60 dB/cm range, even for BaM films which are less than a micron in thickness. Because hexagonal ferrite is not a metal, electromagnetic waves propagating in the filters do not have the significant eddy current losses which can be found in filters using ferromagnetic metals. This allows devices to be made with thicker hexagonal ferrite layers and can lead to larger attenuations at the notch and to larger phase shifts. As a phase shifting device, the out-of-plane hexagonal ferrite geometry requires strong fields, between 6 kOe and 8 kOe. However phase shifts on the order of 360o/cm are possible for materials which are about one micron in thickness and which have an FMR linewidth less than 200 Oe at 40 GHz. An in-plane hexagonal ferrite phase shifting 68 device with the same thickness and damping parameters can also show a significant tunable phase shift at lower fields between 3 kOe and 6 kOe. CHAPTER V Non-reciprocal Ferromagentic Devices using Attenuated Total Reflection Polaritons are electromagnetic oscillations in a material strongly coupled with magnetic or electric dipole excitations. There has been a good deal of research done on surface polaritons.47-51 A well known and widely used technique for studying the properties of both bulk and surface polaritons is attenuated total reflection (ATR). ATR is also used in material analysis and biological analysis.52-55 Additional work has shown the connection of polaritons and plasmons with ATR.56-59 There have been a number of experimental studies of antiferromagnetic materials using ATR or reflectivity as a probe of both bulk and surface polaritons.60-62 Previous work for ATR considered an electromagnetic wave interacting with a three layered planar structure as shown in Fig. 5.1.56 For the three layer case, the first layer is generally a dielectric prism guiding electromagnetic waves. The waves encounter an interface with a second layer, which we will call the gap layer (in previous work, the gap layer was often taken to be a vacuum). If the incident angle that the incoming wave makes with the interface is adequately large, and the permittivity of the prisim is greater than the permittivity of the gap, then the electromagnetic wave totally internally reflects off the interface, and the reflected power of the wave is equal to the incident power. However, there is also an evanescent wave in the gap which exponentially decays with distance from the upper interface. If the gap layer is thin enough, a third layer beneath it 70 could interact with the evanescent wave. If the third layer is a semi-infinite magnetic material, the evanescent wave can excite a magnetic polariton in this layer. This takes energy away from the reflected wave, typically in a limited frequency range. Figure 5.1: Three-layer ATR geometry. The prism layer is treated as semi-infinite. The gap layer has a finite thickness d. The magnetic layer is also semi-infinite. M is the magnetization of the magnetic layer and Ho is an applied magnetic field. This chapter explores the physics of attenuated total reflection (ATR) in a multiple layered heterostructure in order to find possible alternative methods for creating non-reciprocal devices. A non-reciprocal device is a system where the reflection coefficient is different if the directions of the incident and reflected waves are reversed. The origin of non-reciprocal ATR comes from nonreciprocal polaritons traveling in opposite directions on the surface of a magnetic material. These polaritons attenuate the reflection of an incoming electromagnetic wave, resulting in a possible isolator or 71 circulator device application. In fact, we propose a device with an effective ATR-like structure which can be built on a planar geometry using microstrip-like structures. In addition to the Fig. 5.1 geometry, this chapter examines a different geometry. This different case, illustrated in Fig. 5.2, involves a fourth layer, as also investigated by Ruppin.61,62 The fourth layer is a semi-infinite dielectric layer. This four-layer geometry is necessary for our proposed device. Figure 5.2: Four-layer ATR geometry. The incoming electromagnetic wave reflects off the interface between the prism layer and the gap layer at an incident angle (θ). Permitivities ε 1, ε2, and ε4 are all for dielectric materials. The magnetic layer has permittivity and permeability given by ε3, μ1, and μt . The gap thickness and magnetic layer thickness are given by d 1 and d2. There is an applied static magnetic field in the z-direction (H0). The key result of our calculations is that the ATR geometry exhibits strong nonreciprocal behavior even for thin magnetic films. For relatively thick gap and magnetic layers (0.05 cm and higher), the non-reciprocity follows the results for surface polaritons defined by the gap material and the magnetic material. For thinner magnetic films, the 72 non-reciprocal behavior is complex. Results are presented for yttrium iron garnet (YIG) and M type barium hexagonal ferrites (BaM). The frequency range of non-reciprocal behavior can be quite large and is tunable both by an external field and by changing the thicknesses of the gap layer, the magnetic film, or both. For example, nonreciprocal behavior is found in BaM from 45–80 GHz. ATR geometries display a wide range of possible applications, such as notchfilters, bandpass fileter, and isolators. The type of application possible depends not only on the materials, but also on choosing the incident wave angle, the gap thicknesss, and the magnetic layer thickness. Surface Polaritons and Three-layer ATR We examine solutions for electromagnetic waves with the electric field in the z direction, transverse to the planar structure, and parallel to a static magnetic field Ho. These transverse electric (TE) modes propagate is the xy plane. Before we calculate the ATR results, we obtain the dispersion relation for a surface polariton on a magnetic medium. We start with the permeability tensor of a given magnetic material: i 1 i 0 t t 1 0 0 0 1 (5.1). This particular tensor is for a ferromagnetic material with an external magnetic field applied in the z-direction (see Fig. 5.1 or Fig. 5.2). The factors μ1 and μt are well-known functions of the applied field in the z-direction, Ho, the frequency of the interacting electromagnetic wave, ω, the magnetization of the material, M, and the linewidth of that material, ΔH.60 By using Maxwell’s equations, a wave equation can be made and 73 evaluated at the boundary between two different materials (one of which is the magnetic material defined by permeability). From this evaluation the dispersion relationship of a magnetic spin wave confined to the interface between a semi-infinite dielectric and a semi-infinite magnetic medium has been found to be : 2 kx 2 1 c ( v 1)( 1 v d m 2 ) 2 2 t 2 d ( v t 1) 2 1 1 ( 1( 2 d 2 m ) d ( m v 1 1)) 2 4 t (5.2). The dielectric material’s permittivity is εd and the magnetic material’s is εm. The components of the magnetic material’s permeability tensor are μ1 and μt. The propagation wavevector is kx (which is parallel to the surface), c is the speed of light, and ω is the wave’s frequency. The factor μv is the voight permeability and is defined as 2 2 1 t (5.3). v 1 Eq (5.2) is an extension to the result found in Harstien et al.48 In that work they considered only the case where εd is 1. In our device, we will need to consider different values of εd. The dispersion relationship for a surface mode is nonreciprocal. Notice that there are four possible solutions for Eq. (5.2). From the initial conditions used for this equation we find that only two of the four solutions are physical. The two physical solutions are: kx kx 1 ( v 1)( 1 v d m ) 2 c 2 t ( 1 c ( v 1 1)( v d m ) 2 2 d v ( 1 2 d v 1 1) 2 2 t t 1 ( 1( 2 2 d m ) d m ( v 1 1)) 2 4 (5.4), t ( 1( t 1 1) 2 4 2 t 2 d 2 m ) d m ( v 1 1)) (5.5). 74 Eqs. (5.4) and (5.5) show that there is a different dispersion relationship depending on which direction the wave is propagating (notice the different sign at the beginning of Eq. (5.5) and in its radical). Surface spin waves are non-reciprocal and therefore we expect non-reciprocal behavior from interactions with them. Eq. (5.4) and (5.5) establish what polaritons can be excited on the surface of a magnetic material. In addition to the dispersion relationship for spin waves on the surface of a magnetic material, we also consider spin waves in the bulk of the material. This dispersion relationship is much simpler, kx 2 ky 2 2 c2 v (5.6). Where kx and ky are the components of the propagation wavevector k in the magnetic material. The point of this is that the bulk mode spin waves do not exhibit any directional preference. Solutions for k in one direction are the same as for k in the opposite direction. Now we consider the ATR of the three layer case (Fig. 5.1). The prism contains a TE electromagnetic wave traveling within it. Both the prism and magnetic layers are semi-infinite. The prism layer is defined by the permittivity ε1 and has a scalar permeability of 1. The gap layer has a permittivity of ε2, where ε2 < ε1, and a permeability of 1. The third layer has a permittivity of ε3, and a permeability tensor defined by Eq (5.1). The angle incidence for the incoming electromagnetic wave in the prism is defined by θ. For θ greater than or equal to the critical angle established by ε 1 and ε2, an evanescent surface wave travels along the interface of the prism and the gap layers. Depending on the gap’s thickness, the evanescent wave will interact with the semi-infinite magnetic layer and create polaritons. The energy required to make the 75 polaritons must come from the incident electromagnetic wave, therefore the reflected wave will have energy less than the incident. The solution for the reflection coefficient in the prism has been found in previous work56 by finding wave-like solutions using the boundary conditions of the layers. We present the results for the reflection amplitude here R3 A 1 A 1 layer (5.7). Where, A Fe Fe k y2 2ik y 2 d 2ik y 2 d 1 k y3 v F k y2 1 v 1 i 1 sin c 1 k y2 t 1 sin 1 k y3 t 1 1 sin c (5.8) (5.9) c The subscripts in Eqs. (5.8) and (5.9) refer to the layer which each value is associated with. The quantities ky2, and ky3 are wavevectors in the y-direction in layers 2 and 3 as shown in Fig. 5.2. These values will be defined explicitly later on in this paper; for now we shall say that the ky’s are functions of the electromagnetic wave’s frequency, permeability, and permittivity of the layer being considered. The thickness of the second layer is d (this notation will change in the four-layer case). The results for the three-layer case will provide a check for the solutions to the four layer case, which we will now discuss. 76 Four-layer ATR We present solutions for the reflection coefficients of the four-layer case in more detail. The four-layer geometry is shown in Fig. 5.2.We again consider a TE electromagnetic wave in each layer of the geometry. If we construct a wave solution in each layer that is a superposition of an incident wave and a reflected wave. Using the coordinates in Fig 5.2, we propose a position and time dependant solution for the zcomponent of the E field (since we are only considering a TE mode) in each of the four layers to be a summation of two waves: E zn ik y n y A 2n 1e A 2n e ik y n y eik x x e i t (5.10). In Eq. (5.10), n is the layer index, the A’s are the coefficients for individual waves being summed, kx is the propagation wavevector parallel to the interfaces, ω is the angular frequency of the wave, and kyn is the y-component of the wavevector in layer n. The z-component of the wave vector has been set to zero. For the next few steps, we will need to consider Maxwell’s two curl equations: E H 1 c H t (5.11) E t 1 c (5.12) The H-fields in each layer can be derived from the E-field using Eq. (5.11). H xn c vn H yn c vn E zn y E zn x tn i 1n i tn 1n E zn x E zn y (5.13) (5.14) 77 The solutions for the kyn components of the wave vector in each layer can be solved in terms of the kx component, the angular velocity ω, the material’s permittivity εn, and the material’s permeability. To do this, first take a curl of Eq. (5.12). Then substitute the left side of Eq. (5.11) into the right side of the double curl of H in order to arrive at a wave equation H H 2 2 c2 t (5.15). Using Eqs. (5.9) and Eqs (5.13)-(5.15), we obtain a connection between the ky values, the parallel wavevector kx, the wave frequency, and the material’s properties: k yn kx 2 2 c2 n vn (5.16). Given an incident wave, which specifies ω, ε, kx and μv , the ky values in each layer can be uniquely determined..We set kx based on the angle of incidence (θ) and the wave vector magnitude (k) of the electromagnetic wave in the prism layer. The magnitude of the wave vector in the prism would be: k c 1 (5.17). This gives the x-component of the wave vector in all layers: kx k sin (5.18). We can now represent the E-field and H-field of an electromagnetic wave in every layer of the geometry with the only unknown quantities being the incident and reflected coefficients in each layer. In the first layer, we have two unknown coefficients, A1 and A2. The second and third layers each have another two unknowns. The fourth layer only has one unknown coefficient because the wave will continue on infinitely in 78 the y-direction and therefore does not have a reflected part. This gives us a total of seven unknowns. The boundary conditions are used to solve for these unknowns. The z-component of the E-field must be continuous across the boundaries. From this we produce 3 equations, one for each boundary. Three more equations can be produced from the xcomponent of the H-field also being continuous at the boundaries. We now have six equations with seven unknowns. In order to reduce the number of unknowns, we divide each equation by A1. This will give us solutions for six unknown ratios. The first ratio, (A2/A1) tells us how much of the original incoming wave is reflected from the layered structure. This is the reflection amplitude. All of the boundary condition equations at the interfaces of y = 0, d1, and d1+d2 can now be written in a 6 1 k y1 1 k y2 ik y 2 d1 0 e 0 k y2 e 0 0 ik y 2 d1 6 matrix form: 1 k y2 e ik y 2 d1 k y2 e 0 0 0 ik y 2 d1 e k y3 0 0 0 e k y3 i ik y 2 d1 t i 0 0 e e kx 1 ik y3 ( d1 d 2 ) t 1 kx e 0 0 ik y 2 d1 ik y 3d1 t k y3 i v3 1 e k y3 i ik y 3d1 0 v3 ik y3 ( d1 d 2 ) ik y 3 ( d1 d 2 ) v3 0 e kx t kx e 1 e ik y 4 ( d1 d 2 ) ik y 3 ( d1 d 2 ) k y4 e v3 ik y 4 ( d1 d 2 ) A2 A1 A3 A1 A4 A1 A5 A1 A6 A1 A7 A1 1 k y1 0 0 0 0 (5.19). This set of linear equations can be solved numerically. The reflection coefficient we’ve defined as the ratio of the coefficients A2 and A1. R A2 A1 (5.20). 79 However, in the results section we will be examining the return loss of the geometry, which is the log of the modulus squared of the reflection coefficient: * A A L 10 log( R ) 10 log 2 * 2 A1 A1 2 (5.21). Results and Discussion In our calculations we considered two different magnetic materials as examples, yittrium iron garnet (YIG) and M-type barium hexagonal ferrite (BaM). The permittivity used for YIG was ε3 = 15.0, the saturation magnetization was M = 141.6 G and the gyromagnetic ratio was |γ| = 2.801 GHz/kOe. The linewidth used for the YIG was about 100 Oe at 10 GHz. The linewidth was assumed to change linearly with frequency at a rate of 5 Oe per GHz. For the BaM, the permittivity used was ε3 = 10.0, the magnetization was M = 334 G and the gyromagnetic ratio was also |γ| = 2.801 GHz/kOe. The BaM also had a strong crystalline anisotropy constant of K = 2.74x106 egr/cm3, which results in an anisotropy field of 16.4 kOe. The damping used for BaM was 200 Oe at 40 GHz and changed with frequency at a rate of 2.5 Oe per GHz. All figures have an applied field of 3 kOe in the z-direction, unless specifically stated otherwise. Other unique parameters will be given for each particular example. Fig. 5.3 shows frequency versus propagation wavevector (kx) for surface and bulk polariton solutions for a two layer, semi-infinite geometry. The solid blue lines are the solutions for the surface polaritons, given by Eqs. (5.4) and (5.5). The bulk polariton modes are the blue hatched regions, given by Eq. (5.6). The permittivity for the semiinfinite dielectric layer was εd = 1.5, and the semi-infinite magnetic layer was YIG. This 80 demonstrates the non-reciprocal nature of the surface polaritons. The surface modes for + kx are clearly different than the surface modes for – kx. scan lines Frequency (GHz) 20 18 16 14 12 surface modes bulk modes 10 -15 -10 -5 0 5 10 15 Propagation Vector kx (1/cm) Figure 5.3: Dispersion relationship for allowed bulk polaritons in YIG and surface polariton modes for two semi-infinite materials, one dielectric and one magnetic, based on equations 4, 5, and 6. The dielectric material has a permittivity of εd = 1.5, and the magnetic layer uses YIG parameters. Also included are scan lines, k x /c 1 sin , based on an incident EM wave in an ATR geometry for a prism with a dielectric constant of ε 1 = 11.6. The incident angle is θ = ± 70o. The black lines in Fig. 5.3 maps out scan line indicating the dispersion relation for a light wave in the prism. This is given by k x /c 1 sin . One expects a transfer of energy from the incident beam to the magnetic polariton when the scan line overlaps with the surface or bulk modes. In Fig. 5.3 the black scan lines show results for a prism with permittivity of ε1 = 11.6 and incident angles of θ = ±70o. The positive scan line intersects 81 the surface polariton mode at about 17.9 GHz. On the other side of the graph, the negative scan line intersects the surface mode at about 10.4 GHz. We expect that the lowest reflectivity from an ATR geometry with these incident angles would be at these frequencies. Fig. 5.4 shows the ATR plot of the return loss versus frequency for incident angles θ=±70o in the three layer geometry, with a prism permittivity of ε =11.6, gap permittivity of ε2=1.5, and YIG as the semi-infinite magnetic material. The gap permittivity matches the value for the dielectric in Fig. 5.3. The thickness of the gap is 0.05 cm. The black line is for θ = +70o and the red line is for the opposite direction of propagation, θ = -70o. The lowest return loss values occur about where we would expect. For the positive propagation vector, θ = +70o, the lowest return loss is at 18.5 GHz. For the negative propagation vector, θ = -70o, there is a sharp decrease in return loss at around 10.4 GHz. We note that the lower frequency region of nonreciprocity is near the ferromagnetic resonance frequency (near 9 GHz). The nonreciprocity for the higher frequencies (near 17 GHz) is related to the surface mode and the upper bulk band. The ATR values do not match exactly with the results for the surface polariton, 18.5 GHz compared to 17.9 GHz, but this should also be expected. The Fig. 5.3 values are based on a different geometry than the ATR geometry, therefore the surface polariton modes excited should not be exactly the same. 82 -kx direction 0 Return Loss (dB) -5 -10 +kx direction -15 -20 -25 -30 5 10 15 20 25 Frequency (GHz) Figure 5.4: A plot of the return loss verses frequency for a three-layer ATR case with YIG as the magnetic material. The prism has a permittivity of ε1 = 11.6. The gap is 0.05 cm thick and has a permittivity of ε2 = 1.5. The incident angles are θ = ±70o and the applied field is Ho = 3 kOe. There are two regions of strong nonreciprocity, one near the usual ferromagnetic resonance frequency (near 9 GHz) and one substantially higher in frequency related to the upper bulk polariton band. To get a more complete picture, Fig. 5.5 shows a return loss map, which is the return losses for all possible frequencies and propagation wavevectors in a three-layer ATR geometry. The layer materials and their thicknesses are the same as in Fig. 5.4. In this figure, we only include physical results. Regions for non-physical results are shown with a hatched pattern. The outer hatched region represents incident angles that are greater than 90o kx c 1 (5.22). 83 The inner hatched region represents propagation phase velocities greater than the speed of light, kx c (5.23). White Lines = surface modes from Fig. 5.3 20 Return 18 Frequency (GHz) Loss (dB) 0 16 -3.000 -6.000 14 -9.000 -12.00 12 -15.00 10 -15 -10 -5 0 5 10 Propagation Vector kx (1/cm) 15 Figure 5.5: A return loss map for frequency versus propagation wavevector kx for a three-layer ATR geometry with YIG as the magnetic material. In addition, the dispersion relationships for surface polaritons between a semi-infinite YIG layer and a semi-infinite dielectric layer with εd = 1.5 are shown as white lines. We see strong nonreciprocity occurs near the intersection of the surface polariton with the upper bulk polariton band. The prism has a permittivity of ε 1 = 11.6, the gap is 0.05 cm thick and has a permittivity of ε2 = 1.5. The applied field is Ho = 3 kOe. The return loss map of Fig. 5.5 outlines both the bulk and surface polariton regions seen in Fig. 5.3. The lowest return loss features in Fig. 5.5 seem to correspond to 84 where the surface modes and bulk modes intersect. The white lines in Fig. 5.5 show the dispersion relation for surface modes between a semi-infinite dielectric with permittivity εd = 1.5 and semi-infinite YIG from Fig. 5.3. The white lines are there to draw attention to the correlation between the surface modes and points of low reflectivity. The range of nonreciprocal behavior is immediately obvious in Fig. 5.5. Using these types of plots, we were able to see that as the gap became thicker the structure became more reflective over all due to the weaker interaction of the evanescent wave with the magnetic layer. The regions of lowest reflection were closely related to the surface polariton between the gap material and the magnetic layer. With a thinner gap, the regions of lowest reflection are due the surface polariton between the prism and the magnetic layer rather than the gap and the magnetic layer. We now consider the four layer geometry, where the magnetic layer has a finite thickness, followed by a semi-infinite dielectric with a relatively high permittivity of ε4 = 15.0. In Fig. 5.6, we show the evolution of the reflected signal as the thickness of the magnetic film is increased from 0.05 cm to 3 cm. Each graph in Fig. 5.6 contains the semi-infinite YIG results, as dotted lines and the forward and reverse propagation return loss for the finite YIG film as solid lines. As the thickness is increased, we see a gradual convergence with the semi-infinite case except with oscillatory deviations that could be attributed to Fabry-Perot interference. Even for the thinnest films, however, one can find substantial nonreciprocal behavior. 85 0 -10 -kx -20 Return Loss (dB) -30 -40 0 Three-layer case +kx +kx a) b) YIG = 0.05 cm -kx YIG = 0.1 cm -10 -kx -20 -30 -40 0 c) +kx YIG = 0.3 cm -kx d) +kx YIG = 0.5 cm -10 -kx -20 -30 -40 e) 5 +kx YIG = 1 cm 10 15 20 Frequency (GHz) -kx f) 25 5 +kx YIG = 3 cm 10 15 20 25 Frequency (GHz) Figure 5.6: A series of four-layer ATR return loss plots, each with a different thickness for the YIG magnetic layer. The dashed lines, which are the same in a) through f), show the results for a infinitely thick YIG film. The incident angles are θ = ± 70o for all cases. The gap thickness is 0.05 cm. We now look at results involving barium hexagonal ferrite (BaM) as the magnetic material. This is motivated by the fact that BaM has a large internal anisotropy and the resulting excitations are therefore at a much higher frequency than what is found for YIG. Fig. 5.7 shows a return loss map of frequency vs. wavevector for a four-layer structure. The prism permittivity is ε1 = 11.6, the dielectric gap is 0.03 cm thick with a permittivity of ε2 = 1.5, the BaM layer is 0.1 cm thick, and the semi-infinite 4th layer has a 86 permittivity of ε4 = 15. The two white lines show the surface modes for a structure based on a semi-infinite BaM and semi-infinite gap dielectric according to Eqs. (5.4) and (5.5). White lines are the results for surface modes Frequency (GHz) 85 Return Loss (dB) 80 75 70 0 -3.000 65 -9.000 60 55 -15.00 -40 -20 0 20 40 Propagation Vector kx (1/cm) Figure 5.7: A return loss map for frequency versus propagation wavevector kx for a four-layer ATR geometry with BaM as the magnetic material. There is strong nonreciprocal behavior in the 57-59 GHz and in the 71-75 GHz ranges. The dielectric gap is 0.03 cm thick, the BaM layer is 0.1 cm thick. The applied field is Ho = 3 kOe. The two white lines show the surface modes for a structure based on semi-infinite BaM and semi-infinite dielectric εd = 1.5. The overall results for Fig. 5.7 display clear non-reciprocal behavior. On the positive kx side of the graph, there is a relatively strong signature along the path of the surface polariton, in the 70-75 GHz range. On the negative side, there are regions of low reflection where we would expect the negative propagating surface polariton in the 57 to 59 GHz frequency range. 87 Fig. 5.8 shows results for return loss as a function of frequency for 40 o with the same geometry and parameters as Fig. 5.7. The black line shows the results for forward propagation, and the red line is for the reversed signal. We can observe the acute non-reciprocity at 73 GHz, where there is over a 45 dB return loss difference between the positive and negative propagation. Around the dip, ranging between 72 to 75 GHz, there is a band of frequencies where the positive propagation has return losses below -10 dB and the negative propagation has return losses above -1 dB. This region behaves as an isolator, allowing propagation in the negative direction, but not the positive direction. There is another frequency region of 53 to 60 GHz where propagation in the negative direction is strongly attenuated while propagation in the forward direction is allowed. -kx direction Return Loss (dB) 0 -10 +kx direction -20 -30 -40 50 60 70 80 Frequency (GHz) 90 Figure 5.8: A plot of the return loss verses frequency for a four-layer ATR case with BaM as the magnetic material. There are two distinct frequency regions that exhibit nonreciprocal behavior. The dielectric gap is 0.03 cm thick and the BaM layer is 0.1 cm thick. The incident angles are θ = ± 40o and the external field is H0 = 3 kOe. 88 The properties of these surface polaritons can be modified and tuned by the applied magnetic field Ho. Fig. 5.9 uses the same parameters of Fig. 5.8, except the applied field has been reduced from Ho = 3 kOe to Ho = 2 kOe. The 3 kOe case is represented by dotted blue lines, while the 2 kOe case uses the black and red scheme of previous graphs. With the reduction of field, the isolator bands have shifted down in frequency. Ho = 3 kOe Return Loss (dB) 0 -10 -20 -30 -40 50 -kx Ho = 2 kOe +kx Ho = 2 kOe 60 70 80 Frequency (GHz) 90 Figure 5.9: A plot of the return loss verses frequency for two 4-layer ATR cases with BaM as the magnetic material but with different magnetic fields. A change of 1 kOe in the magnetic field essentially shifts all features of the reflectivity, including the nonreciprocity, by a 2.8 GHz. The BaM layer is 0.1 cm thick. The dashed lines represent the results where the applied field is H 0 = 3 kOe. The solid lines are for an applied field of H0 = 2 kOe. The incident angles are θ = ± 40o. The previous plots using BaM have used a relatively thick gap and magnetic layers. When these parameters are reduced, the return loss landscape becomes modified and looks different from the bulk and surface polariton dispersion curves implied in Fig. 89 5.7. This can be seen in Fig. 5.10. For Fig. 5.10, the gap thickness is now 0.02 cm and the BaM layer is 0.01 cm thick. White lines are the results for surface modes. 85 Frequency (GHz) 80 Return Loss (dB) 75 0 -3.000 70 65 -9.000 60 55 -15.00 -40 -20 0 20 40 Propagation Vector kx (1/cm) Figure 5.10: A return loss map for frequency versus propagation wavevector k x for a 4-layer ATR geometry with BaM as the magnetic material. The dielectric gap is 0.02 cm and the BaM layer is 0.01 cm thick. The applied field is H0 = 3 kOe. The two white lines show the surface modes for a structure based on a semi-infinite BaM and semi-infinite dielectric εd = 1.5. The strongest nonreciprocal behavior occurs near the intersections of the surface modes with the bulk bands. Despite the thin BaM film, Fig. 5.10 still displays significant nonreciprocity. We see this more clearly for a particular example in Fig. 5.11 which uses all the parameters of Fig. 5.10 but only shows the return loss for incident angles 75 o . There is a large dip in return loss in the negative direction at about 58 GHz, but very high reflection in the 90 positive direction. As we saw earlier, the position of the isolation frequency can be tuned by changing the applied field Ho. Return Loss (dB) 0 +kx -5 -10 -15 -20 -kx -25 -30 50 60 70 80 Frequency (GHz) 90 Figure 5.11: A plot of the return loss verses frequency for a four-layer ATR case with a BaM layer of 0.01 cm in thickness. We note a distinct nonreciprocity even for this thin BaM film. The incident angles are θ = ± 75o and the applied field is H0 = 3 kOe. Depending on the parameters chosen, the importance of the surface modes can be lessened and one may even be able to take advantage of the just the bulk modes of the ATR geometry. Fig. 12 give results for a possible band pass filter. The gap thickness was 0.001 cm, the thickness of the BaM was 0.1 cm, and the angles of incidence were θ = ± 30o. There was non-reciprocal behavior, but it was restricted to below 60 GHz and above 70 GHz. Between those frequencies the device has high reflectivity and outside of that region the return loss did not go over -5 dB. In fact, outside of 50 GHz and 75 GHz, the 91 return loss didn’t go over -10 dB. Essentially, the region between 60 and 70 GHz has high reflectivity because there are no surface or bulk magnetic polaritons in this region. 0 Return Loss (dB) -5 +kx -kx -10 -15 -20 -25 -30 40 50 60 70 80 Frequency (GHz) 90 Figure 5.12: A plot of the return loss verses frequency for a four-layer ATR case with BaM as the magnetic material. In this case the dielectric gap is very thin (0.001 cm thick) and the BaM layer is 0.1 cm thick. With a small gap, the incident wave couples well to the bulk waves in the magnetic material, and the resulting reflection looks like a band-pass filter with limited nonreciprocal behavior. The incident angles are θ = ± 75o and the applied H-field is H0 = 3 kOe. Two factors that play big roles in the dramatic changes of the reflection landscape are the thickness of the gap and the thickness of the magnetic layer. As the gap becomes thinner, the overall reflection is reduced because the evanescent wave is much stronger in the magnetic material encouraging a transfer of energy from the electromagnetic wave in the prism to that in the magnetic material. A reduction in the thickness of the magnetic layer generally reduces the nonreciprocity. For example, starting with the angle of incidence, gap thickness, and 92 BaM thickness used in Fig. 5.11, we present Table 5.1 for additional calculations for different BaM thicknesses, where the first column gives the thickness of the BaM, the second column gives the frequency where the greatest difference occurs between positive and negative propagation, and ΔL is the difference in return loss between the two propagation directions at that frequency. When the BaM has reached a micron in thickness, the signals only differ by about ½ a dB at the frequency of greatest difference. Magnetic Layer Thickness and Return Loss Differences BaM (cm) 0.01 0.005 0.001 0.0005 0.0001 Frequency (GHz) 58.7 59.3 59.8 59.8 59.8 ΔL (dB) 25 .9 15.1 4.1 2.3 0.5 Table 5.1 It may be possible to create an effective ATR like geometry in a planar microstrip structure. This is illustrated in Fig. 5.13. We propose that it may be possible to use lithography to create planar devices that uses a v-shaped signal line to guide an electromagnetic signal close to a multiple layer interface. Fig. 5.13 b) shows a possible device where a microstrip imports a signal to a gap layer near a magnetic layer. For a device like this to produce non-reciprocal ATR results, it would need the magnetic material to have its magnetization oriented out of the plane. This is one of the reasons we have presented results using BaM, which has a high crystalline anisotropy creating the out of plane orientation. In both geometries the oscillating E field is parallel to the static magnetization, and the oscillating H field is perpendicular to the magnetization. 93 We note that the planar device is different from the structure we have considered here, and we do not believe it will exactly reproduce our results. However, based on the physical principles we have explored, it should exhibit non-reciprocal behavior which could be adjusted by many of the parameters we have discussed. Figure 5.13: Figure a) is a side view of the typical ATR geometry - a prism separated from a magnetic layer by a gap. The incident wave creates an evanescent wave in the gap which can interact with the magnetic material. Figure b) is a top view of a planar geometry proposed which would utilize the physics of ATR. A microstrip waveguide directs an EM wave to close proximity of a magnetic material, with the magnetization directed into the page, parallel with the applied magnetic field. Conclusions The four-layer ATR geometries using YIG and BaM magnetic layers exhibit strong non-reciprocal behavior for magnetic films of thicknesses between 0.001 cm and 3 cm. The range of non-reciprocity can be tuned by changing the applied magnetic field, 94 gap layer thickness or magnetic layer thickness. For a YIG magnetic layer, the range of non-reciprocal frequencies can be adjusted between 5 and 20 GHz. For a BaM magnetic layer, the non-reciprocal range is between about 50 and 80 GHz. With the parameters used here, we find the isolation in terms of return loss can be 30 to 40 dB, with the minimum loss being only -3 dB. We also calculate the properties of a reciprocal band pass filter based on BaM in a similar geometry. The band pass filter used a very thin gap layer compared to the nonreciprocal devices discusser earlier. We also propose a possible planar device, designed to utilize the physics involved in the ATR geometry. This device could be adjusted to accommodate different reciprocal and non-reciprocal devices, such as notch filters, band pass filters, isolators, or circulators. CHAPTER VI Background: Micromagnetic Simulations The true magnetization of a magnetic structure, particularly in nanosized objects, can be quite complicated. In real materials, local regions of magnetization can change orientation throughout the structure. It is difficult to analytically model such a landscape. Computer simulations are a way to evaluate these complicated structures. There are two main techniques currently used in dynamic computer modeling of magnetic structures: 1) The Finite Difference Method, and 2) the Finite Element Method.64 In this chapter, we will talk about the principles involved in the Finite Difference Method. Then in Chapters VII and VIII we will use it to study the effects oscillating magnetic fields have on nanosized magnetic squares and on magnetic bilayer structures. The Finite Difference Method This method relies on dividing a magnetic structure up into individual cells, as illustrated in Fig 6.1. The main approximation is that each cell’s magnetization is uniform. The LL equation is then used to evaluate the motion for each cell dM dt M H M M [ M H] (6.1) . M is the magnetization of a single cell and H is the average effective magnetic field in the cell. This equation of motion is solved in a time integration scheme to find how the 96 magnetization of each individual cell evolves in time. Our results are based on either a 2nd order Runga-Kutta method or an Adams-Bashforth method.65 Figure 6.1: Magnetic structure divided into cells. Each cell has a uniform magnetization. In order to evaluate the equation for each individual cell we must be able to account for the average effective field acting on each cell. The contributions of the effective field are given by H H 0 H A H ex H dip (6.2). H 0 is an applied field, H A is the local anisotropy field of the material, H ex is the effective exchange field produced by neighboring cells, and H dip is the dipolar field created from all the cells in the structure. The applied field, H 0 is chosen. The anisotropy 97 field H A , exchange field H ex , and the dipole field H dip must be calculated based on the magnetization and geometry of the cells. The Anisotropy Field H A depends only on the magnetization of the cell that is being considered. It is often based on the spin-orbit coupling and the interaction of neighboring atoms. Because of this, the details of the lattice structure are important, and this interaction can make particular directions of magnetization preferred. We can talk about it in terms of the energy density due to the orientation of the magnetization. For different types of crystalline anisotropy, the total energy density has different forms. For a single preferred axis, or uniaxial anisotropy, it has the form U K cos2 (6.3). K is the anisotropy constant, which is determined experimentally for a material.27 The angle θ is the angle between the local magnetization and the easy axis (the preferred axis) of the material. The H-field for interaction can be found by considering how the energy density changes with the magnetization H U M (6.4). The vector derivative is M x̂ mx ŷ my ẑ mz (6.5). 98 If the easy axis is the z-axis, then the field along that axis would be HA HA mz [K cos 2 ] mz m K z M 2K cos M 2 2K m z M M (6.6) In the calculations done in later sections, this is the form we use for the average uniaxial anisotropy field of a single cell. The Effective Exchange Field H ex depends on the relative orientation between the cell under consideration and that of its neighboring cells. This short-range field accounts for the quantum mechanical effect that allows for the existence of permanent magnets. This is due to overlapping wave functions of the electrons orbiting atoms. Because it is short range, only the nearest atomic neighbors are usually considered. In its most basic interaction form, the exchange energy for two spins is given by E Here J S1 S2 (6.7) S1 and S 2 are the spins of the two electrons, and J is the exchange constant. J is dependent on what the two atoms are and how far apart they are spaced. Because this interaction depends on electron orbital overlap, J becomes very weak over large spacing. It is because of this that when evaluating the exchange energy associated with a single spin, one often uses only nearest neighbor atoms. 99 In the finite difference method, we are not considering interactions between single atoms, we are dealing with volumes of spins. Even though the interaction is short ranged, it still has an effect at much greater distances. From one spin, there is an effective exchange on a spin located at a far distance (L) through the exchange interactions of all the spins in between, see Fig. 6.2. Because our method does not deal with just one spin, we consider the effective interaction between the average spins of two neighboring regions and the length between the middle of each region (the exchange length). From this idea comes the effective exchange constant. Instead of J, which is how the energy scales with the difference between two spins, we have the exchange stiffness constant A. The constant A describes how the energy scales with the change of magnetization over a length (cgs units of erg/cm). Figure 6.2: A row of spins. The circled spins influence each other through the spins in between over a length L, giving rise to an effective exchange between the two. We can represent the energy density due to exchange as: U A mx M 2 my M 2 mz M 2 (6.8). The exchange stiffness constant A, for a simple cubic lattice, relates to J and S by the atomic lattice spacing a: 100 A JS2 2a JM 2 a 5 2 (6.9). With the energy density, we follow the same method used to find the field due to the anisotropy by calculating the variation in energy density with respect to the magnetization H ex U M 2A M 2 M M (6.10). In the computer simulation, this derivative is turned into a difference equation. A single component of the exchange field can be written as H ex x ( x, y, z) 2A M x ( x 1) x 2 M( x 1) 2 M x ( x 1) 2 M( x 1) (6.11). Where Hex-x(x,y,z) indicates the x-component the exchange field in some cell defined by its location with x, y, and z. The Mx(x+1) term is the x-component of magnetization for the nearest neighbor cell in the positive x-direction and Mx(x-1) term is the x-component of the nearest neighbor cell in the negative x-direction. The denominators for the respective terms are the modulus squared of the total magnetization in those cells. The Dipole Field and the FFT method The dipolar field in a cell being considered is dependent on all cells in the structure. It is for this reason that this field is the most time consuming calculation in the simulation. In Chapter II we established how to calculate the demagnetizing field for a single cell.29 The demagnetizing field of a volume is its dipolar interaction with itself. 101 This method can also be used to evaluate the average dipolar field produced by one volume acting on another. Recall from Eq. (2.33): Hd 1 V dV M V V r 1 dV r N M (6.12) . This double integration evaluates the dipole field at a point in volume V created from every point in volume V’, and then taking an average of all field values across the entire volume V, see Fig. 6.3. The Newell tensor is then dependant on the vectors locating the x two volume ( x ). Recalling Chapter II, we rewrite Eq. (2.35) as N( x x ) 1 dA dA VA A r r (6.13). Where A and A’ are the surface areas of the respective volumes. The vectors x and x’ are the relative locations of the volumes. Figure 6.3: Average magnetic field inside volume V generated by volume V’. x and x’ are vectors indicating the locations of the two volumes. r and r’ are vectors are vectors indicating specific instances contained in the integral from Eq. (6.12). 102 This allows us to write the dipolar field in a given cell as summation of the products of its relative position to all cells (including itself) and their magnetization. Considering Fig 6.4, if we index the cells and ask for the average dipolar field in cell “j”, we would have to sum up the relative products of N and M for all cells H j (x j ) N ij ( x j x i ) M i ( x i ) (6.14). i Figure 6.4: Magnetic structure divided into cells. The average effective field Hj(xj) is due to the contribution of the magnetization from all other cells, M i(xi). The time cost in the simulation comes from the double iteration that must be performed. The average dipolar field must be found for each individual cell, and that calculation requires summing through each individual cell, shown in Eq. (6.14). The amount of time for a computer to perform these operations increases rapidly with the number of cells being considered. It is for this reason that an alternate way of evaluating Eq. (6.14) is used. 103 A more efficient method of performing the calculation of Eq. (6.14) is done by utilizing the convolution theorem to transform the summation into something that can be evaluated faster and then transform the result back.66 As an example, we will consider a simpler case. We will switch from summations of discrete cells to an integration over a continuum, and we reduce the problem down to one dimension. Consider the H-field in one dimension at a single point due to the products of the relative Newell tensor and position dependant magnetization H(x) N(x x ' ) M ( x ' )dx ' (6.15). x' The Newell tensor can be rewritten as an inverse Fourier transformation from some function n(q) 1 2 N(x x' ) n(q )e iq ( x x ') dq (6.16). q Plugging Eq. (6.16) back into Eq. (6.15) and regrouping some terms gives us H(x ) q 1 2 M( x ' )e iqx ' dx ' n(q )e iqx dq (6.17). x' We now recognize that the function in the brackets is only of x’ and it is a Fourier transformation of another function m(q) 1 2 M( x ' )e iqx 'dx ' m(q ) x' (6.18). 104 Substituting m(q) into Eq. (6.17), we get H(x ) m( q ) n ( q )e iqx dq (6.19). q It might be easier to identify the product of m(q) and n(q) simply as some function F(q) H(x) F(q ) e iqx dq (6.20). q We again recognize this as an inverse Fourier Transformation. 1 2 F(q ) e iqx dq h( x ) (6.21). q So the result is H(x ) 2 h( x ) (6.22). This says that we can find the H-field at a particular point by taking the inverse Fourier transformation of F(q), which is a product of the Fourier transformation of the Newell tensors and the magnetization of the entire structure. We will now change back to a discrete format and include three dimensions. Using a different notation, we define T(f) as the discrete finite Fourier transformation of a function and T-1(g) is a discrete finite inverse Fourier transformation. We can now write a component of the average H-field for all cells as Hi T ( T( N ii ) T (M i ) ) 1 T ( T( N ij ) T (M j ) ) T 1 ( T( N ik ) T(M k ) ) 1 (6.23). 105 H i is a matrix containing all of the average i-components of the H-fields in every cell of the structure. The subscripts i, j, and k denote vector components and tensor elements (if i = x, then j = y and k = z). The Newel tensors N nm indicate the geometric relations for all the cells surrounding a single cell. The magnetization matricies M n represent a single directional component of magnetization for every cell in the structure. Please note that the 2 factor from Eq. (6.21) is not obvious in Eq. (6.22). That constant is based on the choice of how the discrete Fourier transformations are defined. The transformation can be defined to do away with it. Performing these transformations and inverse transformations may seem complicated and one would question the benefit of performing the calculation in such a way. The advantage comes in two ways: 1) The Newell tensor transformations are constant throughout the simulation. The standard practice is to perform them once before the simulation begins, store them, and then call them when needed. 2) Taking advantage of symmetries in the transformed space allows for a discrete Fourier transformation to include redundant calculations. This reduces the number of unique calculations that need to be done. A Discrete Fast Fourier Transformation (FFT) scheme reduces the number of calculations to be done from C2 (C = total number of cells) to approximately C ln(C). 66 106 These factors result in a significant run-time improvement depending on how many cells are being included in the simulation. Fig. 6.5 shows the time taken to perform dipole calculations on a particular computer using the two different methods. The double iteration method quickly increases in time cost as the number of cells increases; the FFT method increases at a much slower rate. From Fig. 6.5, for a 1024 cell structure the double iteration method took 1459 seconds to perform 1000 cycles while the FFT method only took 39 seconds. Figure 6.5: Time to complete 1000 iterations of dipole calculations versus the number of cells being considered. This graph compares the double-iteration, Eq. 6.12, and the FFT method, Eq. 6.21, used to calculate dipole fields. For large numbers of cells, the FFT method is clearly advantageous. These methods of calculating the effective anisotropy, exchange, and dipolar fields will be used in the next two chapters to evaluate the time-dependant behavior of magnetic nano-structures in microwave fields. CHAPTER VII Accessing Multiple States in Magnetic Squares with Microwaves Magnetic reversals13-15, microwave assisted magnetic reversal (MAMR)16-20, and nano-scale bit patterned media21-26 have been investigated recently because of their potential use in high density magnetic storage systems. MAMR uses microwaves to induce large angle precessions in the magnetization and dislodge it from a stable state. To do this efficiently, one generally employs a frequency that is in resonance with one of the normal modes of the magnetic structure. A particularly interesting example that combines the ideas of MAMR bit patterned media has recently been reported. S. Li et al67 proposed a scheme with two films made of different materials. Each film has a different resonant frequency so it is possible to flip the magnetization of one structure but not the other with an oscillating field that has a frequency that matches the resonance of only one of the layers. The frequency-matched layer will experience large angle precessions and will switch, while the other layer will not. This paper demonstrates both a multistate micromagnetic structure68-72 and the use of MAMR to change an initial magnetic state into one of two other states, or to let it remain in its current state. We consider a thin, square magnetic structure with an applied static magnetic field directed nearly opposite to the magnetization as shown in Fig 7.1. There are three important magnetic orientations for this structure that are stable, for both negative fields and zero fields: an initial state, an intermediate state and a 108 reversed state. We find that the strength of the oscillating microwave field can control which one of the three allowed states emerges as the final configuration. As we will see, the transition from one state to another occurs at a negative field, sometimes with the addition of a microwave signal. After the orientational transition has occurred, the microwave field can be removed and the static field reduced to zero, and all three states remain stable at zero field. Figure 7.1: A 10 nm x 160 nm x 160 nm structure of Fe-Ti-N with an in-plane average magnetization (M). An external magnetic field (H0) is applied in the negative z-direction. An oscillating magnetic field (hd) is applied in the y-direction. The Calculation We consider a thin, square-shaped magnetic element with an in-plane uniaxial anisotropy. It is 10 nm thick and its sides are 160 nm long. The material, one form of Fe-Ti-N 73-74, will be described by the following parameters: the saturation magnetization (Ms = 1.106 kG), the uniaxial anisotropy constant for the z-axis (K = 11500 erg/cm3), the 109 gyromagnetic ratio (|γ| = 2.897 GHz/kOe), the exchange stiffness parameter (A = 5e-7 erg/cm) and the damping of the material (α = 0.04). For the micromagnetic calculation, the structure is divided into 4096 cells. The cells are square in the y and z plane, with dimensions of 10 nm thick and 2.5 nm sides. The dynamic micromagnetic calculation applies the Landau-Lifshitz equation (LL equation) to the magnetization of each of the cells to evolve them through time. The LL equation is: M t (M H eff ) Ms (M M H eff ) (7.1) where M is the magnetization of the cell and Heff is the averaged effective magnetic field present in the cell. The effective magnetic field for each cell (Heff) is made up of an external applied field (H0), the effective exchange fields between nearest neighbor cells (Hex), dipole fields created from all cells including the single cell’s own demagnetizing field ( Hdip), the driving field from the microwave field when it is present (hdcos(ωt)), and the field due to the uniaxial anisotropy (HA). H eff H0 H ex H dip h d cos( t ) ŷ HA (7.2) The Multi-State Results We are interested in a structure that holds a stable magnetic state even if an applied static magnetic field is opposite to the magnetization of the structure. To find such a state, we started with Ho positive and gradually brought it to a negative value. The resulting equilibrium state was an S shaped state, as seen in Fig. 7.2a. This state was then 110 used as the starting point for all following simulations. The external magnetic field was applied opposite to the magnetization, with a slight angle off the z-axis (0.01o). The slight angle is included to avoid any non-physical symmetry, and we find that the rotation of the S state is in the direction of this small angle. For a negative field up to about -320 Oe the initial state configuration was stable. We will simply refer to this as the initial state, Fig. 7.2a. For fields stronger than -320 Oe, the state flipped to an S shaped state, with magnetization almost anti-parallel to the zaxis (about 128o off the positive z-axis). We will refer to this as the reversed state, seen in Fig. 7.2c. Figure 7.2: Three graphical representations of the magnetization in the y-z plane of the structure for the three distinct states (with an applied field of H0 = -220 Oe). The large arrow represents the average magnetization and the small arrows indicate locally averaged magnetization. (a) the initial state, (b) the intermediate state, and (c) the reversed state. In order to study the microwave assisted switching, we used fields in the initial state’s stability range (0 to-320 Oe) and added an oscillating magnetic field oriented along the y-axis. To choose the oscillation frequency, we slightly disturbed the equilibrium state, recorded My(t), and did a Fourier transform which resulted in a set of 111 peaks at different frequencies representing bulk and localized modes75-78. The frequency for the most prominent mode and set it to the driving frequency for our oscillating field. We found an approximately linear relationship for the driving microwave frequencies as a function of the applied field Ho. f 7.97 GHz H0 kOe 4.025 GHz (7.3) This linear relationship is not valid for fields within 25 Oe of the switching field (-320 Oe). In the MAMR study, we found a third stable state, with the magnetization oriented in an S shape close to 900 relative to the z-axis. We refer to this as the intermediate state, and it is shown in Fig. 7.2b. An interesting point is that this state may or may not be seen if transitions are induced by a static applied field, e.g. in a standard hysteresis curve measurement. For the chosen parameters in the hysteresis calculation, the intermediate state did not appear. However, for cases where the saturation magnetization was lower we did in fact see stable intermediate states for certain applied fields. For example, using a saturation magnetization of 850 G (a value close to Permalloy), and applying an opposing field of -80 Oe an intermediate state was produced. To study the behavior of the system, we applied oscillating driving fields of varying amplitudes (hd) for a fixed static field, Ho = -220 Oe (opposing the initial magnetization). The frequency of the driving field was determined by Eq. (7.3) and was held constant. Fig. 7.3 shows the results for the normalized time evolution of the average z-component of magnetization for different values of hd. The driving field was turned on at 3 ns in the simulation and removed at 7 ns. For driving fields of hd = 30 Oe, the normalized mz component remains at about 0.82 after the 112 driving field is removed and the structure remains in the initial state. For driving fields of hd = 50 Oe, the average mz component changes rapidly and then oscillates around mz = 0.14, indicating the structure is in the intermediate state. The highest driving field of hd = 85 Oe finishes with the mz component ending at -0.95, this means the structure is in the reversed state. Figure 7.3: The average normalized mz component of magnetization for the entire structure versus time. There is an applied external field (H0) of -220 Oe. The different lines represent the results for different amplitudes of the driving field (hd). The oscillating driving field starts at a time of 3 ns and ends at 7 ns. An interesting question is - why does adding a microwave of a given strength cause the magnetic configuration to leave the initial state, but possibly become trapped in the intermediate state? The answer is that the switching depends on the orientation and frequency of the microwave field. The microwave is set to pump the y-component of 113 magnetization. In the initial state, this is effectively the transverse component of magnetization. In the intermediate state the y direction is close to that of the longitudinal component and the pumping is less effective. Furthermore, the driving frequency is set to that of the principal resonance of the initial state. At this frequency, enough energy is added so that the structure can get out of its local energy minimum and have a transition to the intermediate state. The driving frequency is then no longer in resonance with the normal modes of the new state. Evidence for this can be seen in the amplitude of the oscillations in Fig. 7.3. Even when the initial state is stable, the oscillations in <mz> are fairly large in the time range of 4 – 6 ns. In contrast, when the intermediate state is stable the oscillations are considerably smaller even though hd is larger. Of course if hd is sufficiently large, enough energy can be added to cause a transition to the reversed state. We collected data for a range of applied fields and driving fields. Fig. 7.4 shows the ending state results as a function of driving field amplitude (hd) at different static applied fields, H0. In general larger driving fields lead to transitions to the intermediate or reversed state. As Ho is made more negative, smaller hd values are needed to change the initial state. One parameter we found to be an issue was the angle of the applied field. Changing the angle to 1o altered the range where the intermediate state occurred. The intermediate state is now found for oscillating fields between 60 Oe to 65 Oe, in contrast to the values of hd = 45 – 75 Oe for the 0.01o case. Nonetheless, all the main features of the microwave driven switching remain true. 114 Figure 7.4: Phase diagram showing the final magnetic state of the structure after applying a driving field (hd) for 4 ns in an external field (H0). The final state is shown as a function of the driving field versus the applied field. We also performed trials on a smaller square structure with a thickness of 10 nm and side lengths of 80 nm. The results again were similar. The smaller structure flipped to the reversed state at a lower static field (at about -280 Oe). The resonance frequencies in the small structure were higher. For the smaller structure and with a static field of -160 Oe directed at a 1o angle away from the –z axis, the initial state was found to be stable up to oscillating fields of 47 Oe. The intermediate state occurred for 48 Oe < hd < 56 Oe, and the reversed state is found if hd is larger than 57 Oe. 115 Conclusions In summary, we have explored a structure with several stable magnetic configurations at zero magnetic field. We found that an added microwave field can change the initial state to one of two other states. The final state is determined by the frequency and strength of the driving field. CHAPTER VIII Microwave Assisted Switching in an Exchange Spring Bilayer In recent years exchange spring structures79-81, typically layered structures comprising a hard magnetic material and a soft magnetic material, have been proposed as a method to increase magnetic data storage densities82-83. In these materials, a small external static magnetic field can reverse the magnetization in the soft material. Because of the high anisotropy, however, the magnetization in the hard material does not change, and a domain wall is created near the interface between the two materials. Increasing the static magnetic field can then move the domain wall through the hard material and lead to its reversal. This field is smaller than the field normally required to switch the hard material in isolation and the hard layer reversal is therefore referred to as “domain wall assisted.” Composite recording media may be designed in this way with a large anisotropy (high thermal stability) but with a lower switching field than is normally associated with hard materials. Another way proposed to lower the energy required for magnetization reversal is through so-called “microwave assisted reversal” .84-85 The main idea of microwave assisted reversal is that applied microwave fields can resonantly transfer energy to a magnetic system thereby inducing large amplitude dynamics. This may allow the spin system to escape a metastable energy minimum. In contrast, a static applied field can cause a reversal by altering the energy landscape and destroying an energy minimum. A comparison of the switching time of single phase media to that for 117 exchange spring media can be found in Ref. [86]. Since the domain wall in the exchange spring represents a metastable state, it therefore seems plausible that microwave magnetic fields may be used to help move the domain wall and switch the hard material. In fact, microwave fields have been found to move domain walls in a past experiment.87 In that experiment, the resonant frequency of the domain wall was very different from the resonant frequency of the domains. The application of a driving field with frequency that matched the domain wall resonance was seen to most efficiently move the domain walls. In this chapter we examine whether domain wall and microwave assisted reversal can work together to lower the energy required for magnetization switching. We consider a bilayer containing a magnetically hard and a soft thin film. A similar work has been carried out by Li et al67 but for a small bit with magnetization perpendicular to the interface. Here we consider in-plane magnetization due to the thin film geometry. Also, the microwave fields used in Ref. [67] are too large (on the order of several kOe) to be useful for applications so we limit our study to lower microwave powers. In our results, we find that there are a number of different eigenmodes for the exchange spring structure, each with a different frequency. We study the microwave driven switching for each of these frequencies and find that a driving field set to the frequency of the lowest eigenmode is most effective at switching the hard material. The efficiency of switching is correlated with the profile of the eigenmode and the microwave orientation. 118 Theoretical Calculations The geometry of the exchange spring is illustrated in Fig. 8.1. Both layers have an in-plane uniaxial anisotropy in the z direction, but the hard layer’s anisotropy constant is 32 times larger than that of the soft layer. Both layers start in a state of mutual alignment in the +z direction. A static external field is then applied to both layers in the –z direction. As the external field increases in strength, the magnetization in the soft layer turns to align with the field while the hard layer remains pinned in the +z direction. Away from the interface, the turn is complete, but near the interface the magnetization in each atomic layer is at a different angle with respect to the –z axis. Thus a domain wall is formed near the interface between the layers. The domain wall produces an additional torque on the hard layer, encouraging it to switch. This allows the hard layer to switch and align with the field at a lower static field strength than it would without the presence of the soft layer. Figure 8.1: The geometry of the exchange spring bilayer. The magnetization is initially oriented in the +z direction in both materials. A negative static applied field is large enough to reorient most of the magnetization in the soft layer but the hard layer remains pinned in the positive z direction. An oscillating microwave field hd is added parallel to the y-axis. 119 For the simulations in this chapter we have chosen a set of material parameters. For the saturation magnetization of both materials, 4πMs = 18.1 kOe. The uniaxial anisotropy constant for the hard material is Kh = 5.3x106 erg/cm3, making the constant for the soft material Ks = 0.166x106 ergs/cm3. The effective exchange constant for both materials is A = 1.03x10-6 erg/cm. This is also taken to be the exchange constant across the hard/soft interface. For the micromagnetic calculation, the structure is divided into 64 cells in the x direction. The cells are square in the y and z plane. The side lengths of the cells are 100 microns, which is quite large compared to the cell thickness of 1 nm. This is done so that the calculation is consistent with earlier models which assumed a single spin in each layer could represent the entire layer.89, 90, 80 The dynamic micromagnetic calculation applies the Landau-Lifshitz (LL) equation to the magnetization of each of the cells to evolve it through time. The LL equation is as follows: M t (M H eff ) Ms (M M H eff ) (8.1) where M is the magnetization of the cell and H eff is the effective magnetic field present in the cell and is given by. H eff H0 H ex H dip h d cos( t ) ŷ H A Here H 0 is the external static magnetic field, (8.2) H ex is the effective exchange field between nearest neighbor cells, H dip includes the dipole fields created from all cells including the single cell’s own demagnetizing field , h d cos( t ) ŷ is the driving field 120 from the microwave field when it is present, and the field due to the uniaxial anisotropy is H A . The dipole field is calculated efficiently through a standard FFT method. To find the exchange spring state, with the magnetization in the soft film nearly opposite to that in the hard material, we started with H0 positive and gradually brought it to a negative value. The final magnetic field orientation was close to the –z direction, with a slight angle off the negative z-axis (0.01°). The slight angle is included to avoid any nonphysical symmetry. Results and Discussion In Fig. 8.2 we show two hysteresis curves. One curve is for an 8 nm film of the hard material. The second curve is for an exchange spring bilayer with a 56 nm thick film of the soft material, exchange coupled to the 8 nm thick film of the hard material. The dashed loop shows the result for the hard material by itself, the solid line loop shows the behavior of the exchange spring system. The addition of the soft layer allows the structure to completely switch at an applied field of 1.12 kOe as opposed to an applied field of 7.37 kOe. This is a reduction of the required switching field by 85%. The question is whether the application of a microwave field may further reduce the switching field. The hysteresis loop maps out a region of static applied field strength where the structure is in an exchange spring state. This range runs from about -0.38 kOe to -1.12 kOe. To explore how this structure is affected by a microwave field, we choose an applied field in the middle of this range, H0 = -0.7 kOe. In order to drive the structure 121 effectively, the applied microwave frequency should match that of an eigenmode of the bilayer.91 1.0 Mz (a.u.) 0.5 0.0 -0.5 -1.0 -10 -8 -6 -4 -2 0 2 4 6 Applied Field H0 (kOe) 8 10 Figure 8.2: Hysteresis loops for an 8 nm thick film of hard material (red dashed line) and a 64 nm bilayer structure (blue solid line) with 56 nm of soft material and 8 nm of the hard material. Both materials have 4πMs = 18.1 kOe. The hard material has an anisotropy constant of Kh = 5.3x106 erg/cm3. The soft material has an anisotropy constant of Ks = 0.166x106 ergs/cm3. To find the frequencies of the eigenmodes, we numerically started the system in an equilibrium state and added a small magnetic field pulse in the y-direction for 0.0005 ns. At the end of the pulse, we recorded m( t ) for each cell. We then did a Fourier transform for the time evolution of magnetization in each cell, and summed the resulting power spectrum over all cells. This results in a set of peaks at different frequencies representing bulk and localized modes.75- 78 Because the chosen pulse was spatially uniform, it is expected that the mode with the strongest response will be a mode with a spatial profile that is also close to uniform. We will see this is in fact the case below. 122 Fig. 8.3 shows the frequency spectrum of the oscillations after the pulse is applied. The peaks correspond to the resonant frequencies of the structure. In Fig. 8.4(a), (b), (c) and (d) the spatial profiles of the first four modes are drawn as a function of thickness through the bilayer. Figure 8.3: Frequency spectrum of the magnetic eigenmodes in the 64 nm bilayer structure with an applied field of -700 Oe. The eigenmode profiles are constructed by first taking the Fourier transform of mx(t) in each cell. From this one finds the amplitude A(i) and phase a function of frequency. i) in each cell i as One can find a spatial profile of a mode at a particular frequency by plotting A(i) cos( i)) as a function of the cell index i. This is equivalent to an instantaneous “snapshot” of the out-of-plane precession for a particular mode at frequency 75 123 Figure 8.4: Mode profiles for the four lowest frequency modes in the 64 nm bilayer structure. The frequencies are taken from the frequency spectrum in Fig. 8.3. The darker shading shows the location of the 8 nm hard film at the bottom of the 64nm thick bilayer. It can be seen that the lowest resonant frequency corresponds to an excitation which is mainly localized in the soft layer, with a maximum near the domain wall at x =15 nm. As discussed earlier, the profile of the lowest frequency mode is excited with the largest amplitude since it is the closest to a spatially uniform mode. In contrast, the standing wave mode at 29.17 GHz is only weakly excited by the spatially uniform pulse (see Fig. 8.3). Our calculations were compared to those found using a linear spin wave theory89 with the dipolar contributions on each layer estimated as 4πMs and the frequencies and mode profiles agreed quite well. Next a spatially-uniform oscillating magnetic field (hd) is applied to the structure, as shown in Fig. 8.1, with the frequency matching one of the resonant modes. The driving field is oriented along the y axis, which is in-plane and perpendicular to the static applied 124 field. Fig. 8.5 shows an example of the time evolution of Mz (the average z component of the magnetization of the whole structure) when the driving frequency is 10.74 GHz and hd = 120 Oe. The microwave field is off from 0 ns to 1 ns and M z is initially near -0.5, indicating that the structure is in the exchange spring mode. At 1 ns the driving field is turned on. At this point, the magnetization undergoes a cycle and a half of oscillation before the hard material flips over at around 1.25 ns. After the switching point, the magnetization in the entire structure is almost completely aligned with the applied field along the –z axis. This state no longer has a resonance at 10.74 GHz, which is made apparent by the decay of the oscillation amplitude, even though the driving field is still pumping the structure all the way up to 2 ns. 0.00 Mz (a.u.) -0.25 -0.50 -0.75 -1.00 0.5 1.0 1.5 2.0 time (ns) Figure 8.5: Time evolution of the average z-component of magnetization for the bilayer structure in a -700 Oe applied field and a 120 Oe driving field. The driving field has a frequency of 10.74 GHz, matching the lowest magnon mode depicted in Fig. 4. The driving field is started at 1 ns. 125 We drove the structure at several resonant frequencies with different strengths of the driving fields. Fig. 8.6 is a plot of the switching results at each of the resonant frequencies. Depending on the strength of the driving field, the structure either switched so that both layers lined up with the static field (solid), or it stayed in the exchange spring state (hatched). The primary observation from Fig. 8.6 is that the microwave induced switching occurs at lower hd values for lower frequencies. The lowest driving field required for switching was 105 Oe when the frequency was 10.74 GHz. Exchange Spring State 10.74 GHz 14.28 GHz 21.12 GHz 29.17 GHz Switched State 37.96 GHz 47.85 GHz 0.00 0.25 0.50 0.75 Driving field strength (kOe) 1.00 Figure 8.6: Final spin configuration as a function of the magnitude of the driving field. The exchange spring state is illustrated by the hatched area and the solid area shows for what values of the driving field the hard layer has switched (uniform state). Results for driving fields with frequencies corresponding to that of different eigenmodes are shown. The static field is -700 Oe. 126 One can imagine at least two possible factors governing the switching of the hard layer. 1) If the amplitude of the mode is largest near the domain wall then the switching should occur at lower field strengths. The basic idea is that the larger amplitude precession helps the magnetization jump over an energy barrier. 2) If the spatial profile of the eigenmode is close to that of the driving field, the energy in the driving field will be more effectively transferred to that of the eigenmode, leading to a larger amplitude precession and switching at a lower hd value. From Fig. 8.4, it is apparent that, in this case, all the modes have a large amplitude near the domain wall. In contrast, only the lowest frequency mode is close to being spatially uniform, and this is why pumping this mode leads to switching at the lowest power level. Mathematically, the aerial density of power absorbed in the structure should be given by x d Aerial Power Density = h d (x) x 0 dm y ( x ) dt x d dx h d ( x )m y ( x )dx . (8.3) x 0 Generally as the power absorbed is increased, the amplitude of the mode also increases. x d If hd is uniform, the expression for the power absorption reduces to hd m y ( x )dx and x 0 thus the mode with frequency 10.74 GHz will absorb significantly more power than the mode at f = 29.17 GHz because its profile has a larger value of the integral (see Fig. 8.4). We also examined the case where the microwave strength was held constant, at 120 Oe, and the static field, Ho, was changed. The frequency of the driving field at each Ho was set at the frequency of the lowest eigenmode for that value of Ho. The 127 frequencies ranged from almost 9 GHz at -400 Oe to almost 16 GHz at -1000 Oe. Fig. 8.7 is a hysteresis loop constructed from these driven results (solid line) and compared to the undriven hysteresis loop (dashed line). The static switching field in Fig. 8.6 is -550 Oe. If we sum the magnitude of the static switching field and the magnitude of the driving field, we create an effective switching field (HES). H ES Hs hd (8.4) Thus for the driven case, the effective switching field is 0.67 kOe. In the undriven case, the switching field is 1.12 kOe. This gives a reduction in the effective switching field of about 40%. 1.0 Mz (a.u.) 0.5 0.0 -0.5 -1.0 -2 -1 0 1 2 Applied Field H0 (kOe) Figure 8.7: Hysteresis loops for the undriven bilayer structure (dashed line) and the bilayer structure driven with a 120 Oe oscillating field (solid line). The frequency of the driving field is the frequency of the lowest magnon mode in the structure, which varies as a function of static applied field. 128 We also explored driving the structure with the microwave field oriented along the x-axis and z-axis. The results were similar to those for hd along the y-axis in that driving at the lowest frequency eigenmode caused switching at the lowest strength of driving field. However, in both cases (hd along x, and hd along z) the amplitude of the driving field required to switch the structure was almost twice what was required for the y-axis oriented microwave. This again can be understood by considering the aerial power density absorbed by the structure. In Eq. (8.3) the integrand is actually a dot product between the driving field h d and the dynamic magnetization m . Therefore, the relative orientation between the two vectors is important. The dynamic magnetization in the bilayer has its largest component in the y direction due to thin film demagnetizing effects (see Fig. 8.1) and so the largest power absorption occurs when the microwave field is applied in this direction to maximize the dot product in Equation (8.3). We also investigated using a spatially non-uniform driving field in an attempt to excite the non-uniform modes seen in Fig. 8.4b, 8.4c, and 8.4d more efficiently and thereby cause switching at a lower value of hd. The results for a driving field, that changes linearly through the thickness of the structure and has a sign change at the midpoint, did not show an advantage for any mode. The driving field required to switch the structure was increased for all the modes. In particular, the lowest frequency mode, the most uniform eigenmode, became much more difficult to excite as opposed to being the easiest when driven with a uniform microwave field. This is consistent with Eq. (8.3) because now hd(x) is odd about the midplane of the film while my(x) is close to even about the midplane and the resulting integral is small. In order for there to be an 129 advantage to driving the higher frequency modes, the non-uniform field profile must closely match the profile of the eigenmode that it is trying to excite. There are methods to adjust the spatial profile of the driving field92 but adjusting the field profile to match the profiles in Fig 8.4 (b-d) would be difficult. Conclusions We find that domain wall assisted and microwave assisted switching can be used together to decrease the power required to reverse the magnetization in a hard material. The applied field required to switch a hard thin film is decreased by 86% (for our material parameters) when it is coupled to a soft material and an exchange spring is formed. By adding a spatially-uniform, oscillating magnetic field (hd), and setting its frequency to match the lowest frequency magnon mode in the structure, we show that the switching field is reduced further by 40%. The construction of an exchange spring and the application of a driving microwave field together lead to a reduction in the total effective switching field of 91%. The reduction in the total field required for switching, requires the application of both a static and microwave field. The additional complexity that this creates might be mitigated by several factors. For example, one could saturate a larger region with an oscillating field and localize a static applied field to switch particular bits. The inverse situation, where the static field extends over a larger region and the microwave field is localized could also be considered. CHAPTER IX Summary This dissertation deals with issues that arise in signal processing and magnetic storage. The majority of Chapters III, IV, VII, and VIII have been adapted from our publications in scientific journals.93-96 The work is theoretical in nature, but the results depend on a range of parameters that can be used in experiments. Indeed, some of the predictions made in the theoretical calculations were later confirmed by experiments.46 In general, the dissertation is concerned with how a microwave signal is changed by a magnetic material and the inverse problem of how a magnetic material can be changed by a microwave field. The advent of nanosized structures makes the second topic particularly interesting. In Chapters III, IV, and V we dealt with how a microwave signal is altered as it passes through a layered magnetic material. The calculations were done using Maxwell's equations and the appropriate boundary conditions. This gave results for microwave absorption and phase shifts in a magnetic material as a function of frequency and applied magnetic field. The main results are the following: 1) Positioning a magnetic film directly in the middle of the dielectric spacer in a microstrip geometry increases the signal attenuation at the resonant frequency and also narrows the width of the absorption, optimizing notch filter results. 2) Using hexagonal ferrites as the magnetic element in notch filters or phase shifter devices accommodates operational frequencies in the 50-70 GHz range for relatively low 131 static magnetic fields. Relatively thin hexagonal ferrite films (less than 1 micron) can produce usable device results. 3) Using attenuated total reflection, a variety of non-reciprocal devices can be made from multi-layered structures which include a magnetic layer. The methods used in these sections do not account for non-linear effects and thus may not be appropriate for high power microwaves. Despite this, the theoretical results often come quite close to experimental results, as seen in Chapter IV. Chapters VII and VIII dealt with how magnetic nano-structures are influenced and even changed by an applied microwave field. The calculations were done with a locally-developed micromagnetics program that is appropriate for both linear and nonlinear effects. The key results in this portion of the dissertation were: 1) The geometry of a nano-structure can influence which magnetic states (C state, S state, flower state, etc.) are stable. Driving the structure with an oscillating magnetic field at a resonant frequency can influence which of these initially stable states remain stable and effectively cause a transition from one magnetic state to another. 2) When trying to switch the orientation of a bilayer magnetic structure, the total switching field can be substantially reduced by adding an oscillating magnetic field matching the appropriate resonant frequency. 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We had 0 G 11 which is explicitly given by G 12 G 13 G 14 0 0 0 0 A1 0 G 21 G 22 G 23 G 24 0 0 0 0 A2 0 G 31 G 32 G 33 G 34 0 0 G 37 G 38 A3 0 G 41 G 42 G 43 G 44 G 45 G 46 0 0 A4 0 G 51 G 52 G 53 G 54 G 55 G 56 0 0 A5 0 G 61 G 62 G 63 G 64 0 0 G 67 G 68 A 6 0 G 78 A 7 0 0 0 0 0 0 0 G 77 0 0 0 0 G 85 G 86 0 0 A8 0 (A1) The nonzero elements of G1n are associated with Ex at z = 0 and are given by G1n c [ n ky n k zn ] (A2) 1 The nonzero elements of G2n are associated with Ey at z = 0 and are given by G 2n c k zn (A3) 1 The nonzero elements of G3n are associated with Ex at z = d1 and are given by G 3n c eik zn d1 [ n ky n k zn ] for n = 1, 2, 3, 4 (A4) 1 G 37 c e 2 ik z d1 [ ( k z )k y ( kz ) ] (A5) 140 c G 38 e ik z d1 [ ( k z )k y ( kz ) ] (A6) 2 The nonzero elements of G4n are associated with Ey at z = d1 and are given by c G 4n eik zn d1 k zn for n = 1, 2, 3, 4 (A7) 1 c G 45 e ik z d1 ( kz ) (A8) e ik z d1 ( kz ) (A9) 2 c G 46 2 The nonzero elements of G5n are associated with hx at z = d1 and are given by G 5n eik zn d1 G 55 e G 56 e for n = 1, 2, 3, 4 ik z d1 (A10) (A11) ik z d1 (A12) The nonzero elements of G6n are associated with hy at z = d1 and are given by G 6n ne ik zn d1 G 67 e ik z d1 G 68 e ik z d1 for n = 1, 2, 3, 4 (A13) (A14) (A15) The nonzero elements of G7n are associated with Ex at z = d1+d2 and are given by G 77 c e 2 ik z ( d1 d 2 ) [ ( k z )k y ( kz ) ] (A16) 141 G 78 c e ik z ( d1 d 2 ) [ ( k z )k y ( kz ) ] (A17) 2 The nonzero elements of G8n are associated with Ey at z = d1+d2 and are given by G 85 c e ik z (d1 d 2 ) e ik z ( d1 d 2 ) ( kz ) (A18) 2 G 86 c 2 ( kz ) (A19)

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