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An apparatus for the measurement of the surface resistance of high temperature superconducting thin films Knobel, Robert 1995

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A N A P P A R A T U S F O R T H E M E A S U R E M E N T O F T H E S U R F A C E R E S I S T A N C E O F H I G H T E M P E R A T U R E S U P E R C O N D U C T I N G T H I N F I L M S By . Robert Knobel B.Sc. (Honours, Engineering Physics), Queen's University A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of PHYSICS We accept this thesis as conforming to the required standard * THE UNIVERSITY OF BRITISH COLUMBIA 1995 © Robert Knobel, 1995 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) A b s t r a c t A n apparatus to measure the millimeter-wave surface resistance of thin films of high temperature superconductors has been buil t . The apparatus consists of a flow cryostat, an open resonator and a swept-frequency millimeter-wave source/detector pair. The flow cryostat is used to cool the experiment to cryogenic temperatures (either 2 K with l iquid helium, or 70 K with l iquid nitrogen), while keeping a very short distance between the experiment and room temperature. The cryogenic fluid is drawn through the cryostat, and circulates through heat exchangers, absorbing incident heat. The open resonator allows sensitive measurement of the surface resistance of the film at high frequency. The resonator built operates at 117 G H z , and has a Q for copper at 77 K of 39000. The millimeter-wave source and detector are taken from a fixed frequency apparatus, and were modified to provide a frequency sweep capability. The source and detector operate in a band between 116.8 and 117.6 G H z . Measurements were performed on both metallic and superconducting samples. The metallic samples were used as calibration of the geometric factors of the resonator, in order to extract the value of the surface resistance. Four thin films of high temperature superconductors were measured, as a demonstration of the capabilities of the apparatus. n Table of Contents Abstract n List of Tables < vi List of Figures vii Acknowledgement ix 1 Introduction 1 1.1 Surface Impedance 3 1.2 Electrodynamics of Superconductors . . , 6 1.3 Motivation • . . . . : . 9 2 The Flow Cryostat 14 .2.1 Introduction ' . . . . . . . " . * . • . . . . • . . ' . , . 14 2.2 Cryogenic Theory 1*7 2l3 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . '. 21 3 Theory of the Open Resonator . , v •) 33 3.1 Introduction 33 3.2 History • • • 33. 3.3 Beam-Wave Theory 35 3.4 Cavity Losses . : ; 39 3.5 Beyond the Beam-Wave Theory 46 iii \ 4 Millimeter-Wave System 50 4.1 Introduction 50 4.2 Millimeter-Wave System •. , . 50 4.3 Resonator Design • • 54 4.4 Coupling : • • • 56 4.5 Microwave measurements 59 5 Experimental Method 62 5.1 Introduction 62 5.'2 Metallic Samples • • •. .63 5.3 Measurement Procedure ' . : . . - . .64 6 Data 76 6.1 Introduction : . . . . . . . 76 6.2 Data : . . . 77 6.2.1 Furukawa Electric Bi2Sr2CaiCu208 Thick Fi lm . . . . . . . . .' 77 6.2.2 U . B . C . Y B a 2 C u 3 0 7 film on LaAlOg 79 6.2.3 U.B-.C. Y B a 2 C u 3 0 7 film on S r T i 0 3 . . . . . . . . . .... . .;". \ 80 6.2:4 McMaster Y B a 2 C u 3 0 7 _ s l.OOOOA film.on LaAlOg . . . . . . . . 81 6.2,5 Comparison . . . . . . . / . . . • . . . . • . . . . . ••. 84 6.3 Finite Thickness Effects . . . . . . . . . . . 86 7 Conclusions and Discussion 91 7.1 Introduction : : . . . . 91 7.2 Equipment Performance . 91 7.3' Surface Resistance Data . . 92 7.4 Future Work 93 iv Bibliography 97 v List o f Tables 2.1 Heats and temperatures for the numerical model of the final flow cryostat design. The flow of liquid helium is 0.5 L/hr, and the pressure is 0.1 atm. 29 5.1 Fitting parameters for various functions . . 73 vi List of Figures 2.1 Photograph of the flow cryostat . 15 .2.2 Photograph of the flow cryostat with shields removed 16 2.3 Schematic cut-away drawing of the flow cryostat 20 2.4 Isometric cut-away drawing of the flow cryostat. • 23 2.5 Heat conduction as a function of inner pipe length 25 2.6 Temperatures as functions of inner pipe length 26 2.7 Heat conduction as a function of outer pipe length . .' 27 2.8 Temperatures as functions of outer pipe lengths ' 27 2.9 Temperature of inner pipes in final model . 29 2.10 Temperature of gas in the exchanger in the final model 30 2.11 Temperature of gas in the outer pipes in the final model 30 3.1 Terminology and geometry of open resonators 34 3.2 The semi-planar hemispherical geometry of the open resonator as used in the experiment 38 3.3 Field amplitude numerical simulation 40 3.4 Relative intensities of the cartesian components of the electric field in the complex point-source theory 48 4.1 Block diagram of the millimeter-wave system . . 51 4.2 Schematic diagram of waveguide circuit 53 4.3 The design of the resonator and sample positioning apparatus . . . . . . 55 .4.4 Mode chart for the resonator as built 57 vii 4.5 Numerical simulation of Q for copper at 77K . . . 58 4.6 Numerical simulation of the resonator Q as a function of sample loss . . . 59 4.7 Lumped-element equivalent circuit of a resonator connected between a , generator and a detector. 60 5.1. Surface resistance of copper at 117 GHz calculated from d.c. resistivity data 64 5.2 Comparison of the detected signal amplitude for direct transmission (ie. no resonator) for standard flanges and choke flanges. . . . . . . . . . . . . 67 5.3 Q vs. temperature for brass and copper . . ' 69 5.4 Resonant frequency as a function of temperature for brass and copper. . 70 5.5 F i t t ing of resonant peak 71 5.6 F i t ted residuals for a sample resonant peak - 72 5.7 Q vs. temperature for various fitting functions ' 74 5.8 Rs of copper and brass at 117 GHz as a function of temperature . . . . . 75 6.1 Furukawa Electr ic B S C G O thick film surface resistance 77 6.2 U . B . C . Y B C O on L a A l O substrate thin film surface resistance 79 6.3 U . B . C . Y B C O on S r T i O substrate thin film surface resistance 81 6.4 McMaster Y B C O on L a A l O substrate thin film surface resistance . . . . 82 6.5 Comparison of surface resistance measurements on two McMaste r films at different frequencies . 83 6.6 Comparison of the surface resistance of superconducting films. . . . . . . 84 6.7 Comparison of the surface resistance of superconducting films from the literature. 89 6.8 Effective surface resistance for infinite substrate 90 6.9 Effective surface resistance for a 1mm thick substrate backed by copper. . 90 vm Acknowledgement I would like to thank, first of al l , my supervisor Walter Hardy. His guidance and expertise have made this project possible. I 'm indebted to him for his patience and advice that have helped me grow as a scientist and a person. I would like to thank as well Doug Bonn, who has been a leader by example in the laboratory as well as a friend. The example of what a scientist should be presented by these two wi l l be a goal to which I wi l l strive. The high temperature superconductivity group as a whole has helped me immensely, through encouragement, help and by making the lab an enjoyable place in which to work. Thanks, go to Dave Morgan, K u a n Zhang and Saeid K a m a l who have helped with friendship and scientific insight since I began, and to Chris Bidinost i , Andre Wong who have helped since they began. Thanks to Dave Baar, Ruix ing Liang and Pinder Dosanjh who have shown me the entrepreneurial side of science - encouraging me to open my mind to opportunities. Q . Y . M a and Andre Wong are to be thanked for develloping the collaboration of f i lm growth and film characterization which is the center of this project. I am very much indebted to the U.B . .C . physics machine shop. The whole staff have helped through advice and conscientious work. I owe a special debt of gratitude to George Babinger, who built most of the cryostat, and provided essential advice for the rest of the project. Thanks to Pinder Dosanjh for great technical help throughout the project, and (with Levi Waldron) for measuring the D . C . resistivity of metals. I owe a huge thanks to Wendy Braham, whose support and faith in me never failed through the hard times. Thanks to Andrew H i l l , who has been a good friend and room-mate during the last year. Thanks to my parents and sisters.who have always encouraged ix me, and, despite the miles apart, I've always-counted ori.them. Thank you to the Natural Sciences and.Engineering Research Council, for the financial support of the PGS scholarship.' The U . B . C . physics department also gets my thanks, for financial support thrqugh scholarships and teaching assistantships. x Chapter 1 Introduction In 1986, Bednorz and Muller discovered superconductivity in lanthanum barium copper oxide at 35 K [1]. This discovery opened a new chapter and renewed interest in the study of superconductivity. Kamerlingh Onnes in Leiden discovered the first superconductor, mercury, in 1911 shortly after he developed the techniques of liquifying helium [2]. Many other metals were subsequently found to undergo a transition to superconductivity within a few degrees of absolute zero. The phenomenon has been studied both theoretically and experimentally in the intervening years and was thought to be a well understood phenomenon until the recent discoveries of high temperature superconductivity and other exotic superconductors. The theory of superconductivity developed in 1957 by Bardeen Cooper and Schrieffer [3] (called the BCS theory) has been a remarkable success, both in terms of the intuitive understanding of the mechanism that it imparts, as well as in its predictions of experimental results. This theory, with extensions by many other researchers, explains very well the behaviour of so-called "conventional superconductors" (circularly defined as those materials that are well described by the theories). The discovery of superconductivity in the LaBaCuO system was soon followed by the discovery of other cuprate compounds with superconducting transition temperatures (Tc) even higher (YBa 2Cu 307 has a Tc of 93 K for example). With T^ 's higher than the boiling point of liquid nitrogen (77 K), the refrigeration needed to achieve supercon-ductivity has become relatively inexpensive, compared to the cost of the liquid helium needed to cool conventional superconductors. This latter requirement has limited the 1 Chapter 1. Introduction 2 applications where superconductors can be cost effective. W i t h high temperature su-perconductors, the possibility exists for many applications which, up unt i l now, have been far too expensive to pursue. It may soon be possible to have magnetically levitated trains, low loss powerlines, and high speed superconducting computers. However the new cuprate high temperature superconductors are not understood very well, and are difficult to fabricate due to the complicated chemical structure. The simplest method of fabrication, to simply mix and bake the ingredients, is capable of producing a superconductor, but the material is polycrystalline with very poor properties. The ma-terials are britt le ceramics, which cannot be simply drawn to form wires. These material problems have l imi ted the insight into the fundamental properties given by experiments, since many extrinsic effects mask the true intrinsic nature of the superconducting mech-anism. Single crystals of superconductors are being made successfully [4], but are small and difficult to grow. They are currently useful for determining the basic properties of the materials, but not for applications. Most applications of high Tc superconductors use thin films. These are made by depositing layers of the superconducting material (by various methods including sputtering, laser ablation, chemical vapour deposition, etc.) on a dielectric substrate material whose lattice parameters closely match those of the su-perconducting crystal. These films can be patterned using methods similar to those used in the semiconductor industry, and a variety of devices can be made. These processing methods are becoming very highly developed, and have enabled some very complicated devices to be envisioned. Many different types of transistor are being devised, and S Q U I D elements for magnetic sensing as well as Josephson junction switching elements have all been produced using thin films of superconducting materials. Perhaps the most likely prospect for the first commercially successful applications of high Tc superconductors is in passive microwave devices [5]. These filters, delay lines, Chapter 1. Introduction 3 interconnects, etc. all can be made to have superior properties if the conductors used have low loss. Good quality high Tc superconducting thin films at 77 K have lower loss than any standard conductor at microwave frequencies [6] (though not zero loss, cf. section 1.2). The quantity quoted most often for the electromagnetic loss at microwave frequencies is the surface resistance Rs- This quantity is an effective measure of the "quality" of the film, both in terms of its direct application in microwave use, as well as a general indication of the number of defects in the film (grain boundaries, lattice strain etc.). Such defects are common in films, and must be eliminated through tuning of the many parameters used in film growth. The purpose of the project described in this thesis is to build an apparatus to measure the surface resistance of films of high temperature superconductors. Such measurements are very commonly made [6], and have been made in this laboratory. However, the plan for this project is to have a device which is particularly suitable for thin films, and allows rapid, non-destructive testing for film-growth optimization. 1.1 Su r face I m p e d a n c e The concept of surface impedance is important for normal metals as well as superconduc-tors. It will be instructive to outline where this quantity comes from in both cases. The microwave loss in the best normal metals and the best high temperature superconductors (at 77 K) is about the same at 100 GHz — though the mechanism is quite different. In this section I will define the concept and look at normal metals. Consider a good conductor for which Ohm's law applies (this derivation follows that in [7]): 3 = *E. (1.1) Chapter 1. Introduction 4 We can use Maxwell ' s equations (assuming harmonic time variation of the fields e3Ujt): V D = p (1.2) V B = 0 (1.3) V x E = -juB (1.4) V x H = J+juD. (1.5) Inserting equation 1.1 into equation 1.5, we get V x H = (a + ju>e)E. (1.6) Equations 1.3 and 1.1 imply p = 0. If we can neglect the displacement current (cr cue, which is a good approximation up to optical frequencies in metals), then we can write (in various ways): V 2 E = jufiaE (1.7) V 2 H = ju^o-H (1.8) or V 2 J = jup.aJ. (1.9) The solutions to these equations depend on the particular boundary conditions, the simplest of which is to consider a semi-infinite slab of conductor of infinite depth (z > 0). This case, though artificial, is actually very important because at high frequencies most bodies are much larger than the depth of penetration of the electromagnetic fields. For a uniform field pointing along the plane of the conductor (say in the x direction), the differential equation becomes: d2Er = junaEx = (32EX (1.10) Chapter 1. Introduction 5 Which has the general solution E ^ C i e - ^ + cV* (1.11) Where the propagation constant 8 is most often written as p = 1 - ^ 1 ( L 1 2 ) S = J ^ . (1.13) For Ex to remain finite as z —> oo, C2 = 0. Writing C i as i£ 0, the field at z = 0, we can write: = £ 0 e _ / 3 2 = E 0 e - z / s e - l z / s (1.14) with similar equations for both the magnetic field H and current density J. The quantity <5 is the skin depth in the material, and is in general a complex quantity (since a is a complex quantity). However, in a normal metal the real part of the conductivity dominates at millimeter-wave frequencies, and thus S is, to a very good approximation, real. The value of 6 is the depth at which the fields have decayed to 1/e their value at the surface. This description is valid, and the quantity of the skin depth is appropriate, for the situations where: a >^ u>t; where the dimensions of the sample are large in comparison to the skin depth; and where the mean free path of the electrons in the conductor is short in comparison to the skin depth (cf. chapter 5). The total amount of current carried in such a semi-infinite conductor (of unit width) is given by integrating the current density (given by the equivalent equation to 1.14): Ix = r Jxdz = r Joe-t+M'Mdx = ~~——~:• (1.15) Jo Jo 1 + J The surface impedance Zs is defined as the ratio of the electric field on the surface to the current carried in the conductor (here for unit width): Chapter 1. Introduction 6 The surface impedance is often separated into its real and imaginary parts Zs = Rs+jXs (1.17) = H . r l ± i ) + I m f l ± i ) (1.18) which, for a normal metal where Im(<5) w Im(cr) ~ 0, gives Zs-_Rs + jXsKLs + ]Lr_Jfa+,Jfa (,19) where the real part Rs is the surface resistance and the imaginary part Xs is called the surface reactance. The surface resistance and surface reactance are measured in the same units as resistance (Ohms). The surface impedance measured across two opposing edges of a square is independent of the size of the square. For this reason, the units used for surface reactance and impedance are often Cl/O, or Ohms per square. For more general geometry, where the above definition isn't obvious, the surface impedance is defined as [8] n x E = Z 5 n x H x fi (1.20) along a surface wi th normal n, where Zs w i l l in general be a complex-valued tensor for anisotropic materials (where a is a tensor). The power dissipated i n the conductor is proportional to the surface resistance P = \Rs f I |n x H x n\2dS. (1.21) Z J J Surface 1.2 Electrodynamics of Superconductors In this section a brief overview of the electrodynamics of superconductors w i l l be given. This work has been the subject of much effort over the years. For a more complete discussion of particular points, see the references which wi l l be cited throughout the Chapter 1. Introduction 7 text. For more complete summaries, see the reviews of T inkham, Waldram, or K l e i n ([9][6][10]). There are a number of theoretical models used to describe the electrodynamic be-haviour of superconductors, both microscopic and empirical [11] [12] [13]. The model described here is a "generalized two-fluid model", based on a number of papers from this lab [14] [15], but the basic idea was originally proposed, i n a more specific way, by Gorter and Casimir [16] and by London [17]. The basis of this model is that the conduction electrons in a superconducting material can be thought of as being in two types: the normal electrons and the superconducting electrons. The normal electrons behave like electrons do in a non-superconducting material: dissipating energy by scattering. The superconducting electrons (often referred to as the superfluid) however do not dissipate any energy. The dynamic behaviour of these two electron types can be described through transport equations [18]: m ^ L + m ^ = N n e 2 E ( L 2 2 ) dt T = Nse2E (1.23) dt where JV n and Ns are the number density of normal and superconducting electrons re-spectively, r the scattering time of the normal electrons, and J n = i V n e v n , J s = Nsevs are the current densities (for electron drift velocities v n and vs). The total electron density, N = Ns + Nn, is constant. When there is only a d.c. field, the superconducting electrons carry a l l the current. However, for an a.c. field, the inductive response of the superconducting electrons (caused by their inertia) allows some response by the normal electrons which dissipate energy. The conductivity can be derived from equation 1.22 and 1.23 to be: a = an + a, Chapter 1. Introduction 8 Nne2r . I Nne2ur2 Nse2\ m(l +LO2T2) ~ J \m(l+u2T2) + 7 ^ 7 / = o-i-jo-2 (1.25) where o~i and <T2 are the real and imaginary parts of the conductivity respectively, and are the fundamental quantities related to theories. The solution of the differential equation for the electric field given in equations 1.10 and 1.11 hold as well for a superconductor as for a normal metal. However, in this case, the approximation of real conductivity is no longer valid. The solution of the equation can thus be writ ten as: Ex = E0e-pz (1.26) wi th 8 = yfjupLa, as before. Using the separation of the complex conductivity as <r = o~\ — J < T 2 , 8 can be rewritten as: 8 = + ufia2. (1-27) The penetration depth of a superconductor, A, is defined through 1 ( r°° H\\(z) \ In a superconductor at millimeter-wave frequencies, a2 ^> 01 [it. most of the current is carried by the superfluid). Thus the fields at the surface of the superconductor are suppressed over a length scale set by A. A t a depth of A from the surface, the fields wi l l be 11e their value at the surface. Thus the values for the surface resistance and reactance may be derived in this two-fluid model as: Rs « iorVcnA 3 (1.29) Xs = tofxX. (1.30) Chapter 1. Introduction 9 This derivation assumes local electrodynamics, where the penetration depth (scale over which fields change in the material) is much greater than both the mean free path of the electrons, as well as their coherence length [9]. Bo th of these conditions are well satisfied for high temperature superconductors [6]. The surface resistance Rs is directly proportional to the real part of the conductivity, and to the cube of the penetration depth. The surface reactance is directly proportional to the penetration depth. The temperature dependence of the penetration depth is a very important quantity, since it can be related directly to the fraction of superconducting electrons [18]: N1_m X s ' N ~ A 2 ( T ) ' 1 j In terms of practical uses for superconductors in the millimeter-wave regime, an im-portant difference occurs between equations 1.19 and 1.29. For normal metals, the surface resistance increases wi th frequency according to a*1/2, whereas for superconductors it goes as to2. There is therefore a crossover regime where the losses are comparable for metals and superconductors at the same temperature. For the high temperature superconduc-tors currently used, this crossover frequency (at 77K) is approximately at 100-150 G H z . 1.3 Motivation There are a number of ways of measuring the surface resistance of a conductor. The most common way of making these measurements is to form a resonant circuit, and measure the width of the resonance in frequency. The Q of the resonance Q = ( Resonant frequency \ y F u l l width at half maximumy is related to the surface resistance of the material forming the resonator (c/. chapters 3 and 4). Chapter 1. Introduction 10 There are three general ways in which the surface resistance of high temperature superconductors may be measured by resonant techniques: • patterning a stripline resonator out of thin films of the material. • creating a resonant cavity where some or al l of the body is made from the material. • inserting a piece of the material into the interior of a resonant cavity. The first method [19] is of the most practical use, since it most resembles the final form of microwave circuit elements. A thin fi lm of the material deposited on a substrate is patterned into a resonant structure, and the Q of the resonator is evaluated subject to other variables (ie. power, temperature, material properties). This is, however, a destructive technique, since the material must be patterned. It is not very useful as a diagnostic tool. As well , it is unclear what effects the patterning has on the materials. It would be better to measure the films non-destructively beforehand. The third method is the one used most often in this laboratory [14] [15] [20] [18]. Here a cavity resonator is constructed out of a material wi th low loss (a conventional supercon-ductor such as lead for best results) wi th an opening through which the sample of interest is inserted. Again , the change in Q with various variables can be measured, giving the surface impedance of the material. This method can be very accurate, since the filling factor can be made quite high with proper choice of resonator geometry [14]. Al though it is an appropriate technique for single crystals, it is not generally suitable for thin films. The films must be made of the correct size to fit in the resonator yet give high filling factor — which might mean breaking the film into small pieces. T h i n films are grown on substrates whose electromagnetic properties (loss tangent and dielectric constant) w i l l affect the measurement. Final ly, at millimeter-wave frequencies, the resonant cavities become very small , and are difficult to fabricate and work with . Chapter 1. Introduction 11 The most common way to implement the second method is to replace one wall of a resonant cavity (normally a cylindrical T E o n cavity) with a piece of the material ([21] [22] and many others). This is a convenient method, although the results obtained are not very accurate in general. This is because the loss due to the superconductor is only part of the loss in the cavity, and separation of the two terms can be difficult. The superconductor must be in close contact wi th the rest of the cavity, and so the cavity as a whole must be subject to changes in the same external variable(s) as the fi lm (ie. temperature, magnetic field, etc.). The cavity material w i l l have some dependence in its loss wi th temperature, and wi l l be non-trivially included in the measurement. Forming the cavity body from conventional superconducting material would give no advantage in terms of resolution, since most measurements w i l l be made above the Tc of any conventional superconductor. The fi lm itself must be of the same size as the resonator, and the contact wi th the resonator may in itself damage the fi lm. A very successful technique is the parallel plate resonator [23]. This technique has yielded some of the most sensitive measurements of surface resistance around 10 G H z . Here two films of the material are pressed together, separated by a thin teflon sheet. The assembly forms a resonator, whose Q may be measured as a function of temperature. The difficulty wi th this technique is that two films are required, and their shape is quite important. For f i lm characterization, it is desirable to have a technique not dependent on having two similar films. The objections raised for this method apply as well to dielectric resonators [24] [25], where a bulk dielectric crystal forms the body of the resonator. The method described in this thesis is the open resonator. This method is particularly appropriate for the measurement of surface resistance of thin films of high temperature superconductors at millimeter-wave frequencies. It is not a method invented here, it has been described in numerous papers [26] [27] [28], and it was the advantages of this method as described in these papers that first attracted our interest. The open resonator Chapter 1. Introduction 12 technique wi l l be described in detail further on in the thesis (cf. chapters 3 and 4), but a quick overview wi l l be appropriate here. The open resonator consists of a spherical mirror separated from the sample under test by a distance less than the radius of curvature of the mirror (see figure 2.2). The electromagnetic field is focussed by the mirror on to the sample surface. The resonator is open to the environment, unlike conventional closed cavity resonators, this allows the resonator to be a larger size than would be possible for other geometries. As well, the sample is physically separated from the rest of the resonator, so the temperature of the sample can be varied independently of the rest of the apparatus. This apparatus is ideally suited for the measurement of thin films because only the top surface is in contact wi th the electromagnetic field, avoiding dependence on the substrate material. As well there is no dependence on the shape of the superconducting sample (as long as it is larger than some m i n i m u m size), accommodating any shape of sample. In fact, the focus of the resonator can be scanned across the surface, giving information on the surface resistance over the area of the sample [28]. Such an apparatus (working at 77 K ) is being sold commercially by Conductus[29]. The open resonator is the only method capable of doing this non-destructively. Though this project has drawn on the ideas of other researchers i n terms of using an open resonator, there are problems which have not been dealt with before in this context: • The frequency used is quite high (117 G H z ) , involving complicated source and detector techniques (cf. chapter 4). • The samples measured are quite small , forcing the resonator to be used at its l imi t in terms of spot size (cf. chapter 3). • The temperature dependence is being investigated, and a novel cooling technique has been developed particularly for this apparatus (cf. chapter 2). Chapter 1. Introduction 13 This thesis describes the development of the apparatus, as well as giving background on the theory. There is some data described, though it is secondary, since no exhaustive study has been done. Without systematic study of thin films under various growth conditions, subject to various variables, the information is more qualitative i n nature. Chapter 2 The Flow Cryostat 2.1 Introduction The cooling of the experiment becomes problematic when one wishes to use millimeter-waves. A t these frequencies, the signal is carried through waveguides, which consist of hollow rectangular tubes. Standard waveguides become quite lossy as the frequency increases, wi th losses of about 1.5 dB per foot near 100 GHz[30]. As well, the standing waves that arise from discontinuities in the circuit can seriously degrade measurement accuracy, especially for long transmission lines. Thus, one wants to have as short a length of waveguide as possible. O n the other hand, standard cryostats involve the lowering of the experimental ap-paratus into a long dewar filled wi th l iquid helium. Here the heat conducted along the supports and electrical connections (wires, coaxial cable, waveguides, etc.) is reduced by having long lengths of material wi th low thermal conductivity. This is exactly the opposite of the ideal mi Hi meter-wave system having short, highly conductive connections. For this experiment, we have designed a flow cryostat that avoids the long trans-mission lines (figures 2.1 and 2.2). In this apparatus, cryogenic fluid (l iquid hel ium or l iquid nitrogen) is drawn up from a storage dewar and circulated through a series of heat exchangers, thus cooling the experiment. This contrasts with standard immersion cryostats, where the experiment is immersed in a large volume of l iquid . The present design allows for a very short length of waveguide (~5 cm) to go from room temperature 14 Figure 2.1: The outer view of the flow cryostat. The l i d has been removed to show the heat shield. The plates on the sides are for adjusting the choke flanges (see chapter 4) Chapter 2. The Flow Cryostat 16 Figure 2.2: The flow cryostat wi th both the vacuum and heat shields removed. The two heat exchangers are clearly visible, wi th the resonator and sample-holder attached to the inner one. Chapter 2. The Flow Cryostat 17 to low temperature (~2 K wi th l iquid helium, ~70 K wi th l iquid nitrogen). As well, it allows rapid thermal cycling of the whole experiment (~ 1/2 day) and can accommodate a variety of experiments. However, unlike the standard cryostat, where many researchers have refined the techniques [31], the design of the flow cryostat requires quite careful study and modeling. In this chapter I outline the basis for the design, and show the theoretical performance of the final setup. 2.2 Cryogenic Theory In designing a cryostat, it is important to determine how the heat wi l l be transferred to the experimental chamber. For a flow cryostat in an evacuated container, heat can be transferred to the experimental chamber by radiation across the vacuum, or by con-duction along the solid materials either used for support of the apparatus, or used in the experiment (eg. wires, waveguides, adjusting screws, etc.). B y minimiz ing the heat transfer through both of these mechanisms, the amount of cryogenic fluid needed can be reduced. The rate at which a surface of area A at temperature T emits thermal radiation Q is given by the Stefan-Boltzmann equation [32]: Q = o-eAT4. (2.1) The value cr = 5.67 x 1 0 - 8 W m ~ 2 K - 4 is the Stefan-Boltzmann constant. The emissivity e is the ratio of the radiative energy emitted by an object at temperature T divided by the radiative energy emitted by a black body at the same temperature. It is a simplified model, in which one uses an average of the frequency dependent emissivities that exist in real materials, but should be adequate for the present use. The net rate of radiant energy exchange between surface 1 of area A at temperature T i and surface 2 at the (hotter) Chapter 2. The Flow Cryostat 18 temperature T 2 is Qrad21 = aEA(T24-T14) (2.2) where E is a factor involving the emissivities of the two surfaces, and depends upon whether the reflections are diffuse or specular. We have polished the surfaces to give specular reflection, since this gives lower transfer of heat. In this case E = (2.3) e2 + ( l - e 2 ) e i v ' For the materials used in this apparatus (aluminum, copper and brass), the emissivities ei and e 2 are approximately 0.04-0.06 at room temperature and w i l l decrease wi th de-creasing temperature [32]. Thus E = 0.025 w i l l give a conservative estimate for the rate of radiative heat transfer in the design. To reduce heat transfer by radiation, one can insert radiation shields. B y inserting a shield at an intermediate temperature Ts between T\ and T 2 much of the radiation can be intercepted. This shield may be floating, ie. thermally disconnected, or actively cooled. Such shielding is very important in the flow cryostat, since the experiment wi l l be surrounded by material at room temperature, not l iquid nitrogen temperature as i n a standard cryostat. The rate of heat conduction Qcond along a solid material of thermal conductivity k dT dx  and cross sectional area A wi th a temperature gradient —— is given by the relation [33] • , k dT , . Qcond = fcA^T- (2-4) B y using materials with poor thermal conductivity and small cross-sectional area (eg. thin wall stainless steel) this conduction can be reduced. However, the thermal conduc-tion can be further reduced, and perhaps made insignificant, by using the flowing gas to cool the supports. The rate of heat absorption by a warming gas of heat capacity Cv Chapter 2. The Flow Cryostat 19 flowing at a rate of h moles/second is dQ = tiCpdTgas (2.5) or, dQ dTgas te = ( 2 - 6 ) if one can use distance as the independent variable (as in gas traveling along a pipe). The approximation that Cv is constant wi th temperature wi l l be used without significant reduction in accuracy. Heat is transferred between the wall of the containing vessel and the gas it contains wi th a temperature difference of AT over an area A according to Q = hATA. (2.7) The value heat transfer coefficient, h, depends upon whether the flow of fluid is laminar or turbulent. The turbulence of a flow of fluid at velocity v wi th viscosity 77 and density p through a pipe of diameter d is described by a dimensionless quantity called the Reynolds number Re: Re = P~. (2.8) A flow wi th Reynolds number above 2300 is turbulent. For laminar flow, the heat transfer coefficient for a fluid of thermal conductivity kgas is given by [34]: h oc kgas/d. (2.9) For turbulent flow [35] h - T ^ o ^ - (2-10) nC where the dimensionless Prandt l number is Pr = 4—E and G = vp. "'gas Chapter 2. The Flow Cryostat 20 Heat Shield Access Ports Needle Valve Vacuum Can. sOuter Heat Exchanger Needle Valve Control Figure 2.3: Schematic cut-away drawing of the flow cryostat. Chapter 2. The Flow Cryostat 21 2.3 Design As described earlier, the flow cryostat, instead of using a large reservoir of cryogenic l iquid surrounding the experiment, draws a small constant flow from a separate storage dewar. This flow of l iquid, which rapidly becomes gas, is brought through a series of heat exchangers, absorbing incident heat through both the latent heat and the heat capacity of the gas. The chief advantage of this design is that it allows short distances from the cryogenic region to the outside. W i t h a standard cryostat, this is difficult, and would contribute a large heat conduction path. Extending into the storage dewar is a small capillary (of th in wall stainless steel) which enters into the flow cryostat (see figure 2.3). Cryogenic l iquid is drawn up this line and enters a heat exchanger through the control of a needle valve. The heat exchanger consists of a hollow copper block, with copper plates arranged inside so that the fluid must take a circuitous path. Upon entering this chamber, the l iquid boils — at the equil ibr ium temperature for the pressure set. B y lowering the pressure (increasing the pumping rate on the gas lines), the temperature at which the l iquid boils can be lowered. This temperature becomes the temperature of the experiment, since the experimental apparatus is attached to the top of this heat exchanger. It is important that as much of the l iquid changes to gas here as possible, since small droplets of l iquid are not as efficient in heat transfer as is the gas. To accomplish this, a finely corrugated disk of copper is inserted into the first stage of the exchanger, ensuring very good thermal contact immediately. The gas, upon leaving the inner exchanger, is carried through three thin wall stainless steel tubes to another, larger exchanger (see figure 2.4). This heat exchanger, which works in a manner very similar to the first one, is connected to a radiative heat shield surrounding both the experiment and the inner exchanger. B y cooling this exchanger wi th Chapter 2. The Flow Cryostat 22 the flowing gas, the radiative heat load on the experiment is reduced. As a compromise between simplici ty of design and efficiency, it was decided to use only two heat exchangers (and one heat shield). The gas leaving the exchanger is carried through three more th in-wal l stainless steel tubes, exiting the vacuum shield through ports on the underside. These tubes physically support the experiment, and since they conduct gas, give a reduced conductive heat load to the apparatus. The base temperature of the experiment is controlled through two variables: the pumping rate on the exit tubes, and the flow impedance provided by the needle valve at the inner exchanger. B y tuning these two variables, we attempt to minimize the amount of coolant required. This is of course dependent upon proper design of the heat exchangers and connecting tubes. The flow cryostat was designed to be used wi th l iquid helium as the coolant. However all of the work described in this thesis uses l iquid nitrogen as the coolant. The design was based on the assumption of l iquid hel ium both because it is quite expensive and because its low boil ing temperature imposes more stringent design criteria. L i q u i d nitrogen, though it has different properties and thus the cryostat w i l l not operate at peak efficiency, is relatively cheap. Thus the design criteria for the flow cryostat are: • It should have the ability to reach 2 K wi th 1 L / h r of l iquid He flow. • The experimental chamber should have a m i n i m u m volume of approximately 7cm diameter by 5cm high. • The experimental chamber should have the ability to accommodate various exper-iments. • The flow cryostat should work on top of a storage dewar (either l iqu id helium or Chapter 2. The Flow Crvostat 23 Figure 2.4: Isometric cut-away drawing of the flow cryostat. l iquid nitrogen). • There should be only a short distance from room temperature to cryogenic tem-perature. Computer modeling was used to determine the design of the cryostat. Initially, each portion of the design was examined separately, wi th assumed values for important pa-rameters. Once the design was set (which was affected at least as much by ut i l i ty and ease of fabrication as it was by efficiency of the design), a complete simulation was used to check the performance. Conservative values were used ini t ia l ly for the design, since it is difficult to predict what sort of experimental apparatus might eventually be used in this cryostat. Chapter 2. The Flow Cryostat 24 The modeling was done by numerically solving the coupled differential equations describing the flow of the gas, and the heat exchange associated wi th the gas. For the flow of gas through the pipes (between the exchangers and exiting the apparatus), the temperature of the two ends of the pipes, as well as the in i t i a l temperature of the gas, were assumed. The differential equations were integrated using a Runge-Kut ta method unt i l the computed final temperature of the pipe matched the set final temperature (essentially a "shooting method"[36]). The flow was assumed to be turbulent, and any pressure change along the length of the pipe was ignored (after assuring that both assumptions were justified). The heat capacity of the gas was assumed constant, and the heat exchange between the gas and the pipe was taken from equation 2.10. In figures 2.5 and 2.6, the dependence of the heat conduction and temperature of the gas vs. the length of pipe connecting the two heat exchangers is shown. As can be clearly seen, the amount of heat transported down the pipes drops off strongly as the length is increased. It was decided that a length of 2-3 cm was sufficient to make the heat load along these pipes negligible. Similarly, i n figures 2.7 and 2.8, the dependence on the length of pipes exiting the heat exchanger is shown. Here, any length of pipe longer than 5 cm gives essentially zero heat conducted to the outer exchanger. In fact, even at 5 cm, the heat load along the exit pipes should be even smaller than calculated here. This is due to fact that the outer tube temperature is much less than the assumed 300 K , and in fact becomes frosted wi th ice, even with a moderate flow rate. The design of the heat exchangers involves maximizing the heat transfer between the gas and the body without imposing too large a pressure gradient. To increase the heat transfer, the surface area to which the gas is exposed should be maximized. The heat exchangers use plates of copper wi thin a copper block to cause a circuitous flow: the first plate has a hole in the center; the next plate has no hole in the center but has a gap at its outside edge (see figure 2.3). This sequence is repeated, and provides a large increase Chapter 2. The Flow Cryostat 0.20 25 0.20 0.10 h 0.05 0.00 Figure 2.5: Heat conducted (in Watts) along the pipes joining the inner heat exchanger with the outer heat exchanger, as a function of the length of pipes (in cm). The 3 pipes are assumed to be 3/8" diameter, .012" wall thickness stainless steel. The inner exchanger temperature is assumed to be 4.2 K , and the outer exchanger temperature 30 K . The gas flow is set to be that corresponding to a flow of 1 L / h r of l iquid helium. Chapter 2. The Flow Cryostat 30 26 3 4 5 6 Figure 2.6: Temperature of the helium gas and the pipe wall i n K e l v i n as a function of pipe length in cm. A l l parameters are as assumed in the previous figure. Chapter 2. The Flow Cryostat 27 Figure 2.7: Heat conducted (in Watts) along the pipes joining the outer heat exchanger wi th the vacuum shield, as a function of the length of pipes. The 3 pipes are assumed to be 3/8" diameter, .012" wall thickness stainless steel. The heat exchanger temperature is assumed to be 30 K , and the outside temperature to be 300 K . The gas flow is set at Figure 2.8: Temperature of the hel ium gas and the pipe wall as a function of pipe length. A l l parameters are as assumed in the previous figure. Chapter 2. The Flow Cryostat 28 in gas contact area over a simple pipe. Model ing of the heat exchangers proved to be simpler than for pipes, i n that the thermal conductivity of copper is so high that even .020" thick copper plates are essentially a thermal "short" to the heat flows encountered. Thus the problem reduces to determining the best design for the gas to absorb the required amount of heat while passing through a constant temperature body, which involves integrating a set of coupled differential equations. Unl ike the pipe problem, there is no fixed end-point that must be satisfied, so no "shooting method" is required. The heat absorbed is maximized by having turbulent flow of the gas which is accomplished by throttl ing the gas at the plates. As well , the pressure must not drop significantly through the exchanger or we w i l l not have good control over the temperature at the interior of the cryostat. In the end, it was determined that having three plates i n the exchangers, separated by approximately 1/10" gives negligible pressure drop, and allows the gas to reach thermal equil ibrium wi th the exchanger body before exiting (see figure 2.10). Note that a number of simplifying assumptions have been made here chiefly in terms of assuming symmetric geometry, and uniform turbulent flow. 0 I , , , , 1 , , , , I , , , , 1 0 1 2 3 Distance along Tubes (cm) Figure 2.9: Temperature of gas and pipe for the inner pipes of the flow cryostat in a self-consistent numerical model. The pressure is 0.1 atm, and the flow rate 0.5 L / h r of l iquid nitrogen. The inset shows the amount of heat (in m W ) flowing along the pipes. Temperatures Inner exchanger 2.488 K Heat Shield 16.263 K Gas at inner exch. 2.488 K Gas at end of inner pipe 2.618 K Gas at beg. of outer pipe 16.262 K Gas at exit 39.985 K Heats Radiat ion on inner chamber 9.7 l O " 7 W Conduction to inner exch. 0.00961 W Radiat ion on shield 0.586 W Conduction to shield 0.709 W Absorption in inner exch. 0.00961 W Absorption i n outer exch. 1.273 W Table 2.1: Heats and temperatures for the numerical model of the final flow cryostat design. The flow of l iquid helium is 0.5 L / h r , and the pressure is 0.1 atm. Chapter 2. The Flow Cryostat 30 20 5 10 15 Distance along Exchanger (cm) Figure 2.10: Temperature of the gas in the outer exchanger of the final design of the flow cryostat for pressure of 0.1 atm and flow rate of 0.5 L / h r , along the circuitous path inside the exchanger. The shield temperature reaches an equil ibrium temperature of 16.26 K . The inset shows the integral of the heat absorbed by the gas (in W). 300 g 200 4-> d i-i <D CL, S H 100 0 o.o J I I 1 I L _ l I I I l_ 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 o1-o Tube Wall Temperature ° Gas Temperature ^ • • • • • • • • • • • • • • • • • • • • • • • • n n n n D D D D [ : i n a D G n n D D _i i i_ j i i_ 0 1 2 3 4 5 6 7 Distance along Tubes (cm) Figure 2.11: Temperature of the gas and pipe for the outer pipes of the flow cryostat in its final configuration. The inset shows the amount of heat conducted along the pipe (in W). See the previous two figures. Chapter 2. The Flow Cryostat 31 The final design of the cryostat is shown in the figures. The full performance of the set design was modeled in a self-consistent solution for various flow rates and pressures (figures 2.9 to 2.11 show a particular set of parameters). This modeling entailed setting the flow rate of l iquid , pressure of the exit gas, and external temperature; unlike for the in i t ia l models, where each aspect of the cryostat was examined independently. The solution of these coupled differential equations was done iteratively, ending when a self-consistent solution was found. For the condition shown in the figures, 0.5 L / h r of l iquid hel ium flow at 0.1 atm, the latent heat of the l iquid is greater than the amount of heat incident on the inner exchanger, meaning that not al l the l iquid boils before leaving the exchanger. This neglects the heat conduction along waveguides, etc., and thus is an underestimate of the temperatures reached in various portions of the cryostat. To partially account for this, it was assumed that any remaining l iquid is transformed into gas by the excess heat (a crude approximation). As can be seen from figure 2.11, the exiting gas is st i l l very cold (about 50 K ) . This suggests two things: first of a l l , that a slower flow of gas could probably be used. This is true up to the l imi t that the inner heat exchanger should reach a low enough temperature (as determined by the needs of the experiment). Secondly, that one would probably want to use another heat exchanger/heat shield combination. These exchangers are (theoretically) very efficient, as can be seen from figure 2.10, where the gas reaches equil ibrium quite well. Unfortunately, another level i n the cryostat would entail greater complication in both the use and fabrication of the apparatus, and was not included in the design for that reason. The modeling done in this chapter has been solely for the steady state — once the experiment has reached operating temperature. In fact, a significant amount of cryogenic l iquid would be needed to cool the apparatus to the operating point. For this reason, it is expected that i n practice the cryostat w i l l be pre-cooled wi th l iquid nitrogen before Chapter 2. The Flow Cryostat the liquid helium cooling is beg Chapter 3 Theory of the Open Resonator 3.1 Introduction Unlike closed cavity resonators, which have been a key tool for microwave engineers and physicists for decades, open resonators have been exploited only relatively recently. Whi l e closed resonators can be regarded as an extension to higher frequency of lumped element R L C circuits, open resonators resemble more a scaling down in frequency of optical mirrors. In this chapter I wi l l outline the theory pertaining to the existence and characteristics of stable resonant modes of such structures. 3.2 History In the late 1950's, Fabry-Perot structures were shown to be useful as resonators at optical frequencies for use in lasers. The Fabry-Perot resonator consists of a pair of plane-parallel mirrors facing each other, between which light reflects over several passes, storing electromagnetic energy. These structures, ini t ial ly analysed using optical techniques, were later studied in terms of the resonant modes of the electromagnetic field [37]. In related work, Goubau and Schwering showed that an electromagnetic field can be described in terms of a complete set of cylindrical waves. If these waves are propagating largely along one axis, then they can be resolved into an elementary set of beams. These beams can be confined along the axis through a repeated series of guiding structures (ie. lenses). [38] 33 Chapter 3. Theory of the Open Resonator 34 P l a n e - P a r a l l e l , Confocal Hemispher ica l or F a b r y - P e r o t H a l f - C o n f ocal Concentr ic Figure 3.1: Terminology and geometry of open resonators. Boyd and Gordon[39] developed the confocal resonator, in which the plane parallel mirrors are replaced by spherical mirrors separated by their common radius of curvature (figure 3.1). This geometry has the advantage of requiring less precision in the positioning and machining of the reflectors to give low loss. As well, the losses due to diffraction are significantly minimized. Subsequent work generalized the solutions to reflectors of different radius of curvature, and an arbitrary separation. In an important review paper, Kogelnik and Li[40] reviewed the properties of such resonators, incorporating the beam-wave theory of Goubau and Schwering. I wi l l mainly follow their development of the theory in this chapter. As well , I wi l l make use of the work of Bucc i and D i Massa[8] who use the eigenvalue approach of Kurokawa[41]. These methods al l involve approximations which may not be valid for the experiment described in this thesis. Cullen and coworkers, in a series of papers[42] [43] [44], have attempted to determine the accuracy of these approximations, as well as to develop a more precise theory. These aspects w i l l be described later in this chapter. Chapter 3. Theory of the Open Resonator 35 3.3 B e a m - W a v e T h e o r y The behaviour of the electric and magnetic fields in a resonator can be obtained by solving the wave equation for the magnetic field H or the electric field E V 2 H + fc2H = 0 (3.1) V 2 E + A; 2E = 0 (3.2) subject to the particular boundary conditions of the problem. Here k = 2ir/A is the propagation constant for wavelength A in the medium, and is the eigenvalue of the differ-ential equation. The field components in cartesian coordinates of such a coherent wave satisfy the scalar wave equation V 2 u + fc2u = 0. (3.3) For a wave traveling in the positive z direction, wi th a harmonic time dependence, we can put u(x,y,z) = ^(x,y,z)e-^e^t (3.4) where the function ip(x,y,z) represents al l deviations from a plane wave. Substituting equation 3.4 in equation 3.3, we get d2j> d2V> g-fc^-O dx2 dy2 dz2 dz ' If most of the variation of u wi th z is taken up by the exponential, ie. tp is slowly ying in the z 3.5 becomes [8] var  direction then we can neglect ^ ~ in comparison wi th k^-, and equation ozl dz This is the parabolic or paraxial approximation often used for the solution of the open resonator problem. The validity and accuracy of this approximation wi l l be discussed later. Chapter 3. Theory of the Open Resonator 36 The differential equation 3.6 has a form similar to the time dependent Schroedinger equation, suggesting that one can try a solution in cartesian coordinates of the form [40] k tp = g(x,z)h(y,z) exp < -j 27O (x2 + y (3.7) where g is a function of x and z, h is a function of y and z. P(z) represents a complex phase shift along the axis, and 'j(z) a complex beam parameter, which describes the intensity and curvature of the beam. Substituting we find that For convenience we define d1 dP — = 1 and —— dz dz 1 7 A (3.8) (3.9) 7 R(z) J KW2(Z) where one can see that R(z) is the radius of curvature of the wavefront (surface of constant phase) at z, and w(z) is a measure of the decrease of the field amplitude wi th distance from the axis. Since this decrease is Gaussian in nature, w(z) is the distance from the axis at which the amplitude is 1/e times that on the axis — often called the beam radius or spot size. A t the point where the beam contracts to its min imum diameter, called the beam waist, the beam parameter is purely imaginary 7o = JZQ w here z0 = _ 7Twl kwl (3.10) (3-11) A 2 Here w0 is the min imum spot size, s t i l l to be determined. Measuring z from this point, we have 7 = jzo + z = ——Si + z. A Substituting equation 3.12 into equation 3.9, we get w2(z) = w2 l + l -(3.12) (3.13) Chapter 3. Theory of the Open Resonator 37 and R(z) z (3.14) Trying functions of the form g ( ) and h I ] in the differential equation 3.6 \w(z)J \w{z)J gives equations of the following form for both functions d2g j 2 -2x^- + 2mg = 0 (3.15) which defines the Hermite polynomial of order ra [45] g • h = Hm (V2^-) HV(V2 w(z) w(z) / For the fundamental mode (ra = p = 0), one can just integrate dP(z) dz (3.16) =• ~Jh to get the phase shift, although it is more complicated for the higher order modes, yielding P(z) = (ra + p + 1) t a n - 1 ( — ) - j l n \Z0J V w Combining all these factors, and suppressing the time dependence, we get (3.17) WQ w(z) exp w(z) t w exp < -j kz - <S>mp(z) + kp2 2R(z)\ where we define $ m p = (ra + p + 1) tan 1 p2 = x2 + y2. (3.18) (3.19) (3.20) Ant ic ipat ing later normalization following Bucc i and D i Massa[8], we write for mirror separation D (see figure 3.2) 2 / u. mp (x,y,z) = 1 w(z)\j Dir2m+Pmlp\ e x P fziT^l e x p ( - J HmV2 w(z) t w2(z) t kz - $mp(z) + kp2 2R{z)\ (3.21) Chapter 3. Theory of the Open Resonator 38 T a —1 Figure 3.2: The semi-planar hemispherical geometry of the open resonator as used in the experiment. For the semi-planar geometry used in this experiment (figure 3.2), the curvature of the beam at the spherical mirror is given by the curvature of the mirror . R(z)U=D = Ro (3.22) A t the planar mirror (the sample), the beam should have no curvature R(0) = oo. These geometrical constraints give the values for the constants z0 and w^: zo = ^D(R0 - D) (3.23) wl = l^D(R0 - D). (3.24) In a resonator, neglecting any losses due to the size of the mirrors or finite conduc-tivities, a standing wave wi l l be developed of the form ^mpq = umpq ^ Umpq (3.25) where the (+) and (—) indicate waves traveling in the positive and negative z direction. If u is a transverse component of the electric field (say u = ey), then the boundary Chapter 3. Theory of the Open Resonator 39 condition that must be satisfied for perfectly conducting mirrors is ^ | Mirror surface — 0. (3.26) This boundary condition is approximate, and corrections to it w i l l be discussed in the following section. Using equations 3.25 and 3.21 this condition gives the propagation constant in the lossless case k^ = ^ ±l±ltm-^(£) + 1^p. (3.27) where the label q refers to the number of nodes along the z axis. The standing wave field can thus be writ ten Vmpg(x,y,z) = : J W ^ i ,„, , Hm {V^^r ) Hp (v^^r 1 e x p ' P " w(z)\l D7r2m+Pm\p\ m \ v w(z)J p\v w{z)) y \w2(z)t s i n (3.28) The mult ipl icative constants have been chosen such that, following Bucc i and D i Massa, the eigenfunctions ^ m v q of the wave equation are normalized (to the same order as the approximation used i n its derivation) wi th r r r 1 if (mpq) = (nst) J j j ^mp^nstdv = (3.29) Cavity 0 otherwise. The mode of interest in this experiment is the mode ^>ooq, often called T E M o o ? since, to a first approximation, both the electric and magnetic fields are transverse. 3.4 Cavity Losses The important quantities of the resonance for microwave measurements are the resonant frequency and the Q, or quality factor of the resonator. For the lossless derived above, the resonant frequency of the mode mpq is given by -m + p + 1 A _ ! (D\ , 7T ^mpq — ckmpq — C D t a n _ 1 ( - ) + ^ ( ? + l )] (3-30) Chapter 3. Theory of the Open Resonator 40 z (m) Figure 3.3: The amplitude of the transverse electric field (e y) using the beam-wave ap-proximation for the mode [m,n,p] = [0,0,14] and dimensions as used in the experiment. and the Q is infinite, since it is defined as 1 Total Power Loss P — = = (3.31) Q u (Total Stored Energy) LOU and we have neglected al l power losses up to now. When the losses of the resonator are introduced, the resonant frequencies of the modes wi l l shift, and the resonances wi l l develop a finite width. These losses come from: • finite conductivity of the mirror and sample surfaces. • diffraction from having non-infinite mirrors (ie. some of the beam is radiated away). Chapter 3. Theory of the Open Resonator 41 • coupling between the resonator modes and the modes of the feeding waveguide. 1/Q can be written as a sum of contributions to the total loss of the cavity: 1 1 1 1 1 , o = 7)—+n + 7T-—~+n (3-32) ^mirror sample diffraction coupling A n y electromagnetic field inside the cavity, including the losses, can be expressed as an eigenfunction expansion of the modes of the lossless case (except possibly at a finite number of points[41]): ' mpq^mpq E = £ Vm m,p,q •H ^ ] fmpq^mpq- (3.33) m,p,q For the y-polarized mode, let 2 ' w{z)y DK2™+V m\p\"my " w{z)) "p\v"w{z)J ^t'\w2(z)i empq(x,y,z) = W ^ ^ j J v ^ j - HJy/2-Z-r) exp - f sin k z - $ 4- k m p q p 2 4- ^ m p q m p + 2R(z) + 2 These modes form a complete orthogonal set. Since the differential equation used is scalar, there are two degenerate modes corresponding to the x and y polarisations. A n y inhomogeneity without rotational symmetry i n the xy plane of the resonator w i l l tend to split the frequency of these modes. The resonator as designed has no method of distinguishing the two modes. Following the argument of Kurokawa[41], V x empq is a function similar to the magnetic field, since V x E = —jupH.. Let us the define the functions hmpq such that V X dmpq — kmpq\lmpq. (3.35) This gives j . (3.34) cos k m p q Z _ d>mp + hmP + TL pq m p 2R(z) 2 W\Z) 2 tJ'/l I (3.36) Chapter 3. Theory of the Open Resonator 42 for the same polarization. B y substituting equation 3.35 into V x hmpq we get the symmetric result that V X h m p 7 — krnpqGmpq. (3.37) The constants Impq and Vmpq in the expansion are so named due to their similarity to the current and voltage in transmission line equations: [41] Vmpq = J j J E • empqdv (3.38) Impq = J J J B.- hmpqdv. (3.39) We can expand the function V x E in terms of the functions hmpq for the same reasons as given above, V x E = £ h - P 9 / / / ( V x E ) • hmpqdv. (3.40) m,p,q J J J Expanding the integral, using vector calculus identities and Gauss' theorem, V x E = Y h » w / / / ( V • ( E x h " w ) + E • ( V x hmpq))dv = n ' « P 9 (/y n • ( E x hmpq)dS + kmpq J J j E • empqdvj . (3.41) Similarly, we can expand V x H , V x H = e™Pqkmpq / / / H • empqdv (3.42) m,p,q J J J where the surface integral vanishes because fi x empq = 0 on the surfaces of the cavity. These eigenfunction expansions can be then substituted into Maxwell ' s equations: V x E = - j w / / H V x H = jueE (3.43) Y^ h m p ? n . • ( E x hmpq)dS + kmpq J J J E • empqdv = -juft Yl h ™ p ? / / / H ' h m p ^ u Chapter 3. Theory of the Open Resonator 43 ^ , ^mpqkmpq Iff ' h mpg dv = jue / / / E • empqdv. (3.44) m,p,q J J J m,p,q J J J Since the functions hmpq and empq are orthogonal, we can equate the coefficients of each vector, and solve for the constants Impq and Vmpq: I m P * = U2 JUJL 11 hmPq • ( n x E)dS (3.45) Vm™ = J J hmP, • (n x E)dS. (3.46) mpq Here kmpq is the propagation constant for the lossless mode labeled (mpq), and k is the propagation constant for the actual field. The surfaces over which the integration is done all have different boundary conditions on the field, so j j h m p g • (n x E)dS = J J hmpq • (n x E)dS Sample + J J h m p g • (n x E)dS + J J hmpq • (n x E)dS + J J hmpq • (n x E)d& Curved Mirror Couphng Apertures Everywhere Else Each of these contributions is related to the terms in the expansion of — i n equation 3.32. The expression for the electric field in equation 3.33 is not correct up to the cavity bound-ary, since it forces the tangential component of E to zero. For a conductor wi th finite conductivity, we can use the boundary condition of surface impedance (cf. Introduction) [8]. n x E = Zsh x H x i i . (3.48) So for the mirror and sample surfaces, J J hmpq • (n x E) dS = Zs jJ hmpq • n x H x hdS = X ] Im'p'q'Zs / / hmpq • hmipiqidS. (3.49) m'p'q' Chapter 3. Theory of the Open Resonator 44 This total stored electromagnetic energy can be expressed in terms of either the electric or magnetic fields: U = e/2JJjE-E*dv = p/2 J J Jn-Wdv 12 Lmpq | mpq mpq (3.50) The power loss on the surface is P = Re = Re Zs J J H - H*dS 2^2 2\2 Irnpqlmip>g' / / " m p g • hmip. m.v.a m l n< ni •* iqidS (3.51) ,p,g m',p',q' If losses are low, then we would expect the field present in the cavity to be similar to the field in a lossless cavity. Thus, the dominant term in the sum wi l l be for (mpq) = (m'p'q'), and we can rewrite the equations for power energy and ohmic power loss as: Pmpg = Re [Zs] \Impq\2 J J |h m p 9 | 2 dS (3.52) Umpq = nl'l\Impq\ . (3.53) Thus, we can write expressions for the Q of the cavity due to losses on the curved mirror and on the sample as Y^— = R e [ Z w ] A [ f\hmpq\2dS (3.54) QSample U[1 J JS T^— = MZsmtrrJ— Ii \hmpq\2dS. (3.55) Mirror UljJL J JM As written, this assumes that hmpq is parallel to both mirror surfaces — which is not true for the curved mirror. This is another instance where the paraxial approximation has been invoked. The fourth integral in equation 3.47 deals with the losses due to diffraction. This term can be dealt with by assuming that the field is an outgoing locally-plane wave, so Chapter 3. Theory of the Open Resonator 45 that n x E = ( 0 n x H x n (3.56) where the surface of integration is to simply extend the mirror surfaces unt i l they intersect [8] and Co is the impedance of free space. This solution would be very rough, since the approximations and assumptions made are quite extravagant. However, this term should be small for the fundamental mode if the mirror and sample have a diameter greater than twice the spot size. This follows from an empirical discussion by Cul len [44] based on analytic results by Weinstein [46] for the confocal geometry. Weinstein expresses the loss due to diffraction as Q = kD/A (3.57) where A is the loss per pass, given empirically by A ~ 200exp( -4a 2 / u ; 2 ) (3.58) where a is the radius of the mirror (not radius of curvature) and w is the spot size of the beam at the mirror. For the geometry of the experiment, this gives Q's on the order of 10 8 — much higher than the Q's due to ohmic losses. The area of the coupling apertures is small, and can be neglected in many cases. How-ever by using the boundary condition that the fields must be continuous wi th the feeding waveguides, one can determine the amount of energy coupled into the resonant mode [8] [47]. It is likely, however, that any coupling determined theoretically wi l l disagree with the actual values, due to the cumulative effect of approximations and non-idealities in the geometry (eg. non-circular coupling holes, finite thickness of the coupling holes, etc). A more empirical approach to the coupling wi l l be discussed i n the following chapter (cf. experimental data). Chapter 3. Theory of the Open Resonator 46 The resonant frequency of the cavity, including losses, can be found by a variational formula [41] J J J [(V x E ) 2 + (V • E) 2] dv-2J | f i x E - V x EdS k — 7 7 7 • (3.59) / / / E ' E < t o The shift in frequency due to the change of the penetration depth of the supercon-ductor with frequency is an important quantity measured in this laboratory [20] [15], however it is inaccessible to measurement in this apparatus since the resonant frequency is not stable enough in general. 3.5 Beyond the Beam-Wave Theory The beam-wave theory, as developed in the preceding section, depends upon the validity of the parabolic approximation 7T7 « (3-«0) OZ OZ which can be shown to be equivalent to saying « 2 = ^ » 1. (3.61) This constraint is definitely satisfied for the laser resonators for which this theory was developed and for which the resonator dimensions are much greater than the wavelength of the light, however its validity is less obvious for millimeter-wave systems. The approx-imation is even less valid in the experiment described here, since the objective is to have a spot size of the same order of magnitude as the wavelength of the radiation. Although Cullen has shown that the theory is actually more accurate than this in terms of the resonant frequency (to approximately 0 ( o T 4 ) [42]), the expressions for the fields are not accurate to this order and a more accurate theory is desirable. Chapter 3. Theory of the Open Resonator 47 A second problem wi th the theory is that, as can be seen from equations 3.21 and 3.14, the surfaces of equal phase are not spherical, but have a parabolic shape. This difference wi l l be small near the axis, however, and can be treated wi th perturbation techniques [42] [48]. A third remaining defect wi th the theory lies i n the fact that it was developed using a scalar method. Thus the boundary condition that u = 0 on the mirror surfaces is not true for al l cartesian components of e. What actually is involved is ^tani the tangential component of the electric field. This wi l l be close to a combination of ex and ey for large radius of curvature, but w i l l strictly involve ez as well. A vector theory, incorporating the relative intensities of the components, but satisfying the actual boundary conditions, is important for improved accuracy. A vector field theory of the fundamental mode of open resonators has been developed by Cul len and Y u [43]. The derivation of this theory wi l l be sketched here, and behaviour of the fields outlined, though most of the calculations from the preceding section wi l l not be repeated. The starting point of this theory is to consider an infinitesimal electric dipole located at the origin and directed along the x-axis. The vector potential of such a system is A = i ( l / r ) exp(-jkr) (3.62) where r = ^Jx2 + y2 + z2. (3.63) If this dipole is shifted along the z axis a distance ZQ the change in the field is t r iv ia l . If, however, one considers a shift of the field —jz0, the interpretation is less obvious, but the mathematics st i l l simple (since Maxwell ' s equations wi l l be satisfied in the same way). Thus r = \J x2 + y2 + {z + j z 0 ) 2 . (3.64) Chapter 3. Theory of the Open Resonator 48 4 r s 0 . 001 0 . 0 0 2 0 . 003 0 . 004 0 . 0 0 5 Figure 3.4: Relative intensities of the cartesian components of the field along the y = x line using the complex point-source theory. Solid: \ey\, Dashed: A^-iOole^, Dotted: &u;o|e 2|. The parameters were chosen for the geometry used in the experiments of this thesis. If we now assume that z0 is large and expand equation 3.62 binomially, we find, except for a constant factor i exp z + z0 -jkz — jk x2 + y2 (3.65) "2(z+jz0)_ which is identical to equation 3.18 for m = p = 0 and the substitutions for R and w0 [44]. The physical interpretation is now more clear, in that the solution to Maxwell 's equations in the paraxial approximation, corresponds to a Gaussian beam. However, without making the paraxial approximation, we have a solution that satisfies Maxwell 's equations exactly, and that is an inherently vector solution (all six components of the fields can be found explici t ly) . These fields can be written down exactly[43], although they are quite complicated. Using a consistent order of approximation, 0(a~4), Cul len Chapter 3. Theory of the Open Resonator 49 and Y u give tractable equations for the fields i n an open resonator of even mode number 9-too I f 2 \ \ • (, - kp>\ 2 . /, ^ kp' ey = — exp - ,,2 w \ wz • o , sin \kz - 3$ + - Q sin ^ - 4$ + (3.67) e, = A n d the h field can be derived from this. Again , this theory does not take spherically shaped mirrors for boundary conditions, but this can be corrected through perturbation theory, and the correction to the resonant frequency is 0 ( a - 4 ) , and can be ignored [43]. These equations can be used in the same way that the beam-wave theory equations are used in the development of Q, but the extra accuracy obtained is not useful in most applications. This is because the 1/Q contribution from the loading of the resonator by the waveguide circuit is likely to mask any deviation from the simple beam-wave picture. The complex-point source theory does give some idea of the relative intensities of the components of the field. It shows that the approximation that both the electric and magnetic fields are transverse is only good to order 1/a (since ex ~ aez). Chapter 4 Millimeter-Wave System 4.1 Introduction One of the important results from the preceding chapter is the expression for the spot size on the sample (equation 3.24): w20 = l^D(R0 - D). To measure small samples, a small w0 is required. This quantity scales inversely with the frequency, of course, so we wi l l want to work with high frequencies for good spatial resolution. The highest frequency millimeter-wave source and detector pair available in the lab is at ~ 117 G H z , which is the frequency used i n our experiments. As well, the decision to work wi th a flow cryostat, and the final design of this cryostat, impose restrictions on the size of the resonator to be used. In this chapter, the design of the resonator and the millimeter-wave detector/source pair are outlined. 4.2 Millimeter-Wave System The source and detector for the experiment are a modification of a 115 G H z spectrometer previously used for experiments on atomic hydrogen and deuterium [49]. The millimeter-wave source is a klystron (Varian VRT2123A7) which operates nominally at 114.5 G H z . The klystron is actually quite old, and the lowest operable frequency has increased wi th age. It is currently operated at about 115.6 G H z . This frequency is stabilized by locking to the 8th harmonic of a reference signal provided by the H P 83620A synthesizer (JREF = 50 Chapter 4. Millimeter-Wave System _ _ T _ 51 386 Computer 48B BUS ~1 A/D Converter HP 8663A Synth 0.B-1.B GHz +8 dBm 10 MHz Kef. ampl i tude B u t PAR 5204 Look-in Amplifier Signal I D 200_MHz + 2DD.05 MHz 300.05 MHz + PED 619A Klystron Power Supply V„ = 2430V V„ = ~260V Klystron 115.80868 GHz Pre-Amp D C B i a s Detector Diode f + 200.05 MHz Microwave Systems MOS-5 "Lock-Box" *8 HP B349B Harmonic Amplifier Mixer HP 83620A Synth. ' E E F +8.6 dBm 14.471085 GHz D/A Converter Input Sweep Control Sweep] Input Weinschel 438A 1-2 GHZ Mixer Diode K L Y MOD RESONATOR AND K L Y HOD Figure 4.1: Block diagram of the millimeter-wave system Chapter 4. Millimeter-Wave System 52 14.471085 G H z for al l work described here). This locking is accomplished by controlling the reflector voltage using a Microwave Systems M O S - 5 frequency stabilizer (or "lock-box") which keeps the klystron operating at 40 M H z from JREF-IKLY = 8 * fREF ± 4 0 M H z = 115.80868GHz. (4.1) This is a stable baseline frequency from which the measurements are based, however it is desirable to have a swept frequency source for accurate measurements of Q [50]. The original spectrometer operated at a fixed frequency, mixing the stabilized klystron signal wi th 1480 M H z from a SSB generator, and using a heterodyne system [49]. Since there is no easy way to sweep the klystron frequency, we have used a modification of this system, the main difference being that we scan the intermediate frequency, providing a l imited sweep capability. A swept frequency from the H P 8663A synthesizer (fsYN — 0.8 to 1.8 G H z ) is mixed wi th both a 50 k H z source and a 200 M H z source (figure 4.1) giving a number of har-monics. In the original spectrometer, the unwanted sidebands were filtered out. In the current design, because /SYN is swept, such filtering is impractical . To obtain a clean single frequency to mix with the klystron, a separate 1-2 G H z source (Weinschel 438A) is locked to the desired sideband. This locking is accomplished by setting the frequency of the oscillator roughly to the desired point through the sweep input of the Weinschel, and fine, fast adjustments through the F M input. A sweep control circuit has been designed so that if lock is lost, the frequency is swept rapidly unti l lock is regained. B y keeping the locking time shorter than the frequency step rate, erroneous measurements are avoided. This clean, variable frequency /MOD, is fed to a millimeter-wave diode which also receives a large signal from the klystron. The diode operates in the non-linear region, so it mixes the two signals, sending a combination of /KLY, IKLY+IMOD and JKLY—/MOD to the experimental resonator. The open resonator acts as a narrowband filter, so essentially Chapter 4. Millimeter-Wave System ^ Varian VRT2123A7 Klystron Cooling "Water Control Data TRG Mixer Un To Modulation =] Hughes 47436H-1000 Harmonic Mixer To Lockbox] and Synth. =fll=H Hughes 44346H-310 Dir. Coupler 45° Twist Control Data TRG Dir. Coupler Tee -See below for continuation Variable Attenuator) To ^ Demodulator| Avantek TRG 21-102 Mixer Pre-Amp Diode Variable Attenuator flHZHE*-Tee Figure 4.2: Schematic diagram of waveguide circuit Chapter 4. Millimeter-Wave System 54 only one of these sidebands wi l l be near the resonant frequency /RES, the others w i l l be filtered out. This signal is detected through the reverse of the input heterodyne system, finally giving a 50 k H z signal which is detected wi th a P A R 5204 two-phase lock-in amplifier. This complicated heterodyne system is used because the signal to noise ratio is quite low, and coherent phase sensitive detection must be used. As the experiment currently stands, the phase information is not used in determining the Q of the resonator, however the modifications to do so would certainly be possible. It turned out to be difficult to convert the fixed frequency SSB spectrometer to a swept source, but the final configu-ration yields an adequately level signal over the range of 116.8 G H z to 117.6 G H z (see the following chapter). This frequency is obtained by taking the positive polarity for al l mix ing frequencies, the mode used for al l measurements described here. 4.3 Resonator Design Once the frequency of the source and the size of the experimental chamber have been set, it remains to determine the actual design of the resonator. The criteria are essentially as follows: 1. spot size wo as small as possible (l imited by number 3 below). 2. diffraction minimized: keep the diameter of the mirror a > 2wi, the spot size on the curved mirror. 3. keep the assumption of k2w^ ^> 1 satisfied. To satisfy these constraints we would like to operate in the regime where D ~ R — the hemispherical geometry. Other researchers using open resonators for Rs measure-ments are split on which geometry to use: either half-confocal or hemispherical [26] [27]. Chapter 4. Millimeter-Wave System 55 Vertical Positioning Screw Quartz Glass Tube I  I II L J 7 Copper -jr4^-----'Mounting Tj Block Stainless ^ Steel Plates Sample / Mirror Surface I I I Y Waveguide Trough Heat Exchanger Figure 4.3: The design of the resonator and sample positioning apparatus. See also 2.2 for a photograph. For large samples or very high frequency, the semi-confocal setup is preferable, since diffraction losses are minimized. In this experiment, the total size of the resonator, and the size of the sample make the hemispherical resonator the ideal choice. The mirror to sample separation in the apparatus, D, must be variable for two reasons. First ly, samples are attached to a support facing the curved mirror (see figure 4.3 ), so any variation of substrate or sample thickness wi l l change the effective resonator dimensions, and thus its electromagnetic characteristics. In moving this block vertically D can be kept constant. Secondly, because of the l imited frequency sweep range, and the fact that theoretical calculations of the resonant frequency are approximate, the mirror separation must be variable to ensure that the /RES lies wi thin this accessible range. Addit ionally, as the cryostat cools, the mirror separation distance changes due to thermal contraction which changes /RES- Thus D must be variable at low temperature. Chapter 4. Millimeter-Wave System 56 This is accomplished by having the sample-mounting block attached to two parallel flexible stainless steel plates which allow vertical travel. The height is set by pushing down from above with a finely threaded screw. A retractable rod engages this screw so that there is no heat conduction when disengaged (figure 4.3). The sample mounting block is formed of copper (about 1 cm 3 ) and is cooled v ia conduction down the stainless steel plates. To this block is attached a thermometer and heater, to control the temperature of the sample. In early versions of the apparatus, the block was solid copper, but temperature changes were found to cause changes i n D due to thermal expansion and thus /RES- B y inserting a quartz glass tube in the main part of the block (see figure 4.3), problems with thermal expansion were reduced to a more acceptable level. The actual frequency drift wi th temperature wi l l be discussed in the following chapter, but is now stable to wi thin 5 M H z over the course of an experiment. The resonator as fabricated has radius of curvature of R = 2.08 cm and radius of the mirror aperture of a = 1.91 cm (see figure 2.2). The theoretical performance of a resonator wi th these characteristics is shown in the accompanying figures. These calcu-lations neglect all losses other than the ohmic resistance of the sample and mirror. 4.4 Coupling The microwave energy used for the experiment must be coupled both into and out of the resonator, in order that measurements be made. We have decided to work with a transmission mode cavity (in which energy is coupled in at one spot, and out another) since this configuration is better suited to situations where the loss has a wide variation with temperature and sample. Coupling is accomplished by connecting waveguides to small apertures near the axis of the resonator on the curved mirror. Because there must be two holes, slightly off center, the transmission method does have some disadvantages Chapter 4. Millimeter-Wave System 57 1.40 1.60 1.80 2.00 2.20 Mirror Separation(cm) Figure 4.4: Mode chart of the resonator as built . These modes depend on the beam-wave theory and are approximate. As well losses are ignored. Hence, this chart is only a guide as to the behaviour of /RES wi th mode number and D. compared to reflection methods. These two imperfections in the curved mirror cause scattering and lower the Q of the resonator. It is difficult to determine a priori what size of holes are needed to give adequate signal strength. There are theoretical and phenomenological models [47] [8], however the assumptions made often negate their ut i l i ty in this problem. In this apparatus, the size of the coupling holes was determined empirically, by starting wi th small holes and increasing their size unt i l an adequate signal strength was observed. The final size of the holes is ~ 0.025" diameter, which is not negligible in comparison to the wavelength of radiation, and thus w i l l lower the Q of the resonator. Chapter 4. Millimeter-Wave System 58 160000 \-150000 -o 140000 -130000 \-I—i 1 1 1 — — • - — i 1 1 . 1 1 1 . . i i . i i i • 0.016 0.017 0.018 0.019 0.02 Mirror Separation (m) Figure 4.5: Numerical simulation of the resonator Q for copper at 77 K as a function of the mirror separation in the complex-point-source model (cf. Chapter 3). The thick line corresponds to the mode TEM0013 a n d the thin line to T E M 0 0 i 4 . The data on this graph becomes increasingly inaccurate as D approaches R = 2.08cm, due to neglect of diffraction losses which dominate i n this regime. A related matter concerns the problem of bringing the microwave power to the res-onator without a large heat load. This is accomplished by having low thermal conductiv-i ty stainless steel waveguides, as well as non-contacting choke flanges. These flanges have been used in microwave work for years for other reasons ([51] and other volumes of the M.I.T. Radiation Laboratory series], and also have been exploited in low temperature work [52]. B y having a A/4 shorted stub around the waveguide, the gap between waveg-uide sections is minimized as a discontinuity for the fields. Radiat ion from the joint, as well as reflection down the waveguides are minimized. In the experiment, these flanges are used on both the input and output waveguides. Their performance w i l l be discussed in the following chapter. Chapter 4. Millimeter-Wave System 59 300000 -250000 -200000 • 150000 • 100000 • 50000 -" I i , , , i , • • • • , . , i , i , . i • 1 0.01 0.05 0.1 0.5 1 5 Sample Surface Resistance Figure 4.6: Numerical simulation of the resonator Q at 117 G H z and 77K, as a function of Rs of the sample in the complex-point-source model. This corresponds to a mirror separation of 1.986 cm. 4.5 Microwave measurements Following the derivations of the previous chapter, the basic relation in the measurement of a microwave resonator is: -!- oc ^2 All Losses. (4.2) When measuring the Q of a resonator, one must have some method of coupling energy into and out of the resonator, wi th a circuit including source and detector. This contributes to the losses for the resonator. Near a particular resonant mode of the resonator, it is possible to represent the energy stored in the fields of the resonator by the energy stored in the lumped parameters of an equivalent circuit [53]. The equivalent circuit of a resonator can be described by three parameters: Chapter 4. Millimeter-Wave System 60 Qo = u0L R Ro = LOQLQO (4.4) (4.5) where LO0 is the resonant frequency, Q0 is the unloaded quality factor of the resonator (called simply Q unt i l this point) and R0 is the shunt resistance of the resonator . When joined to the rest of the system through coupling, the equivalent circuit can be represented as shown i n figure 4.7 [50]. Figure 4.7: Lumped-element equivalent circuit of a resonator connected between a gen-erator and a detector. The method i n which the coupling is represented can be shown to have no effect on the final result, and is shown here as two ideal transformers. The loaded Q of the system, Qi, can be found by examining this circuit, giving: LO0L QL = (4.6) R + nlZr + njZ2 where Z\ and Z2 are the characteristic impedances of the input and output transmission lines respectively [50]. Defining the coupling constants 2 Z\ (4.7) and similarly for 82, the relation between the unloaded Q and the loaded Q of the system can be writ ten QL QO Chapter 4. Millimeter-Wave System 61 Qo <3o = i + i + i (4.8) Qo Ql 0,2 where Qi and Q2 are the coupling Q's. These two terms are simply two additional terms to equation 4.2. Thus the measured Q of a resonator w i l l be lower than the Q if it were isolated from the circuit. To make good measurements of the Q of the resonator, it is desirable to work wi th very weak coupling (small /3's). There is no way of modifying the coupling of the system as it is currently constructed, and hence no method of determining Q0 on its own. The loss due to coupling, as well as due to all factors except the sample, should be constant between experiments and during an experiment at different temperatures. Thus the relation between the Q of the cavity and the surface resistance of the sample can be writ ten, using equations 4.2 and 3.55 as: = Rs , — f f \h\2dS + constant x (Other Losses). (4.9) Q L S a m p l e Up J JSample* ' V J \ J In this equation the relationship between the surface resistance and the measured quantity of the Q of the resonator is clearly noted. The main idea of the experiment is that this equation can be rewritten simply as (writing Q instead of QL again) ^ = <*Rs.anpU + P (4.10) where a and 8 are two constants, independent of the sample. Though both of these constants are related to values from the theory (in the case of a the theoretical value is shown above), the approximations made in their derivation would make the final result suspect. In the following chapter the values of these constants are found experimentally. Chapter 5 Experimental Method 5.1 Introduction In this chapter, I will describe the techniques used in taking, calibrating and analyzing the surface resistance data as measured by the apparatus previously described. Measurement of the surface resistance of normal metals will be described here, although the techniques would be identical for superconducting thin films. The basic formula used in analyzing the data is the relation where Q, the resonator quality factor, is the measured quantity, and Rs, the surface re-sistance of the sample, is the desired quantity. This equation comes from the theoretical discussion in chapter 3, as well as the section on microwave measurements in chapter 4. The losses of the resonator each contribute independently to l/Q, allowing a simplifica-tion of the form of equation 5.1. The parameter a relates the fraction of the loss of the resonator due to the surface resistance of the sample. The parameter 8 encompasses all other losses in the resonator (surface resistance of the curved mirror, scattering losses, etc.). It is hoped that the product aRs is not small in comparison with 8 for accurate measurement. Although it is possible to make estimates for the values of a and 8 from theory (chapters 3 and 4), such values are only approximations, and ignore such losses as the finite size of the mirrors and the effect of coupling holes. Therefore, the parameters will Q = l aRs + /3 62 Chapter 5. Experimental Method 63 be determined empirically by measuring the surface resistance of known samples. 5.2 Me t a l l i c Samples Since the surface resistance of high Tc superconductors at 117 GHz and ~ 77K is not much different from that of metals (such as copper) at similar temperatures, it is con-venient to use good metals as references for calibration of the resonator. It is, however, important to realize that the simple relation for the surface resistance of a conventional metal given in the introduction (equation 1.19) does not necessarily apply for a good metal at cryogenic temperatures and high frequencies. This is due to the anomalous skin effect, where the mean free path in the material is comparable to the classical skin depth. Hence conduction is no longer a local phenomenon, and more complicated effects are introduced [54]. The criterion for these effects to be important is given by the product COT, the fre-quency of the radiation times the scattering time for electrons in the metal. When LOT <C 1, then the standard formula applies, Rs = —-. In Dingle's two papers [55] [56], (TO the quantitative theory of Reuter and Sondheimer [54] is evaluated numerically, and the results for various metals are tabulated. The surface resistance for copper as a function of temperature at 117 GHz is shown in figure 5.2. The surface resistance of brass is used as a calibration as well. Here, the literature is not very much help when finding the resistivity, since there are many different alloys, all given the name brass. The d.c. resistivity of a small sliver of the same material used in the open resonator experiment was measured directly [58]. From this the surface resistance was calculated as Rs = 7. Both the copper and brass samples used for surface resistance 0 measurements were 1 x 1cm 2. They were mechanically polished to ~ 5//m grit, then chemically polished (using a solution of 1/3 each of nitric acid, orthophosphoric acid and Chapter 5. Experimental Method 0.10 i 64 0.00 1 i • • i • I 0 50 100 150 200 250 300 Temperature (K) Figure 5.1: Surface resistance of copper at 117 G H z calculated from d.c. resistivity data from the literature [57], using the standard skin depth formula (squares); and using a numerical solution of the full integral equations accounting for the anomalous skin effect [56]. glacial acetic acid at 70 d e g C [59]) to remove damage from the mechanical polishing. The brass surface became noticeably shinier after the chemical polish, though results were only mediocre with copper, despite repeated trials wi th various methods (including electrochemical polishing [59] [60]). 5.3 Measurement Procedure The general procedure followed for al l measurements w i l l be described here. The proce-dure was the same whether the sample was superconducting or metallic. The sample was attached to the underside of the mounting block of the sample gantry in the cryostat (see figure 4.3) wi th a small amount of vacuum grease. The cryostat was pumped to ~ 1 0 - 6 torr and mounted onto a l iquid nitrogen storage dewar. The cryostat Chapter 5. Experimental Method 65 was cooled ini t ia l ly by opening the needle valve and drawing l iquid nitrogen through the heat exchangers rapidly by pumping wi th a large mechanical pump. After about 45 minutes, when the cryostat had reached approximately 77 K , the flow of the nitrogen was reduced by closing the needle valve and regulating the pressure of the escaping nitrogen gas. This controlled the temperature of the upper exchanger (and hence the curved mirror of the resonator) to wi th in 69-69.6 K (for al l measurements used here). Having three independent controls on the temperature (pressure of the exit gas, throttle valve outside the cryostat, and needle valve inside the cryostat) meant that there were more controls than actually needed, and the same temperature could be reached for various flow rates. In most runs, the min imum flow rate was used. The temperature of the sample was measured and controlled using a resistance bridge (made by the U . B . C . physics electronics shop) connected to a Lakeshore C E R N O X resis-tance temperature sensor. The thermometer, and another resistor used as a heater, were attached to the sides of the sample mounting block. The temperature gradient between this block and the rest of the experiment was maintained across the two stainless steel plates supporting the block. There were no noticeable temperature gradients between the sample and the block, which was verified by mounting a diode on the sample block in the same manner as a sample. The temperature as measured by the diode corresponded closely to the temperature of the Lakeshore thermometer. Once the apparatus had cooled, the frequency of the source was set to a constant value (normally JMOD ~ 1200 M H z ) and the vertical position of the sample was varied unt i l a resonant mode was found (by noting both a rapid change in phase and an increase in the amplitude of the received signal by the lock-in amplifier). It was difficult to determine in advance what setting of the sample-mirror separation would give a resonant frequency within the bandwidth of the apparatus. This is due to the samples having varying thicknesses, and due to the thermal contraction of the gantry. Once a resonant mode was Chapter 5. Experimental Method 66 found, the choke flanges were adjusted to give the best signal. In the current apparatus, it is difficult to do this prior to cooling, since the transmitted signal for a high temperature superconductor sample at room temperature was undetectable (the Q was so low). The procedure was repeated, scanning for other modes of the cavity, once the choke flanges have been adjusted to give good transmission. The same mode of the resonator has been used for al l the experiments described here. It is identified by various methods: its "distance" from other modes (both in frequency and mirror separation), its peak signal strength (largest of any mode observed), and the behaviour with respect to changing the frequency components making up the feeding signal (since I use the sideband where /RES = 8 X fsYN + JMOD, then increasing fsYN should lower /MOD at resonance, and vice versa). It is, nevertheless, difficult to make a precise identification with the modes shown in figure 4.4 for two reasons: firstly, because the measurement of the separation between the sample and the curved mirror is relatively imprecise; secondly, the equations defining the resonant frequency do not take into account such perturbations as the finite size of the surfaces, the effect of the coupling holes or ohmic losses of the mirrors on the resonant frequency. The result is that one usually doesn't know the exact resonator mode. Using the resonator at room temperature and probing the field with a point dielectric perturbation (a small drop of epoxy on a monofilament thread), we were able to show that there were no nodes in the transverse field profile for the mode used. From the evidence, we judge that the resonator-is most likely operating in the T E M 0 o i 4 or T E M 0 0 1 5 mode. A more conclusive study of the mode pattern could be accomplished by scanning the resonator across the surface of an object with a clearly defined transition in surface resistance, such as a bi-metallic (brass / copper) surface. This was attempted early on, but the results were inconclusive due to l imi ted resolution (the resonator was at room temperature, and the experimental techniques not yet refined). Chapter 5. Experimental Method 67 1.0 I — 1 — i — ' — i — • — i — 1 — i — 1 — i — 1 — i — 1 — i — 1 — i — 1 — i 1 — 0.8 - ^ — ~ ~ — N O 0.6 " / / \ . - J 3 / Cu / Q 4 _ / / With Standard Flanges / / With Choke Flanges 0.2 j / 0 o ' ' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 '116.9 117.0 117.1 117.2 117.3 117.4 117.5 117.6 117.7 117.8 117.9 Frequency (GHz) Figure 5.2: Comparison of the detected signal amplitude for direct transmission (it. no resonator) for standard flanges and choke flanges. A comparison of the signal obtained wi th no resonator or choke flanges (direct waveg-uide connection) against the signal obtained wi th no resonator but wi th choke flanges is shown in figure 5.3. As can be seen, the transmission amplitude is quite flat, wi th only small ripples as a function of frequency. The choke flange transmission is considerably poorer, and is very sensitive to misalignment. This is something that must be remedied in future versions of the apparatus. Once the proper mode of the resonator was found and the flanges aligned, the Q of the resonator is found as a function of sample temperature. The vertical adjustment is set so that, at the lowest temperature (~ 70 K ) , the resonant frequency /MOD is 1210 to 1215 M H z . The same approximate frequency was used to avoid any sytematic errors which may occur when operating at different regions wi thin the bandwidth of the Chapter 5. Experimental Method 68 millimeter-wave system. The source frequency is swept over the resonant frequency under computer control in 50 to 100 steps wi th the amplitude signal from the lock-in amplifier being recorded at each point. It was found that the sweep time (how long the computer paused with each data point) affected the value received for the Q. Fast sweeps gave low Q's, which increased as the sweep t ime lengthened. The experiment was operated in the regime where the Q's received were asymptotically close to the l imi t ing value. This effect was due to the slow response time of the lock-in amplifier. Though a slow scan of frequency as performed here wi l l be more susceptible to drift i n the apparatus (due to changing temperature etc.), this effect was not noticeable wi thin one scan of the frequency when the nitrogen pressure was regulated (though there is a significant drift in the frequency over the course of an experiment - see later). The data was collected on an Intel 286 computer (as described in chapter 4) and transferred to a Sun Sparc workstation for analysis. The Q of the resonator was found by fitting the amplitude vs. frequency data to an appropriate function through a least-squares method. If we were measuring the power transmitted through the resonator, we would expect a Lorentzian relation between power and frequency about the resonant frequency. In this experiment, however, one is measur-ing the amplitude of the microwave signals which are proportional to -v/Power. There was a significant background noise associated with the signal; including random fluctuations, as well as directly transmitted signal (leakage through isolators, direct transmission be-tween coupling holes in the cavity, and/or leakage from lower frequency portions of the circuit). In addition, the gain of the source/detection circuit was not necessarily uniform over the frequency range. Thus the amplitude vs. frequency was fitted to a function of Chapter 5. Experimental Method 40000 69 39000 4F- * =F O 38000 37000 - i — I— I— r -o o Copper 36000 30000 29000 28000 h 27000 J i _ i . • i 80 100 105 110 85 90 95 Temperature (K) Figure 5.3: Q vs. temperature for brass and copper. The lines jo in the points in the order the data was taken. the form: V A + Bv + D + Ev. (5.2) 'i + (^y This was intended to account for both the background (linear) as well as variation in the gain source/detector pair with frequency (linear as well). The fitted results for the resonant frequency and Q of the resonator wi th copper and brass as a function of temperature are shown in figures 5.3 and 5.4. In both cases, the lines join points taken in chronological order. Thus one can follow the drift in frequency over t ime i n figure 5.4, although there is a much smaller corresponding drift in Q. The error bars shown are from the least-squares fitting, assuming equal error for each data Chapter 5. Experimental Method 70 117.224 117.222 N g o fl cr & 117.220 fl c3 fl O * 117.218 117.216 G—© C o p p e r • — • Brass 70 75 80 85 90 95 Temperature (K) 100 105 110 Figure 5.4: Resonant frequency as a function of temperature for brass and copper. Again , lines jo in points chronologically, and show clearly the drift i n frequency wi th t ime of about 5 M H z . Error bars from the fits are smaller than the symbols on this graph. point [36]. Various functional forms were compared in the fitting both for a single resonant peak, and also over a full experiment (see figures 5.5, 5.6, 5.7 and table 5.1). The final function chosen has the lowest y2 in most cases. Perhaps even more importantly, the Q displays li t t le change wi th frequency (drifting over time), unlike the other functions (see figure 5.7). This is due to the asymmetric gain in the fitting function corresponding more closely to the actual behaviour of the experiment. Clearly, it would be better to have a flat gain over the bandwidth of the system, but modeling the asymmetry of the gain is an inter im solution. The earlier version of the experiment where thermal expansion effects were rather large produced results of dubious validity. Chapter 5. 0.5 Experimental Method 71 o Measured Data — Const. Background | — Linear Bkgd. j — Quadratic Bkgd. j Lin. Gain + Const Bkgd. f — Lin. Gain * (Sqrt(Lorentzian) + Const. Bkid.) Lin. Gain + Lin. Bkgd. p — Lin. Gain * (Sqrt(Lorentzian) + Lin. Bkgd.) Q 0 I i i i I i i i I i i i I 1.1699e+ll 1.1701e+ll 1.1703e+ll 1.1705e+ll Frequency (Hz) Figure 5.5: Comparison of the various fitting functions wi th the measured data. The data in question is for copper at 70 K and represents a "good" peak — where the amplitude and Q are both quite high. The functions are described in more detail in table 5.1. The data in figure 5.3 are fit to the equation 1 1 T) 7j = a i R S B r a s s - RScopper) (5-3) tyBrass ^SCopper to get the proportionality constant a. The result is then inserted into equation 5.1 to find the sample-independent loss 3. The result of the fit is shown in figure 5.8. The values of the fitting constants received are a = (7.45 ± 0 . 3 2 ) x 1 0 - 5 f T 1 (5.4) and, 6 = (2.334 ± 0.024) x 10~ 5 . (5.5) Chapter 5. Experimental Method 72 O Const. Background • Linear Bkgd. -0.010 1 1 1 1 1 ^ 1 • 1 1 1.17e+ll 1.1701e+ll 1.1702e+ll 1.1703e+ll 1.1704e+ll Frequency (Hz) Figure 5.6: Residuals of the fitted peak for the various functions used. The curve used for data analysis is shown with the points joined. The uncertainty in these values includes both the uncertainty in the measurement of Q for each point, and the uncertainty i n fitting the two experiments to the d. c. data. No systematic error is included in the uncertainties. Chapter 5. Experimental Method 73 Function ^0 x 2 Q 1 + [ V - U 0 y 2 / V 2 V B + Cv 117.0231090 * 10 9 0.0003902 38481 ± 194 A 1 p 117.0229941 * 10 9 0.003456 37759 ± 5 6 0 i + ^ - ^ ) 2 / r 2 + B + Cv 117.0229815 * 10 9 0.002665 37759 ± 499 A , 2 / P 2 + B + Cv + Dv2 117.0229815 * 10 9 0.002665 37999 ± 496 A+Vv , p i + ( ^ 0 ) 2 / r 2 -° 117.0229809 * 10 9 0.003364 37792 ± 554 117.0229941 * 10 9 0.003456 37758 ± 554 ( 1 + ^ ) ( 1 + ( , - t w r 2 + - g + ^ ) 117.0229941 * 1 09 0.003456 37759 ± 564 Table 5.1: Comparison of (some of) the fitting parameters for various functions, as shown in the accompanying figures. This is for a particular scan of copper at 71 K . The first line of the table shows the function used in all subsequent work. Chapter 5. Experimental Method 42000 40000 38000 36000 74 42000 40000 38000 36000 42000 40000 38000 36000 42000 40000 38000 36000 75 80 85 90 95 Temperature (K) 100 105 42000 40000 38000 36000 42000 -3 40000 38000 36000 42000 40000 38000 36000 110 F i gure 5.7: Comparison of Q vs. temperature for copper using various fitting functions, wi th the points connected in the order the data was measured. The functional form used is in the top graph, the other graphs use the functions in the same order as table 5.1. Note how the uncertainty is much smaller, and the drift wi th time is reduced i n the first graph. Chapter 5. Experimental Method 75 0.20 0.15 a Pi 0.10 0.05 0.00 - 1 — 1 — I — I -a = (7.45+.32)10 O. P = ( 2 . 3 3 4 ± . 0 2 4 ) 1 0 ' s OBrass from measurement OCopper from measurement — Copper from Literature - Brass fromD.C. meas. 70 75 80 85 90 95 Temperature (K) 100 105 110 Figure 5.8: The cavity perturbation data measurements of Rs of copper and brass at 117 G H z fitted to values obtained from the literature and d.c. measurements. The fitted values of the constants a and 8 as determined from this data are shown in the figure, and their uncertainty has been added to the error bars as shown. Chapter 6 Data 6.1 Introduction In this chapter the measurements of surface resistance performed to date using the open resonator w i l l be presented. These samples were available to our lab i n November and December 1994, and are actually a good survey of the capabilities of the apparatus. There are, however, no "state of the art" thin films yet available in the group, which might push the l imits of the resolution of the apparatus. In fact the problem is the opposite: the surface resistances of the films are high enough that the Q of the resonator is quite low, and the transmission through the resonator very small. Measurements above the transition temperature are lacking for two reasons: first of all the loss of a high temperature superconducting thin fi lm i n the normal state is quite large, which reduces the Q of the resonator, and the transmitted signal becomes too small to measure. A related problem, of the enhancement of loss when the skin depth is large, w i l l be discussed later in the chapter. The second reason is that the sample thermometer had been calibrated incorrectly, and when the sample was thought to be above To, it was in fact 4 K lower. This problem has since been corrected, and the existing data re-analysed, however no new data has been taken. 76 Chapter 6. Data 77 80 Temperature (K) 85 90 F i gure 6.1: The temperature dependence of the surface resistance of a 6 /ITTI thick B S C C O thick fi lm on an M g O substrate. The surface resistance of copper and brass are included for comparison. 6.2 Data 6.2.1 Furukawa Electric Bi 2Sr 2CaiCu208 Thick Film Figure6.1 shows the surface resistance vs. temperature of a thick f i lm of B i2Sr2CaiCu20s. The film is deposited through a silk-screen-like method on an M g O substrate by the Furukawa Electr ic Company. Unlike the other films studied here, this film is not of Y B C O , but of B S C C O , a material which hasn't been studied as extensively in the microwave/millimeter-wave region. The film is also quite thick, at 6 fim, and has visible patterns on the surface, suggesting multiple grains as well as roughness. To avoid mea-suring this very rough layer, which would scatter the radiation and lower the resolution, Chapter 6. Data 78 the sample was placed with the film side towards the mounting block (ie. away from the curved mirror) . As a consequence the fields must penetrate through the substrate. This change wi l l have a number of effects: • the dielectric constant of the M g O (er = 16 at 10GHz [6]) w i l l cause the effective size of the resonator to be larger, possibly shifting modes. • the dielectric loss of the substrate (tan S = 6.2 x 10~ 6 [6]) w i l l lower the Q of the resonator. • reflection wi l l occur from the substrate-air interface as well as from the substrate-film interface. The effective resonator length for the substrate-air surface w i l l not be resonant, and so wi l l scatter the energy and lower the measured Q. • any change of these parameters with temperature wi l l give a contribution to the Q which wi l l be indistinguishable from properties of the superconductor itself. To avoid, or at least account for, these problems the substrate should be measured with a known surface below (such as copper). This has not yet been done. The surface resistance as measured is very high. This may be due to the substrate, but it is very likely due to the properties of the film itself. To have low surface resistance, the crystallinity of the superconductor needs to be very high. This measurement is a non-local probe of the material, so a film which shows zero d.c. resistivity may only have small connected superconducting sections, and show high surface resistance. The appearance of the film, and the manner in which it was fabricated, suggest problems of this sort. Chapter 6. Data 79 a oi 0.5 0.4 0.3 0.2 h -3TT 0.1 0.0 Las® 0 a s 70 75 80 o U.B.C. Y B E L , C U 3 C V 5 on LaA103 — Brass - - Copper 0 ± 70 _»_<L » »t-t_t- - t-*- -I- - -r- ~ 75 80 Temperature (K) 85 90 Figure 6.2: The temperature dependence of the surface resistance of a thin film of YBa2Cu 3 07_5 on an LaA103 substrate. 6.2.2 U . B . C . Y B a 2 C u 3 0 7 film on LaA10 3 This YBa2Cu307_5 film was grown by Andre Wong in the U . B . C . physics department using pulsed laser ablation ablation [61] (film AW1107). This technique produces highly epitaxial films on a number of substrates, in this case on LaA103. There are a large number of variables and procedures which must be optimized to give this good epitaxy and crystallinity of the film. The films measured here which were grown at U . B . C . have not reached optimal properties, and show a Tc of approximately 85-88 K . This film shows a rapid decrease in the surface resistance with temperature, reaching 0.1 ri surface resistance at 71 K . This is the lowest surface resistance of the four films measured here, though it is still about twice the loss of copper at the same temperature. As mentioned before, this film was not measured over a wide enough temperature range Chapter 6. Data 80 to find the transition temperature — though the resolution of the apparatus was very poor at ~86 K where the measurements stopped. It is encouraging that the surface re-sistance was still decreasing when the lowest temperature was reached, but its high value and the low transition temperature suggest further work must be done to improve the film growth procedure. The thickness of this film has not been measured, but the growth time and conditions used give a value of about 3500A. This thickness is comparable to the penetration depth of the superconductor in the temperature range of the experiment. The derivation of the surface resistance in chapter 1 assumed that the thickness super-conductor was much greater than the penetration depth, and thus is not appropriate in this case. The effective surface resistance of such thin films will be discussed in section 6.3. 6.2.3 U . B . C . Y B a 2 C u 3 0 7 film on S r T i 0 3 This sample is another thin film of YBa2Cu307_§ (figure 6.3) grown by Andre Wong at U . B . C . using pulsed laser ablation. This sample, however, was grown on a substrate of SrTiOs , which probably accounts for the unusual behaviour observed for the temperature dependence of the surface resistance. Strontium Titanate (SrTiOs) , is a ferroelectric material, which exhibits change in the relative permittivity between 300 at room temperature and 20000 at 4.2 K [62]. If any radiation is transmitted through the film then resonances can be established between the film and the copper mounting block. This will be discussed in section 6.3. It is, therefore, difficult to estimate the actual surface resistance of this sample. For this reason superconducting films on S r T i 0 3 are not often used for high frequency applications. This is unfortunate, since S r T i 0 3 is a substrate with very good lattice match to Y B C O , and does not exhibit twinning (unlike LaA103)[6]. Chapter 6. Data 81 g o -1 1— A U.B.C. YBa2Cu3075 on SrTi03 — Brass - - Copper 11 70 75 80 Temperature (K) 85 Figure 6.3: The temperature dependence of the surface resistance of a thin film of Y B a 2 C U 3 0 7 _ 5 o n an SrTi03 substrate. 6.2.4 McMaster Y B a 2 C u 3 0 7 _ 5 10000A film on LaA10 3 This is another fi lm of YBa2Cus07_5 grown by pulsed laser ablation on LaAlC>3 (see figure 6.4). This f i lm was grown at McMaste r University through a process which has been well optimized to produce films of high epitaxy and crystallinity. This f i lm, however was quite thick (1 /«m), and there were problems associated wi th this. This f i lm shows a sharp drop in Rs below Tc, but levels out before 80K at a level of about 0.15 ft. The growth of thicker films is currently the subject of substantial research, since thicker films are able to carry a larger amount of current, useful for high power applica-tions. A l l substrates currently in use have a lattice parameter that is slightly different from that of YBa 2 Cu307_5. As the f i lm thickness is increased, the strain that is induced by this lattice mismatch causes dislocations, grain boundaries and other defects in the Chapter 6. Data 82 a y 1 0 0.5 0.4 0.3 0.2 .0.1 0.0 s2 * 70 75 80 85 V McMaster YB a^Cu^ on LaA103 — Brass - - Copper _ m - | * * 1 5 5 70 75 80 85 Temperature (K) 90 95 Figure 6.4: The temperature dependence of the surface resistance of a 10000A thick film of YBa2CU307_i grown at McMaste r Universi ty on an SrT iOs substrate. f i lm. These defects tend to increase the surface resistance. Work is progressing both in the development of new substrates wi th a reduced lattice mismatch, as well as in the techniques needed to grow films of a thickness of about 1 fxm. The surface resistance of this f i lm was measured by D r . Doug Bonn at U . B . C . as well, using a 3.7 G H z resonant cavity method [14]. Unfortunately, the surface resistance showed a strong dependence on the power of the excitation, suggesting weak links in the sample and precluding further measurements. The presence of weak links is consistent with current problems in growing thick films, as described above. Unfortunately, the open resonator described here does not have the capability of examining the power dependence of the surface resistance, in order to determine if such weak links exist. Another fi lm grown at McMaste r University by the same process was measured by Chapter 6. Data 83 2.0 1.5 1.0 o c CT1 (D V-l 0.5 0.0 -1 1 1 r— I f 1 D3 70 80 90 Temperature (K) 100 Figure 6.5: The temperature dependence of the surface resistance of two McMaste r films (solid squares: 3000A thick, open circles: 10000A thick). The thi nner fi lm was measured at 3.7 G H z and the thicker at 117 G H z using the open resonator. The data are scaled by the frequency squared, according to the two-fluid model. This scaling is not appropriate for the normal state, which accounts for the discrepancy near Tc-Dr. Bonn. This film had a thickness of 3000 A, and showed a very low surface resistance (approximately 10 - 4 Cl below 70K at 3710 M H z ) . A comparison of this f i lm wi th the data in figure 6.4 is shown in figure 6.5. The data are scaled by the frequency squared, to account for the dependence of R$ on frequency. The data are quite comparable, with the (scaled) Rs of the thin fi lm only slightly lower than that of the 10000A f i lm. The frequency scaling may actually be weaker than u; 2 , since the two-fluid model used to derive it is only approximate. Further study is needed to determine the frequency response of both films and crystals. The McMaste r f i lm measured by the open resonator, does however show a clear Tc, Chapter 6. Data 6 84 g Pi 70 O Furokawa BiSrCaCuO on MgO — Brass - - Copper OU.B.C. YBa 2 Cu 3 0 7 . 5 on LaA10 3 V McMaster Y B a j C ^ O ^ on LaA10 3 o v 75 80 85 Temperature (K) 90 95 Figure 6.6: The surface resistance of the superconducting films described earlier shown on the same plot for comparison. The surface resistance of the fi lm grown on SrTiOs is not included, both for clarity and because of the systematic substrate problem. unlike the others. This is most likely due to the fact that the fi lm is thick, thicker in fact than the skin depth of the material in the normal state. This allows measurements above the transition temperature to be made wi th acceptable accuracy. In fact, it was the in i t ia l measurement of a transition at 96K that alerted us to the mis-calibration of the sample thermometer. 6.2.5 Comparison Chapter 6. Data 85 As mentioned earlier, the l imited data obtained so far cannot be considered to be the basis of an exhaustive study. Trends in the data can be seen quite clearly, however: • The surface resistance of the thick B S C C O film is very high, and this f i lm is not suitable for microwave work. • The films grown on LaA103 show the best microwave performance. • The effect of the changing dielectric constant of S r T i 0 3 makes its use as a substrate for high-frequency superconducting circuits l imited. • The surface resistance of the McMaste r film is lowest at high temperature, but reaches a l imi t ing value that is quite high (.15 fl) at about 75 K . • The surface resistance of the U . B . C . film grown on L a A 1 0 3 drops rapidly around 85K, and continues to drop below the temperature range accessible in the measure-ment. The continuing decrease in the surface resistance of the U . B . C . film on L a A 1 0 3 sug-gests that there are less defects present than in the McMaster film, though its low Tc may be due to reduced oxygen concentration. It is for measuring properties such as these that this apparatus has been designed, and a systematic study would be fruitful. The work done by many researchers in improving the crystallinity of thin films in order to reduce the surface resistance contrasts wi th the studies done in this lab suggesting a certain level of defects reduce the surface resistance [63]. The cause of this discrepancy, that the lowest Rs is obtained in films by striving to remove all defects, while i n crystals by adding defects, is currently unknown. It is possibly due to the nature of the defects; that dislocations and twins increase the surface resistance, while substitutional disorder can lower it . Chapter 6. Data 86 Included here for comparison is a graph showing the surface resistance of thin films of Y B C O at 77K vs. frequency (figure 6.7). This graph was taken from a paper published in 1992 [6], wi th much of the data older than this, however data from more recent publications is quite similar. 6.3 Finite Thickness Effects The surface resistance of the films as measured is likely to be dominated by extrinsic effects such as defects. There is, however, an extrinsic effect for which we can correct: the effect of the finite thickness of the film. T h i n films of superconductors are grown to a few thousand angstroms thick — about the same size as the penetration depth i n the material. Thus the assumption of infinite thickness in the derivation of the surface resistance in chapter 1 is not justified. Thus the effect of the reflection of the electromagnetic energy off the backside of the film, as well as of transmission completely through the film, can no longer be ignored. The effective surface impedance of a film of thickness d for penetration depth A on a substrate of infinite thickness of relative permit t ivi ty er is given by [62]: # e f r = Rsf{d/\) + R trans (6.1) X e f f = Xscoih(d/\) (6.2) where fWX) = c o t h W A ) + - ^ _ 1 / 2 ( a y * A ) 2 1 t r a n s ~ r Z0 smh2(d/X) where Z0 is the impedance of free space (377 fl). These equations are derived by con-sidering both the reflection and transmission from both the top and bottom of the film. This is done conveniently through impedance transformations, as described thoroughly Chapter 6. Data 87 in the reference [62]. The function f(d/\) is the enhancement of the loss in the fi lm. This occurs because the current density excited in a f i lm of finite thickness are greater than for an infinitely thick superconductor. The transmitted energy is described by the term i ? t r a n s . This term, i n most cases, is smaller than the enhancement due to increased current. The situation is actually improved significantly by having the substrate backed by metal, as is the case in this apparatus (see figure 6.9). Here, almost almost al l of the transmitted energy is reflected back by the metallic backing, rendering the -Rtrans term negligible. When the thickness of the substrate (in between the film and the metallic support) resonates, the transmission which is normally negligible can have a significant effect. Such a resonance occurs when the thickness of the substrate D satisfies [62]: D = n —^ ,n = 1,2,3... (6.3) where A 0 is the free-space wavelength of the radiation, and e r (T) is the relative permit-t ivi ty of the substrate at temperature T. For most substrates, this would be a problem at only isolated frequency/thickness combinations. However, for SrTiOs the temperature dependence of the permit t ivi ty is dramatic. Here a series of resonances can occur within a temperature sweep. Off resonance, the transmitted radiation is reflected back into the resonator, as described above. A t resonance, the radiation wi l l bounce back and forth between the conducting surfaces unt i l it is dissipated. This explains the peaks and valleys observed in the temperature dependence of Rs of the Y B C O film grown on strontium titanate (see figure 6.3). K l e i n et. al. [62] calculate the relative permit t ivi ty of S r T i 0 3 using the positions of the min ima of these resonances. This is only really possible at temperatures above Tc of the film, since the rapid change in Rs below Tc w i l l distort the curve. Near, and above Tc, where the penetration depth becomes large (and is replaced by Chapter 6. Data 88 the skin depth), the finite thickness effect becomes more pronounced. The skin depth for a high temperature superconductor in its normal state is given by as shown in equation 1.13. A typical d.c. resistivity in the normal state just above T c of p = 100/iOcm gives a skin depth of about 15000 A. This is thicker than most thin films, causing the spurious increase in surface resistance noted before. It is for this reason that the apparent surface resistance becomes unmeasurably high in thin films of superconductor above Tc- This l imi ta t ion is inherent in the measurement, and precludes accurate normal state measurements on thin films in this apparatus. Chapter 6. Data 89 f {GHz] Figure 6.7: The frequency dependence of Rs of untextured (circles) and c-axis tex-tured (triangles) polycrystalline bulk or thick fi lm samples, as well as for epitaxial thin films (squares) and single crystals (rhombuses) of Y B a 2 C u 3 0 7 at 77K. This plot is taken from reference [6]. The cross in the upper right corner is the data for the McMaste r f i lm from this thesis. Chapter 6. Data 90 Figure 6.8: The effective surface resistance as a function of the intrinsic surface resistance for a f i lm 3500A thick at 117 G H z assuming A = 2600A and e r = 16 [62]. Figure 6.9: The calculated effective surface resistance for various thicknesses of f i lm. This figure is reproduced from [62], and assumes / = 87 G H z , A = 2600A and the substrate wi th er = 16 is 1mm thick, backed by copper. Chapter 7 Conclusions and Discussion 7.1 Introduction The project described in this thesis has been successful in creating an apparatus to routinely evaluate the surface resistance of superconducting films. These measurements can be used as feedback to those involved in the deposition of these films, wi th the goal of lowering the microwave loss of the materials. 7.2 Equipment Performance The apparatus as constructed, consisting of the cryostat, the millimeter-wave source / detector pair, and the resonator assembly, has been tested under a variety of conditions. The cryostat's performance has been numerically modeled thoroughly for l iquid he-l i u m cooling, and was designed based on this data. The cryostat has been operated numerous times using l iquid nitrogen cooling, wi th good control of both the sample and base temperature from 70 to 120 K . This control was achieved using pressure regula-tion on the exit gas, as well as active control of the sample temperature using electronic regulation. A swept millimeter-wave source and detector have been assembled using a previously buil t fixed-frequency spectrometer. The resulting system has a flat transmitted power over a 500 M H z frequency range around 117 G H z , wi th a peak detected signal of 200 m V , and a background noise level (due to leakage or random noise) of < 1 m V . 91 Chapter 7. Conclusions and Discussion 92 The resonator as designed has an unloaded Q wi th copper at 70K of 39000, and the system can handle a min imum Q of ~ 2000. The resonant frequency is stable to wi th in 4 M H z over 50 K changes in the temperature; this stability has required minimizat ion of the effect of thermal contraction. W i t h a data acquisition and analysis system that accounts for much of the asymmetry present in the gain of the millimeter-wave system, the Q and resonant frequency are fitted wi th good precision. 7.3 Surface Resistance Data The apparatus has been used to measure the surface resistance of superconducting films as well as normal metals. The surface resistance of these normal metals has been used as a calibration of the geometrical factors and of the parasitic losses of the resonator. The fitted results show scatter, but well within the error bars as given by the x2 from the fit done to extract the Q. The measurements on superconducting thin films give the variation of surface resis-tance vs. temperature from 70K to Tc of the film. Measurement of the normal state properties is only possible for thick films where the skin depth of the material in the normal state is less than the thickness of the film. The films measured show a steep drop in Rs wi th temperature below the supercon-ducting transition. This drop levels out in al l the films below 80K, though Rs does continue to decrease slowly. None of the films measured have surface resistance lower than copper at the same temperature, as is expected at this frequency. None of the films measured show outstanding performance in comparison wi th the best of the literature data, indicating that further development work is needed. The films grown on LaA103 show the lowest surface resistance of those measured. The YBa 2Cu307_5 films, which were epitaxially grown, show much better microwave Chapter 7. Conclusions and Discussion 93 performance than the thick B i 2 S r 2 C a 1 C u 2 0 8 fi lm. The fi lm grown on SrTiOa shows a complicated effect due to the temperature dependence of the permit t iv i ty of the substrate causing resonances i n between the f i lm and the copper mount. This dependence makes such films difficult to use for high frequency applications. The effect of the finite thickness of the films is to increase the effective surface resis-tance depending on the ratio d/X (the thickness of the film divided by the penetration depth). This effect is due to reflection off the rear surface of the film increasing the loss in the film. This effect is magnified near Tc where the penetration depth increases, and can lead to an effectively broadened transition. There is a small amount of transmission through the film, but it is negligible unless resonance occurs (as above for SrTiOs) . The thickness of the film is also an important quantity in terms of growth: since thinner films seem to have better crystallinity and thus lower microwave loss. Thicker films have an inherently higher current carrying ability, and thus are desirable for high power applications. The improvement of film growth techniques to have thick films with low Rs is a goal of many researchers. Currently, the surface resistance of films has a min imum as a function of thickness, wi th the Rs enhancement l imi t ing the ut i l i ty of the thinnest films, and non-epitaxy l imi t ing the ut i l i ty of the thickest films. 7.4 Future Work Future work on this apparatus wi l l be greatly affected by a new millimeter-wave vector network analyzer which is expected to be delivered to the lab very soon. This network analyzer w i l l allow more accurate measurements over a much wider range of frequencies. This should avoid any systematic shift in the Q wi th frequency due to asymmetry of the gain of the source/detector pair. However, many of the problems wi th the current Chapter 7. Conclusions and Discussion 94 apparatus are due to the poor performance of the choke flanges, these should be re-designed for better uniformity of transmission over frequency, as well as for greater ease of use. A new sample-mounting gantry should be built to provide: • A smaller thermal conduction to the base, so that higher temperatures can be reached simply. This could also incorporate active sample cooling to reach the lowest temperatures (using l iquid helium) without this large conduction currently present. • A carrousel, so that multiple samples can be measured during the same run. This w i l l allow both reference samples, and multiple unknown samples to be measured rapidly. This is important for routine measurements. • Very small thermal contraction effects. This property of the resonator was an after-thought in the current design, and hasn't been fully perfected. W i t h very small thermal contraction, systematic errors in measurement can be reduced. The current apparatus has not been used at l iquid hel ium temperatures, due to the unsuitability of the current sample gantry. Such measurements are not expected to be done very often, since the high temperature (70-90 K ) properties are most important for applications of films. Studies of fundamental properties, however, w i l l require a greater temperature range than currently used. The use of a conventional superconducting spher-ical mirror could increase the sensitivity of measurements made at all temperatures (at 4.2 K and 117 G H z , the surface resistance of Nb is about 0.001 O, while for copper it is 0.04 Cl) [64]. The flow cryostat wi l l never be as efficient as a standard cryostat, though improvements to the current design can be made (chiefly in the addition of more radiation shielding). These changes may be made in the interest of using less helium. Chapter 7. Conclusions and Discussion 95 Since the new network analyzer w i l l be capable of working over a number of modes of the resonator, a study of the frequency dependence of Rs w i l l be possible. Such a study can probe how well the prediction of Rs oc u>2 holds, as predicted by the simple two-fluid model. As well, a device to allow the study of the spatial dependence surface resistance of large area films at 77K can be constructed. A simple version of this was built previously, but suffered from instability due to temperature drifts. This wi l l allow the routine measurement of the homogeneity of large-area films, which is required for many applications such as microwave filters. It is hoped that studies of fundamental properties of superconductors w i l l be possible with the new millimeter-wave network analyzer. Since higher frequencies and greater resolution w i l l be possible, a goal of measuring single crystals is feasible. Currently single crystals of Y B a 2 C u 3 0 7 _ 5 are too small for measurement, but the l imi ta t ion due to the spot size approaching the wavelength of the radiation becomes more forgiving at higher frequencies. Comparison of the behaviour of single crystals of Y B a 2 C u 3 0 7 _ 5 , and other materials, wi th films is important work for both practical, as well as fundamental reasons. The effect of defects on the electromagnetic properties of these materials gives insight into the still-unknown mechanism for the superconductivity. Defects are currently suspected to give the electromagnetic properties of thin .films their strikingly different behaviour from single crystals. Investigation of these differences could aid in improving the films for practical purposes. There are a number of materials which currently show the promise of becoming useful for microwave work, though most work has focussed in the past on Y B a 2 C u 3 0 7 _ 5 . Studies of the mercury, thal ium and bismuth copper oxides wi l l help in the development of these for commercial purposes, as well as in the understanding of the whole family of high temperature cuprate superconductors. These materials w i l l require the same systematic study as has been done in Y B a 2 C u 3 0 7 _ 5 both for optimization, as well as to determine Chapter 7. Conclusions and Discussion any similarities and differences in their properties. Bibliography [1] J . G . Bednorz and K . A . Mul le r . Z. Phys., B64, 1986. [2] H . Kamerl ingh Onnes. Leiden Comm., (120b, 122b, 124c), 1911. [3] J . Bardeen, L . N . Cooper, and J . R . Schrieffer. Physical Review, 108, 1957. [4] Ruix ing Liang, P. Dosanjh, D . A . Bonn, D . J . Baar, J . F . Carolan, and W . N . Hardy. Physica C, 195, 1992. [5] Glor ia B . Lubk in . Applications of high-temperature superconductors approach the marketplace. Physics Today, March 1995. [6] N . K l e i n . Electrodynamic properties of high temperature superconductor films. In J . J . Pouch, S. A . Alterowitz , and R. R . Romanofsky, editors, Synthesis and Characterization of High-Temperature Superconductors, volume 130-132. Materials Science Forum, 1993. [7] Simon Ramo, John R . Whinnery, and Theodore Van Duzer. Field and Waves in Communication Electronics. John Wi ley and Sons, New York, 1965. [8] 0 . M . Bucc i and B . D i Massa. Open resonators powered by a rectangular waveguide. IEE Proceedings H, 139(4):323-329, August 1992. [9] Michael T inkham. Introduction to Superconductivity. M c G r a w - H i l l , New York, 1975. [10] J . R . Waldram. The surface impedance of superconductors. Advances in Physics, 13(49), 1964. [11] J . Halbrit ter. On surface resistance of superconductors. Z. Phys., 266:209-217, 1974. [12] D . C. Mat t is and J . Bardeen. Theory of the anomalous skin effect in normal and superconducting metals. Physical Review, 111(2), July 1958. [13] J . P. Turneaure, J . Halbrit ter, and H . A . Schwettman. The surface impedance of superconductors and normal conductors: the Mattis-Bardeen theory. Journal of Superconductivity, 4(5):342, 1991. 97 Bibliography 98 [14] D . A . Bonn, Ruix ing Liang, T . M . Riseman, D . J . Baar, D . C . Morgan, K u a n Zhang, P. Dosanjh, T . L . Duty, A . MacFarlane, G . D . Morr is , J . H . Brewer, W . N . Hardy, C . K a l l i n , and A . J . Berlinsky. Microwave determination of the quasiparticle scattering time in Y B a 2 C u 3 06.95. Physical Review B, 47, 1993. [15] W . N . Hardy, D . A . Bonn, D . C. Morgan, R . Liang, and K . Zhang. Physical Review Letters, 70:3999, 1993. [16] C . J . Gorter and H . B . G . Casimir . Phys. Z., 35, 1934. [17] H . London. Nature, 133:497, 1934. [18] K u a n Zhang. Studies of the Microwave Surface Resistance of Pure, Zinc and Nickel Doped YBCO single crystals. P h D thesis, University of Br i t i sh Columbia , 1995. [19] T. Kuhlemann and J . H . Hinken. Measurement of the surface resistance Rs of structured Y - B a - C u - 0 films on different substrates in the lower gigahertz range. Supercondutor Science and Technology, 5:37-40, 1992. [20] S. K a m a l , D . A . Bonn, N . Goldenfeld, R. Liang, and W . N . Hardy. Penetration depth measurements of 3D X Y crit ical behavior in Y B a 2 C u 3 0 6.95- Physical Review Letters, 73(13):1845-1848, September 1994. [21] N . M c N . Alford, T . W . But ton , M . J . Adams, S. Hedges, B . Nicholson, and W . A . Phi l l ips . Low surface resistance in Y B a 2 C u 3 O x melt-processed thick films. Nature, 349, February 1991. [22] S. Sridhar. Microwave response of thin-film superconductors. Journal of Applied Physics, 63(1):159-166, 1988. [23] R . C . Taber. A parallel plate resonator technique for microwave loss measurements on superconductors. Reviews of Scientific Instruments, 61(8):2200, August 1990. [24] 0 . Llopis and J . Graffeuil. Microwave characterization of high Tc superconductors wi th a dielectric resonator. Journal of the Less-Common Metals, 164 and 165:1248-1251, 1990. [25] C . J . Edgcombe. Measurement of surface resistance of high-Tc superconductor by use of dielectric resonator. Electronics Letters, 27(10), M a y 1991. [26] J . S. Martens, V . M . Hietela, D . S. Ginley, T . E . Zipperian, and G . K . G . Hohen-warter. Confocal resonators for measuring the surface resistance of high-temperature superconducting films. Applied Physics Letters, 58(22):2543-2545, June 1991. Bibliography 99 [27] Bao Jia-shan, Zhou Shi-ping, W u Ke-qing, Lou Wei-gen, Ding A i - l e i , and Wang Shu-hong. Microwave properties of highly oriented Y B a 2Cu307_ 5 superconducting thin films. Journal of Superconductivity, 4(4), 1991. [28] W . L . Holstein, L . A . Paris i , Z . - Y . Shen, C . Wi lker , M . S. Brenner, and J . S. Martens. Surface resistance of large-area Tl2Ba2CaCu20$ thin films at microwave and m i l -limeter wave frequencies measured by three noncavity techniques. Journal of Super-conductivity, 6(3):191-200, 1993. [29] Conductus, Inc. 969 West Maude Avenue, Sunnyvale C a 94086. [30] The Microwave Engineers' Handbook and Buyers' Guide. Horizon House Inc., 1964. [31] Low Temperature Techniques. Materials Science Center, Cornell University, Ithaca, New York, Spring 1981. [32] Russell B . Scott. Cryogenic Engineering. D . Van Nostrand Company, Inc., Princeton New Jersey. [33] F . E . Hoare, L . C. Jackson, and N . K u r t i . Experimental Cryophysics. Butterworths, London, 1961. [34] Octave Levenspiel. Engineering Flow and Heat Exchange. P lenum Press, New York, 1984. [35] Guy Kendal l Whi te . Experimental Techniques in Low-Temperature Physics. Claren-don Press, Oxford. [36] W . H . Press, B . P. Flannery, S. A . Teukolsky, and W . T . Vetterling. Numerical Recipes in C, The art of Scientific Computing. Cambridge University Press, 1989. [37] A . G . Fox and Tingye L i . Resonant modes in a maser interferometer. Bell System Technical Journal, 40, 1961. [38] G . Goubau and F . Schwering. On the guided propagation of electromagnetic wave beams. IRE Transactions on Antennas and Propagation, pages 248-256, M a y 1961. [39] G . D . B o y d and J . P. Gordon. Confocal multimode resonator for mill imeter through optical wavelength masers. Bell System Technical Journal, pages 489-508, March 1961. [40] H . Kogelnik and T . L i . Laser Beams and Resonators. Proceedings of the IEEE, 54(10), October 1966. [41] K . Kurokawa. An Introduction to the Theory of Microwave Circuits. Academic Press, 1969. Bibliography 100 [42] A . L . Cul len, P. Nagenthiram, and A . D . Wi l l i ams . A variational approach to the theory of the open resonator. Proceedings of the Royal Society of London, A., 329:153-169, 1972. [43] A . L . Cul len and P. K . Y u . Complex source-point theory of the electromagnetic open resonator. Proceedings of the Royal Society of London, A., 366:155-171, 1979. [44] A . L . Cul len. Millimeter-Wave Open-Resonator Techniques, volume 10, pages 233-280. Academic Press Inc., 1983. [45] P. M . Morse and H . Feshbach. Methods of Theoretical Physics, volume 1. McGraw-H i l l , 1953. [46] L . A . Weinstein. Open Resonators and Open Waveguides. The Golem Press, 1969. [47] R . K . Mongia and R . K . Arora . Equivalent circuit parameters of an aperture coupled open resonator cavity. IEEE Transactions on Microwave Theory and Techniques, 41(8):1245-1250, August 1993. [48] C . W . Erickson. High order modes in a spherical Fabry-Perot resonator. IEEE Transactions on Microwave Theory and Techniques, MTT-23:208, 1975. [49] Ichiro Shinkoda. Studies of Spin-Polarized Hydrogen and Deuterium at Temperatures Below 1 K using E.S.R. P h D thesis, University of Br i t i sh Columbia , Ju ly 1990. [50] E . L . Ginzton. Microwave Measurements. M c G r a w - H i l l , New York, 1957. [51] G . L . Ragan. Microwave Transmission Circuits, volume 9 of M.I.T. Radiation Lab-oratory Series. M c G r a w - H i l l , 1948. [52] H . Abe and K . Koga. Japanese Journal of Applied Physics, 13:1145, 1974. [53] C . G . Montgomery, R . H . Dicke, and E . M . Purcel l . Principles of Microwave Circuits, volume 8 of M.I. T. Radiation Laboratory Series. Dover Publications Inc., New York, 1965. [54] G . E . H . Reuter and E . H . Sondheimer. The theory of the anomalous skin effect in metals. Proceedings of the Royal Society of London, A., 195:336, 1948. [55] R . B . Dingle. The anomalous skin effect and the reflectivity of metals I. Physica, XIX:311-347, 1953. [56] R . B . Dingle. The anomalous skin effect and the reflectivity of metals II. Physica, XIX:348-364, 1953. Bibliography 101 [57] L . A . Ha l l . Survey of Electrical Resistivity on 16 Pure Metals in the Temperature Range 0 to 273 K. N . B . S . Technical Note 365, February 1968. [58] Lev i Waldron and Pinder Dosanjh. Private communication. [59] W . J . M c G . Tegart. The Electrolytic and Chemical Polishing of Metals in research and industry. Pergamon Press, London, 1959. [60] P. V . Shigolev. Electrolytic and Chemical Polishing of Metals. Freund Publishing House, Tel A v i v Israel, 1974. [61] Q. Y . M a , P. Dosanjh, J . F . Carolan, and W . N . Hardy. Inhibit ion of superconductiv-ity in Y - B a - C u - 0 films by Si ion implantation. Applied Physics Letters, 63(26):3633-3635, 1993. [62] N . K l e i n , H . Chaloupka, G . Mul ler , S. Orbach, H . P ie l , B . Roas, L . Schultz, U . K l e i n , and M Peiniger. The effective microwave surface impedance of high-Tc thin films. Journal of Applied Physics, 67(ll):6940-6945, June 1990. [63] D . A . Bonn, S. K a m a l , K u a n Zhang, Ruix ing Liang, D . J . Baar, E . K l e i n , and W . N . Hardy. A comparison of the influence of N i and Zn impurities on the electromagnetic properties of Y B a 2 C u 3 0 6 . 9 5 -[64] H . P ie l and G . Mul ler . The microwave surface impedance of high-Tc superconduc-tors. IEEE Transactions on Magnetics, 27(2), March 1991. 


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