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Studies of the microwaves surface resistance of pure, zinc, and nickel doped YBCO crystals Zhang, Kuan 1995

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STUDIES OF THE MICROWAVE SURFACE RESISTANCE OF PURE, ZiNC,AND NICKEL DOPED YBCO CRYSTALSbyKUAN ZHANGB.Sc., Fudan University, 1984M.Sc., Institute of Technical Physics, Chinese Academy of Science, 1987A THESIS SUBMIfI’ED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDWSDepartment of PhysicsWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAAugust 1995© Kuan Zhang, 1995In presenting this thesis in partial fulfillment of therequirements for an advanced degree at the University of BritishColumbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission forextensive copying of this thesis for scholarly purposes may begranted by the head of my department or by his or herrepresentatives. It is understood that copying or publication ofthis thesis for financial gain shall not be allowed without mywritten permission.Department of___________(Signature)The University of British ColumbiaVancouver, CanadaDate 2Q, qqcAbstractAn apparatus utilizing a superconducting cavity resonator has been developed for thesystematic study of the microwave surface resistance of high purity(twinned and untwinned),zinc and nickel doped YBa2Cu3O74single crystals at 35 GHz. The objective was to obtainthe intrinsic microwave properties of high temperature superconductors. By using our highlysensitive apparatus to study the effects of twinning and doping, we have been able to confirmthat what we have obtained is indeed the intrinsic properties ofYBa2Cu3O78.The results show that the intrinsic microwave properties of YBa2Cu3O78 are verydifferent from that of conventional s-wave BCS superconductors. Firstly, the quasiparticlescattering rate is strongly temperature dependent, and drops rapidly below the transitiontemperature, which implies a quasiparticle-quasiparticle scattering mechanism. Secondly,both the surface resistance and the real part of the conductivity vary linearly with T below35K (except for the zinc doped crystals), suggesting an unconventional superconductingpairing state with line nodes in the order parameter. Thirdly, the real part of the conductivitywhen extrapolated to the OK is very small, in contrast to previously existing data, andapproaches the limit predicted for a d-wave superconductor.A practical result of our studies is that we have shown that the microwave loss of hightemperature superconductors can be reduced by doping with certain impurities. This is animportant result in the area of microwave applications, where one generally wants as low asurface resistance as possible.11Table of ContentsAbstract.iiTable of Contents iiilist of Figures vNotation viiiAcknowledgments xChapter 1 Introduction 1Chapter 2 Properties ofYBa2Cu3O7 92.1 Structure of YBa2Cu3O78 112.2 Some Basic Features ofYBa2Cu3O7 Superconductors 18Chapter 3 Electromagnetic Properties of Superconductors at MicrowaveFrequencies 263.1 Microwave Surface Impedance 273.2 The Two Fluid Model 323.3 The Electromagnetic Properties of a BCS Superconductor 37Chapter 4 The Microwave Circuit 48ifi4.1 The Resonant Circuit 494.2 Cavity Perturbation Method 60ChapterS Experimental Preparation 695.1 Samples 705.2 The Experimental Apparatus 745.3 Determination of the Surface Resistance 825.4 Circuit Diagram and Instrumentation 85Chapter 6 Data Analysis and Discussion 926.1 Background Runs 936.2 Microwave Surface Resistance and Conductivity of Pure YBa2Cu3O7Crystals. ..1056.3 Microwave Surface Resistance and Conductivity of Zinc and Nickel Doped YBCOCrystals 1196.4 Anisotropic Properties in the ab Plane 132Chapter 7 Conclusions 141Bibliography 149Appendix A 156Apenx B 158Appendix C 160ivList of FiguresFigure 2.1 The crystal structure of YBa2Cu3O7 12Figure 2.2 The transition temperature of YBCO versus oxygen content 14Figure 2.3 The twin boundary caused by the change of orientation of CuO chains in the CuObasal layer 15Figure 2.4 The variation of lattice constants a and b versus doping concentration 17Figure 2.5 T versus doping concentration 20Figure 2.6a Resistivity versus temperature for the pure and 0.75% nickel doped samples inthe ab plane 22Figure 2.6b Resistivity versus temperature for pure and zinc doped samples in the abplane 23Figure 2.7 The resistivity versus temperature in a, b, and c directions for a YBCO singlecrystal 24Figure 2.8 Temperature dependence of the thermal conductivity in the a and b directions fora YBCO single crystal 25Figure 3.1 The definition of the surface impedance 28Figure 3.2 A circuit representing the two-fluid model 34Figure 3.3 The ratio A(0)2/(7, i.e., the superfluid fraction, x versus T for YBCOcrystals 44Figure 3.4 The microwave absorption of a Nb sample in the superconducting state 47Figure 4.1 The equivalent series RLC circuit 50vFigure 4.2 The coupled resonant circuit 52Figure 4.3 The power transmission through a resonant circuit 58Figure 4.4 Field and current patterns inside the cavity(a, b), and on the cavity walls(c) 63Figure 4.5 Samples of the shapes of a plate and an ellipsoid in magnetic fields 67Figure 5.1 Specific Heat at Transition Temperature for One of the pure YBCO Crystals... 71Figure 5.2 A picture of one of the pure YBCO crystals taken with a polarizingmicroscope 72Figure 5.3 The 34.8 GHz cavity 75Figure 5.4 The lower end of the experimental apparatus 77Figure 5.5 The lower end of the sample holder 79Figure 5.6 Measurement circuit diagram 86Figure 5.7 (a) HP83620A Synthesized Sweeper, (b) HP8349B Microwave Amplifier, and (c)HP83554A Source Module 89Figure 6.1 The frequency shift of the resonator due to perturbation by the bare sapphirerod 94Figure 6.2 The loss due to bare sapphire 96Figure 6.3 The loss due to the silicone grease and the sapphire rod 98Figure 6.4 The frequency shift(a) and the loss(b)due to a sample versus its position in thecavity 100Figure 6.5 The loss due to a series of Pb:Sn samples 104Figure 6.6 The surface resistance of a sample from batch RL1095 106viFigure 6.7 The Comparison of the surface resistance at two frequencies, 3.8 GHz and 34.8GHz 107Figure 6.8 The real part of the conductivity of a sample at two frequencies 109Figure 6.9 The scattering rate in the superconducting state derived from the two fluid modelfor a pure YBCO sample 112Figure 6.10 The calculated real part of the conductivity 113Figure 6.11 Calculated scattering rate versus temperature 115Figure 6.12 The surface resistance of YBCO crystals with various oxygen contents 118Figure 6.13 The surface resistance of various YBCO samples measured by three groups.. 121Figure 6.14 The superconducting state surface resistance of zinc doped YBCO crystals. .. 123Figure 6.15 The superconducting state surface resistance of nickel doped YBCO crystals. 124Figure 6.16 The effect of zinc doping on the real part of the conductivity 127Figure 6.17 The effect of nickel doping on the real part of the conductivity 128Figure 6.18 The calculated curves of the real part of the conductivity at 34.8 GHz 130Figure 6.19 The scattering rate derived from the two fluid model 131Figure 6.20 The surface resistance of two untwinned samples 133Figure 6.21 The normal state surface resistance of sample 1 in the a and b directions 135Figure 6.22 The surface resistance of sample 1 in the superconducting state 136Figure 6.23 The real part of the conductivity in the two directions at 34.8 GHz 139VuNotationA potential vector of electromagnetic fieldB magnetic inductionE electric fieldH magnetic fieldJ surface current densityj current densityk or q wave vectorkF Fermi wave vectorc, c electron creation, annihilation operatore electron chargef Fermi Dirac functionf microwave frequencyf, c, resonant frequencyh Plank’s constantkB Boltzmann’s constant1 mean free pathm mass of the charge carriersN density of statesN number of charge carriersn charge carrier densityQ quality factorR3 surface resistanceviiiT superconducting transition temperatureW energy stored in a cavity resonatorX surface reactanceXs supeffluid fractionZ surface impedanceZ impedancef3 coupling constant between a resonant circuit and external circuitsuperconducting order parameterSci skin depthmagnetic flux7* or y quasiparticle creation or annthilation operatorA magnetic penetration depthpermeabilityCo or £2 microwave angular frequencyU) quasiparticle frequencyp resistivityconductivityreal part ofimaginary part of oquasiparticle scattering timecoherence lengthxAcknowledgmentsI would like to extend many thanks to my supervisor, Professor W. N. Hardy, whocontinuously inspires me to approach physics problems with a scientific style, and hasprovided help with many experimental details throughout the whole Ph.D. project. I alsowould like to thank Professor D. A. Bonn, who actually led me into the area of microwaveexperimental techniques, and who constantly helps me with his broad knowledge in physics.I would like to extend my gratitude to Dr. D. Morgan for many discussions from which Ilearned a lot about microwave techniques, to Dr. R. Liang, who has grown high qualitysingle crystals for the studies, and to J. Bosma, who offered much help during the building ofthe experimental apparatus. Finally, I would also like to express my gratitude to all of theother people in the group, Dr. D. Baar, R. Knobel, S. Kamal, A. Wong, C. Bidinosti, and P.Dosanjh, who have helped to make this lab a great place to study and work.Funding of this work by the Natural Sciences and Engineering Research Council ofCanada, the Canadian Institute of Advanced Research, and CTF systems, Inc. is gratefullyacknowledged.xChapter 1IntroductionThe history of superconductors goes back to 1911, when Kamerlingh Onnesdiscovered that mercury became superconductive below 4.2 K 1 In 1934, a two-fluid modelwas proposed by Gorter and Casimir2 to explain the phenomena. In this model it is assumedthat there are two kinds of electrons in a superconductor. One kind, called normal electrons,acts as those of a normal metal, while the other ldnd, called superconductive electrons, formsa frictionless superfluid, which is responsible for the screening of magnetic fields from thesuperconductor at low frequencies. The penetration depth of the field into thesuperconductor is determined by the superfluid density through the London equation, firstproposed by F. and H. London in 1935 and which can be expressed as0Aj=—A, 1.1where A is the vector potentiaL j is the current density. A is a function of the superfluiddensity in the two fluid model: A = m /(,i0Ne2).As long as A is positive, the field decaysexponentially within the superconductor, with the characteristic penetration length A =Therefore the London equation implies the Meissner effect, i.e., the ability of asuperconductor to expel the magnetic flux when it is cooled below the transitiontemperature, provided that the field is not too strong. The two fluid model provides areasonable qualitative macroscopic description of some of the thermodynamic and1Chapter 1. Introductionelectrodynamic properties of superconductors. However, the first satisfactory microscopictheory was given by Bardeen, Cooper and Schrieffer (BCS) in 1957 . The theory is basedon electron-phonon interactions and provides explanations for most of the properties of thesuperconductors known at that time. In 1958 Mattis and Bardeen5 derived an expressionrelating current density and an applied field for superconductors using this theory. From theirresult a relation similar to that of the London equation can be derived, where a quantityanalogous to the superfluid density in the two-fluid model can be defmed. The qualitativebehavior of the temperature dependence of the penetration depth in conventionalsuperconductors can be explained very well by this relation. A key feature of the theory is anenergy gap at the Fermi surface, with the density of states sharply enhanced right below andabove the gap. This enhancement of the density of states gives rise to a peak (coherencepeak) right below the transition temperature for the electromagnetic absorption. This peakwas in fact experimentally observed in the real part of the conductivity6 and in the NMRrelaxation rate7.The coherence peak is a feature not found in the usual two fluid model.The discovery of high temperature superconductors in 19868 has inspired a wave instudying the fundamental mechanism of the new system. A number of basic experimentalresults for these new superconductors are found to be not described by the BCS theory, atleast not by the conventional s-wave BCS theory. Difficulties in making samples of sufficientquality further complicates the matter. Recently, several experiments suggest that the orderparameter of the high temperature superconductors have d-wave symmetry. For such asymmetry the energy gap is not isotropic, and instead it has nodes on the Fermi surface.2Chapter 1. IntroductionThese experiments include NMR9”°, penetration depth 11 13 photoemission 14Josephson tunneling’5,etc. The possibility of a d-wave pairing state was explored by anumber of theorists. Scalapino and co-workers have successfully analyzed the temperaturedependence of the Cu(2) and 0(2,3) nuclear relaxation rates of YBa2Cu3O7,in terms of aweak-coupling antiferromagnetic Fermi-liquid theory 16,17,18 They have also analyzed theresults from microwave impedance measured mainly in our group’9,and found that the datafit better to the theoretical result for a superconductor with a d2_ symmetry. Based on theexperimental result on the NMR relaxation rate ofYBa2Cu3O73,Millis, Monien and Pines 20proposed a phenomenological model which assumes that the spin-spin correlation functionstrongly peaks at ( ±tIa, ±tIa ) positions in the Brillouin zone. This assumption isqualitatively consistent with the result from the neutron scattering experiment2’,22,23• Usingthese results Monthoux, Balatsky and Pines proposed a phenomenological theory24,whereit is argued that the antiferromagnetic copper spin fluctuations in the system are responsiblefor the superconductivity. An interaction between the quasiparticles via spin fluctuationsprovides pairing of the quasiparticles and favors an energy gap with ad2_z symmetry.Measurements of the microwave surface impedance of classical superconductors werecarried out by H. London, A. B. Pippard and other researchers in the forties and fifties andplayed an important role in the development of BCS theory25 ,26,27 The imaginary part ofthe surface impedance, the surface reactance, is directly related to the magnetic penetrationdepth, A, which provides a measure of the superfluid density. Its temperature dependence3Chapter 1. Introductionreflects the quasiparticle density of states near the Fermi surface. The real part of the surfaceimpedance, the surface resistance, R, provides information about the real part of theconductivity a, which is proportional to the electromagnetic absorption inside thesuperconductor in a given electric field. The real part of the conductivity involves both thedensity of states and the quasiparticle scattering rate. For the high temperaturesuperconductors, the surface impedance has also received extensive scrutiny, both in crystalsand films, reflecting its importance for both the fundamental and application aspects of thesematerials 28,29, 30,31 Techniques used to study the surface impedance include that using aparallel plate resonator (Stanford 29.32), a solenoid coil(U. of Tokyo33), and cavityperturbation (Northeastern34,UBC 11,35, UCLA31 Maryland36). Early results on thepenetration depth of YBa2Cu3O7thin films using the parallel resonator technique haveshown an exponential temperature dependence at low temperatures, taken as an evidence fora finite minimum gap in the material. The measurements made at UEC on high qualityYBa2Cu3O78 single crystals have shown a linear temperature dependence at low T, givingstrong evidence for a gap with line nodes. This linear temperature dependence was alsoconfirmed by Mao et al.36, but with a much larger slope. Ma et al.32 reported their data onBi2SrCaCuO8single crystals and YBa2Cu3O73 thin films, also using the parallel plateresonator technique. They obtained a quadratic temperature dependence at lowtemperatures. This I behavior can be explained by a calculation done by Hirschfeld andGoldenfeld for a d-wave superconductor with strong impurity scattering37.Data obtainedfor the surface resistance on those materials varies enormously, depending on the quality of4Chapter 1. Introductionthe materiaL Taking YBa2Cu3O74 as an example, some of the data shows that the surfaceresistance decreases monotonically with temperature, whereas other data show a smallplateau or even a peak in the region near 40 K. In addition to the different qualitativefeatures, the magnitudes vary drastically32’89.For poor quality samples the surfaceresistance can be dominated by extrinsic factors. In samples with grain boundaries, forexample, weak links at the boundaries which are partially normal can dominate the surfaceresistance. For samples of good quality, the surface resistance reflects the intrinsic propertiesof the material and is determined by the real part of the conductivity crj. Bonn et at., with ageneralized two fluid model in mind, have taken o to be proportional to the product of thenormal fluid density and quasiparticle scattering time40. Recently, Scalapino et at. haveshown that for layered materials, this ansatz is in fact appropriate, with the use of anappropriately averaged scattering time41.In this thesis the microwave surface resistance of YBa2Cu3O74 has been systematicallyinvestigated at 34.8 GHz using cavity perturbation methods, in an apparatus designed andbuilt as part of the Ph.D. project. The central part of the apparatus is a superconductingcavity resonator operating in the TE011 mode. The samples in this study were all very highquality crystals grown in our group by Liang et att2. We have measured the surfaceresistance of pure crystals, and as well as those with zinc and nickel doping. Usinguntwinned crystals we were also able to measure the anisotropy of the surface resistance inthe ab plane. Finally, the real part of the conductivity was derived from our R data bycombining it with measurements of the penetration depth, obtained on the same crystals by5Chapter 1. Introductionour group and the McMaster group43.Before this Ph.D. project began, a broad peak in the surface resistance had beenobserved by Bonn et al in a 3.8 GHz apparatus40.This broad peak was claimed to be anintrinsic property of the material, resulting from a peak in the real part of the conductivity. Itwas proposed that the peak in the conductivity was due to a rapid drop of the quasiparticlescattering rate below T. A good method to confirm these claims would be to obtain thefrequency dependence of the surface resistance and hence the frequency dependence of thereal part of the conductivity for the material: a low scattering rate combined with a higherfrequency would eventually cause the product co’r to be close to or even greater than one,and accordingly this would alter the overall shape of the electromagnetic absorption in thematerial. This was one of the motivations for developing the 35 GHz apparatus. Theexistence of the broad peaks in R and 0i were confirmed at this frequency with a shift of thepeaks toward higher temperatures. The direction of this shift is exactly that expected for asuperconductor with a rapid drop in the quasiparticle scattering rate35’. The broad peak inthe surface resistance was also confirmed by Achkir et al.45 and Mao et al.36The surface resistance data of some state-of-the-art thin films have raised questions onwhether the broad peak in R is intrinsic or extrinsic. The surface resistance of some films notonly lacks the broad peak, but is also found to be lower than that of the crystals at lowtemperatures32’839 Needless to say, it was very important to decide which data representedthe intrinsic properties of the material. In order to confirm that what we were measuring was6Chapter 1. Introductionindeed intrinsic, and also to further explore the influence that impurities have on the surfaceresistance, we decided to work with crystals with zinc and nickel impurities deliberatelyintroduced. The measurement of the surface resistance of these doped samples has led tovery convincing results and provides important information on the quasiparticle scatteringmechanism 46,47 It also suggests a method to obtain lower surface resistance in thin films,something that is important for many applications.It is well known that the amount of oxygen in YBa2Cu3O76 strongly affects theproperties of the material. We were interested in whether oxygen vacancies on the Cu-Ochains would also act as scattering centers and thus change the scattering rate of thequasiparticles in the system. Crystals with a range of oxygen concentrations have beenproduced and their surface resistance studied .In common with all of the copper oxide superconductors, YBa2Cu3O75 is a layeredmaterial with atomic layers stacked along the c-axis. This structure results in the c axistransport being very different from that in the ab plane. Apart from this anisotropy, there areCu-O chains along the b direction, causing the structure to be orthorhombic and leaving thetransport properties also different in the a and b directions. Friedmann et al.48 and R. Gagnonet al.49 reported that the normal state resistivity in the a direction is more than twice as largeas that in the b direction. Cohn et aL5° and Yu et at.51 found a quite remarkable anisotropyin the thermal conductivity in both the normal and superconducting state. However,microwave measurements performed on YBCO crystals before this study all obtained only7Chapter 1. Introductionaveraged values in the a and b directions. We were particularly interested in the role, if any,that the Cu-O chains play in the unusual temperature dependence of the surface resistance.The anisotropic microwave surface resistance of YBCO system in the ab plane has beeninvestigated and found to be quite large. However, the qualitative temperature dependenceof the surface impedance is the same in the a and b directions52.The mechanism giving rise to high temperature superconductivity is not yet known.We believe that our measurements of the microwave surface resistance, and thus the realpart of the conductivity and the quasiparticle scattering rate of pure and impurity dopedcrystals, provide important information for understanding the YBCO system. The study alsoreveals the anisotropic electromagnetic response of the YBCO material in thesuperconducting state. We hope that in the very near future, with the continuous efforts ofthe community, the nature of high temperature superconductivity wifi eventually be revealedto us.8Chapter 2Properties ofYBa2Cu3O74The high temperature superconductors generally refer to a family of superconductorscontaining copper oxides in the form of Cu02 planes. The most exciting feature of thesematerials is the high superconducting transition temperatures, for some of them well above77 K, the boiling temperature of liquid nitrogen under atmospheric conditions. Because ofthe relatively easy access to liquid nitrogen temperatures, the discovery of thesesuperconductors opens tremendous opportunities in the area of applications. Consequently,the materials have been intensively studied by a large number of groups world wide. The firstcopper oxide superconductor was found by Bednorz and Muller in 1986; it had the formulaLa2BaCuO4 and its T was greater than 30 K2. Fairly soon after, YBa2Cu3O7Bi2SrCaCuO54,Tl2BaCa..lCuO2+455,etc. became members of the family. Among themYBa2Cu3O73 is the most studied, probably because high quality samples are more easilyobtained. For microwave experiments, results are very sensitive to the sample quality, in partdue to the fact that the microwave field only penetrates a thin layer of the material so thatdefects in the surface layer can qualitatively change the features of the surface impedance.Consequently, YBa2Cu3O73 crystals, with their high crystalline perfection and smoothsurface, became the best choice for studying the intrinsic properties of the copper oxidesuperconductors.9Chapter 2. Properties ofYBa2Cu3O74In the following two sections we present the structure and the electromagneticproperties of YBa2Cu3O73.We wifi only discuss properties which are directly related to ourwork, since the purpose of this short chapter is to provide a necessary introduction to laterchapters.10Chapter 2. Properties ofYBa2Cu3O72.1 Structure ofYBa2Cu3O7The crystal structure of YBa2Cu3O7is orthorhombic for values of 6 less than 0.6. Theconventional unit cell of the crystal can be thought of as consisting of several layers stackedalong the c-axis. Sequentially, they are layers of: CuO, BaO, Cu02 Y, Cu02BaO, CuO. Infigure 2.1 we show such a unit cell for a value of 6 equal to zero. On the CuO basal plane,CuO forms Cu-O chains along the crystal’s b axis, and causes this material to have anorthorhombic symmetry. Ideally, if 6 = 0, all the oxygen sites are occupied, leaving perfectCu-O chains. An increase in 6 produces vacancies in the chain oxygen sites (labeled 0(1)). Itis well known that the properties of the system can be strongly affected by the oxygencontent. A reduction in hole density in the CuO plane, for example, is observed withincreasing 856. it is also found that the transition temperature is quite sensitive to the valueof 6. In figure 2.2 we show a plot of T versus x (x=l-S) taken from Schleger’s Ph.D.thesis57.The data were from various groups. The maximum T can be seen to occur when xis very close to one. For our samples we have found that the transition temperature changedfrom 93.5K to 89K in a small range of 6 (0.02’—0.15). When 6 is roughly 0.05, T is highestaround 93.5K. Crystals with a 6 of this value are often said to be optimally oxygen doped.Most of the crystals studied in this project have optimal oxygen doping.The chains cause the length of the unit cell along the b direction to be slightly longerthan that in the a direction. From x-ray and the electron diffraction they are 3.89A and11‘3 Ct p 00,.—z CDt (0c-JI-”—‘0(0tdU-<UCp‘-4’C03,-no-‘R)UUC/0nPChapter 2. Properties ofYBa2Cu3O73. 82A respectively. In the c direction, the unit length is 11 .68A, with the distance betweenthe two Cu02 planes being 3.36A, and that between the CuO and Cu02 planes being4. 16A58 During the oxygenation of the crystals, the CuO chains are formed as the samplepasses through a tetragonal to orthorhombic transition, and the chains can be oriented ineither of two perpendicular directions in the crystal. As a consequence, in certain regions theCuO chains are along one direction and in other regions the chains are along a perpendiculardirection. A region where all the CuO chains are in one direction is called a single domain,and the boundary between these two regions is called a twin boundary. The twin boundary isusually along a direction having an angle of 45° with respect to the crystal’s a or b axis, andacross the boundary the orientation of the CuO chains changes by 90°. An idealized pictureof the region at the twin boundary is shown in figure 2.3. Because of the mismatch of thelattice constants in the a and b directions, properties may slightly change in the boundaryregion comparing with that in a single domain. For the twinned crystals we have studied, thedomains have widths varying from a few 4urn to 100 urn.The charge state of the copper ion on the plane is Cu2(3d9),with one 3d orbit beinghalf filled. This configuration leads to a spin of 1/2 for each Cu2. it is now generallybelieved that zinc and nickel prefer to substitute for the copper on the Cu02 plane59 60,61 Itis interesting to notice that the Zn and Ni both carry a charge 2+ in the system. For thecopper sites on the basal plane, the substitutions usually have a valence 3+, as for Co, Al, orFe. For small concentrations of Zn2 or Ni2 in the Cu02 plane, the oxygen content, the13Chapter 2. Properties ofYBa2Cu3O731008060402000.0 0.2 0.4 0.6 0.8xinYBaCuO2 3 6+xFigure 2.2. The transition temperature of YBCO versus oxygen content. Data were collectedform Jorgensen et al.62’3, Cava et al.M, Brewer et al.65, and PouLsen et al. (fromSchleger57).1.014Chapter 2. Properties ofYBa2Cu3O7•o•o•o•o•o• •Cu(1)• •o•c•o•o• 00(1)oo\O • •ooo•oo0 0-tw;n boundarybFigure 2.3 The twin boundary caused by the change of orientation of CuO chains in the CuObasal layer.15Chapter 2. Properties ofYBa2Cu3O7chain structure and the orthorhombic phase remain unchanged; whereas a small amount ofCo, Al and Fe quickly causes an orthorhombic to tetragonal phase change67, as shown infigure 2.4.16Chapter 2. Properties ofYBa2Cu3O75CoI I€.0VCCu3. c=b•=aI I I I/ AlIZn4. .. — -I I INiI I Io 0.2 0.4 0.6 0.8 1x in YBaCu3MO7Figure 2.4. The variation of lattice constants a and b versus doping concentration afterTarascon et a167. For crystal structure with the tetragonal symmetry, a equals to b. One cansee that a small amount of Co, Al and Fe quickly causes an orthorhombic to tetragonal phasechange.17Chapter 2. Properties ofYBa2Cu3O782.2 Some Basic Features ofYBa2Cu3O7As expected from its structure, the system is very anisotropic. In the normal state, thec-axis resistivity is about 60 times as large as that in the ab plane. From magnetizationexperiments, the c-axis coherence length in the superconducting state is found to be about3A and the ab plane coherence length to be roughly 16A68. The magnetic field penetrationdepth 2 is also very anisotropic. The c direction penetration depth is found to be about1 1,000A69 and in the ab plane, Basov et al.7° report values of 1000A in the b direction and1600A in the a direction from far infrared spectroscopy measurements. Thus YBCO is a so-called type II superconductor, with the penetration depth much larger than the coherencelength . The coherence length is related to the upper critical fields H2 byH22r’where t0 is the flux quanta equaling to 2.07xlW15 Wb. The short coherence length means avery high upper critical field for the material. From Hall effect experiments71,72, 1/RH wasfound to decrease linearly in T in the normal state, which is rather anomalous. The ab planemean free path is estimated to be of order 104A at low temperature in the superconductingstate. When the current is in the ab plane, the penetration depth into the c direction (It iscalled the ab plane penetration depth because it is determined by the conductivity in the abplane.) is believed to be much larger than the c-axis mean free path. This ensures that thescattering process occurs in a range of almost constant field. As a result, the electrodynamic18Chapter 2. Properties ofYBa2Cu3O73properties can be described in the local limit, to be discussed in the following chapter.Although there are no substantial structural changes accompanying zinc or nickeldoping, the transport properties change drastically, reflecting the crucial importance of theCu02 plane to the superconductivity in the system. In figure 2.5 a plot of T versus thedoping concentrations for YBa2(Cui.M)3O6.95 from Markert et al.73 is displayed. Thecurves may vary somewhat from group to group, but it is evident that zinc dopingsuppresses T more than nickel does. In particular, 5% zinc (the percentage is compared withthe total copper content) results in a drop of the transition temperature from 93.4K to about45K. The NMR experiment on zinc and nickel doped YBCO show that 1% zinc doping intothe Cu02 plane causes a strong suppression of the normal state Cu nuclear relaxation ratenear zinc impurities, whereas 5% nickel doping does not have such an effect74. It wasargued that spinless zinc impurity acts as magnetic impurity and leads a local collapse ofantiferromagnetic spin correlation in the system, and results in a suppression in T.In figure 2.6 we compare the resistivity curves of pure crystals with that of zinc andnickel doped ones. Data in figure 2.6a were obtained by Baar et al. the crystals grownwithin the group, and figure 2.6b was copied from a paper published by T. R. Chien et al.76For a pure crystal, we see that the resistivity is remarkably linear in Tin the normal state, andextrapolates close to zero at zero temperature. This feature is another unusual property ofthe high temperature superconductors and is not yet understood. Some attempts have beenmade to explain the phenomenon. For examples, Moriya et al.77 and Monien et al.78 have19Chapter 2. Properties ofYBa2Cu3O73100• YBa2(Cuj_M)3O7Zn o200 5 10x (%)Figure 2.5. T versus doping concentration after Markert et al.73 Open diamonds are for zincdoped crystals and filled diamonds for nickel doped crystals.20Chapter 2. Properties ofYBa2Cu3O73obtained a quite linear resistivity for these materials using an antiferromagnetic-Fermi-liquidtheory combined with the NMR experimental data; nonetheless, the calculated curvesinevitably have a negative intercept at zero temperature. The value of the resistivity in the abplane at 120K is about 75 p.lcm. The surface resistance corresponding to this value is 0.32£2 at 35 GHz. For the doped crystals, the resistivity is found to be much larger in the normalstate, as indicated in figure 2.6.In figure 2.7 we show the normal state resistivity curves in the a and b directions byT. A. Friedmann et a148. From their experiment they concluded that the Pa/Pb anisotropy is2.2±0.2 between 150K to 275K. This ab plane anisotropy has been also found in the thermalconductivity. We show in the next figure the thermal conductivity in the two directions in areport by R. C. Yu et a15. The anisotropy persists all the way from the normal state into thesuperconducting state, with the thermal conductivity in the b direction being larger than thatin the a direction. In addition, they have also observed in both directions a broad peak in thethermal conductivity versus T in the superconducting state, which they interpreted as due tothe contribution of the electrons, rather than the phonons in the system.This concludes our basic introduction to the structure and relevant electromagneticbehavior of the YBCO material.21Chapter 2. Properties ofYBa2Cu3O730.150.10‘00C-0.050.0070.0 170.0Figure 2.6a. Resistivity versus temperature for the pure and 0.75% nickel doped samples inthe ab plane, taken by Baar et al.7590.0 110.0 130.0 150.0T(K)22Chapter 2. Properties ofYBa2Cu3O78— IYBa Cu Zn 02 3—x x 7—6 -x=Ox=O.O1O ..• ..‘- —- x=O.O2 •.- ‘Fci ——— x=O.091 ..•.‘0-x=O.108 .- —2 •..‘.“. ‘ — —0 100 200 300Temperature (K)Figure 2.6b Resistivity versus temperature for pure and zinc doped samples in the ab plane,after reference (76).23Chapter 2. Properties ofYBa2Cu3O712001 751501 251 ‘‘C-)37550250225 250 275Figure 2.7. The resistivity versus temperature in a, b, and c directions for a YBCO singlecrystal, after Friedmann et al. The resistivity in both a and b directions is quite linear.5432075 100 125 150 175 200T(K24Chapter 2. Properties ofYBa2Cu3O73T(K)Figure 2.8. Temperature dependence of the thermal conductivity in the a and b directions fora YBCO single crystal. Plot is from reference (51).30252010500 50 100 150 20025Chapter 3Electromagnetic Properties of Superconductors atMicrowave FrequenciesThe aim of this chapter is to provide an overview of the general electromagneticproperties of conventional superconductors at microwave frequencies. The chapter isdivided into three sections. In the first section we will introduce the surface impedance,which can be measured by microwave techniques, and the relationship between the surfaceimpedance, the complex conductivity, and the penetration depth. In the second section thephenomenological two fluid model will be discussed. The complex conductivity has a simpleform and interpretation within this model. In the fmal section we wifi review theelectromagnetic properties derived from weak coupling BCS theory.26Chapter 3. Electromagnetic Properties of Superconductors at Microwave Frequency3.1. Microwave Surface ImpedanceAn important measurable quantity describing the properties of a normal metal or asuperconductor in the presence of microwave radiation is the surface impedance Z, which isdefined as:Z=I---” 3.1.1LHI\ ‘Jz=Owhere E and H are the components of the electric and magnetic fields in the plane parallelto the surface, with z=O at the surface of the material to be considered, as shown in figure3.1. To take a perfect metal as an example, where the electric field can only be perpendicularto the surface, the surface impedance is obviously zero. We have assumed in (3.1.1) that thematerial is isotropic in the X-Y plane, as in this case the electric and magnetic fields in theplane are perpendicular to one another. If the material is anisotropic in the plane, one has todecompose the fields along the two principal axes and defme the anisotropic surfaceimpedance for each direction.For the purpose of studying high T superconductors, we need only discuss theelectrodynamics in the local limit. In this limit the field varies slowly in space on the scale ofthe mean free path 1 of a normal metal or the coherence length of a superconductor. In thislimit the current density at a point in the material is determined locally by the electric field atthat point, i.e.,j=oE, where cris the conductivity. At a microwave frequency co, the electricfield inside the material may be written in the exponential form, E = Exoe_03e, with27Chapter 3. Electromagnetic Properties of Superconductors at Microwave FrequencyFree Spo.ceMetoi.SFigure 3.1 The definition of the surface impedance.xz-.. Hyy28Chapter 3. Electromagnetic Properties of Superconductors at Microwave FrequencyE0 being the amplitude of the electric field at the surface and z the depth into the material.Using Maxwell’s equations and Ohm’s law, one obtains k = 4iu0coo, and the surfaceimpedance is (see appendix A):=R+iX. 3.1.2The real part of the surface impedance R8 (called the surface resistance), is essentiallythe sheet resistance of the surface layer into which the fields penetrate, and has the units ofOhms (Ohms per square). The imaginary part X, the surface reactance, is the reactance ofsuch a layer. In the case when the material is a normal metal, the thickness of the surfacelayer is the classical skin depth Si ; and in the case when the material is superconductive, it isthe penetration depth AThe surface resistance is proportional to the total electromagnetic loss inside thematerial. Let us consider the real part of the normal component of the complex Poyntingvector at the surface, which is equal to the power absorbed per unit square by the metal orthe superconductor:ReSt =Re! ExH =Re-ZH03.1.3—U 1,7.72_iD 72—‘ssO29Chapter 3. Electromagnetic Properties of Superconductors at Microwave Frequencywhere H0 is the amplitude of the magnetic field along y axis and J.o is the surface currentdensity. Due to the strong reflection from the metal, the total magnetic field at the surfaceH0 is about twice as large as the amplitude of the incident field. The square of Ho is thenproportional to the incident power. Therefore from equation (3.1.3) we have the ratio of thepower absorbed to the incident power to be proportional to R.In general, the magnetic penetration depth is defined as:3.1.4B (0)Using Maxwell’s equations and the definition of the surface impedance(3.l.1), one obtainsthe relationship between A. and Z (also see Appendix A):) z 3.1.5IloColThus the magnetic penetration depth is a complex quantity. For a normal metal unless thefrequency is very high, the conductivity is real, and the surface resistance is then equal to thesurface reactance. The magnetic penetration depth can be calculated:A=s—,s. =———i-1—=(1 ) / 1 , 3.1.6Po0 2jioxwhere 6 and 6,. are the reactive skin depth and resistive skin depth respectively. Theclassical skin depth öci is defmed as the depth in which the amplitude of the field drops30Chapter 3. Electromagnetic Properties of Superconductors at Microwave Frequencyto lie of its original value. It can be easily found that is actually 2ö or 23r.If the metal is superconductive, one usually has the relation X. >> R, and the magneticpenetration depth A. becomes the reactive skin depth & In this local limit the reactive skindepth 5 is called the London penetration depth. It is independent of the frequency and isthus also the d.c. magnetic penetration depth, A.L= (o) = 0).In general, the conductivity o is complex, 0= 0i — io. The surface resistance can becalculated from equation (3.1.2) in terms of the real part 01 and the imaginary part o of it:R =/0+—2) 3.1.72(+o)If o >> 01, which is true for a superconductor at relatively low frequencies and attemperature not too close to T, one can simplify equation (3.1.7) and obtain:R =______3.1.8—__________2where we have used equation (3.1.6) to write 01 in terms of A. The surface resistance isproportional to the real part of the conductivity for a superconductor.31Chapter 3. Electromagnetic Properties of Superconductors at Microwave Frequency3.2. The Two Fluid ModelThe two fluid model was proposed by Gorter and Casimir2 as a tool to describe someof the properties of superconductors. When combined with a London equation , it was usedto explain the electrodynamic properties, before the more sophisticated microscopic BCStheory was developed. It is stifi very useful now, since one can borrow some simple conceptsfrom the model to help in interpreting the BCS theory, particularly when working in the areaof applied science. In the two fluid model picture, it is considered that there are two types ofelectrons in the superconductor, the normal electrons (the normal fluid), and thesuperconducting electrons (the superfluid). In a given field, the normal electrons are assumedto behave like electrons in the normal state and therefore dissipate energy by scattering.Their motion is described by the usual transport equation:m4+=Ne2E 3.2.1dt rwhere j=Nev , with f being the normal current density, N the density, v,, the driftvelocity of the normal electrons, and ‘r the relaxation time. The superconducting electrons donot dissipate energy as they move; however, they do show inertial effects due to their mass.The equation describing their motion is:m.4!= N e2E 3.2.2dt32Chapter 3. Electromagnetic Properties of Superconductors at Microwave Frequencywherej3 is the supercurrent density and N is the superconducting electron density. The totalelectron density is then N-i-N, and the total current density is the sum ofj +j.A schematic circuit may be used to represent the idea of this model as shown in figure3.2. When there is only d.c. current, the superconducting electrons carry all the current andthere is no loss. When there is an ac current, a voltage builds up across the inductance Lbecause of the inertia of the superconducting electrons. This voltage causes normal electronsto flow through the resistance R and causes loss.Assuming an applied electric field varies with time at a frequency 0) as: E=E0 e’ °,both real and imaginary parts of the conductivity can be derived from equation 3.2.1 and3.2.2,Ne2t0 = 3.2.3Im(l+w22)andN 2 N 2yr+‘. 3.2.4mU) m(1+U)2r2)The normal electrons are responsible for the real part of the conductivity. If the frequency isnot too high, such that on<<1, one has cy2>>o. The surface reactance can be calculated interms of the conductivities from equation (3.1.2). Using equation (3.1.6) and (3.2.4), themagnetic penetration depth becomes A = X/ = (mlpoNse2) , which is just the London33Chapter 3. Electromagnetic Properties of Superconductors at Microwave FrequencyFigure 3.2 A circuit representing the two-fluid model.L R34Chapter 3. Electromagnetic Properties of Superconductors at Microwave Frequencypenetration depth L’ originally derived from London equation. At absolute zero, it isconsidered that all the carriers are superconducting electrons. We therefore have thesuperfluid fraction of the total carriers asN3(T) 2(O)___= 3.2.5N0 2(T)The penetration depth is an important quantity because it is a measurable quantity thatis related to the density of the superconducting electrons. Microwave teclmiques provide adirect measure of the penetration depth, and thus the superfluid density.In the model, the electric field at the surface and inside the superconductor is mainlydetermined by the density of superconducting electrons. Because X> >R5, the electric fieldat the surface isE0=ZH iXHo, with I Ho 2 proportional to the incident power. Inside thematerial, the amplitude of the field varies with depth as IEI= IEoIe. The loss per unitsquare is:= EoI2edz = 2iXS2IHI.3.2.6— 2 2gii0ox1 j322(4J)3 235Chapter 3. Electromagnetic Properties of Superconductors at Microwave FrequencyHere we have expressed H0 in terms of J and X in terms of A.. Comparing this result withequation (3.1.3), we found the same expressions for the surface resistance as equation(3.1.8).The surface resistance is also an important quantity, since combined with thepenetration depth, one is able to get the real part of conductivity o, which is proportional tothe electromagnetic absorption inside the superconductor in a given field. The absorptioncan be due to the relaxation process such as inelastic or elastic scattering, or due to thedirect excitations from superconducting electrons to normal electrons. Therefore the realpart of the conductivity provides information on the scattering process.36Chapter 3. Electromagnetic Properties of Superconductors at Microwave Frequency3.3 The Electromagnetic Properties of a BCS Superconductor.Following most of the literature, we will use c.g.s. units in this section. We will startwith the perturbation Hamiltonian describing the applied field, and then describe thederivation of the penetration depth and microwave absorption in the transition region.3.3.1 The Perturbation HamiltomanIn the BCS theory the external microwave field can be treated as a perturbation, whichderives from the change in the canonical momentum in the presence of a field, and the BCSpotential term of the electron-phonon interaction remains unchanged. The field can beexpressed by a vector potential A(r), which is supposed to be the total real field seen by theelectrons(i.e., A(r) is not merely the external field). The kinetic energy term of theHamiltonian is then (p-eAJc)2/2m. If the field is weak, the non linear term containing A2 canbe ignored. If the system is clean, the impurity scattering is weak, and its contribution to theperturbation can be ignored. The perturbation term for the total system can then bewritten:792inc=___JdJf*(rj1(r)Jdq a(q)e .(2k+q) 3.3.12mce *, a(q).(2k+q)ck÷qck,2mc k,q,j37Chapter 3. Electromagnetic Properties of Superconductors at Microwave Frequencywhere we have used the Fourier transform of the vector potentiala(q) =1JdrA(r)e”,and the free electron field operatorsr(r)= cke.In the BCS scenario, the excitations are described by quasiparticles rather than electrons.The relation between the quasiparticle and electron operators are as follows80:Ckt _Uk’YkO+2kYkl 332Ck = Vk7ko + UkYkl.When one substitutes (3.3.2) into (3.3.1), one gets terms of the form or ,corresponding to creating or annihilating a pair of quasiparticles, and terms of the form ‘/y,corresponding to quasiparticle scattering from one state to another. For a fmite gap, as longas hco of the microwave fieldis less than 2A, the processes of creation and annihilation maybe ignored, and we will only be interested in the scattering term (The issue is morecomplicated in the case of a d-wave ground state where the gap goes to zero at certain lineson the Fermi surface.In terms of the quasiparticle operators, the perturbation Hamiltonian (3.3.1) can thenbe written as (omitting pair annihilation and creation terms) 81:38Chapter 3. Electromagnetic Properties of Superconductors at Microwave Frequencyeh * *H =— , a(q).(2k+q)( C_k,_jC_k_q,_o.)2mc k,q,o 333eh * *= — a(q )• (2k + q )(Uk+qUk + Vk+qVk )(Yk+q,cr2’k,ci —2mc k,q,oThe factor in front of the quasiparticle operators is called the coherence factor, and here it isUkUk+q+ VkVk+q for the particular Hamiltonian under consideration. In general, for aninteraction Hamiltonian of the form ‘=Bk(,k(Ck,(,. c ±C7 ck,), the positive signcorresponds to what BCS called case I, and a coherence factor of the form UkUk+qVkVk+qappears once it is written in terms of the quasiparticle operators. This type of interaction isappropriate to describe processes such as ultrasonic attenuation. The negative sign (case II)applies for interaction of the electron with the electromagnetic field, as for the microwaveabsorption and for NMR relaxation, and a coherence factor of the form UkUk+q+ VkVk+qappears. The different coherence factors result in very different behaviors near T forultrasonic attenuation, as opposed to that expected for microwave absorption and nuclearrelaxation. A correct description of the different coherence effects was one of the biggestsuccesses of the BCS theory.At microwave frequencies, the wavelength is much larger than both the coherencelength and mean free path, and the vector potential A may be considered uniform in space,i.e., we make the approximation q=O. The Haniiltonian (3.3.3) can then be simplified. Theusual quantum mechanical formula for the current density operator can be calculated interms of electron operators39Chapter 3. Electromagnetic Properties of Superconductors at Microwave Frequencye . eA eAj(r)=—{Vi (p——)ip—[(p+—)ir ]‘y)2m c ce * e=—{iip’-(piy )iy}-——2A1iiVf 3.3.42m 2mceh * e *= ckGcklY(2k) ——Ai.I 1.2mV k mcBecause A varies slowly in space, the second term of the equation can be treated as theproduct of a constant and the number density operator. Therefore the expectation value ofthe second term is -ne2A!mc, where n is the density of electrons. This term is sometimescalled the diamagnetic term since it tries to cancel the applied magnetic field. Theexpectation value of the first term can also be calculated and is called the paramagneticterm since it opposes the diamagnetic current(If we ignore this term, equation 3.3.4 is inanalog with equation 3.2.1 for a normal metal, with the scattering rate equaling zero.). Thediamagnetic term is dominant in the superconducting state. However, the paramagnetic termis associated with the fraction of excited quasiparticles, and its temperature dependenceprovides important information on the low energy density of states, and thus the pairingstate.3.3.2 The penetration DepthAs long as the relation between the current density j and the potential vector A islinear: j = -A/cA, the magnetic field would vary exponentially with distance inside the40Chapter 3. Electromagnetic Properties of Superconductors at Microwave Frequency1 A ico 4,rojc2isuperconductor. Because E = ——— = ——A, o= 2 (equation 3.1.2), andcat c zZ2 = —(4,r&)2(equation 3.1.5), one can obtain:j=oEC 3.3.5C A.4,2This relation can be considered as a defmition of the penetration depth in a special case, validin the local limit, of the more general defmition of (3.1.4), where the variation of the fielddoes not have to be of the exponential form.Assuming we work with a single microwave frequency and that the wave vector q istaken to be 0, the vector potential A becomes its q=0 Fourier component, a(0). Theparamagnetic term in the current response involves the electron operators which should bewritten in terms of quasiparticle operators. Its expectation value was calculated fromequation (3.3.4). The total current density is 81j(q =0)= 2eh [a.k]k(—-.-)—-—a, 3.3.6mc k c9Ek mcwhere is the Fermi function and Ek is the k dependent quasiparticle energy, Ek=(e.+z12)112with =(hk)2I2m-(h F)2m measured from the Fermi surface. The sum is over all thequasiparticle states above the Fermi surface. One can express the penetration depth using41Chapter 3. Electromagnetic Properties of Superconductors at Microwave Frequency(3.3.5) and (3.3.6) as= l_2(O)2eh 3.3.72(T) mc2 k dEkwhere A2(O) —mc2/4nne is the penetration depth at T=O, and a is the angle between k andthe external field. The valueA2(O)/A7) is defmed as the superfluid fraction, first defmed inthe two fluid model (equation 3.2.5).Equation (3.3.7) can be evaluated for different gap structures. If the gap function isisotropic, the sum becomes$ dk 14 cos2a(——)a = !!_afdEN(E)3.3.8= .LafdEN(E)(_.L)where one assumes that k is not far from kF . N(E) is the density of states of thequasiparticles, and N(e) is the density of states of electrons at the Fermi surface in the normalstate. If the gap function is not isotropic, the quasiparticle energy Ek will be dependent onthe angle, and the transformation of the integrating parameter from fdkcos2a(-df/dE) tofdEN(Ex-af,’aE) involves angular integration. In principle, the superfluid fraction can bewritten in both cases as2(O)l—cfdEN(EX—--). 3.3.9A2(T) dE42Chapter 3. Electromagnetic Properties of Superconductors at Microwave FrequencyTherefore the temperature dependence of the penetration depth is dependent only on thequasiparticle density of states. A relevant case for high T superconductors is a gap functionwith line nodes on the Fermi surface, where it is well known that the density of states islinear with energy for the low excitations. At low temperatures, the largest contribution tothe integral comes from the low energy excitations because of the factor (-df/dE), and thedensity of states in the integral may be approximated by a linear term c’E. The fractionalnormal fluid density is then proportional to T Jxdr, where x=E/T. Therefore the normalfluid fraction is linear in T at low temperature. In figure 3.3 we display the penetration depthdata on YBCO crystals obtained by our group11. The graph was plotted asversus T, i.e., the superfluid fraction x versus temperature. From the graph it can be foundthat the normal fluid fraction (x = l-x) is linear in T, suggesting that there are line nodes inthe gap.3.3.3 The AbsorptionOne of the great successes of the BCS theory was an explanation of the coherencepeak, which is predicted by the perturbation Hamiltonian of case II as in (3.3.3). TheHamiltonian isH’= —----. a(q). (2k + q)(uk+qu + Vk+qVk )(Y+q,GYk— Y-k,-r7-k-q,-cr)mck,q,, 3.3.10= Mk,q (Uk+qUk + Vk+qVk )(Y+q,rYk,cr — Y—k,—crY—k—q,_7)k,q,cr43Chapter 3. Electromagnetic Properties of Superconductors at Microwave FrequencyFigure 3.3 The ratio (O)2/A.(7),i.e., the superfluid fraction, Xy versus T for YBCO crystals(open squares). At low temperatures the curve is approximately linear, suggesting that thereare line nodes in the gap function. The solid line is from a calculation using the s-wave BCStheory.1.00.8F—C40C’40.20.00 20 40 60 80Temperature (K)10044Chapter 3. Electromagnetic Properties of Superconductors at Microwave FrequencyAt microwave frequencies, q is taken to be zero. The coherence factor becomesu2+v=l.For the scattering from states with energy E to states with energy E+hü, one can use theGolden rule to estimate the net transition rate, W, which is proportional toW = fiMI2N(E)N(E + ho.))[f(E)— f(E + ho)]dE. 3.3.11In the normal state, the gap is zero, and the integral reduces to hcoIMI2N(O). Thereforearound the transition region the ratio of absorption just below T to that above T is:j N(E)(df)dE 3.3.12N(O) e9Ewhere o is the conductivity in the normal state. For an isotropic s-wave gap, the density ofstates is highly peaked at the gap edge. Since fN(E)dE= !N()d, one should have1N32(E)dE> >fNfl2()d8. This causes the real part of the conductivity in the superconductingstate to be larger than that in the normal state. Thus an absorption peak (coherence peak) ispredicted just below T. In real superconductors, the coherence peak in s-wavesuperconductors has been observed experimentally just below T, and at lower temperatures0z decreases exponentially. However, the observed peak is usually much smaller than thatpredicted, which has been argued to be due to anisotropy in the gap function and othermechanisms81. An example of an experimentally observed peak82 for a classicalsuperconductor Nb is presented in figure 3.4. For a d-wave superconductor, due to the45Chapter 3. Electromagnetic Properties of Superconductors at Microwave Frequencyhighly anisotropic gap, the density of states is not sharply peaked, and in addition there arealternations in the sign of the gap function. As a result a coherence peak is not expected.A more complete treatment of the absorption process in the presence of impurities wasfirst given by Mattis and Bardeen,5 as well as by Abrikosov et at.83 In the work of Mattisand Bardeen the scattering was treated in zero order of the Hamiltonian, and the externalfield as a perturbation. By a plausible argument, the existence of the scattering centers wasseen as to merely bring into the current expression a range factor which exponentiallydecreases with the mean free path. The complex conductivity was then evaluated for type Isuperconductors, for which the penetration depth was shorter than the coherence length.Recently, Hirschfeld et at. studied the electrodynamic absorption for unconventionalsuperconductors84,defmed as those for which the symmetry of the superconducting orderparameter is lower than that of the crystal lattice, and have derived a remarkably simpleexpression for in the London limit (q— 0)85[-)s:dojL)N(co) Im[-3.3.13where Q is the microwave frequency, Co is the quasiparticle frequency, and N(w) is thedensity of states normalized to that at the Fermi surface in the normal state. This expressionpredicts a power law behavior of the absorption at low temperatures rather than anexponential behavior for an isotropic s-wave superconductor.46Chapter 3. Electromagnetic Properties of Superconductors at Microwave Frequency12Figure 3.4. The microwave absorption of a Nb sample in the superconducting state. Theabsorption peak right below T is called the coherence peak, after reference (82).0 0.2 0.4 0.6 0.8 1.0T/T —47Chapter 4The Microwave CircuitThe following chapter describes the basic features of resonant circuits, and the cavityperturbation techniques used in surface impedance measurements. In section 4.1, the basicRLC equivalent circuit used to represent the cavity resonator is introduced, and the qualityfactor of the resonator and the coupling of an external circuit to such a resonant circuit isdiscussed. In section 4.2 small perturbations to a cylindrical cavity resonator operated in theTE011 mode, which forms the basis of our microwave measurement techniques, will bediscussed.48Chapter 4. The Microwave Circuit4.1 The Resonant Circuit86In our experiments the cavity is designed such that only a single mode is excited by theexternal microwave source. For this situation, the microwave circuit can be described by anequivalent low frequency RLC resonant circuit, either in the form of a series or parallel RLCcircuit. For convenience, we will only discuss the series case, shown in figure 4.1. Theproperties of this circuit are completely described by three circuit elements, R, L and C.The usual definition for the quality factor of a resonant circuit isQ=2,r.!- 4.1.1WLwhere W is the energy stored in the circuit, WL is the energy loss per cycle.For a cavity resonator, the quality factor becomesQ..... 111v112(TfrnI , 4.1.2R3H(r)dswhere c is the resonant frequency, H is the magnetic field, R is the surface resistance of thecavity wall, and dV and dS are volume and area elements, respectively.In the equivalent low frequency circuit shown in figure 4.1, the energy stored is thesum of stored magnetic WH and electric energy WE. At resonance,(WE)—(WH) ——LI04wC’4.1.349Chapter 4. The Microwave CircuitFigure 4.1 The equivalent series RLC circuit.LC R50Chapter 4. The Microwave Circuitwhere Jo is the maximum amplitude of the current, and the brackets refer to the valueaveraged over time. The quality factor Q of the cavity is expressed by the parameters of itsequivalent low frequency circuit as!LJ2 4.1.40 01 R R20where XL is the reactance of inductor in circuit. Equation 4.1.4 can be regarded as anotherdefmition of the quality factor.In order to perform measurements, the resonant circuit has to be coupled to anexternal circuit. A typical circuit representing the transmission configuration is shown infigure 4.2. Here the couplings between the resonant circuit and the input and output circuitsare represented by the mutual inductances M1 and M2, respectively. In general, the couplingitself would contain both reactive and resistive parts, shown as L1, L2, R1 and R2 in thefigure. However, it is usually a good approximation to neglect the resistive contributions,which simplifies the analysis of the circuit. The input and output circuits are seen by theresonant circuit through the couplings. The effective series impedances added to theresonant circuit by coupling to the input and output circuits are ok)2Ml/(Zl+icLl) and2M/(Z+iüLz) respectively, with Z1 and Z2 being the loads seen at position aa for theinput circuit and position bb for the output circuit. If one has matched loads, Z1 and Z2would just be the characteristic impedance of the corresponding transmission lines.51Chapter 4. The Microwave CircuitRG-LC RRLFigure 4.2 The coupled circuit. The internal resonant circuit sees the impedance of externalcircuits through the mutual inductances, M1 and M2.a52Chapter 4. The Microwave CircuitAccording to (4.1.4), the quality factor of this coupled circuit is the total XL divided bythe total resistive part:O.)0L-f31RXZ-f32IQT— R(1+13+$2)415—Qo1+13Here 13=131+132 , and 131 and 132 are the coupling constant between the resonant circuit and theinput and output circuits, respectively, and are defined as:13 coMZR(1+X/Z2)416132 Z2R(l+X /Z)For the second step in 4.1.5 we have made the approximation that the second and the thirdterms in the numerator could be ignored, which is usually true since aL is much larger thanR and /3 is smaller than 1. Equation (4.1.5) can also be written as1• 4.1.5’QT Q0 Q0 Q0 QEXTAs shown in the defmition of the quality factor, the total loss of the internal resonant circuitis proportional to l/Qo. Therefore l/QEXT represents the loss due to the coupling with theexternal circuits, and is found to be independent of the resistance of the internal resonantcircuit, R.53Chapter 4. The Microwave CircuitReturning to the microwave resonator, if a small sample is added to the cavity, theperturbation to the field pattern is small enough that the total energy stored in the cavitystays almost the same; however, there will be a change in the internal reactance andresistance of the equivalent circuit due to the presence of the sample. As a consequence,there will be a change in the resonant frequency and quality factor of the internal resonantcircuit. The change of the quality factor is directly related to the loss due to the sample.Because QEXT is independent of R, and provided that the mutual inductance is keptunchanged, QEXT would remain unchanged after the introduction of the sample. Thus thechange in the loss of the whole circuit is the same as that of the internal resonant circuit,LI (l/QT) = LI (1/Qo).The voltage reflection and transmission coefficient F and T, which are defmed as theratio of the reflected voltage at aa and bb to the input voltage respectively, can be found forsuch a circuit. Supposing the characteristic impedance of the input and output transmissionline is Zo, the impedance seen from the input side into the network is:Co2/31M- R + j(oL — L) + Co2$MO)C Z04.1.7= /3ZORR+j(coL_L)+/3RcoCHere L1, L2, R1 and R2 have been ignored. The voltage reflection coefficient at aa by thenetwork is:54Chapter 4. The Microwave Circuit4.1.8z+zowith the relations between the incoming voltage reflected voltage Vre and the totalvoltage Va at position aa beingv_vz+zo v_vz_zoin — a 2Z ‘ re — a 2ZIf f3 is zero and 13i is one, Z=Zo at the resonant frequency and F is zero at aa, or no voltageis reflected from the network. The resonant circuit is then said to be critically coupled.The voltage transmission coefficient is,T =2aM1, 4.1.9n (Z+Zo)[R+jO)L+l/jWC+C02(M121Z0 +M iZ0)]where V is the voltage traveling out of the network at position bb. It can be seen again thatthe effect of the coupling is to add resistive terms in the form th2M,Zto the resonant circuit.The total resistive part of the circuit is then RT, the sum of the internal resistance and theadditional terms caused by the external circuits.In our measurement situation, the coupling between the input wave guide and thecavity mode is quite small compared to one. Z is much less than Z0 and the networkresembles a short circuit termination for the input wave guide. The factor 1/(Z+Z0)in 4.1.9can be written as liZ0. Using equation 4.1.3 and 4.1.9, one has the maximum transmission55Chapter 4, The Microwave Circuitcoefficient at the resonant frequency ci:T(o. ) = 2413113201+131+132 4.1.10—‘,Iaa QT— ‘VP1P20Therefore the voltage transmission coefficient at the resonant frequency is dependent on thecoupling coefficients as 4/31132’From an engineering point of view, the circuit between aa and bb is a two-portnetwork, and can be completely described by a scattering matrix S. In appendix B theelements of S are defined. It was found that 511=522 =F andS12=S21T. Any otherelectrical quantities associated with the network can be derived through S.The power transmission I T2 as a function of the frequency is given by:1T12 -____________________—(i+/3+$)2+Q_-]4.1.11= [0)0))COO (1)In figure 4.3 we plot the power transmission versus frequencyf for such a circuit. The widthof the peak at the half maximum power, Af, can be seen through equation 4.1.11 to be56Chapter 4. The Microwave CircuitA f = . Therefore the quality factor2ir QTQT= =--. 4.1.122irAf AwIn practice, the perturbation due to the sample can sometimes be large enough to alterthe mutual inductance, M1 and M2. For example, the field pattern close to a coupling loop inthe cavity may be slightly distorted because of the introduction of the sample. Then theexternal loss l/QExT is no longer constant and the relation (1 ‘QT)= A(1 /Q0) would not bestrictly valid. Under these circumstances one can still get accurate results by simply makingthe coupling constants sufficiently small. As one can see from equation 4.1.5’, for smallcoupling constants i/QExT approaches zero and Qr approaches Qo. Another method one canuse is to extrapolate the quality factor of the internal resonant circuit by varying the couplingstrength. The voltage transmission coefficient at resonance isT=2wM1M4113[RZ0 + w( M + M )]Introducing a parameter c = (M12+M)/2M,one has the following relation between thequality factors and the voltage transmission coefficient:QT 1Qo 1+J3= ‘O 4.1.14RZ0+o(M+ M)=1— cT.57Chapter 4. The Microwave Circuit1.00.80.61- 4.3 The power transmission through a resonant circuit. The full width at half peakpower is equal to oi2rQ.34793.02 34793.04 34793.0658Chapter 4. The Microwave CircuitIf the coupling is made symmetric, the parameter c becomes 1. The quality factor QT of thecoupled circuit is linear with the transmission coefficient, T. In the experiments one canmeasure the output voltage V0 at the resonant frequency and QT. A plot of QT versus Vshould be a straight line, and the extrapolated V0= 0 intercept should be the quality factorof the internal resonant circuit, Qo. Practically, M1 and M2 will not be exactly equal;however, this is not a requirement to obtain Qo from extrapolation. As long as the ratio ofM1 to M2 is kept reasonably constant, c would be constant, and Qo can stifi be found fromthis method.59Chapter 4. The Microwave Circuit4.2 Cavity Perturbation MethodIn chapter 3 when the electromagnetic properties of a metal were discussed, therelative permeability r was set to be 1, i.e., the material is assumed to have free carriers, butintrinsic magnetism is negligible. In such a treatment, one has:B=u0(H+M)=p# 421n x(H0 — ) = iswhere Js is the surface current, the subscripts ‘out’ and ‘in’ stand for quantities outside orinside the metal, and the n is the normal unit vector of the metal surface. The magnetizationM (11r1)H = 0, and the magnetic field H inside the metal is the sum of the external field(here it means the field when the sample is not present) and the field induced by theconducting currents at the surface. As a result, inside the metal H = 0 and the magneticinduction B = =0. The surface current is determined by FI0 right at the surface. Indiscussing the cavity perturbation method, one may take a different approach toward theproblem. For a perfect conductor(skin depth=O), the relative permeability 4u can be set tozero (perfect diamagnet). In this case the set of equations 4.2.1 becomesB=110(H+M)=IlrIlO422nx(M0—M)=j5 . .The magnetization inside (caused by the conducting current) M=(4u—1 )H= —H, and H is themagnetic field inside the conductor, which is proportional to the external field (again it is the60Chapter 4. The Microwave Circuitfield before the sample is present) by some demagnetizing factor. However, the magneticinduction is still zero, B=u0(H+M)=4u#0H=O. The surface current is determined bywhich is equal to —H0 at the surface. In this approach one can apply the general results ofthe cavity perturbation techniques to a metal by putting p,=O.In our experiment a cylindrical resonant cavity operated at 34.8 GHz is used tomeasure the sample loss. We use the TE011 mode, where the electric field everywhere in thecavity is perpendicular to the axis of the cylinder. The fields for this mode, in polarcoordinates, are:E,. =0E9 = AJ0’(kr)sin=0H =---AJ’(kr)cos 4.2.3T kh hH8 =0H =AJ(kr)sin-and,k2=O0p_.- ., 4.2.4where A (amperes per meter) is a normalizing factor, which is imaginary, b is the radius, h isthe height of the cavity, Jm is the mth Bessel function, and is a zero for Jo’(kr) and is equal61Chapter 4. The Microwave Circuitto 3.832 for this mode. Thus for the TE011 mode, the field is independent of q’, Jo’(kr) (i.e.,IJ) reaches its first zero when kr=, and H has one maximum.In figure 4.4 the field in the cavity and the current patterns in the cavity walls aredisplayed. The total energy stored in the cavity is calculated from equations(4.2.3)87:w = — ft40320V 4.2.52.(l—--)4h2where v0 is the volume of the cavity and A is the wavelength in free space corresponding tothe resonant frequency.If the surface resistance (defmed in chapter 3) of the cavity walls, R, is known, thequality factor of the cavity can be easily calculated through equations 4.2.3 and 4.2.5(Appendix C.):=. 4.2.6ir( )R8As an example, for R= 30 u.Q, the quality factor of the mode is about 5x107.One approximation we are making for the cavity perturbation is that the fields are onlyslightly modified after introduction of the sample. Far from the sample, the fields are thesame as those in the unperturbed case. Since the sample is very small compared to thevolume of the cavity, the total energy stored is approximately the same as that of the62Chapter 4. The Microwave Circuit.xFigure 4.4 Field and current patterns inside the cavity(a, b), and on the cavity walls(c)..Iz(cx) (to)44444444I I I I I I III I I I I II I I I I I I(c)Electric fieldMagnetic fieldCirrent63Chapter 4. The Microwave Circuitunperturbed case. The quality factor of the cavity with sample, Qo’, is then 27CW/(WL+Ws),where Ws is the loss per cycle due to the sample. By comparing with equation (4.1.1), wecan write:S 4272,rW Q0’ Q0 Q0If the sample to be measured is put in a fairly uniform region of field in the cavity, thecalculation of the loss per cycle W is dramatically simplified. (It would be a reasonableapproximation if the sample is small compared to the volume of the cavity and is not near anode in the field of interest.) In such an approximation W and W5 can be calculated andcompared to zl(l/Qo) measured from an experiment.To fmd the loss due to the sample, the magnetic field at the sample surface has to beknown. Because of the strong reflection, the magnetic fields at the surface of a sample arevery close to that of a perfect conductor, although the parallel electric fields will be differentfrom zero. Therefore, we can treat the sample as a perfect conductor as far as the magneticfield at the sample surface is concerned. If a sample is placed in a uniform external magneticfield H0, the final magnetic field H at the surface isH=H0, 4.2.81+(u—1)Nwhere N is the demagnetizing factor, and p is taken to be zero in our case. The64Chapter 4. The Microwave Circuitdemagnetizing factor is a pure number, which is dependent on the shape of the sample, notthe nature of the material. Values of N for samples of various shapes can be found in theliterature88.The conducting current at the surface is calculated from the boundary condition givenin equation (4.2.2). One can get the amplitude of the surface current density, Jo =H, whereH is the parallel component of H obtained from equation (4.2.8) at the sample surface. Theloss per cycle W. due to the sample with a surface resistance R0 is obviouslyw,, =1HR30dS. 4.2.9wherefo is the resonant frequency.In the following we are going to consider two relevant cases. In the first case, a samplein the shape of a thin plate is put at the center of the cavity such that the principal axis of thecylindrical cavity is parallel to the major surface of this sample, as shown in figure 4.5a. Inthe second case a sample in the shape of an effipsoid is considered, with a dimension c<<aand c<<b, and this sample is put at the center of the cavity such that the principal axis of thecylindrical cavity is parallel to c axis of the ellipsoid, as shown in figure 4.5b.a)Sample is a thin plateIn this situation, the demagnetizing factor is 0, and the magnetic field H at the surfaceis just the external field H0. We may take H0 to be the field at the center of the cavity, which65Chapter 4. The Microwave Circuitis A from equation (4.2.3). The loss per cycle is thenSA2RW = 4.2.10fowhere S is the area of one major surface of the sample. The relation between the surfaceresistance and the zl(1/Qo ) can be written (see equation 4.2.5 and 4.2.7.):12A’ W = 4c R. 4.2.11Q0 2iW 0.4032f.LLVFor our 34.8 GHz cavity, we obtainA!_=0.32.10SR,Qwhere S is in mm2 and R is in £2.b)Sample is an ellipsoidIn this situation the surface of the sample satisfies the equationa2 b2 c2One may make the further approximation that a=b. Using equation 4.2.9, the loss per cyclecan then be writtenWs =2nHR ? x I dz 4.2.12= SJ1+(—)dz66Chapter 4. The Microwave CircuitH(a) (b)Figure 4.5 Samples in the form of a plate and an ellipsoid in magnetic fields.zyzHx x67Chapter 4. The Microwave Circuitwhere Idddxl=cx/[a2( -x2/a)”], and H is the field in the center of the cavity determined byequation (4.2.3) and (4.2.8). The demagnetizing factor of such an effipsoid placed this way inthe field is 1- crJ2a, with the condition a>>c. Thus one has H=2aA!c27 The integration(4.2.12) can be calculated numerically for various sizes of samples. For example, for aneffipsoid having dimensions a = 0.5 mm and c = 0.02 mm, we obtain Ws=4.8x1017(m2s)R5ATherefore we can obtain the relation between the surface resistance and A (1/Q) for thisparticular sample= =0.56. 104R. 4.2.13Q 2,rWFrom the above calculations we fmd that if S is 1 mm2 and R5 is around 100 4uQ,A (i/Qo) is around 0.5x108 For a quality factor Qo about 4x106, introduction of such asample into the cavity corresponds to a change of 2% in Q. As we can see, the sensitivity ofthe apparatus depends on having a high, and also stable, quality factor.In both cases of a thin plate and an effipsoid, one has the relation A (1/Qo) = yI?, whereyis called the geometrical factor. It is dependent on the shape and the size of the sample andalso the position of the sample in the cavity, since the field strengh depends on position.Once yis determined, the surface resistance can be calculated straightforwardly.In practice, the geometrical factor is not usually calculated because of the irregularshape of the samples. Instead, it is determined by a calibration method to be discussed in thefollowing chapter.68Chapter 5Experimental PreparationThis chapter deals with experimental details. In the first section the samples we haveinvestigated and their characterization are briefly reviewed. In the second section the designsof the cryostat, the resonant cavity, the sample probe and some technical details aredescribed. In the following section we discuss our methods for determining the quality factorof the resonant circuit, and the calibration constant for different measuring geometries. In thefmal section, we discuss the signal generation and detection, and give a measurement circuitdiagram. We also record basic features of the instruments we have used in this section.69ChapterS. Experimental Preparation5.1 SamplesThe samples that have been investigated are all very high quality YBa2Cu3O singlecrystals, with and without zinc and nickel doping. Most of the crystals had x set at 6.95, butwe have also studied some of the nominally pure crystals with various oxygen contents. Allsuch samples fall under the general label of YBCO single crystals. These crystals have beengrown by a CuO-BaO flux method using yttria stabilized zirconia (YSZ) crucibles42 ]i isbelieved from chemical analysis that YSZ is one of the best crucible materials for YBCOcrystal growth. Pure YBCO samples have been characterized by magnetization, dcresistivity surface impedance and specific heat measurements. The transition temperatureobtained from magnetization measurement for the crystals with oxygen content 6.95 isaround 93.2K and from specific heat measurement is around 93.4K. The bulk transitionwidth is narrower than 0.25K from specific heat measurement, as shown in figure 5.142. Thed.c. resistivity above the transition temperature is about 70 in the ab plane and afactor of sixty larger along the c-axis. For zinc and nickel doped crystals, the transitiontemperatures are 89.5K for 0.31% zinc, 91.5 for 0.15% zinc and 90.8 for 0.75% nickeldoped, respectively. Here the percentages are relative to the total amount of copper in thecrystals. The transition temperatures of pure crystals with oxygen concentrations from 6.80to 6.98 are all around 92K. The typical size of samples in this study is about lxi mm2 in thea and b directions of the crystal axes and 10 pm to 50 pm in thickness (c direction). Thecrystals are black with a mirror like surface. In figure 5.2 we show a picture of one of thetwinned crystals taken with a polarizing microscope. It can be seen that the twinned70ChapterS. Experimental Preparation210 I IB000200 -tCp=6.9mJ/gKIon 208c_)-pC-)00203-0) ..i180rfTc=0.25K198 I92 93 94170 I I I85 90 95Temperature (K)Figure 5.1 Specific heat at transition temperature for one of the pure YBCO crystals, afterR. Liang et al.4271Chapter 5. Experimental PreparationFigure 5.2 A picture of one of the pure YBCO crystals taken with a polarizing microscope.One can easily see the twin boundaries at 45 to the a and b axes.72ChapterS. Experimental Preparationboundaries are clearly exposed in the polarized light.73ChapterS. Experimental Preparation5.2 The Experimental ApparatusA cavity perturbation system for measuring surface resistance around 35 GHz wasdesigned and built as part of the thesis project. The central part of the apparatus is acylindrical cavity resonator operating on the TE011 mode, shown in figure 5.3. The cavityassembly is made of deoxygenated (OFHC) copper. The inside of the cavity is 1.27 cm indiameter and 0.76 cm in height. According to the mode chart given in standard microwaveengineering handbooks89,the TE011 and TM111 modes are degenerate and are at 34.7 (3Hzfor the above dimensions. To separate them, a blind hole in the center of the bottom plate isadded as a perturbation (see figure 5.3). Such a perturbation has little influence on the TE011mode, since the fields are quite weak at the center of the bottom plate. For the TM111 mode,the hole is at an antinode in the magnetic field and consequently has a larger influence on thefrequency. Experimentally, the TE011 mode is at 34.8 (3Hz and the TM111 mode is at 33.5GHz. This separation is much larger than the width of the resonances and we expect nointerference between the two modes.After machining and cleaning, the cavity is electroplated with Pb-Sn alloy. We use analloy of 95% Pb and 5% Sn as the anode in a Pb-Sn fluoborate solution, and control theplating current density at 3x102Am.This alloy is superconducting at about 7K, and at4.2K the quality factor is already quite high, about 2x106.The coupling between the cavity and the input waveguide (and also the outputwaveguide) is realized by making a through hole connecting the two, and putting a piece of74ChapterS. Experimental Preparationsopphire rodrf, E fieldrf H fieldTeflon tubingcoupling loopscxripleFigure 5.3 The 34.8 GHz cavity. The cavity is made of oxygen free copper and then platedwith Pb-Sn alloy.75ChapterS. Experimental Preparationconducting wire partially encased by a Teflon tube inside the hole(see both figures 5.3 and5.4). The conducting wire and the wall of the hole form a short piece of coaxial line. Oneend of it is coupled to the waveguide mode, and the other end is bent and couples to themagnetic field of the cavity mode. The Teflon tube is connected to a piece of phosphorbronze, which can be moved along the longitudinal direction of the Teflon tube. The motionof this piece of bronze via a coupling adjuster changes the length of the conducting wireinside the cavity, and thus adjusts the coupling strength, i.e., /3 and /3.The assembly including the waveguides is situated inside a hollow stainless steelcylinder, with the top and bottom plates made of brass. The cylinder is pumped to a highvacuum before helium is transferred into the dewar. To keep the cavity at a temperatureclose to that of the helium bath, we use a copper braid to connect the bottom of the cylinderto the bottom plate of a block housing the cavity assembly. This block holds the cavityassembly tightly, and its bottom is made of copper to ensure a good thermal link. A thermalcalculation based on the structure, but ignoring the waveguides, indicates that thetemperature of the cavity assembly should be close to that of the bath. However, we havefound that the heat leak down the waveguides contributes significantly to the heating of thecavity assembly, and we have installed a refrigerator consisting of a long thin stainless tubepassing through the top and the bottom plate of the cylinder. During its operation, the upperend is pumped while the lower end is kept in the helium bath. The flow of cold gas in thetube cools whatever is heat-sunk to it. With the refrigerator operating, the temperature of76ChapterS. Experimental Preparation0,511 tubing, stainless s-teelsectionwaveguide0,511 tubing, brass sectionloross fingerscoupUng adjusterthermo.l insulatorhousing for thermometerand heatersample holderphosphorus bronze sheetcoupling ioopTeflon tubingsampleOFHC copperresistive cardcopper braid2,011 stainless steelcylinderFigure 5.4 The lower end of the apparatus. The copper braid is attached the copper part ofthe bottom plate of the cylinder, which is not shown.77ChapterS. Experimental Preparationthe cavity is fairly close to the bath temperature. The quality factor can be as high as 2x107once the bath temperature is at 1.3K.A 0.5 in OD tube is directly connected to the block housing the copper cavityassembly, via a section of thermal insulating material(The imperial unit system is used heresince these materials are manufactured with dimensions given in inches.). The sample holder(figure 5.5) consisting of a long piece of one-eighth inch stainless steel tubing is secured andcentered by a set of brass fmgers inside the above mentioned 0.5 in tube (figure 5.4.). Theupper section of the 0.5 in tube is also made of stainless steel for the purpose of thermalinsulation. The lower section is made of brass and is heat-sunk to the refrigerator. The brassfingers also act as thermal liiik between the sample holder and this section of the 0.5 intubing. The lower end of the sample holder is a copper chamber, where the thermometer andheater are located. The leads for the thermometer and heater run up the inside of the 0.125in stainless steel tube and connect to an electrical feedthrough at the upper end of the holder.The sample is attached to a sapphire rod or strip, depending on the desired orientation of thesample. This piece of sapphire is in turn glued to a copper piece which can be screwed intothe copper chamber assembly.On the top plate of the cavity there is a through hole for sample access. To preventmicrowave radiation loss from the cavity, the diameter of this hole is made small (about2.3mm), which restricts the size of sample that can be measured. During a measurement thesample can be inserted into, and removed from, the cavity.78ChapterS. Experimental Preparationbrass fingers0.125” stainLess steeL tLibingheatercopper chamberthermometersapphire rodFigure 5.5 The lower end of the sample holder. The sapphire rod can be replaced by asapphire plate so that one can orient the sample with the ab plane parallel to the r.f. H field.sample79ChapterS. Experimental PreparationIn order to position the sample in the cavity with its c-axis parallel to the r.f. H field, asapphire rod is used. Sapphire is chosen for its excellent thermal conductivity (—10 W/cmKat 10K, close to OFHC Copper) and low dielectric loss at low temperatures90.The sample isattached to the end of the rod using silicone grease. Since the sapphire rod itself substantiallyperturbs the cavity, it is extremely important to know its loss and what it does to othercavity modes. In fact, we found that a 1mm diameter rod can bring down the frequency ofsome of the high index modes far enough to interfere with the working (TE011) mode. Thethinner sapphire rods are quite expensive, so we have ground down the 1mm rod ourselves,using diamond paste. After grinding and polishing, one end of the rod had a square crosssection about 0.6 mm2.The processed rod was then ultrasonically cleaned in an acetone bathand boiled in a solution of sulfuric and nitric acids for 30 minutes. The frequency shift of thehigh frequency modes was then sufficiently reduced that they did not interfere with theworking mode. It was determined experimentally that the loss due to the sapphire wasessentially negligible compared to the loss produced by the YBCO samples. When a sampleneeded to be placed in the cavity with its c-axis perpendicular to the r.f. H field, a sapphirestrip cut from a 100pm thick plate was used. The sample was attached to the lower end ofone side of the strip by silicone grease. Because of its small volume, the frequency shift dueto the sapphire strip was very small.With no exchange gas present, the lowest sample temperature reached was about 11K.To obtain data below this temperature, a tiny amount of helium exchange gas had to beadmitted to the apparatus. By carefully controlling the amount of exchange gas in the80ChapterS. Experimental Preparationapparatus, any desired temperature could be reached down to a temperature close to that ofthe bath.81ChapterS. Experimental Preparation5.3 Determination of the Surface ResistanceAs we have mentioned in chapter 4, the quality factor of a resonant circuit can bedetermined by measuring the voltage transmission through the resonance. Fromequation (4.1.7), we have:T2=C 2 2 5.3.1(R/2L) +fro—0where C is a constant, and RT is the total resistive part of the coupled circuit. For a constantinput, V is proportional to the voltage transmission coefficient. It is therefore peaked atthe resonant frequency and the shape of V02 versus frequency is a Lorentzian. In ourexperiment a microwave synthesizer steps through the resonance at discrete frequencies andconstant input power. The curve V0jt2 versus 0 can then be fitted to a Lorentziandistribution by using %-square fitting, and the quality factor and the resonant frequency canbe obtained.The measurement position determines the pattern of the currents in a sample. In thecenter of the cavity, the magnitude of the H field of the TE011 mode is maximum and thedirection of the field is parallel to the central axis of the cavity. If a sample is placed with caxis parallel to the H field of the mode (See Figure 4.5a, and this position will be called theparallel position later.), the screening current would be only in the ab plane. For thismeasuring position, the loss is proportional to a mixture of the surface resistance in the a and82ChapterS. Experimental Preparationb directions. Therefore we can measure an averaged surface resistance in the a-b plane,provided that the constant of proportionality, called the geometrical factor y, can beobtained. For twinned crystals, this is the measurement position we have used. If the sampleis oriented with the a axis parallel to the r.f. H field (In this case c axis is perpendicular to Hfield, referred to later as the perpendicular position; and the same holds when b is parallel tothe r.f. H field.), the screening current will be along the b direction in the ab plane and alongthe c direction in the ac plane. The total loss is then proportional to JS2A (BRb+CR,),where J is the surface current density and A, B and C are dimensions of the sample in the a,b and c directions respectively. To obtain Lb accurately, the condition BR,b>>CR isrequired. Samples usually have A, B>>C, and we can fmd untwinned samples with A,B l2OC. In such a case the total loss would be proportional to Lb. provided that R8 is notmuch larger than Lb. After obtaining the geometrical factor, Lb can be calculated. Similarly,Rsa can be measured by orienting the a axis parallel to the r.f. H field.To ensure that contamination by is not a concern in the perpendicular position, wehave estimated R3 for several twinned samples of different thickness. We first measured Rsabin the parallel position, then measured BRsab+CRsc in the perpendicular position. From thenormal state resistivity data and the size of the sample, Rab+(C/B)Rsc can be found. (Thecalibration formula using the normal state resistivity will be discussed in the followingparagraphs.) The c axis surface resistance R8 can then be obtained by subtracting Lab fromRsab+(C/B)Rsc. We found that R, obtained by this method is of the same order as that in thea and b directions in the superconducting state. Therefore it is safe to measure La and Lb in83ChapterS. Experimental Preparationthe superconducting state using thin samples with the perpendicular position, where the losscontributed by R5 can be neglected. In the normal state, however, the dc resistivity in the cdirection, c, is much higher than that in the a or b directions, resulting in a higher R3 thanRsa and Lb. The loss due to R5 usually conthbutes a few percent to the total loss.The geometrical factor y is obtained by measuring the total loss of samples in thenormal state and comparing it to the surface resistance calculated from the measured dcresistivity. In the parallel position for twinned samples, the currents are assumed to be equalin the a and b directions, and the total loss is then given by:_____Pa+ / Pb1Q) 2 2 2po V2p0a)where Pa and Pb are resistivities in the a and b directions. A measurement of A (l/Q), Paand Pb therefore gives 7.For the perpendicular position (See figure 4.5b.), the surface resistance can be foundin a similar way. Taking a IIH as an example, the total loss of a sample in the normal statecan be written as:A(±=y.(Rb+R (j Pb PCjQ) S B Sc) 2#o B24u0oFrom dimensions C, B, the measured normal state resistivities Pa, Pb and the measured valueof A(1IQ), ‘ycan be found. In the superconducting state, for thin samples, the term involvingR5 is neglected, and R5b is obtained from the measured A (1/Q), and84ChapterS. Experimental Preparation5.4 Circuit Diagram and InstrumentationIn figure 5.6 we present the measurement circuit diagram. The microwave source is anHP83620A 0.0 1-20 GHz Synthesized Sweeper, which scans a series of discrete frequenciesunder its own sweep mode or via computer control. An HP8349B 2-20 GHz MicrowaveAmplifier amplifies the signal from the synthesizer and drives a HP83554A Source Module,which doubles the input frequency. There is an interface cable connecting the sweeper andthe Source Module, allowing the sweeper to monitor the output signal from the module andadjust the power to a designated value. The microwave signal is then brought to the inputwaveguide of our apparatus. When the source scans through the resonant frequency of thecavity, the power versus frequency transmitted to the output waveguide of the apparatus isin the shape of a Lorentzian(See equation 5.3.1.). Since in our experiment the output signalwas quite low due to the weak couplings, i.e., small J3s, and was at a frequency where wehad no amplifiers, the signal was brought down to a lower frequency using a microwaveoscillator and a mixer, and then amplified. This oscifiator was set close to 35 GHz, whichproduced a signal at around 0.5 GHz. This signal was amplified by a broad band amplifierand then detected by a crystal detector operated in the square law region. The output of thedetector, which is a d.c. signal proportional to the transmitted power, is digitized by an A/Dconverter and sent to a computer. The computer controls the scanning and collects the signalamplitude at each frequency. A data set consisting of transmitted power versus frequency,typically covering a region about ten times wider than the resonance width at the halfmaximum power is stored and analyzed by the computer, from which the quality factor85ChapterS. Experimental PreparationHP83620A HP83493 HP83554A 356HzSynthes— AripUfier Source Apporotusized ModuleSweeper______________- Source ModuleInterfoceMicrowove HP11517AOsciUtor Mixer-aComputer UBC 88117 HP8473B WJ6201—A to 13 Cr’ysto. 312 LowDetector FrequencyAmplifierFigure 5.6 Measurement circuit diagram.86Chapter 5. Experimental Preparationof the circuit is obtained.In the following, some of the basic features of these instruments will be recorded.a) HP83620A Synthesized Sweeper91The HP83620A Synthesizer Sweeper (shown in figure 5.7a) is a high performance,computer programmable and broadband frequency synthesizer. The frequency range is from10 MHz to 20 GHz. It can be used in continuous wavelength (CW) and start /stop frequencysweeping operations. In CW operation, a single frequency, low noise, synthesized signal isproduced. The leveled power range for this operation is from -2OdBm and +25dBm. Instart /stop frequency sweeping operation, the synthesizer sweeps a frequency span which canbe as wide as the frequency range of the instrument, or as narrow as 0 Hz (equivalent tosweeping CW). A sweep from the selected start frequency to the selected stop frequency isproduced in this mode. The frequency resolution is one hertz. The total sweep time can bechanged from a few milliseconds to more than a minute. The leveled power range is alsofrom -2OdBm to +25dBm. The Power Slope Operation allows the output power to increaselinearly as the frequency increases to compensate for frequency dependent attenuation in asystem when a wide sweep is being used. In the CW operation, the power can also be sweptstarting with a lower level and stopping at a higher level.b) HP8349B Microwave Amphfier92The HP8349B Microwave Amplifier is used as a driver for the millimeter wave source87ChapterS. Experimental Preparationmodule (Figure 5.7b). It amplifies signals in the 11 to 20 0Hz microwave region to levelsgreater than +l6dBm. A built in source module interface controls signals required by thesource module (which is the HP83554A source module in our case). As a general poweramplifier, its frequency range is from 2.0 GHz to 20.0 GHz. The minimum leveled poweroutput for a 5 dBm input is 19 dBm for the frequency from 2 GHz to 18.6 GHz and 17 dBmfrom 18.6 0Hz to 20.0 GHz. For small signals, the gain is 15 dB and 12 dB for the abovecorresponding bands.c)HP83554A Source Module93The HP83544A Source Module (Figure 5.7c) is designed to be used with the HPsynthesized sweeper and HP8349B amplifier. The power required to drive the sourcemodule is about 20 dBm, with an input frequency range of 13.25 to 20.0 GHz. The outputmicrowave signal is doubled to a range from 26.5 0Hz to 40.0 GHz. A source moduleinterface cable is also required to complete the interfacing. The specification of the maximumleveled power output when combined with the HP83620A synthesized sweeper and theHP8349B amplifier is not given. In actual operation we usually get a maximum leveledpower output of around l2dBm, somewhat larger than the specification value of +7 to +8dBm when it is combined with an HP8350 series sweeper and the HP8349B amplifier.d) Other ItemsThe microwave oscifiator mentioned in the beginning of this section is a sweepableBackward Wave Oscillator (BWO) with a frequency range from 26.5 to 40 0Hz. It is88ChapterS. Experimental PreparationFigure 5.7 (a) HP83620A Synthesized Sweeper(a)89C dl00 s. C I 0 00 dl C C CD CI0ChapterS. Experimental Preparationoperated in the CW mode with the frequency about 0.5 GHz away from the outputfrequency of the source module. The mixer is an HP1 15 17A harmonic mixer with sensitivityof -60 dBm and burnout power level of 1mW at 35 GHz. The low frequency amplifier is aWatkins-Johnson 6201-3 12 amplifier with frequency range from 5 to 1000 MHz. The smallsignal gain is > 22dB with noise figure < 6.5dB. The detector is an HP8473B crystaldetector with frequency range from 0.01 to 18 GHz, and the sensitivity is > 0.5 mV4tW. Thesample thermometer used is a Lakeshore Carbon Glass Resistor (CGR-1-1000 series). Thesample heater is home-made using Evanohm wire with a resistance 81.4 £2/ft and a totalresistance of 33 £2. A UBC-made 85003 Temperature Controller and a 83-047 AutomaticResistance Bridge are used to control the temperature of the samples during measurements.91Chapter 6Data Analysis and DiscussionIn this chapter we present the surface resistance data obtained using our 34.8 GHzapparatus. The chapter is separated into four sections. In the first section, backgroundinformation such as the losses due to the bare sapphire rod and strip, the silicone grease usedto attach samples onto the tip of the sapphire rod or strip, and the Pb:Sn reference samplesare given. We also show how the resonant frequency shifts because of the presence of theseitems. In the second and third sections, the microwave surface resistance and the real part ofthe conductivity of YBCO crystals including those with a range of oxygen content, and thatof zinc and nickel doped samples wifi be displayed and discussed. The data in these twosections were all obtained on twinned crystals and thus give averaged values in the ab planeof the crystals. In the fmal section, we present and analyze the anisotropy in the microwavesurface resistance and the real part of the conductivity within the ab plane for pure,untwinned YBa2Cu3O695 single crystals.92Chapter 6. Data Analysis and Discussion6.1 Background RunsIn this section the major experimental error sources are discussed. The error associatedwith our measurements on the surface resistance of the samples has been determined to beless than 100 uQ in the region below the transition temperature and about 5 m.12 in thenormal state.In order to measure the surface resistance of samples using our apparatus, the tip ofthe sample holder, as well as the sample have to be introduced into the cavity. Therefore it isvery important to know the loss and frequency shift due to the tip of the sample holderalone, before a meaningful measurement can be made. As mentioned earlier, sapphire waschosen as the tip material because of its excellent thermal properties and low dielectric loss.The silicone grease used to attach the sample onto the tip has mediocre but adequate thermalproperties; its dielectric loss are also acceptable. In figure 6.1 the change of the resonantfrequency due to the sapphire rod versus its position in the cavity is shown. In thisexperiment the temperature of the cavity was held at 4.2 K, and the quality factor of theresonator was about 2x106. The x coordinate stands for the depth of the sapphire rodinserted into the cavity, but with arbitrary origin. Because of the different thermal expansionsfor the sample holder and the 0.5 in stainless steel tube, it is difficult to determine preciselythe value x=xo corresponding to the position where the rod started to enter the cavity.Roughly, we estimate x= 1.9 cm, although the resonant frequency starts to change at x 1.8cm due to the penetration of the field into the insertion hole. From the height of93Chapter 6. Data Analysis and Discussion0.0 I I I****N-10.0 --20.0**U- *****-30.0 I I I1.6 1.8 2.0 2.2 2.4 2.6Y (cm)Figure 6.1 The frequency shift of the resonator due to perturbation by the bare sapphire rod.94Chapter 6. Data Analysis and Discussionthe cavity, we know the center of the cavity should correspond to x2.3 cm. We can seefrom the graph that the frequency shift due to the sapphire rod is negative, as expected fromcavity perturbation theory. (Physically, the presence of the sapphire as a piece of dielectric inthe cavity makes the space it occupies effectively larger for the electric field. This results inan apparent volume of the cavity that is larger, which shifts the resonant frequency down.)We can see a length of 0.5 cm of this sapphire rod caused a shift about 20 MHz. Comparedto the resonant frequency of 35 GHz, this shift is relatively small. The upwards shift at aboutx=2.5 cm is probably caused by the uneven thickness of the rod, resulting from an imperfectgrinding job.In figure 6.2 we show the extra loss of the resonator due to the perturbation of thesapphire rod and the sapphire strip, as a function of temperature. Here the loss is presentedby the value of IX(1!Q)=1/QL—1/Qo because they are proportional to each other, where QL isthe quality factor in the presence of the sample, sapphire, etc., and Qo is the quality factorwithout any perturbation. The two loss curves for the same sapphire rod are from differentruns. Between the two runs the resonator was disassembled, re-plated with Pb:Sn, and thenreassembled. It can be seen that the loss for both the rod and the strip is in the range 1x108,which is quite low compared with that due to a typical YBCO crystal in the superconductingstate. This low loss is partly a result of the sapphire being at a node of the electric field (antinode of the magnetic field). The increase in the loss is also in the range 1x108 from lowtemperatures to 100 K, and the curve is approximately linear versus temperature.95Chapter 6. Data Analysis and Discussion2.0 I I I • I1.5* 0 *—‘10• * ()o c ** .0.5 *0 0•.•I I..o o.o . •-J• •• • .-0.5—1.0 I I I • I0.0 20.0 40.0 60.0 80.0 100.0T(k)Figure 6.2 The loss due to bare sapphire. The filled and unfilled circles are data points forthe sapphire rod from two different runs. The stars are data points for the sapphire strip.96Chapter 6. Dada Analysis and DiscussionFurthermore, the perturbation losses extrapolated to zero temperature are not equal betweeneach plating and reassembling. We believe that this zero temperature perturbation is relatedto the re-arrangement of the field pattern and currents in some weak links in the jointbetween the top plate and the body of the cavity, caused by the perturbing object. A smallextra loss (it can be either a positive or negative systematic error) is then observed. For eachrun, this systematic error is different due to slightly different conditions for plating andreassembling, and is carefully determined and corrected using pieces of Pb:Sn as referencesamples. The correction procedure will be discussed later in this section.Before making measurements on a sample, we also have to know the loss caused bythe silicone grease used to attach a sample to the tip of the sapphire rod. In figure 6.3 theloss curve versus temperature is displayed. On the tip we have put a tiny amount of highvacuum silicone grease (Dow Corning high vacuum grease), which is just enough to hold asample. Again the loss is found to be in the range 1x108, with a little larger slope than thatof the bare sapphire. The amount of the grease used is hard to control accurately, and mayvary slightly from run to run. In the parallel configuration a superconducting sample tends toshield grease from the electric fields and this loss is not a concern. In the verticalconfiguration, the loss due to the grease may introduce a small systematic error in themeasurements.In figure 6.4a we show the frequency shift versus sample position along the central axisof the cavity for an aluminum contaminated YBCO crystal (batch RL1087) in the parallel97Chapter 6. Data Analysis and Discussion2.0 I I I I*1.0*o ****0** ***U)U) * *0-J 0.0• * **** ******I I I-1.00.0 20.0 40.0 60.0 80.0 100.0T(K)Figure 6.3 The loss due to the silicone grease and the sapphire rod.98Chapter 6. Data Analysis and Discussionmeasuring position (see chapter 4, section 2 for the geometry of this position). The samplesize is 1.2 mm in length and 1.1 mm in width, with thickness unmeasured. We can see thatthe frequency shift is strongly dependent on the sample position. The maximum frequencyshift is about 100 MHz for this sample and occurs when the sample is at 2.31 cm. Thisposition would correspond to the center of the cavity if the perturbation by the sapphire rodwere negligible. Close to the maximum shift region, a change of position by 0.25 mm causeda shift of around one MHz. It is not unusual for the thermal motion of the sample holder tocause a frequency shift of a comparable amount during a wide temperature ramp (from a fewKelvin to more than a hundred Kelvin). The resonant frequency is also dependent on themicrowave penetration depth for the sample. For our experiment, a change in microwaveskin depth of 2 pm causes a change of about 1.0 MHz in the resonant frequency. It is thusevident that the observed frequency shift is completely dominated by motion of the sampledue to thermal effects, and the apparatus in its current configuration is not suitable forprecision measurements of changes in penetration depth of YBCO samples, where aresolution of several angstroms is desired.In figure 6.4b the loss versus position for this sample at a temperature of 12.5 K isdisplayed. A strong dependence of loss on the sample position is observed. The maximumloss position is at 2.34 cm, quite close to the maximum frequency shift position. Themaximum loss occurs when the sample is at the center of the cavity, where the magnetic fieldis maximum and thus the shielding current is the largest. We also notice that in a range about0.05 cm around the center, the variation of the loss is only 2%. Therefore, as99Chapter 6. Data Analysis and Discussion100.0* *80.0*NI60.00)a)3a)U-*20.0*0.0 I I • I • I I1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6Y(cm)(a)60.0 .1.40.0*‘ 20.00.0I I I I I1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6Y(cm)(b)Figure 6.4 The frequency shift(a) and the loss(b)due to a sample versus its position in thecavity. The data were taken when the sample temperature was 12.5K.100Chapter 6. Data Analysis and Discussionlong as the sample is placed at the center of the cavity, the error caused by the thermalexpansion of the sample holder is negligible. The apparatus in its current configuration isthus suitable for good measurements of the surface resistance. We also notice that the loss ofthis particular sample at 12.5K is quite high, with z(1/Q)=5x1O7.This high loss was laterfound to be the result of aluminum contamination of the crystal during fabrication. Highquality YBCO, and zinc or nickel doped samples have losses that are considerably lower atlow temperatures, with A(1/Q) around 5x108 at a few Kelvin. Since this is comparable tothe 1x108systematic errors, it becomes important to know the systematic error for each runwhen measuring very low losses at low temperature.In order to accurately correct for the systematic error, a Pb:Sn reference sample ismeasured either before or after a YBCO sample is measured. The Pb:Sn sample is cut to asize close to that of the YBCO sample such that when it is placed in the cavity, the fieldpattern resembles that when the YBCO sample is present. To determine the surfaceresistance of the YBCO sample, we first obtain the difference in the loss of the YBCOsample and the 1.3 K loss of the Pb:Sn sample of the same size, then use the calibrationmethod discussed in section 5.3 to calculate the difference in R, and fmally add the R3 ofthe Pb:Sn alloy at 1.3K to the data set. The surface resistance of the Pb:Sn alloy wasestimated by determining the surface resistance of Pb:Sn plated on the cavity walls from thequality factor of a fresh plated cavity, using the equation relating the quality factor and thesurface resistance (equation 4.2.6). In this way we obtained a value around 40 iQ for Pb:Sn101Chapter 6. Data Analysis and Discussionat 1.3K. We may also estimate it from the fact that the quality factor of the cavity resonatorincreases by a factor of ten from 4.2 K to 1.3 K; or in other words, from the fact that the Rof Pb:Sn at 1.3 K is at least ten times less than that at 4.2 K. The R of our calibration Pb:Snsample at 1.3 K can then be calculated from its value at 4.2 K, since the R at 4.2 K issubstantially larger than the systematic error and can thus be determined quite accurately.The surface resistance of Pb:Sn at 4.2 K obtained from the experiment is about 300 uQ,which is very close to the value of 340 jiC2 from the BCS calculation for Pb94. Using thismethod, a slightly smaller R, about 30 4uQ at 1.3 K, is obtained. We emphasize that theseestimated corrections are less than the usual scatter of our data. In figure 6.5 we show theloss curves for several Pb:Sn reference samples of various sizes from various runs. We cansee from the figure that the systematic errors are all in the range of 1.5x108,which roughlycorrespond to a value of R around 100 uQ.Other error sources come from temperature variations of the cavity and sample and thepossible mechanical vibration the of sample holder. These errors are responsible for thescatter of the data and are about 30 pQ and 50 4uQ for measurements in the parallel andvertical configuration respectively in the superconducting state, and are about 5 mQ in thenormal state. As pointed out earlier, the possible error caused by the thermal expansion ofthe sample holder should be less than 2% of the R values. We have also checked whether Rdepends on the input microwave power leveL It was found that R of our samples wasessentially power independent within a fairly wide range; however, we have always kept the102Chapter 6. Data Analysis and Discussionpower level at the lower end of such a range during measurements.To summarize, we apply a systematic correction of order 100 uQ to the raw data,which itself is not much larger than the scatter in the data. We believe our results to have anerror of order 50 4u.2 at temperatures below 30 K, at most 100 uQ for the entire region inthe superconducting state, and about 5 mQ in the normal state, for crystals of nominaldimensions 1 mmxl mm.103Chapter 6. Data Analysis and DiscussionI I01 06-7110U)(00-J IE108I I0.0 5.0 10.0 15.0T(K)Figure 6.5 The loss of a series of Pb:Sn samples. The value at 1.3 K typically corresponds toa value of 100 uQ, which consists of two parts, the R of the Pb:Sn and the systematicerror(T of. Pb:Sn is about 7.2 K).104Chapter 6. Data Analysis and Discussion6.2 Microwave Surface Resistance and Conductivity of Pure YBa2Cu3O7..CrystalsIn figure 6.6 we show the surface resistance versus temperature for a pure YBCO(6—(J.05) crystal from sample batch RL1095. The dc resistivity in the ab plane for this batchof samples is about 75 #.2cm at 100 K. This corresponds to a value of 0.3 16 £2 for thesurface resistance at 34.8 GHz. Below the transition temperature, the surface resistancedrops rapidly, has a minimum near 72 K, then rises to a broad peak around 50 K, and fmallydecreases at low temperatures. The broad peak in R is typical for almost all pure crystalsthat we have measured, and was first observed in our lab at around 40 K4° in a lowfrequency (f3. 8 0Hz) apparatus. This peak was attributed to a rapid drop in the scatteringrate of the quasiparticles in this material.To clearly show the features of R3 in the superconducting state, in figure 6.7 we displaythe data on a linear scale. below the broad peak, the surface resistance at 34.8 GHz is quitelinear at low temperatures, which is typical for all pure YBCO crystals that we have studied.The slope of the linear term for these crystals is typically about 50 to 55 uQ K’. Theresidual R extrapolated to 0 K is roughly 1 mQ, which is much larger than our systematicerror. We note that the residual R values in samples of earlier batches tended to be higherthan that of later batches, presumably due to slightly different growth conditions.In the same figure we show the data of Bonn et al.4° taken at 3.8 GHz. Forcomparison, the low frequency data have been scaled to the higher frequency by a factor r =(34.8GHz /3.8GHz)2, on the assumption that R is proportional to the square of the105Chapter 6. Data Analysis and Discussioniü° I I010100l0210 003.8 GHzI I0.0 50.0 100.0 150.0T(K)Figure 6.6 The surface resistance of a sample from batch RL1095. We found the residual Ris relatively high for this sample, which is typical for samples from earlier batches.106Chapter 6. Data Analysis and Discussion60 I I • I I • 5.004.040 •.•• • 0___.. r-\00..0 3.0••0 0Q0.000202.0•3.8 GHz0 034.8GHz 1.00 I I 0.00.0 20.0 40.0 60.0 80.0 100.0T(K)Figure 6.7 The comparison of the surface resistance at two frequencies, 3.8 GHz (Bonn etal.40) and 34.8 6Hz (this work). The error is essentially the scatter in the data.107Chapter 6. Data Analysis and Discussionfrequency. It is found that the scaled data match the higher frequency data quite well fromT down to 50 K, but start to deviate below 50 K. In chapter 3 the relation between thesurface resistance and the real part of the conductivity, y has been derived. As a reminder,the simplified expression is given again:R 3.1.82valid foro2>>, which for our frequencies occurs within a fraction of a degree below T.Provided that there is no much difference in the penetration depth at the two frequencies, weare actually comparing the difference in the real part of the conductivity oi at the twofrequencies. The real part of the conductivity was therefore calculated and is displayed infigure 6.8. To derive 01, we have used penetration depth data from other experiments on thesame crystals. The absolute penetration depth at 0 K, )i(O), was measured using far infraredspectroscopy 70, and its temperature dependence was obtained by our group at 1.0 GHzHere we have assumed that the penetration depth at 3.8 and 34.8 GHz is the same as that at1.0 GHz. This is equivalent to assuming that the screening of the microwave fields is onlycaused by the superfluid. In the two fluid model, this assumption is true if the product of themicrowave angular frequency and the scattering time, or, is much less than one. If co’rapproaches one, then the normal fluid contributes to the screening and an error is introducedif one uses the penetration depth at 1.0 GHz instead of its effective value at higherfrequencies. Within this model, we will fmd later from the calculation of the scattering rate,that for pure crystals co’r does approach unity at 34.8 GHz at temperatures108Chapter 6. Data Analysis and Discussion15 •• • D 34.8GHz•‘3.8GHz10 • flacoo I Ia_•‘ID,I•t11I I • I I0.0 20.0 40.0 60.0 80.0 100.0Temperature (K)Figure 6.8 The real part of the conductivity of a sample at two frequencies. An error of up to15% may be introduced into in the region below 30 K for the higher frequency databecause of the use of the low frequency penetration depth data in the calculation (see text).109Chapter 6. Data Analysis and Discussionbelow 30 K. However, since we do not have penetration depth data at 34.8 GHz, and sinceits frequency dependence is not necessarily well described by the two fluid model, the lowfrequency penetration depth data were used to derive the real part of the conductivity. Forsuch a treatment, we estimate that an error of up to 15% may be introduced into 01 for thepure sample in the region below 30 K at the higher frequency. It is seen in the figure thatthere are strong broad peaks in o. It is worth noting that the peaks are not associated withthe coherence peak expected for s-wave superconductors, which should have risen sharply,immediately below T. At low temperatures, a strikingly linear behavior of the conductivity isobserved.The broad peak in the surface resistance can be traced to the peak in the real part ofthe conductivity. In the two fluid model where the real part of conductivity is expressed inequation (3.2.3), the peak can be seen to result from a competition between the rapid rise ofthe scattering time and the relatively slow drop of the normal fluid density below thetransition temperature. If the frequency is high enough such that orr starts to approach one atlow temperatures, a factor 1/(1+co2r)would play a role in the real part of the conductivity,and would result in a frequency dependent 01. The observed difference for the two data setsat the low temperatures is consistent with the claim that the scattering rate drops rapidlybelow T, such that if r approaches the higher frequency, 2it x 34.8 GHz — 2.2 x 1011 s_i.The sudden rise in 01 close to the transition temperature is due in part to fluctuation110Chapter 6. Data Analysis and Discussioneffects, where the dc oi in the normal state diverges at TC95 It is also a region where R and), which are measured in the different apparatuses, are both varying rapidly, and a smalldifference in the two temperature calibrations can produce spurious effects.Within the two fluid model, the scattering time of all the quasiparticles is consideredthe same, independent of frequency, and it can actually be calculated from equation (3.2.3)and (3.2.4), using the conductivity and penetration depth data. The calculated scattering ratefor 34.8 GHz data is displayed on a log-linear scale in figure 6.9. We can see that thescattering rate drops by a factor of 10 from the transition temperature to 50 K, and starts tosaturate below 35 K. The scattering rate below 40K is approximately 3.5x10’1 s, whichcauses or to be around 0.6 at 34.8 GHz.In a more sophisticated BCS-like microscopic theory however, the density of statesand the dynamic properties of the quasiparticles such as their velocity, vary rapidly withrespect to energy, because of the presence of an energy gap at the Fermi surface. Inparticular, the scattering rate is strongly dependent on the energy for the quasiparticles. Theexpression derived by Hirschfeld et al.85 for the real part of the conductivity for d-wavepairing,1’.—f dco1Z.flN(co)ImI 1 3.3.13XX Lm*J_ do) Q—iI’r(o))was used to estimate the low temperature behavior. They derived expressions for thescattering rate for a d2. state due to different types of impurities and obtained a quadratic111Chapter 6. Data Analysis and Discussion1013.0 I I000012.009010110 I I0.0 20.0 40.0 60.0 80.0 100.0T(K)Figure 6.9 The scattering rate in the superconducting state derived from the two fluid modelfor a pure YBCO sample.112Chapter 6. Data Analysis and Discussion0.0 0.3 0.5 0.8 1.0T/TCFigure 6.10 The calculated real part of the conductivity, after Hirschfeld et a119. The uppercurve corresponds to a frequency at 3.89 GHz and the lower curve corresponds to 38.9GHz.00bb00Co00113Chapter 6. Data Analysis and Discussionor constant behavior for the low temperature real part of the conductivity at the lowtemperatures in the presence of magnetic or non-magnetic impurities. By setting thescattering rate equal to the sum of the unitary impurity scattering (strong scattering withphase shift iW2) rate and the inelastic rate estimated using a spin-fluctuation model byQuinlan et al.96, they calculated the real part of the conductivity and fitted it to our data85.The results are displayed in figure 6.10. It was found that the overall shape of the curvesagrees quite well with the data, however, the linearity of our data at low temperatures clearlydeviates from the calculated curves. This linearity could not be explained by either a simples-wave or a simple d-wave model.The quantity calculated from the two fluid model can be treated as an energy averagedquasiparticle scattering rate, which can be defined as (in the limit co’r <<1)” 85f() =-aE1 6.2.2L dE(_ )N(E)For a pure d-wave superconductor, this averaged rate corresponds to that of a typicalquasiparticle with an energy of order kBT above the Fermi surface. Most of the heavilyoccupied states with such an energy are those in the region of the nodes of the energy gap.The inelastic scattering of such quasiparticles by spin-fluctuations was predicted to droprapidly below the transition temperature96.In figure 6.11 we display the calculation for thescattering rate using the spin-fluctuation model by Quinlan et al.96, which was fitted to our114—2Chapter 6. Data Analysis and Discussion0— —0.50C,C—1.50.5T/TC00- —0.5C—1.5—2Figure 6.11. Calculated scattering rate versus temperature compared to our data, afterQuinlan et. at.96 a) logio(z(T)It(T)) calculated from an s-wave model with 24CJkBTC = 4(dotted), 6 (dashed), and 8 (solid). b) log1o(t(T)/(T)) calculated from a d-wave model with2AJkBTc — 6 (dashed) and 8 (solid). (The sharp features near T are due to a combination offluctuation effects and differences in temperature calibration for the penetration depth andsurface resistance apparatuses. See discussion following figure 6.8.)0 0.5T/T©115Chapter 6. Data Analysis and Discussiondata. From the figure, the d-wave gap results with 2A0 — 6 to 8 kBTC appear to follow theexperimental data fairly well in the inelastic region, which is above T=O.5T.The broad peak in the surface resistance of YBCO single crystals was later confirmedby the Sherbrooke45 and Maryland36 groups, both using a cavity perturbation technique.Shibauchi et al.97 reported their o data obtained on YBCO single crystals, also using thecavity perturbation method. However, the measured 01 appears to increase with decreasingtemperature. They argued that the large residual surface resistance obtained in their samples,which is likely extrinsic, could strongly affect the extracted values of a at low temperatures,and the intrinsic behavior was in fact expected to have a broad peak.The sample used for the data presented above was also used to study the possible rolethat the oxygen vacancy sites on the chain would play in the scattering rate. The sample hadoriginally been annealed in oxygen gas to have an oxygen content x=6.95. After themeasurement we ultrasonically cleaned the sample in a heptane bath to remove the siliconegrease, and annealed it again to have a different oxygen content. In this way a sample withthe same geometry, but different oxygen content was obtained. In figure 6.12, the surfaceresistance of this sample with several different oxygen contents is displayed. Surprisingly, thesurface resistance is found to be quite insensitive to oxygen content for 8 from 0.02 to 0.15,although in this range T varies by 4 degrees. This range of 8 corresponds to a factor of 7change in the number of oxygen vacancies on the chains. The result indicates that the chainvacancies contribute little to the qualitative behavior of the surface resistance, and in116Chapter 6. Data Analysis and Discussionparticular, the quasiparticle scattering process is not strongly affected by vacancy sites in theCuO chain layers.117Chapter 6. Data Analysis and Discussion100.0101.09- 1oo1 -•°1 ü•°0.0 150.0T(K)Figure 6.12. The surface resistance of YBCO crystals with various oxygen contents at 34.8GHz. Fified circles correspond to 6 = 0.05, open circles correspond to 6=0.10, filledsquares correspond to 6 = 0.15, and open squares correspond to 6 = 100.0118Chapter 6. Data Analysis and Discussion6.3 Microwave Surface Resistance and Conductivity of Zinc and Nickel Doped YBCOCrystalsAs pointed out in chapter 1, the microwave surface resistance plays an important rolein understanding the fundamental property of high temperature superconductors; and it isalso one of the key parameters to be understood in terms of various applications of thismaterial. Consequently, it has been extensively studied. However, the results are very sampledependent. In figure 6.13 we show the surface resistance of YBCO samples measured bythree different groups(Klein et al.39, Ma et aL32, and Mogro-Campero et al.98 ,). Most of thedata were obtained on thin films. The magnitudes vary significantly from sample to sample,and most of them decrease monotonically with decreasing temperature in thesuperconducting state, while the surface resistance of a few samples shows a plateau around50 K to 60 K, and then falls at lower temperatures. These data are qualitatively differentfrom what we have observed in our pure crystals. Because the surface resistance isproportional to the square of the frequency in the two fluid model (which neglectingcoherence effects, we expect to be accurate at medium or high temperature, where thescattering rate is not comparable to the frequency), it is usually scaled to some commonfrequency, say 10 GHz for comparison purposes. When all of the data is scaled to 10 GHz,the surface resistance of some state-of-the-art thin films is actually lower than that observedin the crystals at low temperatures.As it is known that the surface resistance is very sensitive to the quality of the samples.119Chapter 6. Data Analysis and DiscussionIn samples of poor quality, grain boundaries and structural defects may be partially normaland may dominate the surface resistance. However we do not believe these sort of defectsplay a role either in our single crystals or in very good films. The intrinsic surface resistanceis proportional to the real part of the conductivity, as shown by equation 3.1.8. For highquality samples one expects that the introduction of a proper amount of impurities wouldincrease the scattering rate, which would reduce the real part of the conductivity, andthereby lower the surface resistance. It is worth pointing out that this property ofsuperconductors is very different from that of normal metals, for which the surface resistanceis inversely proportional to the square root of the conductivity, and the introduction of theimpurities would lead to a higher surface resistance.In order to confirm that the peak in R(T) that we measured was indeed the intrinsicbehavior of the material, and to further explore the influence that the impurities have on thesurface resistance, zinc and nickel doped crystals were grown in our lab. Both nickel andzinc dopants will predominantly substitute for coppers on the Cu-O planes in YBCO59’60These dopants would thus act as impurity scattering centers for the quasiparticles, andproduce a qualitative change in the behavior of the surface resistance in the superconductingstate. In figure 6.14 we present the surface resistance of three samples in thesuperconducting state, with nominal zinc concentrations of 0, 0.15 and 0.3 1%. In figure 6.15we present the surface resistance of two samples with nominal nickel concentrations of 0,and 0.75%. (The percentage is given with respect to the total copper concentration in thecrystals.) All the samples here are twinned. For the pure sample, we found a much lower120T(K)3 I I p02-000. -1.0• •.••I. I I40 60TEMPERATURE (K)Figure 6.13 The surface resistance of various YBCO samples measured by three groups. Theupper graph is from reference 39 with a frequency at 18.9 GHz, the middle one is fromreference 32 with a frequency scaled to 10 GHz using R oc f2, and the lower is fromreference 98 with a frequency at 12 0Hz.121Chapter 6. Data Analysis and DiscussionCUcNI ioQ00-.410In1O STI laser ablated, 90.5 K aO AT&T post annealed, 88.8 KX AT&T in situ, 85.7 K 0a AT&T in situ, 81.5 K at3ct4e0.xx a_oOooo.0000.1... I., • I. • .1.,.0 40 60 80 10020EUC’)C/)UUC-)DU)20 80 1 (JOChapter 6. Data Analysis and Discussionresidual R (value extrapolated to 0 K) than that of earlier batches. At 77K, the value of R3 isabout 2.2 mQ, or about 180 4uQ when scaled to 10 GHz, which is quite a bit lower than thatfor the best films reported. This very low loss at 77K is another indication that extrinsicfactors play little or no role in our crystals.The overall shape of the surface resistance of doped crystals was found to resemblethat of the best films, which may be seen by comparing figure 6.14 and 6.15 with figure 6.13.In the case of 0.15% zinc doping, the broad peak near 50 K is reduced to a plateau at 60 K,and the overall surface resistance is lower than that of the pure crystals. For the sample with0.31% zinc concentration, there is no longer even a plateau apparent, and the surfaceresistance decreases monotonically in temperature, to an even lower value. We also note thatthe linear behavior of the surface resistance at low temperature is modified by zinc doping.Both zinc doped curves exhibit curvature near 10 K, with an obvious departure fromlinearity for the larger concentration. For the 0.75% nickel doped sample, the overall surfaceresistance is similarly much lower than that of pure samples. However, the linearity persistsin a greater range of temperatures, with a small slope of about 12 jiQ K’. This slope is lessthan one quarter of that for the pure samples at low temperatures. It is also interesting tonote that the dosage of 0.75% nickel is not as effective as that of 0.31% zinc in shiftingdown the transition temperature, which agrees with the results of other groups76.Furthermore, we found that a relatively high transition temperature for nickel doped samplemake R particularly low at 77K. When scaled to 10 GHz, the surface resistance is about 130uQ, which is lower than the value 160 pfl for the 0.31% zinc doped sample, and122Chapter 6. Data Analysis and DiscussionFigure 6.14 The superconducting state surface resistance of zinc doped YBCO crystals at34.8GHz. The zinc concentrations compared to that of copper are: open circle, 0; filledcircle, 0.15%; and open square, 0.3 1%. The error is about ±30 u(2.0.0040.0030.0020.0010.0000.0 20.0 40.0 60.0 80.0T(K)100.0123Chapter 6. Data Analysis and Discussion0.0020.0040.003.I I I II0 9C 0C0000.001 g00.000 I I I0.0 20.0 40.0 60.0 80.0 100.0T(K)Figure 6.15 The superconducting state surface resistance of nickel doped YBCO crystals.The nickel concentrations are: open circle, 0%; filled circle, 0.75%. The error is about ±30124Chapter 6. Data Analysis and Discussionsubstantially lower than that of the pure crystals with twins. The residual loss at 0 K is foundto be higher for the nickel doped versus the zinc doped crystals, suggesting that nickel is adifferent type of scattering source than zinc. However, we should keep in mind that theresidual loss is a complicated quantity, since twin boundaries, defects, or surfacecontamination can all contribute to it. We note that the above measurements have beenrepeated on different samples with the same doping concentration, and with resultsessentially the same as displayed here.For the doped crystals, the product cr is much less than one, because of the increaseof the impurity scattering. The assumption of A. being independent of frequency is thusprobably better; however, A.(0) values for these doped samples have not been measured yetand have to be assumed in order to derive the real part of the conductivity. We know frompure samples that A.(0) is around 1300 A7° Because the impurity concentrations are quitelow, it is reasonable to use A.(0)=1300 A for all samples. The impurities may increase A.(0)slightly which leads to an over-estimate of the magnitude of o, but this wifi not affect theoverall shape of the curves.In figure 6.16 and 6.17 we show the real part of the conductivity of the pure sampleand the samples with zinc and nickel doping. When compared to the data for the purecrystal, it is seen that both zinc and nickel impurities suppress the amplitude of the peak ino, and shift the maximum position to a higher temperature. The much lower o is a clearindication that the impurity scattering limits the rapid drop of the scattering rate. The shift of125Chapter 6. Data Analysis and Discussionthe peak towards higher temperature is caused by saturation of the scattering rate at a highertemperature, i.e., the point where the impurities start to dominate the scattering process. Wenote that doubling the amount of zinc impurities roughly halves the real part of theconductivity at low temperatures, showing that the impurity scattering dominates attemperatures below 30 K.Zinc and nickel doping are seen to have different effects on the low temperaturebehavior of oi. Zinc doping changes the linear behavior of pure sample to a nearly quadraticbehavior at 0.3 1% dosage, while 0.75% nickel doping has no such effect. In the simple twofluid model picture, the scattering rate is taken to be independent of energy, and reaches alimit determined by the impurities at low temperatures. The dependence of ai at lowtemperatures can thus be traced to the temperature dependence of the normal fluid density.The linear behavior for the pure and nickel doped samples suggests a linear temperaturedependence for the normal fluid, and the quadratic behavior for zinc doped sample indicatesT2 behavior. Within the simple two fluid picture, these features are consistent with thetemperature dependence of the penetration depth and thus also the normal fluid densitymeasured on these crystals. As indicated earlier, the microscopic theory for superconductorswith line nodes in the gap predicts a quadratic temperature dependence for o for unitaryscattering and a constant T dependence for Born scattering (weak scattering with smallphase shift) at low temperatures, inconsistent with our experimental results on pure samples.Nevertheless, these predictions appear to agree with the experimental data for 0.31% zincand 0.75 nickel doped samples if Zn2 and Ni2 are assumed to be magnetic and non-126Chapter 6. Data Analysis and Discussion20.015.0E0.0 —0.0 80.0 100.0T(K)Figure 6.16 The effect of zinc doping on the real part of the conductivity. In the figure theopen circles stand for that of pure sample, filled circle for 0.15% zinc doped sample, andopen square for 0.31% zinc doped sample.CoO090020.0 40.0 60.0127Chapter 6. Data Analysis and Discussion20.015.0E1:.:0.0 —0.0 20.0 40.0 60.0 80.0 100.0T(K)Figure 6.17 The effect of nickel doping on the real part of the conductivity, open circles aredata points for the pure sample and filled circles are data points for the doped sample.0QooOQp.0I I I I128Chapter 6. Data Analysis and Discussionmagnetic impurities respectively. The actual fitting to our zinc doped data using the modelby Hirschfeld et al.85 are shown in figure 6.18. It was found that the size and position of theprominent maximum in the real part of the conductivity are reproduced qualitatively, but it isclear that the low temperature behavior of the data does not agree in detail with thepredictions of the model being proposed.Finally, the averaged scattering rate versus temperature for each dopant and eachconcentration is displayed in figure 6.19. The two fluid model is again used to derive thisenergy independent scattering rate, which can be considered as an averaged scattering rate.As we see in the figure, the main effect of doping is an increase in the quasiparticle scatteringrate at low temperatures.The doping effect is a very important result. It not only supports the conclusionreached from the pure samples that the scattering rate drops rapidly below the transitiontemperature for this superconductor, but also demonstrates the possibility of reducing thesurface resistance for these materials at a temperature as high as 77K. The surface resistanceis one of the key parameters in terms of applications, especially in the area of microwavedevices, where strong efforts have been focused on the reduction of the surface resistance.The result on crystals indicates that one might be able to produce lower loss hightemperature superconducting thin films by slightly doping the material with impurities.129Chapter 6. Data Analysis and DiscussionC)0—%% Q0.0 0.2 0.4 0.6 0.8 1.0 1.2T/T1/ ICFigure 6.18 The calculated curves (solid line) of the real part of the conductivity at 34.8GHz, which are compared to our data. In the figure, 1 is a parameter proportional to thedensity of impurities, and the frequency should be 34.8 GHz instead of 34.8 Hz, afterHirschfeld et al.85IVTc= .0008f/T=.O1 81 34.8 HzpureI0 I I I I34.8 Hz.15% Znr/T=.009Q/T=.01834.8 Hz.31% Znr/T=.o1 8c2/T=.o18130Chapter 6. Data Analysis and Discussion1015 II I • I1410a 1013. 01012•••lU••II•Iç0008000000000001011 I I0.0 20.0 40.0 60.0 80.0 100.0T(K)Figure 6.19 The scattering rate derived from the two fluid modeL Filled circles are datapoints for the 0.75% nickel doped sample, open square are for the 0.3 1% zinc doped sample,filled squares are for 0.15% zinc doped sample, and open circles are for the pure sample.131Chapter 6. Data Analysis and Discussion6.4 Anisotropic Properties in the ab PlaneThe existence of CuO chains in the structure of YBCO inevitably causes anisotropictransport properties in the ab plane. In the normal state, transport properties in the b(chain)direction have been found to be quite different from that in the a direction 48, and it isimportant to see how this anisotropy manifests itself in the electrodynamics below T. Inaddition, the role of twin boundaries in the qualitative behavior of the surface resistance wasnot known. We particularly wanted to know how the twin boundaries would affect theresidual surface resistance, which is of special importance in d-wave superconductors.Furthermore, there were suggestions in the literature that the qualitative feature of thesurface resistance could be seriously affected by the weak links at the twin boundariesand that the Cu-O chain might play a key role in the qualitative behavior of the surfaceimpedance 100, 101 For these reasons, we set out to measure the microwave properties oftwin free single crystals.Figure 6.20 shows the surface resistance of two untwinned samples on a log scale.Sample 1 is naturally untwinned and has a size l.2x1.2x0.01 mm3. Sample 2 wasmechanically detwinned and has a size 1.lxl.OxO.035 mm3. As discussed in section 4.3, witha ratio of the length to the thickness exceeding 100, the contamination of can be ignoredfor sample 1 for the measurement in the superconducting state. When scaled to 10 GHz, wefound that the surface resistance Rb for sample 1 corresponded to 125 uQ at 77 K, and theaverage of Rsa and Rsb corresponded to 140 11X2 These values are substantially lower than132Chapter 6. Data Analysis and Discussion1001011020.0 50.0 100.0T(K)Figure 6.20 The surface resistance of two untwinned samples. R of sample 2 matches that ofsample 1 at low temperatures but is higher in the normal state, due to the contribution fromThe error is ±50 uQ below T and ±5 mf2 above T.• sample 2, Rsa• sample 1, Rsample 2, RSb0 sample 1, Rth133Chapter 6. Data Analysis and Discussionall reported surface resistance values for YBCO material at 77 K. We note that in the normalstate, the surface resistance of sample 2 is slightly higher due to contamination by In thesuperconducting state, it also has a higher residual resistivity at 0 K and higher values at 77K, perhaps indicating slightly lower quality. Furthermore, its thickness to length ratio wasmore than three times larger than that of sample 1. For these reasons, the rest of the analysisconcentrates on sample 1.In figure 6.21 we show the surface resistance of sample 1 plotted to emphasize thenormal state surface resistance. Because the sample is approximately square, it is reasonableto assume the same geometrical factor for the measurement of Rsa and Lb. Therefore weobtained the ratio of the dc resistivity values in the two directions from the ratio of thesurface resistance in the normal state in this measurement. Knowing the averaged dcresistivity in the plane from the twinned samples, the d.c. resistivity values in the a and bdirections can be derived. The geometrical factor can thus be determined from the loss in thea and b directions and their respective d.c. resistivity values. It can be seen from the figurethat Rsa is about 1.5 to 1.6 times as large as R3b at 121 K. This corresponds to a ratio of 2.4for the resistivities in the a and b directions, which agrees with the highest ratios reported sofar for the d.c. resistivities48’In figure 6.22, the surface resistance is shown on a linear scale in order to highlight thelow temperature behavior. The center curve is the averaged surface resistance. It wasobtained using a measurement method similar to that used for the a and b directions, asdiscussed in chapter 4, except that the current ran at 45° to the a or b directions. This curve134Chapter 6. Data Analysis and Discussion0.50 I I II040 0 b direction• a direction0.30 o00.20 -0.10 -60.0 80.0 100.0 120.0 140.0T(K)Figure 6.21 The normal state surface resistance of sample 1 in the a and b direction. Thesurface resistance in the a direction is roughly 1.6 times as high as that in the b direction.135Chapter 6. Data Analysis and DiscussionFigure 6.22 The surface resistance of sample 1 in the superconducting state. The error is ±504uQ as indicated by the scatter in the data. Rsa is 80% larger than Rj, at low temperatures,and has a more prominent peak.•RsaRsabORSba0a.1’.0.I4. 20.0 40.0 60.0 80.0T(K)100.0136Chapter 6. Data Analysis and Discussionis found to be close to the average of Rsa and Lb. The values of Rsa are seen to have a veryprominent broad peak at about 48 K, whereas the peak for Lb is much smaller. Theminimum, occurring at nearly 72 K in twinned samples, is shifted up to around 75 K and isnow deeper. This could be explained by the twin boundaries producing extra losses, thatvary only slowly with temperature. Below 35 K, both Rsa and Lb are very linear, with the Lacurve about 1.8 times steeper than that of Lb. If we fit the data below 30 K to a straight line,and scale the values to 10 0Hz, we obtain Rsa(0) = 2.5 ± 2 uQ and Rb(0) = 0.8 ± 1 iQ.Given the fact that the error on each point is of order ±5 jiQ, these extrapolated valuesshould not be taken too seriously. However, these appear to be the lowest residual surfaceresistance values ever reported for YBCO material. For comparison, the typical residualsurface resistance of our twinned crystals is about 25 4uQ when scaled to 10 0Hz.The absolute penetration depths in the two directions have been measured on sample 2at McMaster university70.They obtained a value of 1000 A for 2,(0) and a value of 1600 Afor Aa(0). The temperature dependence of 2a and )Lb was measured on sample 1 by our owngroup at 1 GHz’°2.Using these data we are able to derive the real part of the conductivity inthe two directions. We should keep in mind that an error up to 15% in the region 10 to 30 Kmight be introduced, for the same reasons discussed in section 6.2. The result is displayed infigure 6.23. It is interesting to note that the large anisotropy in A.(0) has caused Rsa to belarger than Rib, in spite of the fact that 01a is smaller than 1b. At temperatures just belowT, Oib is about 2.4 times as large as Oia, consistent with the corresponding normal state137Chapter 6. Data Analysis and Discussionconductivities estimated from the surface resistance data. Both Oia and 01b rise 6 to 7 timesfrom their values at T to the peak values around 42 K, showing again the rapid drop ofquasiparticle scattering rates in both directions. Throughout the range down to the lowtemperatures, Oib remains about a factor of two larger than cTia. At low temperatures, bothcurves are linear up to 15 K and somewhat sub-linear up to 30 K.It has been pointed out that the conductivity of a d2_ state superconductorapproaches a low temperature limit oo=ne2/m*rA(0) that is independent of the scatteringrate 85, 103 with n being the total carrier density, m* the effective mass, and A(0) themaximum gap at T=0. If one takes hI’r(T0) 2kBT and A(0)— 2kBT, then Ooo 0.3 Dc(TC).If we fit our low temperature conductivities to a straight line we obtain residualconductivities of Oia(T>0) (0.45±0.15) x106 11m’ — 0.45±0.15 Oia, DC (Ta) andOib(T—*0) — (0.7±0.2) x106 £2m’ — 0.35±0.10 oib, DC (Ta). The residual conductivity is nearour resolution limit and close to the predicted Ooo for a d-wave superconductor.The anisotropy in the penetration depth implies a value of 2.4 for the ratio (n/m*)b to(n/m*)a. This is the simplest explanation for the anisotropy in the conductivity. Othercontributions may come from the difference in the quasiparticle scattering rate in the twodirections or a favored position of the nodes in the gap toward one of the a and b directionsin k space, where the momentum a typical quasiparticle carries would be different in the twodirections. At the present stage, the error transferred to the real part of the conductivity fromthe measurement of A(0) and also the use of low frequency (1 GHz) A2(I) at 35 GHz138Chapter 6. Data Analysis and Discussion30.020.0-4‘I“ 1000.00.0 100.T(K)Figure 6.23 The real part of the conductivity in the two directions at 34.8 GHz. Note that itis the larger anisotropy in A that causes La > R3b,in spite of the fact that olb is larger that01a.20.0 40.0 60.0 80.0139Chapter 6. Data Analysis and Discussionoverturned any chance to distinguish the difference in the scattering rate in the twodirections. Therefore the role that ‘r plays in the anisotropy is not yet clear. The positions ofthe nodes in the gap are not easily probed by microwave techniques. Direct or indirectmeasurements involving large momentum (order of kF) but small energy may help determinethose positions.140Chapter 7ConclusionsWe have systematically studied the surface resistance of YBCO based materials. Themeasurements were performed using a 34.8 GHz cavity perturbation apparatus constructedas part of the Ph.D. project. For a sample of typical size (1.0 mm in length and 1.0 mm inwidth), the apparatus is capable of measuring the surface resistance from 1.3 K to 130 K,and has a sensitivity of about 30 4uQ for measurements of the average R in the plane and 50uQ for the R3 along a particular direction. An important part of the design is the ability tomove the sample, which is mounted on a slender sapphire rod or strip, in and out the cavityat the operating temperatures. This allows an in-situ measurement of the unperturbed cavityquality factor Q.The samples investigated were both twinned (with and without impurity doping) anduntwinned YBCO single crystals. In the next three sections, we summarize themeasurements on the various kinds of samples, and at the end of the chapter we presentconclusions for the whole thesis project.1) Undoped, twinned samplesThe surface resistance of undoped YBCO crystals at 34.8 GHz just below T and thebroad peak around 50K indicate a rapid drop of the quasiparticle scattering rate and a141Chapter 7. Conclusionsrelatively slow decrease in the number of quasiparticle excitations in the superconductingstate. This slow decrease of the excitation density at low temperatures indicates theexistence of low lying states, which suggests the presence of nodes in the gap function. Thelinear behavior of the surface resistance at low temperatures has no satisfactory explanationat the current stage.The lack of the coherence peak in the real part of the conductivity in the transitionregion suggests a non BCS s-wave behavior, a picture consistent with a d-wave gap, wherethe density of states has only a logarithmic singularity at the maximum gap position and alsothe coherence factor vanishes for the quasiparticle scattering across a line node on the Fermisurface’8”°4.The source of the broad peak around 50 K in the real part of the conductivityis the same as that for the surface resistance. Just as R does, o decreases linearly in T at lowtemperatures, a behavior which is not consistent with the exponential behavior of an s-wavesuperconductor nor with the quadratic law behavior expected for a simple d-wavesuperconductor.Our data have been fitted by Hirschfeld et al.85 They have obtained results for oi byusing the numerical results of Quinlan et al.96 for an inelastic scattering rate based on a spinfluctuation model with a d22 state, and an impurity scattering rate in the unitary limit. Theoverall shape of the fitting and the data agree reasonably well, but there are obviousdifficulties for the fit with regard to the linear behavior in the low temperature region. Theaveraged quasiparticle scattering rate was extracted from our data through the two fluid142Chapter 7. ConclusionsmodeL The data were compared to the quasiparticle scattering rates calcuiated for an s-waveand a d-wave superconductor associated with spin-fluctuation scattering by Quinlan et a196.The results from the d-wave model appear to follow the experimental data in the region 0.5T < T < T, where the dominant scattering process is expected to be inelastic.The measurement on YBCO with various oxygen contents indicates that both theoxygen vacancies in the CuO chains and the changes in doping of the holes in the Cu02plane have no large effect on the surface resistance of the material, although the effect on thenormal state resistivity and the transition temperature is more obvious. Therefore thequasiparticle scattering process remains mainly within the plane and is not very sensitive tothe small amount of oxygen vacancies existing on the chains.2) Zinc and nickel doped samplesThe experiments performed on zinc and nickel doped samples show that bothdopants generally suppress R below T and in particular suppress the broad peak in thetemperature dependence of the surface resistance, a result indicating that the inelasticscattering drops rapidly below T and thus the impurity scattering plays an important role inthe magnitude of R. The extreme sensitivity of the surface resistance to impurities explainsthe large variation of the R values in the literature, and the similarity of both magnitude andthe shape of the temperature dependence of R of doped crystals to that of the best filmssuggests that extrinsic factors no longer dominate R3 for the highest quality films. A surfaceresistance value of 130 uQ at 77K when scaled to 10 GHz was achieved for the 0.75%143Chapter 7. Conclusionsnickel doped sample, the lowest R value for YBCO materials of which we are aware. Thisresult suggests that the microwave loss of high temperature superconducting thin films mightbe improved by slightly doping the material with impurities.The doping effect on the real part of the conductivity, given that the doping levels donot appreciably change the penetration depth A, directly reflects the changes in R. Asexpected, the huge peak in the conductivity is suppressed by doping, with a stronger effectwith the larger doping concentration. The qualitative features at low temperatures remain thesame as that for the surface resistance. The data for zinc doped samples presented in thiswork have also been fitted to the d-wave model by Hirschfeld et a185 Although overallqualitative agreement is obtained, there are also obvious difficulties remaining with regard tothe details at low temperatures, as for pure crystals.3) Uniwinned crystalsA large anisotropy in the surface resistance within the ab plane of the untwinnedcrystals was revealed for the first time by our study. The qualitative features in the twodirections are similar. However, the surface resistance in the a direction is found to beremarkably larger than that in the b direction, and at low temperatures, the surface resistancein both directions is very linear, with the Rsa curve about 1.8 times steeper than that of Lb.The data on untwinned YBCO crystals also show a very small residual surface resistance,about 100 jiQ at 1.3K at 35 GHz. This result indicates that the twin boundaries mightcontribute to the R of twinned crystals, where a typical residual R is about 300 4uQ.144Chapter 7. ConclusionsThe real part of the conductivity was extracted using recently obtained values ofAa(7) and A.b(T). We found oib to be about twice as large as 1a, similar to the normal stateconductivity anisotropy. It is interesting to note that the large anisotropy in the penetrationdepth causes Rsa>Rsb, in spite of the fact that 1a< Oib. At low temperatures the oi’s alsoexhibit linear behavior in both directions. If we fit our averaged low temperatureconductivity to a straight line, we obtain residual conductivities ofo1(T—O) 0.6±0.2xlO0.4±0. 15o, DC (Ta). The residual conductivity is near our resolution limit and closeto the predicted ci(0) O.3o1,DC (Ta) for a d-wave superconductor by Lee andHirschfeld85’103 The experiments on twin free crystals not only allowed a determination ofthe anisotropic microwave properties of YBa2Cu3O6.95in the ab plane for the first time, butthey have also clearly demonstrated that the unusual features of the surface resistance in ourtwinned samples were not caused by twin boundaries. However, we concluded that much ofthe residual conductivity previously observed was in fact caused by twin boundaries.The broad peak in the surface resistance was later confirmed by other groups onYBCO single crystals, also using the cavity perturbation techmique(Mao et al.36, Achkir etal.45). The plateau in the surface resistance observed in some thin film samples (Ma et al. 32)and also in 0.15% zinc doped crystals can be well explained as due to a somewhatsuppressed peak in,caused by impurity scattering. Even stronger impurity scattering cancompletely suppress the broad peak, as observed in some thin film samples(Ma et al.32,Klein et al.39) and also in our 0.3 1% zinc and 0.75% nickel doped crystals.145Chapter 7. ConclusionsFinally, let us conclude the project by the following:1. For undoped crystals there is a broad peak in both the surface resistance and thereal part of the conductivity. The peak is attributed to the rapid drop of thequasiparticle scattering rate in the superconducting state.2. The position of the peak in i at 34 GHz is at higher temperature in comparisonwith that at 3.8 GHz, which further confirms the rapid drop of the scattering rate.3. As a result of the low scattering rate in the superconducting state, the R3 and oare extremely sensitive to a small amount of impurities, which has also beenconfirmed by our measurements on doped crystals.4. The temperature dependence of a clearly does not show an isotropic s-wavebehavior, for which the absorption would exponentially decrease at lowtemperatures, and would also have a coherence peak right below T. The overallshape of the real part of the conductivity agrees with the calculation from a dwave model; however, there are obvious difficulties for the theory with regard tothe details for the low temperature behavior. The averaged quasiparticlescattering rate derived from the two fluid model was explained by a d-wavemodel of spin fluctuations in the region 0.5 T < T < T.5. The residual real part of the conductivity extrapolated to zero temperature isquite small, and is roughly equal to two-fifths of its value at T, close to the valuepredicted by Lee103 and Hirschfeld85for a d-wave superconductor.6. Our study also provides information on how one might produce low loss, high146Chapter 7. Conclusionsquality thin films for commercial applications. In particular, the microwave lossof high temperature superconducting thin films may be improved by slightlydoping the material with impurities. Because the doping effect is most effective ata temperature relatively far away from the transition temperature, materials witha T higher than that of YBCO such as the Tl based compounds may be bettercandidates for reduction in R by doping.7. A large anisotropy in electrodynamic responses exists within the ab plane. Thedifferences in magnitude can largely be subsumed under an anisotropy in n/m’,with m* being the effective mass. Such an anisotropy explains most of thedifference in the magnitudes of Oia and 01b below T and is also consistent withthe overall size of the dc resistivity anisotropy above T.8. The linearity in both the surface resistance and the conductivity at lowtemperatures in both a and b directions indicates that the CuO chains are notlikely to be the sole source of the low-lying states responsible for the linear Tdependencies. 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Dworsky, Modern Transmission Line Theoiy andApplications, John Wiley & Sons, Inc. 1979.106)A more general description can be found in reference 87.155Appendix AIn this appendix some basic relations between the surface impedance Z and theconductivity, a; the penetration depth, ), are derived.Starting with Maxwell’s equations (omitting displacement current which is negligiblecomparing toj):aBVxE=-—,at AlVxH=j,plus Ohm’s law, j cyE, we obtain2V E=io-—.The solutions are therefore of the form1? — 17 —kz+io)tLJ—k A2H0 E0,/1(01where k = g/1cooi and Eo and Ho are the amplitudes of the fields at the surface of thematerial.The surface impedance according to the definition 3.1.1 is then:156=--=‘-‘°°Ii/10) A3I1 kwhere we let p =o for non-magnetic metal. A3 is just equation 3.1.2.The magnetic penetration depth is:B1laE‘=I—-c1= A4ojwdz ZoBdBAbove we have used Al, A2 and —- = iwB . A4 is just equation 3.1.5.dt157Appendix BWe would like to relate the elements of the scattering matrix of the network discussedin chapter 4 to the voltage transmision coefficient T and the reflection voltage ratio F’.For the resonant network between aa and bb in Figure 4.2, we denote the voltagesand the currents flowing to the network at aa and bb to be Va, Vb, Ia, and lb. The scatteringmatrix is defined by the equation105(Ca”1 (S11 S12”(ea”1 Bi4b)1S21 S22)fb)’where Ca and db are the reflection parameters and ea and fb are the incident parameters atports aa and bb correspondingly, and are defined byCa‘4={i-.fIb]ea =_f.+4JIa}=[q+.JIbJwhere Z0 is the characteristic impedance of the input and output transmission line. If thereflection parameter at a certain port is zero, then there is no reflection from this port.Similarly, a zero incident parameter at certain port indicates that there is no incident voltage158at the port. For example, if the incident parameter fb= 0, one has Vb= —Zoib. Noting that lb isdefmed flowing into the network at bb, this condition corresponds to the situation where thetransmision line connected to port bb is terminated by its characteristic impedance, i.e., thereis only an outgoing wave at position bb. Si1 can then be expressed as11eafbOB2-T_z+zo_where Z-VaIIa is the input impedance at aa. S21 can be found in a similar way:S21 =eafb=0B3= 2Vb==T.Above we have used the relation between Va and the incoming voltage Vi,, at port aa,v v Z±Z0 and the relation between Vb and output voltage V at port bb, VbVout.From the symmetry of the network, one hasS22=S11T, andS12=S21T.159Appendix CHere we would like to fmd the relation between the quality factor and the surfaceresistance of the walls of a cylindrical cavity operated at TE011 mode.The loss per unit time of the cavity isL=j!IHI2Rsds, Clwhere R is the surface resistance and H is the amplitude of the magnetic field given byequation 4.2.3. First let us consider one of the bottom plates with z=0. In this case theamplitude of the magnetic magnetic field isHr =AJ0’(kr),H9 = H =0.Using the formula for Bessel functions:fji2 (kr)rdr =the loss due to the bottom plate per unit time Lb isLbP= IAI2J(z). C2160The amplitude of the magnetic field on the cylindrical wall is= Il_-s =H =AJ0()sin.The loss due to the cylindrical wall is then= !Rs7rbhIAI2J(x). C3The total loss is 2Lb-i-L. The quality factor of the resonant cavity can be derived from theenergy stored and the loss per unit time from equation 4.2.5 , C2, and C3’°6:C00 2 1 3b22(1— ---)(---rbh+ k2h)R8-V0coep—2h 2,rbir(,b h2 )R3Above we have used the relation:22 2 2 /1_Uk =oe0ii—=a)08/.t(1——- .Equation C4 is just equation 42.6.161


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