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UBC Theses and Dissertations

New directions in microwave spectroscopy of high temperature superconductors Musselman, Kevin Philip Duncan 2006

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New Directions in Microwave Spectroscopy of High Temperature Superconductors by Kevin Philip Duncan Musselman B.Sc.Eng., Queen's University, 2004 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E OF Master of Science in The Faculty of Graduate Studies (Physics) The University of British Columbia September 2006 © Kevin Philip Duncan Musselman, 2006 11 A b s t r a c t Two new variations on traditional microwave resonant cavity surface resistance mea-surements of high temperature superconductors are presented. A new cavity per-turbation technique is developed for the measurement of Tl2Ba 2Cu06±<5 (Tl-2201) single crystals, and preliminary measurements are presented for uniquely prepared YBa2Cu306+ y single crystals in which a non-local normal fluid response is expected. The resonant cavity probe designed for measurement of Tl-2201 single crystals operates in the TMoio mode such that the sample is located at a magnetic field node and electric field antinode. A dual-sapphire hot-finger technique was developed for supporting the sample in the electric field of the cavity, and non-perturbative issues associated with this technique were addressed. A dielectric layer resulting from the degradation of sample outer surfaces was identified as a significant source of dielectric loss. The dimensional dependence of this loss was treated theoretically and etching was shown to drastically reduce this unwanted absorption. The factor currently limiting use of this probe to measure Tl-2201 single crystals is absorption by the optical-grade sapphire used to support the sample in the cavity. The loss due to the sapphire is currently on the same order of magnitude as the sample loss to be measured. Employment of premium-grade sapphire plates should significantly reduce this background level and permit accurate measurement of the surface resistance of single crystal Tl-2201. Y B a 2 C u 3 0 6 + v single crystals have been meticulously prepared with large c-axis dimensions. This has permitted microwave surface resistance measurements where a magnetic field is applied perpendicular to the Cu02 planes, while minimizing de-magnetization effects. In this geometry, the electrodynamic response of the normal fluid is expected to be non-local, as the mean free path of the quasiparticles exceeds the penetration depth. RS(T) measurements were obtained for the crystals using a resonant cavity operating in the TEou mode at 13.4 GHz. No increase in the surface resistance, which is expected for a non-local quasiparticle response, was observed at low temperatures. Possible explanations for this result, and recommendations for further study of these samples are discussed. i i i C o n t e n t s Abstract i i Contents iii List of Figures v Acknowledgements vii 1 Introduction 1 1.1 Conventional Superconductivity 1 1.2 High Temperature Superconductivity 2 1.2.1 Cuprate High Temperature Superconductors 3 1.3 Superconductor Electrodynamics • • • 8 1.3.1 The London Equations 8 1.3.2 Superfluid Electrodynamics 9 1.3.3 Normal Fluid Electrodynamics and the Two-Fluid Model . . . 10 1.3.4 Surface Resistance Measurements of High Temperature Super-conductors 13 2 Electric Field Cavity Perturbation Technique 18 2.1 Motivation 18 2.2 Cavity Perturbation Techniques 20 2.3 Design of Microwave Cavity Resonator 22 2.3.1 Probe Body and Microwave Cavity 22 2.3.2 Sample Stage 24 2.3.3 Microwave Circuit 30 2.3.4 Operation of the Resonant Cavity 32 2.4 Performance of Microwave Cavity Resonator 32 2.4.1 Non-Perturbative Effects 32 2.4.2 Sapphire Loss 33 2.4.3 Dielectric Loss 37 Contents iv 3 Non-local Normal F lu id Effects in YBa 2 Cu 3 06 - ) - j / 40 3.1 Non-local Quasiparticle Electrodynamics 40 3.2 Sample Preparation 43 3.3 Experimental Technique 46 3.4 Results 46 4 Future Considerations 50 4.1 Sapphire Loss in Electric Field Cavity Perturbation Measurements . . 50 4.2 Cold Stage of Electric Field Cavity Perturbation Probe 51 4.3 Spectroscopic Measurement of Y B a 2 C u 3 0 6 + y in the Non-local Orien-tation 53 Bibliography 54 A Theoretical Treatment of Lossy Dielectric Layer 59 V L i s t o f F i g u r e s 1.1 Hole-doped cuprate phase diagram 3 1.2 Fully-doped Y B a 2 C u 3 0 7 unit cell 5 1.3 T l 2 B a 2 C u 0 6 ± « 5 unit cell 6 1.4 Rsa(T) of YBa2Cu3C"6.993 measured with 5 cavity resonators 13 1.5 Quasiparticle scattering rate of YBa2Cu3C"6.993 determined from cavity perturbation measurements 14 1.6 Rsa(u>,t) spectra of YBa2Cu306.99 measured using the bolometry ap-paratus 16 2.1 Field distribution and induced currents for a superconducting sample in an electric field 19 2.2 Cross-section of microwave resonant cavity 23 2.3 Sample stage for cavity resonator probe 26 2.4 Loss of sapphire and silicone grease in a microwave electric field . . . 27 2.5 Dual-sapphire sample mounting technique 28 2.6 Contamination of dual-sapphire stage 29 2.7 Microwave loss of dual-sapphire stage and paraffin in an electric field 30 2.8 Microwave circuit for cavity perturbation measurements 31 2.9 Alignment of dual-sapphire stage 33 2.10 Background loss of dual-sapphire stage 34 2.11 Effect of sample etching on electric field measurement 38 3.1 Sample orientation for local quasiparticle response 41 3.2 Sample orientation for non-local quasiparticle response 42 3.3 A YBa 2 Cu 3O6.93 sample prepared for non-local quasiparticle study . . 44 3.4 Sequential etching of YBa 2 Cu 3 06.9 3 for non-local quasiparticle study . 45 3.5 R3(T) measurements of YBa 2Cu 306.93 in non-local quasiparticle orien-tation 47 3.6 RS(T) measurements of YBa2Cu306.998 in non-local quasiparticle ori-entation 48 List of Figures vi A . l Capacitance approximation for the dielectric layer on samples . . . . 60 A.2 Ellipsoidal sample geometry for dielectric loss analysis 61 A.3 Dependence of dielectric layer loss on sample dimensions 63 V l l A c k n o w l e d g e m e n t s From shattering sapphire and an exploding dewar to leaks, lossy sapphire, a faulty microwave synthesizer, crumbling samples, more shattering sapphire, and a sample-eating cavity, I feel as though I have faced more than my share of challenges and adversity over the past two years (let it be recorded in writing that I was not in the lab when the microwave synthesizer broke and not even on the continent when my dewar exploded). As a result, I have many people to thank, both inside and outside the U B C Superconductivity Laboratory for their help and support throughout my degree. It has been a pleasure and privilege to work under the supervision of Doug Bonn and Walter Hardy. They continue to direct some of the world's best research on high temperature superconductors and it has been an amazing learning experience to work in this environment. Doug, thank you for your many ideas and extensive tutelage on sample preparation and experimental techniques. Your willingness to provide advice and entertain my many questions on the theory behind the measurements did not go unnoticed. Never throughout my two years at U B C did a request for help go unanswered. Walter, your experimental abilities and technical know-how truly are amazing. One of the most valuable skills I have acquired these last two years is the ability to better design controlled experiments where every possible sources of error is considered and examined. Thank you for this. In addition to all your help in the lab, thank you for getting me back onto the ice after a 10 year hiatus. The Friday afternoon hockey games are some of my fondest memories of my time at U B C and I hope we will have another opportunity to play the point together when I visit in the coming years. I have been fortunate to work with many wonderful, dedicated people in the Su-perconductivity Lab. Ruixing Liang's commitment to the growth of the highest purity samples is what makes our work possible and is greatly appreciated. Darren Peets has continually been a source of entertainment, advice on all things chemical, and com-pany for many late nights in the lab. His success in growing some impressive Tl-2201 crystals is the motivation for a large portion of the work presented here. Our labora-tory was lucky to have the services of a number of excellent summer students and an excellent Postdoc in Andrea Morello. I greatly appreciate the help of Jordan Baglo, Brad Ramshaw, Alexandre Rousseau, and Elizabeth Ledwosinska with everything from helium transfers to replacing dewars. I would especially like to thank Elizabeth who laid much of the groundwork for the theoretical treatment of microwave losses in dielectric sample layers, which is presented in Appendix A . Last but certainly not least, many thanks to Pinder Dosanjh and Jake Bobowski for all of their help. I likely Acknowledgements viii asked Jake more questions than all of the other members of the lab combined, and constantly drew on Pinder's extensive experimental knowledge. I truly appreciate your patience, company in the lab, and all the help you both provided. I would not have made nearly as much progress these past two years if it were not for you. I am very thankful for the support I have received from many great friends and family throughout my time in Vancouver. Many thanks to Maureen and John Scherebnyj who have been like surrogate parents and made Vancouver feel much more like home. I owe a great deal to my family; their support of my' academic pursuits has been unwavering over the years. Mom and Dad, thank you very much for your constant encouragement and interest in my work. Kristin and Shane, thank you for the advice you have provided about graduate school and all the great visits in Van-couver and Edmonton. I have been very lucky to have a wonderful person to share my experiences in Vancouver with. Katie, our Whistler ski trips, rainy backpacking weekends, and jogs on the beach are my fondest memories of the past two years. I can't thank you enough for the amazing patience you displayed for my late nights, early mornings, and entire weekends in the lab. Your support and encouragement have been remarkable. 1 Chapter 1 Introduction 1.1 Conventional Superconduct ivi ty Superconductivity was discovered in 1911 by H. Kamerlingh Onnes, three years af-ter he first liquified helium [1]. He observed that several metals displayed perfect conductivity (zero electrical resistance) when cooled below a characteristic critical temperature Tc on the order of a few Kelvin. This characteristic alone does not define superconductors, as infinite conductivity is also expected in very high purity materials where scattering mechanisms are not present. The more fundamental hall-mark of superconductors, perfect diamagnetism, was put forward by Meissner and Ochsenfeld in 1933 [2]. Not only did they show that a magnetic field is excluded from entering a superconductor, they also found that a field in a normal state sample is expelled as the sample is cooled through Tc. This phenomena cannot be attributed to perfect conductivity, which would tend to trap flux in the sample. Surface currents circulate within a well-defined depth of the superconductor A, creating a magnetic field that perfectly opposes the external field. It was not until the 1950s that a complete theoretical description for classical superconductivity was formulated. In 1957, Bardeen, Cooper, and Schrieffer put for-ward a microscopic theory that explained conventional superconductivity and came to be known as the BCS Theory [3]. They showed that a weak attractive interac-tion between electrons causes an instability in the ordinary groundstate electron gas, leading to the formation of bound pairs of electrons occupying states with equal and opposite momentum and spin. Furthermore, they showed that it is phonons that pro-vide this weak attraction and mediate the coupling of electrons. A n electron locally polarizes the medium by attracting positive ions and the excess positive charge in turn attracts a second electron. If the retarded ionic attraction is stronger than the screened Coulomb repulsion, a net attraction results between the electrons. These bound pairs of electrons are known as Cooper pairs. The spin singlet (S=0, s-wave) Cooper pairs form a superconducting condensate with a groundstate energy separated from excited states by an energy gap A . A key prediction of BCS Theory is that a Chapter 1. Introduction 2 minimum energy Eg(T) = 2 A ( T ) is required to break a Cooper pair and create two quasiparticle excitations. It follows from the B C S theory that this energy gap A ( T ) increases from zero at Tc to a l imit ing value of Eg(T = 0) = 2 A ( T = 0) = 3.528A;TC. Bardeen, Cooper, and Schrieffer's prediction agreed perfectly wi th calorimetric and electromagnetic absorption measurements of the superconducting energy gap [4, 5, 6] providing convincing verification of their microscopic theory. In the ensuing thirty years, many other materials were discovered to be superconducting, al l wi th transition temperatures around 20 K or less. 1.2 H i g h Temperature Superconduct ivi ty The field of superconductivity was reinvigorated in 1986 when Bednorz and Muller discovered a new class of high temperature superconductors [7]. The materials had a distinct layered structure, which included copper oxide planes, and remarkably, crit ical temperatures on the order of 100 K . Despite their different structures and properties, high and low temperature superconductors are united by the existence of a many-particle condensate wavefunction which has an amplitude and phase, and is phase coherent on a macroscopic length scale. This characteristic many-particle condensate can be loosely compared to other condensates, such as that observed in superfluid helium, wi th Cooper pairs of electrons replacing the bosonic helium atoms. Unlike conventional superconductivity, a satisfactory model describing the physi-cal mechanisms responsible for high temperature superconductivity has not yet emerged Whi le models such as that of Lawrence and Doniach (layers of conventional super-conductors wi th Josephson coupling between them) [8] have been successful in de-scribing a number of the properties of layered high temperature superconductors, they have provided little insight into the underlying mechanisms. Electron pairing is st i l l believed to occur in high-T c materials, however the exchange of bosons other than just phonons likely gives rise to the exotic superconductivity. Hardy and the U B C Superconductivity Group used a cavity perturbation method to show that the superconducting gap of the high temperature superconductor YBa2Cu 3 06 .95 is not spherically symmetric like the conventional gap function obtained from the B C S so-lution, but instead has a d-wave nature [9]. This finite angular momentum of the electron-paring state of unconventional superconductors is believed to result from electron correlations caused by the large Coulomb repulsion at each C u site in the CuC>2 planes. Group theory calculations indicate 4 possible singlet pairing states for a single square C u 0 2 plane and experimental evidence has conclusively shown that Chapter 1. Introduction 3 Hope doping p Figure 1.1: Generic phase diagram of the hole-doped cuprates which displays the antiferromagnetic, pseudogap, superconducting, strange metallic, and normal Fermi liquid phases. dx2_y2 is the relevant gap symmetry in the cuprates [10]. As such, the gap varies as a cosine function around the Fermi surface with a maximum A 0 along ±kx, ±ky and line nodes at kx = ±ky. A n important consequences of this gap structure is the existence of residual quasiparticle excitations at all non-zero energies. Much effort continues to be focused on clearly defining the pairing interactions through which high temperature superconductivity arises and the study of quasiparticle electrodynamics has proven to be an excellent source of clues to this riddle. 1.2.1 Cuprate High Temperature Superconductors The cuprate superconductors are layered perovskites characterized by two-dimensional C u C _ planes in which superconductivity is believed to reside. The copper and oxy-gen atoms form nearly square lattices and the planes are separated by a variety of chemical units whose structures vary among the different cuprate family members. The nature of these materials varies considerably with the electronic doping of the C u C _ planes. A rudimentary hole-doping phase diagram of the cuprate super-Chapter 1. Introduction 4 conductors is shown in Figure 1.1. At low dopings, cuprates are antiferromagnetic (AFM) insulators. In this state, each planar Cu atom has a single 3d hole and the va-lence electrons are strongly localized on the Cu atoms due to Coulomb repulsion. As electrons are removed from the Cu02 planes, the Neel temperature falls as the holes become mobile. As hole-doping is further increased, the material eventually reaches the superconducting state. At higher temperatures and intermediate dopings, the material is characterized by the onset of a partial gap in the spin and charge excita-tion spectra and hence is known as the pseudogap state. Like the superconducting gap in high temperature superconductors, the d-wave nature of the order parameter also holds for the pseudogap and it has become evident that the superconducting gap and pseudogap are intimately related [11]. The strange metallic phase that exists at temperatures above the superconducting region displays variations from Fermi-liquid theory [12] then as the doping is further increased, more conventional metallic behavior is recovered [13]. Doping of the CuC"2 planes can be achieved by annealing in an oxygen atmosphere or by adjusting the cation composition in the layers away from the plane. In general, the mobility of chain oxygen ions is considerably higher than that of cation dopants in cuprate crystal structures, so samples doped by oxygen annealing display greater homogeneity than those of the cation-doped variety. Yttrium Barium Copper Oxide YBa2Cu306+j/ (YBCO) is one of the most widely studied cuprates due to its high chemical purity and the relative ease with which high-quality single crystals can be grown. Fig. 1.2 shows a schematic representation of the fully-doped (y=l) YBa2Cu307 unit cell. Hole-doping of the C u 0 2 planes is reversibly manipulated by changes in the annealing conditions following crystal growth. During these an-nealing steps, oxygen atoms can be added or removed (y=0—>1) from the CuO chains which run along the b direction. The undoped y=0 unit cell is tetragonal, however this tetragonal phase becomes unstable above y = 6.28 and orthorhombic phases prevail [14]. Within the superconducting dome, certain doping levels are know to correspond to specific orderings of the off-plane CuO chains. The ortho-IIF phase, for example, has an ordering of empty-empty-full chains. Ortho-II on the other hand contains alternating full and empty chains. The maximum critical temperature of Y B C O (Tc — 93K) occurs at y = 0.93, which is therefore known as optimal doping. At a doping of y = l , the ortho-I phase (YBa2Cu 3 0 7 ) has complete CuO chains along the b direction. Y B C O is tetragonal at high temperatures and the small orthorhom-Chapter 1. Introduction 5 C u O c h a i n s Figure 1.2: The fully-doped YBa2Cua07 unit cell. Doping is achieved by annealing the material in an oxygen-rich environment, which introduces oxygen atoms into the C u O chains. bic distortion occurring below the tetragonal-orthorhombic transition can result in the formation of twins during sample growth, which must be mechanically removed prior to study. The chain oxygen content alone does not uniquely decide the doping level in the C u 0 2 planes. Charge transfer is dependent on the oxygen coordination number of the chain C u atom, such that the oxygen content and oxygen ordering together determine the number of holes introduced into the planes [15]. As a result, the doping p (number of holes per planar C u atom) does not display a 1:1 relationship with the experimentally-controlled oxygen content y. Recently, Liang et al. used X -ray diffraction studies of ordered Y B C O phases to establish an empirical relationship between the doping p and c-axis parameter (which can be easily measured) that appears accurate across all dopings [16]. This will allow more accurate determination Chapter 1. Introduction 6 F i g u r e 1.3: T h e s to i ch iomet r i ca l l y ove rdoped Tl2Ba2Cu06±<$ un i t ce l l . of the d o p i n g dependence of var ious phys i ca l p roper t ies , a key advancement i n the search for a un ive rsa l t heo ry of h i gh t empera tu re supe rconduc t i v i t y . Thal l ium Bar ium Copper Oxide It is ev ident t ha t the h i g h t empera tu re superconduc to rs d i sp lay a r i ch a n d exc i t i ng var ie ty of phys ics , capab le of e x h i b i t i n g an t i fe r romagne t i c i n s u l a t i n g , supe rconduc t -i ng , a n d a range of me ta l l i c proper t ies . A comp le te u n d e r s t a n d i n g of these exot ic mate r ia l s w i l l therefore requi re the i r s t u d y i n di f ferent reg ions of the phase d i a g r a m us ing a var ie ty of expe r imen ta l techn iques. T h e me ta l l i c s ta te of h i g h t empera tu re superconduc to rs near o p t i m a l dop ing , for examp le , has been shown to d i sp lay a re-Chapter 1. Introduction 7 sistivity that is linearly dependent on temperature [17]. Interestingly, in some com-pounds this anomalous temperature dependence survives until superconductivity is completely destroyed, suggesting that the origin of this additional scattering may also be a candidate for the pairing mechanism of high temperature superconductivity. This phenomenon must be studied at very low temperatures where thermal effects do not mask the intrinsic properties of the metallic state, and as such requires samples that become non-superconducting at only a few Kelvin. Similarly, Fermi surface mapping techniques that probe bulk quantities of samples, such as cyclotron resonance and de Haas van Alphen studies, require that the sample be in the normal state at low tem-peratures in order to avoid the gap. This translates to a need for samples with a low upper critical field Hc2 (the applied field at which the magnetic flux completely pen-etrates the material). The upper critical fields of optimally-doped cuprates are well beyond those achievable in the laboratory, such that either underdoping or overdoping of the cuprates to low transition temperatures is the most promising route. While underdoped crystals continue to be grown to greater quality and reveal interesting properties [14, 18, 19], sample homogeneity remains an issue and stud-ies of quasiparticle electrodynamics are limited by short mean-free-paths. This has encouraged a dedicated effort to grow high-quality overdoped compounds. With the exception of L a 2 _ x S r x C u 0 4 + 1 / (LSCO), cuprates that can be heavily overdoped con-tain either thallium or mercury, both of which are extremely toxic and volatile in growth conditions. Although toxicity is not a concern in LSCO, it is cation-doped and therefore intrinsically disordered. As a result, the overdoped area of the cuprate phase diagram has been studied relatively little in the first 20 years of high temper-ature superconductivity. However, this void in research has begun to be filled. The Liang-Bonn-Hardy group at U B C have made significant progress in growing high-quality single crystals of Tl 2Ba 2Cu06±«s [20, 21]. T l 2 B a 2 C u 0 6 ± < 5 (Tl-2201) is a single-layer overdoped member of the Tl-Ba-Ca-Cu-0 system [22]. It is overdoped when stoichiometric and also has a simpler crystal structure than most of the commonly-studied underdoped compounds, suggesting that homogeneous samples may be more easily prepared. Since it can be doped without introducing cation disorder and has a neutral cleavage plane, Tl-2201 is particularly well-suited to angle-resolved photo-emission spectroscopy (ARPES) [23]. Fig. 1.3 shows the Tl 2 Ba 2 Cu06±a crystal structure. Unlike in Y B C O where the oxygen content can vary considerably from y = 0 to y = 1, the oxygen content in Tl-2201 is believed to vary little from 6.00 (5 — 0). The difference between the overdoped Tc = 0 and optimal doping is approximately Chapter 1. Introduction 8 S = 0.09 [24, 25], which is in stark contrast to Y B C O where superconductivity first appears at y = 0.35 and optimal doping occurs at y = 0.92. The dopant oxygen atoms in Tl-2201 are believed to occupy the interstitial sites in the tetrahedral holes between the thallium atoms of the TIO double layer [25, 26, 27]. As with Y B C O , the oxygen content can be set by annealing in a controlled atmosphere after crystal growth. Tl 2 Ba 2 Cu06±<5 crystallizes in both orthorhombic and tetragonal phases, depend-ing on the oxygen excess 5 as well as the cation substitution of copper for thallium. The orthorhombic distortion is believed to result from a lattice mismatch between the C u 0 2 plane layer and the naturally larger T 1 2 0 2 double layer; interstitial oxy-gen dopants increase this distortion, while cation disorder suppresses it [28]. There-fore, stoichiometric or near-stoichiometric crystals are orthorhombic for all oxygen contents, whereas those with significant cation substitution are strictly tetragonal [28, 29, 30]. For intermediate levels of cation substitution, the orthorhombic phase dominates as the oxygen level is increased [31]. Since cation doping generally in-troduces disorder which limits crystal homogeneity, the synthesis of single crystal orthorhombic Tl-2201, which is free of tetragonal phases, is desirable. 1.3 Superconductor Electrodynamics 1.3.1 The London Equations In 1935 F. and H. London described the hallmarks of conventional superconductivity (perfect conductivity and perfect diamagnetism) in terms of microscopic electric and magnetic fields [32]: E m i c r o = ^ ( A J s ) (1-1) h = - c c u r l ( A J s ) (1.2) where Js is the superconducting current density and A is a phenomenological param-eter: A = * ^ t = J 2 l (1.3) c2 nsez Here m* is a superfluid effective mass and ns is the number density of the supercon-ducting charge carriers, which varies continuously from zero at Tc to a limiting value on the order of n (the density of conduction electrons) at T < T c . The perfectly conductive nature of the material is seen in Eq. 1.1 where an electric field acceler-ates the superconducting electrons, rather than simply sustaining their momentum Chapter 1. Introduction 9 against a resistance, as expected from Ohm's law. In a similar manner, Eq. 1.2 can be combined with the Maxwell equation curl h = 4TTJ/C to give: V 2 h = h/A_ (1.4) which describes the perfect diamagnetism of the material. It requires that a magnetic field is exponentially screened from the interior of a sample with penetration depth XL. F. London showed the quantum mechanical origin of these two equations by con-sidering the canonical momentum < p >=< mv + e A / c > of the superconducting particle (Cooper pair) and assuming that the ground state has zero net momentum in the absence of an applied field, as shown in Bloch's theorem [33]. He postulated that the superconducting carrier wavefunction is rigid, and retains its ground state (p) = 0 nature, even in non-zero fields. This leads to an expression for the local average velocity of the superconducting carriers in the presence of a field: (v s) = = ^ and a corresponding current density: J 8 = n s e(v s ) = U s e A = — - (1.5) m*c Ac Taking the time derivative and curl of the current density yields the two London equations, Eq. 1.1 and Eq. 1.2, respectively. If the density of conduction electrons n is taken as an upper limit for the number density of superconducting electrons n s , Eq. 1.3 yields an ideal theoretical limit for the London penetration depth: 1.3.2 Superfluid Electrodynamics A low temperature London penetration value around 200 A is predicted for typical classic metallic superconductors using Eq. 1.6, whereas experimental values for ele-ments such as Pb and Sn are typically closer to 500 A for T <C Tc. This discrepancy is analogous to the larger anomalous skin depths observed in normal metals with large mean free paths £. Non-local electrodynamics cause the difference in both situations. Only electrons within approximately fc_Tc of the Fermi energy are expected to contribute significantly to the superconducting phenomena present at temperatures below Tc. Thus the relevant electrons have a momentum range Ap « kgTc/vp, where vp is the Fermi velocity. A n uncertainly principle argument then gives a corresponding Chapter 1. Introduction 10 range in the position of the electrons Ax > h/Ap « hvp/ksTc, which can be used to define a characteristic length <"> where a is a constant of order unity. This quantity is known as the coherence length and was first introduced by Pippard in 1953 [34]. It represents the size of the smallest wave packet that the Cooper pairs can form and for typical elemental superconductors £o > ^L- Since the penetration depth of an applied field is smaller than the size of the superconducting wave packet, the vector potential A(r ) is not constant over the extent of the wave packet and a weakened supercurrent response is expected. Hence the role of £n in superconductors is analogous to the mean free path £ in the non-local electrodynamics of normal metals. As with the anomalous skin effect, the weakened supercurrent response results in increased field penetration, to depths beyond the London limit A^,. The opposite situation is found for high temperature superconductors. Because of the high Tc and consequent small value of £n (on the order of tens of angstroms), the high-Tc compounds are type II superconductors, characterized by the property that £o < A [35]. As such, there is an instantaneous relationship between the field experienced by the superconducting charge carriers and the superfluid current. This local superfluid response greatly simplifies electrodynamic relations within these ma-terials, permitting extraction of intrinsic properties such as penetration depths A and conductivity spectra o(u,T) from transport measurements. It should be clarified that due to the nodes in the energy gap of d-wave superconductors, £n is actually ^-dependent, with large values in the nodal directions [36]. However, for the majority of fc-states, £n(&) *C A such that the superfluid response can be assumed local for most purposes. 1.3.3 Normal Fluid Electrodynamics and the Two-Fluid Model The London equations disregard the presence of the non-superconducting, dissipative carriers (quasiparticles), which provide a parallel channel for conduction and play an integral role in the electrodynamics of high temperature superconductors. Gorter and Casimir were the first to interpret superconductor electrodynamics in terms of coexisting fluids of normal and superconducting electrons [37], where normal electrons refer to those thermally excited from the superconducting groundstate. In the clean superconductor limit, all charge carriers condense into the superconducting ground Chapter 1. Introduction 11 state at T = 0. At finite temperatures however, the density of normal electrons nn is non-zero due to thermal fluctuations of groundstate electrons into nearby low-energy states, and in the case of high temperature superconductors, due to the presence of nodes in the gap function. While Gorter and Casimir's two-fluid model assumed a particular form for the temperature dependence of A, the generalized two-fluid model widely used today takes the superfluid fraction ns(T)/ns(0) = A 2 (0)/A 2 (T) as a measured quantity. The two-fluid model is an especially useful picture when considering the response of superconductors to high-frequency fields. A l l superconductors, even classical ones, display a finite dissipation when carrying alternating currents. This can be seen from the first London equation (Eq. 1.1); while a constant dc supercurrent will persist in the absence of an applied field, an electric field E m i c r o must be present to accelerate and decelerate the superconducting electrons in order to produce a time-varying supercurrent. This electric field also acts on the normal electrons, which scatter from impurities and thus produce a finite resistance, in accordance with Ohm's law. Within the framework of a generalized two fluid model [38, 39], the electrodynam-ics can be approximated as a superposition of the responses of the superconducting and normal fluids to the high-frequency field. In the clean limit, the total density of charge carriers can be taken as the sum of the superconducting and normal electron densities n — ns + nn, which have characteristic scattering times r s and r„ respec-tively. The ac response of the superconductor to a field Ee t w t is approximated by a superposition of the superconducting and normal fluid complex conductivities: a(ui, T) = Gis — iois + < 7 i „ — %oin (1.8) By assuming r s —> oo (perfect conductivity), a zero-frequency delta function and an associated reactive term are obtained for the complex conductivity of the superfluid: 2 2 o(u,T) = —0{u)-i—— + oln-iO2n (1-9) m* m*u Referring back to Eq. 1.3, one observes that ns(T)e2/m* — l / [ / i 0 A_(T)], so the conductivity at finite frequencies may be rewritten as: a(u, T) = oln(u, T) - i[o2n(u>, T) + (1.10) It is seen that when oin makes a significant contribution to the total conductivity, Chapter 1. Introduction 12 the normal fluid electrons play a role in screening the electromagnetic fields and the measured magnetic penetration depth X = l ^ 7 2 ( L 1 1 ) will differ from the London value AL = 4 ^ c ^ 2 , which only takes into account screening by the superfluid. The surface impedance is the experimental quantity that provides access to o~(u, T). It is defined as the ratio of tangential electric and magnetic fields at the surface of the superconductor and in the limit of local electrodynamics it can be written as: Zs = Exo/Hyo = Rs + iXs = ^ 1 - ^ (1.12) For T < T C , few quasiparticles exist, and as such the low frequency reactive response is dominated by the superfluid component (o2n(w,T) <gC \ — ) . In this case, the surface resistance and reactance can be approximated as: Rs(u,T) = ^//2a , 2A 3(r)ai(w ,r) (1.13) Xs(u,T) =pc0u\{T) (1.14) For the microwave frequencies and low temperatures considered here, Eqs. 1.13 and 1.14 provide an accurate first estimate. However, at higher temperatures and frequen-cies, the quasiparticle contribution to field screening which enters through 02n(u>,T) must be accounted for, and the approximate forms (Eqs. 1.13 and 1.14) of Eq. 1.12 cannot be used. Thus, it is seen that measurement of the microwave surface impedance Zs(u, T) = Rs(u>, T) +iXs(u, T) provides detailed information on the intrinsic properties of high temperature superconductors. The surface reactance Xs provides a route for the measurement of the penetration depth A and hence the superfluid density ns. Mea-surement of the surface resistance Rs, together with knowledge of the penetration depth, reveals the real part of the microwave conductivity oi(u,T), which corre-sponds to electromagnetic absorption by quasiparticle excitations (either thermally excited quasiparticles or those excited out of the condensate by photon absorption). These parameters provide valuable insight into the nature of the pairing mechanism in high temperature superconductors and have therefore been the focus of much ex-perimental effort over the last 20 years. Chapter 1. Introduction 13 T(K) T(K) (a) (b) Figure 1.4: RS(T) measured in d direction of YBa 2 Cu 3 06 .993 by Hosseini et al. [40] 1.3.4 Surface Resistance Measurements of High Temperature Superconductors Precise surface resistance measurements of high-quality single crystals are technically challenging since the crystals exhibit very low absorption when exposed to microwave fields. Moreover, cuprate single crystals are typically anisotropic, small in size, and susceptible to twinning, which further complicates the measurements. Nonetheless, a number of techniques have been developed to accurately measure the microwave surface resistance and conductivity of high-Tc single crystals over broad ranges of temperature and frequency. Initially, the absorption of superconductors in the microwave region was resolved over most temperatures using high-precision cavity perturbation techniques, where the sample in question is brought into a high quality factor Q superconducting res-onator and its surface resistance and penetration depth obtained from measured changes in the resonant frequency and quality factor [40, 41, 42, 43, 44]. Fig. 1.4 (a) shows RS(T) measurements obtained by Hosseini et al. using five microwave cav-ities of various frequencies for currents running in the a direction of a YBa2Cu 306 .993 single crystal. If the u2 dependence indicated by Eq. 1.13 is removed by division, comparison of the different frequencies becomes easier and the general shape of the conductivity spectra can be inferred, as shown in Fig. 1.4 (b). Absorption measure-ments in the ab plane of single crystal Y B C O are characterized by a sharp drop in the Chapter 1. Introduction 14 T(K) Figure 1.5: Quasiparticle scattering rate obtained from Drude fits to resonator con-ductivity spectra [40]. surface resistance at the superconducting transition T c , corresponding to the onset of screening by the superfluid, followed by a broad peak in the 30-40 K range and then a gradual decrease to some non-zero residual resistance at low temperatures. Since no peak is observed in A(T) measurements, the absorption peak must come from a corresponding peak in the quasiparticle conductivity a\(u,T). More specifically, this peak has been attributed to a competition between the rapid increase in quasiparticle lifetime r below Tc and a decrease in the density of thermally-excited quasiparticles nn at lower temperatures [41]. While the extraction of oi(T) from RS(T) is relatively straightforward in the limit of local electrodynamics, accurate determination of r (T) from oi(T) requires knowledge of the shape of the quasiparticle conductivity spectra ai(u,T). Cavity perturbation techniques suffer from the drawback that a resonator is typically re-stricted to a single frequency, so a number of resonators are required to construct a spectrum. Hosseini et al. used the RS(T) measurements from 5 resonators (shown in Fig. 1.4) to map out rough conductivity spectra for YBa2Cu 3 06 .993, which they found to be described by a Drude model [40]. From these Lorentzian fits, the tem-perature dependence of the quasiparticle scattering rate 1/r was determined and is shown in Fig. 1.5. The dramatic increase in quasiparticle lifetime r at low tern-Chapter 1. Introduction 15 peratures is of particular importance to work presented in this thesis. It has been attributed to the suppression of inelastic scattering processes (which are apparent in the large normal state resistivity) as carriers condense into the superconducting state [38]. The quasiparticle scattering rate 1/r decreases rapidly (decreasing as T 4 or even exponentially) until it reaches a limiting value on the order of 10 1 1 s _ 1 in the 20 K range. This limiting value is believed to be due to impurity scattering, which was experimentally verified by Bonn et al. who doped samples with Zn and Ni impurities [42]. These large values of r at low temperatures correspond to mean free paths £ on the order of 4 Lim, more than an order of magnitude larger than typical penetration depth values. This situation where £ 3> A would suggest that the quasiparticle electrodynamics are non-local and invalidate the use of Eq. 1.13 in determining fJi (o>, T) from cavity perturbation measurements. As will be discussed in Section 3.1 however, the anisotropic nature of the cuprate superconductors combined with a judicious choice of experimental geometry has allowed all prior surface resis-tance measurements to be performed such that the quasiparticle electrodynamics are local. Chapter 3 presents the first surface resistance measurements of Y B a 2 C u 3 0 6 + J / single crystals where non-local quasiparticle electrodynamics are expected. As mentioned, flattening of 1/r below ~ 20 K was attributed to an impurity-limited elastic scattering regime, however this temperature independence was incon-sistent with simple models of d-wave superconductivity [45]. This failure of simple models to describe the observed temperature and frequency dependent conductivity o\ (u, T) motivated the development of a non-resonant bolometric technique for high-resolution measurements of surface resistance over a continuous range of microwave frequencies [46]. In this method, the surface resistance of a single crystal is obtained via measurement of power absorption in a magnetic field. For any conductor, the power absorption in a microwave magnetic field Pabs is directly proportional to the surface resistance Rs: where Hrf is the rms magnitude of the tangential magnetic field at the surface of the conductor. It is on this principle that the bolometric technique is based. The su-perconducting sample in question, a resistive thermometer, and a resistive heater are held in thermal equilibrium, and attached to a base temperature via a weak thermal link. A microwave magnetic field, amplitude modulated at a low frequency (of order 1 Hz), is applied to the sample, which absorbs some of the radiation and heats up the ensemble accordingly. This temperature modulation is detected using the resis-tive thermometer as an appropriately amplified voltage signal. The resistive heater is (1.15) Chapter 1. Introduction 16 i i 1 i • — i — • — i — i — i i Freauencv (GHz) F i g u r e 1.6: S u r f a c e r e s i s t a n c e o f Y B a 2 C u 3 0 6 . 9 9 m e a s u r e d i n t h e a d i r e c t i o n w i t h t h e b r o a d b a n d b o l o m e t r i c a p p a r a u t s . F i g u r e p r o v i d e d c o u r t e s y o f P . T u r n e r . u s e d t o c o r r e l a t e t h i s c h a n g e i n t h e r m o m e t e r v o l t a g e t o a n a s s o c i a t e d p o w e r a b s o r p -t i o n . It w a s f o u n d t h a t t h e p o w e r d e l i v e r e d t o t h e s a m p l e v a r i e s c o n s i d e r a b l y w i t h f r e q u e n c y , d u e t o t h e p r e s e n c e o f s t a n d i n g w a v e s i n t h e m i c r o w a v e c i r c u i t . T o a c -c o u n t for t h i s , a n e q u i v a l e n t h e a t e r - t h e r m o m e t e r - s a m p l e a r r a n g e m e n t is a d d e d w h e r e t h e s a m p l e is a n o r m a l - m e t a l r e f e r e n c e o f k n o w n s u r f a c e r e s i s t a n c e , w h i c h a c t s a s a n a b s o l u t e p o w e r m e t e r . T h i s t e c h n i q u e a l l o w e d r e s o l u t i o n o f s u r f a c e r e s i s t a n c e s p e c t r a i n u n p r e c e d e n t e d d e t a i l , a s s h o w n i n F i g . 1.6. T h e b r o a d b a n d s u r f a c e r e s i s t a n c e m e a s u r e m e n t s r e v e a l e d c u s p - s h a p e d c o n d u c t i v -i t y s p e c t r a ( p o o r l y fit b y D r u d e l i n e s h a p e s a t l o w f r e q u e n c i e s ) , c o n s i s t e n t w i t h w e a k i m p u r i t y s c a t t e r i n g o f n o d a l q u a s i p a r t i c l e s [47, 48] . A s s u c h , t h i s g r e a t e r s p e c t r o -s c o p i c d e t a i l r e c o n c i l e d m i c r o w a v e s u r f a c e i m p e d a n c e m e a s u r e m e n t s w i t h t h e o r e t i c a l m o d e l s o f d - w a v e s u p e r c o n d u c t i v i t y . A n e w p h e n o m e n o l o g i c a l f o r m i s u s e d t o fit t h e s e c u s p - s h a p e d s p e c t r a , r e s u l t i n g i n s l i g h t v a r i a t i o n s t o t h e t h e s c a t t e r i n g r a t e v a l u e s p r e s e n t e d i n F i g . 1.5. H o w e v e r , t h e a b s o l u t e v a l u e s o f t h e l o w t e m p e r a t u r e i n - p l a n e q u a s i p a r t i c l e l i f e t i m e s a r e s t i l l o n t h e o r d e r o f 1 0 - 1 1 , s u c h t h a t t h e s i t u a t i o n w h e r e £ 3> A r e m a i n s f o r i n v e s t i g a t i o n . It is a p p a r e n t f r o m t h i s p r e c e d i n g d i s c u s s i o n t h a t Y B C O h a s b e e n t h e m a j o r Chapter 1. Introduction 17 focus of microwave spectroscopy in recent years and has led to major advancements in our understanding of superfluid and quasiparticle behavior in the cuprate high temperature superconductors. While Y B C O will continue to receive much attention due to its high chemical purity, advancement of this science to new cuprate members and new regions of the phase diagram will most certainly shed more light on the mystery of high temperature superconductivity. Chapter 2 of this thesis presents the design of a microwave resonator probe for the measurement of surface resistance of Tl-2201 single crystals. The ultimate goal is detailed study of normal and superfluid dynamics, as has been described for Y B C O , however new measurement challenges are confronted when dealing with this highly anisotropic overdoped compound. Chapter 3 presents, to our knowledge, the first microwave surface resistance measurements on Y B C O for which a non-local quasiparticle response is expected. Chapter 2 18 Electric Field Cavity Perturbation Technique 2.1 Mot iva t i on In Tl2Ba2Cu06±<5 single crystals, the Cu02 planes are widely separated by thick T l - 0 bilayers, resulting in highly anisotropic transport properties. The ratio of c-axis resistivity to in-plane resistivity pc/pab has been reported to be 1000 at 300 K, increasing up to 2500 at 30 K [49]. This is significantly larger than in YBa 2 Cu 3 06+ y where pc/pab is on the order of 10-100, depending on the oxygen content. As a result, traditional microwave cavity perturbation and bolometric techniques, where the magnetic field H is applied in the d or b direction, are not expected to provide reliable measurements of in-plane surface resistances and penetration depths. For YBa2Cu306+y, interlayer coupling is sufficiently strong that quasiparticle transport in the c direction is nearly coherent and the problem can be treated as quasi-three dimensional. Currents are driven both in the ab planes and in the c direction, but the contribution in the c direction is assumed small for sufficiently thin samples and can be neglected. This assumption was verified experimentally for Y B C O by Hosseini et al. who successively cleaved and polished samples (creating larger c-axis contributions) in order to extract Rsc [50, 51]. For the highly-anisotropic Tl-2201 on the other hand, the interlayer coupling is very weak, such that the transport cannot be treated as quasi-three dimensional. Here quasiparticle transport between adjacent layers is analogous to tunneling in superconductor-insulator-superconductor junctions [52]. Prior microwave cavity perturbation measurements on Tl-2201 and other highly-anisotropic single crystal cuprates such as Bi 2 Sr 2 CaCu208 have therefore been per-formed with the magnetic field parallel to the c-axis. In this configuration, purely ab plane screening currents are induced, eliminating c-axis components from the mea-sured values [53, 54, 55]. While this experimental geometry was successful in elimi-nating c-axis currents, it likely introduces significant demagnetization effects (typical Tl-2201 single crystal dimensions were on the order of 1mm x 0.1mm x 0.01mm in Chapter 2. Electric Field Cavity Perturbation Technique 19 Figure 2.1: Field distribution and induced currents for a superconducting sample in an electric field. these studies) and hence calls into question the accuracy of the results obtained by this method. It would be desirable to perform cavity perturbation measurements on single crys-tal Tl-2201 in an experimental geometry that precludes c-axis currents and avoids these large demagnetization effects. It is suggested that a cavity perturbation method be used where the crystal is located at a magnetic field node and electric field antin-ode, with the field parallel to the ab plane. This geometry would induce currents pre-dominantly in a single direction in the plane, as shown in Fig. 2.1, greatly reducing the contribution of c-axis currents. Unaided by the Meissner effect, this experimen-tal arrangement results in a non-uniform electric field at the surface of the crystal and therefore a non-uniform current density as well. However, we believe that the depolarizing effects are under better control than the demagnetization effects of the H || c geometry, especially in regards to the avoidance of c-axis effects. The remain-der of this chapter details the design, construction, and testing of a microwave cavity perturbation experiment for electric field measurements of single crystal Tl-2201. Chapter 2. Electric Field Cavity Perturbation Technique 20 2.2 Cav i ty Per turbat ion Techniques Microwave resonant cavities are a powerful tool for measuring the surface impedance of low-loss samples such as the high temperature superconductors. A n instructive starting point is to consider a resonant cavity containing a sample of infinite con-ductivity. A cavity mode can be excited by an input coupling loop and detected by an identical loop placed, for example, on the opposite side of the cavity. Since the conductivity is infinite, the penetration depth is zero (A=0) and there is no loss in the sample. The measured resonant frequency and quality factor (resonant frequency divided by the full-width of the resonance peak at half of it maximum) in this situ-ation are defined as u> = U>Q and Q = Q0. The resonant frequency can be described more completely in complex form: where elU}Qt = e^ote-(u>0/2Q)t -ls t n e f r e e s o i u ^ i o n f o r the cavity mode. In reality however, the sample has finite conductivity and therefore causes a perturbation of the cavity's resonant frequency, the magnitude of which is determined by the surface impedance of the sample: 6u = u — cJrj oc i I ZsJ2dS (2.2) J sample where Js is the integrated surface current density. For highly conductive samples where A remains small, it is reasonable to assume that J3 is essentially unchanged from the case of infinite conductivity. Under this assumption, division of the real and imaginary parts of Eq. 2.2 by the frequency components in the case of infinite conductivity leads to the following expressions: 5u = u-u* = _ K X g ( 2 3 ) < { h ) = h - k - 2 K R - ( 2 4 ) In Eq. 2.4 it has been assumed that SU/LUQ <C 1, an accurate assumption for microwave cavity resonators where UQ is on the order of GHz and 8u on the order of MHz. The constant of proportionality K depends on the shape of the sample and resonator and can be determined experimentally using a reference sample of known resistivity. Expressions 2.3 and 2.4 have been derived by conceptually perturbing a sample already inside a resonant cavity from infinite conductivity to a large but finite con-Chapter 2. Electric Field Cavity Perturbation Technique 21 ductivity. This situation is of course experimentally inaccessible, but removing the sample while keeping the resonator temperature fixed can give an accurate value of Qo- In other words, it is assumed that the quality factor of an empty cavity is the same as the quality factor of the cavity with a sample of infinite conductivity. A sample of infinite conductivity does result in a shift from the empty cavity resonant frequency due to its finite size, and may cause a non-perturbative correction to Qo-However, as long as the sample does not change the current distribution in the cavity walls to any great extent, this correction is small and Qempty = Qo is an accurate assumption. Eq. 2.4 can therefore be re-written more explicitly as: where again T is a constant that depends on the geometry of both the sample and the cavity and Q0 is now taken as the Q of the unloaded cavity. Thus it is seen that the small change in quality factor that results when a sample is inserted into a microwave resonator can be used to determine the surface resistance of the sample. The assumption that an empty cavity is representative of a cavity containing a sample of infinite conductivity does not hold for the change in resonant frequency however, since there is a large frequency shift associated with the presence of a sample. As a result, measurements of the surface reactance and hence the penetration depth (via Eq. 1.14) are restricted to changes from some base value at low temperatures. Moreover, the change in resonance frequency is very sensitive to the exact position of the sample, such that the sample must be rigidly fixed in the cavity without the ability to be retracted. As a result, it is difficult to use the same experimental arrangement to measure both RS(T) and AXS(T) over the entire superconducting temperature range without substantially compromising precision. As the focus of this work is accurate measurement of the microwave surface resistance, a cavity resonator has been designed that allows insertion and removal of the sample. When Zs is anisotropic, as is the case for the cuprates, the various components of the surface resistance can generally be extracted by making measurements with different sample orientations and aspect ratios. In past studies of Y B C O , crystals have been oriented with their broad ab planes parallel to an applied magnetic field such that surface currents run in the planes and in the c direction. In this configuration, the samples can be rotated so the in-plane surface current is almost entirely in the a or b direction and for sufficiently thin platelet samples, the c component of the surface resistance Rsc can generally be neglected, as was mentioned earlier. For T l -(2.5) Chapter 2. Electric Field Cavity Perturbation Technique 22 2201 crystals however, cavity perturbation fails in this geometry. Given the large anisotropy of Tl-2201, the assumption that currents flow almost entirely parallel to the surfaces (Eq. 2.2) is not accurate for transport in the c direction. Thus the need to limit induced currents to the superconducting planes is again emphasized. 2.3 Design of Microwave Cav i ty Resonator 2.3.1 Probe Body and Microwave Cavity The main body of the probe designed for electric field measurements of Tl-2201 single crystals is a retrofit to a probe previously constructed for split-ring resonator measurements of single crystal Y B C O [41]. It contains two movable coupling coaxial cables that are sealed to vacuum by O-rings. The rods have coupling loops soldered to their ends that excite and detect resonant modes in the cavity. The coaxes are brought to low temperatures by 3 sets of brass fingers and can be adjusted vertically and rotated in the cavity during operation to alter the coupling. The probe also houses a sample rod, which allows the sample being measured to be adjusted vertically and rotated in the cavity during operation of the experiment. The ability to retract the sample is of course paramount to measurement of the unperturbed quality factor of the cavity Qo and hence the surface resistance. The sample rod is also sealed to vacuum by O-rings and consists of a stainless steel tube which runs from the room-temperature flange of the probe to the low temperature end. A 5-inch-long copper cold stage is attached to the end of the stainless steel portion of the sample rod and a copper braid connects the cold stage to a large brass face-plate on the exterior of the probe body. This provides a thermal link between the sample rod and the pumped liquid helium bath (1.2 K) in which the probe resides, while maintaining the sample rod's ability to be moved within the probe. Six insulated brass leads (for a sample thermometer and heater) run along the length of the stainless steel portion of the sample rod to an external connection at room temperature. At the low temperature end of the stainless steel, six 34-gauge brass wires are soldered to the copper leads and are attached along the length of the copper cold stage using G E electrical varnish for good thermal contact to the base temperature. At the end of the cold stage, the brass leads are terminated by a pin connector which attaches to a matching connector located on the sample stage. The sample stage supports the thermometry of the experiment, as well as the sample, and is discussed further in Section 2.3.2. Chapter 2. Electric Field Cavity Perturbation Technique 23 Figure 2.2: Cross-sectional view of the microwave resonant cavity used. The cutoff hole for sample insertion is located at the centre of the supporting body and two access holes for the coupling loops are located on either side. A Pb-Sn plated copper resonant cavity is attached to the bottom flange of the probe with an indium seal to maintain vacuum. The resonant cavity employed is one from previous microwave surface resistance studies [40], which was operated in the TEon mode at 13.4 GHz. The central hole of the supporting body of the resonant cavity was increased to a diameter of 0.47" to allow insertion and retraction of the cold and sample stages. This access hole for the sample stage is reduced to a 0.138" cutoff hole through which the sample can be inserted into the microwave cavity. Two 0.21" diameter holes are located on either side of the central hole, allowing the coupling loops to be placed in proximity to magnetic fields in the cavity. A cross-sectional view of this arrangement is shown in Figure 2.2. The actual resonant cavity is comprised of the bottom face of the supporting body, which contains the access holes, and a 1.40 inch diameter cylindrical copper cavity that is 0.70 inches in length. The relevant surfaces which comprise the cavity are superconducting in order to obtain large Q Chapter 2. Electric Field Cavity Perturbation Technique 24 values. This is achieved by electroplating them with an alloy of lead containing 5 percent tin [56]. A film of this composition has a high critical temperature and low microwave loss. The tin prevents the lead from oxidizing, prolonging the lifetime of the cavity before it must be plated again. Once plated, the cylindrical cavity is tightly screwed onto the bottom face of the supporting body (also plated) where a knife edge forms a conducting joint between the two. The cavity is then sealed in a brass vacuum can which attaches onto the bottom of the supporting body with an indium seal. This maintains vacuum within and surrounding the cavity. When the cavity is cooled to liquid helium temperature (4.2 K) it becomes superconducting, resulting in a significant increase in the quality factor of the cavity. Further cooling to 1.2 K, achieved by pumping on the helium bath, results in a further increase in Q and therefore permits more accurate measurement of the microwave surface resistance. The resonant mode selected for this work was the TMoio mode, which resonates at approximately 6.49 GHz when the cavity is empty. There is a strong, uniform electric field and a magnetic field node along the central axis of the cylindrical cavity, providing an ideal arrangement for electric field measurements. It is important to note that the field distribution of this mode is such that the superconducting sur-face currents run across the knife-edge joint between the cylindrical cavity and the supporting copper base. Therefore, in order to obtain Q's comparable to those for modes where the surface currents do not cross this boundary, the cylindrical cavity must be attached to the supporting base extremely tightly, such that the two super-conducting surfaces make intimate contact. Typically, quality factors over 3 x 107 were obtained in the T M 0 i o mode immediately after plating. This mode is more sus-ceptible to degradation over time than the TEon mode due to the presence of these boundary-crossing currents. However, by keeping the cavity under constant vacuum (except for when loading or unloading a sample), it was found that the Q levels out to a reasonable value on the order of 5 x 106 and hence can be used a number of times before replating is necessary. 2 . 3 . 2 S a m p l e S t a g e The sample stage comprises the main body of the experiment and is screwed onto the end of the cold stage along with a small amount of silicone vacuum grease for thermal contact. Two such copper stages were fabricated with the intention that one can be cleaned and loaded with a sample while the other is being used in the experiment. Fig. 2.3 shows the sample stage and identifies a number of the key components. The stage Chapter 2. Electric Field Cavity Perturbation Technique 25 employs a sapphire hot linger technique [57]. The sample to be measured is mounted onto the end of a thin sapphire plate, which is used to insert the sample into the centre of the resonant cavity. The plates are .0055" thick, approximately 0.75 mm wide and 23 mm long. This length is such that the sample can be inserted into the centre of the cavity, while keeping the rest of the sample stage sufficiently recessed from the cutoff hole. The sapphire used was Crystal Systems Hemlite optical grade sapphire [58]. The plates were cut using a K . D . Unipress WS-22 wiresaw with 60 LOXI diameter wire and boron-carbide grit paste, and using an UltraTec Ultraslice 2000 precision diamond saw. Since the sapphire being cut was extremely thin, considerable effort was required to obtain clean cuts with sufficiently small widths. Debate exists as to whether sapphire actually cleaves along certain crystallographic planes or just parts preferentially along certain directions due to the presence of stress, defects, or other perturbations of the crystallographic structure [59]. However, it was certainly found that the sapphire was prone to cracking along certain directions when cut to small dimensions. After cutting, the edges of the plates were lightly polished to remove contaminants and reduce the roughness of these surfaces. Finally, before use, the plates were etched and cleaned by rinsing and boiling in 3 parts H 2 0 :1 part H 2 S O 4 , followed by a sequence of rinses in distilled water, trichloroethylene, acetone, then ethanol or propanol. The sapphire plate is mounted on a sapphire block using G E electrical varnish for good thermal contact. A sample heater and thermometer are also located on the sapphire block and are used to regulate and monitor the temperature of the sample being measured. Sapphire is characterized by a very short thermal time constant and therefore acts as an isothermal stage. This permits precise control of the sample temperature without the need for any thermometry inside the cavity itself. The sample heater used is a 1500 f2 surface-mount resistor [60] and the thermometer of choice is a Lake Shore Cryotronics Cernox 1050-BC resistance thermometer [61]. The resistance of the Cernox sensor was calibrated as a function of temperature (down to 1.9 K) against the Quantum Design SQUID magnetometer thermometer in the U B C Superconductivity Laboratory. The sapphire block is epoxied (Emerson and Cumming Stycast 1266 epoxy) onto a 1/8" inner-diameter quartz tube, which sustains the temperature gradient between the sapphire block and the rest of the resonator assembly that is held at 1.2 K by a pumped helium bath. The rigidity and low thermal expansion of the quartz also minimizes any sample motion as the temperature of the sample is varied. Two 34-gauge brass wires are connected to the sample heater and four are connected to the Cernox for a true 4-wire measurement Chapter 2. Electric Field Cavity Perturbation Technique Figure 2.3: Stage constructed to insert superconducting samples into the microwave resonant cavity. of the resistance. These leads are wound around the length of the quartz tube using G E varnish to ensure that the thermal gradient is maintained. They are terminated at a male pin connector, which is recessed into the body of the copper sample stage and mates with the corresponding female connector that runs along the length of the cold stage and sample rod to the exterior of the probe. Of fundamental importance to this experimental technique is the manner in which the sample is fixed onto the end of the sapphire plate. Prior cavity perturbation measurements in magnetic fields have used a tiny amount of silicone vacuum grease to attach the sample and provide thermal contact to the sapphire plate. Silicone grease is characterized by a relatively low dielectric loss and sapphire by a very small dielectric loss, such that in magnetic field measurements where the sapphire and sample are located at an electric field node, the sapphire and grease are essentially transparent and result in negligible losses. This is not, however, the case when measurements are performed in an electric field. Fig. 2.4 shows the change in Q that resulted when a sapphire plate at 1.9 K with a small amount of silicone grease on its end was inserted into the resonant cavity operating in the TM 0 io mode. The Q dropped precipitously from an empty cavity value of 3.5 x 107 to value of 5 x 106 when the end of the Chapter 2. Electric Field Cavity Perturbation Technique 27 4.0x10 7 ' 2.5x10 7 ' 05 t - : 2.0x10 7 ' b o 1.5x10N 1.0x10N 5.0x10 H 0.0. A I A i | i 1 ' A A A -A -A -A • A A A -A A A A A A A A A A A A " -0.2 0.0 0.2 0.4 Distance of silicone grease into cavity (inches) Figure 2.4: Q of cavity at 1.9 K in TMnin mode as sapphire and silicone grease are inserted. sapphire plate (where the silicone grease was located) first entered the cavity. This corresponds to a A ^ of 1.7 x 10~7. A similar measurement was performed with a Pb-Sn reference sample mounted on the sapphire plate using silicone grease. The sample temperature was maintained at 15 K such that the sample was in the normal state and would result in a significant loss. For this arrangement, the Q was observed to drop from an empty cavity value of 1.6 x 107 to 3 x 106 upon insertion of the sample. This corresponds to a A ^ of 2.7 x 10~7. Thus, the loss of the small amount of silicone grease in an electric field is on the same order of magnitude as the loss produced by a normal metal sample. This result clearly indicates that silicone grease cannot be used when measuring the surface resistance of low-loss superconducting crystals in an electric field and necessitates a new method for supporting the sample in the cavity. The proposed solution was to use a dual-sapphire sample stage, where two sap-phire plates are epoxied (Emerson and Cumming Stycast black epoxy) together at their base, near the sapphire block. This essentially creates two cantilevers between which a small sample can be wedged, as shown in Fig. 2.5. This dual-sapphire Chapter 2. Electric Field Cavity Perturbation Technique 28 Fi gure 2.5: Two sapphire plates epoxied flush against one another can be used to support a sample in the microwave cavity by pinching the sample between them. Thermal contact between the sample and plates is achieved using a ~ 90% paraffin oil - 10% parafin wax mixture which has negligible dielectric loss. method replaces silicone's role as an adhesive, but another substance is still required to maintain thermal contact between the sample and the sapphire plates. A combi-nation of paraffin oil and wax was selected as an appropriate replacement. An alkane, paraffin is a mixture of long, straight-chained hydrocarbons and is characterized by extremely low dielectric loss. A mixture of both paraffin oil and wax was found to be most effective, as the oil spreads out over the surface of the sample, while the wax helps prevent the oil from being wicked up the length of the adjacent sapphire plates. Proportions of approximately 90 percent oil and 10 percent wax were found to work well. Loading of samples is complicated by this dual-sapphire arrangement, however a satisfactory technique was devised. Small pieces of wax paper then white paper are slid between the plates, creating a gap for a sample to be inserted. A small dab of the paraffin oil-wax combination is then applied to the inner surfaces of both the upper and lower plates. The sapphire plates are epoxied in such a manner that the lower one is slightly longer, allowing the sample to be rested on the end of the lower plate before sliding it between the two. Finally, the paper is removed to clamp the sample in place. The cleaning of the sapphire plates is also complicated by the dual-sapphire ar-Chapter 2. Electric Field Cavity Perturbation Technique 29 O T — < 1 E - 7 1 A Contaminated with GE varnish O Etched 1 E - 8 - ^ 1 E - 9 1 1 1 <~ A A A A A A A A A A A A A A A A O o ooo o o o o o o o o o o o o —r— 2 0 -1 ' 1 -4 0 6 0 Temperature (K) i — 1 — i — 1 — i — 1 — i — 1 — r 8 0 1 0 0 1 2 0 1 4 0 1 6 0 Figure 2.6: Background loss of the dual-sapphire stage (which includes a small amount of paraffin) after the plates have been contaminated with G E varnish and after etching. rangement. Typically a sapphire stage can be gently wiped with a kimwipe soaked in trichloroethylene then acetone between successive experimental runs. However, for the dual-sapphire apparatus it is found that cleaning solvents are wicked up the length of the plates and dissolve some of the G E varnish used to attach the plates to the sapphire block. This spreads a residue of varnish over the length of the plates, causing an additional dielectric loss. Fig. 2.6 shows measurements of A ^ oc Rs as a function of temperature for a newly etched and assembled dual-sapphire stage with a small amount of paraffin on its tip and for the same stage after it had been 'cleaned" with acetone. The varnish residue caused an order of magnitude increase in the back-ground loss, stressing that care must be taken not to contaminate the dual-sapphire stage. Clear Stycast epoxy could be used in place of G E varnish to attach the plates to the sapphire block. However, given the likelihood that they will need to be removed periodically for more extensive cleaning, this solution is unfavorable. The most important result seen in Fig. 2.6 is that the background loss has been reduced by two orders of magnitude from its value of A ^ ( T = 1.2K) = 1.7 x 1 0 - 7 when silicone grease was used. Fig. 2.7 shows A ^ oc Rs for the newly-etched dual-Chapter 2. Electric Field Cavity Perturbation Technique 30 5.0x10 - i ' l — ' 1 ' 1 — — 1 r i o . O 4.0x10'9- -O 3.0x10"9- 0 _ o II t O 2.0x10"9- o -i — < 0 • 1.0x10"9-0 o 0 _ 0 <o 0.0- 0 O A O 1 T — 1 1 1 1 1 - i r- 1 1 I I I I   1 1 1 1 -0.1 0.0 0.1 0.2 0.3 0.4 Distance of paraffin into cavity ( inches) Figure 2.7: A ^ of the cavity in the TMnio mode as the dual-sapphire stage and paraffin are inserted. The temperature of the sample stage is regulated at 40.0 K. sapphire stage with a small amount of paraffin as it is inserted into the electric field of the microwave cavity. No sharp increase is observed when the sapphire and paraffin enter the cavity. Instead a smooth increase in A ^ is seen, indicating that the background loss is almost entirely due to the dielectric loss of the sapphire plates. 2.3.3 Microwave Circuit The microwave circuit used to control the cavity perturbation experiment is shown schematically in Figure 2.8. A Hewlett Packard 83620A microwave synthesizer is used to excite the resonant mode in the cavity. The simplest procedure for finding the Q of the resonance involves sweeping the synthesizer frequency through the resonance window and measuring the location and width of the peak. However, for well-plated cavities with large Q's such as that used here, this method is susceptible to small mechanical vibrations of the sapphire stage (microphonics) which have undesirable effects on the measurements. This is especially true for this experimental design where there is a relatively high-e dielectric in an electric field and the sample stage is not rigidly fixed in place. Microphonic oscillations were observed in preliminary Chapter 2. Electric Field Cavity Perturbation Technique 31 4-8 G H z Microwave Amplifier (Avantek GaAs FET) .01-18 G H z Crystal Diode Detector (HP 8473 B> Detection Amplifier (x 2000) Figure 2.8: Microwave circuit used for cavity perturbation measurements. The microwave synthesizer is pulsed near the resonant frequency to obtain a time domain profile of the transmitted power through the cavity. measurements of the resonance peak, so throughout this work Q values have been obtained via time-domain measurements. Here, a short microwave pulse around the centre frequency is applied to the cavity. The decay of the transmitted power is amplified by an Avantek GaAs F E T 4-8 GHz amplifier, detected by a HP 8473B .01-18 GHz crystal diode detector, further amplified by a detection amplifier (gain set to 2000), then measured on a Tektronix TDS 520B digitizing oscilloscope. The time constant of the decay r gives the quality factor as Q = 27r/r. The decay is digitized by an A / D converter and stored on a computer where it is iteratively fit to obtain Q. Power dependencies are not expected to be an issue as the experiment is operated in a transmission configuration where low signals are applied at the input to the res-onator and amplified at the output for measurement. While some power dependency was observed at certain signal levels, broad regions of signal levels exist that are es-sentially power-independent and therefore suitable for measurement. Generally pulse powers of approximately -10 to 10 dBm were applied and a power ranging from -30 to -35 dBm detected. This detected power range is situated in the square-law region of the crystal diode detector and corresponds to a 250-500 fj,V signal that is further amplified by the detection amp to a 0.5-1.0 V signal observed on the oscilloscope. The power dependency was checked prior to measurements to ensure reliable Q values. Trigger Output Hewlett Packard 83620A Microwave Synthesizer A/D Converter GPIB < — * GPIB Computer Control IGPIB RF output Pulse Output Trigger Tektronix TDS 520B Digitizing Oscil loscope 15 V DC Power Supply Input Chapter 2. Electric Field Cavity Perturbation Technique 32 2.3.4 Operation of the Resonant Cavity Operation of the electric field resonant cavity generally proceeds as follows. The as-sembled probe is first cooled to 1.2 K in a pumped liquid helium bath and allowed to stabilize. The coupling loops are adjusted to the weak coupling regime (~30 dBm loss across the couplings) where the Q is maximized, while maintaining a sufficiently large, power-independent decay signal over the entire temperature range. A value Q0 is obtained for the empty cavity, then the sample is inserted and Q is measured as the temperature of the sample is changed. The experiment operates under computer control, with the sample temperature being set by a Conductus LTC-20 tempera-ture controller and the decay signal saved and fitted by the computer. Periodically throughout the experiment and at the end, Q0 is re-measured to ensure that it is unchanged. Typically the sample stage is first examined without a sample present (paraffin only), then a sample is mounted and measured. The background measurement of the sapphire and paraffin is then subtracted from the sample measurement to obtain the loss due exclusively to the sample. Finally, a Pb-Sn reference sample of similar dimensions is measured in the normal state to obtain the calibration factor V (Eq. 2.5), which is used to convert measured A ^ values into absolute values of surface resistance. 2.4 Performance of Microwave Cav i ty Resonator 2.4.1 Non-Perturbative Effects Non-perturbative effects were found to be a major issue when performing cavity perturbation measurements in an electric field. Sapphire is characterized by a non-negligible dielectric loss and slight misalignments of the plates cause significant varia-tion in the depolarization factor. This is in contrast to the magnetic field arrangement in which \i « 1 for sapphire, such that changes in the depolarization factor are in-significant. Variation of the depolarizing factor in the electric field geometry results in a differing field distribution in the cavity. As a result, it was often found that the Q of the cavity with the sapphire inserted exceeded that of the empty cavity, due to changes in the current pattern in the walls of the resonator. While this effect could sometimes be eliminated by removing and re-inserting the dual-sapphire plate, it often could not be avoided and calls into question the accuracy of the measured differences in Q and QQ. Chapter 2. Electric Field Cavity Perturbation Technique 33 Figure 2.9: Dual-sapphire stage alignment on (a) previous and (b) newly-constructed sample stages. A l l preliminary measurements presented thus far were performed with a prior construction of the sample stage that is currently used for resistivity measurements of single crystals. The sapphire block on this stage was not perfectly flat with respect to the electric field direction. This made alignment of the sapphire plates parallel to the electric field challenging, and often resulted in slight offset angles on the order of a degree or two, as shown in Fig. 2.9 (a). While small, this slight misalignment of the sapphire was sufficient to measurably distort the field distribution. When constructing the new sample stages, care was taken to ensure that the face of the sapphire block was parallel to the electric field direction. This improved the alignment of the dual-sapphire hot finger, as shown in Fig. 2.9 (b), and was found to completely remove these non-perturbative effects in the weak-coupling limit. A newly-etched dual-sapphire plate with a small amount of paraffin was mounted on one of the new sample stages to obtain an accurate value for the background loss in the absence of non-perturbative effects. The background loss ( A ^ ) , shown in Fig. 2.10, is on the order of 1 x 10" 9, with values increasing up to 1 x 10~8 at higher temperatures. 2 . 4 . 2 S a p p h i r e L o s s A n indication of how the background loss compares to that of the Tl-2201 crystals to be measured was obtained by calibrating the cavity using Pb-Sn reference samples. Chapter 2. Electric Field Cavity Perturbation Technique 34 1E-8-^ O 1— ^E-9-^ T 1 1 1 1 1 r A H A A A A A A A A A A A A A A A A A A 1 A — i 1 1 1 1 1 1 1 1 1 1 1 1 1 1— 0 20 40 60 80 100 120 140 160 Temperature (K) Figure 2.10: Background loss (sapphire and paraffin) for newly-constructed sample stage that eliminated non-perturbative effects. Typica l single crystals of Tl 2 Ba2Cu0 6± (5 grown by the U B C Superconductivity Group have dimensions on the order of 1 m m 2 x 0.02 mm thick. A Pb-Sn reference sample of similar dimensions (1mm x 0.73mm x .02 mm) was measured and calibrated against known resistivity values to obtain a calibration factor of T = (6.59 ± .05) x 105Q. The error indicated here is simply that of the experimental fit to resistivity data. When deviation in Q measurements and variation between reference and sample sizes are considered, a systematic uncertainty of up to 10% is associated wi th the calibration factor T. This T indicates that background losses well below Tc would correspond to unsatisfactorily high surface resistances values of several hundred LIQ, on the same order as the sample values that are expected to be measured. Electric field measurements are hampered by the additional complication that the sample significantly alters the field distribution in its vicinity, as shown in F ig . 2.1. Therefore, the loss due to the sapphire and paraffin when the sample is loaded is not expected to be identical to that measured with no sample present. If the background loss is relatively small compared to the sample measurement, it is likely suitable to ignore changes in the background due to the presence of the sample. Then the background level measured with no sample present can simply be subtracted from Chapter 2. Electric Field Cavity Perturbation Technique 35 the sample values. This, however, may not be an accurate approximation when the measured background is on the same order of magnitude as the sample loss, as is observed here. It is imperative to increase the ratio of sample:background losses in order to reduce the error associated with measuring and subtracting a background value. One way to do so is by increasing the size of the sample. Theoretical study of the microwave cavity losses of a long needle-like crystal in an electric field has previously predicted that the loss is particularly sensitive to the length of the sample in the field direction, increasing on the order of the fifth power of sample length [62]. This work was performed in order to study the temperature-dependent microwave conductivity of the one-dimensional conductor tetrathiofulvalene tetracyanoquinodimethan (TTF-T C N Q ) . The loss of an isotropic, ellipsoidal sample in an electric field was calculated to be proportional to: A i OC y N 2 a / 6 ) - l ] " 2 (2.6) where a is the length of the semi-major axis (field direction) and b is the length of the semi-minor axis ( b / a < l ) . It was verified experimentally that the difference in shape between needle-like samples and ellipsoids, as well as the effect of anisotropy, intro-duced only small errors into this expression. The shape dependence was checked using various lengths of cylindrical wire and wire flattened to rectangular cross-sections with base:height ratios of ~3:1. Such a strong dependence of cavity loss on sample length would be extremely useful in increasing the sample background signal ratio in the resonator. Tl-2201 single crystals up to 3 mm long have been produced by the U B C Superconductivity Group and this strong length dependence suggests that using one of these longer crystals could result in more than an order of magnitude increase in the ratio. To test the length dependence of the loss, another PbSn calibration sample of similar width and thickness, but length 1.8 mm instead of 1 mm, was measured. A calibration fit of T = 4.24 ± .01 x 105 was obtained in this case, indicating an increase in the sample loss by a factor of only ~1.54. Thus the strong length dependence is not present here, as a 1.8-fold increase in sample length results in a smaller increase in the sample loss. Moreover, when a sample of slightly smaller width b was used, the loss was lower, contrary to the 1/b term in Eq. 2.6. This is most certainly due to the very different aspect ratio examined. The width of typical platelet samples is 50 times greater than the typical thickness, and thus the Tl-2201 crystals are not well approximated by ellipsoids. Working with long needle-like crystals is ideal, as this Chapter 2. Electric Field Cavity Perturbation Technique 36 induces in-plane currents predominantly in one direction, permitting separation of the surface resistance components. However, given the current level of the background sapphire loss, it is necessary to work with the largest crystals possible in order to maximize the loss in the sample. The most significant improvement in accuracy is likely to come from a reduction in the dielectric loss of the sapphire plates. The dielectric loss tangent is defined as tan 6 — e"/e', and for a sapphire plate extending through the length of a resonant cavity it can be approximated as: where Ares and As are the cross-sectional areas of the resonator and sapphire re-spectively [63]. At 60 K, a background loss of approximately A ^ = 2 x 10~9 was observed, as shown in Fig. 2.10. For the cross-sectional areas of this cavity and sapphire stage, and including an additional factor of 2 to account for the fact that the sapphire only extends half-way through the cavity, a loss tangent of approxi-mately tan 6(60 K) = 5 x 1 0 - 6 is found. This is two orders of magnitude larger than published values for ultrahigh-grade sapphire crystal [64], and more than an order of magnitude larger than past measurements on sapphire rods by the U B C Super-conductivity Group. The larger loss tangent observed for the dual-sapphire stage has been attributed to the particular grade of sapphire being used. Hemlite optical grade sapphire is characterized by a limited density of impurities and lattice distor-tions, which are a likely source of dielectric loss [58, 64]. Use of Hemlux (superior) or Hemex (premium) grade sapphire, in which there are minimal or no impurities and lattice distortions, will likely result in a significant decrease in the dielectric loss of the sapphire and hence a corresponding increase in the sample background ratio. If, for example, a low temperature loss tangent of tan 5 = 2 x 1 0 - 8 is assumed, as was reported for ultrahigh-grade sapphire [64], Eq. 2.7 suggests a background loss of A ^ = 8 x 1 0 - 1 2 due to the presence of the sapphire in the cavity. For a 1 mm 2 x 0.02 mm sample, this corresponds to a background contribution of only a few u.0. , using the calibration factor given previously. This suggests that the use of high grade sap-phire may reduce the background loss in an electric field to only a few percent of the loss expected for high quality Tl-2201 single crystals. In this situation, the change in the background loss due to the presence of the sample can be assumed negligible with reasonable confidence, and accurate single crystal surface resistance measurements can be obtained by subtracting a measured background level (sapphire and paraffin) tan 8 = . (2.7) Chapter 2. Electric Field Cavity Perturbation Technique 37 from the sample values. In addition, a dramatic decrease in the background loss may permit the study of smaller, more one-dimensional needle-like crystals in which the two components of the in-plane surface resistance can be better isolated. 2 . 4 . 3 D i e l e c t r i c L o s s Comparison of electric field cavity perturbation measurements with those taken in a magnetic field in the traditional manner provides a careful evaluation of the perfor-mance of the electric field technique. A platelet sample of optimally doped Y B C O was measured in the electric field as described in Section 2.3.4, then re-measured in a magnetic field by operating the cavity in the T E 0 n mode. This is the mode for which the cavity was originally designed, and with negligible dielectric losses, this measurement provides an excellent standard to which measurements in the electric field can be compared. Figure 2.11 shows the measured surface resistance of the as-grown platelet in the magnetic field (diamonds) and electric field configuration (triangles). It is seen that the surface resistance measured is orders of magnitude larger than that in the magnetic field, with even the 1.2 K surface resistance in the electric field being similar to the normal state value measured in the magnetic field. This has been attributed to a lossy dielectric layer on the exterior of the sample. This particular sample was grown over 10 years ago, and although it has been stored is a desiccator, it has most certainly received significant exposure to water vapor in the air. Water is known to significantly degrade the chemical structure of the cuprate superconductors, resulting in a thin layer of non-superconducting material on the exterior of the crystal. While this dielectric layer is essentially transparent to a magnetic field, it absorbs a significant amount of energy when measurements are performed in an electric field. Etching the sample in a 0.5% bromine in ethanol solution for one minute was found to reduce the loss by an order of magnitude, confirming that this excess loss is in fact due to dielectric absorption on the exterior of the crystal. Further etching for a total of 10 minutes is seen to reduce the measured surface resistance to the same order of magnitude as that observed in magnetic field measurements. However, the shape of the low temperature peak is not yet regained and the resistance values diverge at higher temperatures, indicating that some dielectric material remains on the surface. Perfect agreement between the electric field and magnetic field measurements are not expected for the grade of sapphire currently being employed in the cavity, as the background loss subtracted is on the same order of magnitude as the measurements Chapter 2. Electric Field Cavity Perturbation Technique 38 1-J a o c cc w 0.01 • (Ji a> CC <D O CO "t 1E-3-CO 1E-4-J -< r T 1 r T 1 1 • r A A A A A A / K ^ A A A A A A Z A A A A A A A A ^ o o o O o 3 ° - o o o o o j • • • • D o o o o o o o o o o o o o o o o o o • • • • • • • • • • • £ ^ • 0 0 0 0 0 0 0 0 0 0 O T E 0 1 1 A A s grown o 1 min etch • 10 min etch J 20 I 40 I 60 I 80 —I— 100 - 1 — 120 T e m p e r a t u r e (K ) Figure 2.11: Comparison of YE^ surface resistance for measurement in a magnetic field (TEnu mode) and in the electric field configuration with the crystal as grown, etched for 1 minute, and etched for 10 minutes. A dielectric layer on the exterior of the sample causes significant loss in the electric field but can be removed by etching. below Tc. Employment of high grade sapphire in the cavity will allow a more precise comparison between electric field and magnetic field measurements, providing a more rigorous evaluation of the electric field technique. A lower normal state resistance is expected for the electric field measurement due to the absence of c-axis currents. However, the reduction in normal state resistance observed in Fig. 2.11 is larger than that expected from previous measurements of the c-axis contribution to RS(T) [50, 51]. This has been attributed to unusually large errors in the calibration factors used. The Y B C O platelet test sample was of an irregular shape, so it was difficult to produce an appropriate Pb-Sn reference sample and obtain accurate calibration factors. As mentioned previously, the reduction in background loss expected with the use of higher purity sapphire will likely permit measurement of more needle-like single Chapter 2. Electric Field Cavity Perturbation Technique 38 _ 0.1 w 0.01 CO o (0 "E 1E-3 1E-4-J T T A A A A A A A A A A A A A A A A A A A ^ > < > < > 0 < > < > ^ o o o o o o o o o o o o o o o o o o o o o o i • • • LJ • • • • • • * ^ O O O O O O O O O O O O * - O T E m , p o O v A A s g rown o 1 m i n e tch • 10 m in e tch i 20 40 60 80 T e m p e r a t u r e ( K ) 100 120 Figure 2.11: Comparison of YBa 2 Cu306 .95 surface resistance for measurement in a magnetic field ( T E 0 n mode) and in the electric field configuration with the crystal as grown, etched for 1 minute, and etched for 10 minutes. A dielectric layer on the exterior of the sample causes significant loss in the electric field but can be removed by etching. below Tc. Employment of high grade sapphire in the cavity will allow a more precise comparison between electric field and magnetic field measurements, providing a more rigorous evaluation of the electric field technique. A lower normal state resistance is expected for the electric field measurement due to the absence of c-axis currents. However, the reduction in normal state resistance observed in Fig. 2.11 is larger than that expected from previous measurements of the c-axis contribution to RS(T) [50, 51]. This has been attributed to unusually large errors in the calibration factors used. The Y B C O platelet test sample was of an irregular shape, so it was difficult to produce an appropriate Pb-Sn reference sample and obtain accurate calibration factors. As mentioned previously, the reduction in background loss expected with the use of higher purity sapphire will likely permit measurement of more needle-like single Chapter 3 40 Non-local Normal Fluid Effects in Y B a 2 C u 3 0 6 + 2 / 3.1 Non- loca l Quasiparticle Elect rodynamics In prior microwave studies of YBa 2Cu306+j /, thin platelet samples have been oriented with their broad ab plane parallel to an applied magnetic field, such that screening currents are driven near the surface in a closed path along the ab face and in the c direction, as shown in Fig. 3.1. As discussed in Section 2.1, this experimental orientation is such that the mea-sured response is primarily that of transport in the ab planes, with c-axis currents contributing little [50]. A key result of this geometry is that the quasiparticle elec-trodynamics remain local. While measurement of the quasiparticle scattering time indicates low temperature in-plane mean free paths (/„, lb) on the order of several mi-crons for high-quality overdoped Y B C O [40], the weak coupling between C u 0 2 planes [52, 65] restricts quasiparticle motion in the c-direction such that lc is on the order of the lattice spacing [66, 67]. As a result, quasiparticles driven along the ab surface remain within a well-defined region of the penetration depth (/ c < A) and exhibit a local, instantaneous relationship with the electric field. For the current driven along the dc or be planes, A c has been measured to be on the order of a micron for fully-doped Y B C O [68]. Therefore this current includes quasiparticles with in-plane mean free paths larger than the corresponding penetration depth (la, lb > A c ) , but these quasiparticles are coherently hopping from one plane to another. Moreover, any non-local effect due to these quasiparticles would be masked by the c direction's relatively small contribution to the measured overall surface resistance(Rsc <§C Rsa, Rsb)-Different quasiparticle behavior is expected when the orientation of the supercon-ducting sample is changed so that the applied magnetic field runs perpendicular to the C u 0 2 planes as shown in Fig. 3.2. This sample geometry is such that currents in both the cd and cb planes include contributions from quasiparticles with relevant mean-free paths greater than the corresponding penetration depths (la > Xb, h > A a ) . These Chapter 3. Non-local Normal Fluid Effects in YBaqCu^Oe+y 41 Figure 3.1: Traditional experimental geometry in which both the normal and super-fluid responses are local. quasiparticles can move freely in and out of the region where the field penetrates without scattering from anything but the surface. As such, the field experienced by these quasiparticles may change drastically between scattering events and a nonlocal electrodynamic response is expected. Since quasiparticles contribute to the screen-ing current at finite frequencies, small changes in the penetration depth may result. However, this change is expected to be very small and the largest effect should be observed in the surface resistance, since the quasiparticles are entirely responsible for these losses at frequencies below the pair-breaking level. A theoretical treatment of this sample geometry in the limit of weak elastic scat-tering was presented by Rieck et al. and revealed a peak in RS(LU) around 100 GHz, which greatly exceeded RS(UJ) values in the local limit [69]. The peak heights were found to decrease rapidly with decreasing temperature, indicating that this increase in RS is due to photon absorption by thermally excited quasiparticles. The non-local corrections were also found to be dependent on the level of scattering, decreasing as the scattering phase shift is increased. Nonetheless, it was found that non-local corrections should be observable in all but the unitary scattering limit. Some microwave measurements have been performed previously with single crys-tal cuprates in this non-local orientation. Cavity perturbation measurements of the highly anisotropic Tl-2201 and Bi 2 Sr 2 CaCu 2 08 were carried out with the magnetic field applied perpendicular to the planes in order to avoid c-axis screening currents [53, 54, 55]. No significant increase in the surface resistance, attributable to non-local Chapter 3. Non-local Normal Fluid Effects in YBa2Cu'iO&+y 42 Figure 3.2: Experimental arrangement for which non-local quasiparticle electrody-namics are expected. effects, was observed at low temperatures. However, the scattering rates 1/r mea-sured for these samples were an order of magnitude larger than those determined for fully-doped Y B C O [40, 48], such that the low temperature quasiparticle mean free path in Tl-2201 and Bi 2 Sr 2 CaCu 2 08 is only marginally larger than the penetration depth. As a result, significant non-local quasiparticle effects are not expected for these crystals. Y B C O , where the mean free path at low temperatures is an order of magnitude larger than the penetration depth, is the most promising cuprate for observation of these effects. The remainder of this chapter reports on the preparation and measurement of Y B C O single crystals in this geometry (Hrf \\ c) in order to identify a non-local quasiparticle response. Chapter 3. Non-local Normal Fluid Effects in YBa2Cu306+y 43 3.2 Sample Preparat ion Low-temperature measurements of quasiparticle dynamics are highly dependent on the purity of the samples. Bonn et al. doped YBa 2 Cu 3 0 6.95 single crystals with varying concentrations of Ni and Zn, demonstrating that impurities act as strong elastic scattering centres that alter the observed surface impedance [42]. It was this impurity effect that had previously masked the linear temperature dependence of the penetration depth in Y B C O and prevented identification of the superconducting gap's d-wave nature. Therefore, to examine the intrinsic properties of these materials it is necessary to work with homogenous samples of extremely high purity. Dr. Liang of the U B C Superconductivity Group grows the world's highest purity Y B C O single crystals using a self-flux method in BaZr0 3 crucibles [70]. A l l the measurements presented in this chapter were performed on single crystals grown by this technique. For microwave measurements, it is essential that an appropriate sample geometry be chosen to ensure that demagnetization effects do not skew the results. In particular, demagnetization effects are minimized by aligning a broad, fiat face of the sample parallel to the field. Typical dimensions of the Y B C O single crystals grown at U B C are 1 mm 2 x 0.01 mm, with the c direction having the smallest dimension. However, for this study of non-local quasiparticle dynamics, it was essential to align the c direction with the magnetic field and hence a relatively large c dimension was required to minimize demagnetization effects. Therefore several particularly thick optimally-doped samples grown by the self-flux method were selected for study. The samples were mechanically detwinned after cutting them along the 100 or 010 direction using a K . D . Unipress WS-22 Precision Wiresaw to significantly reduce their size in the a or b direction. To further reduce this dimension relative to the c-axis thickness, the samples were sequentially polished using an electric polishing station with diamond grit sizes ranging from 5 fim down to 0.5 Lim. As interplanar bonding is relatively weak in comparison to bonding within the ab plane, the samples are extremely prone to structural failure when cutting and polishing in such a fashion where the size of the planes is actively reduced. The fact that several crystals remained intact throughout this cutting and polishing procedure is a testament to the quality of crystals grown by Dr. Liang. For all samples examined here, cutting and polishing produced an a or b dimension that was less than half the size of the c-axis thickness, ensuring that demagnetization effects would be negligible. Fig. 3.3 shows optical micrographs of a YBa2Cu306.93 crystal whose (100) faces were polished until the crystal was 120 ± 10 Lim wide. The polishing sequence Chapter 3. Non-local Normal Fluid Effects in YBa^CusOe+y 44 Figure 3.3: Optical micrographs of a YBa2Cu306.93 sample whose (100) planes were polished to reduce the a dimension. for this crystal was 5, 27, and 45 minutes with grit sizes of 3.0, 0.3, and 0.05 Lim respectively, and was typical of the polishing times used for the other crystals prepared as well. The coarser 3.0 Lim grit was used for the majority of the crystal thinning, while the long, finer grit polishes were employed to alleviate some of the deeper surface damage that is believed to occur when polishing with large grit sizes. A prior study examined the effect of crystal polishing on the microwave surface resistance of single crystal YBa2Cu306+2/ and found that polishing can reduce the height of the peak in RS(T) by more than a factor of two [71]. However, it was also found that etching the polished surfaces was an effective method for clearing away the damaged material and restoring the peak. The samples here were etched in a 0.5% bromine in ethanol solution to improve the regularity of the outer surfaces and remove regions of surface damage caused by the cutting and polishing procedures. Bromine etchant preferentially attacks dislocations and damaged regions, leaving, for the purpose of Chapter 3. Non-local Normal Fluid Effects in Y B a 2 C113 Oe+y 45 Figure 3.4: Optical micrographs of a YBa2Cu 30 6.93 surface at etching times of (a) 1 minute (b) 3 minute (c) 5.5 minutes microwave surface impedance measurements, a high quality, intrinsic surface. Fig. 3.4 shows optical micrographs of a sample surface at various stages of the etching procedure. Regions damaged by polishing that are etched away over time are evident. Following initial measurements on the optimally doped samples, one polished and etched sample was annealed in an oxygen-rich environment for two weeks until it was fully-doped Y B a 2 C u 3 06.998. Composed of filled CuO chains, Y B a 2 C u 3 06.998 samples are particularly defect-free, without the oxygen vacancy clustering identified by Erb et al. [72]. As a result, they are the most likely cuprate to display non-local effects, which require long mean free paths and hence high sample purity. Chapter 3. Non-local Normal Fluid Effects in YBa^Cu^Oe+y 46 3 . 3 Exper imenta l Technique The microwave resonant cavity discussed in in Section 2.3 was used to measure the surface resistance of the polished and etched samples as a function of temperature. The resonator was operated in the T E 0 u mode at 13.4 GHz so that the sample was located in a uniform magnetic field and at an electric field node. Dielectric losses are negligible in this magnetic field orientation, so a single plate of .0055" sapphire 1 mm in width and 20 mm long was used to support the samples in the cavity, which were attached with a small amount of silicone vacuum grease. The 4-8 GHz microwave amplifier was replaced with a Miteq 12-18 GHz amplifier, but the operation of the experiment was otherwise the same as that detailed in Section 2.3 . At each temperature, several measurements of Q were made and averaged. The standard deviation in these measurements indicates an error of less than 4% at high temperatures, increasing up to 6% at the lower temperatures. Errors associated with measuring the sample dimensions, Pb-Sn reference dimensions, and differences between these values accounted for a systematic error of up to 10% in the calibration factor T. This results in a total error in the surface resistance measurements of up to 16% at the low temperatures. Reference samples were, however, carefully selected and measured to minimize this error and error bars have been left off the graphs for clarity. 3 . 4 Results The microwave surface resistance of a platelet sample of optimally doped YBa2Cu306.93 (from the same batch of crystals as the non-local samples) was first measured in the normal orientation (H \\ ab) for comparison and the results are shown as diamonds in Fig. 3.5. Two samples of YBa2Cu306.93, cut and polished as discussed in Section 3.2 and etched for periods of 2.5 and 5.5 minutes respectively, were then measured in the field orientation for which non-local effects are expected (H \\ c). These measurements are also displayed in Fig. 3.5. A number of consistencies with previous microwave surface resistance studies of Y B C O are observed. A slightly lower peak in RS(T) is seen for the polished sample that was etched for 2.5 minutes. This has previously been attributed to surface damage caused by polishing [71], but the peak is observed to recover with longer etching time as the damaged surface material is removed. A slightly larger normal state resistance is also observed for the platelet sample, consistent with a small c-axis Chapter 3. Non-local Normal Fluid Effects in YBa2Cu3Oe+y 47 0.01 a o c 3 To CD rr o 03 t CO 1E-3-J 1E-4 4 O Platelet Y B a 2 C u 3 O e 9 3 (etched 1 min) o Non-local Y B a 2 C u 3 O e m (etched 2.5 min) A Non-local Y B a 2 C u 3 0 6 m (etched 5.5 min) o o * o O <> o O ft A ft* 20 40 » 8 & & A 60 80 0.25 \ 0.20 0.15 0.10 0.05 0.00 100 Temperature (K) Figure 3.5: Cavity perturbation measurements of a YBa2Cus06.93 platelet and two crystals examined in the non-local orientation (H \\ c). The non-local YBa2Cu306.93 samples were etched for 2.5 min and 5.5 minutes respectively. No increase in surface resistance is observed at low temperatures as is expected for a non-local quasiparticle response. contribution in this orientation. The most important result of Fig. 3.5, however, is the apparent absence of any increase in low temperature surface resistance due to non-local quasiparticle effects. Instead, the low temperature surface resistance appears smaller for the non-local geometries, consistent with the absence of c-axis contributions to the loss [50, 51]. Measurements were repeated on the sample annealed to full-doping in which supe-rior homogeneity is expected. It is shown in Fig. 3.6 that a decrease in the low tem-perature surface resistance was again observed. The peak in Ra(T) was suppressed, indicating that the annealing process had reduced the quality of the sample's surface. The sample was therefore etched for 12 minutes and re-measured. While the peak re-appeared following the etch, the low temperature resistance values are still less than that observed in the local orientation. If non-local effects are present in the measurements, their magnitude is small at the 13.4 GHz frequency examined, and they are masked by the differing c-axis contributions between the local and non-local geometries. Chapter 3. Non-local Normal Fluid Effects in YBa^CuzO^y 4 8 1E-3. a a o c B GO w a> DC a o CD t 1E-<H O Platelet Y B a 2 C u 3 0 6 9 3 (etched 1 min) • Non-local Y B a 2 C u 3 0 6 g g e (annealed) o Non-local Y B a 2 C u 3 0 6 ^ (etched 12 min) O o o 0 o o o • • o o o ° O f t f r i 9 Q o CO o ° • § ^ o D 0 o y A O ° o ° • D O o • • o • 20 40 60 Temperature (K) 120 Figure 3.6: Cavity perturbation measurements of a YBa2Cu 3 06 .93 platelet in the local orientation and a YBa2Cu3 06.998 crystal in the non-local orientation (H || c). The non-local Y B a 2 C u 3 0 6 . 998 sample was annealed to full-doping and etched for 12 minutes. No increase in surface resistance is observed at low temperatures as is expected for a non-local quasiparticle response. Theoretical treatment of this non-local geometry has assumed that the applied field is perfectly perpendicular to the CuC>2 planes [69]. Experimentally, the sam-ple is never perfectly aligned with the field and it is likely that this will affect the non-local quasiparticle response. For slight sample misalignments, a very small c-axis contribution to the screening current results, which induces a corresponding c-axis component in the velocity of the quasiparticles. Since this causes coherent hopping of quasiparticles between the superconducting planes, it limits the in-plane distances traveled by the quasiparticles and hence any non-local effect. As well, the theoretical prediction of this non-local response accounts only for in-plane quasiparticle scat-tering, neglecting out-of-plane contributions. This is a significant omission, as CuO chain disorder, for example, has recently been shown to be a weak-limit scattering source in Y B C O [73]. These observations are in agreement with the results presented here, which appear to indicate that experimental effects associated with microwave Chapter 3. Non-local Normal Fluid Effects in YBa2Cu306+y 49 surface resistance measurement techniques produce a local quasiparticle response for any sample geometry. The most conclusive evidence however, is likely to come from broadband surface resistance measurements of these samples in the non-local orien-tation, as is discussed further in Section 4.3. 50 Chapter 4 Future Considerations 4.1 Sapphire Loss i n Electr ic F i e l d Cav i t y Per turba t ion Measurements A number of design challenges have been confronted and addressed in the development of a cavity perturbation technique for measurement of the surface resistance of highly anisotropic Tl-2201 single crystals in an electric field. The remaining obstacle that currently limits the use of this probe is the large background loss associated with the sapphire plates used to support the sample in the resonant cavity. As discussed in Sec. 2.4.2, the sapphire absorption is expected to be reduced by the employment of premium-grade sapphire plates. With greater quality comes greater cost, and is suggested that a concerted effort be made to identify the most reliable way to cut this expensive material, in order to reduce the amount of sapphire wasted. Little technical information exists on methods for cutting thin strips of sapphire, however a comprehensive literature review should be conducted to ensure that the sapphire is being cut in a direction in which it preferentially cleaves or parts. This should allow more successful cutting of the plates and produce smoother cut edges that are less susceptible to the adherence of contaminants. If the use of premium-grade sapphire does not reduce the level of the background loss sufficiently, it may be necessary to construct a new resonant cavity with a higher resonant frequency. The surface resistance of a superconductor is proportional to the square of the frequency, as shown in Eq. 1.13 so, provided the background loss increases more slowly with frequency, operation at a higher frequency would result in a much larger sample loss relative to that of the sapphire. Reduction of the error associated with subtracting background losses will permit the accurate measurement of smaller Tl-2201 single crystals in the cavity. As discussed throughout this work, one-dimensional needle-like crystals better isolate the in-plane surface resistances, since currents are induced almost entirely in one direction. Cohen et al. showed that the loss of needle-like crystals in an electric field increases rapidly with the length of the crystal [62]. A theoretical treatment in Appendix A indicates Chapter 4. Future Considerations 51 that the loss due to an undesired dielectric layer on the exterior of an ellipsoidal sample (used to approximate the needle shape) also increases with length, but at a much smaller rate than the conduction losses due to the sample. As a result, the ratio of sample to background signals increases significantly with the length of the needle-like crystal examined. One area that remains for examination, however, is the effect of the sample on the background loss of the sapphire, and the dependence of this effect on the size of the sample. The sample distorts the electric field in its vicinity and hence the field experienced by the sapphire close to the sample. It has been assumed that this change in the background loss will be small and can be ignored. This is likely a reasonable assumption for sufficiently small background losses, given that the field distribution will only be significantly altered for the small portion of the sapphire that is close to the sample. However, sapphire exhibits dielectric uniaxial anisotropy, so it is conceivable that changes in the field direction could result in noticeable changes in absorption by the sapphire. Experimental examination of the magnitude of this effect is therefore recommended. Once premium-grade sapphire is employed in the cavity, a Y B C O sample can be measured both in the E and H field configurations. The magnetic field surface resistance values (corrected for background losses and the different frequency) can be subtracted from the electric field measurement to give an indication of the sapphire loss in the presence of the sample. This can then be compared to an electric field measurement of the sapphire with no sample present to determine whether the two agree within experimental uncertainty. If the difference is significant and must be accounted for, theoretical treatment of this experimental arrangement is strongly suggested. Ideally, theoretical examination would indicate what sample geometry and dimensions should be chosen to minimize the correction, while maintaining a large signal to background ratio. 4.2 C o l d Stage of Elect r ic F i e ld Cav i t y Per turbat ion Probe The copper cold stage of the electric field probe connects the stainless steel sample rod to the copper sample sample stage and provides a thermal link to the pumped liquid helium bath. As this stage is is moved when inserting and retracting the sample, it was found that the brass leads along its length were susceptible to shorting (both to the cold stage and to each other) due to rubbing against the brass probe body. This problem was likely amplified by the fact that G E electrical varnish, while necessary Chapter 4. Future Considerations 52 for thermally anchoring the leads, tends to cause deterioration of wire insulation. A groove was therefore machined along the length of the cold stage into which the brass leads were inset and covered with thin paper to prevent rubbing against the probe body. While this was very successful in preventing damage to the brass leads, shorting of the wires to each other (due to varnish-induced insulation degradation) continued to be a problem. As a temporary fix, for a portion of the cold stage length, the brass wires were fed through a small teflon tube which was placed in the cold stage groove, instead of being varnished directly to the stage. While the brass leads were still varnished directly to the copper at the top and bottom of the stage, this arrangement undoubtedly reduces the thermal contact between the thermometry leads and the liquid helium bath somewhat. Nonetheless, this decrease in thermal contact was sufficiently small that the sample temperature could still be reliably monitored and regulated, so this arrangement was deemed suitable for preliminary design, testing, and use of the microwave probe. As the sensitivity of the probe improves with the introduction of superior sapphire and high-precision surface resistance measurements are to be performed, it would ad-vantageous to improve the temperature stability of the copper sample stage. This would likely improve the speed and accuracy of sample temperature regulation, since fewer thermal fluctuations would reach the sapphire stage and thermometry. Faster temperature regulation would permit more accurate and comprehensive surface resis-tance measurements, as more data points could be acquired over a shorter interval of time. It is advantageous to reduce the time period required for a temperature scan, as a number of experimental factors such as the base temperature and empty cavity quality factor Qo tend to drift. The largest improvement in thermal stability would likely be achieved by replacement of the copper cold stage. Thermal fluctuations down the length of the probe could be drastically reduced by using a material with lower thermal conductivity for the cold stage, such as stainless steel. In this situation, the strength of the thermal connection to the base temperature could be maintained by attaching two copper braids along the length of the cold stage. This should be easily achievable with the current design, with slight modifications to the probe body. In addition, larger braids that attach to the cold stage with two screws instead of one could be used. A larger inset surface area should also be added so that the electrical leads can be safely varnished to the stage in order to limit thermal fluctuations down the length of these wires. Chapter 4. Future Considerations 53 4.3 Spectroscopic Measurement of YBa2Cu306+2/ i n the Non- loca l Orientat ion A n important result from the theoretical treatment of the (100) non-local orientation by Rieck at al. was that depending on the temperature, there exists a frequency below which the surface resistance in the local limit exceeds the result from the nonlocal calculation [69]. This was attributed to the particular orientation of the order parameter. At low frequencies, the contribution from quasiparticles moving parallel to the (100) surface is greatest. Since this is the direction in which the order parameter has its maximum, the number of thermally excited quasiparticles that are effective in the absorption process is reduced. The crossover at which non-local effects begin to dominate occurs at higher frequencies as the temperature is increased. According to the weak-limit scattering case examined by Rieck et al., for a 13.4 GHz frequency, the nonlocal surface resistance should be greater than the local value for temperatures up to « 15 K . However, the calculated local and nonlocal values differ by little more than a factor of 2 for this frequency. As such, this theoretical study suggests that a frequency of 13.4 GHz is not well-suited to observation of nonlocal effects and that measurements at frequencies closer to 20 GHz are more likely to display an obvious difference between the local and non-local situations. The most likely method to conclusively determine whether non-local quasiparti-cle effects are present in these samples is spectroscopic measurement of the surface resistance Rs(u,T). The broadband bolometric apparatus discussed in Section 1.3.4 is perfectly suited for this study. It permits measurement of the surface resistance up to 20 GHz, where non-local corrections are expected to be large, and would allow detailed comparison of the shape of the non-local Rs(UJ) spectra with those previously measured for samples with local normal fluid responses. Considerable time has been spent repairing the bolometry apparatus for this purpose. However, these efforts have been hampered somewhat by the lack of sapphire plates that can be easily cut to suf-ficiently small widths. 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Bibliography 58 [73] J.S. Bobowski, P.J. Turner, R. Harris, Ruixing Liang, D.A. Bonn, and W . N . Hardy. To be published, 2006. Appendix A 59 Theoretical Treatment of Lossy Dielectric Layer The relative permittivity of the dielectric can be written as e' o t = e'T + ie" = £ r e l *. Assuming a thickness T for the dielectric layer, a series capacitance treatment, as is shown in Fig. A . l , gives the following expression: I = A + T (A. l ) C c0A eoe^A such that C = ^ l—-r (A.2) Assuming T <C d, one can Taylor expand — — ^ - ~ 1 - to obtain: C ^ f l - ^ ] (A.3) d \ a er J Now f/ = CV = CVQewt such that: e0A T e - * , , , ^ Vbe^ (A.4) Taking the derivative of this expression: dq VJJ€QAVQ T cos(ut) + isin(ut) — -— (cos(uit — (/>) + isin(u>t — 4>)) deT (A.5) dt d The instantaneous power can then be calculated as: P = R e { / } R e { V } = Re | ^ | V0cos(ut) ~ ——sin (cut) cos (ut) H y. — sin (ut) cos (tut) cos<p — cos [uit) smqo d d2Er L Appendix A. Theoretical Treatment of Lossy Dielectric Layer 60 Figure A . l : The electric field normal to a dielectric layer of thickness T and relative permittivity er can be modeled by an equivalent capacitance. Substituting V = Ed gives the following expression for the instantaneous power: Tue AE2 P ~ —ueoAE2dsin (cut) cos (ut) H sin (tot) cos (ut) cos<f) — cos2 (uot) sincj) Er L (A.6) The average power over one period P can then be calculated: 1 f „ , TUCQAE2 , , ¥ j ~2e S i U ( l ) ( A J ) where Tp is the length of one period and a minus sign has been dropped for notational convention. For an ellipsoidal sample in an electric field, the magnitude of the field E and hence the power dissipation are not constant over the surface. The expression of interest is therefore the power dissipation per unit area: P TuoepE2 — = —^ sincf) (A.8) This can be integrated over the surface to determine the loss due to a dielectric layer of thickness T in an electric field. Assuming a dielectric layer of uniform thickness Appendix A. Theoretical Treatment of Lossy Dielectric Layer 61 Figure A.2: The sample is assumed to be a prolate ellipsoid with semi-major axis a, semi-minor axis 6 (5 < a) and a dielectric layer of thickness T. and permittivity, this total power loss can be written: D T U J E ° nnt = ——sin 2er E2dS (A.9) Following from the work of Cohen et al. [62], for an applied electric field in the i-direction, the field at the surface of an ellipsoidal sample can be taken as: EQ E = —n z (A. 10) where nz is the component of the unit surface normal in the field direction and n — (£ ) 2 [/n — 1 is the geometric depolarization factor of the ellipsoid. This expression for E was obtained by considering the ellipsoidal conductor in the limiting case of an identically-shaped insulator, with e —* oo. Thus the dielectric loss can be re-written as: P t o t — Tue0El 2errf sin<j) J n2dS (A.ll) Appendix A. Theoretical Treatment of Lossy Dielectric Layer 62 and the problem is reduced to integration of the component of the surface normal in the field direction over the surface of the ellipsoidal sample. For a prolate ellipsoid with geometry as shown in Fig. A.2, the surface area S can be calculated using: S = 2TT T r(z)^l + [r'(z)]2dz (A.12) where r(z) is the radius of the ellipsoid as a function of z, given by r(z) = 6^1 — . Therefore the surface area of the ellipsoid is: S = 4rr r 6 Jl + —Ab2 - a?)dz = f dS (A.13) J0 V CL Jsurface The component of the surface normal in the direction of the electric field nz was calculated to be: nz(z) = b Z (A.14) + * 2(& 2 - a 2) Referring to Eq. A.ll, the only terms in the expression for the loss that are depen-dent on sample dimensions are the integral / n2dS and the geometric depolarization factor n. Therefore, the dependency of the dielectric loss on the dimensions of the ellipsoidal sample can be expressed as: L oc 1 f 9 47ra2 fa z2dz I", /2a \ / n2zdS = —— / . = ln( — ) V2J b Jo Ja4 _ z2<a2 _ fe2) L V b J -2 (A.15) Since corresponds to the loss L divided by the energy stored in the cavity, which is independent of sample size, Eq. A.15 can be re-written as the dependency of A i , on the dimensions of the ellipsoid: V dielectric A 1 47ra 2 ra z2dz A— a f Jo i - 2 (A.16) 'Q dielectric" b Jo ^/a4 - z2(a2 - b2) We now recall that Cohen et al. [62] derived the following expression for the scaling of the sample loss with the dimensions of the ellipsoid: A l o c ^ [ l n ( 2 a / 6 ) - l ] - 2 (A.17) *V sample u Numerical calculation of Eq. A.16 and Eq. A.17 allows comparison of how the sample loss and the loss due to the dielectric layer vary with ellipsoidal sample dimen-sions. Fig. A.3 shows these two expression evaluated as functions of the semi-major Appendix A. Theoretical Treatment of Lossy Dielectric Layer 63 Semi-major axis length a Figure A.3: Scaling of the dielectric layer and sample losses with dimensional varia-tion of the ellipsoidal sample, calculated numerically using Eq. A.16 and Eq. A.17. axis length of the ellipsoid a for three different values of the semi-minor axis length b. The expressions are scale factors, not absolute values of A ^ . Therefore, the dielectric results have been normalized such that the dielectric and sample values coincide at a=l for each value of b. This allows direct comparison of how the two losses scale with sample length. As expected from the a 5/6 term of Eq. A.17, the sample loss is larger for smaller values of b and increases significantly with the length of the sample. While the dimensional dependence is not as obvious from Eq. A.16, the dielectric layer exhibits similar behavior. The loss due to a dielectric layer on an ellipsoidal sample increases with sample length and decreases with sample width. The major difference between the scaling of these two losses, however, is the much stronger length dependence observed for the loss in the sample. For the b = 0.2 case in Fig. A.3, the ratio of sample:dielectric scale factors increases by a factor of 3.5 when the length of the sample a is doubled. It is clear from this analysis developed with the help of Elizabeth Ledwosinska, Appendix A. Theoretical Treatment of Lossy Dielectric Layer 64 that for needle-like crystals (well-approximated by ellipsoids) in an electric field, it is advantageous to maximize the length of the sample in the direction of the field and limit the sample dimension perpendicular to the applied field. This minimizes the portion of the measured loss that is due to any dielectric material on the exterior of the surface. In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my depart-ment or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. (Signature) Department of Physics and Astronomy The University of British Columbia Vancouver, Canada Date '. 


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