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## Material Information- Title:
- Quasiparticale interference and the local electronic structure of disordered d-Wave superconductors
- Creator:
- Zhu, Lingyin
- Publication Date:
- 2005
- Language:
- English
- Physical Description:
- xiv, 117 leaves : ill. ; 29 cm.
## Subjects- Subjects / Keywords:
- Atoms ( jstor )
Electrons ( jstor ) Energy ( jstor ) Fermi surfaces ( jstor ) Greens function ( jstor ) Impurities ( jstor ) Momentum ( jstor ) Quasiparticles ( jstor ) Superconductivity ( jstor ) Superconductors ( jstor ) Dissertations, Academic -- Physics -- UF ( lcsh ) Physics thesis, Ph. D ( lcsh ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (Ph. D.)--University of Florida, 2005.
- Bibliography:
- Includes bibliographical references.
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- Printout.
- General Note:
- Vita.
- Statement of Responsibility:
- by Lingyin Zhu.
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- University of Florida
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- University of Florida
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- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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QUASIPARTICLE INTERFERENCE AND THE LOCAL ELECTRONIC STRUCTURE OF DISORDERED d-WAVE SUPERCONDUCTORS By LINGYIN ZHU A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005 Copyright 2005 by Lingyin Zhu I dedicate this work to my loyal family. ACKNOWLEDGMENTS I attribute the completion of my dissertation in great part to my wonderful It i I advisor and friends. I would like to express my special gratitude to Prof. Peter Joseph Hirschfeld for his patience, caring guidance and invaluable friendship. He has been not only an academic advisor, but also a mentor to me. If there is a perfect advisor, he can never be closer to that. Special thanks go to Prof. David Tanner for his continuous care and support and Prof. Douglas Scalapino for his instructive discussions with me. I appreciate the invaluable help from Prof. Alan T. Dorsey greatly. I should also thank my committee members Prof. Selman Hirchfield, and Prof. Stephen Pearton for their time and input. I have benefitted greatly from stimulating conversations with my best friends Xu Du, 21,iw..,.- and Tamara, Brian, Ashot, Matthew,Wei, Suhas, Fillippo. Finally, the loyalty and encouragement from my family were the priceless treasure to me in the past 28 years. I can never thank my wife enough for her tolerance, understanding, support and what she has sacrificed for me. I am forever indebted to my ., io.11 Irl and my parents: they seeded this special moment with endless love and have been .- 1 .... for it for so long. I wish my dear grandma could share this happiness with all of us in heaven, peacefully. TABLE OF CONTENTS page ACKNOWLEDGMENTS ........... ................... iv LIST OF TABLES .. .............. .............. viii LIST OF FIGURES ................................ ix ABSTRACT ............................ ........ xiii CHAPTER 1 INTRODUCTION ............ .................... 1 1.1 History of High Temperature Superconductivity .......... 1 1.2 Background About High Temperature Superconductivity ..... 2 1.2.1 Crystal Structure ....................... 2 1.2.2 Phase Diagram .................. ..... 3 1.3 Current Situation in High Temperature Superconductivity Research 5 1.3.1 What We Know About High Temperature Superconductivity 7 1.3.2 What We Do Not Know About High Temperature Super- conductivity .............. ... ..... 7 1.4 Disorder ill Spectroscopic Experiments ............ 8 1.4.1 Scanning Tunnelling Microscopy (STM): In Touch With A tom s .. .. 8 1.4.2 Impurities in Bi2Sr2CaCu120 : A Probe of High Tempera- ture superconductivity ................... 9 2 SINGLE IMPURITY PROBLEM ................. 13 2.1 One Impurity in Normal State ................... 13 2.1.1 Bound State and Resonant State ............... 13 2.1.2 Local Density of States: Friedel Oscillations and Ripples in the Fermi Sea ............. ........ 15 2.2 Green's function for BCS Superconductors .............. 16 2.3 One Impurity in an s-wave Superconductor ............. 18 2.3.1 Nonmagnetic Impurity Problem .............. ..18 2.3.2 Magnetic Impurity Problem .. 19 2.4 One Impurity in a d-wave Superconductor 20 2.4.1 Nonmagnetic Impurity ................... .20 2.4.2 Magnetic Impurity .. 26 2.4.3 Discussion: Characteristics of Impurities ... 28 3 QUANTUM INTERFERENCE BETWEEN IMPURITIES ........ 32 3.1 Two Impurities in a d-wave Superconductor ............ 33 3.1.1 T-matrix and Resonance Frequency ............. 33 3.1.2 Resonance Frequencies ................. .. 35 3.1.3 The Interference of Two One-impurity Wavefunctions 38 3.1.4 Local Density of States, Realistic Band and Standing Wave Condition ....... ..... ........... 41 3.2 DOS of two Impurities and Many Impurities at w = 0 : Local Resonance vs. Cumulative Interference .... 47 3.2.1 Perturbative Prediction: SCTMA and Its Validity 47 3.2.2 Nonperturbative Predictions: .... 48 3.2.3 Divergence in Pure Nested Band with Unitary Scatters .. 49 3.2.4 Discussion: Bridge Between the Two-impurity and the Many- impurity Problems ...... 53 3.2.5 Conclusion ......... .... 56 3.3 Power Spectrum for Many Impurities .... 57 3.3.1 Weak Scattering Limit: Octet Model and Kinematics of Quasiparticles ........................ 58 3.3.2 Power Spectrum for Many Impurities in Bi2Sr2CaCu20O 60 4 DISCUSSION: QUASIPARTICLES, DISORDERS AND THEIR IMPLI- CATIONS FOR STM SPECTROSCOPY .... 68 4.1 Atomic Level Resonances, Quasparticle States and Unitary Scat- ters ................ ..... ...... 68 4.2 Nanoscale Gap Inhomogeneity .................. 70 4.3 Long-range Modulation and Exotic Orders ... 74 5 ANGLE-RESOLVED PHOTOEMISSION SPECTRA OF DISORDERED CUPRATES ....... ............. ...... .. 76 5.1 Angle-resolved Photoemission Spectra of High Temperature Su- perconductors ............................ 77 5.1.1 Spectral Function and Self-energy .............. 77 5.1.2 Self-energy in Normal and '%i.. [* .,i,,. t i.-, States 78 5.2 Elastic Forward Scattering in Bi2Sr2CaCu2Os ... 82 5.2.1 Self-energy ........ 82 5.2.2 Spectral Function ................ ... .. 89 5.3 Discussion and Conclusion ................. .. .. 94 6 CONCLUSION AND REMAINING QUESTIONS .... 97 APPENDIX ................... 99 A GREEN'S FUNCTION FOR THE SUPERCONDUCTING STATE .. 99 B EVALUATION OF THE SELF-ENERGY IN BORN APPROXIMATION 102 B.1 "Yukawa I" Potential ............. ............ 102 B.1.1 Normal State ................... 102 B.1.2 Superconducting State ..... .. 103 B.2 F .... n i l" Potential ................... .. 106 B.2.1 Normal State .................. .... .. .. 107 B.2.2 Superconducting State ........... .. .. .. 107 B.3 Delta-function Type Potentials ..... .. 107 REFERENCES... ..... ......... .......... 110 BIOGRAPHICAL SKETCH ................ .. .. 117 vii LIST OF TABLES Table p 2-1 The imaginary part of self-energies due to non-magnetic impurities in superconductors ................... ......... 21 2-2 The ionization energy of Copper and Zinc atoms ... 30 LIST OF FIGURES Figure page 1-1 The crystal structure of La2CuO4. Red sphere: Cu2+; blue sphere: 02-; yellow sphere: La3+. .................. 2 1 2 Schematic plot of the phase diagram of hole-doped high temperature superconductors. X axis: hole dopant concentration; y axis: tem- perature. . 4 1-3 Current situation in high temperature superconductivity. ...... .. 6 1-4 The crystal structure of Bi2Sr2CaCu2O0 ..... 10 1-5 STM image in Bi2Sr2CaCu2Os showing a) atomic scale native defect resonances; b) Zn resonance. Both were taken at zero bias ... 11 2 1 Diagranunatic representation for the scattering process by single im- purity. Coarse arrow: full Green's function; solid line: pure Green's function; cross: impurity; dashline: impurity potential. ...... .. 14 2 2 Schematic illustration of the bound state in normal metal. Left:solution to the T-matrix; Right: the bound state is only located at high en- ergies, i.e., outside of the band. ............... .. .... 15 2-3 LDOS modification due to single impurity in metals. Right: image on the surface of Ag [36]; left: theoretical calculation of LDOS around a single impurity in a two-dimensional metal. .... 16 2-4 Schematic illustration of the pair breaking effect due to potential scat- ters in a d-wave superconductor. ................. 21 2-5 The fall-off of LDOS around a native defect. ............. .24 2-6 On-site and NN site spectra for tight binding band with one impurity. 25 2-7 Conductance spectra above the Ni atom and at several nearby loca- tions. Solid circle: on-site; open circle: NN-site; square: 2NN-site; triangle: 30A away from Ni; solid line: average of the first four spec- tra ................. .... 27 2-8 Left: the schematic plot of CuO2 plane; middle: experimental LDOS around Zn impurity; right: theoretically predicted LDOS for Vo = 5.3t (rotated by 45 with respect to a) and b)). ... 28 2-9 The resonance frequency as a function of scattering potential Vo; in- set: the resonance frequency as a function of 1/Vo. ... 31 3-1 Two-impurity resonance energies l,2 vs impurity orientation. 38 3-2 The wavefunction at resonance and corresponding LDOS. Impurities are separated by R = (6,6). ..... ...... 40 3-3 Spectra on the nearest sites close to the impurities. See text for de- tails. ................... .... .. ....... 41 3 4 LDOS maps at resonant energies for R 11 (110). Pure nested band, Vo = 10t; Impurity separations are shown on the top of the graph. 43 3-5 LDOS maps at resonant energies for R 1| (100). Pure nested band, Vo = 10t; Impurity separations are shown on the top of the graph. 44 3-6 LDOS spectra for realistic band and Vo = 5.3tl on nearest neigh- bor site. Left panel: impurities at (-R/2, 0) and (R/2, 0) ( R = (R, 0)), spectra taken at r = (R/2, 1). Right panel: impurities at (-R/2, -R/2) and (R/2, R/2) ( R = (R, R)), spectra taken at r =(R/2, R/2 + 1) .................. ........ 45 3-7 Fermi surface of BSCCO-2212 with constant energy surfaces at w = 0.04 shown as small filled ellipses at the nodal points. ql, q2, q3 are wave vectors for which the joint density of states is large. ... 46 3-8 Schematic plot of the self-energy diagram with many impurities. .. 48 3 -9 Schematic figure of the many-impurity DOS (a) in the unitary limit of the half-filled band and (b) in the unitary limit of a generic band. 48 3-10 Schematic plot of bipartite lattice. .................. ..50 3- 11 ( I. ii-. in p(w) due to impurities. Impurity separation: R = (2, 2). 51 3-12 Change in p(w) due to impurities. Impurity separation: R = (2, 1). 52 3-13 (a) DOS for Vo=100t. (b) Scaling of the DOS with V0. (c) Scaling of the DOS with L. (d) Scaling of the DOS with Vo and L=60. .... 53 3-14 LDOS for 2% concentration of impurities and IEn| < 10-5t (20 eigen- values). ...... .......... ....... ........... 55 3-15 LDOS for 0.5% concentration of impurities. Left: E, = 0.0385t; right: averaged over five eigenvalue in energy interval IE, 0.03t\ < 0.02t. 57 3-16 Left panel:the constant energy contours with the q vectors; right panel: the evolutions of q vectors as a function of energy. ... 58 3-17 FTDOS at w =14 meV for weak potential scatters (Vo = 0.67t1): (a) for one weak impurity, with a few important scattering wavevec- tors indicated; (b) for 0.15% weak scatterers. Cuts through the data of (a)(thick line) and (b)(thin line) along the (110) direction and scaled by 1/ J are plotted vs. q, in (c), while (d) shows the weak scattering response function Im A3(q,w). Peaks at q = 0 are re- moved for clarity. In all the figures, the x and y axes are aligned with the Cu-0 bonds ................... .. 62 3 18 The plot of Nambu component of spectral functions at several fre- quencies ......... ...... ..... ....... 64 3 19 Comparison of theory and experiment. Left panel: LDOS; middle panel: calculated FTDOS; right panel: FTDOS from STM exper- iments ............ ...... ....... ........ 67 4 1 Zn impurity resonance, left channel: dI/dV Vs.w; right channel: the LDOS spectrum above the Zn site. .. ...... 69 4 2 The gapmap of overdoped BSSCO. Sample size 500Ax500A ..... 71 4 3 Spectra on different sites along the horizontal cut. ... 71 4-4 Gap map from numerical solution of BDG equations with only smooth disorder. .. ......... .... ....... .. 73 4-5 LDOS along the horizontal cut in the top figure .... 73 5-1 ARPES spectra from overdoped Bi2212 (T,=87K). a) normal- and b), (c) superconducting state data measured at the k point indi- cated in the inset of (c). ................... 79 5 2 Geometry for the forward scattering process in which a quasiparticle scatters from k to k'. ................. ....... 83 5 3 The self-energy terms -Im Eo(k,w), Im El(k,w), and -Im E3(k,w) in the superconducting state at T = 0 for k = kA (top) and kN (bottom), for K = 5 and 0.5 and the same band and scattering pa- rameters as previously used. Here Ak = Ao (cos, cos k,)/2 with Ao = 0.2t .. ......... .. ............... 86 5-4 Scattering rate t,,(k,w) vs. w for k = kA(left) and kN(right) in the superconducting state at T = 0, for K = 5 (top) and K = 0.5 (bot- tom). Here Fo(kA) = 0.21. ................ .... ..88 5-5 Finite temperature spectral function at the antinodal point A and nodal point N on the Fermi surface multiplied by the fermi func- tion. Results for n = 2 and 0.5 with Fo(kA) = 0.2t are shown. 92 5-6 A(k,w) vs. w for n = 2 and 0.5. Results are given for the k points at (Tr, 0), (r, 0.05r), (ir, 0.17), (7, 0.15r). The disorder levels corre- spond to ro(kA)/Ao = 1, 0.5, and 0.025. Note the spectra for dif- ferent k points have been offset for clarity. .... 93 5-7 Comparison between recent ARPES data (left panel) and calculated A(k, w) (right panel). .................. ...... .. 94 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy QUASIPARTICLE INTERFERENCE AND THE LOCAL ELECTRONIC STRUCTURE OF DISORDERED d-WAVE SUPERCONDUCTORS By Lingyin Zhu May 2005 Chair: Peter J. Hirschfeld Major Department: Physics This dissertation studies the electronic structure of disordered cuprate super- conductors. Bi2Sr2CaCu2aO (BSCCO), a typical candidate for high temperature superconductivity(HTS), is believed to be intrinsically nonstochiometric due to the way it is prepared. The anisotropic form of the gap function makes the quasi- particle states extremely sensitive to impurities and therefore the spectroscopy and transport properties of BSCCO are influenced significantly by disorder. This compound is singled out for study because it turns out to be the most suitable sample for surface probes like angle-resolved photoemission (ARPES) and scanning tunnelling microscopy (STM). In this dissertation, two kinds of nonmagnetic impurities are studied: point- like, strong scatters and spatially extended, weak scatters, since both are believed to occur naturally in BSCCO. The quantum interference between two point-like scatters is solved explicitly, with detailed analysis of the resonant energy, local density of states and interference of wavefunctions. Understanding of the low energy DOS for a fully-disordered system is developed at the local two-impurity level. It is pointed out the divergent DOS at the Fermi level proposed by Pepin and Lee [1] is the consequence of a special nesting symmetry possessed by the half-filled tight-binding model as well as two-impurity interference. The Fourier transformed density of states in the presence of impurities is also studied. It is proposed that unitary and extended scatters are both necessary to explain the experimental observations. The implication of weak scatters on nanoscale gap inhomogeneity is discussed afterwards. I! ,i, ll the influence of forward scatters on single particle spectral functions is investigated. It is found that the II- .. ,-.' due to forward scattering collapses in the superconducting state. It is then concluded that a wealth of data from experiments different in nature can be understood in terms of quasiparticle scattering due to impurities of various spatial structure. The limitations of the quasiparticle scattering scenario are also discussed. CHAPTER 1 INTRODUCTION 1.1 History of High Temperature Superconductivity The year of 1986 witnessed one of the most revolutionary discoveries in the history of science -high temperature superconductivity-by Bednorz and Miiller [2]. The critical temperature T,, below which the electrons can pair and superconduct, was unprecedentedly as high as 35K in the perovskite structure. Such a high transition temperature suggested the possibility of superconductivity (SC) even at the liquid nitrogen temperature and immediately created a renaissance in research on superconductivity. The enthusiasm and perseverance invested in this area over the past two decades have been particularly fruitful and have led to the application of related techniques to diverse fields of industry. Besides the previously inconceivable increase in T, ( which is approximately 153K now !), a collection of novel physical effects, such as the anisotropic form of the order parameter, peculiar normal state properties, hidden ordered states and so on, emerged successively, but generally in unexpected manners. Their appearances kept turning over our conventional understanding of SC within the BCS framework [3] and stimulated numerous theoretical concepts aiming at the explanation on the underlying mechanism of HTS [5, 6, 7, 8, 9]. However, until now, a basic yet convincing picture of HTS has not been found, not only because of deficiencies in proposed theories, but also because of the lack of direct experimental tests at the microscopic level (Until 1999, only bulk measurements such as conductivity, heat capacity, etc. were available). Due to the omnipresence of -..-in. ... disorder in all HTS samples, analysis of all such experiments required a model of disorder treated in an effective medium approach. This obstacle was circumvented in 1999, by the notable progress in the scanning tunnelling microscopy technique. The differential conductance map Figure 1-1: The crystal structure of La2CuO4. Red sphere: Cu2"; blue sphere: 02-; yellow sphere: La4. displays an amazing surface structure of superconducting samples with atomic resolution [10]. The improved low temperature STM boosted our observational capacity to an unprecedentedly fine level and enabled us to concentrate on the electronic properties of high temperature superconductors (HTS) in the view of local probes. This has raised a new era of the theoretical study of HTS and contributed to the debate over the nature of the pseudogap regime where electronic states display spontaneous ordering and possibly intrinsic inhomogeneity. 1.2 Background About High Temperature Superconductivity 1.2.1 Crystal Structure Among the various types of HTS, the copper-oxide compounds, generally referred to as cuprates, are of particular importance. They usually possess high critical temperatures and complicated crystal structures. But what really makes them so interesting is the illuminating ubiquity of CuO2 planes. These layers are separated by intervening insulating layers (See Fig. 1-1). LaCCuO4 is a typical candidate, whose structure basically mimics a "sandwich," with one CuO2 plane between two LaO planes. Each Cu atom is surrounded by four in-plane and two out-of-plane oxygen atoms forming a perovskite structure. The electronic structure of parent compounds of HTS is quite unusual. Con- ventionally, transition metal oxides have 3d and 2p orbitals whose bare energies are well separated, and the strong local Coulomb interaction gives rise to superex- change mediating long range antiferromagnetic order in a Mott-insulating state with gap of order 10 eV; in HTS compounds, Cu atoms lose two electrons and become divalent ions, leaving a hole in the 3d,2ry orbit with a net 1/2 spin; the 3d level of copper ions is so close to the in-plane -, level of oxygen ions that they hybridize' strongly with each other to form the so-called charge transfer insulators with a much smaller insulating gap (~ 2ev). 1.2.2 Phase Diagram Superconducting materials are formed from the half-filled antiferromagnetic Mott-type ground state when the parent compounds are doped, leaving mobile carriers in the CuO, plane. The evolution of the electronic and magnetic order upon doping establishes a complicated phase diagram as illustrated in Fig. 1- 2 [11]. The intrinsic antiferromagnetic magnetic order is quickly destroyed by adding a small number of holes, and only occupies a "sliver" region close to hall- :.!II.,. whereas lr .... i 1 r in electron-doped samples is more robust. When the doping increases, a spin-glass behavior may be present before the superconducting phase appears. At a certain minimal hole concentration, we obtain the superconducting state. The critical temperature increases initially with the doping concentration, peaks at about x = 0.16 (optimally doped), and then decreases until it drops to zero. Besides the high critical temperature and the peculiar low carrier density in this domain, the unusual shape of the energy gap in momentum space is another defining property in HTS. And it is ( i1 the reason I HI 1 I.. includes many physical procedures here: the crystal field split- ting lifts the degeneracy of 3d Cu atomic level; Jahn-Teller effect further lifts the degeneracy by crystal distortion; the resultant 3d orbitals then hybridize with the in-plane 2p oxygen orbitals to form Cu-O bands which we may have Mott insulator or charge transfer insulator depending on where the lower anti-bonding Hubbard band lies. T Strange IMtal T(K) Pseudogap Optimally doped Underdoped Overdoped AFM T Halsd r Superconductor 0.16 Holes per CuO, plane Figure 1 2: Schematic plot of the phase diagram of hole-doped high temperature superconductors. X axis: hole dopant concentration; y axis: temperature. we generally refer to them as unconventional superconductors: the order parameter is found to possess a dz_,2 symmetry, lower than that of the Fermi surface itself. The w. .......l. ,.... phase becomes unstable at higher temperatures since thermal fluctuations break the ground state Cooper pairs and a "normal" domain with anomalous attributes emerges. It is the normal phase of HTS since the temperature T > T,, but is anomalous in the sense that the conventional Fermi liquid (FL) theory of metals fails to predict its one particle spectrum and transport properties [12, 13, 14, 15]. Among numerous proposals, the so-called Marginal Fermi Liquid theory" [8] (MFL) was aimed to address the normal state properties of cuprates by reconciling the contradiction between the well-defined Fermi surface and the non-Fermi liquid (but universal) behaviors in a phenomenological fashion. In the MFL theory, electrons are postulated to couple with bosonic excitations whose spectrum is only a scale-invariant function of energy and temperature, i.e., B(w, T) ~ min(w/T, 1). As the consequence, the self-energy for the coherent component of the one particle spectrum, extracted from this proposal displays a peculiar form (See Ch. 5 for details) with logarithmically vanishing Z = (1 - dReE(w)/ddw) at zero temperature on the Fermi surface. The Marginal Fermi Liquid theory has proved to be reasonably consistent with a wealth of unusual experimental observations, such as the linear resistivity I, in several decades of temperature, but has never been convincingly derived from a microscopic theory. Especially, how it yields the momentum-anisotropic d-wave pairing symmetry is the intrinsic deficiency of this theory (although it has been argued that this could come from the vertex corrections). In the optimal- and overdoped region, while the normal state property is fairly depicted by MFL theory, a conventional BCS theory is believed to be qualitatively applicable to the superconducting states, with the order parameter taken to possess d-wave like symmetry. There is one well-established, controversial region which is presently at the heart of the debate of HTS, which lies in the crossover from AF order to SC state, above the possible messy spin glass phase. It is named the "pseudogap" phase because of the spectroscopic identification of strongly depleted density of states (DOS) near the Fermi level. The pseudogap temperature T*, below which this phenomenon develops, declines linearly with increasing doping and merges with T, at roughly optimal concentration. This phase is believed by some physicists to be strongly related to the superconducting state in that it evolves smoothly into the latter and displays a similar d-wave symmetry in the order parameter. So far, numerous theories such as, spin-fluctuation [4], resonant-valence bond [5], precursor scenario [17], and competing states [18] have been proposed to address the underlying mechanism of the pseudogap region with each of them winning support from certain experiments [19]. Nevertheless, there is yet no consensus about the origin of the pseudogap to date. 1.3 C un ir i ; _,i ,, ;, |T;, iT i, ,i, ,. ,,, i. n i I. II ,i. h h n l The research on HTS in the past twenty years attracted an enormous amount of theoretical and experimental attention, generating interest from physicists, chemists, material scientists, and even mathematicians because of its fascinating Figure 1-3: Current situation in high temperature superconductivity. anomalous behavior observed and enticing potential applications. However, despite all this work, no consensus on the origin of unconventional superconductivity has been reached and there is a huge number of interpretations on HTS, based on the diverse observations collected from different groups, with different techniques, and on different physical properties. In fact, the current situation in the study of high temperature superconductivity can be "figuratively" depicted by Fig. 1 3 -:- which comes from a famous Indian Buddhist parable about the blind men and the elephant. And the following paragraph from this famous parable reflects how intensive the debate among them over the shape of the elephant ( according to everyone's own understanding) is: "Oh, how they cling and wrangle, some who claim For preacher and monk the honored name! For, quarrelling, each to his view they cling. Such folk see only one side of a thing." There is such an analogy happening in the high-T, community! I am sure that one day the "Buddha" ( a correct and comprehensive theory) will come for the sake of our piety and opens our eyes to the rest of the "elephant" (the nature of high temperature superconductivity), but I would like to, with the modest expectation that I could not be the most "blind" disciple, summarize the appearance of the monster after "caressing" it carefully: 1.3.1 What We Know About High Temperature Superconductivity 1. Cuprates have copper oxide planes, and these planes are crucial to high tem perature ii. I..ii.,. t. .r. 2. The electrons pair up in the CuO2 plane in a state with d-wave symmetry; there are nodes in the pairing function which give lower energy excitations in quasiparticle spectra; 3. The carrier density in HTS is considerably lower than that of the conven- tional superconductors; the coherence length is as small as (0 ~ 3 4a; 4. It is the proximity to a underdoped antiferromagnetic Mott insulator that may be essential to understand HTS, and probably indicative of the underly- ing magnetic mechanism of HTS; 5. Cuprates are in general metals; there exists generically a Fermi surface (at least a segment) in these samples. Normal states are anomalous, compared to good metals which can be well described by Fermi liquid theory. 1.3.2 What We Do Not Know About High Temperature Superconductivity 1. What is the mechanism responsible for the formation of Cooper pairs? 2. What is the nature of the normal state and the pseudogap regime? 3. What do the disorders in HTS tell us? What is the source of the local inhomogeneities observed in the tunnelling experiments? The collection of known and unknown aspects of cuprates paints the big picture about high temperature superconductivity but also imposes rather constraints on any theory regarding the fundamental mechanism. In my dissertation, I do not intend to propose any novel fundamental mechanism or reconcile the discrepancies between already existing theories; instead, I will concentrate on an interesting issue related to the impurities in d-wave superconductor, which has been the focus in the recent spectroscopic experiments on cuprates and discussed intensively by a huge number of illuminating works, within the extended version of conventional BCS quasiparticle theory in high temperature superconductivity. 1.4 Disorder in Spectroscopic Experiments 1.4.1 Scanning Tunnelling Microscopy (STM): In Touch With Atoms In 1981 Binnig and Rohrer revolutionized the field of surface science by inventing the scanning 1 .ni. IIn;,., microscopy [21]. When a fine pointed tip of metallic needle is brought extremely close to the sample surface (in general a few Angstroms) and a voltage is applied in between, the wavelike properties of electrons allow quantum tunnelling to happen. The probability of finding the I ...I. lih.. electrons decays exponentially when the distance between the tip and the sample surface increases. The topographical image of the surface is registered as the variation in the current when the tip scans over the surface. First-order time-dependent perturbation theory gives out the tunnelling current as function of sample bias V and position r: I(r, V) 27re r, V)= {f(p) [1-f(l-eV)-f(v)[1-f(E,-eV)]}lMJ(r)'28(E,-E,+eV). rv (1.1) where A,,(r) is the -.II... matrix element and f the Fermi function. The 6 function conserves the energy as it does in the usual cases. With several appropri- ate assumptions2 we can write I as I(r, V) = e-k dep(r, EF + e), (1.2) where r is the coordinate of points sampled and z is the vertical distance between the tip and the plane: the local density of state (LDOS) is defined as p(r, E) '. I, .,I E,). (1.3) 2 1) the -i.,. lii... event is extremely localized (this ensures the ,.[I Ui cur- rent is proportional to the local density of states): 2) it is spin-independent; 3) for small V, we can using WKB approximation such that AI(k, z) Aloe ", where k'2 = 4mr(Aw + V)/h' for a rectangular barrier of height A,,. Assiuning that p(r, E) is roughly a constant, the the differential tunnelling con- ductance dl/dV is proportional to the LDOS (this can be obtained by moving the tip away or towards to the surface to maintain a constant current and then apply an ,i. i!,, .i. voltage). This separation enables us to compare the theoretically calculate LDOS with the STM measurement qualitatively. The invention of STM technique opened an new era of surface science by extending our "fingers" to atoms and obtain atomic resolution images of sample surfaces for the first time. It was such an important contribution that Binnig and Rohrer were awarded with the Nobel Prize of physics in 1986. In the study of high temperature superconductivity, STM also contributed in a unique way, especially from 1999, by producing stun- ning local electronic images of the surface of BSCCO and providing local spectral information. 1.4.2 Impurities in Bi2Sr2CaCu2Os : A Probe of High Temperature superconductivity ,.i ', ., I is a typical candidate of high temperature superconductors. It is well known that the Van der Waals coupling between the BiO layers in BSSCO makes it easy to cleave (Fig. 1 4) and obtain the image of its surface by STM. However, this attribute also provides space where dopant oxygen atoms can reside and hence result in interesting experimental observations. Five years ago, the first high-resolution STM experiment on BSCCO [10] displayed fascinating atomic- scale resonances (as shown in Fig. 1-5), whose spatial and spectroscopic features agreed crudely with the theory of strong quasiparticle scattering from impurities in a d-wave superconductor. This agreement underscored the possibility of using impurities of different electronic properties to probe the superconducting phase [22, 23]. Publications about STM images on the ir, ? I....ii. impurity-doped BSCCO soon appeared, supporting the simple theoretical impurity scattering models by their qualitative agreement with predicted resonance energies and fourfold spatial structure [24, 25. 26]. Subsequent STM measurements revealed further the existence of nanoscale inhomnogeneities in the order parameter map [33,either to interactiondriven effects such as stripe formation [31, 32] or to the34]. [33, 34]. Figure 1 5: STM image in Bi2Sr2CaCu2O8 showing a) atomic scale native defect resonances; b) Zn resonance. Both were taken at zero bias. Appreciable differences in the spectroscopic and transport properties between BSCCO and YBCO, another popular HTS material (T, 2 93K), were reported extensively [34, 35], despite the fact that both are good high temperature supercon- ductors with well-identified d-wave order parameters as well. These discrepancies were not successfully reconciled until it was realized that unlike YBCO, BSCCO is ,irr,,.... l,!." contaminated by disorder through the way it is prepared; this was gradually established by the accumulated facts from STM and ARPES exper- iments. The charge inhomogeneity introduced by doping is poorly screened by the states in the CuO2 plane and induces spatially smooth potentials from which quasi- particles scatter. We have proposed that the dopant disorder should be responsible for the observed unusual properties of BSCCO. Hi.--r... ol. disorder studies have played essential roles in the comprehension of superconductivity: we can understand the mechanism of superconductivity by understanding how it could be destroyed by perturbations. However, until recently, research on impurity effects was limited to their impact on bulk properties such as transport coefficients or heat capacity, in which a finite concentration of disorder is generally assumed and treated in a statistical manner. It is the improvement in STM image resolution that enables us for the first time to focus on the local response from the superconducting state to one single impurity individually, from 12 which we can attempt to understand inter-impurity correlations and macroscopic disorder phenomena from a microscopic point of view. CHAPTER 2 SINGLE IMPURITY PROBLEM The study of the impact of impurities or atomic-scale disorder on the physical properties of materials has been of great theoretical and practical significance. In reality, no samples are pure. As Pauli said: "solid state physics is the physics of dirt." Sometimes impurities are destructive to desired physics; sometimes they are substituted intentionally to bring out special features and help identifying the underlying physics of the pure system. For example, impurity induced Friedel oscillations can aid to probe the kinematics of quasiparticles in metals [36, 37]. In this chapter, I will discuss the effect of a single impurity in an otherwise clean sample. The cases in normal metals and conventional superconductors are reviewed, followed by detailed analysis of the case of a d-wave superconductor. I will then discuss the discrepancies between recent STM experiments and theoretical predictions. 2.1 One Impurity in Normal State 2.1.1 Bound State and Resonant State It is of pedagogical significance to review the problem of single impurity in a metallic state before we begin to discuss its effect in a superconductor. The simplest grand canonical haniltonian for the normal state problem is i = (c(k) u1)ck,, + Y V+OCkCk (2.1) k k,k' where Ck, ctk are the creation and annihilation operators of particles respectively The first term in Eqn. 2.1 describes the single particle dispersion of the non- interacting ground state and the latter denotes the on-site energy change due to the perturbation (The single impurity is assumed to be located at R=0). The + 2 + Figure 2 1: Diagrammatic representation for the scattering process by single im- purity. Coarse arrow: full Green's function; solid line: pure Green's function; cross: impurity; dashline: impurity potential. single particle propagator is defined as G(k, k') = -(Tck(r)ct,), with 7 being the imaginary time. In the case of free fermions, it takes the simple form: G"(k,w) [w (k)]-' (2.2) The single impurity problem is ... i. i i!i solved by the so-called T-matrix method which provides immediate access to spectral properties [38]. The perturbed Green's function can be expressed in terms of the pure propagator G and the T-matrix as G = Go + GoTGo, (2.3) where the T-matrix is defined as T = V + VGT (2.4) and V is the potential matrix. Fig. 2-1 shows the perturbative expansion of the full Green's function in a geometric series in V. In the case of isotropic point-like scattering, i.e., V(r) = Vo6(r r') (Vo is the impurity strength), the T-matrix is momentum- independent and reduces to a simple form: vs T(w) 1 V k ) (2.5) 1 Vo Ek Go(k, W) The poles of T-matrix signify the formation of new quantum states, since they are not poles of Go, i.e., the eigenstates of the pure system. There are two kinds of states introduced in general, as shown in Fig. 2-2: a discrete, real bound or anti-bound state located outside of the band where the imaginary part of G() = Gk OG(k, w) vanishes, i.e., where the density of states (DOS) is zero (the -- -- _fl) no low-E BS Figure 2-2: Schematic illustration of the hound state in normal metal. Left:solution to the T-matrix; Right: the hound state is only located at high energies, i.e., out- side of the band. right panel), and an overdamped state in the continuum, which has a finite lifetime because of its overlapping with the background (the left panel). A real bound state is generically localized on an atomic scale if the impurity potential is of short range. 2.1.2 Local Density of States: Friedel Oscillations and Ripples in the Fermi Sea Apart from the determination of bound state energies, the correction to the local density of states due to the impurity is interesting as well. The introduction of the imperfection does break the stillness of the degenerate Fermi sea, inducing "ripples" in the space referred to as the Friedel oscillations. The change of LDOS in the bound state is 6N (r, ) = l,'. ,,r 1 .. o). (2.6) In a two-dimensional space, the real space Green's function is evaluated as GO(r,w) .- .. I ,'', .... r], k(w)= kF 1+ (2.7) where 1 is the chemical potential and kF is the Fermi momentum. When k(w)r is large, the Green's function has the following asymptotic form: CGO(r, ) -~ i'i- ,. r- /4J. (2.8) The LDOS falls off from the impurity with an envelope of 1/(kFr) and oscillates with a period of the inverse of the Fermi vector. .- Figure 2 3: LDOS modification due to single impurity in metals. Right: image on the surface of Ag [36]; left: theoretical calculation of LDOS around a single impurity in a two-dimensional metal. In Fig. 2-3, the theoretical prediction on the distribution of LDOS around an repulsive impurity (right panel) at zero energy in a two dimensional square lattice is shown. The four-fold symmetry of LDOS is the signature of the underling lattice symmetry. In the left panel, the Fourier component of LDOS ( obtained from the experiment performed by Crommie et al.[36] on the surface of Ag), which is defined as p(q, w) = e"rp(r, w) (2.9) is plotted: the radius of the bright feature in the FT-STS spectroscopy is 2kp. 2.2 Green's function for BCS Superconductors In the conventional BCS theory, the ground state of superconducting systems is a condensation in momentum space in which electrons pair up into "Cooper pairs." The mean field hamiltonian for the BCS superconducting state is : NHcs = tkcfk k + -(c'ctki*Ak + h.c), (2.10) k,, k with Ak = V(cklC-kl). The first term describes the motion of free band electrons and the second term describe how two electrons pair up to form the superconduct- ing order. It is convenient to define the column vector Dk and its conjugate' as the following: 4k Ckt) = (C C-kj), (2.11) and work within this framework. The single particle Green's function is then a 2x2 matrix: Go(k,T) = -(Tk(T7)I) S(TrCk(T)C4k) (Trkt(T)-k) (Trtkl ()Ckt) (T-rCtk(T)C-ki) For a time-invariant system, we can further introduce the Fourier transform: Go(k. iw) = e""''Go(k, T), (2.13) where uw, is the Matsubara frequency. With the gap Ak being the exact self-energy due to the phonon-mediated electron-electron interaction, the mean field single particle Green's function for the clean SC state (after analytical continuation to the real axis) is 0(O(k, w) = ,T7 + AkT1 + CkT3 (2.14) 2 E(2.14) where the quasiparticle energy defined as Ek = "-, + A and T, T,T 7 are the Pauli matrices. The imaginary part of the real space Green's function (G(r, v) = ,k e'kG(k. w)) is related directly to the local tunnelling density of states: p(r, ) = p,(r, ) (2.15) SThis spin and particle-hole resolved matrix structure is conventionally referred to as Nambu representation. with the spin-resolved LDOS, Pr(r,) = -r-llm Gjj(r,r,w+iO+) (2.16a) pI(r,w) = +T-Im G22(r,r,- iO*) (2.16b) With the general property of the retarded Green's function: ImG(w + i0 ) = -ImG(w iO') (2.17) the LDOS may also be written as : 11 p(r,w) = -n- Im [GI (r, r,r + iO ) + G22(r,r, -w+ ]1 (2.18) 2.3 One Impurity in an s-wave Superconductor 2.3 .1 ...... ... r.. I,, ., p F,,|; r i,., .. The non-magnetic disorder problem in an isotropic s-wave superconductor was studied by Anderson [39]. In the conventional Nambu notation, the hamiltonian with the presence of impurity (at R = 0) is formulated as 7 = Hu[s + Y VcCk', (2.19) kk',. where Vo is the strength of the impurity. It was pointed out that in dilute concen- tration limit, nonmagnetic impurities introduce :. ...I.I.I effects on bulk properties (such as T,) of isotropic superconductors, because the quasiparticle energy should be conserved in elastic scattering and the momentum transfer of quasiparticles in scattering process cannot disrupt the phase coherence established if the pair- ing is isotropic, i.e., Ak = A0. This conclusion goes under the name "Anderson Theorem." As a trivial test, the self-energy of quasiparticles can be treated with Abrikosov-Gorkov formalism [40] (AG) yielding: 1 IIk F(k, ) = 2- +A (2.20) 2TN Vk _+A 19 where TN is the normal state scattering rate. It is easy to see that the quasiparticle elastic scattering rate is suppressed from its normal state value 1/rN upon the opening of a gap, and it is zero at the Fermi level, reflecting Anderson's theorem. 2.3.2 Magnetic Impurity Problem When a :, .. ,.. i impurity is injected into the superconducting condensate, it couples with the spin density of conduction electrons by means of the exchange interaction. In the classical limit, 2 the impurity term in the momentum space may be written as: li,,, = J(k, k')ct 6io3 Sck,/3, (2.21) k,k' with J(k, k') the exchange energy, S the impurity spin and 5 the spin operator of electrons. Abrisokov and Gorkov [40] first treated the magnetic impurity in Born approximation, followed by a treatment in the unitary-scattering limit by Yu [41] and Shiba [42]. It is found that the magnetic moment breaks the time-reversal sym- metry upon which the Cooper pairs are formed and hence lifts the spin degeneracy of quasiparticle excitations.3 Consequently, an intragap bound state appears for spin down excitations if the local moment is spin up and vice visa. Increasing the concentration of magnetic moments leads to gapless superconductivity and finally destroys the superconducting state. The T, suppression is obtained in AG form [40] as T, 1 In(t) = (1/2) + p,), (2.22) 2 The classical limit is obtained by taking J 0 and S oo, but keeping JS = constant. In that sense, the spin can be viewed as a local magnetic field. 3 This can be verified by working on the BCS hamiltonian in the Bogoliubov for- malism. One caln that the magnetic term couples in the Nambu To channel and changes sign for spin up and down excitations. A potential scatter couples in the T channel, introducing nothing but level shifts. where i is the digamma function4 T, and Te are the actual and disorder-free transition temperatures respectively, and c, = (27rTbT)-' is a constant determined by the self-energy due to magnetic interaction, where r71 = rTco/2y. 2.4 One Impurity in a d-wave Superconductor Because of the doping procedures through which superconducting BSCCO samples are prepared, atomic scale imperfections are inevitably introduced. Recently, the differential conductance map obtained by STM experiments on the cuprates BSCCO-2212 with astonishing resolution have verified the existence of point-like atomic defects, and further investigated the behavior of Zn or Ni atoms in the d-wave superconducting bath by replacing Cu atoms with them on purpose [43, 44]. I will mainly review the works on single impurity in d-wave superconductors and discuss extensively their implications for STM experiments. 2.4.1 Nonmagnetic Impurity Isotropic Scattering and Pair-breaking Effect. It is speculated that when a Zn atom (with atomic configuration '...t'" i replaces the on-site Cu atom in the copper oxide plane, it will lose 2 electrons and the resulting Zn2+ should behave like a classical nomnagnetic impurity (if we neglect its quantum nature). This is because the Zinc cation has a closed 3d"0 shell which will gives a zero net spin S = 0. SThe digamma function is defined as v(x.) = lnr(x) = -y+ 1-(7,+ ,x -1, -2, -3...... (2.23) 7(n + x) where 7 is the Euler constant. There are also other points of view which are suggestive of the magnetic nature of ZIn2 impurity because magnetic moment sensitive experiments [45] revealed that Zn atoms actually induce local moments and hence one probably has to also take the local correlation or dynamic effect (Kondo resonance) into consideration [46]. S-A( ) clean + + fermi sea angle on FS -- Figure 2-4: Schematic illustration of the pair breaking effect due to potential scat- ters in a d-wave superconductor. However, unlike the case of nonmagnetic impurities in s-wave superconductors, quasiparticle scattering due to Zni+ ions in cuprates violates Anderson's theorem without breaking time-reversal symmetry. The physical process is illustrated in Fig. 2 4: scattering of quasiparticles mixes the initial and final states with different momenta k, thus mixing order parameters with different phases. When the scattering is strong enough, it mixes the states of complete opposite signs of order parameters with high probability and suppresses the condensate. However, I emphasize there that the gap structure for any T < T, is still unrenormalized since Ek Ak = 0 and the angular structure of the gap function remains unchanged [47]. To illustrate the effect of impurities in superconductors with different order parameter symmetries, I hereby tabulate the corresponding Nambu components of self-energies (in Born approximation, i.e., E(w) = n, Vo2 Ek G(k, )) in the table below: Table 2-1: The imaginary part of self-energies due to non-magnetic impurities in superconductors. d-wave Fo( ) 0 s-wave -rFu r where rF is the impurity scattering rate in the normal state, with j = w Eo(w) and Ak =A k E1. The symbol (.) denotes angle averaging around the Fermi surface. Gas Model: Circular Fermi Surface and Parabolic Band. The single impurity is analytically solvable in a d-wave superconductor with a circular Fermi surface and a planar continuum form of the gap function, i.e., Ak = Ao cos(20). With the T-matrix formalism introduced in Sec 2.1.1 and the BCS formalism of the Green's function, the full Green's function can be constructed as G(k, k', w)= ("0(k. a)kk, + G(;(k, w)T,.., .'" k', w). (2.24) While its off-diagonal component vanishes because Zk Ak = 0, the T-matrix reduces to the simple form: T = Toro + TT3r To = Vig1/(S+S-) T, = v (,c- l,,, i i SY T..",. i- (2.25) (2.26) where 1/, .,,., = c is the cotangent of the s-wave scattering phase shift r1o, and No is the density of states at the Fermi level. 9 is the momentum integrated Green's function. This expression has resonances when S = 1 Vo(G3a F Go) = 0. (2.27) In the special case of a particle hole symmetric system gs = 0, and the resonances are entirely dominated by gs(w). In the unitary limit, i.e., lo -> 7r/2 (this is equal to c < 1), the resonance energy f0L and scattering rate F are f = a (2.28a) 2 log(8/rc) f = ..c (2.28b) I I '. 7rC) These results were first obtained by Balatsky et al. [24]. Clearly, the two symmet- ric resonances are the signature of electron-hole duality nature of BCS quasipar- tides; furthermore, there is a finite damping rate for those resonances since they emerge from a continuum background, namely p(w) $ 0. However, a salient feature of those resonances should be pointed out: when c -- 0, the resonances will be tuned towards to the Fermi level and the damping rate diminishes simultaneously, i.e, the resonant states become "marginally" bound. These midgap states are ..... II referred as virtual bound states because of their asymptotic behavior described above, which can be attributed to the linearly-vanishing DOS (p('i) ~ w) at low energies. The spatial distribution of the impurity induced states conveys additional information about the nature of HTS and can be measured directly by STM experiments. The theory anticipates that the LDOS spreads in an anisotropic manner [24. 26]: it decays as the inverse second power of the distance from impurity along the nodes of the gap and exponentially in the vicinity of the extrema of the gap, and the decay length of the latter is characterized by (o = '-,. The spatial patterns of these "native" defects [10] agree with the naive theory to the extent that the observance of a bright spot of the diameter of 2- 3nin6 with p(r) decaying s 1/r2 for r > o, i.e., the Friedel oscillations at the periphery of the central bright image, as illustrated in Fig. 2 5. The coherence length extracted from the size of the atomic resonance is around 15 A. This again coincided with the ( known for Bi2Sr2CaCu2() being a convinc- ing evidence for the quasiparticle scattering scenario. However, the anisotropic con- figuration of LDOS distribution within a distance of o0 from the impurity, namely, the structure inside the bright spots, was not discernible due to the poor spatial resolution. Later improvement in STM yielded differential conductance map with subatomic details, and the fine structure of LDOS around impurities were obtained eventually, which invoked another quarrels upon the t1. ..,, II..,,, mechanism and the nature of impurity itself. 24 01 1 - I ; \ Distance from scattering center (nm) Figure 2-5: The fall-off of LDOS around a native defect. To study the spatial modulation of LDOS, we need the explicit form of the Green's function. These have been worked out by Joynt and Balatsky in some special cases [26, 48]. I also evaluated the Greens function when for the situation w/Ao < 1/ke.r < 1/ ..,, The results are tabulated as follows for future adoption: cO(R,w) (2.29) ,.-- R II (110) S N [(i--- -[,, + T- + T3) cos kF?+ R (100) (i a To + r T3) sin kFRI Lattic :I1...-. i ..ir _.. ,rr, _,,, i i ,r ,,. Li.,,,. The copper-oxide plane of cuprates materials consists one Cu atom and two oxygen atoms in one unit cell. A square lattice with copper atoms on the commensurate sites is the simplest reasonable model for practical computations. While the real-space components of the Green's functions are analytically unobtainable in the SC state, I hereby present the numerics for both a simple tight binding model Ck = -2t(cos k,. + cos ky) p and a realistic 6-parameter tight-binding model proposed by Norman et al. [49], both having the corresponding d-wave order parameter Ak = Ao(cos k, - cos k). Note the maximum value of the order parameter in the half-filled lattice system with the current convention is 2Ao. LDOS on-site LDOS nearest neighbor o0 02 0 0 1 0 2 02 -01 0 0 electron hole Figure 2-6: On-site and NN site spectra for tight binding band with one impurity. Fig. 2-6 shows the results for the nearest neighbor tight binding model. The impurity potential is taken to be Vo = 5.311, which generates two resonances at f = 0.013t, : l.5,mev using the Norman band, chosen to reproduce the observed resonance energy of Zn. The LDOS on the impurity site and its neighbor site are plotted. Note that the weight is almost excluded from the impurity site due to the strong potential 7 and the peak at f = -0.013tl captures the essence of resonant physics. Because the off-diagonal components of the integrated Green's function vanish, i.e., Gi12(r = 0, ) = 0, only the 11 element of the T-matrix contributes to the LDOS and this yields only one peak in the on-site spectrum. The situation is completely inverted on its nearest neighbor sites, where one observes a pronounced peak at positive sample bias. The coherence peak heights are also strongly suppressed, implying that superconductivity is locally suppressed as well. There are two distinct ranges for the falloff of LDOS: an exponential 7 The integrated spectrum weight on arbitrary site is still conserved f dwp(r, w) = 1. The sum rule is verified by the excessive weight trapped at high energy regime, i.e., the real bound or antibound states. However,we focus on the low energy bound states within the gap. envelope along the antinodal direction and a power law decay along the gap nodes. Additionally, the alignment of LDOS around the scatters shows a four-fold symmetry. 2.4.2 Magnetic Impurity Unlike the nonmagnetic impurities, even for a s-wave superconductor, mag- netic impurities are pairbreaking since they violate time-reversal symmetry, lift the Kramers degeneracy of the quasiparticle states and hence introduce intragap bound states.8 While the theoretical expectation for a magnetic impurity in a s-wave superconductor (for example, Nb) has been experimentally verified by Yazdani et al. [50], Hudson et al. [44] performed the first atomically resolved STM experiment on Ni doped Bi2Sr2CaCu2() (d-wave superconductor). As shown in Fig. 2-7, two distinguishable peaks were reported, at Q1 = 9.2 1.1 meV and Q2 = 18.6 0.7 meV separately in both the particle and hole channels. The spectral weight of the impurity state decayed in an *.. .;1 .i 1 manner: these peaks were particle-like (i.e., on positive sample bias) on the impurity site, then became hole-like on the next nearest sites and again particle-like at the 2-NN sites. Therefore, the LDOS around the impurity at positive and negative biases were rotated by 450 with respect to each other. The spatially complementary feature led to overall nearly particle-hole symmetric site-averaged spectra; moreover, the coherence peaks remained robust. A conclusion which was apparently in conflict with our naive intuition was then drawn: superconductivity was not: ... il disrupted substantially by the Ni impu- rity. Detailed analysis even revealed that the potential scattering is the dominant agent in Ni doped cuprates since the associated energy is of roughly 5-10 times bigger than those of other channels. s We neglect the Kondo physics here. The Kondo screening in HTS supercon- ducting state is novel in sense that it takes place on the background with depleted DOS rather than on the background with a roughly constant DOS around the Fermi level. 2 -"2 +0, b 075 -0 -25 0 25 0 75 Sample bas (mY) F ........ 2 7: Conductance spectra above the Ni atomni and at several nearly loca- tions. Solid circle: oil-site; open circle: NN-site; square: 2NN-site: triangle: 30A away flom Ni; solid line: average of the first four spectra. S3- ^ l2- A 1 !i* ^ |0 I '-- away from Ni; solid line: average of thie first, four spectra. Figure 2 8: Left: the schematic plot of Cu02 plane; middle: experimental LDOS around Zn impurity; right: -i .... I. i. predicted LDOS for Vo = 5.3t (rotated by 450 with respect to a) and b)). 2.4.3 Discussion: Characteristics of Impurities Classic impurities are generally categorized into non-magnetic and magnetic ones, with the Zn and Ni as the representatives. Although experiments exploiting them as probes for the superconducting state have achieved crude agreement with the quasiparticle scattering theory, characterization of their own nature is far from complete yet. While the measurements on the bulk properties such as resistivity [51, 52]. microwave conductivity [53] and T,: suppression display qualitatively similar dependence of Zn and Ni. the magnetically sensitive probes such as NMR (nuclear magnetic resonance) [45], INS (inelastic neutron scattering) exhibit appreciable differences between Zn- and Ni-doped samples. On one hand. the standard Abrikosov-Gorkov formalism predicts that T, is only suppressed about 20% faster by Zn than Ni if the magnetic nature of Ni impurity is neglected: on the other hand. the superfluid density is known to be strongly depleted by Zn rather than Ni. The image of the immediate surroundings of Zn and Ni provided by the STM experiments now provides a coherent microscopic picture: while Zn is fatal to local superconductivity, Ni has a much weaker impact, consistent with earlier transport measurements suggesting Zn is a unitary scatterer and Ni is an intermediate strength scatterer. Besides those differences between Zu and Ni impurities, the sign of Zn poten- tial itself is a matter of controversy. It was first claimed that, according to Hudson 29 et al.. Zn atoms are attractive potentials for electrons since "the on-site (impurity site) spectrum only has one peak at the negative sample bias around Do = -1.5 meV 0.5 meV [10]. However, theoretical calculations predict that an attrac- tive potential should break the pairs and bind the electrons, yielding prominent tunnelling DOS peaks at positive sample bias [24, 54]. Beyond the position of the resonance peaks, there are other discrepancies between the STM differential conductance map and the predicted LDOS. In Fig. 2 8, the schematic plot of the structure of CuO2 plane is given, where the orange spots and green spots stand for copper atoms and oxygen atoms respectively. The middle panel shows the experimental results around a Zn impurity with the same lattice orientation as the left panel and the rightmost panel displays the theoretical calculation of LDOS for a repulsive potential (Vo = 5.35t), but rotated by 45 degrees with respect the first two panels. Strong potentials scatterers are expected to exclude weight significantly from the impurity site and produce an on-site LDOS minima with a relatively bright spots on the next nearest neighbor sites, as illustrated by the right panel in Fig. 2-8, whereas the experiments observe completely reversed patterns with a local maxima on the impurity site and darker spots on the NN sites (the middle channel). Comparison between the middle and right panels elucidates these discrepancies clearly. Is Zn a repulsive or attractive potential in BSCCO-2212? A hand-waving argument can be given by considering the atomic configurations of Zn and Cu atoms. After losing two electrons, the energy of the closed 3d shell of Zn should be a prior far below the Fermi level, as derived from tlhe tabulated ionization energies (Table 2 2) of Copper and Zinc atoms, since the energy to create a Cu' is less than that required to create Zn' '. It is then speculated that Zn atoms should be attractive potentials relative to background Cu2I ions with strength of approximately -2ev. Table 2 2: The ionization energy of Copper and Zinc atoms. Cu Zn first ionization energy(eV) 7.726 9.934 second ionization I i.. r,,i \ I 20.292 17.964 third ionization energy(eV) 36.83 39.722 However, calculations based on the criterion for uiitarity in the realistic band structure of BSCCO [49] yield an opposite result. It has been well established by experiments that Zn atoms are indeed hard-core, unitary scatters, for example, the phase shift *m7 extracted from the STM data of Zn impurity [44] is 0.487r, close to r/2, i.e.. unitary limit. One can also perform computations on the phase shift with the conventional T-matrix fashion hIn detT rj(w) = tan det (2.30) Re detT It is found that only a repulsive potential can possibly approach to this limit." In Fig. 2 9, I show the resonance energy as a function of the impurity potential. as the resonance energy is defined to be the position of the local maxima in the on-site (the impurity site) spectrum. It is clear that to drive the resonant frequency towards to the Fermi level, a repulsive potential with Vo 1- 201t I is needed. Finally, a recent ab initio calculation exploiting density functional ,,..*. on the band structures and effective potentials of different defect-doped BSCCO samples [55] claims that Zn atoms are actually short range repulsive scattering centers. Recently, two similar arguments [56, 57] were addressed to reconcile the discrepancy between the result of attractive potentials and the spectra from experiments. It was contemplated that the spectral weight on the nearest neighbor Cu sites in the CuO2 planes appears above the surface Bi atoms directly over the 9 The conclusion is rigorously robust regardless of the detailed band structure as long as the particle-hole asymmetry is introduced and a hole-doped sample is discussed. VEt Figure 2-9: The resonance frequency as a function of scattering potential Vo; inset: the resonance frequency as a function of 1/Vo- impurity sites since STM tips are indeed probing the wave functions or LDOS of the nearest sites rather than the impurity site itself, the so called "filter effect." The Zn impurity is then identified as an attractive scatter, since it is the "filter effect" that allows us to observe a pronounced on-site peak at negative energy (an attractive potential is expected to yield a peak on the positive sample bias) However, it doesn't explain why the calculations based on the realistic band structure aiming to the unitary limit necessitate a repulsive potential. dO -XI CHAPTER 3 QUANTUM INTERFERENCE BETWEEN IMPURITIES Images of the LDOS around impurities have confirmed the existence of resonant quasiparticle states near strong scatters as Zn, but have given rise to new questions regarding the microscopic model for impurities as well. While the debate on single impurity is unsettled yet, HTS materials usually contain finite concentrations of impurities and the impurity wavefunctions do interfere with each other. The most remarkable consequence of the quantum interference between many impurities is the formation of a impurity band at low energies (the nodal quasiparticle states), which is believed to dominate the transport properties such as microwave conductivity. The many-impurity problem has been treated with numerous methods, including the self-consistent T-matrix approximation (SCTMA) [48, 58, 59, 60], exact diagonalization of Bogoliubov De-Gennes equations (BDG) [61, 62, 63], nonperturbative techniques [1, 64, 65] and perturbative treatment including weak localization effects [66]. None of these approaches offers insight into the mechanism of interference itself and we are still unable to answer the question of how the macroscopic i;-....i., ... .1 properties of a d-wave superconductor actually arise from the local properties of individual impurity states. The lack of the connection between the understandings of physics at these two different scales initiated the studies on the two-impurity problem [54, 67, 68], which (.- .i, the quantum interference effect at the simplest level. In this chapter, I will report my study on the two-impurity problem with a nongeneric band. I will then generalize the conclusion from this special example to generic cases and discuss how we can make the bridge connecting the single impurity result to the STM experiments on the :-iil, .- 1~ .1. !. I..1 system s. 3.1 Two Impurities in a d-wave Superconductor 3.1.1 T-matrix and Resonance Frequency With the formalism introduced in the previous chapter, the T-matrix for two isolated point-like impurities can be obtained by iterating the procedure for single-impurity T-matrix, in a 4 x 4 basis of spin and impurity site labels, T,,,( )= (3.1) T,,,(-R)T, fT,, where R = R, R, and where T, T,,, are the single impurity T-matrices associated with the two impurities. For identical impurities, TF = P,, = T(w), the single impurity T-matrix defined previously. The quantity f is defined as: f/(w) = [1 '"(-R,w ,i ....' ", R ,7 1 i-', (3.2) where G(R, w) = Ek exp[ik R]G(7(w) is just the Fourier transformation of G0(w), the unperturbed Nambu Green's function. For systems with inversion symmetry G((R, u) = G(-R.w). Note that in Eqn. 3.1, the physical processes are clearly identifiable as multiple scatterings from each impurity e and m, individually, plus interference terms where electrons scatter many times between f and In. In k-space, we can write the T-matrix in the more usual 2 x 2 notation as Tw(w)= [eikTRO R e 'RT (3.3) e-ik',R,,, where To is the Pauli matrix. Provided the resonance energies are distinct, peaks in the total density of states correspond to minima of the T-matrix denominator: V det[1 G(-R, )T(uw)GO(R,w)T , Si.1. .,l V)= DID2/(S S' ) with DI = D'1 + V2G (R,w) DV D2= D + VoCG(R,w) (3.5) where D~ = [1 VoG(0.) VoGo(O,w)] +(-1 l.,[F.....R -)+G3(R,w)]. (3.6) The factors Di, DI determine the four 2-impurity resonant energies. Here G,,(R, w) is the T. component of the integrated bare Green's function Go(R, w) = Tr (T,,Go(R, w)) (3.7) In certain special configurations, e.g. if the two impurities are located at 45 with respect to one other, it is easy to check that the off-diagonal Green's function Gi(R, w) = 0 V R. In this case the entire resonant denominator factories D= D14D1 _D2iD2-. The T-matrix then takes the simple diagonal form Tk,k'() = 2V cos(k. R) cos(k'- + 2 2 [1 Un \ R R r + 2Vosin(k. -)sin(k'.) +- 2 2 [ 152+ (3.8) where T (T3 0o)/2. When two identical impurities with resonance energies So are brought together, the bound state wavefunctions interfere with one another, in general split- ting and shifting each resonance, leading to four resonant frequencies fi-+, fi, I- and Q where the subscript indicates which factor in Eqn. 3.6 is resonant. If splitting are not too large, the electron and hole resonances are related in a sim- ilar way as in the 1-impurity problem, ; l -Q' and Qf 2 -Q+. Again the weight of each resonance may be quite different or even zero on any given site. A large splitting may be taken as evidence for strong hybridization of quasiparticle wavefunctions. If we take the interimpurity distance R as a parameter and keep impurity potentials and other parameters fixed, there are two obvious limits where this splitting vanishes. In the case of separation R = 0, the two impurities combine ii, ,i,. ,,I. ,ii .. to create a single impurity of strength 2Vo, so both 1,2 approach the Q1(2Vo) appropriate for the double strength potential. In the case of infinite separation R -> oo, we must find fl,2 approaching the Qo(Vo) appropriate for isolated single impurities. To illustrate this argument explicitly, I will present both the analytical calculation with a gas model and numerical results on a tight binding lattice model. 3.1.2 Resonance Frequencies Gas Model. Eqn. 3.4 is a general result for two 6-function potentials embedded in a host described by an arbitrary G. We would like to derive analytical results for the resonance energies obtained therefrom to get some sense of the appropriate length scales and symmetries in the problem. At large distances, the resonance energies must approach the single impurity values, so the splitting can be cal- culated perturbatively. To do so one must first obtain analytical expressions for the large-distance behavior of the unperturbed Green's functions. This is difficult for the superconducting lattice ],,i.I -..,....., model on which most of this work is based, but much insight can be gained by studying the equivalent gas model, with spectrum (k = k2/2m. In this case expressions have been obtained by Joynt and Balatsky et al. [24, 48] for the d-wave integrated Green's functions G,(R,w = 0) at large distances, both for R making an angle 45 or 0 with the x axis. We have evaluated the real space Greens function in the last chapter for the range of fre- quencies w/Ao < 1 r < 1/kFo. With these expressions, the resonance energies may now be found by inserting these expressions for frequencies WJ = fl + 6 into (3.4) and solving for the shifts 6. We find f+12 if 6, with (3.9) S (o .... .R- II (110) A.r .. r..-/o ,,,P *' + 7r/4) R (100) These expressions are valid for 6/112 < 1. Clearly the decay of the splitting cxp(-r/l,,I i r- is much more rapid for distances larger than the coherence length along the antinode (100) than for along the nodes, where it falls as ~ 1/r. This anisotropic form of the splitting is the signature of the marginal bound (nearly localize) single impurity state in a d-wave superconductor, which, by allowing :-i..I..- ,n..- low-energy nodal quasiparticles, has extended tails along the gap nodes. It is the lack of a scale along this particular direction in the impurity wavefunctions that allows the strong overlapping even over long-distances. We will see later that how this attribute and its consequence questions the STM analysis of "isolated" impurity patterns. Lattice Model. In the following sections, I will briefly sum up the numerical results on two types of lattice band structure: the nearest neighbor hopping model (i = 0, pure nested band) and the 6-site tight binding hand [49], with Ak = Ao(cos k, cos ky). A = 0.11 and impurity strength Vo = 10t for the former band and Vo = 5.3t for the latter. The solutions of resonant frequency corresponding to each factor in Eqn. 3.5 can be tracked as a function of separation R by minimizing Di,2 separately. In Fig. 31 1 show the result for the pure nested band : the pentagons and open circles identify different branches of the splitting resonance frequencies: the left panel illustrates the result of R I| (110) inter-inpurity separation and the right panel the R 11 (100) case. It is seen that each factor D, corresponds to an i. 'l.r .-, function of R. with the factor determining, e.g., 12, changing from site to site according to whether the site is even or odd. This is due to the strong R dependence of the components Go; in the simplest case, R || (110) and p = 0, G3(R,w) = G,(R,w) = 0 but Go(R,w) Ek cos(kR/v/2)cos(kyR/v -: .*, ) oscillates rapidly. At R = 0, the problem reduces to the double-strength single impurity case; the factor il gives the resonant frequency to (2Vo) and the factor D2 is 1. At large separation the l2 and Ojf "envelopes" are seen to converge to QO (Vo) with a length scale of a few so 10a for the parameters chosen. In the R | (100) case, the oscillations of the bound state energies with increasing R are not so simple, as seen in Fig. 3-1. The one obvious simple difference from the (110) case is that the energy splitting vanish much faster with distance, as expected hrom the discussion in Sec. 3.1.2. Otherwise the short distance behavior of the bound state energies is complicated. One can check that the resonant frequency closest to the Fermi level is QL1 when R = 2 + 4n, n integer, and Wf otherwise. In general, short inter-impurity distance behavior is difficult to analyze ana- lytically and it is found that in neither the (110) or (100) direction do resonances appear at all for R = 1. The hybridization between wavefunctions is so strong in these cases that the picture of perturbatively split 1-impurity states breaks down. More importantly, the splitting are significant up to quite large distances. Param- eters in Fig. 3-1 are chosen such that So 10a, as seen from the right panel of Fig. 3-1 where we indeed expect an e-r/9t falloff according to the previous section. On the other hand, the left panel of Fig. 3 1 indicates strong interference out to separations of 30a or more! This means that if two impurities are oriented along 45 degree with each other, the interference effect will be long range, due to the delocalized nature of the single impurity states along gap nodes. 0 035 0.03 0025 ! -- ---05-- 0,015 upp rsonanc lower 2 ~onc 2a. * o io 20 .. R a Figure 3 1: Two-impurity resonance energies Q.,2 vs impurity orientation. 3.1.3 The Interference of Two One-impurity Wavefunctions In the 1-impurity case, the T-matrix is given by Eqn. 2.26 and it is easy to show (from Eqn. 2.3) that the change of LDOS due to the single impurity is: ( .llrl (G (r))2 -6Gf,(r,r,'u) = -Vo Im r + ( ). (3.10) Quite _. 1 ,il1 the Green's function can also be expressed in terms of the exact eigenfunction (,,(r) of the system (and its conjugate component r I i, with the presence of the impurity [24, 38] G(r,r,) = (r),,(r) S-' (3.11) where the final approximation is valid for a true bound state with w very close to a particular bound state energy n,, and will be a good approximation in the present case to the extent the resonances are well defined, in the sense discussed previously. Comparing with the form (3.10) allows us to identify the positive and negative energy wavefunctions of the single-impurity resonances (assuming Vo > 0): S GoI(r, ) = O ' (r) 0 Z (3.12) ;,.i ') wu=ttQ where Z are non-resonant wave function normalization factors. Note that the electron-like bound state eigenfunction is directly related to the off-diagonal bare Green's function, while the hole-like wave function is proportional to the diagonal bare Green's function. We can follow the same procedure for the two-impurity Green's function, and ask how the eigenfunctions at a particular resonant energy are related to the single impurity wave functions we have just found. Since the single-impurity resonant energies are different from the 2-impurity energies, this analysis will be valid to the extent the splitting are small compared to Sos. The Green's function iG(r. r) can now be constructed from Eqn. 3.8 and the wave functions read off by comparing with the spectral representation in the same way as in the 1-imlpurity case. By examining Eqn. 3.8, it may be shown that, depending on whether D1 or D2 is resonant, the wave functions thus extracted will be of definite spatial parity, ,,(r) = +p,,(-r). We find (R)Vo' ',f = (f +- (R)Vo ''1 ,h' = Z (c1, + jv R) ) =11- where G,) -= ((r R/2) G0(r + R/2), and the Z'P are normalization coefficients. These are the two-impurity odd (p) and even-parity (s) resonant state eigenfunctions expressed directly as linear combinations of the corresponding one- impurity eigenfunctions ip given in Eqn. 3.12 With the aid of the explicit forms of those wavefunctions, we can study how the Friedel oscillations induced by single impurity interfere. In general. impurity wavefunctions include particle and hole components because of the anomalous scattering processes with amplitude G, (R). However, in some special cases where GI (R) = 0, for example. R 1] (110), the ReW ImV I 12 LDOS 1 (r, c A are separated by R = (6, 6). eigenfunction become simpler and do not mix particle and hole degrees of freedom. L;(r) = Z; G ,(r, ) 0 = Oi- (3.14) and (r) = Z j(r) (3.15) / lP(r, L,)) U) = ,_ In order to illustrate to what extent the approximated wavefunction is effective, I show the real part (1st column) and imaginary part (2nd column) of those wavefunctions at energies equal to the four resonant frequencies (in this separation, 2l/t = 0.0195, l-/t = -0.0075) explicitly in Fig. 3 2. The density probability, i.e., 1 p21 (3rd column) and the actual LDOS (4th cloumn), defined as G(r,r,w) = Y (k, w) (3.16) k k,k' I .-.i e I 03 03 Figure 3 3: Spectra on the nearest sites close to the impurities. See text for details. are plotted as well. It is clearly seen that there is a good agreement between the spatial pattern of the I|',|2 and the exact LDOS calculated at each resonant energy, implying that near each resonant energy the nonresonant contributions are quite insignificant. The distribution of LDOS with different inter-impurity orientations for pure nested band is plotted in Fig. 3 4 and Fig. 3 5. There are several novel features which need to be addressed. First, It is clear that some states involve constructive and some destructive interference between the 1-inpurity wavefunctions in different regions of space, but the spatial patterns are, not unexpectedly, considerably more intricate than the I. -,,. ,, molecule" type states one might first imagine would form, with electrons living either directly between impurities or completely expelled from this region. This is of course due to the d-wave character of the medium in which the quasiparticles propagate. For example, the LDOS is zero at the point halfway between the two impurities for the p-wave states, but it is quite small in the s-wave states as well. It is furthermore clear from the figure that both s I-.. ,i 1.,- 1 and p .nI l....i. i.,_ I functions can have either constructive or destructive character, in the molecular sense. Note that the states are shown arranged vertically according to their eigenenergies, but recall that the ordering of the s and p (D) and D2) states changes according to whether R is even or odd, as indicated in Fig. 3 1. In addition, one can look into the spectra on some particular sites close to the impurities. I -.- 3 3 displays the spectra on several sites whose positions are illustrated as the numbers in the insets and the impurities positions are R1 = (-3,0) R2 = (3, 0) (left), R1 = (-3, -3), R2 = (3,3) (right). It is surprising to observe that there are some surprisingly sharp peaks, which always happen at high energies and are far sharper than a single-impurity resonance at the same energy. This is counterintuitive based on our knowledge of the one-impurity problem: the T-matrix denominator S possess an imaginary part which vanishes linearly when Fermi surface is approached, hence a resonance with lower energy naively has narrower width (longer lifetime). This novel peculiarity must then stem from the quantum interference effect. Indeed, this can be understood by examining the structure of the denominator of T-matrix, with one impurity at origin: D =det[Vo (G'""P) R R I i.1 I (w), where T is the one-impurity T-matrix and 0(-"r"p) is the Green's function with one impurity at the origin. Thus sharp two-impurity resonances occur for exactly the same reason as in the one-impurity case, but because the one-impurity DOS at R is nonmonotonic in uw, the resonance broadening is not i ..* i1i proportional to the resonance energy. One might attribute the unexpected "trapped" (weight primarily sitting between impurities) or I. .i n,'' (weight is populated outside of the region between impurities rather than inside) patterns to the special symmetries of the half-filling -I.i.r-i .i..i. band structure, for example, the pure nesting of the Fermi surface. It is this nesting which by allowing a considerable phase space for scatterings with a fixed, momentum vector Q ( in general, Q = (, 7r)), leads to a static or dynamic spatial modulation in terms of periodic arrangement of particle density, i.e., charge density wave (CDW), spin density wave (SDW) etc. Upon the study on a realistic band, we demonstrate that it is the commensuration of dominating scattering (2,2) (5,5) (6,6) P-M sM I -, -,' 3 4: LDOS maps at resonant energies for R (110). Pure nested band, Vo = 10t; Impurity separations are shown on the top of the graph. wave vectors at the bound-state energy rather than the peculiarity described above that determines the standing-wave like modulation, a typical argument for wave interference. Fig. 3-6 illustrates the spectra on the adjacent sites of one of the two impu- rities, with impurity separation R = (R, 0) and R = (R, R) for a more realistic tight-binding coefficients fitted by Norman et al. [49] from ARPES data, t(k) = to + 2t[cos(k, ) + cos(k,)] + 4t2 cos(kf,)cos(k,) +2t3[cos(2k,) + c..-, : I +2t4[cos(2k, ) cos(ky) + c... I i..... : ., I I + 1. .. .. .. .,I (3.17) with to... t = 0.879, -1,0.275, -0.087, -0.1876, 0.086 and I|i 0.1488eV. All the curves have been offset for optical clarity. The impurity resonance dependence on separation is transparent: the hybridization is strong when they are too close (no resonance pattern for R=l); then the -.i, ,-.. shows up and persists over a certain (2,0) (5,0) (6,0) n P mms-m Figure 3-5: LDOS maps at resonant energies for R I| (100). Pure nested band, VO = 10t; Impurity separations are shown on the top of the graph. range of inter-inmpurity distance for each case. The sharp high energy peaks emerge sporadically and those along (100) direction occur more frequently than that of (110) direction. In wave mechanics, when two waves are Ir II.. along a string in opposite directions, a standing wave can he established as the position and time depen- dence of the resulting wave could be separated(when the two waves are 180 out-of-phase with each other they cancel, and when they are in-phase with each other they add together). In our case, the formation of these trapped states is surprisingly explicable' in a similar fashion. For a real bound-state, the equation (G'(R, w))tFO(R, w)T = 1 should be satisfied strictly for both real and imaginary parts at particular energies. The product (;'(R. w)T((R, w)T in Enq. 3.4 could 1 It is a bit surprising in that this native picture doesn't take into consideration the w dependence of the T-matrix, or the Nambu structure of Green's functions. Moreover, the quantitative justification of the analysis is unexpectedly obtained although there are some fundamental difference between the one-impurity and two- impurity problems as discussed below. -003 -002 1 0 001 002 003 FreuencvO Frequency (( Figure 3 6: LDOS spectra for realistic band and Vo = ". :, on nearest neighbor site. Left panel: impurities at (- 1/2, 0) and (R/2, 0) ( R = (R, 0)), spectra taken at r (H/2, 1). Right panel: impurities at (-R/2, -1/2) and (R/2, R/2) ( R - (R, R)), spectra taken at r = (R/2. R/2 + 1). be equivalently written as eiq' (G(k, w)T(,)G(k + q, )T(w). (3.18) k,q It was argued by Hoffman et al. [69, 70]. that the characteristic wavevectors in the spatial Fourier transform of the LDOS of disordered BSCCO samples are determined by peaks in the joint density of states (See Sec. 3.3.1 for details), i.e., Ek Im G11(k, w)Im G (k + q, w), as illustrated in Fig. 3-16 Although the above argument was proposed for single impurity and in Eqn. 3.18 we have a separation dependent phase factor e"'R which is deleterious to this argument by averaging the joint density of states with random phase factors, it is still found that the long- lived two-impurity bound-states are primarily derived from a a few selected wave vectors; in order to have standing waves, the phase accumulated from scattering off from two impurities, together with the contribution from space propagation should sum up to nr, i.e., q R + 2j0o = nor, with q being the dominating vectors of scattering process and rio the phase shift from one impurity. In the unitary limit, 2r0 = wr, so the -.... n, i.-.i i.- requires qi R + 20r0 = q2 R + 20/o = nr, qc R + 210o = mr to be satisfied simultaneously. It is easy to check for our present band structure that (3.0), (7,0), (11.0). (14,0) and (3.3), (11,11) roughly cI/2 q, n nt2 0 i/2 I !,-.' 3 7: Fermi surface of BSCCO-2212 with constant energy surfaces at w = 0.04 shown as small filled ellipses at the nodal points. ql, q2, qs are wave vectors for which the joint density of states is large. balance the equation and we indeed see sharp resonant states for those specific I...-f,,.io ,i.... Also, it is observed that the sharp resonant states occur much frequently in the (100) direction than in the (110) direction as the criterion holds. We also noticed that the splitting surprisingly persists up even to a relatively large distance R = 13 in (100) direction, while the exponential decay of wavefluc- tions in the gas model would predict weak interference and negligible splitting at this separation. This results suggests that even with a .. i r. 1, dilute concen- tration of impurities, the spatial LDOS around impurities should not be identical because of the random distribution of impurities and the interference between their wavefunctions; however, STM experiments apparently observed point-like, well isolated, and nearly identical impurities patterns embedded in the superconducting bath. In the following sections, I will show how this contradiction can be relaxed. F... il it is also noteworthy that while the gas model or pure nested band an- ticipates four distinctive resonant peaks, there are usually only 2 visible resonant peaks which can be observed for the realistic band. 47 3.2 DOS of two Impurities and Many Impurities at w = 0 : Local Resonance vs. Cumulative Interference 3.2.1 F. ,, 1 ; F ,,, I; ... -,-T 'IN ,i;, , The problem of low-energy d-wave quasiparticle excitations has been treated I, 1..,. .,11 ,! with a so called self consistent T-matrix approximation (SCTMA) [48, 58, 59, 60] which partially sums up the perturbation series and models the impurities as strong point-like scatters. As illustrated in Fig. 3-8, the self-energy is approximated as E(u) a n,T(w), i.e., the product of the concentration of impurities and the single impurity T-matrix, by dropping all the crossed diagrams. It is then clear that SCTMA incorporates arbitrary potential -r. i-.!-b appropriately but neglects inter-impiuity correlations. SCTMA predicts that the impurity states broaden with the increase of impurity concentration and finally evolve into a subgap impurity Iand. The corresponding roughly constant low energy region of the DOS is referred to as the impurity plateau, as illustrated in Fig. 3 9. While the existence of excess quasiparticle states (the impurity band) has been confirmed by several transport and thermodynamic measurements, the lifetime of those nodal quasiparticles is not consistent with what the simplest SCTMA conjectures. For example, the absorptive part of the microwave ..k. l i. r a is significantly larger than the disorder-independent "universal limit" proposed by Lee [71]. Moreover, the expected T2-dependence of a at low temperatures (one factor of T comes from the density of states and another one comes from the self-energy which is inversely proportional to T) [72] arc not borne out by experiments as well; instead, a displays a linear T dependence in YBa2Cu107_a [73. 74, 75]. Among the various explanations proposed for these discrepancies, one crit- icism casts doubts on the validity of the SCTMA itself by arguing that crossed interference diagrams neglected in this approximation leads to a second-order r.a log' 2, divergence in 2D coming from the gap nodes [64], to be compared with the SCTMA, where one has 2nd order contribution of hi ,. Thus for any fixed n,, the crossed diagrams dominate at sufficiently low energy. SCTM (B) U., + ,. + ... .' * G + + miad diagmm Figure 3 8: Schematic plot of the self-energy diagram with many impurities. (a) Zero-energy (b) peak Sero-energy suppression Impurity hand F r-i-. 3-9: Schematic figure of the many-impurity DOS (a) in the unitary limit of the half-filled band and (b) in the unitary limit of a generic band. 3.2.2 Nonperturbative Predictions: When casting doubt on the effectiveness of SCTMA, Nersesyan et al. [64] accordingly performed a nonperturbative calculation for scattering within a single node and predicted a vanishing DOS at the Fermi level, p(w) w L with exact hosonization. After this work, several i.i......ii11 exact nonperturbative theories made diverse conclusions: p(O) may vanish with different power laws [61, 76], saturate at a finite value [65], or even diverge[l, 62, 77, 78]. Fig. 3 9 displays the band-dependent p()I = 0) .. ,. ,, i,. .11. All these nonperturbative works claimed to yield exact results for the problem of low-energy DOS of a d-wave superconductor. These apparently conflicting results were reconciled by Hirschfeld and Atkinson [63] and Yashenkin et al. *-,.*. who pointed out the d-wave disorder problem was very sensitive to symmetries of both the disorder model and the normal state band. Among all the theories, the one proposed by Pepin and Lee [1] is of great interest to me, which claims that the result for a N-impurity T-matrix is essentially n, times the single-impurity result. However, a closer inspection reveals that this peculiarity is tied to the disorder interference rather than the single impurity resonance at the Fermi level. Hirschfeld and Atkinson [63], Yashenkin et al. [66]. and Atkinson et al. [79] further pointed out that this divergence is the consequence of a global particle-symmetry particularly possessed by the half-filled tight binding model. 3.2.3 Divergence in Pure Nested Band with Unitary Scatters Hirschfeld and Atkinson [63] have expended considerable effort to reconcile the disagreement among the nonperturbative approaches. However, including their works, none of the studies above have addressed this problem from a local point of view. The two-impurity problem offers a possibility to study this problem, to observe how the divergent DOS in the perfectly nested case might arise. As we know from Sec. 3-1, the Green's function for two a-like impurities can be written as G(r,r', w) = G;(r r', w) + ("(r R .. i i., o'(R, r, s). The 2-impurity T-matrix is a 4 x 4 matrix (in the Hilbert space expanded by site and spin indices) satisfies VU 4-C (^O .) -Go(Rw) T= "(. ) (R (3.19) -G (R,1 ) to-'3 (G" (0,) It is technically expensive to evaluate of the local Green's functions ("(R. 1 ) and we can only approximate then under some specific circumstances. For the A F,-.!. 3 10: Schematic plot of bipartite lattice. -i-i.,,ii.,,Lh.. hlall-filled band and the limit w -- 0, we obtained an asymptotic form (see Appendix A): (o, ) = o. (3.20) where a = N I i _, N = 4 is the number of nodes, vF is the Fermi velocity and v is the anomalous quasiparticle velocity IVkAk|. and the cutoff A is of order of An. The expansion in w for r = (mn, ) depends on whether n and m are odd or even. For the (even, even) case, we have G"(r,w) (-1) i' ) + C'o(r)] to, (3.21) where Co(r) is a real function of r. We find similar leading-order expressions for (m, n) = (odd, odd), ( ... ... ^) Co(r)o,. (3.22) while for (m, n) = (odd. even) or (even. odd). G o(m, n, L) Cl(r)f + (.(r)3s, (3.23) where C1 (r), and C3(r) are real coefficients as functions of r as well. This dis- tinction between even and odd sites accounts for the oscillatory nature of the wavefunctions for the special case that the Fermi wavevector is commensurate with the lattice. A concept of bipartite" lattice should be introduced here, which divides the square lattice into two interleaved subsets, as illustrated in Fig. 3 10, where V102 2x10' 001 Vo= .0. 002 0 Vo 5xl0 vo=100 V o=1000 S-0.002 -0.01 0 .o001 0.002 Figure 3-11: ( I, i... in p(w) due to impurities. Impurity separation: R = (2,2). red and blue sports belong to different sub-lattices respectively. A stunning effect happens when the two impurities are in the same sub-lattice, i.e., R = (even, even) or R = (odd, odd). In that case, we found det T diverges as det :-T .. R .::,. i R = (even, even) SGo(0, )- R= (odd, odd) and the correction to total density of states: S1/[wIn2\ -" R = (even, even) (3.24) S2/[wh l2(A/w)] R (odd, odd) It is worthy to pinpoint the origin of the divergence since the similar result was also obtained by Pepin and Lee (PL)[1], which claims that by averaging over all the possible impurity separation using the approximate form G0(R, w) ~ 1/R out to a cutoff t/R, p(w 0) diverges as p(n)= (3.25) p(w) i(A/w) + (7/2)2] where A is a cutoff. In our model, for the (odd, odd) case, the local Green's functions dominate over the nonlocal ones and the physics of the low-energy resonance is essentially that of two noninteracting impurities, which is verified by the factor of 2 in front of the single impurity result( see Eqn. 3.24). For the (even, 10 0. 10V1000 0 04 Figure 3 12: ( I .... in p(~) due to impurities. Impurity separation: R = (2, 1). even) case, the interference effect seems to wipe out the spectral weight and move it to high energies but the divergence survives, as its magnitude reduces to half of that of two isolated resonances. In Fig. 3 11, I show the change of DOS due to two impurities with separation R = (2, 2) for different impurity strengths. It is observed for intermediate impurity strengths, four resonant peaks are clearly defined as the consequence of -i .i t ..... of single impurity resonances. When the unitary limit is approached, a single divergent peak is emergent. However, it should he stressed that, in both cases of separation above, our results are intrinsically different from PL result which is inherently nonlocal. When impurities belong to different sublattices and are separated by a distance with site indices parity as R =(even, odd). The two-impurity T-matrix defined in Eqn.3.19 is: 1 -i, )|) C,R Rl-,+C3(R)fc D' Ct(R)Tl + C3(R)f3 -( i, .-, with )'= G(0, w)2 C (R)2 -(C3(R)2. It follows easily that det T = D'-2 and that 6p( I 0) cx d In -1 A 0 (3.26) A similar result holds for R =(odd. even). Physically, the fact that bp vanishes at the Fermi level indicates that bound state energies must always arise at nonzero 1*5 | I. I l t | 1.5 Vl(X) (a) (b) 2 .Vo=250 0 I 05 I V r'- :o-0.5 -0.02 0 0.02 -0.02 0 0.02 (1) =20 (c) (d)- 8 4- 62 V,)=250 -2 -0.01 0 0.0] -2 0 2 Figure 3 13: (a) DOS for V0=100t. (b) Sealing of the DOS with Va. (c) Scaling of the DOS with L. (d) Scaling of the DOS with Vo and L=60. energies. Numerical calculations of the DOS shown in Fig. 3 12 demonstrate that there is no remnant of the single impurity w 0 divergence for this orientation, and that the resonance energies scale very little with Vo. In this case, it is the dominance of the nonlocal terms which shifts the resonance to finite energy. 3.2.4 Discussion: Bridge Between the Two-impurity and the Many-impurity Problems We have been aiming to show how the two-impurity problem is suggestive of the fully disordered system. To explore their correspondence, two basic issues need to be addressed here: 1) what gives the divergence at a = 0 in the pure nested tight-binding model? 2) how does the impurity band evolve away from w=0? 6-like Divergence at w = 0. In Fig. 3 13, I show the scaling of the integrated DOS as a function of the impurity strength and the system size. The impurity concentration is fixed to be in = 0.1%. The nonperturbative PL result from Eqn. 3.25 is also plotted as dash lines for direct comparison. By setting A = 1, we achieved a good fit the numerics at Vo = 100t. However, when the impurity strength increases, the weight keeps concentrating at the center peak and saturates slowly. This finite-size effect requires us to work in the limit where the level spacing is smaller than the center peak width. The limit Vo -r oo is illustrated in Fig. 3 13 (d). Generally, the peak becomes sharper when Vo is bigger. It is indicated that the divergence is -II,. a delta function, i.e., liimvoi, p(w) ~ 6(w) since the peaks scales as p(w) z VoF'(wVo). What gives the divergence at zero energy and why it is not observed in any other many-impurity case with a generic band? It is pointed out by Yashenkin et al. [66] that the half-filled tight-binding band possesses a particular global nesting symmetry 2G0o(k + Q)f2 = Go(k), which in the unitary limit, i.e., U = oo and p = 0 (in this this special band) gives additional poles with moment Q = (7T, 7) to the particle-hole and particle-pariclcle propagators. This diffusive mode is gapless and hence controls the singularity at zero energy. Any distortion on the band or deviation from unitarity will destroy this nesting symmetry and yield zero DOS at the Fermi level. We have performed careful examinations on the scaling of the peak area with respect to the system size and extracted that the localization length -o x 40a. When system size is bigger than o, the localization effect makes the effective system size finite and the peak weight saturates. This is distinctively different from the Pepin and Lee's result in origin. The divergence of the latter arises from the cumulative effects of interference between a large number of distant impurities. In Fig. 3 14, I show the real space structure of those states in the divergent peak. The LDOS from the eigenstates with energy |E,, < 10-s is plotted, where impurities on different sublattices are indicated with open and filled circles respectively. A remarkable discovery is that only part of the impurities have pronounced structures while others are weakly visible. It is further noticed that + + oo 4 + Fi-.,!, 3 14: LDOS for 2% concentration of impurities and E1,, < 10- t (20 eigen- values). those which "light up" or "dim out" are located in the same sublattice A or B as denoted in the previous section. While this finding can be tmderstood as reminiscent of the two-impurity problem, it is still puzzling in that every impurity is expected to have some close neighbours which belong to a different sublattice and hence contribute to the destruction of the zero-energy peak, but this is not observed apparently. Away from uJ = 0: A Real Space Perspective for STM Experiments. An amazing pattern observed in STM is that almost all impurities, regardless of their local environment, appear to be"resonant" all through the impurity band [10, 43, 44]. Furthermore, the number of the Zu atoms matches the number of resonances, which likely implies that there are no impurity atoms "missing." These collective evidences above were exactly the impetus to the proposal of single impurity quasiparticle scattering scenario. However, the disagreement between the width of resonance peaks observed in experiments (which is of the order of ~ 10 meV and is surprisingly as the same order of the impurity band width in the dilute unitary limit 7y ~ ~1 !) and that of theoretical prediction on single-impurity (7 <1 meV) questioned the preciseness of the single-impurity scenario. Fig. 3 15 displays the spatial distribution of LDOS, with a 0.2% concentration of impurity, a value close to the experimental observations. In the left panel, LDOS is plotted at a particular eigenstate whose energy lies in the impurity band but far enough away from the Fermi level; in the right panel, LDOS is plotted by averaging over a final energy window within which five eigenstates are included. It is interesting to note that while in the left panel, only a portion of impurities "light up" (resonant) while others are "dark" (not resonant) simultaneously; the right panel recovers the experimentally observed single impurity pattern I. ,i ,1tr i' .. This possibly implies that the DOS plateau for wu > 0 is formed by summing over many impurities with inhomogeneouss broadening" (different impurities become resonant and turn off at different energies, the basic fact is that the resonant lifetimes are different for different energies, since we have a nontrivial w-dependent DOS for the pure system with which the resonant states overlap). We are then inspired to formulate such a hypothesis: the exact image obtained by STM tips could be the summation of many N-impurity eigenstates, some of which are resonant at a given energy and others are not. over a finite energy window, and the "single" impurity pattern is visually recovered by piling up the snapshots at different energies. Notice that even though the energy resolution in STM is as small as 1meV, there are still numerous eigenstates within such a window size for a typical sample size (L ~ 500A) and impurity concentration (ni, 0.2 0.5W) with which the tips may bin over to produce the nearly isolated, four-fold symmetric, localized impurity patterns. 3.2.5 Conclusion I would like to finish the discussion on two-impurity problem by summing up the physics of most importance we learnt from this specific model: The commensurability of the nodal wave vectors and the tight-binding lattice classifies the two-impurity configuration into two sublattices: the impurity pairs on the same sublattice contribute to a divergence in p(w). At a = 0. The extended tails are blurred by the incommensurability between the lattice and the wavevectors of eigenstates, and this also helps to set up the single impurity pattern. Apart from the Fermi level, this phenomenon is even more pronounced. C4: +****. SI * X po2it2n Xposi-n Figure 3-15: LDOS for 0.5% concentration of impurities. Left: E, = 0.03851; right: averaged over five eigenvalue in energy interval IE,, 0.03t| < 0.02t. with the global constructive interference by the particular T2 symmetry in hall-filled tight binding band, the DOS of the many-impurity system diverges and contains a 6-function form. However, I should emphasize that this sharp peak at w = 0 is the consequence of the special feature of hamiltonians with 72 symmetry only, and is not a generic attribute of d-wave superconductors. SAway from the Fermi level, the quantumn interference between impurities still exists. Incommensurate scattering wavevectors allowed by the increase in the available kinematics phase space smears the nodal network and distorts the resonant states, which are inhomogeneously distributed. However, the finite energy window of STM technique, although it is quite small in laboratory terms, still introduces considerable homogeneous broadening to the resonant states by essentially averaging over a large number of eigenstates of the macroscopic system. The isolated impurity pattern with classic fourfold symmetry is visually recovered. 3.3 Power Spectrum for Many Impurities When pebbles are thrown into water, they cause ripples. Impurities induce "ripples" in the electron sea in the similar way. If the images of the impurity induced "ripples" are Fourier transformed (FT) into momentum space, they generally select out some defining wavevectors from which we can map out the S (1.1) ; (0.I) (1.1) a 56 10 16 20 30 3 R,(a) u ntV) i ,._i,. 3 16: Left panel:the constant energy contours with the q vectors; right panel: the evolutions of q vectors as a function of energy. dispersion of eigenstates in the pure system. In normal metals, they are mainly Fermi wavevectors [36, 37]; in a d-wave superconductor, those Fourier transformed DOS (FTDOS) patterns contain fruther useful information on the SC state. 3.3.1 Weak Scattering Limit: Octet Model and Kinematics of Quasiparticles The Cornell STM group [69, 70] first obtained the Fourier transformed density of states for optimal-doped disordered BSCCO. which is defined as p(q) = r e,"rp(r). They plotted its absolute value as a function of momentum q and bias energy w and observed some distinct peaks. An explanation based on quasiparticle scattering identified those peaks as the consequence of a disturbance due to atomic scale disorder. It was speculated [80] that the scattering processes which have a momentum shift q = kfj,,,,, kiltel connecting the regions with substantial density of states will dominate others, because there is large available phase space both in the initial and final states. In a hand theory, the quasiparticle DOS p(w) at w is proportional to J Vk' I k|k (3.27) E(k) and the area with the smallest energy gradient contributes to p(w) mostly. If there exist well defined quasiparticles in the superconducting state, then certain segments of the Fermi surface will be gapped out and the constant energy contour (CEC. the surface of which all the points have the same energy {k : E(k) = Jo}) deforms in the d-wave case into "bananas", as schematically plotted in Fig. 3 16. The quasiparticle DOS around the tips of bananas is higher than the other domains and we should be able to observe a pronounced |p(q)[ for the q vectors connecting these tips. In addition, these q vectors should disperse with STM sample bias w in a characteristic way as the quasiparticle CEC's change. This proposal is called the "octet" model for quasiparticle scattering since for each binding energy there are eight identical q vectors connecting the "banana" tips in the first Brillouin zone. We can work in momentum space within the T-matrix formalism and obtain the explicit form of p(q) (after taking off the homogeneous component) as well: p(q) Im[ G(e)(k + q,)T(k + q k ,.."'k (3.28) k Of course the p(q) derived from the complete T-matrix includes not only the octet peaks but also additional structures that may hinder the visibility of the desired ones. However, in the weak scattering limit, i.e., T(k', k) V(k', k), may scattering of high order be neglected and the disorder potential be factored out as well: p(q) h~Im{V(q) G(O)(k + q, w)G(O)(k, w)}, (3.29) k where V(q) is the Fourier component of the real space potential V(r). The term k G(')(k + q, w)G(O)(k, w) is called the response function (also named A function) which is only determined by the kinematics of the pure system. The physical interpretation of Eqn. 3.29 is transparent: the imaginary parts of the Green's functions, namely, density of states, p,,i,,,l and Pfnal determine the probability of scattering events jointly. Furthermore, the magnitudes of those q's display two distinct dispersions: while some of them increase with increasing bias, others decrease monotonically (see Fig. 3 16). The comparison between the experimental and theoretical w-dependence of those q peaks is presented in Fig. 3 16, showing good agreement with each other! The theoretical prediction is calculated from a standard BCS theory with quasiparticle dispersion Ek = ,+ A. (3.30) where Ak takes d,2_,2 form. It should be noted that Eqn.3.3.1 has contributions also from the real parts of the Green's functions as well. The result of FT-STS analysis has several implications: it suggests that opti- mally doped BSCCO is qualitatively well described by conventional BCS quasipar- tides. Moreover, because the unperturbed Green's function contains the dispersion of free quasiparticles in a homogeneous system, we then can trace the energy dependence of those vectors and map out the kinematics of the clean system! In fact, the q vectors extracted from FT-STS have been used to construct the Fermi surface of BSCCO and a comparison with that from angle-resolved photoemission experiments displayed a good agreement [70, 81] which confirms our knowledge of the electronic structure of BSSCO and, probably more importantly, reinforces our confidence in both the real space (STM) and momentum space (ARPES) tech- niques because the matrix elements for these physical processes are quite different. It is interesting to note that the "signal" used to determine the Fermi surface of he clean system via this technique vanishes in the absence of disorder. Finally, it can also be used to measure the momentum-dependence of the gap function Ak. A result performed by Hoffman et al. gave: Ak = Ao[Acos(20k) + Bi w-i", with A0 = 39.3 meV, A = 0.818, B = 0.182, which reflects approximately the proposed d,2-y. symmetry of the pairing function in BSCCO. 3.3.2 Power Spectrum for Many Impurities in Bi2Sr2CaCu2Os The "octet" model demonstrates that effectiveness of the quasiparticle scattering scenario within the conventional BCS theory for the superconducting state of optimal- or over- doped samples. The study on the Fourier transform of the Friedel oscillations due to impurities further provides important information on the kinematics of pure systems. However, there are also some distinct features in experimental data which is not explicable within this simple model, for examples, the resolved q,.q7 peaks are much broader than expected; furthermore, they have roughly equal weight while the octet model predicts the peak intensity of q7 should be much stronger that of ql. Is the quantum interference between the Friedel oscillations responsible for these discrepancies? In the study of two-impurity problem [54], I have shown that the interference pattern can persist up to a relatively long separation; practically. concentration of impurities in samples generally yields an average inter-impurity distance over which the impurities are still quite "visible" to each other. If we further recall 1) PL's result of the low energy DOS obtained by averaging over the randomly distributed impurity configurations: the interference gives an expression for N-impurity T-matrix which is essentially n, times the single-impurity result; 2) the nearly isolated impurity pattern could be the artifact of homogeneous broadening by the energy window size of STM. We should then doubt the precision of the "octet" model and ask the following questions: 1) is it appropriate for us to a prior assume that one-impurity pattern survives from many impurities and manipulate the experiments data on fully-disordered samples within the frame of this theory? 2) to what extent will its predictions such as peak positions and peak widths be modified qualitatively and quantatively by the interference effect? Many Random Weak Scatters. Capriotti et al. [82] and Zhu et al. [83] an- swered the first question by studying the power spectrum of N randomly dis- tributed point-like impurities. In these works, the N-impurity T-matrix is expressed in terms of the 1-impurity T-matrix 1, = [1 VG(r =I L by N, T, i = ij + t ,[1 ,' "i R -R,,.)T,,,, (3.31) where the impurity potential at R, is V = Vo, and t, are the Pauli matrices. In the dilute concentration limit, only the leading order in i, of Eqn 3.31 needs to be Figure 3 17: FTDOS at u =14 meV for weak potential scatters (V0 = 1 '* I: (a) for one weak impurity, with a few important scattering wavevectors indicated; (b) for 0.15% weak scatterers. Cuts through the data of (a)(thick line) and (b)(thin line) along the (110) direction and scaled by 1/V/'V are plotted vs. q, in (c), while (d) shows the weak scattering response function In A3(q,w). Peaks at q = 0 are removed for clarity. In all the figures, the r and y axes are aligned with the Cu-O bonds. considered: 6p(q,w) qi1,,,," q (3.32) S00,3 where t,(q) = I,, i e-iqR and A,,(q, ) = Ek [G(k, w)ioGo(k + q "," In the weak potential limit, Eqn. 3.32 reduces to 6p(q,i) 2- -V(q)Im A;,(q,w)/7r, (3.33) which is also valid for finite range V(r). It is realized that both of the limits afford a complete separation between degrees of freedom associated with the disorder potential and those of the pure system. There are several consequences of this separation: 1) the "octet" peaks are not broadened or shifted by disorder and it is A_,, the response function of the clean system that determines the peak positions and the widths in the FTDOS; 2) the randomness of disorder introduces noise which is actually as large as the signal since the disorder average gives ',,i.q I-: N, and (16p(q,w)j4) - SI '*- -'I 1- ~ N ,(NI 1); consequently, some one impurity peaks will not be robust in many-impurity case (for example, the weak qi peaks in Fig. 3-17 (a) are lost in Fig. 3-17 (b)); 3) technically, we may also compare the response function and the many-impurity FTDOS at different energies to map out the shape of the potentials V(q) if-l.. are spatially extended. Actually, since there is still lack of the knowledge on the spatial form of the disorder in BSCCO, this proposes in principle a method to resolve the impurity structure from LDOS measurement directly. Finally, it should be pointed out that the imaginary part of the single particle self-energy due to the point-like scatters does not broaden the "octet" peaks but suppresses their magnitude in momentum space (at any energies) since it is momentum-isotropic, i.e., 6E/dk = 0. Point-like impurities with either dilute concentration or weak strength prove to be inadequate to address on the relevant peak widths and peak intensities of ql and q7 vectors in experiments. Probably, the former fails because it neglects the interference effect and the latter further excludes the contribution from other channels of the response function. In Fig. 3 18, I show To, T and T7 components of An at several energies. It is observed that, interestingly, while Aa resolves q7, q,, q6, Ao resolves qi, q5 quite clearly (q3 and q4 are present in both of the two channels). This difference between Ao and Aa seems to necessitate the inclusion of unitary impurities as possible scattering centers and is supported plainly from the structure of the T-matrix for point-like scatters, 9a( )ra cr3 T(w) = g(a- CT (3.34) c- .(W) ....,- ( where go(w -> 0) ~ w. In the weak scattering limit the second term in Eqn 3.34 dominates and ql and qs peaks are missing. However, in the unitary limit, i.e., W=0.045455 w=0.090909 w0.13636 0 1 2 3 0 1 2 3 0 1 2 IT 0 1 2 3 0 1 2 3 0 1 2 3 MEN 0 1 2 3 0 1 2 3 0 1 2 3 Figure 3-18: The plot of Nambu component of spectral functions at several fre quencies. c = g(w 0) -' 0, To and r, channels will have roughly equal weight and therefore bring up all the q peaks at lower energies. Additionally, the presence of unitary scatters is also consistent with the finding of the plainly evident local resonances in the LDOS from STM for w < 15meV.3 Another constraint on the impurity potential imposed by the experiments is the peak widths of those vectors. If we allow that experiments may be seeing background features as well as (or instead of) the true "octet" peaks, then the FTDOS in Fig. 3-17 is fairly compatible with experimental data at -14 meV (as shown in Fig. 3-19). However, comparisons at other energies are significantly worse. As revealed by the experiments, besides the observance of unitary scatters, the energy gap extracted from STM measurements of BSCCO is inhomogeneous at the nanometer scales [84, 85], ranging from underdoping to slightly overdoped. With the lack of understanding of the source of those inhomogeneities, we modified them as smooth random potentials, probably originating from charge inhomo- geneities from BiO layer, with the structure as V(r) = Ej V(i) exp(-ri/A)/ii and fi = [(r Ri)2 + (~]1/2, where R, + zd, are the defect locations, V(i) are the defect potentials and A is a screening length. These extended impurities do not induce any resonance feature at all as they never make real poles for the denominator of T-matrix, but their momentum-dependent potentials V(q) assign a considerable probability to the "off-shell" scattering (the process involved with the k's which do not satisfy the poles equations for the Green's functions of initial and final states simultaneously) which act to blur the octet peaks. 3 This is also true for magnetic impurities since their spin resolved matrix is written as To in Nambu notation. But we focus on the potential scatters here. 66 In Fig. 3 19, I show the comparison between theoretically calculated FT-DOS and experimental results. The theoretical results were obtained by solving the Bo- golingov De-Gennes (BDG) equations 4 including both unitary and weak, extended scatters. The q vectors from the "octet" model are also plotted. A fair agreement between the calculated and experimental FT-DOS is achieved at low energies. The qi and qr, peaks are well-resolved theoretically, but most of the structure comes from a set of broader background features which disperse along with the octet vec- tors simultaneously. In general, the background structures disperse qualitatively as one expects in single impurity model. Meanwhile, theoretically calculated FTDOS display an asymmetry between positive and negative energies where experimental ones do not. This is maybe the result of the large asymmetry of the model band, in which there is a Van hove singularity at w 5 50 meV that is not observed in STM experiments. While ql, q5 are identified as "remnant" of octet peaks together with the dispersing back ground, qr seems to be inisidentified as octet peaks in experimental data, as we argued. since the measured magnitude is as twice as the predicted value from nearest-neighbor d-wave model. Because the q7 peaks comes from intranodal scattering, they are a direct measure of the k-dependence of the superconducting gap and scales with ~ 1/vI, where Ca is the gap velocity at the nodes. McEhoy et a. [70] have exploited this mechanism to map out the gap function Ak and were forced to introduced a significant subleading cos 60 harmonic to fit their data. This is in sharp contrast to the pure near-neighbor d-wave form at optimal doping resolved from ARPES spectroscopy. Recognizing that the observed SIn BDG formalism, the hamiltonian for a superconductor on a square lattice is, where the angle brackets indicate that site indices i and jare nearest neighbors, Ui is the impurity potential, and A,a = -V(cjlcii) is the mean-field order parameter. I ,. ,. 3 19: Comparison of theory and experiment. Left panel: LDOS: middle panel: calculated FTDOS; right panel: FTDOS from STM experiments. feature at roughly twice the true q7 is in fact the background feature found in our calculations may enable one to bring the two experiments in closer agreement. At energies Iw, > 15 meV, Fig. 3 19 shows a qualitatively similar nanoscale variation of LDOS as seen in experiments. However, the (110) peaks associated with forward scattering are stronger than (100) peaks. This can be understood since the unitary scatters are only resonant at low energies and do not have noticeable effects at high energies. Therefore, the difficulty to bring up qi peaks places further constraint on the impurity potential. Alternatively, it has also been proposed that q, peaks could be attributed either to the exotic orders such as stripe formation [30, 31] or to the scattering from gap inhomogeneities, i.e., the contribution from Ti channel. CHAPTER 4 DISCUSSION: QUASIPARTICLES, DISORDERS AND THEIR IMPLICATIONS FOR STM SPECTROSCOPY The notable improvement in the resolution of STM experiments has shifted the focus of the study of high-temperature superconductivity from bulk, macroscopic property to nanoscale, specifically, local quantum states and electronic structures. a level from which we believe the fundamental mechanism of high temperature superconductivity may eventually be uncovered. The lii.... ....... results of STM experiments have established several undisputed facts: a) the spatial modula- tions of the LDOS and order parameters are inhomogeneous at the nanoscale in Bi2Sr2CaCu2Os ; b) disorder with unknown origin and structure is ubiquitous in this sample; c) optimal- and over-doped cuprates may be well described by the conventional BCS theory with unusual order parameters. I attempt to elucidate to what extent we can relate such atomnic-scale electronic phenomena to disorder induced wave-like quasiparticle states and the quantum interference effects be- tween them. I here briefly summarize my i.i. 1 .1 ,-,.i... on the implications of the quasiparticle scattering model for the recent discoveries in STM experiments. 4.1 Atomic Level Resonances. Quasparticle States amd Unitary Scatters The zero-bias differential conductance map on the surface of BSCCO displays localized resonant features of size of 3 nanometers and four-fold symmetric distribu- tion of LDOS. In addition, the spectra on the centers of those bright spots exhibit a zero-bias sharp peak, as shown in ..- 4 1 [43]. There has been great deal of effort ascribing these observations to the quasi- particle scattering from atomic-scale defects. While the identities of these "native" defects are still not clear ( the possible candidates include defects such as Cu and O vacancies, crystal defects etc), tIhe spectroscopic signature are quite similar to I Figure 4 1: Zn impurity resonance. left channel: dI/dV Vs.w; right channel: the LDOS spectrum above the Zn site. those observed when Zn and Ni are explicitly substituted for Cu. Theoretical cal- culations, in which Zn and Ni were modelled as potential and magnetic scatterers respectively, seemed to support this conjecture by yielding qualitative agreement with experimental results and further substantiated the possibility of using local- ized defects as atomic probes for the superconducting state. The validity of this scenario was afterwards reinforced by its successful application to the analysis of Fourier transformed LDOS, through which we mapped out the kinematics of the pure superconducting state (such as the Fermi surface, the band dispersion and the momentum-dependence of the order parameter) that was confirmed by the results from momentum sensitive techniques as well. Additionally, transport measurements [86, 87] also demonstrated the existence of excess low-energy quasiparticle states, which may result from the collective interference between impurity wavefunctions. However, there remain a few difficulties in the quasiparticle scattering scenario: The expected spectral weight distributions around Zn impurities are com- pletely reversed in experiments ( see Chapter 2 for details). A proposal emphasizing the "filter" effect arising from the coupling between the orbitals with zero in plane components and the 3d,.2 _, wavefunctions of the nearest neighbor Cu reconciles this disagreement but raises another debate on the sign of the Zn impurity itself. So far. hand-waving arguments based on the atomic configuration suggests Zn is an attractive potential for electrons while calculations determining the unitary limit by adopting realistic band structure and ab initio DFT calculations find a repulsive potential. This, together with the fact that Zn (presumably a potential scatterer) atoms are more destructive to local coherence than Ni (presumably a magnetic scatter), remains an unsettled issue. The ability of one-impurity quasiparticle scattering model of any kind to explain the details of local STM spectra in samples with percent level disorder is severely limited. Despite the fact that conventional scattering theory might be intrinsically deficient since it is completely blind to the possible strong correlations between the impurity states and the surrounding electrons, it is inadequate to address several experimental observations. For example, although the analysis of Fourier transformed LDOS (within the single impurity scattering scenario) resolved the characteristic peaks associated with the kinematics of pure system, the actual heights and widths of these peaks are inconsistent with the one-impurity model. With the inclusion of extended impluities, proposals based on the quantum interference between impurities yielded better agreement with the experimental data but still could not reproduce all the observations consistently. 4.2 Nanoscale Gap Inhomogeneity In addition to aforementioned zero bias resonances due to planar defects. a further fascinating aspect of the local electronic structure is the observation of large, nanoscale fluctuations of the LDOS. In Fig. 4-2, I show the plot of gapmap in an overdoped BSCCO-2212. In this plot, each spot is a real lattice point and the color represents the relative magnitude of the local gap maxima which is defined as the position of coherent peak in the local spectral weight on the chosen site. Generally, this nanoscale inhomogeneity appears at biases close to the bulk gap edge in BSCCO accompanied by homogeneous low energy (IwI < Ao) spectra across Figure 4 2: The gapmap of overdoped BSSCO. Sample size 500A x500A. fb I , -10 -50 0 50 100 Sample has (mV) Figure 4-3: Spectra on different sites along the horizontal cut. the sample (See Fig. 4-3). This observance was interpreted as the local fluctuation of the order parameter and divided the system into "patches" with fluctuating phase coherence, which motivated the study of "granular -.[" 1 ...i.i+i. 1i. %T, . However, we should keep in mind that the STM tips measure qiuasiparticle excita- tions directly rather than the order parameter; additionally, in a fully-disordered superconductors, quantum interference effects can also give birth to the local gap inhomogeneity. While we should question whether this inhomogeneity is intrinsic in BSCCO, we may also ask how for can we proceed with quaiparticles and disorder alone, i.e., is this phenomenon completely explicable in "QP+disorder" scenario, or is it some intrinsically exotic state that can be distinguished from a highly disor- dered superconductor? There was an attempt [29] to study the second i .. 11I.1, by using Ni to induce local resonance in superconducting regions and discriminate them from regions of other identities. It was found that Ni atoms only "light up" in the region with relatively small gap value and higher coherent peak (i.e., good superconducting grains) but how the Ni resonances correlated with the inhomo- geneity was not resolved explicitly yet. As far as the first : -,l..j is concerned, the homogeneous shape of the low energy spectra excludes a large concentration of short-range, strong scatters since they should scatter the nodal quasiparticles and result in excess weight around the Fermi level presumably; furthermore, the fact that these inhomogeneities take place at energies close to the bulk gap edge suggests that it is the antinodal quasiparticles that suffer intense scattering from weak, smooth potentials. This could be particularly true in the optimal- to overdoped BSCCO where we have a better global .i ..-..1....i. 11... phase coherence. In Fig. 4 4 and Fig. 4-5. I show the simulation on the gapmap and the spectra along a horizontal cut by solving the Bogoliubov De-Gennes equations. The superconducting patches are not granular in a structural sense. We call them "granular" to refer the fluctuation of the order parameters which characterize the phase condensate in each "grain." Figure 4-4: Gap map from numerical solution of BDG equations with only smooth disorder. Figure 4-5: LDOS along the horizontal cut in the top figure. Finally, there is another piece of evidence supporting the significant scattering of antinodal quasiparticles regardless of the origin of the disorder [70]: the con- ductance map shows that for a given w, the observed Umklapp scattering induced LDOS modulation is extremely localized to the region whose local gap value is approximately equal to w. This indicates that for k s (7, 0), the quasiparticle dispersion experiences strong nanoscale variation and the antinodal quasipaticle lifetime is remarkably influenced by the scattering. Nevertheless, this is in con- tradiction with the fact that the antinodal quasiparticles are well-defined in the superconducting state, indicated by the the sharp (Tr, 0) quasiparticle spectra in ARPES. 4.3 Long-range Modulation and Exotic Orders The electronic structure of cuprates has been proven to be extremely sensitive to transitions into a variety of ordered states. Doping increases the mobility of the itinerant holes and they redistribute among the disordered potential landscape to establish ......h. r..1 electronic modulations. Various spatial modulations, such as inhomogeneous gap distribution, incommensurate periodicity oriented along the copper-oxygen bond, "checkboard" pattern were consecutively reported in various underdoped samples [88, 89]. Identification of those spatial modulations generally falls into two categories: it could be the ordering either tied to the lattice or tied to the Fermi surface (interference effect). The first possibility tries to attribute the experimental observations as the consequence of competition between the superconducting state and exotic orders such as d-density-wave [18], antiferromagnetic spin density-wave [90] and centers the debate onto the exploration on the underlying mechanism of the "pseudogap" state, the region where the observations described above are prominent. Several authors have studied the impurity effect in those hypothesized orders (or a mixed states of the exotic orders and the ..w. '.. ..1.,. I.. state) in the "pseudogap" state [91, 92, 93] and proposed that the spectroscopic feature, such as the spatial distribution and energy of the resonant states could be used to i. I. wii the different types of orders. The second scenario, although depending on the detailed information about the quasiparticle excitations for the proposed order, generally predicts energy-dependent characteristic scattering vectors, which is in conflict with the energy-independent wavevectors extracted from the Fourier transformed STM DOS at q = 4.5ao in underdoped samples. While it might be premature to reject the quasiparticle interference scenario entirely, this contradiction at least imposes a constraint on the Green's function by asking for a dispersive imaginary part and a non-dispersive convolution between two imaginary components (similar to the argument of the joint density of states ) simultaneously. As for now, the content 75 in this section still remains the most intriguing puzzle in the understanding of the phase diagram of cuprates. CHAPTER 5 ANGLE-RESOLVED PHOTOEMISSION SPECTRA OF DISORDERED CUPRATES In the previous chapters, I have discussed how the recent improvements in STM technique facilitated our comprehension of the cuprates by providing detailed real space image of their surfaces and extracting local, subatomic electronic structures of superconducting materials directly. However, spectroscopies in momentum space provide complementary information and are sometimes even more important because they interpret the dynamics and kinematics of the systems in a more "physically transparent" way. The angle-resolved photoemission (ARPES) technique is such one desirable method since the coherent part of its momentum-resolved spectrum elucidates the dispersion and self-energy of single particle unambiguously in quasi two dimensional systems. ARPES has been empirically useful in determining the electronic states of cuprates. for example, the energy distribution curves (EDC) of ARPES follow the hypothesized d-wave like momentum-dependence of the order parameter Ak. Recently, the dramatic improvement in the angular and momentum resolution of ARPES, as of 2 meV and 0.2" respectively, makes this technique a leading tool in the study of high-T,. superconductors. The one particle spectral function resolved from the advanced ARPES experiments reveals novel and perplexing line shapes [33, 94, 95, 96, 34] in the low energy regime and each has ushered in a new round of investigation in high-T, superconductors. I will concentrate, in this chapter, on the impact of elastic forward scattering and its implications on the contemporary dispute, regarding the energy- and momentunm-dependence of the single particle self-energy [97, 98]. 77 5.1 Angle-resolved Photoeinission Spectra of High Temperature Superconductors 5.1.1 Spectral Function and Self-energy When photons hit metal surface, they will be absorbed by electrons and hence pump the electrons up into higher energy levels. The excited electrons will then propagate in the metals and eventually be emitted through the surfaces if they have sufficient energy to overcome the work function. The excitation of electrons is governed by the Fermi's golden rule. The intensity of the photoelectrons are measured by the detectors. If we assume that the relaxation time of the excited electrons is much longer than the time for them to reach the surface (this is the so called sudden approximation). we could roughly formulate the intensity of outgoing electrons beams as: I(k, ) = lo(k)f(a()A(k, )., (5.1) where lo(k) is determined by the momentum dependent matrix elements and the Fermi function f(w) illustrates that this process can only measure the unoccupied states. The one particle spectral function A(k,w) describes the probability of creating or i..... iii .1 one particle in an interacting many-body system and is associated with the imaginary part of the retarded Green's function, as defined in Eqn. 5.2: A/(k, ) = .; I .I +'- '"+ ) + + | 1 | t .*- E + ') u=Im(k., + iO ). (5.2) In a non-interacting system, A(k, )) is nothing but a delta function, 6(. (k), everywhere, due to the infinite lifetime of electrons in the absence of inter-body interactions. When the many-body interaction is turned on adiabatically, the bare electrons will be "dressed" by the medium and can be described by the Landau Fermi liquid theory (FL) provided they still live long enough to be considered as approximately independent objects. The FL theory painted the big picture of interacting-electron dynamics and introduced an important theoretical concept. i.e., the "quaisparticles." With this concept, we can treat the dressed "electrons" as bare entities with finite but considerable long lifetimes. The one particle Green's function is generally modified as G-'(k, w) = Go1 E(k, ), where E(k, w) = E'(k,w) + iE"(k, w), E'(k, w) = ReE(k, )), E"(k, w) = ImnE(k, ) and E is called the "self-energy." Its real part describes the renormalization of the bare electronic dispersion due to many-body effects and its imaginary part is related to the one particle lifetime by r(k, w) = -1/2Im(k, w). The general expression for the spectral function is then: 1 E"(k, J) A(k, w)) --- ( (5.3) i I k The self energy E(k, w) arises from many-body and impurity effects and encapsulates all the information about interactions. Generally, we can plot the measured intensity (oc A(k, )) either as a function of w I. I.lhiI- energy) for a fixed momentum k (EDC), or as function of k with a fixed value of w (MDC). The detailed forms of self-energies extracted for the cuprates from these two methods exhibit quite unusual features which are in conflict with the traditional FL theory and will be the focus point in the discussions below. The spectral properties of single particles in the normal states obtained from ARPES data are extremely unusual. Despite the existence of well-defined Fermi surfaces in various samples I the energy and momentum dependence of the spectral function deviates considerably from the predictions of the FL theory. The most striking discovery is that there exist no well-defined quasiparticles in the normal state of optimally doped or slightly overdoped HTS's near the (7r, 0) point of the Brillouin zones [99, 100, 101], while in superconducting states of these samples quasi-particles are well-defined over a certain range in the 1st BZ, 0.4 0.2 0 Binding energy (eV) Figure 5 1: ARPES spectra from overdoped Bi2212 (T,=87K). a) normal- and b), (c) superconducting state data measured at the k point indicated in the inset of (c). as shown in Fig. 5 1 [100]. The photoemission intensity exhibits (panel (b)) a pronounced peak, followed immediately by a "dip" and then a "hump" feature at higher binding energies in the -. ......-l. i,.... i ,,. state' When the temperature increases, the coherent low energy quasiparticle peak evolutes -..I IiriI.., .i and finally melts into a quite broaden background (panel (a)). indicating that a considerable self-energy develops across the phase transition and the quasiparticles are no longer well-defined. We might cast a doubt on the applicability of the FL theory to the normal state of high T,. superconductors since the normal state temperature is of the order of 100K ( it is quite high compared to the general definition of "low" temperature), I It is called the "peak-dip-hump" feature. The high energy humps are specu- lated generally to he the consequence of phonon modes, magnetic, collective mode or I ilayer ..J.r r.,. effects. but we should also not abandon it unthinkingly because 1) it has assisted our understanding in various crystalline solids and been useful over a wide range of temperatures 2) there are still well-defined quasiparticles in certain segments of Fermi surfaces when the gap opens, which is consistent with the conventional BCS theory. However, an estimate of the full width-half maximum (FWHM) of the broadened peak yields a value of ~ 100meV, which is one order of magnitude bigger than the thermal broadening. This, together with the continuous evolution of the qp peak suggests that temperature cannot be responsible for such a large peak width directly. While there are numerous scenarios concerning the origin of the one particle self-energy such as electron-phonon coupling, spin-fluctuations, magnetic resonance etc, a Marginal Fermi liquid hypothesis proposed by Varma et Ua. [8] seems to describe the transport and thermodynamic properties and the anomalies observed in ARPES phenomenologically.2 It was further speculated [9] that in order to interpret the behavior of self-energy, we need include a term independent of frequency and temperature, which is probably associated with elastic forward scattering of quasiparticles by impurities located away from the CuO2 plane, and a MFL component as defined in Eqn 5.4. The scattering rate in the normal state then reads: F(k,w) -I ik .,)+r ,... .k ), (5.5) 2 The MFL theory was postulated to reconcile the conflict between the anoma- lous (but universal) properties in the normal state of cuprates and the well-defined Fermi surface (in optimally doped samples). It assumes that electrons couple with a bosonic excitation whose spectrum has a scale-invariant form B(w) minr(, 1). The electron self energy due to the scattering from this bosonic spectrum has a form: E(k.,w) = A( wlog .r, (5.4) a', 2' where : =. ... I | T). This singular behavior of self-energy leads to the absence of the FL-like quasiparticles even on the Fermi surface when T=0. The possible 'l,.i;. .iii of elastic forward scattering is further reinforced by the f. -1..- --.- fact: in Born approximation, the self-energy due to impurity scattering is written as E(k,w) = n, IVk,k'l2o(k',w). (5.6) k' Qualitatively, if Vk,k' is peaked at certain direction, then E(k, w) is roughly proportional to the imaginary part of the integrated Green's function over a narrow range centered around k, i.e., G(k, ) ~ Ek'-kk Go(k', w). When 16k is extremely small, namely, in forward limit, E(k,w) is proportional to vF(k)-l (the inverse of group velocity of hand electrons), which generally takes the maximum value at antinodes and minimuun value at the nodes on the Fermi surface. This momentum anisotropy of i k I coincides with the experimentally extracted elastic component Fr,(k, w) by assuming the forward scattering scenario, which also displays the similar behavior with a maximum value of 120 meV at antinodal points and 40 meV at nodal points. However, there are two obvious difficulties with this scenario. The first is that the spectral peak measured by ARPES near the (7r, 0) point is known to sharpen dramatically when one goes below T,, a phenomenon interpreted as the formation of a coherent quasiparticle in the superconducting state. This sharpening has normally been attributed to the well-known collapse of the inelastic scattering rate below T, due to the opening of the superconducting gap, but it is hard a priori to guess why something similar should happen in the presence of an elastic scattering rate of order 100meV. The second problem is that recently increased momentum resolution [34] and the use of different photon energies [102, 103, 104, 105] has resolved a .I, splitting which has its maximum effect near the (7r, 0) point. Some of the previously observed "elastic broadening" is therefore certainly due to this as well as to pseudogap effects [106], but exactly how much is not clear. On the other hand, one can certainly not ignore the out-of-plane disorder. The BSCCO material is thought to be doped by excess oxygen in the SrO and BiO planes, and even the best single crystals are believed to contain significant amounts of cation ., ;. iT .. and other out-of-plane defects [107]. It is therefore reasonable to assume that quasiparticles moving in the CuO2 planes of this material must experience a smooth potential landscape due to these defects, and useful to pursue the question of the effect of this type of scattering in the superconducting state. In fact, fits [83] to Fourier transformed-scanning tunnelling spectroscopy measurements [69, 70] on similar samples to those used in the ARPES studies have recently been shown to require both a strong (near-unitary limit) scattering component, attributed to native defects in the Cu02 planes, as well as a weaker. smooth scattering potential component attributed to defects away from the plane. Recently, Markiewicz has also attempted to relate STM and ARPES data in the .ii" ......1.. r .... state assuming a smooth potential [108]. 5.2 Elastic Forward S ..... .1 ,* , r. i : ; .i ... Normal State. Consider a model system including elastic scatters of finite range K-1, with a concentration ni. Assuming the single impurity takes the form 3 V(r) = Voe-", we can calculate its two dimensional Fourier component: 27r Vo Vkk' = 27 (5.7) Iuk- k I .I where Vo sets the strength of the potential. The -. il... ,. in Born approximation is then defined through Eqn 5.6 As the range of the potential ;-' increases, the scattering of a qnasiparticle from k to k' becomes peaked in the forward direction. As shown in Fig. 5 2, 3 There is a consensus that quasiparticle moving in the CuO2 plane experience a smooth potential landscape due to defects whose detailed structure is unclear yet. However, a concrete impurity model is needed for any computations. We adopt the exponential form rather than any other particular structure such as Yukawa law for the sake of analytical convenience. Our qualitative picture is robust against any specific form of forward scatters. 0 In Fr. ,w.' 5 2: Geometry for the forward scattering process in which a quasiparticle scatters from k to k'. when k is close to k' and both are not too far from the Fermi surface, we may parameterize them as k = k. + k-k (5.8) k' = kF + q1 + l'k (5.9) where q = k k' is the momentum transfer and q|| its component parallel to the Fermi surface. The unit vectors kj and k' are the projections of k and k' onto the Fermi surface, respectively, such that, e.g., tk' = iF(k')k'. The imaginary part of the retarded self-energy Eqn. 5.6 becomes ,, I. i i+ Y"(k, ) = ,(w k') "" [+ 2+(k -)2 + ] S(5.10) Eqn. 5.10 shows explicitly that in the limit of small K, the -, iI I.. becomes more and more sharply peaked "on the Imass shell" ; = tk, as a generic feature of long-range potentials. The angular dependence of the self-energy in the limit of -> 0 can be verified exactly as the same as that of 1/vF(k). Furthermore, I should point out the self-consistent treatment (by requiring E[Go] -. 11 is important in forward scattering case since it eliminates the van Hove singularity in the spectrum, which may account for the absence of such peculiarity in STM and other -,-..i.- I!1_... experiments on BSCCO. while in the case of point-like scatters, the self-consistency (in Born approximation) only introduces a correction of the order of w'2/E to the non-selfconsistent result, which may be neglected. Superconducting State. In the superconducting state, the self-energy is approxi- mated similar to Eqn. 5.6 as: E = n, ViVkk'I2T3GO(k',w)T, k' = F.S0 (5.11) with Nambu components E, defined as following; (5.12) Eo(k,w) = n, |V(k,k') |2 (5.12) SE(k,w) = n, lV(k,k '2 2 (5.13) k' k' ~ k' and ,o ( 2 Ak, (5.14) El(k,w) = -n E |V(k,k, Ak (5.14) k' k ~ k' In the limit of << 1, the asymptotic forms of self-energies can be derived analytically, which read: 2, Ak,, 1 [ ',(k) (5.15) Here s, I = wl. -Ak sgnw, and v .* ... K -, for the Nambu components a = 0, 1 and 3 respectively, and ro(k) is the normal state scattering rate. E' vanishes on the Fermi surface (k = 0 in this limit. Specifically, when the momentum k is close to the Fermi surface and the energies u; are small, such that I(tk + V-w ,1 i k i < 1, the self-energies reduce to the following: :.,,k ) ~ -ro(kF) K(. V2 (5.16) ) ~Fo(k k sgn w (5.17) ,i'k I 1 0, (5.18) but are strongly suppressed due to energy conservation when I 2 A - (kh becomes greater than KvI, as one may observe in Fig. 5-34 The detailed derivation of superconducting self-energies are included in the appendix. It is noteworthy that for k's away from the node, these results are identical to those expected from an s-wave superconductor [109] (even when self-consistency is included) since the extreme forward scattering eliminates the momentum averaging over the Fermi surface. We therefore expect to recover Anderson's theorem, the insensitivity of bulk thermodynamic properties to nonmagnetic scattering. The physics here is that the for k away from the nodal direction, forward peaked scattering doesn't mix the order parameter of different signs, and hence doesn't break the Cooper pairs. In Fig. 5 3, I also present the numerical results for the self-energy components at nodal and antinodal points on the Fernmi surface. The bare electron band 4 Roughly speaking, the change of momentum 6k in the scattering is cut off by K. Therefore for scattering processes close to the Fermi level, the corresponding variation of energy is avp(k). if we linearize the electron dispersion. On one hand, the quasiparticle energy should be conserved in scattering and kinematics only gives a finite available phase space, i.e., the constant energy contour as discussed in Chapter 3. On the other hand, the linearization in analytical computation may violate the exact energy conservation. But the energy difference between the initial (tk) and final electron (V Ak) component should still be cut off as well due to the forward nature of scattering. 0.5 0.2 0.4 0.15 X 1 < 0.3 N * S- 0.110. 0.2 c=5 0 0 -1 -0.5 0 -1 -0. 0 I T 0 8 0.8 0.6 I 0.6 0.4 0.4 0.2 0.2 0 0- -1 -0.5 0 -1 -0.5 0 (0 O0 0.5 0.1 0.4 0.3 N I 'I 0 0.2 0.1 ic=5 00 0 o -0.1 -1 -0.5 0 -1 -0.5 0 {) (0. Z /1 I - 9 I=0.5 -0.5 0 00 0.2 4 0.15 / 0 0.1 * % 0.05 - -1 -0.5 0 0) 0.2 0.15 -0.05 -1 -05 0 -I 0.02 -1 -0.5 0 (0 0) 0 -1 -0.5 co Figure 5-3: The self-energy terms -Im :",, k Im i (k, ), and -Im E3(k,w) in the -.i ..-...,.1,1. I.... state at T = 0 for k = kA (top) and kN (bottom), for N: 5 and 0.5 and the same band and scattering parameters as previously used. Here Ak = Ao (cos, cos k)/2 with Ao = 0.21. |

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