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Absolute value of the magnetic penetration depth and field profile in the Meissner state of exotic superconductors.. Hossain, Md Masrur 2012-10-18

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Absolute Value of the Magnetic Penetration Depth and Field Profile in the Meissner State of Exotic Superconductors YBa2Cu3O6+x and Ba(Co0.074Fe0.926)2As2 by Md Masrur Hossain M.Sc., The University of British Columbia, 2006 A Thesis Submitted In Partial Fulfillment Of The Requirements For The Degree Of Doctor of Philosophy in The Faculty Of Graduate Studies (Physics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) October 2012 © Md Masrur Hossain, 2012ABSTRACT One of the fundamental quantities of a superconductor is the magnetic penetration depth, 𝜆, which is the characteristic length scale that amagnetic field penetrates into the surface of a super- conductor while in the Meissner state. In the clean limit the absolute value of 𝜆 is directly related to the superfluid density ns via 1/𝜆2 =  0e2ns/m*, wherem* is the effective mass. Consequently, its variation as a function of temperature, doping and orientation are of central importance in testing microscopic theories of exotic superconductors. A low energy beam of spin polarized muons has recently been developed, at the Paul Scher- rrer Institute, to directly measure 𝜆 in a superconductor. When a muon ( +) decays, it emits a fast decay positron preferentially along the direction of its spin due to the parity violating decay. The time evolution of statistical average direction of the spin polarization of the muon ensemble depends sensitively on the local magnetic field which can bemonitored as a function of themean depth of implantation. In this way it is possible to measure the field profile near the surface of a superconductor and extract the magnetic penetration depth in a direct manner which is not otherwise possible with conventional bulk methods. In this thesis, accurate measurements of 𝜆 and its anisotropy (︀ ” 𝜆a/𝜆b )︀ have been made for three different oxygen (x = 6.52, 6.92 and 6.998) contents of YBa2Cu3O6+x as well as in Ba(Co0.074Fe0.926)2As2. The measured values of 𝜆 and the in-plane anisotropies are consider- ably different from that reported in the literature, using less direct methods. The a–b anisotropy is surprisingly insensitive to x in YBa2Cu3O6+x . We observe an exponential decay of the mag- netic field and corresponding supercurrent density deep inside the crystals. Small deviations from the London model are observed which indicate there is a suppression of the supercurrent density close to the surface. The measured (𝜆) values are also found to depart substantially from the Uemura relation (︀ Tc ∝ 1/𝜆2 )︀ . iiPREFACE Results presented in section 4.1.1 has been published by Kiefl, R. F., Hossain, M. D., Wojek, B. M., Dunsiger, S. R., Morris, G. D., Prokscha, T., Salman, Z., Baglo, J., Bonn, D. A., Liang, R., Hardy W. N., Suter, A. Morenzoni, E. in Direct measurement of the London penetration depth in YBa2Cu3O6.92 using low-energy  SR. Phys. Rev. B, 81, 180502 (2010). The design of research methods, literature review, data analysis, were done by myself in consultation with my supervisor R. F. Kiefl. The manuscript of this paper was written primarily by R. F. Kiefl. The co-authors have been partly involved in taking the data and reviewing and commenting on the manuscripts, or supplying the studied samples. Sections 4.1.3 and 4.1.4 are currently in the process of being published. The results presented in section 4.2 is published by Ofer, O., Baglo, J. C., Hossain, M. D., Kiefl, R. F., Hardy, W. N., Thaler, A., Kim, H., Tanatar, M. A., Canfield, P. C., Prozorov, R., Luke, G. M., Morenzoni, E., Saadaoui, H., Suter, A., Prokscha, T., Wojek, B. M. and Salman, Z. in Absolute value and temperature dependence of the magnetic penetration depth in Ba(Co0.074Fe0.926)2As2. Phys. Rev. B, 85, 060506 (2012). A significant of part of data analysis was done by me. Manuscript was written by O. Ofer and microwave analysis was done by J. C. Baglo. iiiTABLE OF CONTENTS Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Brief History of Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Brief Review of Superconducting Properties . . . . . . . . . . . . . . . . . . . . 3 1.3 High Temperature Superconductivity: A Review . . . . . . . . . . . . . . . . . . 5 1.3.1 Cuprates: CuO2 Layer Based High Temperature Superconductor . . . . 5 1.3.2 Pnictide: A New Type of High Temperature Superconductor . . . . . . . 10 1.4 Pairing Symmetry and Magnetic Penetration Depth Measurement . . . . . . . . 12 2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1 BCS Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 London Penetration Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3 Pairing Mechanism and Order Parameter Symmetry . . . . . . . . . . . . . . . . 19 2.4 Critical Temperature and Bose Einstein Condensation . . . . . . . . . . . . . . . 22 3 Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.1 Introduction To  SR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.1.1 Properties and Production of Muons . . . . . . . . . . . . . . . . . . . . 26 3.1.2 General  SR Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2 Low Energy  SR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2.1 Principles of LE- SR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Spin Evolution in Quantum Mechanics . . . . . . . . . . . . . 32 iv3.2.1.2 Classical Spin Precession . . . . . . . . . . . . . . . . . . . . . 34 Spin Polarization and Asymmetry Spectrum . . . . . . . . . . 35 3.2.2 Stopping Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2.3 Maximum Entropy (ME) Fourier Analysis . . . . . . . . . . . . . . . . . 37 4 Results and Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.1 LE- SR Experiments on YBCO Single Crystals . . . . . . . . . . . . . . . . . . 41 4.1.1 YBa2Cu3O6.92 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.1.2 Temperature Dependence of Penetration Depth and Estimation of En- ergy Gap in YBa2Cu3O6.92 . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.1.3 YBa2Cu3O6.998 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.1.4 YBa2Cu3O6.52 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.2 Ba(Co0.074Fe0.926)2As2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.3 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.4 Discussion on ``Deadlayer'' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 vLIST OF TABLES Table 3.1 Properties of Muon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Table 4.1 Summary of results in YBa2Cu3O6.92 . . . . . . . . . . . . . . . . . . . . . . 50 Table 4.2 Summary of results in YBa2Cu3O6.998 . . . . . . . . . . . . . . . . . . . . . . 58 Table 4.3 Summary of results in YBa2Cu3O6.52 . . . . . . . . . . . . . . . . . . . . . . 66 Table 4.4 Summary of results in Ba(Co0.074Fe0.926)2As2 . . . . . . . . . . . . . . . . . 70 Table 4.5 Measurements of the absolute value of themagnetic penetration depth (︀ 𝜆a/b )︀ in YBa2Cu3O6.52, YBa2Cu3O6.92 and YBa2Cu3O6.998 . . . . . . . . . . . . . 72 Table 4.6 Measurements of the absolute value of themagnetic penetration depth (︀ 𝜆a/b )︀ in Ba(Co0.074Fe0.926)2As2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 viLIST OF FIGURES Figure 1.1 Approximate number of publications on superconductivity . . . . . . . . . . 2 Figure 1.2 External field expulsion in the Meissner state . . . . . . . . . . . . . . . . . 2 Figure 1.3 History of the superconducting critical temperature (Tc) . . . . . . . . . . . 3 Figure 1.4 Simplified phase diagram of type I and type II superconductors . . . . . . . 4 Figure 1.5 A schematic model of the electronic/magnetic structure of theHTSC vortex core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Figure 1.6 Stoichiometric crystal structure of YBCO . . . . . . . . . . . . . . . . . . . 7 Figure 1.7 Schematic phase diagram of cuprate superconductors . . . . . . . . . . . . . 8 Figure 1.8 Two models for high-Tc superconductivity . . . . . . . . . . . . . . . . . . . 9 Figure 1.9 Crystal structure of Pnictide . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Figure 1.10 Phase diagram for BaCoxFe2–xAs2 . . . . . . . . . . . . . . . . . . . . . . . 11 Figure 2.1 BCS model of electron interactions in superconductors . . . . . . . . . . . . 15 Figure 2.2 Density of electronic states in s and d-wave superconductivity . . . . . . . . 16 Figure 2.3 Temperature dependence of the superconducting energy gap in the weak coupling limit of BCS interaction . . . . . . . . . . . . . . . . . . . . . . . . 16 Figure 2.4 Modified London model of exponential decay of external magnetic field . . . 19 Figure 2.5 Angular dependence of a d-wave gap . . . . . . . . . . . . . . . . . . . . . . 21 Figure 2.6 A schematic 2D representation of the superconducting (s, d and s– wave) order parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Figure 3.1 Positively charged pion decay into muon and a neutrino . . . . . . . . . . . 27 Figure 3.2 Angular probability distribution of positrons emitted from muons . . . . . . 28 Figure 3.3 Schematic of the arrangement for a TF- SR experiment . . . . . . . . . . . . 29 Figure 3.4 LE- SR experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Figure 3.5 LE- SR asymmetry spectrum of a YBa2Cu3O6.92 mosaic in a transverse field of 9.5mT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Figure 3.6 Histogram of raw counts and asymmetry for a ``forward'' positron detector . 37 Figure 3.7 Muon implantation profiles and average depth (Æz æ) in YBCO . . . . . . . 38 Figure 4.1 Quick outline of fitting procedure in normal and superconducting states . . . 41 Figure 4.2 YBCO mosaics of small crystals . . . . . . . . . . . . . . . . . . . . . . . . . 41 Figure 4.3 Measured surface roughness of YBCO crystals . . . . . . . . . . . . . . . . . 42 Figure 4.4 Magnetic field distribution inside YBa2Cu3O6.92 . . . . . . . . . . . . . . . 44 Figure 4.5 Comparison of Meissner model and ME fits . . . . . . . . . . . . . . . . . . 45 viiFigure 4.6 The muon spin precession signal in the normal and superconducting state of YBa2Cu3O6.92 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Figure 4.7 The average magnetic field (︀ ÆBæa/b )︀ versus mean stopping depth in an ap- plied field of 9.5mT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Figure 4.8 Energy and temperature dependence of 𝜆, 𝜙 and 𝜎 and A0 in YBa2Cu3O6.92 49 Figure 4.9 Temperature dependence of 𝜆(T) in YBa2Cu3O6.92 . . . . . . . . . . . . . . 52 Figure 4.10 Temperature dependence of 𝜌(T) in YBa2Cu3O6.92 . . . . . . . . . . . . . . 52 Figure 4.11 The muon spin precession signal in the normal and superconducting state of YBa2Cu3O6.998 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Figure 4.12 Magnetic field distribution inside YBa2Cu3O6.998 . . . . . . . . . . . . . . . 54 Figure 4.13 Broadening parameter for experiments on YBa2Cu3O6.998 . . . . . . . . . . 55 Figure 4.14 YBCO 6.998 average magnetic field ÆBæa/b inside YBa2Cu3O6.998 . . . . . . 56 Figure 4.15 Temperature dependence of magnetic penetration depth in YBa2Cu3O6.998 . 56 Figure 4.16 Temperature dependence of 𝜆(T) in YBa2Cu3O6.998 . . . . . . . . . . . . . 57 Figure 4.17 Temperature dependence of 𝜆(T) in YBa2Cu3O6.998 . . . . . . . . . . . . . 58 Figure 4.18 Summary of results for YBa2Cu3O6.998 . . . . . . . . . . . . . . . . . . . . 60 Figure 4.19 YBa2Cu3O6.52 spectra in –1.45mT . . . . . . . . . . . . . . . . . . . . . . . 61 Figure 4.20 The muon spin precession signal in the normal and superconducting state of YBa2Cu3O6.52 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Figure 4.21 YBa2Cu3O6.52 broadening parameter (𝜎) are shown for an external applied field of 1.45mT, 4.6mT and 7.8mT . . . . . . . . . . . . . . . . . . . . . . . 63 Figure 4.22 Magnetic field distribution inside YBa2Cu3O6.52 . . . . . . . . . . . . . . . 63 Figure 4.23 Average magnetic field ÆBæa/b inside YBa2Cu3O6.52 . . . . . . . . . . . . . . 64 Figure 4.24 Temperature dependence of 𝜆 in external fields of 1.5mT and 4.6mT . . . . 65 Figure 4.25 Asymmetry, phase (𝜙), broadening parameter (𝜎) and the goodness of fit 𝜒2/DF for three different external fields . . . . . . . . . . . . . . . . . . . . 66 Figure 4.26 Ba(Co0.074Fe0.926)2As2 magnetization measurement . . . . . . . . . . . . . 67 Figure 4.27 Muon precession signals in Ba(Co0.074Fe0.926)2As2 in an applied field of  0H = 4.7mT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Figure 4.28 Broadening parameter (𝜎) in Ba(Co0.074Fe0.926)2As2 in external applied fields of  0H = 2.5mT and 4.7mT . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Figure 4.29 The average magnetic field versus the muon energy in Ba(Co0.074Fe0.926)2As2 69 Figure 4.30 Magnetic field distribution inside pnictide . . . . . . . . . . . . . . . . . . . 70 Figure 4.31 The temperature dependence of 𝜆 plotted versus t = T/Tc . . . . . . . . . . 71 Figure 4.32 Tc and anisotropy versus ns . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Figure 4.33 Simulated external field [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 viiiLIST OF SYMBOLS Δ(k,T) Momentum dependent superconducting energy gap, page 15 eV 𝜅 Ginzburg-Landau parameter, page 4 Tc Superconducting critical temperature, page 1 K 𝜉 Ginzburg-Landau coherence length, page 5 m 𝜆 London penetration depth, page 1 m 𝜆ab Average magnetic penetration depth, page 72 m Φ0 Unit of flux quantum, page 4 2.067833667 · 10−15Wb 𝜓 Superconducting order parameter, page 1 𝜎 Optical conductivity, page 9 𝜏 Relaxation time, page 17 Hc Critical magnetic field, page 4 Tesla ns Superfluid density, page 5 T* Strange metallic phase temperature, page 6 K   Muon gyromagnetic ratio, page 26 MHz/Tesla A(t) Time dependent muon asymmetry, page 40 R Ratio of magnetic penetration depth, ie, 𝜆a𝜆b , page 58   Muon lifetime, page 26 s ixGLOSSARY UBC University of British Columbia PSI Paul Scherrer Institute, Villigen, Switzerland CIFAR Canadian Institute for Advanced Research, Canada TRIUMF TRI-University Meson Facility, Canada NSERC Natural Sciences and Engineering Research Council of Canada YBCO Yttrium Barium Copper Oxide/YBa2Cu3O6+x FE-PNICTIDE Fe-As Based Superconductors MUSR Muon Spin Resonance/Rotation/Relaxation QCP Quantum Critical Point HTSC High Temperature Superconductor ODLRO Off Diagonal Long Range Order xACKNOWLEDGEMENT First and foremost, I would like to thank my supervisor Rob Kiefl and his support and advice throughout the course of my PhD. I am very grateful for his responses to my inquiries and avail- ability during my entire stay at UBC. I would also like to thank my committee members: D. A. Bonn, M. Franz, and A. Damascelli for their comments, and for reading the thesis. I also thank Zaher Salman and Gerald Morris who have always been very helpful in provid- ing software expertise and acting as general UNIX/LINUX gurus. Zaher's super-fast response to queries for technical help has been invaluable in many occasions. Special thanks to D. Ar- seneau for letting me use the fastest computer in  SR group in TRIUMF. This has saved many precious hours of analysis time and possible frustrations. Many thanks to R. Liang for providing the YBCO samples with three different oxygen contents. Also thanks to AMES laboratory for providing the FE-PNICTIDE samples. I wish also to thankmy colleagues in βNMR group. Special thanks to SusanQ. Song for help- ing taking magnetization data at AMPEL. A big thanks to J.C. Baglo for providing supplemen- tary microwave analysis. My colleagues Terry Parolin, Dong Wang, Micheal Smadella, Hassan Saadaoui, Susan Q. Song, J.C. Baglo and others have spent many nights taking the actual data; for that and for the helpful discussions and good times we had together, I am very grateful. The data in this project was taken over many years and a lot of people, other than the ones that have been already mentioned above, have helped to take shifts and helped doing supple- mentary analysis. I will also like to thank  SR support staff and colleagues at PSI, B. M. Wojek, T. Prokscha, A. Suter. Special thanks goes to B. M. Wojek for his critical and thorough analysis of data and valuable insight. I would like to mention the βNMR technical support staff R. Abasalti, D. Arseneau, K. H. Chow, S. Dunsiger, B. Hitti, C. D. P. Levy, R. Miller, M. R. Pearson, and D. Vyas. The  SR measurements were performed at PSI. These works were supported by NSERC, CIFAR and the U.S. Department of Energy. Finally, thanks to the writers of LATEX and many accompanying packages, used in typesetting this thesis. xiCHAPTER 1 INTRODUCTION 1.1 Brief Histor y of Superconductiv ity Historically, superconductivity has played an important role in condensed matter physics. With the discovery of superconductivity in Hg [2], about 100 years ago, it remains a very active area of research with continuing surprises. Figure 1.1 roughly shows the number of publications on the phenomenon of superconductivity in the last decade. Before the discovery of the phenomena of superconductivity, it was known that the resistivity of ametal dropswith decreasing temperature. In a perfect crystal, resistivity inmetals is generally attributed to electron-phonon scattering (and to a lesser extent electron electron interactions) the rate of which is proportional to the thermally excited phonons. However, the number of thermally excited phonons is finite above absolute zero and thus the resistivity is expected to be non zero at any finite temperature. Consequently, K. Onnes' discovery of virtual absence of resistivity in Mercury below 4.15K, in 1911 [3] was rather surprising. Soon after, in 1913, Lead was found to be superconducting below 7.2K and after 17 years of this discovery, niobium was found to be superconducting at 9.2K. The virtual absence of resistance in a superconductor has been demonstrated by experiments with persistent currents in superconducting rings. Such currents have a decay time of on the order of 105 years. The other important characteristic beyond zero resistivity is the Meissner effect in which magnetic field is expelled ([4] figure 1.2) out of a sample when it's cooled below the critical temperature Tc. The Meissner effect is different from perfect diamagnetism in a perfect conductor. In perfect diamagnetism, currents are generated to oppose any change in the applied field such that, if the sample already had non-zeromagnetic flux through it, cooling through Tc wouldn't result in any flux expulsion. However, in theMeissner effect, the field is expelled from the samplewhen cooled below Tc. This Meissner effect led the London brothers [5] to propose equations to predict how the field is excluded from the sample and in particular, the field penetration near the surface. Londons' theory was later (1950) derived from the phenomenological theory of Ginzburg and Landau [6] (GL), who described superconductivity in terms of a macroscopic complex order parameter 𝜓 which is a rough measure of the extent to which a system is ordered. In the case of superconductivity, the amplitude of the order parameter is proportional to superconducting electron density or condensate. One of the most remarkable aspects of a superconductor is that the electrons remain coherent over macroscopic distances. The QM phase at one position is correlated with the phase everywhere else in the sample. Although the phenomenological GL theory had been successful, an accepted microscopic theory only came in 1957 from J. Bardeen, Leon Cooper and John Schrieffer [7, 8], now famously known as BCS theory. BCS theory explained superconductivity in terms of an electron-electron interactionmediated by short wavelength phonons andMcMillan et al. predicted that supercon- 1SC HTSC YBCO 𝜆M 𝜆L P-HT N-SC 104 105 Shorthand of search terms Number of publications Figure 1.1 : Number of publications (year 2000 onwards) in log scale, for different search terms from a prominent search engine’s scholar edition, done on August 7, 2012. Expansion of the shorthand terms: SC: superconductiv- ity; HTSC: high temperature superconductivity; YBCO:YBCO; 𝜆M : magnetic penetration depth; 𝜆L: London pene- tration depth; P-HT: pseudogap in high temperature superconductivity; N-SC: normal state in high temperature superconductivity. As may be seen, an enormous scholarly interest in the phenomenon of superconductivity exists in contemporary condensed matter physics. T > Tc T < Tc B Figure 1.2 : Meissner effect for a type I superconductor. When a superconductor is placed in an external magnetic fieldH and cooled below its superconducting temperature Tc, the magnetic flux is abruptly expelled. For B < Bc , it penetrates the surface of the superconductor within the penetration depth 𝜆. ductivity may be found with critical temperatures Tc ≤28K [9] (for Nb3Sn). It was believed that the electron phonon interaction was strong enough to create lattice instablility and thus there was a fundamental limit to Tc from electron phonon interaction. The carriers of supercurrents were shown to be a pair of electrons (“Cooper pairs” [10]) with opposite spin and momentum. Many new metals and alloys with superconducting properties, at low temperatures, were found by 1980, with the noted exceptions of ferromagnets such as Fe, Ni. It was later realized that magnetic order is antagonistic to the conventional so called low-Tc superconductivity. In 1986, J.G. Bednorz and K.A. Muller [11] discovered superconductivity in La2–xBaxCuO4 at 35 K, thus initiating the era of high-temperature superconductivity. Building on that, Maw- Kuen Wu and his graduate students, Ashburn and Torng [12] at the University of Alabama dis- 21900 1920 1940 1960 1980 2000 2020 2040 0 20 40 60 80 100 120 140 LaOFePHg Ba(Co0.074Fe0.926)2As2 YBa2Cu3O7 Tb,N2 Tb,He MgB2 Pb NbN Nb3Sn Nb3Ge La2–xSrxCuO4 Bi2Sr2Ca2Cu3O10 Ti2Ba2Ca2Cu3O10 HgBa2Ca2Cu3Ox LaO1–xFxFeAs CeO1–xFxFeAs SmO1–xFxFeAs Year of discovery T c (K ) Figure 1.3 : Superconducting critical temperature ([13]) Tc has risen almost linearly with time, from 4K to 40K till about 1986. Around 1987, one of the CuO based high temperature superconductor family was found. In 2008, one new family of Fe-based superconductors were discovered. Due to the typical antagonistic relationship of superconductivity and magnetism, this was a significant surprise for science. covered YBCO has a Tc of 93 K. Their work led to a rapid succession of new high temperature superconducting materials, ushering a new era in material science and chemistry. YBCO was the first family of materials to become superconducting above 77 K, the boiling point of liquid nitrogen. All materials developed before 1986 became superconducting only at temperatures near the boiling points of liquid helium (︀ Tb,He = 4.2K )︀ or liquid hydrogen (︀ Tb,H = 20.28K )︀ - the highest being Nb3Ge at 23 K. Although met with initial skepticism, the observations were validated when Uchida et. al. and Chu et. al reproduced original results in 1987. In 2008, a new family of Fe-based superconductorswere discovered. Due to the typical antagonistic relationship of superconductivity and magnetism, this was quite surprising. Remarkable progress has been made in discovering high-Tc superconductors as shown in the figure 1.3. As superconductivity is found in somany different material families, it is considered an almost “common” phenomenon for metals which have a partially filled conduction band. Nevertheless the discovery of high-Tc superconductivity has lead to many open questions regarding the microscopic mechanism. 1.2 Brief Rev iew of Superconducting Properties Besides having a critical temperature Tc, superconductors also have critical magnetic fields (Hc), above which their properties change. In this respect, superconductors are classified in two broad categories (figure 1.4), i) Type I, in which the material becomes normal above a critical magnetic field Hc,0. ii) Type II, in which the material has two critical magnetic fieldsHc1 andHc2. In type II, at H < Hc1, the material remains in the Meissner state and at Hc1 < H < Hc2, magnetic field 3penetrates the material in the form of vortices carrying a quantized amount of magnetic flux (for a very detailed review, consult [14]) and forH > Hc2, it becomes normal. Two other parameters characterize superconductivity in general, namely the coherence length 𝜉 and the magnetic pen- etration depth 𝜆. The coherence length 𝜉 is the distance over which order parameter 𝜓 varies appreciably and penetration depth 𝜆 is the depth over which shielding currents circulate to expel the applied external field. 𝜆 and 𝜉 are two fundamental length scales in superconductivity. Other parameters of interest such as Ginzburg-Landau parameter 𝜅 = 𝜆𝜉 , two critical fields Hc1, Hc2, thermodynamical critical field Hc may be derived from them: Hc1 = Φ0 4𝜋𝜆2 ln (︂ 𝜆 𝜉 )︂ , (1.1a) Hc2 = Φ0 2𝜋𝜉2 , (1.1b) Hc = √ 2Φ0 4𝜋𝜆𝜉 , (1.1c) whereΦ0 = h 2e is unit of flux quantum. The thermodynamic critical fieldHc is related to the free energy difference between the normal and superconducting states. Meissner Hc,0 H T Tc Normal Meissner Vortex Hc,2 Hc,1 H T Tc Normal Type I Type II Figure 1.4 : Superconductivity is destroyed when the external field is too large or temperature too high. Super- conductors are divided in two classes depending on the manner of this destruction. For type I superconductors, superconductivity is abruptly destroyed in a first order phase transition if H > Hc or T > Tc. Type II super- conductivity has a complete Meissner region (below H < Hc) in H – T phase diagram, however, in the “vortex” (Hc1 < H < Hc2) state, laminar vortices with normal state cores enter into superconductor and superconductiv- ity is destroyed in a continuous 2nd order phase transition to a normal state. Most high-Tc superconductors are type II. When the external magnetic field exceeds Hc1 in a type II superconductor, the material en- ters into a “mixed state” in which laminar vortices with normal state cores enter into the su- perconductor. At the center of a vortex, superconductivity is completely destroyed, i.e, the order parameter |𝜓|2 vanishes (figure 1.5). However, the velocity of the carriers tends to increase as one 4approaches the core. Beyond a radial distance of 𝜉, the superfluid density ns and order parameter reaches its bulk value. The radius 𝜉 is known as “vortex core”. The supercurrent flowing around the vortex produces a magnetic field which is maximum at the center and decays approximately exponentially, with a length scale of 𝜆 in the radial direction. The vortices are usually arranged in a periodic lattice known as the Abrikosov lattice, the flux lattice or the flux line lattice. Vor- tices may also be dynamic and interacting depending on the level of doping and the magnetic field [15]. |𝜓(x)|2 v(x) Distance from vortex center (a) 0 0.5 1 x y |𝜓 (x ,y )| 0 0.2 0.4 0.6 0.8 (b) Figure 1.5 : Left: A schematic model of the electronic/magnetic structure of theHTSC vortex core. Superfluid ve- locity v(x) rises and theHTSC order parameter |𝜓(x)| falls as the core is approached. Right: The superconducting order is suppressed at the cores of the vortices. The colored surface shows the envelope of this order parameter, superimposed on the vortex lattice. This type of order can be static or dynamically fluctuating depending on the level of doping and the magnetic field. 1.3 High Temperature Superconductiv ity : A Review 1.3.1 Cuprates : CuO2 Layer Based High Temperature Supercon- ductor Discovery of superconductivity in the ceramic cuprate materials has led to a pursuit to under- stand this new phenomenon. This new type of superconductivity is considerably different from the “conventional” (i.e, BCS)-type superconductivity and the exact microscopic mechanism is still being debated. However, significant inroads have been made in understanding different aspects of this “unconventional” superconductivity. A traditional description of electronic behavior in solids is modeled after Drude, Sommer- field, Wiedemann and Franz, where heavier positively charged cores of atoms form a periodic lattice and are immobile and upper valance electrons are almost free as in gas molecules in a jar, aptly named as “free electron gas”. This is also the basis for Landau's “Fermi-liquid theory” (FL) where weak short range interactions between electrons are taken into account similar to a normal liquid. The Wiedemann-Franz (WF) law (an empirical observation) is one of the basic properties of a Fermi liquid, reflecting the fact that the ability of a “free electron” to transport 5heat is the same as its ability to transport charge and is written as k 𝜎T = 𝜋2 3 (︂ kB e )︂2 = constant (1.2) where k and 𝜎 are the heat/electrical conductivity, respectively. In high-Tc cuprate superconduc- tor (Pr,Ce)2CuO4, it has been reported that the the WF law is violated in the normal state [16]. The standard theory of electrons, including FL and band theory, is highly successful in explaining metallic, insulating and semiconducting behavior, however fails to account for unconventional superconductivity where electron-electron interaction is too strong. Strong Coloumb repulsion among electrons leads to antiferromagnetic Mott insulating behavior in CuO materials at a com- position where “free electron gas” theory predicts a metal (e.g YBa2Cu3O6.0, This should be a metal but is an AF insulator). Changes in composition (O doping) leads to many exotic phenomena such as superconduc- tivity, charge ordering, strangemetallicity, quantum criticality and Fermi liquid behavior. AMott insulator is very different from a regular (band) insulator. In a band insulator, lack of conduc- tivity arises due to Pauli exclusion principle as the highest occupied band contains two electrons per site and all the bands are filled. In a Mott insulator, there is one electron per atom and thus the band is exactly half filled. However conduction is blocked by the strong Coulomb repulsion which prevents double occupancy. Typically the ground state is an antiferromagnet (figure 1.8). Doping (hole/electron) restores some electrical conductivity by creating sites to which electrons can jump without having to gain additional Coloumb energy. The resulting motion of the elec- trons is highly correlated. High-temperature superconductivity arises in a family of layered copper oxides that all fea- ture weakly coupled square-planar sheets of CuO2. Structure of one of themember of this family, YBa2Cu3O7-𝛿 (hereafter “YBCO”, possibly the most studied) is shown in figure 1.6, as this ma- terial was a subject of this research. For YBa2Cu3O7, three copper-oxygen layers are stacked along the tetragonal c^ axis. Two of these layers have oxygen atoms between the copper ions in both the a^ and b^ directions, and are called CuO2 plane layers. The third layer, called the CuO chain layer, has oxygen ions only along the b^ direction [19]. The phase diagram for YBCO, de- pendent on oxygen(hole) doping, is shown schematically in figure 1.7. Asmay be noted from the phase diagram, with increased hole doping, the antiferromagnetic insulating state evolves into a superconducting state. The dependence of critical temperature Tc(p) on doping is given by an empirical relationship [20], Tc(p) = Tc,max [︁ 1 – 82.6(p – 0.16)2 ]︁ , (1.3) where doping level p varies from 0.05 to 0.27. The proximity of antiferromagnetism and super- conductivity gives rise to the conjecture that superconductivity is driven bymagnetic interactions 6a b c CuO2 Plane CuO Chain Y3+ Ba2+ Cu2+ O2– a = 0.382113 nm b = 0.38828 nm c = 1.16741 nm Figure 1.6 : YBCO consists of CuO2 planes and CuO chains. Each plane layer consists of a single Cu atom sharing with four Oxygen vertices. The CuO chains run parallel to the planes along the b direction shown in the figure. Each Cu atom in the chain in the fully oxygenated crystal of YBa2Cu3O7 shares twooxygen vertices. Other phases have a reduced number of oxygens in the chain layer. The Yttrium atoms are found between CuO2 planes, while the Barium atoms are found between CuO2 planes and CuO chains. Non-stoichiometric compounds have some disorder in the CuO chains. In YBa2Cu3O6+x , with x = 0, the structure is tetragonal and insulating. For x < 0.35, CuO chains along b-axis start to form and the structure becomes orthorhombic. Maximum Tc ~ 95 K occurs for x ~ 0.92 [17, 18]. between electrons rather than pairing via phonons. Also it is important to note that, a signature of phonon driven superconductivity, ie, the “isotope effect” has been found to be small [21] in Cuprates. However there has been renewed interest in the possible role of electron-phonon cou- pling [22--27] in high-Tc superconductivity, although the role is suggested to be indirect [27] and small [24, 28]. A possible mechanism proposed by P. W. Anderson [29] is that “quantum fluctuations” may create instability in the antiferromagnetic order and give rise to a resonat- ing valence bond state [30--32] in which the spins form a “spin-liquid” phase of singlet(s = 0) pairs. “Spin liquid” is defined to be aggregation of pairs of antiparallel spins. The motion of such singlet pairs is similar to the resonance of 𝜋 bonds in benzene, and thus is the origin of the 7T* Superconductivity Temperatur e (K ) Pseudo-gap Strange Metal Fermi Liquid Holes (per planar Cu) Tc AF Insulat or M agnetism Oxygen (x) YBa2Cu3O6+x 0.0 0.1 0.2 0.3 6.0 7.0 8.0 0 100 200 300 Figure 1.7 : Schematic phase diagram: At very low levels of electron-hole doping, cuprates are insulating and anti- ferromagnetic (the materials’ neighboring spins point in opposite directions). With increased hole doping, YBCO shows disordered magnetism, superconductivity (at temperatures below Tc) and Fermi liquid behavior. At tem- peratures above Tc but below T *, YBCO shows a pseudogap phase. The “strange metal” region, above Tc, shows anomalous (︀ 𝜌(T ) ∝ T )︀ temperature dependent electrical resistivity. The boundary of the pseudogap region at low doping levels is unknown. The transition between the Fermi liquid phase and the strange metal phase occurs gradually. term “resonating valence bond” (RVB), schematically shown in figure 1.8. In this picture, elec- trons are paired up in antiparallel spin-formation but cannot move due to Coloumb repulsion. Anderson argued, reducing average occupancy, from one, will make these singlet pairs mobile, giving rise to superconductivity. In contrast, YBCO was found out to be an antiferromagnet and not a spin-liquid [33--35]. It has been established that in cuprate systems, antiferromagnetic or- dering resides predominantly on the CuO2 planes [36, 37], with a three-dimensional magnetic transition dictated by weaker coupling between planes. As seen in the phase diagram in figure 1.7, Tc varies (peaks at “optimal” doping) as a func- tion of doping and is a commonly observed phenomenon in all CuO2 layer based supercon- ductors [38]. With increased doping 3D antiferromagnetic ordering gives way to a disordered magnetic state with short range correlations [38], thereby retaining some magnetism. At T > Tc, metallic behavior is observed for a broad range of dopings and d.c electrical resistance is T-dependent rather that T2-dependent as would have been expected from a normal metal Fermi liquid behavior. In the overdoped regime, the copper oxides behave more like ordinary metals with aT2 dependence of d.c resistivity [39]. A naturally overdopedCopper oxide TlBa2Cu3O6+x has been observed to show polar angular magnetoresistance oscillations [40, 41], in high mag- 8Antiferromagnet Spin Liquid = 1√ 2 ( + ) Figure 1.8 : Left: In the antiferromagnetic state, spins on alternating sites are antiparallel and the net spin magne- tization is zero. Right: an example of a short range “resonating valence bond” (an aggregation of antiparallel neighboring spins). An oval represents a superposition of different possible spin configurations. This is a “spin- liquid” since there is no static order but their motions are highly correlated. Motions of singlet pairs are hindered due to Coloumb repulsion. Reducing average occupancy from one may make these singlet pairs mobile. netic fields, establishing the existence of a 3D Fermi surface, consistent with the prediction from band theory, i.e, as in a conventional metal. In underdoped regime, copper oxides have been found to show quantum oscillations [42], an indication of metallic behavior, in de Haas-van Alphen spectra. As may be noted in the phase diagram, Cu-O based superconductors includes a “pseudogap” region [43, 44], a precursor to the superconducting state. It has been shown that this phase originates in CuO2 planes and not in the CuO chains [45]. Whether this is a dis- tinct state of matter is still under debate [46]. It is metallic, however some parts of the Fermi surface show gaps [43, 44, 47]. A quantum critical point (QCP) [48--50] in the cuprates is spec- ulated to be the end point of a line (T*) [51] that separates the pseudogap and “strange metal” regions. It has also been speculated that the phase diagram is controlled by such a quantum crit- ical point [52, 53]. A QCP develops in a material at zero temperature when a new form of order emerges from it's ground state as function of some external parameter such as pressure, doping or electric field etc. The quantum critical points are of great interest because of their ability to influence the finite temperature properties of materials. The “normal” region (“strange metal”) above the transition temperature Tc has very unusual properties [54--56] (thermal conductivity k(T), optical conductivity 𝜎(𝜔), the nuclear relaxation rate T–11 (T)), with a large temperature- dependent resistivity implying a scattering rate linear in T [57, 58]. Although there are significant differences compared to conventional superconductivity, su- percurrents are still carried by electron pairs, as shown by quantization of magnetic flux in units of h2e [59--61]. Most of the physical properties of the CuO2 superconductor have experimentally been establishedwith a high degree of reliability and advances in preparing thematerials are such that spurious effects and uncertainties in materials compositions, homogeneities and impurity content may be eliminated as hindrance to the understanding of the phenomenon of high-Tc su- perconductivity. In spite of substantial efforts, both experimental and theoretical, there remain 9a b c Ba Fe As a = b = 0.395702(4) nm c = 1.29685(2) nm Figure 1.9 : Pnictide crystal structure [69] shows the antiferromagnetic alignment and magnetic moment (red ar- rows) both along the longer a axis in the FeAs plane of the parent compound BaFe2As2. The magnetic unit cell is the same as the orthorhombic chemical unit cell. It is widely believed that superconductivity originates in the FeAs plane. Although this parent compound BaFe2As2 is antiferromagnetic, another pnictide parent compound LaOFeAs reportedly shows spin-density-wave (SDW) [70] type magnetic order. many open questions regarding the underlying microscopic mechanism for high-Tc supercon- ductivity. 1.3.2 Pnictide: A New Type of High Temperature Superconductor The antagonistic relationship between superconductivity and magnetism has led researchers to avoid using magnetic elements (eg. Fe), as potential building blocks of new superconducting materials. The recent (2008) discovery of superconductivity at Tc's up to 55K in iron pnictide systems [62--66] has sparked enormous interest in this new class of materials. Even more sur- prising is that pnictides are the only class of material other than cuprates (CuO2 layer based superconductors) to have a Tc higher than 40K (~BCS theoretical maximum). The crystal struc- ture of the parent compound (of the so called 122 system) BaFe2As2 is shown in the figure 1.9. Co-doped pnictide (Ba(Co0.074Fe0.926)2As2) (grown using self-flux methods [67, 68]) is a sub- ject of this work. Cuprates and pnictides show similar behavior, such as (i) both are layered structures, (ii) in both cases, the parent (non-superconducting) compounds are antiferromagnets (iii) both mate- 100.00 0.02 0.04 0.06 0.08 0.010 0.012 0 50 100 150 AFM TN Ort Ort SC Tet TS x Temperatur e (K ) Tc BaCoxFe2–xAs2 Figure 1.10 : Phase diagram [81, 82] for BaCoxFe2–xAs2. Yellow indicates the superconducting phase, which ap- pears below the superconducting transition temperature Tc. A structural transition occurs at Ts from the tetrag- onal phase (Tet) at higher temperature to the orthorhombic phase (Ort). Blue represents the antiferromagnetic order (AFM), which appears at TN, slightly below Ts. The stripes of enhanced superfluid density are observed only in the regime 0.04 < x < 0.06. rials show superconducting order upon doping. There are also striking dissimilarities (i) parent compound for cuprates are Mott insulators while for pnictides, they are semimetals, (ii) cuprates have a single band crossing the Fermi surface while iron pnictides have multiple bands at the Fermi energy, (iii) the superconducting gap order parameter has dx2–y2-symmetry in cuprates whereas for pnictides, the strong contender is an “extended s-wave”, also called s–-symmetry. Two of the families (parentmaterials) of pnictides originate fromRFeAsO [65] (R=rare earth, abbreviated as 1111 for its 1:1:1:1 ratio of the four elements) and AFe2As2 [71] (A=alkaline,the 122 compounds) earth metal, which are tetragonal at room temperature but undergo an or- thorhombic distortion in the range 100K to 200K that is associated with the onset of antiferro- magnetic order [69, 70, 72--75]. Tuning the system via element substitution [76, 77] or oxygen deficiency [78, 79] suppresses the magnetic order and structural distortion, in favor of super- conductivity with Tc up to 55K, with an overall behavior strikingly similar to the high-Tc cop- per oxide family of superconductors. However, the induced superconductivity by Co doping or other transition metals into the Fe site indicates that atomic disorder in the superconducting Fe layer ostensibly does not suppress superconductivity, contrary to the behaviors of layered cuprate high-Tc superconductors where doping onto the Cu sublattice is always detrimental [80] to Tc. A preliminary phase diagram [81, 82] of pnictide superconductors is shown in the figure 1.10. It may be noted that the parent compound of the superconducting iron arsenides exhibits a spin density wave (SDW)-type magnetic ordering at low temperatures [70, 83]. It appears that in high-Tc superconductivity, AF order needs to be suppressed before superconductivity may ap- 11pear. This has led many to the propose that dynamic AF fluctuations rather than static antiferro- magnetism is favorable for high-Tc superconductivity. A recent neutron-scattering experiment found that, in BaFe1.85Co0.15As2, the AF fluctuations are as strong as that in YBa2Cu3O6+x [84] and electron-phonon coupling is not the primary driver of superconductivity in pnictides, as in the case of cuprate.The origin of antiferromagnetic ordering in the pnictide parent compounds is a hotly debated topic, largely owing to its implications for the pairing mechanism: the elec- tronic structure suggests that the same magnetic interactions that drives the AFM ordering also produces the pairing interaction for superconductivity [85]. Regardless of the exact nature of magnetic order, it is believed that magnetostructural coupling is prevalent throughout the Fe- based superconductors in the form of coupled magnetic and structural transitions [86, 87]. The presence of both superconductivity and AF spin fluctuations has led to suggestions that quantum criticality may play an important role [88--90], however, prominence of quantum critical behav- ior in iron pnictides is disputed elsewhere [91]. Due to the large number of pnictides and the nature of chemical substitution, one limitation so far is that many experiments have been car- ried out on different systems or different chemical compositions of the same crystalline system, which makes comparisons difficult. However their generic features enable general conclusions to be drawn from several experiments. For instance, NMR Knight shift measurements indicate that the superconducting state has a singlet spin symmetry [92--94], suggesting an even order parameter symmetry (eg. s wave, d wave). 1.4 Pairing Symmetr y and Magnetic Penetration Depth Measurement One fundamental quantity in any superconductors is the London penetration depth 𝜆, which is closely related to superfluid density (𝜌s ” 1𝜆2 ∝ ns m* ). In general, the penetration depth 𝜆 is given as a function of the superfluid density ns, effective massm*, Ginzburg-Landau coherence length 𝜉 and the mean free path l as [95] 1 𝜆2 = 4𝜋nse2 m*c2 · 1 1 + 𝜉/l (1.4) In the clean limit, 𝜉l → 0 and the second term in (1.4) becomes unity. In this case, the variation in 𝜆 as a function of temperature, doping and orientation are of central importance in testing micro- scopic theories of exotic superconductors. For example, the linear variation of 1/𝜆2 with respect to temperature was a key finding confirming the d-wave nature of the pairing in YBa2Cu3O6+x [96, 97]. Also, early  SR studies of the vortex phase in polycrystalline samples found a linear correlation between 1/𝜆2 and Tc in the under-doped region [98, 99]. The resulting Uemura plot has played a prominent role in theoretical efforts to understand high Tc superconductivity [100]. Departure fromUemura scaling and the decline of the slope as the Tc = 0 quantum critical point is approached can be understood in terms of a 3D-QCP model [101]. Scaling of Tc with ns(0) in underdoped cuprates may also be due to quantum fluctuations near a 2D quantum critical 12point [102]. It is widely believed that cuprate high-Tc superconductivity originates in the twodimensional CuO2 layers [35, 36, 103]. YBa2Cu3O6+x also has one dimensional (1D) CuO chains which may contribute to superconductivity, the mechanism for which is not fully understood [104]. However, it is clear that CuO chains act as a quasi-1D system and charge reservoir [19]. Due to differences in band structures between planes and chains [105], one natural explanation for the same transition temperature is the proximity effect [106--108] whereby electron hopping be- tween chains and planes contribute to superfluidity along chain direction. The 1D chain induces, a–b anisotropy which has been observed in dc resistivity [109, 110] and optical conductivity and penetration depth measurements [111, 112]. In this simple model of multiband superconductivity in YBa2Cu3O7-𝛿, there is an intrinsic pairing interaction in the plane, but the chains are intrinsically normal, which means that the superconducting order parameter is nonzero in the plane layer only [106]. The pairing mecha- nism of YBa2Cu3O6+x is thought to be predominantly d-wave type. Other possible pairing states involving complicated gap functions, have been suggested [113--116]. The pairing symmetry in the recently discovered BaFe2As2 family of superconductor is still unclear. An accurate determi- nation of 𝜆(T) is one way to probe the symmetry of the pairing state. It has been theorized [117] that only the temperature dependence of 𝜆(T) at very low T is sensitive to pairing state of the superconductor. It is clear that accurate measurements of 𝜆 and a – b anisotropy are essential in clarifying central questions in YBa2Cu3O6+x . Unfortunately, accurate measurements of 𝜆 are difficult due to many possible systematic un- certainties. For example, in any bulk measurement the assumption of an exponential decay of the field in the Meissner state is only valid in the local London limit of a perfect surface [118]. Significant non-local effects are expected if the coherence length is comparable to the penetra- tion depth [119] or if there are nodes in the superconducting gap function [120]. Even within the London limit, there may be a non-exponential decay of the field, arising from any depth de- pendent change in the magnitude or symmetry of the order parameter. These add uncertainty to all conventional bulk measurements where the field profile is assumed and not measured. Al- ternatively, one can determine the absolute value of 𝜆 from  SR studies in the vortex state where the muon acts as a sensitive probe of the local magnetic field distribution. However, an accurate determination of 𝜆 requires there to be a well ordered vortex lattice with known symmetry. Also, there are substantial non-local and non-linear effects associated with vortices which complicate the theory and make it difficult to extract the true 𝜆 [120--123]. One approach is to fit the ob- served field distribution to a simple Ginzburg-Landau model involving an effective 𝜆 and then to extrapolate to zero magnetic field (or vortex density) [124]. Until now, the penetration depth has been measured in the vortex state via muon spin rotation [125] and using microwave tech- niques [96, 126--128] and using tunnel diode resonator method. In vortex state measurement, Sonier et al. used a GL model for magnetic field distribution to extract 𝜆 as a function of applied 13magnetic field. However, it was mentioned that 𝜆ab measured is an effective penetration depth which is model dependent. Consequently, one may expect some difference in 𝜆measured in the Meissner state where there are no vortices. The microwave techniques used in [96, 126--128] reported London penetration depth for a number of high-Tc superconductors. Microwave tech- niques arewell-suited tomeasuring temperature dependence of 𝜆 but generally not very sensitive to the absolute value of 𝜆. This is also true for the TDR method - which is closely related to the microwave techniques. In this thesis, we present measurements of the London penetration depth in detwinned crys- tals of YBCO for three oxygen dopings and a Co doped FE-PNICTIDE using low energy  SR. The field profiles are obtained directly by measuring the muon frequency spectrum - which con- tains most of the relevant information about muons' interaction with the local magnetic envi- ronment. The thesis is organized as follows: chapter 2 briefly discusses BCS pairing, London model and order parameter symmetry; chapter 3 discusses low energy  SR technique, stopping distri- bution of muons and Fourier analysis of asymmetry spectra; chapter 4 contains all the results and discussion followed by summary in chapter 5. 14CHAPTER 2 THEORY 2.1 BCS Theory The basic idea behind the BCS superconductivity is that an attractive interaction between elec- trons, regardless of their strength, can bind the electrons into pairs [10]. We consider a case for only two electrons added to the Fermi sea. The first electron attracts positive ions and these ions, in turn, attract the second electron giving rise to an effective attractive interaction between elec- trons. Since the lattice distortion can be expressed in terms of phonons, the interaction between electrons is said to be phonon mediated. The total energy of the electron system is minimized when there are Cooper pairs compared to a Fermi gas with no such pairing. The center of mass of a Cooper pair is zero since the electrons have opposite momenta and spin |ħk, ↑æ and |–ħk, ↓æ. Due to this opposite momenta and spin, it is labelled s-wave pairing since the relative angular momenta of the two electrons is zero. The electron-phonon system is described by the single complex order parameter 𝜓. A schematic of “in-phase” motion of the system is shown in the figure 2.1. vpair phonon =–k k Spin up electron Spin down electron Figure 2.1 : In the superconducting state, electrons pair up in zero-spin composites. They all move “in phase” and are said to be “coherent”. This is considered to be a ordered state and the whole electron-phonon system may be described by a single wavefunction. One important consequence of the BCS theory is the presence of a momentum dependent energy gapΔ(k) at the Fermi surface so that an amount of 2Δ(k,T) energy is required to create an excitation which loosely amount to breaking up a Cooper pair and removing it from the ground state. The energy gap is schematically shown in the figure 2.2. The gap is opened at the Fermi energy as the temperature is lowered below the critical temperature. A d-wave density of states is also shown in the figure 2.2. Unlike s-wave superconductors, some states are always available at the Fermi surface or where it was prior to the transition to superconductivity. In the weak coupling limit, the gap Δ(0) is much smaller than the characteristic phonon energy ħ𝜔D, 2Δ(0) kBTc = 3.52. (2.1) The numerical factor 3.52 is well tested in experiments and found to be reasonable, in the weak 15N(E) N(EF) -1 0 1 E Δ 1 EF (a) Fully G apped s- w av e -1 0 1 E Δ 1 EF (b) Reduc ed G ap d- w av e Figure 2.2 : (a), s-wave: Density of states in a conventional superconductor such as Nb. Note there is a region near EF where there are no available states and that excitations require a finite amount of energy given by the gap. This region comprises the superconducting energy gap. On either side of this gap there are singular peaks in the density of states. (b), d-wave: Density of states in a high-Tc superconductor likeYBCO. The density of states is reduced at the Fermi surface, however there is no true energy gap since there are certain directions in k space (the so called nodes) where it is possible to create an excitation with an arbitrarily small energy. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Δ(T )/Δ(0) = (︂ 1 – (︁ T Tc )︁4 )︂1/2 Δ(0) » 1.76kBTc T /Tc Δ (T )/ Δ (0) Figure 2.3 : Temperature dependence of the superconducting energy gap in the weak coupling limit of BCS inter- action. The superfluid densityΔ(T ) ∝ n1/2s in a two-fluid model [129] implies the (T /Tc) dependence of energy gap. This gap model is also experimentally verified [130]. coupling limit. Δ(T) remains fairly constant until the thermal energy becomes enough to excite quasiparticles, at low temperature. As will be shown shortly, the superfluid density depends on effective penetration depth as ns (︁ ~ Δ2(T) )︁ = 𝜆–2 mc2 4𝜋e2 (2.2) 16Inserting the empirical approximation, 𝜆–2 ∝ (1 – t)4 (︂ where t ” T Tc )︂ (2.3) into (2.2) ns (︁ ~ Δ2(T) )︁ = 1 – t4 = (1 – t)(1 + t)(1 + t2) ⇒ Δ2(T) » 4(1 – t) (at T ~ Tc) Δ(T) ~ (1 – t)1/2 (2.4) Near the transition temperature Tc, a more accurate form of Δ(T) may be given as Δ(T) Δ(0) ~ 1.74 (︂ 1 – T Tc )︂ 1 2 , T ~ Tc. (2.5) This is shown graphically in the figure 2.3. The most important manifestation of the electron-phonon interaction is the superconduct- ing state itself. According to our present understanding of Cooper pairing, the electron-phonon induced attraction between two electrons would not overcome their direct Coulomb repulsion, except for the fact that the former is retarded (where the second electron interacts with the lattice distortion after some time the first electron leaves that location), whereas the latter is not. 2.2 London Penetration Depth We consider the penetration depth in the Meissner state of a type II superconductor. BelowHc1, the London equations provide a good description of the electromagnetic properties. According to Maxwell's equations,  ⃗ · E⃗ = – 1 c 𝜕B⃗ 𝜕t . (2.6) In the classical Drude model of electrical conductivity, we have F⃗ = –m v⃗ 𝜏 – eE⃗ = m dv⃗ dt , (2.7) where v⃗ is the average drift velocity of the electrons, m is the mass of an electron, E⃗ is the electric field the electrons are in and 𝜏 is the relaxation time, i.e, roughly the time required to bring the drift velocity to zero if the electric field was suddenly set to zero. In a normal metal, the competition between the scattering and the acceleration in (2.7) leads to a steady state average (or drift) velocity v⃗ = eE⃗𝜏 m . (2.8) 17Assuming n conduction electrons per unit volume, we get the electric current density via Ohm's Law, J⃗ = nev⃗ = (︂ ne2𝜏 m )︂ E⃗ = 𝜎E⃗. (2.9) To describe superconductivity, London assumed that a certain fraction of electron density ns experience no relaxation i.e., letting 𝜏s in (2.7) go to infinity. This leads to d⃗Js dt = (︂ nse2 m )︂ E⃗, (2.10) where ns is density of the superconducting carriers. Taking curl on both side of the (2.10), we get m nse2 (︃  ⃗ · d⃗Js dt )︃ =  ⃗ · E⃗. (2.11) Substituting Maxwell's equation (2.6) into (2.11), we obtain the second London equation mc nse2 (︃  ⃗ · d⃗Js dt )︃ + dB⃗ dt = 0. (2.12) London postulated mc nse2 (︁  ⃗ · J⃗s )︁ + B⃗ = 0, (2.13) which is consistent with (2.12). Assuming no time varying electric field, another of Maxwell's equations connects J⃗s with B⃗ J⃗s = c 4𝜋 (︁  ⃗ · B⃗ )︁ . (2.14) Substituting (2.14) into (2.13), we get 𝜆2 (︁  ⃗ ·  ⃗ · B⃗ )︁ + B⃗ = 0, 𝜆2 2B⃗ + B⃗ = 0, (2.15) where 1 𝜆2 = 4𝜋nse2 mc2 . (2.16) In a vacuum-superconductor interface (which is also the case in our experiment), the solution of (2.15) is given by B(x) = B0exp (︁ – x 𝜆 )︁ (2.17) where B0 is the magnitude of the external applied field. The quantity 𝜆 is known as the London penetration depth. The most important success of the London equations (2.14) and (2.15) is that a static magnetic field is screened from the interior of a bulk superconductor over a characteristic penetration depth 𝜆. A simple estimate shows that this distance is mesoscopic and much larger 180 0.2 0.4 0.6 0.8 1 0.4 0.5 0.6 0.7 0.8 0.9 1 d Meissner state D ead La yer B(z) B0 = e–(z–d)/𝜆 z/𝜆 B( z)/ B 0 Figure 2.4 : External magnetic field drops exponentially ((2.18)) as it enters a superconductor inMeissner state. The characteristic distance 𝜆 is called the London penetration depth. Close to the surface, a reduction of superfluid density has been observed (cf. figure 4.23). With all of our experimental observations for energies (E ≥ 2 keV i.e. average muon stopping depth Æz𝜇 > 12 nmæ), a simple model of the presence of normal state close to the surface suffices to fit the data. than the mean distance r between electrons in the superconductor. As T → Tc from below, ns → 0 continuously and as a consequence, 𝜆(T) diverges according to (2.16). While (2.17) may be valid for a superconductor with an atomically flat surface, a real surface where the translations symmetry is broken and is non-flat might give rise to a suppressed order parameter for several nanometers and a modified London model (figure 2.4) B(z) = ⎧ ⎪⎨ ⎪⎩ B0 exp (︂ – z – d 𝜆a/b )︂ if z ‡ d B0 if z < d (2.18) may be more appropriate. Here B0 is the magnitude of the applied field, 𝜆a/b is the magnetic penetration depth in the a or b direction, respectively, z is the depth into the crystal, and d is an effective dead layer inside of which the supercurrent density is suppressed. Naively one might expect d to be the coherence length since the order parameter must be zero at the surface. However there are many other effects that might contribute to a d larger than the coherence length as will be discussed later. 2.3 Pairing Mechanism and Order Parameter Symmetr y The critical temperatureTc is the onset of long-range,macroscopic phase coherence in theCooper pairs. Long range correlations between pairs are described by off-diagonal long-range order (ODLRO) [131, 132], with no classical analog, which implies non-zero value of the pair correla- 19tion function 𝜌(⃗r, r⃗′) = ⟨ 𝜓†↓(⃗r)𝜓 † ↑(⃗r)𝜓↓(⃗r ′)𝜓↑(⃗r ′) ⟩ = ⟨ 𝜓†↓(⃗r)𝜓 † ↑(⃗r) ⟩ ⟨︀ 𝜓↓(⃗r ′)𝜓↑(⃗r ′) ⟩︀ (2.19) in the limit the pair separation |⃗r – r⃗′| is infinite. Here, 𝜓†↑(⃗r) and 𝜓↑(⃗r) are the particle field op- erators1 for creation and annihilation at a coordinate r⃗, with momentum k⃗ and spin ↑. From the finite value of the pair correlation function ((2.19)), the local pair amplitude ⟨ 𝜓†↓(⃗r)𝜓 † ↑(⃗r) ⟩ must be non-zero, which is, in essence the amplitude squared of the GL order parameter, |𝜓(⃗r)|2 ∝ ns, the superfluid density. It's important to note that, although ns is a “local pair” amplitude, pair- ings are non-local, ie, partners of a single pair are in macroscopic distance. Given macroscopic ODLRO, it's possible to derive the Meissner effect [131, 133] and flux quantization [131]. It is thought that ODLRO is a property not only of BCS superconductors but also of high-Tc super- conductivity [133, 134], including the recently discovered Pnictide superconductors [135]. As in any appearance of order, superconducting order lowers the symmetry of the system. In case of a 2nd order, continuous, superconducting transition, the order parameter is a measure of the “amount” of symmetry breaking in the ordered state. The symmetry group H describing the superconducting state is generally a subgroup of the normal state symmetry group G: G = X · R · U(1) · T for T > Tc (2.20) and H  G for T < Tc (2.21) whereX is the symmetry group of the crystal lattice, R the symmetry group of spin rotation,U(1) the one dimensional global gauge symmetry, andT the time reversal symmetry operation. Nearly all group-theoretic classifications of superconducting states are based on point-group symmetry. Point-group symmetry classification of pair states has been extensively studied in cuprate super- conductors [136--138]. Order parameter symmetry can give insight into the mechanism/nature of the pair conden- sate and limit the possible interactions that are possible. Crystal structures that have mirror symmetry (eg, CuO2 layer based superconductors) can be described by parity of the pair state. It has been argued that the complex phase diagram of high-Tc superconductors can be deduced from a symmetry principle that unifies antiferromagnetism and superconductivity [139]. The superconducting energy gapΔ(⃗k,T) for s-wave superconductors, is believed to be sym- metric in momentum space (figure 2.6a). One other very important pairing symmetry is of 1𝜓↑ (⃗r) = ⟨︀ 𝜓 ⃒ ⃒ 𝜓↑ (⃗r) ⃒ ⃒ 𝜓 ⟩︀ is the real space representation, where |𝜓æ is the ground state wavefunction. 20dx2–y2-wave where the energy gap is thought to be of the form Δ(⃗kF ,T) = Δ0g(T)(cos kx – cos ky) u Δ0 2 g(T)(k2x – k 2 y ), for (kx , ky) close to the nodes, (2.22) where momentum k⃗F is measured from the Fermi surface. It may be noted that for |k^x| = |k^y|, the gap is zero (figure 2.6b) meaning thermal excitations can be created with an arbitrarily small energy, implying there is significant T dependence in the superfluid density even at very low T . The temperature and angular dependence of the energy gap function,Δd(𝜑,T), may be obtained by integrating over momentum: Δd(𝜑,T) = Δ0,d gd(T) ∫︁ kmax k=0 (cos (cos k𝜑) – cos (sin k𝜑)) dk (2.23a) ” Δ0,d gd(T) gd(𝜑) (2.23b) ∝ Δ0,d gd(T) cos (2𝜑) (2.23c) 0 𝜋 2 𝜋 3𝜋 2 2𝜋 −1 −0.5 0 0.5 1 𝜑 g d (𝜑 ) Figure 2.5 : Angular dependence of a dx2–y2 -wave gap. High-Tc superconductors such asYBCO are believed to be primarily of dx2–y2-wave charac- ter [140--142]. Establishment of predominantly d-wave order in Tc materials, over a wide range of doping and temperature ranges implies that d-wave symmetry is a feature of the cuprates. This also suggests that d-wave pairing in cuprates has a common origin. The newly discovered family of Fe based superconductors may have s– (figure 2.6c) symmetry in the order parameter from band structure calculations [85, 144]. Although a superconducting mechanism isn't determined by the order parameter, the pairing Hamiltonian must obey the point-group symmetry of the gap function Δ(⃗k). 21(a) (b) (c) Figure 2.6 : A schematic 2D representation of the superconducting order parameter Δ(k) (where energy gap = |Δ(k)|) in different cases:(a) a conventional, s wave superconductor (eg. Nb); (b) a d wave, as is the case in cop- per oxides; (c) an s– wave, as is thought to be the case in iron-based superconductors. In (a) and (b), the two- dimensional Fermi surface is approximatedbyone circle. In (c), the Fermi surface is approximatedby a small circle in the center (the first band) surrounded by four larger circles (to comply with the tetragonal symmetry [143]; the second band). In all cases, the height of the “rubber sheet” is proportional to the magnitude of the order parameter (including its sign). 2.4 Critica l Temperature and Bose Einstein Condensa- tion In a conventional superconductor, the formation ofCooper pairs2 and that coherence set in at the same temperature. In cuprates, there is some evidence that Cooper pairs form above the critical temperature and that Tc represents the temperature where phase coherence (LRODO) sets in. This is similar to Bose-Einstein condensation (BEC). A simple model for a superconductor in this limit is to assume a gas of non-interacting bosons, which are Cooper pairs. The bosons are assumed to be confined to a box of volume V (︀ = L3 )︀ . The time-independent Schrodinger equation in such a case, – ħ2 2m  2𝜓(r) = E𝜓(r) (2.24) may be solved with periodic boundary conditions of 𝜓(x + L, y, z) = 𝜓(x, y + L, z) = 𝜓(x, y, z + L) = 𝜓(x, y, z) (2.25) yielding kx,y,z = (︂ 2𝜋 L )︂ nx,y,z (2.26) and E(k) = ħ2k2 2m = ħ2 2m (︂ 2𝜋 L )︂2 (︁ n2x + n 2 y + n 2 z )︁ , (2.27) where nx , ny , nz are non-negative integers. In otherwords, allowed k- wave vectors form a simple cubic lattice with a lattice constant 2𝜋/L. Therefore, each state occupies a volume in k-space equal 2The arguments in this section are largely from Prof. Rob Kiefl and has been included with his permission. 22to Δk = (︂ 2𝜋 L )︂3 = 8𝜋3 V (2.28) A summation of any smooth function F(k) may be written as  k F(k) = V 8𝜋3  k F(k)Δk (2.29a) ⇒ lim Vfi∞ 1 V  k F(k) = 1 8𝜋3 ∫︁ dkF(k) (2.29b) In the independent electron approximation, the Cooper pair density may be written as n = N V = 2 ∑︁ k f (E(k)) = 1 4𝜋3 ∫︁ dkf (E(k)) (2.30) using (2.29b), where f (E(k)) is the Bose-Einstein distribution function and the multiplication factor of 2 is due to spin degeneracy of electronic energy level. Changing the integration variable from k to E, (2.30) may be written as n = ∫︁ ∞ k=0 k2dk 𝜋2 f (E(k)) = ∫︁ ∞ E=0 dEg(E)f (E) , (2.31) where the density of states g(E) in three dimensions is given by g(E) = 1 4𝜋2 (︂ 2mE ħ )︂3/2 . (2.32) The first excited state has the energy Δ ” ħ2 2m (︂ 2𝜋 L )︂2 . (2.33) In general, kBT >> Δ and as such large number of states have non-zero Bose-Einstein distribu- tion function f (E) = 1 exp ( (E –  )) – 1 , (2.34) where  = kBT and  is the chemical potential. The number of Cooper pairs may be written as N » V ∫︁ ∞ 0 g(E) exp ( (E –  )) – 1 dE, (2.35) With a change of variable x =  E, (2.35) may be written as N = V 4𝜋2 (︂ 2m ħ2 )︂3/2 ∫︁ ∞ 0 x1/2 exp (x –   ) – 1 dx. (2.36) 23The maximum value of the integral in (2.36) is 1.36𝜋1/2, as  → 0. However, an unlimited num- ber of Cooper pair can condense into the ground state. Thus with a suitable choice of chemical potential  → 0, the occupation number diverges f (0) = 1 exp ( (E –  )) – 1 » 1 (1 –   + · · · ) – 1 = kBT | | . (2.37) For excited states, (E– ) > Δ, no such divergence can occur in f (E). Therefore, the ground state contribution to N is treated separately and the total number of Cooper pairs is written as N = f (0) + V 4𝜋2 (︂ 2m ħ2 )︂3/2 ∫︁ ∞ 0 x1/2 exp (x –   ) – 1 dx » N0 + 2.612V (︂ mkBT 2𝜋ħ2 )︂3/2 (2.38) Rearranging and dividing by N , the fraction of particles in the ground state may be written as N0 N = 1 – 2.612V N (︂ mkBT 2𝜋ħ2 )︂3/2 = 1 – (︂ T TBE )︂3/2 , (2.39) where the Bose-Einstein condensation temperature may be written as TBE = 2𝜋ħ2 mkB (︁ 𝜌 2.612 )︁2/3 , (2.40) with 𝜌 = N/V being the density of total number of pairs. Identifying TBE as the temperature at which pairs start to condense into the ground state, it is indeed the critical temperature Tc. Rewriting (2.40), the superfluid density (N0/N) may be written as 𝜌s(T) = 1 – (︂ T Tc )︂2/3 . (2.41) When c-axis effective carrier massmc is different than the a – b-plane carrier massm: The energy equation (2.27) may be written as E(k) = ħ2k2 2m = ħ2 2m (︃(︂ 2𝜋 L )︂2 (︁ n2x + n 2 y )︁ + (︂ 2𝜋 L𝛼 )︂2 n2z )︃ , (2.42) where 𝛼 = √︂ mc m . (2.43) In essence, the effective mass in all three axis directions may be taken equal to m, with the con- stant energy surface being an ellipsoid. For such a case, density of states g(E) in (2.31) may be 24shown to be g(E) = 𝛼 4𝜋2 (︂ 2mE ħ )︂3/2 . (2.44) Using the same arguments and calculations as before, the fraction of bosons in the ground state may be written as N0 N = 1 – 2.612V 𝛼 N (︂ mkBT 2𝜋ħ2 )︂3/2 = 1 – (︂ T TBE )︂3/2 , (2.45) where TBE = 2𝜋ħ2 mkB (︂ 1 2.612 𝜌 𝛼 )︂2/3 . (2.46) In the limit, the effective mass ratio 𝛼→∞, Tc → 0 - which is expected as the system becomes two dimensional and the Bose-Einstein critical temperature is zero in such a system. It is also clear that anisotropic effective mass will tend to suppress TBE. It will be seen in section 4.3 that dimensionality of possible BEC interaction may be influential in setting critical temperature, Tc, of phase coherence. 25CHAPTER 3 EXPERIMENTAL TECHNIQUES 3.1 Introduction To  SR Muons were discovered in the 1930's and their properties continue to be studied even now. Par- ity violation in muon decay was discovered by Garwin et al. in 1957. Wu et al. discovered parity non-conservation in  -decay that same year [145]. The meson factories such as TRIUMF and PSImade SR experimentsmuch easier in the 1970's. Lowenergy- SRwas only developed about 15 years ago.  SR/MUSR refers tomuon spin Rotation/Relaxation/Resonance techniques which use the anisotropic decay of almost 100% spin-polarizedmuons to investigate the local magnetic environment of matter, both in bulk and in thin films. This is very different compared to other magnetic resonance probes such as nuclear magnetic resonance (NMR) and electron spin reso- nance (ESR) methods which rely on thermal equilibrium spin polarization in a magnetic field to generate polarization and electrical circuits are used to detect the time evolution of polarization. Typically this implies relatively high magnetic fields which are not required in  SR. When a positivemuon decays, it emits a fast decay positron preferentially along the direction of its spin at the time of decay due to the parity violation. Froma single decay positron one cannot be certain which direction the muon spin is pointing in the sample. However, by measuring the anisotropic distribution of the decay positrons from a large number of muons, the statistical average direction of the spin polarization of themuon ensemble as a function of time after arrival in the sample can be determined. The time evolution of the muon spin polarization depends on the static and dynamic local magnetic field at the muon site. As such, the muon acts as sensitive probe of the local magnetic environment. 3.1.1 Properties and Production of Muons Muons are leptons, 207 times more massive than electrons. Muon properties are briefly men- tioned in the following table 3.1. Muons are generated from the decay of charged pions ( –), typically produced via energetic collisions of high-energy protons with target nuclei, such as car- Table 3.1 : Properties of Muon [146] Mass, m 105.658389(34) MeV/c2 Lifetime,   2.197034(21)  s Charge, q –e Spin ħ2 Magnetic moment 4.49044786(16)·10–26 J/T Spin g-factor, g 2.0023318414(12) Gyromagnetic ratio,   = g   /h 2𝜋 · 135.69682(5) MHzT−1 26bon or beryllium. The positively charged pions have a mean lifetime of about 26 ns and decay into a muon and muon neutrino (antineutrino)  + →  + +    – →  – +   (3.1) and is schematically shown in figure 3.1. The most common source of muons is from positive  + S p  + S = 0   p S Figure 3.1 : Positively charged pions live for about 26ns and then decay into amuonand a neutrino. Both themuon and neutrino are 100% spin-polarized along their momentum direction. The muons carry a kinetic energy of 4.12MeV in the rest frame of pion. Muons, originating from pions decaying near the surface of the production target, are called “surface muons”. pions those stop near the surface of the production target and decay at rest into a positive muon. The resulting muons have an energy close to 4.1MeV since they do not lose much energy before escaping from the target. This is called a ``surface'' muon beam. Negative pions, those stopping in the production target, behave as heavy electrons and are captured by the nucleus before they can decay into negative muons. Positive pions occupy interstitial positions in the target material and subsequently decay into a positive muon and a neutrino with a lifetime equal to that in free space. Therefore, positive muons are most commonly used to investigate local magnetic envi- ronments in materials. Because of the parity violating nature of weak beta-decay, the positron in a  + decay is correlated with direction of muon spin at the time of decay. The time evolution of the ensemble average polarization may be determined from the decay pattern measured from a large ensemble of muons. The probability per unit time, that the positron is emitted in decaying at an angle 𝜃 with respect to the  + spin polarization is given by dW(𝜖, 𝜃) dt = e–t/𝜏 𝜏 [1 + a(𝜖) cos(𝜃)] n(𝜖) d𝜖 d(cos(𝜃)) (3.2a) ” e–t/𝜏 𝜏 K(𝜖, 𝜃) n(𝜖) d𝜖 d(cos(𝜃)) (3.2b) where the “reduced energy” 𝜖 = E/Emax, the asymmetry a(𝜖) = (2𝜖 – 1)/(3 – 2𝜖), the energy density function n(𝜖) = 2𝜖2(3–2𝜖). The maximum positron energy Emax ≃ 52.8MeV, is slightly less than half of the muon rest mass. Integrating over reduced energy (𝜖), we get dW(𝜃) = e–t/𝜏 𝜏 [︀ 1 + Æa(𝜖)æ cos(𝜃) ]︀ d(cos(𝜃))dt (3.3) 27Ee+ = 52.8MeV Ee+ = 26.4MeV Ee+ = 39.6MeV, Æa(𝜖)æ = 1/3 μ+ 𝜃 Figure 3.2 : Angular probability distribution (K(𝜖, 𝜃), between 26.4MeV and 52.8MeV) of positrons emitted from muons with polarization along the red arrow direction is shown. The maximum average asymmetry for such a  -decay process is 1/3 (the dotted curve). where, the average of the asymmetry function is Æa(𝜖)æ = ∫︁ 1 0 a(𝜖) n(𝜖) d𝜖 = 1 3 (3.4) The probability function K(𝜖, 𝜃) as a function of polar angle 𝜃 is plotted in figure 3.2 for reduced energy values of 𝜖 =0.5, 0.625, 0.75, 0.875 and 1.0. The average theoretical -decay asymmetry is therefore 1/3. The asymmetry is determined in our experiments in a transverse field (TF- SR) arrangement as shown in figure 3.3. TF- SR refers to the case of incomingmuonpolarization be- ing perpendicular to the external field direction For more details of muon production the reader is referred to references [147, 148]. 3.1.2 General  SR Techniques Detailed descriptions of the various techniques may be found in the following references: the book of Schenck [149], the review article of Cox [150] & of H. Keller [151]. For general tech- nical and statistical details, readers may consult these theses: Riseman [147], Chow [152] and Luke [153]. While conventional surface muon beams can be used to investigate rather small samples, there is a desire for still lower energy muons those can be stopped near sample surfaces (for example to determine depth dependent magnetic field), in thin films and near multi-layer interfaces (to determine exotic magnetic phenomena). Several innovative methods have been developed to produce ultra slow muon beams. The results, in this thesis, were obtained using the most successful method mentioned in the next section. 28Spin-polarized muon beam Muon detector Positron detector Sample + νe ν e + H X Y Z Electronic clock Start Stop Figure 3.3 : Schematic of the arrangement for a TF- SR experiment. The muon spin Larmor precesses about the local magnetic field B at its stopping site in the sample, and subsequently undergoes the three-body decay  + → e+ +  e +   . The time evolution of the muon spin polarization is accurately determined by detection of the decay positrons from ~ 106 muons implanted one at a time. 3.2 Low Energy  SR The experiments detailed in this thesis were done in the LowEnergyMuon (≤ 30 keV)LEM- SR beamline ( E4 [154]) at Paul Scherrer Instituit (PSI). The LEM group has developed a technique of moderating a surface muon beam of 4MeV down to about 15eV and then accelerating these muons to form a beam with a tunable energy between 2 keV to 30 keV. Remarkably, these low energymuons remain 100% spin polarized. Figure 3.4 shows the schematic low energy beamline  E4. The 4MeV beam passes through a moderator consisting of a thin layer (~ 100 nm) of rare gas solid or solid nitrogen deposited on 125 m silver substrate. A small fraction (︀ 10−5 to 10−4 )︀ of the muons escape the moderator with a mean energy of about 15 eV with an energy spread (FWHM) of ~ 20 eV. The dominant fraction of the beam exits the moderator target as fast (de- 29graded but not moderated) muons with a mean energy of 500 keV and a FWHM of the same order. These fast muons are separated from the slow ones by an electrostatic mirror composed of a fine wire grid, to deflect the low energy muons by 90 degrees. This deflection, of slow muons, necessarily changes the momentum direction of muons while keeping the spin direction unal- tered. After deflection, the muon spin and momentum directions are perpendicular. Fast muons are unaffected by the electrostatic mirror and are monitored by a multi channel plate (MCP) detector. The low energy muons are clearly identified by a time-of-flight (TOF) measurement between the start scintillator and the trigger detector. The trigger detector, made of an ultra- thin carbon foil (︀ 2.2 gcm−2 )︀ , is used to set time-zero, t0 for the incoming low energy muons. The muons, traversing the foil, emit a few electrons, which are deflected by 90° and detected by a MCP to give the start signal for the  SR measurement. The trigger detector causes an energy loss of the muons ~ 1.6 keV with a Gaussian energy spread ~ 500 eV. Detection efficiency of the trigger detector is ~ 80 %. After passing the trigger detector, the  + beam is focused on the sample by an Einzel lens (L3) and a conically shaped electrostatic lens. The ring anode labelled RA in figure 3.4 is the final element which also acts to steer and focus themuons onto the sample. The sample and its Ni-coated1 Al holder are electrically insulated by a thick sapphire crystal and can be biased -12.5 keV to +12.5 keV. The sapphire crystal also provides a good thermal contact between the cold finger and the sample. The bias voltage (between 12 keV to 20 keV), coupled with the voltage at the trigger detector, determines themuon energy at the sample which is typically in the range 1 keV to 30 keV. A small coil is used to generate a small external mag- netic field of up to 20mT applied parallel to the sample surface and perpendicular to the beam direction. Extensive details on low energy muon beamline may be found elsewhere [155, 156]. 3.2.1 Principles of LE- SR For each detector one creates a histogram of events as a function of time after arrival of themuon in the sample. The resulting time histograms have the following form: N(t) = Nbg + N0e –t/  (1 +A(t)) , (3.5) where N0 is the normalization, Nbg is a time-independent background and A(t) is the time de- pendent asymmetry for the detector along n^ direction, defined as A(t) = A0P(t) = A0n^ ·P(t) (3.6) where A0 is directly related to the theoretical maximum asymmetry a(𝜖) referred in (3.4) The 1The sample holder is coated with ~ 1 g of Ni. Since Ni is ferromagnetic, muons, those miss the sample, strike Ni-coated sample plate and experience a large hyperfine field and disappear from the frequency window of interest. This very effective background suppression method was the critical step which allows low energy  SR to be applied to crystals much smaller than the beam diameter. 30MCP1 Mirror L1 S1 Surfacemuon beam E ≃ 4MeV S p Mod. L2 Trigger Compensa- tion coil L3 RA Helmholtz coils H ^ p , H ^ S e+ det. e+ det. Sample Cryostat S1 Surface muon scintillator Mod. Moderator (moderation & subse- quent re-acceleration) L1 Electrostatic Einzel lens (cooled by liquid N2) Mirror Electrostatic mirror (separation of LE muons) MCP1 Fast muon detector L2 Electrostatic Einzel lens Trigger LE muon start detector for μSR measurements L3 Electrostatic Einzel lens (cooled by liquid N2) RA Electrostatic conical lens (main fo- cusing element) e+ det. Scintillators for positron detection Figure 3.4 : Adapted from [155]: A small fraction of incoming muons incident on the cold moderator come out as slow muons with a mean energy of about 15 eV. They are subsequently accelerated to energies up to 20 keV. The positrons, from the muons decaying in the sample, are detected by four pairs of scintillators surrounding the vacuum tube. 310 0.5 1 1.5 2 2.5 3 3.5 4 −30 −20 −10 0 10 20 30  0H = 9.5mT T = 110 K E = 14.1 keV Depolarization from Ni Time (µs) A (︀ 10 −2 )︀ Figure 3.5 : LE- SR asymmetry spectrum of a YBa2Cu3O6.92 mosaic in a transverse field of 9.5mT. The muon im- plantation energy is 14.1keV and the temperature is 110K. Note that the first two points are off the curve; these are attributed to a fast precessing signal in the Ni film which is rapidly dephased in a time t < 150 ns. The almost undamped oscillation with a frequency corresponding to the applied field at higher times resulting from muons in the YBa2Cu3O6.92 mosaic. Consequently all the subsequent analysis cuts out the first 150ns of data. muon polarization P(t) reflects time-dependent spin-polarization and its modulus defined as P(t) = ÆS(t) · S(0)æ ÆS(0)2æ (3.7) where P(0) = –n^ and n^ refers to the direction of observation (detector). The observed asymmetry at t = 0 in our samples is typically about 0.12. The reduction com- pared to theoretical maximum 1/3 is mainly due to the fact that emitted positrons are collected in a finite solid angle and about half of the muons miss the sample. Those that strike the Ni- coated sample plate precess outside the the frequency window of interest and effectively reduce the amplitude of precession close to the Larmor frequency. Spin Evolution in Quantum Mechanics This discussion follows largely from [157]. In quantum mechanics of a spin-12 system, the spin operator is given by S = ħ 2 𝜎, (3.8) where 𝜎x , 𝜎y , 𝜎z are Hermitian Pauli matrices, given as 𝜎x = (︃ 0 1 1 0 )︃ , 𝜎y = (︃ 0 –i i 0 )︃ , 𝜎z = (︃ 1 0 0 1 )︃ . (3.9) 32The eigenstates (︀ | ↑æ, | ↓æ )︀ of 𝜎z is conventionally chosen as basis states to represent any general spin state (wavefunction) |𝜓æ = 𝛼| ↑æ + 𝛽| ↓æ, (3.10) with the normalization condition, |𝛼|2 + |𝛽|2 = 1. (3.11) A spin pointing in direction n^ = (cos 𝜙 sin 𝜃, sin 𝜙 sin 𝜃, cos 𝜃), will be an eigenstate of the pro- jection of the spin operator (3.8), along n^, with eigenvalue of 1, i.e, n^.𝜎 (︃ 𝛼 𝛽 )︃ = (︃ 𝛼 𝛽 )︃ (3.12) Solving (3.12), one arrives at the normalized wavefunction |𝜓æ = ⎛ ⎝ exp (︀ –i 𝜙2 )︀ cos (︁ 𝜃 2 )︁ exp (︀ i 𝜙2 )︀ sin (︁ 𝜃 2 )︁ ⎞ ⎠ (3.13) The time evolution of a spin system is dictated by the Zeeman interaction Hamiltonian, H = –𝜇 · B, (3.14) where 𝜇 = 𝛾S with 𝛾 being the gyromagnetic ratio of the relevant particle (or system). Time evolution of the spin system is given by the unitary operator U(t) = exp (︂ – i ħ Ht )︂ = exp (︂ i 2 𝜎 · Bt )︂ . (3.15) For simplicity, the applied field may be assumed to be in the z direction, i.e, B = Bz^ and the time evolution operator may be written as U(t) = ⎛ ⎝ exp (︁ i 2 𝛾Bt )︁ 0 0 exp (︁ – i2 𝛾Bt )︁ ⎞ ⎠ (3.16) At time t, the initial wavefunction (3.13) evolves to |𝜓(t)æ = U(t)|𝜓(0)æ = ⎛ ⎝ exp (︁ – i2 (𝜙 – 𝛾Bt) )︁ cos (︁ 𝜃 2 )︁ exp (︁ i 2 (𝜙 – 𝛾Bt) )︁ sin (︁ 𝜃 2 )︁ ⎞ ⎠ , (3.17) 33which clearly shows the initial spin gets rotated around the applied field direction by an angle 𝛾Bt, with a Larmor precession frequency 𝜔L = 𝛾B. (3.18) If the initial spin is in the +x (𝜙 = 0, 𝜃 = 𝜋/2) direction, (3.13) becomes |𝜓æ = 1 √ 2 (︃ 1 1 )︃ (3.19) and wavefunction at t may be written as |𝜓(t)æ = 1 √ 2 ⎛ ⎝ exp (︁ i 2 𝜔Lt )︁ exp (︁ – i2 𝜔Lt )︁ ⎞ ⎠ . (3.20) The expectation value of the spin operator Sx = ħ2 𝜎x may be written as Æ𝜓(t)|Sx|𝜓(t)æ = (︂ ħ 2 )︂ (︂ cos2 (︂ 𝜔Lt 2 )︂ – sin2 (︂ 𝜔Lt 2 )︂)︂ = (︂ ħ 2 )︂ cos (𝜔Lt) (3.21) Similarly, the expectation values of spin operators Sy and Sz may be written as Æ𝜓(t)|Sy|𝜓(t)æ = (︂ ħ 2 )︂ sin (𝜔Lt) (3.22a) Æ𝜓(t)|Sz |𝜓(t)æ = 0 (3.22b) In other words, ensemble average spin precesses in the x – y plane when the applied field is in the z direction. Classical Spin Precession A similar result can be obtained classically as follows. The ensemble average spin changes ac- cording to the Bloch equation may be written as, dS dt = μ · B (3.23) 34where, B = Bext + Bint is the sum of external and internal field. For total magnetic field along z^, Bext = Bz^, dSx dt = 𝛾SyB (3.24a) dSy dt = 𝛾SxB (3.24b) dSz dt = 0 (3.24c) The above equations (3.24a) and (3.24b) are solved by Sx(t) = S(0)sin(𝛾Bt + 𝜙) Sy(t) = S(0)cos(𝛾Bt + 𝜙) (3.25) where S(0) and 𝜙 are determined by spin direction at t = 0. Spin Polarization and Asymmetry Spectrum Here we discuss the effects of (random) internal fields those are present in any sample due to randomly oriented magnetic moments: P(t) = ∫︁ ∫︁ ∫︁ P(B)PB(t)dB (3.26) where B is the local magnetic field muon experiences, B = |B|, PB(t) is the time-dependent (oscillating) muon polarization and P(B) is the normalized probability distribution of magnetic field inside the sample. Inside the sample, randomly oriented magnetic moments can generate a distribution of magnetic field, at any given muon site, and may be approximated by a three- dimensional Gaussian distribution, P(B) = (︂ 𝛾𝜇√ 2𝜋𝜎 )︂3 exp (︃ – 𝛾2𝜇B 2 2𝜎2 )︃ , (3.27) where 𝛾𝜇 is the muon gyromagnetic ratio. For an external field perpendicular to muon spin direction, e.g. Bext = Bextz^, the Gaussian probability distribution (3.27) is centered on field B = (0, 0,Bext), and has the form P(B) = (︂ 𝛾𝜇√ 2𝜋𝜎 )︂3 exp ⎛ ⎝– 𝛾2𝜇 (︁ B2x + B 2 y + (Bz – Bext) 2 )︁ 2𝜎2 ⎞ ⎠ (3.28) 35With P(0) ‖ x^, the muon polarization may be written as, P(t) = (︂ 𝛾𝜇√ 2𝜋𝜎 )︂3 ∫︁ ∫︁ ∫︁ exp ⎛ ⎝– 𝛾2𝜇 (︁ B2x + B 2 y + (Bz – Bext) 2 )︁ 2𝜎2 ⎞ ⎠ cos(𝛾𝜇Bt)dB ≃ 𝛾𝜇√ 2𝜋𝜎 ∫︁ ∞ Bz=–∞ exp (︃ – 𝛾2𝜇 (Bz – Bext) 2 2𝜎2 )︃ cos(𝛾𝜇Bzt)dBz = (︂ 𝛾𝜇√ 2𝜋𝜎 )︂ (︂√ 2𝜎 𝛾𝜇 )︂ ∫︁ ∞ n=–∞ exp (︁ –n2 )︁ cos (︂ 𝛾𝜇 (︂√ 2𝜎n 𝛾𝜇 + Bext )︂ t )︂ dn = (︂ 1 √ 𝜋 )︂ ∫︁ ∞ n=–∞ exp (︁ –n2 )︁ cos (︁√ 2𝜎tn + 𝛾𝜇Bextt )︁ dn = exp (︂ – 𝜎2t2 2 )︂ cos (︀ 𝛾𝜇Bextt )︀ (3.29) As may be seen from (3.29), random moments in a sample cause damping in the muon polar- ization. Additionally, when muons are implanted into a sample with inequivalent magnetic sites (e.g. superconducting state), there will be a distribution (𝜌(B)), discussed in detail in the follow- ing section, of fields (B(z) ” Bext(z)) inside the sample & the polarization (3.29) takes the form P(t) = exp (︂ – 𝜎2t2 2 )︂ ∫︁ ∞ 0 𝜌(z)cos (︀ 𝛾𝜇B(z)t )︀ dz (3.30) The asymmetry A(t) = A0P(t) is fitted to the experimentally observed asymmetries to obtain physical parameters A0, 𝜎, d, 𝜆, where d and 𝜆 are deadlayer and London penetration depth in the Meissner state. An example histogram of raw data, from the forward counter and asymmetryA(t) are given in figure 3.6. In this experiment, a transverse external field of 9.5mT was applied and the result- ing sinusoidal oscillations can be taken into account by the theoretical model in (3.30). For the purposes of this thesis, the most important characteristic of the muon polarization decay spec- trum is the average precession frequency - which is a direct measure of the local magnetic field averaged over the stopping distribution. 3.2.2 Stopping Distribution Even at a single muon implantation energy, ions bombarded onto a material, end up at differ- ent depths due to randomness in the collision process. The energy dependent stopping profiles can be generated using “Transport of Ions in Solids” (TRIM) codes introduced by Ziegler et al [158] based on the ideas of Eckstein [159]. The accuracy of TRIM, in calculating ion range distributions in various materials, is well established and they are routinely used in similar depth controlled experiments such as Low-Energy  SR. By specifying the energy, charge, and mass of the probe ions ( + in this thesis), and the mass density and atomic numbers of the elements of the probed material, one is able to simulate the implantation profile using TRIM. Generated 36200 400 600 YBa2Cu3O6.92 T = 90 K E = 22 keV Coun ts/ns 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −10 0 10 Time (µs) A (︀ 10 −2 )︀ Figure 3.6 : Top: A histogram from “forward” positron detector in a time-differential measurement on the sample of YBa2Cu3O6.92 crystal mosaic, depicting the exponential decay of the muons and muon precession. Bottom: Larmor precession of muons’ spin magnetic moment may be noticed from the time dependence of asymmetry A(t), in an external applied field of 9.5mT with an implantation energy of 22keV, at the temperature of 90K. energy-dependent profiles of YBCO are shown in figure 3.7. The average depth of muons in- creases with energy and an almost linear relationship (Æz æ (E) = 3.26 + 4.37*E nm) is obtained as shown in figure 3.7. For Ba(Co0.074Fe0.926)2As2, a different energy dependent linear relation- ship (Æz æ (E) = 4.26 + 4.53*E nm) is obtained. The non-zero mean depth at E = 0 is due to the Gaussian distribution (of width 0.45keV) of any energy that has been set. The stopping profiles are only relevant for the  + ions stopping in the probed material. A fraction of incoming  + ions stop at Ni-coated sample plate where it quickly depolarizes (see figure 3.5) and do not affect the frequency spectrum from the sample. A review of depth resolved studies of materials may be found in [156, 160] and references therein. 3.2.3 Maximum Entropy (ME) Fourier Analysis The stopping distribution 𝜌(z) introduced in the previous sections, in conjunction with the Lon- don model field decay, can be used to calculate a theoretical muon frequency spectrum in the Meissner state. Typically, we fit the observed precession frequency to the above model to extract the parameters of interest such as the magnetic penetration depth. However, sometimes it is useful to compare the theoretical frequency distribution with an experimental frequency distri- bution with minimal modeling. In principle, a fast Fourier transform is one way to approximate the experimental frequency distribution. However there are distortions due to ringing etc. in a 370 50 100 150 0 2 4 6 E = 3.5 keV 14.1 24.6 YBa2Cu3O6+x z (nm) 𝜌 (︀ 10 −2 nm −1 )︀ (a) 0 5 10 15 20 25 0 20 40 60 80 100 120 3.26+4.37* E YBa2Cu3O6+x TRIM.SP Simulation E (keV) Æz 𝜇 æ(nm) (b) Figure 3.7 : Muon implantation profiles inYBCO: (a) Implanted muons stop at different depths even if the incom- ing beam energy is the same. By specifying the energy, charge and mass of the + ions, and the mass density and atomic numbers of the elements, profiles are simulated via Monte-Carlo algorithm using TRIM.SP. The accu- racy of TRIM.SP, in calculating ion range distributions in various materials, is well established and TRIM.SP is rou- tinely used in similar depth controlled experiments. (b)Average muon implantation depth (︀ Æz𝜇æ = ∫︀ 𝜌(z)dz )︀ is shown. A linear fit (3.26 + 4.37*E) well describes the average implantation depth as a function of muon energy. Fourier transform. A better way is to generate the experimental frequency distribution by using the principle of maximum entropy (ME). In the ME method, a trial field distribution function PME(bj) is proposed where the the amplitudes of histogram bin j, are the fitting parameters. Fourier transforming the trial field distribution, one obtains the theoretical complex asymmetry Acomplex(tk) = bj,max∑︁ bj,min PME(bj)exp (︀ i2𝜋  bjtk )︀ 𝛿bj ” Ar(tk) + iAi(tk), (3.31) whereAr(tk) andAi(tk) correspond to fictional detectors with phases of 0 and 90 degrees with re- spect to the initial muon polarization. The actual ME asymmetry for a counter is then calculated as AME(tk) = cos(𝜑)Ar(tk) + sin(𝜑)Ai(tk), (3.32) where 𝜑 is the detector phase relative to the incoming muon spin direction. The theoretical function above can then be used to calculate a 𝜒2: 𝜒2 = M∑︁ k=1 (︀ A(tk) – AME(tk) )︀2 𝜎2(tk) , (3.33) whereA(tk) is the measured asymmetry amplitude in k-th bin and 𝜎(tk) is the error in measured asymmetry. Unfortunately, the (3.33) has many fit parameters and additional information is required to obtain the most likely solution. The ME method introduces an entropy term, in this 38regard: S = – ∑︁ j P(bj) e(bj)Pd ln P(bj) e(bj)Pd , (3.34) with the default value Pd is related to statistical noise in the spectra and e(bj) are ``independent errors''. Adding the 𝜒2 term using a Lagrange multiplier 𝛼, a constrained entropy Sc is defined as Sc = 𝛼S – 1 2l2 𝜒2, (3.35) with the constraint 𝜒2 ” l2M, (3.36) where the ``looseness'' factor l is introduced to account for any imperfect timing in the exper- imental electronics and to account for the ME Fourier transformation being overly sensitive to errors 𝜎(t). In practice, a l few percent above 1 is good enough to estimate field distributions for most data described in this thesis. The goal of the ME fitting routine is to maximize the con- strained entropy given in the (3.35). Extensive details of ME analysis may be found elsewhere [161--164]. 39CHAPTER 4 RESULTS AND ANALYSES In this chapter, we present the first direct measurements of 𝜆 in the Meissner state of high qual- ity crystals of cuprates and pnictides. Measurements of 𝜆 and its anisotropy in the a – b plane are presented for three different oxygen (x = 6.52, 6.92 and 6.998) contents of YBa2Cu3O6+x . In addition we present measurements of the in-plane 𝜆 on a single crystal of Co doped Ba(Co0.074Fe0.926)2As2. Measured values of 𝜆 and the anisotropies are, in some cases, consid- erably different from that of literature, often found with less direct methods. The local magnetic field decreases inside a superconductor, on a length scale 𝜆, according to the London model, described in section 2.2. Close to the surface, there may be deviations from a purely exponential B(z) as conjectured in (2.18). The magnetic field profile can be detected using LE- SR since the muon precession frequency is proportional to the local magnetic field. A schematic representation of the fitting procedure to obtain 𝜆 and other parameters is shown in figure 4.1. In the normal state, the external field ( 0H) penetrates the sample fully yielding an average muon precession frequency of (︀    0H )︀ /2𝜋. A distribution of the local fields, from randomly oriented nuclear moments (see (3.29)), is accounted for by the broadening parameter 𝜎. In the superconducting state, the muon precession frequency depends on implantation depth (z). For the muons landing between depth z and z + dz, the time dependent asymmetry may be written as A(t) = A0 exp [︁ –𝜎2t2/2 ]︁ · ∫︁ 𝜌(z) cos [︀  𝜇B(z, d, 𝜆)t + 𝜙 ]︀ dz. (4.1) A slightly modified form (4.2) of the above equation (4.1) is used to fit all the data presented in this chapter and thereby to extract the relevant parameters of interest A0, d, 𝜎, 𝜆, 𝜙. Average magnetic fields ÆBæ are computed from the above mentioned parameters for various externally appliedmagnetic fields ranging from 1.5mT to 10mT. All measurements in the superconducting state were carried out under zero-field-cooled conditions in order to avoid flux penetration into the sample. Note in a field-cooled experiment it is possible for vortices to become trapped inside the sample even below Hc1. The data analyses, in this chapter, are presented chronologically: (i) YBa2Cu3O6.92, section 4.1.1. (ii) YBa2Cu3O6.998, section 4.1.3 (iii) YBa2Cu3O6.52, section 4.1.4 (iv) Ba(Co0.07Fe0.93)2As2, section 4.2 400 2 4 6 8 −0.1 0 0.1 Time (µs) A symmetr y A (t ) Normal State 0 2 4 6 8 −0.1 0 0.1 Time (µs) A symmetr y A (t ) Superconducting State Fit to: A0 exp [︀ –𝜎2nuct 2/2 ]︀ cos (︀   B0 + 𝜙0 )︀ Fit to: A0 exp [︀ –𝜎2t2/2 ]︀ ·∫︀ 𝜌(z) cos (︀   B(z, d, 𝜆)t + 𝜙 )︀ dz Figure 4.1 : Quick outline of the fitting procedure: Left: In the normal state, oscillatory signal in muon polarization is fitted to a cosine function with a small Gaussian broadening (𝜎nuc) originating from randomly oriented local internal fields. Right: In the superconducting state, polarization is a sum of frequency (︀   B )︀ dependent oscil- lation and an additional broadening 𝜎 taking into account expelled flux from neighboring crystals or from any other sources of random fields. 4.1 LE- SR Experiments on YBCO Single Crysta ls 8.6 mm 8.3mm YBa2Cu3O6.92 ~ 71mm2 13.3 mm 9.6mm YBa2Cu3O6.998 ~ 129mm2 Figure 4.2 : Mosaics of detwinned crystals: The total areas of the boxes surrounding the mosaics are about 71mm2 and 129mm2 for YBa2Cu3O6.92 and YBa2Cu3O6.998, respectively. Each of the crystals was approximately 1mm to 3mm in the a – b plane and a c-axis thickness of 0.1mm to 0.3mm. An ultra high vacuum compatible Ag epoxy was used to attach each crystal to the sample holder of high purity Al coated with 1 m of Ni. The LE- SR experiments on YBa2Cu3O6+x were conducted on a mosaic of small freshly grown single crystals using the self-fluxmethod [165]. The purity of the crystals is the same as for 410 2 4 0 2 4 µm µm 0 0.5 1 1.5 2 0 0.5 1 1.5 2 µm 0 20 40 60 nm Figure 4.3 : YBCO roughness: Left figure shows a 5 m·5 m area of a YBa2Cu3O6.92 crystal being scanned by tapping AFM. Right figure shows another region of a smaller size and even smaller sized box. A 2nm to 3nm of average roughness is found from these measurements. crystals in which quantum oscillations in resistivity have recently been reported [166]. Each of the crystals was approximately rectangular in shapewith lateral dimensions in the a-b plane rang- ing from 1mm to 3mm and a thickness in the c-direction ranging from 0.1mm to 0.3mm. They were detwinned to a level greater than 95%. The mosaic of YBa2Cu3O6.92 and YBa2Cu3O6.998 on the coldfinger is shown in the figure 4.2. Two crystal mosaics with different O concentrations were mounted in a very similar method. The crystal faces were mirror-like in appearance and atomic force microscopy indicated the roughness of the surface was few nm on a short length scale (cf. figure 4.3). A UHV compatible Ag epoxy was used to attach each crystal to the sample holder made of high purity Al coated with 1  m of Ni. Muons that miss the sample stop at the polycrystalline Ni coated sample holder and experience a large (~ 64mT, see [167]) hyperfine field from Ni moments; thereby removing them from the frequency window of interest. Control experiments on a Ag disc the same size as the sample showed that such a thin layer of Ni has no effect on the precession signal in the Ag sample. The amplitude of the precession signal in all the experiments was typically about 0.11 which is considerably smaller than the instrumental maximum asymmetry of about 0.28. As mentioned previously in chapter 3, this reduction is due to the fact that only about 40% muons land in the YBa2Cu3O6+x crystals; the remaining muons which land in the Ni do not contribute to the precession signal but still produce positrons that are detected by the telescopes. 424.1.1 YBa2Cu3O6.92 The precession signal in the superconducting state was fit to the following form which is equiv- alent to (4.1) A(t) = A0 exp (︂ – 𝜎2t2 2 )︂ ∫︁ 𝜌(B) cos [︀   B(z, d, 𝜆)t + 𝜙 ]︀ dB, (4.2) The difference compared to (4.1) is that the integration is over the field distribution 𝜌(B) instead of the stopping distribution 𝜌(z). 𝜌(B) is defined as 𝜌(B) = 𝜌(z) ⃒ ⃒ ⃒ ⃒ dB dz ⃒ ⃒ ⃒ ⃒ –1 (4.3) Although equations (4.1) and (4.2) are mathematically equivalent for the analysis presented in this thesis, the latter offers a more general approach for analyzing data using models where a specific field distribution is believed to be present. To take into account any random local field present at a muon site, a Gaussian probability model is assumed as, P (︀ B′,B )︀ = (︂ 1 √ 2𝜋ΔB )︂ exp (︃ – 1 2 (︂ B′ – B ΔB )︂2 )︃ (4.4) where P (︀ B′,B )︀ is the probability function of the total field experienced by muon and ΔB is the width of the probability function. The total probability density is obtained by convoluting (4.4) with the field distribution from the pure London model in the (4.3) 𝜌′ (︀ B′ )︀ ” ∫︁ ∞ B=0 P (︀ B′,B )︀ 𝜌(B) dB (4.5) Since the stopping distributions (𝜌(z)) are obtained as discrete numbers from TRIM.SP simula- tions, the integral in (4.5) is replaced by a summation to calculate the total probability density. Asmay be seen from the figure 4.4a, the field distributions from amaximum-entropy (ME) anal- ysis [162--164] are wider than that expected from a pure London model (𝜎nuc = 0). For the ME analysis a Gaussian apodization (tapod = 2.0 s) is used for all the energy dependent spectra. In apodization, the statistical errors in asymmetry are enhanced, by multiplying with an increasing function in time, for the later part of a spectrum, as 𝜎apodized(t) ” exp (︃ + t2 t2apod )︃ 𝜎(t). (4.6) The apodization time is chosen such that at √ 2tapod, the asymmetry signal either fully relaxes or disappears in noise. Apodization leads to a very small symmetric broadening and smoothing of 434 6 8 10 0 0.2 0.4 0.6  0H YBa2Cu3O6.92 T = 8 K E = 14.1 keV j ‖ a B (mT) 𝜌( B) (︀ m T−1 )︀ (a) 2 4 6 8 10 12 14 0 0.1 0.2 0.3 0.4 0.5 2.0  0H 10.0 E = 24.0 keV j ‖ a T = 8 K B (mT) (b) Figure 4.4 : (a) Magnetic field distribution as seen by muons at an implantation energy of E = 14.1 keV and at T = 8 K in an external magnetic field 0H = 9.5mT, applied parallel to the b axis of the crystals. The field distribution corresponds to the asymmetry spectrum ( ) in the figure 4.6. The red line ( ) is model field from pure London model with d = 10.3 nm, 𝜆a = 128.9 nm and 𝜎nuc = 0. The spike in 𝜌(B) at the applied field is from muons stopping in dead layer. The black line ( ) is the theoretical distribution convoluted with a Gaussian with a second central momentΔB = 0.72mT corresponding to the average 𝜎nuc = 0.61 s−1 from the figure 4.8a). The blue line ( ) is obtained from maximum-entropy analysis. The the slight difference between the ME results and the London model may, in part, be due to the simplified model we are using for the deadlayer. The higher weight at high fields in the actual field distribution may be due to from trapped vortices close to the surface and is not taken fully into account by our symmetrically broadened field distribution. (b) Maximum-entropy field distributions are shown at three different energies. At the lowest energy, there is significant broadening which may be due to muons close to the surface where there may be broadening due to vortices or demagnetization affects. the field distribution but is not expected to change the averagemagnetic field ÆBæ. It is interesting to note that there is some weight at fields (cf. figure 4.4b) higher than the applied field  0H which cannot be accounted for by (4.3). However, it should be noted that 𝜒2/DF is close to 1 (cf. table 4.1) in the Meissner model fit (4.3) and as such actual field distributions cannot be very different from the ones realized from the Meissner model (4.3). Also, the phase (𝜙) of the incoming muon beam is not properly taken into account in the ME analysis. Yet, tails in ME field distribution (figure 4.4b) are indication of higher fields than predicted from our simple model, possibly due to a small fraction of muons seeing a large fields at the edges of the sample. Also the fast reduction (due to Ni hyperfine interactions) of asymmetry in earlier times (cf. figure 4.5) contributes to the long tail in field distribution. In one sense, ME gives a better representation of the field distribution since it does not involve any particular model. However, for the purposes of extracting useful parameters such as 𝜆, fits to the Meissner model in the time domain are more useful. Figure 4.5 shows a comparison of Meissner model fit and prediction frommaximum-entropy Fourier analysis. As has been argued before, fast decay [168] in asymmetry, in the first [0-0.15] s, is due to muons' interaction of large random field from Ni moments. From the asymmetry spectra figure 4.5 (top panel), it is clear that no discernible 44−10 −5 0 5 10 (a)E = 2.0 keV  0H = 9.5mTt1 = 0.15 µs A (︀ 10 −2 )︀ 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −10 −5 0 5 10 (b) YBa2Cu3O6.92, j ‖ a, T = 8 K Maximum Entropy Meissner Model Time (µs) A (︀ 10 −2 )︀ Figure 4.5 : Comparison of Meissner model fits and maximum-entropy Fourier results: (a) The red ( ) and black ( ) points represent the muon asymmetry spectra in YBa2Cu3O6.92 at energy 2keV in an external field of 9.5mT. The solid red ( ) and black ( ) lines are maximum-entropy fits (0.15 s to 8.0 s) to the spectra. Fits exclude [0 – 0.15] s - which primarily contains precession from Ni (sample plate). The arrows at 2.5 s point to the time where the experimental asymmetry signal is lost in noise. (b) A comparison of theoretical fits of maximum- entropy ( ) and London model ( ) is shown. It is clear that no discernible experimental asymmetry is left after 2.5 s; while ME predicts a small oscillating asymmetry for an additional ~ 2.0 s. This suggests that the ME prediction of field distribution should be viewed with caution. experimental asymmetry is left after 2.5  s; while ME predicts a small oscillating asymmetry for an additional ~ 2.0  s. This suggests that the ME prediction of field distribution is to be viewed with some caution. The longer tails in figure 4.4 are indicative of higher local magnetic field experienced by muons landing close to the sample surface. This may in part be due to ME transformation which can produce signal where there are not any. In the equations (4.1) and (4.2), 𝜙 should depend only on geometric parameters and the average muon arrival time on the sample. Thus, one way to analyze the spectra is to fix 𝜙 to values obtained at some temperature above Tc, as has been described in [157]. However, 𝜙may be different in the superconducting state compared to the values forT > Tc, as wewill see shortly. 𝜙may also depend on muon implantation energy in the following manner: in the normal state (T > Tc), the angular part of the asymmetry is given by A(t) ~ cos (𝜔(t – t0)) (4.7) assuming 𝜙 = 0, for simplicity. Here 𝜔 =   B is the frequency and t0 being the time muon 45enters the sample. If an error is made in determining t0 → t0 – dt, then (4.7) becomes A(t) ~ cos (𝜔(t – t0) + 𝜔dt) (4.8) Equation (4.8) implies that the fitted phase will be 𝜔dt. However, in the superconducting state and at a high implantation energy, muons deep inside the sample will see a reduced field and a corresponding reduced frequency 𝜔′E < 𝜔, assuming for simplicity that all the muons precess in a single frequency. This implies that, in superconducting state, (4.8) may be written as A(t) ~ cos (︀ 𝜔′E(t – t0) + 𝜔 ′ Edt )︀ (4.9) implying a phase shift of (𝜔′E – 𝜔)dt, relative to the normal state, will be detected. The possibil- ity of the actual frequency distribution being more asymmetric than our theoretical frequency distribution one can also introduce an apparent phase shift. To summarize, imperfections in the fitting function, time-of-flight distribution and detector geometry may all contribute to an energy dependent phase shift. For this reason, phases for individual energies were kept free (individual phase) in the global fit. A shared phase global fit analysis (shared phase) was also done which yields 𝜆a/b values within the systematic errors ~ 3 nm. This is also a strong indica- tion that the subtleties of muon arrival time, detector geometry and thereby the phases (𝜙) and broadening parameter (𝜎) do not significantly influence absolute values of magnetic penetration depth reported from these measurements. Figure 4.6a shows themuon precession signal in the normal state at 110K in a small magnetic field of 9.5mT applied along the a axis of the crystals with an implantation energy at 14.1 keV. The fitted precession signal in figure 4.6a also includes a damping rate of 0.086(11) s−1 which is consistent with that expected from randomly oriented Cu nuclear dipole moments. Meissner screening of the external field is apparent by comparing the normal state (figure 4.6a) with the superconducting state (figure 4.6b). As mentioned earlier, the superconducting state data are fitted to (4.2) with shared A0, d and 𝜆 for all the spectra at 8K in both a and b directions. The depth dependence of the average internal field is evident by comparing the two spectra ( and ) in figure 4.6b which have different implantation profiles. Examples of the field profiles are shown in figure 3.7a. Figure 4.7 shows the average local field [ÆBæ = ∫︀ 𝜌(z)B(z)dz] determined from fits at a single energy as a function of beam energy (bottom scale) and corresponding mean implan- tation depth (top scale). The filled circles ( ) and filled squares ( ) are from data taken with the shielding currents flowing along the a and b axes respectively (or equivalently the magnetic field along the b and a axes respectively). The corresponding Londonmodel curves are generated from a global fit of runs taken at 8K for both orientations and all energies using the calculated trim.sp implantation profiles. The common parameters are 𝜆a = 128.9(12) nm, 𝜆b = 108.4(10)nm and d = 10.3(4) nm. In the shared phase fits, d is fixed to 10.3 nm and the obtained 𝜆a/b differs by only about 1nm as shown in the table 4.1. Statistical uncertainties are determined from the 46−10 0 10 (a) A (︀ 10 −2 )︀ 110 K, 14 keV 0 1 2 3 4 5 6 7 8 −10 0 10 (b) YBa2Cu3O6.92, j ‖ a (” H ‖ b), T = 8 K  0H = 9.5mT Time (µs) A (︀ 10 −2 )︀ 14.1 keV; 24 keV Figure 4.6 : (a) The muon spin precession signal in the normal state of YBa2Cu3O6.92 at 110K in an external field of 9.5mT applied parallel to the b direction. The mean implantation energy is E = 14.1 keV which corresponds to a mean implantation depth of 65nm. In the normal state, external field ( 0H) penetrates the sample fully and all implanted muons experience the same magnetic field - which is reflected in the precession frequency of the asymmetry spectrum. A small damping of asymmetry is due top randomly oriented Cu nuclear dipole moments. (b) The same conditions as top panel, except in the superconducting state at T = 8 K for two energies 14.1keV and 24 keV. H ‖ b results in shielding current flowing parallel to a direction (j ‖ a). Notice the drop in aver- age frequency in the superconducting states. Higher energy muons experience a corresponding distribution of lower magnetic fields due to London decay and due to muons stopping at different depths. The distribution of fields causes rapid dephasing of muon polarization and consequently a loss of signal at ~ 2.5 s. global 𝜒2 surface and take into account the correlation between 𝜆a/b and d. Since there is almost no correlation between d and 𝜆a/𝜆b, this ratio is determined more accurately than the absolute values of 𝜆a and 𝜆b as may be seen in the table 4.1. As may be evidenced from figure 4.7, the average field falls exponentially from the surface for higher implantation energies however, close to the surface, deviation from a simple exponential is modeled from the curvature at the low- est energies. This implies that the shielding current is suppressed near the surface relative to a London model. This model of a simple deadlayer is idealized but the nature of the data does not warrant more complicated picture. It is unclear to what extent this suppression of the super- current density is intrinsic as a result of the discontinuous nature of electronic properties near a surface. A suppression of the supercurrent density was also reported in a previous low energy  SR study of the field profile in a thin film of YBa2Cu3O6.92 and attributed to surface rough- ness [169]. It is difficult to exclude such extrinsic effects which could also lead to a suppression of the supercurrent density near a surface. Measurements on atomically flat cleaved surfaces are 470 5 10 15 20 25 4 6 8 10 j ‖ a j ‖ b T = 8 K YBa2Cu3O6.92  0H = 9.5mT E (keV) ÆB æ(m T) 20 40 60 80 100 Æz𝜇æ (nm) Figure 4.7 : (a) The average magnetic field (ÆBæ) versus mean stopping depth in an applied field of 9.5mT such that the shielding currents are flowing in the a direction (j ‖ a, ) and b direction (j ‖ b, ). The curves are the average fields generated from a global fit of all the spectra at 8 K taken at all energies and for both orientations. The common parameters are 𝜆a, 𝜆b and d. The individual points are from a fit to the same model but at a single depth. The differences between the data points and curves reflect how close the data at a single energy agrees to the global fit. needed to resolve the origin of d. There are no othermeasurements of electromagnetic properties as a function of depth in YBa2Cu3O6.92 crystals to compare with. Surface sensitive techniques such as STM and ARPES are only sensitive to the top few unit cells where the properties can be very different than in the bulk. It may also be noted from figure 4.8a, that the fitted asymmetry drops from 0.10 at 2keV to 0.05 at 2 keV. This is expected since at low energies, incoming muons have an angular distri- bution as they approach the sample surface and at lower energy, many of them undergo large angle scattering such that they can be back scattered off the surface. Such muons form muonium (︀  +e– )︀ or stop in the walls of the stainless steel vacuum vessel. In either case, they do not con- tribute to the precession signal at the Larmor frequency. The average phase from the individual phase fits is close to that from the shared phase fits, indicating that the latter model determines some “effective average phase” as one would expect. The energy dependence of the broadening parameter 𝜎 reflects the random local fields and is also temperature independent below 0.8Tc, as evidenced from data shown in figure 4.8b. This suggests there may be bulk magnetization effects, whereby flux expelled from neighboring crystals broadens the magnetic field distribu- tion at the surface of any given crystal. However, it will be seen in the next section that 𝜎 did not change appreciably when the crystals were separated (cf. figure 4.2). This indicates that each crystal of YBCO experiences a distribution of fields independent of the other crystals. This in turn suggests the broadening may be related to the bulk demagnetization effects and/or surface 484 6 8 10 T = 8.0K  0H = 9.5mT j ‖ a j ‖ b A 0 (︀ 10 −2 )︀ 20 40 60 80 100 Æz𝜇æ (nm) 25 30 35 YBa2Cu3O6.92 𝜙 (∘ ) 0 5 10 15 20 25 0.5 0.6 0.7 0.8 E (keV) 𝜎 (︀  s−1 )︀ (a) 0 20 40 60 80 100 120 120 140 160 107.64 + 0.358*T Tc = 94 K T (K) 𝜆 b (nm) 25 30 35 40 E = 22.0keV  0H = 9.5mT 𝜙 b (∘ ) 0 20 40 60 80 100 120 0.2 0.4 0.6 0.8 YBa2Cu3O6.92 𝜎nuc T (K) 𝜎 b (︀  s−1 )︀ (b) Figure 4.8 : (a) The asymmetry, phase and broadening parameter versus mean stopping depth in an applied field of 9.5mT in YBa2Cu3O6.92 in the global fit of all asymmetry spectra in both directions (a, b). The asymmetry and phase were kept as free parameters at each energy dependent spectra. Reduced asymmetry at lower energies is due to backscattering of muons. The dipolar broadening 𝜎 is shared in both directions. (b) Temperature de- pendence of 𝜆, 𝜙 and 𝜎 when j ‖ b. The slope from the linear fit of 𝜎b(T ) is used to extrapolate 𝜆a/b(T = 8 K) to their zero kelving values 𝜆a/b(T = 0 K). 49Table 4.1 : Summary of results in YBa2Cu3O6.92: Measured magnetic penetration depths at 8K are shown for two models: “phase individual” and “phase shared”, for individual energies. The errors are reported here are purely statistical. An additional ~ 2% (~ 2 nm) error is due to uncertainty in stopping distribution. It may be noted that the two phase models yield penetration depths within a nm of each other for both a and b axis, stressing that the determination of phase does not affect the measurement of absolute 𝜆. B d 𝜙 𝜆a(8 K) 𝜆b(8 K) R ” 𝜆a/𝜆b 𝜒2/DF (mT) (nm) (∘) (nm) (nm) 10.3(5) 32.2(6)a 128.9(12) 108.4(10) 1.19(1) 1.058 9.46 10.3 (fixed) 33.0(8)b 130.2(14) 109.2(12) 1.19(1) 1.069 a Average of energy specific phases from “individual phase” analysis. b Global phase from “shared phase” analysis. roughness. Since 𝜎 reflects the broadening of an effective field at a specific energy, it is kept as a free parameter both in individual phase and shared phasemodels. Table 4.1 gives our results for 𝜆a and 𝜆b measured at 8K in YBa2Cu3O6.92. The errors re- ported here are purely the statistical uncertainties. Systematic errors due to uncertainties in the muon stopping distribution and the extrapolation to 0K are estimated to be about 3 %. At the moment, the latter dominates the overall uncertainty but should improve with refinements of the stopping distribution calculations. 4.1.2 Temperature Dependence of Penetration Depth and Estima- tion of Energy Gap in YBa2Cu3O6.92 The absolute value of 𝜆b as a function of temperature is shown in figure 4.8b. The data points are the fitted values of 𝜆b determined from a fit to the model at a single implantation energy of 22keV. Since d is not temperature dependent it was fixed at 10.3nm. The solid line is a linear fit of our data below 30K and gives a slope of 0.357(67)nmK−1. This was used to extrapolate our measurement of 𝜆b at 8K down to zero temperature. The slope and extrapolated value depend slightly on the fitted temperature range, adding an additional systematic error of about 1nm. To obtain 𝜆a(0), normalized superfluid density in a and b axis directions are taken as approximately equal, for T < 8 K, as in the (4.10) (︂ 𝜆a(0) 𝜆a(T) )︂2 ≃ (︂ 𝜆b(0) 𝜆b(T) )︂2 ⇒ 𝜆a(0) = 𝜆b(0) (︂ 𝜆a(T) 𝜆b(T) )︂ , (4.10) where 𝜆a/b(T) are the finite temperature penetration depths measured by our modified London model. For low temperatures (4.10) is justified as may be seen from the 0–40K fits of superfluid density in figure 4.10. This is reasonable since 1/𝜆2 ∝ ns/m* and ns is a scalar quantity inde- pendent of orientation. Thus to the extent that the effective mass tensor is T-independent one would not expect (𝜆(0)/𝜆(T))2 to be orientation dependent. 50The temperature dependence of the magnetic penetration depth and the corresponding su- perfluid density are shown in the figures 4.9 and 4.10. For determination of 𝜆a/b(T), the dead layer has been fixed to 10.3nm as d is not expected to change with temperature. The obtained 𝜆a/b are fitted according to the semiclassical model of a cylindrical Fermi surface [170] where the normalized superfluid density is given by (︂ 𝜆(0) 𝜆(T) )︂2 a = 1 – 1 2𝜋kBT ∫︁ 2𝜋 0 cos2 𝜑 ∫︁ ∞ 0 cosh–2 (︃√︀ 𝜖2 + Δ2(𝜑,T) 2kBT )︃ d𝜖d𝜑 (4.11a) (︂ 𝜆(0) 𝜆(T) )︂2 b = 1 – 1 2𝜋kBT ∫︁ 2𝜋 0 sin2 𝜑 ∫︁ ∞ 0 cosh–2 (︃√︀ 𝜖2 + Δ2(𝜑,T) 2kBT )︃ d𝜖d𝜑 (4.11b) where √︀ 𝜖2 + Δ2(𝜑,T) is the quasiparticle (pair) energy. The temperature dependence of the gap is approximated by Δ(T) = Δ0(0) tanh (︃ 𝜋kBTc Δ0(0) √︃ a (︂ Tc T – 1 )︂)︃ g(𝜑), (4.12) where Δ0(0) is zero temperature gap energy and gd = cos(2𝜑), ad = 4/3 for a dx2–y2 gap sym- metry. In the weak-coupling limit,Δ0(0) » 2.14 kBTc [171]. The model in the equations (4.11a) and (4.11b) is employed to fit the data in the figure 4.9 and also shown in the figure 4.10. The fit parameters in the model are Δ0,a/b(0), 𝜆a/b(0) while Tc is fixed to 94 K. The fit is done for data T ≤ 40 K. Note that the applied magnetic field exceeds Hc1 [174] at T ≃ 55 K. The fits yield 𝜆a(0) = 125.9(19) nm, 𝜆b(0) = 108.5(10) nm, Δ0,d,a(0) = 23.3(60)meV = 2.87(74) kBTc and Δ0,d,b(0) = 23.7(34)meV = 2.92(42) kBTc and thereby |Δa/Δb| (defined as1 |Δ0,d,a(0)/Δ0,d,b(0)|) = 0.98(29). Fits to the microwave( W) data, using the absolute val- ues of 𝜆a/b(0 K), yields |Δa/Δb| = 2.73 kBTc/2.62 kBTc = 1.04. The  SR gap values are higher than the weak coupling predictions. The gap anisotropy is close to that found via other methods: ARPES [175] (|Δa/Δb| = 0.86), Raman scattering [176] (|Δa/Δb| = 0.83), phase sensitive mea- surement [177] (|Δa/Δb| = 0.82). Although YBCO is thought to be primarily of dx2–y2-wave gap superconductor, there are measurements suggesting an additional s-wave gap [178] from an upturn in superfluid density for T ≤ 10 K. In YBa2Cu3O6.92, as shown in figure 4.10, there is no such upturn. Based on the current data we may conclude that inclusion of an additional gap is not warranted. In summary of this section, we have used low energy  SR to measure the magnetic field profiles in the Meissner state of a mosaic of detwinned single crystals of YBa2Cu3O6.92. The comparison of 𝜆a/b values obtained here with those from different methods is deferred until a later section. Since the data analysis method for the next three sections will be very similar to 1Note that the fitted energy gapsmay be considered as “effective d-wave gaps” in a and b directions. An additional isotropic gap (eg. s-wave) may give rise to an anisotropic overall gap. 510 10 20 30 40 50 60 70 80 100 150 200 250 300 𝜆a, E = 14.1 keV 𝜆b, E = 22.0 keV YBa2Cu3O6.92 Tc = 94 K 𝜆a(0) = 125.9(19) nm 𝜆b(0) = 108.5(10) nm T (K) 𝜆 (T )(nm) 𝜆a 𝜆b µW, YBa2Cu3O6.95 Figure 4.9 : Temperature dependence of the penetration depth 𝜆(T ) for the two crystal orientations (a/b ) at ener- gies 14.1keV and 22.0keV, respectively. 𝜆(T ) should be independent of muon implantation energies. The solid lines ( , ) are fits to the equations (4.11a) and (4.11b), with a pure dx2–y2 -wave order parameter symme- try, of the temperature region 0 ≤ T ≤ 40 K (↑). Penetration depth data from microwave (µW : , ) measurements [172, 173], shifted to agree to the 𝜆a/b(0 K) from the dx2–y2 -wave phenomenological fits, are also shown. Microwave data measuresΔ𝜆(T ) which requires the absolute value of the 𝜆 at some temperature before it can be converted to a T -dependence of the superfluid density. Interestingly, microwave data deviates from LE- SR measurements for T > 50 K - the temperature at which 0H exceeds Hc1 and vortices start to enter. 0 20 40 60 80 0 20 40 60 80 𝜆–2a 𝜆–2b YBa2Cu3O6.92 Tc = 94 K T (K) 𝜆 –2 (T ) (︀ µm −2 )︀ 𝜆–2a 𝜆 –2 b µW, YBa2Cu3O6.95 (a) 0 20 40 60 80 0 0.2 0.4 0.6 0.8 1 j ‖ a j ‖ b 𝜆a(0) = 125.9(19) nm Δ0,d,a = 23.3(60) meV 𝜆b(0) = 108.5(10) nm Δ0,d,b = 23.7(34) meV YBa2Cu3O6.92 Tc = 94 K T (K) 𝜆 2 (0)/ 𝜆 2 ( T ) (b) Figure 4.10 : (a) Temperature dependence of 𝜆–2a and 𝜆 –2 b . With the solid lines are fits to the equations (4.11a) and (4.11b), with a pure dx2–y2 -wave order parameter symmetry, in the temperature region 0 ≤ T ≤ 40 K. (b) Normalized superfluid density. It may be noted that 𝜆2a(0)/𝜆 2(T ) ≃ 𝜆2b(0)/𝜆 2(T ) for T £ 50 K for both the  SR and the µW data. The fitted values of 𝜆a/b(0 K) from these temperature dependence is slightly different from the global fit measurements given in table 4.1. Penetration depth data from microwave measurements [172, 173], shifted to agree to the 𝜆a/b(0 K) from the dx2–y2 -wave phenomenological fits, are also shown. Note that there is close agreement between the current data and the microwave data for T < 50 K. 52this one, we will primarily focus on discussing differences. 4.1.3 YBa2Cu3O6.998 In this section, we present the analysis of a total of four sets of data: (i) in an external field of 4.7mT at three temperatures 4K, 5K and 12K (ii) in an external field of 9.5mT at the temperature of 5K. Examples of the muon precession signals in the crystal mosaic may be seen in figure 4.11. Figure 4.11a shows the muon precession signal in the normal state at 100K in a small magnetic field of 4.7mT applied along the a axis of the crystals with an implantation energy at 23.1 keV. In figure 4.11a, the average frequency corresponds to the applied field with a damping rate of −10 0 10 (a) A (︀ 10 −2 )︀ 100 K, 23.1 keV 0 1 2 3 4 5 6 7 8 −10 0 10 (b) YBa2Cu3O6.998, j ‖ b, T = 5 K  0H = 4.7mT Time (µs) A (︀ 10 −2 )︀ 14.1keV; 23.1keV Figure 4.11 : (a) The muon spin precession signal in the normal state of YBa2Cu3O6.998 at 100 K, 23.1 keV in an external field of 4.7mT applied parallel to the a direction.(b) The same conditions as (a) except in the supercon- ducting state at T = 5 K with an two implantation energies 14.1 keV and 23.1 keV. The solid lines are fits to a London model profile described in the previous section. 0.110(10) s−1, which is slightly larger than obtained in YBa2Cu3O6.92, where the fitted 𝜎 was 0.086(11)  s−1. All measurements in the superconducting state were carried out under zero- field-cooled conditions in order to avoid flux trapping at the surface. Meissner screening of the external field is apparent by comparing the normal state in figure 4.11a with the superconducting states in figure 4.11b. The depth dependence of the average internal field(𝜔) is also evident by comparing the two spectra in the figure 4.11b which have different implantation profiles. The curves in figure 4.11b are generated from fits to a London model profile as described in the pre- vious section. 530 2 4 6 8 0 0.2 0.4 0.6 0.8 2.0 10.0 E = 23.1 keV YBa2Cu3O6.998 j ‖ b T = 5 K B (mT) 𝜌( B) (︀ m T−1 )︀ (a) 0 5 10 0 0.1 0.2 0.3 0.4 0.5 2.0 10.0 E = 23.1 keV j ‖ b T = 5 K B (mT) (b) Figure 4.12 : Magnetic field distributions generated by maximum-entropy method at various implantation ener- gies and at T = 5 K in an external applied magnetic field (︀  0H )︀ of (a)4.7mT and (b)9.5mT, applied parallel to the a axis. Maximum entropy field distributions corresponding to the two external fields of 4.7mT and 9.5mT are shown in the figure 4.12. As may be noticed, 𝜌(B), determined at the higher applied field (9.5mT), is more asymmetric. However, in both cases, there are long tails, in field distribu- tions, even at higher implantation energies possibly due to trapped vortices (dead layer) close to the surface. At low temperatures, significantly below the critical temperature Tc, the broadening pa- rameter 𝜎 is about 0.4  s−1 (figure 4.13) which is about half of that observed (figure 4.8a) in YBa2Cu3O6.92 in 9.5mT. Themosaic geometry has virtually no effect on 𝜎 - which indicates that muons experience broadened field distribution in each crystal, possibly due to surface roughness and/or demagnetization effects at the edges of each crystal. However, simulations done by Mike et al. [1] suggest that 𝜎(E) having a peak at some energy - which is not observed. Figure 4.14 shows the average local field ÆBæ = ∫︀ 𝜌(z)B(z)dz determined from fits at a single energy as a function of beam energy (bottom scale) and correspondingmean implantation depth (top scale). The open circles and filled squares are from data taken with the shielding currents flowing along the a and b axes respectively (or equivalently the magnetic field along the b and a axes respectively). As may be noticed in figure 4.14, the theoretical global fitted ÆBæ curves are in excellent agreement with the individually measured ÆBæ. The common parameters are all shown in the table 4.2 and 𝜆a/b along with 𝜒 2 per degrees of freedom are plotted in the figure 4.18. Since there is almost no correlation between d and 𝜆a/𝜆b, this ratio is determined more accurately than the absolute values of 𝜆a and 𝜆b as shown in the seventh column of the table 4.2. Slightly different penetration depths 𝜆a/b and other parameters are obtained in the individual phase fits compared to the shared phase fits. Differences in averagemagnetic field between the twomodels 540 2 4 6 8 10 12 14 16 18 20 22 24 0.3 0.4 0.5 0.6 0.7 0.8 T = 5 K T = 4 K, 5 K, 12 K YBa2Cu3O6.998  0H = 9.5mT 4.7mT E (keV) 𝜎 (︀  s−1 )︀ 𝜎a 𝜎b 𝜎b 10 20 30 40 50 60 70 80 90 100 110 Æz𝜇æ (nm) Figure 4.13 : Broadening parameter as function of mean energy or depth for various magnetic fields: The three important parameters for individual energy runs in the global fit, with shared 𝜆a, 𝜆b and d are shown. The broadening parameter (𝜎) shows some random variation with averages close to 0.42 s−1 and 0.7 s−1 at the external fields of 0H = 4.7mT and 9.5mT , respectively. are negligible. Figure 4.15 shows the temperature dependence of penetration depth 𝜆a at 4.7 mT and 23.1keV. The 𝜆a's are obtained via the method of “individual phase” analysis of measurements with fixed d = 14.79 nm, obtained from global fit of measurements in both a and b direction. The measured 𝜆a's are fitted to a line giving 𝜆a(0 K) = 112.26(146)nm and slope of the line (︀ Δ𝜆/ΔT = 0.495(78)nmK−1 )︀ . Since these measurements were done at a single energy, 𝜆a(0 K) is not as good a measure of penetration depth at zero temperature. Nevertheless, the fitted mag- netic penetration depth (𝜆a(5 K) ~ 115nm) is close to the absolute 𝜆a's found from global fits as shown in the table 4.2. The Δ𝜆/ΔT is a good measure of low temperature dependence of 𝜆a and can be used to extrapolate zero temperature penetration depths from the finite temperature ones. It may be noted that 𝜆a/b increases slightly as the temperature is increased, from 4K to 5K, which is consistent with ~ 0.5 nmK−1, found from the temperature dependence in the fig- ure 4.15. It may also be noted that the ratio between the two penetration depths, R, varies only slightly (~ 4%) depending onwhether the individual phase or the sharedphasemodel was used. Furthermore, the difference is within the statistical (~ 1% to 2 %) and systematic uncertainties (~ 3 %) in 𝜆a/b. The goodness of fit, 𝜒 2/DF, becomes slightly worse (~ 10%) with the shared phasemodel, which is reasonable since there are fewer degrees of freedomwith the shared phase 550 5 10 15 20 25 2 3 4 5 j ‖ a j ‖ b T = 5 K YBa2Cu3O6.998  0H = 4.7mT E (keV) ÆB æ(m T) 20 40 60 80 100 Æz𝜇æ (nm) (a) 0 5 10 15 20 25 0.4 0.6 0.8 1  0H = 4.7mT  0H = 9.5mT T = 5 K YBa2Cu3O6.998 j ‖ b E (keV) ÆB æ/  0 H 20 40 60 80 100 Æz𝜇æ (nm) (b) Figure 4.14 : (a) The average magnetic field (ÆBæ) versus mean stopping depth in an applied field of 4.7mT. The curves are the average fields generated from a global fit of all the spectra at 5 K taken at all energies and for both orientations (cf. figure 4.7). (b) Relative average magnetic fields in this sample for two temperatures and twomagnetic fields. It is apparent that the absolute magnetic penetration depths measuredat the lowdifferent magnetic fields are almost the same. 0 5 10 15 20 25 30 35 40 110 115 120 125 130 135 YBa2Cu3O6.998 j ‖ a  0H = 4.7mT E = 23.1 keV T (K) 𝜆 a (nm) Figure 4.15 : Temperature dependence of magnetic penetration depth below 40K. The muonbeam energy for this set of runs is E = 23.1 keV and an external field of 4.7mT is applied parallel to b-axis. A linear fit of 𝜆a(T ) yields 𝜆a(0 K) = 112.26(146) nm and (︀ Δ𝜆/ΔT = 0.495(78) nmK−1 )︀ . 56fits. Using the phenomenological models equations (4.11a), (4.11b) and (4.12) introduced in the previous section, figures 4.16 and 4.17 shows the temperature dependence of penetration depths, 𝜆–2(T) and normalized superfluid density, respectively. The penetration depths aremeasured by 0 10 20 30 40 50 60 70 80 100 150 200 250 300 𝜆a 𝜆b YBa2Cu3O6.998 Tc = 90 K E = 23.1 keV 𝜆a(0) = 114.6(6) nm 𝜆b(0) = 93.7(9) nm T (K) 𝜆 (T )(nm) 𝜆a 𝜆b µW, YBa2Cu3O6.99 Figure 4.16 : Temperature dependence of the magnetic penetration depth 𝜆(T ) for the two crystal orientations in at 23.1 keV implantation energy. The solid lines are fits to the (4.10) with a pure dx2–y2 -wave order parameter symmetry, of the temperature region 0 ≤ T ≤ 40 K (↑). However, unlike in figure 4.9 the penetration depths are measured by keeping the phase fixed to 23.1° (cf. table 4.2). Penetration depth data from microwave (µW) measurements [172, 173], shifted to agree to the 𝜆a/b(0 K) from the dx2–y2 -wave phenomenological fits, are also shown. Interestingly, microwave data agree with LE- SR measurements for the entire temperature range upto Tc. This is unlike the scenario in YBa2Cu3O6.92 (cf. figure 4.9) where the agreement, between our data and microwave measurements, is observed for 0 ≤ T ≤ 50 K. Vortices most likely entered in YBa2Cu3O6.92 for T > 55 K. keeping the phase fixed to 23.1° (cf. table 4.2). In the temperature dependence of YBa2Cu3O6.92 (cf. figure 4.9), phases for individual temperature measurements were kept free. This method of a common phase is employed to limit the correlation between phase and frequency at the low magnetic field of  0H = 4.7mT. Theoretical fits to a dx2–y2-gap are obtained for T ≤ 40K ( 0H ~ Hc1(80 K)). The zero temperature energy gaps are found as Δ0,d,a = 19.7(13)meV = 2.53(17) kBTc and Δ0,d,b = 23.4(43)meV = 3.01(55) kBTc - yielding a gap anisotropy 0.84(16). Fits to the microwave( W) data yields |Δa/Δb| = 2.71 kBTc/2.70 kBTc = 1.00. The  SR gap values are higher than the weak coupling prediction as was observed in YBa2Cu3O6.92, whereas the gap anisotropy is close to other measures of gap anisotropy in YBCO. Although the gap anisotropy in this section is consistentwith someothermeasurements, there is a large uncertainty in the measured value primary due to 𝜆–2a not being a very good fit to a dx2–y2-gap model. Results reported in this section are summarized in the table 4.2 and figure 4.18. It is inter- esting to note that the dead layer d (the layer close to the surface of the crystals where super- current is suppressed) varies between 15nm to 17nm; whereas in YBa2Cu3O6.92, it was found 570 20 40 60 80 0 20 40 60 80 100 120 𝜆–2a 𝜆–2b YBa2Cu3O6.998 Tc = 90 K E = 23.1 keV T (K) 𝜆 –2 (T ) (︀ µm −2 )︀ 𝜆–2a 𝜆 –2 b µW, YBa2Cu3O6.99 (a) 0 20 40 60 80 0 0.2 0.4 0.6 0.8 1 j ‖ a j ‖ b 𝜆a(0) = 114.6(3) nm Δ0,d,a = 19.7(13) meV 𝜆b(0) = 93.7(9) nm Δ0,d,b = 23.4(43) meV YBa2Cu3O6.998 Tc = 90 K E = 23.1 keV T (K) 𝜆 2 (0)/ 𝜆 2 ( T ) (b) Figure 4.17 : (a) Temperature dependence of 𝜆–2a and 𝜆 –2 b (b) Normalized superfluid density. Penetration depth data from microwave measurements [172, 173], shifted to agree with the 𝜆a/b(0 K) from the dx2–y2 -wave phe- nomenological fits, are also shown. Table 4.2 : Summary of results for YBa2Cu3O6.998: Measured magnetic penetration depth in external fields of 4.7mT and 9.5mT are shown, for “phase individual” and “phase shared” models. Note, there is some 𝜙-dependence (which contributes to the systematic uncertainty) of measured penetration depths. It’s interesting to note that the “dead layer”d is within 10nm to 20nmindicating that it is a temperatureand magnetic field invariant quantity. The shared phase for 4.7mT measured is ~ 25° and slightly higher for the data at 9.5mT. The 𝜒2/DF is slightly less for “individual phase” models, which indicates better fits. B T d 𝜙 𝜆a 𝜆b R ” 𝜆a/𝜆b 𝜒2/DF (mT) (K) (nm) (∘) (nm) (nm) 27.0(16)a 115.6(18) ˘ ˘ 1.323 4.66 4 14.8(9) 25.9(13)b 119.2(24) ˘ ˘ 1.385 4.67 5 17.2(15) 27.0(16)a 107.5(39) 84.0(35) 1.28(2) 1.257 23.0(13)b 114.2(26) 92.7(22) 1.23(1) 1.382 27.9(15)a 117.2(25) 87.7(21) 1.34(3) 1.448 4.66 12 16.8(6) 25.4(6)b 120.3(18) 93.9(14) 1.28(1) 1.586 9.49 5 16.5(8) 31.1(20)a ˘ 91.9(22) ˘ 1.105 30.1(16)b ˘ 94.4(17) ˘ 1.135 a Average of energy specific phases from “individual phase” analysis. b Global phase from “shared phase” analysis. 58to be 10.3 nm. Regardless of the temperature, oxygen doping and applied external magnetic field, there seems to be a thin outside layer on the order of 10nm where the shielding currents are suppressed compared to a simple London model. Regardless of the phase models, the de- termination of phase is not very sensitive to whether we have data along both axis or just one axis. Determination of phase is important since the fitted values of 𝜆a/b depends on whether we allow for individual phases or assume a common phase for the whole energy scan. To discern between phase and frequency (and thereby average field), more oscillations in the signal is better, which happens closer to the surface. Deeper in the sample, field drops by a significant fraction and thereby the frequency decreases which results in the phase and the average frequency being somewhat correlated. Nevertheless it is important to note that the correlation between 𝜙 and 𝜔 has less effect on global parameters such in d and 𝜆a/b as they are determined from a range of implantation energies (i.e global fit). The fitted phases are found to be almost temperature independent, but vary weakly with applied magnetic field, as 𝜙 increases ~ 5° if the applied field is doubled. With the mosaic of YBa2Cu3O6.92, the global 𝜙 was found to be 33.0(8) at an exter- nal magnetic field of 9.5mT. While the origin of dependence of phase on applied magnetic field remains to be understood, the determination of the magnetic penetration depths 𝜆a/b and their ratio R can still be determined with a few percent uncertainty. With our simple model of the magnetic penetration depth, the dead layer d and penetration depths 𝜆a/b are inversely corre- lated, such that as d increases, 𝜆a/b decreases. However, the ratioR is insensitive to the variations just mentioned and thereby is more robust and less susceptible to systematic effects. Although measured at finite temperatures (4 K, 5 K and 12 K), we expect R to be weakly dependent on temperature at low temperatures since it depends on the effective mass tensor. As may be noticed the from table 4.2 and figure 4.18, the 𝜒2/DF depends on applied external magnetic field and only slightly on the (phase) model. The 𝜒2/DF also is smaller (better fit) for a fit with energy specific phases compared to a global phase. This is expected as more free (fit) parameters mean more degrees of freedom for the model and thus a better fit, however it in- troduces correlations among variables 𝜙, 𝜎 and asymmetry. Nonetheless, statistical uncertainty in the “shared phase” model is better as 𝜒2/DF is inversely correlated with the number of pa- rameters. Model dependent systematic uncertainties can be avoided if better measurements of the phase are done for a specific set of measurements. In the following section, we describe a better method to determine the phase, which avoids the uncertainty introduced by leaving it a free parameter. However, it should still be noted that any phase dependence of 𝜆a/b introduces an additional error of the order of statistical uncertainty in either shared or independent phase global fits. 594 6 8 10 12 80 90 100 110 120 𝜆a 𝜆b YBa2Cu3O6.998 T (K) 𝜆 (nm) 𝜆a,s= shared 𝜙 𝜆b,s 𝜆a,i= individual 𝜙 𝜆b,i 𝜆b,s 𝜆b,i (a) 1.25 1.3 1.35 4.7 mT YBa2Cu3O6.998 1.23 1.281.28 1.34 𝜆 a /𝜆 b 4 6 8 10 12 1.2 1.4 1.6 4.7 mT 9.5 mT YBa2Cu3O6.998 T (K) 𝜒 2 /DF 𝜙s= shared 𝜙 𝜙i= individual 𝜙 𝜙i 𝜙s (b) Figure 4.18 : Summary of results for YBa2Cu3O6.998: Measured magnetic penetration depth in external fields of 4.7mT and 9.5mT are shown, for individual(shared) phase models. There is some 𝜙-dependence (of the order of systematic uncertainty) of measured penetration depths. The 𝜆a/𝜆b and 𝜒 2/DF is slightly less for individual phase, which indicates better fits. 604.1.4 YBa2Cu3O6.52 Ashas been discussed in the sections 4.1.1 and 4.1.3, statistical uncertainties in the determination of magnetic penetration depths 𝜆a/b are on the order of a few percent. However, the shared phase vs individual phase models suggests one should add to this a systematic uncertainty of about 3%. To have a simpler model having one effective phase (depending on whenmuon enters into sample and partially on other geometric factors), a novel method was used to determine 𝜙. Two normal state runs were taken at a single field (to be determined from fit) but one of them having it's direction reversed. The resulting spectra is plotted in the figure 4.19. These normal −8 −6 −4 −2 0 2 4 6 8 −10 −5 0 5 10  0H = +1.516mT 0H = –1.516mT E = 14.1keV T = 80 K YBa2Cu3O6.52 Time (µs) A (︀ 10 −2 )︀ Figure 4.19 : YBa2Cu3O6.52 in an applied field of 1.45mT and −1.45mT at 80K. Field reversing is like taking a mea- surement in negative time. 𝜙 = 19.8(10)° and 𝜎 = 0.14(1) s−1. states runs were taken, as a standard procedure, after the superconducting state runs. Having a negative magnetic field –B is nearly identical to having a precession in negative time, as 𝜔t ”   (–B)t = (  B)(–t). Combining the two precession signals as shown in figure 4.19 for negative and positive times allows one to, in effect, double the time window of observation and also to determine the phase unambiguously. The data in the normal state are fit to a damped cosine: A(t) ” A0 exp (︁ –𝜎2t2/2 )︁ cos (︀    0Ht + 𝜙0 )︀ one can determine  0H , 𝜙0 and 𝜎. The fitted values from figure 4.19 were found to be  0H = 1.514(6)mT , 𝜙0 = 19.8(10)° and 𝜎 = 0.14(1) s−1. This unique determination of phase is spe- cially crucial at low field since, muon oscillation frequencies (𝜔 =   B) are also lower and fewer 61full oscillations are available, introducing a large correlation between 𝜙 and 𝜔, in figure 4.20. Note the clear reduction in precession frequency (thereby, internal field) in figure 4.20 from nor- −10 0 10 (a) T = 80 K E = 14.1 keV  0H = 1.5mT A (︀ 10 −2 )︀ 0 1 2 3 4 5 6 7 8 −10 0 10 E = 14.1 keV E = 24.6 keV (b) j ‖ a YBa2Cu3O6.52T = 5 K Time (µs) A (︀ 10 −2 )︀ Figure 4.20 : Top: The muon spin precession signal in the normal state of YBa2Cu3O6.52 at 80K in an external field of 1.5mT applied parallel to the b-direction. The mean implantation energy is E = 14.1 keV - which corresponds to a mean implantation depth of 65nm. Bottom: The same conditions as above except in the superconducting state at T = 5 K with energy E = 14.1keV and 24.6keV . mal state to superconducting states. It should be stressed that treating 𝜙 as a geometric parameter equal to the normal state phase is a deviation from the analyses done so far. However at such low magnetic fields this is the best option since the phase and frequency are so strongly correlated. Fixing 𝜙 to the normal state values yields a better estimate of average frequencies with only a slightly higher 𝜒2/DF. However, it may be noticed from table 4.3 that set of runs at 1.5mT yields the lowest 𝜒2/DF as well. The phase determined for measurements at 7.8mT is similar to the “field reversed” method while the phase in 4.6 mT set of runs is estimated from interpolating the phases in 1.5mT and 7.8mT set of measurements, assuming linear variation of phase with applied field. The fitted broadening parameter, 𝜎, as a function of energy, obtained from the global fit, is shown in figure 4.21 for  0H = 1.5mT, 4.6mT and 7.8mT. At 1.5mT, 𝜎 is almost independent of implantation energy, except close to the surface - where there appears to be an upturn in the higher magnetic field. This suggests there may be some vortex penetration near the surface which would lead to a field distribution which is broader than expected from just the stopping distribution 𝜌(z) alone. One possible explanation is that vortices enter close to surface which 620 5 10 15 20 25 0.2 0.4 0.6 0.8 1 1.5mT 4.6mT, 𝜎a  0H = 7.8mT, 𝜎a YBa2Cu3O6.52T = 5 K E (keV) 𝜎 (︀  s−1 )︀ 𝜎a 𝜎b 20 40 60 80 100 Æz𝜇æ (nm) Figure 4.21 : YBa2Cu3O6.52 broadening parameter are shown for an external applied field of 1.45mT, 4.6mT and 7.8mT such that the shielding currents are flowing in the a direction (j ‖ a). The common parameters are 𝜆a, 𝜆b and d. The upturn in 𝜎 at lower energies are possibly due to vortex entrance close to the surface. The solid lines are guides to the eye. would introduce broadening in the magnetic field while maintaining an average field close to the external applied field. Deeper in the sample, the diamagnetic shift is larger and therefore, 𝜎(E) ∝ E - seen in the measurements at 4.6mT and 7.8mT (cf. figure 4.22). 0 1 2 3 4 0 0.5 1 1.5 2.0 keV 10.0 keV 21.5 keV YBa2Cu3O6.52 j ‖ a T = 5 K B (mT) 𝜌( B) (︀ m T−1 )︀ (a) 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 j ‖ a j ‖ a j ‖ b YBa2Cu3O6.52 T = 5 K E = 14.1 keV ·2 1.5mT 4.6mT 7.8mT B (mT) (b) Figure 4.22 : Maximum-entropy magnetic field distribution, in YBa2Cu3O6.52, at various implantation energies and at T = 5 K in an external applied magnetic field (︀  0H )︀ of (a)1.5mT and (b)4.6mT applied parallel to the b axis. The arrows (↓,↓,↓) in (b) indicate the applied external fields for the corresponding ME field ( 0H) distributions. The superconducting state runs at 1.5mT were fit to the London model, as described earlier, 63yielding global fit values for 𝜆a/b and dead layer d. The corresponding global and individual average magnetic fields are shown in the figure 4.23a. As before the difference between the the 0 5 10 15 20 25 1 1.1 1.2 1.3 1.4 1.5 j ‖ a j ‖ b T = 5 K YBa2Cu3O6.52  0H = 1.5mT E (keV) ÆB æ(m T) 20 40 60 80 100 Æz𝜇æ (nm) (a) 0 5 10 15 20 25 0.8 0.9 1 j ‖ a  0H = 1.5mT j ‖ a 4.6mT j ‖ b 7.8mT T = 5 K YBa2Cu3O6.52 E (keV) ÆB æ/  0 H 20 40 60 80 100 Æz𝜇æ (nm) (b) Figure 4.23 : (a)The average magnetic field ÆBæ versus energy (mean stopping depth) in an applied field of 1.45mT such that the shielding currents are flowing in the a direction (j ‖ a, open circles) and b direction (j ‖ b, filled squares). (b)Relative average local magnetic field with respect to the applied field as a function of muon implantation energy for the two fields of 4.7mT and 9.5mT applied parallel to the b axis of the YBa2Cu3O6.52. The depicted fields have been calculated in the same way as in figure 4.7 and normalized to the applied fields. global fit and the individual points reflects how well the data at individual energies agrees with the global fit. A comparison of the relative average magnetic field as a function of energy for 1.5 mT, 4.6 mT and 7.8 mT is shown in the figure 4.23b. It is evident that the lowest 𝜆a/b for this sample results from the 1.5mT set of runs. As may be noticed the average field (ÆBæ) in the figure 4.23 doesn't drop until ~ 5 keV, corresponding to an implantation energy ~ 25nm which is also reflected in the dead layer d in the table 4.3. One interesting phenomenon consistently seen in all analysis is that the d is between 10 nm to 25 nm, irrespective of the external field magnitude  0H or orientation, suggesting that some intrinsic mechanism being responsible for the reduced supercurrent close to the surface. To extract 𝜆a/b (0 K), temperature dependentmeasurements of penetration depths have been obtained as shown in the figure 4.24. It may be noticed that at low temperatures, 𝜆 is linear only at the external field of 1.5 mT. The non-linear behavior in 4.6 mT is likely due to vortex penetration. This is consistent with the fact that 𝜎(E) increases almost linearly, with energy, as shown in the figure 4.21. Note the unusual temperature dependence of 𝜆a(T) in the 4.6mT set of measurements. These prompted us tomakemeasurements at a significantly lowermagnetic field of 1.5mT. The resulting Δ𝜆/ΔT = 6.30(96)nmK−1 from the 1.5mT temperature dependence 645 10 15 20 25 30 35 200 400 600 800 𝜆 a , E = 16.1 keV 𝜆b, E = 14.1 keV  0H 4.7mT 1.5mT YBa2Cu3O6.52 𝜆b, Microwave, YBa2Cu3O6.60 Global fits T (K) 𝜆 (nm) 𝜆a, 1.5mT 𝜆b, 1.5mT 𝜆a, 4.6mT Figure 4.24 : As may be noticed, 𝜆 sharply increases with temperature at  0H = 4.6mT. Possible entrance of vortices is likely the reason of increase of “effective 𝜆”. This prompted us to take low field (1.5mT) set of runs. The fit parameters at 1.5 mT external field are, 𝜆a(0 K) = 191.0(134) nm. Δ𝜆/ΔT = 6.30(96) nmK−1. The slope is an order of magnitude bigger than that found in other oxygen dopings of YBCO. Penetration depth data from microwave measurements [172, 173], shifted to agree with the 𝜆b(2 K) from the linear fit, is also shown. As may be noticed clearly,Δ𝜆/ΔT in our data is significantly higher than the slope (︀ ~ 0.82 nmK−1 )︀ from microwave data. The difference may be partly due to the difference in oxygen concentration (O=6.52 vs O=6.60). Measurements in even lower magnetic fields (~ 0.5 mT) may be needed to compareΔ𝜆/ΔT in the two methods. has been used to extrapolate 𝜆b(5 K) to 𝜆b(0 K). Although 𝜆(T) is linear at low temperatures, the slope Δ𝜆/ΔT is an order of magnitude larger compared to the other two oxygen dopings of O=6.998 and O=6.92. It suggests that measurements at even a lower field may be required to test if the value at 1.5mT is intrinsic. The magnetic penetration depth in the other orientations 𝜆a(0 K) has been obtained using the equation 𝜆b(0 K) = 𝜆a(0 K) R , (4.13) where R is the ratio of penetration depths measured at 5K. Figure 4.25a shows effective penetration depth 𝜆b's dependence on external field. The phase was left fixed at the respective normal state values. As argued before, data taken at the higher fields (> 1.5mT) may involve vortex penetration and obtained 𝜆a should be regarded as an effective penetration in the presence of vortices. Note that the effective 𝜆as from the global fit (figure 4.23) involving many energies are much more reliable than those in figure 4.25a which are obtained using the spectrum at a single energy. Nevertheless, the almost linear field depen- dence is strong evidence for vortex penetration at the higher fields. This is not too surprising for x = 6.52 since the lower critical field is expected to be much smaller than in the x = 6.92 652 4 6 8 10 12 500 1000 𝜆a, global fit 𝜆b, individual fit E = 14.1 keV YBa2Cu3O6.52, T = 5 K  0H (mT) 𝜆 (nm) (a) 2 4 6 8 10 12 0 0.2 0.4 0.6 0.8 1 𝜎nuc 𝜎b T = 5 K YBa2Cu3O6.52  0H (mT) 𝜎 (︀  s−1 )︀ (b) Figure 4.25 : (a) Single measurements of effective penetration depths in various external magnetic fields are shown here. The almost linear rise in 𝜆b with magnetic field stresses that lower external field must be used and also be made sure that the samples are indeed in the Meissner state. (b) The rise of 𝜎 with 0H indicates broadening of field distribution due to vortex penetration. Table 4.3 : Summary of results in YBa2Cu3O6.52: Measuredmagneticpenetration depths at 5K areshown. The errors reported here are just statistical errors. An additional (~ 2 nm) error is due to uncertainty in stopping distribution. The lowest penetration depthswere obtained for bothaandb axis in 1.5mTexternal field. The penetration depths at higher fields of 4.6mT and 7.8mT include contributions from vortices entering the sample, as may be observed from 𝜎 ∝ E (see figure 4.21) at higher energies. In this sense, they are an effective penetration depth in the presence of vortices. B T d 𝜙 𝜆a 𝜆b R ” 𝜆a/𝜆b 𝜒2/DF (mT) (K) (nm) (∘) (nm) (nm) 1.52 5 22.9(2) 19.8(10)a 303.2(144) 231.3(155) 1.31(6) 0.999 4.63 5 27.0(12) 22.0(13)b 333.7(99) ˘ ˘ 1.117 7.79 5 22.9(2) 24.1(16)a ˘ 605.7(165) ˘ 1.075 a Measured via reversing of field. b Interpolated. or x = 6.998 sample. Also, it may be noted from figure 4.25b, the broadening parameter 𝜎 increases with  0H . This also corroborates our assertion that vortices are present in all other measurements except possibly at 1.5mT. Although the single fit 𝜆bs in figure 4.25a are not very reliable, they serve to emphasize that cautionmust be taken tomeasure themagnetic penetration depth in the Meissner state in a low enough field to be sure there is no vortex penetration. Oth- erwise, 𝜆 obtained is some effective penetration depth which depends on the degree of vortex penetration at the edges. From themeasured 𝜆a/b in table 4.3, it is very likely only minor vortex penetration happened in 4.6mT set of runs whereas a significant number of vortices (cf. asymmetric 𝜌(B) in figure 4.22) entered in a 7.8mT set of runs, making the global 𝜆 very long. 664.2 Ba(Co0.074Fe0.926)2As2 In this section, we present measurements of the magnetic field profile and magnetic penetration depth on single crystals of Ba(Co0.074Fe0.926)2As2. In the current study, we measure the field profile directly on a freshly cleaved surface of Ba(Co0.074Fe0.926)2As2 using themodified London model as described in the section 4.1.1. The single crystal of optimally-dopedBa(Co0.074Fe0.926)2As2 was grown using a self-fluxmethod [179]. The crystal was approximately square-shaped with di- mensions 5.6mm · 4.5mm · 0.3mm and exhibited a transition at Tc = 21.7K as measured by SQUID magnetometry (figure 4.26). The crystal was cleaved to a thickness of about 0.3 mm 13 14 15 16 17 18 19 20 21 22 23 −3 −2 −1 0 Ba(Co0.074Fe0.926)2As2 SQUID B = 1 G 21.7K T (K) M (︀ 10 −3 emu )︀ Figure 4.26 : Data fromO.Ofer: Magnetization of Ba(Co0.074Fe0.926)2As2 as a function of temperature, showing that the superconducting transition starts at Tc = 21.7 K. under flowing N2 gas just prior to loading it into the ultra-high-vacuum sample chamber. Figure 4.27 shows typical  SR precession signals obtained in a 4.7mT field applied parallel to the ab face of the crystal. The top panel shows the precession signal in the normal state where the mean internal field is equal to the applied field obtained via fitting the spectrum to a Gaussian damped precession signal. The normal state at  0H = 4.7mT yields 𝜎 of 0.12  s−1. Signals taken below Tc are shown in the bottom panel. All the measurements in the Meissner state were made in zero field cooled method as detailed earlier. The reduction of precession frequency, for a higher muon implantation energy, is clearly visible comparing the two spectra ( and ) in the figure 4.27. The broadening parameters obtained in the superconducting state are shown in the figure 4.28. The Æ𝜎æ in the Meissner state of  0H = 4.7mT is slightly higher (0.28(1) s−1) than that (0.21(1)  s−1) corresponding to that obtained at  0H = 2.5mT. The increase of 𝜎 with external field may be due to demagnetization effects, arising from a slight misorientation of the field with respect to a – b plane. Any bulk demagnetization effect is linearly dependent to the external field. However, the present pnictide sample is a single crystal. The additional (not accounted for by the stopping distribution 𝜌(z)) broadening increases by only ~ 25% with the 67−10 0 10 (a) A (︀ 10 −2 )︀ 25 K, 17.3 keV 0 1 2 3 4 5 6 7 −10 0 10 (b) Ba(Co0.074Fe0.926)2As2, j ‖ a, T = 4 K  0H = 4.7mT E = 14.1 keV E = 23.1 keV Time (µs) A (︀ 10 −2 )︀ Figure 4.27 : Muon precession signals in Ba(Co0.074Fe0.926)2As2 in an applied field of  0H = 4.7mT. (a) In the normal state at T = 25 K. (b) In the superconducting state at T = 4 K with E = 14.1 keV and E = 23.1 keV. 0 5 10 15 20 25 0.2 0.4 0.6 0.8 4.7mT 14.6mT After additional cleaving  0H = 2.5mT 4.7mT T = 4 K Ba(Co0.074Fe0.926)2As2 E (keV) 𝜎 (︀  s−1 )︀ 20 40 60 80 100 Æz𝜇æ (nm) Figure 4.28 : Broadening parameter (𝜎) in Ba(Co0.074Fe0.926)2As2 in external applied fields of  0H = 2.5mT and  0H = 4.7mT. The solid lines are guides to the eye. 68approximate doubling of 0H . Also itmay be noted from the figure 4.28, that 𝜎 does not increase sharply close to the surface as has been seen previously in the two YBCOmosaics shown in the figures 4.13 and 4.21. This suggests that there are no vortices close to the surface and yet there is a suppression of superfluid density ~ 15 nm, close to the sample surface, as may be seen from the curvatures of the global fits in the figure 4.29. Figure 4.29 shows the average local field ÆBæ as a function of the beam energy at T = 4 K. The data are consistent with an exponential decrease as a function of increasing depth or implanta- tion energy, as expected from a London model. The global fit parameters are 𝜆(T = 4 K) = 0 5 10 15 20 25 2 3 4 Additionally clea ved T = 4 K Ba(Co0.074Fe0.926)2As2  0H = 2.5mT  0H = 4.7mT E (keV) ÆB æ(m T) 20 40 60 80 100 Æz𝜇æ (nm) (a) 0 5 10 15 20 25 0.7 0.8 0.9 1 After additional cleaving 4.7mT 14.6mT 2.5mT  0H = 4.7mT T = 4 K Ba(Co0.074Fe0.926)2As2 E (keV) ÆB æ/  0 H 20 40 60 80 100 Æz𝜇æ (nm) (b) Figure 4.29 : (a) The average magnetic field versus the muon energy at T = 4 K with  0H = 2.5mT, 4.7mT and 14.6mT. The dotted line indicate the applied field and the solid line indicates a fit to (2.18). (b) Relative aver- age magnetic field in the above mentioned external fields. Notice that there is little field dependence of 𝜆 - measured in 2.5mT and 4.7mT; however, after additional cleaving ( ), the deadlayer seems to get larger (~ 27 nm). This suggests that d is not due to demagnetization - which would have predicted a smaller d for a thinner sample. The average magnetic field in 14.6mT possibly includes significant vortex contribution (cf. figure 4.30b). 251.7(19) nm and d = 14.5(9)nm where the uncertainties are purely statistical. There is also a 3% systematic uncertainty in 𝜆 due to uncertainties in the muon stopping distribution. Similar results were obtained at a magnetic field of 2.5mT, where 𝜆 = 252.2(45) nm, indicating there is little field dependence in 𝜆, which is also reflected in the ÆBæ in the figure 4.29b. Results of analyses done on two phase (independent and shared) models are shown in the table 4.4. No- tice that the average of phases from individual phase is within ~ 1°, while shared phase yields a ~ 3°. Yet with all the model dependence of A0, 𝜎 and 𝜙, there is only ~ 2nm difference in the measured 𝜆 in the external fields of 2.5mT and 4.7mT. This suggests the robustness of our 690 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 1.2 4.0 keV 11.0 keV E = 20.0keV Ba(Co0.074Fe0.926)2As2 T = 4 K  0H =2.5mT B (mT) 𝜌( B) (︀ m T−1 )︀ (a) 0 5 10 15 0 0.1 0.2 0.3 0.4 0.5 20.0 ·2.5 T = 4 K Ba(Co0.074Fe0.926)2As2 E = 20 keV B (mT) (b) Figure 4.30 : (a)Magnetic field distribution as seen by muons at various implantation energies and at T = 5 K in an external applied magnetic field (︀  0H )︀ of 2.5mT, applied parallel to the ab plane. (b)Same as in (a) except field distributions are shown for 0H = 2.5mT, 4.7mT and 14.6mT, indicated by the respective arrows (↓,↓,↓). Table 4.4 : Measured London penetration depths at 4 K are shown. The errors reported here are just statistical errors. An additional (~ 2 nm) error is due to uncertainty in stopping distribution. 4.7 mT 2.5 mT State 𝜆 (nm) 𝜙(∘) 𝜒2/DF 𝜆 (nm) 𝜙(∘) 𝜒2/DF Normal – 25.5(18) 0.932 – 21.4(17) 0.972 Superconducting 252.8(19) 30.9(12)a 1.118 254.5(41) 27.6(7)a 1.030 251.7(19) 28.3(6)b 1.052 252.2(45) 27.8(5)b 1.029 254.4(52) 27.9(9)a 1.198Superconducting and cleaved 253.9(51) 27.5(2)b 1.178 a Global phase from shared phase analysis. b Average of energy specific phases from individual phase analysis. method in obtaining the absolute London penetration depth in this pnictide. The temperature dependence of 𝜆measured with LE- SR ( ) and with  W ( ), MFM ( ) and tunnel-diode-resonator (TDR) ( ) are shown in the figure 4.31. The data points were ob- tained with a single energy of 21.3 keV with d fixed to the value determined from the global fit at T = 4K (see figure 4.29). Note the temperature dependence of 𝜆 at low temperatures is sim- ilar to recent TDR results [180] on a thin sample but considerably weaker than previous studies on thicker crystals [181]. This suggests early studies may have been affected by an anomalous temperature-dependent field penetration from the c-axis edges. It is interesting to note that all of these results are stronger than found by MFM also shown in figure 4.31 [182]. However, there are recent published results showing agreement between TDR and MFM [183]. One difference is that the present measurements, as well as previous TDR results, measure an average over the 700 2 4 6 8 10 12 14 16 18 250 300 350 400 Ba(Co0.074Fe0.926)2As2  0H = 4.7mT d = 14.5 nm (fixed) E = 21.3 keV T (K) 𝜆 (nm) MFM μSR µW TDR Figure 4.31 : The temperature dependence of 𝜆 plotted versus t = T /Tc . The red squares ( ) are measurements of the absolute value of 𝜆 from LE- SR whereas the green line ( ) are from microwave cavity perturbation on a piece of the same crystal shifted to overlap with the LE- SR at low temperature. For comparison, we also show recent TDR ( ) and MFM ( ) results forΔ𝜆, all shifted to agree at T = 0 K. surface whereas MFM is a point-like probe. Such differences between methods and crystals in- dicate there are considerable variations in the spectrum of low energy excitations depending on doping and/or surface quality. In conclusion, we have investigated the absolute value of the magnetic penetration depth in the Meissner state of freshly-cleaved Ba(Co0.074Fe0.926)2As2 using LE- SR. The absolute value of 𝜆 extrapolated toT = 0 is 250(8)nm, wheremost of this uncertainty originates from themuon stopping distribution. There is no evidence of sharp rise in field broadening close to the sample surface. Also there is weaker field dependence of 𝜎 which may have a different origin other than bulk magnetization effect. It has been reported that a two-gap s-wave model and a weak power law model describes the temperature dependence of the superfluid density equally well [184]. The latter model is characteristic of any non s-wave gap. 4.3 Summary of Results Table 4.5 shows the summary of measured London penetration depths, extrapolated to T = 0 K, for three oxygen dopings of YBa2Cu3O6+x . The first column is the measured 𝜆a,b extrapolated to T = 0 K. The error in brackets is the statistical uncertainty and the second error is an estimate of the systematic uncertainty. For comparison, 𝜆a,b values from other techniques are also given. In YBa2Cu3O6.92, “individual phase” and “shared phase” models yield 𝜆a,b within a nm of 71each other and the anisotropies are the same. This implies that there is little dependence on the model used to fit the data. The 𝜆ab values obtained here are in good agreement with bulk  SR measurement in vortex state extrapolated to zero field. It may be noted that conventional  SR is very different type of measurement done in the presence of vortices. The 𝜆ab in YBa2Cu3O6.92 obtained here is shorter than that obtained in a thin film. The difference is understandable con- sidering the measurement in the film was done at 20K and the Tc of the film (87.5K) is less than in crystals possibly indicating a different doping level than in crystals. Row two of table 4.5 represent our results of 𝜆a,b in Ortho-I YBa2Cu3O6.998, measured at 5 K and 4.7mT, extrapolated to 0 K by the method described above. Note, there is very close agreement between our result and ESR on Gd-doped Ortho-I YBa2Cu3O6.95 [185]. Row three of table 4.5 represent our results of 𝜆a,b in YBa2Cu3O6.52, measured at 5K and 1.5mT, extrapolated to 0 K. 𝜆a,b's were measured in 1.5mT, 4.7mT and 7.8mT. However, at the higher magnetic fields of 4.7mT and 7.8mT the unusual temperature dependence is clear evidence for vortex penetration. In 1.5mT external field, a unique determination of phase was made by reversing the field, essentially doubling the range of muon polarization's oscillation time. Determination of the phase is crucial especially at low field since the frequency is low and very few full oscillations are observed in the the muon polarization. It is also interesting that the low temperature linear dependence of 𝜆 in YBa2Cu3O6.52 is an order of magnitude Table 4.5 : Measurements of the absolute value of the magnetic penetration depth (︀ 𝜆a,b )︀ in YBa2Cu3O6.52, YBa2Cu3O6.92 and YBa2Cu3O6.998. Average magnetic penetration depth 𝜆ab = √ 𝜆a𝜆b. Vortex state measure- ments are quoted without systematic errors. 𝜆a (nm) 𝜆b (nm) 𝜆ab (nm) 𝜆a/𝜆b Comment 125.6(17) – 3 105.5(11) – 3 115.1(10) – 3 1.19(1) YBa2Cu3O6.92 a 105.0(39) – 3 82.0(33) – 3 92.8(26) – 3 1.28(2) YBa2Cu3O6.998 a 261.7(212) – 3 199.8(162) – 3 228.7(131) – 3 1.31(6) YBa2Cu3O6.52 a 118.0(4)  SR in vortex stateb[124] 146(3) LE- SR in thin film at 20K [169] 160 100 126.5 1.6 IR reflectivity at 10K [111] 103(8) 80(5) 91(7) 1.29(7) ESR on Gd-doped YBa2Cu3O6.995 [185] 202(22) 140(28) 168(19) 1.4(3) ESR on Gd-doped YBa2Cu3O6.52 [185] 150(10) 1.16(2)  SR at 10K [186] 138(5) 1.18(2) SANS at 10K [187] a This work b Conventional  SR 72Table 4.6 : Measurements of the absolutevalue of the magnetic penetration depth (︀ 𝜆a/b )︀ in Ba(Co0.074Fe0.926)2As2. Average magnetic penetration depth 𝜆ab = √ 𝜆a𝜆b. 𝜆ab (nm) Comment 250.0(26) – 5 Ba(Co0.074Fe0.926)2As2 a 325(50) MFM [182] in Ba(Co0.05Fe0.95)2As2 190(10) Estimated for a range of dopings in BaCoxFe2–xAs2 [188] 250(36) MFM [189] on Ba(Co0.074Fe0.926)2As2 a This work higher than those found in other two oxygen dopings. Note that 𝜆ab in LE- SR is significantly longer than that measured with ESR onGd-doped YBa2Cu3O6.52. This is surprising considering the other two oxygen dopings have produced very similar results. It would be interesting to do measurements at even lower magnetic fields with low energy  SR to see if the 𝜆 is in closer agreement with the ESR results. The first row in table 4.6 gives the measured 𝜆ab, extrapolated to 0K for the pnictide. Mea- surements were done in external field of 2.5mT and 4.7mT. The 𝜆ab s obtained for both external field in “shared phase” and “individual phase”models were within ~ 4nm i.e, introducing ~ 2% systematic uncertainty. There is broad agreement in the temperature dependence of 𝜆 between LE- SR, TDR &  W, at least at this one Co concentration. However, the MFM results look sig- nificantly different. This method is unique in that it probes Δ𝜆 on much smaller length scale. For comparison, some results from other methods, especially MFM are also shown. Among the MFM results, one experiment made on the exact same Co-doping as ours, yield the same 𝜆, however with a much larger uncertainty. Figure 4.32 shows the critical temperature versus 1/(𝜆a𝜆b) –1 = ns/ √︁ m*am * b. A linear rela- tionship between the two quantities was first suggested by Uemura et al. LE- SR results signifi- cantly differ from a linear relationship. Recent ESR measurements on Gd-doped YBCO [185] arrived at the same conclusion - although, themeasured penetration depths are somewhat differ- ent than this work. It is clear the true relationship seems to be sub-linear dependence of Tc on 𝜆–2ab. The linear relationship has been widely regarded as an evidence of preformed Cooper pairs that form above Tc at a temperature T*. The superconducting Tc occurs when the order pa- rameter phases become coherent. Interestingly Bose-Einstein condensation in 3D also predicts a sub-linear relationship of Tc on 𝜆–2ab. The sub-linear relationship of Tc versus 𝜆 –2 ab is clear in- dication of other mechanisms being influential in determining Tc. One of the interesting as- pects from figure 4.32 is that optimally doped cuprate and Co-doped pnictide almost falls on a straight line going through the origin to optimally doped YBCO. Measurements on a range of dopings will be needed to determine the exact relationship of Tc on 𝜆–2ab. The dependence of the anisotropy (︀ 𝜆a/𝜆b )︀ on O-doping in YBa2Cu3O6+x is very surprising. In YBa2Cu3O6.995, 𝜆c/𝜆ab ~ 7.0 [185] - which indicates this system is much more three dimensional BEC system 730 2 4 6 8 10 12 14 0 20 40 60 80 100 Uemur a Line Tc Tc ~ n 2/3s ~ n z/(d –2+ z) s x = 6.52 x = 6.92 x = 6.998 Co = 0.148 ns ∝ (︀ 𝜆a𝜆b )︀–1 (︀ 10−5 nm−2 )︀ T c (K ) 0.6 0.8 1 1.2 1.4 1.6 Anisotropy 1.31 1.19 1.28 𝜆 a (0)/ 𝜆 b (0) Figure 4.32 : Tc and anisotropy vs superfluid density ns are shown. Tc vs ns in YBCO ( ) appears to be sub-linear. The theoretical models of 3D-XY critical fluctuation (︁ ~ nz/(d–2+z)s , d = 3, z = 0.53 (fitted) )︁ and3D-BEC (︀ ~ n2/3s )︀ are also used to fit points (x = 6.52 and 6.92). It is interesting that Tc for optimally doped YBa2Cu3O6.92 and Ba(Co0.074Fe0.926)2As2 ( ) almost fall on a (Uemura) line. The penetration depth anisotropy in YBa2Cu3O6+x ( ) is surprisingly weakly dependent on superfluid density. than in YBa2Cu3O6.52 where 𝜆c/𝜆ab ~ 45.0 [185]. This drop in dimensionality could also play a role in the reduction of Tc going from Ortho-I to Ortho-II since BEC Tc is suppressed in a more 2D superconductor (cf. section 2.4). The in-plane anisotropy inmagnetic penetration depthmay be influenced by several factors such as lattice anisotropy - which may cause static charge stripes [190]. From in-plane resistivity anisotropy measurements [191], these spin stripes are claimed to be intrinsically conducting and transport is easier along the direction of the stripes. It has been argued that CuO2 planes, not the CuO chains, are responsible [45] for these stripe states. Although CuO chains can contribute superfluid carriers, for T < 80 K, carriers in the chains are thought to be localized [166] and do not contribute to the electrical transport properties for this temperature range. However, CuO chains may exert a potential on the planes - thereby indi- rectly influencing the effective mass tensor and electrical transport as well as the anisotropy in the magnetic penetration depth. The measured anisotropies (︀ 𝜆a/𝜆b )︀ in YBa2Cu3O6.998 (every CuO chain is full) and YBa2Cu3O6.52 (every second CuO chain is full) are very similar; however, YBa2Cu3O6.92 (not fully ordered chains) yields a very different anisotropy. This suggests chain ordering may be important in determining in-plane anisotropy. Measurements of anisotropies for a number of dopings will be required to elucidate the anisotropy's dependence on chain or- 74dering and superfluid density. 4.4 Discussion on ``Deadlayer '' As mentioned in earlier sections, a superconductor belowHc1 expels any external magnetic field and the internal decays exponentially inside the sample close to the surface, according to the London equations. However, very close to surface, there is apparently a distance over which magnetic field essentially remains constant - suggesting an effective dead layer. One possible explanation is surface roughness: small perturbations from a perfect flat geometry. Recently the problem has been studied theoretically using a combination of analytic and numerical methods [1]. Without loss in generality, the rough surface was modeled as one which varies harmonically in space: z = 𝜖cos(𝜔xx)cos(𝜔yy). (4.14) Lindstrom et al. [1] has found via linear analysis, that average magnetic field |B|avg does indeed differ from the London solution by as much as 1 %. These simulations (figure 4.33) suggest that for surfaceswith roughness amplitudes (𝜖) in the ballpark of 𝜆/10whose spatial frequencies aren't too high, the dead layer is no bigger than 𝜆/20. This is consistent with observation. However, one difference between theory and experiment is that the gradient in the average field in the direction normal to the surface is always finite in theory whereas a simple deadlayer model used in this thesis assumes d|B/dz| is zero inside the deadlayer. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 0.5 0.6 0.7 0.8 0.9 1 z/𝜆 B/  0 H 𝜔y𝜆 = 10𝜋 𝜔x 𝜆 = 10𝜋 Figure 4.33 : Simulated external field [1] for the depth range 0nm to 10𝜆. Only the region of 0-𝜆 is shown. The blue and red curves are for sinusoidal variation in surface height along y and x axis, respectively, while the external field is along x-direction. The fits to simulations suggest an average ~ 5% longer 𝜆. 75Another possibility is thatmagnetic flux can penetrate near the surface in the formof vortices which is favored if there is suppression of the order parameter near twin or grain boundaries [192, 193]. Surface roughness will also tend to favor vortex penetration due to demagnetization effects associated with a rough surface. For example it has been proposed that surface roughness is the source of vortex nucleation at fields H ≤ Hc1 [194]. Surface vortices have been observed in YBCO in fields as minute as 4G applied parallel to a – b plane [195]. Thus it seems possible that the higher magnetic fields required for low energy  SR will favor vortex penetration near the vacuum-surface boundary. Such vortex penetration and surface roughnesswould also contribute to the additional line broadening we observe in the Meissner state. Field inhomogeneities may result from local variations of the supercurrent close to surface due to surface roughness, twin boundaries and grain boundaries [196]. Further experiments on atomically flat surfaces may help elucidate the origin of the reduction of screening supercurrents near the surface. 76CHAPTER 5 CONCLUSIONS AND OUTLOOK In this thesis, the most direct measurements of the magnetic penetration depth in the 𝜆a/b and its anisotropyies (︀ ” 𝜆a/𝜆b )︀ have been made for three different oxygen (x = 6.52, 6.92 and 6.998) contents of YBa2Cu3O6+x as well as in Ba(Co0.074Fe0.926)2As2. A novel method, using low en- ergymuons, was used tomeasure the field profile directly in theMeissner state. Using the London model and simulated stopping profiles in the Meissner state, one is able to extract a precise mea- surement of themagnetic penetration depth as a function of observable parameters such doping, magnetic field, and temperature. The measured penetration depths depend only slightly on the details of the model used. For example, the uncertainty in the manner, by which the phase is treated, is included as a systematic error in the final results. The field profiles are obtained directly by measuring the muon frequency spectrum - which contains most of the relevant information about muons' local magnetic environment, as a function of the implantation energy. All mea- surements in the superconducting state were carried out under zero-field-cooled conditions in order to avoid flux trapping in the sample. This is the first direct observation of exponentially decaying field profiles in high quality crystals of an exotic superconductor YBa2Cu3O6+x and Ba(Co0.074Fe0.926)2As2. The measured values of 𝜆 are somewhat different from previous works using bulk methods. The fitted 𝜆 is independent of magnetic field at sufficiently low magnetic fields, as one expects from a simple Londonmodel ignoring any non linear effects. However, care must be taken to ensure that the field is low enough to avoid vortex penetration. Surprisingly, the field below which the fitted 𝜆 is independent of field, is substantially belowHc1, which is not fully understood at present. Small deviations, from the London model, are observed, below the surface of the crystals, which indicate that there is a suppression of the supercurrent density close to the surface, on the scale of d around 10nm to 20nm. Also there is a substantial broadening of the field distribution beyond what one would expect from the range straggling and Cu nuclear dipole moments. The mosaic geometry has virtually no affect on this broadening which indicates that muons experi- ence broadened field distribution in each crystal. It seems likely that surface roughness and/or demagnetization affects at the edges of each crystal are at least partially responsible for the devi- ations from the London model. A suppression of the supercurrent density was also reported in a previous low energy  SR study of the field profile in a thin film of YBa2Cu3O6.92 and attributed to surface roughness. Also, this is confirmed by recent theoretical studies on the affect of surface roughness. Measurements on atomically flat cleaved surfaces may be needed to resolve the true origin of d and deviations from the London model we report. Our most important result is that the measured 𝜆a/b(0 K) values do not follow the widely reported Uemura relation (︀ Tc ∝ 𝜆–2ab )︀ . Interestingly Bose-Einstein condensation in 3D and 773D-XY critical fluctuation also predict a sub-linear relationship of Tc on 𝜆–2ab. It is also impor- tant to note that YBCO becomes more two dimensional at lower doping. This could partially be responsible for the reduction in Tc at low doping. For example, in a 2D system, the critical temperature Tc = 0, so that there cannot be any superconductivity in such a system at any finite temperature. One of the interesting aspects of the Tc-vs - (︀ 𝜆–2ab )︀ is that optimally doped cuprate and Co- doped pnictide almost falls on a straight line going through the origin to optimally doped YBCO. Thus while the variation of Tc versus doping in YBCO does not follow the Uemura relation, we do confirm that the Tc at optimal doping does seem to follow this relation when comparing different families. Clearly many more measurements are needed in a variety of supercondcutors to see how valid this empirical relationship really is. We also report the a – b anisotropy in the penetration depth for all three oxygen concentra- tions of YBa2Cu3O6+x . The a–b anisotropy, being the ratio of 𝜆 (between the a and b directions), is determinedwithmore accuracy than the penetration depths themselves. There is a ratherweak dependence of 𝜆a/𝜆b on oxygen concentration - which is surprising considering the significant changes in the oxygen chains between Ortho-I and Ortho-II. A low temperature linear dependence of 𝜆–2a/b has also been observed in all the oxygen dop- ings in YBa2Cu3O6+x , which is an important signature for a d-wave superconducting gap func- tion. The temperature dependence of the London penetration depth 𝜆a/b can be fit to a semi- classical model of a cylindrical Fermi surface in YBCO. TheT-dependence is consistent with µW measurements. Our measurements of 𝜆(0) are much more accurate and are needed to convert Δ𝜆(T) (from µW measurements) into the T-dependence of the superfluid density. The fitted su- perconducting gapsΔa/b(0 K) are consistent with a dx2–y2-type order in YBCO. Gap anisotropy in YBCO is close to 1, suggesting that effective mass tensors in two directions are independent of temperature. A two-gap s-wave model and a weak power law model describe the temperature dependence of the superfluid density equally well in Ba(Co0.074Fe0.926)2As2. Perhaps most importantly, it is shown that low energy  SR can be used to probe the near surface region of small single crystals which are much smaller than the beam diameter. FutureWork: It should be possible to test the origins suppressed superconductivity near the surface, by depositing thin metallic layers on top of the crystals. By controlling the thickness of themetallic layer, it should be possible to implantmuons close to the surface of a superconductor using a muon energy around 2 keV. Basically the overlayer would act as a degrader. Very low energy muons will stop close to the interface between the superconductor and normal metal interface. In this way, it may be possible to probe the magnetic field much close to the surface of the superconductor than is currently possible. In this way, it may be possible to discern between demagnetization effects and a true dead layer from surface roughness. We note that, these experiments require a considerable amount of beamtime since the am- plitude of the signal is reduced compared to studies on large samples. One way to increase our 78signal to noise ratio would be to make the sample plate just big enough to hold the sample. This would get rid of the non precessing background signal coming frommuons landing on the nickel sample plate. Also, current positron counters may be replaced by incorporating position sensi- tive detectors to ensure the positron is coming from the sample and not elsewhere. Despite these limitations, the technique has proven to work extremely well. It has provided the most accu- rate measurements of the London penetration depth so far. We expect this method will have a significant impact on the field of exotic superconductivity. 79BIBLIOGRAPHY [1] Lindstrom, M. 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