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Studies of the flux flow resistivity in YBa₂Cu₃O₆ 95 by microwave techniques Morgan, David Craig 1993

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STUDIES OF THE FLUX FLOW RESISTIVITY IN YBa2Cu3O695 BYMICROWAVE TECHNIQUESByDavid Craig MorganM.Sc. (Physics), University of British ColumbiaB.Sc. (Physics), University of WaterlooA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESPHYSICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIADecenibar1 1993© David Craig Morgan, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature__________________________________Department of Ph’sLc5The University of British ColumbiaVancouver, CanadaDate D€cembr 21., ()c)3DE-6 (2/88)AbstractMeasurements of the microwave surface resistance at 5.4, 27 and 35 GHz on a highquality single crystal of YBa2Cu3O695 in magnetic fields up to 8T (applied parallel tothe c-axis) and at temperatures from 20K to ]00K are presented. The Coffey-Clemexpression for the surface impedance of a superconductor in the mixed state is used to fitthe data in terms of a pinning frequency and a free flux flow resistivity (thermal hoppingof vortices is not included since it is not important for YBa2Cu3O6•95 at microwavefrequencies). A temperature dependent pinning frequency is found that varies from ‘-. 20GHz at 20K to effectively zero by 80K. The flux flow resistivity is strongly temperaturedependent and consistent, down to at least 50K, with the Bardeen-Stephen expressionpff/p(T) H/H2(T) if an extrapolation of the linear normal state resistivity fromabove T is used for p(T). This suggests that the traditional picture of a vortex core asa cylinder of normal material may be valid in these materials. In contrast, the normalfluid quasiparticle scattering rate, as determined by zero field microwave measurementson the same single crystals1,drops precipitously below T and, thus, is probably not therelevant scattering rate for the charge carriers in the vortex core.‘D.A. Bonn et al. Phys. Rev. Lett., 68:2390, 1992.11Table of ContentsAbstract iiList of Figures viAcknowledgements ixNotation xi1 Introduction 12 Introduction to YBa2Cu3O7_5 72.1 Some basic features 72.2 Introduction to vortex motion in YBa2Cu3O7_ 163 Flux Flow in Conventional and High-Ta Sup erconductors 213.1 Basic Properties of the Mixed State 213.2 Transport Properties. 223.2.1 Simple treatment of the viscous flow of vortices . . 223.2.2 Viscous flow including the Hall Effect 253.3 The Bardeen-Stephen Model 283.4 Tinkham’s time dependent order parameter and TDGL . . 333.5 Experiments on Flux Flow 353.6 Theory of Flux Flow Resistivity in High-Ta Superconductors 453.6.1 Surface Impedance in the Mixed State 451113.6.2 Microscopic description of vortex motion in terms of core states3.7 Experimental work on YBa2Cu3O7_4 The Microwave Experiment4.1 The Resonant Cavity / Cavity4.2 The Split-Ring Resonator4.3 The 27 and 35 GHz Cavities4.4 Probe Design4.5 The Low Frequency Cryostat4.6 The 27 and 35 GHz Cryostat7676778390909696100114125136142515357576065697374Perturbation5 The Experimental Procedure5.1 The YBa2Cu3O695 single crystal sample5.2 5.4 GHz Measurements5.3 The 27 and 35 GHz measurements6 The Analysis6.1 Overview of the Data6.2 Extraction of the pure flux-flow resistivity6.2.1 Preliminary discussion6.2.2 Fitting with a field dependent and w6.2.3 Fitting with a frequency and field independent r and6.3 Scaling the Data7 Discussion and ConclusionsAppendicesivA The Surface Impedance of Metals and Superconductors 142B Rf Current Distribution in Samples 147Bibliography 150VList of Figures2.1 Temperature dependence of the a-b plane dc resistivity of YBa2Cu3O695 92.2 Temperature dependence of R8 at 3.9 GHz in zero field for YBaCu695 132.3 Temperature dependence of o at 3.9 GHz 142.4 Temperature dependence of the quasiparticle scattering rate 152.5 Temperature dependence of z()) close to T = 0 172.6 a-b plane resistivity of YBa2Cu3O7_5in a magnetic field 193.1 Vortex motion in response to a uniform, superfluid, transport current 233.2 I-V characteristics for Nb05Ta in a magnetic field 373.3 Normalized flux-flow resistivity vs. magnetic field for Nb05Ta5 383.4 Normalized flux-flow resistivity vs. field and reduced field: PbIn 403.5 Frequency dependent crossover from a flux-pinned to a flux-flow state 413.6 Microwave determination of pj in conventional superconductors 443.7 ii dependence of parameters in the Coffey-Clem model 493.8 R3 at 10 GHz in a magnetic field in a YBa2Cu3O7_5thin film 554.1 Split-ring resonator and field geometry 604.2 5.4 GHz split-ring resonator 624.3 Schematic of circuit used for the 5.4 GHz reflectance measurements . 644.4 27/35 GHz cylindrical cavity 664.5 Schematic of circuit used for the 27/35 GHz transmission measurements 684.6 Detail of the lower end of the probe 704.7 Overview of the apparatus used in the 5.4 GHz experiments 72vi5.1 Dependence of z\(1/Q) on z(f) at 100 K for the 5.4 GHz copper split-ring5. 785.2 Dependence of L\(1/Q) on z(f) for zero-loss sample in the 5.4 GHz coppersplit-ring resonator 80R3 in fields up to 4T at 5.4 GHzSystematic error in 35 GHz copper cylindrical cavityTemperature dependence of R3 in a magnetic field at 35 GHzTemperature dependence of R3 in a magnetic field at 27 GHz(5.4/frequency)” 4T for 5.4, 27 and 35 GHz 91Magnetic field dependence of R8 for all three frequencies 92Frequency dependence of R3 at low and high temperature 94Magnetic field dependence of R at 27 GHz 95Normalized R versus reduced field at 27 GHz 97Frequency fits at 20K 101Frequency fits at 30K 102Frequency fits at 40K 103Frequency fits at 50K 104Frequency fits at 60K 105Frequency fits at 70K 106Frequency fits at 78K 107Field dependence of the vortex viscosity at 82K 109Temperature dependence of the pinning frequency at 4 T 111Temperature dependence of the flux-flow resistivity at 4 T 112Effect of pinning on R3 at 27 GHz in a 4T field 1136.17 Field dependence of the flux-flow resistivity at all temperatures 115vii6J8. .. 1166.19. 1176.20. 1186.21. 1196.22. 1206.23. 1216.24 1226.25 1236.26 1266.27 1276.28 1286.29 1306.30 1316.31 1336.32 134B.1 Surface current distribution in uniformly magnetized ellipsoid 148as a function of frequency and field atas a function of frequency and field atas a function of frequency and field atas a function of frequency and field atFits 20KFits 30KFits 40KFits 50KFits as a function of frequency and field at 60KFits as a function of frequency and field at 70KFits as a function of frequency and field at 78KFits as a fullction of field at 82K and 27 GHzFits as a function of frequency and field at 70K, AL(0) p--’ 0Fits as a function of frequency and field at 20K, )L(0) ‘-‘ 0Comparison of different fitting procedures . .Field dependence ofp11/p(T)pff/p(T) as a function of the reduced field H/H2Predicted H2 using pff dataPredicted p(T) using PH dataVII’AcknowledgementsI would like to thank, first and foremost, my research supervisor, Walter Hardy, whothrough his enthusiasm for physics and expertise in so many areas has created a productive environment in which it is a pleasure to work. He has treated me with the utmostrespect and deference, and in return he has my deepest respect and admiration. He hasbeen directly involved in all phases of this work and without his help and focus, it wouldnot have been possible. Many thanks to Doug Bonn who has helped and encouraged methroughout my Ph.D. work. His unique blend of practical ability, communication skills,physical insight and integrity is a standard to which I will aspire for the rest of my career.A big thank-you to Kuan Zhang who actually built the high frequency cryostat whichwas used for many of the measurements described in this thesis. He has been a pleasureto work with always so friendly and helpful. Warm thanks also go to Pinder Dosanjh,Dave Baar, Rob Knobel, Saied Kamal, Ruixing Liang and Q.Y. Ma who have helpedmake the lab such a full and stimulating place to be. I owe a special debt of gratitude toRuixillg Liang who grew the spectacular YBa2Cu3O695 single crystals on which all themeasurements were made in this thesis.I’d like to thank all the members of my Ph.D. committee: Ian Affleck, Jim Carolan,Gordon Semenoff and Tom Tiedje. They have helped bring the important issues intofocus and have always treated me with kindness and consideration.Special thanks go to Catherine Kallin and John Berlinsky who have been involvedin this project since its inception. John Berlinsky, in particular, has always shown greatinterest and enthusiasm in this work. His ideas and suggestions have played an importantrole as have his warm words of encouragement during difficult times.ixLastly, I’d like to thank Mary Flesher whose love and support I not only cherish buthave been able to count on during the course of this Ph.D. project.xNotationh Plank’s constante charge of electrollkB Boltzmann’s constant4o flux quantumw angular frequency of rf fieldsm mass of the charge carriersm inertial mass associated with the vortexJ current densityJ normal fluid current densityJ8 superfluid current densityJT transport current densityE electric fieldB magnetic inductionH magnetic fieldH thermodynamic critical fieldH1 lower critical fieldH2 upper critical field1 mean free pathcoherence length in the c-direction= a-b plane coherence length\L London penetration depthxi) penetration depth in a dirty superconductor.A complex penetration depthS normal state skin depthT superconducting transition temperatureLi superconducting energy gapEF Fermi energyVp’ Fermi velocity,I2 superconducting order parametern3 density of superconducting electronsx normal fluid fractionZ3 surface impedanceR3 surface resistanceX surface reactanceQ qualityfactorfo resonant frequency5 complex conductivityo1 real part of the complex conductivity2 absolute value of the imaginary part of the conductivityn number of vortices per unit area,On resistivity associated transport due to normal carrierspff flux flow resistivityvortex viscosityf linear/angular pinning frequencyi pinning force constantr scattering timevL vortex velocityxiiv velocity of charge carriers in the core of the vortexVT velocity of charge carriers in superfluid transport currentv velocity of charge carriers ill the superfluid circulating the vortexP power dissipated per unit volumeW power dissipated per unit lengthUo barrier height of pinrnng potentiala coefficient of magnus force term in the vortex equation of motion°H Hall anglew, cyclotron frequencyxiiiChapter 1IntroductionIn the 1960’s, during dc transport measurements on conventional type-IT superconductorsin a magnetic field[1], it was observed that for current densities greater than a so-calleddepinning critical current density, a finite resistance to current flow developed. Indeed,above this critical current density, the I-V characteristics took on an ohmic character.This finite resistance was attributed to viscous flow of vortices and the slope of the I-Vcurve was used to define a flux flow resistivity, pff . The superconductor was thus actinglike a normal metal with a resistivity given by pff instead of the usual resistivity of a normal metal (which at the low temperatures associated with conventional superconductorswould be the temperature independent impurity limited value). The basic picture of theprocess was that the vortices move in response to a Lorentz force per unit length, J x,where J is the transport current density and the magnetic flux quantum that threadsthrough the vortex. The vortices remain immobile until this force exceeds the pinningforce (typically due to defects that act to pin the vortices at particular locations). Oncethey start to move, an electric field is induced along the direction of the applied currentcausing power dissipation and a finite resistance.It was postulated that the vortices are subject to a viscous retarding force per unitlength, F = —ivL, where vL is the vortex velocity and i a vortex viscosity. It wasfound from dc transport measurements on a number of superconducting alloys[1, 2, 3]that for low temperatures and magnetic fields the flux flow resistivity obeys a law of1Chapter 1. Introduction 2corresponding states— HPnalthough there there seems to be some confusion as to whether the H2 factor in thedenominator on the RHS should be H2(0) or H2(T). The flux-flow resistivity wasalso found to determine the energy dissipation at microwave frequencies[4, 5, 6, 7, 8]even for current densities much less than the depinning critical current densities at dc.The power absorbed was found to exhibit a crossover{4, 5] as a function of frequencyfrom a low dissipation regime at low frequency to a regime at high frequency wherethe superconductor effectively behaves like a metal with resistivity given by . Thecrossover was characterized by a so-called pinning frequency, f’,, the magnitude of whichwas typically in the 10 MHz range. Thus the microwave surface resistance could be usedto extract the flux flow resistivity and the results were also found to be consistent withequation 1.1 (with H2(O) in the denominator) for low temperatures and fields.Equation 1.1 also comes out of a theory developed by Bardeen and Stephen[9] whocalculated the dissipation in the clean limit (mean free path much greater than thecoherence length, 1 > ) and at zero temperature, to a vortex moving in response to aLorentz force per unit length, J x,and retarded by a viscous force per unit length,—llvL. They modelled the vortex in terms of two components: a totally normal core ofradius the coherence length, , and a transition region where the order parameter changesover a length scale from zero at the core boundary to its equilibrium value in the bulkof the superconductor. They also found that close to TH12p7, — H2(T)More rigorous calculations based on time dependent Ginzburg-Landau theory[10, 11, 12,13, 14, 15] were able to obtain solutions for Pif but only in certain limits. In the dirtylimit and for T << T and H << H2 an expression close to equation 1.1 was obtained[16].Chapter 1. Introduction 3One of the interesting aspects of the flux-flow resistivity is that it provides information about the scattering rate of the normal quasiparticles in the core of the vortex.In conventional superconductors, the quasiparticle scattering rate is temperature independent and the same in the normal core as it is in the bulk superconductor. In thehigh-Ta superconductor, YBa2Cu3O695 , the normal state resistivity is strikingly linearover a large temperature range above T until the pronounced rounding that occurs dueto two dimensional fluctuations near the superconducting transition[1 7]. Thus the scattering rate of the charge carriers is temperature dependent at T and not temperatureindependent as it is for conventional superconductors. Moreover, the scattering rate forthe quasiparticles in the normal fluid in zero field has been observed to drop rapidly asa function of temperature below T [18, 19]. However, it is not obvious whether or notthis scattering rate is the relevant one for the charge carriers in the core of the vortex.If we retain the picture of a normal core, it is tempting to extrapolate the linearly decreasing scattering rate of the charge carriers from above T down to low temperatures.The Bardeen-Stephen[9] picture of moving vortices has a contribution to the dissipationfrom the transition region outside the completely normal core equal to the one from thenormal core itself. This raises the possibility that both of the above scattering ratesmight be relevant in determining the flux flow resistivity. In addition, the Hall angle, OH,in the mixed state is intimately bound up with the lifetime of the quasiparticles in thecores[9, 20]. Due to strong pinning, OH in YBa2Cu3O695 has not been measured belowabout 70K. Thus a possible effect on the measurement of p due to a non-negligible Halleffect must be kept in mind.The situation is made more complicated by the possibility that the picture of a cylinder of normal material may not be a very good approximation for vortex cores in high-Tasuperconductors. The basis for the idea of a normal core in conventional superconductors(apart from the fact that the order parameter goes to zero at the centre of the core) wasChapter 1. Introduction 4the demonstration that there is a set of low-lying closely spaced energy levels localizedin the vortex core that is comparable to a cylinder of normal metal of radius [21]. Thelevel spacing is of order/2EF where Li is the energy gap and EF the Fermi energy.For low-Ta superconductors, this spacing is lmK whereas for high-Ta materials it isestimated to be 10K or more[22, 23]. Thus there is the possibility of the discrete natureof the energy level spectrum playing a role.Another layer of complexity has become apparent with the growing body of evidencefor an unconventional pairing state in the high-Ta superconductors[24]. A possible candidate is d-wave pairing and the effect of such a state, whose gap function has line nodeson the Fermi surface, on the models of vortex motion is unclear. It is thus important toretain a broad view of flux flow resistivity measurements since it is unlikely, given thecurrent state of uncertainty as to the underlying mechanisms involved, that we will beable to arrive at a definitive understanding of flux flow.The microwave technique is especially helpful in trying to measure the flux flow resistivity of YBa2Cu3O695 because of the very strong pinnrng of vortices in this material.Extremely high current densities are required to exceed the depinning critical currentdensity. Practically speaking, this limits the dc technique to only a 10—15K temperature range below T and, even here, very large current densities (‘-- 106A/cm2)arerequired [25]. Also, to measure pS,-, directly by applying a magnetic field greater thanH2 is only possible close to T because of the large H2 ‘s in YBa2Cu3O695 (H2 (0) isestimated to be ‘—‘120T[26}). Such strong pinning means that the pinning or crossoverfrequency might be much higher than that observed for conventional superconductors.We cannot assume, therefore, that frequencies in the range 5—35 GHz will be in the highfrequency, flux flow limit. In addition, due to the higher temperature scale associatedwith high-Ta materials, thermally activated vortex motion[27] plays a more prominentrole than in the vortex dynamics of conventional superconductors, although later, we willChapter 1. Introduction 5argue that for YBa2Cu3O695 at our frequencies, the dynamics are not contaminated bythis latter effect.We will rely mainly on a model derived by Coffey and Clem[28, 29] (but in thelimit of no thermally activated flux motion) to fit our microwave data. In this limit theirexpression for the surface impedance is just a slight generalization of the one used[4, 5] toextract pff in the microwave experiments on conventional superconductors. It should bemade clear from the outset that the use of this model in no way assumes a temperatureor field dependence of the flux flow resistivity. It is a vehicle to get from the surfaceresistance data to the flux flow resistivity: we use it to fit the data at three differentfrequencies (5.4, 27 and 35 GHz) in terms of a vortex viscosity and a pinning frequency.We are principally interested in the viscosity since this gives pff directly; the fit to theprnnrng frequency removes pinning effects that otherwise might be attributed to the fieldor temperature dependence of pj . With only three frequencies, we are not able tounequivocally demonstrate the existence of the high frequency, flux flow limit. However,fitting to the Coffey-Clem model shows that the data is consistent with a temperaturedependent pinning frequency that is --20 GHz at 20K and that decreases to zero by--80K. Thus we believe that the vortex viscosity and flux flow resistivity that we extractfrom the surface resistance is independent of the effects of pinning. We find a flux flowresistivity that decreases rapidly below T and is reasonably well described down to about50K byfiff— H13p(T) — H2(T)where p(T) is given by the extrapolation of the linear dc resistivity from above T andH2 (T) is given by an estimate based on the measured slope of H2 (T) near T . Below50K, the extracted is higher than what we would have expected from the aboveequation.Chapter 1. Introduction 6To our knowledge, this is one of the oniy measurements of the flux flow resistivitydown to lower temperatures in high quality single crystals of YBa2Cu3O695 We willdiscuss some of the possible implications of these measurements but must ultimatelyleave the questions open until there is a better understanding of vortex motion in thehigh-Ta superconductors.Chapter 2Introduction to YBa2Cu3O7_In this chapter, we will introduce some of the essential features of high-Ta superconductors, specificallyYBa2Cu3O7_8.It is lot meant as a comprehensive review of the currentstate of research in the field. Rather, it is meant to set the context for the main topic ofthis thesis: the free flux-flow resistivity in the mixed state.Before embarking on this discussion, we should briefly mention a matter of convention. The units used in this thesis will be MKS. However, there has been a strongtradition in the literature of using cgs to discuss theory and experiments concerning superconductivity. In particular, the terminology of H, H1and H2 as referring to thecritical fields instead of B, etc. is deeply ingrained. To adhere to this convention whilestill maintaining consistency with MKS units, we adopt the following compromise: in ageneral discussion of, say, the upper critical field we use, H2 , but in formulae we insertthe appropriate factors of [to; the reader can take for granted that, in this thesis, B canalways be obtained from H simply by multiplying by p.2.1 Some basic featuresOne of the most basic features of high-Ta superconductors is their anisotropy: conductionoccurs primarily in two dimensional Cu02 layers which are weakly coupled together. Wecan imagine the superconductor as a stack of such layers in what we label, conventionally,as the c-direction; the Cu02 layers lie correspondingly in the a-b plane. Figure 2.1 showsthe dc electrical resistivity in the a-b plane as a function of temperature measured by7Chapter 2. Introduction to YBa2Cu3O7_6 8Baar et al[17J on high quality single crystals of YBa2Cu3O695 made at U.B.C. by Lianget al[30]. Pc has also been measured on these crystals and shows the same T but has anoverall magnitude about fifty times greater than Pab aild exhibits a slight upturn beforeTc - this thesis deals almost exclusively with a-b plane currents and so c-axis transportwill not be discussed further. The superconducting transition at —93K is dramaticallyevident T ‘s of up to 93.4K have been observed in these crystals.There are two other features of interest. First is the striking linearity of the resistivity in the normal state above about 120K and that it extrapolates to very close tozero at zero temperature. The Debye temperature in YBa2Cu3O7_is estimated to be400K[31] and so a linear resistivity in this temperature range is highly anomalous.Indeed, this linear resistivity is a powerful constraint on theories that try to explainhigh-Ta superconductors. The other interesting feature of the resistive transition is thepronounced rounding starting at about 120K. This rounding has been interpreted interms of fluctuations associated with the two-dimensional nature of the material. Thelarge temperature range above T where these fluctuation effects are observable is characteristic of the Cu02 superconductors. The usual fluctuations associated with deviationsfrom mean field theory near the transition would only be expected to play a role within‘1KofT.Estimates of the coherence lengths in YBa2Cu3O7_5can be made by magnetizationmeasurements close to T. Weip et al[26] performed such measurements on single crystalsof YBa2Cu3O7_5down to about 8K below the transition and found their data consistentwith a linear dependence of H2 on temperature with uodH2/dT ITc being -1.9T/K and-10.5T/K for the field oriented parallel and perpendicular to the c-axis respectively. Theyuse the Werthamer-Helfand-Hohenberg formula{32, 33]H2(0)=0.7 (8Hc2)T (2.1)9C)C-0 100 200 300T(K)Figure 2.1: Temperature dependence of the a-b plane dc resistivity of YBa2Cu3O695Inset: a close-up of the transition. This measurement was made by Baar et al[17] 011 theU.B.C. single crystals (Liang et al[30]).Chapter 2. Introduction to YBa2Cu3O7_ 910090 100 110300250200150100500T (K120Chapter 2. Introduction to YBa2Cu3O7_ 10to obtain estimates of iuoHc2c(0) = 122T (field parallel to the c-axis) and it0H2(0) =674T (field parallel to the a-b plane). An empirical formula consistent with0dH2/dT ITcbeing -1.9T/K and H2(0) being 125T is given by(1 _t2)0H2(T)= 125(1 +t2)’/ ‘ (2.2)(t = T/T) and we will use it for the temperature dependence of0H2(T) throughoutthis thesis. Using the formulae[34}[LOHc2c= ; = (2.3)aban a-b plane coherence length, tab, of 16.4 A and a c-axis coherence length, , of 3.OA canbe found. Both and ab are considerably shorter than the coherence lengths associatedwith conventional superconductors. In this thesis, we are concerned principally with tab,and so we set = ab for convenience.The zero temperature value of the London penetration depth, )..L(0), is also of greatinterest. However, considerable debate still exists about its correct value. Callin andBerlinsky[24] have reviewed and evaluated the experiments that claim to measure thisquantity and we follow their discussion here. Typically, microwave experiments measureonly the change in the penetration depth and not its absolute value. Nevertheless, Pondet al[35] used a transmission line resonator consisting of a 2000A thick YBa2Cu3O7_layer and found )L(0) = 13001; however, they have had difficulty in reproducing themeasurement. Muon spin resonance is a bulk probe that measures the distribution ofmagnetic fields in the mixed state. This can be related to the penetration depth viaGinzburg-Landau theory. Estimates of .L(0) from this technique on the U.B.C. singlecrystals give 1400A although there is still some field and sample dependence in the data.Infrared measurements can also be used to obtain a value for \L(0). Since the complexChapter 2. Introduction to YBa2Cu3O7_5 11conductivity for frequencies well below the gap is given by— i2 (2.4)/10 ALAL can be obtained by measuring the coefficient of the 1/w divergence in the imaginarypart of the conductivity as w— 0. It is also possible to measure the missing oscillatorstrength in a1 (cü) from above to below T to determine AL [34]. These two methods werefound to give the same value for AL. Basov et al[36] foulld a value of 1440A for theYBa2Cu3O695 single crystals from U.B.C. . Measurements of the lower critical field canalso be used to obtain a value for AL through the expression42 In () . (2.5)Umezawa et al[37] obtained values in the range 900—950A while Liang et al[38] obtained800)1 (on U.B.C. crystals). The significant difference between these numbers and thevalues obtained using other techniques has not yet been explained.Clearly YBa2Cu3O7_is an extreme type-TI superconductor with i’ = AL/C ‘ 70—80and therefore a system in which a local treatment of the electrodynamics makes sense.Thus, the two London equationsE = t0 A2 (2.6)andH = —AL2V x J (2.7)might be expected to do a reasonable job of describing the electrodynamics. We shouldalso note that because the coherence lengths are so small, we are almost certainly in theclean limit (1 > ).Given the linear nature of the temperature dependence of the scattering rate of thecharge carriers above T , it is natural to wonder about the temperature dependence ofChapter 2. Introduction to YBa2Cu3O7_8 12the quasiparticle scattering rate below T . The zero-field microwave surface resistancemeasured by Bonn et al[18, 19] on the U.B.C. single crystals of YBa2Cu3O695 addressesthis issue rather directly. Figure 2.2 shows R3 as a function of temperature on a semi-logscale. (Incidentally, the sharpness of the transition as well as the magnitude of the dropof the surface resistance just below T is testimony to the quality of the U.B.C. singlecrystals). To analyze the data, they used a generalized two-fluid picture with the complexconductivity givell by equation 2.4. The surface impedance is given by (see Appendix A)= (iILOLL)h/2 (2.8)When o <<cr2 (which is a good approximation until very close to T ), we obtainR8 = 1 2 XL3. (2.9)Taking‘L from the penetration depth measurements of Hardy et al on the same singlecrystals, they obtain a1 as a function of temperature and this is shown in figure 2.3.Assuming a Drude type form for the conductivity, theno cx x(T) T(T) (2.10)if cr << 1 where x(T) is the normal fluid fraction and T(T) is the normal quasiparticlelifetime. In this two-fluid approach x(T) = 1— )‘L2(0)/)(T) and so they extract-r(T). This is given in figure 2.4, which shows a quasiparticle scattering rate that dropsprecipitously below T . The characteristic bump in the temperature dependence of thesurface resistance is understood as follows: the initial drop below T is due to the onsetof enhanced screening as‘L decreases very quickly; below 77K, the increase in a1 dueto the rapidly increasing r wins out as the temperature depeildence of ‘\L slows down sothat R8 starts increasing with decreasing temperature; at about 40K, we reach a peakin the conductivity below which the decreasing normal fluid fraction takes over as l/rChapter 2. Introduction to YBa2Cu3O7_ 131001 01-2a)Cl)io1120 40 60 80 100T(K)Figure 2.2: Temperature dependence of R3 at 3.9 GHz in zero field for YBa2Cu3O695This was measured by Bonn[39] on U.B.C. single crystals (Liang et al[30]).80000cP00609000Chapter 2. Introduction to YBa2Cu3O7_5 143e+072e+07le+07Oe+000T(K)00000080 100Figure 2.3: Temperature dependence of o at 3.9 GHz. This is obtained by Bonn[39]from the surface resistance using equation 2.9. The spike near T is thought to be dueto slight broadening of the transition (the transition width is slightly wider than wouldbe expected on the basis of the width of the specific heat jump at T ). It is sampledependent and not thought to be part of the intrinsic physics.0000o000 00 000000000000000020 40 6000Chapter 2. Introduction to YBa2Cu3O7_5 15001013 0000000 000 00000 0012 010 0 00000”cpO 000010110 20 40 60 80 100T(K)Figure 2.4: Temperature dependence of the quasiparticle scattering rate[39]. The spikeat T is again due to sample dependent broadening of the transition. Below about 40 K,the scattering runs into an impurity limit and levels off.Chapter 2. Introduction to YBa2Cu3O7_5 16begins to run into an impurity limit, and thus R3 starts decreasing again. It should benoted that the broad peak in the real part of the conductivity has nothing to do with acoherence peak. The rapidly decreasing quasiparticle scattering rate with an onset at Tsuggests the dominance of electron-electron scattering above T which is rapidly quenchedonce the charge carriers freeze out into the superconducting condensate. Certainly, anelectron-phonon picture for the scattering seems unlikely.Another aspect of the R5 temperature dependence which has provoked intense interestis its rather slow variation as T —* 0 (approximately linear). In a conventional BCS swave picture, an exponential temperature dependence is expected. Also suggestive ofunconventional superconductivity in YBa2Cu3O7_is the linear temperature dependenceof zX(T) at low temperatures measured by Hardy et al[40] (they measure AA and takea value of )L(0) from the literature to obtain )L(T)). This is shown in figure 2.5. Sucha linear temperature dependence is predicted by calculations based on clean d-wavesuperconductors. For d-wave pairing, the gap function could be of the form= L (cos k — cos k) (2.11)and this has line nodes on a cylindrical Fermi surface. Regions of the Fermi surface wherethe gap vanishes imply sets of low lying energy levels which have dramatic effects on thelow temperature variation of a wide variety of properties. There is a growing bodyof evidence that is consistent with d-wave or at least some sort of effectively gaplesssuperconductivity. For a more complete review of the evidence for and against a d-wavemodel of the high-Ta superconductors, the reader should consult Kallin et al[24].2.2 Introduction to vortex motion in YBa2Cu3O7_5One of the principal differences in the vortex dynamics between high-Ta and conventionalsuperconductors is the more prominent role of thermal effects. This is a vast subjectChapter 2. Introduction to YBa2Cu3O7_ 17200150Cl)E0•1-’Cl)D)C50020 40T(K)Figure 2.5: Temperature dependence of /A) =— )(1.3K) close to T = 0 as measured by Hardy et al[40]. This linear behaviour is strikingly different from the exponentialdecrease observed for conventional superconductors.A00000000 10 30Chapter 2. Introduction to YBa2Cu3O7_ 18about which we give oniy a very general description, our main concern being to motivatethe picture of vortex dynamics which we use in this thesis.Paistra et al[27j have performed detailed studies of the dc resistivity of YBa2Cu3O7_5single crystals (T 88K) in magnetic fields both perpendicular and parallel to the a-bplane. Figure 2.6 shows their measurements of Pdc as a function of T on a semi-log plotwith the applied field perpendicular to the a-b plane. It is immediately apparent thateven in fields up to 12T, the resistivity decreases very rapidly below T . For example,in a field of iT, the resistivity has decreased at least five orders of magnitude by 80K(t = T/T = 0.91). They allalyze their data in terms of the Anderson-Kim flux creeptheory[41, 421. In this scenario, vortices are pinned at the minima of a periodic pinningpotential. They can move in response to the Lorentz force of the transport current eventhough the applied current is less than the depinning critical current due to thermalactivation over the barrier height, Uo, between the neighbouring pinning sites. Based onthis type of analysis, they find that the barrier heights or activation energies have alreadyincreased to approximately 500K in YBa2Cu3O7_5by the time t has dropped to 0.93(whereas activation energies as high as this are only found at temperatures, T < 50K,inBi2SrCaCuO8).It is natural to postulate the existence of a boundary in H—T space above whichthe vortices are thermally depinned and flow freely for any transport current aild belowwhich a non-zero depinning critical current density still exists and must be exceeded toachieve flux flow. Such irreversibility or depinning lines have indeed been observed in. bothYBa2Cu3O7_andBi2SrCaCuO8[43]. There is considerable debate as to whether or notthese are in fact ‘melting’ lines and representative of thermodynamic phase transitionsfrom a vortex solid to a vortex liquid[44]. This is an interesting and complex subjectbut a discussion of it is beyond the scope of this thesis. The important point for ourpurposes is that while forBi2SrCaCuOg this line (at low frequencies) is at ‘—i 30K forChapter 2. Introduction to YBa2Cu3O7_ 19i0T(K)100Figure 2.6: a-b plane resistivity of YBa2Cu3O7_8in a magnetic field. This is from thepaper by Paistra et al[27]. Also shown is the a-b plane resistivity forBi2SrCaCuO8which decreases much more slowly than in YBa2Cu3O7_510210I1020 20 40 60 80Chapter 2. Introduction to YBa2Cu3O7_ 200 < H < lOT, for YBa2Cu3O7_it is within a few degrees of T [43]. Moreover, this lineis frequency dependent[27] and shifts to higher fields aild temperatures as the frequencyis increased. Paistra et al[27] remark that even forBi2SrCaCuO8, if the frequency isgreater than 1GHz, thermally activated flux motion only plays a role very close to T.Within this picture, the reason for this frequency dependence is that at high enoughfrequencies the vortices never get a chance to wander far enough away from the pinningminima during the rf cycle to thermally hop to the next pinning site. In fact, at suchhigh frequencies we must modify our picture[4, 5] to include explicitly a vortex viscosity,, and a pinning force constant, is,, in our description of the dynamics. Together, theydefine another frequency scale, w = called the pinning frequency. We defer adiscussion of this until Chapter 3.We conclude this introductory discussion of YBa2Cu3O7_5in the mixed state bybriefly mentioning the work of Welp et al[45] who analyzed their magnetization data nearT on single crystals of YBa2Cu3O7_5in terms of thermodynamic fluctuations. They findthat in the temperature region corresponding to the upper half of the resistive transitionin a magiletic field, their data scales according to Ginzburg—Landau theory for a 3-Dsuperconductor. Thus, they argue that the characteristic shape of these dc transitionsin a magnetic field as well as the apparent lack of field dependence on the the onset ofsuperconductivity is a result of fluctuations. This must be kept in mind when we cometo look at the surface resistance in a magnetic field close to T or H2Chapter 3Flux Flow in Conventional and High-Ta SuperconductorsIn this chapter, we will review the current understanding of flux-flow both in conventionaland high-Ta superconductors. We will first describe the basic phenomenology of flux-flowand then summarize the theories that have been suggested to model the dissipation andHall effect in conventional superconductors. These range from the semi-phenomenologicalapproaches of Bardeen and Stephen[9] and Tinkham[46] to the more rigorous methodsbased on time dependent Ginzburg-Landau (TDGL) theory[lO, 11, 12, 13, 14, 15]. Wewill then review some of the most important experiments aimed at understanding flux-flow. These experiments basically fall into two categories: dc and high frequency. Inthe second part of the chapter, we will summarize some of the work done on high-Tasuperconductors, both theory and experiment.3.1 Basic Properties of the Mixed StateIn a type-Il superconductor, the magnetic flux, instead of being screened out until thethermodynamic critical field, H, has been reached, starts to penetrate in the form ofvortices once the applied field has exceeded a lower critical field, H1. Each vortex isthreaded by only one quantum of flux, o = h/2e. If the number of vortices per unitarea is given by n, then the average magnetic field is given by B = n40. As the appliedfield is increased, the superconductor is penetrated by greater and greater numbers ofvortices until at an applied field H2 > H, the superconductivity is quenched and atransition into the normal state occurs.21Chapter 3. Flux Flow in Conventional and High-Ta Superconductors 22A vortex consists of a core of radius equal to the coherence length, , in which the orderparameter (or equivalently, the density of superconducting electrons) decreases to zero atthe centre from its bulk, equilibrium value at the perimeter. The microscopic magneticfield is maximum at the centre of the vortex and decays to zero outside the core due tocirculating supercurrents that screen out the field over a length scale characterized bythe penetration depth, )‘L. For typical type-TI superconductors, )‘L >> , and so a typicalvortex has an essentially normal inner core where the density of the superconductingelectrons goes to zero and a much larger region outside this core where the supercurrentsand the field decays. The concept of a ‘normal’ core was put on a more rigorous footingby Caroli, de Gennes and Matricon[21] who found that although the order parameter isstrictly zero only at the centre of the vortex, there is a sea of low lying energy levels withspacing of order L2/EF (f.1 1 mK for conventional superconductors) centred on the axisof the vortex that acts essentially like a cylinder of normal material with radius .3.2 Transport Properties3.2.1 Simple treatment of the viscous flow of vorticesWe now consider such a lattice in the presence of a uniform, superfluid, transport curreiltJ= as shown in figure 3.1. In the presence of the transport current, each vortex issubjected to a Loreritz force per unit length, J x o which in the absence of otherforces causes them to move in a direction perpendicular to the direction of the transportcurrent [34]. In a real material, there may be defects and other structures which teild topin the vortices and stop them from moving in response to the Lorentz force. For themoment, we consider an ideal material where there is no pinning and the vortices are thustotally free to move. The moving vortex gives rise to a time dependent magnetic field ona microscopic scale and so we might expect an induced electric field in the direction ofChapter 3. Flux Flow in Conventional and High-Ta Superconductors 23nevT= nevTFigure 3.1: Vortex motion in response to a uniform, superfluid, transport current. Theupper figure shows the response considering only the conventional Lorentz force. Thelower figure includes the magnus force which is thought to be responsible for the Halleffect.VIVs>>>>>>>>>>V3XEChapter 3. Flux Flow in Conventional and High-Ta Superconductors 24the transport current coming from the Maxwell equation V x E —8B/ãt. It has beenargued[47], however, that in a steady state experiment with vortices moving across thesample (perpendicular to the transport current) and leaving on one side while enteringon the opposite side at the same rate, there is no net change of flux through a circuitconsisting of the superconductor and a voltmeter, and thus there should be no inducedelectric field. This question was resolved by Josephson[48] who found that the electricfield created by moving vortices with velocity vL is indeedE=—VLXB (3.1)as we might guess based on a simple induction mechanism. The gist of this argumentcan be gleaned by considering a closed loop, C, consisting of a segment, C2, that residesentirely in a purely superconducting region and a segment, C1, that completes the loopin a superconducting regioll in the mixed state. From Faraday’s law[49] we have thatE.d1=—,cvLXB.d1 (3.2)where E is the electric field around the ioop and vL is the velocity at which the magneticflux crosses the boundary. No flux can cross the boundary, C2, in the superconductingregion and E is everywhere zero inside the pure superconductor; therefore, equation 3.2becomesJE.dl= Jc.’L xBd1. (3.3)If we assume that the field distribution at a point in the mixed state is only determinedby the flux lines etc. in the immediate vicinity, then equation 3.3 holds for any path Ci,not just one completed through a purely superconducting region. The differential formof equation 3.3 is given by equation 3.1.Since there is an electric field parallel to the transport current, there is energy dissipation per unit volume given byP=EJ (3.4)Chapter 3. Flux Flow in Conventional and High-Ta Superconductors 25The situation is similar to a normal material where a voltage, V, is developed along thepath of the current, I, and we get energy dissipation VI. This energy dissipation can betaken into account in a phenomenological fashion by introducing the parameter , calledthe vortex viscosity. We imagine that the energy dissipation comes from the work doneagainst a viscous force per unit length of vortex, ?lvL. Then in the steady state with nopinning, we haveJ0 = ?7VJ . (3.5)Using E = VLB, we getE 0BfJff7= (3.6)where we have defined to be the flux-flow resistivity— the ratio of the induced electricfield to the applied transport current. The work done per unit time per unit length bythe viscous force to dissipate energy can be writtenW=—FvL=vL2. (3.7)Equivalently, we can write the power per unit volume asP=E•J=pffJ2=n(ivL) (3.8)Since n is the density of vortices per unit area, we find again that the energy dissipatedper unit length is given by 7v2.3.2.2 Viscous flow including the Hall EffectAlthough the treatment given in section 3.2.1 is a good starting point for understandingflux flow and, indeed has been used extensively to model experiments, it ignores the Halleffect where there is a component of vortex velocity not strictly perpendicular to theapplied superfluid transport current. It has been argued, originally by de Gennes andChapter 3. Flux Flow in Conventional and High-Ta Superconductors 26Matricon[50j, and subsequently by Nozières and Vinen[20}, that this motion is the resultof the Magnus force similar to that felt by vortices in a classical uncharged fluid. In thisscenario, the force exerted on the vortex is not simply J x 4o =n8evT x o butn3 e (VT — vL) x o (3.9)where it is now the relative velocity of the vortex with respect to the applied superfluidvelocity that determines the Lorentz force. We can clearly see from this expression thata vortex moving in a direction perpendicular to the transport current will see a force inthe direction of the transport current. The force balance equation for the vortex becomesf+n8e(vT—vL) x o =0 (3.10)where f is the frictional drag term. This equation is appealing because in the limit ofa pure superconductor, f = 0 and so VT = vL i.e. the vortices move along with thesuperfluid as in superfluid helium II. However, writing f = —7lvL leads to significant disagreement with experimerit[51]. It was argued by Bardeen[52} that the term r evL X oshould be dropped while Nozières and Vinen assert that it is the form of the frictionaldrag force that must be modified. This question has never been elltirely resolved (indeed,a complete theory of flux-flow valid in all limits of physical interest has yet to be developed), but as a reasonable compromise we can write a force balance equation in termsof phenomenological parameters to be determined either by experiment or subsequenttheory [53]:71VL—aVLXz=---4oJXz, (3.11)where we have now taken o = —o in order to be consistent with figure 3.1. J =as before is the superfluid transport current density (in the x-directiori),n3eVT, and a isa parameter that together with determines the Hall angle (see figure 3.1). The avL Xterm comes from the second term on the LHS of equation 3.10, with its coefficient nowChapter 3. Flux Flow in Conventional and High-Ta Superconductors 27given by a. The 0J x term (conventionally thought of as the Lorentz force) is preciselythe first term on the LHS of equation 3.10. Since we now have a component of the vortexviscosity in both the x and the y direction, the derivation of the longitudinal flux-flowresistivity is less trivial. Writing down the x and y components of the equation, we have7iVLa — avL, = 0 (3.12)7lvLy+avLx= FoJ (3.13)Using equation 3.1 for the induced electric field, we haveE=vLB ; E=—vLB (3.14)and sinceEv Ea VLy B (3.15)andtanOHI= E = (3.16)VLy(OH is the Hall angle) we find that0B(i +(3.17)tanOH = . (3.18)To first order in a/a, we recover equation 3.6, the previous result for the flux-flow resistivity. We learn from this that for small Hall angle, we get the same flux-flow resistivitythat we would get in the absence of Hall fields.There have been two main approaches’ to modelling this power dissipation in termsof parameters describing the structure of the vortex. The most widely known approach is‘There is also the model proposed by Clem[54] who showed that dissipation can arise from irreversibleentropy flow. His ideas were used to explain the existence of a flux flow resistivity minimum as a functionof temperature that was observed in certain high , alloys such as Ti-V[1].Chapter 3. Flux Flow in Conventional and High-Ta Superconductors 28the one due to Bardeen and Stephen. The other involves the more rigorous calculationsbased on time dependent Ginzburg Landau theory originally inspired by Tinkham’s ideason the time relaxation of the order parameter. We consider first the Bardeen-Stephentheory.3.3 The Bardeen-Stephen ModelGiven the picture of a vortex core as a cylinder of normal material, a natural approachto modelling the dissipation due to a moving vortex would be to somehow relate it tojoule heating of the normal electrons in the core. This is in essence the approach takenby Bardeen and Stephen in their solution of the problem[9}. They model the vortex interms of a totally normal core (superconducting order parameter is zero) of radius thecoherence length, , and a transition region outside the normal core where the orderparameter goes from zero to its equilibrium value in the bulk. Their theory is derived atzero temperature and in the clean limit where the mean free path, 1, is greater than thecoherence length.Following Tinkham[34], we can quickly reproduce the calculation of the dissipationdue to the electrons in the core (in the slightly more simplified case that, outside thecore, the material is completely superconducting). We imagine a vortex at the origin (itsmagnetic flux directed in the —z-direction) with a velocity, vL, in the y-direction (seeupper diagram in figure 3.1). We use the first London equation, equation 2.6, to relatethe electric field outside the normal core to the circulating superfluid current density:E = (m:s) (3.19)where we have used that J(r — vLt) =n3ev(r — VLt). We have by the chain rule that= —(vL . V) v, (3.20)Chapter 3. Flux Flow in Conventional and High-Ta Superconductors 29and so we get thatE=—vLV () . (3.21)For >> (,>> 1), we have that [341v = —0 , (3.22)m rwhere 0 is measured from the x-direction and 0 is a unit vector in the direction ofincreasing 0. Substituting this into equation 3.21 and recalling that vj. is in the ydirectionE =— (VLo) 8 (ö =— Vl(sin0 +cos0 ). (3.23)2r öy \rj 2KrRequiring continuity of the tangential component of E at the boundary of the normalcore gives a uniform electric field in the coreE *. (3.24)We can see this because at r = and 0 = ir/2, 3r/2, the electric field is given by theabove expression, whereas, in the core, we have both V E = 0 and V x E = 0, andso a constant field given by equation 3.24 certainly satisfies Maxwell’s equations andalso satisfies the boundary condition on the tangential componeilt of E. Therefore, itis the unique solution for the electric field in the core. Of course, discontinuity of thenormal component of E implies a surface charge density at the core boundary. This isan unrealistic feature of the model due to the simplified view being taken of the vortexcore; ill reality, this charge density would be smeared out.We can now easily calculate the energy dissipation per unit length of vortex in thecore. Since the core is normal with a conductivity, o, we have J = uE and so thedissipation per unit length is given byw=2JE=2uE2=’l0. (3.25)Chapter 3. Flux Flow in Conventional and High-Ta Superconductors 30This calculation of the dissipation relies on local equilibrium of the electrons in the normalcore with the lattice. It was remarked by Nozières and Vinen[20] that this does not makesense for the situation where the mean free path is bigger than the core size (1 > ).However, Bardeen et al[9] argued that even if an individual electron does not suffer acollision in the core, the friction force involves an average over all electrons and so thepicture still makes sense on the average.It turns out that an equal amount of dissipation occurs due to normal currents in thetransition region outside the core (where the order parameter rises back up to its bulk,equilibrium value), but this is considerably more complicated to calculate[9]. The totaldissipation is thereforeW = ° . (3.26)2irEquating this to W = and using= 27r (3.27)we get= 00H2 (328)PmIn terms of the flux-flow resistivity, this becomes=--. (3.29)p. H2This is an appealing result because H/H2 is roughly the fraction of material in thenormal cores of the vortices, and so it is as if the curreilt flows right through them. Infact, we can calculate the velocity of the electrons in the cores. If we consider the coresto be normal material with an electron scattering time, r, thenv = (er/m)E . (3.30)Chapter 3. Flux Flow in Conventional and High-Ta Superconductors 31Equating the viscous force with the Lorentz force on the vortices, we get‘7 Vj = J F0 = Ti e VT (h/2 e) = (n h/2) VT . (3.31)Combining equations 3.30, 3.24 and 3.31 we findV VT . (3.32)In other words, the normal current density in the cores equals the superfluid transportcurrent density and so we see that in this picture the current does indeed flow rightthrough the moving cores. It is important to realize that the motion of the cores isessential otherwise the material in the superconducting bulk would simply short out thenormal material in the vortices.Close to T0 we can explicitly verify that the dissipation due to the normal currentsoutside the core contributes as much to the total dissipation as the normal currents insidethe core. Close to T , the real part of the conductivity is approximately equal to theconductivity of the normal material above T0 (see figure 2.3 for example) and so thedissipation can be writtentoo p2rP = I I o, E(r) . E(r) r dr dO . (3.33)J JoUsing equation 3.23 for the electric field outside the core, we haveuvbgjoo J27—rdrdO= UflVL1J (3.34)47r o r4and this is the same as the dissipation from equation 3.25. Thus close to T0 , we haveH335Pfl C2( )In treating the Hall effect, Bardeen and Stephen replace the simple expression for thecore velocity, equation 3.30, withv = (er/m)E + (v x B). (3.36)Chapter 3. Flux Flow in Conventional and High-Ta Superconductors 32This is just the expression we would use for the drift velocity of electrons in a normalmaterial in a field H 2 (and in this model the Hall effect due to the currents in thetransition region is the same as that due to the normal currents in the core[55]). Using= VT from equatioll 3.32 we can write this asvT=(er/m)E+—(vTxB). (3.38)Resolving this equatioll into x and y-components as we did with equation 3.11, we fluid!MLIL (339)°flas before, andtanOH=zL,Cr (3.40)where w is the cyclotron frequency, eB/m. For wr >> 1, we find that VL = (H/H2)vTwhich does not reduce to VL = VT as expected for a pure superconductor. Nozières andVinen[20] suggested a slight modification of the Bardeen-Stephen model which yields thesame flux-flow resistivity buttanOH=w2r (3.41)where w2 = eB2/m. In the limit w,2r>> 1, their equation reduces to vL = VT.For conventional superconductors, L.’T and w2r are both small and so only small Hallangles are expected. In any event, the longitudinal flux-flow resistivity is left unchanged.2The force acting on a charge carrier of charge e with velocity v (and scattering time r) in an electricfield E and magnetic field B can be written as a differential equation for the momentum p = my:= e (E + v x B) — . (3.37)In the steady state, dp/dt = 0, and we get equation 3.36.Chapter 3. Flux Flow in Conventional and High-Ta Superconductors 333.4 Tinkham’s time dependent order parameter and TDGLTinkham’s idea[46] was that there is energy dissipation associated with the time dependence of the order parameter at a particular location in the superconductor due to thepassage of a moving vortex through that point. The order parameter, [bI2, at this pointchanges from its equilibrium value, 1oI2, (before the arrival of the vortex) to zero (theaxis of the vortex lies on top of the point) back to boI2 again (after the vortex has passedthrough). Following Gor’kov and Kopnin[15], we give a rough calculation of the flux-flowresistivity that we would get due to this mechanism valid for dirty superconductors atlow temperature.We can estimate the time required to reestablish equilibrium in the superconductorafter the passage of a vortex as the time taken for an electron to move a distance of theorder of the coherence length (for a dirty superconductor ‘- 1) and since(VF is the Fermi velocity and L the energy gap) we getTO-—. (3.42)The time it takes for the vortex to pass through any given point in the superconductorwill be of the orderto -f-. (3.43)Since 1/ro is the gap frequency, To/to << 1 for most processes and so, if F is the free energydensity, the amount of energy dissipated will be approximately the fraction (ro/to) <F>(< F > is the time average of the free energy density). Tinkham interprets this energydissipation as the result of generation and heating of normal quasiparticles as the electronsleave and condense back into the superconducting condensate during the 11011-equilibriumsituation that causes the finite rate of change of I,I2. The power per unit volume in aChapter 3. Flux Flow in Conventional and High-Ta Superconductors 34superconductor with vortex density, n, aild an average flux density B = n4o isW=<F> 2 (344)Substituting for to and n and using < F >-- H2 we findw= Tçj vL2H B (345)SinceW = (l/pff) E2 = H E2 = Uff VL2B2, (3.46)we have using H1 -. Fo/X2 (X is the penetration depth in a dirty superconductor and isdifferent from the London penetration depth) and H2 H1H2 thatr0 r0 H2= B 2 H1 = B )2 (3.47)For a dirty superconductor, )2 /\L24o/l and so we can writene2l 1 h= mvF = VFAL2(O) (O)2 (3.48)Together with equation 3.47 this gives(349)Pfl C2in qualitative agreement with the Bardeen-Stephen result.A more rigorous approach to the effects of a time dependent order parameter inthe context of moving vortices was attempted using the time dependent Ginzburg Landau theory originally formulated by Gor’kov and Eliashberg[56]. The difficulties insolving a time dependent version of the Ginzburg Landau equations are such that thesolutions obtained were mostly restricted to gapless superconductors. Following the workof Schmid[57j and of Caroli and Maki[1O, 11, 12, 13], Thompson and Hu[14] were able toChapter 3. Flux Flow in Conventional and High-Ta Superconductors 35obtain expressions for pff in a high i superconductor with a large number of paramagnetic impurities (to satisfy the condition that the superconductor be gapless). Among otherthings, their solutions contained supercurrents in the core of the vortex. Tinkham[34]has pointed out how this highlights the oversimplification involved in considering thecore to be a cylinder of normal material. Nevertheless, in the limit as T —* 0 for a dirtysuperconductor with no paramagnetic impurities, Gor’kov and Kopnin[16] obtailled theexpression= 1.1H(0) (3.50)which is in good agreement with the simple Bardeen-Stephen result.3.5 Experiments on Flux FlowWe will discuss first the dc measurements made mostly on superconducting alloys such asPbIn and NbTa. Real materials have defects which tend to pin the vortices and stop themfrom moving in response to some driving force. Indeed, this is a very desirable featurewhen fabricating magnets out of superconducting wire. Such magnets are designed torun in the so-called persistent mode where they are disconnected from the power supplyafter having been charged to the desired current. Because the wires are sitting in theirown high magnetic field, they are threaded by many vortices. These vortices will movein response to the Lorentz force and hence cause energy dissipation and degradation ofthe current unless they are held in position by even stronger pinning forces. However,for doing experiments probing intrinsic physics, it means that a large enough transportcurrent (called the depinning critical current) must be applied to overcome the pinningforces. The I-V characteristics of such an experiment (performed by Kim et al[l]) whena dc current is applied to PbIn alloys is shown in figure 3.2. To define the flux-flowChapter 3. Flux Flow in Conventional and High-Ta Superconductors 36resistivity under such circumstances requires a slightly different approach from the zero-pinning case. Now we have?lvL=FL—Fp for FL>Fp (3.51)where F is the pinning force and FL is the Lorelltz force J cTo. Since E = VLB, we getdE 0Bfjff= —j = (3.52)and here we have generalized the definition of the to be the derivative of the electricfield with respect to the current. Experimentally, this means that the flux-flow resistivityis obtained from the slope of the I-V characteristic once the depinning critical currenthas been exceeded. This slope was found to be independent of the critical current asit would have to be for this definition of p to be consistent. A set of p curves as afunction of field is shown in figure 3.3 for a Nb05Ta alloy. For low temperatures andfields, we see thatH3pt-, — H2(O)As the temperature gets closer to T , this empirical law is violated at lower and lowerfields; nevertheless, for small enough fields the slope of the Pff/Pn is 1/H2(O) for alltemperatures (note that the pt-, used to scale the p data is independent of temperatureand equal to p(T) since an alloy is in the impurity limit by T ). This behaviour isclaimed to be representative of low-field (H2 (0) iT), intermediate (t 5) superconducting alloys such as NbTa and PbIn. A similar result was obtained by Vinen andWarren[2] on Nb and NbTa alloys. Thus the data at low fields seem to agree with thezero temperature Bardeen-Stephen result even though this result was derived for a pure,clean-limit superconductor whereas the alloys used in the experiments are in the dirtylimit. The result is also in agreement at with the TDGL result, equation 3.50, for dirtysuperconductors for T << T and H << H2.Chapter 3. Flux Flow in Conventional and High-Ta Superconductors>>-J0I (AMP)37Figure 3.2: I-V characteristics for Nb05Ta in a magnetic field. This is reprinted fromthe paper by Kim et al[l]. The Lorentz force must overcome the pinning force beforethere is any dissipation.30 I 2 3 4Chapter 3. Flux Flow in Conventional and High-Ta Superconductors 380.40.2H(kG)Figure 3.3: Normalized flux-flow resistivity vs. magnetic field for Nb0.5Ta5 This isreprinted from the paper by Kim et al [1]. At low temperatures and fields, the dataobeys the empirical law pj/p,, = H/H2(0).0.80.600 I 2 3 4 5 6 7 8 9Chapter 3. Flux Flow in Conventional and High-Ta Superconductors 39Another phenomenon observed[1] for high-field superconducting alloys (.c ‘-‘. 10— 100)such as NbZr, NbTi and VTi. was the paramagnetic effect on p . The effect is due toaligurnent of the electron spins along the direction of the applied magnetic field. Thislowers the free energy of the corresponding normal state by Xn H2/2 (where Xn is thenormal state susceptibility) and leads to a transition into the normal state as the fieldis increased even for H < H2 once this energy reduction becomes comparable to thegap energy. What Kim et al found was that the empirical relation 3.53 holds not forthe actual H2 (0) but rather from the H2*(0) that we would calculate based on theGinzburg-Landau theory in the absence of the paramagnetic effect. Since it is H/H2*(0)and not H/H2(0) that represents the volume fraction of normal material, it is intuitivelyappealing that it is the latter ratio which appears in the equation 3.53. However, tobe totally consistent, it is H/H2(T) (for a given temperature) and not H/H2(0) thatrepresent the volume fraction of normal material. Rather, we might have expectedp HPm — H2(T)as can actually be derived from the Bardeen-Stephen model for T —+ T.Behaviour more in line with equation 3.54 was actually seen in later dc experiments ona Pb076In24 alloy by Fogel et al[3], see figure 3.4. Reasonable agreement with equation3.54 was seen for H/H2(T) < 0.5 above which increases more rapidly as itcrosses over in to the normal state (slightly anomalous behaviour is seen close to theorigin, however). In fact, Fogel claims that, based on a re-examination of all existingdata at that time, there are departures from the empirical relation of Kim et al[1] for allconventional type-Il superconductors!We now turn to the experiments performed at high frequencies. Central to theseexperiments was the early work of Gittleman and Rosenblum[4, 5]. They found that theflux-flow state is accessible for high-frequency transport currents even with J << J asChapter 3. Flux Flow in Conventional and High-Ta Superconductors 40“:6j7 4’tl.217Figure 3.4: Normalized flux-flow resistivity vs. field and reduced field: PhIn. This isreprinted from the paper by Fogel [3]. The different curves represent field sweeps atdifferent temperatures the higher numbers represent lower temperatures. It is thetemperature dependent H that seems to be the relevant parameter for scaling the data.Pt iPI/ 2 3 4 5HkOctDi8.2 D.Q 8.6 8.8 1 n/Ijt)Chapter 3. Flux Flow in Conventional and High-Ta Superconductors 41— —IL (mcs) (mcs)SAMPLE Io ‘ocMEAS. cA(.c.Pbtn—L79( .9 7.00 PbIn—L7K 5.1 taoX NbTa—4.2 15 49(mcs)‘opCALC.az3.815.6Figure 3.5: Frequency dependent crossover from a flux-pinned to a flux-flow state forsome conventional superconducting alloys. This graph is reprinted from the paper byGittleman and Rosenblum[4, 5]. It shows that above the characteristic pinning frequency,= the superconductor acts as if it were totally unpinned.long as the frequency is greater than the so-called pinning frequency, w,,, = i,/i where.‘c1, is an effective pinning force constant, see figure 3.5. For >> w, the material actslike an ideal, defect-free material with the vortices totally unpinned. Their picture of thedynamics is of a vortex sitting at the minimum of a pinning potential energy well which,close to the minimum, can be thought of as a parabola ix2/2. If the vortex is acted on bythe viscous retarding force nv = i and the Lorentz force J10 in the opposite direction,we can write down an equation of motion for the vortexmi —ic,x—rith+J40 (3.55)where m0 is the effective mass of the vortex. This mass term is typically thought to benegligible[58]. If we write J = J0e”t and = = voei(t, we can solve for J0 in theI I411414 I 1141111 II I 11114 I 4414fl1.00.60.6gC0.402• I 14411 4 I 44444I I 144144II. l lull-I lIlt. I 41111111 4 I 1411110--0- Io_2 10I I 4114411 I 4 14444(1LO (0’f/f0I l1IIIItChapter 3. Flux Flow in Conventional and High-Ta Superconductors 42above equationJ0 = ( — i (3.56)Since E0 v0B (see equation 3.1), we get for the effective ac resistivityE0 F0BPeff== 1 (1 — iw/w) (3.57)For w << , p3 becomes purely imaginary and there is no energy dissipation. For(.) >> L’)p,0BPeff = P11 = (3.58)the pure flux-flow resistivity that would be measured directly in a dc experiment. Anexample of the transition from the flux-pinned to the flux-flow regime is shown in figure3.5. It can be seen from the values of fo in this figure (3.9—15 MHz) that at microwavefrequencies their samples are well in to the flux-flow limit. This result has been understood as follows: at high enough frequencies, the vortex spends the entire cycle of the rffield in the neighbourhood of the potential minimum (where the restoring force, ic,,x, isvery small); therefore, the vortex only samples that part of the pinnrng potential which isessentially flat, and thus it responds as if it were totally unpinned. In fact, this argumentis flawed in that it implies a dependence upon the amplitude of the motion. We ca seefrom equation 3.55 that this is not so: the effective viscous force increases proportiollal tothe frequency and will eventually dominate the pinning force at high enough frequencies.These observations about the vortex dynamics were later confirmed and investigatedin more detail by Le Gilchrist and Monceau[6, 7, 8] who found that the rf responsewas virtually unaffected by pinning for frequencies greater than 1 MHz for appropriatelyprepared samples of PbIn, NbTa and Nb. To relate the flux-flow resistivity to the surfaceresistance, they used the expressionR3(B) — (\1/2R pn)Chapter 3. Flux Flow in Conventional and High-Ta Superconductors 43The reasoning was as follows: if the classical skin effect expression applies to the mixedstate as well as the normal state, thenR8(B) = (PffILow)h/2 (3.60)andR(PnItOW)h/2 (3.61)and dividing these two equations, we get equation 3.59. The classical skin effect theorymight be expected to apply if the skin depth, 8, of the rf currents in the mixed state arelarge compared to all the length parameters that characterize the mixed state such asa (the vortex lattice parameter), and (the coherence length). ). is typically the largestof these. If the skin-effect theory is approximately correct and Pff/Pn ‘-. H/H2 then=IL (3.62)w H2 H2Using 6 3 x iO’ cm and X i—’ iO cm, thell= = 0.03 \/rii;:;- (3.63)which is small except for H close to zero. The field dependence of R3/R and Pff/Pn 5shown in figure 3.6. The latter is obtained by squaring the former (in accordance withequation 3.59). As can be seen from the plots, they also made measurements at dc andthe results are plotted on the surface resistance plot. The agreement is quite good andconfirms that at high frequency, the ideal flux-flow state is recovered. At low fields, thedata seem to scale with a temperature independent H2 in qualitative agreement withKim et al[l].Chapter 3. Flux Flow in Conventional and High-Ta Superconductors 44/0•-po P0’ O•• 4Co0• 2_tooVC0 2 4B (kG)Figure 3.6: Microwave determination of P1f in conventional superconductors. Thesefigures are reprinted from the paper by Gilchrist and Monceau[8]. The upper figureshows R3 and the lower figures pff extracted using equation 3.59. In the lower figure, thesmooth curves are obtained from the microwave data and the discrete points are from dcmeasurements of the differential resistivity.I-00.510—I IH0 (kO€)4 6.44+() (b)I I6 8 0 2B (kG)3 4Chapter 3. Flux Flow in Conventional and High-Ta Superconductors 453.6 Theory of Flux Flow Resistivity in High-Ta Superconductors3.6.1 Surface Impedance in the Mixed StateThe advent of high-Ta superconductors has stimulated considerable new work on derivingmore general expressions[28, 59] for the surface impedance of a superconductor in themixed state. Coffey and Clem[28, 29], in particular, propose a solution to the surfaceimpedance problem taking into account the vortex viscosity, pinning, thermally activatedflux motion and contribution of the normal fluid from the bulk of the superconductor.They consider the response of the superconductor in the mixed state to an applied acfield, b, parallel to the surface along the z-axis. In the simplest version of their theory,the superconductor is taken to occupy the half-space x > 0. They treat the problem ofa static field H >> H1 applied parallel to the z-axis. This establishes the flux latticecorresponding to an average field B0 = = H/1t0 (where n is the number of vorticesper unit area); the amplitude of the rf driving field is small, b = ,tt0h << Bo. The theoryalso assumes B/t0 >> H1 which implies that the vortex lattice parameter, a0, is muchless than the penetration depth, )‘L. The surface impedance is given by= iw,to.(w,B,T) (3.64)where ) is the effective skin depth and is in general complex. The ac b field gives riseto oscillating supercurrents that shake the vortices in their pinning potential wells andcarry field perturbations farther into the superconductor than, say, simply the Londonpenetration depth. The approach is basically two-fluid in nature withJ = J + J (3.65)whereJ=u1E (3.66)Chapter 3. Flux Flow in Conventional and High-Ta Superconductors 46(o is the real part of the conductivity due to the normal fluid) and Js is given by amodified second London equationV x J = —(toAj’ (B —40n&) (3.67)(a is a unit vector in the direction of the field in the core of a vortex). This equation takesinto account variations in the flux density from the average, macroscopic value and canbe seen as a generalization of the equation for the current and field distribution about asingle vortex[34]0VxJ8+B=oS2(r)&. (3.68)They supplement these equations by an equation of motion for the vortex identical tothe one assumed by Gittleman and Rosenbiumiñ(r,t) + iu(r,t) = J x o& (3.69)(they ignore the possibility of Hall fields here) where the forces in this equation are perunit length of vortex. The vortex displacement from its equilibrium position, u(r, t),and its velocity, ñ(r, t), are both dependent on position, here, because the ac fields decayinside the superconductor due to screening deep inside the superconductor, u = ii = 0.Using the ansatz,— *uo e”3’et (3.70)for the vortex displacement and ignoring (for the moment) thermally activated flux motion, they derive the following expression for ).B T— (B, T)— i/2 B, T) 1/23 71— 1 +2i(B,T)/Sf(w,B,T) .\L (B, T) is the temperature and field dependent London penetration depth, 6, is givenby2PV — 0BPv. (. )0wChapter 3. Flux Flow in Conventional and High-Ta Superconductors 47(w = i/j is the pinning frequency as before) and Snf is the normal state skin depthcoming from the normal fluid= 2 (373)P0 °i C’They also showed [29] that for the slab geometry, we get the same expression for ).whether the static field is strictly perpendicular or parallel to the broad faces of the slab(provided that the rf f field is parallel to B0 in the latter case).As T —* T or H — H2, )L(B,T) diverges and ) becomes82 1/2 6x= [-a] —(1—i) (3.74)and so for the surface resistance we getR8 = Re(ipo) = P06nf = Pm PO (3.75)which is the expression for the surface resistance of a metal in the normal state. In zerofield, the Snf term is an essential part of the physics: setting 6 to zero, equation 3.71becomes11/2(3.76)1/) +Z[tOJnfCL’and so the expression for the surface impedance is1/2Z3=ipo)= 0 = 0 (3.77)U Uflf 2where ö- is the previously discussed generalized two-fluid expression for the complexconductivity of a superconductor in zero field.At finite fields, the term including in the denominator of the expression in equation3.71 is, practically speaking, rather unimportant apart from the above limiting behaviour.This is because (XL/8f)2 << 1 except very close to II2 or T . Therefore, over most ofthe range of temperatures and fields of interest1/2=—(3.78)Chapter 3. Flux Flow in Conventional and High-Ta Superconductors 48An important observation is that for )/6 << 1 (and typically this is a good approximation especially for frequencies of the order of a few GHz) ). cx B”2 and in particularo B’12 no matter how strong the pinning as long as the i and w, are independent ofthe field. For w >>,(free flux-flow limit), equation 3.78 becomes1/2 1/2= [_jo] [2_2] (3.79)where is the free flux-flow skin depth. If )/S << 1 i.e. flux-flow skin depth muchlarger than the London penetration depth, then1/2= —i ° = (3.80)[L0LUT/ ,U0Wand so proceeding as before, we getR3 =(Pffo)1/2. (3.81)Thus, to the extent that we can ignore the London penetration depth term, the superconductor acts like an effective metal with resistivity pff. We can also see that this is theapproximation used in analyzing the flux-flow microwave experiments on conventionalsuperconductors.Equation 3.71 can be generalized to take into account thermally activated flux motion. This necessitates introducing an additional parameter U0 which corresponds to anenergy barrier height to thermal hopping of vortices between adjacent pinning sites. Themodified expression for is___— e+(wr)2+i(1—e)wr (382)pjj — 1+(r)2where— 1 I(v)—1— U02 ‘ ,anuv— .10(v) I,(v)Io(z) JcBTChapter 3. Flux Flow in Conventional and High-To Superconductors 491.21.00.81kI-i %.. 3.7: ii dependence of parameters in the Coffey-Clem model. This is reprintedfrom the paper Coffey and Clem[28J.100io2U“Iicr41 2 3 4 5 •6 7 8 9 10VChapter 3. Flux Flow in Conventional and High-Ta Superconductors 50The I are modified Bessel functions of the first kind of order p. The ii dependence of eand ‘rw are shown in figure 3.7. For low temperatures or large activation energies, Uo,v>> 1 and so T.4J —* 1 and € —* 0 and so equation 3.82 becomes____— (/w)2+ i (w/cü)—.w (3 84ff — 1+(w/w) —In the limit, >> wi,, the RHS is simply equal to 1 and we recover the result for freeflux-flow that we had in the absence of any thermal activation. In the high T or low Uolimit, e —* 1, and equation 3.82 becomes____1 + (cr)2 + 0= 1 (3.85)ff 1+(r)2and we again recover the free flux-flow result independent of how the operating frequency compares to the pinning frequency. It is important to understand the distinctionbetween these two ways (for small driving currents) of achieving free flux-flow. At lowenough frequencies (w <<,), 1ô,, —k pff provided U0 << kBT. At high enough frequencies,( >> wi,) we get free flux-flow even for U0 >> kBT. Thus the shift of the irreversibilityline to higher fields and temperatures as we increase the frequency (see Chapter 2) is undoubtedly related to thermally activated flux motion. At very high frequencies, thermalhopping of vortices is inhibited but we recover free flux-flow again because the vorticesare spending all of their time in the essentially flat bottom of the pinning potential well.When we come to use the Coffey-Clem theory, we shall set Uo/kBT >> 1 and neglectthermal hopping entirely.Coffey et al[60, 61] have also investigated the size of the vortex inertial mass term(finding results qualitatively similar to those calculated previously by Suhl[58]). Theyestimate that for T close to Tm(T) =e00B2(T) . (3.86)Chapter 3. Flux Flow in Conventional and High-Ta Superconductors 51Using the expression from Chapter 2 for H2 , we find that at, say, t 0.96, B02 = 6.6T.From equation 3.86, we find that rn,, Cs’ 1025kg/m. If we include a mass term rn,,ii inthe equation of motion for the vortex, the resulting resistivity isnv = (3.87)—zwrn,,/I +ico/w)and so 77/rn,, = Wm defines another characteristic frequency. Using the Bardeen-Stephenformula we can estimate ‘7 at t ‘—‘ 0.96 to be r-’ io. This givesWm== lo’ (3.88)At microwave frequencies of the order of 30 GHz, w ‘-s 2 x 1011 and so W/Wm l0 .It seems likely, therefore, that the effect of the vortex mass is not significant at thefreqnencies of interest. In the subsequent analysis, we shall ignore it.3.6.2 Microscopic description of vortex motion in terms of core statesHsu[22, 23] has modelled vortex motion microscopically in the low temperature, lowfield, clean, extreme type-Il limit in terms of the quasiparticle states inside the vortex.As we have already briefly mentioned, the nature of these states was first elucidated byCaroli, de Gennes and Matricon for conventional type-IT superconductors who showedthat there is a spectrum of energy levels for quasiparticle excitations localized inside thevortex with a very small energy gap given by4 o = &/EF (‘-.‘lmK for conventionalsuperconductors). For high-T0 materials this is estimated to be rs’ 10K and thus it is3Colfey and Clem also treat the case of a layered superconductor and are able to derive a low temperature expression for the thertial mass of a vortex parallel to the planes. They find m —‘ 3 x 1022kg/mfor YBa2Cu3O75 over 100 times bigger than the estimate given above. At low temperature, ij 10(using the Bardeen Stephen formula) and this gives w,,, ‘s’ 4 x i0’. Thus, W/Wm n.j and we stillfind that the effect of the mass term is not important at frequencies 30 GHz.4This energy gap can be estimated by considering the energy levels of a particle confined to a boxof radius . The lowest level has energy n., 112/me Using llvF/tX[341 and Pip n.j mv, we findSE L12/Ep.Chapter 3. Flux Flow in Conventional and High-Ta Superconductors 52important to understand what effect such a relatively large spacing might have on thedynamics of moving vortices.Hsu uses the Bogoliubov-de Gennes equations for s-wave superconductors and supplemeilts them with a local gap equation. He considers the response of a vortex to a uniformtime-varying electromagnetic wave and finds that the time evolution of the quasiparticlestates in the vortex corresponds to vortex motion provided that the velocity involved issmall. A nice feature of the equations is that for a clean system with no pinning and nodissipation the vortex moves along with the background superfluid as we would expect(see section 1.2.2). He is able to derive an equation of motion for the vortex including adissipation term which is characterized by the lifetime, r, of the low energy quasiparticlestates. The dissipation due to a single moving vortex is calculated and in the limit as—* 0 a core resistivity corresponding to=-- (3.89)Pfl C2is recovered. This is roughly consistent, then, with the experimental work on conventional superconductors where 1l is small. For finite o (and for a given polarization ofthe electromagnetic wave) there is an antiresonance in the dissipation at w= Theresulting conductivity has a shallow minimum near ü 1o correspondillg to this antiresonance. For high-TC superconductors, these features are probably located at much higherfrequencies than the maximum frequency used for the measurements in this thesis. Inthe limit as —* 0, l0r —+ co and H/HC2 —* 0 equation 3.89 is again recovered. However, it is not obvious that this clean limit applies to YBa2CuO695 : in the normal stater 10_14 sec while- 2 x 1011. Even if we allow for a one hundred fold increase in rbelow TC , the clean limit seems of doubtful validity. This work is sensitive to the natureof pairing and the effects of a gapless superconductor on the microscopic structure of thevortex have yet to be calculated. Nevertheless, it seems that Hsu’s approach is the bestChapter 3. Flux Flow in Conventional and High-Ta Superconductors 53starting point for a complete understanding of vortex motion in high-Ta superconductors.3.7 Experimental work on YBa2Cu3O7_5There has not been a great deal of experimental work done that directly addresses thequestion of the temperature and field dependence of the flux-flow resistivity. The commontendency is simply to assume that Pff/Pn = H/H2(T) or p/pn = H/H2(0) and ignorethe question of what is the most appropriate value for p, below T. To our knowledge, theonly real attempt to directly measure this quantity in high quality samples has been byKunchur et al[25] who worked with c-axis oriented epitaxial films ofYBa2Cu3O7_5.Theytake a novel approach by supplying a pulsed transport current density comparable to thedepinning critical current density (106A/cm2). Nevertheless, they were still restricted tomeasurements close to the irreversibility line and above. The films had Ta’s of 88.5Kand 91.1K and the lowest temperature measured was 76K. Lower temperatures requiredexcessively high current densities. They found good agreement with the Bardeen-StephellformulaH390p(T)— H2(T)with p(T) obtained by a linear extrapolation of the linear resistivity from above T andH2(T) = (T — T) (3.91)was obtained as a fittillg parameter. Values for dH2/dT T were 1.85T/K and 2.2T/Kill good agreement with values obtained from the literature[26]. They argue that thisresult points to the conventional nature of flux-flow in the high-Ta materials. However,they have only checked the Bardeen-Stephen formula down to t ‘—‘ 0.84 which leavesopen the question of the low temperature behaviour. Also there is some question as tohow much YBa2Cu3O7_films are affected by intrinsic defects such as grain boundariesChapter 3. Flux Flow in Conventional and High-Ta Superconductors 54which might create superconducting weak links, i.e. it is unclear how meaningful it is tocompare thin-film and single crystal data.Owliaei et al[62] have studied YBa2Cu3O7_epitaxial films at microwave frequencies.Figure 3.8 shows their surface resistance data at 10 GHz in magnetic fields up to 7T. Itis clear that by 65K the surface resistance has decreased to a value close to zero. Theyclaim to see a crossover as a fullction of magnetic field and extract and i within theframework of the Coffey-Clem theory from the field profiles in the temperature range78K to 85K. They find a pinning frequency that increases from 3 to 15GHz from 85Kto 78K and a viscosity that increases rapidly in this temperature range. They assumeequation 3.90 with poH2(T) = 115(1— t) T and find a p(T) that decreases roughlylinearly with temperature but which extrapolates to 0 by about 75K. Again, it is ullclearwhat effect the defect structure has on the surface resistance in a magnetic field.Marcon et al[63j have performed even higher frequency measurements (23 and 48GHz) in an attempt to deduce information about the vortex viscosity. Unfortunately,the samples used were not single crystals or films but ceramic samples. The microwaveresponse of such materials are usually dominated by grain boundary effects such as supercoilducting weak links and other sources of anomalous loss associated with the granularnature of the material.That a single pinning frequency model governed by an equation of motion such asequation 3.55 (the Gittleman and Rosenblum or Coffey-Clem models for example) isprobably not true in detail is indicated by the swept frequency measurements (1—600MHz) of the impedance of c-axis oriented epitaxial thin films of YBa2Cu3O7_by Wu etal[64]. Their technique allows them to extract the complex resistivity very simply fromthe impedance data ill a model independent way. They found in the field range (0.5—8T) and the temperature range (80—86K) that their data was consistent with a vortexglass to vortex liquid transition[65, 66] characterized by a critical field Hg. Away fromChapter 3. Flux Flow in Conventional and High-Ta Superconductors 55C 0+o °i*0+ +oaC0 +00 E(i’l0iø0.8 I0.o0’10.100.050QO01’11’4T+C70 9080T(K)Figure 3.8: R,, at 10 GHz in a magnetic field in a YBa2Cu3O7_thin film. This isreprinted from the paper by Owliaei et al[62].100Chapter 3. Flux Flow in Conventional and High-Ta Superconductors 56the transition line, the frequency dependence of the phase angle was consistent with thescaling model of Fisher, Fisher and Huse[67, 68] rather than the more simple behaviourpredicted by an equation such as 3.55.Chapter 4The Microwave ExperimentIn this chapter, we describe in some detail the experimental apparatus used to makethe surface resistance measurements on the single crystals of YBa2Cu3O695. We firstquickly review the basics of resonant cavities and then describe the principles of cavityperturbation. Since, the data presented later in this work was taken at three distinctfrequencies, we describe in detail the so-called split-ring resonator used to make the 5.4GHz measurements and the right circular cylindrical cavities used to make the higherfrequency measurements at 27 and 35 GHz. Finally, we discuss the overall design of thetwo cryostats used to house the low and higher frequency resonators. The heart of thedesign is the probe at the end of which the sample is mounted; this piece of apparatus iscommon to both the low and high frequency cryostats.4.1 The Resonant Cavity / Cavity PerturbationA resonant cavity is often modelled as an RLC circuit, the resonant frequency of a modein the cavity being thought of as the resonant frequency of the corresponding circuit.Just as for the circuit, if we sweep through the resonant frequency in the cavity, theabsorbed power has a Lorentzian line shape. The resonant frequency, and the qualityfactor, Q, can both be determined from the Lorentzian line shape, fo being the frequencyat the peak of the line shape and(4.1)57Chapter 4. The Microwave Experiment 58where /f if the full width at half maximum power. The Q of the mode is also given byQ = 2 ( energy stored ) (4.2)energy dissipated per cycleIt is clear from this equation that 1/Q is proportional to the power dissipated in thecavity.A real cavity has many resonant modes some of which are degenerate. Consider asingle non-degenerate mode far enough in frequency from all other modes so as to beunaffected by them. As we have already seen, l/Q is a measure of the power dissipationinside the cavity; in a empty cavity, it will be due to the energy losses in the walls ofthe cavity plus the losses due to the coupling holes which are needed to couple microwavepower into and out of the cavity. Thus, the Q of the cavity can be written as=-+- (4.3)where Qo is the intrinsic Q of the cavity and Q is the coupling Q. It is clear that if weintroduce a sample into the cavity, then we will get another term in the above equationdue to the power dissipation in the sample,1 1 1 1—=+—+. (4.4)Q Qo Q QWe can therefore measure the loss ill the sample by measuring the change in the 1/Qafter inserting the samplez\(l/Q) = c power dissipation in the sample. (4.5)The proviso here is that the sample is only a small perturbation of the rf fields in thecavity. If the change ill the geometry of the fields due to the insertion of the sample isnot small then the modified fields at the walls and at the coupling holes will give rise todifferent wall and coupling losses. Consequently, the terms 1/Qo and 1/Q in equationsChapter 4. The Microwave Experiment 594.3 and 4.4 can no longer be considered constant when we perform the subtraction toget 1/Q3. The net effect is to introduce systematic error into the measurement of thesample losses. The essence of the cavity perturbation technique, therefore, is to introducea sample whose losses are large enough to be measurable but whose effect on the overallfield configuration is negligible.From Appendix A, we know that the surface resistance, R3, of a sample is proportionalto the power dissipated per unit area, or explicitlyPA — R,. H02. (4.6)Therefore, we know thatR3 cx PA cx A(1/Q) or R3 = Cz(1/Q) (4.7)where C is a calibration constant. Once we have measured (1/Q), the task of flildingR is reduced to determining the calibration constant. For some standard cavities andsamples of particularly convenient shape, C can be calculated. However, much of thetime this is not possible, and we are left with two options: we can set the sample at atemperature where we know the dc resistivity and (providing the sample is metallic andsufficiently thick) use the expression= (Pdco)hI2 (4.8)to find R5 at that temperature. If the dc resistivity of the high-Ta sample is not known,or, if for any reason, the simple skin effect formula is not applicable, we can measurez(l/Q) for an appropriate reference sample of the same dimensions as the sample ofinterest.Chapter 4. The Microwave Experiment 60rf E fields4.2 The Split-Ring Resonatorrf B fieldsThe measurements at 5.4 GHz were made with a so-called split-ring resonator[69, 70]shown in figure 4.1. In principle, it is a single-turn inductor tuned by the capacitance ofthe gap. The mode of interest is shown in the figure. It consists of axial B fields (exceptwhere the field lines turn around at the ends of the resonator) and E fields everywhereperpendicular to the resonator axis and for the most part confined to the gap. Theresonant frequency of this mode depends primarily on the gap thickness and the radiusFigure 4.1: Split-ring resollator and field geometry.Chapter 4. The Microwave Experiment 61of the central bore of the resonator. It is a relatively simple procedure to machinethe resonator from oxygen free high conductivity (OFHC) copper and then use a sparkerosion machine to cut the gap to the desired thickness. Typically, the resonator sits on asupport structure within an outer cylinder which is beyond cutoff for the mode of interest.Figure 4.2 shows the arrangement used in the 5.4 GHz measurements. Particularly forhigh frequencies, this outer cylinder helps to confine the fields and to maintain a high Q.The main advantage of the split-ring resonator over a conventional cavity (a rectangularor cylindrical cavity, for example) at frequencies 1 GHz is that it is considerably lessbulky (the dimensions of the conventional cavity are proportional to the wavelength— at5 GHz, A = 6 cm), and that the filling factor (magnetic energy stored in the sample/totalstored magnetic energy) is typically much higher.The sample is moved in towards the resonator from above as shown in figure 4.2. Asit begins to interact with the B fields emerging from the central bore of the resonator(these are roughly perpendicular to the broad surface of the sample), screening currentsare induced in the plane of the sample and dissipate energy thus reducing the Q. For ametallic sample, this will also have the effect of increasing the resonant frequency. In thisgeometry, this is due to the sample reducing the effective size of the cavity (a metallicsample acts so as to confine the B fields in the space below it) and thus increasing theresonant frequency. This increase in frequency effectively measures the position of thesample with respect to the resonator and can be used to set the position of the sample ina reproducible manner. A change in skin depth of the currents in the sample also has theeffect of changing the resonant frequency; indeed, this is the principle by which we canmake surface reactance or skin depth measurements using cavity perturbation. However,in this setup, the frequency shift due to a chailge in the skin depth is very small comparedto the frequency shift due to a change in the overall position of the sample. Since thisposition is affected by thermal length contraction and expansion effects, it is clear thatChapter 4. The Microwave Experiment 62sapphiresamplerodresonatorteflon holderSMA feedthroughconnectorbrassmatingflangepieceFigure 4.2: 5.4 GHz split-ring resonator and field geometry. During a measurement,the sample is positioned ill the axial B fields that emerge from the central bore of theresonator.OFHCsplit—CuringCu coupling loopOFHCouterCucylinderChapter 4. The Microwave Experiment 63this arrangement is not well suited to measuring surface reactance. By moving the samplecloser to the top of the resonator, we increase the filling factor because we have movedthe sample in to a region where the B fields are more concentrated (equivalently, a regionwhere the energy density is higher). The fact that we can increase the sensitivity merelyby moving the sample closer to the resonator is a considerable advantage of the split-ring resonator. As already mentioned, the currents are induced in the sample principallyby screening the B fields from the metal or superconducting sample. The E fields areconfined to the region of the gap and do not play a significant role in generating currentsin the sample (the bevelling of the top of the resonator shown in figure 4.1 and 4.2 helpskeep electric fields away from the sample— this is not so important for single crystals,but it ca be for thin films having dielectric substrates).Figure 4.3 shows a schematic diagram of the circuit used for reflectance measurementsat 5.4 GHz. A Hewlett-Packard 83620A frequency synthesizer is the source and deliverspower to the cavity through the main arm of a 20 dB directional coupler. The signal isthen transmitted by semi-rigid co-axial cable to the cavity. The coupling to the cavityis made inductively with a ioop of wire beneath the resonator and co-axial with itscentral bore (see figure 4.2). The coupling strength is set by adjusting the position ofthe coupling ioop beneath the resonator. This must be done at room temperature duringassembly and cannot be changed once the apparatus is cooled. Care must be taken inchoosing the length of this wire to avoid )/4 self resonances. One percent of the reflectedsignal is coupled to the crystal diode detector. Operation of the calibrated detectorwithin its square-law regioll yields a dc output voltage proportional to the power in thereflected signal. The output from the detector is then amplified before being sent to ananalog—to—digital converter for storage and subsequent analysis on the computer.The resonant frequency, fo, and the quality factor, Q, are determined by sweepingthrough the resonance with the synthesizer and recording the reflected power at eachChapter 4. The Microwave Experiment 64couplerdirectionalFigure 4.3: Schematic of circuit used for the 5.4 GHz reflectance measurements.Chapter 4. The Microwave Experiment 65of a discrete number of frequency points (100, for example). The line shape of thereflected signal is fit to a Lorentzian plus a third order polynomial to allow for backgroundvariations. The latter is due to standing waves that are created in the circuit betweenthe cavity and the directional coupler. These result from constructive and destructiveinterference between the incoming and outgoing sigilals. The power in any part of thecircuit becomes position and frequency dependent. The problem tends to get worse asthe overall frequency is increased, the net result sometimes being a background signalwhich can be a significant fraction of the power variation associated with the real signalof interest— the amount of power absorbed at the resonant frequency in the cavity andsample. This affects our ability to reliably extract the Lorentzian from the backgroundin the fitting procedure and introduces an uncertainty in the fitted values of fo and Q.4.3 The 27 and 35 GHz CavitiesThe measurements at 27 and 35 GHz were obtained by moving the sample into the centreof right circular cylindrical cavities. In this frequency range, conventional cylindrical cavities work well because, since the cavities are smaller, the filling factor is much larger thanat low-frequency (also, the losses in the sample have typically increased with the squareroot of the frequency or even faster). The TE011 mode was used, the field configurationof which is shown in figure 4.4. During the measurement, the sample is positioned at thecentre of the cavity which corresponds to a node in the E field and a maximum in theB field. Just as for the split-ring resonator, the sample is placed in an rf B field that isperpeildicular to the broad face of the sample, inducing screening currents in the plane ofthe sample. Thus for a YBa2Cu3O695 single crystal, we induce only a-b plane screeningcurrents. This field geometry is slightly different from that of the split-ring resonatorin that there are strong B fields at both top and bottom surfaces of the crystal thusChapter 4. The Microwave Experiment 66sapphire rodsamplerf B fieldscoupling loopteflon spaghettirf E fieldsOFHC CuFigure 4.4: 27/35 GHz cylindrical cavity showing the geometry of the E and B fields.The well machined into the bottom surface of the cavity is to remove the degenerateTM111 modes.Chapter 4. The Microwave Experiment 67sampling both sides equally; in the split-ring resonator experiment, more of the currentruns on the bottom side of the sample (the side closest to the resonator). It turns out(see Appendix B) that most of the current runs near the edges of the sample, and so thedifference in current distribution for the two geometries is not thought to be significant.A separate cavity was built for each frequency in order to have good isolation betweenthe mode of interest and all other modes. By fine-tuning the radius and height of thecavity, we can locate the TE011 mode at the desired frequency and place all other modesfar enough away to avoid any mode-crossing when the sample is moved to the centre ofthe cavity. Special precaution must be taken to remove the two degenerate TM111 modesfrom the vicinity of the TE011 mode. This was achieved by machining out an on-axis wellat the bottom surface of the cavity (see figure 4.4). The axial B fields of the TE011 modehave a node at the top and bottom surfaces of the cavity (while the E fields have a lodealong the central axis), and thus, it is not shifted much in frequency. The TM111 modeson the other hand have high field density on the bottom surface. The well is thereforevery efficient in increasing the effective size of the cavity for these modes and shiftingtheir resonant frequency down and out of the neighbourhood of the TE011 mode.The electronic circuit for making measurements using either of the two cavities isshown in figure 4.5. In contrast to the 5.4 GHz measurements, these higher frequencymeasurements were made in transmission. The 83620A synthesizer was again used as themicrowave source, but since the unit can generate frequencies no higher than 20 GHz,the output is amplified (HP 8349B) and doubled (HP 83554A) to give frequencies in the26.5 GHz— 40 GHz range. The signal is again coupled into the cavity inductively usinga loop of wire to link the H-fields from the TE011 mode. Allother loop and coupling holeis used to couple the signal to a separate output waveguide. This waveguide takes thetransmitted signal to a microwave detector. The voltage signal is then amplified and sentto an A/D converter and computer, as for the reflectance measurement.Chapter 4. The Microwave Experiment 68detectorwaveguideFigure 4.5: Schematic of circuit used for the 27 and 35 GHz transmission measurements.Chapter 4. The Microwave Experiment 69In transmission, the non-trivial problem of the standing waves manifests itself differently from the sloping background signal observed in reflectance. Away from resonance,no signal is transmitted through the cavity and so there is not a sloping background.Instead, the standing waves present in the input and output waveguide circuits will tendto distort the line shape, and this again introduces uncertainty into the fitted values offo and Q.4.4 Probe DesignFigure 4.6 shows the design of the lower end of the probe. Both the low (100 MHz— 6GHz) and high (26.5 - 40 GHz) frequency inserts use this design. The sample is mountedon the end of a rod made from sapphire, which has two important properties in the thetemperature range of interest: it has a very high thermal conductivity and extremely lowloss at microwave frequencies. When the cavity is loaded with the sample, only the sampleitself and the sapphire rod are exposed to the high frequency E and B fields. This isensured by having a sapphire rod long enough to keep the copper housing assembly for thethermometer and heater away from the fields. Since the sapphire has negligible loss, it isthus only the losses in the sample itself that are measured. In good thermal contact withthe sapphire rod is a Lakeshore Cryogenics carbon-glass thermometer (a tiny amount ofsilicone grease is used to ensure good thermal contact between the thermometer and thecopper housing). Thus, because the thermal conductivity of the sapphire is so high, thisthermometer accurately measures the temperature of the sample. At typical operatingtemperatures, a gradient of no more than a few tenths of a kelvin was observed betweenthe two ends of the rod.Evanohm heater wire of total resistance approximately 200 is wrapped around thetop part of the copper heater/thermometer assembly. It is this heater that is used toChapter 4. The Microwave Experiment 70L.. thin—wallstainless steel tube[j of brass11>1 LII outer viewspringbrass spring/sample heaterCu thermom./heaterhousing assemblysample thermometersapphire rodFigure 4.6: Detail of the lower end of the probe. There is another brass spring positionedapproximately halfway between this end of the probe and the room-temperature end.There are two brass radiation baffles placed along this length as well.Chapter 4. The Microwave Experiment 71adjust the temperature of the sample from its low temperature value of approximately20K up to temperatures of the order of lOOK. This copper assembly is connected via athin wall stainless steel tube to a brass “spring” (see figure 4.6) through which a centralhole has been drilled. The stainless steel tube is soldered to this brass spring and also toanother identical brass spring farther up the probe. Both outer and cross-sectional viewsof the brass spring are given in figure 4.6. There are eight spring-like fingers on each endof the piece. The probe is designed to slide inside a 0.5” O.D. stainless steel tube whichconnects the microwave cavity assembly to the top of the cryostat and forms the vacuumchamber. Figure 4.7, for example, shows the probe inserted into the apparatus used forthe 5.4 GHz experiments. The brass springs have two important functions: they holdthe probe in aligilment with the vertical axis of the experiment; and, since the outer 0.5”tube is in good thermal contact with the helium bath, they provide the main thermalconnection between the sample and 4.2K. The length of the stainless steel tube betweenthe copper housing assembly and the solder joint at the lower brass spring sets the timeconstant for cooling of the sample. In choosing the length of this piece, a compromisehas to be reached between the power needed to raise the temperature of the sample toits upper limit and the thermal time constant. In the present apparatus, approximately100 mW is required to heat the sample above lOOK and the cooling time is about twohours.The two wires from the heater and the four wires from the thermometer are enclosedwithin teflon spaghetti which is threaded through the stainless steel tubing on its wayto the top of the probe. Connection to a four-lead resistance bridge and a temperaturecontroller is made via the electrical feedthrough. The six brass wires are heat sunk ateach of the brass springs to reduce the heat leak down the wires. Small baffles are alsoused on the outside of the stainless steel tubing to reduce thermal radiation down the0.5” stainless steel tube.Chapter 4. The Microwave Experiment 72outer 0.5”stainless steel tubeinsert used forhigh frequenciesouter OFHC Cu tube4T superconductingmagnetbrass flangeFigure 4.7: Overview of the apparatus used in the 5.4 GHz experiments.Chapter 4. The Microwave Experiment 734.5 The Low Frequency CryostatBecause of the modular design, the cryostat used for the 5.4 GHz measurements couldbe used (with the appropriate insertion of a particular resonator) anywhere in the rangefrom a few hundred MHz up to about 6 GHz. In fact, over most of this frequellcy range,all that is required is the fabrication of a split-ring resonator with appropriate innerbore and gap to obtain the desired resonant frequency. At the higher frequencies (e.g.5.4 GHz) a special insert was used (see figure 4.2) to reduce the diameter of the outershield and so keep it well beyond cut-off. Otherwise, there was the possibility of themicrowave fields reaching the copper thermometer/heater housing assembly. Figure 4.7gives an overall picture of the 5.4 GHz experiment. The resonator sits in a teflon holderwhich must be screwed down on to the flange piece. An indium 0-ring is used to make avacuum seal at the lower flange. An hermetically sealed SMA feedthrough connector isused on the insert piece to allow the microwave signal in to the cavity. The feedthroughconnector is screwed into the insert piece during construction and then sealed around thethreads with black 2850 FT epoxy. One end of the coupling loop wire is soldered to theinner conductor on the feedthrough while the other end is screwed down onto the insertpiece itself to provide a path to ground.The outer copper cylinder that houses the resonator assembly sits within the boreof a superconducting magnet that is able to generate a field of up to 4T (correspondingto a current of 40A). The magnet was home-made using NbTi wire with provision forpersistent mode operation. The two ends of the NbTi wire that emerge from the magnetare joined together using a technique[71] to give a high critical current. Evanohm heaterwire is wound around a 6cm stretch of the NbTi wire. This section of wire is then pottedin Emerson and Cummings Stycast 1266 epoxy to form the persistent switch. Connectionto two tinned brass shimstock magnet leads are made via short stretches of tinned copperChapter 4. The Microwave Experiment 74wire. These are joined to the superconducting magnet wire via simple soft solder joints.These joints are not critical as they are only important during charging and dischargingof the magnet. In persistent mode (after the external power supply is switched off) thecurrent only runs through the loop of superconducting wire and the high critical currentjoint. When a current is applied to the heater wire in the persistent switch, the stretchof NbTi wire potted in epoxy is driven normal, thus allowing the magnet to be chargedor discharged. Two protection diodes on the top of the cryostat at room temperatureprovide a safe, dissipative current path in case of a quench of the magnet during chargingor while in persistent mode.4.6 The 27 and 35 GHz CryostatA detailed discussion of the cryostat used for the 27 and 35 GHz measurements will begiven in the Ph.D. thesis of Kuan Zhang. Here, we shall discuss only its most importantfeatures and any conceptual differences from the low-frequency cryostat. Modularity ofdesign is a convenient feature of this high frequency cryostat as is evidenced by the easewith which we can switch between the 27 and 35 GHz measurement. All that is requiredis to replace the 35 GHz cavity block (made from OFHC copper) with the 27 GHz cavityblock. The frequency limits are set by the 26.5 —40 GHz waveguide, two sections of whichrun from each side of the cavity up to the top plate of the cryostat. Unlike the split-ringresonator set-up, there is variable coupling (on both input and output) and this allowscoupling adjustment at low temperature. This is an absolute necessity if the OFHC cavityis plated with a superconducting PbSn alloy for high-Q (> 106) usage at low temperature.The Q of the cavity increases so dramatically, once its temperature drops below the T ofthe PbSn, that it is very difficult to set a coupling at room temperature that is suitablefor the high Q at low temperature. The high field measurements of the present studyChapter 4. The Microwave Experiment 75preclude the use of PbSn plating (since the field would destroy the superconductivity inthe PbSn); however, variable coupling is still of considerable practical convenience.A high homogeneity, 8T superconducting magnet is used to generate the static magnetic fields for the 27 and 35 GHz measurements. It is mounted on a separate supportin which the entire high frequency cryostat can be inserted.The other important difference in this cryostat is that the main stainless steel tube inwhich the probe is situated is not in direct contact with the helium bath. The couplingholes through which the coupling wires must be free to move while adjusting the couplingare not leak-tight. Therefore, everything is enclosed within a large stainless steel tubeclosed at the low temperature end and sealed to the top-plate of the cryostat with arubber 0-ring seal at the room temperature end. It is this tube that is immersed directlyin the helium bath, the main apparatus itself is in vacuum. In order to provide increasedthermal contact between the cavity and the bath, two large copper braids join the cavityto the bottom of the outer tube. To use the PbSn superconducting cavity, it is importantto lower the temperature of the cavity to well below the 7K transition temperature.For this purpose, an open stainless steel tube, the lower end of which sits directly in theliquid helium, is placed in good thermal contact with the cavity. By pumping on the topend of this tube, cold helium can be drawn up into the tube and used to cool the cavityto the temperature of the helium bath.Chapter 5The Experimental ProcedureIn this chapter we first present some of the general characteristics of the YBa2Cu3O695single crystal sample used in the experiments and then describe the experimental procedure used for collecting the data. Since the low-frequency 5.4 GHz measurements and thehigh frequency 27 and 35 GHz measurements were made in two totally different cryostats,they each had their own specific set of problems and are described separately.5.1 The YBa2Cu3O695 single crystal sampleThe YBa2Cu3O6g5sample used in the measurements at all frequencies was a very highquality single crystal grown at U.B.C.[30]. Crystals from all batches have Ta’s greaterthan 93K. The transitions can be characterized by techniques such as dc resistivity,magnetization, specific heat and microwave surface resistance. All show extremely sharptransitions; in particular, the specific heat jump at T has a width of less than 0.25K- the narrowest yet reported. The crystals are typically of uniform thickness and thebest ones have optically smooth surfaces and clean, unfractured edges. The crystalscan be cleaved in the a—b plane quite easily and so often it is possible to cleave awayparts of a crystal that are contaminated by flux or that include a damaged edge. Thelatter is important for the microwave technique described in this thesis since the highestcurrent density exists near the edges of the sample and a crack can cause anomalouslyhigh losses. Microwave surface resistance in zero field indeed provides a stringest testof sample quality. R8 data measured at 3.8 GHz on one of the single crystals is shown76Chapter 5. The Experimental Procedure 77in figure 2.2. The sharpness of the transition and the low residual loss are indicative ofvery high sample quality. Although the occasional crystal can be found with no twinboundaries, the majority of the crystals are not twin-free. The single crystal used in theexperiments described in this thesis was of dimensions 1.5mm xl .5mmx 1 Om and wastypical of the best crystals; however, twin boundaries were observed to be present.5.2 5.4 GHz MeasurementsAn important characteristic of the split-ring resonator is how (1/Q) depends on L(f) asthe sample is moved towards the top of the resonator. Such a curve for the YBa2Cu3O695single crystal at a temperature of lOOK (this is in the normal state where the losses arehigh) is shown in figure 5.1. It is basically linear through the origin with a small verticaloffset. The linear behaviour is to be expected since both the loss in the sample and thefrequency shift measure the overall perturbation to the cavity. The slight offset is due to asmall negative dielectric shift from the sapphire rod which quickly becomes overwhelmedby the positive shift due to the sample. This is only observed for small samples; largersamples not only screen the sapphire more effectively but also give rise to larger positivefrequency shifts (for such samples, the zS.(1/Q) versus z(f) curve extrapolates through theorigin). The z(1/Q) versus A(f) curve is universal for a given sample in the sense that nomatter what the loss, the negative offset is always the same and the behaviour is linear. Itis the slope that changes, the line having a steeper slope when there is more loss. Clearly,the slope of this line is proportional to the loss in the sample. However, since the curveis always linear with the same negative frequency shift offset, we don’t need to measurethe whole curve but only one point. This is in fact how the measurements were made:an operating LS.(f) was chosen, 7 MHz for example, the sample was moved in towardsthe resonator until a frequency shift of 7 MHz was achieved, and then measurementsChapter 5. The Experimental Procedure 78le-048e-05I,—6e-05-o4e-051’2e-05Oe+OO I I0 2 4 6 8A(f) (MHz)Figure 5.1: Dependence of L(1/Q) on (f) at 100 K for the 5.4 GHz copper split-ringresonator. The dashed line is a linear fit. The data is fit well by a line with a smallvertical offset.Chapter 5. The Experimental Procedure 79were made by varying the temperature while leaving the sample in this position. Due tothermal expansion and contraction of the sapphire rod (and to a lesser extent, the entireprobe), the position of the sample with respect to the resonator can change slightly andthe resonant frequency wanders away from the nominal value. However, this is simplycorrected for by moving along the universal curve. For example, if the nominal frequencyshift is given by L\(f)’°m and the measurement gives the pair of values ((f) 4(l/Q) ),nomthen we can calculate z(l/Q) as follows: the universal curve isL\(l/Q) = aA(f) + b (5.1)and so we get=b) (f)flOm+ b. (5.2)It is clearly important for the the operating point for the measurements to be on alinear portion of this curve. If the operating point corresponds to a physical positiontoo close to the top of the resonator, it is possible to introduce non-linearity due to thefringing electric fields from the gap interacting with the sapphire and causing anomalousnegative frequency shifts. This may be accentuated if the sapphire rod is not perfectlycentrally aligned and descends towards the resonator on the side where the gap is. Caremust be taken before a series of measurements is started to ensure that the /X(l/Q)versus z(f) curve is linear over the frequency range of interest.Another important check to be made is the /(l/Q) versus z(f) curve for the sample inthe superconducting state in zero field. At 5.4 GHz, the loss in zero field ofYBa2Cu3O695is orders of magnitude below the sensitivity of the OFHC copper resonator. The L(1/Q)versus z(f) curve should therefore be a line of zero slope along the L(f) axis (within thenoise). Figure 5.2 shows what was actually measured for the YBa2Cu3O695 single crystal.It is a line with negative slope meaning that the Q increases as we move the sample intowards the resonator! This type of non-perturbative effect was discussed in section 4.1.Chapter 5. The Experimental Procedure 80Oe+OO I600-2e-06 608-4e-068-6e-06 00 0-8e-06-le-05 I0 2 4 6 8 10A(f)(MHz) VFigure 5.2: Dependence of LSl/Q) on z(f) for zero-loss sample in the 5.4 GHz coppersplit-ring resonator. The Q of the cavity increases when we insert a sample with effectively no loss. This is due to the rearrangement of the fields as the sample is movedtowards the resonator. The two points at each value of z(f) reflects the fact that twomeasurements were made at each nominal frequency shift.Chapter 5. The Experimental Procedure 81It is due to a rearrangement of the rf fields when the sample and sapphire rod are movedclose to the resonator (the rearrangement of the fields is insensitive to any temperaturedependent changes in the sample since these are dwarfed by the effect caused by just itsphysical position). This background systematic effect will be independent of temperatureand present in all of the data tending to give an apparent loss less than the actual loss.Therefore, we must be careful to correct for this in the raw L(1/Q) data. This is doneas follows: if the systematic effect is described by—a’/X(f)— c’ (53)then the L(1/Q) corrected for the systematic effect will be= (a + a’) (f) + (c + c’), (5.4)and we then get the A(1/Q)cb0m corresponding to the nominal frequency shift as inequation 5.2(1/Q)cnom = ((a + + c) (f)flOm+ (c + c’). (5.5)We thus have all the ingredients to convert the raw data into a set of L(1/Q) valuescorrected for the negative background loss and any small changes in the position of thesample during the course of the temperature sweep. The following was the step by stepprocedure used to measure the L\(1/Q) as a function of temperature for a givell field:• the LS1/Q) versus z(f) curves for the sample in the normal and superconductingstate in zero field were measured to ensure a linear dependence of z(1/Q) on zX(f)and to measure the background systematic effect.nom• an operating point (or nominal frequency shift) was chosen. A L(f) was choselllarge enough to produce easily measurable (1/Q) values but not so large so as tobe out of the linear portion of the zS(1/Q) versus (f) curve.Chapter 5. The Experimental Procedure 82• the sample was heated above T• the static magnetic field was applied.• with the sample pulled completely away from the resonator, the unloaded fo andQ o were measured.• the sample was moved in to the nominal frequency shift position chosen.• the heater was turned off and the sample cooled to low temperature.• the resonant frequency and Q of the cavity were measured as a function of temperature right up through T . The highest temperature measured was typicallylOOK.• at the highest temperature, the sample was again pulled back from the cavity, andthe unloaded fo and Qo were measured again.• the L(1/Q) (T) and z(f) (T) data were corrected for the negative background lossand any change in position of the sample usillg equation 5.5.In the above procedure, great care was taken not to change the field or move thesample unless T > T. The point is that very strong pinning forces can result when theflux density in the sample is changed. In an earlier experiment, a large YBa2Cu3O6•95crystal was totally shattered by moving it in the inhomogeneous part of the magneticfield when T was below T.Once the corrected z(l/Q) values have been obtained, the data was calibrated byusing the L(l/Q)(lOOK), taking Pdc =77.8 [ta—cm, and using the classical skin effectformula for the surface resistance of a metal. Although the dc resistivity on this particularcrystal was not measured, this value for pa(lOOK) was typical of other crystals madeChapter 5. The Experimental Procedure 83with the same technique. It is probably good to +10%. This gives a calibration factorc— pdc(100K) to cü/2— 1686 5 6- A(l/Q)(100K) - (.)However, because the crystal had a thickness of only 10—12 tm and the microwave skindepth at 5.4 0Hz with a dc resistivity of 77.8 t—cm is 6gm, it was not clear whetheror not the sample was accurately in the classical skin depth limit. For this reason, a PbSncalibration sample was cut’ to close to the same dimensions as the single crystal sample.The resistivity of the PbSn was taken from the literature for Pb and has been found(by Bonn, iii independent measurements with a superconducting split-ring resonator at 3GHz) to reproduce the temperature dependence of the PbSn to +1%. Using the coppersplit-ring at 5.4 0Hz the temperature dependences were found to agree to +5%. Thecalibration constant determilled using the reference sample was found to be 1898. Asexpected, this number was greater than that obtained using the classical skin effectformula (a sample comparable in thickness to the skin depth, gives enhanced losses), butconsidering the uncertainty in the PbSn calibration procedure, the difference is not large.It was the latter number that was used to convert the (1/Q) values to surface resistance,R. The temperature dependence of R5 for applied static magnetic fields ranging between0.5T and 4T is shown in figure The 27 and 35 GHz measurementsThe high frequency measurements made in the circular cylindrical cavities are slightlyless complicated in practice because the sample is located at a maximum in the microwavefield strength (see Chapter 4) rather than at a steep gradient as is the case with the splitring resonator. Thus, if we introduce a iossy sample into the cavity and measure the Q asa function of position, it exhibits a broad maximum at the centre of the cavity where we‘This was done by D.A. BonnChapter 5. The Experimental Procedure 84a)Cl)U) 30 40 50T(K)60 70+0Ii0 20 40 60T(K)80 100Figure 5.3: R5 in fields up to 4T at 5.4 GHz. Q O.5T; A 1.OT; l.5T; D 2.OT; + 2.5T;3.OT; x 3.5T; * 4.OT. Inset: close-up of the lower temperatures. R8 begins to becomedifficult to measure below about 60K.Chapter 5. The Experimental Procedure 85locate the sample for measurements. This means that the sample is insensitive to smallchanges in its position during the course of a temperature sweep. However, systematic,non-perturbative effects on the Q are more difficult to correct for because the zero-fieldloss in the sample is not negligible as it was at 5.4 GHz. The reason is that the zero-fieldlosses in the sample have increased by a factor close to 2 and are therefore measurableat 35 aild 27 GHz even with copper cavities. We thus have no ‘zero-loss’ sample to easilyidentify the changes in Q that result from rearrangement of the field pattern.This problem was solved in the 35 GHz measurements by making an identical measurement of the zero-field L(l/Q) as a function of temperature in the copper cavity andalso in an identical but PbSn plated, superconducting, high-Q, cavity. The two A(1/Q)curves are shown in figure 5.4. Subtracting off a temperature independent constant fromthe curve obtained using the copper cavity, causes the two curves to be essentially identical. This constant was then subtracted off all the data taken with the copper 35 GHzcavity to correct for this systematic effect. The reasoning here is that when the sampleis inserted into the centre of the cavity, it perturbs the ac fields enough so that extraloss is generated either in the walls of the cavity or most likely in the coupling loops andholes. The effect of this will be much less ill the superconducting cavity because of thegreatly reduced loss in all current-carrying surfaces. Thus, we take the superconductingcavity as the zero-loss reference; certainly, on the scale of the losses measured in a finitemagnetic field, any further corrections are negligible. Once we have determined this constant for measurements in the copper cavity for a certain coupling strength on the inputand output, we must not adjust the coupling throughout the rest of the experiment.As far as acquiring all of the magnetic field data was concerned, the same basicsteps were used as for the split-ring measurements. Only the subsequent corrections forsystematic effects and changes in sample position were different.Calibration of the 35 GHz data was done using the classical skin effect formula andChapter 5. The Experimental Procedure 862.Oe-06 I I I98088 61.5e-0601.Oe-06a5.Oe-070.Oe+00 I0 20 40 60 80T(K)Figure 5.4: Systematic error in 35 GHz copper cylindrical cavity. The circles are thedata obtained with the Cu cavity and the squares are the same sample measured in anidentical cavity plated with PbSn.Chapter 5. The Experimental Procedure 87Pdc = 77.8jtQ—cm. The skin depth at 100 K is approximately 2 ,um at 35 GHz — wellinto the bulk limit. R versus T for fields ranging from 0 to 6T is shown in figure 5.5.The 27 GHz R data was obtained in the same manner as the 35 GHz measurementsapart from the correction for the non-perturbative change in the Q upon insertion of thesample. Since a PbSn plated version of the 27 GHz cavity was not available, the followingprocedure was adopted as the next best alternative: the 27 GHz data was first calibratedignoring the possibility of a systematic error; zero-field measurements by Doug Bonnand Kuan Zhang on YBa2Cu3O6•95 single crystals suggest a w19 scaling of the surfaceresistance at 70K, and so a constant was subtracted from the 70K, 0-field point at 27GHz to make it consistent with the 35 GHz zero-field data and the above scaling relation;this constant was then used to subtract from all the R data measured at 27 GHz.The resulting R data plotted versus temperature for fields ranging from 0 to 8T areshown in figure 5.6.Chapter 5. The Experimental Procedure 880.4 I I I0.30)asDC,)oo e e0 20 40 60 80 100T(K)Figure 5.5: Temperature dependence of R in a magnetic field at 35 GHz. Q OT; A iT;x 2T; K 3T; D 4T; * 5T; L’ 6T.Chapter 5. The Experimental Procedure 890.3 I I I I$0.2a)as0D*WCl)crcD0.1vp008Ox00.0______o00 20 40 60 80 100T(K)Figure 5.6: Temperature dependence of R3 in a magnetic field at 27 GHz. Q OT; A iT;x 2T; 3T; D 4T; * 5T; L 6T; + 7T; V 8T.Chapter 6The Analysis6.1 Overview of the DataFigure 6.1 shows the measured surface resistance as a function of temperature in a magnetic field of 4T for all three frequencies. The 27 and 35 GHz data have been scaled byfactors of (5.4/27)1/2 and (5.4/35)1/2 respectively to facilitate the comparison. Since fora metal in the classical skin-effect limit R3 is proportional to ,1/2, this operation shouldresult in the curves being brought into coincidence above T . However, at 5.4 GHz thesample is not much thicker than the normal state skin depth, and so above T there areenhanced losses as is evidenced in the figure. Below T , however, and certainly below78K, the skin depth is much reduced and we can therefore directly compare the 5.4 GHzdata and the 27 and 35 GHz data. It is immediately apparent that below 70K, the 5.4GHz surface resistance is greatly suppressed compared to the 27 and 35 GHz surfaceresistance. The two higher frequencies scale reasonably well with the square root of thefrequency although at low temperatures the 27 GHz data begins to drop below the 35GHz result. This behaviour suggests the picture of flux-flow at high temperatures wherethe data at all three frequencies scale roughly as 1/2 like an effective metal. It alsosuggests temperature dependent pinning frequencies in the range of 5 - 35 GHz whichare large at low temperatures and which decrease with increasing temperature.This basic picture is reinforced by lookillg at the data in a slightly different way.Figure 6.2 shows R8 versus B at all three frequencies at low and high temperature.90Chapter 6. The Analysis 910.05 I I I0.150.04 0i0 000.05 0a) -LJtv.03(00.000 20 40 60 80 100T(X0.02L0.01 60000.00 I I0 20 40 60 80T(K)Figure 6.1: (5.4/frequency)h/2R3at 4T for 5.4, 27 and 35 0Hz. Q: 5.4 GHz; D: 27 0Hz;: 35 0Hz. The 5.4 GHz data drops clearly below the higher frequency data at lowtemperatures indicating that pinning is playing a role at this frequency.Chapter 6. The Analysis 920.04 I IT=30K0.03 D0U00.02--0.01--I °°T=78K 4o0.10• U0.05--00I I I0.0 2.0 4.0 6.0 8.0B(T)Figure 6.2: Magnetic field dependence of R3 for all three frequencies at 30 K and 78 K.Q: 5.4 GHz; 0: 27 GHz; : 35 GHz. Clearly, at 30K, the loss at 5.4 GHz is greatlysuppressed with respect to the higher frequency data.Chapter 6. The Analysis 93Again, we see that at 30K, the 5.4 GHz data is greatly suppressed with respect to the 27GHz and 35 GHz data while at 78K it has increased to the point that it is scaling roughlyas wh/2 with the two higher frequencies. Again, this suggests a temperature dependentpinning frequency that is probably larger than 5.4 GHz at 30K and probably less than5.4 GHz at high temperature.Another way of seeing the difference at low and high temperature is to plot R3 against.‘ for a given field and temperature. Figure 6.3 shows this for low and high temperatureon a log-log plot. At low temperature, the low frequency point is well below the lineof slope 1/2 which represents scaling, whereas at high temperature, the scaling isapproximately square-root like.In figure 6.4, R2 versus B at 27 GHz has been plotted and indicates that R3 hasa roughly square-root dependence on the magnetic field. Ignoring the term in theCoffey-Clem expression (which we reproduce here for convenience)2 1/2LPu/PO F0BA1+2i/Sf , Pu— q(1—iw/w)we showed in Chapter 3 that R oc in both the flux-flow and flux-pinned regime(ignoring any field dependence in , and ). Thus, the field dependence is consistentwith this. However, it is not clear that the A, term is completely negligible especially atthe higher frequencies where the mixed state skin depth decreases. Thus the non-linearportions at low temperature in the R curves for low fields could be due to the ) in thenumerator of the Coffey-Clem expression 6.1 The ) term increases in importance forlow fields and low temperatures where B is small and pj is likely to be small also.Another feature of the data is the role played by fluctuations in determining R3as H —* H2. We have already discussed the pronounced rounding of the dc resistivetransition starting at about 20K above T . Thus as H — H2, R3 takes on a valuegiven by the fluctuation dominated Pdc. This fact must be kept in mind when we lookChapter 6. The Analysis 941 01.01ci)crDCl)10121 01.610Figure 6.3: Frequency dependence of R at low and high temperature at 4T. At lowtemperature, the 5.4 GHz point is well below a square-root frequency dependence.frequency (MHz)Chapter 6. The Analysis 95le-03 I I020K8e-04 I30K6e-0494e-0402e-04- Q0-0e+00°°8e-03 I I I060K6e-03 70K4e-03002e-03o 00Oe+000.0 2.0 4.0 6.0 8.0B(T)Figure 6.4: Magnetic field dependence of R at 27 GHz. At low temperatures the curvature for small fields is due, presumably, to the presence of the ) term. At highertemperatures, R8 is looking more square-root like.Chapter 6. The Analysis 96at the field dependence of R3 and related quantities when H is not small compared toH2 . For example, figure 6.5 shows R8 at 27 GHz scaled by a normal state surfaceresistance obtained using a resistivity extrapolated from the linear portion of the normalstate resistivity from above T and plotted as a function of H/H2(T) where H2 (T) isgiven by the empirical expression in Chapter 2. It seems reasonable (we will discuss thisin greater detail in section 6.3) to use the linear resistivity to set the scale for the surfaceresistance for temperatures away from T and H2. Indeed, for temperatures above 70K,scaling the data iii this way does a good job of mapping the field sweeps onto a universalcurve (for temperatures 70K and below, we are presumably seeing suppressed loss andsurface resistance due to pinning). However, R3/R does not go to 1 as H/H2 goes to 1nor should we expect it to, because as H/H2 —* 1, fluctuation effects start to enter. Thiscan also be seen in the (R/R)2versus H/H2 plot (inset in figure 6.5) where the datalooks roughly linear for low fields but begins to curve over at higher fields as fluctuationstake over.Clearly, we would like to get a view of the data uncontaminated by pinning effects.What we are really interested in, after all, is the flux-flow resistivity or equivalently thevortex viscosity. This process will be described in the next section.6.2 Extraction of the pure flux-flow resistivity6.2.1 Preliminary discussionWe will be using the Coffey-Clem expression for the surface impedance from equation6.1. However, before we do, we must address an important issue: Does it make sense touse this expression given the geometry of the sample and the rf fields in our experimentalsetup? The Coffey-Clem expression is derived with the boundary condition h parallelto the surface of the sample. However, in both the 5.4 GHz split-ring resonator and theChapter 6. The Analysis 971.0Figure 6.5: Normalized R8 versus reduced field at 27 GHz. This scaling does a reasonablejob of mapping the data onto a universal curve00.4H/HC2Chapter 6. The Analysis 9827 and 35 GHz cylindrical cavities we place the sample into perpendicular h fields. Tosolve this problem in general is quite complicated (involving demagnetizing factors etc.)but we do know the qualitative solution to the problem. The h fields will bend aroundthe sample and the current distribution will be non-uniform and concentrated principallyaround the edges of the sample on the top and bottom surfaces near to the edges (seeAppendix B). The current is distributed so as to screen the h field or, equivalently,to maintain the boundary condition that h be always parallel to the surface. So asfar as the physics of the material is coilcerned, the two geometries are equivalent; itis the current distribution which is different. We argue that this non-uniform currentdistribution does not produce different physics for the following reasons: the screeningcurrents all run in the a-b plane and the relevant quantity is the a-b plane penetrationdepth— c-axis currents are not involved; the single crystals are of high quality andthe superconducting properties are not expected to vary over the surface of the sample(this has been explicitly verified in numerous zero-field experiments on many differentcrystals — in addition Kuan Zhang, Walter Hardy and Saied Kamal[72] have measuredR3 with h tangential to the surface and similar results are always observed); the surfaceresistance has been found to be independent of power for the range of low power used inthese experiments, and so the varying power in the surface currents near and around theedge of the sample should always give the same value for the surface impedance (which,after all, is just the ratio of the e11 and h at the surface).One other point that must be mentioned is that on the edges of the sample thevortices are parallel to the surface while on the top and bottom surfaces the vorticesare perpendicular. However, we noted in Chapter 3 that Coffey aild Clem have shownthat these two situations give rise to the same expression for the surface impedance. Notincluded in the Coffey-Clem calculation is the possibility of vortex bending. We assumethat this does not play a role. Given the anisotropic nature of high-Ta supercoilductorsChapter 6. The Analysis 99and the small displacements of the vortices from their equilibrium position with veryhigh frequency currents, this is probably not a bad assumptioll.Equation 6.1, for the surface impedance has four unknowns: AL, u,, , and i,. ALis the London penetration depth, and for this we use the penetration depth of Hardyet al measured on single crystals made with the same technique (assuming AL(O) =1500A). For the field dependence we use the Ginzburg Landau expression AL(B) =AL(O)/\/l — B/B2. u is the real part of the conductivity for the quasiparticles in thenormal fluid in the bulk of the superconductor. This can be simply extracted from thezero-field R5 versus T curve using AL(T) (see discussion of the zero field data of Bonn etal in Chapter 2). We do this for only one frequency although it is likely the u is mildlyfrequency dependent in the range 5 to 35 GHz. Such details are in fact almost irrelevant.The value of u only enters in the denomillator as part of the expression 1 + 2iA/S,which approximately equal to 1 until we are within a degree of T (where AL diverges).We are thus left with two unknowns: and ic. We can write the surface impedanceexpression in terms of ‘ii and = (see equatioll 6.1) and we choose to do the fittingin terms of these parameters.The remaining question is then how to do the fitting. However, before we discussthe fits in detail, our perspective on this whole process should be reiterated: we arereally only interested in w or ic to the extent that it affects our ability to reliablyextract the flux-flow resistivity. The pinning strengths and frequencies are very likelyto be sample dependent and influenced by factors such as twin density and impurityconcentration. Also, the concept of a single pinning frequency is itself highly questionablesince in general we might expect a distributioll of pinning strengths and frequencies.should be regarded as an effective pinnillg frequency. It represents our best effort, giventhe limited frequency information available, at modelling the pinning and removing itseffects from the data. Thus our discussion of the pinning strengths and frequencies willChapter 6. The Analysis 100be restricted to qualitative features and orders of magnitude.6.2.2 Fitting with a field dependent r andWe expect both i and to be temperature dependent. The nature of any field dependence is less clear; however, assuming a field independent , for example, is equivalent toenforcing a flux-flow resistivity that depends linearly on the field aild given the previousexperiments on other superconductors this hardly seems justified, (there is no compellingprecedent for assuming , to be independent of field either). It seems reasonable to assume that both and are illdependent of frequency. Indeed, the whole phenomenologyof ac effects in the mixed state makes this assumptioll. Thus for every field and temperature at which we have three frequency points, we can fit for and w. For B > 4T, andcertain temperatures such as 20K and 78K < T < 90K, data at all three frequencies isnot available; in fact, at some fields and temperatures there is only one frequency point(the most complete data set was obtained at 27 GHz). At 20K, we simply solve for wand ij using the 27 and 35 GHz points while for T> 78 K, the pinning frequency is smallenough (see figure) that we can just set it to zero and use the 27 GHz point to determine‘7.The fits were performed using the MINUIT fitting program and are given in figures 6.6— 6.12. The error bars on the data points result almost entirely from the uncertaintyin the overall calibration constants used at each of the three frequencies. As described inChapter 5, the 27 and 35 GHz data were both calibrated by using the classical skin effectformula and assuming a dc resistivity at lOOK typical of the U.B.C. YBa2Cu3O695 singlecrystals. There is a variation from crystal to crystal of ‘-.-‘ +10% in pd(100K)[l7]. The5.4 GHz data was calibrated usillg a reference sample whose dc resistivity was known.There is thus an uncertainty of at least +10% in the relative value of R8 at 5.4 GHzwith respect to R3 at 27 and 35 GHz. Because the 27 and 35 GHz data were calibratedChapter 6. The Analysis 1010.040.03cE30Co0.010.00frequency (GHz)60Figure 6.6: Frequency fits at 20 K. Q: iT; K: 2T; D: 3T; A: 4T; *: 5T; x: 6T. Ill fact,these are not fits but solutions of the Coffey-Clem expression for the two parameters riand using the two frequency points at each field.0 20 40Chapter 6. The Analysis 1020.040.03G)ctsc3-Co0.010.000frequency (GHz)60Figure 6.7: Frequency fits at 30 K. Q: iT; O: 2T; D: 3T; A: 4T; *: 5T; x: 6T. Thesefits allow both i and w, to be field dependent.20 40Chapter 6. The Analysis 103a)0Cof 0.02Figure 6.8: Frequency fits at 40 K. Q: iT; K’: 2T; D: 3T; A: 4T; *: 5T; x: 6T. Thesefits allow both r and w to be field dependent. 20 40frequency (GHz)60Chapter 6. The Analysis 104DFigure 6.9:fits allow both ri and , to be field dependeilt.Frequency fits at 50K. Q: iT; : 2T; El: 3T; A: 4T; *: 5T; x: 6T. These0. 20 40frequency (GHz)60Chapter 6. The Analysis 1050.080.06(I)0.020.000 60frequency (GHz)Figure 6.10: Frequency fits at 60 K. Q: iT; O: 2T; D: 3T; A: 4T; *: 5T; x: 6T. Thesefits allow both ri and w to be field dependent.20 40Chapter 6. The Analysis 106ci)I—:3Cl)Cl)ci0.020.00Figure 6.11: Frequency fits at 70 K. Q: iT; : 2T; D: 3T; A: 4T; *: 5T; x: 6T. Thesefits allow both ri and w, to be field dependent.0.100.080 20 40frequency (GHz)60Chapter 6. The Analysis 1070.150.10ci)I...D0,Co0.050.000 60frequency (GHz)Figure 6.12: Frequency fits at 78 K. Q: iT; O: 2T; 0: 3T; A: 4T; *: 5T; x: 6T. Thesefits allow both r, and to be field dependent.20 40Chapter 6. The Analysis 108assuming the same value for pa(10OK) they are not subject to this +10% uncertainty -we estimate their relative uncertainties to be +3%. To represent these uncertainties inthe absolute value of the surface resistance we have put +10% error bars on the 5.4 GHzdata and +3% error bars on the 27 and 35 GHz data. Given these error bars, the fitsto the data are satisfactory. It should be kept in mind that an error in any one of thecalibration constants has the potential to globally affect the look of the fits.The field dependence of the fitted vortex viscosity values at a given temperature aretypified by figure 6.13 which shows , vs B at 82K. At 82K, the pinning frequency isclose to zero and so this curve is derived from only the 27 GHz data. However, it isfairly representative of the field dependence at all the temperatures (except possibly forT < 40K where we can fit the data reasonably well with a field independent r, seesection 6.2.3): it changes most quickly at low fields and levels off at the higher fields.This suggests the possibility of a genuinely field independent q at higher fields. It alsoraises the possibility that insufficient knowledge of the London penetration depth andits magnetic field dependence is affecting, at low fields, the quality of the fits and theresultant fitted values of . At low fields, we are most sensitive to errors in )‘L sincethe flux-flow term in the numerator is proportional to B whereas the term probablyincreases more slowly. Thus, at higher fields, the flux flow term dominates. For example,at 82K, ) is 9.7 x l0’ and the flux-flow term is 13.9 x l0_14 at 1T (still comparable)while at 8T, the pj term becomes 111 x l0’. The effect of a field dependent onpff is to introduce curvature in its field dependence at low fields, gradually straighteningout as the field increases. Typically, we expect a linear p (or field independent ) forfields, B << B2 with r, being solely a property of an isolated vortex and independentof the density of vortices. To show the temperature dependence of w, and , or pwe will use the 4T data as representative of all fields. It will become evident that thefield dependence of and w, is a relatively small effect compared to their temperatureChapter 6. The Analysis 1098e-08 I I0 006e-O800____8e-084e-080)7e-08 o1 0Cl) 06e-0802e-08 5e-0804e-080 2 4 6 8B (T)Oe+OO I I I0 2 4 6 8B(T)Figure 6.13: Field dependence of the vortex viscosity at 82K. Inset: close-up. i appearsto level off as the field increases.Chapter 6. The Analysis 110dependence. We will address this issue once again in section 6.2.3.Figure 6.14 shows,as a function of temperature for B = 4T and this curve isrepresentative of the temperature dependence at all fields (with the possible exceptionof iT). This is the result which we anticipated in the qualitative overview of the datagiven in section 6.1. The error bars were generated by the MINUIT fitting program afterhaving been provided with the experimental uncertainties on the individual frequencypoints. The error bars are large because there are only three data points and the 27 and35 GHz points are providing information mainly about the flux-flow resistivity. Nearlyall the information on the pinning frequency comes from the overall difference in R8between low and high frequency. We see that the pinning frequency is of the order of20 GHz at 20K and goes to zero at T 80K. Thus at low temperature, all threefrequencies are affected by pinning: the 27 and 35 GHz data are slightly reduced fromwhat they would have been in the absence of pinnillg and the 5.4 GHz data is almosttotally suppressed. As the temperature is increased the surface resistance at the twohigher frequencies quickly become only slightly affected by pinning while at 5.4 GHz thesurface resistance experiences a transition from a flux pinned to a flux-flow regime.Figure 6.15 shows the temperature dependence of the flux-flow resistivity at 4T. It ishighly temperature dependent especially above 50K where it increases rapidly towardsits value near T . It is not hard to understand how this curve comes out of the fittinggiven the high frequency R curves. Since R8 pff1/2, it is clear that this curve mirrorsthe surface resistance except for the fact that the pinning has been removed. We cansubstitute this pj back into the Coffey-Clem equation to generate the R8 that we wouldhave measured in the absence of pinning. This is shown in figure 6.16 for B = 4T at 27GHz. We can see that higher loss would have been measured at low temperatures if ithadn’t been for pinning effects.In figure 6.17, we have plotted pff against B for the whole range of temperaturesChapter 6. The Analysis 11130 I I20>C)Ca)a)0)CCClO00 I I0 20 40 60 80T(K)Figure 6.14: Temperature dependence of the pinning frequency at 4 T. At temperaturesmuch above 80 K, we are seeing almost pure flux-flow at all frequenciesChapter 6. The Analysis 11240 I I I654or E%JlJ 0o 0I 0 20 40 60 80T(K)010a00 0 000 20 40 60 80 100T(K)Figure 6.15: Temperature dependence of the flux-flow resistivity at 4 T. Inset: a close-upof the low temperature behaviour.Chapter 6. The Analysisa)Coct0. 6.16: Effect of pinning on R3 at 27 GHz in a 4T field. R3 is not affected at hightemperature but at low temperature it is suppressed by pinning. The circles show thecurve as measured. The squares are R3 generated from the pj data so that the effect ofthe pinning has been removed and this curve is what we would have measured if therehad been no pinning. 0C!a030O40T (K)50 60BDO0D0D0D0 020 40 60 80T(K)Chapter 6. The Analysis 114studied in these experiments. We can see from these sets of curves that not only is theflux-flow resistivity highly temperature dependent but that it is so in a manner verydifferent from that observed by Kim et al[1] (see figure 3.3). In their data, they only sawtemperature dependence once they departed from the low-field linear region. Our datacompares better with that of Fogel[3] (see figure 3.4).6.2.3 Fitting with a frequency and field independent and wThe problem, of course, with fitting to three data points with two parameters is that weare extremely sensitive to error in any given data point (as we have seen in the previoussection especially with the low field data). It is instructive to try fitting the data atall three frequencies and all fields at a given temperature in terms of a frequency andfield independent vortex viscosity and pinning frequency. In this situation, we are muchless sensitive to fluctuations in the individual data points: at temperatures where wehave information at all three frequencies, 22 data points can be included in the fitting interms of only two parameters. The fits at temperatures 20 - 82 K are shown in figures6.18—6.25. At the lowest temperatures, 20 and 30 K, the fits are reasonably goodand this gives us some more confidence that the Coffey-Clem model is applicable to ourdata. At low temperatures, we might indeed expect to fit in terms of a field independentvortex viscosity (equivalently, a flux-flow resistivity that is linear in the magnetic field)especially for B << B2. Note that this good fit in terms of field independent parametersmeans that the model is correctly describing the characteristic kilee in the R versus Bcurve that we have seen in figure 6.4 and that we discussed also in Chapter 3. This isthe signature of the A term and would seem to indicate that our value for \L(0) is nottoo far off. This further strengthens our conviction that the Coffey-Clem expression isdoing a good job of describing the data at these temperatures.However, the fits are not as good at the higher temperatures. It is important at thisChapter 6. The Analysis 11540 I30 VVVV +20:±U+V +10 +V + ** I,* C* C04B(T)Figure 6.17: Field dependence of the flux-flow resistivity at all temperatures. Q: 20K;A: 30K; x: 40K; D’: 50K; 0: 60K; *: 70K; I: 78K; + : 82K; : 86K; 1: 90K. Thereis strong temperature dependence even at low fields.Chapter 6. The Analysis 1160.040.03a):50CoaC))0.010.000 10B(T)Figure 6.18: Fitting as a function of frequency and field at 20K. K: 27 GHz; D: 35 GHz.Both r and are field independent.2 4 6 8Chapter 6. The Analysis 117a)ctsCl) 10B(T)Figure 6.19: Fitting as a function of frequency and field at 30K. Q: 5.4 GHz; K’: 27GHz; 0: 35 GHz. Both j and w are field independent.2 4 6 8Chapter 6. The Analysis 1180.050.040) -DCl) 6.20: Fitting as a function of frequency and field at 40K. Q: 5.4 GHz; G: 27GHz; D: 35 GHz. Both ri and w, are field independent.0 2 4 6 8Chapter 6. The Analysis 1190.060.04ciC,)0.020.0010B(T)Figure 6.21: Fitting as a function of frequency and field at 50K. Q: 5.4 GHz; K: 27GHz; D: 35 GHz. Both r, and w, are field independent.0 2 4 6 8Chapter 6. The Analysis 120Figure 6.22: Fitting as a function of frequency and field at 60K. Q: 5.4 0Hz; K: 270Hz; D: 35 GHz. Both r and w are field independent.0.080.06ci) 2.0 4.0 6.0 8.0B(T)10.0Chapter 6. The Analysis 121Figure 6.23: Fitting asGHz; 0: 35 GHz. Botha function of frequency and field at 70K. 0: 5.4 GHz; K: 27and are field independent.0.100.081) -1ctSDc3-Cl) 2 4 6 8B(T)10Chapter 6. The Analysis 1220)CtsDCl) 10B(T)Figure 6.24: Fitting as a function of frequency and field at 78K. Q: 5.4 GHz; : 27GHz; 0: 35 GHz. Both r, and c’, are fieki independent.2 4 6 8I-DCl)Chapter 6. The Analysis 1230. 10B(T)Figure 6.25: Fitting as a function of field at 82K and 27 0Hz. r is field independent,2 4 6 8and has been set to zeroChapter 6. The Analysis 124point to distinguish between two separate issues: first, the poor overall fit to the 5.4 GHzdata at intermediate temperatures; second, the poor fit to the field dependeilce of, inparticular, the high frequency data. The first point is most probably due to limitationsof the single pinning frequency model. At low temperatures, the pinning frequency islarge compared to 5.4 GHz and at high temperatures it starts to be small compared to 5.4GHz. At these temperatures, therefore, we are less sensitive to the shape of the crossoverfrom the flux-pinned to the flux-flow regime. At the intermediate temperatures where thepinning frequency is comparable to 5.4 GHz is where we are most sensitive to the exactshape of the crossover: it will be relatively sharp for a sillgle pinning frequency modelbut broadened out if there is a distribution of pinning frequencies. The second point ismore difficult and one about which we will be able to come to no definite conclusion. Itis unclear whether or not the shape of the field dependence and its systematic deviationfrom the fitted curve is due to genuine field dependence of or perhaps w (certainly,canilot alone be responsible because the anomalous field dependence persists even after,,has become very small) or our lack of knowledge about the true field depeildence ofthe London penetration depth. Comparing the 78 K field profiles of the 35, 27 and 5.4GHz data, figure 6.24, suggests the possibility of a answer. The field depeildence of the5.4 GHz profile is not inconsistent with the fit whereas the 27 and 35 GHz curves clearlydeviate systematically from the fitted curves. Since we expect and wi,, to be frequencyindependent, we might therefore conclude that field dependence of these parameters isnot responsible for the systematic deviations at the higher frequencies. The effect of the) term on the other hand is quite frequency dependeilt; as we have already seen, atlow frequencies its importance is diminished because of the in the denominator of theflux-flow skin depth. However, the large error bars on the 5.4 GHz data preclude anydefinite conclusions based on the shape of this curve. It is also quite possible that fielddependence of the vortex viscosity due to interactions or a non-zero Hall effect could beChapter 6. The Analysis 125responsible. It is interesting to note that a good fit to the data at, for example, 70 Kcan be obtained by setting )L(0) to effectively zero as in figure 6.26. In other words, thefield profiles are almost purely square-root like at this temperature. However, figure 6.27shows the same procedure tried at 20 K only here the fit is poor. Evidently, at least atlow temperature, the presence of the ) term is reflected in the data.Figure 6.28 compares the temperature dependence of at 4T obtained by the fielddependent and field independent fitting procedures. The curves are qualitatively similar,the largest discrepancies occurring at intermediate temperatures (see inset)— an effectanticipated by our previous discussion. It is clear that on the scale of the changes observedas a function of temperature, the field dependence is a minor effect. When we discussthe temperature dependent scaling of the data in the next section, we will use the pjcurves obtained in section 6.2.2 and shown in figure 6.17. This is more appropriate foran overall discussion of the data because close to T and H2 we have already seen thatthe field dependence of the surface resistance is affected by fluctuations.6.3 Scaling the DataIn order to try and uncover the mechanism that gives rise to the data set, the obviousstarting point is to try scaling the data by p,-, and H2 as was done in the experiments onconventional superconducting alloys. We face two immediate problems. First, in scalingwith H2 , do we use the temperature dependent H2 (T) or its zero temperature valueH2 (0). Second, what should we use for p,-, ? This is perhaps the most fundamentalquestion that we will address in this thesis because this is equivalent to asking about thescattering time for the electrons in and possibly around the cores. In other words, weare trying to deduce information about the nature of dissipation in the moving vortexcores. With the possibility of discrete, widely spaced electronic levels in the cores, isChapter 6. The Analysis 1260Cl,0.04Figure 6.26: Fitting as a function of frequency and field at 70K, )L(O) “-i 0. Q: 5.4 GHz;K: 27 GHz; D: 35 GHz. Both,and w are field independent. 2 4 6 8B(T)10Chapter 6. The Analysis 127Cl)C,)cr0. 10B(T)Figure 6.27: Fitting as a function of frequency and field at 20K, AL(0) 0. K: 27 GHz;U: 35 GHz. Both ri and are field independent.2 4 6 8Chapter 6. The Analysis 12825 I I I6 I20E0I15C.) 0 I20 40T(K)60 805.0-20 40 60 80 100T(K)Figure 6.28: Comparison of different fitting procedures. The circles were generated bythe field dependent fits and the triangles by field independent fits. Inset: a close-up ofintermediate temperatures where there is a slight discrepancy.Chapier 6. The Analysis 129it even appropriate to think of a core as a normal material with an effective normalresistivity ? How would the picture change if the gap has d-wave symmetry and isthus effectively gapless ? In conventional superconducting alloys, the p used was thetemperature independent impurity limited value. For YBa2Cu3O695 , the normal stateresistivity above T is not constant but linearly decreasing until fluctuations set in about20-30K above T . As a first attempt at a guess for p, we will use an extrapolation ofthe linear portion of p,-, from above Tp(T) = 0.869 T + 0.85j—cm . (6.2)This was obtained from dc resistivity data[17] and is consistent with the dc resistivityvalue used to calibrate the surface resistance (pd(100K) = 77.8 tf—cm). The vortex coreis thus being treated like a region of normal material: its resistivity is determined by thenormal state properties with a linear scaling factor to take into account the temperaturevariation. We would not expect the core resistivity to be affected by the fluctuationcontribution to the resistivity because over most of the temperature range of interest, itis well away from T (except possibly when H gets close to H2 ). Scaling pjj by just thislinear p. and a temperature independent H2 (0) gives the set of curves shown in figure6.29. Comparing with figure 3.3 from Kim et al[1] shows that we are seeing qualitativelydifferent behaviour i.e. a law of corresponding states as given by equation 3.53 does notfit the data.If we use the H2 (T) function given ill Chapter 2 (equation 2.2) to scale the magneticfield we get the result shown in figure 6.30. It is clear that this scaling of the flux flowresistivity and the magnetic field does a reasonable job of bringing all the field curvestogether onto a universal curve suggesting the relationfJff — H63p(T) — H2(T)Chapter 6. The Analysis 1300.5 I I0.4 VVV0.3-+C V I.V +O2-+V+ ***0.1- v * H0.02 4 6 8B(T)Figure 6.29: Field dependence of pff/pfl(T). Q: 20K; A: 30K; x: 40K; O: 50K; D: 60K;*: 70K; L: 78K; + : 82K; : 86K; ci: 90K. p(T) is given by a linear extrapolation ofthe normal state resistivity from above T : p(T) = 0.869T + 0.85l-cm.Chapter 6. The Analysis 1311.0H/H(T)Figure 6.30: pff/p(T) as a function of the reduced field H/H2. p(T) is again givenby the linear extrapolation of the normal state resistivity from above T as in figure 6.29and H2 (t) is given by the empirical expression of Chapter’0.00.0 0.2 0.4 0.6 0.8Chapter 6. The Analysis 132This would favour a picture in which the amount of dissipation is proportional to thevolume fraction of normal material. It is also more in line with the behaviour observedby Fogel[3] in his experiments on PbIn. Again we see that for H close to H2 , pff beginsto curve over as it become contaminated by the fluctuation effects that affect the surfaceresistance close to T and H2 (see the discussion in section 6.1).We can get a better appreciation for how well this empirical expression is describingthe data by assuming that it holds and using the p data and say a linear p,. to predictH2(T). Alternatively, we can assume an equation for H2 (T) and predict p(T). Theresults are shown in figure 6.31 and figure 6.32. The picture seems to hold down toabout 50K, but, below that, significant deviations occur. The flux-flow resistivity is notdecreasing fast enough at low temperatures to maintain consistency with equation 6.3.One possibility that might account for the above discrepancy is that our formula forH2 (T) is using an inflated value for 11C2 (0). Indeed, there is some experimental supportfor this possibility. Two different Japanese groups[73, 74], using pulsed magnetic fields,have reported values of about 40T for H2 (0) (field parallel to the c-axis) in YBa2Cu3O7_5single crystals. The reason for the discrepancy between this value and the one estimatedfrom theory is not well understood. In any event, such a value for 11C2 (0) would meanthat pff is below what we would expect from this formula.It is also possible that the low temperature behaviour in figure 6.31 has something todo with Pauli spin paramagnetic limiting of H2 as discussed in section 3.5 in connectionwith the flux flow resistivity measurements of Kim et al[1] on high field superconductingalloys such as Ti-V. For conventional type-Il superconducting alloys in the dirty limit,Werthamer et al[32] calculated how Pauli spin paramagnetism competed with another effect — spin-orbit impurity scattering’ in its effect on H2 . They found that‘Spin-orbit scattering refers to a process whereby an electron (due its spin-orbit interaction) canscatter off an impurity in the superconductor with a flip of its spin.Chapter 6. The Analysis 1331.0Figure 6.31: Predicted H2 using Pff data. Equation 6.3 was used along with the expression for p(T) used in figures 6.29 and 6.30 to calculate H2 from pe.12510075502500.0 0.2 0.4 0.6 0.8t=T/TChapter 6. The Analysis 13480600402000.0 1.0Figure 6.32: Predicted p(T) using pff data. Equation 6.3 was used along with theexpression for H2 used in figure 6.30 to calculate p(T) from0.2 0.4 0.6 0.8t=T/TChapter 6. The Analysis 135spin-orbit scattering renders the spin paramagnetism less effective in reducing H2 . InYBa2Cu3O695 , which is in the clean limit, this scattering mechanism is presumably lessimportant than in dirty, conventional type-IT superconductors, and thus we might expectPauli spin paramagnetism to be an important factor in limiting H2 at low temperaturesin this material. However, it is difficult to quantify this because the standard calculationsfor the reduction of H2 due to paramagnetic limiting are for BCS s-wave superconductorsin the dirty limit rather than the clean limit. Another important consideration is that,as we saw in section 3.5, Kim et al[1] found that the flux-flow resistivity at low fieldsand temperatures was related not to the paramagnetically limited H2 but rather to theexpected value ignoring paramagnetic effects— the volume fraction of normal materialbeing the physically relevant quantity. If we adopt a similar position with regards theflux flow resistivity in YBa2Cu3O695 , then we would lot expect a value of H2 based onpj to contain information about paramgnetic limiting.Chapter 7Discussion and ConclusionsBased on the temperature dependence of the flux flow resistivity and a reasonable estimate for H2 (T), we conclude that the temperature dependence of the scattering rate ofthe charge carriers in the vortex cores below T is qualitatively similar to a continuationof the linearly decreasing normal state resistivity from above T . This is suggestivethat the the picture of a vortex as having a ‘normal’ core has some applicability forYBa2Cu3O695Using the Bardeen-Stephen formula[9] at 20K with H2 (20K) 120T and the fluxflow resistivity determined from the microwave surface resistance, we obtain a value ofp(20K) 30t1—cm a decrease of a factor of —2—3 from its value ill the normalstate above T . This is in sharp contrast to the rapid decrease in the scattering rate ofthe quasiparticles in the normal fluid of the bulk superconductor away from the vortices.The surface resistance data in zero field measured on similarYBa2Cu3O695 single crystalsby Bonn et al[18, 19] indicate a drop in this normal quasiparticle scattering rate by afactor of -.50—100 between T and 20K. That there is qualitative similarity between thetemperature dependence of this scattering rate and that of the flux flow resistivity isprobably misleading. It seems more likely that the rapid drop in p is associated withthe steeply rising H2 with decreasing temperature below T . In fact, we have foundthat, for T> 50K, the flux-flow resistivity is in reasonable agreement with the equationfiff— H71p(T) — H2(T)136Chapter 7. Discussion and Conclusions 137where we use an extrapolation of the linear dc resistivity from above T given byp(T) = 0.869T + 0.85 tf—cm (7.2)and a temperature dependence for H2 given by(1 —t2)uoH2(T)= 125(1 +t2)1/ (7.3)This is in qualitative agreement with Kunchur et al[25] in their high current dc measurements of YBa2Cu3O7_8epitaxial films, although in our measurements the surfaceresistance (and thus the determination of pjj ) is affected by fluctuations close to T andH2 . To the extent that pj seems to scale with the inverse of the temperature dependentII2 , our results are also similar to Fogel’s[3] pff obtained from measurements of the dcresistivity of a PbIn alloy. However, our results differ markedly from those of Kim etal[1] and others[2] who find that pj can be described byH74p.— H2(0)If we try to adopt equation 7.4 to describe our data by incorporating all of the temperaturedependence of p into a rapidly temperature dependent p, this leads to absurdly largevalues of p(T) (>> p(T)) at higher temperatures. Therefore, the picture of the vortexdynamics, at least for T > 50K, seems to be a conventional one: the cores are regionsof normal material with a scattering rate for the charge carriers given by a simple linearextrapolation of the normal state material from above T ; the resistivity of the materialis given by approximately the volume fraction of the material in this ‘normal state’(H/H2(T)) multiplied by the resistivity of the normal material. However, it should beappreciated that this is at best an incomplete picture. In the Bardeen-Stephen model[9],the transition region outside the normal core contributes as much to the dissipation asthe normal core itself. In conventional superconducting alloys, the scattering rate isChapter 7. Discussion and Conclusions 138characterized in both regions by the same 1/r (because the scattering is in the impuritylimit). For YBa2Cu3O695 , we might expect different scattering rates for the two regionsbecause in the transition region it is possible that the ilormal fluid quasiparticle scatteringrate is a factor. It is not obvious therefore that a simple Bardeen-Stephen like model ofvortex motion should make sense in such a situation.Below 50K, p is larger than what we would expect on the basis of equation 7.1 aildour estimates of p(T) (equation 7.2) and H2(T) (equation 7.3). Although a large Hallangle can affect the flux flow resistivity,0B_0B= (1 +a2/) (7.5)(see equation 3.17), the result would be to decrease, not increase p . The possibilitythat a value of 125T is in fact an overestimate of H2 (0) has been discussed. Given thisuncertainty in H2 (0) it is possible that the validity of equation 7.1 can be extended tolow temperatures.It is worth reiterating at this point that, at low temperatures, the data analysis is mostsensitive to the limitations of a single pinning frequency model. The pinning frequenciesobtained from the fits, while consistent with estimates [75] for high-Ta superconductorsand with the data of Owliaei et al[62] (see figure 3.8— their surface resistance at 10GHz is greatly suppressed by 70K indicating a pinning frequency greater than 10 GHzat lower temperatures), should not be taken too seriously. At lower temperatures, whenthe pinning frequency is a substantial fraction of 27 or 35 GHz, we are most sensitiveto errors in pff caused by the possible inability of the model to properly fit the shapeof the crossover from the flux pinned to the flux flow state. We can certainly conclude(as above) that the p data at low temperature is inconsistent with a rapidly decreasingscattering rate. However, given the limitations of the model used to fit the data, theuncertainty in H2(0) and the uncertainty in the exact form for p(T), it is probablyChapter 7. Discussion and Conclusions 139unwise to make any other definitive statements about the low temperature behaviour ofpffWe are able to get good fits in terms of a field independent and at 20 and 30K.At these temperatures, we are fitting the data at all frequency and field in terms ofjust two parameters. This reinforces our faith in the applicability of the Coffey-Clemsolution[28, 29] to our data. A field independent corresponds to a linear dependenceof pff on the magnetic field and this is what we might expect at low temperature andfield. The functional dependence of the R3 field profiles deviate from a simple square-root like behaviour because of the presence of the screening term, , in the expressionfor the surface impedance. The effect of this term is made quite apparent by plottingversus B: at low fields, there is non-linear behaviour which gradually gives way to alinear dependence at higher fields where the flux flow contribution dominates. At highertemperatures, it is difficult to fit in terms of a field independent . Its field dependence isstrongest at low fields where the effect of the ) term is most important. This suggeststhe possibility that our lack of knowledge of the field dependence of )L is affecting thefitted value of . Other sources for the field dependence of could be interactions withother vortices or possibly Hall effects. While it is not possible to resolve these issues atpreseilt, we should remember that these are relatively minor effects compared to the scaleof the temperature dependence of or . Our conclusions based on this temperaturedependence are unaffected by the above discussion.It is unclear what bearing the large spacing between the energy levels of the quasiparticle states in the core of the vortex have on our results. The main effects of thediscrete nature of the energy spectrum happen, according to Hsu[22, 23], at a frequencycorresponding to the level spacing, 2/EF, which is 200—300 GHz for high-Ta superconductors. It is possible that we are in some sense in the low frequency limit of histheory in which, in the clean limit, he recovers an expression like that of Bardeen andChapter 7. Discussion and Conclusions 140Stephen[9].At present, it is unknown what effect the existence of a d-wave instead of an s-waveground state might have on the picture of moving vortices. We have seen that, withsome reasonable assumptions, the flux flow resistivity, at least in a qualitative sense, canbe understood in YBa2Cu3O695 within the same framework used for conventional s-wavesuperconductors. A better understanding of the vortex core energy level spectrum withinthe context of d-wave superconductivity is needed before the possible relevance of eitherto the flux flow resistivity can be ascertained.In summary, then, the central conclusions of this work are as follows:• The Bardeen-Stephen like expression, equation 7.1, is in reasonable agreement withthe data down to at least 50K using a linear extrapolation of the normal state resistivity from above T for p(T) and an estimate of H2 (T) based on magnetizationmeasurements and theoretical estimates from the literature.• The rapidly temperature dependent normal fluid quasiparticle scattering rate observed in zero field microwave measurements on the same crystal is not the relevantscattering rate for charge carriers in the vortex core.• The fact that a linear extrapolation of the normal state resistivity from above Tis consistent with the data in our analysis is perhaps suggestive that the idea of a‘normal’ vortex core is valid for YBa2Cu3O695Of great interest would be to extend these measurements to even higher frequencies.This would push the flux dynamics further into the high frequency, flux flow limit andmake the analysis less sensitive to any uncertainty incurred by fitting to the fairly crude,single pinning frequency model. A study of the sensitivity of the pinning frequency tothe density of twin boundaries would also be useful. It is possible that by measuringChapter 7. Discussion and Conclusions 141nearly twin-free crystals, experiments performed at the existing frequencies could betterprobe the high frequency limit.Appendix AThe Surface Impedance of Metals and SuperconductorsWe consider a material subject to a time dependent magnetic field at its surface. Thematerial occupies the half space x > 0 and we apply the field in the z-direction along itssurfaceH=H0e””t. (A.1)The material has conductivity & which is in general complex.We start by taking the curl of the Maxwell equationVxH=J+E0 (A.2)we getVxVxH=VxJ+Eo(VxE). (A.3)Using the Maxwell equationVxE=- (A.4)and the vector ideiltityVxVxH=V(V.H)-V2H=- (A.5)(V H = (l/10)V. B = 0) andJ=ó-E, (A.6)equation A.3 becomes2 oH 02HVH=ou--+eoiio--. (A.7)142Appendix A. The Surface Impedance of Metals and Superconductors 143Substituting equation A.1 into equation A.7, we havea2H9x2 =i,aow&H—eotow (A.8)The second term on the RHS of equation A.8 is the well-known displacement currentterm and is negligible at our frequencies provided that ö >> 2 (fm)1 (u(lOOK)106 (lm)1 for YBa2Cu3O695 ). If we let(A.9)then the solution to equation A.8 (having dropped the displacement current term) canbe writtenH = H0 ea (A.1o)The surface impedance is defined byz All8fg°Jdwhere E0 is the value of the tangential electric field at the surface of the material (ydirection). That the electric field and the current density are both in the y-direction caneasily be seen from the Maxwell equation A.2 (after dropping the displacement currentterm) which can be written(9IIz)= = (A.12)(we know that öH2/öy is zero by symmetry). By integrating equation A.12 we can alsoget thatj J dx = H x-0= H0 (A.l3)aild so equation A.l1 can be writtenZ= (A.l4)Appendix A. The Surface Impedance of Metals and Superconductors 144We can express E0 in terms of H0 using the Maxwell equation A.4 which, with ourboundary conditions, becomes-— =—--—. (A.15)Integrating this equation and setting x = 0 to give the fields at the surface, we findE0=i1tw;\H (A.16)and so the surface impedance can be writtenZ3 = = (A.17)Since is related to & by equation A.9, the surface impedance can be written in termsof the complex conductivity asz=(iow)h/2 (A.18)The surface impedance is often writtenZ3=R+iX (A.19)where the real and imaginary parts are referred to as the surface resistance and surfacereactance respectively. To get a better feeling for R3 we can calculate the power dissipatedper unit area of the surface. This is given simply by the magnitude of the time averagedPoynting vectorISavi IRe(E x H*)I (A.20)evaluated at the surface (x = 0). Substituting for E and H in equation A.20 usingequation A.14 givesSay = Re(EoH) = .ZRe(Z3HoHo*) = (ReZ3) Ho2 =R3H02. (A.21)Thus the power per unit area dissipated is proportional to the surface resistance.Appendix A. The Surface Impedance of Metals and Superconductors 145We conclude this appendix by deriving the expressions for the surface resistance of anormal metal and a superconductor (for the case of local electrodynamics).In a ilormal metal the conductivity is purely real and given by ö = o where isthe usual conductivity of a metal and p = 1/un is the resistivity. Substituting for inequation A.18 gives= (jItopnW)1/2 (1 +i)(/IoPw)h/2(A.22)and so the surface resistance for a normal metal isR3= (A.23)By substituting ä = o into equation A.9, we fluid1i (A.24)where/ 2 \1/2S= ( . (A.25)\ fLj n JFrom equation A.1O, we can see that this is the characteristic decay leuigth for the Hfield in the metal. Equation A.25 is, of course, the familiar expression for the skin depthin a metal.In a superconductor, the total current density has both a normal and a superfluidcomponentJ = J, + J8. (A.26)The normal component is given byJ=o1E (A.27)where o is the conductivity of the normal fluid. The superfluid component is determinedby the first London equation(A.28)Appendix A. The Surface Impedance of Metals and Superconduciors 146which gives (assuming an eu1t time dependence for the supercurrent)= —i a2 E ; a2=2 (A.29)w ALTherefore the total conductivity is complex and given by2 (A.30)jO WTo calculate the surface resistance for a superconductor, we substitute equation A.30 intoequation A.18 and take the real part. Except very close to the transition temperature,a1/a2 << 1, and so we do the calculation to first order in ui/a2. Equation A.18 thenbecomes= (L)’2 (_i +)“2 . (A.31)To take the real part we use the identityRe(A+iB)= (A2+B2+A)”. (A.32)The real part of equation A.31 is then1/2R8= ([Lw)u/2 (_i + 1 + (A.33)Expanding the square root to first order and substituting for a2 we finally get the approximate form valid for a1 <<a2,118 = ‘.2a2A3 (A.34)Similarly, we find that the surface reactance for a superconductor is given byX8=powAL (A.35)in the limit o <<u2.Appendix BRf Current Distribution in SamplesWe wish to find the current distribution in a thin, plate-like, metallic or superconductingsample in an rf magnetic field applied perpendicular to its broad face in the limit thatthe skin depth or penetration depth is much less than the thickness. A sample of thisshape with thickness 2c can be well approximated by an oblate spheroid with a cwhere a, b and c are the lengths of the semi-axes. The rf current distribution in suchan ellipsoid can be determined by analogy with the magnetostatic problem of a perfectlydiamagnetic ellipsoid in a uniform magnetic field, Hext, applied parallel to its minoraxis (of length 2c). The boundary conditions for the two situations are approximatelythe same: for the magnetostatic case, the magnetic field inside the sample, is zeroeverywhere, and for the rf case, is zero except in a thin layer (characterized by eitherthe skin depth or the penetration depth) near the surface. Thus, in the magnetostaticcase,B1 = km Hint = 0 (B.l)because km = 0 for a perfect diamagnet. For the internal H field, we haveext extHlflt=l+(km_l)N=l_N (B.2)where N is the demagnetizing factor. Therefore, the internal magnetization isextMint = —— Hint = — ,110 1—N147Appendix B, Hf Current Distribution in Samples 148Figure B .1: Surface current distribution in uniformly magnetized ellipsoid.i.e. is uniform and everywhere antiparallel to Hext. The surface current density,K (curreilt per unit length perpendicular to the direction of flow) is given byK = M1 x ñ = sinO (B.4)where ñ is a unit vector perpendicular to the surface, see figure B.1. It is clear from thediagram that the surface current runs in concentric rings centred about the minor axis(of length 2c) of the ellipsoid. The sample used for the measurements described in thisthesis corresponds to a ratio a/c ‘—‘ 150; therefore, for a sample of this shape most ofthe current runs very close to the edges of the crystal where sinO . 1. Thus, we havedetermined the corresponding rf current distribution for a metal or a superconductorand also, therefore, for a superconductor in the mixed state. The latter is threaded by avortex lattice, but to the rf fields these are just regions where the material looks more likea normal metal as opposed to a superconductor— the macroscopic current distributionis virtually unchanged.= MTfltXflAppendix B. 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