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
Physical Description:
viii, 214 leaves : ill. ; 29 cm.
Quijada, Manuel Alberto, 1962-
Publication Date:


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Thesis (Ph. D.)--University of Florida, 1994.
Includes bibliographical references (leaves 200-213).
General Note:
General Note:
Statement of Responsibility:
by Manuel Alberto Quijada.

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University of Florida
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Full Text






It is with great

pleasure that I thank my advisor, Professor

David B.


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.


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


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


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




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




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




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


Anisotropy in the ab Plane

YBa2Cu3O _
Pbrnia~rhin C'


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



Superconducting State Models

Symmetry of the Order Parameter .

Evidence for Proposed Pairing States

Determination of Gap by Optical Spectroscopy


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



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


Single-Domain Crystal

4 4 4 5 S 86

Bi2Sr2 CaCu2 08 Single-Domain Crystals

La2Cu04+6 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


* 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


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


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


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


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

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

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





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




Concluding Remarks


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






S S 5 5 S S S S 5 S 173

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


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


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





Manuel Alberto Quijada

April 1994


David B. Tanner

Major Department: Physics

The optical


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.


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.







a strong,




band in

the midinfrared in



temperature-dependent narrow

Drude-like band

the far infrared.


is also anisotropy in

the infrared



a and


of both

YBa2Cu3 07_6





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


simple Drude term combined with a more broad midinfrared

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


be explained

by an anisotropic second midinfrared component in

optical conductivity.


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-


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


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


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



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



be addressed:

how this




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


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-


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


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


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.


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


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


The organization

of this thesis is

as follows.


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



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


VI and

VII respectively.


VIII is for presentation of results regarding dc-transport

measurements of the resistivity tensor of Bi2Sr2CaCu20s samples.

Finally, Chapter

IX contains concluding remarks.



This chapter will be


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



occur mainly in


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


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


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.


, 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


compound. In this nearly stoichiometric material, the oxidation

state of the individ-

ual species is La3+


, 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


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.


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


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


They argued the Bmab symmetry would be


by the presence

of interstitial oxygen in the


Following up on

their conclusion,

they car-

ried out a refinement of the

structural parameters for the single-crystal data in the

Fmmm space



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 _



the first

copper-oxygen-based material

showed super-



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-


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


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



There are two CuO2

planes per unit cell.

Unlike the tetragonal


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


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.

O Cul(2)
0 '

o 0(4)


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

Special growing t

techniques 0 in recent years have pro

ded large enough single-





possible the study

ssue in

more detail.

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



The discovery of 20

K supercondu


n Bi

r9 C U 0 r 2


up the race

u u v

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



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


are separated



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


the ideal crystal structure for the case when


this structure the



are seen

to be separated

by sheets of Bi202


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




also contributes to



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


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


oxygen in the B



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


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


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


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.


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

crystals of these compounds.2


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



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


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


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


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:



-= 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.


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




Photon Energy (eV)

I .0I
I I C I4


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


The results of the reflectance measured at


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


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

weak dc transport.


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


n the

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


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









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


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


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


Other materials such as


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-


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





as a function


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.


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


Anisotropy in the ab Plane

YBa Cu.O,


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


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.


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.


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



et al. interpret the a-axis conductivity

as intrinsic to the Cu02


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







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


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


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


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


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


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


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-


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


Because of

their short coherence length (

~o10 A)

the high-Tc superconductors are considered


II material.

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

the mean free path

b 1000


v (cme)







YBa2Cu307 _6 in the a direction.

The weak peaks are calculated phonon


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


that of the superconducting gap,

observation of this gap

by optical methods


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.


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







.1~ LI J


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


The second

interband peak, which appears at 30,000 cm


eV), is found to show some sam-

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

mostly in the Bi20O


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.,


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-


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




Photon Energy
2 3






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


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



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


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.)


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.


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


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


0 2p ---UHB

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

Hamiltonian can be written explicitly

= tpdp(di + h.c.)


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

d +
fl1 +62,

+ Ud

nd .d


1i,0 1,-Uc

+ Vdp,



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


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


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




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


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


the Hamiltonian is defined as

dt d,,

+h.c +U


where the dt are fermionic operators that create holes at site

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


the strong



the previous model

can be


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.


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],

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


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 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.


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


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



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> -






to the


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-1 )

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


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





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

be written as

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


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


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

function of dopi:

2 holes in the 4


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.


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


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



. 0.10

- 0.05


5 10 15


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


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


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
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


oscillators with wyj,

being the resonant frequency,


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

The last term,

is the high frequency limit of


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.


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-


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



the Kramers-Kronig relations,

they must obey causality.


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(





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


to the coupling constant A.


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.


-Im YE(w) is nearly constant for cw


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


1000 cm

This is in conflict with the high

cutoff frequency suggested


experiments where


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


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


Unfortunately, there have been contradictions in some of the


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



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 -


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

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


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.


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



can be

derived from

the irreducible representations of



x D4h,


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-


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


The results show the average distance


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

It has



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



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


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



perature value.55'128

Most available results are best fitted129 by using the empirical

Gorter-Casimir formula

L (T) =

AL (O)



t = T/Tc.

Likewise, in more recent penetration depth experiments on single

crystals of YBa2Cu307_6


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



as consistent


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) -

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



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


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

to T2 in less than ideal samples.131"132


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.





the same


as the one



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


, 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


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-


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-


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


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-


acting state

reflectivity of the high-Tc materials with superconducting energy



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.


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



To explain the absence of a gap feature, it has


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.





the measurement of the

transmittance or the

reflectance of a sample as a function

of the incident light frequency.


is done over a

very wide frequency range,



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,


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


p(w) = B Aw2

where kB and c are the Boltzmann's constant and speed of light respectively.

the ratio of the emitted to the total power up to frequency


w is 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.



j Fixed Mirror

/ 1I






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

the ideal


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


I(S, v) = AA*

= 2a2


cos2ivS) 2S(v)(1+

cos 27rz


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) =


0 o

S(v)dv +



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,



I- -F


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


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


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


problem is sol


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


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


a Hg and


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



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


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


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


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,


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-



The diagram showing the details of the instrument is

in Fig.

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


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






grating filters

---slits glo rb

sample lamp


vacuum tan






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.


allows access to


wavelengths sequentially.

The now


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



are pos

tioned at

the point

The reflected

or transmitted light


a detector that


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


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.


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


Perkin-Elmer Grating Monochromator Parameters

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
























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



the substrate is polyethylene.


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.








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



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


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


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


The pressure inside of this shroud was kept below


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


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.


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.


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


the phase shift can be obtained from the Kramers-

Kronig integral

0 (3)-

w /3: In R(w')


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



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


by interband transitions of the inner core electrons to excited


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)

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


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-


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


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)


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


= 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


= 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


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


El= N2 = (, i.)2


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.


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


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


in all the crystals

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

superconducting transition


as determined b

y cooling the sample in a

field of 10 G,


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



(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)






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


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





1 -


100 125 150 175 200 225 250

-- 200

a 175

- 150






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,


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


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



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


is around 91

with a transition


Results of the

resistivity tensor along the a, b, and







c axes



on these crystals are reported in Chapter


Field Cooled

field Cooled


Fig. 21.

Meissner effect measurements on a Bi2Sr2CaCu2Os single crystal.

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


* *


La9Cu04+A Single Crystal




an optical



on an oxygen-doped

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

Chou and D.C.



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


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


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.


The other face that

studied only contained the twinned ab-plane direction.

I Cu I
La CuC

, I I i I I' 1


H //ab

r 1- I

crystal B

ti/Ic U

I11I I It .J

rr -

W .i



Oe _










*I I I T1 II
La CuO



I f 1 1

staid B

H //ab

H //c





T (K)

Fig. 22.

Meissner effect measurements on L

a9CuOAlS single crystal (After

- --

r r I r-r


1 r-r 1 1

I.I~ t I


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.


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).


I allowed us,


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

E E aab
E flab


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


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.


, 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


1.0 -

0.8 -

0.6 -

0.4 -

0.2 -







Fig. 24.

ab pl



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


Reflectance of La9 CuOal

Room Temperature Spectra


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


r-doped superconducting samples derived from the stoichiometric parent



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,


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.
















al (W)



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-



Low Temperature c-Axis Reflectance

Figure 26 displays the reflectance in the

nfrared region along the



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


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




Temperature dependence in


c-axis reflectance of La2 Cu04+

single crystal.

of this condensate to the dielectric function can be written


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


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


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


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


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




















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


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.


The formula is


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.


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




100 150




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


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


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


ionic vibrations in

the crystal.

Since the

TO frequencies involve transverse

vibrations of the atoms,


are obtained directly from

the peak position in


, 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


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-






La2 Cu04+5.

Oscillator# wTOj WLOj wipj y

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

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


[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



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

solving Eq.

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


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


7500 cm

This energy is not much different than the

one observed in Sr-doped

superconducting samples


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


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-


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



Based on a tetragonal structure for the unit cell,


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


-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






Fig. 29.


Measured temperature dependence in the ab-plane reflectance (face

II) of La2CuO48+ sample.


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


the visible and above.

A similar temperature dependence has


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


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.


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-


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




0.6 0.9 1.2




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


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

frequency measured in

the experiment as explained in


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


which is the free-electron behavior


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


much higher than



It should

be pointed

out that


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







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


In ordinary metals, the peak position of this

function gives the longitudinal

(LO) mode


for plasma excitations of the charge carriers.

The width

this peak is related to the lifetime of these excitations.

As will be discussed later,


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-



Estimates for the screened plasma w,

eM Wp /

can also be deduced

from the position of this peak.




5000 10000 15000




Fig. 31.


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













500 1000





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


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



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


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


after subtracting the phonons

free-carrier contribution at zero frequency, are shown in Fig.


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












500 1000 1500 2000




Fig. 33.


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


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


We also observe that a bump-like structure,

which at 300

K appears at

around 1000

about 200 K


, 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.







100 150 200


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