Ion-size effects on cuprate High Temperature Superconductors
IIon-size effects on cuprate High TemperatureSuperconductors byBenjamin Patrick Pennington Mallett
A thesissubmitted to the Victoria University of Wellingtonin fulfilment of the requirements for the degree ofDoctor of Philosophyin PhysicsThe MacDiarmid InstituteIndustrial Research LimitedandVictoria University of Wellington2013 a r X i v : . [ c ond - m a t . s up r- c on ] O c t i Abstract
The cuprates are a family of strongly electronically-correlated materials which exhibithigh-temperature superconductivity. There has been a vast amount of research into thecuprates since their discovery in 1986, yet despite this research effort, the origins of theirelectronic phases are not completely understood. In this thesis we focus on a little knownparadox to progress our understanding of the physics of these materials.There are two general ways to compress the cuprates, by external pressure or by internalpressure as induced by isovalent-ion substitution. Paradoxically, they have the oppositeeffect on the superconducting transition temperature. This thesis seeks to understand thesalient difference between these two pressures.We study three families of cuprates where the ion size can be systematically altered;Bi (Sr . − x A x )Ln . CuO δ , ACuO and LnBa − x Sr x Cu O − δ where Ln is a Lanthenide orY and A={Mg,Ca,Sr,Ba}. We utilise a variety of techniques to explore different aspectsof our paradox, for example; Raman spectroscopy to measure the antiferromagnetic su-perexchange energy and energy gaps, Density Functional Theory to calculate the densityof states, Muon Spin Relaxation to measure the superfluid density as well as a variety ofmore conventional techniques to synthesize and characterise our samples.Our Raman studies show that an energy scale for spin fluctuations cannot resolve thedifferent effect of the two pressures. Similarly the density of states close to the Fermi-energy, while an important property, does not clearly resolve the paradox. From oursuperfluid density measurements we have shown that the disorder resulting from isovalent-ion substitution is secondary in importance for the superconducting transition temperature.Instead, we find that the polarisability is a key property of the cuprates with regardto superconductivity. This understanding resolves the paradox! It implies that electronpairing in the cuprates results from either (i) a short-range interaction where the polar-isability screens repulsive longer-range interactions and/or (ii) the relatively unexploredidea of the exchange of quantized, coherent polarisation waves in an analogous fashion tophonons in the conventional theory of superconductivity. More generally, we have alsodemonstrated the utility of studying ion-size effects to further our collective understandingof the cuprates. cknowledgements Throughout this thesis ‘we’ is the pronoun of choice which reflects my deep gratitude toall those who helped with experiments and shared ideas. First and foremost I thank mysupervisors, Dr. Jeffery Tallon, Prof. Alan Kaiser and Dr. Grant Williams for their vitalrole in my PhD and career.In addition Dr. Evgeny Talantsev selflessly guided me through thin-film synthesis.Without Dr. Nicola Gaston I could not have hoped to undertake DFT studies and withoutProf. Christian Bernhard the µ SR studies could not have happened. Mr. Thiery Schnyderis acknowledged for starting the Bi2201 work. My thanks to Prof. Thomas Wolf for singlecrystal Ln123 samples and Dr. Edi Gilioli for polycrystalline YSCO samples preparedunder high-pressure and high-temperature.I could not have completed this work without the support of the MacDiarmid Instituteand Industrial Research Limited (‘Callaghan Innovation’ from Feb. 2013).Finally, I am extremely grateful for indispensable support and advice from Dr. ShenChong and Dr. James Storey throughout my PhD.iiiv ontents
List of Figures ixList of Tables xii1 Introduction 1 T c , and the superfluid density . . . . . . . . . . . . . . . . . . . . 402.7 Pairing mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 CONTENTS T c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.7 Muon Spin Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.8 Raman Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.8.1 Two-magnon measurements . . . . . . . . . . . . . . . . . . . . . . 723.8.2 High-pressure measurements . . . . . . . . . . . . . . . . . . . . . . 733.8.3 Variable-temperature measurements . . . . . . . . . . . . . . . . . . 75 T c . . . . . . . . . . . . . . . . . . . . . . . . . 784.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.1.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 794.2 Investigation of Raman phonon modes . . . . . . . . . . . . . . . . . . . . 834.2.1 Low frequency modes . . . . . . . . . . . . . . . . . . . . . . . . . . 854.2.2 O(2) Sr mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Cu O . . . . . . . . . . . . . . . . . . . . 1066.1.1 Comparison of external pressure and ion-size effect on J . . . . . . 1096.2 Is J related to T c ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.3.1 Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.3.2 Foray into the literature for low- T c superconductors . . . . . . . . . 118 ONTENTS vii6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1206.5 Appendix: Interesting extensions - A g and B g spectra . . . . . . . . . . . 120 λ − and T c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1357.2.4 YSr Cu O y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1377.2.5 Zn substitution in YBaSrCu O − δ . . . . . . . . . . . . . . . . . . 1417.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 estimates from phonon shifts . . . . . . . . . . . . . . . . . . . 1568.3.3 2∆ estimates from ERS . . . . . . . . . . . . . . . . . . . . . . . . 1648.4 Summary and discussion of results . . . . . . . . . . . . . . . . . . . . . . 170 T c . . . . . . . . . . . . . . . . . . . . . 1779.2.1 Pseudogap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1779.2.2 Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1789.2.3 Polarisability: Resolution of a paradox . . . . . . . . . . . . . . . . 1789.3 Implications for pairing mechanisms . . . . . . . . . . . . . . . . . . . . . . 1819.3.1 Pairing from repulsive interactions . . . . . . . . . . . . . . . . . . 1819.3.2 Incoherent polarisation and longer-range interactions . . . . . . . . 1849.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1859.5 Further studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1869.5.1 Quantities to measure . . . . . . . . . . . . . . . . . . . . . . . . . 1869.5.2 Specific Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1879.5.3 RIXS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1879.5.4 Ellipsometry and Reflectometry . . . . . . . . . . . . . . . . . . . . 1889.5.5 YSr Cu O − δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189iii CONTENTS E P G and J ion-size study . . . . . . . . . . . . . . . . . . . . . . . 1909.6 Closing remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Bibliography 194 ist of Figures layer . . . . . . . . . 172.7 Phase diagram of the cuprates . . . . . . . . . . . . . . . . . . . . . . . . . 202.8 Generic electronic dispersion for the cuprates . . . . . . . . . . . . . . . . . 212.9 Generic Fermi-contour for the cuprates . . . . . . . . . . . . . . . . . . . . 222.10 Doping and angular dependence of the superconducting gap and pseudogap 232.11 Illustrated pressure dependence of T c . . . . . . . . . . . . . . . . . . . . . 272.12 Correlation of T c with the composite bond valence sum V + . . . . . . . . . 292.13 B g and B g scattering geometry factors . . . . . . . . . . . . . . . . . . . 332.14 Theoretical electronic Raman scattering efficiency of superconducting Y123 352.15 Schematic diagram of the two-magnon scattering process . . . . . . . . . . 362.16 Typical Raman spectra showing two-magnon scattering . . . . . . . . . . . 363.1 Standard sample synthesis procedure. . . . . . . . . . . . . . . . . . . . . . 463.2 Raman spectrum and magnetisation data for a thin-film of YSCO. . . . . . 533.3 T c vs S (295K) or annealing temperature for YBaSrCu − z Zn z O y . . . . . . . 573.4 T c vs S (295K) for polycrystalline Ln123 with Ln=Yb, Eu, Nd. . . . . . . . 593.5 Typical XRD patterns for Bi2201 and YBaSrCu O y . . . . . . . . . . . . . . 633.6 Powder XRD patterns of Nd123 attempts. . . . . . . . . . . . . . . . . . . 643.7 An illustration of two ways to estimate T c . . . . . . . . . . . . . . . . . . . 683.8 A schematic diagram a µ SR experiment. . . . . . . . . . . . . . . . . . . . 693.9 Schematic diagram of Raman active symmetries and the required polarisa-tion configurations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.10 Schematic diagram of the experimental set up for variable-temperature Ra-man spectroscopy measurements. . . . . . . . . . . . . . . . . . . . . . . . 764.1 Schematic diagram of the Bi2201 crystal structure. . . . . . . . . . . . . . 794.2 T c vs. p for Bi2201 from the literature for La, Sm and Eu substitutions. . . 814.3 T c vs. S (295K) for Bi (Sr . − x A x )Ln . CuO δ with A=Ba or Ca. . . . . . 82ix LIST OF FIGURES T max c , disorder and ion-size in Bi2201. . . . . . . . . . . . . . . . . . . . . . 834.5 Typical Raman spectrum of Bi2201. . . . . . . . . . . . . . . . . . . . . . . 844.6 Dependence of the low frequency Raman modes on various ion substitutions. 864.7 Ion-size dependence of apical oxygen phonon mode in Bi2201 . . . . . . . . 884.8 Sketch of van Hove singularity hypothesis for the ion-size effect in Bi2201. . 895.1 The infinite-layer cuprate, ACuO , crystal structure. . . . . . . . . . . . . 925.2 An illustration of the procedure followed to calculate the electronic structurewith VASP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.3 Calculated total energy of CaCuO as a function of the density of k -spacesampling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.4 Calculated free energy vs. unit-cell volume for ACuO with A=Mg, Ca, Sr,Ba. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.5 Calculated electronic dispersion of CaCuO . . . . . . . . . . . . . . . . . . 985.6 Calculated band structure of all ACuO . . . . . . . . . . . . . . . . . . . . 1005.7 Band structure and density of states for all ACuO . . . . . . . . . . . . . . 1016.1 Representative raw Raman spectra from a Ln123 single crystal in all fourscattering geometries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.2 The B g two-magnon scattering peak for all Ln123. . . . . . . . . . . . . . 1076.3 The position of the two-magnon peak as a function of internal pressure. . . 1086.4 Grüneisen scaling analysis of J for internal and external pressure. . . . . . 1096.5 Comparison of the effect of internal and external pressure on J . . . . . . . 1106.6 T max c plotted against J for Ln(Ba,Sr) Cu O y under internal and externalpressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.7 Raw two-magnon scattering data for YBa . Sr . Cu O thin films and forYSr Cu O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.8 Possible doping dependence of J for Ln123 with Ln=Nd, Eu and Yb. . . . 1176.9 T c plotted against the Debye temperature for superconducting elements andbinary alloys. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.10 A g Raman two-magnon scattering spectra of Ln123. . . . . . . . . . . . . 1216.11 B g Raman spectra of Ln123. . . . . . . . . . . . . . . . . . . . . . . . . . 1227.1 A schematic diagram a µ SR experiment. . . . . . . . . . . . . . . . . . . . 1257.2 Typical Zero-Field µ SR data. . . . . . . . . . . . . . . . . . . . . . . . . . 1267.3 Representative zero-field µ SR data for YSCO. . . . . . . . . . . . . . . . . 1307.4 Representative raw data in transverse field mode for YBaSrCu O − δ aboveand below T c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1327.5 The TF- µ SR relaxation rate for YBaSrCu O − δ samples studied. . . . . . . 1337.6 A ‘Uemura plot’ of T c vs. superfluid density. . . . . . . . . . . . . . . . . . 1367.7 Representative TF- µ SR data for YSr Cu O − δ at T = 5 K. . . . . . . . . . 1387.8 TF- µ SR relaxation rate from a two-component fit for all YSr Cu O − δ sam-ples studied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 IST OF FIGURES xi7.9 Normalised superfluid density plotted against relative decrease in T c due toZn doping in YBaSrCu − z Zn z O y . . . . . . . . . . . . . . . . . . . . . . . . 1428.1 Examples of the Bose-Einstein correction factor at various temperatures. . 1488.2 A typical fit to a Raman spectrum using a Greens function approach . . . 1528.3 Temperature-dependent A g + B g Raman spectra of Yb123. . . . . . . . . 1548.4 Calculated temperature dependence of the shift phonon frequency solely dueto lattice contraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1558.5 Reproduction of Fig. 1 from Nicol et al. showing the renormalisation ofphonon energy and lifetime in the presence of a 2∆ . . . . . . . . . . . . . . 1588.6 Temperature dependence of phonon mode energies and HWHM for opti-mally doped Yb123, Eu123 and Nd123. . . . . . . . . . . . . . . . . . . . . 1598.7 Temperature dependence of more phonon mode energies and HWHM foroptimally doped Yb123, Eu123 and Nd123. . . . . . . . . . . . . . . . . . . 1608.8 A g + B g spectra of optimal doped Eu123 at various temperatures. . . . . 1628.9 A g + B g spectra of optimal doped Nd123 at various temperatures. . . . . 1638.10 B g electronic Raman scattering of Yb123 and Nd123. . . . . . . . . . . . . 1658.11 B g electronic Raman scattering of Eu123. . . . . . . . . . . . . . . . . . . 1678.12 A g + B g electronic Raman scattering of Yb123 and Eu123. . . . . . . . . 1688.13 Fitting of Yb123 and Eu123 A g + B g electronic Raman scattering. . . . . 1698.14 B g ERS spectra for Eu123 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1698.15 Possible ion-size dependence of the B g and B g spectral gap . . . . . . . . 1719.1 T c vs J and T c vs the refractivity sum (4 π/ P i n i α i for Ln123 under internaland external pressures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1809.2 Illustration of the higher-order hopping integrals relevant to Hubbard model. 1839.3 The suppression of T c in Zn-doped YBaSrCu O − δ at optimal doping. . . . 192ii LIST OF FIGURES ist of Tables ” annealing process . . . . . . . . . . . . . . . . . . . . . . . . . . 553.2 Annealing conditions, y , S (295K) and T c values for YBaSrCu − z Zn z O y . . . 583.3 Annealing conditions, y , S (295K) and T c values for Yb123 and Eu123. . . . 607.1 Samples studied by µ SR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1288.1 The Raman-active phonon modes in Ln123. . . . . . . . . . . . . . . . . . 1508.2 Summary of measured spectral gaps for Yb123, Eu123 and Nd123 . . . . . 170xiiiiv
LIST OF TABLES hapter 1Introduction
Superconductivity is a fascinating phenomenon. Below the superconducting transitiontemperature, T c , electrons suddenly overcome their mutual repulsion, pair together withopposite momenta and spin and with phase coherence between pairs, adopt a single quan-tum mechanical wave function of macroscopic dimensions! Superconductors transportelectric charge with zero resistance when cooled below T c and as such have a broad rangeof important applications, from compact and powerful magnets to fault-current limiters.Until recently, superconductors have needed to be cooled to liquid Helium temperatures( T = 4 . liquid Helium is veryexpensive. Furthermore, designing the liquid Helium cryogenic system is very complicated.This point was highlighted when a Helium leak in 2008 between two of the Large HadronCollider’s 1232 magnets contributed to the entire facility having to be shut down.However in 1986, Berdnoz and Müller discovered the first [1] of a whole family ofcuprate High Temperature Superconductors (HTS) in Switzerland. These materials have T c ’s as high as 160 K (at elevated pressure) and can be operated without the need for liquidHelium . Making commercial products from HTS has proved very challenging due to theircomplexity, novel physics and the difficulty in making the material into wire. Nevertheless,there are now some commercially viable companies selling a broad range of HTS products- for example the New Zealand company HTS-110 Ltd They also have other advantages over the conventional ‘Low Temperature Superconductors’ CHAPTER 1. INTRODUCTION mance of the superconductor itself. Here, we are interested in the superconductor itself.The cuprates are a family of strongly electronically-correlated materials which exhibithigh-temperature superconductivity. The cuprates are built up from metal-oxide ‘layers’,Sec. 2.1. Common to all cuprates is at least one corner-shared, square-planar CuO layeras this is where superconductivity originates below T c . The electronic properties of thecuprates are strongly modified by doping charge carriers into the CuO layer and theyhave a phase diagram rich in competing and co-existing electronic phases, Fig. 2.7.A vast amount of research into HTS has been done since their discovery in 1986. Oneresult is approximately 115000 papers published , another is an acknowledgment of thecomplexity of these materials and the associated physics, e.g. see [2]. Despite the researcheffort spent on HTS, the origins of their electronic phases are not completely understood,which is perhaps shown most starkly by our inability to theoretically predict a new super-conductor [3] or to find ‘the next higher T c ’ - the record has remained at T c = 164 K since1994 [4].A wide range of experimental probes has been used to study these materials ; elec-tronic transport [5], magnetic structure from neutron studies [6], optical studies [7, 8],tricrystal experiments sensitive to the phase of the superconducting wavefunction [9], NMR[10], high-pressure studies [11, 12], scanning tunneling microscopy [13], polar-Kerr effect[14]. . . Some techniques, such as Angular Resolved Photo-emission Spectroscopy (ARPES)and its derivatives such as Laser ARPES [15] or time-resolved ARPES [16], have beenrefined especially to study the cuprates. Key to progressing our understanding of HTS isthe correct interpretation of these high quality and wide ranging data.There are some important outstanding questions, such as;1. What mechanism(s) cause superconductivity at such high temperatures (Sec. 2.7)?2. Perhaps the more interesting question is; what theoretical framework can be usedto describe the region between the Mott-Hubbard physics of the un-doped cuprate( p = 0, where p is the number of doped holes per Cu in the CuO layer) and theFermi-liquid-like overdoped region ( p > . The number of papers searched by the superconductivity papers database,http://riodb.ibase.aist.go.jp/sprcnd_etl/DB013_eng_top_n.html. The references cited are recent examples from the literature of each probe being used to study thecuprates or are review articles. .2. SYNOPSIS .Four years ago a new class of HTS, the pnictides, which contain Fe, were discovered[19]. The pnictides are only the most recent addition to a long list of strongly-correlated,superconducting materials for which we do not have a satisfactory theoretical understand-ing [20]. Evidently, the problem of HTS is not limited to several, unusual, materials but isa general unexplained phenomenon in condensed matter physics. The superconducting transistion temperature, T c , in the cuprates can be modified by thesubstitution of isovalent ions of differing ion-size. This work seeks to understand how theion-size relates to T c . §2: Theory The purpose of this chapter is to introduce and explain the results and concepts necessaryto understand the research presented in this thesis. We do not attempt to summarize thevast body of literature relating to high-temperature superconductivity in the cuprates andinstead cite some especially good reviews of the topic. §3: Techniques and Samples
Chapter 3 explains the techniques and methodologies used throughout this thesis. Sam-ple synthesis and processing methods are presented in the first sections and then variousmeasurement techniques are presented. The chapter also presents some basic data on thesamples that we have made and studied so as to not encumber the later chapters. §4: Ion-size and Raman spectroscopy studies of Bi2201
Using a simple materials variation approach, we study the effect of ion-size and disorderon T c and on Raman modes in the single-layered cuprate Bi Sr CuO − δ (Bi2201). We findthat ion substitution can increase T c despite increasing disorder and conclude that both See [17] and [18] and related literature for attempts on this problem.
CHAPTER 1. INTRODUCTION the ion-size and disorder significantly affect T c in Bi2201. We then resolve phonon modeassignments in Bi2201 using simple arguments based on material variation experiments.We consider the possibility that ion-substitution affects T c by altering the density ofstates and explore this possibility in the next chapter. §5: Density Functional Theory study of the ion-size effect We perform DFT calculations on undoped ACuO for A={Mg, Ca, Sr, Ba} to investigatethe effect of ion-size on the electronic properties in this model cuprate system. Where thesematerials have been synthesised we find good agreement between our calculated structuralparameters and the experimental ones. There is a peak in the density of states ∼ T c via the densityof states.In addition to the density of states, another purportedly important energy scale is theantiferromagnetic superexchange energy and in Chapter 6 we measure the ion-size effectand external pressure effect on this energy scale. §6: Two-magnon scattering In this chapter we measure the antiferromagnetic superexchange energy, J , by two-magnonRaman scattering in undoped cuprates while systematically altering the internal pressureby changing ion-size. We then compare the internal pressure dependence of J with datain literature for the external pressure dependence of J . From these data we can showthat J is likely unrelated to T max c : J and T max c anti-correlate with internal pressure as theimplicit variable and correlate with external pressure as the implicit variable. Thus it ismost probable some other physical property is dominant in setting the value of T max c .The results from this chapter have various interpretations and consequences whichwe explore in the following chapters. For example, many consider that ion-substitutiondisorder significantly weakens superconductivity. We explore this suggestion by measuringthe superfluid density by Muon Spin Rotation ( µ SR ) in an ion-substituted sample. §7: Muon Spin Rotation and the superfluid density
Here we present µ SR measurements of the superfluid density in the YBaSrCu O − δ cuprate.We find the nice result that YBaSrCu O − δ is a boring, well-behaved material! The su- .2. SYNOPSIS Cu O − δ cuprate. Furthermore the suppression of the superfluid density due to Zndoping is fully consistent with that found for YBa Cu O − δ . The immediate consequenceof these experiments is that disorder does not play a significant role in YBaSrCu O − δ ascompared with YBa Cu O − δ . Consequently, the lower T max c seen in this compound, andthose with lower Sr content, is a genuine ‘ion-size effect’ (or ‘internal pressure effect’). Thisis an important conclusion to have demonstrated and it is nice to know the effects studiedin one’s thesis are genuine!We also report measurements on the fully Sr substituted YSr Cu O − δ material. Theinterpretation of these results is less clear. §8: Gap estimates from Raman Spectroscopy We seek to measure energy gaps in the single-electron levels of our ion-substituted cupratematerials. To the best of our knowledge these measurements have not been done as afunction of internal pressure before. At the level of accuracy available, we did not findany systematic internal-pressure dependence of the energy gap. We did find however thatthe gap values for our ion-substituted samples are consistent with the well-characterisedYBa Cu O − δ cuprate. §9: Discussion In the final chapter we discuss some broader interpretations of our results and concludethat dielectric properties play a key role in superconductivity in the cuprates. Also, wepropose studies to further elucidate ion-size effects on a variety of superconducting andnormal-state properties of the cuprates.
CHAPTER 1. INTRODUCTION hapter 2Theory
Notes
The purpose of this chapter is not necessarily to summarize the vast body of literaturerelating to HTS in the cuprates, but rather to introduce and explain the results andconcepts we feel necessary to understand the research presented in this thesis.An excellent introduction to superconductivity and High Temperature Superconduc-tivity is the Chapter by Hott et al. in reference [2] and the proceeding chapters in thatreference are generally excellent reviews of their respective topics. A useful handbook toHTS, but especially to the materials themselves is the handbook by Poole et al. [21]. Anearly, yet detailed overview of the structure of these materials is the book chapter by Hazen[22].
The cuprate family of materials are layered perovskite-oxides with at least one corner-shared, square-planar CuO layer per unit cell. For a compilation of the members thislarge and diverse family see [21].Fig. 2.1 illustrates the basic components of these materials. The CuO layers arestructurally and electronically quasi-2dimensional. The electronic bands associated withthis layer reside closest to the Fermi energy, E F , hence superconductivity occurs withinthese layers . The CuO layer is sometimes carelessly called the CuO ‘plane.’ There is An obligatory caveat - for any statement about the cuprates there is usually a counter example - is CHAPTER 2. THEORY
Figure 2.1: A cartoon of the generic cuprate structure. Single, bi- and tri- CuO layercompounds are shown. ‘A’ represents a group II ion, Ln represents a member of theLanthanide series. ‘BL’ is the metal-oxide blocking layer and ‘CRL’ is the metal-oxidecharge-reservoir layer.in fact a gradation of planarity, or flatness, amongst cuprate materials with some authorsbelieving this to be crucially correlated with T max c [23].A cuprate can be distinguished by the number of CuO perovskite oxide layers (“n”)per structural repeat unit and if n > or Y , as indicated in Fig. 2.1. Either side of the outer CuO layer is ametal-oxide layer, labelled ‘BL’ (it is sometimes called the ‘blocking layer’) in the figure.The oxygen in this layer, called the ‘apical oxygen’ or O(1), plays an important role inthe charge transfer between the charge-reservoir layer and CuO layer it sits between. Theapical oxygen further lifts the degeneracy of the Cu- d orbitals, Sec. 2.2.1, and may play animportant role in the materials dependence of T c in the cuprates, e.g. [24, 25].‘CRL’ represents the metal-oxide charge-reservoir layer which may have variable oxygencontent and can act as a doping site for the material. The ability to electronically dopethe CuO layer is crucial to the plethora of interesting electronic properties the cupratesexhibit. Finally, not all of these layer-types are necessarily present in a given cuprate.A significant research effort was needed for the synthesis of these complicated, defectprone, materials to be reliable and reproducible [26, 2] and it is not uncommon for phaseseparation or cation disorder to be a major complication in the interpretation of data, seee.g. [27]. ‘Ion-size’ is a concept for the physical space occupied by a nucleus and its “core” electrons- those electronic states which are orthogonal to electronic states associated with neigh-bouring ions. This is a ‘region of exclusion’ as no orbital overlap is permitted in this region proximity induced superconductivity that occurs on the Cu-O chains in LnBa Cu O , see Sec. 2.1.3. .1. CRYSTALLOGRAPHIC STRUCTURE or Y (VIII) r [ × − m] A (IX) r [ × − m]Lu 0.977 Mg 0.90Yb 0.985 Ca 1.18Ho 1.015 Sr 1.31Y 1.019 Ba 1.47Dy 1.027 Ra 1.59Gd 1.053Eu 1.066Sm 1.079Nd 1.109La 1.160Table 2.1: The ‘ion-size’ of various ions of valence and co-ordination (VIII or IX) relevantfor the materials we study. Data are from Shannon et al. [28].(electrons associated with neighbouring ions are excluded from occupying this region). Analternate view of ion-size is what I call ‘cavity size’ which views ion-size as the crystallo-graphic volume occupied by the ion. This fine distinction in conception is probably bestleft to the theorists and does not affect the results of this thesis.The concept of ion-size is important for targeted materials engineering. Normally iso-valent ions of similar size can be readily substituted in a given material without affectingthe material’s stability. A variation in ion size however will induce an effect somewhatanalogous to pressure, as discussed below Sec. 2.3.2.Ion-size is dependent on the valence of the ion, its atomic number and its co-ordination.Empirical values for most ion sizes have been tabulated by Shannon [28]. In Table 2.1 wetabulate some ion sizes relevant to this work. Most of the work in this thesis relates to LnBa − x Sr x Cu O − δ where 0 ≤ δ ≤
1, 0 ≤ x ≤ . The chemical formula initially looks a mess but it is nothing more than thewell-known high-temperature superconductor YBa Cu O with the possibility of severalion substitutions and variable oxygen content. The crystal structure of this ‘system’ or‘family’ is shown in Fig. 2.2. Barring Promethium and Holmium! CHAPTER 2. THEORY
Figure 2.2: The crystal structure of Ln123. Space group 123.1 (or P4/mmm) for small δ (the exact value of δ depends on the Ln ion and Sr concentration) and the lower symmetryorthorhombic space group 47.1 (or P/mmm) at larger δ . Generally, oxygen does not occupythe O(5) site. Some terminology
YBa Cu O − δ is often written in short hand as YBCO or Y123 (reflecting the 1:2:3 ratioof Y:Ba:Cu). Throughout this work we use the latter because it can be transparentlygeneralised to Ln123 = LnBa Cu O − δ , or for example Nd123 = NdBa Cu O − δ . This no-tation also makes it easy to distinguish Y123 from YBa Cu O (Y124) and Y Ba Cu O − δ (Y247), although we have not studied these materials in this work. Both δ = 0 and δ = 1are special cases and so are given their own short hand; Y123O6 and Y123O7.The case where x > x = 2, YSCO is somewhat common. Structural features
Fig. 2.2 shows the crystal structure of Ln123. With reference to the generic cuprate crystalstructure shown in Fig. 2.1 we identify the role that each layer of ions takes;CuO layers: Composed of Cu(2), and O(2) and O(3) sites with Wycoff positions 2 q ,2 r and 2 s respectively for the space group P/mmm. The valence of these ions depends onthe hole carrier concentration per Cu (the doping level p ) in the layer. .1. CRYSTALLOGRAPHIC STRUCTURE layers. The energies associated with quasi-particles inthis layer reside closest to the E F . Doping holes, p , into this layer leads to rich electronicstructure and the magnetic-moments associated with this layer at low p give rise to staticand dynamic magnetic order, see Sec. 2.2.2.Ln ions: The Y site has Wycoff position 1 h for the space group P/mmm. Y ,Ln ={La,. . . ,Lu} or Ca occupies this site between the two CuO layers. Initially itwas a surprise how little effect the Ln species had on the superconducting properties ofthe CuO layer, for example see reference [29], especially given the strong magnetism ofsome Ln ions which in a BCS picture effectively dephases Cooper-pairs. As such, thecoupling between the Ln magnetism and CuO layer has received quite some interest[30, 31, 32, 33]. These effects are observed at low temperatures and do not appear toinfluence the superconducting properties. In fact, the only observable effect of the Ln f -electrons in this thesis (other than via the ion-size), is that of the Nd crystal-field in lowtemperature Raman spectroscopy studies, Sec. 8.3.2.Cationic substitution of Ca for Y or Ln is a method of hole doping the CuO layers. There is a slight decrease in the highest obtainable T c with this method [34] at-tributable to disorder or ion-size effects or both.Other than the valence, the Ln property of key importance is the ion-size. This system-atically alters the crystal structure (precisely how is discussed in detail below) via whatcan be thought of as “internal pressure” or the ion-size effect. We discuss these conceptsin more detail in Sec. 2.3.2.BaO layers and apical oxygen: Composed of Ba and O(1) sites with Wycoff positions2 t and 2 q respectively for the space group P/mmm. It is possible to partially or fullysubstitute Sr for Ba [35, 36] to effect changes in “internal pressure”. O(1) is called theapical oxygen capping as it does the square pyramid co-ordination of the Cu. The apicaloxygen has an important [37, 24, 25], but not fully understood, role in the electronicproperties of the cuprates.CuO − δ layers: Composed of Cu(1) and (1- δ )O(4) and O(5) (a defect) sites with Wycoffpositions 1 a , 1 e and 1 b respectively for the space group P/mmm. This layer acts as the‘charge reservoir’. Doped O onto the O(4) site removes negative charge from the CuO layer, via the apical oxygen, increasing the hole density if successive, linear lengths ofO-Cu-O form [38]. This additional structural feature of one-dimensional CuO ‘chains’gives the name “chain layer”. Conventionally the chains are defined to lie along the b -axis.Chains are formed if the doped O occupies the O(4) site. Occupation of the O(5), ‘chain2 CHAPTER 2. THEORY disorder’, reduces the hole concentration in the CuO layer. Single crystals of this familyare ‘twinned’ if some lengths of CuO chains are perpendicular to others, or put anotherway, the a and b axes are swapped within regions of the crystal. ‘De-twinned’ crystalsindicate the chains run parallel throughout the whole crystal and this can be achieved byapplying uniaxial stress.With sufficiently long chain-lengths (small δ and low O(5) occupation), µ SR stud-ies have shown proximity-induced superconductivity occurs on these Cu-O chains [39].Amongst other effects, this significantly increases the superfluid density (or condensatedensity), λ − , measurable by µ SR .LnBa Cu O , or Ln123O6, has no oxygen in the chain layer. In this state the chainsare sometimes referred to as ‘empty’ and ‘inert’. To remove oxygen from the chain layerwe anneal our samples at 600 ◦ C in a pure Ar gas and then quench, see Sec. 3.3.
We owe much to the excellent study by Guillaume et al. [40]. Many of the results presentedin this section are from their systematic investigation of Ln123O7 and Ln123O6 by neutrondiffractometry.In Fig. 2.3 we plot the dependence of lattice parameters on the Ln/Y ion-size, r Ln , inLn123. Larger r Ln result in larger lattice parameters. This is what underlies what we callthe “internal-pressure” effect, or simply the “ion-size effect”.Fig. 2.4 shows the dependence of the apical oxygen bond length (a,d), inter-cell CuO separation (b,e) and intra-unit cell CuO separation (c,f) on the Ln ion-size for undoped(a,b,c) and fully oxygen-loaded crystals (d,e,f). In Fig. 2.5 we also plot these bond lengthsfor the YBa − x Sr x Cu O − δ materials using additional data from [35, 36]. These dataserve to show that the crystallographic strain introduced by the ion substitution is not‘hydrostatic’ in the sense some intra-unit cell dimensions do not simply scale with thechange unit-cell parameters shown in Fig. 2.3. We now discuss in more detail the effect ofthe ion substitution on various structural parameters.The intra-unit-cell CuO layer separation, d [Cu(2) − Cu(2)] , is shown in panels (b,e) ofFig. 2.4. This distance between the CuO layers shortens with decreasing r Ln . The Joseph-eson coupling between the two CuO layers in the Ln123 unit cell is strongly influenced by d [Cu(2) − Cu(2)] . Larger d [Cu(2) − Cu(2)] results in weaker coupling meaning the CuO becomes“more 2-dimensional”. This energy scale can be estimated from the Josephson-like plasmamode measured by infrared c-axis spectroscopy [41]. .1. CRYSTALLOGRAPHIC STRUCTURE N dN d L aL a V (x10-30m) L n B a C u O Y bT m E rH oY D y S mE uG d P rN d L a a or b (x10-10m) Y b T m E rH o Y D y S mE uG d P r c (x10-10m) r L n ( x 1 0 - 1 0 m )
L n B a C u O Y bT m E rH oY D y S mE uG d P rN d L a a bY bT m E rH oY D y S mE uG d P r r L n ( x 1 0 - 1 0 m )
Figure 2.3: Unit-cell parameters, a , b and c and the unit-cell volume V for Ln123O6(tetragonal) and Ln123O7 (which is orthorhombic because oxygen occupation the O(4)site breaks the a - b symmetry) as a function of Ln (or Y) ion-size, r Ln .4 CHAPTER 2. THEORY
On the other hand, the bond length between the in-layer Cu (Cu(2) and Cu(3)) andapical O (O(1)), d [Cu(2) − O(4)] , is lengthened with decreasing r Ln as shown in panels (a,d) ofFig. 2.4. Similarly, the inter-unit-cell CuO layer separation also increases with decreasing r Ln as shown in panels (c,f). This can be pictured by separating Ln123 into two blocks;a CuO -Ln-CuO block and a BaO-CuO − δ -BaO block. Decreasing r Ln is can then bethought to increasingly separate these two blocks from each other.The Cu(2)-O(2,3) bond angle does not change significantly with Ln substitution [40].This angle is sometimes called the ‘buckling angle’. There are suggestions the bucklingangle in the CuO layer is correlated with T c in the cuprates, see [23] and referencestherein. Out of interest, there is also evidence T c depends on the As-Fe-As bond angle inthe pnictide superconductors [42]. Apparently the optimum angle there is 109 ◦ .As the r Ln decreases, charge transfer between the CuO layer and CuO chains is en-hanced. This has been ascribed to the decrease in c -axis length and concurrent decreasein separation between these layers [43] but we feel that the primary cause is the that thesmall Lanthenide ions promote better long-range ordering in the chains. Occupying of theotherwise vacant O(5) sites is promoted by larger Lanthenide ions. This results in a lowerdoping, p , for given δ for larger Ln ions. Pressure-induced charge transfer is an importanteffect to consider for which a separate section is devoted below, see Sec. 2.3.1. Because of their layered structure the cuprates are often electronically highly anisotropicand are some of the best experimental approximations for 2D magnetic systems that weknow [44].Fig. 2.6 shows how we arrive at the electronic configuration of the Cu ion in the CuO layer of undoped cuprates [45]. The valence of Cu in the CuO layers of undoped cupratesis 2+ with an electronic d configuration giving nine electrons to place in the d -orbital shell.The degeneracy of the Cu d orbital is firstly split by the oxygen octahedral crystal fieldinto upper e g (d x − y and d z ) and lower t g (d xy , d xz , d yz ) bands. A strong Jahn-Tellereffect, caused by apical oxygen distortion of the oxygen octahedra, then splits these two This reference concludes that the buckling of the CuO layer may be a response to lower the freeenergy as the Fermi-energy passes through a peak in the DOS. .2. ELECTRONIC STRUCTURE, PHASE DIAGRAM AND GAPS sepa-ration (b,e) and inter-unit cell CuO separation (c,f) on the Ln ion-size for undoped (a,b,c)and fully oxygen-loaded crystals (d,e,f).6 CHAPTER 2. THEORY ( b ) ( a ) x = 0 . 2 5x = 0 . 5x = 1x = 1 . 2 5
Y B a
S r x C u O d [Cu(2)-O(1)] (x10-10m) L n B a C u O x = 2 ( c ) intra-cell Cu(2)-Cu(2) (x10-10m) inter-cell Cu(2)-Cu(2) (x10-10m) V ( x 1 0 - 3 0 m )
Figure 2.5: The dependence of the apical oxygen bond length (a), intra-cell CuO separa-tion (b) and inter-unit cell CuO separation (c) on the unit cell volume, V , for Ln123O7and, using data from references [35, 36], YBa − x Sr x Cu O − δ . .2. ELECTRONIC STRUCTURE, PHASE DIAGRAM AND GAPS E F Upper6Hubbard6bandLower6Hubbard6band O c t a h e d r a l c r y s t a l f i e l d s p litti ng J a hn - T e ll e r d i s t o r ti on o f a p i ca l oxyg e n O n - s it e c o l ou m b r e pu l s i on e g t d yz d xy d xz d z d x -y ~ Cu63d O62 p E Figure 2.6: Various degeneracy lifting distortions that give rise to the splitting of electroniclevels associated with the undoped CuO layer.bands further and lowers the energy of the d z orbital. As each orbital can accommodatetwo electrons of anti-aligned spins, the d x − y band is left half filled with one electron.Conventional band theory predicts that a half-filled band at the Fermi-level resultsin a metal. However with these strongly correlated materials it is energetically a poorchoice to have two electrons occupy the same d x − y orbital because of strong on-siteColoumb repulsion between the electrons. Mott-Hubbard physics is more appropriate forthis situation and the band is split by quite a large energy, a Mott-Hubbard gap of energy U ≈ d x − y electrons are no longer free to move resulting in an insulatingmaterial.The p σ band of the oxygen in the CuO layer is now nearest to the Fermi-energy. Thelobes of the Cu d x − y electron wavefunction align with the O p σ lobes. The Cu electronscan virtually hop, with significant probability, to the O p σ band. Hence, Cu electrons lowertheir energy by anti-aligning their spins to allow both adjacent Cu electrons to virtuallyhop to the oxygen band.We can discuss this last idea in terms of exchange interactions. The exchange interactionis a purely quantum mechanical effect which comes from the Pauli exclusion principle foridentical particles. For interacting Fermions the total wavefunction must be anti-symmetric(the Pauli exclusion principle), thus, when the spatial (or orbital) wavefunction symmetryis symmetric for a two Fermion system, the spin symmetry must be antisymmetric (i.e.the spin-singlet state; S = 0 where S is the total spin of a two particle system) and vice8 CHAPTER 2. THEORY versa (i.e. the symmetric spin-triplet state; S = 1). The energy difference between thesetwo spin-symmetry states is usually denoted J , the exchange energy. The interaction isextremely short ranged in space as it occurs only where the wavefunctions of the twoFermions overlap ‘appreciably’.Superexchange [46] is where the exchange interaction is mediated through an inter-mediary ion, usually a ‘non-magnetic’ anion (e.g. O) between two cations (e.g. Cu), seee.g. [47]. In the cuprates the magnetic moment associated with the spin of the unpaired d x − y electron interacts with the neighbouring Cu d x − y electron via the O p σ orbital.As mentioned, this interaction is possible due to the spatial overlap of the d x − y and p σ orbitals.Thus, rather than a normal metal the ground state of a cuprate with no doped chargecarriers in the CuO plane is an antiferromagnetically ordered, charge-transfer insulator. J is the energy difference between a parallel and anti-parallel alignment of spin moments onneighbouring Cu(2) ions in the CuO layers and is of order 0 .
12 eV. This can be comparedwith the charge-transfer gap, ∼ ∼ .
025 eV.
The physics of the cuprates really comes alive once the materials are electronically doped.This can be either electron-doping or hole-doping. The hole-doped cuprates are moreextensively studied and are what we consider exclusively in this work.Fig. 2.7 illustrates an indicative phase diagram for hole doped cuprates, sometimescalled the “universal phase diagram” to emphasize the scaled quantitative similarity of itbetween the members of the hole-doped cuprates . The figure has been compiled fromvarious sources [38, 50, 51]. Clearly the phase behaviour is rich (and this is a partlysimplified phase diagram as well)! The complicated phase diagram is partly a result ofthese materials having been studied in such detail, but mostly a result of strong electroniccorrelations allowing an array or electronic orders that jostle with each other for dominanceat a particular T and p .At low doping the cuprates are charge-transfer insulators with the Cu(2,3) magneticmoments antiferromagnetically ordered, as discussed above.As p is increased the long-range, inter-unit cell antiferromagnetic order is rapidly de-stroyed. At low temperatures, below T G , spin fluctuations may freeze out forming a spin- There is however good evidence now that some members of the family have distinct doping-phasediagrams, such as Bi2201 [48, 49]. .2. ELECTRONIC STRUCTURE, PHASE DIAGRAM AND GAPS µ SR) [52].Superconductivity (dark blue shaded region) first appears around p ≈ .
06 with theonset temperature of this macroscopic phase coherence, T c , increasing to its highest tem-perature, T max c by p = 0 . T c then decreases again as indicated making a ‘dome’. Inthe region of T max c almost all cuprates follow T c /T max c ≈ − . p − . [53]. Above T c superconducting fluctuations are observed [54, 55], sometimes up to remarkably hightemperatures (180 K) [56].Around p = / a particular type of charge ordering results as a compromise between(dynamic) 2D antiferromagnetic order and Coloumb repulsion. This fascinating physicsis given the name ‘stripes’ [57]. In some choicely doped cuprates these charge and spindensity waves become static and destroy superconductivity [57].The ‘pseudogap’ phase is a ground state property that competes with superconductivity.In the underdoped region of the phase diagram ( p < .
16 as annotated in Fig. 2.7) thepseudogap ‘opens’ at a temperature higher than T c , T ∗ > T c and has a significant effecton the superconducting properties. It is strongly doping dependent with the approximatebehaviour E P G ≈ J (1 − p/p crit ) [58]. p crit = 0 .
19 is the so-called critical doping where thepseudogap closes, E P G ∼ k B T ∗ →
0. The pseudogap is discussed in more detail below2.2.4.
Using Angular Resolved Photo-emission Spectroscopy (ARPES) it is possible to measurethe electronic dispersion of the CuO layers, see e.g. [61]. As the CuO layers are quasi-2Dso their dispersion can be easily pictured, as shown in Fig. 2.8. The dispersion has 90 ◦ symmetry reflecting the square-planar, 4-fold symmetry of the CuO layer. The “nodes”are at ( π / , π / ) and symmetry related points, the “antinodes” at (0 , π ), see Fig. 2.9 fora simplified schematic diagram of the Fermi-contour. With increasing hole concentrationthe Fermi-level moves lower in energy relative to this dispersion - a so call ‘rigid band’shift. At p ≈ .
27 the Fermi-level for the single-layer cuprates reaches the saddle pointin the electronic dispersion at the anti-nodes resulting in a van Hove singularity (vHs)in the density of states, see e.g. [62]. Increasing hole doping further, the Fermi-surface,counter-intuitively, becomes electron-like [63].0
CHAPTER 2. THEORY T o r k B T hole doing, p [holes/Cu in CuO layer] ( M a r g i n a l ) F e r m i L i qu i d SCPseudogap SC fluctuationsStripesSpin glass A n ti -f e rr o m a gn ti c QCP? "Overdoped""Underdoped""Undoped" "Optimally doped" T c T * T g T N Figure 2.7: An indicative phase diagram for hole doped cuprates compiled from varioussources [38, 50, 51]. At low doping the cuprates are charge-transfer insulators with theCu(2,3) magnetic moments antiferromagnetically ordered. The long-range, inter-unit cellAF order is rapidly destroyed as p is increased. At low temperatures a spin glass phaseis observed. Superconductivity (dark blue shaded region) first appears around p ≈ . T c , increasing to itshighest temperature, T max c by p = 0 . T c then decreases again as indicated making a‘dome’. In the region of T max c almost all cuprates follow T c /T max c ≈ − . p − . [53].Above T c superconducting fluctuations are observed [54, 55], sometimes up to remarkablyhigh temperatures (180 K) [56]. At still higher charge carrier concentration, p , Fermi-liquidbehaviour is observed. Around p = / “stripe” charge ordering results as a compromise be-tween (dynamic) AF order and Coloumb repulsion [57]. The ‘pseudogap’ phase dominatesthe underdoped region. It is a ground state property that competes with superconductiv-ity. The temperature below which it is observed, T ∗ = E P G /k B decreases with increasing p . By p ≈ . T ∗ ≈ T c and as p → . E P G = k B T ∗ →
0. The pseudogap is discussed inmore detail below in section 2.2.4. QCP? marks the location of putative quantum criticalpoint [59, 51, 60] where the pseudogap closes at p = 0 .
19. In a ‘cone’ above QCP? theresistivity, ρ , above T c is linear in temperature up to at least T = 1000 K [2] - a signatureof self-similarity associated with critical phenomenon. Note the terminology annotated atthe top of the diagram to describe the doping level of a cuprate. .2. ELECTRONIC STRUCTURE, PHASE DIAGRAM AND GAPS k x ( π / a ) k y ( π / b ) E − E F ( e V ) Figure 2.8: Electronic dispersion of the CuO layer calculated using tight binding param-eters derived from ARPES data [63]. Superconducting energy gap
In the superconducting state an energy gap of width 2∆ opens in the single-electron statessymmetrically about E F . 2∆ represents the energy required to break a Cooper-pair. Inweak-coupling BCS theory for a d -wave gap (see below) the magnitude of 2∆ at T = 0 Kis proportional to the (mean-field) transition temperature, T mf c , as;2∆ = 4 . k B T mf c (2.1)where k B is the Boltzmann constant. The prefactor 4 .
28 relates only to weak-coupling,for strongly-coupled superconductors the prefactor is larger. The appropriate value for thecuprates is still a topic of debate which is partly obscured by the fact that T mf c can besignificantly larger than T c in the cuprates [56].In the cuprates the superconducting gap is observed to be anisotropic around the Fermi-surface. 2∆ is the amplitude of the gap which has d x − y symmetry;∆ ( k ) = 12 ∆ [cos( k x a ) − cos( k y b )] (2.2)where x = π/a and y = π/b are unit vectors in the 2D Brillouin zone and a , b unit celllengths, parallel to the Cu-O bonds in the CuO layer. Normally the a = b simplification2 CHAPTER 2. THEORY ( π/a ) k x ( π / a ) k y θ Anti-node( θ =0 ◦ )Node( θ =45 ◦ ) Γ MX Figure 2.9: Schematic plot of the Fermi-contour in the 2D Brillouin zone. The red lineindicates how the angle θ is defined.is made. Alternatively we can write the angular dependence of the d x − y gap as ∆ ( θ ) =∆ cos(2 θ ) where tan( θ ) = k x / k y is the angle along the Fermi surface as shown in Fig. 2.9.In this case the nodes are at 45 ◦ and anti-nodes at 0 ◦ .An illustration of Equation 2.2 is shown in Fig. 2.10 (this Figure is reproduced from[54]) by the red curve as a function of doping and θ . In this figure the blue curve is ourphenomenological understanding of the pseudogap, as is discussed in the next section. Themaximum energy of both ∆ ( k ) and E P G ( k ) on the (normal state) Fermi-surface is at theantinodes, whilst the d x − y symmetry of ∆ ( k ) means ∆ ( k ) → near T c in the simple BCStheory and Ginzberg-Landau theory is;∆ ( T ) = ∆ [1 − t ] / (2.3)where the reduced temperature t is defined as t ≡ T /T c . Pseudogap
This pseudogap has been the topic of much debate for many years. Originally it was seen asa downturn in the temperature dependence of the NMR spin susceptibility at temperature .2. ELECTRONIC STRUCTURE, PHASE DIAGRAM AND GAPS et al. [64]. Phenomenologyof the doping and angular dependence of the pseudogap energy, E P G (blue curve) andsuperconducting gap energy, ∆ (red curve). θ = arctan[ k x / k y ] measures an angle aroundthe normal-state Fermi-surface with 45 ◦ equivalent to the nodes and 0 ◦ the antinodes.As is customary, p is holes per Cu in the CuO layer. Dotted curves show the expectedprogression of one gap in the absence of the other. Thus the green line indicates theboundary where the pseudogap vanishes. Note that in the cuprates the ∆ ( k ) has d -wavesymmetry (Equation 2.2) so that ∆ ( k ) → CHAPTER 2. THEORY well above T c [65] and soon after in specific heat [66]. We can now locate signatures ofthe pseudogap in most physical properties; e.g. resistivity [67], magnetic susceptibility[68], ARPES [69], STM [70], Raman spectroscopy [71], µ SR [72], thermopower [73], c -axistransport [8], infrared ellipsometry [74], and the nature of the pseudogap state is still antopic of active research and debate [75, 76, 14, 77, 70].Some view the pseudogap as phase-incoherent Cooper-pairs [78, 79], a “preformedpairs” or SC-precursor-state idea. However, the experimental evidence, that I consideroverwhelming, rules in favour of the pseudogap co-existing and competing with the super-conducting state. As a selection see [38, 80, 8, 69, 74, 81, 82] and references therein. Notethat superconducting fluctuations also exist above T c [45, 74, 56] which probably leads toconfusion between precursor pairs and the pseudogap. The effect of a pseudogap howevercan be observed in ground-state SC properties such as the superfluid density, λ − , in thelimit T → E F . This understanding results from the successof the YRZ model [75].The phenomenological k -dependence of the pseudogap to the best of our knowledgeis illustrated in Fig. 2.10. It is similar to ∆ ( k ) being largest at the anti-nodes, howeverit drops to zero more quickly leaving an ungapped region around the nodes above T c -the so-called ‘Fermi arcs’. Below T c the SC gap, ∆ , opens solely on these Fermi arcs.The Fermi arcs as observed in ARPES (e.g. [84]) are demarked by the green contour inFig. 2.10. This figure also shows the doping dependence of the pseudogap as does Fig. 2.7.The pseudogap is only weakly temperature dependent, if at all. It is also relativelyinsensitive to moderate disorder, for example Ca doping for Y in Y123 [68], Zn doping onthe Cu site [67, 85] or electron irradiated samples studied by Alloul et al. [81]. E P G ishowever enhanced by Ni doping, a magnetic ion, on the Cu site [86].The emerging physical picture of the pseudogap is that of a Fermi-surface reconstruc-tion, possibly caused by short-range, fluctuating AF correlations which strongly scatterquasi-particles [67, 87, 88, 75]. In this case, the anti-ferromagnetic exchange energy, J , at p = 0 is the key energy scale for the pseudogap [58]. The effect of Ni doping then wouldbe to pin these spin fluctuations to promote Fermi-surface reconstruction. .3. PRESSURE EFFECTS AND A DICHOTOMY! Spectral gap
In spectroscopy, there is depletion of spectral-weight below a certain energy in the super-conducting state relating to the opening of the superconducting energy gap, which we callthe spectral gap, 2∆ . In a d -wave superconducting state and in the absence of a pseudo-gap, 2∆ = 2∆ at the anti-nodes but still vanishes, 2∆ →
0, at the nodes [64]. Theserelations are modified by a finite pseudogap energy, E P G , [89, 64] so that;2∆ [ B g ] = 2 q ∆ + E P G (2.4)2∆ [ B g ] ≈ q ∆ − ( / E P G ) (2.5)Here B g and B g represent Raman scattering geometries that measure scattering fromregions of the Brillouin zone around the anti-nodes and nodes, respectively, as shown inFig. 2.13. There is a contribution to 2∆ from both ∆ and E P G but because of their distinctangular dependence around the Fermi-surface their respective contributions vary aroundthe Fermi-surface. Equation 2.5 is approximate only due to the different θ -dependence of∆ and E P G . If they were to have the same θ -dependence, then the term in the squareroot is replaced by ∆ − E P G . (La,Ba) CuO was the first superconducting cuprate to be discovered by Bednorz andMüller and it had T c = 35 K. A very important early observation was that under externalpressure, this T c increased further [90]. This is in fact a very general property of thecuprates: T max c increases under external pressure.Paul Chu and his group took this idea and experimented with simulating this externalpressure with ion substitution - the so called ion-size effect or ‘internal pressure’. The Lain (La,Ba) CuO is a prime target for such an experiment. Because of its f -electrons theion-size of La is much larger than Y (see e.g. Fig. 2.3), which sits above it on the periodictable of elements and so they began exploring Y O , BaO, CuO combinations. What theyaccidentally discovered doing this is the famous YBa Cu O − δ which has T max c = 93 . CuO they wereguided by; it has extra CuO chains, two CuO layers and the Y ion in a different site com-pared with La. Although they had not increased T c simply with internal-pressure, it was6 CHAPTER 2. THEORY that idea that led them to the discovery of the extremely significant Y123 superconductor.A vast amount of research into HTS has been carried out since their discovery in 1986[1]. Correctly interpreting the high quality data from this research is key to progressingthe understanding of HTS. In particular we focus on a little known but central puzzle: T max c decreases with ‘internal pressure’, where ionic substitution is used to decrease thelattice parameters. However, T c increases under external pressure [12, 11, 43, 34]. Whatis (are) the important difference(s) affecting T max c between these two types of ‘pressure’in the cuprates? Answering this question will lead us to a better understanding of theimportant material properties affecting T c and to new insights into the HTS puzzle. A comprehensive discussion of pressure effects in HTS can be found in the review bySchilling [11].Pressure-induced charge transfer (PICT) is the first of many complications caused bythe application of pressure that one must consider [11]. Negative charge is transferredfrom the CuO layers to the charge reservoir layers because of better electronic coupling.This causes an increase in p on the CuO layers with increasing pressure. This is seenexperimentally and in DFT calculations [91].In addition to PICT, the T max c of cuprates is also universally increased under externalpressure [12, 11, 43, 34]. At P = 0 optimal doping the initial increase in the T max c withpressure for the cuprates [11, 92] is dT max c /dP ∼ − . For example, for Y123 itis estimated [93] as 0 .
96 K.GPa − whereas for single- bi- and tri-layer Hg-based cupratesit is 1 . ± .
05 K.GPa − [11]. At higher pressures, the T c of samples that are optimallydoped at P = 0 will decrease as PICT starts to overdope the CuO layer. To counteractthe effect of PICT and measure the pressure dependence of T max c , it is necessary to usesamples that are progressively more underdoped at P = 0 as one goes to higher pressures.We illustrate these ideas in Fig. 2.11. Fig. 2.11 shows the inferred change in T c withpressure for several P = 0 doping states from underdoped to overdoped. Because of PICTthe horizontal axis also measures ∆ p . Although this behaviour has yet to be systematicallymapped out experimentally, it can be inferred, especially for Ln123, from published data[94, 95, 96, 11].An important question is what causes this intrinsic increase in T c , as distinct froma variation in T c because of a PICT doping variation? One possibility is a reduction influctuations due to stronger c -axis coupling. An alternate possibility is that the enhanced .3. PRESSURE EFFECTS AND A DICHOTOMY! T c ( P ) P (GPa) I n c r e a s i n g p a t P = Overdopedat P=0Optimally dopedat P=0 Underdopedat P=0 ambient pressure T cmax ~10 Figure 2.11: An illustration of the pressure dependence of T c for several P = 0 dopingstates from underdoped to overdoped. Doping of the CuO layer increases with pressurebecause of PICT but there is also an intrinsic increase in T max c .polarisability of the material is important. This idea, which results partly from the researchpresented in this thesis, is discussed further in Sec. 9.2.3. There are two cuprate systems we have studied which show this decrease in T c due to‘internal pressure’, Bi Sr . − x (Ba,Ca) x Ln . CuO δ (Bi2201) and LnBa Cu O − δ (Ln123)where Ln represents a member of the Lanthanide series (La, Ce, . . . , Lu). In Ln123 theincrease in T max c as Lu123 goes to La123 has long been known [97, 98]. The Ln ion-sizeincreases from Lu to La [28] and results in shorter in-plane bond lengths [40] and largereffective internal pressures on the CuO layers [99].A recent paper by Gao et al. reports that T c is increased by Ba substitution for Srin Sr − x Ba x CuO δ [100]. They find T max c = 98K for x = 0 . T c with the Cu(2)-O(2,3) bond lengthacross a wide variety of single-layered cuprates and find their new compound fits the trend;materials with higher T c values have longer Cu-O bond lengths. Soon afterwards an articleby Geballe and Marezio questioned these conclusions, citing issues with determining the8 CHAPTER 2. THEORY correct lattice from these multi-phase materials [101]. Nevertheless, the work by Gao etal. appears to be a demonstration of the same ion-size effect that we are discussing .This ‘internal pressure’ effect can been seen more broadly in the cuprates by consideringbond valence sums [102]; Fig. 2.12 shows T max c plotted against the composite bond valencesum (BVS) parameter, V + = 6 − V Cu(2) − V O(2) − V O(3) , from reference [102] (green squares).BVSs are a strategy that crystallographers use to describe valence states of ions basedon the degree of nearest-neighbour co-ordination around that ion. A BVS is calculatedusing the formula; V i = X j exp (cid:18) r − r ij . (cid:19) (2.6)where r is a constant for a given anion-cation pair and tabulated by Brown and Altermatt[103], while r ij is the inter-atomic distance. Summing over all of the co-ordinated anions,for example, should return the formal oxidation state of a given cation. Departures fromthis value indicate mixed valence but can also indicate stress on a bond. V + is actually a composite BVS - combining BVS of O and Cu from the CuO layer.In this case, V Cu(2) , V O(2) and V O(3) are the planar copper and oxygen BVS parametersand the plot reveals a remarkable correlation of T max c across single-, two- and three-layercuprates. We may write V + = (2 − V O(2) ) + (2 − V O(3) ) − ( V Cu(2) − V + is ameasure of doped charge distribution between the Cu and O orbitals [103]. But V + is alsoa proxy for the stress on the Cu-O bond in the CuO plane [104], as noted at the top of thefigure. Broadly, a stretched CuO plane has more positive V + and higher T max c . However,as shown by Fig. 2.12(c) it is more subtle than a simple dependence on volume or CuO bond length as V + also includes contributions from the apical oxygen.Crucially, this plot reveals that all cuprates follow a systematic behaviour. Thereare no anomalous outliers. It is common to regard La − x Sr x CuO as anomalous due toits propensity for disorder. But the left-most data point in Fig. 2.12 shows that it isentirely consistent with the other cuprates. It remains then to determine just what this V + parameter encapsulates so systematically.To this plot we now add new data for the compounds, LnA Cu O y , as Ln is variedand, in the case of Ln = Y, A = Ba − x Sr x where x = 0, 0.5, 1, 1.25 and 2 as blackcrosses. We use the structural refinements of Guillaume et al. [40], Licci et al. [35] andGilioli et al. [36] and calculate V + in the same way as previously [102]. Notably, the globalcorrelation is also preserved across this model system, reflecting the progressive expansion It would appear a PICT explanation of the increase in T c cannot be excluded. .3. PRESSURE EFFECTS AND A DICHOTOMY! T max c , plotted as a function of the bond valence sum parameter V + =6 − V Cu(2) − V O(2) − V O(3) as discussed in the text. Green squares are as previously reportedin [102]. Black crosses are for LnBa Cu O y (Ln = La, Nd, Sm, Gd, Dy and Yb) andYBa − x Sr x Cu O y ( x = 0, 0.5, 1.0, 1.25 and 2) and these same data are shown in moredetail in panel (b). (c) These same data are shown as a function of unit-cell volume ratherthan V + .0 CHAPTER 2. THEORY of the lattice as ion size is increased. These same data are shown in more detail in panel(b) of Fig. 2.12. Using the bulk compressibility these volume changes may be convertedto an effective internal pressure and Fig. 2.12 thus summarises a general feature of thecuprates, namely that internal pressure decreases T max c .In contrast, as noted, it is well known that external pressure increases T max c [11]. Itshould be explicitly stated the analogy between external pressure and internal pressureis imprecise. For example, large ion-size changes do not isotropically shorten lattice pa-rameters as discussed previously, 2.1.4. Nevertheless, such differences allow us to pose ourquestion again; what is (are) the salient difference(s) affecting T max c between these twopressures in the cuprates?For illustrative purposes, this question may be discussed within a weak-coupling BCSframework [105], which recent work has shown can describe the cuprates once the com-peting pseudogap phase, see e.g. [80, 2], and superconducting fluctuations are considered[55, 64]. Specifically, the weak coupling relation for d-wave symmetry is; k B T mf c = 0 . (cid:126) ω B exp " − N ( E F ) V (2.7)where T mf c is the mean-field superconducting transition temperature, ω B is the pairingboson energy scale, N ( E F ) is the density of states (DOS) integrated around the Fermisurface at the Fermi-level and V the pairing potential.On the underdoped side the DOS is progressively depleted by the opening of the pseu-dogap [106, 87] (with energy scale E g ), and, on the overdoped side, is enhanced by theproximity of the van Hove singularity (vHs) [107]. A full treatment of the problem wouldtherefore explore the comparative effects of internal and external pressure on the key vari-ables, ω B , V , N ( E F ), E g . Internal pressure effects on fluctuations have already beeninvestigated in the LnBa Cu O − δ system and found not to be significant compared withchain disorder [5]. Raman scattering is the inelastic scattering of light. Raman spectroscopy measures theintensity of this inelastic scattered light as a function of energy difference relative to theexcitation source energy. This energy difference is usually expressed as a frequency, ω , andis conventionally quoted in units of inverse wavelength cm − = 0 .
124 meV. Lasers are used .4. RAMAN SPECTROSCOPY AND TWO-MAGNON SCATTERING d σ / dΩ d ω .This expression for the scattering efficiency involves the scattering cross-section, σ , per unitsolid angle of detected light, dΩ, per unit frequency, d ω , of a particular excited state.There are three basic components to d σ / dΩ d ω that we will discuss;1. Scattering from phonons. Phonons generally result in sharply defined ‘peaks’ in thespectra.2. Electronic Raman scattering (ERS). ERS from intra-band excitations is generally abroad continuum reflecting the density of states. However, features in the DOS, suchas the pile up of states either side of a superconducting gap, or a van Hove singularitycan be observed.3. Two-magnon Raman scattering. In the cuprates this has a distinctive broad, asym-metric line shape.Generally, the scattering efficiency is expressed in terms of a susceptibility tensor, χ ,that can be derived from, or at least reflects, the phononic, electronic or magnetic structureof the material. In general it is frequency dependent. The Raman tensor, R , is one exampleof such a susceptibility tensor. A feature of Raman spectroscopy that is important tothis work is the ability to measure different elements of the susceptibility tensor, whichcorrespond to different symmetries of the material, by adjusting the polarisation of boththe incident and detected scattered light with respect to the crystal;d σ dΩ d ω ∝ | e i · χ · e s | (2.8)where e i and e s are the polarisation unit vectors of the incident and scattered light respec-tively.2 CHAPTER 2. THEORY
The scattering efficiency for phonons can be expressed as [108];d σ dΩ d ω ∝ | e i · χ · e s | [ n ( ω, T ) + 1] F ( ω ) (2.9)where n ( ω, T ) = [exp( (cid:126) ωk B T ) − − is the Bose-Einstein thermal occupation factor [109]. F ( ω ) is the line shape of the phonon excitation. Generally it is Lorentzian. However, theFano effect occurs when discrete excitations, such as a phonon with well-defined energy,coherently couple to continuous excitations [109]. It is due to interference between the twoexcitations. In our particular case the phonons, particularly the phonon with B g characterat ≈
330 cm − in Ln123, are coupled to the electronic continuum.The effect results in a modified line shape of the discrete excitation. This Fano lineshape is described by; ( q + η ) η (2.10)where q is the ‘Fano parameter’ and η ≡ ( ω p − ω ) / Γ. ω p is the peak centre, which is shiftedfrom the energy shift of maximum intensity, ω max , due to the asymmetry of the Fano lineshape as ω max = ω p + Γ / (cid:126) q , and Γ is the half-width-at-half-maximum (HWHM) in unitsof energy. η is dimensionless.This can be compared with the Lorentzian line shape;11 + η (2.11)If | q | (cid:29) | η | the Fano line shape reduces to a Lorentzian shape. q is inversely proportionalto the coupling between the discrete and continuous excitations. Devereaux and Hackl [110] have written a comprehensive review covering the theory ofelectronic Raman scattering and exemplary results from this technique.The scattering efficiency is related to the imaginary part of the Raman response function χ γγ ( q , ω ) by [108]; d σ dΩ d ω = − ω s ω r (cid:126) π ! [ n ( ω, T ) + 1] χ γγ ( q , ω ) (2.12) .4. RAMAN SPECTROSCOPY AND TWO-MAGNON SCATTERING (π/ a)k x ( π / b ) k y B w e i gh t i ng f a c t o r Γ XM e s e i ab CuO (π/ a)k x ( π / b ) k y B w e i gh t i ng f a c t o r Γ XM ab CuO e s e i Figure 2.13: The B g and B g scattering geometry factors superimposed on a representativeFermi-contour of the cuprates. The inset of each plot sketches the incident and scatteredpolarisation direction with respect to the Cu and O ions in the CuO layer. B g measuresscattering preferentially from the anti-nodes, whilst B g measures nodal scattering.where χ γγ ( q , ω ) is the imaginary part of the susceptibility tensor χ = χ + iχ and is thequantity of interest for probing the redistribution of spectral weight due to the opening of agap, 2∆ . Thus, when comparing spectra we must multiply them by [ n ( ω, T ) + 1] − usingthe appropriate temperature for the spectra. γ is the so called Raman vertex that weightswhat region of the electronic dispersion is measured and depends on the scattering geome-try. Two important examples are B g and B g scattering geometries and are illustrated inFig. 2.13;The B g vertex has the form γ B g ( k ) = γ B g (cos k x − cos k y ) (2.13)and the B g vertex has the form γ B g ( k ) = γ B g sin k x sin k y (2.14)These two functions are shown in Fig. 2.13 as false-colour plots superimposed on arepresentative cuprate Fermi-surface.When including a superconducting gap function, 2∆ ( k ) , into the expression, theimaginary part of the Raman response at T = 0 is the density of states weighted by a4 CHAPTER 2. THEORY scattering geometry factor, γ , and scaled by a factor containing 2∆ ( k ) [89]; χ ( ω ) = Z d k π ∂ ( ω − E ( k )) | ∆( k ) | E ( k ) | γ ( k ) | (2.15)where E ( k ) = q (cid:15) ( k ) + 2∆ ( k ) and (cid:15) ( k ) is the bare normal-state electronic dispersion.Scattering efficiencies in various geometries in the superconducting state were calculatedby Strohm and Cardona [111] and their Fig. 2 is reproduced here in Fig. 2.14. A magnon is a propagating spin-excitation. Alternatively in a classical picture they canbe thought of as spin-waves. A magnon can be thought of as the spin analogue to phonon-related ion-displacements. A damped magnon, as would be the case resulting from spin-flipscattering from charge carriers, is called a paramagnon. As discussed, in the cuprates thesespins are coupled and in the case of nearest-neighbour coupling with energy ∼ J . Suchcoupling is necessary for a spin-excitation to propagate through the spin-lattice. J is ofinterest to us here and it can be measured using Raman spectroscopy.The technique of Raman two-magnon scattering is illustrated in Fig. 2.15. An absorbedincident photon from the laser excites a spin to doubly occupy an adjacent site. Relaxationoccurs by the opposite spin filling the empty site accompanied by the emission of a red-shifted photon. Two magnons are created from this two-spin flip process with a rotation inpolarisation of the Raman scattered photon and an energy loss proportional to J . Becausetwo magnons are created only the sum of their crystallographic momenta k must satisfythe conservation of momentum requirement that k i ≈ k s for Raman spectroscopy, where k i and k s are respectively the incident and scattered light wave vectors. This is the reasonwhy we are able to measure spin excitations that predominantly occur around ( π,
0) in theBrillouin zone.The most common and tractable theoretical treatment of Raman scattering from magnonsis the Loudon-Fleury theory [113]. An extremely useful result for us relates the Ramanshift, commonly denoted ω , of the peak maximum with the antiferromagnetic exchangeconstant, ω max ≈ . J eff ≈ . J . A similar relation is found from several, more modern,theories of two-magnon scattering [114, 115] and so we use the position of the two-magnonscattering peak to estimate J in our work. This is also referred to as the Fleury-Loudon-Elliott theory[112] .4. RAMAN SPECTROSCOPY AND TWO-MAGNON SCATTERING
CHAPTER 2. THEORY bac E S c-axis E S c-axis Magnons (i) (ii) (iii)
Cu(2)O(2)O(3) p x p y ' ~ ' ~ ~ j` ? J d x -y Figure 2.15: Schematic diagram of the simplified B g two-magnon scattering process; (i)the undoped CuO layer has long-range anti-ferromagnetic order. (ii) An absorbed photonexcites a spin to doubly occupy an adjacent site. (iii) Relaxation occurs by the oppositespin filling the empty site accompanied by the emission of a red-shifted photon. Twomagnons are created from this two-spin flip process with a rotation in polarisation of theRaman scattered photon and energy loss proportional to J . Counts w ( c m - 1 ) A + B B A + B B Figure 2.16: Typical Raman spectra showing two-magnon scattering. The material is theundoped SmBa Cu O and the scattering geometries A g + B g , B g , A g + B g and B g are indicated in the legend. . .4. RAMAN SPECTROSCOPY AND TWO-MAGNON SCATTERING − are shown in Fig. 2.16 for the undoped Sm123O6. Also visible areseveral phonon modes below 1500 cm − which are not of immediate interest to our two-magnon studies. A cuprate two-magnon peak has the following basic features;(i) Position
The maximum intensity is ω max ∼ − with the ω max dependingmainly on the doping (presumably by spin-flip scattering) and super-exchange path length:the Cu(2) to O(2,3) to Cu(2) bond length. Using the result ω max ≈ . J , this correspondsto a nearest-neighbour superexchange energy of the order J ∼ − = 1570 K= 0 .
135 eV.(ii)
Laser wavelength dependence
Raman experiments commonly use lasers in the visi-ble light frequencies whose energy, hc / λ i ≈ . ≈ . Asymmetry
Higher frequencies are better represented leading to a generally asym-metrical peak-shape. The peak-shape changes little with incident photon wavelength be-tween 666 nm and 454 nm [114, 112] .(iv)
Width
The two-magnon peak is very broad, extending ∼ − either side ofthe maximum.(v) Polarisation selection
In both A g and B g scattering geometries the two-magnonpeak is observable. The intensity in A g was measured by Lyons et al. to be ∼ B g intensity in La CuO [120]. Our own results on Ln123O6 do not show suchintense two magnon scattering in A g however it is still observable, see Sec. 6.5. Herethe intensity is the measured counts per-second and the A g spectra are constructed fromsubtracting either B g or B g spectrum from combined A g + B g or A g + B g spectrum (seeSec. 3.8.1) respectively that have been measured under identical experimental conditions.Two-magnon scattering is not seen for B g geometry.The Loudon-Fleury theory has several deficiencies in the description of two-magnon It is possible to talk about a cuprate two-magnon peak as the position, width and shape of the peakis similar for all cuprate materials, see [112] and references therein. CHAPTER 2. THEORY scattering in the cuprates. Firstly it cannot describe the fact that comparable two-magnonintensity is seen in A g geometry as well as B g . It cannot describe the large width ofthe peak and its asymmetry is also outside the scope of this theory. Also curious is thatin the incident photon frequency dependence of the two-magnon peak intensity, a singlemaximum is located ∼ et al. [115] and Piazza et al. [121]. Density Functional Theory (DFT) is an exact method for calculating the ground state ofa system ab initio. Practical implementation of DFT however requires several approxima-tions which we will now discuss.We want to calculate how the ground state (lowest energy) electronic and crystal struc-ture is affected by ion-size substitution. To do this we need to solve Schrödinger’s equationfor the specific materials to find the wavefunction, Ψ, that is a solution for the lowestenergy, E min ; H Ψ = E Ψ (2.16)We use the result from the Hohenberg-Kohn-Sham theorem which proves that theground-state energy of a physical system is a unique functional of the particle density, n ( r ). The particle density is given by n ( r ) = | h Ψ | r i | , hence n ( r ) ⇔ Ψ and the originof the name Density Functional Theory. However, solving the equations for the many-bodyinteractions is intractable for all but the simplest systems. The Kohn-Sham approach toovercome this difficulty is to reduce the interacting many-body system to a single-electronproblem with an effective exchange-correlation functional of n ( r ), E ex [ n ] [122, 123, 124].The energy of the system, E , can then be expressed as a functional of the particle density, n ( r ), which can be iteratively refined to find the lowest energy (within a specified range). E [ n ] is written as a sum of various terms; E [ n ] = T [ n ] + E H [ n ] + E ex [ n ] + Z V ( r ) n ( r ) d r (2.17) .5. DENSITY FUNCTIONAL THEORY T [ n ] is the kinetic energy functional, E H [ n ] an electron-electron repulsion energyterm, V ( r ) is an external potential and E ex [ n ] is the exchange-correlation energy whichis included to account for reformulation of the many-bodied Schrödinger’s equation to asingle-electron one. Of these four components of the Hamiltonian the exact form of thekinetic energy and exchange energy is not known.Next a method called the Local Density Approximation (LDA) is used. The LDAmethod is to divide the system into small enough regions such that each region canbe thought of as non-interacting, homogeneous electron gas. For example, for a non-interacting, homogeneous electron gas the kinetic energy functional is known from theThomas-Fermi theory; T [ n ] = Z αn ( r ) / d r (2.18)where α = (cid:126) / m (3 π ) / and m is the mass of a free electron.There are many possible forms of E ex [ n ], each with the purpose of exactly reformulatingthe many-electron Schrödinger’s equation. One of the simplest is an LDA approach suchthat E ex [ n ] is an integral over space of the exchange-correlation energy of a homogeneouselectron gas of density n ( r ), η ex [ n ( r )] [123, 124]; E ex [ n ] = Z n ( r ) η ex [ n ( r )] d r (2.19)where the exchange correlation energy can be expressed as η ex [ n ( r )] = − e π (3 π n ( r )) / .A further correction to this is the L(S)DA+U approach, which includes a Hubbard-likeon-site repulsion between electrons.Equation 2.18 and 2.19 show that even if the local detail of n ( r ) is not faithfullyreproduced in our calculations, the LDA can still predict the correct ground state providedthat the integrals over space are accurate. Put another way, ‘first-order’ errors in the exactform of η ex become ‘second-order’ errors in the energy E ex [ n ] because of the integrationover space.In the Kohn-Sham approach Ψ is written as a weighted sum of (orthogonal) basisfunctions, ψ i ; Ψ = P i c i ψ i . There are a variety of options for the appropriate choice ofbasis functions, for example one could choose basis functions that are believed to representphysical states in the crystal, such as atomic orbital-like basis functions, e.g. d x − y . Suchbasis functions may not be easiest to work with computationally - leading to slower code -but may have a clearer physical interpretation. Computationally the easiest basis functions0 CHAPTER 2. THEORY to work with for a periodic crystal are plane waves, ψ i = exp( − i k i . r ). Plane waves areimplemented in the Vienna ab initio Simulation Package (VASP), which is a computerprogram that implements DFT calculations [125, 126, 127, 128].An issue with using plane-waves as a basis set is that a large number are needed toaccurately describe Ψ close to ion sites because Ψ changes rapidly there. A large basis setleads to long computation times. One solution - and the one we use here - is to replacethe ‘true’ potential around an ion site with a pseudopotential that nevertheless reproducesthe ‘true’ potential outside of a specified cut-off radius from the ion site.In particular we use the GGA-PW91 pseudopotentials in the VASP library [129, 130].These are Projector Augmented Wave (PAW) pseudopotentials (see e.g. [131, 132, 124]),which are a development of the basic pseudopotential idea that more accurately accountsfor the potential inside the cut-off radius, with the Generalised Gradient approximation.PAW+GGA is a good compromise between accuracy and calculation speed. T c , and the superfluid density The superfluid density, n s , is the density of Cooper-pairs in a superconductor. n s is directlyrelated to the London penetration depth, λ , which is the length scale over which magneticflux can penetrate into the superconductor. These quantities are related as [62]: n s = m ∗ µ e λ − ⇒ λ − ∝ n s m ∗ (2.20) e is the charge of an electron, µ is the permeability of free space and m ∗ is the effectivemass of a carrier and cannot be determined from µ SR. Also, the cuprates are extremelyisotropic so that the c -axis penetration depth is much greater than the ab -axis penetrationdepth, λ c (cid:29) λ ab . In polycrystalline samples where one gets and average over the c and ab directions, we measure λ − ≈ λ − ab and this is the quantity we refer to throughout as the‘superfluid density’, as is common in the literature, though strictly it is only proportional to n s . Finally, often we discuss the superfluid density in the low temperature limit, λ − ( T =0) ≡ λ − .The superfluid recruits its electrons from a fraction of the normal-state electrons. In- VASP is able to do other simulations as well, such as Molecular Dynamics (MD), see the VASPhomepage for details http://cms.mpi.univie.ac.at/vasp. .6. DISORDER, T C , AND THE SUPERFLUID DENSITY p = 0 . λ − = µ e D v x N ( E ) E (2.21)where v x is the projection of the Fermi-velocity supercurrent direction and we averagethe density of states, N ( E ), over an energy ± ∆ about E F . Modifications to the densityof states are thus reflected in λ − and therefore both stripe-order [133], the pseudogap[133, 83, 72] have observable effects on λ − and are discussed below.In addition, disorder and pair-breaking effects suppress λ − . In fact, it is well estab-lished that λ − is especially sensitive to disorder [134, 135, 136] in the cuprates due to thephase symmetry of the superconducting order parameter.The pioneering work of Uemura et al. [137] found a correlation between T c and λ − in the underdoped cuprates; ρ s ∝ T c with the same gradient for six different cuprates.This linear relation has come to be known as the Uemura line. Since this 1989 paper,Uemura has shown a correlation between T c and λ − for a wide variety of unconventionalsuperconductors, e.g. [138, 139, 140] and he argues these observations imply real-space-paired bosons rather than the k -space-paired Cooper pairs.However, ρ s ∝ T c is only accurate for well underdoped cuprates and the proportionalityconstant differs between members of the cuprates [83]. Indeed, the relationship between T c and λ − as a function of increasing doping proves to not be linear [83, 141]. Close tooptimal doping λ − increases faster than T c and in the overdoped region both λ − and T c decrease again (the exception is YBa Cu O − δ due to a λ − contribution from oxygenrich, superconducting CuO chains). Plotting T c against λ − results in a horseshoe, orboomerang, shape [142].The authors of [83] show that instead, across three disparate cuprates from under- toover-doping, λ − T c ∝ S ( T c ) ∝ z crit where S ( T c ) is the electronic entropy at T c and z crit is the Zn concentration required to justsuppress T c to zero. This succinct result shows the importance of the pseudogap energyon the superconducting properties by tying λ − to the electronic entropy, a ground stateproperty that is suppressed by the pseudogap [66]. The ‘Uemura line’ of proportionalitybetween T c and λ − is then understood rather as a consequence of the decreasing pseudogapenergy with increasing doping, which results in sublinear behaviour.Transverse-field muon spin relaxation (TF- µ SR ) is an ideal technique to accurately2
CHAPTER 2. THEORY determine the London penetration depth, λ [139], and this technique will be discussed inSec. 3.7. A fascinating physical aspect to superconductivity is that below T c , electrons suddenlyovercome their mutual repulsion, pair together and with phase coherence between pairs,adopt a single quantum mechanical wave function of macroscopic dimensions! The physicsthat causes them to pair is referred to as the pairing mechanism.In the low-temperature superconductors it is (almost [3]) universally believed thatphonons mediate a retarded attractive interaction between metallic electrons as laid outby Bardeen, Cooper and Schrieffer (BCS theory) [105]. For the cuprates there are com-plicated and sophisticated theories covering a wide variety of alternative possible pairingmechanisms as well as extensions to the Eliashberg-BCS theory, see for example the reviewarticle by Wolf et al. [2].In the same year as Bednorz and Müller discovered the first high- T c cuprate [1], threepapers were published showing models in which spin-mediated electron pairing occurredwith a resulting d -wave superconducting state [143, 144, 145]. The authors did not mentionthis could be the mechanism in the just discovered cuprates but it was soon realisedthat this spin-wave mediated pairing was a good candidate for superconductivity in thecuprates, e.g. [146]. These papers find a positive pairing potential (by convention, that isone that leads to electron pairing) close to an anti-ferromagnetic, but not ferromagnetic ,spin density wave (SDW) state. Superconductivity occurs in the even-parity, aniostropicsinglet channel and the gap function has d x − y symmetry.In most unconventional superconductors, superconductivity occurs close to a magneticinstability - that is a point in phase space close to where some form of magnetic orderingoccurs. These magnetic-to-superconducting transitions are tuned in different materials byaltering either the doping state, pressure or the magnetic field [60]. The similarity of themagnetic excitation spectrum for both the cuprates and pnictides has been interpreted asevidence for a pairing mechanism of magnetic origin [140, 147, 6, 148].There appears to be a broad consensus now that the pairing mechanism in the cupratesis probably mediated by a spin-fluctuations [149, 150, 140, 147, 6, 115, 148]. However, a They were instead seeking to understand superconductivity in the heavy-Fermion superconductors andorganic superconductors, the “Bechgaard salts”. Apart from some obscure possibilities, see footnote in [143]. .7. PAIRING MECHANISMS et al. calculate the T c ofYBa Cu O − δ to be ≈
170 K using their experimentally determined paramagnon disper-sion - easily accounting for the experimental value T c = 90 K, especially if one recognisesthat T mf c may be significantly above T c .In this situation, a key energy scale for pairing is, to leading order, the antiferromagneticexchange interaction, J [150, 6].Pairing mechanisms are discussed further at the end of the thesis, Sec. 9.3.4 CHAPTER 2. THEORY hapter 3Techniques and Samples
The purposes of this chapter are to (i) explain the techniques and methodologies usedthroughout this thesis and (ii) present some basic data on the samples we have studied sothat it does not encumber the later chapters.Fig. 3.1 illustrates the standard synthesis and characterisation procedure used for mostof our samples. Each of the steps is discussed in the following sections.
Solid-state synthesis is a common technique for making polycrystalline material. It involvesmixing and then reacting solid precursor materials to form your material. For the cuprates,the precursor materials are almost always metal- oxides, nitrates, or carbonates as theseare stable at room temperature. To make these materials react, one must (i) increase thetemperature (i.e. increase their thermal energy) (ii) decrease the separation between theprecursor agents (which reduces the energy barrier for reaction) and (iii) perhaps alter thegas under which they are reacted. As implied by the name of the technique, the temperatureshould be kept (just) below the melting temperature of the constituent materials so thatthe precursor does not melt into the sample holder and upset the stoichiometry of theconstituent metals. Thus to decrease the separation between the precursors we grind themtogether, in a mill or with mortar and pestle, to a fine powder and then press the powder456
CHAPTER 3. TECHNIQUES AND SAMPLES
Figure 3.1: An illustration of the standard procedure followed when making new samples- especially polycrystalline samples. .2. SAMPLE SYNTHESIS content in the reacting-gas increases the reaction rate - whichis not always desirable and sometime leads to unwanted solid-phases. From experience,the melting temperature of a cuprate is approximately 1200 ◦ C in O and lower in loweroxygen partial-pressure.By way of example, polycrystalline YBaSrCu O − δ pellets are synthesized by startingwith high-purity Y O , BaCO , SrCO , CuO powder .1. The powders are weighed out so that there is the desired 1:1:1:3 molar ratio ofY:Ba:Sr:Cu. The oxygen content is controlled later by the annealing process, Sec. 3.3.2. The powders are then thoroughly mixed with mortar and pestle. Isopropanol can beadded at this stage to aid mixing of the powders. We have also used acetone insteadof isopropanol, particularly for the synthesis of NdBa Cu O − δ , with no observablechange in the results. The powder is then dried and pressed at 20-30 MPa in astainless-steel die to produce pellets of 10 mm diameter.3. Next the pellets are ‘decomposed’ at 750 ◦ C under dry air, a process which removesmuch of the C and excess O. Normally the pellets are placed on a Yittria-stabilisedZirconia container when put into the furnace and preferably seated on an inert sub-strate such as MgO.4. Next the pellets are broken and ground to a fine powder with mortar and pestle.As before, isopropanol can be added at this stage. The powder is pressed into pel-lets again and this time reacted at 950 ◦ C in dry air. For YBaSrCu O − δ there isno particular temperature ramping sequence. Other materials can require carefultemperature sequences, which are generally found in the literature. Even so, there isalmost always an period of trial-and-error before the appropriate synthesis conditionsare found.5. The previous step is repeated 2-4 more times. Each repetition increases the homo-geneity of the final product. Sometimes these reactions are at incrementally highertemperatures and for incrementally longer times.Generally the synthesis process - reaction temperatures, times, O partial pressure -for our materials, or similar materials, can be found in the literature. As mentioned above Which are commercially available from e.g. Sigma Aldrich. CHAPTER 3. TECHNIQUES AND SAMPLES however, there is almost always a period of trial-and-error before the appropriate synthesisconditions for us to make the material are found. These depend on such things as theparticle sizes in the precursor powders.For Bi2201 we were in such a position.We prepared samples of Bi − l Pb l Sr . − x A x Ln . CuO δ for A = Ba or Ca and Ln =La, Nd, Sm, Eu, Gd. For l = 0 we make Ba with x = 0 . . x = 0 . x = 0 and Ln = La we also make samples with l = 0 .
0, 0 .
2, 0 .
35 and 0 .
45. Typicallywe react at 840 ◦ C in a dry air atmosphere. We vary the doping state, p , via the oxygencontent, δ , by annealing at various temperatures and oxygen partial pressures. By doing sowe can achieve a maximum T c of 30 K (the onset of diamagnetism is 2 K higher than this).The highest reported T c for Bi Sr . La . CuO δ is T c = 35 . et al. [152]. Despite replicating their synthesis conditions, several times, we still only achieve a T max c ≈
30 K. Doubtless, replicating their higher T c result is a matter of fine tuning themany variables that are involved in the synthesis process of such a complicated materialas Bi2201.Finally, a note on impurity substitution; Zn is considered to substitute onto the Cu(2)site preferentially to the Cu(1) site [153] and so a 0.1:1 Zn-Cu ratio implies a concentra-tion of / × . layer for bi-layer Ln(Ba,Sr) Cu O y . Moderate Znsubstitution does not affect the average doping state either [39, 67]A typical example of a sample synthesis and characterisation process is given in Fig. 3.1 Nd123 synthesis
Here, we wish to synthesize polycrystalline NdBa Cu O − δ , which is reported to haveideally T max c = 96 K, and then to carry out a set of thermopower and T c measurementsin order to determine the annealing conditions for optimal doping. See Sec. 3.3.1 for anexample relating to a similar material.We believe that the large ion-size of Nd (with co-ordination number VIII), 1 . × − m, and concurrent smaller Cu(2)-O(1) bond length mean that the energy differencefor the Nd ion occupying the ‘Y’ site or the ‘Ba’ site becomes small (see Fig. 2.2 and Fig. 2.4for an illustration). Nd occupation on the Ba-site is a known issue when synthesising thismaterial [5] and this impurity will donate electrons to the CuO layer causing the materialto be more underdoped than desired. The issue is even more severe in La123 and is theprimary reason why we choose to make Nd123 rather than La123 polycrystalline samples.Sintering Nd123 in low partial O pressure discourages Nd occupation of the Ba site. .2. SAMPLE SYNTHESIS et al. [5] and MacManus-Driscoll et al. [154].We did not have much success following these, see Fig. 3.4. Instead after many attemptswe describe below what we found to be the optimal synthesis conditions.We use high purity Nd O , BaCO and CuO precursor powders in a 1:2:3 ratio ofNd:Ba:Cu. Before being weighed out the Nd O powder was fired at 280 ◦ C for 3 hours inorder to remove any adsorbed H O. The precursor powders were weighed out, thoroughlymixed in acetone and pressed into pellets. These pellets are placed in an Au containerbasket and placed into the furnace. At 350 ◦ C the furnace is evacuated and backfilled with0.1%O in N gas. The temperature is then increased to 850 ◦ C , without any overshoot,and sintered for five hours with a moderate flow rate of 0.1%O in N gas.The pellets are then reground in acetone, pressed into pellets again and sintered at865 ◦ C in 0.1%O in N overnight. The pellets are again reground, but not in acetone thistime, pressed into pellets again and sintered at 872 ◦ C in 0.1%O in N for 72 hours.After suitable annealing, see 3.3.1, this produced a Nd123 sample with T c = 94 K, T onset c = 96 . S (295K) = 15 . ± . µ V.K − , indicating significant underdoping andhence the reduced T c .Some experiments which did not produce good quality Nd123 for us include:1. Providing excess Ba by starting with stoichiometry Nd:Ba:Cu = 1:2.25:3.2. Providing deficient Nd by starting with the stoichiometry Nd:Ba:Cu = 0.9:2:3.3. Using metal-nitrate precursors rather than metal-carbonate precursors.4. Adding a small amount of Zn in the ratio Nd:Ba:Cu:Zn = 1:2:3:0.006. Zn acts asa fluxing agent and is reported to reduce the melting point of YBa Cu O − δ byapproximately 2 K/(% of Cu) [155].5. ‘Pure’ N sintering atmosphere, with a concurrent lower temperature of 830 ◦ C (thematerial melts at higher temperatures in the lower O partial pressures). Although I did not attempt to grow any single crystals in this work, single crystals grownby Thomas Wolf at the Karlsruhe Institute for Technology (KIT) of Ln123 were usedextensively. These high-quality LnBa Cu O − δ (Ln=Lu, Yb, Dy, Gd, Eu, Sm, Nd, La) CHAPTER 3. TECHNIQUES AND SAMPLES single crystals were flux grown in Y-stabilized zirconia crucibles under reduced oxygenatmosphere, where necessary, to avoid substitution of Ln ions on the Ba site.
Bulk, polycrystalline, fully Sr substituted YSr Cu O − δ cannot be grown by the usual solid-state synthesis techniques described above, Sec. 3.2.1, presumably because the YSr Cu O − δ phase is only meta-stable. Instead, polycrystalline samples of YSr Cu O − δ must be pre-pared in O at 3 GPa of pressure and at 1050 ◦ C with KClO as an oxidant. In order toachieve such high pressures (we are capable of 2 MPa only in our labs) a multi-anvil ap-paratus is needed - similar in concept to the Diamond Anvil Cell. As such, these sampleswere prepared by Dr. Edi Gilioli at IMEM, CNR in Parma, Italy. The ratio of precursormaterials (Y O , SrO , CuO) to KClO is 1:0.45. A description of the synthesis method isfound in references [35, 36].YSr Cu O − δ is clearly difficult to synthesize. Unfortunately a range of attractive mea-surements (e.g. Resonant Inelastic X-ray Scattering, Raman spectroscopy, Ellipsometry)require single crystal samples, or at least aligned thin films. Making such samples willrequire a big materials-chemistry effort which was beyond the scope of this work. Thatbeing said we were able to prepare some YSr Cu O − δ in a film on a SrAlLaO substrate atatmospheric O pressure. This particular substrate was chosen for its close lattice matchto YSr Cu O − δ . The method used for making these films is described below, Sec. 3.2.4. We make thin films using a metal-oxide deposition (MOD) process. Here salts, e.g. Y-TFA , Cu-OHP , Sr-acetate, are weighed out then dissolved in dry methanol. Each newsalt is added to the solution and completely dissolved before the next salt is added. Weadd ∼
3% of volume propionic acid once to assist dissolution. A sonicator is used to fullymix the solution and dissolve the salts after each new salt is added. For our samples, themaximum length of time needed in the sonicator fully to dissolve the salts is 20 minutes.The most common substrate used is the trademarked RABiTS substrate which is anindustry standard for ‘2G’ (‘second generation’ - YBa Cu O ) wire. It is a Ni-W alloy basewith three 75 nm buffer layers, Y O , YSZ and CeO to ensure good lattice matching. The Yittrium tri-fluoro acetate. The chemical formula is an industrial secret! Although we note that the Cu-acetate salt may be usedinstead of Cu-OHP. .2. SAMPLE SYNTHESIS ∼
800 nm, although this depends on thespin speed and solution concentration. The samples are transported from the glove box tothe decomposition furnace inside a sealed container and the decomposition process startedimmediately. Although the un-decomposed films are briefly exposed to air whilst beingloaded into the decomposition furnace, we were able to good quality films with this processand so take this method to be satisfactory.The film is decomposed in flowing O and H O vapour (a source of H) up to 450 ◦ Cto remove all organic material. The optimal specific temperature-time profile will changedepending on the solution. After the decomposition process the only non-oxide remainingshould be Ba-OF (strictly speaking (Y,Ba)-OF). Ba-OF converts to BaF at T = 575 ± ◦ C, a process which takes place during the initial phase of the next process.The sintering sequence (or reaction sequence) - where the desired Y123 phase is formed- is done under a partial H O pressure. During the initial stage of the sintering Ba-OF con-verts to BaF which in turn reacts above 720 ◦ C with the water present; BaF +H O → BaO+ 2HF (which is vented to the atmosphere). This allows the remaining BaO to participatein the Y123 phase formation. The temperature ramp rate, O and H O pressure and sin-tering temperature (standard is 788 ◦ C) are critical for growing epitaxial films (i.e. wherethe Y123 c-axis aligned vertically). The temperature ramp rate is controlled by the speedat which the sample is moved into the hot zone of the furnace. It is important to ensure theoptimal BaF :BaO ratio at each temperature in the sequence, which is in turn importantto ensure the YBCO crystal growth is seeded from the correct (001) face off the (aligned)substrate to result in c -axis aligned epitaxial films.There is an additional complication in our material. To make YBa − x Sr x Cu O − δ we useSr-acetate and Ba-TFA salts in the solution. After the decomposition process Ba-OF andSrO are present. This is a problem as during the reaction sequence the SrO is immediatelyavailable to react with Y O and CuO to form a Sr-rich impurity phase whereas the BaF must first decompose into BaO before it can react with Y O , SrO and CuO. Neverthelesswe are able to synthesize good quality, well aligned YBa . Sr . Cu O − δ thin films forRaman spectroscopy measurements. The quality of films with higher values of x werenot as good. Presumably a concerted effort to optimise the sintering conditions wouldsufficiently improve the thin film quality and epitaxy.A summary of this synthesis process can be found in a review by Obradors et al. [156]. And for critical current densities this process can be fine-tuned so that there is good a and b axisalignment also. CHAPTER 3. TECHNIQUES AND SAMPLES
YSr Cu O − δ thin film synthesis on SrLaAlO substrates This section describes how we grow thin films of YSr Cu O − δ on single crystal substrates,in both (100) and (001) orientations, of SrLaAlO .SrLaAlO has a tetragonal unit cell with c = 12 . × − m a = b = 3 . × − m,very similar to the YSr Cu O − δ a and b unit cell parameters reported as a = b = 3 . × − m [36]. This close lattice matching promotes the growth of the desired YSr Cu O − δ phase. It is hoped this will also promote epitaxial growth of the YSr Cu O − δ .Y-TFA, Cu-OHP and Sr-acetate salts are mixed, spin coated on the substrate anddecomposed as per our standard MOD process (see above, 3.2.4). The standard reactionconditions used to grow the Y123 phase did not produce YSr Cu O − δ , or indeed anycrystalline phases. This is perhaps not too surprising as it was found that high temperaturesand O pressures are required to grow polycrystalline YSr Cu O − δ [36]. We thereforecarried out a series of reactions at 950 ◦ C, 1000 ◦ C and 1050 ◦ C in pure O with each followedby XRD analysis, to determine suitable reaction conditions.It was found that processing a decomposed YSr Cu O − δ film on SrLaAlO (001)at 1000 ◦ C in pure O for 20 minutes, followed by a rapid cool did indeed produce theYSr Cu O − δ phase! The presence of the YSr Cu O − δ phase was confirmed by Ramanspectroscopy as shown in Fig. 3.2(Left). Only phonon modes associated with the (unorien-tated) YBa Cu O − δ structure are present - the fact that the 550 cm − mode is relativelyintense indicates the YSr Cu O − δ phase is multi-domain and unaligned. Interestinglythey are at similar energies to the equivalent YBa Cu O − δ modes. Note however that itis not a YBa Cu O − δ sub-phase we observe as might be the case with a partially Sr-Basubstituted film. This film has no Ba, only Sr.Furthermore, a weak diamagnetic transition was observed (as might be expected from athin film) for this film at the low temperature of T c = 14 K upon zero-field-cooling (ZFC) asshown in Fig. 3.2(Right). Upon field-cooling (FC) the diamagnetism is destroyed. Theseobservations are consistent with a weak superconducting transition at T c = 14 K. Webelieve this to not originate from the SrLaAlO substrate for two reasons (i) at T = 4 Kwe measure no magnetism from the bare substrate down to the resolution of the SQUIDmagnetometer; 10 − emu (ii) there is no reported superconductivity in this material. with I4/mmm space group, similar to the P4/mmm space group of Y123O . .3. SAMPLE PREPARATION AND ANNEALING a r e a 1 a r e a 2 Raman intensity (counts/min) w ( c m - 1 ) - 6 - 6 - 6 Long Moment (emu)
T e m p e r a t u r e ( K )
F i e l d - c o o l e dZ e r o - f i e l d - c o o l e d
Figure 3.2: (Left) Raman spectra from two different, representative, regions of aYSr Cu O − δ film on a SrLaAlO (001) substrate after being conditioned at 1000 ◦ C inpure O for 20 minutes. Each is made from two spectra joined at ≈
550 cm − - as canbe seen by the slight intensity discrepancy there. Taken on the LabRam spectrometerwith the 514.5 nm laser and a x100 objective. Note that the phonon spectrum here isentirely consistent with the (unorientated) YBa Cu O − δ phase. (Right) Zero-field-cooledand field-cooled DC magnetisation of the same film as measured in the SQUID magne-tometer. H = 20 Oe (2 mT). These data show a likely transition to a superconductingstate at T c = 14 K when zero-field cooled. After our samples have been synthesized they are taken through a process of annealing.The materials we work with have variable oxygen content. In the case of YBa Cu O − δ and related materials O can be incorporated at the O(1) or, undesirably, at the O(5) site,see Fig. 2.2. In Bi2201, O can be incorporated in the BiO − δ layer or interstitially, seeFig. 4.1. To control the level of oxygen we anneal our samples at carefully controlledtemperatures (or a sequence of temperatures) in a controlled O partial pressure thenquenched. The standard procedure and two special cases are described below. Standard annealing process
For this task, we always use vertically orientated furnaces, the reason for which will beclear soon. The sample is held in a YSZ container which is suspended from a hooked wireinside the furnace tube, held in place by strong rare-earth magnets on the outside of thefurnace tube. When annealing in an Ar atmosphere an Au basket is used instead of a YSZcontainer as Au does not contain any O that may be taken up by the sample.Initially the sample is not raised to the centre of the heating coils (the ‘hot zone’) - this4
CHAPTER 3. TECHNIQUES AND SAMPLES must wait until the correct gas has been backfilled into the furnace tube.The furnace tube is sealed and pumped on to a moderate vacuum. The vacuum valveis then closed and the desired gas (e.g. 2% O in N , or pure O ) introduced into thetube. The tube is evacuated again and then again, the desired gas introduced. This is theprocess of ‘backfilling’, or as it is called when the PPMS and SQUID fill the sample spacewith He, ‘purging’. This step is repeated three times. After the last backfill, the desiredgas is pushed through the tube with a slow flow rate (the purpose being to mitigate airleaking into the sealed furnace tube).Only now is the sample raised to the centre of the heating coils, using the strong rare-earth magnets outside the tube to pull the basket up. Typically the sample is held at theannealing temperature overnight (for polycrystalline samples) or over three days (for singlecrystals) to give sufficient time for oxygen to diffuse from/into the sample.When removing the sample one tries to avoid exposing it to an O partial pressuredifferent to that in which it was annealed, at elevated temperatures. The solution forpolycrystalline samples is dramatic! Once the furnace tube is unsealed, the polycrystallinesamples are dropped rapidly from the centre of the heating coils and into liquid N , a processcalled ‘quenching’. This rapidly cools the samples, preventing any further significant oxygendiffusion. Finally the sample is dried once removed from the liquid N to prevent H Ocondensation on the surface.For single crystals there is a significant chance a quench into liquid N will destroy thecrystal, and so in this case the sample is instead rapidly lowered as far as possible fromthe hot-zone and the gas flow-rate increased. The furnace is not unsealed until the crystalcools to room-temperature. “O ” anneal With this process we aim to incorporate as much O as possible. The process is outlined inTable 3.1 described below;The samples are placed on Au foil and placed inside the pressure furnace - called the‘Bomb’. The Bomb is firmly sealed, flushed with O several times and heated to 570 ◦ C overtwo hours. The temperature is then decreased to 400 ◦ C over two hours. The O pressureis then increased to 10 or 12 bar. A conventional pressurized gas cylinder is the sourceof the above atmospheric pressure O . We then cool the sample to 300 ◦ C over 72 hoursbefore turning off heating coil. The sample not removed or O pressure released until thefurnace is close to room-temperature. .3. SAMPLE PREPARATION AND ANNEALING
55T ( ◦ C ) 0 →
570 570 →
400 400 →
300 300 → ∼
18 hrs)O pressure* 1 bar (100kPa) 1 bar 10-12 bar 10-12 barTable 3.1: An annealing process designed to incorporate as much O as possible into thesample. For the purposes of this work it is referred to as the “O ” anneal. *Aboveatmospheric pressure. Ar anneal
With this process we aim to remove as much O as possible from the material. An alternativename for the process is the ‘O ’ anneal.The sample, held in a Au basket, is introduced into the furnace at ≈ ◦ C . Nextthe furnace is sealed and pumped to a moderate vacuum. The vacuum valve is closed andAr admitted into the chamber (Ar is less likely to have trace O impurities than N gas).The furnace is again evacuated and then refilled with Ar. This is known as ‘backfilling’and the process is repeated 3-4 times to ensure the furnace is filled with Ar only. Afterbackfilling we maintain a slow flow rate of Ar through the furnace to mitigate air leakinginto the sealed furnace (the inside of the furnace will be at a slightly higher pressure thanoutside). The temperature is then increased to 600 ◦ C and held there for 24 hours (forpolycrystalline samples where diffusion is faster) to 72 hours (for single crystals). Duringthe longer anneals the furnace is backfilled every day.At the end of the anneal, the Ar flow rate is significantly increased, the sample rapidlylowered from the centre of the furnace (the ‘hot-zone’) and then left to cool to roomtemperature before the furnace is opened.Samples annealed this way must be stored in a desiccated environment under weakvacuum.
Samples of YBaSrCu − z Zn z O y for z = 0 .
00, 0.02 0.04 and 0.06 were synthesized by themethod described above in Sec. 3.2.1 and followed the sintering temperatures reportedby Licci et al. [35]. Zn is known to substitute preferentially for Cu(2,3) [157] - the Cusite in the CuO layers. Hence we have a set of 1%, 2% and 3% Zn doped samples. InTable 3.2 and Fig. 3.3 we show the results of a systematic annealing study of these samples.From these data we know the annealing conditions to obtain specific p and corresponding T c values. For example, these data show the annealing conditions required for ‘optimal’6 CHAPTER 3. TECHNIQUES AND SAMPLES doping, T c = T max c , for each Zn concentration. T c , y and p values for z = 0 . z = 0 .
04 sample to determine T max c . T c plotted against the number of holes per Cu in the CuO layer, i.e. the ‘doping’ p ,has a maximum shaped like a parabola at p ≈ .
16 We call this doping, where T c = T max c ,‘optimal doping’ and it gives us a common point at which to compare Ln123. Sometimes p is estimated from this parabolic relation, T c /T max c = 1 − . p − . (which assumes,amongst other things, optimal doping at p = 0 .
16 holes/Cu) [53]. However, determiningoptimal doping solely by measuring T c is problematic as the small decrease in T c from T max c could imply either under- or over-doping. The thermo-electric power (TEP) at roomtemperature, S (295K), is however a monotonic function of doping [158], being positive forunder-doping and going negative for over-doping - see Sec. 3.5. For optimal doping in thecuprates S (295K) is typically +2 µ V.K − .To determine the annealing conditions to optimally dope each of Nd123, Eu123, Yb123we measure T c vs. S (295K) on polycrystalline samples made by standard solid-state reac-tion methods 3.2.1 using known sintering conditions [5]. Our results are shown in Fig. 3.4and tabulated in Table 3.3. For cuprates without Cu(1)-O(4) chains, close to optimal dop-ing S (295K) is linearly proportional to p [158] but for Y123 we expected an additional, andpositive, contribution to S (295K) from the metallic chains as they are filled. The chaincontribution cannot be easily distinguished in polycrystalline samples.As can be seen from Fig. 3.4 Nd123 has high S (295K) values indicating our Nd123 isunderdoped, which in turn implies Nd occupation of the Ba site - a known issue. SeeSec. 3.2.1 for discussion of our attempts to synthesize Nd123 with low Nd occupationof the Ba site. The method to identify optimal doping used for Yb123 and Eu123 willobviously not work in this case. We know from the literature that Nd123 is difficult tooverdope [97] in general. Our approach to ‘optimally dope’ Nd123, and La123, then is touse the “O ” annealing process 3.1 to incorporate as much oxygen as possible. We notehowever that for our best Nd123 polycrystalline sample T c = 94 K and T onset c = 96 . X-ray diffraction (XRD) is a common technique for examining crystal structure. A colli-mated beam of (parallel) X-rays are directed onto the sample at a specific angle of incidence.This incident radiation is scattered from electrons in the material which are most densely .4. X-RAY DIFFRACTION T c (K) S ( T = 2 9 5 K ) ( m V . K - 1 ) Z n = 0 % Z n = 2 % Z n = 1 % Z n = 3 % ( a ) - 2 0 2 4 6 85 56 06 57 07 58 08 5
T c (K) S ( T = 2 9 5 K ) ( m V . K - 1 ) Tc / Tc(O7 anneal)
A n n e a l t e m p e r a t u r e ( d e g C i n O ) Z n = 0 % Z n = 1 % Z n = 3 % ( b )
Figure 3.3: (a) T c plotted against the room temperature thermoelectric power, S (295K), forYBaSrCu − z Zn z O y with z = 0 .
00, 0 .
02, 0 .
04, 0 .
06. Systematically increasing the annealingtemperature, see Table 3.2, is used to reduce the oxygen content with a correspondingincrease in S (295K). T c is estimated from magnetisation measurements, see Sec. 3.6.1.The inset shows the region around optimal doping in closer detail. (b) A plot of T c normalised to the T c value measured after an ‘O ’ anneal, see Table 3.1, plotted againstthe subsequent annealing temperature in O for three Zn concentrations indicated in thelegend. Note that for higher Zn concentration the highest T c value is obtained after lowerannealing temperatures and therefore presumably higher oxygen content and larger p .8 CHAPTER 3. TECHNIQUES AND SAMPLES
Zn Annealing temperature ( ◦ C ) T c (K) S (295K) ( µ V.K − ) y p z=0.0 “O ” 81 -1.0 7.00 0.184350 81.1 -1.2 6.99 0.183400 82.0 -1.0 6.98 0.180425 82.5 -0.8 6.97 0.179450 83.5 -0.4 6.96 0.174475 84.0 0.1 6.95 0.171500 84.4 0.7 6.93 0.168525 84.9 1.5 6.91 0.160550 80.9 2.8 6.88 0.135575 71.4 5.0 6.95 0.115600(Under air.) 32 19.0 6.68 0.075600(Under 2% O in N ) 32 21.8 6.67 0.073z=0.02 “O ” 76.9 -1.4 6.99 0.167350 75.5 -0.94 7.00 0.176375 77.2 -0.55 6.98 0.160425 76.7 0.65 6.95 0.151450 76.2 1.0 6.94 0.147500 76.8 2.2 6.90 0.152550 74.8 3.4 6.87 0.141600 67.1 7.2 6.83 0.120z=0.04 “O ” 56.4 3.8 6.98* -350 57.7 3.3 7.00* -400 53.8 4.2 6.97 -425 50.0 5.2 6.96 -450 47.4 6.2 6.95 -475 44.0 7.1 6.93 -500 40.4 8.2 6.92 -525 36.1 9.6 6.90 -550 31.5 11.3 6.88 -575 30.7 12.5 6.88 -z=0.06 “O ” 61.1 -0.3 7.00 0.173350 62.0 -0.5 6.99 0.160400 61.3 1.5 6.94 0.148425 60.8 1.9 6.92 0.144475 58.0 3.2 6.90 0.132525 50.0 5.2 6.86 0.112575 26.0 10 6.80 0.076Table 3.2: Annealing temperatures under pure O for polycrystalline YBaSrCu − z Zn z O y (unless otherwise noted). T c is estimated from magnetisation, see Sec. 3.6.1, S (295K) is anaverage of measurements on different pellets. y is estimated from the mass change after eachanneal and assumes the “O7” anneal results in y = 7. p is estimated from the parabolicrelation T c /T max c = 1 − . p − . (which assumes, amongst other things, optimaldoping at p = 0 . ± . T c , ± . µ V.K − in S (295K), ± .
03 in y and ± .
005 holes/Cu in p . *Because of the high thermopower valuesand lower than expected T c values, we believe this is not fully oxygenated, or if y = 7, thensignificant O(5) site occupation causing underdoping. .4. X-RAY DIFFRACTION - 5 0 5 1 0 1 5 2 0 2 5 3 0 3 56 57 07 58 08 59 09 5 Y b 1 2 3 E u 1 2 3 N d 1 2 3
Tc [K] S ( 2 9 5 K ) [ m V . K - 1 ] Figure 3.4: T c plotted against the room temperature thermoelectric power, S (295K), forYb123, Eu123 and Nd123. We reduce the oxygen content by systematically increasing theannealing temperature, see Table 3.3, which causes an increase in S (295K). T c is estimatedfrom magnetisation measurements, see Fig. 3.7.0 CHAPTER 3. TECHNIQUES AND SAMPLES
Ln Annealing T ( ◦ C ) T c (K) S (295K) ( µ V.K − ) y p Yb “O ” 90.3 2.0 7.00 0.168350 90.1 2.5 6.99 0.169400 90.7 2.8 6.99 0.160450 90.1 3.8 6.97 0.151475 89.8 4.5 6.96 0.149500 89.6 5.0 6.95 0.148525 88.7 6.5 6.93 0.144550 87.9 6.8 6.91 0.141575 86.1 7.9 6.88 0.135Eu 120 kbar O ” 95.1 -0.3 7.00 0.160350 95.1 -0.8 6.99 0.160400 94.7 0.6 6.98 0.153450 94.6 2.0 6.97 0.152475 94.2 3.6 6.96 0.149500 93.2 4.5 6.94 0.144525 90.1 7.5 6.91 0.135550 87.2 8.7 6.89 0.128575 81.7 14.0 6.86 0.119Table 3.3: The annealing temperatures under pure O for LnBa Cu O y as described insection 3.3.1. T c is estimated from magnetisation (see section 3.6.1), S (295K) measuredon a home-built rig (each value represents an average of typically 3 measurements, eachwith different heating conditions or on different pellets). y is estimated from the masschange after each anneal and assumes the “O7” anneal (3.3.1) results in full oxygen loading( y = 7). p is estimated from the parabolic relation T c /T max c = 1 − . p − . (whichassumes, amongst other things, optimal doping at p = 0 . ± .
5K in T c , ± . µ V.K − in S (295K), ± .
03 in y and ± .
005 holes/Cuin p . Together, these data show the annealing conditions required for ‘optimal’ doping, T c = T max c . T c values and annealing conditions for various Ln123 can also be found here[159, 160]. .4. X-RAY DIFFRACTION k − k = G , where G = h A + k B + l C is areciprocal lattice vector, ( hkl ) are the Miller indices and k and k are the scattered andincident X-ray wave vectors. This condition combined with the momentum conservation(since the photon has such a tiny momentum) | k | = | k | , gives the Laue formulation forconstructive scattering (reflection) from a crystal lattice, 2 k · G + G = 0. This is equivalentto the more commonly quoted Bragg condition; 2 d sin( θ ) = nλ , where d is the distancebetween crystal planes, λ the wavelength of the radiation, θ the angle between the crystalplane and incident radiation and n is some integer.The resulting diffraction pattern reflects the periodicity, and internal structure, in thecrystal. These are some excellent resources describing this common technique, from thebasic [161], to the more advanced [162].In this work, XRD is used to characterise the samples we make, as illustrated in Fig. 3.1.Polycrystalline samples are ground to a fine powder which is then placed level in an Alu-minium holder. There are several reasons why it is better to measure a powder than asintered pellet; (i) signal-to-noise is better (ii) from experience the composition and phasesat the surface of a pellet may not be representative of the bulk whereas the powder fromground pellet ensures an average composition is being measured and (iii) if prepared prop-erly the crystallites are known to be randomly-orientated. On the other extreme, XRD onsingle-crystals requires much more geometrical care when setting up the measurement. Be-cause they are very well-orientated, constructive scattering peaks can be easily missed withthe point detectors we use. All XRD measurements are carried out at room temperatureand pressure. A typical measurement takes 1 hour.We use Cu-K α X-rays and a Bruker “D8” diffractometer in parallel beam (Debye-Scherrer) geometry.
Fig. 3.5 shows typical XRD data used to check the phase purity of samples we synthesized.Bi2201 has especially weak X-ray diffraction reflecting poor long-range crystallinity.We primarily draw from a database of previous refinements on similar materials (e.g. For example, there is no database entry for Bi Ba . Sr . La . CuO δ as we are the first to synthesize CHAPTER 3. TECHNIQUES AND SAMPLES
Y123 or pure Bi2201). From these patterns we can identify diffraction peaks with particularreflections from the unit cell, e.g. (001) reflection or (115) reflection, refine the latticeparameters and identify additional diffraction peaks. With this method however we do notget any quantitative information from the intensity of the diffraction peaks - for that wemust perform a Rietveld refinement.If there are impurity phases visible in the XRD pattern, they can often be identified bysearching the database for appropriate possibilities. An example of a diffraction patternwith impurity phases is shown in Fig. 3.6 for (mostly) NdBa Cu O y . Crystallographically,the material is very similar to YBaSrCu O − δ shown in Fig. 3.5, displaying the samereflections, but with larger lattice parameters. However it is difficult to prevent the Ndion occupying the Ba site. When this occurs there is an excess of Ba, Cu and O to formthe ‘123’ phase and so impurity phases such as CuO and Ba CuO form which we thenobserve in the diffraction pattern. Thermopower, or thermo-electric power (TEP), is the voltage developed across a materialdue to a temperature gradient. More formally, the thermopower is the voltage differencebetween point A and point B, ∆ V = V A − V B , divided by the temperature differencebetween point A and B, ∆ T = T A − T B , in the limit where ∆ T /T is small; S AB = − ∆ V ∆ T Conventionally S is quoted in the convenient units µ V.K − as ∆ V is usually ∼ µ V. Room-temperature thermopower, S (295K), measurements are made on a custom-built rig.The sample is placed between two polished Cu plates. By applying pressure we makea good thermal and electrical contact between the sample and the Cu plates. Cu leadsconnect the Cu plates to a nano-Volt amplifier, which is powered by batteries to minimiseelectrical noise. A heater coil is wound around the bottom Cu plate and a thermocouplemeasures the temperature difference between the plates. it, thus we compare our diffraction patterns with Bi Sr . La . CuO δ . .5. THERMOPOWER Ba . Sr . La . CuO δ and (b) YBaSrCu O y (note the y -axis square-root scale now de-spite both having 10 s count-times per step). Light blue lines mark the expected crystalreflection 2 θ and approximate relative intensity for similar materials. The signal to noiseratio of these XRD scans is not good enough for a Rietveld analysis. Appropriate data forthis analysis requires more than a day of counting time (and careful sample preparation)if using a lab based source (whilst on a synchrotron only a few minutes would be needed).Instead, these data are used to measure the lattice parameters of the material and identifyany crystals of different composition and symmetry (impurity phases).4 CHAPTER 3. TECHNIQUES AND SAMPLES
Figure 3.6: Powder XRD patterns from two attempted NdBa Cu O y samples. Lightblue lines mark the expected crystal reflection 2 θ and approximate relative intensity forYBa Cu O y , while the other colour lines represent possible impurity phases. From these weidentify Cu and Ba-rich phases such as Ba CuO δ and BaCuO δ as impurities - probablyforming as a result of excess of Ba and Cu after Nd occupation of the Ba site in the ‘Y123’phase. Interestingly, the Nd123 sample sintered under the fully reducing atmosphere ofpure N (red curve) had lower T c and larger S (295K) values than the samples with moreBa- and Cu-rich impurity phases. The reflection in this sample at 2 θ ≈ . ◦ is from anunidentified phase which must have a large repeat unit of ∼ . .6. MAGNETISATION V are made without a current to the heatercoil giving ∆ T ≈ T is measured). The heater is then left onfor 2 minutes followed by another set of 10 measurements of ∆ V and ∆ T . The current tothe heating coil is adjusted so that ∆ T ≈ . S (295K) a linear fit is made to all the data points and a 1 . µ V.K − correction for the room-temperature Cu voltage leads subtracted from the best fittinggradient. DC magnetisation is the linear response of a material to a static applied magnetic field, B = µ ( H + M DC ) = µ H (1 + χ ), where B is the total magnetic flux, H is the appliedmagnetic field and M DC is the DC magnetisation, χ the magnetic susceptibility and µ the permeability of free-space. The superconducting state is characterised by the Meissnereffect whereby magnetic flux is expelled from the superconductor. Thus, for the case ofweak fields , χ = −
1. This provides a nice, clear signal that we use to characterise T c .The magnetic moment of a sample is often measured in units called ‘emu’ - electro-magnetic units. The following relation is used to convert from these units to a volumesusceptibility; χ V = µ m [ emu ] ρ × mass × H where ρ is the sample density in g.cm − , mass is measured g, H in Tesla, and the magneticmoment, m , in emu.We measure the magnetic susceptibility of our samples using either Vibrating Sam-ple Magnetometry (VSM) or for more sensitive measurements we use a SuperconductingQUantum Interference Device (SQUID). These are now both standard techniques for mag-netisation measurements and we use commercial Quantum Design SQUID and VSM sys-tems and LakeShore Design VSM systems. The Quantum Design SQUID Magnetometeris capable of measuring magnetic moments down to 10 − emu at temperatures between2 K and 400 K in magnetic fields up to 7 T.We use magnetisation as the primary technique to measure T c . We note here the poten- Such that the applied field is less than the lower critical field of the material, H ≤ B c , assuming thatthere are no demagnetisation effects (which are related to the geometry of the sample). The Quantum Design VSM is part of a Physical Properties Measurement System (PPMS) that we usealso for resistivity measurements. CHAPTER 3. TECHNIQUES AND SAMPLES tial difficulties with extracting reliable T c values from resistivity data. In polycrystallineand multi-domain single crystal HTS the resistivity will fall to zero only when there is anunbroken path of superconducting material between the voltage probes. In inhomoge-neous samples, for example, oxygen may not be evenly distributed throughout the sampleleading to a path through material of varying doping and higher T c than most of the rest ofthe sample; an overestimation of the bulk T c would result. More often however, the resis-tivity has a tail below T c due to weak links between grains. This tail grows with increasingcurrent density or applied field. In this case, T c defined as where the electrical resistancefalls to zero would underestimate the bulk T c .Measuring T c by magnetisation can avoid these issues as the magnitude of the diamag-netic signal from the Meissner effect shows when the bulk material has become supercon-ducting. In this respect, weak link behaviour between grain boundaries in polycrystallinesamples can be clearly seen if present in the magnetisation data. In polycrystalline mate-rials, smaller single crystallites are weakly linked to their neighbouring crystallites withinthe ceramic sample [163]. Pressing the powders into pellets before sintering at high tem-peratures is an attempt to strengthen the links between grains and to aiding the growth insize of the crystallites through diffusion-assisted grain-growth during sintering [164]. Weaklink behaviour can be seen in DC susceptibility vs. temperature data by two distinct gra-dients, a higher temperature intra-grain superconductivity onset and lower temperatureinter-grain currents shielding the external magnetic field .There are some magnetic effects arising from the superconducting state which mayconfuse the determination of T c . The thermal energy at temperatures around T c for HTSis often large enough to significantly suppress SC below its mean-field value, see e.g. [54].Consequently SC fluctuations are significant above T c and manifest, in this case , inremnant diamagnetism above T c [166]. These effects however are generally subtle and werenot an issue in this work.Cuprates, being type II superconductors, display irreversible magnetisation at temper-atures below T c . Different magnetic behaviour is seen if one’s sample is cooled below T c in no magnetic field, referred to as zero-field cooled (ZFC), relative to if it is cooled below Of sufficient quality and size so that the probing current does not exceed the critical current density- though this is not an issue for HTS with the low currents used to measure resistivity. In AC magnetisation measurements two distinct peaks in χ show these same two crossovers [165]. Superconducting fluctuations also lead to a decrease in resistivity above T c which must be carefully dis-tinguished from a pseudogap related downturn in resistivity by suppressing the SC fluctuations in a strongmagnetic field. Alternatively superconducting fluctuations at a given temperature may be suppressed byunitary scatterer substitution for Cu (e.g. Zn) which suppresses T c as well, but not the pseudogap [67, 85] .7. MUON SPIN RELAXATION T c in a non-zero magnetic field, referred to as field-cooled (FC). This difference is due tothe pinning of vortices as they attempt to enter the superconductor below T c . In ZFC,there is no flux within the material when the external magnetic field is initially turned on.Supercurrents form on the surface of the sample to repel magnetic flux from entering thebulk superconductor. We determine our T c estimates from ZFC magnetisation data. T c There are several ways of estimating T c from magnetisation measurements;1. From the onset of the diamagnetic downturn in magnetisation vs temperature dataunder a small DC field, typically 20 Oe (0 . T onset c .2. From taking the intercept of the extrapolated normal state background magnetisationwith the extrapolated steepest slope of the diamagnetic signal in M ( T ) data. This isthe method used to measure T c throughout this work. The first method listed leadsto higher T c values but can be misleading; detecting a downturn in magnetisationdepends on the sensitivity/noise in the measurements.3. From the peak in the imaginary part of the susceptibility from an AC magnetisationmeasurement.An illustration of the first two methods listed above is shown in Fig. 3.7.Resistivity measurements are another common way of estimating T c . T c can be definedas where ρ ( T ) →
0, or by extrapolating to zero the steepest slope in ρ ( T ), or where ρ ( T ) is10% of its normal-state value before the onset of the SC transition. On samples with sharpsuperconducting transitions all these methods of measuring T c give very similar results,but for poorer quality transitions they differ. In Muon Spin Relaxation ( µ SR) one directs a polarised beam of positively charged muonsonto the sample. The implanted positive muons are repelled by the positively charged ionsin the crystal which causes them to preferentially occupy an interstitial site in the lattice This is sometimes called Muon Spin Resonance instead. CHAPTER 3. TECHNIQUES AND SAMPLES
40 60 80 100-0.05-0.04-0.03-0.02-0.010.00 M ( e m u ) T (K) T c T conset Figure 3.7: An illustration of two ways to estimate T c . The solid red curve is ZFC mag-netisation data and the green lines are illustrative extrapolations.(although we cannot control which interstitial site is preferentially occupied, it is believedto be adjacent to the apical oxygen). The muons are strongly coupled to the local magneticfield, B loc , causing the muon’s magnetic moment to precess at a frequency proportional tothe local field. This is Larmor precession and the angular frequency is given by ω µ = γ µ B loc where γ µ = 0 . × rad.s − T − is the gyromagnetic ratio of a muon [167]. Note thesimilarities between the fundamental concepts of the µ SR and the NMR technique.When the muon decays, with a half-life of 2 . µ s, it emits a positron with momentumpreferentially along the muon’s spin direction. It is these positrons we detect. Conveniently,they have a kinetic energy large enough to not interact with the sample or holder so byobserving the direction in which the positron was emitted we can infer, with sufficientstatistics, what was the B loc that the muon experienced. The direction of emission isdetermined by comparing the positron count rate from detectors on opposing sides of thesample, e.g. above and below, or in front and behind, see Fig. 3.8. Then, a depolarisationfunction P α is used to represent the probability of emission in a certain direction α . Thisis the parameter we discuss throughout. The asymmetry function is defined as P α ( t ) = F ( t ) − B ( t ) F ( t ) + B ( t ) (3.1) .7. MUON SPIN RELAXATION Sample and holder DownUp B a ck F o r w a r d F o r w a r d B ext Muon spin polarisation (with spin rotator on) Transverse-Field muon spinprecession Positrondectectors p + Coincidence timerstart
Figure 3.8: A schematic diagram of the experimental set-up at the GPS beamline (usedto make our µ SR measurements). Normally, the spin of the muon is anti-parallel to itsmomentum, however, GPS is equipped with spin-rotating magnets which flip the muonspin near perpendicular to its momentum - as annotated. A magnetic field up to 6000G (0 . t between the Up andDown detectors. t = 0 is when a muon is detected passing through the coincidence timerin the Front positron detector. At the bottom a sample (black circle) is sketched taped(light grey square) to a silver holder.where F ( t ) and B ( t ) are the raw counts (positron detection events) measured by theforward and backward detectors (or more generally, the opposing detectors) after a time t from the muon injection. At t = 0, P α ( t ) takes the value A which is typically ≈ . α = Z as the direction of the external magnetic field B ext . In zerofield however, P Z is usually the direction of the muon spin polarisation. See Fig. 3.8 for apicture of these geometries.By way of example, if all muons experience exactly the same, static field, B loc at anangle θ to the muon spin S µ , then the Larmor equation gives; P α ( t ) = cos θ + sin θ cos( ω µ t )This is of course not generally the case. Generally there is a spatial distribution of staticfields B loc that the muons experience and/or fluctuations which cause a time variation0 CHAPTER 3. TECHNIQUES AND SAMPLES in B loc . In either case, our muons are depolarised. Depolarisation because of a fielddistribution D ( B loc ) is described as dephasing, whilst from fluctuations it is described asrelaxation. Our depolarisation function in these cases looks like; P α ( t ) = Z S µ,α ( t, B loc ) D ( B loc ) d B loc where S µ,α ( t, B loc ) is a projection of S µ along α and is a function of time (Larmor precession)and B loc (via the Larmor frequency). One of the simplest field distributions, D ( B loc ) , andthe one we use to analyse our data, is the Gaussian distribution.Now, let us consider the case where B ext = 0T and h B loc i = 0 and we measure thedifference between the ‘up’ and ‘down’ dectectors relative to the muon spin polarisation, P Z ( t ). If D ( B loc ) is isotropic and Gaussian then there is an analytical formula for P Z ( t )called the Kubo-Toyabe function [168, 169]: P Z ( t ) = 13 + 23 (1 − σ t ) exp (cid:18) − σ t (cid:19) (3.2)where σ /γ µ is the variance of the magnetic field distribution which depolarises the muons.Generally we expect this form of P Z ( t ) from magnetic fields caused by the weak nuclearmoments in a sample.I have found a useful way to think of the asymmetry P α ( t ) is as a Fourier transformof the local field distribution. A broad distribution leads to a rapid decay of asymmetryin the time domain (in which we measure), on the other hand a single local field value,as you might get from a large external field (in the absence of vortices), will lead to anoscillating signal (depending on the relative orientation of P and B ). Within this picturethe amplitude of a Fourier component (field value or frequency ω µ ) is understood as thefraction of muons experiencing that component - which can then be taken as a volumefraction of the sample with that local field value. The local field distribution is often whatwe are interested in and is the physically intuitive quantity to grapple with.We are primarily concerned with measuring the superfluid condensate density, λ − ,and so the key step in all of this is that the depolarisation rate is due to D ( B loc ), which ina superconductor is in turn related to the vortex lattice. Outside the vortex core the fielddecays as B ( r ) = Φ πλ K ( r/λ ) where Φ is the flux quantum and K ( r/λ ) the zeroth-orderBessel function, see pg. 553 of reference [21]. While the vortex core radius is r c = ξ (cid:28) λ (for the cuprates), the London penetration depth scales the distance over which the fielddrops to zero outside of the vortex core. For the cuprates the separation between adjacent .8. RAMAN SPECTROSCOPY λ for B ext > . B = D ( B − h B i ) E is proportional to the penetration depth [137].For a static, Gaussian distribution of B loc in sufficiently large B ext , then σ ∝ ∆ B loc where σ is the depolarisation rate [167]: P X ( t ) = A exp − σ t ! cos( γ µ B loc t + δ ) (3.3)Here A is the volume fraction with this magnetic phase. Note the convention we haveadopted where a factor of 1 / σ reportedin the literature needs to be corrected by a factor of √ γ µ B loc = ω µ is a fre-quency. This is valid for polycrystalline samples of YBa Cu O − δ (where the vortex latticeis 3D) but for single crystals, or Bi-based HTSs , more care must be taken to accountfor the vortex lattice structure and perhaps a more sophisticated model would be required[167]. Indeed, from the Fourier transform of our P ( t ) time spectra we see a nice Gaussiandistribution of local fields with its mean value below T c slightly shifted below B ext (seeFig. 7.4).In detail, we have the following relation between λ − ab and the depolarisation rate σ fromBernhard et al. [170]; λ − ab = 12 . σ (3.4)Here λ ab has units of µ m and σ has units of µ s − . Thus for a typical value of λ = 160 nmat optimal doping, σ is about 3.1 µ s − .So, let us now follow the string all the way out of the labyrinth; σ ∝ ∆ B loc ∝ λ − ab ∝ n s /m ∗ We use commercial T64000 and LabRam Raman spectrometers. For lasers we use the514.5 nm (green) and 458 nm (blue) lines of an Ar ion laser and the 633 nm (red) laserline from a HeNe laser. Measurements are performed in backscattering geometry. Exceptfor variable-temperature measurements, we use confocal microscopes (with x50 or x100 Depending on the anisotropy of the compound, which is Pb doping and hole doping dependent, Bi-based SCs have a pancake vortex lattice above a certain magnetic field which can take quite low values(100-300 Oe) anisotropic samples. CHAPTER 3. TECHNIQUES AND SAMPLES objective lenses) to focus the laser and collect the Raman-scattered light. A λ / half-plate is used to orientate the polarisation of the incident laser beam. Liquid N cooledCCDs count the diffracted photons where the diffraction grating is chosen depending onthe measurement. For example, to measure the broad two-magnon scattering signal it wasdesirable to use a low, 300 lines/mm grating in order to measure a large range of ω within asingle spectrum. These spectrometers detect light of a certain polarisation more efficiently,the T64000 has more than a factor of 10 difference between the efficiency of horizontal-and vertically-polarised light detection. Therefore the experiments are set up to have theexit polariser in the appropriate alignment. The relative polarisation of the incoming laser and detected scattered light must be con-sidered in a two-magnon experiment. As mentioned earlier, in the cuprates appreciabletwo-magnon response is seen in both the A g and B g channels but not in the B g channel.Selecting the polarisation configuration allows a check on whether a genuine two-magnonpeak is present in the data and can be manipulated to select out the background for sub-traction. Magnons are excited from incident light traveling parallel to the c -axis so thatthe electric field gradients are in the a, b plane. Following the notation of Sugai et al. [7],let a and b denote polarisation parallel to the a and b crystallographic axes, which lie alongthe Cu-O-Cu directions, respectively, while x and y denote polarisation rotated 45 ◦ from a and b . Next, we represent incident polarisation parallel to a and scattered polarisationparallel to b as ( a, b ). The Raman-active symmetries and polarisation configurations arethus ( x, y ) for B g , ( a, b ) for B g , ( x, x ) for A g + B g and ( a, a ) for A g + B g . These aresketched in Fig. 3.9.Single-crystal samples have the feature of well-defined crystallographic axes on lengthscales that the x50 objective microscope probes. Although there are clearly different singlecrystal domains visible under the microscope, it is possible to find a homogeneous regionto measure from. To change between B g and B g scattering geometry, for example, thepolarisation of the incident laser beam must be rotated from 45 ◦ to 0 ◦ relative to the Cu-Obond length direction. In practice the orientation of the single crystal, both the c-axisdirection and orientation of the polarisation with respect to the Cu-O bond length, can beascertained by inspecting the phonon modes with reference to polarisation-specific spectra Charge-Coupled-Detectors S ≡ E × B is parallel to the c -axis unit-cell vector. .8. RAMAN SPECTROSCOPY bac Incident polarisation: e i Detected polarisation: e s e i e i e s e s CuO layers Cu O B B e i e i e s e s A +B A +B Figure 3.9: Schematic diagram of various Raman active symmetries and the polarisationconfigurations to obtain them. The two-magnon peak is manifest in B g and to a lesserextent in A g . These relations are strictly correct only for tetragonal material.reported in the literature, for example [171, 7]. We found it was sufficient to manuallyrotate the crystal approximately 45 ◦ to swap between B g and B g (or A g + B g and A g + B g ).The most reliable way to unambiguously detect a two-magnon peak was by measuring B g and then B g geometries. Both have detection polarisation orthogonal to the incidentpolarisation resulting in weak ‘background’ electronic scattering and most phonon peaksare greatly reduced in intensity. The two-magnon peak, if present, is therefore a majorfeature of the spectrum. Most other experimental parameters can easily be kept constant bythis modification, such as: spot position on the sample, temperature, incident laser power.In practice though all four configurations are measured to confirm the (non)existence ofthe two-magnon peak, see Fig. 6.1 for an example. In the underdoped Ln123 samples thetwo-magnon peak is easily observed at room temperature. A Diamond Anvil Cell (DAC) is capable of producing very high hydrostatic pressures, wellbeyond 10 GPa , whilst allowing spectroscopic measurements to be carried out becauseof the transperency of diamond. A pressure medium surrounds the sample and is used to In an ideal situation up to 200 GPa CHAPTER 3. TECHNIQUES AND SAMPLES ensure pressure on the sample is isotropically (hydrostatically) distributed.Background subtraction is vital for our two-magnon measurements when using a DAC.Diamond has a strong Raman active phonon mode at 1332 cm − [172] which is pressuredependent and can even be used as a pressure calibrant, though less reliably than an in situ ruby chip. Higher-order phonon modes are significant and can be seen between2100 cm − and 2670 cm − , precisely the area of interest for two-magnon scattering. Inaddition, the greater the impurities or C vacancies, the stronger the fluorescence seenbetween ∼ − and 4000 cm − . Again, precisely the area of interest for two-magnonscattering. We use a 4:1 methanol:ethanol pressure medium but due to the vibration energyof H modes coinciding with the two-magnon scattering energy range, Ar or He would bepreferable [173] if the facilities are available. Some possible methods for subtracting thesebackground contributions, and why they did not work, are;• Location variation; Take a spectrum with the laser spot on sample and then subtractoff a spectrum taken with the laser focused on a blank part of the chamber. It isdifficult to reliably find blank part within the chamber which is commonly ∼ µ m in diameter and shrinks (initially) under pressure. Also the chamber becomes opaqueto optical wavelengths above about . In addition there is a huge signal fromstray ruby crystals which are used as pressure calibration within the chamber. Thisis an issue when coming to measure the two-magnon peak in cuprates ( to − ) as the signal from the tail of the ruby peaks (centred ≈ − with the . laser) swamps the two-magnon peak and/or changes the character of thebackground signal compared with other parts of the chamber. • Polarisation variation; Take a spectrum first with a parallel polarisation configura-tion, and then with crossed polarisation. Subtract one from the other (dependingon the relative orientation of the sample’s CuO layer). Unfortunately, the responsefrom diamond seems to also be polarisation dependent .• Rotation variation; Take a spectrum and then rotate the DAC 45 ◦ and re-take thespectrum. Subtract one from the other. Again, diamond unfortunately seems tohave a different response depending relative orientation of the diamond to the laserpolarisation, just as for two-magnon scattering .For the reasons listed above we were not able to subtract the large fluorescence signalfrom our DAC data. .8. RAMAN SPECTROSCOPY
For these measurements the sample is mounted on a Cu-plate which is in good thermalcontact with a Cu-cold finger cooled by a closed-cycle cryogenic system (a fridge). Thisallows us to access temperatures between 8 . . µ Torr by a turbo-pump .With the cryogenic apparatus in the way we can no longer do micro-Raman because thefocal lengths of microscopes are too short. Instead, the polarised incident beam is focusedby a conventional 15 cm focal length lens and an intermediate mirror reflects the beamonto the sample in as close to backscattering geometry as possible.A schematic diagram of the experimental set-up for variable-temperature Raman mea-surements is shown in Fig. 3.10.Next, we need to collect scattered light from our sample and create an image of it tosend into the spectrometer. We use a lens of focal length f = 10 cm to collect some ofthe divergent scattered light and focus it into a parallel path. A second lens captures thisparallel beam and focuses it through an entrance slit into the spectrometer. The numericalaperture of this second lens, f = 17 . is matched to the size of the collection mirrorin the spectrometer. The magnification of our image is easily calculated, m = f /f . If m is too large, reducing the entrance slit size, which is done to reject stray light and excessRayleigh scattered light, will cut off signal from the sample. It really is a fine balancingact so that although we collect very little of the scattered light with this experimental setup, it is close to the optimal set up (barring radical changes to the entire system).By using a λ / -half plate to rotate the polarisation of the incident laser beam we canmeasure Raman spectra in the usual B g , B g , A g + B g , A g + B g symmetries. Notethat switching between B g and B g geometries, for example, requires manually rotatingthe single crystal ex-situ , whereas switching between B g and A g + B g geometries, forexample, requires only a rotation of the λ / -half plate. The detection polarisation directionis kept fixed because the T64000 triple monochromator is approximately ten times more With diffusion pump systems oil-droplets can back-diffuse into the sample chamber. which is related to the angle of the cone of scattered light collected, θ , and the refractive index, n , by, NA = n sin θ . In our case n = 1. CHAPTER 3. TECHNIQUES AND SAMPLES half-platelens (a)mirrorcryostat and sample lens (b) polariser 1 st intermediate slitsto triple-stage monochromatorlens (c) Figure 3.10: Schematic diagram of the experimental set up for variable-temperature Ramanspectroscopy measurements.sensitive to vertically-polarised light than horizontally-polarised light. hapter 4Bi2201: Ion-size and Ramanspectroscopy studies
Summary
Using a simple materials variation approach, we study (i) the effect of ion-size and disorderon T c and (ii) Raman modes in the single-layered cuprate Bi Sr CuO − δ (Bi2201).We apply negative internal pressure by Ba substitution for Sr in Bi2201. We find thatthis substitution can increase T c despite increasing disorder. That is, T max c is in fact bettercorrelated to the average ion-size, rather than disorder, on the Sr site. We conclude boththe ion-size and disorder significantly affect T c in Bi2201. This is a new and importantinterpretation of the data and will be built on in subsequent chapters.We also present Raman spectroscopy measurements on Bi − x Pb x Sr . Ln . CuO − δ wherewe alter Ln and x and compare the results with measurements on the bi- and tri-layer Bi-based cuprates, Bi2212 and Bi2223. We argue from the simple interpretation of these ma-terial variation experiments that the 120 cm − mode in Bi-based cuprates has a significantBi vibration contribution and insignificant Sr vibration contribution. Neither vibrationsignificantly contributes to the lower 70 cm − mode in Bi2201 however we do see a con-sistent shift to lower frequencies with increasing CuO layers of this mode in the Bi-basedcuprates. Motivation
In Sec. 2.3 we introduced the differing effects of internal pressure (as caused by isovalent ionsubstitution) and external pressure on superconductivity in the cuprates. As mentioned, it778
CHAPTER 4. BI2201: ION-SIZE AND RAMAN SPECTROSCOPY STUDIES is unclear what the salient difference between these two sources of unit-cell compression is.In this section however we merely use this observation to increase T max c through negativeinternal pressure effected by substituting Ba for Sr in Bi2201. By doing so we elucidate anunexplored contribution to the magnitude of T c in the Bi2201 system.The hypothesis that we wish to explore here is that it is the positive internal pressurewhich contributes mainly to the reduction in T max c upon Ln substitution in the Bi2201system. A simple test on this hypothesis is to substitute the larger Ba ion for Sr to induce“negative internal pressure” (i.e. increase the unit cell volume) and measure the resultingdoping dependence of T c . In this case disorder, as commonly characterised in this systemby the variance quantity σ = h r i − h r i [174] where r is ionic radius of the Sr-site ion from[28], is actually increased because of the difference in ion-size, r , between the Ba and other(Sr and Ln) ions on the Sr-site (see Fig. 4.1 for diagram of the Bi2201 crystal structure).If disorder plays a dominant role T max c should decrease, while T max c will be increased if ourinternal pressure hypothesis is correct and significant. T c Bi Sr − y Ln y CuO δ (Bi2201), where Ln may be any lanthanide rare-earth element, hasreceived much attention recently [174, 85, 48, 175, 27, 82, 79, 176, 177]. Bi2201 has asingle CuO layer residing between SrO layers which are, in turn, situated between BiO δ layers with variable oxygen content, see Fig. 4.1. Doping, p , is often controlled by varyingthe Ln ratio, y , on the Sr site and it was observed that smaller Ln ions suppress the T c forany particular doping [178]. Many authors have attributed this decrease in T c to disorderon the Sr site (the “ A -site”) [85, 175, 179, 174] resulting from the difference in ion-sizesbetween Sr and Ln and quantify this disorder using σ .Superconductivity in the cuprates occurs on the CuO layer and as such intra-layerimpurities or disorder (e.g. Zn or Ni ions on the Cu site in the CuO layer) rapidlysuppresses superconducting properties such as T c [180, 85] and the superfluid density [136,152]. Inter-layer disorder is generally less effective at suppressing these properties and itseffectiveness is dependent on the type of disorder [181]. For this reason we seek to the testhypothesis that the ion-size, rather than disorder, is the variable with the most importanteffect on T c . As we have discovered, the Bi2201 material is amenable to such an experimentas the large Ba ion can be partially substituted for Sr, as indicated in Fig. 4.1. .1. ION SUBSTITUTION EFFECTS ON T C OCuSr, Ba, Ca, LnBi { bac O ( ) O(3)O(1)
Figure 4.1: Schematic diagram of the Bi2201 crystal structure.It has also been found that smaller Ln ions enhance the pseudogap in Bi2201 as mea-sured by ARPES [182, 179, 82], possibly because of a larger J arising from a shortersuperexchange path-length . The pseudogap competes with superconductivity [80] and soa larger pseudogap energy will further suppress T c . We believe this must be considered asan additional (rather than primary) effect because T max c itself is reduced by decreasing Lnion size. Bi2201 has two doping channels, the Sr /Ln ratio and excess oxygen in the BiO δ blocking layer. Unlike others, we anneal at different temperatures and oxygen partialpressures to reversibly alter the doping state through the oxygen content. Most otherstudies in this system have simply utilised the as-prepared oxygen content and altered theSr/Ln ratio to control doping.The experimental procedure follows standard techniques asdescribed in the previous chapter.Room-temperature thermopower measurements, S (295K), are used as a proxy for thedoping state p - which is very tricky to directly measure in these systems. The well-knownObertelli-Cooper-Tallon (OCT) relation between p and S (295K) [158] is invalid for Bi2201[49, 183] and in general the precise relation between S (295K) and p is not known for Bi2201 As extensively discussed in Chapter 6. CHAPTER 4. BI2201: ION-SIZE AND RAMAN SPECTROSCOPY STUDIES - although it is known that S (295K) monotonically decreases with increasing p as is familiarfrom the OCT relation. In the special case of Ln=La (Bi Sr − y La y CuO δ ) however, wehave derived a working relation between these two quantities from the work of Ando et al. [49]; p = − .
026 + 0 .
18 exp − S (295K)110 ! (4.1)Furthermore, this relation is verified by the independent measurement technique of p ofSchneider et al. [48].Once this relation is applied to the smaller Ln ions, the materials describe a morenarrow parabolic doping dependence of T c as shown in Fig. 4.2. This is likely becauseof a genuinely smaller doping range over which superconductivity is observed - despitepossible pressure-induced-charge-transfer (PICT) y seems to be a good measure of p [174],though it may also be because Equation 4.1 is not valid for Ln =La. As mentioned, we usethermopower measurements, rather than p estimates, throughout.Fig. 4.3 shows T c values for our model system with various Ln substitutions with dif-ferent oxygen content. The data is plotted as T c versus S (295K) for a range of dopingstates from underdoped (right-hand side, S (295K) > − µ V/K) to overdoped (left-handside, S (295K) < − µ V/K). As other groups have noted T max c falls rapidly with decreas-ing ion size. To this we add our new data for Ba and Ca substituted samples of compositionBi (Sr . − x A x )La . CuO δ , where A = Ba or Ca. Samples of the Bi (Sr . Ba . )La . CuO δ and Bi Sr . Eu . CuO δ materials were made by Thierry Schnyder, a joint Victoria Uni-versity of Wellington - École Polytechnique Fédéral de Lausanne (EPFL) Master’s studentwho studied in our group, and the data presented here for these samples were taken byhim. We find that T max c is in fact slightly increased by moderate Ba substitution on the Srsite despite the increasing disorder, while higher Ba substitution then decreases T max c .We note however that our samples did not display superconducting transition temper-atures as high as others have reported [85] despite replicating their sintering conditions.Unfortunately this leaves the possibility that the best possible quality Ba-free sample mightstill have a higher T max c than best possible quality Ba-doped sample. We would need to ofcourse synthesize a Ba-doped Bi2201 with a T max c higher than the highest reported valuefor Ba-free Bi2201 to make the conclusions of this section more reliable. One possibility inthis regard is 10-25% substitution of Bi by Pb. The Bi-O x rock salt layer size mismatchwith the rest of the unit cell causes an incommensurate modulation over several unit celllengths, see e.g. [38, 184, 178]. The unit-cell repeatability can be returned when over- .1. ION SUBSTITUTION EFFECTS ON T C L a , A n d o H a l l d a t a L a , S c h n e i d e r X A S d a t a L a , O k a d a S ( 2 9 5 K ) S m , O k a d a S ( 2 9 5 K ) E u , O k a d a S ( 2 9 5 K ) T c / T c m a x = 1 - 2 5 0 ( p - p o p t i m a l ) T c m a x = 3 5 K , p o p t i m a l = 0 . 1 6 5 T c / T cmax p ( h o l e s p e r C u ) Figure 4.2: A collection of normalised T c values of Bi Sr − y Ln y CuO δ (Ln=La, Sm or Euas annotated in the legend) vs. p taken from literature. Ando et al. (black square datapoints) determine p from Hall-effect measurements, testing the validity of their results bycomparing with La214 Hall-effect data [49]. From this work we also derive the workingrelation between p and S (295K) shown in Equation 4.1. Independent confirmation of these p values come from the X-ray Absorption Spectroscopy measurements of Schneider et al. [48] (Red line). From these two data sets we derive the modified parabolic dependenceof T c /T max c with parameters shown in the top left of the figure. Applying Equation 4.1to data from Okada et al. [182] results in the three other data sets plotted. The Ln=Ladata are all consistent, but not the smaller ion-size data. The discrepancy for smaller ionsis probably because of a genuinely smaller doping range over which superconductivity isobserved but may also be because Equation 4.1 is not valid for Ln =La.2 CHAPTER 4. BI2201: ION-SIZE AND RAMAN SPECTROSCOPY STUDIES - 1 6 - 1 2 - 8 - 4 0 4 8 1 21 21 62 02 42 8 u n d e r d o p e d Tc (K) S ( 2 9 5 K ) ( m V / K ) o v e r d o p e d
Figure 4.3: T c plotted against the room temperature thermopower, S (295K), forBi (Sr . − x A x )Ln . CuO δ where A=Ba or Ca and in the legend the Ln ion is speci-fied. Note the comparable T c values of the Ba substituted samples with the standardBi Sr . Ln . CuO δ material. Higher S (295K) values represent underdoping as annotated.The onset of superconductivity is typically observed between 1 and 3 K higher than thevalues plotted here.doped with oxygen or when Pb is partially substituted for Bi in Bi − x Pb x Sr . La . CuO δ as the (Bi,Pb)-O x unit increases in size and fits into the unit cell [185]. This tends to im-prove the sample quality, for example, sharper T c transition widths and less super-latticemodulations, see below 4.2.That ion-size affects T c through internal pressure can be more clearly seen by the dataplotted in Fig. 4.4. Recall that a common measure for disorder used in the literature is thequantity σ = h r i − h r i where r is the ion-size at the Sr site. As an example, for the caseof Bi Sr . La . CuO δ , h r i = 1 . r Sr + 0 . r La and h r i = 1 . r + 0 . r where r La and r La are the ionic radii of Sr and La respectively from Shannon [28]. The superfluid densityis more sensitive than T c to disorder and so a more direct measure of the effect of disorderon the superconducting properties could be obtained by measurements of the superfluid .2. INVESTIGATION OF RAMAN PHONON MODES Tc max (K) D i s o r d e r m e a s u r e , (cid:1) , ( x 1 0 - 5 n m ) ( a ) Tc max (K) A v e r a g e i o n i c r a d i u s a t S r s i t e , < r S r > ( n m ) ( b )
Figure 4.4: (a) The maximum T c , T max c , of our Bi2201 samples plotted against a standardmeasure of disorder in this system, σ . (b) The same T max c data plotted against the averageion-size at the Sr site, h r i . Collectively these plots show the importance of ion-size on thevalue of T max c in this system. Apparently, with negative internal pressure one can increasethe T max c of a cuprate.density using, for example, the muon spin-relaxation technique. We use this approach tostudy disorder effects in the YBaSrCu O − δ system in Chapter 7. If disorder were the onlymechanism suppressing T c , we would expect the plot of T max c vs σ shown in Fig. 4.4a toshow a clear correlation between the two quantities. It does not. As Fig. 4.4b shows thereis in fact a clearer relation between T max c and the average ion size at the Sr site. Fig. 4.4thus demonstrates the influence and importance of internal pressure on T c .It is likely that both disorder and internal pressure have an effect on T c upon ion-sizesubstitution in our samples; disorder weakly suppressing T c and negative internal pressureincreasing T c in the Ba doped samples. Indeed decreasing the internal pressure furtherby going to 0.2Ba substitution does not further increase T max c in this system, as shown inFig. 4.4b. Introduction
Raman spectroscopy is an important tool for studying the vibrational properties andcharacterizing the superconducting order parameter [7] and pseudogap [107] of high tem-perature superconducting cuprates. For example, it can be used to characterize theirdoping level and in Bi-based cuprates the Bi:Pb ratio (where Pb is substituted for Bi)4
CHAPTER 4. BI2201: ION-SIZE AND RAMAN SPECTROSCOPY STUDIES
S r ? O
B i
Intensity (arb. units) (cid:2)(cid:1) ( c m - 1 ) O S r ( a p i c a l o x y g e n )I n c o m m e n s u r a t eu n i t - c e l l m o d u l a t i o na d d i t i o n a l p e a k sf r o m s y m m e t r y r e d u c t i o nB i
Figure 4.5: An annotated spectrum for optimally doped polycrystallineBi Sr . La . CuO − δ indicating the predominant nature of each phonon mode.[185, 186, 187]. However there are several conflicting assignments of the low frequencyphonon modes in the Bi Sr CaCu O − δ (Bi2212) system [186] and these modes have notbeen extensively studied in the Bi Sr CuO − δ system.Our approach to studying the nature of these modes is simple; we make ion-substitutionsand observe the effect on the low-frequency modes. Using standard synthesis and char-acterisation techniques described earlier, we prepare Bi − x Pb x Sr . La . CuO − δ for x =0 . ≤ Sr . La . CuO − δ constructedfrom two high-resolution frames joined at ω ≈
410 cm − . The Raman signal was weak forall samples studied which necessitated long integration times.The peaks at ≈
450 cm − and ≈
620 cm − are predominantly due to vibrations of .2. INVESTIGATION OF RAMAN PHONON MODES − δ layer, O Bi , and the apical-oxygen ions in the (Sr,Ln)O layer,O Sr , respectively [185, 186, 188] - refer to Fig. 4.1 for a picture of the crystal structure. Theshoulder peak at ≈
650 cm − is likely due to incommensurate modulation of the Bi2201‘unit cell’ [185, 171] and it disappears when we substitute Pb on the Bi site ( x = 0 . − and 150 cm − are due to local symmetryreduction from Ln substitution, oxygen non-stoichiometry and the orthorhombicity ob-served with XRD [186, 171]. The un-labelled peak at 65 cm − is discussed in the followingsection along with the assignment of the 120 cm − mode.This work has been published here [189]. As mentioned, the two most prominent low-frequency modes observed in Bi2212, labelled ω and ω , have received conflicting assignments [186]. In Bi − x Pb x Sr . La . CuO − δ theyoccur at ω = 117 . ± − for x = 0, ω = 115 ± − for x = 0 . ω = 110 ± − for x = 0 . ω = 105 ± − for x = 0 .
45 and ω = 70 ± . − independentof x . These data are plotted in Fig. 4.6. We note Sato et al. report Raman data on singlecrystals of optimally-doped Bi − x Sr . Ln . CuO − δ which show this peak slightly higher at ω = 121 ± − . In fact, we find their peak positions are consistently ∼ − higherthan ours.The peak at ω = 70 ± . − does not shift with doping, Pb concentration x , or Lnion substitution, as shown in Fig. 4.6a (inset). We compare these data with the positionof this peak in the bi-layer Bi2212 and tri-layer Bi2223 members of the Bi-based cupratefamily (the general chemical formula for this family is Bi Sr Ca n − Cu n O n +4+ δ where n isthe number of CuO layers) as reported in [187]. These data are shown in Fig. 4.6a andreveal that the frequency of this mode decreases with increasing number of CuO layers.Thus, changing the local geometry about the Bi site (through oxygen content in the BiO − δ layer, Ln ion size and Pb substitution) does not affect the frequency of this mode in Bi2201.However the larger Bi2212 and Bi2223 unit cells do. These two observations discourage anassignment of this peak to a mode with a strong Bi contribution.The mode at ω ∼
120 cm − does not shift with doping level or Ln. Increasing Pbconcentration however does shift the mode to lower frequencies. Again we can comparethis result with that found in Bi2212 and Bi2223 where the position of this mode was found6 CHAPTER 4. BI2201: ION-SIZE AND RAMAN SPECTROSCOPY STUDIES
B i 2 2 2 3( B i S r C a C u O d ) ( a ) w (cm-1) n , n u m b e r o f C u O l a y e r s p e r u n i t c e l l L a ( x = )
B i 2 2 0 1B i 2 2 1 2( B i S r C a C u O d ) w (cm-1) B i 2 2 0 1 ( b ) w (cm-1) P b c o n t e n t , x
B i 2 2 0 1 p o l y c r y s t a l l i n e B i 2 2 1 2 p o l y c r y s t a l l i n e B i 2 2 1 2 s i n g l e c r y s t a l B i 2 2 2 3 p o l y c r y s t a l l i n e
Intensity (a.u.) w ( c m - 1 ) x = 0 . 2 x = 0 . 4 5 Figure 4.6: (a) The position of the ω = 70 ± . − peak for Bi2201 with data forBi2212 and Bi2223 from [187]. Inset; Peak positions for the 70 cm − mode for Bi2201with x = 0, Ln={La, Nd, Sm, Eu} and various Pb concentrations. (b) Positions of the ω ∼
118 cm − mode in Bi2201 are plotted against Pb content, x , as red circles. Alsoplotted are reproduced data for Pb substituted Bi2212 (black squares and downward-triangles) and Bi2223 (green diamonds) from reference [187]. The line is a fit to ω = a + a h − tanh (cid:16) x − x a (cid:17)i and a guide to the eye [187]. Inset; Raw spectra for Bi2201 withtwo different x values. .3. DISCUSSION x = 0 . x = 0 .
35 areslightly higher than for Bi2212 and Bi2223 of equivalent Pb concentrations, the overalltrend is replicated. Hence this mode would appear to have an insignificant Sr contributionin Bi2201 but significant Bi contribution in the three Bi-based cuprates Bi2201, Bi2212and Bi2223.These results also further demonstrate the utility of Raman measurements in rapidcharacterization of Bi-based cuprates. Sr mode The O(2) Sr site, also known as the apical oxygen, is the nearest neighbour to the Srsite and its relative position to the CuO plane can be related to the hole concentration[190]. We therefore measured the O(2) Sr mode frequency for the series of optimally-dopedBi Sr . Ln . CuO − δ for Ln=La, Nd, Sm and Eu. The results are plotted in Fig. 4.7. AsFig. 4.7b shows there was no systematic shift in the peak frequency due to the Ln ion-size.We compare the small shifts of this peak, ∼ − , where the FWHM is ∼
40 cm − , withthose expected from mode Grüneisen scaling: γ i = − δ ln( ω i ) δ ln( V ) where V is the unit cell volumeand γ i = 0 . The next step in these investigations would be to collect appropriate XRD scans for a fullRietveld refinement analysis across the series. From these data one can determine intra-cellbond lengths and from these bond lengths the V + bond-valence-sum parameter introducedin Sec. 2.3.2. Recall from Fig. 2.12 that V + has been shown to correlate with T max c acrossall known cuprates [102] and so it would be interesting to test whether the correlation stillholds for our ion-substituted Bi2201 materials. In addition, the full structural refinementof our materials would provide information on the ion-substitution induced changes ofimportant bond-lengths, such as the apical oxygen bond length, and structural parameters8 CHAPTER 4. BI2201: ION-SIZE AND RAMAN SPECTROSCOPY STUDIES ( a )
Intensity (arb. units) w ( c m - 1 ) E uS mN dL a L a N d S m E u6 1 66 1 86 2 06 2 2 ( b )
M e a s u r e d p e a k p o s i t i o n E x p e c t e d p o s i t i o n f r o m G r u n e i s e n s h i f t w (cm-1) L n
Figure 4.7: (a) Representative high-resolution spectra for optimally-dopedBi Sr . Ln . CuO − δ with, from bottom to top, Ln=La, Nd, Sm and Eu. (b) Themeasured position of the O(2) Sr , apical oxygen mode plotted as squares and the predictedshift from mode Grüneisen scaling is plotted as triangles.such as the CuO layer buckling angle. These structural parameters potentially have asignificant effect on the superconducting properties, as discussed in Sec. 2.1.3. Furthermore,the apical oxygen phonon mode shown in Fig. 4.7 would be expected to closely follow theapical oxygen bond-length and this would be useful to test.Another useful data set for our novel Bi2201 materials that could be gathered usingreadily accessible equipment would be resistivity measurements for a range of dopings.From the resistivity, it may be possible to reliably estimate the pseudogap energy as hasbeen done recently by Kim et al. on their ion-substituted Bi2201 samples [85]. It would beinteresting to measure what effect negative internal pressure has on the pseudogap energy, E P G = k B T ∗ .We now briefly consider the measurements of T c as a function of ion-size in our novelBi2201 materials. If we take the view that the ion-size effect is significant, how can weexplain its effect on T c ? One hypothesis is indirectly via the density of states (DOS). Thehypothesis is that ion-size substitution results in a distortion of the Fermi-surface suchthat the van Hove singularity (vHs) in the (electronic) DOS moves closer in energy tothe Fermi-level, that is ( E F − E vHs ) →
0, at optimal doping for larger ion-size. Visually,this hypothesis is sketched in Fig. 4.8. This figure shows measured T c values as symbolsconnected by dotted lines corresponding to the left axis for Bi Sr . Ln . CuO δ withLn={La, Nd and Eu} Also on shown in this figure is an hypothesised DOS at the Fermi- .4. CONCLUSIONS T c values for Bi Sr . Ln . CuO δ with Ln={La,Nd and Eu}. The solid lines correspond to the right-hand axis and are an hypothesisedDOS at the Fermi-energy, N ( E F ), for each Ln. These lines show that at optimal doping N ( E F ) increases for larger ion-size due to the closer proximity of the vHs. In the simpleBCS theory T c is very sensitive to N ( E F ).energy, N ( E F ), shown as solid lines. The solid lines corresponding to the right-hand y -axis and show that at optimal doping N ( E F ) increases for larger ion-size due to thecloser proximity of the vHs.The associated, perhaps dramatic, changes in the DOS can have a significant in-fluence on T c . For example, in a simple, weak-coupling BCS picture we have T c ∝ exp[ − / ( N ( E F ) V )] where V is the pairing potential as shown previously Equation 2.7.Thus, within this framework at least a small increase in N ( E F ) could significantly increase T c . We explore this idea further in the next chapter using density functional theory cal-culations and also note that a fuller discussion of ion-size effects must wait until we haveconsidered our investigations on the Ln(Ba,Sr) Cu O y system. In this chapter we used a simple materials variation approach to study (i) the effect of ion-size and disorder on T c and (ii) Raman modes in the single-layered cuprate Bi Sr CuO − δ (Bi2201).We simulated negative internal pressure by Ba substitution for Sr in Bi2201 and found0 CHAPTER 4. BI2201: ION-SIZE AND RAMAN SPECTROSCOPY STUDIES that this can increase T c despite increasing disorder. That is, T max c is in fact better corre-lated to the average ion-size, rather than disorder, on the Sr site, although it is possiblethat ion-substitution induced disorder is a significant additional effect that suppresses T c .We also presented Raman spectroscopy measurements on Bi − x Pb x Sr . Ln . CuO − δ where we alter Ln and x and compared the results with measurements on the bi- andtri-layer Bi-based cuprates, Bi2212 and Bi2223. We argued from the simple interpretationof these material variation experiments that the 120 cm − mode in Bi-based cuprates hasa significant Bi vibration contribution and insignificant Sr vibration contribution. Neithervibration significantly contributes to the lower 70 cm − mode in Bi2201 however we dosee a consistent shift to lower frequencies with increasing CuO layers of this mode in theBi-based cuprates. hapter 5Density Functional Theory study ofthe ion-size effect Summary
We perform DFT calculations on undoped ACuO for A={Mg, Ca, Sr, Ba} to investigatethe effect of ion-size on the electronic properties in this model cuprate system. Where thesematerials have been synthesised we find good agreement between our calculated structuralparameters and the experimental ones. There is a peak in the density of states ∼ T max c via the density of states. In this chapter we seek to correlate structural distortions from ion-size substitution withchanges in the normal-state electronic structure of the materials in an effort to further un-derstand the observed effects of ion substitution. Electronic structure and crystallographicstructure are intimately related and one could imagine that structural distortions inducedby isovalent ion substitution can have a significant effect on the former.The family of materials we choose to study are the so-called infinite layer cuprates.These are the simplest HTS cuprates known. They have the chemical formula ACuO with A={Mg,Ca,Sr,Ba} and an illustration of their crystal structure is shown in Fig. 5.1.This system is ideal for a computational study: it has a simple tetragonal crystal structure,912 CHAPTER 5. DENSITY FUNCTIONAL THEORY STUDY OF THE ION-SIZE EFFECT
OCubac MgCaSrBaRa { Figure 5.1: Idealised crystal structure of ACuO where A={Mg,Ca,Sr,Ba,Ra}. The spacegroup is P4/mmm and atomic positions are A= ( a / , a / , c / ), Cu= (0 , , a / , , , a / ,
0) where a = b and c are the lattice parameters.there are comparatively few electrons per unit-cell and a wide range ion-size variation ispossible.As such there have been many previous computational studies of ACuO , the majorityon CaCuO [192, 193, 194, 195, 196, 197, 198]. In distinction, we are primarily interestedin systematic trends in the electronic structure as A varies from Mg to Ba.We characterise the electronic structure by the electronic dispersion, (cid:15) ( k ) = E ( k ) − E F under the reasonable assumption that k is a good quantum number for our materials. Weare also interested in the density of states. The density of states of the n th band is relatedto E n ( k ) directly; N n ( E ) = Z k π δ ( E − E n ( k )) (5.1)where δ ( E − E ( k )) is the delta function. Summing over all bands will give the total densityof states, N ( E ). Within a BCS theory of superconductivity the density of states at theFermi-level, N ( E F ), is a key electronic parameter [105, 199, 200], irrespective of the specificpairing mechanism [201]. N n ( E ) can also be represented as a function of E n ( k ) as [62]; N n ( E ) = Z S ( E ) d S π |∇ E n ( k ) | (5.2)where S ( E ) is the surface of E ( k ) of energy E and ∇ E ( k ) is the gradient of E ( k ) (orequivalently (cid:15) ( k ) ). The purpose of expressing N ( E ) in this way is to see that at extremes(and inflection points) of (cid:15) ( k ) the integrand of Equation 5.2 diverges. These points arecalled “van Hove singularities” (vHs) and at such points, depending on dimensionality ofthe system, N ( E ) will diverge. In practise such a divergence may be cut-off by disorder or .2. EXPERIMENT layers along the c -axis.To calculate the electronic structure of our materials we use density functional theory(DFT) implemented with the Vienna Ab-Initio Simulation Package [202, 125, 126, 127, 128](VASP). Recent reviews of the use of DFT implemented with VASP in condensed matterphysics are given by Hafner [123, 124]. Density Functional Theory (DFT) is a method tofind the charge density, n ( r ), corresponding to the minimal energy of a physical system.To perform the calculation considering the interaction of every electron in the system iscomputationally infeasible and so some approximations need to be made. Firstly the Kohn-Sham approach replaces the many-electron problem by a single-electron problem with anexchange-correlation potential between electrons. We next use a Generalised Gradient Ap-proximation (GGA) to derive the form of the exchange-correlation potential and kineticenergy in the single-electron Hamiltonian. In particular, we use the GGA-PW91 schemedeveloped by Perdew et al. system (espe-cially for A =Ca) is another reason to opt for a DFT study; it is very difficult to synthesizethese materials with good quality for experimental studies. We summarize the calculation procedure with a flow chart in Fig. 5.2.A critical step is to perform convergence tests on the total energy vs. k space samplingof the Brillouin Zone (BZ) to determine a sufficient k space sampling interval for accuratecalculations. The results of this procedure are shown in Fig. 5.3. From this convergencetest we find that a 16 × × k -space mesh is sufficient. This mesh corresponds to a4 CHAPTER 5. DENSITY FUNCTIONAL THEORY STUDY OF THE ION-SIZE EFFECT
61 62 63 64 65-22.044-22.040-22.036-22.032-22.028-22.024-22.020 ‘ F r ee / ene r g y ’ / s e V ) Volume/sx10 -30 m) sd) ‘ F r eeene r g y ’ s e V ) c/a F r ee / ene r g y / s e V ) Number/of/irreducible k -points
200 400 600 800-22.850-22.848 F r eeene r g y s e V ) Numberofirreducible k -points Γ X (0, π ) M ( π , π ) Γ (0,0) E ne r g y [ e V ] CaCuO k x k y E ne r g y [ e V ] (a) Figure 5.2: An illustration of the procedure followed to calculate the electronic structure.The various calculation parameters that should be altered for each step can be found inthe VASP manual [203]. The inputs to the calculation are Projector Augmented Waves(PAW) pseudopotentials for Cu, O and A={Mg,Ca,Sr,Ba}. Indicated in the flow-chart aretypical data-plots obtained from each step. .2. EXPERIMENT Free energy (eV)
N u m b e r o f i r r e d u c i b l e k - p o i n t s Free energy (eV)
N u m b e r o f i r r e d u c i b l e k - p o i n t s Figure 5.3: Calculated total energy of CaCuO as the density of k -space sampling isincreased. A larger number of irreducible k -points returns more accurate calculations atthe cost of increased computational time. We find a 16x16x16 k -space mesh is sufficientfor our system. The green data point is from a ‘high precision’ calculation.calculation over 288 irreducible k -points for a Monkhorst scheme.Before attempting any calculations of the electronic structure we must first determinethe most stable crystal structure, as characterised by a minimum in the “free energy”, F , of the material, F min . Note that we use the term “free energy” here to be consistentwith the nomenclature used by VASP, however, because the calculations are carried outat T = 0 they really represent the ground-state internal energy, U . We can then comparewith the experimentally determined cell volume and lattice parameters. For CaCuO theexperimental values are | a | = 0 . | c | = 0 . c / a = 0 . V = 4 . × − nm [204]. A more accurate way to determine F in VASP is for afixed unit cell volume. We therefore calculate F for various, fixed, unit cell volumes andlocate the minimum.For a fixed volume, the cell shape, i.e. c / a , and ion positions were first relaxed, forwhich VASP has sophisticated in-built algorithms, and then an accurate calculation of F was performed without further relaxation. The cell volume is then specified to be another Or, ‘inequivalent’, as in not related by the system’s particular symmetry to other k -points. CHAPTER 5. DENSITY FUNCTIONAL THEORY STUDY OF THE ION-SIZE EFFECT ‘Free energy’ (eV)
V o l u m e ( x 1 0 - 3 0 m ) ( a ) ‘Free energy’ (eV) c / a ‘Free energy’ (eV)
V o l u m e ( x 1 0 - 3 0 m ) E x p e r i m e n t a l v a l u e . D V = 0 . 9 x 1 0 -3 0 m = 1 . 9 % ( b ) ‘Free energy’ (eV) c / a ‘Free energy’ (eV) V o l u m e ( x 1 0 - 3 0 m )( c ) ‘Free energy’ (eV) c / a ‘Free energy’ (eV)
V o l u m e ( x 1 0 - 3 0 m ) ( d ) ‘Free energy’ (eV) c / a
Figure 5.4: The results of fixed-volume, ion relaxation calculations on ACuO for (a)A=Mg, (b) A=Ca, (c) A=Sr and (d) A=Ba. In the main panels the free energy is plot-ted against various fixed unit-cell volumes while insets show the corresponding c / a latticeparameter ratios. Annotated blue arrows/boxes represent reported experimental values ofthe unit-cell volume or lattice parameters. These plots show that there is a satisfactoryagreement between our DFT+LDA calculations and experimental results.value and F calculated again. The results of this procedure are shown in Fig. 5.4(b). Withthis process we find F min occurs at V = 4 . × − nm for CaCuO which is 1.9% higherthan the experimentally determined unit cell volume . On the other hand, the ratio c / a isexactly the experimentally determined one at F min .These two results demonstrate good agreement between our VASP calculations andreality in the CaCuO .Similarly we have calculated the most stable unit cell parameters for A=Mg, Sr andBa. These results are presented in Fig. 5.4. On these panels, the annotations in blue markexperimentally determined values of structural parameters where they are available.To our knowledge there are no reports that MgCuO has been synthesized. Other It is common for these DFT calculations to overestimate cell volumes by ∼ .2. EXPERIMENT has not been synthesized suggesting this compound is un-stable - up to 30% vacancies on the Cu site were reported by de Caro et al. [205], seealso citations of this paper. What BaCuO they could make had the lattice parameter | c | ≈ .
42 nm and | a | = 0 . sub-strate [205]. The most stable crystal structure from our calculations has | c | = 0 . | a | = 0 . c : a ratio compared with these experimental results.Super-lattices of intercalated Ba-, and Ca- infinite layer compounds have been reported tobe superconducting [205, 206] although even numbers of Ba- unit-cells are needed [205].In general, the comparison in Fig. 5.4 reveals a good correspondence between our LDAcalculations and experimental results. Once the crystal structure with F = F min has been found, we can use this structure for aprecise calculation of the electronic structure. VASP is run with a high density of k -points(24 × ×
24 giving 936 irreducible k -points in most cases) to accurately calculate theelectronic dispersion, (cid:15) ( k ) = E ( k ) − E F . We wrote Python code to recover the k -pointswhere the energy was not calculated because of their symmetry with another point. Thecode can be thought to ‘unfold’ the symmetries in order to recover the full dispersion forvisualisation.Fig. 5.5 shows an example of such a calculation. For CaCuO with k z = 0, we plotthe full dispersion of the band which crosses E F in panel (a) and (cid:15) ( k ) = 0 in panel (b).For comparison, we also plot on this panel the same (cid:15) ( k ) = 0 as parameterised fromARPES measurements on Bi2212 [63, 107] (red dashed curves). We note that there isa k z dispersion to this band that becomes more pronounced for small ion-size. By theLuttinger rule, the area enclosed by the Fermi contour should equal (1 + p ) where p is thehole concentration. So, in Fig. 5.5, the area enclosed by our computed blue curve shouldbe half of the area of the Brillouin zone. It is within the uncertainties from finite samplingof the Brillouin Zone and after taking into consideration the k z dispersion (which meansone is measuring the volume enclosed by the Fermi-surface).It is also possible to calculate the electronic dispersion along specified k directions ofhigh symmetry, for example k = (0 ,
0) to (0 , π ) to ( π, π ) to (0 ,
0) for k z = 0 is a commonlychosen path and is illustrated in the lowest panel of Fig. 5.6. In other notation this pathis Γ- X - M -Γ.8 CHAPTER 5. DENSITY FUNCTIONAL THEORY STUDY OF THE ION-SIZE EFFECT k x k y E n e r g y [ e V ] (a) k x k y (b) k y MgCaSrBa k x (c) Figure 5.5: (a) The dispersion of the band that crosses E F in CaCuO at k z = 0. Energyis represented by colour from dark red at +2 eV to dark blue − (cid:15) ( k ) whilst dashed lines separate negative (cid:15) ( k ). Note the saddle point in (cid:15) ( k )at the anti-node and that ∂ k (cid:15) ( k ) = 0 results in a peak in the density of states [62]. (b) (cid:15) ( k ) = 0 for CaCuO shown as a solid blue line and for comparison (cid:15) ( k ) = 0 derivedfrom ARPES measurements [63, 107] on Bi Sr CaCu O δ as the dashed red curve. (c) (cid:15) ( k ) = 0 for ACuO with A={Mg,Ca,Sr,Ba} indicated in the panel. Here k z = 0 .
27 sothat all Fermi-contours enclose half the Brillouin zone. .2. EXPERIMENT (cid:15) ( k ), for such restricted regions of the Brillouin zone itis essential to perform the calculation in a different way, involving a non-self-consistentcalculation where the n ( r ) cannot be modified. The optimal procedure is described in theVASP manual [203].In Fig. 5.6 we plot (cid:15) ( k ) along the path discussed for each ACuO . With respect toCaCuO the resulting band structure shows three nearly degenerate bands at Γ with (cid:15) = − . k = (0 , π ) only two are degenerate with one higher in energy. Between k = (0 , π ) and k = ( π, π ) the degeneracy is completely lifted with a strongly dispersiveband crossing the Fermi-level close to k = ( π / , π ) and k = ( π / , π / ) (compare these k values with the Fermi-contour plotted in Fig. 5.5). From other cuprates we know thatundoped cuprates have a charge transfer gap of approximately 2 eV [45], in contrast tothese calculations showing the undoped CaCuO is a metal. This same metallic band isseen in MgCuO , SrCuO and BaCuO .For ease of comparison between the four materials, in the upper panel of Fig. 5.6 weplot the band structures of MgCuO , CaCuO , SrCuO and BaCuO together.With an accurate calculation of (cid:15) ( k ) it is straight forward to calculate the electronicdensity of states (DOS). The results are plotted in the lower panel of Fig. 5.7 and indeedshow a finite DOS at E F confirming that we have calculated a metallic ground state.Nevertheless, the calculated evolution of the DOS with ion-size shows a vHs like featuremoving progressively closer to E F . The vHs feature is a result of the weakly dispersive (cid:15) ( k ) around X clearly visible in the upper panel of Fig. 5.7.There are several reasons why a metallic ground state is to be expected from these LDAcalculations. The LDA often under-estimates band gaps. Furthermore, in the cuprates theCu d x − y orbital (hybridised with the O p σ orbitals) is the closest occupied band to E F as shown in Fig. 2.6. We also know that strong on-site Coloumb interaction leads tolong-range antiferromagnetic ordering of the Cu d x − y electron moments in the undopedcuprates. More generally, the overlap of Cu d x − y and O p σ orbitals means that cupratesare strongly-correlated materials. As such we cannot expect a simple LDA calculation tocalculate a faithful band structure for ACuO . One must at least perform a calculation thatdistinguishes electron spins and such a scheme is called LSDA. Because undoped cuprateshave strong electronic correlations it will also be necessary to add a further correction toour calculations.However, the dispersion reflects the rigidly shifted band structure observed using ARPESat finite doping where the strong-correlations are screened out by mobile carriers. Further,00 CHAPTER 5. DENSITY FUNCTIONAL THEORY STUDY OF THE ION-SIZE EFFECT Γ X (0 ,π ) M ( π,π ) Γ(0 , E n e r g y [ e V ] MgCuO Γ X (0 ,π ) M ( π,π ) Γ(0 , E n e r g y [ e V ] CaCuO Γ X (0 ,π ) M ( π,π ) Γ(0 , E n e r g y [ e V ] SrCuO Γ X (0 ,π ) M ( π,π ) Γ(0 , E n e r g y [ e V ] BaCuO ( π/a ) k x ( π / b ) k y Γ MX Figure 5.6: The calculated electronic dispersion of for each ACuO along certain pathsin the Brillouin zone as illustrated in the lowest panel. The energy is relative to theFermi-energy, E F . .2. EXPERIMENT Γ X (0 ,π ) M ( π,π ) Γ(0 , E n e r g y [ e V ] MgCaSrBa - 2 . 0 - 1 . 5 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0 1 . 50 . 00 . 20 . 40 . 60 . 81 . 0
DOS
E - E F ( e V ) M g C a S r B a
A C u O Figure 5.7: The top panel shows the calculated electronic dispersion around the Fermi-level for (undoped) ACuO where A={Mg,Ca,Sr,Ba} are colour differentiated in the plot,as annotated. Experimentally we know these materials (where synthesisable) are insulatorswith long range antiferromagnetic ordering of the Cu d x − y electron spins, however theband crossing the Fermi-level close to k = ( π / , π,
0) and k = ( π / , π / ,
0) shows that thesecalculations predict our material to be a metal. This is also evident from the density ofstates shown in the lower panel. The strong electronic correlations involving d orbitalsrequires some corrections are made to the DFT method with the LDA.02 CHAPTER 5. DENSITY FUNCTIONAL THEORY STUDY OF THE ION-SIZE EFFECT the saddle-point vHs is known to reside below E F and is crossed in the overdoped regionleading to a change in the Fermi-surface topology [107]. So these calculations do reveal thedispersion features that are known to exist when correlations are suppressed and we maytherefore assume that the systematic changes with ion-size do reflect real band structureevolution. At the start of this chapter we discussed the correspondence between the results of ourDFT calculations and reality. We concluded that our calculations satisfactorily reproducedexperimental lattice parameters where the materials have been reported to be synthesized.The calculated evolution of the DOS with ion-size provides a validation of the hypothesisthat increase of T max c with larger ion size is a result an enhanced N ( E F ). There are twocaveats that must be mentioned;1. The doping level, p : If we are to compare ion-size effects on T max c we should compareACuO at optimal doping. Our calculations however have been performed on un-doped ACuO and the DOS and band structure evolution with doping is not clear.A rigid-band approximation, where E F is modified by doping but not the electronicdispersion E ( k ), is often used in the literature. However we note that some recentwork has called into question the validity of this approximation for the cuprates [207].It is therefore desirable to repeat these calculations at different p .2. Is ACuO a representative cuprate? The infinite layer ACuO represents the fun-damental component of a cuprate superconductor. However the apical oxygen andother metal-oxide layers clearly play some role in the superconducting characteristics.Therefore it is desirable to repeat these calculations on a more complicated systemand suggest the LnA Cu O system, where Ln is a member of the Lanthanide seriesand A={Ba,Sr}, as it supports a wide range of ion-size variations.There is considerable scope for further studies in this direction. With ACuO we havea simple system for which these types of calculations can be performed accurately andrelatively quickly. There is a large range of ion-size that A can take between Mg and Ra.On the other hand, there is not the same quality of experimental evidence to compare withas for e.g. Ln123 or the Bi-based compounds. Furthermore, it is quite possible MgCuO Or conversely, the decrease of T max c with internal pressure .3. DISCUSSION cannot be synthesised in the real world. It would be interesting though toemploy high-pressure/high-temperature synthesis in attempts to stabilise at least the Mgand Ba members.We identify two key avenues of further study;• Vary the electronic doping state. The easily accessible states are p = ( / ) n for aninteger n (because only integer numbers of electrons can be added/removed from the‘unit cell’ in VASP). For intermediate doping states the number of CaCuO groupsin the ‘unit cell’ must be increased.• Vary the external pressure. This would provide an interesting comparison with theion-size variation calculations and could be effected by performing the DFT cal-culations on a unit-cell volume restricted to a value smaller (for positive externalpressure) or larger (for negative external pressure) from the calculated F min valueshown in Fig. 5.4.A similar breadth of calculations could be done for Ln(Ba,Sr) Cu O . In this case wewould be able to more directly compare the results of the computational studies with theexperimental studies. A DFT study of YBa Cu O and YSr Cu O has been carried outusing VASP by Khosroabadi et al. [91] with a purpose of exploring the difference between“mechanical” pressure (external pressure) and chemical pressure (i.e. ‘internal’, or ion-size)- as we hope to do. The range of mechanical pressures they use is −
15 to 15 GPa in 5 GPaintervals.We firstly note that their study does not explore ion-size variations as systematically ascould be hoped. Intermediate Ba:Sr concentrations would require a larger unit cell for thecomputations which would make them significantly more time-consuming. With this inmind LaBa Cu O , LuBa Cu O and YSr Cu O (and perhaps YBaSrCu O ) representsa better set of materials for exploring the ion-size effect in the Ln123 family of materials.Their calculations show quite distinct mechanical pressure dependences of the YBCO(YBa Cu O − δ or Y123) and YSCO structures. The pressure-induced charge transfer(PICT) is qualitatively similar for both YBCO and YSCO, but less dramatic in YSCOattributable to a less compressible z -axis. Furthermore, under chemical pressure (smallerion-size) they calculate an increase in the apical oxygen bond length, which agrees with theexperimental situation summarised in Sec. 2.1.4, and in particular Fig. 2.4, in distinction tothe shorter apical bond length resulting from mechanical pressure. Thus, they identify theCu(2)-O(1) (the apical oxygen bond length) as the primary difference between mechanical04 CHAPTER 5. DENSITY FUNCTIONAL THEORY STUDY OF THE ION-SIZE EFFECT and chemical pressures and the cause of their opposite effect on T c .As mentioned in Sec. 2.1.3, the apical oxygen likely has an important [37, 24, 208, 25,209], but not fully understood, role in the electronic properties of the cuprates. Indeed,the apical oxygen bond length is a parameter in the composite bond valence sum V + thatcorrelates with T max c for all known HTS cuprates, Fig. 2.12 [102]. For example, the authorsof [24, 25] discuss the role of the apical oxygen in terms of its influence on the electronicenergy levels of states in the CuO layer, that being in addition to its role in charge-transferfrom the charge reservoir layer to the CuO layer. For a different perspective, Raghu etal. argue that the charge reservoir layers themselves are the important parameter for T max c [210]. In our closing discussion we suggest a possible unified understanding of the physicalrole of the apical oxygen and charge reservoir layer. We performed DFT calculations on undoped ACuO for A={Mg, Ca, Sr, Ba} to investigateion-size effects the electronic dispersion, (cid:15) ( k ), in this model cuprate system. Where thesematerials have been synthesised we found good agreement between our calculated structuralparameters and the experimental ones. Our calculations show a peak in the density ofstates ∼ T max c via the density of states. hapter 6Two-magnon scattering Summary and Introduction
In this chapter we seek to understand the opposite effects of internal and external pressureon T max c by (i) measuring J while controlling the internal-strain through isovalent ionsubstitution in LnA Cu O and (ii) comparing it to data in literature for the externalpressure dependence of J and T max c . We find no resolution, J and T max c anti-correlatewith internal pressure as the implicit variable and correlate with external pressure as theimplicit variable.Many of the systematics of Ln123 have been explored, but until now not the nearest-neighbour, antiferromagnetic, superexchange energy, J . This is an important energy be-cause it is widely believed that superconductivity in the cuprates is caused by a magneticinteraction. Many theoretical arguments [146, 149, 140, 148] favour a pairing mechanism inthe cuprates of magnetic origin. For a long time it was not clear that magnetic correlationswere present across the entire superconducting phase diagram but recently damped spinwaves (paramagnons) have been detected using Resonant Inelastic X-ray Scattering, ex-hibiting a similar dispersion for many different cuprates across a broad doping range [6]. Inthis case the energy scale of ω B is J [6]. Even more recently, beautiful Raman data of Li etal. shows a correlation, or in their words a ‘feedback effect’, between magnetic fluctuationsas detected by two-magnon scattering and 2∆ as measured by electronic Raman scatter-ing in HgBa CuO δ [211]. J can be deduced from Raman spectroscopy measurements ofthe two-magnon scattering. In the following we present our two-magnon scattering studiesof the model system LnA Cu O y as ion size is varied.For a full introduction to the techniques and theory used in this chapter, see Sec. 2.4.3and 3.8.1. 10506 CHAPTER 6. TWO-MAGNON SCATTERING
Counts w ( c m - 1 ) A + B B A + B B Figure 6.1: Raw Raman spectra from single-crystal SmBa Cu O in A g + B g , B g , A g + B g and B g symmetries. Possible small ‘leakage’ B g and A g + B g signal is observed inthe B g channel as a result of imperfect polarisers and alignment. Experimental set up;514 . ∼ ∼ µ m with a x100 objective(NA=0.90). A 300 lines/mm diffraction grating is used to capture the entire spectrum ina single frame. Cu O Cu O in A g + B g , B g , A g + B g and B g symmetries are plotted in Fig. 6.1. The broad, asymmetric peak around 2600 cm − in B g , A g (to a lesser extent - see Sec. 6.5) but not in B g spectra is the result of two-magnonscattering. The position of its maximum intensity is linearly related to J ; ω max ≈ . J [114].Also visible from these spectra are phonon modes below ∼ − , especially the B g phonon mode at ≈
330 cm − that is due to out-of-phase vibration of the O(2) andO(3) ions in the CuO layer [212], and a two-phonon mode at ≈ − [7].These data, and those reported below, were taken with the 514 . The estimation of J can also be made from a more elaborate fitting procedure [117]. .1. TWO-MAGNON SCATTERING IN LNBA CU O Normalised intensity w ( c m - 1 ) L a N d S m E u G d D y Y b L u
D e c r e a s i n g L n i o n - s i z e
Figure 6.2: Raman spectra of LnBa Cu O showing a systematic increase in the B g two-magnon peak position as the Ln ion-size is decreased (each Ln is indicated in the legend).Data have had a constant fluorescence contribution subtracted and have been normalizedby the peak intensity.power ∼ ∼ µ m with a x50 (NA=0.75) or x100 objective(NA=0.90). A 300 lines/mm diffraction grating is used to capture the entire spectrum ina single frame of the CCD.In Fig. 6.2 we plot normalised Raman spectra for the entire LnBa Cu O series, focusingon just the ω -range of two-magnon scattering. Quite nicely, we see a systematic increase inthe two-magnon peak frequency as the Ln ion size decreases. This is almost entirely becauseof a variation in the structural parameters: smaller Ln ion-size results in shorter in-planebond lengths, particularly on the CuO layers [40], and thus a shorter superexchange path-length. This is the “internal pressure” effect which can also be seen altering the phononmode frequencies, see for example Fig. 8.6. J is extremely sensitive to the superexchangepath-length in the transition metal oxides [213] and to a lesser extent in the cuprates [214].The shorter in-plane bond lengths result in increased overlap between Cu(2) d x − y andO(2,3) p σ orbitals that increases the superexchange energy.This previous statement can be illustrated by considering an expression for J based onthe two-band Hubbard model; J ≈ t pd ∆ − [214, 45] where ∆ is the charge-transfer gapenergy and t pd the hopping integral between Cu d and O p orbitals. t pd is itself inversely08 CHAPTER 6. TWO-MAGNON SCATTERING
I n c r e a s i n g L n i o n - s i z eY bD yG dE uS mN d (cid:1) max (cm-1) D P e f f ( G P a ) L a
Figure 6.3: The B g Raman two-magnon peak position, ω max , for each Ln123 sampleplotted against the effective internal pressure using ∆ P eff = − B. ∆ V /V where B = 78 . V = V − V is referenced toLa123.proportional to the superexchange path-length, t pd ∼ d − n with 2 . < n < . J is shown to be strongly dependent on the superexchange pathlength.We can estimate the effective internal pressure, ∆ P eff , from the change in unit cellvolume, ∆ V = V − V where V is referenced to La123, using ∆ P eff = − B. ∆ V /V . B = 78 . et al. [215].In Fig. 6.3 we plot ω max against ∆ P eff . Again, the observed positive gradient can beunderstood as an increased exchange interaction arising from increased overlap betweenCu(2) d and O(2,3) p orbitals.If J is dependent on structural parameters, then we can quantify the shift in J withLn size using an area Grüneisen parameter , γ A = − ln( J/J )ln( A/A ) (6.1)where A is the area of the basal (CuO ) plane. Note the Grüneisen parameter is unit-lessand so is not sensitive to the constant relating the maximum two-magnon peak frequency, ω max to the anti-ferromagnetic exchange constant J . It has been suggested from uniaxial high pressure studies and substrate strained-lattice studies thatCuO plane compression is the critical parameter [11]. .1. TWO-MAGNON SCATTERING IN LNBA CU O - 0 . 0 4 - 0 . 0 3 - 0 . 0 2 - 0 . 0 1 0 . 0 00 . 0 0 00 . 0 2 50 . 0 5 00 . 0 7 50 . 1 0 00 . 1 2 5 L n B a C u O L a C u O N d ln( w / w
0) or ln(J/J0) l n ( A / A ) L aS mE uG dD yY b - 0 . 1 5 - 0 . 1 0 - 0 . 0 5 0 . 0 00 . 00 . 10 . 20 . 3
Y B a C u O ln(J/J0) l n ( A / A ) Figure 6.4: To determine the Grüneisen parameter for our Ln123 system and for externalpressure data on La214 from [214], we plot the natural log of the CuO basal plane area, A , against the natural log of J where both values are referenced to La123 (for Ln123) ortheir value at atmospheric pressure (for La214). We find γ A = 3 . ± . γ A = 3 . ± . J is consistent withthat of internal pressure. Inset: Data for YBa Cu O . from [216]. The dotted line is aguide to the eye. Again, the effect of external pressure on J appears consistent with thatof internal pressure.The area dependence of J is plotted in log scale in Fig. 6.4, where A is calculated fromGuillaume et al. [40] and the zero subscript refers to La as reference. From this plot wefind γ A = 3 . ± . γ A = 3 . not included in the fitting whereas we obtain γ A = 2 . γ A is not constant acrossthe range, as can be seen by the non-linearity of the plot and as indicated by externalpressure studies. J It is also instructive to plot the dependence of J on absolute CuO area, A . These data areplotted in Fig. 6.5. Plotting the data in this way allows us to readily compare the effect ofinternal and external pressure on J . Thus in Fig. 6.5 we also show the effect of external10 CHAPTER 6. TWO-MAGNON SCATTERING P = 0 L a 2 1 4 L n 1 2 3 N d J (meV) A ( (cid:1) ) L aS mE uG dD yY b J (meV) A ( (cid:1) ) Figure 6.5: The basal area dependence of J determined from our two-magnon scatteringstudies on LnBa Cu O (red squares) where internal pressure (or ion size) is the implicitvariable. This is compared with the same plot for La CuO where external pressure (0to 10 GPa) is the implicit variable [214]. Remarkably a single behaviour for J ( A ) ispreserved across a 50% change in J irrespective of whether the effective pressure is internalor external. Inset: J versus A for pressure-dependent two-magnon scattering data forYBa Cu O . to 80 GPa [216] (green diamonds) compared with our ion-size-dependentdata for LnBa Cu O (red squares). Again external and internal pressures appear to havequantitatively similar effects on J . The dashed lines are guides to the eye.pressure on J in the related system La CuO [214] (blue diamonds) ranging from ambientpressure to 10 GPa, as annotated.Remarkably, the dependence of J on A is preserved across the entire range includingtwo quite structurally disparate cuprates, irrespective of whether the pressure is internal or external . Collectively these cover a 50% increase in the magnitude of J arising fromsimple structural compression.We also plot these data of Aronson et al. on a log-log scale to further compare withthe ion-size data, Fig. 6.4. Using Equation 6.1 we determine γ A = 3 . ± . CuO under external pressure (in this case the zero subscript in Equation 6.1 refers to the valueof J and A at atmospheric pressure). Within uncertainties, this value agrees with what wefind for the internal pressure ion-size effect for Ln123. .2. IS J RELATED TO T C ? Cu O . [216] and this is plotted by the green diamonds in the inset to Fig. 6.5along with our data for Ln123 (red squares) as well as in the inset to Fig. 6.4. The residual0.2 oxygens in the chain layer introduce uncertainties around possible pressure-inducedcharge transfer and we are therefore cautious in the use of this data. Nonetheless, theyreveal a trend which is fully consistent with the ion-size effect for Ln123.Furthermore, Kawada et al. have shown that the pressure dependence of J for a varietyof cuprate-oxides is very similar [173]. This means we can be even more confident thatinternal and external pressure have the same effect on J in the cuprates.In an attempt to improve on the available data for two-magnon scattering under exter-nal pressure we undertook Raman scattering studies using a Diamond Anvil Cell to achievehigh pressures. Although we were able to measure the temperature dependence of the 150cm − , 310 cm − and 475 cm − phonon modes, which have already been extensively studied[217], we could not detect two-magnon scattering from our sample in the Diamond AnvilCell. These are tricky measurements. See Sec. 3.8.2 for more details.Nevertheless, from Fig. 6.4 and Fig. 6.5 we can conclude that external pressure has aquantitatively similar effect to internal pressure on J in the cuprates. This is an importantresult.Our conclusion that external pressure has a similar effect to internal pressure on J inthe cuprates contrasts with what Aronson et al. find comparing their data to Cooper [218]. J related to T c ? Our motivation for these two-magnon measurements was that it is widely believed that amagnetic interaction between electrons is the pairing mechanism leading to Cooper pairsand superconductivity. To leading order the energy scale of these magnetic interactions, ω B , is set by J [6] and so we can use these Raman measurements to characterise magneticinteractions as a function of ion size. In a fuller treatment of magnetic interactions howeverwe must also consider longer-range interactions, see Sec. 9.3. Recalling the weak-coupling, d -wave BCS equation for T c k B T mf c = 0 . (cid:126) ω B exp [ − / ( N ( E F ) V )]we might expect our systematic shift decrease of J with ion-size to be reflected in the valueof T c . Even when T c is determined by solving the k -dependent gap equation it will be12 CHAPTER 6. TWO-MAGNON SCATTERING
Figure 6.6: T max c is plotted versus J for single crystals of LnBa Cu O (red squares) andfor YBa − x Sr x Cu O (blue diamonds) with x = 0 . .
0. This reveals a systematicanticorrelation of T c with J where ion size or “internal pressure” is the implicit variable.Greens squares show T max c versus J under external pressure, revealing a behaviour whichis orthogonal to that for internal pressure.determined to leading order by J .We thus plot in Fig. 6.6 T max c versus J for the Ln123 single-crystal series (red squares).We have used T max c = 98 . T max c = 96 K for Nd123 [97] as these arethe highest reported values of T max c in these compounds (where Ln occupation of the Basite is minimised). Contrary to expectation from Equation 2.7, T max c anticorrelates with J when ion-size or “internal pressure” is the implicit variable.To push out to higher J values, we repeated the Raman measurements on a c -axis-aligned thin film of YBa . Sr . Cu O and on individual grains of polycrystalline YSr Cu O ,prepared under high-pressure/high-temperature synthesis by Edi Gilioli [36]. The data isplotted by the blue diamonds in Fig. 6.6 and the raw data can be seen in Fig. 6.7.Significantly, the same anticorrelation between T max c and J is preserved, and in this See Sec. 3.2.4 for details of our thin-film synthesis process. .2. IS J RELATED TO T C ? cts. w ( c m - 1 ) A + B A + B B B Intensity (cts.s-1) w ( c m - 1 ) m W )x 5 0 o b jc r o s s p o l a r i s e d
Figure 6.7: (Left) Raw Raman scattering for epitaxial thin films of YBa . Sr . Cu O madeusing the MOD-spin coating method on RABiTs substrates, Sec. 3.2.4. (Right) Raw micro-Raman scattering (x50 microscope objective) from a single crystallite in a polycrystallinepellet of YSr Cu O . Because it is not possible to check B g and B g sample geometrieson the single crystallite, we cannot be as confident this is true two-magnon scattering.However, the peak does have the expected asymmetry, the characteristic disorder-mode at ∼
600 cm − and sharply reduces in intensity with longer exposure time - probably due tolaser heating of the grain.case now out to a more than 60% increase in the value of J . This is a very large increasein J and it is surprising indeed to not be reflected in the value of T max c if indeed magneticinteractions alone set the energy scale for pairing.Furthermore, to this plot we add data summarising the effect of external pressure on T max c and J in YBa Cu O (green squares: for 1 bar and 1.7, 4.5, 14.5 and 16.8 GPa).The shift in J with pressure is taken from the inset to Fig. 6.5 and the values of T max c at elevated pressures are from references [94, 96]. We discuss the analysis to obtain thesedata points in more detail in the following paragraphs.Under ambient pressure T max c for YBa Cu O y is close to 93 K. To find T max c at higherpressures it is important to note that the application of pressure has two effects: (i) toraise the magnitude of T max c and (ii) pressure-induced charge transfer which increases thehole doping state as was discussed in Sec. 2.3.1 and summarised in Fig. 2.11. Thus todetermine T max c at elevated pressure one must investigate underdoped YBa Cu O y . Thecloser is the system to optimal doping (on the underdoped side) the lower the pressureneeded to attain optimal doping and T max c rises little above its ambient pressure value.14 CHAPTER 6. TWO-MAGNON SCATTERING
But as doping is decreased, higher pressures are required to reach optimum doping and T max c is raised further. This applies until the 60 K plateau is attained around p ≈ . T max c are then encountered [96].Unfortunately no report has yet been presented of these systematics in small increments ofdoping. This is a gap that needs to be filled. However there are enough reports at severaldoping states to confirm the pattern. We then have the following data: ( P = 1 bar, T max c = 93 K); ( P = 1 . T max c = 93 . P = 4 . T max c = 97 . P = 14 . T max c = 107 . P = 16 . T max c = 107 K [96]).For each of these pressures the value of J is determined for YBa Cu O y from the insetto Fig. 6.5, interpolating on a simple proportional basis. We obtain: ( P = 1 bar, J = 106 . P = 1 . J = 107 . P = 4 . J = 109 . P = 14 . J = 114 . P = 16 . J = 116 . T max c areplotted versus J (with P as the implicit variable) by the green squares in Fig. 6.6.Fig. 6.6 now reveals that the shift in data under external pressure is orthogonal to thatfound under internal pressure. This, again, reflects our central paradox and shows that theobserved shifts in J simply do not correlate with T max c .This is the second main result from this chapter. We have two main results to discuss; (i) external pressure and internal pressure havequantitatively similar effects on J and (ii) we find J to anti-correlate with T max c whereinternal pressure is the implicit variable, whereas a correlation is observed when externalpressure is the implicit variable.There are several reasons why the observed shift of J with ion size in Fig. 6.2 is not adoping effect:1. For low oxygen concentration in the chain layer of Ln123, δ ≈
1, the CuO planes areundoped and, in fact, insensitive to small changes in the oxygen concentration [39].2. Recent measurements of the pressure dependence of the thermopower of deoxy-genated Y − x Ca x Ba Cu O show no evidence of pressure-induced charge transfer[219]. Typically when δ < δ ≈
1) the thermopower becomes pressureindependent. .3. DISCUSSION δ < δ the doping on the CuO planes increases with decreasing ionsize in the Ln123 system [221]. Given that J decreases as doping is increased [222, 7],this would have the effect of J decreasing with decreasing ion size, the opposite towhat we measure.Our results appear to differ from Ofer et al. [52]. They correlate T max c with the Neéltemperature, T N , in the (La − x Ca x )(Ba . − x La . x )Cu O y system (CLBLCO !). Usinga model, J ∝ T N ln( α ), they relate T N to J . Here α is a measure of magnetic anisotropywhere α = 0 is the 2D limit. After performing a scaling analysis that utilises estimatesof α from temperature-dependent local-magnetisation data, they are able to collapse thephase diagrams for x = 0 . , . , . , and 0 . J controls T N , the spin-glass transition temperature and T c . In other words, they argue J and T max c correlate.Their work is very interesting. However, T N (what they directly measure) is not directlyrelated to J , thus requiring a model to estimate the anisotropy as mentioned above. More-over, this complex co-substituted sexenary system has large Nuclear Quadrupole Resonance(NQR) linewidths reflecting a high degree of disorder. It is our view that two-magnon Ra-man scattering in our model Ln123 system more direct and reliable in its implications.On the other hand, several groups that have also observed J does not correlate with T max c [223, 224, 225, 115] although none have shown, as we have, a systematic anti-correlation between J and T max c with internal pressure is the implicit variable. Now there are several interpretations of the internal pressure data presented in Fig. 6.6and this possible anti-correlation between T max c and J ;1. We must first note that we are of course comparing two different doping states. J measured at p = 0 and T max c at optimal doping (possibly p = 0 . et al. [7] show a roughly linear decrease of J with doping as well as asuppression of two-magnon scattering intensity. Perhaps the doping dependence of J for La123 is stronger than say for Lu123 so that, at between zero doping and optimaldoping they have the same J and by optimal doping the ordering of J has reversed sothat T max c indeed increases with J . This hypothesis is sketched immediately below;See Sec. 1 for details of our measurements investigating this possibility.16 CHAPTER 6. TWO-MAGNON SCATTERING pJ optimal p Lu123La123
On one level, if J is primarily dependent on the superexchange path-length, as ev-idence suggests Sec. 6.1.1 and [226], it is unlikely that this is the case. There is nosuch ‘cross-over’ behaviour in the superexchange path-length with doping in Ln123(see Sec. 2.1.4). Nevertheless this is an important consideration so we set out to trackthe doping dependence of the two-magnon scattering in Nd123, Eu123 and Yb123single crystals.Our results are unexpected. Representative raw spectra showing possible two-magnonscattering at different doping levels is shown in the left panel of Fig. 6.8 for Nd123.The possible two-magnon peak at finite doping does not uniformly lower in energyas expected nor does its intensity dramatically weaken at higher doping levels (asindicated by T c ). Furthermore the profile of the peak at finite doping is different tothe undoped peak profile.If we nevertheless plot the position of the ‘two-magnon’ peak maximum intensity, ω max , for B g symmetry as a function doping for Nd123, Eu123, Yb123 we obtainthe right-panel in Fig. 6.8. From these data, it would appear possible that there isa cross-over in the magnitude of J with doping between Nd123 to Yb123, althoughthere is considerable scatter. Because of this scatter and some issues discussed above,I am very cautious about the validity of these estimates of J .However, we must also cite the recent, puzzling, Resonant Inelastic X-ray Scattering(RIXS) study of le Tacon et al. [6]. This study shows, in contradiction to the Ramanresults [7], paramagnons with spectral weight similar to the undoped cuprates and anessentially doping independent J . To our knowledge this surprising RIXS result, andits discrepancy with the Raman results, is not well understood. If we are to believe J is in fact (essentially) doping independent from p = 0 to p > .
16 - and indeedNd123O6, Nd123O7, Y123O6.6, Y123O7 are materials used in le Tacon’s study - itonly further validates our anti-correlation between T max c and J when ion-size is theimplicit variable. .3. DISCUSSION Intensity w ( c m - 1 ) T c = 0 K T c = 4 7 . 5 K T c = 5 2 K T c = 6 3 . 5 K T c = 7 0 K w max (cm-1) T c ( K ) N d 1 2 3 b 1 g E u 1 2 3 b 1 g Y b 1 2 3 b 1 g B Figure 6.8: (Left) Doping dependent B g Raman spectra of Nd123. The 458 nm laserline is used for all spectra except the undoped Nd123 spectrum (top). Note that once thecrystal loses tetragonal symmetry with doping (due to a preferential direction of oxygenordering on the chains) it is now no longer pure B g and instead the lower A g symmetry.(Right) The doping dependence of the supposed B g two-magnon peak positions for Nd123,Eu123 and Yb123. It appears possible that there is a cross-over in the magnitude of J withdoping between Nd123 to Yb123. However, it is also not expected that J increases withdoping and together with a strangely shaped ‘two-magnon scattering peak’ shown in theleft panel, suggests these rather erratic data may not be actual two-magnon scattering.2. J might set the energy scale of an electronic order that competes with supercon-ductivity. The most likely candidate here is the pseudogap as there is evidence frominelastic neutron scattering [87] and specific heat data [58] that the pseudogap energyscale is set by J . However, when we consider the effect of external pressure on J and T c shown in Fig. 6.6, this interpretation becomes less likely. For example, if wetook the view that the suppression of T c with ion-size was due to an enhancementof the pseudogap phase (or, similarly a Spin Density Wave or stripe phase) from anincreasing J , we would need to explain why both T c and J increase under externalpressure.3. Perhaps J does set energy scale for ω B , but other physical properties are affected byion substitution which have a larger effect on T c . We consider this the mostly likelyinterpretation of Fig. 6.6. Specifically, we consider the electronic density of states(Sec. 6.3.2) and dielectric properties (Sec. 9.3) in the following chapters.4. Disorder due to our ion substitutions could be responsible for the suppression of T c and so we are not comparing ‘clean samples’. We consider this unlikely for two18 CHAPTER 6. TWO-MAGNON SCATTERING reasons. (i) La123 and Nd123 are difficult to synthesize without disorder, but havea T max c higher than the most pure Y123 . (ii) With our Bi2201 studies we found T c correlated more closely with ion size than disorder. Nevertheless, as disorder is anoft-cited explanation for unconventional results, to test this possibility we carried outextensive µ SR experiments. This is the topic of the next chapter.We note that B g two-magnon scattering only probes nearest-neighbour magnetic inter-actions [117] while recent RIXS studies [227] reveal the presence of extended interactionsinvolving next-nearest-neighbour, and higher, hopping integrals, t and t [24]. The as-sociated additional exchange interaction is about half the magnitude of the changes thatwe have imposed by ion-size variation. It is remotely possible that inclusion of extendedinteractions might reverse the systematic anti-correlation between T max c and J shown inFig. 6.6. However, it is our view that variations in t and t will have a stronger influencevia N ( E F ) by distorting the Fermi surface and shifting the vHs, presumably in a similarway to that found in the ACuO system we studied using DFT. T c superconductors Given the anti-correlation between J and T max c across a wide range in our model cupratesystem where internal pressure is the implicit variable, it is interesting to compare with theanalogous energy scales in the low-temperature superconductors: the Debye temperature [228, 199], Θ D , and T c . In the McMillan formula T c and the Debye temperature are relatedby [228] T c = Θ D .
45 exp " − . λ ) λ − µ ∗ (1 + 0 . λ ) (6.2) µ ∗ accounts the screened, or ‘renormalised’, Coulomb repulsion between electrons . λ = 2 R ∞ Ω − α F (Ω) dΩ is the electron-phonon coupling constant which is the integral ofthe phonon density of states, F (Ω), weighted by the square of the electron-phonon couplingmatrix, α . There are various modifications to Equation 6.2 found in the literature [199].The results are interesting. As shown by Fig. 6.9, T c and Θ D in general also anti-correlate , even in alloy systems. T c = 94 . Superconductivity in these materials is described by the BCS(+Eliashberg) theory with phonon-mediated pairing. A value difficult to experimentally measure. . . I use the notation T c rather than T max c as, in general, the T c of LTS materials is not optimised through .3. DISCUSSION B eA lZ n a - G aC dI n S nH gT l P b T iVZ rN b M o R uH fT a W R e O sI r a - L a b - L aT h Tc (K) Q D ( K ) M o ( 1 - x )
R e x : x = 5 %x = 1 0x = 2 0x = 3 0x = 4 0x = 5 0 V ( 1 - x ) C r x : x = 1 0 %x = 2 0 x = 2 5 x = 4 0 x = 5 0 Tc (K) Q D ( K ) Figure 6.9: T c plotted against Θ D for (Top) the elements known to be superconductingat ambient pressure (in fact, the second lightest element known to be superconducting atambient pressure, Be, with Θ D = 1390 K and T c = 0 .
026 K, is not plotted) and (Bottom)alloy systems. The Debye temperature is a prefactor in the McMillan formula for T c ,Equation 6.2, and so such a consistent anti-correlation between the two is noteworthy. Theelectron-phonon coupling constant is the most important additional parameter needed todescribe the T c . Data is from [228, 21].20 CHAPTER 6. TWO-MAGNON SCATTERING
Indeed, it is the electron-phonon coupling constant, λ , that has been found the mostimportant additional parameter needed to describe the T c values observed [199] (althoughan anti-correlation T c and Θ D over such a wide range is noteworthy).Given T c and Θ D anti-correlate in LTS materials, should we be surprised that J and T max c anticorrelate in our model LnA Cu O system if there is a magnetic pairing mecha-nism? Perhaps not, but internal and external pressure have opposite effects on T c , while wefound they have quantitatively the same effect on J . If we look at the analogous situationfor some LTS materials we do not find this opposite behaviour. For example, turning to theMo-Re alloy system: ∂T c ∂P < ∂ Θ D ∂P ≈ Θ D B > B = 230 GPa is the bulk modulus. Thus, underexternal pressure, Θ D increases as T c decreases similarly to the effect of alloying (althoughthe gradient is greater with external pressure as the implicit variable). In other words, wedo not see the opposite behaviour between internal and external pressure as we do withHTS cuprates. In summary, we have sought to understand the opposite effects of internal and externalpressure on T max c by (i) measuring J while controlling the internal-strain through isovalention substitution in LnA Cu O and (ii) comparing it to data in literature for the externalpressure dependence of J and T max c . We find no resolution: J and T max c anti-correlatewith internal pressure as the implicit variable and correlate with external pressure as theimplicit variable. It is therefore most probable some other physical property plays a moredominant role in setting the value of T max c . In the following chapters we explore furtherthe various interpretations and consequences for the results presented above. A g and B g spectra As mentioned in the theoretical introduction (2.4.3), the cuprates have the non-conventionalfeature of showing two-magnon Raman scattering in the A g scattering geometry. An ex-ample of this is shown in Fig. 6.10 for Sm123 (left-panel) and for the entire Ln123 se- doping as it is in the cuprates (and pnictides). Hence while T max c is the material-specific value of interestfor the cuprates, T c is the material-specific value of interest for LTS. .5. APPENDIX: INTERESTING EXTENSIONS - A G AND B G SPECTRA A g is determined by subtracting the B g spectrum from the A g + B g spectrum or the B g spectrum from the A g + B g spectrum. For all spectraplotted in Fig. 6.10 we find A g + B g - B g = A g + B g - B g . Counts w ( c m - 1 ) S m 1 2 3 A 1 g [ ( A 1 g + B 2 g ) - B 2 g ] S m 1 2 3 A 1 g [ ( A 1 g + B 1 g ) - B 1 g ] S m 1 2 3 B 1 g counts w ( c m - 1 ) L a 1 2 3 S m 1 2 3 Y b 1 2 3 L u 1 2 3
Figure 6.10: (Left) SmBa Cu O A g spectra determined two different ways; firstly bysubtracting B g spectra from A g + B g spectra, secondly from subtracting the B g spectrafrom the A g + B g spectra. Note the two overlap almost perfectly - this nice overlap is seenin most, but not all of the Ln123 family. Also shown, for comparison, is the B g spectrumand a line marking the estimated peak maxima, ω max (which coincides for B g and A g spectra). (Right) A g spectra for the Ln123 series. B g Raman two-magnon scattering contains information of the nearest-neighbour su-perexchange energy, J , and so it would be nice to know if something in the Raman-spectrathat tells us about further-neighbour interactions. For example, perhaps A g spectra maycontain information about longer-range interactions? One published result suggests thismay be possible. It is the early work of Singh et al. [117] (a great read). They argue A g two-magnon scattering (and B g scattering) arise from diagonal nearest-neighbour (next-nearest-neighbour) excitations (represented in the t - J model by including the t hoppingparameter). These are Raman active due to quantum fluctuations. However, they onlyuse their model of A g (and B g ) scattering as a further estimate of J . On the other hand,an exact numerical treatment of the 2D Hubbard model by Tohyama et al. [116] finds A g and B g scattering to be a consequence of treating fully both spin and charge degrees offreedom.It may be fruitful to pursue this line of investigation further. As I understand, one takesthe Hubbard Hamiltonian on a 2-D square-lattice with nearest neighbour hopping only andthen project out the spin part of the Hamiltonian [117, 114, 112]. Doing this one gets the22 CHAPTER 6. TWO-MAGNON SCATTERING
Heisenberg Hamiltonian which can then be solved to find the Raman response using theLoudon-Flurey Hamiltonian to describe the interaction of light with the spins. Thus, apossible approach to this question is to add a next-nearest-neighbour interaction on our 2-Dsquare lattice (i.e. spins along a diagonal interact via a hopping integral, usually denoted t in a Hubbard Hamiltonian). Perhaps it would then be possible to calculate what effect thishas on the Raman-response from two-magnon scattering following a similar procedure?In any case, elucidating the interplay between longer-range interactions and two-magnonscattering would be useful for studying other materials as well. Counts w ( c m - 1 ) Counts w ( c m - 1 ) L a 1 2 3 N d 1 2 3 S m 1 2 3 G d 1 2 3 D y 1 2 3 L u 1 2 3
Figure 6.11: (Left) SmBa Cu O B g spectrum. (Right) B g spectra for the Ln123 series.There is also an interesting ‘edge’ feature in the B g spectra at ω ∼ − whichis shown in Fig. 6.11. At first we suspected this was a fluorescence signal because itdisappears when switching from the 458 nm to the 514.5 nm laser. There is also no reportedexplanation of this feature, again leading us to believe it is not interesting. However frompersonal correspondence with Prof. Sugai it appears this feature may contain interestinginformation about the sample. A paper on this feature may be published soon by hisgroup . We also see this ‘edge’ feature at finite doping levels. The data in Fig. 6.11 showthis ‘edge’ moves to higher ω as the ion size decreases for undoped samples. It is difficult tosay what this means given the origin of this Raman feature is not known, but it certainlywould be an interesting aspect to look into further. Unfortunately he did not reveal anything else! hapter 7Muon Spin Rotation and thesuperfluid density
Summary
From our superfluid density measurements of YBaSrCu O − δ by µ SR we find the niceresult that this is a boring, well-behaved material! The superfluid densities, λ − , areconsistent with those previously reported for pure and Ca-doped YBa Cu O − δ [137, 83].These data are plotted in Fig. 7.6. Furthermore the suppression of λ − due to Zn dopingis fully consistent with that found for YBa Cu O − δ [135] as shown in Fig. 7.9. Theimmediate conclusion is that disorder does not play a significant role in YBaSrCu O − δ as compared with Y123. Consequently, the lower T max c seen in this compound, and thosewith lower Sr content, is a real ‘ion-size effect.’ Its systematic variation should thereforeprovide important insights into the SC mechanism and ultimately to the prediction of themagnitude of T c . Motivation
In such complicated, multi-ion compounds as the cuprates, the effect of disorder mustalways be considered. There may be deviations from perfect stoichiometry, e.g. sayY . Ba . Cu . O . rather than YBa Cu O , or partial substitution of an ion, e.g.Y . Ca . Ba Cu O . Both are types of compositional disorder and belong to the art of thematerials scientist, who may also have an eye on opportunities for flux pinning. A goodreal example of such is the Y − x Ca x Ba − x La x Cu O cuprate, where the NMR line-widthsbroaden significantly as x is increased [230]. There may also be distortions of unit cell12324 CHAPTER 7. MUON SPIN ROTATION AND THE SUPERFLUID DENSITY symmetry, bond lengths or bond angles which may be caused by ion-size mismatch. Aninstructive attempt to classify types of disorder in the cuprates and quantify their effectswas made by Eisaki et al. [181].There remains the possibility that the decreasing T max c with ‘internal pressure’ inLnA Cu O y is primarily a disorder effect, as is the orthodox view of the situation for Bi2201(Chapter 4). Alternatively, the suppression of T c is possibly the result of the growth ofa competing stripe-ordered phase e.g. because of the larger J [57, 2]. We consider bothscenarios unlikely, however this can and should be tested.Therefore we seek to ascertain whether the suppression of T max c in our model LnA Cu O y system is a disorder effect. The superfluid density is very sensitive to disorder in d -wavesuperconductors (e.g. see Fig. 2 of reference [135]), as well as to stripe-order, making it anideal quantity to measure to test this hypothesis. If T max c were reduced because of disorderone would see a concurrent, and larger, reduction in the superfluid density. Recall that the basic principles of the µ SR technique were discussed in Sec. 3.7 and thesuperfluid density in Sec. 2.6. The experiment was performed on the GPS beam line atthe PSI muon source (Villigen, Switzerland). Because there is a muon-spin rotator magnetinstalled on this beam line, it is capable of both transverse and longitudinal geometries,see Fig. 7.1 (which is a reproduction of Fig. 3.8).A coincidence counting method is used. An incoming muon starts the timer. If apositron is detected within 10 µ s, the event and where it was detected, is recorded. Ifinstead a second incoming muon is detected within 10 µ s, then the next two positrondetections must be discarded. Otherwise it could not be determined from which muon thedetected positron came and hence how long those muons spent precessing in their B loc before decaying. The remarkably intense GPS line has this luxurious problem and so weuse shutters to reduce the rate of muons incident on the sample.Our samples are mounted between the prongs of a silver holder, held in place by thintape. This is sketched on the bottom of Fig. 7.1 where the black circle represents thesample. The tape is thin enough that the muons have too much energy to stop inside it.The sample is cooled by a He-flow cryostat and the temperature monitored by a nearbythermocouple.In zero-field (ZF), relaxation of the muon polarisation results purely from local magnetic .1. EXPERIMENT Sample and holder DownUp B a ck F o r w a r d F o r w a r d B ext Muon spin polarisation (with spin rotator on) Transverse-Field muon spinprecession Positrondectectors p + Coincidence timerstart
Figure 7.1: A schematic diagram of the experimental set-up at the GPS beamline (usedto make our µ SR measurements). Normally, the spin of the muon is anti-parallel to itsmomentum, however, GPS is equipped with spin-rotating magnets which flip the muonspin near perpendicular to its momentum - as annotated. A magnetic field up to 6000G (0 . t between the Up andDown detectors. t = 0 is when a muon is detected passing through the coincidence timerin the Front positron detector. At the bottom a sample (black circle) is sketched taped(light grey square) to a silver holder.fields. Omni-present nuclear magnetic moments are generally weak and isotropic in whichcase a ZF asymmetry plot is well described by the Kubo-Toyabe function, Equation 3.2[169]. An example of such is shown in Fig. 7.2. We perform zero-field µ SR measurementsto determine if there is additional static, or dynamic , magnetism in our samples. Thesignature of such magnetism is an initial exponential suppression of the asymmetry, orit may be so rapid (large field distribution, fast dynamical processes) that it is insteadinferred by a low initial asymmetry, which should be A ≈ .
21 from our calibration on Agfoil. ZF measurements are performed in longitudinal geometry which at the GPS beamlinemeans measuring with the up/down detectors if the muon spin rotator is on Fig. 7.1.Generally, we fit TF data using a single-frequency cosine oscillation modulated by asimple Gaussian dephasing, Equation 3.3, discussed in Sec. 3.7. The fits are excellent and With respect to the time scale relative to the muon - which is between ∼ ns at GPS (instrumentalresolution limited) and ∼ µ s. CHAPTER 7. MUON SPIN ROTATION AND THE SUPERFLUID DENSITY
5 K K u b o f i t
Asymmetry (%)
T i m e ( m s ) ( b ) ( c )( d ) ( a ) U n d e r d o p e d T c = 3 0 KU n d e r d o p e d T c = 8 0 K Figure 7.2: Typical Zero-Field µ SR data (red circles with grey error bars) onYBaSrCu O − δ ; In the first column are data for UD80K ((a) and (b)), in the secondcolumn data for UD30K ((c) and (d)). In the first row are data taken at 90 K, in thesecond row data are taken at 5 K. Each data set can be fitted well to the Kubo-Toyabefunction (Equation 3.2), as shown by the blue lines. In addition, the initial asymmetryis close to the calibrated value, A ≈ T = 5 K and T = 90 K. We conclude that these samples are do not have significantinternal magnetic fields. .1. EXPERIMENT χ ≈ . P X ( t ) = A exp ( − Λ t ) cos( γ µ B loc t + δ ) (7.1)or to the sum of two Gaussian components.To perform the fits we use the freely-available WiMDA software package written byFrancis Pratt [231]. We plan to study NdBa Cu O y (Nd123), YBaSrCu O − δ and YSr Cu O − δ . These, alongwith previous measurements on YBa Cu O y [39, 232, 135, 233], span the internal pressurerange in our model Ln(Ba,Sr) Cu O y system. With our beam-time allotment we had timefor µ SR measurements of six YBaSrCu O − δ samples and three YSr Cu O − δ samples.They are listed, along with a short reference code, in Table 7.1.The polycrystalline YSr Cu O − δ samples were grown by our collaborator Dr. Gilioliat IMEM in Parma, Italy. Dr. Gilioli has refined the technique of YSr Cu O − δ synthesiswhich involves specialised high-pressure, high-temperature equipment [36]. The first sam-ple was an as-prepared, overdoped T c = 60 K sample and is labelled Edi35. The second,with T c = 66 K, had been slightly de-oxygenated by annealing in 10 bar O while slowcooling from 400 ◦ C to 300 ◦ C and is labelled Edi22. Unfortunately, the YSr Cu O − δ werefound to contain a considerable fraction of magnetic impurities (see the ZF measurementssection 7.2.1). In an attempt to dissolve the KCl out of the pellet, which we thought may bethe magnetic impurity, we ground pellets of the T c = 66 K sample (Edi22) to a powder indry Methanol (KCl is soluble in Methanol). We then poured the mixture onto “Kim Wipe”tissue paper, flushed with more Methanol, and then evaporated excess Methanol with ahair drier. The YSr Cu O − δ recovered from this process would have spent 10 minutes inMethanol. The powder, which is named Edi22w , was then immediately re-measured withmuons.Unfortunately, there was insufficient time to take measurements on Nd123. InsteadNd123 and further YSr Cu O − δ measurements are the subject of a new µ SR proposal,now awarded at the time of writing.28
CHAPTER 7. MUON SPIN ROTATION AND THE SUPERFLUID DENSITY
Code Material formula T c (K) S(295K)( µ V.K − ) p OD80K YBaSrCu O
80 -1.0 0.184OP84K YBaSrCu O . O .
80 2.8 0.135UD30K YBaSrCu O .
30 20.0 0.0751% Zn OP77K YBaSrCu . Zn . O .
77 -0.55 0.1673% Zn OD61K YBaSrCu . Zn . O .
61 -0.50 0.1603% Zn OP62K YBaSrCu . Zn . O .
62 1.50 0.160Edi35 YSr Cu O y
62 -6.1 -Edi22 YSr Cu O y
66 -9.5* -Edi22w YSr Cu O y
64 - -Table 7.1: A table of the samples studied by µ SR along with the code used to refer tothem throughout this section. See Sec. 3.3.1 for more details on sample preparation andannealing. *We consider this measurement less reliable due to poor electrical and thermalcontact between the sample and probe.
We perform zero-field (ZF) measurements ( B ext = 0 T) at low temperature ( T = 5 K)and high temperature ( T = 90 K) before the transverse-field measurements (TF). Thesemeasurements are to check for additional magnetic order in our samples - the signal ofwhich may be very small compared with the SC signal in a TF measurement. In Fig. 7.2we plot P Z ( t ) (Equation 3.1 ) in ZF for UD80K (underdoped, T c = 80 K) and UD30K(heavily underdoped , T c = 30 K) YBaSrCu O − δ at both T = 90 K (top panels) and T = 5 K (lower panels). For high and low dopings we see very little change between ZFdata at T = 5 K and T = 90 K. This tells us D ( B loc ) is the same at both temperatures,which in turn means no magnetic phase transition between these two temperatures. At90 K YBaSrCu O − δ is a well-behaved paramagnet and so at 5 K in zero-field the onlylocal magnetic fields are from, presumably, nuclear moments. Furthermore, the initialasymmetry is high, A ≈ Z is usually defined as the direction parallel to the muon spin polarisation. This doping is only just above the spin-glass magnetic phase which forms at low temperatures anddoping. .2. RESULTS . ∼ P Z ( t ) in Fig. 7.2 for YBaSrCu O − δ fit well to a Kubo-Toyabe function (Equa-tion 3.2) as shown by the solid blue lines. The extracted widths of the field distributionsat 5 K are σ = 0 . µ s − , 0 . µ s − for underdoped and heavily-underdoped YBaSrCu O − δ respectively. These dephasing rates are a consequence of the isotropic nuclear moments inthe sample.We conclude from these data that our YBaSrCu O − δ are free from an additionalmagnetically ordered phase which could have resulted from magnetic impurities or from anintrinsic property of YBaSrCu O − δ due to e.g. its larger super-exchange energy, J . Thisconclusion is also supported by our bulk magnetisation measurements. YSCO
In Fig. 7.3 we show representative ZF data for our YSr Cu O − δ samples. We find thatthe time dependence of the ZF data is more appropriately fitted to a Gaussian dephas-ing expression, P Z ( t ) /P Z (0) = exp( − σ t /
2) + 1 /
3, rather than a Kubo-Toyabe function(Equation 3.2). While the distinction is not clear for T ≥
60 K, it is clear for
T <
60 K.Typical fitting values are σ ≈ . µ s − (Gaussian fit) and σ ≈ . µ s − (Kubo-Toyabe fit)for T <
40 K, and σ ≈ . µ s − (Gaussian fit) and σ ≈ . µ s − (Kubo-Toyabe fit) for T >
40 K.Also evident from these data is a significant ‘missing fraction’ at low temperatures. Theinitial asymmetry measured is A ≈
19% whereas from calibration we expect A = 21%.For Edi22w this drops to A ≈ P Z ( t ),most pronounced at low temperatures and for Edi22w, is not captured by the Kubo-Toyabeor Gaussian profiles. Subsequent magnetisation measurements reveal a magnetic transitionat T ≈
20 K (the actual temperature is field dependent), in addition to the SC transition.In light of these complications, we quarantine the YSr Cu O − δ data for now andinstead discuss it in its own separate section, Sec. 7.2.4. Given the nice ZF data for YBaSrCu O − δ we expect the TF measurements to not throw upany surprises. To conduct a TF measurement we always apply the field above T c (typically30 CHAPTER 7. MUON SPIN ROTATION AND THE SUPERFLUID DENSITY
Asymmetry (%)
T i m e ( m s ) Figure 7.3: Zero-field µ SR data from Edi35 (as prepared YSr Cu O − δ , slightly over-doped, T c = 62 K) at three representative temperatures. At T ≥
60 K the datais well fitted by either the Kubo-Toyabe function (as shown by the solid red line for σ = 0 . µ s − ) implying an isotropic Gaussian distribution of B loc with h B loc i = 0, orby P Z ( t ) /P Z (0) = exp( − σ t /
2) + 1 / σ = 0 . µ s − . For T <
20 K however the datais more appropriately fitted by the latter expression, plotted red line, with σ = 0 . µ s − . Adotted black line annotates 1/3 of the initial asymmetry value. Note the lower asymmetryat t = 0 µ s at T = 5 K. .2. RESULTS B ext = 0 . µ SR data for YBaSrCu O − δ above and below T c . Above T c one sees the precession of the muon in B loc = B ext with only a small dephasingdue to inhomogeneity in B loc . This is shown by the sinusoidal oscillation of P X ( t ) with asmall decrease in amplitude over time. Below T c one sees the precession of the muon spinin B loc < B ext with the amplitude of the oscillation decreasing as exp( − σ t / D ( B loc ) is determined by taking the Fourier transform of P X ( t ) and is plotted inFig. 7.4b. The centre of the peak at T = 5 K is B loc = 0 .
992 T. As expected fromthe good fits of P X ( t ), we consistently see a nice Gaussian distribution of B loc when look-ing at the Fourier transform. Fig. 7.4c shows the mean of D ( B loc ) is less than B ext below T c and this is because of diamagnetic screening of B ext in the SC state.If there were also a non-SC, paramagnetic volume fraction within our sample we wouldsee remnant sinusoidal oscillation which would relax at the much slower rate observed inthe T > T c TF data. Within uncertainties (which could be reduced by better statistics- i.e. longer counting times) we do not see an additional paramagnetic volume fraction.Furthermore the initial asymmetry, P X ( t = 0) is close to 0 .
21. Thus we conclude that oursamples have close to 100% superconducting volume fraction.To make the discussion above more quantitative we now discuss the fits of theYBaSrCu O − δ data to a Gaussian ‘relaxation’ function Equation 3.3. Over all temperatureranges and for all YBaSrCu O − δ samples the fits are excellent. The ‘relaxation’ rate σ is proportional to the field distribution and the superfluid density by Equation 3.4.In Fig. 7.5 we plot the relaxation rate versus temperature for all YBaSrCu O − δ sampleswe had time to measure before the muon beam crashed . To improve the quality ofthe Gaussian fits, we perform a global fit for all temperatures and fields for a particularYBaSrCu O − δ composition which allows us to fix the phase δ and A as these should notvary for a given sample and set-up (they are related only to the experimental set up ifthere are no additional magnetic phases). Recall that the ZF measurements rule out a magnetically ordered inclusions. Technically it is a dephasing of the muon spins due to the vortex-lattice-induced field distributionsrather than a relaxation, which would be caused by time-varying local magnetism. This is being pedanticthough and σ is often happily called a relaxation rate anyway. Half-way through our allotted beam time the carbon target broke (three weeks before the Christmasshutdown period). This section of the beam line is heavily radiation shielded by huge 1000+kg concrete‘lego’ blocks and these had to be removed before the C target could be replaced. Needless to say this is nominor undertaking and the beam was not on before our time ran out. At the time of writing, more beamtime had just been allotted following the next proposal round. CHAPTER 7. MUON SPIN ROTATION AND THE SUPERFLUID DENSITY
Asymmetry (%) t ( m s ) ( a ) - 2 0- 1 001 02 0 Spectral Intensity
B ( G )
01 02 03 04 05 06 0 ( b ) s ( m s-1) Average Local Field (G)
T ( K )
A v e r a g e L o c a l F i e l d B e x t ( c ) s Figure 7.4: (a) Representative raw data in transverse field mode for YBaSrCu O − δ above( T = 90 K) and below ( T = 5 K) T c . For clarity the data after t = 2 µ s is not shown. (b)Amplitude of the components of the Fourier transform of the data shown in (a), i.e. thelocal magnetic field distribution. (c) The temperature dependence of the average local field, h B loc i - a result of diamagnetic screening of the external field. Also shown for comparisonis the temperature dependence of σ as defined in Equation 3.3. .2. RESULTS ( a ) s ( m s-1) T ( K )
O D 8 0 K O P 8 5 K U D 8 0 K U D 3 0 K 1 % Z n O P 7 7 K 3 % Z n O P 6 2 K 3 % Z n O D 6 1 K s / s T / T c O D 8 0 K O P 8 5 K U D 8 0 K U D 3 0 K 1 % Z n O P 7 7 K d - w a v e ( b ) Figure 7.5: (a) The TF- µ SR relaxation rate σ versus temperature for various dopingstates for pure and Zn-doped YBaSrCu O − δ . “OD” refers to overdoped samples, “OP”to optimally doped and “UD” to underdoped. These letters are followed by the T c of thatdoping state as measured from bulk susceptibility. σ ( T ) is determined from Gaussian fits(Equation 3.3) to the raw P X ( t ) data - an example of which is shown in Fig. 7.4. The slowrelaxation rate above T c of σ ≈ . µ s − comes from the nuclear moments. Solid curvesare fits of the data with σ = σ (1 − t α ) where t = T /T c . α and T c are free parameters,except in the case of Zn doping where the susceptibility derived T c values are used. The T c determined from fitting these µ SR measurements are generally lower than the bulksusceptibility values, but by no more than 2 K. α values range between 3 . .
7. (b) Herethe data is normalised to T c and to the lowest temperature σ SC ( T ) value, σ SC (0). σ SC ( T ) istaken to be σ SC ( T ) = σ ( T ) − σ ( T > T c ) . All except the most underdoped YBaSrCu O − δ scale well with σ = σ (1 − t α ). The faster suppression of σ SC with temperature for themost underdoped YBaSrCu O − δ is discussed in the text. The yellow dashed line is thetheoretical temperature dependence of σ for a d -wave superconducting gap [234].34 CHAPTER 7. MUON SPIN ROTATION AND THE SUPERFLUID DENSITY
In addition we measure σ with low ( B ext = 0 .
03 T) and high fields ( B ext = 0 . . D ( B loc ) going from low to high fields if the vortex lattice were to undergoa phase transition to a less ordered, or lower dimensional, state. That we see very littlevariation in D ( B loc ) shows the vortex lattice is quite robust in these conditions - as itis for YBa Cu O − δ . In fact, a robust lattice gives us further confidence that σ ∝ λ − .Correspondingly we find only a small variation in the calculated value of σ ; for exampleon optimally-doped YBaSrCu O y we measure σ = { . ± . , . ± . , . ± . } µ s − for B ext = { . , . , . } T respectively.We now discuss the temperature dependence of σ for YBaSrCu O y shown in Fig. 7.5.For T (cid:28) T c , a linear suppression of λ − ∝ σ with temperature is characteristic ofa SC order parameter with nodes. This is a result of small scattering vectors at lowthermal excitation being able to connect regions of the Fermi-surface close to where theorder parameter changes sign, for example across the nodes in d x − y symmetry [134, 167].If there were significant impurity scattering, σ would change little with temperature for T (cid:28) T c , much like for an s -wave superconductor, because the scattering potential of theimpurities is enough to break Cooper-pairs at the nodes in analogy to the effect of a finitetemperature [134, 135]. We see a linear depression of σ ( T ) and this indicates, again, goodsample quality and the absence of disorder scattering.Closer to T c we expect σ = σ (1 − t α ) where t = T /T c . We use this relation to fit ourdata and estimate T c , fitting only data close to T c . These fits are shown as solid lines inFig. 7.5(a). We generally find values T c values ∼ −
2K lower than those obtained frombulk susceptibility measurements, although in the latter case they are zero-field-cooled T c estimates. Fitted values for α range between 3 . .
7. The paucity of data points closeto T c reflects that our primary interest was σ in the low temperature limit.The two-fluid model has α = 4 and our values are of this order. Closer to T c theGinzberg-Landau theory requires α = 1 so that a single α value in the fit, though common,is not rigorous. The cross over to linear suppression of λ − may well account for the lower T c values that we recover from the fits. The full theoretical d -wave temperature-dependenceof λ − [234] is shown by the dashed curve in Fig. 7.5(b) for T c = ∆ / . k B . The matchis not good in the mid-temperature range and this is typically found for polycrystallinesamples, while single crystal microwave measurements recover the d -wave temperaturedependence. This failure is not well understood. However, it is also clear that this simple .2. RESULTS d -wave T -dependence is not expected where there is a pseudogap present which does nothave d -wave symmetry, wiping out only those states near the antinodes [72]. This gives astronger T -dependence with α rising above 1, as observed.Hence, the temperature dependence of σ SC is generally as we would expect. The excep-tion is the UD30K data which decreases more rapidly, as can be clearly seen in Fig. 7.5(b).Note this is not merely the result of an incorrect T c , but rather a different temperature de-pendence as can be seen by the poor fit to (1 − t α ) in Fig. 7.5(a). To test the possibility ofan inhomogeneous doping state the raw data were fitted again with multiple components.The poor resulting fit with additional components however discredits this possibility. It isinstead likely a consequence of a large pseudogap energy at this doping [72] or competingcharge order, as observed at this doping state [235]. Many other physical properties showanomalous temperature dependence in this doping region, e.g. the strong pseudogapped T -dependence of S ( T ) /T , where S ( T ) is the electronic entropy, becomes rather flat at thisdoping [66]. In addition, the estimated p = 0 .
075 of this sample places it close to thespin-glass state, see Sec. 2.2.2, and this could be expected to additionally contribute to thedephasing of muons. λ − and T c We now discuss the relation between the superfluid density in the low temperature limitof YBaSrCu O − δ and the well-characterised YBa Cu O − δ . In Fig. 7.6 we reproduce thedata presented in Fig. 1 of [83] showing T c plotted against the superfluid density λ − inY . Ca . Ba Cu O y , Bi Sr CaCu O δ and La − x Sr x CuO . This is the so called ‘Uemuraplot’ [137], see Sec. 2.6 for a discussion of its features. Although not included in [83],we add similar data for the pure YBa Cu O − δ compound from [137]. Unlike the Ca-doped material, oxygen filling of the chains is required to optimally dope YBa Cu O − δ and these filled chains become superconducting themselves. This contributes significantlyto the superfluid density and leads to the ‘plateau’ at high doping in this material where λ − doubles with almost no change in T c [170].To this published data we add our results on YBaSrCu O − δ presented above. We useEquation 3.4 to convert the measured relaxation rates σ into λ − values and take T c to bethe values determined from µ SR with uncertainties estimated from their difference to thebulk susceptibility measurements. Note that the λ here is really the ab -plane value λ ab . Pending thorough optical reflectometry and ellipsometry measurements we do not know the effectivecarrier mass m ∗ seen in Equation 2.20. CHAPTER 7. MUON SPIN ROTATION AND THE SUPERFLUID DENSITY Y C a
B a C u O y L a
S r x C u O B i S r C a C u O y Y B a S r C u O y Y B a C u O y Tc (K) l ( m m - 2 ) Y S r C u O y Figure 7.6: T c plotted against the superfluid density, λ − , in the low temperature limit.The plot is adapted from [137] and [83]. Open symbols indicate the doping states p = 0 . .
16 and 0 .
19. To these published data we add our own data from YBaSrCu O − δ . Increas-ing doping proceeds in a clockwise direction and gives rise to the ‘boomerang’, althoughfor pure YBa Cu O − δ contributions to λ − from the chains lead to a plateau. Presentedin this way the YBaSrCu O − δ superfluid density is comparable with both pure and Ca-doped YBa Cu O − δ for similar doping and T c values. If Sr-substitution-induced disorderwere significant we would have expected a suppression of λ − relative to YBa Cu O − δ . .2. RESULTS λ − for YBaSrCu O − δ is similar to λ − for Ca-dopedY123 of similar doping states. If T c were suppressed by disorder there would be a char-acteristic more rapid suppression of λ − than T c . For example, see [134, 83] or Fig. 7.9where Zn substitution is seen to suppresses λ − twice as fast as T c . This is not observed.Consequently the lower T c in YBaSrCu O − δ with respect to Y123 can be attributed todisorder no more so than the small decrease in T max c of Ca-doped Y123.Our YBaSrCu O − δ data is also consistent with the pure Y123 of similar doping stateswith the UD30K data even falling on the so called ‘Uemura line’ of [137]. The pure Y123 λ − does become larger than our most overdoped YBaSrCu O − δ sample, but presumably this isdue to an additional chain contribution rather than a suppression of λ − in YBaSrCu O − δ (which may not be fully oxygen loaded, or be more susceptible to chain disorder [236] - wedo not know enough about the effect of Sr substitution to say which).We conclude from these data that it is not disorder, or a developed stripe phase, causingthe 8K suppression of T max c upon Sr substitution for Ba .The nuclear quadrupole resonant (NQR) line-widths in YBaSrCu O − δ have been mea-sured by Ying et al. [236]. They find a broadening of the line-widths with Sr substitutionand a blue-shift of the Cu(2) peaks and red-shift of the Cu(1) peaks. The peak frequencyshifts are consistent with an increasing hole concentration with Sr substitution (the inter-nal pressure effect). The broadening line-widths are attributed to Sr substitution induceddisorder . It is interesting that while these authors clearly see a pronounced increase in dis-order in YBaSrCu O − δ , our µ SR results show no appreciable suppression of λ − from thisdisorder. Note that NQR measures electric field gradients whereas muons do not couple toelectric field gradients (they are spin 1/2 particles). That the superfluid density remainsconsistent with the pure Y123 material, which has sharp NQR line widths, shows the su-perconducting properties are less sensitive to disorder causing broad NQR line-widths onthe Cu sites. The same conclusion can be drawn from NMR [230] and NQR [237] studiesof the co-substituted Y − x Ca x Ba − x La x Cu O cuprate. Cu O y We also spent considerable time measuring the YSr Cu O − δ samples Edi35, Edi22 andEdi22w. In Fig. 7.7 we show representative TF asymmetry data on YSr Cu O − δ at A weaker, more acceptable, conclusion would sound like; disorder does not have a significant effect onthe superconducting properties of YBaSrCu O − δ as compared with YBa Cu O − δ . In particular, the authors conclude a Sr-substitution induced O(4) (chain oxygen) b -axis ‘buckling’leads to the observed suppression in T c with Sr substitution. CHAPTER 7. MUON SPIN ROTATION AND THE SUPERFLUID DENSITY
Asymmetry (%) t ( m s )- 2 0- 1 001 02 0 ( b ) ( a ) Figure 7.7: Representative TF asymmetry data on YSr Cu O − δ at T = 5 K. Shown in(a) is the fit to a single Gaussian relaxation (blue line), Equation 3.3, as well as to a single‘Lorentzian’ relaxation (red line). In (b) the data is fitted to the sum of two Gaussians.Data are of the Methanol-washed, T c = 66 K powder YSr Cu O − δ sample. T = 5 K. To the data in Fig. 7.7 we show a single (Gaussian) component fit in panel(a) and a two component fit in panel (b) and solid lines. We have not presented these datawith the YBaSrCu O − δ data as the YSr Cu O − δ were found to contain a considerablefraction of magnetic impurities (see the ZF measurements section 7.2.1).In an attempt to partially account for this we fit the TF data to two Gaussians, Equa-tion 3.3: component 1 is slowly dephasing component while component 2 is more rapidlydephasing, i.e. σ > σ . Also, from the raw TF data shown in Fig. 7.7(a), it is clear that asingle Gaussian dephasing component is not sufficient, although a better fit to the decreas-ing oscillation amplitude can be obtained from the ‘Lorentzian’ function; σ ( t ) ∼ exp( − Λ t )- see Sec. 7.1.We have two possible ways to proceed; .2. RESULTS D ( B loc ) and a single component will not dis-tinguish between this and any flux-line lattice field broadening. With this approachwe will not gain any useful information about the superfluid density.• Fit with more components, of which using only two is the simplest (e.g. the sum oftwo Gaussians or a Gaussian and Lorentzian), and then see if one of these componentsis consistent with what we expect from a superconducting phase.In an attempt to extract some useful information from the YSr Cu O − δ µ SR data, weproceed with the latter.The Gaussian function is chosen to keep the model simple and to introduce as fewadditional parameters as possible. The free parameters in the fit are now { A (1) , A (2) , σ (1) , σ (2) , B (1)loc , B (2)loc , δ (1) = δ (2) , A B } where the superscript refers to component 1 or 2. Theratio A ( i ) / ( A (1) + A (2) ) gives the volume fraction of the i th component. The ‘baseline’asymmetry, A B , as a free parameter is typically small, ≈ .
0, for the two-component fits.Thus, for the fit results displayed it has been set to 0 . T = 5 K from the Methanol-washed T c = 66 K sample. Note that the more-rapidly decaying oscillations for t < . µ sare better reproduced.In Fig. 7.8 the results of this two-component fitting approach for all TF- µ SR YSCOdata are presented. Only parameters for component 2 are plotted because this componentis possibly related to the superconducting volume fraction.This approach is at least partially successful as shown by several encouraging featuresin Fig. 7.8 for the Edi22w and Edi35 samples (green and black respectively);1. The fraction of this phase goes to 0 at T c (above T c the errors for this componentbecome very large and the TF data are reasonably fitted with one component only).2. B (2)loc < B ext for T < T c which implies diamagnetic screening.3. The temperature dependence of σ (2) is what we would expect for a SC phase with T c ≈
60 K, especially for Edi22w.The volume fraction of the second component is ≈ . σ (2) ( T →
0) = (2 . ± . µ s − for threeYSCO samples and so we tentatively include this data point in Fig. 7.6. The data point40 CHAPTER 7. MUON SPIN ROTATION AND THE SUPERFLUID DENSITY
T ( K ) s (2) ( m s-1) E d i 3 5 O D 6 0 K E d i 2 2 O D 6 6 K E d i 2 2 w O D 6 6 K component 2 fraction
T ( K ) B loc(2) (G) A p p l i e d f i e l d
Figure 7.8: Plotted in the top panel is the temperature dependence of relaxation rate, σ (2) , for the second component, for all three YSr Cu O − δ samples measured. Shown inthe inset of this panel is the volume fraction of this second component, also as a function oftemperature. In the lower panel is the average local field, h B loc i , of the second component.The externally applied field, B ext = 1003 ± .2. RESULTS T max c (=70 K in the best samples)despite their high pressure O anneals? This seems unlikely, especially given a thermopowerof − µ V.K − in Edi35, but it could be due to significant O(5) occupation. Alternatively,this may reflect a real superfluid suppression from disorder and/or from an enhancedpseudogap or stripe-phase (enhanced by large J ). To investigate further requires samplesfree of magnetic impurity phases, which in turn requires considerable materials-chemistryeffort (now underway), followed by characterisation of the material with other techniquesand a range of µ SR measurements from over- to underdoping. O − δ We turn now to our initial λ − data for Zn-doped YBaSrCu − z Zn z O y with z=0.02 (“1%”)and z=0.06 (“3%”) shown in Fig. 7.5. The substitution of only a small amount of Zn on theCuO layer rapidly lowers T c as can be seen in Table 7.1, but even more rapidly suppresses λ − (in the former case it is linear [238] while in the latter case it is super-linear [134]).Here is a situation where disorder really does matter!Bernhard et al. have measured λ − for a range of Zn concentrations in overdoped,optimally-doped and underdoped Ca-doped Y123 [135]. From these data, reproduced inFig. 7.9, they could conclude the SC order parameter has d x − y (or similar) symmetry.To these published data we add our results from YBaSrCu − z Zn z O y on overdoped andoptimally-doped samples . In taking the ratio of λ − at z = 0 and z > m ∗ changes negligibly with Zn substitution, which is most likely justified [135] but couldbe tested by reflectometry and ellipsometry measurements, for example. As was found inthe previous section, YBaSrCu O − δ and YBa Cu O − δ behave in the same way, this timeunder Zn substitution. This verification is additionally important because it shows a con-sistency not only between YBaSrCu O − δ and Y123 at different oxygen concentrations, butalso a reproducibility of that consistency between different ‘batches’ of YBaSrCu − z Zn z O y .These data also confirm YBaSrCu O − δ has a SC order parameter with d x − y symmetryand that Zn acts as a unitary scatterer. Which, unfortunately, was all we had time to measure before the beam failed. CHAPTER 7. MUON SPIN ROTATION AND THE SUPERFLUID DENSITY s s D T c / T c ( z = 0 ) Figure 7.9: The normalised superfluid density plotted as a function of the relative decreasein T c . The data are been normalised to the Zn free material values. The plot is adapted from[135]. Data for Y . Ca . Ba Cu − z Zn z O y optimally-doped (black squares) and overdoped(black triangle) is from Bernhard et al. [135] and data for YBa Cu − z Zn z O y near optimallydoped (black crosses) is from Bucci et al. [239]. The solid line show the predicted superfluidsuppression for a system with d x − y order parameter symmetry due to isotropic scatteringin the unitary limit [134]. To this plot we add our data for YBaSrCu − z Zn z O y optimallydoped (open orange squares). From these initial data YBaSrCu O − δ appears entirelyconsistent with Y123. We have used µ SR to measure the superfluid density, λ − , of YBaSrCu O − δ and we findthe nice result that this is a boring, well-behaved material. Fig. 7.6 showed that the λ − areconsistent with those previously reported for pure and Ca-doped YBa Cu O − δ [137, 83].Furthermore, the suppression of λ − due to Zn doping shown in Fig. 7.9 is fully consistentwith that found for YBa Cu O − δ [135] and confirms that YBaSrCu O − δ has a SC orderparameter with d x − y symmetry and that Zn acts as a unitary scatterer. The immediateconclusion is that disorder does not play a significant role in YBaSrCu O − δ as comparedwith Y123. Consequently, the lower T max c seen in this compound, and those with lower Sr .3. CONCLUSIONS λ − by µ SR for YSCO are less certain as it appears our samplecontain additional magnetic impurities. The values of λ − extracted from a two-componentfit of the data revealed a comparatively low λ − . This indicates that disorder may sig-nificantly suppress the superfluid density in the fully Sr substituted material. Furthermeasurements on pure YSr Cu O − δ samples are desirable to confirm or refute this result.44 CHAPTER 7. MUON SPIN ROTATION AND THE SUPERFLUID DENSITY hapter 8Gap estimates from RamanSpectroscopy
Summary
The purpose of the work presented in this chapter is to measure the spectral gap in B g and B g symmetry for Yb123, Eu123 and Nd123 single crystals at optimal doping. To thebest of our knowledge these measurements have not been done before. We find B g spectralgap energies similar to Y123 but no clear systematic ion-size dependence which was theobject of our study. We were unable to unambiguously identify spectral gap features in B g scattering geometry. In BCS theory, the superconducting energy gap, ∆ , is proportional to the mean-fieldtransition temperature T mf c , 2∆ = α.k B T (mf) c . For d -wave ∆ ( k ) symmetry in the weak-coupling limit, α = 4 .
28 while for the strong-coupling limit the value is larger. Observationof an energy gap in Raman spectra provided early estimates of α for the cuprates. Forexample, the early work of Friedl et al. [240] on Ln123 found α = 4 . ± .
10, fromwhich they concluded that the cuprates are strongly-coupled superconductors . Recentwork by Guyard et al. [241] on Hg1201 illustrates a symmetry and doping dependentvalue of α ; at the nodes α ≈ . p , whilst at the anti-nodes α decreasesfrom ≈
12 in the under-doped regime to ≈ . At the time an s -wave symmetry 2∆ ( k ) was popular, in which case the weak-coupling BCS result is α = 3 . CHAPTER 8. GAP ESTIMATES FROM RAMAN SPECTROSCOPY results highlight two important facts about the cuprates: (i) The difference between T c and T mf c can be significant [56, 55], but is often overlooked. (ii) Both the pseudogap andsuperconducting energy gap contribute to the energy of the spectral gap, 2∆ , observed byRaman spectroscopy [64], Sec. 2.2.4. In addition, the pseudogap modifies the temperaturedependence of the 2∆ from that expected from a mean-field superconducting gap [241].There are two ways by which 2∆ is manifest in Raman spectra. Firstly in the Elec-tronic Raman Scattering (ERS) continuum and secondly, via the renormalisation of phononenergy and life-time [242, 243].There have been many studies exploring the doping dependence of the B g and B g ERS features, e.g. [244, 71, 245, 246, 7, 247, 248], as well as recent measurements of thetemperature dependence of these features [241, 248, 211].An early study made measurements on Ln123 with the purpose of estimating 2∆ [240, 249]. Here they made use of the renormalisation of phonon life-time to estimate 2∆ .In distinction to this work, they assume that 2∆ is the same in these materials and insteaduse the variation in phonon life-time with Ln ion-size to provide more data points for a fitto the Zeyher-Zwicknagl model [242].With our measurements however we are primarily interested in comparing 2∆ betweenmaterials at a given doping state , namely, between Yb123, Eu123 and Nd123 at optimaldoping. Optimal doping is chosen for two reasons; (i) it is a suitable doping state forinter-comparison between materials and (ii) T c = T max c here. These samples were chosenbecause they span the ion-size range available and because polycrystalline samples of thesame material can be synthesized in the lab [5]. At best, these measurements can provideestimates of the pseudogap energy and symmetry [66, 241], the superconducting gap energyat the nodes and anti-nodes [246, 241] and information on the quasi-particle dynamics[71, 247] as the ion-size varies across our Ln123 series. A description of the experimental set up for these variable temperature Raman spec-troscopy measurements is given in Sec. 3.8.3. Or to put it another way, the renormalisation of the real and imaginary parts of the phonon self-energy,Σ, respectively. .2. EXPERIMENT
Optimally-doped Ln123 is distorted from a tetragonal unit cell into a slightly orthorhombicunit cell due to the formation of Cu-O chains along the b -axis direction. This orthorhom-bic distortion leads to a mixing of the B g , A g and B g symmetries [109]. Because thedistortion is small, the approximation of tetragonal symmetry is made giving the usualscattering geometries: A g + B g , B g etc. [250, 212, 108].Early studies found no significant ERS resonance effects between the 647 nm and 458 nmlaser lines [245], that is, there is no significant change in the B g spectra below T c forincident laser wavelengths between 647 nm and 458 nm. However, more recent studies ofle Tacon et al. show a resonant enhancement of the B g gap and phonon modes [247, 251]that may be related to the electronic band structure via inter-band transitions [251]. Weuse the 514.5 nm laser line as this is the most common wavelength used by others in thefield.The temperature of the sample should be as close as possible to that measured by thethermocouple. The two main reasons why this may not be the case are:1. The laser heats the sample. To mitigate, the laser power is kept at less than 10 mW- an experimentalist’s ‘rule of thumb.’ In this regard we are also helped by the largelaser-spot size, ∼ , reducing the intensity of light on the sample.2. There is poor thermal conductivity between the sample and cold finger. A highsurface area contact was made with silver paint or vacuum grease to ensure goodthermal contact.Fortunately for these measurements, phonon thermal conductivity dominates over elec-tronic thermal conductivity in the cuprates (at least in the normal state) [21] which meansthat the thermal conductivity remains large, and is in fact enhanced in the superconductingstate due to the decrease in phonon scattering. From a simplified model of the experimentalsituation we estimate ∼ CHAPTER 8. GAP ESTIMATES FROM RAMAN SPECTROSCOPY
50 100 150 200 250 300 350 400 450 500 ω (cm − ) [ n ( ω ) + ] − T = T = T = T = T = T = Figure 8.1: Examples of the correction factor used to scale spectra at various temperatures. n ( ω, T ) = [exp( (cid:126) ωk B T ) − − is the normal Bose-Einstein distribution.Another issue in this regard is our use of vacuum grease to attach the sample to thecold-finger. Despite our most conscientious efforts, it is possible some of this grease foundits way to the surface of our Eu123 sample. We speculate this is the cause of a linearcomponent to the Eu123 B g and B g spectra shown in Fig. 8.11 and Fig. 8.14 respectively.After subtraction of a suitable ‘dark count’ , every spectrum is multiplied by the Bose-Einstein factor, [ n ( ω, T ) + 1] − where n ( ω, T ) = h exp( (cid:126) ωk B T ) − i − is the normal Bose-Einstein distribution (see Chapter 1 and 5 of [108]). Fig. 8.1 shows the correction factorfor the range of interest, [50 , − , for several representative temperatures. There are six Raman-active phonon modes in the Y123 structure which will show up inaddition to any electronic Raman scattering (ERS). They are listed in Table 8.1. The‘names’ given to them are the approximate energy shift at which they are observed. Foreach mode there is a cartoon of the main ion displacements involved in the mode. Alsolisted in the table are experimentally determined, mode-specific Grüneisen parameters, γ = − d ln( ω ) / d ln( V ), from reference [252]. These Grüneisen parameters are used todetermine the dependence of the phonon mode energy on the volume of the unit-cell.Assignment of the modes are documented, for example, here [212]. The classification ofthe symmetry of each phonon mode is listed and is important with regard to which region These are the counts recorded by the CCD with the shutters to the spectrometer closed. It is calibratedat the start of each measurement. .2. EXPERIMENT − A g - OCuBaLn bac O ( ) O(3)O(1)(1 - d )O(4) LaNd...YbLu {
150 cm − A g OCuBaLn bac O ( ) O(3)O(1)(1 - d )O(4) LaNd...YbLu {
330 cm − quasi- B g OCuBaLn bac O ( ) O(3)O(1)(1 - d )O(4) LaNd...YbLu {
430 cm − A g OCuBaLn bac O ( ) O(3)O(1)(1 - d )O(4) LaNd...YbLu { CHAPTER 8. GAP ESTIMATES FROM RAMAN SPECTROSCOPY
Name Symmetry Grüneisen parameter Ion displacements500 cm − A g OCuBaLn bac O ( ) O(3)O(1)(1 - d )O(4) LaNd...YbLu {
580 cm − A g - OCuBaLn bac O ( ) O(3)O(1)(1 - d )O(4) LaNd...YbLu { Table 8.1: The Raman-active phonon modes in optimally-doped Ln123 named by theirapproximate energy in Y123. For each mode a cartoon shows the main ionic displacementsinvolved. The Grüneisen parameters, γ = − d ln( ω ) / d ln( V ), are from [252]. Many physical properties of these materials are doping dependent and 2∆ is no exception.Measuring this doping dependence has been the focus of previous Raman studies. Herehowever we are interested in Ln substitution effects and so we compare samples at thesame doping state. When measuring two-magnon scattering we compared undoped Ln123.When measuring the superconducting energy gap we should compare optimally dopedLn123.Having determined the annealing conditions for optimal doping on polycrystalline sam-ples, Table 3.3, we co-anneal Ln123 single crystals with polycrystalline pellets of the samematerial under the conditions required for optimal doping. As a check, S(295K) is measuredon the polycrystalline sample after the anneal to confirm that we indeed have optimally .2. EXPERIMENT . The exception is Nd123which we were not able to overdope. Instead, we use the most fully oxygenated sample wecould prepare .The T c values of the single crystals measured below are T c = 90 . , . We use the ‘Python’ coding language and libraries to analyse much of the following data.In order to characterize the phonon modes, we first subtract a linear background overa suitable range, which we determine for each peak or pair of peaks, before fitting thepeaks themselves. The peaks are fitted, individually or as pairs, to a Fano profile asspecified in Equation 2.10. Next the ‘minimize’ package [254, 255] uses a least-squaresminimization algorithm to determine the best fitting parameters within the physicallysensible bounds that we specify. Every fit is visually inspected, and rejected if necessary,before the calculated parameters are recorded.A more sophisticated fitting procedure is outlined in [212] and the earlier papers [244,256, 257]. Below we outline the function and parameters involved, but firstly note thatthe reason for using this alternative procedure is to obtain an alternate estimate of theERS component of the spectra. The fit parameters of the phonons are just as accuratelyobtained by the procedure described above.The entire spectrum is fitted to a coupled phonon (the 330 cm − mode) and electronicterm plus five independent, Lorentzian line-shape phonons. Each independent phonon isdescribed by three parameters; an intensity A i , HWHM Γ i and energy Ω i (following similarnotation in [212]); I i ( ω ) = A i Γ i Γ i + ( ω − Ω i ) The coupled phonon and electronic term we take from [212] which is based on a Greenfunction operator to relate the electronic and phonon Raman tensor matrix elements to It has been known for p resulting from some given annealing conditions to be dependent on the historyof the sample. This involved high O pressure, low temperature anneals - see Table 3.1. CHAPTER 8. GAP ESTIMATES FROM RAMAN SPECTROSCOPY datafit (a)
Figure 8.2: A typical fit using the procedure described in the text, Sec. 8.2.4. The data isfrom Yb123 in A g + B g symmetry at T = 20 K.the imaginary part of the susceptibility tensor χ ( ω ) [244]; I ( ω, T ) = A [1 + n ( ω, T )] g ( ω ) g ( ω )1 + g ( ω ) (cid:16) S g ( ω ) g ( ω ) + ( ω − Ω) (cid:17) Γ (1 + g ( ω )) + ( ω − Ω) − (8.1)This complicated looking expression has the following free parameters; intensity A , a scaled ω -independent phonon-electron coupling term S , un-renormalised HWHM of the 330 cm − mode Γ , renormalised energy of the 330 cm − mode Ω and g ( ω ) = V ρ ( ω ) / Γ where V isthe electron-phonon coupling and ρ ( ω ) is the imaginary part of the electronic response, seeSec. 2.4.2. After Limonov et al. we fit to the following form of g ( ω ) with five parameters C , C and C , Γ e and D ; g ( ω ) = C + C ω + C ω Γ e + ( ω − D ) We are interested in the frequency where g ( ω ) is maximum as it can be identified withthe renormalisation peak associated with 2∆ .In practice, fewer than 6 parameters are refined at a time. In general the independentphonon parameters are easily deduced and many other parameters have restrictive sensiblebounds; e.g. Γ e ≈
200 cm − , Γ ≈
10 cm − , D ≈
500 cm − , Ω ≈
330 cm − and S ≈ − A g + B g symmetry at T = 20 K. .3. RESULTS In Fig. 8.3 we plot spectra from optimally-doped Yb123 in A g + B g symmetry at varioustemperatures as indicated in the legend. The only manipulation of these spectra has beenmultiplication by the appropriate Bose occupation factor, [ n ( ω, T ) + 1] − . These datanicely demonstrate some really interesting physics;1. ‘Hardening’ of phonon modes with decreasing temperature due primarily to latticecontraction with temperature.The temperature dependence of the i th phonon mode’s energy due to lattice contrac-tion can be expressed as [258, 259, 260]; ω i ( T ) = ω i (0) exp " γ i Z T α ( T ) d T (8.2)where γ i is the Grüneisen parameter and α is the linear coefficient of thermal expan-sion. α is anisotropic in Ln123 [261], so we replace 3 γ i R T α ( T ) d T in Equation 8.2with γ i X j = a,b,c Z T α j ( T ) d T Fig. 8.4 shows ω ( T ) calculated with ω (0) = 340 cm − , thermal expansion data fromMeingast et al. [261] and a range of experimentally and theoretically estimated valuesof γ for the relevant Y123 modes: γ = − . ± . γ were from Syassen et al. [252] were used in the fitting. Only T > T c data points were included in the fitting, and although there are sometimesfew data points to determine the fit from, generally ω i (0) is the only free parameter.As can be seen in Fig. 8.6, the fits are generally satisfactory and serve to clearly showwhere ω i ( T ) is renormalised by a gap in the electronic density of states, as discussedbelow. Nomenclature used in the Raman community meaning increasing Raman shift, ω , when an experi-mental parameter is varied e.g. an increase in energy of the feature. Occasionally it seemed appropriate to also fit γ within the physically sensible range [ − . , − .
0] whenthere was sufficient data points above T c . As can be seen from Equation 8.2 and shown in Fig. 8.4, γ determines the gradient of ω i ( T ) . CHAPTER 8. GAP ESTIMATES FROM RAMAN SPECTROSCOPY c ’’ (comparable between spectra) w ( c m - 1 ) Figure 8.3: Temperature-dependent Raman spectra of optimally-doped Yb123 in A g + B g scattering geometry. The temperature of the cold finger is shown in the legend and eachspectrum has been multiplied by the appropriate [ n ( ω, T ) + 1] − factor. We estimate thatthe sample temperature may be ∼ .3. RESULTS T (K) ω ( c m − ) γ = -2.2 γ = -2.0 γ = -1.8 γ = -1.6 γ = -1.4 α = const., γ =-1.7 Figure 8.4: Calculated temperature dependence of the shift in a T = 0 K, 340 cm − phonon frequency solely due to lattice contraction [258, 259, 260]. This is calculated fromEquation 8.2 with the thermal expansion data of [261] and for various sensible values of theGrüneisen parameter, γ , as indicated in the legend [252]. For comparison the temperaturedependence is also shown if thermal expansion is taken to be a temperature independentvalue, α = 15 . × − K − . Note that the Grüneisen parameter determines the gradientof the curve.2. The asymmetric line shape of the peaks is due to the Fano effect and indicatescoupling between phonons and the electronic continuum, Sec. 2.4.1.We use the Fano line shape in the fitting routine noting that more complicated fittingroutines give similar results [212].3. ‘Softening’ of some phonon modes with decreasing temperature - contrary to thebehaviour described above. The theory behind this effect is explained by Zeyherand Zwicknagl [242] in the case of an s -wave superconducting order parameter andgeneralised by Nicole, Jiang and Carbotte [243] to the case more relevant for thecuprates, of a gap with nodes. An opening of a gap in the electronic density of states(e.g. due to the opening of a superconducting gap) renormalises the energy and life-time (as revealed by the HWHM) of a mode via electron-phonon coupling and viamodified scattering channels.In the simplest model, a particular mode will ‘anomalously’ soften if its unrenor-malised energy is below 2∆ while harden if its unrenormalised energy is above 2∆ .This is illustrated in Fig. 8 and 9 from Zeyher and Zwicknagl [242].56 CHAPTER 8. GAP ESTIMATES FROM RAMAN SPECTROSCOPY
4. The conventional temperature dependence of the half-width-at-half-maximum (HWHM)of a phonon mode (which is related to the half-life of that phonon excitation) takesthe form [262, 108, 212];Γ( ω p , T ) = Γ( ω p ,
0) [1 + 2 n ( ω p / , T )] + Γ imp (8.3)Γ imp results from phonon scattering off impurities, similar to the residual resistivity inmetals as T → n ( ω, T ) is the Bose-Einstein distribution function. Physicallythis represents a phonon decaying into two phonons of opposite momenta, each withenergy of ω p /
2. Fits to this function, with similar comments relating to the phononenergy fits, are shown as solid lines in Fig. 8.6.The HWHM is also renormalised by the 2∆ as shown in Fig. 8.5. This manifestsas an increased HWHM when the energy of the mode is close to 2∆ . Physically itresults from additional scattering opportunities available to the phonon, e.g. Cooperpair breaking.5. Depression of the ‘background’ at low temperature and low ω . The backgroundis due to Raman scattering from the electronic continuum in the material (ERS)as opposed to phonon modes which have well-defined energies. The reduction inintensity at low energies arises from the opening of a gap in the electronic density ofstates at temperature T < T c . estimates from phonon shifts Fig. 8.6 shows the energy and half-width-at-half-maximum (HWHM) as a function oftemperature for various phonon modes. These parameters were determined from fitting toa Fano line-shape with a linear background.In the simplest model the renormalised energy of a particular mode will be lower if itsunrenormalised energy is below 2∆ but larger if its unrenormalised energy is above 2∆ .This is illustrated in Fig. 8 and 9 from Zeyher and Zwicknagl [242]. In our data we clearlysee renormalisation of both the energy and HWHM of phonon modes. There are howeverseveral further considerations which we now discuss.We will actually be measuring some combination of the superconducting gap and pseu-dogap, namely 2∆ . Again, we note that 2∆ is a pair-breaking energy. The 2∆ energyobtained will depend on the symmetry of the particular mode. For example, calculations .3. RESULTS et al. [253] show the different responses of an A g and B g symmetry phononto a spectral gap with nodes. In these calculations the transition from renormalised hard-ening to softening for A g phonons broadens but remains centred close to 2∆ . The rangeof A g phonon energies whose HWHM are renormalised by the energy gap increases, andthe energy of maximum broadening decreases with respect to B g phonons. Physically,these A g phonons are providing pair-breaking excitations in regions of the Fermi-surfacewhere 2∆ ( k ) vanishes at the nodes.Impurity scattering will further broaden out in energy the transition between renor-malised and unrenormalised phonon excitations. The main concern for us however is thatimpurity scattering also alters the magnitude of the relative phonon shifts that would haveallowed a more accurate estimate of 2∆ , as shown in Fig. 8 and 9 from Zeyher andZwicknagl [242].However, the chief consideration, comes from the extension from the s -wave to d -wavegap model described by Nicol et al. [243]. Their findings are more applicable to ourmaterials as they consider the effects of anisotropic gaps with nodes rather than an isotropic s -wave gap.Nicol et al. introduce a normalised chemical potential (or ‘filling factor’), ¯ µ ≡ µ/ t where t is the hopping integral, that alters the ‘crossover energy’ from renormalised soften-ing to renormalised hardening. In Fig. 8.5 we reproduce Fig. 1 from their paper [243] whichillustrates this point (our annotations). The solid line corresponds to ¯ µ = 0 (half-filling or p = 0), the long dashed curve is ¯ µ = − s -wave case the crossing energy is ω = 2∆ but for the d x − y case this energynow becomes ω = 4∆ − | ¯ µ | . At half-filling the crossing is in fact at 4∆ ! This reducesto 2∆ predicted by Zeyher Zwicknagl if ¯ µ = − µ for our compounds? Firstly, note ¯ µ ∼ t − . The hopping integralin the Hubbard model, t , is related to the hopping integral between p and d orbitals in atwo-band Hubbard model presumably as t ∼ t pd . From Aronson et al. [214], t pd ∼ r − n where r is the superexchange path length and 2 . < n < .
0. ¯ µ would appear to be verysensitive to our ion-size substitutions that result in lattice parameter variations (Sec. 2.1.4): δ ¯ µ ∼ − r − δr . To the best of our knowledge, ion-size effect on t remains to be tested andwe discuss this issue in more detail later in Sec. 9.3.1.With sufficient data all of these parameters could be extracted from a fitting procedure58 CHAPTER 8. GAP ESTIMATES FROM RAMAN SPECTROSCOPY the phononhardens the phononsoftensthe phononbroadens
Figure 8.5: Fig. 1 from [243] supplemented with our annotations. The figure presents theresults of calculations of the phonon self-energy, ∆Σ( υ ), as a function of energy υ scaledby the spectral gap in the low-temperature limit, 2∆ . As annotated the real part of theself-energy results in a shift (renormalisation) of the phonon energy, whilst the imaginarypart relates to the width (life-time or scattering rate) of the phonon mode. The solid linecorresponds to ¯ µ ≡ µ/ t = 0 where µ is the chemical potential and t the hopping integralin the Hubbard model, which is extremely sensitive to the super-exchange path-length.The long dashed curve is ¯ µ = − = 39 . ± . s -wave modelof Zeyher and Zwicknagl - their work was published before the Nicol paper and at thattime, 1990, it was far from clear whether the superconducting gap had s -wave symmetryor otherwise. Nevertheless, they achieve good agreement between their measured phononwidths and the model using mostly experimental or calculated parameters. .3. RESULTS Peak centre (cm-1)
Y b 1 2 3 E u 1 2 3 N d 1 2 3
T ( K )
HWHM (cm-1)
Peak centre (cm-1)
Y b 1 2 3
681 01 21 41 61 8
T ( K )
HWHM (cm-1)
Peak centre (cm-1)
Y b 1 2 3 E u 1 2 3 N d 1 2 3
T ( K )
HWHM (cm-1)
Peak centre (cm-1)
Y b 1 2 3 N d 1 2 3
T ( K )
HWHM (cm-1)
Figure 8.6: Phonon mode energies and HWHM determined from fitting to a Fano lineshape. Similar data for the 110, 150 and 675 cm − modes are presented in Fig. 8.7. Solidlines are fits of the data above T c to the conventional temperature dependence for the modeenergy, given by Equation 8.2, or HWHM, given by Equation 8.3.60 CHAPTER 8. GAP ESTIMATES FROM RAMAN SPECTROSCOPY
Peak centre (cm-1)
Y b 1 2 3 E u 1 2 3 N d 1 2 3
T ( K )
HWHM (cm-1)
Peak centre (cm-1)
Y b 1 2 3 E u 1 2 3 N d 1 2 3
T ( K )
HWHM (cm-1)
Peak centre (cm-1)
Y b 1 2 3
T ( K )
HWHM (cm-1)
Figure 8.7: Phonon mode energies and HWHM determined from fitting to a Fano lineshape. Solid lines are fits of the data above T c to conventional temperature dependences. .3. RESULTS betweenNd123 and Yb123. However, we can place bounds on 2∆ from these data. For example,renormalised softening only occurs for ω < - from which we can place a strict lowerbound on 2∆ for Yb123 and Eu123. Such a lower bound would be overly conservativebecause we know that ¯ µ < µ ≈ − et al. [240] described above and so we use the result that the transition fromrenormalised softening to hardening occurs at ≈ . HWHM renormalised-broadeningwill be largest at ≈ , but present at lower and higher energies.We note the renormalisation of the 330 cm − B g phonon shown in Fig. 8.6 is largestonly at T ≈
60 K, which is 30 K lower than T c where the SC gap opens. Depending onthe relative energy of the mode and the SC gap, this is not too surprising, see for example[212]. Yb123
For Yb123 spectra we observe a conventional temperature dependence (solid line) of the‘500 cm − mode’ energy. At low temperature there appears to be additional broadening ofthis mode, which signals a proximate energy gap. The two higher energy phonon modes arecompletely conventional. On the other hand the 465 cm − (57.5 meV) mode hardens at lowtemperatures, by 4 ± − more than one would expect lattice contraction alone (solidline). In contrast, the ‘330 cm − (41 meV) mode’ softens by − . ± . − . With theassumption the transition from softening to hardening occurs at ≈ these observationslead to the follow plausible energy range for 2∆ : 39 meV < <
58 meV.
Eu123
The A g + B g spectra for Eu123 are shown in Fig. 8.8. There are fewer phonons ofsignificant intensity in these data. Of real interest is the 4 ± − renormalised softeningof the 500 cm − (63 meV) mode, immediately suggesting 63 meV < . The 330 cm − mode also shows a 6 ± − softening (the large uncertainty given to this value reflectsthe poor fit from the unexpected softening of the phonon mode above T c ). In contrast tothe 500 cm − mode, there is negligible additional broadening of the 330 cm − mode at lowtemperatures.Unfortunately, the higher energy phonon modes so clearly visible in the Yb123 spectrado not make an entrance in our Eu123 spectra which prevents us from putting an upper62 CHAPTER 8. GAP ESTIMATES FROM RAMAN SPECTROSCOPY c ’’ (comparible between spectra) w ( c m - 1 ) Figure 8.8: Representative A g + B g spectra of optimal doped Eu123. Temperature isindicated in the legend.limit on 2∆ based on these data. Nd123
The Nd123 spectra shown in Fig. 8.9 have several strange features which will be discussedshortly. To begin with however we discuss our estimate of the gap energy.There is no anomalous temperature dependence of the 500 cm − mode energy (or forthat matter the 610 cm − mode, although there is considerable scatter in fitting resultshere). It is possible however that there is some additional broadening at low temperatures.That places an upper bound of 2∆ ≈
63 meV.With a larger unit cell than both Eu123 and Yb123, one would not expect the 330 cm − mode of Nd123 (at 320 cm − ) to be higher in energy than that of Eu123 (at 312 cm − ) ina simple picture whereby the restoring force for the ion-displacement goes as Hooke’s law, F ∝ − k ( x − x ). The higher energy cannot be explained by the Nd123 being underdoped(which is entirely possible and in fact likely) because the energy of this mode changes littlewith doping - for Nd123O we measure this mode at 316 cm − at room temperature.There is however an even bigger anomaly; the 330 cm − peak splits into two distinctpeaks as the temperature is lowered! Below T ≈
90 K a second, lower energy peak becomes .3. RESULTS c ’’ (arb. units) w ( c m - 1 ) Figure 8.9: Representative A g + B g spectra of oxygen loaded Nd123. Temperature isindicated in the legend and the spectra have been vertically offset for clarity.clearly distinguishable (at higher temperatures it is suggested by the asymmetric profile ofthe 330 cm − mode). As the temperature is further lowered the energy difference of thesemodes increases and the HWHM decreases from 15 cm − to 10 cm − . At T = 10 K wemeasure the two peaks to be centred on energies 330 ± − and 285 ± − respectively,a 45 cm − difference. For comparison only, a conventional temperature dependence fittedto the high temperature data points is plotted as a solid line. By T = 9 K the lower modeis softer by 40 ± − whilst the other is harder by 7 ± − .Both these observations are now understood as the result of coupling between andmixing of the out-of-phase O(2)-O(3) phonon mode and an Nd crystal-field excitation[263, 264].The crystal field (CF) of the almost tetragonal Nd123 lifts the degeneracy of the Nd f electrons into five doublets. A particular transition between two of these has an energyof 304 cm − and symmetry B g (or A g in an orthorhombic crystal field) [264]. In order forCF excitation and phonon to couple several requirements must be met. The CF electronsand ion involved in the phonon must be in close spatial proximity, in this case the Nd andO(2,3) ions are adjacent, separated by 2 . ± .
04 Å [40]. Next both excitations must havesimilar symmetry, in this case both have quasi- B g symmetry. Finally, the renormalisation64 CHAPTER 8. GAP ESTIMATES FROM RAMAN SPECTROSCOPY effects are inversely proportional to the difference (or higher power of the difference) oftheir un-renormalised energies (perturbation theory). In this case that difference happensto be small; Heyen et al. observe these peaks at 334 cm − and 282 cm − , a splitting of52 cm − , at T = 10 K. From these energies, which agree with our data, and relative peakintensity data they compute un-renormalised energies of the phonon and CF excitationsas 308 ± − and 304 ± − respectively [264], a difference of 4 ± − . Theun-renormalised phonon energy is slightly softer than the Eu123 mode.The coupling between phonon and CF results in a mixed phonon-CF character to the334 cm − and 282 cm − excitations. As the temperature is increased, it appears theeffective coupling between the CF excitation and phonon decreases, as (1 − κT ), resultingin a weaker renormalisation of both energies and a greater phonon component to the higherenergy mode [264].In fact, other than via ion-size variation, this represents the only observable effect ofthe variable Ln f -electron number in this thesis! estimates from ERS B g The inset to Fig. 8.10 shows the results of very long count-time spectra on Yb123 at T = 9 K (cid:28) T c , T ≈ T c = 94 . T = 250 K in B g scattering geometry. The Ramanshifts are shown in both cm − and meV units. A constant dark count was subtracted priorto correction by the Bose-factor. Next, each spectrum was scaled by a constant so thattheir intensities are equal at large ω .The top main panel of Fig. 8.10 shows the difference between the 9 K and 90 K spectra.At low ω there is less spectral weight at 9 K with some of this weight shifting to the broadpeak around 500 cm − (65 meV). We identify this broad peak of enhanced ERS as the‘renormalisation peak’ associated with an increase of states above the superconductinggap . These two features result from a redistribution of spectral weight from low to higherfrequencies due to an opening of a gap in the electronic density of states. Similar, butmuch weaker, features can be seen comparing 20 K data with the 250 K data.Normally, the energy of 2∆ is identified as position of this feature in the ERS spectra(see for example [246, 7, 247]). Thus, our estimate of the B g for Yb123 from this ERSdata is 65 ± We note however that this feature coincides with a phonon mode. .3. RESULTS c ’’S - c ’’N w ( c m - 1 ) ( m e V ) c ’’ (arb. units) w ( c m - 1 ) c ’’S - c ’’N w ( c m - 1 ) ( m e V ) c ’’ (arb. units) w ( c m - 1 ) Figure 8.10: (Top panel) The difference between long count-time B g spectra at 9 K and 90K of optimally doped Yb123. Inset shows these spectra as well as data taken at T = 250 K.(Bottom panel) The difference between long count-time B g spectra, plotted individuallyin the inset, at 9 K and 175 K for optimally-doped Nd123.66 CHAPTER 8. GAP ESTIMATES FROM RAMAN SPECTROSCOPY
Fig. 8.11 shows similar data for Eu123. Here a ‘renormalisation peak’ candidate is notas clear, although spectral weight is lost in the 20 K data below ≈
600 cm − . The figurealso shows the data at 9 K on its own. In this spectrum, spectral weight is clearly lostbelow ≈
600 cm − as indicated by a sudden change in gradient . From these data weestimate 2∆ ≈
600 cm − , or 70 ±
10 meV.Finally, we discuss the B g ERS data for Nd123 which are shown in bottom panel ofFig. 8.10. Other than multiplication by the Bose-Einstein factor, ( n ( ω, T ) + 1) − , thesedata have not been manipulated in any way. From the raw data shown in the inset, orthe difference plot shown in the main panel, a ‘renormalisation peak’ candidate is not asclear. The feature at 400 cm − is possibly a weak phonon mode, slightly softened withrespect to the equivalent mode in Yb123 due to Nd123’s larger unit-cell, that has only beenobservable in this long count-time spectra. Note also the mixed phonon-CF excitationsat 290 cm − and 330 cm − . With similar reasoning to that used for the Eu123 data, weestimate 2∆ ≈
400 cm − , or 50 ± − (70 meV) may be the renormalisation peak, in which case thesedata would give us the estimate 2∆ ≈
70 meV. A g + B g Similar to the long count-time B g spectra discussed above, the A g + B g spectra in Fig. 8.3clearly show a suppression of the ERS at low ω for T < T c . Fig. 8.12 plots the differencebetween the T = 9 K data and several higher temperature data. In this geometry thereare many more phonon modes complicating the analysis, nevertheless a peak feature canbe identified in the Yb123 data presented in panel (a), as annotated with an arrow. Thisgives us the estimate for Yb123 of 2∆ = 55 ± A g + B g ERS scattering of Eu123 in Fig. 8.12b we estimate 2∆ is between 40 meV and 50 meV - lower than expected from the previous two estimates forthis material. The phonon at 303 cm − is significantly more intense at low temperatures,Fig. 8.8, and its extra spectral weight may be the cause this feature we see in the differencedata around 45 meV.Another approach is to fit the entire spectrum using a Green’s function method de-scribed above (Sec. 8.2.4) that incorporates both phonon and electronic contributions tothe Raman intensity as well as the interaction between them [244, 257, 212]. Fig. 8.13 shows With these data there is a significant background linear in ω . It is largest in the 9 K data and wesuspect that it may be from vacuum grease on or around the sample. .3. RESULTS c ’’S - c ’’N w ( c m - 1 ) ( m e V ) c ’’ (arb. units) w ( c m - 1 ) c ’’ (arb. units) w ( c m - 1 ) Figure 8.11: (Top) The difference between spectrum long count-time B g spectrum at 20 Kand 90 K of optimally-doped Eu123. The inset shows these spectra as well as data takenat T = 295 K. (Bottom) B g spectrum at 9 K on optimally-doped Eu123.68 CHAPTER 8. GAP ESTIMATES FROM RAMAN SPECTROSCOPY c ’’ K- c ’’ T > K w ( c m - 1 ) ( a )Y b 1 2 3 ( m e V ) ( m e V ) c ’’ K- c ’’ T > K w ( c m - 1 ) ( b )E u 1 2 3 Figure 8.12: A g + B g difference spectra of (a) Yb123 and (b) Eu123 (with representativeraw spectra shown in Fig. 8.3 and Fig. 8.8 respectively). These data clearly show thereduction in spectral weight for small ω at low temperatures due to the opening of anenergy gap. In (a) an arrow indicates a peak in the ERS. No such feature is clearly visiblefor the Eu123 data.fitted data and the g ( ω ) parameter derived from this procedure for Yb123 and Eu123 dataat various temperatures. The signal-to-noise ratio of Nd123 A g + B g data was not goodenough for this analysis. The energy where g ( ω ) is maximum can be interpreted as 2∆ .As such, we obtain an estimate of the 2∆ = 50 ± B g Fig. 8.14 shows the results of long count-time scans in B g scattering geometry on Eu123.There is a weak suppression of spectral weight below 180 cm − which could be indicativeof the gap, but it is not very compelling. Unlike B g scattering, in B g we do not observeclearly identifiable evidence of an energy gap in any of the materials we have studied.These observations are similar to those of others on near-optimally-doped Y123 [71, 7].This is a disappointing result as it means we are not able to disentangle the contributionsto 2∆ from 2∆ and E P G using Raman spectroscopy alone. The B g gap feature is much more prominent, for some reason, in optimally doped Bi2212 [250, 71,246, 7] and Hg1201 [247, 241] .3. RESULTS ( a )Y b 1 2 3( m e V ) g( (cid:1) ) (arb. units) w ( c m - 1 ) ( b )E u 1 2 3 Figure 8.13: The g ( ω ) parameter from the fitting procedure described in the text, Sec. 8.2.4,for data at various temperatures for (a) Yb123 and (b) Eu123. g ( ω ) is the imaginary partof the electronic response (Sec. 2.4.2) and the energy where g ( ω ) is maximum can beinterpreted as the energy of 2∆ [212]. B c ’’ (arb units) w ( c m - 1 ) ( m e V ) Intensity (Cts) [n( w ,T) +1] w ( c m - 1 ) Figure 8.14: Scaled B g spectra for optimally-doped Eu123 at T = 175 K (black) and T = 9 K (blue) and their difference (red). The raw data is shown in the inset. Any ERSfrom these data, and all other B g data, are too weak to unambiguously identify.70 CHAPTER 8. GAP ESTIMATES FROM RAMAN SPECTROSCOPY
Table 8.2 presents a summary of the various gap estimates on optimally-doped Yb123 andEu123 and oxygen-loaded Nd123 single crystals. For comparison, literature results fromoptimally-doped Y123 are also included in the table. xx and yy for the two A g + B g Y123values indicate the anisotropy in 2∆ found by Limonov et al. [212] and also indicates thepotential precision of these measurements.Material T c (K) 2∆ (meV) Symmetry Method (Reference)Yb123 90.5 39 < < A g + B g Phonon shifts60 ± B g ERS55 ± A g + B g ERSEu123 94.5 60 < A g + B g Phonon shifts70 ± B g ERS50 ± A g + B g ERSNd123 88.5 2∆ < A g + B g Phonon shifts57 ± B g ERSY123 93 52.7 ( xx ) 55.8 ( yy ) A g + B g [212]67 ± B g [7]67 . ± . B g [265]40 . ± . A g [265]Table 8.2: A summary of estimates of the spectral gap, 2∆ , for Yb123, Eu123 andNd123. Recall 2∆ is a pair-breaking peak, see Sec. 2.2.4. The estimates are derived eitherfrom phonon energy renormalisation (labelled ‘Phonon shifts’) or from electronic Ramanscattering (labelled ‘ERS’). The symmetry condition by which the spectra used to estimatethese values were taken is also listed. For comparison, literature values of 2∆ for optimallydoped Y123 are included. For a given material, there is general agreement between variousestimates 2∆ .We find that for a given material, our various estimates of 2∆ are consistent with eachother with the exception of the Eu123 A g + B g ERS estimate. In our view the value fromthe B g ERS data is more reliable. Further, our estimates are also similar to the equivalentvalues measured in Y123 [212, 265, 7]. Nd123 is approximately 10 meV lower than Y123,however it is our view that the precision of these estimates is generally inadequate, and inthis case too poor to draw conclusions about the relevance of this discrepancy. Moreover,we are unable to comment on any systematic variation of 2∆ with the Ln ion size. Forexample; for Ln123 ∆ T c ≈
10% at most and if we assume T c ∝ T mf c ∝ the precision in .4. SUMMARY AND DISCUSSION OF RESULTS Cu(2)-O(2,3) Bond Length [nm] E n e r g y [ m e V ] E PG ∆ q ∆ + E PG Cu(2)-O(2,3) Bond Length [nm] E n e r g y [ m e V ] E PG ∆ / q ∆ − (2 / E PG ) Figure 8.15: An estimate for the dependence of ∆ , E P G and ∆ [ B g ] (Left) and 2∆ [ B g ](Right) on the Cu(2)-O(2,3) bond length, r , (the super-exchange path length). Thesevalues are relevant to Ln123 at p = 0 . measurement must be better than ≈ consider the possible effect of the pseudogap.2∆ is expected to scale with T c (or more correctly, T mf c ) whereas the pseudogap isthought to originate from AF fluctuations and thus has energy scale J . To see how thesetwo quantities are affected by Ln substitution in Ln123 we express each as a function ofthe Cu(2)-O(2,3) bond length, r (the superexchange path-length).There is some evidence that J sets the energy scale for E P G as E P G ≈ J (1 − p/ . J and r determinedin Chapter 6, to arrive at the following estimate for the Ln123 series, E P G ( r ) ≈ m P G r + c P G with m P G = − ±
30 meV/nm and c P G = 105 ± p = 0 .
16. More recent datasuggests the doping dependence is sub-linear with E P G ( p ) ≈ (1 − p/p crit ) . [54], in whichcase m P G = − ±
40 meV/nm, c P G = 162 ± p = 0 .
16. We use the former valuesof m P G and c P G in Fig. 8.15. We can estimate the dependence of ∆ on r by assumingthat a similar relation to 2∆ = 4 . k B T mf c holds across the Ln123 series [64]. Note thatat optimal doping T c ≈ T mf c [56]. Next we use the experimental dependence of T c on r (refer to Sec. 2.1.4) to estimate ∆ ( r ) = m ∆ r + c ∆ with m ∆ = 330 ±
30 meV/nm and c ∆ = − ± r relevant to the Ln123 series.The energies of E P G and ∆ have opposite dependences on r such that, depending on thevalidity of the assumptions above, the spectral gap in B g symmetry, 2∆ = 2 q ∆ + E P G , It is interesting that a pseudogap like state has been seen in pnictide superconductors and is alsocorrelated to anti-ferromagnetic fluctuations [266]. CHAPTER 8. GAP ESTIMATES FROM RAMAN SPECTROSCOPY is expected to be approximately independent of r and therefore Ln ion substitution.These ERS measurements have the potential to determine both the superconduct-ing and pseudogap energies. By a simple change in the polarisation selection and rel-ative sample alignment, it is possible to measure 2∆ either at the anti-nodes where2∆ = 2 q ∆ + E P G , or at the nodes where 2∆ ≈ / q ∆ − ( / E P G ) . With B g data toaccompany the B g data, the ∆ and E P G contributions to 2∆ can in principle be isolated.Furthermore, 2∆ in the B g geometry is expected to be strongly dependent on ion-size asshown in Fig. 8.15. Unfortunately, we were not able to observe ERS in the B g geometry.These polarisation selection rules were used in the same way to measure and unam-biguously identify two-magnon scattering. There is the possibility to combine ERS andtwo-magnon scattering measurements to study, in a very direct way, the relation between J and the pseudogap, and indeed with the superconducting energy gap also. One can en-visage then a combined two-magnon scattering and ERS study where doping and ion-sizeare systematically varied. Such a study would provide valuable evidence of the nature ofthe pairing interaction and the pseudogap.What this chapter shows, in combination with the two-magnon scattering data for p > et al. have very recently carried out such measurementson a single crystals of Hg1201 at three doping states [211]. Their study is exemplary andtheir results are fascinating. In B g geometry they clearly observed temperature dependenttwo-magnon scattering from underdoped to a p = 0 .
19 sample. The two-magnon scatteringintensity and energy appears to be correlated with the suppression in ERS at low ω thatthe authors associate with the pseudogap. Furthermore, both these features are evidentabove T c in the underdoped samples. The authors describe the correlation between thesetwo features as a “feedback effect” such that magnon excitations are modified in the su-perconducting state. Similar behaviour is observed for the prominent ERS renormalisation .4. SUMMARY AND DISCUSSION OF RESULTS J with ion-size in Sec. 9.5.6 in the followingchapter. The method we describe can be readily extended to study the doping dependenceof J and E P G as well.74
CHAPTER 8. GAP ESTIMATES FROM RAMAN SPECTROSCOPY hapter 9Conclusions and Discussion
Summary
We begin this chapter by summarising key results from previous chapters. We then discussthree possibilities for the salient effect of ion size on T max c . The most attractive possibilityis that the polarisability plays an important role in determining T max c because this cannaturally resolve the opposite effect of internal pressure and external pressure.We then discuss how ion-size, polarisability and T c might be related within the frame-work of the Hubbard model where superconductivity emerges from a strongly correlatedelectronic system. An alternative connection between polarisability and T c is also proposedwhere Cooper pairing is based on the exchange of coherent, quantized waves of electronicpolarisation (polarisation-waves).In Sec. 9.5 we discuss and suggest some studies that would utilize ion-size effects tofurther elucidate the nature of the pseudogap, parameters relevant to T c and the pairingmechanism in the cuprates.The chapter concludes with a very brief summary of some key findings from this workand how we see them relating to the wider field of HTS research. We have presented the argument that internal pressure, as induced by isovalent ion-substitution, decreases T max c in the cuprates. In contra-distinction, external pressure in-creases T max c . This work seeks to understand the salient physical difference between thesetwo pressures by studying ion-size effects on various important physical parameters.17576 CHAPTER 9. CONCLUSIONS AND DISCUSSION
In Chapter 4 we used a simple materials variation approach to study the effect of ion-size and disorder on T c Bi2201. We found that ion substitution can increase T c despiteincreasing disorder and concluded that while disorder does play a role in decreasing T c there is a comparable effect arising from changing ion-size.In Chapter 5 we used DFT calculations for undoped ACuO for A={Mg, Ca, Sr, Ba} toinvestigate the effect of ion-size on the electronic properties in this model cuprate system.We found that larger ions move the van Hove singularity (vHs) in the DOS closer to theFermi-level. This finding is consistent with an interpretation of the ion-size affecting T c via the density of states.In Chapter 6 we measured the antiferromagnetic superexchange energy, J , by two-magnon Raman scattering in undoped cuprates while systematically altering the internalpressure by changing ion-size. We then compared the internal-pressure dependence of J with data in literature for the external pressure dependence of J . From these data weshowed that J is likely un related to T max c : J and T max c anti-correlate with internal pressureas the implicit variable and correlate with external pressure as the implicit variable. Weconcluded that some other physical property is dominant in setting the value of T max c .In Chapter 7 we tested the idea that ion-substitution-induced disorder is responsiblefor the variation in T max c with ion-size. We presented µ SR measurements of the superfluiddensity in YBaSrCu O − δ showing the nice result that YBaSrCu O − δ is a boring, well-behaved material! The superfluid densities are consistent with those previously reportedfor pure and Ca-doped Y123. Consequently, the lower T max c seen in this compound, andthose with lower Sr content, is a genuine ‘ion-size effect’ (or ‘internal pressure effect’).In Chapter 8 we sought to measure energy gaps in the single-electron dispersion ofour ion-substituted cuprate materials. Within the accuracy of our estimates, we did notfind any systematic internal-pressure dependence of the energy gap. We did find howeverthat the gap values for our ion-substituted samples are similar to the well-characterisedYBa Cu O − δ cuprate. A better approach to addressing this important question needs tobe found and our proposal for such an approach will be discussed later in this chapter.As will be discussed in detail below, the cumulation of our studies leads us to believethat the polarisability plays a central role in superconductivity in the cuprates. .2. DISCUSSION: THE ION-SIZE EFFECT ON T C T c What is the salient effect of the ion size on T c ? We identify several possibilities that wewill discuss in turn below;1. Enhancement of a competing electronic state, e.g. the pseudogap.2. Modification of the DOS.3. Altering the dielectric properties of the material (polarisability). This in turn couldplay a role in altering the screening or modifying the spectrum of polarisation waveexcitations. As discussed in Chapter 6, the effect of ion-size on T c may be indirect, such as via anelectronic order that competes with superconductivity. The most likely candidate hereis the pseudogap as there is evidence from inelastic neutron scattering [87] and specificheat data [58] that the pseudogap energy is set by J . As we have shown in Chapter 6and Sec. 2.1.4, J is in turn sensitive to the ion-size-modified superexchange path-length.Alternately, it is possible at lower doping, that a charge ordered phase is the competingphase [57] especially modified by ion size.However, when we consider the effect of external pressure on J and T c shown in Fig. 6.6and reproduced below in Fig. 9.1 (a), this interpretation becomes less likely. For example,if we took the view that the suppression of T c with ion-size was due to an enhancementof the pseudogap phase (or, similarly a stripe phase) from an increasing J , we would needto explain why both T c and J increase under external pressure. To put it another way; ifone expects ∆ to scale with J (magnetic pairing scenario [6]) and E P G scales with J asobserved, then in the competing gap scenario after Bilbro and McMillan [267] we wouldexpect T c ∼ q ∆ − E P G so that T c will still scale with J as both ∆ and E P G themselvesdo. We have shown that it does not for the Ln123 system: T max c anti-correlates with J when ion-size is the implicit variable and only correlates with J when external pressure isthe implicit variable.78 CHAPTER 9. CONCLUSIONS AND DISCUSSION
Another hypothesis is that these ion-size substitutions introduce structural distortion whichin turn distorts the electronic dispersion in such a way to increase the DOS by shifting thevHs closer to E F . In a BCS-like framework, a high DOS is an important requirement forhigh T c , irrespective of the pairing mechanism [105, 201, 199]. We tested this idea withDFT calculations of the band-structure for the infinite-layer cuprate ACuO where A ={Mg, Ca, Sr, Ba} and the results are consistent with this hypothesis.On the experimental side, Kim et al. showed that for optimally-doped Bi2201 N ( E F )was suppressed with smaller ion-size [85]. However they also demonstrated that the de-crease in N ( E F ) could be understood to result from an enhanced pseudogap as deter-mined from E P G = k B T ∗ with T ∗ from resistivity measurements. We speculate that thechanges in E P G in their Bi2201 system are caused by changes in J which result from thesmaller superexchange path-length with reduction in ion-size. From our own preliminaryZn-substitution studies on YBaSrCu O − δ presented in Fig. 9.3 it would also appear thatsmaller ion size depletes N ( E F ). But again this may be a pseudogap effect and furthersystematic studies, such as described below in Sec. 9.5.6, would clarify this.It is possible then that the increase of T max c with larger ion size is a result of an enhanced N ( E F ). But does this offer a clear understanding of why internal and external pressurehave opposing effects on T max c ? We do not see how it could, but this is yet to be testedin our calculations. Of course there remains the possibility that external pressure affects T c in a manner distinct to internal pressure. Furthermore, the subtle relationship betweencrystallographic and electronic structure prevents us from commenting more broadly oncorrelations such as that between T max c and the bond valence sum V + shown in Fig. 2.12. However, in all this we must bear in mind the simplicity of the systematic evolution of T max c with V + , shown in Fig. 2.12, that suggests we may have to look elsewhere to locateits underlying origins. One arena where ion-size plays a central role is in the dielectricproperties, where the ionic polarisability varies roughly as the cube of the ion size [268].This suggests two possibilities (i) the material dependence of T c results mainly from screen-ing of longer-range repulsive interactions in a magnetic pairing scenario [269, 210] (ii) analternative and relatively unexplored idea that pairing in the multi-ion cuprates might be Or conversely, the decrease of T max c with internal pressure .2. DISCUSSION: THE ION-SIZE EFFECT ON T C h − π P i n i α i i − where the sum is over all ions i , n i arethe volume densities of these ions and α i their polarisabilities. The factor π is ultimatelydependent on structure, the dielectric constant more generally being replaced by a dielec-tric matrix and, in a fuller treatment, the enhancement factor is replaced by frequency-and momentum-dependent terms [270]. The dielectric constant for bound electrons can bewritten as; (cid:15)(cid:15) = 1 + 4 π P i n i α i − π P i n i α i (9.1)The product n i α i is the ionic refractivity and we call π P i n i α i the refractivity sum . Herewe focus just on the contributions to the refractivity sum from the non-cuprate layers andin Fig. 9.1 (b) we plot T max c versus π P i n i α i for Ln(Ba,Sr) Cu O where the sum is overLn, Ba, Sr, and the apical O(1) oxygen. Red squares summarise the effects of changing Lnand blue diamonds the effects of progressively replacing Ba by Sr. These are the internalpressure effects. Polarisabilities are from Shannon [268].Not only does T max c correlate with the π P i n i α i with internal pressure (or ion-size) asthe implicit variable, but moreover, this also resolves the paradox of the opposing effectsof internal and external pressure. On the one hand, increasing ion size (decreasing internalpressure) increases the polarisability whilst, on the other hand, increasing external pressureenhances the densities n i , in both cases increasing the dielectric enhancement factor. Totest this we plot in Fig. 9.1(b) T max c versus π P i n i α i for YBa Cu O at 1 bar and 1.7, 4.5,14.5 and 16.8 GPa (green squares) where we have assumed to first order that only the iondensity n i , and not the ionic polarisability, α i , alter under pressure. The correlation withpolarisability is now preserved over a range of T max c from 70 K to 107 K, including bothinternal and external pressure. This is a key result and it also now links to the correlationwith V + shown in Fig. 2.12. The additional role of the apical oxygen bond length (which alsocontributes to the value of V + ) has yet to be clarified, but it possibly plays a supplementaryrole in controlling the large polarisability of the Zhang-Rice singlet [274]. Inclusion of therefractivities from the CuO layers adds a further 0.4 to the refractivity sum bringingthese systems close to the conditions for polarization catastrophe where π P i n i α i → CHAPTER 9. CONCLUSIONS AND DISCUSSION
Figure 9.1: (a) T max c plotted versus J for single crystals of LnBa Cu O (red squares)and for YBa − x Sr x Cu O (blue diamonds) with x = 0 . . T max c anticorrelateswith J where “internal pressure” (blue trajectory) is the implicit variable. The greensquares show that T max c versus J under external pressure (green trajectory) is effectivelyorthogonal to the behaviour for internal pressure. (b) T max c plotted against the refractivitysum (4 π/ P i n i α i for the ions in the non-cuprate layer for LnBa Cu O y (red squares, Ln= La, Nd, Sm, Eu, Gd, Dy, Yb) and YBa − x Sr x Cu O y (blue diamonds, x = 0 , . , . , . T max c versus (4 π/ P i n i α i for YBa Cu O y under external pressure, revealing a correlation whichremains consistent with that for internal pressure. .3. IMPLICATIONS FOR PAIRING MECHANISMS α Ra = 8 . for the Radiumion, Ra , we deduce an implied T max c of 109 ± Cu O x and about 117 ± Cu O x . On similar grounds T max c for HgRa Ca Cu O should be about 150 K.These inferences can be tested, though not without their challenges! We have presented the idea that the opposite effects of internal and external pressure on T max c can be understood by considering the polarisability. This implies that the polarisabil-ity of the material plays an important role in determining T max c . But, if there is a causalrelationship between the two, how are we to understand it?In this section we present two broad ways in which to understand how the polarisabilitymight effect T max c .In the first approach, Sec. 9.3.1, we begin with the currently most successful micro-scopic model for superconductivity in the cuprates which is based on the Hubbard modelof strongly correlated systems. We then discuss how, via screening, the polarisability (mod-ified by ion-size and external pressure) might impact relevant parameters in the model. Alarger polarisability may enhance T max c by screening repulsive longer-range interactions inthe CuO layers.The second approach invokes the alternative, and relatively unexplored, idea that pair-ing in the cuprates might be attributable to the exchange of so-called polarisation-waves.In this scenario, discussed in Sec. 9.3.2, polarisation-waves are the boson that mediatesCooper pairing and the overall polarisability will set the energy scale of the pairing boson.In the following section, Sec. 9.5, we propose further studies to measure various param-eters discussed here and, more generally, to further investigate ion-size and pressure effectsin the cuprates. It was first shown by Kohn and Luttinger [275] that the superconducting state can be theground state of a homogeneous electron liquid with purely repulsive short-range interactions[276]. Substantial subsequent investigation has validated the idea of Kohn and Luttingerand extended it to more complex and realistic systems, see for example [270, 269] and82
CHAPTER 9. CONCLUSIONS AND DISCUSSION references therein. The idea is that particle-hole fluctuations screen (or renormalise) thebare repulsive interaction and can lead to an effective attractive interaction. The effectiveattractive interaction is weaker than the bare repulsive interaction, but if the attractiveinteraction is longer-range while the repulsive interaction is only short-range (or on-site),it can lead to pairing.This can be called intrinsic superconductivity as it does not involve any mediatingbosons, e.g. phonons, or even the periodic potential of a lattice - simply the presence of‘valence’ electrons.Intrinsic superconductivity has been found as a solution to the Hubbard model with onlynearest-neighbour interactions - see [210, 277, 278] and references therein. The Hubbardmodel is a model to treat correlations between electrons on a lattice. The Hamiltonian canbe written as; H = − t X h i,j i ,σ c † i,σ c j,σ + U X i n i, ↑ n i, ↓ (9.2) i and j denote adjacent lattice sites and σ denotes the spin of the quasi-particle - spinup, ↑ , or spin down, ↓ . c † ( c ) is a quasi-particle creation (annihilation) operator and n a number operator. U is an on-site Coulomb repulsion term - the energy cost associatedwith doubly occupying a single lattice site. t is a hopping parameter and represents theenergy associated with moving a quasi-particle, without a flipping its spin, to its nearest-neighbour site. The Hamiltonian in Equation 9.2 can be extended to include hoppingbetween next-nearest-neighbour sites and even longer-range interactions. These hoppingparameters are represented by t , t , etc. and are illustrated in Fig. 9.2. Note that herethe nearest-neighbour superexchange energy is J = 4 t /U and an expression for the next-nearest-neighbour superexchange energy is J = 4 t /U .Gull et al. [278] calculate exact numerical solutions to the Hubbard model on finite-sizeclusters of up to 16 ×
16 sites (which is about the practical size limit for exact numeri-cal solutions). They control the ratio
U/t (the interaction strength) and the doped holeconcentration, choosing parameter values relevant to the cuprates. The results of theircalculations qualitatively capture most of the experimentally observed behaviour in thecuprates: with increasing doping there are transitions from an antiferromagnetic Mott in-sulator phase to a pseudogap phase, to a co-existing (and competing) superconducting andpseudogap phase, to a pure superconducting phase and finally to a Fermi-liquid-like phase . Refer to Fig. 2.7 for a schematic phase diagram of the cuprates. .3. IMPLICATIONS FOR PAIRING MECHANISMS bac
Cu(2)O(2)O(3) p x p y d x -y t''tt'U Figure 9.2: Illustration of the higher-order hopping integrals relevant to Hubbard model.Additionally, they find T max c ≈
100 K and an increasing 2∆ /T c ratio with underdoping.Thus the essential aspects of the phase behaviour of the cuprates are captured by theproper treatment of electron-electron interactions within the Hubbard model with nearest-neighbour interactions only ( t = 0). However, this leaves unanswered the question of whythere is such large variability in superconducting properties between families of cuprates,and indeed within a single family of cuprates via ion-size and pressure effects. This is nottoo surprising given the many simplifying assumptions of a Hubbard model of the cuprates.If this significant materials variability is to be captured by the Hubbard model, it is clearlynecessary to at least include longer-range interactions via t and t or the related J and J . The idea that materials variability of T max c results from t is not new. For example,Pavarini et al. [24] correlated T max c with a composite parameter proportional to t /t for avariety of cuprates. In their work, the apical oxygen bond length was shown to significantlyaffect t /t . t , t , etc. alter the electronic dispersion (see e.g. [279, 121]) and thus reshapethe Fermi-surface. The work presented in this thesis indirectly shows the importance ofthese extended interactions in providing an understanding of the systematic variation in T max c with ion-size. But what is the role of ion-size (and polarisability) and can it beunderstood within this Hubbard-model framework?Perhaps the role of t , t , etc. is merely to modify the DOS by shifting the vHs -as discussed earlier and in Chapters 4 and 5. Alternatively, these parameters relating to Which are also called extended interactions . CHAPTER 9. CONCLUSIONS AND DISCUSSION longer-range interactions may be related to T max c in a more fundamental way and this willbe subject of our discussion in the following section. A long-standing perceived difficulty with the Kohn and Luttinger picture has been thelength scale of the repulsive interactions. For example, recently Raghu et al. [269] showqualitatively that longer-(real-space)range interactions are always detrimental to supercon-ducting pairing resulting from short-range (on-site) interactions.When we consider core electrons, or in the case of the cuprates, the additional ionsoutside of the CuO layers, we introduce a complementary channel that adds additionalscreening and quite possibly a new pairing channel that involves excitations of this polar-isable media as discussed below in Sec. 9.3.2.Firstly we discuss implications of additional screening. The isolated CuO planar arrayis electrostatically uncompensated and as such it cannot constitute a thermodynamic sys-tem. It is necessary to include the compensating charges lying outside of the CuO planein any thermodynamic treatment and these will also contribute to electron-electron inter-actions within the plane. These charges also reside on ions that are notably polarisable ,and they contribute to the high background dielectric constants observed in the cuprates[280].The highly-polarisable non-CuO layers screen longer-range interactions in the CuO layer. Recently, the effect of the charge reservoir layers on T max c via screening has beenexplored Raghu et al. [210]. Thus we can possibly understand the correlation of T max c with the refractivity sum shown in Fig. 9.1 as resulting from more effective screening oflonger-range interactions in the CuO layer [210] that are detrimental to intrinsic super-conductivity [269]. This is an attractive explanation for our data, especially as it connectswith a large body of theoretical work on pairing interactions specific to the cuprates.We do not yet have any measurements of how U , t , t and t vary with ion-size (orindeed with external pressure) and hence how the interaction strength U/t varies. The roleof future studies should focus on measuring and understanding the systematic variationof these parameters. In Sec. 9.5 below we propose studies to measure these importantparameters. In general, ions with larger atomic numbers, and hence a greater number ‘core’ electrons which occupya larger volume, are more polarisable [268]. .4. OUTLOOK
Coherent polarisation excitations
Building on the intrinsic pairing mechanism discussed above, it may be necessary to con-sider the more realistic inhomogeneous electron liquid where there is now a distinctionbetween localised ‘core’ electrons and the delocalised ‘valence’ electrons. As such, Atwaland Ashcroft include the effect of the lattice not by way of collective excitations of the ions(phonons) but instead by way of the coherent dynamics of the ‘core’ electrons associatedwith each ion [270]. This alternative, and relatively unexplored, idea is that pairing inthe cuprates might be attributable to the exchange of bosons comprising coherent, quan-tised excitations of polarisation associated with the core electrons [270, 271] and/or thecharge-reservoir-layer ions - so-called polarisation waves.This idea is analogous to the BCS-like phonon-mediated pairing of conventional su-perconductors, but with polarisation waves in place of phonons. The energy scale of thepolarisation waves is governed by the effective plasma frequency of the total ensemble ofcore electrons in and out of the CuO layers and is of order 1 . ≈
30 meV [281]. WhatAtwal and Ashcroft find is that the exchange of polarisation waves, associated with thecore electrons, by valence electrons can give rise to Cooper pairing and high transitiontemperatures, especially in the d -wave channel. For example, they make a comparison of T c values for homogeneous (valence only) electron system and inhomogeneous (valence andcore) electron system calculated with BCS-Eliashberg theory and find T c is up to 50 timeslarger in the latter case [270].While there is considerable scope to develop this idea, it is expected that the polaris-ability will set the energy scale of the pairing boson.Note the distinction between T c enhancement from screening by incoherent polarisationfrom our first discussion, with the enhancement of T c by coherent (quantised) polarisationwaves in the second. In both cases however the polarisability of the material is a keyproperty. The challenge remains to distinguish these. The cuprates are complicated but do show some clear systematics. This thesis has shownthere is good reason to believe that dielectric properties lie behind the systematic vari-ations in T max c associated with pressure and ion-size effects. This is not to say that thepolarisability is the only salient parameter, indeed the DOS is likely to set an additionally86 CHAPTER 9. CONCLUSIONS AND DISCUSSION important energy scale, but it should prompt two responses;• To consolidate, or disprove, this interpretation with further experimental and theo-retical studies. We briefly discuss some ideas in this regard below.• To use this understanding in the search for quite novel superconducting materials.Perhaps superconductivity is to be found in a metallic medium with large N ( E F )coupled with a highly polarisable medium. The role of quasi-two-dimensionality maysimply be to enhance the DOS through the introduction of a vHs.This thesis has also shown that by studying ion-size and pressure effects we can gainadditional insights into the physics of the cuprates and unconventional superconductivity. At the end of each chapter we discussed possible follow-on work. We will not repeat thathere, but instead discuss some additional studies we believe would be instructive and/orimportant.
Earlier in this chapter we discussed the recent work of Gull et al. [278] which demonstratesthe ability of the Hubbard model in capturing essential aspects of the phase behaviour.Here
U/t , the interaction strength, is the key variable, yet we do not know the systematicion-size effect on U or t . From the work of Sawatzky et al. [282] we expect enhancedscreening from the more highly-polarisable larger-ions to lower U , but this has not beenmeasured. We see this as a priority and plan to undertake UV absorption measurementsto determine the charge-transfer gap ( ∼ U ( ∼ J = 4 t /U in the one-band Hubbard model, we can use our two-magnon scatteringmeasurements of J together with the UV absorption data to estimate t as a function ofion-size as well.Furthermore, that discussion also made apparent the need to better understand the ion-size (and external pressure) effect on longer-range interactions as characterised by t , t or J , J . One possibility we are currently exploring is whether Raman B g spectra contain This idea is motivated by the separate region of the Brillouin Zone, the nodal region, probed by the B g geometry. .5. FURTHER STUDIES and E P G as a function of ion-size. In this regard we discuss specific heat (Sec. 9.5.2),ellipsometry (Sec. 9.5.4) and impurity scattering (Sec. 9.5.6) studies below. Finally theDOS could be investigated (in additional to a computational DFT approach) using specificheat (Sec. 9.5.2), NMR and/or impurity scattering studies (Sec. 9.5.6).
Collaborators at Cambridge University have a unique differential heat capacity rig. The fulltechnical description of their differential method can be found in reference [284]. With thismethod the phonon contribution can be very accurately subtracted so that the electronicspecific heat can be measured to within 0 . − .K − . Their rig is capable of sampletemperatures between 1 . γ , and from this the N ( E F ) and any enhancement of γ fromthe so called ‘bare band structure’ value, γ b , that results from electron-boson coupling, (ii)the condensation energy U , (iii) E P G [58] and (iv) the mean-field transition temperature T mf c [54] is believed to scale with 2∆ . Resonant, Inelastic, X-ray Scattering (RIXS) measurements [227, 6] allow one to explorethe magnetic excitation spectrum across a wide area of the Brillouin zone. The aim wouldbe to probe with RIXS the full dispersion of paramagnon excitations (thus elucidatinglonger-range magnetic interactions J , J , etc.) to see if it is compatible with the observedion-size effects on T c . Working from the full dispersion, T c may be calculated under certainassumptions as shown by le Tacon et al. [6]. A key question is whether the variationin dispersion with ion size follows the observed T c variation. High quality (though notnecessarily large, 1 × is sufficient) single crystals with clean surfaces, or high-quality88 CHAPTER 9. CONCLUSIONS AND DISCUSSION epitaxial thin films are needed for these measurements. In addition, there is the possibilityof detecting polarisation waves with RIXS at the barium edge. Such a measurement ofpolarisation waves would be pioneering and would necessarily draw on a collaboration withRIXS experts.
Infrared optical spectroscopy has previously provided crucial information about the chargeand lattice excitations of the cuprate HTSC [285, 286]. It has the advantage of a large probedepth of the light which ensures the bulk nature of the observed phenomena. Significantly,the energy resolution and accuracy in the determination of the optical constants of infraredspectroscopy in combination with powerful sum rules enables one to identify the specificinteractions of the charge carriers and their underlying energy scales. In particular, theellipsometry technique, which is self-normalizing and does not require a Kramers-Kroniganalysis procedure, allows one to determine with high accuracy even small changes of thedielectric function, (cid:15) , for example versus temperature or doping [287, 288] or as we proposeto do, ion-size. c -axis conductivity measurements on Ln123 The Bernhard group has previously established that measurements of the far-infrared (FIR) c -axis response allow one to identify the superconducting energy gap and pseudogap sepa-rately [86, 41, 74, 56]. They have shown that these two ordering phenomena have distinctspectral features. The superconducting energy gap yields a spectral weight shift toward adelta function at zero frequency that describes the response of the superconducting (SC)condensate. Its energy scale, 2∆ , can be identified from the onset of the suppression ofthe optical conductivity. In contrast the pseudogap (PG) gives rise to a spectral weightshift to higher energy into a very broad peak above the gap edge at E P G .The c -axis conductivity is also a direct measure of the electronic coupling between themetallic and superconducting CuO layers. In the normal state the anisotropy can be easilydetermined from the absolute value of the electronic part of the c -axis conductivity (which isusually only weakly frequency dependent). In the superconducting state it can be deducedfrom the position of the so-called Josephson plasma edge (and for bi-layer materials suchas our model system, of the transverse Josephson-plasma mode) which is proportional tothe c -axis component of the superfluid density and the c -axis magnetic penetration depth. .5. FURTHER STUDIES layer, has long been known [24] and may yet play an important role in controllingthe huge polarisability of the Zhang-Rice singlet [274]. Knowledge of systematics of theelectronic coupling between the CRL and CuO layers in our model system may prove animportant piece in the puzzle.Thus, it is would be desirable to undertake systematic measurements of the c -axisconductivity in our model LnA Cu O y system to reveal the effect of ion-size substitutionon the important parameters of ∆ and E P G and c -axis coupling. In-plane conductivity on crystals and thin films
The in-plane conductivity represents the coherent dynamics of the charge carriers thatare confined to the CuO layers. From the normal-state data one can directly deduce thefrequency-dependent scattering rate and mass enhancement due to the interaction of thecharge carriers. Also one can get the screened and, with some additional information,the unscreened plasma frequency of the charge carriers. These parameters are importantmeasures of the interaction of the charge carriers, which may be strongly affected by ion-sizesubstitution via the varying polarisability of the ions; e.g. screening due to large Ln-ionsor reduced screening because of Sr substitution for Ba. As discussed above, it has beensuggested screening plays an important role in raising T c and improving SC properties bysuppressing longer-range, repulsive interactions [269]. To our knowledge, no systematicmeasurements in this direction have been done.In the superconducting state, one can directly obtain the superconducting condensatedensity and its a - b anisotropy (on de-twinned single crystals) from the in-plane data. Thesemeasurements would complement the further µ SR studies discussed at the end of Chapter 7. Cu O − δ Sample quality was an issue in our µ SR studies on YSr Cu O y (YSCO). There were mag-netic impurities from K-Cl rich inclusion phases derived from the KClO oxygen sourceused in the synthesis process. The magnetism from these phases masked the componentarising from the vortex lattice in the superconducting YSCO phase in µ SR. For our future µ SR studies on YSCO it is clearly desirable to significantly improve the sample quality.In fact, it would be highly desirable to have well-oriented (preferably with the c -axisaligned perpendicular to the substrate) thin films of pure YSCO. To this end our MOD90 CHAPTER 9. CONCLUSIONS AND DISCUSSION synthesis of YSCO on a (001) SrLaAlO substrate at atmospheric pressures is an encour-aging start, see Sec. 3.2.4. YSCO is a member in the Ln(Ba,Sr) Cu O y system under alarge ‘internal-pressure’ (or rather, it has one of the smallest ion-sizes, LuSr Cu O y beingthe extreme). Such extreme cases are important for determining systematics.Such samples would facilitate two important measurements of the YSCO material (i)Two-magnon Raman scattering, not only in the un-doped YSCO6 sample, but also dopingdependent studies, and (ii) RIXS, a variant Raman-like technique involving synchrotronradiation that allows k -space selection and thus has the ability to map out the dispersion ofmagnetic excitations over almost the entire Brillouin zone [6]. Such thin films would also beamenable for precise resistivity measurements to investigate superconducting fluctuationsabove T c , from which the c -axis coherence length can be inferred. High quality resistivitymeasurements can also provide an estimate of the pseudogap, T ∗ = E P G /k B , where T ∗ isthe temperature below which the pseudogap opens marked by a deviation from linearityin the resistivity.As such it is desirable and most likely feasible to grow well-oriented YSCO thin-films foron-going ion-size studies. Clearly internal and external pressure effects on both fluctuationsand the pseudogap should prove highly instructive. Whilst the MOD synthesis processesmight be tuned to grow high-quality, aligned films of YSCO, we believe it would be worthtrying pulsed laser deposition (PLD) as well. PLD is a technique where one can have veryfine control over the growth conditions. However it requires a sophisticated set up andan experienced technician. The attraction of PLD is the fine control of growth conditionsand that, in principle, PLD can produce nearly atomically perfect thin films [289]. Analternative to PLD we may wish to try is Pulsed Electron Deposition for which we woulduse the expertise and equipment of our collaborator Dr. Gilioli at IMEM. E P G and J ion-size study The previous section mentioned three important follow-up questions; (measurements of)the effect of ion-size on fluctuations, the pseudogap and on the doping dependence of J (which we believe to be related). In principle, two-magnon scattering (and hence J ) and thepseudogap can both be measured by Raman spectroscopy, along with 2∆ and the densityof states as discussed in Chapter 8. However, as we found, these are delicate measurementsof weak effects. Below we describe a simpler study to determine E P G and J . The study wedescribe requires only readily-available sample preparation and measurement apparatus.The study involves both good quality polycrystalline (PC) and single crystals or well- .5. FURTHER STUDIES O y and YSCO would represent ideal candidates that spana wide range of internal pressures (together with literature values from the Y123 mate-rial). The polycrystalline samples, Ln(Ba,Sr) Cu − z Zn z O y , would be prepared for severalconcentrations of Zn impurities; z = {0.0, 0.04, 0.08, 0.12} would be sufficient, althoughmore values of z will lead to statistically more accurate results. The doping level of thesesamples will be controlled by the oxygen concentration, y , by way of co-annealing all thePC samples with the SC/F samples. Together with T c from magnetisation or resistivitymeasurements, reliable measurements of S (295K) can be readily made on the PC sam-ples which will reflect the doping state of both the PC and SC/F samples. In this way p -dependent studies can be effected.A variety of measurements can now be made, but it would be of interest to measureboth J and the pseudogap energy E P G as a function of p and ion-size. With this setof samples J can be determined from two-magnon Raman scattering using micro-Raman.Meanwhile E P G = k b T ∗ can, in principle, be determined from resistivity measurements, seee.g. [85, 290]. Normally if T ∗ ≤ T c then it is not possible to determine T ∗ from resistivitymeasurements. However, Zn is known not to effect the doping state or pseudogap in thequantities we propose [39, 67, 290, 85] but will suppress the T c from its z = 0 value, T c .Thus, it would be possible to determine T ∗ down to reasonably low temperatures (probably ∼
25 K) or conversely, E P G to doping levels quite close to p = 0 . E P G scales with J [58] and that J ( p ) ∼ J ( p = 0) [1 − p / . ],irrespective of ion-size, for the Ln123 system. But this remains to be tested!Furthermore, such a study will provide quantitative estimates of the normal-state den-sity of states (DOS) at the Fermi-energy, N ( E F ) , as a function of both ion-size and p byuse of the previously-validated [180] Abrikosov-Gorkov model . The linearized Abrikosov-Gorkov equation for small scattering rates, Γ, is [238, 134]; T c T c = 1 − .
69 ΓΓ c (9.3)The scattering rate, Γ, is inversely proportional to the DOS as Γ = n/ [ πN ( E F )] where n = z ab /V is the density of impurity scatterers in the CuO layers, V is the unit cellvolume and z ab the concentration of Zn in a CuO layer. Γ c is the critical scattering raterequired to fully suppress superconductivity and can be expressed as Γ c = 0 . . As STM studies showed scattering from Zn in the CuO layers is close to the unitary limit [291] CHAPTER 9. CONCLUSIONS AND DISCUSSION
Y B a S r C u O y Y C a
B a C u O y Y B a C u O y T c (K) p l a n a r f r a c t i o n o f Z n Figure 9.3: The T c suppression of optimally doped YBaSrCu O − δ (red squares), Ca-dopedYBa Cu O − δ (open circles) and YBa Cu O − δ (blue triangles) due to Zn substitution inthe CuO layer.such the suppression of T c due to Zn substitution is inversely proportional to the DOS.The temperature coefficient of the normal-state heat capacity, γ , is related to N ( E F ) as γ = γ b (1 + Λ) = (1 + Λ) / π k B N ( E F ). Here γ is enhanced by a factor (1 + Λ) from the socalled ‘bare band structure’ value, γ b , by (screened) electron-electron interactions. Theseinteractions can be mediated by spin-fluctuations or phonons or more generally any otherrelevant bosonic interaction. For conventional superconductors, Λ is a function of the sameelectron-phonon interaction that pairs electrons, see [62] or the Chapter 13 by Gladstone,Jensen and Schrieffer in [292] . (1 + Λ) = m ∗ /m = 1 + N ( E F ) V ph + N ( E F ) V SF where m ∗ is the effective mass of the electrons and V ph , V SF are matrix elements of electron-electroninteraction mediated by phonons and spin-fluctuations respectively.The suppression of T c with impurity content for optimally-doped YBaSrCu − z Zn z O y is plotted in Fig. 9.3 along with similar data for Y . Ca . Ba Cu O y [135]. Because thesetwo data sets fit reasonably well to the same AG parameters , these initial measurementsimply the (1 + Λ) enhancement in γ is the same for these two systems with comparable T c .To this plot we add data for pure YBa Cu O − δ from [293]. T c = 93 . T max c is It appears possible that the T c of the Ca-doped Y123 is suppressed more rapidly by Zn substitutionthan YBaSrCu − z Zn z O y . A more thorough study is needed to explore this possibility. .6. CLOSING REMARKS T c / d z , is reduced implying thatthe total specific heat γ = γ b (1 + Λ), is larger. This may be intrinsic to optimally dopedYBa Cu O − δ and indicate an enhancement of Λ because of its more highly polarisablematrix. This is indicative of how a comprehensive study such as this might proceed. Itwould involve similar measurements for a number of different doping states (under- andover-doped) for different ion sizes in order to separate the ion-size effect from the dopingeffect on the DOS. But this is much easier than specific heat measurements and would bea powerful prelude to the latter if they were embarked upon.Because the DOS changes dramatically around optimal doping whilst T c does not, itis also possible the different slopes in Fig. 9.3 are a consequence of not comparing likedoping states. If the authors had also reported thermopower values of their YBa Cu O − δ we could say more. In any case, with such small changes in the slope it is important to seeif this trend continues for the two end members Nd123 and YSCO, where the effects willbe greatest. From such studies we ought to be able to extract (1 + Λ) and explore how m ∗ varies with ion size and the refractivity sum. In principle this will be measurable with wellhomogenised samples and with closely spaced z values for good statistics. The study of ion-size effects on the cuprates presented in this thesis was motivated bythe paradoxical opposite effect of external pressure and internal pressure, as induced byisovalent ion substitution, on T max c .Motivated by this observation and wishing to further test its applicability, we synthe-sised a new variants of Bi2201 that were under positive and negative internal pressure.We argued that in both Bi2201 and Ln(Ba,Sr) Cu O − δ disorder introduced by the ion-substitutions studied here cannot solely explain the variation in T c observed.We also explored several ideas relating to ion-size effects on;• The density of states. Primarily this idea was explored via DFT calculations althoughin Sec. 9.5 we suggest possible experimental routes to study ion-size effects on th DOS.• The anti-ferromagnetic superexchange energy, J . From these Raman-based studieswe argued that J cannot explain the opposite effects of internal and external pressureon T max c .94 CHAPTER 9. CONCLUSIONS AND DISCUSSION • The superconducting gap and pseudogap. Here, our Raman data were not preciseenough to resolve any systematic ion-size effect on these two energies.In this final Chapter we described how by considering the polarisability one can natu-rally understand the seemingly paradoxical opposite effects of external and internal pressureon T c . Thus, this thesis highlights the relatively unexplored idea that the polarisabilitypotentially plays a central role in superconductivity in the cuprates.These results do not necessarily conflict with the conventional understanding of Cooperpairing in the cuprates based on spin-fluctuations (although this picture should find dif-ficulty with the Raman results presented in Chapter 6). For example as discussed abovethe primary role of the polarisability could be via the screening of longer-range Coulombinteractions which are detrimental to Cooper pairing.Alternatively, these results could be initial experimental evidence for a novel pairingmechanism based on the exchange of quantised, coherent waves of polarisation. Here theenergy scale for such excitations is ∼ eV which is an order of magnitude larger than J andtwo larger than the Debye energy. These two scenarios can be tested and this statementleads us an important aspect of this work: the ion-size variation approach we have utilisedhere, especially for Ln123, represents an under-utilised, systematic method for supercon-ducting and pseudogap phenomenon to be explored theoretically and experimentally . Forexample, we discussed the recent paper of Gull et al. that showed how a proper treat-ment of the Hubbard model could reproduce several key experimental observations in thecuprates. Now it is likely the Hubbard parameters U , t , t etc. systematically vary in theLn123 and this would provide for a presumably insightful comparison between theory andexperiment.It appears that there remains many interesting, and most likely fruitful, avenues forfurther systematic ion-size studies such as those presented here in this thesis. Considering all valence electrons, this may in fact be more like ∼
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