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Resistivity in Y Ba₂Cu₃O₆₃̣₃₃: DeBenedictis, Jennifer Jean 2005

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Resistivity in YBa2Cu^0^^z'-by Jennifer Jean DeBenedictis B . S c , The University of Brit ish Columbia, 1997 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in The Faculty of Graduate Studies (Physics) T H E U N I V E R S I T Y O F BRIT ISH C O L U M B I A Apr i l 6, 2005 © Jennifer Jean DeBenedictis, 2005 11 A b s t r a c t The resistivity of two high quality YBa,2CuzO§.zzz single crystals was measured, the first via the Montgomery method, which allows the determination of all three components of the resistivity tensor, and the second via a colinear arrangement of contacts that allows the determination of pa. A comparison of pa between the two crystals allowed the intrinsic behaviour to be separated from the extrinsic behaviour. The intrinsic behaviour of the resistivity was found to be a quadratic temperature dependence, subject to two abrupt and doping independent changes in the fitting parameters, which occurred at 45 K and 206 K (these changes were most visible in the derivatives of the resistivity). A low temperature Ln(^) behaviour, reported elsewhere in the literature, was found to be sample dependent, and thus extrinsic. The intrinsic behaviour of the resistivity showed a very weak upturn at low temperatures, but the ground state appears to be essentially metallic. pc was fit to a model having two components added in series, one component being of the form of one over a quadratic expression. This inverse quadratic term contained almost all the doping dependence of the c-axis resistivity, with the remaining term (dominant at low temperatures) proving to be almost doping independent. The derivative of one over the inverse quadratic term showed a remarkable similarity to the derivatives of the intrinsic in-plane resistivities, implying that the inverse quadratic term is also intrinsic behaviour. Most notably, the derivative of the inverse quadratic showed discontinuous changes of slope at 45 K and 206 K again. However, a further two changes of slope, occurring at 172 K and 239 K, were seen as well. It is suggested that all four of these changes in behaviour affect the resistivity along all three principal axes, but that the ones at 172 K and 239 K are too subtle to be seen above the experimental noise in the in-plane data. These changes of behaviour cannot be due to the charge carriers alone or associated with the C11O2 planes only. It is conjectured that they may be due to subtle structural changes in the unit cell that alter either the charge carrier-phonon coupling in the crystal, or the shape of the Fermi surface. A model due to Rojo and Levin was found to fit both the in-plane and out-of-plane resistivity data very well, provided that the low temperature (possibly extrinsic) contribution to the resistivity is neglected, and the temperature dependence of boson-assisted hopping along the c-axis is taken to be proportional to T 2 . This model is consistent with the metallic ground state suggested by the in-plane data, and further suggests that the pa ~ -f- behaviour seen in the data is to be expected P c for underdoped YBa2CuzOQ+x-i i i C o n t e n t s A b s t r a c t i i Contents i i i List of Tables v List of F igures vi Acknowledgements x i 1 Introduct ion 1 1.1 The Phase Diagram 1 1.2 The Compound Studied: YBa2Cu306.333 2 1.3 Resistivity 4 1.4 The Motivation 4 2 T h e o r y 6 2.1 Modeling the Electric Potential Inside a Sample 6 2.1.1 Colinear Contacts . , 6 2.1.2 The Montgomery Method 7 2.1.3 Rescaling the Axes 9 3 E x p e r i m e n t a l 11 3.1 Sample Preparation 11 3.2 Preparation and Anneal Times 15 3.3 Measuring Contact Spacings 16 3.3.1 Montgomery Contacts 16 3.3.2 Colinear Contacts 17 3.4 The Probe 17 3.5 The Electronics 19 3.5.1 Controlling Temperature 19 3.5.2 Measuring Resistance 20 Contents iv 3.5.3 The Computer Program . . . 21 3.6 Deconvolving the Raw Data For the Montgomery Method 23 3.7 Uncertainties 23 3.7.1 The Colinear Method 23 3.7.2 The Montgomery Method 24 3.8 Miscellaneous 25 4 Resu l t s 27 4.1 Temperature Dependent Resistivity for YBa2CuzO6.333; Montgomery Method . . . 27 4.2 Temperature Dependent Resistivity for YBa.2Cu3O6.333; Colinear Method 31 4.3 Uncertainties in the Resistivity Values for YBa.2Cu3O6.333 33 4.3.1 The Montgomery Method 33 4.3.2 The Colinear Method 36 5 Discussion 37 5.1 Comparison of the Two YBa2Cu30e.333 Results 37 5.2 Comparing U B C Data to Relevant Data From The University of Toronto 38 5.3 Fit t ing the a- and b-Axis Data 42 5.3.1 Montgomery Data 42 5.3.2 Colinear Data 46 5.3.3 Analysis of the a-b Plane Resistivity 53 5.4 Fit t ing the c-Axis Data 58 5.4.1 Analysis of the c-axis Resistivity 63 5.4.2 The Connection Between pa and P c ^ m p ) 69 5.4.3 The In-plane Resistivity Revisited 73 6 Conc lus ions 75 B i b l i o g r a p h y 78 V L i s t of T a b l e s 4.1 Effective dimensions of the underdoped YBa2CuzO§.zzz crystal used for the Mont-gomery method 27 4.2 Effective dimensions of Y Ba^CuzO^.zzz crystal used for the colinear method 31 5.1 Pseudogap temperatures from the literature 55 vi L i s t o f F i g u r e s 1.1 General phase diagram for the cuprate superconductors. A F M refers to the the antiferromagnetic phase, SC refers to the superconducting phase, P G refers to the pseudogap region, and F L ? refers to the region that may be Fermi liquid-like. . . . 1 1.2 The unit cell for YBa2CuzO-r.Q. Image provided by D. Peets [3] 3 2.1 The colinear arrangement of contacts. Current is injected through the sample via the end contacts, and the voltage drop is measured between the two middle contacts. This configuration only allows the determination of pa 7 2.2 The Montgomery arrangement of contacts (where the bottom surface of the sample is identical to the top, visible here). Current is injected through the sample via one pair of adjacent contacts, while voltage is measured across the two other adjacent contacts located on the same face of the crystal. The configuration of current and voltage contacts pictured here corresponds to the resistance measurement denoted by R\2, where "1 " refers to current flowing nominally in the " 1 " , or a- direction, and "2" refers to voltage measured across contacts displaced from the current contacts in the "2" or b- direction 8 3.1 The "cradle" mask, viewed from the bottom. The mask has two components: a 25 micron thick mask with a rectangular hole of the size of the sample cut into it, and a 10 micron thick "x" shaped mask that leaves small square areas of the sample's corners exposed to the evaporation source 12 3.2 The strap mask, for putting on colinear style contacts. Three long straps are cut into a rectangle of 10 micron thick foil, and this mask is folded, and then wrapped around the sample to expose the contact areas 13 3.3 Sample with Montgomery style contacts, mounted in such a way as to allow electrical contact to be made to all eight contacts. Wires are attached directly to the four exposed contacts with silver epoxy. The four hidden contacts are attached with silver epoxy to four gold foil squares that are in turn attached to the substrate. Wires are then attached to the gold foil squares to allow electrical contact to the hidden contacts. 14 List of Figures vii 3.4 The probe used to measure resistivity 18 3.5 Temperature regulation circuit for the probe. The heater and thermometer are at-tached to the sapphire block that the sample is mounted on; the rest of the circuit is kept at room temperature 21 3.6 Sample circuit for the probe. Current is passed through a 1000 ft resistor, then through the sample. Voltages are measured across both the known resistor and the sample, the former to determine the current through the sample 22 4.1 Resistivity along the a-axis for YBa2Cu-$0§.zzz as a function of temperature. The different curves were obtained for various anneal times. (From top to bottom) Red circles: nominally no anneal, Green crosses: 6 hours anneal, Purple squares: 19 hours anneal, Blue diamonds: 68 hours anneal, Orange stars: 328 hours anneal 29 4.2 Resistivity along the b-axis for FBa2Cu3<96.333 as a function of temperature. The different curves were obtained for various anneal times. (From top to bottom) Red circles: nominally no anneal, Green crosses: 6 hours anneal, Purple squares: 19 hours anneal, Blue diamonds: 68 hours anneal, Orange stars: 328 hours anneal 30 4.3 Resistivity along the c-axis for YBa2CuzO§.zzz as a function of temperature. The different curves were obtained for various anneal times. (From top to bottom) Red circles: nominally no anneal, Green crosses: 6 hours anneal, Purple squares: 19 hours anneal, Blue diamonds: 68 hours anneal, Orange stars: 328 hours anneal 30 4.4 Resistivity along the a- and b-axes for two anneals. Red circles: pa with no anneal, Orange squares: pt, with no anneal, Purple diamonds: pa with 328 hours anneal, Blue stars: pb with 328 hours anneal 31 4.5 Resistivity along the a-axis for YBa.2CuzOe.z33 as a function of temperature, mea-sured using the colinear technique. The different curves were obtained for various anneal times. (From top to bottom) Red circles: nominally no anneal, Green crosses: 6 hours anneal, Purple squares: 1 day anneal, Blue diamonds: 3 days anneal, Orange stars: 7 days anneal 32 4.6 Resistivity along the a-axis for YBa2Cuz06.z33 as a function of temperature, mea-sured using the Montgomery technique, and showing the limits of uncertainty. The red curve (circles) is the data after nominally no anneal, with its limits of uncertainty shown in orange. The blue curve (squares) is the data after 328 hours of anneal, with its limits of uncertainty shown in light blue. Note that the uncertainties for the two curves overlap 33 List of Figures vii i 4.7 Resistivity along the b-axis for YBa.2Cu3O6.333 as a function of temperature, mea-sured using the Montgomery technique, and showing the limits of uncertainty. The red curve (circles) is the data after nominally no anneal, with its limits of uncertainty shown in orange. The blue curve (squares) is the data after 328 hours of anneal, with its limits of uncertainty shown in light blue 34 4.8 Resistivity along the c-axis for YBa2Cu306.333 as a function of temperature, mea-sured using the Montgomery technique, and showing the limits of uncertainty. The red curve (circles) is the data after nominally no anneal, with its limits of uncertainty shown in orange. The blue curve (squares) is the data after 328 hours of anneal, with its limits of uncertainty shown in light blue 35 4.9 The effect on the a- and b-axis resistivity of increasing the effective b-axis length by its uncertainty. Red circles: a-axis. Blue squares: b-axis 35 4.10 The effect on the a- and b-axis resistivity of decreasing the effective b-axis length by its uncertainty. Red circles: a-axis. Blue squares: b-axis 36 4.11 Resistivity along the a-axis for YBa\Cu3 06.333 as a function of temperature, mea-sured using the colinear technique, and showing the limits of uncertainty. The red curve (circles) is the data after nominally no anneal, with its limits of uncertainty shown in orange. The blue curve (squares) is the data after 7 days of anneal, with its limits of uncertainty shown in light blue 36 5.1 A comparison of the resistivity along the a-axis measured using the Montgomery method (red circles) and the colinear method (blue squares) for YBa2Cu30,6.333- • 37 5.2 A comparison of the resistivity along the a-axis for the U B C and the UofT samples. Purple squares: U B C colinear sample with no anneal, Orange crosses: U B C colinear sample with 6 hour anneal, Red circles: U B C Montgomery sample with no anneal, Green stars: UofT sample " L " , no anneal, Blue diamonds: UofT sample " J " , no anneal. 39 5.3 A comparison of the resistivity along the a-axis for the U B C sample measured by the colinear method, and the UofT samples. Purple squares: U B C colinear sample with no anneal, Green stars: UofT sample " L " , Blue diamonds: UofT sample " J " 40 5.4 A comparison of the resistivity along the a-axis for the U B C sample measured by the Montgomery method, and the UofT samples. Red circles: U B C Montgomery sample with no anneal, Green stars: UofT sample " L " , Blue diamonds: UofT sample " J " . . 41 5.5 Low temperature comparison of the resistivity along the a-axis for the U B C sample measured by the Montgomery method, and the UofT sample " L " . Red circles: U B C Montgomery sample with no anneal, Green stars: UofT sample " L " 41 List of Figures 5.6 A log-plot comparison of the resistivity along the a-axis for the U B C sample measured by the Montgomery method, and the scaled UofT sample " L " . The approximately Ln(if) behaviour at low temperatures is evident. Red circles: U B C Montgomery sample with no anneal, Green boxes: UofT sample " L " 5.7 ^ for the a-axis for the Montgomery sample, fit to Equation 5.2, showing three linear sections with a t n (_ r ) term overlaid upon the entire data set 5.8 _p for the b-axis for the Montgomery sample, fit to Equation 5.2, showing three linear sections with a Ln{^) term overlaid upon the entire data set 5.9 The a-axis resistivity for the Montgomery sample, fit to Equation 5.3. (From top to bottom) Red circles: nominally no anneal, Green crosses: 6 hours anneal, Purple squares: 19 hours anneal, Blue diamonds: 68 hours anneal, Orange stars: 328 hours anneal 5.10 The b-axis resistivity for the Montgomery sample, fit to Equation 5.3. (From top to bottom) Red circles: nominally no anneal, Green crosses: 6 hours anneal, Purple squares: 19 hours anneal, Blue diamonds: 68 hours anneal, Orange stars: 328 hours anneal 5.11 The a-axis resistivity for the U B C Montgomery sample (red circles), and scaled low temperature data from a UofT sample (green stars), with the fit for the U B C data shown to low temperatures 5.12 ^£fr for the colinear data, fit to Equation 5.2, showing similar behaviour to the Mont-gomery a-axis data. The temperatures at which the graphs suddenly change slope are the same as for the Montgomery data 5.13 The a-axis resistivity for the colinear sample, fit to Equation 5.3. (From top to bottom) Red circles: nominally no anneal, Green crosses: 6 hours anneal, Purple squares: 1 day anneal, Blue diamonds: 3 days anneal, Orange stars: 7 days anneal. 5.14 ^f- for both the colinear (blue squares) and Montgomery (red circles) samples with the extrinsic Ln{^) term subtracted off, demonstrating the good agreement between the two samples 5.15 The a-axis resistivity for the colinear sample (blue squares) and the Montgomery sample (red circles) with the extrinsic Ln{^) term subtracted off 5.16 The ratio of pa for the colinear sample to pa for the Montgomery sample (red circles), with the scaled down anisotropy shown alongside it (blue line) 5.17 pc for the Montgomery data, scaled along the vertical axis to agree at 40 K. (From top to bottom) Red circles: nominally no anneal, Green crosses: 6 hours anneal, Purple squares: 19 hours anneal, Blue diamonds: 68 hours anneal, Orange stars: 328 hours List of Figures x 5.18 $ for the c-axis of the Montgomery sample, showing no discontinuity at 45 K, but (arguably) three discontinuities at 172 K, 206 K and 239 K. (From top to bottom) Red circles: nominally no anneal, Green crosses: 6 hours anneal, Purple squares: 19 hours anneal, Blue vdiamonds: 68 hours anneal, Orange stars: 328 hours anneal. . . 61 5.19 Close-up view of $f for the c-axis of the Montgomery sample, with curves shifted vertically to agree at 210 K, showing three discontinuities at 172 K, 206 K and 239 K. (From top to bottom) Red circles: nominally no anneal, Green crosses: 6 hours anneal, Purple squares: 19 hours anneal, Blue diamonds: 68 hours anneal, Orange stars: 328 hours anneal 62 5.20 pc for the Montgomery data, fitted by Equation 5.5. (From top to bottom) Red circles: nominally no anneal, Green crosses: 6 hours anneal, Purple squares: 19 hours anneal, Blue diamonds: 68 hours anneal, Orange stars: 328 hours anneal 63 5.21 The semi-conducting-like contribution to pc for the Montgomery data. (From top to bottom) Red circles: nominally no anneal, Green crosses: 6 hours anneal, Purple squares: 19 hours anneal, Blue diamonds: 68 hours anneal, Orange stars: 328 hours anneal 64 5.22 The anomalous hump contribution to pc for the Montgomery data. (From top to bottom) Red circles: nominally no anneal, Green crosses: 6 hours anneal, Purple squares: 19 hours anneal, Blue diamonds: 68 hours anneal, Orange stars: 328 hours anneal 64 5.23 The derivative of j- for the Montgomery data. (From bottom to top) Red circles: nominally no anneal, Green crosses: 6 hours anneal, Purple squares: 19 hours anneal, Blue diamonds: 68 hours anneal, Orange stars: 328 hours anneal 65 5.24 The derivative of j- for the Montgomery data, close-up view. (From bottom to top) Red circles: nominally no anneal, Green crosses: 6 hours anneal, Purple squares: 19 hours anneal, Blue diamonds: 68 hours anneal, Orange stars: 328 hours anneal. . . 73 xi A c k n o w l e d g e m e n t s I acknowledge that graduate school was a horrible, horrible mistake. Acknowledgements xii A c k n o w l e d g e m e n t s II Okay, okay, I wil l be a soft old codger and actually thank people. I would like to thank my husband Andrew for being supportive, encouraging and faultlessly loving through what proved to be some extremely difficult years for me. You da best, Sweetie. I would like to thank Pinder Dosanjh for building the probe with which I took my data, as well as for his invaluable help in getting me on track with my project. Pinder is one of the most amazingly useful people I've ever met. And I might still be polishing things if he hadn't laughed at me. I would like to thank Darren Peets for not only being the first person in the lab who would talk to me, but also the one who could consistently make me laugh. Darren also has one of the finest minds I've ever encountered, as does one Richard Harris: I thank both Darren and Richard for so graciously putting up with me always picking their brains. Richard and his wife Adeline Chin also provided valuable stress relief by always being willing to drop everything and go rock climbing with me. Darren, Richard and Addy are the people I met here who really became my friends. This has proved to be a very vibrant and stimulating lab group that I came to be a part of, and the one thing about this whole experience that has really been worth it is the wonderful people I've had the honour of working with. In no particular order, I would like also to thank Tami Pereg-Barnea, Patrick Turner, Mart in Kurylowicz, Geoff Mull ins, Jake Bobowski, David Broun, Guillaume Chabot-Couture, Grant Minor and Ricky Chu for being so damned lively and fun to work with. Finally, I would like to thank Drs. Doug Bonn, Walter Hardy and Ruixing Liang for their guidance through my degree, for being available as knowledge resources, and for always being willing to help when things went suddenly wrong. The quality of the work in this thesis is completely due to the quality of the crystals I was working with, and for this reason I am very grateful to have become a member of Doug, Walter and Ruixing's lab group. And I would like to compliment the individual faculties and departments at U B C for making sure that this remains a very fine academic institution in spite of the greedy short-sighted people who are currently mismanaging the place, treating the undergraduate students as cash cows and the graduate students as slave labour. Chapter 1 i I n t r o d u c t i o n 1.1 The Phase Diagram Figure 1.1 represents a commonly reported version of the doping phase diagram for the high temperature cuprate superconductors. doping Figure 1.1: General phase diagram for the cuprate superconductors. A F M refers to the the antifer-romagnetic phase, SC refers to the superconducting phase, P G refers to the pseudogap region, and F L ? refers to the region that may be Fermi liquid-like. The undoped parent compounds for the cuprates are Mott insulators - materials where simple band calculations predict metallic behaviour, but strong Coulomb repulsion between electrons on different sites prevents the electrons from being mobile within the crystal. As holes are doped onto the copper-oxygen planes of the cuprates, the charge carriers in the planes become mobile, and the crystals become more metallic. Moving along the T = 0 axis in Figure 1.1, one sees that with increasing doping, the materials cross out of the Neel region, where long-range antiferromagnetic order exists, and become superconducting. As doping is further increased, superconductivity is eventually suppressed again, and the materials apparently begin to behave as Fermi liquid metals. Chapter 1. Introduction 2 On the underdoped side of the superconducting dome, a partial gap in the spectrum of the quasiparticles opens up for both the spin and charge degrees of freedom. This partial gap is known as the pseudogap. The exact form of the phase diagram shown in Figure 1.1 is still disputed. The location of the pseudogap transition is not well defined, and there is argument over whether it intersects the superconducting dome near optimal doping (as shown), or smoothly merges into the overdoped side of dome. Another point of contention is the proximity of the antiferromagnetic phase to the superconducting phase. In the diagram above, the two phases are shown essentially touching one another at T = 0 , but there is the possibility that the two domes overlap slightly, or perhaps have a small gap between them. Since the samples studied in this thesis cover dopings very close to the underdoped side of the superconducting dome, the results presented here will help to clarify this matter. 1.2 The Compound Studied: Y Ba2Cu2,0%,zzz The unit cell for (fully doped) YBa^Cu^Oj is shown in Figure 1.2. This compound has two planes of copper and oxygen ions [CuOi planes) that lie parallel to the ab plane, straddling the yttrium ions. Superconductivity takes place primarily on the Cu02 planes, although the materials are three dimensional superconductors. YBazCuzOs+x (YBCO) also has chains of copper and oxygen ions [CuO chains) that run parallel to the b-axis. The oxygen ions present on these chains dope the planes with holes. This is preferable to chemical substitution (which is the doping method used in many other cuprates) because these substitutions occur directly adjacent to the planes, while the chains are farther removed from the planes. Thus, by being able to dope the planes with holes via the chains, less disorder is introduced directly into the Cu02 planes. Y B C O can be doped from the parent Mott insulator right into the slightly overdoped region of the superconducting dome. However, the actual number of holes doped onto the planes as a function of the number of oxygens in the chains is not a straightforward relationship. A n isolated oxygen on a chain will not be capable of pulling an electron up out of the planes, but two or more oxygens sitting adjacent to one another on a chain will do so (for a nice discussion of this, please see Reference [2], Chapter 2 ) . The fact that the oxygens in the chains must organise into "chainlets" to pull electrons out of the plane allows an experimentally tidy method of altering the doping. The oxygens in the chains are randomly distributed when the sample first comes out of the furnace after having its overall oxygen content set. Wi th annealing time at room temperature, the oxygens will begin self-organising into Chapter 1. Introduction 3 longer and longer segments of chain. Thus, by simply leaving the sample at room temperature for periods of time, different dopings are accessed in a single sample without altering the overall oxygen content. The ends of the chainlets are possible sources of scattering for the charge carriers in the planes. Some dopings of Y B C O can avoid this problem by having the chain-oxygens ordered into periodic structures, for example, every chain completely filled (YBa2Cus07, perfect Ortho I ordering), or alternating completely empty and completely full chains (YBCLICU-SOQ.S, perfect Ortho II ordering). In principle, the samples studied in this thesis (YBa.2Cu-iOG.333) can also be coerced into such a periodic arrangement, with two chains completely empty, alternating with one chain completely full (Ortho III* ordering). However, this ordered phase is not stable at room temperature and atmospheric pressure, and the pressures needed to form the ordered phase are not currently accessible by our lab group. Thus, the samples studied here will be subject to the disorder introduced by chain ends, and these samples will essentially be Ortho II samples with many oxygen vacancies, i.e. the oxygens in the crystal attempt to form the Ortho II ordered phase, but are thwarted due to the insufficient number of oxygens. Chapter 1. Introduction 4 1.3 Resistivity Resistivity measurements provided one of the first indications that charge transport in the high temperature cuprates was unlike anything described theoretically up to that point. The most striking example of this is the linear temperature dependence of the resistivity seen in optimally doped cuprates, which cannot be understood in terms of Fermi liquid theory. On the overdoped side of the superconducting dome, the resistivity tends toward a T2 temperature dependence, which implies that the materials are behaving like Fermi liquids. On the underdoped side of the superconducting dome, the effect of the pseudogap is seen. Below an ill-defined crossover-temperature, the resistivity begins to deviate away from a linear temperature dependence in this region. The samples studied in this thesis are underdoped, and should be well below the cross-over temperature for the pseudogap for the entire temperature range studied. There is no expectation of seeing a linear temperature dependence in the resistivity presented here, for the data are all well within the pseudogap region of the phase diagram. 1.4 The Motivation It is thought that high temperature superconductivity may have a close relationship to anti-ferromagnetism. For this reason, the study of compounds doped close to the superconducting-antiferromagnetic boundary is of importance, as it may shed light on how the two states are related. What is really being sought, is an understanding of how the charge carriers organise themselves at low temperatures, as this will hopefully help illuminate the mechanism that allows the electrons to form Cooper pairs at the very high temperatures that they do in the high Tc cuprates. A shocking number of theories exist for these materials, and determining what the ground state of the charge carriers is will help cull the number of potentially correct theories, as well as direct future work. As mentioned, the hole-doping of the planes in. Y B C O when the chain-oxygens are completely disordered is less than the doping when the chain-oxygens have ordered into short chainlets. By allowing a sample to anneal a,t room temperature after its oxygen content has been set, different dopings can be studied in the same crystal without subjecting the sample to any further alter-ation of its overall chemical composition. The YBa-zCu-iOn.^i samples studied in this thesis were measured as a function of annealing time, and thus as a function of hole doping as the oxygens organize into longer chainlets. When the samples were initially measured, they showed the start of a superconducting transition, but were not, fully superconducting yet. This places them close to Chapter 1. Introduction 5 the antiferromagnetic-superconducting boundary. When the samples were measured for their final time, they had well defined critical temperatures. The formation of electronic instabilities such as charge stripes and charge density waves has been suggested for these materials at low dopings and low temperatures. Since charge carriers tend to localise about any source of disorder, especially at low temperatures, it is extremely important that the samples used be very clean and well-ordered, or it will be very difficult to sort out possibly intrinsic charge-ordering or localisation from disorder-driven localisation. The U B C samples studied in this thesis are of exceptionally high chemical purity, and crystalline perfection [1], making them ideal for such a study. 6 Chapter 2 T h e o r y 2.1 Modeling the Electric Potential Inside a Sample The samples used in this work were all right rectangular prisms, whose sides were aligned with the principal directions of the resistivity tensor. Two methods were used to obtain the resistivity components. The so-called Montgomery method [7], based on the series calculations of Logan, Rice and Wick [6], is the more complicated of the two and will be dealt with in more detail. The other method used was the method of colinear contacts. To obtain the resistivity of the samples, electrical contacts of gold were first evaporated onto the surface of the samples, and then annealed to reduce the contact resistance. Details of this can be found in the Subsection 3.1 of this thesis. Current was injected into the sample through one contact and removed through another, while the voltage drop across a separate pair of contacts was measured. A resistance for the sample was then calculated by dividing the measured voltage by the current. This four-point measurement removes the resistance of the wires and contacts from the final resistance calculated. However, to obtain the components of the resistivity tensor from a set of such resistance measurements requires a suitable model for the electric potential function within the sample. 2.1.1 Colinear Contacts In this method, four relatively large contacts are evaporated onto the sample. The contacts are as pictured in Figure 2.1, with two current contacts encasing the ends of the sample, and two voltage contacts encircling the sample at some distance from the ends. In this configuration, only the resistivity along the long dimension of the crystal, which in this case was the a-axis, can be measured. Current flow in the b and c directions is assumed not to occur, an assumption that relies on the contacts having exactly the geometry shown. Care is taken to completely cover the ends and edges of the sample, so that both the b and c axes will be completely shorted out. This allows the experimenter to assume that the current flows only in the a-direction, and is homogenous through any b-c slice of the crystal, particularly between Chapter 2. Theory 7 Figure 2.1: The colinear arrangement of contacts. Current is injected through the sample via the end contacts, and the voltage drop is measured between the two middle contacts. This configuration only allows the determination of pa. the two voltage contacts. The voltage contacts also short out the c-axis of the crystal, to try to prevent any contamination of the measured a-axis resistivity by (the relatively large, in YBCO) c-axis resistivity. In this configuration of contacts, the measured resistance need only be multiplied by a geometric factor to convert it into the a-axis resistivity. The geometric factor is simply the distance between the voltage contacts divided by the cross-sectional area of the sample (the area of a slice in the b-c plane), i.e. where L is the distance between the voltage contacts, w is the width of the sample measured along the b-axis, and t is the thickness of the sample measured along the c-axis. 2.1.2 The Montgomery Method The so-called Montgomery method makes use of the method of images, and is based heavily on the series calculations of Logan, Rice and Wick (LRW) in Reference [6]. Their groundwork was used by Montgomery in Reference [7] to demonstrate how the components of the resistivity tensor could be obtained. However, their solution assumes a sample of isotropic resistivity, while the samples studied here were anisotropic, i.e. the resistivity tensor has three different, non-zero components along its diagonal. In Reference [5], van der Pauw introduced a rescaling of the sample dimensions that maps an anisotropic sample onto an equivalent isotropic sample, and in Reference [8], Friedmann et al. combined van der Pauw's rescaling with LRW's series calculations to obtain an expression that models the electric potential function inside a rectangular sample with anisotropic resistivity. Chapter 2. Theory 8 The Montgomery method (without rescaling) assumes eight infinitesimal contacts, located exactly on the corners of a rectangular sample of isotropic resistivity, as shown in Figure 2.2. Current flows into the sample through one contact, and out through one of its nearest neighbour contacts. Voltage is measured across the other pair of contacts on a face of the sample that contains the two current contacts. The sample dimensions are taken to be of length li along the i-axis, lj along the j-axis and Ik along the k-axis. The current-in contact is located at ( ^ - , 0 , 0 ) and the current-out contact at ( ^ , 0 , 0 ) . The voltage is measured between the contacts located at (+^-,^,0). Figure 2.2: The Montgomery arrangement of contacts (where the bottom surface of the sample is identical to the top, visible here). Current is injected through the sample via one pair of adjacent contacts, while voltage is measured across the two other adjacent contacts located on the same face of the crystal. The configuration of current and voltage contacts pictured here corresponds to the resistance measurement denoted by i ? 1 2 , where "1" refers to current flowing nominally in the "1", or a- direction, and "2" refers to voltage measured across contacts displaced from the current contacts in the "2" or b- direction. The current flow in the sample is modeled by tiling all of space with a three dimensional array of current sources and current sinks, which satisfy the boundary conditions of the sample. There is a correspondence between the current flow in this scenario and the lines of force in an equivalent space tiled with positive and negative charges. L R W exploit this correspondence in [6]. The voltage measured between the two contacts is 4 V - - I p M(xi,Xj,xk,li,lj,lk) where M(xi,Xj,Xk,h, lj, h) is equal to the electrostatic potential at (xi,Xj,Xk) from an array of +1 positive charges located (where n, m and p are integers) at positions: (2n/j - ^ , 2mlj , 2plk) Chapter 2. Theory 9 and an array of-1 negative charges located at positions: (2nk + - , 2mlj , 2plk) L R W found this function to be given by [6]: where and M C2 °ijk 2-KL E E Uh J fto^0 Sijk s inh(5 i j A ) 7T (2n + l)f + 7T p (2.2) (2.3) co = 1 for p = 0 ep = 2 for p > 0 In other words, the measured resistance is given by: R •ijk - t i f E E ilk J ^ Q p ^ 0 S i j k smh(Sijk) (2.4) Friedmann et al. [8] applied van der Pauw's rescaling technique to obtain an expression valid for anisotropic samples: ' Q I \ oo oo ° { j Pj \ £P ijk <jfi\yy_ Uh ) ^t^Sijk sinh(Sijfc) where now: c 2 _ Dijk — 7r (2n + 1) n 2 + 7T p (2.5) (2.6) 2.1.3 Rescaling the Axes This discussion follows closely that given by van der Pauw in [5]. The sample is assumed to be a parallelopiped with its edges parallel to the axes of the coordinate system in which the resistivity tensor is diagonal. P - I 0 pn 0 (2.7) / The x, y and z axes are rescaled in the following way: x' = ax y' = fiy (2-8) Px 0 0 \  Py  0 0 Pz I Z = JZ Chapter 2. Theory 10 The electric potential at equivalent points on the scaled and not-scaled samples is taken to be the same, and thus V = V, and: dV ' dx' 1 " (2-9) 1 E' = - — - -E a dx a x E'v ~ -pEv K = -EZ 7 The amount of current entering or leaving the surface of the sample must be unchanged by the rescaling, so: i'z dx1 dy' = i'z afi dx dy = iz dx dy Therefore: a/3 (2.10) 0:7 Substituting E' and i' into Equation 2.7 obtains: ( E ' \ If the following is now defined: And: 0 0 0 Py p 0 0 0 fi'A {PxPyPzY a = Pm _ Py Pz_ Pm Then Equation 2.11 becomes: ( E > \ to (2.11) (2.12) (2.13) (2.14) (2.15) (2.16) Equation 2.16 is Ohm's Law for an isotropic sample of resistivity pm. Thus, by scaling the axes by the factors given in Equations 2.8, the anisotropic crystal has been mapped onto an equivalent isotropic one. 11 Chapter 3 E x p e r i m e n t a l 3.1 Sample Preparation As-grown crystals were selected for shape and to a lesser extent surface quality. Resistivity, being a bulk measurement, is less sensitive to surface states, so it sufficed that the samples had no cracks or serious inclusions. The model of the electric potential assumes perfectly rectangular samples, and so when necessary, the samples were reshaped to make them more rectangular. This could be done by cleaving with a razor blade in the case of very thin crystals, or cutting with a wire saw and fine alumina abrasive in the case of very thick crystals. For very small projections on an otherwise rectangular thick sample, the sample could be affixed to a square metal block, with the projection extending out from the edge of the block, and gently sanded with fine emery paper. Once the samples were shaped, the electrical contacts were evaporated onto the crystals through a mechanical mask. The masks were hand-cut by either razor blade or microscissors from either 25 micron thick copper foil or 10 micron thick aluminum foil. Gold was used as the contact material. In order to cut precise masks, the foil was first attached to a smooth rigid metal plate with Crystal Bond thermal plastic. The foil could then be cut into shape with a razor blade, and the final mask gently pulled away from the plate. If the mask was robust, this could be done with the metal plate cold, but if the mask was delicate, it was better to heat the plate to soften the Crystal Bond. After detaching the mask from the plate, the mask was washed in acetone to remove the Crystal Bond. The mask was then flattened by lying it on a glass slide, covering it with a sheet of 25 micron foil, and burnishing the foil with the flat of a fingernail. To obtain masks with the correct dimensions for a given sample, the sample was laid on top of the foil, and a sharpened sewing needle mounted on a pencil was used to gently scribe guidelines onto the foil. This work was done under a x 4 microscope. The sample was then removed and the mask cut. In the case of the "x" shaped mask used in the Montgomery method (discussed in the next paragraph), the inside corners of the "x" had to be cut quite precisely, and the razor blade was not adequate for this. For this reason, the inside corners of the mask were left uncut until after the mask had been removed from the metal plate, cleaned and flattened. Then the mask was Chapter 3. Experimental 12 pinched firmly against a glass slide using a small block of malleable plastic (specifically, a slice of Mars plastic eraser), and the inside corners were carefully cut using microscissors. Microscissors give a much cleaner and more precise cut, but are not well suited to making long straight cuts, and for this reason, were only used for very detailed work. For the Montgomery style contacts, the procedure was as follows: a rectangular hole of roughly the same dimensions as the a-b face of the crystal was cut into 25 micron thick foil, and a second mask of 10 micron thick foil was affixed to this with small beads of Crystal Bond thermal plastic, forming a "cradle" that the sample could sit in, held in place either by gravity alone or by an additional weight provided by a thin slice of Viton rubber laid atop the crystal. The 10 micron mask was "x" shaped, and exposed small sections of the corners of the sample. Figure 3.1 shows a bottom view of the cradle mask. For stability and ease of handling, the mask for the Montgomery style contacts was attached to a metal washer with Crystal Bond thermal plastic. Once the sample was mounted in the mask, a second washer with a mylar window attached to it was placed over top of the mask's washer, and both of these were clamped to a metal plate with a hole drilled in it. The purpose of the metal plate was simply to make it easier to handle and orient the sample/mask combination within the evaporator. The evaporated gold passed through the drilled hole to reach the sample/mask combination. The clamps and the second washer with the mylar window ensured that the sample could not be lost in the event that the mask was dropped or jolted sharply during transport between the microscope and the evaporator. Figure 3.1: The "cradle" mask, viewed from the bottom. The mask has two components: a 25 micron thick mask with a rectangular hole of the size of the sample cut into it, and a 10 micron thick "x" shaped mask that leaves small square areas of the sample's corners exposed to the evaporation source. Chapter 3. Experimental 13 To obtain the eight contacts required for a Montgomery measurement, gold was evaporated di-rectly onto the a-b face of the crystal through the mask, resulting in small rectangular gold pads located at the four corners of the sample. Two evaporations were done in the same cradle mask, one on each a-b face of the crystal. For the colinear arrangement of contacts, a series of narrow parallel straps were cut into a rectangle of 10 micron thick foil, and this mask was then folded around the sample such that the contact areas were exposed. Figure 3.2 shows how a sample is held in this type of mask. To obtain good gold coverage on the edges of the samples, two evaporations were done. In the first evaporation, one corner of the sample was pointed directly at the evaporation source, so that three faces of the sample were exposed to gold. For the second evaporation, the opposite corner of the sample was pointed at the evaporation source, exposing the other three sides. This method had the drawback that the mask itself could cast a shadow on the sample and prevent good coverage, but this problem could be usually be overcome by tucking the mask snugly around the sample. The fact that the 10 micron foil was so thin helped in this regard, because it was very ductile and could be manipulated to fit tightly against the sample, and also because the thinner mask didn't cast as much of a shadow as a mask made of 25 micron foil would have in the same arrangement. Figure 3.2: The strap mask, for putting on colinear style contacts. Three long straps are cut into a rectangle of 10 micron thick foil, and this mask is folded, and then wrapped around the sample to expose the contact areas. Y B C O , particularly underdoped Y B C O , does not accept contacts readily. It forms a Shottky barrier at the interface between the sample and contact, which leads to very resistive and non-Ohmic contacts. Also, there is anecdotal evidence to suggest that the surface of Y B C O degrades in air, and becomes more insulating with time. For this reason, the crystals were etched in a 0.5% bromine in ethanol solution immediately before the gold was evaporated onto the crystals. Generally the gold was on the samples within 1.5 hours of the etch (where time spent in vacuum is not counted). Chapter 3. Experimental 14 After the gold was evaporated onto the samples, they were annealed at 390° C for a few days to diffuse the gold into the surface of the crystal. During this anneal, the samples were sealed inside a quartz glass ampoule with a small volume of dead space and a large volume of YBCO ceramic having the same oxygen content as the samples. This is to ensure that the oxygen content of the samples is not altered by the annealing process, as the oxygen would tend to diffuse out of the crystals at these temperatures if sealed in vacuum. Once the contacts were "baked in" in this manner, electrical contact to the sample had to be made. For the eight-point Montgomery measurements, the first step was to attach small squares of gold foil to an insulating substrate, such as lanthanum aluminate or magnesium oxide, using Epo-Tek H20E silver epoxy. The squares needed to be close enough together to allow the sample to sit with each of its four corners resting on a, separate square of foil. At this stage, 30 micron gold wires were attached to the gold foil squares using silver epoxy. Next, tiny dabs of silver epoxy were placed on the top of the gold squares with a sharpened sewing needle, and then the sample was carefully placed on top of this to allow each corner to come into contact with the wet epoxy. Figure 3.3 shows the sample mounted up to this point. Figure 3.3: Sample with Montgomery style contacts, mounted in such a way as to allow electrical contact to be made to all eight contacts. Wires are attached directly to the four exposed contacts with silver epoxy. The four hidden contacts are attached with silver epoxy to four gold foil squares that are in turn attached to the substrate. Wires are then attached to the gold foil squares to allow electrical contact to the hidden contacts. The sample was manipulated by means of a small wedge cut from a thin sheet of teflon, attached to the end of a paintbrush handle. The teflon wedge was given a weak static electric charge by-brushing it against cotton fabric, and the YBCO sample, being a dielectric, would stick to this wedge. Care was taken to avoid having the wet epoxy short two gold pads together when the sample was lowered into it; this could generally be avoided by using sufficiently small amounts of epoxy. The Chapter 3. Experimental 15 sample/substrate arrangement was then cured on the hot plate at 180° C for two minutes before the next stage of preparation. It should be noted that these four silver epoxy connections provided the only thermal contact between the sample and its substrate, but because the sample is so small, and the thermal conductivity of silver and gold so high, this was considered to be adequate. After the arrangement had cooled, 10 micron gold wires were attached to each of the four contacts located on the top face of the sample. This was accomplished by placing a tiny amount of silver epoxy on a contact with a sharpened sewing needle, and then carefully lying the end of the wire onto the epoxy. A second dab of epoxy on top of the wire helped ensure a strong bond between the wire and the sample. The sample/substrate arrangement was cured again, and the process repeated for the other three contacts. In principle, it would be possible to attach all four wires at the same time, and cure al l four of these bonds at the same time, but in practice it is difficult to avoid nudging one wire out of alignment while working on another. For the colinear arrangement of contacts, the sample was attached directly to the substrate with G .E . Varnish and the wires then epoxied onto the exposed top face of the sample. This was also done by dabbing silver epoxy onto the contacts with a sharpened sewing needle and then lying the gold wires into the wet epoxy, but less care was needed because the contacts are larger in the colinear arrangement. The small piece of insulating substrate onto which the sample was anchored was attached to the probe tip (described in detail in Section 3.4) using a small amount of silicone grease, which provides good thermal contact between the substrate and the probe tip. The 10 micron wires that provided electrical contact to the sample were then soldered onto the probe's electrical connections with indium. 3.2 Preparation and Anneal Times The time taken to prepare a sample for data taking is important because it affected what one calls the total anneal time of the sample. After the contacts had been annealed in the furnace to diffuse the gold into the surface (as described in the previous Section), the sample was quenched to ice water temperature, and then stored at approximately —10° until it was time to begin a data run on it. Storing the sample at low temperatures prevented it from annealing. Summarising what happened to the sample next: the sample was first warmed to room tempera-ture, then photographed in order to allow its contact spacings to be measured (this will be discussed in Section 3.3), and then mounted on its substrate. Its wires were attached with silver epoxy, which Chapter 3. Experimental 16 involved heating the sample repeatedly to 180° on the hotplate. Once the wires were attached, the sample/substrate arrangement was mounted on the probe tip, and its wires soldered onto the probe's electrical connections. A n electrical check was performed, and if no problems were found, the probe was sealed and evacuated of air, and the electrical check was repeated. If the second check was also clear of problems, then the probe was placed in liquid nitrogen to start it cooling in preparation for a data run. As noted in Section 1.4, the area of the phase diagram between the antiferromagnetic phase and the superconducting phase is scientifically interesting. Y B C O with the oxygen content studied here, before its room temperature anneal, often shows no superconducting transition. As the crystal is allowed to anneal at room temperature, the sample becomes a superconductor, and its Tc rises with further annealing until it eventually saturates at some value, usually around 15 K for this doping. Because this transition from insulator to superconductor is of such interest, the preparation of the sample for a data run was performed as quickly as possible so that the sample might be studied before Tc appeared. Generally, the time between taking a sample out of the freezer and putting it into its liquid nitrogen pre-cool was around six hours. Thus, every sample has had some anneal time at room temperature before it is measured for the first time. There is the matter of the curing of the silver epoxy however. It is possible that the heat treatments used to cure the epoxy scramble the oxygen ordering of the sample anew, so that the real anneal time prior to measuring a sample should perhaps only be counted from the time of the last epoxy curing to the liquid nitrogen pre-cool. 3.3 Measuring Contact Spacings 3,3.1 Montgomery Contacts The Montgomery method assumes infinitesimal contacts located on the exact corners of a rect-angular sample, but physical contacts can obviously only approximate this. The contacts for the samples studied here were generally about 50 microns by 50 microns, or less, and were located as close to the corners of the sample as possible, usually within 10 microns or better of the sample edges. Because the theory assumes tiny corner contacts, the spacing between contacts, rather than the actual physical dimensions of the sample, are used in Equation 2.5. To measure contact spacings as accurately as possible, an optical microscope and digital camera were used. The graticule of the microscope's eyepiece was first carefully calibrated for various Chapter 3. Experimental 17 magnifications by photographing an accurate distance scale. The photographs were then examined on a computer, where they could be enlarged to the limit of the camera's resolution, for better accuracy. By repeatedly altering the microscope's focus and other settings, and performing the calibration over again, it was determined that distances measured in this manner were accurate and that the calibration was not subject to any noticeable drift due to microscope usage between photographs. The samples were photographed just before wires were epoxied on, and their contact spacings determined from, these photographs. The uncertainty in the spacing of the contacts was given by the size of the gold pads. Because the gold was located only on the a-b faces of the crystals, the uncertainty in distances in the a-b plane was dominated by the contact size rather than by the measurement method. However, for distance measurements along the c-axis (where there was no gold), contact spacing was taken to be the physical thickness of the sample, and the uncertainty was the uncertainty of the method of measurement. There are eight contacts on the corners of the rectangular sample. For every direction parallel to a sample edge, there are four pairs of contacts that lie displaced from one another along that direction. The effective size of the sample is defined to be the average of the distances between the centres of these four pairs of contacts. The uncertainty in one of these effective distance measurements was taken to be plus or minus one half of the size, (along the direction in question) of the largest contact. As noted in the previous paragraph, in the case of measurement along the c-axis, the "effective" size was simply the actual thickness of the crystal, and the uncertainty was the uncertainty of the method of measurement. 3.3.2 Colinear Contacts For the colinear arrangement of contacts, the cross sectional area of the crystal is found using the actual dimensions of the crystal, so the uncertainty was given by the method of measurement. The effective length of the crystal was taken to be the distance between the centres of the two voltage contacts, and so the uncertainty of this was taken to be plus or minus one half of the widest voltage contact. 3.4 T h e P r o b e A schematic for the probe used to measure resistivity in this experiment is. shown in Figure 3.4. The sample on its piece of substrate was mounted onto the probe tip. The probe tip consisted of a sapphire block mounted on a quartz tube. The heater and thermometer for the probe were mounted Chapter 3. Experimental 18 Figure 3.4: The probe used to measure resistivity. on opposite sides of the sapphire block, just above the sample. Sapphire has excellent thermal con-ductivity, and so the sample, heater and thermometer were all kept in thermal equilibrium. Quartz has very poor thermal conductivity, so the quartz tube that the sapphire was mounted on estab-lished only a weak thermal link to the rest of the probe. This helps in temperature regulation in that the heater is, for the most part, only required to heat the sapphire block and the sample/substrate combination. Both sapphire and quartz are electrically insulating, so the sample was electrically isolated from the rest of the probe. The thermometer was a calibrated Cernox chip, and the heater a small resistor through which current was passed. The wires for making electrical contact to the sample were made of brass, which has a lower thermal conductivity than that of a wire made from a pure metal, and this helps to minimize the thermal leak from the probe tip through the wires. The 10 micron gold wires that attach to these brass wires were purposefully chosen to be thin, also to minimize this thermal leak. The probe tip was attached to the probe body, a large (relative to the tip) volume of copper that sits at the end of three lengths of thin-walled stainless steel tubing. A brass cap fits over top of the probe tip, and the space where the sample resides could be made vacuum-tight by means of an indium seal between the brass cap and the probe body. Copper plates were mounted on the stainless steel tubing at intervals, to acts as cooling fins. This will be discussed in more detail presently. One of the pieces of stainless steel tubing provided the pumping line, so that the space inside the brass Chapter 3. Experimental 19 cap (where the sample is located) could be evacuated. One of the remaining two lengths of stainless steel tubing contained the wiring for the heater and thermometer, and the other contained the wires for the electrical contacts to the sample. Because the sample was in vacuum, the only thermal leaks from the sample stage are through the quartz tube that connected it to the probe body, through the sample wires, and through radiative losses. The sealed and evacuated probe sat in a bath of liquid helium or liquid nitrogen (depending on the temperature range desired), such that the probe body was held at the boiling temperature of the liquid for the duration of the experiment. The sample temperature could be varied independently from the bath temperature by means of the heater on the probe tip. Temperatures between the bath temperature and 300 K were accessible. Despite the low thermal conductivity of the thin-walled stainless steel tubing, it still provided a heat leak to the probe tip from the electrical connections at the top of the probe (which are kept at room temperature). The copper plates attached to the tubing helped minimize this effect, by thermally anchoring the tubes to the bath temperature when the fins were beneath the level of the bath liquid, and by increasing the ability of the cold gases flowing past the fins to cool the tubes when the fins were above the level of the bath liquid. The bath was contained inside a silvered glass dewar which was nested inside a second silvered glass dewar. The inner dewar contained either liquid helium or liquid nitrogen, while the outer contained liquid nitrogen. The top of the probe housed connectors that attached the probe to the electronics needed for taking data. These connectors, and the top of the probe itself, were all kept at room temperature. The probe was capable of sealing against the inner dewar so that the bath could be pumped on, to lower the base temperature further. With a probe bath of liquid helium, a base temperature of 1.2 K could be thus obtained. Because several temperature sweeps are necessary to obtain the raw data for a Montgomery analysis, data taking was done in two separate stages to conserve helium. Low temperature data, from 1.2 to 100 K, were taken with the probe sitting in a liquid helium bath, while high temperature data, from 80 K to 300 K were taken with the probe in a liquid nitrogen bath. 3.5 The Electronics 3.5.1 Controlling Temperature The temperature of the sample was controlled by a Conductus LTC-20 Temperature Controller (referred to hereafter as the LTC-20, or the temperature controller) into which had been programmed Chapter 3. Experimental 20 a calibration table for the thermometer and a table of PIDs (described presently) for the probe tip. PID stands for Proportional, Integral and Derivative, and refers to the user-set parameters (P, I, and D) that allow the instrument to stabilise on a target temperature quickly. Power is sent to the heater according to the following equation: where k and C are constants set by the manufacturer, Tset is the target temperature, and p(t) is the power output of the controller over the last I seconds. The LTC-20 has a range of power outputs that it can deliver to the heater between 0.05 W and 50 W. However, while the maximum output of the temperature controller is 50 W , the heater could only tolerate a maximum of 1 W of power without burning out. Rather than restricting the LTC-20 to just the lowest power outputs, a ladder of resistors was inserted between the LTC-20 and the heater. This reduced the power delivered to the heater to less than 1 W even when the LTC-20 was supplying its full 50 W. This allowed the entire range of the power settings on the LTC-20 to be used, which gave greater flexibility and a much finer control over low temperature regulation. The temperature regulation circuit is given in Figure 3.5. 3.5.2 Measuring Resistance The current was provided to the sample by the internal oscillator of a Stanford Research Systems SR850 D S P Lock In Amplifier (referred to hereafter as the SR850, or the lock-in) at a frequency of 587.5 Hz for the Montgomery sample and 497.5 Hz for the colinear sample. The current first passed through a resistor of 1000 ±0.25% Cl, then through the sample, and back to the SR850. The voltage drop across the sample was measured differentially by the lock-in detector, and the lock-in's signal was also echoed to a Hewlett-Packard HP3478A multimeter for data collection purposes. The voltage drop across the 1000 Cl resistor was also measured, by means of a second Hewlett-Packard HP3478A multimeter, and the current through the sample was calculated from this. A diagram of this circuit is given in Figure 3.6. Because the signals measured by the lock-in are so small, care was taken to shield the circuitry from sources of noise. A l l of the coaxial cables taking current to or from the sample, or measuring voltage across it, were sheathed in copper braid and grounded to the SR850 lock-in. In addition, the two coaxial cables used to measure the voltage drop across the sample were, inside their sheath, wound tightly together to avoid any open space between them that might serve as a pickup loop for magnetic fields. Power = k x P x (T - Tset) + ^- / p(t) dt - D^-Chapter 3. Experimental 21 LTC-20 Temperature Controller 35.2 Q. 69.3H I—WV—*—VW—1 Thermometer Heater 33 "0 i CD Q 152. $ 1 CO CD i o => ' CD 5. ' CL I CD Figure 3.5: Temperature regulation circuit for the probe. The heater and thermometer are attached to the sapphire block that the sample is mounted on; the rest of the circuit is kept at room temperature. Because the Montgomery method involves sending current through and measuring voltage across various pairs of contacts, a switching box was built to allow the experimenter to quickly select through which contacts current entered and left the sample, and across which contacts the voltage was measured. A l l the wiring for the switching was contained inside an aluminum box, which was itself in good electrical contact with the metal braids of the cables, and thus was also grounded to the SR850 lock-in. 3.5.3 The Computer Program The data taking was done by a Lab V I E W 6.0 (National Instruments) program. This program first initiated G P I B sessions with, and adjusted the settings of, the LTC-20 temperature controller, the SR850 lock-in amplifier, and the two HP3478A multimeters. Then the program sent the first of a series of set temperatures to the LTC-20 temperature controller, and waited until the probe tip had stabilised at that temperature. Chapter 3. Experimental 22 HP3478A Multimeter SR850 Lock-in Amplifier (A-B differential measurement of voltage) 1000 Q. Sample SR850 Lock-in ' Amplifier Current f r\ Source \f > Figure 3.6: Sample circuit for the probe. Current is passed through a 1000 Cl resistor, then through the sample. Voltages are measured across both the known resistor and the sample, the former to determine the current through the sample. Once the temperature had stabilised to within user-established thresholds, the program took ten readings from the multimeter that measured the voltage drop across the 1000 Cl resistor. Then the program calculated the average of, and the standard deviation of that average for these values. It then calculated the current through the sample and the uncertainty of this value. Only ten readings were required because the voltage across the 1000 Cl resistor tended to be a very stable number. The voltage measured across the sample by the lock-in amplifier was echoed to a multimeter, and the computer program took two hundred readings of the voltage from this multimeter. The program then calculated an average for these values and the standard deviation of that average. The much larger sample of readings for the voltage across the crystal was taken because this voltage was very small and fluctuated much more than was acceptable for the desired accuracy level of the experiment. Although the SR850 lock-in amplifier does provide the means for users to average the signal, and thus reduce the noise seen a.t the output, these averaging algorithms can sometimes introduce a systematic error to the data. Because the voltages measured in this experiment were so small that any artificial offset could seriously compromise the quality of the data, it, was decided that, having the computer do a brute force averaging of a large number of raw data would be the safest course of action. The standard deviation of the averaged voltage value was generally quite a bit, less than 1%, so the amount of averaging done appeared to be quite sufficient to obtain good data,. Chapter 3. Experimental 23 The program then calculated the resistance of, and uncertainty for the resistance of, the sample. It wrote these values to file along with information on the temperature of the sample, the current through the sample and the voltage across the sample. The resistivity of the sample was calculated later in a separate program. Once the data point had been acquired, the program provided a new set temperature to the LTC-20 temperature controller, and the process repeated itself until all the set temperatures in its file had been exhausted. Then the program let the sample drop back to base temperature, and closed its open files and GPIB sessions with the other instruments. 3.6 Deconvolving the Raw Data For the Montgomery Method The model for deconvolving the raw data of the Montgomery method, in order to obtain the com-ponents of the resistivity tensor, is that used by Friedmann et al. in Reference [8]. The deconvolution was done using a program in Mathematica 4.0 (Wolfram Research). The resistivity components were found by minimising, using the Levenberg-Marquardt algorithm, the following x2 function: 2 _ y- (Rijk- R[i,j,k])2 J£ (R[i,j,k})2 where Rijk is given by Equation 2.5, R[i,j,k] are the raw data, and i, j, and k label the contact configurations. The term in the denominator normalises the data, which is important given that the signal for one configuration of contacts may be several orders of magnitude smaller than that of another, and all the data must be given equal weight. Twenty terms were used for each of the two sums in Equation 2.5, it having been determined that adding or subtracting one term from that number did not alter the final values by more than 1%. As will be seen, the uncertainty in the resistivity values due to the uncertainty in measuring the sample dimensions tends to swamp this effect anyway. 3.7 Uncertainties 3.7.1 T h e C o l i n e a r M e t h o d Calculating the uncertainty of the a-axis resistivity for a sample measured using the colinear method of measurement is straightforward. The resistivity is obtained from the measured resistance values Chapter 3. Experimental 24 by the following expression: wt Pa = T R So the uncertainty is given by the total derivative of pa: , t R \ 2 j 2 fwR\2 J j 2 fwt\2 - (wtR\2 JT., (3.2) The resistance values, R, were temperature dependent, so the uncertainty in pa was also slightly temperature dependent. However, this temperature dependence was found to be very weak compared to the temperature dependence of pa itself, and thus does not significantly distort the results. Section 4.3 gives the quantitative uncertainty in the magnitude of the measured pa, and discusses the limiting factors in the measurement's accuracy. 3.7.2 The Montgomery Method Calculating the uncertainty in the p values obtained via the Montgomery method is a much more complicated procedure than in the case of the colinear measurement. The total derivative of the resistance given by Equation 2.5 is computed, with the values Zj, lj, lk, Pi, Pj and pk considered to be variables, since these are the quantities that have uncertainties associated with them. This gives: The uncertainties on the k, lj and lk values are set by the size of the contacts or the accuracy of measuring the c-axis, and are thus known values. The uncertainty of the measured resistance, dRijk, is also known, from the standard deviations calculated for the current and voltage data. Thus, the first line of Equation 3.3 contains only known values, while the second line contains the unknown (and desired) quantities dpi, dpj and dpk. The sign of the various terms in Equation 3.3 isn't important, so a x2 function can be formed by subtracting the unknown quantities: ((^ )2*Kff)2*HffH <"> from the known quantities: (<•»++(^f )2"'?+(^ f )^ ) • <"> squaring the result, and dividing by the square of Equation 3.4, to give all the data equal weight. Minimising this x2 function allows values for the uncertainties dpi, dpj and dpk to be obtained. Chapter 3. Experimental 25 This calculation was performed by a program in Mathematica 4.0 (Wolfram Research). The uncertainties are shown with the data in Section 4.3 of this thesis. 3.8 Miscellaneous The resistivity components were measured using an eight-point Montgomery style technique, with van der Pauw's rescaling technique incorporated as in the work of Friedmann et al. [8]. In this method, measurements are taken using a variety of lead configurations. Although a minimum of three measurements are needed, in an ideal situation measurements would be taken on every possible configuration to provide a consistency check on the data. It proved difficult to do this. In certain configurations, the voltage drop across a pair of contacts might be so small as to be lost in the experimental noise. Also, if one contact proves to be unusable, this severely restricts the contact configurations possible. More than one unusable contact on a sample can make a Montgomery-style measurement impossible. Since the gold pad might establish a good electrical connection with the Y B C O sample in only a few locations, or not at all, (rather than everywhere that gold sits on the surface of the crystal, which is what one might normally assume), the contact resistances were always tested at room temperature first. If it was determined that at least seven of the contacts were in good electrical contact with the sample, then the probe was cooled to base temperature. A common problem with contacts is for them to become extremely resistive at low temperatures. For this reason, the contact resistances were checked again as a function of temperature once the sample was at base temperature, to ensure that there were still a minimum of seven usable contacts. At the same time, the phase shift between the current injected into the sample and the voltage drop measured across the sample was checked, to ensure that it was small. A large phase shift usually indicates that a contact is pulling away from the sample at low temperatures, introducing a capacitance. The power dissipated across the heater while the LTC-20 temperature controller maintained cer-tain temperatures was also checked prior to taking data. Anomalously high power being dissipated across the heater signals that there is a poor vacuum inside the probe, and this usually indicates a leak. Chapter 3. Experimental 26 Because the samples were measured at several times during their room temperature anneal, and because several data sweeps must be taken using various lead configurations in order to obtain the three components of the resistivity tensor, care must be taken to ensure that the sample has no opportunity to anneal during a data run. For this reason, data was only collected up to 265 K. It would likely have been safe to collect data up to 273 K. Chapter 4 27 R e s u l t s 4.1 Temperature Dependent Resistivity for YBa2Cu^0^_zzz'') Montgomery Method A YBCL2CU3O6.333 single crystal was studied using the eight-contact Montgomery method, with van der Pauw's rescaling technique incorporated into it as done by Friedmann et al. [8]. The sample dimensions and effective dimensions were measured as described in Section 3.3 of this thesis. The effective dimensions that were used to deconvolve the data into the three components of the resistivity tensor are tabulated in Table 4.1. Effective a axis length 781 ± 20pm Effective b axis length 372 ± 26u.m c axis length 70.91 ± 0.50/xm Table 4.1: Effective dimensions of the underdoped YBa2C7u306.333 crystal used for the Montgomery method. The sample had one unusable contact, but its seven other contacts had contact resistances under 100 fi at 5 K, which is acceptable for underdoped Y B C O . Five of the contacts had resistances of about 30 Cl at 5 K, and none of the usable contacts showed any signs of capacitance at low temperatures. For several configurations of voltage and current contacts, difficulty was encountered in obtaining a signal larger than the experimental noise. In the end, there were only three configurations of contacts for which the resistance was measured. This is the minimum number of data sets required for the Montgomery method. Measuring the resistance over more configurations of the contacts provides a consistency check on the data, which would have been useful. However, as will be seen in Chapter 5, the pa data obtained from this sample were found to agree extremely well with those found in a second sample measured using a colinear arrangement of contacts. This offers strong reassurance that the data obtained for this sample were of good quality. Chapter 4. Results 28 Denning Rij to be the resistance found by passing current through two contacts oriented along the i-direction and measuring the voltage drop across the two contacts displaced from the current contacts along the j-direction, the configurations studied for this sample were R12, R21 and R31. The three data sets were measured once with nominally no room temperature anneal time (beyond what occurs during the preparation of the sample for data taking), and then the probe was warmed up to room temperature for 6 hours ± 10 minutes. The probe was cooled again, and the data sets remeasured. The sample was warmed up to room temperature for 13 hours ± 5 minutes, for a total nominal anneal time of 19 hours ± 1 5 minutes, then cooled and measured again. The next room temperature anneal brought the total nominal anneal time up to 68 hours ± 20 minutes, and the last to 327 hours, 45 minutes ± 25 minutes. The data sets were deconvolved as detailed in Section 3.6 to obtain the components of the resis-tivity tensor, and although this process was successful above Tc, an artifact appeared in the data just below Tc. The cause of this artifact is the fact that for different configurations of the voltage and current contacts, Tc did not always occur at the same temperature. For configurations where pc is being heavily sampled, the resistance went to zero at a higher temperature and over a smaller temperature range than for other configurations. This caused a large peak to appear just below Tc in the small (relative to the c-axis) resistivities pa and pb- The peak is not present in the raw data (although it could conceivably be hidden by the large c-axis contribution) and so is considered to be an artifact. The data taken on a second y = 6.333 doped crystal with the colinear configuration of contacts showed no hint of this peak, which helps confirm that the peak is without physical meaning. The fact that different configurations of current and voltage contacts produced different T c values is troubling. It may be due to sample inhomogeneity producing filamentary superconductivity that shorts out the measurement prematurely in certain configurations but not in others. The isotropic sample that the anisotropic crystal scales onto (see Section 2.1.3) is very long in the c-direction, so it is possible that when currents are run nominally in this direction, they sample the bulk of the crystal much more effectively than in other configurations, and thus are more likely to find a superconducting path through the crystal. The components of the resistivity tensor are obtained by minimising Equation 3.1. A small x2 indicates that the data are in close agreement with the theoretical expression given by Equation 2.5. For this crystal, x2 was always very small, on the order 1 0 - 3 1 . This excellent x2 value became significantly worse (x2 ~ 2) when a fourth R^ configuration was added, which was one reason why only three configurations were used. It may be that for different configurations of contacts, Chapter 4. Results 29 the effective sample dimensions differ, making good agreement between two partially redundant measurements impossible. The temperature dependence of the resistivity along the a-axis, pa, for the various anneal times, is shown in Figure 4.1. The temperature dependence of the resistivity along the b-axis, pb, for the various anneal times is shown in Figure 4.2, and the temperature dependence of the resistivity along the c-axis, pc, for the various anneal times is shown in Figure 4.3. p a F o r V a r i o u s A n n e a l T imes g u I Cl « 2 0 0 0 • 1 7 5 0 1 5 0 0 • a • 1 2 5 0 + -• i • • 1 0 0 0 • 7 5 0 ' - TIL. 5 0 0 • 2 5 0 • J 50 100 150 T e m p e r a t u r e ( K ) 2 0 0 2 5 0 Figure 4.1: Resistivity along the a-axis for YBa2Cu3 06.333 as a function of temperature. The different curves were obtained for various anneal times. (From top to bottom) Red circles: nominally no anneal, Green crosses: 6 hours anneal, Purple squares: 19 hours anneal, Blue diamonds: 68 hours anneal, Orange stars: 328 hours anneal. Figure 4.4 shows the a- and b-axes for the nominally unannealed, and fully annealed cases on the same scale. The surprising thing about this graph is that the b-axis resistivity is larger than the a-axis resistivity for part of the temperature range. This is an odd result, for it is naively thought that the a- and b- plane conductivities for Y B C O are the same, and the the b-axis chain conductivity accounts for the fact that the b-axis resistivity is always measured to be smaller than the a-axis resistivity. If this picture is realistic, then one would never expect to see the b-axis resistivity exceed the a-axis resistivity. As will be shown in Section 4.3, this oddity can be safely attributed to uncertainty in measuring the a- and b- axis dimensions. Chapter 4. Results 30 Figure 4.2: Resistivity along the b-axis for YBa2Cu306.333 as a function of temperature. The different curves were obtained for various anneal times. (From top to bottom) Red circles: nominally no anneal, Green crosses: 6 hours anneal, Purple squares: 19 hours anneal, Blue diamonds: 68 hours anneal, Orange stars: 328 hours anneal. F o r V a r i o u s A n n e a l T i m e s 2 . 5 x 1 0 2 x 1 0 I 1 . 5 x 1 0 : O 3 . o 1x10* Q. 5 0 0 0 0 0 50 1 0 0 1 5 0 T e m p e r a t u r e ( K ) 2 0 0 2 5 0 Figure 4.3: Resistivity along the c-axis for YBa2Cu3 06.333 as a function of temperature. The different curves were obtained for various anneal times. (From top to bottom) Red circles: nominally no anneal, Green crosses: 6 hours anneal, Purple squares: 19 hours anneal, Blue diamonds: 68 hours anneal, Orange stars: 328 hours anneal. Chapter 4. Results 31 Figure 4.4: Resistivity along the a- and b-axes for two anneals. Red circles: pa with no anneal, Orange squares: pi, with no anneal, Purple diamonds: pa with 328 hours anneal, Blue stars: pb with 328 hours anneal. 4.2 Temperature Dependent Resistivity for YBazCuzO^m,', Colinear Method A YBaiCuzOs.zzz single crystal was studied using the four-contact colinear method. The sam-ple dimensions and effective dimensions were measured as described in Section 3.3. The effective dimensions that were used to calculate the geometric factor that converts the raw resistance into pa are given in Table 4.2. Effective a axis length 698 ± 59p,m b axis length 219.4 ± 4.1pm c axis length 7.25 ± 0.40/um Table 4.2: Effective dimensions of underdoped YBa2Cu306.333 crystal used for the colinear method. A l l of the contacts were usable, and - as might be expected for the larger contacts used in the colinear arrangement - the contact resistances were lower than those of the sample discussed in the previous Section. The resistances of three of the contacts were less than 12 f i at 2 K, and the fourth was below 40 at 2 K. Chapter 4. Results 32 The resistance of the sample was measured once with nominally no room temperature anneal (beyond that involved in preparing the crystal for data taking), then the probe was warmed to room temperature for 6 hours, ± 5 minutes. The sample was cooled and measured again, then warmed for eighteen hours, for a total nominal anneal time of 24 hours, ± 10 minutes. The sample was measured again, then brought up to room temperature to give the sample a total nominal anneal time of 72 hours, 17 minutes ± 20 minutes. After another period of data taking, the total nominal anneal time was brought to 168 hours, 17 minutes ± 25 minutes. The sample's a-axis resistivity was obtained as detailed in Subsection 2.1.1. The temperature dependence of the resistivity along the a-axis, p a , for the various anneal times, is shown in Figure 4.5. p a F o r V a r i o u s A n n e a l T i m e s 2000 1750 g 1500 o £3 1250 3 . £ 1000 750 500 0 50 100 150 200 250 T e m p e r a t u r e ( K ) Figure 4.5: Resistivity along the a-axis for Y Ba.2Cu3Os.333 as a function of temperature, measured using the colinear technique. The different curves were obtained for various anneal times. (From top to bottom) Red circles: nominally no anneal, Green crosses: 6 hours anneal, Purple squares: 1 day anneal, Blue diamonds: 3 days anneal, Orange stars: 7 days anneal. Chapter 4. Results 33 4.3 Uncertainties in the Resistivity Values for YBa2Cu3OG.3M 4.3.1 The Montgomery Method The uncertainties in the data (for both samples) were calculated as detailed in Section 3.7. Figures 4.6, 4.7 and 4.8 show representative data, with their calculated limits of uncertainty, for the sample measured by the Montgomery method. The magnitudes of these uncertainties are typical for all the data sets, including those that, for clarity, are not shown on these graphs. Specifically, the graphs show the uncertainties for the sample with nominally no room temperature anneal, and with 328 hours of anneal. p a W i t h U n c e r t a i n t y 2500 i — i 2000 0 50 100 150 200 250 T e m p e r a t u r e ( K ) Figure 4.6: Resistivity along the a-axis for YBa2Cu30e.333 as a function of temperature, measured using the Montgomery technique, and showing the limits of uncertainty. The red curve (circles) is the data after nominally no anneal, with its limits of uncertainty shown in orange. The blue curve (squares) is the data after 328 hours of anneal, with its limits of uncertainty shown in light blue. Note that the uncertainties for the two curves overlap. Note that, within the limits of uncertainty, the a-axis resistivities overlap one another for much of the temperature range. This should not be interpreted as the two curves being indistinguishable from one another. The uncertainties reflect an uncertainty in the sample dimensions, and thus an uncertainty in where on the sample's surface current is actually being injected. Since these points of injection are unlikely to change much with anneal time, only the absolute magnitudes of the curves Chapter 4. Results 34 p b W i t h U n c e r t a i n t y 2500 r 1 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 T e m p e r a t u r e ( K ) Figure 4.7: Resistivity along the b-axis for YBa2Cu-sO§.3zz as a function of temperature, measured using the Montgomery technique, and showing the limits of uncertainty. The red curve (circles) is the data after nominally no anneal, with its limits of uncertainty shown in orange. The blue curve (squares) is the data after 328 hours of anneal, with its limits of uncertainty shown in light blue. are uncertain, but not their relative displacement from one another. By purposefully increasing or decreasing the lengths of the axes by their uncertainties, and re-calculating values for pa, p^ and pc, it was determined that the uncertainty in measuring the a-and b-axes of the crystal is responsible for the majority of the the uncertainty in the calculated resistivities. Altering the value of the b-axis length had the greatest effect, which is perhaps not surprising as the uncertainty in the b-axis measurement was 7.0% of the total measurement, while the uncertainty in the a-axis was only 2.6% of the total measurement. Oddly however, altering the c-axis by four times its determined uncertainty, which is a change of 2.8%, had almost no effect on the data. The calculated resistivities were quite insensitive to uncertainty in the c-axis, probably because van der Pauw's rescaling of the crystal makes the c-axis extremely large compared to the mapped a- and b-axes, and so making this axis somewhat shorter or longer doesn't noticeably alter the path taken by the majority of the current through the sample. The effect of the uncertainty in the b-axis length is shown in Figures 4.9 and 4.10. As can by seen, expanding the b-axis gives relative magnitudes of the a- and b-axes that are more in keeping with the expected result, where pa exceeds pb everywhere in the temperature range studied. It is Chapter 4. Results 35 Figure 4.8: Resistivity along the c-axis for y.Ba2C7u306.333 as a function of temperature, measured using the Montgomery technique, and showing the limits of uncertainty. The red curve (circles) is the data after nominally no anneal, with its limits of uncertainty shown in orange. The blue curve (squares) is the data after 328 hours of anneal, with its limits of uncertainty shown in light blue. therefore assumed that the matter of the calculated pb being larger than pa is not physical, but due to uncertainty in measuring the crystal's dimensions. p a and p b With 'h' Dimension Increased 1800 1600 O 1400 i a 3 1200 « 1000 a 800 600 0 50 100 150 200 250 Temperature (K) Figure 4.9: The effect on the a- and b-axis resistivity of increasing the effective b-axis length by its uncertainty. Red circles: a-axis. Blue squares: b-axis. Chapter 4. Results 36 p a a n d Pb W i t h ' b ' D i m e n s i o n D e c r e a s e d 2000 1800 g 1600 o G 1400 a. B 1200 A I O O O 800 600 0 50 100 150 200 250 T e m p e r a t u r e ( K ) Figure 4.10: The effect on the a- and b-axis resistivity of decreasing the effective b-axis length by its uncertainty. Red circles: a-axis. Blue squares: b-axis. 4.3.2 The Colinear Method Representative uncertainties for pa in the sample measured by the colinear method are shown in Figure 4.11. Pa W i t h U n c e r t a i n t y 2000 1500 i o a. 1000 Q. 500 0 0 50 100 150 200 250 T e m p e r a t u r e ( K ) Figure 4.11: Resistivity along the a-axis for YBaiCu$0§.Ms as a function of temperature, measured using the colinear technique, and showing the limits of uncertainty. The red curve (circles) is the data after nominally no anneal, with its limits of uncertainty shown in orange. The blue curve (squares) is the data after 7 days of anneal, with its limits of uncertainty shown in light blue. Chapter 5 37 Discussion 5.1 Comparison of the Two YBa2Cu30^^z Results It should first be noted that the data for the sample measured by the colinear method and for the sample measured by the Montgomery method agree qualitatively, but not quantitatively. This can be seen in Figure 5.1. C o m p a r i n g p a F o r T w o D i f f e r e n t S a m p l e s T e m p e r a t u r e ( K ) Figure 5.1: A comparison of the resistivity along the a-axis measured using the Montgomery method (red circles) and the colinear method (blue squares) for YBa2CusOe.333. In the colinear method of measurement, it is assumed that no current flows along either the b- or c-axes of the crystal. If the resistivities in the b- and c-directions are comparable in magnitude to the resistivity along the a-axis (and much smaller than that of the contact material), and if the contacts cover the ends and edges of the crystal well, this is a safe assumption. However, in YBC12CU3O6.333, as can be seen in Figures 4.1, 4.2 and 4.3, the c-axis resistivity is approximately three orders of magnitude larger than the a- and b-axis resistivities. For this reason, any contamination of the measurement by c-axis resistivity is likely to have a noticeable effect on the measured value of pa. Chapter 5. Discussion 38 The crystal measured by the colinear method was almost ideal in shape for this sort of measure-ment. Its effective a-axis length was over three times the b-axis length, and the c-axis was very thin - only about 1% of the effective a-axis length. Also, it was thought that the contacts had good gold coverage on the ends and edges of the sample. However, given the discrepancy between the sample measured by the colinear method and the sample measured by the Montgomery method, one must consider the possibility of c-axis contamination. As will be shown in Subsection 5.3, the discrepancy between the two samples is due to both a small amount of c-axis contamination, and a sample-dependent low temperature resistivity term of the form Ln(^). 5.2 Comparing U B C Data to Relevant Data From The University of Toronto Three samples from the same crystal growth run as the two Y B a2C U3OS.333 samples studied in this thesis were studied by researchers at the University of Toronto (UofT). The three UofT samples had colinear contacts, and both resistivity and thermal conductivity were measured on these samples. The researchers were mainly interested in the low temperature characteristics, and obtained data down to 80 mK in their dilution refrigerator. The non-annealed pa data for the UBC Montgomery sample, and the non-annealed and six hour annealed data for the UBC colinear sample, are plotted in Figure 5.2, alongside the resistivity data for two of the University of Toronto samples [9]. The third UofT sample was found to be inhomogenous, with a small transition at ~ 45 K, and so its data has not been included here. The researchers at UofT had a thermometry problem as they were cooling the samples down from room temperature: the thermometer temperature tended to lag behind the sample temperature. This was not considered to be a problem once the probe had been cooled all the way to base temperature, and the sample was being warmed in a controlled manner. For this reason, the dense UofT data between 80 mK and 22 K should be considered reliable, while the sparser resistivity data between 9 K and 300 K will tend to be lower than they should be. It should also be noted that the UofT data at the very highest temperatures are likely more accurate than at intermediate temperatures, because the sample had to be heated to reach 300 K. However, heating the sample to this temperature likely accelerated the room temperature anneal process. This is less of an issue for the UofT researchers, since they only take one data sweep on a sample for a given anneal, and the time spent at 300 K happens before the rest of the data is taken. Their time spent at room temperature will not necessarily be comparable to the same anneal in a UBC sample however, because our samples were Chapter 5. Discussion 39 not warmed above room temperature in this manner. p a ; UBC and U . of T . Data compared 2000 g- 1500 o i Cl 3 . X I O O O Q. 500 0 50 100 150 200 250 300 Temperature (K) Figure 5.2: A comparison of the resistivity along the a-axis for the UBC and the UofT samples. Purple squares: UBC colinear sample with no anneal, Orange crosses: UBC colinear sample with 6 hour anneal, Red circles: UBC Montgomery sample with no anneal, Green stars: UofT sample "L", no anneal, Blue diamonds: UofT sample "J", no anneal. As can be seen in Figure 5.2, the sample measured by the Montgomery method agrees with the UofT data within its limits of uncertainty below about 100 K, although the UofT data has no apparent Tc, while the UBC sample does. Note that if the UBC sample were of higher doping than the UofT data (thus having a higher T c ) , one would expect its resistivity to be lower than the UofT's crystal, not higher. However, given the limits of uncertainty on the data sets (the uncertainty on the UBC data is shown in Figure 4.6; the UofT data should have a comparable uncertainty to that of the UBC colinear sample, see Figure 4.11), this may not be an issue. By adding a constant to the resistivity values such that the UofT's data agree with the non-annealed data for the UBC colinear sample at 100 K, the two data sets can be made to overlap in certain temperature ranges. This was done in order to directly compare the low temperature UofT data to the UBC data, and see if there was good enough agreement between the two data sets to allow conclusions to be drawn about the very low temperature behaviour of these samples (since the UBC data is missing such information). The comparison can be seen in Figure 5.3. The data sets agree well at high temperatures, but clearly do not agree at low temperatures, where the slopes and curvature differ significantly. Chapter 5. Discussion 40 P a ; UBC a n d U . o f T . D a t a C o m p a r e d 2000 0 50 100 T e m p e r a t u r e (K) 150 200 Figure 5.3: A comparison of the resistivity along the a-axis for the U B C sample measured by the colinear method, and the UofT samples. Purple squares: U B C colinear sample with no anneal, Green stars: UofT sample " L " , Blue diamonds: UofT sample " J " . By adding a constant such that the data for the UofT's samples agree with that of the U B C Mont-gomery sample at 20 K, the two data sets can be made to overlap well over the entire temperature range. This can be seen in Figure 5.4. At higher temperatures, where the UofT data is considered to be less accurate, there is only a reasonable agreement between the two data sets. However, there is good agreement between the data sets over the temperature range where the UofT data is considered to be accurate, although the overlap region is small. A close-up view of this temperature range can be seen in Figure 5.5. It can be seen in Figure 5.6 that a log-plot of this data reveals that the very lowest temperature data show a Ln (^) temperature dependence (although they also shows signs of the saturation mentioned by Ando et al. in [13]). The UofT data show this clearly, while the U B C data do not go low enough in temperature to confirm the behaviour, although they are in keeping with it. This L n ( ^ ) behaviour will be discussed in great detail in the next section. Chapter 5. Discussion 41 p a ; U B C a n d U . o f T . D a t a c o m p a r e d 2 0 0 0 1 8 0 0 1 6 0 0 O 1 4 0 0 i a — 1 2 0 0 Q. 1 0 0 0 8 0 0 i. , , , , , , , , , i , , i , , , i , i 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 T e m p e r a t u r e ( K ) Figure 5.4: A comparison of the resistivity along the a-axis for the UBC sample measured by the Montgomery method, and the UofT samples. Red circles: UBC Montgomery sample with no anneal, Green stars: UofT sample "L", Blue diamonds: UofT sample "J". p a ; U B C a n d U . o f T . D a t a c o m p a r e d 1 4 0 0 ~ 1 2 0 0 u i C l 3 I O O O III c . 8 0 0 0 1 0 2 0 3 0 4 0 5 0 6 0 T e m p e r a t u r e ( K ) Figure 5.5: Low temperature comparison of the resistivity along the a-axis for the UBC sample measured by the Montgomery method, and the UofT sample "L" . Red circles: UBC Montgomery sample with no anneal, Green stars: UofT sample "L". Chapter 5. Discussion 42 Pa , U B C a n d U . o f T . D a t a C o m p a r e d -p 1 4 0 0 ° " 1 0 0 0 • ^ 1 2 0 0 1 8 0 0 1 6 0 0 8 0 0 0 . 1 0 . 5 1 L 5 1 0 T e m p e r a t u r e ( K ) 5 0 1 0 0 Figure 5.6: A log-plot comparison of the resistivity along the a-axis for the U B C sample measured by 5.3 Fitting the a- and b-Axis Data 5.3.1 Montgomery Data It was determined that the data for the sample measured by the Montgomery method could be fit well with a very simple function: where A, B and C are fitting parameters. Although the simple function in Equation 5.1 fit the a- and b-axis data very well, the residuals showed a systematic deviation. The reason why was found when the derivatives of the data were studied. As can be seen in Figures 5.7 and 5.8, the derivatives of the a- and b-axis data show three distinct behaviours over different temperature ranges. There is a sub-linear low temperature section up to approximately 50 K, an approximately linear section between ~50 K and ~200 K , and another apparently linear section with a different slope above ~200 K. the Montgomery method, and the scaled UofT sample " L " . The approximately L n ( ^ ) behaviour at low temperatures is evident. Red circles: U B C Montgomery sample with no anneal, Green boxes: UofT sample " L " . (5.1) Chapter 5. Discussion 43 d o . / d T , no anneal d p , / d T , 68 hour anneal - 1 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 Texperature (K) Figure 5.7: j £ for the a-axis for the Montgomery sample, fit to Equation 5.2, showing three linear sections with a Ln(j;) term overlaid upon the entire data set. The derivatives of the resistivity were fit by assuming that the three sections (below ~50 K, between ~50 K and ~200 K, and above ~200 K) were all linear, and that overlaid on this is a curvature due to the Ln(^) term in the resistivity. Note that the derivative of the logarithmic term Chapter 5. Discussion 44 50 ICO 150 200 Temperature (K) apt, /d~, 6 nour annea. 2S0 50 100 150 200 Temperature (K! d j D b / d T , 328 hour annea l 50 IOC 150 20C Temperature (K; d p 8 'dT, 19 nour anneal 250 5 0 100 150 200 Temperature (K) 250 Figure 5.8: | | for the b-axis for the Montgomery sample, fit to Equation 5.2, showing three linear sections with a I n ( ^ ) term overlaid upon the entire data set. is This gives the following fitting equation: $L = <A1T + A3) [ l -e (r -Ti) ] + ( f t T + B2) [Q(T - f i l l P - ®( T - T*)\ + (C 1T + C 2 ) [6 (T -r 2 ) ] - I (5.2) Chapter 5. Discussion 45 where 0 is the Heaviside function, 7\ = 45 K and T-z — 206 K (allowing Tv and T 2 to vary resulted in values within 1 K of these temperatures, with no obvious trend as a function of doping/anneal time). To reduce the number of free parameters in the equation, the following equalities were used to remove the constants A2 and B2 from the expression: Ax Tx + A2 •- Bi Ti + B2 BiT2 + B2 = CiT2 + C2 The fits shown in Figures 5.7 and 5.8 were obtained using Equation 5.2. If the derivative of the resistivity is linear, i.e. ^ — A\ T + A2, then the resistivity itself should be fit by a quadratic, p — 4fT2 + A2T + A3. Informed by the shape of the derivative graphs, the three sections of the a- and b-axis resistivities were fit separately (making use of the Heaviside function again) to an equation of this form, with an extra H Ln(^) term overlaid upon the entire data set. The explicit form of this fitting equation is given by: P = T'2 + A2T + A-^j [1 - 0 ( T - T,.)] + (T T 2 + ° 2 T + C3) L1-0^ -1^ )] + H L N where T\ — 45 K and T2 = 206 K again. Note that the number of fitting parameters in Equation 5.3 can be reduced from ten to six by using the following equalities to remove A2, A3, B2 and B3. Ai Ti + A2 = Bi Ti + B2 Bi T2 + B2 = Ci T2 + C2 (^j T2 + A2 Ti + A3 = (^j T2 + B2 Ti + B3 (B±y2 + B 2 T 2 + B 3 = ( ^ ) T 2 + C2T2 + C3 It was decided to remove A2 and A3 in lieu of C2 and C3 because the high temperature part of the graph is least affected by the L n ( ^ ) term, and so the fitting parameters should be most accurately determined for that section. Note that because of the Heaviside functions, the high temperature data have four fitting param-eters (three for the quadratic term, and one for the LTI(Y) dependence), while the mid- and low temperature data each have just two independent parameters (one for the quadratic term and one for the above-mentioned logarithmic term, which, affects the entire data set). Chapter 5. Discussion 46 The a- and b-axis data are replotted with the fits obtained from Equation 5.3 in Figures 5.9 and 5.10 respectively. The residuals for these fits were very good, of the order of 2 pQ • cm, which is 0.5% or less of the absolute value of the resistivity. p a W i t h F i t s 50 100 150 T e m p e r a t u r e ( K ) 200 250 Figure 5.9: The a-axis resistivity for the Montgomery sample, fit to Equation 5.3. (From top to bottom) Red circles: nominally no anneal, Green crosses: 6 hours anneal, Purple squares: 19 hours anneal, Blue diamonds: 68 hours anneal, Orange stars: 328 hours anneal. As mentioned in the previous Section, some of the low temperature data from a measurement performed at the University of Toronto could be, by means of adding a constant to the data, grafted onto the data for the U B C sample measured by the Montgomery method. The function which fit the U B C data also turned out to be a good fit for the low temperature UofT data as well. Note that UofT data was not used in the fitting, so the excellent agreement between the fit for the U B C data and the UofT data confirms that the low temperature logarithmic behaviour is present, and can be determined even with the limited amount of low temperature U B C data. The UofT and U B C data are shown together, with the fit obtained for the U B C data alone, in Figure 5.11 5.3.2 Colinear Data The derivatives for the pa data obtained by the colinear method were calculated, and can be seen in Figure 5.12. Again, three distinct and presumably linear sections of the graph can still be seen in every case, with a curvature overlaid upon the entire graph. The temperatures at which the linear behaviour changes slope appear to be exactly the same as in the Montgomery data, 45 K and 206 K. Chapter 5. Discussion 47 Figure 5.10: The b-axis resistivity for the Montgomery sample, fit to Equation 5.3. (From top to bottom) Red circles: nominally no anneal, Green crosses: 6 hours anneal, Purple squares: 19 hours anneal, Blue diamonds: 68 hours anneal, Orange stars: 328 hours anneal. Figure 5.11: The a-axis resistivity for the UBC Montgomery sample (red circles), and scaled low temperature data from a UofT sample (green stars), with the fit for the UBC data shown to low temperatures. Chapter 5. Discussion 48 d p a /dT, No Anneal 15- , . _ 0 5C 100 150 200 Temperature (K) d p a / d T , 6 Hour Anneal £ 10 0 50 100 150 200 250 Temperature (K) d p a / d T , 1 Day Anneal 1 5 _ 0 50 100 150 200 25C Temperature (K) d P a / d T , 3 Day Anneal 15r ' •o -10 0 50 100 150 200 250 Temperature (K) d p . /dT, 1 Day Anneal 1 5 ( — 1 x 0 50 100 150 200 250 Temperature IK) Figure 5.12: for the colinear data, fit to Equation 5.2, showing similar behaviour to the Mont-gomery a-axis data. The temperatures at which the graphs suddenly change slope are the same as for the Montgomery data. The colinear data were fit using the same equations as the Montgomery data: Equation 5.3 for the resistivity and Equation 5.2 for the derivative of the resistivity. Figure 5.13 shows the fits to the resistivity data, and Figure 5.12 shows the fits to the derivative of the resistivity. The residuals for the fits to the colinear data were of the order of 2 /zfi • cm, which is 0.2% or less of the absolute magnitude of the resistivity. Chapter 5. Discussion 49 p a With F i t s 3. u Q. 2000 1500 lOOOf 500 5 0 100 150 Temperature (K) 200 250 Figure 5.13: The a-axis resistivity for the colinear sample, fit to Equation 5.3. (From top to bottom) Red circles: nominally no anneal, Green crosses: 6 hours anneal, Purple squares: 1 day anneal, Blue diamonds: 3 days anneal, Orange stars: 7 days anneal. Comparing the data for the Montgomery and colinear a-axis resistivity, it becomes clear that the linear behaviour seen in j£ is intrinsic. This is demonstrated in Figure 5.14, where the derivatives of the a-axis resistivity for the two samples are shown, with the logarithmic terms subtracted off. This Jf plot demonstrates the excellent agreement between the two samples better than a resistivity plot would, because the A3, B3 and C3 parameters for the two samples are not in good agreement. The reason for this is discussed presently. The fact that the changes in the slope of % happen at the same temperatures in the colinear sample as they did in the Montgomery sample further supports the conclusion that this is intrinsic behaviour. The Ln(^) term is clearly not intrinsic behaviour. The good agreement between the resistivity derivatives for the two samples verifies that the samples are of sufficient purity and perfection to reveal intrinsic behaviour, however the logarithmic term in the colinear sample is two or more times larger than it is in the Montgomery sample for every doping. Such a strong sample dependence argues strongly against the low temperature logarithmic behaviour being intrinsic. This matter wil l be discussed further in Subsection 5.3.3. Chapter 5. Discussion 50 dpi, / d T , Two Samples, No Anneal dp„ / d T , Two Samples, Four th Anneal 0 50 100 150 200 250 Temperature (K) dP„ /dT , Two Samples, 6 Hour Anneal • • • f • V: • • 0 50 100 150 200 250 Temperature (K) dp, / d T , Two Samples, T h i r d Anneal 5 0 50 100 150 200 250 Temperature (K) 100 150 200 Temperature (K) 250 d p , / d T , Two Samples, F i f t h Anneal 100 150 200 Temperature (K) Figure 5.14: for both the colinear (blue squares) and Montgomery (red circles) samples with the extrinsic I « ( f r ) term subtracted off, demonstrating the good agreement between the two samples. Subtracting off the logarithmic terms allows a direct comparison of the a-axis resistivities of the two samples. This is shown in Figure 5.15. The first two data sets for the colinear sample (no anneal and six hours anneal), which had originally shown no hint of Tc, now show a clear downturn at the lowest temperatures. The strong logarithmic term in this sample had apparently been masking the onset of superconductivity. Chapter 5. Discussion 51 Figure 5.15: The a-axis resistivity for the colinear sample (blue squares) and the Montgomery sample (red circles) with the extrinsic Ln(^) term subtracted off. One thing that can be seen in Figure 5.15 is that the a-axis resistivities are still not in exact agreement, although the qualitative agreement is greatly improved. The reason for this lack of exact agreement is discussed next. If one assumes that the a-axis resistivity of the colinear sample has some c-axis contamination, but that the a-axis resistivity of the Montgomery sample does not (which is sensible, given that the Montgomery analysis is designed to extract all three of the components of the resistivity tensor, and makes no assumptions about certain dimensions being shorted out), then by taking the ratio of the two, one expects: ..contaminated n i an n Pa _ Pa ' PPc _ j _|_ p Vc Pa Pa Pa where /? is a factor that quantifies how much c-axis contamination is present. Wi th the logarithmic terms subtracted off, the ratio of the a-axis resistivities for the fully annealed cases of the colinear sample and the Montgomery sample was taken. This ratio, minus one, is shown in Figure 5.16. Alongside the ratio of pa for the colinear sample to pa for the Montgomery sample is pictured the scaled-down anisotropy (£*•) for the fully annealed Montgomery case. The anisotropy has been divided by a constant to make it the right order of magnitude, and then another constant was added Chapter 5. Discussion 52 R a t i o o f p a F o r T w o S a m p l e s , w i t h S c a l e d A n i s o t r o p y 2 - 5 , • • • , 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 T e m p e r a t u r e ( K ) Figure 5 .16: The ratio of pa for the colinear sample to pa for the Montgomery sample (red circles), with the scaled down anisotropy shown alongside it (blue line). to shift it vertically, such that the high temperature tail of the two curves agree. There is not an exact agreement, but the shape of the ratio of colinear pa to Montgomery pa clearly mimics the shape of the anisotropy. Much of the remaining discrepancy between the colinear pa data and the Montgomery pa data is certainly due to a very small amount of c-axis contamination in the colinear sample's data. The vertical shift that the scaled anisotropy needed to agree with the ratio is not understood. This might be due to the samples having different dopings, or uncertainty in the measurement of contact spacings. The c-axis contamination would tend to artificially increase the logarithmic term found for the colinear sample, and since the relative magnitudes of the logarithmic terms were the basis for discounting the Ln(j:) term as intrinsic behaviour, it is important to quantify by how much the c-axis contamination would distort the fit parameter for the logarithmic term. pa for the Montgomery sample was subtracted from pa for the colinear sample, and fit to the form H IM$)+G. This was done to give an indication of how much the parameter accompanying the Ln(^) term might be distorted by the c-axis contamination. This fitting never gave an H parameter that was more than 1 3 , while the discrepancy between the fit parameters accompanying the logarithmic terms for the two samples was always at least 100 . Therefore, although the small amount of c-axis contamination in the colinear sample would tend to inflate the size of the logarithmic Chapter 5. Discussion 53 term, it should never do so by more than 13%, and so the conclusion that the L n ( ^ ) term is not intrinsic to YBa2Cu30e.333 stands. The size of the term in the two samples is still different by a factor of two or more. 5.3.3 Analysis of the a-b Plane Resistivity It wil l be assumed hereafter that the intrinsic behaviour of the in-plane ^ is linear throughout the temperature range, although this is arguable in the region below 45 K. As can be seen in Figure 5.14, particularly in the less annealed/more underdoped samples, there may be some curvature in this region, although the presence of the logarithmic term makes the low temperature fitting problematic, and that may account for the apparent curvature. The bipolaron model of Alexandrov and Mott, [25], is one model that may capture the quadratic temperature dependence seen in the a- and b-axis resistivities. This model predicts the following for the in-plane resistivity: Ci T + C2T2 P = n - n L + bnLT ' (5"4) where C\ and C2 are constants, n is the total density of bosonic (bipolaron) carriers, ni is the number of localised bosons, and b n^T gives the number of unoccupied impurity wells. It is explicitly assumed that the residual resistivity is zero. If one makes the assumptions that the residual resistivity is not zero, and that b is negligible in the samples studied here, then this model is quadratic, and in keeping with the form of the intrinsic part of the in-plane resistivity presented here. However, from the discussion of the model given in [25], it does not appear possible for C i , the term accompanying T, to be negative, and this was found to be the case for the mid- and low temperature data. Thus, while the bipolaron model of Alexadrov and Mott holds promise of being a good description of the data presented here, it apparently has some inconsistencies with the data as well. The fact that ^ shows discontinuous changes in slope is difficult to explain. Although 45 K is close to the superconducting transition temperature for the Ortho II phase of Y B C O , the change in slope at 45 K cannot be due to contamination by small domains of Ortho II ordered Y B C O , because the resistivity increases at this temperature. Contamination would tend to decrease the resistivity at the critical temperature of the unwanted Ortho II ordered domains. It is unlikely that this change in behaviour is due to the Neel transition either, as will be discussed in detail in Section 5.4. Chapter 5. Discussion 54 The change in slope at 45 K may be due to a spin glass transition, but 45 K is much higher than one would expect for this. The spin glass transition in YBa2Cu30a+x, seen by Sanna et al. in [27], was never more than 20 K, with the highest values seen at an estimated 0.05 holes per CuOi plane (well inside the antiferromagnetic dome). The spin glass transition temperatures steadily decreased with increased doping, going to 0 K at approximately 0.08 holes per CuO-i layer (well inside the superconducting dome). Given these much lower temperatures, it is unlikely that the 45 K change is due to a spin glass transition. The 45 K change is not due to the formation of charge stripes oriented along the either the a- or b-axis, because the effect is seen in the data for both axes, and appears roughly equal in magnitude for the two cases. If the charge stripes orient along the diagonal, then the 45 K change might signal their formation, but in a system such as Y B C O , which is orthorhombic and has chains along the b-axis, diagonal charge stripes seem unlikely. The formation of some kind of two-dimensional ordering of the charges, such as a charge density wave, is a possible explanation for the 45 K change, but, the insensitivity to doping that, the 45 K change exhibits argues against this. The temperature at which the charge carriers within the sample order should be sensitive to the density of those carriers, not completely insensitive to it. The change in slope at 206 K might be related to the pseudogap transition, but here the temper-ature seems quite a, bit too low for this to be the case. Table 5.1 lists some pseudogap temperatures from the literature, and based on these values, the pseudogap temperature for this material should be in excess of 300 K, and thus not visible in the data presented here. Also, the temperature at which the slope of ^ abruptly changes does not, appear to be increasing with doping/anneal time, which would be expected if this change were associated with the pseudogap temperature. Further-more, the high-temperature region of ^ appears to be linear with a non-zero slope. For resistivity measurements, the pseudogap temperature is defined to be the temperature below which p begins to deviate from linear behaviour. This would imply that if the change at 206 K marks the pseudogap temperature, that the slope of ^ above this temperature should be zero. Instead, the slope is non-zero (although small). It may be that the high temperature data is in fact a slow sub-linear rollover to zero slope that cannot be seen, due to the relatively small number of data points in this region (about 30 points). However, the residuals of the fit given by Equation 5.3 do not show any increase in magnitude with increasing temperature in this region. They remain small and in keeping with the residuals throughout the entire temperature range. This argues in favour of the linear behaviour with non-zero slope being the correct form of ^ , which argues against the 206 K change being associated with the pseudogap. Chapter 5. Discussion 55 A glass-like transition due to the freezing of the oxygen ordering processes found by Nagel et al. in in optimally doped Y B C O [29] occurs at 280 K, and one would expect this transition to occur at a lower temperature for a sample with a lower density of oxygens in the chains. However, the transition seen by Nagel et al. is quite broad, covering several tens of Kelvins, while the transition seen at 206 K in the data presented here occurs over a temperature range no broader than approximately 5 K. For this reason, the change in behaviour seen at 206 K is unlikely to be due to a glass-like transition like the one seen by Nagel et al. Material: To (K) T* (K) Method of Measurement: YBa2Cu3O6.90 [19] 90 ~110 Resistivity YBa2Cu306.85 [19] 90 ~150 Resistivity YBa2Cu3 06.78 [19] 76 ~190 Resistivity YBa2Cu306.68 [19] 63 • ~260 Resistivity YBa2Cu306.58 [19] 61 ~300 Resistivity YBa2Cu306A5 [19] 52 >300 Resistivity YBa2Cu3 06.95 [21] 92 110 Knight shift from N M R YBa2Cu408 [22] 81 240 Knight shift from N M R Y2Ba4Cu70i5 [23] 93 190 Knight shift from N M R Table 5.1: Pseudogap temperatures from the literature. The fact that the intrinsic behaviour of ^ appears to be linear throughout the temperature range, and that the changes in behaviour at 45 K and 206 K are the same (simply a change of slope), suggests that these two discontinuites might be due to similar processes. Furthermore, the lack of any doping dependence in both the 45 K and 206 K changes implies that while the charge carriers respond to whatever causes these two changes, the carriers are not themselves responsible for it. If they were, a doping dependence would be expected, since the density of the charge carriers affects most quantities that are due to the carriers themselves. In [28], Zhai et al. find two dielectric transitions in nonsuperconducting YBa2Cu306.o at 60 K and 110 K. They attribute these transitions to subtle structural changes in the unit cell of the material. The two changes in behaviour that Zhai et al. observed were seen in undoped antiferromagnetic Y B C O , so if these two structural changes occur at different temperatures when doping is increased, then they may be responsible for the two changes in the slope of -J^  seen in the data presented here. However, the fact that the 45 K and 206 K changes show absolutely no doping dependence may Chapter 5. Discussion 56 argue against this. On the other hand, since the doping in the UBC samples is accomplished by allowing the oxygens in the chains to order into chainlets, the overall chemical composition of the sample is not changed - only the doping of holes onto the planes. It is possible that one must change the total number of oxygens in the chains before the positions of the 45 K and 206 K transitions would shift, i.e. the number of oxygens in the chains would be more likely to change the structure of the overall unit cell than the simple rearranging of oxygens there would. A detailed study of the structure of YBa^CuzO^^s, focusing on any changes that occur in it. as a function of temperature, would be a very valuable complement to the work presented here, but such information is not yet available for this doping of YBCO. The discrepancies in temperature between the two transitions seen by Zhai et al. and the two seen in the data presented here are troubling, but the proposed mechanism of small structural changes is an intriguing one, for if correct, it would demonstrate that the charge carriers in YBCO respond to changes in the lattice. There are two mechanisms by which this could occur: the change in. the unit cell could affect the phonon modes in the sample, implying a coupling between phonons and charge carriers, or it might change the dimensions of the Brillouin zone, thereby changing the shape of the Fermi surface for the charge carriers. As will discussed in Section 5.4, the latter possibility-can probably be discounted. As already mentioned, the bipolaron model of Alexandrov and Mott may be in agreement with the data presented here. If it is found that the 45 K and 206 K are due to structural changes that alter the charge carrier-phonon coupling, this may be further support for their model. The final matter to address is the presence of the Ln(jt) term in the resistivity. Yoichi Ando and coworkers have reported, [12], [13], [14], seeing pab for a variety of cuprates change from "metallic" (Jr > to "insulating" 0) behaviour at low temperatures, with the low temperature data, following a Ln(^) dependence. These measurements were taken by applying a 60 T magnetic field along the c-axis in order to suppress superconductivity in a variety of relatively high Tc materials. Ando and coworkers argue that this metal-insulator (M-I) transition is intrinsic behaviour, and that it implies that the ground state of the cuprates is insulating. As the comparison of the UBC data to the UofT data in Section 5.2 shows, the low temperature behaviour of the in-plane resistivity for Y BaoCusOosss does have a Ln(^) temperature dependence. However, as stated in Section 5.3, the data presented here argue strongly against this logarithmic term being intrinsic, because it is sample dependent. When the logarithmic term is subtracted off entirely, as in Figure 5.15, the resistivity still passes through a minimum at T — =^-, where Ay and Chapter 5. Discussion 57 Ai are the constants given in Equation 5.3, but this upturn is an extremely weak one. The ground state is essentially metallic, with the resistivity approaching T = 0 with almost zero slope. The most likely explanation for the L n ( ^ ) term is that it is caused by localization due to impurity scattering or disorder, and if this is the case, then it is not surprising that the logarithmic behaviour was shown to be sample dependent. However, the two samples studied here came from the same crystal growth run, and are very pure with a high degree of crystalline perfection. One would not expect them to have very different impurity concentrations or levels of disorder. It is possible that because the samples were affixed to a rigid substrate using a heat treatment, the colinear sample - which was very thin in the c-direction - would be under considerably more strain than the rather thick Montgomery sample. This might account for the difference in the logarithmic terms. There are several suggested models for what could bring about a L n ( ^ ) temperature dependence, but no agreement as yet on which is most likely the explanation for the behaviour. In Reference [15], X . F. Sun et al. suggest that the Ln{^) divergence in resistivity reported in [12], [13] and [14] might be caused by localization due to magnetic-field-induced ordering of the quasiparticles around vortices. However, given that the Ln(^) behaviour was seen in both the U B C and UofT samples in the absence of any magnetic field, this proposed mechanism is apparently incorrect. The fact that the in-plane resistivity approaches T = 0 with almost zero slope even in samples right at the underdoped edge of the superconducting dome (see Figure 5.15) is important. There is no hint in the intrinsic behaviour that the samples become insulating when they cross out of the doping range that produces superconductivity, which would be expected if the samples immediately became antiferromagnetic at that point. The implication of the metallic behaviour at T = 0 is that there is a gap between the superconducting dome and the antiferromagnetic dome in the phase diagram, which is a deviation from the common view implied by the phase diagram shown in Figure 1.1. As the doping is decreased, the samples cease to be superconductors, but remain strange metals. The samples would become antiferromagnetic with further decreases in doping, but judging when a sample becomes insulating based on resistivity measurements alone must be approached carefully, given that the L n ( ^ ) resistivity term that Ando et al. have shown to be ubiquitous in the cuprates is not intrinsic, and would tend to mask the onset of such behaviour. Chapter 5. Discussion 58 5.4 Fitting the c-Axis Data The most distinctive feature of the c-axis resistivity is the broad "hump" in the data that has its maximum at approximately 100 K. The c-axis resistivity in highly anisotropic cuprates at low temperatures usually shows a "semi-conducting" temperature dependence, with pc ~ T~a (where (0 < a < 2) [18] - this is a bit different from thermally activated behaviour across a gap, i.e. pc ~ e - E s / k B T y - p m s semi-conducting-like behaviour can also be seen in the data presented here, but it is partially obscured by the anomalous hump feature. As can be seen in Figure 5.17 however, where the c-axis data have been scaled along the resistivity axis such that all the curves agree at 40 K, with increased doping, the position of the hump's maximum is rapidly decreasing both in magnitude and in temperature. It is reasonable to assume that with a further increase in doping, the hump feature would completely disappear, leaving just the familiar "semi-conducting" behaviour. p c F o r V a r i o u s A n n e a l T i m e s , S c a l e d 2 . 5 x l 0 6 r 1 • •— — — ' — ' — ' — 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 T e m p e r a t u r e ( K ) Figure 5.17: pc for the Montgomery data, scaled along the vertical axis to agree at 40 K. (From top to bottom) Red circles: nominally no anneal, Green crosses: 6 hours anneal, Purple squares: 19 hours anneal, Blue diamonds: 68 hours anneal, Orange stars: 328 hours anneal. A similar feature to this broad peak has been seen by other groups in other systems. In Reference [10], Lavrov, Kameneva and Kozeeva ( L K K ) study pc and pb for underdoped TmBa2CuzO§+x and LuBa2CuzOz+x, and find a similar hump feature for both underdoped superconducting samples and highly underdoped non-superconducting samples. L K K note that the behaviour had not at that point been seen in underdoped Y B C O . Chapter 5. Discussion 59 The hump seen by L K K is more pronounced than the hump seen in the YBa2CuzOtj.zzz data presented here, and the superconducting samples studied by L K K do not show the sharp, semi-conducting-like upturn below this hump (and above Tc) that the YBazCuzOs.wz data do. This may be partly because their samples had higher Tc values than the samples studied here (on the order of 20 K , as opposed to approximately 12 K ) , but it seems more likely that the "semi-conducting" behaviour of Y Ba^CuzO^.333 is simply much more prominent than in their systems. After studying the superconducting TrnBa2CuzOe+x sample, L K K altered the oxygen content of the sample such that the sample would not be superconducting. Wi th no Tc, the sample did show the sharp low temperature upturn seen in the c-axis data presented here. L K K associate this peak with the Neel temperature, rather than "semi-conducting" behaviour, however. Data presented in Reference [11] by Lavrov, Ando and Segawa (LAS) for highly underdoped, non-superconducting Y Ba2CuzO$+x do not support this interpretation. The highly underdoped Y B C O c-axis resistivity studied there shows a sharp upturn at low temperatures, and a broad hump feature at ~ 125 K. Most notably however, the data also shows a clear discontinuity in slope when the sample crosses the Neel temperature (and the temperature of the Neel transition for the sample is confirmed by a separate experiment), and this discontinuity is not associated with the sharp, semi-conducting-like upturn in resistivity at low temperatures. As can be seen in Figure 4.3 or Figure 5.17, the YBa2Cv.zOQ.zz3 data presented here show the sharp upturn in pc below the anomalous hump feature, but they also show a superconducting tran-sition. Since the sharp upturn happens above Tc, it seems unlikely that this feature is associated with the Neel temperature, as this would involve a coexistence of superconductivity and antiferro-magnetism. Such a coexistence is a possibility, but it is an exotic one, and the metallic behaviour of in-plane data presented in Section 5.3.3 tends to refute this. It, is much more likely that the upturn is due to the semi-conducting-like behaviour seen in highly anisotropic cuprates. Note that the discontinuity in slope that L A S see at the Neel temperature does not show up in the U B C data at all, implying both that the sample studied here is above the Neel temperature, and that it is relatively homogenous in doping. L K K suggest in [10] that the c-axis resistivity has two contributions, and given the shape of the U B C c-axis data - with its combination of broad hump and semi-conducting-like divergence - this seems a reasonable suggestion. The model used by L K K in [10] consists of a conductivity with two components: an exponentially activated term taken from a variable range hopping model, and a metallic resistivity that tracks the Chapter 5. Discussion 60 in-plane resistivity. L K K fit their anistropy to a form based on this model, with excellent results except at low temperatures, but they do not fit their c-axis data to the model. The author of this thesis found that L K K ' s model fit the anisotropy data for the Montgomery sample very well also, but that the corresponding form for the c-axis resistivity obtained very poor fits. Good fits to the data could be obtained from L K K ' s model by not forcing the metallic contribution to track the a-axis resistivity, and by adding a constant term to the conductivity. However, this complicated model, with its many fitting parameters, did not leave the author with much confidence that real physics was being extracted. For this reason, the c-axis data has been fit to a much simpler model, with good results and a much more straightforward interpretation. The low temperature peak in the data can be fit to more than one type of function. The simplest two functions found to work were a pi ~ e~ST form and a ^ 7^+g 2 f ° r m > w ^ t n * n e l a t t e r working slightly better. The high temperature data can be fit to an inverse quadratic of the form p2 ~ A i TI+\* T+A3 • Thus, the data was fit to a function of the form: 1 * H1T2 + H2 ' A1T2 + A2T + A3 One notable aspect of Equation 5.5 is that if there are two channels for the charge to be carried through, one would expect the resistivities for these two channels to add like parallel resistors, rather than in series as in Equation 5.5, i.e. they should add as p = _i_+x, rather than p = pi + p2. Adding PI PI the two contributions in parallel was how L K K formed their model. A model with the two contributions added in parallel was tried, and good fits to the data could be obtained only by adding an extra constant term to the denominator of the expression. Also, the model had the same ambiguity as L K K ' s model in terms of what meaning could be extracted from it. As will be discussed in detail in Subsection 5.4.1, the simple model given by Equation 5.5 provides a straightforward interpretation, and reveals obvious parallels between the behaviour seen in the c-axis resistivity and that seen in the a- and b-axis resistivities. This offers reassurance that Equation 5.5 is the correct model to be using. Equation 5.5 is a continuous function. Given that the in-plane resisitivities showed discontinuous changes in the slope of -Jf, it is prudent to look for similar features in the out-of-plane resistivity before attempting to fit the data to this function. The derivatives of the c-axis data are shown in Figure 5.18. Chapter 5. Discussion 61 d p c / d T F o r V a r i o u s A n n e a l T i m e s o - 1 0 0 0 0 • . - 1 2 0 0 0 -- 1 4 0 0 0 • 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 T e m p e r a t u r e ( K ) Figure 5.18: % for the c-axis of the Montgomery sample, showing no discontinuity at 45 K, but (arguably) three discontinuities at 172 K, 206 K and 239 K. (From top to bottom) Red circles: nominally no anneal, Green crosses: 6 hours anneal, Purple squares: 19 hours anneal, Blue diamonds: 68 hours anneal, Orange stars: 328 hours anneal. Like the a- and b-axis data, the derivative of the c-axis resistivity has discontinuous changes in slope. The surprising thing is that while one discontinuity occurs at 206 K (the same temperature where one was seen in the in-plane ^ graphs), there is no discontinuity visible at 45 K. Also, there is a third discontinuity at 172 K, and arguably a fourth at 239 K, neither of which was visible in the a- and b-axis data (although arguably the much cleaner data for the colinear sample do show the discontinuity at 239 K - see Figure 5.12 or Figure 5.14; if true, this is interesting, for the discontinuity has again shown up at the same temperature in two different samples). The three high temperature discontinuties are shown in close-up in Figure 5.19, where all the data have been shifted to agree at 210 K. At high temperatures, the ^ data appear roughly linear and map on top of one another down to the discontinuity at 206 K. Between 172 K and 206 K, the data still appear linear, but the slopes differ in a clearly doping-dependent manner. Below 172 K, the slope changes again, although this change becomes more and more subtle with increasing doping. As mentioned, there is no sign of the discontinuity at 45 K. Chapter 5. Discussion 62 H Q. •4000 -4500 -5000 d p c / d T F o r V a r i o u s A n n e a l T i m e s e o Ci - 5500 -6000 -6500 -7000 ************** • i TT* '••V 125 150 175 200 225 250 T e m p e r a t u r e ( K ) Figure 5.19: Close-up view of % for the c-axis of the Montgomery sample, with curves shifted vertically to agree at 210 K, showing three discontinuities at 172 K, 206 K and 239 K. (From top to bottom) Red circles: nominally no anneal, Green crosses: 6 hours anneal, Purple squares: 19 hours anneal, Blue diamonds: 68 hours anneal, Orange stars: 328 hours anneal. In light of the discontinuities in jfe along the c-direction, the model used to fit the data employs the Heaviside function again. Demanding that both of the contributions to the resistivity be continuous functions having continuous derivatives again allowed the removal of superfluous parameters from model. In the case of the low temperature peak, these demands require that the fitting parameters for the expression remain unchanged across the transition temperatures, i.e. the semi-conducting-like behaviour dominant at low temperatures is unaffected by whatever causes the abrupt changes in slope in Jf. Only the inverse quadratic term responsible for the anomalous hump at higher temperatures is affected by these. Because of the extra two discontinuous changes in ^fr, the explicit form of the model is quite complicated and has been omitted. Suffice it to say that it is based on Equation 5.5 with only the inverse quadratic term subject to abrupt changes in behaviour at 45 K, 172 K, 206 K and 239 K. The c-axis resistivity is shown in Figure 5.20, fit to this model. The residuals of the fits were of the order of 0.4% of the magnitude. The two contributions to the c-axis resistivity were then graphed separately. The semi-conducting-like contribution is shown in Figure 5.21, and the anomalous hump contribution is shown in Figure 5.22. Chapter 5. Discussion 63 W i t h F i t s 2 . 5 x l 0 6 2 x l 0 6 .< I 1 . 5 x l 0 6 3 . l x l O 6 5 0 0 0 0 0 1 0 0 150 T e m p e r a t u r e (K) Figure 5.20: pc for the Montgomery data, fitted by Equation 5.5. (From top to bottom) Red circles: nominally no anneal, Green crosses: 6 hours anneal, Purple squares: 19 hours anneal, Blue diamonds: 68 hours anneal, Orange stars: 328 hours anneal. 5.4.1 Analysis of the c-axis Resistivity As can be seen in Figures 5.21 and 5.22, the contribution that provides the semi-conducting-like part of the c-axis resistivity has almost no doping dependence, while the contribution that forms the anomalous hump accounts for nearly all the doping dependence seen. Note that although the in-plane resistivity of the sample was found to be quadratic and the out-of-plane resistivity has a contribution of the form of an inverse quadratic, the fitting parameters for the two cases do not appear to be closely related, although if one multiplies the fitting parameters for the hump contribution by a large constant, the values are at least of the correct orders of magnitude and the right sign to agree with the in-plane data. It may be a reasonable approximation to say that pa ~ - ( h „ m p ^ , but the relationship is not exact. This runs counter to L K K ' s claim that this contribution to the c-axis resistivity tracks the a-axis resistivity. There are several groups that have suggested that in certain doping ranges, pa ~ j-. The models of two of these groups will be discussed in the next Subsection. Although the pa ~ p ( f t „ m p ) relationship is not exact, there are still striking similarities between the behaviours for the a- and b-axis resistivities and the hump-like contribution to the c-axis resis-tivity. This is best shown in Figure 5.23, where the derivative of the inverse of this contribution is shown. Chapter 5. Discussion 64 Semi-conducting-like C o n t r i b u t i o n t o pc 1.4xl0 6 1.2xl0 6 _ l x l O 6 E V 800000 a — 600000 £ 400000 200000 0 0 50 100 150 200 250 Temperature (K) Figure 5.21: The semi-conducting-like contribution to pc for the Montgomery data. (From top to bottom) Red circles: nominally no anneal, Green crosses: 6 hours anneal, Purple squares: 19 hours anneal, Blue diamonds: 68 hours anneal, Orange stars: 328 hours anneal. 2 x l 0 6 1.75xl0 6 1.5xl0 6 o 1.25xl0 6 i 3 l x i o 6 „ 750000 Q. 500000 250000 0 50 100 150 200 250 Temperature (K) Figure 5.22: The anomalous hump contribution to pc for the Montgomery data. (From top to bottom) Red circles: nominally no anneal, Green crosses: 6 hours anneal, Purple squares: 19 hours anneal, Blue diamonds: 68 hours anneal, Orange stars: 328 hours anneal. A n o m a l o u s H u m p C o n t r i b u t i o n t o p c Chapter 5. Discussion 65 Figure 5.23: The derivative of j- for the Montgomery data. (From bottom to top) Red circles: nominally no anneal, Green crosses: 6 hours anneal, Purple squares: 19 hours anneal, Blue diamonds: 68 hours anneal, Orange stars: 328 hours anneal. Figure 5.23 shows a remarkable similarity in behaviour to the derivatives of the a- and b-axis data. There is a clear discontinuity in slope at 206 K, and an apparent one at 45 K (which was not visible in the raw data, as it was obscured by the semi-conducting-like contribution), just as there was in the in-plane resistivities. The sections between all of the discontinuities appear to be linear, although perhaps subject to a small amount of low temperature curvature - because of the low temperature semi-conducting-like peak, fitting the data in this region is problematic. This linear behaviour is also completely in keeping with the behaviour of the in-plane resistivity derivative data. Note that if the discontinuities in the slope of the in-plane resistivity derivative at 45 K and 206 K are due to subtle structural changes, then it is only reasonable to suggest the same for the two new discontinuities seen in the out-of-plane resistivity derivative at 172 K and 239 K. However, one would expect to see these two new discontinuities in the a- and b-axis data as well. This will be discussed more in Subsection 5.4.3. The doping dependence of the data in Figure 5.23 proceeds in an opposite sense to that seen in Figures 5.7 and 5.8 (presumably because Figure 5.23 shows the derivative of the inverse of Chapter 5. Discussion 66 the resistivity contribution). Also, as mentioned, the fitting parameters between the in-plane and out-of-plane behaviour are not exactly proportional. However, the implication of Figure 5.23 is clear: behaviour seen in the in-plane resistivity is also affecting the behaviour of the hump-like contribution to the out-of-plane resistivity (or vice versa). The semi-conducting-like contribution to the c-axis resistivity appears to be associated only with transport along the c-axis direction, i.e. this behaviour is not seen to have any effect on the a- and b-axis data. Note that the serni-conducting-like contribution seen at low temperatures in the c-axis data is not well fit by a Ln(^) form, nor does it show a significant doping dependence, which indicates that it is not related to the sample dependent Ln(^) term that was seen in the a- and b-axis data. However, it is not known whether the low temperature semi-conducting-like behaviour is intrinsic or not. Resistivity data along the c-axis of a second sample of similar doping would be required to determine this. However, the correlation between the intrinsic in-plane behaviour and the be-haviour of the anomalous hump contribution to the out-of-plane resistivity strongly suggests that this contribution is also intrinsic. There are two reasonable explanations for why the two contributions to the c-axis resistivity add in series, rather than in parallel. The first explanation is that in traveling through the crystal, a charge carrier experiences one type of scattering mechanism while in one part of the unit cell, and a different sort of scattering mechanism while in another. For example, the charge carrier might experience one sort of scattering when in the presence of the Cu02 bilayer, and another when it is travelling between bilayers. Effectively, the charge carrier can be thought of as traveling through two resistors connected in series. The second scenario is in a sense a more general case of the first. The two contributions to the c-axis resistivity may simply be a manifestation of Matthiessen's rule, which states that if two distinguishable scattering mechanisms exist, but do not interfere with the function of one another, then their resistivities simply add. Matthiessen's rule does require that the relaxation times for the two mechanisms be k-independent (where fik is crystal momentum). Given the clear interplay between the in-plane data and the anomalous hump contribution to the out-of-plane data (which will be discussed in more detail presently), this condition might seem to be obviously violated, but it, is worth noting that the Montgomery method extracts the resistivity for charge carriers traveling only in the c-direction. This removes most of the information about the k-dependence of contributions to this resistivity. In one dimension, Matthiessen's rule is more likely to hold. Chapter 5. Discussion 67 If the second scenario, having to do with Matthiessen's rule, is correct, then the two contributions to the resistivity are due to two different forms of scattering. For example, one contribution might be due to boson-assisted scattering while the other is due to scattering from static impurities. The first scenario, involving the charge carrier seeing the unit cell as two resistors in series, is also a case of two physically distinguishable types of scattering, but with the added condition that the scattering mechanisms are present only in specific locations within the unit cell. The data probably does not contain enough information to determine which of these two scenarios is correct, so the two cases will be kept on an equal footing, and discussed separately below. The correlation between the behaviour of pa and that of p ^ m p \ is of particular interest, and will be discussed in the next Subsection in some detail. Resistors in Series If the two contributions to the c-axis resistivity are due to scattering mechanisms that are found in two distinct areas of the unit cell, then one can attempt to identify which contribution is associ-ated with which area. The two. obvious candidate locations for seeing distinctly different transport behaviours are in the vicinity of the Cu02 bilayer, and far from the bilayer. Transport that takes place in the vicinity of the bilayer wil l be referred to as inrra-bilayer hopping in what follows, and transport that takes place between bilayers will be referred to as mier-bilayer hopping. Given the similarities in behaviour between pa and p m p j , both in terms of the quantities' derivatives and their doping dependence, the anomalous hump contribution to the c-axis resistivity would most likely be associated with intra-bilayer hopping. The semi-conducting-like contribution to the resistivity would then have to be associated with inter-bilayer hopping. The fact that semi-conducting-like contribution is relatively doping-independent tends to reinforce the idea that it cannot be associated with transport near the bilayer, since it is known that the doping of holes onto the CuO-2. layers profoundly influences transport on the planes. Presumably it would also influence transport across, but still in the immediate vicinity of, the planes. The relative doping independence of the semi-conducting-like contribution would also imply that the "blocking" layer [18] that the charge carriers must tunnel across during inter-bilayer hopping does not change strength much as the doping is increased, at least not in the case of the time-anneal method of doping that is being used in this study. This runs counter to claims in the literature [30] that the inter-bilayer hopping rate increases exponentially with doping, however most of these studies do not separate out or even recognize that the c-axis has two distinct contributions, so this discrepancy is understandable. Also, as noted earlier, the time-anneal method for doping holes into the planes might not affect the structure of the unit cell particularly, and so it might be that the Chapter 5. Discussion 68 overall oxygen content of the sample has to be changed in order to increase or decrease the size of the "blocking" layer. A study done over a wider range of dopings would be required to settle this matter. There is also the possibility (based on ideas put forth in the literature regarding the nature of the pseudogap), that the charge carriers are composite bosons that must split apart into their constituent fermions in order to tunnel between the bilayers. Unless the forms for the two contributions to the c-axis resistivity can be clearly associated with models for how such objects would behave within Y B C O , there is probably not enough information in the presented data to confidently make such a conclusion. However it remains a possibility, and if correct, one point becomes particularly interest-ing: the low temperature semi-conducting-like contribution (which would be presumably associated with the fermions tunneling between bilayers) does not react to whatever causes the discontinuities in the slope of Since the anomalous hump contribution (which would be associated with the composite bosons) does react to whatever causes these discontinuities, this implies that the source of the discontinuities must be somehow related to the pairing mechanism for the composite bosons, since the unpaired fermions are not affected by it. For example, if the discontinuities are caused by small changes in the unit cell that affect the charge carrier-phonon coupling, then the pairing mechanism must be either due to phonons, or must couple to phonons. Since any insights on the nature of such a pairing mechanism in the cuprates would be of considerable interest to the field, this picture is one worth keeping in mind. Matthiessen's Rule If the two contributions to the c-axis resistivity are due to two distinct scattering mechanisms that do not interfere with one another, but which are not associated with specific locations in the unit cell, then one can attempt to identify what each contribution is due to. The relative insensitivity to doping exhibited by the semi-conducting-like contribution to the c-axis resistivity suggests that this contribution could be due to static impurity scattering. Since the density of static impurities would not change with doping, one would hot expect to see a large increase in the number of scattering events from these impurities even if the density of charge carriers increases. The interplay of pa and pc(hurnp) implied by the similarities in their derivatives suggests that the anomalous hump contribution may be due to boson-assisted scattering. This will be discussed in the next Subsection in detail. Chapter 5. Discussion 69 5.4.2 The Connection Between pa and 1 , Several groups have suggested a possible pa ~ j- relationship. The models of Rojo and Levin [32] and Xiang and Wheatley [31] will be discussed here as being particularly relevant to the data presented. The model of Rojo and Levin studies the effect of both static and dynamic scattering (physical examples of these might be impurity and phonon-assisted scattering) on transport, and finds not only that the in-plane scattering rate affects the out-of-plane scattering, but also proposes a model that (with one change from what is suggested in [32]) fits the behaviour of both the a- and c-axis resistivity presented here. The model of Xiang and Wheatley studies superfluid density rather than resistivity, but finds that the out-of-plane hopping has a component that is influenced by the in-plane momentum. Combined with other experimentally determined behaviours seen in the cuprates, it has been suggested [33] that this k a 6 dependent c-axis hopping might account for the effect of the pseudogap on both the a- and c-axis resisitivites. The Model of Rojo and Levin The model of Rojo and Levin proposes that in the limit where the out-of-plane disorder is much greater than the in-plane disorder, aab ~ r and ac ~ ^ , where r is the lifetime of plane waves in the a-b direction. Thus, the resistivity in the c-direction will look roughly like the inverse of the resistivity in the a-b plane, in agreement with the Montgomery data presented here. The model of Rojo and Levin proposes that the resistivities for the ab-plane and the c-axis will have the following forms: P a b = a + dT + d'I(T) (5.6) P° = r T(rr\ , 1 , i — ( 5 - 7 ) b I{T) +c+ 5 7 5 ^ where I(T) is a temperature dependent function due to dynamical (boson-assisted) out-of-plane scattering, which Rojo and Levin estimate goes as I(T) ~ T. It is obvious from the form of Equation 5.6 that this will not fit the data presented here unless I{T) ~ T2. Rojo and Levin do note that neglected vertex corrections in their calculations could alter the form of I(T), particularly at low temperatures, although they suggest that this would tend to cause I(T) to be different for pab and pc. It is not made clear in [32] how the temperature Chapter 5. Discussion 70 dependence of I(T) is determined, so it is not known immediately whether I(T) ~ T 2 might be reasonable in some circumstances. If I(T) ~ T 2 is assumed to be physically reasonable, and is substituted into Equations 5.6 and 5.7, then these two models fit the data presented here with as much success as the phenomenological models already introduced. Equation 5.6, for the a-axis, is identical to Equation 5.3, which was used to fit the presented data. Equation 5.7, for the c-axis, only fits the anomalous hump contribution to the resistivity and does contain one more fitting parameter than the second term of Equation 5.5 (which was used to fit the anomalous hump contribution). However, in the fitting, the b' term in Equation 5.7 always turns out to be much smaller than c', which makes Rojo and Levin's model for pc nearly equivalent to the second term in Equation 5.5. Thus, the model of Rojo and Levin accurately describes the data presented here, provided that the substitution of I(T) ~ T2 is physically reasonable. If the model of Rojo and Levin is taken to be an accurate description of the data presented in this thesis, then the data presented here is fully within the limit where boson-assisted c-axis hopping dominates the bare c-axis hopping (which Rojo and Levin identify as being the appropriate limit for underdoped Y B C O ) . In the opposite limit (identified by Rojo and Levin as being appropriate for optimally doped Y B C O ) , the c-axis resistivity becomes linear with a positive slope, which is behaviour that has been seen experimentally. Rojo and Levin find that their model is consistent with a metallic ground state, which the in-plane data presented in this thesis implies the presence of. Note that if the low temperature semi-conducting-like contribution to the c-axis resistivity is in fact extrinsic (which could be determined with the study of another similarly doped crystal), then the anomalous hump contribution to the c-axis resistivity (which is certainly intrinsic, given it's clear relation to the intrinsic a- and b-axis behaviour) also clearly implies a metallic ground state for the cuprates. It is noted that difficulty was had in modeling both the in-plane and out-of-plane resistivity at low temperatures. This was seen to be due to the presence of low temperature terms that complicated the fitting. In the case of the in-plane transport, it was the extrinsic L n ( ^ ) term, and in the case of the out-of-plane transport, it was the (possibly extrinsic) semi-conducting-like term. Rojo and Levin do note however that the form of I{T) in their model may change at low temperatures due to vertex corrections that they have neglected. Specifically, I(T) may become different for the in-plane and the out-of-plane resisitivities at low temperatures. They suggest that if phonons are the bosons responsible for the dynamic scattering, at low temperatures I(T) ~ T3 for the in-plane resistivity and I(T) ~ T 5 for the out-of-plane resistivity. Therefore, part of the difficulty had in modeling the Chapter 5. Discussion 71 data at low temperatures could be due to the onset of this effect. Based on the graphs of -Jf, the fit to the a-b plane data only begins to be problematic below about 50 K, while the fit to the c-axis data becomes problematic at a higher temperature, perhaps around 100 K. This would be in keeping both with the temperature dependences of the model changing slightly at low temperatures, and with the c-axis temperature dependence changing more than the a-axis temperature dependence does. The Model of Xiang and Wheatley The model of Xiang and Wheatley addresses the superfluid density in the superconducting state primarily, but their prediction for the c-axis hopping term has some interesting implications when combined with certain experimental facts [33]. Xiang and Wheatley find that the out-of-plane hopping term contains a strong dependence on the in-plane momentum. The mechanism for this is proposed to be that the primary channel for hopping between bilayers along the c-axis is along the Cu(l)-0(4)-C7u(2) bonds. Since the holes in the planes tend to sit primarily on the 0(2) sites, this means that for a charge carrier to hop between bilayers, it must first move along the 0{2)-Cu(2) bond, which means that the c-axis hopping should depend on the in-plane momentum. Xiang and Wheatley quantify this in the expression tj_(k\\) = where Pk = \/cos2(kx/2) + cos2(ky/2) x (cos kx — cos ky). This expression is for tetragonal materials, and has zeroes when kx = ±ky. Xiang and Wheatley generalize their model to t±.(k\\) = t°±u-l + £™ode, where tn°de denotes interlayer hopping at the gap nodes when a mechanism for this exists. Xiang and Wheatley note that a noticeable deviation from t± oc (cos kx — cos A^) 2 and a finite contribution from the nodes would be expected for a material such as Y B C O , where hybridization between the planes and chains is expected. The interesting thing about Xiang and Wheatley's model is the dependence of the c-axis hopping on the in-plane momentum when this idea is considered in combination with certain experimentally observed behaviours having to do with the pseudogap. This synthesis of ideas is due to Bonn [33]. A good summary of the experimental observations discussed below can be found in [20]. The first relevant observation is from optical conductivity measurements, which find that the pseudogap manifests itself very strikingly in measurements along the c-axis. The pseudogap appears as a depression in the spectrum at low frequencies that becomes deeper with decreasing temperature, but does not really change its characteristics otherwise. The depressed area remains flat in frequency and fairly uniform in size regardless of the temperature. Chapter 5. Discussion 72 Optical conductivity measurements on the a-b plane do not show such a distinct depression due to the pseudogap. The pseudogap first shows up in the frequency dependent scattering rate calculated from these spectra as a subtle break in the otherwise linear scattering rate seen at high frequencies and temperatures. This break then deepens and becomes more noticeably gapped (flatter in frequency) with decreasing temperature. The size of the pseudogap is still largely constant in frequency, but its effect on the scattering rate is not as pronounced for the a-b plane as it is for the c-axis. The final observation of relevance is the fact that A R P E S studies have found that the pseudogap is anisotropic, having d-wave symmetry with a minimum along the (7r,7r) direction and a maximum along the (7r,0) direction. The (?r,0) direction lies along the Cu{2)-0{2) bonds in the planes, so that charge carriers moving along this direction are most affected by the pseudogap. As already mentioned, Xiang and Wheatley suggest that c-axis transport takes place along the Cu{2)-0(4)-Cu(\) bonds. Since the holes in the planes sit primarily on the 0(2) sites, travel in the c-direction would usually need to be preceded by travel along the Cu(2)-0(2) direction. Since the pseudogap is largest for carriers traveling in this direction, this would provide a tidy explanation for why the c-axis transport is so strongly affected by the pseudogap. Furthermore, this also provides an explanation for why the pseudogap affects transport in the a-b plane only as a subtle decrease in the scattering rate of the charge carriers. If a primary scattering mechanism for in-plane transport is the one responsible for sending charge carriers out of the planes along the c-direction, then if transport in the c-direction is suppressed by the opening of the pseudogap, a suppression of a-b plane scattering rate is expected at the same time. This may explain the approximate relationship pa ~ P c ( f e „ m p ) seen in the Montgomery data, as an increase in pc should cause a corresponding decrease in pa. The lack of an exact correspondence between the in-plane and anomalous hump contribution to the out-of-plane resistivity could be due to the modifications that must be introduced to Xiang and Wheatley's model when the material is orthorhombic rather than tetragonal. In summary, the model of Rojo and Levin [32] describes the data well if the low temperature semi-conducting-like contribution to the c-axis resistivity is neglected, and if the temperature dependence of the boson-assisted hopping goes as I(T) ~ T2 rather than I(T) ~ T as originally proposed in [32]. The model of Xiang and Wheatley, when considered in conjunction with certain experimental facts, provides an explanation for the way in which the pseudogap affects the a-b plane and c-axis Chapter 5. Discussion 73 transport. Both models provide a link between transport in the planes and transport out of the planes that may explain the approximate p a ~ p ^ u m p ^ relationship found in the Montgomery data. 5.4.3 The In-plane Resistivity Revisited The striking similarities between Figure 5.23 (of which an expanded version is shown in Figure 5.24 below) and Figures 5.7 and 5.12 are worth a second mention. The shape of Figure 5.23 is extremely reminiscent of those for the in-plane resistivity derivatives. However, there appear to be three discontinuities in the slope of Figure 5.23 at high temperatures, rather than just one at 206 K. The discontinuities at 172 K and 239 K, seen in Figure 5.24, are subtle enough that if they occurred in the a- and b-axis data, which have considerably more experimental noise, they would be undetectable. As mentioned, the much cleaner a-axis resistivity for the colinear sample potentially shows the transition at 239 K, although there is still nothing visible at 172 K. Derivative of the Inverse of the Hump-like Contribution 1 . 2 x 1 0 V 1 x 1 0 " u 6 x 1 0 T , > 1 J _ • • .... • * * * * • * m. • • • • • • • • •••• a . • • .* • " * • * ••• «•• • * * . 1 4 0 1 6 0 1 8 0 2 0 0 2 2 0 2 4 0 2 6 0 Temperature (K) Figure 5.24: The derivative of j- for the Montgomery data, close-up view. (From bottom to top) Red circles: nominally no anneal, Green crosses: 6 hours anneal, Purple squares: 19 hours anneal, Blue diamonds: 68 hours anneal, Orange stars: 328 hours anneal. It is the opinion of the author that there are indeed four of these transitions (at 45 K, 172 K, 206 K and 239 K ) , and that the a- and b-axis resistivities, and the inverse quadratic (anomalous hump) contribution to the c-axis resistivity, are all affected by them. The inverse quadratic contribution to the c-axis resistivity appears to be brought about by the same charge carriers that cause the a- and b-axis resistivity, and if the c-axis data shows four of these transitions, then it would only make sense for the a- and b-axis data to do so as well. A cleaner data set (which would be difficult Chapter 5. Discussion 74 to obtain) would be required to know for certain if the in-plane resistivity is affected by all four transitions, but preferable to this would be an experiment that determines precisely what causes (any of) the changes in the slope of as well as independently determining at what temperatures they occur at in clean YBa.2CuzO&.333. As noted in the previous Subsection, there is a possibility that determining what causes these transitions may help determine the pairing mechanism for the charge carriers in the cuprates. The identification of the changes in slope in ^ as being possibly due to subtle structural changes in the unit cell was based upon a comparison to work performed by Zhai et al. in Reference [28]. However, this paper only reported seeing two dielectric transitions (at 60 K and 110 K - not the same temperatures as the transitions seen here), not four, so although small structural changes in the unit cell of Y B C O are still a possible explanation for what causes the transitions in the data presented here, the comparison with the work of Zhai et al. becomes even more tenuous with the identification of another two transitions. However, the conclusion that all of the transitions are brought about by a similar mechanism is strengthened, for it becomes less plausible that four such similar-looking transitions could be due to different causes. Finally, there is no ambiguity about the fact that the changes seen at 45 K and 206 K were found to affect the resistivity along all three of the principal axes. This fact, combined with the doping independence of the temperatures at which the behavioural changes are seen, severely restricts what might be responsible for the effect. For example, a two-dimensional ordering of the charge carriers in the planes (or any other effect that is strictly confined to the planes) fails to explain why the effect is seen along the c-axis as well. Also, the changes cannot be due to the charge carriers alone, or a doping dependence would be expected. Therefore, the behavioural changes seen at 45 K and 206 K (and possibly seen at 172 K and 239 K) are very likely lattice related effects. Chapter 6 75 C o n c l u s i o n s The resistivity was measured in two high quality YBa2CuzOe.333 single crystals. A l l three com-ponents of the resistivity tensor were determined for one sample via a full Montgomery treatment, and pa was determined for the other sample using a colinear arrangement of contacts. The doping of the samples was such that the temperature range studied is presumed to be completely within the pseudogap region of the doping phase diagram. The intrinsic behaviour of the in-plane resistivity was found to follow a quadratic temperature dependence throughout the temperature range studied, subject to two discontinuous changes in behaviour, which are conjectured to be associated with small structural changes in the unit cell of the material. The changes occurred at 45 K and 206 K in both samples studied, and were most easily seen in the derivatives of the data. A Ln(lp) temperature dependence was found to affect all of the in-plane resistivity data. Such a temperature dependence has been reported in the literature for a variety of cuprate superconductors, and analysed as being intrinsic behaviour, but in this study, the logarithmic term was convincingly shown to be sample dependent, and thus not intrinsic. Wi th the extrinsic logarithmic term subtracted off, there was excellent agreement in pa between the two samples, with the discrepancy remaining found to be due to a small amount of c-axis contamination in the data from the sample with the colinear arrangement of contacts. The intrinsic behaviour of the in-plane resistivity showed a very weak upturn at low temperatures, but the ground state appears to be essentially metallic for all of the dopings studied, including those right on the underdoped edge of the superconducting dome. This implies that there is a gap between the antiferromagnetic and superconducting domes on the phase diagram. As doping is decreased to the point where samples are no longer superconductors, they remain strange metals, rather than becoming insulating antiferromagnets immediately. The resistivity along the c-axis was fit to a model consisting of two components added in series, one term being an inverse quadratic expression. There did not appear to be a precise correlation Chapter 6. Conclusions 76 between the fitting parameters for the quadratic in-plane resistivity and the inverse quadratic out-of-plane resistivity, although there were strong similarities in behaviour between the two. Most notably, the derivative of one over the inverse quadratic term in the c-axis resistivity showed the same discontinuous changes in slope at the same temperatures (45 K and 206 K ) that the in-plane data did, and had similarly linear behaviour between its discontinuities. The c-axis data also revealed two other discontinuous changes in the slope of at 172 K and 239 K. It is believed that these transitions also affect the in-plane data, but are too subtle to be seen above the experimental noise. There is some suggestion of the transition at 239 K in the much cleaner data set taken from sample measured by the colinear technique, but the change at 172 K is still not visible in that case. The clear similarities in behaviour between the intrinsic in-plane resistivities and the inverse quadratic term in the out-of-plane resistivity was taken to be evidence that the inverse quadratic term is also intrinsic behaviour. It cannot be determined from the presented data alone whether the second term in the c-axis resistivity, dominant at low temperatures, is intrinsic or not. The low temperature term was found to have an extremely weak doping dependence compared to the inverse quadratic term. It is suggested that the inverse quadratic term must be associated with scattering on or near the CuOi bilayer, due to its close relationship with the scattering seen in the plane. The low temperature term may be associated with scattering when the charge carriers are tunneling between bilayers. Two models were investigated in regard to explaining the approximate pa ~ Pc(humP) behaviour seen in the data. The model of Rojo and Levin [32] modeled the data as well as the general phenomenological models introduced in this thesis did, provided that the low temperature (possibly extrinsic) term in the c-axis data was neglected, and that the temperature dependence for boson-assisted scattering was taken to have a T2 temperature dependence, rather than the T-linear temperature dependence suggested in [32]. The model is consistent with the metallic ground state implied by the intrinsic a-b plane resistivity data, and furthermore suggests that pa ~ j- is an appropriate behaviour to see in underdoped Y B C O . The model of Xiang and Wheatley, taken in conjunction with certain behaviours seen experimen-tally by other researchers, provides an explanation for how the pseudogap can create an interplay between transport in the c-direction and scattering in the a-b plane. This interplay is in keeping with the pa ~ p ( f t „ m p ) behaviour seen in the data. Chapter 6. Conclusions 77 It was conjectured that the discontinuities in ^ are due to subtle structural changes that affect the charge carrier-phonon coupling in the crystal. These behavioural changes cannot be associated with the charge carriers alone, or a doping dependence would be expected in the temperatures at which they occur. They also cannot be an effect associated only with the CuOz planes, for two of them (at least) are seen along all three of the principal axes. This argues in favour of the behavioural changes at 45 K, 172 K, 206 K and 239 K being lattice related effects. 78 B i b l i o g r a p h y [I] Ruixing Liang, D. A . Bonn, W. N. Hardy "Growth of high quality Y B C O single crystals using BaZr03 crucibles". Physica C, 304, 105, 1998. [2] Richard Harris, Ph .D. thesis, University of Brit ish Columbia (unpublished). [3] Darren Peets, Masters thesis, University of Brit ish Columbia (unpublished). [4] L. J . van der Pauw. " A Method of Measuring Specific Resistivity and Hal l Effect of Discs of Arbitrary Shape". Philips Res. Repts, 13, 1, 1958. [5] L. J . van der Pauw. "Determination of Resistivity Tensor and Hall Tensor of Anisotropic Con-ductors". Philips Res. Repts, 16, 187, 1961. [6] B. F. Logan, S. 0 . Rice, and R. F. Wick "Series for Computing Current Flow in a Rectangular Block". J. Appl. Phys., 42:7, 2975, 1971. [7] H. C. Montgomery "Method for Measuring Electrical Resistivity in Anisotropic Materials". J. Appl. Phys., 42:7, 2971, 1971. [8] T. A . Friedmann, M . W. Rabin, J . Giapintzakis, J . P. Rice, and D. M . Ginsberg "Direct measurement of the anisotropy of the resistivity in the a-b plane of twin-free, single-crystal, superconducting YBa2Cu307s" • Phys. Rev. B, 42:10, 6217, 1990. [9] Dr. Michael Sutherland, Dr. Louis Taillefer; Resistivity data for three crystals of YB2C3O6.333; Private Communication [10] A . N. Lavrov, M . Yu . Kameneva, and L. P. Kozeeva "Normal-State Resistivity Anisotropy in Underdoped RBa2Cu306+x Crystals". Phys. Rev. Lett, 81:25, 5636, 1998. [II] A . N. Lavrov, Yoichi Ando, and Kouj i Segawa "Antiferromagnetic correleations and the normal-state transport in heavily underdoped YBa2Cu30e+x" • Physica C, 341—348, 1555, 2000. [12] G . S. Boebinger, Yoichi Ando, A . Passner, T. Kimura, M . Okuya, J . Shimoyama, K. Kishio, K. Tamasaku, N. Ichikawa, and S. Uchida "Insulater-to-Metal Crossover in the Normal State of La2-xSrxCuOi Near Optimum Doping". Phys. Rev. Lett, 77:27, 5417, 1996. Bibliography 79 [13] S. Ono, Yoichi Ando, T. Murayama, F. F. Balakirev, J . B. Betts and G . S. Boebinger "Metal-to-Insulator Crossover in the Low Temperature Normal State of Bi2Sr2-xLaxCu06+s". cond-mat/0005459, 2000. [14] X F. Sun, Kouj i Segawa, and Yoichi Ando "Metal-to-Insulator Crossover Y Ba2Cu3Oy Probed by Low-Temperature Quasiparticle Heat Transport". Phys. Rev. Lett, 93 :10, 107001, 2004. [15] X F. Sun, Seiki Komiya, J . Takeya and Yoichi Ando "Magnetic-Field-Induced Localization of Quasiparticles in Underdoped La2-xSrxCuO/i Single Crystals". Phys. Rev. Lett., 90 :11, 117004, 2003. [16] Yoichi Ando, G . S. Boebinger, A . Passner, N. L. Wang, C. Geibel, F. Steglich, I. E. Trofimov and F. F. Balakirev "Normal-state Hal l effect and the insulating resistivity of high-T c cuprates at low temperatures". Phys. Rev. B, 56 :14, R8530, 1997. [17] Yoichi Ando, G . S. Boebinger and A . Passner "Logarithmic Divergence of both In-Plane and Out-of-Plane Normal-State Resistivities of Superconducting La2-xSrxCuO± in the Zero-Temperature L imi t" . Phys. Rev. Lett, 75 :25, 4662, 1995. [18] S. L. Cooper, K. E. Gray, "Physical Properties of High Temperature Superconductors IV" Edited by D. M . Ginsberg (World Scientific, 1994) [19] T. Ito, K. Takenaka, and S. Uchida "Systematic Deviation from T-Linear Behavior in the In-Plane Resistivity of YBa2Cu307-y: Evidence for Dominant Spin Scattering". Phys. Rev. Lett., 70 :25, 3995, 1993. [20] T. Timusk and B. Statt "The pseudogap in high-temperature superconductors: an experimental survey". Rep. Prog. Phys., 6 2 , 61, 1999. [21] J . A . Martindale, P. C. Hammel "Oxygen N M R on the 90K plateau of YBa2Cu307-y,:. Phil. Mag. B, 7 4 , 573, 1996. [22] R. L. Corey, N. J . Curro, K. O'Hara, T. Imai, C. P. Slichter, K. Yoshimura, M . Katoh andK. Ko-suge " 63Cu(2) nuclear quadrupole and nuclear magnetic resonance studies of YBa^Cu^O^, in the normal and superconducting states". Phys. Rev. 5 , 5 3 , 5907, 1996. [23] R. Stern, R. Mal i , J . Roos and D. Brinkmann "Spin pseudogap and interplane coupling in Y2BaiCu70\e,: A 6 3 C u nuclear spin-spin relaxation study". Phys. Rev. B, 5 1 , 15478, 1995. [24] A . S. Alexandrov, V . V . Kabanov and N. F. Mott "Coherent ab and c Transport Theory of High-T c Cuprates". Phys. Rev. Lett, 77, 4796, 1996. [25] A . S. Alexandrov and N. F. Mott "Bipolarons". Rep. Prog. Phys., 5 7 , 1197, 1994. Bibliography 80 [26] Y . F. Yan, P. Mat l , J . M . Harris and N. P. Ong "Negative magnetoresistance in the c-axis resistivity of Bi2Sr2CaCu208+s and YBa2Cu306+x" • Phys. Rev. B, 52, R751, 1995. [27] S. Sanna, G . Al lodi , G . Concas, A . D. Hiller and R. De Renza "Nanoscopic Coexistence of Magnetism and Superconductivity in Y BCL2CUZOQ+X Detected by Muon Spin Rotat ion". Phys. Rev. Lett., 93, 207001, 2004. [28] Z. Zhai, P. V . Parimi, J . B. Sokoloff and S. Sridhar "Onset of dielectric modes at 110 K and 60 K due to local lattice distortions in nonsuperconducting YBa2Cu3Oe.o crystals". Phys. Rev. B, 63, 092508, 2001. [29] P. Nagel, V . Pasler, C. Meingast, A . Rykov and S. Tajima "Anomaslouly Large Oxygen-Ordering Contribution to the Thermal Expansion of Untwinned YBa2Cu-$0§.$$ Single Crystals: A Glasslike Transition near Room Temperature". Phys. Rev. Lett., 85, 2376, 2000. [30] P. Nyhus, M . A . Karlow, S. L. Cooper, V . W. Veal and A . P. Paulikas "Dynamically assisted interlayer hopping in YBa2Cu306+x" • Phys. Rev. B, 50, 13898, 1994. [31] T. Xiang and J. M . Wheatley "c Axis Superfluid Response of Copper Oxide Superconductors". Phys. Rev. Lett., 77, 4632, 1996. [32] A . G. Rojo and K. Levin "Model for c-axis transport in high-T c cuprates". Phys. Rev. B, 48, 16861, 1993. [33] Dr. Douglas A . Bonn; Private Communication 

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