Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Measurement and analysis of the in-plane electrodynamics of Y Ba2Cu3O6.993 at microwave frequencies Harris, Richard Graydon 1999

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata


831-ubc_1999-0520.pdf [ 7.37MB ]
JSON: 831-1.0089157.json
JSON-LD: 831-1.0089157-ld.json
RDF/XML (Pretty): 831-1.0089157-rdf.xml
RDF/JSON: 831-1.0089157-rdf.json
Turtle: 831-1.0089157-turtle.txt
N-Triples: 831-1.0089157-rdf-ntriples.txt
Original Record: 831-1.0089157-source.json
Full Text

Full Text

M E A S U R E M E N T A N D A N A L Y S I S O F T H E I N - P L A N E E L E C T R O D Y N A M I C S O F YBa2Cu306.9g3 A T M I C R O W A V E F R E Q U E N C I E S By Richard Graydon Harris B. Sc. McMaster University, 1997 A THESIS S U B M I T T E D IN PARTIAL F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in T H E F A C U L T Y O F G R A D U A T E STUDIES D E P A R T M E N T O F PHYSICS A N D A S T R O N O M Y We accept this thesis as conforming to the required standard T H E UNIVERSITY O F BRITISH C O L U M B I A June 1999 © Richard Graydon Harris, 1999 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at ' the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics and Astronomy The University of British Columbia 6224 Agricultural Road Vancouver, B.C. , Canada V6T 1Z1 Date: Abstract Measurements of the electromagnetic absorption in ultra high purity YBa.2Cu3OQ.g93 at a frequency of 75 GHz are presented. When presented in conjunction with measurements taken at 1, 2, 13 and 22 GHz from other microwave surface impedance experiments performed at U B C , the low temperature data below 20 K reveals a regime which can be characterized by an impurity limited quasiparticle scattering rate and a pure d-wave density of states. This regime is analyzed within two theoretical pictures. A simple two-fluid model with a frequency independent scattering rate can simultaneously match the magnitude of all data sets, but does not account for the observed evolution of the temperature dependence from sublinear in T at 1 GHz to super linear at 75 GHz. A generalized BCS formalism with isotropic elastic point scattering does provide the correct curvature, but in order to do so it requires a frequency dependent one body scattering rate function that cannot be derived from within the isotropic point scattering picture. ii Table of Contents Abstract ii Table of Contents iii List of Tables v List of Figures vi Glossary of Symbols viii Acknowledgements xi 1 Introduction 1 2 Theory 4 2.1 Generalized Two Fluid Model with Drude Quasiparticle Scattering . . . . 5 2.2 Microscopic Isotropic Point Scattering Model 7 3 Experiment 19 3.1 Samples 20 3.2 Measurement Principle < 23 3.3 Measurement Technique 24 3.4 Implementation of the 75 GHz Apparatus 31 4 Results and Analysis 39 4.1 Measurement Procedure 39 iii 4.2 Experimental Results 43 4.3 Theoretical Analysis 59 4.4 Theoretical Results 63 5 Conclusions 77 B ib l iography 80 iv List of Tables 4.1 Summary of error estimates for measurements of Rs(a) 53 4.2 Summary of power law fits to Rs(a) i n the pure regime 61 4.3 A consistency check between the simple Drude and point scattering model. 63 4.4 Best fit parameters to the point scattering model 64 4.5 Best fit parameters using the modified fitting algorithm 67 v List of Figures 2.1 Near perfect diamagnetism in a superconducting ellipsoid 4 2.2 Feynman diagram for electromagnetic absorption 9 2.3 Diagram representation of the quasiparticle self energy 11 2.4 The d-wave density of states 15 2.5 A comparison of the theoretical expressions of Berlinsky et al. and Hirschfeld et al 17 3.1 The crystal structure of YBa2Cu307-$ 22 3.2 Measurement of Zs{£l) in the a direction of a thin YBCO crystal 23 3.3 A cross section of the 75 GHz TEQH mode right cylindrical resonator. . . 25 3.4 The hot finger assembly used in the 75 GHz apparatus for sample temper-ature regulation 30 3.5 A cross section of the sample gantry and the resonator block 32 3.6 A cross section of the microwave circuitry inside the dewar 33 3.7 A detailed view of the microwave couplings 34 3.8 A detailed view of a mechanical feedthrough 35 3.9 Schematic diagram of the 75 GHz apparatus 37 4.1 A depiction of the sequence of measurement geometries 41 4.2 Raw measurements containing an admixture of o and c axis contributions. 44 4.3 Raw measurements containing an admixture of b and c axis contributions. 45 4.4 The extracted a, b and c axis losses 46 4.5 The extracted background signal 47 vi 4.6 Calibration of the 75 GHz apparatus using a PbSn sample 48 4.7 The surface impedance of YBCO measured at 75 GHz 50 4.8 A scaled plot of Rs(a) measured at five microwave frequencies 51 4.9 The normal and superfluid densities in the a axis orientation 52 4.10 A scaled plot of Rs(b) measured at five microwave frequencies. . . . . . . . 56 4.11 A comparison of the a:b anisotropy of R$ at 13 and 75 GHz 57 4.12 A scaled plot of Rs(c) measured at two microwave frequencies 58 4.13 Fitting of the pure regime Rs(a) data with a power law expression 62 4.14 Independent fits of each Rs(a) data set to the point scattering model. . . 65 4.15 Modified independent fits of the Rs(a) data sets to the point scattering model 68 4.16 Global fit of all five Rs(a) data sets to the point scattering model 70 4.17 Global fit of all five Re\agp\ data sets to the point scattering model. . . . 74 4.18 A plot of the extrapolated cr*0 75 vii Glossary of Symbols Sample surface area traversed by screening currents. Electromagnetic vector potential. Speed of light. Inverse of the tangent of the s wave scattering phase shift. Free electron annihilator/creator. Nambu electron-hole pair annihilator/creator. Elementary electronic charge. Electric field. G(u>) = GQTQ + G\T\ + G2T2 + G 3 T3 Integrated Green's function. Q,QM Superconducting/normal state quasiparticle Green's function. h Decaying magnetic field inside a superconductor. H Magnetic field. 1 Square root of -1. j Surface current density. k A momentum variable. K Electromagnetic absorption tensor. m*,m* Normal/superconducting quasiparticle effective masses in two fluid model. nf Fermi function. rii Number density per unit volume of impurities. nQ Number density per unit volume of charge carriers. Na Density of states at the Fermi level. N(u>) Density of states function. p A momentum variable. viii 1/c = tan 60 c £ ' 4 c * 4 e E q A momentum variable. Q,Q0 Loaded/unloaded quality factor of a resonator. TT Imaginary time (Matsubara) ordering operator. Vjr £, Electron-electron interaction. Zs = Rs + lXs Complex surface impedance. 0 = 1/ksT Inverse temperature. 60 S-wave scattering phase shift. <5(0) Dirac delta function centred at O = 0. A^r Superconducting gap parameter. A Q Magnitude of the superconducting gap parameter. e0 Permittivity of free space. 7 a , 7 C Geometric factors for comparing sample dimensions. r Geometric calibration factor. ro Normal state scattering rate. Tu, Td, Tb Unitary, Drude and Born scattering rate parameters. A Magnetic penetration depth. A; London magnetic penetration depth. fj,a Permeability of free space. l/i/ Volumetric factor for summation in k space. £0 Superconducting coherence length. £jr Bare/renormalized quasiparticle energy. p DC resistivity. a Electrical conductivity tensor. an,as Normal/super fluid electrical conductivity. aoa, a*QO Bare/renormalized T = 0 quasiparticle electrical conductivity. aqp Electrical conductivity due to quasiparticles. ix E(w) = EoTo + SiTi + E 2 r 2 + E 3 T 3 Quasiparticle self-energy. r Quasiparticle scattering time. ro, , R 2 , 7 3 Pauli matrices. to, to Bare/renormalized quasiparticle frequency. O Frequency variable. 0 o Natural resonant frequency of a resonator. AD, The full width at half maximum of a resonance. x Acknowledgements The author would like to recognize a sincere debt to all of the members of the U B C superconductivity laboratory, and especially those who shared their microwave surface impedance data: Saeid Kamal (1 GHz), Pinder Dosanjh (2 GHz) and Ahmad Hosseini (13 and 22 GHz). The experiments were conducted under the infallible guidance of Doug Bonn and Walter Hardy at U B C . Ruixing Liang of U B C deserves many accolades for producing the superior quality Y B C O samples used in this research. The theoretical work presented herein was directed by John Berlinsky of McMaster University, to whom I am very grateful. Final editing was performed by Jim Carolan of U B C , who provided key support in having this thesis submitted on schedule. As a proud Canadian, I would like to thank NSERC for funding all aspects of this research. I must also recognize those who assisted me personally during the course of my research at U B C : Benny's Bagels for supplying the Miscella Dark Roast during those nocturnal experiments, Jackie Seidel for the metaphysical conversations and skiing, the members of my band, the members of the SJC Beer Brewing Society, Patrick Turner for the friendship and more skiing, Nellie Harris for being fun, and my resilient companion through life, Adeline Chin. x i Chapter 1 Introduction Superconductivity is an ordered state manifested by many condensed matter systems. However, due to the initial absence of adequate means for studying many body physics, there was a significant chronological gap between the first observation of superconduc-tivity in Hg [1] and the development of the electron pairing picture by Bardeen, Cooper and Schrieffer [2]. It now appears that the saga of high temperature superconductivity (HTSC) will unfold in a similar manner. The unconventional nature of HTSC in the ceramic oxide materials has been unques-tionably demonstrated via a battery of condensed matter probes. This statement can be justified by examining the history of Y Ba^CuzOj-s (YBCO), first discovered by Wu et. al in 1987 [3]. In particular, the absence of a well defined gap edge in the infrared [4], phase sensitive tunneling measurements [5], the large gap anisotropy observed via an-gular resolved photoemission spectroscopy (ARPES) [6] and the observation of a linear temperature dependence of the magnetic penetration depth at low temperatures [7] all conclusively demonstrated that YBCO must have an unconventional superconducting energy gap, the most plausible being one with d-wave symmetry [8]. Despite the plethora of data, neither the superconducting nor the normal state prop-erties of HTSC have been explained within a universally accepted theoretical framework. One significant reason is the unfinished nature of key matters such as electromagnetic absorption measurements on pure single crystals, which will certainly provide key pieces 1 Chapter 1. Introduction 2 of information. Infrared (IR) and electron energy loss spectroscopy (EELS) provide re-liable data in the superconducting state at frequencies above 300 GHz [9], below which the losses become too small to detect via reflectometry methods. One must then resort to microwave probes, but these are typically single frequency apparatuses. Hence, there are considerable technical difficulties that one must overcome in order to assemble a com-plete absorption spectrum for a material such as YBCO. This is a very serious problem since it is known via oscillator sum rules that a significant portion of the spectral weight resides in the microwave range within the superconducting state of HTSC [10]. If a complete electromagnetic absorption spectrum were available, one would then have a very powerful means of testing the validity of any theory of HTSC. In particular, one can consider predictions concerning the nature of the fundamental excitations out of the superconducting groundstate, with particular emphasis on the following questions: • Are these fundamental excitations well defined electronic quasiparticles? • If the above is confirmed, what interactions govern the quasiparticle lifetime? These questions are particularly relevant when examining the microwave portion of the absorption spectrum, as one anticipates that at sufficiently low frequency the ab-sorption will be dominated by electronic thermal excitations, as opposed to phonons and plasmons. Initial single frequency microwave measurements by Bonn et. al. on pure single crystals of YBCO indicated the presence of thermally excited quasiparticles with a rapidly decreasing scattering rate below Tc that saturated at a low temperature impu-rity limit [11]. This has proven to be a very significant result, but without spectroscopic information, details of the scattering could not be accurately assessed. However, given a sufficient number of single frequency microwave measurements, it should be possible to quantitatively test electromagnetic absorption models in the mi-crowave regime. With the recent introduction of the 75 GHz apparatus to complement Chapter 1. Introduction 3 the 1, 2, 13 and 22 GHz apparatus in the U B C Superconductivity repertoire, this feat has now become feasible. This thesis will concentrate firstly upon the experimental details of the 75 GHz ap-paratus and secondly on subsequent data analysis using results from all five microwave systems simultaneously. Chapter 2 will outline two theoretical pictures of the electromag-netic properties of superconductors; the simple two-fluid model and a generalized BCS picture obtained via Green's function methods. The sample and experimental details will be presented in Chapter 3. Finally, the confluence of the theoretical and experimental results will be presented in Chapter 4, in an examination of the measured electromagnetic absorption in the low temperature impurity limit. Chapter 2 Theory The electromagnetic response of a superconductor is intimately connected to a macro-scopic phenomenon known as the Meissner effect. This involves the almost complete expulsion of a magnetic field applied to a superconducting sample. It is anticipated that those electrons within the superconducting groundstate will contribute to a macroscopic screening current that will extinguish the applied field over a length scale known as the penetration depth, A. • i i i i i i Figure 2.1: Near perfect diamagnetism in a superconducting ellipsoid. 4 Chapter 2. Theory 5 The magnitude of the screening current j is related to the strength of the external field H = V x A , where A is the vector potential. For a good metal, one can invoke a materials property known as the electrical conductivity, o(D), which quantifies charge —* carrier mobility in response to an applied vector potential A($V) at frequency fi. The required relation then has the following form [12]; J(Q) = %(n)A(n) (2.i) By definition, those electrons contributing to the supercurrent shall provide com-pletely lossless response, however any electronic excitations within the superconductor will allow for electromagnetic absorption. Therefore, a theory of the electrodynamics of superconductivity must provide a conductivity tensor <r(Q) which embodies the two distinct types of behaviour cited above. The only caveat that need be introduced at this point is the assumption that the electromagnetic response can be considered as being local. This is believed to be a valid assumption for the highly two dimensional HTSC cuprates since the reduced dimensionality ensures that the penetration depth, A, will be much larger than both the coherence length, £ 0 , of the superconducting electrons and the mean free path, £, of any electrons not contributing the the supercurrent [13]. Two approaches to this problem will be examined; the generalized two fluid model and a more rigorous microscopic picture based upon a generalized BCS formalism. 2.1 Genera l ized Two F l u i d M o d e l w i th Drude Quasipart ic le Scatter ing The generalized two-fluid model is an empirical approach to modeling superconductivity rooted in the London equations [14, 15]. Though it may be regarded as an excessively simplified model, among its virtues is its ability to serve as a good working model with a simple physical interpretation. In fact, it is within this formalism that magnetic pene-tration depth measurements are typically interpreted [7]. Chapter 2. Theory 6 To begin, one assumes that the electromagnetic response of a superconductor can be divided into 2 components due to normal and superconducting electron densities n n and ns, respectively. <r(fi) = <7n(fi) + as(n) (2.2) The two types of electrons are assumed not to interact. It is further assumed that the superfluid will accelerate unimpeded under an applied field, and by Newton's equations of motion dJ,q nse2E dt m s 9 - / J 0 V x Js = -=—h (2.3) Use of Eqn. 2.3 and Maxwell's equations then shows that the magnetic field h decays exponentially into the superconductor over a characteristic distance A; [10]. A i _ w ) W ( 2 4 ) Ill's Given this information and Kramers-Kronig relations for the electrical conductivity [16], one can rewrite Eqn. 2.2 as follows; a(Q) = an{n) + WO) - 1 ) (2.5) One should note that the London equations provide no insight into the behaviour of on{Q). A somewhat arbitrary yet useful ansatz is that the normal electrons behave like almost free electrons in a metal, with a Drude shaped electromagnetic response due to a single particle scattering rate 1/r [11]. / 0 n .... aQ _nne2 r f0Rs Thus the complete expression for the electrical conductivity becomes a(Q) = (^7n5(f t ) + ^ T——) - %(^- + ^ Q t [ J (2.7) Chapter 2. Theory 7 One can extract three significant details from this simple picture. First and foremost, by the oscillator sum rule j Re\cr(Q)\aTl = constant, one can see that if nse2/m*s grows as T —• 0, then it must do so at the expense of n n e 2 /m*. In particular, if all of the oscillator strength condenses into the superfluid at T = 0, then nA ! ^ ( r ) = !^£f!(T = Q ) ( 2 8 ) ms* v ' mn* K ' m* v ' v 1 Second, the inductive response Im,\o(Q)\ is dominated by the superfluid at low fre-quency; however there is a normal fluid component that cannot be neglected under all circumstances. Finally, one should note that this picture involves a frequency and temperature inde-pendent scattering rate 1/r, hence Re\o(Q)\ should scale with nne2/m*n, provided 1/r is independent of temperature. 2.2 Microscop ic Isotropic Poin t Scattering M o d e l A more rigorous model of the electromagnetic response of a superconductor requires the use of many-body techniques, the most common being Green's function methods. This is a perturbative approach which employs free particle properties as the zeroth order approximation. The derivation to be presented herein is based upon arguments assembled by Hirschfeld, Scalapino and Puttika [17]. In many respects, the derivation may be considered to be a generalization of BCS theory by allowing the order parameter A to be A; dependent. The reader is referred to Abrikosov-Gor'kov for a complete derivation of the BCS theory using Green's function methods [18]. To begin, one must make a subtle adjustment to the parlance used in the previ-ous section. The so called normal electrons in the two fluid model are now viewed as thermally excited quasiparticles that have been wrested from the superconducting Chapter 2. Theory 8 groundstate. Since an excited electron must be accompanied by a hole, it will prove con-venient to enforce particle-hole symmetry by employing Nambu notation for particle-hole creation/annihilation operators. 4 = (4> c -£) (2-9) A general electrical conductivity expression, known as the electrical Kubo formula, may be derived using the current-current correlation function, [ja(q, t), q, 0)j [19]. oaP{q, ty = -^JQ dt eiat{ [ja(q, t),jp{-q, 0)] ) + ^y*<W (2-10) The brackets (...) denote that the quantity contained within is a thermal average calculated via Boltzmann statistics, and the symbols a and (3 denote orientations in real space. The electrical current due to quasiparticles can be written as follows; ' P,<r Combining Eqns. 2.10 and 2.11 and then resorting to complex time (Matsubara for-malism) yields the following useful expression; • ( r r C ^ W ^ a W C ^ ^ ' ^ W ) + (2.12) where TT denotes imaginary time ordering of the Nambu operators. By momentum conservation p' = p — q and by spin conservation a' — a. One can now discern a pair of quasiparticle propagators, (Cp>(T)Ct. ] £ r(0)) and (^_ g >(0)Ct_- C T (T)^. These represent excitations with a finite lifetime r due to the absorption at t — 0 and expulsion at t = IT of a photon of momentum q, as can be represented in a single Feynman diagram. Chapter 2. Theory 9 Figure 2.2: Feynman diagram for electromagnetic absorption, as dictated by the electrical Kubo formula. By shifting the momentum variable p —>• p + q/2 and lett ingp± = p±q/2, one obtains an expression for the conductivity in terms of particle/hole propagators, Q = (C&). o-aP(q,iQm) = ^ fj^ E PaPl3tr(p(p+,tpn)g(p-.,ipn - i f i m ) ) + (2.13) P,*Pn Equation 2.13 is a general result that contains no assumptions about the nature of the electron-electron interaction. However, it is inherently a local electrodynamics theory for conduction by well defined quasiparticles. At this point one must resort to a model dependent form for the quasiparticle prop-agators. A convenient means of doing so is to introduce a generalized form of the BCS electron-electron interaction. n = E & 4 « c £ , a + ( E 4 , A (2.14) k,a k,a,fi k,f,5 Here, ^ represents the quasiparticle kinetic energy and V£ £, the electron-electron interac-tion of an unspecified nature. The subscripts a, ft, 7, 8 take the values + or —, indicating the spin of the excitation. It can be shown that Eqn. 2.14 then gives Green's function Chapter 2. Theory 10 propagators of the form [18]; (2.15) where the effective interaction has been absorbed into the self consistent definition of the superconducting gap parameter. A j = i E % - ( ^ | £ M ) A J , (2.16) ti ^k Thus Eqn. 2.13, in conjunction with Eqns. 2.15 and 2.16, provide a complete formal-ism for calculating the electrical conductivity by quasiparticles in a superconductor. One is free to adjust the pairing interaction via Vgg,; however, given the significant amount of data which supports a d-wave pairing symmetry in the HTSC cuprates [5, 6, 7], it will be assumed henceforth that = A ocos(20) is valid, where 0 is the polar angle on a cylindrical Fermi surface. To proceed any further one must include the effect of scattering centers which will limit the quasiparticle lifetime. A simple picture can be constructed if one assumes that these are randomly distributed, isotropic and elastic point scatterers. The situation becomes even more manageable if one assumes that the scatterers are dilute enough that quasiparticles are not subject to multiple scattering. Thus, one should consider this to be a viable model for a dilute concentration of atomic impurities in an otherwise perfect crystal lattice. The self energy corrections to Q(k,Q,) in this scenario can be represented using an infinite sum of Feynman diagrams. Chapter 2. Theory 11 Figure 2.3: Diagram representation of the quasiparticle self energy due to elastic point scattering interactions with dilute impurities. Note that for elastic scattering there is no energy transfer and for a dilute population of scattering centers there are no crossed boson lines. Figure 2.3 can then be translated into the following equation for the quasiparticle self energy. •G(k - 9i, w) • • • Q(k - qn, w)] Z{k,uj) = r0T(k,k,u) (2.17) Here rij represents the number density of impurities, n0 the number density of electrons and N0 the density of states at the Fermi surface. These quantities have been incorporated into the definition of a normal state scattering rate T0 = nin0j'n'No. It follows from the above expression that T is defined by a self consistent Lippman-Schwinger equation. T(k, k, u) = 1/(0) + Y^V(k-k')g(k', u)T(k', k, u) (2.18) k' Therefore, the present objective is to evaluate T via Eqn. 2.18, thus allowing one to determine the quasiparticle self energy via Eqn. 2.17. This approach is known as the self consistent T-matrix approximation [20]. Chapter 2. Theory 12 To begin, Eqn. 2.18 is rewritten by invoking the normal state propagator Qff(k, ui) = Q(k,uj)\A-_o- The purpose of doing so is that one can then take advantage of the partial wave expansion formalism for describing the scattering of free particles. Neglecting the summation signs in Eqn. 2.18 momentarily, T = V + V{G + gN-GN)T (2.19) and then exploiting the self consistent nature of T one obtains the following; T = [v + vgNv + vgNvgNv + • • • ) • (1 + (g - Gn)T) - TN(k, k, u) [l + Y , - gN(k',u;))T(k', k, u)] (2.20) k' If one also assumes that the scattering potential is isotropic, then the angular depen-dencies of TN and T may be neglected. Since it has been assumed throughout that the scattering is elastic, then = \k'\. Given these two statements, one can now neglect the k dependence of the T-matrices. T{u) = TN{u) + TN{u) £ ( £ ( £ » - gN{k\u))T{uj) (2.21) k' One should recognize that Ylfc ^ ( ^ ' i 1 ^ ) = 0 exactly, as can be seen from Eqn. 2.15. The final simplification is to replace TN by an appropriate quantity obtained from a partial wave expansion [19]. By virtue of the assumption of an isotropic scattering center, it is quite natural to conclude that can be replaced by tan <5rj = 1/c, where 80 is the phase shift in the s wave channel [20]. This substitution allows one to explicitly solve Eqn. 2.21 for T(ui). From Eqn. 2.17 one can then define the quasiparticle self energy. S ( " ) = ~L , (2-22) c + G(w) G{LO) = E Q(k, u) = G0r° + G ^ 1 + G2r2 + C7 3r 3 it' (2.23) Chapter 2. Theory 13 From the form of G(k',u>) given by Eqn. 2.15, one would suspect that G\ = G2 = 0 for a pairing mechanism with odd parity (A^ = —A_^), such as a d wave model. It can also be argued that G 3 = 0 based upon particle-hole symmetry {t\^paTticle + €k>hole = 0). Hirschfeld et al. provide a rigorous proof of these conclusions, showing that they are justified within any self consistent calculation of £(w) [20]. Based upon these conclusions, one can rewrite Eqn. 2.22 in the following manner. E ( w ) = c ^ r 0 - c ^ r 3 = s ° r ° + E a r 3 ( 2 - 2 4 ) From Eqns. 2.15 and 2.24, one can see that the only quantities which will be subject to renormalization are UJ = u) + E0(aj, A , ^ ) and ^ = ££ + £ 3 ( 0 ) , A , ^ ) . The renormalization of u> physically corresponds to a shift in the quasiparticle energy and the introduction of a finite quasiparticle lifetime, while the renormalization of ^ corresponds to a shift in the chemical potential relative to the Fermi surface. The full solution to the quasiparticle scattering problem may be obtained by solving Eqns. 2.15 and 2.24 self consistently for the renormalized quantities ui and The latter renormalization is not significant for c —> 0 (strong or unitary scattering) or c >^ 1 (weak or Born scattering), as can be seen from the definition of £ 3 in Eqn. 2.24. Thus, one can obtain reasonable results in the Born and unitary limits by neglecting £ 3 . The only renormalized quantity to be considered hereafter will be CJ. Armed with the conclusions concerning the renormalization of G(k\ to), one can return to the general conductivity expression given by Eqn. 2.13. The sum over the complex fermionic frequencies can be performed exactly and the momentum integral can be sim-plified for long wavelength radiation (q —> 0). Keeping only the leading singular terms for low energies (Q —• 0), Hirschfeld et al. provide the following key result [17]. Chapter 2. Theory 14 o , n ) - 4 - f ^ ( " W ) - u n h W „ - » ) / 2 ) ) ( ^ M } m A G 7-oo v 2S2 ' vf2 — I/T{U))J (2.25) 9 r _ 1 (u;) = -2 im(E 0 (u ; ) ) <r00 = — 7rmA 0 This expression for the electrical conductivity due to thermally excited quasiparticles is valid in the limits D, -C T and Q,T <C A 0 . Given the reasonable estimate A 0 ~ 200 K [21], then one anticipates that Eqn. 2.25 will be valid for T < A o / 1 0 ~ 20 K and Q < T/10 ~ 2K (40 GHz) for T = 20 K . Note that the self consistent calculation of T,0(uj) has been terminated at EO(CJ), which should be valid in the so called 'pure regime', above the temperature T* [17]. Hence, Eqn. 2.25 will not provide an accurate description of the electrodynamics at very low temperatures, where low energy bound states are anticipated to give rise to the universal f2 —> 0 Lee limit, a00 [22]. This issue will be examined in more detail in Chapter 4. One should also note that ReT,o(u) has been neglected in Eqn. 2.25. This quantity corresponds to shifting of the quasiparticle energies, however its effect is anticipated to be relatively negligible in the pure regime [23]. Equation 2.25 provides a very simple and appealing interpretation if one employs the definition of the low energy pure d-wave density of states N(u>) = u>/A0 and the small O form for the hyperbolic terms; oxx(q - 0,fi) - — / (Lj(--f-)N(u)(- - - - ) +a00 (2.26) one can clearly see that the electrical conductivity is a thermally weighted sum of non-interacting quasiparticle responses with a frequency dependent one body scattering rate. In particular, it is the convolution of N(u) with — which distinguishes between the Chapter 2. Theory 15 'pure' and 'gapless' regime mentioned earlier. To illustrate this distinction, consider the following sketches of N(u>) and — Figure 2.4: The d-wave density of states for both an ideally pure system (dashed curve) and a system containing a small concentration of impurities (solid curve). On the right is a sketch of the derivative of the Fermi thermal factor, demonstrating its half width ~ 2T. Both a sketch of N(u>) for an ideally pure d-wave system and for a system containing a small concentration of impurities are shown above [24]. The addition of impurities quells the logarithmic divergence at the gap maximum, adds low energy states below an energy ~ T* and generates a nonzero DOS at u> = 0. This latter modification effectively eliminates the influence of the superconducting gap at low energy. However, above T* but below A 0 the impure DOS remains linear in cu, which is characteristic of a pure system. Now consider the effect of convolving the impurity altered N(u>) with — The derivative of the Fermi thermal factor is essentially a gaussian centred at u = 0 and whose half width ~ 2T at any given temperature T. Therefore, the integral in Eqn. 2.26 Chapter 2. Theory 16 is effectively truncated at ±2T. If T < T*, then the low energy bound states in N(to) will tend to dominate the integral, thus giving 'gapless' behaviour. Conversely, if T > T* then the linear portion of N(UJ) will dominate the integral, thus giving behaviour in accord with a 'pure' system. These conclusions demonstrate how the electrical conductivity of a d-wave superconductor may be divided into a 'pure' and 'gapless' regime with a crossover energy scale T* which will be dependent upon the impurity strength and concentration. Further discussion of T* will be reserved until Chapter 4. It is important to note that if r is considered to be frequency independent, then Eqn. 2.26 reduces to the same form as the normal electron conductivity, Eqn. 2.6. This startling correspondence thus gives merit to the simple two fluid approach, suggesting that in some respects it is consistent with a more rigorous quasiparticle picture. Hence the only remaining task is to examine the truncated self consistent calculation of the quasiparticle scattering rate, r _ 1 ( ( x ; ) . Hirschfeld et al. provide a relatively sim-ple expression for T~1{UJ) in both the Born and unitary limit, however a simple explicit expression valid for all scattering strengths remains elusive [25]. For numerical investi-gations, the reader is referred to Graf et al. [26] and Hensen et al. [27]. One possible means of coping with these circumstances is to attempt to mimic intermediate scattering strength by an appropriate linear combination of limiting forms for r _ 1 (o;) , as suggested by Berlinsky and Kallin [23]. r Hu)= , r " A - ° . + F d + ^ (2.27) + TUA0 A Here, Tu,Fd and Fb adjust the relative contributions of the unitary, Drude and Born terms, respectively. A n important consequence of the relatively straightforward forms of Eqns. 2.25 and 2.27 is that they can be easily manipulated using a P C and commercially available mathematics software. Chapter 2. Theory 17 It is worth noting that more general expressions for <JXX(Q) can be derived using the Dirac cone approach, as shown by Berlinsky [28] and Durst [29]. However, due to limited computation power, these expressions have proved difficult to examine. Fortunately, it has been observed that the predictions of Hirschfeld et al. [17] and Berlinsky [28] agree remarkably well over the range of temperatures and frequencies to be investigated herein. 8 r 4 E at a e 2 Berlinsky Hirschfeld et al. \ \ 20 K \ l 0 K \ \ 4 K ^ N N N > < ^ ^ ^ ^ ^ ^ ^ i J 1 ' 1 ' 1 ' 1 1 1 1 1 1 1 ' 1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Q / K Figure 2.5: A comparison of the theoretical expressions of Berlinsky et al. [28] and Hirschfeld et al. [17], using an arbitrary scattering rate function r _ 1 = 0.5 K . The results demonstrate that Eqn. 2.25 agrees with the more rigorous expression derived by Berlinsky et al. over the range of temperatures and frequencies of interest. Chapter 2. Theory 18 To summarize, two pictures of the electrical conductivity by thermally excited quasi-particles have been presented. In the simple two fluid approach, the quasiparticles are regarded as being almost free electrons, as in a normal metal, thus giving a Drude shaped electrical conductivity, Eqn. 2.7. A more rigorous approach built upon the electrical Kubo formula for isotropic elastic point scatterers provides a picture of non-interacting quasiparticles subject to a frequency dependent scattering rate, Eqns. 2.25 and 2.27. These expressions are valid in the limits O < T and Q,T <C A G . The two models coincide if a purely frequency independent scattering rate is inserted into Eqn. 2.25. Chapter 3 Experiment The observable quantity in an electromagnetic absorption experiment on a metallic sur-face is the surface impedance, which is defined as Zs(£l) — E±/Hrf. Hrj is an applied rf magnetic field which drives a surface current j± in a direction perpendicular to Hrj and parallel to the surface of the sample. The field Ej_ is then a result of charge carrier motion in the material. In the limit of local electrodynamics, the surface impedance can be related to the conductivity as follows [10]: Therefore the ideal measuring apparatus would be capable of probing a sample with an applied field Hrf over a broad range of frequencies and temperatures. The conventional means of performing such measurements has been through a combination of reflectometry techniques and Kramers-Kronig analysis. However, it was observed very early in the HTSC saga that extremely small sample losses in the FIR and experimental limitations would prevent these methods from probing below ~ 300 GHz [9]. Nonetheless, such measurements have provided several significant conclusions, one of which is that, when in the superconducting state, the bulk of the spectral weight associated with the mobile carriers in the HTSC's must reside at frequencies below 300 GHz. Early measurements performed by Bonn et al. at 4 and 35 GHz clearly demonstrated the presence of significant structure in Re\a(Q)\ at microwave frequencies [11]. This provided the justification to build a number of single frequency microwave systems to (3.1) 19 Chapter 3. Experiment 20 resolve the low frequency portion of the electrical conductivity spectrum of a HTSC. The measurements discussed herein were taken at 1, 2, 13, 22 and 75 GHz using cavity perturbation techniques by several members of the U B C HTSC group on ultra-high purity (99.995% atomic purity) single crystal samples of YBCO. Details concerning the samples and a brief outline of cavity perturbation methods will be presented. Specific details concerning the implementation of the newest addition to the U B C microwave spectroscopy repertoire, the 75 GHz apparatus, will also be discussed. The reader is referred to the appropriate references for the details concerning the implementation of the other apparatuses [7, 30, 31]. 3.1 Samples A necessary prerequisite for determining the fundamental electrodynamics of the HTSC's is the production of exceptionally pure samples, since trace amounts of chemical impuri-ties or crystal defects are known to substantially alter the observed electrodynamics [11]. Hence a significant effort has been made to optimize the growth technique for YBCO, culminating in the successful production of extremely high purity samples by Erb et al. [32] and Liang et al. [33]. The samples used in this study were prepared by the latter at U B C . Figure 3.1 shows a unit cell of the HTSC cuprate Y Ba2Cu30t-s. This material has been the subject of intense research due to the high quality with which samples can be prepared as compared to many of the other cuprate materials. However, it is perhaps an understatement to say that obtaining sufficiently pure samples for measuring intrinsic properties has presented unforseen challenges. The key elements of this perovskite crystal structure are the ab oriented Cu02 planes and the b oriented CuO chains. In particular, the Cu02 plane is a common structure Chapter 3. Experiment 21 amongst all of the HTSC cuprates and is thought to be the critical component in the HTSC of these materials. Hence, one is mainly interested in ab plane electrodynamics. Due to practical considerations, this is most easily accomplished by using samples with broad ab faces. Single crystals of YBa2Cu307^s were grown using the self-flux method. The Y2Os — BaO — CuO melt was prepared using starting materials with greater than 99.995% atomic purity. In order to maintain such high purity, it was necessary to ensure that the crucible in which the melt was prepared did not react with the melt. This stringent requirement led to the development of the new BaZrOz crucibles, which have proven to be both inert and impervious to the melt after repeated crystal growths [33]. Subsequent chemical analysis of these growths revealed atomic purities between 99.99 and 99.995%. This con-clusively demonstrated that the crucibles were no longer a significant limiting factor in growth quality, as contrasted to previous generation growths produced in yttria stabi-lized Zr02 (YSZ) crucibles [34]. Furthermore, X-ray (006) rocking curves measured for crystals grown in BaZrOz crucibles proved to be a factor of 3 narrower than those made on the best crystals grown in YSZ crucibles, thus standing testament to the high degree of crystalline perfection. For complete details of the crystal growth and characterization, the reader is referred to Liang et al. [33]. The as-grown crystals were then detwinned under uniaxial stress and annealed under appropriate conditions to set the oxygen content at O6 .993 which yielded Tc « 88.7K, as obtained from magnetization measurements. These crystals were intentionally overdoped (optimal oxygen content being, Tc = 93.7K) to circumvent an unexpected charac-teristic of the high purity crystals, namely enhanced oxygen defect mobility in optimally doped BaZrOz grown crystals as compared to YSZ grown crystals [33, 35]. From mea-surements of flux pinning for various annealing treatments, Erb et al. concluded that in extremely pure crystals oxygen vacancies tended to cluster at temperatures typical of the Chapter 3. Experiment 22 annealing process. Thus the high purity crystals are extremely susceptible to the devel-opment of an inhomogeneous distribution of oxygen, which in turn produces behaviour that mimics samples of inferior quality [32, 35]. Setting the oxygen content very close to the stoichiometric level 07 reduces the number of vacancies, thus minimizing these anomalous effects. Figure 3.1: The crystal structure of YBa2Cu307-i [36]. The key components to note are the ab oriented Cu02 planes and the b oriented CuO chains. Chapter 3. Experiment 23 3.2 Measurement Principle The surface impedance of YBCO was probed by driving surface currents with a field Hrf applied parallel to the surface of the sample. However, the orientation of this magnetic field must be carefully controlled during experiments on single crystals because of the anisotropic properties of YBCO. This circumstance limits the use of many conventional techniques used to measure the surface impedance of thin film samples [37]. One possible solution to this problem is to adjust the sample geometry according to the direction in which one wishes to measure Zs(f2). Figure 3.2: Measurement of ZS(Q) in the a direction of a thin YBCO crystal. In the example shown in Fig. 3.2, a sample is immersed in a microwave magnetic field applied parallel to the b direction and the corresponding screening current density J is driven in the a direction. The disadvantage to this type of measurement is that J must also run across the be oriented edges of the sample. By choosing the sample dimensions x a , xb 3> xc, o n e c a n minimize the admixture of c axis response. Note that one can probe Chapter 3. Experiment 24 the b axis of such a sample by rotating it 90° about the c direction. YBCO crystals with broad ab surfaces are frequently produced during growth since this material has a growth instability in these two directions. Methods for extracting the c axis response will be discussed below. 3.3 Measurement Technique Given the circumstances described thus far, one can discern three key criteria that must be met by a microwave HTSC surface impedance measuring system. 1. The sample must be immersed in a homogeneous microwave magnetic field Hrf. 2. The sensitivity of the apparatus must be sufficient to measure very low losses. 3. The sample temperature should be adjustable from above TC to as low as possible, thus fully resolving the pure regime in which elastic scattering of quasiparticles prevails. It will be argued that these criteria can all be met using microwave cavity perturbation techniques. Each of the above will be addressed accordingly, with particular emphasis on how the 75 GHz apparatus was implemented. A convienient means of producing a relatively homogeneous microwave magnetic field is to employ an appropriate resonance mode inside a resonant cavity. There are many possible geometries capable of producing potentially useful resonances at any given fre-quency, but the TEQH resonance mode in a right cylindrical cavity has proven to be quite practical at frequencies greater than ~ 10 GHz [31]. The field profiles can be found in any standard microwave circuitry textbook [38], and the key features of the 75 GHz TEQU resonator are sketched in Fig. 3.3. Chapter 3. Experiment Cavity Lid, OFHC copper Sample Insertion I I —I I— 0.06" I l \ Radiation Coupling Hole • 0.04 i . L Cavity Body, OFHC Copper /i • • i I I I I r £ = 0.1394" ._!_ (3.54 mm) T i i i — | j — 0.075" 2a = 0.2325" (5.9 mm) z 4f> -» y Figure 3.3: A cross section of the 75 GHz TEQU mode right cylindrical resonator. I represents the length and a the radius of the cylindrical structure. The field Hrf is maximized in the vertical direction at the centre of the resonator. Two holes were created for coupling radiation into and out of the resonance structure, and a third hole along the axis of the resonator allows a sample to be inserted. Chapter 3. Experiment 26 The TEQH resonance can be characterized by the following set of equations for E and H. Er = Q (3.2) Ee = -yfioBJo(kr) smiirz/Z) (3.3) Ez = 0 (3.4) Hr = ^-BJ'0{kr)cos{iTz/l) (3.5) Hz = BJo(kr) sin(irz/e) (3.6) Hg = 0 (3.7) k2 = Q20e0^0 - n2/f = X2/o2 X = first root of J'Q = 3.832 Note that near the centre of the resonator, Hrf « Hzz and E « 0. B y employing an appropriate aspect ratio of ^ / 2 a ~ 1.7, one can locate the TE0n resonant frequency Q,0 well away from other resonant modes, an exception being the TMm mode, which wi l l be discussed below [39]. From Eqns. 3.2 3.7 and the dimensions in F ig . 3.3, it is predicted that Q0 w 75.1 G H z . One should also notice the small indentation in the bottom of the resonator, as in-dicated in F ig . 3.3. The purpose of this modification is to remove the degeneracy of the T ^ o i i a n d T M m modes [39]. The latter resonance mode contains a magnetic field maximum at the centre of the top and bottom faces of the resonator. The indentation effectively increases the volume of the resonator for the T M m mode, thus moving its resonance to a lower frequency. This alteration should have a negligible effect upon H in the TEQH mode. Chapter 3. Experiment 27 Nearly lossless resonator walls are essential to obtain the required sensitivity. By coating the interior of the resonator with a sufficiently thick superconducting PbSn alloy (95% Pb, 5% Sn, Tc ~ 7 K) and cooling the resonator structure with a liquid 4He bath regulated at 1.2 K , this condition can be comfortably met. It has been determined from experience within the U B C superconductivity laboratory that a coating approximately 1 pm thick effectively shields the copper walls of the resonator from the impinging fields. Losses due to the resonator walls gives rise to a broadened frequency response by the resonator, and rather than a simple 8 function centered at f20, the resonance becomes Lorentzian with a width characterized by a dimensionless parameter (J, _ 2-7T x energy stored in the resonator fi0 energy lost per cycle A f i where VtQ denotes the centre frequency and A Q is the F W H M [40]. Thus, 1/Q is a measure of the real losses inside of the resonator. If the PbSn coating has a uniform characteristic surface resistance Rs(pbSn), then it can be shown that the field profiles given by Eqns. 3.2 —> 3.7 give the following expression for the intrinsic quality factor of the resonator, Q0 [38]: Qo v0nie0fi 1 j where V0 is the volume of the resonator cavity and all other factors have been defined previously in Eqns. 3.2 —> 3.7 and Fig. 3.3. This expression is valid if the penetration of the fields into the resonator walls can be regarded as a slight perturbation of the fields to the ideal field profile given previously. Since \(1.1K) of the PbSn alloy is several orders of magnitude smaller than the resonator dimensions, the perturbative approach is justified. Now consider the scenario when a thin superconducting sample is positioned at the centre of the resonator, such that H is oriented parallel to the broad surface, as shown in Fig. 3.2. According to the definition of Q there will be additional terms oc ARs(samPie)> Chapter 3. Experiment 28 where A is the total external surface area of the sample being traversed by the screening currents and Rs(sampie) is the surface resistance of the sample. If one assumes that the sample is sufficiently thin such that it does not alter the field profiles considerably, then a perturbative calculation yields [31] A(l/0) = — - — = ARS{sample) , . V / V J ~ Q Qo 2Q0p0V0J§(ka)(l + (^) {°^) A(l/Q) = RS{samPle)/T (3.10) Therefore, a measurement of A{l/Q) provides one with a means of determining the surface resistance of a thin sample via cavity perturbation methods. From Eqn. 3.9 and the dimensions of the resonator in Fig. 3.3, one obtains a predicted value T = (2.74 x 10~ 2fi • m3)/A. For a sample with a broad surface area of A/2 = (0.5mm)2 this gives T ~ 105£1 • m. In practice, the calibration factor T may be determined more accurately by measuring A(l/Q) for a well characterized material of the same size as the superconducting sample to be studied. To demonstrate that such a microwave cavity perturbation device has a sensitivity that far exceeds that of any available FIR reflectometry device, consider a sample with a complex reflectivity r = 0.9999e^ 0"") 7 r; through the use of appropriate optics relations, it can be shown that this corresponds to Rs ~ 4mO. Given the above value of V and an unloaded Q0 ~ 107, the loaded Q for such a sample would be Q ~ 7 x 106. A 30% change in Q can easily be resolved experimentally, as compared to the corresponding near perfect optical reflectance. Therefore, one can conclude that microwave cavity perturba-tion devices are capable of surpassing the resolution of most FIR apparatuses presently available. The reader should be aware that discussions concerning the imaginary portion of Zs{£l) have been tacitly avoided thus far. In the parlance of cavity perturbation meth-ods, Im\Zs{®)\ = Xs{Q) is manifested through a shift in the centre frequency of the Chapter 3. Experiment 29 resonance from the value it would have if the sample screened Hrf perfectly [35]. One can intuitively understand this connection by recognizing that the sample will exclude magnetic flux via the Meissner effect in the superconducting state, thus reducing the amount of electromagnetic energy stored in the resonator. The amount of flux exclusion by the sample will be particularly sensitive to any changes in the sample position during the course of an experiment. Therefore, one should avoid the use of moving parts inside of the resonator in order to obtain accurate measurements of X§. However, precise mea-surements of Rs require one to measure A(l/Q) - a procedure which necessarily involves loading and unloading the sample from the centre of the resonator. The inescapable conclusion is that accurate measurements of Rs and Xs on single crystal samples are difficult to carry out in a single experiment. For a more thorough analysis of measuring Xs(£l) of thin YBCO single crystal samples grown at U B C , the reader is referred to Kamal [41]. The final criterion that needs to be addressed is that of sample temperature regula-tion. Fortunately this objective and that of supporting a thin sample at the centre of the resonator can be met simultaneously, whilst maintaining the validity of the cavity perturbation approach. This is accomplished via the hot finger technique [42, 43] and is sketched in Fig. 3.4. A 0.004" thick sapphire (AIO3) plate is secured to an O F H C copper block using #2303 Stycast epoxy and the sample is mounted on the opposite end with a minute drop of NonAq stopcock grease. Sapphire was chosen for its high thermal conductivity [44] and its very low losses when subjected to an rf magnetic field [45]. The thermometer and heater are secured to the copper block and a weak thermal link to the 4He bath, regulated at 1.1K, is established through the thin walled stainless steel tube. This de-sign provides minimal intrusion of components into the resonator and almost complete thermal isolation of the sample from the resonator. However, it must be recognized Chapter 3. Experiment 30 that the sapphire and grease will introduce dielectric losses due to the field Eg given in Eqn. 3.3. Therefore, one must account for a background signal when determining Rs from Eqn. 3.10; A ( l / Q ) = A(l/Q)sample + A{l/Q)background (3.11) By this means it is possible to regulate sample temperature in a manner that does not invalidate the principles of cavity perturbation. Methods for accounting for the background signal will be discussed in due course. Figure 3.4: The hot finger assembly used in the 75 GHz apparatus for sample temperature regulation. The sample is mounted on the end of the sapphire plate and the temperature regulated by a feedback loop containing a Cernox thermometer and a heater. A thermal gradient is maintained across the thin walled stainless steel tube. The 3 :' long copper braid, secured by a stainless steel clamp to the brass tongue, provides a heat sink to the liquid He bath on the opposite end of the stainless steel tube. Chapter 3. Experiment 31 3.4 Implementat ion of the 75 G H z Appara tus The measurement techniques used in the 75 GHz apparatus are very similar to those used in the 13 and 22 GHz surface impedance apparatus at U B C [31], however the implemen-tation details differ considerably. In particular, the small size of the resonator, the high microwave frequency and a plethora of thermal issues posed significant challenges. As discussed previously, objects being inserted into the resonator must be sufficiently small such that they do not alter the field profiles considerably. Hence it proved necessary to restrict the cross section of the sapphire to 0.4mm x 0.004" and sample size to less than 0.5mm x 0.5mm x 50/OT?,. These dimensions were deemed suitable after repeated experimentation with various YBCO and PbSn samples. Difficulties concerning the propagation and coupling of a 75 GHz signal required this experiment's microwave circuitry to be rather different from the other microwave devices presently being used at U B C . At such high microwave frequencies, one is restricted to using hollow metallic waveguides since coaxial cables are too lossy, however even the waveguides will attenuate signals very rapidly since Rs(metai) oc yff. The problem of standing modes in long lengths of waveguide becomes severe at 75 GHz where the free space wavelength is 2.5 mm - the density of the standing wave pattern increases proportional to the length of waveguide and the frequency. The solution to this problem involved placing the radiation source and detector as close as possible to the resonator, thus minimizing the length of waveguide required. This in turn prompted the need for alternative cryogenics since the conventional immersion dewar and modular probe design used for the 1, 2, 13 and 22 GHz apparatus require microwave conduction paths approximately 1 metre long [31, 41]. A n Infrared Laboratories HDL-8 dewar with a AHe cold plate was chosen for the task, and the layout of the dewar contents is shown in Figures 3.5 —• 3.7. Chapter 3. Experiment 32 N y l o n R o d z X 2. 4700" Fi b r e g l a s s R o d r o s s T h r e a d e d Rod B r a s s Samp le G a n t r y H o t F i n g e r A s s e m b l y C o p p e r R e s o n a t o r B l o c k S a n p l e G a n t r y F o o t 2 5 3 3 " Figure 3.5: A cross section of the sample gantry and the resonator block. The nylon and fiberglass rods are secured to the threaded brass rod with epoxy and these parts rotate freely. Threads in the top of the hot finger assembly allow the sample to be moved vertically along the axis of the resonator rotation of the fiberglass rod. Chapter 3. Experiment 33 H e l i u n C o o l e d C o p p e r B a s e p l a t e <z = 0 ) • urtl i n e S a n p l e ( z = 0. C o o l e d hi e l d M e c h a n F e e d i h R u b b e r • - r i n g v e r e g u i d e s s S t e e l 0 . 0 0 2 " T h i c k M y l a r W indow Figure 3.6: A cross section of the microwave circuitry inside the dewar, at z = 1" below the helium cooled baseplate (the dewar is inverted during operation, refer to Fig. 3.9). Shown in this diagram are the sections of W band waveguide, microwave coupling adjust-ment mechanisms and the resonator in the centre. The resonator block, sample gantry, microwave couplings and those section of waveguide near the resonator all reside on a 0.625" thick copper plate, which can be easily removed from the dewar. Chapter 3. Experiment 34 Figure 3.7: A detailed view of the microwave couplings. PbSn coated wires are en-cased in teflon tubes, which can be translated horizontally via the coupling mechanical feedthroughs. The PbSn wires use capacitive coupling in the waveguides and inductive coupling to the TEQH mode in the resonator. A n obvious issue with the design in Figures 3.5 —* 3.7 is the inevitable heat leakage through the waveguides, as they support large thermal gradients from room temperature to 1.1K over a distance of only 10 cm. To minimize the thermal load on the He bath, the critical parts of the W band waveguide were manufactured from 0.015" thick stainless steel. Additional heat leaks through the microwave coupling feedthroughs and the sample position feedthrough were minimized by employing mechanisms that allow for complete thermal isolation of these components when they are not being adjusted, as shown in Figure 3.8. The experimenter can engage the mechanical feedthrough by rotating the dial sufficiently far to bring both fibreglass rods into contact with the rotating feedthrough on the A^ 2 cooled shield, and then disengage it by an appropriate sequence of partial Chapter 3. Experiment 35 rotations of the external dial. The heat leak through the electrical feedthrough for the thermometry was assumed to be negligible. Figure 3.8: A detailed view of a mechanical feedthrough. The same mechanism is em-ployed for both microwave coupling adjustments and the sample position adjustment. When disengaged, as shown above, the thermal contact to the room temperature dials on the dewar exterior are then broken. Since the microwave circuit elements are not immersed in the coolant, it was necessary to take extra precautions when heat sinking components to the cold plate. Wakefield Engineering thermal compound and a grease loaded with O F H C copper, known as Cry-Con grease, were employed. Finally, the entire assembly surrounding the resonator and the sample gantry was encased in a copper radiation shield, with the interior coated in black Stycast epoxy. This proved to be necessary in order to shield these components from the 77 K thermal radiation emanating from the N2 cooled radiation shield. Chapter 3. Experiment 36 A schematic diagram of the complete apparatus is shown in Fig. 3.9. The reader should note that there are four subsystems associated with this experiment: the mi-crowave circuit, thermometry circuit, cryogenics and a computer to coordinate the data acquisition and thermometry. The critical components of the microwave circuit are the 75 GHz resonator, Schottky diodes for generating and detecting the 75 GHz signal, an HP 83620A microwave synthe-sizer and HP 83498 microwave amplifier for generating a 15 GHz signal and an ABmm millimeter vector network analyzer (MVNA) for signal processing [46]. Isolators were used to improve signal quality and a directional coupler was needed to feed a sample of the 15 GHz signal into the M V N A . The diode for generating the input signal, known as the harmonic generator (HG), and the diode for detecting the signal transmitted through the cavity, known as the harmonic mixer (HM),were tuned to optimize the response to the 5th harmonic, with the 15 GHz fundamental frequency supplied by the HP synthe-sizer. Though the M V N A was provided with an internal synthesizer (resolution of 50 kHz), it proved necessary to use an external synthesizer (resolution of 1 Hz) in order to resolve the high Q resonances encountered in this experiment. A l l measurements of Q were made in the frequency domain, with the key advantage that the M V N A provides one with a measure of both amplitude and phase of the transmitted signal. Both the sample and resonator module temperature were monitored using a 4-probe measurement of Cernox 1050 resistors, calibrated to an accuracy of ±0.1 K . The sample temperature was regulated through a feedback circuit consisting of the sample thermome-ter, a metal film heater (R ~ 300 Q) and a Conductus LTC-21 PID controller. A similar feedback circuit was available for regulation of the resonator module temperature, how-ever it proved to be unnecessary. The heater and thermometer were all secured to their respective surfaces with G E varnish. Chapter 3. Experiment 37 V a c u u m L i n e t o S t o k e s P u m p C o n d u c t u s L T C - 2 1 T e n p e r a t u r e C o n t r o l I e r HP 8 3 4 9 B Mi c r o w a v e Anpl i f i e r HP 83620A S y n t h e s i z e d S w e e p e r A B n n M V N A Figure 3.9: Schematic diagram of the 75 GHz apparatus. The sample module is mounted on the underside of the He reservoir. The JV2 cooled radiation shield is secured to a toroidal N2 reservoir. The key components to note are the microwave circuitry, ther-mometry electronics, cryogenics and the P C which controls the experiment. Chapter 3. Experiment 38 As stated previously, the dewar used for the 75 GHz apparatus was a customized Infrared Laboratories HDL-8. This dewar consists of a N2 cooled radiation shield and a AHe cold plate to which the resonator module and an additional radiation shield are mounted. During the course of an experiment the 4 H e bath is pumped by a Stokes model 212-11 Microvac pump, resulting in a base temperature of 1.2 K . A 4.3 L charge of 4.2 K AHe provides approximately 12 hours of running time after reducing the vapour pressure from atmospheric to 0.4 mmHg, the minimum attainable pressure using the present configuration of the apparatus. A customized computer software package was written to automate the resonance char-acterization as a function of temperature. Automated temperature regulation, frequency sweeps and resonance curve fitting routines were incorporated into the software supplied by A Brum,. Resonances were fit to a single Lorentzian peak parameterized as a circle in the complex amplitude plane. Mf) = T - 7 ^ % - X e l ( ' ( / ° ) + ( / " / o ) * U + + (3-12) 1 + ^ /o / / ' A/ The free parameters and <j>(f0) correspond to the amplitude and phase at the resonance frequency /„, while Af represents the F W H M . Thus Q = f0/Af is an implicit fitting parameter. Any slowly varying background signals are accounted for by the free parameters xleak, yieak and d(f>(f0)/df. To summarize thus far, the development of a cavity perturbation apparatus capable of probing the real part of the surface impedance of the high temperature superconductor YBCO at 75 GHz has been described. This technique has been shown to be suitable for measurements on samples that are thin plates. Measurements of the change in the cavity's Q upon loading a sample at the centre of the resonator are related to the surface resistance Rs via Eqn. 3.10. Electrical conductivity expressions derived previously can then be compared to experiment through Eqn. 3.1. Chapter 4 Resul ts and Ana lys is 4.1 Measurement Procedure The discussion in the previous chapter detailed the mechanism by which one can extract Rs from measurements of A(l/Q). However, as noted previously, the measurement inherently involves an admixture of the surface resistance in two crystal directions. It was also noted that YBCO crystals preferentially grow faster in the a and b directions, hence crystals with broad ab faces and comparatively thin in the c direction are frequently found in growths. Therefore, the prescribed technique results in measurements of Rs(a) and Rs(b) of YBCO with a slight admixture of Rs(c)-Nonetheless, with a careful series of experiments, one can disentangle the admixture of Rs from different crystal orientations. Furthermore, one can also determine the back-ground signals alluded to in Eqn. 3.11 by altering the crystal geometry in an appropriate manner. The measurement procedure described herein was inspired by the work of Hos-seini et al. [47], in which the authors successfully managed to extract Rs(c) by measuring Rs(b) before and after cleaving their YBCO sample parallel to the ac plane. The first two measurements were performed on a single crystal of YBCO with di-mensions (xa,xb,xc) = (380 ± 6/im,395 ± 6/im, 23.3 ± 2.3//m). The first measurement yielded an admixture of Rs(a) a n d Rs(c)- The sample was then rotated by 90° about the c axis, and a measurement containing an admixture of Rs(b) and Rs(c) was obtained. The uncalibrated results may be expressed as follows; 39 Chapter 4. Results and Analysis 40 Mla = A{l/Q)a + A ( l / Q ) c + A ( l / Q ) 6 a c f c g r o u n d (4.1) M l b = A(l/Q)a + 7 C A(1/Q) C + A ( l / Q ) 6 a c f e f f r o u n d (4.2) The slight difference between the ac and 6c surface areas on the sides of the sample is accounted for by the factor jc = 0.96 ± 0.02. The sample was then cleaved parallel to the be plane into 3 fragments and then prepared for another a axis measurement. However, as shown in Fig. 4.1, the screening currents must now traverse 6 be faces, which multiplies the c axis contribution to the loss by a factor of 3. Finally, the largest of the three fragments was measured again in the a axis orientation. This portion of the crystal had dimensions (x'a, x'b, x'c) = (172 ± 3(J,m, 395 ± 6/xm, 23.3 ± 2.3//m), thus the ab surface was 7 a = 0.43 ± 0.03 of the area of the entire' crystal. Assuming that the background signal was dominated by the sapphire and grease and therefore relatively independent of the change in sample size, Equations 4.1 —> 4.4 form a set of 4 independent equations for 4 unknowns. One can then simultaneously solve for the unknown quantities by diagonalizing a 4 x 4 matrix. M 2 = A{l/Q)a + 3A(1/Q) C + A(l/Q)background (4-3) M 3 = 7 o A ( l / Q ) a + A ( l / Q ) c + A(l/Q)background (4.4) A ( l / Q ) a Mla-i 1 - 7 , M 3 a (4.5) A(l/Q)b = Mlb - 7 c A ( l / Q ) c - A(l/Q)background (4.6) (4.7) &{1/Q)background = Mla ~ A ( l / Q ) a - A(l/Q)c Chapter 4. Results and Analysis 41 J Geometry for measurement M. J Geometry for measurement M Figure 4.1: A depiction of the sequence of measurement geometries. The first two measurements were performed upon a single piece of YBCO with crystal dimensions (xa, Xf,, xc) = (380 ± 6/m?,,395 ± 6pm,23.3 ± 2.3pm). The crystal was rotated by 90° about the c axis to obtain both a and b oriented measurements. The sample was then cleaved into three pieces and arranged for another a axis measurement, thus multiplying the c axis contribution by a factor of 3. Finally, the largest section of the crystal with dimensions (x'a, x'b, x'c) = (172 ± 3pm,, 395 ± 6pm,, 23.3 ± 2.3pm) was measured. Chapter 4. Results and Analysis 42 The only remaining task is that of calibration, which was accomplished by measuring a sample of PbSn alloy (~ 5% Sn) cut to (a;, y, z) = (442 ± 7pm, 465 ± 8pm,, 20 ± 3pm) which is approximately the same size as the YBCO crystal. The dc resistivity p(T) was determined independently by a four probe measurement and the normal state impedance at 75 GHz was calculated via the skin depth relation; RS(Q, T) = y/p0 ft p(T)/2 (4.9) where fl is the angular frequency in units of rad • s - 1 and p has units of Qm. The calibration constant is then determined by matching calculated values of Rs{T) with A(l/Q)p0sn over a broad temperature range by varying the multiplicative constant T defined in Eqn. 3.10. This was accomplished by minimizing the function X2(T) = ^(RsiTA - rA(l/Q)PbSn | T i ) 2 (4.10) Geometric factors were calculated to account for the difference in surface areas be-tween the PbSn and YBCO samples. One can assume that the surface impedance of the PbSn sample is isotropic and then scale by the ratio of the total surface area of the PbSn sample traversed by screening currents to the combined area of the two broad ab faces of the YBCO sample to yield both Rs(a) and Rs(b)-RsaW=T 2-^^A{l/Q)a[b) (4.11) ZXaU,b Scaling by the ratio of the total surface area of the PbSn sample traversed by screening currents to the combined area of the two be edges of the YBCO sample yields Rs(c)-R S * = T 2 { 1 X + J C Z ) ^ 1 / Q ) C ( 4 - 1 2 ) This procedure provides one with calibrated measurements of the surface impedance of YBCO in the 3 crystallographic orientations. Thus, by a careful measurement and cleaving programme one can account for all calibration factors and background signals. Chapter 4. Results and Analysis 43 4.2 Exper imenta l Resul ts Figures 4.2 and 4.3 display the four measurements of A(l/Q) prior to any data processing. Error bars have been neglected on these plots since the scatter in any measurement below 80 K proved to be less than 1%. Above 80 K the error estimates never exceeded 5%. The individual a, b and c axis contributions are depicted in Fig. 4.4. Due to the use of the geometrical factor 7 a in Eqns. 4.5 and 4.7, the error estimates for A(l/Q)a and A(l/<5)c have increased to ~ 5% at any point below 80K. Shown in Fig. 4.5 is the extracted background signal. Below 20 K, this quantity remains relatively constant at —5.5 x 10~9, which is approximately the same size as the low temperature c axis signal. At 80 K the c axis signal exceeds the background by only a factor of 2. Therefore, A(l/'Q)background has a negligible effect upon measurements of A ( l / Q ) a and A(l/Q)( , , however it does complicate the extraction of A(l/Q)c. Also shown in Fig. 4.5 is a direct measurement of A(l/Q) from sapphire with a minute quantity of grease. Note that this measurement yielded a positive signal, while the background signal extracted from the 4 sets of measurements is negative; a priori, one would expect these quantities to be the same. Therefore, the presence of a sample inside of the resonator alters the fields in a nonperturbative manner, and the assumption that the background signal is only weakly dependent upon sample size may be questionable. Figure 4.6 displays the results of the PbSn calibration procedure. The circles represent the measured values of A(l/Q)pbs n multiplied by the fitting parameter Y = 1.72 x 104 ± 10%. The solid curve is a spline fit to the surface impedance calculated via Eqn. 4.9 from the independently measured values of p(T) - it is not a fit to the A(l/Q) data. The 10% relative error in Y due to uncertainty in the dimensions of the sample used to measure p(T) far exceeds the error from any other measured quantity in this experiment. Thus, a more accurate means of obtaining p(T) for the PbSn sample is required. Chapter 4. Results and Analysis 44 1.0 0.8 CO 0.6 g T— < A O o o o • 20 40 80 100 0.4 h o A(1/Q) a + 3 c , 3 Fragments D A(1/Q) a + c, Single Piece A A(1/Q) y a + c,1 Fragment o • o • o • o • 0.2 h o • o a A A A A A A 9 e O - A O • A o a A QO A 8 8 8 0.0 ^mm£A4AAA A A A A A A 20 40 60 Temperature /K 80 Figure 4.2: Raw measurements containing an admixture of a and c axis contributions. Chapter 4. Results and Analysis 45 1.0 0.8 C D i o g < 0.6 0.4 h 0.2 \-0.0 1 1 1 1 1 1 1 -' 1 1 1 10 • • • 1 • • • • — 0.1 • p . i . • ( 20 40 60 80 100 - • -D A(1/Q)b+c, Single Piece • a • • — • — • • - • -o • • • • 0 n n i 1 • D I 1 . 1 • 0 20 40 60 Temperature /K 80 Figure 4.3: Raw measurements containing an admixture of b and c axis contributions. Chapter 4. Results and Analysis 1.0 0.8 C D I o g < 0.6 0.4 0.2 0.0 1 1 1 1 ' 1 ' aE5 • 0 A ; • _ n g 0 ° A ^ 1 20 40 60 60 100 o~5~o o A(1/Q) • A(1/Q) A A(1/Q) I o o 8 • • o 2: o • 0 o • • -J i t tWAAAAAA A A A —A— A A A A A A A - A - A A A - + - A A A ~ f - , _ 20 60 80 Temperature /K Figure 4.4: The extracted a, 6 and c axis losses. Chapter 4. Results and Analysis 47 0.000 -0.025 CD o O -0.050 -0.075 -0.100 bniQinnaic • n n U T o • • o 1 1 1 1 1 1 1 o a a n : a o o • • • a 3 1 ° a o • • • n j £ > o 0 o o 0 o 0 Extracted Background Signal Measured Sapphire and Grease Losses 1 i i i i 20 40 60 Temperature /K 80 Figure 4.5: The extracted background signal and the directly measured losses due to the sapphire plate and the NonAq stopcock grease used to secure the sample. The dashed line indicates the low temperature nonperturbative correction ~ —5.5 x 10~9. Chapter 4. Results and Analysis 48 0.16 0.00 h 20 40 60 Temperature /K 80 100 Figure 4.6: Calibration of the 75 GHz apparatus using a PbSn sample. The solid curve is a spline fit to the surface impedance calculated from the independently measured dc resistivity - it is not a fit to A(l/Q). The circles represent T x A ( l / Q ) , where T = 1.72 x 104 was determined by a least squares fit as described in Section 4.1. Chapter 4. Results and Analysis 49 Subjecting the data in Fig. 4.4 to Eqns. 4.11 and 4.12 then yields Rs(a), Rs(b) and Rs(c), a s shown in Figure 4.7. Representative error bars have also been included, which range from ~ 10 —> 14% over 1.7 to 80K, but can be as large as 30% on Rs(c) at 94K. Recall that the experimental accuracy has been limited by errors in the measurement of sample dimensions - in particular xc of the YBCO crystal, and all dimensions of the PbSn calibration sample used for the dc resistivity measurement. It is instructive to plot the 75 GHz data in conjunction with other microwave mea-surements performed on crystals from the same high purity YBCO growth in order to comment upon the frequency dependence of the surface resistance. Such data are avail-able for Rs(a) and Rs(b) at 1, 2, 13 and 22 GHz. Unfortunately, there are no other Rs{c) measurements on ultra high purity crystals available at the time of writing. Figure 4.8 shows a plot of Rs(a) scaled by / 2 , a scaling suggested by Hosseini et al. [48]. There are two remarkable features that should be noted: the consistency of the scaling above 60K and the broad low temperature peaks. Both of these features may be qualitatively understood in terms of the simple Drude picture suggested in Chapter 2. As T rises up to Tc, it is reasonable to assume that the quasiparticle scattering time r approaches its normal state value. Thus in the range of microwave frequencies studied here, one can assume that O T <g; 1 for T near Tc. From Eqn. 2.7 one can then see that .Re | cr is independent under such conditions and an approximate low frequency form of Eqn. 3.1 then reveals RS(Q.) oc 0 2 i?e |a | [10]. The peaks were interpreted by Bonn et. al. [49] as being generated by a competition between a rapidly increasing r and a decreasing nn as T —^  0 in Eqn. 2.7. Eventually r must encounter a low T impurity limit, but nn continues to decline essentially linearly due to the d-wave density of states. The evolution of nn(T) has been plotted in Fig. 4.9. Here, A 0 (T) data of Kamal et al. has been translated into nn(T) and ns(T) via Eqns. 2.4 and 2.8, assuming a zero temperature value A 0 = 1600 °A [35]. Chapter 4. Results and Analysis 50 20 18 16 14 12 a E 1 0 l _ 1000 t-8 6 4 2 0 _l . l_ n 0 40 60 o- - R S(a) • - • R S(b) A- - R S(c) .o-o-a-o. . • o 0 P 1 . • - • D ; o - o - B ' A J o > a : f l A A . A . X 20 40 60 Temperature /K 80 Figure 4.7: The surface impedance of YBCO measured at 75 GHz. Note the crossing of Rs(a) and Rs(b) at 26 K , thus indicating a reversal of the low temperature a:b anisotropy observed in lower frequency measurements. Chapter 4, Results and Analysis 51 10 t-" • D D D ° 0 • O, • o rf o o V o 1.139 GHz 2.2 GHz 13.373 GHz 22.71 GHz 75.416 GHz Inelastic Scattering A A A A 0 A A 0 • • • o.8 20 40 60 80 Temperature / K Figure 4.8: A scaled plot of Rs(a) measured at five microwave frequencies. One should note the consistency of the l / / 2 scaling above 65 K and the obvious evolution of the peak height and position with increasing frequency. The plot has been divided into three scattering regimes based upon the hypothesized evolution of the scattering rate. Chapter 4. Results and Analysis 52 1.0 0.8 CD -f—» S 0.6 T 3 CD 0 4 N u n 0.2 h-0.0 l D o D r ' • O r 'Da, ns(T) ns(0) *0T • o n s(°) m n * X(T)2 fp CO ,00' ± ± 20 40 60 Temperature IK 80 Figure 4.9: The normal nn(T) and superfluid ns(T) densities derived from penetration depth measurements in the a axis orientation [35]. The key element to note is the linearity of nn(T) below ~ 40 K , which reflects the low energy d-wave density of states, N(u)~u/A0. Chapter 4. Results and Analysis 53 From a less empirical standpoint, the evolution of r in the cuprates is dictated by inelastic scattering from bosonic excitations such as antiferromagnetic spin fluctuations [17]. However, at sufficiently low temperatures an elastic scattering l imit wi l l be en-countered, thus giving rise to the 'pure regime' discussed in Chapter 2 - the region from ~ 1 —> 20 K shown in F ig . 4.8. A t very low temperatures it is anticipated that the so called 'gapless regime' wi l l also appear. However the penetration depth data in F ig . 4.9 implies a pure d-wave density of states down to the lowest temperature obtained in these experiments, 1.1 K [35]. It w i l l therefore be assumed that the gapless regime resides below a value of T* < 1 K , and the theoretical conductivity expressions derived in Chap-ter 2 wi l l be valid from ~ 20 K to the lowest attainable temperature in each surface impedance experiment. Since the a axis data wi l l be subject to intense analysis in the forthcoming section of this treatise, it is worthwhile cataloging the error estimates for each set of data shown in F i g . 4.8. Recal l that there are three key types of errors which one must consider when examining calibrated surface resistance data: systematic errors due to the non-perturbative correction 6Rs(background), systematic errors from the determination of the multiplicative factor V from PbSn calibration and the stochastic noise in measurement of A(l/Q). These have been summarized in Table 4.1 below for T < 20K. Frequency / G H z 6T 3Rs(background) / 3Rs (stochastic) 1.139 10% 0.7 < 1% 2.2 10% 0.2 < 1% 13.373 10% 10 < 1% 22.71 10% 20 < 1% 75.416 10% 345 < 1% Table 4.1: Summary of stochastic and systematic error estimates for measurements of Rs(a) presented in F ig . 4.8. Chapter 4. Results and Analysis 54 A plot of RS(b)/f2 is shown in F ig . 4.10. As for Rs(a), one can employ the simple Drude picture to understand why Rs(b) has a broad peak at each frequency, but the 1 / / 2 scaling does not work above 65 K , especially for the 75 G H z data. From F ig . 4.7 one can see that experimental errors do not account for this discrepancy. Also notice the crossing of Rs(a) and Rs{b) near 26 K in F ig . 4.7, thus indicating that the low T a:b anisotropy at 75 G H z is opposite to that found at lower frequencies [31]. For comparison, the 13 and 75 G H z surface impedance in both planar directions is plotted in F i g . 4.11. The difference in the T —> 0 anisotropy is significant; Rs(a)/Rs(b) ~ 1.6 at 13 G H z and ~ 0.6 at 75 G H z . Preliminary Rs measurements with a microwave bolometry apparatus operating at 1 K suggest that the a : b anisotropy of Rs may very well reverse at frequencies greater than 25 G H z [50]. The interpretation of b axis electrodynamics in YBCO is further complicated by the fact that both Cu02 planes and CuO chains oriented parallel to b w i l l contribute to the observed Rs (refer to F ig . 3.1). Given that it may be necessary to invoke two conduction mechanisms to fully understand b axis electrodynamics in YBCO, Rs(b) is not an ideal candidate for performing quantitative analysis using the models described in Chapter 2. Since microwave measurements of Rs in the c direction are particularly scarce, it wi l l only be possible to make a qualitative comparison between measurements of Rs(c) at 75 G H z shown in F i g . 4.7 and those of Hosseini et al. at 22 G H z [47]. Note that any quantitative interpretation is complicated by three factors; the difference in measurement frequency, slightly different oxygen content (6.993 vs. 6.95) and the purity of the samples. The results of Hosseini et al. were obtained from an optimally doped sample grown in a YSZ crucible. Figure 4.12 shows the two above mentioned results scaled by l / / 2 . A s can be seen, the two curves are of comparable magnitude over most of the temperature range, although the 75 G H z data does not display the upturn below 20 K as reported by Hosseini et al. [47]. Chapter 4. Results and Analysis 55 The fact that the l / / 2 scaling works reasonably well over most of the temperature range is quite remarkable, and suggests that the c axis conductivity is frequency independent over a broad temperature range, and is relatively insensitive to doping and impurity levels. Since there are 3 key differences between the two measurements, any speculation concerning fine details such as the presence or absence of the upturn would be difficult to justify. Chapter 4. Results and Analysis 56 1.139 GHz • 2.2 GHz A 13.373 GHz v 22.71 GHz o 75.416 GHz o n Temperature / K Figure 4.10: A scaled plot of RS(b) measured at five microwave frequencies. Again, i t is possible to interpret the peaks in terms of well defined quasiparticles, however one should note the obvious breakdown of the 1 / / scaling of Rs in this orientation. Chapter 4. Results and Analysis 57 3 r-2 1 r-1 1 1 r o a axis, 13.373 GHz • b axis, 13.373 GHz • a axis, 75.416 GHz • b axis, 75.416 GHz ° ° ° • D D D n a • D 0 - J O • o • * • • • < * • • • • * * 0 • • o • • • o y • °£> • ° o • • n a n • D • • u • _ * • • • fe£. ' • • i • x 20 40 60 Temperature / K 80 Figure 4.11: A comparison of the a:b anisotropy of R s at 13 and 75 GHz. The T —>• 0 limits of Rs{a)/Rs(b) are ~ 1.6 at 13 GHz and ~ 0.6 at 75 GHz. Chapter 4. Results and Analysis 58 10 \n 6 a CM o co 0 T T • BaZr0 3 Grown, Over Doped, 75 GHz (Tc = 88.7 K) o YSZ Grown, Optimally Doped, 22.71 GHz (Tc = 93.5 K) 0 D O O 0.0 • o • • o I o o • o • • • • • • D ° D D 0 D o n o D o D ^ 0 ° • o • o 0.2 0.4 0.6 0.8 1.0 T / T . Figure 4.12: A scaled plot of RS(C) measured at two microwave frequencies. As scaled, the curves are of comparable magnitude. However, sample differences prevent one from drawing any quantitative conclusions from this plot. Chapter 4. Results and Analysis 59 4.3 Theoret ica l Ana lys is A wealth of experimental surface impedance data has been presented thus far, how-ever the intrinsic quantity of interest is the electrical conductivity, cr(Q,,T). Hence, the traditional approach employed by microwave spectroscopists has been to extract cr by using a low frequency approximation in conjunction wi th the two-fluid model and the assumption that i"m |cr| ^> Re\o~\ over a broad temperature range below TC. Upon ap-plying these approximations to Eqn. 2.7, one can then conclude that on « Re\cr\ and Im\o\ ~ nse2/m*SQ = 1/U.0LO\2(T). Equation 3.1 then yields the following simple rela-tions [10]. 2 R ^^-iwm (4-13) ' " ^ - j s s b r S ( 4 1 4 ) These approximations have been extremely useful at frequencies below ~ 20 G H z , however one must acknowledge that the normal fluid contribution to 7ra|er| in Eqn. 2.7 cannot be neglected at higher frequencies. In response to this difficulty Hosseini et al. devised a self consistent correction procedure to account for normal fluid screening, but it did so in a model dependent manner [48]. Therefore, a truly general means of extracting cr(Q, T ) from measurements of Rs{£l, T ) over a broad range of frequencies and measurements of Xs(l G H z , T ) has yet to be realized. Fortunately, when fitting to models it is possible to circumvent the problem dis-cussed above by using Eqn . 3.1 in conjunction wi th a complete theoretical expression for the quasiparticle conductivity, and by measurements of XS(Q, T) at a sufficiently low frequency. Given either the simple Drude expression, Eqns. 2.6 and 2.8, or the more rig-orous point scattering model Eqns. 2.25 and 2.27, one has a prediction for both the real and imaginary components of aqp(fl,T). The conductivity due to the superconducting Chapter 4. Results and Analysis 60 groundstate was given previously in Eqn. 2.5. If one assumes that the 1 GHz penetration depth measurements presented in Fig. 4.9 were obtained at a sufficiently low frequency to consider them equivalent to the London penetration depth, A^ in Eqn. 2.5, then the surface impedance can be expressed as follows: ( 4 i 5 ) Equation 4.15 provides a direct means of comparing model conductivity functions to surface resistance data. This can be accomplished by treating r in Eqn. 2.6 or ( r „ , Td, Tb) in Eqn. 2.27 as free parameters in a least squares fit to experimental results. By fitting each available Rs using the minimization function X2(n, Tu, Td, Tb) = E * m..2(Re\Zs(Sl, Ti, Tu, Td, Tb)\- RS(Q, Ti))2 (4.16) i = i (AKs{il, li)) one can then compare amongst the best fit parameters to locate trends. Here, AR$(fl, Ti) represents the absolute error in the measurement of Rs at frequency ft and temperature Ti, and N represents the number of measurements at frequency f2 between T = 1 and 20 K . Given this information, one can then attempt to fit all five available data sets simul-taneously to determine if one set of best fit parameters can account for all observations. This was accomplished by minimizing the following function; x 2 ( r u , r d , r b ) = E E , A P L ^ J R ^ Z S ^ T ^ Y ^ - R S { O , T 3 ) ) 2 (4.17) The minimization procedure was implemented using Brent's method with Mathe-matica. Though this minimization procedure is not as robust as a simplex or a gradient descent algorithm, it proved to be adequate for the task at hand. Thus, through a carefully designed analysis, the absence of measurements of Im\Zs(£l,T)\ at each of the frequencies that Re\Zs(Q, T)\ has been measured is no longer a hindrance. Equation 4.15 provides a means of translating an expression for <x(f2, T) and measurements of Xt(T) into an expression for Rs(Q,T), which can then be compared to experimental results. Chapter 4. Results and Analysis 61 As argued in the previous section, the measurements of Rs in the a direction are the best candidates to be subjected to theoretical analysis since the pure regime has been clearly identified in this case and there is only one likely conduction mechanism, namely the CuOi planes, which can contribute to the observed electrodynamics. Therefore, it would be useful to first quantify the characteristics that a successful model must provide. From a careful examination of the pure regime in Fig. 4.8, one should note that Rs appears to be sublinear at 1 and 2 GHz, linear at 13 and 22 GHz, and almost quadratic at 75 GHz. Hence, a useful parameterization would allow one to quantify this observation. Shown in Figs. 4.13a—>e are power law fits to the pure regime data using a function of the form Rs = C + ATa (4.18) and the results have been summarized in Table 4.2. Frequency / G H z C / o A j ( f i /K) a 1.139 5.0 x 10-7 7.2 x 10-7 0:88 ± 0.05 2.2 3.1 x 10-6 2.2 x 10-6 0.91 ± 0.04 13.373 3.0 x 10-5 1.0 x 10-5 1.13 ± 0 . 0 9 22.71 6.0 x 10-5 9.4 x 10-6 1.27 ± 0 . 0 5 75.416 5.7 x 10-4 9.9 x 10-6 1.5 ± 0 . 1 Table 4.2: Summary of power law fits to Rs in the pure regime. One can clearly identify the sublinear to superlinear evolution of the exponent a with increasing frequency. Error estimates on a were generated by forcing the fitting parameters to extreme, values such that the resulting curve need only remain within the error bars shown in Figs. 4.13 a—>e. In Figs. 4.13 a—>e, the error bars on the data represent the ~ 10% errors due to the calibration factor T and stochastic noise. The systematic errors due to nonperturbative corrections have not been included in this figure. Rather, one can adopt the values of 8 Rs (background) in Table 4.1 as error estimates for C in Table 4.2. The above results then constitute a list of criteria that a successful theory must meet. Chapter 4. Results and Analysis 62 10 15 Temperature / K 20 Figure 4.13: Fitting of the pure regime Rs(a) data with a power law expression. One can clearly see the evolution of the power from sublinear at 1 GHz to super linear at 75 GHz. Chapter 4. Results and Analysis 63 4.4 Theoret ica l Resul ts Before proceeding with the analysis involving the point scattering model Eqn. 2.25, it was essential to perform some consistency checks. This was required since the point scattering expressions are quite complicated, thus introducing many opportunities for programming errors. As noted in Chapter 2, both the simple Drude picture and the point scattering picture should be in accord if one assumes r = constant in both cases and disposes of o0o in Eqn. 2.25. Therefore, a convenient means of validating the implementation of Eqn. 2.25 was to fit each of the five surface resistance data sets independently using both of the above mentioned models and a constant scattering parameter. In order to fit with Eqn. 2.25 one must first determine a reasonable value for the amplitude factor ne2/mA0. Given a value of A 0 = 1600 °A in the a direction [9] and A 0 = 23QK [21], the constant ne2/m,A0 was determined by assuming that all electrons condense into the superconducting groundstate at T = 0. Thus, >2 ns(T = 0)e2 _ ne mAr 1 ~ i x io6 fr1™ -1 m*A 0 / i 0 A 2 A 0 The results of the consistency check are compiled in Table 4.3, and the fits using the constant scattering rate in the point scattering model are shown in Figs. 4.14a —>e. Frequency / G H z 1/r (Simple Drude) jK 1/r (Point Scattering)/^ 1.139 0.41 0.39 2.2 0.41 0.39 13.373 0.28 0.31 22.71 0.34 0.37 75.416 1.39 1.12 Table 4.3: A comparison of the best fit constant scattering rates determined by fitting each data set independently to both the simple Drude model and the point scattering model. Only the fits to the 75 GHz data exhibited a substantial discrepancy. Chapter 4. Results and Analysis 64 One can see from Table 4.3 that the values of 1/r extracted by these two methods are comparable, except at 75 GHz. However, a recurrent theme throughout this section of the treatise will be the absolute failure of both models to accurately describe the high frequency data. Nonetheless, the results of the consistency check do confirm that the complicated point scattering formulae have been implemented in a manner that is in accord with the simple Drude expressions. The next logical step was to fit each data set independently to the point scattering model using all three independent parameters Tu, Td and Tb in Eqn. 2.27. The results are shown in Figs. 4.14a —>e, and the best fit parameters are presented below in Table 4.4. Figs. 4.14a —->d demonstrate that the point scattering model is capable of providing the correct curvature, however in order to do so a mixture of scattering types and relative contributions is required. Frequency / G H z Tu/K Td/K Tb/K 1.139 5.1 x n r 5 0.37 0.87 2.2 0 0.31 0.84 13.373 2.1 x lCT 3 0.20 0.31 22.71 4.0 x 10~5 0.30 0.11 75.416 4.3 x IO" 2 0 2.07 Table 4.4: Best fit parameters to the point scattering model using Eqn. 2.27 as the model scattering rate function. Chapter 4. Results and Analysis 65 Figure 4.14: Independent fits of each Rs(a) data set to the point scattering model, using a constant scattering rate (dashed line) and a frequency dependent scattering rate (solid line). Chapter 4. Results and Analysis 66 In examining the results of the point scattering fits it was noted that the fitting function had to contend with the clearly nonzero residual values of Rs at T — 0. Hence, the fits were severely influenced by attempts to match the residuals, as opposed to the temperature dependence of the data. This was found to be particularly true at the highest frequency, 75 GHz. Note from Table 4.4 that this data set required an unusual mixture of unitary and Born terms in order to fit the data, while all other data required a significant Drude term and a negligible unitary term. One can elucidate the source of the problem by examining Eqn. 2.25 in the limit T —> 0. At sufficiently low temperature, one will inevitably violate the approximation Q, <C T; in fact, in the limit Q > T Eqn. 2.25 gives the following erroneous result (assuming r is roughly constant for UJ < fl): ]imcrav(Q,T) —» —T- I dfl^-(——^4— ) + cr0 T^O GPK ' ; mA0J-n Q ^ Q - z / W 3 2 fn ,„ 1 / - M 1 ne2 ( Q \ , 2 ^ ( r r = ^ ) + < r - ( 4 ' 1 9 > The integral in Eqn. 2.25 is clearly nonzero at T = 0, therefore the model does not provide an accurate description of the electrodynamics at temperatures T < Q. In particular, at high frequencies (Or >^ 1) Eqn. 4.19 has the following limit: h m f l e M « , T ) l - 5 ^ ^ + ^ (4.20) Assuming that Eqn. 4.13 is approximately correct at higher frequencies one can see that the theoretical prediction for the residual Rs(D,T —• 0) ~ 1/r. Note that a large unitary term in Eqn. 2.27 provides large values of 1/r at very low UJ. Hence, the fitting of the 75 GHz data proved to be particularly sensitive to the measured T —> 0 residual losses. Chapter 4. Results and Analysis 67 To further complicate the issue, thermal [51] and microwave [31] conductivity mea-surements on YSZ grown crystals provide values of 0~oo that are significantly larger than the predicted value, which is ne2j'nmAo ~ 3 x 105 ft_1m_1. Recent theoretical investi-gations suggest that aOQ may be renormalized by vertex corrections to Fig. 2.2 [29], thus giving a quantity referred to hereafter as u*OQ. Therefore, fitting the data with Eqn. 2.25 which contains the disputed quantity aOQ may lead one to erroneous conclusions. A more fair fitting algorithm was devised for performing the global fits in which the T = 0 residuals of both the theoretical curves at the data were forced to be equal at each frequency prior to fitting. This was accomplished by removing a00 from Eqn. 2.25 and subtracting the power law fit constants C (Table 4.2) from the data sets. The data were then fit via Eqn. 4.16, the nonzero residuals of the theoretical fits were used to update the quantities subtracted form the data sets, and the fitting procedure performed again. Within two iterations of this procedure, the T = 0 residuals of the theoretical curves and the data agreed within 5%. The resulting fits to individual data sets, illustrated in Figs. 4.15a —+e, are representative of fits to the observed temperature dependences as opposed to the magnitude of the T = 0 values of Rs- The new best fit parameters have been summarized below. Frequency / G H z Tu/K Td/K rb/K 1.139 0 0.34 1.12 2.2 0 0.38 1.22 13.373 0 0.19 0.42 22.71 0 0.19 0.63 75.416 3.0 x IO" 3 (0) 0 (0.189) 2.61 (1.7) global 0 0.33 0.69 Table 4.5: Best fit parameters to the point scattering model using the modified fitting algorithm. Note that only the 75 GHz data set required a nonzero Fu. The quantities in brackets indicate the best fit parameters obtained by forcing Tu = 0. Chapter 4. Results and Analysis 68 16 12 8 o 30 9. 20 r> 10 400 | - c) o 1 3 . 3 7 3 G H z D a t a P o i n t S c a t t e r i n g M o d e l 300 h 1200 h m 800 400 a) o 1 .139 G H z D a t a P o i n t S c a t t e r i n g M o d e l |_ e ) o 7 5 . 4 1 6 G H z D a t a P o i n t S c a t t e r i n g M o d e l , B e s t F i t P o i n t S c a t t e r i n g M o d e l , r = 0 10 15 20 Temperature / K Figure 4.15: Modified independent fits of the five Rs(a) data sets to the point scattering model using a frequency dependent scattering rate (solid line). Chapter 4. Results and Analysis 69 Upon comparing the solid curves in Figs. 4.14 and 4.15 one can discern very few improvements in the quality of the fits. However, a comparison of Tables 4.4 and 4.5 reveals one significant difference: by forcing the T = 0 intercepts of the data and the fit curves to be equal, the unitary contribution has been essentially eliminated from the 1, 2, 13 and 22 GHz fits. Therefore, one should regard the unitary terms in Table 4.4 as artifacts from fitting to a function with an incorrect T —> 0 limit. It should also be noted that the unitary term in the 75 GHz fit has been reduced by an order of magnitude, but it has not been eliminated. This is most likely due to the extreme sensitivity of the theoretical T —• 0 residual to the unitary fitting parameter, as argued previously. Slight errors in the quantity subtracted from the 75 GHz data prior to the first iteration of fitting may bias the fit towards a local minimum with a nonzero Tu. To test this hypothesis, the 75 GHz data were also fit with a scattering rate in which Tu was defined to be zero, as shown in Fig. 4.15e (dashed curve) and the best fit parameters are shown in brackets in Table 4.5. The fit is quite acceptable over the range 4 —> 20 K, which is reasonable since the approximation O <C T will be violated f o r T < f 2 ^ 3 . 6 1 K . The logical conclusion from this discussion is that all nonzero values of Fu in Tables 4.4 and 4.5 are artifacts from using a fitting function with an incorrect T —> 0 limiting form due to the violation of the f2 <C T approximation. Hence, the trend to note in Table 4.5 is that the all data sets can be fit by a scattering rate of the form l/r(u) — Td + Tbco/AG. A global fit was performed using the modified fitting procedure described above and Eqn. 4.17. As shown in Table 4.5, the global fit value of Td is roughly the average of the individual best fit values. Given that the simple Drude model has been shown to provide a reasonable fit to the data [48], this is to be expected. The results have been plotted in Figs. 4.16a —>e along with the linear fits generated by the simple Drude model using 1/r = 0.426 K —> 5 x 1 0 1 0 s _ 1 . This is very close to the temperature independent quasiparticle scattering rate of 0.382 K extracted by Hosseini et al. [48]. Chapter 4. Results and Analysis 70 Figure 4.16: Global fit of all five Rs(a) data sets to the point scattering model using a frequency dependent scattering rate (solid line). Also shown is the simple Drude model proposed by Hosseini et al. (dashed line). [48]. Chapter 4. Results and Analysis 71 A more intuitively appealing means of plotting the data and best fit results is in the form of electrical conductivity. One can use the values of Irn\<jqv\ obtained from the theo-retical fits and Eqn. 4.15 to convert measured values of Rs into Re\aqp\exp. These results have been plotted with the best fit theoretical Re\oqp\theory curves in Figs. 4.17a—>e. The best fit theoretical curves include estimates for the renormalized T = 0 limit of the con-ductivity, fj*0, which were obtained from the extrapolated T —• 0) intercepts shown in Table 4.2. Error estimates for these quantities were generated by determining the max-imum and minimum possible value of a*a given the error estimates on Rs(Q,T = 0) in Table 4.1. Figs. 4.17a—>e clearly show that the model function is capable of creating the correct concavity at each frequency, and the inclusion of more details may improve the fits. In particular, the reader should note that in the derivation of Eqn. 2.25 only the imaginary portion of E c was kept, however Kramers-Kronig relations demand that l/r(u>) must be accompanied by an effective mass factor, manifested through the renormalization of the real quantity UJ [52]. Unfortunately, a scattering rate of the form constant + terms linear in to is non-integrable, thus preventing the calculation of i?e |E 0 | without introducing an ad hoc high frequency cutoff. Given that the best fit form of 1/T(UJ) must have a high frequency cutoff, one must then address the question of the range over which the scattering rate function could have the global best fit form - ^ = 0.30 + 0.72^- (4.21) One can estimate this range by examining the low frequency form of Eqn. 2.25. -df Re\oqp(Q - 0,T)| ~ duj{-J-)r{uj) J—oo auj Recall that —df/duj is a roughly Gaussian shaped function centered at UJ = 0 and ap-proximately 2T wide. Therefore, if one deems the theoretical fits to be acceptable up to Chapter 4. Results and Analysis 72 T « 10K, then the scattering rate function Eqn. 4.21 must be valid up to w « 20 K , or 400 GHz. Now that a functional form for 1/T(U>) and the potential range of its validity have been established, one can now return to the two questions posed in the introduction. First, can the in-plane electrical conductivity of Y B C O be described in terms of well defined quasiparticles? The answer is yes, but in order to do so an unconventional scattering rate function, Eqn. 4.21 had to be invoked. The second question then follows quite naturally; can an interaction or an admixture of interactions be identified which gives a scattering rate of the form Eqn. 4.21? One must examine three distinct possibilities: • Eqn. 4.21 emulates l/r(u) for an intermediate isotropic point scattering strength. This possibility can be eliminated based upon two pieces of evidence. Investigations by Berlinsky et al. to find an intermediate scattering strength which yields a slowly rising imaginary part of the self energy over the range 0 —* 20 K have revealed that no such scattering strength exists [23]. However, the most compelling evidence comes from recent attempts by Rieck et al. to fit the same surface impedance data used in this treatise with the full self consistent T-matrix formulation [53]. These authors concluded that an s-wave phase shift 80 = 0 . 4 4 7 T (c = 0.19) best fits the data, however their fits clearly show structure in Rs at temperatures below 5 K that is not observed in the data. Therefore, one must conclude that the scattering rate function given by Eqn. 4.21 is not emulating an intermediate isotropic point scattering strength. • Eqn. 4.21 represents the cumulative effect of many types of scattering. As argued by Berlinsky et al. [23], the ultrapure BaZrOz grown Y B C O crystals may be so pure that no single type of scatterer dominates. This possibility must be regarded as viable since impurity studies using YSZ grown crystals have demonstrated Chapter 4. Results and Analysis 73 that common impurities such as Zn behave as a unitary scatterer while Ni and Ca seem to be significantly weaker [54]. One may also need to include scattering from extended defects such as residual twin boundaries [23]. Given a sufficient variety of scattering types and relative concentrations, it is entirely possible their cumulative effect will be manifested through a scattering rate of the form Eqn. 4.21. • Eqns. 2.25 and 4.21 may not be the appropriate model. This final possibility should not be regarded as an advocation of scientific nihilism, rather one should recognize that some of the essential physics may be missing from the isotropic point scattering model. As argued by Franz et al, it may be important to include localized suppression of the d-wave order parameter near impurities [55]. Thus the point scatterers shall become centers of order parameter holes, which do not have an isotropic structure in k space. The consequence shall be the introduction of off diagonal components into the calculation of the self energy E. This possibility is currently under investigation by Hettler and Hirschfeld, and the preliminary results look promising [56, 57]. Chapter 4. Results and Analysis 74 Figure 4.17: Global fit of all five Re\oqp\ data sets to the point scattering model using a frequency dependent scattering rate (solid line) and assuming a renormalized T = 0 residual conductivity a*0. Chapter 4. Results and Analysis 75 The plot of <r*0 shown in Fig. 4.18 may prove to be a contentious issue for future research. It should be noted that this plot does not agree with the low temperature behaviour predicted by Lee [22]. Rather, the microwave surface impedance measurements indicate a frequency dependent residual conductivity that has a high frequency limit a factor of 3 larger than o00. A concrete means of confirming this conclusion would be to measure the losses at a temperature below T*. Such an experiment is presently being devised at U B C [50]. 5.0 r i 1 | i | r 4.5 - - -4.0 - -3.5 - -E 3.0 - c ) -a 2.5 T -2.0 -1.5 1.0 f k — 0.5 . L J 0.0 i . i . i . i 0 20 40 60 80 Frequency / GHz Figure 4.18: A plot of the extrapolated o*0, with a spline curve (dotted) fit to the data. The data clearly disagrees with the predicted constant o00 ~ 3 x 1 0 5 O _ 1 m _ 1 (long dashed). Chapter 4. Results and Analysis 76 Despite all of the difficulties encountered, two very concrete conclusions can be drawn from this analysis. First, the best fit parameters in all cases shown here produce scatter-ing rate functions that cannot possibly be generated from the full self consistent T-matrix approximation for isotropic elastic point scattering. At the time of writing, there is no known single mechanism which can generate a scattering rate function in the HTSC's which contains a constant Drude-like term. Second, the T —». 0 limit of the conduc-tivity must be recalculated as the predicted frequency independent value o00 is clearly incorrect. Chapter 5 Conclusions Throughout the saga of HTSC, it has become increasingly apparent that having a com-plete electromagnetic absorption spectrum will be a necessary condition for understand-ing the phenomenon. This treatise marks a significant step forward by demonstrating how the confluence of transport theory and microwave frequency measurements can yield new information about electromagnetic absorption in high temperature superconductors. Data obtained from the complete U B C microwave cavity perturbation repertoire, consisting of five apparatus operating at 1, 2, 13, 22 and 75 GHz, have been used to resolve the low frequency surface impedance in both the a and b directions of ultra high (99.995 %) purity y 5 a 2 C u 3 0 6 . 9 9 3 - In both cases it was concluded that one could model the data in terms of a simple quasiparticle picture, in which a rapidly increasing one body scattering time competes with a decreasing quasiparticle density as T —> 0. However, the reversal of the low temperature Rs(a)/Rs(b) anisotropy at 75 GHz (~ 0.6) as compared to that measured at lower frequencies (~ 1.6) suggests that b axis transport may be more complicated than anticipated. A qualitative comparison was made between c axis losses from an overdoped ultrahigh purity sample measured at 75 GHz and those from a previous generation (99.95 % purity) optimally doped crystal measured at 22 GHz. It was noted that the data scaled roughly as l / / 2 , thus suggesting that c axis transport may be relatively insensitive to differences in doping and impurity levels. 77 Chapter 5. Conclusions 78 The key factor limiting the resolution of all surface impedance measurements proved to be sample geometry. Errors in measuring the YBCO dimensions contributed less than 5% uncertainty to measurements of Rs, but the errors in measuring the PbSn calibration sample dimensions contributed approximately 10% uncertainty to the final results. A more accurate means of quantifying the dimensions of both samples is clearly necessary. The a axis surface impedance data were further analyzed within two theoretical frame-works; a simple 2-fluid-Drude model in which thermally excited quasiparticles behave like almost free electrons in a normal metal, and a more rigorous generalized BCS theory in which the low temperature quasiparticle lifetime is limited by isotropic elastic point scat-tering. In the first case, it was found that a simple expression of the form with a frequency independent scattering rate 1/r = 0.38K could match the relative magnitude of the data, however it could not describe the evolution of the temperature dependence of the measured losses from sublinear at 1 GHz to superlinear at 75 GHz. From a generalized BCS formalism the electrical conductivity was determined to be a thermal averaged sum of quasiparticle responses with a frequency dependent one body scattering rate 1/r(u>). A simple scattering rate formula was constructed from a sum of limiting forms for weak (Born) and strong (unitary) scattering from an isotropic elastic point scattering centre plus a constant (Drude) term. oxx(Q, T) nn(T)e2. -i ) m Q — I/T Chapter 5. Conclusions 79 u , , T u n i t a r y A 0 _ u i-/T{LO) — . + 1 Drude + 1 Born^— yju2 + runitary A c A ° The data were then subjected to a least squares fitting algorithm, and the scattering rate which best described the data had Y u n i t a r y = 0, Tj^rude = 0.33 K and TBorn = 0.68 K . The above form for 1/T(U) is expected to be valid over the frequency range 0 —> 400 GHz. However, a function of this form cannot be derived from the isotropic point scattering framework. At present, there is no known scattering mechanism in the HTSC cuprate materials which can generate a scattering rate function of the form shown above. The T —> 0 limit of the quasiparticle conductivity, oao, was also extrapolated from the raw surface impedance data. The results of this analysis do not agree with the predicted frequency independent value oao ~ 3 x 1 0 - 6 Qm. Rather, they indicate that the residual conductivity has a strong frequency dependence at frequencies < 10 GHz and a high frequency tail approximately a factor of 3 larger than the predicted value of o00. Bib l iography [1] H.K. Onnes. Leiden Comm., pages 120b, 122b, 124c, 1911. [2] J . Bardeen, L.N. Cooper, and J.R. Schreiffer. Phys. Rev., 108:1175, 1957. [3] M.K. Wu, J.R. Ashburn, C.J . Torng, P.H. Hor, R.L. Meng, L. Gao, Z.L. Huang, Y .Q. Wang, and C.W. Chu. Phys. Rev. Let, 58:908, 1987. [4] D.A. Bonn. Infrared Reflectance of Exotic Superconductors. Ph.D. thesis, McMaster University, 1988. [5] D.A. Wollman, D.J. Van Harlingen, J . Giapintzakis, and D.M. Ginsberg. Phys. Rev. Let, 74:797, 1995. [6] Z.X. Shen, D.S. Dessau, B.O. Wells, D.M. King, W.E. Spicer, A . J . Arko, D. Mar-shall, L.W. Lombardo, A. Kapitulnik, P. Dickinson, S. Doniach, J . DiCarlo, A . G . Loeser, and C.H. Park. [7] W.N. Hardy, D.A. Bonn, D.C. Morgan, R. Liang, and K. Zhang. Phys. Rev. Let, 70:3999, 1993. [8] D.J. Scalapino, E. Loh, and J.E. Hirsh. Phys. Rev. B, 34:8190, 1986. [9] D.N. Basov, R. Liang, D.A. Bonn, W.N. Hardy, B. Dabrowski, M. Quijada, D.B. Tanner, J .R Rice, D.M. Ginsberg, and T. Timusk. Phys. Rev. Let., 74:598, 1995. [10] D.A. Bonn and W.N. Hardy. Microwave surface impedance of high temperature superconductors. In D.M. Ginsberg, editor, Physical Properties of High Temperature Superconductors, Vol. 5. World Scientific, Singapore, 1996. [11] D.A. Bonn, R. Liang, T .M . Riseman, D.C. Morgan, K. Zhang, P. Dosanjh, T.L. Duty, A. MacFarlane, G.D. Morris, J.H. Brewer, W.N. Hardy, C. Kall in, and A . J . Berlinsky. Phys. Rev. B, 47:11314, 1993. [12] D.J. Griffiths. Introduction to Electrodynamics. Prentice Hall, New Jersey USA, 1989. [13] J-J Chang and D.J. Scalapino. Phys. Rev. B, 40:4299, 1989. [14] F. and H. London. Proc. Roy. Soc. (London), A149:71, 1935. 80 Bibliography- Si [15] The reader is cautioned not to confuse the generalized 2-fluid model with the Gorter-Casimir 2-fluid model. The former considers the penetration depth A to be a mea-sured quantity, whilst the latter ascribes a particular temperature dependence to A. [16] N.W. Ashcroft and N.D. Mermin. Solid State Physics. W.B. Saunders Company, Philedelphia USA, 1976. [17] P.J. Hirschfeld, W.O. Putikka, and D.J. Scalapino. Phys. Rev. B, 50:10250, 1994. [18] A .A. Abrikosov, L.P. Gorkov, and I.E. Dzyaloshiniski (trans. R. A. Silverman). Methods of Quantum Field Theory in Statistical Physics. Dover Publications, New York, 1963. [19] G.D. Mahan. Many Particle Physics. Plenum Press, New York, 1993. [20] P.J. Hirschfeld, P. Wolfle, and D. Einzel. Phys. Rev. B, 37:83, 1988. [21] I. Maggio-Aprile, Ch. Renner, A. Erb, E. Walker, and 0 . Fisher. Phys. Rev. Let, 75(2754), 1995. [22] P.A. Lee. Phys. Rev. Let, 71:1887, 1993. [23] J . Berlinsky and C. Kallin. Private Communication. [24] W.N. Hardy, S. Kamal, and D.A. Bonn. Magnetic penentration depth in cuprates: A short review of measurement techniques and results. In Bok et al., editor, The Gap Symmetry and Fluctuations in High-Tc Superconductors. Plenum Press, New York USA, 1998. [25] P.J. Hirschfeld, W.O. Puttika, and D.J. Scalapino. Phys. Rev. Let, 71:3705, 1993. [26] M.J. Graf, M. Palumbo, D. Ranier, and J.A. Sauls. Phys. Rev. B, 52:10 588, 1995. [27] S. Hensen, G. Muller, C T . Rieck, and K. Scharnberg. Phys. Rev. B, 56:6237, 1997. [28] A . J . Berlinsky, D.A. Bonn, R. Harris, and C. Kallin. (unpublished), 1999. [29] A. Durst and J . Berlinsky. Private Communication. [30] B. Gowe, P. Dosanjh, and D.A. Bonn. To be published. The development of the 2 GHz Nb split-ring resonator continues to this day. [31] A. Hosseini. M.Sc. thesis, University of British Columbia, 1997. This thesis focuses upon the 22 GHz resonator, however all of the same principles apply to the 13 GHz apparatus. Bibliography 82 [32] A. Erb, E. Walker, and R. Flukiger. Physica C, 245:245, 1995. [33] R. Liang, D.A. Bonn, and W.N. Hardy. Physica C, 304:105, 1998. [34] R. Liang, P. Dosanjh, D.A. Bonn, D.J. Baar, J.F. Carolan, and W.N. Hardy. Physica C, 195:51, 1992. [35] S. Kamal, R. Liang, A. Hosseini, D.A. Bonn, and W.N. Hardy. Phys. Rev. B, 58:8933, 1998. [36] Procured from A. Hosseini by Machiavellian means. [37] N. Klein. Materials Science Forum, 373:130, 1993. [38] R.A. Waldron. Theory of Guided Electromagnetic Waves. Van Nostrad, New York, 1969. [39] E. Hadge(ed.). The Microwave Engineers' Handbook. Horizon House Inc. Dedhan M A , USA, 1964. [40] S. Ramo, J.R. Whinnery, and T. VanDuzer. Fields and Waves in Communication Electronics. John Wiley and Sons, Toronto, 1994. [41] S. Kamal. (to be written). Ph.D. thesis, University of British Columbia, 1999. An extensive analysis of magnetic penetration depth using the 1 GHz resonator. [42] S. Sridhar and W.L. Kennedy. Rev. Sci. Instrum., 59:531, 1988. [43] D.L. Rubin, K. Green, J . Gruschus, J . Kirchgessner, D. Moffat, H. Padamsee, J . Sears, Q.S. Shu, L.F. Schneemeyer, and J.V. Waszczak. Phys. Rev. B, 38:6538, 1988. [44] WADD technical report. 60-56. [45] D. Kajfez and P. Guillon. Dielectric Resonators. Noble Publishing Corporation, Atlanta USA, 1989. [46] Phillipe Goy. ABmm 8-350 MVNA Operation Manual. 52 Rue Lhomond, 75005 Paris, France, 1994. [47] A. Hosseini, S. Kamal, D.A. Bonn, R. Liang, and W.N. Hardy. Phys. Rev. Let., 81:1298, 1998. [48] A. Hosseini, R. Harris, S. Kamal, P. Dosanjh, J . Preston, R. Liang, and W.N. Hardy. Phys. Rev. B, (accepted May 1999). Bibliography 83 [49] D.A. Bonn, P. Dosanjh, R. Liang, and W . N . Hardy. Phys. Rev. Let, 68:2390, 1992. [50] P. Turner, (to be written). M . Sc. thesis, University of British Columbia, 1999. [51] L. Taillefer, B. Lussier, R. Gagnon, K . Behnia, and H. Aubin. Phys. Rev. Let, 79:483, 1997. [52] T. Timusk and W. Tanner. Infrared reflectance of high temperature superconduc-tors. In D . M . Ginsberg, editor, Physical Properties of High Temperature Supercon-ductors, Vol. 1. World Scientific, Singapore, 1994. [53] C T . Rieck, D. Straub, and K . Scharnberg. unpublished, 1999. [54] D.A. Bonn, S. Kamal, A . Bonakdarpour, R. Liang, W . N . Hardy, C.C. Homes, D . N . Basov, and T. Timusk. Proc. of the 21st Low Temp. Phys. Conf, Czech J. Phys., 46:3195, 1996. [55] M . Franz, C. Kallin, A . J . Berlinsky, and M.I. Salkoa. Phys. Rev. B, 56:7882, 1997. [56] M . H . Hettler and P.J. Hirschfeld. Phys. Rev. B, 59:9606, 1999. [57] M . H . Hettler and P.J. Hirschfeld. unpublished, 1999. 


Citation Scheme:


Citations by CSL (citeproc-js)

Usage Statistics



Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            async >
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:


Related Items