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Quasiparticale interference and the local electronic structure of disordered d-Wave superconductors

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Quasiparticale interference and the local electronic structure of disordered d-Wave superconductors
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Zhu, Lingyin
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xiv, 117 leaves : ill. ; 29 cm.

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Atoms ( jstor )
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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 )
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Thesis (Ph. D.)--University of Florida, 2005.
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Includes bibliographical references.
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Printout.
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Vita.
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by Lingyin Zhu.

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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,


a ie i.j

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.




Full Text
QUASIPARTICLE INTERFERENCE AND THE LOCAL ELECTRONIC
STRUCTURE OF DISORDERED rf-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
family, 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, Zhihong 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 grandfather and my parents: they seeded this special moment with
endless love and have been waiting for it for so long. I wish my dear grandma could
share this happiness with all of us in heaven, peacefully.
IV

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 I
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 in Spectroscopic Experiments 8
1.4.1 Scanning Tunnelling Microscopy (STM): In Touch With
Atoms 8
1.4.2 Impurities in B^S^CaC^Og : 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
v

3QUANTUM 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 u = 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 BUS^CaCu/L . 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 Superconducting States 78
5.2 Elastic Forward Scattering in B^S^CaC^Os 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.l “Yukawa I” Potential 102
vi

B.1.1 Normal State 102
B.1.2 Superconducting State 103
B.2 “Exponential” 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 page
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
viii

Figure
1-1
LIST OF FIGURES
page
The crystal structure of La2Cu04. Red sphere: Cu2+; blue sphere:
O2-; 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 Bi2Sr2CaCu208 10
1-5 STM image in Bi2Sr2CaCu20g showing a) atomic scale native defect
resonances; b) Zn resonance. Both were taken at zero bias 11
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 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 CuO^ plane; middle: experimental LDOS
around Zn impurity; right: theoretically predicted LDOS for Vq =
5.3£ (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
IX

3-1 Two-impurity resonance energies 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 || (110). Pure nested band,
Vo = 10i; Impurity separations are shown on the top of the graph. . 43
3-5 LDOS maps at resonant energies for R || (100). Pure nested band,
Vo = 101] Impurity separations are shown on the top of the graph. 44
3-6 LDOS spectra for realistic band and Vb = 5.3¿i on nearest neigh¬
bor site. Left panel: impurities at {—R/2,0) and (R/2,0) ( 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 u =
0.04 shown as small filled ellipses at the nodal points, qi, q2, q3
axe 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 Change in p(uj) due to impurities. Impurity separation: R = (2,2). . 51
3-12 Change in p{uj) due to impurities. Impurity separation: R = (2,1). . 52
3-13 (a) DOS for Vb=100t. (b) Scaling of the DOS with Vb. (c) Scaling of
the DOS with L. (d) Scaling of the DOS with Vb and L=60 53
3-14 LDOS for 2% concentration of impurities and \En\ < 10~5t (20 eigen¬
values) 55
3-15 LDOS for 0.5% concentration of impurities. Left: En = 0.0385Í; right:
averaged over five eigenvalue in energy interval \En — 0.03£| < 0.02L 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
x

3-17 FTDOS at uj =14 meV for weak potential scatters (Vo = 0.67*i): (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 /y/Nj are plotted vs. qx in (c), while (d) shows the weak
scattering response function Im As(q, uj). 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: dl/dV Vs.u;; 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 BÍ2212 (Tc=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, cj), Im £j(k, uj), and -Im Es(k, uj)
in the superconducting state at T = 0 for k = k^ (top) and kjv
(bottom), for k = 5 and 0.5 and the same band and scattering pa¬
rameters as previously used. Here A*, = A0 (cosx — cosky)/2 with
A0 = 0.2* 86
5-4 Scattering rate re/(k,u;) vs. uj for k = k^(left) and k#(right) in the
superconducting state at T = 0, for k, = 5 (top) and k = 0.5 (bot¬
tom). Here r0(k^) = 0.2* 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 k = 2 and 0.5 with Toik^) = 0.2* are shown. ... 92
xi

5-6 A(k,u;) vs. u for k = 2 and 0.5. Results are given for the k points
at (ir, 0), (%, 0.05ir), (n, 0.l7r), (7T, 0.157t). The disorder levels corre¬
spond to ro(kyi)/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, to) (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. Bi2Sr2CaCu208 (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
xiii

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. Finally, the influence of forward scatters on single particle
spectral functions is investigated. It is found that the self-energy 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.
xiv

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 Müller [2].
The critical temperature Tc, 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 Tc ( 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 significant 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
1

2
Figure 1-1: The crystal structure of La2Cu04. Red sphere: Cu2+; blue sphere:
O2-; yellow sphere: La3+.
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 Cu02 planes. These layers are
separated by intervening insulating layers (See Fig. 1-1). La2Cu04 is a typical
candidate, whose structure basically mimics a “sandwich,” with one Cu02 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

3
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 3dxz-y2 orbit with a net 1/2 spin; the 3d
level of copper ions is so close to the in-plane 2pXtV level of oxygen ions that they
hybridize1 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 Cu02 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 liall-
filling, whereas antiferromagnetism 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 exactly the reason
1 “Hybridize” includes many physical procedures here: the crystal field split¬
ting lifts the degeneracy of 3d Cu atomic level; Jahn-Teller effect further hits the
degeneracy by crystal distortion; the resultant 3d orbitals then hybridize with the
in-plane 2p oxygen orbitals to form Cu-0 bands which we may have Mott insulator
or charge transfer insulator depending on where the lower anti-bonding Hubbard
band lies.

4
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 dx2_y2 symmetry, lower than that of the Fermi surface itself.
The superconducting 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 > Tc, 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(u,T) ~ min(uj/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 —

5
dRe£(a;)/du)~x 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 [16], 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
Tc 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 whining
support from certain experiments [19]. Nevertheless, there is yet no consensus
about the origin of the pseudogap to date.
1.3 Current Situation in High Temperature Superconductivity Research
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

6
High Tc Superconductivity
Palrlmf Defect |
EORY
m
â– 
*
‘5
n
Li
Neutron
1 Scattering 1
Transport
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 [20],
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-Tc 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:

7
1.3.1 What We Know About High Temperature Superconductivity
1. Cuprates have copper oxide planes, and these planes are crucial to high
temperature superconductivity;
2. The electrons pair up in the CuC>2 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 £o ~ 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 revolutionised the field of surface science by
inventing the scanning tunnelling 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
tunnelling 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:
/(r, V) = ^ ^i{f(ii)[l—f(Eiy—eV)—f(i')[l—f(El,—eV)}}\Mlív(r)\2ó(Ell—E„+eV),
(1.1)
where Mm„(r) is the tunnelling matrix element and / the Fermi function. The <5
function conserves the energy as it does in the usual cases. With several appropri¬
ate assumptions2 , we can write / as
(1.2)
where r is the coordinate of points sampled and 2 is the vertical distance between
the tip and the plane; the local density of state (LDOS) is defined as
(1.3)
2 1) the tunnelling event is extremely localized (this ensures the tunnelling 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 M(k, z) = Moe~kz, where
k2 = 4mn(Aw +V)/h2 for a rectangular barrier of height A^.

9
Assuming 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 alternating 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 Bi2Sr2CaCu208 : A Probe of High Temperature superconductivity
Bi2Sr2CaCu20s 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 intentionally 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 inhomogeneities in the order parameter map

10
&Q
ScQ
CuQj
Ca
CuQ,
StQ
fiiQ
SrQ
QaQ,
Ca
QuQi
sea
Figure 1-4: The crystal structure of B^S^CaC^Os .
[27, 28, 29, 30] which are currently the subject of intense debate, being attributed
either to interaction-driven effects such as stripe formation [31, 32] or to the
Friedel oscillations [28, 29] of weakly interacting quasiparticles due to disorder.
Those results were then Fourier transformed (FT-STS) to successfully extract
some characteristic wave-vectors which reflect kinematics of quasiparticles in clean
samples; but other nondispersive wave-vectors, which are not explicable in terms
of the quasiparticle scattering scenario, were found as well and provoked diverse
proposals concerning the ground state of cuprates.
Besides STM experiments, angle-resolved photoemission (ARPES) is another
leading technique in the study of HTS. ARPES provides the momentum dependent
single particle spectrum and information regarding the self-energy which is inherent
from many-body effects. In the ARPES spectra of BSCCO, the width of the
quasiparticle peak also suggested a significant elastic component in the self-energy
due to the scattering from spatially extended potentials with unknown identity
[33, 34].

11
Appreciable differences in the spectroscopic and transport properties between
BSCCO and YBCO, another popular HTS material (Tc ~ 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 “intrinsically” 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 Cu02 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.
Historically, 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 hamiltonian for the normal state problem is
n = +52 i2-1)
k k,k'
where c^, cj, 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
13

14
> = 1 + —i :— + -L—'—+
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 (7°(k, k') = — (Tt-c^t)^,) , with r being the
imaginary time. In the case of free fermions, it takes the simple form:
<7°(k,u;) = [uj — e(k)]—1
(2.2)
The single impurity problem is generally 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 = G° + G°TG°, (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) = Vo¿(r — r') (Vo is
the impurity strength), the T-matrix is momentum- independent and reduces to a
simple form:
7» =
Vo
1 - M,EkG°(k,w)'
(2.5)
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°(uj) = vanishes, i.e., where the density of states (DOS) is zero (the

15
1
0)
no low-E BS
Figure 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 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,ui) = Vq2 |G0(r,<¿)|2<5(u/ — w0)-
(2.6)
In a two-dimensional space, the real space Green’s function is evaluated as
(2.7)
where ¡i is the chemical potential and kp is the Fermi momentum.
When k(u))r is large, the Green’s function has the following asymptotic form:
G°(r,ui) ~ -¿W(0)[—JLlVVM")’-*/4!.
k(u))r
(2.8)
The LDOS falls off from the impurity with an envelope of l/(kpr) and oscillates
with a period of the inverse of the Fermi vector.

16
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) = ^Vqrp(r,w)
(2.9)
r
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 :
(2.10)
with Ak = F(ck|C_k|). 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.

17
It is convenient to define the column vector <3?^ and its conjugate1 as the
following:
V c-ki
.n = (4T.c-ki),
(2.11)
and work within this framework.
The single particle Green’s function is then a 2x2 matrix:
G°(k,r) = -{TAfr)*!)
(TTCkt(r)4T> (TrCktii-jc-ki) \
^ (TVcLyirJckf) (TVclyMc-w) J
For a time-invariant system, we can further introduce the Fourier transform:
(2.12)
G°(k,iu„) = ^ei""TG°(k,r),
(2.13)
where ujn 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
G°(k,u) =
UlTo + AfcTi + ekT3
(2.14)
where the quasiparticle energy defined as Ek = y/tk + A|, and To, r,, T3 are the
Pauh matrices. The imaginary part of the real space Green’s function (G(r,ur) =
(2.15)
1 This spin and particle-hole resolved matrix structure is conventionally referred
to as Nambu representation.

18
with the spin-resolved LDOS,
Pf(r,u>) = —n 'im Gn(r,r,ii) + ¿0+)
pj(r,a>) = +7r_1Im G22(r,r, —n; — j0+)
(2.16a)
(2.16b)
With the general property of the retarded Green’s function:
IinGfoi + ¿0+) = — ImG(o; - ¿0+)
(2.17)
the LDOS may also be written as :
2.3 One Impurity in an .s-wave Superconductor
2.3.1 Nonmagnetic Impurity Problem
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
(2.19)
k,k' ,(t
where Vo is the strength of the impurity. It was pointed out that in dilute concen¬
tration limit, nonmagnetic impurities introduce negligible effects on bulk properties
(such as Tc) 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 = Aq. 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:
(2.20)

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 /tn 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 magnetic 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:
(2.21)
with J(k, k') the exchange energy, S the impurity spin and 3 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 Tc suppression is obtained in AG form [40]
as
ln(^) = ^(1/2) — + Pc),
1 cO ¿
(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 can verify that the magnetic term couples in the Nambu ro channel
and changes sign for spin up and down excitations. A potential scatter couples in
the 73 channel, introducing nothing but level shifts.

20
where i¡) is the digamma function4 , Tc and T& are the actual and disorder-free
transition temperatures respectively, and pc = (27rr^Tc)_1 is a constant determined
by the self-energy due to magnetic interaction, where r6_1 = kTcq/2').
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 3d104s2) 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 nonmagnetic impurity (if we neglect its quantum nature). This is
because the Zinc cation has a closed 3d10 shell which will gives a zero net spin
S = 0.5
4 The digamma function is defined as
^(x)
shr'*l
OO
-7+ ^
“ n(n + x)
,x± -1,-2,-3..
(2.23)
where 7 is the Euler constant.
5 There are also other points of view which are suggestive of the magnetic nature
of Zn2+ 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].

21
A(«)
O
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 Zn2+ 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 < Tc is still unrenormalized
since X]k A* = 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(o)) = n¡V¿ X)k£r°(k>w)) m the table
below:
Table 2-1: The imaginary part of self-energies due to non-magnetic impurities in
superconductors.
Eo Ei
d-wave r0( _2) 0
s-wave
where To is the impurity scattering rate in the normal state, with Si =
ui — E0(u>) and Ak = Ak - Sp The symbol (•} denotes angle averaging around the
Fermi surface.

22
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., A¿ = Aocos(2). 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',o;) = Gü(k,ü;)<5kk< + G0(k,iu)TV)G°(k',in). (2.24)
While its off-diagonal component vanishes because Ak = 0, the T-matrix
reduces to the simple form:
f = T0t0 + T3t3
To = V02g0/(S+S-)
t3 = v0\c-g3 )/(s+s_)
«» = ¿5>GV,k)T0, (2.25)
k
(2.26)
where l/(nN0Vo) = c is the cotangent of the s-wave scattering phase shift r/o,
and No is the density of states at the Fermi level. Q is the momentum integrated
Green’s function. This expression has resonances when
s± = 1 - v0(g3 t So) = 0.
(2.27)
In the special case of a particle hole symmetric system g3 = 0, and the resonances
are entirely dominated by ).
In the unitary limit, i.e., r)0 —* 7t/2 (this is equal to c energy í2q and scattering rate V are
±7rcAo
21og(8/7rc)
7T2cAq
41og2(8/7Tc)
(2.28a)
r
(2.28b)

23
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¬
ticles; furthermore, there is a finite damping rate for those resonances since they
emerge from a continuum background, namely p(uj) ^ 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 nfidgap states are
generally referred as virtual bound states because of their asymptotic behavior
described above, which can be attributed to the linearly-vanishing DOS (p(u^) ~ cu)
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 £0 =
vp/hAo. 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-
3nm6 , with p(r) decaying s 1/r2 for r > £0, ie., the Friedel oscillations at the
periphery of the central bright image, as illustrated in Fig. 2 5.
6 The coherence length extracted from the size of the atomic resonance is around
15 A. This again coincided with the £0 known for B^S^CaC^Og , being a convinc¬
ing evidence for the quasiparticle scattering scenario. However, the anisotropic con¬
figuration of LDOS distribution within a distance of £0 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 tunnelling mechanism and the
nature of impurity itself.

24
21 ° o o
o
I
cí,
c*
c
0.1
; «r
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
u;/A0 1 /kpr l/kp^Q. The results are tabulated as follows for future adoption:
G°(R,u)
(2.29)
Lattice Model, Strong scatters and Unitary Limit. 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 ek = —2t(coskx +
cos ky) — fi 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 kx —
cos ky). Note the maximum value of the order parameter in the half-filled lattice
system with the current convention is 2A0.

25
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 = 5.3¿i, which generates two resonances at
= ±0.013¿i ~ ±1.5mev 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 Q = —0.013¿i captures the essence of
resonant physics. Because the off-diagonal components of the integrated Green’s
function vanish, i.e., Gi2(r = 0, cu) = 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
/ dujp(r,uj) = 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.

26
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 BUS^CaC^Os (d-wave superconductor). As shown in Fig. 2-7, two
distinguishable peaks were reported, at Qi = 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 oscillating 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 45° 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 locally 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.
8 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 rathern than on the background with a roughly constant DOS around the
Fermi level.

27
Figure 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 spectra.

28
Figure 2-8: Left: the schematic plot of CUO2 plane; middle: experimental LDOS
around Zn impurity; right: theoretically predicted LDOS for Vo = 5.3£ (rotated by
45° 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 Tc suppression display
qualitatively similar dependences 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 Tc 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 Zn 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 fio = —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 Cu02 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 = 5.35f), 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 prioñ far below the Fermi level, as derived from the 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 Cu2+ ions with strength of
approximately -2ev.

Table 2-2: The ionization energy of Copper and Zinc atoms.
30
Cu
Zn
Atomic configuration
3dlu4s'
3<¿lu4s'¿
first ionization energy(eV)
7.720
9.934
second ionization energy(eV)
20.292
17.964
third ionization energy (eV)
36.83
39.722
However, calculations based on the criterion for unitarity 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 770 extracted from the STM data of Z11 impurity [44] is 0.487T, close to
7t/2, i.e., unitary limit. One can also perform computations on the phase shift with
the conventional T-rnatrix fashion
, , Im det f
V(u) = tan . -â–  (2.30)
Re detT
It is found that only a repulsive potential can possibly approach to this limit.9
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 ~ 20|/i| is needed. Finally,
a recent ab initio calculation exploiting density functional theory 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 Cu02 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.

31
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.

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 disorder-averaged 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 displays 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 fully-disordered systems.
32

3.1 Two Impurities in a d-wave Superconductor
33
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,
' ffe ffeG°(R)fm
K ffmG°(-R)ft ffm
(3.1)
where R = R< — R„( and where 7), Tm are the single impurity T-matrices associated
with the two impurities. For identical impurities, 7) = Tm = T(in), the single
impurity T-matrix defined previously. The quantity / is defined as:
/M = [1 - G°(-R, in)7f(ai)G°(R,in)f'm(aj)]_1, (3.2)
where G° (R, uj) = X]kexP[?^ ' is just the Fourier transformation of
(7£(ü;), the unperturbed Nambu Green’s function. For systems with inversion
symmetry G°(R,¿j) = G°(—K,uj). Note that in Eqn. 3.1, the physical processes are
clearly identifiable as multiple scatterings from each impurity Í and m individually,
plus interference terms where electrons scatter many times between i and m. In
k-space, we can write the T-matrix in the more usual 2x2 notation as
Tkk'(in) = [e'kR|ro eik'R"T0]Ttm
p-ik'-R;
T0
(3.3)
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[l - G°(-R, in)T(u;)G0(R, in)T(u;)].
(3.4)

Explicitly, V = V\Vi/{S\S^_) with
34
Vi = VfV^+ V¿Gl(R,u>)
V2 = V¡V2+V2G\( R,w)
(3.5)
where
= [1-VoG3(0,w)±VoGo(0,w)]
+(-1)"V,o[tG0(R.u;) + G3(R,o>)].
(3.6)
The factors V\, V‘¿ determine the four 2-impurity resonant energies.
Here Ga(R, u) is the rQ component of the integrated bare Green’s function
(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, uj) — 0 V R. In this case the entire resonant denominator factorizes
D = £>!+£>! _X>2+X>2-. The T-matrix then takes the simple diagonal form
Tk,k'M = 2V'ocos(k-5:)cos(k'â–  y) +
+ 21/0sin(k.5)sin(k'.5)[^: + ^_
(3.8)
where r± = (r3 ± r0)/2.
When two identical impurities with resonance energies Hq , are brought
together, the bound state wavefunctions interfere with one another, in general split¬
ting and shifting each resonance, leading to four resonant frequencies ÍÍ+, £1%
and > where the subscript indicates which factor in Eqn. 3.6 is resonant. If

35
splittings are not too large, the electron and hole resonances are related in a sim¬
ilar way as in the 1-impurity problem, Í2]- ~ — and f^2 • 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
(mathematically) to create a single impurity of strength 2Vo, so both iif2 approach
the Í2q (2Vo) appropriate for the double strength potential. In the case of infinite
separation R —> 00, we must find Q|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 ¿-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 splittings 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 tight-binding model on which most of this work is
based, but much insight can be gained by studying the equivalent gas model, with
spectrum ek = 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 <7Q(R,u; = 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 u;/A0 1/kpr «C l/kp^o- With these expressions, the resonance energies

36
may now be found by inserting these expressions for frequencies = í2q + <5 into
(3.4) and solving for the shifts d. We find 11/ 2 — Qq ± S, with
(3.9)
R || (110)
1 sinfep-fi kp£o
log(i!„/Am„) kFR 4+nH‘kj.
»k)“sfr« + '/4) «- II (100)
These expressions are valid for <5/ÍÍq Clearly the decay of the splitting ~ exp(—r/^a)/V^Fr 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 localized) single impurity state in a d-wave
superconductor, which, by allowing propagating 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 (p = 0, pure nested band) and the 6-site tight binding band [49], with
Ak = Ao(cos kx - cosky), A0 = O.lf and impurity strength Vo = 10i for the former
band and Vo = 5.3f 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 V¡ 2 separately. In
Fig. 3-1 I 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 || (110) inter-impurity separation and the right
panel the R || (100) case. It is seen that each factor Va corresponds to an
oscillating function of R, with the factor determining, e.g., il2 , changing from site
to site according to whether the site is even or odd. This is due to the strong R

37
dependence of the components Ga; in the simplest case, R || (110) and = 0,
G3(R,u>) = Gi(R,u) = 0 but G0(R,u) = Ek cos(kxR/y/2)cos(kyR/y/2)G°k(uj)
oscillates rapidly. At R = 0, the problem reduces to the double-strength single
impurity case; the factor V\ gives the resonant frequency Qq (2Vo) and the factor
V2 is 1. At large separation the and ilj “envelopes” are seen to converge to
ft0+ (Vo) with a length scale of a few £o — 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 splittings vanish much faster
with distance, as expected from 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 when R = 2 + 4n, n integer,
and ill 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 splittings are significant up to quite large distances. Param¬
eters in Fig. 3-1 are chosen such that £o ~ 10a, as seen from the right panel of
Fig. 3-1 where we indeed expect an e~R^° 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.

38
o.c
o
o
10
20
O
5
io R/a
Figure 3-1: Two-impurity resonance energies ílf2 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:
(3.10)
Quite generally, the Green’s function can also be expressed in terms of the exact
eigenfunction ip„(r) of the system (and its conjugate component ip*(r))in with the
presence of the impurity [24, 38]
C(r)lMr)
uj
(3.11)
where the final approximation is valid for a true bound state with u very close to a
particular bound state energy fin, 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):
u = Qq
U = f¡ó
(3.12)

39
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 splittings are small compared to Qq±. The Green’s function
6G(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-impurity
case. By examining Eqn. 3.8, it may be shown that, depending on whether T>\ or
V2 is resonant, the wave functions thus extracted will be of definite spatial parity,
ipn(r) = ±^„(-r). We find
(3.13)
where G°M = G°(r - R/2) ± G°(r + R/2), and the Zs¿r 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 Gi(R).
However, in some special cases where Gi(R) = 0, for example, R || (110), the

40
|V|/|2 LDOS
HI
HI
m
li H
Figure 3-2: The wavefunction at. resonance and corresponding LDOS. Impurities
are separated by R. = (6,6).
eigenfunction become simpler and do not mix particle and hole degrees of freedom.
GiiJr.w)
G°n.(r,cj)
f <%,(r,w)
( G?ip(r,^)
UJ — ÍÍ i_|_
UJ = ill-
UJ — Í124-
UJ = ÍI2—
(3.14)
(3.15)
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, = ±0.0195. £l\±/t = ±0.0075) explicitly in Fig. 3-2. The density
probability, i.e., |>f2| (3rd column) and the actual LDOS (4th cloumn), defined as
G(r, r, u) = ^G°(k,a;)
k
+ e'i k.k'
(3.16)

41
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 \ip\2 and the exact LDOS calculated at each resonant energy,
implying that near each resonant energy the nonresonant contributions are quite
insignificant.
3.1.4 Local Density of States, Realistic Band and Standing Wave Condition
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-impurity wavefunctions in different
regions of space, but the spatial patterns are, not unexpectedly, considerably
more intricate than the “hydrogen 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 (bonding) and p (antibonding) 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 (V\ and T>‘¿) states changes according to whether R is even or odd, as
indicated in Fig. 3-1.

42
In addition, one can look into the spectra on some particular sites close to
the impurities. Fig. 3-3 displays the spectra on several sites whose positions
are illustrated as the numbers in the insets and the impurities positions are
Ri = (—3,0), R-2 = (3,0) (left), R, = (-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:
V = det^-V-, - G)] det T(u),
where T is the one-impurity T-matrix and Cilrmv> 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 u>, the resonance broadening is not necessarily proportional to the
resonance energy.
One might attribute the unexpected “trapped” (weight primarily sitting
between impurities) or “leaking” (weight is populated outside of the region between
impurities rather than inside) patterns to the special symmetries of the half-filling
tight-binding 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 = (jr,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

43
(2,2) (53) (6,6)
Figure 3-4: LDOS maps at resonant energies for R || (HO). Pure nested band,
Vo = 10Í; 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 = (fi, 0) and R = (R, R) for a more realistic
tight-binding coefficients fitted by Norman et al. [49] from ARPES data,
e(k) = io + 2/., [cos(!:„.) + cos(fc¡,)] + 4Í2 cos(kx) cos(ky)
+2Í3[cos(2A:i) + cos(2fcy)[
+2Í4[cos(2fcI) cos (ky) + cos (kx) cos(2 ky)}
+4Í5 cos(2fcr) cos(2 ky) (3-17)
with to... t5 = 0.879, -1,0.275, -0.087, -0.1876,0.086 and |fi| = 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 splitting shows up and persists over a certain

44
Figure 3-5: LDOS maps at resonant energies for R || (100). Pure nested band,
Vo = lOt; Impurity separations are shown on the top of the graph.
range of inter-impurity 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 travelling along a string in opposite
directions, a standing wave can be 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 explicable1 in a similar fashion. For a real bound-state, the equation
G°(R, ui)TG°(R, ui)T = 1 should be satisfied strictly for both real and imaginary
parts at particular energies. The product G°(R, u;)TG0(R, ui)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.

45
Figure 3-6: LDOS spectra for realistic baud and Vó = 5.3fi on nearest neighbor
site. Left panel: impurities at (—if/2,0) and (if/2,0) ( R = (if,0)), spectra taken
at r = (if/2,1). Right panel: impurities at (-if/2, —if/2) and (if/2, if/2) ( R =
(if, if)), spectra taken at r = (if/2, if/2 + 1).
be equivalently written as
Y e‘qRG(k,iu)7'(w)G(k + q,u)f(a>). (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.,
bn Gn(k, o>)Im Gn(k + q, u>), 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 mr, i.e., q R + 2r/0 = rwr, with q being the dominating vectors
of scattering process and r/o the phase shift from one impurity. In the unitary
limit, 2r/o — it, so the commensurability requires qi ■ R + 2r/o = q2 • R + 2r/o =
mr, q3 • R + 2r/o = rmr 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

46
Figure 3-7: Fermi surface of BSCCO-2212 with constant energy surfaces at
¡11 = 0.04 shown as small filled ellipses at the nodal points, qi, qj, q3 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
configurations. 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 wavefunc-
tions in the gas model would predict weak interference and negligible splitting at
this separation. This results suggests that even with a relatively 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.
Finally, 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 Perturbative Prediction: SCTMA and Its Validity
The problem of low-energy d-wave quasipaxticle excitations has been treated
traditionally 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(a;) « n¿T(a;), 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 strength appropriately but
neglects inter-impurity correlations. SCTMA predicts that the impurity states
broaden with the increase of impurity concentration and finally evolve into a
subgap impurity band. 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 conductivity 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] are not borne out by experiments as well; instead,
a displays a linear T dependence in YBa2Cu307_¿ [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
n2u2 log2 a; divergence in 2D coming from the gap nodes [64], to be compared with
the SCTMA, where one has 2nd order contribution of iiiuJ2. Thus for any fixed n*,
the crossed diagrams dominate at sufficiently low energy.

SCTM
48
(B)
I -
x X
u/ \ + ,*T\ + ...
X x
, * V * 4.
' L I )
crossed diagrams
Figure 3-8: Schematic plot of the self-energy diagram with many impurities.
(a) Zero-energy (b)
peak
Figure 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(ui) ~ uia with exact
bosonization. After this work, several supposedly exact nonperturbative theories
made diverse conclusions: p(0) 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(ui = 0) schematically. 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. [66], who pointed out the d-wave disorder

49
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
rii 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 ¿-like impurities can
be written as
G(r, r',ui) = G°(r-r',w)
+ ^G°(r - R,,oi)Ty(u;)G°(Rj - r,u;).
The 2-impurity T-matrix is a 4 x 4 matrix (in the Hilbert space expanded by site
and spin indices) satisfies
Vq'tz - G°(0,u;) —G°(R,tn)
T = (3.19)
—G°(R,o>) Vo-1T3 — G°(0,u>)
It is technically expensive to evaluate of the local Green’s functions G°(R, ui)
and we can only approximate them under some specific circumstances. For the

50
•
•
A
•
•
•
•
/
•
•
•
/B
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Figure 3-10: Schematic plot of bipartite lattice.
tight-binding hall-filled band and the limit ui —► 0, we obtained an asymptotic form
(see Appendix A):
G0(0,u) = -^ln^fo, (3.20)
where a = NI(2i\vfv^), N = 4 is the number of nodes, Up is the Fermi velocity
and Ua is the anomalous quasiparticle velocity and the cutoff A is of order
of A0. The expansion in ui for r = (m, n) depends on whether n and m are odd or
even. For the (even, even) case, we have
G°(r,w) -* (-1)^ [G°(0, (u) + uiCo(r)] f0, (3.21)
where Go(r) is a real function of r. We find similar leading-order expressions for
(m, n) = (odd, odd),
G°(m,n,ui) -* iuC0(r)fo, (3.22)
while for (m,n) = (odd, even) or (even, odd),
G°(m,n,ui) —* Ci(r)fi + C3(r)f3, (3.23)
where Ci(r), and G;¡(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

51
Figure 3-11: Change in p(u) 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
[2wCo(R)G(j(0,uj)] 2 R = (even, even)
G°( 0, u)_4 R = (odd, odd)
and the correction to total density of states:
Sp(u) -
l/[o>ln2(A/u>)] R = (even, even)
2/[u>ln2(A/u;)] R= (odd,odd)
(3.24)
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 G°(R, u) ~ 1/R out to a
cutoff ~ t/R, p(u —> 0) diverges as
M[lna(A/w) + (7r/2)s]’ <3'25)
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,

52
150
100
3
£
50
0
Figure 3-12: Change in p(cj) 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 splitting of single impurity resonances. When the unitary limit is
approached, a single divergent peak is emergent. However, it should be 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:
-<7°(0, cj)t0 C, (R)fi + C3(R)f3
C1(R)fi + C3(R)f3 - with D' = (7(0, a;)2 — Ci(R)2 — C^R)2. It follows easily that det T = D'~2 and that
Sp(uj —» 0) oc -j- fa;2In— ^ —► 0 (3.26)
duj \ uj )
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

53
Figure 3-13: (a) DOS for Vo=100t. (b) Scaling of the DOS with Vo. (c) Scaling of
the DOS with L. (d) Scaling of the DOS with V0 and L=60.
energies. Numerical calculations of the DOS shown in Fig. 3-12 demonstrate that
there is no remnant of the single impurity u-*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 ui = 0 in the pure nested
tight-binding model? 2) how does the impurity band evolve away from iu=0?
¿-like Divergence at. uj — 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 n¡ = 0.1%. The nonperturbative PL result from

54
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 = lOOi. 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 —> oo is illustrated in Fig. 3-13
(d). Generally, the peak becomes sharper when Vo is bigger. It is indicated that the
divergence is actually a delta function, i.e., lim^o-*» p(w) ~ <5(w) since the peaks
scales as p(u) « VqF(ljVq).
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 k + Q)t2 = G°(k), which in the unitary limit, i.e., U = oo and
p = 0 (in this this special band) gives additional poles with momenta Q = (7r, 7r)
to the particle-hole and particle-particle 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
£0 ~ 40a. When system size is bigger than £0, 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 \En\ < 10-5 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

55
Figure 3-14: LDOS for 2% concentration of impurities and \En\ < 10 5t (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 understood 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 Zn 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 7 ~ y/ñ¡ErA0!) 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

56
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 qualitatively. This possibly implies
that the DOS plateau for uj > 0 is formed by summing over many impurities with
“inhomogeneous 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 (¿-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 ImeV, there are still numerous
eigenstates within such a window size for a typical sample size (L ~ 500/1) and
impurity concentration (n¿ ~ 0.2 — 0.5%) with which the tips may bin over to
produce the nearly isolated, four-fold symmetric, localized impurity patterns.2
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 coinmensurability 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(uj). At uj = 0,
2 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.

57
Figure 3-15: LDOS for 0.5% concentration of impurities. Left: En = 0.0385£; right:
averaged over five eigenvalue in energy interval \En — 0.03£| < 0.02£.
with the global constructive interference by the particular 72 symmetry in
hall-filled tight binding band, the DOS of the many-impurity system diverges
and contains a ¿-function form. However, I should emphasize that this sharp
peak at u = 0 is the consequence of the special feature of hamiltonians with
72 symmetry only, and is not a generic attribute of d-wave superconductors.
• Away from the Fermi level, the quantum 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

58
Figure 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) =
e?qr/9(r). They plotted its absolute value as a function of momentum q and bias
energy u 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 = kfinai — kinitial 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 band theory, the quasiparticle DOS p(uj) at uj is
proportional to
(3.27)
and the area with the smallest energy gradient contributes to p{uj) 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) = cjq}) deforms

59
in the d-waive 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 u 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 coimecting 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<0)(k + q,u;)r(k + q,k,u;)G<0)(k,a;)]. (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'fk', k), may
scattering of high order be neglected and the disorder potential be factored out as
well:
p(q) ~ ImiV'tq^G^k + q.i^G^k,^)}, (3.29)
k
where F(q) is the Fourier component of the real space potential Ffr). The term
£kc(0)(k + q, a;)(7(0)(k,a;) 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, Pintuai and p final 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 (¿-dependence of those q peaks is presented in Fig. 3-16, showing
good agreement with each other! The theoretical prediction is calculated from a

60
standard BCS theory with quasiparticle dispersion
Ev = (3.30)
where Ak takes dx2_ya 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¬
ticles. 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) + £cos(60k)L with
A0 = 39.3 meV, A = 0.818, B = 0.182, which reflects approximately the proposed
dx2_y2 symmetry of the pairing function in BSCCO.
3.3.2 Power Spectrum for Many Impurities in B^S^CaC^Os
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

61
experimental data which is not explicable within this simple model, for examples,
the resolved qi, qr 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 qj.
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 priori 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 — VjG°(r = 0)]-1Vj by
N,
Tij = iAj + ¿¡[1 - (W]G°(R, - R™)tm„ (3.31)
m=l
where the impurity potential at R, is V¿ = V0T3, and f¿ are the Pauli matrices. In
the dilute concentration limit, only the leading order in U of Eqn 3.31 needs to be

62
Figure 3-17: FTDOS at u =14 meV for weak potential scatters (Vo = 0.67¿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 l/y/N] are plotted vs. qx in (c), while
(d) shows the weak scattering response function Im A3(q,a;). Peaks at q = 0 are
removed for clarity. In all the figures, the x and y axes are aligned with the Cu-0
bonds.
considered:
sp(q,w) - -- y; ia(q)Im[e'*“A0(q,w)] (3.32)
where ta(q) = «<,£ie_iq'R' and A0(q,in) = [G°(k, u)faG°(k + q, w)]u .
In the weak potential limit, Eqn. 3.32 reduces to
Sp(q, ui) ~ — K(q)Im A3(q,ü;)/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

63
signal since the disorder average gives (|¿p(q, a;)|2) ~ N¡ and (|£p(q, w)|4) —
(|¿p(q,uj)\2)2 ~ 7V/(iVj — 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 they 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., 6T,/dk = 0.
Unitary Scatters and Extended Weak Scatters.
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
qi 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 T3 components of
Aa at several energies. It is observed that, interestingly, while A3 resolves q7, q2, qe,
Ao resolves qi, qs quite clearly (q3 and qi are present in both of the two channels).
This difference between Ao and A3 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,
T M =
goMro ct3
C2 - flg(w) C2 - 0¡j(ü>)
(3.34)
where go(u> —> 0) ~ a;. In the weak scattering limit the second term in Eqn 3.34
dominates and qi and qs peaks are missing. However, in the unitary limit, i.e.,

64
w=0.045455 w=0.090909
w=0.13636
Figure 3-18: The plot of Nambu component of spectral functions at several fre¬
quencies.

65
c = g(uj —> 0) —> 0, To and 73 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 uj < 15rneV.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) = JT V(i) exp(—f¿/A)/r¿ and
= [(r — Rj)2 -|- d2]1/2, where R, -I- zdz 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-
goliugov 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 qs 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 uj « 50 meV that is not observed in STM
experiments.
While qi, qs are identified as “remnant” of octet peaks together with the
dispersing back ground, q7 seems to be misidentified 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/^a, where va is the gap velocity at the
nodes. McElroy et al. [70] have exploited this mechanism to map out the gap
function Ak and were forced to introduced a significant subleadiug cos 6(9 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
4 In BDG formalism, the hamiltonian for a superconductor on a square lattice is,
n=-,Y,micW- 12^-5Z{a¡íc!tc]i}+(3-35)
o i,(T i,j
where the angle brackets indicate that site indices i and jare nearest neighbors, U¡
is the impurity potential, and Ai3- = — K(CjjC¡,) is the mean-field order parameter.

67
LDOS FTDOS EXPT
Figure 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 |u>| > 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 qi 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 t\ 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 illuminating 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
B^S^CaCujOg ; 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 atomic-scale electronic phenomena to disorder
induced wave-like quasiparticle states and the quantum interference effects be¬
tween them. I here briefly summarize my understanding on the implications of the
quasiparticle scattering model for the recent discoveries in STM experiments.
4.1 Atomic Level Resonances, Quasparticle States and 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 Fig. 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), the spectroscopic signature are quite similar to
68

69
Figure 4-1: Zn impurity resonance, left channel: dl/dV Vs.a;; right channel: the
LDOS spectrum above the Zn site.
those observed when Zn and Ni axe 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 3dx2_y2 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

70
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 impurities, 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 (|u;| < Ao) spectra across

71
Figure 4-2: The gapmap of overdoped BSSCO. Sample size 500Ax500A.
-100 -50 o ’ so Too
Sample bias (mV)
Figure 4-3: Spectra on different sites along the horizontal cut.

72
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 superconductivity.” 1
However, we should keep in mind that the STM tips measure quasiparticle 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 possibility
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 possibility 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 presumabl; 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 superconducting 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.
1 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.”

73
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 regar dless of the origin of the disorder [70]: the con¬
ductance map shows that for a given u, the observed Umklapp scattering induced
LDOS modulation is extremely localized to the region whose local gap value is
approximately equal to u. This indicates that for k ~ (7r, 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 (7r, 0) quasiparticle spectra in
ARPES.

74
4.3 Long-range Modulation and Exotic Orders
The electronic structure of cuprates lias 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 significant 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 superconducting 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 identify 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 A|<. 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-Tc
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-Tc 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 momentum-dependence of the single particle self-energy [97, 98].
76

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:
/(k,u;) = /0(k)/(u;)/l(k,u;), (5.1)
where /o(k) is determined by the momentum dependent matrix elements and the
Fermi function f(u) illustrates that this process can only measure the unoccupied
states. The one particle spectral function /l(k,u;) describes the probability of
creating or annihilating one particle in an interacting inany-body system and is
associated with the imaginary part of the retarded Green’s function, as defined in
Eqn. 5.2:
d(k,w) = ^2\K+'\4\^\2S(cj - E^+1 + £f)
m
= —-ImG(k,u; + ¿0+). (5.2)
7T
In a non-interacting system, /Uk,lj) is nothing but a delta function, 6(u) — et),
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

78
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_1(k, u;) = Gq1 — S(k, a;), where
E(k,(j) = E'(k,w) + ¿E"(k,u>),E'(k,u;) = ReE(k,u;), E"(k,a;) = ImE(k,u/) and
S 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 T(k, u) = — l/2ImE(k, cj). The general expression for
the spectral function is then:
A(k,u) = -
E"(k,w)
(5.3)
7r [w — tic — E'(k, in)]2 + [E"(k, a;)]2
The self energy E(k, u) arises from many-body and impurity effects and
encapsulates all the information about interactions. Generally, we can plot the
measured intensity (oc A(k, u>)) either as a function of uj (binding energy) for a
fixed momentum k (EDC), or as function of k with a fixed value of u; (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.
5.1.2 Self-energy in Normal and Superconducting States
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 [34], 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 (7T, 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,

79
T-I05K ^
imliiiilimíiiiiH
0.4 0.2 0 0.4 0.2 0 0.4 0.2 0
Binding energy (eV)
Figure 5-1: ARPES spectra from overdoped BÍ2212 (TC=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 superconducting state1 . When the temperature
increases, the coherent low energy quasiparticle peak evolutes continuously 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 Tc superconductors since the normal state temperature is of the order
of 100K ( it is quite high compared to the general definition of “low” temperature),
1 It is called the “peak-dip-hump” feature. The high energy humps are specu¬
lated generally to be the consequence of phonon modes, magnetic, collective mode
or bilayer splitting effects.

80
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 ~ lOOmeV, 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 al. [8] seems to
describe the transport and thermodynamic properties and the anomalies observed
in ARPES phenomenologically.2 ft 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 Cu02 plane, and
a MFL component as defined in Eqn 5.4. The scattering rate in the normal state
then reads:
r(k,o;) = re/(k,u;) + rA/F¿(k,u;), (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(lS) ~ 1).
The electron self energy due to the scattering from this bosonic spectrum has a
form:
£(k, uj) = Afu; log —— (5.4)
wc 2
where x = max{\uj\, T). This singular behavior of self-energy leads to the absence of
the FL-like quasiparticles even on the Fermi surface when T=0.

81
The possible applicability of elastic forward scattering is further reinforced by the
following fact: in Born approximation, the self-energy due to impurity scattering is
written as
E(k,w) = ntJ2 |Vklkf C0(k',u;). (5.6)
k'
Qualitatively, if Vk,k' is peaked at certain direction, then £(k, proportional to the imaginary part of the integrated Green’s function over a
narrow range centered around k, i.e., G(k, uj) ~ X^k'~k± extremely small, namely, in forward limit, £(k,ó;) is proportional to v¿?(k)-1 (the
inverse of group velocity of band electrons), which generally takes the maximum
value at antinodes and minimum value at the nodes on the Fermi surface. This
momentum anisotropy of £(k, uj) coincides with the experimentally extracted
elastic component Te¿(k, uj) 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 (7T, 0) point is known to sharpen
dramatically when one goes below Tc, 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 Tc 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 lOOmeV. The second problem is that recently increased momentum
resolution [34] and the use of different photon energies [102, 103, 104, 105] has
resolved a bilayer splitting which has its maximum effect near the (n, 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

82
planes, and even the best single crystals are believed to contain significant amounts
of cation switching and other out-of-plane defects [107]. It is therefore reasonable
to assume that quasiparticles moving in the Cu02 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 Cu0-2 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
superconducting state assuming a smooth potential [108].
5.2 Elastic Forward Scattering in Bi2Sr2CaCu2Q8
5.2.1 Self-energy
Normal State. Consider a model system including elastic scatters of finite
range k-1, with a concentration n¡. Assuming the single impurity takes the form 3
V(r) = Voe~Kr, we can calculate its two dimensional Fourier component:
14k'
27r/tVó
((k - k')2 + /t2)3/2’
(5.7)
where Vo sets the strength of the potential. The self-energy in Born approximation
is then defined through Eqn 5.6
As the range of the potential re-1 increases, the scattering of a quasiparticle
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 Cu02 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.

83
Figure 5-2: Geometry for the forward scattering process in which a quasiparticle
scatters from k to k'.
wheu k is close to k' and both are not too far from the Fermi surface, we may
parameterize them as
kp + k±k±
(5.8)
kj? + q« + k'Lk'L
(5.9)
where q = k — k' is the momentum transfer and qj| its component parallel to the
Fermi surface. The unit vectors and k'± are the projections of k and k' onto the
Fermi surface, respectively, such that, e.g., = vf(k')fci- The imaginary part of
the retarded self-energy Eqn. 5.6 becomes
£"(k,u/) =
[<72
—n¡(2jr/íVo)2 f dk'^dk'^
(27T)2
n¡K2Vg
M*h)I
37r/i¡ Vq
J
I Ju
fií á(^ - ek')
b J
dq[.
[?f + (fcx - £)2 + k2]
(5.10)
Eqn. 5.10 shows explicitly that in the limit of small k, the self-energy becomes
more and more sharply peaked “on the mass shell” lj = e*, as a generic feature
of long-range potentials. The angular dependence of the self-energy in the limit

84
of /c —► 0 can be verified exactly as the same as that of l/vjr(k). Furthermore,
I should point out the self-consistent treatment (by requiring £[(7°] —► £[£?]) 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 tunnelling 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 uj2/Ep to the non-selfconsistent result, which may be neglected.
Superconducting State. In the superconduting state, the self-energy is approxi¬
mated similar to Eqn. 5.6 as:
£ = n/£|Vkk'|27sG0(k',w)r3,
k'
= aTa (5-11)
a
with Nambu components £a defined as following;
E„(k,w) = n, ^|\/(k.k')|2 J a . (5.12)
S»(k,w) = n;^|K(k.k')|2 — 3 . (5.13)
k/ U — Ck' “ ^k'
and
=i(k,«) = -Bf ^|y(k,ior. y A2 ■ (5.14)
k/ u ~ ek' ~~
In the limit of k « 1, the asymptotic forms of self-energies can be derived
analytically, which read:
S£(k,w)
To(kp)
. > sn x
2v^A
tk ~ vs/u1 - A‘^
KDjr(k)
(5.15)
Here sa = |cu|, —AkSgnu), and v sgntn — A)) for the Nambu components a =
0,1 and 3 respectively, and To(k) is the normal state scattering rate. E3 vanishes
on the Fermi surface ek = 0 in this limit. Specifically, when the momentum

85
k is close to the Fermi surface and the energies u are small, such that | (e* ±
sju2 — A2k)/KVp(k)\ E!(k,w)
(5.16)
Ej(k, sc)
~ r"“ $=*
(5.17)
~ 0,
(5.18)
but are strongly suppressed due to energy conservation when | \/~j2 — —
eic| becomes greater than kvf, 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 Fermi surface. The bale electron band
4 Roughly speaking, the change of momentum ák in the scattering is cut off by
k. Therefore for scattering processes close to the Fermi level, the corresponding
variation of energy is kvf(k), if we linearize the electron dispersion. On one hand,
the quasipart.icle 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 (\Jui2 — A£) component should still be cut off as well due to
the forward nature of scattering.

86
Figure 5-3: The self-energy terms -Im Eo(k, u), Im Ei(k,a;), and -Im £3(k,(n) in
the superconducting state at T = 0 for k = k,i (top) and k/v (bottom), for k = 5
and 0.5 and the same band and scattering parameters as previously used. Here
Ak = A0 (cos* — cos ky)/2 with A0 = 0.2t.

87
dispersion is taken as:
tk = —2i(cos kx + cos ky) — 4i' cos kx cos ky — p
(5.19)
and t/t' = —0.35, p/t = -1.
The quasiparticle properties near the Fermi surface ¡are determined by some
particular combination of the Nambu self-energy components, by examining the
denominator of the full Green’s function:
(5.20)
where ü = u — E0, tk = ek + S3, Ák = Ak + Ej. The total elastic scattering rate
will be:
(5.21)
provided one can neglect S3 (see Fig. 5-3).
In the forward scattering limit where Eqn. 5.16-5.18 hold, the effective elastic
scattering rate ( Eqn. 5.21) becomes
(5.22)
For k along the nodal direction, the elastic broadening in the superconducting d-
wave state is equal to its value in the normal state. However, for k in the antinodal
region, the broadening vanishes as ui —> Ak and approaches the normal state
value only when in becomes large compared with Ak- The elastic contribution
re|(k,,,u;) to the broadening at the antinodal k.4 point versus in is shown in
Fig. 5-4. Physically, the individual contributions to the normal E0(k,w) and
anomalous E^k, ui) self-energies are both enhanced by the density of states factor
(in2 - A2(k))“^ (Fig. 5-3). However, the normal contribution describing the
scattering out of state k into k' is compensated by the anomalous contribution
scattering into k from the pair condensate. This gives rise to the suppression of the

88
Figure 5-4: Scattering rate rej(k, ui) vs. ui for k = k.r(left) and kjy (right) in the su¬
perconducting state at T = 0, for k = 5 (top) and k = 0.5 (bottom). Here Tofka) =
0.2t.

89
elastic scattering rate seen in Fig. 5-4 relative to Fig. 5-3 as uj approaches -|Ak,,|
from below.
5.2.2 Spectral Function
General Spectrum. In the forward scattering limit, with the self-energy given
by Eqns. (5.16)-(5.18), one obtains a result for the Green’s function previously
discussed by Markiewicz [108],
G(k,u)
(iurQ + AkTi)z(k,tu) + ekT3
(in2 — A£)z(k.u>)2 — tl
(5.23)
Here z(k,w) = 1 + ¿ro(k)sgnu>/y,u;2 - A£. The electron component of the spectral
function is then
A(k, uj)
——ImGji(k,u)
7T
= —— Im
uiz( k, oj) + tk
(5.24)
tr (tu2 - A£)z(k,iu)2 - tl
It is useful to consider a few special cases of (5.24) more closely. In particular, on
the Fermi surface tk = 0 one has the simple expression
r„(k) 1 M
^U) * “2 - Ak + To(k)2 ’
while near the gap edge, e.g., uj < — |Ar|,
A(k, uj) r.
r„(k)
M
(5.25)
(5.26)
7T e? + r5(k) ^/w2 - A]l '
At the nodal point k/v, where the gap vanishes, the spectral weight is given by
the simple Lorentzian form
r0(ky)/7r
A(kw,in) = â– 
(5.27)
w2 + r0(kN)2’
and at low temperatures where the elastic scattering is dominant one can determine
r0(kjv). However, for k at the antinodal point, k^ such that —Ak,, - Su < ui <
~Ak„,
A(k^,u;) = i Ak"
1
tr r0(kA)
(5.28)

90
where the “width” 6co depends upon the ratio A^/Tof^). If Ak4 is large com¬
pared with the width 6u) ~ Toik^/pAk^). If one integrates Afk^u;)
from — Ak4 — Su to — AkA to define a “peak intensity”
í~AkA l
7(k^) = / A(Ila,u) du ~ , (5.29)
7-Ak/1 -Suj k
which is independent of Ak^/rofk^). However, when the system is sufficiently
dirty such that Ak¿ < Fofk^), the falloff of A(\ía,uj) as cj decreases below — A^a
varies as (a;2 — A^)~1//2. In this case, the scale is set by &kA and if one takes
Suj = Ak¿, the peak intensity varies as / (k^) ~ Ak^/rofk^). This is quite different
from the usual DCS quasiparticle result which is proportional to the quasiparticle
renormalization factor ^(k^) times a coherence factor which is 1/2 on the fermi
surface. It should be possible to test the foward scattering scenerio by comparing
the variation of 7(k^) with Akj/r0(kyl).
Nodal and Antinodal Spectra. To compute a realistic spectral function and
compare it with experiments, one should include a small concentration of unitary
scatters as well as inelastic electron-electron collisions. The former is verified
by STM experiments and has been intensively discussed in Chapter 2-3. With a
concentration roughly ~ 0.2% decided from the number of zero bias resonances in
LDOS map, it gives a broadening in the self-consistent T-matrix approximation:
£el'u = -£kG,(k,u;)T0' (530)
This gives a scattering rate of r.(t ~ 10~3i in the normal state, leading to an
impurity bandwidth 7U ~ \/ruAo ~ 10_2i (of order 1 to 2 meV). As far as
the inelastic self-energies are concerned, we use a numerical interpolation of
the uj— and T— dependence obtained from the spin-fluctuations performed by

91
[110, 111, 112].5 Combining the contributions from these three channels, the total
scattering rate is then:
Sot = Sz,/ + Sl.u + —inel • (5.31)
The spectral functions at antinodal point A and nodal point N with the
scattering rate defined in Eqn. 5.31 and the Fermi function cutoff f(u/T) are
plotted in Fig. 5-5. In the antinodal direction, a square-root-like behavior develops
when uj approaches — Ak„ from below. This should be contrasted with the broad
Lorentizian peak in the normal state and is an intrinsic consequence of elastic
forward scattering. It is also noteworthy to point out that this anomaly at Ak is
stable against the many-body interactions since at lower temperature, when uj is
equal to the real paid, of the gap at the gap edge, the inelastic broadening vanishes
as T:i and the elastic components dominate.
Along the nodal direction, since the scattering always mixes the initial and
final states with order parameters of different signs, a Lorentizian centered at Fermi
level is observed for zero temperature. This peak will be further cut off by the
Fermi distribution and broadened by inelastic effects when temperature increases.
In the limit of k, = 0, we recover the normal state spectral function, even at zero
temperature
Quasiparticle Dispersion Near the Antinodal Point. In a clean superconductor,
there is a peak in the spectral function A(k, uj) at the quasipart.icle pole u = =
y/c^ + A£. In particular, as the momentum moves along the cut from (k¿, 7r) to
(0,7r), one expects to see a dispersion of this peak to higher energies.
5 The spin-fluctuation self-energies behave like follows: at the nodal point, at low
temperatures, the scattering rate initially increased as the third power of uj or T
depending upon which is larger. At other k points on the Fermi surface, the scat¬
tering rate varies approximately as the third power of this energy measured relative
to Ak- The reduction of the inelastic scattering rate at low excitation energies re¬
flects the suppression of the low energy spin-fluctuations due to the opening of the
d-wave gap.

A(kA, (ú)f(co) o A(kA’
92
(0
Figure 5-5: Finite temperature spectral function at the antinodal point A and
nodal point N on the Fermi surface multiplied by the fermi function. Results for
ft = 2 and 0.5 with ro(ka) = 0.2t are shown.

93
k=2 ks0.5
Figure 5-6: A(k,uj) vs. uj for k, = 2 and 0.5. Results are given for the k points
at (7T, 0), (7T, 0.057r), (it, 0.l7r), (7t, 0.157t). The disorder levels correspond to
Tofk^J/Ao = 1, 0.5, and 0.025. Note the spectra for different k points have been
offset for clarity.
However, if the forward elastic scattering strength Toik^) > Ak^, then the
peak in A(k,o;) remains at -Ak rather than dispersing. Fig. 5-6 shows plots of
A(k,u;) for different values of k between the M and A points for k = 2 and 0.5, and
several values of the scattering rate To(k4).
As samples improve, there is a natural tendency in this model for the spectrum
for k not too far from the antinode to cross over from one characterized by a
nondispersive peak at Ak in the dirty limit where To ~ Ao to one characterized
by a dispersive quasiparticle peak at Ek when To is small compared to A0. This
crossover is due to the way in which the forward elastic scattering rate for a d-wave
superconductor is reduced as the gap edge is approached and is analogous to the
same effect discussed analytically (in Appendix B) for an s-wave superconductor.
In a system with To > Ao, no quasiparticle peak is observed, but a sharp feature
does appear at — Ak, representing simply the spectral weight in the overdamped
quasiparticle peak piling up at the gap edge as in the s-wave case. Only when To
becomes small compared to A0 does one see a true quasiparticle peak dispersing

94
Figure 5-7: Comparison between recent ARPES data (left panel) and calculated
A(k,o>) (right panel).
as — Ek- In the most strongly forward scattering case, n = 0.5, one can see that,
depending on the strength of the scattering rate, one can have simultaneously a
broadened dispersing feature as well as a gap edge feature.
In Fig. 5-7, the measured ARPES spectra along a special cut ( (0,0.64-it) —»
(kf, 0.647r) ) (data from private communication with Stanford ARPES experi¬
mental group (2004)) and the theoretically calculated spectra adopting elastic
impurities with 8% and k = 0.5 are compared. While the variation of the calculated
order parameter is smaller than the exact measurement, other details, such as the
dispersion of the hump feature and the gap peak intensity, are in good agreement
with each other.
5.3 Discussion and Conclusion
Elastic forward scattering of quasiparticles due to spatially smooth potentials
is prevailing in BSCCO. We have discussed how this scattering can fit with the
BCS framework to invalidate out naive intuition on the insensitivity of elastic
scattering self-energy to entering the SC state and hence argued that elastic
forward scattering does not necessarily prevent the formation of quasiparticles
across the normal-superconducting transition.

95
While the bulk and thermodynamics properties such like Tc is [113] and the
transport properties such as conductivity are not significantly modified by extended
impurities because of their forward nature (there is not significant momentum
transfer in scattering), the spectroscopic properties are modified significantly by
the elastic forward scattering: the near-cancellation of the two Nambu components
of the self-energy near the gap edge in the forward scattering limit leads to a
dramatically reduced elastic scattering rate in the superconducting state, which
sharpens the spectral features of quasiparticles which are not too close to the
nodes. Besides the collapse of self-energies, a square-root-like singularity local is
also found to develop at the local gap edge uj = Ak-
The local gap anomaly is of particular interest since it is inherent from forwar d
scattering. This singular structure remains when k is below the Fermi surface
and is speculated to be robust against many-body effect because the latter, in
general, is suppressed when temperature decreases. To date, ARPES experiments
generally mapped out the momentum anisotropy of Ak along the Fermi surface,
with assumed information of bare electron dispersion e*. However, this gap edge
anomaly seems to propose an alternative method to determine the gap function for
k values away from Fermi surface by taking use of forward scatterers. Successful
realization of this technique may complete the momentum dependence of Ak in
the whole first Brillouin zone, which definitely contributes to the understanding of
cuprates. Furthermore, another possible advantage of this method is that it doesn’t
necessitate any knowledge of £k in advance at all!
Moreover, in samples of less disorder, we found a dispersive peak associated
with the bare quasiparticle dispersion. It is tempting to speculate that this phe¬
nomenon is related to the peak-dip-hump features observed generically below Tc in
cuprate ARPES experiments. When the system becomes dirtier, the quasiparticle
peak will be broadened from d'-like function into Lorenzian form. However, far away
enough from the nodal direction, since we physically recover Anderson’s theorem

96
and Ak remains unchanged, the weight of the Lorenzian tail cannot leak into the
subgap regime, namely, |u;| < |Ak| and will keep piling up at the leading gap
edge instead. We then expect this anomaly is qualitatively more pronounced in
dirties samples. This could possibly explain the excessive peak height observed by
STM experiment at the coherent gap edge, which is even much higher than the
theoretically calculated peak height for pure systems.
The last concern of this chapter is the lifetime of nodal quasiparticles. By
extracting the half-width AA: of the peaks in MDC, we obtained a scattering
rate which is linear in temperature above Tc but varies nonlinearly below Tc and
saturates to a value determined by the small-angle scattering. However, it was
reported by Valla et al. [16] that the width of MDC has a linear dependence on
temperature across the normal-superconducting transition, which suggested the
irrelevance of the nodal quasiparticles in the superconductivity and a quantum
critical behavior. This discrepancy is rather interesting and definitely worthy of
further investigation.

CHAPTER 6
CONCLUSION AND REMAINING QUESTIONS
I have studied impurity induced quasiparticle interference and the related
electronic structure in fully disordered BSCCO samples. I adopted two types of
nonmagnetic impurities: short-ranged, unitary scatters and spatially extended,
weak scatters, in conjunction with the d-wave version of the mean field BCS
theory (which is believed to be appropriate to optimally to overdoped cuprates), to
calculate the spectroscopic properties such as local density of states, local gap map
and spectral function.
I discussed how the theoretical predictions on Zn and Ni impurities mesh with
the details of the atomic scale resonances observed in the STM image of BSCCO.
I addressed on the dispute regarding the characteristics of Zn, which is not fully
pinned down yet.
There are divergent theories concerning how quantum interference between
many impurities modifies the low energy quasiparticle states and the Fourier
transformed DOS. I joined into this discussion by solving the two impurity problem
exactly. I concluded that the divergent p(u) at the Fermi level for half-filled nearest
neighbor band is the consequence of the special symmetry of the nearest tight-
binding model seen in the analogous the two impurity interference effect. I further
proposed a real space perspective regarding the homogeneous broadening by finite
STM resolution and explained why the impurities appear nearly isolated in the
STM image while they are expected theoretically to interfere with each other even
separated by long distance.
I further examined the “octet” single impurity model as a way understanding
the Fourier transformed DOS in BSCCO. I argued that while its on-shell approx¬
imation captures the fundamental scattering processes, it fails to explain other
97

98
details such as the peak widths and peak height for certain scattering wavevectors.
I showed why it is necessary to include both of the two kinds of disorder above to
reconcile these discrepancies. Additionally, I discussed the implications of extended
impurities to the nanoscale gap inhomogeneities in STM measurements.
The extended weak potentials have a remarkable influence on the spectral
function. From my computation, I realized that the quasiparticle scattering rate
actually collapses in the superconducting state due to the near cancellation between
the normal and anomalous scattering processes, yielding well-defined antinodal
quasiparticles. This interesting finding invalidates the argument that elastic
scattering is intuitively robust against the opening of a gap upon lowering the
temperature.
In my dissertation, I have established, within the frame of quasiparticle
scattering due to impurities of different spatial nature, a reasonably coherent
picture subsuming recent observations in STM and ARPES. However, despite this
modest success, there are still remaining discrepancies between STM and ARPES
experiments from this point of view, such as, 1) why we have nanoscale electronic
disorders at the gap edge but quasiparticles are well defined in momentum space?
2) why the ARPES spectra for the nodal quasiparticle display a kink while STM
does not reflect any collective mode features. Full comprehension of these puzzles
may require physics beyond the quaisparticle scattering scenario. However, I would
like to put a period here, as I realized that I’ve been too talkative.

APPENDIX A
GREEN’S FUNCTION FOR THE SUPERCONDUCTING STATE
In this section, we try to evaluate the real space component oí the Green’s
function for a pure nested band, i.e., tk = —2t(cos kT + cos ky). If we define,
r = (m, n), where m, n are in unit of lattice constant a, the real space Green’s
function is defined through the Fourier transform as:
G°(r,u)
£yk'rG°(k,w),
k
E cos (kxm) cos (kyn)
wfb + (tk - p)f3 + Akfi
Start from Eq. (A.l), and we express
(A.l)
cos (kxm)
2ra_1 cosm k¡
, Ifl
3=1
j\(m - 2j)\
where [...] refers to the integer part of the argument. We focus on the half-filled
case fi = 0 and write Eq. (A.l) as the sum of terms of the form
fa = Ecos'-tMco S'(ky)“f°+fj + Akfl
k ^ Ljk
where p = m, m — 2,... and q = n, n — 2,.... We proceed by linearizing the
dispersion near the node at (tt/2, 7t/2) and making the coordinate transformation
E2 = ek + A£, tand = Ak/ek-
a f2lT d6 / sin# cos9\q /sin6 cas8\p
2^ ]0 27t \ ~Aq 2t~)
x /A + E(COS + sinfr^)
Jo ' ÍU2 - E2
99

100
The prefactor is a = N/(2ttvfVa) where N = 4 is the number of nodes, vf is
the Fermi velocity and Va is the anomalous quasiparticle velocity |VkAk|, and the
cutoff A is of order Ao- The integrals over E and 8 are easily done and
= ■^iujFp+i(ul)Pnf° + + P¿ f,)]
where P3pq are constants given by the angular integrations, and
Ea
W = [
Jo
EdE-
E*- u2'
The constants P¿ vanish for j = 1,3 when p + q = even and vanish for j = 0 when
p + q = odd. The first few nonzero elements are
P0 — 1
MM) — 1
P1 — — P1 — 1
^1° ~ 01-2Ao
p3 _ p3 — L
- 'In- 4t
p0 _ p0 _
Ml “ Ml “ ~Ta2
Only even moments of Fa(cj) are needed:
n u2j A2(n^) w2n, A2
* â–  gTynTT + T1"^
Since we are interested in the leading order behavior of G(R, uj) we note that for
small uj ,
Pit !, A2
F„H - j In ^
A2"
F^u) - -
For R = (2m, 2n), the leading order contribution to (7(R,u;) comes from the
single term in the expansion containing gooâ–  To second order in uj:
\ 2
G(R, uj) = -(-l)”+n*^ln—jio + u/CoWfo,
(A.2)

101
where Co(R) is real, and is the sum of several terms. The largest term contributing
to C(R) is of order
a\ui\ / A_\2(m+B)
16(m + n) VA0/
from which we estimate a range of validity
(A/An)a(”»+")
|0l| < Ae 16<"'+"> ,
For other R, there is no single dominant term in the expansion for the Green’s
function, and the leading order behavior comes from the sum over a large number
of real nondivergent terms. For our purposes, it is sufficient to note that when
R = (2m + 1,2n + 1), the sums take the form
G(R,aO=wC0(R)Tb, (A.3)
and when R = (2m + 1,2n) or (2m, 2n + 1)
G(R,o>) = CjíRJÍr + C3(R)f3, (A.4)
where Gq(R), Cr(R), and C3(R) are real constants.

APPENDIX B
EVALUATION OF THE SELF-ENERGY IN BORN APPROXIMATION
In this section, I give out the evaluation of self-energy in Born approximation
for several different types of impurities.
B.l “Yukawa F Potential
Assume that V (r) = Qexp(—nr)/r, and its Fourier transform in 2D takes the
form Vq = 2nQ/ \/q2 + k2. According to Fig. 5-2, we obtain:
= kF + k±k±
(B.l)
= kF + q„ + k'±k'±
(B.2)
for momentum close to the Fermi surface, we can linearize the dispersion and we
have ek' = Vp(k')
B.1.1 Normal State
The self energy in Born approximation can be then evaluated:
)l [q|| + k±k± — —tjjt¿x]2 +
Ttn,i Q2
nntQ2
TTTliQ2
h
J <*4
fall + *l(*l - + k2
1
M*ii)I yck
L Mi*,[)>
Tr2n.iQ2
1 + K2
y/(k±.vF(ky) - u)2 + it2vF(k||)2’
where in the middle steps the difference between v¡.'(k') and vp(k) is neglected
since in the limit of small k, the two approach one another, and we need to retain
(B.3)
102

103
only the most singular dependence on k — k'. In addition it is reasonable to set that
dk\\ = dq\\, since k is a constant in the integration.
Special Cases:. • k = kp (on Fermi surface)
(B.4)
• k = kF, u> = 0
(B.5)
Note this diverges in forward scattering limit as K —* 0. If we normalize to E¿, this
gives
S"(kF,u; = 0) |nF(kj4)|
£"(k„,u; = 0) K(k)| '
(B.6)
• u = 0, k kF
(B.7)
Note in forward scattering limit k —> 0, self-energy is a very singular function of
either k± or u.
B.1.2 Superconducting State
In the main Nambu channels, i.e., tallo, tau¡,tau3, the imaginary parts of the
Green’s function are the summation of two 5 functions:
ImGo(k', u¡) -- Im
w2 - El 2
[% - Ev) + S(u + Bk.)] (B.8)
(B.10)
(B.9)

104
where Ek> ~ yJvp(k/^)‘2k'A2 + A£,. The 6 functions in the above equations can be
further evaluated as follows:
0(w ± Ek>)
E
9Ek
dk\
S fe',
sju2 - Ai,
| — —
vfW
0(Tu)
(B.ll)
^v2Fk'x2 + A2,
= Sí «**1
= M
Wffk')^2 - A'¿,
The sum of 2 delta-functions is therefore
(B12)
5
¿(o; + a/Jk’)
Q=±
M
uF(k')\/w2 - A£, ,
, M
vF( k'Jyi*;2 - A£,
M
wF(k')^/w2 - A£, (
> x- t s/w2 — A J,
> <5 fc, - a—
Sí \ Mk')
. X' &ik' - a ~ A*'
¿í r Mko
Sí \ uf(k)
*(-«)
|«M
rp Component of the Self-energy.
£0(k,w)
U'Q^ Jdk¡J
fdky E
J Q=±
Mk)| 2
n¡Q2\u;\ tt
vf(k) 2
riiQ2|a)| 7r
|fp(k)| 2
dk
Ea=±¿(fcl-a^£)
/ \ /_
/ - - \2
Íq|| + A:_lA;x - k'Lk'L\ + k?
^li'
¡ d(¡wE
- otyj* k’± + K2
* <7¡ + (fea. -
(B14)
The integral as it stands is not trivial due to the dependence of Ak> on q\\. Since
k' is close to k'. q\\ is small so we may write Ak' ~ Ak + v&q\\. The most singular
factor in the limit k —» 0 is the («2 + a2)-1 factor, so we are justified in neglecting
the <7|| dependence in the square root. The self-energy then takes the form:

105
£¡>'(k,u;) a:
niQ2n\uj\
2i>F(k)vV! - A2
/ rf J a—±
(B.15)
7r2n¡Q2 M
2 \A>2 - Ak
x E
\J(kíVF(k\\) - ay/ui2 - A£)2 + K2vF(k\\Y
Obviously the result, is valid only for u > A^; otherwise the Im part of E is zero.
On the Fermi surface, with k = kp, this reduces to
(B.16)
£"(1cf,í¿) — -7T 2n¡Q2
M
\Ju/2 - A£ yjul2 — A¿ + K2VF(k¡\)2
Note this is NOT the form we had earlier in Chapter 5,
Eóíkftw)» -r(k)|w|/^ - A2k,
(B-17)
(B.18)
unless u)2 - A£ « K.2vF(k\\)2.
T3 Component of The Self-energy. For the 73 component, proceeding similarly,
we get
E3(k,u>)
ntQM* [d¡J fdl.,VFk 1 ^«=±á(^
Mk)|- “ '
n¡Q2sgn ui 7r
vF(k)
1W
(q« + k±k± - k'Jc'^j + K2
i/*i,E
q« -
+ K¿
tt2tiíQ¿
sgno;
»=± \! (ArM^ll) - ay/ü2- A£)2 + fc2uF(A:||)2
Here the presence of the two contributions for the two different signs of k± is
crucial, as we see that on the Fermi surface, k± = 0, E3 vanishes.
. (B.19)

106
Ti component of the self-energy.
Ei(k,iv) ~ sgnu
* V“2 - Ak
x Y. , 1 =â– 
»=± \j(k±VF(k¡¡) - ctsJJ1- A£)2 + /t2uP(fc||)2
(B.20)
Normal State Limit. It is instructive to take the limit Aj, —> 0 of the above
expressions and attempt to recover Eqn. B.3. Eo and E3 become
So-
w2UiQ2
£
1
y/(k±vF(k\\) - <*M)2 + /t2np(fc||)2
^ 7r2n¿Q2sgnuj y
(B-21)
(B.22)
2 - a|u;|)2 + k2M*I|)2
Now the normal state self energy is the Nambu 11 component, i.e., £o + S3, which
goes to
£jv = lim £0 + £3 = -7T2n.iQ2 * (B.23)
Ak_*° y/(k±VF(k\\) - cj)2 + K?vF(k\\)2
Note odd frequency dependence of £3 is necessary to recover the correct result.
B.2 “Exponential” Potential
Consider
V(r)/V0 = e~Kr
which gives in momentum space (neglecting the periodicity)
14,k- = V0 J rdrd e^e*â„¢'**
= 2x14 [°° rdre-KrV0J0(qr)
Jo
2itk,Vo __ 2-kkVq
(1q2 + /C2)3/2 ” ((k - k')2 + «2)3/2 '
(B.24)
(B.25)
(B.26)

107
B.2.1 Normal State
Self energy in Born approximation:
SuíK2Vq
E"(k,u/) = -
(B.27)
• Special case k = kF, uj = 0:
E"(kp,u = 0)
3ir2n,iVg 1
(B.28)
16|t!jr(fc||)| K3’
Clearly the above form is more singular than Yukawa I potential, but still varies as
|t)f(k)|-1 around the Fermi surface for any nonzero k.
B.2.2 Superconducting State
If we are allowed to perform analogous operations as we did in the last step
and neglect the q dependence of the gap, we find
sS(k,«) * —y r i
uF(k)-/iv- - „=±
37r2rc¡K2V()2 |oi|
+ (*-L - aV^?) + K*
16t>F(fc|) sJuj2 - Ak
E
i
k a=±
(fcx-a^S)2 + «2
n s/2
„ 3ir2fii/i2Vo2 v-^
Ea(klW) * -ic^ysgnwE
37r2nift2V¡)2 Ak
(fci-a^P)2 + «2
5/2
£"(k,a>) — sgnw
16"f(*ll) ~ Ak !r,
E
i
(k^-CC^^Y + K2
5/2
(B.29)
(B.30)
(B.31)
B.3 Delta-function Type Potentials
What type of potential gives a self energy as in Chapter 5, e.g.,
C"(kF, w = 0) ~ r0(k) \ui\/yju2 - A2 (B.32)

108
It’s not a pure 2D delta function. If we simply assume |Vkk'|2 = V^ák.k', the self
energy in the superconducting state is
£¡¡(k,w) = niVtfi'ftG"(k,w) (B.33)
= n-^Im^ _ ^ _ A2 (B.34)
= -riiV^Mu-EiJ + Siu + E*)}, (B.35)
which is much more singular on the Fermi surface than Eqn. B.32. To recover the
smoother result we actually need to assume
I Vick'|2 = VtfVi (B-36)
and then we get (see Eqn. B.13):
sjj1 - AjpN
o=± \
mVg
uF(k')
M
(B.37)
47r|uF(k)| - Al'
It would be nice to ask what potential would result in real space from a ID delta
function like this. One way to regularize the delta function would be to write
vw =
k/tt
- fcf,)2 + K2
(B.38)
but of course we see immediately that this is not a function of |k — k'| and therefore
can’t correspond mathematically to a function of r if we reverse Fourier transform.
The problem is that to get the special form of the self-energy near the Fermi
surface, we only need the k-space form of the matrix element near the Fermi
surface, but this does not suffice to determine the form in real space everywhere.
We would need to guess a real space form which gives this result in k space for
a given band, which seems difficult. Therefore the only connection one can make

109
with the simple form in Chapter 5 is an approximate one. Not in the full forward
scattering limit n —* 0 but in an intermediate regime given by
vF
« K « 1
(B.39)
do we actually get exactly the anticipated result. It is interesting to note that
this works actually for any form of the scattering potential. The only thing that
changes is the To(k) describing the normal state scattering.

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BIOGRAPHICAL SKETCH
I was born in Yichang, one of the most beautiful cities in China, where the
world-famous Three-Gorge Dam is located. I went to primary school in 1983 when
I was 6 and graduated in 1989. I spent 3 year afterwards in No. 3 middle school of
Yichang, and then was matriculated into the No. 1 high school in 1992. In 1995, I
was proudly recruited as the first freshman into the Department of Physics, Nankai
University. In 1999, I received my bachelor’s degree in physics and was accepted
as a graduate student in the Department of Physics, University of Florida. In May
2005, I finished my doctorate study and was awarded the Ph.D. degree in physics.
117

I certify that I have read this study ahd that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor''3f Philosophy. / ,
JtlUAAk
Peter J. Hirschfeld, Chair
Professor of Physics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
David B. Tanner
Distinguished Professor of Physics
I certify that I have read this study arid that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Professor of Physics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor o| Philosophy.
Selman P. Hershfield
Professor of Physics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.

Stephen J. Pearton
Distinguished Professor of Materials
Science and Engineering
This dissertation was submitted to the Graduate Faculty of the Department of
Physics in the College of Liberal Arts and Sciences and to the Graduate School and
was accepted as partial fulfillment of the requirements for the degree of Doctor of
Philosophy.
May 2005
Dean, Graduate School

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