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Magnetic properties near the surface of cuprate superconductors studied using Beta-Detected NMR Saadaoui, Hassan 2009

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Magnetic Properties Near the Surfaceof Cuprate Superconductors StudiedUsing Beta-Detected NMRbyHassan SaadaouiB.Sc., Universit´e Mohammed Premier, 2001M.Sc., Laurentian University, 2004A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December, 2009c© Hassan Saadaoui 2009AbstractBeta-detected Nuclear Magnetic Resonance (β-NMR) uses highly spin po-larizedβ-emitting nuclei as a probe. Besides its use in nuclear physics, it hasalso become a powerful and sensitive tool in condensed matter physics andmaterials science. At TRIUMF,β-NMR of 8Li+ has been developed to studymaterials in a depth-resolved manner, where the implantation depth of 8Li+is controlled via electrostatic deceleration. In this thesis, β-NMR of 8Li+has been used to study high-Tc cuprate superconductors (HTSC). The ob-jective of this work is to search for spontaneous magnetic fields generated bya possible time-reversal symmetry breaking (TRSB) superconducting statenear the surface of hole-doped YBa2Cu3O7−δ (YBCO), and study the natureof the vortex lattice (VL) in YBCO and electron-doped Pr2−xCexCuO4−δ(PCCO). For several advantages, our measurements were carried out byimplanting 8Li+ in thin silver films evaporated on the superconductors.In our TRSB studies, the magnetic field distribution p(B) is measured 8nm away from the Ag/YBCO interface in magnetic fields B0 = 5 to 100 G,applied parallel to the interface. p(B) showed significant broadening belowthe Tc of ab- and c-axis oriented YBCO films. The broadening signals theexistence of weak disordered magnetic fields near the surface of YBCO. Fromthe broadening’s temperature and field dependence we draw an upper limitof 0.2 G on the magnitude of spontaneous magnetic fields associated withTRSB.To study the VL,p(B) is measured at average implantation depths rang-ing from 20 to 90 nm away from the Ag/YBCO or Ag/PCCO interface inB0 = 0.1 to 33 kG, applied perpendicular to the surface. p(B) showed adramatic broadening below Tc as expected from the emerging field lines ofthe VL in the superconductor. In YBCO, p(B) is symmetric and the de-iiAbstractpendence on B0 is much weaker than expected from an ideal VL, indicatingthat the vortex density varies across the face of the sample on a long lengthscale, likely due to vortex pinning at twin boundaries. In PCCO, a 2D VLis established due to the high anisotropy of the superconductor leading to anearly symmetric p(B).iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . viiStatement of Co-Authorship . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The β-NMR Technique . . . . . . . . . . . . . . . . . . . . . 41.1.1 Production of Spin-Polarized 8Li+ . . . . . . . . . . 51.1.2 The High-Field Spectrometer . . . . . . . . . . . . . . 81.1.3 The Low-Field Spectrometer . . . . . . . . . . . . . . 91.1.4 Implantation Profiles . . . . . . . . . . . . . . . . . . 111.1.5 β-NMR Resonance Spectra . . . . . . . . . . . . . . . 121.1.6 Comparison of β-NMR with NMR and µSR . . . . . 141.2 Generic Properties of HTSC . . . . . . . . . . . . . . . . . . 171.2.1 Crystal Structure . . . . . . . . . . . . . . . . . . . . 171.2.2 Electronic Configuration . . . . . . . . . . . . . . . . 191.2.3 Doping Phase Diagram . . . . . . . . . . . . . . . . . 201.3 Theoretical Foundations . . . . . . . . . . . . . . . . . . . . . 211.3.1 London Theory . . . . . . . . . . . . . . . . . . . . . 221.3.2 Ginzburg-Landau Theory . . . . . . . . . . . . . . . . 24ivTable of Contents1.3.3 BCS Theory . . . . . . . . . . . . . . . . . . . . . . . 261.4 Time-Reversal Symmetry Breaking in HTSC . . . . . . . . . 281.4.1 Which Order Parameters Break TRS? . . . . . . . . . 281.4.2 Theories . . . . . . . . . . . . . . . . . . . . . . . . . 311.4.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . 351.5 Vortex Lattice in HTSC . . . . . . . . . . . . . . . . . . . . . 401.5.1 Regular VL Characteristics . . . . . . . . . . . . . . . 401.5.2 Disorder . . . . . . . . . . . . . . . . . . . . . . . . . 471.5.3 Anisotropy and Thermal Fluctuations . . . . . . . . . 491.5.4 Temperature Dependence of λ . . . . . . . . . . . . . 531.5.5 Proximal Detection of the VL . . . . . . . . . . . . . 541.6 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . 55Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572 Search for Broken Time-Reversal Symmetry Near the Sur-face of (110) and (001) YBa2Cu3O7−δ Films . . . . . . . . . 662.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 662.2 Experimental Details . . . . . . . . . . . . . . . . . . . . . . 692.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722.4 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . 75Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783 Vortex Lattice Disorder in YBa2Cu3O7−δ Probed Using β-NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.2 The Magnetic Field Distribution p(B) in the Vortex State . 843.3 Experimental Details . . . . . . . . . . . . . . . . . . . . . . 923.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953.5 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . 102Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105vTable of Contents4 Vortex Lattice Near the Surface of Pr1.85Ce0.15CuO4−δ . . 1114.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.2 Experimental Details . . . . . . . . . . . . . . . . . . . . . . 1124.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . 113Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1195 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . 122Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127AppendicesA RF Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129A.1 CW RF Mode . . . . . . . . . . . . . . . . . . . . . . . . . . 129A.2 Pulsed RF Mode . . . . . . . . . . . . . . . . . . . . . . . . . 130B β-NMR Resonance in Ag . . . . . . . . . . . . . . . . . . . . . 133B.1 Field and Temperature Dependence . . . . . . . . . . . . . . 133B.2 Effect of the Dewetting Transition in Ag . . . . . . . . . . . 134C Obtaining Zero-Field . . . . . . . . . . . . . . . . . . . . . . . . 140D Spin-Lattice Relaxation . . . . . . . . . . . . . . . . . . . . . . 143D.1 Spin-Relaxation Signal . . . . . . . . . . . . . . . . . . . . . 143D.2 Ag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145D.3 Ag/YBCO . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150D.4 PCCO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152E Sample Characteristics . . . . . . . . . . . . . . . . . . . . . . 154viList of Tables1.1 Example of isotopes suitable for β-NMR compared to µ+.Intrinsic characteristics of nuclei are given by nuclear spin,I, half life T1/2, gyromagnetic ratio γ, and asymmetry. Pro-duction of rate µ+ used in Low-energy µSR, and of isotopesproduced at TRIUMF and world-wide labs are given. . . . . . 51.2 Summary of the characteristics of high and low-field spec-trometers. Both spectrometers have an ultra high vacuum(UHV) cryostat with a temperature range ∼ 3-300 K, andare mounted on high voltage platforms allowing the variationof 8Li+ energy. . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Electron configurations of common atoms in cuprate HTSC. . 19E.1 Characteristics of the samples used in thesis thesis. TW:twinned crystal, DTW: detwinned crystal. d is the nominalthickness of the films, dAg the thickness of deposited Ag ifapplicable, andRa the RMS surface roughness of the samplesafter Ag was deposited found measured by AFM. . . . . . . 156viiList of Figures1.1 A simplified mean-field magnetic field-temperature phase dia-gram of a type II superconductor. The superconductor expelsthe applied field from the bulk belowBc1, and is in the vortexstate for fields up to Bc2. . . . . . . . . . . . . . . . . . . . . 21.2 (a) Layout of the polarizer and β-NMR spectrometers. (b)Polarization of 8Li+ using optical pumping. (c) Sodium cellscan shows the dependence of the asymmetry on Na cell bias. 61.3 Experimental setup used to measure the spin polarization of8Li+ in (a) the high-field, and (b) the low-field β-NMR spec-trometers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 TRIM simulation of the stopping distribution of 20000 ionsof 8Li+ implanted in a 1000 nm Ag layer. The energy of 8Li+is varied from 1 keV to 28 keV leading to mean depths from 5nm to 104 nm. A significant fraction of 8Li+ is back scatteredby the Ag surface. . . . . . . . . . . . . . . . . . . . . . . . . 101.5 (a) Angular distribution, W(θ), of the electrons emitted af-ter the 8Li+ β-decay: when all energies are sampled withequal probability, the asymmetry parameter has the valuea = −1/3. (b) Energy distribution of the emitted electronsafter the 8Li+ decays. . . . . . . . . . . . . . . . . . . . . . . 12viiiList of Figures1.6 (a) Asymmetry of the betas emitted after implanting a 28 keV8Li+ beam into a 120 nm Ag film. Both helicities are shown.(b) The difference of the asymmetries of the two helicities isplotted. The solid line is a fit to a Lorentzian, and the dashedline refers to the Larmor frequency. The spectra are taken inB0 = 152.6 G, and T = 100 K. . . . . . . . . . . . . . . . . . 151.7 TypicalcrystalstructureofYBa2Cu3O7 (left)andPr2−xCexCuO4(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.8 Thetypicaldopingphasediagramsofhole(right)andelectron-doped (left) high-Tc cuprates are shown. The various phasesare explained in the text. . . . . . . . . . . . . . . . . . . . . 201.9 Idealized spatial variation of the magnetic field B and orderparameter ψ for κ ≪ 1 (type I) and κ ≫ 1 (type II) super-conductors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.10 Normalized amplitude and phase of the main candidate or-der parameters in HTSC are shown. The order parameter iswritten as ∆ = |∆|eiφ = |∆|g(θ). Here θ is a polar anglein momentum space k = k(cosθ,sinθ); g(θ) = cos(2θ +α),where θ = 0 for s-wave and finite for d- and dxy-wave; α = 0(π/2) for s-wave and d-wave (dxy-wave). . . . . . . . . . . . 301.11 Left: the µSR relaxation rate in Sr2RuO4 showing the ap-pearance of spontaneous magnetic fields below Tc [3]. Right:absence of spontaneous magnetic fields in YBCO (Tc=90 K)and BSCCO (Tc=85 K) systems [65]. In µSR, the relaxationrate is a measure of the spread of the internal fields. . . . . . 361.12 Top: conductanceinYBCO/Cutunneljunctionshowingsplit-ting of the ZBCP as a function of external field [56]. Bottom:absence of the splitting of the ZBCP in the normalized con-ductance spectra taken on the (110) surface of YBCO with aPt-Ir tip at 4.2 K for an STM tunnel junction (main panel)and for a point contact (left inset). Right inset shows theexpected curves from d+s and d+is symmetries. [54]. . . . . 37ixList of Figures1.13 (a) Spontaneous magnetic field generated by a thin YBCOfilm plotted versus temperature, as measured by Carmi etal. [78]. (b) Scanning SQUID microscope image of an areawith a 45o asymmetricc-axis YBCO bicrystal grain boundaryof 180 nm thickness. The flux through the grain boundaryis shown [79]. (c) Scanning SQUID images of (100)/(103)YBCO cooled in 3mG. The corresponding flux distributionis shown. A broader field distribution is observed in (001),although it has the same average flux as in (103) [80]. . . . . 391.14 (a) Simulated field distribution using Eq. (1.45) at depthsz =2000 nm andz = −10 nm, andB0 = 1 T. Parameters relevanttoYBCOatT ≪Tc havebeenused: λab = 150nm,ξ = 2nm.A Gaussian cutoff is used whereb = 0. Solid (dotted) lines arelineshapes of triangular (square) lattice. Top inset shows thepositions of the low-field cutoff (A), most probable field (B)and high-field cutoff (C) in a triangular lattice. Bottom inset:field distribution measured in a YBCO crystal by µSR at 0.5T fitted by the London model convoluted with a Gaussiandistribution [88]. . . . . . . . . . . . . . . . . . . . . . . . . . 431.15 Variance of the field distribution as a function of depth, withz > 0 (z < 0) corresponding to inside (outside) the super-conductor. The parameters λab = 150 nm, ξ = 2 nm, and aGaussian cutoff (b = 0) have been used. . . . . . . . . . . . . 451.16 Variance of the field distribution inside the superconductorat z = 50 nm for (a) and (c), and outside at z = −50 nmfor (b) and (d). Parameters relevant to YBCO (PCCO) havebeen used where λab = 150 (300) nm and ξ = 2 (6) nm. Also,plotted is the variance for ξ = 0,20 nm. A Gaussian cutoff isused in all, where b = 0 is taken for YBCO, and b = B/Bc2(Bc2=9 T) is used for PCCO. . . . . . . . . . . . . . . . . . . 46xList of Figures1.17 Top panel: the vortex configuration in YBCO crystals ob-tained using Bitter decoration technique [93]: images of (a)a twinned and (b) a twin-free area are taken after coolingin 20 G. The marker is 10 µm. Bottom panel: STM imagesof twinned YBCO single crystals at 4 K, with the magneticfield applied parallel to the c-axis, and perpendicular to thesurface [94]. (c) Image taken after field cooling in 3 T. (d)Topographic images of the YBCO surface showing the twinboundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481.18 The field distribution in BSSCO crystals measured by µSRin an external field of 1.5049 T applied along the c-axis. Themeasured lineshape is not well fitted by a 3D model (dashedline), and well fitted by a 2D model [13]. . . . . . . . . . . . . 511.19 (a) The µSR field distribution measured in low and high ap-plied fields. The low field lineshape is that of a regular VL,but the high field is due to a melted lattice. (b) The skewnessparameter α = 〈∆B3〉1/3/〈∆B2〉1/2 of the field distributionis shown [101]. . . . . . . . . . . . . . . . . . . . . . . . . . . 521.20 (a) T-dependence of the second moment σ ∝ 1/λ2 in a sin-tered YBCO sample measured in a field of 350 mT. The datais well fitted by the two-fluid model suggesting an s-wavepairing symmetry [104]. (b) T-dependence of 1/λ2 in a highquality YBCO single crystal in 5 kG external field. The datais well fitted by the d-wave order parameter, and stronglydeviates from s-wave [105]. . . . . . . . . . . . . . . . . . . . 532.1 Geometry of the experiment where the field is applied alongthe surface. The orientation of the YBCO films is shown:Ag/YBCO(110) on the right and Ag/YBCO(001) on the left. 68xiList of Figures2.2 Typical β-NMR spectra in Ag taken above and below Tc at2 keV, about 8 nm from the (110)-oriented YBCO interface,in an external field of B0 = 10 G (FC) applied along thesurface of the film. Solid lines are fits to a Lorentzian ofamplitude A, HWHM ∆ and resonance frequency ωL, i.e.L(ω) = A4(ω−ωL)2+∆2. Inset: simulated implantation profilefor the 8Li+ in a 15nm Ag layer on YBCO from TRIM.SP[28]. The 8Li+ stops at an average depth of 8 nm away fromthe Ag/YBCO interface. . . . . . . . . . . . . . . . . . . . . 702.3 The T-dependence of the Lorentzian HWHM of the 8Li+resonance at 2 keV, i.e. 8 nm from the Ag/YBCO(110),Ag/YBCO(001), and Ag/STO (open circles) interface. ThedataontheAg/YBCO(110)andAgfilmwastakenwhilecool-ing in B0 = 10 G. The data on Ag/YBCO(001) was collectedinB0 = 10 after ZFC to 5 K. The widths were independent offield-cooling conditions. The dashed lines represent the aver-age Ag width, the arrows point to the Tc of the YBCO films,and the solid lines are guide to the eye. . . . . . . . . . . . . 732.4 The 10 K linewidth (HWHM) of the NMR resonance versusthe applied field B0 taken in Ag on (110)- and (001)-orientedYBCO films. The data were taken after ZFC and gradu-ally increasing the field, except in the (110) films which wasFC. The solid lines are fits to the linewidth in the Ag onYBCO(001) at 10 and 100 K. . . . . . . . . . . . . . . . . . . 742.5 Extra broadening, ∆(10 K) − ∆(100 K), versus the averagedepth of 8Li+ into a 50nm thick Ag on (001) YBCO film,ZFC in 10 G and 100 G. . . . . . . . . . . . . . . . . . . . . . 75xiiList of Figures3.1 (a) Simulation of p(B) in an applied field of 52 mT at a dis-tance 90 nm from the superconductor using Eq. (3.5) convo-luted with a Lorentzian of width ∆D. (b) The broadening ofp(B) versus the applied field. Solid lines representσ from Eq.(3.6) in the bulk, and 40 and 90 nm away from the supercon-ductor. Long dashed line shows σ at 90 nm from Eq. (3.9)for D = 4 µm and f = 0.1. Inset: sketch of a possible vor-tex arrangement including vortex trapping at twin boundariesspaced byD and a regular triangular vortex lattice elsewhere.In all figures λ(0) = 150 nm. . . . . . . . . . . . . . . . . . . . 873.2 σ(T) (normalized at T = 0), from Eq. (3.9) for B0 = 52mT, z = −90 nm, and f = 0.1, is plotted against T/Tc fordifferent values of D. A d-wave temperature dependence ofλ(T) is used [37], where λ(0) = 150 nm. . . . . . . . . . . . . 883.3 Implantation profiles of 8Li+ at energies of (a) 5 keV into 60nm of Ag with the mean at 40 nm away from YBCO film,and (b) 8 keV into 120 nm of Ag with the mean at 90 nmaway from YBCO crystals, as calculated via TRIM.SP (Ref.[45]). Solid lines are phenomenological fits. . . . . . . . . . . 933.4 βNMR resonances in Ag/YBCO (crystal I) at temperatures100 K, 80 K, 20 K, and 4.5 K measured in a magnetic fieldB0 of 52.3 mT applied along YBCO c-axis. The solid linesare best fits using a Lorentzian. . . . . . . . . . . . . . . . . 963.5 The vortex-related broadening below Tc, ∆sc(T) = ∆(T) −∆ns of the twinned (full symbols) and detwinned (opaquesquares) YBCO crystals in an applied field B0. ∆(T) is thelinewidth at temperature T of the Lorentzian fits and ∆ns isthe constant linewidth in the normal state. Solid lines repre-sent a fit using ∆DVL = 2.355σ where σ is given in Eq. (3.9)and D and f are varied to fit the data. A d-wave tempera-ture dependence of λ(T) in YBCO is used, ( Ref. [37]) whereλ(0) = 150 nm. . . . . . . . . . . . . . . . . . . . . . . . . . 98xiiiList of Figures3.6 Comparison of the field distributions in the three samplestaken at temperatures 100 K (top panel), 5 K (crystals) and10 K (film). The x-axis is shifted by B0, the applied fieldwhich is 52.3 (51.7) mT for the crystals (film). Solid lines areLorentzian fits and dashed lines are simulation described inthe text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 993.7 Superconductingbroadening∆sc(T)ofβ-NMRresonancespec-tra at temperatures ∼ 4.5-10 K in the three samples. Theexperimental broadening is compared with the linewidth ofan ideal VL ∆VL (dashed lines), and ∆DVL of a disorderedVL (solid lines), which are both weighted by the 8Li+ profilegiven in Fig. 3.3. . . . . . . . . . . . . . . . . . . . . . . . . . 1014.1 TRIM simulation of 8Li+ stopping profile into 40 nm of Agon 300 nm of PCCO, at energies of 5 keV and 28 keV. . . . . 1124.2 Spin relaxation spectra in Ag/PCCO at 5 K, taken in an ap-plied static magnetic fieldBapp = 200 G, and 8Li+ energies of5 and 28 keV. The solid line is a fit using a phenomenologicalbiexponential function for beam on (0 to 4 s) and beam off(4 s to 8 s) [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . 1144.3 The βNMR asymmetry spectra of 8Li+ implanted at 5 keVin the Ag/PCCO at different temperatures. The dashed lineshows the Larmor frequency in the applied magnetic field ofBapp = 216 G. The solid lines are Lorentzian fits. . . . . . . 1164.4 (a) The resonance frequency, Bres, versus T in an appliedfield Bapp = 216.62 G (dashed line), fitted by Bapp +0.8(1−(T/22)4)2 (solid line). (b) The linewidth (FWHM) versusT. The solid line represents FWHMNS +4.2(1−(T/22)2.2)2,where FWHMNS = 0.74 G is the normal state broadening(dashed line). All data are extracted from Lorentzian fits ofthe resonances. . . . . . . . . . . . . . . . . . . . . . . . . . . 117xivList of FiguresA.1 (a) Pulsed RF sequence where RF is on during a time intervaltp = 80 ms. The excited frequencies have a bandwidth ∆ω =200Hz. Thefunctionalformisln-sech, andthepulsesequencecontrols when the RF is delivered. (b) The spectra in Ag(15nm)/YBCO at E = 2 keV, T = 10 K, and B0 = 100 G takenusing CW and pulsed RF modes with similar power (B1) areshown. The width is broadened by a factor of two using CWRF. The width of the resonance taken with the pulsed RF is348 Hz, which is close to the ideal intrinsic dipolar broadeningin Ag of 250 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . 131B.1 The temperature dependence the Larmor frequency, FWHM,amplitude and baseline of Lorentzian fits of β-NMR reso-nances in an 15 nm film thick Ag evaporated on SrTiO3. Thedata is taken with a 2 keV 8Li+ beam in an external fieldB0 = 10 G, applied parallel to the surface of the substrate. . 135B.2 (a) β-NMR resonances in Ag(120 nm) on YBCO twinnedcrystal at B0 = 3.33 T and E = 8 keV. Solid lines are fits toone (two) Lorentzian(s) at 100 (85) K. (b) The shift and am-plitude of the S and O peaks are plotted against temperatureat 3 T. Copied from G. D. Morris et al., Phys. Rev. Lett.93, 157601 (2004). . . . . . . . . . . . . . . . . . . . . . . . . 136B.3 Theβ-NMR spectra in Ag(15 nm)/STO at room temperaturewhere B0 = 10 G, and ELi = 2 keV. The linewidth increasesfrom 314 Hz, when measured in December 2007, to 504 Hz inAugust 2008. . . . . . . . . . . . . . . . . . . . . . . . . . . . 137B.4 AFM of silver films showing an increase in the surface rough-ness with time. (a) AFM is done immediately after evapo-rating 15 nm of Ag onto an STO substrate. (a) AFM done8 months after 15 nm of Ag was evaporated onto an STOsubstrate (this sample is different from the one used in (a)). . 138xvList of FiguresB.5 Auger depth profile of (a) fresh Ag(15 nm)/STO, and (b) fewweeks old sample. Ion bombardment time is given in seconds.The sample was sputtered by Ar, removing layers at a rateof ≈ 0.1 nm/s. Carbon, oxygen, and Sulfur atoms have beenidentified in the first top layers. . . . . . . . . . . . . . . . . . 139C.1 Variation of the Larmor frequency, ωL, with the currents Ix,Iy, Iz which run into three Helmoltz coils and create fieldsalong x, y, and z direction, respectively. . . . . . . . . . . . 141D.1 SpinrelaxationspectrarecordedinB0 = 150G. Theasymme-try recorded with a Laser of positive (A+) and negative (A−)helicity are shown. The overall asymmetry a = A+ −A− isalso plotted. Solid lines are fits explained in the text. . . . . . 144D.2 (a) T-dependence of the spin relaxation in 25 µm silver foilat B0 = 150 G. The 28 keV 8Li+ beam is implanted intothe sample in 4 s pulses every 20 s. Solid lines are fits usingEq. (D.5) with two components. (b) λ = 1/T1 (slow compo-nent) extracted from (a) compared with the baseline of theresonance in an Ag film (50 nm) taken at 3 T. . . . . . . . . 147D.3 TheB0 andT-dependenceoftheamplitude,A, andrelaxationrate, 1/T1, of 8Li+ polarization implanted at full energy into asilver foil. The data was extracted by fitting the asymmetryto a pulsed exponential function. A and 1/T1 of the slowrelaxing part of the function is shown here. The solid linesare fits using a Lorentzian form described in the text. . . . . 149D.4 Spin relaxation of 8Li+of energy (a) 13 keV and (b) 28 keVimplanted into Ag(50 nm)/YBCO(600 nm) at 100 G appliedfield under zero field conditions. Solid lines are single expo-nential fit to the beam off asymmetry. (c) Spin-relaxationspectra versus energy at 10 K. . . . . . . . . . . . . . . . . . . 151xviList of FiguresD.5 Temperature dependence of 1/T1 in Ag(50 nm)/YBCO(600nm), at two different energy extracted from Fig. D.4. Also,shown is 1/T1 in a Ag foil extracted using single exponentialfit to the beam off asymmetry. Inset: energy dependence of1/T1 in Ag/YBCO at 10 K. . . . . . . . . . . . . . . . . . . . 152D.6 The Spin relaxation of 8Li+ at full energy implanted into a300 nm thick PCCO film, in B0 = 150 G. . . . . . . . . . . . 153xviiAcknowledgmentsFirst and foremost I would like to thank my co-supervisors Rob Kiefl andAndrew MacFarlane for their support and advice throughout the course ofmy PhD. Their enthusiasm combined with patience and accessibility arewhat made this work possible. I am very grateful for their responses to myinquiries and availability during my entire stay at UBC. I would also like tothank my committee members: D. A. Bonn, M. Franz, and J. H. Brewer fortheir comments, and for reading the thesis.This thesis has benefited a lot from the help and expertise of GeraldMorris who was always willing to help and go an “extra mile”. Gerald’sgreat passion for what he does makes him a brilliant collaborator. I wouldalso like to thank Zaher Salman who taught me the first lessons of how touse β-NMR and analyze the data. Zaher’s involvement and encouragementsnever stopped even after he moved to another laboratory.I wish also to thank my colleagues in β-NMR group. Special thanks toSusan Q. Song for helping with the characterization of samples at AMPELand Masrur Hossein for helping with computer-related issues. My colleaguesTerry Parolin, Dong Wang, Micheal Smadella, Masrur Hossein, Susan Q.Song, and others have spent many nights taking the actual data; for thatand for the helpful discussions and good times we had together, I am verygrateful.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 helpedto take shifts, tune the spectrometers,...etc. I would like to mention R.Abasalti, D. Arseneau, K. H. Chow, S. Dunsiger, B. Hitti, I. Fan, T. Keeler,C. D. P. Levy, A. I. Mansour, R. Miller, M. R. Pearson, and D. Vyas. Manythanks to AMPEL staff for showing me how to use the characterizationxviiiAcknowledgmentstools. I would especially like to thank Ruixing Liang, Pender Dosanjh,Micheal Whitwick, and Mustafa Masnadi.My friends at UBC have always been a source of encouragement. To all,I say thank you very much for your support and encouragement, and formaking my stay in Vancouver so enjoyable and memorable. The supportof my family has been and is still exceptional, and I would not have gottenhere without that; I don’t have the words to thank them enough.xixStatement of Co-AuthorshipChapters 2, 3, and 4 of this thesis have been written as journal papers bymyself as the first author. The co-authors have been partly involved in tak-ing the data and tuning the β-NMR spectrometers and optical polarizer,reviewing and commenting on the manuscripts, or supplying the studiedsamples. The design of research methods, literature review, data analysis,and manuscript preparation were all done by myself in consultation with mysupervisors: Drs. R. F. Kiefl and W. A. MacFarlane. The pulsed RF tech-nique employed in chapter 2 was a collaborative effort of S. R. Kreitzman,R. F. Kiefl, Z. Salman, R. I. Miller, W. A. MacFarlane, myself, and others.xxChapter 1IntroductionFollowing the discovery of zero resistance in mercury below 4.19 K by H.Kammerlingh Onnes in 1911, the superconducting materials showed anotherstriking phenomenon by expelling the magnetic field from the bulk as seenby Meissner and Ochsenfeld in 1933. Because of the negligible power dissipa-tion, Onnes predicted the use of superconductors for high-field electromag-nets which was realized in the 1960s. These magnets, among other uses, arepivotal for magnetic resonance imaging. Today, there are many applicationsof superconductivity. The study of this exotic state has not only led to newapplications, but was also a driving force for new experimental techniquesand theoretical methods for studying systems with strong electron-electroninteractions, generating significant concepts in physics and related fields.After the discovery of superconductivity, it took almost 50 years until itwas understood microscopically, when in 1957 Bardeen, Cooper and Schrief-fer published their BCS theory. This theory, however, predicted that super-conductivity can only be realized below low critical temperatures Tc∼ 23 K.In 1986, Bednorz and M¨uller discovered a new material La2−xBaxCuO4−δthat superconducts at Tc= 35 K. A year later they were awarded the NobelPrize in physics. Within a few months, new materials were made to super-conduct at temperatures as high as 93 K in YBa2Cu3O7−δ. Because of theirhigh Tcs, these materials are called high-Tc superconductors (HTSC).Up to the present, the mechanism behind HTSC remains a major fo-cus of condensed matter research. The unique properties of HTSC haschallenged and keeps challenging, not only well-established theories, butalso experimental detection techniques. HTSC are type II superconductorswhere an external magnetic field, above a lower critical field Bc1, penetratesin the superconductor in quantized vortices each carrying a flux quantum1Chapter 1. Introduction0Temperature0Magnetic fieldTcBc2Bc1Meissner StateVortes stateNormal StateFigure 1.1: A simplified mean-field magnetic field-temperature phase dia-gram of a type II superconductor. The superconductor expels the appliedfield from the bulk below Bc1, and is in the vortex state for fields up to Bc2.Φ0 = h2e = 2.067×107 G nm. These vortices form cylindrical normal stateregions surrounded by superconducting regions, a state known as the mixedor vortex state. In contrast, in type I superconductors, an applied magneticfield aboveBc1 penetrates uniformly and bulk superconductivity disappears.The field-temperature phase diagram of type II superconductors is shown inFig. 1.1.Investigations of the magnetic properties of HTSCs have been central touncover the existence of striking new phenomena. In the Meissner state, al-though the flux expulsion suggests that the bulk material is free of magneticfields, some magnetic order may develop near the surface, grain bound-aries, or impurities where the magnetic shielding is not perfect. As a re-sult, a spontaneous magnetic field may appear, indicating a broken time-reversal symmetry state analogous to ferromagnetism [1]. Such a statehas been observed in several superconductors such as heavy Fermion su-2Chapter 1. Introductionperconductors U1−xThxBe13 [2], and Ru-based superconductors Sr2RuO4[3], and PrOs4RuSb12 [4]. The most direct evidence of spontaneous fieldsin these superconductors was provided by conventional muon-spin rotation(µSR), which is a bulk probe. In HTSC the effect is thought to be morepronounced near the surface of materials such as YBa2Cu3O7−δ and theassociated magnetic fields may be local and small in magnitude. Severalexperimental investigations have been done reaching different conclusions,where the effect is apparent in some and absent in others. Therefore, theneed is clear for a sensitive magnetic probe to detect such magnetic fieldsnear the surface.In the vortex state, the vortices often form a regular lattice of flux linesof triangular symmetry. This leads to a periodic variation of the magneticfield in space, which is maximum at the core, and decays exponentially overa length scale called the magnetic penetration depth λ. The latter is a fun-damental length scale of superconductivity because 1/λ2 is proportional tothe modulus of the superconducting order parameter, and its temperaturedependence is determined by the pairing symmetry of the superconduct-ing wave-function. Impurities, grain and twin boundaries, and anisotropyaffect the vortex lattice, leading to a disordered lattice and modifying themagnetic field variation from that of a regular lattice. This makes the accu-rate extraction of the penetration depth difficult, as disorder may stronglymodify the spatial variation of the magnetic field. Thus, it is important tounderstand the effect of disorder on the vortex lattice.In this thesis, I have used β-detected nuclear magnetic resonance (β-NMR) as a local probe to detect any spontaneous magnetic fields nearthe surface of YBa2Cu3O7−δ HTSC superconductors in the Meissner state,and to study the features of the field distribution of YBa2Cu3O7−δ andPr2−xCexCuO4−δ superconductors in the vortex state. The β-NMR tech-nique is unique in its capability to measure directly the field distribution ina depth resolved manner, on a 2-500 nm length scale. In this chapter, I willintroduce the reader to the topics presented in this thesis. In Section 1.1,the β-NMR technique will be described and compared to other techniques.In Section 1.2, some generic properties of HTSC materials will be briefly in-31.1. The β-NMR Techniquetroduced. In Section 1.3, some established theories of superconductivity willbe discussed. In Section 1.4, the notion of time-reversal symmetry breakingand its manifestation in HTSC will be reviewed. In Section 1.5, the physicsof the vortex lattice and its dependence on disorder will be discussed. InSection 1.6, an outline of the thesis will be given.1.1 The β-NMR TechniqueLike most other nuclear methods, β-NMR is a sensitive local technique of-fering a wealth of information gained via the radioactive decay of the probenuclei. The technique has made great contributions to the understanding ofnuclear structure of unstable nuclei [5, 6]. It has also become a powerful andsensitive tool in condensed matter physics and materials science for inves-tigating micro-structures, local magnetism, Knight shifts, spin relaxation,impurities and defects [7]. The technique is similar to NMR, but more sen-sitive; NMR needs in general ≈ 1018 probe nuclei, while β-NMR requiresonly 107 probe nuclei. The high sensitivity of β-NMR is due primarily tothe method of detecting the spin polarization through the properties of theradioactive decay of highly spin polarized short-lived nuclei. Once implantedinto the material, and during their lifetime, the probe nuclei are interactingwith internal and external magnetic fields within the sample. All these in-teractions can affect the spin polarization of the probe nuclei, which emitβ-particles with an anisotropic angular distribution that is correlated withthe nuclear spin direction. The polarization can be destroyed by applyingan RF field with a frequency matching the spin resonance condition, leadingto isotropic emission of β electrons. By placing two detectors facing eachother with the sample in between, one can measure the asymmetry of theemitted β electrons, therefore determining the polarization of the probe atthe moment of its decay, which is a function of the local environment. Inthe following I will describe the technique in more details.41.1. The β-NMR TechniqueIsotope I T1/2 γ Asymmetry Production(s) (MHz/T) rate (ions/s)µ+ 1/2 2.2×10−6 135.5 0.33 1048Li 2 0.84 6.013 0.33 10811Be 1/2 13.8 22 ∼0.2 10715O 1/2 122 10.8 0.7 108Table 1.1: Example of isotopes suitable for β-NMR compared to µ+. In-trinsic characteristics of nuclei are given by nuclear spin, I, half life T1/2,gyromagnetic ratio γ, and asymmetry. Production of rate µ+ used in Low-energy µSR, and of isotopes produced at TRIUMF and world-wide labs aregiven.1.1.1 Production of Spin-Polarized 8Li+A 500 MeV proton beam is produced at the 9 m radius TRIUMF cyclotron,and is used as a driver for the Isotope Separator and Accelerator (ISAC)facility. At ISAC, the high intensity proton beam strikesa specialproductiontarget, such as tantalum or SiC, which is heated to 2000 ◦C and is sittingon a high-voltage (HV) platform. The radioactive ions created by nuclearreactions diffuse out of the target with thermal energies and are acceleratedas they leave the HV platform. A high resolution mass separator is usedto select the isotope of interest. In β-NMR, any β emitting isotope is inprinciple usable, provided it meets some basic requirements such as: (i) highproduction rate and facile ionization, (ii) ability to have polarized nuclearspins, and (iii) high β-decay asymmetry and small nuclear spin for simplespectra [8]. 8Li+ meets many of these requirements (see Table 1.1) and iseasily produced at ISAC with high intensity (108 ions/s). The ions are thenaccelerated to 28 keV (± 1 eV) in a low energy beam transport (LEBT) lineand delivered to the polarizer.A schematic view of the polarizer and low and high-fieldβ-NMR stationsis given in Fig. 1.2-(a). a high nuclear spin polarization of 8Li+ is achievedby optical pumping with circularly polarized light from a single frequencyring dye laser (300 mW CW power). The first step in the procedure is toneutralize the ion beam by passing it through a Na vapor cell. The neutral51.1. The β-NMR Technique(a)(b)(c)Figure 1.2: (a) Layout of the polarizer and β-NMR spectrometers. (b)Polarization of 8Li+ using optical pumping. (c) Sodium cell scan shows thedependence of the asymmetry on Na cell bias. 61.1. The β-NMR Techniquebeam then drifts 1.9 m in the optical pumping region in the presence of asmall longitudinal magnetic field of 1 mT. The remaining charged fractionis removed electrostatically by two deflection plates and dumped onto ametallic cup known as Faraday cup.The wavelength of the circularly polarized laser is tuned to 671 nm corre-sponding to the D1 atomic transition of neutral 8Li; 2s2S1/2 →2p2P1/2. Thepolarization scheme is shown in Fig. 1.2-(b). The ground and first excitedatomic states are split by the hyperfine coupling between total (s = 1/2 elec-tron+nucleus I = 2) spin states F = 5/2, and 3/2. For circularly polarizedlight with positive helicity, only MiF +1 →MfF (∆MF = 1, MF is the totalatomic magnetic quantum number) transitions are allowed during excita-tion, whereas, the atom decays spontaneously with ∆MF = 1,0,−1. Thequantization axis is established by the helicity direction and maintained bya small magnetic field produced by coils co-axial with the beam line. Afterabout 10-20 cycles of absorption and emission, a high degree of electronicand nuclear spin polarization is achieved. About 70 % of the spins are po-larized in an atomic state F = 5/2, MF = 5/2 for the positive helicity andF = 5/2, MF = −5/2 for the negative helicity of the laser. The nuclear po-larization p ≡ (1/I)summationtextmpmm (m is the nuclear magnetic quantum numberm=±2,±1,0 for I = 2, and pm is the normalized occupation of sublevel m)is p ∼ 70 % for spins in a state m = 2 for the positive helicity, or m = −2for the negative helicity. The highest polarization is tuned by making smalladjustment of the ion energy via the Na cell bias voltage (Fig. 1.2-c), whichchanges the Doppler shift and brings the atom into resonance with the laserlight. Calibration using an unpolarized beam is easily carried out simply byblocking the laser.By passing the neutral beam through a He gas cell, the valence electronis stripped. The charged polarized beam exits the polarizer, and is thenpassed through 45o electrostatic bending elements and delivered to the β-NMR low- or high-field spectrometers. The remaining neutral fraction goesundeflected into a neutral beam monitor placed outside the polarizer. Topreserve the polarization only electrostatic elements are used in the beamline optics after the polarizer. The polarization of 8Li+ is parallel to the71.1. The β-NMR TechniqueStation Field and Detector Magnetic RF freq. EnergyPolarization positions Field (T) (MHz) (keV)High-field ⊥ to surface B, F 0.01 - 6.5 0.05 - 45 1 - 60Low-field bardbl to surface L, R 0.0 - 0.02 DC - 0.2 1 - 28Table 1.2: Summary of the characteristics of high and low-field spectrome-ters. Both spectrometers have an ultra high vacuum (UHV) cryostat with atemperature range ∼ 3-300 K, and are mounted on high voltage platformsallowing the variation of 8Li+ energy.momentum at the high-field spectrometer and perpendicular at the low-field station. The characteristics of both stations are summarized in Table1.2, and discussed next.1.1.2 The High-Field SpectrometerA schematic of the detectors in the high-field spectrometer is shown in Fig.1.3-(a), where the beam enters from the left, passing through a small hole inthe back (B) detector, and lands in the sample to be studied. The beam spoton the sample depends primarily on focusing Einzel lenses, three adjustablecollimating slits placed before the spectrometer, and on the applied magneticfield, and the beam energy at the spectrometer. Pictures taken with a chargecoupled device (CCD) camera show a beam spot of 2-4 mm in diameterdepending on the above conditions. As mentioned above, the polarizationis along the beam axis, which is also parallel to a magnetic field B0 = B0ˆz(100 G < B0 < 6.5 T) generated by a high homogeneity superconductingsolenoid. A small Helmoltz coil is used to apply a transverse radio-frequency(RF) field B1(t) (B1,max ∼ 1 G) at frequency ω in the horizontal direction,perpendicular to both the beam and initial polarization. See Appendix Afor more details about the type of RF field used in this thesis.Two plastic scintillators are placed in front and behind the sample todetect the outgoingβ electrons emitted after the 8Li+ β-decay. The emittedelectrons of an average energy 6 MeV can easily pass through thin stainlesssteel windows in the UHV chamber to reach the detectors. The B detectoris located outside the bore of the magnet, since the emitted electrons, when81.1. The β-NMR Technique(a) (b)Figure 1.3: Experimental setup used to measure the spin polarization of8Li+ in (a) the high-field, and (b) the low-field β-NMR spectrometers.inside, are confined to the magnet axis in high magnetic fields above about 1Tesla. Thefocusingeffectofthehighmagneticfieldleadstosimilardetectionefficiencies in both detectors although they have different solid angles asthe forward detector is closer to the sample. The sample is mounted on aUHV cold finger cryostat. UHV is critical to avoid the buildup of residualgases on the surface of the sample at low temperature. The pressure inthe main chamber can be reduced to 10−10 torr using differential pumping.Temperatures from 300 K to 3 K are obtained by cooling with cryogenicliquid He.1.1.3 The Low-Field SpectrometerThis spectrometer is complementary to the high-field one, and allows for lowfield measurements (up to 220 G) in a transverse geometry. A schematicof the the low-field spectrometer is shown in Fig. 1.3-(b). A set of threeorthogonal magnetic coils are wrapped around the sample chamber, wherethe main coil allows one to apply a static uniform magnetic field B0 =B0ˆywith 0 < B0 < 220 G, applied parallel to the initial polarization directionand parallel to the surface of the sample. The other two coils are used tocancel the x, z components of residual magnetic fields (see Appendix C).A small Helmoltz coil is used to apply an RF oscillating magnetic fieldB1(t)(B1,max ∼1G)atafrequencyωintheverticaldirection, perpendicular91.1. The β-NMR Technique0 100 200 300Depth (nm)Stopping  Distribution of 8 Li28 kev, BackS. ions=21%, mean=104 nm20 kev, BackS. ions=24%, mean=75 nm10 kev, BackS. ions=28%, mean=38 nm 1 kev, BackS. ions=40%, mean=5 nmFigure 1.4: TRIM simulation of the stopping distribution of 20000 ions of8Li+ implanted in a 1000 nm Ag layer. The energy of 8Li+ is varied from 1keV to 28 keV leading to mean depths from 5 nm to 104 nm. A significantfraction of 8Li+ is back scattered by the Ag surface.toboththebeamandinitialpolarization. ThesampleissittinginsideaUHVchamber at a pressure ∼ 10−10 torr. Four plastic scintillators measuring 10cm × 10 cm × 0.3 cm are placed outside the UHV chamber: two coincidencedetectors on the (R) right and two on the left (L) of the sample. The emittedβ electrons can easily pass through thin stainless steel windows in the UHVchamber and reach the detectors. A CCD camera placed outside the UHVchamber is used to image the beam spot (2-4 mm in diameter) centered onthe sample. The sample can be cooled from 300 K to 3 K. Currently, acryostat is under design to allow cooling with 3He to lower temperatures(T < 1 K), and to add an oven to reach higher temperatures above 300 K.101.1. The β-NMR Technique1.1.4 Implantation ProfilesThe most important aspect of the high and low-field spectrometers at TRI-UMF is the ability to decelerate the beam. Both spectrometers are mountedon separate HV platforms which are electrically isolated from ground. Theimplantation energy is controlled by adjusting the platform voltage bias.Using this setup, the energy of 8Li+ can be varied from 28 keV down to 1keV corresponding to implantation depths from 500 nm to 2 nm, allowingfor depth-resolved surface and interface studies.The profile of the implanted ions in materials can be simulated usingthe Stopping and Range of Ions in Matter (SRIM) or Transport of Ions inSolids (TRIM) codes introduced by Ziegler et al. [9, 10]. Both programs useMonte Carlo algorithms based on the binary collision approximation modelfor atomic collision processes in solids. The accuracy of both programs incalculating ion range distributions in various materials is well established,and they are routinely used in similar depth controlled experiments such asLow-Energy µSR [11]. By specifying the energy, charge, and mass of theprobe ions, and the mass density and atomic numbers of the elements ofthe probed material, one is able to simulate the implantation profile usingSRIM or TRIM, which both lead to comparable results. An example ofstopping profiles of 8Li+ in a thin Ag layer is given in Fig. 1.4, whichshows the characteristic asymmetric and positively skewed distribution of8Li+ ions into Ag. The mean depth in Fig. 1.4 increases from as low as 5nm to about 100 nm at full energy (28 keV). At low energy, a significantfraction of 8Li+ is back scattered from the sample and deposited mainly onthe thermal radiation shield surrounding the sample. This is mainly copperin the high-field spectrometer and aluminum in the low-field. The signalfrom 8Li+ stopping outside the sample does not affect the resonance, sincethese ions do not see the RF field. This background signal, however, mayslightly shift the baseline, i.e. the off-resonance asymmetry, which has noeffect on the resonance features that are presented in this thesis.111.1. The β-NMR Technique0 5 10 15E (Mev)N(E)(a) (b)θW(θ)8Li+a=−1/3Figure 1.5: (a) Angular distribution, W(θ), of the electrons emitted afterthe 8Li+ β-decay: when all energies are sampled with equal probability, theasymmetry parameter has the value a = −1/3. (b) Energy distribution ofthe emitted electrons after the 8Li+ decays.1.1.5 β-NMR Resonance Spectra8Li+ β-decays after a lifetime of τ = 1.209 s according to83Li →84Be+e− + ¯νe. (1.1)This decay occurs predominantly to an excited state in 84Be at approximately3 MeV rather than to the ground state [12]. The parity violation in weakinteractions leads to an angular distribution W of the emitted electrons,W(θ) = 1+apvc cosθ, (1.2)where v is the average velocity of the β electrons, c is the speed of light (ingeneral v ≈c), a= −13 is the asymmetry factor of the beta-decay of 83Li, p isthe nuclear polarization, andθ is the angle between theβ-emission directionand spin polarization axis (Fig. 1.5-(a)). Thus, the βs are preferentiallyemitted along the direction of the arrow of W(θ) in Fig. 1.5-(a). Thehighest probability is for electrons emitted opposite to the spin polarization121.1. The β-NMR Techniqueof 8Li+ nuclei at the moment of their decay. Theβ particles have an averageenergy of ∼ 6 MeV, with an end-point energy of Emax = 12.5 MeV (Fig.1.5-(b)). This energy is different from the Q-value of the β-decay because83Li decays, as mentioned above, mostly to an excited state in 84Be [12].The nuclear polarization has an initial value of p0 and evolves in timedue to interactions with the local environment. In two scintillators facingeach other at 180◦, the ratio, R, of the right detector counting rate NF tothat of the left detector NB yieldsR = NRNL= ǫRW(θ = 0◦)ǫLW(θ = 180◦) =ǫR(1+ap)ǫL(1−ap). (1.3)Here ǫR (ǫL) is the detection efficiency of the right (left) detector. If oneassumessimilardetectionefficienciesandsolidanglesofbothdetectors(ǫR ≈ǫL), one easily findsap = NR −NLNR +NL≡A(t). (1.4)This means that the final polarization, p, of 8Li+ just before it decays canbe deduced by measuring the asymmetry A(t) of the β rates as a functionof time.In our experiment, we place the two detectors either at the left/right(low-field station) or at the back/front (high-field station) of the samplesurface. The measured asymmetry is smaller than expected due to scatteringof the electrons in the sample, background count rates, and the geometricalinequalities between the two detectors. To minimize these instrumentaleffects, we measure the asymmetry for the positive (A+) and negative (A−)helicities of the laser, i.e. 8Li+ with nuclear spins polarized in a state ofm = 2 or −2. The final asymmetry is given bya(t) = A+(t)−A−(t). (1.5)To measure the spin resonance we use a similar method to NMR. InNMR, a small degree of nuclear spin polarization of some atoms is achievedby a large static external magnetic field B0. By applying a continuous131.1. The β-NMR Techniquewave (CW) RF field B1 of frequency ω perpendicular to B0, one can inducea transition between the spin states when ω equals the Larmor frequencyωL =γB0, 1 where γ is the gyromagnetic ratio of the polarized nuclei, thusdestroying the polarization. A pickup coil is used to detect the inducedvoltage from the ensemble’s nuclear spin polarization. In β-NMR, one ap-plies the same principle, except one measures the NMR signal using theasymmetry of the emitted electrons rather than a pick-up coil. One recordsthe β-decay asymmetry as a function of the RF frequency, while the con-tinuous beam of polarized 8Li is implanted into the sample. Two modes ofRF have been used in this thesis: a CW mode, and a pulsed mode. Moredetails about these modes are provided in Appendix A. When ω matchesthe Larmor frequency ω8Li = γ8LiBlocal, determined by the gyromagneticratio γ8Li = 0.63015 kHz/G and the local field Blocal, the 8Li spins precessabout Blocal, causing depolarization, thus a reduction in the measured av-erage asymmetry. Here, the resonances are conventionally plotted pointingdownward in contrast to NMR.The Larmor frequency is a local property determined by the appliedfield B0 and the internal magnetic field at the probe site. Thus, Blocalmay be distributed over a range of values, broadening the nuclear magneticresonance. The resonance offers insightful information about the magneticand electronic properties in the material. The linewidth of the resonancemeasures the inhomogeneities in the static magnetic field sensed by the 8Li.While, a relative shift of the resonance frequency indicates the presence of ahomogeneous static field causing paramagnetic enhancement or diamagneticreduction of the external field by the electrons surrounding the nucleus. Anexample of β-NMR resonance is given in Fig. 1.6.1.1.6 Comparison of β-NMR with NMR and µSRβ-NMR is similar to both NMR and µSR techniques; NMR is the oldestof these three. In all these nuclear techniques, one can measure the staticmagnetic field distribution and nuclear spin-relaxation. Since the distances1Zeeman energy is Emag = −µ·B0 = −mγB0, where µ = γI is the magnetic moment,I the nuclear spin, and m spin quantum number. The energy splitting is ∆Emag = γB0.141.1. The β-NMR Technique−0.2−0.15−0.1−0.050A=(NR−NL)/(NR+NL)94 95 96 97 98RF frequency ω (kHz)00.050.10.150.2a=A+ −A−Positive helicityNegative helicityFigure 1.6: (a) Asymmetry of the betas emitted after implanting a 28 keV8Li+ beam into a 120 nm Ag film. Both helicities are shown. (b) Thedifference of the asymmetries of the two helicities is plotted. The solid lineis a fit to a Lorentzian, and the dashed line refers to the Larmor frequency.The spectra are taken in B0 = 152.6 G, and T = 100 K.between the probe nuclei in a material are small relative to the source of thefield inhomogeneities, the magnetic field distribution is sampled by measur-ing the fields at the sites of the probe. As long as the probe is uniformlydistributed, the sampling is volume-weighted and the field distribution israndomly measured.There are limitations and difficulties associated with each technique.For example to measure the field distribution of the vortex lattice (VL) inthe bulk using NMR, magnetically aligned powders are often used due tothe limited sensitivity [13]. However, powders have strong pinning of thevortices at the crystallite surface. It is also challenging to obtain a completealignment of the crystallites. The penetration of the RF field also limits therange over which the VL can be sampled. Active NMR nuclei in HTSC suchas copper and oxygen have quadrupole moments and chemical shifts which151.1. The β-NMR Techniquecomplicate data analysis. 8Li+ nucleus of spinI = 2 is also quadrupolar andiscoupledtotheelectricfieldgradient(EFG).Thelatteriszerobysymmetryfor ions in cubic sites. Thus, the quadrupolar interaction is only significantwhen 8Li+ occupies sites of non-cubic symmetry. One can overcome thiscomplication using a probe with zero nuclear quadrupole moment (I = 1/2).Currently β-NMR with 11Be of I = 1/2 is under development.In a µSR experiment, positively charged muons are implanted one at atime into the sample. The positive muon has a spin 1/2, a mass that is206 times the mass of an electron, a gyromagnetic ratio γµ/2π = 135.5342MHz/T, and a magnetic moment of 4.84×10−3µB. The muons are naturally100% polarized due to parity violation. Once implanted into the sample,the muon spin precesses about the local magnetic field Blocal with a LarmorfrequencyωL = γµBlocal. (1.6)After a life of τµ = 2.2 µs, the positive muon β decays according toµ+ →e+ +νe + ¯νµ. (1.7)Similartoβ-NMR,thedistributionofthedecaypositronsisasymmetricwithrespect to the spin polarization of the muon, and the highest probability isalong the direction of the muon spin. Thus, the time evolution of the muonspin polarization is monitored by measuring the count rates in scintillatorsplaced around the sample. InµSR, however, one only measures the spectrumin the time domain, and a Fourier transform is needed to find the fielddistribution in the frequency domain. Measuring the spectrum in the timedomain is also an advantage though because one samples all frequenciesat once. Another difference between µSR and β-NMR is the time scale ofthe probe: 8Li+ decays after 1.2 s, and µ+ after 2.2 µs. Thus, β-NMR issensitive to spin relaxation processes on much longer times scales than µSR.The smaller beam spots in β-NMR (2-4 mm) compared to µSR (1-2 cm) isan advantage which allows one to study small samples. Conventional µSRis typically done with muons at a high implantation energy of 4.1 MeV,making it a bulk probe with a range ≈ 120 mg/cm2. Similar to β-NMR,161.2. Generic Properties of HTSClow-energy muons can be also produced in the lab; thus depth-controlledµSR is possible. Low-energy µSR (LE-µSR), developed at Paul Scherrerinstitute (PSI) [14], has a lower efficiency for achieving low-energy muons(104/s) compared to low-energy 8Li+ in β-NMR (108/s). Another differencebetween LE-µSR and β-NMR is the field range: LE-µSR is limited to fieldsbelow 2 kG, whileβ-NMR can be done with fields as high as 6.5 T. The abovetechniques have been very successful in studying HTSC. Some properties ofHTSC will discussed in the next section.1.2 Generic Properties of HTSCAfter the discovery of superconductivity in mercury in 1911, many othersuperconductors have been discovered (Pb, Nb...etc), but until 1973 thehighest Tc was only 23 K in Nb3Ge. In 1986, a new kind of superconductorswas discovered, when Bednorz and M¨uller detected superconductivity inLa2−xBaxCuO4 (LBCO) [15]. Rapidly within months, Chu et al. wereable to drive the initial transition temperature of 35 K up to 50 K usinghigh pressure [16]. One year later another high-Tc material, YBa2Cu3O7−δ,was discovered with a transition temperature of 90 K [17]. Since then, byvarying the pressure, crystal structure...etc, higher Tc’s have been achieved.Recently, in 2008, new HTSC materials were discovered, containing FeAsas the active layers rather than CuO2 [18]. A wide variety of FeAs-basedmaterials have been discovered in the last few months, and some hold strongsimilarities to the CuO2-based superconductors.1.2.1 Crystal StructureThe crystal structure of all cuprate HTSC can be viewed as a stacking ofCuO2 layers sandwiched between planes containing atoms like Cu, O, Ba,La...etc, yielding highly two-dimensional (2D) electronic properties. Themobile superconducting electrons reside in the CuO2 planes. In Fig. 1.7, thecrystal structures of YBa2Cu3O7−δ (YBCO) and Pr2−xCexCuO4−δ (PCCO)are shown (for δ = 0). YBCO has an orthorhombic unit cell with nearly171.2. Generic Properties of HTSCFigure 1.7: Typical crystal structure of YBa2Cu3O7 (left) andPr2−xCexCuO4 (right).equal in-plane lattice constants a≈b, i.e. it is almost tetragonal, with twoCuO2 planes per unit cell. The CuO1−δ plane involves Cu-O chains alongthe b direction, where the oxygen is linear with Cu(1) atoms. It is themissing oxygen between two Cu(1) atoms along the a direction that leads toan orthorhombic distortion of the crystal structure [19]. Forδ = 1, the Cu-Ochains are fully depleted of oxygen and the material becomes an insulatingantiferromagnet. PCCO contains one CuO2 per unit cell, and has the so-called T’ tetragonal structure, where the oxygen environment of each Cuatom is in the form of a planar square. In all cuprates, the non-CuO2 planesare called charge reservoir planes, which capture or give away electrons fromor to the CuO2 planes upon doping [20]. Since it is possible to change thenumber of copper planes per unit cell and the composition of the layeredplanes by doping, a large number of compounds has been discovered [21, 22].181.2. Generic Properties of HTSCSymbol Atom Atom No. of Valence Ion Ionnumber Configuration electrons configurationO 8 [He]2s22p4 4 O1− 2p5O2− 2p6Cu 29 [Ar]3d104s1 11 Cu1+ 3d10Cu2+ 3d9Cu3+ 3d8Sr 38 [Kr]5s2 2 Sr2+ −Y 39 [Kr]4d15s2 3 Y3+ −Ba 56 [Xe]6s2 2 Ba2+ −La 57 [Xe]5d16s2 3 La3+ −Ce 58 [Xe]4f5d6s2 4 Ce4+ −Pr 59 [Xe]4f36s2 5 Pr3+ 6s2Table 1.3: Electron configurations of common atoms in cuprate HTSC.1.2.2 Electronic ConfigurationThe electronic configuration of some common atoms in cuprates is givenin Table 1.3. The most common atom, Cu, has an unusual oxidationstate, and assumes a mixed valence state. For example, in YBCO, as-suming that Y, Ba, and O have states of 3+, 2+, and 2−, respectively,then for neutrality, Cu must be an average of 2.33+, which could be under-stood as a mixture of 3+ and 2+ states. The generic oxidation states ofthe layers in the superconducting YBa2Cu3O7 compound is (CuO)+ (BaO)(CuO)2−Y3+(CuO)2−(BaO) with zero net charge per unit cell. The Cu(2)atoms in Cu-O chains appear to be the most oxidized (Cu3+), whereas Cu2+are primarily involved in the superconducting CuO2 planes. Thus, the Cu(2)atoms are regarded as a “sink” which can accept electrons from the super-conducting planes in a charge transfer process [23]. It is, however, debatablethat instead of oxidation state Cu3+ of Cu(2), some of the oxide ions areoxidized to O− rather than O2−. Upon reduction of the oxygen in Cu-Ochains, the average oxidation state of Cu is reduced and consequently re-ducing or destroying superconductivity. The parent non-superconductingYBa2Cu3O6 compound (of a tetragonal structure) has no oxygen in the Cu-O chains and the Cu(2) is unaffected (Cu2+), whereas Cu(1) is reduced from191.2. Generic Properties of HTSCFigure 1.8: The typical doping phase diagrams of hole (right) and electron-doped (left) high-Tc cuprates are shown. The various phases are explainedin the text.Cu3+ to Cu+.1.2.3 Doping Phase DiagramThe metallic and superconducting states in most cuprate superconductorsare achieved by altering the chemical composition of the parent compounds.For example, in La2−xSrxCuO4−δ, one replaces La3+ with Sr2+ (cation sub-stitution), leading to the loss of one electron by a CuO2 layer creating moreholes in the CuO2 layers. This is known as hole-doping, which may also bedone by oxygen intercalation (in the case of YBCO) or by a combinationof these [24]. Superconductivity in PCCO is achieved by substituting triva-lent Pr atom by tetravalent Ce, which dopes more electrons into the CuO2planes. This is known as electron-doping.The typical phase diagram of HTSC’s is shown in Fig. 1.8. Near zerodoping, the material is an antiferromagnetic (AFM) insulator, where the201.3. Theoretical Foundationsspins are arranged anti-parallel to one another on adjacent copper atomsin the CuO2 layers. In hole-doped materials, by increasing the doping,the AFM ordering is destroyed. Below Tc and for adequate doping, thesuperconducting (SC) state is achieved. The doping level that yields themaximum Tc for a given HTSC material is called optimal doping (po). Theregion below (above) po is called the underdoped (overdoped) region. In theunderdoped (UD) phase with p<po, the material is in a poor metallic stateshowing strong 2D anisotropy. This phase is characterized by an intriguingfeature of the HTSC’s, namely the so-called pseudogap state (PG), wherea partial gap opens up in the excitation spectrum below a characteristictemperature T∗ [25]. The normal-state of the strongly overdoped regimeis believed to be more or less a “Fermi liquid” (FL), i.e. a conventionalmetallic state. The applicability of the FL theory is questionable in theregion close to the optimal doping where the so-called marginal FL theoryseems more appropriate [26]. The phase diagram of the electron-doped sideof Fig. 1.8 looks similar to hole doping, however the AFM long-range orderis robust up to higher doping than the hole-doped side, and the maximumTc is substantially lower.There is great interest in the region near the underdoped-overdopedboundary, which is characterized by sharp maxima in a variety of propertiesnear zero temperature [27]. It has been suggested that this is a quantumcritical point (QCP) where the quantum critical fluctuations are the originof strong superconducting pairing and unusual normal state properties inthis region such as the PG state [26]. Below the proposed QCP, the sys-tem is believed to be in a state of co-existence superconductivity with othercompeting phases like AFM or PG below Tc, and in a PG state above [27].1.3 Theoretical FoundationsThe search for HTSC’s has been and is still empirical, since there is no pre-dictive theory for this type of superconductivity [20]. The famous Bardeen-Cooper-Schrieffer (BCS) [28] theory of conventional superconductivity doesnot explain the physics of HTSC as it failed to predict their high Tc’s. In211.3. Theoretical Foundationsaddition to BCS, London [29], and Ginzburg-Landau [30] theories are widelyused to discuss the physics of HTSC. These theories will be reviewed in thissection.1.3.1 London TheoryThe application of a time-dependent magnetic field to a superconductor gen-erates an electric field. In an ordinary metal this creates the eddy currents;whereas, in a superconductor persistent currents are established [21]. Thecurrents in turn generate a magnetic field of their own, which opposes theapplied field. Using Newton’s law, mdvdt = −eE, the induced current densityJ = −ensv obeys the equation,dJdt =nse2m E, (1.8)wherens isthedensityofconductionelectrons, andmandearethemassandthe charge of an electron, respectively. Using Faraday’s law ∇×E = −1c ∂B∂t ,and Maxwell’s equations ∇×B = 4pic J (cis the speed of light) and ∇·B = 0,Eq. (1.8) leads toddtbracketleftBig∇2B− 4πnse2mc2 BbracketrightBig= 0. (1.9)Initially B = 0, hence the field inside the material would remain zero whenan applied field is turned on. The induced magnetization can be obtainedvia B = µ0H + µ0M = µ0(H + χH) = 0, with µ0 being the magneticpermeability of free space. Therefore, in addition to its property as a perfectconductor, a superconductor is also a perfect diamagnet (χ = −1), since theinduced magnetization completely cancels the applied field.F. London and H. London explained the Meissner effect by proposingthat the term between brackets in Eq. (1.9) must vanish [21]. This leads tothe London equation∇2B− Bλ2 = 0, (1.10)221.3. Theoretical Foundationswhere λ is the London penetration depthλ = ( mc24πnse2)1/2. (1.11)For an external field B0 applied parallel to the z-axis, and to the surface ofa superconductor occupying half space (z > 0), the London equation has asolution of the formBz = B0e−z/λ. (1.12)The decrease of the magnetic induction inside the sample in Eq. (1.12) is aresult of the screening of the external field by the superconducting currentsgiven byJy =cB0e−z/λ/4πλ, (1.13)which flow within a surface layer of thickness λ.The empirical temperature dependence of λ can be easily found usingthe phenomenological two-fluid model. One can approximate the free energyof the conduction electrons as [22],F(x,T) = √xfn(T)+(1−x)fs(T), (1.14)which depends on the fraction x = nn/n of the density of the normal stateelectrons (nn) to the total density of electrons (n = nn + ns). The freeenergy of normal electrons is fn(T) ∝ −T2/2, and the superconductingcondensation energyfs is a constant belowTc and zero above. Minimizationof the total free energy with respect to x for fixed T yieldsx= nn/n = (T/Tc)4. (1.15)Using Eqs. (1.11) and (1.15), one finds the temperature dependence of thepenetration depth,λ(T) = λ(0)[1−(T/Tc)4]1/2. (1.16)Reasonable agreement of measuredλ(T) with the above form is seen in someconventional superconductors like lead [31], although deviations are found231.3. Theoretical Foundationsin other materials especially HTSC.1.3.2 Ginzburg-Landau TheoryGinzburg and Landau introduced a pseudo-wavefunction ψ(r) = |ψ(r)|eiφ,which is a complex order parameter of phaseφ, to represent the local densityof the superconducting electrons ns(r) = |ψ(r)|2. The free energy of thequantum state proposed by Ginzburg-Landau (GL) theory in 1951 takes onthe formF = Fn +α|ψ(r)|2 + β2|ψ(r)|4 + 12mvextendsinglevextendsinglevextendsingleparenleftbigg−i¯h∇− 2eAcparenrightbiggψvextendsinglevextendsinglevextendsingle2 + B28π, (1.17)where Fn is the free energy of the normal state, A the vector potentialrelated to the external field B = ∇× A, and B28pi is the magnetic energydensity. By minimizing this free energy with respect to ψ and A, it is foundthat the parameter α must change sign at Tc, while β assumes a constantvalue. The GL equations determining the spatial variations of the orderparameter and the vector potential are given byξ2bracketleftBig∇+ 2πiΦ0AbracketrightBigφ+parenleftBigg1− β|Ψ|2|α|parenrightBiggψ = 0, (1.18)λ2 βα|ψ|2∇×∇×A+A = −Φ02π∇φ. (1.19)The length scales entering these equations are the London penetration depthλ(T) (Eq. (1.11)) and the coherence length ξ(T),ξ2(T) = ¯h22mα = ξ20(1−T/Tc)−1, (1.20)λ2(T) = mc2β16πe2|α|2 =λ20(1−T/Tc)−1, (1.21)determining the length scale of variations in the order parameter and vectorpotential, respectively. The GL results only apply very close to Tc where ψ241.3. Theoretical Foundationsλ λξ ξB Bψ ψκ>>1 κ<<1Figure 1.9: Idealized spatial variation of the magnetic field B and orderparameter ψ for κ≪ 1 (type I) and κ≫ 1 (type II) superconductors.is small. The dimension-less Ginzburg-Landau parameter is defined byκ = λ(T)ξ(T). (1.22)The boundary between the normal and superconducting region involves asurface energy. A crossover from positive to negative surface energy occursfor κ = 1/√2. For κ>1/√2 the negative surface energy causes penetrationof quantized vortices in type II superconductors, whereas κ < 1/√2 intype I superconductors. The spatial variations of the order parameter andmagnetic field inside type I and II superconductors is shown in Fig. 1.9.The GL equations are only applicable near Tc as the order parameter inthat region is weak with small variations in space. However if one ignores thespatial variation of the order parameter, one can construct an approximatefree energy valid at all temperatures from the magnetic density and kineticenergy of the currents,F = 18πparenleftBigB2 +λ2(∇×B)2parenrightBig. (1.23)Variation of this free energy with respect to B leads to the London equationgiven in Eq. (1.10). Gor’kov showed that the Ginzburg-Landau theory couldbe also derived from BCS theory, and leads to similar results close toTc [32].251.3. Theoretical Foundations1.3.3 BCS TheoryThe superconducting state in conventional superconductors has been ex-plained microscopically by the BCS theory [28]. The latter is based onan effective attractive interaction between electrons induced by phonon ex-change. This attraction dominates the repulsive Coulomb interaction forelectrons at the Fermi level. In the superconducting ground state, the elec-trons are virtually excited in pairs of electrons with opposite spin and mo-mentum known as Cooper pairs. The BCS theory uses a many-body theoryto construct an explicit wave-function for the ground state, which is thenused to calculate different quantities. The BCS Hamiltonian takes on theform [28]H =summationdisplaypσǫpc†pσcpσ + 12 summationdisplayqpp′ss′V(q)c†p+q,sc†p′−q,s′cp′,s′cp,s. (1.24)Here, ǫp = p22m is the single-electron kinetic energy and cp,σ (c†pσ) is thecreation (annihilation) operator of an electron of momentum p and spin σ.The attractive interaction V(q) between electrons in the neighborhood ofthe Fermi surface is essential to establish the formation of Cooper pairs.Solving the BCS Hamiltonian leads to the energy spectrum Ep =radicalBigǫ2p +∆2where ∆ is the energy gap with a spherical symmetry (s-wave symmetry).The BCS theory successfully predicts a transition temperature of theformkBTc = 1.14¯hωc exp(− 1NFV0), (1.25)where NF is the density of states near the Fermi level. The phonon cutofffrequency ωc is related to the Debye frequency ωD ∝ 1/√M, where M isthe atomic mass, which explains why Tc ∝ 1/√M, i.e. the isotope effect.The energy gap in the weak limit ∆ ≪ ¯hωD is found to be proportional toTc and is given by the ratio2∆(0)kBTc ≈ 3.5. (1.26)261.4. Time-Reversal Symmetry Breaking in HTSCThe temperature dependence of the gap is given implicitly by∆(T) = ∆(0)ef(∆(T)/kBT), (1.27)where f is a universal function of the ratio ∆(T)/kBT. Near Tc,∆(T) = 1.74∆(0)(1−T/Tc)1/2. (1.28)The penetration depth in BCS at low temperatures followsλ(T)−λ(0)λ(0) =radicalBigπ∆(0)/2kBTe−∆(0)/kBT. (1.29)The BCS gap function also provides the microscopic interpretation ofthe Ginzburg-Landau order parameter. Gor’kov showed that the BCS gapfunction ∆ is simply proportional to the Ginzburg-Landau parameter ψψ ∝ ∆. (1.30)This is somewhat expected since both ψ and ∆ are complex functions, andboth vanish above Tc. Below Tc, when the gap opens up at the Fermi level,the order parameter becomes non-zero. Thus, these two parameters arerelated to each and have the same symmetry, and both reflect the symmetryof the pairing interaction. The latter, in HTSC, is now widely accepted tobe of a dx2−y2-wave (d-wave) nature in contrast to the s-wave symmetryof conventional superconductors like Nb. However, near the surface anddefects, order parameters of other symmetries may compete with the d-wave symmetry leading, to co-existence of more than one order parameterthat could potentially break time-reversal symmetry (TRS) [1]. This will bediscussed in the next section.1.4 Time-Reversal Symmetry Breaking in HTSCUnder time-reversal one transforms time t, spin S, and momentum p to−t, −S, −p, respectively. Time-reversal symmetry also transforms an order271.4. Time-Reversal Symmetry Breaking in HTSCparameter ψ = |ψ|eiφ to its complex conjugateT[ψ] = ψ∗ = |ψ|e−iφ. (1.31)This symmetry is considered broken if the state under this operation isdifferent from the original one (not only by a phase factor). For example,for a combination of two order parameters ψ+ψ′, TRS is broken ifT[ψ+ψ′] negationslash= eiγ(ψ+ψ′), for γ ≡φ−φ′ negationslash= 0,π. (1.32)Thus, the order parameterψ+ψ′ and its time inverseψ∗+ψ′∗ are degenerate,i.e. they have the same free energy.1.4.1 Which Order Parameters Break TRS?There are many allowed symmetries of the pairing state of superconductivity[33]. The amplitude and phase of the leading candidates are plotted in Fig.1.10. The simplest is the isotropics-wave pairing state which occurs in mostconventional superconductors, where the order parameter is independent ofthe wave-vector k and has a spherical symmetry,∆(k) = ∆0. (1.33)In real metals, the crystal structure leads to a small anisotropy of theisotropic s-wave, giving rise to an anisotropic s-wave. In HTSC, the d-wavesymmetry is now well established, with a gap function∆(k) = ∆0[cos(kya)−cos(kya)]. (1.34)Here ∆0 is the maximum gap value and a is the in-plane lattice constant.This gap has strong anisotropic magnitude with nodes (∆(k) = 0) along the(110) direction in k space and a sign change in the order parameter betweenthe lobes in the kx and ky directions. Physically, this sign change indicatesa relative phase of π in the superconducting condensate wave function forCooper pairs with orthogonal relative momenta. In the cuprates, this state is281.4. Time-Reversal Symmetry Breaking in HTSCbelieved to describe the order parameter in the CuO2 planes, with the lobesbeing aligned with the in-plane lattice vectors a and b. Another candidatesymmetry is the dxy-wave which is similar to the d-wave but rotated by 45o,and has the gap function∆(k) = ∆0 sin(kya)sin(kya). (1.35)A variety of experiments have been performed to determine the symme-try of the order parameter. One class of experiments considers the propertiesof the quasi-particle excitations in the superconducting state which modi-fies the low temperature dependence of various thermodynamic quantitiessuch as the penetration depth. The microwave measurements of the tem-perature dependence of the penetration depth were the first to confirm thed-wave nature in YBCO crystals [34]. Later angle resolved photo emissionspectroscopy (ARPES) [35], NMR [36], Raman scattering [37], µSR [38],neutron scattering [39], and other experiments have all reached the sameconclusion. Another class of experiments probes the pairing symmetry bystudying the relative phase of the gap between different points on the Fermisurface. Studies of the magnetic flux modulation of DC superconductingquantum interference devices (SQUID) provided direct evidence of d-wavefrom the π shift between pairs tunneling along the a and b directions [40].All conventional superconductors conserve time-reversal symmetry be-cause of the single isotropic s-wave pairing symmetry. In the bulk, HTSC,also appears to conserve TRS due to the single component d-wave symme-try. However, near surfaces, interfaces, vortices, impurities, or structuraldefects, local pair breaking effects can suppress the d-wave order parameterand lead the way for an order parameter of different symmetry, likedxy-waveor s-wave. The new pairing component together with the dominant d-wavecomponent can form a complex order parameter. Complex mixtures of theabove symmetries namely the d+is and the d+idxy states are respectivelygiven by∆(k) = ∆0braceleftBig(1−ǫ)[cos(kya)−cos(kya)]+iǫbracerightBig, (1.36)291.4. Time-Reversal Symmetry Breaking in HTSC0 90 180 270 360θ0101010101Magnitude |∆|0 90 180 270 360θ00000Relative phase φpipipipipiFigure 1.10: Normalized amplitude and phase of the main candidate or-der parameters in HTSC are shown. The order parameter is written as∆ = |∆|eiφ = |∆|g(θ). Here θ is a polar angle in momentum spacek =k(cosθ,sinθ); g(θ) = cos(2θ+α), where θ = 0 for s-wave and finite ford- and dxy-wave; α = 0 (π/2) for s-wave and d-wave (dxy-wave).301.4. Time-Reversal Symmetry Breaking in HTSC∆(k) = ∆0braceleftBig(1−ǫ)[cos(kya)−cos(kya)]+iǫsin(kya)sin(kya)bracerightBig, (1.37)where ǫ is the fraction of the sub-component s or dxy. The ferromagneticstates d+ idxy can directly couple to an external magnetic field via its netorbital momentum or spin [41, 42].These complex order parameters represent a TRS broken state, and arealso associated with the opening of a gap in the quasi-particle density ofstates which lowers the local free energy density [1]. A direct evidence ofa TRSB state is the presence of spontaneous supercurrents and magneticfields if the Cooper pairs carry a finite angular momentum of either the spinor orbital (or both) parts of the pair wave-function [1]. TRSB superconduc-tors must still exhibit the Meissner effect where the compensating screeningcurrents are set up to ensure thatB = 0 in the bulk of the sample. Althoughthese effects mean that no large bulk magnetic moment is to be expected, thesample will always contain surfaces and defects where the Meissner screen-ing of the TRSB fields is not perfect, and a small magnetic signal is thusexpected locally. Other features include unusual tunneling spectra and theappearance of fractional vortices [1]. The occurrence of TRSB in HTSCis still controversial as different studies have lead to contradictory results.The findings of a few leading experimental and theoretical studies will besummarized in the rest of this section. Due to the large number of paperspublished about this topic, I will only discuss those most relevant to theexperiments described in Chapter 2.1.4.2 TheoriesLet me briefly review some of the leading theories that addressed the possibleoccurrence of TRSB in HTSC. Several theories have been formulated, butone can classify most of them into those that: (i) considered the effectof intrinsic and extrinsic structural defects such as grain boundaries, twinboundaries, Josephson junctions, and domains walls, (ii) proposed a locallybroken TRS near the interface of d-wave superconductors, and (iii) studiedthe effect of magnetic impurities on the order parameter which may induce311.4. Time-Reversal Symmetry Breaking in HTSCspontaneous magnetic moments in the vicinity of the impurities.Structural DefectsSigrist and co-workers have studied the current and field distribution neargrain boundary junctions between two superconductors with different orderparameters on each side of the boundary [43, 44, 45]. Their calculations,in the framework of GL theory, showed the appearance of fractional vor-tices at the boundary below temperatures T∗ ≪ Tc. This is generated bycircular currents flowing in the x-y plane which cancel in the homogeneoussuperconducting phase but survive in the inhomogeneous region near theinterface [45]. The magnetic field generated by these currents has a peaknear the interface with a width of the order of the coherence length. Awayfrom the interface, the field changes sign and decays on the length scale ofλ. It was shown that the positive and negative parts of the field cancel eachother leading to zero net magnetization [45]. Amin et al. studied the currentand field distribution in systems with grain boundaries involving a mixtureof s and d-wave symmetries by numerically solving the self-consistent quasi-classical Eilenberger equations [46]. The magnitude of the calculated fieldis of the order of a mG, and decays on the length scale of few coherencelengths. The total flux generated by these TRSB states is thus very smalland difficult to detect.Several papers considered the role of twin boundaries [47, 48, 49, 50,51]. Feder et al. used the Bogoliubov-deGennes formalism to study TRSBnear the twin boundaries in d-wave superconductors [47]. They assumedan induced s-wave pairing potential below a temperature T∗ which dependsstrongly on the electron density in the twin boundary. This leads to signsof TRSB below T∗ ∼ 0.1Tc. Belzig et al. used a quasi-classical formalism ofsuperconductivity to study the effect of twin boundaries on the electronicstructure [49]. They found that at low temperatures (∼ 5 K), a localizedTRSB state near the twin boundary appears, and spontaneous currents flowparallel to the twin boundary. In Ref. [48], it was found that an admixture ofs- and d-wave near the twin boundaries due to the orthorhombic distortion321.4. Time-Reversal Symmetry Breaking in HTSCin materials such as YBa2Cu3O7−δ may lead to spontaneous fields of theorder 5 − 50 G which decay on the length scale of the twin inter-spacing.Yang and Hu also suggested in Ref. [52], that a small s-wave componentmay arise due to the orthorhombic distortion in YBCO and may lead toTRSB.Quasi-Particle Reflection from the SurfaceC.-R. Hu was the first to suggest that a quasi-particle reflecting from a sur-face of a d-wave superconductor, with nodes of the order parameter beingnormal to the sample surface, experiences a sign change in the order param-eter leading to zero energy states (ZES) at the surface [53]. These states arealso expected if the superconductor is coated with a normal metal, and areabsent in s-wave superconductors as well as at the (100) surface of a d-wavesuperconductor [52]. The consequence of the ZES is an increase in the lo-cal density of states at the Fermi level at the surface, resulting in zero-biasconductance peak (ZBCP) anomaly observed in tunneling spectroscopy ofHTSC [54]. Under the influence of an applied magnetic field or a strongspontaneous field, the spectrum of surface states acquires a Doppler shiftwith energy given by vf ·ps, where the vf is the Fermi velocity, and ps isthe momentum of the bound states which depends on the local magneticfield. This leads to a splitting of the ZBCP into two states above and belowthe Fermi level [56, 57, 58]. In this way, the quasiparticle contribution tothe free energy can be lowered, with the appearance of TRSB spontaneouscurrents near the surface. The splitting of the zero-energy level leads to animbalance in the occupation between electrons states with momentum com-ponents parallel and antiparallel to (1,¯1,0). Thus, there is a finite currentalong the surface whose direction depends on which of the two degeneratetime-reversal symmetry states is realized.Fogelstr¨om, Rainer, and Sauls extended Hu’s work and showed that alarge ZBCP is possible for all orientations if the interface is microscopicallyrough [55]. The authors found that surface states are induced in (100) as aresult of Andreev scattering by the rough surface. Their calculations for a331.4. Time-Reversal Symmetry Breaking in HTSC(110) surface showed the current density approaches the London limit (seeEq. (1.13)) for ξ ≪ x ≪ λ. At x ≤ ξ, a current is carried by the ZESand counter-flows relative to the Meissner currents. This ZES current scaleslinearly with the applied field and leads to the splitting of the ZBCP. TheZES lead to pair breaking and the electrons may paired by a sub-dominantpairing channel, resulting in a spontaneous current which splits the ZBCPeven in zero external field below a transition temperature Ts. This sponta-neous current is confined within few coherence lengths of the surface. TheZES splitting varies non-linearly with increasing field, and saturates at fieldsof the order of 3 T. The authors also indicated that surface roughness sup-pressesTs, and also decreases the intensity of the ZBCP aboveTs. The sameconclusion was reached by Asano and Tanaka in Ref. [59], when consideringthe interface between a normal metal and a d-wave superconductor.Near the surface, one may also consider the role of Abrikosov vorticeswhich generate an essentially inhomogeneous superfluid velocity field, whichleads to a nontrivial electronic structure of the surface-bound state [60, 61].TRSB States Induced by ImpuritiesLocalspontaneouscurrentscanbealsoinducedbymagneticandnon-magneticimpurities. Asano et al. found that the scattering by impurities near theinterface of a normal metal and a (110) d-wave superconductor splits theZBCP in the same way that a d+is order parameter does [59, 62]. Thesplitting disappears at sufficiently high temperature and increases with ex-ternal applied fields. In the bulk, Balatsky suggested that a TRSB complexorder parameter is “generated around a magnetic impurity in the presenceof coupling between the orbital moment of the condensate and impurity spinSz” [63]. He argues that a sub-dominant order dxy develops simultaneouslywith the impurity spins. He also noted that a magnetic field applied parallelto the layers H ≫ Bc1,ab ∼ 1 G suppresses the dxy order as the couplingbetween the impurities and dxy vanishes.Okuno has studied the effect of impurities in a TRSB superconductor,and found that a spontaneous current can be induced near the impurities341.4. Time-Reversal Symmetry Breaking in HTSCwith patterns that reflect the nature of the pairing channels and vanishes afew lattice constants from the impurity [42]. Choi and Muzikar have alsodrawn a similar conclusion where the magnetic impurities strongly perturbthe order parameter on a length scale of ξ, but that leads to zero net mag-netization [64].1.4.3 ExperimentsµSRFigure 1.11: Left: the µSR relaxation rate in Sr2RuO4 showing the ap-pearance of spontaneous magnetic fields below Tc [3]. Right: absence ofspontaneous magnetic fields in YBCO (Tc=90 K) and BSCCO (Tc=85 K)systems [65]. In µSR, the relaxation rate is a measure of the spread of theinternal fields.The most direct consequence of TRSB would be the appearance of spon-taneous magnetic fields, which can be resolved by a sensitive local magneticprobe such as µSR. TRSB violated throughout the whole material is clearlyobserved in some unconventional superconductors such as heavy Fermion351.4. Time-Reversal Symmetry Breaking in HTSCsuperconductors U1−xThxBe13 [2], and Ru-based superconductors Sr2RuO4[3], and PrOs4RuSb12 [4]. However, bulk studies of HTSC showed little ev-idence of spontaneous magnetic fields. Kiefl et al. studied YBCO powdersand Bi2Sr2Cu2O8+δ (BSCCO) thick films using µSR and observed internalmagnetic fields consistent with the nuclear dipolar fields with no signs ofTRSB fields below Tc [65]. The authors concluded that the spontaneousfields due to TRSB, if existent, are less than 0.8 G.Contrary to Kiefl’s finding, Sonier et al. measured weak magnetism inYBCO single crystals using zero field µSR, with the onset being well belowTc for optimally doped and above Tc for underdoped samples [66]. Thiseffect may be due to TRSB above Tc as proposed by Varma [67], or theCu-O chains in YBCO [68, 69]. Recent similar studies of single crystalsLa2−xSrxCuO4−δ of different doping using zero-fieldµSR have concluded theabsence of measurable strong TRSB fields above Tc in both the underdopedand overdoped regimes in contradiction to Varma’s predictions [70].Tunneling SpectroscopySpontaneous splitting of the ZBCP, in zero magnetic field, was seen by Cov-ington et al. below 7 K in ab-oriented YBCO optimally doped thin films byplanar tunneling spectroscopy as shown in Fig. 1.12 [56]. External appliedmagneticfieldsparalleltothesurfaceinducestheZBCPtosplitmorebeyondits zero field value, varying non-linearly with increasing field. This splittingof ZBCP was attributed to a transition to a state of broken time-reversalsymmetry with a d+is order parameter. Deutscher and collaborators haveshown that such splitting takes place at 4.2 K only beyond some criticaldoping level, close to optimal, and increases with strong magnetic field (1-5T) applied parallel to the surface of the (110)-oriented films [71, 72]. Thesame effect was also observed by Sharoni et al. [73] below a critical tem-perature ≈ 10 K in optimally doped and overdoped thin (110) YBCO filmsand absent in underdoped films. Deutscher and collaborators measured theAndreev reflections between Au STM (scanning tunneling microscopy) tipand Ca doped YBCO thin films, and found a conductance spectra in (100)361.4. Time-Reversal Symmetry Breaking in HTSCFigure 1.12: Top: conductance in YBCO/Cu tunnel junction showing split-ting of the ZBCP as a function of external field [56]. Bottom: absence ofthe splitting of the ZBCP in the normalized conductance spectra taken onthe (110) surface of YBCO with a Pt-Ir tip at 4.2 K for an STM tunneljunction (main panel) and for a point contact (left inset). Right inset showsthe expected curves from d+s and d+is symmetries. [54]. 371.4. Time-Reversal Symmetry Breaking in HTSCbest fitted with d+is complex order parameter, with a ratio of the energygaps ∆d/∆s ∼ 0.1−0.8 [74].Other tunneling studies have led to different conclusions. In tunnelingexperiments on Ag/YBCO junctions, a strong ZBCP was observed in (110)junctions with no splitting of the ZBCP in external fields up to 5 T [76].Similarly, directional tunneling and point contact spectroscopy on (001),(100), and (110) faces of YBCO single crystal done by Wei et al. showed ano splitting of the ZBCP in (110) and (100) as shown in Fig. 1.12. Quan-titative spectral analysis done by the authors suggested a dominant d-wavesymmetry with less than 5% s-wave in mixed d+s or d+is [54]. Yeh et al.scanning tunneling measurements of a YBCO single crystal showed signs ofZBCP splitting only in Ca, Zn, Mg doped samples while absent in underand optimally doped samples along the (110) and (100) directions [77].SQUID MicroscopyMacroscopic magnetometry is not expected to yield a signal from TRSB andwill be dominated by the Meissner signal of the bulk. In certain geometries,sensitive SQUID magnetometers can be used to seek fields near surfaces.Carmi et al. have measured a spontaneous weak magnetic field (≈ 30 ×10−6 G) picked up by a SQUID near c-axis oriented YBCO epitaxial thinfilms [78]. By cooling the films in zero field, a small magnetic flux appearsabruptly at Tc with weak temperature dependence below as shown in Fig.1.13-(a). This signal was found to be thickness and orientation independent,and did not change appreciably if measured at the edges of the sample.Tafuri and co-workers have also detected spontaneous magnetization us-ing scanning SQUID microscopy near c-axis films [80]. Using a square pickup coil a few microns across, images of flux were taken near (001)/(103)grain boundaries junctions. A broader flux distribution was imaged in (001)but absent in (103) as seen in Fig. 1.13-(b). The total flux appears to be afraction of the superconducting flux quanta. Spontaneous magnetic flux wasalso measured near asymmetric 45o grain boundary of c-axis YBCO filmsin zero field by Mannhart et al. [79]. The authors attributed the result to381.4. Time-Reversal Symmetry Breaking in HTSC(a) (b)(c)Figure 1.13: (a) Spontaneous magnetic field generated by a thin YBCO filmplotted versus temperature, as measured by Carmi et al. [78]. (b) ScanningSQUID microscope image of an area with a 45o asymmetric c-axis YBCObicrystal grain boundary of 180 nm thickness. The flux through the grainboundary is shown [79]. (c) Scanning SQUID images of (100)/(103) YBCOcooled in 3mG. The corresponding flux distribution is shown. A broaderfield distribution is observed in (001), although it has the same average fluxas in (103) [80].391.5. Vortex Lattice in HTSCfluctuations of the d-wave order parameter through the faceting of the grainboundaries, over a length scale of few microns.The TRSB superconducting state may also exist in the vortex phase.However, the magnetic field inhomogeneities in this phase will be dominantat high fields. Next, we discuss the response of the cuprates to high appliedmagnetic fields above Bc1.1.5 Vortex Lattice in HTSCIncreasing the external field beyond Bc1, the formation of vortices becomesenergetically favorable and we enter the vortex state. Four energies competein this state and determine the arrangement of the vortices [81]. (i) The re-pulsive interaction between the vortices leads to a three-dimensional (3D)lattice where the vortices are straight lines arranged in a triangular lattice.(ii) Vortex pinning by defects may lead to a random amorphous glass state.(iii) The CuO2 planes decoupling modifies the 3D lattice to 2D or pancakevortices. (iv) The thermal fluctuations compete with the tendency to formthe lattice and cause thermally activated depinning of the vortices and melt-ing the VL to a liquid phase. The energies from (i) to (iv) are controlled byexternal magnetic field, defects, anisotropy, and temperature, respectively.The competition between these energies gives rise to a VL structure withfeatures that could be identified by several techniques. The features of thefield distribution due to a regular VL, the effect of energies (i) to (iv), andexamples of some measurements will be covered in this section.1.5.1 Regular VL CharacteristicsProvided that the repulsive interaction is dominant, the magnetic vorticesform a regular vortex lattice with a lattice constant a. Numerous theoriesand experiments have confirmed that the vortices are typically arranged intoa triangular VL [82]. The area occupied by each vortex in a triangular unitcell is A =√32 a2, and for an average field B0 = Φ0A , the vortex spacing is401.5. Vortex Lattice in HTSCthus,a=radicalBigg2√3Φ0B0 =1546 nmradicalbigB0 (mT). (1.38)The magnetic field profile of the VL can be found using different methods:GL theory, Bogoliubov-deGennes theory,...etc. Unfortunately most theoriesare only applicable at certain temperatures/fields, and have too many un-known parameters to be used to fit the experimental data [82]. Because ofits simplicity, the London model provides the most convenient way of calcu-lating the magnetic fields inside a HTSC. The model is only applicable forextreme type II superconductors with λ ≫ ξ, and is valid at all tempera-tures below Tc, and fields B0 ≪ BC2. All these conditions are satisfied inβ-NMR and µSR experiments.For magnetic fields applied along thez-direction parallel to the crystallo-graphicc-axis of the superconductor, the spatial dependence of the magneticfield inside the superconductor (z ≥ 0), can be found by modifying the Lon-don equation (Eq. (1.10)) to account for vortices,B(r)−λ2∇2B(r) = ˆzΦ0summationdisplayRδ(r⊥ −R). (1.39)Here λ = λab = √λaλb, δ(r⊥) is a 2D delta function, r⊥ = xˆx+yˆy is a 2Dvector in the xy plane, r = r⊥ +zˆz, and R = a[(m+ n2)ˆx+n√32 ˆy] are thevortex positions. Outside the superconductor (z ≤ 0), the magnetic fieldfollows the equation−∇2B(r) = 0. (1.40)Throughout this work we are only working with the z-component of themagnetic field Bz = ˆz·B, and refer to it for simplicity as B. The transversefield B⊥ can be neglected as it contributes a second order correction tothe spin Hamiltonian, while the longitudinal field results in a first ordercorrection. The longitudinal component can be found by solving Eqs. (1.39)and (1.40) using the Fourier transform,B(r) = B0summationdisplaykF(k)eik·r, (1.41)411.5. Vortex Lattice in HTSCF(k) = 1Φ0integraldisplaycelldr⊥B(r)e−ik·r. (1.42)The Fourier components F(k) take on the formF(k) = 1λ2bracketleftBigΘ(−z)ekzΛ(Λ+k) +Θ(z)Λ2 (1−kΛ+ke−Λz)bracketrightBig. (1.43)Here Λ2 = k2 + 1λ2, and k = 2pia [nx+ 2m−n√3 y] are the 2D reciprocal latticevectors of the triangular VL, and Θ(z) = 1 for z ≥ 0 and zero otherwise. Inthe bulk (z ≫a,λ), the magnetic field simplifies toB(r) =B0summationdisplaykeik·r1+k2λ2. (1.44)This result diverges on the axis of the vortex line at R. To account forthe finite size of the vortex core, and correct the unphysical divergence ofB at the vortex cores, the Fourier components are multiplied by a cutofffunction C(k) which is approximated by a simple Gaussian C(k) ≈e− ξ2k22(1−b),or Lorentzian C(k) ≈ e− ξk2(1−b), where b = B0/Bc2 [83, 84]. This functionserves as a cutoff for the reciprocal lattice vectors at k ≈ 2π/ξ which yieldsa finite value for the magnetic field at the vortex cores. The term (1 −b)reflects the field dependence of the superconducting order parameter, whichfor constants B0 and λ causes a reduction in the broadening of the fielddistribution with decreasing Bc2 [85]. For our range of applied fields (0 to6.5 T), the (1−b) term is negligible in YBCO where Bc2 ∼ 100 T, and canbe significant in PCCO where Bc2 ∼ 9 T [86, 87]. The magnetic field bothinside and outside the superconductor is given byB(r) =B0summationdisplaykF(k)C(k)cos(k·r). (1.45)The magnetic field distribution, p(B) = 〈δ[B(r) − B]〉 (〈...〉 denotesspatial averaging over r) strongly depends on the depthz, penetration depthλ, magnetic field, and to a lesser extent on the coherence length of the421.5. Vortex Lattice in HTSCFigure 1.14: (a) Simulated field distribution using Eq. (1.45) at depthsz = 2000 nm and z = −10 nm, and B0 = 1 T. Parameters relevant toYBCO at T ≪ Tc have been used: λab = 150 nm, ξ = 2 nm. A Gaussiancutoff is used where b = 0. Solid (dotted) lines are lineshapes of triangular(square) lattice. Top inset shows the positions of the low-field cutoff (A),most probable field (B) and high-field cutoff (C) in a triangular lattice.Bottom inset: field distribution measured in a YBCO crystal by µSR at 0.5T fitted by the London model convoluted with a Gaussian distribution [88].superconductor, ξ. An example of the magnetic field distribution, p(B),calculated using Eq. (1.45) is plotted in Fig. 1.14 for a triangular andsquare lattice with z = 2000 nm, and B0 = 1 T, and using parametersrelevant to YBCO: λ = 150 nm ξ = 2 nm. p(B) shows four characteristics:(i) a low-field cutoff due to the minimum field at the center of a triangularformed by three vortices, (ii) a cusp corresponding to the most probable fieldBsad between two nearest-neighbor vortices, (iii) a tail towards high fieldarising from regions close to the vortex cores, and (iv) a high field cutofffor the maximum field at the core of a vortex. These features are sharedby the triangular and square lattices. However the lineshape in the latter is431.5. Vortex Lattice in HTSCbroaderduetoasmallerlowfieldcutoff, asthedistancebetweenthecentreofa square lattice from each vortex is smaller than in the triangular lattice [89].For clarity, we will limit our discussion to the triangular lattice which is morecommon in HTSC superconductors. An example, of a measured lineshapeis shown in the inset of Fig. 1.14, which is well fitted by the London modelin a triangular VL. To fit the experimental data, the theoretical p(B) wassmeared out by a Gaussian distribution which accounts for disorder of theVL [88].Near the surface, p(B) is strongly depth-dependent, where the broad-ening is drastically reduced as plotted in Fig. 1.14 at a depth z = −10nm outside the superconductor. This is because the Fourier components inEq. (1.43) outside the superconductor vary as exp(kz), where k has valuesalways equal to or larger than 2π/a. At high field, the vortex spacing in Eq.(1.38) is very small, thus the Fourier components decay on a length scale of1k ≈a2pi. For example, at B0 = 1 T, a≈ 50 nm, and1k ≈ 7 nm.One can quantify the field and depth dependence of the field distributionby computing the second moment, σ2 = 〈B2〉−〈B〉2, which takes on theformσ2 =B20 summationdisplayknegationslash=0F2(k)C2(k). (1.46)The second moment versus z is plotted in Fig. 1.15 for various fields. Thebroadening outside the superconductors decays faster at high fields thanlow fields as the vortices are closely spaced, and the field inhomogeneityoutside the superconductor is thus smaller than at low fields. Inside thesuperconductor, the variance recovers to the bulk limit as exp(Λz), whereΛ2 =k2 +1/λ2.For implantation depths inside the superconductor, comparable or largerthan 2π/k, the field distribution is nearly field independent for 2Bc1 ≤B0 ≤Bc2 (for a≪λ), and the second moment of p(B) follows the formula [90],σ ≈ 0.0609Φ0λ2(T) , (1.47)neglecting the cutoff field. This can be seen in Fig. 1.16-(a), where the441.5. Vortex Lattice in HTSC−200 −100 0 100 200z (nm)0246σ (mT)B0= 1 TB0= 0.1 TB0= 0.01 TFigure 1.15: Variance of the field distribution as a function of depth, withz > 0 (z < 0) corresponding to inside (outside) the superconductor. Theparameters λab = 150 nm, ξ = 2 nm, and a Gaussian cutoff (b = 0) havebeen used.variance σ is field independent for ξ = 0 at high fields. Using the lattermakes σ field-dependent, and the correction is significant inside the super-conductor. The correction, however, is smaller outside the superconductor;where the second moment is strongly field-dependent and vanishes at ∼ 0.5T for z = −50 nm. At this field, the field inhomogeneity goes to zero overthe short distance a/2π ∼ 20 nm. Thus, the field recovers to uniformity atdistances higher than 20 nm away from the superconductor.1.5.2 DisorderDefects can pin the vortices at energetically favorable locations in the sam-ple, where the order parameter is already suppressed. Having a vortex cen-tered on the defect is favorable since it saves the core energy. In HTSC, thevortices are susceptible to pinning because of their short coherence lengths,and weak coupling between CuO2 layers. Vortex pinning leads to a vortexdensity gradient which modifies the current density in the material. Vortex451.5. Vortex Lattice in HTSC0 1 2B (T)00.10.20.30.4σ (mT)ξ=0ξ=6 nmξ=20 nm00.51 ξ=0ξ=6 nmξ=20 nm012 ξ=0ξ=2 nmξ=20 nm0246ξ=0ξ=2 nmξ=20 nm(a)z=50 nm, λ=150 nm(b)z=−50 nm, λ=150 nm(c)z=50 nm, λ=300 nm(d)z=−50 nm, λ=300 nmFigure 1.16: Variance of the field distribution inside the superconductor atz = 50 nm for (a) and (c), and outside at z = −50 nm for (b) and (d).Parameters relevant to YBCO (PCCO) have been used where λab = 150(300) nm and ξ = 2 (6) nm. Also, plotted is the variance for ξ = 0,20nm. A Gaussian cutoff is used in all, where b = 0 is taken for YBCO, andb= B/Bc2 (Bc2=9 T) is used for PCCO.461.5. Vortex Lattice in HTSCpinning also causes distortions of the regular VL and has direct implicationson potential applications.The disorder can be classified into three categories based on (i) strength:either weak and strong disorder, (ii) nature: uncorrelated point-like de-fects such as impurities and correlated disorder from naturally occurringgrain boundaries or artificially induced columnar defects by irradiation withheavy ions. (iii) geometry: bulk disorder or surface disorder such as energybarriers and surface roughness. In cuprates, oxygen deficiencies and twinboundaries are the two leading sources of pinning. The strong pinning sit-uation naturally appears due to the layered structure. The strength of thepinning can be easily revealed in µSR by slightly changing the applied fieldand measuring the resulting change in the field distribution, especially thecusp frequency [82]. If that produces no change, then the vortices are highlypinned. This has been observed by µSR in highly twinned YBCO samples[82]. In contrast, in conventional NbSe2, the pinning is weak as the cuspfield easily responds to a small change in the applied field [91].The correlated disorder is dominant in orthorhombic systems such asYBCO due to the existence of twin and grain boundaries. Point-like de-fects from oxygen deficiencies and impurities are weak pinning centers, andmay also perturb the VL. These random point-like pinning centers are moresignificant at low fields where the interaction energy between the vorticesis weak [92]. At high magnetic fields, the vortex-vortex energy overcomesthe weak random pinning centers, and only strong pinning sites will keepthe vortices localized [82]. Rough surfaces can be significant in thin filmsor powdered samples, and can dominate the vortex structure of the bulk.Under zero field conditions, vortices cannot form spontaneously within thebulk sample but have to penetrate from the sample edge. This edge pro-duces a barrier against vortex entry, which vanishes for fields above Bc1 foran ideal surface. The surface barrier vanishes at the first penetration fieldBp ≈ Φ0/4πξλ. At this field, the order parameter is strongly suppressed atthe surface, allowing the vortices to penetrate. Real samples, however, usu-ally have inhomogeneities at the surface producing local field enhancementsand reducing the first penetration field to a value below Bp [19].471.5. Vortex Lattice in HTSC(a) (b)(c) (d)Figure 1.17: Top panel: the vortex configuration in YBCO crystals ob-tained using Bitter decoration technique [93]: images of (a) a twinned and(b) a twin-free area are taken after cooling in 20 G. The marker is 10 µm.Bottom panel: STM images of twinned YBCO single crystals at 4 K, withthe magnetic field applied parallel to the c-axis, and perpendicular to thesurface [94]. (c) Image taken after field cooling in 3 T. (d) Topographicimages of the YBCO surface showing the twin boundary.481.5. Vortex Lattice in HTSCThe attraction of vortices to twinning planes leads to an enhanced vortexdensity along the twin boundaries, as clearly observed by Bitter decoration[93] and STM [94] (see Fig. 1.17). This leads to a locally distorted VL,a large scale distorted VL, or a completely random vortex structure. Thedisorder of the VL due to twin boundaries and other defects strongly mod-ifies the ideal magnetic field distribution of the VL. The latter is smearedout by the distortions and becomes more symmetric and broader [95]. Forexample, the second moment of randomly positioned stiff parallel vorticesis proportional to the applied field [96]σ2rand = B0Φ04πλ2 . (1.48)The ratio,σ2rand/σ2 ≈ 0.6ln(κ)B0/Bc1, is typically≪ 1, therefore the secondmoment of randomly positioned vortex lines is always larger than that of theperfect VL. This disorder in the VL is often accounted for by convoluting theideal VL lineshape by a Gaussian of width σD, which is thus a quantitativemeasure of the disorder. Such disorder smears out the Van Hove singularitiesof p(B) and renders the lineshape symmetric when σD ≥ σ. So, the widthof the field distribution increases from the theoretical second moment σ toa width dominated by σD [97].1.5.3 Anisotropy and Thermal FluctuationsAllcupratesuperconductorsconsistofsuperconductingCuO2 planesofspac-ing d which interact with each other by weak Josephson coupling. Thelayered structure causes two novel phenomena: pancake and Josephson vor-tices. When the magnetic field is applied along thec-axis, 2D point vortices;which have a zero order parameter only in one layer, are established insidethe material. Point vortices in the same layer repel each other, and those indifferent layers attract each other to form pancake vortices, with the lowestenergy achieved when straight line vortices are formed. The field of a singlepoint vortex with centre at ri = 0 is confined to a layer of thickness 2λab,491.5. Vortex Lattice in HTSCwhere the z and in-plane components take on the form [98]Bz(r) = (sΦ0/4πλ2abr)e−r/λab (1.49)B⊥(r) = (sΦ0z/4πλ2abr⊥)[e−|z|/λab/|z|−e−r/λab/r], (1.50)where s is the CuO2 layer spacing. For a straight vortex line, the in-planecomponent of the point-vortex fields (if s ≪ λ) cancel; and only the z-component survives. The flux of a point-vortex in the plane z = zn isφ(zn) = (sΦ0/2λab)exp(−|zn|/λab) ≪ Φ0, with the sum of the flux at allpoints along a stack yields Φ0. When the applied field is along the ab-plane,the vortex core prefers to run between the CuO2 layers, and these vorticesare called Josephson vortices. The width of Josephson core is λJ = γs andits thickness is s.HTSC are characterized by two magnetic penetration depths for currentsin the ab-plane λab,2 and along the c-axis λc, and by two coherence lengthsξab and ξc. The anisotropy ratio γ = λc/λab = ξab/ξc, is γ ≈ 5 in YBCO,γ ≈ 25 in PCCO, and γ > 125 in BSSCO [99]. For fields applied parallelto the c-axis, the Josephson length λJ = γs determines how effective is thetunneling of currents between the superconducting planes is in connectingthe point-vortex flux. The ratio λJ/λab determines the dimensionality ofthe vortex configuration. In optimally doped YBCO, λJ/λab ≪ 1, thus thevortices are straight lines, and have a 3D vortex configuration over most ofthe field-temperature phase diagram; similar to isotropic superconductors.This justifies the use of the London model to study the magnetic field ofthe vortex state. In contrast, for optimally doped BSSCO and underdopedYBCO, λJ/λab ≥ 1, and the coupling between point-vortices in adjacentCuO2 planes is very weak [98]. As a result, the flux structure may consist ofpancake vortices which may couple at low fields via Josephson and magneticinteractions to form straight vortices.The field distribution of a 2D VL structure has different features thanthe 3D structure discussed earlier. For a 2D VL, the field variance σ given2A small anisotropy exists between the a and b direction in YBCO, where λa ≈ 1.19λb.501.5. Vortex Lattice in HTSCFigure 1.18: The field distribution in BSSCO crystals measured by µSRin an external field of 1.5049 T applied along the c-axis. The measuredlineshape is not well fitted by a 3D model (dashed line), and well fitted bya 2D model [13].in Eq. (1.46) reduces to a much smaller value [90]σ2D = 1.4(s/a)1/2σ ≪σ. (1.51)The field distribution of a 2D VL is nearly symmetric, and the cusp fieldis equal to the applied field [100, 101]. An example is shown in Fig. 1.18.Monte Carlo simulations have shown that p(B) of a 2D structure has a lowfield tail rather than a high field tail in a 3D VL [100]. This was observed inµSR measurements (Fig. 1.19-(a)) [13, 101]. In highly anisotropic BSSCOsystems, µSR studies have found evidence of a crossover from a 3D VL (pos-itively skewed) to a 2D VL (nearly symmetric or negatively skewed) as afunction of the field. When the magnetic field is increased, the interactionbetween point vortices within a layer will eventually exceed the interlayerelectromagnetic coupling, and random pinning sites will lead to a misalign-ment of the pancake vortices from 3D to 2D structure. In BSSCO, wheres ∼ 1.5 nm and γ ∼ 150, the crossover field is B2D ∼ Φ0/(γs)2 ∼ 40 mT.In Ref. [101], B2D was found to be ≈ 50 mT. Similar effects have beenobserved in a more isotropic system, LSCO [102]. The origin is, however,511.5. Vortex Lattice in HTSC(a) (b)Figure 1.19: (a) TheµSR field distribution measured in low and high appliedfields. The low field lineshape is that of a regular VL, but the high field is dueto a melted lattice. (b) The skewness parameter α = 〈∆B3〉1/3/〈∆B2〉1/2 ofthe field distribution is shown [101].due to a phase transition from a Bragg glass where the vortices have a longrange order (with weak disorder), to a more disordered vortex glass of shortrange order.At low temperatures, the vortices are frozen into a vortex structure.As the temperature increases, thermal fluctuations of the vortices becomesignificant. These are more important in HTSC due to the higher values ofTc and the layered structure of these systems. Strong thermal fluctuationsovercome the pinning potential and greatly reduce the effect of pinning.If the fluctuations are large, and the coupling is very weak, the VL maymelt into a vortex liquid. A melting of the vortex structure was detectedby µSR in BSSCO where the field distribution changes asymmetry aboveTm ∼ 58 K, as seen in Fig. 1.19-(a) at 45.4 mT. The liquid phase aboveTm resembles a 2D structure, and has a noticeable low field tail where theskewness parameter, α = 〈∆B3〉1/3/〈∆B2〉1/2 is negative (Fig. 1.19-(b)),where 〈∆Bn〉 = integraltext∞−∞dB(B − 〈B〉)np(B), is the n-th central moment ofp(B), and 〈B〉 is the first moment of p(B).A very useful phenomenological theory of layered superconductors is theLawrence-Doniach (LD) theory [103], where the superconducting layers areseparated by an insulating layer of thickness d. The LD model approaches521.5. Vortex Lattice in HTSC(a) (b)Figure 1.20: (a) T-dependence of the second moment σ ∝ 1/λ2 in a sinteredYBCO sample measured in a field of 350 mT. The data is well fitted bythe two-fluid model suggesting an s-wave pairing symmetry [104]. (b) T-dependence of 1/λ2 in a high quality YBCO single crystal in 5 kG externalfield. The data is well fitted by the d-wave order parameter, and stronglydeviates from s-wave [105].the anisotropic GL and London results when the c-axis coherence lengthξc(T) exceeds the layer spacing d. In this case, there is no phase differencein the order parameter between adjacent CuO2 layers, and the VL assumesthe 3D structure in the absence of pinning. When ξ/d < √2, there is aphase difference and a 2D VL takes place.1.5.4 Temperature Dependence of λThe temperature dependence of λ is routinely used to determine the na-ture of the pairing of the superconducting condensate. Experimentally, thepenetration depth can be directly extracted from the second moment of thefield distribution ( see Eq. (1.46) ). In many experiments on the VL struc-ture in HTSC it has been difficult to determine how much of the observedstructure is directly attributed to the symmetry of the pairing state andhow much is due to disorder of the VL caused by extrinsic effects. Only afew experiments such as µSR, SANS, and NMR can quantitatively extractλ and ξ. Here I will only discuss the temperature dependence of the lengthscales from µSR, which is the most direct experiment to extract λ, and itstemperature dependence [82].531.5. Vortex Lattice in HTSCThe first µSR experiments conducted on powdered and less homoge-neous HTSC samples, such as YBCO, mistakenly concluded that the orderparameter is of an s-wave nature (see Fig. 1.20-(a)). The second momentextracted from Gaussian fits of the field distribution showed a weak temper-ature dependence at low temperature consistent with BCS theory and ans-wave superconducting gap. This could be due to: (i) random orientation ofthe c-axis and strong pinning in the powdered and less homogeneous HTSCsamples modifying the temperature dependence of the broadening from thatof the penetration depth; (ii) extracting the second moment from Gaussianfits rather than fitting the data to a more realistic model (where 1/λ2 is afree parameter) that takes the features of the field distribution of the VLinto account. Later, once high quality single crystals were available, it wasfound that the temperature dependence of the broadening resembles thatof the d-wave [38]. Similar µSR experiments on PCCO single crystals havebeen unable to determine the nature of pairing symmetry [106]; identifiedby other techniques as d-wave [107].1.5.5 Proximal Detection of the VLAs we discussed earlier, features of the VL can be also studied outside thematerial. The imaging techniques such as Lorentz force, Bitter decoration,scanning tunneling microscopy, magnetic-force microscopy, are widely usedto image the VL structure at the surface [108]. These techniques allowedvisualization of both the VL and its distortions [108]. The imaging tech-niques are, however, unable to quantify the field distribution and extract λ.Contrary to this, LE-µSR has been able to study the field distribution ina depth-resolved manner both inside and outside the superconductor [14].Niedermayer et al., measured p(B) in an Ag layer of 70 nm deposited onYBCO films, and estimated a value of λ = 155 nm at 10.4 mT [109]. Themeasured field distribution showed features that are consistent with a regu-lar VL inside YBCO [109].NMR or Electron-Spin Resonance (ESR) are suitable for the determi-nation of the flux distribution in the bulk of the sample, but no intrinsic541.6. Thesis Outlineresonance can be measured in a HTSC sample. This is overcome by de-positing spin labels onto HTSC’s surface. Due to narrow intrinsic lines ofusual spin labels, these methods could be useful in determining the fielddistribution outside the sample and therefore λ near Tc [110]. Bontemps etal. [111] proposed a method for determination of the spatial length scale ofthe field distribution to distinguish the intrinsic and extrinsic effects on thefield distribution. The method is based on varying the distance of the spinlabels from the superconducting surface by a suitable insulating layer. Thismethod measured field inhomogeneities with a linewidth 30 ± 10 G, char-acterized by a long length scale of 7±0.1 µm in a YBCO superconductingsintered ceramic [111]. This long scale is not due to a regular VL, and thelinewidth is rather attributed to the disorder of the VL inside the YBCOceramic. Similar measurements on YBCO powders have led to a temper-ature dependence of λ more consistent with the two-fluid model than thed-wave symmetry [112, 113]. The above methods are, however, limited tolow fields, and cannot be used at high field where the broadening of p(B)outside the sample is sensitive to the disorder of the VL (see Fig. 1.16).1.6 Thesis OutlineThis thesis presents β-NMR studies of two families of superconductors, thehole-doped cuprate YBCO and electron-doped cuprate PCCO, and is orga-nized as follows. In Chapter 2, I report the results of a new method to seekevidence of TRSB order near the surface of YBCO films. Several theorieshave suggested that YBCO may develop a TRSB order parameter near thesurface of (110)-oriented films. Theβ-NMR technique is an ideal tool to testthis scenario, as magnetic fields as small as 0.1 G could be detected a fewnm from the surface. The field distribution is measured by implanting 8Li+ions into a silver layer (15-120 nm thick) deposited onto YBCO. The silverlayer is used to stop the 8Li+ ions and vary the average distance betweenthe probe and the YBCO. In principle, one can use other metals or insu-lators, however, silver is preferred as it has a narrow 8Li+ spin resonancewhich is temperature independent below 100 K at low fields, and for other551.6. Thesis Outlinereasons as discussed in Appendix B. I will present measurements of the fielddistribution above and below Tc for films of (110) and (001) orientations,with the probe ions are stopping at an average distance of 8 to 40 nm fromthe Ag/YBCO interface with the superconductor in the Meissner state andthe applied magnetic field is parallel to the surface. I will quantitativelycompare these resonances and identify any changes upon cooling below Tcwhich could signal the existence or absence of TRSB spontaneous fields.In Chapter 3, I will present measurements of the field distribution outsideYBCO (in the vortex state) with the applied field perpendicular to thesurface in an Ag overlayer evaporated onto the superconductor. This allowsone to isolate the contribution from long length scale disorder of the VL,since the VL disorder that occurs on such scales cannot be probed insidethe superconductor where the features of the short range ordered latticeare dominant. β-NMR studies were done on three different near-optimallydoped YBCO samples: a twinned single crystal, a partially detwinned singlecrystal, and a 600 nm thick film. I will show measurements carried out inapplied fields from 150 G to 3.33 T as a function of temperature. 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Monod, and R. Even, Phys. Rev. B 43,11512 (1991).[112] A. Steegmans, R. Provoost, R. E. Silverans, and V. V. Moshchalkov,Physica C 302, 159 (1998).[113] A. Steegmans, R. Provoost, V. V. Moshchalkov, H. Frank, G.G¨untherodt, and R. E. Silverans, Physica C 259, 245 (1996).65Chapter 2Search for BrokenTime-Reversal SymmetryNear the Surface of (110)and (001) YBa2Cu3O7−δFilms2.1 IntroductionIn addition to the broken electromagnetic gauge symmetry common to allsuperconductors (SC), the SC order may break other symmetries [1]. Aparticularly interesting case occurs when the order parameter (OP) is com-plex, breaking time-reversal symmetry (TRS),1 e.g. analogous to the orderof superfluid 3He-A [2]. A characteristic feature of TRS-breaking (TRSB)superconductors is spontaneous magnetization; however, Meissner screeningcancels this in the bulk, limiting the associated fields to within the magneticpenetration depth from defects and interfaces [3]. The magnetic signaturesof TRSB superconductivity are thus very subtle, and few techniques are suf-ficiently sensitive to detect them. To observe the magnetic fields directly, onerequires a sensitive local magnetic probe, such as the positive muon in µSR,1A version of this chapter will be submitted for publication. H. Saadaoui, G.D. Morris,K.H. Chow, M.D. Hossain, C.D.P. Levy, T.J. Parolin, M.R. Pearson, Z. Salman, M.Smadella, Q. Song, D. Wang, P.J. Hentges, L.H. Greene, R.F. Kiefl, and W.A. MacFarlane.Search for broken time-reversal symmetry near the surface of YBa2Cu3O films using β-detected NMR.662.1. Introductionwhich has found evidence of TRSB superconductivity in several systems,notably Sr2RuO4[5]. In this chapter we use a novel technique based on beta-detected nuclear magnetic resonance (β-NMR) to seek evidence for TRSBorder near the surface of the high-Tc cuprate superconductor YBa2Cu3O7−δ(YBCO).In the high-Tc cuprates in general, and YBCO in particular, there is noevidence for TRSB in the bulk [6, 7] and OP-phase-sensitive measurementshave established spin-singlet dx2−y2-wave order [8], but there are also indi-cations of weak magnetism [9, 10], some of it related to the CuO chains inYBCO [11], or to vortex cores above the lower critical field [12]. Such resultshave motivated new theories, e.g. for a TRSB pseudogap state above thesuperconducting Tc [13].Interface scattering of the d-wave Cooper pairs may also stabilize TRSBsuperconductivity [3]. Scattering from most interfaces perpendicular to theCuO2 planes frustrates dx2−y2-wave order within a few coherence lengthsof the interface, leaving a high density of mobile holes (as evidenced by thezero bias conductance peak (ZBCP) found in many tunneling measurements)that may condense into a superfluid of different symmetry than the bulk[14], e.g. s-wave, or TRSB states such as dx2−y2+is and dx2−y2+idxy [15].Experiments to detect TRSB near surfaces have yielded controversial results.Carmi et al. measured a weak spontaneous magnetic field using SQUIDmagnetometry near the edges of epitaxial c-oriented YBCO thin films belowTc [16], while Tafuri and co-workers detected fractional vortices in c-axisfilms using scanning SQUID microscopy [17]. Spontaneous magnetic fluxwas also measured near asymmetric 45o grain boundaries in c-axis YBCOfilms in zero field by Mannhart et al. [18]. Magnetism is also apparentin some tunneling measurements, as a spontaneous Zeeman-like splitting ofthe ZBCP. Some tunneling experiments have found such a splitting [19], andothers did not [20], while phase sensitive measurements showed no evidencefor such a TRSB state [21]. This diversity of results calls for more studies ofinterface magnetism in cuprates using a sensitive local magnetic probe thatcan locate the origin and distribution of such fields on the atomic scale.In this chapter, we present measurements of the magnetic field near672.1. IntroductionFigure 2.1: Geometry of the experiment where the field is applied along thesurface. The orientation of the YBCO films is shown: Ag/YBCO(110) onthe right and Ag/YBCO(001) on the left.the interface of silver and (110)- and (001)-oriented YBCO films using β-NMR. We measure the field distribution using a highly spin polarized 8Li+beam implanted into a thin silver overlayer deposited on the YBCO. Wefind an inhomogeneous broadening of the field distribution below Tc forboth orientations, with the probe ions stopping at an average distance of 8nm from the Ag/YBCO interface. However, the magnitude of these fields issmall(∼ 0.2G),andseemstooriginatefrominhomogeneousfluxpenetrationnear the interface rather than TRSB order.682.2. Experimental Details2.2 Experimental DetailsThe experiment was performed using β-NMR of highly-spin-polarized 8Li+at the ISAC facility at TRIUMF in Vancouver, Canada. For details, seeSection 1.1 and Refs. [23, 24, 25]. Similar to NMR, to measure the spinresonance signal, we apply a field along the spin polarization (here in theplane of the films, see Fig. 2.1)) B0 = B0ˆy (with 5 ≤ B0 ≤ 150 G), andfollow the polarization of 8Li+ as a function of the frequency ω of a smalltransverse radio-frequency (RF) field of amplitude B1 ∼ 1 G, applied alongthe ˆx-axis. The resonance condition is ω =γB0, where for 8Li+ γ = 0.63015kHz/G. At thisω, the polarization, initially parallel to the ˆy-axis, is averagedby precession in the oscillating field. The resulting resonance lineshape isgenerally broadened by any inhomogeneity in the local magnetic field. In theabsence of other effects, the lineshape thus offers a detailed measurement ofthe distribution of local magnetic fields in the sampled volume determinedby the beam-spot (∼ 2 mm in diameter) and the implantation profile (seebelow).A novel pulsed RF mode was used in this study. The RF field is appliedin 90o pulses randomized in frequency order, instead of the continuous wave(cw) mode commonly used [24]. In pulsed RF, the polarization at eachfrequency is measured three times: before, during, and after the RF pulse,and the signal is the step in polarization caused by the short RF pulse.In this way, one obtains a high signal to noise with minimal contributionfrom both variations in the incoming 8Li+ rate and cw power broadening.Because of the limited B1 in the broadband tank circuit, this pulsed modeis suitable for narrow lines up to a few kHz in width. Fig. 2.2 shows theresonance spectrum at 100 K exhibiting a linewidth, i.e. half width at halfmaximum (HWHM), of approximately 140 Hz ∼ 0.2 G. Similar cw spectracan be at least twice as broad [23], making it difficult to resolve the smalladditional broadening we find at low temperature. The pulsed mode alsoeliminates history effects of the slowly recovering off-resonance polarization.See Appendix A for more details about this mode.Major advantages of β-NMR in detecting TRSB are the abilities (i) to692.2. Experimental Details2 4 6 8 10 12ω(kHz)−0.04−0.020Asymmetry0 10 20 30 Depth (nm)0100020003000Stopping density−0.04−0.020 T=100 KAgT=4.3 KYBCO2∆B0=10 GFigure 2.2: Typical β-NMR spectra in Ag taken above and below Tc at 2keV, about 8 nm from the (110)-oriented YBCO interface, in an externalfield of B0 = 10 G (FC) applied along the surface of the film. Solid linesare fits to a Lorentzian of amplitude A, HWHM ∆ and resonance frequencyωL, i.e. L(ω) = A4(ω−ωL)2+∆2. Inset: simulated implantation profile for the8Li+ in a 15nm Ag layer on YBCO from TRIM.SP [28]. The 8Li+ stops atan average depth of 8 nm away from the Ag/YBCO interface.702.2. Experimental Detailsimplant the probe 8Li+ at low energy into thin layered structures and (ii)to control this implantation depth on the nanometer scale. In this study,8Li+ is preferentially implanted into the thin silver overlayer evaporatedonto YBCO, instead of the superconductor itself. Stopping the probes inthe overlayer eliminates the possibility that the probe perturbs the super-conductivity. Also, the 8Li+ nucleus carries an electric quadrupole moment,so the spectrum in non-cubic YBCO is complicated by quadrupole splittings[23]. In contrast, 8Li+ in Ag, below 1 Tesla, exhibits a single narrow reso-nance with T-independent linewidth [24]. From basic magnetostatics, anyinhomogeneous fields in the YBCO layer will decay exponentially outsidethe superconductor as exp(−2pia z) where a is the length scale of the inho-mogeneity in YBCO [26, 27]. Thus we can only detect such fields providedour probe-YBCO stopping distance z is ≤ a2pi. Any magnetic field inhomo-geneities arising in this way will broaden the intrinsic resonance of the Aglayer.The measurements presented here were carried out on (110) and (001)-oriented YBCO films capped with 15 to 50 nm of Ag. The (110) film ofTc =86.7 K was grown by RF magnetron sputtering on a (110) SrTiO3 (STO)substrate measuring 8×6 mm. Three (001) films were also studied, (i) Tc =88.7 K grown on a (001) STO substrate under similar conditions asthe (110),(ii) the others are grown by thermal co-evaporation on 8×10 mm LaAlO3and have Tc ∼ 88.0 K. The Ag (99.99% purity) was deposited ex-situ on thefilms at room temperature by DC sputtering in an Ar pressure of 30 mtorrat a rate of 0.5 ˚A/s while rotating the sample to ensure uniformity. The8Li+ implantation energy was varied so that the probe ions are implantedat average depths ranging from 8 to 43 nm. The inset of Fig. 2.2, shows thestopping profile of 2 keV 8Li+ ions in 15 nm of Ag calculated using TRIM.SP[28]. Here the average probe-YBCO distance is ∼ 8 nm. At 2 keV, about20% of 8Li+ ions stop in the YBCO, yielding no associated NMR signal dueto fast spin lattice relaxation at low magnetic fields. The measurementswere taken in the Meissner state by field-cooling in a small magnetic fieldB0 (FC) or in zero field (ZFC). Residual magnetic fields were reduced toless than 30 mG normal to the surface (FC) or in all directions (ZFC) (see712.3. ResultsAppendix C). To measure the resonance in Ag, an applied field (above ∼ 5G) is required, which was applied at 10 K (≪ Tc) in the ZFC case (seeAppendix B).2.3 ResultsFig. 2.2 shows two resonances at 100 K and 4.3 K in the Ag on the c-axisYBCO film. Above Tc, the resonances are all identical and show negligibledifferences in amplitude and linewidth, and are indistinguishable from thoseintrinsic to Ag. Below Tc, the resonance broadens, therefore reducing theamplitude. The HWHM, ∆, of a single Lorentzian fit to the data in both(110) and (001) samples is plotted in Fig. 2.3. It is nearly T-independentabove Tc, consistent with the nuclear dipolar broadening in Ag, and it is thesame in both samples and comparable to a control sample of an Ag grown onaninsulatingSTOsubstrateundersimilarconditions(opencircles, Fig. 2.3).Below Tc, the resonance broadens, signaling the appearance of disorderedstatic magnetic fields from the underlying YBCO. ∆ in Ag on the (110) filmbelow Tc is larger and reaches 0.4 kHz (0.6 G) at 5 K, while the maximumwidth in the (001) is approximately 0.3 kHz (0.45 G). The excess broadeningat low temperatures for the (110) film is about 0.25 kHz and 0.15 kHz inthe (001) film.The additional broadening below Tc is not accompanied by a resonanceshift, as seen in Fig. 2.2. The resonance frequency is constant from 300K to 5 K in all films, independent of FC or ZFC cooling. This rules out asuperconducting proximity effect in the Ag layer where an induced Meissnershielding of the applied field leads to a diamagnetic resonance shift, e.g.as seen recently in Ag/Nb heterostructures [29]. ∆ versus B0 is displayedin Fig. 2.4. At 100 K, ∆ is constant from 100 G to 5 G as expected.At 10 K (ZFC), ∆ increases linearly with the applied field with no sign ofsaturation. The broadening at 10 K extrapolates to ∆(B = 0) ≈ 0.2 kHzin the (001) and ≈ 0.33 kHz in the (110), which are comparable to thenormal state broadening ∆ns(B = 0) ≈ 0.18 kHz. Thus, the net internalfield in the superconducting state extrapolated to zero applied field is less722.3. Results0 100 200 300 T (K)0.10.20.30.4∆ (kHz) Ag/YBCO(001) Ag0.10.20.30.40.5 Ag/YBCO(110)Ag.TcTcB0=10 GB0=10 GFigure 2.3: The T-dependence of the Lorentzian HWHM of the 8Li+ reso-nance at 2 keV, i.e. 8 nm from the Ag/YBCO(110), Ag/YBCO(001), andAg/STO (open circles) interface. The data on the Ag/YBCO(110) and Agfilm was taken while cooling in B0 = 10 G. The data on Ag/YBCO(001)was collected in B0 = 10 after ZFC to 5 K. The widths were independent offield-cooling conditions. The dashed lines represent the average Ag width,the arrows point to the Tc of the YBCO films, and the solid lines are guideto the eye.732.3. Results0 50 100 150B0 (G)00.20.40.60.81∆ (kHz) Ag/YBCO(110)Ag/YBCO(001)T=10 KT=100 KFigure 2.4: The 10 K linewidth (HWHM) of the NMR resonance versusthe applied field B0 taken in Ag on (110)- and (001)-oriented YBCO films.The data were taken after ZFC and gradually increasing the field, except inthe (110) films which was FC. The solid lines are fits to the linewidth in theAg on YBCO(001) at 10 and 100 K.742.4. Discussion and Conclusions0 10 20 30 40 50zavg (nm)00.40.8∆(10 Κ)− ∆(100 Κ)[kHz]B0 =100 GB0 =10 GFigure2.5: Extrabroadening, ∆(10 K)−∆(100 K), versus the average depthof 8Li+ into a 50nm thick Ag on (001) YBCO film, ZFC in 10 G and 100 G.than 0.15 kHz ≈ 0.2 G in both orientations. This value is partially due tothe slight broadening of the Ag resonance upon cooling from 100 K to 5 K,as seen in Fig. 2.3. In addition, the broadening in Ag on the c-axis film isfound to be independent of the average probe-interface distance from 8 to43 nm as shown in Fig. 2.5, indicating that the field inhomogeneity mustoccur on a length scale longer than 43 nm.2.4 Discussion and ConclusionsWe turn now to discuss possible origins of the line broadening, which isclearly caused by the superconducting YBCO, since it is absent above Tcand in the Ag film (Fig. 2.3). First, it may be due to a TRSB superconduct-ing order at the (110) interface [14]. Spontaneous fields could also arise inthe c-axis films due to twin and grain boundaries [30]. The broadening we752.4. Discussion and Conclusionsobserve here has an onset close toTc, in contrast to tunneling measurementswhere the ZBCP splitting was observed only below 7 K [19]. In theory, itis also expected that TRSB order condenses at a second transition temper-ature Tc2 ≪ Tc [3, 14, 15]. The amplitude of the inhomogeneous magneticfields detected outside YBCO at low temperature is very small (≤ 0.2 G),and is comparable in both orientations even though the studied YBCO filmshave different thicknesses and surface morphology. Moreover, this internalmagnetic is depth independent from 5 - 50 nm, so we expect it to have a sim-ilar magnitude at the Ag/YBCO interface. The extrapolated broadening atzero field is an estimate of the spontaneous magnetic field at the Ag/YBCOinterface, and has an upper limit of 0.2 G. This is inconsistent with tun-neling experiments where the spontaneous fields must be of a fraction of aTesla [19]. The magnetic field inhomogeneity is enhanced by the appliedfield as reflected in the linear increase of the linewidth ∆ in Fig. 2.4. Thelong length scale of the broadening rules out the role of impurities whichcould frustrate the order parameter; inducing spontaneous fields on smallerlength scales of the order of few coherence lengths [31]. Because of the onsettemperature at Tc, the linear-field dependence, the small magnitude of theextrapolated broadening at zero field, and long length scale of the signal,TRSB spontaneous fields are unlikely to be the origin of these internal fields.An alternative explanation is that the broadening is due to inhomoge-neous penetration of the external magnetic field in the form of flux vortices.Penetration of vortices should not occur at the fields employed here thatare well-below the lower critical field Hc1[32], especially since the demagne-tization factor for the field parallel to the thin film is very small. Moreover,surface barrier effects may even enhance the field required for vortex pene-tration under ZFC conditions [33]. However, at the interface the flux maypenetrate more easily due to suppression of the d-wave order. Twin orgrain boundaries, where the OP is already suppressed, may favor vortexnucleation [34]. Interfacial vortices have been observed in YBCO crystalsin fields as small as 4 G applied parallel to the surface [35]. At such lowfields, the vortex spacing, D, is of the order of few microns [36]. Outsidethe superconductor, the resulting field inhomogeneity leads to a broaden-762.4. Discussion and Conclusionsing which would appear depth independent, since z ≪ D, consistent withour results. This is also consistent with the T-independence of the averagefield (resonance frequency). The T-dependence of the linewidth in Fig. 2.3maybe arising from an effective penetration depthλeff due to vortices insidethe samples, where λeff(T) is modified from the characteristic temperaturedependence due to a d-wave order parameter by the crystal orientation andpolycrystallinity of the films [37]. It is possible that such vortices could alsocontribute to apparent dead layer seen in HTSC and other superconductorsusing low-energy µSR [38].In summary, we have conducted a depth resolved β-NMR study of thefield distribution near the interface of Ag with YBCO(001), and YBCO(110)films, where the 8Li+ probes stop in the Ag. In both orientations we findadditional broadening of the NMR belowTc, signaling the appearance of dis-ordered internal static fields in YBCO. These fields are likely due to vorticespenetrating at the interface below the bulk Hc1. Attributing the linear-with-field term in the line broadening to vortex penetration, we establish an upperlimit on TRSB fields, from the extrapolation to zero field, of 0.2 G.77Bibliography[1] For reviews see L. P. Gor’kov, Sov. Sci. Rev. A Phys. 9, 1 (1987); M.Sigrist and K. Ueda, Rev. Mod. Phys. 63, 239(1991);[2] A. J. Leggett, Rev. Mod. Phys. 47, 331 (1975).[3] M. Sigrist, Pro. The. Phys. 99, 899 (1998).[4] J. R. Kirtley and C. C. 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Yeshurun, and F. Holtzberg,Phys. Rev. B 43, 13707 (1991); S. Salem-Sugui Jr., A. D. Alvarenga,O. F. Schilling, Supercons. Sci. Technol 10 284 (1997).[34] R. P. Huebener, Magnetic flux structures in superconductors, p. 86,Springer (2001).[35] G. J. Dolan, Phys. Rev. Lett. 62, 2184 (1989).[36] For a review see S. J Bending and M. J. W. Dodgson, Condens. Matter17, R955 (2005).[37] C. Panagopoulos, J. L. Tallon, and T. Xiang, Phys. Rev. B 59, R6635(1999).[38] A. Suter, E. Morenzoni, N. Garifianov, R. Khasanov, E. Kirk, H.Luetkens, T. Prokscha, and M. Horisberger, Phys. Rev. B 72, 024506(2005).81Chapter 3Vortex Lattice Disorder inYBa2Cu3O7−δ Probed Usingβ-NMR3.1 IntroductionThe vortex state of cuprate superconductors is of central importance in un-derstanding high-Tc superconductivity (HTSC)1. One the most well studiedquantities is the internal magnetic field distribution p(B) associated withthe vortex lattice (VL) [1, 2, 3, 4]. As discussed below, several methods canbe used to measure p(B), which depends on the London penetration depthλ, the coherence length ξ [5, 6], and, to a lesser extent, the internal struc-ture of the vortices [7], and non-linear and non-local effects [4, 8, 9]. Theform ofp(B) has a distinctive asymmetric shape due to the spatial magneticinhomogeneity characteristic of an ordered two-dimensional (2D) lattice ofvortices. One basic feature in p(B) is a prominent high field tail associatedwith the vortex cores, which depends on the magnitude of ξ. There is alsoa saddle point in the local field profile located between two vortices. Thisgives rise to a Van Hove singularity or sharp peak in p(B) below the averagefield. The overall width or second moment ofp(B) depends primarily on theLondon penetration depth λ, the lengthscale over which the magnetic fieldis screened. Anisotropy of the Fermi surface or the superconducting order1A version of this chapter has been published. H. Saadaoui, W. A. MacFarlane, Z.Salman, G. D. Morris, Q. Song, K. H. Chow, M. D. Hossain, C. D. P. Levy, A. I. Mansour,T. J. Parolin, M. R. Pearson, M. Smadella, Q. Song, D. Wang, and R. F. Kiefl, Phys. Rev.B 80, 224503 (2009). Vortex lattice disorder in YBa2Cu3O7−δ probed using β-NMR.823.1. Introductionparameter can result in a different VL but the main features are similar forany ordered lattice [10, 11].Another general feature associated with any real VL is disorder arisingfrom vortex pinning at structural defects and impurities [12, 13]. Structuraldefects are present in all superconductors to some degree, but may be moreprevalent in structurally complex compounds such as YBCO. For example,YBCO’s slightly orthorhombic structure facilitates crystal twinning, i.e. ina single crystal, there are generally domains with the nearly equal a andb directions interchanged. Separating such twin domains are well defined45◦ grain boundaries or twin boundaries which have been shown to be ef-fective extended vortex pinning sites [14, 15, 16, 17]. Scanning tunnelingmicroscopy (STM) imaging of a twinned YBCO crystal show that the arealvortex density is strongly modified by the twin boundaries [18]. Small-angleneutron scattering (SANS) studies of YBCO confirm that the twin bound-aries strongly deform the VL [15, 19]. Understanding the influence of suchstructural defects on the VL has been the subject of intense theoretical work[20, 21, 22], and is important for two main reasons. Firstly, it affects p(B)and thus adds uncertainty to measurements of fundamental quantities likeλand ξ, since it can be difficult to isolate such extrinsic effects from changesin fundamental quantities of interest. Secondly, the degree of pinning ofvortices determines the critical current density which is important for manyapplications [23].Measurements of the vortex state field distribution p(B) are most oftendone using SANS [15, 19], nuclear magnetic resonance (NMR) [24], andconventional muon-spin rotation (µSR) [25]. All these methods probe theVL in the bulk and can be applied over a wide range of magnetic fields. It isalso possible to probe the magnetic field distribution p(B) near the surfaceof the sample using low energy-µSR (LE-µSR) in low magnetic fields [26].Recently we have demonstrated that similar information on p(B) near asurface can be obtained using β-NMR [2]. This has the advantage that itcan be applied over a wide range of magnetic fields.In this chapter, we report measurements of the VL above the surface ofthe cuprate superconductor YBa2Cu3O7−δ using β-NMR [27, 28, 29, 30, 31,833.2. The Magnetic Field Distribution p(B) in the Vortex State32]. The 8Li+ beam was implanted into a thin silver overlayer evaporatedonto several YBCO samples. Measuring in the Ag allows one to isolate thecontribution to p(B) from long wavelength disorder, i.e. disorder that oc-curs on length scales much longer than the vortex spacing and λ, and is dueto structural defects such as twin and grain boundaries. This is possiblebecause the field distribution broadening just outside the superconductordue to the VL inside has a very distinctive field dependence. In particu-lar it vanishes in high magnetic fields where the VL spacing becomes lessthan the characteristic distance of the probe from the superconductor. Onthe other hand, long wavelength disorder has a much weaker dependenceon magnetic field and dominates the observed p(B) in the high field limit.Our results show evidence for significant broadening of p(B) from such longwavelength disorder on the scale of D ≈ 1 µm, which is attributed to pin-ning at twin or other grain boundaries. The magnitude of the broadeningis similar to that observed in bulk µSR measurements, suggesting that thesame broadening contributes to p(B) in bulk µSR measurements. There isa crossover such that near Tc, where λ≫D, the broadening scales with thesuperfluid density, whereas at lower temperatures, whereλ≪D, the broad-ening does not track the superfluid density. We discuss the consequences ofthis for the inference of λ(T) from measurements of p(B) in polycrystallinesuperconductors.The chapter is organized as follows: section II reviews the theory forthe field distribution, and its second moment near the surface of a super-conductor. Section III contains all the experimental details. In Section IV,we present the results. Finally in section V we discuss the results and drawconclusions.3.2 The Magnetic Field Distribution p(B) in theVortex StateIn a type II superconductor, above the lower critical field Bc1, the magneticfield penetrates the sample inhomogeneously forming a lattice of magnetic843.2. The Magnetic Field Distribution p(B) in the Vortex Statevortices, each carrying a flux quantum, Φ0 = h/2e. In a perfect crystal,intervortex interactions lead to a long-range ordered 2D lattice of vortices,usually of triangular (hexadic) symmetry [1]. At the core of each vortex (acylinder of radius approximately the superconductor’s coherence length ξ),the local magnetic field is maximal. Outside the core, concentric circulatingsupercurrents partially screen the field which thus falls exponentially witha lengthscale λ. The average magnetic field in the VL is the applied fieldB0 for flat samples, where demagnetization effects are negligible [33, 34, 35].At a given field, the average vortex spacing, i.e. the lattice constant of theVL, a is fixed. For the triangular lattice this isa=radicalBigg2Φ0√3B0 ≈1546 nmradicalbigB0(mT). (3.1)If one considers the profile of the magnetic field along a line in the lattice(perpendicular to the direction of the applied field), it is thus corrugatedwith a period determined by a. This inhomogeneity in the magnetic fieldcauses a characteristic broadening in local magnetic resonance probes suchas the muons in µSR or the host nuclei in NMR. Since the muon (or hostnuclear spin) is at a well-defined lattice site(s), it samples the VL with agrid spacing given by the lattice constant of the crystal. Since this is muchsmaller than the VL constant, the resulting field distribution p(B) providesa random sampling of the spatially inhomogeneous field B(r) over the VLunit cell:p(B) = 1AintegraldisplayAδ[B−B(r)]dr, (3.2)where the integral is over a unit cell of the VL of area A. In this chapterwe are concerned with the z-component of the magnetic field Bz (parallelto the c-axis of YBCO samples), and refer to it simply as B.For an ideal triangular VL, the spatial dependence of thez-component ofthe magnetic field in or outside a type II superconductor follows the modified853.2. The Magnetic Field Distribution p(B) in the Vortex StateLondon equation,−∇2B− ∂2B∂z2 +Bλ2Θ(z) =Φ0λ2 Θ(z)summationdisplayRδ(r−R). (3.3)Here λ = λab when the applied field is along the c-axis and the screeningsupercurrents flow in the ab plane, ∇2 is the 2D Laplacian, Θ(z) = 1 forz> 0 and zero otherwise, r is a 2D vector in the xy plane, R are the Bravaislattice vectors for the VL. We define the z axis as the normal to the surfaceof a superconducting slab with negative z outside the superconductor. Thesolution of Eq. (3.3) is easily obtained using the Fourier transform,B(r,z) =B0summationtextkeik·rF(k,z), where the dimensionless Fourier components, F(k,z),are given by [36],F(k,z) = 1λ2bracketleftBigΘ(−z)ekzΛ(Λ+k) +Θ(z)Λ2 (1−kΛ+ke−Λz)bracketrightBig. (3.4)Here Λ2 = k2 + 1λ2, and k = 2pia [nx + 2m−n√3 y] are the reciprocal latticevectors of the triangular VL, where m,n = 0,±1,±2.... A cutoff functionC(k), approximated by a simple Gaussian C(k) ≈ e−ξ2k22 , can be used toaccount for the finite size of the vortex core, where F(k,z) is replaced byF(k,z)C(k) [2, 4]. However, the corrections due to C(k) are very small inour case, so it will be omitted. An approximate solution for the magneticfield along z (both inside and outside the superconductor) is given byB(r,z) =B0summationdisplaykF(k,z)cos(k·r). (3.5)The second moment of p(B) at a depth z, σ2 = 〈B2〉−〈B〉2, where 〈..〉 isthe spatial average, is given byσ2 =B20 summationdisplayknegationslash=0F2(k,z). (3.6)The field distribution for a perfectly ordered triangular VL calculatedfrom Eqs. (3.2) and (3.5), at B0 = 52 mT and outside the superconductor863.2. The Magnetic Field Distribution p(B) in the Vortex State50 52 54 00.020.040.060.08p(B) ∆D=0∆D=0.3 mT∆D=0.6 mT0 100 200 300 400 500B(mT)00.511.5σ (mT)VL (Bulk)VL (z=−40nm)VL (z=−90nm)DVL (z=−90nm)(a)D(b)aa‘6Figure 3.1: (a) Simulation ofp(B) in an applied field of 52 mT at a distance90 nm from the superconductor using Eq. (3.5) convoluted with a Lorentzianof width ∆D. (b) The broadening of p(B) versus the applied field. Solidlines represent σ from Eq. (3.6) in the bulk, and 40 and 90 nm away fromthe superconductor. Long dashed line shows σ at 90 nm from Eq. (3.9)for D = 4 µm and f = 0.1. Inset: sketch of a possible vortex arrangementincluding vortex trapping at twin boundaries spaced by D and a regulartriangular vortex lattice elsewhere. In all figures λ(0) = 150 nm. 873.2. The Magnetic Field Distribution p(B) in the Vortex State0 20 40 60 80 100T(K)00.20.40.60.81σ(T)/σ(0)D=0D=1000 nmD=2000 nmD=4000 nmλ2(T)/λ20Figure 3.2: σ(T) (normalized at T = 0), from Eq. (3.9) for B0 = 52 mT,z = −90 nm, and f = 0.1, is plotted against T/Tc for different values of D.A d-wave temperature dependence of λ(T) is used [37], where λ(0) = 150nm.atz = −90 nm andλ = 150 nm (relevant to YBCO atT ≪Tc), is presentedin Fig. 3.1(a) (∆D = 0, defined below). It shows the characteristic high fieldskewness with a cutoff corresponding to the field at the core of the vortices.The sharp peak corresponds to the most probable fieldBsad at saddle pointsinB(r) midway between adjacent vortices. The low field cutoff occurs at thecenter of an elementary triangle of vortices. As we move farther from thesuperconductor,Bsad moves towards the applied field as the field approachesuniformity for z → −∞, i.e., p(B) → δ(B−B0). This crossover occurs asexp(2pia z), as the z variation of the Fourier components F(k,z) in Eq. (3.4)is controlled by k which takes values equal to or larger than 2π/a. However,if instead we consider a simple non-superconducting overlayer instead of freespace, then the limiting p(B) will be the intrinsic lineshape in the overlayermaterial.As mentioned above, p(B) is also affected by disorder in the VL due to883.2. The Magnetic Field Distribution p(B) in the Vortex Statepinning at structural defects in the crystal, where the superconducting orderparameter is suppressed. Such disorder causes broadening of the magneticresonance, obscuring the features expected from an ideal VL [25], addinguncertainty to parameters of interest such as λ and ξ. Relatively little isknown about the detailed characteristics of this disorder. Accounting forthe disorder of the VL is most often done by smearing the ideal lineshapewith a Gaussian or Lorentzian distribution of width ∆D, where the latteris a phenomenological measure of the degree of disorder [5, 38]. Calculateddistributions for an applied field B0 = 52 mT are shown in Fig. 3.1(a) for∆D = 0.3 and 0.6 mT, together with the ideally ordered VL (∆D = 0).Such disorder is more pronounced outside the superconductor and rendersthe lineshape symmetric when the depth dependent intrinsic VL broadeningis smaller than ∆D.One major difference between conventional µSR and β-NMR or LE-µSRis the stopping range of the probe. In conventional µSR the µ+ stoppingrange is ≈ 120 mg/cm2, yielding a fraction of a mm in YBCO. In contrast,in β-NMR or LE-µSR, the mean depth of the probe can be controlled on anm lengthscale from the surface. For implantation depths inside the super-conductor, comparable or larger than λ, the µSR lineshape (proportional top(B)) is nearly field independent for 2Bc1 ≤B0 ≤Bc2 (for a≪λ), and thesecond moment of p(B) follows the formula [39],σ ≈ 0.00609Φ0λ2(T) , (3.7)neglecting the cutoff field. Using the latter makesσ slightly field-dependent,but the corrections are small for fields B0 ≪Bc2. Outside the superconduc-tor, the magnetic field inhomogeneity of the VL vanishes over a lengthscalethat depends on the spacing between vortices, a. In particular, the recoveryto a uniform field occurs on a lengthscale of a2pi [31]. The field distributionis thus strongly field dependent when a(B0) is of the order of |z|. This isshown in Fig. 3.1(b), where σ due to the VL given in Eq. (3.6) is plottedagainst the applied field at a distance of 90 nm and 40 nm above the sur-face. In low magnetic fields, the magnetic resonance lineshape outside the893.2. The Magnetic Field Distribution p(B) in the Vortex Statesuperconductor is sensitive to both the intrinsic inhomogeneity of the VL aswell as any additional broadening from disorder. However, in high magneticfields the linewidth is dominated by VL disorder.Taking the view that the dominant source of disorder is due to twin orgrain boundaries [40], one can model the effect of disorder on the regular VLin different ways. The simplest is to assume that, in addition to the regulartriangular lattice, a fraction of vortices is trapped along the structural de-fects such as twin or grain boundaries as shown in the inset of Fig. 3.1(b).The local field in real space will be the superposition of both contributionsB(r,z) = Bvl(r,z)+Bdis(r,z),=Bvl0 summationdisplaykF(k,z)eik·r +Bdis0 summationdisplaygF(g,z)eig·r, (3.8)where Bvl(r,z) is the field due to the regular VL, Bdis(r,z) the field dueto the vortices pinned by disorder and g is some generally incommensuratewave vector related to the pinning, which for simplicity we take to be ofthe form g = 2pia′ nx + 2piDmy, where a′ is the spacing between vortices ina boundary, and D is the separation between boundaries as drawn in theinset of Fig. 3.1(b). The average field is then B0 =radicalBig(Bvl0 )2 +(Bdis0 )2,where Bdis0 =fB0 and f is the fraction of pinned vortices (0 ≤f <1). Thesecond moment of B(r,z) from Eq. (3.8) can be then easily calculated:σ2 = B20bracketleftBig(1−f2)summationdisplayknegationslash=0F2(k,z)+f2 summationdisplaygnegationslash=0F2(g,z)bracketrightBig. (3.9)It is clear from Eqs. (3.4) and (3.9), that the broadening from the VL(first term) at a distance z outside the superconductor becomes small athigh magnetic fields where |z| ≫ a. However the broadening outside thesuperconductor due to disorder (second term) remains large provided |z| isnot much larger than D2pi. Since D and f depend on the arrangement oftwin boundaries we expect them to be sample dependent. In addition, onemay also anticipate that f will decrease at high magnetic fields where theincreased repulsive interaction between vortices overcomes vortex-pinning.903.2. The Magnetic Field Distribution p(B) in the Vortex StateTherefore, we assume a simplified phenomenological parameterization f =δB−γ0 , where δ is temperature and sample dependent and γ ≥ 1.The broadening of the field distribution due to a regular VL can be sig-nificantly larger when introducing the effect of disorder due to the twin andgrain boundaries. When taking the disorder into account, σ of Eq. (3.9)is no longer zero at high magnetic fields as seen in Fig. 3.1(b). This isbecause the broadening has a disorder component which decays on a lengthscale of D rather than a, where D ≫ a (we also assume D ≫ a′ and thusignore the effect of the spacing within the twin boundaries). Consequentlyσ shows a strong deviation from the ideal VL result as seen in Fig. 3.2as D increases. In this case the second moment from Eq. (3.9) no longerscales with 1/λ2 as predicted for an ideal VL (see Eq. (3.7)). In partic-ular, at low T, the broadening is almost T-independent irrespective of thesuperconducting gap structure. It is interesting to note that the first µSRstudies on powder samples of cuprates showed a very flat variation in thelinewidth [41, 42]. This was taken as evidence for s-wave superconductiv-ity. Later measurements on high quality crystals of YBa2Cu3O7−δ showeda much different low temperature behaviour [43], and, in particular, a linearvariation in 1/λ2(T) consistent with d-wave pairing [44]. Although the line-shapes in powders are expected to be more symmetric than in crystals dueto the additional disorder and random orientation, the different temperaturedependence is surprising since it was thought that the line broadening fromdisorder should also scale with 1/λ2(T) [25]. The current work providesa clear explanation for the discrepancy between powders and crystals. Inpowders, the line broadening is dominated by long wavelength pinning ofvortices at grain boundaries. Consequently the resulting broadening at lowtemperature reflects variations in the vortex density and is thus only weaklydependent on temperature. In later work on crystals, the contribution fromsuch long wavelength pinning is much less important. This is evident frombulk µSR in crystals where one observes the expected characteristic line-shape associated with a VL [25].913.3. Experimental Details3.3 Experimental DetailsThe measurements were carried out on three different near-optimally dopedYBCO samples, two flux-grown single crystals and a thin film. I) Thetwinned single crystal in the form of a platelet ∼ 0.5 mm thick with anarea ∼ 2 × 3 mm2 had Tc = 92.5 K. It was mechanically polished with0.05 µm alumina, then chemically etched with a dilute (0.8%) Bromine so-lution followed by annealing at 200◦ C in dry N2 to improve the surfacequality. It was then sputter coated with a 120 nm thick Ag film (99.99 %purity) at room temperature in an Ar pressure of 30 mtorr. The deposi-tion rate was 0.5 ˚A/s, and to ensure Ag uniformity, the crystal was rotated.II) The optimally doped detwinned single crystal had Tc = 92.5 K, ∼ 0.5mm thickness, and area ∼ 3×3 mm2. The crystal was cleaned, annealed,and mechanically detwinned. A 120 nm thick Ag, from the same source asabove, was sputtered onto the prepared surface under similar conditions.III) The film of Tc = 87.5 K, critical current density Jc = 2.106 A/cm3 and600 nm thickness, supplied by THEVA (Ismaning, Germany), was grown bythermal co-evaporation on a LaAlO3 substrate of area 9×8 mm2. The filmwas coated in situ with a 60 nm silver layer (99.99% purity).The experiments were performed using the β-NMR spectrometer at theISAC facility in TRIUMF, Canada, where a highly nuclear-spin-polarizedbeam (intensity ∼ 106 ions/s) of 8Li+ is produced using collinear opticalpumping with circularly polarized laser light [27]. The beam is directedonto the sample which is mounted on the cold finger of a He flow cryostatand positioned in the centre of a high homogeneity superconducting solenoid.The beamline and entire spectrometer are maintained in ultrahigh vacuum(10−10 torr). Inβ-NMR measurements, the 8Li+ nuclear spin polarization ismonitored via its asymmetric radioactive beta decay (lifetime τ = 1.203 s),where the high energy (several MeV) beta electron is emitted preferentiallyopposite to the nuclear spin direction. The experimental asymmetry, definedas the ratio F−BF+B of the count rates in two plastic scintillation detectorsplaced in front (F) and at the back (B) of the sample, is proportional to theprobe’s spin polarization [27, 28].923.3. Experimental Details−150 −120 −90 −60 −30 0 30Depth (nm)010002000300040005000Implantation Profile ρ(z)−150 −120 −90 −60 −30 0 30AgAg filmcrystalVacuum5 keV8 keVVacuum(a)(b)YBCOYBCOFigure 3.3: Implantation profiles of 8Li+ at energies of (a) 5 keV into 60nm of Ag with the mean at 40 nm away from YBCO film, and (b) 8 keVinto 120 nm of Ag with the mean at 90 nm away from YBCO crystals, ascalculated via TRIM.SP (Ref. [45]). Solid lines are phenomenological fits.933.3. Experimental DetailsThe whole spectrometer can be biased at high voltage, allowing one totune the implantation energy of 8Li+ ions and their implantation depthbetween 5-200 nm. Therefore, the implanted 8Li+ can monitor the depthdependence of the local magnetic field distribution in materials at nm scaleby measuring the NMR lineshape in a manner analogous to conventionalNMR [2, 28, 29]. In this work, the 8Li+ ions are decelerated to stop in theAg overlayer deposited on each of the three YBCO samples. Implantationprofiles of 8Li+ were calculated using the TRIM.SP code [45], examples ofwhich are shown in Fig. 3.3. The implantation energies used in this study(8 keV in the crystal samples and 5 keV the film), were tuned to stop allthe 8Li+ within the Ag. The mean distances are 90 and 40 nm from theAg/YBCO interface in the crystals and film, respectively.Theβ-NMR measurement is carried out by monitoring the time averagednuclear polarization through the beta decay asymmetry, as a function ofthe radio frequency (RF) ω of a small transverse oscillating magnetic fieldB1 = B1 cos(ωt)ˆx, where B1 ∼ 0.01 mT. When ω matches the Larmorfrequency ωLi = γLiBlocal, where γLi = 6.3015 kHz/mT is the gyromagneticratio andBlocal is the local field, the 8Li+ spins precess aboutBlocal, causinga loss of polarization. To establish the vortex state in the YBCO samples,they are cooled in a static magnetic field B0 ≥ Bc1 applied parallel to thec-axis of YBCO (normal to the film and platelet crystals). B0 is also parallelto both the initial nuclear spin polarization and the beam direction. Thelocal field sensed by the 8Li+ is determined by the applied field and theinternal magnetic field generated by the screening currents associated withthe vortex lattice. Thus, Blocal is distributed over a range of values, whichcan be calculated usingp(B) =integraldisplay 0−ddzρ(z) 1AintegraldisplayAdrδ[B−B(r,z)]. (3.10)where ρ(z) is the implantation profile calculated using TRIM.SP given inFig. 3.3.When 8Li+ is implanted in Ag (with no superconducting substrate) attemperatures below 100 K, it exhibits a single narrow resonance at the Lar-943.4. Resultsmor frequency [28]. The resonance should yield an approximately Gaussiandistribution caused by nuclear dipolar moments [46]. However, continuouswave RF leads to a power-broadened Lorentzian lineshape, whose linewidthis small (∼1 kHz ≈ 0.15 mT) and corresponds to the dipolar broadeningdue to the 107,109Ag nuclear moments and RF power broadening [47]. Inthe presence of any additional magnetic inhomogeneity in the Ag, due forexample to a VL associated with a superconducting substrate, the observedresonance lineshape will be a convolution of the narrow RF power broad-ened Lorentzian of Ag with the (depth dependent) field distribution due tothe VL in the substrate. There are unique aspects of measuring the fielddistribution in the Ag overlayer compared to the superconductor itself. Asmentioned above, in high magnetic fields it is possible to isolate and studythe broadening due to VL disorder that occurs on a long length scale. Alsoin low magnetic fields, where the broadening is dominated by the VL, itshould be possible to measure λ in magnetic superconductors since the fielddistribution above the sample is free of any internal hyperfine fields thatmake a bulk measurement impossible [32].3.4 ResultsThe β-NMR resonances were measured as a function of temperature underfield-cooled conditions at fields ranging from B0 = 20 mT to 3.3 T in eachone of the three samples. Fig. 3.4, shows typical resonance lineshapes atvarious temperatures in sample I with B0 = 51.7 mT. Above Tc, the linebroadening is small and temperature independent as expected from nucleardipolar broadening. Below Tc, the field distribution in the Ag overlayerbroadens dramatically from the VL in the underlying superconductor. Suchbroadening was observed in all samples and at all magnetic fields, althoughthere are significant variations as a function of both magnetic field andsample as discussed below. The first thing to note is that the lineshape isvery symmetric and fits well to a simple Lorentzian. This is much differentfrom the asymmetric lineshape observed with conventional µSR in samplessimilar to I and II [25, 43]. The other significant difference between the953.4. Results46 48 50 52 54 56 58Blocal(mT)Asymmetry100 K20 K4.5 K80 KFigure 3.4: βNMR resonances in Ag/YBCO (crystal I) at temperatures100 K, 80 K, 20 K, and 4.5 K measured in a magnetic field B0 of 52.3 mTapplied along YBCO c-axis. The solid lines are best fits using a Lorentzian.963.4. Resultscurrent results and previous bulkµSR measurements [4, 9, 25, 43] on crystalsis that the broadening at low temperatures is only weakly dependent ontemperature, as may be seen by comparing the resonances at 20 K and4.5 K. In contrast, the broadening from an ordered VL lattice scales with1/λ2 and consequently in YBCO shows a strong linear T-dependence at lowtemperatures due to the d-wave superconducting order [43, 44].The observed lineshape fits well to a single Lorentzian which is a convolu-tion of two Lorentzians, one from vortices in the superconducting state witha full width at half maximum (FWHM) ∆sc, and one from other sources de-termined from the normal state of FWHM ∆ns. The width of a convolutionof two Lorentzians is the sum of the individual widths: ∆(T) = ∆sc(T)+∆ns.Therefore, thecontributionfromvorticesinthesuperconductingstatecanbeobtained by simply subtracting the temperature independent normal statewidth. Fig. 3.5 shows the resulting ∆sc(T) as one enters the superconduct-ing state in samples I and II. At low field, the measured width (∼ 2.2 mT)at low temperature is larger than expected from a regular VL and decreasessignificantly in the detwinned crystal to about ∼ 0.6 mT. For comparison,simulations using Eq. (3.10) and the 8Li+ stopping profile in Fig. 3.3(b);indicate that the broadening due to a regular VL is only ∆VL ∼ 0.3 mT.At 3.33 T, the discrepancy between the observed width (see Fig. 3.5) andthat expected from a regular VL is even more dramatic. At this high fieldthe vortices are spaced so closely (a ≈ 27 nm), that there should be nodetectable broadening from a regular VL for our stopping depths. This canbe seen clearly from the simulation in Fig. 3.1(b), where the VL broadeningapproaches zero at high fields. In contrast, the data at 3.33 T shows sig-nificant broadening below Tc which is therefore solely attributed to vortexdisorder on a long length scale.The temperature dependence of the broadening is also much weaker thanexpected from a regular VL in YBCO, where 1/λ2 has a strong linear termdue to the d-wave order parameter [43, 44]. The observed temperaturedependence fits well to our model of disorder, where ∆sc(T) is compared toan estimate of the FWHM given by ∆DVL ≈ 2.355σ, where σ is given in Eq.(3.9). This leads to an estimate of D of the order of a micron, consistent973.4. Results0 20 40 60 80 100 120T(K)0123Linewidth ∆ sc (mT)B0=52.3 mT, D=1 µm, f~0.08  (TW crystal)B0=3.33 T, D=1 µm, f~0.0006 (TW crystal)B0=52.3 mT, D=2 µm, f~0.01 (UTW crystal)Figure 3.5: The vortex-related broadening below Tc, ∆sc(T) = ∆(T) −∆ns of the twinned (full symbols) and detwinned (opaque squares) YBCOcrystals in an applied field B0. ∆(T) is the linewidth at temperature T ofthe Lorentzian fits and ∆ns is the constant linewidth in the normal state.Solid lines represent a fit using ∆DVL = 2.355σ where σ is given in Eq. (3.9)and D and f are varied to fit the data. A d-wave temperature dependenceof λ(T) in YBCO is used, ( Ref. [37]) where λ(0) = 150 nm.983.4. ResultsAsymmetryUTW crystalFilm−8 −6 −4 −2 0 2 4 6 8TW crystalB−B0 (mT)AsymmetryFigure 3.6: Comparison of the field distributions in the three samples takenat temperatures 100 K (top panel), 5 K (crystals) and 10 K (film). Thex-axis is shifted by B0, the applied field which is 52.3 (51.7) mT for thecrystals (film). Solid lines are Lorentzian fits and dashed lines are simulationdescribed in the text.993.4. Resultswith the the separation between twin boundaries or grain boundaries [48].In the detwinned crystal, D is found to be larger but not infinite since thedetwinning is not complete. The fraction of vortices f pinned by structuraldefects in the twinned crystal is about ∼ 0.1 at low field (52 mT) anddecreases considerably at high field (3.33 T). Thus, the amplitude of theenhanced vortex density at the defects, fB0, varies between 2-4 mT atall fields. In the detwinned crystal, f ∼ 0.01, is an order of magnitudesmaller than in the twinned crystal at the same field, with small variationin the vortex density (0.5 mT) compared to the twinned crystal. Theseresults are consistent with expectations from pinning at twin boundaries.In particular, one expects the fraction of vortices pinned will decrease inthe partially detwinned crystals. Also, it is reasonable to expect that inhigh magnetic fields the fraction of vortices pinned will decrease due tothe smaller separation between vortices, and the resulting increase in therepulsive interaction.In Fig. 3.6, the spectra in all three samples above and belowTc are com-pared with the corresponding simulated field distributions. The observedlineshapes are all symmetric and significantly broader than expected, show-ing little or no sign of the characteristic VL field distribution. Simulationof the VL lineshape (dashed lines) was done using Eqs. (3.4), (3.5), and(3.10), for λ = 150 nm, and was convoluted with a Lorentzian represent-ing the normal state spectra with ∆ns = 0.3 mT. The theoretical lineshapefor the film is broader and asymmetric because it is weighted by the 8Li+stopping distribution which was on average closer to the superconductor.The lineshapes for the crystals are almost symmetric as the ideal VL line-shape at the depths of an average 90 nm away from the superconductor arenarrower than the Lorentzian they are convoluted with. The magnetic fielddependence of the superconducting linewidth ∆sc(T) at low temperatures ∼4.5-10 K is plotted in Fig. 3.7. In all samples, we find that the broadeningis largest at low field and decreases gradually with increasing field. Also, inall cases the broadening remains large and well above the prediction froma regular VL, approximated by ∆VL ≈ 2.355σ, where σ of an ideal VL isgiven in Eq. (3.6) and weighted by the 8Li+ profile given in Fig. 3.3. The1003.4. Results10 100 1000B0 (mT)01234Linewidth ∆ sc (mT)TW crystalUTW crystalFilmVL (zavg=−40nm)VL (zavg=−90nm)DVL, D=1µm, f~85B0−1.14DVL, D=1µm, f~25B0−1.1DVL, D=2µm, f~8.5B0−1.14Figure 3.7: Superconducting broadening ∆sc(T) of β-NMR resonance spec-tra at temperatures ∼ 4.5-10 K in the three samples. The experimentalbroadening is compared with the linewidth of an ideal VL ∆VL (dashedlines), and ∆DVL of a disordered VL (solid lines), which are both weightedby the 8Li+ profile given in Fig. 3.3.1013.5. Discussion and Conclusionsbroadening is substantially smaller in the detwinned crystal compared tothe other samples. One can account for all of the data using a linewidth dueto a disordered VL, ∆DVL ≈ 2.355σ, where σ is now given in Eq. (3.9)) andweighted by the 8Li+ profile plotted in Fig. 3.3. The data is well fitted (seeFig. 3.7) by assuming that the twin/grain boundaries spacing D is sampledependent of the order of a few microns, and by assuming a phenomenolog-ical form for the fraction of pinned vortices f = δB−γ0 with γ ∼ 1.1 and δsample dependent.3.5 Discussion and ConclusionsIt is clear that theβ-NMR lineshapes in the Ag overlayer differ substantiallyfrom that expected from a well-ordered VL field distribution. This has littleto do with the method of observation. For example, in the conventionalsuperconductor NbSe2, β-NMR shows the expected VL lineshape [2]. Thelineshapes reported here in the YBCO film are also qualitatively differentthan that seen with LE-µSR in a YBCO film coated with a 60 nm thickAg layer [36]. In that experiment the authors found a more asymmetriclineshape in the Ag overlayer which was closer to that of a regular VL.Some of this difference may be due to the different pinning characteristics ofthe samples, although the YBCO film used by Niedermayer et al. was fromthe same source as sample III. Also, the LE-µSR experiment was probingthe VL closer to the interface and in a lower applied field where the disorderis less important compared to the contribution from the ordered VL. Thesymmetry and large broadening of the lineshape at low fields cannot beaccounted for by VL melting (at a reentrant vortex liquid state near Bc1)which would instead yield a motional narrowing of the field distribution [49].The observed resonances in the current experiment are dominated bylong range variations of the vortex density across the face of the sample[18]. Such disorder in the VL can produce a symmetric lineshape [38, 50],and can broaden the field distribution significantly compared to that ofthe corresponding ordered state [51], Weak random pinning or point-likedisorder due to oxygen deficiency may slightly distort the VL, and may also1023.5. Discussion and Conclusionsbroaden the lineshape [39]. However, the correlated disorder due to thetwin and grain boundaries is dominant at long wavelengths [40, 52], andtherefore we are mostly sensitive to the twin/grain boundaries. Indeed, theposition of the probe outside the superconductor enhances its sensitivityto long wavelength disorder, as the proximal fields fall off with distanceas exp(−2π|z|/D) where D is the wavelength of the inhomogeneity of thefield [53]. The broadening is reduced in a detwinned crystal where the twinboundaries are more sparse as shown in Fig. 3.6, thus the vortex densityvariation across the face of the sample is smaller than in the twinned crystalsas f is largely reduced.The extrinsic broadening due to disorder at low temperature, ∆D =∆sc −∆VL, reported here is between 0.5 mT and 2.5 mT. This is remark-ably close to the additional Gaussian broadening required to explain line-shapes in bulk µSR measurements on crystals [25, 38, 54]. In the bulk,this extrinsic broadening is small compared to the intrinsic VL broadening,whereas outside the sample the reverse is true. It is important to note theT-dependence of the extrinsic broadening in Fig. 3.5 does not follow thesuperfluid density (∝ 1/λ2) which varies linearly at low-T because of thed-wave order parameter [44, 55]. Instead, we observe a much weaker T-dependence. This is expected from our model of disorder which occurs on along length scale D. For example, at low-T where λ is short compared to D,the flux density outside the sample is determined solely by inhomogeneitiesin the vortex density, and is independent of the superfluid density as the vor-tices are static and well-pinned in the twin boundaries. The current resultsmay also explain early µSR work on HTSC powders and sintered sampleswhich mistakenly indicated an s-wave T-dependence of 1/λ2 [41, 42]. It islikely in these cases the linewidth was dominated by extrinsic VL disorder ona long length scale. This tends to flatten the T-dependence of the linewidthand the effective λ obtained from the analysis [41, 56]. Therefore we con-clude that although the linewidth obtained from powders can be useful inmaking rough estimates of λ, one cannot extract accurate measurements ofλ or its T-dependence without additional information about the source ofbroadening and in particular VL disorder.1033.5. Discussion and ConclusionsIn conclusion, we have measured the magnetic field distributions due tothe vortex state of YBCO using β-NMR. We find a significant inhomoge-neous broadening of the NMR attributed to the underlying VL in YBCO.However, the observed resonances have several unexpected properties. Inparticular, they are broader and more symmetric than for an ideal VL. Theanomalous broadening is most evident in high fields where there is no sig-nificant contribution from the regular VL. These effects are attributed tolong wavelength disorder from pinning at twin or grain boundaries. Thetemperature dependence of the disorder-related broadening does not scalewith 1/λ2, suggesting there is a contribution to the linewidth in the bulk ofthe vortex state that does not track the superfluid density. This is likely tohave only a minor effect on the interpretation of data on crystals where theobserved lineshape is close to that expected from a well ordered VL. 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B 42, 8019 (1990).110Chapter 4Vortex Lattice Near theSurface of Pr1.85Ce0.15CuO4−δ4.1 IntroductionIn contrast to the hole-doped high-Tc cuprate superconductors, less is knownabout the corresponding electron-doped materials.1 Some of these sys-tems such as Pr2−xCexCuO4−δ (PCCO) and Nd2−xCexCuO4−δ have re-ceived special attention because of their similarities with the hole-dopedLa2−xSrxCuO4−δ. One important issue is the correlation between mag-netism and superconductivity in the vortex state [1, 2, 3]. In this state,the internal magnetic field distribution p(B) associated with the vortex lat-tice holds important insights about the phase diagram. For example, p(B)has special features that identify the nature of the vortex state. Only afew experiments can measure p(B), such as µSR and small-angle neutrondiffraction (SANS) [3, 4]. Such techniques are sensitive not only to fieldinhomogeneities of the vortex state, but also to magnetic fields of the rareearth moments and magnetic impurities. However, as bulk techniques, theycannot be used to study thin films, which for the electron-doped cuprates areoften more homogeneous in oxygen content than single crystals. Until now,only Low-Energy µSR (LE-µSR) has had the capability to measure p(B)in a depth-resolved manner both above and below the surface [5]. How-ever, LE-µSR is currently limited to relatively low magnetic fields. In this1A version of this chapter has been published. H. Saadaoui, W.A. MacFarlane, Z.Salman, G.D. Morris, K.H. Chow, I. Fan, P. Fournier, M.D. Hossain, T.A. Keeler, S.R.Kreitzman, C.D.P. Levy, A.I. Mansour, R.I. Miller, T.J. Parolin, M.R. Pearson, Q. Song,D .Wang, and R.F. Kiefl, Physica B 404, 727 (2009). β-NMR investigation of the vortexlattice near the interface of silver and Pr1.85Ce0.15CuO4−δ thin films.1114.2. Experimental Details−40 40 120 200 280 360depth (nm)8 Li+  profile [a.u.]E= 5 keVE=28 keVAg PCCOFigure 4.1: TRIM simulation of 8Li+ stopping profile into 40 nm of Ag on300 nm of PCCO, at energies of 5 keV and 28 keV.chapter, we use the β-NMR technique, to measure p(B) near the interfacebetween a PCCO film and a thin silver overlayer by stopping the probe ionsin the Ag.4.2 Experimental DetailsOur measurements were carried out on a 300 nm thick PCCO film grown byPulsed Laser Deposition (PLD) at Sherbrooke on a 10×8 mm SrTiO3 sub-strate. The PCCO film was near optimally doped (x ∼ 0.15) with Tc ∼ 22K as found by SQUID measurements. X-ray diffraction of the PCCO filmrevealed the existence of inter-penetrating layers of the insulating impurityphase CeO2. This is also confirmed using high-resolution transmission elec-tron microscopy (HRTEM) showing intergrowths with thicknesses ∼ 5nm,1124.3. Results and Discussionsandwiched between much thicker (50 nm) superconducting PCCO layers[6]. In this case, a stack of correlated vortex states may occur between thedisconnected PCCO layers. In order to measure p(B) near the PCCO sur-face, a 40 nm thin film of Ag was thermally evaporated onto the PCCO. The8Li+ β-NMR resonance in Ag films below 100 K is known to be temperatureindependent and narrow [7].The experiments were performed at TRIUMF’s ISAC facility in Van-couver, Canada. In β-NMR, highly spin-polarized 8Li is obtained in-flightvia optical pumping with circularly polarized laser light [9]. The time andfrequency evolution of the implanted 8Li spin polarization can then be usedto monitor magnetic fields near the surface of a material. The 8Li+ nuclearmagnetic moments precess in the local field Blocal, with a Larmor frequencyωLi = γLiBlocal, where γLi = 0.63015 kHz/G.A deceleration system is in place to control the kinetic energy of the im-planted 8Li+ beam, i.e. controlling the depth of the beam into the studiedmaterial. This is particularly powerful as it allows depth-resolved measure-ments of p(B) in materials at nm depths [7, 10, 11]. This is an advantageover conventional µSR and other existing β-NMR spectrometers which arelimited to bulk studies. In this study, the 8Li+ ions are decelerated froman initial kinetic energy E = 28 to 5 keV and stop in the Ag layer, withan implantation profile (Fig. 4.1) peaked 28nm from Ag/PCCO interfaceaccording to TRIM simulations[12].Thevortexstateisobtainedbyfield-cooling(FC)instaticmagneticfieldsBapp >Bc1 applied perpendicular to the film surface and parallel to the c-axis of the oriented PCCO film. Two types of measurements were done, (i)spin-polarization versus frequency to measure p(B), where the polarizationis subjected to a pulsed RF, and (ii) spin-polarization against time with noapplied RF to yield the spin-lattice relaxation rate.4.3 Results and DiscussionTypical spin relaxation curves are shown in Fig. 4.2. In this experiment,the beam is implanted into the sample for 4 s, and is off for 8 s. The1134.3. Results and Discussion0 2000 4000 6000 8000 Time(ms)00.020.040.06Asymmetrya) 5 keVb) 28 keVFigure 4.2: Spin relaxation spectra in Ag/PCCO at 5 K, taken in an appliedstatic magnetic field Bapp = 200 G, and 8Li+ energies of 5 and 28 keV. Thesolid line is a fit using a phenomenological biexponential function for beamon (0 to 4 s) and beam off (4 s to 8 s) [8].average nuclear spin polarization of 8Li in the sample, measured in this way,approaches an equilibrium value, determined by a balance between the rateof incoming polarized 8Li+ and the spin-lattice relaxation rate 1/T1. Afterthe beam is turned off, the polarization relaxes towards zero. At an energyof 5 keV, about 90% of 8Li+ stop in Ag, so the relaxation is expected to bedominated by the Korringa relaxation in Ag [7]. At 28 keV, most of the ionsstop in PCCO, where at this low magnetic field, T1 is much shorter thanthe 8Li lifetime and almost no polarization is observed. Therefore, althoughsome of the 8Li+ does stop in the PCCO at 5 keV, the resonance signal isexclusively due to 8Li+ in the Ag layer.1144.3. Results and DiscussionResonance measurements in low applied fields of Bapp = 216 G as afunction of temperature show a dramatic broadening below Tc, while theline is narrow and temperature independent above Tc. The spectra shownin Fig. 4.3 are quite different than expected for the ideal vortex lattice. Thehigh-field tail (corresponding to high magnetic fields at the vortex cores)is absent, and instead the line is nearly symmetric with a marginal lowfield tail, suggesting a departure of the vortex structure from a perfect 3-dimensional (3-D) lattice. Our data, from a wide range of applied fields:100 G to 6.5 T, show a similar behavior of the lineshape. The low fieldtail may be associated with the tendency of vortices to re-arrange into a 2-dimensional (2-D) lattice in anisotropic materials [13]. A similar lineshapewas observed by µSR in Bi2Sr2Cu2O8+δ samples and was attributed to a2-D vortex structure [13]. Although PCCO is known to be quite anisotropic,a 2-D like response may also be due to the presence of CeO2 intergrowthsleading to very thin decoupled superconducting PCCO layers. A disorderedlattice may also lead to a symmetric or negatively skewed p(B) as shown byU. Divakar et al. [14]. Similar β-NMR experiments on YBa2Cu3O7−δ singlecrystals and films have shown no evidence of the low field tail and ratherexhibit a symmetric lineshape with broadening determined largely by vortexdisorder [15].Another striking feature in Fig. 4.3, is the small positive shift of the res-onance frequency upon cooling belowTc. The resonance frequency is plottedin Fig.4.4(a). The net shift is of the order of 1 Gauss. SQUID measurementsof the susceptibility on this sample in 1 G and 100 G fields applied parallel toc-axis, under both FC and zero-FC, have shown no paramagnetic Meissnereffect [16]. Impurity phases such as (Pr,Ce,La)2O3 are ruled out by neutronscattering below 7 T [17], while pure CeO2 is non-magnetic.A similar paramagnetic shift was measured byµSR in bulk single-crystalPCCO samples by Sonier et al., where the net shift at 5 K varied between10 G at low applied fields (100 G) to 1 G at high field (2 kG) [3]. Thiswas attributed to copper moments (∼ 0.44 µB) induced by the externalmagnetic field. A similar study by Kadono et al. on Pr0.89LaCe0.11CuO4also found a paramagnetic shift in low fields of about 1-5 G associated with1154.3. Results and Discussion200 210 220 230 240Blocal (G)Asymmetry25 K18 K12 K4 KFigure 4.3: The βNMR asymmetry spectra of 8Li+ implanted at 5 keV inthe Ag/PCCO at different temperatures. The dashed line shows the Larmorfrequency in the applied magnetic field of Bapp = 216 G. The solid lines areLorentzian fits.1164.3. Results and Discussion0 10 20 30T (K)02468FWHM (G)217217.5B res (G) (a)(b)Figure 4.4: (a) The resonance frequency, Bres, versus T in an appliedfield Bapp = 216.62 G (dashed line), fitted by Bapp + 0.8(1 − (T/22)4)2(solid line). (b) The linewidth (FWHM) versus T. The solid line representsFWHMNS + 4.2(1 − (T/22)2.2)2, where FWHMNS = 0.74 G is the normalstate broadening (dashed line). All data are extracted from Lorentzian fitsof the resonances. 1174.3. Results and Discussionthe weak (van Vleck) magnetization of the Pr ions, due to interaction withantiferromagnetic Cu moments [1]. However, with increasing field, the shiftdecreased and became negative above 1 kG. We have measured a similareffect in 4.1 T with a shift ∼ −8 G, which is consistent with the work ofKadono et al..Thefullwidthathalfmaximum(FWHM)oftheLorentzianfitsisplottedin Fig. 4.4. It is temperature independent above Tc, where the width is dueto nuclear dipolar broadening in Ag, and increases dramatically below Tc,due to the field inhomogeneity of the vortex state. In the bulk, and for anideal 3-D arrangement of vortices, the second moment, i.e. FWHM/2.355,is proportional to λ−2 with the temperature dependence sensitive to thesymmetry of the order parameter [3]. However, because of the large errorbars on the measured FWHM at low temperatures, and absence of databelow 3.5 K, it is hard to draw a conclusion about the nature of the orderparameter in PCCO. The FWHM follows the phenomenological form (1−(T/Tc)a)2, where a≈ 2.2. This unexpected temperature dependence of theFWHM, could be related to the fact that the 8Li+ probes are outside thePCCO, where the intrinsic vortex lattice broadening is suppressed and p(B)is more sensitive to long-range fluctuations of the vortex density [18].In conclusion, we have made the first direct measurements of the mag-netic field distribution near the surface of an electron-doped superconductorin the vortex state. We have observed an unusual asymmetry ofp(B), incon-sistent with the ideal vortex lattice, and probably due to both the artificialand intrinsic anisotropy of the sample. We have also observed a paramag-netic shift of the resonance frequency in low fields.118Bibliography[1] R. Kadono, K. Ohishi, A. Koda, W. Higemoto, K. M. Kojima, M.Fujita, Shin-ichi Kuroshima and K. Yamada, J. Phys. Soc. Jpn. 732944 (2004).[2] K. M. Kojima, K. Kawashima, M. Fujita, K. Yamada, M. Azuma, M.Takano, A. Koda, K. Ohishi, W. Higemoto, R. Kadono, and Y. 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Wang, andW. A. MacFarlane, J. Magn. Reson. 191, 47 (2008).121Chapter 5Summary and ConclusionsIn this work, we have studied the magnetic properties of the cuprate super-conductors using the β-NMR technique. In particular, we have addressedtwo problems: (i) the occurrence (or lack) of time-reversal symmetry inHTSC near the surface, and (ii) the disorder of the vortex lattice. To an-swer these questions, the magnetic field distribution was measured near thesurface of the superconductor. The measurements were done outside thesuperconductor rather than inside for a few reasons: (i) The Meissner stateinside the superconductor shields the magnetic field due to any magnetic or-der, and therefore any spontaneous fields due to TRSB are greatly reducedin the bulk. (ii) Vortex lattice disorder that occurs on a long length scalecannot be probed inside the superconductor where the features of the shortrange ordered lattice are dominant. (iii) Fast spin-relaxation of 8Li+ insideYBCO and PCCO at low magnetic fields leads to weak or no resonance, andthe resonances at high field are also complicated by the nuclear quadrupo-lar interactions. (vi) Implanting 8Li+ ions inside the superconductor mayperturb the superconductivity.The measurements were done in silver evaporated onto the superconduc-tors. The Ag capping layer is used as a stopping material for low energy8Li+ ions. The separation between the mean stopping depth of 8Li+ and theinterface of the superconductor is varied by changing the thickness of the Aglayer and/or the energy of 8Li+ ions. Silver is an ideal material for measur-ing a simple NMR resonance, and exhibits a small linewidth in comparisonwith the field inhomogeneities expected from the vortex lattice at high ap-plied fields, or TRSB fields. Silver is also an ideal normal metal for coatingYBCO, as it does not chemically react or diffuse into YBCO, unlike othermetals (e.g. Nb, Cu) which are very reactive with the oxygen in YBCO, and122Chapter 5. Summary and Conclusionscan form an insulating oxide layer at the interface [1]. However, a limitationof Ag is its surface degradation and agglomeration with time [2] (see Ap-pendix B.2). This has been overcome here by taking our data shortly aftercoating the superconductors with silver. Studying the normal-metal/HTSCinterface is also of importance for technological applications, where under-standing the interface is critical for manufacturing superconducting devicesbased on proximity or Josephson effects [3, 4, 5].To seek evidence of TRSB, the magnetic field distribution was measuredin thin silver layers with thicknesses ranging from 15 to 50 nm depositedon YBCO films. The measured field distribution broadens upon coolingbelow Tc, signaling the appearance of disordered magnetism from the un-derlying superconductor. The linewidth of the field distribution, above Tc,is consistent with the temperature independent nuclear dipolar broaden-ing from silver moments. Below Tc, the linewidth increases almost linearlywith decreasing temperature without saturation or any sign of the secondarytransition observed by some tunneling experiments. The broadening of thefield distribution occurred in Ag deposited on YBCO films of both (110) and(001) crystal orientation. Theories suggest that time-reversal symmetry canbe broken near the surface of (110)-oriented YBCO, but not in the (001)-oriented films [6]. Hence, the measured broadening is unlikely of a TRSBorigin. This conclusion is further supported by the field dependence of thebroadening at 10 K, which increases linearly with the applied field. Fromthis, we estimate the net broadening extrapolated to zero field is small ∼ 0.2G, and nearly equals the normal state broadening. We conclude that anyspontaneous fields generated near the Ag/YBCO interface must be smallerthan 0.2 G at a distance of 8 nm.TRSB has been observed by only a few studies. In some tunnel junc-tions, it was estimated that spontaneous fields of a fraction of Tesla could begenerated near the surface of YBCO films. Our results are inconsistent withthese measurements. In our experiment, we detected no strong magneticfield, but rather small randomly oriented magnetic fields near the surface.It is possible that the spontaneous fields of TRSB have a very short decaylength scale. For example, the TRSB fields may order into antiferromagnetic123Chapter 5. Summary and Conclusionsdomains of short length scale compared to the distance where our measure-ments are done (≥ 5 nm), and thus cannot be resolved. Our estimate islarger than SQUID microscopy which measured weak spontaneous fields of≈ 10−5 G near c-axis oriented YBCO epitaxial thin films [7].The field inhomogeneities detected by our probe cannot be ascribed tothe proximity effect, where superconductivity “leaks” into Ag. For thisphenomenon to take place the superconductor has to be in a good electricalcontact with the normal metal. However, the Ag layers were deposited ex-situ on all the YBCO samples, where we do not expect a good ohmic contactand the resistivity of the SN contact is high [8]. The proximity effect is wellestablished for metals in contact with thes-wave superconductors because oftheir isotropic order parameter, and large coherence lengths [9]. To rule outcompletely this effect, experiments should be done on YBCO coated in-situwith silver or gold, and YBCO/insulator/Ag junctions. For comparison,β-NMR experiments should be also conducted on s-wave superconductorswith in-situ and ex-situ deposited Ag. A recent β-NMR experiment on Nbcoated in-situ with Ag showed clear manifestation of the superconductingproximity effect in Ag with features of the field distribution and spin-latticerelaxation that are very different from the results presented in this thesis.Spin-relaxation experiments are also needed to address the question ofTRSB. Preliminary results given in Appendix D have proved the feasibilityof such measurement in YBCO (without Ag), although the signal is fastrelaxing on the 8Li+ time scale. Spin-lattice relaxation could be performedas a function of a weak external field (0−200 G) and temperature. Zero fieldmeasurements are also possible, provided that the spin-relaxation is not toofast. These could detect dynamic magnetic fields associated with possiblefluctuations of the order parameter in YBCO rather than the static fieldsthat have been measured in Chapter 2.Similar to the TRSB measurements, studies of the vortex lattice ofYBCO and PCCO superconductors have been conducted. In principle, β-NMR is able to probe the vortex lattice near the surface of type II supercon-ductors, e.g. NbSe2 [11]. The technique of proximally probing the vortexlattice field inhomogeneities outside (rather than inside) the superconduc-124Chapter 5. Summary and Conclusionstor has been previously demonstrated. For example, it has been used byLow-energy µSR to probe the field distribution in a silver film associatedwith the emerging vortices from the underlying YBCO [12]. Similarly, NMRand ESR techniques have used spin labels deposited onto the superconduc-tors to measure the field distribution [13, 14]. Our results clearly show thatmagnetic flux penetration into YBCO may be dominated by grain and twinboundary effects in many instances. Disorder of the vortex lattice occur-ring on a long (µm) length scale leads to strong modifications of the fielddistribution of a long range ordered vortex lattice. Surprisingly, the temper-ature dependence of the measured penetration depth, if extracted from thefield distribution, reflects the temperature dependence of the vortex latticedisorder more than that of the intrinsic penetration depth of the material.Our results thus provide a clear explanation of why early experiments onpowders or highly twinned YBCO crystals have measured an effective pen-etration depth with a temperature dependence close to that expected fromthe s-wave order parameter. To measure the penetration depth of HTSC,one must use high quality detwinned crystals. In addition, for measure-ments outside the superconductor, one must probe the field distribution atdistances of few nm away from the surface, where the field inhomogeneitiesof the regular vortex lattice are dominant.Theexperimentsconductedontheelectron-dopedsuperconductorPCCOfilms have shown a similar, but narrower, field distribution than in YBCO.However, the effect may be related to an intergrowth of insulating CeO2layers separating PCCO superconducting layers, which leads to a weaklycoupled 2D vortex lattice. We observed a paramagnetic shift of the mostprobable field. This finding is consistent with bulk µSR, and attributed tofield induced antiferromagnetic ordering of the Cu moments [15]. A newgeneration of PCCO films are now grown free of the insulating phase, andshould be investigated. The order parameter in PCCO cannot be resolvedin our measurements due the extrinsic insulating phase as well as the limitedtemperature range of β-NMR cryostat. This could be overcome once a newlow temperature cryostat (under design) is constructed.In all these experiments, the measured field distribution broadens upon125Chapter 5. Summary and Conclusionscooling below Tc. The micro-structure of the SN boundary and surfaceroughness of the superconductors could also contribute. It is known thatprecipitates and chemical inhomogeneities including surface roughness leadto a local variation of the order parameter and flux-line energies, giving riseto pinning forces in the normal conductor [16]. However that would not bethe leading source here, as the YBCO films had an almost atomically smoothsurface. For example, the YBCO film studied in Chapter 3 is atomically flatas confirmed by STM measurements, but displayed higher broadening thanthe untwinned crystal which has relatively a larger surface roughness. In theTRSB studies, it could be possible that such measurements are influencedby the nm scale roughness, but regardless, the magnitude of the broadeningis so small that TRSB can be ruled out safely.In conclusion, we have ruled out the occurrence of TRSB or any othersource of spontaneous magnetic fields exceeding 0.2 G at distances of 8 nmfrom the YBCO surface, and confirmed that the disorder in the vortex latticecan modify the temperature dependence of the field distribution linewidth,and hence the penetration depth extracted from it. This work demonstratesthe applicability of β-NMR to study unconventional superconductors. Forexample, it would be interesting to probe magnetism near the surface ofSr2RuO4, and compare with the µSR results which detected bulk TRSB.One can also use β-NMR to study the vortex lattice in the newly discoveredFeAs-based HTSC.126Bibliography[1] Siu-Wai Chan, Lie Zhao, and C. Chen, Qi Li and D. B. Fenner, J.Mater. Res. 10, 2428 (1995).[2] K. Sugawara, M. Kawamura, Y. Abe, K. Sasaki, Microelectronic En-gineering 84, 2476 (2007).[3] P. A. Rosenthal, E. N. Grossman, R. H. Ono, and L. R. Vale, Appl.Phys. Lett. 63, 1984 (1993).[4] R.Kalyanaraman, S.Oktyabrsky, andJ.Narayan, J.ofAppl.Phys.85,6636 (1999).[5] V. Pendrick; FL Brown, and J. R. Matey, A. Findikoglu, X. X. Xi,and T. Venkatesan, A. Iham, J. Appl. Phys. 69, 7927 (1991).[6] C. R. Hu, Phys. Rev. Lett. 72, 1526 (1994).[7] R. Carmi, E. Polturak, G. Koren, and A. Auerbach, Nature 404, 853(2000)[8] S. C. Sanders, S. E. Russek, C. C. Clickner, and J. W. Ekin, Appl.Phys. Lett. 65, 17 (1994).[9] S. Gu´eron, H. Pothier, Norman O. Birge, D. Esteve, and M. H. De-voret, Phys. Rev. Lett. 77, 3025 (1996).[10] I. Asulin, A. Sharoni, O. Yulli, G. Koren, and O. Millo, Phys. Rev.Lett. 93, 157001 (2004).[11] Z. Salman, D. Wang, K. H. Chow, M. D. Hossain, S. Kreitzman, T. A.Keeler, C. D. P. Levy, W. A. MacFarlane, R. I. Miller, G. D. Morris,127Chapter 5. BibliographyT. J. Parolin, H. Saadaoui, M. Smadella, and R. F. Kiefl, Phys. Rev.Lett. 98, 167001 (2007).[12] Ch. Niedermayer, E. M. Forgan, H. Gl¨uckler, A. Hofer, E. Morenzoni,M. Pleines, T. Prokscha, T. M. Riseman, M. Birke, T. J. Jackson, J.Litterst, M. W. Long, H. Luetkens, A. Schatz, and G. Schatz, Phys.Rev. Lett. 83, 3932 (1999).[13] A. Steegmans, R. Provoost, V. V. Moshchalkov, H. Frank, G.G¨untherodt, and R. E. Silverans, Physica C 259, 245 (1996).[14] N. Bontemps, D. Davidov, P. Monod, and R. Even, Phys. Rev. B 43,11512 (1991).[15] J. E. Sonier, K. F. Poon, G. M. Luke, P. Kyriakou, R. I. Miller, R.Liang, C. R. Wiebe, P. Fournier, and R. L. Greene, Phys. Rev. Lett.91, 147002 (2003).[16] C. Ciuhu and A. Lodder, Phys. Rev. B 64, 224526 (2001).128Appendix ARF ModesIn NMR, one applies the RF in two commonly used methods: (i) the contin-uous wave (CW) where a weak RF oscillating magnetic field is continuouslyapplied, and (ii) the pulsed RF mode, where a short RF pulse is used. Inβ-NMR we employ similar methods. The pulsed RF is convenient for narrowresonances and has been used in all the TRSB measurements described inChapter 2, while the CW RF is useful for measuring broader lineshapes inthe vortex lattice. In this appendix, I would like to introduce the reader tothese two different modes.A.1 CW RF ModeIn this mode, the RF is on at all times and has a sinusoidal formB1(t) = B1 cos(ωt)ˆx, (A.1)where B1 ∼ 0.1 − 1 G, applied perpendicular to B0 and the initial spinpolarization. To measure the asymmetry, the frequencyf =ω/2πis changedin steps of ∆f in a frequency range ∆R . At each frequency, the counts aremeasured in N bins of time per bin tb. Each scan is recorded for a timetscan = Ntb∆R/∆f. For example, for ∆R = 100 kHz, ∆f = 250 Hz,N = 100, and tb = 10 ms, the time spent to take each scan is tscan = 400s. After each scan, the direction of the frequency sweep is reversed andanother scan is taken with the same helicity. After two scans with onehelicity, the latter is flipped and new scans will be taken, and so on. Afterabout 3-5 good scans for each helicity (10-60 min), one generates the finalasymmetry by averaging the scans of each helicity, and taking the difference129A.2. Pulsed RF Modeof the asymmetry of the two helicities. This mode is convenient for broadlines where the measured width is approximately,1σ =radicalBigσ2int +(γB1)2 +(1/T1)2. (A.2)The intrinsic width of the sample is σint ≥ γB1, and σint ≫ 1/T1. Usingthis mode, we can measure resonances as broad as 20 kHz as reported inChapter 3. For narrow lines, the resonance has a width comparable to theartificial broadening γB1. To overcome this, we use the pulsed mode wherethe RF is on only a fraction of time.A.2 Pulsed RF ModeIn order to excite a rectangular frequency band of bandwidth ∆ω, a shapedRF pulse at the required frequency is applied. The excited frequency rangeis ω ± ∆ω/2, and the RF is on periodically for short times tp ∝ 2π/∆ω.A common shape in today’s pulsed NMR is the frequency and amplitudemodulated hyperbolic secant pulse,2B1(t) =B1sech(βt)expeiφ(t)ˆx (A.3)where B1 is the amplitude (B1 ∼ 0.1−1 G), β is a constant proportionalto the bandwidth, t is truncated between ±tp/2, φ(t) = µ(ln(sech(βt))) isthe phase, and µ is a constant. In β-NMR, short RF 90o pulses are appliedperiodically to suppress the polarization while the beam is continuous. Anexample of the pulsed RF is shown in Fig. A.1-(a), where the RF is onduring tp = 80 ms corresponding to a bandwidth of ∆ω = 200 Hz.The RF pulse is shaped to destroy all the spins in a frequency intervalω±∆ω/2. The difference in the polarization after and before the pulse isproportional to the number of spins in that interval. To find the asymmetry,the counts are measured in a time bin tp before the RF pulse, and after theRF pulse. The asymmetry for each helicity (h = ±) isAh(t) =Aha(t)−Ahb(t),1T. Parolin et al., Phys. Rev. B 77, 214107 (2008).2M. Garwood and L. DelaBarre, J. of Mag. Res. 153, 1577 (2001).130A.2. Pulsed RF Mode(a)0 80 160 240 320 400 480 560 640 720 800Time (ms)−40−30−20−10010203040Amplitude (mV)01Pulse sequence0 20 40 60 80−20−1001020(b)4 5 6 7 8Frequency (kHz)−0.03−0.010.010.030.05AsymmetryCW RF, width=651(2) HzPulsed RF, width=348(2) HzFigure A.1: (a) Pulsed RF sequence where RF is on during a time intervaltp = 80 ms. The excited frequencies have a bandwidth ∆ω = 200 Hz. Thefunctional form is ln-sech, and the pulse sequence controls when the RF isdelivered. (b) The spectra in Ag(15 nm)/YBCO at E = 2 keV, T = 10 K,and B0 = 100 G taken using CW and pulsed RF modes with similar power(B1) are shown. The width is broadened by a factor of two using CW RF.The width of the resonance taken with the pulsed RF is 348 Hz, which isclose to the ideal intrinsic dipolar broadening in Ag of 250 Hz. 131A.2. Pulsed RF Modewhere Ahb is the asymmetry before the pulse, and Aha the asymmetry afterthe pulse for helicity h. The final asymmetry is again the difference of thepositive and negative helicity asymmetries A(t) =A+(t)−A−(t).In pulsed RF mode, scanning a frequency range takes few seconds ratherthan minutes as in the CW mode. The time spent at each frequency isdetermined by the bandwidth, and is typically 10 ms - 160 ms, i.e. at least10 times smaller than the CW mode. For example, for tp = 160 ms, therepetition rate is 1/160 ms = 6.25 Hz (see Fig. A.1-(a)), and for frequencysteps ∆f = 100 Hz, one would cover a range ∆R = 10 kHz in tscan=16 s.To accumulate enough statistics each run (final asymmetry) takes about anhour. Because the repetition rate is high, the frequencies of the RF pulseare generated randomly to minimize history effects where slowly recoveringpolarization after a pulse may contribute to the asymmetry before the nextplanned pulse.The pulsed RF has proved useful in the TRSB measurements where theextra broadening below Tc is small and comparable to B1. A comparisonof the asymmetry measured using the CW and RF pulsed modes is givenin Fig. A.1-(b). At the same experimental conditions, the linewidth hasbeen reduced by a factor of two using pulsed RF. Thus, our resolution ishigher in pulsed RF than CW RF. The pulsed mode is more convenient fornarrow lines of linewidth smaller than ≈ 5−10 kHz. At higher widths, theamplitude of the signal is weak and CW RF is more practical. Note thatthe net broadening due to the vortex lattice in Chapter 3 is similar usingpulsed or CW for resonances of width smaller than 8 kHz.132Appendix Bβ-NMR Resonance in AgAll the results presented in this thesis involved Ag films as a capping layerevaporated on the superconducting materials. The β-NMR resonance inpure Ag will be discussed in this appendix.B.1 Field and Temperature DependenceThe resonance in Ag is a single Lorentzian at fields below ≈ 1 T, and is adouble Lorentzian at higher fields. The two peaks of the resonance above1 T are due to the 8Li+ ions stopping in two different sites as discussedin Appendix D. At low fields, since the magnetic splitting is small the twopeaks are unresolved, leading to a single resonance.The single Lorentzian has a weak dependence on field and temperature.The resonance in a silver film of 15 nm grown on an SrTiO3 substrate hasshown no temperature dependence in a low field of 10 G applied parallel tothe substrate surface. Shown in Fig. B.1 are the Larmor frequency (ωL),the full width at half maximum (∆), the amplitude (A), and the baseline(B) of a Lorentzian fit given byL(ω) = B+A ∆(ω−ωL)2 +(∆/2)2. (B.1)The linewidth of the resonance is attributed to the nuclear dipolar momentsof the two Ag isotopes. Estimates using the Van Vleck method of momentsare found to be between 0.2-0.4 kHz.1 The β-NMR resonance in Ag hasa Lorentzian form independent of the type of RF mode (see Appendix A).1A. Abragam, Principles of Nuclear Magnetism (Oxford University Press, 1961).133B.2. Effect of the Dewetting Transition in AgIn Fig. B.1, the data was taken by a pulsed RF mode where the powerbroadening is minimal.The absence of any temperature dependence of the resonance spectraproves the advantage of using Ag as a capping layer in this thesis. It alsoconfirms that the broadening observed in Ag/YBCO (see Chapter 2) is ex-clusively due to the underlying YBCO.The resonance spectra in a 120 nm Ag film on YBCO, plotted in Fig.B.2-(a), shows the unresolved peaks of the S and O sites at 3.33 T and 100K. At an energy of 8 keV, all 8Li+ ions are stopping in the 120 nm Ag layer,and above Tc the lineshape is unrelated to YBCO. It should be noted thatthe amplitude and resonance shift of the resonance due to 8Li+ stopping inthe S and O is constant below 100 K, as shown in Fig. B.2-(b). Thus, belowTc in the vortex state, the double peaks are almost unresolved, leading to asingle resonance as shown in Fig. B.2-(a).B.2 Effect of the Dewetting Transition in AgThe β-NMR spectra at low fields in Ag showed an aging effect. An Ag filmthat was studied one month after it was grown has shown a width of 314 Hz.Measured 8 months later, the same film showed twice the original broad-ening. A comparison of both resonances is given in Fig. B.3. Fortunately,this effect is only significant if the silver is few months old. All our TRSBexperiments were done within 1 to 2 months after the YBCO was cappedwith Ag. The aging effect leads to a small broadening of the signal, thereforeit does not affect our vortex lattice measurements where the broadening is5-30 kHz. While, the extra broadening measured in Chapter 2 is still twicebigger than the extra age-related broadening.The extra broadening is likely due to the dewetting transition in Ag.2Using atomic force microscopy (AFM), as shown in Fig. B.4, the surfaceof a freshly grown Ag (15 nm thick) is smooth, with a root mean square(RMS) roughness between 2-5 nm. The roughness, however, increases forolder samples to about 10-20 nm. Impurity atoms that condense onto the2e.g. M. M. R. Evans, B. Y. Han, J. H. Weaver, surface science 465, 90 (2000).134B.2. Effect of the Dewetting Transition in Ag0 100 200 T (K)66.26.46.66.8ω L (kHz) 00.20.40.60.8∆ (kHz)                          00.020.040.060.08Amplitude00.0010.0020.003BaselineFigure B.1: The temperature dependence the Larmor frequency, FWHM,amplitude and baseline of Lorentzian fits of β-NMR resonances in an 15 nmfilm thick Ag evaporated on SrTiO3. The data is taken with a 2 keV 8Li+beam in an external field B0 = 10 G, applied parallel to the surface of thesubstrate.135B.2. Effect of the Dewetting Transition in Ag(a)20970 20980 20990 21000 21010 21020Frequency (kHz)00.040.080.120.16Asymmetry0.080.10.120.140.16 Ag/YBCO crystal, B0=3.33 T, E= 8 keVOST=85 KT=100 K(b)Figure B.2: (a)β-NMR resonances in Ag(120 nm) on YBCO twinned crystalatB0 = 3.33 T andE = 8 keV. Solid lines are fits to one (two) Lorentzian(s)at 100 (85) K. (b) The shift and amplitude of the S and O peaks are plottedagainst temperature at 3 T. Copied from G. D. Morris et al., Phys. Rev.Lett. 93, 157601 (2004). 136B.2. Effect of the Dewetting Transition in Ag−0.05−0.03−0.010.01 Asymmetry2 4 6 8 10 12ω (kHz)−0.05−0.03−0.010.01AsymmetryGrown on 28/11/2007, β−NMR on: 14/12/2007Same film as above, β−NMR on: 28/08/2008FWHM=504 HzFWHM=314 HzFigure B.3: The β-NMR spectra in Ag(15 nm)/STO at room temperaturewhere B0 = 10 G, and ELi = 2 keV. The linewidth increases from 314 Hz,when measured in December 2007, to 504 Hz in August 2008.surface could also contribute. Auger spectroscopy has shown the existenceof carbon and sulfur atoms in the first few atomic layers of the Ag films (Fig.B.5). These parasite atoms have probably precipitated onto the sample afterit was exposed to ambient air and may broaden the lineshape if they aremagnetic. However, from our measurements it is very clear that in fresh Agfilms the dewetting effect and impurity atoms have no contribution to thesignal.137B.2. Effect of the Dewetting Transition in AgFigure B.4: AFM of silver films showing an increase in the surface roughnesswith time. (a) AFM is done immediately after evaporating 15 nm of Agonto an STO substrate. (a) AFM done 8 months after 15 nm of Ag wasevaporated onto an STO substrate (this sample is different from the oneused in (a)).138B.2. Effect of the Dewetting Transition in AgFigure B.5: Auger depth profile of (a) fresh Ag(15 nm)/STO, and (b) fewweeks old sample. Ion bombardment time is given in seconds. The samplewas sputtered by Ar, removing layers at a rate of ≈ 0.1 nm/s. Carbon,oxygen, and Sulfur atoms have been identified in the first top layers.139Appendix CObtaining Zero-FieldThe objective of our TRSB measurements is to search for small spontaneousmagnetic fields generated near the surface of YBCO superconducting films.In such experiments, the field is applied parallel to the surface of the super-conductor, B0 bardbl ˆz, and varied between 10 to 150 G (see Fig. 2.1). However,the applied field may also have x and y-components which originate mainlyfrom stray fields in the Helmoltz coils around the sample. The x-componentof the applied magnetic field B0x, perpendicular to the surface (in the lowfield spectrometer), should be smaller than the lower critical field Bc1⊥ ∼ 1G to reach the Meissner state and avoid inducing a vortex state in YBCO.Minimizing this component also diminishes the demagnetization corrections.It is not possible to accurately measure the components of the appliedfield at the sample position due to the lack of sensitive Hall probes that areUHV compatible and can fit in the limited empty space surrounding thesample in the cryostat. However, B0x and B0y, can be accurately set to zerousing Morris and Heffner’s routine.1 This method depends on measuringthe resonance frequencies in Ag or any other suitable sample with a largenarrow resonance in a set of x, y, and z field scans, and then extracting thecurrents needed to achieve zero field. Here, we set the main component toB0z = 10 G, and vary the second component around zero covering a rangeof few Gauss, while keeping the third component fixed. The exact routineis as follows:• (i) Set the current Iz to 4.3 A which corresponds to about 10 G.• (ii) Measure the resonance as a function of Ix, while keeping the othercurrents fixed: Iy = 0 and Iz = 4.3 A.1G. D. Morris and R. H. Heffner, Physica B 326, 252 (2003).140Appendix C. Obtaining Zero-Field0 2 4 6 8Iz (A)−50510ω L (kHz)−10 −8 −6 −4 −2 0 2 4 6 8 10Iy (A)6.86.9ω L (kHz)−10 −8 −6 −4 −2 0 2 4 6 8 10Ix (A)6.77.17.5ω L (kHz)Iz=4.3 A, Ix=−0.7 AIx=−0.7A, Iy=2.6 AIz=4.3 A, Iy=0.0 AFigure C.1: Variation of the Larmor frequency, ωL, with the currents Ix,Iy, Iz which run into three Helmoltz coils and create fields along x, y, andz direction, respectively.• (iii)FittheresonancetoextracttheLarmorfrequencytoωL =γLiradicalBigB20z +(aIx +b)2to find the constants a, b, and B0z.• (iv) Set the current Ix to the value that minimizes ωL: Ix = −ba.• (v) Repeat (iii) and (iv) for Iy.• (vi) Measure the resonance versus Iz and fit the Larmor frequency toωL = γLi(aIz +b).• (vii) If cooling in zero field is necessary, the current Iz must be set toIz = −ba above Tc before cooling.141Appendix C. Obtaining Zero-FieldThis routine has been used to produce the results in Fig. C.1. The reso-nances were taken in a gold foil. This procedure minimizes the field alongx and y direction to less than 0.1 G which can be estimated from the un-certainty on the minimum of the Larmor frequency in Fig. C.1. Our TRSBmeasurements required cooling below Tc in zero field, so it was necessary toset the current Iz to a value that leads to zero field.142Appendix DSpin-Lattice RelaxationIn β-NMR, in addition to measuring the resonance, spin relaxation can alsobe measured. This offers additional information about the behavior of thespin polarization, which is influenced by the fluctuating magnetic fields inthehostmaterial, onatimescaleofω−1L ; theinverseoftheLarmorfrequency.This information is complementary to µSR, as these fields may be static ona time scale of the muon’s lifetime τµ+ = 2.2 µs. In a material with largefluctuating fields at ωL, the spin polarization, i.e. asymmetry, is destroyedquickly. Hence, these measurements are also of importance for the resonancemeasurements, where the polarization is strongly dependent on fluctuatingfields and may be destroyed even in the absence of an RF field. In thiswork, we have made several measurements to study the spin relaxation inAg, YBCO, and PCCO. Ag/YBCO and Ag/PCCO heterostructures werealso investigated to look for the influence of the underlying superconductoron the spin-relaxation signal in Ag. These measurements are discussed inthis appendix after a short review of the spin-relaxation method.D.1 Spin-Relaxation SignalIn the the absence of an RF field, the spin-lattice relaxation of the implanted8Li is measured by pulsing the beam, and measuring the time dependence ofthe spin polarization during and after the pulse. To find the asymmetry, thecount rates are measured in a time resolution of 10 ms during the beam-onperiod (typically 0.5 − 4 s) and the beam-off period (8 − 12 s). A typicalspectrum is measured in about 30 min with an incoming 8Li+ rate R0 ∼ 106ions/s. The polarization of 8Li+, initially being p0, measured at a later time143D.1. Spin-Relaxation Signal0 4000 8000 12000 16000Time (ms)−0.100.10.2a=A+ −A−   00.10.2A=(NR−NL)/(NR+NL) Positive helicityNegative helicityFigure D.1: Spin relaxation spectra recorded inB0 = 150 G. The asymmetryrecorded with a Laser of positive (A+) and negative (A−) helicity are shown.The overall asymmetry a = A+ −A− is also plotted. Solid lines are fitsexplained in the text.t yieldsp(t) =p0e−t/T1, (D.1)where a single mechanism of relaxation with a spin-lattice relaxation rate1/T1 is assumed. During the beam-on, and for a constant incoming beamrate of R0, the polarization at t averaged over all times depends on the 8Li+life time and 1/T1,p(t) = R0p0integraltextt0 dt′e−t′/τLie−t′/T1R0integraltextt0 dt′e−t′/τLi , (D.2)= p0 τ′τLi1−e−t/τ′1−e−t/τLi, (D.3)where 1τ′ = 1τLi + 1T1. During the beam-on period, the average polarization144D.2. Agapproaches the equilibrium value¯p= R0p0integraltext∞0 dt′e−t′/τLie−t′/T1R0integraltext∞0 dt′e−t′/τLi =p01+τLi/T1. (D.4)From Eqs. (D.1) and (D.3), the asymmetrya(t) ∝p(t) during (ad) and after(aa) the beam pulse of a period ∆, can be written asad(t) = a0 1−e−t/τ′1−e−t/τLi, 0 <t≤ ∆aa(t) = ad(∆)e−(t−∆)/T1, t> ∆. (D.5)wherea0 is the maximumβ-decay asymmetry att = 0. The above equationsare often used to extract 1/T1 of 8Li+ in the host material. An example ofthe spin-relaxation spectra is given in Fig. D.1. The asymmetry was fitto Eq. (D.5) with two functions a(t) = as(t) +af(t): as(t) the asymmetryof a slow relaxing component of rate 1/Ts1, and as(t) the asymmetry of afast relaxing component of rate 1/Tf1 (1/Ts1 ≪ 1/Tf1 ). The fast componentseen in some measurements is likely due to the backscattered 8Li+ stoppingin other materials surrounding the sample, such as Cu, Al. The fast com-ponent has almost no temperature dependence and is about two orders ofmagnitude higher than 1/T1. The component that we are interested in hereis the slow component and we refer to it by 1/T1. Note that, β-NMR doesnot distinguish between 1/T1 and 1/T2; the spin-spin relaxation; thus allmechanisms are included in 1/T1.D.2 AgThe relaxation of nuclear spin polarization to equilibrium is generally dueto transverse magnetic fields fluctuating at the Larmor frequency. The spin-relaxation rate depends on the temperature and the external magnetic field.In a metal, 1/T1 is due to Korringa relaxation of spin flip scattering fromthe conduction electrons. A feature of this law is the linear variation of 1/T1145D.2. Agwith temperature as1T1 = K2T, (D.6)where the Korringa constant K depends on the density of the conductionelectrons at the Fermi levels, and the hyperfine coupling of the conductionelectrons to the nucleus. This electronic 1/T1 is independent of the magneticfield.The asymmetry in Ag is strongly temperature dependent due to theKorringa relaxation. This can be seen in Fig. D.2-(a), where the spectraare more slowly relaxing as T decreases. The T-dependence of 1/T1, plot-ted in Fig. D.2-(b), confirms the validity of Korringa law both at low andhigh temperature and reflects the coupling of 8Li+ spins to the conductionelectrons in Ag. The “bump” in 1/T1 at intermediate temperatures is dueto the site change of 8Li+ in Ag. In the FCC lattice of Ag, 8Li+ couldoccupy three possible sites of cubic symmetry: substitutional (0,0,0), octa-hedral (1/2,1/2,1/2) and tetrahedral (1/4,1/4,1/4). The non-cubic sites areexcluded due to the absence of quadrupolar splitting in Ag. The substitu-tional site vacancies are made during the implantation process. From severalβ-NMR measurements, 8Li+ at low temperature are trapped in interstitialsites and thermal activation at high temperature leads to a transition to asubstitutional site.1 In Fig. D.2-(b), the relaxation rate is fit to a modelwith thermally activated O to S transitions.2The relaxation rate 1/T1 is of great importance to the resonance mea-surements as well. When measuring the resonance, the polarization of acontinuously implanted beam approaches the equilibrium value given in Eq.D.4, which defines the baseline of our resonance spectra, i.e. the differencebetween the off-resonance asymmetry of positive helicity to the negativehelicity. Thus, for very fast spin-lattice relaxation (1/T1→ ∞), the equi-librium polarization of 8Li+ approaches zero, making measurements of theresonance impossible. In Fig. D.2-(b), one can see that the baseline of theresonance spectra tracks the same temperature dependence of 1/T1. Note1G. D. Morris et al., Phys. Rev. Lett. 93, 157601 (2004).2M. D. Hossain et al., Physica B 414, 419 (2009).146D.2. Ag(a)0 4 8 12Time(s)00.10.20.3Asymmetry4 K100 K150 K250 K(b)0 100 200 3000.00.10.20.30.40.5 Foil 150 G 50 nm film, 3 T, Morris et al.  50 nm film, 3 T, Scaled Baseline  λ (s-1)Temperature (K)SilverFigure D.2: (a) T-dependence of the spin relaxation in 25 µm silver foil atB0 = 150 G. The 28 keV 8Li+ beam is implanted into the sample in 4 s pulsesevery 20 s. Solid lines are fits using Eq. (D.5) with two components. (b)λ = 1/T1 (slow component) extracted from (a) compared with the baselineof the resonance in an Ag film (50 nm) taken at 3 T.147D.2. Agthat the plotted 1/T1 and baseline are extracted from a thin silver film anda bulk Ag foil under different experimental conditions, confirming that thesignal is an intrinsic property of Ag and independent of field (see Fig. D.2).The Korringa law is dominant at high field, where the applied field po-larizes a large fraction of the conduction electrons. At very low fields, thepolarization approaches zero as the energy levels in 8Li+ and Ag are de-generate, favoring flip-flop processes between spins.3 The Zeeman splittingcreated by applying higher magnetic fields lifts this degeneracy and theasymmetry approaches its high field value. This can be seen in Fig. D.3,where at fields below 10 G, the asymmetry falls rapidly to zero, and satu-rates at fields above 20 G. The enhanced relaxation at low field in Fig. D.3is attributed to cross relaxation with the Ag spins which is driven by lowfrequency fluctuations in the magnetic dipolar interaction. This interactionis dominant at low field, while high magnetic fields quenches this interac-tion leaving only the Korringa relaxation mechanism. For Ag, the 1/T1 andamplitude can be fitted to a phenomenological Lorentzian formc1 +c2 B2d4B20 +B2d, (D.7)where Bd ≈ 12 (3) G at 293 (5) K (shared between 1/T1 and A), is anestimate of the magnitude of the fluctuating field needed to quench thedipole-dipole interaction. These values are comparable to those of 8Li+ ingold.4 The constants c1 and c2 are both field and temperature dependent.c1 values are consistent with Fig. D.2-(b).From the above discussion, we conclude that one needs to apply anexternal magnetic field higher than 5 G to measure a signal in Ag withlarge amplitude. This was taken into account when conducting the TRSBmeasurements. Ideally, one should do these measurements in zero field.However the loss of polarization in Ag at very low fields requires one toapply a static field that is at least 5 G to obtain a good signal to noise ratio.3C. P. Slichter, Principles of Magnetic Resonance, 2nd ed., Springer-Verlag (1980).4T. Parolin et al., Phys. Rev. B 77, 214107 (2008).148D.2. Ag0 50 100 150B0 (G)00.2A0.060.1A Asymmetry0.020.031/T1 (s−1)0.40.51/T1 (s−1) T=293 KT=5 KT=293 KT=5 KFigure D.3: The B0 and T-dependence of the amplitude, A, and relaxationrate, 1/T1, of 8Li+ polarization implanted at full energy into a silver foil.The data was extracted by fitting the asymmetry to a pulsed exponentialfunction. Aand 1/T1 of the slow relaxing part of the function is shown here.The solid lines are fits using a Lorentzian form described in the text.149D.3. Ag/YBCOD.3 Ag/YBCOThe spin-lattice relaxation rate in NMR is a versatile tool inprobing the elec-tronic structure of HTSC cuprates. For example, 1/T1 of 63Cu have showedevidence of the pseudogap state below the characteristic temperature T∗.5The nuclear and electronic moments of Cu, Y, O atoms in YBCO lead tofast spin-lattice relaxation of 8Li+at low magnetic fields. Consequently the8Li+ ions stopping in the YBCO do not contribute to the resonance at theLarmor frequency. Thus we are unable to measure the β-NMR resonancesin YBCO at low field, but at high field they are seen.6Preliminary measurements of the spin-relaxation of 8Li+in Ag on YBCOhave shown a fast relaxing signal in YBCO. This sample grown by Theva(Ismaning, Germany), was used in Chapter 2 to study the length scale of theextra broadening observed in Ag below the Tc of YBCO. The asymmetrymeasured in an applied field of 100 G with a beam of energy 13 keV is shownin Fig. D.4-(a). This shows fast relaxation of the signal at all temperaturesdue to YBCO with a slow relaxation attributed to Ag. At 13 keV, about30% of 8Li+ stops in Ag, 35% in YBCO, and 25% is backscattered.The spectra at full energy are shown in Fig. D.4-(b) as a function oftemperature. At full energy, 15% stops in Ag, 65% in YBCO, and 20% isbackscattered. The amplitude of the asymmetry and its temperature depen-dence look inconsistent with the Ag results (with no YBCO). By plottingthe 1/T1 versus T as shown in Fig. D.5, one finds a slight upward shift ofthe relaxation rate from the intrinsic 1/T1 of Ag. The extrapolated 1/T1 tozero temperature is inconsistent with Ag, and is either due to the Ag/YBCOinterface or bulk YBCO. One also notices that the upward shift peaks up atthe Tc of YBCO and approaches 1/T1 of Ag above Tc.The energy dependence of the spin relaxation at 10 K is shown in Fig.D.4-(c), and confirms the fast relaxation of the spin polarization as more8Li+ stops in YBCO. The energy dependence of 1/T1 at 10 K from theabove measurements is shown in the inset of Fig. D.5. This shows a linear5See the review: T. Timusk and B. Statt, Rep. of Prog. in Phys., 62, 61 (1999).6R. F. Kiefl et al., Physica C 326, 189 (2003).150D.3. Ag/YBCO0 5 10Time (s)00.10.200.10.210 K50 K85 K87.2 K87.9 K100 K00.10.2Asymmetry10 K50 K85 K87.2 K87.9 K100  K(a) E=13 keV(b) E=28 keVE=2 keVE=13 keVE=20 keVE=28 keV(c) T=10 KFigure D.4: Spin relaxation of 8Li+of energy (a) 13 keV and (b) 28 keVimplanted into Ag(50 nm)/YBCO(600 nm) at 100 G applied field underzero field conditions. Solid lines are single exponential fit to the beam offasymmetry. (c) Spin-relaxation spectra versus energy at 10 K.151D.4. PCCO0 100 200 300T (K)00.20.41/T1(s−1)Ag/YBCO, 28 keVAg/YBCO, 13 keVAg foil, 28 keV0 10 20 30E (keV)00.10.21/T1(s−1)Figure D.5: Temperature dependence of 1/T1 in Ag(50 nm)/YBCO(600nm), at two different energy extracted from Fig. D.4. Also, shown is 1/T1in a Ag foil extracted using single exponential fit to the beam off asymmetry.Inset: energy dependence of 1/T1 in Ag/YBCO at 10 K.increase in the relaxation rate as energy increases, i.e. more 8Li+ stops inYBCO. It may be of interest to study this signal as a function of field andtemperature in a YBCO film or crystal without Ag to draw conclusionsabout the origin of this signal.D.4 PCCOThe spin relaxation measured in a 300 nm thick PCCO film grown on STOshows a very small signal, and a fast relaxation. This is plotted in Fig.D.6. This suggests the existence of large dipolar fluctuating fields in PCCO,likely due to the magnetic moments associated with Pr or Cu atoms.152D.4. PCCO0 2000 4000 6000 8000 10000Time(ms)−0.0500.05Asymmetry  T=290 K   150 K    50 K    26 K    13 KB0=150 GFigure D.6: The Spin relaxation of 8Li+ at full energy implanted into a300 nm thick PCCO film, in B0 = 150 G.153Appendix ESample CharacteristicsIn this thesis, we have studied several near optimally-doped cuprate super-conductors. In Chapter 2, four YBCO samples were studied. The (110)-oriented film was grown on an SrTiO3 substrate using off-axis RF magnetronsputtering by P. J. Hentges and L. H. Greene at the University of Illinois.The film has a Tc of 84.5 K, an area of ∼ 4×8 mm, and a thickness of 100nm. Another sample from the same group was studied. It is (001)-oriented,of Tc=88.7 K, size 4 × 8 mm, and thickness 100 nm. Two other c-axisfilms were studied. They are provided by a commercial supplier (Theva,Ismaning, Germany). The films are grown using thermal co-evaporation onLaAlO3 substrates, have Tc of 88 K, and thickness of 600 nm.In Chapter 3 we studied three YBCO samples. The YBCO crystalswere flux-grown by Ruixing Liang, D. A. Bonn, and W. N. Hardy at theUniversity of British Colombia. These crystal are ∼ 2×3 mm, ∼ 0.5 mmthick, and have aTc of 92.5 K. The results presented in Chapter 3 were takenon these crystals which were capped with 120 nm of Ag (99.99% purity).Also studied in Chapter 3 is a YBCO film (Theva) of 600 nm thickness andTc = 88 K. The film was covered in-situ with 60 nm of Ag (99.99% purity).The PCCO film (300 nm thick) studied in Chapter 4 was grown byP. Fournier at the University of Sherbrooke using Pulsed Laser Deposition(PLD) on an SrTiO3 substrate. The film was covered with 40 nm of Ag.The Ag deposition on all these samples is done ex-situ (except one Thevasample) at room temperature using DC sputtering in an Ar pressure PAr =30 mtorr. A calibrated thickness monitor is used, with a deposition rate of0.5 to 1 ˚A/s, and the growth is done while rotating the sample. A summaryof the characteristics of all these samples is given in table E. More detailsare given in the chapters. Note that the AFM on the YBCO crystals was154Appendix E. Sample Characteristicsdone more than two years after the Ag was deposited. Hence, because ofthe aging effect in Ag (see Section B.2 ), the estimated roughness is higherthan the actual roughness at the time of experiments, which were done fewweeks after the silver was deposited.155AppendixE.SampleCharacteristicsSample Type Substrate Lab Tc (K) d Area (mm) dAg(nm) Ra(nm)YBCO TW xtal - UBC 92.5 ≈0.5 mm 3×3 120 12.8YBCO DTW xtal - UBC 92.5 ≈0.5 mm 2×3 120 47YBCO (001) SrTiO3 Theva 87.5 600 10×8 60 -YBCO (110) SrTiO3 Urbana 86.7 100 8×6 15 -YBCO (001) SrTiO3 Urbana 88.7 100 5×4 15 -YBCO (001) LaAlO3 Theva 88 600 10×8 15 2.4YBCO (001) LaAlO3 Theva 88 600 10×8 50 -PCCO (001) SrTiO3 Sherbrooke 22.5 300 10×8 40 -Ag – SrTiO3 AMPEL - - 10×8 15 2.7Ag – SrTiO3 AMPEL - - 10×8 15 5.5Ag – SrTiO3 AMPEL - - 10×8 15 17.9Table E.1: Characteristics of the samples used in thesis thesis. TW: twinned crystal, DTW: detwinned crystal.d is the nominal thickness of the films, dAg the thickness of deposited Ag if applicable, and Ra the RMS surfaceroughness of the samples after Ag was deposited found measured by AFM.156

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