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Optical properties of lamellar copper oxides with in-plane magnetic and charged impurities

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Optical properties of lamellar copper oxides with in-plane magnetic and charged impurities
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Moore, Sean
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ix, 250 leaves : ill. ; 29 cm.

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Conceptual lattices ( jstor )
Conductivity ( jstor )
Cuprates ( jstor )
Domain walls ( jstor )
Electrons ( jstor )
Impurities ( jstor )
Magnons ( jstor )
Phonons ( jstor )
Raman scattering ( jstor )
Reflectance ( jstor )
Copper oxide superconductors ( lcsh )
Dissertations, Academic -- Physics -- UF ( lcsh )
Physics thesis, Ph.D ( lcsh )
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Thesis (Ph.D.)--University of Florida, 1999.
Bibliography:
Includes bibliographical references (leaves 240-249).
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Printout.
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Vita.
Statement of Responsibility:
by Sean Moore.

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OPTICAL PROPERTIES OF LAMELLAR COPPER OXIDES WITH IN-PLANE MAGNETIC AND CHARGED IMPURITIES














BY
SEAN MOORE














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


1999















ACKNOWLEDGMENTS


Graduate school.. .what a ride. What a convoluted path to which I now find myself at the end. Now that I have weathered the storm, there are several people I need to thank before I embark on future endeavors. First and foremost, I would like to thank my advisor, John Graybeal, for his guidance, patience, and his willingness to entertain my ideas and observations. His intellectual openness laid the ground work for stimulating and illuminating discussions which at times extended beyond the narrow confines of condensed matter physics. He infused me with a drive to succeed in the lab at times when I was defeated by repeated failures. What is more, he gave me the personal space to find my own way through the problems that typically plague graduate research work, yet he was willing to assist me at a moment's notice. Such working environments are rare and I can not begin to overstate my gratitude. Without being afford this freedom to mend the messes that I made, I doubt very much that I would be the better for pursuing graduate work.

I would not be writing these acknowledgments if it were not for David Tanner and his eternally patient and amenable graduate students with whom I worked. Although I was not officially a member of Dr. Tanner's group, I believe that I was the adopted son they were never sure how they got. My defacto status as a group member was due in large part to the numerous friendships that I struck with David's graduate students over the four years that I worked in his lab. They bestowed upon me the same rights and responsibilities as any other member of the group and gave me unbridled access to their spectrometers and software. Furthermore, I enjoyed several useful and illuminating discussions with David which helped immeasurably with my










research. Without exception, David and his graduate students were always fair and patient to a fault. I am also compelled to thank them specifically for their empathy and understanding when, in the face of experimental disaster, I was less than patient, if not livid, with my research. I hope that in hindsight my occasional eruptions become a source of amusement. If not, well, they can always ridicule me in the years to come. I can not overstate my gratitude to them all.

I would also like to thank my parents and fiancee for their vital support and encouragement on my journey through graduate research en route to a Ph.D. Without their confidence in my ability, I would have had no faith in my own. They endured my bouts of abject misery and frustration without complaint when they were well within their rights to to infuse me with a new perspective with a proverbial kick in the rear. Thank you for handling my sensibilities with kid gloves.

I must also single out a few individuals and groups for special thanks. I would like to thank Steve Thomas for giving me access to his polishing wheel and rescuing me from the ravages of hand polishing and from all the attendant frustration and exasperation that it brings. I would also like to thank Joe Simmons and his group for performing fluorescence measurements vital to our pressure-dependent transmission studies. Many thanks also go to Graig Prescott and Charles Porter for helping me navigate way through a plethora of computer woes. Finally, I thank all my colleagues in the UF Tae Kwon Do Club for their dedication to the art and for their high standards of martial arts training. They are zealous practitioners and good friends. I will miss them.













TABLE OF CONTENTS





ACKNOW LEDGMENTS ..................................... ii


A BST R A CT ............................................ vi


1 INTRODUCTION ......................................... 1
1.1.1 In the Beginning ...................................... 1
1.2.1 Why Study La2CUl-_LiO4 and Sr2CU1-xCoxO2Cl2? .............. 3
1.3.1 A bout this Thesis ..................................... 5


2 REVIEW OF EXPERIMENTAL WORK .......................... 8
2.1.1 Introduction .. .. ... .... . .. . .. ... ... ....... . .. . .... ... 8
2.2.1 La2C uO 4 . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.2 Electronic Configuration and Magnetism: La2CuO4 .............. 10
2.2.3 Optical Conductivity: La2CuO4 ........................... 14
2.2.4 Raman Scattering: La2CuO4 ............................. 19
2.3.1 Lattice Structure: Sr2CuO2Cl2 ............................ 21
2.3.2 Electronic Configuration and Magnetism: Sr2CuO2C12 ............ 22
2.3.3 Reflectivity and Optical Conductivity: Sr2CuO2Cl2 .............. 25
2.3.4 Raman Scattering: Sr2CuO2Cl2 ........................... 27


3 T H EO RY ..................... ......................... 30
3.1.1 Models for Carriers in the CuO2 Planes: Normal State ............ 30
3.1.2 Three-Band Hubbard Model ............................. 31
3.1.3 t-J M odel . . . . .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.4 Cluster Calculations ................................... 36
3.2.1 Models for the Midinfrared Band .......................... 38
3.2.2 Sm all Polaron M odel .................................. 40










3.2.3 Magnetic Excitations: Zhang-Rice Singlets .................
3.2.4 D om ain W alls ....................................
3.3.1 The Charge-Transfer Band ............................
3.5.1 Phonon-Assisted Multi-Magnon Absorption ................


4 EXPERIMENTAL TECHNIQUES ..........................
4.1.1 Introduction .. . ....... . ... . . . .. . ..... ... ..... . ...
4.2.1 Fourier Transform Infrared Spectroscopy ..................
4.2.2 Bruker Fourier Spectrometer ..........................
4.2.3 Bolom eter Detector ................................
4.3.1 The Perkin-Elmer Monochromator ......................
4.4.1 Sample Mounting and Low Temperature Measurements .........
4.5.1 Data Analysis of the Spectra: The Kramers-Kronig Transformations 4.5.2 High and Low-Frequency Extrapolations ..................
4.5.3 Optical Constants .................................
4.5.4 Analysis Procedure for Transmission .....................
4.6.1 Raman Scattering: Experimental Technique ................
4.6.2 2-Magnon Raman Scattering: Theory and Analysis ...........
4.7.1 Sample Preparation ................................


5 OPTICAL PROPERTIES OF La2Cul-,LiO4 ............
5.1.1 Introduction ...............................
5.2.1 Li-doping La2CuO4 ..........................
5.3.1 Overall Reflectance of the a-b Plane ...............
5.4.1 Optical Properties of the Far Infrared: Phonon Assignment 5.4.2 Phonon Assignment in La2Cul-.LiO4 .............
5.5.1 The Midinfrared Band in La2CU1-i Li.O4 ............
5.5.2 Polaron M odel ..............................
5.5.3 Zhang-Rice Singlets ..........................
5.5.4 Dom ain W alls ..............................
5.5.5 M agnetic Strings ............................
5.6.1 Effects of Li-Doping in the Charge-Transfer Region .....
5.6.2 Temperature Dependence of the Charge-Transfer Band ...
5.7.1 Concluding remarks ..........................


in La2CuO4


96 96 97 98 100 110 117 119 126 130 136
140 146 148










6 OPTICAL PROPERTIES OF Sr2Cul-,CoxO2Cl2 ...............
6.1.1 Introduction .. ..................................
6.2.1 Co-doping Sr2CuO2Cl2 .............................
6.3.1 Overall Reflectance of Sr2Cul-,CoxO2Cl2 ................
6.4.1 Phonon Assignment in Sr2Cul-.CoxO2Cl2 ................
6.5.1 Phonon Assisted Multi-Magnon Scattering in Sr2Cul-XCoO2Cl2 �


6.5.2
6.5.3
6.6.1
6.6.2
6.7.1


... 152
... 152
... 153
... 154
... 155
... 164


Pressure Dependence of the MIR Excitations .................
Effects of Co-Doping on the MIR excitations ..................
Temperature Dependence of the Charge-Transfer Region ..........
Effects of Co on the Charge-Transfer Region ..................
Excitations in the Near Infrared ..........................


7 2-MAGNON RAMAN SCATTERING .......................... 195
7.7.1 Introduction ....................................... 195
7.2.1 2-Magnon Raman Scattering Data: La2CuO4 .................. 195
7.2.2 2-Magnon Raman Scattering: La2Cul-xLiO4 .................. 197
7.3.1 Data Analysis: Curve Fitting the 2-Magnon Band ............... 201
7.3.2 Domain Walls vs. Bound Holes ........................... 203
7.4.1 2-Magnon Scattering in Sr2CUlxCoxO2Cl2 ................... 213
7.4.2 Analysis and Interpretation: Sr2Cul-xCoXO2Cl2 ............... 215
7.4.3 Phonon-Assisted Mutimagnon Absorption Revisited .............. 218
7.5.1 Concluding Rem arks .................................. 221


8 CONCLUSION ..............................
8.1.1 Sum m ary ..............................
8.2.1 Future Experiments .......................


........... 223
. . . . . . . . . . . 223
. . . . . . . . . . . 226


APPENDIX: RAMAN SCATTERING ........................... 228
Introduction . ..... .. ...... . . . .. .. ... . .... ... .... . .. . .. . 228
The N-Particle Radiation Hamiltonian ......................... 229
Scattering Cross Section and Perturbation Theory ................... 231
Calculation of One-Magnon Raman Scattering Cross Section ........... 232
2-Magnon Raman Scattering in Antiferromagnetic Systems ............. 235










REFERENCES .......................................... 240


BIOGRAPHICAL SKETCH ................................. 250









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


OPTICAL PROPERTIES OF LAMELLAR COPPER OXIDES WITH IN-PLANE MAGNETIC AND CHARGED IMPURITIES


By
Sean Moore

December 1999

Chair: Dr. John M. Graybeal
Major Department: Physics

The optical properties of the superconducting cuprates are unusual, and many of their spectral features are still not well understood. This is especially true in the midinfrared, where a heavily weighted band grows with charge-doping concurrent with a rapid loss of spectral weight in the charge-transfer band. To date, this redistribution of spectral weight is not well understood.

We have extensively studied the optical properties, including the 2-magnon Raman spectra, of La2Cu1-LiO4 and Sr2Cul-CoO2Cl2, two materials that do not have a superconducting phase. Li adds a hole carrier to the CuO2 plane in much the same way as divalent Sr in La2-xSrxCuO4. However, Li goes into the CuO2 plane with a closed shell and so introduces a S=O impurity. La2Cu1-iLi-O4 undergoes the aforementioned redistribution of spectral weight prototypical of all of the superconducting cuprates. However, the charge-transfer band remains comparatively robust. This suggests that the additional weight in the mid-infrared must come from spectral regions above 4eV. Furthermore, no Drude tail is observed. This is consistent with d.c. conductivity measurements that place La2Cu1-iLi.O4 in the insulating regime up to x = 0.50. Our two-magnon Raman scattering studies suggest that the 2D









AF spin order of La2Cul-LiO4 remains relatively intact up to x = 0.10, in stark contrast to what is found in the under-to-overdoped regimes of the superconducting cuprates. Despite this, 3D long-range spin order is lost at Li concentrations very similar to those found La2-xSrxCuO4.

The absence of charge carriers in Sr2Cul-xCoxO2C12 precludes the formation of a heavily weighted MIR band and subsequent redistribution of spectral weight. Co substitutes for Cu in the CuO2 plane and is believed to introduce a spin-3/2 impurity into the spin-1/2 AF background. Modest variations in the phonon-assisted bimagnon absorption band suggest that the spin-3/2 impurity sites are, to a first approximation, magnetically decoupled from their spin-1/2 AF host. This is consistent with 2-magnon Raman scattering measurements which indicate that the 2D in-plane spin correlation length changes little with the introduction of cobalt. We find that the charge-transfer band is relatively impervious to the presence of Co in the light-to-moderately doped regime.













CHAPTER 1
INTRODUCTION


1.1.1 In the Beginning

The world of condensed matter physics changed forever in 1986 when Bednorz and Miiller1 reported a superconducting phase in La2_xBa.CuO4 at T- 30K. This discovery initiated an enormous amount of research to push the superconducting transition to ever higher temperatures. These efforts were not without reward. In 1987, Wu et al.2 synthesized YBa2Cu307-6 and soon thereafter found a T, of _ 90K. One year later, Parkin et al.3, along with Sheng and Hermann, discovered a Tc of 125K in T12Ba2CaCu208. Not to be outdone, Schilling et al.4 synthesized tri-layer HgBa2Ca2Cu308+6 in 1993 with a critical temperature of 133K. Over a time span of less than two years, superconductivity transcended a subfield of condensed matter physics and emerged a headline story on everything from "Physics Today" to CNN. By 1987 it seemed that every armchair scientist in the nation, if not the world, was talking about energy efficient levitating trains that could zip along superconducting rails at enormous speeds. Star Trek technology was just around the corner we were told, it was just a matter of time until the critical temperature topped a tepid 300K. Unfortunately, just how long said time would be was unclear. It still is. The intelligence, dedication, and sweat of a relative few may have gotten us to a critical temperature of 133K, but it will most likely take the collective efforts of the entire condensed matter community to get us to room temperature. Efforts to manufacture materials that superconduct above 133K have, in the absence of enormous applied hydrostatic pressures5, failed. After only a few memorable years, the specter of room temperature superconductivity, and all of its attendant novel technology, faded from










the headlines and eventually from the public consciousness. Besides, by this time we had cold fusion and the promise of cheaper utility bills. Depressing? A little. Is this how the story ends? Hardly. Now that the fervor over high temperature superconductivity is over we can make a sustained and deliberate effort to figure out what makes these things tick.

The superconducting mechanism of the high temperature superconductors is not understood at this time and it is not the principal intent of this dissertation to elucidate it. We chose to focus on the higher energy optical properties of the superconducting planes in and well above the far-infrared. These excitations have received comparatively little attention since they are so far removed from the characteristic superconducting transition energy around 20-30 cm-1 . We feel that, despite the large scale differences in energy, these excitations are important in their own right and, at some future point, may shed some light on the elusive nature of high temperature superconductivity.

High temperature superconductivity is achieved in the spin-1/2 magnetically ordered lamellar cuprates by introducing charge carriers, typically holes, into the CuO2 planes. The superconducting phase is highly anisotropic in these systems. The coherently paired electrons superconduct primarily in the CuO2 plane with only a weak perpendicular component. Optical excitations which probe the CuO2 plane in and above the far-infrared region change dramatically with the introduction of chargecarriers. There are three principal deviations in the optical spectra from the undoped (insulating) phase. A Drude tail grows in the far-infrared, consistent with the insulator-to-metal transition observed above doping concentrations of 2-3%. This is expected. However, a heavily weighted mid-infrared (MIR) band which scales roughly linearly in the carrier concentration emerges around 0.50eV (- 4000cm-1). This is concurrent with a rapid erosion of the charge-transfer band near 2eV. This redistri-










bution in spectral weight is not well understood at this time, and it is the hope of this dissertation to help resolve this mystery. There is no doubt that part of the difficulty in understanding these systems stems from their complexity. La2CuO4 and Nd2CuO4, two of the simpler host high T, superconductors, contain 14 atoms per unit cell. In fact, the T, generally scales with the complexity of the unit cell, increasing from -, 30K to - 125K from La2_.SrCuO4 to T12Ba2CaCu2O8. Early measurements on these materials were designed to see if the superconducting mechanism could be explained in the context of the Bardeen-Cooper-Shreiffer (BCS) theory6 for conventional superconductors. Indeed, some similarities to the conventional superconductors were found. Flux quantization7 and A.C. Josephson effect8 measurements showed that the elementary unit of charge in the superconducting condensate was 2e while photo-emission9-11 and tunneling12'13 measurements suggested the presence of a superconducting gap. However, at the same time unconventional properties of the high-Tc materials were found. Some of the principal differences between the high T, superconductors and their conventional brethren aside from obvious deviations in their respective critical temperatures include a linear d.c. resistivity with temperature in the normal state14'15, extremely small coherence lengths6'17, and anisotropy in many of their physical properties. The anisotropy is especially evident in the predominately 2D nature of the superconductivity within the CuO2 planes.

1.2.1 Why Study La2Cui_,LiO4 and SrgCul_,CoO2CI2?

The discovery of the high T, superconductors certainly spawned enormous theoretical and experimental efforts to unveil the superconducting mechanism in these materials. From an experimental standpoint this typically involved synthesizing and measuring new materials in an effort to push the high-T, envelope to ever higher temperatures. Much of the optical work performed on these systems focused on excitations well into the far-infrared. However, any materials stress engineer will tell










you that the best way to make something better is to break it. Once we know how and why it breaks, we are in a position to build a superior device. So that is what we did: We hit high temperature superconductivity with a proverbial hammer and measured what we got in the process. Furthermore, to make matters simple, we examined the simpler single layer copper-oxide superconductor hosts La2CuO4 and Sr2CuO2C12. In the latter, a superconducting phase has not been observed. However, it is isostructural to La2CuO4 and shares many optical properties. The "hammers" used in this case were charge and magnetic impurities introduced into the CuO2 superconducting planes. Our measuring devices were optical spectrometers and a Raman scattering setup on an optical table. Specifically, we looked at the optical properties of La2Cu1-.LiO4 and Sr2Cul-,CoxO2Cl2, both insulators, in the mid-infrared and charge-transfer spectral regions and compared our results to existing studies on one of the original high-Tc materials, La2-xSrxCuO4. Superconductivity in the high T, superconductors is achieved by substituting charge donors outside of the spin-1/2 antiferromagnetic (spin-1/2 AF) CuO2 plane. In La2CuO4, for example, this is done by replacing divalent Sr for trivalent La which adds holes to the CuO2 planes. By contrast, monovalent Li is substituted for divalent Cu in La2Cul-.Li.O4. Since Li goes into La2CuO4 with a closed electronic shell configuration each adds both a hole and a spin-zero impurity to the CuO2 plane. The presence of an in-plane spin-zero impurity coupled with the strong binding energy of each hole carrier to its Li host precludes superconductivity. In Sr2Cu1XCoxO2Cl2, cobalt substitutes for Cu in the CuO2 plane. Since both incorporate into the lattice as divalent entities, no carriers are added to the system and, thus, there is no possibility for a superconducting phase. However, cobalt is believed to go into Sr2CuO2Cl2 with a 3d7 electronic configuration in a high spin state. Therefore, each cobalt site introduces a spin-3/2 impurity into the spin-1/2 AF background. Hence, we can examine the effects of adding charge










and/or magnetic impurities to the CuO2 plane by charting the optical properties of these two systems. This will hopefully contribute to an understanding of the optical properties of the high T, superconductors.


1.3.1 About this Thesis

In writing this dissertation I attempted to compartmentalize the chapters as much as possible according to material and experimental technique. This was done in hopes of making the individual chapters as independent of each other as possible. The decision to format this dissertation in this manner was arrived at after considerable consternation and second (and third and fourth) thought. There are both advantages and disadvantages to this structure, but it is my reasoned belief that the former eclipses the latter and I hope that after reading this dissertation you agree. There are, I believe, two principal advantages to the modular approach. One is that the reader can read any of the chapters in no particular order without constantly being referred back to some earlier or later chapter, although this dissertation is not completely devoid of that sort of future/past referencing. When it is reasonable and proper I have tried to reproduce figures and data rather than to send the reader backwards or forwards 50-100 pages. I realize that this can be very annoying. The second advantage of a modular structure is that it allows the researcher to decouple experiment from theory by placing each in separate chapters. Although this defeats the first principal advantage to some degree, I believe it more logical and find it less cluttered to separate theory from experiment than to bury theoretical models in different chapters. In this way the reader is referred only to a particular subsection of a single chapter when the data is analyzed in the context of a given theoretical construct.

This dissertation is arranged in the following order. First, a brief review of the work done on previous high-T, systems is presented in Chapter 2. The structural, elec-










tronic, magnetic, and optical properties of some the single layer copper-oxide superconductors are discussed and comparisons to La2Cu1-LiO4 and Sr2Cul- Co.O2CI2 are made. In Chapter 3 many of the theoretical models used to interpret the observed optical excitations are presented in modest detail. One- and three-band Hubbard models, polaron excitations, Zhang-Rice singlets, domain wall structures, and phonon-assisted magnon absorption are discussed. The intent here is not to provide a complete tutorial on these theoretical constructs but rather simply to motivate the physics behind them. It is recommended that this chapter be read prior to reading Chapters 5-7 where the experimental data for this dissertation is reported. Chapter 4 discusses the experimental apparatuses and techniques used to perform optical measurements over a range of temperatures. Notes concerning sample manufacturing and preparation are also presented here. Chapter 5-7 are the experimental crux of this dissertation. In Chapter 5 the optical data for La2Cu1-.Li.O4, including reflection, optical conductivity, and sum rule calculations, are presented over a wide range of temperatures. It is found that La2Cul-,Li.O4 bears many similarities to and a few critical differences from its superconducting counterpart La2_,SrxCuO4. Many of its optical features are discussed in the context of the theoretical models of Chapter 3. Chapter 5 is structured in a similar fashion for the experimental results of Sr2CuI-.Co1O2C12. Lastly, Chapter 6 is devoted to 2-magnon Raman measurements on both La2Cu1-pLiO4 and Sr2Cul-xCo.O2Cl2. Since 2-magnon Raman data require substantially less analysis than optical transmission and reflection data, the 2-magnon Raman scattering cross sections for both La2Cul-,LiO4 and Sr2Cul-,Co.O2Cl2 were compiled into one chapter without, I believe, violating the aforementioned principle of compartmentalization. Chapter 8 summarizes the previous three chapters, offers some perspective on the relevance of this dissertation, and speculates about possible related future projects. And lastly, the Appendix pro-







7

vides a short tutorial on 1- and 2-magnon Raman scattering in magnetically ordered materials. I hope this format makes for a coherent and stimulating presentation.














CHAPTER 2
REVIEW OF EXPERIMENTAL WORK



2.1.1 Introduction

This chapter will briefly review the physical and optical properties of the materials studied for this dissertation. It is divided into two sections. Each is devoted to the physical and optical properties of one of the two materials investigated. These properties include lattice and spin structure, electronic configuration, magnetism, optical conductivity and 2-magnon Raman scattering. Some of the properties discussed may seem tangential to this dissertation, but they are important in their own right. For example, while this dissertation is not an examination of the crystal or spin structure of the high-T, cuprates, it is nonetheless vital to have a working understanding of these structures if the peculiar and poorly understood optical properties of the high T, superconductors are to be tamed. The lattice structures of all the high T, layered cuprates, from La2- SrCuO4 to Bi2Sr2Ca2Cu3O10 are similar and share many physical properties. Charge transport and superconductivity, for example, occur mainly in the nearly identical 2D CuO2 basal planes of these materials. The planes consist of Cu sites surrounded by four oxygen sites with a Cu-O separation of roughly 1.9A. This separation changes little from material to material. Furthermore, the CuO2 planes of this class of materials are separated by layers of other atoms which, when properly doped, provide the charge carriers responsible for superconductivity in the CuO2 planes. Also common to this family of cuprates is the bulk anisotropic antiferromagnetic spin order (AFO) of the parent insulators. Many speculate that it is the spin order, especially in the CuO2 planes, that drives the superconductivity in the











Lo2CU04-6 (CmcO)


Figure 2.1. Crystal and AF spin structure of La2CuO4. From Reference 20.


layered cuprates. For concision, lattice structures and optical properties discussed in the following sections will be limited to materials covered in this dissertation.


2.2.1 La2CuO4

Ostensibly the simplest of the high T, materials, La2CuO4 was discovered by Bednorz and Muiller1 to have a superconducting phase when divalent Ba was substituted for trivalent La. Doping the system with Ba added hole carriers to the system and subsequent measurements which mapped the Fermi surface determined that these holes resided on oxygen sites in the CuO2 planes. Bednorz and Miller speculated that La2CuO4 had a tetragonal K2NiO4 structure and this was later confirmed by










Takei et al.18. Further study19 revealed that stoichiometric La2CuO4 is an antiferromagnetic insulator with a 2D superexchange energy in the CuO2 plane of roughly 0.125eV. Fig. 2.1 shows the crystal structure of undoped La2CuO4.20 Each Cu is surrounded by four oxygen atoms with a Cu-O separation of 1.9A. There are two more distant apical oxygens above and below the central copper site at a distance of 2.4A. Hence, each Cu site in the compound is surrounded by an octahedron of oxygens. The larger Cu-O separation for the out-of-plane oxygens suggests that the dominant bonds are those in the CuO2 plane. Several experiments have buttressed this prediction. The octahedral oxygen configuration which shrouds each Cu2+ ion implies that the lattice has a tetragonal 14/mmm space group structure. However, neutron scattering measurements show that the apical oxygens tilt from their high symmetry points21'22. The tilting produces a tetragonal-to-orthorhombic distortion in the crystal lattice that sets in below a specific characteristic temperature which depends on the stoichiometry of the compound. For nearly stoichiometric La2CuO4 (no excess oxygen or impurity substitutions) the characteristic temperature is close to 800K, while for a Sr concentration of roughly 15% (Lal.s5Sro.15CuO4) it drops to about 240K.

2.2.2 Electronic Configuration and Magnetism: La9CuO4

Undoped La2CuO4 is a charge-transfer spin-1/2 antiferromagnetic insulator (spin1/2 3D AFO). The valences of La, Cu, and 0 when introduced into La2CuO4 are +2, +2, and -2 respectively. Hence, charge neutrality is preserved. Lanthanum and oxygen go into La2CuO4 with closed 2p and 5f valence shells respectively. Copper, however, enters the matrix with a half-filled 3dX2_y2 orbital. Crystal field splitting removes the degeneracy of the 3d orbitals on the copper sites. Figure 2.2b illustrates the splitting of the electronic energy levels at the copper sites while Fig. 2.2a gives the orbital configurations of the Cu 3dX2_Y2 and 0 2px, 2py orbitals in the CuO2 plane.23










The energies of the remaining La2+ 5f, apical 02-, and in-plane 0 2pz orbitals lie far below the in-plane Cu2+ 3d and 02- 2pz, 2py energy states and are not presented here. Since the 3dx2_y2 orbital is directed at neighboring 0 2px, 2py orbitals in the CuO2 plane, crystal field effects dictate that it must have the highest energy of the five Cu3d orbitals. It is followed in order by the 3d3z2-r2, 3d,, 3dxz, and 3d,,, orbitals. Cu2+ enters the matrix with a 3d9 configuration, so there is one unpaired spin per Cu2+ site. These residual spins couple to nearest-neighbor Cu2+ sites via a superexchange interaction to give La2CuO4 3D antiferromagnetic order. Although the spin order of the Cu2+ ions is three dimensional, the superexchange interaction is dominant along the Cu-0 bonds in the CuO2 planes. While there is only a modest discrepancy in the bond distance between neighboring in-plane and out-of-plane Cu2+ sites (1.915A vs. 2.219A), neighboring Cu2+ sites in adjacent planes are displaced laterally by one-half a lattice spacing (see Fig. 2.1). This lateral displacement coupled with the longer Cu-0 bonds along the b-axis results in an appreciable difference between the in-plane vs. out-of-plane superexchange energies (J11 vs. JI). J11 is roughly 1200K while JI is on the order of 200K.19 Hence, it is anticipated that 3D AFM will be lost with the introduction of charge or spin impurities well before the 2D spin correlations are eliminated in the CuO2 planes. Indeed, a variety of studies24'25 have determined the 2D spin correlation length in the CuO2 plane to be on the order of 1000A. From Fig. 2.1, it is seen that for stoichiometric La2CuO4 the spin directions and the antiferromagnetic propagation vectors are in the [001] and [100] directions respectively. The substitution of Sr2+ for La3+ induces a small canting26'27 (- 0.170) towards the b-axis due to an antisymmetric exchange interaction originating from the orthorhombic distortion of the lattice with doping.

Figure 2.3 shows the phase diagram for La2_,Sr.Cu04.23 La2CuO4 can be holedoped by substituting Ca, Sr, or Ba for La to produce a superconducting phase







12

z

... X




X27L
y z z





2 2

Al
Xoy


S- B2
x2. XZ. yz
T2 XZ0 Yz2 T "-. xz. yz
E


Spherical Cubic 1 0 J TctragonaJ D4

Figure 2.2. (a) Cu 3d orbital configurations in the cuprates. (b) Crystal field splitting of the Cu 3d energy levels under D4h symmetry. From Reference 23.


between 30 and 40K.1 Alternatively, a superconducting phase may be realized by altering the oxygen stoichiometry to a number greater than four28. In the latter case electrons are added to the CuO2 plane and serve as the charge carriers in the superconducting phase. In either case, the compound undergoes three distinct phases as carriers are added to the system. For Sr concentrations between x = 0.00 and x = 0.03, La2-xSrxCuO4 remains a 3D antiferromagnetic insulator. For concentrations between x = 0.02 and x = 0.07 3D antiferromagnetic order is lost and the system becomes a weakly metallic spin glass. However, as doping proceeds, starting at x = 0.07 and continuing through x = 0.23, La2-xSrxCuO4 becomes superconducting with











T [K] La2.xSrxCuO4 530



325
Antifcrromagnct Tetragonal


Orthorhombic


40 Si-ls
10 .G .. Ovcrdoped 10 Superconductor

0.02 0.05 0.2 0.25X Figure 2.3. Phase diagram for La2_xSrCuO4. From Reference 23.


optimal doping at x = 0.175 (T, - 39K). For Sr concentrations in excess of x 0.24, La2-xSrxCuO4 is metallic with a d.c. conductivity that scales roughly linearly with the temperature29. This unusual temperature-dependent behavior of the d.c. resistivity in the normal state has inspired many in the condensed matter community to dubb the high T, superconductors "strange metals". Although Fig. 2.3 is specific to La2-xSrxCuO4, it is prototypical of all of the doped high T, cuprates and requires only modest changes in the critical concentrations from one material to the next.

This dissertation investigates, in part, the effects of Li-doping La2CuO4. Contrary to Sr-doping, Li is substituted for Cu in the CuO2 plane. However, like its Sr counterpart, each Li adds a single hole the CuO2 plane which resides on an oxygen site. Li+ goes into La2CuO4 with a closed ls shell, and so introduces a spin-zero impurity in the AFM spin-1/2 matrix. X-ray diffraction measurements indicate that the addition of Li to the CuO2 plane modestly suppresses the orthorhombic distor-










tion of the CuO6 octahedron, particularly along the Cu-O axes in the CuO2 plane30. By x _ .30, the symmetry of the lattice increases from orthorhombic to tetragonal. Unfortunately, to date no comprehensive neutron scattering measurements have been performed on La2Cul-xLixO4 to corroborate these findings or elucidate the effect of the spin-zero impurity on the 2D spin order of the Cu2+ lattice. NQR studies on La2Cul-LixO4 demonstrate that3l, much like La2_xSrxCuO4, 3D AFM is lost at x = 0.03. However, since the superexchange coupling is dominant in the CuO2 plane, this gives little information concerning the 2D spin correlations of the Cu2+ ions in the CuO2 plane. For Li concentrations between x = 0.03 and x = 0.50, La2Cul-xLixO4 remains insulating with no superconducting phase, in stark contrast to what is observed in La2-xSrxCuO4. This implies that holes introduced by Li+ ions in the CuO2 planes are tightly bound their impurity hosts. This insulating character makes La2Cul-xLixO4 an exciting and relevant material to study as it eliminates contributions to the optical properties from free carriers which would otherwise dominate in the far- and mid-infrared. Furthermore, by comparing the optical properties of La2-XSrxCuO4 and La2Cul-xLixO4 some light may be shed on the elusive nature of the superconducting mechanism in the doped cuprates.


2.2.3 Optical Conductivity: La2CuO4

The optical properties of all the superconducting cuprates are remarkably similar despite disparities in their crystal structure and chemical composition. All are anisotropic in the optical conductivity in the doped and undoped phases, and all demonstrate the formation of a mid-infrared in the a-b plane with doping. Since trends in the optical conductivity are relatively invariant from compound to compound, we will focus only on the optical properties of La2_xSrxCuO4 as it closely resembles the chemical composition of La2Cu1xLixO4.









One of the more striking features of the lamellar cuprates is the anisotropy of the optical conductivity in the a-b axes vs. c-axis directions in the insulating, superconducting, and metallic phases. Figures 2.4a and 2.4b show the reflectivity of single crystals of La2-,SrxCuO4 for light polarized parallel and perpendicular to the CuO2 plane32. There is little change in the c-axis reflectivity (Fig. 2.4b) from x = 0.00 to x = 0.34, from the insulating to metallic regime. Only at x = 0.34 are there qualitative changes in the reflectance at low frequencies. In this regime, the spectrum has a weak free carrier component, although the lowest energy optical phonon is not screened out entirely. This weak metallic character is consistent with d.c. transport measurements along the c-axis on single crystals of La2_xSrxCuO4 in the overdoped (metallic) regime.29 Two major optical phonons with A21 symmetry dominate the c-axis spectrum at 0.029eV and 0.074eV. There are actually three infrared active phonon modes, but the lowest energy phonon at .017eV has considerably less spectral weight and is effectively absorbed by the A2,, phonon centered at 0.029eV.

Far more structure can be found in the a-b plane reflectivity of La2CuO4 as can be seen in Fig. 2.4a. In the far-infrared below 0.10eV, the spectrum is dominated by four infrared active modes centered at 0.018eV, 0.045eV, 0.05eV, and 0.086eV.33 Between 0.10eV and 1.5eV, the spectrum is featureless while between 1.5eV and 4eV, the reflection drops precipitously. The reflectivity edge centered near 2eV is attributed to a charge-transfer excitation of an electron from an oxygen 2p, (2px,y) orbital to a copper 3dX2_y 2 orbital in the CuO2 plane. This charge-transfer transition at roughly 2eV is common to all the lamellar cuprates and its position scales in energy with the oxygen coordination number of the Cu2+ sites34. While the c-axis reflectivity of La2-xSrxCuO4 remains relatively unchanged up to x = 0.34, the reflectance in the a-b plane dramatically changes as the Sr concentration varies from x = 0.00 to x = 0.34. The reflectivity and corresponding optical conductivity are characterized by the
















H

C-)
NI 00.2.

0.10



1. 0 .02

MI .0 I C.1
0.50.15
0.1I 032 ( . 20
R T 0.30.5
0.05 0.1 0.2 0.5 1 2
PHOTON ENERGY (eV)



I.C
,~~~ ~~ 0. .p_,,.. -0.,s4


4.C.
0.5 "xO.1 --*.. .0.5
0 . 0. 3 1.4 tn Phon energy (eV)
0 t x=0.34
. - -0.5


0.5 X=o. is 0 w

La2_xSrxCu04- -0

00 0. 1 0 .2 - 0 .3 0.4
Photon Energy (eV)

Figure 2.4. Reflectivity of La2-xSrxCuO4 at T=300K for (a) Ell and (b) E I to the a-b plane. From Reference 32.









growth of a mid-infrared band between 0.1eV and 0.6eV accompanied by the erosion of spectral weight in the charge-transfer region as can be seen in Fig. 2.4a. With only a small amount of doping (x < 0.1), the spectral weight of the charge-transfer band is sharply reduced while a reflectivity edge forms near 0.8eV. As doping increases, the reflectivity edge sharpens while the reflectivity below the edge increases rapidly. for 0.00 < x < 0.25, the edge does not shift appreciably in energy. For x > 0.25, just above the superconducting phase, the edge moves to lower energies. This is not consistent an increase in the number of carriers as the system is progressively doped.

The aforementioned trends in the reflectivity of La2-xSrxCuO4 are even more apparent in the optical conductivity. In Fig. 2.5, the two salient phenomena associated with Sr-doping may be readily observed. There is a shift in spectral weight from the region above 1.5eV to the region below 1eV. A band between 0.1eV and 0.6eV grows with Sr content while the charge-transfer edge erodes rapidly. Below 0.1eV, phonons that once dominated the spectrum in the undoped phase are now buried in a Drude tail. For x < 0.15eV the mid-infrared band peak shifts to lower energy with increasing Sr content and eventually merges with the Drude band above x = 0.2. Other superconducting cuprates such as YBa2Cu3O7-6 and Nd2-xCexCuO4-y mimic this behavior in the low-to-moderate doping regimes.

Despite the lack of a discernible band maximum in the optimally doped regime the presence of a mid-infrared band is evident by a decay rate of the optical conductivity that is slower than the typical - Drude response of metals. Evidence of this non35-40
Drude response has been found in nearly all of the copper based superconductors - . Explanations have been a matter of heated controversy. Despite this, some irrefutable facts about the unconventional nature of the optical response persist. One is that the region between 0.1 and 0.5eV shows little temperature dependence. This is concurrent with an increase in the d.c. conductivity by a factor of three from 300K to 10K.














'E 0.20
u 1.0
X=0
o) 0.02
0.10 .0 0.10
3 O.15 ' 0.5 0.06 -0.20

0.34
0.02
0 0
0 1 2 3 4 hw ( eV )
Figure 2.5. a-b Plane optical conductivity of La2-,SrCuO4 at T=300K.
From Reference 32.



Clearly, a Drude response with a single relaxation rate for the charge carriers can not explain such behavior. Also, there is a definite temperature dependence of the low frequency conductivity that is consistent with d.c. measurements37-39,41,42


Explanations and models of the mid-infrared band vary. Some have speculated that these excitations are the photo-induced transitions of bound-holes (electrons) from their ground state to the continuum. On the other hand, since the band is centered near the antiferromagnetic coupling energy, J, other theories attribute this band to magnetic excitations associated with charge impurities hopping from site to site in the CuO2 plane. Still another model relates the excitations in the mid-infrared to polar coupling of the charge impurities to the lattice [electron(hole)- phonon coupling]4344. Recently, charged domain walls have been introduced to model the electronic and spin behavior of the doped cuprates. These issues will be addressed in the following chapters.









The erosion of the charge-transfer band with impurity doping is also not well understood in these materials. Each Sr2+ introduces one hole into the CuO2 plane. In a three-band Hubbard model, this removes one electron removal state since there is one fewer oxygen site available from which an electron may be photo-excited to a neighboring Cu2+ site. Based on this argument, one would expect the spectral weight loss of the charge-transfer region to scale linearly with doping concentration. As readily observed in Fig. 2.5, this is not the case. The charge-transfer band erodes at a rate which rapidly exceeds x, particularly above x = 0.05. One possible explanation for this anomalous behavior in the charge-transfer region is that the excitation of an electron from an oxygen to a copper site is a spin-dependent transition. Holes or electrons added to the CuO2 plane break superexchange bonds between neighboring CU2+ ions and frustrate the 2D spin lattice. Cu2+ spins near the charge impurity which were once anti parallel to their neighbors are now distorted. Since chargetransfer from oxygen site to neighboring Cu2+ site is only possible in an anti-parallel spin channel, spin frustration may obscure or smear the charge-transfer excitation in the region surrounding a charged impurity. More will be said concerning this in Chapter 5.


2.2.4 Raman Scattering: LaCuO4

Two-magnon Raman scattering data for the lamellar cuprates is remarkably similar. From single layer La2CuO4 to YBa2Cu3OT7_ all have the signature two-magnon band centered between 3300 cm-1 and 2600 cm-1 associated with the charge(spin)

-transfer excitations in the CuO2 plane. Common to these compounds is the rapid suppression of the 2-magnon band with impurity doping. This suggests that impurities introduced into the parent insulators not only provide charge carriers for the superconducting condensate, but also dramatically impact the 2D spin correlations.










As with the optical conductivity, only Raman scattering data pertaining to materials studied in this dissertation will be discussed.

Two-magnon Raman scattering via the exchange mechanism is unique to antiferromagnetic systems. It occurs when an electron with, say, spin-up located at one site is photo-excited and exchanges position with an electron of opposite spin on an adjacent site. In the antiferromagnetically ordered cuprates, this represents electrons on neighboring 3dX2_y2 orbitals in the CuO2 plane exchanging positions and creating two spin-flips in the 2D antiferromagnetic background. Since this transition reverses the spins on two neighboring sites in the 2D Cu2+ lattice, it breaks six antiferromagnetic bonds and costs roughly 3J (ignoring magnon-magnon interactions). This creates two magnons at the antiferromagnetic Brillouin zone boundary with equal and opposite wave vectors (momentum) to preserve momentum conservation. Two-magnon Raman scattering provides a measure of the local 2D spin order of antiferromagnetic systems and, when properly analyzed, can yield information about the magnon scattering length of the spin lattice.

Two-magnon Raman scattering in the lamellar cuprates is sensitive to the presence of doped carriers. In Fig. 2.6, the spurious effects of Sr-doping La2CuO4 are readily apparent. Even for Sr concentrations below x = 0.1 the two-magnon Raman band broadens rapidly and shifts to lower energy with increasing Sr content19. In the optimally doped regime (0.1 < x < 0.25), the two magnon peak intensity is flattened into a broad spectral background. While it is difficult to extract an accurate estimate of the magnon scattering length from the Raman data in the optimally-to-overdoped regime (x > 0.1), some conclusions about the spin order in the optimally doped materials can be reached. In this regime, the two-magnon band is obscure, merging into a featureless continuum. This implies that the 2D in-plane magnon scattering length, which in the undoped regime is on the order of 25-30A, has been shaved down to


















* 1
La2.xSrxCuO4
F 2 Magnon Rama n

-U.O10
x-.0.034
x-0.070
x-0. 116 (uhfLFB,46
0
0 low0 2000 30,00 4000 R~ma asdfi (=7')

Figure 2.6. 2-Magnon Raman scattering peak in La2-..Sr..CUO4 as a function
of Sr content at T=300K. From Reference 19.


one-to-two lattice spacings. Doped holes residing on the oxygen 2p, orbitals severely weaken the superexchange coupling constant J turning the 2D antiferromagnetic lattice into a quasi spin-glass. This suggests that the holes are weakly bound to the Sr2+ sites and are essentially free to hop from site to site. However, it is difficult to ascertain precise estimates of the 2D spin correlation length from two-magnon Raman scattering. Neutron scattering proves a more useful tool for probing local and long-range spin order.

2.3.1 Lattice Structure: Sr9CuO9Cl2

Sr2CuO2CI2 is a parent insulator of the high T, cuprates. X-ray and neutron scattering studies45,46 have demonstrated that Sr2CuO2C12 is a layered perovskite with tetragonal (I/mmm) K2NiF4 structure and is isostructural to the high temperature form of La2CuO4. In composition, however, Sr2CuO2C12 is dissimilar to its superconducting counterparts. Sr2CuO2CI2 consists of basal CuO2 planes intercollated with Sr-Cl layers as shown in Fig. 2.7a. Instead of a single buffering layer, there









are now two Sr-Cl planes partitioning the CuO2 planes. Each copper site is flanked by four oxygens in the CuO2 plane and by two chlorines along the c-axis forming a CUO2C12 octahedron. Unlike La2CuO4, there are no apical oxygens in the buffering layer between adjacent CuO2 planes. Neutron scattering measurements by Vakin et al.47 determined the Cu-O interatomic distance to be 1.985A and estimated the CuCl spacing to be 2.8A. The presence of two chlorines, each translated one-half lattice spacing with respect to other, increases the c-axis parameter by 18% in Sr2CuO2CI2 with respect to La2CuO4. The c-axis measures 15.618A while the a- and b- axes parameters measure 3.975A, making the Wigner-Seitz unit cell considerably larger than in La2CuO4. Sr2CuO2CI2, unlike its lanthanum-based counterpart, is stoichiometric as grown and no orthorhombic distortion of the crystal lattice has been observed down to 10K.


2.3.2 Electronic Configuration and Magnetism: Sr2CuO2 Cl9

Sr2CuO2C12 is a 3D antiferromagnetic insulator with a N~el temperature between 250 and 310K. As with the superconducting lamellar cuprates, the anisotropic superexchange interaction between neighboring copper sites binds the spins. Sr, Cu, 0, and Cl incorporate into the Sr2CuO2C2 matrix with +2, +2, -2, and -1 valences respectively. And, as with its superconducting cousins, the highest energy electronic bands in Sr2CuO2Cl2 lie in the CuO2 plane (see Fig. 2.2a with electrons in the Cu 3dX2_Y2 orbital having the greatest energy. As with La2CuO4, Cu+2 goes into Sr2CuO2Cl2 with a 3d9 shell configuration. The 3d3z2_r2, 3dxy, My, and 3dx, shells are filled while the 3dX2_Y 2 orbital is half-filled, giving each Cu2+ site a net spin of 1/2 (see Fig. 2a). These spins couple to neighboring Cu2+ sites antiferromagnetically to form a spin-1/2 antiferromagetic insulator. Since neighboring Cu2+ sites are closer in the basal CuO2 plane than in adjacent planes, the superexchange constant, J, is










MAGNETIC STRUCTURE


0 Cu 0 0
0 CI


Figure 2.7. (a) Crystal and (b) spin structure of Sr2CuO2C12. From Reference 47.


highly anisotropic with the in-plane value, J11, many orders of magnitude larger than its out-of-plane counterpart, J1. Hence, while Sr2CuO2CI2 is a 3D spin-1/2 insulator below ,- 300K, the spin system and its excitations are often treated as 2D. Magnetic suseptibility and Raman measurements (see Chapter 7) estimate J11 at - 860K. The magnetic structure is similar to that found in La2CuO4. The spins are aligned on the [110] direction and are perpendicular to the antiferromagnetic propagation vector, as shown in Fig. 2.7b.47 However, the absence of an orthorhombic distortion down to 10K precludes the spins from canting towards the c-axis and forming a weak bond. The fact that Sr2CuO2Cl2 remains tetragonal raises the question as to how the CuO2 planes couple to give 3D antiferromagnetic order. Recall that in La2CuO4 and La2NiO4 it is believed that the orthorhombic distortion of the crystal lattice drives










the anti symmetric interplane coupling between the spins48. In tetragonal symmetry, these antisymmetric spin interactions sum to zero. Some have speculated that, in the absence of this interplane superexchange, spins in neighboring planes couple via magnetic dipole-dipole interactions47. Preliminary calculations based on this speculation estimate the the 2D spin correlation length in the CuO2 plane to - 3000A at TN. This model is buttressed by the fact that La2CuO4 has a spin correlation length of roughly the same order of magnitude (- 1000A at TN). However, the nature of the interplane binding mechanism of the spins in Sr2CuO2Cl2 is still contentious.


To date, comparatively little work has been done on non-stoichiometric compositions of Sr2CuO2C12. As previously mentioned, no superconducting phase has been reported in Sr2CuO2C12. This is not surprising since divalent strontium has replaced trivalent lanthanum, and chlorine has replaced oxygen in the rock salt layer buffering the CuO2 planes. The high electron affinity of chlorine traps charge carriers introduced, for example, by substituting Sr2+forLa+3 and precludes a superconducting phase. This same instability also makes doping Sr2CuO2Cl2 prohibitively difficult in most cases. One of the few impurities to be successfully doped into Sr2CuO2Cl2 is cobalt. Co2+ replaces Cu2+ and so preserves the valency of all the elements in the matrix. Thus, the introduction of cobalt does not add charge carriers to the CuO2 plane. However, each should incorporate into the CuO2 plane with a 3d7 shell configuration. Referring to Fig. 2.2b, crystal field splitting dictates that the 3dy, and 3d._ orbitals are filled while the 3d1y, 3d3,2,2 , and 3d,,,2_2 orbitals are half-filled. Hund's rule coupling is believed to bind the unpaired spins ferromagnetically in a high spin state (S=3/2). Thus, each Co2+ ion introduced into the CuO2 plane is a spin-3/2 impurity embedded in a spin-1/2 3D antiferromagnetic background. This dissertation will examine the effects of Co-doping in Sr2CuO2CI2 on such interactions









and parameters as the charge-transfer gap, the superexchange coupling constant, and the phonon-magnon coupling strength in the Cu(Co)02 plane.


2.3.3 Reflectivity and Optical Conductivity: Sr2.CuO9 Cl)

Measurements on Sr2CuO2C12 have not been pursued as aggressively as they have been on its superconducting brethren. As mentioned in the previous section, no superconducting phase has been found in Sr2CuO2C12, and doping carriers into this system has proven difficult. Nonetheless, Sr2CuO2C12 is nearly isostructural to La2CuO4 and, as are the other superconducting cuprates in the undoped regime, is a 3D antiferromagnetic insulator. These similarities to its superconducting cousins make Sr2CuO2C12 a relevant material to study.

Figure 2.8 shows the reflectivity, transmittance, and corresponding optical conductivity of Sr2CuO2C12 for light polarized parallel to the CuO2 planes49. Below .leV (1000 cm-1), the spectra are dominated by four infrared optical phonons in much the same way as La2CuO4. Between 1000 and 10000 cm-1, a new excitation emerges just below 3000 cm-1 with two weaker sidebands at - 4200 cm-1 and - 6000 cm-1. These excitations were not evident in the optical conductivity of La2CuO4 presented earlier. However, this is not due to fundamental differences in the electronic structure, rather it is related to the enhanced sensitivity of transmission measurements. In fact, these same excitations have been observed in many of the cuprates (La2CuO4, Pr2CuO4, Nd2CuO4, PrBa2Cu3OT76). Unfortunately, most cuprate single crystals as grown are too opaque and too thick for transmission studies. To obtain transmission data on La2CuO4, for example, it is necessary to polish the sample down to a thickness no greater than 10 microns. Nonetheless, transmission measurements have been performed on the aforementioned cuprates5'51. While the precise energy positions and relative weights of these mid-infrared bands vary from material to material,











Photon Energy (eV)
0.1 1


1.0
08Sr2Cu02C2 u T =300K L"
O 0.6 0 0.4
'
0.2
0.0
4)
c- 0.8
0
~0.6
S0.4
C
Lo 0.2 0.0
1000

E 100 i t ' I 10

b 0.1
100 1000 10000
Frequency (cm-') Figure 2.8. Transmission, reflection, and optical conductivity of Sr2CuO2Cl2
at T=300K. From Reference 49.



the spectra are otherwise identical. Lorenzana and Sawatzky52 proposed that these

relatively weak bands observed in the mid-infrared are phonon-assisted magnon pair

excitations. It is now widely held that this interaction is responsible for the strongest

band centered at 2900 cm-1. However, the nature of the higher energy sidebands is

still a contentious issue at present. More will be said about this in Chapter 5.


Above 10000 cm-1, the optical conductivity is dominated by a charge-transfer

absorption (0 2p,y-Cu 3d12-y 2 charge transition) centered just below 2eV. The optical conductivity in this region has roughly the same magnitude as that found in

other lamellar cuprates suggesting that there is little variation in the dielectric con-


0.01









stant from cuprate to cuprate. The dominance of the charge-transfer absorption is manifest by the fact that it is roughly one order of magnitude larger than the phonon conductivities and three orders of magnitude greater than the absorption in the midinfrared. Though not presented here, the charge-transfer edge sharpens and moves to slightly higher energy when the temperature is lowered49. Falck et al. proposed that this weak redistribution of spectral weight is associated with the temperature dependence of phonons dipole-coupled to the charge-transfer excitation44. This will be discussed in Chapters 5 and 6.

2.3.4 Raman Scattering: SrqCuO9Cl2

Since Sr2CuO2CI2 is an antiferromagnetic insulator, 2 magnon Raman absorption should be evident in the mid-infrared. Figure 2.9 shows the 2-magnon Raman shift for incident light polarized in the CuO2 plane for several cuprates courtesy of Tokura et al.34. As will be discussed in greater detail in Chapter 4, two-magnon Raman scattering in a 2D AF ordered material occurs when an electron on spin sublattice A is photo-excited to an adjacent spin sublattice B. To avoid double occupancy, the electron originally on sublattice B hops back onto sublattice A, thereby introducing a spin-flip in the antiferromagnetic background. The incident light backscatters with B1 symmetry and breaks six antiferromagetic bonds which, neglecting magnon-magnon interactions, should cost 3JII. Effects of interplanar coupling are small and are not taken into account here. When magnon-magnon interactions are taken into account, this value drops to 2.7J. As can be seen from Fig. 2.9, there is little change in the 2-magnon energy and only modest discrepancies in the lifetime (width of the two-magnon Raman shifted peak). The two-magnon peak for Sr2CuO2C12 is located at 2890 cm-1. The estimated Jii from this is 1070 cm-1, comparable to what is observed in La2CuO4. The relative invariance of the two-magnon Raman shift in the cuprates is not surprising. The CuO2 planes in the cuprates listed in Fig. 2.9 are










nearly identical. Differences in the Cu-O spacing are less than 1%. Discrepancies in the charge-transfer energy are modest with ACT ranging from 2.0 to 1.5eV. Since J
t 4
scales as c (-T + U), with tpd proportional to the Cu d,2_y2-O 2px, overlap, only
CT CIT
modest variation is expected from cuprate to cuprate.

As discussed in the previous section, no charge impurities have been successfully doped into Sr2CuO2C12 to date. Consequently, two-magnon Raman data for carrier-doped Sr2CuO2C12 is not available. However, magnetic impurities have been successfully introduced into the CuO2 plane. This dissertation will examine the effects of substituting cobalt for copper on the optical and two-magnon Raman scattering properties of Sr2CuO2C12. Cobalt is believed to go into the CuO2 plane with a 3d7 shell configuration in a high-spin state and so would introduce a spin-3/2 impurity in the spin-1/2 antiferromagnetic background. Hence, while little change is expected in the conductivity, it is very possible that significant changes may be observed in the two-magnon Raman scattering. More concerning this will be discussed in Chapters 6 and 7.


































RAMAN SHIFT (cmj
Figure 2.9. 2-Magnon Raman scattering data host insulators. From Reference 34.


(B1 mode) of assorted cuprate













CHAPTER 3
THEORY


3.1.1 Models for Carriers in the CuO9 Planes: Normal State

The superconducting mechanism of the high T, cuprates is not understood at this time. Many contend that understanding the normal state properties, ostensibly less enigmatic, will provide knowledge of the pairing mechanism of the carriers in dopedcuprates. Unfortunately, the wisest and seemingly simplest course of action is seldom easy. The normal state properties of the copper based high T, superconductors are unusual and, to date, have not been entirely accounted for with existing models. Most outstanding of the normal state properties is the linear temperature dependence of the resistivity above T, . This behavior is not characteristic of a Fermi liquid where p - T2, nor is it akin to metals where p is less temperature sensitive. For this reason, the superconducting cuprates are often referred to as "strange-metals" in the normal state. Other unusual properties of the normal state are a temperature dependent Hall coefficient, proximity of superconductivity to a magnetic phase, and unusual behavior of the optical conductivity with charge-doping. Most theoretical models for the normal-state properties start with the three-band Hubbard model. The following sections will review this model and discuss its limiting form in the strong coupling limit, the so-called one-band Hubbard model. Results from numerical calculations will then be presented for the optical conductivity. This will be followed by a discussion of small polarons and the Zhang-Rice singlet, two models which have been proposed to explain the mid-infrared band in the high T, cuprates and nicklates. Lastly, phononassisted multi-magnon absorption will be discussed in modest detail.









3.1.2 Three-Band Hubbard Model

The starting point for understanding the normal state properties of the cuprates is constructing a Hamiltonian to describe the motion of carriers in the CuO2 plane. This two dimensional approach is justified since conduction anisotropy favors carrier transport in the CuO2 plane. If we naively begin by charge-counting in La2CuO4, for example, we find that lanthanum goes into the system as a +3 ion, while copper goes in with +2 and oxygen with -2 valency, respectively. Thus, La3+ and 02- go into the matrix with closed shells while Cu2+ enters with an unpaired electron in the 3d shell. Crystal field splitting removes the degeneracy of the copper 3d orbitals and places the unpaired electron in the 3dx2_y2 orbital. Each unpaired electron, in turn, is antiferromagnetically coupled to its neighbor. This leaves the system with one hole per tetragonal unit cell. Hence, one would expect this system to be metallic with a half-filled conduction band in the undoped phase. However, measurements have determined that La2CuO4, as well as the other high T, cuprates, is insulating. The missing piece is, of course, correlation of the unpaired electrons. For a system to conduct, charge carriers must hop from site to site in the CuO2 plane. There are two avenues to this end in La2CuO4. One is that an electron on an oxygen site hops to a neighboring partially filled 3d.2_y2 band, leaving a hole on the oxygen site. This process costs A (charge-transfer energy) and generates a mobile hole in the oxygen band. The second possibility is that an unpaired electron on a 3d,2_y2 site, or in the lower Hubbard band (LHB), promotes directly to a nearest neighboring 3d"2_y2 site, or to the upper Hubbard band (UHB) at energy cost U (Mott-Hubbard insulating limit). This leaves a mobile hole in the lower Hubbard band.

A good starting point for mapping the lowest energetic excitations is the threeband picture diagrammed in Fig. 3.1. Here, two possible electronic configurations are shown. On the left side of Fig. 3.1, the Cu 3d9 band lies above the 0 2p ,y









band and the Fermi energy is located at the top of the former. The least energetic excitation of the system would therefore be the transfer of an electron from the 3d9 band to the 3d10 states at cost U, the electron-electron repulsion on the Cu sites. In this case the upper and lower Hubbard bands (UHB and LHB) are the 3d1� and 3d9 bands respectively. This is the Mott-Hubbard limit for the three-band picture. Mott-Hubbard behavior characterizes the early transition metal-oxides such as V203, Ti203, and Cr203. Contrasting to Mott-Hubbard insulators are chargetransfer systems. The charge-transfer limit is depicted on the right side of Fig. 3.1. Here, the 0 2p band lies above the Cu 3d9 so the lowest lying electronic excitations would be the transfer of electrons from the 02px,y band (LHB) to the Cu 3d10 band (UHB). Here, the Fermi energy lies at the top of the 0 2p band. Charge-transfer behavior is typically found in the late transition metal-oxides such as CuO, La2CuO4 and Sr2CuO2C12 , the latter two being the parent insulating systems studied for this dissertation. In the charge-transfer limit the ground state of the system lies in the lower Hubbard band where the highest lying electrons are unpaired in the 3d2_y 2 orbitals. The lowest lying excitation would therefor be the transfer of an electron from the 0 2pz,y orbital to a neighboring partially filled Cu 3d,2_y2 orbital, denoted by A in Fig. 3.1. This has been corroborated in optical studies of La2Cu04 where a charge-transfer excitation has been observed in the optical conductivity centered near 2.0-2.25eV32'53.

Having successfully worked out the dynamics of the unpaired electrons in the undoped phase, our next task is to construct a Hamiltonian to describe the motion of holes introduced by doping. This has been accomplished with a 2-dimensional tight binding model introduced by Emery et al.54'55 and Varma et al.56. This model, contrary to the previous discussion of carriers in the undoped phase, introduces a hybridization parameter, tpd, to allow for the hopping of charge carriers from the 0












g(E) d1O



u


g(E)








A

Fermi Level


p6



Mott-Hubbard Insulator U Figure 3.1. Three Band Hubbard Model. Left: Charge-transfer limit.


Fermi Level


Charge-Transfer
Insulator
A Mott-Hubbard limit. Right:


2p to Cu 3d orbitals. Other parameters included to account for charge interactions are site energies Ed and ep, Coulomb energies Ud and Up between two holes of opposite spin on the same Cu and 0 sites, respectively, and Vpd which accounts for interactions between holes on neighboring Cu and 0 sites. For electron-doped cuprates, such as Nd2-xCeCuO4-y, these hole interaction parameters are simply replaced by electron interaction terms, usually without changes in notation. The Hamiltonian may then be written as


H ~ tpdPjp(di +h.c.) -tppP(pJi,+h. c.) +'EdZn T �EP flP'�
(i j) 0j1'Y)

Udn fn,,n,+UpflnPn + Vp E n P (3.1)
i i (ij)









The first term represents hybridization or hopping between nearest neighboring copper and oxygen sites in the CuO2 plane. The pj are fermionic operators that destroy holes at the oxygen site labeled j, while the dj are corresponding annihilation operators at the Cu site i. (ij) appearing in the first sum indicates that the sum is only over nearest Cu-O pairs. Note that only on-site and nearest neighbor interactions are accounted for. Interactions at larger distances, to first order, are screened by the electronic background. For cp - Ed > 0 and in the undoped regime, Eq. 3.1 suggests that there will be one hole on every Cu site. In the limit that Ud > Ep Ed, any additional holes introduced by doping should predominately reside on the 0 2p orbitals. This has been verified by numerous experiments. Band structure57 and cluster calculations58 have estimated Ep - Ed - 3.6eV, Ud '- 8-11eV, and Up - 4eV. The values of tpd, Upd, and tpp are in the neighborhood of 0.5-2eV. The relatively large value of Ud, in comparison to the other terms, suggests that the strong to intermediate coupling limit is appropriate to describe the physics of the charge carriers in the high temperature superconductors.

3.1.3 t-J Model

It is unclear that a model as cumbersome as Eq. 3.1 is necessary to correctly account for the low energy physics of the high T, superconductors materials. To simplify matters, Anderson58 proposed that it may be possible to reduce Eq. 3.1 to a one-band Hubbard model while still retaining an effective theory. Here, the Hamiltonian is defined as


H t c, c + h.c) + U n n 3.2
(ij)
where dif are fermionic operators that create electrons at site i with spin a. In this
t ,gs
picture, the 02p orbitals have been effectively absorbed into the Cu sites. So site











g(E) U>>t

LHB UHB





t >U




U E
Figure 3.2. One band limit for insulators.


i refers to a plaquette consisting of a central Cu site surrounded by four oxygens. As before, t is a hybridization parameter between neighboring Cu-O pairs. The parameter U is the on-site Coulomb repulsion of two holes on the same plaquette. Figure 3.2 shows a schematic for this truncated model. Note that now there are only two bands: an upper and lower Hubbard band (UHB and LHB). For t > U, the LHB and UHB overlap and the system becomes metallic.

In the strong coupling limit (U > t), the one-band Hubbard model may be reduced to the so called t- J model, first derived from the Hubbard model by canonical transformation by Hirsch59 and Gros et al.60 As before, this is a single-band model, but now the state of the doped hole is represented only by the spin of the Cu site (plaquette) on which it resides. That is, the fermion operators in Eq. 3.1 which create and destroy holes on the oxygen and copper sites are replaced by boson spin operators which couple neighboring copper spin sites. The spin is zero on a given site if a single doped hole resides there, and is either spin-up or spin-down if a doped hole is not present. In the strong coupling limit, the Hamiltonian is










JE(Si. Sj - 1 t [ctcj,, + h.c] 3.3 (t j) (ij>,o

where J is the antiferromagnetic coupling between nearest Cu-Cu neighbors (ij) and is defined as
4t2 3
J = -4. 3.4
U
This expression is valid in the limit that J
3.1.4 Cluster Calculations

As discussed in the previous section, the one-band Hubbard and t - J models are the starting points in many calculations concerning the dynamics of carriers in the CuO2 planes. Such calculations include the response of the carriers to electromagnetic fields. Unfortunately, these models lack analytic solutions in the strong coupling limit. Consequently, numerical techniques on finite clusters have been employed to extract physical parameters. Here, there will be a brief summary of the numerical studies relating to the optical conductivity, al, generated in the context of the one-band Hubbard and t - J models.

The conductivity tensor is obtained from linear response theory. It is used to relate the current density operator, jx(q, w), to the electric field vector, E,(q, w), that induced it. Here 4 and w are the wave vector and energy of the electric field vector respectively. In the limit of j -. 0, jx is given by


j(0w)= o, ,(,w









where ux is the complex optical conductivity at zero temperature. The real part of o, is given by the Green's function

11
Reoa. = 1 Im[(0 Ij, IX I 0)]. 3.6 7rW H - E0 - if


Here, H is the total Hamiltonian with eigenenergy E0, w is the frequency, and f is a small number that moves the poles of the Green's function into the complex plane. In the Hubbard model, the current operator jx in the x direction at zero wave vector is given by,

= itY(4 tc ~x,, - h. c.), 3.7


where ct and c are fermion creation and destruction operators, I labels the sites onto which the carriers may hop, and 1 + x labels the site displaced by one lattice spacing from 1. Numerical solutions for the one-band Hubbard model on a 4 x 4 cluster performed by Dagotto61 employing the Lanczos62 technique extracted the relative strengths of t, U, and J. This was done for hole concentrations between 0.000 (halffilled system) and 0.375. The results for the optical conductivity are presented in Fig. 3.3. At half filling, x = 0.000, there is an accumulation of spectral weight around 6t. Using previous estimates63 of t this band can be correlated with the charge-transfer gap observed near w - 2eV. In the context of the one-band Hubbard model, these are excitations from the lower to upper Hubbard band. As doping proceeds, there is marked shift in spectral weight from the charge-transfer band to lower energies. By x = 0.125 (two holes on the 4 x 4 lattice), two distinct features emerge below the charge transfer gap. The first is a Drude or free carrier response centered at w = 0. The second is an accumulation of spectral weight near 0.3-0.4eV. This has been associated61,64,65 with the mid-infrared band observed in optical conductivity of the doped cuprates. For increased doping concentrations, the Drude component









grows considerably while the mid-infrared spectral weight appears to increase only modestly. However, appearances may be deceptive. The modest growth of the midinfrared band with doping relative to the free carrier band centered at w - 0 may in fact be due to strong overlap with the more heavily weighted Drude component which overshadows it. The growth rate of the mid-infrared band as a function of doping is still a contentious issue. Dagotto et al.61 contend that the results of Fig. 3.3 indicate that the appropriate coupling limit for the cuprates is the intermediate coupling regime, U - 8t. In the strong coupling limit, a gap develops between the chargetransfer and mid-infrared bands, whereas for small to intermediate coupling the two bands merge, making separation difficult. Similar calculations using the t - J model, or in the strong coupling limit, have been performed by Stephan and Horsh65 for different values of J and their results are presented in Fig. 3.4. As can be seen, the Drude and mid-infrared bands are clearly discernible and there is qualitative agreement with one-band Hubbard model. Similar results for cases near half-filling were also obtained by Chen and Schiittler66 using the one-band Hubbard model in the strong coupling limit.


3.2.1 Models for the Midinfrared Band

As discussed in the previous section, there is a shift in the experimentally observed spectral weight from high to low energy in the high T, superconductors cuprates as the systems are doped with charge carriers. In particular, a mid-infrared band and Drude tail grow with increasing doping concentrations accompanied by a sharp loss of spectral weight out to 3-4eV in the charge-transfer band. Numerical solutions employing the one-band and t - J models have modeled this behavior, but the precise mechanism responsible for the bound excitations in the mid-infrared is still elusive. To simply chalk them up to intraband excitations in the lower Hubbard band does































5 10
0)/t


Figure 3.3. Cluster calculations for model. From Reference 62.


the optical conductivity under the t - J


0.8


0.6


O Q) 0.4



b
0.2



0.0


1 2 3 4 5 6
co/t


Figure 3.4. Cluster calculations for the optical conductivity under the t - J model. From Reference 65.


0.151


-5 0.10



0.05


0.00









not explain the essential physics of the bound excitations. The fact that the midinfrared band is centered near 0.5eV, or roughly 3J, has led some to suggest that the excitations are magnetic. As a hole is photo-excited to a neighboring site, it creates a spin-flip in the antiferromagnetic background and breaks three antiferromagnetic bonds. Others contend that the band is the result of the charge impurities coupling to the crystal lattice. In this scenario, small polaron hopping is the mechanism responsible for the mid-infrared band. Still, others have proposed that novel charge structures in the CuO2 planes such as domain walls are driving the bound excitations in the mid-infrared. These models, and some observations concerning the anomalous loss of spectral weight in the charge-transfer band, will now be discussed.



3.2.2 Small Polaron Model

The small polaron model was first introduced by Reik67 in the 1960's to explain the mid-infrared band found in semiconducting lanthanum cobaltite, La2CoO3. Small polaron hopping was later revisited by Bi et al.68 nearly thirty years later to account for the mid-infrared band observed in La2-,SrNiO4. The basic premise is that charge carriers introduced by doping distort the crystal lattice and become selftrapped (bound) in a potential well, thus creating a polaron. The polarons then hop from site to site in the lattice when photoexcited. Starting from Kubo's expression for the a.c. conductivity67 given by


o 3-Hi
a(w) = TrZ f d( dAexp[(iw+c)(] j exp[-- ((-ihA)] j exp[ ((-ih[A-])], 3.9

an expression for the complex optical conductivity is found. Here, Z is the partition sum, ( is an integration variable in the complex time plane, e is a small real frequency









introduced to ensure convergence of the integral, A is an inverse-temperature integration variable, /3 = 1/kT, and j is the current density operator, similar to that defined in Eq. 3.7. H is the Hamiltonian of the system and is the sum of three parts:


H = H, + Hph + H,ph. 3.10



H, and Hph are the Hamiltonians of the charge carriers and phonons (lattice distortions), respectively, while He,ph couples the doped charges to the lattice (i.e.,chargephonon coupling term). It is the latter term, He,ph, that generates excitations in the mid-infrared. H, is modeled after the one-band Hubbard model and is given by,



H = (o- p)c~c, + YZt,(csIcs+i +CSI+iCS). 3.11
8 S,i


Here, s labels the sites where the charge carriers sit and i labels all sites onto which the charge may hop. In the same vein as the one-band Hubbard model discussed previously, t,,i is the electronic resonance integral and (co - P) is the on-site energy of the doped carriers. Only singly occupied sites are considered in this model which is reasonable when one considers that the energy cost of two charge impurities (electrons or holes) residing on the same site is roughly 8-9eV in the cuprates. As before, c8 and c, are fermion creation and destruction operators, respectively. The phonon Hamiltonian is given in the usual way:


Hph hwxqjbt,\bi,,. 3.12



is the phonon wave vector, w,\(q) the phonon energy, and A the phonon branch index. Here, b$ and b,A are boson operators that create and destroy phonons in










branch A with wave vector 7, respectively. The final and most important term, the charge-phonon interaction, has the structure

He,ph -e w 1/2 - bt e-iq'fRs)cc 3.13
Hph hw,\(q aA (q- i(b ,eiR q, IAs 4, S


where fs is the position vector of site s and a(q- is the phonon coupling constant. After inserting Eqs. 3.11-3.13 into Eq. 3.9 and integrating, the following expression for the real part of the optical conductivity is found in the limit that the temperature is greater than one quarter of the Debye temperature67: sinh(l h/w) exp[-w2T2r(W)] 31 Re u(w,T) = (OT) hw/ [1 + (wrA)2]1/4'


where r(w) is a frequency dependent function given by,


r(w) = 2(wrA)-llog{wTA� [1+ (wTA)2]1/2}- 2(wrA)2- {[1 +(wTA)2]1/2 1}. 3.15


Here, T and A are defined as,

2 sinh( hwof3) 3.16 2w77

and

A = 2w07 3.17 where wo is the average phonon energy to which the charge carriers couple and 7 is number of phonons associated with each polaron, or the number of phonons in the polaron cloud. As usual, 3 . The dc part of the conductivity is given by the expression
22 20-r 2 1
o(O, T) = 2e2j2a /fr1h -exp{-7tanh(hwo/3)} 3.18
4

where a is the lattice constant, J is the bare electronic resonance integral (hopping integral between neighboring sites), and N is the number of doped carriers. Despite









the complicated form of Eq. 3.15, the frequency and temperature dependence of Rea(w, T) is well behaved. This will be discussed further in Chapter 5 when the mid-infrared band observed in La2Cul-.LiO4 is discussed in the context of small polaron theory. Note the sensitivity of Rea(w, T) to both temperature and ri. This will be important in Chapter 5 when the temperature dependence of the mid-infrared band is discussed for La2Li.Cu-xO4.

The average phonon frequency to which the charges couple is a thorny issue. Eklund et al.68 proposed that the doped carriers couple to the bending modes of La2NiO4 in the NiO2 planes centered around 50meV. Using this energy for wo and setting 7 - 10, they closely modeled Eq. 3.15 to the mid-infrared band observed in the optical conductivity at 300K. However, it is not clear that the doped carriers couple solely to the bending modes in the basal planes. It seems reasonable that the stretching modes of Ni(Cu)-O pairs in the basal plane, centered near 90meV, may be activated by the presence of charge impurities. In this case, perhaps the best approach would be to consider weighted contributions from both modes. More concerning this model will be discussed in Chapter 5 when the mid-infrared band is investigated for La2Li1Cul-XO4.


3.2.3 Magnetic Excitations: Zhang-Rice Singlets

Other theories have cast an eye on a magnetic origin for the mid-infrared band in the doped cuprates. As mentioned before, it has been established that holes introduced by doping reside predominately on the oxygen sites in the CuO2 plane. Zhang and Rice69 proposed each of these holes is strongly bound on a square of 0 2p sites to a central Cu2+ ion and forms a singlet. This singlet then moves through the lattice of Cu2+ ions in a similar way to a hole in the single-band effective Hamiltonian. It is important to note that the Zhang-Rice singlet differs from an ordinary two-site









singlet in that it is formed by a central Cu2+ ion and four neighboring oxygen sites. As will be shown, it is the phase coherence of the four 0 2p orbitals that leads to the the anomalously large binding energy of the singlet holes. The starting point for this model is the three-band Hamiltonian constructed in the hole representation:


H =E ddtadia +Z EppTpl + U U ditdd di + H' 3.19 ?,a. l,o


In Eq. 3.19, Zhang and Rice have defined the vacuum as the filled Cu 3d1� and 0 2p6 states. The operators dt create Cu 3dX2_ 2 holes at site i, and pf create 0 2px,y holes at site 1. Ed and e are the on-site energies of holes on the Cu and 0 sites respectively. The last term, H', represents hybridization of neighboring Cu-0 pairs and is given by69

H' E E Vildtpi, + h.c. 3.20 i,a 1E{i}

where the sum over I runs over the four 0 sites around a given Cu site i. The hybridization matrix Vil is proportional to the wave function overlap of the Cu and 0 holes. At this point it is crucial to take the phase of the hole wave functions into account. This is accomplished by writing the hybridization matrix as69


Vi = (-1)Miat0 3.21


where to is the amplitude of the hybridization and Mi,t = 2 if 1 = i - 1i or i - !9, and Mi,L = 1 if 1=i+ liori+!1.

For to < U (strong coupling limit) and in the absence of any doped holes, Eq. 3.19 reduces to a S 1 Heisenberg Hamiltonian on the square lattice of Cu2+ sites:



H = i �










J=40+ 4 3.22 2U 2E3

where E = Ed - Ep (charge-transfer energy). When holes are introduced, they have predominately 0 2p character. However, each can lower its energy by hybridizing with a central Cu+2 site and by spreading out over the attendant square plaquette of four oxygens. Thus, the wave function of the 0 2p holes should be constructed from a combination of a four oxygen sites surrounding a central Cu2+ ion. Doing so, a given set of four 0 2p orbitals may form either a symmetric or antisymmetric state with respect to the central Cu ion69:


p S,A = 1 (1)1 3.23
IE~i}


where -(+) corresponds to the S(A) state, and the phase of the p- and d-state wavefunctions is defined in Fig. 3.5.70 Both S and A may combine with the d-wave Cu hole to form either a singlet- or triplet-spin state. Using Eq. 3.19 as the unperturbed Hamiltonian of the system with H' = 0, it is found to second order in perturbation theory that the energies of the singlet and triplet states for S are -8(tl + t2) and 0,69 respectively, where t, = t2 and t2 = t while A has energy -4t1. The large CP (U-EP)
binding energy of the S singlet state is due to the phase coherence of the 0 2p and Cu 3d hole orbitals. It is instructive to compare this energy to an 0 hole sitting at a fixed site 1. In the latter case, the binding energy of a singlet combination of an oxygen hole and neighboring Cu hole is only -2(tl + t2), one-fourth that of the square S state. Since the energy separation of the S singlet and A states is much larger than tl and t2, the antisymmetric states are projected out of the lowest lying energy states of the system. Also, the energy of two holes residing on the same square is -(6tl + 4t2), much higher than the energy of two separated 0 holes. For this reason, contributions









from states with two holes bound to the same Cu site are neglected for the lowest lying excitations.

The localized states of Eq. 3.23 are, however, not orthogonal since neighboring squares share a common 0 site. This problem is circumvented by constructing a set of Wannier functions with a method similar to that used by Anderson to treat isolated spin quasiparticles71. The wavefunctions for the oxygen holes are now given by69


Oi, = Ns1/2E Pka exp(ik" Ri), 3.24



P- =NSl/23 O 5P exp(-i.R), 3.25
i
where 3 is a normalizing factor

1
- [1 - -(coskx + cosky)]-12, 3.26


and NS is the number of sites in the system. The functions Oi, are orthogonal and complete in the symmetric 0-hole subspace. �i, combines with the central Cu hole at site i to form a spin singlet(-) or triplet(+): � =1
V- (Oitdj � Oildi ) 3.27


with energies in second order perturbation theory


E� = j (0� I H' I w) 12 /AE 3.28 {w}

where w runs over all possible intermediate states, and AEw is the 0th-order energy difference between Oi, and w. Summing over all possible states numerically, it is found that69


E� = -8(1 :: A2)t


3.29






















Pi+ I
2


P.
2


. -f
i+ A


/
/
/


Figure 3.5. Orbital configuration for Zhang-Rice singlets. From Reference 69.


I I I III
* -071eV
* 035 ev .0185ev " 495 cm-I " U9 eV " 12cm1


I I III[


9 f0




5
3


0
Lo185 Sro 15 CuO4.8




EXPERIMENT







)2 iO3 104
w (cm-1)


Figure 3.6. Numerical results for the optical conductivity of phonon mediated Zhang-Rice singlets70. co = E(Cu3dx2-y2) - E(02p,), V is the Cu3d.2_y 2-2p, hopping integral, Ie and Lp = phenomenological natural linewidths, wo is the average phonon energy, and "/is the phonon-electron coupling strength. From Reference 70.


f z m I 1









where

A = N,-1 Z fl 0.96. 3.30


Since E+ - E_ "_ 15t > t, transitions between {)-} and {+} are ignored and the system may be treated on the singlet {4'-} subspace. The singlet-bound holes may then dipole-couple to radiation and be photo-excited to more energetic linear combinations of O2px,y and Cu3d,2-_2 orbitals in the singlet subspace. Once again, it is the phase coherence of the Cu 3d and 0 2p hole states that produces the large energy separation between the singlet and triplet states. Prior to the work of Zhang and Rice69, the importance of phase coherence went unrecognized save for the work of Hirsch72 who considered the S combination of 0 states in the case of fixed spin direction on the Cu site. Rice et al.70 modeled the optical conductivity in the midinfrared with a similar construct based on the phase coherence of the oxygen orbitals. However, in this model charge modulation via coupling to the phonon modes drove the bound hole states and so introduced an additional band in the far-infrared. Their results are presented in Fig. 3.6.


3.2.4 Domain Walls

One of the more novel models introduced in recent years to explain some of the magnetic properties of the cuprates and nicklates is the formation of domain walls or stripes. In this picture, holes (electrons) doped into the CuO2 plane align linearly in a particular direction, forming parallel domain walls. If the walls are filled (one hole per site along the wall), the antiferromagnetic order parameter reverses direction across the wall, dividing the CuO2 plane into regions with sequentially alternating spin-order parameters. Domain walls in the [1,0] and [1,1] directions are shown in Fig. 3.7.73 At first glance, there is no obvious reason why the charge should spontaneously arrange in linear formations and, in fact, it would seem that the Coulomb energy of two









electrons or holes residing on neighboring sites would preclude this. However, there is mounting evidence for the existence of stripes in both the cuprates and nicklates, at least for specific doping concentrations. Tranquada et al. reported a striped phase in both Sr-doped La2NiO4 and Lal.6-.SrNd.4CuO4 found in magnetic suseptibility measurements74,75. However, in the case of Lal.6-SrNd.4Cu04, it is believed that it is the excess Nd that pins the doped charge and leads to the formation of domain walls. To date, no static stripes have been observed in La2_xSrxCuO4 for x > .03. Hammel et al. have also proposed that stripe formation may be responsible for the anomalous behavior of the 139La resonance splitting observed in NQR measurements of La2LiCu1_xO431 below 30K.

Numerical calculations76 of the correlation function for two or more doped holes in a spin-1/2 antiferromagnetic background have pointed toward the formation of stripes for intermediate to strong J. Figures 3.8 and 3.9 show the 2-hole correlation function for two and four holes in a spin-1/2 antiferromagnetic background, respectively, as a function of the superexchange coupling. This was done for the t - J model and at zero temperature but it should be kept in mind that the holes actually reside on the neighboring plaquette of four oxygens sites around a given copper site. Here, r is the spacing between holes measured in lattice spacings and N is the number of sites in the lattice. For the case of two holes, the charge will form pairs along [1,1] direction for J/t > 0.25. For J/t > 0.4, pair formation along the [1,0] direction becomes more stable with respect to separated holes. Note, however, that the correlation function for filled-walls along the [1,1] direction is the most robust for all values of t/J. Care must be taken when interpreting the four-hole correlation function. The arrangement as well as the mean correlation length must now be accounted to correctly interpret the charge configuration. Figures 3.10a and 3.10b shows the four-hole correlation function, G, for four different configurations for N = 18 and N = 20. The sites are







0
0
0
0


0404

Figure 3.7. Domain wall structures tions. From Reference 73.


(a)
+ 4' 4, 4'


(b)
co



(stripes)


0
0
0
0


4,4' 4+ 4,4' 4+


in the a) [1,0] and b) [1,1] direc-


numbered in Fig. 3.11. For N = 18 (Fig. 3.9a), the configurations are: (a) (3 5 9 12), the holes being as far apart as possible, (b) (1 4 6 12), the holes forming two separate pairs with intrapair distance vf2, (c) (5 6 7 8) with holes forming a stripe along the [1,0] direction, and (d) (1 4 6 9) with three holes forming a domain wall along [1,0] direction. In Fig. 3.9a it can be seen that configuration (d), a stripe along the [1,1] direction, dominates for intermediate coupling, 0.3 < J/t < 1.2, while for strong coupling configuration (c), a domain wall in the [1,0] direction, wins out. For N = 20, the results for the four hole correlation function are similar to the N = 18 case and the




























0.0' 1 1 1 0.0 1'
0.0 0.2 0.4 0.6 0.2 0.3 0.4 0.5 0.6 0.7 ]A ]A


Figure 3.8. 2-Hole correlation function for a) N = 20 and b) Reference 76.


N=26 sites. From


1.0 2.0 3.0 0.0 1.0 2.0


Figure 3.9. 4-Hole correlation function sites. From Reference 76.


for a) N = 20 and b) N=26


0.4 0.2









hole configurations are listed as follows: (a) (3 7 13 17) representing two separated hole pairs, (b) (2 8 13 19), with holes forming a line along the [1,1] direction, (c) (3 8 12 17), with holes along the [1,01 line, and (d), a staggered domain wall stretching over the system with periodic boundary conditions. As before, configuration (b), a stripe in the [1,1] direction, dominates for intermediate coupling 0.4 < J/t < 1. For J/t > 1, configuration (d), a staggered domain wall, dominates and is surprisingly robust even in the intermediate coupling regime. It is curious to note that this configuration is more stable than (c), a stripe in the [1,01 direction, for all coupling strengths J/t above roughly 0.2. Zaanen et al.73 also demonstrated the stability of partially filled and completely filled domain walls in [1,0] and [1,1] directions using a semiclassical approach. Of course, it should be kept in mind that these 2- and 4-hole correlation functions do not include potentially important contributions from electron-phonon coupling which could change the direction and/or energetics of the stripes in key ways.

Having realized that striped phases are instabilities of doped Mott-Hubbard insulators, our next task is to make an estimation of the binding energy of the charge on the domain wall. This was done by Nayak et al.77 treating the spin-1/2 background in the strong coupling limit and using the t - J model. The energy per hole, if they are localized to walls at filling fraction f, is77 ET 3AJ tsin27rf' 3.31, Nh 2f' 7rf'

where f' = 1 - f is the hole filling fraction and A - 3/4. To find energy minimum, Eq. 3.31 is redefined as
r sinx
h(x) = - 3.32 X X
where x = 27rf' and r = 37rAJ/2t and x is in the interval 0 < x < 7. For very small r, the strong coupling limit, the minimum occurs at x -, (3r)1/3, a nearly filled wall;










53





0.012
(a)

N=18
/
0.008 //






0.004

~dd



0.0I
0.0 0.5 1.0 1.5 2.0 J/A


(b) d
0.012
N=20

,M / b






00
JA t / , #






0.0 0.5 1.0 1. 2,0 i/t


Figure 3.10. 4-Hole correlation function for different configurations for a) N

18 and b) N=20 sites. From Reference 76.








N=I8 1 N=20 1


15 16 7 5 l1 14 1 10 1 1 13 14 4 9 13 18


5 7 8 3 8 12 172

2 42 7 11l






Figure 3.11. 4-Hole correlation function key for a) N = 18 and b)

N=20 sites. From Reference 76.









for r = 1, the intermediate to strong coupling limit, it occurs at x = 7/2, a "minimal domain wall"; and for r > 7r the minimum occurs at x = 7r, an empty wall. It should be noted that Eqs. 3.31 and 3.32 are far from rigorous and are only intended to give an approximate energy of a hole bound to a domain wall. The antiferromagnetic background is treated as static, while it should properly be regarded as composed of dynamical electrons on the same footing as those on the domain walls. Despite these limitations, Eqs. 3.31 and 3.32 are reasonable first approximations for the energies of holes bound to domain walls in a spin-1/2 antiferromagnetic background. This will be revisited in Chapter 5 when stripe formation in La2Cul-Li.O4 is discussed.





3.3.1 The Charge-Transfer Band


Heretofore, little has been said concerning the doping dependence of the chargetransfer band save that there is a rapid loss of spectral weight in this region as the high T, systems are doped with carriers. While this phenomenon has been readily observed, it has not been well explained. That there should be some loss of spectral weight in this region as the antiferromagnetic background is hole and/or electron-doped is expected as states in the upper Hubbard band are removed. The shift in spectral weight from the charge-transfer to mid-infrared spectral regions can be qualitatively understood by considering the band structure of charge-transfer systems, shown in Fig. 3.12 for a Cu-O chain. Here, N on the left-hand side of the figure represents the band energy of electrons on singly occupied Cu 3d,2_y2 sites while N on the righthand side gives the band energy of two electrons residing on the same Cu 3d.2_y2 orbital, the upper Hubbard band78. 2N, located in the middle, represents the filled 0 2p, bands and the Fermi level is centered between this and the upper Hubbard












Charge Transfer : Insulating


U


2N N


PES EF -- IPES





Figure 3.12. Schematic diagram of charge-transfer systems. From Reference
78.


band. There are two doping scenarios to consider: One for electrons and the other for holes. For charge-transfer systems, each is unique.

For hole-doping, electrons are removed from the 0 2p, sites and the Fermi level is pushed down into the oxygen band. Neglecting hybridization between the copper and oxygen sites, introducing a hole will leave one electron-addition state near the Fermi level while the upper Hubbard band retains N states. Each hole creates only one electron-addition state near the Fermi energy since any electrons which hop onto a single-occupied O2p site must do so with the proper spin. Hence, intraband (midinfrared) excitations in the oxygen band should scale linearly with x, the concentration of holes. This situation resembles a semiconductor where the spectral weight of an impurity band is expected to scale as the doping concentration. Although the number of electron-addition states in the upper Hubbard band (filled Cu3d shell) for a chargetransfer excitation is not affected by hole-doping since the holes reside in the oxygen









band, for each hole added there is one less electron available for charge-transfer. Thus, to a first approximation, one would expect the spectral weight of the charge-transfer band to decrease as x.

Similar trends in the charge-transfer region are found in electron-doped systems, but fundamental differences exist in the MIR intraband excitations. Electrons added to the system must reside on the previously singly-occupied Cu 3dX2_Y 2 sites, which push the Fermi level into the upper Hubbard band. Contrary to hole-doping, each electron added creates two electron-removal states in the upper Hubbard band since there are now two electrons on the doped sites available to hop to neighboring Cu sites, provided of course the excitations are consistent with the Pauli exclusion principle. Hence, the spectral weight of the intraband excitations, now located in the upper Hubbard band, is expected to scale as 2x. The situation for the charge-transfer excitations is much the same as with hole-doping, but the reasoning is reversed. For every electron added, one electron-addition state is removed in the upper Hubbard band while the oxygen band remains unaffected. Now there is one fewer state available in the upper Hubbard band into which an electron may hop. Thus, as with holedoping, the spectral weight of the charge-transfer region is expected to decrease as X.

Of course, the above discussion is only qualitative and does not include treatment of important details. For example, it is vital to include the effects of hybridization between neighboring copper and oxygen sites if a quantitative comparison is to be done with experimental results. Eskes et al.78 performed cluster calculations for the integrated spectral weight below the charge-transfer gap of a doped chain of Cu-O cells as a function of electron/hole doping concentrations and of Cu-O hybridization, Jpd. Their results are presented in Fig. 3.13 for a cluster of four unit cells. For tpd = 0.00, the behavior mimics the previous discussion with the spectral weight










3.0
N U N- 2 N -l

- .EF EF
2.0



1.0 V\

S..\\

0.0 I
1.0 o 0.0 0.5 1.0
hole doping electron doping
Figure 3.13. Integrated MIR spectral weight below 1 eV as a function of
electron and hole concentration away from half-filling. From Reference 78.


growing twice as fast with electron-doping compared to hole doping. However, as tpd is increased to 1.5eV (a value not inconsistent with other estimations) there is little distinction between the effects of electron and hole doping below concentrations of 0.5, a doping level well above the superconducting phase or the onset of the metallic phase in the high T, cuprates. Most of the optical studies to date on the high T, cuprates have been done on optimally doped, underdoped, and modestly overdoped samples and little asymmetry has been reported in the mid-infrared band between electron and hole-doped systems.

Unfortunately, this does not close the book on the shift in spectral weight from high to low energy observed in the high T, materials. While the above analysis is consistent with the observed growth of the mid-infrared band with doping, it does not adequately explain the anomalous loss of spectral weight in the charge-transfer









region. From Fig. 2.5, it is seen that for hole concentrations in excess of 0.02, the loss of spectral weight in the charge-transfer region scales more rapidly than x contrary to what is naively expected. The situation is much the same with electron-doped systems such as Nd2-xCe.CuO4-y. It is curious to note that the hole concentration at which the spectral weight of the charge-transfer begins to degrade more rapidly than x roughly coincides with the concentration at which 3D antiferromagnetic order is lost. This raises speculation that perhaps the charge-transfer excitations, in addition to being dependent on the number of electron-removal/addition states available, are also dependent on the spin-order of the Cu2+ lattice. This possibility and the discrepancy between theoretical expectation and experimental reality in the charge-transfer region will be discussed in more detail in Chapter 5.



3.5.1 Phonon-Assisted Multi-Magnon Absorption

As discussed in subsequent sections, the introduction of charge carriers to the CuO2 plane of the layered cuprates induces a broad, heavily weighted band in the mid-infrared. However, sharper, more discreet excitations are also found in the same spectral region in the undoped phase of the cuprate systems. The absorption coefficients of a few these materials are presented in Fig. 3.14. It is obvious upon inspection that these excitations are roughly three orders of magnitude weaker than the bound excitations associated with the introduction of charge carriers in the optimally doped regime ('- 1-2 f-1cm-1 vs. 500-800 Q-1cm-1). Consequently, it is not possible to observe these excitations in reflection mode. They can be observed in transmission studies of thin samples or in 2-magnon Raman scattering measurements. In the latter case, however, a phonon is not needed to make direct two-magnon coupling weakly dipole allowed. The spectrum consists of three absorptions at "- 2900, 4000, 6000 cm-1 overlayed on background which grows linearly up to the charge-transfer









edge. The first is distinct and sharp while the latter two are broader and weaker. The mechanism responsible for these excitations has been a contentious issue but it is now generally accepted that the lowest energy band is due to phonon-assisted two-magnon absorption. The suggestion that magnons play a role in these excitations is motivated by the fact that the bands are associated with the AF phase and are centered at roughly 3J, 4J, and 6J, the energies needed to flip two adjacent spins and four spins on a plaquette and in a row in the AF background. Nonetheless, the debate still rages concerning the origin of the two higher absorption features. However, when the pressure dependence of these excitations in Sr2CuO2C12 is discussed in Chapter 6, it will become apparent that the absorptions near 4000 and 6000 cm-1 are most likely associated with phonon-assisted four magnon excitations. Lorenzana and Sawatzky79 introduced the phonon-assisted multimagnon model in 1994 to explain the weak midinfrared excitations observed on the undoped phase of the superconducting cuprates. A brief synopsis of the model is now presented with the intent to simply motivate the theory behind it. For a detailed account of the model, consult Reference 79 and the references therein.

Assuming that the mid-infrared excitations in the parent insulators are associated with the creation of magnons in the CuO2 plane, a problem immediately surfaces: IR absorption of magnons is not allowed in the tetragonal structure of the cuprate materials. This is due to the presence of a center of inversion in the unit cell which inhibits any asymmetric displacement of charge (and thus spin) and, hence, quenches the dipole moment. However, this restriction is lifted if symmetry-breaking phonons are nearby. The presence of a phonon effectively lowers the symmetry of the surrounding local environment and, consequently, allows magnon absorption processes. Although a similar theory was introduced by Mizuno and Koide8� decades earlier to explain magnetically related IR absorption features in NiO, Lorenzana and Sawatzky were







60



150





- 0/
0



r0




0 * I I
0.0 0.2 0.4 0.6 0.8 1.0
Photon Enery [ eV I

Figure 3.14. Phonon-assisted multi-magnon absorption for various cuprate
systems at T=10K. From Reference 23.



the first to explicitly calculate the coupling constant for phonon-assisted absorption of light by multimagnon excitations and the line shape for two-magnon absorption. The motivation for this model stems from the large value of J in the cuprates. The basic idea is that phonon-induced modulations of the Cu-O lattice parameter will lower the symmetry of the lattice and lead to large changes in the magnon-phonon coupling energy. The starting point for this model of the charge and spin dynamics of the CuO2 plane is the three-band Peierls-Hubbard model in the presence of an electric field (f ) associated with an incident photon. For simplicity, the Cu atoms are kept fixed and the 0 ions are allowed to move with displacements 1149J2 . Here 1" labels Cu sites and 6 = ,, , so that i+ d/2 labels 0 sites. 6 is a unitless displacement vector linking neighboring Cu2+ sites. Holes have an on-site (hole-hole) interaction









Ud on Cu and Up on 0, a Cu-O repulsion Upd, on-site energies on Cu Ed, and on 0 Ep, and Cu-O hopping t. For later convenience, the parameters A = Ep - Ed - Upd and E = 2(Ep - Ed) + Up are introduced. When an 0 ion moves in the direction of a Cu with displacement I u I via dipole-coupling of the incident radiation to lattice, the corresponding on-site energy of a charge sitting on a Cu site changes to first order by -03 1 u 1, and the corresponding Cu-O hopping of said charge by a I u 1. Opposite signs apply when the 0 moves in the opposite direction. The coupling constants of light with the one-phonon-multimagnon processes are found from a perturbative expansion of the photon-phonon-magnon system valid when t < A, C, Ud and when the phonon field and the electric field vary slowly with respect to typical gap frequencies79,81"


H = I J(E{ ;4)B+g/2 + Hph - E -Pph. 3.33



Here, = and S are the fermionic spin operators. Hph is the phonon Hamiltonian containing spring constants and masses for the 0 ions, Pph is the phonon dipole moment, and J is the superexchange energy, or the energy difference between the singlet and triplet states of two spins located on adjacent Cu sites (CUR and CuL in Fig. 3.15). For light incident on this Cu207 configuration, only three polarization directions of f (A,B,C in Fig. 3.15) need be considered. Expanding J to first order in f and ul;+4/2 yields79'81:



J =Jo + ?(UL - UR) - E[qluo + AqA(2uo - UL - UR)]. 3.34


A is a polarization dependent parameter, equaling 1 for configuration A and 0 for configurations B and C. In each configuration the displacements of the central 0









and the electric field are parallel. UR and UL are only relevant in configuration A with UR = -UR1 + UR2 + UR3 and UL = UL1 + UL2 - UL3 (see Fig. 3.15). The first term in Eq. 3.33 is the superexchange in the absence of electric and phonon fields, Jo (4t2/A2)[l/Ud + 2/E]. The second is the magnon-phonon coupling constant, 77 (-4t2/A2)3[A-1(1/Ud + 2/E) + 2/E2] associated with local modulations in J due to the presence a phonon. The last term gives the effective charges, or dipole moments, associated with the one-phonon and multimagnon processes79:


8t4 1 1 2 2
qj e- + -) + J2]' 3.35



4t4 2 1 2 1 1 2
qA = -e 2apd[2A (Ud + ) + Ud( + Ud)2] 3.36 where apd is the Cu-O distance. The dipole moment of the magnon-phonon system is obtained from P =- . Using Eq. 3.33 this yields up to fourth order in t, OE"


P = Plph + Plph+mag" 3.37


The first term describes conventional phonon absorption. However, the second term in Eq. 3.37 gives rise to a new dipole moment associated with the phonon modulation of the charge and spin on the Cu sites. By defining 6B;412 = B+4/2 - (B-g/2), its Fourier transform JB6, and the Fourier transform of u;+g/2, u-4, the one-phonon and multimagnon processes for an in-plane field in the x direction has the form79:


Px,lph+mag N[q A4qA sin( PX )sin( P )6Bu6-.] 3.38 2-P2 i


with A = 1. N is the number of unit cells. u6- is the the Fourier-transformed xP
displacement vector in the x direction of the oxygen atom at site i + '/2 . For the










case of an electric field polarized perpendicular to the Cu02 plane, A is set to zero and u6- is replaced by u6. The first term in Eq. 3.38 is isotropic in the polarization and is a spin-dependent correction to the charge on the central 0 (call it O0) in Fig. 3.15. This is illustrated in Fig. 3.16a where the electric field, polarized parallel to the yaxis, couples to a bending mode and lowers the Madelung potential around the 00 site. The reduction of local potential induces the spins (charges) on CUR and CUL to both hop to the central oxygen. However, this is a spin-dependent process since both charges are forbidden by the Pauli exclusion principle from transferring to the same site unless they have opposite spins. This argument also holds for incident light polarized perpendicular to the Cu02 plane. The second term in the dipole moment is anisotropic and is present only for light polarized in the Cu02 plane. It is referred to as a "charged-phonon-like" effect. This contribution to the dipole moment can be understood by considering the configuration in which the electric field and the displacement of 00 are both parallel to the CuL-CuR bond direction and a phonon mode in which the O's around the CUL breathe in and the O's around the CUR breath out (A in Fig. 3.15). In this breathing mode, the Madelung potential in CUR decreases while in CUL it increases, creating a displacement of charge from left to right that contributes to the dipole moment. Figure 3.16b illustrates such a process.

The real part of the optical conductivity is given by the dipole-moment-dipolemoment correlation function. By decoupling the phonon system from the magnetic system which is valid in the lowest order magnon-phonon coupling, the real part of the optical conductivity is given by79,81 (h = 1), 7rLO 2
-w E Im[-(-6B -p_; 6Bp) w'15


+ 16A2q2sin2 (p,/2)sin2(py) + [4AqAsin2(px/2) - ql]2
+ _.Sx..; 6B )] 3.39











U U


CU L


IU 3 CUR


IL I Uo R2
L2 U L U R




E 1


A B C
Figure 3.15. Ion displacement scheme used for calculations. The full dots represent the Cu's and the open dots O's. Thick arrows represent the spin, thin arrows the lattice displacements, and the thin long arrows represent the direction of the electric field. From Lorenzana and Sawatzky. From Reference
79.


w11f is associated with the in-plane Cu-0 stretching mode while w_, is linked to the Cu-0 bending mode phonons.

Assuming weak absorption, the absorption coefficient is obtained from the expression a = 4" The magnon-magnon Green functions, ((6Bxr; 6B )), are then computed using interacting spin wave82 theory with a Holstein-Primakoff transformation. The results of Lorenzana and Sawatzky are presented in Fig. 3.17. The dashed-dotted line is the theoretical line shape from the full expression in Eq. 3.38, whereas the dashed line, which yields a superior fit to the data, is found by assuming that the anisotropic processes dominate and by setting ql = 0 in Eq. 3.38. Note that the bimagnon disperses around the saddle point (7r, 0). At first glance, this would seem to violate momentum conservation, but it should be remembered that it is the total momentum of the system, magnons plus phonons, which must be conserved





















Cu L CU R
0

E b

Figure 3.16. Typical processes contributing to the (a) isotropic and (b) effective charges. From Lorenzana and Sawatzky. From Reference 79.


(assuming that the incident photon has zero momentum). Within this model, the sidebands at higher energy found in the experimental data are assigned to higher multimagnon processes. Specifically, the bands observed at roughly 4000 cm-1 and 6000 cm-1 are assigned to 4-magnon + phonon excitations in a plaquette and row, respectively. The Green function calculation shown in Fig. 3.17 matched computations from the exact diagonalization of a small cluster83. The exact diagonalization calculations produce the higher sidebands at 4000 cm-1 and 5000 cm-1, but the relative weights are substantially smaller than what is experimentally observed. Thus, the origin of the higher sidebands is still a contentious issue. Nonetheless, it is generally accepted at this time that the strong resonant absorption centered near 2900 cm-1 is due to phonon-assisted 2-magnon excitations.






66



r1200

E) T~t0.37r(i,O)
S 15 (- ,.(�)
,o0 (Oo.4 (0.81To)

(0.77T.0)
o
(- 100

02 1.2


0 0

0.2 0.4 0.6 0.8 1.0 1.2 Photon Energy [eV] Figure 3.17. Experimental data (solid line) and theoretical line shape for two-magnon absorption (dashed line) in La2CuO4. The dash-dotted is the contribution to the line shape from the bimagnon at fi = (7r, 0). The insert shows the density of states of the bimagnon from different values of the total momentum. From Reference 79.













CHAPTER 4
EXPERIMENTAL TECHNIQUES

4.1.1 Introduction

Optical experiments measure the transmittance and/or the reflectance of samples as a function of incident light frequency. Unfortunately, when this is done over a broad frequency range, as was done for materials studied for this thesis, one must use multiple spectrometers, light sources, and detectors. This chapter discusses experimental techniques used in this work. This first section describes the Fourier transform spectroscopy technique employed to cover the far and mid-infrared spectral regions. This is followed by a section on the Bruker 113v fast-scan Fourier spectrometer. Following this, the Perkin-Elmer monochrometer which covered frequencies between 1000 and 40000cm-1 is discussed. The last experimental section discusses the Raman scattering technique employed to study two-magnon excitations in antiferromagnetic materials. A discussion of the detectors, polarizers, sample mounting, and data analysis is also presented. Finally, a brief section is devoted to sample manufacture.

4.2.1 Fourier Transform Infrared Spectroscopy

Fourier transform infrared spectroscopy was developed to overcome the inaccessibility of the far-infrared region to single-grating spectrometry. Grating spectrometers fail in the far-infrared due to the reduced power available from light sources at low frequencies. This is readily seen by simply ratioing the power emission spectrum of a blackbody to that of a light source which obeys the Raleigh-Jeans Law. For any source, the total blackbody power spectrum is given by


P0 aT4A (4.1)









where A and T are the area and temperature of the source, respectively, and a is the Stefan-Boltzmann constant. For a typical mercury arc lamp, the radiated power available from zero to a frequency w obeys the Raleigh-Jeans Law, kBT T2
P(w)- 2T Aw2, (4.2)


where kB and c are the Boltzmann's constant and speed of light respectively. The ratio of the emitted to the total power up to frequency w is then given by,


P(w) 5( hw 3
- P0 - 4(k- ) "(4.3)
PO 704 kBT~

For w = 100 cm-1 and T = 5,000K 77 is only 1.2 x 10-6. Such reductions in power make optical grating spectroscopy techniques impractical.

Fourier transform spectroscopy overcomes the aforementioned energy deficiencies with the use of a Michelson interferometer. Figure 4.1 shows a generic diagram of such an interferometer. The Michelson interferometer works on the principal of linear superposition of light waves and has the advantage that it measures an entire spectrum (typically sections of the far- and mid- infrared) in a single sweep of the movable mirror seen in Fig. 4.1. Light emitted from a source strikes a beam splitter which (ideally) transmits half the light and elastically backscatters the remaining half. The transmitted light is forwarded to the moveable mirror M2 while the reflected light is directed to the stationary mirror labeled M1. After reflecting from their respective mirrors, the two light beams recombine at the beam-splitter. If both mirrors are equidistant from the beam-splitter, then the optical paths for the reflected and transmitted light are identical and the two beams add constructively when they recombine at the beam-splitter: The intensity of the recombined beams will be a maximum. However, if M2 moves away from this optimal position by an amount 5,


























Focusing Lenses


://J Fixed Mirror






Beamsplitter / i


Movable Mirror


Detector
D


Figure 4.1. Generic schematic of Michelson interferometer.


then the two beams will be out of phase by a factor 0 = 27r~v and the intensity of the sum of the two beams at the beam-splitter will no longer be a maximum. Here v is the frequency of the light measured in cm-1 . If the beam-splitter transmits half the light while reflecting the remainder (no absorption losses), then the sum of the amplitudes of the light beams reaching the beam-splitter is


A(v) = a(v)(1 + e'2'Tv)


(4.4)


Multiplying this quantity by its complex conjugate yields the intensity at the beam splitter as a function of frequency and path difference, 6:


M 2









1
I(V, 6) - 2S(v)(1 + cos27r6) (4.5)
2

where S(v) = 4A(V)2 and is referred to as the spectral density of the source. Integrating the intensity from v = 0 to v = co gives the interferogram 1(6), 1(6) = j S(u)dv + f S(v)cos27wv~dv (4.6). v can be determined by taking the Fourier transform of 1(6), S =f) I(6)e2"'6. (4.7) Hence, if the moving mirror is scanned over an infinite range of path differences, 6, and 1(6) is recorded at each 6, the spectrum can be extracted by performing the integral in Eq. 4.7. From the standpoint of signal strength, little is gained by using interferometry in the far-infrared: it is the quality and type of detector that determines the strength of the signal. The chief advantage of using interferometric techniques to measure in the far-infrared is that many measurements can be performed in a relatively short amount of time. Each complete scan (half-oscillation of the moveable mirror) measures a given frequency region and takes about 1 second. This allows many measurements to be taken (typically 256-512 scans) in a relatively short span of time which reduces the signal to noise ratio. This is referred to as the Fellgate advantage.

In reality, of course, an interferogram cannot be measured over infinite path differences and this limits the resolution of the measured spectrum. By varying beam splitters and the maximum path difference, 6max, an array of regions can be scanned in the far and mid infrared. However, since 1(6) is a discreet function in practice, this changes the Fourier integral in Eq. 4.7 to a sum. Also, the maximum path difference, 'Ymax, introduces side-lobes near sharp features in S(v). Fortunately, this effect is mitigated by apodization and aliasing.





































OI A 1"W
*w f b.W .*"Wft ,
* cT4.caI ffomftnd
I CW O.WW ftt ".4 PM a% coma .4Q'-A*


Figure 4.2. Diagram of Bruker Fast Scan Interferometer.



4.2.2 Bruker Fourier Spectrometer

A Bruker 113v fast-scan Fourier transform spectrometer was employed to measure materials in the far and mid-infrared (30-4000 cm-1). Its principal of operation is essentially identical to that of the Michelson interferometer discussed in the previous section. A schematic diagram of the spectrometer is shown in Fig. 4.2. The system is equipped with two sources. A mercury lamp covers the far-infrared spectral region









while a glowbar covers the mid-infrared. Each source is paired with a separate detector: a He-cooled detector for the far-infrared and a DTGS detector which operates at room temperature for the mid-infrared. There are also two channels for measurements. The first is for transmission measurements and the second is for reflection studies. This is necessitated by a reflection stage which must be placed in the sample chamber to achieve near-normal incidence of the incident light on the sample/mirror. Having two chambers precludes the need for drastic realignment when switching from reflection to transmission modes and vice versa.

In the interferometer chamber, the optical dynamics are similar to those found in the Michelson interferometer, Fig. 4.1. Light emitted from a source focuses on a beam splitter which transmits and reflects half the light intensity. Both beams, after reflecting from stationary mirrors on opposite sides of the chamber, are sent to a two-sided moveable mirror which reflects both beams back to be recombined at the beam-splitter. The combined beams are then forwarded to the sample chamber (either transmission or reflection) where they reflect from or transmit through the sample, mirror, or aperture at near normal incidence. After this, the interfering beams are directed to the detector chamber where the signal is measured. It is important to note that the entire system is evacuated to a pressure of roughly 50mTorr to avoid absorption and dispersion of the light by water and CO2 present in the air. The two-sided mirror moves at a constant speed v so the optical path difference is 4vvt, where t is the time measured starting from 6 = 0. Hence, the detector receives a modulated signal at the detector as a function of the light frequency and velocity of the mirror,

D(t) = Docos27rfat. (4.8) Here, fa = 4vvt where v is the light frequency. The signal is then amplified and digitized before being sent to an Aspect computer. At this point, the signal is apodized









and phase corrected. Finally, the signal is Fourier transformed over the frequency to produce the interferogram and subsequently inverse Fourier transformed over the path difference, 6, to yield the spectrum. Table 4.1 lists the parameters and settings used in FIR and MIR measurements on the Bruker 113v fast-scan interferometer.



Table 4.1. Bruker FTIR Operating Parameters

Range Beam Splitter Opt. Filt. Source Pol. Detector

(cm-1) Material Material Material

20-90 Mylar Black PE Hg Arc 1 Bolometer 80-400 Mylar Black PE Hg Arc 1 Bolometer 100-600 Mylar Black PE Hg Arc 1 Bolometer

450-4,000 Germanium on KBr None Glowbar 2 DTGS


PE = polyethylene. Polarizer 1 = wire grid on oriented polyethylene; Polarizer
2 = wire grid on AgBr.



For cryogenic measurements, the sample/mirror/aperture was attached to the tip of a cryostat and inserted into the samples chamber (see section on low temperature measurements and sample mounting). A polyethylene window in the far-infrared and a KC1 window in the mid-infrared sealed the cryostat vacuum and allowed incident light to transmit through the cryostat. For mid-infrared studies (800-4000 cm-1), a DTGS (pyroelectric deuterated triglycine sulfate) detector was used to measure signal while in the far-infrared a bolometric (temperature sensitive) detector was employed. While the bolometer is equipped with a channel for the mid-infrared and provides superior signal-to-noise ratio than the DTGS detector, measurements consistently showed that the DTGS was less erratic in the mid-infrared. The reason for this is











DEWAR. MODEL HD-3
OUTUNE SKETCH


Figure 4.3. Bolometer detector used with Bruker Fast Scan Interferometer.


still unclear. The inferior signal-to-noise ratio of the DTGS was compensated by simply increasing the number of scans (typically to 512). This yielded clean results up to 4000 cm-1. The bolometer will be discussed in more detail in the next section.

4.2.3 Bolometer Detector

The intensity spectrum of sources in the far-infrared is poor (see the discussion under Fourier Transform Infrared Spectroscopy). While interferometric techniques mitigate many of the difficulties associated with weak signal, generally more needs to be done to increase the signal to noise ratio. This is accomplished with the use of bolometric detectors. These detectors operate at liquid helium temperatures (1-2K) and so eliminate the thermal noise associated with room temperature detectors. For






























Figure 4.4. Diagram of Perkin-Elmer spectrometer system used for measuring
transmission and reflection of samples.


example, the intensity spectrum of a blackbody at 300K is centered at 1000 cm-1, near the center of the infrared regime. A diagram of the bolometer used in this study is presented in Fig. 4.3. The detector element is Si-based and detects the signal by bolometric means. That is, the bolometer detects the intensity of the incoming signal by measuring changes in temperature of the Si element as the light strikes it. Changes in the temperature are amplified and recorded as voltage signals which are then sent to the Aspect computer for processing.


4.3.1 The Perkin-Elmer Monochromator

For frequencies above roughly 1000 cm-1, a Perkin-Elmer Monochrameter was used for reflection and transmission measurements. In this frequency regime, the aforementioned difficulties encountered in the far-infrared are not at issue and the simpler, more traditional technique of grating spectroscopy may be employed. For this









reason, a modified Perkin-Elmer 16U monochromator was used for studies between the near infrared and UV regions (1000 cm-1- 40000 cm-1). A diagram of this spectrometer is presented in Fig. 4.4.

The system uses three sources to cover the entire spectrum: a glowbar for the mid-infrared (1000-4200 cm-1), a tungsten lamp for the near infrared and visible (3800-21000 cm-1), and a deuterium source for the UV (17000-40000 cm-'). The optical dynamics of the PE system are straightforward (see Fig. 4.4). After the radiation is emitted from a given source, it is modulated, or chopped, by a rotating motor blade before passing through a set of high and low optical filters to select a particular frequency region. Next, it is forwarded to a grating where different wavelengths are diffracted according to the formula 2d sin0 = nA, where n is the nth order of the diffracted light, A is the the wavelength, 9 is the angle of incidence, and d is the spacing between grating lines. For measurements in this study, n = 1. A variety of gratings with different line spacings were used to cover the entire frequency region between 1000 and 40000 cm-1. These are listed in Table 4.2. Next, the light backscatters and exits the monochrometer through slits before being focused on the sample/mirror located at R?. For transmittance measurements, a mirror is placed at R and the sample and reference (aperture) is placed at 9. The reflected or transmitted light is finally focused onto a detector at the bottom left-hand corner of Fig. 4.4. The modulated detector output is fed into a lock-in amplifier which then amplifies the ac signal. A LED diode and photodetector located on opposite sides of the chopper supply the reference signal for the lock-in amplifier. The output of the lock-in is fed to a digital voltmeter remotely controlled by a PC computer. The PC also controls the stepping motor for the gratings and optical filters. Data are collected by recording the single-beam spectrum (frequency vs. signal) for the sample and reference sequentially and then by ratioing the two spectra to obtain the true












Table 4.2 Perkin-Elmer Grating Monochromator Parameters


Frequency

( cm-1) 801-965

905-1,458 1,403-1,752 1,644-2,612 2,467-4,191 4,015-5,105 4,793-7,977 3,829-5,105 4,793-7,822 7,511-10,234 9,191-13,545 12,904-20,144 17,033-24,924 22,066-28,059 25,706-37,964 36,386-45,333


Grating (line/mm)

101 101 101

240 240 590 590 590 590 590

1200 1200 1200 2400 2400 2400


Slit Width (micron)

2000 1200 1200 1200 1200 1200 1200 225 75 75 175 225 225 700

700 700


a G: Globar; W: Tungsten Lamp; D2: Deuterium Lamp. b TC: Thermocouple; Pbs: Lead Sufite; 576: Silicon photocell.



reflectance or transmission of the sample. During measurement, as with the Bruker 113v, the spectrometer chamber is pumped down to a pressure below 200mTorr to eliminate CO2 and water absorption. For cryogenic measurements, the samples were


Sourcea


GB GB GB GB GB GB

W

W

W

W

W

W

W D2 D2 D2


Detectorb


TC TC TC TC TC TC TC Pbs Pbs Pbs Pbs Pbs 576 576 576 576









attached to the tip of a Hansons and Associates cryostat and inserted into the PE chamber at R (reflection) and T (transmission). This cryostat and details of the sample mounting procedure will be discussed in the next section. A KCI window in the mid-infrared and a quartz window in the near infrared through UV regions permitted transmission of the incident light through the cryostat.


4.4.1 Sample Mounting and Low Temperature Measurements

Sample mounts for reflection measurements are shown in Fig. 4.5. The mount is simply a rectangular piece of copper with a conical section removed in the center surrounding a circular aperture. Samples are attached to the back side of these mounts (on the side opposite the conical wedge) with varnish or ordinary paper glue. Aperture sizes varied from sample to sample but were typically 1.5-2.5mm in diameter. The conical section surrounding the central aperture where the sample/mirror is placed serves to scatter residual light from the source which strikes the sample mount and should not contribute to the detector signal. Removal of unwanted stray light in this way was also enhanced by painting the conical section with non-glossy black paint. Transmission sample mounts are similar to their reflection counterparts. For these measurements, the conical section is removed in Fig. 4.5 (A = B) and the rectangular piece is thinner. The sample mounts were then attached to a sample holder assembly that, in turn, was mounted to the cryostat. All sample mounts and holder assemblies are interchangeable between the Bruker 113v and Perkin-Elmer Grating Spectrometer for easy transfer of samples from one instrument to the other.

Low temperature measurements were achieved by attaching the sample holder assembly to the tip of a Hansen and Associates High-Tran cryostat. A flexible transfer line flowed liquid helium from a storage dewar to the cryostat. The temperature of the sample was regulated and stabilized by a Scientific Instruments 5500 temperature

















* "A = Outer Radius

B -0- B = Inner Radius


t = Thickness





Front Side

Figure 4.5. Diagram of sample mounts used for reflection and transmission studies. For reflection scans, t - 3-4mm and B < 1/2A. For transmission, t
- 1mm and A = B.


controller connected to 25 Watt Menko Heater coils and a calibrated LI diode sensor (thermometer) at the tip of the cryostat. This combination of helium flow and heating of the cryostat tip permitted stable temperature measurements between 10-370K. For cold measurements, the sample holder and cryostat unit were placed inside a shroud equipped with optical windows. The shroud/cryostat system was then evacuated to a pressure below 10-6 Torr to prevent ice formation on the optical window and sample. Since the sample and mirror were on opposite sides of the sample holder in reflection mode, measurements of the sample and mirror were possible by simply rotating the cryostat assembly by 180'. For transmission studies, the sample and aperture were mounted at 90�angles, so sequential measurements of the sample and aperture were possible by a 900 rotation of the cryostat. The two spectra were then ratioed to obtain the true reflectance/transmittance.









4.5.1 Data Analysis of the Spectra: The Kramers-Kronig Transformations

With the exception of Sr2Cul.XCoXO2Cl2, all optical measurements were restricted to single bounce reflectance. By performing a Kramers-Kronig analysis of the full reflectance spectrum, a wealth of information about the optical properties of a given material can be extracted. For normal incidence, the reflectivity amplitude is given by,

r(w) = (n- 1) + i; (4.9) (n + 1) + i ,'

where n and r, are the real and imaginary parts of the complex refractive index. The reflectance power (measured reflectance), 6? (w) is given by the multiplying Eq. 4.9 by its complex conjugate. Thus, r(w) can be written as the product of a real amplitude and a phase factor:

r(w) = p(w)expiA(w). (4.10)


Here, 8 is the phase shift in the reflected light relative to the incident phase and p(w) is related to the reflectance power, R (w), by the expression:



_(W) = (p(w))2. (4.11)


The amplitude, p(w), and the phase shift, 1(w) are, respectively, the real and imaginary parts of ln[r(w)], and thus can be related by means of a Kramers-Kronig transformation:
) ln4 (w') - In4 (w)
- w . 4.12

Unfortunately, this transformation requires that the full frequency spectrum be measured. This of course is not practical. In reality, 9(w) is measured over a finite frequency region, in this case from 20-40000 cm-1, and then merged with existing









data above 40000 cm-1 and up to 380000 cm-1. Below 20 cm-1 the data is extrapolated according to the d.c. and low frequency transport properties of the material under study (metal, insulator, superconductor, etc.), while at higher energies above 40eV a free electron response is assumed ( l/w4). It is important to note that for w1 < w, w > w, and for regions where 4 (w) is flat there are negligible contributions to the integral in Eq.12. This implies that making extrapolations in regions where data is not available has little effect on results in regions where V(w) is measured. More on extrapolations will be discussed in the next section.

4.5.2 High and Low-Frequency Extrapolations

For metals and insulators the high frequency reflectance is usually dominated by interband transitions of the inner core electrons to excited states. Only at frequencies above 100000 cm-1 does the free-electron response of the system become important. In this study, samples were measured out to roughly 40000 cm-1. Above this, the data were merged with existing data on the same samples in the undoped regime which typically covered frequencies up to 300000 cm-1. However, it is only reasonable to assume that the free-electron response dominates at energies higher than 106 cm-1. The reflectance in the interband region between 300000 and 106 cm-1 was modeled using the formula

q(W) = (4.13)
W
where Vf and wf are the reflectance and the frequency of the last measured data point respectively. The exponent s can have values between 0 and 4. Above 106 cm-1, free-electron behavior sets in and the reflectance may be approximated by ...,(Wf)4. (4.14)
W

The low frequency (< 20 cm-1) extrapolation depends on the material investigated. For this study, all the materials measured, including hole-doped La2Cu1-,Li.O4, were









insulating. Consequently, the reflectance was extrapolated to 0 cm-1 at a constant value.

4.5.3 Optical Constants

Once the data has been measured and the proper extrapolations have been made, a Kramers-Kronig transformation can be performed to extract the phase shift 6(w). From this, and 9, (w) one can obtain a variety of optical constants for a given material starting with the real and imaginary parts of the complex refractive index, n(w) and 1(W):
n M = V (W) (4.15)
1 + V (w) - 2 V'! -w) cos O (w)' and
K(W) = 2VI-N" (w) sin O(w) (4.16) 1 + q(w)- 2 V"-(w) cos E(w) respectively. By squaring the refractive index, the complex dielectic function is obtained:

E(w) = (n(w) + iK(w))2. (4.17) Hence, the real and imaginary parts of the dielectric constant are given by:


el(w) = n(w)2 - rn(w)2 (Real) (4.18)


and

E2(w) = 2nr(w) (Imaginary). (4.19) From n, El, and E2 other important optical properties such as the real and imaginary parts of the optical conductivity, skin depth, and absorption coefficient can be deduced. These relations are listed below:


oM(w) WE2
47r









Or2 (W) W(1 - 'E)
47r



=2wK (4.20)



4.5.4 Analysis Procedure for Transmission

For samples that transmit as well as reflect, it is no longer reasonable to use Eq. 4.12 to extract the phase shift of the incident light upon reflection from the sample surface since the measured signal will now contain contributions from the front and rear surface of the sample. A portion of the transmitted light through the front surface will partially reflect from the rear surface and contribute additional signal to the single-bounce reflection spectrum. Remedying this problem requires combining the reflection and transmission data for the full measured spectrum with the aid of numerical techniques. This yields a single-bounce reflectance spectrum from the multi-bounce data. However, numerical processing imposes limits on the results and serves only as an approximation, albeit a very good one, in regions where the transmission is not small.

The aforementioned numerical techniques were not used to extract a single-bounce reflectance spectrum from transmittive samples. Rather, an approximation scheme was employed. First, the unmodified reflectance spectrum was Kramers-Kronig transformed with the appropriate high and low energy extrapolations and all relevant optical constants were extracted as per Eq. 4.20. In regions where the absorption coefficient was large (a > 10000 cm-1), these results served as very good approximate solutions for the optical constants of the single-bounce spectrum. The reasoning is as follows: Since the material is strongly absorptive in these regions, little incident light transmitted through the medium reaches the rear surface, and of the small fraction










of incident light which reflects from the rear surface, an even smaller percentage of this will return to the front surface and contribute to the single-bounce spectrum. To prove this point consider the charge-transfer region in Sr2Cu1-.CoO2Cl2. In this frequency regime (2-3eV), the absorption coefficient is on the order of 105 cm-1. For samples measured in this study, the average sample thickness was roughly .007cm. Hence, only e-1000 < 0.1% of the incident intensity reaches the rear surface and, if roughly 15% of this is backscattered to the front surface, virtually 0% of the transmitted incident light not reflected from the front surface will contribute to the singlebounce spectrum. The same reasoning may be employed in the phonon frequency region (50-700 cm-1). Although the absorption coefficient in this spectral range varies strongly (a - 100-50000 cm-1) and the reflectance is, on average, higher (-. 50%), the above argument given above shows that the contribution from the rear surface to the single-bounce is still less than 12%. These measurable effects from rear surface contributions manifest as fringes in the reflectance spectrum below 5000 cm-1. With a little diligence and care, the fringes were removed by Fourier transforming the spectra and removing the spurious frequency componets. The spectra were then inverse Fourier transformed to yield the filtered spectra.

The above approximation for the charge-transfer and phonon regions is reasonable provided the absorption is large. However, this technique fails in the mid-infrared and near infrared spectral regions in Sr2Cul-xO2Cl2 where the absorption coefficient is comparatively small (a ,- 250 cm-1). However, there is an alternate approach. The absorption coefficient may be written as

47ro
a =- (4.21) ElV

Here, o is the optical conductivity and v is the energy velocity of the light in the medium. For weak absorption, E = n2 - k 2 n2, and v ,. c/n, so that Eq. 21 may









be written as
47ror
a =(4.22) nc

Eq. 4.22 may then be inverted to find or(w):


onc (4.23) 4ir


A similar expression may be found for E2:


2 = anc (4.24)
E2


It should be noted that while E / n2 for strong absorption, Eqs. 4.22-4.24 still holds for the case of strong absorption84.

For transmittive samples, the absorption coefficient may be found by inverting the expression,

T = exp(-ad), (4.25) where d is the medium thickness and T is the measured transmission to solve for a. From this the a and E2 may be found from Eqs. 4.23 and 4.24. While it was noted that Eqs. 4.22-4.24 are valid even for strong absorption, these expressions were not used to extract a and E2 in this limit for practical reasons. In regions of strong absorption, the transmitted signal is either very weak (< 10pV) or non-existent making for a poor signal-to-noise ratio.

The optical constants derived in the limits of strong and weak absorption in their respective spectral regimes were then merged together. However, while practical, this procedure was less than perfect. There was poor overlap in the interstitial regions where the optical constants derived from the Kramers-Kronig transformation and









from the transmission data were merged. This was due primarily to the poor signalto-noise ratio in the transmission data in these interstitial regions. Values for the transmission-derived optical constants in this regime were invariantly a full order of magnitude smaller than their Kramers-Kronig derived counterparts. This was 'corrected' by deleting both sets of data in the interstitial region and patching the weak and strong absorption regimes with a polynomial fit. Fortunately, the interstitial regions were few and relatively narrow. In Sr2Cul-..Co.O2C12, these were located between 125 and 135 cm-1, 550-600 cm-1, and 11000-130000 cm-1. In regions where spectral overlap was not an issue, there was excellent agreement with existing data.

4.6.1 Raman Scattering: Experimental Technique

Raman scattering is the inelastic scattering of light from a medium. In the Raman scattering process, the energy of incident light shifts by either absorbing energy from (Anti-Stokes scattering) or imparting energy to (Stokes scattering) excitations in the scattering medium. For Raman measures in this study, only Stokes scattering is investigated. This contrasts to optical spectrometry where only the elastically back-scattered light is measured for a single frequency. Single-beam spectroscopy predominantly measures symmetry allowed dipole transitions in materials such as phonon and charge-transfer excitations. Quadrupole and higher order interactions can, in principle, be measured with this technique but are at least two orders of magnitude weaker than their dipole counterparts. Furthermore, depending on the crystal symmetry of the sample and orientation of the carrier orbitals, particular dipole transitions are forbidden or severely attenuated by symmetry. Many of these inherent weaknesses in single-beam spectroscopy are remedied by Raman scattering. Raman scattering measures shifts in frequency of light of known energy, invariantly provided by a He-Ne Argon or other energy intensive laser capable of delivering several mWatts to the sample surface. Hence, transitions once to feeble to observe in











He-Ne Laser (514.Snm) >L xJ Band I'ss Fler











DilterMicoe LenLe (X40)

Magni ying Lense Bean] Spliuer

Figure 4.6. Raman scattering setup. CP - circular polarizer; LP =linear
polarizer BS = Beam Splitter; CCD = charged coupled device detetector.



optical spectroscopy may now be measured. Raman scattering may also verify dipole transitions observed using spectroscopic techniques. Hence, Raman scattering is an excellent compliment to optical spectroscopy.

Figure 4.6 shows the experimental set up for Raman scattering measurements used in this study . The principal equipment consisted of a He-Ne Argon laser, polarizers and notch filter, a CCD (charge-coupled device) detector, and a grating to diffract the inelastically back-scattered light from the sample surface. The CCD was interfaced to a PC to record the data. 514.5 nm light was first emitted from He-Ne Argon laser. The light then passed through a non-dispersive band pass filter to eliminate plasma lines from the laser. Typical laser output was roughly 300mW. After reflecting from a second mirror, the now attenuated laser beam passed through a broad-band linear polarizer on a rotating mount. This polarized the beam before reaching the beam splitter at near normal incidence (a - 40). While this did not completely eliminate









mixing of the s- and p-polarizations in the partially reflected beam, the small angle of incidence insured that this effect would be strongly suppress. The partially reflected light from the beam-splitter then passed through a 40 x microscope lens before striking the sample. The microscope piece was placed inside a vacuumed sealed housing unit threaded for translational movement of the lens. This translational degree of freedom was needed to squarely focus the incident light onto the sample surface. The housing unit was capped with a transmittive quartz window and attached to the cryostat shroud. The beam intensity and ocular piece provided 2-5 mWatts/cm2 at the sample surface. The sample was attached to the tip of a nitrogen cooled cryostat tip. Polarization of the incident light on the sample surface was controlled by rotating the aforementioned broad-band polarizer. It should be kept in mind in subsequent discussions of the Raman data that there is a small elliptical distortion in the incident beam to the sample surface due to weak mixing of the s- and p-polarizations at the beam splitter. The elastically and inelastically back-scattered light from the sample surface then transmitted back through the glass beam-splitter before going through an adjustable aperture to help remove stray laser light and to align the back- scattered beam with the grating slits. Next, the back-scattered beam passed through a second (back) broad-band linear polarizer. Front and back-end polarizers were needed to separate the symmetry sensitive, inelastically back-scatterd light. More concerning this will be discussed in the next section. The beam then passed through a 514.5 nm holographic notch filter to remove the elastically backscatterd light from the sample surface which would have 'swamped' the detector signal and overshadowed weaker, more physically relevant interactions. Finally, the now solely inelastically scattered light entered a Perkin-Elmer spectrometer where it diffracted from a fixed 300 lines/mm grating. The diffracted radiation was partitioned in space as a function of wavelength according to the formula:









dsinO = nA (4.26)


where d is the line spacing, 0 is the angle of reflection, and n is the order of diffraction (n=1 here). The frequency separated, inelastically scattered light was then forwarded to a CCD detector equipped with 1200 horizontal and 500 vertical pixels. This provided 4.5 cm-1 data resolution. Although it is possible to perform Raman scattering with a moveable grating and detector, a CCD was employed for two reasons. First, it is a nitrogen cooled detector and thus provides a superior signal-to-noise ratio than room temperature detectors commonly used with single-beam moveable grating techniques (e.g., thermocouples and PbS chips). Second, the CCD measures an entire spectral region instantaneously. There is no need to step a grating the 200-300 increments needed to cover a frequency region that the CCD can measure in an instant. To reduce signal-to-noise levels to a suitable level, the measuring time of the CCD is simply increased. For Raman data presented here, 5 minute measuring intervals were used. Finally, the data was digitized and forwarded to a PC where it was recorded for further analysis.

4.6.2 2-Magnon Raman Scattering: Theory and Analysis

Raman scattering can measure transitions in materials that are too weak to be adequately observed in reflection studies. With the exception of Sr2CoxCul-xO4C12, all samples prepared for this study were too opaque for transmission studies. For light to transmit, these samples would have had to have been polished down to a thickness no greater than 5M. The brittle nature of La2CuO4 virtually precludes this. While still possible, the resources needed to polish samples down to such a fine thickness were not at our disposal. This is where Raman scattering proves especially useful. Since it is only the back-scattered radiation that is measured, transmission is no longer an issue and, in fact, becomes a hindrance. Hence, two-magnon scattering in antiferromagnetic









systems, a process far too feeble to be observed in reflection, can readily be observed in Raman measurements. For mediums such as Sr2CoxCul-XO4Cl2 where two-magnon scattering can be observed in transmission, two-magnon Raman scattering is a useful tool for verification.

Two-magnon Raman scattering in antiferromagnetic systems can be understood as an exchange-scattering mechanism. This differs from two-magnon scattering in magnetically ordered materials (ferromagnetic and antiferromagetic) which results from the spin-orbit coupling of the charge carriers. In this process, two-magnon Raman scattering is a fourth order interaction in perturbation theory. Two-magnon scattering via the exchange mechanism involves the flipping of spins on interlinked sublattices in antiferromagnetic systems and has no application to ferromagnetically ordered systems. In the present case, alternating half-filled Cu3d,2_v2 orbitals in the CuO2 lattice make up each spin sublattice. In the exchange mechanism process, an incident photon dipole-couples to an electron on spin sublattice A (spin-up). The electron is subsequently excited to the neighboring sublattice B (spin-down) provided the incident photon is polarized along the Cu-Cu bond. However, this excitation comes at an energy cost U, the Coulomb repulsion energy of two electrons residing on the same site. This induces the spin-down electron on sublattice B to 'relax' by transferring to the now vacant site on sublattice A. Hence, dipole-coupling of the incident light to the charge carriers induces a spin-flip, or exchange, in the antiferromagnetic background. In second-quantized form, this is viewed as creating a separate magnon quasiparticle on each sublattice. An approximate form of the exchange scattering mechanism may be written as:



H attct c (4.27) Hem~~~~ ~~ 4: k b:( Z "'f+, k"









Here, /3 and Ot are magnon creation operators on the two interlinked sublattices, c and ct are electron creation and destruction operators, respectively, and -I'k, is the associated coupling tensor and is directly related to the antiferromagnetic coupling energy J. For more information on physical dynamics of Raman scattering, see the Appendix.



Table 4.3. Scattering Symmetries for 2-Magnon Raman Scattering.

Symmetry Geometry

A, + B1 xx, yy A2 + B2 xy, yx A, + B2 XIX , y'y' A2 + B1 xy', yx

Symmetries accessed with various combinations of incident and scattered light polarizations. x and y are along the Cu-O inplane bond directions. x' and y' are oriented 450 to x and y.





The 2D CuO2 antiferromagnetically ordered planes have C4 symmetry. Consequently, the Raman scattering tensor in Eq. 4.27, 4k, for the in-plane electric field vectors may be separated into four irreducible representations: A1, A2, B1, and B2. Under C4 symmetry, these four representations transform as the polynomials x + Y x3y - y3x, x - y2, and xy respectively. To isolate each of these representations, a variety of polarization combinations of the incident and scattered light was employed. These are listed in Table 4.3 The first column in Table 4.3 indicates the symmetry representations contained in the scattering for the polarization combinations given in the second column. The first letter in the notation of the second column indicates the




Full Text

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OPTICAL PROPERTIES OF LAMELLAR COPPER OXIDES WITH IN-PLANE MAGNETIC AND CHARGED IMPURITIES BY SEAN MOORE 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 1999

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ACKNOWLEDGMENTS Graduate school... what a ride. What a convoluted path to which I now find myself at the end. Now that I have weathered the storm, there are several people I need to thank before I embark on future endeavors. First and foremost, I would like to thank my advisor, John Graybeal, for his guidance, patience, and his willingness to entertain my ideas and observations. His intellectual openness laid the ground work for stimulating and illuminating discussions which at times extended beyond the narrow confines of condensed matter physics. He infused me with a drive to succeed in the lab at times when I was defeated by repeated failures. What is more, he gave me the personal space to find my own way through the problems that typically plague graduate research work, yet he was willing to assist me at a moment's notice. Such working environments are rare and I can not begin to overstate my gratitude. Without being afford this freedom to mend the messes that I made, I doubt very much that I would be the better for pursuing graduate work. I would not be writing these acknowledgments if it were not for David Tanner and his eternally patient and amenable graduate students with whom I worked. Although I was not officially a member of Dr. Tanner's group, I believe that I was the adopted son they were never sure how they got. My defacto status as a group member was due in large part to the numerous friendships that I struck with David's graduate students over the four years that I worked in his lab. They bestowed upon me the same rights and responsibilities as any other member of the group and gave me unbridled access to their spectrometers and software. Furthermore, I enjoyed several useful and illuminating discussions with David which helped immeasurably with my ii

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research. Without exception, David and his graduate students were always fair and patient to a fault. I am also compelled to thank them specifically for their empathy and understanding when, in the face of experimental disaster, I was less than patient, if not livid, with my research. I hope that in hindsight my occasional eruptions become a source of amusement. If not, well, they can always ridicule me in the years to come. I can not overstate my gratitude to them all. I would also like to thank my parents and fiancee for their vital support and encouragement on my journey through graduate research en route to a Ph.D. Without their confidence in my ability, I would have had no faith in my own. They endured my bouts of abject misery and frustration without complaint when they were well within their rights to to infuse me with a new perspective with a proverbial kick in the rear. Thank you for handling my sensibilities with kid gloves. I must also single out a few individuals and groups for special thanks. I would like to thank Steve Thomas for giving me access to his polishing wheel and rescuing me from the ravages of hand polishing and from all the attendant frustration and exasperation that it brings. I would also like to thank Joe Simmons and his group for performing fluorescence measurements vital to our pressure-dependent transmission studies. Many thanks also go to Graig Prescott and Charles Porter for helping me navigate way through a plethora of computer woes. Finally, I thank all my colleagues in the UF Tae Kwon Do Club for their dedication to the art and for their high standards of martial arts training. They are zealous practitioners and good friends. I will miss them. iii

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TABLE OF CONTENTS Page ACKNOWLEDGMENTS ii ABSTRACT viii 1 INTRODUCTION 1 1.1.1 In the Beginning 1 1.2.1 Why Study La2Cui_xLix04 and Sr2Cui-xCox02Cl2? 3 1.3.1 About this Thesis 5 2 REVIEW OF EXPERIMENTAL WORK 8 2.1.1 Introduction 8 2.2.1 La2Cu04 9 2.2.2 Electronic Configuration and Magnetism: La2Cu04 10 2.2.3 Optical Conductivity: La2Cu04 14 2.2.4 Raman Scattering: La2Cu04 19 2.3.1 Lattice Structure: Sr2Cu02Cl2 21 2.3.2 Electronic Configuration and Magnetism: Sr2Cu02Cl2 22 2.3.3 Reflectivity and Optical Conductivity: Sr2Cu02Cl2 25 2.3.4 Raman Scattering: Sr2Cu02Cl2 27 3 THEORY 30 3.1.1 Models for Carriers in the Cu02 Planes: Normal State 30 3.1.2 Three-Band Hubbard Model 31 3.1.3 t-J Model 34 3.1.4 Cluster Calculations 36 3.2.1 Models for the Midinfrared Band 38 3.2.2 Small Polaron Model 40 iv

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3.2.3 Magnetic Excitations: Zhang-Rice Singlets 43 3.2.4 Domain Walls 48 3.3.1 The Charge-Transfer Band 54 3.5.1 Phonon-Assisted Multi-Magnon Absorption 58 4 EXPERIMENTAL TECHNIQUES 67 4.1.1 Introduction 67 4.2.1 Fourier Transform Infrared Spectroscopy 67 4.2.2 Bruker Fourier Spectrometer 71 4.2.3 Bolometer Detector 74 4.3.1 The Perkin-Elmer Monochromator 75 4.4.1 Sample Mounting and Low Temperature Measurements 78 4.5.1 Data Analysis of the Spectra: The Kramers-Kronig Transformations ... 80 4.5.2 High and Low-Frequency Extrapolations 81 4.5.3 Optical Constants 82 4.5.4 Analysis Procedure for Transmission 83 4.6.1 Raman Scattering: Experimental Technique 86 4.6.2 2-Magnon Raman Scattering: Theory and Analysis 89 4.7.1 Sample Preparation 92 5 OPTICAL PROPERTIES OF La2Cui-:,Li^04 96 5.1.1 Introduction 96 5.2.1 Li-doping La2Cu04 97 5.3.1 Overall Reflectance of the a-b Plane 98 5.4.1 Optical Properties of the Far Infrared: Phonon Assignment in La2Cu04 100 5.4.2 Phonon Assignment in La2Cui_xLix04 110 5.5.1 The Midinfrared Band in La2Cui_xLia;04 117 5.5.2 Polaron Model 119 5.5.3 Zhang-Rice Singlets 126 5.5.4 Domain Walls 130 5.5.5 Magnetic Strings 136 5.6.1 Effects of Li-Doping in the ChargeTransfer Region 140 5.6.2 Temperature Dependence of the ChargeTransfer Band 146 5.7.1 Concluding remarks 148 V

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6 OPTICAL PROPERTIES OF Sr2Cui_xCox02Cl2 152 6.1.1 Introduction 152 6.2.1 Co-doping Sr2Cu02Cl2 153 6.3.1 Overall Reflectance of Sr2Cui_xCox02Cl2 154 6.4.1 Phonon Assignment in Sr2Cui_a;Coi02Cl2 155 6.5.1 Phonon Assisted Multi-Magnon Scattering in Sr2Cui_xCox02Cl2 .... 164 6.5.2 Pressure Dependence of the MIR Excitations 167 6.5.3 Effects of Co-Doping on the MIR excitations 174 6.6.1 Temperature Dependence of the Charge-Transfer Region 181 6.6.2 Effects of Co on the Charge-Transfer Region 182 6.7.1 Excitations in the Near Infrared 187 7 2-MAGNON RAMAN SCATTERING 195 7.7.1 Introduction 195 7.2.1 2-Magnon Raman Scattering Data: La2Cu04 195 7.2.2 2-Magnon Raman Scattering: La2Cui_xLia;04 197 7.3.1 Data Analysis: Curve Fitting the 2-Magnon Band 201 7.3.2 Domain Walls vs. Bound Holes 203 7.4.1 2-Magnon Scattering in Sr2Cui_iCox02Cl2 213 7.4.2 Analysis and Interpretation: Sr2Cui_xCox02Cl2 215 7.4.3 Phonon-Assisted Mutimagnon Absorption Revisited 218 7.5.1 Concluding Remarks 221 8 CONCLUSION 223 8.1.1 Summary 223 8.2.1 Future Experiments 226 APPENDIX: RAMAN SCATTERING 228 Introduction 228 The N-Particle Radiation Hamiltonian 229 Scattering Cross Section and Perturbation Theory 231 Calculation of One-Magnon Raman Scattering Cross Section 232 2-Magnon Raman Scattering in Antiferromagnetic Systems 235 vi

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REFERENCES BIOGRAPHICAL SKETCH

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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 OPTICAL PROPERTIES OF LAMELLAR COPPER OXIDES WITH IN-PLANE MAGNETIC AND CHARGED IMPURITIES By Sean Moore December 1999 Chair: Dr. John M. Graybeal Major Department: Physics The optical properties of the superconducting cuprates are unusual, and many of their spectral features are still not well understood. This is especially true in the midinfrared, where a heavily weighted band grows with charge-doping concurrent with a rapid loss of spectral weight in the charge-transfer band. To date, this redistribution of spectral weight is not well understood. We have extensively studied the optical properties, including the 2-magnon Raman spectra, of La2Cui_xLix04 and Sr2Cui_xCox02Cl2, two materials that do not have a superconducting phase. Li adds a hole carrier to the Cu02 plane in much the same way as divalent Sr in La2-iSrxCu04. However, Li goes into the Cu02 plane with a closed shell and so introduces a S=0 impurity. La2Cui_xLix04 undergoes the aforementioned redistribution of spectral weight prototypical of all of the superconducting cuprates. However, the charge-transfer band remains comparatively robust. This suggests that the additional weight in the mid-infrared must come from spectral regions above 4eV. Furthermore, no Drude tail is observed. This is consistent with d.c. conductivity measurements that place La2Cui_j;Lix04 in the insulating regime up to X = 0.50. Our two-magnon Raman scattering studies suggest that the 2D viii

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AF spin order of La2Cui_xLia;04 remains relatively intact up to x = 0.10, in stark contrast to what is found in the under-to-overdoped regimes of the superconducting cuprates. Despite this, 3D long-range spin order is lost at Li concentrations very similar to those found La2-xSra;Cu04. The absence of charge carriers in Sr2Cui_a;Coa;02Cl2 precludes the formation of a heavily weighted MIR band and subsequent redistribution of spectral weight. Co substitutes for Cu in the Cu02 plane and is believed to introduce a spin-3/2 impurity into the spin-1/2 AF background. Modest variations in the phonon-assisted bimagnon absorption band suggest that the spin-3/2 impurity sites are, to a first approximation, magnetically decoupled from their spin-1/2 AF host. This is consistent with 2-magnon Raman scattering measurements which indicate that the 2D in-plane spin correlation length changes little with the introduction of cobalt. We find that the charge-transfer band is relatively impervious to the presence of Co in the light-to-moderately doped regime. ix •s

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CHAPTER 1 INTRODUCTION 1.1.1 In the Beginning The world of condensed matter physics changed forever in 1986 when Bednorz and Miiller^ reported a superconducting phase in La2_xBaj;Cu04 at 30K. This discovery initiated an enormous amount of research to push the superconducting transition to ever higher temperatures. These efforts were not without reward. In 1987, Wu et al} synthesized YBa2Cu307_5 and soon thereafter found a Tg of ~ 90K. One year later, Parkin et al.^, along with Sheng and Hermann, discovered a Tc of « 125K in Tl2Ba2CaCu208. Not to be outdone, Schilling et al.^ synthesized tri-layer HgBa2Ca2Cu308+5 in 1993 with a critical temperature of 133K. Over a time span of less than two years, superconductivity transcended a subfield of condensed matter physics and emerged a headline story on everything from "Physics Today" to CNN. By 1987 it seemed that every armchair scientist in the nation, if not the world, was talking about energy efficient levitating trains that could zip along superconducting rails at enormous speeds. Star Trek technology was just around the corner we were told, it was just a matter of time until the critical temperature topped a tepid 300K. Unfortunately, just how long said time would be was unclear. It still is. The intelligence, dedication, and sweat of a relative few may have gotten us to a critical temperature of 133K, but it will most likely take the collective efforts of the entire condensed matter community to get us to room temperature. Efforts to manufacture materials that superconduct above 133K have, in the absence of enormous applied hydrostatic pressures^ , failed. After only a few memorable years, the specter of room temperature superconductivity, and all of its attendant novel technology, faded from 1

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2 the headlines and eventually from the public consciousness. Besides, by this time we had cold fusion and the promise of cheaper utility bills. Depressing? A little. Is this how the story ends? Hardly. Now that the fervor over high temperature superconductivity is over we can make a sustained and deliberate effort to figure out what makes these things tick. The superconducting mechanism of the high temperature superconductors is not understood at this time and it is not the principal intent of this dissertation to elucidate it. We chose to focus on the higher energy optical properties of the superconducting planes in and well above the far-infrared. These excitations have received comparatively little attention since they are so far removed from the characteristic superconducting transition energy around 20-30 cm" ^ . We feel that, despite the large scale differences in energy, these excitations are important in their own right and, at some future point, may shed some light on the elusive nature of high temperature superconductivity. High temperature superconductivity is achieved in the spin-1/2 magnetically ordered lamellar cuprates by introducing charge carriers, typically holes, into the Cu02 planes. The superconducting phase is highly anisotropic in these systems. The coherently paired electrons superconduct primarily in the Cu02 plane with only a weak perpendicular component. Optical excitations which probe the Cu02 plane in and above the far-infrared region change dramatically with the introduction of chargecarriers. There are three principal deviations in the optical spectra from the undoped (insulating) phase. A Drude tail grows in the far-infrared, consistent with the insulator-to-metal transition observed above doping concentrations of 2-3%. This is expected. However, a heavily weighted mid-infrared (MIR) band which scales roughly linearly in the carrier concentration emerges around 0.50eV (~ 4000 cm~^). This is concurrent with a rapid erosion of the charge-transfer band near 2eV. This redistri-

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3 bution in spectral weight is not well understood at this time, and it is the hope of this dissertation to help resolve this mystery. There is no doubt that part of the difficulty in understanding these systems stems from their complexity. La2Cu04 and Nd2Cu04, two of the simpler host high Tc superconductors, contain 14 atoms per unit cell. In fact, the Tc generally scales with the complexity of the unit cell, increasing from « 30K to ~ 125K from La2-xSrxCu04 to Tl2Ba2CaCu208. Early measurements on these materials were designed to see if the superconducting mechanism could be explained in the context of the Bardeen-Cooper-Shreiffer (BCS) theory^ for conventional superconductors. Indeed, some similarities to the conventional superconductors were found. Flux quantization^ and A.C. Josephson effect^ measurements showed that the elementary unit of charge in the superconducting condensate was 2e while photo-emission^"^ ^ and tunneling^^'^^ measurements suggested the presence of a superconducting gap. However, at the same time unconventional properties of the highTc materials were found. Some of the principal differences between the high Tc superconductors and their conventional brethren aside from obvious deviations in their respective critical temperatures include a linear d.c. resistivity with temperature in the normal state^^'^^, extremely small coherence lengths^^'^^, and anisotropy in many of their physical properties. The anisotropy is especially evident in the predominately 2D nature of the superconductivity within the Cu02 planes. 1.2.1 Why Study La9Cui-xLi..04 and Sr^Cui-.Co^O^Ch? The discovery of the high Tc superconductors certainly spawned enormous theoretical and experimental efforts to unveil the superconducting mechanism in these materials. From an experimental standpoint this typically involved synthesizing and measuring new materials in an effort to push the high-Tc envelope to ever higher temperatures. Much of the optical work performed on these systems focused on excitations well into the far-infrared. However, any materials stress engineer will tell

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4 you that the best way to make something better is to break it. Once we know how and why it breaks, we are in a position to build a superior device. So that is what we did: We hit high temperature superconductivity with a proverbial hammer and measured what we got in the process. Furthermore, to make matters simple, we examined the simpler single layer copper-oxide superconductor hosts La2Cu04 and Sr2Cu02Cl2. In the latter, a superconducting phase has not been observed. However, it is isostructural to La2Cu04 and shares many optical properties. The "hammers" used in this case were charge and magnetic impurities introduced into the Cu02 superconducting planes. Our measuring devices were optical spectrometers and a Raman scattering setup on an optical table. Specifically, we looked at the optical properties of La2Cui_a;Lix04 and Sr2Cui_3:Cox02Cl2, both insulators, in the mid-infrared and charge-transfer spectral regions and compared our results to existing studies on one of the original high-Tc materials, La2-xSra;Cu04. Superconductivity in the high Tc superconductors is achieved by substituting charge donors outside of the spin-1/2 antiferromagnetic (spin-1/2 AF) Cu02 plane. In La2Cu04, for example, this is done by replacing divalent Sr for trivalent La which adds holes to the Cu02 planes. By contrast, monovalent Li is substituted for divalent Cu in La2Cui_xLia;04. Since Li goes into La2Cu04 with a closed electronic shell configuration each adds both a hole and a spin-zero impurity to the Cu02 plane. The presence of an in-plane spin-zero impurity coupled with the strong binding energy of each hole carrier to its Li host precludes superconductivity. In Sr2Cui_a;Cox02Cl2, cobalt substitutes for Cu in the Cu02 plane. Since both incorporate into the lattice as divalent entities, no carriers are added to the system and, thus, there is no possibility for a superconducting phase. However, cobalt is believed to go into Sr2Cu02Cl2 with a Zd^ electronic configuration in a high spin state. Therefore, each cobalt site introduces a spin-3/2 impurity into the spin-1/2 AF background. Hence, we can examine the effects of adding charge

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5 and/or magnetic impurities to the Cu02 plane by charting the optical properties of these two systems. This will hopefully contribute to an understanding of the optical properties of the high Tc superconductors. 1.3.1 About this Thesis In writing this dissertation I attempted to compartmentalize the chapters as much as possible according to material and experimental technique. This was done in hopes of making the individual chapters as independent of each other as possible. The decision to format this dissertation in this manner was arrived at after considerable consternation and second (and third and fourth) thought. There are both advantages and disadvantages to this structure, but it is my reasoned belief that the former eclipses the latter and I hope that after reading this dissertation you agree. There are, I believe, two principal advantages to the modular approach. One is that the reader can read any of the chapters in no particular order without constantly being referred back to some earlier or later chapter, although this dissertation is not completely devoid of that sort of future/past referencing. When it is reasonable and proper I have tried to reproduce figures and data rather than to send the reader backwards or forwards 50-100 pages. I realize that this can be very annoying. The second advantage of a modular structure is that it allows the researcher to decouple experiment from theory by placing each in separate chapters. Although this defeats the first principal advantage to some degree, I believe it more logical and find it less cluttered to separate theory from experiment than to bury theoretical models in different chapters. In this way the reader is referred only to a particular subsection of a single chapter when the data is analyzed in the context of a given theoretical construct. This dissertation is arranged in the following order. First, a brief review of the work done on previous high-Tc systems is presented in Chapter 2. The structural, elec-

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6 tronic, magnetic, and optical properties of some the single layer copper-oxide superconductors are discussed and comparisons to La2Cui_xLi2;04 and Sr2Cui_xCoj;02Cl2 are made. In Chapter 3 many of the theoretical models used to interpret the observed optical excitations are presented in modest detail. Oneand three-band Hubbard models, polaron excitations, Zhang-Rice singlets, domain wall structures, and phonon-assisted magnon absorption are discussed. The intent here is not to provide a complete tutorial on these theoretical constructs but rather simply to motivate the physics behind them. It is recommended that this chapter be read prior to reading Chapters 5-7 where the experimental data for this dissertation is reported. Chapter 4 discusses the experimental apparatuses and techniques used to perform optical measurements over a range of temperatures. Notes concerning sample manufacturing and preparation are also presented here. Chapter 5-7 are the experimental crux of this dissertation. In Chapter 5 the optical data for La2Cui_a;Lix04, including reflection, optical conductivity, and sum rule calculations, are presented over a wide range of temperatures. It is found that La2Cui_xLix04 bears many similarities to and a few critical differences from its superconducting counterpart La2-xSrxCu04. Many of its optical features are discussed in the context of the theoretical models of Chapter 3. Chapter 5 is structured in a similar fashion for the experimental results of Sr2Cui_xCoa;02Cl2. Lastly, Chapter 6 is devoted to 2-magnon Raman measurements on both La2Cui_a;Lia;04 and Sr2Cui_xCox02Cl2. Since 2-magnon Raman data require substantially less analysis than optical transmission and reflection data, the 2-magnon Raman scattering cross sections for both La2Cui_a;Lij;04 and Sr2Cui_a;Cox02Cl2 were compiled into one chapter without, I believe, violating the aforementioned principle of compartmentalization. Chapter 8 summarizes the previous three chapters, offers some perspective on the relevance of this dissertation, and speculates about possible related future projects. And lastly, the Appendix pro-

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vides a short tutorial on 1and 2-magnon Raman scattering in magnetically ordered materials. I hope this format makes for a coherent and stimulating presentation.

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CHAPTER 2 REVIEW OF EXPERIMENTAL WORK 2.1.1 Introduction This chapter will briefly review the physical and optical properties of the materials studied for this dissertation. It is divided into two sections. Each is devoted to the physical and optical properties of one of the two materials investigated. These properties include lattice and spin structure, electronic configuration, magnetism, optical conductivity and 2-magnon Raman scattering. Some of the properties discussed may seem tangential to this dissertation, but they are important in their own right. For example, while this dissertation is not an examination of the crystal or spin structure of the high-Tc cuprates, it is nonetheless vital to have a working understanding of these structures if the peculiar and poorly understood optical properties of the high Tc superconductors are to be tamed. The lattice structures of all the high Tc layered cuprates, from La2-xSra;Cu04 to Bi2Sr2Ca2Cu30io are similar and share many physical properties. Charge transport and superconductivity, for example, occur mainly in the nearly identical 2D Cu02 basal planes of these materials. The planes consist of Cu sites surrounded by four oxygen sites with a Cu-0 separation of roughly 1.9A. This separation changes little from material to material. Furthermore, the Cu02 planes of this class of materials are separated by layers of other atoms which, when properly doped, provide the charge carriers responsible for superconductivity in the Cu02 planes. Also common to this family of cuprates is the bulk anisotropic antiferromagnetic spin order (AFO) of the parent insulators. Many speculate that it is the spin order, especially in the Cu02 planes, that drives the superconductivity in the 8

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9 La2Cu04-A (Cmca) Figure 2.1. Crystal and AF spin structure of La2Cu04. From Reference 20. layered cuprates. For concision, lattice structures and optical properties discussed in the following sections will be limited to materials covered in this dissertation. 2.2.1 La9Cu04 Ostensibly the simplest of the high Tc materials, La2Cu04 was discovered by Bednorz and Miiller^ to have a superconducting phase when divalent Ba was substituted for trivalent La. Doping the system with Ba added hole carriers to the system and subsequent measurements which mapped the Fermi surface determined that these holes resided on oxygen sites in the Cu02 planes. Bednorz and Miiller speculated that La2Cu04 had a tetragonal K2Ni04 structure and this was later confirmed by

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10 Takei et aO^. Further study^^ revealed that stoichiometric La2Cu04 is an antiferromagnetic insulator with a 2D superexchange energy in the Cu02 plane of roughly 0.125eV. Fig. 2.1 shows the crystal structure of undoped La2Cu04.^° Each Cu is surrounded by four oxygen atoms with a Cu-0 separation of 1.9A. There are two more distant apical oxygens above and below the central copper site at a distance of 2.4A. Hence, each Cu site in the compound is surrounded by an octahedron of oxygens. The larger Cu-0 separation for the out-of-plane oxygens suggests that the dominant bonds are those in the Cu02 plane. Several experiments have buttressed this prediction. The octahedral oxygen configuration which shrouds each Cu2+ ion implies that the lattice has a tetragonal I4/mmm space group structure. However, neutron scattering measurements show that the apical oxygens tilt from their high symmetry points^^'^^. The tilting produces a tetragonal-to-orthorhombic distortion in the crystal lattice that sets in below a specific characteristic temperature which depends on the stoichiometry of the compound. For nearly stoichiometric La2Cu04 (no excess oxygen or impurity substitutions) the characteristic temperature is close to 800K, while for a Sr concentration of roughly 15% (Lai.85Sro.i5Cu04) it drops to about 240K. 2.2.2 Electronic Configuration and Magnetism: La?!Cu04 Undoped La2Cu04 is a charge-transfer spin-1/2 antiferromagnetic insulator (spin1/2 3D AFO). The valences of La, Cu, and O when introduced into La2Cu04 are -1-2, +2, and -2 respectively. Hence, charge neutrality is preserved. Lanthanum and oxygen go into La2Cu04 with closed 2p and 5/ valence shells respectively. Copper, however, enters the matrix with a half-filled 3d^2_y2 orbital. Crystal field splitting removes the degeneracy of the Sd orbitals on the copper sites. Figure 2.2b illustrates the splitting of the electronic energy levels at the copper sites while Fig. 2.2a gives the orbital configurations of the Cu Mj.2_y2 and O 2px, 2py orbitals in the Cu02 plane.

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11 The energies of the remaining La^^ 5f, apical 0^", and in-plane 0 orbitals lie far below the in-plane Cu2+ 3d and 0^" 2px, 2py energy states and are not presented here. Since the 3d^2_y2 orbital is directed at neighboring 0 2px, 2py orbitals in the Cu02 plane, crystal field effects dictate that it must have the highest energy of the five CuSd orbitals. It is followed in order by the 3d2z2_j.2, Sdxy, Mxz, and 3dyz orbitals. Cu^+ enters the matrix with a 3d^ configuration, so there is one unpaired spin per Cu^+ site. These residual spins couple to nearest-neighbor Cu^"*" sites via a superexchange interaction to give La2Cu04 3D antiferromagnetic order. Although the spin order of the Cu^+ ions is three dimensional, the superexchange interaction is dominant along the Cu-0 bonds in the Cu02 planes. While there is only a modest discrepancy in the bond distance between neighboring in-plane and out-of-plane Cu^"*" sites (1.915A vs. 2.219A), neighboring Cu^"*" sites in adjacent planes are displaced laterally by one-half a lattice spacing (see Fig. 2.1). This lateral displacement coupled with the longer Cu-0 bonds along the b-axis results in an appreciable difference between the in-plane vs. out-of-plane superexchange energies (Jy vs. 3±). Jy is roughly 1200K while is on the order of 200K.^^ Hence, it is anticipated that 3D AFM will be lost with the introduction of charge or spin impurities well before the 2D spin correlations are eliminated in the Cu02 planes. Indeed, a variety of studies^^'^^ have determined the 2D spin correlation length in the Cu02 plane to be on the order of lOOOA. From Fig. 2.1, it is seen that for stoichiometric La2Cu04 the spin directions and the antiferromagnetic propagation vectors are in the [001] and [100] directions respectively. The substitution of Sr"^"*" for La^"*" induces a small canting^^'^^ (~ 0.17°) towards the b-axis due to an antisymmetric exchange interaction originating from the orthorhombic distortion of the lattice with doping. Figure 2.3 shows the phase diagram for La2-xSra;Cu04.^^ La2Cu04 can be holedoped by substituting Ca, Sr, or Ba for La to produce a superconducting phase

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12 z A, ( Y2„ I / «y »y, xz. yr Tj »J.yz E Sphexical Cubic ( O | Tctragonil P4 ] Figure 2.2. (a) Cu 3d orbital configurations in the cuprates. (b) Crystal field splitting of the Cu 3d energy levels under D^h symmetry. From Reference 23. between 30 and 40K.^ Alternatively, a superconducting phase may be realized by altering the oxygen stoichiometry to a number greater than four^^. In the latter case electrons are added to the Cu02 plane and serve as the charge carriers in the superconducting phase. In either case, the compound undergoes three distinct phases as carriers are added to the system. For Sr concentrations between x = 0.00 and x = 0.03, La2_xSrxCu04 remains a 3D antiferromagnetic insulator. For concentrations between x = 0.02 and x = 0.07 3D antiferromagnetic order is lost and the system becomes a weakly metallic spin glass. However, as doping proceeds, starting at a; = 0.07 and continuing through x = 0.23, La2-xSra;Cu04 becomes superconducting with

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13 0.02 0.05 0.21 0.25 ^ Figure 2.3. Phase diagram for La2-a;Sra;Cu04. From Reference 23. optimal doping at x — 0.175 (Tc = 39K). For Sr concentrations in excess of x = 0.24, La2_xSriCu04 is metallic with a d.c. conductivity that scales roughly linearly with the temperature^^. This unusual temperature-dependent behavior of the d.c. resistivity in the normal state has inspired many in the condensed matter community to dubb the high Tc superconductors "strange metals". Although Fig. 2.3 is specific to La2-zSrxCu04, it is prototypical of all of the doped high Tc cuprates and requires only modest changes in the critical concentrations from one material to the next. This dissertation investigates, in part, the effects of Li-doping La2Cu04. Contrary to Sr-doping, Li is substituted for Cu in the Cu02 plane. However, like its Sr counterpart, each Li adds a single hole the Cu02 plane which resides on an oxygen site. Li"^ goes into La2Cu04 with a closed Is shell, and so introduces a spin-zero impurity in the AFM spin1/2 matrix. X-ray diffraction measurements indicate that the addition of Li to the Cu02 plane modestly suppresses the orthorhombic distor-

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14 tion of the CuOe octahedron, particularly along the Cu-0 axes in the Cu02 plane . By X ~ .30, the symmetry of the lattice increases from orthorhombic to tetragonal. Unfortunately, to date no comprehensive neutron scattering measurements have been performed on La2Cui_xLix04 to corroborate these findings or elucidate the eff"ect of the spin-zero impurity on the 2D spin order of the Cu^"*" lattice. NQR studies on La2Cui_xLix04 demonstrate that3\ much like La2-xSrxCu04, 3D AFM is lost at a; = 0.03. However, since the superexchange coupling is dominant in the Cu02 plane, this gives little information concerning the 2D spin correlations of the Cu^+ ions in the Cu02 plane. For Li concentrations between x = 0.03 and x = 0.50, La2Cui_xLix04 remains insulating with no superconducting phase, in stark contrast to what is observed in La2-xSra;Cu04. This implies that holes introduced by Li"*" ions in the Cu02 planes are tightly bound their impurity hosts. This insulating character makes La2Cui_xLix04 an exciting and relevant material to study as it eliminates contributions to the optical properties from free carriers which would otherwise dominate in the farand mid-infrared. Furthermore, by comparing the optical properties of La2-xSrxCu04 and La2Cui_xLix04 some light may be shed on the elusive nature of the superconducting mechanism in the doped cuprates. 2.2.3 Optical Conductivity: La?!Cu04 The optical properties of all the superconducting cuprates are remarkably similar despite disparities in their crystal structure and chemical composition. All are anisotropic in the optical conductivity in the doped and undoped phases, and all demonstrate the formation of a mid-infrared in the a-b plane with doping. Since trends in the optical conductivity are relatively invariant from compound to compound, we will focus only on the optical properties of La2-xSrxCu04 as it closely resembles the chemical composition of La2Cui_xLix04.

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15 One of the more striking features of the lamellar cuprates is the anisotropy of the optical conductivity in the a-b axes vs. c-axis directions in the insulating, superconducting, and metallic phases. Figures 2.4a and 2.4b show the reflectivity of single crystals of La2-xSra;Cu04 for light polarized parallel and perpendicular to the Cu02 plane^'^. There is little change in the c-axis reflectivity (Fig. 2.4b) from x — 0.00 to a: = 0.34, from the insulating to metallic regime. Only at rr = 0.34 are there qualitative changes in the reflectance at low frequencies. In this regime, the spectrum has a weak free carrier component, although the lowest energy optical phonon is not screened out entirely. This weak metallic character is consistent with d.c. transport measurements along the c-axis on single crystals of La2-xSrxCu04 in the overdoped (metallic) regime. Two major optical phonons with symmetry dominate the c-axis spectrum at 0.029eV and 0.074eV. There are actually three infrared active phonon modes, but the lowest energy phonon at .017eV has considerably less spectral weight and is effectively absorbed by the phonon centered at 0.029eV. Far more structure can be found in the a-b plane reflectivity of La2Cu04 as can be seen in Fig. 2.4a. In the far-infrared below O.lOeV, the spectrum is dominated by four infrared active modes centered at O.OlSeV, 0.045eV, 0.05eV, and 0.086eV.^^ Between O.lOeV and 1.5eV, the spectrum is featureless while between 1.5eV and 4eV, the reflection drops precipitously. The reflectivity edge centered near 2eV is attributed to a charge-transfer excitation of an electron from an oxygen 2p.,^ {'^Px,y) orbital to a copper 3dj.2_y2 orbital in the Cu02 plane. This charge-transfer transition at roughly 2eV is common to all the lamellar cuprates and its position scales in energy with the oxygen coordination number of the Cu^"*" sites^^. While the c-axis reflectivity of La2-xSrxCu04 remains relatively unchanged up to a: = 0.34, the reflectance in the a-b plane dramatically changes as the Sr concentration varies from a: = 0.00 to x = 0.34. The reflectivity and corresponding optical conductivity are characterized by the

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16 0.34 La2_j(Srj^Cu04 0.05 R T J L 0.34^ 0.05 0.1 0.2 0.5 1 2 PHOTON ENERGY (eV) 1.0 > o 0.5 0.5 l\ A 0 ,l .. ^g 0 0.1 0.2 0..; 0..I ^ Phi>li)n l;ncrgy (eV) x=0.34 La2_xSr/u04. 0.5 0 0.1 0.2 0.3 0.4 Photon Energy (eV) Figure 2.4. Reflectivity of La2-xSrxCu04 at T=300K for (a) E\\ and (b) E ± to the a-b plane. From Reference 32.

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17 growth of a mid-infrared band between O.leV and 0.6eV accompanied by the erosion of spectral weight in the charge-transfer region as can be seen in Fig. 2.4a. With only a small amount of doping {x < 0.1), the spectral weight of the charge-transfer band is sharply reduced while a reflectivity edge forms near 0.8eV. As doping increases, the reflectivity edge sharpens while the reflectivity below the edge increases rapidly, for 0.00 < X < 0.25, the edge does not shift appreciably in energy. For x > 0.25, just above the superconducting phase, the edge moves to lower energies. This is not consistent an increase in the number of carriers as the system is progressively doped. The aforementioned trends in the reflectivity of La2-xSrxCu04 are even more apparent in the optical conductivity. In Fig. 2.5, the two salient phenomena associated with Sr-doping may be readily observed. There is a shift in spectral weight from the region above 1.5eV to the region below leV. A band between O.leV and 0.6eV grows with Sr content while the charge-transfer edge erodes rapidly. Below O.leV, phonons that once dominated the spectrum in the undoped phase are now buried in a Drude tail. For x < 0.15eV the mid-infrared band peak shifts to lower energy with increasing Sr content and eventually merges with the Drude band above x = 0.2. Other superconducting cuprates such as YBa2Cu307_j and Nd2-xCea;Cu04-y mimic this behavior in the low-to-moderate doping regimes. Despite the lack of a discernible band maximum in the optimally doped regime the presence of a mid-infrared band is evident by a decay rate of the optical conductivity that is slower than the typical ^ Drude response of metals. Evidence of this nonDrude response has been found in nearly all of the copper based superconductors^^"^°. Explanations have been a matter of heated controversy. Despite this, some irrefutable facts about the unconventional nature of the optical response persist. One is that the region between 0.1 and 0.5eV shows little temperature dependence. This is concurrent with an increase in the d.c. conductivity by a factor of three from 300K to lOK.

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18 1.5 0 ; 1 1 La2_xSrxCu04 1 J9.34 |Lo.20 i._0.15 x=0 fo^oz^ — ^^^^^^^ /Mr O-io \ \ o.06 ,.' 0.34 / 0 1 2 3 4 ho) ( eV ) Figure 2.5. a-b Plane optical conductivity of La2-xSrxCu04 at T=300K. From Reference 32. Clearly, a Drude response with a single relaxation rate for the charge carriers can not explain such behavior. Also, there is a definite temperature dependence of the low frequency conductivity that is consistent with d.c. measurements^^"^^'^^'^'^. Explanations and models of the mid-infrared band vary. Some have speculated that these excitations are the photo-induced transitions of bound-holes (electrons) from their ground state to the continuum. On the other hand, since the band is centered near the antiferromagnetic coupling energy, J, other theories attribute this band to magnetic excitations associated with charge impurities hopping from site to site in the Cu02 plane. Still another model relates the excitations in the mid-infrared to polar coupling of the charge impurities to the lattice [electron(hole)phonon coupling]'*^''*'*. Recently, charged domain walls have been introduced to model the electronic and spin behavior of the doped cuprates. These issues will be addressed in the following chapters.

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19 The erosion of the charge-transfer band with impurity doping is also not well understood in these materials. Each Sr^"*" introduces one hole into the Cu02 plane. In a three-band Hubbard model, this removes one electron removal state since there is one fewer oxygen site available from which an electron may be photo-excited to a neighboring Cu^"*" site. Based on this argument, one would expect the spectral weight loss of the charge-transfer region to scale linearly with doping concentration. As readily observed in Fig. 2.5, this is not the case. The charge-transfer band erodes at a rate which rapidly exceeds x, particularly above x = 0.05. One possible explanation for this anomalous behavior in the charge-transfer region is that the excitation of an electron from an oxygen to a copper site is a spin-dependent transition. Holes or electrons added to the Cu02 plane break superexchange bonds between neighboring Cu^"*" ions and frustrate the 2D spin lattice. Cu^"*" spins near the charge impurity which were once anti parallel to their neighbors are now distorted. Since chargetransfer from oxygen site to neighboring Cu^"*" site is only possible in an anti-parallel spin channel, spin frustration may obscure or smear the charge-transfer excitation in the region surrounding a charged impurity. More will be said concerning this in Chapter 5. 2.2.4 Raman Scattering: La9!Cu04 Two-magnon Raman scattering data for the lamellar cuprates is remarkably similar. From single layer La2Cu04 to YBa2Cu307_j all have the signature two-magnon band centered between 3300 cm~^ and 2600 cm~^ associated with the charge(spin) -transfer excitations in the Cu02 plane. Common to these compounds is the rapid suppression of the 2-magnon band with impurity doping. This suggests that impurities introduced into the parent insulators not only provide charge carriers for the superconducting condensate, but also dramatically impact the 2D spin correlations.

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20 As with the optical conductivity, only Raman scattering data pertaining to materials studied in this dissertation will be discussed. Two-magnon Raman scattering via the exchange mechanism is unique to antiferromagnetic systems. It occurs when an electron with, say, spin-up located at one site is photo-excited and exchanges position with an electron of opposite spin on an adjacent site. In the antiferromagnetically ordered cuprates, this represents electrons on neighboring 3dj.2_y2 orbitals in the Cu02 plane exchanging positions and creating two spin-flips in the 2D antiferromagnetic background. Since this transition reverses the spins on two neighboring sites in the 2D Cu^+ lattice, it breaks six antiferromagnetic bonds and costs roughly 3 J (ignoring magnon-magnon interactions). This creates two magnons at the antiferromagnetic Brillouin zone boundary with equal and opposite wave vectors (momentum) to preserve momentum conservation. Two-magnon Raman scattering provides a measure of the local 2D spin order of antiferromagnetic systems and, when properly analyzed, can yield information about the magnon scattering length of the spin lattice. Two-magnon Raman scattering in the lamellar cuprates is sensitive to the presence of doped carriers. In Fig. 2.6, the spurious effects of Sr-doping La2Cu04 are readily apparent. Even for Sr concentrations below x = 0.1 the two-magnon Raman band broadens rapidly and shifts to lower energy with increasing Sr content^^. In the optimally doped regime (0.1 < a; < 0.25), the two magnon peak intensity is flattened into a broad spectral background. While it is difficult to extract an accurate estimate of the magnon scattering length from the Raman data in the optimally-to-overdoped regime {x > 0.1), some conclusions about the spin order in the optimally doped materials can be reached. In this regime, the two-magnon band is obscure, merging into a featureless continuum. This implies that the 2D in-plane magnon scattering length, which in the undoped regime is on the order of 25-30A, has been shaved down to

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21 ' I— 1 — I — " — i — 1 — ' — > — — ' — — ' 0 IPQQ 2000 3000 A/OOO Raman flhjft (cm') Figure 2.6. 2-lVIagnon Raman scattering peak in La2-xSra;Cu04 as a function of Sr content at T=300K. From Reference 19. one-to-two lattice spacings. Doped holes residing on the oxygen 2pa orbitals severely weaken the superexchange coupling constant J turning the 2D antiferromagnetic lattice into a quasi spinglass. This suggests that the holes are weakly bound to the Sr^"*" sites and are essentially free to hop from site to site. However, it is difficult to ascertain precise estimates of the 2D spin correlation length from two-magnon Raman scattering. Neutron scattering proves a more useful tool for probing local and long-range spin order. 2.3.1 Lattice Structure: Sr9Cu09Cl9 Sr2Cu02Cl2 is a parent insulator of the high Tc cuprates. X-ray and neutron scattering studies'*^'^^ have demonstrated that Sr2Cu02Cl2 is a layered perovskite with tetragonal (I/mmm) K2NiF4 structure and is isostructural to the high temperature form of La2Cu04. In composition, however, Sr2Cu02Cl2 is dissimilar to its superconducting counterparts. Sr2Cu02Cl2 consists of basal Cu02 planes intercollated with Sr-Cl layers as shown in Fig. 2.7a. Instead of a single buffering layer, there

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22 are now two Sr-Cl planes partitioning the Cu02 planes. Each copper site is flanked by four oxygens in the Cu02 plane and by two chlorines along the c-axis forming a CUO2CI2 octahedron. Unlike La2Cu04, there are no apical oxygens in the buff"ering layer between adjacent CUO2 planes. Neutron scattering measurements by Vakin et al.^"^ determined the Cu-0 interatomic distance to be 1.985A and estimated the CuCl spacing to be 2.8A. The presence of two chlorines, each translated one-half lattice spacing with respect to other, increases the c-axis parameter by 18% in Sr2Cu02Cl2 with respect to La2Cu04. The c-axis measures 15.618A while the aand baxes parameters measure 3.975A, making the Wigner-Seitz unit cell considerably larger than in La2Cu04. Sr2Cu02Cl2, unlike its lanthanum-based counterpart, is stoichiometric as grown and no orthorhombic distortion of the crystal lattice hcis been observed down to lOK. 2.3.2 Electronic Configuration and Magnetism: Sr^jCuO^Cl?! Sr2Cu02Cl2 is a 3D antiferromagnetic insulator with a Neel temperature between 250 and 310K. As with the superconducting lamellar cuprates, the anisotropic superexchange interaction between neighboring copper sites binds the spins. Sr, Cu, 0, and CI incorporate into the Sr2Cu02Cl2 matrix with -1-2, -1-2, -2, and -1 valences respectively. And, as with its superconducting cousins, the highest energy electronic bands in Sr2Cu02Cl2 lie in the Cu02 plane (see Fig. 2.2a with electrons in the Cu 3rfj.2_y2 orbital having the greatest energy. As with La2Cu04, Cu+2 goes into Sr2Cu02Cl2 with a 3(^ shell configuration. The 3d^22_r2, 3dxy, Sdyz, and 3dxz shells are filled while the 3dj.2_y2 orbital is half-filled, giving each Cu'^"'" site a net spin of 1/2 (see Fig. 2a). These spins couple to neighboring Cu^"*" sites antiferromagnetically to form a spin-1/2 antiferromagetic insulator. Since neighboring Cu^"^ sites are closer in the basal Cu02 plane than in adjacent planes, the superexchange constant, J, is

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23 MAGNETIC STRUCTURE Figure 2.7. (a) Crystal and (b) spin structure of Sr2Cu02Cl2. From Reference 47. highly anisotropic with the in-plane value, Jy, many orders of magnitude larger than its out-of-plane counterpart, J±. Hence, while Sr2Cu02Cl2 is a 3D spin-1/2 insulator below ~ 300K, the spin system and its excitations are often treated as 2D. Magnetic suseptibility and Raman measurements (see Chapter 7) estimate Jy at ~ 860K. The magnetic structure is similar to that found in La2Cu04. The spins are aligned on the [110] direction and are perpendicular to the antiferromagnetic propagation vector, as shown in Fig. 2.7h^^ However, the absence of an orthorhombic distortion down to lOK precludes the spins from canting towards the c-axis and forming a weak bond. The fact that Sr2Cu02Cl2 remains tetragonal raises the question as to how the Cu02 planes couple to give 3D antiferromagnetic order. Recall that in La2Cu04 and La2Ni04 it is believed that the orthorhombic distortion of the crystal lattice drives

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24 the anti symmetric interplane coupling between the spins'^^. In tetragonal symmetry, these antisymmetric spin interactions sum to zero. Some have speculated that, in the absence of this interplane superexchange, spins in neighboring planes couple via magnetic dipole-dipole interactions'*''. Preliminary calculations based on this speculation estimate the the 2D spin correlation length in the CuOa plane to ~ 3000A at T^vThis model is buttressed by the fact that La2Cu04 has a spin correlation length of roughly the same order of magnitude (~ lOOOA at Tjv)However, the nature of the interplane binding mechanism of the spins in Sr2Cu02Cl2 is still contentious. To date, comparatively little work has been done on non-stoichiometric compositions of Sr2Cu02Cl2. As previously mentioned, no superconducting phase has been reported in Sr2Cu02Cl2. This is not surprising since divalent strontium has replaced trivalent lanthanum, and chlorine has replaced oxygen in the rock salt layer buffering the Cu02 planes. The high electron affinity of chlorine traps charge carriers introduced, for example, by substituting Sr^"*" /or La"*"^ and precludes a superconducting phase. This same instability also makes doping Sr2Cu02Cl2 prohibitively difficult in most cases. One of the few impurities to be successfully doped into Sr2Cu02Cl2 is cobalt. Co^"*" replaces Cu^"*" and so preserves the valency of all the elements in the matrix. Thus, the introduction of cobalt does not add charge carriers to the Cu02 plane. However, each should incorporate into the Cu02 plane with a 3(f shell configuration. Referring to Fig. 2.2b, crystal field splitting dictates that the 3dyz and 3dxz orbitals are filled while the 3dxy, 3d^22_j.2, and 3dj.2_y2 orbitals are half-filled. Hund's rule coupling is believed to bind the unpaired spins ferromagnetically in a high spin state (S=3/2). Thus, each Co^+ ion introduced into the Cu02 plane is a spin-3/2 impurity embedded in a spin-1/2 3D antiferromagnetic background. This dissertation will examine the effects of Co-doping in Sr2Cu02Cl2 on such interactions

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25 and parameters as the charge-transfer gap, the superexchange coupUng constant, and the phonon-magnon coupling strength in the Cu(Co)02 plane. 2.3.3 Reflectivity and Optical Conductivity: Sr^CuO^Cb Measurements on Sr2Cu02Cl2 have not been pursued as aggressively as they have been on its superconducting brethren. As mentioned in the previous section, no superconducting phase has been found in Sr2Cu02Cl2, and doping carriers into this system has proven difficult. Nonetheless, Sr2Cu02Cl2 is nearly isostructural to La2Cu04 and, as are the other superconducting cuprates in the undoped regime, is a 3D antiferromagnetic insulator. These similarities to its superconducting cousins make Sr2Cu02Cl2 a relevant material to study. Figure 2.8 shows the reflectivity, transmittance, and corresponding optical conductivity of Sr2Cu02Cl2 for light polarized parallel to the Cu02 planes'*^. Below .leV (1000 cm~^), the spectra are dominated by four infrared optical phonons in much the same way as La2Cu04. Between 1000 and 10000 cm~\ a new excitation emerges just below 3000 cm~^ with two weaker sidebands at ~ 4200 cm"^ and ~ 6000 cm~^. These excitations were not evident in the optical conductivity of La2Cu04 presented earlier. However, this is not due to fundamental differences in the electronic structure, rather it is related to the enhanced sensitivity of transmission measurements. In fact, these same excitations have been observed in many of the cuprates (La2Cu04, Pr2Cu04, Nd2Cu04, PrBa2Cu307_j). Unfortunately, most cuprate single crystals as grown are too opaque and too thick for transmission studies. To obtain transmission data on La2Cu04, for example, it is necessary to polish the sample down to a thickness no greater than 10 microns. Nonetheless, transmission measurements have been performed on the aforementioned cuprates^^'^^. While the precise energy positions and relative weights of these mid-infrared bands vary from material to material.

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26 Photon Energy (eV) 0.01 0.1 1 T • SrzCuOzClj 100 1000 10000 Frequency (cm"') Figure 2.8. Transmission, reflection, and optical conductivity of Sr2Cu02Cl2 at T=300K. From Reference 49. the spectra are otherwise identical. Lorenzana and Sawatzky^^ proposed that these relatively weak bands observed in the mid-infrared are phonon-assisted magnon pair excitations. It is now widely held that this interaction is responsible for the strongest band centered at 2900 cm~^ However, the nature of the higher energy sidebands is still a contentious issue at present. More will be said about this in Chapter 5. Above 10000 cm"^, the optical conductivity is dominated by a charge-transfer absorption (O 2pa;_y-Cu 3d^2_y2 charge transition) centered just below 2eV. The optical conductivity in this region has roughly the same magnitude as that found in other lamellar cuprates suggesting that there is little variation in the dielectric con-

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27 stant from cuprate to cuprate. The dominance of the charge-transfer absorption is manifest by the fact that it is roughly one order of magnitude larger than the phonon conductivities and three orders of magnitude greater than the absorption in the midinfrared. Though not presented here, the charge-transfer edge sharpens and moves to slightly higher energy when the temperature is lowered'*^. Falck et al. proposed that this weak redistribution of spectral weight is associated with the temperature dependence of phonons dipole-coupled to the charge-transfer excitation^^. This will be discussed in Chapters 5 and 6. 2.3.4 Raman Scattering: Sr^CuO^Cl^ Since Sr2Cu02Cl2 is an antiferromagnetic insulator, 2 magnon Raman absorption should be evident in the mid-infrared. Figure 2.9 shows the 2-magnon Raman shift for incident light polarized in the Cu02 plane for several cuprates courtesy of Tokura et a/.^^. As will be discussed in greater detail in Chapter 4, two-magnon Raman scattering in a 2D AF ordered material occurs when an electron on spin sublattice A is photo-excited to an adjacent spin sublattice B. To avoid double occupancy, the electron originally on sublattice B hops back onto sublattice A, thereby introducing a spin-flip in the antiferromagnetic background. The incident light backscatters with Bi symmetry and breaks six antiferromagetic bonds which, neglecting magnon-magnon interactions, should cost 3J||. Effects of interplanar coupling are small and are not taken into account here. When magnon-magnon interactions are taken into account, this value drops to 2.7J. As can be seen from Fig. 2.9, there is little change in the 2-magnon energy and only modest discrepancies in the lifetime (width of the two-magnon Raman shifted peak). The two-magnon peak for Sr2Cu02Cl2 is located at 2890 cm~^ The estimated Jy from this is 1070 cm~\ comparable to what is observed in La2Cu04. The relative invariance of the two-magnon Raman shift in the cuprates is not surprising. The Cu02 planes in the cuprates listed in Fig. 2.9 are

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28 nearly identical. Differences in the Cu-0 spacing are less than 1%. Discrepancies in the charge-transfer energy are modest with Act ranging from 2.0 to 1.5eV. Since J scales as + with tpd proportional to the Cu d^2_y2-0 2px,y overlap, only modest variation is expected from cuprate to cuprate. As discussed in the previous section, no charge impurities have been successfully doped into Sr2Cu02Cl2 to date. Consequently, two-magnon Raman data for carrier-doped Sr2Cu02Cl2 is not available. However, magnetic impurities have been successfully introduced into the Cu02 plane. This dissertation will examine the effects of substituting cobalt for copper on the optical and two-magnon Raman scattering properties of Sr2Cu02Cl2. Cobalt is believed to go into the Cu02 plane with a 3(f shell configuration in a high-spin state and so would introduce a spin-3/2 impurity in the spin1/2 antiferromagnetic background. Hence, while little change is expected in the conductivity, it is very possible that significant changes may be observed in the two-magnon Raman scattering. More concerning this will be discussed in Chapters 6 and 7.

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29 -1 r 1 1 1 1 1 300K 0 2000 4000 6000 8000 RAMAN SHIFT (crn*) Figure 2.9. 2-Magnon Raman scattering data (Bi mode) of assorted cuprate host insulators. From Reference 34.

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CHAPTER 3 THEORY 3.1.1 Models for Carriers in the CuO?. Planes: Normal State The superconducting mechanism of the high Tc cuprates is not understood at this time. Many contend that understanding the normal state properties, ostensibly less enigmatic, will provide knowledge of the pairing mechanism of the carriers in dopedcuprates. Unfortunately, the wisest and seemingly simplest course of action is seldom easy. The normal state properties of the copper based high Tc superconductors are unusual and, to date, have not been entirely accounted for with existing models. Most outstanding of the normal state properties is the linear temperature dependence of the resistivity above Tc This behavior is not characteristic of a Fermi liquid where p ^ T^, nor is it akin to metals where p is less temperature sensitive. For this reason, the superconducting cuprates are often referred to as "strange-metals" in the normal state. Other unusual properties of the normal state are a temperature dependent Hall coefficient, proximity of superconductivity to a magnetic phase, and unusual behavior of the optical conductivity with charge-doping. Most theoretical models for the normal-state properties start with the three-band Hubbard model. The following sections will review this model and discuss its limiting form in the strong coupling limit, the so-called one-band Hubbard model. Results from numerical calculations will then be presented for the optical conductivity. This will be followed by a discussion of small polarons and the ZhangRice singlet, two models which have been proposed to explain the mid-infrared band in the high Tc cuprates and nicklates. Lastly, phononassisted multi-magnon absorption will be discussed in modest detail. 30

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31 3.1.2 Three-Band Hubbard Model The starting point for understanding the normal state properties of the cuprates is constructing a Hamiltonian to describe the motion of carriers in the Cu02 plane. This two dimensional approach is justified since conduction anisotropy favors carrier transport in the Cu02 plane. If we naively begin by charge-counting in La2Cu04, for example, we find that lanthanum goes into the system as a +3 ion, while copper goes in with +2 and oxygen with -2 valency, respectively. Thus, La^"*" and O'^" go into the matrix with closed shells while Cu'^"'' enters with an unpaired electron in the 3d shell. Crystal field splitting removes the degeneracy of the copper 3d orbitals and places the unpaired electron in the 3c?j.2_y2 orbital. Each unpaired electron, in turn, is antiferromagnetically coupled to its neighbor. This leaves the system with one hole per tetragonal unit cell. Hence, one would expect this system to be metallic with a half-filled conduction band in the undoped phase. However, measurements have determined that La2Cu04, as well as the other high Tc cuprates, is insulating. The missing piece is, of course, correlation of the unpaired electrons. For a system to conduct, charge carriers must hop from site to site in the Cu02 plane. There are two avenues to this end in La2Cu04. One is that an electron on an oxygen site hops to a neighboring partially filled 3dj.2_y2 band, leaving a hole on the oxygen site. This process costs A (charge-transfer energy) and generates a mobile hole in the oxygen band. The second possibility is that an unpaired electron on a Zdj.2_y2 site, or in the lower Hubbard band (LHB), promotes directly to a nearest neighboring Mj.2_y2 site, or to the upper Hubbard band (UHB) at energy cost U (Mott-Hubbard insulating limit). This leaves a mobile hole in the lower Hubbard band. A good starting point for mapping the lowest energetic excitations is the threeband picture diagrammed in Fig. 3.1. Here, two possible electronic configurations are shown. On the left side of Fig. 3.1, the Cu 3d® band lies above the O 2px,y

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32 band and the Fermi energy is located at the top of the former. The least energetic excitation of the system would therefore be the transfer of an electron from the band to the 3d^° states at cost U, the electron-electron repulsion on the Cu sites. In this case the upper and lower Hubbard bands (UHB and LHB) are the 3d}^ and 3(f bands respectively. This is the Mott-Hubbard limit for the three-band picture. Mott-Hubbard behavior characterizes the early transition metal-oxides such as V2O3, Ti203, and Cr203. Contrasting to Mott-Hubbard insulators are chargetransfer systems. The charge-transfer limit is depicted on the right side of Fig. 3.1. Here, the O 2p band lies above the Cu 3(fi so the lowest lying electronic excitations would be the transfer of electrons from the 02px,y band (LHB) to the Cu 3d^° band (UHB). Here, the Fermi energy lies at the top of the O 2p band. Charge-transfer behavior is typically found in the late transition metal-oxides such as CuO, La2Cu04 and Sr2Cu02Cl2 , the latter two being the parent insulating systems studied for this dissertation. In the charge-transfer limit the ground state of the system lies in the lower Hubbard band where the highest lying electrons are unpaired in the 3dj.2_y2 orbitals. The lowest lying excitation would therefor be the transfer of an electron from the O 2px,y orbital to a neighboring partially filled Cu 3dj.2_y2 orbital, denoted by A in Fig. 3.1. This has been corroborated in optical studies of La2Cu04 where a charge-transfer excitation has been observed in the optical conductivity centered near 2.0-2.25eV32'53. Having successfully worked out the dynamics of the unpaired electrons in the undoped phase, our next task is to construct a Hamiltonian to describe the motion of holes introduced by doping. This has been accomplished with a 2-dimensional tight binding model introduced by Emery et al.^^'^^ and Varma et al.^^. This model, contrary to the previous discussion of carriers in the undoped phase, introduces a hybridization parameter, tp^, to allow for the hopping of charge carriers from the O

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33 Fermi Level Fermi Level Mott-Hubbard Charge-Transfer Insulator Insulator U < A A
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34 The first term represents hybridization or hopping between nearest neighboring copper and oxygen sites in the Cu02 plane. The pj are fermionic operators that destroy holes at the oxygen site labeled j, while the dj are corresponding annihilation operators at the Cu site i. (ij) appearing in the first sum indicates that the sum is only over nearest Cu-0 pairs. Note that only on-site and nearest neighbor interactions are accounted for. Interactions at larger distances, to first order, are screened by the electronic background. For > 0 and in the undoped regime, Eq. 3.1 suggests that there will be one hole on every Cu site. In the limit that Ud > ep ej, any additional holes introduced by doping should predominately reside on the O 2p orbitals. This has been verified by numerous experiments. Band structure^*^ and cluster calculations^^ have estimated ep — 3.6eV, Ud ~ 8-lleV, and Up ~ 4eV. The values of tpd, Upd, and tpp are in the neighborhood of 0.5-2eV. The relatively large value of Ud, in comparison to the other terms, suggests that the strong to intermediate coupling limit is appropriate to describe the physics of the charge carriers in the high temperature superconductors. It is unclear that a model as cumbersome as Eq. 3.1 is necessary to correctly account for the low energy physics of the high Tc superconductors materials. To simplify matters, Anderson^^ proposed that it may be possible to reduce Eq. 3.1 to a one-band Hubbard model while still retaining an effective theory. Here, the Hamiltonian is defined as where d]^^ are fermionic operators that create electrons at site i with spin a. In this picture, the 02p orbitals have been effectively absorbed into the Cu sites. So site 3.1.3 t-J Model 3.2 {ij)

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35 g(E) U»t \ LHB UHB t»U U E Figure 3.2. One band limit for insulators. i refers to a plaquette consisting of a central Cu site surrounded by four oxygens. As before, i is a hybridization parameter between neighboring Cu-0 pairs. The parameter U is the on-site Coulomb repulsion of two holes on the same plaquette. Figure 3.2 shows a schematic for this truncated model. Note that now there are only two bands: an upper and lower Hubbard band (UHB and LHB). For t >U, the LHB and UHB overlap and the system becomes metallic. In the strong coupling limit {U ^ t), the one-band Hubbard model may be reduced to the so called t—J model, first derived from the Hubbard model by canonical transformation by Hirsch^^ and Gros et al.^^ As before, this is a single-band model, but now the state of the doped hole is represented only by the spin of the Cu site (plaquette) on which it resides. That is, the fermion operators in Eq. 3.1 which create and destroy holes on the oxygen and copper sites are replaced by boson spin operators which couple neighboring copper spin sites. The spin is zero on a given site if a single doped hole resides there, and is either spin-up or spin-down if a doped hole is not present. In the strong coupling limit, the Hamiltonian is

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36 H = J Y,{Si Sj ^Tiirij )-tY^ [c^Cj^a + h.c] 3.3 where J is the antiferromagnetic coupling between nearest Cu-Cu neighbors {ij) and is defined as J=— . 3.4 U This expression is valid in the limit that J <^ t or t <^ U. The Si are spin-1/2 operators and the ct^ create electrons with spin a on site i. Hence, the electrons move in a 2-dimensional lattice with hopping amplitude, t, constrained such that there are no doubly occupied sites {U oo). It is unclear as to whether or not the three-band Hubbard and reduced t J models will lead to the same low energy physics on a temperature scale near Tc . 3.1.4 Cluster Calculations As discussed in the previous section, the one-band Hubbard and t — J models are the starting points in many calculations concerning the dynamics of carriers in the Cu02 planes. Such calculations include the response of the carriers to electromagnetic fields. Unfortunately, these models lack analytic solutions in the strong coupling limit. Consequently, numerical techniques on finite clusters have been employed to extract physical parameters. Here, there will be a brief summary of the numerical studies relating to the optical conductivity, cri, generated in the context of the one-band Hubbard and t — J models. The conductivity tensor is obtained from linear response theory. It is used to relate the current density operator, jx{Q,'^), to the electric field vector, Ex{q,u}), that induced it. Here q and to are the wave vector and energy of the electric field vector respectively. In the limit oi q ^ 0, jx is given by jx(0, uj) = axxEx{0, Lj) 3.5

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37 where Uxx is the complex optical conductivity at zero temperature. The real part of Cxx is given by the Green's function Reaxx = — /m[(0o | Jxtt ft-Jx I M3.6 TTu; H — tQ — ijj — le Here, H is the total Hamiltonian with eigenenergy £'0, a; is the frequency, and e is a small number that moves the poles of the Green's function into the complex plane. In the Hubbard model, the current operator jx in the x direction at zero wave vector is given by, jx = it ^{c]aCi+x,G h.c), 3.7 l,a where and c are fermion creation and destruction operators, / labels the sites onto which the carriers may hop, and / + x labels the site displaced by one lattice spacing from /. Numerical solutions for the one-band Hubbard model on a 4 x 4 cluster performed by Dagotto^^ employing the Lanczos^^ technique extracted the relative strengths of t, U, and J. This was done for hole concentrations between 0.000 (halffilled system) and 0.375. The results for the optical conductivity are presented in Fig. 3.3. At half filling, x = 0.000, there is an accumulation of spectral weight around 6t. Using previous estimates^^ of i this band can be correlated with the charge-transfer gap observed near 00 ~ 2eV. In the context of the one-band Hubbard model, these are excitations from the lower to upper Hubbard band. As doping proceeds, there is marked shift in spectral weight from the charge-transfer band to lower energies. By a; = 0.125 (two holes on the 4x4 lattice), two distinct features emerge below the charge transfer gap. The first is a Drude or free carrier response centered at a; = 0. The second is an accumulation of spectral weight near 0.3-0.4eV. This has been associated^^'^^'^^ with the mid-infrared band observed in optical conductivity of the doped cuprates. For increased doping concentrations, the Drude component

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38 grows considerably while the mid-infrared spectral weight appears to increase only modestly. However, appearances may be deceptive. The modest growth of the midinfrared band with doping relative to the free carrier band centered at a; ~ 0 may in fact be due to strong overlap with the more heavily weighted Drude component which overshadows it. The growth rate of the mid-infrared band as a function of doping is still a contentious issue. Dagotto et al.^^ contend that the results of Fig. 3.3 indicate that the appropriate coupling limit for the cuprates is the intermediate coupling regime, U ^ St. In the strong coupling limit, a gap develops between the chargetransfer and mid-infrared bands, whereas for small to intermediate coupling the two bands merge, making separation difficult. Similar calculations using the t — J model, or in the strong coupling limit, have been performed by Stephan and Horsh^^ for different values of J and their results are presented in Fig. 3.4. As can be seen, the Drude and mid-infrared bands are clearly discernible and there is qualitative agreement with one-band Hubbard model. Similar results for cases near half-filling were also obtained by Chen and Schiittler®^ using the one-band Hubbard model in the strong coupling limit. 3.2.1 Models for the Midinfrared Band As discussed in the previous section, there is a shift in the experimentally observed spectral weight from high to low energy in the high Tc superconductors cuprates as the systems are doped with charge carriers. In particular, a mid-infrared band and Drude tail grow with increasing doping concentrations accompanied by a sharp loss of spectral weight out to 3-4eV in the charge-transfer band. Numerical solutions employing the one-band and t — J models have modeled this behavior, but the precise mechanism responsible for the bound excitations in the mid-infrared is still elusive. To simply chalk them up to intraband excitations in the lower Hubbard band does

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39 0.8 Figure 3.4. Cluster calculations for the optical conductivity under the t — J model. From Reference 65.

PAGE 49

40 not explain the essential physics of the bound excitations. The fact that the midinfrared band is centered near 0.5eV, or roughly 3 J, has led some to suggest that the excitations are magnetic. As a hole is photo-excited to a neighboring site, it creates a spin-flip in the antiferromagnetic background and breaks three antiferromagnetic bonds. Others contend that the band is the result of the charge impurities coupling to the crystal lattice. In this scenario, small polaron hopping is the mechanism responsible for the mid-infrared band. Still, others have proposed that novel charge structures in the Cu02 planes such as domain walls are driving the bound excitations in the mid-infrared. These models, and some observations concerning the anomalous loss of spectral weight in the charge-transfer band, will now be discussed. The small polaron model was first introduced by Reik^^ in the 1960's to explain the mid-infrared band found in semiconducting lanthanum cobaltite, La2Co03. Small polaron hopping was later revisited by Bi et al.^^ nearly thirty years later to account for the mid-infrared band observed in La2-2rSra;Ni04. The basic premise is that charge carriers introduced by doping distort the crystal lattice and become selftrapped (bound) in a potential well, thus creating a polaron. The polarons then hop from site to site in the lattice when photoexcited. Starting from Kubo's expression for the a.c. conductivity^^ given by an expression for the complex optical conductivity is found. Here, Z is the partition sum, C is an integration variable in the complex time plane, e is a small real frequency 3.2.2 Small Polaron Model a{uj) = Tr dC J dXexp[{iu)+e)C] j exp[^^{C-ih\)] j exp[^-^{C-ih[X-P])],

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41 introduced to ensure convergence of the integral, A is an inverse-temperature integration variable, /3 — 1/kT, and j is the current density operator, similar to that defined in Eq. 3.7. H is the Hamiltonian of the system and is the sum of three parts: H = He + Hph + He^ph3.10 He and Hph are the Hamiltonians of the charge carriers and phonons (lattice distortions), respectively, while i/g.p/i couples the doped charges to the lattice (i.e.,chargephonon coupling term). It is the latter term, H^^ph, that generates excitations in the mid-infrared. He is modeled after the one-band Hubbard model and is given by, He = Xl(^o ~ /^)4cs + ^ts,i{clcs+i + 3.11 s s,i Here, s labels the sites where the charge carriers sit and i labels all sites onto which the charge may hop. In the same vein as the one-band Hubbard model discussed previously, ts,i is the electronic resonance integral and (eo — /x) is the on-site energy of the doped carriers. Only singly occupied sites are considered in this model which is reasonable when one considers that the energy cost of two charge impurities (electrons or holes) residing on the same site is roughly 8-9eV in the cuprates. As before, cj and Cs are fermion creation and destruction operators, respectively. The phonon Hamiltonian is given in the usual way: /fp;, = ^/ia;A(g)6|;,6,-A3.12 q,X q is the phonon wave vector, u!x{q) the phonon energy, and A the phonon branch index. Here, fet^ and b^x are boson operators that create and destroy phonons in

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42 branch A with wave vector q, respectively. The final and most important term, the charge-phonon interaction, has the structure where Rg is the position vector of site s and a{q) is the phonon coupling constant. After inserting Eqs. 3.11-3.13 into Eq. 3.9 and integrating, the following expression for the real part of the optical conductivity is found in the limit that the temperature is greater than one quarter of the Debye temperature^*^: Re a{u,T) = a{0,T) [i + (,,A)2]i/4 ' ^'^^ where r{w) is a frequency dependent function given by, r{w) = 2{ujTA)-Hog{uTA + [l + {ujTA)^]^/^}-2{ujTA)'^{[1 + {ujtA)^]^^^ -1}. 3.15 Here, r and A are defined as, ^ ,• 2 sinhilhuJoP) and A = 2ujQT 3.17 where uq is the average phonon energy to which the charge carriers couple and r/ is number of phonons associated with each polaron, or the number of phonons in the polaron cloud. As usual, /5 = The do part of the conductivity is given by the expression a{0,T) = 2Ne'^J^a^pTnh^~exp{-T]tanh{^huop)} 3.18 where a is the lattice constant, J is the bare electronic resonance integral (hopping integral between neighboring sites), and is the number of doped carriers. Despite

PAGE 52

43 the complicated form of Eq. 3.15, the frequency and temperature dependence of Rea{uj,T) is well behaved. This will be discussed further in Chapter 5 when the mid-infrared band observed in La2Cui_xLix04 is discussed in the context of small polaron theory. Note the sensitivity of Rea{u),T) to both temperature and rj. This will be important in Chapter 5 when the temperature dependence of the mid-infrared band is discussed for La2Lij;Cui_x04. The average phonon frequency to which the charges couple is a thorny issue. Eklund et al.^^ proposed that the doped carriers couple to the bending modes of La2Ni04 in the Ni02 planes centered around 50meV. Using this energy for (Jq and setting T] ~ 10, they closely modeled Eq. 3.15 to the mid-infrared band observed in the optical conductivity at 300K. However, it is not clear that the doped carriers couple solely to the bending modes in the basal planes. It seems reasonable that the stretching modes of Ni(Cu)-0 pairs in the basal plane, centered near 90meV, may be activated by the presence of charge impurities. In this case, perhaps the best approach would be to consider weighted contributions from both modes. More concerning this model will be discussed in Chapter 5 when the mid-infrared band is investigated for La2Lia;Cui_j;04. 3.2.3 Magnetic Excitations: ZhangRice Singlets Other theories have cast an eye on a magnetic origin for the mid-infrared band in the doped cuprates. As mentioned before, it has been established that holes introduced by doping reside predominately on the oxygen sites in the Cu02 plane. Zhang and Rice^^ proposed each of these holes is strongly bound on a square of O 2p sites to a central Cu^"*" ion and forms a singlet. This singlet then moves through the lattice of Cu2+ ions in a similar way to a hole in the single-band effective Hamiltonian. It is important to note that the Zhang-Rice singlet differs from an ordinary two-site

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44 singlet in that it is formed by a central Cu^"*" ion and four neighboring oxygen sites. As will be shown, it is the phase coherence of the four O 2p orbitals that leads to the the anomalously large binding energy of the singlet holes. The starting point for this model is the three-band Hamiltonian constructed in the hole representation: H = Y1 'd^U^'^ + E 'ppIp^+ ^ E 4t^it4i^4 + ^' 3.19 »,ff l,a i In Eq. 3.19, Zhang and Rice have defined the vacuum as the filled Cu 3d^^ and O 2p^ states. The operators d]^ create Cu 'idj.2_y2 holes at site z, and p]^ create 0 2px,y holes at site /. and Cp are the on-site energies of holes on the Cu and O sites respectively. The last term, H' , represents hybridization of neighboring Cu-0 pairs and is given by^^ //' = ^^F,,4aP/a + /l.C. 3.20 where the sum over / runs over the four O sites around a given Cu site i. The hybridization matrix V^; is proportional to the wave function overlap of the Cu and O holes. At this point it is crucial to take the phase of the hole wave functions into account. This is accomplished by writing the hybridization matrix as^^ Vu = {-1)^'Hq 3.21 where to is the amplitude of the hybridization and / = 2 if / = i — |f or i — and Mi^i = 1 if I = i + or i + ^y. For to
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45 J = 3.22 where e = — €p (charge-transfer energy). When holes are introduced, they have predominately 0 2p character. However, each can lower its energy by hybridizing with a central Cu+2 site and by spreading out over the attendant square plaquette of four oxygens. Thus, the wave function of the O 2p holes should be constructed from a combination of a four oxygen sites surrounding a central Cu^"*" ion. Doing so, a given set of four O 2p orbitals may form either a symmetric or antisymmetric state with respect to the central Cu ion^^: where -{+) corresponds to the S{A) state, and the phase of the p— and d— state wavefunctions is defined in Fig. 3.5.^° Both S and A may combine with the d-wave Cu hole to form either a singletor triplet-spin state. Using Eq. 3.19 as the unperturbed Hamiltonian of the system with i/' = 0, it is found to second order in perturbation theory that the energies of the singlet and triplet states for S are —8{ti + t2) and fiQ t'^ 0, respectively, where = and t2 = 777-'^) while A has energy —4ti. The large tp (u ep) binding energy of the S singlet state is due to the phase coherence of the O 2p and Cu 3d hole orbitals. It is instructive to compare this energy to an O hole sitting at a fixed site /. In the latter case, the binding energy of a singlet combination of an oxygen hole and neighboring Cu hole is only -2{ti + <2), one-fourth that of the square S state. Since the energy separation of the S singlet and A states is much larger than ti and t2, the antisymmetric states are projected out of the lowest lying energy states of the system. Also, the energy of two holes residing on the same square is -(6
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46 from states with two holes bound to the same Cu site are neglected for the lowest lying excitations. The localized states of Eq. 3.23 are, however, not orthogonal since neighboring squares share a common O site. This problem is circumvented by constructing a set of Wannier functions with a method similar to that used by Anderson to treat isolated spin quasiparticles^^ . The wavefunctions for the oxygen holes are now given by^^ ia = N'^^^ Yl Pka^Mik Ri), 3.24 it Pka = ^s"^^k E • R^). 3.25 t where is a normalizing factor /5jt = [l-^(cosA;x + cosA;j^)]-^/2^ 3.26 and Ns is the number of sites in the system. The functions i(^ are orthogonal and complete in the symmetric 0-hole subspace. i^dii^4>iidi^) 3.27 with energies in second order perturbation theory E± = Y.\ ^"^t \H'\w) |2 /A^^, 3.28 where w runs over all possible intermediate states, and iS^Eyj is the Oth-order energy difference between (j)icr and w. Summing over all possible states numerically, it is found that^9 E± = -8(1 T A^)* 3.29

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47 Figure 3.5. Orbital configuration for Zhang-Rice singlets. From Reference 69. 15 q e" 10 7 5 o I 5 o 9 I 10^ 10^ 10'' (cm"') Figure 3.6. Numerical results for the optical conductivity of phonon mediated Zhang-Rice singlets^°. cq = E{Cu3d^2_y2) E(02p^), V is the Cu3dj.2_y2-2pff hopping integral, Fg and Tp = phenomenological natural linewidths, is the average phonon energy, and 7is the phonon-electron coupling strength. From Reference 70. T I T I I I I ll| 1 1 — I I I I I I [ J I I r ] I 1 1 1 I I

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48 where A = AT^i^^ji 0.96. 3 3Q k Since E+ E~ Ibt > t, transitions between {ip~} and {tp^} are ignored and the system may be treated on the singlet {i^^} subspace. The singlet-bound holes may then dipole-couple to radiation and be photo-excited to more energetic linear combinations of 02px^y and Cu3dj.2_y2 orbitals in the singlet subspace. Once again, it is the phase coherence of the Cu M and O 2p hole states that produces the large energy separation between the singlet and triplet states. Prior to the work of Zhang and Rice^^, the importance of phase coherence went unrecognized save for the work of Hirsch'^^ who considered the S combination of O states in the case of fixed spin direction on the Cu site. Rice et al7^ modeled the optical conductivity in the midinfrared with a similar construct based on the phase coherence of the oxygen orbitals. However, in this model charge modulation via coupling to the phonon modes drove the bound hole states and so introduced an additional band in the far-infrared. Their results are presented in Fig. 3.6. ' 3.2.4 Domain Walls One of the more novel models introduced in recent years to explain some of the magnetic properties of the cuprates and nicklates is the formation of domain walls or stripes. In this picture, holes (electrons) doped into the Cu02 plane align linearly in a particular direction, forming parallel domain walls. If the walls are filled (one hole per site along the wall), the antiferromagnetic order parameter reverses direction across the wall, dividing the Cu02 plane into regions with sequentially alternating spin-order parameters. Domain walls in the [1,0] and [1,1] directions are shown in Fig. 3.7P At first glance, there is no obvious reason why the charge should spontaneously arrange in linear formations and, in fact, it would seem that the Coulomb energy of two

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49 electrons or holes residing on neighboring sites would preclude this. However, there is mounting evidence for the existence of stripes in both the cuprates and nicklates, at least for specific doping concentrations. Tranquada et al. reported a striped phase in both Sr-doped La2Ni04 and Lai.6-iSrxNd.4Cu04 found in magnetic suseptibiUty measurements^^''^^ . However, in the case of Lai.6_xSrxNd.4Cu04, it is believed that it is the excess Nd that pins the doped charge and leads to the formation of domain walls. To date, no static stripes have been observed in La2-xSrxCu04 for x > .03. Hammel et al. have also proposed that stripe formation may be responsible for the anomalous behavior of the ^^^La resonance splitting observed in NQR measurements of La2Lij;Cui_x04^^ below 30K. Numerical calculations''^ of the correlation function for two or more doped holes in a spin-1/2 antiferromagnetic background have pointed toward the formation of stripes for intermediate to strong J. Figures 3.8 and 3.9 show the 2-hole correlation function for two and four holes in a spin-1/2 antiferromagnetic background, respectively, as a function of the superexchange coupling. This was done for the t — J model and at zero temperature but it should be kept in mind that the holes actually reside on the neighboring plaquette of four oxygens sites around a given copper site. Here, r is the spacing between holes measured in lattice spacings and N is the number of sites in the lattice. For the case of two holes, the charge will form pairs along [1,1] direction for J/t > 0.25. For J/t > 0.4, pair formation along the [1,0] direction becomes more stable with respect to separated holes. Note, however, that the correlation function for filled-walls along the [1,1] direction is the most robust for all values of t/ J. Care must be taken when interpreting the four-hole correlation function. The arrangement as well as the mean correlation length must now be accounted to correctly interpret the charge configuration. Figures 3.10a and 3.10b shows the four-hole correlation function, G, for four different configurations for N = 18 and N — 20. The sites are

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50 (a) o t 1 f o O 1 1 t i o O f j t o O 1 f 1 o t I t (b) O f { f O M I f M O } f I O M O I f I O f f O M I O M Figure 3.7. Domain wall structures (stripes) in the a) [1,0] and b) [1,1] directions. From Reference 73. numbered in Fig. 3.11. For N = 18 (Fig. 3.9a), the configurations are: (a) (3 5 9 12), the holes being as far apart as possible, (b) (14 6 12), the holes forming two separate pairs with intrapair distance \/2, (c) (5 6 7 8) with holes forming a stripe along the [1,0] direction, and (d) (1 4 6 9) with three holes forming a domain wall along [1,0] direction. In Fig. 3.9a it can be seen that configuration (d), a stripe along the [1,1] direction, dominates for intermediate coupling, 0.3 < J/t < 1.2, while for strong coupling configuration (c), a domain wall in the [1,0] direction, wins out. For N = 20, the results for the four hole correlation function are similar to the N = 18 case and the

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51 Figure 3.8. 2-Hole correlation function for a) N = 20 and b) N=26 sites. From Reference 76. Figure 3.9. 4-Hole correlation function for a) N = 20 and b) N=26 sites. From Reference 76.

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52 hole configurations are listed as follows: (a) (3 7 13 17) representing two separated hole pairs, (b) (2 8 13 19), with holes forming a line along the [1,1] direction, (c) (3 8 12 17), with holes along the [1,0] line, and (d), a staggered domain wall stretching over the system with periodic boundary conditions. As before, configuration (b), a stripe in the [1,1] direction, dominates for intermediate coupling 0.4 < J/t < 1. For J/t > 1, configuration (d), a staggered domain wall, dominates and is surprisingly robust even in the intermediate coupling regime. It is curious to note that this configuration is more stable than (c), a stripe in the [1,0] direction, for all coupling strengths J/t above roughly 0.2. Zaanen et alJ^ also demonstrated the stability of partially filled and completely filled domain walls in [1,0] and [1,1] directions using a semiclassical approach. Of course, it should be kept in mind that these 2and 4-hole correlation functions do not include potentially important contributions from electron-phonon coupling which could change the direction and/or energetics of the stripes in key ways. Having realized that striped phases are instabilities of doped Mott-Hubbard insulators, our next task is to make an estimation of the binding energy of the charge on the domain wall. This was done by Nayak et alP treating the spin-1/2 background in the strong coupling limit and using the t — J model. The energy per hole, if they are localized to walls at filling fraction /, is^^ Et _ SXJ tsin2TTf' where /' = 1 — / is the hole filling fraction and A ~ 3/4. To find energy minimum, Eq. 3.31 is redefined as , , , r sinx h(x) = 3.32 X X where x = 2%/' and r = 3TrXJ/2t and x is in the interval 0 < x < tt. For very small r, the strong coupling limit, the minimum occurs at a; ~ (3r)^/^, a nearly filled wall;

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Figure 3.11. 4-Hole correlation function key for a) N N=20 sites. From Reference 76. = 18 and b)

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54 for r = 1, the intermediate to strong coupling limit, it occurs at x = 7r/2, a "minimal domain wall"; and for r > tt the minimum occurs at x = tt, an empty wall. It should be noted that Eqs. 3.31 and 3.32 are far from rigorous and are only intended to give an approximate energy of a hole bound to a domain wall. The antiferromagnetic background is treated as static, while it should properly be regarded as composed of dynamical electrons on the same footing as those on the domain walls. Despite these limitations, Eqs. 3.31 and 3.32 are reasonable first approximations for the energies of holes bound to domain walls in a spin1/2 antiferromagnetic background. This will be revisited in Chapter 5 when stripe formation in La2Cui_a;Lix04 is discussed. 3.3.1 The Charge-Transfer Band Heretofore, little has been said concerning the doping dependence of the chargetransfer band save that there is a rapid loss of spectral weight in this region as the high Tc systems are doped with carriers. While this phenomenon has been readily observed, it has not been well explained. That there should be some loss of spectral weight in this region as the antiferromagnetic background is hole and/or electron-doped is expected as states in the upper Hubbard band are removed. The shift in spectral weight from the charge-transfer to mid-infrared spectral regions can be qualitatively understood by considering the band structure of charge-transfer systems, shown in Fig. 3.12 for a Cu-0 chain. Here, N on the left-hand side of the figure represents the band energy of electrons on singly occupied Cu 3d^2_y2 sites while A'^ on the righthand side gives the band energy of two electrons residing on the same Cu 3dj.2_y2 orbital, the upper Hubbard band*^*. 2N, located in the middle, represents the filled O 2pff bands and the Fermi level is centered between this and the upper Hubbard

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55 Charge Transfer : Insulating U A PES IPES Figure 3.12. Schematic diagram of charge-transfer systems. From Reference band. There are two doping scenarios to consider: One for electrons and the other for holes. For charge-transfer systems, each is unique. For hole-doping, electrons are removed from the O 2pa sites and the Fermi level is pushed down into the oxygen band. Neglecting hybridization between the copper and oxygen sites, introducing a hole will leave one electron-addition state near the Fermi level while the upper Hubbard band retains A'^ states. Each hole creates only one electron-addition state near the Fermi energy since any electrons which hop onto a single-occupied 02p site must do so with the proper spin. Hence, intraband (midinfrared) excitations in the oxygen band should scale linearly with x, the concentration of holes. This situation resembles a semiconductor where the spectral weight of an impurity band is expected to scale as the doping concentration. Although the number of electron-addition states in the upper Hubbard band (filled Cu3d shell) for a chargetransfer excitation is not affected by hole-doping since the holes reside in the oxygen 78.

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56 band, for each hole added there is one less electron available for charge-transfer. Thus, to a first approximation, one would expect the spectral weight of the charge-transfer band to decrease as x. Similar trends in the charge-transfer region are found in electron-doped systems, but fundamental differences exist in the MIR intraband excitations. Electrons added to the system must reside on the previously singly-occupied Cu 3dj.2_y2 sites, which push the Fermi level into the upper Hubbard band. Contrary to hole-doping, each electron added creates two electron-removal states in the upper Hubbard band since there are now two electrons on the doped sites available to hop to neighboring Cu sites, provided of course the excitations are consistent with the Pauli exclusion principle. Hence, the spectral weight of the intraband excitations, now located in the upper Hubbard band, is expected to scale as 2x. The situation for the charge-transfer excitations is much the same as with hole-doping, but the reasoning is reversed. For every electron added, one electron-addition state is removed in the upper Hubbard band while the oxygen band remains unaffected. Now there is one fewer state available in the upper Hubbard band into which an electron may hop. Thus, as with holedoping, the spectral weight of the charge-transfer region is expected to decrease as X. Of course, the above discussion is only qualitative and does not include treatment of important details. For example, it is vital to include the effects of hybridization between neighboring copper and oxygen sites if a quantitative comparison is to be done with experimental results. Eskes et al7^ performed cluster calculations for the integrated spectral weight below the charge-transfer gap of a doped chain of Cu-0 cells as a function of electron/hole doping concentrations and of Cu-0 hybridization, JpdTheir results are presented in Fig. 3.13 for a cluster of four unit cells. For tpd = 0 00, the behavior mimics the previous discussion with the spectral weight

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57 3.0 2.0 a-* O ^ 1.0 UN A 2N-1 : 1 A p 0.0 1.0 -T — I — I — I — I — r^0.5 0.0 hole doping electron doping Figure 3.13. Integrated MIR spectral weight below 1 eV as a function of electron and hole concentration away from half-filling. From Reference 78. growing twice as fast with electron-doping compared to hole doping. However, as tpd is increased to 1.5eV (a value not inconsistent with other estimations) there is little distinction between the effects of electron and hole doping below concentrations of 0.5, a doping level well above the superconducting phase or the onset of the metallic phase in the high % cuprates. Most of the optical studies to date on the high Tc cuprates have been done on optimally doped, underdoped, and modestly overdoped samples and little aisymmetry has been reported in the mid-infrared band between electron and hole-doped systems. Unfortunately, this does not close the book on the shift in spectral weight from high to low energy observed in the high Tc materials. While the above analysis is consistent with the observed growth of the mid-infrared band with doping, it does not adequately explain the anomalous loss of spectral weight in the charge-transfer

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58 region. From Fig. 2.5, it is seen that for hole concentrations in excess of 0.02, the loss of spectral weight in the charge-transfer region scales more rapidly than x contrary to what is naively expected. The situation is much the same with electron-doped systems such as Nd2-xCexCu04_y. It is curious to note that the hole concentration at which the spectral weight of the charge-transfer begins to degrade more rapidly than x roughly coincides with the concentration at which 3D antiferromagnetic order is lost. This raises speculation that perhaps the charge-transfer excitations, in addition to being dependent on the number of electron-removal/addition states available, are also dependent on the spin-order of the Cu^"*" lattice. This possibility and the discrepancy between theoretical expectation and experimental reality in the charge-transfer region will be discussed in more detail in Chapter 5. 3.5.1 Phonon-Assisted Multi-Magnon Absorption As discussed in subsequent sections, the introduction of charge carriers to the Cu02 plane of the layered cuprates induces a broad, heavily weighted band in the mid-infrared. However, sharper, more discreet excitations are also found in the same spectral region in the undoped phase of the cuprate systems. The absorption coefficients of a few these materials are presented in Fig. 3.14. It is obvious upon inspection that these excitations are roughly three orders of magnitude weaker than the bound excitations associated with the introduction of charge carriers in the optimally doped regime (~ 1-2 f2~^cm~^ vs. 500-800 f2~^cm~^). Consequently, it is not possible to observe these excitations in reflection mode. They can be observed in transmission studies of thin samples or in 2-magnon Raman scattering measurements. In the latter case, however, a phonon is not needed to make direct two-magnon coupling weakly dipole allowed. The spectrum consists of three absorptions at ~ 2900, 4000, 6000 cm~^ overlayed on background which grows linearly up to the charge-transfer

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59 edge. The first is distinct and sharp while the latter two are broader and weaker. The mechanism responsible for these excitations has been a contentious issue but it is now generally accepted that the lowest energy band is due to phonon-assisted two-magnon absorption. The suggestion that magnons play a role in these excitations is motivated by the fact that the bands are associated with the AF phase and are centered at roughly 3J, 4 J, and 6 J, the energies needed to flip two adjacent spins and four spins on a plaquette and in a row in the AF background. Nonetheless, the debate still rages concerning the origin of the two higher absorption features. However, when the pressure dependence of these excitations in Sr2Cu02Cl2 is discussed in Chapter 6, it will become apparent that the absorptions near 4000 and 6000 cm~^ are most likely associated with phonon-assisted four magnon excitations. Lorenzana and Sawatzky*^^ introduced the phonon-assisted multimagnon model in 1994 to explain the weak midinfrared excitations observed on the undoped phase of the superconducting cuprates. A brief synopsis of the model is now presented with the intent to simply motivate the theory behind it. For a detailed account of the model, consult Reference 79 and the references therein. Assuming that the mid-infrared excitations in the parent insulators are associated with the creation of magnons in the Cu02 plane, a problem immediately surfaces: IR absorption of magnons is not allowed in the tetragonal structure of the cuprate materials. This is due to the presence of a center of inversion in the unit cell which inhibits any asymmetric displacement of charge (and thus spin) and, hence, quenches the dipole moment. However, this restriction is lifted if symmetry-breaking phonons are nearby. The presence of a phonon effectively lowers the symmetry of the surrounding local environment and, consequently, allows magnon absorption processes. Although a similar theory was introduced by Mizuno and Koide*° decades earlier to explain magnetically related IR absorption features in NiO, Lorenzana and Sawatzky were

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60 Photon Energy ( eV ) Figure 3.14. Phonon-assisted multi-magnon absorption for various cuprate systems at T=10K. From Reference 23. the first to explicitly calculate the coupling constant for phonon-assisted absorption of light by multimagnon excitations and the line shape for two-magnon absorption. The motivation for this model stems from the large value of J in the cuprates. The basic idea is that phonon-induced modulations of the Cu-0 lattice parameter will lower the symmetry of the lattice and lead to large changes in the magnon-phonon coupling energy. The starting point for this model of the charge and spin dynamics of the Cu02 plane is the three-band Peierls-Hubbard model in the presence of an electric field {E) associated with an incident photon. For simplicity, the Cu atoms are kept fixed and the O ions are allowed to move with displacements u^j^^j^. Here i labels Cu sites and 8 — x,y, so that x-\-6/2 labels O sites. 5 is a, unitless displacement vector linking neighboring Cu^"*" sites. Holes have an on-site (hole-hole) interaction

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61 Ud on Cu and Up on O, a Cu-0 repulsion Upd, on-site energies on Cu Ed, and on 0 Ep, and Cu-0 hopping t. For later convenience, the parameters A = Ep Ed — Upd and e = 2{Ep — Ed) + Up are introduced. When an 0 ion moves in the direction of a Cu with displacement ] u \ via dipole-coupling of the incident radiation to lattice, the corresponding on-site energy of a charge sitting on a Cu site changes to first order by \ u \, and the corresponding Cu-0 hopping of said charge by a | u |. Opposite signs apply when the O moves in the opposite direction. The coupling constants of light with the one-phonon-multimagnon processes are found from a perturbative expansion of the photon-phonon-magnon system valid when t <^ A,e, Ud and when the phonon field and the electric field vary slowly with respect to typical gap frequencies''^'^^ : ^ = E •^(^' {^r+5})^r+^72 + Hph-EPph. 3.33 Is Here, B^^^j^ = S-^S^_^^ and are the fermionic spin operators. Hp}^ is the phonon Hamiltonian containing spring constants and masses for the O ions, Pph is the phonon dipole moment, and J is the superexchange energy, or the energy difference between the singlet and triplet states of two spins located on adjacent Cu sites (Cu/^ and Cul in Fig. 3.15). For light incident on this CU2O7 configuration, only three polarization directions of E (A,B,C in Fig. 3.15) need be considered. Expanding J to first order in E and u^^^^^ yields^^'^^ : J = Jo + r){uL ur) E[qiUo + XqAi'^UQ ui ur)\. 3.34 A is a polarization dependent parameter, equaling 1 for configuration A and 0 for configurations B and C. In each configuration the displacements of the central 0

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62 and the electric field are parallel, ur and ul are only relevant in configuration A with Ur = -URi + UR2 + urz and ul = un + ml2 ml3 (see Fig. 3.15). The first term in Eq. 3.33 is the superexchange in the absence of electric and phonon fields, Jq = (4t^/A^)[l/J7d + 2/e]. The second is the magnon-phonon coupling constant, = {-4t^ / A^)/3[A~^{1/Ud + 2/e) + associated with local modulations in J due to the presence a phonon. The last term gives the effective charges, or dipole moments, associated with the one-phonon and multimagnon processes*^^: 2, 2, where Opd is the Cu-0 distance. The dipole moment of the magnon-phonon system is obtained from P = Using Eq. 3.33 this yields up to fourth order in t, oE P — Plph P\ph+Tnag3.37 The first term describes conventional phonon absorption. However, the second term in Eq. 3.37 gives rise to a new dipole moment associated with the phonon modulation of the charge and spin on the Cu sites. By defining ^B^^^^^ = B-t^^j^ — {B-^_^^i^, its Fourier transform 5B^^, and the Fourier transform of u^^^^j^, m^, the one-phonon and multimagnon processes for an in-plane field in the x direction has the form^^: Px,iph+mag = iV[gi J] <5Bjt.f . + A4g^ J] sin(^) sin(^)5S?tzy 3.38 p,6 p,S with A = 1. A'' is the number of unit cells, ^ is the the Fourier-transformed xp displacement vector in the x direction of the oxygen atom at site 1 + 6/2 . For the

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63 case of an electric field polarized perpendicular to the Cu02 plane, A is set to zero and is replaced by u^-. The first term in Eq. 3.38 is isotropic in the polarization and is a spin-dependent correction to the charge on the central 0 (call it Oq) in Fig. 3.15. This is illustrated in Fig. 3.16a where the electric field, polarized parallel to the yaxis, couples to a bending mode and lowers the Madelung potential around the Oq site. The reduction of local potential induces the spins (charges) on Cxxr and Cu^, to both hop to the central oxygen. However, this is a spin-dependent process since both charges are forbidden by the Pauli exclusion principle from transferring to the same site unless they have opposite spins. This argument also holds for incident light polarized perpendicular to the Cu02 plane. The second term in the dipole moment is anisotropic and is present only for light polarized in the Cu02 plane. It is referred to as a "charged-phonon-like" effect. This contribution to the dipole moment can be understood by considering the configuration in which the electric field and the displacement of Oq are both parallel to the Cu^-Cuj^ bond direction and a phonon mode in which the O's around the Cu^ breathe in and the O's around the Cu/j breath out (A in Fig. 3.15). In this breathing mode, the Madelung potential in Cur decreases while in Cu^ it increases, creating a displacement of charge from left to right that contributes to the dipole moment. Figure 3.16b illustrates such a process. The real part of the optical conductivity is given by the dipole-moment-dipolemoment correlation function. By decoupling the phonon system from the magnetic system which is valid in the lowest order magnon-phonon coupling, the real part of the optical conductivity is given by''^'^^ {h = 1), 16A^g^sm^(pj;/2)sm^(py) + [4XqAsin'^{pa:/2) gi]^ (((5B^-;<55|))]. 3.39

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64 ABC Figure 3.15. Ion displacement scheme used for calculations. The full dots represent the Cu's and the open dots O's. Thick arrows represent the spin, thin arrows the lattice displacements, and the thin long arrows represent the direction of the electric field. From Lorenzana and Sawatzky. From Reference 79. wy^ is associated with the in-plane Cu-0 stretching mode while uj±^ is linked to the Cu-0 bending mode phonons. Assuming weak absorption, the absorption coefficient is obtained from the expression a = -^7^. The magnon-magnon Green functions, {{5B^g]5B^)), are then computed using interacting spin wave^^ theory with a Holstein-Primakoff transformation. The results of Lorenzana and Sawatzky are presented in Fig. 3.17. The dashed-dotted line is the theoretical line shape from the full expression in Eq. 3.38, whereas the dashed line, which yields a superior fit to the data, is found by assuming that the anisotropic processes dominate and by setting = 0 in Eq. 3.38. Note that the bimagnon disperses around the saddle point (tt.O). At first glance, this would seem to violate momentum conservation, but it should be remembered that it is the total momentum of the system, magnons plus phonons, which must be conserved

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65 o o o E 6 a 9 oCUl -o-o E b 6 Figure 3.16. Typical processes contributing to the (a) isotropic and (b) effective charges. From Lorenzana and Sawatzky. From Reference 79. (assuming that the incident photon has zero momentum). Within this model, the sidebands at higher energy found in the experimental data are assigned to higher multimagnon processes. Specifically, the bands observed at roughly 4000 cm~^ and 6000 cm~^ are assigned to 4-magnon + phonon excitations in a plaquette and row, respectively. The Green function calculation shown in Fig. 3.17 matched computations from the exact diagonalization of a small cluster^^. The exact diagonalization calculations produce the higher sidebands at 4000 cm~^ and 5000 cm~\ but the relative weights are substantially smaller than what is experimentally observed. Thus, the origin of the higher sidebands is still a contentious issue. Nonetheless, it is generally accepted at this time that the strong resonant absorption centered near 2900 cm ~^ is due to phonon-assisted 2-magnon excitations.

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66 0.2 0.4 0.6 0.8 1.0 1.2 Photon Energy [eV] Figure 3.17. Experimental data (solid line) and theoretical line shape for two-magnon absorption (dashed line) in La2Cu04. The dash-dotted is the contribution to the line shape from the bimagnon at p = (7r,0). The insert shows the density of states of the bimagnon from different values of the total momentum. From Reference 79.

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CHAPTER 4 EXPERIMENTAL TECHNIQUES 4.1.1 Introduction Optical experiments measure the transmittance and/or the reflectance of samples as a function of incident light frequency. Unfortunately, when this is done over a broad frequency range, as was done for materials studied for this thesis, one must use multiple spectrometers, light sources, and detectors. This chapter discusses experimental techniques used in this work. This first section describes the Fourier transform spectroscopy technique employed to cover the far and mid-infrared spectral regions. This is followed by a section on the Bruker 113v fast-scan Fourier spectrometer. Following this, the Perkin-Elmer monochrometer which covered frequencies between 1000 and 40000 cm^^ is discussed. The last experimental section discusses the Raman scattering technique employed to study two-magnon excitations in antiferromagnetic materials. A discussion of the detectors, polarizers, sample mounting, and data analysis is also presented. Finally, a brief section is devoted to sample manufacture. 4.2.1 Fourier Transform Infrared Spectroscopy Fourier transform infrared spectroscopy was developed to overcome the inaccessibility of the far-infrared region to single-grating spectrometry. Grating spectrometers fail in the far-infrared due to the reduced power available from light sources at low frequencies. This is readily seen by simply ratioing the power emission spectrum of a blackbody to that of a light source which obeys the Raleigh-Jeans Law. For any source, the total blackbody power spectrum is given by Po = (^T^A 67 (4.1) 1

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68 where A and T are the area and temperature of the source, respectively, and a is the Stefan-Boltzmann constant. For a typical mercury arc lamp, the radiated power available from zero to a frequency u obeys the RaleighJeans Law, n-) = Mr^A-'^ (4-2) where and c are the Boltzmann's constant and speed of light respectively. The ratio of the emitted to the total power up to frequency uj is then given by, For ui = 100 cm~^ and T = 5,000K r] is only 1.2 x 10~^. Such reductions in power make optical grating spectroscopy techniques impractical. Fourier transform spectroscopy overcomes the aforementioned energy deficiencies with the use of a Michelson interferometer. Figure 4.1 shows a generic diagram of such an interferometer. The Michelson interferometer works on the principal of linear superposition of light waves and has the advantage that it measures an entire spectrum (typically sections of the farand midinfrared) in a single sweep of the movable mirror seen in Fig. 4.1. Light emitted from a source strikes a beam splitter which (ideally) transmits half the light and elastically backscatters the remaining half. The transmitted light is forwarded to the moveable mirror M2 while the reflected light is directed to the stationary mirror labeled Mi. After reflecting from their respective mirrors, the two light beams recombine at the beam-splitter. If both mirrors are equidistant from the beam-splitter, then the optical paths for the reflected and transmitted light are identical and the two beams add constructively when they recombine at the beam-splitter: The intensity of the recombined beams will be a maximum. However, if M2 moves away from this optimal position by an amount 6,

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69 M 1 7^ A V F ocusing Lenses V Fixed Mirror Beamsplitter Detector I M. Movoble Mirror D Figure 4.1. Generic schematic of Michelson interferometer. then the two beams will be out of phase by a factor 9 = 27t5i' and the intensity of the sum of the two beams at the beam-splitter will no longer be a maximum. Here w is the frequency of the light measured in cm~^ . If the beam-splitter transmits half the light while reflecting the remainder (no absorption losses), then the sum of the amplitudes of the light beams reaching the beam-splitter is A{u) = a{u){l + e'^"""^) (4.4) Multiplying this quantity by its complex conjugate yields the intensity at the beam splitter as a function of frequency and path difference, 5:

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70 (4.5) where S{u) = 4>l(i/) and is referred to as the spectral density of the source. Integrating the intensity from // = 0 to i/ = oo gives the interferogram I (5), Hence, if the moving mirror is scanned over an infinite range of path differences, 6, and I{6) is recorded at each 6, the spectrum can be extracted by performing the integral in Eq. 4.7. From the standpoint of signal strength, little is gained by using interferometry in the far-infrared: it is the quality and type of detector that determines the strength of the signal. The chief advantage of using interferometric techniques to measure in the far-infrared is that many measurements can be performed in a relatively short amount of time. Each complete scan (half-oscillation of the moveable mirror) measures a given frequency region and takes about 1 second. This allows many measurements to be taken (typically 256-512 scans) in a relatively short span of time which reduces the signal to noise ratio. This is referred to as the Fellgate advantage. In reality, of course, an interferogram cannot be measured over infinite path differences and this limits the resolution of the measured spectrum. By varying beam splitters and the maximum path difference, 5max, an array of regions can be scanned in the far and mid infrared. However, since I{5) is a discreet function in practice, this changes the Fourier integral in Eq. 4.7 to a sum. Also, the maximum path difference, 7max, introduces side-lobes near sharp features in 5(f). Fortunately, this effect is mitigated by apodization and aliasing. (4.6). u can be determined by taking the Fourier transform of 1(5), (4.7)

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71 • Two •
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72 while a glowbar covers the mid-infrared. Each source is paired with a separate detector: a He-cooled detector for the far-infrared and a DTGS detector which operates at room temperature for the mid-infrared. There are also two channels for measurements. The first is for transmission measurements and the second is for reflection studies. This is necessitated by a reflection stage which must be placed in the sample chamber to achieve near-normal incidence of the incident light on the sample/mirror. Having two chambers precludes the need for drastic realignment when switching from reflection to transmission modes and vice versa. In the interferometer chamber, the optical dynamics are similar to those found in the Michelson interferometer, Fig. 4.1. Light emitted from a source focuses on a beam splitter which transmits and reflects half the light intensity. Both beams, after reflecting from stationary mirrors on opposite sides of the chamber, are sent to a two-sided moveable mirror which reflects both beams back to be recombined at the beam-splitter. The combined beams are then forwarded to the sample chamber (either transmission or reflection) where they reflect from or transmit through the sample, mirror, or aperture at near normal incidence. After this, the interfering beams are directed to the detector chamber where the signal is measured. It is important to note that the entire system is evacuated to a pressure of roughly 50mTorr to avoid absorption and dispersion of the light by water and CO2 present in the air. The two-sided mirror moves at a constant speed v so the optical path difference is Auvt, where t is the time measured starting from 5 = 0. Hence, the detector receives a modulated signal at the detector as a function of the light frequency and velocity of the mirror, D{t) = Docos2nfat. (4.8) Here, /„ = Avvt where u is the light frequency. The signal is then amplified and .\ digitized before being sent to an Aspect computer. At this point, the signal is apodized 1 i i 1 1

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73 and phase corrected. Finally, the signal is Fourier transformed over the frequency to produce the interferogram and subsequently inverse Fourier transformed over the path difference, 5, to yield the spectrum. Table 4.1 lists the parameters and settings used in FIR and MIR measurements on the Bruker 11 3v fast-scan interferometer. Table 4.1. Bruker FTIR Operating Parameters Range Beam Splitter Opt. Filt. Source Pol. Detector (cm-l) Material Material Material 20-90 Mylar Black PE Hg Arc 1 Bolometer 80-400 Mylar Black PE Hg Arc 1 Bolometer 100-600 Mylar Black PE Hg Arc 1 Bolometer 450-4,000 Germanium on KBr None Glowbar 2 DTGS PE = polyethylene. Polarizer 1 = wire grid on oriented polyethylene; Polarizer 2 — wire grid on AgBr. For cryogenic measurements, the sample/mirror/aperture was attached to the tip of a cryostat and inserted into the samples chamber (see section on low temperature measurements and sample mounting). A polyethylene window in the far-infrared and a KCl window in the mid-infrared sealed the cryostat vacuum and allowed incident light to transmit through the cryostat. For mid-infrared studies (800-4000 cm~^), a DTGS (pyroelectric deuterated triglycine sulfate) detector was used to measure signal while in the far-infrared a bolometric (temperature sensitive) detector was employed. While the bolometer is equipped with a channel for the mid-infrared and provides superior signal-to-noise ratio than the DTGS detector, measurements consistently showed that the DTGS was less erratic in the mid-infrared. The reason for this is

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74 DEWAR, MODEL HD-3 OUTUKE SKETCH Figure 4.3. Bolometer detector used with Bruker Fast Scan Interferometer. still unclear. The inferior signal-to-noise ratio of the DTGS was compensated by simply increasing the number of scans (typically to 512). This yielded clean results up to 4000 cm~^ The bolometer will be discussed in more detail in the next section. 4.2.3 Bolometer Detector The intensity spectrum of sources in the far-infrared is poor (see the discussion under Fourier Transform Infrared Spectroscopy). While interferometric techniques mitigate many of the difficulties associated with weak signal, generally more needs to be done to increase the signal to noise ratio. This is accomplished with the use of bolometric detectors. These detectors operate at liquid helium temperatures ( 1-2K) and so eliminate the thermal noise associated with room temperature detectors. For

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75 Figure 4.4. Diagram of Perkin-Elmer spectrometer system used for measuring transmission and reflection of samples. example, the intensity spectrum of a blackbody at 300K is centered at 1000 cm~^, near the center of the infrared regime. A diagram of the bolometer used in this study is presented in Fig. 4.3. The detector element is Si-based and detects the signal by bolometric means. That is, the bolometer detects the intensity of the incoming signal by measuring changes in temperature of the Si element as the light strikes it. Changes in the temperature are amplified and recorded as voltage signals which are then sent to the Aspect computer for processing. 4.3.1 The Perkin-Elmer Monochromator For frequencies above roughly 1000 cm~\ a Perkin-Elmer Monochrameter was used for reflection and transmission measurements. In this frequency regime, the aforementioned difficulties encountered in the far-infrared are not at issue and the simpler, more traditional technique of grating spectroscopy may be employed. For this

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76 reason, a modified Perkin-Elmer 16U monochromator was used for studies between the near infrared and UV regions (1000 cm~^40000 cm~^). A diagram of this spectrometer is presented in Fig. 4.4. The system uses three sources to cover the entire spectrum: a glowbar for the mid-infrared (1000-4200 cm~^), a tungsten lamp for the near infrared and visible (3800-21000 cm-^), and a deuterium source for the UV (17000-40000 cm"^). The optical dynamics of the PE system are straightforward (see Fig. 4.4). After the radiation is emitted from a given source, it is modulated, or chopped, by a rotating motor blade before passing through a set of high and low optical filters to select a particular frequency region. Next, it is forwarded to a grating where diflFerent wavelengths are diffracted according to the formula 2d sin ^ = nA, where n is the n*'* order of the diffracted light, A is the the wavelength, d is the angle of incidence, and d is the spacing between grating lines. For measurements in this study, n = 1. A variety of gratings with different line spacings were used to cover the entire frequency region between 1000 and 40000 cm~^ These are listed in Table 4.2. Next, the light backscatters and exits the monochrometer through slits before being focused on the sample/mirror located at . For transmittance measurements, a mirror is placed at ^ and the sample and reference (aperture) is placed at ^. The refiected or transmitted light is finally focused onto a detector at the bottom left-hand corner of Fig. 4.4. The modulated detector output is fed into a lock-in amplifier which then amplifies the ac signal. A LED diode and photodetector located on opposite sides of the chopper supply the reference signal for the lock-in amplifier. The output of the lock-in is fed to a digital voltmeter remotely controlled by a PC computer. The PC also controls the stepping motor for the gratings and optical filters. Data are collected by recording the single-beam spectrum (frequency vs. signal) for the sample and reference sequentially and then by ratioing the two spectra to obtain the true

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77 Table 4.2 Perkin-Elmer Grating Monochromator Parameters Frequency Grating Slit Width Source'' Detector*" (cm-i) (line/mm) (micron) 801-965 101 2000 GB TC 905-1,458 101 1200 GB TC 1,403-1,752 101 1200 GB TC 1,644-2,612 240 1200 GB TC 2,467-4,191 240 1200 GB TC 4,015-5,105 590 1200 GB TC 4,793-7,977 590 1200 W TC 3,829-5,105 590 225 W Pbs 4,793-7,822 590 75 W Pbs 7,511-10,234 590 75 W Pbs 9,191-13,545 1200 175 W Pbs 12,904-20,144 1200 225 W Pbs 17,033-24,924 1200 225 w 576 22,066-28,059 2400 700 D2 576 25,706-37,964 2400 700 D2 576 36,386-45,333 2400 700 D2 576 " G: Globar; W: Tungsten Lamp; D2: Deuterium Lamp. ^ TC: Thermocouple; Pbs: Lead Sufite; 576: Silicon photocell. reflectance or transmission of the sample. During mccisurement, as with the Bruker 113v, the spectrometer chamber is pumped down to a pressure below 200mTorr to eliminate CO2 and water absorption. For cryogenic measurements, the samples were

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78 attached to the tip of a Hansons and Associates cryostat and inserted into the PE chamber at R (reflection) and T (transmission). This cryostat and details of the sample mounting procedure will be discussed in the next section. A KCl window in the mid-infrared and a quartz window in the near infrared through UV regions permitted transmission of the incident light through the cryostat. 4.4.1 Sample Mounting and Low Temperature Measurements Sample mounts for reflection measurements are shown in Fig. 4.5. The mount is simply a rectangular piece of copper with a conical section removed in the center surrounding a circular aperture. Samples are attached to the back side of these mounts (on the side opposite the conical wedge) with varnish or ordinary paper glue. Aperture sizes varied from sample to sample but were typically 1.5-2. 5mm in diameter. The conical section surrounding the central aperture where the sample/mirror is placed serves to scatter residual light from the source which strikes the sample mount and should not contribute to the detector signal. Removal of unwanted stray light in this way was also enhanced by painting the conical section with non-glossy black paint. Transmission sample mounts are similar to their reflection counterparts. For these measurements, the conical section is removed in Fig. 4.5 (A = B) and the rectangular piece is thinner. The sample mounts were then attached to a sample holder assembly that, in turn, was mounted to the cryostat. All sample mounts and holder assemblies are interchangeable between the Bruker 113v and Perkin-Elmer Grating Spectrometer for easy transfer of samples from one instrument to the other. Low temperature measurements were achieved by attaching the sample holder assembly to the tip of a Hansen and Associates High-Tran cryostat. A flexible transfer line flowed liquid helium from a storage dewar to the cryostat. The temperature of the sample was regulated and stabilized by a Scientific Instruments 5500 temperature

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79 A = Outer Radius B = Inner Radius t = Thickness Front Side Figure 4.5. Diagram of sample mounts used for reflection and transmission studies. For reflection scans, t ~ 3-4mm and B < 1/2A. For transmission, t ~ 1mm and A = B. controller connected to 25 Watt Menko Heater coils and a calibrated LI diode sensor (thermometer) at the tip of the cryostat. This combination of helium flow and heating of the cryostat tip permitted stable temperature measurements between 10-370K. For cold measurements, the sample holder and cryostat unit were placed inside a shroud equipped with optical windows. The shroud/cryostat system was then evacuated to a pressure below 10~^ Torr to prevent ice formation on the optical window and sample. Since the sample and mirror were on opposite sides of the sample holder in reflection mode, measurements of the sample and mirror were possible by simply rotating the cryostat assembly by 180°. For transmission studies, the sample and aperture were mounted at 90°angles, so sequential measurements of the sample and aperture were possible by a 90° rotation of the cryostat. The two spectra were then ratioed to obtain the true reflectance/transmittance.

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80 4.5.1 Data Analysis of the Spectra: The Kramers-Kronig Transformations With the exception of Sr2Cui_xCoj;02Cl2, all optical measurements were restricted to single bounce reflectance. By performing a Kramers-Kronig analysis of the full reflectance spectrum, a wealth of information about the optical properties of a given material can be extracted. For normal incidence, the reflectivity amplitude is given by, {n-l) + in , ^ = (4-9) {n+ 1) + IK where n and k are the real and imaginary parts of the complex refractive index. The reflectance power (measured reflectance), ^{oo) is given by the multiplying Eq. 4.9 by its complex conjugate. Thus, r(u} ) can be written as the product of a real amplitude and a phase factor: r{oj) = p{uj) exp (4.10) Here, 0 is the phase shift in the reflected light relative to the incident phase and p{u)) is related to the reflectance power, ^{u), by the expression: ^{co) = ip{oj))\ (4.11) The amplitude, p{uj), and the phase shift, 6(a;) are, respectively, the real and imaginary parts of ln[r{u))], and thus can be related by means of a Kramers-Kronig transformation: ew) = I r Jo Lip— Up' -dJ. 4.12 Unfortunately, this transformation requires that the full frequency spectrum be measured. This of course is not practical. In reality, ^(w) is measured over a finite frequency region, in this case from 20-40000 cm~\ and then merged with existing

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81 data above 40000 cm~^ and up to 380000 cm~^ Below 20cm~^ the data is extrapolated according to the d.c. and low frequency transport properties of the material under study (metal, insulator, superconductor, etc.), while at higher energies above 40eV a free electron response is assumed ( l/w^). It is important to note that for uj' <^ u) , oj' ^ oj , and for regions where <^ (a; ) is flat there are negligible contributions to the integral in Eq.l2. This implies that making extrapolations in regions where data is not available has little effect on results in regions where ^{oj) is measured. More on extrapolations will be discussed in the next section. 4.5.2 High and LowFrequency Extrapolations For metals and insulators the high frequency reflectance is usually dominated by interband transitions of the inner core electrons to excited states. Only at frequencies above 100000 cm~^ does the free-electron response of the system become important. In this study, samples were measured out to roughly 40000 cm~^ Above this, the data were merged with existing data on the same samples in the undoped regime which typically covered frequencies up to 300000 cm~^. However, it is only reasonable to assume that the free-electron response dominates at energies higher than lO^cm"^. The reflectance in the interband region between 300000 and 10^ cm~^ was modeled using the formula ^{uj)=^f{^y, (4.13) where and ujf are the reflectance and the frequency of the last measured data point respectively. The exponent s can have values between 0 and 4. Above 10^cm~\ free-electron behavior sets in and the reflectance may be approximated by ^{uj) = ^f,{'^)\ (4.14) The low frequency (< 20cm~^) extrapolation depends on the material investigated. For this study, all the materials measured, including hole-doped La2Cui_xLix04, were

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82 insulating. Consequently, the reflectance was extrapolated to 0 cm at a constant value. 4.5.3 Optical Constants Once the data has been measured and the proper extrapolations have been made, a Kramers-Kronig transformation can be performed to extract the phase shift ©(w). From this, and one can obtain a variety of optical constants for a given material starting with the real and imaginary parts of the complex refractive index, n(a;) and K{iij): n{.) = '-^^ , (4.15) and 2v/^Hsine(a;) K[u}) = , (4.16) respectively. By squaring the refractive index, the complex dielectic function is obtained: e(a;) = (n(a;) + iK{uj))^. (4.17) Hence, the real and imaginary parts of the dielectric constant are given by: ' ei{oj) = n{ojf (Real) (4.18) and e2(w) = 2nK{u!) (Imaginary). (419) Prom K, ei, and €2 other important optical properties such as the real and imaginary parts of the optical conductivity, skin depth, and absorption coefiicient can be deduced. These relations are listed below:

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83 (72 (w) An c 2U!K (4.20) c 4.5.4 Analysis Procedure for Transmission For samples that transmit as well as reflect, it is no longer reasonable to use Eq. 4.12 to extract the phase shift of the incident light upon reflection from the sample surface since the measured signal will now contain contributions from the front and rear surface of the sample. A portion of the transmitted light through the front surface will partially reflect from the rear surface and contribute additional signal to the single-bounce reflection spectrum. Remedying this problem requires combining the reflection and transmission data for the full measured spectrum with the aid of numerical techniques. This yields a single-bounce reflectance spectrum from the multi-bounce data. However, numerical processing imposes limits on the results and serves only as an approximation, albeit a very good one, in regions where the transmission is not small. The aforementioned numerical techniques were not used to extract a single-bounce reflectance spectrum from transmittive samples. Rather, an approximation scheme was employed. First, the unmodifled reflectance spectrum was Kramers-Kronig transformed with the appropriate high and low energy extrapolations and all relevant optical constants were extracted as per Eq. 4.20. In regions where the absorption coefficient was large {a ^ 10000 cm"^), these results served as very good approximate solutions for the optical constants of the single-bounce spectrum. The reasoning is as follows: Since the material is strongly absorptive in these regions, little incident light transmitted through the medium reaches the rear surface, and of the small fraction

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84 of incident light which reflects from the rear surface, an even smaller percentage of this will return to the front surface and contribute to the single-bounce spectrum. To prove this point consider the charge-transfer region in Sr2Cui_xCo3;02Cl2. In this frequency regime (2-3eV), the absorption coefficient is on the order of 10^ cm"^ For samples measured in this study, the average sample thickness was roughly .007cm. Hence, only e~^°°°
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85 be written as 47r(T , , Q = . (4.22) nc Eq. 4.22 may then be inverted to find a{u}): A similar expression may be found for €2: anc e2 = It should be noted that while t ^ v? for strong absorption, Eqs. 4.22-4.24 still holds for the case of strong absorption^^ . For transmittive samples, the absorption coefficient may be found by inverting the expression, T = exp{-ad), (4.25) where d is the medium thickness and T is the measured transmission to solve for a. Prom this the a and €2 may be found from Eqs. 4.23 and 4.24. While it was noted that Eqs. 4.22-4.24 are valid even for strong absorption, these expressions were not used to extract a and €2 in this limit for practical reasons. In regions of strong absorption, the transmitted signal is either very weak (< lO/xV) or non-existent making for a poor signal-to-noise ratio. The optical constants derived in the limits of strong and weak absorption in their respective spectral regimes were then merged together. However, while practical, this procedure was less than perfect. There was poor overlap in the interstitial regions where the optical constants derived from the Kramers-Kronig transformation and (4.24)

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86 from the transmission data were merged. This was due primarily to the poor signalto-noise ratio in the transmission data in these interstitial regions. Values for the transmission-derived optical constants in this regime were invariantly a full order of magnitude smaller than their Kramers-Kronig derived counterparts. This was 'corrected' by deleting both sets of data in the interstitial region and patching the weak and strong absorption regimes with a polynomial fit. Fortunately, the interstitial regions were few and relatively narrow. In Sr2Cui_xCox02Cl2, these were located between 125 and 135cm~\ 550-600 cm~\ and 11000-130000 cm~^ In regions where spectral overlap was not an issue, there was excellent agreement with existing data. 4.6.1 Raman Scattering: Experimental Technique Raman scattering is the inelastic scattering of light from a medium. In the Raman scattering process, the energy of incident light shifts by either absorbing energy from (Anti-Stokes scattering) or imparting energy to (Stokes scattering) excitations in the scattering medium. For Raman measures in this study, only Stokes scattering is investigated. This contrasts to optical spectrometry where only the elastically back-scattered light is measured for a single frequency. Single-beam spectroscopy predominantly measures symmetry allowed dipole transitions in materials such as phonon and charge-transfer excitations. Quadrupole and higher order interactions can, in principle, be measured with this technique but are at least two orders of magnitude weaker than their dipole counterparts. Furthermore, depending on the crystal symmetry of the sample and orientation of the carrier orbitals, particular dipole transitions are forbidden or severely attenuated by symmetry. Many of these inherent weaknesses in single-beam spectroscopy are remedied by Raman scattering. Raman scattering measures shifts in frequency of light of known energy, invariantly provided by a He-Ne Argon or other energy intensive laser capable of delivering several mWatts to the sample surface. Hence, transitions once to feeble to observe in

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87 Magnifying Unsc Beam SpUlIer Figure 4.6. Raman scattering setup. CP = circular polarizer; LP =linear polarizer BS = Beam Splitter; CCD = charged coupled device detetector. optical spectroscopy may now be measured. Raman scattering may also verify dipole transitions observed using spectroscopic techniques. Hence, Raman scattering is an excellent compliment to optical spectroscopy. Figure 4.6 shows the experimental set up for Raman scattering measurements used in this study . The principal equipment consisted of a He-Ne Argon laser, polarizers and notch filter, a CCD (charge-coupled device) detector, and a grating to diffract the inelastically back-scattered light from the sample surface. The CCD was interfaced to a PC to record the data. 514.5 nm light was first emitted from He-Ne Argon laser. The light then passed through a non-dispersive band pass filter to eliminate plasma lines from the laser. Typical laser output was roughly 300mW. After reflecting from a second mirror, the now attenuated laser beam passed through a broad-band linear polarizer on a rotating mount. This polarized the beam before reaching the beam splitter at near normal incidence (a ~ 4°). While this did not completely eliminate

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88 mixing of the sand p-polarizations in the partially reflected beam, the small angle of incidence insured that this effect would be strongly suppress. The partially reflected light from the beam-splitter then passed through a 40x microscope lens before striking the sample. The microscope piece was placed inside a vacuumed sealed housing unit threaded for translational movement of the lens. This translational degree of freedom was needed to squarely focus the incident light onto the sample surface. The housing unit was capped with a transmittive quartz window and attached to the cryostat shroud. The beam intensity and ocular piece provided 2-5 mWatts/cm^ at the sample surface. The sample was attached to the tip of a nitrogen cooled cryostat tip. Polarization of the incident light on the sample surface was controlled by rotating the aforementioned broad-band polarizer. It should be kept in mind in subsequent discussions of the Raman data that there is a small elliptical distortion in the incident beam to the sample surface due to weak mixing of the sand p-polarizations at the beam splitter. The elastically and inelastically back-scattered Hght from the sample surface then transmitted back through the glass beam-splitter before going through an adjustable aperture to help remove stray laser light and to align the backscattered beam with the grating slits. Next, the back-scattered beam passed through a second (back) broad-band linear polarizer. Front and back-end polarizers were needed to separate the symmetry sensitive, inelastically back-scatterd light. More concerning this will be discussed in the next section. The beam then passed through a 514.5 nm holographic notch filter to remove the elastically backscatterd light from the sample surface which would have 'swamped' the detector signal and overshadowed weaker, more physically relevant interactions. Finally, the now solely inelastically scattered light entered a Perkin-Elmer spectrometer where it diffracted from a fixed 300 lines/mm grating. The diffracted radiation was partitioned in space as a function of wavelength according to the formula:

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89 rfsin^ = nA (4.26) where d is the Hne spacing, 9 is the angle of reflection, and n is the order of diffraction (n=l here). The frequency separated, inelastically scattered light was then forwarded to a CCD detector equipped with 1200 horizontal and 500 vertical pixels. This provided 4.5 cm~^ data resolution. Although it is possible to perform Raman scattering with a moveable grating and detector, a CCD was employed for two reasons. First, it is a nitrogen cooled detector and thus provides a superior signal-to-noise ratio than room temperature detectors commonly used with single-beam moveable grating techniques (e.^., thermocouples and PbS chips). Second, the CCD measures an entire spectral region instantaneously. There is no need to step a grating the 200-300 increments needed to cover a frequency region that the CCD can measure in an instant. To reduce signal-to-noise levels to a suitable level, the measuring time of the CCD is simply increased. For Raman data presented here, 5 minute measuring intervals were used. Finally, the data was digitized and forwarded to a PC where it w£is recorded for further analysis. 4.6.2 2-Magnon Raman Scattering: Theory and Analysis Raman scattering can measure transitions in materials that are too weak to be adequately observed in reflection studies. With the exception of Sr2CoxCui_a;04Cl2 , all samples prepared for this study were too opaque for transmission studies. For light to transmit, these samples would have had to have been polished down to a thickness no greater than 5^. The brittle nature of La2Cu04 virtually precludes this. While still possible, the resources needed to polish samples down to such a fine thickness were not at our disposal. This is where Raman scattering proves especially useful. Since it is only the back-scattered radiation that is measured, transmission is no longer an issue and, in fact, becomes a hindrance. Hence, two-magnon scattering in antiferromagnetic

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90 systems, a process far too feeble to be observed in reflection, can readily be observed in Raman measurements. For mediums such as Sr2Coa:Cui_x04Cl2 where two-magnon scattering can be observed in transmission, two-magnon Raman scattering is a useful tool for verification. Two-magnon Raman scattering in antiferromagnetic systems can be understood as an exchange-scattering mechanism. This differs from two-magnon scattering in magnetically ordered materials (ferromagnetic and antiferromagetic) which results from the spin-orbit coupling of the charge carriers. In this process, two-magnon Raman scattering is a fourth order interaction in perturbation theory. Two-magnon scattering via the exchange mechanism involves the flipping of spins on interlinked sublattices in antiferromagnetic systems and has no application to ferromagnetically ordered systems. In the present case, alternating half-filled Cu3dj.2_y2 orbitals in the Cu02 lattice make up each spin sublattice. In the exchange mechanism process, an incident photon dipole-couples to an electron on spin sublattice A (spin-up). The electron is subsequently excited to the neighboring sublattice B (spin-down) provided the incident photon is polarized along the Cu-Cu bond. However, this excitation comes at an energy cost U, the Coulomb repulsion energy of two electrons residing on the same site. This induces the spin-down electron on sublattice B to 'relax' by transferring to the now vacant site on sublattice A. Hence, dipole-coupling of the incident light to the charge carriers induces a spin-flip, or exchange, in the antiferromagnetic background. In second-quantized form, this is viewed as creating a separate magnon quasiparticle on each sublattice. An approximate form of the exchange scattering mechanism may be written as: (4.27)

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91 Here, l3 and 0^ are magnon creation operators on the two interlinked sublattices, c and ct are electron creation and destruction operators, respectively, and is the associated coupling tensor and is directly related to the antiferromagnetic coupling energy J. For more information on physical dynamics of Raman scattering, see the Appendix. Table 4.3. Scattering Symmetries for 2-Magnon Raman Scattering. Symmetry Geometry Ai + Bi XX, yy A2 + B2 xy, yx Ai + B2 x'x', y'y' A2 + Bi x'y', y'x' Symmetries accessed with various combinations of incident and scattered light polarizations, x and y are along the Cu-0 inplane bond directions, x' and y' are oriented 45° to x and y. The 2D Cu02 antiferromagnetically ordered planes have C4 symmetry. Consequently, the Raman scattering tensor in Eq. 4.27, for the in-plane electric field vectors may be separated into four irreducible representations: A\, A2, B\, and B2Under C4 symmetry, these four representations transform as the polynomials ar^-l-y^, x^y — y^x, — y"^, and xy respectively. To isolate each of these representations, a variety of polarization combinations of the incident and scattered light was employed. These are listed in Table 4.3 The first column in Table 4.3 indicates the symmetry representations contained in the scattering for the polarization combinations given in the second column. The first letter in the notation of the second column indicates the

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92 incident polarization and the second letter indicates the scattered light polarization. X and y denote axes directions along the Cu-0 bonds in the plane, while the x' and y' are rotated 45° with respect to the x and y. Hence, by selecting appropriate incident and backscattered polarizations, various combinations of scattering symmetries may be measured {Ai + Bi, A2 + B2, etc.). Note that due to the nature of the representation sums for the different polarization combinations in Table 4.3, it is not possible to extract a single representation from linear polarizations alone. Two-magnon scattering in the Cu02 transforms as the Bi representation. While it is not possible to solely extract the Bi contribution to the Raman spectra using linear polarizers alone, advantage of the fact that the A2 and B2 contributions are more than one order of magnitude weaker than the Bi scattering mode may be taken. Hence, by measuring the Raman scattered light in the x'y' polarization configuration and then in the xy configuration, the latter may be subtracted from the former to obtain the Bi B2 contribution. Since B2 scattering is better than an order of magnitude weaker than B\ scattering, its contribution may be neglected to first order. The relative scattering strengths of the various symmetry channels are presented in Fig. 4.7 for Gd2Cu04.^^ 4.7.1 Sample Preparation All the crystals studied in this thesis were grown by J. Sarrao and Z. Fisk at the National High Magnetic Field Laboratory in Tallahassee, Florida. For x < 0.10, single crystal La2Cui_a;Lia;04 samples were grown by top-seeded growth in Pt crucibles using CuO fluxes^°'^^'^^ . For x > 0.10, single crystals were grown using a Li20-B203 flux^'^. Polycrystalline samples of La2Cu. 511.504 were synthesized using standard ceramic preparation techniques. The appropriate stoichiometric amounts of La203, SrCOa, and Li2C03 were mixed and fired at 900° C over several days with

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93 80 60 40 20 0 0 8000 0 2000 4000 6000 ENERGY SHIFT (cm' ) Figure 4.7. Raman scattering modes for Gd2Cu04. From Reference 85. several intermediate regrindings^°. By contrast, single crystals of Sr2Cu02Cl2 and Sr2Cui_xCoa;02Cl2 samples were grown by cooling a stoichiometric melt in a Pt crucible^^'^^. For optical studies samples with large lateral dimensions are desired. The La2Cui_xLia;04 single crystals grown by the top-seeded solution method form square Cu-0 layers parallel to the surface of the melt. The resultant crystals have large [100] faces, l-3cm on a side, and are several millimeters thick. As grown La2Cu04 and La2Cui_a;Li2;04 single crystals contain excess interstitial oxygen sites which contribute excess holes to the Cu02 plane. These excess holes spin frustrate the spin lattice and lower the Neel temperature to 240K. Since we are interested in determining the impact of the Li holes on the optical properties of La2Cui_xLij;04 it is critical to remove as many of these interstitial oxygen sites, and hence excess holes, as possible. Normally excess oxygen sites are removed from La2Cu04 samples by annealing them

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I 94 at 900°C under high vacuum (P < 5 x lO'^ Torr) for 45 minutes to 1 hour. However, in the case of La2Cui_xLix04 it was feared that such an aggressive procedure would also remove volatile Li sites near the sample surface. To avoid this, the selected La2Cui-a;Lix04 samples were annealed at 650°C for 48 hours under an ultra pure (99.998%) N2 flow. By performing the anneal at a lower temperature over a longer time period in a nitrogen based latm environment, Li losses near the surface could be minimized if not entirely eliminated. After annealing, the samples were encrusted in a plastic mold for polishing on multi-speed polishing wheels. The samples were first polished with 10// diamond paste followed by a 1/i diamond paste treatment. This yielded flat, smooth, mirror-like a-b plane faces suitable for reflection measurements. The samples were removed from their plastic molds by dissolving them in an acetone solution for several hours. Lastly, the polished samples were etched in a 1% methylbromide solution for 20 minutes to remove any residual impurities from the mold and polishing pastes. The samples were then mounted on sample holders. The sintered samples of polycrystalline La2Cuo.5Lio.5O4 were pressed into dense pellets but were still too porous to be successfully polished. To compensate for scattering losses from the course surface, all La2Cuo.5Lio.5O4 samples used for reflection measurements were coated with a 2000A aluminum layer and remeasured over the entire spectral regions available on the PE and Bruker spectrometers. The reflection data of the unadulterated sample was then ratioed to its aluminum coated counterpart to determine the reflectivity. Fortunately, the Sr2Cui_j;Coa;02Cl2 samples were much easier to prepare for transmission and reflection measurements. Since Sr2Cui_xCox02Cl2 is stoichiometric as grown, there was no need to anneal the crystals. Also, these crystals are very micaceous. This made them easy to cleave with a sharp scalpel or razor blade. There was no need to polish the crystals and, if fact, their micaceous nature would have

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95 made that prohibitively difficult. As with La2Cui_xLix04 the as grown crystallized Sr2Cui_xCoa;02Cl2 melts yielded reflective, flat, thin, smooth facets only along the [100] direction, or parallel to the Cu02 plane. For this reason, optical measurements were performed only for incident polarization parallel to the a-b plane. By successive cleaving or with the aid of scotch tape, sample thicknesses of 10/xm can be achieved. However, crystal thicknesses of 40-80^m were suitable for our needs. It was hoped that by using slightly thicker samples the effects of fringing (multiple internal reflections from the back and front side of the crystals) could be minimized. Disappointingly, this tactic enjoyed only modest success. While the Sr2Cui_iCox02Cl2 crystals were comparatively much easier to prepare for transmission and reflection measurements, there was one egregious drawback: Sr2Cu02Cl2 and its cobalt-doped cousin are very hygroscopic. The chlorine sites in the Cl-0 buflfering layer, due to their high electron affinity, act as magnets for stray water molecules in the atmosphere. Consequently, several layers of H2O chemiabsorb on the crystal surface in the span of a few minutes. Unfortunately, these water layers have proven difficult to remove. The spurious H2O layers manifest in the transmission data as sharp, relatively strong absorption features between 3100 cm~^ and 3600 cm~^ which, to our dissatisfaction, are in a spectral region of vital interest. This problem is exacerbated by the high humidity common to Florida, especially in the summer and fall months. However, these undesirable bands were suppressed, with mixed results for different samples, by heating the samples to 330-340K under high vacuum for 1 hour after mounting them in the cryostat and prior to performing room temperature and low temperature measurements. Higher temperatures were not used for fear of damaging the cryostat and/or melting the sample mounting varnish. This seemed to control the magnitude of the H2O bands but did not entirely remove them.

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CHAPTER 5 OPTICAL PROPERTIES OF La2Cui_xLi:,04 5.1.1 Introduction The optical properties of the a-b plane of La2Cui_a;Lix04will be discussed in this chapter. The chapter is divided into three parts. The first reviews the reflectance and attendant optical constants in the far infrared. The characteristic energies, splittings, and temperature dependence of the phonons observed in this region will be discussed and analyzed. The second section presents and discusses the optical features of an observed mid-infrared band which grows with Li content. This phenomenon is significant in its own right, but is especially important as it mimics the behavior of its superconducting cousin, La2-xSrxCu04. The existence of this mid-infrared band in both insulating La2Cui_a;Lix04 and superconducting La2_xSra;Cu04 may eventually help illuminate the elusive mechanism of high temperature superconductivity, or may simply dispel any lingering notions that the mid-infrared band observed in all of the doped-high Tc superconductors materials is relevant to high temperature superconductivity. To this end, a few theoretical models for the origin of this mid-infrared band are discussed and analyzed, starting with the small polaron model and working our way through domain walls. The final third of the chapter discusses the eflFects of Li-doping on the charge-transfer band. Surprisingly, despite the similarities between Liand Sr-doping on the optical conductivity in the mid-infrared, the two systems exhibit vastly different behavior in the charge-transfer region. Although the nature of this deviation is still open for debate, we speculate that the charge-transfer absorption may depend on the local 2D spin order of the Cu^^ lattice in the vicinity of doped holes. 96

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97 5.2.1 Li-doping La9Cu04 Contrary to Sr-doping where Sr^"^ is substituted for La^"*", Li replaces Cu in the basal plane of La2Cu04. Since it goes into the matrix as a +1 ion, each Li introduces one hole into the Cu02 plane. Since carriers are added with the introduction of Li, one might expect that the system undergoes an insulator-to-metal transition at some critical concentration. The fact that La2Cui_xLix04 remains insulating up to X = 0.50 suggests that, in contrast to the effects of Sr-doping which produce both a superconducting and metallic phase, the holes introduced by Li are tightly bound to their impurity sites. Li"*" goes into La2Cu04 with a spherical closed shell (2s^) configuration. Hence, each Li"*" introduces a spin-zero impurity into the spin1/2 antiferromagnetic background. 3D antiferromagnetic long-range order is lost at X ^ 0.02-0.03, as is the case with Sr-doping. However, the effects of the spin impurity on the long range 2D spin order of Cu^"*" lattice is still unclear. More will be said concerning this in subsequent sections and Chapter 7 where 2-magnon Raman scattering is discussed. In the undoped phase and in the absence of any excess interstitial oxygen La2Cu04 has D^^ tetragonal point group symmetry. Typically, however, as grown single crystals of La2Cu04 possess non-stoichiometric oxygen content. The excess oxygen resides on interstitial sites in the La-0 layer buffering adjacent Cu02 planes. Annealing the crystals at 600-800C in flowing N2 gas for several days can remove many of these excess sites, but invariably some residual oxygen remains. These additional oxygen sites enhance a small orthorhombic distortion to the tetragonal unit cell. To a first approximation, however, the unit cell may be classified under point group symmetry. Li substitution in La2Cu04 has a similar effect. Li is comparable in size to Cu and introduces a net +1 charge into the Cu02 plane. Consequently, neighboring sites distort and relax around the charged impurity. X-ray diffraction

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98 studies by Sarrao et al.^^ reveal that the orthorhombic distortion is relatively weak. For Li concentrations investigated here, the difference between the in-plane aand faaxis lattice parameters is about 1% (5.32 vs. 5.39A). For Li concentrations exceeding X — 0.30, the orthorhombic distortion in the a-b plane vanishes. The c-axis parameter remains essentially unchanged for x < 0.3 (13.12A). 5.3.1 Overall Reflectance of the a-b Plane 100 1000 10000 Frequency (cm~^) Figure 5.1. Reflectance of La2Cui_a;Li2;04 at T=300K. In Fig. 5.1, the reflectance of La2Cui_a;Lij;04 up to 40000 cm~^ is presented for X 0.00, 0.01, 0.02, 0.05, and 0.10. In the undoped phase, there are two spectral

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99 regions of interest: The far-infrared where phonons dominate the spectra and the visible near 2eV where the charge-transfer edge can be observed. As the Li content is increased, two trends manifest. A mid-infrared band grows between 1000 and 8000 cm~^ accompanied by a broadening of the charge-transfer edge. This behavior is both similar to and different from the effects of Sr-doping. As with Sr-doped crystals, carriers introduced by Li induce a mid-infrared band. Also, the Li impurities broaden the charge-transfer edge in much the same way as Sr erodes the charge-transfer region in La2-xSra;Cu04 (see Fig. 5.13b). However, a comparison of Fig. 5.13b and Fig. 5.1 shows that the effect of Li on the charge-transfer edge does not extend as high in energy as its Sr counterpart. As will be discussed in subsequent sections, the relatively modest broadening of the charge-transfer edge observed in the reflectivity of La2Cui_iLia;04 generates a charge-transfer band which remains relatively sharp up to a; = 0.10. This contrast starkly with the rapid broadening of the charge-transfer band observed in La2-iSra;Cu04 (Fig. 5.13b). In the far-infrared, the spectra are vastly different as anticipated. Li-doped La2Cu04 is insulating and therefore has only a weak tail associated with the bound carriers below 1000 cm~^. Consequently, the phonon absorptions remain well defined and are not 'swamped' by Drude tail contributions as is the case with metallic La2-xSrxCu04. This difference in the farinfrared spectra is significant. There has been some speculation that the mid-infrared band in all of the high Tc superconductors is simply an energy-dependent scattering rate within the Drude tail introduced by additional scattering sites when the parent insulators are doped. The existence of a mid-infrared band in La2Cui_xLij;04, an insulator, would seem to make this school of thought implausible. This will be discussed in more detail in the subsequent sections. But first we turn to the far-infrared optical properties of La2Cui_a;Lix04.

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100 Frequency (cm ) Figure 5.2. The a-b plane reflectance of La2Cu04 at T=300, 200, 90, 15K. 5.4.1 Optical Properties of the Far Infrared: Phonon Assignment in La9Cu04 Figure 5.2 shows the a-b plane reflectivity of La2Cu04 below 1000 cm~^ for four temperatures. Since La2Cu04 possesses symmetry, there are seven infrared active modes. Three are out-of-plane excitations along the c-axis (not seen here) and four are in-plane excitations. The former transform as the Ef^ irreducible representation while the latter transform as A^. Typically, only three of the four main in-plane excitations can be observed^^. The fourth is believed to be obscured by the broader, more heavily weighted mode near 360 cm~^ However, it is readily apparent that there are no fewer than seven modes, with considerable fine structure below 200 cm~^. The strength of the orthorhombic distortion increases as the crystal is cooled as evidenced by the fine structure in the reflectance between 100 and 200 cm~^ below 300K. Hence,

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101 as discussed in the previous section, the assumption of tetragonal symmetry serves as only a first approximation. Nonetheless, if we ignore for the moment the splittings of the various modes, the three principal modes at ~ 140, ~ 360, ~ 690 cm~^ agree well with previous measurements^^"^^ . Decreasing the temperature increases the reflectivity in the phonon gaps and narrows the width of the in-plane excitations. The introduction of Li into La2Cu04 broadens and shifts the phonon peaks as shown in Fig. 5.3. Also, despite the insulating nature of La2Cui_xLix04 there seems to be a weak background that adds additional spectral weight to the far-infrared spectrum. This is not surprising since each Li impurity adds a hole to the system. The d.c. conductivity suggests, however, that these carriers are tightly bound to their impurity hosts. Figure 5.3. The a-b plane reflectance of La2Cui_a;Lix04 at T=300K.

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102 Assignment of the three principal in-plane phonon modes has been done by Tajima. A complete listing of the atomic displacements for all the infrared active modes for symmetry is presented in Fig. 5.4.^^ The lowest lying mode near 140 cm~^ is an external mode. In this figure, the out-of-plane La slides against the CuOe unit as shown in Fig. 5.4d. The next highest lying mode near 380 cm~^ is a bending mode. In this mode the Cu-0 bond angle modulates as shown in Fig. 5.4b and 5.4c. Note that in Fig. 5.4b the two in-plane O sites slide against or shear the central Cu atom while in Fig. 5.4c it is the apical oxygens that shear against the Cu02 plaquette. Hence, splitting of the bending mode is expected. Evidence of this splitting is seen in Fig. 5.1 where two distinct peaks are observed at 400 cm"^ The highest lying excitation near 680 cm"^ is the Cu-0 stretching mode. The eigenvector of this mode is diagrammed in Fig. 5.4a. Here, the central Cu atom and two adjacent O sites move against a third O atom in the Cu02 • The transverse and longitudinal optical phonon (TO and LO) modes can be extracted from the real part of the optical conductivity, ai, and the loss function, /m(^), respectively. These are obtained from a Kramers-Kronig transformation of the reflectivity and subsequent inversions to retrieve a variety of optical constants as discussed in Chapter 4. The results for the a-b plane optical conductivity and loss function of La2Cu04 are presented in Figs. 5.5a and 5.5b respectively, for different temperatures. Much like the phonon absorptions observed in the reflectivity, the vibrational modes sharpen as the temperature is lowered. As shown in Fig. 5.5, the three main modes at 140, 360, and 680 cm~^ are split by the orthorhombic distortion of the lattice. This splitting is not surprisingly enhanced at lower temperatures. Note that the splitting of the lowest Efj^ mode centered at 140 cm"^ is especially egregious. The

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103 ! (e) Aau (0 Aju (9) (h) Aju (inactive) Figure 5.4. Atomic displacements of the infrared active phonon modes for K2NiF4 structure with D\J^ symmetry^^. Modes (a)-(d) correspond to the vibrational modes perpendicular to the c-axis.(£'^) and modes (e)-(g) correspond to those parallel to the c-axis {A2^). (g) is an inactive mode. From Reference 96. TO phonons were fitted by a set of Lorentzian oscillators for each temperature point according to the formula: EN 2 2 where ujpj is the oscillator strength, ujtj the eigenfrequency, and the line width of the jth oscillator. Due to the large number of vibrational modes, the far-infrared region was divided into two parts for the purpose of fitting the phonon spectra: a

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104 E o I E o 400 300 200 100 (a) La2Cu04 Frequency (cm 200 400 600 800 Frequency (cm"') 1000 Figure 5.5. (a) The a-b plane optical conductivity (log scale) and (b) loss function of La2Cu04at T==300. The optical conductivity is plotted on a log scale. region below 200 cm~^ and a region between 200 and 800 cm~^. Each region was fitted separately. The spectrum below 200 cm~^ was fitted with four oscillators, while the higher lying region was fitted with six oscillators. The results as a function of temperature for the fitted eigenfrequencies are presented in Fig. 5.6a and 5.6b

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105 while line widths are shown in Figs. 5.7a and 5.7b. Below 200 cm"\ many of the modes harden (move to higher energy) as the material is cooled. Conversely, there is little apparent change in the eigenfrequencies of the bending and stretching mode. While the bending mode is not expected to be very temperature sensitive, the weak temperature dependence of the stretching mode is a bit surprising. Of the three inplane modes observed, the stretching mode is the most sensitive to the Cu-0 distance since the Cu-0 bond contracts as the temperature is lowered. Here, however, the eigenfrequency of the stretching mode only increases from 676 cm" ^ to 681 cm~^. This can be explained in part by the appearance of shoulder around 640 cm~^ that steals spectral weight from the stretching mode as the temperature is lowered below 300K. This spurious feature is most likely associated with the orthorhombic distortion of the lattice which is enhanced at lower temperatures. It has the unfortunate effect of 'pulling' the phonon eigenfrequency of the stretching mode down to account for the additional spectral weight now below it. This shift in weight was compensated for by adding two additional oscillators near 620 and 640 cm~^ This improved the fit of the stretching mode, but the overlap of the main 680 cm~^ oscillator with the two weaker, broader oscillators at 620 and 640 cm~^ still rendered a low estimation of the stretching mode eigenfrequency at lower temperatures. Although the parameters for these supplemental oscillators were erratic and so not presented in Fig. 5.6a, it is apparent from Fig. 5.5a that the spectral weight of the 620 cm~^ mode grows, taking from the 680 cm~^ stretching oscillator, and softens from ~ 620 cm~^ to ~610cm~^ Note also the enhancement of distinct weak modes at 400, 275, and 255 cm~^ as the temperature is lowered from 300 to 15K. As with the aforementioned mode splitting, these are most likely related to the orthorhombic distortion of the lattice. The line widths, as expected, narrow overall as the temperature is lowered. The main exception is the 680 cm~^ stretching mode for reasons akin to those given above.

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106 180 120 100 1 ' ' ' ' 1 ' ' ' ' 1 ' ' ' ' 1 ' ' ' 1 ' ' ' 1 ' (a) LOzCuO^ _ •> « 6 TO Phonons 0 0 1 00-200cm-1 <> 0A AB_ « 0 . . . . 1 9 ' 50 too 150 200 250 SOO Temperature (K) < 600 E u I E o (b) La2Cu04 TO Phonons 200-800cm" ^ ^ ^ t= = = = = ii^ = =ir=r — — — =g QG — -O 100 150 200 Temperature(K) Figure 5.6. The TO eigenfrequencies of La2Cu04VS. temperature, (a) 0200 cm-^ (b) 200-800 cm-^ The LO phonon modes in Fig. 5.5b were fit by a set of oscillators similar to Eq. 5.1 with an appropriately chosen value of CqoThe results of the eigenfrequencies and line widths are shown in Fig. 5.8a,b and Fig. 5.9a,b respectively. As with the TO modes, the far-infrared was split into two regions for the purpose of phonon fitting. In the region below 200 cm~^ four oscillators were used to fit the LO spectrum. The energies of these LO modes are too low, and the oscillator strengths too weak, to be

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107 40 30 10 0 20 o TO Phonon Width 0-200cm"' ...... o tjQ=1 1 5cm~' Wo=140cm~' ^ Wo=165cm~' 0 0)0=1 75cm~ — * cJo=1 80cm~^ ^^.--'-''^ — a" ^ ^™-=4--^^^^ — A . . , 1 . . D 100 200 T(K) 300 : (b) La^CuO^ ' TO Phonon Widths 200-800cm"' r — — " i-i — me-—,"*! Q O CJq— z/3cm A Wo=350cm~ Q WQ=360cm~' 4 «o=400cm~' * tjQ=680cm~ / / . » ^ ^ / 0 01 1 . . . 1 . . 0 100 200 300 T(K) Figure 5.7. The TO line widths of La2Cu04 vs. temperature, (a) 0-200 cm . (b) 200-800 cm-^ associated with a split external mode. This suggests that these modes may due to excess oxygen content in the sample. The in-plane Cu02 plaquette may dipole couple with the out-of-plane interstitial oxygens not removed when the sample was annealed to form an additional external mode. From Fig. 5.8a(inset), it should be noted that these low lying modes are two orders of magnitude weaker in oscillator strength than

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108 2ZD 1 .... 1 . . (a) LqjCuO, ' ' 1 ' ' 200 LO Phonons 1 00-200cm"' 180 0 — — 0 0 ^ — & e T £ B A ^ o 160 B Q o 3 140 — e ©—-—______ 120 100 0 — . 1 . . . 1 . , 3 SO 100 150 200 250 300 Temperature (K) (b) LojCuO^ LO Phonons 200-800cm" 100 150 200 Temperature (K) Figrue 5.8. The LO eigenfrequencies of La2Cu04VS. temperature, (a) 0200 cm-^ (b) 200-800 cm-^ the higher lying bending and stretching modes. The lowest lying mode hardens from 130 to 136 cm~\ while the modes at 172 and 178 cm"^ pull out of the continuum and edge up 2-3 cm~^ in energy as the temperature is lowered from 300 to 15K. Not surprisingly, the linewidths all narrow as the sample is cooled (Fig. 5.9a). This is accompanied by an appreciable growth in the overall spectral weight below 200 cm In the region spanning 200-800 cm~\ there is substantial change in the three main

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109 25 20 15 10 (a) LqjCuO^ LO Phonon Width 0-200cm"' a;n=1 35cm" _ cJo=160cm~' cjQ=170cm~' _4 cjo=1 80cm"' T(K) 200 300 T(K) Figrue 5.9. The LO linewidths of La2Cu04VS. temperature, (a) 0-200 cm ^ (b) 200-800 cm-i. modes. A total of seven oscillators were used to fit the phonon excitations in this region (six for room temperature). Note that the lowest lying external modes near 270 cm~^ split from two to three oscillators as the sample was cooled below room temperature with the highest of the three increasing from 275 to 287 cm"^ and the lowest decreasing from 259 cm~^ to 250 cm~^ The LO bending mode near 460cm~\

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110 much like the TO mode, is split into two. The higher branch hardens from 453 to 461 cm~^ while the lower edges up more modestly from 452 cm~^ to 458 cm""^ as the sample is cooled from 300 to 15K. However, due to the close proximity of the two branches, these numbers are suspect. Surprisingly, the LO stretching mode shows only a modest increase in energy, from 709 cm"^ to 714 cm~^ There is, however, a rapid increase in the spectral weight of this mode as the temperature is lowered below 300K. Note also the presence of a weak mode just below 500 cm~^ which grows in oscillator strength from 300 to 200K. As with the spurious features below 200 cm~^ this is most likely associated with the orthorhombic distortion of the lattice. The linewidths of the principal LO modes and their orthorhombic branches generally decrease as the temperature is lowered. The exception to this is the lower lying branch of the split bending mode near 450 cm~^. However, much like the eigenfrequency for this split mode, the widths are suspect since the lower branch is only ~ 5-6 cm~^ removed from its comparably weighted counterpart. 5.4.2 Phonon Assignment in La9Cui_TLiT04 The substitution of Li for Cu in La2Cu04 modestly suppresses the orthorhombic distortion^° of the lattice. It is difficult to discern this from Fig. 5.2 where the only effect of Li substitution would seem to be the broadening of the three principal ab plane modes. However, the optical conductivity and loss function tell a different story. In Fig. 5.10a and 5.10b ai and /m(^) are presented for La2Cui_xLix04 at T=15K. As is readily seen both spectra change appreciably with the introduction of Li. Three fundamental changes are observed in the optical conductivity (TO modes). The first and perhaps most obvious change is that the number of distinct modes increases between 200 and 800 cm~^ This may be associated with additional phonon modes localized around the Li sites. The second trend is the growth and the continued splitting of the external mode centered around 140 cm~^. Note in particular the rapid

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Ill growth in spectral weight of the lowest lying of these modes near 110 cm" ^ The third obvious effect of Li-doping is the overall growth in spectral weight in the far-infrared. This is expected since each Li introduces a charge carrier into the Cu02 plane, even if transport measurements suggest that these carriers remain tightly bound to their host. Similar trends are observed in the loss function (LO modes), with the additional observation that the oscillator strengths are now a full order of magnitude weaker than those found in undoped La2Cu04. For this reason, the loss function of La2Cu04 is not shown in Fig. 5.10b. Here, contrary to what is observed in cr(w), doping reduces the overall spectral weight of the LO modes. Note that the bending mode, split into two in the optical conductivity (TO), is now divided into no fewer than four oscillators. It is also curious that the lowest lying modes of La2Cui_iLia;04 below 200 cm~^ (Fig. 5.10b insert) have spectral weight comparable to the corresponding modes in the undoped phase (Fig. 5.5b inset), although the overall spectral weight weakens with Li content in this region. This lends to the argument that these additional modes are related to excess interstitial oxygen sites. As with undoped La2Cu04, the far-infrared region was divided in two regions, one below 200 cm~^ and the other above, to fit the TO and LO phonon modes. The results for the eigenfrequencies of the TO and LO modes are presented in Fig. 5.11a and 5.11b and Fig. 5.12a and 5.12b, respectively, at T=15K. The TO eigenfrequencies show unusual behavior as a function of Li content below 200 cm~^ . Li"*" is comparable in size to Cu2+ (O.esAvs. 0.72A^'') but is nearly an order of magnitude Hghter (6.941 vs. 63.55 amu^*). Hence, the eigenfrequencies of the principal modes and their orthorhombic satellites should generally scale (increase) with Li concentration since the reduced mass of adjoining Cu-0 pairs reduces by 62% for every Li substitution. However, the overall trend does not bear this out. Many of the eigenfrequencies below 200 cm do

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112 Figure 5.10. The optical conductivity (a) and loss function (b) of La2Cui_xLia;04 at T=15K. not scale with Li content. From Fig. 5.11, the four modes observed in La2Cui_a;Lia;04 increase markedly in energy from a; = 0.00 to x = 0.02, but then relax from a; = 0.02 to a; = 0.10. This may be due, in part, to fitting errors associated with the additional spectral weight introduced by the impurity-bound hole carriers. As readily seen in Fig. 5.10a, the presence of Li increases the spectral weight below (and above) 1000 cm~\ especially below 200 cm~^ These charge contributions to the optical conductivity increase the oscillator strengths and linewidths of the fitted spectra.

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113 This obscures some of the modes and may lead to artificially low values of the fitted eigenfrequencies. Above 200 cm"^ similar trends are found. The TO bending mode at 400 cm"^ initially edges up in energy from 398 cm~^ to 402 cm"^ as x increases from 0.00 to 0.02, but then goes unchanged from x = 0.02 to x = 0.10. The two lower lying satellite modes at ~ 350 cm~^ and 360 cm~^ show a similar pattern with the higher of the two actually being lower in energy at x = 0.10 than at x = 0.00 (~ 361 cm~^ vs. 360 cm~^ ). The TO stretching mode at 680 cm~^ jumps from 680 cm~^ at X = 0.00 to 685 cm~^ at x = 0.02 before easing back to 682 cm"^ at x = 0.10. Two of the satellite modes near 495 cm~^ and 600 cm~^ undergo a reverse pattern, decreasing in energy from x = 0.00 to x = 0.02 before edging back up slightly above the undoped energies at x = 0.10. The aforementioned unexpected behavior of the principal TO modes may be resolved when the effects of Li on the effective spring constant of adjoining Li-0 pairs are taken into account. More concerning this will be discussed in the next paragraph where the LO modes are analyzed. The energy trends of the LO modes differ from their TO counterparts but still diverge in many ways from what might be expected when a light atom is substituted for a heavier one. Yet below 200 cm~^ the pattern is very much as one might anticipate. From Fig. 5.12a, all of the mode energies increase, some more steadily that others, when Li is substituted for Cu. If these weak modes below 200 cm~^ are indeed the result of interstitial oxygens sliding against the Cu(Li)02 plane, then it is reasonable that they would be more sensitive to the reduced mass of Cu(Li)04 plaquette than to changes in the a-b plane parameters that result from the substitution of Li for Cu. Hence, one may expect these modes to increase steadily if not linearly from X = 0.00 to X = 0.10. This should be contrasted with the a-b plane bending and

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114 600 7 E o 500 o 3 (b) 'La2(:;u,.jLi,0< T=!15K TO Phonons 200-800cm"' 0.04 0.05 0.08 Li Concentration (x) Figure 5.11. The TO eigenfrequencies vs. Li concentration at 15K. (a) 0200cm-i. (b) 200-800 cm-i. stretching modes which are sensitive to changes both in the reduced mass and in the bond distances. The external mode near 265 cm~^ and its two attendant orthorhombic modes all behave differently with Li-doping. The eigenenergy of the principal or 'true' D4'' external mode is invariant with respect to Li content. This is surprising. Recall that this mode corresponds to the out-of-plane La atom sliding against the CuOe unit. The presence of Li lowers the mass of the octahedral unit from 160 to 110

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115 3 140 1 1 ' 1 ' 1 ' (a) La2Cu,_,Li,0, 1 T=1 5K LO Phonons 1 00-200cm"' — * — « « — & II A * & B B lb B B O e 1 I.I.I. 1 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Li Concentration (x) 3 1 ' (b) 'La^Cui^^'Li^O^ LO Phonons 200-800cm"' T=il5K' « — « j « 1>— » 0 it— « -H> * 9 =4 r 0— — 0 -A— — & r A — — A B Q — — e 1 e i : < 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Li Concentration (x) Figure 5.12. The LO eigenfrequencies vs. Li concentration at 15K. (a) 0200cm-i. (b) 200-800 cm-^ amu. Hence, from a mass standpoint, an increase in the eigenenergy of this mode is expected. Furthermore, X-ray diffraction studies"^^ reveal that the c-axis parameter is invariant under Li substitution. Hence, Uttle change is expected in the bond distances perpendicular to the a-b plane. However, changes in the bond overlap, or the effective spring constant for the phonons, in this direction are more difficult to gauge as Li2s2-02p^ bonds replace the Cu3d322_r2-02p2 overlap. These bond substitutions

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116 may offset the effects of a lighter reduced mass (i.e. change the local spring constant for the phonons). The lower and higher lying structures adjacent to the external LO mode behave entirely differently with the substitution of Li for Cu. The lower branch decreases markedly from 231 cm~^ to 215 cm~^ while the upper branch edges up linearly from 295 cm~^ to 305 cm~^ Like the external mode, the bending mode at 452 cm~^ changes little with Li content. Once again, this may result from competing forces between the reduction of the a-b plane lattice parameters and the changes in the bond overlap energies when Li replaces Cu. Note that as with the external mode, the split oscillators at 395 cm~^ and 415 cm~^ below the bending mode soften while the excitation above the bending mode near 505 cm~^ hardens with Li-substitution. The energy of the LO stretching mode shows little dependence on Li content. In fact, it is Li-invariant to within experimental error. This is perhaps the most surprising result concerning the three a-b plane infrared modes. Of these modes, the stretching mode is the most sensitive to both the Cu-0 bond distance and the reduced mass. Li substitution reduces the a-b lattice parameters by roughly 1-2% and the reduced mass of Cu-0 pairs by 62%. Both reductions should serve to increase the eigenfrequency of this mode. The fact that the stretching mode eigenfrequency is impervious to the presence of Li strengthens the argument that changes in the bond overlap energy should be taken into account. While Li"*" is nearly identical in size to Cu^"^, the orbital overlap between in-plane Cu-0 and Li-0 pairs is entirely different {CuSd^2_y2-02px,y vs. Li2s2-02px,y )• Once again, this may offset the effects of a smaller reduced mass and of reduced a-b lattice parameters.

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117 5.5.1 The Midinfrared Band in La9Cui-rLiT04 One of the optical features common to all high Tc superconductors is the growth of a mid-infrared band with charge-doping. This is also the case with La2Cui_xLix04. In Figs. 5.13a and 5.13b, the optical conductivities of La2Cui_xLij;04 and La2-xSrxCu04 are reported, respectively, at T=300K. A mid-infrared band that grows with Li and Sr content emerges below leV. This is concurrent with spectral weight loss in the charge-transfer region. However, there are two notable differences in the spectra below 40000 cm"^ One is the appearance of a Drude tail in La2-xSr2;Cu04 which is absent in La2Cui_xLix04. This is consistent with the metallic phase reported in La2-xSriCu04 for x > 0.03. For concentrations exceeding X — 0.02, the mid-infrared band merges with the Drude tail. This pulls the MIR band center down and makes it difficult to separate the two bands at higher Sr concentrations. This is not an issue with La2Cui_xLia;04. The absence of a Drude tail in La2Cui_xLix04 should put to rest any lingering notions that the mid-infrared band is simply spill over from the Drude tail. The second deviation between the spectra is the effect of doping on the charge-transfer band. In La2-xSrxCu04, the charge-transfer band erodes quickly with the introduction of Sr and merges with a higher, broader lying band near 12eV. This is prototypical of all the high Tc superconductors. This contrasts with La2Cui_xLix04 where the charge-transfer band remains robust up to X = 0.10 and where a secondary broader band grows just below 3eV (Fig. 5.13a). These disparities in the charge-transfer region will be addressed in Section 5.6.1. To further quantify the redistribution of spectral weight from high to low energy it is necessary to turn to the sum rules. The sum rules for La2-xSrxCu04 and La2Cui_xLix04 at T=300K are presented in Fig. 5.15. The two sum rules are nearly identical below 2eV for x = 0.00 to a; = 0.10. At roughly 3.25eV (26200 cm'^), the sum rules merge in La2-xSrxCu04. This indicates that the MIR band steals

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118 I 600 g 500 u I E 400 300 (a) La2Cu,_,L . T=300K ix04 1 x=.01 K=.02 x=.05 / \ -' ^ /"\ A 1 r . 1 2 3 Energy (eV) Figure 5.13. T=300K optical conductivity of (a) La2Cui_a;Lij;04 and (b) La2-xSrxCu04 (from Reference 32). spectral weight predominately from the charge-transfer band. By contrasts, the sum rule steadily increases above 3eV as a function of Li content which suggests that MIR band found here borrows from excitations above 4eV. This is not surprising since Cu charge-transfer states have been replaced by Li states which appear at much higher energy. We track the spectral weight growth of the MIR band in Fig. 5.16(top). Here, the sum rules for La2-xSrxCu04 and La2Cui_j;Lia;04 evaluated at 9000 cm ^ are shown. Phonon and Drude tail contributions below 1000 cm~^ have been subtracted. This effectively eliminates contributions from the phonons below 1000 cm~\ but it

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119 serves as only as a first approximation for the removal of free carrier contributions. In Fig. 5.16 the growth rates of the two mid-infrared bands are nearly identical. Numerical results for the integrated spectral weight below leV as a function of carrier concentration^^ are given at the bottom of Fig. 5.16. A comparison of the two figures demonstrates that the experimental results closely match theoretical expectations. The presence of nearly identical MIR bands in La2-xSrxCu04 and La2Cui_j;Lix04 coupled with the absence of a Drude tail in La2Cui_a;Lij;04 suggests that the midinfrared band observed in all the high Tc superconductors is independent of the carrier mobility. But this conclusion raises an important question: If the mid-infrared band is invariant with respect to carrier mobility, then what is the mechanism responsible for it? A few models for the mid-infrared band will be discussed in subsequent sections. The mid-infrared band in La2Cui_a;Lij;04 is weakly temperature sensitive. The optical conductivity of La2Cuo.9Lio.1O4 is reported in Fig. 5.14 at four temperatures. Temperature trends in the optical conductivity of this sample are representative of all the Li-doped samples studied. As can be seen, cooling weakly increases the overall conductivity of the mid-infrared band, but it neither narrows the band width nor shifts the band center. This will have dire consequences for polaron theories that attempt to model this band. 5.5.2 Polaron Model Since it would appear that the mid-infrared band can not simply be interpreted as bound excitations of holes trapped by their impurity hosts, some have turned to polaron models to explain this band (Section 3.2.2). The basic premise is that charge carriers introduced by impurities locally distort the lattice and become self-trapped in polaron wells. The charge carriers are often said to be dressed by a 'phonon cloud' in their surrounding vicinity. These carriers can then be photo-excited at some

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120 400 2000 4000 6000 8000 Frequency (cm~^) 1 0000 Figure 5.14. Optical Conductivity of La2Cuo.9Lio.1O4 at T=300, 200, 90, 15K. characteristic band of energies from well to well. From section 3.3.1, the optical conductivity of such polaron hopping is given by Eq. 3.14-3.18: where r{w) is a frequency dependent function given by, r{w) = 2{ujTA)-Hog{uTA + [l + {uTA)^f/'^}-2{ujTA)-^[l + {ujrAf]'^/'^-l}. 3.15 Here, r and A are defined as. 2 sinh{^h(jjQP) T = 3.16 and A = 2a;or 3.17 where uq is the average phonon energy to which the charge carriers couple and 7/ is number of phonons associated with each polaron, or the number of phonons in the

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121 O.E _^Sr^Cu04 1 0.6 -0.4 ,0) 0.20 0.15 X^^' fjf ' ^-^''0.34 0.2 0.10^^^^ ' 0.06 0.02 0 . — X J — 1 =0 0 1 2 3 4 (eV) (b) Lo2Cu,_,Li,04 (.o — — 0 12 3 Energy (eV) Figure 5.15. Sum rule of La2Cui_xLia;04 and La2-xSra;Cu04 (Reference 32) at T=300K. polaron cloud. As usual, /5 = The d.c. part of the conductivity is given by the expression
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122 Spectral Weight (MIR) 0.20 0.18 0.16 0.14 + _ 0.12 + \ 0.10 -t UJ ^ 0.08 0.06 + 0.04 -7 0.02 0.00 Li Sr i + + 0.00 0.02 0.04 0.06 0.08 Li, Sr Cone. 0.10 0.12 3.0 _1 i_ < UN A 2N-1 : I 2N N-1 hole doping electron doping Figure 5.16. (Top) Sum rule of La2Cui_;cLi^04 and Sr2Cu02Cl2 evaluated at 9000 cm (phonon contributions subtracted). (Bottom) Cluster calculations of the integrated spectral weight below leV as a function of carrier concentration (from Reference 78).

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123 1.2 1.0 0.8 . 0.6 Z 0.4 0.2 0.0 0 10000 20000 30000 40000 Frequency (cm"') Figure 5.17. Sum Rule for La2Cuo.9Lio.1O4 at T=300, 200, 90, and 15K. which the carriers couple. It seems plausible that the carriers would couple to all three modes, especially the highest lying stretching mode. Thus, a weighted average of the contributions by the individual phonon modes to the conductivity would seem more reasonable when fitting the mid-infrared band. To illustrate this point, the optical conductivity is plotted in Fig. 5.18b for the case of the charge carriers coupling solely to the stretching modes (a;o=0.09meV). To ensure that the overall energy of the phonon cloud was unaltered, the number of phonons that couple to the charge carriers was reduced from 10 to 5.6. In comparison to Fig. 5.18a there are obvious differences between the two conductivities. The overall weights of the two bands are roughly the same but the band center of the stretching mode band is roughly half its bending mode counterpart 0.16eV vs. ^ 0.3eV). This shift in spectral weight from high to low energy significantly increases contributions to the d.c. conductivity which is not consistent with the insulating character of La2Cui_xLia;04. Also note the loss of spectral weight as the material is cooled from 400 to 200K. It should be remembered that Eq. 3.14 is valid only in the temperature regime greater than one-

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124 Figure 5.18. Polaron model for the optical conductivity vs. energy and temperature. Plotted from Reference 67. quarter the Debye temperature. For a;o=0.05, this is T>145K while for a;o=0.09 it is T>260K. A second point of contention is that the optical conductivity, while fit for a variety of Sr concentrations, was fit at only one temperature point. The optical conductivity, as given by Eq. 3.14, is clearly a temperature sensitive quantity. To better illustrate the temperature dependence of the polaron model, traces of the optical conductivity are plotted for T=365, 300, and 200K (260K) for wo=0.05 (bending mode) and a;o=0.09eV (stretching mode) in Figs. 5.19a, 5.19b, and 5.19c. For both mode cou-

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125 Cond. Cond. E=.05 N«10 T-365. 300. 200K 0.2 0.4 0.6 0. E=.09eV N.5.6 T-365, 300, 200K Cond T=36S.300,200K Mixed Modes 0.2 0.6 Figure 5.19. 2D traces of Fig. 18 for different phonon coupling strengths, (a) Phonon energy = 0.055eV (bending modes) and 7]= 10; (b) Phonon energy = 0.09eV and r;= 5.6; (c) Mixed coupUng modes: M±M. Plotted from Reference 67.

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126 plings there is an obvious loss of spectral weight accompanied by a downward shift in energy of the band center as the material is cooled. This behavior contrasts starkly with what is observed in La2Cui_xLix04. From Fig. 5.14, the spectral weight of the mid-infrared band slightly increases as the sample is cooled below room temperature. Furthermore, the changes are modest at best with the integrated spectral weight below the charge transfer gap increasing by only 5% from T=365 to T=200K. Not only is this behavior far less dramatic than the predictions of the polaron model, but it is opposite to expectations. Although Fig. 5.14 only reports the temperature dependent conductivity of La2Cui_xLix04 for x = 0.10, the same temperature dependent behavior is found for x — 0.02 and x = 0.05. The x = 0.10 sample was selected for presentation on the basis that it had the strongest and most discernible mid-infrared band of the four Li-doped samples investigated. The optical conductivity for the averaged contributions of both phonon modes is shown in Fig. 5.19c. As expected, this does not resolve the conflict between prediction and observation. The conflict between experimental results and theoretical prediction concerning the temperature dependence of the mid-infrared band raises serious doubts that a polaronic mechanism can explain this band in the high Tc superconductors cuprates. This doubt is reinforced by the sensitivity of Eqs. 3.14-3.17 to the coupled phonon modes. This raises the specter of magnetism as a possible mechanism for the midinfrared band observed in the doped cuprates. 5.5.3 Zhang-Rice Singlets The shortcomings of the polaron model have led some to pursue magnetic theories to explain the mid-infrared band in the high temperature superconducting cuprates. One such magnetic picture is that holes introduced into the Cu02 planes form ZhangRice singlets. In this scenario, hole carriers added to the oxygen sites in the Cu02

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127 > -i O 'i; /u=-5.25 D -i; M=-5.1 . ' I o o Figure 5.20. Numerical results for the optical conductivity of mixed singlettriplet bound states, /x is the Fermi energy given in eV. From Reference 99. planes form singlets with a central Cu^"*" site. Consult Section 3.2.3 for a detailed discussion of Zhang-Rice singlets. What is crucial here is not only that each hole introduced into the a-b plane resides on a plaquette of four oxygens, but also that each distributes itself among the four sites in a manner that promotes constructive interference of the oxygen orbitals and thus a negative binding energy. The resulting optical conductivity of such bound magnetic excitations is presented in Fig. 3.6. This particular model, however, contains phonon contributions which add a sharp peak in the optical conductivity near « 250 cm ~^ It seems that, despite the shortcomings of the polaron model outlined in the previous section, polarons may still play an important role in the mid-infrared excitations, even in the context of a magnetic theory. Zhang-Rice singlets were also investigated by Kabanov et al.^^ as a possible mechanism for the mid-infrared band in the doped cuprates. Starting from the Emery model^^ and working within a linear spinwave approximation, it was determined that the lowest energy state of an oxygen hole with = 0 in the antiferromagnetic

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128 background was not a straight ZhangRice singlet, but rather a mixed singlet-triplet state. Nonetheless, the bound excitations were magnetic in origin. The results of Kabanov et al. for the optical conductivity are presented in Fig. 5.20a. There are three principal excitations below 3eV. The first, just below 0.2eV 1500 cm~^), corresponds to the (coherent) excitation of the mixed singlet-triplet ground state to the first excited state which in this case is the next energetically highest linear combination of oxygen orbitals. Note that this band grows rapidly at first with hole concentration before decreasing due to the reduced number of unoccupied coherent states. That is, at higher concentrations it is no longer sufficient to treat the mixed hole states as independent quasi particles and it becomes necessary to account for hole-hole interactions. The second peak near 0.8-l.OeV corresponds to excitations from the mixed singlet-triplet state to the incoherent part of the spectrum and grows continuously with hole content. The third peak in the optical conductivity spectrum is located between 1.5 and 2.0eV. This is a charge-transfer type excitation associated with interband transitions of the holes from states of predominately oxygen character to states of mixed oxygen and copper character. The mixed state model is enticing for the obvious reason that it mimics the growth of a mid-infrared band around 1 500-2000 cm~^. However, the two weaker side bands at 0.8-l.OeV and 1.5-2.0eV that emerge from excitations out of the mixed singlet-triplet state are important in their own right. From Fig. 5.13b, it can be seen that a weak secondary band near 1.5eV grows in La2-iSrj;Cu04 as x increases. This is consistent with the third peak observed in Fig. 5.20 associated with charge fluctuations. Such a band is not observed in La2Cui_a;Lia;04. However, this may be due to the robust nature of the charge-transfer band in La2Cui_2;Lix04 which may overshadow the weaker charge-fluctuation excitations. Although a second distinct band around 0.8l.OeV is not found in La2-xSra;Cu04 or La2Cui_xLia;04 it is possible that these

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129 Figure 5.21. Optical conductivity of electron-doped Nd2-xCexCu04_y at T=300K. From Reference 32. excitations merge with the broad, more heavily weighted coherent excitations centered near 0.2eV. The merging of the two lowest bands at 0.2eV and 0.8-l.OeV may also account for why the tail of the mid-infrared band in the doped cuprates and in La2Cui_iLii04 extends out to « leV. Unfortunately, all is not well with the Zhang-Rice picture or the closely related singlet-triplet mixed state model. This is because both are contingent on holes being added to the antiferromagnetic background. Holes added to the cuprates reside on the oxygen orbitals and may form either a S=0 or S=l spin states with the central Cu^"'" . By contrast, doped electrons must reside on the half-filled Cu Mj.2_y2 orbitals. This fills all five orbitals on a given CUO4 plaquette and precludes the possibility of singlet/triplet formation. Hence, a mid-infrared band should not be present in electron-doped systems if its origin is rooted in Zhang-Rice singlet/triplet formation. However, this is not the case. In Fig. 5.21 the optical conductivity of Nd2-iCexCu04_j, is reported for T=300K where it is readily apparent that a midinfrared band grows with Ce content. Trivalent Ce^+ replaces divalent Nd^"*" and

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130 so introduces an electron into the Cu02 plane. A mid-infrared band is also found in electron-doped La2Cu02+i. These disquieting similarities in the mid-infrared for holeand electron-doped systems suggests two possibilities. One is that there are two separate mechanisms at work in the mid-infrared for holeand electron-doped systems that, coincidentally, yield virtually identical conductivity spectrums. The other is that the Zhang-Rice singlets (triplets) are not the principal mechanism underpinning the mid-infrared band observed in the superconducting cuprates. Of these two possibilities, the latter seems more probable. That two separate mechanisms should generate nearly identical bands in the mid-infrared is too serendipitous to be plausible. Since is appears that both the polaron and Zhang-Rice models fall short of the mark in explaining the origin of the mid-infrared band in the doped cuprates, stripes should be considered as a possible mechanism. This will be discussed next. 5.5.4 Domain Walls Since the discovery of a possible striped phase in Lai.6-xNdo.4Sra;04^^ with a; = 0.12 there has been renewed interest in the possibility of domain wall formation in the doped cuprates. The striped phase consists of an ordered pattern of antiferromagnetic spin domains separated by antiphase boundaries on which the holes are localized. Holes introduced into the Cu02 plane form charged domain walls in either the [1,0] or [0,1] directions that have a filling fraction (number of holes per Cu site) which varies with the material in question. In Lai.8Sro.2Ni04 and La2Ni04.i25 (electron doped) static stripe modulation has been observed^'* in the [1,1] with a filling fraction of one while in Lai.6-xNdo.4Sra;04 domain walls have been reported in the [1,0] direction with a filling fraction of one-half. For a review of domain walls, consult Section 3.2.4. Inelastic neutron scattering measurements performed on La2-xSrxCu04^°° suggest that, at a;=0.075 and 0.14, antiferromagnetic domains persist with a correlation length, ^, on the order of 15-18A (4.5-5 Cu-Cu lattice spacings)

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131 which depends weakly on x. However, it is important to note these neutron scattering studies were inelastic, meaning that only magnetic fluctuations, and not frozen spin correlations, were measured. Hence, these measurements only off"er evidence of dynamic or fluctuating stripes in the superconducting regime. Finally, magnetic susceptibility measurements performed on lightly-doped La2-xSra;Cu04+j by Cho et al}^^ suggest the possibility of static stripe formation. They report the antiferromagnetic domain size, L, or spacing between domain walls, to scale as ^. However, since the hole density is small, x < 0.025, the filling fraction was estimated to be possibly as low as one hole for every 5-10 Cu sites along the domain wall. It should be noted that static stripes have only been observed in La2-xSrxCu04 samples with some additional impurity such as Nd or excess oxygen incorporated into the lattice. These stray impurities suppress the low temperature orthorhombic phase, raising the symmetry from octahedral to tetragonal. Tranquada et alJ'^ cited this phase transition in the lattice structure as the the mechanism responsible for pinning the stripes into a static phase. Recent NQR measurements performed on La2Cui_3;Lia;04^*'*' suggests the possibility of stripe formation. In Fig. 5.22, NQR measurements of La2Cui_xLia;04 are reported for x < 0.025 as a function of temperature. Note that, as with the magnetic susceptibility measurements of Cho et ai, NQR measurements were performed only on lightly doped samples (x < 0.03). Above roughly 30K, the data is uneventful. The splitting of the ^^^La line is suppressed with increasing Li content. However, below 30K, 6 recovers its x = 0.00 value around 10-15K. This is accompanied by the formation of a peak in the ^^^La nuclear spin-relaxation rate, 2W, which is indicative of a non-zero sublattice magnetization. Hammel et a/.^^ cited this is as evidence for the formation of a weakly mobile, well defined domain walls. From these measurements they determined the spacing between stripes to be between 30 and 50 lattice

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132 250 X 0.0?5 , T « K 2.2 12.4 12.6 12.8 13 a Frequency (MHz) O A1 (X . 0.019) J A B1 (X 0.020) B B2 (X > 0.02S) o o 50 100 150 200 Temperature (K) 250 300 Figure 5.22. ^^^La NQR data from Hammel et al.^^ .(a) A = 1/1—1/2 VS. T. The solid curves are fits to the critical behavior in the absence of a spinordered phase transition, (b) ^^^La nuclear spin-lattice relaxation rate. From Reference 136. spacings, with a hole density of 0.5 per Cu site along the domain walls separating the antiphase antiferromagnetic regions. These antiferromagnetic domain sizes, and their dependence on doping concentration, are in good agreement with estimations determined from magnetic susceptibilty measurements on La2-xSrxCu04-|-5.^°^ While it seems that stripe formation, be it static or dynamic, is increasingly probable in the cuprates, it is unclear what its effect might be on the mid-infrared band. From Eq. 3.31, the binding energy of a hole coupled to a static domain wall was determined to be,

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133 2000 4000 6000 8000 Frequency (cm~^) 10000 Figure 5.23. Temperature dependent optical conductivity of 2% Li-doped La2Cu04. E 3AJ fsin(27r/') = — — 3 31 where ^ is the energy per hole, J is the superexchange energy, t is the hopping integral, and /' is the filling fraction measured with respect to the number of electrons per Cu site in the undoped phase. A is a parameter that accounts for the 2D nature of the singlet bonds in the Cu02 plane. Its value is typically estimated between 0.5 and 0.7. If J is taken to be 0.125eV, t ~ LOeV^, A=0.6, and f = \ (one hole per two Cu sites along the domain wall), then Eq. 3.31 yields a binding energy of approximately (-)0.82eV (ss 6600 cm"^). This is about twice the experimentally observed band center in 5% Li-doped La2Cu04(Fig. 5.13a). However, it should be noted that this estimation is sensitive to the values of t and A chosen. Estimations for t in the t J model for the doped cuprates range from 0.25eV to 1.25eV so it may be possible to tune t to fit the observed MIR band centers. For example, choosing t = 0.7eV and keeping the remaining parameters fixed yields a binding energy of (-)0.44eV,

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134 in close agreement with the observed band centers. So the static spin model of Nayek (Eq. 3.31) produces band energies in the neighborhood of the experimentally observed MIR band. Thus, it is at least fair to say that this static spin model is roughly consistent with experiment. Of course, spin and charge fluctuations will soften Eq. 3.31 making it more difficult to reconcile stripe formation with experimental observations. Recent numerical studies by White and Scalapino^^^ have addressed the question of domain wall stability. Specifically, the calculated the energy of a domain wall in the 2D i J model {J/t = 0.35), as a function of hole filling, in the low to moderately doped {x < 0.2) regime using density matrix renormalization group (DMRG) techniques ^"^^ on 16 x6 and 17 x 6 clusters. For x < 0.125 domain walls formed in the [1,0] direction with a filling fraction, p/, of 0.5 (one hole per 2 sites). The binding energy per hole was estimated to be 1.74^ and the domain wall separation, d, scaled as ^. Domain walls also formed in the [1,0] direction for x > 0.17. However, here the energy per hole was lowest for a p = 1 and the charged wall separation scaled as ^. For 0.125 < x < 0.17 the domain walls phase separated into regions with p = 1 and p = 0.5. If J is set to the (approximate) experimentally observed 0.125eV, t = 0.357eV and the binding energy per hole for x < 0.125 is ~ .36eV (~ 2900 cm"^), roughly 75% of the observed MIR band center in Figs. 5.13 and 5.14. The close proximity of this binding energy to the MIR band center lends support to a connection between domain walls and the MIR band in the doped cuprates. The lack of significant change in the optical conductivity below 30K, reported in Fig. 5.23 for 2% Li-doped La2Cu04, would seem to weigh against a correlation between the MIR band and domain walls. The reasoning being that if domain walls exist as dynamic entities above 30K for x < 0.03 and freeze into static stripes below this temperature, then it may be reasonable to expect a measurable transition

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135 in the optical conductivity for 2% Li-doped La2Cu04 as the sample is cooled below 30K. In Fig. 5.23, this is not the case. However, the lack of an observable transition may not testify against domain wall formation. It may be that there is not so much an electronic phase transition below 30K as there is a temperature crossover in the electronic phase of the holes. This is supported by the absence of a discernible phase transition in the d.c. conductivity of La2Cui_xLia;04. The NQR measurements were performed on a much longer time scale compared to the optical measurements. If the domain walls diffuse through the spin-1/2 background at some temperature dependent diffusion rate D{T), then the distance a stripe travels over the time span (r) of the measurement is roughly sJD{T)t. Should this distance exceed the average separation between walls, L, then the stripe signature will be lost. Since NQR measurements are performed in the 100-200kHz range (r ~ 5 — lO^sec), whereas reflectance measurements occur on a timescale of 10~^^ — 10"^'* sec, the stripe signature will be lost in the NQR measurements well before than in the optical conductivity. Stripes may persist well above 30K (and above x = 0.03) but can not be observed in the NQR data because the diffusion rate exceeds the time scales of the measurement. Rough agreement between the binding energies of charge carriers to domain walls and the experimentally observed MIR band centers in La2Cui_xLia;04 is by no means unequivocal evidence of a correlation between the two. Equation 3.31 is sensitive to the values of /', f, and A selected and the DMRG numerical studies of White and Scalapino were limited to one coupling strength {J/t = 0.35). Equation 3.31 was calculated for a filling fraction of 0.50 (/' = 1/4), a value consistent with magnetic susceptibility studies on Lai.6-a:Ndo.4Sra:Cu04''^'''^ in the moderately doped regime {x > 0.07) and with NQR studies on La2Cui_xLi3;04 for x < 0.03. However, for smaller filling fractions and low estimations of t it may no longer be energetically favorable for the holes to form domain walls. The numerical results of White and

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136 Scalapino hold promise for drawing a definitive connection between the MIR band and domain walls. However, before this can be done with reasonable assurance more AF coupling strengths need to be investigated. 5.5.5 Magnetic Strings Another potential mechanism for the MIR band in the doped cuprates is the creation of 'magnetic strings' when photons couple to hole carriers. In this picture, holes photo-excite from one Cu^"*" site to a neighboring one, in a single band picture, flipping two spins in the spin-1/2 AF background and breaking six bonds. This costs roughly 3J|j in the context of a simple Ising model. The induced magnon-magnon interactions soften the energy cost to 2.7 J. For typical values of J in the cuprates (~ 0.125eV), this yields an excitation energy of 0.375eV, or roughly 3000 cm~^ This agrees reasonably well with the MIR band centers observed in La2Cui_3;Lii;04. The charge carriers may also hop multiple sites, flipping two spins per hop and breaking 2n + 4 bonds if traversing in a straight line where n is the number of hops. A twosite hopping process is sketched in Fig. 5.24. For example, the energy needed to hop two and three sites is O.SOeV (~ 4000 cm~^) and 0.625eV (~ 5000 cm^^). These latter hopping processes may account for the extension of the MIR tail into the near infrared. Since the energy needed for an electron or hole to hop increases by J per hop it is effectively tethered to its host site. By retracing its path, each carrier in this simple bond counting picture returns to its original energy. There are of course more novel excitations in this 'spin on a magnetic string' model. One such example is cyclic excitations, where hole carriers are photo-excited through a complete loop.

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137 -oA " ' No Bond Broken Bond (J) Figure 5.24. Schematic representation of a hole hopping in a spin-1/2 AF background. The dashed circle is the starting site for the hopping process. However, these excitations are complex, requiring a minimum of four hops, and will be not be addressed here. The magnetic string picture would seem to be a prime candidate for the driving mechanism behind the MIR band in the doped cuprates. All that is needed is a 2Dspin correlation length on the order of 3-4 lattice spacings (10-15A) in the Cu02 plane to ensure that there are indeed AF bonds to break. In the-low to-moderately doped regime (x < 0.05 in the Li-doped regions investigated here), this prerequisite is met. Furthermore, contrary to ZhangRice singlet formation, these magnetic excitations are invariant with respect to the sign of the charge carriers. Also, since the excitations depend to a large measure on J and the 2D spin correlation length, neither of which is temperature sensitive at or below room temperature, little temperature dependence is anticipated. This too is consistent with observation. Magnetic strings would therefore seem to fit the bill, so to speak, for the MIR band and offer an avenue for closing the book on the elusive nature of these excitations in the doped cuprates. However,

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138 as is often the case with explanations that are incisive in their simpUcity, there are problems. 0 0 I I I 1 0 1000 2000 3000 4000 5000 6000 7000 SOOO Frequency (cm"') Figure 5.25. Reflectance of 5% and 50% Li-doped La2Cu04 at T=300K. In Fig. 5.25 the reflectivity of La2Cui_j;Lij;04 is reported for x = 0.10 and x = 0.50 at 300K. La2Cuo.5Lio.5O4, unlike the lightly and moderately doped samples measured, possesses no AF order. As grown, La2Cuo.5Lio.5O4 is polycrystalline so the reflectance in Fig. 5.25 contains contributions from both the a-b plane and the c-axis. La2Cuo.5Lio.5O4 is a structurally ordered material with Cu and Li forming a checkerboard pattern in the Cuo.5Lio.5O2 plane. Each Cu^"*" site is surrounded by four Li"*" sites, and vice versa. As with La2Cui_a;Lia;04 for x < 0.10, each Li site introduces one hole into the Cu(Li)02 plane. Thus, La2Cuo.5Lio.5O4 forms a dense array of Zhang-Rice singlets. Each hole is trapped in a CUO4 well bounded by four Li sites. The absence of Cu-Cu nearest neighbors eliminates the possibility of superexchange coupling in or perpendicular to the Cu02 plane. This precludes any magnetic excitations associated with the holes hopping to neighboring CUO4 wells. Hence, if magnetic strings are responsible for the mid-infrared band observed in the doped cuprates, then no such band should be found here. From Fig. 5.25 this

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139 is obviously not the case. There is a robust band centered at ^ 2600 cm"^ in the reflectivity that bears resemblance to the mid-infrared bands found in La2Cui_xLix04 for X < 0.10. It is unlikely that this optical feature is associated with excitations along the c-axis since the c-axis reflectance of doped La2Cu04 is generally impervious to the presence of charge carriers. Thus, the broad mid-infrared excitations observed in La2Cuo.5Lio.5O4 must occur in the a-b (Cuo.5Lio.5O2) plane. From Fig. 5.25, it is unclear if the narrow MIR band observed in La2Cuo.5Lio.5O4 is related to the broader MIR band. For one, the MIR band observed in the reflectivity of La2Cuo.5Lio.5O4 is symmetric, centered near 2600 cm~^, and bears a relatively narrow bandwidth ^ 1500 cm~^ In comparison, the MIR band in La2Cuo.9Lio.1O4 is broad and would appear to be centered near 1300 cm~^ Considerably more spectral weight between 1000 cm~^ and 2500 cm~^ is found in La2Cuo.9Lio.1O4 in comparison to La2Cuo.5Lio.5O4. This may be associated with the weak background found in the light-to-moderately doped regime of La2Cui_xLix04. No such tail is observed in La2Cuo.5Lio.5O4. Unfortunately, our inability to decouple the a-b plane and caxis contributions to the reflectivity precludes us from performing a Kramers-Kronig transformation bn the single bounce reflection data and extracting a frequency dependent optical conductivity for the a-b plane. This would provide detailed knowledge of the line shape and the position of the MIR conductivity band. If the MIR band observed in La2Cuo.5Lio.5O4 is indeed the remanent of the MIR bands observed in its more lightly doped counterparts, then it seems that a simple magnetic string picture is insuflficient to explain its origin. This revives the idea that Zhang-Rice singlets are driving the MIR band in La2Cui_xLix04. Cu nuclear spin relaxation and NQR measurements performed by Yoshinari et al}^'^ on La2Cuo.5Lio.5O4 reveal low lying magnetic excitations with an energy of ISOmeV, which is strikingly similar to the superexchange energy, J, of coupled Cu spin moments reported in

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140 the lightly doped regime of La2Cui_xLia;04 and La2-xSrxCu04. Although these excitations were reported only above 170K, it was suggested that they persist below this temperature but are obscured by charge-fluctuation interactions which dominate at lower temperatures. Given the isolated nature of the Cu spin moments in La2Cuo.5Lio.5O4 it is difficult to understand how an energy similar to J could be found in this material. Consequently, these excitations, while inconsistent with magnetically coupled hopping of holes in the Cuo.5Lio.5O2 planes, breath new life into the Zhang-Rice singlet model. Yoshinari et al. suggested that the ISOmeV excitation was the result of Li-holes of predominately 02px,y character localized on CUO4 plaquettes forming spin singlets with the central Cu^+ sites (polaron-mediated Zhang-Rice singlet formation). Since it is not possible to extract an optical conductivity for La2Cuo.5Lio.5O4 from the reflection data, it is diflftcult to say if the optical data is consistent with the observed magnetic excitations near ISOmeV found in the NQR data and with the subsequent interpretation that they are linked to polaron-mediated Zhang-Rice singlets in the Cuo.5Lio.5O2 planes. This is especially disappointing since Yoshinari et al. had hoped that spectroscopic measurements capable of directly probing the ISOmeV excitation would lend support to the Zhang-Rice singlet picture. 5.6.1 Efl^'ects of Li-Doping in the ChargeTransfer Region Charge-transfer excitations in the near infrared to visible regime are found in all of the parent insulators to the high Tc superconductors and in isostructural materials such as La2Ni04 . In Figs. 5.26 and 5.27 the reflectance and optical conductivity of La2Cu04 are reported in the charge-transfer region at four temperatures. In both spectra, a gap (band peak) is observed at roughly 2.0eV. This band corresponds to the photo-excitation of an electron from an oxygen to copper site in the Cu02 plane. Consequently, this band is only observed for incident light polarized parallel to the Cu02 plane.

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141 0.20 0.18 0) ^ 0.16 O u _2 % 0.1 4 La2Cu04 T=300K T=200K T=90K T=15K 0.12 0.10 8000 13000 18000 23000 Frequency (cm~^) Figure 5.26. Temperature dependent reflectance of La2Cu04. Figure 5.27. Temperature dependent optical conductivity of La2Cu04.

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142 The reflectance and optical conductivity of La2Cui_xLix04 are reported at T=300K in Fig. 5.28 and Fig. 5.29 respectively. The eflPect of Li on the reflectance edge and charge-transfer peak is appreciable, yet it is not as dramatic as what is found in La2-xSrj;Cu04. Sr-doping flattens the charge-transfer edge by x = 0.10 with a dramatic eff'ect on the optical conductivity extending up to 4eV. Li-doping La2Cu04 has comparatively little impact on the charge-transfer band with spectral weight loss extending only to 2.5-3.0eV. The reason for this disparity is not clear since both Li and Sr add the same number of hole carriers to the Cu02 plane. Both impurities remove electron states from which carriers can hop and so should have essentially the same effect on the charge-transfer band. In fact, one might reason that Li should have a more devastating effect on the charge-transfer band since each Li removes both an initial and a final state for a photo-excited electron. Of course, one key diff'erence is that the Sr-doped holes are mobile (strong spin frustration) for x > 0.02 while the Li-bound holes are localized (weak spin frustration). This implies that the discrepancies in the doping dependence of the charge-transfer bands in La2-xSrj;Cu04 and La2Cui_j;Lix04 may be tied to the local 2D spin order of the Cu02 plane. This will be discussed shortly. But first, to get a better quantitative handle on the effects of Liand Sr-doping in the charge-transfer region, it is instructive to examine their respective sum rules. The sum rules for La2Cui_a;Lix04 and La2-xSra;Cu04 are reported in Figs. 5.15a and 5.15b, respectively. It is seen that sum rule converges at 3.2eV for x < 0.2 (the over-to-underdoped regime) in La2-xSrxCu04 , where as for La2Cui_xLix04 the sum rule increases steadily with Li substitution beyond 4eV. The additional spectral weight above 4eV in La2Cui_xLij;04 is expected since Li adds charge-transfer states that appear well above the Cu-0 charge-transfer energy. Since the spectral weights of

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143 0.07 8000 13000 18000 23000 Frequency (cm~^) Figure 5.28. Temperature dependent reflectance of La2Cui_a;Lix04. 8000 13000 18000 23000 Frequency (cm"') Figure 5.29. Optical conductivity of La2Cui_2:Lia;04 at T=300K. the mid-infrared bands in La2-xSra;Cu04 and La2Cui_xLix04 are virtually identical (Fig. 5.16), the observed charge-transfer band spectral weight in La2Cui_a;Lia:04

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' 144 ' ' ^-i^ Charge Transfer Spectral Weight 0.90 0.85 -U Sr 0.80 0.50 -7 0.45 r 0.40 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 X Figure 5.30. T=300K sum rules for La2Cui_a;Lia;04 and La2-xSrxCu04 (from Reference 32) evaluated at 4eV (32000 cm~^ ). Contributions to the sum rules below 9000 cm~^ (including Drude tail) have been removed. should remain comparatively robust. This is readily observed in Fig. 5.30 where the sum rules for La2Cui_a;Lij;04 and La2-xSrxCu04 are evaluated at 32000 cm~\ with contributions below the charge-transfer band (9000 cm~^) subtracted. The spectral weight of the charge-transfer band in La2-xSrxCu04 decreases linearly with x in this regime, consistent with the rapid erosion of the charge-transfer band. By contrast, the integrated spectral weight is nearly invariant with respect to Li content, decreasing by less than 10% from x = 0.00 to x = 0.10. The disparity between La2Cui_xLix04 and La2-xSrxCu04 in the charge-transfer region may lie in the 2D spin order of the Cu02 plane. The introduction of Sr into La2Cu04 strongly spin frustrates both the 3D and 2D long range spin order of the system. This is evidenced by a rapid suppression of Neel temperature and a loss of 3D spin order at Sr concentrations as low as 2-3%. A similar pattern is found in La2Cui_xLix04 where the 3D spin order is lost at 2-3%. However, in La2Cui_xLix04

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145 c) Figure 5.31. Schematic representation of AF spin frustration in the Cu02 plane, (a) No frustration, (b) Strong frustration, (c) Weak frustration. Raman scattering measurements (see Chapter 7) indicate that the 2D spin correlation length in the Cu02 plane remains relatively well preserved up Li concentrations as high as a; = 0.10 and possibly beyond. This is consistent v^rith d.c. conductivity measurements that place La2Cui_xLia;04 in the insulating regime. The argument for a connection between the charge-transfer band and spin order is motivated by the potential effects of spin frustration on the local value for Jcu-0 for bonds connecting nearest neighboring Cu-O pairs in the Cu02 plane. This is shown schematically in Fig. 5.31. In the absence of holes on the oxygen sites, the local 2D spin order is intact and the local exchange energy takes on an undoped value of Jcu-o as diagrammed in Fig. 5.31a. Here, electron A is free to hop to site 1 but is spin blocked from hopping to site 2. The same is true for electron B with respect to site 2. If a hole is added to the central oxygen site which strongly frustrates the spin lattice, as in La2-xSrj;Cu04, then the local spin order may be as

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146 shown in Fig. 5.31b. J'cu-o should differ from Jcu-O since electron A (B) is partially spin blocked from hopping to site 1 (2). However, at the same time electron A (B) has acquired a probability amplitude for hopping to site 2 (1) since it now has some spin down (up) character. Hence, it is possible that the charge-transfer edge may 'smear' by order J'(^^_q with the introduction of hole carriers which strongly spin frustrate the system. Should the holes weakly perturb the spin order of the Cu02 plane, as with La2Cui_2;Li3;04, then the local spin order may resemble Fig. 5.31c. Here, the spin correlation of the Cu sites differs little from its undoped character and the local bond coupling strength, J'cu-o^ comparable to that found in Fig. 5.34a {Jcu-O ~ J'cu-o ^ Jcu-O ~ J'cu-o)Consequently, weak spin frustration may only modestly smear the charge-transfer edge and may account for the robust character of the charge-transfer band observed in La2Cui_j;Lix04. However, the argument given here is cursory and is intended only to motivate a possible connection between the 2D spin order of the Cu02 plane and the charge-transfer excitations. Not accounted for were important issues such as hybridization of the Cu-0 pairs and the 2-dimensional nature of the Cu^"*" lattice, not to mention the need to make fully antisymmetric the ground state and accessible excited states. Clearly, more work needs to be done in this area. 5.6.2 Temperature Dependence of the ChargeTransfer Band In Fig. 5.26 the reflectance of La2Cu04 at four different temperatures is reported. Decreasing the temperature redistributes the spectral weight near the charge-transfer edge, shifting the gap to higher energy while sharpening the excitation edge. In the optical conductivity these effects translate into a redistribution of spectral weight as can be seen in Fig. 5.27. As with the band gap observed in the reflectivity, the band edge in the optical conductivity shifts to higher energy as the material is cooled. This is concurrent with a redistribution of spectral weight from the low energy to the high

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147 energy side of the charge-transfer band center. This phenomenon is observed in all of the parent insulating cuprates but has received relatively little attention. On the basis of a purely electronic excitation, one would expect little temperature dependence of the charge-transfer band. The narrowness of the charge-transfer band at 2.25eV indicates that the interaction between the photo-excited electron and the hole (bound exciton) should not be neglected. In this picture the charge-transfer spectrum is a superposition of an bound exciton band, which accounts for the peak near the band edge, and a broad continuum above 2eV. This is illustrated schematically in Fig. 5.33. Falck et al.*^'^^^ proposed that the observed redistribution of spectral weight in the charge-transfer region is the result of phonons coupling to the bound electron-hole exciton pairs created when an electron is photo-excited from an 02px,y to a neighboring Cu3(i^2_j^2 orbital. Measurements by Chen et al}^^ show that the static dielectric constant in La2Cu04 is large enough that the electron-hole pairs can form Frohlich polarons. The electron-hole pairs couple to clouds of LO optical phonons which distort the lattice and self-trap the charge carriers. For such polarons, the temperature dependence of the polaron self-energy causes a shift of the band edge given by^°^: where n is the Bose-Einstein occupation number, or the number of phonons to which the electron-hole pairs couple, ujq is the LO phonon frequency, and a is the electron(hole)-phonon coupling strength given by^*^^ Eg{T) = El 2hijjQap[n{ ) + l] 5.2 1 ^^ 2ma;o Cs h 5.3 Here, is the static dielectric constant and Cqo is the electronic contribution. The corresponding imaginary part of the dielectric function e2{oj) for this model is plotted

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148 in Fig. 5.32 along with the experimental results of Falck et al.A comparison of theory to experiment shows that there is good agreement. It is interesting to note that the temperature dependent redistribution of spectral weight extends to the Li-doped samples. In Fig. 5.34, the temperature dependent optical conductivity of La2Cui_j;Li2:04 is shown for x = 0.00, 0.01, 0.05, and 0.10. This is especially true in the 1% and 2% Li-doped samples. Here, the charge-transfer band center increases by ^ 240 cm~^ and 145cm"\ respectively, as the materials are cooled from 300 to 15K. However, contrary to what is observed in La2Cu04, there is not so much a redistribution of spectral weight as opposed to an overall rise in the optical conductivity in the charge-transfer region. This is readily apparent in the 5% and 10% Li-doped samples where the top of the charge-transfer band is less discernible. However, even here a hardening of the band with decreasing temperature is apparent. 5.7.1 Concluding remarks To summarize our findings, the a-b plane optical properties of La2Cui_xLij;04 are at once similar and dissimilar to those found in its superconducting counterpart, La2-xSra;Cu04. La2Cui_a;Lij;04 is insulating and so does not bear the Drude tail common to all the high Tc superconductors. The absence of a Drude tail allows the in-plane phonon modes to be tracked as a function of Li content and temperature. MIR bands emerge and grow with Li and Sr content at nearly identical and linear rates in the low-to-moderately doped regime. The presence of a MIR band in insulating La2Cui_3;Lia;04 should dispel any lingering notions that this band is simply spill over from the Drude tail below 1200 cm"^ Unfortunately, none of the theoretical constructs investigated here unequivocally model the behavior of the MIR band. The small polaron model succeeds in modeling the evolution of the MIR band with

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149 0 0 . Experiment ^ II CuO, -y^^ ^^^^^^^^^^^^ T=122K ^ T=295K — • — 1 — • — t — i — I — T=378K — • — t — • — 1 — • — i — . T=447K it ti f / Energy (eV) Energy (eV) Figure 5.32. ti data for La2Cu04 at several temperatures. From Falck et al. Reference 105). a(co) Exciton CT Band 2.25eV Figure 5.33. Schematic representation of bound exciton contributions to the charge-transfer band.

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150 •000 13O00 IMOO 23000 MOO 13000 tSOOO 23000 Fr«qu«ocy (cm"') Froquvncy (cm"') (b) La,C*i^,Li^,0, (d) LaiCujLI.,04 13000 laooo Fr»qu*ncY (cm"') 13000 10000 Froquoncy (cm"') Figure 5.34. The optical conductivity of La2Cui_xLix04at T=300, 200, 90, 15K. (a) x=0.01 (b) x=0.02 (c) x=0.05 (d) x=0.10. charge-doping at T=300K, but it does not reproduce the experimentally observed temperature dependence. Zhang-Rice singlet formation between the doped holes and the Cu^+ sites is at this point also an unlikely candidate for the MIR band in the lightto-moderately doped regime of La2Cui_xLij;04 and La2-xSrxCu04. Singlet models coupled with charge-modulating phonons replicate, to a fair degree, the MIR band. However, Zhang-Rice singlet formation applies only to hole-doped systems. Electrondoped systems, such as Nd2-xCea;Cu04_y, also bear the impurity sensitive MIR band prototypical of all of the hole-doped systems. Since it seems improbable that there could be two separate mechanisms responsible for the nearly identical MIR bands observed in the holeand electron-doped systems, Zhang-Rice formation seems unlikely in either La2Cui_xLix04 or the high Tc superconductors. However, Zhang-Rice singlet formation is a viable possibility in La2Cuo.5Lio.5O4, and is supported in this structurally ordered material by Cu nuclear spin relaxation measurements. Domain

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151 walls may play a role in the formation of the MIR band and are the leading candidate in the models investigated here, but nothing definitive can be said at this time. The existence of domain walls in Hghtly-doped La2Cui_j;Lia;04 and La2-iSriCu04 has been proposed to explain their unusual NQR properties below 30K. Unfortunately, binding energy estimations of the hole carriers to the domain walls, which can place the bound excitations in the MIR, are very sensitive to the filling fraction and t and are presently limited to static domain walls. Clearly, more theoretical work needs to be done in this emerging and novel field. The charge-transfer bands in La2Cui_a;Lia;04 and in the high Tc superconductors, by contrast to the MIR properties, behave vastly diff'erently under charge doping. In La2Cui_a;Lix04, the charge-transfer band remains robust up to a; = 0.10, broadening slightly and losing only a modest amount of spectral weight which scales roughly with X. By comparison, the charge-transfer band in La2-xSri;Cu04 erodes rapidly with the introduction of Sr, so that by a: = .10 there only remains a broad continuum in the charge-transfer spectral region. The disparities between the two systems are difficult to understand and are, in fact, counterintuitive when one considers that Li removes both an electron-removal and an electron-addition state in the Cu02 plane. On this basis alone, ignoring hybridization effects, the charge-transfer band in La2Cui_j;Lij:04 might be expected to degrade twice as rapidly as that of its Sr-doped counterpart. This deviation from expectation may be rooted in the 2D spin order of their respective Cu02 planes. 2-Magnon Raman scattering data, to be presented in Chapter 7, indicates that the 2D spin order of the Cu^+ lattice in La2Cui_a;Lia;04 remains intact compared to that found in La2-xSra;Cu04.

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CHAPTER 6 OPTICAL PROPERTIES OF Sr2Cui_xCox02Cl2 6.1.1 Introduction Having examined the effects of adding charge carriers on the optical properties of the cuprates in the previous chapter, it is instructive to next investigate the effects of introducing magnetic impurities. This is relevant, in part, since it is known that adding magnetic impurities to many of the superconducting cuprates suppresses and eventually destroys superconductivity. In this chapter, the optical properties of Sr2Cui_a;Cox02Cl2 will be examined. As with Chapter 5, this chapter is divided into three parts. The first will discuss the phonon spectra for Sr2Cui_xCoa;02Cl2. The phonon energies, splittings, and temperature dependencies will be analyzed. Part two will present transmission and absorption coefficient data in the mid-infrared for Sr2Cui_j;Coa;02Cl2. As was the case with La2Cui_a;Lii04, a mid-infrared band is observed in Sr2Cui_xCox02Cl2However, unlike the broad, doping-induced band found in La2Cui_a;Lix04, its origin is believed to be well understood. It is generally accepted that this narrow band and attendant higher lying side bands represent phonon-assisted multi-magnon scattering processes in the Cu02 plane. The pressure dependence of Sr2Cu02Cl2 will be presented as testimony to the validity of this model. And lastly, the effects of Co on the charge-transfer band will be discussed. Here, the temperature dependent reflectance and optical conductivity of Sr2Cui_xCox02Cl2will be presented for x = 0.00, x = 0.06, and x = 0.12. It will be shown that magnetic impurities differ substantially from charge carriers in their effect on the charge-transfer region. Also presented in this section are transmission and absorption coefficient data in the near-infrared. Here, a weak excitation centered near 0.9eV is observed which 152

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153 grows with Co-content. Finally, there is a sharp band in the transmission data between 3000 cm~^ and 3500 cm~^ associated with water absorption on the surface. As discussed at the end of Chapter 4, a water layer chemiabsorbs on the sample surface even when exposed to atmosphere for brief amounts of time when loading and unloading the samples from the spectrometer chamber. With a little diligence and care, we were able to contain these spurious features. 6.2.1 Co-doping Sr9Cu09Cb An examination of the optical properties of Sr2Cui_j;Cox02Cl2 provides an opportunity to investigate the effects of adding magnetic impurities to the cuprate parent insulators. Like Li in La2Cui_a;Lix04, Co replaces Cu sites in the Cu02 plane. However, Co incorporates with a +2 valency (3d^ electronic configuration) and so does not add carriers to the the system. Hund's rule dictates that the three unpaired spins should reside on different orbitals in a high spin state. Thus, we anticipate that each Co introduces a spin-| impurity into the spin-| antiferromagnetic background. In tetragonal crystals with octahedrally coordinated Co sites, one expects the three unpaired electrons to lie on the 3d^2_y2, 3rfj.2_3^2, and 3dxy orbitals. Co was selected for substitution of Cu over other transition metal ions by default. Attempts to introduce Ni, Fe, Zn, and Li into Sr2Cu02Cl2 failed. The identical ionic radii of Co2+ and Cu'^+ (0.721^'') may explain why Co can be successfully substituted into tetragonal Sr2Cu02Cl2. The ionic radii of Ni and Fe (0.69A and .74A), by contrast, differ appreciably from that of Co^+ and may induce an orthorhombic distortion too great for the crystal to bear. It should be noted that the ionic radii quoted here were estimated from lattice spacing studies on various salts. Consequently, Co may be expected to induce a small orthorhombic distortion in the tetragonal lattice. A discussion of sample preparation is discussed in Chapter 4.

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154 0.B o 0.4 0.2 0.0 " 1 • • • 1 ' ' ' Sr2Cu,_,Co,02Cl2 T=300K "A .= nn x=.06 / -/ I' ... 1 if r 1 too 1000 10000 Frequency (cm"') 100 1000 10000 Frequency (cm"') Figure 6.1. (a) Reflectance and (b) transmission of Sr2Cu02Cl2 at T=300K. 6.3.1 Overall Reflectance of Sio.Cu-i-T.CoTOo.Ch. In Fig. 6.1a the reflectance of Sr2Cui_a;Co2;02Cl2 is reported at T=300K up to 40000 cm~^ Below 1000 cm~\ the spectra are dominated by four E^^ modes centered near 140 cm-\ 175 cm~\ 350 cm~\ and 525 cm~^ while just below 2eV 16000 cm"^) the charge-transfer edge is observed. The sharp, modulated fringing below 100 cm~^ are due to reflection from the rear surfaces of the samples. Conspicuously absent here is the presence of a strong mid-infrared band. This is not surprising since Co does not add charge carriers to the system and so should not contribute

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155 to the sum rule. This precludes polaron or Zhang-Rice singlet formation discussed in the Chapters 3 and 5. In fact, at first glance, Co substitution would appear to have little effect on the optical spectra of Sr2Cu02Cl2However, this in not entirely true. In Fig. 6.1b, the transmission data for Sr2Cui_xCox02Cl2 is shown. Differences are now discernible, especially in the mid-infrared. A sharp excitation, associated with phonon-assisted two-magnon scattering in the Cu(Co)02 plane is observed near 2900 cm~^ (~ 2.7J). Not surprisingly, this absorption broadens with increasing Co concentration (number of spin-impurity sites). Also evident in the undoped phase, and to a lesser extent in the doped regime, are two higher lying side bands around 4000 cm~^ and 5500 cm~^. These are believed to be higher fourmagnon excitations. Note also the sharp, spurious absorption just above the two magnon band between 3100 and 3600 cm~^. The is the result of H2O absorbed onto the surface and, unfortunately, was difficult to avoid. To ensure that this unwelcome absorption did not significantly eff'ect our results, the spectral region between 3100 and 3600 cm~^ was not included in the fits to the mid-infrared spectra. Also present in the doped samples is a weak absorption just below the charge transfer edge around 7300 cm~^. It is speculated here that it may result from bound excitations on the Co sites. It can also be seen from Fig. 6.1b that the charge-transfer edge softens with the introduction of Co. This, it will later be shown, is consistent with expectations. 6.4.1 Phonon Assignment in Sr^Cui-rCorO^Cl^ Unlike La2Cu04, Sr2Cu02Cl2 has a true tetragonal structure (D^^ symmetry) with only a small distortion, associated with weak xy anisotrophy^''^"^^^, setting in below the Neel temperature. The introduction of Co may induce a small orthorhombic distortion as discussed in the previous section. Figure 6.2 and 6.3 show the temperature dependent a-b plane reflectance of Sr2Cu02Cl2 and the T=15K spectra of Sr2Cui_xCoa;02Cl2, respectively, in the far-infrared. The dominant features are

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156 0 200 400 600 800 Frequency (cm~^) Figure 6.2. Far-infrared reflectance of Sr2Cu02Cl2 at T=300, 200, 90, 15K. 0 200 400 600 800 Frequency (cm"') Figure 6.3. Far-infrared reflectance of Sr2Cui_j;Coa;02Cl2 at 15K. the four infrared-active modes near 138cm~\ 173cm~\ 343cm~\ and 525 cmAbsent here is the mode splitting observed in La2Cui_xLia;04 and La2Cu04. This

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157 15 SrjCuOjClj T=300K T=200K T=90K T=t5K 1 200 300 400 500 Frequency (cm"') Figure 6.4. (a) Far-infrared optical conductivity and (b) loss function of Sr2Cu02Cl2. consistent with the true D^^ symmetry of the Sr2Cu02Cl2 lattice. The three remaining infrared A^j, active modes can only be observed for incident light polarized along the c-axis and so are not seen here. The phonon assignments are nearly identical to those found in La2Cu04. The two lowest lying excitations are a split external mode. The 138 cm~^ mode is an apical bending mode of the chlorine atom against the Cu-0 unit. The 173 cm~^ mode is a translation vibration of the Sr-atom layer against the whole CuOe octahedron. The 343 cm~^ vibration is a Cu-0 bending mode, which

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158 Sr2Cu,_^Co^02Cl2 T=1 5K x=.00 0 100 200 300 400 500 600 Frequency (cm"') 20 19 9 1 1 1 Sr2Cu,_,Co,02Cl2 1 1 T=15K «= nn x=.06 -. .. x=.12 ^ , 1 , 1 ^ . y ......... 0 100 200 300 400 500 600 Frequency (cm"') Figure 6.5. (a) Far-infrared optical conductivity and (b) loss function of Sr2Cui_a;C0a;02Cl2. modulates the bond angle, while the 525 cm~^ vibration is a Cu-0 stretching mode, i.e., a, bond distance modulation mode. At lower temperatures the phonons sharpen and the overall reflectivity increases. This is accompanied by an enhancement of the fringing both below and above 100 cm~^ These fringes were removed by Fourierfiltering the reflectance spectra prior to performing a Kramers-Kronig transformation to extract the optical conductivity and loss function.

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159 1 0— --0— -0 TO Phonon Eigenfrequencies 100 200 Temperature(K) I ' ' ' ' I SrjCuOjClz TO Phonon Linewidths Q ii)o=l*Ocm"' Wo=175cnn~' A Uo=345cm~' 0 Wo=525cm"' 0 I I , I . . I 0 50 100 150 200 250 300 Temperature(K) Figure 6.6. TO eigenfrequencies (a) and line widths (b) of Sr2Cu02Cl2. The effects of temperature on the phonon modes in Sr2Cu02Cl2 are readily apparent from the optical conductivity and loss function data shown in Fig. 6.4a and 6.4b. Recall that the optical conductivity and loss function yield the transverse and longitudinal optical phonon modes, respectively. The phonon modes were fit by a set of Lorentzian oscillators as prescribed in Eq. 5.1. The results for the mode energies and line widths of the TO and LO modes are reported in Fig. 6.6 and 6.7. The TO modes, save for the Cu-0 bending mode, harden with decreasing temperature. This is

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160 0--0 500^ SrjCuOjClj LO Phonon Eigenf requencies E o — ' S 300 100 150 200 Tempera ture(K) Sr2Cu02Cl2 LO Phonon Linewidths a u„=140cm"' tyo=200cm~' A WQ=395cm~' 0 (yo=560cm~' E o 10 100 150 200 Temperature(K) 250 JOO Figure 6.7. LO eigenfrequencies (a) and line widths (b) of Sr2Cu02Cl2. especially apparent in the highest lying stretching mode which hardens from 524 cm~^ to 533 cm before relaxing back to 530 cm ^. That the stretching mode energy increases with declining temperature is expected since the lattice spacing contracts, and hence the Cu-0 in-plane force constant increases, as the material is cooled. The bending mode, contrary to expectations, softens from 347 cm"^ to 342 cm~^ These effects are even more apparent in the loss function data (LO modes). From Fig. 6.7a, the LO stretching mode shifts 5 cm~^ from 560 cm~^ to 565 cm~^ from T=300K to

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161 15K, while the bending mode, which softened modestly in the optical conductivity, now softens by 9 cm~^ over the same temperature range. The higher lying branch of the external mode hardens appreciable from 201 cm~^ to 205 cm~^ while the lower lying branch edges up in energy only slightly from 141 cm~^ to 143 cm~^ All of the line widths narrow with decreasing temperature for both the TO and LO modes as anticipated. The substitution of Co for Cu has an appreciable effect on the phonon spectra as shown in Figs. 6.5, 6.8, and 6.9. It can be seen here that Co broadens and shifts the bending and stretching modes. This may be a reflection of a small orthorhombic distortion which sets in with Co substitution. For this reason one additional oscillator was added to the fitting routine near 302 cm~^ for the x = 0.06 sample, while two additional oscillators were added to Eq. 5.1 at 302 cm~^ and 510 cm~^ for the 12% Codoped sample. Contrary to the temperature dependent data for Sr2Cu02Cl2, there is no systematic trend in the eigenenergies of the modes in Sr2Cui_j;Coa;02Cl2 as a function of temperature. However, a pattern in the eigenfrequencies emerges as a function of Co content. Prom Fig. 6.8a, the TO eigenenergies of the two lowest lying external modes remain fixed at 140 cm" ^ and 176 cm~^ for all Co concentrations while both the bending and stretching modes first harden, and then soften, with increasing Co content. The former nudges up from 341 cm~^ at x = 0.00 to 342 cm~^ at x = O.06 before sliding back to 339 cm~^ The stretching mode undergoes a similar but more dramatic transition, increasing from 529 cm~^ to 536 cm~^ from x = 0.00 to a: = 0.06 before declining below the undoped value to 527 cm~^ These anomalous shifts in the bending and stretching modes are enhanced in the loss function data (Fig. 6.9a). The bending LO mode increases from 391 cm~^ to 394 cm~^ prior to softening to 393 cm~^ at a; = 0.12. The LO stretching mode oscillates more dramatically, shifting up 11 cm-i in energy from x = 0.00 to x = 0.06 before relaxing back to 567 cm~^

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162 at X = 0.12, just 2 cm~^ above its x = 0.00 energy. The two external LO modes, as with the corresponding TO modes, are stationary at 143 cm~^ and 205 cm~^ 600 500 ^ 400 I E o . — ' 3° 5°° 200 Sr2Cu,_„Co,<02Cl2 T=15K TO Phonon Eigenf requencies 1 — e 1 .... 1 o 0.00 0.05 0.10 0.15 Co Concentration(x) 22 20 18 16 E '2 o 10 o 8 6-: 4'i Sr2Cu,_,Co„02Cl2 T=15K TO Phonon Linewidths Q w= 140cm"' 0 cjo=175cm~' ? Uo=345cm~'
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163 9— 0 LO Phonon Eigenf requencies 0.05 0.10 Co Concentration(x) SrjCui.^Co^OjClj T=15K LO Phonon Linewidths _Q Wo=140cm" . o cjo=200cnn"' _i (j(,=395cm"' _0 W(j=560cm'' 0 (1^ ==^ 0.05 0.10 Co Concentration (x) Figure 6.9. LO eigenfrequencies (a) and line widths (b) of Sr2Cui_a;Cox02Cl2. bond overlap should be comparable to that of Cu-0. A rudimentary estimation of the phonon energy shifts can be found be fixing the effective spring constant between adjoining Cu(Co)-0 pairs and computing the eigenfrequencies from the classical expression for the energy of an harmonic oscillator, oj = .IJ. Using the reduced masses for Cu-0 and Co-0 pairs (12.78g/mol and 12.58g/mol respectively) the ratioed eigenfrequency, is ~ 1.01. Thus, the eigenfrequencies may be expected to harden by about 1% (3-5 cm"^). This is roughly observed in the 6% Co-doped sample, but

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f 164 the stretching and bending modes appear to soften as Co substitution increases to 12%. This deviation from expectation in the more heavily doped material may be due an enhanced orthorhombic distortion at higher Co concentrations which broaden the Ef^ modes. As with the Li-doped samples, this has the deleterious effect of pulling down the eigenenergies of the fitted oscillators to the stretching and bending modes. To compensate for this, additional fitting oscillators were added at 302 cm~^ and 510 cm~i to bolster the principal fits at 343 cm"^ (390 cm~^ LO) and 525 cm~^ (565 cm~^ LO). However, while this mitigates the apparent softening of the phonon spectra, it does not eliminate it completely. The remaining difference may be rooted in the local rescaling of the effective spring constants of the bending and stretching modes in the neighborhood of the Co^+ sites. Little change is anticipated in the principal 3dj.2_y2-02p^ overlap integral since the 3dj.2_j^2 orbitals are partially filled on both the Cu^"*" and Co^"*" sites. However, significant differences may possibly be found in the 3rf3^2_r2-02p^ and 3dxy-02py overlap integral energies since the 3d^2^_j.2 and Zdxy orbitals are filled on the Cu^+ sites, but are only partially filled on the Co^"*" sites. The absence of an extra electron in these orbitals on the Co^"*" sites may lower the overlap energies and, in the process, soften the bending and stretching modes. 6.5.1 Phonon Assisted Multi-Magnon Scattering in Sr^Cui-rCorO^jCb Sr2Cui_j;Cox02Cl2 does not bear the broad, heavily weighted mid-infrared band observed in La2Cui_xLij;04 or the superconducting cuprates, but it is not devoid of excitations in this spectral region. Figure 6.10 shows the absorption coefficient of Sr2Cu02Cl2 from 1000 cm^i to 7000 cm^^ at T=300, 200, 90, and 35K. There is a sharp absorption just below 2900 cm~^ coupled with two weaker, broader excitations near 4000 cm~^ and 5500 cm~^ A comparison of these absorptions with the magnitude of the charge-transfer gap (Fig. 6.2a) shows that they are weaker by nearly three orders of magnitude. The absorptions were first observed by Perkins

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165 350 I I I I I I . I I . . I . . I I 1000 2000 3000 4000 5000 6000 7000 8000 Frequency (cm~^) Figure 6.10. Absorption coefficient of Sr2Cu02Cl2 at T=300,200,90,15K. et al}^^ in a comprehensive study of the mid-infrared optical properties of cuprate parent insulators. At the time, they were attributed to 3d^22_j.2 excitons with 1and 2-magnon side bands. An alternate explanation of the MIR excitations was later provided by Lorenzana and Sawatzky*"^. They proposed that the sharp excitation near 2900 cm~^ was a quasi-bound two-magnon plus phonon "bimagnon" state. They further speculated that the two higher-energy sidebands were 4-magnon plus phonon and 6-magnon plus phonon excitations, with the magnons (flipped spins) arranged in either a row or a 'plaquette. Using a simple Ising-bond counting argument, the bimagnon state consists of two adjacent spin flips in the antiferrromagnetic background and costs ~ 3 J. The reduction from 3 J is due to magnon-magnon interactions. This is consistent with the measured J of ^iillOOcm"^ (2.7J?ii2950 cm~^). Likewise, the energies of the 4-magnon bands should be located at 4 J (5500 cm" ^) for a plaquette and 5 J (5500 cm~^) for a row of 4-spin flips. The numerical results of Sawatzky and Lorenzana are presented in Fig. 6.12. For a review of phonon assisted multi-magnon

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166 scattering, consult Section 3.5.1. A comparison of Fig. 6.10 with Fig. 6.12 demonstrates the near perfect match of theoretical prediction with experimental results for both the energy and intensity of the lowest excitation near 2900 cm~^. However, numerical calculations predict that the two higher-energy sidebands should be ~ 2 orders of magnitude weaker than what is actually observed, despite their excellent agreement with the experimentally observed band energies. The discrepancy in magnitude of the two sidebands has led some to attribute these features to other novel excitations, such as exciton-magnon and exciton-phonon coupling^^^. Nonetheless, it is generally accepted at this time that the 2900 cm~^ excitation is attributable to a bimagnon state. Each MIR excitation (absorption coefficient) was fit by a set of 2 Lorentzian oscillators, one narrow and centered near the excitation in question, and the other broad, heavily weighted, and placed at 10000 cm~^ The latter was introduced to simulate the linear tail that slopes downward from the charge-transfer gap to 0cm~^ For consistency, the parameters of this oscillator were fixed for each temperature point. The results of the fitted spectra are presented in Fig. 6.11. The bimagnon band increases linearly from 2890 cm~^ to 2920 cm~^ from 300 to 35K. This is consistent with a contraction of crystal lattice which increases the Cu-0 overlap in the Cu02 plane and, consequently, the superexchange energy, J. A similar temperature trend is found in the first sideband near 4000 cm ~\ but no discernible pattern is found in the higher lying 5500 cm~^ sideband. This latter band is too broad and too weak to be tracked with precision. As anticipated, the line widths of these excitations, except the 5500 cm~^ sideband, narrow with cooUng.

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167 I 1500 E ^ 1000 MIR Energies 100 150 200 Temperature(K) 100 150 200 Temperature ...... ^ °-a ' SrjCuOjClj MIR Linewidths O 7— Ungnrtn+Phnnnn D 4— Magnon+Phonon G e ' . . . . 1 . , . . 1 , . . . 1 . . . . 1 . . .... 1 . Figure 6.11. (a) 2-, 4-, 6-magnon + phonon band energies vs. temperature, (b) 2and 4-magnon line widths vs. temperature. 6.5.2 Pressure Dependence of the MIR Excitations One experimental method to either bolster or lay to rest the multi-magnon plus phonon model is to squeeze the Cu02 lattice and see what happens to the MIR excitations. Compressing the Cu02 plane should increase the orbital overlap, t, between neighboring Cu-0 pairs. This, in turn, should increase the superexchange energy, J, which scales as ~ + Hence, the MIR bands, if they are tied to phonon assisted multi-magnon excitations, should harden as the material is com-

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168 0.2 0.4 0.6 0.8 1.0 1.2 ' Photon Energy [eV] Figure 6.12. Numerical results (dashed line) and experimental data (solid line) for phonon-assisted 2-magnon absorption. The dashed-dotted line is the contribution to the line shape from the bimagnon at p = (tt, 0). The inset shows the spectral weight at the energy of the bimagnon for different values of the total momentum. From Reference 79. pressed. Hydrostatic compression was performed with the aid of a diamond anvil cell and transmission measurements were performed at several different pressures. Freon, optically inert in the MIR, was used as a fluid medium to ensure that the applied pressure was hydrostatic. The pressure dependent absorption coefficient at T=80K for Sr2Cu02Cl2 reported by our group is shown in Fig. 6.13. The low signal of the tungsten source in this spectral region coupled with the small size 250//m x350/i) of the sample used for measurement allowed only the sharp 2-magnon band to be tracked as a function of pressure. The bimagnon band was fitted with a Lorentzian oscillator plus a linear background to simulate the tail from 1000 cm~^ to 0 cm~^ The results are shown in

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169 Sr2Cu02Cl2 Absorption Coefficient Pressure=0,1. 4,2.0,4.0,1 3 kb T=80K 400 ^ 350 g 250 150 2600 2700 2800 2900 3000 3100 3200 Energy (cm ') Figure 6.13. Absorption coefficient for Sr2Cu02Cl2 at T=80K. Figs. 6.14a and 6.14b. As anticipated, the 2-magnon band hardens from 2875 cm~^ to 2908 cm~^ as the pressure is increased from 0 to 13kb, or ss23cm~VGPa. (Ikb = 0.1087GPa). However, this should only be taken as a rough approximation. Experimental error precludes any definitive statement concerning the linearity of pressure vs. band energy. Nonetheless, it is clear from Fig. 6.14 that there is an overall shift in energy of the 2-magnon band with increasing static pressure. There is no discernible in the pressure dependence of the 2900 cm~^ excitation line width. While weak signal, small sample size, and large optical distances all conspired to undermine our results for the pressure dependence of the MIR excitations, other groups have enjoyed more success. In collaboration with our group, Struzhkin et al}^"^ performed the same experiment at the Geophysical Laboratory and Center for High

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170 1 Phonon + 2 Magnon Interaction Peak Position vs. Pressure (^a^ T=80K 1 Phonon + 2 Magnon interaction Peak Width vs. Pressure T=80K 5 10 Pressure (kb) Figure 6.14. (a) 2-Magnon peak position and (b) peak width vs. pressure. T=80K. Pressure Research using a Nicolet 750 FT-IR spectrometer and MCT detector. They circumvented the aforementioned experimental difficulties by using a custom-built Ceissegrainian objective with x8 magnification and a 45mm working distance which was ideally matched to the fnumber of the diamond anvil cell. The small chamber size of the diamond cell allowed pressures up to 30GPa to be investigated, better than a full order of magnitude greater than the maximum pressure of ~2GPa generated by our group's diamond anvil cell. The results of Struzhkin et al}^"^ for the absorption coefficient and 2-magnon Raman scattering of Sr2Cu02Cl2 are shown in Fig. 6.15. The relatively high signal-to-noise ratio permitted both the 2and 4-magnon MIR bands to be tracked with pressure. Clearly both bands edge toward higher energy as the material is compressed. This is quantified in Fig. 6.16a where the 2and 4-magnon plus phonon band energies are plotted as a function of pressure.

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171 0) o c CO JO o < Ja) 4-M| 7.0 5.4 I 2.2 4000 6000 Wavenumber (cm"^) 3 CO c 0) 2000 3000 Raman shift (cm' ) 4000 Figure 6.15. (a) Optical and (b) phonon-assisted 2-magnon scattering in Sr2Cu02Cl2 at several pressures. From Reference 112. The pressure dependence of the 2-magnon Raman scattering band is also reported in Fig. 6.16b to corroborate the optical results. They report a linear relationship between the pressure and the multimagnon band energies in the absorption coefficient with pressure coefficients of ~ 30 cm~^/GPa and ~ 50 cm~^/GPa at room temperature for the 2and 4-magnon MIR excitations, respectively. Since both the 2900 cm~^ and 4500 cm~^ bands scale in energy with the applied pressure, they are consistent with an increase in t, the Cu-0 overlap integral in the Cu02 plane, and hence in J. But should the 2and 4-magnon bands scale linearly with pressure? To answer this question a crude calculation of tpd between neighboring Cn3d^2_y2 and 02px orbitals was performed over the pressures range 0 12GPa. This was accomplished

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172 with the aid of neutron diffraction measurements on La2Cu04^^^ that tracked the lattice constants as a function of hydrostatic pressure. Although these measurements were only performed up to 0.612GPa, the linearity of the data suggested that the lattice parameters could reasonably be extrapolated well beyond iGPa. The standard functional forms^^^ of the 3d^2_y2 and 2px orbitals, corrected for shielding effects of the core electrons using Slater's rules^^^, were used to compute the Cu3dj.2_y2 02px overlap integral, tp^, as a function of lattice spacing (i.e., pressure). From this, the superexchange energy, J, was computed from the expression, as a function of pressure. As usual, A is the charge-transfer energy (~ 1.85eV for Sr2Cu02Cl2), Upp is the oxygen on-site hole-hole repulsion energy (~ 4eV), and Uad is the hole-hole potential on the Cu sites (~ 8eV). The charge-transfer band in MBa2Cu306 (M= Y, Sm, Gd) has been shown to soften by a scant 2meV/GPa.^^^ Based on this finding the effects of pressure on Udd and Upp were ignored in Eq. 6.1. The results for J ratioed to its OGPa value are reported in Fig. 6.16b. As readily is seen, J grows linearly with pressure. A comparison of Fig. 6.16b to Fig. 6.16a shows that they agree well with respect to the 2-magnon band. The theoretical estimations report a 17% increase in the 2-magnon band while the experimental results indicate about a 14% hardening of the band. It should be kept in mind that the computations reported in Fig. 6.16b are only a first estimation of the effects of pressure on the superexchange energy. Not taken into account were important effects such as covalency of the Cu-0 pairs and ligand field splitting. Furthermore, tpd was computed only over the a-b plane (x, y directions) of the unit cell. This was due to the limitations of the software (Mathmatica) used

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173 5000 _ 4500 E r 4000 E i 3500 CO ^ 3000 2500 Pressure (GPa) 2 4 6 8 10 12 3.5 -4000 cm' absorption IR 2800 cm ' absorption . 2-M Raman, 295 K 2-M Raman, 80 K --2800 cm ' IR, 295 K -4000 cm ' IR, 295 K 10 15 20 25 Pressure (GPa) 30 35 Figure 6.16a. Bimagnon band center vs. pressure. Inset: 4-Magnon scattering band position vs. energy (Reference 112). to/cOg VS. Pressure Bimagnon Energy Shift T=300K 1.25 1.20 1.15 -i 1.10 -1.05 1.00 I ' ' ' ' I ' ' ' ' I ' ' ' ' I ' ' I I ' ' I 0 2 4 6 8 10 12 Pressure (GPa) Figure 6.16b. The superexchange energy as a function of pressure ratioed to its uncompressed (OGPa) value.

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174 to compute the resonance integrals. However, the 3dj.2_y2 and 02px orbitals lie predominately along a-b plane so the contributions to the resonance integral along the c-axis should change comparatively little with contraction of the lattice. Despite the cursory theoretical treatment given here Fig. 6.16b lends to the argument that the 2900 cm~^ and 4500 cm~^ bands are associated with the creation of two and four magnons in the Cu02 plane. While this is not unequivocal proof that the two lowest MIR excitations are related to magnon-phonon interactions, it does buttress the multi-magnon plus phonon picture, especially for the 2900 cm~^ bimagnon state. 6.5.3 Effects of Co-Doping on the MIR excitations 350 300 250 E o S 200 150 100 Sr2Cui_,
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175 hardens and broadens the bimagnon state and introduces additional spectral weight between 1500 cm~^ and 2750 cm~^ The effects on the sideband energies are difficult to gauge since both bands, which are comparatively weak and broad relative to the bimagnon excitation to begin with, disperse with increasing cobalt content. In fact, Co-doping decimates the highest lying sideband near 5500 cm~^ by a; = 0.06 and so this was not tracked as a function of cobalt concentration. As with the temperature dependent data of Sr2Cu02Cl2, two Lorentzian oscillators were used to fit, separately, the 2900 cm~^ and 4500 cm~^ excitations: the first narrow and centered at the fitted band and the second broad, heavily weighted, and placed at 10000 cm~^ to simulate the MIR-NIR tail running from high to low energy. The results for these fits are presented in Fig. 6.18. The bimagnon state hardens substantially from 2882 cm~^ to 2949 cm~^ as the cobalt concentration is increased from 0.00 to 0.12. By contrast, the 4-magnon plus phonon excitation appears to soften with cobalt content, sliding from 4032 cm~^ at x = 0.00 to 3877 cm~^ at a; = 0.12. However, fits to this sideband were hampered by Co-induced band broadening and by the H2O absorption band. Note as well the shift in spectral weight from the region between 3000-6000 cm~^ to the region above 6000 cm~^. This excitation will be discussed in the next section. Cobalt substitution has a discernible effect in the MIR magnon excitations, yet it does not have as dramatic effect as one might anticipate when a spin impurity is introduced into the spin5 background. But is this really surprising? Cobalt goes into the Cu02 plane with the same valency as copper and so does not introduce charge carriers which generally disrupt the 2D spin order of the Cu^+ lattice. Furthermore, atomic Cu and Co are comparable in size so little change is expected in the transition metal-oxygen overlap integral, tpdTo get a handle on the effects of Co-substitution on the magnon spectrum it is first necessary to understand how Jy, the in-plane superexchange, varies in the 3d transition metal series. Zaanen and Sawatzky^^^ per-

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176 5000 4500 4000 E ^ 3500 3000 2500 Sr2Cu,_^Co^02Cl2 T = 35k 600 I 500 400 300 0.00 —a — — 0.00 0.05 0.10 o.i; Co Concentration (x) _ — — — 0.05 0.10 Co Concentration (x) 0.15 Figure 6.18. 2-(circles) and 4-(square) magnon + phonon band energies vs. Co concentration at T=35K. Inset: 2-magnon + phonon line widths vs. Co concentration. formed a detailed study of the superexchange interactions in the late 3d transition metal monoxides and found that J generally decreases from right to left across the periodic table. This phenomenon is tied to the exchange stabilization, or Hund's rule coupling, of the unpaired electrons on the divalent transition metal sites. For example, the unpaired 3dj.2_y2 and 3d^^2_j.2 electrons in NiO are bound in a high spin state (S=l) by Hund's rule coupling. Since the number of unpaired spins, and thus Hund's coupled spin bonds, on the divalent late transition metal sites increases from right to left across the period table, the corresponding charge-transfer energies associated with superexchange mechanism increase in kind. Consequently, since Jn scales as ^(1/A5£ + 1/Use) where A^^ and Use are the charge-transfer and 3d-3d double occupancy energies associated with 2D superexchange, the superexchange energy should decrease rapidly to the left of Cu in the transition metal series. The results for A^^;, Use, and J from Zaanen and Sawatzky are tabulated in Table 6.1 for some

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177 of the late transition metal monoxides. Although Sr2Cui_a;Cox02Cl2 has a lamellar structure whereas the structure of the transition metal monoxides is square planar, the relative strengths of the superexchange energies across the periodic table should be similar. It should also be noted that these estimations for the charge-transfer and double occupancy energies were calculated in the context of a superexchange mechanism and will differ from spectroscopic estimations of A and U (dipole optical transitions) discussed in Section 6.6.2. The superexchange energies in Table 6.1 are calculated from theoretical estimations of the microscopic parameters of the threeband Peirls-Hubbard Hamiltonian and are intended only to show the trend in the superexchange energy across the periodic table. Table 6.1 ChargeTransfer Energies, Double Occupancy Potentials, and Superexchange Constants of the Late Transition Metal Monoxides. Element ^SE Use J (MO) eV eV eV Cu 2.75^^* 9.7 .1659 Ni 6.0 9.5 .026 Co 7.2 9.6 .016 Fe 7.8 8.9 .014 Mn 9.9 8.8 .008 A^e = charge-transfer Energy, U — 3d ~ 3d double occupancy potential, J = ^{ 2AsE+Upp + i)"^ (superexchange energy). Upp is the on-site oxygen hole-hole interaction ^ 4.0eV^^. tpd is set to l.OeV {AsE and Use are taken from Zaanen and Sawatzky^^''.) From Table 6.1, it can be seen that the superexchange energy drops off rapidly to the left of CuO in the periodic table. The superexchange constants of NiO through

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178 MnO are, for the most part, nearly one order of magnitude weaker than their copper based counterpart. This is consistent with experimental estimations^^^'^^*^ of J in La2Ni04 and CoO of 15meV and 6meV, respectively. The substitution of cobalt for copper introduces a spin-3/2 impurity into the spin1/2 AF background that couples via superexchange to its four nearest neighbors with a binding energy of J' as diagrammed in Fig. 6.19. The superexchange energy of CoO is better than one order of magnitude weaker than that of CuO 16meV/160meV = 0.1) so it is reasonable to conclude that J', associated with (in-plane) Co-O-Cu exchange, will be much smaller than J, or Cu-Cu 2D exchange. Just how much smaller is very difficult to gauge, but a rough approximation may be attempted by averaging A^^; and Use from Table 6.1 for CuO and CoO. Using Eq. 6.1 with tpd = l.OeV and Upp — 4eV, J' is roughly 0.039eV. This is about 23% of the superexchange constant of CuO from Table 6.1. For simplicity, we have used the value of tpd from the cuprates in estimating J and J'. This is reasonable to first order since copper and cobalt are very similar in size. The magnitudes of the in-plane superexchange constants will differ from the transition metal monoxides to the transition metal planes with MO2 structure (M = transition metal) of the host insulators, but the relative strengths of the superexchange energy should scale about the same across the periodic table. Hence, we may expect the Cu-O-Co superexchange energy to be ?s 20 % of its host Cu-O-Cu value. Since J'/ J « 0.20 "C 1 it may be reasonable, as a first approximation, to treat the spin-3/2 Co sites as decoupled from the strongly correlated spin-1/2 AF background. By doing so, the doped Co^"*" sites act as static vacancies. The spin dynamics of a 2D spin-1/2 AF Heisenberg model in the presence of static vacancies and magnetic impurity sites has been investigated by Brenig and Kampf By applying linear spin wave theory to the spin-1/2 AF host they calculated the configurationally averaged spin wave excitations of magnetically doped systems. By setting J' = 0, the spectral

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179 AW y t t Figure 6.19. Schematic representation of a spin impurity with coupling strength J' in the spin-1/2 background. function of the one magnon spin wave propagator was computed at the Brillouin zone center (|^, |). Their results are shown in Fig. 6.19 for impurity concentrations of 2%, 7.5%, and 15%. Here, the spectral band positions and widths give the energies and lifetimes, respectively, of the one-magnon excitations. The presence of static vacancies redistributes the spectral weight, shifting the principal band center at 0.71J to higher energy while introducing a weaker side band whose position scales negatively with impurity concentration. The hardening of the principal band is consistent with the numerical studies of Bulut et al}^^ on the quantum spin fluctuations of a spin 1/2 quantum AF. Here, it was found that the local AF order is enhanced on the four nearest neighboring Cu sites around a static vacancy. The side band ~ 0.4 ^ is associated with spin excitations adjacent to the vacancies. The sharp, less heavily weighted bands at and just below OeV are zero energy free spin contributions from the decoupled sites. The results of Brenig and Kampf are consistent with what is observed in the absorption coefficient data of Sr2Cui_xCox02Cl2 in Fig. 6.17. Here, the bimagnon peak shifts roughly 2% (2887 cm~^ to 2954 cm~^)

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180 Imp. cone. Z% 7.5% 15% 0 -5 -10 — s_ O E " -15 -20 -25 0 0.5 1 CJ/J Figure 6.20. Imaginary part of the diagonal element of the 1-magnon spin wave propagator matrix at the magnetic Brillouin zone center for various impurity concentrations and J' = 0. Note: the " J" used here is equal to 2 J in the text. From Reference 120. to higher energy as the Co concentration is increased from x = 0.00 to x = 0.12 at T=35K. This is accompanied by a steady increase in spectral weight between 1500 cm and 2700 cm and by a loss of spectral weight in the bimagnon band, again consistent with the redistribution of spectral weight from high to low energy in Fig. 6.20. Of course, this is not unequivocal proof that the bimagnon state is in agreement with the numerical study of Brenig and Kampf. The spectral function computations were performed inside the Brillouin zone whereas the phononassisted bimagnon state creates magnons at the zone boundary. Furthermore, while there is little doubt that J' is much less than J, there is no assurance that it is acceptable to magnetically decouple the impurity sites from the spin-1/2 AF host. This was done above only as a first approximation since little theoretical work has been done

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181 on the effects of magnetic impurities in a spin 1/2 AF host in the absence of doped charge carriers. In fact, for J' = ij, Brenig and Kampf determined that the principal one magnon band in Fig. 6.20 near 0.7 J actually softened with increasing impurity concentration. A more proper treatment would take into account the contribution of J' to the magnon dispersion curve for several coupling strengths and would evaluate the spectral function at more points in the Brillouin zone, particularly at the zone boundary. This would be a useful future project for theoretical study. 6.6.1 Temperature Dependence of the Charge-Transfer Region In Figs. 6.21 and 6.22 it is seen that the charge-transfer absorption is similar in magnitude and in shape to that found in La2Cu04. As with La2Cu04 this excitation represents the transfer of an electron from an 02px^y to a Cu3dj.2_y2 site in the Cu02 plane and can only be observed for incident light polarized perpendicular to the c-axis. As with La2Cu04 there is a redistribution of spectral weight in the optical conductivity from below to just above the charge-transfer edge as the material is cooled. The charge-transfer excitation is believed to soften by coupling to acoustical phonons in the a-b plane. This coupling of the charge excitation (exciton) to the lattice vibrations is believed to induce the spectral weight redistribution as the temperature is lowered. For a review of charge-phonon coupling in the charge-transfer region, consult Section 5.6.2. From Fig. 6.22 it is seen that the charge-transfer band center edges from 15250 cm"! to 15800 cm-^ from T=300 to T=15K. This is associated with the hardening of the in-plane acoustical phonons to which the charge-transfer exciton couples and is consistent with the results of Falck et al. (See section 5.6.2).

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182 1 ' 1 ' Sr2Cu02Cl2 1 ' 1 ' 1 ' 1 ' 7=300,200,90,1 5K T=30nK T=200K T=90K w W \ 1.1. 1 . 1 . 1 . 1 . 9000 11000 13000 15000 17000 19000 21000 23000 Frequency (cm"') Figure 6.21. Reflectance of Sr2Cu02Cl2 at T=300,200,90,15K. J400 b Sr2Cu02Cl2 T=200K 13000 15000 17000 19000 21000 23000 Frequency (cm"') Figure 6.22. Optical Conductivity of Sr2Cu02Cl2 at T=300,200,90,15K. 6.6.2 Eflfects of Co on the Charge-Transfer Region The effect of Co-doping on the charge-transfer band in Sr2Cu02Cl2 is vastly different than the effect of Li-doping in La2Cu04. Figure 6.21 and 6.22 show the reflectance and optical conductivity of Sr2Cui_a,Cox02Cl2 at T=300K. Co substitution, relative

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183 to Srand Li-doping in La2Cu04, has little effect on the charge-transfer excitation aside from a modest loss in spectral weight and a slight softening and broadening of the charge-transfer edge. This is not surprising since cobalt incorporates into the Cu02 plane with the same valency as Cu and so adds no charge carriers, and thus no electron removal or addition states, to the system. Zaanen and Sawatzky^^^, based on photo-emission and inverse photo-emission studies of the transition-metal oxides, determined that A for the transition-metal oxides should increase from right to left across the periodic table. The reason for this is rooted in the exchange stabilization, or Hund's rule coupling, of the partially filled 3d orbitals. The more partially filled 3d orbitals on a transition metal site, the more Hund's coupled bonds are broken when an electron is added to the site from a neighboring oxygen. Thus, the charge-transfer energy is smallest for Cu^"*" since it has only one unpaired 3dj.2_y2 spin and is greatest for Mn^"*" (3c?^) since it has five partially orbitals. This trend is bolstered by photoemission spectra performed by Lee et al}'^^ on the late transition metal monoxides. A listing of the charge-transfer estimations from Zaanen and Sawatzky for the 3dtransition metal oxides is given in Table 6.2. It should be remembered that these calculations were done explicitly for transition metal monoxides with square planar symmetry. However, the materials under investigation here have tetragonal structure. Consequently, the crystal field splitting effects will differ. While the charge-transfer energy generally increases from right to left across the transition-metal oxides, the opposite trend is found in the d — d correlation energy (double-occupancy energy), U. The 3d orbitals increase in size from right to left, so the electron-electron repulsion of decreases in the same direction. The numerical results of Zaanen and Sawatzky for U are also presented in Table 6.2. Based on the aforementioned reasoning and the photo-emission spectra some of the spectral weight of the Cu-0 charge-transfer band, Acu-0, should transfer to the

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184 Table 6.2. ChargeTransfer and Double Occupancy Potentials of the Transition Metal Monoxides. Element A U (MO) eV eV Cu 4.0 5.1 Ni 6.0 7.3 Co 5.4 4.9 Fe 6.1 3.5 Mn 8.9 7.8 Cr 6.3 3.3 V 9.9 4.8 Ti 8.3 2.9 M refers to the transition metal element in MO. Crystal field effects have been taken into account. Co-0 band, A^o, as Co is substituted for Cu. Thus, the spectral weight in the chargetransfer region should broaden and redistribute itself with the introduction of cobalt. Prom Fig. 6.24 the charge-transfer band broadens weakly concurrent with a modest loss of spectral weight. There is no discernible trend in the position of the band center as a function of Co content. The charge-transfer peak first hardens up to a: = 0.06 and then softens below its undoped value at x = 0.12eV. From Table 6.2 this bucks expectations if the charge-transfer excitations in the Cu(Co)02 plane are treated as a weighted average or superposition of Cu-0 and Co-0 contributions. However, the charge dynamics may be considerably more complicated than can be described by such a simple approach. For example, a Cu3dxy exciton near 1.5eV, symmetry forbidden in the pure cuprates, may become locally allowed due to nearby Co sites

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185 which break the local inversion symmetry of the lattice. This may steal spectral weight from Aqu-O s^nd soften the charge-transfer peak at higher Co concentrations as more of these excitons become locally allowed. 0.25 Sr2Cu,_^COj<02Cl2 T=300K 13000 18000 Frequency (cm~') 23000 Figure 6.23. Reflectance of Sr2Cui_xCox02Cl2 at T=300K. Si'2Cu,_^Co^02Cl2 T=300K 200 0 I 1 1 . > 1 I 13000 15000 17000 19000 21000 23000 Frequency (cm"') Figure 6.24. Optical Conductivity of Sr2Cui_j;Cox02Cl2 at T=300K.

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186 200 I • 1 1 • 1 1 1 1J000 19000 17000 19000 21000 23000 Frequency (cm"') 200 ' ' 1 • 1 • ^ ' 1 ' 1 13000 15000 17000 19000 21000 23000 Frequency (cm"') Figure 6.25. T=300K optical conductivity of (a) Sr2Cui_xCox02Cl2 x = 0.06 at and (b) x = 0.12. On a final note before moving on to excitations in the NIR, it is worth mentioning that the charge-transfer band in Sr2Cui_i:Cox02Cl2 undergoes the same redistribution of spectral weight observed in the undoped material as the temperature is lowered. This can be seen in Fig. 6.24 where the optical conductivities of 6% and 12% Co-doped Sr2Cu02Cl2 are plotted for four different temperatures. As observed in Sr2Cu02Cl2, there is a shift in spectral weight from low to high energy in the chargetransfer region concurrent with an overall narrowing of the charge-transfer band. To a close approximation the redistribution of spectral weight ceases below 90K as the temperature sensitive phonons believed to be coupled to the charge-transfer excita-

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187 tion freeze out. Since Co substitution in the light-to-moderately doped regime has little effect on the phonon structure, there is no reason to believe that there should be any significant changes in the temperature dependence of the charge-transfer gap. This is indeed what is observed. 6.7.1 Excitations in the Near Infrared 350 300 3 a Sr2Cu,_,Co,02Cl2 T=35K x=.00 x=.06 x=.12 / / 6000 7000 8000 9000 10000 11000 Frequency (cm"') Figure 6.26. NIR absorption in Sr2Cui_xCoa:02Cl2at T=35K. Before leaving this chapter the weak excitations observed just below leV in the transmission and absorption coefficient data alluded to in the previous section need to be addressed. In Fig. 6.26 a weak, relatively broad band in the absorption coefficient grows near 7300 cm~^ with increasing Co concentration and merges with the more heavily weighted charge-transfer band centered near 15800 cm~^ (1.91eV). The absence of this band in the undoped material suggest that its origin is tied to the Co sites. An obvious explanation for these Co-induced bands is that they are localized 3d 3d excitons on the Co sites. Co^"*" incorporates into the Cu02 plane with a ZdJ configuration with partially filled 3d^2_y2, 3(^3^2 .^2, and M^y orbitals. A schematic

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188 Tetragonal , Symmetry Co 3d Ground State Spherical Symmetry 2 2-22 X y , 3z r, xy, xz, yz 1 1 x"-y" 3z r xy xz, yz Figure 6.27. Hund's rule ground state of Co sites under D4/J symmetry in Sr2Cui_a;COx02Cl2. diagram of the crystal field splitting for Cu-O-Co neighbors is given in Fig. 6.27. On the Co sites there are three available partially filled orbitals to which a lower lying electron from either the filled Zdxz or 3dyz states may be photo-excited. It should be noted that the there is one partially filled 3rfj.2_y2 state available on each Cu^"^ so the absence of a NIR band in the undoped material implies that it is not the result of transitions from the lower lying 3d filled states to the 3dj.2_y2 band. Any local transitions on the Co sites must of course obey the Pauli exclusion principle meaning that an electron with a given spin can only be photoexcited to partially filled states with the opposite spin. This will be tacitly assumed in the following discussion. For the NIR band to be linked to interband transitions between the partially filled 3d shells on the cobalt sites it is first necessary to determine whether or not the excitations are allowed under tetragonal point group symmetry. This is readily accomplished with the aid of group theory. Since this section is relatively short and group theory will not be used in any other segment of this thesis, the reader is referred

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189 to "Introduction to Group Theory with Applications" for a complete guide to the basics of group theory and its applications. We will only be interested in using group theory as a tool to determine which excitations are allowed under the symmetry of the crystal in question. The basic idea here is that under a given set of symmetry operations, any given function of the coordinates will transform as an "irreducible" representation, or linear combination thereof. With the exception of spherical symmetry, there is a finite number of operations, and thus irreducible representations, for a given point group symmetry. The direct products of irreducible representations for several sets of symmetry operations have been painstakingly tabulated by several authors (see, for example, Reference 124). The direct product table for D4/, symmetry along with some relevant functions and variables which transform as the given irreducible representations is reproduced in Table 6.3 along with a sketch of the Oe tetragonal unit centered on each Cu (Co) site in Fig. 6.28. What is important here is to realize that for an optical transition to occur, via some perturbation or other interaction Hamilitonian, the direct product of the irreducible representations of the initial and final states and the interaction Hamiltonian must have some Ai^ character. The Aig irreducible representation is effectively an identity representation where all functions of the coordinates that fall into this category transform into themselves under all the symmetry operations of the group. This is just a fanciful way of saying that for a transition between different electronic states to be permissible its matrix element must have the same symmetry as the lattice. We are now ready to calculate all the possible matrix elements, in the language of group theory, between different 3d states. Dipole excitations of the form {de \ E x,y \ dxz,dyz) 6.2

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190 Figure 6.28. CuOe structure of unit cell in Sr2Cui_xCoj;02Cl2. Table 6.3. Direct Product Table for the D4 Point Group. Orbitals, coordinates Ai A2 Bi B2 E Ai Ai A2 Bi B2 E Rz,z,pz{u) A2 A2 Ai B2 Bi E Bi Bi B2 Ai A2 E dxy{g) B2 B2 Bi A2 Ai E dxz{g),dyz{g),x{u),y{u) E E E E E Ai+ A2+B1+B2 Direct product table for the D4 group, (g) and (u) are for even and and odd inversion symmetry respectively. The product rules for inversion symmetry are given simply by: uxu = gxg = g and g x u = u. are symmetry forbidden by the inversion symmetry of the lattice so we turn next to quadrupole excitations. These have the form,

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191 1 dE -(3de I X^<5»i-^(0) I 3(/a:z,3dyz), 6.3 dxi where Qij = Sx'^x^j 6.4 and the sum over j in Eq. 6.5 runs over the x and y directions for incident radiation polarized parallel to the Cu02 plane. {3de \ are the accessible excited states. Quadrupole interactions generally have non-zero matrix elements under the D4/1 point group. As an example, consider dE i^^x^-y^ I 'of^y^ I " ^ EgXEg = Alg + A^g + Big + B2g . 6.5 This quadrupole term has Ai^ character and so is allowed under D4/1. By contrast, a quadrupole matrix element of the form dEx {3dj.2_y2 I "o^Qxy I ^dxz) = Big X B2g X Eg = Eg, 6.6 is zero under D^ih since it has no Ai character. The results for the quadrupole matrix elements between the high spin ground state and higher states are tabulated in Table 6.5. These correspond to excitations from the filled Co Mxz,yz states to the partially filled states on the same site (3(ij.y 322-^2 _a;2_y2). As readily seen, three of the quadrupole transitions have the requisite Ai^ character needed for photo-excitation. The intensity of the NIR band is roughly consistent with quadrupole excitations. The conductivity at the NIR band center increases from ~ l.lfi^^cm"^ to ~ 1.4 n~^cm~^ from Co = 0% to 12%, so the magnitude of the NIR band is roughly 0.3 ri~^cm~^ . By comparison, the charge-transfer band, a dipole excitation, is ~ 1000f]~^cm~\ or 3 xlO""* orders of magnitude greater strength than the NIR band.

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192 Table 6.4. Direct Product Table for the Quadrupole Excitations under D/^hMatrix Elements D4/. {3dj.2_y2 1 Qxy 1 ^dxz,yz) E, {3dj.2_y2 1 Qxz,yz 1 3dxz,yz) A1P+A25+B1P+B23 (^^3z-^— 1 Qxy \ 3dxz,yz) E, i^^Sz'^—r'^ 1 Qxy,yz \ ^dxz,yz) {3dxy 1 Qxy 1 3dxz,yz) E, {Sdxy 1 Qxz,yz 1 3dxz,yz} Aip+A2p+Bi^+B2p {3dj.2^y2 1 Qxx 1 3dxz,yz) Ep (3c(j.2_y2 1 Qyy 1 3dxz,yz) E, (3c^^2_j^2 1 Qzz 1 3dxz,yz) E, (^'^Sz^— 1 Qxx 1 3(ij;z,yz) E, (3^3z2-r2 1 Qyy 1 Srf^z^y^) E, (3^322 1 Qzz 1 3dxz,yz) E, (3(ixy 1 Qxx 1 3(ixz,yz) E, (3(ixy 1 Qyy 1 3c?xz,yz) E, (3c?xy 1 Qzz 1 3c?xz,yz) E, Quadrupole matrix elements between the high spin ground state (under Hund's coupling of the unpaired electrons) on Co sites and the excited states. Quadrupole-to-dipole optical absorption strength scales^^^ approximately as j, where a is the interatomic spacing and A is the wavelength of the light associated with absorption. Taking a ~ 2A(Cu-0 spacing) and X llOOnm (~ 7300 cm"^) this ratio is 2 xlO~^, in approximate agreement with the comparative strengths of the NIR and charge-transfer bands. The weakness of the NIR absorption feature relative to

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193 the charge-transfer band coupled with the inversion symmetry of the lattice therefore implies that the 7300 cm~^ feature just below the charge-transfer edge is either a quadrupole excitation or perhaps is a result ofad — d exciton coupled to a symmetrybreaking process. The latter possibility will now be addressed. There are other avenues for photoexcitation on the Co sites. A phonon or magnon could break the local inversion symmetry of the CoOe octahedron and make a d — d exciton weakly dipole allowed. This may be thought of as introducing some odd parity, such as a px character, to the Co3d orbitals. The initial and final states in Eq. 6.2 should then be replaced with the ligand forms | 3dg + a-iPx + ot2Py + otzPz) and (3de + Oi\Px + OL2Py -h 0:3^2 I, respectively. This will introduce and A2u character into the dipole interaction term which may make some previously forbidden dipole terms weakly allowed. For example, consider the case where some 2px odd character is mixed into the 2>dxz and Zdxy orbitals. Then the in-plane dipole matrix elements are of the form, {dxy + Oi\Px I ExX I dxz + ot\Px)6.7 Expanding Eq. 6.3 and computing the direct products for all the terms using Table 6.3 yields a{kig + A25 + Bi^ + 62^ + Ai„ + k2u + Bi^ + B2«) + a^{^u + Ep). Thus, adding some odd character to the ground state makes some previously forbidden dipole excitations weakly allowed. This is also true for the the forbidden quadrupole excitations in Table 6.5. Our conclusions on the nature of NIR impurity band in Sr2Cui_xCox02Cl2 are consistent with transmission studies on La2Ni04 by Perkins et alP^ La2Ni04 is isostructural to Sr2Cui_a;Cox02Cl2 with the exception that the CUO4CI2 octahedron surrounding each Cu site is replaced by the more common CuOe structure. As with Co^+ in Sr2Cui_a;Cox02Cl2, each Ni^+ site incorporates into the La2Ni04 lattice

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194 with more than one partially filled 3d orbital so transitions from the lower lying filled 3d states become quadrupole allowed under D4/1 point group symmetry. The Ni^"*" sites have 3d^ electronic configurations with half-filled 3dj.2_y2 and 3(^3^2 orbital states. Specifically, Perkins et al. found a weak absorption band very comparable in magnitude and in energy to the impurity band observed in Sr2Cui_xCox02Cl2. What is more, they observed the band for incident radiation polarized both parallel and perpendicular to the Cu02 plane. As with our findings in Sr2Cui_a;Cox02Cl2, they attributed the band to 3d — 3d excitons on the Ni sites. To summarize, the weak absorption band observed in the NIR when cobalt is substituted for copper is most likely associated with local 3d — 3d exciton transitions on the former which are either quadrupole coupled or which become weakly dipole allowed when odd character is mixed into the ground state. The absence of this band in the undoped material further suggests that this is not the result of transitions from the lower lying filled 3d states to the highest lying 3d^2_y2 states on the copper sites. Our findings and conclusions agree well with those of Perkins et al. where a similar NIR band was found in La2Ni04 just above leV and was attributed to 3d — 3d exciton transitions on the Ni sites.

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CHAPTER 7 2-MAGNON RAMAN SCATTERING 7.7.1 Introduction As discussed in chapters 3 and 4, Raman scattering is a complementary technique to optical spectroscopy for examining fundamental excitations. An intense power source, typically an He-Ar laser, is used to probe the excitations in the material under study. Only frontand back-end polarizers are needed to tune the polarization of the source beam to the correct symmetry of the excitation. For this dissertation, Raman measurements were the only tool at our disposal to measure the 2-magnon spectrum of La2Cui_xLii04. Thus, 2-magnon Raman scattering provides information concerning the 2D spin order of the Cu02 plane which otherwise could not be ascertained from the reflection data. In the case of Sr2Cui_xCox02Cl2, in addition to providing 2D spin order, 2-magnon Raman spectroscopy bands can be compared to the phonon-assisted 2-magnon bands observed in the absorption coefficient to check for consistency. The experimental procedure was outlined in chapter 4 and a brief tutorial on Raman scattering is given in the appendix. 7.2.1 2-Magnon Raman Scattering Data: La^CuOa The 2-magnon Raman spectrum of undoped La2Cu04 at T=300 and 80K is presented in Fig. 7.1. The band center is located near 3100 cm~^ for both temperature points, but grows in intensity and narrows in breadth as the material is cooled. Within interacting spin wave theory, this band should be centered at 2.72J, where J is the in-plane superexchange energy52,i27 Hence, the 2-magnon Raman spectrum yields an experimental value for J. For the data presented in Fig. 7.1, J is 0.123eV, in 195

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196 8000 7000 La2Cu04 2— Magnon Raman Scattering T=300,80K I c 5000 v> 6000 T = 300K T=80K I. 4000 i 3000 c "c 2000 1000 0 1500 2000 2500 3000 3500 Frequency (cm"') 4000 4500 Figure 7.1. 2-Magnoii Raman Band {Bi — B2 symmetry) in La2Cu04. agreement with other experimental values determined from susceptibility and neutron scattering measurements. It is important to note that the spectra presented here and in subsequent sections have both B\ and B2 character. Bi symmetry gives the energy needed to flip two nearest neighbor spins in the spin 1/2 2D antiferromagnetic background while B2 symmetry has to do with higher order excitations involving the spin exchange of diagonal nearest neighbors. Unfortunately, with linear polarized light it is not possible to separate these two symmetries: it is only possible to find their difference, B\ — B2. Information from circularly polarized measurements is needed to separate all of the symmetries. However, from Fig. 4.7 it is evident that the B2 symmetry channel is better than one order of magnitude weaker than the B\ channel. Thus, to first order, the 2-magnon Raman spectrums presented heretofore may be thought of as arising simply from the exchange of two nearest neighbor spins in the 2D antiferromagnetic background.

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197 (a) La2Cu,.,Li,04 2— Magnon Raman Scattering T=300K K=.00 2500 3000 3500 Frequency (cm"') 7000 •! sooo c sooo *, 4000 = 3000 c C 20O0 1000 0 IS Figure 7.2. 2-Magnon Raman Scattering in {B\ — B2 symmetry) (a) T = 300K (b) T = 80K. 7.2.2 2-Magnon Raman Scattering: La9Cui-.rLiT04 Substituting Li for Cu in the Cu02 plane dramatically reduces the intensity of the 2-magnon Raman signal, as can be seen from Fig. 7.2a and 7.2b. Despite this, the 2-magnon peak width broadens only modestly and the peak position changes little. This can be seen in Fig. 7.3a and 7.3b where all the data for x > 0.00 has been rescaled to match to 2-magnon scattering intensity in the undoped phase at T=300K and 80K. The additional spectral weight between 3500 cm~^ and 4000 cm~^ in the 5% Li-doped sample is due to spurious contributions just of the optical axis in the (b) La2Cu,l,Li',04 , . 2— Magnon Raman Scattering 2000 2500 3000 3500 4000 4500 Frequency (cm"')

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198 BOOO » 5 6000 "S c • 2000 0 1500 2000 2S00 3000 SSOO 4000 4900 Frequency (cm"') Figure 7.3. Rescaled 2-magnon Raman Scattering in {B\ — symmetry) (a) T=300K and (b) T=80K. CCD pixel array which could not be removed by binning techniques. Consequently, the high end of the 2-magnon band observed in 5% Li-doped La2Cu04 was not included in the fitting routines. The relative invariance of the 2-magnon bandwidth to the substitution of Li"'' for Cu^"*" differs substantially from what is observed when divalent Sr is substituted for trivalent La. In Fig. 7.4, the rescaled 2-magnon spectrum for La2-iSr2;Cu04 is presented for T=300K courtesy of Sugai et al}"^^. Here, the intensity of the 2-magnon Raman band broadens dramatically concurrent with a rapid softening of the peak center. For Sr concentrations exceeding x = 0.07 the (b) L02Cu,..Li.O, 2-Magnon Raman ScaWering (Rescaled) T=80K I

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199 i 1 1 1 I ' 1 1 T" 1 1 ' / jSw IN. 2'Magnon Raman • / ^ \ ^ \ / / X V x=0.010 — x-0,034 [Sugii. tf o/L, PRD 643^1 X . 1 1 1 d 1 I.I... Raman shift (cm~') 3000 4000 Figure 7.4. 2-Magnon Raman scattering band vs. Sr concentration at T=300K. From Reference 19. 2-magnon peak energy softens rapidly. This starkly contrasts to what is observed in La2Cui_a;Lix04 where the 2-magnon signal broadens only modestly and the peak energy remains essentially fixed. These key differences are quantified in Fig. 7.5a and Fig. 7.5b where the 2-magnon Raman peak positions and widths of La2Cui_xLix04 and La2-iSrj;Cu04 at T=300K are plotted as a function of doping concentration. Here, the data were fit to a simple Lorentzian for the purpose of illustrating the general trends of the 2-magnon band with Li and Sr doping. Having taken note of the saUent features of the 2-magnon data in La2Cui_a;Lix04 and La2-xSra;Cu04, the next task is to fit the data to a theoretical model and to interpret the results. 1

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200 La2Cu^.j(Li^04, La2.xSrj(Cu04 YO (T=300K) 0 I I ' I ' ' I ' I I I 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Li, Sr Concentration (x) La,Cu,.,Li,04, La^.^Sr^CuO, 1000 -0.00 0.02 0.04 0.06 0.08 0.10 0.12 Li, Sr Cone, (x) Figure 7.5. Lorenztian fit to 2-magnon Raman scattering data of La2_xSrj;Cu04 (Reference 19) and La2Cui_a;Lij;04 . (a) Peak width and (b) peak position vs. x.

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201 7.3.1 Data Analysis: Curve Fitting the 2-Magnon Band When properly analyzed, the 2-magnon Raman band observed in La2Cui_xLix04 and La2-iSra;Cu04 can yield information about the 2D spin order of the Cu02 plane. The 2-magnon line shape has been modeled by several groups^^^"^^^. The most thorough and detailed work was done by Canali and Girven^^^ and so this section will follow their lead. By assuming only nearest-neigbor coupling of the spins in the Cu02 plane and a lattice with square symmetry only the Big mode is Raman active and the scattering Hamiltonian is given by A = 5 ^ P{Einc, Esc] S)Si Si+s, 7.1 i,S where Einc and Esc are, respectively, the incident and scattered polarization vectors and P{Einc, Esc',S) is a site dependent function given by P{E^nc, Esc] S) = [^E,nc " ^5C {S E^nc){S Esc)]7.2 The Raman intensity is determined from the relation 2 ~ I{u}) = ~-ImGA{uj) 7.3 n where G\{u) is the frequency dependent Green's function defined as /-Hoo d{t-t')e''-^'-''^GA{t-t'). 7.4 -oo Here, Ga(^ t') is the zero temperature time-ordered Green's function defined as iGAit t') = (^0 I TA{t)A{t') I V'o) 7.5

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202 By performing a Dyson-Maleev transformation on the spin operators in Eq. 7.5, expanding the result, and keeping terms to first order in the magnon-magnon interaction, the 2-magnon scattering Green function can be written as^^^ Ga,{u) = N{B/2fg{E,ncJsc)z[Sa{S)f ""4 + \{Jih){m + L(2)) i(j/n)2(L(i)L(i) mmy where 1 1 L(^)(a;) = ^ yJji ^ . 7.7 k " Here, = cos{kxa) — cos{kya) is a symmetry factor appropriate for Bi scattering, g{Einc, Esc) = i^inc^sc~ ^inc^sc)"^ , oc{S) is the an Oguchi factor, z is the coordination number of the Cu2+ lattice, Ctmax is the magnon energy at the antiferromagnetic zone boundary, and is a dispersion term given by the expression ^ (1 co5(fc^a) + cos{kya) ^if2 Here, F determines the width of the 2-magnon band and so is a measure of the magnon pair lifetime. By using T as a fitting variable in Eq. 7.6 and Eq. 7.7, we can extract the magnon pair lifetime^^^ xhe magnon pair lifetime, r, is given by r = (h/TJSz) where J is the superexchange energy, S is the sublattice magnetization (1/2), and z is the coordination number (4). Similarly, the magnon mean-free path is A = a/T\/2. The imaginary part of Eq. 7.6 was plotted for several values of T and the full width at half maximum (FWHM) was measured for each. These results were then compared to the FWHM of the 2-magnon spectrums of La2Cui_j;Lia;04 scaled to the intensity of the undoped sample. By careful measurement and comparison of the theoretical and experimental spectrums, experimental values for F were then extracted. The

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203 results for F are presented in Fig. 7.6 as a function Li content for T=300K and 80K. Likewise, X is reported in Fig. 7.7 for different Li concentrations and temperatures. Contrary to what is observed in La2-xSrxCu04 where the 2-magnon band broadens rapidly accompanied by substantial spectral weight loss as hole carriers are added to the system, the (rescaled) 2-magnon band in La2Cui_xLia;04 remains robust. Indeed, the magnon mean-free path only decreases modestly from 22A to 13. 5A in La2Cui_j;Lix04 at room temperature and even less at 80K, from 24A to 16A. 7.3.2 Domain Walls vs. Bound Holes The modest broadening of the 2-magnon Raman band in La2Cui_xLia;04 in the low to moderately doped regime must ultimately reflect the behavior of the holes in the Cu02 plane and, as will be seen, this presents a quandary. If the holes are localized around the Li"*" sites, and the Li+ sites are homogeneously distributed in the Cu02 plane, then one expects magnon scattering length to be ay/l/x. This is based on the premise that holes localized to Li+ sites only weakly spin frustrate the Cu^"*" lattice and that each Li+ site acts as a scattering center. On the other hand, if the holes are able to free themselves to form domain walls, as diagrammed in Fig. 7.8 (Li sites not indicated), the scattering length in the x direction should scale 2 , as L = ^{-), where Sh is the separation between holes along the domain wall. In this simplistic picture, the magnons propagate freely in the phase-separated AF domains until they scatter off domain walls which are separated by an average distance L (see Fig. 7.8). For Sh ~ 2a, the scattering lengths for domain walls and a uniform distribution of charge are plotted in Fig. 7.7 along with the experimental results. From this figure it is obvious that the experimental magnon scattering lengths for La2Cui_a;Lia;04 fall off much slower than predicted by domain wall formation and a uniform charge distribution. Of course, both models treat the spin 1/2 lattice as static when a proper model should take into account the quantum fluctuations

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204 La,Cu,.,Li,0, Fq (Canali-Girvin) T=300, 80K 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Li Concentration (x) Figure 7.6. T for La2Cui_^Lij;04at T=300, 80K. Fit from Reference 131. Magnon Scattering Length T=300, 80K -#300K 80K -ADomain Wall Uniform Dist. 80 70 -r 1" 60 g m 50 D) C 20 10 0 Figure 7.7. Magnon mean free path length for La2Cui_a;Lij;04 at T=80K. The scattering lengths for domain walls and a uniform distribution of charge are also given for a filling fraction of 1/2.

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205 X X X X X X X X X X X Figure 7.8. Domain wall structures (from Reference 137). which are important in spin 1/2 antiferromagnetic systems. Spin fluctuation effects substantially broaden magnetic excitations as is evident by the finite width of the 2-magnon Raman band in La2Cu04. The 2-magnon Raman peak in La2Cu04 is 5-6 times broader that the band observed in its spin 5=1 AF counterpart, K2NiF4. In the latter 5 = 1 system quantum fluctuations are not as dominant and the magnetic excitations can be adequately described by a classical spin model^^^. Much care is needed when comparing the aforementioned theoretical scattering lengths to what is actually observed. The overly simplistic models for charge structures predict an infinite scattering length at rc = 0 since they treat the spin lattice as a static 2D AF system. In fact, the predicted scattering lengths for domain walls far exceed the experimentally observed scattering length in the lightly doped region for X < 0.1 (Fig. 7.7). For a uniform distribution of Li"*" sites, the predicted scattering lengths are still much larger than what is experimentally observed but they are less strongly x-dependent. The large discrepancy between the predicted and experimen-

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206 tally observed magnon scattering lengths indicates that there are additional scattering mechanisms in the 2D AF system which shorten the lifetimes of the magnons. Spin1/2 quantum fluctuations substantially broaden the 2-magnon band in the undoped phase, as is experimentally observed, and subsequently decrease the magnon pair lifetimes. In this sense, the total scattering rate for the magnons may be thought of as sum of independent processes of the form ' ' +1^.... (7.9) Ttotal TAF Tx where taf is the contribution from the quantum spin fluctuations in the undoped phcise and is the scattering rate associated with the charged impurity structures. In this case, A = \{Ttotai) and in principle the scattering rate for the AF fluctuations could be extracted from the data. It is t^f which substantially depresses the experimentally observed scattering lengths below the predictions of a simple static 2D spin model. Another method to investigate the effects of Li"*" on the 2-magnon peak is to examine the derivative of the data as a function of impurity concentration. In this way the curvature of the 2-magnon band may be closely scrutinized. Any redistribution of spectral weight within the 2-magnon peak associated with the growth of sidebands should then be readily apparent. The presence of additional magnon modes will manifest as a shoulder in the 2-magnon peak. This will show up as a notch in the spectrum derivative. The derivative of the 2-magnon Raman scattering spectra for La2Cui_a;Lij:04 at T=80K is shown in Fig. 7.9. The data has been Fourier-filtered to eliminate spikes associated with high frequency noise in the unprocessed data. It is readily apparent that there is no consistent trend in the growth of a discernible notch outside of fringing, particularly egregious above ~ 3300 cm~\ due to Fourier filtering. This suggests that the 2-magnon Raman data are inconsistent with both

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207 Raman shift (cm' ) Figure 7.9. Derivative of T=80K Raman Spectra for La2Cui_xLia;04. the uniform charge distribution (holes bound tightly to their Li"*" hosts) and domain wall pictures. This may be understood within the context of a simple Ising bond counting argument. In both pictures the spin order of the 2D lattice in the regions buffering the separate holes or domain walls is well preserved. The energy need to create two magnons at the magnetic Brillouin zone boundary near the center of the AF regions is roughly 3 J since this is the energy required to flip two spins in a (static) spin-1/2 AF background (Fig. 7.10(top)). However, the energy cost to flip two spins with one adjacent to a static vacancy or domain wall drops to 5 J/2 since there is now one fewer Ising bond to break (Fig. 7.10 (bottom)). Hence, we would expect to see a sideband in the 2-magnon Raman data red shifted roughly J/2, or ~ 500 cm~^ from the principal band at ~ 3J. The absence of a discernible feature in Fig. 7.9 near ~ 2500 cm~^ suggests that static charged structures are not responsible for the

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208 O i 1 : 1 1 o \ 1 I 1 ^ ^ A/ O t I t 1 ! O t 1 Ay Hole (s=0) I Cu (s=l/2) J Cu (spin frustrated) ^ Broken AF Bond (J/2) 1 O i 1 I I 1 ! O I o I o I Ay tiff 1 O t 1 t t 1 ! 1 O t 1 Figure 7.10. Schematic representation of 2-magnon Raman Scattering process away from a domain wall (top) and with one spin flip adjacent to a hole (bottom). robust character of the 2-magnon Raman peak in La2Cui_xLia;04. The insulating nature of La2Cui_xLia;04 further implies that the domain walls are not mobile. Since there is no evidence for a second peak below ~ 3J in Fig. 7.9 the modest broadening of the 2-magnon peak in La2Cui_xLij;04 is most likely tied to the spin frustration of the 2D Cu^"*" lattice. One possibility is that the Li holes form Zhang-

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209 X X X X X X X Figure 7.11. Schematic diagram of pinned stripes in the Cu(Li)02 plane. Rice singlets with the four Cu^"*" sites neighboring Li sites^^'*. In the lightly-doped regime {x < 0.02) this will moderately spin frustrate the Cu^"*" lattice and weakly broaden the 2-magnon peak which is consistent with what is observed. However, in the moderately doped regime (0.05 < a; < 0.10) the average separation between the Li sites drops to 3-5 lattice spacing, so the Zhang-Rice singlets would begin to push up against each other. In this case one would expect strong frustration of the spin lattice and a rapid broadening of the 2-magnon peak. Aside from a loss of spectral weight with increasing Li content, the 2-magnon band broadens little up to x = 0.10. Thus, Zhang-Rice singlets are an unlikely candidate in the moderately doped regime. An intriguing model to explain the nature of the 2-magnon peak in La2Cui_xLia;04 is the idea that Li holes form pinned stripes with transverse chargefluctuations^^^. In this picture the holes form stripes that link neighboring Li sites as diagrammed in Fig. 7.11. The Li sites effectively anchor the domain walls and prevent conduction of the charge structures in the Cu02 plane. However, the holes

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210 12 2 12.4 126 12.S 13 a Frequency (MHz) O A1 (M 0.019) J A Bl(x 0.020) B2 (K 0.025) so 100 ISO 200 2S0 300 Temperature (K) Figure 7.12. ^^^La NQR data from Hammel et al.^^ .(a) A = z/i 1/2 vs. T. The solid curves are fits to the critical behavior in the absence of a spin-ordered phase transition, (b) ^^^La nuclear spin-lattice relaxation rate. From Reference 136. are not rigorously bound to the domain walls and may lower their kinetic energy by perpendicularly hopping a few lattice spacings as sketched in Fig. 7.11. These transverse charge-fluctuations locally spin frustrate the lattice near the domain walls while leaving the spin order of the AF region near the center comparatively intact. The energy cost to flip two adjacent spins, in the context of a simple Ising picture, near the center of the AF regions is still ~ 3 J, but it is no longer well defined in the regions adjacent to the stripes. Consequently, the 2-magnon peak at 5/2 J would be smeared into a continuum and no peak signature would be observed in Fig. 7.9 below 3J. This is consistent with the observations of Figs. 7.3 and 7.9. Domain walls have been introduced in recent NQR measurements^^^ of lightly doped La2Cui_xLij;04 to account for the anomalous behavior of the ^^^La resonance line below 30K, so it would be interesting to reconcile the NQR results with the 2-

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211 magnon scattering data. However, care must be taken since the time scales of the two measurements are orders of magnitude apart. NQR measurements are performed with alternating magnetic fields on the order of 50 300kHz (?s 5 — 10//s) while Raman scattering occurs on a time scale of roughly 10~^^s. Furthermore, NQR studies are more local probes of the spin order than Raman scattering. The results of Hammel et al. are presented in Fig. 7.12 for x = 0.019, 0.020, and 0.025 (3D AF phase). The ^^^La splitting, 5 is uneventful at high temperatures for x > 0.00, decreasing with increasing Li content as expected since the Li holes should introduce some spin frustration. Little spin frustration is needed to break the AF coupling between the planes since J±, the superexchange energy between neighboring Cu02 planes, is better than two orders of magnitude weaker than its in-plane counterpart. Below 30K, however, S begins to increase and at T=0 recovers its undoped value for all the Li concentrations in the 3D AFO phase. The same phenomenon^"^^ was observed by Borsa et al. in the 3D AFO phase of La2-xSra;Cu04. Hammel et al. attributed this unexpected recovery to the formation of the charge carriers into domain walls with a filling fraction of 0.5 (one hole every 2 lattice spacings along the domain wall). Domain walls preserve the 2D AF spin order in the (antiphase) regions separating neighboring walls and thus should also preserve the interplane spin coupling. In this way the ^^^La resonance signal is preserved. Above 30K, Hammel et al. argue that either (1) charge fluctuations transverse to the domain wall, fast enough now in NQR measurements to be observed, suppress the resonance signal, or (2) the domain walls "evaporate". Case (1) requires only a modest diffusion rate, D, for the holes. For y/Drgxp — Ix, where Tgxp is the characteristic experimental time for NQR measurements (~ 5 — 6//sec and Ix is the mean distance between domain walls. For square domains of size l^ and a wall filling fraction of two — ^ where a is the Cu-Cu lattice spacing and x is the Li content. Thus, d ~ x'^°rexp ~ 10~^cm^/sec. The combination of such a low carrier density with

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212 such a low rate of diffusion implies an unobservably small a{uj) for our measurements. Hence, this does not disagree with our results. Case (2) implies a transition between bound and mobile carriers, and would imply a redistribution of spectral weight in the FIR as the bound carriers are thermally activated off of their respective Li hosts above 30K. We see no such change in the FIR spectral weight above vs. below 30K, and hence find this explanation inconsistent with our data. The relative invariance of the 2-magnon band to the substitution of Li+ for Cu^"*" , and the subsequent deduction that the Li holes do not substantially frustrate the spin lattice, raises an quandary: If the Li-holes act as site vacancies from a magnetic standpoint, then why is 3D LAFO destroyed by a; ~ 0.03? That is, if the Li-holes have little effect on the 2D spin order of the Cu02 plane, then what accounts for the abrupt loss of 3D AFO in La2Cui_a;Lia;04. There appears at the moment to be no simple answer to this question. In La2Cui_xZna;04, another cuprate system where the substitution of Zn'^'^ for Cu^"*" introduces a spin-zero impurity into the Cu02 plane (but no charge), 3D spin order is not lost until roughly x = 0.30. This is ten times the Li concentration needed to destroy 3D AFO. Since the most outstanding difference between these two systems is the presence of charge (hole) carriers in La2Cui_xLia;04 this is the most likely candidate for the discrepancy. Since Zn^"*" goes into the Cu02 plane with the same valency as Cu^"*", it does not introduce charge carriers into the system. The presence of charge carriers, even if localized, is clearly detrimental to the interplane spin coupling. The superexchange coupling energy between adjacent Cu02 planes is nearly 3 orders of magnitude weaker than its in-plane counterpart (3-5meV vs. ~ 0.125eV). The interplane magnetic coupling depends on sensitive anisotropic exchange interactions which may degrade rapidly in the presence of charge impurities and lattice distortions^''. To further obscure matters, Li+ goes into La2Cu04 with a closed 2s shell configuration whereas Zn^^ has a 3rf* configuration comparable to

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213 Cu^"*" {3(fi). It is reasonable to suspect that the substitution of a closed 2s shell for a partially filled Cu3d^ shell may disrupt the complex anisotropic exchange interactions binding adjacent Cu02 planes. 7.4.1 2-Magnon Scattering in Src>Cui_rCoTO?!Cb (b) St'zCui.^Ci.bzClz' ' 2-Magnon Raman Scattering T=80K 1500 2000 2500 3000 3500 4000 4500 Frequency (cm"') Figure 7.13. 2-Magnon Raman scattering in Sr2Cui_a;Coa;02Cl2. (a) T=300K. (b) T=80K. Having seen the eflfects of charge carriers on the 2-magnon Raman band, the next task is investigate the effects of magnetic impurities on the 2D spin lat-

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214 (b) Sr'2Cu,_,Ci^02Cl2 . 2-Mognon Raman Scattering (Rascaled) T=80K 1500 2000 2500 5000 3500 4000 4500 Frequency (cm~') Figure 7.14. Rescaled 2-magnon Raman scattering in Sr2Cui_xCoa;02Cl2. (a) T=300K. (b) T=80K. tice. In Fig. 7.13a and Fig. 7.13b the 2-magnon Raman scattering intensity of Sr2Cui_j;Coj;02Cl2 is reported for T=300K and 80K, respectively. Cobalt apparently goes into the Cu02 plane with the same valency as copper but with a 3(f shell configuration in a high spin state (S=3/2). Thus, each Co should act as a spin-3/2 impurity in the spin1/2 AFO background. In the absence of additional charge carriers which customarily spin frustrate 2D and 3D long range spin order, little change is expected in the 2-magnon Raman band with the substitution of Co^"*" for Cu^^. From Fig. 7.13a and Fig. 7.13b this is to a large measure what is observed. Aside from a steady decrease in spectral weight from a: = 0.00 to x = 0.12, the 2-magnon band

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215 broadens little. To better illustrate this, in Fig. 7.14a and Fig. 7.14b the 2-magnon bands of the two Co-doped samples are rescaled to the intensity of the pure (x = 0.00) material. The results of a simple Lorentzian fit to the bands are shown in Fig. 7.15a (band center) and Fig. 7.15b (band width). Much like 2-magnon Raman scattering in La2Cui_xLij;04 the band center remains fixed, to a first order approximation, while the band width broadens modestly. 7.4.2 Analysis and Interpretation: Sr^^Cui-rCoxO^Ch Closer inspection of Fig. 7.13b reveals that a shoulder grows in the Co-doped samples below 3000 cm"~^ This is readily apparent in Fig. 7.16 where the derivatives of the T=80K Raman spectrums are shown. A notch in the derivative spectrum centered at ~ 2640 cm~^ grows for x > 0.00. We speculate that this sideband is associated with two nearest neighbor spin flips adjacent to a Co site as discussed in the previous section. If the spin-3/2 Co^"^ sites are taken to be decoupled from the spin-1/2 AF background (J' = 0), then the energy cost this process should be J/2 down from the 2-magnon peak centered at ~ 2900 cm~^ In Sr2Cu02Cl2 J ^ 1075 cm~^ so, if this model is correct, the sideband should appear at 2900 cm^^ cm~^ ~ 2360 cm~^ From Fig. 7.16 the secondary peak emerges at ~ 2635 cm~^ The difference between the expected peak position and the observed peak position should provide a rough measure of the Cu-Co superexchange coupling energy. Hence, J' ~ 2635 cm-i 2360 cm'^ ~ 275 cm-\ or roughly 25% of J. This is in close agreement with the estimation of J' extracted from Table 6.1 in Section 6.5.3 . As with La2Cui_a;Lia;04, the 2-magnon Raman bands in Sr2Cui_a;Coi;02Cl2 were fit by Eq. 7.6. The band width, F, was determined for each sample at T=300K and 80K and estimations for the magnon lifetimes and scattering lengths, A, were then extracted. The results for F and A are plotted in Fig. 7.17 and Fig. 7.18 respectively.

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216 Sr2Cui.xCOx02Cl2, La2.xSrxCu04 ©0 (T=300K) 4000 Figure 7.15. Lorenztian fitting parameters for 2-magnon Raman band for Sr2Cui_j;Coa;02Cl2. (a) Band center and (b) band width vs. x.

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217 o o E c/i n u VI JD ta "« c ,M 'ui c I 0.00 n a o u > a 1500 2000 2500 3000 3500 4000 ' Raman shift (cm ') Figure 7.16. Derivative of T=80K Raman spectra for Sr2Cui_iCox02Cl2. The results are consistent with expectations. Since Co impurities do not introduce charge carriers and act only as localized scattering centers, there is no possibility of domain wall formation and the magnon scattering lengths associated Raman scattering should be consistent with a uniform distribution of localized scattering sites. As such, the magnon scattering lengths, in the absence of spin fluctuations, should scale as a/y/x. From Fig. 7.18, this is observed to a rough approximation. The experimental scattering lengths fall off slightly more slowly than theoretical predictions and are well below expected scattering lengths. The latter observation is associated with the quantum spin fluctuations of the spin-1/2 lattice which severely curb magnon lifetimes. As with La2Cui_xLia;04 this suggests that the scattering rates should be treated as a sum of individual contributions from the Co^"*" scattering centers and the quantum spin fluctuations of the lattice as per Eq. 7.9. Since the experimental scattering lengths are all well below theoretical expectations for a static spin lat-

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218 tice, our results are consistent with a uniform distribution of scattering centers in spin-1/2 AF background. Finally, the striking similarities in the scattering lengths of Sr2Cui_xCoa;02Cl2 and La2Cui_j;Lij;04 corroborates the idea that the holes introduced in the latter do not have extended character. That is, for the magnon scattering lengths in La2Cui_j;Lix04 to mimic those observed in Sr2Cui_xCox02Cl2 where there are no charge carriers, the holes must be localized to Li04 plaquettes or domain walls so as not to strongly frustrate the spin lattice. 7.4.3 PhononAssisted Mutimagnon Absorption Revisited In the previous chapter, the weak, sharp absorption bands observed in the midinfrared in Sr2Cui_a;Coa;02Cl2 were interpreted as phonon-assisted 2or 4-magnon excitations. Since the lowest and fortunately strongest excitation creates two magnons in the Cu02 plane, it would be useful to reconcile the phonon-mediated 2-magnon optical band with the 2-magnon Raman scattering data. In Fig. 7.19 the MIR optical conductivity (rescaled) and Raman bands of Sr2Cui_a;Cox02Cl2 are plotted at T=90K and 80K respectively. It is readily apparent that the two bands overlap nicely aside from a 300 cm~^ offset in band position due to the absence of phonons in the 2-magnon Raman scattering process. This close agreement in the band centers buttresses the phonon-assisted multi-magnon scattering model discussed in Section 3.5.1. The 4-magnon processes, prominent in the optical conductivity for reasons still not well understood, are not observed in Bi — Raman scattering symmetry. Hence, the agreement between the two 2-magnon bands is obscured above 3200 cm~^ and the data were not fit above this threshold. In Fig. 7.20 the bandwidths, extracted from Lorentzian fits to the two bands, are plotted as a function of Co concentration. As was done in Chapter 6, a fixed linear background was added to the optical conductivity fit to simulate the linear tail extending from « 1000 cm~^ to the charge f transfer band. From Fig. 7.20, the phonon-mediated 2-magnon optical band and

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219 0.24 (Canali and Girvin) T=300, 80K 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Co Concentration (x) Figure 7.17. F vs. x for Sr2Cui_xCoa;02Cl2. Fit from Canali and Girvin (Reference 131). SrXu. Co OnCL Magnon Scattering Length T=300, 80K 100 300K 80K Uniform Dist. Domain Walls 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Co Cone, (x) Figure 7.18. Magnon mean free path length for Sr2Cu02Cl2. The scattering lengths for domain walls and a uniform distribution of charge are also shown for a filling fraction of 1/2. Fit from Canali and Girvin (Reference 131).

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220 Figure 7.19. 2-Magnon Raman scattering (T=80K) and 2-magnon + 1 phonon optical absorption (T=90K) of Sr2Cu02Cl2. 2-Magnon Scattering Bandwidth 800 700 -600 -500 i 400 300 -200 -100 0 Opt. Abs. (2 Magnons+1 Phonon) T=90K 2-Magnon Raman Width T=80K + 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Co Cone. (X) Figure 7.20. 2-Magnon Raman scattering and phonon-assisted 2-magnon absorption bandwidth vs. Co concentration. the 2-magnon Raman scattering band broaden at roughly the same rate for the Co concentrations investigated. The Raman band, however, is broader than the optical

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221 band by more than a factor of two. The comparatively long lifetime of optical band is due to the nature of the phononassisted 2-magnon interaction. The bimagnon state is driven by phonons at the Brillioun zone edge where the phonon dispersion is flat. Hence, the are no states nearby in A;-space into which a phonon can decay and impart a large fraction of its energy to the system. Consequently, it takes a long time for the phonon-assisted 2-magnon process to evaporate into the lattice and the bimagnon band is sharp. By comparison, the 2-magnon Raman scattering lifetime is limited by quantum spin fluctuations which severely curtail the magnon scattering lengths to 4-5 lattice spacings. Consequently, the magnon lifetimes (scattering lengths) observed in Raman scattering are significantly shorter than those observed in the optical data. Hence, the 2-magnon Raman scattering band should be significantly broader than its phonon-assisted optical counterpart. This is consistent with our observations. ,^ 7.5.1 Concluding Remarks In summary, we found that the 2-magnon Raman scattering bands in La2Cui_xLia;04 and Sr2Cui_a;Cox02Cl2, aside from a loss of spectral weight with impurity doping, remained robust over the Li and Co concentrations investigated. This contrasts dramatically with La2-xSrj;Cu04 where the 2-magnon Raman band decays rapidly for x > 0.03. We associate the discrepancy with differences in the effects the impurity sites have on the spin order of the Cu02 plane. Sr adds holes to the Cu02 plane which are known to strongly spin frustrate the lattice, so a rapid broadening of the 2-magnon band in the moderately-to-optimally doped regime is not surprising. By contrast, the robust character of the 2-magnon Raman bands in La2Cui_a;Lia;04 and Sr2Cui_a;Coa;02Cl2 suggests that neither Li nor Co severely frustrates the 2D spin lattice. Since Li substitution adds hole carriers to the Cu02 plane, we speculate that the holes form pinned stripes with large transverse charge

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222 fluctuations. This model should preserve the 2D spin order in the centers of the sequentially alternating AF regions while introducing strong spin fluctuations adjacent to the stripes which obscure the secondary sideband centered at ~ J/2 down from the principal 2-magnon band. The scattering lengths extracted from the data are consistent with such a model. Substituted Cobalt, by contrast, adds no holes to the Cu02 plane and should incorporate into the lattice with a 3(f configuration in a high spin state. Thus, a sideband may be expected to splinter off at ~ "^-f^ down from the principal 2-magnon band, where J' is the Cu-Co superexchange energy. Evidence of a secondary peak in this energy region was observed in the derivative of the Raman spectra and J' was estimated at ~ 275 cm~^ (35meV). This is consistent with estimations for the Cu-Co superexchange coupling energy from the optical data. Finally, a comparison of the 2-magnon Raman data to the transmission data of Sr2Cui_xCoa;02Cl2 in the MIR supports the increasingly probable picture that the sharp bands observed in the latter are due to phonon-assisted multi-magnon processes.

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CHAPTER 8 CONCLUSION 8.1.1 Summary This dissertation has been devoted to the study of the optical properties of La2Cui_a;Lia;04 and Sr2Cui_a;Coa;02Cl2 with the hope that it may explain some of the anomalous spectral features found in the superconducting cuprates. Reflectance measurements on La2Cui_xLix04 revealed that, depending on the spectral region, it both resembled and starkly contrasted with its superconducting brethren La2-xSra;Cu04. Contrary to what is found in La2-xSra;Cu04, no Drude tail was observed in La2Cui_xLix04 in the far-infrared. This is consistent with d.c. transport measurements that place La2Cui_j;Lix04 in the insulating regime. Significant differences were also noted in the charge-transfer region. The charge-transfer band eroded quickly in La2-xSrxCu04 and yet remained comparatively robust in La2Cui_xLix04. A similar trend was observed in the 2-magnon Raman scattering data. The 2-magnon Raman band broadened rapidly in La2-xSrxCu04 suggesting that the holes introduced by Sr strongly spin frustrated the 2D Cu+^ spin lattice. By contrast, the 2-magnon band in La2Cui_xLix04, aside from a loss in spectral weight, broadened only modestly. This suggested that the Li holes remained tightly bound to their host or formed domain walls in the Cu02 plane. The 2-magnon Raman spectra of La2Cui_xLix04 was consistent with pinned domain walls with transverse charge fluctuations. In the hghtly doped regime (a: < .03) the 2-magnon Raman spectra was consistent with mobile stripes thought to be observed in the NQR data. It was also speculated that differences in the relative 2D spin disorder between La2-xSrxCu04 223

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224 and La2Cui_xLix04 may also account for variations in their respective charge-transfer spectral weights. Despite their vastly different behaviors in the far-infrared and charge-transfer spectral regions, La2-xSra;Cu04 and La2Cui_2;Li2;04 were astonishingly similar in the midinfrared. Specifically, a midinfrared band grew nearly linearly with Sr and Li content in the low-to-moderately doped regime. The origin of this band has been a contentious issue since it was observed in the superconducting cuprates more than a decade ago. In this dissertation we investigated several models that have been introduced over the years to explain this band, including polaron coupling, Zhang-Rice singlet formation, stripes, and magnetic strings. Unfortunately, none of these mechanisms adequately unraveled this quandary in the midinfrared. The polaron model lacks the correct temperature dependence while a Zhang-Rice picture is inconsistent with the presence of a nearly identical MIR band in electron doped systems. Furthermore, the existence of a similar, albeit narrower, band in La2Cuo.5Lio.5O4 would seem to rule out the possibility that the MIR band is tied to flipped spins in the 2D AF background as doped carriers hop from site to site. Ordered charge and spin structures, be they static or dynamic, were not ruled out entirely but it is clear that more work needs to be done in this field before anything definitive can be said concerning their relationship, if any, to the MIR band. Lastly, the presence of a MIR band in insulating La2Cui_a;Lix04 came as something of a surprise and would seem to preclude any direct relationship between this band and the highTc superconducting phase found in the cuprates. The optical properties of Sr2Cui_j;Co2:02Cl2 were less enticing than those of La2Cui_a;Lia;04 but by no means any less important. The Co-doped species diflFered only modestly from its insulating host since magnetic impurities and not charge carriers were added to the Cu02 planes. Aside from variations in the phonon energies

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225 in the doped and undoped systems, little change was observed in the charge-transfer spectral weight and, in the absence of excess charge, no broad, heavily weighted MIR band was found. However, weaker and narrower MIR bands, chalked up to phononassisted multi-magnon absorption, were observed in the transmission data. The least energetic and most heavily weighted of these bands was found to harden and broaden with increasing Co concentration. This behavior was consistent with AF magnon dispersion in the presence of a decoupled spin impurity, or vacancy site. This suggested that the Cu-Co superexchange bond in the Cu02 planes was roughly 20% that of the host Cu-Cu bonds. The weakening of local AF bonds when Co is substituted for Cu was also found to be consistent with trends in the superexchange parameter across the periodic table noted by Zaanen and Sawatzky^^^. The 2-magnon Raman data demonstrated that the magnon lifetimes increased only modestly with Co substitution. Here, a secondary band associated with a two spin-flip process adjacent to a Co site suggested that the Co sites behaved as static vacancies and had little effect on the 2D spin order of the Cu02 plane. It was hoped at the beginning of this thesis that optical studies of La2Cui_a;Lia;04 and Sr2Cui_a;Coi02Cl2 might shed light on some of the perplexing spectral features of the superconducing copper oxides. However, it seems that we have failed in this endeavor. We say this with some measure of pride. This is especially true for the MIR band observed in the high Tc superconductors and now in La2Cui_xLia;04 as well. The presence of this band in La2Cui_j;Lia;04 coupled with a conspicuous lack of a Drude tail dispelled any lingering theories that it was simply spill over from the free carrier contributions in the far-infrared. Unfortunately, it also raised some uneasy questions about a myriad of models introduced in recent years to explain its origin. Hopefully, our disquieting findings will spur others to devise new theoretical constructs, or perhaps just alter the existing ones, to slay this dragon. Also, to our

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226 great surprise, the spectral behavior of La2Cui_xLix04 in the charge-transfer region was very different from expectation. For now, all we can do is speculate that the relative robustness of the charge-transfer band in this material is related to the 2D spin order of the Cu02 planes. Conversely, the near invariance of the charge-transfer band to Co-doping suggests that it is insensitive to magnetic impurities and static vacancies. This was supported by the phonon-mediated 2-magnon band observed in Sr2Cui_a:Coa;02Cl2 which suggested that the Co-sites were nearly decoupled from their AF host. Numerical studies of the effects of vacancies on the spin order of spin-1/2 AF host have consistently shown that they enhanced the local 2D spin order parameter on neighboring sites but that they also introduced long range spin disorder. So, despite our efforts, the enigmas persist. 8.2.1 Future Experiments There are some experiments that could conceivably lift the fog that currently clouds our understanding of the two systems investigated in this thesis and possibly even of the superconducting cuprates. Since it seems increasingly likely that spinorder has a hand in many of the spectral features observed in La2Cui_xLia;04 and Sr2Cui_xCoi02Cl2, neutron scattering measurements, both elastic and inelastic, on these materials would prove useful. This may, for example, help explain why 3D long range spin order is destroyed at 2-3% doping concentration in both La2Cui_a;Lix04 and La2-xSra;Cu04 but 2D spin order, at least locally, seems to be preserved in the former. Such measurements may also support or dispel the possibility of stripe formation in La2Cui_xLix04 and may yield direct information on the magnetic coupling strength of Co to its AF host in Sr2Cui_j;Coj;02Cl2. High pressure transmission and reflection studies of La2Cui_xLix04, particularly in the MIR, may also prove useful. The former would reveal the effects of pressure and of a spin zero impurity on the phonon-assisted multimagnon band while the latter would determine how the heavily

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227 weighted MIR band tracks with pressure. Any changes in the transmission and/or reflection would support theories that the MIR spectral properties are related to the superexchange energy and hence to magnetism. Not undertaken for this dissertation was a study of the c-axis properties of La2Cui_xLi3;04 and Sr2Cu02Cl2. This too may be a worthwhile future endeavor. Another interesting and potentially important set of projects would involve optical and magnetic measurements on isostructural La2Nii_a;Lix04. This would offer a glimpse at the effects of Li-doping on the spin-1 equivalent of the single layer superconducting cuprates. It was hoped that optical measurements on this spin-1 counterpart to La2Cui_xLix04 could be included in this dissertation, but a faulty furnace coupled with a temporary lack of funding to make repairs stifled this effort. Since it is unlikely that I will have a hand in these measurements, I hope that good fortune finds whoever chooses to undertake these studies, and finds them well. . ^ ' .

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APPENDIX: RAMAN SCATTERING Introduction The interaction of light with solid materials generally falls into three categories. The first is absorption, where incident light releases all of its energy to excite various processes in the material. The second is elastic scattering. In this process the incident radiation may change its direction of propagation but suffers no energy loss. And lastly there is Raman scattering. In a Raman scattering process the energy of the incident light shifts by either absorbing energy from (Anti-Stokes scattering) or imparting energy to (Stokes scattering) excitations in the scattering medium. Raman scattering generally involves second, third and fourth order terms in perturbation theory and so is a multi-step process that couples initial and final states with one or more intermediate states. The theoretical underpinning of Raman scattering is far too complex and riddled with too many nuances to be elucidated in the short span of a few pages. The purpose here is simply to outline the principles of Raman scattering and apply the theory to the specific cases of oneand two-magnon Raman absorption in antiferromagnetic systems. In the interest of space and concision the classical and semi-classical approaches to Raman scattering, while useful and illuminating in their relative simplicity, will be skipped in favor of devoting the limited space afforded here to the more general quantum mechanical method. We begin with a brief review of the interaction of light with a charged particle. 228

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229 The N-Particle Radiation Hamiltonian For an N-particle system, the full radiation Hamiltonian for light coupled to a charged particle takes the form N ^ N t=l 1=1 where qi and mj are the charge and mass of the i"* particle, fi and pi are its coordinate and momentum operators, Ai and (p are the vector and scalar potentials of the field, and V represents all the other interaction terms in the Hamiltonian not associated with the radiation field. By expanding the bracket in Eq. 1 and collecting terms, we get H = Ho + Hi + H2 (2) where ^ 1 N i N 2 ^2 = I ^>-^) I' +-^) (7)

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230 H2 = Y}^\A{n,t)\\ (8) i Since Hq describes only the kinetic motion of the charged particles in the absence of an externally applied electromagnetic field, the interaction Hamiltonian for the externally applied field reduces to Hint=Hi+H2. (9) — Recasting the vector field, A, in second quantized form and working within the interaction representation^^® yields k,i V k,i Substituting Eq. 9 into Eqs. 7 and 8 gives the final second quantized forms of Hi and H2 which we will use: H, = ^(f, t) = f x: 1; JS^K i)lH/''+ « ^-''-'V(10) '=1 k,i V '^'^ k,i N ,2 The two terms with double creation or double annihilation operators result in a twophoton absorption or emission. These processes are important in non-linear optics and higher order Raman scattering, but they are not relevant to magnon interactions discussed in this thesis and so will not be addressed. Therefore, Eq. 11 reduces to

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231 Scattering Cross Section and Perturbation Theory To calculate the scattering cross section it is first necessary to use Fermi's Golden Rule which specifies the number of optical transitions per second into all possible final states: W = y XI I T |2 6[{Ef + hujs) {Eq + huji)\. (12) / Here, T is the so-called 'T matrix' factor that depends on the order of the timedependent perturbation theory that we are using, ujj and ujs are the frequencies of the incoming and outgoing radiation, respectively, and Ef and Eq are the final and initial states of the system. T is given by the expressions T={f\H,nm (l^'Order) (13) ^^y {f\H.nt\m){m\H,r.t\0) ^^n^Order) (14) T= (/l-^md^i) jmilHrntlm) {mlHjntlO) (s'-'^Order) (15) vn^2 {Eo Em,){Eo Em^) Equation 12 gives the total transition probability per photon into all final states. To get the scattering diff"erential cross section, which is actually measured, we must multiple Eq. 12 by the number of allowed final photon states per unit frequency per unit solid angle, divide by the incident photo flux, and then take the thermal average. The result is ^-'^^i^i^^'m^^s)-(EM]), (16) where V is the volume of the system, nj and ris are the indices of refraction at and UJS respectively, Nj is the photon occupation number, and c is the speed of light. The

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232 thermal average in Eq. 16 can be expressed as a Fourier transform in uj — ujj ljs of a correlation function, 1 r°° + (Tt(f)r(0))e"*^'d<, (17) oo by using the density matrix p, the integral form of the delta function, 1 />oo 27r/i J_oo and expressing the operators of the T matrix in the interaction representation. The correlation function of the T matrix can be split into a correlation of electron operators and a term called the Raman tensor. The electron correlation function can be solved using Green's function techniques^^^ . The Raman tensor contains the symmetry information of the scattering process and depends on the polarizations of the incoming and outgoing photons. Calculation of One-Magnon Raman Scattering Cross Section To calculate the scattering cross section for one-magnon absorption in a magnetic medium, we must employ third-order perturbation theory. The reason for this will become clear shortly. The first step is to fashion an electron-magnon interaction Hamiltonian. This is accomplished by assuming that the electrons and magnons interact via the electric-dipole spin-orbit coupling mechanism. The interaction Hamiltonian takes the form, Hem = E M,^.(at^ + aj^_^.)cl^s^^, (19) k,q where the coupling tensor M^^^ represents the electron-magnon spin-orbit coupling, at and a are the magnon creation and destruction operators obeying boson computation rules, ct and c are the electron creation and destruction operators obeying

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233 fermion anti-commutation rules, and k and q are the momentum of the coupled electron and photon respectively. The total interaction Hamiltonian to be used in the T matrix is now taken to be Hi + i/em, where Hi couples the photons to the electrons and Hem couples the electrons to the spin lattice. There are now a total of three possible non-vanishing terms in the T matrix. Each term must have Hi appearing twice (once each to couple to the incoming and outgoing photons to the electrons) and Hem appearing once to couple the electrons to the magnons. Hence, the need to use third order perturbation theory in Hi and HemThe total T matrix therefore takes the form {f\Hi\mi){mi\Hep\m2) {m2\Hi\0) mi,m2 {Eo Em,){Eo Em2) _l_ {f\Hi\mi) (mi|gi|m2) (m2|iJem|0) (£"0 Emi){Eo — Em2) ^ {f\Hep\'mi) {mi\Hi\m2) (m2|//i|0) j ^^q) {Eq — Emi){Eo — Em2) where | mi) and | 1712) are the total electronic and magnetic intermediate states of the system. Since we are interested in Stokes scattering the following discussion will be limited to the case where a photon is absorbed and a magnon is emitted. The matrix elements in Eq. 20 are evaluated by introducing many-body Bloch states for the electrons, magnetic spin states for the magnons, and then converting both to an occupation-number representation to take advantage of the creation and destruction operators appearing in Hi and HemFurthermore, the initial and final electronic states are the same since all energy and momentum changes are transferred to the magnons. By re-expressing J^j e^'''^''^^'^^ which falls out of Hi and H2 from the photon field contribution (Eq. ) as

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234 and grouping this with the magnon operators of Hem, the first term in Eq. 20, for example, will have products of the electron operators of the form 0 1712) {m2 'k'+q,b'^k'r After summing over the intermediate states, Eq. 22 simplifies to = 5(22) (23) k"+ki "-Q,ki—ks' Using Eq. 23 to account for contributions from the photon field and to clean up the momentum indices, Eq. 20 becomes he y/Wl _ y^ j^ ^^^. j / y J ^ ^ 2 y eo m n / n 5 y/ZJ/oTg where Rg{u}M',IS) is the Raman tensor alluded to earlier: (24) k,b b'r x{[ + [ bk\pis b'k)(b'k b"k)(b"k p-€l bk'^ [Eb{k) Eb>{k) + huji hujM][Eb{k) Eb^'{k) + hujr] bk p-er b'k)(b'k P-^kg y'it^ (b"k bk [Eb{k) Eh>{k) hus hujM][Eb{k) Eb^^{k) hus] + [ + [ + [ bk b'Pj (b'k pii 6"it^ (b"k Mj^^ bk"^ [Eh{k) Ef,'{k) + hui hujM][Eb{k) Eb"{k) hcvM] (bk p-ei b'k) (b'k p-es b"k)(b"k M, kq bk^ [Eb{k) Eb'ik) hus hujM][Eb{k) Eb^^{k) hujM] bk ^kq b'k) ( b'k p-ej b"k)(b"k P-^S b"k [Eb{k) Eb'ik) + hour hujs][Eb{k) Eb"{k) + hcoj] bk ^kq b'k) ( b'k P-^s b"k) (b"k p-ei bk [Eb{k) Eb'ik) + huJi hujs][Eb{k) Eb^^{k) hcjs]^^' ^^^^ Here, | ruf) and | mo) are the final and initial magnon states in the occupation representation and | b'k^) are the electronic states with momentum k^ and band

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235 index i to which the magnons couple. The sensitivity of the Raman scattering cross section to the incident and scattered photon polarizations now becomes apparent from Eq. 25. The total scattering cross section for the magnons can now be calculated by substituting Eq. 25 into Eq. 16 and putting the thermal average in explicitly as in Eq. 17. The result is (26) where {b^{t)bi{0)) is the magnon correlation function. This can be evaluated using Green's function techniques^^*^. 2-Magnon Raman Scattering in Antiferromagnetic Systems The extension of the above discussion to 2-magnon Raman scattering processes is straightforward. However, we now must use fourth order perturbation theory with the spin-orbit coupling interaction Hamiltonian given in Eq. 19. After operating with the photon operators as prescribed in Eqs. 22-24, the six possible arrangements of the four interaction matrix elements yield a total of twelve terms to the T matrix. This is a mess. What is more, the two-magnon scattering processes will be smaller than their one-magnon counterparts by a factor of Mj^^/Ey For ferromagnets, this mechanism is the only one available for two-magnon scattering because there is only one acoustic magnon branch. Fortunately, we are interested in two-magnon Raman scattering in antiferromagnetic systems. Antiferromagnetic systems offer an additional scattering channel through an exchange-scattering mechanism. Antiferromagnetic systems consist of two interlinked spin lattices pointing in opposite directions. Two magnon scattering via the exchange mechanism involves the flipping of two adjacent spins on each sublattice. This is accomplished when a spin-up electron on sublattice A is photoexcited to

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236 sublattice B. The site to which the spin-up electron hopped is now doubly occupied since a spin-down electron already exists on that site. To lower the Coulomb energy associated with double occupancy, the spin-down electron hops back to the site on sublattice A. This creates two spin flips in the AF background. In second quantized form, this is viewed as creating a separate magnon quasiparticle on each sublattice. The approximate form of the exchange scattering Hamiltonian is^'*^ ^em = E ^^^("-9^-9 + 4^U)4+/fc (27) in which the a and P operators create magnons on the two interlinked sublattices. As usual, c operators are for electrons and E is the associated coupling tensor. The and quasiparticle operators are related in the usual way to the magnon creation and destruction operators by the expression ' a\ = u^\-v^\ 4 = '^^-'"^^ (28) where and are constrained to 4+4=1. (29) at, and ftt, are the destruction (creation) operators on the A (B) sublattices. Since the exchange mechanism involves flipping spins in the AF background, E^is directly related to the superexchange energy between adjacent magnetic sites. Using Eq. 27 in third order perturbation theory as done previously yields the Stokes scattering cross section. The result is = 7:r-l2^ — — Y Rqi.'^^MIS) 2 5.r r dudu {4ireo)^2Tr (jjj nj ^ ^ > /i q,kj-ks

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237 X r {a,it)P,iO)almlme^^'dt, (30) J —OO where now R^u!2m'JS) is a function of the two-magnon frequency, uj2m, which is twice the one-magnon frequency. The form of the Raman tensor, with the exception that U2M replaces ojm, is the same as in Eq. 25. J.B. Parkinson^'*^ demonstrated that the form of the electron-magnon interaction Hamiltonian given in Eq. 27 can be expressed as Hr = "^{Einc • gStj){Esc • dij)Si Sj, (31) {ij} where Einc and Egc are the incident and scattered electric field vectors for the photons, aij are unit vectors connecting neighboring spin sites i and j, and Si, Sj are the spin operators on sites i and j. The Raman scattering polarization channels from Table 4.2 now start to become clear. Figure A-1 is a representation of the Cu02 plane with the X, y directions chosen along the Cu-0 bonds and with the x', y' directions rotated 45° with respect to the Cu-0 bonds. Under D4 point group symmetry, the lattice structure of Sr2Cui_xCoa;02Cl2 and La2Cui_a;Li2;04, xy transforms as B2g and x'y' transforms as Big. The latter can be understood by decomposing x' y' into unit vectors of x and y and collecting the terms: x'-y'= -^{x + y) {-X + y) = ^{-x^ + y^), (32) where y^ — x'^ transforms as B\g under D\. Hence, cross polarized incident and scattered radiation along the x' and y' directions will produce 2-magnon scattering in the B\g channel with the restriction that oij connects nearest neighboring sites^"*^. Using the same method of Eq. 32, the A\g + B^g modes may be accessed with incident

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238 y o 45" o ° o Fig. A-1 Sketch of x, y, directions in the Cu02 plane and scattered radiation polarized along the x'x' directions (not cross polarized) while the Big mode is activated by cross polarized radiation in the x, y directions. Here, however, dij connects nearest neighbor diagonal sites^'*^ (diagonal exchange) and so does not contribute as strongly as the B\g mode. The scattering Hamiltonians introduced to describe the B\g, B^g, A\g, and Aig scattering modes in Eq. 32 all appear in a recent theory of Raman scattering in the Hubbard Model devised by Shastry and Shraiman^^^. They isolated the contributions to the T-matrix from the four symmetries modes as follows: (33) (34) (35)

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239 ^4 1 ^ ^ = _^^3 ^ e^^r (5i {Si+^ + X , (36) where t is the hopping parameter in a one band picture, U is the on-site repulsion, = 3^:, ^y, and e^,^ = — Ci.,^ = and where terms of higher order in the spin operator and terms beyond diagonal nearest neighbors have been neglected. As is apparent, the Big contribution is the leading order term in the jj^^ expansion and so dominates the Raman spectra. Since A2g is higher order in both the expansion and in the spin-operator expansion it bears the weakest scattering intensity. There are a plethora of books and tutorials on Raman scattering that far exceed in quality and in depth the treatment given here^^'*"^'^^. For more specific information and theoretical rigor, consult the suggested book list at the end of the References.

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REFERENCES 1. J.G. Bednorz and K.A. MuUer, Z. Phys. B 64, 189-193 (1986). 2. M.K. Wu, J.R. Ashburn, C.J. Torny, P.H. Hor, R.L. Meng, L. Gao, Z.J. Huang, Y.Q. Wang and C.W. Chu, Phys. Rev. Lett. 28, 908-910 (1987). 3. S.S.P. Parkin, V.Y. Lee, E.M. Engler, A.I. Nazzel, T.C. Huang, G. Gorman, R. Savoy and R. Beyers, Phys. Rev. Lett. 6, 2539-2542 (1988). 4. A. Schilling, M. Cantoni, J.D. Guo and H.R. Ott, Nature 363, 56 (1993). 5. L. Gao, Y. Y. Xue, F. Chen, Q. Xiong, R. 1. Meng, D. Ramirez, C. W. Chue, J. H. Eggert, and H. K. Mao, Phys. Rev. B 50, 4260-3 (1994). 6. J. Bardeen, L.N. Cooper and J.R. Schreiffer, Phys. Rev. 108, 1175 (1957). 7. C.E. Gough, M.S. Colclough, E.M. Forgan, R.G. Jordan, M. Keene, CM. Muirhead, A.I. M. Rae, N. Thomas, J.S. Abell, and S. Sutton, Nature 326, 855 (1987). 8. R.H. Kock, CP. Umbach, G.J. Clark, P. Chaudhari, and R.B. Laibowitz, Appl. Phys. Lett. 51, 200 (1987). 9. Z.X. Shen, D.S. Dessau, B.C. Wells, D.M. King, W.E. Spicer, A.J. Arko, A. Marshall, L.W. Lombardo, A. Kapitulnik, P. Dickinson, S. Doniach, J. DiCarlo, and T. Loeser, Phys. Rev. Lett. 70, 1553 (1993). 10. Y. Hwu, L. Lozzi, M. Marsi, S. La Rosa, M. Winokur, P. Davis, M. Onellion, H. Berger, F. Gozzo, F. Levy, and G. Margaritondo, Phys. Rev. Lett. 67, 2573 (1991). 11. R. Kelley, J. Ma, G. Margaritondo, and M. Onellion, Phys. Rev. Lett. 72, 2567 (1994). 12. H.J. Tao, A. Chang, F. Lu, and E.L. Wolf, Phys. Rev. B 45, 10622 (1992). 240

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BIOGRAPHICAL SKETCH Sean Moore was born and raised in the San Francisco-Oakland Bay Area. He earned his bachelor's degree in physics with a mathematics minor from the California State University at Hayward in 1991. After working in the private sector for one year he came to the University of Florida in 1992 to pursue a Ph.D. in physics. 250

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fliUy adequate, in scope and quality, as a dissertation for the degree of Doctoral of Philosophy. John Graybeal, Chairman Associate Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctoral of Philosophy. David Tanner Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctoral of Philosophy. Associate Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctoral of Philosophy. J. K. Ingersent ^ Associate Professor of Physics

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctoral of Philosophy. Joseph H: Simmons jfessor of Material Science and Engineering This dissertation was submitted to the Graduate Faculty of the Department of Physics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December, 1999 . Dean, Graduate School