# Design, fabrication and modeling of microwave devices based on metallic ferromagnetic materials

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Design, Fabrication and Modeling of Microwave Devices Based on Metallic Ferromagnetic Materials by NICHOLAS KIPPLAN CRAMER B.S., University o f Colorado at Colorado Springs, 1996 M.S., University of Colorado at Colorado Springs, 1998 Technical Report EAS_ECE_2002_4 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 o f Philosophy Department o f Electrical and Computer Engineering 2002 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3042223 ___ __® UMI UMI Microform 3042223 Copyright 2002 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Copyright By Nicholas Kipplan Cramer 2002 I Rights Reserved Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ill This dissertation for the Doctor of Philosophy by Nicholas Kipplan Cramer has been approved for the Department o f Electrical and Computer Engineering by Thottam S. Kalkur (Chair) Robert E. Camley Zbigniew J. Celinski John D. Norgard Jibsj dU^c/LGerald M. Oleszek fth-d a q Date i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. aoo'j Cramer, Nicholas Kipplan (Ph.D., Electrical Engineering) Design, Fabrication and Modeling of M icrowave Devices Based on Metallic Ferromagnetic Materials Dissertation directed by Associate Professor Zbigniew J. Celinski The use o f ferromagnetic metals in microwave device applications has been studied by only a few research groups and only during the past fifteen or so years. This is surprising, however, because ferromagnetic metals possess several properties that are desirable for these applications. These properties include high saturation magnetization, which allows high operating frequencies, ease of fabrication, which allows monolithic integration with solid-state devices, and other properties. Previous experimental studies of ferromagnetic conductor applications have primarily focused on tunable band-stop filter effects in microstrip transmission lines. This study seeks to improve on these effects in this geometry, but also includes two additional effects and one additional geometry. A tunable phase-shift effect near ferromagnetic resonance and a tunable band-pass effect at ferromagnetic anti-resonance are presented. Band-stop and phase-shift effects in coplanar waveguide transmission lines, which hold several advantages over microstrip lines, are also presented. Very few theoretical models of ferromagnetic metal-based devices have been reported by researchers over the past fifteen years and among those reports only two, by Robert Camley and his co-workers, have investigated the phase-shift and band-pass effects. In contrast with Dr. Camley’s work, which was based on first-principles and was limited to the microstrip geometry, a model based on the phenomenon of surface impedance, which is a valid approximation for a variety of geometries and which is easier Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to calculate, is described. This model is used to derive the limits o f operation for various devices and effects and the results o f this model are compared with experimental results. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGEMENTS I thank the National Institute of Standards and Technology (NIST) in Boulder, Colorado for the use of their network analyzer and probe station. I especially thank David K. Walker of NIST for his time and advice on microwave measurements. I thank the Electrical and Computer Engineering Department here at the University of Colorado at Colorado Springs for making the M icroelectronics Research Laboratory available. I am also very grateful for Steve Jem igan’s assistance with equipment in the MRL and for his maintenance of the laboratory. I appreciate the support of Claus M. Schneider and Detlev Tietjen of the Institut fur Festkorper und Werkstofforschung in Dresden (IFW-Dresden), who produced a sputtered permalloy film on GaAs that I used to fabricate coplanar waveguide devices. Finally, I thank the support from the Army Research Office, grant numbers DA AD 19-00-1-0146, DAAG55-98-0294 and DAAG55-97-1-0232. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. vii CONTENTS CHAPTER I INTRO D U CTION .................................................................................................. 1 I . I Purpose of the S tu d y ...........................................................................................................2 1.2 Focus of the Study............................................................................................................... 2 1.3 Overview of Current Technology.....................................................................................3 1.3.1 MEMS D evices..........................................................................................................4 1.3.2 Dielectric Resonator D evices..................................................................................5 1.3.3 Semiconductor D evices............................................................................................6 1.3.4 Ferroelectric D evices................................................................................................ 6 1.3.5 Superconductive D evices........................................................................................ 8 1.3.6 Ferrite D evices...........................................................................................................9 1.3.7 Devices Using Ferromagnetic Conductors..........................................................11 1.4 Arrangement of this R eport............................................................................................. 13 CHAPTER II GENERAL T H E O R Y ........................................................................................ 17 2.1 Permeability of a Ferrom agnet........................................................................................17 2.1.1 “FMR Geometry” and Corresponding Permeability T ensor........................... 18 2.1.2 Voigt Permeability, FM R and FM A R................................................................. 21 2.2 Transmission L in e s ...........................................................................................................25 2.2.1 Power Transmission at High Frequencies.......................................................... 26 2.2.2 LRCG m o d el............................................................................................................27 2.2.3 Characteristic Im pedance...................................................................................... 29 2.2.4 Propagation C o n stan t.............................................................................................31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.2.5 Scattering Parameters.......................................................................................... .32 2.3 High-Frequency Measurements and Calibration....................................................... 34 2.3.1 Measurement of Scattering Parameter with a Vector Network Analyzer ... .35 2.3.2 TRL C alibration.................................................................................................... .36 CHAPTER III DEVICES AND EFFECTS.............................................................................. 39 3.1 Microstrip D evices.......................................................................................................... 39 3.1.1 Basic G eom etry..................................................................................................... .40 3.1.2 Fabrication Techniques........................................................................................ .42 3.1.3 Practical Considerations....................................................................................... .45 3.2 Coplanar W aveguide (CPW) D evices.......................................................................... 46 3.2.1 Basic G eom etry...................................................................................................... .47 3.2.2 Fabrication Techniques........................................................................................ .49 3.2.3 Practical Considerations....................................................................................... .51 3.3 Band-Stop E ffect............................................................................................................. 51 3.3.1 Basic Effect and Definitions of T erm s.............................................................. .52 3.3.2 Design Considerations........................................................................................... .53 3.3.3 A pplications........................................................................................................... .54 3.4 Phase-Shift Effect............................................................................................................ 55 3.4.1 Basic Effect and Definitions of T erm s.............................................................. .56 3.4.2 Design Considerations........................................................................................... .57 3.4.3 A pplications........................................................................................................... .58 3.5 Band-Pass E ffect............................................................................................................. 59 3.5.1 Basic Effect and Definitions o f T erm s............................................................... .59 3.5.2 Design Considerations........................................................................................... .60 3.5.3 A pplications............................................................................................................ .61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ix CHAPTER IV DETAILED TH EORY.......................................................................................63 4.1 Simulations Based on First Principles.......................................................................... 63 4.1.1 Description o f “allguide.f’ ....................................................................................64 4.1.2 Overview o f R esults............................................................................................... 65 4.1.3 Advantages and Lim itations................................................................................. 66 4.2 Simulations Based on Surface Im pedance................................................................... 67 4.2.1 Description o f Surface Impedance.......................................................................68 4.2.2 Calculation o f Surface Im pedance.......................................................................69 4.2.3 Effect o f Surface Impedance in Transmission L in e s........................................71 4.2.4 Results for Non-M agnetic Conductors................................................................ 74 4.2.5 Effects for Ferromagnetic Conductors................................................................ 77 4.3 Equivalence o f the Two Techniques......................................... 79 4.3.1 Adaptation o f Surface Impedance to “allguide.f’ G eo m etry .......................... 79 4.3.2 Comparison o f R e su lts...........................................................................................81 4.4 Calculation o f Limits o f Operation................................................................................ 83 4.4.1 Stop-Band R ejection .............................................................................................. 85 4.4.2 Stop-Band Frequency R ange................................................................................ 88 4.4.3 Stop-Bandwidth and Insertion L oss.....................................................................89 4.4.4 Phase-Shift Tunability............................................................................................90 4.4.5 Phase-Shifler Figure o f M e rit...............................................................................92 4.4.6 Pass-Band Insertion L o s s ......................................................................................94 4.4.7 Band-Pass Selectivity............................................................................................. 96 4.4.8 Pass-Band Frequency R an g e.............................................................................. 100 4.4.9 V S W R ..................................................................................................................... 101 CHAPTER V RESULTS.............................................................................................................104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. X 5.1 Band-Stop E ffects........................................................................................................... 105 5.1.1 Results in M icrostrip D evices.............................................................................106 5.1.2 Comparison with T heory..................................................................................... 108 5.1.3 Results in CPW Devices.......................................................................................108 5.1.4 Comparison with T heory..................................................................................... 109 5.2 Phase-Shift E ffects.......................................................................................................... 110 5.2.1 Results in M icrostrip D evices............................................................................ 111 5.2.2 Comparison with T heory..................................................................................... 112 5.2.3 Results in CPW Devices.......................................................................................113 5.2.4 Comparison with T heory..................................................................................... 113 5.3 Band-Pass Effects............................................................................................................ 114 5.3.1 Results in M icrostrip D evices............................................................................ 115 5.3.2 Comparison with T heory..................................................................................... 116 5.3.3 Results in CPW Devices.......................................................................................117 5.3.4 Comparison with T heory..................................................................................... 118 5.4 VSWR and Insertion L o ss............................................................................................. 118 CHAPTER VI CONCLUSIONS............................................................................................. 122 6.1 Sum m ary...........................................................................................................................122 6.2 Fulfillment of Study Purpose and Focus.....................................................................124 6.3 Final N otes....................................................................................................................... 125 6.3.1 Device Structure and Composition.....................................................................125 6.3.2 Comparative Perform ance................................................................................... 126 APPENDIX A EPITAXIAL GROW TH OF FE ON GAAS................................................ 128 A .l Deposition o f Ag(001) on GaAs(OOl)........................................................................129 A.2 Deposition of Fe(001) on A g(001)............................................................................. 130 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. xi A.3 Details o f Deposition and Characterization...............................................................131 APPENDIX B FILM CHARACTERIZATION W ITH F M R ..............................................136 B.l FMR A pparatus.............................................................................................................. 136 B.2 Data Collection and A nalysis.......................................................................................138 APPENDIX C DIELECTRIC AND RADIATIVE L O S S E S ............................................ 143 C .l Dielectric Losses in Microstrip.................................................................................... 143 C.2 Dielectric Losses in Coplanar W aveguide................................................................. 144 C.3 Radiation Losses in M icrostrip.................................................................................... 145 C.4 Radiation Losses in Coplanar W aveguide................................................................. 145 APPENDIX D SURFACE IMPEDANCE IN THIN FILM S...............................................147 D.l Calculation for Thin F ilm s........................................................................................... 147 D.2 Comparison o f Finite and Infinite F ilm s....................................................................150 D.3 Implications for Ferromagnetic D evices................................................................... 153 APPENDIX E DESIGN EQUATIONS FOR CPW AND M ICROSTRIP........................156 E. 1 Characteristic Impedance of C PW ...............................................................................156 E.2 Characteristic Impedance of M icrostrip......................................................................158 APPENDIX F M ATERIAL PROPERTIES.............................................................................160 APPENDIX G PHOTO MASKS AND TRL CALIBRATION SETS............................... 162 APPENDIX H SURFACE IMPEDANCE SIM U L A TIO N ................................................. 167 REFERENCES.............................................................................................................................. 175 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. xii FIGURES Fig. 1-1 Tunable Dielectric Filter..................................................................................................5 Fig. 1-2 Commercial YIG Filters..................................................................................................9 Fig. 1-3 M icrostrip Comparison.................................................................................................. 12 Fig. 2-1 Arrangement of Magnetic Fields and M om ents........................................................18 Fig. 2-2 Voigt Permeability vs. Frequency...............................................................................22 Fig. 2-3 Resonance and Anti-resonance Frequencies..............................................................23 Fig. 2-4 Voigt Permeability with Dam ping...............................................................................25 Fig. 2-5 Equivalent Circuit for a Transmission Line...............................................................28 Fig. 2-6 Relation between Travelling Waves and S-param eters........................................... 33 Fig. 2-7 Agilent 8 5 IOC VNA and Cascade M icrotech P robe............................................... 35 Fig. 2-8 VNA System at N IST.................................................................................................... 36 Fig. 2-9 TRL Calibration Set and DUT......................................................................................37 Fig. 3-1 M icrostrip Cross Section.............................................................................................. 40 Fig. 3-2 M icrostrip Device Structure.........................................................................................41 Fig. 3-3 Position o f Probes and Electrom agnet........................................................................42 Fig. 3-4 M icrostrip Fabrication...................................................................................................43 Fig. 3-5 M icrostrip Shadow and Photo M asks......................................................................... 44 Fig. 3-6 Photograph o f Completed Microstrip D evice........................................................... 45 Fig. 3-7 Coplanar Waveguide Cross S ectio n ........................................................................... 47 Fig. 3-8 Coplanar Waveguide Device Structure.......................................................................48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. xiii Fig. 3-9 CPW Comer and Width Change..................................................................................48 Fig. 3-10 Coplanar W aveguide Photo M ask............................................................................ 50 Fig. 3-11 Bandstop Device Param eters.....................................................................................53 Fig. 3-12 Tunable Filter Application......................................................................................... 55 Fig. 3-13 Modulation of Phase S h ift......................................................................................... 56 Fig. 3-14 Insertion Loss in a Phase-Shifter.............................................................................. 57 Fig. 3-15 Tunable Phase-Shifter Application........................................................................... 59 Fig. 3-16 Bandpass Device Param eters.....................................................................................60 Fig. 3-17 Bandpass A pplication................................................................................................. 62 Fig. 4-1 allguide.f Structure........................................................................................................65 Fig. 4-2 Effect of Skin Depth on G eom etry............................................................................. 75 Fig. 4-3 Added Impedance for Ferromagnetic Conductors....................................................78 Fig. 4-4 Comparison of Attenuation...........................................................................................82 Fig. 4-5 Comparison of P hase.....................................................................................................83 Fig. 4-6 Stop-Band Rejection......................................................................................................87 Fig. 4-7 Stop-Band Frequency R ange....................................................................................... 88 Fig. 4-8 Stop-Bandwidth and Insertion Loss............................................................................ 90 Fig. 4-9 Phase-Shifter Figure of M erit......................................................................................93 Fig. 4-10 Pass-Band Insertion Loss........................................................................................... 96 Fig. 4-11 Band-Pass Selectivity............................ :................................................................... 98 Fig. 4-12 Pass-Band Frequency R an g e................................................................................... 100 Fig. 4-13 VSWR at Resonance..................................................................................................103 Fig. 5-1 Stop-Band Frequency vs. Applied Field...................................................................106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. xiv Fig. 5-2 Band-Stop in Py M icrostrip........................................................................................ 107 Fig. 5-3 Band-Stop in Fe M icrostrip........................................................................................ 107 Fig. 5-4 Band-Stop in Py C P W ................................................................................................ 109 Fig. 5-5 Simulation and Experiment Com parison................................................................110 Fig. 5-6 Phase-Shift in Py M icrostrip..................................................................................... 111 Fig. 5-7 Phase-Shift in Fe M icrostrip....................................................................................... 112 Fig. 5-8 Phase-Shift in Py C PW .............................................................................................. 113 Fig. 5-9 Simulation and Experiment Com parison................................................................114 Fig. 5-10 Band-Pass in Ni M icrostrip.................................................................................... 116 Fig. 5-11 Band-Pass in Py C P W ............................................................................................. 117 Fig. 5-12 VSWR and Insertion Loss vs. M ism atch..............................................................119 Fig. 5-13 VSWR and Insertion Loss In M icrostrip............................................................ 120 Fig. 5-14 VSW R in Py CPW ......................................................................................................121 Fig. A -l Fe(OOl) film on aGaAs(OOl) surface...................................................................... 129 Fig. A-2 Interface between Fe(OOl) and Ag(OOl) film s........................................................130 Fig. A-3 Orientation o f Crystal and Anisotropy D irections.................................................131 Fig. A-4 GaAs Surface after Sputtering and A nneal............................................................. 132 Fig. A-5 Fe Seed L a y er...............................................................................................................133 Fig. A-6 Ag Template after Anneal...........................................................................................134 Fig. A-7 Surface o f Thick Fe F ilm ..........................................................................................135 Fig. A-8 Surface o f Ag Capping F ilm ...................................................................................... 135 Fig. B -l FMR System Waveguide and Resonance Cavity................................................... 137 Fig. B-2 Equivalent Circuit for Coupled C av ity .................................................................. 139 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. XV Fig. B-3 Reflected Power vs. Cavity Im pedance................................................................... 139 Fig. B-4 Typical FMR Spectrum.............................................................................................. 140 Fig. B-5 Resonance Field vs. Field Angle.............................................................................. 141 Fig. D -l Added Inductance for Finite and Infinite Film s.................................................... 151 Fig. D-2 Added Resistance for Finite and Infinite F ilm s.................................................... 152 Fig. D-3 Skin Depth in Ferromagnetic M etal.......................................................................... 153 Fig. E -l CPW D im ensions......................................................................................................... 156 Fig. E-2 M icrostrip Dimensions............................................................................................... 158 Fig. G -l Photo Mask for CPW on G aA s.................................................................................. 164 Fig. G-2 Photo Mask for CPW on SiC> 2 ................................................................................... 165 Fig. G-3 Shadow Mask for M icrostrip.................................................................................... 166 Fig. G-4 Photo Mask for Microstrip........................................................................................ 166 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. xvi TABLES Table 1-1 Ferroelectric Phase-Shifters........................................................................................ 7 Table 1-2 Tunable YIG F ilters.................................................................................................... 10 Table 4-1 Values Used in Sim ulations...................................................................................... 81 Table 4-2 Sample Rejection V alues...........................................................................................87 Table 4-3 Sample Phase R an g es................................................................................................ 92 Table 4-4 Sample Insertion Loss Values...................................................................................95 Table 4-5 Sample Selectivity V alu es........................................................................................ 98 Table 4-6 Band-Pass Technology Com parison........................................................................99 Table 4-7 Sample VSWR V alues............................................................................................. 103 Table 5-1 Sample Overview..................................................................................................... 104 Table B -l FMR Curve Fit P aram eters.................................................................................... 142 Table F-l Dielectric Properties.................................................................................................160 Table F-2 YIG and Ferromagnetic M etals..............................................................................161 Table H-l Input Param eters...................................................................................................... 167 Table H-2 Output File “skinout.dat” ........................................................................................168 Table H-3 Output File “zpc.dat” .............................................................................................. 169 Table H-4 Output File “sparas.dat” ..........................................................................................170 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER I INTRODUCTION Microwave applications have undergone explosive growth in the past decade. Much of this growth is due to the advent of personal microwave devices, such as cellular telephones. However, cellular communication is not the only application of microwaves in telecommunications— both terrestrial and satellite communication links have used microwaves for decades. In communication applications, higher frequencies allow for greater data transmission speeds. This equates to more voice channels in a telephone system, more television channels in a broadcasting system and more subscribers in an internet service provider. Another common application of microwaves is in radar systems. This includes navigation and remote sensing, as well as military applications such as electronic warfare. In these systems, higher frequencies allow access to “windows” in the atmosphere— frequency bands of low attenuation. Higher frequency also decreases antenna size. In electronic warfare, a wider range of available frequencies makes a communications or radar system more difficult to interfere with. All microwave systems have one thing in common: they are all constructed of basic microwave devices. M icrowave devices range widely in both function and in technology. This chapter includes a brief overview of these technologies, but the majority Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. o f this report concerns one technology: metallic ferromagnets and their use in tunable filters and phase-shifters. 1.1 Purpose of the Study Researchers have studied the use of ferromagnetic conductors in tunable microwave devices very little over the past decades. This is surprising because this class o f materials possesses several desirable qualities. Perhaps the most desirable characteristic of ferromagnetic metals is their generally high saturation magnetization. As shown later, high saturation magnetization leads to effects at higher frequencies than effects in devices with low magnetization materials such as ferrites. The purpose o f this study is to investigate the possibility of applying ferromagnetic conductors in tunable microwave devices. These devices were fabricated and their effects were measured. A model is presented here that describes the effect of ferromagnetic metals in devices and allows one to simulate measured data from material properties and device geometry. This model, in turn, allows one to design devices that maximize desirable characteristics. 1.2 Focus of the Study This study generally concerns the use of ferromagnetic conductors in transmission line devices in order to create tunable microwave devices. In particular: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • There is a focus on three ferromagnetic metals: Fe, Ni and Ni7 gFe22 (permalloy, often abbreviated “Py”). The deposition process of these materials and their characteristics are described. • Two transmission line structures are presented: microstrip and coplanar waveguide. The fabrication of these structures is outlined and the implication their structure has on device design is discussed. • Three tunable device effects are studied: band-stop filtering, band-pass filtering and phase shifting. A theoretical model for devices is derived that predicts these effects and allows one to determine analytically the influence that a particular design parameter has on a particular device characteristic. Each of these effects is demonstrated and experimental results are compared with this model. 1.3 Overview of Current Technology Research on tunable filters and phase-shifters has pursued a wide range of technologies. Ferrimagnets (or ferrites) is one of the oldest technologies and is therefore one o f the most widely used. These materials operate in much the same way as ferromagnetic materials: their permeability is a function of an applied magnetic field. Ferroelectric materials are another similar class of materials; however, it is their permittivity that changes in response to an applied electric field. Another technology is semiconductor technology, which perhaps is best known for its application in digital computing. Other classes o f devices include those based on Micro-Electro-Mechanical Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Systems (MEMS), those consisting of mechanically-positioned dielectrics and those utilizing High-Temperature Superconductors (HTS). This section contains a discussion o f all o f these technologies, including their advantages and drawbacks. 1.3.1 MEMS Devices Micro-Electro-Mechanical Systems (MEMS) can form a wide variety of microwave devices, including switches, filters, inductors, capacitors and phase-shifters. The benefits o f these devices include reduced cost, size and power consumption [1], MEMS filters use either tunable resonators (constructed with tunable capacitors) or switches that connect to fixed resonators [2], Researchers have reported band-pass filters that are tunable by 3.8% at 20 GHz with 3.6 dB insertion loss and filters that have only 1.7 dB of insertion loss at 33.2 GHz [3,4]. MEMS phase-shifters work much the same way as filters: they are based on tunable capacitors. MEMS phase-shifters benefit from small size, light weight, wide operating bandwidth, low insertion loss, excellent reliability and easy integration with other microwave devices [5]. One group reported a tunable phase-shifter with 2 dB of insertion loss and a tunability of 118° at 60 GHz [6]. Another group applied BaSrTiO? (BST), a paraelectric material, to increase the capacitance of tunable capacitors in a tunable phase-shifter [7]. It is very likely that MEMS microwave devices will be used more and more in commercial applications in the near future. Researchers have only recently begun to study these devices, but MEMS devices already show many advantages over competing technologies. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 1.3.2 Dielectric Resonator Devices Dielectric resonators operating at a fixed frequency are a mature technology for filter applications. These devices usually consist simply of a ceramic disk that resonates at a given frequency. O ne example of a tunable filter based on dielectric resonators is shown in Fig. 1-1. Produced by Coleman M icrowave Company, this device relies on the mechanical movement o f dielectric pieces to produce different filter frequencies [8]. Motor Control I s l a n d Pow er Stepper *1 Motor Filter Unit Fig. 1-1 Tunable Dielectric Filter The advantages and drawbacks of such a filter are rather straightforward. The primary advantages of these filters are their frequency accuracy, narrow bandwidth and high power capability [9]. The device in Fig. 1-1 has a bandwidth of only 2 MHz and is tunable from 2.2 to 2.3 GHz in 1024 frequency steps (0.1 M Hz per step). The drawbacks o f such a filter are the size and the slow-speed tuning. In order to allow electronic control, this device includes a stepper m otor for the mechanical adjustment. The filter, motor and the power supply and control for the motor create a device that is 19.5 inches long. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 1.3.3 Semiconductor Devices The primary advantage of semiconductor filters and phase-shifters is the fact that one can construct Monolithic Microwave Integrated Circuits (MMICs). MMICs contain a variety o f semiconductor microwave devices and therefore one can produce an entire system on a single wafer, using a single technology. A second advantage of semiconductor tunable devices is their fast tuning speed; they can modulate at frequencies sim ilar to their operating frequency. Yet another advantage is the possibility of using direct-gap semiconductor materials to integrate microwave devices with optical communications [10]. The most basic tunable semiconductor element is a Schottky diode, which is a voltage-variable capacitor (varactor) [11,12]. This tunable capacitor, when combined with lumped inductors, creates tunable resonators that form the basis o f filters and phaseshifters. M ore advanced tunable semiconductor devices utilize Pseudomorphic High Electron Mobility Transistors (PHEMTs) [13,14], Two major disadvantages of semiconductor devices are their low power-handling capability and their low frequency selectivity in filters [15]. These devices also generally have a small tuning range and high insertion loss. 1.3.4 Ferroelectric Devices Ferroelectric and paraelectric materials have a permittivity that varies with applied electric field. Thus, they are a natural candidate for tunable devices. Although there are some studies of tunable filters based on ferroelectric materials, most research Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. has focused on tunable phase-shifters [16]. Table 1-1 presents a comparison of tunable phase-shifters based on BaxSri.xTiCh (BST) and SrTi0 3 (STO). The Figure of Merit (FOM) is the tunable phase range divided by the insertion loss. Table 1-1 F erroelectric Phase-Shifters M aterial O p eratin g Frequency (GHz) Phase R ange (°) Insertion Loss (dB) F igure o f M erit (°/dB) R eference BST 10 360 6 60 [17] BST 18 200 2.7 74 [18] BST 2.4 165 <3 >55 [19] STO 2-10 55 0.5 110 [20] A major advantage of ferroelectric or paraelectric devices is the fact that the tuning is driven by application of voltage across the material. This is somewhat easier to implement than a tunable magnetic field, which requires an electromagnet. Ferroelectric phase-shifters have two major drawbacks, however. First, the materials themselves are rather lossy at microwave frequencies— this increases insertion loss [21]. Second, the phase-shift effect is accomplished by changing the capacitance per unit length of the transmission line that comprises the device. impedance o f the line and increases VSWR. This also changes the characteristic A solution to this problem is the incorporation o f ferrite material into a device [22], The ferrite provides a variable permeability to complement the variable permittivity of the ferroelectric; this creates a variable phase-shift with nearly constant line impedance. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.3.5 Superconductive Devices High Temperature Superconductors (HTS) hold great promise for a variety o f microwave devices. Below the critical temperature, superconductors such as YBa^Cu.^O? (YBCO) have infinite conductivity and therefore devices fabricated from these materials have very low insertion loss [23,24]. HTS devices, such as resonators, filters, couplers, antennas and delay lines, could one day be used in satellite-based communication systems, in base stations for mobile communications and in radar systems [25,26]. Researchers have produced HTS filters with low insertion loss and other favorable characteristics. O ne group constructed YBCO-based filters with 2 to 3 dB insertion loss [27]. Another used TfCaBaiCuiOg to create filters with 2.5 dB insertion loss that could be tuned from 1 to 21 GHz [28], Studies of HTS phase-shifters have relied on a wide range of technologies. One study reported a phase-shifter based on a combination o f YBCO with a ferrite substrate that demonstrated 0.1 dB insertion loss at 10 GHz [29]. Another group reported 0.8 dB insertion loss in a HTS/ferrite device [30]. Researchers have also combined HTS materials with ferroelectric materials and semiconductors to produce tunable phaseshifters [31,32]. Yet another technique for producing tunable phase in HTS devices relies on the Superconducting-Normal (S-N) state transition, eliminating the need to incorporate a second material technology [33]. HTS devices offer a variety of desirable characteristics, but are limited by the critical temperature o f the material. The highest critical temperatures to date are still only tens of degrees above liquid nitrogen (77 K). Thus, any terrestrial application of HTS materials must either use liquid nitrogen cooling or mechanical compressor cooling. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Space applications, however, may use the ambient temperature to cool below the critical temperature. HTS devices are therefore limited to satellite applications and terrestrial applications where the inconvenience of cooling is offset by the advantages of device performance. 1.3.6 Ferrite Devices The first ferrite-based microwave filter was reported by deGrasse in 1958 [34]. Since that time, researchers have developed a wide range of materials and devices [35]. Non-reciprocal devices, such as isolators, circulators and directional couplers are a common application o f ferrite materials [36,37]. Tunable filters and phase-shifters are other common applications. Fig. 1-2 Commercial YIG Filters Yttrium-Iron-Gamet (YFesOu, often called “Y IG ”) is the most common material for tunable ferrite filters and is capable of being tuned from 0.5 to 40 GHz [38,39]. Tw o commercially available YIG filters are shown in Fig. 1-2 and typical characteristics of these filters are listed in Table 1-2 [40,41], Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 1-2 T unable YIG F ilters M a n u fa c tu re r T un ing R ange (GHz) P eak R ejection (dB) 3 dB B andw idth (M Hz) Insertion Loss (dB) VSW R Filtronic 4.5-18 >40 150 2.0 2.0:1 Filtronic 8.9-9.6 >70 300 1.5 2.0:1 Omniyig, Inc. 2-8 >40 125 1.5 2.5:1 Omniyig, Inc. 8-18 >40 150 1.5 2.5:1 In order to achieve higher frequencies, some filters make use of highly anisotropic materials such as barium hexaferrite (BaFenOig). One such filter achieved frequencies from 50 to 75 GHz with 6 dB of insertion loss [42]. Hewlett-Packard uses a similar device in the 26 to 50 GHz range in their spectrum analyzers [43]. Researchers have reported tunable phase-shifters based on a variety of materials. One study used a lithium ferrite (Lio.e^Zno.iTicusSno.iFei^sO-t), which operated from 12.5 to 18 GHz [44], Another used ZmBaoFenO?? to produce 450° of phase shift at 20 GHz [45], The primary benefit of ferrite devices is the wide range of frequencies available due to the wide range o f ferrite materials. Another benefit of these devices is the tuning speed, w'hich is on the order of milliseconds— in contrast to mechanically tuned filters that have tuning speeds on the order of seconds. The primary drawback of ferrites is their low power capability. These materials exhibit non-linear characteristics at fairly low powers; these characteristics inhibit proper operation o f filters and phase-shifters, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 although these effects are the basis of other ferrite devices such as signal-noise enhancers and power limiters. 1.3.7 Devices Using Ferromagnetic Conductors Ferromagnetic conductors for device applications have only recently been studied by a limited number of researchers. Investigators at Raytheon first presented a study of epitaxial Fe films on GaAs substrates, which they suggested could be used as part of a memory element, in 1988 [46]. The same group, in 1991, published a study of microstrip devices based on these samples for use as tunable band-stop filters [47]. These devices achieved about 4.4 dB/cm stop-band rejection with 0.7 dB/cm of insertion loss. The thickness of the dielectric, which consisted of a 100 pm thick GaAs wafer, greatly reduced the stop-band rejection. Investigators at the University of California at Irvine also reported devices of almost identical design in 1999 and 2000 [48,49]. These also only produced stop-band rejections of about 5 dB/cm. Also in 1999, a research group in Belgium produced microstrip structures with the dielectric film containing ferromagnetic nanowires [50]. These structures produced larger stop-band rejection (up to 20 dB/cm), but suffered from very large stop-bandwidth, insertion loss and impedance mismatch. One group, in France, studied effects in a non-microstrip geometry [51,52], Their studies concerned coaxial cable structures, with the dielectric replaced by a rolled dielectric/ferromagnetic metal film laminate. These structures produced good band-stop effects, but the fabrication process restricted the range of practical materials. Our group published its first report of experimental data from a stop-band filter based on Fe films in 2000 [53], This device demonstrated a very large stop-band Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 rejection o f over 70 dB/cm, but suffered from a high degree of impedance mismatch and insertion loss. The higher rejection of this device, shown in Fig. 1-3 (b), compared with the microstrip devices produced by the Raytheon and Irvine groups, shown in Fig. 1-3 (a), was due to the thickness o f the dielectric, which was two orders of magnitude thinner in our devices. In 2001, we reported the phase-shift effect in the microstrip device and presented a new device structure: coplanar waveguide [54], The phase-shift in the microstrip device was impressive; it could be modulated over a range of 450°/cm at 9 GHz, but the device still showed high insertion loss. The coplanar waveguide device, which was based on permalloy rather than Fe, demonstrated much smaller insertion loss, but also sm aller rejection. 0.1 ju n Fe capped with thick high-conductivity layer 100 or 350pm thick GaAs(100) substrate ground plane x. (a) 2 fim Ag layer 0 .2 p m F e ll001 capped with thin Ag layer Ag(100) layer GaAs(100) substrate Fig. 1-3 Microstrip Comparison In all, five research groups have created devices based on ferromagnetic conductors. O ur group has studied both microstrip and coplanar waveguide devices and both band-stop filters and phase-shifters. Three groups have only studied microstrip devices and only measured band-stop effects and the fifth group studied band-stop effects Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13 in coaxial cable. As of yet, no one has produced devices that show that ferromagnetic metal technology is mature enough to incorporate in commercial devices. Even fewer groups have studied the theoretical modeling o f devices that incorporate ferromagnetic conductors. Researchers at the Massachusetts Institute of Technology and Northeastern University suggested a tunable band-pass filter based on ferromagnetic anti-resonance in a metal film in 1994 and 1995 [55,56]. Robert Camley from our group, along with Douglas Mills from UC Irvine, reported a model of parallel plate structures containing ferromagnetic metal and dielectric films [57]. Their study showed the effect of band-stop filtering at resonance and band-pass at anti-resonance. Soon after, Dr. Camley and Robert Astalos published a study of a simplified device structure, which contained one dielectric film and one ferromagnetic film [58]. This study used the FORTRAN program “allguide.f’, which is discussed in Chapter 4. As in the previous study, this report presented the band-stop and band-pass effects at resonance and anti-resonance; however, Astalos and Camley also presented simulation o f the phaseshift. In 2001, this author, along with David Walker from the National Institute of Standards and Technology (NIST), proposed a model of transmission lines with ferromagnetic conductors and general geometry [59]. 1.4 Arrangement of this Report Chapter 2 is titled “General Theory”. Its purpose is introduce the concepts of ferromagnetic permeability, transmission line theory and microwave measurements. The theory of permeability in ferromagnets is key to the understanding o f the interaction Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. between electromagnetic waves and ferromagnetic materials. Later, this theory will be expanded to describe various effects in microwave devices. The theory of transmission lines is also important in that it provides a method for modeling experimental data from material and geometric characteristics of a device. The details of microwave measurements impact experimental set up and device design. The next chapter, “Devices and Effects”, introduces the devices structures studied and the ferromagnetic effects of interest. The two device structures studied are discussed: microstrip and coplanar waveguide. This discussion includes the design of the devices, how they are fabricated and what practical issues had to be considered. The three effects are band-stop, phase-shift and band-pass. The general behavior o f each effect, along with its characteristic parameters and its applications, is discussed. In Chapter 4, entitled “Detailed Theory”, the theories presented in Chapter 2 is expanded and those theories are applied to the devices and effects discussed in Chapter 3. Chapter 4 begins with a discussion of the FORTRAN simulation program “allguide.f’, written by Astalos and Camley [58]. allguide.f is based on a numerical solution to the wave equation and applies M axwell’s equations along with electromagnetic boundary conditions. Thus, it is based on first-principles and is an accurate method for modeling certain types of structures. Next, a technique for modeling devices based on the phenomenon of surface impedance is presented. This technique is developed for materials with arbitrary permeability and therefore it can be applied to both magnetic and non-magnetic structures. It is then shown that this technique does indeed produce the correct equations for the case of non-magnetic conductors. The next section contains a comparison between the results o f an allguide.f simulation and a simulation using the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i5 surface impedance technique to show that the surface impedance technique is accurate for the case of magnetic conductors. Finally, the chapter concludes with a derivation, based on the surface impedance method, o f the limits of operation of the devices studied. The next chapter is “Results”, Chapter 5. It contains experimental data demonstrating the three effects in the two device structures presented in Chapter 3. The limits of operation observed in the experimental data are compared with the limits derived in Chapter 4. The sixth and final chapter is “Conclusions”, which concludes and summarizes this study. In addition, eight appendices are included that provide background and further detail on some of the concepts discussed in the main paper. Appendix A presents the method of epitaxial growth of Fe on GaAs, which were used to produce some o f the microstrip devices. Appendix B describes the use of a Ferromagnetic Resonance system (FMR) to measure the properties o f a ferromagnetic film. Such a system was used to characterize epitaxial Fe films used in microstrip devices. Appendix C discusses non conductor loss mechanisms in microstrip and coplanar waveguide transmission lines. These mechanisms are not related to ferromagnetic conductor effects and therefore they are not discussed in detail in the main paper. However, one must consider these effects when calculating the limits o f operation of devices. Appendix D concerns the surface impedance of finite-thickness ferromagnetic metal films. This appendix summarizes the implications of thin ferromagnetic films in devices. Appendix E contains equations for the characteristic impedance o f microstrip and coplanar waveguide lines. These equations are used with the surface impedance technique to model devices. Appendix F Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. lists material properties that are used throughout this report. Appendix G presents the photo masks and shadow masks used to fabricate devices. Appendix H contains the source code for a FORTRAN program that uses the surface impedance technique to simulate effects in coplanar waveguide structures. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 17 CHAPTER II GENERAL THEORY In order to understand the effects of ferromagnets in microwave devices, one must understand three general theories. The first is the theory of how a ferromagnet interacts with electromagnetic waves. This describes, for example, the role of material properties such as saturation magnetization in determining device capabilities. The second is the theory o f transmission lines. This theory allows the incorporation of material effects into equations that describe the device as a whole. Finally, the theory behind microwave measurements is important to understand. This allows the comparison of experimental data to theoretical simulations. 2.1 Permeability of a Ferromagnet The most common technique used to describe the interaction o f electromagnetic waves with ferromagnetic materials is through the permeability of the ferromagnet. The permeability is the effect a magnetic field (from the wave) has on magnetization (in the ferromagnet). The characteristics o f the permeability determine the magnetic effects in actual devices, a theory that is developed further in Chapter 4. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.1.1 “FMR Geometry” and Corresponding Permeability Tensor The starting point for determining the permeability is the Landau-Lifshitz equation, named for those who first presented it in 1935 [60]: ^ L = y ( M x B eff) ( 2 - 1) The theory behind this equation is quite simple: the cross-product of the magnetization, M, and the magnetic field, B, produces a mechanical torque. Newton’s 2nd law, in rotational terms, defines a torque as that quantity that produces a change in angular momentum over time. Scaling this torque by the gyromagnetic ratio, y, produces a change in the magnetic moment (rather than the mechanical moment) over time. Thus, we have an equation that relates the dynamic effects in a material to the external magnetic fields. K RF magnetic field _L to applied field Applied F (DC) m Fig. 2-1 Arrangement of Magnetic Fields and Moments The next step is to consider a more specific case to apply to equation (2-1), such as the geometry shown in Fig. 2-1. A large magnetic field, Bo, causes the magnetic Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. moments in the material, m, to align with it. b represents an RF magnetic field, which has a peak magnitude much smaller than Bo- This RF field causes the moments to precess about the axis determined by Bo (indicated by dotted lines). This is the basic configuration o f fields and moments that describes ferromagnetic resonance. The mathematical description of the arrangem ent in Fig. 2-1 is as follows. Start with a large, constant magnetic field in the z-direction. Bo, and add small magnetic Fields in all directions: bx, by and bz, which are allowed to vary in time sinusoidally: B'lj = bxx + byy + ( 5 0 + b. )z (2-2) If Bo is large enough to saturate the ferromagnet, and if the b-fields are small enough not to upset this saturation, then the resulting magnetization is simply the sum o f a large constant magnetization in the z-direction, Ms, plus small oscillating magnetizations: mx, niy and mz: M = m Ix + myy + ( M s + m . ) z (2-3) Assume that the magnetization has sinusoidal tim e dependence and oscillates with an angular frequency o f co. This implies the following for the time-derivative o f the magnetization: M = M aeJa* -> = jcoM (2-4) Applying equations (2-2), (2-3) and (2-4) to equation (2-1) produces the dependence of M on B. This dependence is linearized by dropping terms with more than one small quantity (that is, the m’s and b ’s). This linearization also allows the RF field Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and the dynamic magnetization to have the same time dependence. A tensor, %, quantifies the relation between the dynam ic magnetization and the RF field: (2-5) / V ” = x.t> This tensor, the magnetic susceptibility, is called the Polder susceptibility tensor and has the following form [61]: ' Xi x = -iXi 0 iX i 0 Xx 0 0 ( 2- 6 ) 0 Note that the entire z row and column are filled with O's. This is due to the linearization, which eliminates dynamic effects from the z-direction— only the constant relation of the applied field saturating the magnet exists in the z-direction. The two values in the diagonal and off-diagonal elements are: Xi 7HoMsyB0 f (2-7) *» (2- 8 ) ‘>„■> Y'B q ~ co and _ aW 0M S /C 2 “> r%2 The following relation relates the permeability to the magnetic susceptibility: f i = v 0( i + x ) This produces: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2-9) 2i Mi JMi Oy -jMi 0 Mi 0 0 1 ( 2 - 10) where: ( 2 - 11) and ( 2 - 12 ) Thus, we have a tensor for the permeability, which is a function of frequency and applied field as well as the material properties Ms and y. 2.1.2 Voigt Permeability, FMR and FMAR The tensor permeability defined above in equation (2-10) affects electromagnetic waves in such a way that we can define an effective scalar permeability for the case of waves propagating in the z-direction (the direction of the applied field, Bo). The literature generally refers to this effective permeability as the “Voigt permeability” and defines it in terms of the real tensor components, pi and p 2 , which are defined by equations (2-11) and (2-12). p VOigt is the relative permeability defined by: (2-13) which is a real scalar. Upon substitution and simplification, this becomes: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 _ CD2 - y 2( B0 + / i qM s )2 W ig! m,t' (2-14) co2 - Y 2B Q( B Q+ f i QM s ) The general behavior o f the Voigt permeability as a function of frequency is shown in Fig. 2-2. 20 56 0CO CD E i_ CD CL - 10- -20 0 20 40 60 80 100 Frequency (arb. units) Fig. 2-2 Voigt Permeability vs. Frequency The Voigt permeability has one pole and one zero in terms of positive frequency, as shown in Fig. 2-2. These two frequencies define the resonance and anti-resonance conditions in the ferromagnet. The frequency for ferromagnetic resonance, the “FMR frequency”, corresponds to a pole in the permeability [62]: (2-15) The “FMAR frequency” is the frequency for ferromagnetic anti-resonance and it occurs at: 0 3 FMAR = y {Bq + H qM s ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2-16) 23 Fig. 2-3 shows the effect of applied field strength and the saturation magnetization o f a material on the resonance and anti-resonance frequencies. Py is a common abbreviation for permalloy, which is a ferromagnetic alloy of 78% Ni and 22% Fe. The saturation magnetization o f permalloy is roughly half that of Fe; thus, the FMAR frequency for Fe is roughly twice that of Py for the same applied field. The FMR frequency o f Fe is also larger than that of Py, but the difference is not as large. 80 70N X CD C E < 2 Fe f,FMAR 6050- Py f,FMAR U_ 40tr 2 30- u . Fe f,FMR 20- Ry f.FM R 10- 0 100 200 300 400 500 Applied Field BQ(mT) Fig. 2-3 Resonance and Anti-resonance Frequencies The entire analysis above is for the case of zero damping in the oscillations. O f course, a physical system includes damping. One method for describing the effect of damping on the perm eability is to add an imaginary term to the applied field terms: yB0 - >yB0 - j T a ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2-17) 24 The quantity T is a unitless parameter called the damping coefficient. After the substitution shown in (2-17), and after approximation for T much less than unity, the Voigt permeability is complex with a real part equal to: ' _ (gU & FM AR X ® FMAR ®FM R ) (2-18) voiot and an imaginary part of: FMAR (2-19) 0 3 FMR ^ F M A R where gda is + ®FM R — COf MR ) + CQ± a frequency related to the damping coefficient by: ( 2 - 20 ) <uA = r yji0M s Note that the real part of the Voigt permeability given in equation (2-18) no longer has a real pole and now has two real positive zeros at cofmr and cofmar. Note also that the imaginary part o f the Voigt permeability given in equation (2-19) is a Lorentzian peak with a full width at half maximum (FWHM) of coA. Fig. 2-4 shows the general behavior of the Voigt permeability with the effect of damping. The real part, the solid line, has a similar shape to the permeability shown in Fig. 2-2 only now it remains finite for all frequencies. The real part changes rapidly near co fm r (which is 30 in arbitrary units) and passes through zero at c o fm r- The maximum and minimum in the real part are separated in frequency by coA, which has a value of five in this plot. centered on The imaginary part, represented by the dotted line, is a Lorentzian peak c o fm r with a FWHM of coA. Thus, one can determine the dam ping coefficient Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 empirically by measuring either the real or the imaginary part o f the Voigt permeability in a material. 20 c zi 10- 0.Q CO CD E <5 -10Q_ -20 0 20 40 60 80 100 Frequency (arb. units) Fig. 2-4 Voigt Permeability with Damping As discussed further in Chapter 4, the real part of the Voigt permeability plays a role in the phase-shift of waves in devices that include ferromagnetic materials. The imaginary part of the permeability leads to attenuation of waves in such devices. Thus, the general shapes shown in Fig. 2-4 are similar to those seen in experimental results. 2.2 Transmission Lines A general transmission line is any two-port device that allows electromagnetic power to travel from one port to the other. In basic circuit theory, such a device may consists of an ideal wire, which transmits electrical voltage from one end to the other Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. without time delay, loss or, in the case of oscillating voltages, phase shift. A real wire or other construction for transmitting power becomes less like an ideal wire as lengths become longer or as operating frequencies increase. In this case, one must use the general theory of transmission lines in order to account for the deviations from ideal behavior. 2.2.1 Power Transmission at High Frequencies There are three major effects in general transm ission line theory that one can ignore in the case o f ideal wires. The first is the finite speed of electromagnetic waves. In time-domain terms, this means that a signal at the beginning of a transmission line will arrive at the end o f the line after a finite period of time. In frequency terms, this means a wave at the end o f a line has a different phase than it did at the beginning of the line. The second effect is loss in the line. The power delivered at the end of the line will be somewhat less than that supplied at the beginning of the line. Finally, the third effect is power reflection from a transmission line port. Impedance mismatch between a source and a transmission line, for example, causes some of the pow er incident on the line to be reflected back into the source. This of course also causes less power to be available for transmission to the end o f the line. A general rule o f thumb for the transition between from ideal behavior to general behavior is that transmission line behavior starts to play a role when the length of the line is greater than about 1% of a wavelength [63]. At m icrow ave frequencies, this length is extremely short; for example, 1 mm at 3 GHz. The devices in this report operate at Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. frequencies greater than 3 GHz and measure longer than 1 mm. Hence, transmission line theory is a requirement for understanding device operation. 2.2.2 LRCG model In order to apply circuit theory to transmission lines, engineers developed the “LRCG” model [64]. It models the transmission line by a series impedance per unit length: Z senes=j coL + R (2-21) and by a shunt admittance per unit length: Y ,^ ,= jo * : + G (2-22) As seen in (2-21) and (2-22), and in Fig. 2-5, there are four circuit elements that model the line: L, R, C and G. The series impedance consists of a real and an imaginary part represented by a resistor, R, and an inductor, L, respectively. Together, they define the effect on the voltage from one port to the other; L changes the phase and R attenuates the magnitude. The shunt admittance also has a real and an imaginary part. The capacitance, C, forms the imaginary part of the admittance and describes how the line changes the phase of the current. The shunt conductance, G, is the real part of the admittance and it describes how the line attenuates the current. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 L R —W V c i o O |G o Fig. 2-5 Equivalent Circuit for a Transmission Line Each of the four circuit elements arises from different physical effects. The inductance comes from the interaction of currents in the conductors and the magnetic fields in the wave. The resistance is caused primarily by ohmic losses in the conductors. The capacitance describes the relationship between the charge on the conductors and the electric fields in the wave. The conductance arises from losses in the dielectric between the conductors that simulate a current flow between the conductors. The following set of equations defines the circuit parameters in terms of the fields in the wave, the frequency of the wave and the material parameters [65]: (2-23) (2-24) (2-25) (2-26) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 In the above equations, io is the peak current and vo is the peak voltage. The wave has an angular frequency o f co and the electric field in the wave is represented by a tangential vector, eu and an axial component, et. Likewise, the tangential and axial components of the magnetic field in the wave are hx and hz. The integrals over S are integrals of an area perpendicular to the propagation axis. The material has a permittivity of e and a permeability o f |i— primes and double-primes o f these quantities represent the real and imaginary parts, respectively. Thus, equations (2-23) through (2-26) provide a method to use electromagnetic and material theories to produce quantities useful for circuit theory. 2.2.3 Characteristic Impedance The LRCG model is most useful in defining two quantities: the characteristic impedance and the propagation constant. The formula for the impedance of a lossy transmission line is: (2-27) As noted above, the R and G elements arise from two loss mechanisms in the line. Setting these two quantities to zero produces the lossless case: 0 .lossless (2-28) One important property of the characteristic impedance is its role in power reflection from an interface between two different impedances. Consider a wave with a voltage phasor o f Vq+ traveling in a transmission line of characteristic impedance Zo- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 Upon meeting a line with impedance Z l, the wave will be partially reflected, creating a wave with voltage phasor V0\ The ratio of these two voltages is the voltage reflection coefficient, T [66]: V~ Z —Z T=^ = L -2v; z L+ z 0 (2-29) This is a voltage ratio and therefore the ratio of reflected power to incident power is the squared magnitude o f T. Note that the reflected wave goes to zero as the two impedances become equal; maximum power transmission requires this “impedance matching”. By conservation o f energy, the transmitted power into the second line is equal to the incident power minus the reflected power, or in relative terms: |r |2 = i - | r f = i z —z (2-30) Z[_ + Z0 where T is the ratio o f transmitted voltage to incident voltage. Another representation of the reflected wave is the “voltage standing wave ratio”, or VSWR [67]: i+ n VSWR = — L i i-r (2-31) Unlike T, which is a phasor containing the magnitude and the phase of the reflected wave, VSWR is a real quantity. Also note that the VSW R is greater than or equal to one; it is equal to one for a matched load and increases with increasing impedance mismatch. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3i 2.2.4 Propagation Constant The characteristic impedance defines the behavior of power reflection, one of the three major effects in transmission lines that are not found in ideal wires. The other two effects, attenuation and phase-shift, are described by the propagation constant, y [68]: (2-32) 7lossy. = y !U o Z T W [jo jc T G ) As before, setting R and G to zero defines the lossless case: r to,teI = y W Z c (2-33) The real and imaginary components of y play very different roles and are represented by a and 0: (2-34) y = a + jj3 a is the attenuation constant and it determines the attenuation of the wave as it travels down the line.Note that above, in equation (2-33), a is zero forthe lossless case. P is the phase constant;itdetermines the change in phase of the wave. For a wave with an initial voltage o f Vo, the voltage at a distance of z from the beginning of a transmission line is: (2-35) V(z) = V0e~r: From this usage, it is clear that the units of a are Nepersper unit length and the units of P are radians per unit length.From equation (2-35), the voltage at adistance z down a line relative to the voltage at z equals zero is: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. |7l = ^ i = e ~ az = -4 .3 4 {dB/ Np) a z (2-36) This is the “insertion loss” in dB for a device of length z. In Chapter 4, a model is presented for devices that begins with equations for lossless lines and then adds small additional inductance and loss. To approximate these effects, consider equation (2-32) and alter the inductance by adding a small additional inductance, AL: jcoL —» jcoL + jcoAL (2-37) If the loss elements R and G are small compared to jcoL and jcoC, then equation (2-32) is approximated by: Y ~ r-rzz jcotJ jcoAL R GZa -i + 22Z0 2Z0 2 LC+- . ( 2- 3 8 ) The first term corresponds to the lossless case shown in equation (2-33). The second term is the additional phase shift due to the added inductance and the last two terms are the losses due to R and G. 2.2.5 Scattering Parameters Any one of a variety o f 2x2 matrices can represent a transmission line with two ports. There are several different conventions for defining the format of these matrices, but the one most often used in network analyzer measurements is the scattering matrix, which has elements called S-parameters. The S-parameters define the ratios of magnitude and phase of “travelling waves” represented by the a’s and b’s in Fig. 2-6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 [69]. The travelling waves have units of root power and therefore their ratios are similar to voltage ratios (such as the reflection coefficient). The subscripts of the S-parameters derive from their ratio definition; for example, S21 is the ratio o f the wave exiting port two, bo, to the wave entering port one, ai. Hence, the parameters S 21 and S 12 describe the forward and backward transmissions through the line and S u and S22 describe the reflections from the two ports. Fig. 2-6 Relation between Travelling Waves and S-parameters In order to extract the characteristic impedance and propagation constant of the line from the S-parameters, one must consider the effects due to both ports. S u is not simply the reflection coefficient due to the mismatch between the external impedance and the characteristic im pedance of the line because the total reflected wave consists not only of the reflection o ff port 1, but also of the reflection off port 2 and multiple reflections back and forth inside the line. Likewise, S21 is not simply due to a single transmitted wave, but the sum o f the “straight through” wave and waves that make multiple internal reflections. However, the situation is simplified if the loss in the line is sufficient to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 attenuate these additional reflected waves to a relatively small amount. In that case, S 11 (and S 22 in the case o f the second port) is the voltage reflection coefficient defined in (2-29), where Zo is the reference impedance of the measuring instrument (usually 50 G) and Z l is the characteristic impedance of the line. In addition, S 21 is the transmitted voltage magnitude into the line multiplied by the proper function o f the propagation constant: (2-39) The measured quantities, the S-parameters, can be expressed in terms o f the propagation constant and the characteristic impedance, which are themselves functions of the LRCG parameters. The LRCG parameters can be modeled using knowledge of material properties and device structure. Thus, we have a theoretical link between fundamental device properties and the measured data. 2.3 High-Frequency Measurements and Calibration There are various measurement tools and methods for characterizing transmission lines. One may use a swept frequency source and a spectrum analyzer to characterize the reflection or transmission properties of a line. Another example is a function generator and a high-speed oscilloscope, which produces similar results. The preferred method for characterization o f transmission lines, however, is with a network analyzer. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 2 3 .1 Measurement of Scattering Parameter with a Vector Network Analyzer A modem vector network analyzer is the best tool for measuring the magnitude and phase of the entire scattering matrix for a two-port network. A network analyzer system consists of four major parts. The main network analyzer unit, such as the Agilent 8 5 IOC shown in Fig. 2-7, consists o f a computerized system controller and microwave detectors. The main unit controls the microwave source, which produces power (usually limited to a maximum of 0 dBm) at the desired frequency. The main unit also connects to the test set, which contains switches and directional couplers that direct the source and detector signals to and from the proper measurement ports. The fourth part of a network analyzer system is the probe station, which serves as the connection between the Device Under Test (DUT) and the measurement system. Fig. 2-7 Agilent 85IOC VNA and Cascade Microtech Probe The probe station is connected to the test set by two coaxial cables. A probe such as the one produced by Cascade Microtech shown in Fig. 2-7 connects to the coaxial cable by the SMA connector on the top of the probe. The extension from the lower left part o f the probe terminates in three coplanar strips of metal; the outer two are grounded to the shield of the coax and the center strip connects to the center conductor of the coax. Thus, the probe creates a contact between planar devices and the coax cable from the test set. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Microscope Camera 8510C VNA.. ftSgjProbe [CBS* =r*fstation Microwave Source (under table) Test Set (behind probe station) Microscope Video Display i Microscope Controls Fig. 2-8 VNA System at NIST A vector network analyzer system at the National Institute of Standards and Technology (NIST) in Boulder, CO is shown in Fig. 2-8. This system contains all the components discussed above, plus a video camera microscope to aid in probe placement and a personal computer system for data acquisition (not shown). This system can produce and measure frequencies from 45 MHz to 40 GHz with a maximum of 801 frequency steps. 2.3.2 TRL Calibration A vector network analyzer, like any measurement tool, measures data containing errors. A proper calibration can remove the systematic errors in such a method. Due to the complexity of a network analyzer measurement, which contains four complex quantities, there are in fact 12 error variables. Once determined, these error terms allow an uncalibrated S-matrix to be calibrated. Various methods exist for calibrating network analyzer measurements. These include the “Thru-Reflect-Match” (TRM), the “Short-Open-Load-Thru” (SOLT) and many others. Calibrations such as TRM and SOLT are sufficient for measurements of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 devices that connect directly to the coaxial cables (or waveguide) from the test set. If the device under test (DUT) is embedded in a larger structure with access lines connecting the cables to the DUT, then a TRM or SOLT calibration is not the best choice; the preferred choice is the “Thru-Reflect-Line” (TRL) calibration. R T -o o- DUT -o o- Fig. 2-9 TRL Calibration Set and DUT In a TRL calibration, one measures the effect of the access lines and includes them in the calibration. Thus, their effect is canceled out of the measurement and the resulting data only includes the effect o f the DUT itself. A TRL calibration set consists of the three elements in Fig. 2-9 labeled T, R and L [70]. The “Thru” element, labeled “T” , consists o f only access lines connected directly together. The dotted line in the middle of the thru-line represents the location of the two reference planes, which are each a distance o f D from their respective ports. The two reference planes are at the same location and therefore the length of the thru-line, minus the length of the access lines (2D), is zero. The “Reflect” element, label “R”, is similar to the thru-line but there is an impedance discontinuity at the reference planes. This discontinuity may either be an open-circuit or a short circuit— short circuit discontinuities are preferred due to the fact that they are usually closer to ideal shorts than opens are to ideal opens. The element labeled “L” is a “Line” element. It is also similar to the thru-line, except it is longer by a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. length Lo and therefore this distance separates the reference planes. A TR L calibration consists o f one or more line elements, each with different length; additional line elements improve the accuracy of the calibration. The fourth element in Fig. 2-9 is the device under test (DUT). Note that the access lines o f length D position the reference plane at the immediate beginning and end o f the DUT. When the calibration is used to calibrate raw data, the effects of everything outside o f the reference planes are removed by the calibration and the S-parameters represent only the effect of the DUT itself. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 CHAPTER III DEVICES AND EFFECTS This chapter presents the two devices studied in the project, microstrip and coplanar waveguide, and the three general effects in the devices: band-stop, phase-shift and band-pass effects. The sections covering devices describe the structure o f the devices, their design and fabrication and the practical considerations involved in the devices. The sections discussing device effects present how the effect arises from material properties, what the parameters are that quantify device performance, what factors must be taken into consideration in order to optimize the effect and what applications exist for such a device. 3.1 Microstrip Devices Consider a sheet of dielectric material with strips of metal that are used as wires on one side. On the other side o f the dielectric is a continuous metal film that is electrically grounded. This is the basic structure known as microstrip, which has been studied and implemented for the past seven decades. One o f the oldest, most studied and most familiar transmission line structures, microstrip has a number of qualities that make it easy to analyze theoretically and easy to implement in the laboratory. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 3.1.1 Basic G eom etry The basic cross-sectional geometry of a microstrip is shown in Fig. 3-1. direction of wave propagation is normal to the plane of the cross-section. The The two conductors labeled “S” and “G” are the signal line and ground plane, respectively. This model assumes that the ground plane and dielectric sheet extend far on either side of the signal line. The equations for the characteristic impedance o f this structure are presented in Appendix E. G Fig. 3-1 M icrostrip C ross Section An important point to note is the field structure of a microstrip. If the signal line width, w, is much larger than the dielectric thickness, h, then the structure strongly resembles a parallel plate capacitor. In that case, the electric field would be strictly vertical and uniform and it would be normal to the conductor surfaces. The magnetic field in the wave would then be uniform and perpendicular to the electric field and it would be parallel to the conductor surfaces. This field structure makes the geometry very easy to analyze; this analysis is explored further in Chapter 4. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4i Applied Field Direction jk i Ground Plane 11 Fig. 3-2 Microstrip Device Structure An example of a microstrip structure from this study is shown in Fig. 3-2. This structure differs from the one shown in Fig. 3-1 in two important ways. First, the dielectric film does not completely cover the ground plane; this is necessary to allow probing of the signal line and the ground plane from the same side o f the sample. Although the dielectric film extends only a finite distance from the signal line, the effect on the impedance o f the line is small; this topic is discussed further in Appendix E. The second difference is the two right-angle bends in the line. These are necessary to allow probing from the sides o f the device while the center of the line remains aligned with the applied field from the electromagnet. As seen in Fig. 3-3, the electromagnet coils prevent the probes from being positioned on the axis o f the applied field. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 3-3 Position of Probes and Electromagnet A TRL calibration set o f five elements was created: one “thru” line, one reflect line and three longer lines. Fig. 3-5 shows part of the masks used to create these lines; the actual fabrication of the lines is covered in the next section. The reflect line is an open-type reflect due to the fact that shorting the signal line to the ground plane would require an additional processing step. The through line and the three longer lines all resemble the device in Fig. 3-2, while the reflect line has a break in the signal line to produce an open circuit. The three longer lines are longer than the through line by 2.135, 3.200 and 6.565 mm and these extra lengths are in the portion of the line parallel to the applied field. 3.1.2 Fabrication Techniques The construction of a microstrip device consists of three steps; these steps are shown as (a), (b) and (c) in Fig. 3-4. Step (a) consists of depositing a ground plane on a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43 substrate. Step (b) is the deposition o f the dielectric layer and the signal line through a shadow mask. Step (c) is the etching of the signal line (and part of the dielectric layer) by a non-reactive ion etch (Ar sputtering). A photolithography process defines the etched pattern in this final step. Fe(001) (Ag cap) Ag(001) Ag GaAs(OoV) (a) Epitaxial Growth reduced width ~LT (b) Evaporation through shadow mask (c) Width reduction via sputter etch Fig. 3-4 Microstrip Fabrication The first step, the deposition of the ground plane, can consist of a variety of methods that depend on the device desired. In order to produce single crystal Fe, a GaAs (001) wafer must be used. In this case, an epitaxial layer of Ag creates the ground plane and provides a template for the growth of single crystal Fe. This technique, which requires evaporation in a Molecular Beam Epitaxy system (MBE), is discussed further in Appendix A. For devices that do not require single crystal Fe, one can use any substrate material. For example, one can begin with a Si wafer and deposit a Ag layer for the ground plane. The deposition of a ferromagnetic layer can then occur during this step or Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. through a shadow mask in the next step. A sputtering system can perform this non- epitaxial ground plane fabrication. The next step, step (b) in Fig. 3-4, defines the basic structure of the line by use of a shadow mask. The shadow mask, part of which is shown in Fig. 3-5 (a), consists of a thin sheet of metal with etched holes that correspond to the black regions in the figure. This step consists o f e-beam or sputter deposition of the dielectric layer (SiCh) and the signal line (Ag). One can add ferromagnetic layers on top the ground plane, between the dielectric layer and the signal line or inside the dielectric layer. In addition, adhesion between metal and dielectric films usually requires Films of Ti between the two. (a) Shadow Mask -i H (b) Photo Mask Fig. 3-5 M icrostrip Shadow a n d P hoto M asks Before step (c), the process has already created a microstrip line. However, the shadow mask, due to the limits of its construction, can only produce lines with greater than approximately 100 pm width. In order to produce lines of 50 Q impedance, the dielectric layer needs to have thickness o f the same order o f magnitude as the line width. Depositing a layer o f such thickness places great strains on the deposition system: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45 extremely large sources must be used and the sheer volume of evaporant substantially dirties the deposition chamber. Ground Plane talUne Fig. 3-6 Photograph of Completed Microstrip Device Step (c) in Fig. 3-4, the narrowing of the signal line via photolithography and etching, allows thinner dielectric layers that still produce 50 Q impedance lines. A portion of the photo mask is shown in Fig. 3-5 (b); it corresponds to the shadow mask shown in (a). Etching occurs in the white regions, while the black regions leave the signal line and ground plane intact. The alignment marks in the two masks, the four squares in the shadow mask and the cross in the photo mask, allow one to align better the photo mask with the structures created with the shadow mask. An example of a completed device is shown in Fig. 3-6. 3.1.3 Practical Considerations There are a number of disadvantages to this device structure, as well as some advantages. O ne advantage is the theoretical basis for this device design; this structure is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 very similar to that modeled by allguide.f, a simulation program discussed in Chapter 4. Disadvantages include the complexity of device fabrication and the difficulty of integrating this structure with other devices. The biggest difficulty in device fabrication is the thick dielectric film that one must evaporate through the shadow mask. It is difficult to maintain adhesion between this film and the conductor films. In addition, this thick film requires a large amount of evaporant and necessitates frequent refilling of the evaporator. The three processing steps, with two of those steps requiring vacuum deposition, create a complex timeconsuming process. In all, the fabrication process includes a number of characteristics that would add great expense to any commercial production. The fact that the signal line is isolated from the substrate by the dielectric film and the ground plane makes it difficult to integrate this device with other devices constructed on the substrate. In order to create a contact between the signal line and the substrate, one must construct a via. This would necessitate more fabrication steps and complicate an already difficult process. 3.2 Coplanar Waveguide (CPW) Devices Coplanar Waveguide (CPW) is a useful structure for transmission lines for a number of reasons. The fact that it is formed from a single metal layer makes it easy to produce and test. The design of these structures is made easier by the fact that the characteristic impedance is primarily controlled by the line’s horizontal dimensions— one does not have to vary dielectric layer thickness, as in the case of microstrips. CPW is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 also compatible with integrated devices; one can fabricate the waveguide on top o f solidstate devices, without the need for vias between multiple metal layers. 3.2.1 Basic Geometry The basic structure of CPW is a signal line flanked by two ground planes, as shown in Fig. 3-7. The characteristic impedance is controlled primarily by the substrate permittivity and by the ratio of the signal line width, a, to the spacing between the ground planes, b. The thickness o f the conductors, t, and the thickness of the substrate, h, have a small effect on the impedance. T he influence of all these factors on impedance is presented in Appendix E. b t i Substrate lllBlllPI Fig. 3-7 Coplanar Waveguide Cross Section As in the case o f microstrip devices, two right-angle bends allow probing of the device from the sides while the middle portion of the line runs parallel to an applied magnetic field, as shown in Fig. 3-8. Referring to Fig. 3-3, it is clear that these bends are required by the electrom agnet poles that block the probes along the axis of the applied field. The probes, which have three coplanar tips, connect easily to this coplanar structure. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Substrate Fig. 3-8 Coplanar Waveguide Device Structure GaAs S u b stra te Py Center Conductor 148 micron initial ground plane spacing ,18 micron narrow signal line width 64 micron iniSaistgnai Sne width Py Ground Piane 42 micron narrow ground plane spacing Fig. 3-9 CPW Corner and Width Change The effects of ferromagnetic conductors scale inversely with the distance between conductors. Therefore, a structure with smaller features produces larger effects than one with larger features. In order to measure the effect of smaller features, some CPW Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. devices with a narrower line structure in the middle of a larger scale line were designed. An example of a right angle bend and the transition from wide to narrow line is shown in Fig. 3-9. The center conductor width/ground plane spacing are 64/148 pm and 18/42 pm for the wide and narrow lines, respectively. These dimensions correspond to a 50 Q line when the substrate is GaAs; note that the ratios 64:148 and 18:42 are nearly equal. The transition between the two widths forms a very short (much less than 1% of a wavelength for the frequencies of interest) length of mismatched impedance. This forms a lumped capacitor at the transition, but one can remove the effect of this capacitor through a proper TRL calibration and de-embedding. 3.2.2 Fabrication Techniques The fabrication o f CPW devices is much simpler than the fabrication of microstrip devices. The deposition process consists of a single step, rather than the two deposition steps for microstrip. The deposition of the conductor may be done by sputtering or ebeam evaporation. Sputtering is easier and faster, but e-beam evaporation in a M BE system allows one to fabricate single crystal Fe films. In the case of microstrip, the ground plane must be thicker than about three or four times the skin depth at the frequency o f operation (about 2 |im of Ag, for example), or else radiation losses increase. In the CPW structure, however, the conductor thickness may be less than the skin depth (thicknesses of 250 nm seem sufficient for most conductors) with little effect on line attenuation. A single photolithography and sputter etch step defines the CPW structure. The conductor thickness in CPW is much less than the thickness of the top conductor in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. microstrip; therefore, the etching process takes much less time. An additional benefit is that there is no need to align the CPW photo mask, as there was in the case o f microstrip. SC1 Fig. 3-10 Coplanar Waveguide Photo Mask An example of two CPW structures is shown in Fig. 3-10. The pattern labeled “T C I” is a “thru-line” and the pattern labeled “S C I” is a “short-reflect-line”. Other structures required for a TRL calibration are the longer line structures, which resemble the thru-line with the exception that the length o f the narrow region is longer by 2.135, 3.200 and 6.565 mm. In addition, the photo mask includes patterns for CPW lines with no narrowed region— these lines form a separate TRL calibration set that resembles the narrow line set. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 3 .2 3 Practical Considerations The fact the coplanar waveguide is planar leads to two important advantages. First, one can easily integrate the waveguide with solid-state devices in the substrate; there is no need for via interconnects, as there was in the case o f microstrip. Second, the characteristic impedance of CPW is controlled primarily by its in-plane dimensions (a and b in Fig. 3-7). One can narrow or widen the center conductor while maintaining 50 Q impedance, so long as the ground plane spacing is the proper value. This too aids in device integration because the CPW line can match devices of arbitrary size. In addition to the advantages discussed above, the simple fabrication of CPW is another definite advantage. The deposition of the entire device consists of a single step and the conductor may be as simple as a single metal film. The device structure is defined by a single photolithography and etch step. CPW is therefore a practical device structure for commercial products. There is one final point to consider. Unlike the microstrip device, the substrate plays a role in the electromagnetic properties of the CPW line. Hence, it is important to consider the microwave quality of the substrate— especially the loss tangent of the material, which determines dielectric losses. 3.3 Band-Stop Effect A band-stop effect is simply one that attenuates a certain band of frequencies while allowing others through. A device with this effect is known as a “band-stop” or Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 “notch” filter. This section defines the parameters of such a device, discusses what features are desirable and undesirable and presents some applications. 3.3.1 Basic Effect and Definitions of Terms The parameters o f a band-stop filter are best defined in terms of the transmission, S2 1 , as a function of frequency. Such a plot is shown in Fig. 3-11. The insertion loss is the attenuation of the device outside of the notch; in this case, about 2 dB. The width of the notch is called the “3 dB bandwidth” or “stop-bandwidth” and is defined as the width in frequency of greater than 3 dB attenuation (referenced to the insertion loss). In Fig. 3-11, the bandwidth is about 2 GHz and corresponds to the crossing o f the 5 dB line (3 dB below the 2 dB insertion loss). The maximum attenuation of the notch, about 15 dB, is called the “stop-band rejection”. The position o f the notch, about 14 GHz in this case, is called the “stop-band center frequency” or simply “center frequency” . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 Insertion Loss B Bandwidth 3 dB -5 - m T3 Maximum Attenuation is Stopband Rejection CM CO - 10- -15 10 15 20 Frequency (G H z) Fig. 3-11 Bandstop Device Parameters There are two important parameters not shown in Fig. 3-11. O ne is the VSWR of the device, defined by (2-31). The other is the tuning range, given by the minimum and maximum center frequencies possible in the device. 3.3.2 Design Considerations Some of the parameters above should be minimized and others maximized while still others should simply be varied to fit a certain application. Insertion loss and VSWR are two parameters that one should always minimize. Reducing the attenuation due to losses outside o f ferromagnetic resonance minimizes insertion loss. These losses include conduction loss, dielectric loss and radiation loss and well as the effective loss due to impedance mismatch between the device impedance and the system impedance. Reducing the impedance mismatch also reduces VSWR. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The stopband rejection is a param eter that should be maximized. This parameter is controlled by altering the material properties of the ferromagnet and by changing the length of the device. A stopband rejection o f 40 dB is sufficient for most applications. If the stopband rejection per unit length of the device is increased, then the length of the device can be decreased while still creating the desired total stopband rejection. Thus, we can define a Figure of Merit for a device by: F.O-M - St° pband reJeCli0n Per Unit 'ength (unitless) insertion loss per unit length (3-1) The bandwidth and the tuning range of the device are two param eters that should be varied according to application. In some applications, such as blocking an interfering signal that is near in frequency to a desired signal, the bandwidth should be minimized. In other applications, it is desirable to block a large range of frequencies— this calls for a larger bandwidth. The tuning range must simply match the requirem ents of a given application. 3.3.3 Applications Applications of a band-stop filter include any case where a signal at a certain frequency should be blocked while others should pass through. transmitter/receiver system, such as that shown in Fig. 3-12 [71]. O ne example is in a In such a system, a signal received by an antenna passes to a low noise amplifier (LNA) and then to other signal processing elements. If a large interference signal exists near the desired signal, it could damage the LNA or distort its characteristics. A tunable band-stop filter could Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 block an interfering signal at an arbitrary frequency while allowing the desired signal to pass. Tunable Notch Filter Low Noise Amp Circulator T/R Antenna Power Amp Fig. 3-12 Tunable Filter Application 3.4 Phase-Shift Effect All transmission lines of finite length induce a relative change in phase angle between their two ports. For most lines constructed of passive elements, this phase shift is constant at a given frequency and varies linearly with frequency. A device that modulates the phase shift at a given frequency is known as a “tunable phase-shifter”, “phasor” or simply “phase-shifter” . This section describes such a device, discusses its desirable and undesirable qualities and presents some of its applications. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.4.1 Basic Effect and Definitions of Terms Understanding the effects in a phase-shifter is best accomplished by studying both the phase and the magnitude of transmitted waves. The phase-shift as a function of frequency is shown in Fig. 3-13. The solid, dashed and dotted lines are for the cases of low, moderate and high applied fields, respectively. The total change in phase, A0, is shown for three frequencies. The largest change, an increase of phase with increasing applied field, occurs near 15 GHz. The changes at 7 GHz and 20 GHz are smaller and both are decreases in phase with increasing applied field. 45 co CD C i_D cn o "O o a) c 15 G H z Aq> -4 5 - CO CD C/5 CO Q. -9 0 - 20 G H z Ac? CM CO -135 5 10 15 20 25 Frequency (G H z) Fig. 3-13 M odulation o f Phase Shift Although the phase change is greatest at 15 GHz, this may not be the best operating frequency for a phase-shifter. Another important factor is the insertion loss, which is the maximum attenuation at the operating frequency for all applied field values. The insertion losses at 7, 15 and 20 GHz are noted in Fig. 3-14. Note that while the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 greatest phase-shift occurs at 15 GHz, this is also the frequency with the greatest insertion loss. - CQ 2- -4 - T3 CM CO - 6- - 8- -10 5 10 15 20 25 Frequen cy (G H z) Fig. 3-14 Insertion Loss in a Phase-Shifter A final device characteristic is not depicted in the figures. This is the VSWR of the device, which is caused by impedance mismatch between the device and system impedances. 3.4.2 Design Considerations The primary consideration in phase-shifter design is the balance between total phase-shift and insertion loss. Both increase with device length, but total phase-shift is desirable and insertion loss is undesirable. Hence, we have a Figure o f Merit for tunable phase-shifters that is independent of device length: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 F.O.M s phase shift per unit length insertion loss per unit length (3-2) As seen above, this Figure of M erit varies according to the operating frequency chosen relative to the applied field range used to modulate the phase-shift. As in the case of most all devices, the VSWR should be reduced. Decreasing impedance mismatch primarily does this. 3.4.3 Applications One application of tunable phase-shifters is a phased-array radar system, shown in Fig. 3-15. In such a system, an antenna consists of several smaller emitters. Each emitter is provided with a separate signal that varies in phase with respect to the other source signals. The resulting interference pattern creates a focused beam of microwave energy. By adjusting the relative phases o f the emitters by using tunable phase-shifters, one can direct the beam in various directions without physically moving the radar array. This is a valuable capability in applications such as airborne radar systems, where physical movement of the array would be difficult. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 Steered Beam Array Emitters Tunable Phase-Shifters A© Acb Acp Acb Power Feeds Fig. 3-15 Tunable Phase-Shifter Application 3.5 Band-Pass Effect A band-pass effect is essentially the opposite of a band-stop effect; rather than stopping a finite range o f frequencies, a band-pass allows only a finite range of frequencies through. A device operating with this effect is simply called a “band-pass filter’. This section defines the parameters of such a device, discusses what features are desirable and undesirable and presents some applications. 3.5.1 Basic Effect and Definitions of Terms The terms used to describe a band-pass filter are similar to those used to describe a band-stop filter. The insertion loss, as shown in Fig. 3-16, is the attenuation through the filter for frequencies in the band-pass region (the “passband”). The “3 dB bandwidth” or ju st “bandwidth” is the width in frequency of the range where the attenuation is less than 3 dB greater than the insertion loss and the center o f this frequency range is simply the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. “center frequency”. The attenuation outside the passband is called the “rejection”. This value changes greatly with frequency, so either it must be specified for a certain frequency or it can be described by the change of the attenuation with frequency— the “roll-off’ or “selectivity”, which is usually expressed in dB/octave. Insertion Loss - 2- -3 - 3 dB -4 - CQ T3 -5 - CM co - 6- 3 dB Bandwidth -7 - 8- -9 40 45 50 55 60 65 70 75 80 F re q u en cy (G H z) Fig. 3-16 Bandpass Device Parameters There are two important parameters not shown in Fig. 3-16. One is the VSWR of the device, defined by equation (2-31). The other is the tuning range, given by the minimum and maximum center frequencies possible in the device. 3.5.2 Design Considerations As in the case o f most devices, the insertion loss and VSWR should be minimized. The rejection or selectivity of the filter should be maximized. If the insertion Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ol loss and selectivity both increase with device length, then we can create a Figure of Merit that is independent of length: _ _ ., rejection per octave, per unit length / \ F.O.M = — --------- - --------------2— (octave 1) insertion loss per unit length (3-3) As in the case of stop-band filters, the bandwidth and tuning range desired varies according to the specific application. 3.5.3 Applications Applications of band-pass filter are often similar to those o f band-stop filters. For example, a band-pass filter can replace the band-stop filter in Fig. 3-12 in order to protect the receiver amplifier. One application to which a band-pass filter is particularly well suited is in a frequency synthesizer. In such a system, a tunable frequency source with a maximum frequency of fmax uses a frequency multiplier and a tunable filter to produce frequencies up to an integer m ultiple of fmax. The operation of such as system is shown in Fig. 3-17. The signal source produces power at a single frequency fo. The frequency multiplier, a non-linear device such as a diode, produces harmonics of fo. A tunable band-pass filter then tunes to the desired frequency, which is an integer multiple n of the fundamental frequency. Thus, the effective frequency range of such a system may be several times the range of the signal source. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Signal Source atfn Frequency Multiplier ^0 • • • Band-pass Filter at nfn nfg nfg Fig. 3-17 Bandpass Application Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 CHAPTER IV DETAILED THEORY This section has two major goals. The first is to develop a general model for devices based on ferromagnetic conductors. The second is to apply this model in order to determine the limits of operation for various effects. In order to achieve the first goal, a numerical technique based on first principles that was developed by Astalos and Camley is discussed [58]. Then, a technique based on the property of surface impedance is developed. Next, these two models are compared in order to verify the surface impedance technique. The second goal, to determine limits of operation, follows from Chapter 3, where the desirable and undesirable characteristics of various devices are described. In Chapter 5, measured data from these devices is presented and these results are compared to the limits calculated here. 4.1 Simulations Based on First Principles The allguide.f program, written in FORTRAN by Astalos and Camley, applies first principles to a simulation of a ferromagnetic device. It uses M axwell’s equations, the electromagnetic wave equation and electromagnetic boundary conditions to produce Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 analytical equations for the propagation constant. Unfortunately, these analytical equations are transcendental and therefore the propagation constant at a given frequency must be found by a numerical root-finding technique. One may use allguide.f to model the magnetic effects in certain devices and, by varying parameters, design devices that maximize desirable effects (or minimize undesirable ones). In this section, the operation o f the allguide.f program is explained along with the effects it predicts and its advantages and limitations. 4.1.1 Description of “allguide.f” The structure modeled by allguide.f is shown in Fig. 4-1. It consists of a single dielectric film and a single ferromagnetic metal Film between two perfect conductors. The structure is infinite in the horizontal direction (x) and the wave propagates primarily in the direction perpendicular to the cross-section plane (z-direction). allguide.f determines the permittivity and permeability matrices for the two films in terms of usersupplied material parameters, which includes the relative permittivity of the dielectric. Parameters for the ferromagnet include the conductivity, the applied field, the resonance linewidth and the saturation magnetization. allguide.f applies the permittivity and permeability matrices to the electromagnetic wave equation and M axwell’s equations. Thus, it determines the E and H fields in the two materials as a function of the wave vectors in the y- and z-directions (vertical and perpendicular directions). Next, it applies electromagnetic boundary conditions, which reduces the problem to a system of equations with the z-direction wave vector as the only variable. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 W Perfect Conductor allguide.f structure is infinite in this direction Fig. 4-1 allguide.f Structure The wave vector in the z-direction, kz, is simply the propagation constant of the structure multiplied by j. allguide.f finds the roots of the kz equation, which correspond to the propagation constants of various modes. For dielectric thicknesses much less than a wavelength at the frequencies of interest, only the lowest-order mode is physical, allguide.f iterates over a series of frequency steps and finds the kz roots at each frequency— it lists these frequency points and kz’s in output files. Thus, the simulation creates data for the attenuation and phase as a function of frequency. 4.1.2 Overview of Results The general results from allguide.f are what one would expect. The attenuation reaches a maximum at the resonance frequency and a minimum at the anti-resonance Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. frequency. The phase varies linearly with frequency (as in the case of non-magnetic lines), but deviates from this linear background near resonance and anti-resonance. The resonance and anti-resonance features increase in frequency with applied field, according to the equations for the resonance and anti-resonance frequencies. Astalos and Camley varied several parameters in order to determine their effect on results. By varying D, the dielectric thickness, they found that the attenuation in the structure had a 1/D dependence. By varying the ferromagnet thickness, d, they found that the band-stop rejection increased with increasing d up to a point and then remained constant. The phase increased with increasing d, but this improvement was hampered by the increasing attenuation. They also found that the anti-resonance attenuation minimum depended on the ferromagnet thickness. Below a certain value of d, the band-pass region was nearly undetectable. The conductivity of the ferromagnet also played a role in the band-pass effect; increasing the conductivity greatly narrowed the band-pass region. For lower conductivity, the band-pass region broadened and the phase-shift in this region increased. They were encouraged by this result, pointing out that both these qualities would make a good phase-shifter. 4.1.3 Advantages and Limitations The advantage o f the allguide.f simulation program is that the researchers derived it from first principles. This ensures that it is accurate. To quote from the report by Astalos and Camley, “We have been able to verify the behavior of our device in the band Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 pass region, without resorting to the approximation which, in past analyses, was suspect precisely in the region of interest.” [58] There are, however, several limitations of this simulation. The first is that the simulation is limited to only one geometry, which is not a real device structure. Hence, it is not an exact solution, although it is a fairly good approximation for the microstrip structure. Other geometries, such as a coplanar waveguide structure, would be difficult to approximate. The second limitation is that it does not calculate the impedance of the structure. Calculation o f the propagation constant (jkz) is only half of the task for determining the transmission line properties of the structure. As shown latter, there are changes in the impedance, driven by the ferromagnet, that one must consider. The third limitation is the fact that allguide.f relies on numerical solutions to equations. It is therefore only possible for one to determine the effect of a single parameter by iterating it over a series o f values and observing its effect on a certain feature. The final limitation is due to the assumption that the conductivity of the perfect conductors is infinite. This assumption therefore ignores the additional attenuation due to real conductors with finite conductivity. This especially affects the band-pass region, where the losses in the ferromagnet become small compared to losses in the non-magnetic conductors. 4.2 Simulations Based on Surface Impedance The effects o f ferromagnetic conductors in transmission lines, or any conductor for that matter, are easily described by the concept of surface impedance. The surface impedance is the effect o f the conductor’s finite conductivity and must be considered Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. along with the series impedance of the rest of the line. The standard technique for evaluating these effects is to first analyze a transmission line for the case of infinite conductor conductivity and then to proceed to include the surface impedance as a small correction. Most analyses o f surface impedance in transmission lines are unconcerned with ferromagnetic metals and their dynamic properties; these analyses usually assume that the conductors have a permeability equal to |io, or at least assum e the permeability is strictly real and frequency invariant. The concept of surface impedance has been applied to ferromagnetic properties; however, there seems to be no study that has applied these properties to transmission line structures. Hence, the goal o f this section is to methodically develop an expression for surface impedance that allows for a general conductor permeability and then to consider the special case of ferromagnetic permeability. 4.2.1 Description of Surface Impedance An electromagnetic wave cannot penetrate a conductor with infinite conductivity. However, conductors at room temperature have finite conductivity. Waves therefore do penetrate and decay exponentially from the surface. produces current in the conductor. The electric field in the wave In addition, the electric field at the surface of the conductor produces a voltage drop per unit length in the line. Thus, we can create the definition for surface impedance by simply taking the voltage per unit length and dividing it by the current in the conductor. The surface impedance is a complex quantity and therefore produces both losses and phase-shifts in the transmission line. In terms of the LRCG model, the surface Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 impedance adds both inductance and resistance per unit length. In the case o f non magnetic conductors, this inductance and resistance behave similarly; however, in ferromagnetic conductors they behave very differently. 4.2.2 Calculation of Surface Impedance The surface impedance is calculated, as discussed above, by calculating the current in the conductor due to an electric field penetrating into the conductor. The voltage drop per unit length at the surface is an electric field with magnitude o f Eo. The electric field in the conductor varies exponentially with x according to the propagation constant, y: E: = E 0e-* (4-1) The real part of y, which causes the wave to attenuate exponentially, defines a characteristic length for the penetration of the wave. This characteristic length is called the “skin depth” and is represented by 6: The propagation constant is a function of the frequency, co, and the material properties of permittivity and permeability (e and p., respectively): (4-3) Ignoring the permittivity’s real part and setting it equal to the familiar expression approximates the permittivity of a good conductor: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 (4-4) By Ohm ’s law, the current density in the metal is simply proportional to the electric field: J =oE (4-5) The next step is to determine the current, I, in the conductor. The direction of the electric field is in the z-direction and therefore the current is the integral over an area in the x-y plane: (4-6) where the y-direction is parallel to the conductor surface and the x-direction is normal to the sample surface. The result is proportional to the voltage drop per unit length, Eo: / = — £0 7 (4-7) The surface impedance is therefore the impedance per unit length defined by dividing the voltage drop per unit length by the current: (4-8) The general features of surface impedance are clear in (4-8). The surface impedance increases with frequency and decreases with conductivity. It varies according to the square root of the permeability and, if the permeability is not complex, then the surface impedance’s real and imaginary parts are equal. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71 4.2.3 Effect of Surface Impedance in Transmission Lines In order to include properly the influence of surface impedance on a transmission line, its role must be weighted relative to the role of the regions outside of the conductors. A method to achieve this is to define a unitless geometric factor, g, which is proportional to the inductance per unit length: L = HrVog (4-9) where p.r is the relative permeability of the dielectric in the line (usually equal to unity). By the complementary nature of the inductance and capacitance, the capacitance per unit length is inversely proportional to g: (4-10) g where Ecff is the effective relative permittivity of the dielectric regions. Substituting the above expressions for L and C into the definition of Zo for a lossless line, (2-28), and setting |ir to unity produces: (4-11) where qo is the impedance of free space. Expressions for Zo and £cff are available in the literature for most every transmission line structure. These expressions for CPW and microstrip structures are listed in Appendix E. This geometric factor, g, is calculated for a given geometry using the existing equations for the characteristic impedance—this produces a g for the case of perfect conductors. By retracting the conductors by half the skin depth, a new set of dimensions Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. is formed that allows the effect of finite conductivity to be modeled. The impedance for these new dimensions determines a corresponding geometric factor called g \ The difference of the two g ’s is Ag and it is always greater than zero: (4-12) &g = g'~g This change in the geometric factor creates an additional complex inductance, AL*: L + AL* = n rn 0g + n Kfffi0Ag (4-13) In this case, the first term is simply the inductance defined above in (4-9). The second consists of an effective relative permeability in the conductor (p^rr, which is not yet defined) and the change in the geometric factor due to the skin depth, Ag. The complex AL* is comprised o f two parts containing real quantities AL and R, which are the added inductance and resistance, respectively: jcoAL* = jcoAL + R (4-14) AL = Re(jieIjrn 0A g) (4-15) R = lm{co/2 effn 0A g) (4-16) These values are: and The task is now to derive an expression for the effective relative permeability, (icff. This quantity is not simply the permeability of the conductor; both the permeability of the conductor and the conductivity of the conductor affect the effective permeability, while the permeability o f the conductor is a strictly magnetic effect. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The effective 73 permeability is calculated by setting the surface impedance equal to the impedance of the additional complex inductance: Z, = jcoAL* = jcofJt]f[i()Ag (4-17) Substituting from (4-8) yields: Vv V2(7 V ? = jW .ffU 'A g (4-18) Solving for Pen- produces: Heir = ------------------------------------------------------------ (4-19) W AgjcofdQ V 2<t Hence, the effective permeability is a function frequency, material properties and the geometric quantities of Ag and W. Noting that the change in g for a surface o f width W results from the half skin depth retraction can eliminate these two geometric quantities— the change in g is simply the ratio o f half the skin depth to the width: = (S/2) W ( 4 _2 0 ) Using previous definitions for the skin depth and simplifying produces: H'f f = H c + M c \ (4-21) where pc isthe relative permeability of the conductor. Labeling the real and imaginary parts o f pc with a prime and a double prime, respectively, creates: AL = HcHQAg and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4-22) 74 R = co(nc+ K | K*g (4-23) Thus, we have a theory that defines how to create a weighting function for the added inductance and resistance by using existing equations for the characteristic impedance and effective permittivity. This weighting function Ag, along with the generally complex conductor permeability of pc, defines the added inductance and resistance due to the surface impedance of the conductor. These circuit elements, when added to the existing LRCG parameters, define the characteristic impedance and propagation constant of the transmission line. These quantities produce the scattering matrix o f the line, which one can then compare to experimental results measured by a network analyzer. 4.2.4 Results for Non-Magnetic Conductors Most analysis of surface impedance in transmission lines concerns non-magnetic conductors. In this case, we assume that the relative permeability of the conductor pc is strictly real. This section demonstrates that this assumption, combined with the technique described in the preceding section, produces equations for added inductance and resistance that are equivalent to those found in the literature. To achieve this, two commonly analyzed transmission line structures with analytical solutions for their properties are studied: the parallel plate and coaxial line structures. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 f w— (a)parallel plate (b) coaxial Fig. 4-2 Effect of Skin Depth on Geometry The two structures studied in this section are shown in Fig. 4-2. The parallel plate structure, shown in (a), somewhat resembles a microstrip structure; in fact, the two become equivalent if the thickness of the dielectric, D, is much smaller than the conductor width, W. The inductance per unit length for the parallel plate structure is rather simple: (4-24) and therefore, from (4-9), the geometric factor is simply D divided by W. The coaxial line shown in (b) is also fairly simple— its inductance per unit length is: L = -^-ln(b/a) (4-25) and, as in the case o f the parallel plate structure, its geometric factor is simply the inductance divided by ji. Fig. 4-2 also shows the effect of the skin depth on the two geometries. All conductor surfaces where fields exist retract by half the skin depth (6/2). Note that the outer-most edges of the conductors in both halves of the figure are not retracted; we Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 assume that the fields on these edges are zero. In the case of the parallel plate structure, the retraction of the two inner conductor surfaces produces a change in D: D (4-26) D + 8 and therefore the change in the geometric factor is: A g D + 8 D 8 W w w = (4-27) For the coaxial line, the outer conductor’s inner radius changes by: b ^ b + 8 /2 (4-28) while the inner conductor’s radius changes by: a —» a —8 / 2 (4-29) Combining these two changes calculates the change in the geometric factor: 1 J b + 8/ 2 A g= -m a -8/2 f u\ 2k In 8/2 ( 1 — 2k a + n b — (4-30) where the final result assumes that the skin depth is much less than either of the dimensions a and b. The next step is to calculate the effective permeability and to apply this and the change in the geometric factor to the equations for added inductance and resistance. As noted above, the permeability of a non-magnetic conductor is assumed to be strictly real. Considering this assumption along with (4-21) produces: =A*c (i + y) Combining this result with (4-15) and (4-27) results in: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4-31) 77 AL = R e f r ^ o A s ) = (4-32) w which is the added inductance for a parallel plate structure. Using (4-16) with (4-27) creates: R = lm {c o n ^ 0A g) = - cf^rC°- which isthe added resistance for a parallel plate structure. (4-33) Using change in the geometric factor for a coaxial line, (4-30), with the equation for the the equations for added inductance and resistance, (4-15) and (4-16), leads to: AL = R e ^ ^ A g ) = n cnQ 2K ^ 1 bj (4-34) n b (4-35) and: R /O i R = \m{cofieffn 0A g)= n cn 0co^— 2K a (4-32) through (4-35) are indeed equivalent to their corresponding equations in the literature; hence, the general surface impedance technique developed here does produce the proper results for non-magnetic conductors. 4.2.5 Effects for Ferromagnetic Conductors For the case o f ferromagnetic conductors, the Voigt permeability is simply substituted for the conductor permeability in (4-21): Hr# = / W + ./K „ J Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4-36) Clearly, the real part o f the effective permeability is equal to the real part of the Voigt permeability. However, the imaginary part of the effective permeability is always greater than or equal to twice the imaginary part of the Voigt permeability. In addition, the real part of the Voigt permeability plays a role in the imaginary part of the effective permeability. 1.0 o> c 0) C .5- coaL and R (non-m agnetic) CD Q_ ct DC "O d 0 .0 - CO _l 3 -0.5 0 10 20 30 40 50 Frequency (G H z) Fig. 4-3 Added Impedance for Ferromagnetic Conductors The behavior of the effective permeability, combined with the effect of the skin depth on the change in the geometric constant, produces the data shown in Fig. 4-3. According to (2-38), the quantities R and coAL are proportional to the attenuation and phase-shift, respectively. In non-magnetic conductors (dashed line), these two effects are equal and increase according to f I/2. In magnetic conductors, the two effects behave very differently. R (dotted line), which creates attenuation, has a maximum at ferromagnetic resonance (10 GHz) and a minimum at anti-resonance (34 GHz). This creates the band- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 stop and band-pass effects. coAL (solid line), which creates phase-shift, varies rapidly about resonance— there is also a smaller effect about anti-resonance. 4.3 Equivalence of the Two Techniques Given that the allguide.f simulation is based on first principles, it is desirable to show that the surface impedance technique produces the same results as allguide.f. This section compares results from the two methods and shows that they are indeed equivalent for certain structures. First, however, the surface impedance is adapted to the geometry on which allguide.f is based. 4.3.1 Adaptation of Surface Impedance to “allguide.f’ Geometry Begin by noting that the allguide.f structure, shown in Fig. 4-1, resembles a parallel plate structure. If the structure is not infinitely wide, but rather has a width of W, then its characteristic impedance is: Z --^ 0 (4-37) Starting with the approximation for the attenuation constant given in (2-38) and then substituting in the above expression and (4-16) yields: cofi^Ag lm (ju eJT ) Likewise, the phase becomes: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4-38) (4-39) The change in the geometric factor is very similar to that given in (4-27) for the parallel plate structure, only in this case only one of the two conductors is ferromagnetic and therefore Ag is half the original value: Substituting this equation into (4-38) produces: (4-41) and substituting it into (4-39) creates: (4-42) Analytical equations for the skin depth and the effective permeability have already been listed and therefore the above equations are completely analytical. Note that 1/D scales the magnetic effects, given by the real and imaginary parts of the effective permeability. This matches the conclusion by Astalos and Camley, which is referred to earlier in this chapter. Another important note is that W does not appear in either expression and thus both allguide.f and the surface impedance technique produce results that are independent o f W for this geometry. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 4.3.2 Comparison of Results Table 4-1 lists the parameters that were used in the two simulations, allguide.f is based on CGS units, but MKS units are used in the surface impedance calculations. Note that allguide.f uses the magnetic field linewidth AH to model damping effects; one can easily determine this value at a certain frequency by using a FMR system, as described in Appendix B. Table 4-1 Values Used in Simulations allguide.f Value Surface Impedance Value Ho, Bo 1000 Oe 0.1 T Applied field Ms, MoMeff 1714 G 2.15 T Saturation Magnetization D 10‘2 cm 10^ m Dielectric thickness d 10'3 cm 10*5 m Fe thickness a 107 S/m 107 S/m AH, ©a 100 Oe 1.35 GHz 12.9 12.9 6r Notes Fe conductivity Field/frequency linewidth Dielectric relative permittivity Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 3 .5 3.02.52.0 oo ■o s Surface Impedance allguide.f - 1.51.0 - 0.50.0 0 10 20 30 40 50 60 70 80 Frequency (GHz) Fig. 4-4 Comparison of Attenuation Fig. 4-4 shows the result o f this comparison in terms o f the attenuation “constant” a . There are only tw o minor manipulations to the data. First, allguide.f produces a in units o f Np/cm. Therefore it is multiplied by 4.34 to produce dB/cm. Second, the surface technique produces MKS units and therefore the units o f a are Np/m. The surface impedance output is multiplied by 4.34 and divided by 100 to convert to dB/cm. results from these tw o techniques are almost identical. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The 83 0 .7 8 0.77- Surface Impedance allguide.f 0.76- 0.75- Ll_ CL 0.74- Q- 0.73 0 10 20 30 40 50 60 70 80 Frequency (GHz) Fig. 4-5 Comparison of Phase Fig. 4-5 shows the phase effects in the two techniques. Again, the two are almost identical. As above, some m inor alterations are made to the data that only involve changes in units. In both cases, the phase constant 3 is divided by the frequency in GHz in order to show better the effect o f phase. In the case o f the surface impedance technique, the data is divided by 100 to convert the length scale to cm. 4.4 Calculation of Limits of Operation As seen above, the added inductance and resistance describe the effects due to ferromagnetic conductors. These two quantities, in turn, describe the change in the propagation constant that creates attenuation and phase-shift. One important quantity in these calculations is the change in the geometric factor, Ag. In microstrip, which is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 similar to the parallel-plate case, Ag is rather straightforward. In coplanar waveguide, however, this calculation is more complicated. In CPW, Ag is approximated by: ( K ( k ') dk S dS 2 (4-43) where: a -2 8 dk S a+b k = ----------- > --------* ------ — 8 b + 2S dS 2 b (4-44) a and b are the geometric dimensions o f the CPW line and 5 is the skin depth. For GaAs substrates, two different CPW geometries are used: a wide line and a thin line. The dimensions o f the wide line are 64 and 148 pm for a and b, respectively. This creates: - 0° .4 f -| - ^ 5 j W 3 . 9 6 x I 0 J<J - 4009 9 (4-45) For the thin line, a is 18 pm and b is 42 pm. This produces: A g W ~ l - 4 0 x l 0 4£ (4-46) N ote that the thin line factor is larger than that of the wide line. The ratio of these two factors is similar to the inverse o f the ratio o f a in the two cases— once again there is a 1/D effect. Another important quantity is the skin depth, especially at the frequencies o f resonance and anti-resonance. Generally: s = I Vcro)u0 ^7 , Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4-47) 85 At resonance and anti-resonance, the real part o f the Voigt permeability is zero. At resonance, the imaginary part o f the Voigt permeability is, from (2-19): ^ = i ^ - q g *)2 _L = m M s ^ o ( b q + m 0m s_) (4_4g) {2BQ+ fl0M s ) ^FMR if^FTMR + t°FMR) 03\ and at anti-resonance it is approximately: M l* * 2 r (4-49) Hence, the skin depth at resonance is: SfM = J — F^2B° t V s (b 0 + m0m r (4-50) s) and at anti-resonance the skin depth is: 5fM* V < w > 2 r (4 " 51> All o f these expressions are used in the calculations shown below. 4.4.1 Stop-Band Rejection The stop-band rejection is the attenuation at ferromagnetic resonance. For a 50 Cl line, the attenuation for any frequency is: a =— = Im (a - ) 2 z 0 io o n (4-52) K Ag depends on the geometry, but it is always proportional to the skin depth. Hence, a useful expression is: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 This is proportional to the attenuation at the resonance frequency. attenuation increases with decreasing T (narrower linewidth). Note that the Using the above expression with the proper units conversion and with the geometric factors for the two widths o f CPW lines given in (4-45) and (4-46) produces: < W = 3.44^ ° * ^ ' dB /cm (4-54) » » (4-55) and: = 12.2 <&!cm Table 4-2 contains example rejection values for Py and Fe in the wide and thin CPW geometry for various values o f T. A value o f 0.007 for either material is exceptionally low— one must carefully control epitaxial growth in order to achieve this value. The highest value o f attenuation in Table 4-2 is 32.6 dB/cm. A typical requirement for stop-band rejection o f 40 dB would therefore require a 1.23 cm long line. Fig. 4-6 shows the results o f a simulation o f the FORTRAN program in Appendix H. The line in the sim ulation is 1 cm long with lateral dimensions corresponding to the wide CPW structure. The ferromagnetic material is permalloy with T = 0.02. The simulated value o f stop-band rejection is 4.83 dB, which compares well with the value o f 4.84 dB given in Table 4-2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 Table 4-2 Sample Rejection Values Geometry r Rejection (dB/cm) Py Thin 0.007 29.1 Py Wide 0.02 4.84 Py Thin 0.02 17.1 Fe Wide 0.007 9.15 Fe Thin 0.007 32.6 Material -1 m - - 2- T3 co -3- -4- o Stop-Band Rejection = 4.83 dB 5 10 15 Frequency (GHz) Fig. 4-6 Stop-Band Rejection Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 88 4.4.2 Stop-Band Frequency Range The center frequency o f a band-stop filter is the ferromagnetic resonance frequency. This frequency is not affected by device geometry, but only by the material properties o f the ferromagnet and by the applied field: *°fmr = Y M + B4%B0 + B4 -f /UqM s ) (4-56) where Bo is the applied field strength, B4 is the magnitude o f a four-fold magnetocrystalline anisotropy, y is the gyromagnetic ratio (which varies by material) and Ms is the saturation magnetization o f the material. YIG Py ------------------------► —-- 4 Fe (no anisotropy) k— ---------------------— — -a Fe (50 mT anisotropy) - — - —• ---------------- — ------------------------------------------------------— — 1 0 5 - 1 10 15 20 25 1— 1 - * 30 —• ■-1--------35 40 Frequency (GHz) Fig. 4-7 Stop-Band Frequency Range Fig. 4-7 compares the band-stop frequency range for three different materials. The leftmost symbol in each range corresponds to zero applied field, while the middle and rightmost symbols correspond to applied fields o f 100 and 500 mT, respectively. Yttrium Iron Garnet (YIG) has the lowest saturation magnetization o f the three materials and therefore has the lowest and narrowest frequency range. Permalloy (Py) and Fe have substantially higher magnetizations and therefore can achieve higher frequencies for the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 same applied field. The top three ranges are for materials with no (or very little) magnetocrystalline anisotropy, while the bottom range is for Fe with a 50 mT four-fold anisotropy. This anisotropy boosts the maximum frequency slightly and causes the zerofield frequency to be non-zero. Properties o f these materials are in Appendix F. 4.4.3 Stop-Bandwidth and Insertion Loss It is difficult to define the stop-bandwidth and insertion loss for a band-stop filter based on ferromagnetic metals. Unlike other filters, these do not have a flat pass-band. The general behavior near ferromagnetic resonance is shown in Fig. 4-8. Recalling that the imaginary part o f the Voigt permeability is a Lorentzian peak with a FW HM o f ©a, one might expect that the region o f strong attenuation would only extend over a frequency range equal to ©a- However, this is not the case due to the fact that the real part o f the Voigt permeability also leads to attenuation— note the influence o f the real part o f the Voigt permeability on the imaginary part o f the effective permeability. Even in the case o f zero damping, the real part o f the Voigt permeability creates substantial attenuation far from the resonance frequency. The filter in Fig. 4-8, w hich has stop-band rejection o f 40 dB (typical for band-stop filters), has only 1 GHz o f linewidth, but has a 3 dB bandwidth o f alm ost 30 GHz. In addition, it is unclear what the insertion loss is due to the sloped pass-band. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 -3 dB - m 5. co 10- -20- -30- <0 . = 1 GHz -40 0 10 20 30 Frequency (GHz) Fig. 4-8 Stop-Bandwidth and Insertion Loss 4.4.4 Phase-Shift Tunability The total phase-shift tuning range is equal to the total change in the phase constant p. The change in 3 is due to the added inductance, as shown in (2-38). Assuming that the added inductance approaches zero at some applied field value, the total change is therefore: Afl = = 0.573 H 2Z n 100H degrees / cm (4-57) Hence, the tuning range o f the real part o f the effective permeability mostly determines the phase tunability. The real part o f the Voigt permeability reaches maximum and minimum values at the resonance frequency minus and plus h alf o f the linewidth Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 frequency © a - Thus, the above expression should be evaluated for the frequency cdF m r + ©a/2. At this frequency, the effective permeability is: + T ^ a ) « (~ 0 5 +1 (4_58) This creates the expression: Ofiadft^ * 0.643^ ^ * oT (4-59) Using the equations for the geometric factor o f the wide and thin CPW lines produces: and AA mw = 5 1 . 6 degre6% m (4-61) Table 4-3 lists some phase tuning ranges for various materials. As in the case o f stop-band rejection, narrower linewidth or larger magnetization leads to increased effects. The largest range shown, 1387cm, is quite impressive and would lead to a device length o f 2.6 cm to achieve 360° o f tuning. However, the insertion loss o f this device is also large; this fact is discussed further in the next section. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 Table 4-3 Sample Phase Ranges Material Geometry r Phase Range (°/cm) Py Thin 0.007 122 Py Wide 0.02 20.5 Py Thin 0.02 72.5 Fe Wide 0.007 38.9 Fe Thin 0.007 138 4.4.S Phase-Shifter Figure of Merit As described in Chapter 3, the Figure o f M erit for a phase-shifter is the total tuning range (proportional to a change in P) divided by the insertion loss (proportional to a ). From (2-38), these two quantities are proportional to the added inductance and resistance: F.O .M .= A z A = 1 a 3 . 2 R degr ees / / dB (4-62) ' Assuming that the initial added inductance is zero and substituting (4-15) and (4-16) produces: 6)(AZ.2 -A Z ,,) ^ / V R M+ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93 Except for frequencies near resonance and anti-resonance, the real part o f the Voigt permeability is much larger than the imaginary part. Away from resonance and antiresonance, therefore, the real and the imaginary parts o f the effective permeability become nearly equal. This causes the expression above to approach unity and thus the Figure o f M erit approaches 13.2°/dB. 200 50 •45 150- 40 err 100 - 35 50- C fl in <D 0- 25 « -50- 20 3. - <D TJ ,-s c 3 in £ OJ 30 ID eff Figure of Merit 100 - 15 <D O <D 10 -150-200 0 10 20 30 40 60 50 Frequency (GHz) Fig. 4-9 Phase-Shifter Figure of Merit The Figure o f M erit as a function o f frequency is shown in Fig. 4-9, along with the behavior o f the effective permeability for reference. Clearly, the Figure o f Merit (dashed line) is approximately 13°/dB for all frequencies, w ith the exception of frequencies near 10 and 34 GHz (resonance and anti-resonance). There are tw o important issues to note concerning the Figure o f Merit. The first is that the value o f 13°/dB is a maximum value that does not consider non-conduction losses. Other losses would further reduce this figure. In order to reduce the effect of other losses, the conduction loss must dominate the others. One may achieve this by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 94 restricting the operating frequency to values where the magnetic conduction losses are large, such as near the resonance frequency. The second issue is that the theoretical maximum o f 13°/dB is quite a bit less than reported values for other device technologies (see Chapter 1). This diminishes the possibility o f metallic ferromagnetic devices replacing other technologies. 4.4.6 Pass-Band Insertion Loss The calculation for pass-band insertion loss is very similar to that for the stop band rejection. The insertion loss is simply the attenuation at the ferromagnetic anti resonance frequency. Once again, the expression: • « . (4. M ) is proportional to the attenuation. Using the appropriate equations for the wide and the thin CPW lines, (4-45) and (4-46), produces: a mDE = 3 ,4 4 ^^ ^ ° ^ dB / cm (4-65) a TH1N = i2 .2 ^ ° r r ^°M s d B/ c m (4-66) and: Table 4-4 shows values o f insertion loss for the same materials that appear in the other sections. Fig. 4-10 shows a simulation using the program in Appendix H. simulated line is a 1cm long CPW line with the wide geometry. The Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The material is Py with 95 r = 0.02. The simulated insertion loss of 0.155 dB is somewhat larger than the value o f 0.0978 dB shown in Table 4-4. This is because the above equations and the values in the table are based on the assumption that the impedance o f the line is perfectly matched at anti-resonance. The wide CPW geometry does not produce 50 Q impedance when the ferromagnet is in anti-resonance, but one could certainly design it so that it does. Table 4-4 Sample Insertion Loss Values Material Geometry r (unitless) Insertion Loss (dB/cm) Py Thin 0.007 0.203 Py Wide 0.02 0.0978 Py Thin 0.02 0.343 Fe Wide 0.007 0.0652 Fe Thin 0.007 0.228 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96 o.o - m 0.1 Band-Pass Insertion Loss = 0.155 dB - - 0. 2 - T3 CO -0 .3 - -0 .4 - -0.5 30 25 35 40 45 Frequency (GHz) Fig. 4-10 Pass-Band Insertion Loss 4.4.7 Band-Pass Selectivity Band-pass selectivity describes the rate o f change in attenuation verses a change in frequency. In other words, it describes how much transmission o f signal occurs at frequencies near the pass-band. The literature generally quotes selectivity in terms o f dB attenuation per octave frequency. It also increases with line length and is proportional to the frequency-derivative o f a : Selectivity = 0 . m « £ d % ctave . cm (4-67) Assuming that the permeability changes slowly at high frequencies compared to the change in the frequency itself, the variable expression above becomes: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 97 (4 ‘6 8 ) Scaling this expression with the proper constants for the tw o CPW geometries produces: S e le c t = 3 . 6 7 ^ (4-69) d fi/ /o c ta v e -c m (4-70) v ' 6 3 /^ and S e le C jT n y = 13.0 Y (j These expressions for selectivity are very similar to those for the band-pass insertion loss in the preceding section. N ote that, unlike many other operation parameters, the selectivity does not depend on the damping constant. Dividing the selectivity by the insertion loss creates a Figure o f M erit that only depends on the damping constant o f the material: 1 07 F.O.M. = - L= - octave 1 Vr (4-71) As is often the case, reducing the damping constant improves device performance. Table 4-5 shows values o f selectivity for the previously discussed materials. The Figure o f Merit is the selectivities below divided by values o f insertion loss listed above in Table 4-4. Note that the resulting F.O.M. values depend only on the damping constant. Fig. 4-11 shows a simulation o f a 1 cm long CPW line that has the wide geometry and that is constructed o f Py with T = 0.02. The two dotted lines have slope equal to plus and minus 0.73 dB/octave, which corresponds to the value in Table 4-5. Note that the table value does indeed approximate the selectivity o f the simulated device. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 Table 4-5 Sample Selectivity Values Material Geometry Selectivity (dB/octave cm) r (unitless) F.O.M. (octave1) Py Thin 2.58 0.007 12.7 Py Wide 0.730 0.02 7.46 Py Thin 2.58 0.02 7.5 Fe Wide 0.817 0.007 12.5 Fe Thin 2.89 0.007 12.7 o.o +0.73 dB/octave ■0.73 dB/octave -0 . 2 - m TJ -0 . 4 - CN to - 0 .6 - -0.8 - - 1.0 1 0 1 lo92 (f/fFMAR) (octaves) Fig. 4-11 Band-Pass Selectivity The Figures o f Merit values given in Table 4-5 are not far from those o f other band-pass filters [15]. A typical filter may have 3 dB o f insertion loss and 36 dB per octave o f selectivity, which is a F.O.M. o f 12 octave'1. There are a number o f issues, however, that must be taken into account when considering practical application o f Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 ferromagnetic conductor devices. The first is that the F.O.M. o f about 12 octave'1, listed above, corresponds to a practical minimum o f damping in metallic ferromagnetic materials at room temperature. The second is that other factors add to insertion loss, which in turn decreases the F.O.M. These additions to insertion loss include impedance mismatch, dielectric loss and radiation loss. The third issue is that the selectivities per cm listed above are quite small. In order to achieve 36 dB per octave, even the highest selectivity value would necessitate a 12.5 cm long device. Table 4-6 Band-Pass Technology Comparison Dielectric YIG Semi Ferromagnetic 0.5-2.5 3-8 0.3-2.5 2.8 Selectivity (dB/octave) 12-24 12-36 12-24 36 F.O.M. (octave'1) 5-48 2-12 5-80 12 mmWave? No Yes No Yes IL (dB) Table 4— 6 summarizes the differences between different tunable band-pass filter technologies. The data for the insertion loss (IL), selectivity and millimeter wave capability (mmWave) for dielectric, YIG and semiconductor technologies is taken from a review by Uher and Hoefer [15]. The Figure of Merit values above are calculated by dividing the lowest quoted selectivity by the highest insertion loss for the worst-case figure and by dividing the highest selectivity by the lowest insertion loss for the best-case Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 figure. The data for “Ferromagnetic” technology uses the figures calculated above for Fe-based thin CPW structures that are long enough to achieve 36 dB/octave o f selectivity. 4.4.8 Pass-Band Frequency Range The center frequency o f a band-pass filter is the ferromagnetic anti-resonance frequency. As in the case o f the band-stop frequency, this is not affected by device geometry, but only by the material properties of the ferromagnet and by the applied field: ^ fmar = r(B 0 + B4 + MoM s ) (4-72) where Bo is the applied field strength, B4 is the magnitude o f a four-fold magnetocrystalline anisotropy, y is the gyromagnetic ratio (which varies by material) and Ms is the saturation magnetization o f the material. ♦- Py ----------Fe (no anisotropy) Fe (50 mT anisotropy) 30 40 SO 60 70 80 Frequency (GHz) Fig. 4-12 Pass-Band Frequency Range Fig. 4-12 compares the band-stop frequency range for Fe and Py. The leftmost symbol in each range corresponds to zero applied field, while the middle and rightmost Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101 symbols correspond to applied fields o f 100 and 500 mT, respectively. The range o f frequencies achievable with a specific material is driven largely by that material’s magnetization. Fe is shown for the case o f zero anisotropy and for 50 mT four-fold anisotropy. This anisotropy boosts the maximum frequency very slightly. Note that the effect o f anisotropy in this case is much less than the effect on the band-stop frequency. Properties o f these materials are in Appendix F. 4.4.9 VSWR VSWR, the Voltage Standing Wave Ratio, increases with increasing impedance mismatch. It quantifies the undesirable reflected power from a device. It is not possible for a device to have constant impedance over all frequencies, so there is always a frequency at which reflection is at a maximum. In ferromagnetic devices, this frequency will most likely correspond to ferromagnetic resonance. Assuming that a device without added inductance or resistance has perfectly matched impedance of Zo, the actual impedance o f the line becomes: (4-73) If the added resistance is small and if the added inductance is zero (as it is at resonance), then this becomes: (4-74) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. If the impedance mismatch is small, VSWR is simply equal to the ratio of the two impedances: VSWR = 1H— 2coL (4-75) The task is now to develop an expression for the ratio of the added resistance to the inductance. Applying (4-9), (4-11) and (4-16) creates: R 2coL _ _ 2con0g T]0A g ^ (4 76) 2 Z QjE ~ eff t Clearly, the relative permittivity of the dielectric reduces the VSWR. For the case of GaAs in the CPW geometry, the relative effective permittivity is 7.029 (see Appendices E and F) and the VSWR at resonance becomes: V S W R ™ =1 + I.13 x l0 4 /----\ a f i 0T/TB0 (4-77) in the case of the wide line and VSWR„ro =I + 3.9 8 xl04 I V !— (4-78) in the case of the thin line. An odd effect is that the VSWR does not depend on the saturation magnetization of the ferromagnet, but it does depend on the magnitude of the applied field. Table 4-7 shows the VSWR for various materials and geometries and also for two applied field values. Fig. 4-13 shows the case for a permalloy wide CPW line with T = 0.02 and applied field of 100 mT. The maximum VSWR is 1.22, which is very close to the value Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 103 of 1.21 shown in the table. Overall, the VSWR values shown below compare favorably with other devices—VSWR values of 1.5 to 2.5 are common in commercial devices. Table 4-7 Sample VSWR Values Material Geometry VSWR (unitless) r (unitless) Bo = 100 mT Bo = 500 mT Py Thin 0.007 2.25 1.56 Py Wide 0.02 1.21 1.094 Py Thin 0.02 1.74 1.33 Fe Wide 0.007 1.28 1.13 Fe Thin 0.007 1.99 1.44 1.25 V S W R MAX = 1 .2 2 (at FM R) 1 .2 0 - 1 .1 5 - 1. 1 0 - 1 .0 5 - 1.00 0 10 20 30 40 50 Frequency (G H z) Fig. 4-13 VSWR at Resonance Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 104 CHAPTER V RESULTS This chapter discusses two general topics. O ne is the experimental results measured with a Vector Network Analyzer at NIST. The other is the comparison of these results with the limits of operation in the preceding chapter. Table 5-1 Sample Overview Type Num FM D (pm ) d (nm) Notes CPW 102 Py N/A N/A Sputtered Py on GaAs, T = 0.019 Micro 30 Fe 4 200 MBE-grown single-crystal Fe ground plane Evaporated Ag and SiCF films Micro 40 Py 12 500 Evaporated Ag, Py and SiCF films Micro 32 Ni 4 3000 Evaporated Ag, Ni and SiCF films Micro 108 N/A 6 N/A Structure to test new microstrip process Results from five different samples are presented— out of the over sixty samples fabricated over a four-year period. These five samples are listed in Table 5-1. The column D lists the dielectric thickness (in microstrip only) and column d lists the ferromagnet thickness (again, only in microstrip). One o f the samples, number 102, is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 105 coplanar waveguide fabricated from a sputtered Py film on GaAs. The other four samples are microstrips: one Fe, one Py, one Ni and one non-magnetic. The non magnetic sample, number 108, of course shows no tunable effects. It was produced to demonstrate the effect of the photolithographic process on line impedance. The other three microstrip devices did not use this process and therefore have poor impedance match. 5.1 Band-Stop Effects This section contains band-stop results for three different samples: 30, 40 and 102. The stop-band center frequency, which corresponds to the FMR frequency, is shown in Fig. 5-1 as a function of applied field and ferromagnet type. The solid and open circles are the frequencies in sample 30, which contains single-crystal Fe. The solid squares are from sample 102 (Py CPW) and the open squares are from sample 40 (Py microstrip). The solid and dotted lines represent the theoretical FMR frequency for Fe (along an easy axis) and for Py, respectively. Note that the effect of the anisotropy in Fe causes the zero-field frequency to be just over 10 GHz. Frequencies generated by both materials appear to follow closely the theoretical lines. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 Fe (easy axis) N 10- o IX 2 u . 0 40 20 60 80 100 Applied Field (m T) Fig. 5-1 Stop-Band Frequency vs. Applied Field 5.1.1 Results in Microstrip Devices The band-stop effect is shown for Py and Fe devices in Fig. 5-2 and Fig. 5-3, respectively. The large insertion loss shown by both devices is most likely due to impedance mismatch. The stop-band rejection in both microstrips is approximately 10 dB. Both microstrips measure 0.15 cm in length, producing a stop-band rejection of 67 dB/cm. The linewidth of the Fe device is 3 or 4 times greater than the Py device. The Fe device also shows an odd effect; there seem to be three notches rather than just one. One possible explanation is that the Fe film is contaminated (perhaps by oxygen or carbon) and thus there are three regions of differing magnetization in the film. case, the Fe is certainly not of optimum quality. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Whatever the 107 -10 -1 5 - 0 mT CM CO 57 mT 88 mT -2 5 - 83 mT -30 2 3 4 5 6 7 9 8 10 11 12 24 26 Frequency (G Hz) Fig. 5-2 B and-Stop in Py M icro strip -5 - m "O - 10- \ CM CO -1 5 - 0 mT - 8 8 mT 20- 31 mT 65 mT -25 6 8 10 12 14 16 18 20 22 F requency (GHz) F ig. 5-3 B and-Stop in Fe M icro strip Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.1.2 Comparison with Theory The insertion loss in both devices is much larger than one would expect from conduction losses alone. The large insertion loss is most likely due to the high degree of impedance mismatch: the dielectric thickness in both cases is much too thin to produce line impedances m ore than a few ohms. Referring to Table 5-1, one might expect that the stop-band rejection in the Fe sample would be about 3 times greater than that in the Py sample, due to the fact that the dielectric thickness in the Py sample is 3 times greater than the thickness o f the dielectric in the Fe sample. However, there are three other factors in stop-band rejection that one must consider: conductivity, magnetization and damping: rejection «= (5-1) In this case, the Fe device demonstrates much larger damping, which cancels the effect of the thinner dielectric and causes the two rejections to be similar. 5.1.3 Results in CPW Devices Fig. 5-4 shows band-stop effects in the Py coplanar waveguide device. The most noticeable difference between this and the m icrostrip devices is the insertion loss. In this case, the insertion loss is only about 2.5 dB. This is likely due to the better impedance match in the CPW lines compared with the microstrips. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 109 17 mT m CM A CO _4' -5 - 6- 45 mT 4 5 65 mT 6 7 100 mT 83 mT 8 9 10 11 12 Frequency (GHz) Fig. 5-4 B and-Stop in P y CPW There are two other very noticeable effects. One is the narrower linewidth, as compared with the Py microstrip. Although the two sample produce similar FMR frequencies, which indicates that they have similar saturation magnetizations, the sputtered Py in the CPW device clearly has less damping than the e-beam evaporated Py in the microstrip device. The other effect is the rejection, which is about 3 dB. This particular line is 0.2135 cm long, producing a stop-band rejection of 14 dB/cm. The rejection in the Py microstrip is about 67 dB/cm. 5.1.4 C om parison with Theory As measured from the phase data from this device, coa is approximately 600 MHz. This leads to a damping constant of 0.019. Inserting this value into (4-55) yields a stop band rejection of 17.6 dB/cm, which is close to the experimental value of 14 dB/cm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 110 -200 ■o -4 CM CO -5 - -64 53 mT 5 98 mT 6 7 8 9 10 11 12 Frequency (G H z ) Fig. 5-5 Simulation and Experiment Comparison Fig. 5-5 shows a comparison between the simulated transmission (dotted lines) and the experimental measurement (solid lines) for two different applied field values. The shape, depth and position of the notches are well approximated by the simulated data. There is a significant difference between the insertion loss in the simulated and experimental cases, however, which is likely due to an additional loss not considered in the simulation. 5.2 Phase-Shift Effects This section contains phase-shift data from the same three samples discussed in the preceding section: the Py microstrip, the Fe microstrip and the Py CPW samples. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. in 5.2.1 Results in Microstrip Devices The phase-shift effect in Py and Fe microstrip is shown in Fig. 5-6 and Fig. 5-7, respectively. Just as the stop-band rejection in both samples is similar, so too is the maximum phase-shift; the phase tunability in both devices is about 100°. Both devices are 0.15 cm long, so this value corresponds to 6707cm . The total of insertion loss and stop-band rejection in both devices, as shown in Fig. 5-2 and Fig. 5-3, is about 20 dB. This leads to a Figure of Merit o f approximately 57dB. 360 Frequency (G H z) Fig. 5-6 Phase-Shift in Py M icro strip Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i i2 270 0 mT . O) CD "O 65 mT ' ‘N. 180- (D to (0 31 mT 90- cn 83 mT 6 8 10 12 14 16 18 20 22 24 26 Frequency (GHz) Fig. 5-7 P hase-S hift in Fe M icrostrip 5.2.2 C om parison w ith T heory As noted above, the Figure o f Merit for both devices is about 5°/dB. Ignoring the effect o f the insertion loss and dividing the phase-shift of 100° by the 10 dB rejection instead produces a 10°/dB Figure o f Merit. Both these values are below the theoretical limit of 13.2°/dB. Just as in the case of stop-band rejection, the two samples demonstrate similar phase tunabilities. As before, this is due to the large damping in the Fe sample offsetting the effect of thinner dielectric. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 113 5.2.3 Results in CPW Devices As shown in Fig. 5-8, the largest phase tunability is about 40°. The device is 0.2135 cm long and therefore the phase tunability per unit length is 1907cm. Dividing by the stop-band rejection per unit length in the preceding section of 14 dB/cm produces a Figure of Merit of 13.57dB. 315 300 CO <D CD o> 285CD "O CD CD 270CO sz 45 mT 65 mT 17 mT 83 mT 240 100 mT 4 5 6 7 8 9 10 11 12 Frequency (GHz) Fig. 5-8 Phase-Shift in Py C PW 5.2.4 C om parison w ith T heory The measured Figure of Merit of 13.57dB slightly exceeds the theoretical maximum of 13.27dB derived in Chapter 4. However, this experimental result considers only the stop-band rejection, not the insertion loss. Considering the insertion loss halves this figure and creates a Figure of Merit of 6.87dB. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i 14 270 co © © *— CD © 2, 225- 98 mT 180- © © © sz Q_ CM 135- CO 53 mT 90 4 5 6 7 8 9 11 10 12 F requency (GHz) Fig. 5-9 Simulation and Experiment Comparison As shown in Fig. 5-9, the simulated phase (dotted lines) and the measured phase (solid lines) for two different applied field values match well. The shape, magnitude and position of the features near resonance are very similar. The small offset between the experimental and simulated data is most likely due to an error in the length of the thru line used during the de-embed process. This would introduce a constant phase shift to de-embedded experimental data. 5.3 Band-Pass Effects The band-pass effect is much more difficult to observe than either the band-stop or phase-shift effects. The band-stop effect occurs abruptly and causes the transmission through a line to change rapidly near the FMR frequency. The band-pass effect is characterized by a change in transmission verses frequency. This parameter, the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 115 selectivity, is discussed in Chapter 4 and it is noted that it is rather small. Thus, the band pass effect is not noticeable unless the line is very long or the dielectric is very thin. 5.3.1 Results in Microstrip Devices Signs o f the band-pass effect were only observed in one sample, num ber 32. This sample consists o f Ni films that total 3 pm in thickness and dielectric films that total 4 pm in thickness. The band-pass effect requires a very thick ferromagnet (as discussed in Appendix D) and a thin dielectric (due to the 1/D effect). Fig. 5-10 shows a possible band-pass effect in Ni-based microstrip. This effect extends from 15 to 22 GHz and is demonstrated by the increasing transmission with increasing applied field. The band-pass effect changes the transmission by about I dB over the entire applied field range. There is also a much larger effect (about 2.5 dB total change) in the frequency range from 5 to 15 GHz. This effect appears to correspond to band-stop behavior. As in the case of the band-pass effect, this band-stop effect extends over a much larger frequency range than one would expect. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 116 -15 FM AR FM R -16 -17- B =O m T m ■o Ol -18- ' B„ = 76 mT. CO -19- B„ = 100 mT -20 5 10 15 20 25 Frequency (GHz) Fig. 5-10 Band-Pass in Ni M icrostrip 5.3.2 C om parison with T heory The line terminated by open circles denotes the expected range of pass-band center frequency for the applied Field range. The band-pass behavior does not consist of a peak that moves in frequency, as one would expect, but by an increased transmission over a Fixed frequency range. However, the range of band-pass behavior does appear to be centered about the FMAR range. The band-stop behavior also does not consist of a clear peak that moves in frequency, but rather it consists of a reduced transmission over a Fixed frequency range. The strongest band-stop effect occurs near the maximum FMR frequency, which is denoted by the line terminated in the closed circle. One possibility explains the large frequency range of both effects. The e-beam deposition of Ni creates polycrystalline Films. If these films consist o f large anisotropic Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 117 grains with random orientation, then each grain produces a different resonance and anti resonance frequency. Grains with an easy axis nearly parallel to the applied field generate larger than expected frequencies, while grains with a hard axis nearly aligned with the applied field generate smaller frequencies. This “smearing out” of the two effects explains the frequency ranges and the fact that the band-stop effect is much smaller than that observed in the other microstrip devices. 5.3.3 Results in C P W Devices For the sake o f comparison, the lack of a clear band-pass effect in Py CPW is presented here. The three data sets in Fig. 5-11 diverge at lower frequencies due to the band-stop effect. N ear the expected FMAR range (denoted by the line terminated by closed circles), however, there is no noticeable field-dependence. -3.0 B = 17 m T -3 .2 - FM A R m "O -3 .4 B„ = 65 mT CNJ CO -3 .6 - \^t ^ . ■ f / B = 100 m T -3 .8 - -4.0 3.5 4.0 4.5 5.0 5.5 Frequency (octave) Fig. 5-11 Band-Pass in P y C PW Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.3.4 Comparison with Theory The dotted line in Fig. 5-11 has a slope equal to the theoretical band-pass selectivity o f this device. This device is Py-based thin CPW with a length of 0.2135 cm and therefore, by (4-70), the selectivity should be 0.55 dB/octave. Noting that the background insertion loss and the change in transmission at lower frequencies are comparable with this slope, it is clear that one would have difficulty detecting any band pass effects. 5.4 VSWR and Insertion Loss Impedance mismatch leads to two effects: VSWR and insertion loss. At perfect impedance match, the reflection from a transmission line port is zero and the VSWR is equal to unity. The insertion loss may be non-zero due to losses in the line, but the insertion loss due to reflection is zero at perfect impedance match. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 119 0.0 3.0 -0.5-2 .5 - 1. 0 03 - CD ■o - 2.0 CO - 2.0 - - 1 .5 -2.5- -3.0 1.0 0.1 1 10 ZL/ZQ(unitless) Fig. 5-12 VSWR and Insertion Loss vs. Mismatch Fig. 5-12 demonstrates the effect of impedance mismatch on VSWR and insertion loss. Both effects are symmetrical about perfect match (impedance ratio of unity) in terms o f logarithmic impedance ratio. The VSWR approaches undesirably high values faster than the insertion loss. For example, at an impedance ratio of 0.5, the VSWR is exactly 2, which is quite high. At that same value of mismatch, the insertion loss due to reflection is only 0.26 dB, which is not terribly high. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i 20 0.0 3.0 -0.5: - 2.5 - 1. 0 </> - 2.0 3 -1.5 CO - 2.0 - -2.5- -3.0 5 10 15 20 25 30 Frequency (GHz) Fig. 5-13 VSWR and Insertion Loss in Microstrip Fig. 5-13 shows VSWR and transmission through a microstrip device (sample 108) that underwent the photolithographic line-narrowing process described in Chapter 3. This process improves impedance match greatly. The transmission through this device shows that the insertion loss reaches a minimum of about 0.2 dB in the frequency range of 17 to 18 GHz. The VSWR reaches a minimum o f 1.3 in the 22 to 24 GHz range. This value is much less than typical maximum values o f VSWR due to resonance. Fig. 5-14 shows the effect of resonance on the maximum value of VSWR in sample 102. Both devices demonstrate a peak in VSWR at resonance (about 9 GHz in this case). VSWR should be independent of line length, so differences between the two data sets are most likely due to differences in probe contact that lead to additional reflection. The dotted line denotes the maximum value of VSWR of 1.83 calculated from (4-78); this value falls between the two resonance peaks shown. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 121 2 .5 C O cn 22 Thin CPW 0.2135 cm 2.0 - VSWRUi = 1.83 MAX oc CO > 1.5- Thin CPW 0.6565 cm 1.0 o 10 20 30 40 Frequency (GHz) Fig. 5-14 V SW R in Py C PW At high frequencies, the VSWR becomes small in both cases. This is evidence that the coplanar waveguide devices have good impedance match. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 122 CHAPTER VI CONCLUSIONS This final chapter summarizes this study, reviews the study’s goals and concludes with some final comments and notes. 6.1 Summary The title of this study mentions three topics concerning ferromagnetic microwave devices: their design, fabrication and modeling. All of these topics are discussed in this report. Chapter 3 describes the design of microstrip and coplanar waveguide transmission lines. This includes the design of the TRL calibration set necessary for accurate measurements. The equations for the design of these lines are in Appendix E and the masks containing this design are in Appendix G. Chapter 3 also contains a discussion of the three effects o f interest in this report. This includes the desirable and undesirable characteristics of devices that one must consider when designing those devices. The design of ferromagnetic devices requires understanding o f certain issues that do not apply to non-magnetic devices. Two of these issues are the effect of finite-thickness ferromagnetic films, which is presented in Appendix D, and the magnetic properties of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 123 ferromagnets, which are listed in Appendix F. In all, this report contains a comprehensive discussion of the design of microstrip and CPW devices for use as bandstop filters, phase-shifters and band-pass Filters. The fabrication process for microstrip and CPW devices is described in Chapter 3, including the use of the photo and shadow masks shown in Appendix G. Appendix A accounts the construction of single-crystal Fe Films, which are used to construct microstrip devices. Appendix B explains the use of a FMR system to characterize magnetic films. This characterization aids in the fabrication process by allowing one to select good-quality films before completing a sample. Thus, the fabrication processes of both microstrip and CPW devices are fully covered. Chapter 2 introduces the concepts required for understanding of the modeling technique. This includes the theory of dynamic permeability in a ferromagnet, the theory of transmission line structures and the theory of microwave measurements. This theory is developed further in Chapter 4, where the surface impedance technique is compared with the allguide.f simulation. It is also shown that the surface impedance technique produces the proper equations for non-magnetic conductors. Chapter 4 concludes with a derivation of several limits of operation, which is compared to experimental data in Chapter 5. Several appendices address issues in this author’s modeling technique. Appendix C presents equations for losses due to dielectrics and radiation. Appendix D concerns the effects of thin ferromagnetic films on the modeling process. The equations for the characteristic impedance of microstrip and coplanar waveguide lines are listed in Appendix E; these equations are used by the surface impedance technique to calculate the geometric factor. Material characteristics, which are required by the modeling technique, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. are listed in Appendix F. Finally, Appendix H contains a FORTRAN program for modeling effects in CPW devices. In conclusion, a general theory for modeling effects in transmission lines that contain ferromagnetic conductors is developed. 6.2 Fulfillment of Study Purpose and Focus The purpose of this study, as given in Chapter 1, is “to investigate the possibility of applying ferromagnetic conductors in tunable microwave devices.” Chapter 1 also delineates the focus of this study, including what materials, device structures and effects are studied. The purpose and focus of the study are fulfilled by addressing the following issues: • A range of ferromagnetic materials is explored by fabricating devices from Fe, Ni and permalloy. The implications of using each material in various devices are discussed and devices constructed from each material are demonstrated. • Two device structures are studied: microstrip and coplanar waveguide. The practical considerations in the design and fabrication of these structures are explained and experimental results are measured from both structures. • Three tunable effects are described: band-stop, phase-shift and band-pass. A model is presented for arbitrary device structure and composition that accurately predicts these effects and limits of operation are derived for devices that exhibit these effects. The model of these effects is compared Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i25 with results measured from actual devices and thus it is shown that the model is accurate. The design process for tunable devices is presented, including analytical equations for device performance as a function of physical structure and material composition. Thus, the topic of tunable microwave devices constructed o f ferromagnetic conductors is comprehensively explored via a combination of theoretical and experimental research. 6.3 Final Notes This report ends with the following final conclusions concerning the practical structure and composition o f devices and their comparative performance with other technologies. 6.3.1 Device Structure and Composition The following conclusions are drawn concerning device structure and composition: • Coplanar waveguide is a more practical device structure than microstrip. CPW is easier to fabricate because it requires less process steps. CPW is also more easily integrated with other devices. • Fe possesses several advantages over other ferromagnetic metals. It has high saturation magnetization and a moderately high anisotropy field— both these characteristics increase operating frequencies. In its single crystal form Fe has low damping, which improves several device Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. characteristics. Fe is not an alloy and therefore is easy to deposit. Finally, Fe has a high Curie Temperature, which allows it to operate at higher power densities without losing its magnetic properties. • Many effects improve with a reduction in device geometry (the 1/D effect in microstrip). Thus, the “thin CPW” geometry that is presented above is better than wider structures. However, one cannot make the device geometry too narrow, as this decreases power-handling ability and increases VSWR. Thus, the most promising choice for tunable devices is thin CPW structures that incorporate Fe. 6.3.2 Comparative Performance Considering the above recommendation for device construction, the following comparison between Fe-based CPW devices and other technologies for the three device applications is made: Band-Stop Filters Fe-based devices out-perform YIG devices in terms of frequency range. A YIG device that operates at 40 GHz requires 1.3 T of applied field. Fe devices with the same applied field operate at 62 GHz. However, Fe devices cannot achieve the narrow bandwidth and low insertion loss of YIG devices due to effects that are discussed in Chapter 4. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 127 Phase-Shifters Metal-based devices create large phase tuning ranges in short device lengths. However, this large tuning occurs with large insertion loss and leads to a maximum Figure of Merit of 13.2°/dB. This is far less than typical F.O.M. values given in the literature for ferroelectric-based phase-shifters (see Chapter 1). The fact that magnetic phase-shifters require electromagnets for tuning, while ferroelectric devices only require a voltage, is another impediment to practical ferromagnetic conductor-based phase-shifters. Band-Pass Filters Like band-stop filters, band-pass filter based on Fe out-perform YIG filters in terms of frequency. FMR drives YIG band-pass filters, while ferromagnetic metal-based band-pass filters rely on FMAR. With applied fields from 0 to 1.3 T, Fe-based band-pass filters create pass-band center frequencies from 63 to 101 GHz. The primary drawback of such filters is their selectivity per unit length, which requires devices that are 12.5 cm in length. Such a device would exhibit 36 dB/octave of selectivity (a typical requirement) with 2.8 dB of insertion loss. However, the 12.5 cm length makes it difficult to create such a device and difficult to apply a large field to such a device. In conclusion, band-stop filters are practical when one requires high operating frequency, but not narrow bandwidth. Phase-shifters are not practical, compared with ferroelectric devices. Band-pass filters are practical when high operating frequency is a requirement and when small device size is not. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 128 APPENDIX A EPITAXIAL GROWTH OF FE ON GAAS The deposition of single crystal Fe creates films that are desirable for several reasons. First, Fe has a high saturation magnetization, which allows high resonance and anti-resonance frequencies with moderate applied fields. Second, single crystal Fe has narrow resonance linewidth. Third, Fe has a four-fold in-plane anisotropy that increases the resonance and anti-resonance frequencies. Films of single crystal bcc Fe were created on a single crystal fee Ag Films. The single crystal Ag films grew on a wafer of single crystal (zinc sulfide structure) GaAs. These two materials, Ag and GaAs, also have desirable qualities. Ag is the highest conductivity metal at room temperature; this makes it a good choice for inclusion in a ground plane. GaAs is desirable because it is a common semiconductor for active microwave devices— one can envision Fe-based devices grown on a substrate containing electronic microwave devices. This section contains an overview o f our process for depositing Fe(OOl) and Ag(OOl) films on GaAs(OOl) substrates. The physical structure of the films and a discussion of other research on this system are also presented. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 129 A .l Deposition of Ag(OOl) on GaAs(OOl) There are two difficulties with depositing Ag(OOl) films on GaAs(OOl) substrates. First, the preparation of GaAs often leaves the surface depleted o f As. This is a problem because Ag will then alloy with the excess Ga, thus frustrating epitaxial growth [72]. The second problem is the fact that both Ag(OOl) and A g(011) films can form on the GaAs(OOl) surface [73]. The solution to both these problems is to first deposit a “seed layer” o f Fe(OOl) on the GaAs(OOl) surface [74], Fig. A -l Fe(OOl) film on a GaAs(OOl) surface Fe(OOl) deposits well on GaAs(OOl), as shown in Fig. A -l. The large gray and black circles in Fig. A -l represent As and Ga atoms, respectively; the small gray circles represent Fe atoms. Note the unit cells for GaAs (larger square) and for Fe (smaller square)— the lattice constants o f these two materials are 0.565 nm and 0.287 nm, respectively [75], The lattice mismatch is only 1.4%, as opposed to 2.2% in the case of Ag(OOl) [76]. Also note that the Fe[100] and GaAs[100] directions are parallel. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A.2 Deposition of Fe(OOl) on Ag(OOl) Fe(OOl) and Ag(OOl) Films have a 0.8% lattice mismatch and align as shown in Fig. A-2. Note that the unit cell for Ag (large black square) and the unit cell for Fe (small white square) are rotated with respect to each other by 45°. The lattice constant of fee Ag is 0.409 nm [75]. Fig. A-2 Interface between Fe(OOl) and Ag(OOl) films Ag(OOl) deposits well on the Fe(OOl) seed layer and forms a film that can be annealed for better smoothness. Fe(OOl) deposited on this Ag(OOl) template is of better quality than one deposited directly on GaAs(OOl); the Ag(OOl) surface is smoother and does not contain As, which could contaminate the Fe and impair its magnetic qualities. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 131 Fe easy axes GaAs[100] Fe[100] and Fe[010] Sam ple cleaved on GaAs[110] and GaAs[1-10] Fig. A-3 Orientation of Crystal and Anisotropy Directions The Fe(001) film on the Ag(001) template is rotated twice by 45° with respect to the Fe(001) seed layer. The net result is that the Fe[100] direction in the top Fe film is parallel to the GaAs[100] direction. It is important to note, as shown in Fig. A-3, that GaAs(OOl) most easily cleaves along the G aA s[l 10] and GaAs[l-10] directions; this is in contrast to Si(OOl), which cleaves most easily along the Si[ 100] and Si[010] directions. Fe(OOl) has a four-fold magnetocrystalline anisotropy with easy axes along the Fe[100] and Fe[010] directions. Thus, the easy axes of the top Fe(OOl) film can be determined ex-situ without a measurement of the crystal or magnetic properties. A.3 Details of Deposition and Characterization Thanks to the demand created by the high-speed semiconductor industry, GaAs(OOl) wafers are available at a very reasonable price. These wafers are well cut and polished, and often have been further enhanced with an added epitaxial film. However, they require further preparation before they are ready for vacuum deposition. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This 132 preparation consists of chemical processing before the substrate is loaded in the vacuum system and physical processing once inside. There are two goals for the chemical processing: removal of organic contaminates and removal o f the native oxide film. Good results are achieved by boiling substrates in isopropyl alcohol to remove organic contaminates and briefly etching the sample in 10:1 hydrofluoric acid to remove the native oxide. Once inside the ultra-high vacuum system, one can further process the sample with annealing and Ar sputtering. Heating the wafer to about 600 °C removes fluorine and water left on the surface by the chemical processing. A de-focused sputtering gun is used at low angle and a beam energy of 500 eV. This succeeds in sputtering contaminates o ff the GaAs surface while limiting damage to the crystal. A second anneal helps in repairing some of the crystal damage that is done to the substrate. This process of alternating sputtering and annealing proceeds until the surface shows a good Reflection High-Energy Electron Diffraction (RHEED) pattern. Fig. A-4 GaAs Surface after Sputtering and Anneal Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 133 RHEED uses an electron beam with energy typically in the 5 to 20 keV range at grazing incidence to the sample surface and is very sensitive to the surface structure of the sample [77]. The diffraction pattern holds a variety of information about the surface: its roughness, orientation, lattice parameter and structure. The diffraction pattern in Fig. A-4 is from the surface o f GaAs after sputtering and annealing processes. The streaks and dots are much sharper than in earlier images; this corresponds to an improvement in the surface roughness. In addition, the intermediate dots between the main streaks indicate a surface reconstruction. This reconstruction pattern is favorable because it can only occur on a clean, well-prepared GaAs surface. Hence, the pattern in Fig. A-4 shows a GaAs properly prepared for deposition. Fig. A-5 shows the pattern due to a 1.3 nm thick Fe seed layer. The crystal pattern is clear, but the streaks are very broad, indicating a rough surface. This is to be expected, however, because the seed layer is deposited at room temperature and cannot be annealed to improve smoothness. Fig. A-5 Fe Seed Layer Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The Ag template film can be annealed to improve its smoothness. The pattern from such an annealed film is shown in Fig. A-6. This film is 601 nm thick and was annealed overnight (over 12 hours) at 350 °C. By monitoring the streak width during annealing, one can alter the annealing process maximum smoothness (minimum streak width). Fig. A-6 Ag Template after Anneal After annealing the Ag template best smoothness, the sample is allowed to cool to room temperature. Then, the sample is ready for deposition of the thick Fe film. Fig. A-7 shows the diffraction from a 200 nm thick Fe film. Note that the film roughness is less than that o f the seed layer shown in Fig. A-5; the streaks are narrower for the thick film. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 135 Fig. A-7 Surface of Thick Fe Film In order to prevent oxidation o f the Fe film upon exposure to atmosphere, one must add a protective film. A Ag film works well because Ag is a noble metal and one can deposit it epitaxially to create a film with few defects that would allow oxygen to pass through. The pattern from a 7 nm thick Ag capping film is shown in Fig. A-8. Its good crystal quality is further evidence that epitaxy is successful throughout the sample deposition. Fig. A-8 Surface of Ag Capping Film Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX B FILM CHARACTERIZATION WITH FMR Ferromagnetic Resonance Systems (FMR) are very useful for determining a number of magnetic parameters. FMR systems allow one to measure the microwave absorption o f a film as a function of applied field magnitude and direction [78]. Thus, one determines the field corresponding to a fixed resonance frequency for a set of in plane field angles. A curve fit of the resonance field magnitude vs. the field angle produces parameters such as the effective magnetization and the magnitude and direction of anisotropies. A plot of absorption vs. applied field for a fixed angle allows determination of the resonance linewidth. B .l FMR Apparatus A typical FMR system consists of three sub-systems: a microwave system, an electromagnet system and a computerized data-acquisition system. The electromagnet system consists o f an electromagnet and its power supply. Preferably, the electromagnet can rotate to various field angles and the power supply can be com puter controlled. The data-acquisition system usually consists of voltmeters, a lock-in am plifier and a personal computer. This system is discussed further in Section B.2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 137 The microwave system consists o f a source, a waveguide system, a resonance cavity and a detector. In modem systems, the source and detector are solid-state diodes. The resonance cavity links to the waveguide system by a coupling hole. The waveguide system usually includes a variable attenuator, an isolator in front of the source and a frequency meter (a tunable dielectric filter). A directional coupler routes the reflected wave from the cavity into the detector. reflected wave incident wave coupling hole applied field Fig. B -l FMR System Waveguide and Resonance Cavity Fig. B-l shows the basic construction of the resonance cavity. The cavity can be as simple as a piece of waveguide coupled to the rest of the system by a metal plate with a coupling hole in it. In the case of this study, the cavities are cylindrical. In either case, one designs the resonance frequency of the cavity to be a certain, fixed value corresponding to a convenient frequency in the operating band of the waveguide, source and detector. The sample is in the cavity and the cavity is between the magnet’s pole pieces. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 138 B.2 Data Collection and Analysis The effect of the coupling hole is shown in Fig. B-2 [79]. The coupling hole acts as a transformer that transforms the waveguide impedance from Z q to n2Zo. The impedance of the cavity is: Z cav,n f , 1 ~ R + j coL------coC V (B-l) The quantity “R” represents losses in the cavity due to leakage (radiated losses), dielectric losses in the sample and conduction losses in both the sample and the cavity walls. The “L” and “C” elements represent the resonance properties of the cavity. Clearly, this impedance has a resonance frequency, coo, at which the imaginary component vanishes: m° = 7 u : ( B "2) The transformed waveguide impedance is purely real and therefore decreasing the imaginary part of the cavity impedance decreases the reflected power if R is greater than n2Zo and increases the reflected power if R is less than n2Zo. Thus, one determines the resonance frequency of the cavity by tuning the source and monitoring the detector output for a local maximum or minimum. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 139 1:n R C Fig. B-2 Equivalent Circuit for Coupled Cavity 0 .100 c: 0.075 0.050 0.025 0.000 0.50 0.75 1.00 1.25 1.50 R/n Z Q(unitless) Fig. B-3 Reflected Power vs. Cavity Impedance When the cavity is at resonance, the reflected power (the squared magnitude o f O varies due to changes in R: |rf = R - n Z0 R + n2Z n (B -3 ) The dependence o f the reflected power on the ratio o f R to n2Zo is shown in Fig. B-3. If R is less than the transformed waveguide impedance, then the cavity is “overcoupled” and increases in R near ferromagnetic resonance correspond to decreases in the reflected Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. power. If R is greater than n 2Zo, then the cavity is “undercoupled” and the reflected power reaches a maximum at FMR. One adjusts the value of the coupling n by altering the size of the coupling hole. In the m ost basic FMR system, the magnetic field continuously sweeps while the system monitors the detector voltage, which is proportional to the reflected power. One notes the magnetic field magnitude corresponding to the maximum or minimum in reflected power and thus finds the resonance field. 1.00 0 .7 5 - ca £- 0 .5 0 - 0.25 - O # o 0 .0 0 - O _i -0.25 - -0.50 0.20 0.25 0.30 0.35 0.40 Applied Field (T) Fig. B-4 Typical FMR Spectrum A more advanced system improves the results by employing a lock-in amplifier. A second set o f coils added to the electromagnet provides a small magnetic field that oscillates at a frequency and phase provided by the lock-in. The lock-in then monitors the detector voltage for signals at this frequency and thus rejects noise signals at other frequencies. T he end result is a cleaner signal, such as in Fig. B-4, that is the derivative Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 i of the absorption peak seen by the detector. The zero-crossing of such a signal is the extreme of the detector voltage and is therefore the resonance field. 0.35 •O . ■O 0.30 LL. S 0 .2 5 - ■QO 0.20 -90 -45 0 45 90 Angle of Applied Field (degrees) Fig. B-5 Resonance Field vs. Field Angle Fig. B-5 demonstrates the result of several FMR spectra taken with different applied field angles. The angular dependence of the resonance field is due to crystalline anisotropies; one can derive the magnitude and angle of these anisotropies from a curvefit of such data. In this case, the sample is a thick layer of single crystal Fe. Hence, one expects a four-fold anisotropy. The equation for the resonance frequency as a function of applied field and other parameters is [80]: a 2 = 7 2{B0 + B 4 cos 4 (0 -4>)][B0 + i f l 4[3 + cos4(0 - 0 ) ] + fi0M eff} (B-4) Inverting (B-4) to solve for Bo, the resonance field, produces the proper fit equation, co is the angular frequency measured by the frequency meter, y is 2k multiplied by 29.2 GHz/T for Fe and 0 is the applied field angle. The fit therefore produces the four-fold Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. anisotropy magnitude B4 , the direction of an easy axis <f>, and the effective magnetization field poMCff. The result o f fitting the data in Fig. B-5 is summarized in Table B -l. Table B -l FMR Curve Fit Parameters V alue Notes Bo N/A Resonance field, dependent variable 0 N/A Field angle, independent variable CO 2k 24 GHz y 2n 29.2 GHz/T Source frequency, measured by frequency meter Assumed quantity for Fe (see Appendix F) b4 0.0588 ± 0 .0 0 1 8 T Four-fold anisotropy magnitude |io M Cff 2.1386 ± 0 .0 1 2 6 T Effective magnetization field 0 -32.1 ± 0 .5 ° Direction of an easy axis Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 143 APPENDIX C DIELECTRIC AND RADIATIVE LOSSES The losses in magnetic conductors are, o f course, conduction losses. This loss mechanism is analyzed in-depth in Section 4.2 and in Appendix D. There are, however, two other loss mechanisms to note: dielectric losses, which arise from an effective conduction in the dielectric, and radiation losses, which are caused by power radiated in the plane perpendicular to propagation. The derivation of these losses is beyond the focus o f this report and not related to ferromagnetic resonance, but approximations of these effects are important for the analysis of device capabilities. C .l Dielectric Losses in Microstrip Schneider provides the following equation for dielectric loss in dB/unit length [81]: 20.0flr qtang ln(lO.O) \ (C -l) The quantity “q” is the guide filling factor defined by: (C-2) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 144 The unit o f length is defined by A.g, which is the guide wavelength: 2nc (C-3) Ecff is the effective permittivity. Equations for this value are in Appendix E. tan 8 is known as the dielectric loss tangent, which is the ratio of the imaginary part of the dielectric’s permittivity to its real part. For the microstrips discussed in this report, the dielectric is SiOi, which has a relative permittivity o f 3.8 and a loss tangent of 0.0006 (see Appendix F). This results in: t t r = 6 .6 x l0 ~ * / dB/cm (C-4) where f is the frequency in GHz. For the band-pass frequency in Fe (about 65 GHz) this value is about 0.043 dB/cm, which is not negligible when compared with pass-band insertion loss values (see Chapter 4). C.2 Dielectric Losses in Coplanar Waveguide Gopinath uses the following for dielectric losses in CPW [82]: (C-5) where otd is in Np/unit length, q is the guide Filling factor given by (C-2) and Xg is defined in (C-3). For the coplanar waveguides discussed in this report, the dielectric is GaAs with relative permittivity of 12.9 and loss tangent of 0.0003 (see Appendix F). This results in: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 145 a r = l . l x l O -4/ dB/ c m (C-6) where f is the frequency in GHz. For 65 GHz (near band-pass in Fe) this evaluates to 0.0072 dB/cm, which is small enough to ignore. C.3 Radiation Losses in Microstrip Microstrip radiation loss occurs when higher modes are present [70]. Higher propagation modes exist only when the dielectric thickness is comparable to the propagating wavelength. In this study, the dielectric thickness is on the order of 10 |im and therefore only the lowest order mode exists for frequencies below about 100 GHz. For lines such as these, the low-frequency radiation loss is nearly zero. C.4 Radiation Losses in Coplanar Waveguide CPW exhibits a radiation loss per unit length that was examined by Frankel et al. [83]. The expression is: (C-7) Thus, the radiation loss per cm for the GaAs-based CPW varies according to the cube of frequency: a r = l.OxlO-6/ 3 d B/ c m Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (C-8) where f is in GHz. Clearly, this loss can be significant at higher frequencies. At 65 GHz (band-pass in Fe) this loss becomes 0.27 dB/cm, which far exceeds the dielectric loss at that frequency and which is comparable to insertion loss at band-pass frequencies (see Chapter 4). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 147 APPENDIX D SURFACE IMPEDANCE IN THIN FILMS In Section 4.2, the surface impedance is calculated for a sem i-infinite metal with arbitrary permeability. This calculation is the basis of this author’s report with David W alker that simulates resonance effects in coplanar waveguide structures [59]. In that case, the calculation is valid because the metal widths are extremely large as compared to the skin depth. This is not the case in many other structures. In microstrip structures, for example, the thickness of the ferromagnetic film is usually much less than the skin depth. This section covers the calculation of the surface impedance o f films that are much thinner than the skin depth. From this, the effective permeability of the film is derived. Then a comparison is presented between the cases of thin and semi-infinite films. Finally, the implications of film thickness in the case of ferromagnetic films is discussed. D .l Calculation for Thin Films Begin by considering a metal surface in the y-z plane. The metal is semi-infinite in the x direction, filling all positive x. Outside the metal, the wave propagates in the zdirection and a non-zero z-component of the electric field prohibits the wave from TEM Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 148 propagation; TEM modes are not allowed when a waveguide consists of imperfect conductors. Inside the metal, the electric field in the z-direction is Eo at thesurface (x=0) and varies exponentially with x according to the propagation constant, y: (D -l) E: = E 0e ^ Note that Eo also defines the decrease in voltage per unit length in the direction of propagation. By Ohm ’s law, the current density in the metal is simply proportional to the electric field: 7 = gE (D-2) The next step is to determine the current, I, in the film of thickness d and width in the ydirection of W.This is simply an integral of the current density across theappropriate cross-sectional area in the x-y plane: eW pd rd / = Jo d y h J d x = u W )0 E0e~v dx (D-3) The result is: /= — y £„( \ - e - * ) (D-4) Now, consider that the space from d to infinity isfilled with a high-conductivity metal, such as Ag.Using the method above, the current in this metal iscalculated and added to the result in (D-4): / = — E0 (l - e-* )+ ^ r E ^ e '" * Yn Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (D.5) 149 where On is the conductivity of the noble metal and Yn is the propagation constant in the noble metal. Approximating for small d produces: <7 W E0-^ + — — EQ{ \ - y d - Y Nd ) 7s 7 (D-6) The surface impedance is the ratio of voltage drop per unit length, Eo, to the current, I, in a W-wide area o f surface: Z, 1 ______________ 7 s ! G s W l + d {o y N - o n 7 - (D-7) o „7 s )/ o n The conductivity of the noble metal is much larger than the conductivity in the ferromagnet and therefore one may drop the first term in the parenthesis. Another approximation for small d yields: Zs ^ ^ r - b + dY + d y x ) WoN (D-8) The surface impedance is therefore the sum o f three impedances represented by the three terms above. The first term is simply the impedance of a semi-infinite noble metal film. The second term is a small addition that holds the behavior of the ferromagnetic film. The third term is a small correction to the first and does not depend on the behavior of the ferromagnetic film (except for its thickness). In order to find an expression for the effective permeability of the ferromagnetic film, begin with the second term in parenthesis in (D-8) and substitute the definitions for the propagation constants: (D-9) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 150 where jj.n and ji are the permeabilities of the noble metal and the ferromagnet, respectively. Setting d/W equal to the change in the geometric factor, Ag, and assuming that |1n equals po produces: ~ V JMc ~ ^ (V _ ^ c + |^c| + j^ U c + | ^ c | ) (D-10) D.2 Comparison of Finite and Infinite Films The difference between finite and infinite films is best demonstrated by a comparison of the added impedance of the conductors. This impedance consists of two parts: an imaginary part of coAL, which creates phase shifts, and a real part of R, which leads to losses. Calculation of these two values requires two quantities: the effective permeability and the change in the geometric factor. The effective permeability for an infinite film is calculated in Chapter 4 and the effective permeability for a finite film is above in this appendix. The change in the geometric factor for an infinite film is proportional to the skin depth (see Chapter 4). For a finite film, the change in the geometric factor is assumed to be constant and proportional to the film thickness, d. The finite film case also has one more complication: one must add the impedance due to the finite film to the surface impedance o f the noble metal below it. The data in Fig. D-l and Fig. D-2 is calculated with these factors in mind. The finite and infinite thickness data use the sam e Voigt permeability, but other factors such as conductivity and line geometry are arbitrary. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 151 1.0 c 5 0.5- <D infinite thickness finite thickness c 3 (D Q. S . o.o_i < 3 -0.5 0 10 20 30 40 50 Frequency (G H z) Fig. D -l Added Inductance for Finite and Infinite Films Fig. D-l depicts the general difference in the phase shift (which is proportional to ooAL) for the finite and infinite film thickness cases. The influence of the ferromagnet in the finite thickness case (dotted line) is superimposed on a curved background that is due to the noble metal beneath. The influence of the ferromagnet in the infinite thickness case (solid line) is very clear. There are two major changes in phase shift: one about resonance (10 GHz) and a smaller one about anti-resonance (34 GHz). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i52 O) c © infinite thickness finite thickness DC 0.0 0 10 20 30 40 50 Frequency (G H z) Fig. D-2 Added Resistance for Finite and Infinite Films The differences between the finite and infinite cases shown in Fig. D-2 are similar to those in Fig. D -1. Once again, the curved background due to the noble metal disguises the ferromagnetic effects in the finite thickness case (dotted line). The peak in R (and therefore attenuation) at resonance appears in both cases, but the effect in the infinite thickness case (solid line) is clearly larger. resonance is even clearer. The difference in the two cases at anti There is a significant minimum in attenuation at anti resonance in the infinite thickness case, but the effect is almost invisible in the finite thickness case. In fact, R at anti-resonance actually exceeds R at resonance in the finite case, due to the effect of the noble metal that increases R with increasing frequency. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 153 D.3 Implications for Ferromagnetic Devices In the first section of this appendix, the calculation of the surface impedance in the finite thickness case is based on an approximation that requires: 1» d y (D -l 1) This condition concerns the real part o f the propagation constant, which is the inverse of the skin depth, according to (4-2). Upon making this substitution and rearranging terms, the approximation requirement becomes: (D -l 2) 8 » d Hence, the transition from finite thickness behavior to infinite thickness behavior occurs when the skin depth and ferromagnet film thickness are similar. 10 Py with (i = u CL m 1 1 Py with n = n0nvojgt CO 0.1 1 10 Frequency (GHz) Fig. D-3 Skin Depth in Ferromagnetic Metal Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 Due to the behavior of the Voigt permeability, the skin depth in a ferromagnetic conductor varies drastically with frequency. This effect is shown in Fig. D-3. The solid line shows the skin depth in permalloy as a function o f frequency for a given applied field. The dotted line is the skin depth in a conductor with conductivity equal to that o f permalloy, but with permeability simply equal to |io - There are three major differences between the two. The first difference is the significant offset between the two at low frequencies. The lines run roughly parallel, but the skin depth in the “magnetic permalloy” is much less than the skin depth in the “non-magnetic permalloy”. This is due to the fact that permalloy, at low frequencies, behaves as a conductor with a large constant permeability (hence the name “permalloy”). The second difference is the distinct minimum at resonance (10 GHz) in the magnetic permalloy. A ferromagnetic metal at resonance has a skin depth that is much less than non-magnetic conductors, even conductors with very high conductivities. The third major difference occurs at anti resonance (34 GHz), where the skin depth in the magnetic permalloy reaches a maximum that is much greater than the skin depth in the non-magnetic permalloy at that frequency. As a final note, the two skin depths converge at high frequencies. At these frequencies, the fields in the wave oscillate too quickly to interact with the ferromagnetic properties of the permalloy and therefore the magnetic and non-magnetic permalloy behave the same. Taking into account the role of the skin depth in the finite thickness approximation and the behavior o f the skin depth in a ferromagnetic conductor, it is clear that these factors imply certain considerations that must be made when designing a device with ferromagnetic conductors. If a device incorporates a ferromagnetic film with thickness slightly greater than the skin depth at resonance, then the film will behave as an Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 155 infinite film at resonance, but as a finite film outside of resonance. Such a film makes the “best o f both worlds” by maximizing the attenuation at a resonance and minimizing the attenuation away from resonance (the insertion loss). One must also consider the thickness o f the film if a device based on the anti-resonance effect is desired. At anti resonance the skin depth, and therefore the critical film thickness, is at a maximum. If the film does not exceed this critical thickness, then the anti-resonance effect is greatly hampered. This is clear in Fig. D -l and Fig. D-2, where the anti-resonance effects are almost non-existent in the finite film thickness case. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 156 APPENDIX E DESIGN EQUATIONS FOR CPW AND MICROSTRIP In this appendix, the equations used to calculate the characteristic impedance of coplanar waveguide (CPW) and microstrip transmission lines are presented. These equations are primarily used when one wants to design lines of specific impedance. These equations are used for that purpose when designing devices, but also for modeling the magnetic effects in devices. Chapter 4, in particular, draws heavily from the equations that are listed here. E .l Characteristic Impedance of CPW b Fig. E -l CPW Dimensions Wadell offers the following set o f equations for the geometry shown in Fig. E -l [85]. The dark gray regions represent the conductors— the two regions labeled “G” are the ground planes and the region labeled “S” is the signal line, a and b are the width of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 157 the signal line and the spacing between the ground planes, respectively. metallization is t thick and the substrate is h thick. The The ground planes are expected to be at least 5b wide and thus they have little effect on the impedance. The characteristic impedance is: 30. t o K t p where the K-functions are complete elliptic integrals of the first kind. Eero is the effective permittivity, corrected for the thickness of the metallization: £eff- \ . 0 £<ffj ~ £<ff ~ ( b - g ) / 2.0 K {k) ( t 0.7r (E_2) K (k ')+ ' Eefr is the effective permittivity that is not corrected for the metal thickness, t, but is corrected for the substrate thickness, h: £ e ff = !-Q+ £ r — 1.0 K j k ^ K j k , ) 2.0 K(k)K(k') (VI-3) The parameters in the elliptic integral functions in the equations above are as follows: sinh k=— b k ,= ^ ~ ' b ‘ k{ = 1 C Jta, ^ 4.0/z f Tib, smh y 4 .0 h y (E-4) A primed k corresponds to its complement, with the following relation holding for each of the three k-values: k' = -Jl.O —k 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (E-5) kt and ki depend on the thickness-corrected values o f a and b, represented by a, and bt: 1.251 A.Oku 1.0 + In a, = a + K b, = b - 4.0Jta 1.25t 1.0 + In K (E-6) E.2 Characteristic Impedance of Microstrip t ^ Dielectric S t h Fig. E-2 Microstrip Dimensions The following equations refer to the geometry shown in Fig. E-2. The dark gray regions represent the conductors, with “S” and “G” labeling the signal line and ground plane, respectively. The signal line has a width of w and thickness of t. The substrate has a thickness of h and is assumed to extend indefinitely on either side. The microstrip structures that are discussed in this report have fmite-width dielectrics, however. This case was studied by Smith and Chang, who found that the impedance is affected by less than half a percent [84]. Wadell gives a series of equations from various sources [85]. The characteristic impedance is [86, 87]: 4.0 h rio Zn = In 1. 0 vv 0 2.Chfz6n^jeejr +1.0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (E-7) 159 where F is a function providing further accuracy: r 14.0 + 8.0/£fJ 4.0h 11.0 r 14.0 + 8 . 0 / ^ W 4.0ft V I-O-H.O/% , | vv' V 11.0 w ) 2.0 In the above equations, Ecrr is the effective permittivity [88]: C £ + zz — 1 1.0 £ 2.0 - 1.0 12 . 0/1 /". 1 2.0 (E-9) + - w and w ’ is the width o f the line, w, with a correction for the line thickness, t [89]: w = w+ l.O + l .O / ^ l.O r 2.0 7t 4e In \ m f + (E-10) \/it V H ' / f + 1.1 It should be noted that if the thickness, t, is allowed to go to zero, and if the ratio w/h is very large, then the equation for the impedance, (E-7), becomes much more simple: Z„ = Vo h °~ V ^> This is the familiar equation for a parallel-plate structure [90]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (E-l 1) APPENDIX F MATERIAL PROPERTIES This report describes transmission lines containing either SiCF or GaAs as the dielectric material. The relative permittivity er o f the dielectric determines the characteristic impedance of the line, as shown in Appendix E. The loss tangent, tan 8, determines dielectric losses in the line, as shown in Appendix C. These values for S i0 2 and GaAs are below in Table F— 1. T ab le F -l D ielectric P roperties Notes er (unitless) ta n 5 (unitless) S i0 2 [91] 3.8 0.0006 Can be sputtered (thin films) or e-beam evaporated (thicker films possible) Values are for fused quartz, evaporated materials likely have much greater loss tangent GaAs [92] 12.9 0.0003 Requires special equiptm ent (MOCVD) for deposition, primarily used as a substrate Loss tangent likely varies greatly between un-doped, semi-insulating (Cr doped) and doped (n or p-type) samples Various characteristics of ferromagnetic devices depend on the conductivity of the ferromagnet, a, its saturation magnetization field, poMs, its gyromagnetic ratio, y, its Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 161 anisotropy field, BA, and its damping constant, T. These values for YIG and for three ferromagnetic metals appear in Table F— 2. T able F-2 Y IG and F errom agnetic M etals Notes a (M S/m ) |toMs (T) Y (GHz/T) (mT) r (unitless) N/A 0.176 282 0 .000152 Ni 141 0 .6 11 29.24 0 0.063 Difficult to evaporate Py 6.3* 1.081 29.2 0.57 0.0076 Permalloy 78% Ni 22% Fe Difficult to evaporate Fe 101 2.151 29,24 55s 0.0076 Can be epitaxial grown on Ag (001) films YIG ba Dielectric ferrite References: 1. Bozorth [93] 2. Ishak [94] 3. Bloembergen [95] 4. Heinrich [96] 5. Heinrich [78] 6. M oosmuller [97] 7. Szymanski [98] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 162 APPENDIX G PHOTO MASKS AND TRL CALIBRATION SETS Fabrication of coplanar waveguide devices requires only a single photolithography step (see Chapter 3). The dimensions of the devices depend on the substrate permittivity, as is discussed in Appendix E. Fig. G -l and Fig. G-2 show the photo masks for CPW on GaAs and SiCb substrates, respectively. The only difference between the two sets is the signal line width, which compensates for the difference in substrate permittivity. Each mask contains patterns for three different TRL calibration sets, including several redundant devices. The three sets correspond to the “wide” CPW in straight lines, the “wide” CPW in S-shaped lines and the “thin” CPW in S-shaped lines. Fabrication of microstrip devices is more complicated, requiring a shadow mask for defining the general device and a photo mask for narrowing the signal line width (see Chapter 3). Fig. G-3 shows the shadow mask— closed areas correspond to holes in the mask where deposition occurs, while the area outside the closed shapes blocks deposition. The width of the closed areas is 100 Jim, which is small enough to allow 150 |im -pitch probes to straddle the line. This width would require a very thick dielectric for 50 Q impedance, however. Thus, we use photolithography and etching to narrow the signal line. Fig. G-4 shows this photo mask. This mask contains three TRL calibration Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 163 sets corresponding to three different signal line widths: 12, 18 and 26 |im . Each set appears four times on the mask, allowing a fair amount of redundancy. One must align the photo m ask’s alignment marks (crosses) with the alignment pattern formed by the shadow mask (two by two grids of squares). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. G -l Photo Mask for CPW on GaAs Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. G-2 Photo Mask for CPW on SiC>2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ioo =3 ft' ft =3 ft =3 =3 ft oa ft ft [Vj C D = 3 ft ft =3 OO ft = 3 ft ft 00 CO ft “° =a e= ft =3 ft =3 ft ft CO CO =3 =3 ft ft =3 ft = 3 ft oo f t 00 Fig. G-3 Shadow Mask for Microstrip 1111111111 + + Fig. G-4 Photo Mask for Microstrip Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 167 APPENDIX H SURFACE IMPEDANCE SIMULATION This appendix contains the code listing for the author’s FORTRAN simulation “c p w _ Z s.f\ which models ferromagnetic effects in coplanar waveguide using the surface impedance technique in Chapter 4. Table H -l In p u t P aram eters V ariable Sym bol (units) Notes sigmaM ct (S/m) Ferromagnet conductivity ms epsd losstan fioMs (T) £r (unitless) tan 5 (unitless) BO Bo (T) DC T (unitless) fstart, fstop, fincrement a, b, t, h len f (GHz) a, b, t, h (pm) z (cm) Saturation magnetization field of ferromagnet Relative permittivity o f substrate Loss tangent of substrate Applied field strength Damping constant of ferromagnet Frequencies for sweep Start at fstart, stop at fstop Step size is fincrement Dimensions of metalization and substrate Length of line Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 168 The program receives inputs from “cpw_inputs.dat”, which contains values required for the simulation. The program variable names and their meaning are listed in order in Table H - l. The program writes data to three output files. T ab le H-2 O u tp u t File “ sk in o u t.d at” V ariab le Units Notes f GHz Frequency ZOi Q 8 = 0 impedance Li, Capi H/m, F/m 8 = 0 inductance and capacitance ki radians/m 8 = 0 phase constant One output file, “skinout.dat”, contains data for the zero-skin-depth case. It lists the impedance of the line, the series inductance and shunt capacitance and the propagation constant of the line, each as a function of frequency. The meanings of each data column are listed in Table H-2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 169 Table H-3 O u tp u t File “zpc.dat” V ariable U nits Notes f GHz Frequency L, R H/m, Q /m 6 > 0 inductance and resistance Cap, G F/m, S/m 8 > 0 capacitance and conductance dreal(ZO) Q 5 > 0 Re(impedance) dimag(ZO) Q 8 > 0 Im(impedance) dreal(pc) Np/cm 8 > 0 Re(propagation constant) dimag(pc) radians/cm 8 > 0 Im(propagation constant) skin pm Skin depth (magnetic) skinO pm Skin depth (non-magnetic) Table H-3 lists the data columns of the output file “zpc.dat” . It contains several values for the case of non-zero skin-depth as a function o f frequency. Table H-4 lists the data columns of “sparas.dat” . This output file contains the simulated S-matrix as a function of frequency. Each element of the matrix is given by its real and imaginary part and the format of this file matches the standard output format of an Agilent network analyzer, making it easy to compare w ith experimental data. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T able H-4 O u tp u t File “ sp aras.d at” V ariable Units N otes f GHz Frequency dreaI(S 11), dimag(S 11) unitless R e(Sn) and Im(Sn) dreal(S21), dimag(S21) unitless RefSii) and Im(S 2 i) dreal(S21), dimag(S21) unitless Re(S 2 i) and Im(S 2 i) dreal(S 11), dimag(S 11) unitless R e(Sn) and Im(Sn) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. c* cpw_Zs.f - C a l culates skin d e p t h as a function of fr e q u e n c y c* for a f e rromagnetic c o n ductor and uses su r f a c e c* impedance to c a l c u l a t e losses in CPW. ********************»*********,****************** C O M P L E X * 16 uno, ci, muvoigt, epsm, k. Be C0MPLEX*16 denom, ZO, pc, Sll, S21, Lcomp, mueff, Cc o m p R E A L * 8 pi, f, epsO, skin, s k i n O , muO REAL*8 w, fstart, fstop, fincrement R E A L * 8 BO, ms, gyromag, s i g m a M REAL*8 ZOi, Li, Capi, ki !ideal quantities (skin=0) REAL*8 R, L, Cap, G, dL, dg c**** dielectric a n d ge ometric p r o p e r t i e s RE A L * 8 epeff, epefft RE A L * 8 epsd, losstan, EK R E A L * 8 a, b, t, h, at, bt, kl, as, bs, INTEGER IER len c**** C a l culate some constants pi = 3 . 1415926536d0 pi, u s e d for angular f r e q u e n c y i m a g i n a r y number i ci = (O.OdO, l.OdO) like 1, only more co m p l e x uno = (l.OdO, O.OdO) epsO = 8.854d-12 p e r m i t t i v i t y of free space (F/m) muO = pi*4.0d-7 p e r m e a b i l i t y of free space (H/m) gyromag = 29.2d9 g y r o m a g n e t i c ratio (Hz/T) Input OPEN READ READ READ READ READ CLOSE values for this nun (U N I T = 3 , F I L E = 1c p w _ i n p u t s .d a t • , S T A T U S = 'O LD 1) (3,*) sigmaM, ms conductivity of conductor, Ms p e r m of dielectric, loss tangent (3,*) epsd, losstan (3,*) BO, DC ap p l i e d field, d a m p i n g constant (3, ) fstart, fstop, fincrement ! frequency list (3, ) a, b, t, h, len geometry of C P W (3) c**** Convert to microns a = a*1.0d-6 b = b * l .Od-6 t = t * l .Od-6 h = h * l .Od-6 c**** Convert to GHz fstart = 1.0d9*fstart fstop = 1.0d9*fstop fincrement = 1.0d9*fincrement c**** Open an output file and start it w i t h the input data OPEN (UNIT = 1 0 ,F I L E = 1s k i n o u t .d a t ',S T A T U S = 'U N K N O W N ') OPEN (UNIT = 1 1 ,F I L E = 'z p c .d a t ',S T A T U S ='UNKNOWN 1) OPEN (UNIT=12,F I L E = 1s p a r a s .d a t ',S T A T U S = 1U N K N O W N ' ) c**** Loop over w values DO f = fstart, fstop, fincrement c**** Calculate mu's and epsilon's w = 2.0d0*pi*f for this w Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i 72 Be = uno*BO - c i * D C * f / g y r o m a g muvoigt = (f**2-gyromag**2*(Bc+Ms) **2) & /(f**2-gyromag**2*Bc*(Bc+Ms)) epsm = epsO + c i * s i g m a M / w k = w*cdsqrt(epsm*muvoigt*muO) skin = 1 / d a b s ( d i m a g (k ) ) skinO = 1 / d a b s ( d i m a g ( w * c d s q r t ( e p s m * m u O ))) c ***» Caclulate values for skin = 0 at = a + (1.2 5 d 0 * t / p i )*(1.OdO + d l o g (4.O d O * p i * a / t )) bt = b - (1.2 5 d 0 * t / p i )*(1.OdO + d l o g (4.O d O * p i * b / t ) ) kl = d s i n h ( ( p i * a t ) / ( 4 . O d O * h ) )/ dsinh((pi*bt)/( 4 . O d O * h ) ) epeff = 1 . OdO + & ( e p s d - 1 .O d O ) * E K ( d s q r t (1.OdO-(a/b)* * 2 ) ) * E K ( k l ) / & (2.OdO * E K ( a / b ) * E K ( d s q r t (1.O d O - k l * *2))) epefft = epeff - (epef f - 1 ) /(( (b-a)/ (1.4d 0 * t ) ) Sc * (EK(a/b)/EK(dsqrt(1.0d0-(a/b)**2))) + l.OdO) ZOi = (30.O d O * p i / d s q r t (e p e f f t ) ) Sc * (EK(dsqrt (l.OdO-(at/bt) **2) ) / Sc EK(at/bt) ) Li = Z 0 i * m u 0 * d s q r t (e p e f f t ) /(120.0 d 0 * p i ) Capi = d s q r t ( e p e f f t ) * e p s 0 * 1 2 0 .0d0*pi/Z0i ki = w*dsqrt(epefft*eps0*mu0) c**** Adjust a, b and r e s u l t i n g values for skin > 0 IF (a.GT.skin) T H E N as = a - skin ELSE as = 1.0d-10 ENDIF bs = b + skin c**** Combinations of g e ome t r i c factors (skin > 0) at = as + (1.2 5 d 0 * t / p i )*(1.OdO + d l o g (4.0 d 0 * p i * a s / t ) ) bt = bs - (1.2 5 d 0 * t / p i )*(1.OdO + d l o g (4.0 d 0 * p i * b s / t ) ) kl = d s i n h ( ( p i * a t ) / ( 4 . 0 d 0 * h ) )/ dsinh((pi*bt)/( 4 . 0 d 0 * h ) ) epeff = 1 . OdO + (epsd-1.OdO) Sc *EK( dsqrt (1. OdO - (as/bs) * * 2 ) )*EK(kl) / Sc (2 .0d0*EK (a s / b s )*EK (dsqrt (1. OdO-kl* *2) ) ) epefft = epeff - ( e p e f f - 1 .Od O ) / ( ( (bs-as)/ ( 1 . 4 d 0*t)) & * ( E K ( a s / b s ) / E K ( d s q r t (1.OdO-(as/bs)**2))) + l.OdO) c**** Impedance of line (skin > 0) Z0 = u n o * (30.0 d 0 * p i / d s q r t ( e p e f f t ) ) Sc * (EK( dsqrt ( 1 .0d0- (at/bt) **2) ) /EK(at/bt) ) c**** Circuit parameters L = Z 0 * m u 0 * d s q r t ( e p e f f t ) /(120 .0* p i ) dL = (L-Li) dg = dL/muO mueff = muvoigt + Sc ci*dsqrt (dreal (muvoigt) **2 + d i mag (muvoigt) **2) Lcomp = ci*w*(L + dg*mueff*mu0) Ccomp = c i * w * ( C a p i *uno + ci*Capi*losstan) L = dreal(Lcomp/(ci*w)) R = d a b s ( d i m a g ( L c o m p / c i )) Cap = d r e a l ( C c o m p / ( c i * w ) ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 173 G = d a b s ( d i m a g ( C c o m p / c i )) c**** N e w impedance and propagation c o n s t a n t in cm"'-l ZO = c d s q r t ((Lcomp)/ (Ccomp)) pc = c d s q r t ((Lcomp)* (Ccomp)) pc = (dreal(pc) + d a b s ( d i m a g ( p c ) )* c i )/ 1 0 0 .OdO c**** Calc S-parameters Sll = (Z0-(50.OdO,0 . OdO) ) / (Z 0 + ( 5 0 . O d O , O . O d O ) ) S21 = d s q r t ((1.O d O - d r e a l (S l l )** 2 - d i m a g (Sll)**2)) Sc *cdexp (-pc*len) c**** Out p u t files w r i t e (10,99) f/ld9, ZOi, Li, Capi, ki w r i t e (11,98) f/ld9, L, R, Cap, G, dreal(ZO), dimag(ZO), & dreal(pc) , dimag(pc), skin*1.0d6, Sc s k i n 0 * l .0d6 w r i t e (12,97) f/ld9, dreal(Sll), dimag(Sll), dreal(S21), Sc dimag(S21), dreal(S21), dimag(S21), dreal(Sll), Sc dimag(Sll) E N D DO c*** * Clo s e files and format files C L O S E (10) C L O S E (11) C L O S E (12) 97 f o r m a t (9(l p l e l 2 .4)) 98 f o r m a t (11(l p l e l 2 .4)) 99 f o r m a t (5(l p l e l 2 .4)) END c*** C S U B R O U T I N E S A N D FUNCTIONS F U N C T I O N EK(kin) c**** Finds K (k) (C.E.I. of the 1st Kind) c**** Set a c c u r a c y w i t h while loop R E A L * 8 EK, kin R E A L * 8 an, bn, cn, anl, bnl, cnl anl = 1 . OdO bnl = d s q r t (1.OdO-kin* *2 ) cnl = kin an = (anl + b n l ) / 2 . OdO b n = d s q r t (a n l * b n l ) cn = (anl - b n l ) / 2 . OdO DO W H I L E ( c n .GT.l.0d-9) anl = an bnl = bn cnl = cn a n = (anl + b n l ) / 2.OdO b n = dsqrt(anl*bnl) cn = (anl - b n l ) / 2.OdO END DO Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 174 E K = 3 . 1 4 1 5 9 26536d0/(an*2 .OdO) retu r n END Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 175 REFERENCES [1] K. Q. W. Bumman, “MEMS technology moves increasingly toward microwave applications,” Microw. RF, vol. 40, pp. 97-104, 2001. [2] L. P. B. Katehi and G. Rebeiz, “Novel RF circuits using MEMS devices and silicon micromachining,” in Proc. IEEE-APSIS, vol. 3, p. 1246, 2000. [3] Y. Liu, A. Borgioli, A. S. Nagra and R. A. York, “Distributed MEMS transmission lines for tunable filter applications,” Int. J. RF Microw. Comput. Aided Eng., vol. 11, pp. 254-260, 2001. [4] K. Y. Park, J. Y. Park, H. K. Choi, J. C. Lee, B. Lee, J. H. Kim, N. Y. Kim, J. Y. Park, G. H. Kim, D. W. 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