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Disorder and Correlations in Metallic Cuprates

Permanent Link: http://ufdc.ufl.edu/UFE0022100/00001

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Title: Disorder and Correlations in Metallic Cuprates
Physical Description: 1 online resource (110 p.)
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: correlations, cuprates, defect, disorder, high, magnetism, nmr, resistivity, slave, superconductivity
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: High temperature cuprates are compounds which, when doped, become superconducting below a relatively high critical temperature Tc of order 100K. They are already useful for a number of technological applications and finding ways to raise Tc further so that they may be usable at room temperature is one of the great challenges of condensed matter physics. These materials are extremely sensitive to disorder, which controls properties important for applications and is also a proper subject of fundamental scientific study since controlled addition of impurities can reveal the behavior of the pure state. In this thesis we study disorder in high temperature superconductors in the presence of the strong local Coulomb interactions which are known to be present, particularly in the so-called underdoped phases near the Mott transition. We propose that a variety of experiments, heretofore unexplained, may be understood by calculations of the magnetization induced by nominally nonmagnetic disorder due to the magnetic correlations which exist in the pure system. In particular, three related topics are treated: (1) the low-temperature upturns in the resistivity which occur in metallic cuprates, where we argue that the temperature dependence arises from the dynamical screening of the individual impurities due to strong correlations in the host; (2) the influence of defects on the broadening of the Nuclear Magnetic Resonance(NMR) line in recent experiments on ultrapure samples of ortho-II YBCO at low temperatures; (3)the single-impurity problem in comparison with NMR experiments, particularly the doping and temperature dependence of the NMR spectrum and the experimentally extracted Curie-Weiss temperature and magnetic correlation length, treated particularly with slave boson mean field theory. We conclude that effects of impurity-induced magnetism are virtually omnipresent in the underdoped cuprates, and will be important for the understanding of many experiments on these materials.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Hirschfeld, Peter J.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
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Permanent Link: http://ufdc.ufl.edu/UFE0022100/00001

Material Information

Title: Disorder and Correlations in Metallic Cuprates
Physical Description: 1 online resource (110 p.)
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: correlations, cuprates, defect, disorder, high, magnetism, nmr, resistivity, slave, superconductivity
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: High temperature cuprates are compounds which, when doped, become superconducting below a relatively high critical temperature Tc of order 100K. They are already useful for a number of technological applications and finding ways to raise Tc further so that they may be usable at room temperature is one of the great challenges of condensed matter physics. These materials are extremely sensitive to disorder, which controls properties important for applications and is also a proper subject of fundamental scientific study since controlled addition of impurities can reveal the behavior of the pure state. In this thesis we study disorder in high temperature superconductors in the presence of the strong local Coulomb interactions which are known to be present, particularly in the so-called underdoped phases near the Mott transition. We propose that a variety of experiments, heretofore unexplained, may be understood by calculations of the magnetization induced by nominally nonmagnetic disorder due to the magnetic correlations which exist in the pure system. In particular, three related topics are treated: (1) the low-temperature upturns in the resistivity which occur in metallic cuprates, where we argue that the temperature dependence arises from the dynamical screening of the individual impurities due to strong correlations in the host; (2) the influence of defects on the broadening of the Nuclear Magnetic Resonance(NMR) line in recent experiments on ultrapure samples of ortho-II YBCO at low temperatures; (3)the single-impurity problem in comparison with NMR experiments, particularly the doping and temperature dependence of the NMR spectrum and the experimentally extracted Curie-Weiss temperature and magnetic correlation length, treated particularly with slave boson mean field theory. We conclude that effects of impurity-induced magnetism are virtually omnipresent in the underdoped cuprates, and will be important for the understanding of many experiments on these materials.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Hirschfeld, Peter J.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022100:00001


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DISORDER AND CORRELATIONS IN METALLIC CUPRATES


By

WEI CHEN



















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2008





























2008 Wei C'!. i,


































To my family and my friends









ACKNOWLEDGMENTS

I am especially grateful for the instruction and collaboration of Tamara Nunner, Brian

Andersen, Ashot Melikyan, Lingyin Zhu, and Lex Kemper over the past several years,

particularly for assistance with numerical methods, and also for the fruitful discussions

with Greg Boyd, Siegfried Graser, Kevin Ingersent, Daniel Arenas and Vivek Mishra over

the general ideas about my research and the related physical problems. The collaboration

during 2005~2006 in Paris with Marc Gabay could not have been accomplished without

his generous hospitality. And finally, thanks to my adviser Peter Hirschfeld for alvb-- 7

being patient and providing -ii--. -I i. .s and directions for my research, these works would

not have been possible without his beautiful insights.









TABLE OF CONTENTS


page


ACKNOW LEDGMENTS .. . ...........................

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

A B ST R A C T . . . . . . . . . .

CHAPTER

1 INTRODUCTION TO CUPRATES .. ...................


General Idea of the Approach
Universal Phase Diagram ...
Antiferromagnetic State ...
Superconducting State .....
Pseudogap State .. ......
Normal State .. ........


1.7 Impurities and Induced Magnetic Moment
1.8 Proper Choice of a Microscopic Model .


2 DISORDER INDUCED RESISTIVITY UPTURNS IN METALLIC CUPRATES


Transport Properties in Disordered Cuprates .
Model Hamiltonian .. .............
Resistivity Upturns at Optimal Doping .....
Effect of Pseudogap on Resistivity .. .....
Conclusion . . . . . .


3 CHAIN DEFECTS IN YBCO6.5 ORTHO-II SYSTEM .. ........

3.1 63Cu NMR in YBCO C('! .i-Plane System .. ...........
3.2 Broadening of Low Temperature NMR Lines .. ...........
3.3 Explanation of High Temperature Satellite Peaks .. ........
3.4 C('! ,i-Plane Coupling Induced Correlation Effect .. ........
3.5 C conclusion . . . . . . . .

4 EFFECT OF DOPING ON IMPURITY INDUCED MAGNETIZATION
A STRONG COUPLING PERSPECTIVE ................


FROM


Slave Boson Mean Field Theory .. ............
Single Impurity Problem .. ................
Comparison with Semianalytical Approach .. .......
Induced Moments and Influence of Slave Boson Constraint
C conclusion . . . . . . .









5 CONCLUSION ...................... ............ 87

APPENDIX

A DERIVATION OF CURRENT OPERATOR ......... ........... 88

B DETAILS OF CURRENT-CURRENT CORRELATION FUNCTION ...... 89

C REMARK ON THE ABSOLUTE SCALE OF RESISTIVITY ......... 90

D BDG FORMALISM OF ORTHO-II YBCO6.5 SYSTEM ...... ....... 94

E DETAILS OF SLAVE BOSON MEAN FIELD APPROACH .......... 100

REFERENCES ...................... ................. 102

BIOGRAPHICAL SKETCH ................... ......... 110









LIST OF FIGURES


Figure page

1-1 Schematic of universal phase diagram of cuprates. Four phases appear with changing
doping and temperature: antiferromagnetic state(AF), superconducting state(SC),
pseudogap state(PG), and normal state(N). The nature of the transition between
N and PG states is controversial. .................. ..... 16

1-2 Summary of antiferromagnetism related experiments on LSCO. . ... 17

1-3 Summary of superconducting state properties important for this thesis. (a)Gap
versus angle relative to the Cu-O bond direction from ARPES shows a clear
d-wave feature[24]. (b)Real space STM\i image of single impurity embedded in
BSCCO shows 4-fold symmetry of impurity bound state[19]. (c)LDOS on(solid
line) and away(dotted line) from Zn impurity in BSCCO, where impurity bound
state forms inside the d-wave gap[19]. (d)Neutron diffraction pattern centers at
(r, 7) for LSCO under magnetic field, indicating magnetization in and around
vortex cores[22]. .................. ................ ..19

1-4 Summary of important pseudogap state properties. (a)8gY NMR in underdoped
YBCO shows reduction of spin susceptibility starting at temperature above T,[31].
(b)Gapless excitation in the first quartet of Brillouin zone has an arc shape(thick
line) that extends along the expected Fermi surface(dotted line) as temperature
i, 1i ...- -[21-]. (c)Reduction of DOS at Fermi surface above T, as measured by
STM in BSCCO[25]. (d)N. ii,-i signal above T, in various compounds indicates
vortex-like excitations above superconducting state in the phase diagram[41]. 21

1-5 Normal state resistivity in YBCO[44]. (a)Homogeneous ab plane resistivity versus
temperature at different doping. One sees a clear linear relation at high T but
deviation at lower temperature. (b)Using deviation from linear T(color scale) to
identify pseudogap temperature(open circles)[45]. ............. .. .. 22

1-6 Impurity related NMR experiments. (a)170 NMR in superconducting YBCO,
which shows defect concentration broadens the line[54]. (b)Correlation length
deduced from multinuclei method, indicating increasing correlation length as
underdoping and lowering temperature[53]. (c)7Li NMR Knight shift in YBCO
at underdoping(top) and optimal doping(bottom), together with their fit to the
Curie-Weiss form C/(T + ) (solid line)[56]. Notice that 0 changes dramatically
across T,. (d)7Li NMR Knight shift plotted with scale 1/T. The insert figure
shows screening temperature 0 versus doping, which abruptly increases around
optimal dolpii_-'].. .................. .............. .. 25









2-1 In-plane pab and c-axis resistivity p, in LSCO in the presence of impurities[63,
64]. (a)Underdoped region 6 = 0.08 shows divergence of pab at low temperature.
(b)Both Pab and p, show logT divergence when plotted on logarithmic scale.
(c)Phase diagram determined by mapping out dp/dT, which shows a metal-insulator
transition in underdope region. (d)Magnetoresistnance of pab at different temperatures.
. . . . . . . . . . . 3 3

2-2 In-plane disorder resistivity Pab of BSLCO in (a)optimal doping region, and (b)underdope
region[67]. Underdope sample clearly shows a more dramatic upturn in low T.
(c)Both pab and Hall constant RH shows logarithmic divergence at low T. (d)Negative
magnetoresistance of pb in the presence of applied field H//c, for underdope
sample 6 ~ 0.125 with 2.'' impurities. .............. ...... 34

2-3 In-plane resistivity pab of YBCO[71, 78]. (a)Impurity concentration dependence
of Pab for an underdoped sample, showing low temperature divergence on top
of the linear-T resistivity. The divergence is enhanced with increasing disorder.
Insert figure shows the logT form of the divergence. (b)Doping dependence of
Pab at a fixed impurity content, which shows more dramatic divergence as the
sample is underdoped. (c)Details of low temperature region in two different impurity
concentration z = 2.7'. (top) and z = 2.:;' (bottom). Magnetic field clearly
reduces T, and reveals the upturn. (d)Magnetoresistance at different temperature
for underdoped(top) and optimally doped(bottom) sample up to 60T. Underdoped
sample shows positive magnetoresistance while optimally doped sample remains
constant . . . . . . . . . 36

2-4 Comparing magnetization induced by a single nonmagnetic impurity studied
in this chapter for the normal state, and d-wave superconducting(dSC) state
studied in[89]. Real space magnetization pattern for (a)normal state, and (b)dSC
state, both at U = 1.75 and B = 0.01. One sees clearly that dSC state has
more pronounced nearest-neighbor site magnetization due to bound state formation,
and has smaller homogeneous magnetization due to opening of the gap st the
Fermi surface. These effects are shown at one plots (c)total magnetization of
the system Sz and (d)magnetic contrast A versus external field. . ... 42

2-5 Real space magnetization pattern at T = 0.03 and B = 0.001, with 2' impurities.
One sees clearly a transition of magnetization pattern as increasing U, where
we found its critical value to be around U ~ 1.75 above which the magnetization
is no longer characterized as localized around each impurity. . .... 44

2-6 Results at optimal doping. (a)Magnetization and (b)resistivity versus U at T
0.03, with 2' impurities. ............... ......... .. .. 44

2-7 Results at optimal doping. (a)Magnetization and (b)resistivity versus T at U
1.75, with '"' impurities. .................. ........... ..45

2-8 Results at optimal doping. (a)Magnetization and (b)resistivity versus B at T
0.03 and U = 1.75, with '' impurities. ................ ..... 46









2-9 Proposed phenomenoligical model for the reduction of DOS in pseudogap state.
(a)The normal state DOS of dispersion k we begin with, (b)the DOS for proposed
dispersion Ek in Eq. (2-11) with if -0.6 and a s-wave gap Ak A 0.2,
and (c)the extended hopping model after Fourier transform Eq. (2-12) ..... 48

2-10 Comparison of models with(Extended) and without(\. i ii, I!) reduction of DOS
by applying extended hopping Eq. (2-12). (a)Magnetization and (b)resistivity
versus T both at optimal doping, with U = 1.75 and "' impurities. . 49

2-11 Comparing (a)magnetization and (b)magnetoresistance between extended hopping
model Eq. (2-12) and the normal model both at optimal doping, T = 0.03,
U = 1.75, with '"' impurities. .................. ........ .. 51

3-1 A schematic description of the doping process in YBCO by changing the oxygen
content on the chain 1l'-- r[52]. The undoped parent compound YBCO6 contains
Cu+ on the chain li- -r, and the CuO2 plane has effectively one electron per Cu
site. If (.::;i--, i-; are introduced into the chain l-.v-r but their content is sufficiently
low, then the sparse 02- only changes its two .,.i ,i:ent Cu+ to Cu2+. When a
segment of at least two 0's is formed, the excess charge will be transferred to
the CuO2 plane, resulting in the hole doping. A full chain can therefore dope
the CuO2 plane in a more efficient way than a random O distribution. . 53

3-2 Crystal structure of ortho-II YBCO6.5 and related parent compound YBCO6
and YBCO7, with precise definition of chain and planar Cu[94]. . ... 54

3-3 (a)Full NMR spectrum of Ortho-II at 60K in the presence of external field Ho//a,
where the frequency is flexed at 75.75MHz[94]. Top panel: repetition time Trep 250msec,
middle panel: Trp-25msec, lower panel: the difference. Comparison with YBCO6
and YBCO7 confirms the association between lines and the Cu species: A-Cu(2F),
B-Cu(1E), C-Cu(2E), D-Cu(1F). (b)STM image on the chain l1-v- r of YBCO,
which shows oscillation of DOS along chain direction[96]. Notice that the sample
is not Ortho-II and has relatively short segments of full chains(about 10 sites). 55

3-4 Schemetic of broadening of 89Y NMR line in YBa2NiO5 by magnetization induced
around impurities[52, 99, 100]. The right panel shows the amplitude of magnetization
induced on several sites near the impurity, which results in satellite signals near
the main line. A significant amount of such induced magnetization eventually
smears out the spectrum and broadens the line(the insert of left panel). . 56

3-5 Schematic of the hopping part of model Hamiltonian Eq. (3-2). . ... 60

3-6 Numerics in an isolated 1D chain, equivalent to Hamiltonian Hchain + H1)
in an external field 7.7T at 50K. Each curve is normalized such that the area
underneath is conserved. ............... ........... .. 62









3-7 Comparison of (a)experimental 63CU(1F) NMR line of ortho-II YBCO6.5 at external
field 7.7T along a direction, with three models studied: (b)isolated 1D chain
with random impurities Hhain + H (c)isolated 1D chain with a section of
.,I.i ient Cu(1F) erased Hchain + H ,2 (d)chain-plane coupled system with a
section of Cu(1F) eliminated Hpiane+ Hchain +Hinter+f Hp. Correlation strength
Uc 1.2 in these models. ............... ........... .. 65

3-8 Demonstration of anisotropic magnetization due to chain-plane coupling, in which
we plot normalized magnetization si induced by substitution of a Cu(1F) by a
point defect: (a)on the chain, (b)on the plane along chain direction, (c)on the
plane along direction perpendicular to the chain. The solid line is a fit to spherical
Bessel function si oc siKo(i/K)/Ko(1/0). Size of the plane is 81x24, at temperature
70K. The real space pattern of si on the plane is given in (d) and (e), where
one can see the extension of magnetization along chain direction, coinciding with
(f)the magnetization induced on the chain lIV.r. ............... .. 69

3-9 (a)The combined spectrum of Cu(1F), Cu(2F), and Cu(2E) lines, together with
(b)the linear relation found between broadening of chain and planar lines. In
comparison, the experimental data of linewidth correlation is given in (c) for
Cu(2F) and (d) for Cu(2E)[95]. .................. .... 70

4-1 The phase diagram of slave boson mean field theory for homogeneous cuprates.
The three temperature scales represent the Bose-Einstein condensation temperature(TBEC),
the onset of spin liquid order(TRvB), and the spin gap order(Tsc). Under these
interpretation, we identify four phases in the phase diagram: (I)Fermi liquid,
(II)spin gap phase, (III)superconducting state, and (IV)the strange metal phase.
In this chapter we focus on the impurity problem in the phase (IV) where we
have nonzero spin liquid order, but the system is above Bose-Einstein condensation
temperature and has no pair correlations. .................. .... 72

4-2 Spinon effective bandwidth tf for doping 6 = 0.3 in semianalytical approach of
C, '1' et al[125] near the impurity. .................. ..... 78

4-3 Comparison of spinon effective bandwidth tf between semianalytical approach
of Gabay et al[125] and present exact diagonalization. . . 79

4-4 Summary of semianalytical approach of C i1 i, et al[125] with external field 7
Tesla. (a)Magnitude of normalized magnetization s, = (m mo)/mo near the
impurity at T = 25K, overdope 6 = 0.3. Solid line: fit to Isl oc siKo(r/f)/Ko(1/0)
for ( = 3. (b)T-dependence of normalized magnetization at nn site of impurity
at overdoping 6 = 0.3. (c)Effective moment C and screening temperature fit
to the Curie-Weiss form C/(T+) for si, near overdoped region. (d)T-dependence
of screening length at different dopings, fit to the form in (a). . ... 82









4-5 Results of exact diagonalization at 7 Tesla. (a)T-dependence of nn site normalized
magnetization sl in different dopings. (b)Effective moment C and screening temperature
o in the Curie-Weiss fit of sl. (c)Data by Ouazi et al showing T-dependence of
and magnetization versus distance r away from impurity(insert). (d)i v.s. T
from optimal to overdoping by exact diagonalization. . . ...... 83

4-6 Density pattern by exact diagonalization at T = 100K and 6 = 0.15. (a)Spinon
density, (c)holon density, (e)magnetization plot, and their corresponding cut
through impurity site along (1,0,0) direction in (b), (d), (f). Spinon and holon
oscillation shows the same healing length kf associated with the sape of spinon
fermi surface, while magnetization oscillates under which is related to effective
coupling J/tf . . . . . .. . . . 884









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

DISORDER AND CORRELATIONS IN METALLIC CUPRATES

By

Wei Chen

May 2008

C('! i: Peter J. Hirschfeld
Major: Physics

High temperature cuprates are compounds which, when doped, become superconducting

below a relatively high critical temperature T, of order 100K. They are already useful for

a number of technological applications, and finding v- -v- to raise T, further so that they

may be usable at room temperature is one of the great challenges of condensed matter

physics. These materials are extremely sensitive to disorder, which controls properties

important for applications and is also a proper subject of fundamental scientific study,

since controlled addition of impurities can reveal the behavior of the pure state. In this

thesis we study disorder in high temperature superconductors in the presence of the strong

local Coulomb interactions which are known to be present, particularly in the so-called

underdoped phases near the Mott transition.

We propose that a variety of experiments, heretofore unexplained, may be understood

by calculations of the magnetization induced by nominally nonmagnetic disorder due

to the magnetic correlations which exist in the pure system. In particular, three related

topics are treated: (l)the low-temperature upturns in the resistivity which occur in

metallic cuprates, where we argue that the temperature dependence arises from the

dynamical screening of the individual impurities due to strong correlations in the host;

(2)the influence of defects on the broadening of the Nuclear Magnetic Resonance(NMR)

line in recent experiments on ultrapure samples of ortho-II YBa2Cu3O7_6(YBCO7 6)

at low temperatures; (3)the single-impurity problem in comparison with the local









susceptibility determined by Nuclear Magnetic Resonance(NMR) experiments on

disordered YBCO, particularly the doping and temperature dependence of the NMR

spectrum and the experimentally extracted Curie-Weiss temperature and magnetic

correlation length, treated within the framework of slave boson mean field theory.

We conclude that effects of impurity-induced magnetism are virtually omnipresent

in the underdoped cuprates, and will be important for the understanding of many

experiments on these materials.









CHAPTER 1
INTRODUCTION TO CUPRATES

1.1 General Idea of the Approach

The discovery of high T, cuprate materials in the late 80's was a milestone in the

search for novel superconductors[1]. Not only does the transition temperature exceed the

boiling temperature of nitrogen, making a variety of applications possible, but in addition,

the unusual electronic properties have challenged condensed matter physicists over the

past two decades. The key to understanding the mechanism behind superconductivity

relies on the proper description of the CuO2 plane commonly shared in all high T, cuprate

materials. Although its two-dimensional structure seems remarkably simple, no consensus

on a microscopic description or the origin of unusually high transition temperatures is

available at present.

The complicated phase diagram of the cuprates indicates the importance of the

interplay between different correlations coexisting in the CuO plane, and the competition

between them becomes especially manifest when external perturbations are present. This

is because when the order parameter that characterizes a specific phase is diminished due

to perturbations, the competing correlations have the chance to reveal their importance.

In this report we particularly focus on the two most significant instabilities which exist

in all cuprates, namely superconductivity and antiferromagnetism, and discuss the

thermodynamic anomalies induced, particularly by disorder. We will show that anomalies

in a variety of experiments can be explained by a simple mean field theory that properly

includes both superconducting and magnetic instabilities, in which the effect of disorder

is to induce localized magnetic moments that in turn affect the thermodynamic and

transport observables. In the following sections, we briefly introduce the general properties

of each phase obtained by changing doping and temperature, and highlight the important

effects of disorder that are relevant to the study in the later chapters.









1.2 Universal Phase Diagram

The two dimensional nature of the CuO2 plane allows analysis by various spectroscopic

measurements that were not applicable in conventional 3D materials. For example, Angle

Resolved Photoemission Spectroscopy(ARPES) based on the photoelectric effect could be

used to determine the spectral function A(k, w) of quasiparticles, and map out important

excitations in momentum space. Scanning Tunneling Microscopy(STM) which images ionic

structure and gives access to local density of electronic states and effects due to disorder,

vortices, or other inhomogeneity, is a powerful tool to measure local electronic structure.

Different phases, according to their electronic and magnetic properties, are therefore

identified by spectroscopic measurements, and display a variety of thermodynamic

characteristics that belong to different universality classes. Phase transitions are usually

obtained upon doping the CuO2 plane, and the order parameter that characterizes each

phase is measured via external fields that couple to these local or global order parameters.

At higher temperature, thermal fluctuations are expected to destroy the order, and

hence well defined phase transitions may be observed. In some cases, like the so-called

1'-' I'. '1' phase, no thermodynamic discontinuity indicative of a true phase transition

has yet been discovered. Instead, a crossover, whose nature is presently poorly understood,

takes place.

In general, the donation of electrons or holes by other ions in the lattice changes the

effective doping of CuO2 plane, which controls the charge carrier density of cuprates.

For example, in the YBa2Cu307_6(YBCO) family, the CuO2 plane is coupled to

one-dimensional CuO chains, whose structure contains alternating Cu and 0. If each

chain O site is filled, the so called YBCO7 structure, then the valence of the Cu sites on

the plane are found to be 2+, resulting in effectively slightly overdoped CuO plane, where

T, is close to its maximum. Removing the chain O changes the valence of both chain and

planar Cu, which gives a different charge carrier density on the plane. Samples belonging

to the La2-aSrCu04(LSCO) family, for example, are doped by replacing the La by Sr.













-300K -

N


PG

AF

SC

-0.08 -0.3 Doping

Figure 1-1. Schematic of universal phase diagram of cuprates. Four phases appear with
changing doping and temperature: antiferromagnetic state(AF),
superconducting state(SC), pseudogap state(PG), and normal state(N). The
nature of the transition between N and PG states is controversial.


The effective hole doping on the plane could be calculated via bond-valence sum or other

techniques, and despite the variety of cuprates, the dependence of transition temperature

on doping is fairly universal among different compounds. Half-filled "parent compounds"

typically display long range magnetic order, which is destroyed with increasing doping.

Superconductivity occupies a dome shape in the moderate doping and lower temperature

part of the phase diagram, with a metallic state which exhibits exotic properties outside

of the superconducting dome. Figure 1-1 gives a schematic of the phase diagram which is

universal to all the cuprate materials. We now discuss each phase in some detail, together

with open questions which await future research.

1.3 Antiferromagnetic State

At half-filling, when each Cu "site" is occupied by exactly one electron, the sample

exhibits long range antiferromagnetic order, which can be well understood within the

two dimensional Heisenberg model[2, 3]. Although the exact solution to this model

remains unknown at present, various numerical and semiclassical approaches confirm













S0.03
-J
r
" 0.02

o
a) 0.01
e"


0

(a)




o









(c)

Figure 1-2.
? '













Figure 1-2.


.0 0 I I I I I I I I I I I , I
100 200 300 400 500 600
Temperature (K)


Temperature (K)


\ 290 K > TN S- --
2E IC
Si,.* j | E = 0.15 eV
ch = o0.1 eV
20- 740 i q
1210 cmWl =FWHM 0 700 -

\
660 A
Z 620
OP1 2000 4 00 0Ci '
0
0 2000 4000 6000 8000 -0.4 -0.3 -0.2 -0.1 0 0.2 0.1 0.3 0.4
Energy Shift (cm"') (d) q (A4)

Summary of antiferromagnetism related experiments on LSCO. (a)Inverse of
antiferromagnetic correlation length versus temperature shows Neel
temperature TN ~ 195K for LSCO[3]. The solid line is fit by simulation of
spin-i Heisenberg model. (b)Susceptibility versus temperature, where the
divergence indicates Neel temperature TN 245K[2]. The solid and dotted
lines are fit to spin-i Heisenberg model. (c)Raman scattering shows a high
frequency peak which is attributed to scattering from magnon pairs with
opposite momenta[5, 6]. (d)Neutron scattering shows excitations in momentum
space, together with the fit to spin-wave theory which gives spin-wave velocity
he ~ 0.78eV[7].


the existence of finite ordered -1 I--'. red magnetic moment, but strong fluctuations due

to low dimensionality suppress the -1 .--I red magnetic moment from its classical value.

The long range order is destroyed at a temperature scale TN ~ 250K for the undoped

parent compound, and the fit to neutron scattering data gives a reasonable estimate









of the exchange coupling J ~ 100meV within the framework of spin-wave theory and

other possible approaches to the excitation spectrum[4-7]. The long range order is also

rapidly destroyed by itinerant holes introduced upon doping. At low temperature, a

"spin g! i-- phase is found as doping exceeds about 6 ~ 0.05, in which spins fluctuate in

real space and produce no effective long range order, but the correlation length falls off

with a power law dependence. The existence of antiferromagnetic and spin-glass phases

indicates the importance of electron-electron interaction, since magnetic ordering results

from the effective coupling due to strong Coulomb repulsion. Approaching the unified

theory from its half-filling parent compound inspires a whole class of theories based on

the Mott-transition and the superexchange mechanism, where correlations gain their

importance as doping is reduced[8].

1.4 Superconducting State

In the doping range about 0.08 < 6 < 0.3, the celebrated superconducting phase

gradually appears with highest transition temperature T, of order 100K at optimal

doping 6 ~ 0.15, and occupies a dome shape on the phase diagram. The existence of a

Cooper pair condensate that characterizes superconductivity has been proved by different

exp( iii. iii-['1], but important features that make cuprates different from standard BCS

theory have also been observed. The first significant difference is the d-wave symmetry of

the superconducting gap in momentum space. This finite range pairing can be interpreted

as an effect of the strong Coulomb interaction prohibiting double occupancy per site;

hence, superconductivity is actually a secondary effect inherited from the strong Coulomb

repulsion[8]. Instead of a fully gapped Fermi surface as in a conventional superconductor,

cuprates have finite gapless nodal excitations in momentum space along the (7, r)

directions, and fully gapped antinodes(gap maximum) around the (T, 0) directions, which

results in a linear density of states near the chemical potential. The critical temperature

T, is found to scale roughly with superfluid density n,, which increases and then decreases

with doping, while the superconducting gap decreases monotonically with doping, leading














































-V






0 -100 0 100 20(
Sample bias (mV)


47 A08 0.49 0.50 0.51 0.52 0.53
Wavevector (reciprocal lattice units)


Figure 1-3.


Summary of superconducting state properties important for this thesis.
(a)Gap versus angle relative to the Cu-O bond direction from ARPES shows a
clear d-wave feature[24]. (b)Real space STM image of single impurity
embedded in BSCCO shows 4-fold symmetry of impurity bound state[19].
(c)LDOS on(solid line) and away(dotted line) from Zn impurity in BSCCO,
where impurity bound state forms inside the d-wave gap[19]. (d)Neutron
diffraction pattern centers at (7, 7) for LSCO under magnetic field, indicating
magnetization in and around vortex cores[22].


to the proposal that there exist two distinct energy scales in cuprates[10-13], in contrast

to conventional superconductors where T~ and gap are proportional.


30-

25

20
E
1 15-

10-

5-


8k (degrees)


2.5-



C 2.0-
0
C

O 1.5
4--



co
1.


! 0.5-
C









d-wave symmetry also alters the response of the system in the presence of defects.

The thermodynamic properties of conventional superconductors are unchanged under the

influence of dilute nonmagnetic impurities, an effect known as Anderson's theorem[14].

In addition, impurities in these systems do not produce bound states. By contrast,

if the gap exhibits d-wave symmetry, bound states with energies inside the gap are

induced. The bound state wave functions then interfere and eventually smear out the

gap and reduce T,[15-17]. The exotic response to nonmagnetic impurities, plus their

interplay with magnetic correlations will be the main point of this thesis. Secondly,

cuprates have coherence lengths typically about 4 ~ 6 lattice constants, which makes

them very strong type-II superconductors. The short coherence length indicates the

importance of spatial inhomogeneity, in which one expects a strong local modulation of

order parameters by various sources like vortices, impurities, or grain boundaries. Any of

these effects can destroy the local paired state and reveal something about the underlying

correlations[18-23]. The critical field Hc2 ~ 100T is quite beyond current experimental

limits, and hence the extreme high magnetic field region is still a challenge for both

experimentalists and theorists in the near future.

1.5 Pseudogap State

Between the superconducting dome and the antiferromagnetic phase, at temperature

higher than the spin glass, a phase with exotic density of states and unusual Fermi surface

has attracted enormous attention from both theorists and experimentalists. The nature

of this "I' 'i" -, i '' phase is still under debate. It has various unusual properties that

contradict common understanding of conventional ordered or disordered condensed matter

physics. For example, a reduction of the density of states which characterizes electronic

excitations shows up at the Fermi level, -i-'-i. -1-; a similarity to the superconducting

state where density of states in fully suppressed at this energy[25]. A closer analysis in

momentum space using ARPES shows a section of gapless excitations which extends

from the nodal position to antinodes along the Fermi surface[26-30]. This contradicts,









I I I I I I


160


La, SrxCuO0


140
120

100


0.00 0.05 0.10 0.15 0.20 0.25
Sr content x


Figure 1-4.


Summary of important pseudogap state properties. (a)8gY NMR in
underdoped YBCO shows reduction of spin susceptibility starting at
temperature above Tc[31]. (b)Gapless excitation in the first quartet of
Brillouin zone has an arc shape(thick line) that extends along the expected
Fermi surface(dotted line) as temperature ii, i. -[2;]. (c)Reduction of DOS
at Fermi surface above T, as measured by STM:\ in BSCCO[25]. (d)N. i,-i
signal above T, in various compounds indicates vortex-like excitations above
superconducting state in the phase diagram[41].


e.g., Luttinger's theorem, which requires a closed Fermi surface about some part in the

Brillouin zone. The length of this unique 1:, i i:1 arc" increases as temperature increases,

and the complete Fermi surface is recovered only at temperatures above T*. The first


M Y











XF
-- -- -


-200 -100 0 100 200
Vs", [mV]


0.30










indication of pseudogap behavior was identified in NMR which showed the reduction of

spin susceptibility at temperatures lower than T*[31-34].

One early ir-'-i -1 i, ,1 about the origin of pseudogap behavior is that it represents a

gap for the breaking of Cooper pairs, which exist but are not phase-coherent in the region

Tc < T < T*. Theories of this type are referred to "precursor paiiina t]i, ..i l [, 35-38].

Some evidence of the relation of the pseudogap to pairing comes from the observed N. 1 ,-i

effect that sr-'--- -1- the existence of vortex-like excitations above Tc[39-41]. Other theories

- Ir-.- that it arises from a competing order of some sort[42]. Those favoring this picture

point out that it is well known that magnetic correlations are enhanced as the system is

underdoped, and interpret the pseudogap as the onset of spin density wave(SDW) type

order or some other magnetic order parameter. However, it is not clear in these theories

why no thermodynamic signature of the transition has yet been observed.

1.6 Normal State
10 6350
6.68 645 300-
4- pa .
b -250
2, / 6.58
S200
S6.90 200 0
5 0 100200 300/ 6,
(aT(K) /150 AF
6.7
E
6.8 / 100
6.90 50 SC

S(a; 23 0 0.05 0.1 0.15 0.2 0.25
0 100 200 300
(a) Temperature(K) (b) h

Figure 1-5. Normal state resistivity in YBCO[44]. (a)Homogeneous ab plane resistivity
versus temperature at different doping. One sees a clear linear relation at high
T but deviation at lower temperature.(b)Using deviation from linear T(color
scale) to identify pseudogap temperature(open circles) [45].


Conventional Fermi liquid behavior is thought to be recovered at low temperature

as doping exceeds roughly 6 ~ 0.3, which extends up to high temperatures and

occupies the T > T* part of the phase diagram. As temperature is raised above Te,









resistivity versus temperature at optimal doping shows a linear behavior with different

power law corrections above either side of the superconducting d, i.. [! ;, 44]. Such a

linear-T dependence over as much as three decades of temperature is not consistent with

normal metal physics. Although linear-T behaviors are frequently observed over limited

temperature ranges, they are usually found in the classical limit of electron-phonon

scattering well above the Debye temperature. However, the resistivity at optimal doping

in the cuprates shows no indication of any feature at the Debye temperature at all.

A phenomenology to describe the linear-T behavior goes under the name of marginal

Fermi liquid theory, where temperature is assumed to be the only important energy

scale at low frequency[46, 47]. The deviation from linear relation on the underdoped

side coincides roughly with T*, which has been used as another way to identify the

pseudogap temperature. The resistivity on the overdoped side gradually evolves into a

T2 power law, which is well understood within Landau's Fermi liquid theory. Besides

the linear resistivity, disorder pl. i' an important role in identifying the thermodynamic

properties especially at low temperatures. Experimentally, the change in the resistivity

due to disorder is measured by introducing planar defects and sometimes in addition

an applied magnetic field, which suppresses superconductivity and is assumed to reveal

the underlying normal state resistivity. Such measurements confirm Matheissen's rule

at high temperature, whereby resistivity due to disorders is additive with respect to the

ri--li. i us linear term.

In addition, as temperature is lowered at optimal doping, the resistivity decreases

until the superconducting transition. Either disorder or the magnetic field suppresses Tc,

and in clean samples resistivity versus temperature satisfies a power law down to very low

temperatures. However, if the system is underdoped or highly disordered, the resistivity

begins to increase with decreasing temperature below a temperature of order 30K, which

has been taken as a signature of metal-insulator transition. These resistivity upturns have

been interpreted within several different pictures, and will be studied in the present work.









1.7 Impurities and Induced Magnetic Moment

For intrinsically inhomogeneous cuprates like the Bi2Sr2CaCu2O,(BSCCO) family,

defects like out of plane interstitial O or other ions have long range, nonnegligible effects

on the CuO2, and could result in spatial modulation of superfluid density or other

observables[18]. Samples belonging to this category would be very useful in order to

understand the relation between length scales introduced by disorder and the global

variation of i' i ii:.- or demonstrate change of the electronic environment by defects

if a local measurement is available[48, 49]. Although it has been proposed recently

that specific v--iv of introducing out-of-plane disorder could possibly enhance Tc[50],

one generally expects that planar defects destroy superconductivity. For intrinsically

homogeneous samples like the YBCO family, impurities have been introduced in a

controllable way via different sources and techniques. The planar defects, which change

the local environments of the CuO2 plane, usually result from the replacement of planar

Cu with ions of the same or different valences, and may be considered as bare magnetic

or nonmagnetic scattering centers depending on the nature of the impurities. The other

common way of producing controllable nonmagnetic impurities is by electron irradiation,

which knocks out the planar O and produces topological defects on the square lattice.

The response of the system to defects produced by electron irradiation is found to be

qualitatively the same as replacing planar Cu by impurities with the same valence but a

higher local chemical potential. The most common example is Zn, which produces nearly

unitary scattering on the impurity site. The fact that Zn and irradiation-induced defects

produce nearly the same NMR signature sIlr. 1 -i that a good theoretical model for a

planar defect is simply a missing site on a square lattice. The theoretical and experimental

aspects of impurities in cuprates have recently been reviewed by Balatsky et al[51] and by

Alloul et al[52].

In the presence of a magnetic field, the T-dependent shift and broadening of NMR

lines at low temperatures indicate the formation of magnetic moments[53]. The correlation





















40350 40400 40450 40500 40550
v (kHz)


T (K)


S-SO



r (lattie units)

7

80 100 120 140 160
Temperature (K)


0.002 0.004 0.006 0.008 0.010 0.012
1/T (K )


Figure 1-6.


Impurity related NMR experiments. (a)170 NMR in superconducting YBCO,
which shows defect concentration broadens the line[54]. (b)Correlation length
deduced from multinuclei method, indicating increasing correlation length as
underdoping and lowering temperature[53]. (c)7Li NMR Knight shift in YBCO
at underdoping(top) and optimal doping(bottom), together with their fit to
the Curie-Weiss form C/(T + ) (solid line)[56]. Notice that 0 changes
dramatically across T,. (d)7Li NMR Knight shift plotted with scale 1/T. The
insert figure shows screening temperature versus doping, which abruptly
increases around optimal doping[55].


between broadening and impurity concentration further implies the close relation between

low temperature magnetic moment formation and the local defects[54]. The correlation

length deduced from 70 NMR confirms that magnetization is indeed induced locally

around the defects, and gradually decreases with distance from the defect[53]. Knight

shift data from 'Li NMR in particular provides information on the local environment

very close to the defects. Due to the different valence of Li, an extra electron is produced









around the Li substitution. However the 8"Y NMR -, i.-.- -I the same induced moment

and local environment between Li and other impurities that have the same valence, for

example replacing planar Cu by Zn. The advantage of using Li is that the nucleus itself

is NMR-active, and the magnitude of magnetization on the nearest-neighbor(nn) sites of

impurities can be extracted from the Knight shift data and hyperfine coupling between

Li and its nn sites[55, 56]. The susceptibility extracted is found to fit a Curie-Weiss law

C/(T + 0) both above and below T, indicating the presence of a magnetic moment due

to the strong correlated nature of the metallic background. The induced magnetization is

also found to be enhanced in the underdoped region of the phase diagram, -i-i :. -1 ii-; its

intimate relation with antiferromagnetic correlations in the undoped parent compound[56].

Below T, the reduction of density of states near the Fermi surface promotes bound state

formation and hence increases both density modulation and an induced moment, although

the background homogeneous magnetization of the host is reduced by the formation of

singlet pairs. We therefore expect the Curie-Weiss fit across T, to undergo a dramatic

change, which is indeed observed in 7Li NMR[56].

The electron density is also modulated around planar defects, an effect which

is well understood within the context of a quasi-2D Friedel oscillation[57]. When a

magnetic field is present, the splitting between up and down spin density of states

results in an induced magnetization around the impurities. However, the magnetization

resulting from Friedel oscillation of the density of states in a free electron gas is roughly

temperature independent, which can not account for the enhancement of induced

moment as temperature is lowered. Furthermore, its magnetization is much smaller than

observed. Motivated by the NMR experiments in YBCO, one reasonable proposal is that

magnetic correlations coexist with the superconducting instability in the complex phase

diagram, with its strength increasing toward half-filling. In the presence of impurities,

superconductivity is suppressed due to its d-wave symmetry, and magnetization is induced

as a result of magnetic correlations. In this thesis, we show that such an interplay between









multiple instabilities can be formulated within simple mean field theories, in which the

induced moments and their influence on electronic transport and NMR experiments can

be explained. The main goal of the report will be comparing thermodynamic quantities

as obtained by diagonalization of the mean field Hamiltonian in real space with these

measurements, which shows agreement that is sufficient to prove the coexistence of

superconductivity and magnetic correlations in cuprates, and i.-.i-.- -I; that many

experiments in the underdoped samples are strongly influenced by disorder-induced

magnetization.

1.8 Proper Choice of a Microscopic Model

Since the discovery of cuprate superconductors, a significant effort has been spent on

finding a proper microscopic description that can generate the complex phase diagram.

The antiferromagnetism in the undoped parent compound obviously provides some

clue for the phases obtained upon doping. Although the long range magnetization in

undoped system is well described by a Heisenberg model, it is not obvious that a model

that includes only Cu degrees of freedom can account for systems away from half-filling,

especially since various spectroscopic experiments -'.-.: -1 that holes introduced by

doping actually reside on O sites[58-60]. Zhang and Rice proposed that, when the O

holes are present, they pair up with the Cu holes and become mobile through the Cu-O

bond wave function overlap[61]. The basic idea is that one can construct an operator 0i,

corresponding to creating a hole on the square of four O sites centered on a Cu site, and

forms either a singlet(-) or triplet(+) with the hole on the enclosed Cu2+ site(note that

half-filled cuprates have one hole per Cu site)

1(11)
v2

By calculating energies in second order perturbation theory, the singlet state b- is

found to be the ground state. When wave function overlap between 0 and Cu holes are









considered


H' V El dpl~ + h.c. (1 2)
ilc
an effective hopping between nearest neighbor sites is induced


Ht=- tj (Q di,) t (1-3)
ijc7

Notice that a combination of i,-di, is considered because when a singlet pair hops from

site i to j, the hole on Cu site j is destroyed. The hopping amplitude can be calculated

via second order perturbation theory


i d I4 H' w)(w H'|( di,)/t) /AE (1-4)

where w runs over all possible intermediate states. The hopping Hamiltonian Eq. (1-3) of

S-di, is equivalent to


Ht tj (t ni_,) dj, (t nj_,) ,(1-5)
tj'

because is equivalent to an empty state of Cu hole at site j.

Eq. (1-5) is an effective one band model that describes hopping on a square lattice

formed by Cu sites, with the factor 1 ni_, that projects out double occupancy. This

motivates the adequacy of an effective one band model that can capture the basic features

of electronic correlations in cuprates, with strong coupling U > t that prevents electrons

from sitting on the same site. From a strong coupling perspective, higher order terms may

be shown to induce spin-spin interactions, leading to the so-called t-J model, which treats

the hopping term as a perturbation[62]. The t-J model is therefore the strong coupling

limit of the Hubbard model used to understand the Mott transition in a simple way. To

demonstrate the basic idea of this effective spin-spin interaction, we start from half-filling,









and separate the hopping and interaction Hamiltonian


Ho -tij ccj, ,
ij(c
H, -,Un. i ,


(1-6)


where c = di,(l ni_). The eigenstates of HI') = EI') can be used to expand I') in

terms of eigenvectors of H1


1
- H0Ho I)
E H1

S Hol|) + |1a) (I
E H, E El


(1-7)


where HIa) = Elia) and P = 1 Y, I a)(a We can further introduce a new basis I ,)

which satisfies the following equation


P
I'a) I-=a + -- Ho ,a)
E H,


(1-8)


and expand I4) as


Now expand this new basis


4T) Yaaa),
a
(a|Hol )
an=
E E-


to first order in |a), we get


P 1
I'a) I a)+ Ho1|a) ~ a) -H Ha) ,
E H1 U


where we've used (P3HoIa)

matrix equation


0 at half filling. After substitution, the an's satisfy the


(E Ei)aa


(111)


(a H, a')a ,


(1-9)


(1-10)









which is equivalent to substituting the hopping by H = Hg/U, where H' is given by

2t-
Ho Y S, (1-12)
ij i

after considering Pauli exclusion principle. When mobile holes are present, an extra kinetic

term has to be considered in the above half-filling construction, resulting in the t-J model

that contains electron hopping and the exchange interaction between them


H = -t cc, + JS (1 -13)
ij(7 ij

where Jy = 2t2 /IUl with the constraint


n, -1, 0. (1-14)

The non-double occupancy constraint ensures the strong coupling limit of our starting

point, and is especially important toward half-filling.

In summary, the idea of Zhang-Rice singlet i.-.ii-. -1 the adequancy of a one-band

model which contains only Cu degrees of freedom for the low energy physics, and

the second order perturbation theory bridges the Hubbard and t-J model which are

both suitable to describe the CuO plane with strong correlations, with the non-double

occupancy constraint that gains importance as the system is underdoped. A variety of

techniques have been applied to solve either the Hubbard or t J model with or without

the constraint being considered, and give a qualitatively correct description of the phase

diagram and the pseudogap phenomenon. In this report we focus on two particular

approaches: Hartree-Fock-Gorkov mean field theory of the Hubbard model, and the

slave boson mean field theory for the t-J model. The first approach will be particularly

applicable for systems close to optimal doping, where weak coupling theory gives a

correct description for the low energy excitations above and below Te, and the non-double

occupancy constraint is omitted. The advantage of this kind of weak coupling approach

is that response to external perturbation can be treated in a well defined framework,









although the Mott transition toward half-filling cannot be described properly. For studying

the doping dependence of one impurity problem, we adopt the second approach, and the

results of imposing and omitting the constraint are compared. In the rest of the report

we follow this philosophy and study the response to nonmagnetic impurities in different

situations, and provide comparison with experiments to prove the adequacy of these

approaches.









CHAPTER 2
DISORDER INDUCED RESISTIVITY UPTURNS IN METALLIC CUPRATES

2.1 Transport Properties in Disordered Cuprates

The purpose of this chapter will be to further discuss the influence of disorder

on transport properties in the homogeneous metallic state mentioned in Section 1.6.

Experiments in different cuprates confirm the low temperature anomaly in the resistivity

that occurs in all samples that have been measured, ranging from the underdoped to

optimally doped region of the phase diagram. However, among the most commonly

measured LSCO, BSLCO, and YBCO families, one observes differences due to the way

in which the different families are prepared. Stoichiometric crystals of conducting YBCO

can be grown and defects can be introduced in a controllable way, and hence we will

primarily compare with experiments on this material. However, it is instructive to present

the experiments in various samples and compare their differences.

LSCO. The resistivity upturn was first seen in underdoped LSCO family when both

in-plane Pab and c-axis p, resistivity were measured in the presence of magnetic field, and

a logarithmic divergence at low temperature was discovered[63, 64]. A pulsed magnetic

field up to 60T was used to suppress T,, and the resistivity was measured down to close

to zero temperature. When plotted on a logarithmic scale, the divergence is clearly seen

below a temperature Tde ~ 50K which indicates its deviation from linear T resistivity,

as shown in Figure 2-1. The magnetoresistance is shown to slightly decreases as field

increases, although at extreme low temperature the measurements were limited to field

range B < 100T. A change of slope in dp/dT and continued rise to the lowest temperature

measured was taken to indicate the onset of metal-insulator ti il-il i, i[c .1]. Although

this was one of the earliest exciting discoveries of resistivity anomaly, it is important

to remember that the major source of elastic scattering in LSCO comes from out of

plane Sr ions which dope the CuO plane, and produces long range screened Coulomb

potential on the planar electrons. Impurities of this kind give rise to forward scattering










10 100 1 10 100


-x-x- 4 -- d 0, I H/ b x=0.08, I /c

0 |o T 4 0 b
E __________ 3 a 2 C:
0 T aur d
0- o 1 T 1- 2
o o ^M 30 T 2
S2 A 34T
O 0O [U- 36 T 0 -, I .. ..L 0I
.r 0 x 4 T 7 n ....I .. .....I
S- 5 6TE x=0.13, I l ab x=0.13, I / c
o0 1 2 3 4 I, (E 0




0 I I OW 0 .1 '
0 20 40 60 80 100 120 t 10 100 1 10 100
(a) T (K) b) T (K) T (K)

La Sr CuO T=07 K
METALUiCP 2-x x 4 La2-xSrxCu04, x=0.08 1.0 K
40 s --- 3 H // c, I//ab 15K
S CC


METALL" 2
20 Pab ad
S "INSULATING" (dp /dt 0)
2 Pab c c

|- (dp/dt <0)
,E A L 2 ,








(c) Sr CONCENTRATION, x (d) B (T)

Figure 2-1. In-plane pab and c-axis resistivity pc in LSCO in the presence of
impurities[63, 64]. (a)Underdoped region 6 = 0.08 shows divergence of pab at
low temperature. (b)Both pab and pc show logT divergence when plotted on
logarithmic scale. (c)Phase diagram determined by mapping out dp/dT, which
shows a metal-insulator transition in underdope region. (d)Magnetoresistnance
of Pab at different temperatures.


and smaller momentum transfer, as discussed in [65]. How the long range defects and

forward scattering affect normal state transport properties in the underdoped region
remains an open question at present. The influence of planar defects has also been studied

by introducing Zn to the CuO2 plane, in which case one sees more drastic low temperature

uptui ii-
uptil 11 1 [C.C.]














800 x=0.50 // 1000 x=l = 00-
S (p 0.15) (P 1/8)
600 800
0 6 -o

a400 25 0 4 p-05 1 15)
C 400-3"

CL 400 2O15
400 ".
09- Zn-free 15-
S0.5 10 10
200 -
S1.6%b 200 "(p- 18)
S 100 200 300 400 0 .5 1 1.5 2
Temperature (K) Zn concentration (%)
l O
0 100 200 300 400 0 100 200 300
(a) Temperature (K) (b) Temperature (K)
0 126.3 K
I0 ...-... .. 1 .3 K
a) BiSr La Cu ZnO -2 / K
1500 2 2-x x 1-z z 6+ -002 // \\
1500 2 x 066
x=0.66 I -0.04
b.-0.04 / /
z=2.2% 2.0 K
o 1000 5 .06
So 1 -0.0 1.0K

.* R 4 -.
500 H H 4 .
-0.12 -
3 -0.14 (a) H // c
0.1 1 10 100 -0.16 0.4 5 K
1 1 10 100 15 10 5 0 5 10 15
(c) Temperature (K) (d) H (T)

Figure 2-2. In-plane disorder resistivity Pab of BSLCO in (a)optimal doping region, and
(b)underdope region[67]. Underdope sample clearly shows a more dramatic
upturn in low T. (c)Both pab and Hall constant RH shows logarithmic
divergence at low T. (d)Negative magnetoresistance of pb in the presence of
applied field H//c, for underdope sample 6 ~ 0.125 with 2.-'". impurities.


BSLCO. Due to the difficulty of growing samples with more than 1 defects in

BSCCO-2212, as well as the difficulty of obtaining a wider range of Te, the well-controlled

BSLCO(Ba2Sr2-_La1CuO06+6) material with wide range of doping has been available

for resistivity measurement instead[67, 68]. In-plane defects are produced by replacing

planar Cu by Zn. The resistivity again manifests a divergence on a logarithmic scale below

Tdev ~ 50K, with negative magnetoresistance in the field range explored, as shown in

Figure 2-2. Comparing optimal and underdoped samples, the data shows the tendency









of increasing upturn with underdoping, indicating the close relation between the upturn

and strong correlations, which increase near half-filling. Impurities alone are sufficient to

suppress T, down to zero temperature and reveal or induce the same divergence, such that

the external field is essentially unnecessary at high impurity concentration. The study of

BLSCO under external field seems to be consistent with LSCO compound, i.e. a weak

negative magnetoresistance is observed. Although the scattering center is claimed to be

the Zn substitution of planar Cu's, an ST\ study in related BSCCO-2212 compound

shows unavoidable out-of-plane interstitial O in the anealing process, which appears to

cause real space inhomogeneity of pairing and charge density[18]. Such interstitial O's,

as well as a wide class of defects, are studied in the context of long range scatterers,

in which the real space gap pattern and AC conductivity could be well understood

by assuming causing a locally modulated pairing interaction or long range Coulomb

interaction on the planar electrons[48, 49]. The similarity between BSCCO-2212 and

BSLCO slr-- -I- that out-of-plane dopants could also contribute to the scattering rate

in the resistivity measurement reported here, although no direct ST \i evidence has been

provided in BSLCO yet. We conclude that BSLCO and LSCO should belong to the

"intrinsically doped" category that out-of-plane defects have drastic influence on the

transport properties, and the scattering sources are not solely the planar defects.

YBCO 123. NMR provides evidence of intrinsic homogeneity of YBCO and make

it the best choice to study thermodynamic properties in the presence of planar defects[69].

The resistivity under the influence of defects produced by Zn substitution, as well as those

produced by electron irradiation, has been studied in a wide range of <1' pi"- [7-1179]. A

scaling relation between the suppression of T~ and the change of resistivity multiplied by

the charge density Ap x n is found in all YBCO and TBCO samples studied, indicating

that the hole content is the important parameter to describe electronic transport over a

wide range of d11 pi:i [74]. Figure 2-3 summarizes the experimental results in YBCO, where

one sees the enhancement of resistivity upturns with increasing disorder and underdoping,









1000


Temperature [K]


(c)

Figure 2-3.


Temperature [K]


300
E
S 200

100
-" 100


0 10 20 30 40 50
H (T)


In-plane resistivity pab of YBCO[71, 78]. (a)Impurity concentration
dependence of Pab for an underdoped sample, showing low temperature
divergence on top of the linear-T resistivity. The divergence is enhanced with
increasing disorder. Insert figure shows the logT form of the divergence.
(b)Doping dependence of pab at a fixed impurity content, which shows more
dramatic divergence as the sample is underdoped. (c)Details of low
temperature region in two different impurity concentration z = 2.7'. (top) and
z = 2.:;'. (bottom). Magnetic field clearly reduces T, and reveals the upturn.
(d)Magnetoresistance at different temperature for underdoped(top) and
optimally doped(bottom) sample up to 60T. Underdoped sample shows
positive magnetoresistance while optimally doped sample remains constant.


which is an universal feature among all the three cuprate families reported here[71, 73].

However, Ap increases with increasing field in the underdoped region, in contrast to

the negative magnetoresistance of LSCO and BLSCO. Since YBCO is nearly free of

out-of-plane defects, it is not clear if one can attribute the difference in magnetoresistance


Temperature [K]









to effective long range potential and pairing modulations produced by this kind of defects,

or solely a band structure effect produced by lifting spin degeneracy in the presence of

external field. In the following sections we present a model that takes into account the

strongly correlated nature across a wide range of the phase diagram and the unitary

scattering centers produced by Zn or Li substitutions, in which we associate resistivity

upturns with extra scattering produced by inducing magnetic moments around the planar

defects, as discussed in section 1.7. We focus on comparison of numerics with YBCO

data since its scattering rate is attributed to planar defects. The temperature and defect

concentration dependence of the resistivity is well described in the model proposed, where

we also discuss the general features of magnetoresistance within the same framework that

could possibly account for the discrepancy between YBCO and other two families.

2.2 Model Hamiltonian

It has been known that anomalies in thermodynamical observables such as conductivity,

density of states, specific heat, and Hall constant can exist in a correlated disordered

Fermi system. A significant amount of work in higher order perturbation theory has

been done since the resistivity minimum in Kondo effect was found to be attributable

to the corrections beyond the Born approximation[80]. The higher order calculations

in correlated dicordered systems, pioneered by Altshuler and Aronov, showed that

the correction of density of states crucially affects the transport properties, and gives

logarithmic divergent resistivity in 2D[81-83]. Particularly in cuprates, where the strength

of correlations varies with doping, a theory that can cover the anomalies of transport

properties over a wide range of doping has yet to be discovered. In this chapter we focus

on the optimally doped and slightly underdoped samples, where the Fermi liquid picture

properly describes the electronic excitations in the normal state, and the strength of

correlations remains moderate. The resistivity upturns observed in these samples with

small impurity content are not consistent with the weak localization picture, since the

mean free path for unitary scatters like Zn substitution is much larger than the Fermi









wavelength for the experimental impurity concentration of order 1 and -.-. '-. -I that the

main contribution of the transport anomalies comes from the correlation effect. Within

this range of doping, where the Mott transition has yet occurred, one needs to find the

source of extra scattering rate associated with these resistivity upturns. Although it

has been shown that the logT resistivity can come from the granularity of the measured

- ipl1! [I !], and is relevant for the transport measurement in cuprates since most of

the samples are not detwinned, the enhancement of the resistivity upturns as increasing

impurity concentration sir-.;- -1- that the upturns are associated with the intrinsic disorder,

rather than the tunneling between islands. In addition, a theory that can account for

the transport anomalies should also be able to explain the impurity induced -1 I-.-' red

magnetization observed in disordered cuprates. A natural suspicion is that the source

of the extra scattering rate comes from these induced moments, since the conditions

under which these induced moments are enhanced also enhance the resistivity upturns.

Such ideas have been examined by Kontani et al, where the interplay between impurity

scattering and the correlation effect is formulated within the fluctuation-exchange(FLEX)

approach, although certain assumptions about the scattering process and the self-energy

had to be rni 1. [~;87].

Since the logarithmic divergent nature of these anomalies questioned the adequacy

of finite order perturbation theory, we propose a mean field theory plus a real space

diagonalization treatment to deal with the disordered cuprates, where the effects of

localization and perturbations are included to all orders. The conditions in which positive

correlations between impurity induced magnetization and transport anomalies can be

found are examined, which proves our hypothesis that the enhancement of the scattering

rate is due to an enlarged cross sections associated with these induced moments. We

shall first examine the correlation effect in the case of optimal doping, and discuss a

phenomenological Fermi liquid model that highlights the influence of the spectral anomaly

in underdoped samples in a later section. Evidence from experiments has also si ---- -1. 1









that the impurity induced magnetism remains paramagnetic in YBCO, and does not

form static spontaneous long range magnetic moments, nor is antiferromagnetic ordering

observed except possibly in the extremely underdoped samples. Since resistivity upturns

are revealed after Tc is suppressed down to zero, the pairing correlation is ignored in

describing the normal state properties as a first approximation. We therefore start with

the two dimensional Hubbard model to describe the CuO plane


H =- tIjcc + ta P)ia + Un. (2-1)
iju iu i

and define the following order parameters


ri = (hi + n^}

mi = (nh r1 (2-2)


After mean field factorization, the Hamiltonian becomes

H = -tijcc, + (cia, P)ia + U ni -2o-nhi, (2-3)
ija iTa i7

The site energy ci, takes into account the Zeeman splitting when magnetic field is applied,

and also the local chemical potential shift due to impurities


Ci, = 2ag B + Y 6ir, (2-4)

where V, is the impurity potential energy at site r. The Coulomb repulsion U is chosen

small enough such that the magnetization induced by introduction of nonmagnetic

impurities is paramagnetic, which assures no formation of spontaneous magnetization

in either the homogeneous or impurity case at the relevant temperatures. Real space

diagonalization is applied to find the eigenstates associated with the above Hamiltonian


HIn) = E,|n) (2-5)









and the wave function of spin a at site i for eigenstate in) is


aci(E,) = (7i n) (2-6)

which will be used to calculate the renormalized site-dependent order parameters ni and

mi

i {(hi + nh) c i \ (E.) 2f (En)

mi = (hit -hij-Y a'i^Ej2f(E.). (2-7)
ncr
The dc conductivity in linear response theory is given by


Cr im{I (T (o))} o (2-8)

where the 7r(w) is the retarded current-current correlation function. In a inhomogeneous

system, Tr(w) will be site dependent. The retarded current-current correlation function

between site i and j is


,(t) = (t) ([J(t), j(0)]) ,

Sf n f(E,) f(Em)
wo + E, Em. + i

I(JI 7n\ )(mIJj}I\n})F(E.En) (2 -9)
j n,m

in the basis of real space eigenstates In), their eigenenergies E, and the corresponding

Fermi distribution f(E,). The function F(E,, E,) is symmetric under exchange of E, -+

Em and accounts for the frequency and temperature dependent part of coid.ll Ii i il [88].

Details of the current operator Ji(t) and F(E,, Em) will be discussed in Appendix A and

B.

The proper choice of system size in simulating Eq. (2-3) is determined by several

criteria: Because of the artificial broadening Tr introduced when calculating 7 (r), the

homogeneous resistivity phom in the absence of impurities and field is linear in q. The









slope of Phom versus rT converges as system size increases, which serves as the first hint

of minimum system size. Secondly, the resistivity at U = 0 in the presence of impurities

is linear in impurity concentration, as Matthiessen's rule is expected to be satisfied in

the dilute impurity limit. We found that 40 x 40 lattice is able to achieve the above

two criteria down to temperature as low as T = 0.02, and will be the system size

studied in this report. Each data point is then averaged over 10 independent impurity

configurations unless otherwise specified, which we found is sufficient enough to ensure the

randomness of the impurity distribution. The energy unit is chosen to be t = 100meV,

which fixes the temperature scale as T = 0.01 100K, and the magnetic field to

be B = 0.004/(gpB/2) ~ 7Tesla, using the same energy scale as the study in the

superconducting -i i, ['i].

Before the resistivity under the influence of induced magnetization is studied, we feel

obliged to compare the present study in the normal state with the data in the d-wave

superconducting(dSC) state studied by Harter et al[89]. Such a comparison reveals

the importance of finite DOS in the normal state, as well as the bound state formation

in the dSC state. The effect of nonmagnetic impurities in the dSC state is studied

within the framework of d-wave BCS theory plus magnetic correlations, equivalent to

the two-dimensional Hubbard Model Eq. (2-1) with an additional pairing correlations

between nearest-neighbor sites. The authors then applied a Hartree-Fock-Gorkov mean

field factorization and solved the Bogoliubov-de Gennes equations via real space exact

diagonalization. The real space magnetization pattern induced by a single nonmagnetic

impurity is shown in Figure 2-4(a) and (b), in which we found three 1i ii."r differences:

(1)In the presence of a magnetic field, the normal state has homogeneous magnetization

significantly larger than that of dSC state. This obviously is due to the opening of the

gap in the dSC state reduces the DOS at the Fermi surface, and hence exhibits a smaller

susceptibility than the normal state. (2)The magnetization on the nearest-neighbor sites of

the impurity is drastically enhanced, which we found to be consistent with the bound state










Normal, T=0.05


0.04
0.02


Figure 2-4.


0.0008

0.0006

< 0.0004

0.0002


0.002 0.004 0.006 0.008 0.01
B/t


Comparing magnetization induced by a single nonmagnetic impurity studied in
this chapter for the normal state, and d-wave superconducting(dSC) state
studied in[89]. Real space magnetization pattern for (a)normal state, and
(b)dSC state, both at U = 1.75 and B = 0.01. One sees clearly that dSC state
has more pronounced nearest-neighbor site magnetization due to bound state
formation, and has smaller homogeneous magnetization due to opening of the
gap st the Fermi surface. These effects are shown at one plots (c)total
magnetization of the system S, and (d)magnetic contrast A versus external
field.


formation due to the d-wave symmetry. (3)The dSC state has a shorter correlation length,

resulting from the enhancement of nearest-neighbor site magnetization in comparison

with the relatively smaller magnetization on the second and third shell sites away from

the impurity. To give a quantitative description of above features, we defined the total


0.002 0.004 0.006 0.008 0.01
B/t


dSC, T=0.02









magnetization Sz and the magnetic contrast A


A = m mo| (2-10)


where mi is the magnetization at site i and mo is the homogeneous magnetization in

the absence of impurities but in the presence of a magnetic field. The meaning of A is to

estimate the fluctuation of magnetization away from its homogeneous value mo, hence an

indication of locally induced -1 ,--.. red moment. Since interference between impurities is

alv-i-, present and the local environment is different around each impurity, the deviation

from mo of the whole system needs to be considered, hence we summed over i for A in Eq.

(2-10). The behavior of S, and A versus the applied field is shown in Figure 2-4(c) and

(d), where one sees that Sz in the normal state is one order larger than in the dSC state,

which is attributed to the overall larger homogeneous magnetization in the normal state.

However, in the A versus field plot, we see that after the homogeneous magnetization is

subtracted, as in the definition of A, the dSC state has a larger value due to the enhanced

magnetization attributed to the bound state formation. Such a comparison emphasizes

the importance of DOS at the Fermi surface to the impurity induced magnetization, which

motivates us to propose a toy model that captures the effect of reducing DOS in the

pseudogap state, as will be addressed in a later section.

2.3 Resistivity Upturns at Optimal Doping

Motivated by the NMR experiments, we continue to choose U so as to keep the

magnetic response in the paramagnetic region, but close to the magnetic phase boundary.

Figure 2-5 shows the real space magnetization pattern as U is increased. We found a very

narrow range 1.7 < U < 1.75, which we call it the "critical region", where magnetization

increases dramatically before entering the long range magnetic order phase. Note this is

not the long range ordered phase of the homogeneous model, which has a much higher

critical U, but an "order from di-. i i phase as discussed in Andersen et al[90]. The










U=1.75


I040 40 40



30 30 30
40 40 40

Figure 2-5. Real space magnetization pattern at T = 0.03 and B = 0.001, with 2"'.
impurities. One sees clearly a transition of magnetization pattern as increasing
U, where we found its critical value to be around U ~ 1.75 above which the
magnetization is no longer characterized as localized around each impurity.

0.05 0.2

0.04 0.18 B/t=0
B/t=0
0.03 0.16 B/t=0.01

0.02 B/t=0.01 0.14

0.01 0.12


1.5 1.6 1.7 1.8 1.9 2 2.1 1.5 1.6 1.7 1.8 1.9 2 2.1
u/t u/t
(a) U/t (b) t

Figure 2-6. Results at optimal doping. (a)Magnetization and (b)resistivity versus U at
T = 0.03, with 2' impurities.


magnetic contrast A versus U at optimal doping is plotted in Figure 2-6, where a critical

value U ~ 1.75 is found above which A no longer responds linearly to the external

field. This value is found to depend on system size and impurity content[89]. Comparing

Figure 2-5 and Figure 2-6(a), we see the magnetic contrast A is indeed a good index

to characterize the magnetization induced by disorder. The resistivity increases with

increasing U, coinciding with the behavior of A in the region both below and above its

critical value. At extremely large U and zero field, the resistivity increases significantly


U=1.8









although the magnetization remains zero, which we found to coincide with a development

of real space charge density wave(CDW) order, although its magnetization remains zero.

However in the critical region 1.7 < U < 1.75 one can attribute the increasing of resistivity

to extra scattering induced by the magnetic moments, based on the positive correlation

between A and resistivity. In the following discussion we choose U = 1.75 such that it is

close to but slightly less than its critical value, and exhibits paramagnetic response to the

external field.

0.05 0.16

0.04 B/t=0.01, n,mp=2% 0.15 B/t=0.01, n,mp=2%

0.03 B/t=0, nlmp=2% 0.14 B/t=O, nlmp=2%

0.02 0.13

0.01 0.12


0.02 0.03 0.04 0.05 0.02 0.03 0.04 0.05
(a) T/t (b) T/t
(a) (b)

Figure 2-7. Results at optimal doping. (a)Magnetization and (b)resistivity versus T at
U = 1.75, with "' impurities.


Unlike the previous work in superconducting state, there exists a temperature scale

below which the system flows into long range magnetic order phase when impurities

are present, due to the finite DOS of the normal -i ,i1 ['1]. This mean field artifact

hinders our exploration at extremely low temperature region, and hence prevents us from

comparing the present theory with the experimentally observed logarithmic divergence of

resistivity. However the numerics down to as low as T = 0.02 ~ 20K show a significant

upturn in comparison with the zero field case. Figure 2-7 shows both magnetization and

resistivity versus temperature, where one sees again the positive correlation between

them. In the zero field case, which magnetization remains zero at all temperatures

explored, one sees that resistivity slightly increases as temperature is lowered, instead









of remaining constant. This effect is found to be not universal if the system size is

changed, and hence is attributed to the residual resistivity caused by artificial broadening

r, introduced in Eq. (2-9) plus the finite size effect. The lowest temperature explored is

fairly close to the magnetic phase boundary, and also where the resistivity upturn starts

to become significant. The magnitude of the upturn at T = 0.02 in comparison with high

temperature resistivity is about 1'. roughly consistent with the value obtained in YBCO

after the linear inelastic contribution has been subtracted[71].


0.02 0.16


0.015
0.14

S0.01 .

0.12
0.005


0.002 0.004 0.006 0.008 0.01 0.002 0.004 0.006 0.008 0.01
B/t B/t
(a) (b)

Figure 2-8. Results at optimal doping. (a)Magnetization and (b)resistivity versus B at
T = 0.03 and U = 1.75, with "'- impurities.


However, in Figure 2-7 the magnetoresistance in low temperature region seems to

be not monotonic if one compares the zero field with B = 0.004/(gpB/2) ~ 7Tesla and

B = 0.01/(gB/2) ~ 18Tesla curves. This motivates us to do a careful search for the

magnetic response with increasing field. Figure 2-8 shows the magnetization and resistivity

as a function of external field. We found that at temperature T = 0.03 ~ 30K, the region

that magnetization is linear to the field is fairly small, only up to B ~ 0.002/(gpB/2) in

contrast to the result in superconducting state B ~ 0.01/(gpB/2)[89]. Both magnetization

and resistivity increase with field in this region and then saturates to a roughly constant

value in high field. The magnetoresistance of optimally doped YBCO di-p,'iv similar

increasing-saturation behavior, although the temperature scale that this behavior is









observed is lower, which can be attributed to the difference between the realistic energy

scale and the energy scale modeled herein. Nevertheless, the positive correlations found

between magnetic contrast A and resistivity in all cases explored indeed confirms our

hypothesis that the enlarged cross section due to impurity induced magnetization causes

the resistivity upturns, and can be realized in a simple mean field picture that properly

treats the electron screening and transport on equal footing.

2.4 Effect of Pseudogap on Resistivity

Although no microscopic description of the pseudogap state is generally agreed

upon at present, we expect enhancement of resistivity upturns based on two features

at underdoping: First, correlations are more prominent as one goes toward half-filling,

resulting from an increasing of the effective U that enters our model. Although the

Hartree-Fock type mean field theory can not capture the Mott transition induced by

correlations nor the pseudogap phenomenon, the drastic increase of resistivity upon

crossing the critical value of U -i--.-- -I- that correlations indeed affect resistivity as one

approaches the strong coupling region. The large U region in Figure 2-6 demonstrates

that correlation strength U, as well as the magnetic moment induced, are indeed essential

ingredients to determine the magnitude of the upturn.

Secondly, opening of the pseudogap in the quasiparticle spectrum is known to

favor bound state formation, which in turn promotes the impurity induced magnetic

moment[52]. This is similar to the d-wave superconducting state where the pole of

impurity T-matrix falls within the gap, which produces a bound state localized around

the impurity. We expect that the reduction of DOS in the pseudogap state also produces

resonances of T-matrix near the Fermi energy, although the exact form of Green's function

and Dyson's equation remains unknown. The resistivity is then affected by the pseudogap

formation, based on the naive argument that impurity induced moment results in the

upturn. To get a crude idea of the effect of reducing DOS, we introduce a pseudogap in

an ad hoc way. The following form of dispersion and DOS is proposed for homogeneous










0.3
0.2
Z
0.1


-1-0.5 0 0.5 1 1.5


%0.2
0.1
0.1


-1-0.5 0 0.5 1


0.4
0.3
0.2
0.1
0.1


-1-0.5 0 0.5 1 1.5


Figure 2-9. Proposed phenomenoligical model for the reduction of DOS in pseudogap
state. (a)The normal state DOS of dispersion k we begin with, (b)the DOS
for proposed dispersion Ek in Eq. (2-11) with pf = -0.6 and a s-wave gap
Ak = A = 0.2, and (c)the extended hopping model after Fourier transform Eq.
(2-12).


"1 Il' l'- state


Ek = Sgn(k) A

(w) f dk2 r/
4x2 ( Ek )2 2 '


(2-11)









where k = -2t(cos(kx) + cos(ky)) 4t'cos(k1)cos(ky) is the normal state dispersion.

Figure 2-9 shows the DOS of Ek, where one sees reduction at chemical potential if. To

put this momentum space formalism into a real space exact diagonalization formalism, our

next step is to Fourier transform Ek back to real space and find an effective long range

hopping model that gives Ek. The hopping amplitude tij of this extended hopping model

is therefore
I dk2
wj 4 Ekj. 1.. (xi xj)] + cos[ky (i yj)]} (2 12)

We calculate the hopping range upto Xi Xjy = |yi yj\ = N/2, and N = 40 for 40 x 40

sites BdG. For a translational invariant system it means the hopping ranges from site i to

distance (x = N/2, y = N/2) away from site i. A comparison of the DOS with the

normal state dispersion (k, proposed pseudogap form Ek, and real space extended hopping

model Eq. (2-12) are shown in Figure 2-9. The current operator in the presence of long

range hopping still satisfies Eq. (A-3), but the contribution from all hopping terms t = tij

corresponding to ranges 6 = r r' need to be considered.

0.05 0.15

0.04 0.14Extended B/t=0.004

0.03 Extended B/t=0.004
S0.13
0.02
0.12
0.01 -


0.02 0.03 0.04 0.05 0.02 0.03 0.04 0.05
T/t T/t
(a) (b)
Figure 2-10. Comparison of models with(Extended) and without(Normal) reduction of
DOS by applying extended hopping Eq. (2-12). (a)Magnetization and
(b)resistivity versus T both at optimal doping, with U = 1.75 and "'.
impurities.









Figure 2-10 shows the magnetization and resistivity comparing the extended hopping

model with the normal state Hubbard model Eq. (2-1), which contains only nearest and

next-nearest neighbor hopping. To avoid the problem of possible contributions to the

resistivity coming from phase fluctuating Cooper pairs or other possible excitations in the

pseudogap state, we fix both models at optimal doping 6 = 0.15 and examine solely the

effect of reducing DOS. Within the magnetic field region explored 0 < gpBB/2 < 0.01,

the magnetization is found to be enhanced in the extended hopping model, confirming

our hypothesis that reducing DOS at the Fermi level promotes bound state formation,

which also gives slightly larger resistance between temperature range 0.02 < T <

0.045. The magnetization versus field is shown in Figure 2-11, where one sees larger

magnetization comparing to the normal state model, with a smaller linear response region.

The saturation at high field is again revealed. The magnitude of the resistivity is enhanced

overall in both the low and high field region, while one does not see the drastic increasing

of magnetoresistance in the low field linear response region as in the normal state model,

presumably due to the imperfection of simulating the pseudogap by present model, or the

change of Fermi surface in the Fourier transform process Eq. (2-12). Nevertheless, the

assumption that reducing the DOS promotes an induced moment is well supported, which

also drastically affects the magnitude of resistivity and its upturn.

2.5 Conclusion

In summary, our result sI-.-, -1- that the low temperature resistivity upturns in

cuprates can be understood by the extra scattering rate due to impurity induced

paramagnetic moments. The advantage of real space exact diagonalization approach is

that all the cross diagrams and localizations involved in many-impurity calculations are

included. We found a positive correlation between the magnetic contrast A and resistivity,

in the region where correlation strength is close to its critical value. We also examine the

effect of reducing the DOS at the chemical potential by an effective long range hopping

model, which confirms our hypothesis that reducing the DOS at the Fermi level promotes










0.03


Extended T/t=0.03
.-- 0.135 Extended T/t=0.03
0.02

0.13

0.01
0.125



0.002 0.004 0.006 0.008 0.01 0.002 0.004 0.006 0.008 0.01
B/t B/t
(a) (b)

Figure 2-11. Comparing (a)magnetization and (b)magnetoresistance between extended
hopping model Eq. (2-12) and the normal model both at optimal doping,
T = 0.03, U = 1.75, with "' impurities.


bound state formation and enhances the induced moment. This result -ii--.- -1- that the

reduction of the DOS in the pseudogap region could be an important ingredient for the

more dramatic resistivity upturns observed in the underdoped cuprates.









CHAPTER 3
CHAIN DEFECTS IN YBCO6.5 ORTHO-II SYSTEM

3.1 63Cu NMR in YBCO Chain-Plane System

In addition to the planar defects which can affect thermodynamic properties, a

v 1i. i of out-of-plane defects can also have significant influence on the plane. In

this chapter we study defects of this kind produced specifically on the chain 1-v-r of

YBa2Cu307_ (YBCO7-6) system, with a well defined structure and relatively clean CuO2

plane. The ( i -I I1 structure of the YBCO family for different stoichiometric dopings is

shown in Figure 3-2. YBCO6 corresponds to the half-filled CuO2 plane, where all the

chain 0 are missing, causing planar Cu to have valence exactly one electron per site,

and is a Mott insulator. Upon filling O on the vacancies between .,1i i,'ent Cu sites on

the chain lI--r, the local chain Cu valence is changed, which indirectly produces mobile

charge carriers(holes) on the plane. The YBCO7 structure with all O sites filled gives a

slightly overdoped sample, where T, is close to its maximum. Figure 3-1 gives a schematic

description for the effect of O content on the planar d..'1, l, in which we see that in the

intermediate doping range, the distribution of O has an equally crucial influence as does

the total number of O. For example, if chain 0's cluster in the annealing process rather

than distributing randomly, there will be fewer isolated Cu-O-Cu pairs and hence less

influence on the local valence. Such a configuration dependence is observed experimentally,

and it is found that number of Cu-O bonds is the essential ingredient to determine T,[91].

The planar doping can be calculated via bond valence sums once an O distribution is

determined[92, 93].

In this chapter, we study specifically the Ortho-II structure of YBCO6.5, where

the chain li--r has alternating "full" and "empty" chains referring to the O content,

as shown in Figure 3-2(i). The 63Cu NMR spectrum for this particular structure has

been measured[94, 95], where the authors interpreted the broadening and the shape of

spectrum as a signature of impurity induced magnetic moments. Since the O distribution


























Figure 3-1.


] (-'u phnc:'

A schematic description of the doping process in YBCO by changing the
oxygen content on the chain 1-v. r[52]. The undoped parent compound YBCO6
contains Cu+ on the chain 1~V.-r, and the CuO2 plane has effectively one
electron per Cu site. If (.::;i---, i are introduced into the chain l-v -r but their
content is sufficiently low, then the sparse 02- only changes its two .,l-i i,:ent
Cu+ to Cu2+. When a segment of at least two O's is formed, the excess charge
will be transferred to the CuO2 plane, resulting in the hole doping. A full chain
can therefore dope the CuO2 plane in a more efficient way than a random O
distribution.


is well defined, we can label four species of Cu according to their positions: chain Cu

on the full chain Cu(1F), chain Cu on the empty chain Cu(1E), planar Cu connected

to full chain Cu(2F), and planar Cu connected to empty chain Cu(2E). The chain Cu's

connect to planar Cu's by the overlap of their dz orbital and intermediate O pz orbital,

which produces effective tunneling between two 1lv.-ir. Under external field, each Cu

species processes at different frequencies according to the local magnetic environment of

the nuclei, which results in four distinguishable lines in an NMR spectrum as shown in

Figure 3-3. Based on comparison with the same experiment in YBCO6 and YBCO7, the

Cu species associated with each line can be identified, and the change of local magnetic

environment under certain circumstances can be extracted via the change of the line

shape[94].










Full Chain Empty Clihii






CuO,

Y
CU(2F)
CuO,

BaO 2

CuO&
Cu(IF)
Cu(iE)

(i) Ortho-II (YBaCu O6 )






Cu(2) Cu(2)



Cu(1), Cu(1)
(ii) Ortho-I (YBaCuO,) (iii) Tetragonal (YBaCuO6)

Figure 3-2. Crystal structure of ortho-II YBCO6.5 and related parent compound YBCO6
and YBCO7, with precise definition of chain and planar Cu[94].


ST\I images of the chain lI-.-r, which is where the YBCO ( i --1 I generally cleaves,

show LDOS oscillations along the chain direction. The wavelength of this oscillation shows

strong energy(ST:\ bias) dependence, which r-' --- I a Friedel oscillation explanation for

this density modulation, in contrast to a charge density wave(CDW) picture which usually

has a fixed wave vector[96]. Such an oscillating behavior also indicates the importance

of chain length to its electronic properties, especially those directly related to the DOS,

for example magnetization and ti I, -pl1 -i ['17, 98]. In the NMR experiment, the average

chain length of Ortho-II is found to be around 120 lattice constants by estimating the


















Ea







(a)
Figure 3-3.


H (T)


(a)Full NMR spectrum of Ortho-II at 60K in the presence of external field
Ho//a, where the frequency is fiexed at 75.75MHz[94]. Top panel: repetition
time Tyep,250msec, middle panel: Tryp 25msec, lower panel: the difference.
Comparison with YBCO6 and YBCO7 confirms the association between lines
and the Cu species: A-Cu(2F), B Cu(lE), C-Cu(2E), D-Cu(lF).
(b)ST\ i image on the chain li-r of YBCO, which shows oscillation of DOS
along chain direction[96]. Notice that the sample is not Ortho-II and has
relatively short segments of full chains(about 10 sites).


concentration of terminal Cu on the full ( ii,-['` 1]. With this information in mind, the

physical picture for the NMR line associated with chain Cu is clear: the Friedel oscillation

induced by abrupt termination of chains causes the modulation of magnetization in the

presence of an external field, which changes the local magnetic environment and results in

a significant linewidth in the NMR spectrum. Figure 3-4 gives a schematic demonstration

of how magnetization induced in a one-dimensional Heisenberg chain material can broaden

the NMR spectrum, which corresponds to the histogram of magnetization[99, 100].

Although this Friedel oscillation can give a reasonable interpretation of the spectrum, we

found that magnetization resulting from purely uncorrelated electrons is unable to account

for the observed line width, nor the temperature dependence of the line shape. Instead,

one must include the correlation effect in order to explain these features, in which the









magnetization is strongly enhanced compared to the uncorrelated case. The Cu(1F) line

is then broadened accordingly, and the temperature dependence can be reproduced

by properly choosing the correlation strength. Under the influence of correlations,

an anisotropic magnetization pattern is induced also on the plane, which causes the

broadening of plane NMR lines to follow that of chain NMR line, similar to what is

observed experimentally[95]. Comparison with experiments will be made to highlight the

importance of magnetic correlations, and the consequences of coupling between different

dimensionalities in this particular Ortho-II structure will be demonstrated.


14660



14640- --

"Zn Ni
14620



14600


0 2 4 6
chain site (cell units)

Figure 3-4. Schemetic of broadening of "Y NMR line in YBa2NiO5 by magnetization
induced around impurities[52, 99, 100]. The right panel shows the amplitude of
magnetization induced on several sites near the impurity, which results in
satellite signals near the main line. A significant amount of such induced
magnetization eventually smears out the spectrum and broadens the line(the
insert of left panel).


3.2 Broadening of Low Temperature NMR Lines

The proper choice of a model for the Ortho-II system relies on its unique lattice

structure. Due to missing O's on the empty chain, the Cu(1E) sites are highly localized

and do not directly couple to the full chain; we therefore assume they do not affect the

magnetic properties on the full chain and drop these degrees of freedom. The resulting









minimum effective model contains a single square lattice of Cu(2E) and Cu(2F) where

superconductivity occurs, couples to evenly spaced one dimensional chains via interlayer

hopping, as shown in Figure 3-5. The full Hamiltonian consists of


H = Hchain + Hplane + Hinter + Himp (3-1)


where the unperturbed part is defined by


Hchan = t(d-tI(d,6 + h.c.) + Z(a -'c)h;
Xf Xf
2 C


Hpiane t=j P Cj4 + (-C)e
ija iu
+ EUPP hP v

i (ij)

Hinter = t($ d( x + h.c.) (3-2)
(ix)u

We denote by tP, tc, and t' the hopping amplitudes on the plane, chain, and chain-plane

interface, respectively; cw and dX are electron operators on the plane and on the chain,

where i denotes the 2D coordinates and x is the position along chain direction. The

chain hopping is only between nearest neighbors, while planar hopping tj contains both

nearest t, and next nearest neighbor sites tnn, which is necessary to reproduce the

correct Fermi surface of the YBCO system. A magnetic field is included in the Zeeman

term Cep = gpBB/2. The schematic of Eq. (3-2) is shown in Figure 3-5. Notice that the

experimental NMR line contains both data above and below Tc ~ 60K, which could be

modeled by properly choosing pairing correlations V in the plane. A Hartree-Fock-Gor'kov

mean field decomposition is then applied to the unperturbed chain and plane degrees of









freedom, giving


Hchain = t(d-ti d + h..) + C) c

Hplane 'jjitC + Z-+ (C -p)ie


+ (Asic c + h.c.) + UPO (3-3)
ij it i+61 2
i5 ia
with order parameters defined as

ni ('tn + nj)

rnP (ht nI)

Asi = V(cici+6) ,
C
C ^TC ^C

m <(T ) (3-4)

Since the chains are coupled with a possibly superconducting plane, a Bogoliubov

transformation is necessary for both planar and chain electron operators. We therefore

define

Cdt S- U f1, P* t^ t
CiT = n ,it7nT + UnIiTYT
r-, P -, P* n ^t
c Y ni I + Vnni
n
dxT Cnxt7nt + nc7,xt~T
n
dxi Cl t (3 5)
nX I + ,xII









and diagonalize the following Hamiltonian in the presence of magnetic field

/IP A 0 U UP

A* -P* P -_* V PT
0 en (3-6)


0 n0 ctn n











( It = u | f(i ) + | 1- fE)) ,
n>o
eigenvalues correspond to wave functions ( 1 which have eigenenergie
(cic ) 1 u TIf(Ent) + IVT 2(1 f(E-))

n>O
(didCl) = u 2 i2f(En) + vT 12(I -f(En))
n>O
(dcTcd+) E (1 f+ ( )) 2( f (En())
n>O
(dd) 4 2f(ET) + V, +2(1i f ( EEnt)) ,
n>O
n>0
(ddx) Y uxv +(,+S f (En)) + vX* +iu n /(Ent)
n>O
(3-7)

and were used to calculated the order parameters in Eq. (3-4). Further details of

Bogoliubov-de Gennes(BdG) formalism and numerical simplification are demonstrated

in Appendix C.

We first consider the effect of randomly distributed missing O's on the chain lv.r,

which results in the abrupt termination of a continuous full chain. Since the hopping on

the full chain is due to orbital overlap between alternating Cu and 0, the missing O is

assumed to reduce the hopping amplitude between its two .,.li i:ent Cu sites. The impurity









tc






P
c b hnnn

k- a P
at nnn

Figure 3-5. Schematic of the hopping part of model Hamiltonian Eq. (3-2).

Hamiltonian is therefore

H = -6c(ditd,+,, + h.c.) (3-8)

where 6te is the reduction of chain hopping between Cu site I and I + 6. We further assume

a complete elimination of the hopping, 6te = -t, and one impurity per chain. The BdG

equations are solved on a 121 x 12 square lattice with periodic boundary conditions. We

chose one impurity per chain which, with our system size, gives the proper chain length in

comparison with the NMR experiment. To fix the energy scale we choose t~ = 150meV,

and other parameters are expressed in units of tP: t~ = 0.2, tc = 0.7, t = 0.2, UP 1,
V = 0.7, which gives Tc ~ 60K for the homogeneous system. These parameters are similar

to those used in [101], and can result in a proper temperature and field scale, as well as
giving a proper NMR line shape and linewidth, as will be addressed below. The 63Cu

NMR resonance frequency is determined experimentally by, to lowest order

v = B(+ Kb+ K,,) (3-9)
27

where 7 is the gyromagnetic ratio of Cu, B is the applied magnetic field, and Korb is the

temperature-independent contribution from orbits and the inner shells of nuclei. Although

the Zeeman term eP/ remains the same in different field orientations, the value of Korb

is found to be anisotropic[102], and hence will shift the position of the line differently in









different field orientations. In the NMR experiment the field is directed along a direction,

hence Korb = Kb is chosen to calculate the frequency unless otherwise specified. Ksi, is

the Knight shift coming from magnetization of the conduction electrons, and is related to

the spin susceptibility by
= Kspin (3-10)
Ahf
where pB is the Bohr magneton and Ahf is the hyperfine coupling between nuclei and

conduction electrons. The polarization of spin Si and susceptibility Xi at site i are given

by
1 xiB
Si = (n nil) = B (3-11)
2 g1-B

assuming that the magnetization is in the linear response region, and the resonance

frequency for a specific site i is related to the polarization of spin Si by


Vi B[1 + Kob + S( gA (3-12)
27 B

Experimentally, the magnetization on different sites i gives a distribution of via's and

hence broadens the spectrum, since NMR measures the histogram of Vi that sums over all

sites. For a bulk system with the same value of magnetization on each site, for instance a

homogeneous paramagnetic system with constant magnetic field, the NMR line consists of

a single resonance peak in frequency space with very narrow linewidth. Hence the NMR

linewidth is a rough indication of the deviation of magnetization away from its mean

value, as demonstrated in Figure 3-4. Numerically we collect the magnetization on each

site, and then artificially broaden the discrete distribution by a Lorentzian with width rI to

get a continuous spectrum I(v)

1 1 17
I(v) R N( ,) (313)


where N(vi) is number of sites that has frequency vi, with r = 0.04t and R is the proper

numerical factor that normalizes the area under the I(v) curve.











>.

i uc=0.0
U,
C UC=0.8
a
-a

N

E CA\ =1.2
0
Z



87.8 88 88.2 88.4 88.6
w (MHz)

Figure 3-6. Numerics in an isolated 1D chain, equivalent to Hamiltonian Hchain + H1 ) in
an external field 7.7T at 50K. Each curve is normalized such that the area
underneath is conserved.

Before the realistic chain-plane coupled system is studied, we perform numerics on

an isolated 1D chain to show the importance of the chain correlation energy Uc, without

the influence of proximity effect between chains and plane. Figure 3-6 shows NMR lines

given by considering Hamiltonian Hhain + Hi1) where increasing Uc clearly broadens the

line at a fixed temperature and field, coinciding with the enhancement of impurity induced

magnetization in real space due to increasing magnetic correlations. The U =- 0 line

corresponds to Friedel oscillation of free electrons, where the magnetization due to Zeeman

splitting gives an NMR linewidth only half of what is observed experimentally. Our

best fit to the experimental data gives UW = 1.2, as far as line width is concerned. This

value of U' is again close to but smaller than its critical value, above which the induced

magnetization can no longer be characterized as localizing around the missing O. Together

with previous work on planar d, F. i- [ '], we conclude that magnetic correlation exists on









both chains and plane of YBCO system, and is especially significant on the underdoped

side of the phase diagram.

3.3 Explanation of High Temperature Satellite Peaks

Although the line width is well described by the impurity induced magnetization

due to Uc, one important feature regarding the Cu(1F) line shape in the Yamani et al

experiment[95] seems to be outside of this scheme. At temperature higher than 70K or so,

a satellite peak with significant weight gradually develops at frequency slightly lower than

the main line. This satellite peak, which becomes more prominent at high temperatures,

was attributed by the authors to magnetization associated with smaller density oscillations

far .. liv from the chain ends, in contrast to the amplified magnetization close to the

chain ends[95]. In other words, they postulated that a Friedel oscillation around a single

defect would di*pli w a bimodal distribution of magnetization. However, we found that

the idea of separating large and small amplitude oscillations is ambiguous in a system

where magnetization smoothly decays as moving away from the chain ends. Instead, two

features of this high temperature satellite peak motivate us to propose a different scenario

regarding its origin: (1)The weight of the peak is about 1('-. of the whole spectrum,

and (2)the position of the peak remains roughly the same at all temperatures. Feature

(1) -,-. -; that, if one would associate a value of magnetization to a certain satellite

signal, then the weight of the line is a rough indication of how much portion of the sample

di- pl'- this magnetization. Therefore it's reasonable to assume that roughly 1(' of

the sample has the same magnetization, which results in this satellite peak. Feature (2)

implies that this magnetization remains constant at high temperature. Since increasing

temperature should continuously reduce any finite magnetization developed due to the

correlation effect, it is reasonable to assume this observed constant magnetization is

zero. We therefore propose that the high temperature satellite peak is due to a section

of .,l1i i:ent Cu missing their conduction electrons, presumably caused by the alternation

of Cu valence near the missing 0. The change of local chemical environment prohibits









electrons from populating these sites, as well as transport between them, and gives zero

magnetization in all circumstances. Eliminating the conduction electrons on these .I.1i ient

Cu(1F) sites gives, in the one band model, the following perturbation


S-6tc(dt d16, + h.c.)
1
+ > (6 J r
1 c c
+ 26U 2- (3-14)
1
with L = 15 out of 121 sites. Notice that the summation over impurity positions in Eq.

(3-14) is restricted between a section of consecutive sites 1 < 1 < L, which differs from

Eq. (3-8) where the impurity position is totally random and discrete. We again assume

complete suppression of the associated energy scales tc -tI, 6c -e,, 6p = -pc,

which results in all matrix elements involving sites 1 < 1 < L to be zero.

The effect of eliminating a few .,1i ,i:ent Cu(1F) is first examined in a single isolated

chain. In comparison with randomly distributed missing O Hchain + H(), the Cu(1F) line

given by considering Hchain + H () shows a clear ..i-.,iii. 1 i ic line shape with more weight

at lower frequency, as shown in Figure 3-7. The zero magnetization peak appears at all

temperatures, and is specially noticeable at high temperature as the main line narrows.

Broadening of the main line smears out the satellite peak, which could possibly explain its

insignificance at low temperature.

The conclusion from examining the single isolated chain does not alter as the chains

are coupled to the plane. The spectrum obtained by examining eliminating .Idi i.:ent

Cu(1F) sites on the chain-plane coupled system remains qualitatively the same, as

shown in Figure 3-7. Although the distance between main line and satellite peak slightly

increases, which may be due to change of DOS on the chain l-iv-r and can be resolved by

adjusting hopping or chemical potential, the consequence of (1)broadening of spectrum

at low temperature, (2)asymmetric spectral weight, and (3)the high temperature satellite











peak remain the same. The interlayer coupling and the structure of the connection to the

full chain slightly affect the pairing and homogeneous magnetization on the plane, but as

far as chain magnetization is concerned, the interlayer coupling do not significantly affect

its magnitude or spatial distribution.


(a) (b)




200K0K
1 3 0 K T = 1 5 0 K
-150K
=130K

S110K
110K=1




50K 87.8 88 88.2 88.4 88.6
87.4 87.6 87.8 88.0 88.2 88.4
Frequency (MHz) w(MHz)
(c) (d)




=200K 0200K

150K 150K
0K 0K

--T2 -0 | \

=130K -130K


=90K -90K

T 70K =70K

0K 9=50K
87.8 88 88.2 88.4 88.6 87.8 88 88.2 88.4 88.6
w(MHz) w(MHz)


Figure 3-7. Comparison of (a)experimental 63CU(1F) NMR line of ortho-II YBCO6.5 at
external field 7.7T along a direction, with three models studied: (b)isolated 1D
chain with random impurities Hchain + H)p, (c)isolated 1D chain with a
section of .,l1i .ent Cu(1F) erased Hhain + H (d)chain-plane coupled
system with a section of Cu(1F) eliminated Hplae + Hchain + Hinter + H,.
Correlation strength U" 1.2 in these models.









3.4 Chain-Plane Coupling Induced Correlation Effect

Although superconductivity or magnetization in the plane does not pl li an essential

role in the chain magnetization, the ID nature of the chain does have a drastic influence

on the planar magnetization pattern. One expects that since the interlayer hopping

connects the plane with the chain running along the < i l11 ,ii b axis, anisotropic

magnetization may be induced in the planar Cu's. Similar chain-plane coupling induced

anisotropy particularly in the YBCO family has been observed in dc[103, 104] and optical

conductivity[105], as well as Inelastic Neutron Scattering(INS)[106] and penetration depth

measurements[107, 108]. Several authors have proposed that these observed anisotropies

can be explained by proximity models[101, 109-111] similar to the model presented in this

chapter, although anisotropy in the magnetic response is not included in these models. To

address further the issue of anisotropy in impurity induced magnetization, the key relies

on a proper choice of a impurity model that can highlight both the effect of correlations

and the chain-plane coupling. Since the impurity models H) and H() involve more

than one impurity site, we find it ambiguous to define the magnetization along directions

orthogonal to these impurities. In addition, the interference between magnetization

generate by multiple impurity sites is unavoidable. We therefore study a pedagogical

model that contains a single unitary strong scatter located on the full chain, for the sake

of demonstrating the proximity induced anisotropy. The philosophy is that since this kind

of impurity is point-like, it is easy to define two orthogonal directions provided that the

origin is set to be at the projection of impurity site, unlike Hl and H,2 where their

projection on the plane will involve more than 1 site and the origin is ill defined. A strong

unitary scatter on the chain gives

Himp d di, (3-15)


with Uimp > tc. Figure 3-8 compares the magnitude of the magnetization on the full chain,

planar sites along b, and planar sites along a, with normalized induced moment defined









as si = mi mol/mo. One sees that chain has magnetization one order larger than that

of plane along the b direction, which is another order larger than along a direction. On

the other hand, numerics show that if the unitary scatter is located on the plane, then

the anisotropy is much smaller even when interlayer coupling is present. This hierarchy

caused by interlayer coupling and magnetic correlation on each li--r proves that, when

systems with different dimensionalities are coupled, the response of lower dimension drives

the higher dimension, consistent with our expectation that correlation effect becomes more

significant as dimensionality is lowered.

Finally, we address the issue of the correlation between broadening of Cu(1F) and

Cu(2E/F) lines. Experimentally, the identification of each line on the complete NMR

spectrum is made via comparison with YBCO6 and YBCO7 spectra, and the Knight shift

can be extracted by subtracting the orbital contribution Korb associated with each Cu

sp,, i. [' 1]. The reverse process allows the recovery of the NMR spectrum in the present

model, with each line calculated by collection of magnetization and Eq. (3-9). The value

of Korb at external field B//b is taken from Takigawa et al[102]. The spectrum combining

Cu(1F), Cu(2E), and Cu(2F) for model Hpiane + Hchain + Hinter + H is shown in

Figure 3-9, where one sees a significant overlap between Cu(1F) and Cu(2E/F) lines, with

Cu(2E) basically indistinguishable from Cu(2F). The precise position of these lines could

be affected by higher order moments omitted in Eq. (3-9), or deviation of realistic Korb

from the value taken, as well as the empty chain degrees of freedom ignored. Accounting

for these processes could result in separated lines according to experiments. The linewidth

in the present study is therefore calculated by extracting the magnetization of each

Cu species, instead of fitting the full spectrum. The broadening of extracted Cu(2F)

and Cu(2E) lines as temperature is lowered follows that of Cu(1F), as shown in Figure

3-9, with the slope of Cu(2E) smaller than Cu(2F). A similar linear relation between

broadening of planar and chain lines has been observed[95], which serves as another direct

evidence of induced planar magnetization due to defect and magnetic correlations on the









chain. Note the slope of the linewidth correlation is one order larger than the experimental

value, which we found to be possibly due to the difference of definition of linewidth

between experiments and numerics, as well as the value chosen for interlayer hopping,

which can change the influence of magnetic correlations on the plane. Nevertheless, the

linear relation between planar and chain linewidth remains robust in this proximity model.

3.5 Conclusion

In summary, we showed that the broadening of the NMR line at low temperature

in Ortho-II YBCO cannot be explained unless one assumes the existence of magnetic

correlations on the CuO chain. Magnetization is induced at the chain ends, which is

modeled by a reduction of hopping across the missing O, and the broadening of NMR

line corresponds to the enhancement of induced magnetic moments due to magnetic

correlations. The high temperature satellite peak is attributed to a set of Cu(1F) missing

their conduction electrons in the vicinity of the chain end, presumably due to different

valence caused by missing O, which gives zero magnetization and its corresponding

satellite peak in NMR spectrum. Under the influence of magnetic correlations on the

chain, the chain defect can induce anisotropic magnetization pattern on the plane,

where we see planar induced moment along the chain direction one order larger than the

direction perpendicular to the chain. Therefore the YBCO system provides an excellent

example of studying the effect of coupling between systems with different dimensionality,

and correlation in the lower dimension is more prominent and could drive the response of

the higher dimension if coupling between them is present. The possible application of the

present correlated proximity model to the explanation of anisotropy observed in INS[106]

measurements and transport properties[103-105] will be presented in future studies.












6

5

.9 4

(3
-3


2

1


0 10 20
r


30 40


0 10 20
r


0.08

0.06

0.04

0.02


0 10 20
r


30 40 (d)


plane s, density plot


Figure 3-8.


Demonstration of anisotropic magnetization due to chain-plane coupling, in
which we plot normalized magnetization si induced by substitution of a
Cu(1F) by a point defect: (a)on the chain, (b)on the plane along chain
direction, (c)on the plane along direction perpendicular to the chain. The solid
line is a fit to spherical Bessel function si oc siKo(i/f)/Ko(1/f). Size of the
plane is 81 x 24, at temperature 70K. The real space pattern of si on the
plane is given in (d) and (e), where one can see the extension of magnetization
along chain direction, coinciding with (f)the magnetization induced on the
chain l-ivr.


30 40


plane s,


chain s,
























0.0415
I
0.041
"o
- 0.0405
N
- 0.04
o
-J


87 87.2 87.4 87.6 87.8 88
w(MHz)


Cu(2F): 0.046x+0.028





Cu(2E): 0.025x+0.033


0.27 0.28 0.29 0.
Cu(1 F) standard deviation(MHz)


Figure 3-9.


0.012


0.000 0.010 0.020 0.030 0.000 0.010 0.020 0.030
Cu(1F)-M (T) (d) Cu(1F)-M (T)

(a)The combined spectrum of Cu(1F), Cu(2F), and Cu(2E) lines, together
with (b)the linear relation found between broadening of chain and planar lines.
In comparison, the experimental data of linewidth correlation is given in (c)
for Cu(2F) and (d) for Cu(2E)[95].









CHAPTER 4
EFFECT OF DOPING ON IMPURITY INDUCED MAGNETIZATION FROM A
STRONG COUPLING PERSPECTIVE

Portions of this chapter have been published previously as M. C i1 vi et al, Phys. Rev.

B 77, 165110 (2008).

4.1 Slave Boson Mean Field Theory

In this chapter, we study the influence of doping on the impurity induced moment

due to planar defects. The experiments with which we will compare are summarized in

Section 1.7, and we especially focus on the comparison in the normal state. The evolution

of magnetization induced around a single nonmagnetic impurity under the influence

of external parameters, in particular temperature and doping, will be studied. Such a

single impurity problem has received a considerable amount of attention, including weak

coupling approaches, which are similar to the mean field theory constructed previously in

Ch.2 and Ch.3[89, 112-115]. Several authors assumed the existence of induced moments

around the impurity, and study the response of the system within the context of Kondo

screening[116, 117]. Other models that include the effect of pseudogap and a resulting

spinon bound state have also been proposed[118, 119], as well as essentially numerical

works[120-123].

Our discussion in Ch.2 and Ch.3 involves thermodynamic observables in the presence

of induced moments, studied within Hartree-Fock mean field theory at a specific doping.

The phenomenological parameters that enter the Hamiltonian, for instance hopping t

or Hubbard U, could be determined by fitting experiments related to the quasiparticle

spectrum or the magnitude of the induced moment. However, doping affects these

phenomenological parameters in a strongly correlated system like cuprates. This is because

if one starts from the undoped parent compound, which contains one electron per site on

the CuO plane and is a Mott insulator, the doping has two primary effects: The presence

of vacant sites increases the mobility of electrons, as well as diminishes the magnetic

correlations in the parent compound. In addition, since Tc increases upon doping in









T

TBEC



IV
N TRVB


II X I



III SG

Doping

Figure 4-1. The phase diagram of slave boson mean field theory for homogeneous cuprates.
The three temperature scales represent the Bose-Einstein condensation
temperature(TBEC), the onset of spin liquid order(TRvB), and the spin gap
order(Tsc). Under these interpretation, we identify four phases in the phase
diagram: (I)Fermi liquid, (II)spin gap phase, (III)superconducting state, and
(IV)the strange metal phase. In this chapter we focus on the impurity problem
in the phase (IV) where we have nonzero spin liquid order, but the system is
above Bose-Einstein condensation temperature and has no pair correlations.


the underdoped region, it si.-.- -1- the correlation between superfluid density and hole

concentration. The maximum value of T, is therefore a competition between reducing

magnetic correlations needed for singlet pairing and the increase of charge carrier density,

both result from doping. On the other hand, in a strong coupling perspective, one has

to take into account the strong Coulomb repulsion even when mobile holes are present,

which prevents two electrons from populating at the same site. This nondouble occupancy

constraint, which leads to the effective exchange interaction in the strong coupling limit,

lie at the heart of magnetism in cuprates.

To properly describe the evolution of bandwidth upon doping and the nondouble

occupancy constraint, we adopt the slave boson mean field theory for the metallic state

of the cuprates. The schematic phase diagram for this approach to homogeneous cuprates

is shown in Figure 4-1. Here we focus on the "strange metal" phase as the interpretation









of normal metal above the superconducting dome in this particular type of mean field

treatment. These theories, applied to the t J type of models, provided some of the

first qualitatively correct, if crude, accounts of the high T, phase diagram[36-38]. In the

slave boson representation, a projected electron operator is decomposed into a fermionic

"-1i'..''' and a bosonic "iI..!.i operator c = bifi, in which the spinons represent the

spin degrees of freedom, and the holons represent the empty sites. This decomposition

is possible because we start from the half-filled undoped compound, which has exactly

one electron per site, and the effect of doping is to introduce holons with an additional

constraint Y, ff, f + bbi, 1 imposed locally on each site i. This constraint ensures

the non doubly occupancy for the site simultaneously by spin up and down electrons, even

when the system is doped. As discussed in Section 1.8, the strong Coulomb repulsion also

motivates us to start with the t t' J model Eq. (1-13), which describes a single band

of projected electrons with particle-hole .,i- ii,,i. 1 ry, interacting via an exchange coupling

J. Applying the slave boson formalism to the t t' J model gives

Ho K t hhbf Q f. gIIBB y f
ijj iff

+ J(Si S 4nn) Pf fi f
1
iu
b b bi + iA( t-f + bb 1), (4-1)
i i
where the Lagrangian multiplier A, ensures the slave boson constraint, and the Zeeman

term gpBB/2 takes into account the splitting between up and down spins. We chose

t = 400meV, t' = -0.4t, and J = 0.22t which yields a reasonable spinon and holon

effective bandwidth close to optimal d..l1 pin as addressed below. Numerically, the density

interaction term ninj/4 which resulted from the canonical transformation will be omitted

since it can be absorbed into the definition of fermion chemical potential after mean field

factorization, and only the exchange interaction will be considered, with the definition of

the spin operator Sq = .









Since the hole doping affects the mobility of charge, it -, I-.- -; the following mean
field factorization of both fermionic and bosonic effective hopping amplitudes[124, 125]


(f tfj) -= Xl, (bIbi) = Qj ,

(S) mi {(btb}) Nb (4-2)

where mi is the static magnetization and 6 is the hole doping. The mean field Hamiltonian
then becomes

Ho ,(- -- tijj)fitfj



+ Z( 4 -< f
ijac


+ (Y, x- y,x^}f

ij a
+ Z(- tijYXj)btb


ij a
+ ZJ Y Sf -m-tfY
i(
g- IB ftfi u- bY bi. (43)
i(T i
Details of the mean field formalism are discussed in Appendix E. In the unperturbed,
spin-degenerate case all the order parameters are real and constant, which gives the

following spinon and holon dispersion

((k) -(Jx + 2tQo)[cos(k,) + cos(k,)]

4t'Q' cos(k) cos(k,) i ,

,(k) = -4tX[cos(k,) + cos(k,)]

8t'XO cos(k,) cos(k,) b. (44)

In the presence of impurities, local order parameters are renormalized and the correlation

effect is measured by the ratio between exchange coupling J and the local spinon hopping









amplitude tf = (Jx/2 + tQ) and t' = t'Q', which is now a site dependent variable. The
local holon energy scale will be tb = 2tx and t' = 2tx', which is also affected because the

slave boson constraint alters the holon distribution following the density modulation of

spinon, as we address below.
4.2 Single Impurity Problem

The effect of a single strong nonmagnetic impurity, for instance a Zn substitution for

a planar Cu, is to change the local chemical potential of both spinon and holon on the
impurity site 0 by

Hi A(f fo0 + bobo) (4-5)

with A -- oo. This is equivalent to projecting out site 0 and producing a topological defect

on the two dimensional square lattice. We then adopt real space exact diagonalization for

both spinon and holon Hamiltonians

Hlf E ) = E E),

HbE) = E ), (4-6)

where IEL) and IEb) are the spinon and holon eigenstates, respectively. The corresponding

real space wave functions are


af ,(i) ( (i|EL,)

a (i) = (iE) (47)

(4-8)









which can be used to calculate the local order parameters


ijic (f f1,) >




n ( ) -2 1(iff 2)f(f(E)L)







the local shift of chemica potential) such that 1 for each site i except the
Slab () 12 b(E) (4-9)

The non double occupancy constraint is satisfied by adjusting the Lagrange multiplier (effectively

the local shift of chemical potential) A such that n +n +n b 1 tfor each site i except the
impurity. Numerics are done on a 20 by 20 lattice with periodic boundary conditions and

t = 400meV, t' = -0.2t, J = 0.22t, comparable to the parameters in the semianalytical

approach which we briefly introduce in the next section.

4.3 Comparison with Semianalytical Approach

In the presence of an impurity, the density modulation of spinons and holons

results in the spatial dependence of order parameters in the mean field Hamiltonian

Eq. (4-3), and hence affects the propagation of quasiparticles. In a recent paper by C 'l1'

et al.[125], we have first adopted a semianalytical approach to solve the renormalization

of order parameters and the resulting induced magnetization. The exact diagonalization

method is then also implemented and compared with these results. The basic idea of the

semianalytical approach is to construct the unperturbed Green's function in the absence of

impurity, then solve for the order parameters when the impurity is present[125, 126]. With

these renormalized order parameters the full Green's function is constructed and can be

used to calculate the induced moment when spin degeneracy is lifted. Starting from the

dispersion Eq. (4-4), and defining the Hilbert space of the spinon to be f, |0) = i, a), we









can construct the unperturbed spinon Green's function with complex argument by

l ,
z Ho
1 eik'(ri-rj)
0 i2. -- (4 10)
k

Similarly, we can define holon Hilbert space b~ 0) i) and the holon's unperturbed

Green's function


G(z) (il ljb
z HFo
1 eik.(ri-rj)
N Wk(4 11)
N Z Uok
k

Notice that at this level the up and down spins are degenerate so G o G Dyson's

equation for the single impurity Hamiltonian Eq. (4-5) in the A -- oc limit can be solved

exactly, and we denote this Green's function by yij,


G0 o io 'j (4 12)
i Ga Go
00

where 0 is the impurity site. Note that this spin degenerate Green's function vanishes

when either i or j are on the impurity site 0, and gives various renormalized order

parameters. For example, the "spin liquid" order parameter


Xijc (f-itfJ)

lim gi,(r) (4-13)
7-o+

with imaginary argument 7 = it. The next step in the C i, ,v et al approach involves

self-consistency between the following two equations


G + GV(mT)G,

m = f (w) Im (Gi G) (4-14)











1.4-

1.2- T=500 K
ST=75 K
1.0-- AT=20 K -

0.8- -

S0.6-
4-
0.4-

0.2-

0.0
0 2 4

r.



Figure 4-2. Spinon effective bandwidth tf for doping 6 = 0.3 in semianalytical approach of
C li-v et al[125] near the impurity.


where the first equation is Dyson's equation with potential V(mi) in the presence of

magnetization mi, and the second one expresses mi in terms of the full Green's function

G. The potential V(mi) accounts for the magnetic scattering due to induced moments

which couple to their neighboring sites because of the exchange coupling J


V(mi) = J (mi mo)oni, (4-15)
a

where mo is the homogeneous magnetization and j sums over nn sites of i. A numerical

diagonalization of G in real space up to the third shell of sites away from the impurity site

is applied to solve for mi, by assuming that farther away the Green's function recovers

its analytic form g in Eq. (4-12). The effective spinon bandwidth tf = (JX/2 + tQ)

calculated by using Eq. (4-13) is shown in Figure 4-2, where one sees a drastic reduction

over a length scale that increases as temperature is lowered.

The finite size diagonalization approach in the present report, in comparison with

the semianalytical result, has the following two advantages: (1)The slave boson constraint

is strictly satisfied locally on each site except for the impurity, and (2)it includes all











1.0-

0.8-
3 ass a -* a

0.6

---000
-- 0.4- _o
o o Analytic 8=0.3 T=100K
0.2- o SC 8=0.15 T=100K
SC 8=0.15 T=300K
0.0
0 1 2 3 4 5

r.


Figure 4-3. Comparison of spinon effective bandwidth tf between semianalytical approach
of C il, i' et al[125] and present exact diagonalization.


the higher order diagrams in the Dyson's equation. Regarding the first point, in the

semianalytical approach the particle number is imposed globally, in the sense that the

holon population is set to be equal to the doping. However, when solving Dyson's equation

self-consistently via Eq. (4-14), the density modulation may in this case violate the local

slave boson constraint. The real space diagonalization with proper choice of Lagrangian

multiplier Ai resolves this difficulty. Regarding the second point, since Hamiltonian Eq.

(4-3) and Eq. (4-5) are diagonalized exactly, it is equivalent to including all the higher

order diagrams in the single impurity problem.

The comparison between semianalytical approach and the real space diagonalization

therefore provides some insight into how the slave boson constraint and higher order

scattering processes alter the result of magnetization. In Figure 4-3 we show the

comparison of these two methods, which -i--.-- that slave boson constraint mitigates the

reduction of the spinon bandwidth tf near the impurity, and hence reduces the effective

correlation strength J/tf. We therefore expect a smaller magnetization compared to

the semianalytical result. In addition, the temperature dependence is mitigated, and









one expects that the screening temperature 0 in the Curie-Weiss form will increase

accordingly, as we address in the following section.

4.4 Induced Moments and Influence of Slave Boson Constraint

In this section, the magnetization patterns of semianalytical approach and exact

diagonalization are compared. Following the discussion of the last section, we expect

the renormalized fermion bandwidth to have a significant effect on the induced moment.

Defining the normalized -1 .--, red magnetization si = (i mo)/mo where mo is the

homogeneous magnetization, the results of the semianalytical approach are summarized in

Figure 4-4. The screening length is fit to


sr Nx siKo(r/)/Ko(f1/) (4-16)


where Ko(x) is the spherical Bessel function of the first kind. Such a form of spatial

dependence is motivated by theories dealing with similar problems in 1D[52], but is

not rigorously justified. The magnetization at the nn sites of the impurity is fit to a

Curie-Weiss form

S C/(T + 0) (4-17)

with effective moment C slightly decreased and screening temperature O increased in the

overdoped region. In comparison, the results of real space diagonalization are shown in

Figure 4-5. Notice that in the present mean field theory the choice of periodic boundary

conditions enhances the interference of the magnetization produced by the(periodic)

"-is!, impurity, and hence gives rise to an unavoidable phase transition to long range

magnetic order as temperature is lower than TN ~ 100K, below which the theory breaks

down. This mean field artifact hinders the exploration of the extreme low temperature and

small doping region. We also found that the Friedel oscillation in the presence of magnetic

field produces a residual magnetization sl -- constant at high temperature, which affects

the accuracy of the Curie-Weiss fit because sl -- 0 in Eq. (4-17) at high temperature.

The effective moment and screening temperature are therefore determined via a fit over









the intermediate temperature range such that both long range magnetic order and residual

magnetization of sl are avoided.

Comparison of the spatial dependence of the magnetization around the impurity

between the semianalytical approach and the numerical approach again reveals the

influence of the slave boson constraint, and is summarized as follows: (1)Comparing Figure

4-4(a) and the doping 6 = 0.3 curve of Figure 4-5(a), the magnitude of sl is drastically

reduced as the constraint is imposed. This is consistent with the analysis of the effective

spinon bandwidth in the previous section, which shows that imposing the slave boson

constraint mitigates the reduction of spinon bandwidth tf and hence effectively reduces

J/tf, resulting in less magnetic correlation and smaller si. (2)In the exact diagonalization

results, the screening temperature O shows an abrupt increase as the doping is increased

through optimal doping 6 = 0.15, consistent with experimental data. The semianalytical

approach however has such an abrupt increase at higher doping(roughly 6 = 0.27 by

extrapolation), as shown in Figure 4-4(c). This is consistent with previous slave boson

mean field treatments which have been shown to overestimate the doping scale, and we

found that imposing the nondouble occupancy constraint locally can resolve this dilemma.

(3)The screening length v.s. T fits better with experiment when the constraint is

imposed, indicating the importance of the local constraint to both the Friedel oscillation

and the resulting induced moment pattern.

To address further the issue of spatial distribution of magnetization affected by

the nondouble occupancy constraint, we investigated both the spinon and holon density

patterns in the vincinity of the impurity via the exact diagonalization method. Holons,

due to their bosonic statistics, develop an unphysical length scale that depends only

on temperature if they are unconstrained. The length scale that characterizes Friedel

oscillation of spinons is the Fermi wavelength, which is associated with the Fermi

momentum kf and hence the doping. In the present mean field theory, however, spinons

and holons renormalize each other's spectrum via Hamiltonian Eq. (4-3), so density










a) b)
2-
2.0

1.8-

1. -- 1.6-

1.4

1.2-
0-
1 2 3 4 50 100 150 200 250 300
r T[K]

c) d)
4-
1500-
-.-C
3-
1000-
2-

500- -.=0.32
@ -8=0.3
8=0.28
0
0.28 0.30 0.32 0 50 1
0 50 100 150
T


Figure 4-4. Summary of semianalytical approach of C i1' iv et al[125] with external field 7
Tesla. (a)Magnitude of normalized magnetization s, = (mr mo)/mo near the
impurity at T = 25K, overdope 6 = 0.3. Solid line: fit to
Isr, o s1Ko(r/%)/Ko(1/) for = 3. (b)T-dependence of normalized
magnetization at nn site of impurity at overdoping 6 = 0.3. (c)Effective
moment C and screening temperature O fit to the Curie-Weiss form
C/(T + 0) for si, near overdoped region. (d)T-dependence of screening length
at different dopings, fit to the form in (a).


modulations affect order parameters and in turn the spectrum when impurities are

present. If only Fermi and Bose statistics determine the density modulations, numerics

show that such a system prefers to have holons stay far .1i ,- and have density almost

zero around the impurity, which also reduces the spinon bandwidth. This is because

Qij (btbi) (bbbi) = n at low temperature, therefore the spinon bandwidth

tf ~ 2tQ ~ 2tnb is reduced accordingly, resulting in strong magnetic correlation J/tf













1.5


. 1


0.5


T(K)


Figure 4-5.


10n I4n


1400
1200 0(K
1000
800
600
600 C(K)
400 a .
200
EE b)
0.15 0.2 0.25 0.3


4
S6=0.15
o6=0.175
3 6=0.2
6,=0.25
wr2




d)


.'"" 100 200 300 400
Temperature (K) T(K)

Results of exact diagonalization at 7 Tesla. (a)T-dependence of nn site
normalized magnetization si in different dopings. (b)Effective moment C and
screening temperature in the Curie-Weiss fit of si. (c)Data by Ouazi et al
showing T-dependence of and magnetization versus distance r away from
impurity(insert). (d)i v.s. T from optimal to overdoping by exact
diagonalization.


for the spinon, and the system alv--x flows to a long range magnetic order phase. The

semianalytical approach reaches similar conclusions, although renormalized fermion

bandwidth is only calculated up to g, not full self-consistently. One sees that the dramatic

reduction of tf in Figure 4-2, indicating a drastic suppression of holon density in the

vicinity of impurity, and is clearly an artifact of the mean field theory without local

constraint.


h<









(a) (b)

0.89 0.6
0.8 -
0.89 0.4
0.2
nf 0 5 10 15 20
0.85
x

(c) 0.2 (d)
'Q 0.2

10.17 0.1
b 0 5 10 15 20
0. 15

(e) (f)
^0.003 0.002

0.002 E 0.001
0.001 0 5 10 15 20
0 x

Figure 4-6. Density pattern by exact diagonalization at T = 100K and = 0.15.
(a)Spinon density, (c)holon density, (e)magnetization plot, and their
corresponding cut through impurity site along (1,0,0) direction in (b), (d), (f).
Spinon and holon oscillation shows the same healing length kf associated with
the sape of spinon fermi surface, while magnetization oscillates under which
is related to effective coupling J/t/.

The density pattern and magnetization of exact diagonalization method is shown
in Figure 4-6, together with their plot cut through impurity site along (1,0,0) direction.
Comparing (b) and (d) of the figure, it is clear that the holon healing length is now the
same as the spinon length scale kf, beyond which the homogeneous value is recovered.
We conclude that the nondouble occupancy constraint "-! iv the bosonic length scale
to that of the fermions, because the modulation of holons must now compensate the
modulation of spinons. The magnetization, however, exhibits a different length scale than









the healing length of either density modulation, as one compares (f) with either (b) or

(d) in Figure 4-6. This is because the magnetization is governed by the strength of the

exchange energy J/tf and its associated length scale (, which is not directly related to kf.

The mechanism by which kf and ( depend on temperature or doping are also different.

kf remains roughly the same in the relevant temperature range, which is much lower

than the Fermi energy, while doping has a more dramatic effect on kf since it changes the

shape of Fermi surface. The temperature dependence of comes from the reduction of

magnetic order due to thermal excitations, similar to the usual mean field theory of the

Heisenberg model[4]. Doping affects indirectly by changing particle densities and the

associated bandwidth, resulting in larger effective coupling strength J/tf as the system is

underdoped.

4.5 Conclusion

We studied the creation of impurity induced magnetic moments and their doping

and temperature dependence within the framework of slave boson mean field theory.

The spinon and holon degrees of freedom are separated but related to each other via

renormalization of the effective hoppings, and the non double occupancy constraint

which comes from the strong Coulomb repulsion. A comparison with the semianalytical

approach where the constraint is enforced globally is given. Once the constraint is

imposed locally on every site, holons inherit the fermionic length scale of spinons and

their density recovers the homogeneous value far from the impurity with the same spinon

healing length. The reduction of the effective spinon bandwidth around the impurity

was found to be weaker in the fully self-consistent approach, giving a relatively smaller

correlation strength J/tf and hence a smaller induced moment in comparison with the

semicalssical approach. In comparison with experimental results shown in Figure 1-6(b),

we see that the reduction of correlation length as increasing doping is recovered, whose

magnitude is consistent with the 17 NMR result[53]. In addition, the susceptibility at

the nearest-neighbor site of impurity satisfies a Curie-Weiss form, in which the abrupt









enhancement of screening temperature 0 around the optimal doping is consistent with the

7Li NMR result[55] shown in Figure 1-6(d). We conclude that the slave boson mean field

theory plus the exact diagonalization method can capture the impurity induced moment

formation in cuprates, in which the influence of strong correlations can qualitatively

change the temperature and doping scales implied by other treatments of the same

problem, and hence is important to correctly describe the impurity physics in these

materials.









CHAPTER 5
CONCLUSION

We studied the correlation effects in disordered cuprates, and showed that the

-i .-r-. ed magnetization induced around nonmagnetic impurities can be understood

within a simple mean field description. These induced moments, which in the case

studied were found to be paramagnetic, can explain a variety of experiments, including

the anomalies in transport properties due to the enlarged cross sections associated

with these induced moments. The existence of magnetic correlations particularly on

the full chains of YBCO chain-plane system is found to induce nontrivial anisotropic

magnetization pattern on the CuO2 plane, and can account for the broadening of NMR

lines due to the formation of magnetic moments around the chain ends. The adoption

of a slave boson formalism can properly describe the evolution of susceptibility from

Curie to Curie-Weiss as doping is increased, and the effect of strong Coulomb repulsion

and the resulting no double occupancy constraint are found to have crucial influence on

the Fermion band width, and hence can dramatically change the pattern of magnetic

moments in the vicinity the impurities. The generalization of current theory to study

other thermodynamical observables, such as anisotropic structure factor observed by INS

experiments on strongly underdoped YBCO, and the complete evolution of 7Li Knight

shift across the superconducting transition temperature, will be presented in future

studies.










APPENDIX A
DERIVATION OF CURRENT OPERATOR

There are two v--o- we can derive the current operator on a lattice. The first one is

by defining the local polarization operator at site i


Pi A


and the local current will be

Si i i H,we g

and for the model Hamiltonian of interest we get


(A-l)


(A-2)





(A-3)


t6
Z z\-i


Or the other way is to assume a site dependent vector potential Ai, such that the hopping

part of Hamiltonian picks up a phase shift


H,- t, ci (A4)
i+i


and the current operator is, for its x-component


Jj lim 6H
SU6H
AT -oh AT



it u / ^ixu0 ~ ^cr+x,) + itl Y_^{C^_yyii,
it + r + it a
(T (T


,t
^i ci+x ycr)
X


(A-5)


which gives the same thing. Notice that we've used a dummy index for the last step to

convert Jf into a bond variable between site i and site i + x.









APPENDIX B
DETAILS OF CURRENT-CURRENT CORRELATION FUNCTION

Since we are interested in the dc limit u -+ 0 of conductivity, we can simplify the

frequency dependent part in oi by following the approach by Takigawa et al[88]. Starting

from the linear response theory for the conductivity


lim IM (7 (w))}


jwo + w ET E, + iTl
j n,m +

the frequency dependent part can be rewritten as

lim o f(E) f(Em,) 1
Em,,o Im-
w j + E, Em + ir
lim f(En) f (Em) -I
o w (w + E, Em)2 + r2
f(E) f(Em )
S-7 lim f-(Ew) f( 6 + E, E,)
w ->O (a
7 fi f(En) -f(En + ) E )
-7T lim 6,() + E, E,)
w -L
df(E))(E E
dE,

-T d6( E)6( Em)(- () (B-2)


where the delta function is defined as 6,u( E,) = (r/r)/((w E,)2 + r12). So the

conductivity becomes


S( df (a))
,r = Z (n|J'rn) (TnJjln) Jdu6(a E)6(a Em)(E daw
j n,m
S7t f njJinTF)(T,.Jjn)F(En,Emn) ,
j n,mr

and apparently this function F(En, E,) is symmetric under exchange of ET Em.


(B-3)








APPENDIX C
REMARK ON THE ABSOLUTE SCALE OF RESISTIVITY
Here we describe the the procedure of converting numerical resistivity to experimental
units. The current operator in Eq. (A-5) actually has a prefactor e/h, and is in units of

eta Cm
J] [ ] (C-1)
hi s

where a is the lattice constant. The retarded current-current correlation function defined
in Eq. (2-9) has units of
C2m2
T7 )] [J~JJ] = 2 (C-2)

but what we are interested in is its Fourier transform in 2D

T (:", )= j j dreti ,(r) (C-3)

and has units of
1 C2
-(:" ) A] [j ij ] C2 (C 4)

which gives correct units of conductivity in 2D

7 '7(: )] C2 [2
[a] [h] ] (C*5)
[hw] Js h

To give more details, after summing over all sites i and putting in the physical units,
DC conductivity in Eq. (B 1) could be written as, if we only consider nearest neighbor









hopping


-1 (n| ieta
cr hN2a2,m h i t


ieta
( h 2JCjU


-) 1 f(En)
Cjaj+xWn) lim -Im (E
) w-o ) + (E, -


- f(E,)
Em)/h+ iT


(() I | c X+,c, c4,c+z I|m)
n, i

(mn Ct+c j aj+X|n)(- h7F(En, E))


( ) nre
n,m


cv Ctxc
Ny/^^-rc


cii+xm})


(mN S cY c


c cj+xIn)jF(E, E,,) ,


(C-6)


where the new F(E,, E,) is defined by


F (E, E,)


f duwl(w


E)6(wu


EG
h


(C-7)


df(h))
dh)'


as in Eq. (C-6). What is calculated in my code is the following dimensionless quantity,

where every energy is rescaled by t


(7 = t| c +, c c dc+, |m)
nm i


x (m c + xcjc


x hdU 1 -i _
J E(h ) +(Er)2 )2 w
t t t t

= (n c i+ c ~ cci+, I nm)
n,m


x (m| T cj +zc,


x () du6(a -
h
e2


ht
t


t


cj c+xln)


2,


E,
h



df (h))
xtx(-
dhnijj


cjcj+x, 12)


x ( df((hw))
dAw


(C-8)










with the dimensionless differentiation of fermi function defined as
rw/t
df(hw)) t )e


1 e BT
t x ( ha)) (C9)

tx(- d (C-9)


Therefore the true conductivity is the numerical number found by the code multiplied by

e2/h
e2
a = = 7 x (C-10)

However experimentally what the four-point measurement gets is the 3D bulk conductivity

a3D. To convert the above 2D conductivity into 3D, we make the following two assumptions:

(1)both a2D and s3D satisfies Drude form

n2DC2
92D

33D 2
m

and (2)the conducting charge in cuprate comes completely from the copper planes, so the

relation between n2D and n3D is


n3D X V = Nplane X n2D x A, (C-12)


where Npiane is the number of planes per unit cell, V and A are the volume and cross

section area of the unit cell. Since most of our results will be compared with YBCO, we

take Npane = 2 and V/A = 11.68A. Thus the absolute scale of resistivity is

1 V hi
P3D = ( )(
o3D 2A e2 y
1
x 241pQ cm. (C-13)
7










The resistivity plotted is 1/7, and one can multiply by 241/p cm to get the experimental

P3D-









APPENDIX D
BDG FORMALISM OF ORTHO-II YBCO6.5 SYSTEM

Here we provide the details of BdG Formalism for the YBCO chain-plane coupled

system. Similar procedures are applied to description of metallic states in Ch. 2, except

only planar degrees of freedom are considered, and the pairing channel on the plane is

turned off in that case. Hartree-Fock-Gorkov factorization of bare Hamiltonian Eq. (3-2)

consists of a rearrangement of each interaction term. The on-site Coulomb interaction is

decomposed into


i


P Uf)p + ( i

i
> Up[(np )(n )+ + p 1,' + (h )nP ,1]
i
.up I t i +p (ii ii, h p
iUP[T TnP h )^- 1t )]

i(
E TTrn^~ m^P / .*.LU
LU- -- < ^ ^ )


(D-l)


with the definition of order parameters Eq. (3-4). The last term in the last line is a

constant and could be dropped in the diagonalization process. The same procedure

works for the U' term in chain degrees of freedom. The pairing channel is factorized by,









considering nearest neighbor pairing 6 = {x, y}


( Vjr)
(ij)


VCXt t
C iTvcici+ s1ci+X
i6
- Vc vci+t c t
i5
- V +[(ctce+6) + (ci{c+6 (csci+6))]
i6
x t[ ) + (ct 6 t ciCt c))t

V[(ciTi+a} + (c tCTi+)] [<(C+ + 6(r+~ )]
iS
- V[ c c Ti+S) ci +S6 c + (c icS)C sTCi+ (cic)i+ Si c t ]
iS
= V[(citci+s61 )cici+ + (c+i ct )ci+6ciT + (ciTci+ ) (+cct)]
i5
i Ac Tc + h.c. + V(ciTci)+(ct +, (D-2)t
i6


which again generates constant term but could be dropped in the diagonalization process.

The mean field Hamiltonian Eq. (3-3) is therefore recovered.

As magnetic field is applied, the spin degeneracy is lifted and one expects that up and

down Hamiltonian should be considered separately. Under a Bogoliubov transformation

Eq. (3-5), the spin up channel satisfies the following equations


E i, I


E iI IT

-EnvIvn*T


E P -upT Tun + (-pP + A)rj + Ai .i
J 5
> nj -trvc*t + (-pP + A)vQ1* + > Asji1
J 6
t3 ^n'x+65 n<4 (-,c + A)uXt

tCvp* trc* T (-c + ,c*
- n,x+6T ,T + (_pc + A)v X,


(D-3)









where A is the energy scale that splits up and down degeneracy, in present case the

Zeeman term A = gpB/2. Their counterparts in spin down channel are


-Enrt un




E',T %,


E- t-up t + (-p A)up Z A6v.I


j 6
S t ,u+<5i t-u,+l + (- A)uX,

> tC n p* tr c*+ ( c A)vc*
,X+61 n,i + -I *


One observes that, a matrix representation could be constructed for the equations

associated with spin up eigenenergies En\


as well as spin down eigenenergies


where (, (c, ( are terms in Eq. (D-3

chain hopping, and the interlayer part


o U< ul
-r* pT Vp
SEn (D-6)
0 U C U C



.) and Eq. (D-4) associated with planar hopping,

respectively. From BCS theory we know the


eigenenergies of quasiparticles are ahl--iv positive energies above the ground state, hence

we can rewrite the matrix equation Eq. (D-6) of EnT and simplify it by the following


(D-4)


(D-5)









procedure


0I




A*
-




0



0
-0









A*




0


which becomes exactly the n


-A 0 w

0 x

0 0 y

o -F / \ x
0z

x (-1) on both sides

A -4 0 w
0* 0 (r* x

o o y
0 0 y

(r* 0 (C*
o / \ z /
rearrange w x, y -+ z

A* o* 0 x

-( 0 -1 w

0 o C* 0

-1 o -v Y
take complex conjugate

A 0 x*

-* o T* w*

o o Z*
0 0 *

-Ia* o -e E *

latrix equation Eq. (D-5) for


X*

W*

Z*

y/*

spin up eigenenergies Ei,


except the eigenvalues are negative. We therefore diagonalize only the spin up Hamiltonian,

and then attribute the positive eigenvalues c > 0 to the wave function (u~1, v uTi, )T,

which has eigenenergies Ei = e, and negative eigenvalues e < 0 to (v~, u, v, uj)T ,

with eigenenergies E,, = |I = -e.

The particle densities could be calculated by knowing that the eigenstates are a basis

of quasiparticles ,-t 0) = In), and (,- I 7,,f) = f(E,,) since they satisfy Fermi statistics.


(D-7)









Therefore the charge density for each species may be written as


+c P* +, P V / _Oiyj) P /(u + P* ^/ty
m>0 n>0
S p Ij, 2(. t U/ \ vIi', 2+/ \i)
n>0
= ~ 2f(E o) ,i (1- f( )). (D-8)
n>0

According to the arrangement of eigenvectors, we see


(c t1c) ,it 2 f(ET) + vi (1 f (E,))
n>0
uPi 2f(E) + P 1 2f( -E)
n>O
= 1st element of eigenvector n) 2f(C) ,
all n
(c cit) ,iU, 2f(E,t) + vI,V 1 (i f-(E,))
n>O
I f4 2( f( -E1t)) + IVPi12( f (E,(t))
n>0
= Y 2nd element of eigenvector n) 2( f(C))
all n
(dt dt) E u 1 T2f(E) + IT2(1 f(E,1))
n>O

= E u 2 f(E,) + I T 2( E)
n>o
S|3rd element of eigenvector n) 2f(C)
all n
(dtdj) 2f(E) + IV,, ,2(i f(ET))
n>0
~l 2(1 f(-Et)) + I ,i 2(1 (ET))
n>0
S14th element of eigenvector n) 2(1 f(en)) (D-9)
all n
Notice that in the last step this procedure we convert the physical sum over positive

eigenstates to all eigenstates for numerical convenience and to avoid double counting. The

pairing field could be simplified via the same trick, and notice that the pairing amplitude









of chain 1 -i.r is not zero, although it does not renormalize the Hamiltonian


(CiTCi+6) (Y-(<^ ,iT7mT + ,T'yT) ,i+5 I n+ +Vn7))
m>O n>O
>1/1* i v \(':*;1' i^ '-* )
/ T n*i +S7t7t + vniU Tni+S6 $N I)
n>O

> I +(11 (t- f (E.)) + vT<,zi+65i (En.)
n>O

S n.,iT I 11 (n,)) + UT i (1 f (- S))
n>O
S (1st element of eigenvector In))
all n
x(2nd element of eigenvector In))*(l f(Te)) ,
< t +<5() < (y+ t)+5$ t+ n7))
(dTTdns) (-(Urn)xTYmT + n,T"'nT") 16 +
m>O n>O
S c/'^xTnC*'x+S6 (7nt7tT) + V-xTtUCzx+6s (7t,1" I
n>O
uxTt p* +6(1 f (E)) + xvTX*ux(,x+6tf (Ent)
n>O

SnTT + (1 f(Et)) + v .Tiu,+(1( f(-E))
n>O
Y(3rd element of eigenvector In))
all n
x(4th element of eigenvector In))*(l f(T)). (D-10)









APPENDIX E
DETAILS OF SLAVE BOSON MEAN FIELD APPROACH

The details of slave boson mean field treatment will be provided in this chapter.

Starting from the definition of order parameters, we can factorize the exchange term into


S = SIS (E-1)

The saddle point approximation is then introduced for each channel


I dx6(x a) l= t dxdxei- ), (E2)

in the Lagrangian formalism. The unperturbed Lagrangian is therefore

L C oo + xij, (Xj f* fj,) + i ix (x ,-* f f) (E-3)
ija ija
+ 1 iQ(Qi, b bi) + iQ(Q* b bj) (E-4)
ij ij
+ i (Z bb ) + iN)+Zm(m S) + J Z mi m (E-5)
i i


a ija ija
(1 6)2 2N, (E-7)
4

where fi, is Grasmann number and bi is complex, and the time-derivative part is


(oo f~(, Pf)f, + Eb*(a, b)bi (E-8)
io- i
+ iAj(fg*,f + bbi- 1). (E-9)
i7
The saddle point equation for fields Xija, Qij and mi is therefore

J
2 sij 2X- tijQij = o0, (E-10)

=iQl 5s Xt x = 0 (E- 11)

am = im +J m,=0, (E-12)
j n.n i









which are equations of motion that Xyij, Qj and mi have to satisfy. After replacing these

fields and dropping the constant terms in the original Lagrangian, we obtain the mean

field version


L = oo+ (- ,- iQi)f,,fy, (E- 13)
iji




ij Y(
+ -^ tij )6b (E- 14)




+ J Sz-m .. (E-1 7)


Notice that the spin up order parameter X* renormalizes spin down channel f,* fj[ and

vice versa. It's also manifest that spinons and holons renormalize each other, in the sense

that spinon orders enter holon dispersion and vise versa.
that spinon orders enter holon dispersion and vise versa.









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BIOGRAPHICAL SKETCH

Wei C('!, was born in Taoyuan, Taiwan, as the first son of his parents Sung-Jung

C(! i, and Chuan-Chuan C('! i, He got the bachelor degree in National Chung C('!. 1

University, Chiayi, with a major in physics. At the age of 22, he went to National Tsing

Hua University in Hsinchu, studied with professor Hsiu-Hau Lin. The research in this

period of time was mainly about low dimensional correlated systems, such as carbon

nanotube and ladder materials. In 2003, he took the chance of studying in the US

and started pursuing a Ph.D degree in University of Florida, under the instruction of

Professor Peter Hirschfeld, with superconductivity related research topics and the exciting

phenomena of cuprates. As a theoretical physicist, he continues working on the area of

condensed matter physics and dedicate himself to the development of theories regarding to

this discipline.





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IamespeciallygratefulfortheinstructionandcollaborationofTamaraNunner,BrianAndersen,AshotMelikyan,LingyinZhu,andLexKemperoverthepastseveralyears,particularlyforassistancewithnumericalmethods,andalsoforthefruitfuldiscussionswithGregBoyd,SiegfriedGraser,KevinIngersent,DanielArenasandVivekMishraoverthegeneralideasaboutmyresearchandtherelatedphysicalproblems.Thecollaborationduring20052006inPariswithMarcGabaycouldnothavebeenaccomplishedwithouthisgeneroushospitality.Andnally,thankstomyadviserPeterHirschfeldforalwaysbeingpatientandprovidingsuggestionsanddirectionsformyresearch,theseworkswouldnothavebeenpossiblewithouthisbeautifulinsights. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFFIGURES .................................... 7 ABSTRACT ........................................ 12 CHAPTER 1INTRODUCTIONTOCUPRATES ........................ 14 1.1GeneralIdeaoftheApproach ......................... 14 1.2UniversalPhaseDiagram ............................ 15 1.3AntiferromagneticState ............................ 16 1.4SuperconductingState ............................. 18 1.5PseudogapState ................................ 20 1.6NormalState .................................. 22 1.7ImpuritiesandInducedMagneticMoment .................. 24 1.8ProperChoiceofaMicroscopicModel .................... 27 2DISORDERINDUCEDRESISTIVITYUPTURNSINMETALLICCUPRATES 32 2.1TransportPropertiesinDisorderedCuprates ................. 32 2.2ModelHamiltonian ............................... 37 2.3ResistivityUpturnsatOptimalDoping .................... 43 2.4EectofPseudogaponResistivity ...................... 47 2.5Conclusion .................................... 50 3CHAINDEFECTSINYBCO6.5ORTHO-IISYSTEM .............. 52 3.163CuNMRinYBCOChain-PlaneSystem .................. 52 3.2BroadeningofLowTemperatureNMRLines ................. 56 3.3ExplanationofHighTemperatureSatellitePeaks .............. 63 3.4Chain-PlaneCouplingInducedCorrelationEect .............. 66 3.5Conclusion .................................... 68 4EFFECTOFDOPINGONIMPURITYINDUCEDMAGNETIZATIONFROMASTRONGCOUPLINGPERSPECTIVE ..................... 71 4.1SlaveBosonMeanFieldTheory ........................ 71 4.2SingleImpurityProblem ............................ 75 4.3ComparisonwithSemianalyticalApproach .................. 76 4.4InducedMomentsandInuenceofSlaveBosonConstraint ......... 80 4.5Conclusion .................................... 85 5

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.................................... 87 APPENDIX ADERIVATIONOFCURRENTOPERATOR .................... 88 BDETAILSOFCURRENT-CURRENTCORRELATIONFUNCTION ...... 89 CREMARKONTHEABSOLUTESCALEOFRESISTIVITY .......... 90 DBDGFORMALISMOFORTHO-IIYBCO6.5SYSTEM ............. 94 EDETAILSOFSLAVEBOSONMEANFIELDAPPROACH ........... 100 REFERENCES ....................................... 102 BIOGRAPHICALSKETCH ................................ 110 6

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Figure page 1-1Schematicofuniversalphasediagramofcuprates.Fourphasesappearwithchangingdopingandtemperature:antiferromagneticstate(AF),superconductingstate(SC),pseudogapstate(PG),andnormalstate(N).ThenatureofthetransitionbetweenNandPGstatesiscontroversial. .......................... 16 1-2SummaryofantiferromagnetismrelatedexperimentsonLSCO. ......... 17 1-3Summaryofsuperconductingstatepropertiesimportantforthisthesis.(a)GapversusanglerelativetotheCu-ObonddirectionfromARPESshowsacleard-wavefeature[ 24 ].(b)RealspaceSTMimageofsingleimpurityembeddedinBSCCOshows4-foldsymmetryofimpurityboundstate[ 19 ].(c)LDOSon(solidline)andaway(dottedline)fromZnimpurityinBSCCO,whereimpurityboundstateformsinsidethed-wavegap[ 19 ].(d)Neutrondiractionpatterncentersat(;)forLSCOundermagneticeld,indicatingmagnetizationinandaroundvortexcores[ 22 ]. ................................... 19 1-4Summaryofimportantpseudogapstateproperties.(a)89YNMRinunderdopedYBCOshowsreductionofspinsusceptibilitystartingattemperatureaboveTc[ 31 ].(b)GaplessexcitationintherstquartetofBrillouinzonehasanarcshape(thickline)thatextendsalongtheexpectedFermisurface(dottedline)astemperatureincreases[ 26 ].(c)ReductionofDOSatFermisurfaceaboveTcasmeasuredbySTMinBSCCO[ 25 ].(d)NernstsignalaboveTcinvariouscompoundsindicatesvortex-likeexcitationsabovesuperconductingstateinthephasediagram[ 41 ]. 21 1-5NormalstateresistivityinYBCO[ 44 ].(a)Homogeneousabplaneresistivityversustemperatureatdierentdoping.OneseesaclearlinearrelationathighTbutdeviationatlowertemperature.(b)UsingdeviationfromlinearT(colorscale)toidentifypseudogaptemperature(opencircles)[ 45 ]. ................. 22 1-6ImpurityrelatedNMRexperiments.(a)17ONMRinsuperconductingYBCO,whichshowsdefectconcentrationbroadenstheline[ 54 ].(b)Correlationlengthdeducedfrommultinucleimethod,indicatingincreasingcorrelationlengthasunderdopingandloweringtemperature[ 53 ].(c)7LiNMRKnightshiftinYBCOatunderdoping(top)andoptimaldoping(bottom),togetherwiththeirttotheCurie-WeissformC=(T+)(solidline)[ 56 ].NoticethatchangesdramaticallyacrossTc.(d)7LiNMRKnightshiftplottedwithscale1=T.Theinsertgureshowsscreeningtemperatureversusdoping,whichabruptlyincreasesaroundoptimaldoping[ 55 ]. ................................. 25 7

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63 64 ].(a)Underdopedregion=0:08showsdivergenceofabatlowtemperature.(b)BothabandcshowlogTdivergencewhenplottedonlogarithmicscale.(c)Phasediagramdeterminedbymappingoutd=dT,whichshowsametal-insulatortransitioninunderdoperegion.(d)Magnetoresistnanceofabatdierenttemperatures. ............................................. 33 2-2In-planedisorderresistivityabofBSLCOin(a)optimaldopingregion,and(b)underdoperegion[ 67 ].UnderdopesampleclearlyshowsamoredramaticupturninlowT.(c)BothabandHallconstantRHshowslogarithmicdivergenceatlowT.(d)NegativemagnetoresistanceofbinthepresenceofappliedeldH==c,forunderdopesample0:125with2:2%impurities. ...................... 34 2-3In-planeresistivityabofYBCO[ 71 78 ].(a)Impurityconcentrationdependenceofabforanunderdopedsample,showinglowtemperaturedivergenceontopofthelinear-Tresistivity.Thedivergenceisenhancedwithincreasingdisorder.InsertgureshowsthelogTformofthedivergence.(b)Dopingdependenceofabataxedimpuritycontent,whichshowsmoredramaticdivergenceasthesampleisunderdoped.(c)Detailsoflowtemperatureregionintwodierentimpurityconcentrationz=2:7%(top)andz=2:3%(bottom).MagneticeldclearlyreducesTcandrevealstheupturn.(d)Magnetoresistanceatdierenttemperatureforunderdoped(top)andoptimallydoped(bottom)sampleupto60T.Underdopedsampleshowspositivemagnetoresistancewhileoptimallydopedsampleremainsconstant. ....................................... 36 2-4Comparingmagnetizationinducedbyasinglenonmagneticimpuritystudiedinthischapterforthenormalstate,andd-wavesuperconducting(dSC)statestudiedin[ 89 ].Realspacemagnetizationpatternfor(a)normalstate,and(b)dSCstate,bothatU=1:75andB=0:01.OneseesclearlythatdSCstatehasmorepronouncednearest-neighborsitemagnetizationduetoboundstateformation,andhassmallerhomogeneousmagnetizationduetoopeningofthegapsttheFermisurface.Theseeectsareshownatoneplots(c)totalmagnetizationofthesystemSzand(d)magneticcontrastversusexternaleld. ......... 42 2-5RealspacemagnetizationpatternatT=0:03andB=0:001,with2%impurities.OneseesclearlyatransitionofmagnetizationpatternasincreasingU,wherewefounditscriticalvaluetobearoundU1:75abovewhichthemagnetizationisnolongercharacterizedaslocalizedaroundeachimpurity. ........... 44 2-6Resultsatoptimaldoping.(a)Magnetizationand(b)resistivityversusUatT=0:03,with2%impurities. ............................... 44 2-7Resultsatoptimaldoping.(a)Magnetizationand(b)resistivityversusTatU=1:75,with2%impurities. .............................. 45 2-8Resultsatoptimaldoping.(a)Magnetizationand(b)resistivityversusBatT=0:03andU=1:75,with2%impurities. ...................... 46 8

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2{11 )withf=0:6andas-wavegapk==0:2,and(c)theextendedhoppingmodelafterFouriertransformEq.( 2{12 ). .... 48 2-10Comparisonofmodelswith(Extended)andwithout(Normal)reductionofDOSbyapplyingextendedhoppingEq.( 2{12 ).(a)Magnetizationand(b)resistivityversusTbothatoptimaldoping,withU=1:75and2%impurities. ...... 49 2-11Comparing(a)magnetizationand(b)magnetoresistancebetweenextendedhoppingmodelEq.( 2{12 )andthenormalmodelbothatoptimaldoping,T=0:03,U=1:75,with2%impurities. ........................... 51 3-1AschematicdescriptionofthedopingprocessinYBCObychangingtheoxygencontentonthechainlayer[ 52 ].TheundopedparentcompoundYBCO6containsCu+onthechainlayer,andtheCuO2planehaseectivelyoneelectronperCusite.Ifoxygensareintroducedintothechainlayerbuttheircontentissucientlylow,thenthesparseO2onlychangesitstwoadjacentCu+toCu2+.WhenasegmentofatleasttwoO'sisformed,theexcesschargewillbetransferedtotheCuO2plane,resultingintheholedoping.AfullchaincanthereforedopetheCuO2planeinamoreecientwaythanarandomOdistribution. ..... 53 3-2Crystalstructureofortho-IIYBCO6.5andrelatedparentcompoundYBCO6andYBCO7,withprecisedenitionofchainandplanarCu[ 94 ]. ......... 54 3-3(a)FullNMRspectrumofOrtho-IIat60KinthepresenceofexternaleldH0==a,wherethefrequencyisexedat75.75MHz[ 94 ].Toppanel:repetitiontimeTrep=250msec,middlepanel:Trep=25msec,lowerpanel:thedierence.ComparisonwithYBCO6andYBCO7conrmstheassociationbetweenlinesandtheCuspecies:A!Cu(2F),B!Cu(1E),C!Cu(2E),D!Cu(1F).(b)STMimageonthechainlayerofYBCO,whichshowsoscillationofDOSalongchaindirection[ 96 ].NoticethatthesampleisnotOrtho-IIandhasrelativelyshortsegmentsoffullchains(about10sites). 55 3-4Schemeticofbroadeningof89YNMRlineinYBa2NiO5bymagnetizationinducedaroundimpurities[ 52 99 100 ].Therightpanelshowstheamplitudeofmagnetizationinducedonseveralsitesneartheimpurity,whichresultsinsatellitesignalsnearthemainline.Asignicantamountofsuchinducedmagnetizationeventuallysmearsoutthespectrumandbroadenstheline(theinsertofleftpanel). .... 56 3-5SchematicofthehoppingpartofmodelHamiltonianEq.( 3{2 ). ......... 60 3-6Numericsinanisolated1Dchain,equivelenttoHamiltonianHchain+H(1)imp,inanexternaleld7.7Tat50K.Eachcurveisnormalizedsuchthattheareaunderneathisconserved. .............................. 62 9

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.............................. 65 3-8Demonstrationofanisotropicmagnetizationduetochain-planecoupling,inwhichweplotnormalizedmagnetizationsiinducedbysubstitutionofaCu(1F)byapointdefect:(a)onthechain,(b)ontheplanealongchaindirection,(c)ontheplanealongdirectionperpendiculartothechain.ThesolidlineisattosphericalBesselfunctionsi/s1K0(i=)=K0(1=).Sizeoftheplaneis8124,attemperature70K.Therealspacepatternofsiontheplaneisgivenin(d)and(e),whereonecanseetheextensionofmagnetizationalongchaindirection,coincidingwith(f)themagnetizationinducedonthechainlayer. ................. 69 3-9(a)ThecombinedspectrumofCu(1F),Cu(2F),andCu(2E)lines,togetherwith(b)thelinearrelationfoundbetweenbroadeningofchainandplanarlines.Incomparison,theexperimentaldataoflinewidthcorrelationisgivenin(c)forCu(2F)and(d)forCu(2E)[ 95 ]. ........................... 70 4-1Thephasediagramofslavebosonmeaneldtheoryforhomogeneouscuprates.ThethreetemperaturescalesrepresenttheBose-Einsteincondensationtemperature(TBEC),theonsetofspinliquidorder(TRVB),andthespingaporder(TSG).Undertheseinterpretation,weidentifyfourphasesinthephasediagram:(I)Fermiliquid,(II)spingapphase,(III)superconductingstate,and(IV)thestrangemetalphase.Inthischapterwefocusontheimpurityprobleminthephase(IV)wherewehavenonzerospinliquidorder,butthesystemisaboveBose-Einsteincondensationtemperatureandhasnopaircorrelations. ..................... 72 4-2Spinoneectivebandwidthtffordoping=0:3insemianalyticalapproachofGabayetal[ 125 ]neartheimpurity. ........................ 78 4-3ComparisonofspinoneectivebandwidthtfbetweensemianalyticalapproachofGabayetal[ 125 ]andpresentexactdiagonalization. ............... 79 4-4SummaryofsemianalyticalapproachofGabayetal[ 125 ]withexternaleld7Tesla.(a)Magnitudeofnormalizedmagnetizationsr=(mrm0)=m0neartheimpurityatT=25K,overdope=0:3.Solidline:ttojsrj/s1K0(r=)=K0(1=)for=3.(b)T-dependenceofnormalizedmagnetizationatnnsiteofimpurityatoverdoping=0:3.(c)EectivemomentCandscreeningtemperaturettotheCurie-WeissformC=(T+)fors1,nearoverdopedregion.(d)T-dependenceofscreeninglengthatdierentdopings,ttotheformin(a). .......... 82 10

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............... 83 4-6DensitypatternbyexactdiagonalizationatT=100Kand=0:15.(a)Spinondensity,(c)holondensity,(e)magnetizationplot,andtheircorrespondingcutthroughimpuritysitealong(1,0,0)directionin(b),(d),(f).Spinonandholonoscillationshowsthesamehealinglengthkfassociatedwiththesapeofspinonfermisurface,whilemagnetizationoscillatesunderwhichisrelatedtoeectivecouplingJ=tf. .................................... 84 11

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1 ].Notonlydoesthetransitiontemperatureexceedtheboilingtemperatureofnitrogen,makingavarietyofapplicationspossible,butinaddition,theunusualelectronicpropertieshavechallengedcondensedmatterphysicistsoverthepasttwodecades.ThekeytounderstandingthemechanismbehindsuperconductivityreliesontheproperdescriptionoftheCuO2planecommonlysharedinallhighTccupratematerials.Althoughitstwo-dimensionalstructureseemsremarkablysimple,noconsensusonamicroscopicdescriptionortheoriginofunusuallyhightransitiontemperaturesisavailableatpresent.ThecomplicatedphasediagramofthecupratesindicatestheimportanceoftheinterplaybetweendierentcorrelationscoexistingintheCuOplane,andthecompetitionbetweenthembecomesespeciallymanifestwhenexternalperturbationsarepresent.Thisisbecausewhentheorderparameterthatcharacterizesaspecicphaseisdiminishedduetoperturbations,thecompetingcorrelationshavethechancetorevealtheirimportance.Inthisreportweparticularlyfocusonthetwomostsignicantinstabilitieswhichexistinallcuprates,namelysuperconductivityandantiferromagnetism,anddiscussthethermodynamicanomaliesinduced,particularlybydisorder.Wewillshowthatanomaliesinavarietyofexperimentscanbeexplainedbyasimplemeaneldtheorythatproperlyincludesbothsuperconductingandmagneticinstabilities,inwhichtheeectofdisorderistoinducelocalizedmagneticmomentsthatinturnaectthethermodynamicandtransportobservables.Inthefollowingsections,webrieyintroducethegeneralpropertiesofeachphaseobtainedbychangingdopingandtemperature,andhighlighttheimportanteectsofdisorderthatarerelevanttothestudyinthelaterchapters. 14

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Schematicofuniversalphasediagramofcuprates.Fourphasesappearwithchangingdopingandtemperature:antiferromagneticstate(AF),superconductingstate(SC),pseudogapstate(PG),andnormalstate(N).ThenatureofthetransitionbetweenNandPGstatesiscontroversial. Theeectiveholedopingontheplanecouldbecalculatedviabond-valencesumorothertechniques,anddespitethevarietyofcuprates,thedependenceoftransitiontemperatureondopingisfairlyuniversalamongdierentcompounds.Half-lled"parentcompounds"typicallydisplaylongrangemagneticorder,whichisdestroyedwithincreasingdoping.Superconductivityoccupiesadomeshapeinthemoderatedopingandlowertemperaturepartofthephasediagram,withametallicstatewhichexhibitsexoticpropertiesoutsideofthesuperconductingdome.Figure 1-1 givesaschematicofthephasediagramwhichisuniversaltoallthecupratematerials.Wenowdiscusseachphaseinsomedetail,togetherwithopenquestionswhichawaitfutureresearch. 2 3 ].Althoughtheexactsolutiontothismodelremainsunknownatpresent,variousnumericalandsemiclassicalapproachesconrm 16

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(b) (c) (d) Figure1-2. SummaryofantiferromagnetismrelatedexperimentsonLSCO.(a)InverseofantiferromagneticcorrelationlengthversustemperatureshowsNeeltemperatureTN195KforLSCO[ 3 ].Thesolidlineistbysimulationofspin-1 2Heisenbergmodel.(b)Susceptibilityversustemperature,wherethedivergenceindicatesNeeltemperatureTN245K[ 2 ].Thesolidanddottedlinesarettospin-1 2Heisenbergmodel.(c)Ramanscatteringshowsahighfrequencypeakwhichisattributedtoscatteringfrommagnonpairswithoppositemomenta[ 5 6 ].(d)Neutronscatteringshowsexcitationsinmomentumspace,togetherwiththettospin-wavetheorywhichgivesspin-wavevelocity~c0:78eV[ 7 ]. theexistenceofniteorderedstaggeredmagneticmoment,butstronguctuationsduetolowdimensionalitysuppressthestaggeredmagneticmomentfromitsclassicalvalue.ThelongrangeorderisdestroyedatatemperaturescaleTN250Kfortheundopedparentcompound,andthettoneutronscatteringdatagivesareasonableestimate 17

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4 { 7 ].Thelongrangeorderisalsorapidlydestroyedbyitinerantholesintroducedupondoping.Atlowtemperature,a"spinglass"phaseisfoundasdopingexceedsabout0:05,inwhichspinsuctuateinrealspaceandproducenoeectivelongrangeorder,butthecorrelationlengthfallsowithapowerlawdependence.Theexistenceofantiferromagneticandspin-glassphasesindicatestheimportanceofelectron-electroninteraction,sincemagneticorderingresultsfromtheeectivecouplingduetostrongCoulombrepulsion.Approachingtheuniedtheoryfromitshalf-llingparentcompoundinspiresawholeclassoftheoriesbasedontheMott-transitionandthesuperexchangemechanism,wherecorrelationsgaintheirimportanceasdopingisreduced[ 8 ]. 9 ],butimportantfeaturesthatmakecupratesdierentfromstandardBCStheoryhavealsobeenobserved.Therstsignicantdierenceisthed-wavesymmetryofthesuperconductinggapinmomentumspace.ThisniterangepairingcanbeinterpretedasaneectofthestrongCoulombinteractionprohibitingdoubleoccupancypersite;hence,superconductivityisactuallyasecondaryeectinheritedfromthestrongCoulombrepulsion[ 8 ].InsteadofafullygappedFermisurfaceasinaconventionalsuperconductor,cuprateshavenitegaplessnodalexcitationsinmomentumspacealongthe(;)directions,andfullygappedantinodes(gapmaximum)aroundthe(;0)directions,whichresultsinalineardensityofstatesnearthechemicalpotential.ThecriticaltemperatureTcisfoundtoscaleroughlywithsuperuiddensityns,whichincreasesandthendecreaseswithdoping,whilethesuperconductinggapdecreasesmonotonicallywithdoping,leading 18

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(b) (c) (d) Figure1-3. Summaryofsuperconductingstatepropertiesimportantforthisthesis.(a)GapversusanglerelativetotheCu-ObonddirectionfromARPESshowsacleard-wavefeature[ 24 ].(b)RealspaceSTMimageofsingleimpurityembeddedinBSCCOshows4-foldsymmetryofimpurityboundstate[ 19 ].(c)LDOSon(solidline)andaway(dottedline)fromZnimpurityinBSCCO,whereimpurityboundstateformsinsidethed-wavegap[ 19 ].(d)Neutrondiractionpatterncentersat(;)forLSCOundermagneticeld,indicatingmagnetizationinandaroundvortexcores[ 22 ]. totheproposalthatthereexisttwodistinctenergyscalesincuprates[ 10 { 13 ],incontrasttoconventionalsuperconductorswhereTcandgapareproportional. 19

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14 ].Inaddition,impuritiesinthesesystemsdonotproduceboundstates.Bycontrast,ifthegapexhibitsd-wavesymmetry,boundstateswithenergiesinsidethegapareinduced.TheboundstatewavefunctionstheninterfereandeventuallysmearoutthegapandreduceTc[ 15 { 17 ].Theexoticresponsetononmagneticimpurities,plustheirinterplaywithmagneticcorrelationswillbethemainpointofthisthesis.Secondly,cuprateshavecoherencelengthstypicallyabout46latticeconstants,whichmakesthemverystrongtype-IIsuperconductors.Theshortcoherencelengthindicatestheimportanceofspatialinhomogeneity,inwhichoneexpectsastronglocalmodulationoforderparametersbyvarioussourceslikevortices,impurities,orgrainboundaries.Anyoftheseeectscandestroythelocalpairedstateandrevealsomethingabouttheunderlyingcorrelations[ 18 { 23 ].ThecriticaleldHc2100Tisquitebeyondcurrentexperimentallimits,andhencetheextremehighmagneticeldregionisstillachallengeforbothexperimentalistsandtheoristsinthenearfuture. 25 ].AcloseranalysisinmomentumspaceusingARPESshowsasectionofgaplessexcitationswhichextendsfromthenodalpositiontoantinodesalongtheFermisurface[ 26 { 30 ].Thiscontradicts, 20

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(b) (c) (d) Figure1-4. Summaryofimportantpseudogapstateproperties.(a)89YNMRinunderdopedYBCOshowsreductionofspinsusceptibilitystartingattemperatureaboveTc[ 31 ].(b)GaplessexcitationintherstquartetofBrillouinzonehasanarcshape(thickline)thatextendsalongtheexpectedFermisurface(dottedline)astemperatureincreases[ 26 ].(c)ReductionofDOSatFermisurfaceaboveTcasmeasuredbySTMinBSCCO[ 25 ].(d)NernstsignalaboveTcinvariouscompoundsindicatesvortex-likeexcitationsabovesuperconductingstateinthephasediagram[ 41 ]. e.g.,Luttinger'stheorem,whichrequiresaclosedFermisurfaceaboutsomepartintheBrillouinzone.Thelengthofthisunique"Fermiarc"increasesastemperatureincreases,andthecompleteFermisurfaceisrecoveredonlyattemperaturesaboveT.Therst 21

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31 { 34 ].OneearlysuggestionabouttheoriginofpseudogapbehavioristhatitrepresentsagapforthebreakingofCooperpairs,whichexistbutarenotphase-coherentintheregionTcTpartofthephasediagram.AstemperatureisraisedaboveTc, 22

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43 44 ].Suchalinear-Tdependenceoverasmuchasthreedecadesoftemperatureisnotconsistentwithnormalmetalphysics.Althoughlinear-Tbehaviorsarefrequentlyobservedoverlimitedtemperatureranges,theyareusuallyfoundintheclassicallimitofelectron-phononscatteringwellabovetheDebyetemperature.However,theresistivityatoptimaldopinginthecupratesshowsnoindicationofanyfeatureattheDebyetemperatureatall.Aphenomenologytodescribethelinear-TbehaviorgoesunderthenameofmarginalFermiliquidtheory,wheretemperatureisassumedtobetheonlyimportantenergyscaleatlowfrequency[ 46 47 ].ThedeviationfromlinearrelationontheunderdopedsidecoincidesroughlywithT,whichhasbeenusedasanotherwaytoidentifythepseudogaptemperature.TheresistivityontheoverdopedsidegraduallyevolvesintoaT2powerlaw,whichiswellunderstoodwithinLandau'sFermiliquidtheory.Besidesthelinearresistivity,disorderplaysanimportantroleinidentifyingthethermodynamicpropertiesespeciallyatlowtemperatures.Experimentally,thechangeintheresistivityduetodisorderismeasuredbyintroducingplanardefectsandsometimesinadditionanappliedmagneticeld,whichsuppressessuperconductivityandisassumedtorevealtheunderlyingnormalstateresistivity.SuchmeasurementsconrmMatheissen'sruleathightemperature,wherebyresistivityduetodisordersisadditivewithrespecttothemysteriouslinearterm.Inaddition,astemperatureisloweredatoptimaldoping,theresistivitydecreasesuntilthesuperconductingtransition.EitherdisorderorthemagneticeldsuppressesTc,andincleansamplesresistivityversustemperaturesatisesapowerlawdowntoverylowtemperatures.However,ifthesystemisunderdopedorhighlydisordered,theresistivitybeginstoincreasewithdecreasingtemperaturebelowatemperatureoforder30K,whichhasbeentakenasasignatureofmetal-insulatortransition.Theseresistivityupturnshavebeeninterpretedwithinseveraldierentpictures,andwillbestudiedinthepresentwork. 23

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18 ].Samplesbelongingtothiscategorywouldbeveryusefulinordertounderstandtherelationbetweenlengthscalesintroducedbydisorderandtheglobalvariationofpairing,ordemonstratechangeoftheelectronicenvironmentbydefectsifalocalmeasurementisavailable[ 48 49 ].Althoughithasbeenproposedrecentlythatspecicwaysofintroducingout-of-planedisordercouldpossiblyenhanceTc[ 50 ],onegenerallyexpectsthatplanardefectsdestroysuperconductivity.ForintrinsicallyhomogeneoussamplesliketheYBCOfamily,impuritieshavebeenintroducedinacontrollablewayviadierentsourcesandtechniques.Theplanardefects,whichchangethelocalenvironmentsoftheCuO2plane,usuallyresultfromthereplacementofplanarCuwithionsofthesameordierentvalences,andmaybeconsideredasbaremagneticornonmagneticscatteringcentersdependingonthenatureoftheimpurities.Theothercommonwayofproducingcontrollablenonmagneticimpuritiesisbyelectronirradiation,whichknocksouttheplanarOandproducestopologicaldefectsonthesquarelattice.TheresponseofthesystemtodefectsproducedbyelectronirradiationisfoundtobequalitativelythesameasreplacingplanarCubyimpuritieswiththesamevalencebutahigherlocalchemicalpotential.ThemostcommonexampleisZn,whichproducesnearlyunitaryscatteringontheimpuritysite.ThefactthatZnandirradiation-induceddefectsproducenearlythesameNMRsignaturesuggeststhatagoodtheoreticalmodelforaplanardefectissimplyamissingsiteonasquarelattice.ThetheoreticalandexperimentalaspectsofimpuritiesincuprateshaverecentlybeenreviewedbyBalatskyetal[ 51 ]andbyAllouletal[ 52 ].Inthepresenceofamagneticeld,theT-dependentshiftandbroadeningofNMRlinesatlowtemperaturesindicatetheformationofmagneticmoments[ 53 ].Thecorrelation 24

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(b) (c) (d) Figure1-6. ImpurityrelatedNMRexperiments.(a)17ONMRinsuperconductingYBCO,whichshowsdefectconcentrationbroadenstheline[ 54 ].(b)Correlationlengthdeducedfrommultinucleimethod,indicatingincreasingcorrelationlengthasunderdopingandloweringtemperature[ 53 ].(c)7LiNMRKnightshiftinYBCOatunderdoping(top)andoptimaldoping(bottom),togetherwiththeirttotheCurie-WeissformC=(T+)(solidline)[ 56 ].NoticethatchangesdramaticallyacrossTc.(d)7LiNMRKnightshiftplottedwithscale1=T.Theinsertgureshowsscreeningtemperatureversusdoping,whichabruptlyincreasesaroundoptimaldoping[ 55 ]. betweenbroadeningandimpurityconcentrationfurtherimpliesthecloserelationbetweenlowtemperaturemagneticmomentformationandthelocaldefects[ 54 ].Thecorrelationlengthdeducedfrom7ONMRconrmsthatmagnetizationisindeedinducedlocallyaroundthedefects,andgraduallydecreaseswithdistancefromthedefect[ 53 ].Knightshiftdatafrom7LiNMRinparticularprovidesinformationonthelocalenvironmentveryclosetothedefects.DuetothedierentvalenceofLi,anextraelectronisproduced 25

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55 56 ].ThesusceptibilityextractedisfoundtotaCurie-WeisslawC=(T+)bothaboveandbelowTc,indicatingthepresenceofamagneticmomentduetothestrongcorrelatednatureofthemetallicbackground.Theinducedmagnetizationisalsofoundtobeenhancedintheunderdopedregionofthephasediagram,suggestingitsintimaterelationwithantiferromagneticcorrelationsintheundopedparentcompound[ 56 ].BelowTc,thereductionofdensityofstatesneartheFermisurfacepromotesboundstateformationandhenceincreasesbothdensitymodulationandaninducedmoment,althoughthebackgroundhomogeneousmagnetizationofthehostisreducedbytheformationofsingletpairs.WethereforeexpecttheCurie-WeisstacrossTctoundergoadramaticchange,whichisindeedobservedin7LiNMR[ 56 ].Theelectrondensityisalsomodulatedaroundplanardefects,aneectwhichiswellunderstoodwithinthecontextofaquasi-2DFriedeloscillation[ 57 ].Whenamagneticeldispresent,thesplittingbetweenupanddownspindensityofstatesresultsinaninducedmagnetizationaroundtheimpurities.However,themagnetizationresultingfromFriedeloscillationofthedensityofstatesinafreeelectrongasisroughlytemperatureindependent,whichcannotaccountfortheenhancementofinducedmomentastemperatureislowered.Furthermore,itsmagnetizationismuchsmallerthanobserved.MotivatedbytheNMRexperimentsinYBCO,onereasonableproposalisthatmagneticcorrelationscoexistwiththesuperconductinginstabilityinthecomplexphasediagram,withitsstrengthincreasingtowardhalf-lling.Inthepresenceofimpurities,superconductivityissuppressedduetoitsd-wavesymmetry,andmagnetizationisinducedasaresultofmagneticcorrelations.Inthisthesis,weshowthatsuchaninterplaybetween 26

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58 { 60 ].ZhangandRiceproposedthat,whentheOholesarepresent,theypairupwiththeCuholesandbecomemobilethroughtheCu-Obondwavefunctionoverlap[ 61 ].ThebasicideaisthatonecanconstructanoperatoricorrespondingtocreatingaholeonthesquareoffourOsitescenteredonaCusite,andformseitherasinglet(-)ortriplet(+)withtheholeontheenclosedCu2+site(notethathalf-lledcuprateshaveoneholeperCusite) 27

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1{3 )ofjdiisequivalentto 1{5 )isaneectiveonebandmodelthatdescribeshoppingonasquarelatticeformedbyCusites,withthefactor1nithatprojectsoutdoubleoccupancy.Thismotivatestheadequacyofaneectiveonebandmodelthatcancapturethebasicfeaturesofelectroniccorrelationsincuprates,withstrongcouplingU>tthatpreventselectronsfromsittingonthesamesite.Fromastrongcouplingperspective,higherordertermsmaybeshowntoinducespin-spininteractions,leadingtotheso-calledt-Jmodel,whichtreatsthehoppingtermasaperturbation[ 62 ].Thet-JmodelisthereforethestrongcouplinglimitoftheHubbardmodelusedtounderstandtheMotttransitioninasimpleway.Todemonstratethebasicideaofthiseectivespin-spininteraction,westartfromhalf-lling, 28

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whereci=di(1ni).TheeigenstatesofHji=EjicanbeusedtoexpandjiintermsofeigenvectorsofH1 EH1H0ji+XjihjH0ji whereH1ji=E1jiand^P=1Pjihj.Wecanfurtherintroduceanewbasisjiwhichsatisesthefollowingequation EH1H0ji;(1{8)andexpandjias Nowexpandthisnewbasistorstorderinji,weget EH1H0jiji1 (EE1)a=1 29

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30

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31

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63 64 ].Apulsedmagneticeldupto60TwasusedtosuppressTc,andtheresistivitywasmeasureddowntoclosetozerotemperature.Whenplottedonalogarithmicscale,thedivergenceisclearlyseenbelowatemperatureTdev50KwhichindicatesitsdeviationfromlinearTresistivity,asshowninFigure 2-1 .Themagnetoresistanceisshowntoslightlydecreasesaseldincreases,althoughatextremelowtemperaturethemeasurementswerelimitedtoeldrangeB100T.Achangeofslopeind=dTandcontinuedrisetothelowesttemperaturemeasuredwastakentoindicatetheonsetofmetal-insulatortransition[ 64 ].Althoughthiswasoneoftheearliestexcitingdiscoveriesofresistivityanomaly,itisimportanttorememberthatthemajorsourceofelasticscatteringinLSCOcomesfromoutofplaneSrionswhichdopetheCuOplane,andproduceslongrangescreenedCoulombpotentialontheplanarelectrons.Impuritiesofthiskindgiverisetoforwardscattering 32

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(b) (c) (d) Figure2-1. In-planeabandc-axisresistivitycinLSCOinthepresenceofimpurities[ 63 64 ].(a)Underdopedregion=0:08showsdivergenceofabatlowtemperature.(b)BothabandcshowlogTdivergencewhenplottedonlogarithmicscale.(c)Phasediagramdeterminedbymappingoutd=dT,whichshowsametal-insulatortransitioninunderdoperegion.(d)Magnetoresistnanceofabatdierenttemperatures. andsmallermomentumtransfer,asdiscussedin[ 65 ].Howthelongrangedefectsandforwardscatteringaectnormalstatetransportpropertiesintheunderdopedregionremainsanopenquestionatpresent.TheinuenceofplanardefectshasalsobeenstudiedbyintroducingZntotheCuO2plane,inwhichcaseoneseesmoredrasticlowtemperatureupturns[ 66 ]. 33

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(b) (c) (d) Figure2-2. In-planedisorderresistivityabofBSLCOin(a)optimaldopingregion,and(b)underdoperegion[ 67 ].UnderdopesampleclearlyshowsamoredramaticupturninlowT.(c)BothabandHallconstantRHshowslogarithmicdivergenceatlowT.(d)NegativemagnetoresistanceofbinthepresenceofappliedeldH==c,forunderdopesample0:125with2:2%impurities. 67 68 ].In-planedefectsareproducedbyreplacingplanarCubyZn.TheresistivityagainmanifestsadivergenceonalogarithmicscalebelowTdev50K,withnegativemagnetoresistanceintheeldrangeexplored,asshowninFigure 2-2 .Comparingoptimalandunderdopedsamples,thedatashowsthetendency 34

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18 ].SuchinterstitialO's,aswellasawideclassofdefects,arestudiedinthecontextoflongrangescatterers,inwhichtherealspacegappatternandACconductivitycouldbewellunderstoodbyassumingcausingalocallymodulatedpairinginteractionorlongrangeCoulombinteractionontheplanarelectrons[ 48 49 ].ThesimilaritybetweenBSCCO-2212andBSLCOsuggeststhatout-of-planedopantscouldalsocontributetothescatteringrateintheresistivitymeasurementreportedhere,althoughnodirectSTMevidencehasbeenprovidedinBSLCOyet.WeconcludethatBSLCOandLSCOshouldbelongtothe"intrinsicallydoped"catagorythatout-of-planedefectshavedrasticinuenceonthetransportproperties,andthescatteringsourcesarenotsolelytheplanardefects.YBCO123.NMRprovidesevidenceofintrinsichomogeneityofYBCOandmakeitthebestchoicetostudythermodynamicpropertiesinthepresenceofplanardefects[ 69 ].TheresistivityundertheinuenceofdefectsproducedbyZnsubstitution,aswellasthoseproducedbyelectronirradiation,hasbeenstudiedinawiderangeofdoping[ 70 { 79 ].AscalingrelationbetweenthesuppressionofTcandthechangeofresistivitymultipliedbythechargedensitynisfoundinallYBCOandTBCOsamplesstudied,indicatingthattheholecontentistheimportantparametertodescribeelectronictransportoverawiderangeofdoping[ 74 ].Figure 2-3 summarizestheexperimentalresultsinYBCO,whereoneseestheenhancementofresistivityupturnswithincreasingdisorderandunderdoping, 35

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(b) (c) (d) Figure2-3. In-planeresistivityabofYBCO[ 71 78 ].(a)Impurityconcentrationdependenceofabforanunderdopedsample,showinglowtemperaturedivergenceontopofthelinear-Tresistivity.Thedivergenceisenhancedwithincreasingdisorder.InsertgureshowsthelogTformofthedivergence.(b)Dopingdependenceofabataxedimpuritycontent,whichshowsmoredramaticdivergenceasthesampleisunderdoped.(c)Detailsoflowtemperatureregionintwodierentimpurityconcentrationz=2:7%(top)andz=2:3%(bottom).MagneticeldclearlyreducesTcandrevealstheupturn.(d)Magnetoresistanceatdierenttemperatureforunderdoped(top)andoptimallydoped(bottom)sampleupto60T.Underdopedsampleshowspositivemagnetoresistancewhileoptimallydopedsampleremainsconstant. whichisanuniversalfeatureamongallthethreecupratefamiliesreportedhere[ 71 73 ].However,increaseswithincreasingeldintheunderdopedregion,incontrasttothenegativemagnetoresistanceofLSCOandBLSCO.SinceYBCOisnearlyfreeofout-of-planedefects,itisnotclearifonecanattributethedierenceinmagnetoresistance 36

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80 ].Thehigherordercalculationsincorrelateddicorderedsystems,pioneeredbyAltshulerandAronov,showedthatthecorrectionofdensityofstatescruciallyaectsthetransportproperties,andgiveslogarithmicdivergentresistivityin2D[ 81 { 83 ].Particularlyincuprates,wherethestrengthofcorrelationsvarieswithdoping,atheorythatcancovertheanomaliesoftransportpropertiesoverawiderangeofdopinghasyettobediscovered.Inthischapterwefocusontheoptimallydopedandslightlyunderdopedsamples,wheretheFermiliquidpictureproperlydescribestheelectronicexcitationsinthenormalstate,andthestrengthofcorrelationsremainsmoderate.Theresistivityupturnsobservedinthesesampleswithsmallimpuritycontentarenotconsistentwiththeweaklocalizationpicture,sincethemeanfreepathforunitaryscatterslikeZnsubstitutionismuchlargerthantheFermi 37

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84 ],andisrelevantforthetransportmeasurementincupratessincemostofthesamplesarenotdetwinned,theenhancementoftheresistivityupturnsasincreasingimpurityconcentrationsuggeststhattheupturnsareassociatedwiththeintrinsicdisorder,ratherthanthetunnelingbetweenislands.Inaddition,atheorythatcanaccountforthetransportanomaliesshouldalsobeabletoexplaintheimpurityinducedstaggeredmagnetizationobservedindisorderedcuprates.Anaturalsuspicionisthatthesourceoftheextrascatteringratecomesfromtheseinducedmoments,sincetheconditionsunderwhichtheseinducedmomentsareenhancedalsoenhancetheresistivityupturns.SuchideashavebeenexaminedbyKontanietal,wheretheinterplaybetweenimpurityscatteringandthecorrelationeectisformulatedwithintheuctuation-exchange(FLEX)approach,althoughcertainassumptionsaboutthescatteringprocessandtheself-energyhadtobemade[ 85 { 87 ].Sincethelogarithmicdivergentnatureoftheseanomaliesquestionedtheadequacyofniteorderperturbationtheory,weproposeameaneldtheoryplusarealspacediagonalizationtreatmenttodealwiththedisorderedcuprates,wheretheeectsoflocalizationandperturbationsareincludedtoallorders.Theconditionsinwhichpositivecorrelationsbetweenimpurityinducedmagnetizationandtransportanomaliescanbefoundareexamined,whichprovesourhypothesisthattheenhancementofthescatteringrateisduetoanenlargedcrosssectionsassociatedwiththeseinducedmoments.Weshallrstexaminethecorrelationeectinthecaseofoptimaldoping,anddiscussaphenomenologicalFermiliquidmodelthathighlightstheinuenceofthespectralanomalyinunderdopedsamplesinalatersection.Evidencefromexperimentshasalsosuggested 38

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Aftermeaneldfactorization,theHamiltonianbecomes ThesiteenergyitakesintoaccounttheZeemansplittingwhenmagneticeldisapplied,andalsothelocalchemicalpotentialshiftduetoimpurities 2gBB+XrirVr;(2{4)whereVristheimpuritypotentialenergyatsiter.TheCoulombrepulsionUischosensmallenoughsuchthatthemagnetizationinducedbyintroductionofnonmagneticimpuritiesisparamagnetic,whichassuresnoformationofspontaneousmagnetizationineitherthehomogeneousorimpuritycaseattherelevanttemperatures.RealspacediagonalizationisappliedtondtheeigenstatesassociatedwiththeaboveHamiltonian 39

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Thedcconductivityinlinearresponsetheoryisgivenby inthebasisofrealspaceeigenstatesjni,theireigenenergiesEnandthecorrespondingFermidistributionf(En).ThefunctionF(En;Em)issymmetricunderexchangeofEn$Emandaccountsforthefrequencyandtemperaturedependentpartofconductivity[ 88 ].DetailsofthecurrentoperatorJi(t)andF(En;Em)willbediscussedinAppendixAandB.TheproperchoiceofsystemsizeinsimulatingEq.( 2{3 )isdeterminedbyseveralcriteria:Becauseofthearticialbroadeningintroducedwhencalculatingij(),thehomogeneousresistivityhomintheabsenceofimpuritiesandeldislinearin.The 40

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89 ].Beforetheresistivityundertheinuenceofinducedmagnetizationisstudied,wefeelobligedtocomparethepresentstudyinthenormalstatewiththedatainthed-wavesuperconducting(dSC)statestudiedbyHarteretal[ 89 ].SuchacomparisonrevealstheimportanceofniteDOSinthenormalstate,aswellastheboundstateformationinthedSCstate.TheeectofnonmagneticimpuritiesinthedSCstateisstudiedwithintheframeworkofd-waveBCStheoryplusmagneticcorrelations,equivelenttothetwo-dimensionalHubbardModelEq.( 2{1 )withanadditionalpairingcorrelationsbetweennearest-neighborsites.TheauthorsthenappliedaHartree-Fock-GorkovmeaneldfactorizationandsolvedtheBogoliubov-deGennesequationsviarealspaceexactdiagonalization.TherealspacemagnetizationpatterninducedbyasinglenonmagneticimpurityisshowninFigure 2-4 (a)and(b),inwhichwefoundthreemajordierences:(1)Inthepresenceofamagneticeld,thenormalstatehashomogeneousmagnetizationsignicantlylargerthanthatofdSCstate.ThisobviouslyisduetotheopeningofthegapinthedSCstatereducestheDOSattheFermisurface,andhenceexhibitsasmallersusceptibilitythanthenormalstate.(2)Themagnetizationonthenearest-neighborsitesoftheimpurityisdrasticallyenhanced,whichwefoundtobeconsistentwiththeboundstate 41

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(b) (c) (d) Figure2-4. Comparingmagnetizationinducedbyasinglenonmagneticimpuritystudiedinthischapterforthenormalstate,andd-wavesuperconducting(dSC)statestudiedin[ 89 ].Realspacemagnetizationpatternfor(a)normalstate,and(b)dSCstate,bothatU=1:75andB=0:01.OneseesclearlythatdSCstatehasmorepronouncednearest-neighborsitemagnetizationduetoboundstateformation,andhassmallerhomogeneousmagnetizationduetoopeningofthegapsttheFermisurface.Theseeectsareshownatoneplots(c)totalmagnetizationofthesystemSzand(d)magneticcontrastversusexternaleld. formationduetothed-wavesymmetry.(3)ThedSCstatehasashortercorrelationlength,resultingfromtheenhancementofnearest-neighborsitemagnetizationincomparisonwiththerelativelysmallermagnetizationonthesecondandthirdshellsitesawayfromtheimpurity.Togiveaquantitativedescriptionofabovefeatures,wedenedthetotal 42

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Sz=Ximi;=1 wheremiisthemagnetizationatsiteiandm0isthehomogeneousmagnetizationintheabsenceofimpuritiesbutinthepresenceofamagneticeld.Themeaningofistoestimatetheuctuationofmagnetizationawayfromitshomogeneousvaluem0,henceanindicationoflocallyinducedstaggeredmoment.Sinceinterferencebetweenimpuritiesisalwayspresentandthelocalenvironmentisdierentaroundeachimpurity,thedeviationfromm0ofthewholesystemneedstobeconsidered,hencewesummedoveriforinEq.( 2{10 ).ThebehaviorofSzandversustheappliedeldisshowninFigure 2-4 (c)and(d),whereoneseesthatSzinthenormalstateisoneorderlargerthaninthedSCstate,whichisattributedtotheoveralllargerhomogeneousmagnetizationinthenormalstate.However,intheversuseldplot,weseethatafterthehomogeneousmagnetizationissubtracted,asinthedenitionof,thedSCstatehasalargervalueduetotheenhancedmagnetizationattributedtotheboundstateformation.SuchacomparisonemphasizestheimportanceofDOSattheFermisurfacetotheimpurityinducedmagnetization,whichmotivatesustoproposeatoymodelthatcapturestheeectofreducingDOSinthepseudogapstate,aswillbeaddressedinalatersection. 2-5 showstherealspacemagnetizationpatternasUisincreased.Wefoundaverynarrowrange1:7
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RealspacemagnetizationpatternatT=0:03andB=0:001,with2%impurities.OneseesclearlyatransitionofmagnetizationpatternasincreasingU,wherewefounditscriticalvaluetobearoundU1:75abovewhichthemagnetizationisnolongercharacterizedaslocalizedaroundeachimpurity. (a) (b) Figure2-6. Resultsatoptimaldoping.(a)Magnetizationand(b)resistivityversusUatT=0:03,with2%impurities. magneticcontrastversusUatoptimaldopingisplottedinFigure 2-6 ,whereacriticalvalueU1:75isfoundabovewhichnolongerrespondslinearlytotheexternaleld.Thisvalueisfoundtodependonsystemsizeandimpuritycontent[ 89 ].ComparingFigure 2-5 andFigure 2-6 (a),weseethemagneticcontrastisindeedagoodindextocharacterizethemagnetizationinducedbydisorder.TheresistivityincreaseswithincreasingU,coincidingwiththebehaviorofintheregionbothbelowandaboveitscriticalvalue.AtextremelylargeUandzeroeld,theresistivityincreasessignicantly 44

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(a) (b) Figure2-7. Resultsatoptimaldoping.(a)Magnetizationand(b)resistivityversusTatU=1:75,with2%impurities. Unlikethepreviousworkinsuperconductingstate,thereexistsatemperaturescalebelowwhichthesystemowsintolongrangemagneticorderphasewhenimpuritiesarepresent,duetotheniteDOSofthenormalstate[ 89 ].Thismeaneldartifacthindersourexplorationatextremelylowtemperatureregion,andhencepreventsusfromcomparingthepresenttheorywiththeexperimentallyobservedlogarithmicdivergenceofresistivity.HoweverthenumericsdowntoaslowasT=0:0220Kshowasignicantupturnincomparisonwiththezeroeldcase.Figure 2-7 showsbothmagnetizationandresistivityversustemperature,whereoneseesagainthepositivecorrelationbetweenthem.Inthezeroeldcase,whichmagnetizationremainszeroatalltemperaturesexplored,oneseesthatresistivityslightlyincreasesastemperatureislowered,instead 45

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2{9 )plusthenitesizeeect.Thelowesttemperatureexploredisfairlyclosetothemagneticphaseboundary,andalsowheretheresistivityupturnstartstobecomesignicant.ThemagnitudeoftheupturnatT=0:02incomparisonwithhightemperatureresistivityisabout10%,roughlyconsistentwiththevalueobtainedinYBCOafterthelinearinelasticcontributionhasbeensubtracted[ 71 ]. (a) (b) Figure2-8. Resultsatoptimaldoping.(a)Magnetizationand(b)resistivityversusBatT=0:03andU=1:75,with2%impurities. However,inFigure 2-7 themagnetoresistanceinlowtemperatureregionseemstobenotmonotonicifonecomparesthezeroeldwithB=0:004=(gB=2)7TeslaandB=0:01=(gB=2)18Teslacurves.Thismotivatesustodoacarefulsearchforthemagneticresponsewithincreasingeld.Figure 2-8 showsthemagnetizationandresistivityasafunctionofexternaleld.WefoundthatattemperatureT=0:0330K,theregionthatmagnetizationislineartotheeldisfairlysmall,onlyuptoB0:002=(gB=2)incontrasttotheresultinsuperconductingstateB0:01=(gB=2)[ 89 ].Bothmagnetizationandresistivityincreasewitheldinthisregionandthensaturatestoaroughlyconstantvalueinhigheld.ThemagnetoresistanceofoptimallydopedYBCOdisplayssimilarincreasing-saturationbehavior,althoughthetemperaturescalethatthisbehavioris 46

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2-6 demonstratesthatcorrelationstrengthU,aswellasthemagneticmomentinduced,areindeedessentialingredientstodeterminethemagnitudeoftheupturn.Secondly,openingofthepseudogapinthequasiparticlespectrumisknowntofavorboundstateformation,whichinturnpromotestheimpurityinducedmagneticmoment[ 52 ].Thisissimilartothed-wavesuperconductingstatewherethepoleofimpurityT-matrixfallswithinthegap,whichproducesaboundstatelocalizedaroundtheimpurity.WeexpectthatthereductionofDOSinthepseudogapstatealsoproducesresonancesofT-matrixneartheFermienergy,althoughtheexactformofGreen'sfunctionandDyson'sequationremainsunknown.Theresistivityisthenaectedbythepseudogapformation,basedonthenaivearguementthatimpurityinducedmomentresultsintheupturn.TogetacrudeideaoftheeectofreducingDOS,weintroduceapseudogapinanadhocway.ThefollowingformofdispersionandDOSisproposedforhomogeneous 47

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(b) (c) Figure2-9. ProposedphenomenoligicalmodelforthereductionofDOSinpseudogapstate.(a)ThenormalstateDOSofdispersionkwebeginwith,(b)theDOSforproposeddispersionEkinEq.( 2{11 )withf=0:6andas-wavegapk==0:2,and(c)theextendedhoppingmodelafterFouriertransformEq.( 2{12 ). "pseudogap"state 48

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2-9 showstheDOSofEk,whereoneseesreductionatchemicalpotentialf.Toputthismomentumspaceformalismintoarealspaceexactdiagonalizationformalism,ournextstepistoFouriertransformEkbacktorealspaceandndaneectivelongrangehoppingmodelthatgivesEk.Thehoppingamplitudetijofthisextendedhoppingmodelistherefore 2{12 )areshowninFigure 2-9 .ThecurrentoperatorinthepresenceoflongrangehoppingstillsatisesEq.( A{3 ),butthecontributionfromallhoppingtermst=tijcorrespondingtoranges~=~ri~rjneedtobeconsidered. (a) (b) Figure2-10. Comparisonofmodelswith(Extended)andwithout(Normal)reductionofDOSbyapplyingextendedhoppingEq.( 2{12 ).(a)Magnetizationand(b)resistivityversusTbothatoptimaldoping,withU=1:75and2%impurities. 49

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2-10 showsthemagnetizationandresistivitycomparingtheextendedhoppingmodelwiththenormalstateHubbardmodelEq.( 2{1 ),whichcontainsonlynearestandnext-nearestneighborhopping.ToavoidtheproblemofpossiblecontributionstotheresistivitycomingfromphaseuctuatingCooperpairsorotherpossibleexcitationsinthepseudogapstate,wexbothmodelsatoptimaldoping=0:15andexaminesolelytheeectofreducingDOS.Withinthemagneticeldregionexplored0
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(b) Figure2-11. Comparing(a)magnetizationand(b)magnetoresistancebetweenextendedhoppingmodelEq.( 2{12 )andthenormalmodelbothatoptimaldoping,T=0:03,U=1:75,with2%impurities. boundstateformationandenhancestheinducedmoment.ThisresultsuggeststhatthereductionoftheDOSinthepseudogapregioncouldbeanimportantingredientforthemoredramaticresistivityupturnsobservedintheunderdopedcuprates. 51

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3-2 .YBCO6correspondstothehalf-lledCuO2plane,whereallthechainOaremissing,causingplanarCutohavevalenceexactlyoneelectronpersite,andisaMottinsulator.UponllingOonthevacanciesbetweenadjacentCusitesonthechainlayer,thelocalchainCuvalenceischanged,whichindirectlyproducesmobilechargecarriers(holes)ontheplane.TheYBCO7structurewithallOsiteslledgivesaslightlyoverdopedsample,whereTcisclosetoitsmaximum.Figure 3-1 givesaschematicdescriptionfortheeectofOcontentontheplanardoping,inwhichweseethatintheintermediatedopingrange,thedistributionofOhasanequallycrucialinuenceasdoesthetotalnumberofO.Forexample,ifchainO'sclusterintheannealingprocessratherthandistributingrandomly,therewillbefewerisolatedCu-O-Cupairsandhencelessinuenceonthelocalvalence.Suchacongurationdependenceisobservedexperimentally,anditisfoundthatnumberofCu-ObondsistheessentialingredienttodetermineTc[ 91 ].TheplanardopingcanbecalculatedviabondvalencesumsonceanOdistributionisdetermined[ 92 93 ].Inthischapter,westudyspecicallytheOrtho-IIstructureofYBCO6.5,wherethechainlayerhasalternating"full"and"empty"chainsreferringtotheOcontent,asshowninFigure 3-2 (i).The63CuNMRspectrumforthisparticularstructurehasbeenmeasured[ 94 95 ],wheretheauthorsinterpretedthebroadeningandtheshapeofspectrumasasignatureofimpurityinducedmagneticmoments.SincetheOdistribution 52

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AschematicdescriptionofthedopingprocessinYBCObychangingtheoxygencontentonthechainlayer[ 52 ].TheundopedparentcompoundYBCO6containsCu+onthechainlayer,andtheCuO2planehaseectivelyoneelectronperCusite.Ifoxygensareintroducedintothechainlayerbuttheircontentissucientlylow,thenthesparseO2onlychangesitstwoadjacentCu+toCu2+.WhenasegmentofatleasttwoO'sisformed,theexcesschargewillbetransferedtotheCuO2plane,resultingintheholedoping.AfullchaincanthereforedopetheCuO2planeinamoreecientwaythanarandomOdistribution. iswelldened,wecanlabelfourspeciesofCuaccordingtotheirpositions:chainCuonthefullchainCu(1F),chainCuontheemptychainCu(1E),planarCuconnectedtofullchainCu(2F),andplanarCuconnectedtoemptychainCu(2E).ThechainCu'sconnecttoplanarCu'sbytheoverlapoftheirdzorbitalandintermediateOpzorbital,whichproduceseectivetunnelingbetweentwolayers.Underexternaleld,eachCuspeciesprecessesatdierentfrequenciesaccordingtothelocalmagneticenvironmentofthenuclei,whichresultsinfourdistinguishablelinesinanNMRspectrumasshowninFigure 3-3 .BasedoncomparisonwiththesameexperimentinYBCO6andYBCO7,theCuspeciesassociatedwitheachlinecanbeidentied,andthechangeoflocalmagneticenvironmentundercertaincircumstancescanbeextractedviathechangeofthelineshape[ 94 ]. 53

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Crystalstructureofortho-IIYBCO6.5andrelatedparentcompoundYBCO6andYBCO7,withprecisedenitionofchainandplanarCu[ 94 ]. STMimagesofthechainlayer,whichiswheretheYBCOcrystalgenerallycleaves,showLDOSoscillationsalongthechaindirection.Thewavelengthofthisoscillationshowsstrongenergy(STMbias)dependence,whichsuggestsaFriedeloscillationexplanationforthisdensitymodulation,incontrasttoachargedensitywave(CDW)picturewhichusuallyhasaxedwavevector[ 96 ].Suchanoscillatingbehavioralsoindicatestheimportanceofchainlengthtoitselectronicproperties,especiallythosedirectlyrelatedtotheDOS,forexamplemagnetizationandtransport[ 97 98 ].IntheNMRexperiment,theaveragechainlengthofOrtho-IIisfoundtobearound120latticeconstantsbyestimatingthe 54

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(b) Figure3-3. (a)FullNMRspectrumofOrtho-IIat60KinthepresenceofexternaleldH0==a,wherethefrequencyisexedat75.75MHz[ 94 ].Toppanel:repetitiontimeTrep=250msec,middlepanel:Trep=25msec,lowerpanel:thedierence.ComparisonwithYBCO6andYBCO7conrmstheassociationbetweenlinesandtheCuspecies:A!Cu(2F),B!Cu(1E),C!Cu(2E),D!Cu(1F).(b)STMimageonthechainlayerofYBCO,whichshowsoscillationofDOSalongchaindirection[ 96 ].NoticethatthesampleisnotOrtho-IIandhasrelativelyshortsegmentsoffullchains(about10sites). concentrationofterminalCuonthefullchains[ 94 ].Withthisinformationinmind,thephysicalpicturefortheNMRlineassociatedwithchainCuisclear:theFriedeloscillationinducedbyabruptterminationofchainscausesthemodulationofmagnetizationinthepresenceofanexternaleld,whichchangesthelocalmagneticenvironmentandresultsinasignicantlinewidthintheNMRspectrum.Figure 3-4 givesaschematicdemonstrationofhowmagnetizationinducedinaone-dimensionalHeisenbergchainmaterialcanbroadentheNMRspectrum,whichcorrespondstothehistogramofmagnetization[ 99 100 ].AlthoughthisFriedeloscillationcangiveareasonableinterpretationofthespectrum,wefoundthatmagnetizationresultingfrompurelyuncorrelatedelectronsisunabletoaccountfortheobservedlinewidth,northetemperaturedependenceofthelineshape.Instead,onemustincludethecorrelationeectinordertoexplainthesefeatures,inwhichthe 55

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95 ].Comparisonwithexperimentswillbemadetohighlighttheimportanceofmagneticcorrelations,andtheconsequencesofcouplingbetweendierentdimensionalitiesinthisparticularOrtho-IIstructurewillbedemonstrated. Figure3-4. Schemeticofbroadeningof89YNMRlineinYBa2NiO5bymagnetizationinducedaroundimpurities[ 52 99 100 ].Therightpanelshowstheamplitudeofmagnetizationinducedonseveralsitesneartheimpurity,whichresultsinsatellitesignalsnearthemainline.Asignicantamountofsuchinducedmagnetizationeventuallysmearsoutthespectrumandbroadenstheline(theinsertofleftpanel). 56

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3-5 .ThefullHamiltonianconsistsof Wedenotebytpij,tc,andtrthehoppingamplitudesontheplane,chain,andchain-planeinterface,respectively;^ciand^dxareelectronoperatorsontheplaneandonthechain,whereidenotesthe2Dcoordinatesandxisthepositionalongchaindirection.Thechainhoppingisonlybetweennearestneighbors,whileplanarhoppingtpijcontainsbothnearesttpnnandnextnearestneighborsitestpnnnwhichisnecessarytoreproducethecorrectFermisurfaceoftheYBCOsystem.AmagneticeldisincludedintheZeemantermp=c=gBB=2.TheschematicofEq.( 3{2 )isshowninFigure 3-5 .NoticethattheexperimentalNMRlinecontainsbothdataaboveandbelowTc60K,whichcouldbemodeledbyproperlychoosingpairingcorrelationsVintheplane.AHartree-Fock-Gor'kovmeanelddecompositionisthenappliedtotheunperturbedchainandplanedegreesof 57

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withorderparametersdenedas Sincethechainsarecoupledwithapossiblysuperconductingplane,aBogoliubovtransformationisnecessaryforbothplanarandchainelectronoperators.Wethereforedene 58

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andwereusedtocalculatedtheorderparametersinEq.( 3{4 ).FurtherdetailsofBogoliubov-deGennes(BdG)formalismandnumericalsimplicationaredemonstratedinAppendixC.WerstconsidertheeectofrandomlydistributedmissingO'sonthechainlayer,whichresultsintheabruptterminationofacontinuousfullchain.SincethehoppingonthefullchainisduetoorbitaloverlapbetweenalternatingCuandO,themissingOisassumedtoreducethehoppingamplitudebetweenitstwoadjacentCusites.Theimpurity 59

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SchematicofthehoppingpartofmodelHamiltonianEq.( 3{2 ). Hamiltonianistherefore 101 ],andcanresultinapropertemperatureandeldscale,aswellasgivingaproperNMRlineshapeandlinewidth,aswillbeaddressedbelow.The63CuNMRresonancefrequencyisdeterminedexperimentallyby,tolowestorder 102 ],andhencewillshiftthepositionofthelinedierentlyin 60

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2hni"ni#i=iB gB;(3{11)assumingthatthemagnetizationisinthelinearresponseregion,andtheresonancefrequencyforaspecicsiteiisrelatedtothepolarizationofspinSiby 3-4 .Numericallywecollectthemagnetizationoneachsite,andthenarticiallybroadenthediscretedistributionbyaLorentzianwithwidthtogetacontinuousspectrumI() 61

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Numericsinanisolated1Dchain,equivelenttoHamiltonianHchain+H(1)imp,inanexternaleld7.7Tat50K.Eachcurveisnormalizedsuchthattheareaunderneathisconserved. Beforetherealisticchain-planecoupledsystemisstudied,weperformnumericsonanisolated1DchaintoshowtheimportanceofthechaincorrelationenergyUc,withouttheinuenceofproximityeectbetweenchainsandplane.Figure 3-6 showsNMRlinesgivenbyconsideringHamiltonianHchain+H(1)imp,whereincreasingUcclearlybroadensthelineataxedtemperatureandeld,coincidingwiththeenhancementofimpurityinducedmagnetizationinrealspaceduetoincreasingmagneticcorrelations.TheUc=0linecorrespondstoFriedeloscillationoffreeelectrons,wherethemagnetizationduetoZeemansplittinggivesanNMRlinewidthonlyhalfofwhatisobservedexperimentally.OurbestttotheexperimentaldatagivesUc=1:2,asfaraslinewidthisconcerned.ThisvalueofUcisagainclosetobutsmallerthanitscriticalvalue,abovewhichtheinducedmagnetizationcannolongerbecharacterizedaslocalizingaroundthemissingO.Togetherwithpreviousworkonplanardefects[ 89 ],weconcludethatmagneticcorrelationexistson 62

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95 ]seemstobeoutsideofthisscheme.Attemperaturehigherthan70Korso,asatellitepeakwithsignicantweightgraduallydevelopsatfrequencyslightlylowerthanthemainline.Thissatellitepeak,whichbecomesmoreprominentathightemperatures,wasattributedbytheauthorstomagnetizationassociatedwithsmallerdensityoscillationsfarawayfromthechainends,incontrasttotheampliedmagnetizationclosetothechainends[ 95 ].Inotherwords,theypostulatedthataFriedeloscillationaroundasingledefectwoulddisplayabimodaldistributionofmagnetization.However,wefoundthattheideaofseparatinglargeandsmallamplitudeoscillationsisambiguousinasystemwheremagnetizationsmoothlydecaysasmovingawayfromthechainends.Instead,twofeaturesofthishightemperaturesatellitepeakmotivateustoproposeadierentscenarioregardingitsorigin:(1)Theweightofthepeakisabout10%ofthewholespectrum,and(2)thepositionofthepeakremainsroughlythesameatalltemperatures.Feature(1)suggeststhat,ifonewouldassociateavalueofmagnetizationtoacertainsatellitesignal,thentheweightofthelineisaroughindicationofhowmuchportionofthesampledisplaysthismagnetization.Thereforeit'sreasonabletoassumethatroughly10%ofthesamplehasthesamemagnetization,whichresultsinthissatellitepeak.Feature(2)impliesthatthismagnetizationremainsconstantathightemperature.Sinceincreasingtemperatureshouldcontinuouslyreduceanynitemagnetizationdevelopedduetothecorrelationeect,itisreasonabletoassumethisobservedconstantmagnetizationiszero.WethereforeproposethatthehightemperaturesatellitepeakisduetoasectionofadjacentCumissingtheirconductionelectrons,presumablycausedbythealternationofCuvalencenearthemissingO.Thechangeoflocalchemicalenvironmentprohibits 63

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withL=15outof121sites.NoticethatthesummationoverimpuritypositionsinEq.( 3{14 )isrestrictedbetweenasectionofconsecutivesites1lL,whichdiersfromEq.( 3{8 )wheretheimpuritypositionistotallyrandomanddiscrete.Weagainassumecompletesuppressionoftheassociatedenergyscalestc=tc,c=c,c=c,whichresultsinallmatrixelementsinvolvingsites1lLtobezero.TheeectofeliminatingafewadjacentCu(1F)isrstexaminedinasingleisolatedchain.IncomparisonwithrandomlydistributedmissingOHchain+H(1)imp,theCu(1F)linegivenbyconsideringHchain+H(2)impshowsaclearasymmetriclineshapewithmoreweightatlowerfrequency,asshowninFigure 3-7 .Thezeromagnetizationpeakappearsatalltemperatures,andisspeciallynoticeableathightemperatureasthemainlinenarrows.Broadeningofthemainlinesmearsoutthesatellitepeak,whichcouldpossiblyexplainitsinsignicanceatlowtemperature.Theconclusionfromexaminingthesingleisolatedchaindoesnotalterasthechainsarecoupledtotheplane.ThespectrumobtainedbyexaminingeliminatingadjacentCu(1F)sitesonthechain-planecoupledsystemremainsqualitativelythesame,asshowninFigure 3-7 .Althoughthedistancebetweenmainlineandsatellitepeakslightlyincreases,whichmaybeduetochangeofDOSonthechainlayerandcanberesolvedbyadjustinghoppingorchemicalpotential,theconsequenceof(1)broadeningofspectrumatlowtemperature,(2)asymmetricspectralweight,and(3)thehightemperaturesatellite 64

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(a) (b) (c) (d) Figure3-7. Comparisonof(a)experimental63Cu(1F)NMRlineofortho-IIYBCO6.5atexternaleld7.7Talongadirection,withthreemodelsstudied:(b)isolated1DchainwithrandomimpuritiesHchain+H(1)imp,(c)isolated1DchainwithasectionofadjacentCu(1F)erasedHchain+H(2)imp,(d)chain-planecoupledsystemwithasectionofCu(1F)eliminatedHplane+Hchain+Hinter+H(2)imp.CorrelationstrengthUc=1:2inthesemodels. 65

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103 104 ]andopticalconductivity[ 105 ],aswellasInelasticNeutronScattering(INS)[ 106 ]andpenetrationdepthmeasurements[ 107 108 ].Severalauthorshaveproposedthattheseobservedanisotropiescanbeexplainedbyproximitymodels[ 101 109 { 111 ]similartothemodelpresentedinthischapter,althoughanisotropyinthemagneticresponseisnotincludedinthesemodels.Toaddressfurthertheissueofanisotropyinimpurityinducedmagnetization,thekeyreliesonaproperchoiceofaimpuritymodelthatcanhighlightboththeeectofcorrelationsandthechain-planecoupling.SincetheimpuritymodelsH(1)impandH(2)impinvolvemorethanoneimpuritysite,wenditambiguoustodenethemagnetizationalongdirectionsorthogonaltotheseimpurities.Inaddition,theinterferencebetweenmagnetizationgeneratebymultipleimpuritysitesisunavoidable.Wethereforestudyapedagogicalmodelthatcontainsasingleunitarystrongscatterlocatedonthefullchain,forthesakeofdemonstratingtheproximityinducedanisotropy.Thephilosophyisthatsincethiskindofimpurityispoint-like,itiseasytodenetwoorthogonaldirectionsprovidedthattheoriginissettobeattheprojectionofimpuritysite,unlikeH(1)impandH(2)impwheretheirprojectionontheplanewillinvolvemorethan1siteandtheoriginisilldened.Astrongunitaryscatteronthechaingives 3-8 comparesthemagnitudeofthemagnetizationonthefullchain,planarsitesalongb,andplanarsitesalonga,withnormalizedinducedmomentdened 66

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94 ].ThereverseprocessallowstherecoveryoftheNMRspectruminthepresentmodel,witheachlinecalculatedbycollectionofmagnetizationandEq.( 3{9 ).ThevalueofKorbatexternaleldB==bistakenfromTakigawaetal[ 102 ].ThespectrumcombiningCu(1F),Cu(2E),andCu(2F)formodelHplane+Hchain+Hinter+H(2)impisshowninFigure 3-9 ,whereoneseesasignicantoverlapbetweenCu(1F)andCu(2E/F)lines,withCu(2E)basicallyindistinguishablefromCu(2F).TheprecisepositionoftheselinescouldbeaectedbyhigherordermomentsomittedinEq.( 3{9 ),ordeviationofrealisticKorbfromthevaluetaken,aswellastheemptychaindegreesoffreedomignored.Accountingfortheseprocessescouldresultinseparatedlinesaccordingtoexperiments.ThelinewidthinthepresentstudyisthereforecalculatedbyextractingthemagnetizationofeachCuspecies,insteadofttingthefullspectrum.ThebroadeningofextractedCu(2F)andCu(2E)linesastemperatureisloweredfollowsthatofCu(1F),asshowninFigure 3-9 ,withtheslopeofCu(2E)smallerthanCu(2F).Asimilarlinearrelationbetweenbroadeningofplanarandchainlineshasbeenobserved[ 95 ],whichservesasanotherdirectevidenceofinducedplanarmagnetizationduetodefectandmagneticcorrelationsonthe 67

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106 ]measurementsandtransportproperties[ 103 { 105 ]willbepresentedinfuturestudies. 68

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(b) (c) (d) (e) (f) Figure3-8. Demonstrationofanisotropicmagnetizationduetochain-planecoupling,inwhichweplotnormalizedmagnetizationsiinducedbysubstitutionofaCu(1F)byapointdefect:(a)onthechain,(b)ontheplanealongchaindirection,(c)ontheplanealongdirectionperpendiculartothechain.ThesolidlineisattosphericalBesselfunctionsi/s1K0(i=)=K0(1=).Sizeoftheplaneis8124,attemperature70K.Therealspacepatternofsiontheplaneisgivenin(d)and(e),whereonecanseetheextensionofmagnetizationalongchaindirection,coincidingwith(f)themagnetizationinducedonthechainlayer. 69

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(b) (c) (d) Figure3-9. (a)ThecombinedspectrumofCu(1F),Cu(2F),andCu(2E)lines,togetherwith(b)thelinearrelationfoundbetweenbroadeningofchainandplanarlines.Incomparison,theexperimentaldataoflinewidthcorrelationisgivenin(c)forCu(2F)and(d)forCu(2E)[ 95 ]. 70

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89 112 { 115 ].Severalauthorsassumedtheexistenceofinducedmomentsaroundtheimpurity,andstudytheresponseofthesystemwithinthecontextofKondoscreening[ 116 117 ].Othermodelsthatincludetheeectofpseudogapandaresultingspinonboundstatehavealsobeenproposed[ 118 119 ],aswellasessentiallynumericalworks[ 120 { 123 ].OurdiscussioninCh.2andCh.3involvesthermodynamicobservablesinthepresenceofinducedmoments,studiedwithinHartree-Fockmeaneldtheoryataspecicdoping.ThephenomenologicalparametersthatentertheHamiltonian,forinstancehoppingtorHubbardU,couldbedeterminedbyttingexperimentsrelatedtothequasiparticlespectrumorthemagnitudeoftheinducedmoment.However,dopingaectsthesephenomenologicalparametersinastronglycorrelatedsystemlikecuprates.Thisisbecauseifonestartsfromtheundopedparentcompound,whichcontainsoneelectronpersiteontheCuOplaneandisaMottinsulator,thedopinghastwoprimaryeects:Thepresenceofvacantsitesincreasesthemobilityofelectrons,aswellasdiminishesthemagneticcorrelationsintheparentcompound.Inaddition,sinceTcincreasesupondopingin 71

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Thephasediagramofslavebosonmeaneldtheoryforhomogeneouscuprates.ThethreetemperaturescalesrepresenttheBose-Einsteincondensationtemperature(TBEC),theonsetofspinliquidorder(TRVB),andthespingaporder(TSG).Undertheseinterpretation,weidentifyfourphasesinthephasediagram:(I)Fermiliquid,(II)spingapphase,(III)superconductingstate,and(IV)thestrangemetalphase.Inthischapterwefocusontheimpurityprobleminthephase(IV)wherewehavenonzerospinliquidorder,butthesystemisaboveBose-Einsteincondensationtemperatureandhasnopaircorrelations. theunderdopedregion,itsuggeststhecorrelationbetweensuperuiddensityandholeconcentration.ThemaximumvalueofTcisthereforeacompetitionbetweenreducingmagneticcorrelationsneededforsingletpairingandtheincreaseofchargecarrierdensity,bothresultfromdoping.Ontheotherhand,inastrongcouplingperspective,onehastotakeintoaccountthestrongCoulombrepulsionevenwhenmobileholesarepresent,whichpreventstwoelectronsfrompopulatingatthesamesite.Thisnondoubleoccupancyconstraint,whichleadstotheeectiveexchangeinteractioninthestrongcouplinglimit,lieattheheartofmagnetismincuprates.Toproperlydescribetheevolutionofbandwidthupondopingandthenondoubleoccupancyconstraint,weadopttheslavebosonmeaneldtheoryforthemetallicstateofthecuprates.TheschematicphasediagramforthisapproachtohomogeneouscupratesisshowninFigure 4-1 .Herewefocusonthe"strangemetal"phaseastheinterpretation 72

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36 { 38 ].Intheslavebosonrepresentation,aprojectedelectronoperatorisdecomposedintoafermionic"spinon"andabosonic"holon"operatorcyi=bifyi,inwhichthespinonsrepresentthespindegreesoffreedom,andtheholonsrepresenttheemptysites.Thisdecompositionispossiblebecausewestartfromthehalf-lledundopedcompound,whichhasexactlyoneelectronpersite,andtheeectofdopingistointroduceholonswithanadditionalconstraintPfyifi+byibi=1imposedlocallyoneachsitei.Thisconstraintensuresthenondoublyoccupancyforthesitesimultaneouslybyspinupanddownelectrons,evenwhenthesystemisdoped.AsdiscussedinSection1.8,thestrongCoulombrepulsionalsomotivatesustostartwiththett0JmodelEq.( 1{13 ),whichdescribesasinglebandofprojectedelectronswithparticle-holeasymmetry,interactingviaanexchangecouplingJ.Applyingtheslavebosonformalismtothett0Jmodelgives 4ninj)fXifyifibXibyibi+Xiii(fyifi+byibi1); wheretheLagrangianmultiplieriensurestheslavebosonconstraint,andtheZeemantermgBB=2takesintoaccountthesplittingbetweenupanddownspins.Wechoset=400meV,t0=0:4t,andJ=0:22twhichyieldsareasonablespinonandholoneectivebandwidthclosetooptimaldoping,asaddressedbelow.Numerically,thedensityinteractiontermninj=4whichresultedfromthecanonicaltransformationwillbeomittedsinceitcanbeabsorbedintothedenitionoffermionchemicalpotentialaftermeaneldfactorization,andonlytheexchangeinteractionwillbeconsidered,withthedenitionofthespinoperator~Si=1 2fyi~fi. 73

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124 125 ] wheremiisthestaticmagnetizationandistheholedoping.ThemeaneldHamiltonianthenbecomes DetailsofthemeaneldformalismarediscussedinAppendixE.Intheunperturbed,spin-degeneratecasealltheorderparametersarerealandconstant,whichgivesthefollowingspinonandholondispersion Inthepresenceofimpurities,localorderparametersarerenormalizedandthecorrelationeectismeasuredbytheratiobetweenexchangecouplingJandthelocalspinonhopping 74

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wherejEfniandjEbniarethespinonandholoneigenstates,respectively.Thecorrespondingrealspacewavefunctionsare (4{8) 75

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2Xn(jf"n(i)j2jf#n(i)j2)f(Efn);nfi=Xnjfn(i)j2f(Efn);nbi=Xnjbn(i)j2b(Ebn): ThenondoubleoccupancyconstraintissatisedbyadjustingtheLagrangemultiplier(eectivelythelocalshiftofchemicalpotential)isuchthatnfi"+nfi#+nbi=1foreachsiteiexcepttheimpurity.Numericsaredoneona20by20latticewithperiodicboundaryconditionsandt=400meV,t0=0:2t,J=0:22t,comparabletotheparametersinthesemianalyticalapproachwhichwebrieyintroduceinthenextsection. 4{3 ),andhenceaectsthepropagationofquasiparticles.InarecentpaperbyGabayetal:[ 125 ],wehaverstadoptedasemianalyticalapproachtosolvetherenormalizationoforderparametersandtheresultinginducedmagnetization.Theexactdiagonalizationmethodisthenalsoimplementedandcomparedwiththeseresults.ThebasicideaofthesemianalyticalapproachistoconstructtheunperturbedGreen'sfunctionintheabsenceofimpurity,thensolvefortheorderparameterswhentheimpurityispresent[ 125 126 ].WiththeserenormalizedorderparametersthefullGreen'sfunctionisconstructedandcanbeusedtocalculatetheinducedmomentwhenspindegeneracyislifted.StartingfromthedispersionEq.( 4{4 ),anddeningtheHilbertspaceofthespinontobefyij0i=ji;i,we 76

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Similarly,wecandeneholonHilbertspacebyij0i=jiiandtheholon'sunperturbedGreen'sfunction NoticethatatthisleveltheupanddownspinsaredegeneratesoG0ij"=G0ij#.Dyson'sequationforthesingleimpurityHamiltonianEq.( 4{5 )inthe!1limitcanbesolvedexactly,andwedenotethisGreen'sfunctionbyGij withimaginaryargument=it.ThenextstepintheGabayetalapproachinvolvesself-consistencybetweenthefollowingtwoequations 77

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Spinoneectivebandwidthtffordoping=0:3insemianalyticalapproachofGabayetal[ 125 ]neartheimpurity. wheretherstequationisDyson'sequationwithpotentialV(mi)inthepresenceofmagnetizationmi,andthesecondoneexpressesmiintermsofthefullGreen'sfunctionG.ThepotentialV(mi)accountsforthemagneticscatteringduetoinducedmomentswhichcoupletotheirneighboringsitesbecauseoftheexchangecouplingJ V(mi)=JX(mim0)ni;(4{15)wherem0isthehomogeneousmagnetizationandjsumsovernnsitesofi.AnumericaldiagonalizationofGinrealspaceuptothethirdshellofsitesawayfromtheimpuritysiteisappliedtosolveformi,byassumingthatfartherawaytheGreen'sfunctionrecoversitsanalyticformGinEq.( 4{12 ).Theeectivespinonbandwidthtf=(J=2+tQ)calculatedbyusingEq.( 4{13 )isshowninFigure 4-2 ,whereoneseesadrasticreductionoveralengthscalethatincreasesastemperatureislowered.Thenitesizediagonalizationapproachinthepresentreport,incomparisonwiththesemianalyticalresult,hasthefollowingtwoadvantages:(1)Theslavebosonconstraintisstrictlysatisedlocallyoneachsiteexceptfortheimpurity,and(2)itincludesall 78

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ComparisonofspinoneectivebandwidthtfbetweensemianalyticalapproachofGabayetal[ 125 ]andpresentexactdiagonalization. thehigherorderdiagramsintheDyson'sequation.Regardingtherstpoint,inthesemianalyticalapproachtheparticlenumberisimposedglobally,inthesensethattheholonpopulationissettobeequaltothedoping.However,whensolvingDyson'sequationself-consistentlyviaEq.( 4{14 ),thedensitymodulationmayinthiscaseviolatethelocalslavebosonconstraint.TherealspacediagonalizationwithproperchoiceofLagrangianmultiplieriresolvesthisdiculty.Regardingthesecondpoint,sinceHamiltonianEq.( 4{3 )andEq.( 4{5 )arediagonalizedexactly,itisequivalenttoincludingallthehigherorderdiagramsinthesingleimpurityproblem.Thecomparisonbetweensemianalyticalapproachandtherealspacediagonalizationthereforeprovidessomeinsightintohowtheslavebosonconstraintandhigherorderscatteringprocessesaltertheresultofmagnetization.InFigure 4-3 weshowthecomparisonofthesetwomethods,whichsuggeststhatslavebosonconstraintmitigatesthereductionofthespinonbandwidthtfneartheimpurity,andhencereducestheeectivecorrelationstrengthJ=tf.Wethereforeexpectasmallermagnetizationcomparedtothesemianalyticalresult.Inaddition,thetemperaturedependenceismitigated,and 79

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4-4 .Thescreeninglengthistto 52 ],butisnotrigorouslyjustied.ThemagnetizationatthennsitesoftheimpurityisttoaCurie-Weissform 4-5 .Noticethatinthepresentmeaneldtheorythechoiceofperiodicboundaryconditionsenhancestheinterferenceofthemagnetizationproducedbythe(periodic)"single"impurity,andhencegivesrisetoanunavoidablephasetransitiontolongrangemagneticorderastemperatureislowerthanTN100K,belowwhichthetheorybreaksdown.Thismeaneldartifacthinderstheexplorationoftheextremelowtemperatureandsmalldopingregion.WealsofoundthattheFriedeloscillationinthepresenceofmagneticeldproducesaresidualmagnetizations1!constantathightemperature,whichaectstheaccuracyoftheCurie-Weisstbecauses1!0inEq.( 4{17 )athightemperature.Theeectivemomentandscreeningtemperaturearethereforedeterminedviaatover 80

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4-4 (a)andthedoping=0:3curveofFigure 4-5 (a),themagnitudeofs1isdrasticallyreducedastheconstraintisimposed.Thisisconsistentwiththeanalysisoftheeectivespinonbandwidthintheprevioussection,whichshowsthatimposingtheslavebosonconstraintmitigatesthereductionofspinonbandwidthtfandhenceeectivelyreducesJ=tf,resultinginlessmagneticcorrelationandsmallers1.(2)Intheexactdiagonalizationresults,thescreeningtemperatureshowsanabruptincreaseasthedopingisincreasedthroughoptimaldoping=0:15,consistentwithexperimentaldata.Thesemianalyticalapproachhoweverhassuchanabruptincreaseathigherdoping(roughly=0:27byextrapolation),asshowninFigure 4-4 (c).Thisisconsistentwithpreviousslavebosonmeaneldtreatmentswhichhavebeenshowntooverestimatethedopingscale,andwefoundthatimposingthenondoubleoccupancyconstraintlocallycanresolvethisdilemma.(3)Thescreeninglengthv.s.Ttsbetterwithexperimentwhentheconstraintisimposed,indicatingtheimportanceofthelocalconstrainttoboththeFriedeloscillationandtheresultinginducedmomentpattern.Toaddressfurthertheissueofspatialdistributionofmagnetizationaectedbythenondoubleoccupancyconstraint,weinvestigatedboththespinonandholondensitypatternsinthevincinityoftheimpurityviatheexactdiagonalizationmethod.Holons,duetotheirbosonicstatistics,developanunphysicallengthscalethatdependsonlyontemperatureiftheyareunconstrained.ThelengthscalethatcharacterizesFriedeloscillationofspinonsistheFermiwavelength,whichisassociatedwiththeFermimomentumkfandhencethedoping.Inthepresentmeaneldtheory,however,spinonsandholonsrenormalizeeachother'sspectrumviaHamiltonianEq.( 4{3 ),sodensity 81

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b) c) d) Figure4-4. SummaryofsemianalyticalapproachofGabayetal[ 125 ]withexternaleld7Tesla.(a)Magnitudeofnormalizedmagnetizationsr=(mrm0)=m0neartheimpurityatT=25K,overdope=0:3.Solidline:ttojsrj/s1K0(r=)=K0(1=)for=3.(b)T-dependenceofnormalizedmagnetizationatnnsiteofimpurityatoverdoping=0:3.(c)EectivemomentCandscreeningtemperaturettotheCurie-WeissformC=(T+)fors1,nearoverdopedregion.(d)T-dependenceofscreeninglengthatdierentdopings,ttotheformin(a). modulationsaectorderparametersandinturnthespectrumwhenimpuritiesarepresent.IfonlyFermiandBosestatisticsdeterminethedensitymodulations,numericsshowthatsuchasystempreferstohaveholonsstayfarawayandhavedensityalmostzeroaroundtheimpurity,whichalsoreducesthespinonbandwidth.ThisisbecauseQij=hbyjbiihbyibii=nbiatlowtemperature,thereforethespinonbandwidthtf2tQ2tnbisreducedaccordingly,resultinginstrongmagneticcorrelationJ=tf

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Resultsofexactdiagonalizationat7Tesla.(a)T-dependenceofnnsitenormalizedmagnetizations1indierentdopings.(b)EectivemomentCandscreeningtemperatureintheCurie-Weisstofs1.(c)DatabyOuazietalshowingT-dependenceof,andmagnetizationversusdistancerawayfromimpurity(insert).(d)v.s.Tfromoptimaltooverdopingbyexactdiagonalization. forthespinon,andthesystemalwaysowstoalongrangemagneticorderphase.Thesemianalyticalapproachreachessimilarconclusions,althoughrenormalizedfermionbandwidthisonlycalculateduptoG,notfullself-consistently.OneseesthatthedramaticreductionoftfinFigure 4-2 ,indicatingadrasticsuppressionofholondensityinthevicinityofimpurity,andisclearlyanartifactofthemeaneldtheorywithoutlocalconstraint. 83

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DensitypatternbyexactdiagonalizationatT=100Kand=0:15.(a)Spinondensity,(c)holondensity,(e)magnetizationplot,andtheircorrespondingcutthroughimpuritysitealong(1,0,0)directionin(b),(d),(f).Spinonandholonoscillationshowsthesamehealinglengthkfassociatedwiththesapeofspinonfermisurface,whilemagnetizationoscillatesunderwhichisrelatedtoeectivecouplingJ=tf. ThedensitypatternandmagnetizationofexactdiagonalizationmethodisshowninFigure 4-6 ,togetherwiththeirplotcutthroughimpuritysitealong(1,0,0)direction.Comparing(b)and(d)ofthegure,itisclearthattheholonhealinglengthisnowthesameasthespinonlengthscalekf,beyondwhichthehomogeneousvalueisrecovered.Weconcludethatthenondoubleoccupancyconstraint"slaves"thebosoniclengthscaletothatofthefermions,becausethemodulationofholonsmustnowcompensatethemodulationofspinons.Themagnetization,however,exhibitsadierentlengthscalethan 84

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4-6 .ThisisbecausethemagnetizationisgovernedbythestrengthoftheexchangeenergyJ=tfanditsassociatedlengthscale,whichisnotdirectlyrelatedtokf.Themechanismbywhichkfanddependontemperatureordopingarealsodierent.kfremainsroughlythesameintherelevanttemperaturerange,whichismuchlowerthantheFermienergy,whiledopinghasamoredramaticeectonkfsinceitchangestheshapeofFermisurface.Thetemperaturedependenceofcomesfromthereductionofmagneticorderduetothermalexcitations,similartotheusualmeaneldtheoryoftheHeisenbergmodel[ 4 ].Dopingaectsindirectlybychangingparticledensitiesandtheassociatedbandwidth,resultinginlargereectivecouplingstrengthJ=tfasthesystemisunderdoped. 1-6 (b),weseethatthereductionofcorrelationlengthasincreasingdopingisrecovered,whosemagnitudeisconsistentwiththe17ONMRresult[ 53 ].Inaddition,thesusceptibilityatthenearest-neighborsiteofimpuritysatisesaCurie-Weissform,inwhichtheabrupt 85

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55 ]showninFigure 1-6 (d).Weconcludethattheslavebosonmeaneldtheoryplustheexactdiagonalizationmethodcancapturetheimpurityinducedmomentformationincuprates,inwhichtheinuenceofstrongcorrelationscanqualitativelychangethetemperatureanddopingscalesimpliedbyothertreatmentsofthesameproblem,andhenceisimportanttocorrectlydescribetheimpurityphysicsinthesematerials. 86

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87

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~Pi=X~ri^ni;(A{1)andthelocalcurrentwillbe @Axi=itX(^cyi+x^ci^cyix^ci)+it0X(^cyi+xy^ci^cyixy^ci)=itX(^cyi+x^ci^cyi^ci+x)+it0X(^cyi+xy^ci^cyi^ci+xy) (A{5) whichgivesthesamething.Noticethatwe'veusedadummyindexforthelaststeptoconvertJxiintoabondvariablebetweensiteiandsitei+x. 88

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88 ].Startingfromthelinearresponsetheoryfortheconductivity thefrequencydependentpartcanberewrittenas lim!!0f(En)f(Em) wherethedeltafunctionisdenedas(!En)=(=)=((!En)2+2).Sotheconductivitybecomes andapparentlythisfunctionF(En;Em)issymmetricunderexchangeofEn$Em. 89

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A{5 )actuallyhasaprefactore=~,andisinunitsof [Jxi]=[eta s;(C{1)whereaisthelatticeconstant.Theretardedcurrent-currentcorrelationfunctiondenedinEq.( 2{9 )hasunitsof [ij()]=[JxiJxj]=C2m2 [ij(iwn)]=[1 []=[ij(iwn)] [~w]=C2 B{1 )couldbewrittenas,ifweonlyconsidernearestneighbor 90

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wherethenewF(En;Em)isdenedby C{6 ).Whatiscalculatedinmycodeisthefollowingdimensionlessquantity,whereeveryenergyisrescaledbyt =Xn;mhnj1 t1 t tEn t)2~ t tEm t)2 91

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(df(~!) kBT)e~!=t kBT=t kBT=t+1=t(1 kBT kBT+1=t(df(~!) Thereforethetrueconductivityisthenumericalnumberfoundbythecodemultipliedbye2=~ m;3D=n3De2 m; and(2)theconductingchargeincupratecomescompletelyfromthecopperplanes,sotherelationbetweenn2Dandn3Dis 92

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93

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3{2 )consistsofarearrangementofeachinteractionterm.Theon-siteCoulombinteractionisdecomposedinto 4(np2imp2i)]; withthedenitionoforderparametersEq.( 3{4 ).Thelastterminthelastlineisaconstantandcouldbedroppedinthediagonalizationprocess.ThesameprocedureworksfortheUcterminchaindegreesoffreedom.Thepairingchannelisfactorizedby, 94

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XhijiV^npi"^npj#=XiVcyi"ci"cyi+#ci+#=XiVci"ci+#cyi+#cyi"=XiV[hci"ci+#i+(ci"ci+#hci"ci+#i)][hcyi+#cyi"i+(cyi+#cyi"hcyi+#cyi"i)]=XiV[hci"ci+#i+(ci"ci+#)][hcyi+#cyi"i+(cyi+#cyi")]XiV[hci"ci+#icyi+#cyi"+hcyi+#cyi"ici"ci+#hci"ci+#ihcyi+#cyi"i]=XiV[hci"ci+#icyi"cyi+#+hcyi+#cyi"ici+#ci"+hci"ci+#ihcyi+#cyi"i]=Xiicyi"cyi+#+h:c:+Vhci"ci+#ihcyi+#cyi"i; whichagaingeneratesconstanttermbutcouldbedroppedinthediagonalizationprocess.ThemeaneldHamiltonianEq.( 3{3 )isthereforerecovered.Asmagneticeldisapplied,thespindegeneracyisliftedandoneexpectsthatupanddownHamiltonianshouldbeconsideredseparately.UnderaBogoliubovtransformationEq.( 3{5 ),thespinupchannelsatisesthefollowingequations 95

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Oneobservesthat,amatrixrepresentationcouldbeconstructedfortheequationsassociatedwithspinupeigenenergiesEn" D{3 )andEq.( D{4 )associatedwithplanarhopping,chainhopping,andtheinterlayerpartrespectively.FromBCStheoryweknowtheeigenenergiesofquasiparticlesarealwayspositiveenergiesabovethegroundstate,hencewecanrewritethematrixequationEq.( D{6 )ofEn#andsimplifyitbythefollowing 96

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whichbecomesexactlythematrixequationEq.( D{5 )forspinupeigenenergiesEn",excepttheeigenvaluesarenegative.WethereforediagonalizeonlythespinupHamiltonian,andthenattributethepositiveeigenvalues>0tothewavefunction(upn";vpn#;ucn";vcn#)T,whichhaseigenenergiesEn"=,andnegativeeigenvalues<0to(vpn";upn#;vcn";ucn#)T,witheigenenergiesEn#=jj=.Theparticledensitiescouldbecalculatedbyknowingthattheeigenstatesareabasisofquasiparticlesynj0i=jni,andhynni=f(En)sincetheysatisfyFermistatistics. 97

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)Xn>0(upn;in+vpn;iyn )i=Xn>0jupn;ij2hynni+jvpn;ij2hn yn i=Xn>0jupn;ij2f(En)+jvpn;ij2(1f(En )): Accordingtothearrangementofeigenvectors,wesee Noticethatinthelaststepthisprocedureweconvertthephysicalsumoverpositiveeigenstatestoalleigenstatesfornumericalconvenienceandtoavoiddoublecounting.Thepairingeldcouldbesimpliedviathesametrick,andnoticethatthepairingamplitude 98

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99

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2Xyijij:(E{1)Thesaddlepointapproximationisthenintroducedforeachchannel (E{3) +Xiji~Qij(Qijbjbi)+Xiji~Qij(Qijbibj) (E{4) +i~x(XibibiN)+Xii~mi(miSzi)+JXmimj wherefiisGrasmannnumberandbiiscomplex,andthetime-derivativepartis +Xiii(fifi+bibi1): Thesaddlepointequationforeldsij,Qijandmiistherefore 100

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+Xij(J +Xij(tijXij)bjbi +Xij(tijXij)bibj +JXSzimj: Noticethatthespinuporderparameterij"renormalizesspindownchannelfi#fj#andviceversa.It'salsomanifestthatspinonsandholonsrenormalizeeachother,inthesensethatspinonordersenterholondispersionandviseversa. 101

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WeiChenwasborninTaoyuan,Taiwan,astherstsonofhisparentsSung-JungChenandChuan-ChuanChang.HegotthebachelordegreeinNationalChungChengUniversity,Chiayi,withamajorinphysics.Attheageof22,hewenttoNationalTsingHuaUniversityinHsinchu,studiedwithprofessorHsiu-HauLin.Theresearchinthisperiodoftimewasmainlyaboutlowdimensionalcorrelatedsystems,suchascarbonnanotubeandladdermaterials.In2003,hetookthechanceofstudyingintheUSandstartedpursuingaPh.DdegreeinUniversityofFlorida,undertheinstructionofProfessorPeterHirschfeld,withsuperconductivityrelatedresearchtopicsandtheexcitingphenomenaofcuprates.Asatheoreticalphysicist,hecontinuesworkingontheareaofcondensedmatterphysicsanddedicatehimselftothedevelopmentoftheoriesregardingtothisdiscipline. 110


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