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A probe to study the microwave surface resistance of crystals, of the high temperature superconductor… Gheinani, Ahmad Reza Hosseini 1997

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A P R O B E T O S T U D Y T H E M I C R O W A V E S U R F A C E R E S I S T A N C E OF C R Y S T A L S , O F T H E 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 O R Y B C O By Ahmad Reza Hosseini Gheinani B.Sc. (Physics), University of Toronto, 1995 A THESIS S U B M I T T E D IN PARTIAL F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in T H E F A C U L T Y O F G R A D U A T E STUDIES D E P A R T M E N T O F PHYSICS A N D A S T R O N O M Y We accept this thesis as conforming to the required standard T H E UNIVERSITY O F BRITISH C O L U M B I A September 1997 © Ahmad Reza Hosseini Gheinani, 1997 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 of Physics and Astronomy The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: Abstract A superconducting cavity resonator has been developed for the systematic study of the microwave surface resistance of high purity (twinned and detwinned), zinc and nickel doped YBaiCuzO-i-t single crystals at 22.7 GHz. We have successfully tested the resonator on three pure and two 0.15% zinc doped samples; one of the measurement was on a pure crystal grown using the newly developed barium zirconate crucibles. Measurements in the geometry Hrf//c show a crossover in the low temperature behavior of samples as the purity is varied; in one of the doped samples we see a crossover to a gapless regime with a residual conductivity that is close to o00, the predicted zero temperature limit for a d-wave superconductor in 2-D. Measurements with Hrf parallel to the ab-plane have allowed us to measure the ab-anisotropy of a 0.15% zinc doped crystal, which we find to be qualitatively similar to the ab-anisotropy of the pure system [33]. Finally, the measurements are compared in the two geometries. ii Table of Contents Abstract ii Table of Contents iii List of Figures v Acknowledgements vii 1 Introduction 1 1.1 Microwave Surface Impedance 4 1.2 Microwave Electrodynamics and the T w o F l u i d Model 8 1.3 Methods 11 2 The Cavity Resonator 13 2.1 Introduction 13 2.2 Measurement Principles 19 2.2.1 The Resonant Circui t 19 2.2.2 Measurements of Q 22 2.2.3 Determination of the Surface Resistance 23 3 The Experiment 26 3.1 The Samples 26 3.2 The Experimental Procedure 28 3.2.1 Measurements in the Frequency Domain 28 i i i 3.2.2 Measurements in the Time Domain 31 3.3 The calibration procedure 34 3.3.1 Absolute Losses : 34 3.3.2 Non-Perturbative Corrections 35 4 Data Analysis and Discussion 38 4.1 Background Runs 39 4.2 Microwave Surface Resistance and Conductivity of Pure YBCO Crystals . . 43 4.3 Surface Resistance and Conductivity of Zinc Doped Crystals 51 4.4 Anisotropics in the ab Plane 56 4.5 Barium Zirconate Grown Crystals 61 5 Conclusions 64 5.1 Further Studies 67 Bibliography 69 iv List of Figures 1.1 View of a metal sheet in the presence of an incident microwave field 4 1.2 Circuit diagram representing the two fluid model 9 2.1 View of the resonator 15 2.2 View of the main probe 16 2.3 View of the sample probe and tip 18 2.4 Equivalent circuit of resonator connected to external circuit of generator and detector 21 2.5 View of sample in two geometries 24 3.1 Frequency domain setup 29 3.2 Schematic view of microwave set up for time domain measurements 32 3.3 Decay profile of transmitted power with time. Solid line is the 3 parameter fit to a decaying exponential 33 4.1 Frequency shift due to perturbation by the sapphire rod 40 4.2 A(^) due to bare sapphire (open squares), and grease (filled squares) . . . . 41 4.3 The surface resistance of a nominally pure YBCO sample (YLO) at 22.7GHz. 44 4.4 The surface resistance of sample YLO at 22.7GHz and a second sample measured at a frequency of 34.8GHz. The frequencies are scaled to 22.7GHz. 45 4.5 Derived conductivities at two frequencies 47 4.6 The surface resistance of pure and zinc doped (0.15%) crystals. The Tc's of both doped samples were 91.7K and that of the pure sample, 93.2K 53 v 4.7 The conductivities of pure and 0.15% zinc doped crystals 54 4.8 The surface resistance of sample SZD in the superconducting state 57 4.9 Real part of the conductivity in the a and b directions 58 4.10 Comparison of the surface resistance of sample SZD measured in the two geometries . Also seen in the inset is their normal state values 60 4.11 Surface resistance of pure crystals grown in barium zirconate and yttria stabilized zirconia crucibles 63 vi Acknowledgements. First of all I would like to thank Doug Bonn, my research supervisor. Without his guidance and technical expertise this project would not have been possibe. I am indebted to him for his patience and advice that has helped me grow as a researcher. I would also like to thank Walter Hardy for his invaluable advice at the start of the project, as well as the fine example he sets. There are numerous other people that have helped me throughout this period. I would like to thank in particular Saeid Kamal who has helped me a great deal since my start in the lab and has been a great role model and friend. I would like to thank Bruce Gowe for making work with him greatly enjoyable, and for his help with developing the time domain measurements. Thanks also go to Pinder Dosanjh, Chris Bidinosti, Andre Wong, Kaspar Mossman, Arman Bonakdarpour and the whole superconductivity group for making the lab an enjoyable place to work. Special thanks also to Rob Knobel for his help at the start of my work here. Finally I have to thank my parents, sisters and T Elhoss for their encouragement and support for me, despite the far distances. Thanks also go to the Natural Sciences and En-gineering Research Council of Canada for their support through grants, as well as the UBC Department of Physics and Astronomy for their support through teaching assistantships. vii Chapter 1 Introduction The solid state physics and materials science community were entirely unprepared for the Sept. 1986 publication by J. G. Bednorz and K. A. Muller [1] of the discovery of supercon-ductivity near 30 K in a pollycrystalline sample in the LaBaCuO system. Their discovery opened a new chapter and renewed interest in the study of superconductivity, which goes back to 1911, when H. Kamerlingh Onnes discovered that mercury became superconducting below 4.2 K. This phenomenon had been extensively studied and was thought to be well understood until the discovery of Bednorz and Muller. The theory of superconductivity presented by Bardeen, Cooper and Schrieffer in 1957 [2] had been a remarkable success. This theory, based on electron-phonon interactions as the mechanism by which electrons pair, provided explanations for most of the properties of the superconductors known at the time. The discovery of superconductivity in the LaBaCuO system was soon followed by the discovery of other cuprate compounds with even higher superconducting transition temper-atures. For example, the most studied of the family of high temperature superconductors, YBa2Cu307, has a T c of 93 K. Even higher Tc's have been achieved in mercury compounds, up to 160 K under pressure. With these transition temperatures higher than the boiling point of liquid nitrogen (77 K), the cooling required to achieve superconductivity has be-come relatively inexpensive, compared to the cost of the liquid helium that is needed to cool conventional superconductors below their transition temperatures. This has opened 1 Chapter 1. Introduction 2 many potential applications which, up until now, have been far too expensive to pursue. However, wires made of the cuprate superconductors are quite difficult to fabricate. They are polycrystalline ceramics that are brittle and simply cannot be shaped into wires with the same ease that metals are. As a result of such materials difficulties, the most quickly developing practical applications make use of thin films. These are made by deposit-ing layers of the superconducting material (by methods such as laser ablation, sputtering, etc..) on an insulating substrate with lattice parameters close to those of the supercon-ducting material. As is done in the semiconductor industry, these films can be patterned to make a variety of devices. One family of thin film applications involves microwave devices, [3] where patterned thin films are used as passive components for devices such as filters, resonators and antennas. An important property of the cuprate superconductors which makes them attractive, is their low-loss performance at these frequencies. The surface resistance, Rs, characterizes this loss; for example, at 10GHz the surface resistance of high quality thin films of YBCO grown on a low-loss substrate is some hundred times lower than that of copper at 77 K. This low loss allows resonant structures with a sharp frequency response to be constructed, making them ideal for ultra-narrow-band filters used to separate signals with close frequencies. In order to optimize the performance of such devices, a good understanding of the mech-anisms involved in the microwave loss must be developed. This calls for materials whose extrinsic properties do not mask the intrinsic behaviour of the superconductor being stud-ied. Toward this end considerable effort has been comitted to the growth of single crystals that are of very high quality [4]. Measurements of the microwave surface resistance and magnetic penetration depth on these crystals have helped in answering some fundamental questions. For example, Bonn et al [5] have shown that the microwave loss of high purity YBa2Cu30^s samples shows a broad peak at temperatures around 40K, something that Chapter 1. Introduction 3 was not seen in thin film measurements. Furthermore, the surface resistance of some films was i n fact lower than that of crystals at low temperatures. B y purposely doping crystals wi th zinc and nickel impurities, and observing a subsequent decrease of their microwave loss, it was demonstrated that the higher loss in the pure crystals was an intrinsic property. Crystals studies has also allowed measurements to be made that have been of importance to an understanding of these materials at a more fundamental level. For example, mea-surements on these crystals by Hardy et al [6] have shown a linear T dependence of the penetration depth at low temperatures, strongly indicating that the gap function of these superconductors has nodes on the fermi surface. In this project, a probe has been developed to make high precision measurements of the surface resistance of high purity YBa^CuzO-j-^ it utilizes a superconducting cylindrical cavity, resonant in the TEQU mode at 22.7 G H z . The motivation has been to understand the low temperature behaviour of the microwave conductivity. For a superconductor with an order parameter which has nodes on the fermi surface, it is well known that impurity scattering can lead to a finite density of quasiparticle states at zero energy[24]. The re-sulting residual normal fluid should then dominate heat and charge transport properties at low temperatures. A n interesting feature of quasiparticle transport for a d-wave gap in two dimensions (2-D), first predicted by P. Lee [7], is that while the impuri ty scattering rate T can be very different for different levels of impuri ty concentrations and different scat-tering phase shifts, the microwave conductivity saturates at low temperatures to a value, ooo = ne2/7TmA0, ( A 0 being the maximum gap) which is independent of T, the impuri ty concentration. Experimental verification of this universal behaviour would give further evidence that the order parameter in HiTc materials exhibits nodes on the fermi surface. Our new resonator and newly developed measurement techniques allows very precise mea-surements of the surface resistance to be made at low temperatures which can in principle Chapter 1. Introduction 4 resolve such low values of the conductivity. Before we go further into the apparatus design and measurement techniques, we will review the basic electrodynamics relevant to the HiTc cuprates. 1.1 Microwave Surface Impedance The surface impedance is an important measurable quantity that describes the response of metals and superconductors to microwave radiation. It is defined as Z = (Ex/Hy)z=0 (1.1) where Ex and Hy are components of the electric and magnetic fields in the plane parallel to the surface, with z=0 at the surface of the material that is being considered. Figure 1.1: View of a metal sheet in the presence of an incident microwave field. The high temperature superconductors are commonly considered to be well into the local limit. That is, the fields vary slowly in space on the scale of the mean free path (/) for a normal metal, or the coherence length (£) for a superconductor. The local limit holds in most situations when the coherence length £ is much smaller than the penetration depth (£ << A) [8]. The small values of £ and / in most HiTc materials ensures that the Chapter 1. Introduction 5 electrodynamics is local [10]. We will see later that this is certainly true above 50K, but at lower temperatures / ~ A and the surface resistance becomes a non-local quantity [9]. The coherence length however is much smaller than A and the magnetic penetration depth generally remains local, a situation not seen in most superconductors. If we assume local electrodynamics, then the current density at any point is determined by the local electric field at that point: J = oE (1.2) where o is the conductivity. We can use Maxwell's equations, V x £ = - - (1.3) VxH = J + iuD (1.4) V - £ = 0 (1.5) V-D = p (1.6) with equation(1.2) to obtain V x H = (a + icue)E. (1.7) It is easy to see from (1.6) and (1.7) that Ohm's law (eqn (1.2)) implies the absence of charge density. For metals and other good conductors, it is found that the displacement current is negligible in comparison with the conduction current for microwave and millimeter-wave frequencies, and is actually not measurable until the frequencies are well into the infrared. Hence, (^ << 1), and one obtains, V2E = p a ^ . (1.8) Equation (1.8) can be solved quite simply by assuming harmonic time variation of the fields, and a simple choice of boundary condition (a semi-infinite slab of conductor of infinite Chapter 1. Introduction 6 depth). This turns out to be a good approximation at the high frequencies considered, where the body of the conductor is often much larger than the penetration depth of the electromagnetic fields. For a uniform field along the plane (say in the x-direction) and z the depth into the material , equation(1.8) becomes, d2E —j-^- = ico/j,oEx = k2Ex (1.9) which has general solutions of the form Ex = Ae~kz + Be+kz. (1.10) The condition that E remain finite as z —> co implies B = 0, and we can write Ex = E x 0 e k M (1.11) where, k = (—ifiujo)1/2 It is customary to write the propagation constant k in the form k=l-±^ (1.12) where and (1.11) may be written as Ex = Ex0e~zlse-lzl5 • (1.14) with a similar form for the magnetic field and current density. It is clear from the form of (1.14) that the magnitude of the fields and currents decay wi th penetration into the conductor. The quantity S, known as the skin depth, has the significance of being the depth at which the fields decay to 1/e of their surface values. Chapter 1. Introduction 7 Using eqn (1.14) in (1.4) to get HVQ, we see that the surface impedance in definition (1.1) is then, EXQ Exo 1 + i . . S = HJo = (Ex0o/k) = ~oT ( ' 1 5 ) and is usually separated out into its real and imaginary parts. The real part of the surface impedance Rs, is called the surface resistance and is the sheet resistance of the surface layer into which the fields penetrate. The imaginary part Xs, is called the surface reactance. In general the conductivity is complex, o = o\ —ia2, and the surface resistance in terms of the real and imaginary parts of the conductivity becomes: (1.16) 2{a\ + ol) For a normal metal, if the frequency is not too high, then the conductivity is real, and the surface resistance and reactance are equal, i.e. Z = R. + iXa = M + iM. (1.17) V Z(7 V ZcT However, for a superconductor, when in the superconducting state and temperatures up to a fraction of a degree below T c , we generally have 02 » o\ and the propagation constant becomes k — i(fi0coo2)1^2. We find that fields decay as e~zlx, wi th A, the magnetic penetration depth, given by A = l / ( / x o w a 2 ) 1 / 2 . (1.18) Similarly, in this same regime when cr2 >> (1-16) simplifies to R. = &u2\W (1.19) Equation (1.19) shows that the surface resistance is proportional to the real part of the conductivity for a superconductor, and that measurements of the surface resistance and magnetic penetration depth allow one to extract o\. Chapter 1. Introduction 8 1.2 Microwave Electrodynamics and the Two Fluid Model The model described here is a " generalized two fluid model" [22] that follows the basic ideas that were proposed by London [11] and Gorter and Casimir [12]. According to this model, a superconductor behaves as though it contains electrons of two different types, the normal electrons which behave at least approximately like electrons in normal metals, and the super electrons, which have unusual properties. Both types of electrons can carry current; the normal fluid has resistance whereas the superfluid flows without resistance. The normal electrons are also responsible for heat transport. As the temperature is lowered below the transition temperature Tc, the superfluid density rises from zero, while the normal fluid density falls. The motion of the normal fluid is described by the transport equation: mlt + m J " / r = Nn£2E- (1-20) where Nn is the normal fluid density, r is the normal fluid relaxation time, and Jn = NneVn is the normal current density for normal electron drift velocity Vn. The superfluid motion is described by the London equation [11], m ^ = Nse2E (1.21) with Ns being the superfluid density and Js the superfluid current density. It is clear from (1.21) that after a short pulse of electric field the system will be left with a supercurrent which will not decay, and this is of course the property of superconductivity. The total current density is Js + Jn. In a d.c. field, all the current is carried by the superfluid whereas in alternating fields the inductive response of the superfluid allows the normal fluid to have a share of the current, and energy dissipation takes place. We may draw an equivalent circuit of the two fluid model where the current density can be represented schematically by the currents In and Is of Fig. 1.2. Chapter 1. Introduction 9 In Rn Ln A A A A Figure 1.2: Circui t diagram representing the two fluid model . The conductivity cr = o~\ — io2 can be derived from (1.20) and (1.21) as 0 i 0"2 A U 2 r m ( l + o ; 2 r 2 ) / i V „ e 2 o ; r 2 (1.22) + \ m ( l + w 2 r 2 ) m w J The normal fluid is responsible for the real part of the conductivity o\ for temperatures away from T — 0. A t T = 0, for a system in the clean l imi t , where the energy scale of the gap is much greater than h/r, the area under Oi(co) is a delta function at co = 0 of weight ne2/m. This is the response of the superfluid. A t finite temperatures, oi(co,T) increases over the T — 0 value as more and more quasiparticles are thermally excited. Correspondingly, the strength of the c5-function in o\ at co = 0 decreases. For low frequencies of several G H z or lower where cor « 1, o2 in the cuprates is completely dominated by the superfluid condensate and the normal fluid contributes very l i tt le except when its density rises rapidly close to Tc. Away from T c , er2 >> c i and the magnetic penetration depth A = Xs/JJLQCO = (m/^o-^se 2) 1/ 2. The last form is just the London penetration depth XL that is obtained from the London equations. A t zero temperature the normal fluid density is Chapter 1. Introduction 10 essentially zero, and the superfluid density would be equal to the total carrier density. The superfluid fraction of total carriers is then related to the penetration depth by the relation N1_m X s ~ N0~ A2(T)' [ ' A measurement of the penetration depth is then a direct measure of the superfluid density. In principle the penetration depth and surface resistance can be measured simultaneously. What is usually done in this lab is to make each measurement separately using an apparatus optimized for the particular quantity of interest. In the following work, only the surface resistance measurement methods are discussed. Chapter 1. Introduction 11 1.3 Methods Measurements of the surface resistance are important for both high frequency applications of high temperature superconducting (HTS) materials, and also at a more fundamental level, they can provide information about the nature of the superconducting state of these materials. In most laboratories the technique used for Rs measurements are resonant methods, where Rs is calculated from the measured quality factor or Q of a resonating structure. For the purpose of measuring the Rs of a superconductor, several structures can be used: • A dielectric resonator consisting of a parallel plate or a rod of dielectric sandwiched between two HTS plates; • A scanning confocal resonator; • A HTS transmission line resonator; • A split ring resonator made of two rings seperated by a dielectric, and placed near the sample of interest; • A cylindrical resonator made of copper or of a low temperature superconductor, with a HTS sample forming one face, or being inserted into the structure. The first three methods are solely used for measuring the surface resistance of thin films with the third method being of more immediate use since the transmission line closely resembles the final microwave element. The method, however, is destructive in that the film has to be patterned, thus it is not as good a diagnostic tool for thin films as are the first two methods. The last two methods mentioned are the ones mainly used in this laboratory, for both thin film and high purity crystal measurements. The surface resistance can be Chapter 1. Introduction 12 determined from changes in the resonator's Q-factor before and after a sample is inserted into the structure. A compromise has to be made as which method to use, depending on the frequency at which measurements are to be taken. For this work at 22 GHz, a simple cylindrical resonant cavity was used. The split ring resonator [13] is appropriate at frequencies below 10 GHz, where cylindrical cavities become too large for most purposes. Chapter 2 The Cavity Resonator Microwave cavity resonators are enclosed structures that support a resonant electromag-netic mode at microwave frequencies. The simplest cavities are resonant sections of trans-mission lines, such as a rectangular or circular waveguide. The earliest application of microwave cavities to the study of the electrodynamics of superconductors was the work of Pippard [16]. More recently, microwave cavities have been used successfully by Bonn et al. [14] and others [15] to study the oxide superconductors. 2.1 Introduction The cavity resonator built as part of this project is a cylindrical cavity made of copper as shown in Fig. 2.1 The inside dimensions are .74" by .51" for the diameter and height respectively. For a cavity with these dimensions, the TEQH mode is at a frequency of 22.7 GHz. For a cylindrical cavity this mode is degenerate with the TMm mode, and the cutoff hole at the bottom of the cavity is an added perturbation to remove this degeneracy: its placement at an antinode of the H-field of the T M m mode tends to lower the frequency of it below the TE0n mode, since the effective volume of the cavity seen by the TM mode is increased. Experimentally we do not observe the T M m mode, since the transmission lines do not couple well to this mode in their present arrangement. Coupling to the coaxial transmission lines are by means of two loops on opposite sides of the cavity, inside two 0.2" holes that are drilled close to the inside wall of the cavity. Halfway up the cavity on each 13 Chapter 2. The Cavity Resonator 14 side, 0.02" diameter holes have been cut through the wall to connect the cavity with the area where the coupling loops lie. This allows microwave power to reach the cavity from the input loop, and from the cavity to the output loop which monitors the transmitted power and the transient decay. Both input and output couplings to the cavity can be adjusted by moving the loops vertically away or towards the side transmission holes. In principle we may run the resonator as is, with the inside walls being copper, the material from which the body of the resonator is made. Copper however is not a supercon-ductor at any temperature and a resonator made out of copper would have Q values in the hundreds of thousands at helium temperatures. For measurements on small superconduct-ing crystals this would limit our resolution, since we cannot measure the Q to arbitrary accuracy. To improve the resolution a much higher Q is desirable and we coat the inner walls of the resonator by electroplating with an alloy of lead that is superconducting at helium temperature. The alloy typically contains 5 percent tin and 95 percent lead. It is used as an anode in a lead-tin fluoroborate solution with the copper resonator being the cathode. Currents of about 40-50m 4^ per square centimeter are used for plating the walls of the cavity, with half this amount for the side holes. The alloy is superconducting at about 7K and we obtain quality factors in excess of 4 x 106 at 4.2 K. The resonator shown in Fig. 2.1 fits onto the bottom of a probe as shown in Fig. 2.2 The probe is about a meter long with three stainless steel shafts, the outer two housing the coaxial lines that feed microwave power into and out of the resonator, and the middle shaft housing the sliding sample probe. The probe and resonator are pumped out before precooling to liquid nitrogen temperature. The outer resonator wall is in contact with the helium liquid ensuring that the resonator stays close to the liquid temperature. During an experiment, one wants a Q0 that is higher than the value at 4.2K so we normally pump on the helium liquid to cool it further to about 1.2K. The intrinsic cavity wall losses initially Chapter 2. The Cavity Resonator B r a s s f inger. P -0 L P J L C o a x i a l l ine. Ind ium s e a l . TlLr " S a p p h i r e rod. — S a m p l e . - Cav i t y . ~ L e a d cove r . Bo t t om c a n . Figure 2.1: View of the resonator. Chapter 2. The Cavity Resonator 16 Figure 2.2: View of the main probe. Chapter 2. The Cavity Resonator 17 fall off exponentially below 4.2K, with a residual Q0 value in the range 50 — 200 x 106 at 1.2 K. With the bath being constantly pumped, the resonator temperature stays stable at 1.2K and measurements can be taken. On the top of cavity there is a through hole for sample loading. During the measurement the sample can be inserted into, and removed from, the cavity by moving the sample probe vertically. The sample probe shown in Fig. 2.3 consists of a long piece of one-eighth inch stainless steel tubing that is secured and centered by a set of brass fingers inside the middle shaft of the main probe. These fingers also act as a thermal link between the sample probe and the bath. The end of the sample probe is made of copper and has a chamber that houses the thermometer and heater. The electrical leads to the thermometer and heater run up the one-eighth inch stainless steel tube and connect to a male 9-pin connector at the top of the probe. The sample is placed on a tip that consists of a copper piece which can be screwed into the end of the sample probe, and a piece of sapphire rod or strip (depending on the measurement geometry) that the sample is stuck to. Sapphire is chosen, since it has a very low loss tangent and it has very good thermal conductivity at low temperatures, comparable to copper(~ 10W/cmK at 10 K). The sapphire tip is cleaned by being placed in an ultrasonic bath of sulphuric acid, and then boiled in the acid for a few minutes. The acid is replaced with acetone, ethanol, and isopropyl alcohol, with the same routine in each of the subsequent solutions. The tip is assembled by epoxying one end of the sapphire to the copper piece. With the cleaned tip ready, the sample is placed onto the tip by using a small quantity of silicone grease. It is obviously important to check the effects of the sapphire and grease on the measurements, so before an experiment the contribution to the loss coming from the combined sample plus tip is measured. As we will see later, the contributions are very small, but do become comparable to the sample losses at temperatures below 5 K. We usually have to allow some Chapter 2. The Cavity Resonator LJ Brass finger. Heater. Tip. Sapphire rod. Sample. Figure 2.3: View of the sample probe and tip. Chapter 2. The Cavity Resonator 19 helium exchange gas into the probe, because under high vacuum conditions the sample temperature does not fall below about 10K. By carefully controlling the helium gas density we can get to any desired temperature, down to that of the bath. 2.2 Measurement Principles 2.2.1 The Resonant Circuit For the purpose of measurements on single crystals of YBCO, our cavity is excited in the TEQH mode by an external microwave source. Near any particular mode of the resonator, we can describe the situation by an equivalent low frequency LCR resonant circuit. The properties of the circuit are then completely described by the three circuit elements. The Quality Factor is defined as, W Q = 2 T T — (2.1) where W is the energy stored in the circuit, and PL is the energy loss per cycle. It is easy to show that this resonant circuit's Q equals UJQL/R, which is the ratio of the reactance of the inductor to the resistance . For our cylindrical cavity, the Q factor in terms of the appropriate parameters, i.e the fields, is written; JvH\r)dV Q-^SsRsH2{r)dS (2"2) where LU0 is the resonant frequency, H the magnetic field, and RS is the surface resistance of the cavity walls. One can obtain the Q explicitly if the field distribution is known for any mode. For the TE0n mode these are, in cylindrical coordinates, ER .= 0 (2.3) EE = —j^AJQ(kr)sin-- (2.4) Chapter 2. The Cavity Resonator 20 with, EZ = 0 (2.5) Hr = ^-Aj'0(kr)cos^- (2.6) kh h HB = 0 (2.7) TTZ Hz = AJ0(kr)sin— (2.8) k2 = to2e0p0 - T o = - o (2-9) where A is a normalizing factor, a is the cavity radius, h is the cavity height, Jm is the mth Bessel function, and x 1S the first zero of J0{ka) and equals 3.832 for the TEQU mode. We can see that the fields are independent of 9, Hr reaches its first zero when r — a, and that Hz has only one maximum, when z = h/2. Performing the appropriate integrations [17], the energy stored in the cavity is w = A2aj260p2J2(ka)Vo 4 £ 2 ^ - 1 U J and the Q is found to be, where VQ is the cavity volume. For example, an RS of 10 pQ would yield a Q of ~ 108 for the resonator described in the previous section. For us to be able to measure the quality factor of the resonator, we have to couple the internal resonant circuit to an external circuit. When joined to the external system through the couplings, the equivalent circuit can be represented as shown in Fig. 2.4. It is found that for high Q the way the coupling of the external circuit is represented has no effect on the final results, as shown here by two ideal transformers. By examining the circuit of Fig. 2.4, the loaded Q of the system QT , can be found to be, i t + mfZi + 777,2^2 Chapter 2. The Cavity Resonator 21 Ri R C L Co c L U C ( r r r R2 1:m m2:1 Figure 2.4: Equivalent circuit of resonator connected to external circuit of generator and detector. where Z\ and Z2 are the characteristic impedances of the input and output lines. By defining the coupling constants, 1,2 m 1,2 (2.13) we can write down the relation between the unloaded Q0 (without coupling to external circuit) and loaded QT as, QT Qo 1 + A Qo Qo~ Qo 1 Q~0 l + Q~i l where we have characterized the loads by the external quality factors, Q i ; 2 - It is clear that the measured QT of the resonator will be lower than the Qo of the isolated circuit, and that the external quality factors are independent of the resistance of the internal circuit, R. If we now introduce a small sample into the cavity, then the perturbation to the fields Chapter 2. The Cavity Resonator 22 inside the cavity are small enough to keep the total energy stored in the cavity almost the same. The sample however would change the reactance and resistance of the internal circuit, so as to alter its resonant frequency and quality factor. The change in the quality factor is directly related to the extra loss that the sample contributes to the internal circuit's loss. Since the external quality factors are independent of the resistance of the internal circuit, the overall change in the loss of the whole circuit would be the same as that of the internal circuit. This makes it possible to measure the extra loss presented by a small sample by measuring the quality factor of the external circuit before and after a sample is placed in the resonator. It remains to be able to make an accurate determination of QT-2.2.2 Measurements of Q The power tranmission coefficient, which is the ratio of the power delivered through the resonator to the incident power takes the form [17], where T 0 = 4/3I(32QT/QO (from here on we will write for QT just Q). The bandwidth of T, 5u is the frequency interval between half power points. From eqn(2.15), the half power condition is u>/u0 — UJQ/UJ = ±1/Q. With u> — u 0 ± Su/2, we get for the full width at half power, Experimentally we can measure the frequency dependence of power transmission through the resonator for a fixed input power. This is done by means of a microwave synthesizer that steps through the resonance at discrete frequencies and constant input power. The resulting lorentzian curve can be fitted and the Q and resonant frequency, co0 determined. An alternative method to determine the quality factor of the resonator is to measure the T = l + QU (2.15) 5u = CJQ/Q (2.16) Chapter 2. The Cavity Resonator 23 power relaxation time. In its simplest form, this is done by applying power to the resonator for a long enough time that the response is steady, and then terminating the drive power. The power relaxation time is related to the Q through, r = Q/co0 (2.17) 2.2.3 Determination of the Surface Resistance When a sample is introduced into the resonant cavity, its resonant frequency and Q are altered. The Q of the cavity without sample, Q0, is 2TVW/PL- With a small sample present we assume that the perturbation is small enough that the energy stored in the fields would stay almost the same. The frequecy shift would also be very small, i.e several MHz out of a resonant frequency of several GHz. We can then write for the quality factor with the sample, QL, 2irW/(PL + PSL), where PSL is the loss per cycle due to the sample. By comparing to the the Q-factor without a sample, we can write: s l ^ - c s r ^ (2-18) In order to find the loss per cycle due to the sample, the magnetic field at the sample surface has to be known. If a perfectly diamagnetic sample is placed in a uniform field H0, the field at the sample surface, H, is H = J% <2-19) where N is a demagnetizing factor that depends on the shape of the sample, and is not dependent on the material. Values of iV can be found in the literature for samples of various shapes [18]. The surface current density is equal to the tangential component of H. Recall that Chapter 2. The Cavity Resonator 24 (a ) (b) Figure 2.5: View of sample in two geometries js = n x H i.e js = H T , and the loss per cycle due to a sample with surface resistance Rs is PSL = ^ T I fsRsdS = / H2tRsdS. (2.20) 2/o Js 2/o Js In the experiment we place a thin sample into the field in two ways. In the first case, the sample, which is in the shape of a thin plate, is put at the center of the cavity such that the principal axis of the cavity is parallel to the major surface of the sample as shown in Fig. 2.5(a). In the second case the sample is place with the cavity axis perpendicular to the major face of the sample. a) Sample is a thin plate; H parallel to large face. In this case the demagnetizing factor is zero and the surface field is just the external field at the center of the cavity which is just A. The loss per cycle is then _ SA. JF^B . . . PSL = —7-1 (2-21) Jo where S is the area of one major face of the sample. From equations (2.10) and (2.18) we Chapter 2. The Cavity Resonator 25 can write Rs = TA(i) (2.22) where T = ujQeoplJoikajVo/ASk2 If we know T, it is straightforward to determine the surface resistance of the plate. b) Sample is an ellipsoid; H perpendicular to large face. Here, our rectangular thin samples are placed as shown in Fig. 2.5(b). There is currently no known solution of the problem to determine the demagnetizing factor for such a shape. A crude solution can be obtained by approximating the sample with an ellipsoid. We choose to leave out the solution [17], but note that a similar relation between the measured quantity A(l/Q) and the surface resistance is found, namely Rs = rA(l/Q). In practice, due to the sample shape irregularities and the lack of a proper solution for the geometry of Fig. 2.5(b), the geometrical factor T is found using a calibration method that is discussed in the next chapter. This factor is dependent on the the shape, size and the location of the sample in the cavity, since the field stength varies with sample position. Chapter 3 The Experiment This chapter deals with the experimental details of the measurements. First we describe the properties of the samples that were used in the measurements, then review the experimental methods and instrumentation used to measure their surface resistance by the two methods that were outlined at the end of the preceding chapter. 3.1 The Samples The samples that have been studied so far are all high quality YBa2Cu^OTs single crystals, with and without zinc doping. All the crystals that were measured had 5 = .05, so that they were approximately optimally doped. The crystals have been grown by a CuO — BaO flux method using two different types of crucibles [4]. The first set of crystals, which includes the Zn-doped crystals, were grown in yttria stabilized zirconia (YSZ). From chemical analysis, the YSZ crucibles are among the best crucibles for YBCO crystal growth. Pure crystals grown in these crucibles reach the 99.9% purity level. The second set of crystals have been grown in barium zirconate (BZO) crucibles [19]. These crucibles have an advantage over the YSZ crucibles in that they are inert to the melt. In YSZ crucibles on the other hand, during the crystal growth, the inside layer of the crucible reacts with the melt forming the corrosion product barium zirconate which interferes with the crystals being grown. During this process, crucible impurities are released into the melt that end up in the crystals thereby limiting the purity that can be attained in these crystals to about 99.95%. Crystals grown 26 Chapter 3. The Experiment 27 in the new BZO crucibles are believed to be better than 99.995% pure and to the eye have very clean, mirror-like surfaces. It should be noted that these crystals have been grown very recently and the optimization of their preparation is an ongoing effort at the time of writing. The (YSZ) grown crystals have been studied extensively in the last few years and have been characterized by magnetization, dc resistivity, surface impedance and specific heat measurements. They have Tc's of around 93.3K, with a transition width of less than 0.3K. Their dc resistivity above the transition is about 55 pQ, — cm in the ab plane and about sixty times greater in the c-direction. For the zinc doped samples with a 0.15% doping, the transition temperatures were about 91.7K. The typical size of the crystals is about l x l mm2 in the ab directions, and between 30-60 microns in the c direction. The crystals are black with smooth surfaces. Chapter 3. The Experiment 28 3.2 The Experimental Procedure Since the start of this project many new improvements have been made in the measure-ment techniques that allow for fast accurate determination of the surface resistance of small crystals. It is safe to say that earlier results obtained [17] still hold true, but the improve-ments have mainly served to lower the noise in the data and to ease the data taking. The most significant change that has taken place has been the development of time domain measurements to measure the quality factor of the resonator, as described below. Other improvements have been in the measurement of the so called non-perturbative correction that has to be applied to the data. This correction is important since it is of the same order of magnitude as the measured quantities at temperatures below a few Kelvin. 3.2.1 Measurements in the Frequency Domain The experiment starts with the probe being cooled to about 85 Kelvin by placing the res-onator inside a dewar whose outer jacket is filled with liquid nitrogen. Once the resonator cools to about 90K, liquid helium is transferred into the inner dewar cooling the resonator to 4.2 Kelvin. Following the transfer, the helium is pumped on thereby reducing its temper-ature to about 1.2 Kelvin. It takes about 20-30 minutes before the resonators Q levels off to some constant value, typically around 107. During this period the sample temperature is monitored and is found to fall to about 7-10 Kelvin, slightly higher than the resonator which at this time would be at 1.2 K. With the resonator cool, the input and output lines are connected to the external microwave circuit. The arrangement is as shown in Fig. 3.1 The microwave source is an HP 83630B 0.01-26.5 GHz synthesized sweeper, which steps through a series of discrete frequencies under its sweep mode or provides a constant or pulsed frequency in its CW Chapter 3. The Experiment H P 8 3 6 3 0 A -S y n t h e s i z e r . M i c r o w a v e Amp l i f i e r . Sweep Trigger. Crystal Detector \ amplifier. Oscilloscope A to D Computer. CO. D_ O Figure 3.1: Frequency domain setup. Chapter 3. The Experiment 30 mode. The transmitted power through the resonator is amplified by means of a 30dB microwave amplifier, before being detected by a diode detector that has been incorporated into a detector/low frequency amplifier box with further gains of between 103 to 104. The output of the detector box may be viewed on an oscilloscope. In frequency domain, the swept frequency response of the transmitted power is mea-sured, and stored by the computer. The typical profile of the resonator response is a lorentzian which is fitted by using a non-linear Lavenberg-Marquardt alogrithm, from which the resonant frequency and Q-factor are obtained. This method is appropriate when mea-suring the Q of the unloaded resonator or when the resonator is loaded with a high loss sample. However, in most cases, our samples have very small losses which means that the lorentzian is not broadened much over the unloaded response. This, together with the fact that the unloaded Q of the resonator is extremely high, means that small vibrations in the system tend to move the sample enough that the resonance frequency oscillates over more than the half width of the resonance. Such problems however are not encountered when the unloaded Q is not so high or when the resonance is sufficiently broadened, by a HiTc sample close to T c , say. In the latter cases the resonator response is sufficiently stable to allow for the microwave synthesizer to sweep through the resonance (a process which takes typically 10 seconds), and obtain a true image of the resonance. Most of our measurements on YBa2Cus06.g5 are done over a wide temperature range, from about 1.2K to 100K. The temperature regulation is by means of a Conductus LTC-20 temperature controller that is connected to the thermometer and heater which were located at the end of the sample probe. At the lower temperatures the losses are three to four orders of magnitude lower than the normal state loss , so we are faced with the measurement difficulties mentioned earlier, namely during the measurement sweep, sample vibrations are continiously changing the resonance frequency. These oscillations are very Chapter 3. The Experiment 31 small, of the order of a few kHz, much smaller than the frequency shift from the unloaded frequency which was a few MHz. This means that the cavity Q is quite stable even though the frequency is not, which sets the stage for Time Domain measurements. 3.2.2 Measurements in the Time Domain. In the time domain the microwave circuitry is modified slightly as shown in Fig. 3.2 The resonator is irradiated with a short pulse of RF power, and subsequently the amplitude of the transmitted power is measured as a function of time by a Tektronics TDS520B digital oscilloscope. The profile of the transmitted power is a decaying exponential with relaxation time r = Q/UJQ. The curve as shown in Fig. 3.3 is fitted to an exponential using a nonlinear Lavenberg-Marquart algorithm. The approximate resonant frequency is found by looking at the resonance in frequency domain; then by adjusting the frequency of the pulse in the time domain to obtain the largest power output at the start of the decay we refine the value for the center frequency. Although this is done by eye, this gives the resonant frequency to 1 part in 106. The power delivered to the resonator can be varied by changing the amplitude of the pulse or by changing the width of the pulses. A compromise is made with the choice of the pulse width, since a very narrow pulse would not be sufficient to excite the fields in the cavity. On the other hand too broad a pulse would have too narrow a frequency spectrum to hit the resonance when the cavity's frequency is subject to slight oscillations of a few kHz. For our resonator, values of 50 to 100 ps were used for the pulse width, with a pulse rate about 15Hz. Chapter 3. The Experiment H P 8 3 6 3 0 A -S y n t h e s i z e r . Trigger. M i c r o w a v e Amp l i f i e r . Crystal Detector \ amplifier. TDS520B scope. Computer. go. O Figure 3.2: Schematic view of microwave set up for time domain measurements Chapter 3. The Experiment 33 Figure 3.3: Decay profile of transmitted power with time. Solid line is the 3 parameter fit to a decaying exponential. Chapter 3. The Experiment 34 3.3 The calibration procedure So far we have described how to measure the quantity A(^) by the two measurement methods of the last section. In order to calibrate the losses we have to determine the geometrical constant T in equation (2.23). This is done by a repeated measurement on a lead-tin plate that has been cut to the same shape and size. The Pb:Sn sample serves two purposes. It scales the values of A(^) into absolute losses, and secondly it gives a measure of the so called non-perturbative correction that is applied to the data to take care of the perturbation to the unloaded Q-factor of the resonator due to the rearrangement of the field distribution by a zero loss sample. 3.3.1 Absolute Losses For a metal in the normal state where o~\ » o~2, the surface resistance and reactance are equal, and we have the classical skin effect relation (equation (1.17)) Provided that the dc resistivity values are known accurately, the surface resistance in the normal state is determined by the dc resistivity for frequencies well beyond 100 GHz. It is then clear that a normal state measurement on a piece of Pb:Sn sample cut to the same shape and size will enable one to extract the calibration constant. Given that the sample has about 5% tin impurity added to the lead, we assume that the resistivity of the alloy takes the form, (3-1) PPb:Sn(T) — pPb{T) + pimpurity (3.2) Chapter 3. The Experiment 35 where the resistivity due to the added impurity is taken to be temperature independent. The surface resistance can be written in terms of the resistivity as and three measurements of A(^) are done with the sample placed at the center of the resonator at temperatures of 20, 50 and 75 K. Using the ratio of the A(^) 's at 20 and 75 K, a value for the impurity resistivity is obtained and added to the lead resistivity values to correct for the tin's presence. The value of A(^) at 50 K is used to check the assumption that Pimpurity is indeed temperature independent. Typical values of the impurity resistivity are around 0.3±0.1 pQ — cm. The calibration constant is then found from the expression where we usually use the values at 75K, although T is found to be the same up to about 1% when evaluated at the other temperatures. 3.3.2 Non-Perturbative Corrections When we insert a sample inside the resonator it is commonly assumed that the perturbation has no effect on the magnitude and shape of the current distribution in the resonators walls. In other words, if the sample had infinite conductivity then we would expect the loaded and unloaded Q's to be identical. This is however not the case, and the usual observation in this resonator is that the sample tends to alter the field distribution in such a manner so as to reduce the effective Q from its truly unloaded value. This is in contrast to what is usually seen in the split-ring resonator geometry, where the Q increases when a very low loss sample is inserted [13]. The Q change is always very small, seldom a cause for concern at temperatures above 15-20K, but below about 5K the corrections do make a considerable (3.3) Qo (3.4) Chapter 3. The Experiment 36 contribution to the A(^) value when a sample is loaded in the geometry where Hrf//c and the demagnetizing factor is large. To get a feel for the size of the correction that has to be applied, the Pb:Sn sample that is used for the calibration is cooled to 1.2K. This is accomplished by allowing a small quantity of helium exchange gas into the main probe and allowing the temperature to reach an equilibrium value of about 1.2K, the bath temperature. The Pb:Sn sample being a classical BCS type superconductor, has losses that are exponentially activated and when prepared carefully has losses at 1.2K that are several times lower than YBa2Cu3OQ,Q5 at 1.2K. This in effect would be the closest that we can get to a sample of infinite conductivity and zero loss. By inserting the sample to the center of the cavity where the YBa2Cuj,0^,^ crystal would otherwise be, the Q is measured and always seen to be lowered beyond the reduction that one expects from the loss due to the Pb:Sn sample at this temperature. An estimate of this loss is made by the following argument. Given that the sample is plated with the same Pb:Sn alloy that the resonator is plated with, we assume that the loss at 1.2K is that of a freshly plated resonator at 1.2K. Furthermore the loss at 1.2K is taken to be a temperature independent residual loss term since the BCS value is negligible at this temperature. At 4.2K the BCS value is taken to be that for lead at 22 GHz, about 160 p£l, plus the residual loss. By measuring the Q for such a resonator at 4.2K and 1.2K, we can deduce values of between 8 and 15 pfl's for the residual losses of our electroplated Pb:Sn at 1.2K. The reduction in A(^) for our Pb:Sn sample is usually a few times this loss, suggesting that the difference is partly due to the non-perturbative influence of the sample on the resonator rather than just being the loss of the sample. Some of the reduction in A(^) is due to the sapphire rod or plate as well as the silicone grease that is used to hold the sample to the sapphire. All of these additional contributions to the cavity loss must be subtracted in order to obtain the sample's loss. As will be shown in the next chapter, it is Chapter 3. The Experiment important to do background runs to determine the magnitude of these effects. Chapter 4 Data Analysis and Discussion In this section the surface resistance data obtained using the 22.7 GHz resonator is pre-sented. First some background runs on the sapphire rod and plate, the silicone grease, and the Pb:Sn sample are given. We will see the influence of these items on the resonant fre-quency and Q-factor of the cavity and how they compare with typical values from a YBCO sample. The surface resistance and the derived real part of the conductivity are then shown for pure and zinc doped crystals. A comparison is made with newly grown crystals from the barium zirconate crucibles and the older batches grown in yttria stabilized zirconia crucibles. Also presented is the ab-plane anisotropy data on a single untwinned zinc doped crystal. 38 Chapter 4. Data Analysis and Discussion 39 4.1 Background Runs When measuring the surface resistance of YBCO samples or any other superconducting sample for that matter, the tip of our sample holder along with the sample are introduced into the cavity. It is therefore important to know the loss and frequency shift due to the tip of the sample holder alone before a meaningful measurement can be made. As mentioned before, sapphire was chosen for the tip for its excellent thermal properties as well as its very low dielectric loss. The silicone grease that is used to attach the sample to the sapphire has adequate thermal properties and its dielectric loss also acceptable when used in small amounts just to keep the sample attached to the plate. The perturbation of the cavity by our sapphire rod is shown in Fig. 4.1 as it is pushed along the central axis of the cavity where the strength of the electric fields are weakest and the magnetic field strength peaks at the center of the cavity. The position as shown is measured in inches, and is a distance that is measured from a fixed point at the top of the probe to a point on the sliding sample probe. This measured quantity decreases as the sapphire is pushed into the cavity. At a value of about 1.75in the sapphire is completely out of the resonator. Due to the different thermal expansions of the sample probe and the 0.5in diameter stainless steel shaft, it is difficult to determine when the sapphire actually enters the cavity but we estimate it to be at about 1.65in. The resonant frequency however starts to decrease at a position of about 1.7in indicating that the fields creep up the insertion hole. The drop in resonance frequency is because the sapphire, being a dielectric, increases the effective volume of the cavity as seen by the electric fields. Knowing the depth of the cavity at room temperature and ignoring the thermal contraction of the cavity, we estimate the center to be at a position of 1.42in. Fig. 4.2 shows the loss due to the sapphire plate as a function of temperature as well as Chapter 4. Data Analysis and Discussion 40 1 1 1 1 1 1 1 1 1 1 1 1 1 1 o o — o o o o o o o o o o o 0 0 / \ 0 o o o o o o 0 o 1 • 1 . 1 . 1 . 1 . 1 . 1 . 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Position(Inches) Figure 4.1: Frequency shift due to perturbation by the sapphire rod. Chapter 4. Data Analysis and Discussion 1.20E-009 • 8.00E-010 H o 4.00E-010 0.00E+000 • • • • • —r-20 I 40 I 60 80 T(K) Figure 4.2: A(4) due to bare sapphire (open squares), and grease (filled squares) Chapter 4. Data Analysis and Discussion 42 for the plate with a small quantity of silicone grease. In both cases the the plate is loaded to the center of the resonator. The loss is presented in terms of A(l/Q) = A(1/QL) — A( l /Q 0 ) since they are proportional to each other and it is this quantity that adds to the systematic error in A(^) when measuring a YBCO sample. The loss at low temperatures is found to be about 3 x 10 - 1 0 for the sapphire plate right after being cleaned. After adding a small quantity of grease the loss at low temperature is found to rise slightly to about 4x 10~10. The losses rise approximately linearly from their low temperature values for both the sapphire and grease but the grease loss rises more quickly with temperature. Both losses however are quite low compared to YBa2Cu^O^^ losses at temperatures above 5K but turn out to be only a factor of three or four lower at 1.2K when the sample is placed with its a-c or b-c face perpendicular to the field. This means that we have to be careful with measurements in this geometry. For the case where the a-b plane is perpendicular to the field, the A(^) values are of the order of 10~8 for the sample, so the grease and sapphire contribution are relatively very small. Chapter 4. Data Analysis and Discussion 43 4.2 Microwave Surface Resistance and Conductivity of Pure Y B C O Crystals In Fig. 4.3 the surface resistance of a pure twinned sample (called YLO) of YBo,2Cu306.95 is shown as a function of temperature. The surface resistance was measured with the applied field perpendicular to the ab plane. The sample has a Tc of about 93.2K and the surface resistance is seen to drop rapidly below this temperature by a few orders of magnitude. The surface resistance below the transition temperature has a minimum at about 70K and then is seen to peak at about 42K before decreasing as the temperature is lowered. The broad peak observed here was first reported by Bonn et al. [14] in low frequency measurements at about 2GHz. This peak is seen in almost all pure crystals studied and has been attributed to a rapid drop in the quasiparticle scattering rate. In order to see the features better at lower temperatures, the data is shown on a linear scale in Fig. 4.4. The minimum at about 70K and the peak at 42K are clearly seen and the surface resistance is seen to drop linearly below about 30K all the way to 1.2K with a slope of about . The residual loss is about 85u.fl and is several times larger than the predicted zero temperature limit for a d-wave superconductor (more about this limit shortly). Also shown is an earlier measurement by K. Zhang [17] on a crystal from a different batch at a higher frequency of 34.8GHz. The data has been scaled by the square of the ratio of the frequencies to the lower frequency of 22.7GHz. With the crystals being from different batches, we have to be careful with the comparisons. For measurements at one frequency the features seen to vary most among samples are the depth of the minimum around 70K, the details of the temperature dependence and residual loss below 30K, and the transition widths. So comparing the two frequencies we see deviations in the surface resistance below 50K, where a higher loss is observed at 22.7GHz and the broad peak is about 6K lower in temperature. Also seen to vary above 50K in Rs is that the scaled down Chapter 4. Data Analysis and Discussion 44 Figure 4.3: The surface resistance of a nominally pure YBCO sample (YLO) at 22.7GHz. Chapter 4. Data Analysis and Discussion 45 Figure 4.4: The surface resistance of sample YLO at 22.7GHz and a second sample measured at a frequency of 34.8GHz. The frequencies are scaled to 22.7GHz. Chapter 4. Data Analysis and Discussion 46 loss from 34.8GHz is lower than the loss seen in the newer crystal at 22.7GHz. It is difficult to read much into this difference , since as mentioned, the depth of the minimum at 70K is somewhat sample dependent. Going back to chapter 1, we had the relation between the surface resistance and the real part of the conductivity, R. = ^ u f a (4.1) which is accurate for <72 > > o~\, and at these frequencies is true once in the superconduct-ing state and up to a fraction of a degree below Tc. Assuming the penetration depth is the same at the two frequencies, then the comparisons are mainly between the real part of the conductivity o\ at the two frequencies. The real part of the conductivity has been calculated using penetration depth data on the YLO crystal and is displayed in Fig. 4.5. The penetration depth has been obtained by measuring its temperature dependence at 1GHz [6], and then finding the absolute penetration at 1.2K (1300A) by measuring the in-frared reflectance [20]. Kramers-Kronig analysis of the reflectance allows one to extract the imaginary (and real) part of the conductivity which is related to the magnetic penetration depth. We have assumed that this penetration is frequency independent, or in other words only the superfluid is responsible for the screening of the microwave fields. In the two fluid model this is true so long as LOT is much less than one. At 22.7GHz this is true except in the region 10 to 30K, where a slight distortion in o\ could result by using low frequency A(T) data. The error is less than 10% [21]. The broad peaks seen in the surface resistance (Fig. 4.4) can be traced back to the peaks seen in the real part of the conductivity (Fig. 4.5). These peaks cannot be associated with the usual coherence peak that is seen for a BCS superconductor immediately below T c . They are lower in temperature and much larger than a BCS coherence peak. Bonn et al. [22] attributed this feature to a competition between a slowly decreasing normal fluid density below T c and a rapidly rising scattering time. Using data at 2 GHz they estimated Chapter 4. Data Analysis and Discussion 20.0 15.0 5.0 • 1 o 0.0 o 34.8 GHz • 22.7 GHz o 0 O 8o § = • e o • o 0.0 20.0 40.0 60.0 80.0 100.0 T(K) Figure 4.5: Derived conductivities at two frequencies. Chapter 4. Data Analysis and Discussion 48 the quasiparticle scattering time by assuming a drude like frequency spectrum for o~i(cu). Although far infrared measurements show that o\ in the normal state has a l / u tail, rather than 1/LO2 [23], they nevertheless estimated how much r increased below Tc using a drude form as in equation (1.22). By assuming a temperature dependent scattering time they deduced that r increases over its value at Tc by a factor of about a hundred when the peak in o\ is reached at 30K, while the normal fluid density falls to 20% of its value below Tc as seen from penetration depth data [6]. Far infrared mesurements also indicate that LOT ~ 1 at about 2400 GHz in the normal state just above Tc, so a hundredfold increase means that LOT ~ 1 at 24 GHz and 30K. When LOT ~ 1, a rapidly increasing r no longer competes with a decreasing normal fluid density, so at higher frequency o~i(T) gets dominated by Nn(T) at a higher temperature. In other words we should see the peak in o~i(T) diminish and move higher in temperature with increasing frequency. This is as seen in Fig. 4.5. Returning to the low temperature behavior of the surface resistance and conductivity, we discuss the findings of Hirschfeld et al. [24] as found appropriate. He supposes that transport at low temperatures (T/T c < 0.3) is dominated by impurity scattering. Further-more the temperature dependence of transport coefficients could be quite different in two different low temperature regimes that are separated by some crossover temperature T*. For T < T* « Tc , a gapless regime exists where the superconducting properties reflect those of their normal state analogues. At somewhat higher temperatures than T* there is a crossover to a pure regime, where the transport coefficients follow power laws that reflect on the detailed structure of the order parameter. The crossover temperature is very much dependent on the scattering limits and the impurity concentration. In the unitary limit this temperature goes as, T* ~ (rA 0 ) 1 / / 2 , where T is the impurity scattering rate, and A 0 is the maximum gap over the fermi surface. In the Born limit (weak scattering) Chapter 4. Data Analysis and Discussion 49 the crossover temperature varies as T* ~ A0exp(—A0/T). In the Born limit, unless the impurity concentration is high enough that Tc's are substantially reduced, the physics of the gapless state is unobservable. In the gapless regime it is expected that the conductivity varies as T2 with temperature reaching a zero temperature value a"oo = ne 2/(m7rAo) that is independent of scattering rate for a d-wave gap in 2-D [7]. In order to test these claims we would like to probe this regime in the temperatures that are easily accessible experimentally, i.e down to about IK. In the resonant scattering limit this regime is accessible for relatively small impurity concentrations. Results by Taillefer et al. [25] on the electronic thermal conductivity in YBa2CusOe.95 show strong evidence of universal behavior that is consistent with the theory of quasiparti-cle transport in a d-wave superconductor. Measurements on four crystals doped with zinc (0 to 3%) to give very different impurity scattering rates that were estimated from resid-ual dc-resistivities, all showed remarkably close residual thermal conductivities of about 0.19mWK~2cm~l along both the chain and plane directions. Using the Wiedemann-Franz law we estimate a corresponding zero temperature limit of 0.77 x 10 6 Q - 1 m _ 1 for the charge conductivity which translates to values of about 55 and 15pQ in Rsa(T —>• 0) and Rsb(T —>• 0) respectively at 22.7 GHz where we have used A 0 a = 1600A and A0(, = 1030A. Of note is that the value Rsa(T —>• 0) is at least a factor of five times greater than the uncertain corrections that we apply to the data, so the resonator is capable of resolving this loss. Comparing our results for the pure crystals conductivity, we see no indication of a T2 dependence in the conductivity at any temperature down to 1.2K. Furthermore, the residual values are several times larger than what are calculated above for the "Lee-limit". This has implications for the nature of the residual impurity scattering in our nominally Chapter 4. Data Analysis and Discussion 50 pure samples. The residual impurity scattering rate is very small (T < .01TC) but in the resonant scattering limit this would still give a crossover temperature of about 14K (taking A 0 ~ 2TC). Crossover behavior is however not seen, indicating that the scattering is mainly non-unitary and probably we have to probe quite a bit lower in temperature to see a crossover in pure samples. Work by Zhang et al. [17] has shown that doping with zinc alters the low temperature dependence of the conductivity and penetration depth from linear towards a quadratic behavior with increased impurity concentrations suggesting a shift towards a gapless regime that we could probe at temperatures above IK. Chapter 4. Data Analysis and Discussion 51 4.3 Surface Resistance and Conductivity of Zinc Doped Crystals As mentioned in chapter 1, the surface resistance is an important probe of the fundamental properties of high temperature superconductors. It is also one of the key parameters to be well understood when it comes to the applications of HiTc superconductors. When studying these materials it is then important to know what constitutes the intrinsic microwave loss. Microwave losses of samples (mostly films) by different groups [26] [27] [28] show sample dependencies. Their magnitudes vary significantly from sample to sample and most show a monotonically decreasing loss below Tc , while in some there is a small plateau around 50 to 60 K before the loss falls further with temperature. This looks very different from what is seen in pure crystals. From the earlier discussion on the peak seen in the conductivity, one is led to hypothesize that defects present in these samples (impurities, weakly coupled grain boundaries, or other structural defects) affect the broad peak in 0i (T) by limiting the rapid increase in the quasiparticle lifetime r below Tc . This would mean that the normal fluid density dominates the temperature dependence, and a monotonically decreasing loss should be seen. If the earlier conclusions were correct, then we expect with a hundredfold increase in the scattering rate below Tc in pure samples, that the conductivity should be sensitive to low levels of point defects. With mean free paths of a few thousand Angstroms for very high purity crystals it takes very small impurity concentrations of the order of tenths of a percent to substantially affect these mean free paths. Bonn et al. [29] have shown successfully that by doping samples with either zinc or nickel that the conductivity peak is reduced considerably, providing further support that the peak seen around 40K is due to a rapid increase of r below Tc . We have measured two crystals with 0.15% zinc substituted for Cu mainly in the C11O2 Chapter 4. Data Analysis and Discussion 52 planes. The surface resistances shown in Fig.4.6 are for a pure sample (YLO) and the others for the two zinc doped crystals. Both of these doped samples showed a reduced Tc of 91.7K with a transition width less than 0.4K. The measurements were all done with Hrj perpendicular to the ab-plane. The derived conductivities are shown in Fig. 4.7. In order to derive these, we have assumed the same zero temperature value of 1300A as in the pure crystal, although the temperature dependence found for a 0.15% zinc doped crystal has been used. Because the impurity concentration is low we don't expect much variation in A(0), although the change would be towards a larger A(0), so we are overestimating oi& bit. The general shape of (Jiwould stay unaffected though. The derived conductivities of Fig. 4.7 show what has been reported by Bonne/ al. . The samples with 0.15% zinc have a reduced conductivity, about half that seen in the pure crystal. The position of the peak in the conductivity has also moved to a higher temperature. The lower <7iis a clear indication that the impurity scattering is limiting the rapid drop of the quasiparticle scattering rate, and that it saturates at a higher temperature, i.e., when the impurities dominate the scattering process. At low temperatures we observe a marked difference in the temperature dependence of (Tifor the pure crystal and the 0.15% zinc doped ones. The quadratic behavior seen for temperatures below 15K in the SZD sample is indicative of gapless behavior with a crossover around that temperature. The sample SZB shows a low temperature dependence in CTithat is closer to T than T2 at temperatures below 5K and then shows a slight curva-ture. The deviation seen in the low temperature behavior of the SZB sample below about 30K is consistent with a sample of lower doping perhaps indicating that within the first 1500A the zinc concentration is less than 0.15% and we are not in the gapless regime, where a T2 dependence is expected away from T = 0. The low temperature behavior of SZB is Chapter 4. Data Analysis and Discussion 53 Figure 4.6: The surface resistance of pure and zinc doped (0.15%) crystals. The Tc's of both doped samples were 91.7K and that of the pure sample, 93.2K. Chapter 4. Data Analysis and Discussion Figure 4.7: The conductivities of pure and 0.15% zinc doped crystals Chapter 4. Data Analysis and Discussion 55 characteristic of earlier measurements by K. Zhang [17] on 0.15% zinc doped samples. By noting that the loss in both 0.15% samples has been reduced by about a factor of two and assuming all the additional scattering is predominantly resonant, then an estimate of the crossover temperature in the unitary limit is ( T i m p ~ 0.2TC at 0.15% zinc doping ) about 0.2TC or 20K. This is consistent with the temperature range below which we observe gapless behavior in SZD. The residual conductivity of the sample SZD is found to be about 0.8 ±0 .2 x 10 6f2 - 1m - 1 and that for the second doped sample SZB to be about twice that. SZD also shows a slight upturn below about 3K in the conductivity, being consistent with earlier observations by Bonnei al. [30] of an upturn in the microwave loss of zinc doped samples at 3.8 GHz. At the lower frequency however, this effect is quite drastic, and the loss is seen to increase by a factor of about 3 from 4K down to 1.2K. Chapter 4. Data Analysis and Discussion 56 4.4 Anisotropies in the ab Plane The presence of CuO chains in the structure of YBCO inevitably leads to anisotropic transport properties in the ab planes. Normal state transport in the chains (b- direction) has been found to be quite different from the a- direction [32]. Zhang et al. [33] have seen the manifestation of this in the electrodynamics below Tc by measuring the surface resistance in a high purity crystal in the a and b directions. In Fig. 4.8 we show a similar measurement on a 0.15% zinc doped sample (SZD) at 22.7GHz. The first observation that looks very different from earlier anisotropy data of Zhang et al. [33] is that Rsa at T = 1.2K is considerably higher in value than Rsb, by a factor of 4 to 5 times. Both curves then move away from their lowest temperature values with a dependence that is not quite T 2 . The loss in the a-direction however rises with a steeper slope and at about 20K the loss is about twice as large in the a direction. At higher temperatures Rsa shows a plateau around 50K which is a remnant of the much more prominent peak seen in the a-direction loss of a pure crystal. The loss in the b-direction shows no such plateau, consistent with observation of a much smaller peak in Rsb in pure crystals [33]. The loss in the normal state is also seen to show a considerable anisotropy with Ra about 1.3-1.4 times larger than Rb at 100K, corresponding to a-b resistivity ratios of about 1.8. The real part of the conductivity has been derived and plotted in Fig. 4.9. Again we have used penetration depth values as for a pure sample, i.e Aa = 1600A and Xb = 1030A. It is interesting to see that this large anisotropy in X(T) has caused Rsa > Rsb despite the fact that aia(T) < oib(T). At temperatures just below Tc , <T16 is about 1.6 times larger than o~lQ, more or less consistent with normal state conductivities found from the surface resistance. Below 80K and all the way down to the peaks in the conductivities and Chapter 4. Data Analysis and Discussion a en r r 20.0 40.0 60.0 80.0 T(K) Figure 4.8: The surface resistance of sample SZD in the superconducting state Chapter 4. Data Analysis and Discussion 20.0 15.0 a CO o T— to" 10.0 Figure 4.9: Real part of the conductivity in the a and b directions. Chapter 4. Data Analysis and Discussion 59 even lower to 30K, o~ib(T) is about a factor of 2 larger than crla(T) before they begin to converge. In this region both curves decrease approximately linearly with different slopes, before curving in below about 5K to residual values of about 2.8 x 106 and 2.5 x 1 0 6 £ } - 1 m - 1 for the a and b direction residual conductivities respectively. These values are very close and an uncertainty of about 1 0 0 A in A(0), which is not unreasonable, is sufficient to render these conductivities equal within the errors of the analysis. This observation might suggest a close to zero chain conductivity at low temperatures, supporting similar observations in the thermal conductivity [34]. Before we move on to discussing the new crystal runs, we need to address the clear discrepancy that is seen in the loss of sample SZD (Fig.4.10) that was measured in the two geometries where the field is perpendicular to the ab plane or parallel to it. The averaged loss observed in the geometry where Hrf was put perpendicular to the ac face and be face of the crystal in successive runs, was found to be higher by a factor of 1.4 than the loss observed in the other geometry(Hrf //c). Some of this could be attributed to the c-axis contribution where according to data by Mao et al. [35], they find Rc values that are thirty to fourty times higher than what is seen in the other directions. With the area of our SZD sample being about .75mm2 and a thickness about 20 microns, we get a ratio of about 40 to 1 for the width to thickness of the crystal measured. With such high values for Rsc the discrepancy could be partly explained. We must emphasize here that the way Mao et al. have obtained their values for Rsc has been by noticing this precise discrepancy in the two geometries and doing a subtraction . This procedure is not well founded since the current distribution in the geometry where demagnetizing factors are large ( Hrf//c) is something that is not well understood and is not necessarily going to give similar results. Comparing the residual losses, we see that the losses in the ac direction (after subtractions) are about 180/if2, and the be direction about 40/xfi, slightly more than that measured in Chapter 4. Data Analysis and Discussion 60 1.5 200.0 o o 4> So 1.0 a DC 0.0 -o-o R s ab-geometr ic mean O RS(ab) H/C o o . o o <> o 87.0 92.0 97.0 o o o o o 60.0 80.0 [h] Figure 4.10: Comparison of the surface resistance of sample SZD measured in the two geometries . Also seen in the inset is their normal state values. Chapter 4. Data Analysis and Discussion 61 the other geometry where a residual loss of 27/iQ was obtained. This residual loss was obtained from a non subtracted value of 67/ifi. It is possible to do a subtraction from the data from each of the residual values Rsac and Rsbc and match the geometric mean with the unsubtracted value of Rsa(, in the second geometry but there will be no agreement at higher temperatures. We are left to reanalyze our position on measurements done in the geometry where the field is perpendicular to the ab plane and demagnetizing factors large. Certainly further experiments are needed to answer these questions. 4.5 Barium Zirconate Grown Crystals In this section we will briefly show data on two other crystals that were measured. Both crystals were about 30 microns in thickness and had similar ab plane areas. The difference between them was in their preparation, one being from earlier YSZ crucibles and the other a recently grown BZO crucible crystal. They were both annealed in flowing oxygen for about 10 days to set the oxygen to 6.95 and subsequently quenched to room temperature. Fig. 4.11 shows the measured surface resistance versus temperature for both these crys-tals. The data looks very much like that of a typically pure crystal. There are however a few observable differences in the two. The most notable difference has been in the transi-tion width of the two samples. The barium zirconate crystal showed a transition width of about 2K in the surface resistance and subsequent measurements in the penetration depth revealed a double transition starting at about 91.5K, followed by one at 90.5K. This has not been seen in any crystals grown of late in YSZ crucibles. A look at the surface resistance below T c shows the typical peak, however it is a couple of degrees lower in temperature than seen at 22.7GHz for the YSZ grown pure crystals. This indicates that the BZO crys-tal is of higher purity, since the increase in quasiparticle lifetime dominates the drop of the normal fluid density down to a lower temperature. We should also note that at this Chapter 4. Data Analysis and Discussion 62 frequency where LOT ~ 1 at about 40K, even a small decrease of a couple of degrees in the temperature where Rs peaks would be consistent with a much improved sample purity. The depth of the minimum at 70K is also seen to be lower in the new crystal, though we must add that this feature varies even among crystals grown using YSZ crucibles. Both crystals show linear temperature dependence at low temperatures with the YSZ crystal flattening out over the last 5K down to 1.2K. The BZO crystal on the other hand is very linear down to 1.2K. Similar behavior was seen in A(T) by S. Kamal [36] for this particular sample. Returning to the width of the transition, the higher purity of the BZO crystal that is hinted at by the surface resistance measurement could be the cause of the apparent width. If the crystals are in fact purer, then it is argued [37] that the oxygen mobility in these samples is greater and the simple procedure of quenching the crystals in air after annealing is insufficient to ensure a random distribution of oxygens at the optimum doping that was intended. This would mean that in the period when the sample is being quenched to room temperature the mobile oxygens would tend to cluster leaving regions that are both under and overdoped with a Tc that is lower than the expected optimal doping value. Evidence of this non randomness has come when a measurement on the same crystal after another anneal showed a narrower double transition. It is clear that there remains much work to be done on optimizing the growth conditions, and the annealing procedures might have to be tailored for these higher purity samples. Chapter 4. Data Analysis and Discussion 63 1.0 a E, 0.5 0.0 • YSZ i i OBZO « D E O n 0 • • ©• 0 D © D • 0 0 o • • 8 8 @ © o ° O • o 0.0 20.0 40.0 T(K) 60.0 80.0 Figure 4.11: Surface resistance of pure crystals grown in barium zirconate and yttria sta-bilized zirconia crucibles. Chapter 5 Conclusions Over the last 18 months we have developed and used a resonator at 22.7 GHz to study the surface resistance of YBCO superconductors. The resonator was built with a resonance frequency in mind that would enable us to use coaxial coupling lines, as this would greatly simplify the design and make it user friendly. The resonator as it stands is capable of measuring the surface resistance from 1.2K to 120K, and is capable of resolving losses to better than 7//0 when operating with an unloaded Q of about 40 million. This is about a quarter of the expected residual loss of a sample whose residual loss corresponds with that of the Lee limit, making it among the best resonators used so far to study these materials at these frequencies. Since the period where we have been able to put all the electronics together, a period of about six months at the time of writing, numerous measurements have been taken. The samples studied have been twinned pure crystals and a pair of .15% untwinned zinc doped crystals. We will summarize the basic results here and discuss other measurements that are planned and improvements to the resonator itself. 1) The Pure Samples The pure samples were all twinned crystals, two from the YSZ crucibles and one that was prepared in a barium zirconate crucible. All three samples at 22.7GHz showed a minimum about 70K and a broad peak at lower temperatures. The broad peak for the YSZ crystals were at about 42K and that for the BZO crystal at about 39K. These peaks are indicative of 64 Chapter 5. Conclusions 65 a rapid drop in the quasiparticle scattering rate and a slowly falling number of quasiparticle excitations with decreasing temperature. This slow drop hints at the presence of nodes in the gap function. The temperature dependence of all these samples show a stongly linear behavior at temperatures below 30K, with one of the YSZ crystals showing some curvature below 5K. There are currently no satisfactory explanations of the linear behavior in the pure regime of temperatures. The high residual losses seen in these sample, between 3 to 4 times the universal limit predicted by Lee [7] indicates that we are probably not probing low enough in temperature. From the data of Taillefer [25] the crossover temperature is expected not to be unobservably small but is probably accessible at dilution refregirator temperatures of hundreds of mK. From the experimental side, getting that low in temperature requires great effort and modifications to the setup of the present experiments. There are plans for such an endeavor, but for now we have chosen to study the physics of the gapless regime by moving up its crossover temperature by adding dilute but strong scatterers such as zinc to the system. 2)Zinc Doped Crystals. The experiments on two 0.15% zinc doped crystals both showed that the dopants sup-pressed Tc by about 1.5K, and that the losses in these samples were reduced by about a factor of 2 in the superconducting state from those of pure YBCO. The peak seen in the Rs of pure YBCO has been reduced to a plateau, and the size of the conductivity peak also reduced. The low temperature conductivity of the SZD sample showed a quadratic increase from a residual value of 0 .8±0.2 x 1 0 6 n - 1 m _ 1 while the other had twice this resid-ual conductivity with a more slowly increasing temperature dependence. The sample with the lower residual conductivity, has its om value agree to within experimental uncertainties with that, that is derived from the thermal conductivity data of Taillefer et al. [25], but we note that there are concerns about the Hrj//c geometry used. Chapter 5. Conclusions 66 A large anisotropy in the surface resistance within the ab plane of the zinc doped crystal, SZD, was shown to exist, an extension of the earlier data of Zhang et al. [33]. The surface resistance in the a direction was found to be larger than in the b direction, with the greatest difference at low temperatures close to 1.2K, where Rsa was about 4.5 times larger than Rsb. Using zero temperature penetration depth data for a pure crystal, we extracted <7i for both directions to find that at 1.2K these conductivities are remarkably close, though a few times (Too and not quite showing a T 2 dependence that is expected in the gapless regime. The close residual conductivities suggests that the chain conductivity is approacing zero with temperature. This observation is consistent with thermal conductivity measurements, where recent data show no apparent residual ab anisotropy in the thermal conductivity [34]. Derived residual thermal conductivities from this data would however be 4 times that seen by Taillefer et al. In the normal state surface resistance, we see that Rsa is larger than Rsb by a factor of 1.3-1.4 at 100K. This corresponds to a ratio of 1.8 for the resistivities in the a and b directions. 3)BZO Grown Crystal Studies. A single crystal of YBa2Cu30e,g5 that was grown out in a barium zirconate crucible was shown to have no qualitative differences from crystals grown in YSZ crucibles in contrast to work by Srikanth et al. [38] on supposedly similar BZO grown crystals. The surface resistance was seen to peak at about 3 degrees lower in temperature in contrast to the two YSZ grown crystals that showed peaks about 42K. This is indicative of a crystal that is substantially purer. We believe that our crystals are showing the true intrinsic behavior of these materials, and believe the work of Srikanth et al. shows extrinsic effects from a contaminated surface. Chapter 5. Conclusions 67 5.1 Further Studies As already noted in the discussions, there is a small yet significant disagreement between measurements obtained in the two geometries discussed. We should note that this is not a problem of the resonator, but most likely an effect due to very different current distributions in the two geometries. Furthermore if there are inhomogeneities of Rs on the ab faces then this could well result in different measured losses in the two geometries. In order to separate out the contributions, and get a feel for their magnitudes however, we need to know what sort of magnitude Rsc takes relative to those found routinely for Rsab- Although there is data in the literature, we feel that mixing the two geometries to extract Rsc is not well founded as seen with drastic results in the c-axis penetration depth of Mao et al. that is contradictory to results of Kamal et al. [31]. With the improved sensitivity of this resonator, one can in principle measure Rsc as was done for the penetration depth. This involves measurements where the ab plane is parallel to Hrf before and after cleaving the crystal into needles. Other planned measurements are the ab-anisotropy in crystals with oxygen contents that substantially reduce Tc in order to further understand the role of oxygen vacancies on the chains. Naturally there remains measurements to be done on crystals upon availability, with zinc contents that vary the scattering rate considerably. Observation of a more or less constant residual loss in samples with such different impurity scattering rates would provide very strong support for the theory of quasiparticle transport in 2D for a superconductor with a d-wave gap. Finally, we are in the process of designing a new probe that will enable us to exchange samples while the resonator is kept cool. This in principle would remove uncertainties in measurements where there might be a non constant correction that results from the Chapter 5. Conclusions 68 resonator Q varying from run to run. This was noted when studying the anisotropy of sample SZD where a Q drop by 50% resulted in the loss seen in the a-direction to be 10% higher at low temperatures. We are also planning new thermometry for the sample probe so that temperature regulation would be more efficient and we could reach temperatures in the range 1.2- 250K in order to to study the low temperatures without the need for exchange gas to be present and to study the normal state properties at microwave frequencies. Bibliography [1] J. G. Bednorz and K. A. Muller Phys., B64 1986 [2] J. Bardeen, L. N. Cooper, and J. R. Schrieffer. Physical Review,l08, 1957 [3] Gloria. B. Lubkin. Physics Today, 1995. [4] Ruixing Liang, P. Dosanjh, D. A. Bonn, D. J. Barr, J. F. Carolan, and W. N. Hardy.Physica C, 195, 1992 [5] D. A. Bonn, S. Kamal, K. Zhang, R. Liang, D. J. Barr, E. Klein, and W. N. Hardy, Phys Rev Lett, 73, 1845 1994. [6] W. N. Hardy, D. A. Bonn, D. C. Morgan, Ruixing Liang and Kuan Zhang Phys Rev Lett 70, 3999,1993 [7] P. Lee, Phys.Rev.Lett 71, 1887 1993 [8] Michael Tinkham, Introduction to Superconductivity McGraw-Hill, New York, 1975 [9] C. Zuccaro, C. T. Rieck, K. Scharnberg, Physica C 235-240, 1807 1994. [10] N. Klein, in Synthesis and Characterization of High-Temperature Superconductors, volume 130-132. Materials Science Forum, 1993. [11] H. London Nature, 133:497 1934. [12] C. J. Gorter and J. B. G. Casimir, Phys. Z, 35 1934. [13] Bruce Gowe, M.Sc Thesis, University of British Columbia,1997. [14] D. A. Bonn, P. Dosanjh, R. Liang, and W. N. Hardy, Phys Rev Lett 68, 2390 1992. [15] S. Sridhar and W. N. Kennedy, Rev. Sci. Instrum, 59, 531 1988. [16] A. B. Pippard, Proc. Roy. Soc (London) A216, 547, 1953 [17] Kuan Zhang, PhD Thesis, University of British Columbia 1995 [18] J. A. Osborn, Phys Rev, 67, 351 1945. [19] A. Erb, E. Walker and R. Flukiger, Physica C 245, 245 1995. 69 Bibliography 70 [20] D. Basov, R. Liang, D. A. Bonn, W. N. Hardy, B.Darowski, M. Quijada, D.B. Tanner, J. R Rice, D. M. Ginsberg, and T. Timusk Phys Rev Lett 74, 598 1995 [21] D. A. Bonn, S. Kamal, K. Zhang, R. Liang, D. J. Barr, E. Klein, and W. N. Hardy, Phys Rev B 50, 4051, 1994 [22] D. A. Bonn, R. Liang, T. M. Riseman, D.J. Barr, D.C. Morgan, K. Zhang, P. Dosanjh, T. L. Duty, A. MacFarlane, G. D. Morris, J. H. Brewer, and W. N. Hardy. Phys Rev B, 47, 11314, 1993 [23] T. Timusk and D. B. Tanner in Physical Properties of High Temperature Supercon-ductors (World Scientific, Singapore, 1993) vol3. [24] P. J. Hirschfeld and N. Goldenfeld, Phys Rev B, 48, 4219 1993 [25] L. Taillefer et al. Phys Rev Lett, 79, 483, 1997. [26] N. Klein et al. J. Supercond.5, 195 1992. [27] Z. Ma et al. Phys Rev Lett 71, 781 1993. [28] A. Mogro-Campero et al. Appl. Phys Lett. 60, 3310 1992. [29] D.A. Bonn et al. Phys Rev B 50, 4051, 1994. [30] D. A. Bonn et al. Czechoslovak J. of Phys, 46, 3195, suppl. S6, Proc. of the 21st Int. Conf. on Low Temperature Physics. Prauge, 1996. [31] W. N. Hardy et al. , Proc Of the 10th anniversary HTS workshop, ed. B. Batlogg et al. (World Scientific, Singapore) p. 223, 1996. [32] T. A. Friedmann, M. W. Rabin, J. Giapintazakis, J. P. Rice, and D. M. Ginsburg, Phys Rev B, 42, 1990. [33] K. Zhang, D. A. Bonn, S. Kamal, R. Liang, D. J. Barr, W. N . Hardy, D. Basov, and T. Timusk, Phys Rev Lett. 73, 2484 1994. [34] L. Taillefer, Private Communications. [35] J. Mao et al. Phys Rev B 51, 3316 1995. [36] S. Kamal, Private communications. [37] R. Liang, D. A. Bonn, and W. N. Hardy, Private communications. [38] H. Srikanth et al. Phys Rev. B 55, R14733 1997. 

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