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Slow-wave electrode structures for III-V semiconductor based electro-optic travelling-wave modulators Lee, Zachary Ka Fai 1992

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SLOW-WAVE ELECTRODE STRUCTURES FOR^SEMICONDUCTORBASED ELECTRO-OPTIC TRAVELLING-WAVE MODULATORSbyZACHARY KA FAI LEEB. Sc. (Honours), University of British Columbia, 1989A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIES(The Department of Electrical Engineering)We accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAApril 1992©Zachary Ka Fai Lee, 1992In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department of eiec  The University of British ColumbiaVancouver, CanadaDate  A-CY1 1^19 ci DE-6 (2/88)AbstractElectro-optic modulators are important devices in high-speed optical communications andsignal processing. These devices are particularly useful if they are fabricated on semiconductormaterials since they may be monolithically combined with other electronic and/or optoelectronicdevices. Conventional travelling-wave electro-optic modulators fabricated using III-Vsemiconductor materials such as gallium arsenide and indium phosphide suffer from the problemscreated by the velocity-mismatch between the microwave modulating signal and the optical signal(or equivalently, the mismatch between the effective refractive indices of the microwave and theoptical wave). These problems ultimately limit the modulator bandwidth and require largeamounts of modulating microwave power to obtain useful modulation depths. It was ourobjective to design coplanar slow-wave electrode structures allowing the fabrication of wide-bandmodulators with reasonable demands of modulating power levels. This objective was achieved.In this thesis slow-wave coplanar electrode structures for use in these modulators aredescribed. They are periodically loaded with narrow capacitive fins. Small pads may also beadded on the ends of the fins to further increase the capacitance. If the fins are narrow, thecapacitance per unit length can be increased substantially due to the fringing electric fields aboutthe fins (and pads), while the effect on the inductance per unit length can be relatively small.This increase in capacitance per unit length results in the slowing of the microwave. Bycarefully choosing the dimensions, slow-wave electrodes having a prescribed microwave effectiverefractive index as well as 50 or 75 ohm characteristic impedances can be designed. A dielectriciisuperstrate may also be used to help slow down the microwave somewhat, potentially enablingfine tuning, as well as to protect the electrodes. The electrodes may also be partially buried inthe substrate to achieve some additional slowing of the microwave.Design formulas have been derived. Design curves for 50 and 75 ohm slow-waveelectrode structures fabricated on gallium arsenide and indium phosphide based materials arepresented.In order to verify the theory, a large number of slow-wave electrodes as well as aconventional coplanar strip electrode have been fabricated on semi-insulating gallium arsenide.The fabrication was straight forward, with the whole electrode being formed in a single layer ofmetallization using a single-step photo-resist patterning and lift-off technique.Measurements of the microwave effective refractive indices of the electrodes fabricatedclearly indicate that these electrodes are capable of velocity-match in modulators fabricated usinggallium arsenide based materials. The measured values range from 2.84 to 3.46. Goodagreement was found between the measured results and the theoretical predictions. Theelectrodes were also found to have very low microwave losses of a few tenths of a decibel percentimetre, and very low dispersion at frequencies at least up to the 20 GHz limit of themicrowave source.Slow-wave electrode structures, capable of matching the velocities of microwaves to thoseof optical waves in III-V semiconductor travelling-wave electro-optic modulators, that offer lowloss, low dispersion, flexible dimensions, and ease of fabrication have been designed, fabricated,and tested.iiiTable of ContentsAbstract ^  iiTable of Contents ^  ivList of Figures  viList of Tables ^  ixAcknowledgements  xChapter 1. Introduction  ^1^1.1.^Importance of Slow-Wave Electrode Structures and Electro-opticModulators ^  11.2.^Organization of the Thesis  ^4Chapter 2. Background  62.1.^Introduction  ^62.2.^The Linear Electro-optic Effect ^  92.3.^Velocity Matching, Phase Retardation, and Bandwidth  ^122.4.^Current Research in Electro-optic Modulators  ^152.5.^Coplanar Slow-Wave Electrodes in Mach-Zehnder Modulators  ^18Chapter 3. Theory  ^223.1.^Introduction  ^223.2.^The Slow-Wave Electrode Structures  ^233.3.^Physics of the Slow-Wave Electrode Structures  ^283.3.1. Introduction  ^283.3.2. The Transfer Matrix Method  ^303.3.3. Dispersion Characteristics and Group Velocity  ^353.3.4. Low Frequency Approximation and Design Formulas  ^393.3.5. Computation of Capacitance  ^443.3.6. Fringing Fin Capacitance  ^523.3.7. Calculated Results and Design Curves  ^543.3.8. Attenuation Constant and Microwave Loss  ^613.3.9. Scaling of Dimensions ^  633.3.10. Input Sections  ^643.3.11. The Effects of High Dielectric Constant Superstratesand Partially Buried Electrodes  ^643.4.^Conclusion  ^65ivChapter Fabrication  ^67Introduction  ^67Mask Design ^  68Fabrication Procedure  ^744.3.1. Cleaving GaAs Wafers  ^764.3.2. Cleaning GaAs Substrates  ^764.3.3. Photo-resist Patterning  ^774.3.4. Etching GaAs Substrates for Burying Electrodes  ^804.3.5. Evaporation of Aluminum  ^834.3.6. Removal of Excess Metal (Lift-off)  ^844.3.7. Summary of Fabrication Procedure  ^85The Completed Electrodes ^  86Device Testing and Measured Results  ^93Introduction  ^93The Measurement Equipment ^  94The Interference Technique  ^95The Resonance Technique  100Measured Results ^  105Discussion of Results  111Conclusion and Recommendations for Future Research ^ 115Conclusions ^  115Recommendations for Future Research ^  117^  1194.4.Chapter I. Conventional Coplanar Strips ^  127Appendix II. Coefficients of Inverse Capacitance  131Appendix III. Conditions for Resonance ^  135vList of FiguresFigure 2.1.1.^Typical lumped element electrode electro-optic modulator. ^ 7Figure 2.1.2.^Typical travelling-wave electro-optic modulator.  ^8Figure 2.2.1. Principal axes of the perturbed index ellipsoid for 43m crystals ^12Figure 2.4.1. A p-i-n travelling-wave phase/polarization modulator.  ^16Figure 2.5.1. A Mach-Zehnder modulator employing a coplanar strip slow-waveelectrode ^20Figure 2.5.2 . Cross section of a Mach-Zehnder modulator employing a coplanar stripslow-wave electrode and graded index optical waveguides.  ^21Figure 3.2.1 . Plan view of a section of a slow-wave electrode with fins only. .. .^26Figure 3.2.2 . Plan view of a section of a slow-wave electrode with fins and pads. .^27Figure 3.3.2 .1. Model for transfer matrix analysis.  ^34Figure 3.3.3. 1. 6) versus B curves for the lowest two pass bands of a typical slow-waveelectrode ^36Figure 3.3.3 .2. Phase and group velocities as a function of frequency of a typicalslow-wave electrode.  ^37Figure 3.3.3. 3. Characteristic impedance as a function of frequency of a typicalslow-wave electrode.  ^38Figure 3.3.5. 1. Calculated capacitance of fins in air having S2 = 1 Am, t = 0.5 pm,1 = 1, 2, 3, and 4 Am, and W2 ranging from 3 to 10 Am.  ^50Figure 3.3.5. 2. Assumption made in calculating the capacitance of fins with pads. ..^51Figure 3.3.6. 1. Fin capacitance to length ratio for fins in air having S2 = 1 pm,= 0.5 Am, / = 1, 2, 3, and 4 Am, and W2 ranging from 3 to10 pm.  ^53viFigure curves for 50 and 75 ohm velocity-matched slow-waveelectrodes on GaAs substrates for e s = 1, S2 = 1 Am,t = 0.5 Am, and / = 1 Am  ^57Design curves for 50 and 75 ohm velocity-matched slow-waveelectrodes on GaAs substrates for e s = 3.5, S2 = 1 AM,t = 0.5 Am, and / = 1 Am. ^  58Design curves for 50 and 75 ohm velocity-matched slow-waveelectrodes on InP substrates for es = 1, S2 = 1 AM,t= 0.5 Am, and / = 1 Am.  ^59Design curves for 50 and 75 ohm velocity-matched slow-waveelectrodes on InP substrates for e s = 3.5, S2 = 1 AM,t = 0.5 Am, and / = 1 Am.  ^60Figure Input section for slow-wave electrodes showing the variousdimensions.  ^66Figure 4.2.1. Layout of Mask #1.  ^71Figure 4.2.2. Layout of Mask #2.  ^72Figure 4.2.3. Dimensions of the two input sections used in Mask #2.  ^73Figure 4.3.1. Main steps of the fabrication procedure.  ^75Figure 4.4.1. SEM micrograph of electrode #2 of Mask #1 (half-buried).  ^88Figure 4.4.2. SEM micrograph of electrode #2 of Mask #1 (totally-buried).Figure 4.4.3. SEM micrograph of electrode #1 of Mask #2 (half-buried).  Figure 4.4.4. SEM micrograph of electrode #5 of Mask #2 (surface deposited).Figure 4.4.5. SEM micrograph of electrode #7 of Mask #2 (surface deposited).Figure 5.2.1. Picture showing the network analyzer, synthesized sweeper, anddirectional bridge.  ^96Figure 5.2.2. Picture showing the probe jig with probe mounted (left) and thesample platform (right).  ^96Figure 5.2.3. Picture showing the whole setup of the equipment.  ^97vii ^ 8990. 91. 92Figure 5.3.1. Schematic of the setup used for the interference technique.  ^98Figure 5.3.2a. Measured normalized reflected power in dB with the probe nottouching the conventional coplanar strip electrode of Mask #1. .^. 102Figure 5.3.2b. Measured normalized reflected power in dB with the conventionalcoplanar strip electrode of Mask #1 probed. ^ 103Figure 5.4.1. Schematic of the setup used for the resonance technique. ^ 104Figure 5.5.1. Measured normalized reflected power as a function of frequency ofelectrode #1 (half-buried) of Mask #2. ^  108Figure 5.5.2. Measured normalized reflected power as a function of frequency ofelectrode #2 (half-buried) of Mask #2.  109Figure 5.5.3. Measured normalized reflected power as a function of frequency ofelectrode #5 (unburied) of Mask #2. ^  110Figure I.1.^A coplanar strip electrode on a substrate.  129Figure II.1.^Dimensions for cells oriented parallel and perpendicular to oneanother^  134Figure III.1.^Equivalent circuit of an electrode having a characteristic impedanceZo , terminated by a load ZT and connected to a source Z5 withvoltage V. ^  137viiiList of TablesCapacitance of fins in air.  ^48Capacitance of fins with pads in air ^49Calculated Nd s and Z' 's of some slow-wave electrodesfabricated and tested ^56Mask #1 electrode dimensions.  ^7070Table 4.2.1.Table 4.2.2.^Mask #2 slow-wave electrode dimensions.Table 4.3.4. 1. Etching characteristics of various etchants used in thefabrication ^83Table 5.5.1.^Measured and theoretical Neis of the conventional coplanar stripelectrode^  106Table 5.5.2.^Measured and theoretical Nd s of the slow-wave electrodes(Mask #1).  106Table 5.5.3.^Measured and theoretical Neff's of the slow-wave electrodes(Mask #2). ^  107ixAcknowledgementsI would like to thank the members of my family for their support, encouragement, andfaith throughout my education.I would like to express my gratitude to my supervisor, Dr. N. Jaeger, for suggesting thisproject and providing continual guidance and support during my research.My thanks extend to our research engineer, H. Kato, for invaluable assistance during thefabrication of the devices. I am also grateful to all members of the solid state group and thoseindividuals in the Electrical Engineering department who have helped me in various waysthroughout my research.Finally, I would like to acknowledge the support of the Canadian Cable Labs Fund andthe Natural Sciences and Engineering Research Council (NSERC) of Canada.xChapter 1 Introduction1.1. Importance of Slow-Wave Electrode Structures inElectro-optic ModulatorsOne of the fundamental problems associated with conventional integrated electro-optictravelling-wave modulators is the phase and group velocity mismatch between the optical waveand the modulating microwave signals, which limits the achievable bandwidth as well asmodulation depth for a certain amount of modulating power [1,2:pp. 160-163]. In particular,in modulators using III-V semiconductor materials such as gallium arsenide (GaAs) and indiumphosphide (InP), the microwave signals travel faster than the optical waves if conventionalcoplanar strip electrodes are used. These problems, however, can be solved by using slow-waveelectrodes in place of the conventional coplanar strip or coplanar waveguide electrodes [3,4,5,6].The capacitively-loaded slow-wave electrode structures introduced in this work are designed foruse in integrated electro-optic modulators based on III-V semiconductor materials. They offerlow loss, low dispersion, flexible dimensions, ease of fabrication, and most important of all, theyallow one to engineer the velocity of the microwave signal so as to match it to that of the opticalwave, thus achieving the velocity-matched condition.The huge bandwidth, high information-carrying capacity, immunity to electricalinterference, and security offered by contemporary optical fibres make it very attractive todevelop broadband optical modulators, with low power consumption, for use in high-speed1optical communications and signal processing [2:pp. 1-3].There are generally two common ways to impress information onto the optical wave.These include the direct modulation of semiconductor laser diodes and the use of externalmodulators. Direct modulation of semiconductor laser diodes offers simplicity and largeamplitude modulation [7]. Bandwidths in the range of 20 GHz have been obtained in some high-speed Fabry-Perot lasers [8,9]; however, these lasers produce large linewidths [9]. Distributedfeedback lasers offer the advantages of single mode operation and narrow linewidths [7];however, frequency chirping under direct modulation still broadens the linewidth [10,11],resulting in bandwidths of about 13 GHz [12]. Quantum well lasers are excellent sources forproducing narrow linewidths [13,14,15]; however, the modulation frequency depends on manylaser parameters including differential gain, gain saturation, and the highest possible frequencyis limited by the finite trapping time of the carriers [16]. The injection-locking technique[17,18,19] results in significant reduction in linewidth and frequency chirping [20]; however, theuse of two semiconductor lasers increases the complexity of the system. The use of externalmodulators, on the other hand, offers potentially higher performance and is not necessarily morecomplicated than direct modulation of semiconductor laser diodes.Two major kinds of external optical modulators are known. These are the electro-absorptive devices [21] and the electro-optic devices [22]. Electro-optic modulators offer theadvantages of simplicity and ease of fabrication. Electro-optic Mach-Zehnder modulators arealso immune to frequency chirping [23].There are generally two categories of material used to implement these modulators --lithium niobate (LiNbO3), which is a ferroelectric insulating crystal, and GaAs and InP, which2are semiconductors. Although LiNbO 3 generally has superior optical qualities than thesemiconductors (i.e., lower optical loss, higher electro-optic coefficients...etc.) [1], thesemiconductors allow one to monolithically integrate electro-optic, electronic, and opto-electronicdevices on a single substrate. For example, one can fabricate a laser source, an electro-opticmodulator, as well as other control circuitry on the same GaAs or InP substrate. Moreover, theuse of GaAs and InP allows one to take advantage of their low (microwave) dielectric constants.For instance, as is shown in this thesis, one is able to fabricate slow-wave electrodes on GaAsso that the velocity of a microwave can be matched to that of an optical wave, which is one ofthe most important criteria for achieving wide bandwidths, low modulating powers, and highmodulation depths [1,2:pp. 160-163]. However, this is not easily done for LiNbO 3 since thehigh dielectric constant of LiNbO 3 makes the microwave travel slower than the optical wave,although some recent work involving the use of a last-wave electrode' employing a metal shieldoverlying the substrate has been published [24].Motivated by the many advantages of external compound semiconductor electro-opticmodulators, as well as the very high performance possible for these devices when the microwavevelocity is matched to that of the optical wave, a slow-wave electrode structure, being fabricatedon compound semiconductor materials, that enables one to engineer the velocity of a microwaveso as to match it to that of an optical wave, is developed. This thesis provides a detailed accountof the theory, design, fabrication, and measurement of such a novel slow-wave electrodestructure.31.2. Organization of the ThesisIn Chapter 2, the electro-optic effect and other basic principles are reviewed. The basicoperation of a travelling-wave electro-optic modulator is described. The current status ofresearch in this area is provided. The application of slow-wave electrodes to Mach-Zehndermodulators is finally given.Chapter 3 gives a detailed treatment of the theory behind the slow-wave electrodestructures introduced in this thesis. Analysis using a quasi-static approximation is carried out.The basic principles of capacitive loading, the high frequency filter characteristics and dispersion,as well as structures useful in travelling-wave modulators are described. The computation of thecapacitance of the loading fins using finite difference and finite element methods are alsopresented. Improved and simplified design formulas for slow-wave electrodes are derived.Design curves for possible electrode structures for both GaAs and InP modulators are presented.The microwave losses of the slow-wave electrodes are also calculated.Chapter 4 is dedicated to describing the fabrication of the slow-wave electrodes. The basicsteps in the fabrication are given. Various anisotropic etchants are reviewed and thecorresponding fabrication results presented.Chapter 5 deals with the measurements based on the Hewlett Packard HP8757C scalarnetwork analyzer, the HP8341B synthesized frequency source, and the HP85027E directionalbridge. The measurement techniques and the measured results are presented.Chapter 6 gives a conclusion as well as recommendations for the future continuation ofthis research.4Appendix I summarizes the design formulas for conventional coplanar strip electrodes thatare useful in the design of the slow-wave electrodes presented in this thesis.Appendix II summarizes the equations useful for implementing a finite element programfor calculating the capacitance of the loading fins.Appendix III provides a derivation of the reflection coefficient for the electrodes. As willbe shown here, the rapid decrease of the reflection coefficient at resonance allows one tocalculate the microwave effective refractive indices of the electrodes by measuring the frequencyseparations between the resonances.5Chapter 2 Background2.1. IntroductionThe linear electro-optic effect is the basis for most electro-optic devices includingmodulators. When a voltage is applied to an electrode placed over or alongside an opticalwaveguide in a medium having a non-zero electro-optic coefficient, such as most compoundsemiconductors including gallium arsenide (GaAs) and indium phosphide (InP), the electric fieldcreated in the medium causes a change in the refractive index. This index change may result inphase and/or polarization modulations. When an interferometer such as a Mach-Zehnderstructure is used, intensity modulation results [1,2:pp. 174-184].There are generally two major categories of electrodes used in these modulators--lumpedelement electrodes and travelling-wave electrodes [2:pp. 153-163].The lumped element electrode modulators (Figure 2.1.1) have an electrode length of lessthan a quarter of an electromagnetic wavelength [2:pp. 159-160]. The induced refractive indexchange of the medium is substantially constant throughout the whole length of the electrode. Themodulation bandwidth of this type of modulator is the smaller of the inverse of the optical transittime, or the resistance-capacitance (RC) time constant of the lumped-circuit parameters. Thelatter, however, is usually more restrictive. In order to achieve wide bandwidth the capacitanceand therefore the electrode length should be as short as possible. A short electrode, however,results in high power being needed for a particular modulation depth [1,2:pp. 159-160]. This6Figure 2.1.1. Typical lumped element electro-optic one of the greatest disadvantages of the lumped element electrode design. Modulators of thistype having a bandwidth of 4.5 GHz and a half-wave voltage of 13 V have been reported [25].The travelling-wave electrode (Figure 2.1.2) is an extension of the transmission line whichcarries the modulating signal. In modulators employing this type of electrode, the optical andmodulating microwave signals travel alongside each other, but generally at a different velocity.The phase of the optical signal is being modulated as it propagates along the waveguide. The7Figure 2.1.2. Typical travelling-wave electro-optic modulator.modulating bandwidth in this type of modulator is not limited by the RC time constant of lumpedelement electrodes but rather by the velocity mismatch between the optical and microwave signals[1,2:pp. 160-163]. For broad-band operation, the velocity of the microwave should be equal tothat of the optical wave. It is our objective to develop slow-wave electrode structures, beingfabricated on compound semiconductor materials, to match these velocities. These slow-waveelectrode structures are described in this thesis.8This chapter presents a brief review of the electro-optic effect associated with III-Vsemiconductors. The bandwidth and modulating voltage of travelling-wave modulators are given.Finally the current status of research in this area is provided.2.2. The Linear Electro-optic EffectThe linear electro-optic effect associated with cubic crystals having 43m crystal symmetryin crystallographic system is described mathematically by the electro-optic matrix rii as [26:pp.227-229]Ti] =0 0 00 0 00 0 0r41 0 00 r41 00 0 r41.At a free-space optical wavelength of 1 Am, r41 = - 1.4x10-1° cm/V for both GaAs and InP [2:p.364]. When an electric field having components Ex , Ey , and Ez is applied to the crystal, theindex ellipsoid is perturbed and becomes912nor41Ez r41Eyr41Ez12nor41Ex(x y z) r41Ey r41Ex12nowhere no is the refractive index of the optical wave. At a free-space wavelength of 1 Am, no =3.43 for GaAs and no = 3.3 for InP [2:p. 364,27,28]. It is apparent that the principal axes ofthe index ellipsoid are in general not parallel to the x, y, and z directions with an electric fieldapplied. By diagonalizing the matrix one can obtain the perturbed refractive indices and theprincipal axes corresponding to electric fields applied along different directions. The three mostcommon orientations of the applied electric field are the [100], [011], and [111] directions. Theperturbed refractive indices and the principal axes with electric fields along these directions aresummarized as follows (see also [2:pp. 362-364]):(I) electric field oriented along a [100] direction (i.e., E = Ez):n[011] = no - Ann10177 = no + Ann[100] = no(II) electric field oriented along a [011] direction (i.e., Ex = Ey = E/i2)nbr711] = no + Annb12111 = no - Ann[01T] = no10(III) electric field oriented along a [111] direction (i.e., E x = Ey = Ez = E/13)nfir01 = no + An/i3n[1127 = no + An/^3nil 1 1] = no -where1A n= —2n°3r41E .Since the [100] direction is the preferred direction of epitaxial growth [2:p. 364], the orientationsof electric fields in cases (I) and (II) are most commonly used. It can be seen that for electricfields applied along a [100] direction, with an optical wave propagating along a [O1T] direction,only TE-polarized optical fields will be modulated while TM-polarized fields remain unperturbed.This is because TM-polarized fields are parallel to the unperturbed principal axis (i.e., they areoriented along a [100] direction). For electric fields applied along a [011] direction, with anoptical wave propagating along a [O1T] direction, both TE- and TM-polarized fields will bemodulated, since the two principal axes that lie in the plane perpendicular to the direction ofpropagation are oriented 45° with respect to the TE and TM polarizations. Linearly polarizedinputs will result in polarization rotations. The relative directions of the principal axes in thesetwo cases are shown in Figure oA^no-Ann ono - Am+ A m ^7 no +Am]^ I- // [ Z 1 1 ]Figure 2.2.1. Principal axes of the perturbed index ellipsoid for 43m crystals.2.3. Velocity Matching, Phase Retardation, and BandwidthThe phase retardation Al)(1E) of a TE-polarized optical wave, with the microwave electricfield oriented along the [100] direction, having travelled an interaction length L in a waveguide,12is given by [1,2:pp. 160-163] assin(rriyo)Ad:0(TE) a AC.^ (2.3.1)(t.flf0)where32.1c An^nor4i von4:0 „ -^L-1S1(2.3.2)fo_  (cILNeff)1 —n°/Neff(2.3.3)f the frequency of the modulating electric field (usually a microwave), Vo the voltage amplitudeof the microwave electric field, A the wavelength of the optical wave, S1 the gap between theelectrodes (see Figure 2.1.2), Neff the effective refractive index of the modulating wave, and ris the overlap integral between the normalized optical field Eop, and the applied electric field E,as defined by [2:p. 155]sl—ffElEoptdAvo(2.3.4)where the integration is over the cross-sectional area of the optical mode. The value of r forcoplanar strip electrodes is typically about 0.7 [22]. For TM-polarized optical wave, the phaseretardation is zero since the refractive index is not perturbed by electric fields applied along13[100].For a microwave electric field oriented along [011] the phase retardations of both TE- andTM-polarized optical fields are twice that for the TE-polarized optical field when the microwaveelectric field is oriented along [100]. The [011] orientation results in a higher inducedbirefringence and a higher modulation since both TE- and TM-polarized optical fields aremodulated [22].It can be seen from equations (2.3.1) to (2.3.3) that when the velocities are matched (i.e.,Neff = n„) the total phase retardation of the optical wave is maximized. The optical wave thensees the same voltage over the entire electrode length. When this happens, the total phaseretardation is proportional to the product VoL and arbitrarily long electrodes can be used so asto reduce the drive voltage and no frequency limitation results if microwave losses can beignored. On the other hand, if the velocities are not matched, there will be cancellations in themodulation as the optical wave travels a certain distance down the waveguide. Consequently thetotal phase retardation will be smaller for the same value of VoL. In fact, for a fixed L, therewill be no modulation at all if the modulation frequency f is an integral multiple of fo .Moreover, if the modulation frequency is sufficiently high, the total phase retardation willapproach zero. The 3dB bandwidth [1,2:pp. 160-163] of the modulator can be obtained fromequation (2.3.1) and (2.3.3) by noting that Ad) is reduced to 50% of its value at f=0 when2c/Neff1 —no/Neff(2.3.5)For conventional coplanar strip electrodes on GaAs, Neff. = 2.64, no = 3.43, and zifi, = 242 For velocity-matched electrodes, like the ones described in this thesis, 24 istheoretically infinite provided microwave losses may be ignored. At very high frequenciesmicrowave losses become the dominating factors that limit the bandwidth.2.4. Current Research in Electro-optic ModulatorsTwo major electrode configurations for GaAs travelling-wave modulators have beenreported [22]. These are the microstrip configurations using a p-i-n structure [29] (Figure 2.4.1)and the coplanar strip configurations using undoped epitaxial layers grown on semi-insulatingsubstrates [30]. The slow-wave electrode structures described in this thesis are of the coplanarstrip configuration. Shown in Figure 2.5.2 is the cross section of a Mach-Zehnder modulator(to be discussed in the next section) employing these slow-wave electrodes and graded indexoptical waveguides.Although microstrip electrodes fabricated on GaAs can be designed to give a microwavephase velocity very close to that of the optical wave, the metal strips (signal lines) have to bevery close to the ground plane in order to have a high enough electric field for any usefulmodulation. Consequently, this requires a very thin, usually a few microns, optical guiding layer(usually semi-insulating GaAs) sandwiched between the strips and the ground plane.Unfortunately, growing GaAs on metal is not possible let alone handling such a thin layer.Therefore, a p-i-n structure with thick p and n + layers above and below the semi-insulatingoptical guiding layer is necessary [29]. However, the n + substrate causes very high microwavelosses and dispersions, which severely limit the performance of the modulators [22].15/p 1-.7-1 -t\sG aAsn A 1Xn +APo"^P\etoll izoLionFigure 2.4.1. A p-i-n travelling-wave phase/polarization modulator showing a single opticalwaveguide underneath the electrodes.On the other hand, coplanar strip electrodes fabricated on semi-insulating gallium arsenide havevery low loss and dispersion. However, the overlap between the optical and microwave fieldsis smaller (usually in the range of 70%) as compared to that in microstrip modulators (which canbe almost 100%) [22]. Moreover, the phase velocity of the microwave in coplanar stripelectrodes is higher because of the portion of the field travelling on the air side. Consequently,in conventional coplanar strip structures the velocity mismatch between the microwave and theoptical wave limits the achievable bandwidth at moderate power levels of the modulating signal16[1,2:pp. 160-163]. A technique for slowing down the microwave is therefore needed in orderto achieve high performance in coplanar strip modulators.Recently, a velocity-matched planar microstrip structure using an n --i-n configuration withrelatively low loss has been proposed and analyzed [31]. Nees et al. cemented a GaAssuperstrate directly on top of the coplanar strip electrodes to suppress both velocity mismatch andelectrical dispersion [32]. Lee et al. published a velocity-matched GaAs travelling-wave opticalmodulator using a modulated coplanar slow-wave electrode with periodic cross-tie overlays[33,34,35]. This type of slow-wave electrode, however, requires a multi-layer electrode-dielectric structure. Walker reported a GaAs Mach-Zehnder push-pull travelling-wave modulatoremploying a capacitively-loaded coplanar strip electrode with segmented modulator elements[36,37], in which the loading capacitance between the coplanar strips is formed via the modulatorelements and an n + epitaxial layer. Very high figures of merit have been reported for thesemodulators; however, they use isolation trenches, air bridges, as well as other biasing anddecoupling circuitries [36].The capacitively-loaded slow-wave electrode structures described in this thesis can be usedto directly replace the coplanar strip electrodes in the modulators [3,4,5,6]. These slow-waveelectrodes possess the advantages of low loss, low dispersion, flexible dimensions, ease offabrication, and most importantly, they allow one to flexibly engineer the velocity of themicrowave so as to achieve the velocity-match condition. Also, the full length of the electrodescan be used to modulate the optical wave. The electrodes do not require the use of anypotentially lossy doped epitaxial layers and they can be easily formed in a single layer ofmetallization using a standard lift-off technique [38:pp. 115-138]. Moreover, the electrodes will17not introduce any more stress to the substrate than any other coplanar strip or coplanarwaveguide electrodes. The microwave effective refractive indices and characteristic impedanceswill also be insensitive to the parameters of the substrates such as the thickness of the epitaxiallayers.2.5. Coplanar Slow-Wave Electrodes in Mach-ZehnderModulatorsThe coplanar slow-wave electrodes described in this thesis are designed for use in Mach-Zehnder modulators. A Mach-Zehnder modulator is an intensity modulator using two interferingarms [2]. There is one input and one output port for the optical wave. The optical waveguideconsists of a straight input section, a 3dB power splitting Y-junction, two straight phasemodulating sections, a power combining Y-junction, and an output section. Such a modulatoremploying the slow-wave electrodes described in this thesis is depicted in Figure 2.5.1. Figure2.5.2 shows the cross section of a Mach-Zehnder modulator employing these slow-waveelectrodes and graded index optical waveguides. Here, the microwave electric fields are in the[100] direction. The input wave is split into two equal components by the Y-branch powersplitter. Each of these components then propagates and is modulated over one arm of theinterferometer. Since the modulating electric fields are in opposite directions over these twoarms, the phase retardation is in opposite directions. Consequently the total phase differencebetween the two arms is twice that over one arm. The modulating voltage is therefore halved18for the same interaction length L. The optical waves in the two arms eventually recombine atthe output Y-junction. If the guided modes in the two arms are in phase as they recombine, thenthey interfere constructively and excite the lowest order mode of the output waveguide [2:pp.182-183]. The transmitted light is a maximum. If they are 180° out of phase as they recombine,then they excite the first antisymmetric mode which is not supported by the output section. Thismode is cut off and radiated into the substrate and the transmitted light is a minimum [2:pp. 182-183]. In this type of modulator, it is essential that the two interferometric arms be sufficientlyseparated so as to prohibit evanescent field coupling between them.Since the slow-wave electrodes satisfy the velocity-match condition, the bandwidth of theMach-Zehnder modulator will be theoretically infinite if microwave losses may be ignored.Arbitrarily long electrodes may therefore be used. Assuming r = 0.5 (this value is lower thanthat commonly found in modulators employing conventional coplanar strip electrodes due to theuse of loading fins and pads), S1 = 13 Am, and A = 1 Am, a half-wave voltage, i.e., voltageneeded for creating a phase difference of 7r between the two arms of the interferometer andcomplete turn off of the output, of 22 V for a 1 cm interaction length, or 22, will beneeded for modulating TE-polarized optical fields. This voltage, however, can be reduced byusing longer interaction lengths (i.e., longer electrodes). For instance, if 2 cm interactionlengths are used, the half-wave voltage will be reduced to half, or 11 V. Since arbitrarily longinteraction lengths can be used without sacrificing the bandwidths, these modulators have atheoretically infinite bandwidth to modulation voltage ratio (if losses may be ignored). It shouldbe noted that only modulators using velocity-matched slow-wave electrodes will havetheoretically infinite bandwidths that are independent of the interaction lengths. On the contrary,19the bandwidths of modulators employing conventional coplanar strip electrodes are inverselyproportional to the interaction lengths (see equation (2.3.5)); consequently, these modulators willeither have wide bandwidths at the expense of very high modulation powers, or narrowbandwidths at the gain of reduced modulation powers. These modulators typically have abandwidth to modulation voltage ratio of around 1 GHz/V [1]. Modulators employing velocity-matched slow-wave electrodes, such as the ones described in this thesis, will have the benefitsof both wide bandwidths and low modulation powers.PT I CA LWAVEGU I DESMICROWAVEELECTRODES^OC)INPUT Y -BRANCH PHASE MODULATOR Y -BRANCH OUTPUTWAVE- SECTIONS WAVE-GUIDE GUIDEFigure 2.5.1. A Mach-Zehnder modulator employing a coplanar slow-wave electrode. Note thecapacitive loading fins between the coplanar strips.20x.NA 1,<Go,_■'\sN/xGradedIndexRegionS . I . GaAsSubstrate WI S i WIEl ectrodesMeta 1 1 1 zat IonFigure 2.5.2. Cross section of a Mach-Zehnder modulator employing a coplanar slow-waveelectrode and graded index optical waveguides. The optical guiding layer isformed of AlGa i _xAs with a graded Al concentration. The capacitive loadingfins in the gap between the two coplanar strips are shown.21Chapter 3 Theory3.1. IntroductionAs already discussed in Chapter 2, the velocity mismatch between the optical wave andthe modulating microwave signal in a travelling-wave electro-optic modulator limits its bandwidthand the amount of microwave power needed for a certain modulation depth is relatively high[1,2:pp. 160-163]. In III-V semiconductor based modulators employing conventional coplanarstrip or coplanar waveguide electrodes, with the superstrate being air, the microwave signalstravel faster than the optical signals. Therefore, a slow-wave electrode, being fabricated on III-Vsemiconductors such as gallium arsenide (GaAs) and indium phosphide (InP) based materials,that enables one to engineer the velocity of the microwave that travels along it, is highlydesirable.Several slow-wave electrode structures have been reported. All of these structures operateon the principle that by increasing the effective capacitance per unit length the microwave isslowed. These include the metal-insulator-semiconductor (MIS) structures [39,40,41,42], theSchottky contact structures [41,42,43], and the coplanar waveguide structures using periodicallydoped semiconductor substrates [44]. These electrode structures, however, are inadequate foruse in high-speed electro-optic modulators due to the inherently lossy nature of the dopedsemiconductor materials. Recently a coplanar electrode structure with periodic cross-tie overlayshas been proposed and its use in a Mach-Zehnder modulator has been demonstrated [33,34,35].This structure, however, is a multi-layer structure requiring two layers of metalization and one22layer of dielectric, complicating its fabrication.The slow-wave electrode structures introduced in this work not only enable one to engineerthe velocities of the microwaves that travel along them, thus achieving the velocity matchcondition when they are used in III-V semiconductor electro-optic modulators, but they also offerthe advantages of low loss, low dispersion, and ease of fabrication [3,4,5,6]. The structures donot require the use of any potentially lossy doped semiconductor material. The microwavevelocity and characteristic impedance are also insensitive to the parameters of the substrate suchas the thickness of epitaxial layers.3.2. The Slow-Wave Electrode StructuresThe slow-wave electrode is a coplanar strip periodically loaded with capacitive elements.There are two generations of the electrode structure. In the first generation, narrow capacitivefins extending into the inter-electrode gap region are used as the capacitive loading elements [3].Due to the fringing electric fields about the fins, the amount by which the capacitance per unitlength increases is greater than the corresponding amount by which the inductance decreases,resulting in the slowing of the microwave. The second generation electrode structure is basicallythe same as the first except that rectangular pads are added to the ends of the fins to furtherincrease the loading capacitance [4]. The higher capacitance, as will be shown later in thischapter, allows one to increase the fin period as well as the width of the electrodes, thusincreasing the flexibility of the electrode dimensions required as well as reducing the resistiveloss and the number of loading fins and pads needed. The reduction in the number of fins and23pads makes possible a higher yield in the fabrication.The electrode structure of the first generation is shown in Figure 3.2.1. Here capacitiveloading fins of length 1, width W2, gap width S2, and period d are added to the coplanar stripsof width W1 separated by a gap S1 , and both the fins and the coplanar strips are of thickness t.In order to exploit the fringing electric fields produced by the fins, it is essential to keep the finsnarrow (i.e., small length 1 as compared to the fin width W2). This is because narrow fins havea higher capacitance to fin length ratio as compared to wide fins. The decrease in inductanceper unit length due to the narrow fins is also smaller. In other words, the narrower the fin, thecloser it is to a purely capacitive element.The electrode structure of the second generation is shown in Figure 3.2.2. Here pads ofwidth W' and length 1' are added to the ends of the fins. All other dimensions are defined in thesame way as those of the first generation. The function of the pads is to further increase thecapacitance between the signal and ground lines, thus resulting in a more effective capacitiveloading. If pads of proper dimensions are used in conjunction with narrow fins, as will bediscussed, it is believed that their effect on the inductance per unit length will be the same as ifthe pads were absent.Electrode structures of both generations may be partially buried or surface deposited.Burying the electrodes helps increase the capacitance per unit length somewhat so that a higherdegree of slowing may be achieved. A high dielectric constant superstrate such as a polyimidemay also be applied, as it also increases the capacitance per unit length of the electrodes.In this chapter, the analysis of the slow-wave electrode structures is given and designformulas are derived. Design curves, based on GaAs and InP substrates, are calculated for half-24buried electrodes with superstrates having dielectric constants of 1 (i.e., air) and 3.5 (e.g., apolyimide).25//r" A#A#11/^L^/^rllo^16^lo^lo^7 7 7//"77/AtAA///W21 lei 52^W2WIV  _I\^t--IFigure 3.2.1. Plan view of a section of a slow-wave electrode showing the dimensions S 1 , W1 ,52, W2 , d, and 1.d^d^d2627Nd d dVN dFigure 3.2.2. Plan view of a section of a slow-wave electrode with pads showing the dimensionsSD W1, S2, W2, d, 1, W', and l'.S 1V232\i/2W,W 1\/3.3. Physics of the Slow-Wave Electrode Structures3.3.1. IntroductionAccording to the quasi-static (quasi-TEM) approximation, the phase velocity of amicrowave propagating along thin conventional (unloaded) coplanar strip electrodes on aninfinitely thick substrate with an infinitely thick superstrate is constant regardless of the widthof the strips and the gap separating the strips [45:pp. 257-288]. This is because any change inthe dimensions of the electrodes resulting in a change of the shunt capacitance per unit lengthC always causes a corresponding change in the series inductance per unit length L such that thenet effect is that the phase velocity vp° = (LC) -112 remains constant. With minor modificationsof the expressions given in references [45:pp. 257-288,46:pp. 363-381], vp° is given in terms ofthe dielectric constants of the substrate E r, and superstrate e s asO CV =—P No( c is the speed of light in vacuum and No the effective refractive index given byer+esN0=  2,\1 =c LC . ( characteristic impedance Z, is given as287 L NoC = cc •( capacitance Cp in the form of narrow fins (and pads, if applicable), is added to the electrodeat periodic intervals, then, the inductance per unit length will only be minimally reduced and themicrowave will be slowed due to an increase in the effective capacitance per unit length. Theslow-wave electrodes introduced in this work are based on this principle.In the following sections, the slow-wave electrode structures are first analyzed using atransfer matrix method. Their low frequency and high frequency behaviours are discussed.Then design formulas are derived and design curves generated. The computation of the fincapacitance is also outlined.293.3.2. The Transfer Matrix MethodThe transfer matrix method is useful for analyzing periodic structures [46:pp. 363-381,47,48]. In simple terms, it relates the voltage and current at one section of a periodicstructure to the voltage and current at an adjacent section by a 2 X 2 matrix. Here, the slow-wave electrode structures are analyzed using such a method.In the analysis that follows, the fins (and pads where appropriate) are treated as purecapacitive elements; their effects on inductance are ignored. Then in section 3.3.5, their effectson inductance are approximated by introducing a weighted average. The microwave losses ofthe electrode structures are calculated in section 3.3.8.The model of the slow-wave electrode structure is shown in Figure Here theelectrode consists of sections each of which is subdivided into a smaller section of length d/2,a capacitive section, and another section of length d/2. Using standard network analysis, thevoltage V and current I at the nth and (n+1) th sections are related by 8^. 6cosh— Z smh—2^°^2^1 01 ^8—sink— cosh —6 '6.)Cf 1Zo^2^26^. 0cosh— Z smh—2^°^21 . 0^0—smh— cosh—Zo^2^2vIn =(,^V=(a o+jk)d=(a 0+j)d=(a o+j) 0 )d ( Zo , ao, k, and Ao are, respectively, the characteristic impedance, attenuation constant,propagation constant, and microwave wavelength of an unloaded section, cf the capacitance ofa pair of fin (and pad, where appropriate) elements, vp° the phase velocity of an unloadedelectrode, and 6) the frequency of a microwave that travels along the electrode. Note that thefirst and the last matrices on the right hand side of equation ( are the transfer matricesfor sections 'A' and 'C' respectively (see Figure; the middle matrix is the transfermatrix for the admittance ja)C produced by a pair of fin (and pad, where appropriate) elements.Equation ( can be simplified to give the equivalent transfer matrix of the loaded (slow-wave) electrodecosh()^C+j (1) IZ° sink()2^2CO CO C fZosinhO +j(Z, C f^Z,cosh°2 2)2Infvd(4 C^co C—1 sinhe +j(—fcoshe + —J.)co CIZ0 .cosh° + j smite[Iv:+,112Zo^2^2( can be written ascosh() ZisinhO[vnI lsinhO cosh ( „(I)= (a + j (3)d= (a + j^k)d ( a and 13 are, respectively, the attenuation and propagation constants of the slow-waveelectrode; andand Z' are, respectively, the effective refractive index and characteristicimpedance of the slow-wave electrode and N, the effective refractive index of the unloadedelectrode. Comparing equations ( and ( gives,ZcoCcoshcl) =cosh° +j °^ sinhe2( the attenuation constants a o and a are generally very small as compared to the propagationconstants k and 13 (see section 3.3.8), they can be ignored when calculating N eff and Z'. Thenin section 3.3.8 equation ( will be used when the attenuation constants and microwavelosses are calculated. Ignoring ao and a, equation ( becomes/ZRd=coskd-^C'2 0 sinkd (, the characteristic impedance Z' of the slow-wave electrode can be obtained as follows:321-1^w C^(DC—sink 1 + —f coskt1+ fz ° 2^2z 1 . zo co C1 2 ( effective refractive index Neil of the slow-wave electrode can be obtained using equations( and ( Equations ( and ( provide useful information on thecharacteristics of the slow-wave electrodes, such as dispersion, group velocity, and highfrequency filter characteristics. These are discussed in the next section.33Section 8Scion A^I^Sec-Lion C7070V,n < d/2 d/2Figure Model for transfer matrix analysis.343.3.3. Dispersion Characteristics and Group VelocityThe phase velocity vp of the slow-wave electrode, as given by &B, can be found bysolving equation ( The group velocity vg , as given by d&dB, can also be approximatedby 4 6)/ AB. Thew versus B curve for a typical structure having d = 7.3 pm, S2 = 1 pm, W2= 7 pm, t = 0.5 pm, C1 = 0.68 fF, Zo = 65 ohm, e r = 12.9, e s = 1, and Neff = 3.43 isshown in Figure It is apparent that at a very high frequency when the fin spacing dbecomes comparable to a wavelength the electrode begins to behave like a band-pass filter.There is an infinite number of pass-bands and stop-bands; however, the lowest stop-bandfrequency for a typical structure is very high (in the range of 1000's of GHz). For the intendedapplication of the electrode only the lowest pass-band will be used. Therefore in the discussionthat follows only the lowest pass-band will be considered. vp and vg are shown in Figure3.3.3.2. The characteristic impedance Z' is shown in Figure It can be seen that vp =vg at frequencies up to about 200 GHz before they begin to depart. At frequencies around 3000GHz both vp and vg drop rapidly towards zero, exhibiting their filter characteristics. Z' alsodrops off rapidly at this range of frequency. This range of frequency will definitely be abovethe frequency for any foreseable future operation of such electrodes. As a result, this dispersionof the electrodes, due to filter characteristics, will not introduce any problems for their intendeduse and will not be considered further. However, one must keep in mind that this analysis mayfail at very high frequencies at which the wavelengths approach the transverse dimensions of theelectrodes.35STOP BAND1 1 1 1 1 1^1 1 1 1 1^1 1 1 1 .11. 1 ......^11 1 .11.1.11, 1 ......."1 2^ 3^ 4p ( 1 O5 m-1)Figure 6.) versus 13 curves for the lowest two pass bands of a typical slow-waveelectrode. Note that the two pass bands are separated in frequency by aforbidden stop band.361 0.0(f)F8.00 6.0v-6-o 4.0^ Phase Velocity Vp— — — Group Velocity V gtiti2.011^1^1^1^1^1 1 1^1^1^1^1^1^1^1^1^1 1^1^1^1^1^1^1 1 1^1^1^110 100 1000Frequency (CHz)Figure Phase and group velocities as a function of frequency of a typical slow-waveelectrode. Note that the two velocities begin to depart at around 200 GHz.37I^1^1^1 1 1 1 1 1 I^►^1^► 1 1 1 1 1^►^I^►^1^1 1 1 1 1^1^110 100 1000Frequency (GHz)160502010Figure Characteristic impedance as a function of frequency of a typical slow-waveelectrode. Note that losses are not taken into account in the calculation.383.3.4. Low Frequency Approximation and Design FormulasAs shown in section 3.3.3, the intended frequency of operation of the slow-wave electrodeis much lower than the stop-band frequency. In other words, the fin spacing d will be muchsmaller than the wavelength (i.e., 13d < < 1 and kd < < 1). In this regime equations ( ( can be simplified.Using cos x %--, 1 - x2/2 and sin x x for x < < 1, equation ( becomesN2 2d2 k2d2 6.)CiZokei1-  e^-1-2NO2^2^2which can be simplified using equations (, (, and ( asC, \INO N° \11+----Cd^d"-=c L(C+—C',) . ( equation ( can be simplified as1zi=z0 coCfkd caCcoC +—+--f-k2d2.1e Zo^41Eliminating second order terms and using equations ( and ( one obtains39zczo1C L( —f —I.C+ Cd \ dIt can be seen from equations ( and ( that in the low frequency regime, in whichthe electrodes are intended to be used, the introduction of the loading fins (and pads, whereappropriate) is equivalent to simply changing the capacitance per unit length to (C + C/d). Thisproperty enables the equations governing the slow-wave electrodes in the regime of interest tobe greatly simplified. This simplification is justified because the error involved is only secondorder. Moreover, the quasi-static analysis is in itself an approximation. The objective of thisanalysis is to find a simple yet relatively accurate mathematical description of the behaviour ofthe slow-wave electrode structures. For a more accurate description, a full wave analysis willbe inevitable.However, the assumption that the fins (and pads, where appropriate) were pure capacitiveelements as used in the analysis above was not a very valid assumption, even though the resultswould reflect the general trend of the behaviour. In an attempt to yield more accurate yet simpleformulas, an approximation of the inductive effects produced by the fins, being based on aweighted average between the inductances of two extreme cases, is introduced. Thisapproximation results in significantly more accurate formulas that can be easily used to designthe slow-wave electrodes.With some manipulations of equations ( and ( the inductance per unitlength L and capacitance per unit length C of a conventional (unloaded) coplanar strip electrode40can, respectively, be written asN ZL-  ° °c(^ ( that the substrate and superstrate here are assumed to be infinitely thick. This is consistentwith the normally used GaAs substrates, which are typically quite thick as compared to theelectrode dimensions. A dielectric superstrate of thickness much greater than the electrodedimensions could also be easily applied. If air is used as the superstrate (e s = 1), then itsthickness will automatically be infinite. Note also that the analysis used throughout this thesisassumes that the semi-insulating substrates have infinite resistivities. This is justified becausesemi-insulating GaAs substrates typically have resistivities greater than 10 7 and havevery low background carrier densities (on the order of 10 8 cm 3) [49].To approximate the inductance per unit length of a slow-wave electrode, the weightedaverage between the inductance per unit length of coplanar strips having widths IV/ and W1 +W2is taken. The weighting is based on the fin length / as compared to the fin period d. Usingequation (, the weighted inductance per unit length L' becomes41andZ1 = j Li =N C ([Zo(1- -1--)+Z1 1 1d No CfcZo d ( Z, and Z, are the characteristic impedances of coplanar strip electrodes having strip widthsW1 and W1 + W2 respectively. Note that the approximation above assumes that the loading pads,if used, do not contribute to the weighting. In other words, the inductive effects of fins withpads on their ends are assumed to be the same as those of fins alone. This is justified since thefins are generally narrow and the pads, due to their sizes, are mainly capacitive.The capacitance per unit length C' of a slow-wave electrode can also be written asN CC 1=-2-+--f .Zoc d(, the Neff and Z' of a slow-wave electrode become1Z0(1 - -L)+Z1 -1-11 Ni!+ CcjiciNol ( CV CI =Multiplying and dividing the two equations above one obtains the following design formulas:42( = [Z0(1 - 1) +Z, 1 N0 0(1 - ±)N0 .^( term Zyd in equation ( is typically much smaller than Z0 (1 -1/d) and may beignored for convenience.In designing these slow-wave electrodes, by pre-defining Neff, Z', and Cf of the fins (andpads, where appropriate) to be used, one can easily determine Z, and d by solving equations( and ( following section describes how the capacitance of the fins are calculated. Knowingtheir capacitance, design curves can be easily generated using the design formulas ( and( Computation of Fin CapacitanceThe capacitance of half-buried fins are calculated using both finite difference [50,51,52]and finite element methods [53]. Both of these methods yield similar results. Computerprograms have been written. The finite difference method requires a longer computation timeand memory; consequently the finite element method was used to calculate the fin capacitancein most cases. Due to the more complicated geometry of fins having pads at their ends, as wellas their high demands of computation time and computer memory, their capacitances werecalculated using an approximate formula.To obtain the capacitance of a pair of half-buried fin (and pad, if applicable) elements,their capacitance in air C' is first calculated. Then it is multiplied by the factorer+ es ( can be easily obtained by considering the contributions to the electric energies stored inthe substrate and the superstrate. This factor can also be used for fins (and pads) that lie entirelyon the surface provided they are thin. For instance, as will be shown in Chapter 5, if thethickness to gap width ratio is not too large, i.e., less than about 0.3, there will only benegligible differences between the Ne's of surface deposited electrodes and those of half-buriedones.The finite difference method solves Laplace's equation with a unit potential differenceapplied between the fin elements. The electric field distribution and total electric energy U in44the space around the fin elements are then calculated. The capacitance in air CI is obtained fromC1 =2U.The finite element method solves the integral equationk=2=E f G(F,F1)q(e)ds ik=1 ( to a pair of fin elements (labelled k =1 and k =2) in air producing a potential 4)(6located at a field point r, whereG- 1^ ^104 rc e ( the Green's function, c o the permittivity of free space, and q(r) the electric charge densityat location r' on a fin element sk .Upon discretization of the fin surfaces into cells of area Cj the above equations can be putin the formNTi=Ef=1( 1 fc f c G(P,P)dsds 1^( iCi j45and NT is the total number of cells in the pair of fin elements, (D i the potential at cell C, , and Qjthe electric charge at cell From reference [53], the Pi; s are given in two general algebraicforms corresponding to parallel and perpendicular cells. The expressions corresponding to thesetwo cases are given in Appendix II. The capacitance in air is obtained from C = Q/(P, whereQ is the total charge on a fin element and (i) the potential difference between the fin elements.The capacitance of fins in air, as calculated by the finite element method, are given inTable and are plotted in Figure These values are used to calculate the designcurves as given in section 3.3.7.The finite element program was written in FORTRAN. The fins were divided into over200 cells, depending on their dimensions. The average computation time was in the range of 4hours on a 80386 33MHz IBM PC compatible computer using a 80387 math coprocessor. Basedon the comparisons made with some of the structures having similar dimensions, the resultsobtained here are within 5% agreement with those given in reference [54], in which thecomputational accuracy is given as 5%. The finite element program, however, offers moreflexibility in calculating the capacitance of fins having various dimensions.To calculate the capacitance of fins with pads, the assumption depicted in Figure made. Here the capacitance produced by a pair of fin and pad elements having dimensionsS2 , W2 , W', 1', and / is assumed to be equivalent to the capacitance of a pair of fin elementshaving S2, W2, and / plus the capacitance per unit length of a section of a pair of coplanar stripshaving a gap S2 and strip width W' multiplied by the length (/' - 1), or mathematically,cfr,^ cfin(s,,w2,0^ceo„,..„,,,s(svw) [1' -I] ( Ceop/anar strips is the capacitance per unit length of a pair of coplanar strips, as given by[45:pp. 257-288] (see also Appendix I). This is a reasonable assumption since the fringingelectric field along the edges of the pad will be very much like that produced by the fin. Theextra capacitance contributed by the pad will be produced by electric field lines that run beteenthe pads and which do not have any component in the length direction (i.e., /') along the pads.This extra capacitance is best approximated by the capacitance per unit length associated with apair of coplanar strips, having the same gap and width, multiplied by the length of the pad atwhich the electric field has no component along the pads, i.e., (l' - 1).The capacitance of fins and pads (in air) having some chosen dimensions are given inTable These values are used to calculate the Neis and Z' 's of the slow-wave electrodesfabricated (see section 3.3.7 for theoretical results and Chapter 5 for measured results).47Table Capacitance of fins in air with S2 = 1µm and t = 0.5 Am.W2 (-1 m) / = 1 gmC1 (fF)/ = 2 AmC2 (fF)/ = 3 AmCl (fF)/ = 4 AmC7 (fF)3 0.0612 0.0969 0.122 0.1454 0.0721 0.111 0.139 0.1645 0.0820 0.123 0.153 0.1806 0.0910 0.135 0.166 0.1957 0.0994 0.145 0.178 0.2088 0.107 0.154 0.189 0.2209 0.115 0.163 0.199 0.23110 0.122 0.171 0.208 0.24248Table Capacitance of fins with pads in air.S2 = 4 Am, W2 = 28 Am, / = 4 Am, t = 1µm/' = 8 Am /' = 12 Am /' = 16 AmW' = 7 Am 0.408 fF 0.472 fF 0.536 fFW' = 14 Am 0.420 fF 0.496 fF 0.576 fF49 0.25U0.05 I^I^I^I^I^I^I^I^I^iI^11[11111111^I^I ,^I^I^I^I^I^iiii4 5 6^7^8 9 10W2 (pm)Figure Calculated capacitance of fins in air having S2 = 1 pm, t = 0.5 pm, / = 1,2, 3, and 4 gm, and W2 ranging from 3 to 10 pm.50A^I^IE31^C^18I I^I^I131^C^18I IAA— — I —B BBBBAC+CFigure Assumption made in calculating the capacitance of fins with pads. Here, thecapacitance of the pair of fin and pad elements is approximated by the sum ofthe capacitances of the pair of fin elements and the pair of coplanar strips onthe right. The letters A, B, and C show the different areas that contribute tothe total capacitance. Areas having the same letter indicate that theircontributions to the total capacitance are assumed to be the same.513.3.6. Fringing Fin CapacitanceThe slow-wave electrodes are capable of slowing down a microwave sufficiently so thatits phase velocity can be matched to that of an optical wave in a III-V semiconductor electro-optic modulator. This is because the fringing electric fields produced by the narrow fins andpads increase the capacitance to fin length ratio Cy/ significantly as compared to the capacitanceper unit length associated with two coplanar strips having the same width as the fins. In orderto obtain a large C://, / has to be kept as small as possible. In other words, the fins have to beas narrow as possible. The word 'narrow' here refers to a small / as compared to W2.Moreover, a small 1 reduces the inductive effects of the fins, resulting in a more effectivecapacitive loading. The narrower the fins, the more capacitive and less inductive they are.Figure gives the values of Cii/ for fins of Table having various 1. It isclearly seen that the narrower the fins the greater the value of Cji/. For the fins having 1 = 1Am, Cy/ is almost twice that for fins having 1 = 4 gm. In other words, the narrower that wecan make the fins, the more effective the loading. In the limiting case when / = co, Cji/ becomesthe capacitance per unit length corresponding to a pair of coplanar strips. It is obvious from thefigure that wide fins (large 1) will not have a very high CJ/1 as compared to that of coplanar stripsand consequently result in a less effective loading.In practice, 1 is chosen so that the electrodes can be fabricated reliably. Even thoughmodern electron beam lithography techniques make it possible to fabricate sub-micron structures,fins having 1 = 1 Am are sufficient to give good results in terms of amount of slowing and easeof fabrication. The results are given in the next section.520 . 1 30 . 1 1Il lif^IIIIII IIIIIII IfIIIIIII IIIIIII6*-- 0.05  —0.03  —0.014 5^6^7^8^9^10W2 (gm)Figure Fin capacitance to length ratio C.f// for fins in air having S2 = 1µm, t = 0.5Am, / = 1, 2, 3, and 4 Am, and W2 ranging from 3 to 10 Am. / = cocorresponds to a pair of coplanar strips.533.3.7. Calculated Results and Design CurvesBearing in mind the practicalities of modern fabrication, the value of / is limited to about0.5 gm but 1µm should be sufficient to give good results. The separation between the twocoplanar strips in a typical Mach-Zehnder modulator also limits the practical values of W2 to therange of about 3 to 10 gm. For these reasons, design curves for slow-wave electrodes havingthese dimensions are calculated.Design curves for half-buried electrodes with superstrates having e s = 1 (i.e., air) and e s.= 3.5 (e.g., Dupont PI-2525 polyimide [55]) are calculated [3,5]. Also Z' = 50 and 75 ohmare assumed. The Neal' for electrodes designed for velocity-match in GaAs based waveguidesystems is 3.43 [1,2:p. 364] for an optical free-space wavelength of 1 gm while the N eff forelectrodes designed for InP based waveguide systems is 3.3 [2:p. 364,27,28]. The calculationshere are for velocity-matched electrodes based on e,. = 12.9 for GaAs [38:pp. 25-26] and e r =12.4 for InP [56]. The design curves are shown in Figures to is apparent from the figures, as W2 increases d also increases while Z, decreases. Thisis because a larger W2 corresponds to fins having a higher capacitance, which allows a greaterfin period d to obtain the same increase in capacitance per unit length. As the aspect ratio, 1/W2 ,of the fins is decreased, the fins more and more nearly approximate a purely capacitive load.Therefore in the limit, the capacitance of the fins is determined solely by W2 . It is apparent thatfor each of the cases Z' = 50 ohm and Z' = 75 ohm, Z, approaches a limiting value as W2 isincreased. This limiting value corresponds to that Z, that should be used for a purelycapacitively-loaded electrode.54To illustrate the use of the design curves, a 50 ohm slow-wave electrode on GaAs, withes = 3.5, Neff = 3.43, Si = 15 Am, S2 = 1 Am, and W2 = 7 Am, will have d = 8.5 Am, andIV/ = 30 Am (this corresponds to Zo = 69 ohm). The dimension W1 is determined by using thedesign formulas for conventional coplanar strip electrodes [45:pp. 257-288] (see Appendix I).Another illustration is a 50 ohm electrode on GaAs with air being the superstrate (i.e., e s = 1),Neff = 3.43, Si = 15 gm, S2 = 1 Am, and W2 = 7 Am, will have d = 6 earn, and Wi = 23 Am(this corresponds to Zo = 80 ohm).Even though the design curves are for half-buried electrodes, the fact that the electrodesare only 0.5 gm thick makes them usable for surface deposited (unburied) electrodes with littleerror, as the extra contribution to capacitance by half-burying the fins of this thickness is verysmall. In fact, as will be shown in Chapter 5, half-burying electrodes will only increase theirNeff's very slightly, if not negligibly. Moreover, if thicker electrodes are to be designed, thesecurves may still be used by slightly modifying the dimensions in favour of a higher Neff, such asreducing d.The Neff's and Z' 's for a number of slow-wave electrodes have been calculated based onthe design formulas of section 3.3.4 and the capacitances of section 3.3.5. These electrodes havebeen fabricated and tested. The measured results are given in Chapter 5. The calculated N eff'sand Z' 's are given in Table Calculated Neff's and Z' 's of some slow-wave electrodes fabricatedand tested.Si = 60 Am, S2 = 4 Am, W2 = 28 pm, / = 4 Aim, t = 1.0µmElectrode WI(hm)W'(.hm)1'(hm)d(in)Neff Z'(ohm)#1 72 0 4 18 3.50 49.6#2 110 0 4 32 3.16 57.1#3 110 7 8 32 3.28 55.0#4 110 7 12 32 3.39 53.1#5 110 7 16 32 3.49 51.4#6 110 14 8 32 3.30 54.7#7 110 14 12 32 3.44 52.4#8 110 14 16 32 3.56 50.456130—12011.0E 9.0-0 ^ -7.0 ——110- --^0—100—905.0 ——80^ Z' = 50 0_ _ _ Z' = 75 03.0 ^ I I I I I^703.0^4.0^5.0^6.0^7.0^8.0^9.0^10.0vv2 (, 1-171 )Figure Design curves for 50 and 75 ohm velocity-matched slow-wave electrodes onGaAs substrates for E s = 1, S2 = 1 Am, t = 0.5 Am, and 1 = 1 Am.13.0 ^57^ 11020.0../—- 100-_, 7/16.0 — ........ --^ .^___- . 4 -- >E 12.0 — 7N_- 90 0<^ z,.777/8.0 ——70_ _ _4.0 IIIIIIIIIjIIIIIIIIIIIIIIIII^i^i^II^i^i^ 603.0^4.0^5.0^6.0^7.0^8.0^9.0^10.0vv 2 (gmFigure Design curves for 50 and 75 ohm velocity-matched slow-wave electrodes onGaAs substrates for E s = 3.5, S2 = 1 gm, t = 0.5 gm, and / = 1 gm.Z' = 50 0Z' 75 0=587777—105 N- _^o0—9513.0 ^ —125— 1157.07—855.0 —3.0 ^ 653.0^4.0^5.0^6.0^7.0^8.0^9.0^10.0vv2 (,gym)Figure Design curves for 50 and 75 ohm velocity-matched slow-wave electrodes onInP substrates for e s = 1, S2 = 1 Am, t = 0.5 Am, and 1 = 1 Am.Z' = 50 0Z' = 75 0—755924.0 ^—- 105...-...-,"^ --- _--...-95__.^>^__ ...- N_- o— 85_^„..„,_ •.___.__—7520.0 — -_,^,-----,^_E 16.0 --:^..._.^__1-, ,12.0 —_.8.0 — — 65_ _ _Z' = 50 0Z' = 75 04.0 I i 11111111111111111111111111 I ' m,^- 553.0^4.0^5.0^6.0^7.0^8.0^9.0^10.0Figure Design curves for 50 and 75 ohm velocity-matched slow-wave electrodes onInP substrates for E s = 3.5, S2 = 1 pm, t = 0.5 gm, and / = 1 pm.603.3.8. Attenuation Constant and Microwave LossEquation ( can be used to calculate the attenuation constant a of the slow-waveelectrodes. By equating the imaginary parts, one getssinkd+  63 CiZ° coskdsinhad_ ^2 sinha od^sinf3d( can be simplified since ad, aod, 13d, and kd < < 1. Thereforekd+ 6)Cflo 0 k2d2 )a ^2^2 a 0^f3d( second order terms and using equations (, (, and (, equation( can be simplified to givea . 1 {Neff + Noa 0 2 [ No No( is always greater than unity. This is an interesting point, as it shows that the slow-waveelectrodes always have a higher loss than the conventional coplanar strips. However, since theNeff's are not much higher than No , the losses of the slow-wave electrodes and conventionalcoplanar strips are comparable. Using Neff = 3.43, No = 2.64, one gets a/ao = 1.03, indicating61that the losses are almost the same. The attenuation (in dB, i.e., is equal to 8.68 times theattenuation constant [46:pp. 76-79]) due to conductor loss of conventional coplanar strips can becalculated using Wheeler's incremental inductance formulas [45:pp. 257-288] (see also AppendixI). The dielectric loss for electrodes on GaAs is very low [45:p. 73, pp. 257-288] as comparedto the conductor loss and is therefore ignored here. For the slow-wave electrodes described bythe design curves given in section 3.3.7, their losses at 20 GHz are typically around 3 dB/cm,and increase as the square root of the frequency [57:pp. 151-153] (see Appendix I). The scaledup versions of the slow-wave electrodes given in Table have much lower losses of around0.7 dB/cm, due to their greater dimensions. It should be noted that these calculated values maybe over-estimated since the greater widths of the fin sections will reduce the ohmic losses. Thelosses of these slow-wave electrodes, however, are still very low as compared to those of othertypes [33-35,40-45].623.3.9. Scaling of DimensionsSince the capacitance of a pair of fin elements scales as their dimensions, the term Cidwill remain unchanged if all the fin dimensions S2 , W2, 1, and t, and fin period d are scaled bythe same factor. Moreover, as given in Appendix I, since the characteristic impedance Zo of athin conventional coplanar strip electrode depends on the ratio S 1/(S1 + 2W1) alone, scaling thedimensions of the coplanar strips (i.e., S1 and W1) will have little effect on Zo. Therefore, a thinslow-wave electrode will have the same Neff and Z' if all its dimensions are scaled by the samefactor. This property is especially useful for experimental purposes since, as will be shown inChapter 5, scaled-up versions of the slow-wave electrodes having the same Neff and Z' can befabricated using cheaper photo-generated masks.This scaling, however, has certain limits. First, the scaled-up fin period d still has to bemuch smaller than the microwave wavelength in order to avoid the dispersion caused by the filtercharacteristics. Second, if the transverse dimensions of the electrode approach the wavelength,the quasi-static analysis might fail. Third, as given in Appendix I, if the electrode has a non-zero thickness t, Zo does not remain exactly the same for a constant S1/(S1 + 2W1) but decreasesbecause of the higher capacitance due to the finite thickness. These limitations, however, areoften not very restrictive for most practical applications.633.3.10. Input SectionsSince the width W1 and separation S1 of the main strips of the slow-wave electrode areusually too small for direct probing and connector bonding, an input section having a largerseparation or pitch is sometimes needed. The fact that the characteristic impedance dependssubstantially on S1/(S1 + 2W1) alone suggests that one can use a section of coplanar strip havinga gradually increasing pitch while maintaining the same value of S1/(S1 + 2W1). It can be shownusing simple geometry that this ratio is maintained throughout the whole length of the sectionif its two ends, each satisifying the same ratio, are joined by straight lines. This is depicted inFigure In order to keep reflections and discontinuities as low as possible, it isdesirable to keep the input sections as short as possible.3.3.11. The Effects of High Dielectric Constant Superstratesand Partially Buried ElectrodesAs mentioned in Section 3.2, a high dielectric constant superstrate such as a polyimidemay be applied so as to increase the capacitance per unit length, thus resulting in higher Neff's.The electrodes may also be partially buried in the substrate to increase their Neff's. However,this may only result in a marginal increase (see Chapter 5).While it is not essential to use a superstrate with a dielectric constant higher than that ofair, using it has several advantages. It increases the capacitance per unit length so that the periodof the fins (and pads, if applicable) can be increased while maintaining the same N. Increasing64the period of the fins will reduce any unwanted interactions between them, thus allowing a moreeffective capacitive loading. The increase in fin capacitance allows one to use shorter fins.Therefore, the separation between the coplanar strips can be reduced, resulting in an increase ofthe electric field for modulation. Z, can also be reduced, resulting in wider electrodes whichhave lower microwave losses. Tuning of the microwave effective refractive index to achievebetter velocity-match is also possible by applying superstrates having various dielectric constants.Despite of these advantages, surface deposited electrodes with air superstrates require theleast number of fabrication steps and are perhaps more practical in modulator use. In Chapter5, experimental results for both partially-buried and surface deposited electrodes are given.3.4. ConclusionIn this chapter, slow-wave electrode structures employing narrow capacitive loading finsand pads are described. The analysis of these electrodes is given. Design curves, for both 50and 75 ohm slow-wave electrodes, for matching a microwave's effective refractive index to anoptical wave's effective refractive index in modulators fabricated using GaAs and InP basedmaterials, are provided. The calculated microwave effective refractive indices and characteristicimpedances for a number of slow-wave electrodes are presented.65cr)S/( S+ 2W ) = S ' / ( S ' + 2 Ili ' )Figure Input section for slow-wave electrodes showing the various dimensions.663IChapter 4 Device Fabrication4.1. IntroductionIn order to verify the slow-wave effect as predicted by theory, slow-wave electrodes, bothpartially buried and surface deposited, and having various dimensions, have been fabricated on(100) semi-insulating GaAs. Since a high dielectric constant superstrate is not needed in orderto achieve the velocity match condition, as already discussed in Chapter 3, air was used as thesuperstrate (i.e., e r = 1) due to the reduced number of fabrication steps required. The reasonwhy partially buried (especially half-buried) electrodes were fabricated was because the fins wereassumed to be half-buried when their capacitance was computed. Consequently, half-buriedelectrodes would more closely approximate the ideal model as analyzed in Chapter 3. Moreover,fabricating both partially buried and surface deposited electrodes allows us to compare theireffects on the amount of slowing. Such comparison is made in Chapter 5.A total of twenty electrodes were fabricated, all of which had 50 ohm characteristicimpedances. Among them, one was a conventional coplanar strip electrode, seven were slow-wave electrodes loaded only with fins, and twelve were slow-wave electrodes loaded with finsand pads. All of these electrodes had different dimensions and some were buried.The electrodes were intended to be easily fabricated, being formed in a single layer ofmetallization using a standard chlorobenzene lift-off technique [38:pp. 115-138]. Due to the highcost of electron-beam masks, the dimensions of the electrodes were scaled up so that cheaper67photo-generated masks could be used. This is justified since, as discussed in Chapter 3, theeffective refractive index and characteristic impedance of a slow-wave electrode will remain thesame if all of its dimensions are properly scaled.The fabrication procedure for the different electrodes was basically the same except thatthe buried electrodes required that the GaAs substrates be etched. Here, wet chemical etchingtechniques were used. However, wet etchants result in some lateral etching which can create airspaces around the fins and pads and in turn reduces the microwave effective refractive index.As a result, different wet etchants were tried in order to reduce the lateral-etch. This problemwill be discussed in section Mask DesignTwo photo-generated masks were designed using the CAD package ICED by IC Editors,Inc. Both masks were fabricated by Precision Photo Mask (PPM) Inc. of St. Laurent, Quebec,on ultra-flat sodalime glass. The precision as specified by PPM was +/- 0.25 Am. The maskswere dark field, i.e., the areas of the electrode patterns were transparent.There were only four electrodes on the first mask. Two of the them were conventionalcoplanar strip electrodes and the other two were slow-wave electrods loaded only with fins. Thetwo conventional coplanar strip electrodes were the same except that one of them was shortcircuited at one end while the other was open. Similarly, one of the other two slow-waveelectrodes was shorted at one end; other than that, they were exactly the same. The reason whyboth shorted and open electrodes were designed was because at the time of designing the masks68we were not certain whether electrodes shorted or open at one end would enable better and moreaccurate measurements to be made. However, as it turned out, electrodes shorted at one endproved to offer better results since a plane of reflection can be established very easily by a shortcircuit while the open end capacitance of an open circuit produces an equivalent electrode lengththat extends from the end. The mask layout and electrode dimensions are respectively given inFigure 4.2.1 and Table 4.2.1. Input sections were not used in these electrodes since the pitch(i.e., the separation between the two contact points) of the coplanar strip probe used in themeasurements was just right for these electrodes.The second mask contained eight electrodes (#1 to #8), all of which were slow-waveelectrodes having different dimensions. They were all shorted at one end, as this had been foundto give better results. Electrodes #1 and #2 contained only fins while electrodes #3 to #8 hadpads at the ends of the fins. Electrode #1 had a smaller fin period d, and a smaller strip widthW1 , as compared to other electrodes. Dimensions of electrodes #2 to #8 are all the same exceptthat the pad sizes are different. This enables us to compare the effects on the Neal  to differentpad sizes. The mask layout and electrode dimensions are given in Figure 4.2.2 and Table 4.2.2respectively. These electrodes all had input sections to facilitate proper probing. Thedimensions of the input sections used in these electrodes are given in Figure 4.2.1. Mask #1 electrode dimensions.t = 1.75µm, E r = 12.9, e s = 1.0Electrode S1(hm)W1(pm)S2(1/1n)W2(pm)d(pm)/(pm)Shorted/Open#1 17 133 N/A N/A N/A N/A shorted#2 (slow-wave)31.5 120 3.5 14 29 3.5 shorted#3 (slow-wave)31.5 120 3.5 14 29 3.5 open#4 17 133 N/A N/A N/A N/A openTable 4.2.2. Mask #2 slow-wave electrode dimensions.S1 = 60 gm, S2 = 4 gm, W2 = 28 gm, / = 4 gm, t = 1.1µmE r = 12.9, E s = 1.0Electrode W1(pm)W'(1-im)l'(111n)d(pm)InputSection#1 72 0 4 18 A#2 110 0 4 32 B#3 110 7 8 32 B#4 110 7 12 32 B#5 110 7 16 32 B#6 110 14 8 32 B#7 110 14 12 32 B#8 110 14 16 32 B702 cmW i ^91W i- -132 ulli^0.5 cmW2^\AAS2—=- S IW2^WI> <d - *-- 120 ul,iji^0.5 cmW2^WIS2 ^S1'-- —W2^WIFigure^-->VASiWi0.5 cm\./  4.2.1. Layout of Mask #1.71. 5 cm1W2 -S2- SW2^WI100 um) 0 . 3 cm0.3 cm\fir1202120 u:120 us..1„...9 IFigure 4.2.2. Layout of Mask #2.72132 urnV 120 um1 44\um1 20 urn■120 urn ^30  ■,,".-. um^---15 urn132 UM/INPUT SECTION AINPUT SECTION BFigure 4.2.3. Dimensions of the two input sections used in Mask #2.734.3. Fabrication ProcedureA combination of chlorebenzene lift-off and substrate etching techniques were used in thefabrication of the electrodes [38:pp. 115-138, pp. 96-106]. After cleaning and degreasing thewafers, positive photo-resist was spun onto them and was exposed through the dark field masks.After developing the photo-resist, the areas of the electrode patterns became exposed while theother areas were covered by the photo-resist. Then, if partially buried electrodes were to bemade, the exposed areas of the substrate had to be etched to a depth equal to the amount that theelectrodes were to be buried. Here, for simplicity and convenience, wet etching techniques wereused. Aluminum of the desired thickness was then thermally evaporated onto the wafer. Thisproduced wafers in which the exposed areas, including the etched pits for the electrodes if thesubstrate had been etched, were filled with aluminum. Finally the photo-resist, along with theexcess aluminum, was stripped off, leaving the metallized electrodes. The main steps of thisprocedure are schematically shown in Figure 4.3.1. More details are given in the followingsections.74a ) (d)111111111111 22L111111:-21- 111-=-111- 111-111=111=1111111111111111- 111- 111- 111- 111=( e )ALU/A I NUIAPHOTO-RESISTGoAs SUBSTRATEFigure 4.3.1. Main steps of the fabrication procedure showing (a) the unexposed photo-resiston a GaAs substrate; (b) the photo-resist after development; (c) the substrate afterwet etch; (d) the sample after evaporation of aluminum; and (e) the half-buriedelectrodes after stripping the photo-resist and excess metal. The procedure forforming surface deposited electrodes is the same except that the substrate doesnot have to be etched [step (c)].754.3.1. Cleaving GaAs WafersIn order to make better use of the available GaAs wafers, they were cut into smaller piecesso that they would just accomodate the electrode patterns. This was easily accomplished bymaking a small scratch along the edge of the wafer, then pressure was applied to one side of thescratch until the wafer broke along one of its natural cleavage planes (i.e., the (011) planes).However, care had to be exercised to avoid scratching other parts of the wafer.4.3.2. Cleaning GaAs SubstratesIn order to ensure that the GaAs substrates were free from dust and grease, the substrateshad to be cleaned prior to photo-resist patterning. Due to the small dimensions of the structures,which are typically on the scale of microns, any dust particle on the substrate could result in anelectrode that is cut off and render it useless. Any grease on the substrate would also affect theadhesion of the photo-resist, which would very likely peel off during the wet chemical etch thatfollows the photo-resist patterning.Due to the brittle nature of GaAs, the use of an ultrasonic bath was avoided. Instead,manual agitation was used throughout. The wafers were first cleaned in hot (approximately50°C) acetone for 10 minutes. This would allow the acetone to dissolve any grease on thewafers. On removing the wafers from acetone, the wafers were immediately dipped into hot(approximately 50°C) isopropanol for 5 to 10 minutes. This would remove the residue and water76that are usually found after removing the wafers from acetone. As soon as the wafers wereremoved from isopropanol, they were blown dry using a jet of compressed nitrogen. In orderto avoid dust particles landing on the wafers, photo-resist was immediately spun onto them.If needed, the wafers could also be dipped in buffered hydrofluoric acid (BHF) for 1minute, then rinsing them in deionized (D.I.) water for 5 to 10 minutes in a cascade bath, andfinally baking them at 120-130°C for approximately half an hour.4.3.3. Photo-resist PatterningAfter cleaning the wafers, Shipley PR 1400-30 positive photo-resist was spun onto themat 3500 RPM for 35 seconds. This photo-resist was chosen because it could be applied in arelatively thick layer. In an attempt to ensure that the wafers were dust-free, compressednitrogen was first used, followed immediately by spinning the wafers, for about half a minute,without applying any photo-resist. Then the photo-resist was immediately applied. Thisprocedure ensured that the wafers were as dust-free as possible.The photo-resist was pre-baked at 70°C for 30 minutes. The samples were allowed to coolto room temperature after they had been removed from the oven. In the meantime, they had tobe properly handled and covered so that they would not accumulate any dust particles.Since we were not, at this time, fabricating electro-optic modulators, the orientation of thecrystal was not critical. Nevertheless, the electrodes fabricated using the first mask were alignedparallel to the [011] direction, i.e., same direction they would be if they were used in electro-optic modulators. However, in fabricating partially buried electrodes, which required a77subsequent etching of the substrate, this orientation produced non-vertical etched side walls, thusproducing air gaps around the fins and reducing their capacitance and producing lowermicrowave effective refractive indices. In order to fabricate half-buried electrodes that closelyapproximate the ideal structure which the theory is based on, the electrodes fabricated using thesecond mask were oriented along the [001] direction. This direction enabled side walls that weremore nearly vertical to be etched [38:pp. 96-106,58].The masks were aligned and the photo-resist exposed using a Karl-Suss MJB3 contactmask aligner. The samples were rotated on the mask aligner so that the major flat, whichcorresponds to the [01T1 direction, was aligned either parallel to or at 45° with respect to thedirection of the electrodes. The latter direction of alignment produced electrodes that wereoriented along the [001] direction. Since very high resolution can be obtained with positivephoto-resist, the standard contact setting on the mask aligner was used. After proper contact wasmade, the photo-resist was exposed for 25 seconds using a power density of 25 mW/cm 2produced by the mercury lamp with a wavelength of 320 nm.The exposed photo-resist was then treated in chlorobenzene for 8 minutes so that it wouldbe undercut on development, producing a lip overhanging at the top [38:pp. 115-138]. Thiswould subsequently enable a successful removal of the photo-resist and the excess metal. At theend of the chlorobenzene treatment the sample was blown dry using compressed nitrogen. It wasthen allowed to sit for at least 15 minutes so that the chlorobenzene would evaporate completely.This was necessary as the photo-resist would not develop properly if the chlorobenzene was notcompletely gone.The photo-resist was then developed in cooled (below 5°C) 50% Shipley MF-312 photo-78resist developer / 50% DI water for 1 minute. Constant stirring was needed. At the end of 1minute the pattern should be seen clearly. The sample was rinsed in D.I. water for about 1minute so that most of the developer was gone. Then it was checked under a microscope. Ifthe pattern did not form altogether, the sample would have to be put into the developer foranother 15 seconds. This process was repeated until all of the pattern formed. In mostcircumstances the whole pattern should form within a total developing time of 2 minutes. If thepattern looked acceptable, the sample was to be rinsed in a D.I. water cascade bath for a least5 minutes.The electrodes fabricated using the first mask were either half-buried or totally buried,while some of the electrodes fabricated using the second mask were half-buried and some surfacedeposited (i.e., unburied). Due to the fact that most wet chemical etchants attack photo-resistto some extent, all the wafers on which partially buried electrodes were to be formed (i.e.,wafers that had to be etched) had to be hard-baked (or post-baked) so that water would beremoved and the photo-resist become more resistant to the etchants [38:pp. 115-138]. On theother hand, the wafers on which only surface deposited electrodes were to be formed did nothave to be hard-baked. In fact, hard-baking the photo-resist only made them more difficult toremove.Hard-baking was accomplished by putting the samples in a temperature controlled oveninitially set at 90°C. At the end of 5 minutes the temperature was increased to 100°C. Whenanother 5 minutes had elapsed the temperature was increased to 110°C. The temperature wasfurther increased in this fashion (10°C every 5 minutes) until a maximum of about 130°C wasreached. The samples were then removed from the oven and allowed to cool to room79temperature.4.3.4. Etching GaAs Substrates for Burying ElectrodesOnly the substrates on which buried electrodes were fabricated had to be etched. A buriedelectrode is formed by etching the exposed electrode areas of the substrate to a depth equal tothe amount that the electrode is to be buried before the metal is evaporated. Due to its relativeease of use, wet chemical etching was chosen. In this section the etching characteristics ofdifferent wet etchants used in the fabrication are discussed.Wet chemical etchants tend to etch in both the vertical and lateral directions [38:pp. 96-106,58-61]. The lateral etch produces an undercut beneath the photo-resist. They can result inburied fins that are surrounded by air gaps instead of the substrate. Consequently the amountof slowing of the microwave may be less than as expected due to the corresponding lower fincapacitance produced by the air gaps. Wet etchants may also yield non-vertical walls, due to thedifference in etch rate between different crystallographic planes. Non-vertical walls also affectthe fin capacitance and consequently the amount of slowing of the microwave.In choosing a wet etchant, the etch rate, the amount of undercutting, the relativeanisotropy, and the amount of damage to the photo-resist had to be considered.Since the electrodes were relatively thin, the amount that they were to be buried in thesubstrate was only on the order of a fraction of a micron. Therefore, the depth of the substrateto be etched was also on the order of a fraction of a micron. Due to this small etch depth, the80etch rate also had to be small since a high etch rate would make the etch depth difficult tocontrol.The amount of undercutting was the second most important consideration. A small amountof undercutting would enable the sides of the fins to be as close to the substrate as possible sothat the air gaps could be reduced. A small air gap would result in a larger fin capacitance andsubsequently a larger amount of slowing of the microwave. Due to the nature of wet etchants,however, it is impossible to completely eliminate the air gaps. Hence, a compromise would needbe found, i.e., one would need an etchant that gives a relatively small undercut as well as anacceptable etch rate.Due to the anisotropic nature of wet etchants, the etched walls will be inclined eithertowards the centre of the etch pattern or away from it, depending on the orientation of thepattern. In order to obtain more or less vertical walls, the electrodes can be oriented 45° withrespect to the natural cleavage planes (the (011) planes) [38:pp. 96-106,58]. Even though thisis not the orientation for the ultimate use of the electrodes, it provides better fabrication results(i.e., small air gaps and nearly vertical side walls) so that the electrodes will approximate moreclosely to the ideal structure on which the theory is based. Consequently a more meaningfulcomparion between the measured results and theory could be made using the half-buried slow-wave electrodes fabricated in this orientation.A somewhat less significant but important issue to be considered is whether the photo-resist could stand up against the etchants. Even though the photo-resist had been hard-baked,different etchants attacked the photo-resist to varying degrees. For instance, the etchantHC1:H202 :H20 (1:1:9) supposedly produces the least undercutting [59], yet it attacked the photo-81resist so severely that it could not be used.Table summarizes the etching characteristics of some of the etchants that satisfythe criteria mentioned above [38:pp. 96-106]. Except for HCEH 202 :H20 (1:1:9), all other threeetchants were used in the fabrication. All etchant compositions were freshly prepared fromsemiconductor grade starting materials, and all etchings were performed at room temperature.In mixing the etchants, water was always added first, followed by H 202 and the acids or bases.This helped avoid vigorous reactions when the different compositions were mixed. The etchantswere constantly stirred to ensure uniformity of etch rate across the wafers. The etch rates of thevarious etchants were first measured by patterning the GaAs substrates, followed by a timed etch,and finally the etch depths measured using a Tencor Alpha-Step 200 profilometer. The measuredetch rates were very close to those given in reference [59]. The measured etch rates are listedin Table Knowing the etch rates, the desired etch depths could be very accuratelyattained.The scanning electron microscope (SEM) micrographs of the completed electrodesfabricated using the different etchants are presented in section 4.4.82Table Etching characteristics of various etchants used in the fabrication.Etchant MeasuredEtch Rate(100)(Am/min)RelativeAmount ofUndercut(1 = smallestSmoothness ofEtchedSurfaceAmount ofattack tophoto-resist4 =largest)NH4OH:H202 :H20 1.5 4 Very smooth Relatively(1:1:8) littleNH4OH:H202 :H20 0.17 3 Very smooth Relatively(5:2:240) littleHC1:H202 :H20 0.21 2 Smooth Not too(1:4:40) severe,acceptableHC1:H202 :H20 0.20 1 *** Most severe(1:1:9)*** Not found since the photo-resist was damaged y the etc ant and the electrode  pattern dinot form.4.3.5. Evaporation of AluminumThe metal chosen to form the electrodes was aluminum. This is due to the low cost aswell as the thickness of the electrodes required. The electrodes were typically between 1.1 and1.7 Am thick (see Tables 4.2.1 and 4.2.2). For these thicknesses the most efficient way todeposit metal was thermal evaporation of aluminum in vacuum.Aluminum was evaporated using a Carl Herman & Associates (CHA) vacuum thermalevaporation system. Before loading the samples into the system, compressed nitrogen was usedto remove the dust from them. Three coils of aluminum were loaded each time. This wouldallow approximately 0.5 Am of aluminum to be evaporated. This process was repeated until the83desired amount of aluminum was deposited. However, at the end of each evaporation,approximately half an hour was allowed for the evaporated aluminum to cool before letting airinto the evaporation chamber. Otherwise the hot aluminum would oxidize. New coils wereloaded very quickly so that the exposure of the aluminum to air would be minimized.It should be noted that the film thickness monitor (Inficon Model 321) used in ourlaboratory was not very accurate. It had to be calibrated to ensure that aluminum layers of thepredefined thicknesses were evaporated. This was done using a Tencor Alpha-Step 200profilometer as a reference. Aluminum was first evaporated onto a substrate and the thicknessmeasured using the profilometer. Then the aluminum thickness as indicated by the thicknessmonitor was compared with that measured by the profilometer. It turned out that the filmthickness monitor consistently gave a reading that was about 23% smaller than that measuredusing the profilometer. This error was taken into account during evaporation.4.3.6. Removal of Excess Metal (Lift-off)After evaporation, the photo-resist, along with the excess aluminum, was removed. Thiswas accomplished using either heated (above 70°C) Microstrip 2001 or hot (50°C and above)acetone. On some occasions the photo-resist was difficult to remove and Microstrip 2001 wasnecessary. However, in most situations acetone could remove the photo-resist very quickly andeffectively.In using Microstrip 2001, care had to be taken to avoid getting it too hot; if Microstrip2001 got very hot (above 90°C), it would begin to etch the aluminum.84Sometimes the photo-resist (and the aluminum attached to it) would not come off,especially in the inter-electrode gap region. Blowing at it from very close using compressednitrogen often helped. However, care was exercised so as not to scratch the pattern with thenitrogen gun. In situations when it was very difficult to remove the photo-resist, the samples,along with the beaker holding Microstrip 2001 or acetone, were put into an ultrasonic bath verybriefly (less than 20 seconds), although the ultrasonic bath should be avoided if possible as itmay break the samples.Every so often the pattern was examined under the microscope. When all the photo-resisthad come off, the samples were rinsed in hot (70°C) D.I. water if Microstrip 2001 was used,and hot (50°C) isopropanol if acetone was used.4.3.7. Summary of Fabricaton ProcedureThe fabrication of the slow-wave electrodes was relatively easy, being accomplished bya single-step lift-off. Substrate etching is only necessary in forming buried electrodes. The mainsteps can be summarized as follows:(1) Photo-resist patterning,(2) wet chemical etching of substrate (for buried electrodes only),(3) evaporation of aluminum, and(4) removal of the photo-resist and excess aluminum.854.4. The Completed ElectrodesFigure 4.4.1 is a SEM micrograph showing the isometric view of a pair of half-buried finelements of electrode #2 of Mask #1 with the substrate etched using NH 4OH:H202 :H20(5:2:240). The fin elements are parallel to the [011] direction. As is clearly seen, the etchedwells were over-etched, producing relatively large air gaps of over 1 Am at the ends of the fins.Here, the sizes of the air gaps were measured from other SEM micrographs of the sameelectrode. As explained earlier, this over-etch was caused by the undercut beneath the photo-resist.Figure 4.4.2 is a SEM micrograph showing the isometric view of a pair of totally-buriedfin elements of electrode #2 of Mask #1 with the substrate etched using NH 4OH:H202 :H20(1:1:8). The fin elements are also parallel to the [011] direction. It should be noted that eventhough the air gaps look very large, the amount of undercut produced by this etchant is not muchlarger than that produced by NH 4OH:H202 :H20 (5:2:240) since the etch depth in this case istwice that of the above. The anisotropic nature of the etchant, manifested by the wider air gapsat the ends of the fins than along the fins, is clearly seen.Figure 4.4.3 is a SEM micrograph showing the plan view of a pair of half-buried loadingfins, the gaps between them, and the wells containing them. The electrode is #1 of Mask #2.The etchant used to make the wells was HC1:H 202 :H20 (1:4:40). The fins are oriented along[010] (i.e., the electrode oriented along [001]). Nearly vertical side walls were obtained withthis orientation. The use of this etchant also reduced the amount of over-etch. Here, air gapsof less than 0.5 Am were obtained. Also, the air gaps are uniform all around the fins, except86that they are somewhat smaller along [011].Air gaps, caused by the undercut beneath the photo-resist, reduce the effectiveness of thecapacitive loading and are believed to be in part responsible for producing microwave effectiverefractive indices slightly lower than those predicted by our theory. With electrodes orientedalong [001] and using the etchant HC1:H202 :H20 (1:4:40), the best results were obtained. Eventhough [001] is not the ultimate direction to be used in making electro-optic modulators, thisdirection makes it possible to obtain fabrication results that most closely approximate the idealstructure that the analysis is based on. As discussed in Chapter 3, the electrodes do notnecessarily have to be buried when they are applied to electro-optic modulators.Figures 4.4.4 and 4.4.5 are, respectively, the SEM micrographs showing the fins and padsof the surface-deposited electrodes #5 and #7 of Mask #2. Since the substrates were not etchedin making these electrodes, over-etching was not a concern. However, the fact that theelectrodes were deposited on the surface of the substrate would reduce the loading capacitancesas compared to those produced by half-buried fins. Nevertheless, this reduction in capacitanceshould be very small due to the small thickness of the electrodes. This is evidenced by the smalldifferences in their measured Neff's as compared to those of the half-buried electrodes aspresented in Chapter 5.87Figure 4A.1. SEM micrograph showing the isometric view of a pair of half-buried fm elementsof electrode #2 of Mask #1.88Figure 4.4.2. SEM micrograph showing the isometric view of a pair of totally-buried finelements of electrode #2 of Mask #1.89Figure 4.4.3. SEM micrograph showing the plan view of a pair of half-buried loading fins" ofelectrode #1 of Mask #2.90Figure 4.4.4. SEM micrograph showing the fins and pads of electrode #5 of Mask #2 (surfacedeposited).91Figure 4.4.5. SEM micrograph showing the fins and pads of electrode #7 of Mask #2 (surfacedeposited).92Chapter 5 Device Testing and MeasuredResults5.1. IntroductionIn order to experimentally verify the slow-wave effect, as predicted by the theory, themicrowave effective refractive indices Neffof the fabricated slow-wave electrodes were measured.Two measurement techniques were used. The first technique was based on the interferencebetween waves reflected from the electrodes and waves reflected from a reference short circuit.With a frequency sweep, interference maxima and minima were obtained and their separationsmeasured. These separations allowed us to calculate the Neff's of the electrodes. Theinterference peaks obtained using this technique, however, were not very sharp and it wassometimes difficult to locate them precisely. The second technique was based on the resonancemade possible by the probe-electrode mismatch. The Neff's were calculated by measuring thefrequency separations between the peaks of the resonances. This technique enabled us to obtainvery sharp resonance peaks and was, therefore, the preferred technique. In this chapter bothtechniques are described. The results of the measurements are given and compared.935.2. The Measurement EquipmentThe equipment used for the measurements for both techniques was as follows:(1) a Hewlett Packard HP 8757C scalar network analyzer,(2) a Hewlett Packard HP 8341B synthesized sweeper,(3) a Hewlett Packard HP 85027E directional bridge,(4) a Tektronix TMP9215 microwave coplanar strip probe,(5) a Merrimac CWM-6M-10G 6dB directional coupler,(6) a Hewlett Packard HP 0960-0055 SMA coaxial short,(7) a Hewlett Packard HP 9836 computer,(8) an IBM PC compatible computer with an IEEE488 interface card, and(9) a stereoscopic microscope with microscope lamp.The scalar network analyzer, synthesized sweeper, and directional bridge were used tomeasure the normalized reflected power, i.e., 10 log(reflected power/incident power), as afunction of frequency. This system was capable of swept frequency measurements from 10 MHzup to 20 GHz. The output power of the synthesized sweeper could be automatically levelled.Power levels up to 20 dBm could be selected. However, due to the insertion loss ranging from9 dB to 12 dB of the directional bridge, the actual power output was significantly less than thefull 20 dBm. The directional bridge also carried a certified directivity of over 40 dB. Theoverall power resolution of the system was 0.01 dB. All connectors were of Hewlett PackardAPC 3.5 (3.5 mm) standard. The network analyzer, synthesized sweeper, and directional bridgeare shown in Figure coplanar strip probe had a rated DC to 26.5 GHz bandwidth. Its return loss formatched loads as measured by its manufacturer was greater than 15 dB at its lowest point andover 40 dB at its highest point. Its connector was of Wiltron type K (3.5 mm) standard.The 6 dB directional coupler was used in the interference technique measurements only.It had an insertion loss of about 1 dB. Its rated bandwidth was 2 GHz to 18 GHz. Its directivityand isolation were over 12 dB and 18 dB respectively. All connectors were of SMA (3.5 mm)type. All SMA, type-K, and APC 3.5 connectors are compatible.The HP 9836 computer was used for automated system control functions while the IBMPC compatible computer was used primarily for data acquisition.Items used but not listed above were various coaxial cables and adaptors. A sampleplatform and a jig used for attaching the probe to an XYZ micro-positioner were also designedand built. The sample platform and probe jig with the probe mounted on it are shown in Figure5.2.2. A number of system controller and data acquisition programs for both computers werealso written to facilitate the measurements and subsequent analysis. The whole equipment setupis shown in Figure The Interference TechniqueA schematic of the setup used in the interference technique is shown in Figure 5.3.1.Here the directional coupler behaves both as a power divider and power combiner. The signalleaving the output of the directional bridge is split into two by the directional coupler. Thecoaxial short circuit on the 'output' port of the directional coupler reflects the signal back into95a Cy E_ -G^-C41 •Figure 5.2.1. Picture showing the network analyzer on top of the synthesized sweeper. Thedirectional bridge is attached to the output of the synthesizer sweeper.Figure 5.2.2. Picture showing the probe jig with probe mounted (left) and the sample platform(right). The stereoscopic microscope is also shown.96V •-•••-• • • • V• • * 2 0Ir-17„,atii^41:14jp1^VAI11111111^AS,a. Ai° CIFØl. ■--,_^I■ •%4 ' rir.,„411111•-011•1200--gillior%4111L--Figure 5.2.3. Picture showing the whole equipment setup. The two computers used are alsoshown,97COPLANARSTRIPPROBETEKTRONIXTMP9215SLOW-WAVEELECTRODESYNTHESIZEDSWEEPERHEWLETT PACKARDHP834IBRF OUTINDIRECTIONALBRIDGEHEWLETT PACKARDHP85027EOUTINDIRECTIONALCOUPLERMERRIMACCWM-8M-10GCOMPUTER(SYSTEM CONTROLLER)HEWLETT PACKARDHP9836REFLSYHP%NETWOR‹ANALYZERHEWLETT PACKARDHPB757CHPI9SYSHPIB COMPUTER(DATA ACQUISITION)IBM PC COMPATIBLEOUT COAXIALSHORTHEWLETT PACKARDHP0960-0065SAACPLFigure 5.3.1. Schematic of the setup used for the interference technique.98the directional coupler while the signal entering the electrode via the 'coupled' port will travelto the end of the electrode where it is reflected by the short. The signal reflected by theelectrode then enters the directional coupler and combines with that reflected by the coaxial shortcircuit. The phase difference between the reflected signal at the coaxial short circuit and that atthe input end of the electrode is a function of frequency. By sweeping the frequency one is ableto calculate the phase difference and thus the microwave effective refractive index Neff of theelectrode.By equating the phase change in the reflection between two adjacent peaks, one gets2Tc =-47c [6 o+Neip Afc(5.3.1)where c is the speed of light in vacuum, Af the frequency separation between two peaks (ortroughs), it the intrinsic phase difference due to the directional coupler, cables and probe, andL the length of the electrode. Without the probe touching the electrode equation (5.3.1) becomes2Tc =-47c 0 °Aic(5.3.2)where Af is the new frequency separation between two peaks. Subtracting equations (5.3.1) and(5.3.2) one gets the expression giving the NeffN fl.='[ 1- ] •e- 2L Af Al(5.3.3)99Usually many peaks occur for a typical frequency sweep from 5 to 20 GHz. One can thereforetake the average between a number of peaks. Equation (5.3.3) can be written asN c NI N2eff 2L if1 Oft(5.3.4)where N1 and N2 are the number of peaks corresponding to the frequency separations 4f; and 4f2 ,respectively.Due to the sharpness of the peaks, this technique has an estimated accuracy for Neff ofabout 0.08. This technique was used to measure the Neff of the shorted conventional coplanarstrip electrode of mask #1. The results are given in Figures 5.3.2a and 5.3.2b. Using equation(5.3.4) the Neff is calculated to be The Resonance TechniqueA schematic of the setup used in the resonance technique is shown in Figure 5.4.1. Herethe output of the directional bridge is connected directly to the coplanar strip probe. Due to thesmall impedance mismatch between the probe tip and the electrode, a portion of the energy willbe reflected back into the electrode at the probe tip, leading to resonance. The condition isdescribed in Appendix III. Using reasoning similar to that in section 5.3, Neff can be shown tobe given bycL N„=--2  Af(5.4.1)100where N is the number of resonance peaks in the frequency interval df.Since this technique relies on the resonance produced in the electrode, the peaks aregenerally very sharp. This makes locating the peaks easy and results in higher accuracy.Moreover, the elimination of the directional coupler and coaxial short removes the intrinsic phasedifference inherent in the interference technique. Consequently, the accuracy improves due tothe fact that only one frequency sweep is needed to obtain Neip whereas the interferencetechnique requires two measurements: one without lowering the probe to the electrode and onewith the electrode probed. Moreover, this technique enables one to calibrate the networkanalyzer at the probe tip so that more accurate and noise free measurements, as well as the lossof the electrodes, can be made. Due to these advantages, the error in Neff of this measurementtechnique is estimated to be less than 0.05.Measurements were made between 8 to 20 GHz. To calibrate the network analyzer at theprobe tips, the coplanar strip probe was first shorted by lowering it onto a pad of aluminumformed on a GaAs substate. This produced a plane of theoretically perfect reflection (i.e.,reflection coefficient of unity magnitude) at the probe tips. Due to the terminal capacitance atthe open probe tips, a well-defined plane of reflection cannot be realized easily by just leavingthe probe tips open in air. On the other hand, shorting the probe tips can easily establish a verygood plane of reflection. Consequently, calibration was made with the probe tips shorted. Thenetwork analyzer was then calibrated by measuring the normalized reflected power as a functionof frequency and storing the calibration data into the network analyzer's memory. By using thecalibration function of the network analyzer, all subsequent measurements were automaticallycalibrated by subtracting the measured results by the stored calibration data.10113.424 GHz JoP3O0-0• 1 . 1-aEOz5^8^11^14^17^20Frequency (GHz)Figure 5.3.2a. Measured normalized reflected power in dB, i.e., 10 log(reflected power/incidentpower), for the setup of Figure 5.3.1 with the probe not touching theconventional coplanar strip electrode of Mask #1.1025^8^11^14^17^20Frequency (GHz)Figure 5.3.2b. Measured normalized reflected power for the setup of Figure 5.3.1 with theconventional coplanar strip electrode of Mask #1 probed.103COMPUTER(SYSTEM CONTROLLER)HEWLETT PACKARDHP9636 HPIBSYSHPIBSYNTHESIZEDSWEEPERHEWLETT PACKARDHP63416DIRECTIONALBRIDGEHEWLETT PACKARDHP85027ERF OUTINREFLSYS.HPIBOUTNETWORKANALYZERHEWLETT PACKARDHP8757CCOMPUTER(DATA ACQUISITION)IBM PC COMPATIBLECOPLANARSTRIPPROBESLOW-WAVEELECTRODETEKTRONIXIMP9215Figure 5.4.1. Schematic of the setup used for the resonance technique.104The network analyzer was interfaced to an IBM PC compatible computer using anIEEE488 interface card. A data acquisition program was written to transfer the measured (andcalibrated) data to a disk file.Since this technique relies on the reflection at the probe tip, the magnitude of theresonance depends on how the electrodes are probed. Sometimes the resonance peaks weresmall; by intentionally lowering the probe off centre often resulted in larger peaks (in the idealcase when the probe was perfectly matched to the electrode, no resonance would occur). Largerresonance peaks enable one to locate the resonances more accurately and thus improve theaccuracy of the measurement. It was found, as expected, that how the electrodes were probedcould only affect the magnitude of the resonances, and not the frequency at which the resonancesoccur.5.5. Measured ResultsDue to the higher accuracy and ease of locating the peaks, all the results presented herewere obtained using the resonance technique. The measured Neff's for the electrodes fabricatedon semi-insulating GaAs are tabulated in Tables 5.5.1 to 5.5.3. The theoretically predictedvalues are also given. The normalized reflected power for slow-wave electrode #1 (half-buried)of Mask #2, electrode #2 (half-buried) of Mask #2, and electrode #5 (surface deposited) of Mask#2 are, respectively, shown in Figures 5.5.1, 5.5.2, and 5.5.1. Measured and theoretical Ne ff' of conventional coplanar strip electrode.Si(pm)WI(pm)t(pm)Etchant Comments Neff(theory)Neff(measured)17 133 1.7 NH4OH:H202 :H20(5:2:240)Half-buried 2.64 2.60Table 5.5.2. Measured and theoretical Ne ' s of slow-wave electrodes (Mask #1).Si = 32 pm, WI = 120 pm, S2 = 5 pm, W2 = 13.5 pm, d = 30 pm,1 = 3 pm, t = 1.7µmEtchant Comments Neff(theory)Neff(measured)NH4OH:H202 :H20(5:2:240)Half-buried 2.96 2.84HCEH202 :H20(1:4:40)Half-buried 2.96 2.87NH4OH:H202 :H20(1:1:8)Totally-buried - - - 2.95106Table 5.5.3. Measured and theoretical Neff  of slow-wave electrodes (Mask #2).S'1 = 60 Arn, S2 -= 4 gm, W2 = 28 Am, / = 4 Am, t = 1.1µmElectrode WI(AM)W'(pm)1'(gm)d(gm)Neff(theory,half-buried)NeffHalf-buriedNeffSurfacedeposited#1 72 0 4 18 3.50 3.40 3.33#2 110 0 4 32 3.16 3.10 3.03#3 110 7 8 32 3.28 3.24 3.17#4 110 7 12 32 3.39 3.32 3.30#5 110 7 16 32 3.49 3.43 3.38#6 110 14 8 32 3.30 3.25 3.18#7 110 14 12 32 3.44 3.33 3.32#8 110 14 16 32 3.56 3.46 3.43107811111[111 - 1IIIIIIIIIIIIIIIIIII10^12^14^16^18^20111118.835 GHz 1—12Frequency (CHz)Figure 5.5.1. Measured normalized reflected power as a function of frequency of electrode #1(half-buried) of Mask #2.108L_(1)00_CDN. _o - 12—E0z—15 'm i lli8^10II I IIII I IIII I IIII14^16^18^2016.480 GHz — — — —IFrecuency (G1-1z)Figure 5.5.2. Measured normalized reflected power as a function of frequency of electrode #2(half-buried) of Mask #2.10912 -IIJIIL ^ 8.758 0Hz—18 iiii i IIII I IIII I IIII I IIII I IIII8^10^12^14^16^18Frec uency (G Hz)Figure 5.5.3. Measured normalized reflected power as a function of frequency of electrode #5(unburied) of Mask #2.201105.6. Discussion of ResultsThe Neff of 2.60 for the conventional coplanar strip electrode as obtained using theresonance technique is, within two decimal places, the same as that obtained using theinterference technique. This value is slightly lower than the theoretically (quasi-static analysis)predicted value of 2.64, even though the difference is small, being less than the experimentalerror of 0.05. Visual examination of the SEM micrographs indicated that the electrode is, infact, only about 1/3 buried, which would result in a lower Nes,The slow-wave electrode of Mask #1 was fabricated using various etchants (see Table5.5.2). Two were half-buried and one was totally-buried. The measured Neff's for both half-buried electrodes are almost the same except that the one fabricated using HC1:H 202 :H20(1:4:40) has an Neff of 0.03 higher than that fabricated using NH4OH:H202 :H20 (5:2:240). Eventhough this difference is less than the experimental error, the small increase may in part beaccounted for by the smaller air gaps produced by the former etchant. Further evidence tosupport this assumption comes about by considering the Neff for the totally-buried electrode ofTable 5.5.2, which is about 0.1 higher than those of the half-buried ones, indicating that buryingthe electrode in the substrate does in fact increase its Neff. Also, all of the Neff's of the half-buried electrodes of Mask #2 are higher than those of the surface deposited electrodes. Eventhough we cannot conclude definitively that the half-buried electrodes have higher Neff's, as thedifferences in most cases are less than the experimental error, it is expected that this is so.Comparison of the results obtained for the surface deposited electrodes and the half-buriedelectrodes of Mask #2 indicates that half-burying electrodes with their fins (and pads, if111applicable) having a thickness to gap width ratio of less than about 0.3 would only result innegligible increases in Neff. Due to the extra fabrication steps involved, it may not be verypractical to half-bury electrodes since velocity-matched electrodes can be easily designed withouthaving to bury them. Moreover, the design curves of section 3.3.7 may also be used for surfacedeposited electrodes provided the electrodes are thin, even though they were calculated byassuming that the electrodes were half-buried.As a comparison with theory, the predicted Neff of the half-buried slow-wave electrodesof Mask #1 is 2.96 or 0.32 higher than that of the conventional coplanar strip electrode usingthe modified design formulas of section 3.3.5 while the measured value is 2.87 or 0.27 higherthan that measured for the conventional coplanar strip electrode. On the other hand, the Neffpredicted using the equations that do not take inductance into account (section 3.3.4) is 3.18 or0.54 higher than that of the conventional coplanar strip electrode. It is apparent that themodified design formulas provide better predictions of the New. In fact, if the increases in Neffare considered, the measured and theoretical results agree within the experimental error of 0.05.For electrodes #1 and #2 of Mask #2, the theoretically predicted Neff's are, respectively,3.50 and 3.16 using the modified design formulas of section 3.3.5, corresponding to,respectively, increases of 0.86 and 0.52 over the conventional electrode. The measured valuesof 3.40 and 3.10, corresponding to, respectively, increases of 0.80 and 0.50, indicate anagreement of over 93% in the increases of Neff between the experimental results and the modifieddesign formulas of section 3.3.5. For electrodes #3 to #8, the agreements are equally as good,despite the fact that the capacitances of the fins with pads were calculated using theapproximation formulas of section the experimental errors, the measured results are in very good agreement withtheory. As discussed above, processing problems such as the over-etch of the GaAs substrate,which resulted in more air around the fins, could also be in part responsible for the lowermeasured microwave indices. These problems, however, may be alleviated using dry etchingtechniques [38:pp. 173-195].Comparison between the various slow-wave electrodes of Mask #2 shows that the Neff ofa slow-wave electrode can be increased by reducing d, and by correspondingly adjusting for thechange in inductance per unit length due to the smaller fin period by changing Z0. Adding padsto the ends of the fins while keeping the same d could also result in significant increases in theNeff, However, the dimension W' does not seem to have a significant effect on the Neff's, asindicated by almost the same Net's between electrodes having pad widths W' = 7 pm and W'= 14 pm. This suggests that the electric fields are confined to the gap region. This is alsoevidenced by the small changes in the calculated capacitance of the fins with pads havingdifferent W' 's (see Table uniformity of the frequency separations between the resonance peaks (Figures 5.5.1to 5.5.3) suggests that the dispersion in the slow-wave electrodes is very low. In fact, almostno dispersion was detected. Moreover, at frequencies up to 20 GHz, the normalized reflectedpower in all cases is only a fraction of a dB below 0 at frequencies where anti-resonances occur,i.e., when the voltage and current waves are 90° out of phase and the normalized reflected poweris maximum. This indicates that the losses of the electrodes are well within 0.5 dB/cm up to 20GHz, comparable to conventional coplanar strips having similar dimensions [45:pp. 257-288].Moreover, the losses have little dependence on the frequency, at least up to 20 GHz. These113losses are smaller than the 0.7 dB/cm as calculated in section 3.3.8, which was somewhat over-estimated. Even though the calculated losses of 3 dB/cm in the slow-wave electrodes that areto be used in modulators are higher due to the smaller dimensions of the electrodes, they are stillquite small as compared to those reported by others [33].Results of the measurements (in particular, electrodes #1, #5, and #8 of Mask #2) indicatethat we have been successful in achieving the microwave/optical wave velocity-match conditionfor GaAs based electro-optic modulators using the slow-wave electrodes described in this thesis.The electrodes of Mask #2 can readily be scaled down to the dimensions needed in electro-opticmodulators. III-V semiconductor travelling-wave electro-optic modulators incorporating theseslow-wave electrodes are expected to be capable of providing very wide bandwidths, lowmodulating powers, and high modulation depths.114Chapter 6 Conclusions andRecommendations for FutureResearch6.1. ConclusionsCoplanar slow-wave electrode structures capable of matching the velocity of a microwaveto that of an optical wave in III-V semiconductor electro-optic modulators have been invented,analyzed, designed, fabricated, and tested. These electrode structures slow down microwave byexploiting the fringing electric fields produced by narrow capacitive loading fins and pads. Theslow-wave electrode structures also offer low loss, low dispersion, ease of fabrication, and canbe designed to match to transmission lines having 50 or 75 ohm characteristic impedances.Modulators employing these velocity-matched electrode structures should be capable of achievingvery wide bandwidths, low modulation powers, and high modulation depths.The slow-wave electrodes are periodically loaded with capacitive fins; rectangular padsmay also be added to their ends to further increase their capacitance. If the fins are narrow, thecapacitance per unit length between the electrodes can be substantially increased, whereas theinductance per unit length along the electrode will only be minimally affected, thus resulting inthe slowing of the microwave. The change in inductance per unit length due to the fins can becompensated for by changing the width of the coplanar strips. By choosing the fins and pads tobe used, electrodes having a prescribed microwave effective refractive index, as well ascharacteristic impedance, can be designed.115A quasi-static analysis was used to analyze the electrode structures. This was done usinga transfer matrix method. High frequency dispersion characteristics representing the upper limitof operation of the electrodes were calculated.Finite difference and finite element methods were used to calculate the capacitance of theloading fins. Results of the capacitance computation show that the narrower the fins, the higherthe capacitance to fin length ratio. In order to obtain a high enough degree of slowing of themicrowave, a high capacitance to fin length ratio is needed. The design formulas obtained usingthe transfer matrix method were simplified and modified, allowing easy application as well asimproved agreement with experimental results. Finally, design curves for 50 and 75 ohmelectrode structures suitable for use in both GaAs and InP based electro-optic modulators aregiven.Aluminum electrodes, both buried and surface deposited, having various dimensions havebeen fabricated on semi-insulating GaAs substrates using a combination of single-step lift-off andsubstrate etching techniques. The fabrication process for these electrode structures was provedto be straight forward and reproducible.The microwave effective refractive indices of the electrode structures fabricated weremeasured using a microwave scalar network analyzer. Both interference and resonancemeasurement techniques were used. Good agreement was found between the measured resultsand the modified design formulas. Results of the measurements show that 50 ohm electrodeshaving microwave effective refractive indices of 3.43 (i.e., matched to the effective refractiveindex of the optical wave in AlGa i _xAs/GaAs/AlGa l_„As waveguide structures) have beenachieved. These electrodes should enable modulators having very high performance to be116realized.The measurements of the slow-wave electrodes also showed that their losses anddispersions were very low. The uniformity in frequency separation of the resonance peaksindicated that the effective refractive indices of the microwave were constant up to at least 20GHz. Since the synthesized frequency source had a 20 GHz working limit, measurements couldonly be made up to this limit. However, the electrodes are expected to work at frequenciesmuch higher than 20 GHz.While the theoretical analysis presented is relatively simple, being based on the quasi-staticapproximation, the agreement between the predicted and measured results was quite good,especially when fabrication errors were taken into account.6.2. Recommendations for Future ResearchAlthough the losses indicated by the measurements were very low, it is possible to reducethem further. For example, thicker and wider electrodes can be designed so as to reduce theresistive losses; gold instead of aluminum can be used; coplanar waveguide structures, whichhave lower losses than coplanar strips, can also be designed using the same principles.Besides placing the fin elements (and pads) directly in line with one another, as done inthis thesis, capacitive loading fin elements can also be placed side by side to each other with eachof the fin elements attached to each of the coplanar strips so that a section of the pair of finelements overlaps in the interelectrode gap region. This kind of structure is also worthinvestigating.117By computing the electric field distribution around the electrodes, the overlap between themicrowave and optical fields can be found. The electrodes can then be optimized for the highestoverlap while providing the highest modulating electric field, thus allowing furthur reduction ofthe modulating power requirements.Besides optimizing the slow-wave electrode structures, further work should include thebuilding of III-V semiconductor based electro-optic modulators employing these slow-waveelectrode structures. Velocity-matched electro-optic modulators incorporating graded-indexoptical waveguides should also be designed. The mode profiles of these waveguides may beengineered to obtain their optimal overlaps with the microwave electric field as well as couplingwith optical fibres or semiconductor lasers. Modulators employing these velocity-engineeredslow-wave electrode structures should provide very wide bandwidths, low modulation powers,and high modulation depths. The ease of fabricating the electrodes also reduces the cost ofmaking these modulators.118References1. R. C. Alferness, Waveguide Electrooptic Modulators,' IEEE Trans. on MT7', vol. MTT-30, no. 8, pp. 1121-1137, 1982.2. T. Tamir (editor), 'Guided-Wave Optoelectronics,' Springer Series in Electronics andPhotonics 26, Springer-Verlag Berlin Heidelberg, 1988.3. N. A. F. Jaeger and Z. K. F. Lee, 'Slow-Wave Electrode for Use in CompoundSemiconductor Electro-Optic Modulators,' IEEE J. of QE. Accepted for publication.4. Z. K. F. Lee and N. A. F. Jaeger, 'Electrode Structures for Microwave to Optical WaveVelocity-match in III-V Semiconductor Electro-Optic Modulators,' IEEE CLEO Conf. ,Anaheim, CA., U.S.A., May 1992. Accepted for presentation.5. N. A. F. Jaeger and Z. K. F. Lee, 'Velocity-Matched Slow-Wave Electrodes forIntegrated Optic Electro-Optic Modulators,' 20th ICHSPP Conf. , Victoria, B.C., Canada,Sept. 1992. Accepted for presentation.6. N. A. F. Jaeger and Z. K. F. Lee, 'Slow-Wave Electrode,' US patent pending, filedSept. 1991.7. R. Jungerman, C. Johnsen, D. J. McQuate, K. Salomaa, M. P. 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Lee, `Low-Threshold 1.5 gm Compressive-StrainedMultiple- and Single-Quantum-Well Lasers,' IEEE J. of QE, vol. 27, no. 6, pp. 1440-1450, 1991.12016. S. Morin, B. Deveaud, F. Clerot, K. Fujiwara, and K. Mitsunaga, 'Capture ofPhotoexcited Carriers in a Single Quantum Well with Different Confinement Structures,'IEEE J. of QE, vol. 27, no. 6, pp. 1669-1675, 1991.17. S. Piazzolla, P. Spano, M. Tamburrini, 'Small Signal Analysis of Frequency Chirpingin Injection-Locked Semiconductor Lasers,' IEEE J. of QE, vol. QE-22, no. 12, pp.2219-2223, 1986.18. N. A. Olsson, H. Temkin, R. A. Logan, L. F. Johnson, G. J. Dolan, J. P. Van DerZiel, and J. C. Campbell, 'Chirp-Free Transmission Over 82.5 km of Single ModeFibers at 2 Gbits/s with Injection Locked DFB Semiconductor Lasers,' IEEE J. ofLightwave Tech., vol. LT-3, no. 1, pp. 63-67, 1985.19. C. Lin, and F. Mengel, 'Reduction of Frequency Chirping and Dynamic Linewidth inHigh-Speed Directly Modulated Semiconductor Lasers by Injection Locking,' Electron.Lett., vol. 20, pp. 1073-1075, 1984.20. 0. Lidoyne, P. B. Gallion, and D. Erasme, 'Modulation Properties of an Injection-Locked Semiconductor Laser,' IEEE J. of QE, vol. 27, no. 3, pp. 344-351, 1991.21. T. H. Wood, 'Multiple Quantum Well (MQW) Waveguide Modulators,' IEEE J. ofLightwave Tech., vol. 6, no. 6, pp. 743-757, 1988.22. S. Y. Wang, and S. H. Lin, 'High-Speed III-V Electrooptic Waveguide Modulators atA = 1.3 gm,' IEEE J. of Lightwave Tech., vol. 6, no. 6, pp. 758-771, 1988.23. F. Koyama and K. Iga, 'Frequency Chirping in External Modulators,' IEEE J. ofLightwave Tech., vol. 6, no. 1, pp. 87-93, 1988.12124. K. Kawano, T. Kitoh, H. Jumonji, T. Nozawa, and M. Yanagibashi, 'New Travelling-Wave Electrode Mach-Zehnder Optical Modulator with 20 GHz Bandwidth and 4.7 VDriving Voltage at 1.52 p.m Wavelength,' Electron. Lett., vol. 25, pp. 1382-1383, 1989.25. P. Buchmann, H. Kaufmann, H. Melchior, and G. Guekos, 'Broadband Y-BranchElectro-Optic GaAs Waveguide Interferometer for 1.3 gm,' Appl. Phys. Lett., 46(5), pp.462-464, 1985.26. A. Yariv and P. Yeh, 'Optical Waves in Crystals,' Wiley, New York, 1984.27. M. S. Whalen and J. Stone, 'Index of Refraction of n-type InP at 0.663 Am and 1.15 AmWavelengths as a Function of Carrier Concentration,' J. Appl. Phys., 53(6), pp. 4340-4343, 1982.28. J. P. Donnelly, N. L. DeMeo, F. J. Leonberger, S. H. Groves, P. Vohl, and F. J.O'Donnell, 'Single-Mode Optical Waveguides and Phase Modulators in the InP MaterialSystem,' IEEE J. of QE, vol. QE-21, no. 8, pp. 1147-1151, 1985.29. S. H. Lin., S. Y. Wang, and Y. M. Houng, 'GaAs p-i-n Electro-optic Traveling-WaveModulator at A = 1.3 am,' Electron. Lett., vol. 22, pp. 934-935, 1986.30. S. Y. Wang, S. H. Lin, and Y. M. Houng, 'GaAs Travelling-Wave Polarization Electro-optic Waveguide Modulator with Bandwidth in excess of 20 GHz at 1.3 am,' Appl. Phys.Lett., vol. 51, pp. 83-85, 1987.31. I. Kim, M. R. T. Tan, and S. Y. Wang, 'Analysis of a New Microwave Low-Loss andVelocity-Matched III-V Transmission Line for Travelling-Wave Electro-opticModulators,' IEEE J. of Lightwave Tech., vol. 8, no. 5, pp. 728-737, 1990.12232. J. Nees, S. Williamson, and G. Mourou, '100 GHz Traveling-Wave Electro-Optic PhaseModulator,' Appl. Phys. Lett., vol. 54, pp. 1962-1964, 1989.33. H. Y. Lee, and T. Itoh, 'GaAs Travelling-Wave Optical Modulators using ModulatedCoplanar Electrodes with Periodic Cross-Tie Overlays,' Technical Report, Texas Austin Microwave Lab., Report no: MW-89-P-5; ARO-25045.32-EL, 1989.34. T. Wang, and T. Itoh, 'Confirmation of Slow Waves in a Crosstie Overlay CoplanarWaveguide and its Applications to Band-Reject Gratings and Reflectors,' IEEE Trans. onM7T, vol. 36, no. 12, pp. 1811-1818, 1988.35. T. H. Wang, T. M. Wang, and T. Itoh, 'Experimental Confirmations of Slow-Wave ina Crosstie Overlay Coplanar Waveguide and its Application to Band-Reject Gratings,'1988 IEEE MIT-S International Microwave Symposium Digest, pp. 383-386.36. R. G. Walker, 'High-Speed III-V Semiconductor Intensity Modulators,' IEEE J. of QE,vol. 27, no. 3, pp. 654-667, 1991.37. R. Walker, 'High-Speed Electrooptic Modulation in GaAs/GaA1As Waveguide Devices,'IEEE J. of Lightwave Tech., vol. LT-5, no. 10, pp. 1444-1453, 1987.38. R. Williams, 'Modern GaAs Processing Methods,' Artech House, Massachusettes, 1990.39. H. Hasegawa, and S. Seki, 'Analysis of Interconnection Delay on Very High-SpeedLSI/VLSI Chips Using an MIS Microstrip Line Model,' IEEE Trans. on Electron.Devices, vol. ED-31, no. 12, pp. 1954-1960, 1984.12340. Y. Fukuoka, Y. Shih, and T. Itoh, 'Analysis of Slow-Wave Coplanar Waveguide forMonolithic Integrated Circuits,' IEEE Trans. on MIT, vol. MTT-31, no. 7, pp. 567-573,1983.41. H. Hasagawa, and H. Okizaki, `M.I.S. and Schottky Slow-Wave Coplanar Striplines onGaAs Substrates,' Electron. Lett. , vol. 13, pp. 663-664, 1977.42. G. W. Hughes, and R. M. White, 'Microwave Properties of Nonlinear MIS and SchottkyBarrier Microstrip,' IEEE Trans. on Electron. Devices, vol. ED-22, no. 10, pp. 945-956,1975.43. D. Jager, 'Slow-Wave Propagation along Variable Schottky-Contact Microstrip Line,'IEEE Trans. on MTT, vol. MTT-24, no. 9, pp. 566-573, 1976.44. Y. Fukuoka, and T. Itoh, 'Slow-Wave Coplanar Waveguide on Periodically DopedSemiconductor Substrate,' IEEE Trans. on MIT, vol. MTT-31, no. 12, pp. 1013-1017,1983.45. K. C. Gupta, R. Garg, and I. J. Bahl, `Microstrip Lines and Slotlines,' Artech House,Massachusettes, 1979.46. R. E. Collin, 'Foundations for Microwave Engineering,' McGraw-Hill, New York, 1966.47. A. F. Harvey, 'Periodic and Guiding Structures at Microwave Frequencies,' IRE Trans. ,vol. MTT-8, pp. 30-61, 1960.48. A. W. Lines, G. R. Nicoll, A. M. Woodward, 'Some Properties of Waveguides withPeriodic Structure,' Proc. IEE, vol. 97, pt. III, pp. 263-276, 1950.49. Cominco Electronic Materials Data Sheet, no. 204A, November 1982.12450. D. H. Sinnott, G. K. Cambrell, C. T. Carson, and H. E. Green, 'The Finite DifferenceSolution of Microwave Circuit Problems,' IEEE Trans. on MTT, vol. MTT-17, no. 8,pp. 464-478, 1969.51. D. H. Sinnott, 'The Use of Interpolation in Improving Finite Difference Solutions ofTEM Mode Structures,' IEEE Trans. on MIT, vol. MTT-17, no. 1, pp. 20-28, 1969.52. J. W. Duncan, 'The Accuracy of Finite Difference Solutions of Laplace's Equation,'IEEE Trans. on MIT, vol. MTT-15, no. 10, pp. 575-582, 1967.53. A. E. Ruehli, and P. A. Brennan, 'Efficient Capacitance Calculations for Three-Dimensional Multiconductor Systems,' IEEE Trans. on MTT, vol. MTT-21, no. 2, pp.76-82, 1973.54. P. S. Maruvada, and N. Hylten-Cavallius, 'Capacitance Calculations for some Basic HighVoltage Electrode Configurations,' IEEE Trans. on PAS, vol. PAS-94, no. 5, pp. 1708-1713, 1975.55. Dupont PI-2525 polyimide data sheet, 1986.56. M. S. Whalen and J. Stone, 'Index of Refraction of n-type InP at 0.633 Am and 1.15 gmWavelengths as a Function of Carrier Concentration,' J. of Appl. Phys., 53(6), June1982.57. S. Ramo, J. Whinnery, T. V. Duzer, 'Fields and Waves in Communication Electronics,'Wiley, New York, second edition, 1984.58.^S. Iida and K. Ito, 'Selective Etching of Gallium Arsenide Crystals in H 2SO4-H202-H20System,' J. Electrochem. Soc.: Solid State Science and Technology, vol. 118, no. 5, pp.768-771, 1971.12559. D. W. Shaw, 'Localized GaAs Etching with Acidic Hydrogen Peroxide Solutions,' J.Electrochem. Soc.: Solid State Science and Technology, vol. 128, no. 4, pp. 874-880,1981.60. S. Adachi and K. Oe, 'Chemical Etching Characteristics of (001) GaAs,' J. Electrochem.Soc.: Solid State Science and Technology, vol. 130, no. 12, pp. 2427-2435, 1983.61. S. H. Jones and D. K. Walker, 'Highly Anisotropic Wet Chemical Etching of GaAsUsing NH4OH:H202 :H20,' J. of Electrochem. Soc., vol. 137, no. 5, pp. 1653-1654,1990.126Appendix I. Conventional Coplanar StripsAccording to the quasi-static analysis, the characteristic impedance Z, of a pair of coplanarstrips having a width W, thickness t, and separated by a gap S on a substrate of thickness h,dielectric constant € ,., under a superstrate of dielectric constant E s (Figure 1.1) is given by[45:pp. 257-288] as1207E ace) Zo-g Ki(ke ' e)where(1-k2)A ke-k2WS k-S+2W 'A - 1.25/1 -Fin( 4n FV)] ,n^t(IA )(1.3)(1.4)1271.4(ere -1)t/Seeere ^ ; e,O, t/W<0.1 ,[Kl(k)IK(k)]+1.4tIS Erz9, er2e s [tanh(1.785log(h/W)+1.75)+—kW1 0.04-0.7k+0.01[1-0.1(e r+e s-1)](0.25 -4)1 ,h i(1.6)1K(k)  _ 1 in2 1+Vic ; 0.707sks1 ,K'(k) 71 . 1 -IAK(k)  _K'(k) IC k/41 -k2 ; Osks0.707 . (1.8)InFrom the formulas as given above, it can be easily seen that Z, depends only on the ratioS/(S+2W) if the coplanar strips are very thin (i.e., t approaches zero) and the substrate verythick (i.e., h approaches infinity). As discussed in sections 3.3.9 and 3.3.10, this propertyallows thin conventional coplanar strips and thin coplanar slow-wave electrodes to be scaled up(or down) while having little changes in the microwave effective refractive indices andcharacteristic impedances. This property also makes the design of input sections easy.128e SupersEraLe11^  W 0^11 S 141^W 0^17777E r SubstrateiFigure 1.1. A coplanar strip electrode on a substrate. The dimensions S, W, t, and h areshown.The capacitance per unit length C between a pair of coplanar strips is given asC-e o e r +e s r(k)2 K(k)(1.9)where 6 0 is the permittivity of free space. Equation (1.9) is useful for calculating the capacitanceof fins with pads as described in section 3.3.5.For coplanar strips on GaAs, conductor losses are much higher than dielectric losses [45:p.73, pp. 257-288]. For the purpose of our calculations, the dielectric losses are ignored. Theconductor losses a„ (in dB/unit length) can be calculated using Wheeler's incremental inductanceformula [45:pp. 257-288], as given by129where1+ 1.25 in  4Tc W,  1.25tt^TC W —1R 111+Wa cs = 17 .34 ; TES^S) 11 +2 W + 1.254 1 +in  47c WI[^S^ITS^t )(I.10)P,=  K(k)  fp 9K' (k) k for 0.0dcs0.707for 0.707 sks 1.0P=(1 - si 1 - k2)(1 -k2)314r1 ^( (k))2(1 _ k) Na K(k)9(I.12)and Rs is the surface resistivity given by [57:pp. 151-153] asRs=3.26 x 10 -7(f (Ohm)^ (I.13)for aluminum electrodes andRs=3.03 x 10 -711-  (Ohm)^ (I.14)for gold electrodes, where f is in Hz.130Appendix II. Coefficients of Inverse CapacitanceFrom reference [53], the PiIs of section 3.3.6 are given in two general cases for cellsoriented parallel and perpendicular to each other as follows (see Figure II.1):(1) Cells oriented in parallel:k=4 m=4[ 2 2,,,^m-coE E (-1)k- b^akln(ak+ p)+4ne^1o fa fb sa sbk. 1 m=1 22 2ak-cii 1 2 a ,b^bm1n(bm+p)- (b.-2cii+a 2k)p bmcoaktan-1 - m2 6^ pcii(IIA)wherep = (ak2 +bm2 +cii2.) 1/2^ (II.2)fa sa'al = a:,--2 --2 ,(II.3)fa Saa2 = au+ — - — ,i 2 2(11.4)131fa Saa = a.. -F — + — '3^li 2 2(II.5)fa Soa = a-- - + - '4^V 2 2 (II.6)fb Si,b1 = bu--i ---2- , (II.7)sbb2 = biff.b--27 , (II. 8)b3 = bfb sbi . 4--ri- ,I (II.9)(II. 10)132(2) Cells oriented perpendicular to each other:^2 ^24 2 2Pi^ak ci1.'I-  47ce o fa fb sa s_ b r, i ,,E. , E,=, ( - 1 )1 + m + k + 1 2 - —6 clin(am+P)+^(a 2^,22bmcip- 1 „ b millfr 1+ P)+ a kb mc PO k+ P)^3^3 ^2^2a k^1 bmc1 bmak -1, akCI . akCI^-ii  akbm)_tan- ()^ tan k_..._)^tan k6^akp^2^bmp^2^cipwhere p and ak are the same as for parallel cells andSb, = bv-F-1 , (II.12)b2 = bii-1) ,^(II.13)c, = cICe-- ,2(11.14)^c ^fc., = c;;-- .... .,^2 (II.15)133SbCells orientedin porollel^cijFaCells orientedperpendiculorto each otherSbSe,Figure II.1. Dimensions for cells oriented parallel and perpendicular to each other.134Appendix III. Conditions for ResonanceAs discussed in Chapter 5, resonances occur in the electrodes due to impedance mismatchat the probe tip. This happens when waves are partially reflected at the electrode input andoutput ends. Figure III. 1 shows an electrode having a characteristic impedance Zo terminatedby a load of impedance ZT and on the input end is a source V,. having an impedance 4 (thisrepresents the impedance of the probe, cable, and directional bridge).Suppose at a particular instant a voltage wave starts to propagate along the electrode. Theinstantaneous voltage at the input end of the electrode will be VZo/(Z„. + Zo) since the wave hasnot yet travelled to the other end and so there is no reflection. As the wave propagates toposition z the voltage will become [11,20/(4. + Zo)Je- rz, where y is the propagation constant.When the wave reaches the load, it will be reflected by the load reflection coefficient PTZT ZoP T— ZT+ZoThe voltage of the first reflected wave at position z then becomesVsZo -yL^-Y(1,--z)e p 7e Z+Zs o(III.2)This first reflected wave will again be reflected at the input end by the input reflection coefficientPs135Zs—ZoPs— Zs+Zo(III.3)This second-time reflected wave will have a voltage at position z equal toVsZo  e -yLpe -yLpse -yzZs+Zo(III.4)Due to superposition the steady state voltage at position z is the sum of an infinite number ofreflection terms, i.e.,VsZV(z) —^ (e -yz + p 7e -2yLe yzp + p 7.p se -2yL+ ( p 7,p se -2yL)2 + ...] .Zs +Zo(III.5)Substituting (L - d) for z and simplifying one obtainsV(d)-VsZoe -YL eYd+p ie - I'd•^(III.6)Zs+Zo 1 — p rp se -2YLSince the reflection coefficient r of the 'resonant' electrode is related to V(d) by V(d) = V+ (1+ r), r can be obtained by noting that V+ is simply the V(d) when ps and PT are zero, i.e., bothsource and load are perfectly matched to Zo. At the input end, d = L and so r is given by136r- ^2Z0(1+p le -2YL)^1 .^ (III.7)(Zs+Zo)(1-p Tp se -2YL)Equation (III.7) implies resonance whenever e-2YL = 1 or -1, depending on pT and ps. In otherwords the return loss will be maximum (r minimum) at certain resonance frequencies separatedby dfaf_ c 1- 2L Neff(III.8)where c is the speed of light in vacuum. As discussed in Chapter 5, equation (III.8) is usefulfor calculating Neff by measuring the separations Lif between the peaks of the resonances.Source^Electrode^Load Z oZ d^p^0Lki^ 01Figure III. 1. Equivalent circuit of an electrode having a characteristic impedance Zo ,terminated by a load ZT and connected to a source Z, with voltage V.137


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