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Microwave electrodynamics of the high-Tc superconductor Tl2Ba2CuO6+delta Aghigh, Seyed Mahyad 2014

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Microwave Electrodynamics of theHigh-Tc Superconductor Tl2Ba2CuO6+δbySeyed Mahyad AghighB.Sc. (Engineering Physics), Islamic Azad University, Science & Research Branch,Tehran, Iran, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2014c© Seyed Mahyad Aghigh 2014AbstractThe microwave surface resistance, Rs(ω,T), and the magnetic penetrationdepth, λ(T), of a rectangular Tl2Ba2CuO6+δ (Tl-2201) single crystal mea-sured along its different crystallographic axes are reported. The measure-ments of the surface resistance as a function of frequency were made using aprecise broadband bolometric technique, and a loop-gap resonator was em-ployed to measure the temperature dependence of the magnetic penetrationdepth.Disentangling the in- and out-of-plane components of both microwaveproperties was accomplished by comparing the measurement results ob-tained for different orientations of the sample with respect to the appliedmagnetic field, allowing us to report, for the first time, the c-axis componentsof ∆λ(T), and Rs(ω,T).In particular, our results show a quadratic temperature dependence of∆λc(T) in Tl-2201 which is similar to that in other anisotropic cupratessuch as BSCCO, and YBCO. Furthermore, in the case of the surface re-sistance, a sign change in the curvature of Rcs(ω) is observed. The originof this behavior is not yet understood. The ab-plane components of bothmicrowave properties behave similarly to those reported on other dopings ofthis material.The measurements of Rs(ω,T) and ∆λ(T) allow us to determine thecomplex conductivity of this material. Having Tc of 43 K, the sample studiedhere is in the middle of the overdoped side of the superconducting dome,where very few studies have been made. This particular sample possessesrelatively low quasiparticle scattering rates making the interpretation of themeasurement results more straightforward. The reliability of the results,current limitations, and further potential progress are also discussed.iiPrefaceThis thesis is based on the data obtained from two pieces of experimentalapparatus, desiged and developed by different members of the UBC super-conductivity group over the past two decades. The loop-gap resonator wasdesigend by Dr. Walter Hardy, and was made by a former PhD student,Saeid Kamal. The broadband bolometric spectrometer, used in this thesis,was made by Jake Bobowski. This apparatus, which is capable of performingthe measurements at higher friequencies, is a newer version of the previouslymade device by Patrick Turner. The superconducting sample, studied here,was grown by Darren Peets during the course of his PhD program.The disentangling procedure was suggested by the author, Seyed MahyadAghigh, and all the data collection and data analyses, which are presentedin chapter 3, were carried out by himself. The proposed potential researchimprovements, including the replica technique discussed in Appendix C, isalso considered as the author’s contribution to the research problem. Noneof the text of this thesis is taken from previously published or collaborativearticles.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . ixDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Superconductivity at a Glance . . . . . . . . . . . . . . . . . 11.1.1 What Is a Superconductor? . . . . . . . . . . . . . . 11.1.2 Conventional Superconductivity . . . . . . . . . . . . 51.1.3 Unconventional Superconductivity . . . . . . . . . . . 61.2 T l2Ba2CuO6+δ . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.1 Structure and Properties . . . . . . . . . . . . . . . . 101.2.2 Why Tl-2201? . . . . . . . . . . . . . . . . . . . . . . 111.2.3 Our Sample . . . . . . . . . . . . . . . . . . . . . . . 131.3 Microwave Electrodynamics of Superconductors . . . . . . . 151.3.1 Generalized Two-fluid Model . . . . . . . . . . . . . . 151.3.2 Microwave Surface Impedence . . . . . . . . . . . . . 192 Experimental Techniques and Results . . . . . . . . . . . . . 212.1 Magnetic Penetration Depth λ(T ) . . . . . . . . . . . . . . . 212.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . 212.1.2 Cavity Perturbation Technique . . . . . . . . . . . . . 242.1.3 Measurements of ∆λ(T ) . . . . . . . . . . . . . . . . 292.2 Microwave Surface Resistance Rs(ω) . . . . . . . . . . . . . . 33ivTable of Contents2.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . 332.2.2 Broadband Bolometric Technique . . . . . . . . . . . 352.2.3 Measurements of Rs(ω, T ) . . . . . . . . . . . . . . . 373 Disentangling the In- and Out-of-plane Components. . . . 413.1 General Complex Conductivity Tensor, σ(ω, T ), in Tl-2201 . 413.2 Analysis of the Penetration Depth Data . . . . . . . . . . . . 423.2.1 Disentangling Procedure for ∆λ(T ) . . . . . . . . . . 423.2.2 Results and Discussion . . . . . . . . . . . . . . . . . 433.3 Analysis of the Surface Resistance Data . . . . . . . . . . . . 473.3.1 Disentangling Procedure for Rs(ω, T ) . . . . . . . . . 473.3.2 Results and Discussion . . . . . . . . . . . . . . . . . 504 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60AppendicesA Relationship Between the Power Absorption and the Sur-face Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . 65B The Relationship Between the Slope of ∆λab(T ) and Tc of theSample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67C Using a Replica Sample for Disentangling Rs Components. 69vList of Tables1.1 Some conventional superconductors and their transition tem-peratures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Singlet pairing states allowed for a square CuO2 plane. . . . . 81.3 Physical and chemical properties of our Tl-2201 sample. . . . 143.1 Numerical values of the parameters used in analysis of the∆λ experimental data. . . . . . . . . . . . . . . . . . . . . . . 433.2 Fit parameters obtained through a power-law fitting of thein-plane surface resistance. . . . . . . . . . . . . . . . . . . . . 52B.1 Comparison of the ∆λ(T ) slope for two different samples ofTl-2201. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68viList of Figures1.1 Comparison of a superconductor and a perfect conductor. . . 21.2 Phase diagram of a typical Type-II superconductor . . . . . . 41.3 Superconducting gap in the k-space. . . . . . . . . . . . . . . 81.4 Generic temperature versus doping phase diagram of hole-doped cuprates. . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5 Two vertically attached primitive unit cells of Tl-2201 . . . . 111.6 Dependence of the two different crystal structures of Tl-2201on the copper substitution . . . . . . . . . . . . . . . . . . . . 121.7 Optical micrograph of the Tl-2201 sample measured through-out this thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . 141.8 Expected conductivities of a superconductor at different tem-peratures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.1 A platelet superconductor in a uniform magnetic field. . . . . 222.2 Schematic diagram of a cavity perturbation setup. . . . . . . 252.3 Schematic diagram of a loop-gap resonator apparatus and itsequivalent circuit. . . . . . . . . . . . . . . . . . . . . . . . . . 282.4 Cartoon of the sample orientation with respect to the mag-netic field when (→H ||→a ). . . . . . . . . . . . . . . . . . . . . 302.5 Cartoon of the sample orientation with respect to the mag-netic field when (→H ||→b ) . . . . . . . . . . . . . . . . . . . . . 302.6 Temperature dependence of the resonant frequency when (→H|| →a ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.7 Temperature dependence of the resonant frequency when (→H||→b ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.8 Comparison of the temperature dependence of ∆f(T ) in twosample orientations from 5 to 20 K. . . . . . . . . . . . . . . . 332.9 Schematic diagram of the broadband bolometric techniqueapparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35viiList of Figures2.10 Typical measured power spectrums for both the test and thereference samples. . . . . . . . . . . . . . . . . . . . . . . . . 382.11 Effective Rs of the Tl-2201 sample measured at four fixedtemperatures, when→H ||→a . . . . . . . . . . . . . . . . . . . . 392.12 Effective Rs of the Tl-2201 sample measured at four fixedtemperatures, when→H ||→b . . . . . . . . . . . . . . . . . . . 403.1 ∆λab(T ) = λab(T )− λab(5 K) for Tl-2201 (Tc = 43 K). . . . . 443.2 ∆λc(T ) = λc(T )− λc(5 K) for Tl-2201 (Tc = 43 K). . . . . . 453.3 Temperature dependence of the in-plane superfluid fraction. . 473.4 Temperature depencence of the out-of-plane superfluid fraction. 483.5 Low temperature behaviour of the out-of-plane superfluid. . . 493.6 A platelet sample with a width of w in an applied magneticfield along its length. . . . . . . . . . . . . . . . . . . . . . . . 503.7 In-plane microwave surface resistance, Rabs (ω), for Tl-2201. . . 523.8 Out-of-plane microwave surface resistance, Rcs(ω), for Tl-2201. 533.9 Comparison of in- and out-of-plane microwave surface resis-tance components in Tl-2201 at 1.5 K. . . . . . . . . . . . . . 543.10 Comparison of in- and out-of-plane microwave surface resis-tance components in Tl-2201 at 7 K. . . . . . . . . . . . . . . 543.11 Temperature dependence of the in-plane surface resistanceRabs (T ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.12 Temperature dependence of the out-of-plane surface resis-tance, Rcs(T ). . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.13 Frequency dependence of the in-plane conductivity, σab1 (ω, T ). 563.14 Frequency dependence of the cˆ-axis conductivity, σc1(ω, T ). . . 563.15 Comparison of the in- and out-of-plane conductivities at 1.5 K. 573.16 Comparison of the in- and out-of-plane conductivities at 7 K. 57A.1 A choice of the integration path on the surface of a conductor. 66C.1 Cartoon of the sample orientation (→H ||→c ). . . . . . . . . . . 70C.2 Temperature dependence of the resonant frequency in (→H|| →c ) orientation. . . . . . . . . . . . . . . . . . . . . . . . . . 72C.3 Frequency dependence of the ”apparent” Rs. . . . . . . . . . 73viiiAcknowledgementsThe past few years of my life have been quite different mainly due to mygrowing independence. As one could imagine, this process, although reward-ing in the end, involves numerous tough moments. The transition becomeseven more difficult if you happen to find yourself in a situation where al-most everything in your life has suddenly changed: the language in whichyou speak, the culture with which you should comply, the educational sys-tem, the major of study, even the weather, and more. These all probablychange for many international students, and I was one of them. The hard-ships I faced due to all these changes provided me with valuable experiencethat I could have never gained otherwise and, because of this, I would like totake this opportunity to thank everyone who supported me in this period.First of all I would like to thank my supervisor, Doug Bonn, who providedme with the opportunity to be part of his research group. Thank you Dougfor all your support, and for the freedom you gave me over the course ofmy master’s program. I always benefited from your broad knowledge basewhenever we spoke.Thank you, also, to Walter Hardy who helped me every single timeI needed it. Without Walter’s guidance I would not have been able tolearn even half of what I have learned in superconductivity and microwaveelectrodynamics. Thank you, Walter, for being so helpful always, and forbeing so patient when explaining even very basic concepts to me. I willnever forget your assistance.I feel extremely fortunate that I could benefit from the skills, knowledge,and expertise of James Day, Jordan Baglo, and Pinder Dosanjh during mytwo years in the superconductivity group. James, you are one of the bestcommunicators I have ever met in my whole life so far. The way you explaina concept, especially to someone whose first language is not English, is justamazing. Thank you for teaching me how to run the bolometer in sucha way that I could start collecting data the next day, and for the timelyreminders of how a typical graduate experience unfolds (e.g. it’s not alwayseasy!).ixAcknowledgementsJordan, you always amazed me with the detailed knowledge you have.Thank you for answering my questions every time I asked, and for helpingme with different computer issues, even when you didn’t seem to have thetime. Also, Thank you for letting me know how to run the loop-gap ap-paratus. Pinder, the most valuable thing I leaned from you was how to becareful, patient, and thoughtful before and while performing any task. Ivery much appreciate these skills, as well as the technical tricks you taughtme in different steps of my experiments. Moreover, you were the one whomainly dealt with my lack of proficiency in English in the very first months,and helped me learn what I needed to start my research. Thank you foreverything.I would also like to thank Ruixing Liang, Saeid Kamal, David Broun,Ahmad Reza Hosseini, Brad Ramshaw, Darren Peets, Andrew McDonald,Shun Chi, Nathan Evetts and Ludivine Chauviere for helpful discussions. Aspecial thank goes to Jake Bobowski and Patrick Turner whose comprehen-sive PhD theses were great guides for me throughout preparing this thesis.The great help of Amir R. Tamadon in preparing some of the diagramspresented in this thesis is also appreciated.Last but not least, I would like to thank my lovely family: those whohave always supported me in every possible way. Mom, you were the onealways encouraging me to move toward my goals without any hesitation.The self confidence you grew in my personality has been one of the mostimportant factors in my achievements. Thank you Mom. Dad, thank youfor all your support, and the freedom you gave me in my decisions. Mahdad,and Mohanna, you are the best brother and sister I have ever had (well, andthe only ones, of course :) ) Thank you for all your support from the otherside of the globe.xTo the one I love the most...xiChapter 1IntroductionThis chapter serves as the introduction to this thesis, and is meant to discussthe basics of superconductivity including the history, classifications, as wellas the theories related to the field. It is hoped that, by the end of thischapter, the reader will gain enough familiarity with the concepts and thenomenclature to be able to follow the discussions in the following chapters.The chapter has been divided into three sections: first we introduce theterm ”superconductor” and its different types. The more specific case ofthe unconventional superconductor, T l2Ba2CuO6+δ, is then discussed sincethis is the material studied throughout this thesis. Finally a short sectionis dedicated to the microwave electrodynamics of superconductors.1.1 Superconductivity at a Glance1.1.1 What Is a Superconductor?Perhaps one of the most basic questions to be answered in any review ofsuperconductivity is the title of this section: ”What is a superconductor?”.Let us first introduce the idea of a perfect conductor. A perfect conductoris a material whose electrical DC resistivity (electrical resistivity at zerofrequency) drops to zero when it is cooled down below a specific temperaturecalled the transition temperature (Tc). In other words, at temperaturesbelow Tc the conductivity of a perfect conductor is infinite (σ → ∞ forT < Tc).A superconductor is a perfect conductor that not only shows zero electri-cal resistivity, but also expels any magnetic field from its inside to within apenetration depth λ from its surface at T < Tc. The expulsion of magneticfields from the interior of a superconductor is called the Meissner effect,which is the key difference between superconductivity and the hypotheticalconcept of perfect conductivity1. The difference between the two is illus-trated in figure 1.1.1In fact, superconductors are the only perfect conductors. All other materials havefinite resistivity at any given temperature.11.1. Superconductivity at a GlanceFigure 1.1: Comparison of a superconductor (blue) and a perfect conductor(pink). (a) The two conductors are placed in a dc magnetic field while intheir normal state, T > Tc. In this situation, the magnetic field penetratesthrough both conductors without any disturbance. (b) While keeping theexternal magnetic field constant, the two conductors are cooled below Tc.The magnetic field inside the perfect conductor does not change, whereas themagnetic field inside the superconductor is expelled to within a penetrationdepth λ. (c) Now while keeping the two conductors at T < Tc, the externalmagnetic field is switched off. The magnetic field in the superconductorremain zero. In the case of the perfect conductor the internal magnetic fieldremains at B = Bext. This is due to the fact that for a perfect conductor σ →∞. This guarantees that the internal electric field is zero. This combinedwith Faraday’s law results in ∂B/∂t = 0. In other words B remains thesame inside the conductor. Figure provided courtesy of Jake Bobowski [1].21.1. Superconductivity at a GlanceSuperconductivity was first discovered by Heike Kamerlingh Onnes in1911 [2], three years after he successfully liquified helium. Onnes found thatthe resistivity of solid mercury suddenly vanishes when cooled below 4.2 K.The Meissner effect was then discovered in 1933 [3] by the German physicistsWalther Meissner and Robert Ochsenfeld to be the second defining char-acteristic of superconductors. Since then, numerous studies in condensedmatter physics have been devoted to both discovering new superconductorsand understanding what is responsible for the behavior of these materials.What is believed to be common among all superconductors, is the pairingof electrons at low enough temperatures and their condensation into the su-perconducting state. The pairing mechanism is believed to be understood insome superconductors, whereas in some others, it is still being investigated.Superconductors are categorized in different ways based of different cri-teria:By the response to an external magnetic field: Superconductorsare either Type-I, or Type-II. Type-I superconductors are those having a sin-gle critical magnetic field, above which the superconductivity is completelylost. Type-II superconductors, on the other hand, are those having twocritical magnetic fields: a lower one, Hc1, and an upper one, Hc2. Betweenthe two critical fields the material is said to be in the vortex state, wheremagnetic fields partially penetrate in the sample in the form of vortices. Thesuperconductivity in the material is completely lost only above the uppercritical field, Hc2. Figure 1.2 shows a generic mean-field versus temperaturephase diagram of a Type-II superconductor.By the transition temperature: Usually if the Tc of a superconduc-tor is less than 40 K it is called a ”low-temperature” superconductor, and ifthe Tc is more than 40 K, it is called a ”high-temperature” (or High-Tc) su-perconductor. This classification is, however, incomplete since it is possibleto reduce the Tc of a high-temperature superconductor to some value below40 K, while the material is still called a high-temperature superconductor.A better classification is the one based on the nature of electron pairing inthe superconductor, which is discussed below.By the nature of electron pairing: Based on how the electrons ina superconductor pair, the superconductor can fall under ”conventional”,or ”unconventional” categories. The two terms can be attributed to thepairing state, and the pairing mechanism. In the conventional pairing state,the pairs have neither net momentum, nor angular momentum [4]. This issimilar to the spin-singlet (s = 0, l = 0) s-shell of a hydrogen atom, beingthe reason such states are called ”s-wave”. The unconventional pairing statecan, however, possess some non-zero angular momentum corresponding to a31.1. Superconductivity at a Glancehigher value of l. For example, l = 2 represents the so-called d -wave pairingstate. The terms ”conventional” and ”unconventional” can also refer todifferent pairing mechanisms. These are discussed further in the followingsections.Figure 1.2: Phase diagram of a typical Type-II superconductor. Figureprovided courtesy of Jake Bobowski [1].41.1. Superconductivity at a Glance1.1.2 Conventional SuperconductivityConventional superconductivity is commonly taken to mean the type of su-perconductivity well described by the theory developed by Bardeen, Cooper,and Schrieffer in 1957, forty six years after the discovery of the first super-conductor. The three physicists received the Nobel prize in physics for thistheory in 1972.According to BCS theory, in the superconducting state, pairs of electronshaving equal and opposite momenta with zero total spin are formed. Theeffective attraction between the two electrons forming the so-called Cooperpair is achieved through the electron-phonon interactions. A simple pictureof this mechanism is as follows: Imagine an electron moving through a latticeof positive ions. The attraction of the ions to the electron slightly distortsthe lattice, making a temporary region of excess positive charge. This re-gion in turn attracts a second electron and in this sense the two electronsform a bound pair. The theory, and its more elaborate extension in Eliash-berg theory, provides a truly microscopic explanation of the conventionalsuperconductivity mechanism, and is regarded as one of the most successfultheories in condensed matter physics.Elemental superconductors are mostly conventional. Among the conven-tional superconductors discovered to date, the highest Tc, 39 K, was found inmagnesium diboride (MgB2). Table 1.1 presents some of the conventionalsuperconductors and their transition temperatures.Element Tc (K)Magnesium diboride (MgB2) 39.0Niobium (Nb) 9.26Lead (Pb) 7.19Vanadium (V) 5.30Tantalum (Ta) 4.48Mercury (Hg) 4.15Tin (Sn) 3.72Aluminium (Al) 1.20Titanium (Ti) 0.39Table 1.1: Some conventional superconductors and their transition temper-atures.51.1. Superconductivity at a Glance1.1.3 Unconventional SuperconductivityUnconventional superconductivity can deviate from the original ideas of BCStheory in several ways. A fundamental difference between conventional andunconventional superconductors is the symmetry of the superconductinggap function, ∆(k), defined as the energy difference between the supercon-ductor ground state and the lowest quasiparticle excitation. The zero totalangular momentum of Cooper pairs in conventional superconductors impliesrelatively isotropic attractive forces between the two electrons in all spatialdirections. This results in a relatively isotropic superconducting gap overthe Fermi surface as in figure 1.3(a). Details of the crystal structure can,of course, introduce anisotropies, but the gap never goes to zero.In contrast, in an unconventional superconductor, the electron pairingstate has finite angular momentum. This forces ∆(k) to vanish at certaindirections in the kx, ky-plane. The finite angular momentum of the pairingstate is related to the electronic structure of the material. For example, inthe case of the cuprates (a family of unconventional superconductors to beintroduced shortly), this is due to the electron correlations caused by thelarge Coulomb repulsion between electrons in the CuO2 planes [1]. Basedon group-theoretic calculations [5] for a square CuO2 plane, there are fourpossible distinct singlet pairing states. These are mentioned in table 1.2.The experimental evidence of a new pairing state, with nodes in thesuperconducting gap, was first provided by Hardy et al. [6], after the obser-vation of a linear temperature dependence of the London penetration depth,λ(T ), in YBCO. Combined with earlier NMR measurements of the Knightshift [7, 8], which proved that the orbital part of the pairing wavefunctionneed to be symmetric [4], these results suggested a d -wave density of statesin this material. Figure 1.3(b) shows the superconducting gap symmetryfor a d -wave superconductor.Another difference between the unconventional and the conventional su-perconductors is the pairing mechanism. In this context, a phonon-mediatedpairing, discussed above, is conventional, whereas any other kind of pairing,which is not mediated by phonons is considered as unconventional. Thepairing mechanism of unconventional superconductors is still subject of de-bates.61.1. Superconductivity at a GlanceSome of the most well known families of unconventional superconductorsare as follows:Cuprate superconductors: Cuprate superconductors2 have weaklycoupled copper-oxide (CuO2) planes in their structures. Being commonto all members of this family, the CuO2 planes are believed to be respon-sible for the superconducting behavior of these materials. Y Ba2Cu3O7,T l2Ba2CuO6, and Bi2Sr2CaCu2O8 are well known examples, usually re-ferred to as YBCO, Tl-2201, and BSCCO respectively. Cuprate supercon-ductors are either hole-doped or electron-doped. The undoped ”parent”compounds are insulators which have antiferromagnetic order at low enoughtemperatures.Iron-based superconductors: These are superconductors having lay-ers of iron bonded to a pnictide or chalcogenide. LiFeAs is a member ofthis family which superconducts close at the stoichiometric composition,meaning that no chemical substitution is required to make this materialsuperconducting.Heavy-fermion superconductors: Heavy-fermion superconductorsare materials in which the effective mass (m∗) of the conduction electronsis strongly enhanced. It is believed that in heavy-fermion systems, the un-conventional electron interaction is mediated by the spin fluctuations of anearby antiferromagnetic phase [10, 11]. UPt3 is an example of such ma-terials. URu2Si2 is a different heavy-fermion system which has no relatedantiferromagnetic phase. When cooled below 17.5 K, this material reorderits ground state to an unidentified order, the so-called ”hidden order”. Thenature of the hidden order state in URu2Si2 is still an open question. TheTc in heavy-fermion materials is usually much lower than that in cuprates.Focusing on the cuprates, another interesting subject to discuss is theirtemperature vs. doping phase diagram. Figure 1.4 shows the generic phasediagram of a typical hole-doped cuprate. The diagram shows how differ-ent thermodynamic phases of a cuprate can be reached by changing thetemperature and/or doping.At zero hole doping (defined conventionally as exactly one hole per planecopper atom), the material is an antiferromagnetic Mott insulator (AFI)(the so-called ”parent” material). By doping more holes, the AFI’s Ne´el2In the field of superconductivity, these materials are simply called cuprates, eventhough not all the materials referred to as cuprates are superconductors. In this thesis,wherever we talk about cuprates, we mean cuprate superconductors.71.1. Superconductivity at a GlanceInformal Group-theoretic Representative NodesName notation states+ A1g Const. Nones− (g) A2g xy(x2 − y2) Linedx2−y2 B1g x2 − y2 Linedxy B2g xy LineTable 1.2: Singlet pairing states allowed for a square CuO2 plane. Tableadopted from Ref. [1]Figure 1.3: Superconducting gap in the k-space for (a) an s-wave (conven-tional) and (b) a d-wave superconductor. The cylindrical Fermi surfaces aredepicted by the bold circles. The filled electronic states are in the hatchedregion. For an s-wave superconductor, the gap has the same sign in all di-rections, whereas in d-wave superconductors the sign and the magnitude ofthe gap is a function of direction in the kx, ky-plane. Figure taken from Ref.[9].81.2. T l2Ba2CuO6+δtemperature (TN ) is suppressed, and d-wave superconductivity (dSC) startsto appear. As more holes are introduced into the material, the transitiontemperature of the superconducting phase increases (underdoped region),peaks (optimally doped region), and decreases back to zero (overdoped re-gion). This creates the so-called superconducting dome across which someproperties of the superconductors change dramatically, while some othersdo not. An approximate empirical relationship between the transition tem-perature (Tc) of the superconducting state and the hole doping (p) has beendeveloped [12, 13] as follow:1− TcTmaxc= 82.6(p− 0.16)2 (1.1)At higher temperatures, in each of the three mentioned regions, otherphases exist, each possessing interesting physics. The normal state abovethe underdoped region is called the pseudogap phase. The density of statesnear the Fermi level in this phase is suppressed, but not fully gaped as in thesuperconducting state. The relationship of the pseudogap phase with the su-perconducting state is still debated. The phase above the optimal doping ofsuperconduting state (and above all the other phases at high enough temper-atures) can be successfully explained phenomenologically through marginalFermi liquid theory. However, no microscopic explanation is yet availableto describe the behavior of the materials in this region. Finally, at not veryhigh temperatures, as one leaves pseudogap to higher dopings, a large fermisurface consistent with bandstructure appears, and the resistivity becomesmore Fermi-liquid-like. It is believed that we have a microscopic understand-ing on this part of the phase diagram, referred to as Fermi liquid region.1.2 T l2Ba2CuO6+δHaving the concept of superconductivity and the different types introduced,we are now in a position to focus on the material studied in this thesis,T l2Ba2CuO6+δ (or Tl-2201 in abbreviation). In the following sections, firstthe structure and properties of Tl-2201 are introduced, and then the moti-vation for studying this material is discussed. Finally, the characteristics ofthe unique rectangular sample used in the course of this thesis is reviewed.91.2. T l2Ba2CuO6+δFigure 1.4: Generic temperature versus doping phase diagram of hole-dopedcuprates. The d-wave superconductivity (dSC) has a dome shape in thisdiagram.1.2.1 Structure and PropertiesTl-2201 is a member of the hole-doped cuprates having a body-centred crys-tal structure, and is found in two different crystal symmetries: tetragonaland orthorhombic. The material has one CuO2 plane per primitive unitcell as depicted in figure 1.5. The compound is, therefore, a single-layercuprate which is one of the simplest crystal structures in this family. TheCuO2 planes are widely spaced from one another, making the electrodynam-ics properties of this material highly anisotropic, meaning that the proper-ties along the cˆ-axis are very different than those along the aˆ- or bˆ-axis. Forexample, a resistivity anisotropy, (ρc/ρab), of greater than 1000 has beenreported for this material by Hussey et al. [14].Although T l2Ba2CuO6+δ is the nominal composition of this cuprate,in the actual crystal growth there is some substitution of copper for thal-101.2. T l2Ba2CuO6+δFigure 1.5: Two vertically attached primitive unit cells of Tl-2201. Thereare two CuO2 planes associated with the two primitive unit cells (hence, oneCuO2 plane per primitive unit cell). The cˆ-axis is vertical. Figure providedcourtesy of Darren Peets [15]lium in the T lO2 planes. This results in the actual composition to beT l(2−z)Ba2Cu(1+z)O6+δ. Whether the crystal structure of a sample is or-thorhombic or tetragonal is determined by both the copper substitution, z,and the oxygen excess, δ, as shown in figure 1.6.Tl-2201 is intrinsically on the overdoped side of the superconductingdome. The highest Tc in this cuprate (in optimally doped samples) is be-lieved to be at least 93 K [17]. By doping more holes, and therefore pushingthe material deep into the overdoped side, a Tc as low as ∼5 K has beenachieved without significant surface damage [15].Tl-2201 crystals are both difficult and hazardous to grow, owing to thetoxicity and volatility of thallium oxide. This means that single crystals arerare, especially those with the flat surfaces needed for microwave techniquesused in this thesis. This is one of the most important reasons why relativelyfew microwave studies have been done on Tl-2201.1.2.2 Why Tl-2201?In spite of the problems associated with preparing Tl-2201, there are anumber of reasons that tempt one to further investigate this material.Most importantly, Tl-2201 gives access to the overdoped side of the su-perconducting dome. Although many studies have been dedicated to the111.2. T l2Ba2CuO6+δFigure 1.6: Dependence of the two different crystal structures of Tl-2201 onthe copper substitution, (z), and the oxygen excess (δ). The CuO2 plane andits slice through the unit cell (in green) are shown for both systems (the or-thorhombic distortion is exaggerated for clarity). The gray bar correspondsto z=0.080. Figure provided courtesy of Darren Peets [16]underdoped and optimally doped regimes [18–20], relatively little informa-tion is available on the overdoped side. This is largely due to the lackof proper samples meeting the requirements of the different measurementtechniques.The overdoped regime can be also reached through La(2−x)SrxCuO4(LSCO); however, there are fundamental problems in interpreting the mea-surement results of this material. The problem with LSCO is that, as op-posed to most of other cuprates, a wide range of doping in this materialcannot be achieved by changing the oxygen content (adding or removingholes), but rather it is mainly accomplished through chemical substitutionof strontium on the lanthanum site. This type of chemical substitution pro-duces very strong quasi-particle scattering. Although Tl-2201 also suffersfrom some chemical substitution, these take place far away from the CuO2planes, affecting the electrodynamics of the superconductor much less in thismaterial than the case of LSCO.Another property making Tl-2201 (particularly in the tetragonal form)an ideal candidate to be studied, especially by means of microwave tech-121.2. T l2Ba2CuO6+δniques, is the simple crystal structure of this cuprate. Having only a singleCuO2 plane per unit cell and the same transport properties in every di-rection in-plane, tetragonal Tl-2201 is in fact one of the simplest cupratestructures. This is in contrast with the structure of YBCO cuprates, wherethe CuO chain layers make the interpretation of microwave properties moreinvolved [21].Finally, the rareness of this cuprate has left many of its properties unex-plored, and is a motivation to perform more investigations on this material.1.2.3 Our SampleAs mentioned before, to apply the microwave techniques used in this study,one requires single crystal samples having very flat surfaces. The size ofthe sample is another important factor: the larger the sample, the lesssystematic error is introduced into the results. The shape of the samplecan also be another key point when analyzing the data. In particular, thecalculations performed throughout this study to disentangle the in- and out-of-plane components of the measured properties are most accurate when thesample has a rectangular shape.Growing such samples has successfully been achieved in the case ofYBCO [22, 23]. This material cuts easily, allowing one to modify the size ofthe sample without damaging the surfaces significantly.For the Tl-based cuprates, however, the story is quite different. Most ofthe Tl-based samples, which meet the surface requirements of the microwavetechniques, tend to have random shapes. Moreover, these materials do notcut as easily as YBCO, for example. This restricts one to use the as-grownshape of the samples.The Tl-2201 sample studied in this thesis is unique in this aspect. Figure1.7 shows a micrograph of the basal face of our sample. As seen in thispicture, the shape is fairly rectangular. This makes the results of dataanalysis, discussed in Chapter 3, to have quite low systematic uncertainties.Table 1.3 summarizes the properties of our sample3.3The actual composition of the sample was provided by Darren Peets, who grew thesample.131.2. T l2Ba2CuO6+δFigure 1.7: Optical micrograph of the Tl-2201 sample measured throughoutthis thesis. There is some vacuum grease left on the surface of the sample.Even though vacuum grease does not react with the material very quickly,after each measurement the sample must be rinsed in heptane to remove thegrease. This prevents surface degradation which could otherwise take placeover longer periods of.Actual composition T l1.85Ba2Cu1.15O6.08Crystal system TetragonalTransition temperature Tc = 43 KMass m = 19.1 ± 0.4 µgVolumetric Mass Density ρ = 7.96 g/cm3Basal surface area As = 0.178 ± 0.003 mm2Width a = 230.0 ± 3.0 µmLength b = 775.0 ± 5.0 µmThickness c = 14.0 ± 2.0 µmTable 1.3: Physical and chemical properties of our Tl-2201 sample.141.3. Microwave Electrodynamics of Superconductors1.3 Microwave Electrodynamics ofSuperconductorsAs discussed in section 1.1.1, a superconductor is characterized by zero DCresistivity, and magnetic field exclusion from its interior at temperaturesbelow Tc. These properties were first formulated phenomenologically by theLondon brothers [24]. Based on this model, the electromagnetic response ofsuperconducting carriers with number density ns is governed by the Londonequation:∂→Js∂t =nse2m∗→E (1.2)with→J s, m∗, and e being respectively the supercurrent density, effectivemass and charge of superconducting electrons. Equation 1.2 combined withMaxwell’s equations results in a frequancy independent penetration depth,the London penetration depth:λL =√m∗µ0nse2. (1.3)This is the length scale at which the magnitude of the penetrated mag-netic field is reduced to 1/e of its initial value at the surface of the sample,assuming the repulsion of the field is only due to superconducting electrons4.λL for a typical superconductor is of the order of 1000 nanometers (or less)implying that the interior of a bulk superconductor is mainly free of mag-netic fields.1.3.1 Generalized Two-fluid ModelThe generalized two-fluid model is a useful phenomenological model to usewhen interpreting the electrodynamics of superconductors in AC fields. Themodel postulates that the conduction electrons in a superconductor are di-vided into two types: normal electrons with number density nn, and su-perconducting electrons having number density ns. These two act as twointerpenetrating and independent ”fluids”, determining the electromagnetic4In the next section we will see that there are normal electrons excited from the su-perconducting ground state (quasiparticles) which can contribute in the expulsion of themagnetic field at finite frequencies. Hence, the measured value of the penetration depth atfinite temperature and finite frequencies is not necessarily the London penetration depth,which only takes the superconducting electrons into account.151.3. Microwave Electrodynamics of Superconductorsresponse of a superconductor. The total density of electrons, n, is a con-stant (temperature independent) equal to the sum of the two densities ateach temperature:n = ns(T ) + nn(T ). (1.4)Above Tc all the electrons are in the normal state, i.e. n = nn at T > Tc.As the temperature decreases below Tc, normal electrons start to condenseinto the superconducting state, and ns rises from zero. Eventually, at T=0in the clean limit, all electrons are condensed into the superconducting statesuch that ns(T = 0) = n.The starting point to formulate the generalized two-fluid model is theDrude model. Based on this model, the equation of motion in metals isgiven by:md→vdt = e→E −m→vτ (1.5)with m being the electron mass, →v the average drift velocity of the electrons,e electron charge, and τ is the momentum relaxation time accounting for(normal) electrons collision. Equation 1.5 can be written in another equiv-alent form in which m∗ is the effective mass of the electrons, and→J is thecurrent density:d→Jdt = (ne2m∗σ −1τ )→J . (1.6)Assuming a harmonic time dependent applied electric field, equation1.6 is solved for σ = σ1 − iσ2 to give the real and imaginary parts of theconductivity:σ1 = σ011 + ω2τ2 (1.7)σ2 = σ0ωτ1 + ω2τ2 (1.8)where σ0 = ne2τm∗ . Based on the generalized two-fluid model the conductivitycan be written as the sum of the conductivities of the normal fluid and thesuperfluid:σ(ω) = {σ1n(ω)− iσ2n(ω)}+ {σ1s(ω)− iσ2s(ω)} . (1.9)161.3. Microwave Electrodynamics of SuperconductorsFor the superfluid, the relaxation time, τ , goes to infinity. There-fore, equation 1.7 implies that σ1s is a δ-function weighted by a factorof (pinse2/2m∗). This is obtained through integrating σ1 over all frequen-cies and using Kramers-Kro¨nig transformation which relates the real andimaginary parts of a causal response function5:σ1s(ω) =pi2nse2m∗ δ(ω). (1.10)At finite frequencies, however, σ2s determines the superfluid conductiv-ity, σs. Again allowing τ →∞, we obtain:σ2s(ω) =nse2m∗ω =1µ0ωλ2L(1.11)where in the last equality we have used equation 1.3. Thus, at nonzerofrequencies the total conductivity is given by:σ(ω, T ) = σ1n(ω, T )− i{σ2n(ω, T ) +1µ0ωλ2L(T )}. (1.12)The magnetic penetration depth, λ, is defined to be (µ0ωσ2)−1/2, whereσ2 is the term inside the brackets in equation 1.12. Hence, unless σ2n isnegligible at the measurement frequency used, a measurement of λ will notgive λL. Particularly, near Tc where the density of the normal electrons isquite high, σ2n is not negligible, preventing one from experimentally distin-guishing screenings due to the two ”fluids”, unless the measuring frequencyis low enough. On the other hand, well below Tc the condition σ2s >> σ2nis usually met, allowing one to obtain the superfluid density through mea-surements of the magnetic penetration depth:nse2m∗ω (T ) ≈1µ0ωλ2(T ). (1.13)Figure 1.8 taken from reference [25] summarizes the real and imagi-nary parts of the conductivity expected from a superconductor both in thesuperconducting state and the normal state.5A thorough derivation on this can be found in Appendix A of reference [1].171.3. Microwave Electrodynamics of SuperconductorsFigure 1.8: Expected conductivities of a superconductor at different tem-peratures. Figure taken from reference [25]181.3. Microwave Electrodynamics of Superconductors1.3.2 Microwave Surface ImpedenceThe complex surface impedance, Zs(ω, T ), is the experimentally measurablequantity in microwave measurements. Its real part, Rs(ω, T ), is the surfaceresistance, which sets the power absorbed by the sample in an electromag-netic field, and the imaginary part, Xs(ω, T ), is the surface reactance, beingmainly responsible for the exclusion of the fields from the interior of thesuperconductor.The surface impedance is defined as the ratio of the tangential electricfield and tangential magnetic field on the surface of the conductor in anelectromagnetic field. Assuming a conductor filling the half-space z > 0,this expression is written as:Zs(ω, T ) = Rs(ω, T ) + iXs(ω, T ) ≡Ex(z, t)Hy(z, t)∣∣∣∣z=0. (1.14)The two fields are related through the Maxwell’s equation:∇×→E= −µ0∂→H∂t (1.15)which in this case is written as:∂Ex(z, t)∂z = −µ0∂Hy(z, t)∂t . (1.16)In the local electrodynamics limit (where the coherence length6 is gen-erally much smaller than the penetration depth, ξ  λ), and in appliedoscillatory fields equation 1.14 and 1.16 result in:Zs(ω, T ) =iµ0ωκ =√iµ0ωσ (1.17)where σ is the complex conductivity. The last term in equation 1.17 isobtained relying on the fact that κ = √iµ0ωσ for a good conductor.6The coherence length, ξ, is a measure of the distance within which the concentrationof the superconducting electrons is not drastically changed in a spatially-varying magneticfield [26].191.3. Microwave Electrodynamics of SuperconductorsIn order to obtain the expressions for Rs(ω, T ) and Xs(ω, T ), one cansubstitute the general form of the complex conductivity, σ = σ1 − iσ2 intoequation 1.17 and then separate the real and imaginary parts. This givesthe following general expressions being valid for any ohmic conductor:Rs(ω, T ) =√µ0ω(√σ21 + σ22 − σ2)2(σ21 + σ22)(1.18)Xs(ω, T ) =√ µ0ω2(σ21 + σ22)σ1√√σ21 + σ22 − σ2. (1.19)For the case of a superconductor, if the temperature is not very close to Tcand the frecuency is not extremely high (recall figure 1.8) the approximationσ1  σ2 (or σ ≈ iσ2) is typically valid. In this case, the propagation constantis given by κ = √iµ0ωσ2 ≡ 1/λ(T ), and the general expressions of Rs andXs are simplified to the following:Rs(ω, T ) =12µ20ω2λ3(T )σ1(ω, T ) (1.20)Xs(ω, T ) = µ0ωλ(T ). (1.21)As seen in these equations, a single measurement of λ(T ) gives the surfacereactance, whereas measurements of both Rs(ω, T ) and λ(T ) are requiredin order to obtain σ1(ω, T ), the real part of the conductivity. In the fol-lowing chapter the microwave techniques used in this thesis to obtain thesequantities are discussed in detail.20Chapter 2Experimental Techniquesand ResultsThis chapter deals with the experimental part of this thesis. Two uniquemicrowave techniques, well established by the UBC experimental supercon-ductivity group, are introduced: a loop-gap resonator technique to measurethe change in the magnetic penetration depth ∆λ(T ) as a function of tem-perature, and a broadband bolometric technique developed to measure themicrowave surface resistance Rs(ω, T ) as a function of frequency for a rangeof fixed temperatures.The chapter has been divided into two sections each dedicated to oneof the techniques. Each section gives an introduction to the physical prop-erty being measured, the measurement technique, and the results obtained.Finally, preliminary discussions of the results, being a motivation for disen-tangling the in- and out-of-plane components of the measured values, arealso provided.2.1 Magnetic Penetration Depth λ(T )2.1.1 OverviewBefore going through the measurement technique and results, let us firstdiscuss the magnetic penetration depth, and the information contained inthe absolute value as well as the temperature dependance of this quantity.As discussed in section 1.1, a superconductor expels magnetic fields fromits interior, and this exclusion is produced by the supercurrents runningalong the surfaces of the sample. The length scale at which the magnitudeof the penetrating magnetic field is reduced to 1/e of its initial value on thesurface is called the magnetic penetration depth [26] and is usually denotedas λ.212.1. Magnetic Penetration Depth λ(T )The magnetic penetration depth is a fundamental property of a super-conductor. One can obtain the superfluid density from the absolute valueof λ. In particular, for simple models,1λ2 =µ0e2nsm∗ (2.1)where ns is the number density of superfluid electrons, µ0 is the permeabilityconstant,e is the electronic charge and m∗ is the effective mass of the carriers[27]. Information on the superconducting gap structure can also be obtainedthrough the temperature dependance of λ [28].Figure 2.1: A platelet superconductor in a uniform magnetic field appliedparallel to the crystallographic b-axis. In this orientation, the volume of thefield-free reigon only depends on λa, λc and the dimensions of the crystal.Different components of λ (λa, λb, and λc) are named depending onwhich crystallographic axis is along the direction of the supercurrent underconsideration, as shown in Figure 2.1. In the field of high-Tc supercon-ductivity, where the transport properties in the aˆ and bˆ directions are fairlysimilar but substantially different from those in the cˆ direction, it is commonto attribute the so called ”in-plane” and ”out-of-plane” terms to different222.1. Magnetic Penetration Depth λ(T )components of a sample property. In this terminology the component re-lated to the cˆ-axis is the ”out-of-plane” one and those related to aˆ- or bˆ-axisare the ”in-plane” components. In the case of the penetration depth λc isthe out-of-plane component whereas λa and λb are the in-plane components.In the case of tetragonal materials such as our sample the two in-plane com-ponents are equal and therefore no effort is required to distinguish betweenthese two. In other words λa = λb ≡ λab. This notation is used throughoutthis thesis from now on when talking about a component of the magneticpenetration depth in the ab-plane. This, however, cannot be used for or-thorhombic materials such as YBCO where λa 6= λb.While there are several techniques developed to measure the temperaturedependance of the magnetic penetration depth ∆λ(T ) reasonably accurately[29–32], measuring the absolute value of λ is not an easy task. There is a verylimited number of techniques [25] available to perform such measurements,most of which have poor accuracy. One technique used to determine theabsolute value of λ is Muon Spin Relaxation(µSR). This technique has widelybeen used to determine the in-plane components of λ(T) in the cupratesboth on the underdoped side of the superconducting dome [33, 34], and alsoon the overdoped side of the dome [35]. The magnetic penetration depthalong the cˆ-axis however, cannot be determined using (µSR). To measure λc,Kirtley et al. developed a scanning SQUID microscope [36] used to directlymeasure λc(T) in Tl-2201 single crystals [37]. Unfortunately, so far no suchmeasurements have been performed on Tl-2201 samples with Tc close tothat of the sample being studied in this thesis (43 K). This leaves us withan uncertain determination of λc in our sample, as will be discussed in thenext chapter.The temperature dependence of the magnetic penetration depth on theother hand has been widely measured in both conventional and unconven-tional superconductors [6, 38, 39]. Among different available methods, cav-ity perturbation is relatively popular, and is also used in this thesis. Herewe explain this technique only in the context of a customized experimentaldesign used for measuring ∆λ(T ) in very small samples of high-Tc super-conductors. The reader is invited to consult reference [25] to read about theother methods.232.1. Magnetic Penetration Depth λ(T )2.1.2 Cavity Perturbation TechniqueThe method of cavity perturbation can be used to determine both the changein the magnetic penetration depth, and the surface resistance of a sampleas a function of temperature. Figure 2.2 shows a schematic diagram of acylindrical cavity resonating in its TE011 mode before and after the intro-duction of a superconducting sample. In this mode there is an electric fieldnode and a magnetic field antinode along the central axis. In the middle ofthe cavity, magnetic field is fairly uniform. The microwaves are coupled into(TX) and out of (RX) the cavity through the small holes drilled in the wallof the cavity.Changes in two of the resonant properties can be measured and relatedto the change in the magnetic penetration depth, and the surface resistanceof the sample. These properties are respectively the resonance frequency f ,and the width of the resonance, W. The resonance frequency is a functionof cavity dimensions, and the width of the resonance is a measure of totaldissipated energy per cycle. The latter is related to the cavity quality factorQ through the following equation [1]:Q = 2pi peak energy storedenergy dissipated per cycle =fW (2.2)Without any sample inside the cavity, the width of the resonance (W0)is determined by the amount of energy dissipated on the walls of the cavity,and the resonance frequency is set by the cavity dimensions. Once thesuperconducting sample is introduced into the cavity, the exclusion of themagnetic field from the interior of the sample reduces the effective volumeof the cavity resulting in an increase in the resonance frequency to a newvalue of f + δf . Moreover, the extra power dissipation in the sample nowadded to that due to the walls results in an increased width of resonanceW > W0.The easiest formulation of the cavity perturbation problem can be madeusing a complex frequency notation [40]:ωˆ ≡ ω − i ω2Q (2.3)where ω = 2pif is the resonant angular frequency. The relative change in242.1. Magnetic Penetration Depth λ(T )Figure 2.2: Schematic diagram of a cavity perturbation setup. Resonantproperties of an empty cavity change when a sample is introduced. Thechange in the resonant frequency is due to the change in the effective vol-ume of the cavity, whereas the power loss in the sample is responsible forbroadening the resonance and therefore reducing the Q-factor.the complex frequency due to the sample insertion is therefore given by:∆ωˆωˆ ≈δff −i2δ( 1Q)(2.4)It can be shown that for a platelet superconducting sample of thicknessc, the change in the resonance properties is given by [27, 41]:δff −i2δ( 1Q)= Vs2Vc{1− 8pi2∑n=odd2n2(tanh(γna/2)γna/2− tanh(κnc/2)κnc/2)}(2.5)where Vs is the volume of the sample, Vc the effective volume of the cavity,a and c are respectively the width and the thickness of the sample in the252.1. Magnetic Penetration Depth λ(T )geometry shown in figure 2.1. Finally, γn and κn are the propagationconstants of the field through the basal face and the edge of the sample. Inthe superconducting state these are given by:γn = 1λab√(npiλca)2 + 1κn = 1λc√(npiλabc)2+ 1In the limit where λab  c and λc  a it can be shown that:δff =Vs2Vc{1− 2λabc −2λca}(2.6)This is a particularly simple result: the effective size of the sample is reducedby the appropriate penetration depth, λ, for each surface.In the actual experiment, the sample temperature can be set by a sap-phire hot finger, independent of the cavity temperature, as shown in Figure2.2. This allows tracking of the change in δf as the temperature is changed.Defining ∆f(T ) = δf(T ) − δf(T0), where T0 is a reference temperature, tobe the shift in the resonant frequency due to the change in the temperatureof the sample one has:∆f(T ) = −AsfVc{∆λab(T ) +ca∆λc(T )}(2.7)where As is the area of the basal face of the sample. In deriving the aboveformula the thermal expansion effects have been neglected, since such cor-rections are small for thin samples (c < 20µm) for the geometry shown infigure 2.1 at least for temperatures below 100 K [31]. These conditions areperfectly met in the case of the measurements throughout this thesis.The important point here is that the contribution of the cˆ-axis penetra-tion depth in the measurement is reduced by the factor of c/a as seen inequation 2.7. In flux-grown high-Tc superconducting single crystals the ra-tio c/a is only of the order of a few percent. This means that if the width ofa sample, which is not highly anisotropic, is much larger than its thickness,the contribution of ∆λc(T ) into the signal can be safely omitted. Thereforeequation 2.7 is reduced to:∆f(T ) = −AsfVc∆λab(T ). (2.8)262.1. Magnetic Penetration Depth λ(T )The c/a factor, however, cannot be carelessly neglected since ∆λc(T ) may bemuch higher than ∆λab(T ) in highly anisotropic materials such as Tl-2201as will be shown in the measurement results later in this chapter.This thesis is focused on the application of the cavity perturbation tech-nique, in which temperature dependent frequency shift yields ∆λ(T ). Thesurface resistance of the sample was measured by another technique dis-cussed later in this chapter since the low Q-factor of this cavity makes sucha measurement impossible. Another point here is that although for sim-plicity the measurement technique was described for the case of cylindricalmicrowave cavities, the measurements of ∆λab(T ) made in this thesis weredone using a loop-gap resonator. Both methods take advantage of the samemeasuring concepts, yet are different in some practical aspects discussedbelow.Let us return to equation 2.6. Considering this equation one can im-mediately realize that the sensitivity of the frequency shift measurementsis determined by the filling factor Vs/Vc. In other words, the higher thefilling factor a setup has, the higher the sensitivity of the measurements.The sensitivity of measurements in cylindrical cavities is quite acceptableat high frequencies where the cavities are relatively small. However, shouldone decide to perform such measurements at low frequencies, the sensitiv-ity of cylindrical cavities may not be high enough. This is due to the factthat at low frequencies a conventional cylindrical cavity would have to bevery large (e.g. Vc ≈ 103 cm3 at 1 GHz). Using this volume and that ofa typical platelet sample (Vs ≈ 10−5 cm3) results in a very small fillingfactor(≈ 10−8), and therefore very low sensitivity in the measurements.Moreover, in the case of cylindrical cavities, even for the TE011 mode, itis almost impossible to place the sample or sample holder in a region withoutelectric fields. This results in non-negligible background signals due to thedielectric response of the sample or sample holder to such fields.The above limitations were the motivation for designing a loop-gap res-onator operating at 940 MHz by the UBC experimental superconductivitygroup [6, 27]. Figure 2.3 shows a schematic of this apparatus as well as itsequivalent circuit. As shown in this figure, the loop-gap resonator is anal-ogous to an RLC circuit. The advantage here is that the gap (acting as acapacitor) keeps most of the electric field away from where the sample isplaced inside the loop (acting as an inductor). The signal therefore is notaffected significantly by the dielectric response of the sample/sample holderin the electric field. Furthermore, the relatively small effective volume ofthis resonator (≈ 0.178cm3) at its operating frequency allows a penetrationdepth measurement sensitivity of a few angstroms depending on the size of272.1. Magnetic Penetration Depth λ(T )the sample and the Q-factor of the resonator. The details of determiningthe effective volume have been explained in reference[1]Figure 2.3: Schematic diagram of a loop-gap resonator apparatus and itsequivalent circuit. The sample is inserted into the loop far from the gap sothat the fringing electric fields do not affect the signal significantly. Figureprovided courtesy of Jake S. Bobowski [1]Beside the advantages the loop-gap resonator has, there are also a fewdrawbacks associated with this setup. The main drawback is the drop inthe Q-factor due to aging. Getting a high Q-factor ( > 106) in this designrequires the joint between the loop-gap and the support base (”Supercon-ducting joint” in figure 2.3) to be lossless. To accomplish this, three screwsare used to tighten the loop and the support base which have both beenpreviously coated with Pb0.95Sn0.05 together. Pb0.95Sn0.05 is a conventionalsuperconductor with Tc ≈ 7 K. This temperature is well above the operatingtemperature of the apparatus (1.2 K), and therefore the joint is supercon-ducting while measurements are done. A weak link in this joint which couldeasily occur due to the aging of the apparatus results in a dramatic drop inQ-factor. In fact, at the time the measurements in this thesis were done, theQ-factor was as low as 3×105. This is less than one third of the initial valueof Q-factor in this resonator ( > 106). Having a relatively low Q-factor wasthe reason we could not measure the surface resistance of our sample usingthis technique.282.1. Magnetic Penetration Depth λ(T )2.1.3 Measurements of ∆λ(T )In order to disentangle the in- and out-of-plane components of microwaveproperties in Tl-2201, measurements of two different sample orientationswith respect to the magnetic field are required. Namely, the orientation inwhich the field lines are along the short axis of the sample basal face (→H ||→a )as in figure 2.4, and the one in which the field lines are along the long axisof the sample basal face (→H ||→b ) as in figure 2.5. In both orientations thedemagnetization factors are very low and therefore no effort is required tocorrect for such effects[27].Here we show the measurement results of ∆f(T ) and illustrate howdifferent behaviors of this quantity in the two orientations suggest a non-negligible contribution from the cˆ-axis component to the results and there-fore motivate us to perform further data analysis in order to separate thiscontribution from that of the ab-plane. The full discussion of the data anal-ysis however is postponed to the next chapter.Figure 2.6 shows the temperature dependence of the resonant frequency∆f(T ) = δf(T )−δf(5K) when→H ||→a . Figure 2.7 on the other hand showsthe same measurement but in the other orientation→H ||→b . To make thedifference between these two plots clearer, the low temperature behavior of∆f(T ) in both orientations has been plotted together in Figure 2.8.The surface area of the sample is the same in both of these measurementsand therefore the fact that different behaviors are seen in the results suggeststhat the cˆ-axis contribution cannot be neglected in these measurements onTl-2201. In other words, one must use equation 2.7 to relate the frequencyshifts to both the in-plane ∆λab(T ), and out-of-plane ∆λc(T ) contributions.292.1. Magnetic Penetration Depth λ(T )Figure 2.4: Cartoon of the sample orientation with respect to the magneticfield (black lines) when→H ||→a . The screening currents are shown by thered lines.Figure 2.5: Cartoon of the sample orientation with respect to the magneticfield (black lines) when→H ||→b . The screening currents are shown by thered lines.302.1. Magnetic Penetration Depth λ(T )0 10 20 30 4001000200030004000T (K) f   (Hz)0 5 10 15 20050100150200250T (K)  f   (Hz)Figure 2.6: Temperature dependence of the resonant frequency when→H ||→a .T0=5 K is the reference temperature. Inset: An expanded view of the samedata.312.1. Magnetic Penetration Depth λ(T )0 10 20 30 4001000200030004000 f   (Hz)T (K)0 5 10 15 20050100150200250  f   (Hz)T (K)Figure 2.7: Temperature dependence of the resonant frequency when→H ||→b .T0=5 K is the reference temperature. Inset: An expanded view of the samedata.322.2. Microwave Surface Resistance Rs(ω)5 10 15 20050100150200250  f   (Hz)T (K) H || b H || aFigure 2.8: Comparison of the temperature dependence of ∆f(T ) in twosample orientations from 5 to 20 K. The difference comes from the differentamounts of cˆ-axis contribution into the signal, ruling out the applicabilityof equation 2.8.2.2 Microwave Surface Resistance Rs(ω)2.2.1 OverviewAs discussed in section 1.3.2, in order to determine the real part of theconductivity, σ1(ω, T ), for the case of a superconductor, one has to measureboth the surface resistance, Rs(ω, T ), and the magnetic penetration depth,λ(T ), of the sample. In the previous section we discussed λ(T ), and howit is measured using the cavity perturbation technique. Here we focus onthe microwave surface resistance, Rs(ω, T ). As before, we first discuss thephysical interpretation of the quantity under consideration and the techniqueused in this thesis to measure this quantity. The measurement results thenconclude the chapter.332.2. Microwave Surface Resistance Rs(ω)The real part of the surface impedance is the surface resistance, Rs, andthis is the part determining the power dissipation in a sample. The quantityis a type of sheet resistance and is measured in Ohms (or more precisely asOhms per square).The method used in this thesis to measure the absolute value of thesurface resistance exposes the sample to microwave magnetic fields only.The power absorbed by the sample in such fields is related to the surfaceresistance through the following equation:P = 12RsH2surfA (2.9)where P is the power absorbed, A is the area of the basal face, and Hsurf isthe applied microwave magnetic field at the surface of the sample. KnowingA and Hsurf , one can determine Rs by measuring the power absorbed bythe sample in the magnetic field. The derivation of equation 2.9 is givenin Appendix A. There we show that this equation is simply the Ohmicloss. Microwave surface resistance in high-Tc superconductors has been thesubject of numerous studies in the past few decades [42–44]. Different tech-niques have been employed to measure Rs at frequencies in the microwaverange. The cavity perturbation method [45–48] is one such technique asdiscussed in section 2.1.2. In particular, Broun et al. studied the in-planeelectrodynamics of Tl-2201 (Tc=78 K) using this method[49]. Corbino ge-ometry spectroscopy [50] is another method for measuring the microwavesurface resistance of high-Tc superconductors.Although there are a few reports on the in-plane component of the sur-face resistance, Rabs (ω, T ), in the cuprate Tl-2201, to the best of our knowl-edge, no report has been made of Rcs(ω, T ), the out-of-plane component.Disentangling the in- and out-of-plane components of Rs for this supercon-ductor, therefore, could be interesting in that it provides the first reportedvalue of Rcs(ω, T ) for this material. Moreover, the previously performed stud-ies on this superconductor were performed on samples near optimal dopingregion (Tc=78 K) [49], or were highly overdoped (Tc ≈25 K) [51]. The Tcof our sample (43 K) lies between these values.To perform measurements of the microwave surface resistance, we useda unique technique designed and developed by the UBC superconductivitygroup [1, 52]. The technique gives the absolute value of Rs(ω, T ) as a func-tion of frequency for a range of fixed temperatures, given the sample beingmeasured is not too anisotropic. In the case of highly anisotropic materials,342.2. Microwave Surface Resistance Rs(ω)such as Tl-2201, more effort is required in order to extract the value of Rsalong the different crystallographic directions. This is explained in detail inthe following sections.2.2.2 Broadband Bolometric TechniqueThe broadband bolometric technique used in this thesis is a non-resonanttechnique, capable of measuring the very low level of power absorbed inhigh quality single crystals of superconductors. The low temperature endof the spectrometer is constructed from a rectangular coaxial line madefrom copper. The outer conductor of this transmission line is a rectangularchamber which is shorted to a copper blade, centered in this chamber, at anend-wall. The copper blade acts as the inner conductor of the transmissionline. The whole inside of the rectangular coaxial line is covered with lead-tin alloy, a superconductor at the operation temperatures. When operatedbelow 24.37 GHz, which is the cutoff frequency of the lowest transverseelectric (TE) mode in the spectrometer [1], this design produces symmetricalmagnetic fields around the inner conductor.Figure 2.9 shows a schematic cross-section of the rectangular transmis-sion line as well as one of the two stages used to place the probe sample(here Tl-2201), and the reference sample (made from silver:gold alloy with70:30 atomic %) in symmetric locations in the rf magnetic field.Figure 2.9: Schematic diagram of the broadband bolometric technique ap-paratus. The probe and reference samples are placed in symmetric locationswith respect to the copper blade in the center of the rectangular chamberusing two high quality sapphire plates. Figure provided courtesy of PatrickTurner [52]352.2. Microwave Surface Resistance Rs(ω)When exposed to the microwave magnetic fields, due to the absorptionof power, the temperature of the sample increases. The bolometer attachedto the sapphire plate measures the change in the sample temperature. Achip heater is used, before the actual measurement at each temperature, toimitate the microwave power absorbed by the sample. This determines a cal-ibration factor used to convert the bolometer response to the absolute powerabsorbed by the sample during the measurement. Finally, the quartz tubeacts as a thermal weak link isolating the sample from the low-temperaturebath, yet allowing the sample to be cooled to the base temperature givensufficient time. The stage for the reference side (not shown in the figure)has the same elements and acts in the same manner as the sample side. Theexistence of the reference sample is necessary to calibrate the strength ofthe magnetic fields as the frequency is varied. In fact, this is the key pointof this unique measurement technique as discussed below.Microwaves are delivered to the spectrometer using a standard 0.141inch cylindrical semirigid coaxial line. Due to the standing waves existingin the circuit used to deliver microwaves, the measured absolute power ab-sorption of both samples varies considerably from one frequency to another,as depicted in figure 2.10. The top and middle parts of this figure show thepower spectra of a test sample made of a pure tin platelet coated with 1 µmof AgAu alloy, and that of the reference sample respectively. These powerspectra are related to the surface resistances of both samples through equa-tion 2.9. Since both samples experience the same magnetic fields, Hsurf ,taking the ratio of the two measurements results in:P samP ref =RsamsRrefsAsamAref (2.10)where the superscript ”sam” and ”ref” label the sample and the referencerespectively. The reference conductor is a normal metal with strong scat-tering (because it is an alloy)7, and therefore, its surface resistance, Rrefs , isgiven by the classical skin-effect:Rrefs =ρδ =√µ0ωρ2 (2.11)Using this ratio technique, Rsams is easily determined in terms of P sam,P ref , Asam, Aref and Rrefs which are all known values. The bottom partof figure 2.10 shows a smooth curve obtained for Rs ratio (Rsams / Rrefs )7The reference conductor is in the so-called Hagen-Rubens limit where the carriersmean free path is shorter than the skin depth, δ.362.2. Microwave Surface Resistance Rs(ω)which is proportional to P sam/P ref , demonstrating the usefulness of theratio technique.The broadband bolometric technique discussed here is unique in that itprovides values of the surface resistance as a function of frequency on as finea scale as one wishes. This cannot easily be done by the cavity resonatorsdiscussed in the previous sections since each resonator works at a single fre-quency. There is, however, a drawback associated with the bolometric tech-nique compared to cavity perturbation: the broadband technique is limitedin the sense that the measurement sensitivity decreases as the temperatureis increased. This is due to the lack of sensitivity in the bolometer shown infigure 2.9 at temperatures higher than ∼ 10 K. Because of this, one cannotperform precise measurements at temperatures higher than about 10 K onsamples with low surface resistance. Cavity perturbation, on the other hand,can be used in a much wider temperature range. Hence, the cavity pertur-bation and the broadband techniques can be thought of as complementarymethods to obtain both the frequency and temperature dependence of thesurface resistance, Rs(ω, T ), for a sample of interest.There are several practical details which must be considered both in thedesign and the use of this apparatus. Such details are beyond the scope ofthis thesis, and therefore, if interested, the reader is referred to the PhDthesis prepared by Jake Bobowski [1].The surface resistance measurements for the Tl-2201 sample are pre-sented in the following section. There, it will be shown that in order tostudy highly anisotropic materials using the bolometric technique, one hasto perform some extra data analyses.2.2.3 Measurements of Rs(ω, T )The sample orientations with respect to the applied magnetic fields in theRs measurements are the same as those used in the ∆λ experiment:→H ||→aas in figure 2.4, and→H ||→b as in figure 2.5. In calculating the surfaceresistance, one usually assumes that only the basal faces of the sample cancontribute to the power absorbed. In other words, the sample is treated asa two-dimensional surface with negligible thickness. For a typical high-Tcsuperconductor platelet sample which is not too anisotropic, this assumptionis reasonable since the surface area of the basal faces in such platelets ismuch higher than the edge surface areas. Whether or not the sample isisotropic ”enough”, so that one can ignore the out-of-plane (on the edges)power absorption, is determined by performing the same measurement intwo different sample orientations such as those we chose. In particular, for a372.2. Microwave Surface Resistance Rs(ω)Figure 2.10: Typical measured power spectrums for both the test and thereference samples. Top: The absolute power absorbed by the test samplemade of a pure tin platelet coated with 1 µm of AgAu alloy. Middle: Theabsolute power absorbed by the AgAu alloy reference sample. Bottom: Asmooth curve is obtained when the Rs(ω) ratio of the two samples is taken.Figure provided courtesy of Jake Bobowski [1]382.2. Microwave Surface Resistance Rs(ω)material with tetragonal structure, the two measurements must essentiallygive the same results at a given temperature, if the anisotropy is not toohigh.However, this is not the case for highly anisotropic materials such asTl-2201. Figure 2.11 and 2.12 show the effective surface resistance mea-sured, at four different temperatures, in→H ||→a and→H ||→b orientationsrespectively. Considering that the material is tetragonal, and therefore theelectrodynamics are the same in the a and b direction, the only differencebetween these two orientations is the edge surface areas. A clear differ-ence is noticeable as the sample is rotated, which is a clear indication of anon-negligible cˆ-axis contributions to the results.0 4 8 12 16 20030060090012001500 7 K 5 K 3 K 1.5 K R  H || as() (GHz)Figure 2.11: Effective Rs of the Tl-2201 sample, measured at four fixedtemperatures, when→H ||→a . In this orientation the cˆ-axis contribution isless than that of→H ||→b .392.2. Microwave Surface Resistance Rs(ω)0 4 8 12 16 20030060090012001500 (GHz)R  H || bs()  7 K 5 K 3 K 1.5 KFigure 2.12: Effective Rs of the Tl-2201 sample, measured at four fixedtemperatures, when→H ||→b . In this orientation the cˆ-axis contribution ismore than that of→H ||→a .40Chapter 3Disentangling the In- andOut-of-plane Components ofthe Microwave PropertiesThe results shown in the previous chapter indicate a significant contributionfrom the currents running along the cˆ axis to the effective measured values ofboth the penetration depth and the microwave surface resistance. In otherwords, the measured values of ∆λ(T ) and Rs(ω, T ) for the case of Tl-2201are representative of neither the in-plane nor the out-of-plane propertiesof this cuprate, but rather a combination of both. Hence, unless one cansomehow determine the two different contributions, not a lot of informationon the electrodynamics of the material can be obtained.This chapter mainly deals with the process of disentangling the contri-butions, in order to determine the in-plane and the cˆ-axis components ofboth ∆λ(T ) and Rs(ω, T ). The importance of these quantities is discussedin the following short section.3.1 General Complex Conductivity Tensor,σ(ω, T ), in Tl-2201As discussed in section 1.3, the conductivity of a superconductor is a com-plex number which at non-zero frequencies is given by equation 1.12. Ifthe temperature is not close to the transition temperature, screening of thefields is mainly due to the superconducting electrons and the conductivitytakes the following form:σ(ω, T ) = σ1(ω, T )− i( 1µ0ωλ2(T ))(3.1)Considering the conductivity to be complex, one can write Ohm’s law,J=σ E for the case of a superconductor. This equation can also be writtenin the matrix form. In particular, if the applied field is expressed in the413.2. Analysis of the Penetration Depth Dataframe of the crystallographic axes, the conductivity tensor is diagonal andthe equation reads as:JxJyJz =σxx 0 00 σyy 00 0 σzzExEyEz (3.2)In the case of a Tl-2201 sample which has a tetragonal structure, takingthe cˆ-axis to be along the z direction results in σxx = σyy = σab, andσzz = σc, where σab and σc are the in- and out-of-plane components of thecomplex conductivity. Equation 3.2 can then be written as:JxJyJz =σab 0 00 σab 00 0 σcExEyEz (3.3)The two components of the conductivity are related to the associatedcomponents of the magnetic penetration depth and the surface resistancethrough following equations:σab(ω, T ) =2Rabs (ω, T )µ20ω2λ3ab(T )− i( 1µ0ωλ2ab(T ))(3.4)σc(ω, T ) =2Rcs(ω, T )µ20ω2λ3c(T )− i( 1µ0ωλ2c(T ))(3.5)Therefore, obtaining the in- and out-of-plane components of λ(T ) andRs(ω, T ) is important since these determine the complex conductivity of thematerial in the different directions.3.2 Analysis of the Penetration Depth Data3.2.1 Disentangling Procedure for ∆λ(T )In section 2.1.3, we discussed the response of the loop-gap resonator, hav-ing the superconducting sample inside, to a change in sample temperature.There, we showed how changing the orientation of the sample with respectto the applied field changes the measurement results, which is a clear indi-cation of non-negligible cˆ-axis contributions. Here, the in- and out-of-planecomponents of the penetration depth, λab(T ) and λc(T ), are disentangledusing a very simple geometric analysis.423.2. Analysis of the Penetration Depth DataRemember the general equation 2.7 which relates the measured frequencyshift to ∆λab(T ) and ∆λc(T ) for a geometry shown in figure 2.1. For the ge-ometries we performed the measurements in, this takes the following forms:∆f→H||→a (T ) = −AsfVc{∆λab(T ) +cb∆λc(T )}(3.6)∆f→H||→b (T ) = −AsfVc{∆λab(T ) +ca∆λc(T )}(3.7)where ∆f→H||→a and ∆f→H||→b are the measured frequency shifts in the geome-tries shown in figure 2.4 and 2.5 respectively. The important differencein the two equations is the factor by which the contribution of ∆λc(T ) isreduced. Solving equations 3.6 and 3.7 simultaneously, one can determinethe two components of the penetration depth, ∆λab(T ) and ∆λc(T ).Table 3.1 shows the known values used in the data analysis procedure.f 941.0 ± 0.6 MHzVc 177.90 ± 0.64 mm3As 0.178 ± 0.003 mm2a 230.0 ± 3.0 µmb 775.0 ± 5.0 µmc 13.7 ± 1.8 µmTable 3.1: Numerical values of the parameters used in analysis of the ∆λexperimental data.This procedure only works because we are dealing with a tetragonalstructure of Tl-2201. If, on the other hand, we had to perform the sameanalysis on a sample with orthorhombic crystal structure such as YBCO,the first terms in the brackets in equations 3.6 and 3.7 would not bethe same anymore, leaving us with an extra unknown value. In that casemore measurements are required. For example, the cleaving method usedby Hosseini et al. [53] to extract the cˆ-axis contribution in YBCO samples.This, however, cannot be performed in the case of Thallium-based crystalsas they do not cut as well as YBCO samples do.3.2.2 Results and DiscussionFigure 3.1 shows the temperature dependence of the in-plane magnetic pen-etration depth. At low temperatures (T ≤ 0.4 Tc), a strong linear term is433.2. Analysis of the Penetration Depth Dataobserved in ∆λab(T ). This behavior is consistent with the previous reportby Broun et al. [21] on a sample with Tc = 78 K. The slope of the lowtemperature linear term in the case of our sample (Tc = 43 K) is, however,23 A˚/K, which is ∼1.8 times as much as that reported for the sample withTc = 78 K(13 A˚/K). The fact that the slope of ∆λab(T ) scales with Tcis mainly due to the value of λab(0) in each sample, and is expected fromtheory. This is discussed further in Appendix B.Our data show an anomaly around 9.5 K which we found to be related toa weak link in the sample, and has nothing to do with the intrinsic propertiesof the material. This issue is discussed further in Appendix B.0 10 20 30 400200400600ab(T) - ab(5 K) (nm) Temperature (K)4 6 8 10 12 14 16 18 20010203040   T (K)Figure 3.1: ∆λab(T ) = λab(T ) − λab(5 K) for Tl-2201 (Tc = 43 K). Inset:An expanded view of the same data. Below 18 K, the data show a lineartemperature dependence of ∆λab(T ).The out-of-plane component of the penetration depth, on the other hand,shows no linear temperature dependence. Figure 3.2 plots ∆λc(T ) for thecase of our sample. The out-of-plane component follows a T 2 behavior asshown in the inset of figure 3.2. This behaviour of the cˆ-axis penetrationdepth in our Tl-2201 sample is similar to that in HgBa2Ca2Cu3O8+δ (Hg-443.2. Analysis of the Penetration Depth Data1223), YBCO6.7, and YBCO6.57 reported by Panagopoulos et al. [54], andthat in YBCO6.95 reported by Hosseini et al. [53]. The same feature hasalso been observed in La2−xSrxCuO4 [55], and Bi2Sr2Ca1Cu2O8+δ [56].0 5 10 15 20 25 30 35 4005101520c(T) (m) c(T) -c(5 K) (m)Temperature (K)250 500 750 100002468   T2 (K2)Figure 3.2: ∆λc(T ) = λc(T ) − λc(5 K) for Tl-2201 (Tc = 43 K). Inset:∆λc(T ) plotted vs. T 2 showing a quadratic temperature dependence up tonear Tc.An important issue to be pointed out here is that for equations 3.6and 3.7 to be valid one has to be in the limit where λab  c and λc  a, b.These conditions are violated near Tc since the dimensions of the samplecould be comparable to the relevant penetration depths, and therefore onehas to solve the general form of equation 2.5. This requires a much highervalue of the resonator Q-factor than we had at the time of the measurementsin this thesis. The reader should, therefore, be aware that the results of theanalyses in this section are not reliable at temperatures near Tc. To be morespecific, we do not trust these analyses to be true at temperatures higherthan ∼ 35 K.453.2. Analysis of the Penetration Depth DataAnother important quantity to be discussed is the superfluid fraction,λ2(0)/λ2(T ). To determine this quantity one needs to know both the tem-perature dependance of the penetration depth, ∆λ(T ), and its absolute valueat zero temperature, λ(0).λ(0) cannot be measured directly through our measurement technique;hence, in order to determine the absolute value of λab(0) for our sample, weinterpolated the µSR data, reported by Uemura et al. [35]. λab(0) for oursample was calculated in this way to be 195 ± 10 nm.Figure 3.3 shows the in-plane superfluid fraction in our Tl-2201 samplefor different values of λab(0) chosen between 185 nm and 205 nm. For thein-plane case, these values of λab(0) do not change the overal behavior ofλ2ab(0)/λ2ab(T ). Using higher values of λab(0) introduces more curvature tothe in-plane superfluid fraction. Except for the weak link anomaly effect,the in-plane superfluid fraction behaves similar to that reported for othersamples of Tl-2201 with Tc of 78 K [49] and 25 K [51].For Tl-2201, as opposed to several reports on the in-plane component,there is only one report on the absolute value of the out-of-plane penetrationdepth, λc. This was presented by Moler et al. [37], where the absolute valueof the cˆ-axis penetration depth at 4.2 K, λc(4.2 K), was measured to be ≈20 µm for single crystal samples with Tc around 75 K. To the best of ourknowledge, no other information on this quantity exists, which leave us withmore uncertainty in determining λ2c(0)/λ2c(T ), the cˆ-axis superfluid fraction,in our sample.Figure 3.4 shows the out-of-plane superfluid fraction for different valuesof λc(0) chosen between 15 µm and 45 µm. We believe the chance for λc(0) tobe out of this range is quite low, yet no guarantee could be given on this. Asone could clearly notice, the value of λc(0) is quite important in determiningthe behavior of λ2c(0)/λ2c(T ) and therefore unless more measurements ofλc(0) are carried out for this material no solid conclusion could be drawnhere.To see how it behaves at low temperatures, in figure 3.5 λ2c(0)/λ2c(T ) hasbeen plotted vs. T 2 for λc(0)= 20, 25, and 30 µm. Ignoring the weak linkeffect in the data (around T 2 = 90 K2), it seems that the cˆ-axis superfluidfraction changes as T 2 at low temperatures. The coefficient of this quadraticterm, however, depends on the choice of λc(0), which is currently unknownfor our sample. The quadratic temperature dependence of the out-of-planesuperfluid fraction at low temperatures has also been observed in BSCCO[21], another highly anisotropic material from the cuprate family.463.3. Analysis of the Surface Resistance Data10 20 30 40 500.000.250.500.751.00  ab(0) = 185 nm ab(0) = 190 nm ab(0) = 195 nm ab(0) = 200 nm ab(0) = 205 nmab(0) / ab(T)Temperature (K)Figure 3.3: Temperature depencence of the in-plane superfluid fraction, forTl-2201 (Tc = 43 K). Starting from low temperatures, λ2ab(0)/λ2ab(T ) de-creases linearly up to around 18 K where it starts to slightly curve downand decreases all the way to zero at T ∼ Tc. More data analysis are, however,required for the data near Tc to be reliable as discussed earlier.3.3 Analysis of the Surface Resistance Data3.3.1 Disentangling Procedure for Rs(ω, T )The procedure for separating the components of the surface resistance isquite similar to that in the case of the penetration depth. Again, a geometricanalysis was used to separate the ab-plane and cˆ-axis contributions into themeasured effective surface resistances, Reffs (ω, T ).To perform the analysis, we assumed:1. The in-plane component of the penetration depth is much smaller thanthe thickness of the sample. In other words λab(T )  c.2. Also, the out-of-plane component of penetration depth is much smallerthan the length and the width of the sample (λc(T )  a, b).473.3. Analysis of the Surface Resistance Data10 20 30 400.00.20.40.60.81.0  c(0) = 45 m c(0) = 40 m c(0) = 35 m c(0) = 30 m c(0) = 25 m c(0) = 20 m c(0) = 15 mc(0) / c(T)Temperature (K)Figure 3.4: Temperature depencence of the out-of-plane superfluid fraction,λ2c(0)/λ2c(T ), for Tl-2201 (Tc = 43 K). Choosing different values of λc(0),clearly, causes dramatic change in the curve.Recall that the maximum temperature at which we performed the Rsmeasurements is 7 K which is far below the transition temperature of oursample. Using the data from the previous section, the maximum possiblevalue of λab(7 K) is 215 nm. This is much smaller than the thickness of thesample, c = 13.7µm, and therefore we can be sure that the first assumptionis quite reasonable.The validity check for the second assumption, however, is more challeng-ing, since the absolute value of λc is unknown. If this assumption is violated,the sample is said to be in the ”thin limit”, and a correction for this effectwould need to be applied to the data (more information on this situationcan be found in reference [1]). We performed the corrections on the datanumerically using the root-finding routine of Mathematica and found themto be insignificant, so long as the absolute value of λc is not unrealisticallyhigh. In fact, the corrections result in almost no change in the data evenfor λc values up to 200 µm. We do not think that, at the measurement483.3. Analysis of the Surface Resistance Data100 200 300 4000.750.800.850.900.951.00 c(0) = 30 m c(0) = 25 m c(0) = 20 m c(0) / c(T)T2 (K2)Figure 3.5: Low temperature behaviour of the out-of-plane superfluid frac-tion for Tl-2201 (Tc = 43 K) plotted vs. T 2. Below 18 K, it seems thatf cs (T ) depends quadratically on the temperature.temperatures we chose, λc in our sample is anywhere close to or higher thanthis value, suggesting no need to perform the thin limit corrections.If the screening lengths are much smaller than the corresponding dimen-sions of the sample the effective surface resistance measured in the geometryshown in figure 3.6 is to a good approximation given by [21]:R(eff)s =wRabs + cRcsw + c (3.8)where R(eff)s , Rabs , and Rcs respectively denote the effective (measured), thein-plane, and the out-of-plane surface resistance. c is the thickness of thesample, and w, depending on the orientation, is either the length or thewidth of the sample, whichever is perpendicular to the applied field. Thisequation is simply saying that R(eff)s is a weighted average of the in- andout-of-plane surface resistances.For the geometries used in this thesis, equation 3.8 takes the following493.3. Analysis of the Surface Resistance DataFigure 3.6: A platelet sample with a width of w in an applied magnetic fieldalong its length.forms:• For→H ||→a as in figure 2.4R→H||→as =bRabs + cRcsb+ c (3.9)• For→H ||→b as in figure 2.5R→H||→bs =aRabs + cRcsa+ c (3.10)Solving these two equations simultaneously results in separation of Rabs andRcs.In the following section the two components of the surface resistanceare presented. These along with the results of the in- and out-of-planecomponents of the penetration depth are used to extract the real part of theconductivity, σ1(ω, T ) for our Tl-2201 sample.3.3.2 Results and DiscussionThe frequency dependence of the ab-plane and the cˆ-axis components of thesurface resistance, Rabs (ω) and Rcs(ω), for our sample are plotted in figure3.7 and 3.8 respectively.The in-plane surface resistance found to be related to the frequencythrough the following power law:Rabs (ω) = αωβ (3.11)The fit parameters (α and β) at each temperature are given in table 3.2,considering the surface resistance to be measured in µΩ. Both α and βslightly increase with the temperature except for the α(7 K) which is less503.3. Analysis of the Surface Resistance Datathan α(5 K). This anomaly could be due to the lack of precision in themeasurements at the highest temperature (7 K).As seen in table 3.2, the β values obtained here are quite close to two.This almost quadratic frequency dependence of Rabs (ω) is well-understoodfor a superconductor in the limit where ω is much less than the quasiparticlerelaxation rate, and has been reported in other samples of Tl-2201 [49, 51].In contrast to the in-plane component, the frequency dependence of thecˆ-axis surface resistance, seems to be less straightforward to interpret. Asobserved in figure 3.8, Rcs(ω) curves show different behaviors below andabove ∼12 GHz. More specifically, there seems to be a sign change in thecurvature of the Rcs(ω) as one passes ∼12 GHz. At the moment, we do nothave a solid explanation for this behavior.This is, in fact, the first report of Rcs(ω) in Tl-2201. Different experi-mental proposals are still being considered in order to verify whether or notthe observed behavior in this quantity is real and related to the intrinsicelectrodynamics of the material. One such potential study could be to usean apparatus described in reference [47] to obtain only the in-plane surfaceresistance, without any of the assumptions made in our analyses, and com-pare the results with Rabs (ω) presented here. The validity of our assumptionsand therefore our procedure could then be checked.For comparison purposes, figure 3.9 and 3.10 show the two componentsof the surface resistance at the minimum (1.5 K), and the maximum (7 K)measurement temperatures (the other two temperatures, not shown here,also provide quite similar plots).As the last discussion on the surface resistance, let us consider the tem-perature dependence of Rabs and Rcs. These are plotted in figure 3.11 and3.12 respectively, at seven different frequencies. In general, the out-of-planecomponent seems to be more temperature sensitive than the in-plane com-ponent.513.3. Analysis of the Surface Resistance DataTemp. α β7 K 2.53 1.965 K 2.67 1.893 K 2.28 1.871.5 K 2.18 1.85Table 3.2: Fit parameters obtained through a power-law fitting of the in-plane surface resistance data at the four different temperatures.0 5 10 15 2002004006008001000 Rabs() (GHz) 7 K 5 K 3 K 1.5 K Figure 3.7: In-plane microwave surface resistance, Rabs (ω), for Tl-2201 atfour different temperatures: 1.5 K, 3 K, 5 K, and 7 K, showing an almostquadratic dependence of Rabs (ω) on the frequency at all these temperaturesconsistent with earlier reports on the other samples of Tl-2201 [49, 51]. Blacksolid lines are the fits to the data using the fit parameters in table 3.2.523.3. Analysis of the Surface Resistance Data0 4 8 12 16 200200040006000800010000 Rc s() (GHz) 7 K 5 K 3 K 1.5 KFigure 3.8: Out-of-plane microwave surface resistance, Rcs(ω), for Tl-2201at four different temperatures: 1.5 K, 3 K, 5 K, and 7 K. It seems that thereis a sign change in the curvature of Rcs(ω) above 12 GHz.In- and Out-of-plane Conductivities Knowing the values of the sur-face resistance, Rs(ω, T ), and the penetration depth, λ(T ), we can calculateσ1(ω, T ), the real part of the complex conductivity. In the superconductingstate, these quantities are related through the following formula(see equation1.20):σ1(ω, T ) =2Rs(ω, T )µ20ω2λ3(T )(3.12)Figure 3.13 and 3.14 show the frequency dependence of the in- andout-of-plane conductivities respectively. The in-plain component shows avery weak frequency dependence between 5 to 20 GHz range, whereas theout-of-plane component decreases with frequency as ω− 12 .Finally, one can clearly see that the in-plane conductivity is five orders ofmagnitude higher than the out-of-plane conductivity. Figure 3.15 and 3.16compare the two components at 1.5 K and 7 K respectively. In this scale,the frequency dependence of the cˆ-axis conductivities is not noticeable, andboth σab1 (ω) and σc1(ω) look pretty flat above 5 GHz.533.3. Analysis of the Surface Resistance Data0 5 10 15 20010002000300040005000 Rcs Rabs    (GHz)Rs()Figure 3.9: Comparison of in- and out-of-plane microwave surface resistancecomponents in Tl-2201 at 1.5 K.0 5 10 15 20020004000600080001000012000 Rcs Rabs  Rs() (GHz)Figure 3.10: Comparison of in- and out-of-plane microwave surface resis-tance components in Tl-2201 at 7 K.543.3. Analysis of the Surface Resistance Data0 1 2 3 4 5 6 703006009001200   Rabs() 20  GHZ 17  GHZ 15  GHZ 12  GHZ 10  GHZ 7.5 GHZ 5.0 GHZTemperature (K)Figure 3.11: Temperature dependence of the in-plane surface resistanceRabs (T ).0 1 2 3 4 5 6 70200040006000800010000   Rc s() 20  GHZ 17  GHZ 15  GHZ 12  GHZ 10  GHZ 7.5 GHZ 5.0 GHZTemperature (K)Figure 3.12: Temperature dependence of the out-of-plane surface resistance,Rcs(T ).553.3. Analysis of the Surface Resistance Data4 8 12 16 2003691215ab() (106 -1m-1) (GHz)   7 K 5 K 3 K 1.5 KFigure 3.13: Frequency dependence of the in-plane conductivity, σab1 (ω, T ).4 8 12 16 2003691215   7 K 5 K 3 K 1.5 Kc() (-1m-1) (GHz)Figure 3.14: Frequency dependence of the cˆ-axis conductivity, σc1(ω, T ).563.3. Analysis of the Surface Resistance Data4 8 12 16 2003691215  () (106-1m-1) (GHz)ab( ) c( )Figure 3.15: Comparison of the in- and out-of-plane conductivities at 1.5 K.4 8 12 16 2003691215   ab( ) c( ) (GHz)) (106-1m-1)Figure 3.16: Comparison of the in- and out-of-plane conductivities at 7 K.57Chapter 4ConclusionsThe temperature dependence of the magnetic penetration depth, ∆λ(T ),and the frequency dependence of the microwave surface resistance, Rs(ω),were measured for a Tl-2201 single crystal (Tc=43 K), using two differenttechniques: a 1-GHz loop-gap resonator, and a broadband bolometric tech-nique respectively.The fortunate rectangular shape of the sample, along with the tetrago-nal structure of the material, allowed us to separate the in- and out-of-planecomponents of the measured microwave properties, for the first time, in Tl-2201. This was accomplished by performing the measurements in two differ-ent orientations of the sample with respect to the applied field. Combinedwith the absolute values of the penetration depth at ”zero” temperature, wecould determine the superfluid fraction, λ2(0)/λ2(T ), and the conductivityspectrum of the thermally excited quasiparticles, σ1(ω), in the supercon-ducting state.The in-plane magnetic penetration depth showed a linear temperaturedependence below 17 K (0.4 Tc), with the slope of 23 A˚/K. This is anexpected behavior of a d -wave superconductor with line nodes in the en-ergy gap. The out-of-plane component of the penetration depth, however,changes quadratically with temperature. The same behaviors were obtainedfor the in- and out-of-plane components of the superfluid fraction. Tempera-ture dependencies of this type have also been observed in the case of YBCOand BSCCO, on the underdoped side of the superconducting dome. One can,therefore, conclude that the linear temperature dependence of the in-plane,and the quadratic temperature dependence of the out-of-plane componentsof both magnetic penetration depth and superfluid density are character-istics of any d -wave superconductor with a quasi-2-dimensional structurecomprised of cylindrical Fermi surfaces.The in-plane surface resistance of the Tl-2201 sample scales as ∼ ω2.This is expected for a superconductor in the limit where ωτ  1, and hasbeen reported in samples of Tl-2201 with other dopings. The out-of-planecomponent of the surface resistance did not show any power law frequencydependence, but rather a sign change in the curvature of Rcs(ω) was observed58Chapter 4. Conclusionsaround 12 GHz. The origin of this behavior is not yet understood.Finally, the in- and out-of-plane components of the real part of the mi-crowave complex conductivity were calculated using the values of λab(0)=196 nm, and λc(0)= 25 µm, estimated from other reports for the penetrationdepths in Tl-2201. The relative change of the in-plane conductivity with fre-quency is quite weak. Considering a Drude-like conductivity as in equation1.7, this weak dependence of the conductivity on the frequency is anotherimplication of ωτ  1 (or equivalently ω  τ−1), for the frequencies usedin this study. In other words, the scattering rate, τ−1, in Tl-2201 is muchhigher than our maximum measurement frequency (20 GHz).The in-plane component of the conductivity in our sample was foundto be ∼ 107 (Ωm)−1, which is comparable to that reported in the previousstudies. 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R. Cooper, T. Xiang, G. B. Peacock, I. Gameson,P. P. Edwards, W. Schmidbauer, and J. W. Hodby, Physica C 282,145 (1997).[55] T. Shibauchi, H. Kitano, and K. Uchinokura, Phys. Rev. Lett. 2, 2263(1994).[56] T. Jacobs and S. Sridhar, Phys. Rev. Lett. 75, 4516 (1995).[57] A. R. Hosseini-Gheinani, The Anisotropic Microwave Electrodynamicsof YBCO., Ph.D. thesis, University of British Columbia (2002).64Appendix ARelationship Between thePower Absorption and theSurface ResistanceIn this Appendix, equation 2.9, relating the surface resistance to the powerabsorbed by the sample, is derived.When in an electromagnetic field, the total power absorbed per unit areaby any conductor is given by the real part of the complex Poynting vector,S, at the surface of the conductor through the following equation:dPdA =12Re[nˆ.S]= 12Re[nˆ. (E ×H)](A.1)where nˆ is an outward unit vector normal to the surface of the conductor.Considering the same geometry as in section 1.3.2, where the conductorfills the half space z > 0 and the magnetic field is along the y-direction,equation A.1 becomes:dPdA =12Re[Ex0Hy0]= 12Re[ZsH2y0]= 12RsH2y0= 12RsH2surf(A.2)Integrating the power over the whole area of the sample then results inequation 2.9.65Appendix A. Relationship Between the Power Absorption and the Surface ResistanceNow we are going to show that equation A.2 is actually the Ohmic lossper unit area. Consider the differential form of Ampere’s law:∇×H = J (A.3)in which H is the magnetic field and J is the surface current density inA/m2. Integrating A.3 over an area of the conductor filling half-space (asin figure A.1) and using Stoke’s theorem yields:∫∫∇×H.dS =∫∫J .dS∮H.dl = Js.l(A.4)where l is the length along the surface, and Js is the surface current per unitlength (along y) in A/m. Taking the integration path to be like the red loopin figure A.1, we obtain:Hsurf .l = Js0.lHsurf = Js0(A.5)where Hsurf is the applied field, Js0 is the current density integrated intothe interior. Substituting A.5 in A.2 gives:dPdA =12RsJ2s0 (A.6)which is recognized as the Ohmic loss per unit area.Figure A.1: A choice of the integration path on the surface of a conductor.The conductor fills half-space z > 0 with an applied magnetic field alongthe y-direction. The red line shows our choice of the integration path. Theonly non-zero part of the integral in A.4 is along the lower part of the path.66Appendix BThe Relationship Betweenthe Slope of ∆λab(T ) and Tcof the Sample.In this appendix the theory behind the fact that the slope of ∆λab(T ) scaleswith Tc of the Tl-2201 sample is discussed. Here, we only consider thein-plane components, and therefore, the subscript ”ab” is omitted for sim-plicity.Recall figure 3.3 showing the temperature dependence of the in-plane su-perfluid fraction. The superfluid fraction, nsn , and the normal fluid fraction,Xn = nnn are related to each other through the following equation:ns(T )n =λ2(0)λ2(T ) = 1−Xn(T ) (B.1)Taylor expanding λ(T ) around T = 0 up to the first order gives:λ(T ) = λ(0)√1−Xn(T )≈ λ(0) +[λ(0)2dXn(T )dT∣∣∣∣T→0]T + ...(B.2)Therefore ∆λ(T ) is given by:∆λ(T ) = λ(T )− λ(0)≈[λ(0)2dXn(T )dT∣∣∣∣T→0]︸ ︷︷ ︸ηT (B.3)where η is the slope of ∆λ(T ) at low temperatures. One should be awarethat equation B.3 is an approximation, since it ignores the higher powersof temperatures. As one can clearly see, the slope is proportional to thevalue of λ(0), and because in Tl-2201, λ(0) scales with Tc (the lower Tc, the67Appendix B. The Relationship Between the Slope of ∆λab(T ) and Tc of the Sample.higher λ(0)) [35], one would expect that the slope in ∆λ(T ) also scales withTc.Table B.1 compares the slope of ∆λ(T ) in two different samples oftetragonal Tl-2201. As expected from the approximate expression B.3, dueto a higher value of λ(0), the slope in our sample is higher than that in thesample with Tc=78 K. In both cases, the calculated value obtained is ∼25%less than the measured value. This difference is due to the fact that in B.3,we only kept the first term.Tc (K)dXn(T )dT λ(0) (A˚) η (approx.) η (exp.) Ref.43 0.017 1960 17 A˚/K 23 A˚/K This study78 0.012 1650 10 A˚/K 13 A˚/K [49]Table B.1: Comparison of the ∆λ(T ) slope for two different samples of Tl-2201 having different Tc’s. In both cases the measured slope is ∼25% higherthan the value obtained from the approximate expression B.3. The trendis however the same both in the experimental and theoretical results.68Appendix CUsing a Replica Sample forDisentangling RsComponents.This last appendix discusses another possible procedure through which onecan measure the in-plane component of the surface resistance in any tetrag-onal sample without any assumptions or simplifications being made.In the broadband bolometric technique described in section 2.2.2, con-sider the situation in which the sample is oriented such that→H ||→c as infigure C.1. As shown in this figure, this orientation results in only in-planescreening currents, and therefore in the power absorption there is no con-tribution from the cˆ-axis component of the Rs. However, due to the highdemagnetizing factor in this orientation, the strength of the magnetic fieldon the surfaces of the sample is much higher than that experienced by thereference alloy. This means that equation 2.10 is not valid anymore. In-stead, one has to use the following equation:P samP ref =(H ′H)2 RsamsRrefsAsamAref (C.1)As seen in C.1, unless one knows the ratio of the magnetic fields oneither sides, H ′/H, determining the value of surface resistance is impossible.A possible trick to obtain this information is to use an isotropic replicasample. The replica must:• Be a superconductor at the measurement temperatures so that it repelsthe fields in the same manner that the original sample does.• Be isotropic so that the in- and out-of-plane Rs are the same.• Satisfy the local electrodynamic limit as in the case of the originalsample.69Appendix C. Using a Replica Sample for Disentangling Rs Components.Figure C.1: Cartoon of the sample orientation (→H ||→c ). The black linesshow the magnetic field lines. The screening currents in this orientation areonly in-plane as shown by the red line. Due to the high demagnetizing factorin this orientation, the magnitude of the field on the edge surfaces of thesample (H ′) is much higher than that of the applied field (H) experiencedby the silver-gold alloy.• have the same dimensions, and the surface roughness as the originalsample.For now, let us assume that we have such a replica sample. The proce-dure through which one can obtain the ratio H ′/H is as follow:1. Measure the power absorbed by the replica in→H ||→c orientation. Thisis given by the following equation in which Aedge = 2c(a+b) is the areaof the edge surfaces and Rreps is the surface resistance of the replica:PH||c = 12Rreps AedgeH ′2 (C.2)2. Measure the power absorbed by the replica in→H ||→a orientation. Thisis given by:PH||a = 12Rreps AbaseH2 (C.3)where Abase = 2ab is the surface area of the basal faces of the replica.70Appendix C. Using a Replica Sample for Disentangling Rs Components.3. Divide C.2 by C.3 :PH||cPH||a =[RrepsRreps] AedgeAbase(H ′H)2(C.4)As clearly seen, the isotropy of the replica allows us to cancel out theRreps during the division. Knowing the value of the surface areas and themeasured powers allows us to obtain the magnetic field ratio. Once the fieldratio is known, one can substitute this value in equation C.1 to obtain thein-plane component of the surface resistance in the original sample.There is an important point here one must be careful about: The tem-perature of the measurements are not necessarily the same for the originalsample and the replica. In fact, what has to be the same in the two casesis the in-plane magnetic penetration depth. Therefore, knowledge of thetemperature dependence of the magnetic penetration depth in the replica isalso required before performing the bolometric measurements.A good choice of material to make the replica from is NbZr(1%) alloy.Having body-centered cubic structure (therefore being isotropic), niobiumis a superconductor below 9.2 K. The 1% of zirconium in the alloy providesscattering and avoids non-local effects.Making a replica with the same size as the original sample is, however,not an easy task, at least in the case of our sample with the thickness of ∼14µm. What makes the task difficult is the need for parallel, mirror-finish, andnot-mechanically-damaged surfaces. For such a small thickness it is next toimpossible to accomplish this without automated polishing techniques whichare expensive. Even if one polished the alloy to the required thickness, theshape of the original sample must be cut out of it in such a way that theedges are not rounded. This latter task can be done relatively easily usinga Focused Ion Beam (FIB) facility.The problems mentioned prevented us from using the replica techniqueto extract the in-plane Rs. However, we performed the measurements on theoriginal sample in the→H ||→c orientation both in the broadband bolometricand the loop-gap resonator apparatuses which are presented here. In partic-ular, the measurements with the loop-gap resonator revealed the existenceof a weak link in the sample whose effect we first saw as an anomaly in thepenetration depth data in section 3.2.2.Figure C.2 shows the temperature dependence of the resonant frequencyin the case of our Tl-2201 sample in the (→H ||→c ) orientation. In contrastto theoretical expectation, the resonant frequency shift shows a power de-pendence. We attribute this behavior to a weak link in the sample. By71Appendix C. Using a Replica Sample for Disentangling Rs Components.”weak link” we mean some inhomogeneity in a small part of the sample.In fact, this power dependent behavior suggests that a small part of thesample has a lower Tc (∼9 K) than that of the main part of the sample (43K). Applying more power reduces the Tc of the weak link (i.e. ”killing” thesuperconductivity in the weak link), and above a specific power (we foundit to be -4 dBm for our sample in this orientation) no effect of the weaklinkwas observed.0 5 10 15 200200400600   - 4 dBm -10 dBm -17 dBmf (Hz)Temperature (K)Figure C.2: Temperature dependence of the resonant frequency in (→H ||→c )orientation. The measurements shows an unexpected power dependencysuggesting the existence of a weak link in the sample.Finally, figure C.3 shows the frequency dependence of the ”apparent”Rs (=(H′H)2Rabs ) at different temperatures when (→H ||→c ). Comparing thiswith figure 3.7, one can clearly see that H ′/H can be as high as 10.72Appendix C. Using a Replica Sample for Disentangling Rs Components.Figure C.3: Frequency dependence of the ”apparent” Rs (i.e.(H′H)2Rabs )when (→H ||→c ).73

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