Anisotropy in the infrared, optical and transport properties of high temperature superconductors

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Anisotropy in the infrared, optical and transport properties of high temperature superconductors
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Thesis (Ph. D.)--University of Florida, 1994.
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Includes bibliographical references (leaves 200-213).
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Typescript.
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by Manuel Alberto Quijada.

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ANISOTROPY IN THE INFRARED, OPTICAL
AND TRANSPORT PROPERTIES OF HIGH
TEMPERATURE SUPERCONDUCTORS





By
MANUEL ALBERTO QUIJADA


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













ACKNOWLEDGMENTS


It is with great


pleasure that I thank my advisor, Professor


David B.


Tanner,


for his advice, patience and encouragement throughout my graduate career here at


University of Florida.


I feel fortunate to be part of his research group.


I also


thank Professors J. Graybe


Hirschfeld, C.


Hooper, N. Sullivan, and J.H.


Sim-


mons for their interests in serving on my supervisory committee and for reading this

dissertation.

Thanks also go to all my past and present colleagues in Tanner's group for their


friendship, useful conversations and cooperation.


In particular, I would like to thank


C.D. Porter for his assistance with computer software. I am also indebted to Drs. G.L.

Carr, D.B. Romero, and V. Zelezny for many enlightening and useful discussions.


I would like to express my gratitude to Drs. J.P. Rice,


Onellion, F.C.


D.M. Ginsberg, M. Kelley,


Chou and D.C. Johnston for providing good quality single crystals


that were essential to the completion of this work.

The technical support of the staff members in the physics department machine

shop and engineers in the cryogenic group is appreciated greatly.


would also


like to take this opportunity to thank my wife,


Zunilda, and my


daughter, Melissa, for their support and understanding during the countless nights

they stayed alone while I was working in the laboratory.

Finally, I also thank my parents for giving me their support throughout my aca-

demic life.


T7 _._. _- -1 ...-.. P I rT


-L'


rn .-.. ...f-1_ nL -rv r1'7lr .._. ...


.TTCI















TABLE OF CONTENTS


ACKNOWLEDGMENTS


. a a a a a a a a a a a11


ABSTRACT


CHAPTERS


I. INTRODUCTION


* S S a a a a a a aa a a a a a a S 1


REVIEW OF PREVIOUS


EXPERIMENTAL


WORK


a S S S 0S5


Crystal Structure of Copper-Oxide Materials
La2 CuO4+. . . . .
YBa2 Cu3 07_ *


Bi2Sr2 CaCu2s . . . .
Review of Optical Properties Copper-Oxide Materials


c-Axis Response


Midinfrared Absorption in the CuO2


Planes


Anisotropy in the ab Plane


YBa2Cu3O _
Pbrnia~rhin C'


THEORY


S. 19


R"'V -,O a 0 0 5 0 S


Models for Carriers in the CuO2 Planes: Normal State


Three-Band Hubbard Model


t J M odel . S . .S S. ..a a. .
M odels for 1 .(. . . . . . .
T n r Z-\ an 1 1) I 4, -


a S S S S 0 5


* a a S S 0 0 5 0 *
* S S S S S S S S SS8


Pane


Y1I II LIVI








Superconducting State Models


Symmetry of the Order Parameter .

Evidence for Proposed Pairing States

Determination of Gap by Optical Spectroscopy


EXPERIMENTAL TECHNIQUES


Fourier Transform Infrared Spectroscopy

Optical Spectrometers


S S S S S 4 5 4 4 4
S S S S 4 4 4 4 4 8 S 4 4


Bruker Fourier


Transform


trometer


Bolometer Detector


The Perkin-Elmer Monochromator .
Polarizers. *

Sample Mounting and Low Temperature Measurements

Normalization Procedure of the Reflectance .

Data Analysis of the Spectra: The Kramers-Kronig Tra

High-Frequency and Low-Frequency Extrapolations

Optical Constants


Sample Preparation Techniques


a2Cu307


Single-Domain Crystal


4 4 4 5 S 86


Bi2Sr2 CaCu2 08 Single-Domain Crystals

La2Cu04+6 Single Crystal . . .


OPTICAL STUDY OF La2Cu04+, SINGLE CRYSTAL


c-Axis Reflectance of La2CuO4+ . . .


Room Temperature Spectra

Low Temperature c-Axis Re

Assignment c-Axis Phonons

Effective Charge . .

ab-Plane Reflectance .


Assignment ab-Plane Phonons


iflectance


* 4 5 4 5 4 4 4 4 5 4 5 5 5

* S S S S S S S S S S 4 4 5


* S S S 4 S 5 4 5 5 5 5 S

* S S S S S S S S S S 4 S S 4


Low Temperature ab-Plane Reflectance
fl _. i1 TI f .l. r- z. 1 /ftit._I-._


* S S 4 5 4 4 I

* S 4 5 4 5

* S S 4 S 5


ansformations








Midinfrared Component


S S S 9 S SS S S C 5 59


Comparison of ab-Plane Reflectance:


c and q c


S S S 5 9


Concluding Remarks


S S C 9 C S S S C S S S S S 59


ANISOTROPY IN THE AB-PLANE OPTICAL PROPERTIES


OF YBa2Cu30 7_


Room Temperature Spectra


Temperature Dependent Reflectance


Effect of the Chains


S.w S S 103


S S S S S S S S S S S S S S S SS S S S 10?7


ab-Plane Anisotropy in the London Penetration Depth


. 109


ANISOTROPY IN THE AB-PLANE OPTICAL PROPERTIES


C S S S C S C S S S S S S S S S S 111


Results of the Optical Reflectance
Room Temperature Spectra
Temperature Dependent Spectra
Discussion of Optical Constants


Temperature Dependent Optical Conductivity


One-Component Analysis


* S S S S S C S S 113
S 113


S S S S S C S S S S 5 116


* 121
* S S S 125


Two-Component Ar
Drude Component
Midinfrared Absorp
Superconducting Cone


lalysis


S S S S S SS S S S S S S 12'7


129
tion ............. 134
iensate .. ...... 139


ab-Plane Anisotropy in the London Penetration Depth 141
Optical Conductivity and Symmetry of the Order Parameter 143


VIII.


RESISTIVITY TENSOR OF Bi2Sr2CaCu20s


SINGLE-DOMAIN


CRYSTALS


S S S S S S S S S S S C 9 S 5 5 C C S 14/7


Sample Preparation and Measurement Method


Resistivity Analysis


for Anisotropic Materials


U S C S S S S C C 5 5 5 152


Resistivity Tensor


OF Bi2Sr2CaCu20s








Closer Look to the Transition Temperature


Results and Discussion


S C S. . *161


S C S S S S S S C 5 S C P 5 163


Review of Flux-Flow Resistance and Kosterlit


z-Thouless


Transition


.165


Concluding Remarks


CONCLUSIONS


S C S C P C S C S C S C C 168


S S S C P S S S S C S C C S 5 5 5 170


APPENDICES


OPTICAL


TUDY OF BEDT-TTF(C104)2


MICROWAVE CAVITY


APPARATU


S S 5 5 S S S S 5 S 173


S S 5 5 S C S C S C 5 5 186


REFERENCES


S S S C S S P 5 5 5 5 5 5 S S S S S S S 5 5 200


BIOGRAPHICAL SKETCH


p p p p p p p pp p p p p p 2141












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


ANISOTROPY IN THE INFRARED


OPTICAL


AND TRANSPORT PROPERTIES OF HIGH
TEMPERATURE SUPERCONDUCTORS

By

Manuel Alberto Quijada


April 1994


Chairman:


David B. Tanner


Major Department: Physics


The optical


unusual.


properties of the high-temperature superconductors are extremely


We have extensively studied superconducting high-quality single crystals


of YBa2Cu3O7- Bi2Sr2CaCu20


, and La2CuO4+6.


All these materials have CuO2


planes as the entities responsible for the metallic


behavior and superconductivity.


Polarized


optical reflectance measurements were taken


both above and


below the


superconducting state on a wide frequency range.


All these materials display interesting anisotropy in their optical properties. In

particular, optical investigations of the oxygen-doped La2CuO4+6 reveal the out-of-

plane (c axis) spectrum of this material is typical of an insulator with the optical


conductivity dominated by optical phonons.


In contrast, the ab-plane optical spec-








response is measured on the face of the crystal that has


the c axis parallel to the


propagation vector of the light.


normal-state


infrared


conductivity


CuO2


planes


shows


a strong,


nearly


temperature-independent,


broad


band in


the midinfrared in


addition


strong


temperature-dependent narrow


Drude-like band


the far infrared.


There


is also anisotropy in


the infrared


conductivities


between


a and


axes


of both


YBa2Cu3 07_6


Bi2Sr2CaCu2Os.


case


a2Cu307_6,


strong


anisotropy can be mostly attributed to the presence of CuO chains along the b axis.

One striking result is that in spite of the fact that Bi2Sr2CaCu20s does not have


the CuO chains,


we observed anisotropy between the a and b axes infrared conduc-


tivity of this compound as well.


The presence of this


conducting state suggests two possibilities.


superconducting order parameter.


One pos


A second explanati


anisotropy even in the super-

bility could be an anisotropic

on is that the overall conduc-


tivity is composed of a


component.


simple Drude term combined with a more broad midinfrared


The observed higher absorption in the low-frequency region along the


could


be explained


by an anisotropic second midinfrared component in


optical conductivity.












CHAPTER I
INTRODUCTION


The discovery of superconductivity in the copper oxides by Bednorz and Miiller1


in 1986 has revolutionized the field of condensed matter physics.


The importance of


this remarkable discovery can not be overstated.


On the one hand


, it offers promising


technological applications for materials that lose their resistance to the flow of electri-


cal current above liquid nitrogen temperatures.


On the other hand, many experiments


have provided ample evidence of the exciting new phenomena present in these mate-


rials.


Early measurements were designed to learn if the superconducting properties


of these materials could be explained in the context of the Bardeen-Cooper-Schrieffer


) theory2 for conventional superconductors.


Some of those initial results sup-


ed a BCS-like theory.


Among these, flux quantization


and the AC Josephson


effect4 show that the elementary charge in the superconducting state is


rather than


e. In addition, photoemission5-7


and tunnelings,9


experiments suggest the presence


of a superconducting energy


the same time,


there has


been an accumu-


nation of evidence for an


unconventional nature of the high-Tc materials.


Some of


the most important results that have emerged are high superconducting transition


temperature,l,10,11


linear dc resistivity in the normal state,12,13


and extremely small


coherence lengths.14115


Perhaps the second most striking property in these materials,


beside their high-Tc value, is the anisotropy in their physical properties.16-21

As soon as these materials were discovered, there began an intense effort to study


their optical properties.22-24


Soon, it was realized that the strong anisotropy that





2

means has provided some important results but at the same time has raised some


unresolved


questions.


As it


s well known, superconductivity in


these materials is


associated with the quasi-two dimensional CuO2 planes. Most optical studies related

to the anisotropy in these materials have concentrated in the anisotropy between the

directions perpendicular to (c axis) and parallel to the CuO2 planes. Research of the


anisotropy within the CuO2


(ab) planes has been studied to a lesser degree.


In view


of the orthorhombic distortion that exists in these planes there are two important


questions


must


be addressed:


how this


structural


anisotropy


affects


anisotropy of the


d electronic structure in the normal state and (


) what if any is


the anisotropy of the superconducting order parameter?


ince the energy gap plays a central role in the BCS theory,


substantial efforts


have been


devoted


to observing this gap


by optical methods.30'31


One of the ad-


vantage of the optical methods compared


to, for example,


tunneling is that direct


electrical contact to the sample surface is not necessary.


This is especially important


since crystals and films may have dead layers near the surface that make it nearly


impossible for current to tunnel between an electrode and the superconductor.


In an


optical experiment, by contrast, the probing radiation can penetrate a few thousand


A into the sample so the presence of dead layers becomes less of an issue.


This tech-


unique,


which has been used with great success in the past


to study energy gaps


conventional superconductors, has also given valuable information in solids about lat-

tice vibrations, electron-phonon coupling, low-lying excitations, and electronic band


structures.


In the context of the BCS theory, the presence of a gap means that for


photon energies less than 2A, the bulk properties of the superconductor at T


show only an inductive part, with the real or absorptive part being zero. So, in order








quasiparticles in the sample.


At photon energies well above the gap,


the material


behaves as if it was in the normal state.

This thesis is concerned with the subject of anisotropy in the optical properties


of the high-Tc materials.


The materials investigated are single-domain


crystals of


YBa2Cu307_


C T = 9O K),


2Sr2CaCu2s8 (T7 =


85 K), and an oxygen-dop


ed sin-


gle crystal of La2CuO4+8 (Tc


= 40K).


There are two important issues that will be


considered. The first one is the anisotropy in the optical properties of the ab plane

vs. the c axis. A large surface area containing the c axis in the La2CuO4+s sam-


ple allowed the study of the c-axis polarization as well as the ab-plane response in


this material.


This


also allowed us to examine the ab-plane response in this material


when the propagation vector (q) of the light is parallel and perpendicular to the CuO2


planes.


The second issue is the anisotropy in the optical response along the two prin-


cipal axes in the ab plane, and how this is related to the crystal structure anisotropy


in single-domain crystals of Bi2Sr2CaCu20s and YBa2Cu307 _.


One important ob-


servation is the presence of a larger absorption in the superconducting-state optical

conductivity along the b axis below the frequency where a superconducting energy gap


should be observed.


Explanation for this larger absorption in the framework of the


models that have been proposed will be discussed in addition to its


possible connec-


tion with the symmetry of the superconducting order parameter in the copper-oxide

superconductors.


The organization


of this thesis is


as follows.


Chapter


presents a review of


experimental works that have been done related to the crystal structure and opti-


properties of the high-Tc superconductors.


brief theoretical background for


the understanding


the physical


properties will


be presented in


Chapter





4

to anisotropy between the ab-plane and c-axis optical measurements of La2CuO4+6

crystal are presented in Chapter V. The presentation of results and discussion related


to the measurements on


YBa2Cu3076- and Bi2Sr2CaCu2 O8


single-domain crystals,


with emphasis of the anisotropy within the ab plane, is done in


Chapters


VI and


VII respectively.


Chapter


VIII is for presentation of results regarding dc-transport


measurements of the resistivity tensor of Bi2Sr2CaCu20s samples.


Finally, Chapter


IX contains concluding remarks.












CHAPTER II


REVIEW


This chapter will be


OF PREVIOUS EXPERIMENTAL WORK

devoted to a survey of previous experimental works related


to the physical properties of the copper-oxide superconductors.


two major sections.


It will be divided in


The first one is related to investigations of the crystal structures


of the three kind of materials studied in this study.


Information about the lattice


parameters that are relevant to an optical experiment will be


given here.


The second


major section is a brief review of relevant previous work of the optical properties in

the copper-oxide superconductors.


Crystal Structure of Copper-Oxide Materials

Good knowledge of the crystallographic structure in single-crystal materials is


essential to the understanding of the optical properties.


case of the copper-oxide materials.


This is certainly true in the


These materials are considered by many as a good


example of a 2-dimensional system.


The reason for this point of view


s that the char


transport


superconductivity


occur mainly in


dimensional


arrangement of


copper atoms that are strongly bonded to oxygen atoms with an interatomic distance

of 1.9 A. This distance hardly changes from structure to structure. Above and below


the CuO


2 layers,


there are other


layers of atoms that


are believed to provide the


carriers necessary for conductance in these planes, as well as to provide overall charge


neutrality.


The identity of these atoms depends on the system under study.


La9 CuO4+s








the tetragonal K2NiF4 structure was soon corroborated by Takagi


et al.32


Further


research indicated the nearly stoichiometric La2 CuO4 is an antiferromagnetic insula-

tor that can be hole doped by partially substituting the lanthanum site with some


of the alkaline earths Ca, Sr, or


Ba to produce superconducting materials with Tc


in the 30-40 K range.


Later


, it was also realized that superconducting samples in


this system can be obtained by producing samples where the oxygen stoichiometry is


higher than four.33'34


Figure 1 shows the crystal structure of the undoped L


a2CuO4


compound. In this nearly stoichiometric material, the oxidation


state of the individ-


ual species is La3+


Cu2+


, and 02 in order to have charge neutrality.


The oxidation


state of copper leaves this


atom with one unpaired electron in the d-shell; therefore,


the net spin at the copper site is 1/2.


Fig. 1.


Crystal structure of La2CuO4 material (


Each copper atom is coordinated


After Ref. 33).


to four oxygen atoms in


the plane


In addition, there are two more distant apical oxygens above and below the copper








those in the plane.


The Cu ion and its octahedral oxygen configuration would imply


a tetragonal I4/mmm space group in


the structure.


However, neutron scattering


data for this compound indicate that


below a characteristic temperature there is a


lowering of the symmetry due to a slight tilt of the apical oxygens from their high


symmetry points.


This tilting produces


a tetragonal to orthorhombic transformation


at a temperature that is close to 500 oC for the nearly stoichiometric material and


decreases with increasing doping, reaching

15%.

As mentioned above, superconductivity


calating additional oxygen atoms in the structure.

provide hole carriers in the Cu02 planes. The first


K at a Sr concentration of around


in this material can be achieved by inter-


This excess oxygen is thought to


report on the structure of oxygen-


enriched samples was done by Chaillout


et al.35


They reported neutron diffraction


results on a sample that had


3% extra oxygen.


First,


they reported


that at


doping level the material will phase separate into a superconducting (oxygen-rich)


phase and a near


stoichiometric region, just as happens in La2NiO4+6.36


Based in a


two-phase model, the conclusion was reached that the superconducting phase has


Bmab space


group from the observation of peaks in the neutron scattering data that


are forbidden in Fmmm symmetry.


They created a model where one apical oxygen


is replaced by two oxygen sites that form a peroxide with a


short 0-0 bond distance


of 1.64 A.

Most recently, Radaelli et al.37 pointed out the need to perform such experiments


on samples with oxygen concentration well beyond the phase separation regime.


reported neutron diffraction data on three electrochemically-oxidized


They


samples: two ce-


ramics (6 = 0.08,


with Tc = 32 K, and S = 0.12,


with Tc = 42 K) and one single








After a carefully analysis of the neutron scattering data,


they concluded the basic


crystal structure of all samples corresponds to the Fmmm symmetry.


This was ob-


trained from the absence of small diffraction peaks that would be allowed in a Bmab


symmetry.


They argued the Bmab symmetry would be


frustrated


by the presence


of interstitial oxygen in the


structure.


Following up on


their conclusion,


they car-


ried out a refinement of the


structural parameters for the single-crystal data in the


Fmmm space


group


symmetry.


In the analysis,


they relaxed the apical oxygen po-


sition slightly from the high-symmetry points and introdu

site located nearby and adjacent to a LaO layer (see Fig. 2


ced an additional oxygen

). The lattice parameters


obtained from analysis of the neutron scattering data on the single crystal at


are a b


18 K


= 5.34 A and c= 13.22 A.


Model for the crystal structure of La2CuO4+6 (


After Ref. 33).


YBaCuaO7 _


YBa2Cu3O7_76


was


the first


copper-oxygen-based material


showed super-






9


YBaQOz30.,


Fig. 3. Phase diagram for YBa2Cu307_6 as a function oxygen concentration.
AF: antiferromagnet, SC: superconductor (after Ref. 38).


system, the parent compound YBa2Cu306 crystallizes in tetragonal form at high tem-


peratu


but converts by oxygen ordering to an orthorhombic form (YBa2Cu307_6)


on cooling down at room temperatures.


There is also a tetragonal to orthorhombic


transition driven by the oxygen content in the material at xc = 0.3,


which also deter-


mines the Tc in the sample as shown in the phase diagram of Fig.


Figure 4


shows


the essential features of the crystal structure of the orthorhombi

been gathered based on x-ray and neutron diffraction data. In t


c phase that have


his material, each


copper atom in the plane is linked to four oxygens atoms at about 1.94 A, and a fifth


weakly bonded apical oxygen at 2.3 A as in the 2-1-4 compounds.


This arrangement


of the oxygen atoms in


the planes gives a perovskite-like structure with


the space


group


Pmmm.12


There are two CuO2


planes per unit cell.


Unlike the tetragonal


compound,


the orthorhombic phase contains an additional CuO containing layer that


consists of a 1-dimensional network of CuO chains along the b


axis in the unit cell.









contributes 0.25 hole per copper for each of the two planes).


There is a difference


between the a- and b-axis


dimensions in the order of


with the b axis being the


larger on account of the CuO chains present along this direction.


The unit cell has


lattice dimensions of


(1=


b =3


A, and


c = 11.65 A. Effects of the chains


the superconducting properties of the material have been difficult to clarify because

the as-grown samples are usually twinned in the ab plane with alternating strip-like

domains of a- and b-axis oriented material.


CutI)
O Cul(2)
0 '
Ov

0(2)
o0(S)
o 0(4)



zL.


Crystal structure of YBa2Cu3O7-6 material (after Ref. 39).


Special growing t


techniques 0 in recent years have pro


ded large enough single-


domain


crystals,


have


made


possible the study


ssue in


more detail.


More discussion about this will be given later when discussing sample preparation

techniques.

Bi9Sr9CaCu9Og


The discovery of 20


K supercondu


activity


n Bi


r9 C U 0 r 2


opened


up the race


u u v


























I
I I
aI' oi Il
Io I


Fig. 5.


Crystallographic structure of Bi2Sr2 CaCu2 08 sample (after Ref.


The structure of this material was quickly identified


as Bi2Sr2CaCu2Os or


41).


"2-2-1-


The Bi-based compounds form a series of layered material where the CuO2


planes


are separated


alkaline-earth


cations


sheets of Bi202.


The general


formula is (BiO)


r2Can-1 CUnO2+2n


where n is the number of consecutively stacked


CuO2 sheets.


It is found that Tc increases with the number ni, reaching a maximum


value of 110


when n = 3 and decreasing again as n increases further.


Figure 5


shows


the ideal crystal structure for the case when


=2.


this structure the


CuO/Ca/CuO


layers


are seen


to be separated


by sheets of Bi202


SrO.


crystal lattice parameters of the orthorhombic unit cell are a = 5.41 A,b = 5.44 A,

and c = 30.78 A. In site of the relatively simple arrangement of the many atoms, a


2" .43








is not well


defined;


cation


disorder


also contributes to


problem.


contrast


to the Bi202 disorder,


the CuO2 sheets appear to be free of defects as determined


from x-ray and neutron scattering experiments.44


In addition,


the oxygen content


varies for any fixed cation composition, affecting the transition temperature in the


material.


In general, Tc increases as oxygen is removed.


It is still unclear however


how doping of the CuO2 planes occurs in this structure.


It is generally assumed that


doping of the CuO2


planes in this material comes from


excess


oxygen in the B


planes.


X-ray45


and neutron scattering46


experiments have shown the existence of


a superlattice modulation or distortion that resides mostly on the Bi202 layers.

is incommensurate13'47'48 with the b-axis unit cell parameter given above (period


4.7b).


There have been many attempt


s to explain this superlattice by modeling defects


that are observed in the structure of this compound.41'48-51


In one of those models,


the superstructure is claimed48 to be the result of the addition of one in ten oxygen


atoms in the Bi202


layers, that would also cause a displacement of the surrounding


atoms.


Final verifi


cation of this or any model must await further work.


Review of Optical Properties Copper-Oxide Materials


This section is be devoted to a review of the work related to the status of the


infrared response in the copper-oxide superconductors.


properties in these material


Investigation of the optical


s has been very vigorous since their discovery in 1986.23'24


In spite of this, understanding of the infrared properties in these materials remain a


controversial issue.


As mentioned before


the feature common to these materials is


the CuO


plane structure.


Since the optical


properties of these materials are very


different for directions parallel and perpendicular to the CuO2 planes and even show

substantial anisotropy within these planes, the availability of ab-plane oriented films








(c axis) mainly on La2-. Sra Cu04 single crystals.


The second section will touch the


subject of the midinfrared absorption


that starts


off as soon as a few


earners are


introduced in the CuO2 planes of these materials. In the third section, the subject of

the anisotropy within the ab plane will be reviewed in the context of the work done


with single-domain crystals of YBa2Cu307_- and B


2Sr2CaCu208.


c-Axis Response


As a result of the layered nature of the materials,


the in-plane and out-of-plane


optical properties of the copper-oxide superconductors show a remarkable anisotropy.


This


became evident as soon as the first polarized studies were reported on single


crystals of these compounds.2


8,52-54


In one of the earlier reports,54


the reflectivity


of insulating La2CuO4 and of


8% Sr-doped samples were compared for polarizations


parallel and perpendicular to the CuO2 planes.


The results showed that the c-axis


reflectance in


both


cases


was virtually unchanged suggesting that doping


did not


affect the c-axis response in the system.


The spectra were typical of an insulating


material with phonon-like structures due to infrared-active modes (A2u symmetry).


In contrast, the refle


chance parallel to the ab planes showed substantial differences. In


the undoped sample, the reflectance is dominated by the presence of infrared allowed


phonons


symmetry) with


the reflectance approaching a constant value at low


frequencies indicative of insulating


behavior in


the Cu02


planes as well.


other hand, the doped sample showed a rising reflectivity at low frequencies with the

phonons visibility greatly reduced due to screening by the free carriers in the material.


This rising reflectivity implies a metallic conductivity


n the CuO2 planes of this


doped


material, although at this concentration the material is not yet superconducting.


became clear from this study that the anisotropy between the ab-plane and the c-axis





14


A more systematic study of the c-axis and ab-plane optical response as function


of Sr doping in La2-_.SrzCu04


was done by Uchida et al.28


Let us first review the


c-axis data.


Figure 6 displays the


reflectivity for three representative doping levels:


undoped,


superconducting


-= 0.15),


the overdoped


= 0.34).


These re-


suits confirm28 the previous observation about the unchanged character of the


reflectance between the insulating and superconducting phase.


c-axis


The electronic back-


ground appears to be very weak from the almost flat slope of the reflectance in the


far infrared.


The spectrum is dominated by two major optical phonons.


Only in the


overdoped nonsuperconducting samples do there appear to be


qualitative changes in


the reflectance. In this case, the spectrum appears to have a free carrier component,


although the highest phonon is not completely screened out.


This is consistent with


an observed metallic behavior in the resistivity as a function of temperature in the c


axis at this concentration.


In this context, the authors regard the overdoped samples


as an example of anisotropic 3-dimensional metals.


0.1 0.2


0.5
I:


0



0.4


Photon Energy (eV)


I .0I
I>'.-
I I C I4





15

This semiconducting-like behavior for the c-axis conductivity has also been doc-


umented in other cuprate such as


YBa2Cu3O_6 and Bi2Sr2CaCu20s.25'29,57


In the


case YBa2Cu3 07-_ the results show some variation for various oxygen concentrations.

It appears that samples with the highest oxygen content show a nearly metallic be-

havior along the c axis,25'29 signaling that coupling between the CuO2 planes of this


compound is somewhat stronger.


The temperature dependence of the c-axis optical


properties has also been investigated in superconducting samples of La2-zSrzCu04


compound.


The results of the reflectance measured at


several


temperatures for


three doping levels are shown in Fig.


Above Tc,


the far-infrared reflectance


rather featureles


s, approaching


a constant value at


low frequencies for the lowest


doping,


while it shows a slight negative slope for higher doping indicative of some


weak dc transport.


Below


Tc, a sharp edge develops in the reflectance that signals


the onset of superconductivity.


In the analysis, the authors find that this edge does


not scale with Tc and it is not due to a superconducting gap excitation, but rather to a

plasma-edge-like feature associated with the superconducting-state carriers. A direct

consequence of this is the appearance of coherent transport of charge across the CuO2

planes below Tc. Similar results have also been obtained in the superconducting state


c-axis response of YBagCu3O -6 single crystals.


Midinfrared Absorption in the CuO9 Planes


The optical spectrum of undoped parent compounds of the copper-oxide supercon-

ductors such as La2Cu04 and Nd2Cu04 shows that these materials are charge transfer

insulators, with a band gap energy in the range 1.7-2.0 eV (14,000-16,000 cm-1), de-

pending on the material. It is generally accepted that this energy corresponds to tran-

sitions in the CuO2 planes between occupied 0 2p levels and the lowest unoccupied




























100 150 200 250
Frequency ( cm' )


300 350


7. Temperature dependence in c-axis reflectance of La2- SrzCuO4
function of doping (after Ref. 58).


as a


few carriers are introduced in these planes, there is an appearance


excitations


n the


midinfrared24 that are peaked in an energy range from 0.1 eV to 0.5 eV (


00-4000


depending on


the material under study.


These new


excitations are thought


to be


the result of photon assisted transition of bound holes (electrons) from their


ground state to


excited states and the continuum.


Due to the peak energy of one


of these midinfrared modes being approximately equal to the antiferromagnetic


change energy


J (~ 0.12eV), this band has been attributable to a magnetic origin,


the hopping of a charge from site to site involving the flipping of the nearby


spins.62 In other works, these


excitations have been relat


ed to the polaronic binding


energy of an impurity to the lattice.63'64


a function


increased


doping


these


excitations


seem


grow


even-





17

conductivity in La2-xSr3Cu04 samples as a function of Sr concentration by Uchida


et al.


Doping of the CuO2


planes in this system can be achieved by partially re-


placing the lanthanum sites La3+ by strontium Sr2+


Depending on the doping level,


the properties of the samples change from insulating (x


= 0),


to superconducting


(0.10


x < 0.20)


to nonsuperconducting metallic (x


> 0.2


The insulating com-


pound La2Cu04 shows the charge transfer (CT) band at


V (16,000 cm


-1) and no


absorption below this ener


evidence


For the lightly doped (x = 0.02) sample, there is clear


of the maximum that develops in the midinfrared conductivity at 0.5


followed by a reduction of the spectral weight above the CT band.


This transfer of


spectral weight from energies above the CT band is regarded as evidence of the itin-


erant nature of the states near the Fermi surface of these materials.


appears to shift to lower energies as a function of doping.


at 0.14 eV (1100 cm


The maximum


For x = 0.10, it shows up


1); while for x = 0.15, it appears as a shoulder at a even lower


energy.


Other materials such as


YBa2Cu307-


and Nd2CuO4 show similar evolution


of the infrared conductivity as doping proceeds in these systems.24

Photoinduced absorption, a technique that induces doping in the insulating ma-


trials by shining light onto them to excite photo carriers,


has also shown the pres-


ence of this midinfrared absorption


in several samples.


La2-zSr1CuO4 and Nd2Cu04 by Kim et al.66


The samples studied are


, and in La2Cu04,


YBa2Cu306.25 and


Tl2Ba2Cal -Gd Cu208 by Foster et al.67

The optimally-doped samples do not show a discernible maximum in the midin-


frared.


But even in this case, the conductivity (rl(w)) decays more slowly than the


typical Drude-type dependence w


Evidence for this non-Drude response in crl(w)


has been provided for nearly all the copper-oxide-based superconductors.23'24'65'68-70


























1 2 3 4


hw(


Evolution


ab-plane


conductivity


as a function


doping


La2-z Sr, CuO4(after Ref. 28).


up in optimally-doped samples.


In the first place, there is a region (800-4000 cm


where this absorption shows very little temperature dependence.


This happens at the


same time the de conductivity in the material changes by a factor of three between

300 and 100 K. Clearly, a Drude formula with a single relaxation rate for the charge


carriers will not account for such a behavior.


Secondly,


there is a definite temper-


ature dependence in the low frequency conductivity that is in good agreement with

the measured dc value.65168,69,71'72

The two most commonly mentioned models for interpretation of this midinfrared


absorption have taken two rather divergent approaches.


In one approach,


there is


only one type of carrier which is responsible for both the dc transport and the con-








scattering rate and an effective mass enhancement at low frequencies.

are expected to form a superconducting condensate below Tc.


The second approach is called the two-component model.


infrared conductivity in the cuprate is the result of two types


first type of carriers,


These carriers


In this approach,


of charge carriers.


which are considered to be Drude-like in nature, are responsible


for the dc conductivity in the normal state and form the superfluid density below Tc.

The second component, which is formed by bound carriers, has a semiconducting-like


band gap.


In this model, the Drude component is expected to have a scattering rate


independent in frequency and linear in temperature.


More discussion about


this will be done in Chapter III. The following subsection will address the issue of the

anisotropy of the midinfrared absorption of single-domain crystals of YBa2Cu3O7_6


and B


CaCu208.


Anisotropy in the ab Plane


YBa Cu.O,


YBa2Cu307-_


s one of the most studied high-Tc systems.


At the same time,


most of the optical studies


in this material have been done on samples that show


severe


twinning in the ab plane.


Therefore, these measurements only show an average


of the ab-plane optical properties.


Since the presence of the CuO chains along the b


axis is likely to provide substantial conductivity in the midinfrared, polarized optical


measurements of single-domain crystals have become highly desirable.


Fortunately,


recent developments40'73 in making large enough twin-free samples have made possible

optical studies along the two principal axes in the ab plane of this material.26'31'74'75

The first room temperature measurements done on a wide frequency scale were per-








for linearly polarized light parallel and perpendicular to the chain direction.


At low


frequencies, the reflectance for Ra and Rb are nearly equal. At higher frequencies, Ra


falls off faster than Rb, reaching the plasmon minimum at a lower frequency.


This


plasma edge minimum, which signals the zero crossing of e1(w), occurs at 1 eV for the


a axis,


while it is at 1.5 eV for the b direction.


This shows a splitting of the plasma


edge minimum observed in twinned samples.65


At approximately 3 eV


for higher ener


> Rb.


A Kramers-Kronig analysis of this reflectance reveals


a spectral weight in the midinfrared that is roughly a factor of


larger along the b


axis.


Koch


et al. interpret the a-axis conductivity


as intrinsic to the Cu02


planes.


To obtain the chain conductivity, they subtract the a-axis conductivity from the the


total b-axis conductivity.


In this analysis, the chain conductivity is then modeled by


a broad Drude-like peak with strongly damped carriers.


As pointed out by Koch et


al. the additive nature of the conductivities is violated at higher frequencies (wc


> 2.0


eV) where


Cooper


reported


optical


studies


single-domain


crystals


YB a2!Cu O 6+,


as a function of oxygen doping


The doping levels they studied


were x ~


1 (Tc = 90 K)


0.6 (Tc = 66 K), and x 0.1 (insulating).


The spec-


trum of the insulating (x


_ 0.1) phase shows the charge transfer band at w ~ 2.0


This energy has been associated with a charge transfer across the O 2p and the


Cu 3d levels.28'76'77


finishes.


Upon doping, the strength ofthis charge transfer transition di-


The reduction is accompany by an enhancement of the conductivity in the


midinfrared in both the a and b axes.


Moreover, there is a lifting in the degeneracy


of the peak at 4.1


that is present in the tetragonal insulating phase.


The peak


, Ra = Rb and


J: 21






21


Schlesinger et al.31 reported the temperature dependence in the a- and b-axis re-

flectance of mechanically detwinned single-domain crystals (Tc = 90 K). The normal

state data show qualitatively similar results as described above for samples of similar

Tc. In the superconducting state, the authors reported what appears to be 100% re-

flectivity (within 0.5% uncertainty) for the a-axis polarization below an energy of 500


cm


On the other hand, the b-axis reflectance is 2-3% lower in the same energy


range.


This apparent threshold in the a-axis reflectance is interpreted as the BCS


superconducting energy gap (2A


8kBgTc) in the CuO2 planes.


The extra absorp-


tion observed in the b direction is argued as resulting from the stronger midinfrared


absorption due to the presence of the chain excitations in this direction.


dependence of this


The doping


threshold in the a-axis reflectance has also been investigated by


Rotter


et al.


The samples investigated are oxygen reduced single-domain crystals


of YBa2Cu307_- with Tc of 56,


and 93 K.


The results show the structure at 500


cm


does not shift neither with doping nor with a change in temperature for all


three doping levels studied.


The claim that this energy corresponds to a superconducting energy gap is now

considered rather questionable. Most recent direct absorption (bolometric) measure-


ments


by Pham


et al.30


show a finite an fairly large conductivity below this energy


for polarization along the a-axis direction in 90-K single-domain crystals.


The fact


that the accuracy in these measurements is rather high compared to typical uncer-

tainties in reflectance experiments (0.02% vs. 0.5%) makes the results for the a-axis


conductivity shown in Fig. 9 more representative of the real


situation.


Because of


their short coherence length (


~o10 A)


the high-Tc superconductors are considered


type


II material.


In all the analyses of the conductivity, 68'71'72


the mean free path



















b 1000



0


v (cme)


Superconducting


state


optical


conductivity


single


domain


YBa2Cu307 _6 in the a direction.


The weak peaks are calculated phonon


contributions.


No evidence of a


superconducting gap is seen in the spec-


trum (after Ref. 30).


pointed out by Kamaras et al.6S


Since the width of the free carrier band is smaller


than


that of the superconducting gap,


observation of this gap


by optical methods


might


be difficult


because most of the spectral weight of the free carrier part has


condensed into a 6-function at zero frequency, leaving a negligible amount of weight


for transition across the gap.


Notice this argument agrees with a superconducting


gap that


large compared to the scattering rate of the Drude component.


There


have been attempts79


to produce dirty enough samples, but still superconducting,


that would make (


so the gap might be observable using opt


cal methods.


far, these experiments have not been successful.


Bi9Sr2 CaCu9Oa


There


many


studies


concentrating


optical


properties


.1~ LI J






23

of the unpolarized results. Figure 10 displays results of the average reflectance in the

ab plane of this system on a wide frequency range for three samples with different

doping levels.80 Similar results have been reported by other groups.81-83 The low fre-

quency reflectance is characterized by the absence of strong phonon lines. At higher

frequencies, the reflectance falls off in a quasilinear fashion reaching the plasma edge


minimum at around 10,000 cm


(1.2 eV).


The results at high frequency are char-


acterized by a couple of interband transitions.


The first peak,


which is centered at


16.000 cm


-1 (2 eV), is attributed to the charge transfer band between the occupied


O 2p levels and the lowest unoccupied Cu 3d orbitals in the CuO2


planes.


The second


interband peak, which appears at 30,000 cm


-1(3


eV), is found to show some sam-


pie to sample variation and has been interpreted as an interband transition occurring


mostly in the Bi20O


layers.


As mentioned previously,


this material has no chains.


In spite of this,


there is an orthorhombic distortion of the ab plane resulting from


an incommensurate superlattice modulation presents along the b axis of the material.

There have been only a few studies reporting on the anisotropy of the ab-plane optical


properties of this system.571


85,86 The first of those reports, by Kelly et al.,


consisted


of ellipsometric measurements that showed a strong anisotropy in the near-infrared


region and higher frequencies.


In particular,


the peak at 30,000 cm


(3.8 eV) is


found to be sharper and stronger along the modulation direction.

The temperature dependence in the far-infrared optical conductivity has been ob-

tained from a Kramers-Kronig analysis of the unpolarized reflectance84 and transmit-


tance87


studies of free standing single crystals.


The micaceous nature of the Bi-based


material has made possible the preparation of very thin flakes (1000 A) that could be


used for transmittance studies without having to worry about substrates.


The results










0
1.0







u0.5
0.0







0.0-


Photon Energy
2 3


10000


20000


30000


40000


Frequency


Fig. 10.


Room temperature reflectance of B


2Sr2CaCu20g samples with dif-


ferent oxygen doping (after Ref.


dc value.


At the same time, the temperature dependence of oa(w) is less pronounce


in the midinfrared.


This is consistent with the non-Drude behavior observed in the


infrared properties


of the cuprate.


the superconducting state,


there is a broad


maximum at


1000 cm


(0.15


eV) followed by some


weak phonon structures and


a notch-like minimum at w~-400 cm


ab-plane ani


In addition,


sotropy in the midinfrared region were re]


transmittance studies showing

ported by Romero et al.86 The


reported anisotropy


s quite substantial in spite of the fact the difference in the a- and


b-axis dimensions of the pseudotetragonal unit cell is only 3%.


The transmittance in


the midinfrared is found to be lower for the b direction (more absorbing) than the a

axis.











CHAPTER III
THEORY


Models for Carriers in the CuO9 Planes:


Normal State


Superconductivity in the copper-oxide materials arises through

carriers by hole or electron doping the nearly square CuO2 planes. I

properties of these doped planes are very unusual, and so far, the'


fully accounted for.


the addition of


'he normal-state


y have not been


The property that is most frequently mentioned as indicative of


unusual normal-state behavior is the linear temperature dependence of the resistivity


p(T).


This behavior is not the case in, for example a Fermi liquid description, where


T2 is what is expected due to electron-electron scattering.


Other unusual prop-


erties are a temperature dependent Hall coefficient, proximity of superconductivity


to a magnetic phase,


and a very short coherence length.


It is widely believed that


understanding of the normal-state properties will eventually provide knowledge of


the pairing mechanism in the cuprate.


Most theoretical models for the normal-state


properties start with the so-called three-band Hubbard model.


Next sections discuss


the underlying issues of this and other models and the implications for the description

of electron dynamics in the 2-dimensional CuO2 planes.

Three-Band Hubbard Model


The unusual properties of the copper-oxide superconductors prompted the need

to construct a Hamiltonian that properly describes the motion of carriers in the CuO2

planes of these materials. People working in the field soon realized that electron corre-








system leaves one hole (or one electron) per unit cell.


If one neglects interactions,


this material would be expected to be metallic with a half-filled conduction band. In

reality, this material is an antiferromagnetic insulator. Band structure calculations88

showed the available states for this hole could be in either one of the Cu 3d or O 2p


(2p 2py) orbitals.


As mentioned before, each Cu atom in the structure is surrounded


by an octahedron of


six O atoms.


(This number changes for different materials:


YBa2Cu3O7_6 and 4 for Nd2CuO4.)


This


configuration removes the degeneracy


between the 3d orbitals of the Cu atom.


mainly a 3dz2_y2 character.


It turns out the highest energy level has


Therefore, the hole would reside mostly on the Cu site,


giving this atom a net spin of 1/2.


Hence,


the material can


best described


a model with localized spin-1/2 states.

would seem to explain both the insula


An antiparallel arrangement of these spins


ting and antiferromagnetic properties of the


material


as indeed is


the cas


A low-lying optical excitation


would then


transfer of one hole from the O 2p level to the upper Hubbard band (Cu 3d).


gap for thi


s excitation is denoted by


A in Fig.


This has been corroborate


optical studies of La2CuO4 and other insulating parent compounds,


where a charge


transfer excitation has been observed in the optical conductivity that is peaked at


around 1.7-2.0 eV.


Then, the next question to ask is how to construct a Hamiltonian that includes


the motion of additional holes introduced by doping?


An answer to this has been


provided in a


dimensional tight-binding model by Emery


et at.89'90


and Varma et


al.91 The basic feature of this model is the introduction of an hybridization parameter


between the Cu 3d and


0 2p orbitals.


Other parameters that are included


account for all possible interactions are site energies Ed and Ep,


Coulomb energies Ud


















3 bands


LHB


0 2p ---UHB


11. Energetic position of the three bands in the three-band Hubbard
model.


Hamiltonian can be written explicitly


= tpdp(di + h.c.)
(,d)


-ipx,


p(p' + h'.c.) + edZ


d +
fl1 +62,


+ Ud


nd .d


+Up


72,p
1i,0 1,-Uc


+ Vdp,


tip
1-tin


(1)


The first term is the hybridization or hopping between nearest neighbors on Cu and 0


atoms.


The pi are Fermnionic operators that destroy holes at the 0 site labeled j


while


the di correspond to annihilation operators at the Cu site i. Also, (i,j) refers to pairs


of nearest neighbors on i (Cu) and j (0) sites. A term for direct 0-0 hopping


included for completeness.


into ar.r.nnnt


s also


Notice also that only near-neighbor interactions are taken


Tntera.ctions a.t larrper ditfiane..r a.r th onrht to he screened bv a. finite








limit Ud


> ep


- d,


additional holes produced by doping will go mostly into the O


2p orbitals.


There have been band structure calculations9" and most recently cluster


calculations94 that have placed estimates for the paramet


ers in Eq.


The results


2 3.6 eV, Ud


=8 -11 eV


, Up 4 eV; the remaining terms (tpd,


tpp) are all in the range


It is clear from the value of Ud the appropriate limit


for the physics of the high temperature superconductors is the intermediate to the

strong coupling limit.


1 band

Ueff


LHB


UHB


Fig. 12.


Simplified picture of the three-band Hubbard model where the Cu


3d and 0 2p levels are hybridized to form the lower and upper Hubbard
bands (LHB and UHB respectively).




t J Model


are Ep






29

that the correct effective theory might be reduced to a one-band Hubbard model. In


case


the Hamiltonian is defined as


dt d,,


+h.c +U


nr,tni,,




where the dt are fermionic operators that create holes at site
2,0


i with spin


hybridization of the Cu 3dz2_,y2


orbitals is denoted


by the parameter t.


The parameter U is the on-site Coulomb repulsion.


Figure 12 shows a schematic in


this case where the three bands of the three-band Hubbard model are combined into

two bands labeled as the lower Hubbard band (LHB) and the upper Hubbard band

(UHB).


the strong


coupling


limit


the previous model


can be


transformed


into the so called t -


J model


was first derived from


the Hubbard model


canonical transformations by Hirsch95 and Gros et al.96 In the context of the high-Tc

problem, the model was introduced by Anderson94 and derived by Zhang and Rice97


by canonical transformation from the three-band Hubbard model.


essence,


is a one-band Hubbard model, where the state of the doped hole is only represented


by the spin of the Cu site


on which it resides,


i.e., spin up or down if there is no


hole, or the absence of spin if there is one hole at any particular site.


Out of possible


triplet or singlet mixing states, the latter was found to have the lowest energy for the


hybridization of the Cu and doped-hole wavefunctions.97


H = J (S,.Sj -inj)


The Hamiltonian is


S+ [,cC +h.c],
(ij>,a


where J is the antiferromagnetic coupling between nearest neighbors (ij) similar to


1t-_ -- ...__ TT 1 TT *..


ml C 1. i T 1 P .








where the limit of validity is for


The Si are spin-1/2 operators


and ct create electrons of spin a on site i. Hence, electrons move in a 2-dimensional
lattice with hopping amplitude constrained such that there are not doubly occupied
lattice with hopping amplitude t, constrained such that there are not doubly occupied


sites.


There have been some controversies in whether the three-band Hubbard and


the reduced t -


J models will lead to the same low-energy physics on a temperature


e in the order of Tc.


There have been some cluster calculations on a small number


of atoms that have addressed this issue.


In one of those studies.92


a cluster of the


form Cu5 016 is used with the full three-band Hubbard Hamiltonian and parameters


determined a priory.


problem map

number of spi


For the stoichiometric case, it is found the eigenvalues of the


to the corresponding ones of the Heisenberg Hamiltonian for a finite


IS.


is taken as giving some support to the model, although more


work is clearly needed in the area.


Models for al (w)


Numerical Results


The one-band Hubbard and t -


J models have been the starting points in many


calculations for studying the dynamics of carriers moving in the Cu02 planes.


These


include the response of those carriers under the influence of an electromagnetic field.

Results pertaining to the anomalous midinfrared absorption observed in al(w) for


nearly all copper-oxide superconductors were discussed in Chapter II.


Here, there will


be a summary of the theoretical studies related to obtaining oa(w) from numerical

solutions to the one-band Hubbard and the t J models outlined above.

It is generally accepted the one-band Hubbard model is one of the simplest models


which may contain the essential features of the CuO2 planes.


The problem that exists






31

This is the reason why numerical solutions of finite cluster calculations have been

found useful to obtain approximate solutions to the problem.98s-100

The numerical method commonly used to determine the ground state of the clus-


ters is the Lan


czos


technique01


This technique consists


in providing an initial guess


for the ground state


o0) of the system.


The next step is to apply the chosen Hamil-


tonian to this ground state


to obtain a second state


0o> -


with


constraint


second


state


orthogonal


to the


'Ai=o.


The remaining


states corresponding to the Hilbert space of the cluster


under study can be constructed in this way to give a matrix of coefficients an and bn

that are defined by


('An-i


An-1 )


These definitions assure the states that form the basis of the Hilbert space are or-


thogonal


to each other.


The matrix obtained from these coefficients can be later


diagonalized using standard methods.

The conductivity tensor in linear response theory is obtained from the relationship


between the current density operator, j z(q, w)


and the electric field vector, E,(q,w),


in the limit of q ~ 0.


Hence, it can be


shown


a,


b,





32


where azx the absorptive part of the optical conductivity at zero temperature might

be written as


1
azz =~


Im[(#ojz- H


-- ---xo) ,


where H


the Hamiltonian whose energy is E0, w is the frequency, and e is a small


number that moves the poles of the Green's


Hubbard model


function into the complex plane. In the


, the current operator j, in the x direction at zero momentum can be


written as98


z = itZ (c ,ca+x,a


- h.c.).


As discussed previously, the one-band Hubbard Hamiltonian contains three charac-

teristic energies that are expected to give interesting optical excitations; the hopping


term t


, the on-site repulsion term U, and the exchange interaction J, given by Eq.


in the


strong coupling limit.


Dagotto et al.100 reported numerical solutions for the


one-band


Hubbard model on a


x 4 cluster,


and for


a hole concentration in


interval 0


x < 0.375.


At half filling,


the results show accumulation of weight in


acr(w) above an energy which is close to 6t. If previous estimates of t are taken,92 this

energy can then be correlated with the corresponding charge transfer gap of excita-


tions that


observed in the optical conductivity of the insulating compounds around


0 eV. If such correlation is made, the weight in al(w) is the result of charge


excitations from the lower to


the upper one-band Hubbard model.


Figure 13 also


shows the results at dopin


levels away from half filling.


It is evident in the


figure


the redistribution of spectral weight from this charge transfer gap to lower energies as


function of dopi:

2 holes in the 4


n


In particular, the doping level of x = 0.125, which corresponds to


x 4 cluster, shows two major features that occur below the CT band.






33

weight that occurs half way between zero and the CT band has been associated by

Dagotto et al. and others98-100 as the midinfrared band that shows up in the optical


conductivity spectrum of the cuprate.


The results also show that,


with increased


doping, the Drude peak grows considerably,


while the midinfrared band only shows


a modest increase.


Dagotto et al.100 pointed out the result


s shown in Fig. 13 would


correspond to the intermediate coupling regime, i.e., U


gap develop


~ 8t. For large coupling, a


s between the CT excitation and the midinfrared band, whereas for small


values of U/t both


excitations merge, making it difficult to separate them.100


Sim-


ilar calculations in the context of the t -

Horsch99 for different values of J. The res


ones shown in Fig.

these calculations.


C.-X. Chen and H.-E


J model have been done by Stephan and


ults show a qualitative agreement with the


The Drude and the midinfrared peaks are clearly evident in


similar results in the near-half filling case were also obtained by

. Shilttler from solution of the one-band Hubbard model in the


strong coupling limit.102

Two-Component Model


From the previous discussion, it is quite clear the numerical results obtained from


the one-band Hubbard and t -


J models favor the approach followed by many re-


searchers regarding the interpretation of the optical conductivity al(w) obtained in

experiments involving the optical properties of the copper-oxide superconductors. In


this approach,


the infrared conductivity in these materials is considered


to be the


combination of a Drude-like free-carrier component at w = 0,


with a strongly tem-


perature dependent scattering rate, combined with much broader bound excitations


at higher frequencies.


this picture,


the free carriers track the temperature de-


pendence of the dc resistivity above Tc,


while condensing into the superfluid below













0.20




0.15




. 0.10




- 0.05




0.00
0


5 10 15


O/t


Fig. 13. Optical conductivity obtained from solution of the one-band Hubbard






35

The extra component in the infrared has also been argued to be the result of direct

transitions from valence band states close to the Fermi level into empty states of the


conduction band.104


In the absence of a clear physical origin for these


"intraband"


excitations, the natural choice has been to model those absorptions by Drude-Lorentz

oscillators. Hence, the dielectric function e(w) is fitted to an equation of the form


(w) =


w2 + iwl/r


N ,2
wpj
j=1 i wry1


In Eq.


, the first


term represents


the Drude component describe


a plasma


frequency upD and scattering rate 1/r; the second term is a sum of midinfrared and


interband


oscillators with wyj,


being the resonant frequency,


oscillator


strength, and the width of the jth Lorentz oscillator respectively.


The last term,


is the high frequency limit of


e(Wo),


which includes higher interband transitions.


One-Component Model


An alternative approach that has been proposed to explain the anomalous non-

Drude behavior in the infrared conductivity of the copper-oxide materials is to model

al(w) using a generalized Drude model with a frequency dependent scattering rate


In this model


, there is only one type of charge carriers.


Hence,


dielectric function can be written as105


= coo-


w [W- C(W)I


where Wp is the bare plasma frequency for the charge carriers in the far and midin-


frared


defined by 4rNe2/mb, and oo is a constant that includes contributions from


1/,(,)~31,75,78








through


the Kramers-Kronig relations,


they must obey causality.


Then,


to ensure


this causality condition, E(w) is taken to be complex, i.e., E(w) =


with E22(w)


Bi(w) +


Hence, the model requires the introduction of a modified functional


form for both the effective mass and the scattering rate.

and imaginary part of E(w), we arrive at

m*(w) c1 (=)
m~u}_ S~o;


By decomposing the real


1/r*(w) = -E2(w) m
-Y22Y) ()


where now 1/7*(w) is called the renormalizedd" scattering rate, mb is the band mass,


and m* is the frequency dependent effect


ve mass.


Two models that provide a phe-


nomenological justification for this approach are the "marginal Fermi liquid" (MFL)


theory of Varma et al.106'107


and Ruvalds.108

the MFL model


and the "nested Fermi liquid" (NFL) theory of Virosztek


For example, the imaginary part of the one-particle self-energy in

is written as


-- Im(


wL)rs


ii2AXT

irXw


,w
,w>T


where A is a dimensionless coupling constant.


Hence, for w < T the model predicts


a renormalized scattering rate that is linear in temperature,


which is expected from


the linear temperature dependence in the resistivity that is observed in nearly all


copper-oxide superconductors.


As w increases, reaching a magnitude of order of T


or higher, a new spectrum of excitations arises.


This causes 1/r*(w) to grow linearly


with frequency up to a cutoff frequency wc that is introduced in the model.








effective mass enhancement at low frequencies.


This enhancement is expected to be


proportional


to the coupling constant A.


Finally,


the model requires the presence


of an energy gap that opens up at the Fermi surface as the material enters in the

superconducting state. This gap should show up in the spectrum of a i(w). As it will

be discussed later, observation of this gap by optical means is still an open question.


When data obtained by different groups72'75'87


using different high-Tc materials


are analyzed in the context of this model, there seems to be a qualitative agreement


with the predictions of the model.


In first place, the dc resistivity obtained from the


model agrees with the experiment.


Secondly,


-Im YE(w) is nearly constant for cw


while


t increases


linearly with cw for cw


. Thirdly,


there is an


effective mass


enhancement at low frequencies that is larger at low temperatures.

One important argument against the model is that the cutoff frequency deter-


mined by the agreement between the data and


the model is rather low,


i. e.,wc is


below


1000 cm


This is in conflict with the high


cutoff frequency suggested


Raman109


experiments where


WcFM


7000 cm


In addition, the coupling constant


determined for samples with lower Tc is actually larger when compared with the value

obtained with higher Tc samples.75 The problem arises because the Tc is supposed to


be determined by


Superconducting State Models


Experimental evidence for the unconventional nature of the normal-state


proper-


ties in the copper-oxide superconductors has been established without a doubt.


the same time, there is no unambiguous evidence that the properties in the super-


conducting state show anomalous behavior as well.


It was established early on that


pairing of electrons (holes) was indeed present in the superconducting state of these


> T








questions that one can ask about the nature of this pairing.


The first one is what


is the force that media


es the attractive interactions?


In ordinary superconductors,


the pairing is mediated by phonons interacting with free carriers in the material.2

The second question is whether the pairing is accompanied by an energy gap that

opens up on the Fermi surface as happens in conventional superconductors.110 Evi-

dence against the conventional nature of this pairing could then be, for example, the


presence of nodes or states within this gap.


While answer to the pairing mechanism must await the development of a success-

ful theory to explain superconductivity in the high-Tc materials, experiments could,


in principle, provide answer to the presence of a gap.


In fact, many experimental


techniques, such as tunneling, infrared spectroscopy, photoemission and penetration

depth measurements, have tried to demonstrate the existence of the superconducting


gap and its


symmetry.


Unfortunately, there have been contradictions in some of the


results.


The following section will review the many ideas, both theoretical and


perimental, that have been discussed related to the possible symmetry of the pairing

in the superconducting state of the copper-oxide materials.

Symmetry of the Order Parameter


Superconductivity in general is regarded as evidence of a broken symmetry.

transformation involves a change that allows the description of the system, in the


This

ther-


modynamic limit, by a macroscopic wavefunction and a phase. In group-theoretical

calculations, the symmetry of the superconducting state corresponds to one of the

irreducible representations of the total symmetry group of the normal state.111'112

The possible broken symmetries considered are spin and point group rotations as well


as global gauge symmetries.


the high-Tc materials, such as


YBa2 Cu3O -






39

of this distortion may lead to different conclusions regarding the possible symmetry


classifications of the order parameter in the superconducting state. In addition,


since


in BCS theory superconductivity results from pairing of electrons, the coupling of the


electron spins could lead to singlet (


0) or triplet (


1) states for the spin part


of the pairing wavefunctions.


In the following discussion spin-orbit coupling is not considered for brevity.


Also,


the assumed crystal symmetry in the normal state will be the one of the lattice,"13


although others have been proposed in the literature.114


For example, if a tetrago-


nal point group symmetry is considered for the crystal, the possible superconducting


state


symmetry


can be


derived from


the irreducible representations of


group


SO(3)


x D4h,


where


SO(3) is the group rotation in spin space and D4h is the point


group symmetry of the lattice.


In a singlet state with tetragonal point group sym-


metry,


there are


and 2-dimensional irreducible representations corresponding to


the D4h group in the system.


Based on experimental grounds, the two most quoted


possibilities for the high-Tc are the Alg and Big.


The first one


, the Alg, is the only


singlet state with a gap function that is nonzero everywhere on the Fermi surface.

This symmetry corresponds to the so-called s-wave symmetry in the BCS theory for


conventional superconductors.


The gap function has the form


up to a


function with the symmetry of the lattice.


The second possibility that has arisen in


some theoretical models,115 as well as from some recent experimental results5 116 that


will be discussed later, corresponds to the B1g point group symmetry.


This represen-


station gives a gap function that can represented with the d 2_v2 orbital and that is


usually written


as A(k)


= Ao(cos


- cos


On a spherical Fermi surface, this gap


function would have nodes at 450


angles with respect to the lobe maxima.


aa ,








point group and has s-wave symmetry.


irreducible representations,


The rests have the Big, B2g, and B3g


having single-state gap symmetries of the following d-


wave like orbitals:


dzy, cdz, and dyz respectively.


It should be pointed out that the


d 2__y2


will not be realized in this case.


There are many other possibilities that include triplet state spin wave functions


in both orthorhombic and tetragonal point group symmetries.


The p-wave states, for


example, are important in superfluid 3He.


The interested reader is referred


to the


literature for more details.113

Evidence for Proposed Pairing States

In the context of the t J model, there have been numerical results on finite clus-

ters that suggest an attractive channel for binding of two holes in an antiferromagnetic


background with dz2_y2


symmetry.115'117"118


The results show the average distance


between


the two holes decreases as the ratio J/t increases.


It has


been


proposed


this provides evidence that the strong coupling limit is a necessary condition for the


binding of holes


n the t-


J model and that it eventually leads to phase separation in


the system, zi.e.,


the material separates in hole-rich and hole-poor regions.115"119-121


Dagotto et al.'5 extended the calculations to include higher doping in a 4


as a function of the ratio J/t.


x 4 cluster


The numerical results also suggest in this case a signal


for superconductivity that is stronger in the dz2 y2 c

pairing correlation function shows a maximum at J/t


channell

93.


For this channel,


Of course, these results


are not enough evidence to prove a condensate in the bulk of the system. Finite size

effect studies should be carried out before a definite conclusion could be reached. On


the other hand, finite


cluster calculations in the three- and one-band Hubbard mod-


els have not given evidence for binding of holes in the d,_ y2 channel, although the


*.








Let us turn our attention to the experimental data.


The electromagnetic London


penetration length AL(T) is considered one of the basic lengths in superconductivity.


This is


because


its temperature dependence can give information about


the pair-


ing stat


e in a superconductor.


Earlier ab-plane penetration depth measurements on


YBa2Cu3O7-s single crystals and films by Harshman et al.124'125 and Kruisin-Elbaum

et al.126 suggested the gap was nodeless, i.e., the low temperature behavior of AL(T)


thought to be exponential


as in an ordinary BCS superconductor with isotropic


Most recently, a reanalysis of these and more recent kinetic inductance data


y Fiory


et al.127


have yielded a quadratic variation of


AL(T) from its


zero


tem-


perature value.55'128


Most available results are best fitted129 by using the empirical


Gorter-Casimir formula


L (T) =


AL (O)


--t2)1/2'


where


t = T/Tc.


Likewise, in more recent penetration depth experiments on single


crystals of YBa2Cu307_6


using


a stable microwave cavity,


it has


been found


very low temperature dependence of AL(T) is linear rather than quadratic in T.116


The results have


been


explained


as consistent


with


an order parameter that


d 2_y2 symmetry, since this symmetry would introduce a linear power-law variation

in AAL(T) of the form


AL(T) -
AL(O)


caBT
Amar '


where Amaz is the maximum gap value over the Fermi surface.


Morover, the tern-


perature dependence of the penetration depth in Bi2Sr2CaCu20s single crystals and

YBa2Cu307_ films have been done by Ma et al.130 In this study, it is of found that


in both


cases


AAL(T)


oc T2 for T


Thus far, there has been little progress in


the understanding of the observed differences.


The problem that exists is that any


was









attempts to model the effect of impurities that could change the linear T


to T2 in less than ideal samples.131"132


dependence


It is shown in one of those calculations that


strong resonant scattering could account for the differences that are observed in the

temperature dependence of AAL (T) in films and single crystals that have an order pa-


rameter with a dz2_y2 symmetry.1m


Perhaps, other more subtle mechanisms such as


the CuO chains in YBa2Cu307_6 and the superlattice modulation in Bi2Sr2CaCu208

could have some effect in the electrodynamics response of these materials.


Using


Bi2Sr2CaCu2O8


samples


from


the same


batch


as the one


showed


AL(T)


ac T2


Shen et al.


performed angular resolved photoemission spectroscopy


(ARPES) measurements and found a condensate peak that is larger and more pro-


nounced along the symmetry


line F


-X


, i.e., from center of the Brillouin zone to


the x-direction in momentum space, and seems to vanish (within the experimental


resolution of 10 meV) 450


away from the previous direction (P


-M).


Based on this


observation, they conclude the symmetry of the order parameter is compatible with


_y2 symmetry pairing.?

3 first is that the gap i


This conclusion is strongly dependent on two assumptions.


s zer


o in the direction where it is undetectable in the ex-


periment.


The second is the point group symmetry of the crystal is tetragonal rather


than orthorhombic, i.e., a rotation of 900


or 1800 in the plane would give the same


magnitude of the gap with just a change of the phase.

it is not clear whether these checks were made. More


From the results presented,


lover, such data do not seem


incompatible with similar measurements done by Kelley et al.7 where another singlet


d-wave symmetry (dzz) is proposed as an allowed possibility.


symmetry analysis


This is concluded from


of the condensate peak observed in photoemission experiments


as function of photon polarization and photoelectron collection directions.


The au-






43

Finally, the authors concluded that the point group symmetries of both the normal

and superconducting states of this compound show a D2h instead of a D4h character.

Other probes of the pairing state in the high-Tc, such as tunneling, do not reveal


clear evidence of a gap,


but rather show substantial density of states in the range


where a gap might be expected.8'9

Determination of Gap by Optical Spectroscopy


As is well known, a superconductor is a good reflector of light for energies below


where


ducting state.


is the optical energy gap for quasiparticle excitations in the supercon-

This means that in a reflectance experiment a signature of the gap


will be when R = 1 for photon energies less than 2A.


At higher energies, since there


is enough energy to break a Cooper pair, the material will start absorbing light and


the reflectance will show deviation from unity.


A Kramers-Kronig analysis will then


show an optical conductivity Oai(w) that is zero for w =


and finite for w


> 2A.


This has indeed been confirmed in the experiment, where


predicted by the BC


is very close to the value


theory for most conventional superconductors.110


There have been attempts in many optical studies to associate features in the su-


percondu


acting state


reflectivity of the high-Tc materials with superconducting energy


gaps.31


,65,78,133,134


Values in


the quantity 2A/kbTc reported in the literature have


ranged between


for two reasons.


However, these claims must be taken with certain caution


In the first place, there is a stringent requirement that R must be


unity below the threshold for quasiparticle excitations.


With the typic


al uncertainty


AR in the order 0.5%, it is not possible to rule out any small but finite absorption


when the reflectance is very close to 100%.


This is


important in view of the fact that


such uncertainties in R will introduce errors in acr(w) in the order of








It is clear that


the S/N ratio in cl(w) diminishes as


R approaches one.


Secondly,


there have been reports of direct absorption measurements30


that show a nonzero


value for the a-axis absorptivity of single domain crystals of YBa2Cu307_6 down to


100 cm


-1 or so.


Such energy falls well below the range of values of 2A/kbTc reported


in the literature.


To reconcile these results


, it has been proposed the extra absorp-


tion observed in al(w)


below Tc is part of a second component in the midinfrared


conductivity.68,71


,72,103,116,135


To explain the absence of a gap feature, it has


been


argued that since the high-Tc materials are in the clean limit,68 because of the short

coherence length and long mean free path, the rising reflectance in the far infrared is

not due to a superconducting gap but rather to a scattering rate that approaches zero


as the sample enters in the superconducting state.71

a Drude-term conductivity that condenses into a d


This picture is consistent with


elta function at zero frequency.


This leaves very little spectral weight for transitions across the gap that might still


be present in the superconducting state.


Furthermore, it has been argued that some


of these features might be associated with interaction involving longitudinal c-axis


phonons with the ab-plane bound carriers.135'136


An apparent justification for this is


due to the fact that some of these features are still present above Tc.


This issue will


be addressed in Chapter V with results obtained on a La2Cu04+6 single crystal.












CHAPTER IV
EXPERIMENTAL TECHNIQUES


optical


experiment


involves


the measurement of the


transmittance or the


reflectance of a sample as a function


of the incident light frequency.


When


is done over a


very wide frequency range,


typically


between


the far infrared and


the ultraviolet (UV) regions of the optical spectrum, it is necessary to use different


combinations of spectrometers, light sources, and detectors.


the experimental techniques used in this work.


This chapter describes


The first section in the chapter includes


discussion of the Fourier transform spectroscopy technique that was used to cover the


spectral regions in the far and midinfrared.


There is also a discussion on the Perkin-


Elmer monochromator that was used to cover from the midinfrared up to the UV.


Descriptions of the kind of detectors, polari


zation control, sample mountain


and data


analysis procedures will also be presented.


A final section will briefly describe the


preparation techniques of the high-Tc materials that were studied in this work.


Fourier Transform Infrared Spectroscopv


The far infrared


one of the less accessible spectral regions.


The reason for this


is the reduced available power from radiant sources at those low frequencies. For any

given source, the total blackbody power spectrum is given by


PO = aT4 A,





46


available from zero up to a frequency w can be estimated by using the Rayleigh-Jeans

law137


p(w) = B Aw2
127r2c2


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


Then,


w is given by


P (w) 5 hw 3
Po B4 k T


100 cm


-1 and T


= 5000 K the fraction is T7 =


x 10


, i.e., if the total


power of the source is 1 W, only 1/ will be emitted for frequencies below 100 cm
power of the source is 1 W, only 1 pW will be emitted for frequencies below 100 cm


This energy deficiency


troscopy. 38


was overcome by the development of Fourier transform spec-


The principle of operation can be understood in terms of the Michelson


interferometer shown in Fig. 14.


Source
S


Focusing
Lenses


j Fixed Mirror





Beamsplitter
/ 1I


Movable


Detector


M,

Mirror






47

Light coming from a source falls onto a partially transmitting beamsplitter.


the ideal


case,


half of the radiation will be transmitted to the movable mirror M2.


The other half is reflected onto the fixed mirror labeled M1.


and they recombine again at the beamsplitter.


Both beams are reflected


When M1 and M2 are equidistant with


respect to the beam-splitter position, the sum of the two beams will have maximum


intensity. As


M2 moves away from this maximum position, the two beams will be out


of phase by an amount 0


= 2rvS6,


where v is the frequency of the incident light in


units of cm


-1 and S is the distance as measured away from the position of maximum


intensity or zero path difference.


If the two beams have equal amplitudes a(v),


sum of the amplitudes reaching the detector can be written


A() = a(v)(1 +e e6).


The intensity of the radiation reaching the detector


can then be


as a function of path difference


written


I(S, v) = AA*


= 2a2


(1+


1
cos2ivS) 2S(v)(1+
2


cos 27rz


where


S(u) is the spectral density of the source. For a source having frequencies from


v = 0to v


= 00


, the total integrated intensity, or the interferogram I(S),


can be


obtain by integration of Eq. 21.


The result is


I(6) =


[00

0 o


S(v)dv +


+oo

0


S(u) cos 2ruvSdv,


The first term in Eq.


is a constant equal to the total output intensity of the source


The spectrum itself can be found by computing the inverse Fourier transform of


the interferogram,


[+oo


1Q5)e2T&'bdS.


I- -F


v>








the integral in Eq.


The importance of this is that


the information about


the spectrum is being observed continuously.


This advantage was first recognized by


Fellgett.139


In practice,

ited resolution.


1(8) is recorded over a finite range of path differences and with lim-

The discreet nature of I(6) changes the Fourier integral into a Fourier


series. In addition, the maximum finite path difference 6m introduces side-lobes near


sharp features of S(v).


This problem can be minimized by the method of apodiza-


The sampling interval of I(6) determines the maximum cutoff frequency


(vmax) of


3(u).


This will introduce aliasing for frequencies higher than (Vmax).


This


problem is sol


unwant


ved by introducing proper optical filters that will attenuate those higher


ed frequencies.


Optical Spectrometers


Bruker Fourier


Transform Spectrometer


The instrument used to cover the far and midinfrared (30-4000 cm


-1) is a Bruker


113v


fast-scan Fourier transform spectrometer.


The principle of operation is very


similar to the Michelson interferometer.


A schematic diagram of the spectrometer


is shown in Fig.


The instrument comes equipped with two


sources;


a Hg and


globar


lamps for the far


nfrared and midinfrared respectively.


There are also de-


tectors for each of those spectral regions.


the interferometer area,


the emitted


light from the source


is focused to a beamsplitter that sends the transmitted beam


to one mirror and the reflected light to other mirror facing the first one. Both beams

are sent to a two-sided movable mirror which reflects both beams back to be recom-


bined at the beam-splitter site.


The two interfering beams are then directed to the


tion.140,141





49

the reflectance chamber, that allows reflectivity measurements with near-normal in-


cidence.


The final destination of the light is the detector chamber.


The whole area of


the spectrometer is evacuated to avoid absorptions by water and CO2 present in the

air. Because the two-side mirror moves with constant speed v, the path difference 8

between the transmitted and reflected beams at the beamsplitter, is changed accord-


ing to the relation S = 4vt,


where t is the time as measured from the moment S


Hence, this moving mirror produces a modulation of the infrared signal,


in the form


D(t) = Do cos(2 rfat),


where D(t) is the signal as received by the detector and the infrared frequency Vo is


turned into an audio frequency in the formula fa = 4vvo.


The next stage is amplifica-


tion and digitizing of this signal before is sent to an Aspect computer for apodization,

phase correction, and finally application of the Fourier transform to the obtained in-


terferogram to finally get the spectrum.


Table 1 shows the parameters used in the


Bruker spectrometer.

Bolometer Detector


As was mentioned before, the intensity of the source blackbody spectrum becomes


rather weak in the far infrared.


This can be partially overcome by using detectors


that are sensitive enough to give an acceptable signal to noise ratio.


The limitation


that most infrared detectors have is that the sensitivity is limited by the background


noise.


For example, if the detector is operated at room temperature,


the peak of


the blackbody spectrum of this background is centered at around 1000 cm-1


in the middle of the infrared.


, right


This can be overcome by operating the detector at











Table 1.


Bruker FTIR Operating Parameters


Range Beam Split. Opt. Filt. Source Pol. Detect.

cm-1 Material Material Material

35 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 4000 Germanium on KBr none Globar 2 DTGS, photocell


= polyethylene. Polarizer 1


= wire grid on oriented polyethylene; polarizer 2 = wire


grid on AgBr.


is shown in Fig.


The detector element is made out of Si.


It detects


signal


by bolometric means,


i.e., by changing its temperature as the infrared light strikes


This change in temperature is amplified and recorded as a voltage signal,


which


is then digitized and sent to the Aspect computer.

The Perkin-Elmer Monochromator


At higher frequencies,


the Fellgett advantage losses its importance due to the


availability of brighter sources and more sensitive detectors.

grating spectrometer is an excellent choice to cover frequenci


For these reason


es in the near infrared


and up to the UV


(1000-40,000 cm


The instrument used i


s a modified Perkin-


Elmer


monochromator.


The diagram showing the details of the instrument is


in Fig.


Depending on the frequency of interest, there are three sources to chose


from:


a globar for the midinfrared,


a tungsten lamp for the near infrared, and a























































































* but sflnmmf C-sR
* tlaI filWtr
4 Auterathc b-ur)ator etwigr
* TMan icbenuviw cgwq
* Cn-ad nores tear
I cwitis m.a.n.lw
3 Aslern. har
* Rene cate mflmul thurm


-a~ ~~~~ -


t n


rr, Y m








DEWAR,


MODEL


HD-3


OUTLINE SKETCH





53




grating filters
chopper










---slits glo rb
tungst
larr.p


deuteriun
sample lamp









Sample


detector
vacuum tan

\


,e


a


ik


tr









formula 2d sin0 = nA,


where n is the nth order of the diffracted light, A is the wave


length, 0 is the angle of incidence, and d is the spacing between the grating lines.


angle of incidence is changed by rotating the grating with a


step-motor controller.


This


allows access to


different


wavelengths sequentially.


The now


"filtered"


exits the monochromator and it is focused onto the position labeled R in Fig.


The sample and reference are placed there for reflectance measurements.


of transmittance, the sample and reference (in this case an empty, or "blank,"


In the case


sample


holder)


are pos


tioned at


the point


The reflected


or transmitted light


reaches


a detector that


connected


to a lock-in amplifier for


amplification of the signal.


The output of the lock-in is fed


to a digital voltimeter that is remotely controlled


by a PC computer that also controls the step-motor controllers of the grating and


filters.


The collection of the data is done through the computer by recording a single


beam spectrum (signal


vs. frequency) for the reference and the sample sequentially


and taking the ratio of these two spectra (S


transmission of the sample.


s/Sr) to obtain the true reflectance or


During normal operation, the spectrometer chamber is


evacuated down to 150 mTorr or so to prevent any absorption due to water or C02

present in the air. For more details about the operation of this machine the interested


reader is referred to Reference 142.


Table 2 lists the parameters used to cover each


frequency range.

Polarizers


The need to resolve the dielectric tensor along the principal axes of the single

crystals used in this work required us to polarize the electric field of the light. Since the

radiation generated at the source is randomly polarized, the polarization of the light


was accomplished by inserting a polarizer in the path of the beam. Tt


he characteristics









Table


Perkin-Elmer Grating Monochromator Parameters


Frequency Grating a Slit width Sourceb Detector C
(cm-1) (line/mm) (micron)


801-965

905-1458


2000

1200
1200


1403-175
1644-261


2467-4191


1200


-5105


4793


1200
1200


-7977


9-5105


7511-10234


9191-13545
12904-20144


3-24924


22066-28059


706-37964


6-45333


1200


1200


2400


2400
2400

2400


a Note the grating line number per cm should be the same
order of the corresponding measured frequency range in cm1.


GB: Globar;


c TC: Thermocot


W: Tungsten lamp;

iple; PbS: Lead sulfi


D2: Deuterium lamp.

te; 576: Silicon photo-








the midinfrared spectral range (300-4000 cm


-1) a silver bromide substrate is used,


while in


the far


infrared


(30-600


the substrate is polyethylene.


Dichroic


plastic polarizers were used in


the near infrared,


visible and


In both


Bruker and


Perkin-Elmer spectrometers,


the desired polarization of the light


easily accomplished by mounting the polarizer in the path of the beam using a gear

mechanism that also allowed rotation from the outside without breaking the vacuum


in the spectrometers.


This in-situ adjustment of the polarizers greatly reduces the


uncertainty in the relative anisotropy of the reflectance (better than -0.25%).


This is


specially important when the relative anisotropy in the optical response of the crystal

is not very large.


Front


view


Conical
surface





Sample


Side


view


Coplper
wires


Fig. 18.


Diagram of frame used to mount samples.


Sample Mounting and Low Temperature Measurements


Due to the small size of the samples used in this study, special care had to be


was





57

advantage of the whole area, the mounting of the sample was done in the following

way. A copper frame was machined with a small hole at the center (c 2 mm diameter).

One side of the frame (the front side) had a conical surface as illustrated in Fig. 18.

This conical surface was necessary to scatter away any light that may hit the copper


frame when inserted in the beam path inside the spectrometer.


The next step was


to solder two very fine copper wires across the center of the hole on the back side


of the frame.


Then, the sample was carefully mounted on the two wires and glued


there by a good thermal conductor (apiezon grease).


A piece of Al-coated Si with


approximately the same area as the sample was mounted on another frame in the

same way as the sample in order to be used as a reference.


Mounting of the Bi2Sr2CaCu20s


and La2 CuO4+s samples did not require the


the use of wires due to the larger size of these samples.


Instead, each sample was


mounted on


the back side of a frame, as shown in Fig.


smaller diameter than


the size of the sample.


with a hole of slightly


A frame with same hole-size as the


frame of the sample was used to mount a Al-coated piece of glass to be used as a


reference.


Finally, frames containing the sample and


mounted facing


the Al-coated reference were


out in opposite sides of a sample holder assembly that


was later


positioned inside the spectrometer for reflectance measurements.


To facilitate


study


of the polarization dependence in


the reflectance, samples were mounted so


that the principal axes of the measured face could be studied by setting the polarizer

horizontal or vertical with respect to the spectrometer bench.


Low temperature measurements were possible by attaching the

sembly to the tip of a Hansen and Associates High-Tran cryostat.


sample holder


A flexible transfer


line was used to flow liquid helium from a


storage tank to the cryostat.


The tem-








element attached to the tip of the cryostat.


In this set-up, the temperature of the


sample could be lowered by increasing the flow of helium and increased by applying a

current to the heater element. During measurements, the sample holder and cryostat

units were placed inside a shroud equipped with optical windows in the spectrometer


chamber.


The pressure inside of this shroud was kept below


10-7


Torr to prevent


the formation of ice on the cryostat or the surface of the sample.


Since sample and


mirror were on opposite sides of the sample holder, measurements of the sample and


mirror spectra were possible by simply rotating the cryostat assembly by 1800


. The


final step was to take the ratio of these two spectra to obtain the reflectance of the

sample.

Normalization Procedure of the Reflectance


After measuring the temperature dependence in


the reflectance of the sample,


the final normalizing of the reflectance was obtained by taking a final room temper-

ature spectrum, coating the sample with a film 2000 A thick of Al, and remeasuring


this coated surface.


A properly normalized room-temperature reflectance was then


obtained after the reflectance of the uncoated sample was divided by the reflectance


of the coated surface and the ratio multiplied by the known reflectance of Al.


This


result was then used to correct the reflectance data measured at other temperatures


by comparing the two room-temperature spectra taken in the two separate runs.


This


procedure corrected for any misalignment between the sample and the mirror used

as a temporary reference before the sample was coated and more importantly, it pro-


vided a reference surface of the same size as the actual sample area.


In cases where


the sample surface had some roughness, the procedure also corrected for losses due

to a nonspecular sample reflectance.








Data Analysis of the Spectra:


The Kramers-Kronig Transformations


The power reflectance measured in the experiment is related to the amplitude

reflectivity which contains information about the optical absorption in the sample.

For normal incidence, the amplitude reflectivity is given by


- 1) +i


(n + 1) + is


, (25)


where n and n are the real and imaginary parts in the complex refractive index of the


sample under consideration.


Information about these quantities can be obtained by


noting that the power reflectance R(w) is


related to the amplitude reflectance r(w)


in the following way:


r(w) = p(w) exp i(w),


where (wc


) is the phase shift in the light upon being reflected from the sample and


) is related to the power reflectance R(w) by


R(w) = p2 ().


Since the amplitude reflectance and the phase shift are the real and imaginary parts of

a response function respectively, they can be related by means of the Kramers-Kronig


transformations. 143,144


Hence,


the phase shift can be obtained from the Kramers-


Kronig integral


0 (3)-


w /3: In R(w')
""JQ


-lnR(w)d.
-W12


In principle, knowledge of the phase shift is only possible if the power reflectance is


known over an infinite range of frequencies.


In reality


R(ow) is only measured over a


T(W)






60

w, and for the regions where R(w) is flat, there are negligible contributions


to the integral in Eq. 28.


This implies that for the frequencies where R(w) is not


available, it is possible to make extrapolations that would not affect very much the


results for the range for which R(w) is known.


The kind of extrapolations that can


be made depend on the type of material under consideration, as it will be discussed

in the next section.

High-Frequency 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 very


high frequencies (above 100,000 cm


-1) the free-electron behavior becomes important.


In the absence of any published data that can be append to the existing data,

reflectance in the interband region is usually modeled using to the formula


>= Rf LA)
w3


where Rf and wf are the reflectance and frequency of the last data point measured


in the experiment.


The exponent s is a number that can take up values between 0


and 4.


At very high frequencies (wf,),


where the free-electron behavior sets in, the


approximation used is


R(w) = Rf (


It is still expected some dependence on the results for frequencies close to the last


frequency measured on account of the choice of s and Wff.


(for metals this happen


for frequencies above the plasma edge minimum.) At low frequencies, the scheme for


extrapolation depends


on the properties of the material under study. In the situations






61

relation, R(w) = 1 A ~J, where A is a constant determined by the reflectance of the


lowest frequency measured in the experiment.


For high-Tc samples, it is found this


procedure results inadequate and it can only be used as a first approximation. A more

appropriate procedure is to fit the reflectance using a Lorentz-Drude model where the


free-carrier response is assumed to have the the typical Drude form.


frequency excitations are modeled by Lorentz oscillators.


Other higher


The fitted reflectance is


used as an extension below the lowest frequency that was measured in the experiment

before the Kramers-Kronig analysis is finally performed on the data.


the superconducting state, it is expected that


unity for frequencies close to zero.


the reflectance will approach


An empirical formula that has been found to rep-


resent the way R approaches unity is R =


1 Bw4


where B is a constant determined


from the lowest frequency measured.


It should be pointed out that typical uncer-


taint


in R(w) are in the order AR


= 0.5%.


Hence, the propagated error in, for


example, the optical conductivity al (w) obtained from the Kramers-Kronig analysis

of the reflectance is roughly


A01 1 AR
al 1 -R R


It is clear the RHS of Eq.


31 diverges, or the signal to noise ratio is very small as R


approaches unity. Implications of this in the high-Tc materials will be discussed later.


Optical Constants


Once the proper extrapolations are made and


the phase shift is obtained,


optical constants of the material are easily obtained by means of the formulae that


follow.


The frequency dependent refractive index n and extinction coefficient n(w)










K(W) -


2 R(w) sin O(w)


1 + R(w))


-2 R(w))


cos 0(w)


respectively.


These relationships can be re-written in terms of the dielectric function


where the real (e1(


w)) and imaginary (E2(w)) parts are obtained from


ei(W)


= 7t2
11 T


E2(w) = 2nn.


Other important relations are the real and imaginary parts of the optical conductivity

a(w), the skin depth 6(w), and the absorption coefficient c(w):


=w E2/47


=w(l


= 2wtcI


- E1)/47r,


All these equations reflect the fact that the absorptive (real) and inductive (imagi-

nary) parts of a process are all related to each other due to the causality requirement

in the Kramers-Kronig relations.


Sample Preparation


Techniques


This section will provide a brief discussion of the sample-preparation techniques


2
IC


El= N2 = (, i.)2








YBa9CuOO,


Single-Domain Crystal


Untwinned crystals of YBa2Cu307_- that were used in this study were prepared


at the University of Illinois in


Urbana-Campaign by


J.P. Rice


and D.M.


Ginsberg.


The samples were grown using a standard Cu-O flux growth procedure.145


One dis-


advantage of this method is that micro twinning develops in the crystals together with

the transition from the tetragonal phase at high temperatures to the orthorhombic


phase at room temperatures.


This twinning happens because of randomly oriented


mechanical stress present during the slow-cooling segment from temperatures around

850 OC.146 In order to avoid development of twinning the slow-cooling process was


interrupted by pulling


the sample out of the furnace.


Evidence for the quenching in


the tetragonal phase was obtained by looking isotropic extinctions on the ab plane

of the samples under a microscope with crossed polarizers in reflectance mode.147

The tetragonal crystals were then oxygenated during a post-growth procedure that

converted them to the orthorhombic phase having sharp superconducting transitions


near 90


K.40 Since the crystals were not subjected to any mechanical stress in the


process of converting them from tetragonal


to orthorhombic,


there was a reduced


possibility of creating dislocations or defects that otherwise may affect the intrinsic


properties


in the material.


Typical crystals with single-domain regions of 1 x


1 mm2


were obtained every third or fourth successful attempt.


The dimension along the


was approximately 25 pm. Determination of the a and b


axes


in all the crystals


was done by the sample grower before the samples were sent to this author.


superconducting transition


temperature,


as determined b


y cooling the sample in a


field of 10 G,


was


around 90 K. Fig. 19 shows the Meissner fraction measured for a


crystal grown with this technique for the applied field parallel to the a and b


axes.





64

(Meissner effect) is larger when the field is applied along b axis than when it is applied


along the a axis.


This indicates that flux pinning is larger when the field is along the


chain direction.


86 88 90 92


T (K)


Meissner


effect


measurements


single-domain


crystals


YBa2 Cu3 O7- 6(After Ref.


A qualitative analyst


of the data suggests


that the criti


cal current density (Jc) is


larger perpendicular to the chain direction than parallel to it (Jca


> Jcb).


Studies of


the ab-plane anisotropy in the dc resistivity,16 performed on crystals from the


same


batch have given some interesting results regarding the effect of the chains in the dc


transport properties.


These results, which are shown in Fig.


indicate the ab-plane


anisotropy ratio in the dc resistivity is around


with the lower resistivity being


parallel to


the b axis.


This may suggest,


to a first approximation,


the CuO


chains provide an additional electronic channel for conductivity along this direction,


assuming that the chain conductivity can be additive to condu


activity of the Cu02












5

4

E

Q2

1 -


75


100 125 150 175 200 225 250
T(K)


-- 200

a 175

- 150

125

t00
75

50

25

-o0
275


20. Resistivity anisotropy in single-domain
(After Ref. 16).


In addition


crystals of YBa2Cu3O7_s


, there is indication of anisotropy in the Raman-active phonon lines for


the two polarizations in the ab plane.


Bi Sr CaCu 0s Single-Domain Crystals


The Bi2Sr2CaCu20O crystals used in the study were grown by R. Kelley and M.


Onellion at the University of Wisconsin in Madison,


Wisconsin.


The technique used is


a flux method with slow cooling in a temperature gradient.152 In a typical experiment,

the starting materials, Bi203, SrCO3, CaC03, and CuO are ground and placed in an


alumina crucible.


The mixture is then heated to a temperature of 50-70 OC above


the liquidus temperature and equilibrated for 6 hrs.


The temperature is subsequently


lowered to


75-880 OC, and after reaching


equilibrium for 6 hrs, the temperature is


slowly cooled at 0.5-2 C/h to 820 OC where the experiment is terminated.


are subsequently annealed in dry oxygen at 600 OC for


Samples


h an later reannealed in


Argon at 750 OC for a period of 12 hrs.









low electron energy diffraction (LEED) techniques.


The incomemsurate superlattice


modulation pattern was seen along the b axis and not in the perpendicular direction


(a axis), suggesting the samples were single-domain crystals.


The alignment of the


principal axes in the crystal was confirmed by observing the extinction points when


e sample was rotated


under a microscope (Olympus, model BHM) with


crossed


polarizers.


Meissner effect measurements were performed on one of the samples to


determine the superconducting state transition temperature.


The results


which are


shown in Fi


21, reveal the onset of superconductivity is around 86 K. Moreover,


the onset to the superconducting state, as determine by using four-probe resistance


measurements,


is around 91


with a transition


width


Results of the


resistivity tensor along the a, b, and


0.10


0.05


0.00


-0.05


-0.10


-0.15


c axes


Bi2Sr2CaCu;
Z





I U


on these crystals are reported in Chapter


ero
ero


Field Cooled


field Cooled


Temperature


Fig. 21.


Meissner effect measurements on a Bi2Sr2CaCu2Os single crystal.


The orientation of the magnetic field is H i c axis.


SfN@I!@I


* *


@Li..**








La9Cu04+A Single Crystal


This


dissertation


contains


an optical


study


performed


on an oxygen-doped


a2CuO4+5 single crystal prepared at the University of Iowa by F.C.


Chou and D.C.


Johnston.

method.'50


A stoichiometric single crystal of La2(

The as-grown crystal was insulating.


was prepared using a self-flux


The oxidation procedure was carried


out using a electrochemical cell151 with the La2Cu04 sample as one of the working


electrodes. A platinum wire working


as the negative electrode was attached to one side


of the crystal using silver paint and the contact was fully covered with silicone rubber.


The set-up of the electrochemical cell was La2CuO4/ 1N NaOH / Pt.


of the cell


was


Charging up


done by applying an anodic current of 10 pA to the La2Cu04 crystal


for a period of two months. In order to optimized the oxidation current, an constant

electrical potential of 0.6 V (versus a Ag-AgC1 reference electrode ) was maintained


during the charging


ocess


using a potentiostat.


The exact oxygen content of this


particular sample after the oxidation was complete


ed is not known.


Calculations based


on the gained weight could not be used because small pieces of the crystal were lost


during handling. Meissner effect data, which are shown in Fig.


indicate the onset


of superconductivity is at around 40 K. A comparison of this with the onset of other

samples of known oxygen concentration suggests the excess oxygen should be around


~ 0.1


The sample that was provide to us contained two optically-smooth faces.


On face contained the ab plane and the c axis, so with the help of a polarizer, it was


possible to study the anisotropy between these two directions.


was


The other face that


studied only contained the twinned ab-plane direction.





















I Cu I
La CuC


, I I i I I' 1


IIIr


H //ab


r 1- I


crystal B


ti/Ic U


-F
I11I I It .J


rr -


W .i


=2
FC


'~II'


Oe _

1


T(K)


50


0


-50


-100


-150


-200


-250


*I I I T1 II
La CuO


IITW


'Ill-I


I f 1 1


staid B


H //ab


H //c


11.11


L!I I I


'-1--I--I


'I'll


30
T (K)


Fig. 22.


Meissner effect measurements on L


a9CuOAlS single crystal (After


- --


r r I r-r


rrrl


1 r-r 1 1


I.I~ t I












CHAPTER V
OPTICAL STUDY OF La2CuO4+6 SINGLE CRYSTAL


In this chapter, the optical properties of the ab plane and the c axis of supercon-


ducting La2CuO4+4 are described. The ma

state by electrochemical insertion of oxygen.


trial is transformed from its insulating

The sample was prepared at Iowa State


University by F.C. Chou and D. Johnston in collaboration with S-W


AT&T Bell Laboratories.


Cheong from


Although the exact oxygen content is not known in this


sample, it is estimated to be 6 0.11 from comparison of the observed Tc of 40 K

with other samples of known concentration.152


Optical absorption studies of the stoichiometric parent compound La2CuO4 have

revealed this is a charge transfer insulator having marked anisotropy in both phonons

and electronic features for polarization of the light parallel and perpendicular to the


CuO2 planes in the system.52153'691153-155


Most of the optical investigations of super-


conducting materials derived from this parent compound have been on the Sr-doped


_.Sr, CuO4.


28,58,69,72,156


A review of some of these studies


was done in Chapter


III. In this system, a maximum Tc in the order of 35 K is obtained by substituting

15% of the La atoms by Sr. In addition, superconductivity is also obtained when ad-

ditional oxygen is intercalated in the crystal structure of La2 CuO4. Normally, oxygen

intercalation is obtained by annealing the sample at high temperatures (~ 500 oC)


in an oxygen-rich environment.33,34


In spite of its success, there are two major draw-


backs in producing superconducting samples using this technique. In first place, there


is the requirement of extremely high oxygen partial pressures (


kbar) in order to








uniformly oxidized samples could be one of the reasons why reports on


the optical


investigation of oxygen-doped samples of this material have been limited to lightly


doped nonsuperconducting samples.64,69'154


Recent developments in electrochemical


techniques have made possible the synthesis of uniformly oxidized samples with rel-


atively high oxygen content (6


~ 0.12) and transition temperatures near 40 K.152'158


Details of this technique were given in Chapter IV (p.


Fig. 23.


Sketch of the sample with the two faces that were used to measured


the reflectance on the LasCuO4+e single crystal.



Results of reflectance studies of this oxygen-doped sample were obtained from


two faces of the crystal.


One of the faces


(the face labeled I in Fi


23) contains


both the c-axis and either the a- or b-axis


direction (on account of twinning, we were


unable to distinguish).


Face


I allowed us,


with


the use of linearly polarized light,


to probe the optical response of the


c axis


as well as the ab-plane response.


Face II

Face I I
qulc


E E aab
E flab


C








below the superconducting transition temperature.


As in the undoped material, the


spectrum along the c direction is mainly dominated by optical phonons and no evi-


dence of metallic component is found in the optical conductivity.


At the same time,


the ab-plane response shows metallic reflectance in the far infrared and a plasma edge


around w ~ 7500 cm


The second face that was measured,


the face II in Fig.


, provided an aver-


age of the ab-plane response.


Optical reflectance was measured in a frequency range


that extended from the far-infrared and near-ultraviolet spectral regions (80-38,000


cm-).


Temperature dependence measurements, above and below Tc,


were also car-


ried out in the same range of frequencies. A Kramers-Kronig analysis of the ab-plane

reflectance reveals the unusual non-Drude behavior in the midinfrared conductivity

al (w) that is typical of the copper-oxide superconductors. For frequencies in the near

infrared and above, a rather unusual temperature dependence was observed in the op-

tical reflectance of the sample. In addition to the expected sharpening of the plasma


edge at


7500 cm


, the reflectance became gradually lower as


the temperature of


the sample was lowered.


This result may suggest a temperature dependence in some


high-frequency interband transitions.

Moreover, the reflectance displays some structures in the far-infrared region that

are not present when the ab-plane reflectance is measured on face I of the crystal. A

comparison of the sum rule in both cases reveals that the differences in both spectra

are not likely due to different oxygen compositions in both surfaces of the sample.


Instead


, the differences are most likely due to electron-phonon interactions that are


enhanced when the wavevector of the incident light is parallel to the c axis in the

sample.









0.01
1.0 -



0.8 -



0.6 -



0.4 -



0.2 -



0.0


Photon


Energy


(eV)


1000


Frequency


Fig. 24.


ab pl


10000


(cm


Reflectance of La2 CuO4+6 single crystal for light polarized along the
ane and the c axis (T = 300 K).


c-Axis


Reflectance of La9 CuOal


Room Temperature Spectra


Figure


24 shows the room


temperature reflectance for


light linearly polarized


parallel and perpendicular to the c axis of the sample.


The results show a dramatic


anisotropy for the in-plane and out-of-plane optical properties, just as it has been


reported in

compound.


r-doped superconducting samples derived from the stoichiometric parent


28,69,156,158





73

frequencies, the spectrum is almost featureless, showing only a broad electronic-like


feature whose maximum is around w


5500 cm


The spectrum is almost un-


changed with respect to the c-axis spectrum of the undoped material. However,


there


is a weak structure that appears just below the peak of the phonon mode at 512 cm .

This structure, which is not present in the stoichiometric material,52'54 is resolved as


a sec


ond phonon mode as the temperature of the sample is lowered. More discussion


about this will be given later.


Photon


Energy


(eV)


2500



2000


1500


1000


1000


Frequency


10000


(cm


Room


temperature


optical


conductivity


al (W)


obtained


from


Kramers-Kronig analysis of the reflectance shown in Fig. 24.








completely screened out by the free carriers in the CuO2


planes are also visible in


the far infrared.


The real part of the conductivity a<7(w),


obtained from a Kramers-


Kronig analysis of the reflectance, is shown in Fig. 25.

the electronic-like features are more easily seen. The I


Here, the phonon modes and


results presented here indicate


gen doping in the lanthanum cuprate only affects the electronic excitations


related to the copper-oxide planes in the system.


Similar conclusions regarding the


two dimensionality of the electronic properties in the copper-oxide superconductors

have also been drawn from measurements on YBa2Cu307_. and Bi2Sr2CaCu2Os ma-


trials.


25,27,29,57


Low Temperature c-Axis Reflectance


Figure 26 displays the reflectance in the


nfrared region along the


c-axis


direction


as a function of temperature. As the sample is cooled down, the phonon lines become


sharper, as expected.


The structure that appears in the room temperature spectrum


just below 512 cm


1 is more clearly resolved as a phonon mode at 492 cm


This is


not present in the c-axis spectrum of either undoped or Sr-doped samples.52'54

At the low-frequency end, we notice all spectra above Tc approach a constant value


for the reflectance.


This is indicative of semiconducting behavior.


On the other hand,


the inset in Fig. 26 shows the data at the lowest temperature show a downward trend


towards low frequencies that is not present in the data above Tc.


This trend is most


likely correlated with the appearance of a plasmon minimum in the superconducting-


state reflectance as


was


first observed in the c-axis spectrum of La2


_.Sr, CuO4


Tamasaku et al.58


In this study, it was found the minimum forms part of a reflectance


edge that is related to the formation of a superfluid condensate that provides coherent



































400 600 800


1000


1200


Frequency


Temperature dependence in


(cm


c-axis reflectance of La2 Cu04+


single crystal.


of this condensate to the dielectric function can be written


-Wps


w(w+ i+)'


where wp, represents the oscillator strength of the superconducting condensate and

iO+ is the scattering rate that tends to zero as the the mean free path becomes infinite





76


the reflectance edge associated with this condensate moves toward lower frequencies


as the doping level in the sample is decreased.


In order to fit this edge in our data,


we used a condensate oscillator strength with wps

value with the results presented by Tamasaku et al.


indicates that the effective oxygen doping is


~ 85 cm


A comparison of this


for one of their Sr-doped samples


S 0.11), and that the reflectance edge


should appear around w ~-


20 cm


-1 in the present sample.


Due to the small sample


, the lowest frequency measured in the present experiment was around 33 cm


This would explain why the edge was not observed in the present experiment.

Assignment c-Axis Phonons


Let us turn our attention to a quantitative analysis of the phonon modes


shown


in Fig. 26.


In order to perform a Kramers-Kronig analysis,


frequency end was kept constant,


as is customary for insul


the reflectance at the low-

ators. On the other hand,


the positive slope of R(w) below

a low-frequency extension. The


Tc at low frequencies, required us to use Eq. 38 as


procedure followed for high-frequency extrapolation


as described


n Chapter IV.


The results of the Kramers-Kronig analysis performed


on the data are shown in Fig. 27.


The top


panel of this figure shows the optical


conductivity,


while the bottom panel shows Im(-1/e) both at several temperatures.


This


analysis reveals


c-axis conductivity


is dominated


by a total


of four


infrared-active modes centered at 230, 340, 492,


and 512 cm


A comparison of these


frequencies with other optical studies52'54'69 of the c-axis spectrum in undoped and Sr-

doped samples of La2CuO4 indicates good agreement with the first two phonon modes.


However, in those studies only one mode is observed at ~ 501 cm


Group theoretical


analyses16'162 indicate that modes of ionic displacements (q = 0) along the c axis


the nearly tetragonal


structure of La2CuO4+6


will have the


symmetry.









Photon


Energy


(meV)


1000


800


600


400


200


0



1.5


400


600


800


1000


1200


Frequency


Opti


(cm


cal conductivity (upper panel) and loss function (bottom panel)


along the c axis of La2 Cu04.+ at several temperatures.


The peaks in these








The eigenvector for the latter mode at 491 cm


-1 involves in-phase vibrations


of the apical oxygens above and below with respect to the four oxygen atoms in the


plane.


Based on this,


the present data suggest that


the presence of two modes at


492 cm


-1 and 512 cm


-1 is most likely associated with the incorporation of additional


oxygen atoms in


the structure.


These additional oxygens,


which are located in or


between the LaO


layers, may provide two slightly different force constants between


the apical oxygens and the Cu02 layers.

A quantitative analysis of the intensity and linewidth for each phonon mode as

a function of temperature can be done by modeling the reflectance using a dielectric

function model consisting of four Lorentz oscillators plus a core dielectric constant


Eoo to account for contributions at higher frequency.


4(W>


The formula is


.22


where each term in the sum corresponds to an optical phonon


with Wpj, cyj, and 7j


being the intensity,


center frequency and


damping of each mode respectively.


panel


of Fi


shows the temperature dependence of


obtained from a fit


to the reflectance


at each


temperature using Eq. 39.


The bottom panel shows the


corresponding oscillator strength (wpj) for the jth phonon mode.


The results shown


in Fig.


indicate that most of the temperature dependence in


the reflectance


the result of a reduction in 7y


sample is decreased.


(increase in the lifetime) as the temperature of the


This is what should be expected, since at low temperatures, the


thermal motion of the atoms in the structure will be


frozen out.


This diminishes the


chances of scattering among the atomic vibrations in the crystal.


The results show


cm



































800





400


100 150
Temperature


200


250


300


Temperature dependence of the phonon


parameters showing the


linewidth y: (inner na.nel a.nd t h nncilla.tnr strepn h r :. (h nttnm na.neli






80

not show any significant temperature variation in the measured range; only the mode


at 340 cm


-1 exhibit


s some systematic hardening as the temperature is lowered.


center frequency for this mode goes from 340 cm


at T


-1 at room temperature to 347 cm-1


=101K.


Effective Charge

The intensities (wpj) in each phonon line are related to the effective charge that


is carried by each


considered is that each


To establish this relationship, the first thing that should be


phonon mode splits into transverse (TO) and longitudinal


(LO)


ionic vibrations in


the crystal.


Since the


TO frequencies involve transverse


vibrations of the atoms,


they


are obtained directly from


the peak position in


absorption,


, the real part of the optical conductivity ai (w).


On the other hand, the


LO frequencies involve long range changes in the dipole moment along the direction


of propagation in the crystal.


Information about the center frequencies of these LO


oscillations can be obtained, in principle, from the peak positions in the loss function


Im(-1/e) as shown in the bottom panel of Fig. 27


. A direct determination of the LO-


TO splittin

strength of


is also


possible by noting that this splitting is related to the oscillator


the phonon mode in


the context of the Lyddane-Sachs-Teller relation.


The formula is written as


= j 2 2 o)1
pj(WLO, W TOJ)00.


Table 3 displays the


parameters from


a Lorentz fit


to the room


temperature


reflectance along with the TO and the LO frequencies derived from Eq. 40.


Once the








Table 3.


Parameters of a Lorentz fit for the mea-


sured


c-axis


room


temperature


reflectance


La2 Cu04+5.


Oscillator# wTOj WLOj wipj y

(cm-1) (cm-1) (cm-1) (cm-1)

Elc
1 230 498 1052 20

2 342 343 65 15

3 492 502 218 24

4 512 515 174 23

Eoo == 5.3


has been applied successfully to other systems with ionic character.162


The formula


(Ze)2


[we LOj TOj


where j is the sum over all LO-TO splitting,


V is the volume of the unit cell and


k is the sum over all ions with mass mk and effective charge


must obey charge neutrality:


'k. Since the crystal


In general, Eq. 41 can not be solved unless


the number of unknown parameters Zk is less than or equal to two.


In the case of


La2CuO4+s, since oxygen is much lighter compared to


the other atoms,


the right


hand side of Eq. 41 will be dominated, in first approximation, by the term related to


the o


xygen


atoms.


Thus by neglecting all but the oxygen contributions, the result of


solving Eq.


will yield the effective charge of oxygen averaged over all sites.


Such








oxygen in the order of


-~ 1.1.


The slightly higher value obtained here (14%) is most


likely due to the insertion of additional oxygen in the structure.


These results differ


from the nominal effective charge of two expected for the oxygen in the structure and

they indicate the high degree of covalency of the bonds in the structure.


ab-Plane Reflectance


Result of the reflectance for polarization of the electric field parallel to the CuO2

is shown in Fig. 24. As mentioned above, there is a marked contrast for polarization


of the light parallel and perpendicular to CuO2 planes.


hibits metallic


The ab-plane spectrum


behavior with optical phonon modes at low frequencies that are not


completely screened out


by the free carriers.


At higher frequencies,


we observe the


plasma edge minimum at


Wr's


7500 cm


This energy is not much different than the


one observed in Sr-doped


superconducting samples


28,69
IS


For frequencies above the


plasmon minimum, we see the usual charge transfer (CT) peak at


W ri


11,500 cm


followed by higher energy interband transitions.


Assignment ab-Plane Phonons


In view of the fact that the crystal

with only a weak orthorhombic distortion


structure in La2CuO4 is almost tetragonal,


n that occurs at low temperatures, all phonon


lines can be classified under the D17 point group symmetry.


Therefore, the irreducible


representation of the vibrations that involve in-plane atomic displacements will


cor-


respond to the Eu symmetry. Hence, we should expect four infrared-active modes in


the in-plane spectrum.


In our oxygen-doped sample, we observe a total of six major


phonon-like features at 80, 140,


230, 355, 484,


and 680 cm


Previous measurements


and assignment of the ab-plane phonon modes of La2 CuO4 indicate that only infrared-





83

in the present sample suggests that the assumption of a tetragonal symmetry is only


good as a first approximation.


Nonetheless, the close correspondence of three mea-


sured frequencies in our sample with the assigned modes in La2CuO4


conclude they indeed correspond to the Ez symmetry.


140 cm


allows us to


The low-frequency mode at


-1 corresponds to bending vibrations of the out-of-plane atoms (apex oxygens)


against the Cu02 planes.154


The remaining two modes at 355 cm


-1 and 680 cm-1


are related to bending and stretching vibrations of the Cu-O bonds respectively.

This leaves us with the question for the assignment of the three remaining modes.


Various Raman measurements consistently show Raman-active modes at ~


and ~ 445 cm


0 cm


-1 in the undoped material.160'163-165 Moreover, a mode at ~ 90 cm-


been seen in Sr-doped samples


using inelastic neutron scattering166


Raman16


measurements.


Based on a tetragonal structure for the unit cell,


group


theoretical calculations160,161


indicate that the mode at


30 cm


-1 is double degen-


erate with even vibrations that correspond to the Eg and Aig symmetries.


other hand


On the


, the frequency at -445 is related to the A19 irreducible representation.


The assignment of the mode at ~ 90 cm


metry.53,160


-1 is regarded as belonging to the Eg sym-


Hence, if a correlation is made between the additional infrared modes in


this oxygen-doped sample and the Raman-active modes in undoped samples, it can be

argued that doping in the material lowers the crystal symmetry making even (gerade)


vibrations in the unit cell become infrared actives.


One explanation for this could be


that the Alg mode may couple to electronic excited states of Eu symmetry making

the former infrared allowed modes. Similar observations have been made by Shimada


et al.69 where modes at


, ~230, and ~ 460 cm


-1 have been observed to grow with


Sr doping in La2z


SrzCuO4 samples. A second explanation for the mode at 483 cm


-87




































1000


1500


2000


Frequency


Fig. 29.


(cm


Measured temperature dependence in the ab-plane reflectance (face


II) of La2CuO48+ sample.


Inset:


The reflectance at three selected tem-


peratures in the near infrared and the visible.


see later, is enhanced by interactions of the electronic background with the c-axis


LO phonon modes in the sample.

Low Temperature ab-Plane Reflectance


Figure 29 shows the temperature dependence in the ab-plane infrared reflectance


measured on face II of the sample.


the far infrared,


we observe an increase








crystal lattice is also more clear at


low temperatures where weak splitting of the


phonon modes at


140 cm


and 680 cm


are observed.72


In addition,


the weak


modes at ~ 182 cm


1 and ~ 296 cm


1 are more easily resolved at low temperatures.


These infrared-active modes are visible in the spectrum since they are not completely

screened by the electronic background.


The inset in Fi


near-infrared and visible.


9 shows the reflectance at three selected temperatures in the


We observe a marked temperature dependence of the re-


flectance in


this range.


As expected,


there is a sharpening and steepening of the


plasma edge minimum at


7500 cm


as the temperature of the sample is reduced.


At the same time, the reflectance is reduced (sample becomes more transparent) for


frequencies


the visible and above.


A similar temperature dependence has


been


observed in the ab-plane reflectance of La2-zSraCuO4 thin film72 and lightly oxygen-


doped La2CuO4+ single crystals.64


In contrast to the result of the Sr-doped thin


film,


where a rather abrupt change


s observed in the reflectance at 250


K and no


change below this temperature,72


the temperature dependence here is more gradual.


There are two things that could be said about this temperature dependence.


In the


first place, the fact that the reflectance is decreasing at lower temperatures implies

there should be a strong temperature dependence in the opposite direction in some

interband transition at a higher frequency.


This must be the case in order to satisfy the


sum rule.


Secondly,


this temper-


ature dependence could be related to the structural phase transition (tetragonal to

orthorhombic) that occurs at low temperatures in the lanthanum-based superconduc-


tors.


Such transition is known to be produced by a small staggered tilt of the apical


oxygens in the Cu06 octahedron.


There are two effects associated with this.


rnC *h pm ic a r+bnnreltnn' S +b' nn- rail n +bn n,4r nitnh nrY nb!0ao


rp1,P ccrrnn ;~ 2C







Photon


Energy
12


(eV)
15


0.6 0.9 1.2


Frequency


(cm


(10


Fig. 30.


Temperature dependence of sum rule on a wide frequency range to


show transfer of oscillator strength from low to high frequencies.


Results of ab-Plane Optical Constants


This section is devoted to present results of the temperature dependence in the

ab-plane optical properties of La2 CuO4+6 obtained from a Kramers-Kronig analysis


of the reflectance.


The usual requirement of the Kramers-Kronig integrals to extend





87

Drude model and using the fitted results to extent the reflectance below the lowest


frequency measured in


the experiment as explained in


Chapter


high-frequency extrapolation of the room temperature results was done by merging


the data,


which only extend up to 38,000 cm


(4.7 eV),


with published results on


Sr-doped samples by Tajima et al.,


168 which extend up to 40 eV. The range beyond


40 eV was extended with a power law


lRsW


which is the free-electron behavior


limit.


Since there is a temperature dependence in the reflectance all the way up to


38,000 cm


, as the temperature of the sample is reduced, the scaling factor used in


appending the data from Tajimas et al. to our low-temperature results was rescaled

upward in the range 10-15 eV and this was joined with the unchanged data above 15

eV in order to make the sum rule results equal in the range of frequencies above 15

eV. The results of the sum rule, shown in Fig. 30, suggest that what this procedure

does is to transfer some of the spectral weight that is missing in the range where we


measured a lower reflectance to higher frequencies.


This might suggest some strong


temperature dependence that is occurring in some interband electronic transition at


frequencies


much higher than


38,000


cm


It should


be pointed


out that


doing


this extrapolation


procedure did not make a


gnificant effect on the results of the


Kramers-Kronig analysis below 10,000 cm


-1 (1.2 eV).


Loss Function


second


illustration


observed


temperature


dependence


shown


Here the imaginary part of -1/e is plotted vs. frequency at several temper-


atures.


In ordinary metals, the peak position of this


function gives the longitudinal


(LO) mode


frequency


for plasma excitations of the charge carriers.


The width


this peak is related to the lifetime of these excitations.


As will be discussed later,





88

absorption in the midinfrared electronic background is the broad width (- 0.4 eV)

that is observed in the present results and in nearly all copper-oxide superconduc-


tors.


28,72,81


Estimates for the screened plasma w,


eM Wp /


can also be deduced


from the position of this peak.


Photon


Energy


(eV)


5000 10000 15000


20000


25000


Frequency


Fig. 31.


(cm


Temperature dependence in the loss function.


the present sample


~ 6300cm


This frequency is nearly the same as


in other Sr-doped samples with similar doping level.28'72


The intensity of this peak


is greatly


enhanced at


low temperatures,


while the position


does not show much


Jt~OO~








Photon


Energy


(eV)


200


3000


2500


2000


1500


1000


500


0


500 1000


1500


2000


Frequency


(cm


Fig. 32. Real part of the optical conductivity al(w) at several temperatures
in the far and midinfrared.


doped samples, becomes more resolved and the center position shifts slightly towards

lower energies at lower temperatures.

ab-Plane Optical Conductivity


The results of the temperature dependence in the


nfrared opt


cal conductivity


0al (


) are shown in Fig.


These results were obtained from face II of the sample


As mentioned before, any structure that may be present in the reflectance is more


easily resolved in the spectrum of al(w).


We observe that,


as the temperature of the






90

later. Moreover, the electronic far-infrared background has a strong temperature de-


pendence with a steep enhancement as the sample is cooled down.


This is followed


by a more weaker temperature dependence at frequencies in the midinfrared.


is a reduction in the midinfrared conductivity when cooling the


temperature down to 200 K


ature.


There


sample from room


with not further appreciable change below this temper-


This is connected with the temperature dependence of the reflectance in the


near infrared and visible that is observed in the inset of Fig. 29.


this could be


As mentioned earlier,


due to the phase transition from tetragonal to orthorhombic that occurs


in the material at lower temperatures.


Similar results were reported on a Sr-doped


film by Gao e

above 300 K.


't al..


Gao et al. found no further temperature dependence was present


Similar check was not done in the present sample since oxygen could be


driven out of the structure at temperatures above 300 K.


discussed in Chapter IV, there are two ways to analyze the optical condu


activity


al(w) in


the high-Tc superconductors.


component models.


The two approaches are the one-


Since the present data strongly suggest


the presence of two or


more components to Ocrl(c),


we performed an analysis based on a decomposition of the


conductivity in two parts, a Drude or free-carrier part and a midinfrared contribution.

Discussion of the results in this analysis is done in next section.

Midinfrared Component


The results of


the midinfrared


component,


after subtracting the phonons


free-carrier contribution at zero frequency, are shown in Fig.


atures.


33 at several temper-


The scattering rate of the free-carrier or Drude contribution shows a linear


temperature dependence as shown in Fig. 34, v

perature independent with a magnitude of wpD)


vhile the oscillator strength


~ 5800 + 100 cm


is ten-


A fit of the








Photon


Energy


(eV)


200


300


1400


1200


1000


400


200


0


500 1000 1500 2000


2500


3000


Frequency


Fig. 33.


(cm


Midinfrared conductivity of La2 CuO4$+ at several temperatures ob-


trained after subtracting the Drude-like contribution from the data shown
in Fig. 32.


overall oscillator strength of the midinfrared conductivity appears to decrease


as the


temperature of the sample is decreased from room temperature down to 200 K. A


less noticeable change is seen below this temperature.


This temperature dependence


is connected


with


the decrease that is seen in the near-infrared reflectance shown


in the inset of Fig. 29.


Hence, it appears that some spectral weight is being trans-


ferred to higher frequencies,


which is also affecting ao(w


) in the midinfrared.


It is








constant.


We also observe that a bump-like structure,


which at 300


K appears at


around 1000

about 200 K


cm


, diminishes in intensity with a reduction in temperature.


, a second and sharper feature reappears at a lower energy,


Then, at


~ 750cm


growing again in intensity as the temperature of the sample is further reduced. Simi-

lar midinfrared-like modes in the optical conductivity of both lightly doped64'169 and


superconducting samples have also


been reported in


the literature.68'72,87,135


present results underscore the generality of these features in the optical properties of

the copper-oxide-based materials.


400


350


300


250


200


150


100 150 200


Temperature


Fig. 34.


The scattering rate of the Drude-like part from a two-component fit to


the optical conductivity shown in Fig. 35 (Notice the linear temperature
dependence).