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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xiv, 117 leaves : ill. ; 29 cm.
Zhu, Lingyin
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Thesis (Ph. D.)--University of Florida, 2005.
Includes bibliographical references.
Statement of Responsibility:
by Lingyin Zhu.
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Copyright 2005


Lingyin Zhu

I dedicate this work to my loyal family.


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.


ACKNOWLEDGMENTS ........... ................... iv

LIST OF TABLES .. .............. .............. viii

LIST OF FIGURES ................................ ix

ABSTRACT ............................ ........ xiii


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


3.1 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


4.1 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

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


APPENDIX ................... . 99



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


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

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



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



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

Pseudogap Optimally doped

Underdoped Overdoped
Halsd r
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).

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


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

which we can attempt to understand inter-impurity correlations and macroscopic

disorder phenomena from a microscopic point of view.


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


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:
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


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


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

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 :

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)

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

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)
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]

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

Table 2-1: The imaginary part of self-energies due to non-magnetic impurities in

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)


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.


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

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


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 +0,


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

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


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


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)

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)


D~ = [1 VoG(0.) VoGo(O,w)]

+(-1 l.,[F.....R -)+G3(R,w)].


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+

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

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

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 + ( ).


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

',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


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)


(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)


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


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
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


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)

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.

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.


U., + ,. + ...

.' *

+ +

miad diagmm

Figure 3 8: Schematic plot of the self-energy diagram with many impurities.

(a) Zero-energy (b)


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


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


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

2x10' 001

Vo= .0. 002 0
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.

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 |
Vl(X) (a) (b) 2

0 I
05 I V r'- :o-0.5

-0.02 0 0.02 -0.02 0 0.02

=20 (c) (d)- 8

4- 62


-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

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


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

+ +

4 +

Fi-.,!, 3 14: LDOS for 2% concentration of impurities and E1,, < 10- t (20 eigen-

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)
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)
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


p(q) h~Im{V(q) G(O)(k + q, w)G(O)(k, w)}, (3.29)
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

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


6p(q,w) qi1,,,," q (3.32)
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


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

0 1 2 3 0 1 2 3 0 1 2 3


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

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.

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


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.


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


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.


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

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


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

in this section still remains the most intriguing puzzle in the understanding of the

phase diagram of 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].

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

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
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)
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 + ]


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,
= F.S0 (5.11)

with Nambu components E, defined as following;

Eo(k,w) = n, |V(k,k') |2 (5.12)

SE(k,w) = n, lV(k,k '2 2 (5.13)
k' k' ~ k'

,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 0
-1 -0.5 0 -1 -0. 0


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


0.3 N
I 'I 0

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


0.2 4


/ 0
0.1 *

0.05 -

-1 -0.5 0




-1 -05 0


-1 -0.5 0

0 -1 -0.5

Figure 5-3: The self-energy terms -Im :",, k Im i (k, ), and -Im E3(k,w) in
the -.i ..-...,.1,1. I.... state at T = 0 for k = kA (top) and kN (bottom), for N: 5
and 0.5 and the same band and scattering parameters as previously used. Here
Ak = Ao (cos, cos k)/2 with Ao = 0.21.

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