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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
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UMI Number: 3042223
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Copyright By Nicholas Kipplan Cramer 2002
I Rights Reserved
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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
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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
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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.
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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:
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•
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
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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.
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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.
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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
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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.
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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],
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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,
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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
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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
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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
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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
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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
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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.
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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.
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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
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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
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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:
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(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:
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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 )
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(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 )
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(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
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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
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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
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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.
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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)
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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-
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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.
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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:
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|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
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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
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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.
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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.
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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
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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
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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.
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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.
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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.
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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.
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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
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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
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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:
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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
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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
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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.
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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
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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
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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.
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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
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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” .
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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.
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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
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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.
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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
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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:
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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.
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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
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“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
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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.
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Signal
Source
atfn
Frequency
Multiplier
^0
• • •
Band-pass
Filter at nfn
nfg
nfg
Fig. 3-17 Bandpass Application
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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
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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.
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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
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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
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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
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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
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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:
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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.
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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
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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.
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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
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(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.
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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
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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:
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(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
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(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-
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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:
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(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.
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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
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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.
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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
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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
,
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(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:
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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.
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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
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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
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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.
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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
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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.
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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+
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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
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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
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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
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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:
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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.
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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
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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
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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
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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)
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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
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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
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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
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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.
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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.
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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
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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.
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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.
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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
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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.
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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.
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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
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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.
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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
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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
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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.
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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.
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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.
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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
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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,
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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
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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
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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.
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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.
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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.
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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.
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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.
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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.
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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
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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
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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.
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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
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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.
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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.
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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.
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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
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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
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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
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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
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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)
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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:
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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
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(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).
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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
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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
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(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)
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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.
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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).
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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.
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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
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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
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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.
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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
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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
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(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.
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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
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