Temperature dependence of infrared and optical properties of high temperature superconductors

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Temperature dependence of infrared and optical properties of high temperature superconductors
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TEMPERATURE DEPENDENCE OF INFRARED
AND OPTICAL PROPERTIES OF HIGH
TEMPERATURE SUPERCONDUCTORS





By
FENG GAO


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




UNIVERSITY OF FLORIDA


1992














ACKNOWLEDGMENTS


It is my great pleasure to thank my advisor, professor David B. Tanner, for giving

me the opportunity to study the most exciting new area of solid state physics-high-

Tc superconductivity-and for his valuable guidance and advice, constant support,

patience and encouragement all throughout my graduate work at the University of

Florida. I also thank professors N.S. Sullivan, H.A. Van Rinsvelt, J.W Dufty, M.W.

Meisel, and J.H. Simmons for serving on my supervisory committee.

My recent years of academic and research experience in the "Florida Group" have

been interesting, enjoyable, challenging, hardworking and thus the most memorable

years in my life. I would like to extend my thanks to all of my colleagues in this group,

past and present, for their friendship, useful conversation and cooperation which were

essential in developing a pleasant research environment. In addition, I appreciate C.D.

Porter's help in computer software and programming. Another special thanks goes

to professor D.B Tanner for his great help in using the EZ'EX he wrote, which made

it easy for me to type this dissertation. I also want to thank the staff members in the

physics department machine shop and the engineers in the condensed matter group

for their technical support.

I have had many interesting discussions with Drs. S.L. Herr, K. Kamaras, D.B.

Romero, V. Zelezny, T. Timusk and S. Etemad. In particular, I am very grateful to

Dr. G.L. Carr for his valuable help, suggestions and collaboration during my early

work. His kindness and knowledge are admired.

I would like to express my acknowledgment to Dr. S. Etemad at the Bell Com-

munication Research and to Drs. J. Talvacchio and M. Forrester at the Westing-

house Science and Technology Center for providing good quality high-Tc films, which









were the major topics in this dissertation. Also to be thanked are M. Doss and C.

Gallo at 3M company for preparing the polycrystalline samples studied in this work.

Other people that deserve thanks are Drs. G.P. Williams and C.J. Hirschmugl at

Brookhaven National Laboratory for their help in the far-infrared measurement using

the far-infrared beamline at the National Synchrotron Light Source.

I also have special thanks to my parents, brother and sisters, my wife, my daughter

and son for their love, care, encouragement and understanding during my graduate

studies.

A final thanks goes to the organizations which supported this research through

the U.S. Defense Advanced Research Projects Agency Grant MDA-972-88-J-1006,

and the National Science Foundation Grant DMR-9101676.












TABLE OF CONTENTS


ACKNOWLEDGMENTS .......................

ABSTRACT . . . .


CHAPTERS


I. INTRODUCTION .............

II. BRIEF SURVEY OF SUPERCONDUCTIVITY

Fundamental Properties ...........

Crystal Structure and Phase Diagram .

Other Physical Properties . .


. .. 1

. . 6

. .. 7



. 9
. . 8

. . 9


III. THEORY . . .

Optical Theory . . .
Optical Response of the Medium . .
Determination of Optical Constants . .
Reflectance of thick crystals . .
Kramers-Kronig relations . .
Combination of reflectance and transmittance of thin films
Lorentz and Drude Models . .
Sum Rule . . .

Superconductivity . . .
Perfect Conductor ...................
Superconductor .. .. .


IV. INFRARED TECHNIQUES .

Interferometry ............
Infrared Radiation at Low Frequencies
Fourier Transform Spectroscopy .

Optical Spectrometers .
Bruker Interferometer .
Michelson Interferometer .
Perkin-Elmer Monochromator .


13

. 13
. 13
. 16
. 17
. 18
S20
. 24
. 26

. 26
S26
. 27


. 33

. ......... 33
. . 33
. 36

. 38
. 38
. 40
. 40










V. SAMPLE PREPARATION AND CHARACTERISTICS .

La2-.SSrCuO4 Epitaxial Films . . .

YBa2Cu307aO Oriented Films . . .

YBa2-.SrzCua30_- Polycrystalline Samples . .
Reannealing Procedures for YBa2Cu307- . .
Meissner Effect Test and Susceptibility . .

VI. EXPERIMENTS AND LOW TEMPERATURE TECHNIQUES .

Low Temperature Apparatus . . .

Reflectance Measurements and Uncertainties-La2-_Sr cCuO4 Films

Procedures in Kramers-Kronig Analysis . .
High-frequency extrapolation . .
Low-frequency extrapolation . . .

Combination of .A(w) and f(w) Measurements-YBa2Cu3aO7- Films .


Measurement of YBa2-aSrCu307_6 Pellets .

VII. OPTICAL STUDIES OF La2-.Sr.CuO4 FILMS .

Results and Discussion . .
Infrared Phonons ...............
Structural phase transition . .
Frequency shift and lifetime . .
Two-Component Approach . .
The free-carrier component--wpD and r .
The midinfrared absorption . .
Holstein effect .. .. .
Superconducting-to-normal ratios .
Extra absorption below T . .
Sum Rule-Superconducting Condensate .
One-Component Approach . .
Mass enhancement m*/mb and self energy E(w)
Effective scattering rate 1/r*(w) .
Loss Function .. .. .. .
The Superconducting Gap . .

Summary . . .


. 72

. 76

. 76
. 76
. 77
. 78
. 79
. 81
. 83
84
. 85
. 86
. 87
. 89
. 90
. 92
. 93
. 95

. 97










VIII. FAR-INFRARED STUDIES OF YBa2Cu3O7-_ FILMS

Results and Discussion . .
Free-Standing Transmittance f (w) .
Two-Fluid Model Fit ..............
Relaxation Rate and Superfluid Condensate .
"Coherence Peak" ..............
Mid-Infrared Band and Superconducting Gap .

Summary . . .

IX. YBa2Cu307-_ POLYCRYSTALLINE SAMPLES .

Infrared Spectra and Analysis . .
Superconducting Condensate . .
Phonon Frequency and Linewidth .

Effective Medium Approximation . .
Anisotropic Medium . .
EMA and MGT Approaches ...........
Weighted Average ...............

Summary . . .

X. SUMMARY AND CONCLUSIONS . .

Normal State .. .. .. .

Superconducting State . .


APPENDICES

A. DATA FOR OTHER SAMPLES

B. COMPUTER PROGRAMS ....


REFERENCES .............

BIOGRAPHICAL SKETCH .......


. 115

. 116
. 119
. 120
. 122
. 123
. 124

. 126

. 138

. 138
. 139
. 140

. 142
. 143
. 145
. 147

. 147

. 162

. 162

. 163


. . 165

. . 170


. . 185

. . 194














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





TEMPERATURE DEPENDENCE OF INFRARED
AND OPTICAL PROPERTIES OF HIGH
TEMPERATURE SUPERCONDUCTORS

By

Feng Gao

December 1992


Chairman: David B. Tanner
Major Department: Physics

The infrared properties of the newly discovered high temperature super-

conductors are extremely unusual. We have extensively studied two cuprate families:

La2-;SrCuO4 and YBa2Cu307_6. The former is the material where high Tc was

first discovered; the latter is the first substance with a transition temperature above

the liquid nitrogen boiling point. The samples studied were ab-plane-oriented super-

conducting thin films deposited on insulating substrates. Optical transmittance and

reflectance measurements were made with the films in both the normal and super-

conducting states. Other superconducting samples in forms of randomly oriented or

textured polycrystallines and granular films were also measured.

The infrared conductivity of these cuprates in the normal state showed a strong,

nearly temperature-independent broad band in the mid-infrared region in addition to

a strongly temperature-dependent narrow Drude band in the far infrared. Most of the









free-carrier oscillator strength was found to shift into the zero-frequency delta-function
conductivity in the superconducting state as the free carriers condense into Cooper

pairs. The CuO2-plane London penetration depth AL (~2700 A for La2-,SrzCuO4,

~1700 A for YBa2Cu3O7-6) was estimated from the superfluid density. The low-
frequency tail of the midinfrared absorption, a direct particle-hole excitation, re-

mained for T < Tc.

One striking result observed was the linear temperature dependence of the quasi-
particle scattering rate above Tc, followed by a rapid drop just below Tc. Another

was that the T-dependent conductivity of the YBa2Cu307O- films at low frequencies
exhibited a peak just below Tc, resembling the "coherence peak" of ordinary super-

conductors, yet having a different origin. This peak is associated with the dramatic

decrease of the scattering rate rather than with the coherence effect.

Finally, the superconducting gap absorption was invisible in the infrared
spectra, suggesting that these cuprates were "clean-limit" superconductors. The
energy gap for YBazCu3aO7- might be deduced indirectly from the T-dependence

of the far-infrared vibrational features of the polycrystalline samples, suggesting

3.0 < 2A/kBTc < 4.2.














CHAPTER I
INTRODUCTION

This dissertation describes a detailed study of the optical properties-from far
infrared through ultraviolet frequencies--of the newly discovered copper oxide ma-

terials which have high superconducting transition temperatures. It concentrates on
the investigations on La2-zSrzCuO4 (To 30 K) and YBa2Cu307_6 (To 90 K)

thin films and polycrystalline samples through measurement and analysis of optical

transmittance and/or reflectance as a function of incident photon frequency (w) and
sample temperature (T). Much effort in this work has been devoted to study the
anomalous non-Drude response in the mid-infrared region, the behavior of supercon-

ducting energy gap, and the role of phonon and low-lying excitations.

The discovery of high temperature superconductors (HTSC) by Bednorz and

Miiller1 in 1986 and Wu et al.2 in 1987 has stimulated considerable interest in the
scientific world. These new materials are interesting because they present an exciting

new regime for superconductivity and have the potential of valuable practical applica-

tions. The most fundamental properties of these high-Tc cuprates are threefold: high

superconducting transition temperature (Tc), short coherence length (t), and large

anisotropy. The conventional mechanism of pairing via electron-phonon interaction

cannot describe this oxide materials simply because Tc is to high. Attempts have
been made to determine whether these high-Tc superconductors are fundamentally
different from the conventional superconductors, which have been well described by

the Bardeen-Cooper-Schrieffer (BCS) theory.3

Infrared spectroscopy, a powerful and successful technique for studying classical
superconductors,4-8 has been widely used to study such fundamental physical proper-









ties as superconducting gaps, crystal vibrations, electron-phonon couplings, low-lying

excitations, density of states, and electronic band structure.

In the framework of BCS theory, the existence of an energy gap means that a

bulk superconductor at T < Tc is a perfect reflector of electromagnetic radiation

for photon energy (hw) less than the gap energy (2A). Photons with fiw > 2A can

disassociate the Cooper pairs and cause quasiparticle transitions to unoccupied levels

above the gap, making the superconductor behave like a normal metal. This is indeed

the case for conventional superconductors, as first verified by Glover and Tinkham.4

However, the high-Tc superconductors show much more complicated behavior.

A major difficulty with the observability of the superconducting gap in the infrared

spectra is that the reflectance of HTSC can hardly be distinguished from unity at

low frequencies within which a gap is expected. Because the transition temperature

is comparable to the Debye temperature (Tc Te), the superconducting gap ener-

gies of HTSC are expected to lie in the frequency range where infrared active optical

phonons are present. In return, these phonons (if not well screened by free carri-

ers) may obscure the observation of the energy gap. It has been reported9 by direct

bolometric absorption measurements that these materials have a finite absorption

down to the frequencies well below the BCS gap at T < Tc. In both the normal

and superconducting states, a non-Drude low-lying excitation spectrum is present in

the mid-infrared region. Added to the complexity is the anisotropy of these mate-

rials, which means that for polycrystalline and twinned samples, only an effective

response can be measured. Furthermore, thin film measurements are complicated

by contributions from the substrates. Although the optical results generally agree

among different investigators, the interpretation in many ways-particularly about

the infrared determinations of the gap and the origin of the mid-infrared band-still

remains controversial and is not clearly understood.









One feature common to all cuprate superconductors is the existence of quasi-two-

dimensional Cu-O planes, which appear to play a major role in high-To superconduc-

tivity. Therefore, it is of primary importance to investigate the intrinsic electronic

responses of these planes. Most optical studies to date have concentrated on the 90-

K transition temperature YBa2Cu307r- (YBCO) system, which contains both CuO2

planes and CuO chains. (For reviews, see Refs. 10-13.) It has been observed, however,

that the quasi-one-dimensional CuO chains in YBCO have a substantial contribution

to the optical conductivity,14'15 which has complicated the analysis of this material.

In contrast, the La2-.SrzCu04 (LSCO) system, which has the simplest crystal struc-

ture in the cuprate family and contains only single CuO2 layer per formula unit, has

been studied in most cases on the sintered polycrystalline samples.16-23 Because the

LSCO materials are strongly anisotropic, it is difficult to determine the intrinsic na-

ture of the CuO2 layers from measurements of polycrystalline samples. A few optical

measurements, mostly restricted to the composition-dependence studies at room tem-

perature, on La2-zSrzCuO4 single crystals or thin films have been made,24-29 and

most recently, a systematic temperature-dependent optical study on oriented samples

of this material has been reported.30

In no case can the normal-state infrared conductivity be described by a simple

Drude model. In many studies,10-12,30-36 this non-Drude conductivity observed in

ceramics, crystals and thin films has been described by a two-component approach:

a narrow, strongly temperature-dependent Drude absorption centered at the origin

and a broad, nearly temperature-independent mid-infrared (MIR) band. The Drude

absorption is due to the free carriers which are responsible for the dc transport and

which condense into a superfluid below Tc whereas the MIR absorption is due to

the bound carriers which have a semiconductor-like gap. This approach has been

adopted because it is clear that a single, strongly damped, Drude term, used to model

the infrared date in many early measurements, does not work.10'12 An alternative









is a single-component approach: all of the infrared absorption is due to one type

of carriers, with a strong frequency dependence in the scattering rate and effective

mass. This approach also leads to a broad range of optically inactive excitations in

the mid-infrared region while at low frequencies (including dc) the conductivity goes

inversely with the temperature. This approach has been described in the framework

of the "marginal Fermi liquid" (MFL) theory of Varma et al.37 and the "nested Fermi

liquid" (NFL) theory of Virosztek and Ruvalds.38'39

Attempts have been made 10,15,40-46 to assign the superconducting energy gap

either to the edge of a rapid drop in the ratio 5,(w)/S,,(w) of the superconducting

to normal reflectance or to the absorption onset of the conductivity, ai,(w), in the

superconducting state. The values of 2A(O)/kBTc obtained in this way range between

2.5 and 8. There has been a controversy, however, whether these structures are due

to the energy gap or are part of the midinfrared absorption.32-35,47-49

Most of the work to date determines the frequency-dependent conductivity

through Kramers-Kronig analysis of reflectance measurements.10-13,30-35,40-50 The

main advantage of this technique is that a large amount of important information

about the material can be extracted easily once the reflectance over a wide frequency

range is measured. However, there is a drawback of artificial extrapolations beyond

the measured frequency range, which is required by the Kramers-Kronig integral.

Furthermore, the optical properties derived from this method is very dependent on

the accuracy of the reflectance especially for highly reflecting materials. In contrast,

the optical functions derived from transmission measurement are much less sensitive

to the errors than from reflectance. In this work, both techniques are employed for

data analysis. We will present a method of extracting the optical functions directly

from combined measurements of reflectance and transmittance without referring to

the Kramers-Kronig analysis.









The rest of this dissertation is organized as follows. The second chapter is a re-

view of the previous research work on superconductivity and of some fundamental

properties of HTSC. Chapter III discusses the basic theory about the general optical

properties of solids and the phenomenological models in description of superconduc-

tivity. Chapter IV will describe the infrared technique and experimental apparatus

used in this study. Sample preparation and characteristics will be presented in chapter

V. Chapter VI explains the procedures of experimental measurement and low temper-

ature technique along with data collecting and processing. Chapter VII, VIII and IX

are devoted to data analysis and result discussion, in which the experimental optical

spectra for several high-Tc samples are presented and discussed in detail. Various

theoretical models are used to describe the physical properties of the oxide materials

in the normal and superconducting states. Finally, Summary and conclusions are

given in chapter X.














CHAPTER II
BRIEF SURVEY OF SUPERCONDUCTIVITY

Since Kamerlingh Onnes' 1911 discovery of superconductivity in mercury cooled

to 4.2 K, the observed transition temperatures, Tc, have gradually moved upward but

remained strictly a low temperature phenomenon. The most important developments

of this area in the recent decades include: the Ginzburg-Landau theory,51 the Pippard

nonlocal electrodynamics,52 and the discovery of isotope effect53,54 in early 1950's;

the first satisfactory description of microscopic mechanism (in terms of energy gap

and Cooper pairs) by the BCS theory3 in 1957; the prediction of Josephson effect55

in 1962; and, most recently, the discovery of high-Tc superconductors in late 1980's.

The highest record for-To before 1986 was 23.2 K in Nb3Ge found by L. R. Testardi

et al.56 in 1973. The extraordinarily high Tc superconductivity era started in 1986

when J. G. Bednorz and K. A. Miiller1 observed that La2-,BaCuO4 became super-

conducting below 35 K. This opened the way for intensive work on high temperature

superconductors. Another breakthrough soon arrived by the announcement2 in 1987

of YBa2Cu3O7-6, with Tc above 90 K capable of becoming superconducting in liquid

nitrogen. In 1988, two families of copper oxide compounds with even higher values

of Tc, 110 K in Bi2Sr2Ca2Cu30057 and 125 K in Tl2Ba2Ca2Cu3Ol0,58 were found.

The discovery of high-Tc oxides has generated tremendous excitement in public
because of the new technological promises of these materials. Superconductivity can

now be achieved with a simpler coolant-liquid nitrogen. The electronic properties

of HTSC can be exploited to make more efficient microelectronic components such as

passive microwave devices, logic circuits, computer interconnect boards, and infrared

detectors. The fabrication of ultra-sensitive sensors and production of Josephson









microwave mixers with a wider electromagnetic wave windows become possible with

the use of HTSC because of the larger energy gap expected for these materials.


Fundamental Properties

As stated in the introductory chapter, there are three unusual fundamental prop-

erties which distinguish these new materials from the conventional superconductors.

First, the transition temperature Tc is high. Aside from the promising applications as

stated above, high Tc also presents a new challenge to scientific investigators, as many

low-lying excitations are present near Tc. These excitations will affect some funda-
mental properties of the superconductors such as Tc and critical current Je as well as

the energy gap 2A, if such excitonic energies are large enough to break Cooper pairs.

These high-Tc materials have low dc conductivity in the normal state due to lower

conduction electronic concentration, which results in a longer penetration depth. The

high value of Tc has turned out to complicate the behavior of the materials in both

the normal state and superconducting state.

Second, as a consequence of higher Tc or larger superconducting gap compared to

the classical superconductors, HTSC have a shorter coherence length with typical

value of ( = hvF/kTc ~ 10 A, which is comparable to the unit cell dimensions. All

HTSC are type II superconductors because of ( < A, where A is the electromagnetic

penetration depth with a value of the order 1000 A. The shortness of makes the

superconductivity sensitive to small scale structures. In turn the fluctuations play a

much larger role in high-To materials than in classical superconductors. A small (
also leads to a high value of upper critical magnetic field He2.

Thirdly, HTSC show large optical anisotropy. The physical properties such as

optical conductivity and other fundamental physical parameters vary in different di-

rections. The resistivity, for instance, along the c-axis (pc) is larger than within the

CuO-plane (Pab) by a factor of ~ 102. Therefore, high quality crystals and oriented









films are essential for experimental studies, because the anisotropy of the layered cop-

per oxides requires that the samples be measured along different axes of the crystals.

Finally, it has been estimated and will be shown in this dissertation that the mean

free path I = VFr 100 A, making


S<1

It is this condition that places the HTSC in the "clean limit" which is sharply distin-

guished from most conventional superconductors. The latter ones at low temperatures

are usually in the anomalous skin effect limit or dirty limit.

Crystal Structure and Phase Diagram

The crystal structure and the phase diagrams of La2-,SrCuO4 and
YBa2Cu307-. are shown in Figs. 1 and 2. Note that both figures show the structure

in a unit cell which contains two formula units. The parent compounds of both ma-
terials are anti-ferromagnetic semiconductors. When doped with holes, they become
metallic.

For La2_,Sr,CuO4, it has the perovskite K2NiF4 structure and is the body-

centered tetragonal Bravais lattice (I4/mmm). The typical values of lattice constants

are: a b 3.78 A, and c 13.2 A. As the temperature is lowered, the crystal

exhibits a second order structural phase transition from tetragonal to orthorhombic
phase (Cmca). This transition involves a staggered rotation of the CuO6 octahedra

as shown by the arrows in Fig. 1. Upon further cooling the crystal exhibits another
transition from metallic to superconducting phase. The optimum Sr doping for su-
perconductivity lies in the range of 0.1 < x < 0.2. Since La2-.SrzCuO4 dose not
have chains, one expects that the dynamic conductivity r(w) of an oriented crystal

probed to be solely due to the intrinsic response from the CuO2 planes, provided the

electric field vector E is parallel to these planes.









In contrast with La2-SSrzCuO4 crystals, the presence of CuO chains, as shown
in Fig. 2, along the b-axis in the YBa2Cu3aO7- system complicates the analysis of

this materials. A great deal of effort has been devoted to distinguish the role of the

quasi-one-dimensional chains from that of the quasi-two-dimensional planes in recent

years. However, most of the YBa2Cu3O7a- samples are usually microtwinned, making

it difficult to identify the difference. One the other hand, by far YBa2Cu3aOr are
the most studied material in high-T. family. The parent compound YBa2CuasO is

tetragonal while the superconducting YBa2Cu307_ is orthorhombic. Typical values

of the lattice constants are: a = 3.82 A, b = 3.88 A, and c = 11.68 A. The transition

temperature of this material is very dependent on the oxygen doping concentration

as illustrated in Fig. 2.



Other Physical Properties

A good superconductor is usually a poor electric conductor in the normal state.
The reason is that the conventional electron pairing requires a strong electron-phonon

interaction in order to produce a high transition temperature. It is the same inter-

action that causes the large electronic scattering rate hence high resistivity in the

normal state. The HTSC are quite poor electric conductors above Tc, thus one ex-

pects a strong electron scattering mechanism in these materials.

It is widely believed that the electron-phonon interaction plays a minor role in the

superconductivity for YBa2Cu307_6. However, a significant isotope shift (a 0.2)

due to partial substitution of "80 for 160 in La.s85Sro.15CuO4 has been observed

and interpreted as evidence for strong electron-phonon coupling.59,60 This implies

that phonons may still play an important role, if not a key role, in the pairing mech-

anism. On the other hand, the observed linear behavior of the dc resistivity for

La2-aSrzCuO4 up to 1100 K implies a weak electron-phonon coupling for the free









carriers.61 Therefore the La2-ZSrzCu04 system is expected to bridge the classical

superconductors and HTSC.

In optical studies, a lot of effort has been made in recent years to study the

non-Drude response in the mid-infrared region and to discover the superconducting

energy gap. It has been observed that the MIR absorption is absent in the undoped

parent compounds such as La2CuO4 and YBa2Cu306. For La2-aSrzCuO4, Uchida

et al.29 have reported that the MIR absorption band develops with increasing dopant

concentration and then exhibits a saturation in the higher compositional range 0.1 <

x < 0.25. Similar effects are observed in doping of n-type Pr2-sCezCuO4 by Cooper

et al.62 As a consequence of the redistribution of the the O 2p and Cu 3d orbitals upon

doping, spectral weight is rapidly transferred from the in-plane O 2p -+ Cu 3d charge

transfer (CT) excitations above 2 eV to the free-carrier absorption (Drude band) and

the low-energy excitations (MIR band) below 1.5 eV. Therefore both the Drude and

MIR absorptions in HTSC appear to be related to the introduction of holes on the

CuO2 layers (or CuO chains) by doping. For La2-_Sr,CuO4, the CT gap becomes
weaker or fills in and the phonons are obscured as holes are added upon substituting

Sr2+ for La3+. In contrast to these changes, the plasma minimum in the reflectance is

pinned at 0.9 eV and insensitive to the dopant concentration.22,23,63,64 This unusual

behavior is in contradiction with the prediction that the plasma frequency should

increase with increasing carrier concentration.

In summary, the high temperature superconductors are complex and have many

unusual properties that we have been trying to understand. The major issues that
challenge to the spectroscopists include: the normal transport properties, the super-

conducting mechanism and the energy gap, the roles of electron-phonon interaction

and low-lying excitation. These issues will be addressed in the following chapters.












La2- Srx Cu4 Phqse Diagram


0 0.1 0.2 0.3
Sr concentration, x


0.4


Fig. 1. Phase diagram for La2-SSrzCuO4 and crystal structure of the par-
ent compound-La2Cu04 (Ref. 65). (Note that it has been reported
recently that Tc became zero near x = 0.22. See Ref. 66.)



















I-

Superconducting
Ba2Cu3g?-
Slab




tBa


0 Ba


SCu

S0


c

Lb
a


Cu-O

'" Chains
Planes


YBa2306+,


600
500
400
300
200
100


0.0 02 0.4 0.6 0.8 1.0
x


Fig. 2. Crystal structure for YBa2Cu307-6, and phase diagram (Ref. 65) of
YBa2CuaO6+,. AF: antiferromagnet, SC: superconducting.













CHAPTER III
THEORY


Optical Theory

The frequency dependent optical conductivity al(w) and dielectric function eI(w)

are most directly connected to the absorptive and dispersive nature of a material. In

the zero frequency limit, ac(0) becomes the ordinary dc conductivity, and e1(0) is the

static dielectric constant. However, neither al(w) nor cl(w) can usually be measured

directly. Therefore, the optical properties are usually determined experimentally by

measuring the reflectance or transmittance as a function of the energy of the incident

light radiation, from which al(w) and el(w) can be derived. The interaction between

matter and the applied electromagnetic field is described by Maxwell's equations and

the boundary conditions.

Optical Response of the Medium

In the infrared through ultra-violet region, the wavelength of the light radiation is

much larger than the dimensions of the unit cell. The propagation of electromagnetic

wave in a medium can be described by a set of four differential equations (known as

the macroscopic Maxwell's equations):67

V D = 41pf,

V.B=O,
xE= 1 OB (1)
c at '
Vx1_ ID 47r
Sc t c Jf,

where E and H are the electric and magnetic fields, D and B the displacement

field and magnetic induction, pf and Jf the free-charge and free-current densities,









respectively. Gausian units are used throughout this dissertation unless otherwise

specified.

For weak electromagnetic field and in local limit, the response of the medium is

linear and can be written by the constitutive relations:


D = elE, B = pH, Jf = rlE, (2)


where el, i, il are the frequency-dependent dielectric function, permeability, and

conductivity, respectively, of the medium. For simplicity, we take p =1, the case

for most non-magnetic materials. Thus, B can be replaced by H. If the medium is

isotropic and homogeneous, then e1 and a~ are scalar quantities rather than tensors

and have no space variation. Assume the fields have the plane-wave form:


{H { exp[i(q x t)], (3)


where the vectors Eo, Ho and q (wave vector) are in general complex and independent

of space x and time t, then g can be replaced by -iw, and V by iq, causing the curl

equations in (1) become


iq x E = iH, (4)
c
4?r w 4?r l w i
iq x H = (Jd + p) = -i+ i -( -- =) E, (5)
c c c 47r

where the first term in Eq. (5) is the displacement current, the second is free (con-

duction) current, and the third is polarization (bound) current. One can introduce a

complex conductivity a = al + ia2 with 02 = w(l el)/47r, or a complex dielectric

function e = eC + ie2 with e2 = 4~rCr/w such that

.w 47r .w
iq x H= -i -E+- 4 E=-iW-eE. (6)
c c c









Finally, Eqs. (4) and (6) are simplified as



c (7)
q x H= -H,
qxH= eE.
c


Equation (7) implies that these three vectors are mutually perpendicular with one

other (q I E, H) if e is a scalar, the case for isotropic materials. Such a wave is the

well known transverse wave. The solution of Eq. (7) is q2 = (~) c. One can also

define a complex refractive index N, yielding the very useful dispersion relationship:


W W
q = N = (n + i). (8)
c c


Consider the case of normal incidence and q II x, then Eq. (3) has the form:


{ E I Eoe- ie'N ) Ei (9)
Hf Ho; HoJ (9)


This solution is an attenuated wave with a skin depth 6 = c/wK or a power absorption

coefficient a = 2/6 = 2wI/c; the phase velocity is vp = c/n.

In summary, the optical response of a material can be described by various quan-

tities (called optical "constants") which are not independent and are interrelated by


1 4r
N2 -Z = 1 + i4 (10)



where the complex surface impedance Z = R+iY, with R and Y being the impedance

and reactance, has been introduced. Note all the optical "constants" introduced are

(in general) frequency dependent.









We are particularly interested in the real part of the optical conductivity, o1,

because it is directly proportional to the power dissipation of the electromagnetic

field per unit volume by the medium:


dPdip = Re(J E*) = Re [(7E) E*)] = or E|2. (11)
dV 2 2 2LJ JI


Here J = J1 +Jp = OE, defined by Eq. (5), is the total charge current induced by the

electric field E. This indicates that only the in-phase conduction current Jf = o7E

dissipates power, while the displacement current Jd = -i'E and the polarization

current Jp = iOzE do not because they are 900 out of phase with E thus the time

average of energy flow is zero.

Determination of Optical Constants

One useful experimental technique to determine the frequency dependence of the

optical constants is to measure the fraction of power intensity reflected by, A.(w), or

transmitted through, 5(w), by a sample. The incident photons of energy hw inter-

act with the electrons, ions, spinons ..., causing electronic transitions from occupied

states below the Fermi energy (EF) to unoccupied states above EF; or interact with

lattice vibrations (phonons), causing polariton excitations. Transmittance measure-

ments require good quality films with thicknesses (d) being shorter than the electro-

magnetic penetration depths (6), which are not usually feasible. Thus, reflectance

measurements are more frequently adopted in optical experiments. Here we will

describe the background of the theory and the techniques of extracting the optical

constants from .A(w) in a wide frequency range or from a combination of A.(w) and

F(w). The details of the optical measurements and the experimental approaches for

the optical constants will be given in chapter VI.








Reflectance of thick crystals
The expression for the reflectance will be rather simple for normal incidence on
bulk samples with surface dimensions much greater than the skin depth (d > 6).
In this case, both E and H are parallel to the sample surface. In the absence of
the idealized surface current, the boundary conditions require that the tangential
components of E and H are continuous at the interface:

SE +E, = Et, (12)
Hi Hr = Ht,

where the subscripts i, r, and t denote the incident, reflected, and transmitted fields,
respectively, at the interface. Note that, in Eq. (12), Er and Ei are assumed in the
same direction. Thus H, is opposite to Hi to maintain the relation q 1| E x H as
required by Eq. (7) for a plane wave.
The scalar relation between E and H can be simplified as H = NE according
to Eqs. (7) and (8). Thus, when a plane wave is propagating across the interface
between medium a and medium b, it satisfies

Hi = NaEi,
Hr = NE,r, (13)
Ht = NbEt,

where Na and Nb are, respectively, the complex refractive indices in medium a and
medium b. From Eqs. (12) and (13), it is straight forward to find the complex ampli-
tude coefficients of the reflected (r) and transmitted (t) electric field:



(14)
SEr Na Nb
r E, Na + Nb' (14)
Ei 2Na
t-E +r 1-.
Ei Na + Nb









The light is usually incident from vacuum onto a sample surface so that we take
Na = 1, and Nb = N = n + in. The power (intensity) reflectance is then given by


S(1 n)2 + 2 (15)
(1 + n)2 + r2

The measured reflectance A and q, the phase change of the reflected electric field

wave, are related to n and K by


(1 -n)+ic' (16)
VfAei = r = i (16)
(1 + n) + ir.'
and
2K
tan = --1 2. (17)


Note again that all quantities in the above equation are frequency dependent.

Kramers-Kronig relations

If A.(w) is measured over a wide frequency range, the phase dispersion O(w) can
be evaluated using the Kramers-Kronig relations68


w o In ,,(w) In (w') d

= In w'-wI din (w') dw'.
2r Jo w' + w dw'

The second expression, obtained from the first by integrating by parts, indicates that

the far-away spectral regions (w' < w and w' > w) and the regions in which A(w')
is flat (dY'/dw' 0) have very small contributions to the integral. After A(w) and

O(w) are determined, one can invert Eq. (16) to obtain


n(w) = 1 -(w, (19)
1 + (u(w) 2/A() cos O(w)









) = 2(w sin O(w)
1 + a(w) 2 /})cos O(w)


Finally, other optical constants, such as e(w), a(w), skin depth 6, absorption

coefficient a, and electronic loss function Im(1/e).., can be obtained from Eq. (10),

i.e.,
e 2 2
d1 = n2 K"2,
Q2 = 2niK,
a1 = Wf2/4r, (21)
a2 = w(1 e1)/47r,
6 = CIWK ,
a = 2w/Kc.

Other commonly used Kramers-Kronig equations are



()- = dw', (22)
r J0 :12 W2
2w fl w -1(w')- 1 d'
e2() = -'I' (23)
7 o W012 02


where 9 stands for principle part of the integral. These equations relate a dispersive

process to an absorptive process due to the requirement of causality for linear response

functions. In other words, the real and imaginary parts of a linear response function

(such as Inr = In A + i4, N = n + ix, E = C1 + ie, r = oa + ia2, etc.) are not

independent with each other. Rather, they are rigorously related by the Kramers-

Kronig dispersion relations.

Equation (18) shows that 0 at a single frequency must be determined from A. at

all frequencies (and vise versa). In practice, .(w) can be measured only in a limited

discrete frequency range wl < w < w2. Therefore, reasonable extrapolations beyond

the region of the experimental data must be made. This procedure will be discussed

later in detail (see p. 69) for our La2-.Sr,CuO4 thin film data.









A drawback of the Kramers-Kronig technique is obviously the requirement of a

large frequency range measurements, which are not always possible. Errors will be

introduced by the artificial extrapolations. Furthermore, if the sample is thin, the

single bounce assumption will be no longer valid, hence, the determination of the

optical constants become difficult. The situation is even more complicated for a thin

sample (film) deposited on a substrate. In such case, it is possible to extract a1 and a2

(without the Kramers-Kronig analysis) from combined reflectance and transmittance

measurements over any finite frequency range. This technique will be presented below.

Combination of reflectance and transmittance of thin films

For a film of thickness d < A, the wavelength of the far-infrared radiation, and

d < AL, 6, the penetration depth, the film may be treated as a sheet of conductor of

complex admittance yi+iy2. In this case, the transmission through, ff and reflection

from, Af, a film on a supporting substrate with index n can be approximated as


4n
(yl + n + 1)2 +y '
2(24)


and
S(y+ -1)2 +y
-(y + n (25)
(yl + n + 1)2 22

These single-layer equations are generalizations of expressions given by Glover and

Tinkham.69 The dimensionless complex admittance of the thin film is related to the

conductivity a by


y = Zoad or yl + iy2 = Zo(ol + ia2)d. (26)


Here, d is the film thickness and Zo = 377 fl = 4r/c (esu) is the impedance of free

space.






21

The exact expressions for the composite Fresnel coefficients of transmission and
reflection in normal incidence can be derived easily. A thin film on a substrate can
be considered as sandwiched by two media, the air and substrate. Consider a three
medium system with complex refractive indices of N1, N2 and N3, respectively, where
Nj = nj + ixi (j = 1, 2, 3), the transmission coefficient (from medium 1 through
medium 2 and into medium 3) is


tf= t2t2ei6 1 + r23r2ei26 + (r23r21ei26)2 + .]

t42t23eiS (27)
1 r23r21ei26 '

where rij = (Ni Nj)/(Ni + Nj) and tij = 2Ni/(Ni + Nj) as already derived in
Eq. (14). The complex phase depth 6 of medium 2 with thickness d is defined by

27rN2d w
6b- N2-d. (28)
A c

Similarly, the reflection coefficient is


rf = r12 + lt2r23t21ei26 [1 + r2r23e6 + (r21 r23ei2 +2 .]

t12r23t21ei26
= r12 + 1 r2r23e(29)

r12 + r23ei26
1 r23r21ei26 '

Note the identity t12t21 rl2r21 1 has been used in deriving the last expression in
Eq. (29). The power transmittance and reflectance are finally obtained:


S= 3 Itf12 and f = rf12. (30)


The subscript f has been chosen for Eqs. (27)-(30) to consist with the notations of
Eqs. (24) and (25). The denominators in Eqs. (27) and (29) account for the multiple









internal reflections. One can recover the approximations of Eqs. (24) and (25) easily
from the rigorous expressions of Eqs. (27)-(30). This can be done as follows: first,

take medium 1 as vacuum (N1 = 1), medium 2 as a metal film with thickness d and
refractive index N2, and medium 3 as a weakly absorbing semi-infinite slab with index
N3; then, substitute the following approximations into Eqs. (27)-(30),

IN2z > Ni = 1,
IN21 > IN31 n3 = n, (K3 < n3),
ei26 1 + i2 (31)
4ir
-iN26b -ad = y.
c

Here we have applied the long wavelength (or low frequency) limit and assumed that
film is thin enough such that 6 < 1. The the calculation is straight forward and is
left as an exercise to the interested readers.

In reality, the substrate has a finite thickness and it is a four medium problem
with medium 4 being air. For a nearly opaque metal film, the overall reflectance of
film plus substrate in this 4-medium system is


A (32)


Equation (24) gives the transmittance across the film into the substrate. This quantity
is related to the measured transmittance 3 (across the film and substrate into the
air) by
(1- ,)e-ax
3 = -, (33)
1 AS'e-2azff

where x is the thickness and a the absorption coefficient of the substrate (for example,
MgO). The other terms in Eq. (33) are the substrate-incident internal reflection of
the film,
S(yz -n + 1)2 + y2
(yl -n )2+y (34)
(yj + n + 1)2 + y2









and the single bounce reflection of the substrate,

(1 -)2 + 2 1 n\2
S= (l+-+.a nj (35)

The approximation in Eq. (35) holds when c a/2w < n, the case of weakly
absorbing medium. Equation (33) assumes a thick or wedged substrate, so that there
is no coherence among multiple internal reflections within the substrate.

The index n and the absorption coefficient a of a substrate can be obtained by
measuring the overall transmittance sub(w) and reflectance Asub(w) of the substrate.
In general, for normal incidence, the formulas for ffsb(w) and .,sub(w) of an absorbing
substrate of parallel faces with thickness x are given by70


[(1 A,)2 + 4, sin2 ]e-a (36
,ub) = (1 ,e-ax)2 + 4Ae-az sin2(4 + () '


(1 e-ax)2 + 4e-a sin2
(1 Ae-ax)2 + 4Ae-az sin2(o + )

Here ~, and 0 are defined by Eq. (35) and Eq. (17), respectively, and f = nyx. The
expression of these two equations incorporates interference effects due the substrate.
[Equations (36) and (37) can also be derived from the expressions of Eqs. (27)-

(30), taking medium 2 as the substrate and medium 3 the air.] In a low resolution
measurement (for example, see p. 166), the periodic interference fringes are averaged
out. The averages can be found by integrating Eqs. (36) and (37) over df, to be

(1 (1 e-ax
Ysub(w) (1 s)2e- (38)
1- 2e-2a38)

s 1 + (1 2,s)e-2as
1 e-2a s. (39)


Therefore, n and a can be solved by inverting Eqs. (38) and (39).









After measuring 7 and A at each frequency, we can determine yi and y2, hence

oil and o~ by inverting Eqs. (24), (25) and Eqs. (32)-(34). In this procedure, we

can iterate for a self-consistent 5', using the values of a and n measured for the

substrate. This technique has been applied in data analysis of our YBa2Cu307_s

films. The computer programming routines used for this computation are presented

in Appendix B.

Lorentz and Drude Models

Two classical models (Lorentz and Drude) are frequently used to describe the

optical properties of materials. The Lorentz model can be employed for either bound-

carrier interband transitions or lattice vibrations whereas the Drude model is appli-

cable to free-carrier intraband transitions. Thus we can model the dielectric function

by a sum of three terms:


6 = CDrude + ELorentz + Coo (40)


Here coo is the contribution from the high frequency absorption beyond the measured

range.

The Lorentz dielectric function CL can be derived by assuming that the electrons

are bound to their cores by harmonic forces and are subject to viscous damping forces

which represent the energy loss mechanism. CL is then given by

2
W (41)
L=c E 2-- W2 -- i[jW


where wj, 7j, and Wpj are the resonant frequency, plasma frequency and damping

constant, respectively, of the jth Lorentzian. The plasma frequency-defined by w~i =

47rNie2/m with Ni and m* being the number density and effective mass of the bound

carriers- may also be written as wp = VS'wj such that Si represents the contribution

of the jth oscillator to the static dielectric constant.









From the quantum mechanical point of view, hwj is the transition or gap energy

between the initial and excited atomic states, 7y the inverse lifetime of the excited

carriers or the energy width due to energy uncertainties in the initial and final excited

states. The oscillator strength wpj is related to the transition probability.

The Drude model describes the optical response of free carriers in good metals. It

is just a particular case of the Lorentz oscillator with the resonance frequency equal

to zero (no restoring force for "free" carriers):

2
WPD-- (42)
w(w + i/r)

where the Drude plasma frequency is defined by W2D = 4rNe2/m* with N being

the charge concentration (do not be confused with the index of refraction) and m*

the effective mass of the free carriers. The viscous damping mechanism-described

by a relaxation time r-is associated with collisions between electrons (or holes)

and impurities or lattice vibrational phonons in metals. The real part of the Drude

conductivity is
1 WpD 7 .C
1D = = 1 2 a (43)
47F 1 + W272 1 + L272 ~

with the zero frequency limiting value


p2D r Ne2r
To = N= (44)
47 m*

being the ordinary dc conductivity.

Since the reflectance can be calculated using Eqs. (15) and (21), therefore, as an

alternative to the KK analysis, we can in return fit the experimental reflectance data

with a combined Drude-Lorentz model of Eq. (40) to extract the optical parameters.

This procedure turns out to be very successful as will be discussed in our data analysis

later.









Sum Rule

One of the most important sum rules is called the f-sum rule. It states that the

area under the conductivity al (w) is conserved:

Jo0 w r Ne2
Oldw = -8 2m (45)
Jo 8 2 m

Here m and e are the bare mass and electric charge of a free electron. This sum

rule means that the area, or oscillator strength, is independent of factors such as

the sample temperature, the scattering rate, phase transition, etc. The sum rule has

an important impact on a superconductor, in which an energy gap develops below

the transition temperature Tc; the spectral weight at w < 2A shifts into the origin,

causing an infinite dc conductivity.


Superconductivity

Perfect Conductor

A perfect conductor has no scattering, namely 1/7 = 0. This is the case for ideal

metals with perfect translationally invariant periodic lattice described by Bloch's

theorem. The complex conductivity in this case can be obtained by

iw w2[ i1
(w)= lim () + i(46)
r-*oo 4r(w + i/r) 4 rw (46


or

i = -6(w), a 02 (47)
4 4rw(47)

For simplicity, here wp is used instead of WpD to represent the Drude plasma frequency.

Note Eq. (47) satisfies the sum rule required by Eq. (45), considering b(w) is an even

function thus fo'S 6(w) dw = 1/2. Equation (46), or (47), implies that a perfect

conductor has an infinite dc conductivity but al = 0 for w $ 0, and the inductive









response (represented by a2 which goes as 1/w) is dominant at low frequencies. The
dielectric function e is
W2
e2 = Oa = 0, = 1- (w 0) (48)
W2 '
thus
t i( /2 2 1)1/2, (0' <,< N = n + ie = ( /W(2)1/2 ( < W < Wp) (49)
(49)
(1 _- W/2)1. (W > Wp)
Therefore, because A = [(n 1)2 + K2]/[(n + 1)2 + i2], a bulk perfect conductor is
also a perfect reflector (A = 1) for w < wp.

Superconductor

The optical response of a superconductor is similar to that of a perfect conductor.
One major distinction is that a BCS superconductor has an energy gap, 2A, in the
excitation spectrum and the electrons are paired when the temperature is lowered
below To. In the BCS weak-coupling theory, the energy gap for T < Tc is given by71

1 tanh (2 + A2)1/2
VN(0) J (2 + A2)1/2 d, (50)
0
where N(0) is the density of states at the Fermi level, V is the electron-phonon
interaction potential, 3 = 1/kBT, and w, is the typical phonon frequency or Debye
frequency. The transition temperature is predicted as

kBTc = 1.13 hwce-1/Nv(O). (51)

In the vicinity of Tc, the theory gives

A(T) 1.74 A(0) 1( (52)

The limiting value at T = 0 is

2 A(0) = 3.5 kBTc. (53)

[For a Tc = 90 K superconductor, for example, this would give 2 A(0) = 220
cm-1, a range in the far infrared.] Consequently, one expects al = 0 in the range









0 < iw < 2A. This means that photons of energy less than 2A are not ab-
sorbed because their energies are not sufficient to break Cooper pairs. The incident

light is therefore 100% reflected because the impedance mismatch at the interface

(n = 0 in the superconducting sample, and n = 1 in vacuum). This property agrees
with the Meissner effect that the electromagnetic field is zero in the interior of a bulk
superconductor.

However, part of the field still does penetrate into the superconductor and is
exponentially damped within a length scale called the London penetration depth:



AL = 47Ne2 (54)


with N, being the superfluid density. If all free carriers condense completely into
pairs, then AL = c/Wp. The fraction of the transmitted electric field at the sample
surface can be found from Eqs. (14) and (49),


t =-- M ---, (55)
t 2 .2w (55)
1+N wP


for w < 2A < Wp. This indicates that the transmitted electric field Et has a phase
shift related to, and is much less than, the incident field Ei. Note the transmitted
power is zero, namely 5 = nlt 2 = 0, because n = 0 (al = 0, e1 < 0). The induced
inductive current (associated with 02) in the superconductor is 900 out of phase with

Ei and hence does not dissipate energy. The transmitted magnetic field given by


Ht = NEt i Et = 2Hi (56)


is much larger than E1 and is in phase with the incident Hi. This is a consequence of
the continuity of H at the interface and the 1800 reversal of Er, the reflected wave.









At w > 2A (taking h = 1), the photon energy is high enough to disassociate
Cooper pairs, causing quasiparticle excitations across the superconducting gap to the
unoccupied normal levels. The conductivity thus approaches the normal state value.

At finite temperature below Tc, the conductivity al at w < 2A is no longer
zero, shown in Fig. 3, due to the existence of thermally excited quasiparticles. The
temperature dependence of the low-frequency conductivity exhibits a peak below Tc
(illustrated in Fig. 4) which has been explained by the coherence effect. In BCS
theory, the perturbation Hamiltonian can be written as
H = EBk'kCk'Ck (57)
kk'
Here the subscript k represents the quantum state for momentum and spin, Ct and

Ck are the quasiparticle creation and annihilation operators, and Bk'k are matrix
elements of the perturbation operator. In the normal state, each term in the sum is
independent. At T < Tc, however, there exists phase coherence between the wave
functions of the occupied states. This interference leads to Bk'k = +B_k'-k, where
the upper sign for "case I" and the lower sign for "case II" interactions.

The Hamiltonian for interaction of electromagnetic radiation with matter is rep-
resented by a term p A, where p is the momentum of the electrons and A is the
vector potential of the external field. Since this term is odd with p (or k), the in-
teraction obeys the case II process. Tinkham71 has shown that this interference will
result a coherence factor for scattering:

F(A, E, E) = -1 T, (58)

where E is the quasiparticle energy measured from the Fermi level and E' = E + hw.
The superconducting to normal-state conductivity can be expressed as
1 00
i= 1 [ F(A, E, E')N,(E)N,(E + fw) [f(E) f(E + hw)] dE
(59)
1 wj I(E(E + hw) + A [f(E) f(E + iw)]
hw -oo (E2 A2)1/2[(E + hw)2 A2)]1/2









where f(E) is the Fermi distribution function and N,(E) is the the superconducting
density of states given by


IEI
N,(E) = N(0) Re (E (60)


Here Re stands for real part. Equation (60) indicates that N, = 0 for IEI < A. It
diverges near A and approaches the normal state value at IE\ > A. Note the case II
coherence factors have been used in Eq. (59).

The resulting conductivity calculated from Eq. (59) are shown in Figs. 3 and 4.
The low-frequency upturn in Fig. 3 is due to the coherence factor, and the minimum
moves to higher energy as T is lowered, indicating an opening superconducting gap.
At T = 0, oal(w) = 0 up to w = 2A; above this frequency, ria(w) begins to rise due to
the photo-excited quasiparticle absorption. The difference between ol,,(w) and al,(w)
disappears at higher frequencies. The oscillator strength below 2A (or the "missing"
area) shifts into the origin to form the superconducting condensate. Figure 4 shows
that, at small frequency w, the integral in Eq. (59) gives a peak below Tc because
of the divergence of the density of states N,(E) at E A. The peak gradually
disappear with increasing frequency. Such peak due to the case II coherence factor
has been observed in the nuclear relaxation rate72 and the optical conductivity8 for
classical superconductors.










1.5 I I I I
Mattis-Bardeen theory
-1
I 2Ao = 200 cm1
1/r = 100 cm-1
Tc= 90 K

1.0

b
S I = OK
'. --T = 40 K
T= 40 K
__ T = 60 K
ST = 80 K
b \ T = 85 K
0.5 \. ... T = 90 K


\\ \\ .-^.V




0.0
0 100 200 300 400 500
o (cm- )













Fig. 3. The conductivity ratio of a superconductor vs. frequency at T < Tc
in the framework of Mattis-Bardeen theory.









2.5 .
Mattis-Bardeen theory
2Ao = 200 cm-1
1/T -00 cm -r
/T = 100 cm- 5 cm-
2.0 Tc = 90 K W = 10 cm-
= 20 cm-1
o = 30 cm-1
S= 50 cm-1
= 100 cm1

1.5
b

- /\

b 1.0 / / -/' ___
S// /--------------



0.5/ /
/ / / 1



0.0 I
0 50 100 150
Temperature (K)








Fig. 4. Conductivity ratio as a function of temperature at low frequencies,
showing a coherence peak below Tc.














CHAPTER IV
INFRARED TECHNIQUES

This chapter describes the principles of Fourier transform interferometry and

spectrometers used in this work. The optical response is determined experimentally

by measurements of reflectance or transmittance of the sample as a function of a

wide range of incident photon energies. This range extends from about 20 cm-1 to

40,000 cm-1 (2.5 meV-5 eV) using variety of optical spectrometers, light sources and

detectors.

The following conversion factors for energy units are useful:

E: 1 meV = 11.6 K = 8.066 cm-1 = 0.242 THz

f : 1 THz = 4.133 meV = 33.33 cm-1 = 48 K

w : 1 cm-1 = 0.124 meV = 1.44 K =30 GHz

T : 1 K = 0.086 meV = 0.695 cm-1.


Interferometry

The spectrometers used to measure the optical spectra of the samples in the

far-infrared (20-600 cm-1) and mid-infrared (500-3000 cm-1) region are a slow-scan

Michelson interferometer and a fast-scan 113 V Bruker Fourier Transform Interferom-

eter. (A Perkin Elmer 16 U Grating Monochromator, which will be discussed later,

is used to collect data at higher frequencies of 1000-40,000cm-1).

Infrared Radiation at Low Frequencies

Interferometry is widely used for measurements in the far-infrared region primar-

ily due to the fact of energy-starvation for all thermal sources at low frequencies.

This fact can be seen from the Plank law for the spectral distribution of black-body









radiation. The power p(w) emitted per unit area of the light source at temperature
T and in frequency range between w and w + dw is given by:



p(w)dw dw (61)
4r2c2 ehw/kT 1


The radiation spectra are peaked near xw = 2.82 kT (or w w 2 T for w in cm-1 and
T in kelvin). Figure 5 plots the spectrum of Eq. (61) using logarithmic scales. The
intensity is normalized to the peak value at 1000 K. It can be shown that the peak
power pm(T) ~ T3 and is given by



(T) = 1.42 ( )( ) 1.26 T )3 mW/cm. (62)
2rc h 1000 K


Note that the result in Eq. (62) is for per one unit wavenumber (1 cm-1) interval.
To stress and illustrate the strong temperature dependence, the same spectral distri-
butions are also plotted in linear scale, shown in Fig. 6. The total radiation power
Po emitted from a source of area A can be obtained by integrating Eq. (61) over all
frequencies

Po = A p(w) dw = aT4A (63)


with
Vr2 k4
a= 60 2 = 5.67 x 1012 W/cm2 K4 (64)
60 c2%"

being the Stefan-Boltzmann constant.

Consider a mercury arc lamp source with A = 3 cm2 at T = 5000 K, the total
emitted power is Po = 1.1 x 104 W. To estimate the fraction of radiation power in









the far-infrared region, remembering 1 K = 0.7 cm-1, one can approximate Eq. (61)
in the low frequency limit x hw/kT < 1:


kT
p(w)- 422 (65)


This ~ w2 dependence, as seen from the slopes of the curves plotted in Fig. 5, indicates
that p(w) decays rapidly with decreasing frequency, which can also be seen in Fig. 6.
The emission power up to a frequency w is therefore


P(w) = A p(w') dw' = w3A. (66)
Jo 12=ir2c2

Of the total radiation power Po, only a fraction


9 = P(w)/Po = Tx (k (67)


is emitted below w. For w = 100 cm-1 and T = 5000 K, this fraction is r =
1.2 x 10-6, i.e., only a tiny amount of power, 13 mW out of 11 KW (taking A = 3
cm2), is emitted at w < 100 cm-1. Therefore, a dispersion spectrometer such as
grating monochromator, which measures a one resolution width at a time, is obviously
inefficient in the far-infrared measurement.

The situation can be greatly improved if one uses an interferometer, in which
entire radiation power at all frequencies is utilized. Consequently, the signal-to-noise
ratio can be greatly enhanced which is called the Fellgett advantage.73 The details of
interferometry have been described in literature,74-76 and here only a brief discussion
will be given.









Fourier Transform Spectroscopy

The principle of interferometry is based on the idea of the Michelson interfer-
ometer as sketched in Fig. 7. Light radiation from an extended source S is divided
by a semi-reflecting beam splitter B (mylar film or thin Ge layer) into two parts of
approximately equal intensity. These two beams are reflected by a stationary mirror
M1 and a movable mirror M2, and are then recombined to enter the detector D. As
M2 moves a distance of x/2 away from its neutral position, a path difference between
the two beams, x, is introduced before they are combined, yielding a phase difference

6 = 21rx/A = 2rvx. Here A and v are wavelength and wave number, respectively, of
the incident light. Assuming these two beams have an equal amplitude a(v), then
the complex amplitude of the combined beam reaching the detector is


A() = a(v)(1 + ei2rvx). (68)


In the ideal case, a(v) = /S(v2, where S(v) is the spectral intensity of the radiation
source (as modulated by losses due to detector absorptivity, transmission of filters,
lenses, beamsplitter, windows and samples, and reflection of the mirrors or samples,
etc.). From Eq. (68), one can obtain the intensity at the detector as a function of

path difference x at frequency v


I(x, v) = AA* = 2a2(1 + cos 2rvx) = 1S(v)(1 + cos 2rvx). (69)


For a polychromatic source emitting a continuous spectrum from v = 0 to v = oo,
Eq. (69) must be integrated to obtain the total intensity which gives


21(x) = S(v) dv+ S(v)cos 27rvxd. (70)


I(x) is called the interferogram. The first term in Eq. (70) is constant and is the total
intensity, So, emitted from the source. As x--+oo, there is no correlation between the









two beams, the second term, which is just the Fourier transform of S(v), vanishes
because of the rapid oscillation of the cosine function. At x = 0, the interference
is constructive for all frequencies hence the detector receives a maximum signal 1(0)

called centerburst or "white light" (see the upper panel of Fig. 8). It is straight
forward to see from Eq. (70) the relation between these two limits:


I(0) = 21(oo) = So, (71)


where I(oo) is the average or dc value of the interferogram. One can extend the
lower limit in Eq. (70) by noting that S(v) an even function, i.e., S(-v) = S(v). By
defining D(x) = 4[I(x) -I(oo)], which is linearly related to the signal at the detector,
one finds


D(x) = S(v)e-12" dv = FT[S(U)], (72)

S(v) = j o D(x)ei2w dx = FT-[D(x)], (73)


or
FT
D(z) FT S(V), (74)

where FT represents the Fourier transformation and FT-' is the inverse FT. There-
fore, if one knows the interferogram, D(x), for a continuous path difference, the spec-
tral intensity distribution of the radiation, S(v), can be determined by the Fourier
transform of the interferogram. This computation has turned out to be accessible in
practical applications with the development of the fast Fourier transformation (FFT)
and the availability of modern computers.

In practice, however, it is impossible to measure a continuous interferogram over
a infinite path difference. Instead, one samples a finite number of discrete points up









to some maximum path difference zm and replaces the Fourier integral by Fourier

series. The finite maximum path difference approximation introduces side lobes near

sharp spectral structures. This problem can be repaired by applying the apodiza-

tion technique.76 Another problem, caused by the discrete sampling, is the so-called

aliasing effect which must be reduced by using some cut-off filters to suppress the

high frequency components. Figure 8 illustrates spectra of the real time interfero-

gram D(x) and its Fourier transformation-single beam spectrum-measured by the

Bruker interferometer. As we can see, the interferogram is not symmetric about its

central position, which is caused by the phase error and sampling.


Optical Spectrometers

Bruker Interferometer

Most of the measured reflectance and transmittance spectra in this work is ob-

tained by using an IBM-Bruker fast-scan Fourier transform interferometer, the prin-

ciple of which being similar to that of a Michelson interferometer. The frequency

range covered is 20-5000 cm-1.

As illustrated in Fig. 9, the system is divided into four chambers-source, inter-

ferometer, sample and detector. A Hg arc lamp is used for far infrared (20-700 cm-1)

and a globar source for mid infrared (400-5000 cm-1). The sample chamber consists

of two identical channels which can be used for either reflectance or transmittance

measurements. For reflectance measurement, an optical stage, shown in the top part

of Fig. 9, is place into the sample chamber. The entire system is evacuated to avoid

H20 and CO2 absorption during measurements.

Light from the source is focused onto the beamsplitter and is then divided into

two beam-one reflected, and one transmitted. Each beam is imaged onto the faces of

a movable two-sided mirror. These two beams retrace their route back to the beam-

splitter for recombination. The recombined beam is sent into the sample chamber









and then into the detector. When the two-sided mirror moves at a constant speed, v,

a path difference x = 4vt is made, where t is the time since the mirror is at the zero-
path-difference position. Suppose the light is a monochromatic wave of wavenumber

vo so that S(v) = So 6(v Vo) + So 6(v + Vo), [the second term is needed to make

S(v) an even function,] then Eq. (72) gives


D(t) = Docos2rfat, (75)


where Do = 2So and fa = 4vvo = (4v/c)fo. This indicates that the optical frequency

of the radiation, fo, is reduced by a factor of 4v/c. In other words, the detector

sees a signal with an audio frequency fa instead of the much higher optical frequency

fo. This signal is amplified by a wide-band audio preamplifier and then digitized

by a 16-bit analog-to-digital converter. The digital data are transferred into the

Aspect computer system and are Fourier transformed into a single-beam spectrum,

as shown in Fig. 8, after some necessary corrections such as apodization and phase

correction.77,78

Bolometric detectors. The fact of weak infrared signals requires not only the use
of interferometry techniques, as described earlier in this chapter, but also a detector

of high sensitivity. One kind of detector with adequate sensitivity is the He-cooled

bolometer. The detectors used in this work are a 4.2 K Si bolometer for FIR, and a

4.2 K Si:B photodetector for MIR. Pyroelectric deuterated triglycine sulfate (DTGS)

detectors are also available. The cooled detectors have much better signal-to-noise

(S/N) ratio as compared with the DTGS. The bolometer system consists of three main
parts: detector, liquid helium (LHe) dewar, and preamplifier. Figure 10 illustrates a

diagram of the bolometer detector mounting and the LHe dewar (HD-3). After the
dewar is diffusion pumped to a pressure of ~ 10-6 torr, it is pre-cooled with liquid

nitrogen for about an hour. The pre-coolant is then removed and the liquid helium









is transferred in to the helium can to maintain the detector at 4.2 K. (A temperature

as low as 1.2 K can be achieved by reducing the vapor pressure above the liquid

helium.) A thermal radiation shield is placed between the helium can and the case to

reduce the head load on the cold area. The Si detector is mounted on a cold surface

under the helium can. The optical signal is guided by the pipe along the optical axis

through a window (poly for FIR, KRS-5 for MIR) and optical filters before it finally

arrives at the detector. The output electric signal from the detector is amplified and

then sent to the A/D converter of the Bruker interferometer.

Michelson Interferometer

The far-infrared (10-600 cm-') data are also measured with a slow-scan Michelson

interferometer. In comparison with the Bruker, it has an even better S/N ratio at

low frequencies (particularly below 50 cm-1) due to a larger and brighter mercury

source. The disadvantage, however, is that it runs much slower.

As shown in Fig. 11, the light is interrupted periodically by a rotating chopper

in order to allow lock-in detection. A beam splitter with various thicknesses of mylar

films is used in combination with different optical filters to cover the corresponding

frequency range. The detector used is a 4.2 K bolometer as illustrated in Fig. 10.

Like the Bruker interferometer, the whole system is evacuated during measurements.

Perkin-Elmer Monochromator

Optical Spectra from mid infrared through visible and ultraviolet (UV) at fre-

quencies of 1000-40,000 cm-1 (0.12-5eV) are measured using a model 16U Perkin-

Elmer (PE) monochromator. During measurements, the tank is kept under vacuum

to prevent from water absorption, particularly for the mid- and near-infrared re-

gions. As shown in Fig. 12, three sources-globar, quartz-envelope tungsten lamp,

and deuterium lamp-are used to cover this frequency range. A proper source can

be selected by turning M2 from outside the vacuum tank. The light is chopped and









passes through one or two of a set of band-pass filters. These filters reject the un-

wanted higher order diffraction from the grating, which occurs at the same angle as

the desired first-order component. This case can be seen from a simple diffraction

equation: asin 0 = nA = n/v, where a is the grating constant. At an angle 0, the

first-order component of wavelength A satisfying A = a sin 0 is selected. Meanwhile,

any higher order components with wavelengths An = A/n, or v, = nv (n = 2,3...),

which could also pass through the slit are absorbed by the filter.

Light enters the grating monochromator through an entrance slit and leaves

through an exit slit. The dispersed spectrum is scanned across the exit slit as the

grating is rotated. The resolution of the monochromator is determined by the slit

widths. Increasing the slit widths increases the intensity of the emerging radiation

(higher S/N ratio) at cost of lower resolution. Mirror M1 is a reference mirror which

can be rotated or replaced by a sample for reflectance measurements. For transmis-

sion measurements, the sample is mounted in a "sample rotator," as indicated in

Fig. 12. The positions of the sample on the rotator and of the detector are the two

focal points of an ellipsoidal mirror. Three detectors are used to cover the different

photon energy regions: a thermocouple for 0.11-0.9 eV, a lead sulfur (PbS) photo-

conductor for 0.5-2.5 eV, and a silicon photodiode for 2.2-5 eV. Table 1 lists the

parameters used to cover each frequency range.

The electric signal from the detector is sent to a lock-in amplifier (Ithaco model

393). The output signal from the lock-in system is then averaged over a given time

interval and converted into digital data by an integrating digital voltmeter (Fluke

8520A). The data are finally transmitted through the IEEE-48 Bus and a general

purpose interface box to a PDP 11-23 computer and recorded on the hard disk for

subsequent analysis.










Table 1. Pekin-Elmer Grating Monochromator Parameters


Frequency Grating Slit width Sourceb Detectorc

(cm-1) (line/mm) (micron)

801-965 101 2000 GB TC

905-1458 101 1200 GB TC

1403-1752 101 1200 GB TC

1644-2612 240 1200 GB TC

2467-4191 240 1200 GB TC

4015-5105 590 1200 GB TC

4793-7977 590 1200 W TC

3829-5105 590 225 W PbS

4793-7822 590 75 W PbS

7511-10234 590 75 W PbS

9191-13545 1200 225 W PbS

12904-20144 1200 225 W PbS

17033-24924 2400 225 W 576

22066-28059 2400 700 D2 576

25706-37964 2400 700 D2 576

36386-45333 2400 700 D2 576



a Note the grating line number per cm should be the same order of the corresponding
measured frequency range in cm-1.
b GB: Globar; W: Tungsten lamp; Da: deuterium lamp.

c TC: Thermocouple; PbS: Lead sulfite; 576: Silicon photocell.









In the grating monochromator, depicted in Fig. 13, the reflecting grating diffrac-
tion equation is satisfied:


a(sin a + sin/p) = nA, (n = 0, 1, 2...) (76)


where a is the grating constant (cm/line), a and / are angles of the incident and

diffracted rays, respectively, and n is the order of diffraction. When Eq. (76) is
satisfied, the interference is constructive. One can then rewrite Eq. (76) as


nA = 2acos 6sin0, (77)


where 6 = (a /)/2 and 0 = (a + /)/2. In practice, 6 is fixed (26 = 40) regardless of
the grating position because the incident and diffracted light paths are predetermined

by the physical geometry, whereas 0 changes as the grating (or its surface normal)

is rotated. It can be seen from Eq. (77) or Fig. 13 that at 0 = 0, [i.e., / = -a,
the incident and diffracted rays are on both sides of and symmetric to the normal

of the grating surface N(0)] it will give a zero-order diffraction (white light) for all

frequencies. Therefore, 0 is the rotation angle of the grating surface normal, N(0),
with respect to the zero-order position,N(0), as illustrated in Fig. 13.

The first order is the desired one and the higher orders (n > 2) are removed by
the proper optical filters as described earlier. Taking n = 1, one gets


v = 1/A = Ccsc (78)


with C = 1/2a cos 6 being a constant. Equation (78) indicates that the frequency is
linearly related to csc 0. As the grating is rotated, a single component at frequency

v satisfying Eq. (78) is selected and emerges through the exit slit into the sample









chamber. The monochromater is mechanically designed such that the grating, driven

by a stepping motor, is moved linearly with csc0, thus the scanning is linear in

wavenumber. The rotation angle has been designed in the range 150 < 0 < 600, the

optimum quasi-linear range in the cosecant function. To find the resolution of the

monochromator, one simply needs to take the derivatives of Eq. (78) in its logarithm

form:

In v = InC Insin 0, (79)

du
= cot 0 dO, (80)
V

where dO is the angle subtended by the slit (with a width s) at the collimator with a

focal length f = 26.7 cm, i.e., dO = s/f. Equation (80) implies that a larger 0 will give

a better resolution. Consider the worst case at the maximum slit opening, s = 2000

pm, Eq. (80) gives 0.4% < Idv/vl < 2.8%, which is adequate since most of the solid

materials are lack of sharp features at frequencies above the mid-infrared band. A

more detailed description of the grating monochromator can be found elsewhere.79






































100 1000
Co (cm1-


10000


Fig. 5. Blackbody radiation spectra using log-log scales at three temperatures.
The power intensity is normalized to the maximum value at T = 1000 K
[see Eq. (62) for maximum power]. The slopes are equal to 2 at low
frequencies, indicating an w2 dependence.


O
O
O

x
a
E
0-



3
H-


1



0.1



0.01


0.001



0.0001



0.00001






46




Blackbody radiation spectrum
0
0
0 1.0
X

0..





D 0.5

0 1 x P(w,1000K)/Pmax(1000K)
0- __ 10 x P(c, 300K)/Pmax(1000K)
0 / -__ 100 x P(o, 100K)/Pmax(1000K)
N
.0 I \



Z 0.0
0 1000 2000 3000 4000
cw (cm-1)













Fig. 6. Normalized blackbody radiation spectra using a linear scale. Note
that the relative scale is expanded by factors of 10 for T = 300 K and of
100 for T = 100 K.


















Source


Movable mirror


Beam
splitter


Detector


Fig. 7. Schematic of Michelson interferometer.


r







48


-- 1 I
r- Real time spectrum



4Q

o
L
0)
0
4-
L.
a)
- ... I... I ,1 .II ,I
S-100 -50 0 50 100
Path difference x (Arb. unit)



Single beam spectrum
C


L

E

3





0 100 200 300 400 500
w(cm-1)



Fig. 8. The interferogram D(x) (upper panel), and its Fourier transformation
S(v) (lower panel) measured by a fast-scanning Bruker interferometer.
(Note we have used w instead of v to label the frequency axis for all
figures throughout this dissertation for consistency.)








49

























S h Hg Iv







d ;










nb uI ome ChMbe I Ip Chwamr
a Near-, mir-. or tr-IR sources I Sample locus
b Autornadec Aperture I Bl*erence focus
I Iateren erCamber IV Detector Chmn
S*Optical filler I Near-. tor r-IR
(I Automatic b: rmiptM cOtnger dlteors
Two-aide movtle mirror
( Control in1eremn ter
I RPernce laser
SRewnme control & flnmr nemrror












Fig. 9. Schematic diagram of IBM-IR/98 BRUKER interferometer. The top
part of the figure (enlarged scale) is an optical stage setup for measuring
reflectance.











DEWAR, MODEL HD-3
OUTLINE SKETCH


Fig. 10. Bolometer detector. The dimensions are in inches.


SHIELD


I WORK SURFACE



























































Fig. 11. Michelson interferometer.
















vacuum tank


Fig. 12. Schematic diagram of Perkin-Elmer monochromator spectrometer.


5samrple












Zero order position


Rotatable grating


entrance
slit .


Sdiffracted
0'


exit
slit


N(O)


26


N(1)









Fig. 13. Schematic of the grating monochromator showing the incident and
diffracted rays and the operation of the grating. Note that the grating
constant, a, is significantly exaggerated in order to illustrate the path
difference given by Eq. (76).













CHAPTER V
SAMPLE PREPARATION AND CHARACTERISTICS

Various high-Tc superconducting samples have been investigated in this work.
This dissertation will concentrate on the La2-aSrCuO4 and YBa2Cu3O7-6 oriented

thin films as well as YBa2Cu3O7.. polycrystalline samples. Other samples such as
YBa2-2SrzCu307-_ ceramics, YBa2Cu408 textured pellets, YBa2CuaO3-6 granular
films and ultra-thin YBa2Cu3aOr_ film (96 A) were also measured but in less detail.


La9_,Sr,CuO4 Epitaxial Films

Three La2-.SrCuO4 thin films were prepared at Westinghouse Research Center
in Pittsburgh using off-axis dc magnetron sputtering technique.80 Two of them are

deposited on LaAlO3 substrates with dimensions of 6 mm x 6 mm x 270 nm, and the
third is grown on the (100) face of a SrTiO3 substrate with dimensions of 10 mm x

10 mm x 820 nm. Both kinds of substrates have a perovskite structure which makes
a good lattice match with the films. Figure 14 illustrates the diffractometer position

20 and x-ray counts measured at Westinghouse, showing that the films are highly
ab-plane oriented. In addition to the c-axis texture, the films are epitaxial. In other

words, the [100] and [010] directions which lie in the plane of the films are parallel

to the [100] and [010] directions in the substrates. The properties of the samples are

summarized in Table 2.

The CuO2 plane dc resistivity of a La2-.Sr.CuO4 film, shown in Fig. 15, exhibits
a sharp superconducting transition near 30 K. Above approximately 100 K, the re-
sistivity for all films is roughly of the form of p(T) = po + aT, linear in temperature

(a = 1.2 ~ 1.5 fl cm/K), with a nearly zero extrapolated intercept. Deviations from
this behavior are evident in the plateau below ~ 100 K. The inset of Fig. 15 shows










Table 2. La2-sSrCuO4 Thin Films Characteristics.


Sample # Thickness Area x Tc ATc Substrate

(nm) (mm2) (K) (K)

1, 2 270 6.3 x 6.3 0.15 27 1.5 LaAlO3

3 820 10 x 10 0.17 31 1.5 SrTiO3


an expanded view near Tc for p(T) and the inductive transition measured by the

change of inductance of a coil placed against the film. The composition of the films

is x = 1.5-1.7, near the optimum values for superconducting La2-.SrzCuO4 films.
The resistivity is consistent with the published reports of good quality La2-SSrzCuO4

films.28'29 Details of sample preparation and dc transport properties can be found

elsewhere.80


YBagCu2O7- Oriented Films

The far-infrared spectra (both reflectance and transmittance) of three

YBa2Cu307_. thin films have been measured. The samples were prepared by the

research group at Bell Communication Research (Bellcore). The films were de-

posited on 1-mm-thick MgO substrates by pulsed-laser ablation from a stoichiomet-

ric target.sl Other commonly used substrates for high-T7 superconducting films are

LaAlOa, SrTiOa (as used in our La2-,SrCu04 films), LaGaOs, and yttria-stabilized
zirconia (YSZ). These substrates are usually good for epitaxial growth and for re-

flectance measurements of thin films. However, they are not suitable for thin film

transmission studies as all of them are opaque in much of the far-infrared region. In
contrast, MgO is reasonably transparent up to 330 cm-1 at low temperatures and is









also a good substrate for oriented growth. These properties make it the best choice as
a substrate for far-infrared transmission studies of high-Tc films. Film thicknesses for
two films (480 and 1560 A) were measured by Rutherford backscattering (RBS) and
step profilometry. These two techniques agreed to within 100 A. Thickness for the
third film (1800 A) is estimated from growth conditions. Table 3 lists the parameters
of the YBa2Cu30O7_ samples.



Table 3. Characteristics of YBa2Cu3O7-_ films.

Thickness Area Tc ATc ode (at 300 K) Substrate

(A) (mm2) (K) (K) (f-1 cm-1) 1-mm-thick

1800 5 x 5 89 2 -2000 MgO
1560 5 x 5 90 1 2500 MgO
480 5 x 5 83 3 1600 MgO


Figure 16 illustrates the temperature dependence of the dc resistivity, Pdc,
for a YBa2Cu3O7_- film measured by four-probe technique. In comparison with
La2-.SrCuO4, the YBa2Cu3O7-a films have higher Tc's (~90 K) and show a T-
linear resistivity over most of the temperature region above T,. One striking feature
is that, however, the magnitudes of p(T) are close for all high-To films as represented
by Fig. 15 and Fig. 16.

YBa9.,Sr,CuROT Polycrystalline Samples

The samples were first prepared at 3M center.82 The starting compounds were

Y203, BaCO3 and CuO. The powders were combined to make a homogenous mixture.









After a set of lengthy firings, grindings and jet-millings, the compound is pressed (with

carbonwax) into pellets before the final firing and annealing in a slow steady oxygen

flow. The strontium concentration of these YBa2-.SraCu307_6 samples was in the

range 0 < x < 1.3. Trarascon et al. have observed that the upper limit of Sr doping is

at x 1.4, beyond which the materials are multiphase.83 The transition temperature

of this compound decreases gradually from 93 K (x = 0) to 80 K (x f 1.4).83

Typical parameters of the our original samples are listed in Table 4. The table gives

the Sr composition, the dc resistivity at room temperature, the mass density and

unit cell volume. It appears that the sample density is significantly lower than the

theoretical prediction, indicating porous behavior. The detailed information about

the preparation procedures of these ceramic samples can be found in Ref. 82.

We noted that the initial measurements of these samples showed strong vibra-

tional features in the far-infrared region84 and there were differences but no systematic

variations with x in the reflectance and conductivity spectra. This result was most

likely due to oxygen deficiency or a mixed phase in the samples as indicated by a large

number of vibrational phonons in the infrared spectra. The Cu-O plane phonons, if

not well screened, would show up as a result of reduced carrier concentration. The 02

deficiency was further confirmed by placing the samples in a 0.3 Tesla magnetic field,

with the result that most of the samples did not show a magnetic levitation when

they were cooled by liquid nitrogen, implying that they were not superconducting.

We also found there were green grains on some sample surfaces. The presence of this

"green phase" (2112) indicated that the samples were in mixed phase. It was clear

that 02 had to be added into these samples to recover their superconducting phase

for further study.









Table 4. Typical parameters of YBa2-rSrCCuaOTr- polycrys-

talline samples


z p (300 K) Density Density b Volume

(mi cm) (g/cm3) (g/cm3) (A3)

0 10.0 5.741 6.347 173.55

0.13 18.9 4.992 6.332 172.97

0.26 10.8 6.013 6.292 172.38

0.39 1.7 4.819 6.252 171.79
0.52 2.5 4.921 6.211 171.20

0.65 3.4 4.216 6.169 170.62

0.78 5.1 3.498 6.127 170.04

0.91 15.8 5.420 6.084 169.46

1.04 21.0 5.460 6.048 168.90

1.17 2.5 4.191 5.995 168.30


a Measured mass density.
b Theoretical density calculated from the atomic mass and the
volume in one unit cell.

b Unit cell volume (from Ref. 83).


Reannealing Procedures for YBa9CutO7-..


As the Sr doping in YBa2-,ZSr,Cu307_6 hardly affected the transition tempera-

ture, we turned our attention to the fully oxygenated ceramics and their temperature

dependence, which appeared more interesting. Therefore, we tried to reanneal two

YBa2Cu3076_ (a = 0, 6 > 0.5) samples in order to increase their oxygen content in

two step procedures.









First, the green-black pellets were ground into powder using a freezer mill at

liquid nitrogen (LN2). The power was then placed in a platinum foil and an alumina

crucible to fire at 920 OC for 12 hours in order to release any water content. (Care

must be taken that the sample should not be overheated as the melting point is around

1000 OC). After having been cooled gradually to room temperature, the product was

removed from the furnace.

Second, this dark and brittle bulk product was reground and pressed into pellets

with a pressure of 105 psi. These pellets, blackish with very smooth surfaces, were

returned to the oven for a second retiring. Having been sintered at 920 OC for a 12

hour period, the pellets were annealed in a 1 atm 02 steady flow to start the oxygen

doping process with a gradual temperature decrease. The timetable of this firing is

illustrated in Fig. 17.

Meissner Effect Test and Susceptibility.

After the reannealing procedures, we immediately tested the samples in the mag-

netic field. As the samples were cooled to 77 K (LN2) in the field, we found both pellets

levitated. The levitation lasted for more than 25 sec after the LN2 was removed from

the samples, indicating that they had returned to good quality superconductors. This

test, known as the Meissner Effect, showed that the magnetic flux originally present

was expelled from the interior of the bulk samples. The magnetic susceptibility mea-

surements also showed that the samples had an onset of diamagnetism at Tc ~ 92 K

as illustrated in Fig. 18, suggesting that the YBa2Cu307-_ pellets had been greatly

improved and become almost fully oxygenated (6 < 0.1). It turned out later that

they had a very stable transition temperature and showed very little degradation

over time.

The improvement can also be seen in Fig. 19 which shows the reflectance {(w)

and conductivity al(w) of one YBa2Cu307_- sample at room temperature, before






60

and after the reannealing treatment. In contrast to the initial measurement (dashed
curves), the post-reannealing spectra (solid curves) had higher overall levels in both

((w) and a,(w) and just displayed five pronounced peaks which had been clearly

identified as c-axis phonons. Phonons confined in the CuO2 planes are screened by
the conduction carriers thus only those phonons oscillating along the c-axis are visible.







61





820-nm La2_xSrxCu04 film

008 SrTIO3
300
006

0 0010
0.1 ratio,
004 200
Au
0

O
0.01
>-- 002


X

0.00 1



I I I I I .
10 20 30 40 50 60 70 80
28















Fig. 14. X-ray diffraction pattern measured at Westinghouse for a
La2-,SrCu04 film used in this work. The film was grown on a SrTiO3
substrate, and the growth orientation can be seen in this figure.










600 . I. .. .
200 .
0 p (pi cm)
Inductance (orb. unit)

500 -

100

S400 -
E
0
C 300 25 30 35
T (K)


200



100o 820 nm La2-xSrxCuO4 thin film




0 50 100 150 200 250 300
Temperature (K)












Fig. 15. Resistivity in the ab-plane, as a function of temperature, for a
La2-.SrzCu04 thin film (x ~ 0.17) used in this study. The inset fig-
ure shows an expanded view of the region near Tc for the same sample
and compares the resistive transition to the inductive transition.










600 . .
200 .


500

100

400


0 "
300 85 90 95 100
T (K)


200



100 156-nm YBa2CuO3 f




0 50 100 150 200 250
Temperature (K)


300


Fig. 16. Measured dc resistivity in the Cu-O plane for a 156-nm thick
YBa2Cu307-6 film. It demonstrates a sharp superconducting transition
near 90 K. The inset illustrates an expanded view around T,.






64




Sintering and reannealing for
YBa2Cu3076 polycrystallines


100ooo 02 flow begins

0 1


0)
L

00
_- 500
E
I--






0 I I
0 5 10 15 20 25 30 35
Time (hours)


Fig. 17. Reanneal schedule for YBa2Cu307-6 samples used in this study.















E
CD
0
E





r--

>1
4-
an
CU-
(I)


-10

-12


100


Temperature


130


(K)


Fig. 18. Measured ac magnetic susceptibility of a YBa2Cu307-6 pellet after
reannealing. The lower branch is the "zero-field-cooled" curve (ZFC), and
the upper one is "field-cooled" (FC) which characterizes the true Meissner
effect.


-2

-4

-6

-8


I I I I I
YBo2Cu3O7 ceramic simple
/AA A

-A A A A A



A




A

- A A

p I I I










1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

400


300


1
200 -

I'

100 ,A-"v: /),


0 I I
0 100 200 300 400
CO (cm-1)


500 600 700


Fig. 19. Comparison of the IR spectra of a YBa2CuaOT30 pellet before and
after reannealing treatment. The before-reannealing sample shows rich
phonons, an indication of mixed phase.


E





b














CHAPTER VI
EXPERIMENTS AND LOW TEMPERATURE TECHNIQUES

This chapter describes the experimental techniques used to perform the optical

measurements on various samples over wide ranges of frequencies and temperatures.

Measurements of other physical properties and the experimental apparatus are also

discussed.


Low Temperature Apparatus

The cooling system consists of three major parts: Hansen High-Tran refrigerator

cryostatt), transfer line and helium supply dewar. The sample temperature can be
varied from 4 K to 300 K by a controlled operation of liquid helium transfer. Figure 20

illustrates the flow diagram of the experimental set-up. The sample holder is attached

to the cryo-tip end of the refrigerator. An optical spectroscopy vacuum shroud is used
to isolate the cold tip from the outside environment. Optical windows can be installed

on the vacuum shroud to allow the reflection and transmission measurements. The

sample temperature is sensed by a calibrated silicon diode thermometer (Si-410A)

buried into the cold finger. The accuracy of the diode is 1 K. The sample can be

warmed by adding electrical heat to the tip heater and the temperature is controlled

automatically and monitored by a temperature controller (Hansen & Associates 8000).

A thermal radiation shield is attached to the second cold stage to protect the sample

and to absorb the 300 K black body radiation from the vacuum shroud, hence the

heat load near the cold tip can be reduced. All these steps are necessary in order

to minimize the systematic error in temperature recording. Before the helium flow

is started, the cryostat is evacuated to a pressure of 10-4 torr or less in the vacuum

shroud. By pressurizing the He dewar, the liquid helium is transferred from the dewar









through the transfer line to the cryostat. The flow rate can be regulated by two flow

meters with hoses and shut off valves which control the tip flow and shield gas flow.


Reflectance Measurements and Uncertainties-La~- SrCuO4 Films

The reflectance measurements were performed using two spectrometers with a
variety of light sources, beamsplitters and detectors for different overlapping frequency

ranges. The angle of incidence of the incident light was about 110 from the surface

normal, so that the electric field of the infrared radiation was dominantly parallel

to the ab-plane. The reflectance was calibrated with a reference mirror of 2000 A

thick aluminum evaporated on an optically polished glass substrate. The sample and

Al mirror reference were mounted on a helium-cooled cold tip, along with a silicon

thermometer and a resistance heater, to allow the temperature to be varied from 5

K to 350 K. The sample and reference could be exchanged by rotating the cryostat.

As the overall scale of the reflectance is very crucial to the analysis of HTSC, we

carefully tested the stability and measured the absolute reflectance at each temper-

ature. Thermal contraction of the sample holder and position variation between the

sample and reference were also taken into account. In order to study the temper-

ature dependence of the mid-infrared band and the plasma edge, we measured the

reflectance at each temperature up to 4000 cm-1 (0.5 eV), and at selected temper-

atures up to 40,000 cm-1 (5 eV). The coincidence of spectra in each of the overlap

frequency range was usually within 0.5%. As the film thickness (820 nm) was much

greater than the penetration/skin depth (~ 250 nm), features attributable to the

SrTiOa substrate effect were not detected. Because the sample surface was extremely

smooth and shiny, specular reflection was assumed and there was no need to coat

the sample with a metal film to correct for diffuse scattering losses. Also, the large

sample area (1x1 cm2) enabled us to obtain a high signal-to-noise ratio, making it

unnecessary to smooth the data for analysis.









The experimental uncertainty in our reflectance measurements is estimated to be

0.5%. This error arises mainly from the difficulty in establishing precise optical

alignment as the reference and the sample are interchanged, and partly from the

slight temperature dependence of the Al reflectance at low frequencies. This small
uncertainly in A.(w), however, will cause a larger propagated error at low frequencies

in the optical conductivity a(w) generated by the Kramers-Kronig transformation.


Procedures in Kramers-Kronig Analysis

After obtaining satisfactory results for a wide range of reflectance spectra .A(w),

we have confidence in using the Kramers-Kronig (KK) transform to determine the

real part of the optical conductivity al (w), a more fundamental quantity than S(w) in
description of particle-hole excitations of a material by absorption of photons of energy

hw. In principle, the KK integral requires a knowledge of .(w) at all frequencies68 as

described on p. 18 of chapter III. Thus reasonable and careful extrapolations of the

reflectance beyond the measured range must be made.

High-frequency extrapolation

The high-frequency extrapolation usually has a significant influence on the results,

primarily on the sum rule derived from the optical conductivity. This effect has been

reduced by merging our data to the reflectance spectra of Tajima et al.,2 which

extend up to 37 eV (300,000 cm-'). We find their spectra are in excellent agreement

(within 5% in relative difference; 0.8% in absolute reflectance) with our high frequency

data at room temperature.

After careful measurements, however, we observe a significant decrease in the
overall level of A(w) at frequencies above the plasma edge (~ 7000 cm-1) as the

temperature is lowered below 250 K. This decrease persists up to 40,000 cm-1, the
upper limit of our experimental data, the reflectance at 250 K being about 80% of that

at room temperature in this frequency region. However, as the temperature is further









decreased below 250 K, aside from the steepening of the plasma edge, there is very

little temperature dependence down to 5 K in this high frequency region as shown in

Fig. 21. We have carefully repeated the measurements several times and found this

behavior reproducible in both the cooling and warming processes. At the same time,

we have observed no change at all temperatures in the signal level reflected from the

Al reference which has been mounted near the sample. In addition, the reflectance

remains unchanged as the sample is heated up from 300 K to 350 K. These tests have

convinced us that the extraneous influence such as thermal expansion/contraction of

the sample holder or condensation of water on the sample surface can be ruled out.

We therefore have readjusted the high-frequency room-temperature reflectivity given

by Tajima et al.27 with a relative factor of 5% increase in the range of 5 ~ 8 eV, but

no change above this range, before appending it to our data for temperatures below

250 K. After doing so, we have assumed A(w) ~ w4, a free-electron asymptotic

behavior, above 37 eV. These changes preserve the sum rule at 20 eV.


Low-frequency extrapolation

The low-frequency extrapolation is equally important. We find that using the

Hagen-Rubens relation, A(w) = 1 Ax/J, for the normal state leads to a slightly

depressed conductivity near the low frequency end, followed by a sharp rise towards

zero frequency. This distortion may affect the estimate of the dc conductivity and

also of the sum rule, from which we want to find the superconducting condensate

by calculating the missing area below Tc. Since the Hagen-Rubens relation, a good

approximation for ordinary metals, appears to be inappropriate for the HTSC because

of the presence of phonons and of low-frequency (midinfrared) absorption processes,

First, we make a least-square fit to the optical conductivity, al(w), derived from the

initial KK transform of A(w). In this procedure, we use a two-component dielectric

function (Drude plus mid-infrared and phonon oscillators):









W2 N 2
(W) +D i/r+E 2 i + +oo, (81)
+ W / ( ,,,2 jWT7j
j=1

where the first term is a Drude oscillator described by a plasma frequency WpD and a
relaxation time r of the free carriers; the second term is a sum of oscillators for mid-
infrared and phonon absorptions with wi, Wpi, and 7y being the resonant frequency,
strength, and width of the jth Lorentz oscillator; and the last term coo is the high-
frequency limit of f(w). This last parameter is found from a fit to ,(w).

Using the fit parameters, we recalculate the low frequency reflectance for the nor-
mal state. Then, after extending the experimental A(w) with calculated reflectance, a
second KK transform is made. The results of this "second" ol(w) give a more reason-
able low frequency behavior. In the superconducting state, we have used the formula
A = 1 Bw4, as the way that A goes to unity. For temperatures well below Tc,

the low frequency reflectance is nearly constant, with some noise fluctuations around
unity. We have set A = 1 in this region for the KK transformation. As mentioned
earlier, the experimental uncertainty in .(w) is about A. = 0.5%. As A(w) -+ 1

at low w and low T, the KK transform will give propagated error in o (w)-primarily
coming from the propagated error in the real index of refraction n(w)-roughly equal
to
A'I 1 A (82)
al 1-A (82)

namely, the percentage uncertainty in oa is about 1/(1 ~) times higher than that
in A. We will address this issue later.


Combination of A(w) and f(w) Measurements-YBa9CuiOT-_ Films

The far-infrared transmittance F(w) and reflectance A(w) measurements for the
oriented YBa2Cu307-O films deposited on MgO were made at temperatures from 6 K
to 300 K, concentrated around Tc (~ 90 K). The light was incident nearly normal









(~ 100 for reflectance and ~ 00 for transmission) to the sample surfaces. In other

words, the E vector is polarized on the CuO2 plane of the YBa2Cu307-6 films.

Reflectance measurement is similar to that discussed above for the La2-.SrxCuO4

films. For transmission measurements, the reference was a blank opening. The films,

with surface dimensions of 5 x 5 mm2, were circularly masked to reduce their areas

to about 4 mm diameter. Transmittance spectra from 20 to 100 cm-1 for the 480-

and 1560-A films, and 20-375 cm-1 for the 1800-A film, were measured using the far-

infrared beamline at the National Synchrotron Light Source (NSLS).85 Transmittance

over 50-375 cm-1 for the 480- and 1560-A films and reflectance over 20-375 cm-1

for all three films were measured using a Bruker Fourier-transform interferometer.

Sample and reference spectra were measured three times each to estimate the random

noise. In order to deal with the effect due to the underlying substrates, we have also

measured a 1 mm-thick bare MgO at each temperature where film data were taken.


Measurement of YBa9._SrCu.OTO Pellets

Optical reflectance measurements were made with a slow-scan home made Michel-

son spectrometer and a fast-scan Bruker interferometer for the far- and mid- infrared

regions and with a grating monochromator for higher frequencies up to ultraviolet

region. Initial measurements were made on all pellets (x = 0-1.3) at room temper-

ature. After the reannealing treatments, a fully oxygenated sample with x = 0 was

chosen and carefully measured at temperatures between 7 K and 300 K. In order to

study the role of lattice vibrations in the superconducting transition, a number of

measurements were made around T,. Room temperature data above 4000 cm-1 were

used in analysis for all temperatures. This approximation was justified by the fact of

only little T-dependence throughout the midinfrared range.

Surface correction. One important difference between ceramic samples and

smooth thin films is that diffuse scattering from the granular surface will cause a









rapid decrease in reflectance with increasing frequency, particularly when the wave-

length is comparable to the grain sizes. To compensate for the scattering losses, all

pellets were coated with a thin aluminum layer after the initial optical measurement

was finished, and a second measurement was carried out on the coated samples at

room temperature to estimate the losses due to non-specular reflection. Two factors

must be considered in making the coating. First, it is important that the coating layer

be thin enough so that the microstructure of the sample surface remains unaltered.

Second, the layer must be thicker than the penetration depth of the coating material.

In our case, the Al coating was ~ 2000 A thick, smaller than the sample grain size

(~15 pm) but greater than the Al penetration depth (~200 A). The coating was
made by using an ion milling (microetch) equipment.

The final corrected reflectance was obtained by evaluating the ratio of the initial

reflectance (no coating) to the reflectance of the coated sample, then multiplying the

ratio by the aluminum reflectance from the literature.86 After the reflectance spectra

.9(w) over a wide frequency range were measured, the optical conductivity o(w)

was determined by performing the Kramers-Kronig transformation, with reasonable

extrapolations similar to the case of our La2-,SrzCu04 films described above.

















































































Fig. 20. Schematic of low temperature apparatus.


-*--c-DEWAR PRESSURIZATION RjOW
--PRODUCT FLOW
-u HELIUM EXHAUST
--c SHROUO VACUUM


REGULATOR










Photon
0 1
0.6 I


S 820-n
0.5



0.4 -


Energy (eV)


0.3



0.2



0.1



0.0 I
0 10000 20000
w (cm1)


30000


Fig. 21. Temperature dependence of the reflectance in the interband region.
There is a remarkable change in A((w) between 300 K and 250 K but no
appreciable change above or below this temperature range.













CHAPTER VII
OPTICAL STUDIES OF La2-,SrzCuO4 FILMS

In this chapter, we present the in-plane spectra of reflectance 5(w, T) and con-
ductivity a(w,T) of high quality La2-zSr,CuO4 films over a wide frequency range

of 30-40,000 cm-' (4 meV-5 eV) and for temperatures between 5 K and 350 K. We

make an extensive optical study on the infrared dynamics of the films.

Sample preparation and the characteristic transport properties have been de-

scribed in chapter V. The parameters of the samples have been summarized in Table 2

(see p. 55), and the dc resistivity has been illustrated in Fig. 15.

Thinner films (270 nm thickness) were measured but transparent enough that

some features of the substrate could be seen in the reflectance spectra. Consequently,

the work described here will focus on an especially thick film with thickness (820 nm)

greater than the electromagnetic penetration depth (d > 6) to avoid the substrate

complications. Details of optical measurement techniques and the uncertainties in

the Kramers-Kronig (KK) analysis have been discussed in chapter VI. Here we will

present the spectra of reflectance and other optical functions derived from the KK

analysis. Details of the infrared phonons and optical conductivity a(w) in the normal

and superconducting state are discussed. Comparisons of the normal state data to

both two- and one-component descriptions of the optical dielectric function are also

made.

Results and Discussion

Infrared Phonons

Figure 21 has showed the measured ab-plane reflectance A(w,T) of a

La2-,SrzCuO4 thin film on linear scale over most of the measured range. The details









of the low frequency behavior are presented in Fig. 22 at several temperatures. The

inset, which shows data plotted on a logarithmic frequency scale for the entire mea-

sured frequency range at three typical temperatures, illustrates the strongly damped

plasma edge around 0.8 eV (6000 cm-') and the interband features around the visible

region. As we can see from Fig. 22, M(w, T) increases over a broad frequency range

with decreasing temperature, as expected. A few infrared-active phonons in the ab-

plane are visible. These phonons are more obvious in the spectrum than in the case

of YBa2Cu307_6.10-12,34-36 This indicates that La2-_SrCu04 crystals have a lower

free carrier concentration and a higher vibrational oscillator strength. The phonon

parameters can also be extracted from oa(w), the real part of the optical conductiv-

ity, shown in Fig. 23. Of the seven IR-active phonon modes (3A2, + 4E.) expected

at the r point for the body-centered tetragonal D1 I4/mmm symmetry, three

distinct ab-plane E. modes are observed at 126, 359, and 681 cm-1 for T = 300 K.

These eigenenergies are close to those previously reported by Collins et al.,26 132, 358,

and 667 cm-1, from a room-temperature reflectance study of a La2-zSrCuO4 single

crystal. These three phonons have been assigned as external, bending and stretching

modes, respectively.87,88 More details regarding the phonon mode assignment have

been reported in Ref. 89.


Structural phase transition

We note that the lowest phonon mode at w = 126 cm-1, corresponding to an

in-plane translational vibration of the La atoms against the CuO6 octahedron unit,

broadens and splits into two distinct modes as T decreases below 250 K. The split-

ting begins at the tetragonal-to-orthorhombic structural transition which involves a

staggered rotation of CuO6 octahedra. At 200 K, the degeneracy of the two modes is

lifted but their energies are so close that they can barely be resolved. The splitting

develops upon further cooling as depicted in Fig. 24. This splitting is probably asso-









ciated with the folding back of the zone-boundary mode to the zone center because
of the unit cell doubling due to orthorhombic distortion (D18 Cmca symmetry).

Similar results in neutron scattering measurements have been reported and associated
with a soft phonon mode.90 For comparison, the inset in Fig. 24 shows the results
of Keane et al.91 for the in-plane lattice constants of a Lal.85Sro.15Cu04 sample as
a function of temperature. The structural distortion is evident in their data at T <
200 K.

Frequency shift and lifetime

We also observe that the Cu-O stretching mode at 681 cm-1 hardens by 13 cm-1
as the sample cools off from 300 to 100 K, as expected for thermal contraction. It
stops shifting, however, upon further cooling. In contrast, the frequency of the Cu-
O bending mode at 359 cm-1 remains constant at all temperatures yet exhibits a
discernible splitting at low T. We thus conclude that the stretching mode is much
more sensitive to the Cu-O bond length than the bending mode. Tajima et al.87 have

recently found a similar result when they measured the room temperature phonon
frequencies of different cuprates with different lattice constants a but almost the same

reduced mass by substituting the La atom by other rare earth elements. A similar

effect has also been observed in the T'-RE2CuO4 system by Herr et al.92 In our case
the absence of further hardening at lower temperatures is probably due to the fact
that the real part of the phonon self-energy Eph = A + if has three contributions:


A(T) = A(o)(T) + A(1)(T) + A(2)(T) (83)


where A() accounts for thermal expansion, A(1) and A(2) for the cubic and quartic
anharmonic terms in the lattice potential, respectively. These contributions are gen-
erally of the same order of magnitude but may have different signs. Thus A(o) may
be compensated by the sum of A(1) + A(2) at low temperatures. Another possibility









is the saturation of the T-dependence of all three contributions below 100 K. Such

an effect has been found in silver- and thallium-halides.93 Indeed, Tranquada et al.94

and Keane et al.91 have observed that the interatomic distances of La2-,SraCu04

saturate below 100 K.

It has been reported95'96 that the two lower-lying IR active phonons at 149 and

190 cm-' for YBa2Cu307 ceramic samples narrow dramatically but have no softening

upon entering into superconducting state. In contrast, the phonons above 275 cm-1

exhibit opposite behavior (i.e., little change in width but apparently softening below

Tc). The anomalous dramatic narrowing in phonon widths for YBa2Cu30aO7 has

been attributed to the disappearance of interaction between electrons and phonons

with energies less than the superconducting gap when the electrons condense into

Cooper pairs below Tc.10 The phonon lifetime will increase as a result of decreased

probability of colliding with quasiparticles, because the number of quasiparticle ex-

citations decreases rapidly below Tc. This issue will be addressed in more details in

chapter IX, where polycrystalline samples are discussed. In any event, here we do not

see a dramatic T-dependence in the observed ab-plane phonons for La2-,SrzCuO4,

perhaps because the lowest phonon mode at 126 cm-1 is far above the BCS gap

energy, which is ~ 80 cm-1 for a Tc = 31 K sample.

Two-Component Approach

Returning to the conductivity spectra as shown in Fig. 23, we note that the nor-

mal state al(w, T) at the low frequency limit is nearly equal to the dc conductivity

and exhibits a Drude response. A remarkable depression can be seen at 30 K, just

below Tc, for w < 150 cm-1, indicating the shift of spectral weight into the origin

due to the superconducting condensation. The inductive current represented by the

imaginary part of the complex conductivity, o2, is dominant at w < 100 cm-1 and

it diverges as w-+0 for T < Tc, as shown in Fig. 25. Above Tc, a2 changes slope









at low frequencies and heads for the origin, and the maximum moves to higher fre-
quency and decreases rapidly with increasing temperature, as expected. On the other
hand, at w > 300 cm-1, the normal-state ol(w) in Fig. 23 decays much more slowly

than the free carrier o-2-dependence as one expects in a Drude model. Addition-
ally, al(w) has much weaker temperature dependence at high frequencies than at
low frequencies. This "non-Drude" behavior, which is universal for all copper ox-
ide superconductors,10-13,31-36,40,42 can be described in a two-component picture, in
which a narrow (with a width of order kBT) and strongly T-dependent free carrier

(Drude) absorption peaked at w = 0 combines with a broad bound-carrier (MIR)
absorption centered at higher frequencies. According to this picture, the cuprates are
viewed as consisting of two type of carriers: free carriers which track the dc conduc-
tivity above Tc and which condense to superconducting pairs below Tc, and bound

carriers which are responsible for the broad MIR excitation. The dielectric function
is made up of four parts:

C(w) = CD + EMIR + Cphonon + foo (84)

where ED is the free carrier or normal Drude intraband contribution; eMIR is the

bound-carrier contribution; Ephonon is the phonon contributions, a sum of harmonic
oscillators; and coo is the high frequency contribution.

To decompose the total conductivity into two components, we can assume that
the conductivity at 5 K, al(w, 5K), is a good first approximation of a0MIR, namely

1MIR == 1C(w, 5K), for the Drude part is presumed to have collapsed to a 6(w)
function (the optical spectra are dominated by the inductive response). Thus the
Drude conductivity at higher temperatures can be initially estimated by subtracting
l (w, 5K) from the experimental a1(w, T), namely aD) == 0 .(1) Here the

superscripts denote the number of iterations. Since

1 WPD'
D (85)
4?r 1 + w272 '









we can determine WpD and 1/r from a linear fit to 1/1 s. w2. Once WpD and 1/T
are determined from the slope and the intercept of this straight line, we can again
estimate the mid-infrared conductivity from the difference between a calculated Drude
conductivity and the measured conductivity, namely a2MIR = o1 1D, where

clD is calculated according to Eq. (85). By averaging (2) at temperatures above
Tc, we find the average (o2flR) \ MIR [or l(w, 5K)], but there are noticeable
differences. Therefore we repeat the above procedure with (2)i replacing 1MIRi
1MIR 1MIR'
and find convergence after a few iterations.

The free-carrier component---w and r

Figure 26 illustrates the comparison between the free carrier contribution,
al (alMIR), and the calculated Drude conductivity. This figure shows that the
conductivity is in good agreement with the ordinary Drude behavior after the MIR
component is subtracted. The Drude plasma frequency, WpD = 6300 100 cm-1,
obtained from the above analysis is essentially T-independent, whereas 1/r is linear
in T. Writing A/r = 22rAkBT, we obtain a weak-coupling value for the coupling
constant, A = 0.25. This small value of A is consistent with the observed absence of
saturation up to 1100 K for the dc resistivity.61 Taking the Fermi velocity in the basal
plane to be VF = 2.2 x 107 cm/s, as calculated by Allen et al.97 for Lal.85Sro.15Cu04,
and using our relaxation rate we can estimate the mean free path to be

100 K
l = VFr (110 A) (86)


At T = 1000 K, I ~ 11 A, which is still longer than the interatomic spacing a
(here taken to be 3.8 A, the in-plane lattice constant). The resistivity is expected to
saturate if I < a, because the mean free path can no longer be properly defined in
this region.98 On the other hand, at temperature close to Tc, the mean free path I

[e.g., 150K 220 A according to Eq. (86)] is much longer than the coherence length









( (~ 10 A). It is this case that places the HTSC in the "clean limit", which in turn
gives a significant impact on the observability of the superconducting gap.

Figure 27 depicts the temperature dependence of 1/r in comparison with (1/r)d,

calculated from the measured four-probe dc resistivity Pdc and the value of WpD found

above,
2
(1/r)dc = bD (87)


As seen in Fig. 27, (1/r)dc or Pdc decreases quasi-linearly from room temperature

followed by a plateau and then a sudden drop as the temperature approaches Tc

whereas the far-infrared scattering rate shows a quasi-linear T variation followed

by a faster-than-linear drop (1/7 ~ T2) below Tc. This is evident when the same

data are plotted on a log-log scale, as shown in the inset of Fig. 27. The excellent

agreement in both the slopes and overall levels between the dc transport and infrared

measurements strengthens our confidence in the determination of the normal state

plasma frequency wpD and scattering rate 1/r. The sudden drop in 1/7 just below Tc

is interesting and has received considerable attention recently. Such observations on

quasiparticle damping have been reported previously for laser-ablated YBa2Cua307-

films99'36 and a free-standing Bi2Sr2CaCu208 crystal.100'101 Similar behavior has also

been found for YBa2Cu307_ and Bi2Sr2Ca2Cu3010 in femtosecond optical transient

absorption experiments.102 This result may suggest that the excitation that scatters

the free carriers is also strongly suppressed below Tc, or forms its own gap, as the

free carriers condense. Another interpretation is that the number of unoccupied

states available near the Fermi levels decreases rapidly as a result of the depression

of the density of quasi-particle states near EF as the gap opens, causing a dramatic

decrease in the probability of quasiparticle elastic scattering. Nicol et al.103 have

recently calculated the quasiparticle scattering rate and found such a fast drop within

the phenomenological Marginal Fermi Liquid model. However, on account of the









large error bars at low frequencies (below 100 cm-') and the limited number of

temperatures below Tc (31 K) in our data, we are unable to observe a "coherence"

peak in al(T), as has been calculated by Nicol et al.103 and found in YBa2Cu3aO7-

by Nuss et al.,104 and in Bi2Sr2CaCu2Os by Romero et al.101 This "coherence peak"

has also been observed in our YBa2Cu3aO-s thin films and will be discussed in detail

in the next chapter.


The midinfrared absorption

Figure 28 presents the MIR conductivity in the normal and superconducting

states. This quantity is obtained by subtracting the calculated free carrier contri-

bution (shown in Fig. 26 as solid lines) from the total conductivity. Some features

that are common at all temperatures include: an onset near 80 cm-1, a maximum

around 250 cm-1, a notch-like structure at 400 cm-1, and a broad peak around 800

cm-1. As we can see, the MIR conductivity taIMIR(w, T) has a relatively weak tem-

perature dependence. There do appear to be three distinct temperature regimes: >

250 K, T,-200 K, and below Tc. In each, there is a noticeable conductivity increase in

the region of 150-1500 cm-1 with decreased temperature. The enhancement is more

obvious for T < Tc and will be discussed below.

According to the data in Fig. 28, the "two-gap" structure of an onset near 80

cm-1 (3.7 kBTc) and a notch around 400 cm-1 (18 kBTc) is present both below and

above Tc. This structure is shown more clearly in Fig. 29, where we plot the average

of the curves above and below T,. Thus we cannot associated either feature with the

superconducting gap, since that presumably would not appear above Tc. Furthermore,

there is no shift in any feature in the superconducting state as would be expected

for a Holstein sideband associated with condensate. Such features have also been

observed35'100'105 in YBa2Cu307O_ and Bi2Sr2CaCu208 films. The structure at 400

cm-1 (50 meV), which appears common to the cuprate superconductors, has been









explained as due to strong bound carrier/phonon coupling.48 It can not be accepted
as a superconducting gap simply because its magnitude is too large. The value of the
lower-energy onset usually varies for different samples. The presence of this structure
above Tc and the lack of evidence of an energy shift with varying temperature below
Tc make it difficult to associate it with the BCS gap.

Holstein effect

Lee et al.106 have calculated the dynamic conductivity in the framework of strong-
coupling theory, including the Holstein mechanism.107,108 They obtain a two-gap
structure in the superconducting state. The first onset is presumed to be the super-
conducting gap, while the "second gap" is interpreted as the consequence of inelastic
scattering with phonons due to the Holstein effect.

To estimate this effect, we have calculated the conductivity according to the
Holstein theory for our film and find that the enhancement of the MIR strength
below Tc may not be accounted for by the inelastic scattering contribution. In the
Holstein model, the scattering rate at low temperature can be obtained by108

1/r(w) = F(2)(w 0) dil, (88)

where a2F(Q) is the Eliashberg function or electron-phonon spectral density. The
parameters used in our calculation were: Wp = 6300 cm-1 (from the two component
model fit outlined above), A(w=0) = 0.25, and the average boson frequency flo = 75
cm-1. In general, the coupling parameter is given by108

(w) = 2 a2F() In W n Wd2 2 d (89)

with a zero frequency limiting value

A = A(w- ) = 2 f a ) d (90)

For simplicity, we have assumed the Eliashberg function has the form (in an Eien-
stein model) a2F(0) = Ab(fS 1o), where A = 1AI o according to Eq. (90). The
2 A0acrigt Tq 9) h









calculated result is illustrated as the dash-dotted curve in Fig. 29. The size of the the
Holstein side band could be enlarged to match the measured MIR spectral weight by

increasing A and wp, but this would be in disagreement with the values determined

experimentally.

Superconducting-to-normal ratios

Another unconventional behavior is seen in the superconducting to normal-state

conductivity ratio shown in Fig. 30. Ratios of conductivity have been used frequently

in the past to suggest superconducting gap structure.41,42 In Fig. 30, we compare ol,
and "aln" at the same temperature. We note that if al, and ol, are compared at

different temperatures, the result is totally different as shown in the inset, resembling

a BCS-like behavior as seen in Fig. 3 on p. 31. To estimate lan(w, T) below Tc, we

presume that the "normal state" WpD and 1/7 below Tc follow the "normal" behavior,

i.e., WpD remains a constant (6300 cm-1) and 1/7 follows the linear extrapolation of
the relaxation rate above Tc. Then ain below Tc can be calculated as the sum of the

calculated Drude component and the averaged MIR conductivity (alMIR)n, namely


measured al, T > Tc
ln= 1 W DT (91)
4 1 + w22 + (alMIR)n. < Tc


As we can see in Fig. 30, the ratio ai,/lal exhibits a sharp edge near 100 cm-1

and has a peak around 180 cm-1. The peak is suppressed but does not shift as T

approaches Tc from below. al, "overshoots" al, up to 1000 cm-1 and then gradually

joins the normal state conductivity at higher frequencies. This surprising result can
be attributed first to the observed enhancement of the mid-infrared conductivity in

the superconducting state, and second to the observed faster-than-linear decrease in

the quasiparticle scattering rate as demonstrated in Fig. 27.









Extra absorption below T.

We turn to the differences between the MIR conductivity above Tc and the below-
Tc conductivity. The enhancement is evident in the raw data of Fig. 23, in which we

can see the conductivity at 5 K is higher than that at 50 K, above Tc, for w Z 360
cm-1. By calculating the difference between the averaged mid-infrared conductivity
in the superconducting state, (alMIR),, and the one in the normal state, (UIMIR),,

we find an extra absorption below Tc in the MIR region which counts for roughly
15% of the Drude oscillator strength. This difference is shown in Fig. 29. (Note that
the actual fraction may be smaller for the reason of large error bars in o1 at low w

below Tc, as will be discussed below; thus the difference, (aiMIR)s (aIMIR)n, may
be exaggerated at low frequencies.) This anomalous behavior suggests the existence

of another type of excitation visible in the superconducting state, with the normal

Drude carriers not completely condensing into the superfluid below Tc. However, this
argument can not be taken as rigorous, since our approach of extracting the Drude
component has neglected the w-dependence of the electronic scattering rate, though

it may be weak as suggested by the small value of coupling constant A ~ 0.25.

To confirm our observation of the extra absorption below Tc in the MIR con-

ductivity obtained by the two-component analysis, we use two other independent

methods to estimate the oscillator strength of the superconducting condensate: the

dielectric function and the f-sum rule. According to the clean limit picture, when

2A > 1/7 the Drude oscillator strength will condense into a w = 0 delta function for
T < Tc. Thus the real part of the dielectric function at low frequencies is


w2
fe(w) = elb W (92)


where wp, is the superconducting plasma frequency defined as ws, = 47rne2/ma with

n, being the density of superfluid carriers; and elb is the bound carrier contribution to









e (w), i.e., the low-frequency sum of all finite frequency absorption. In principle, qb
is w-dependent. It is constant only at frequencies well below the lowest bound-carrier
resonant frequency.

Figure 31 shows the plot of el(w) [obtained from KK transform of A(w)] as a
function of w-2. The data fall on a straight line, as predicted by Eq. (92), in the low
frequency range. The slope obtained from the linear regression fit at T = 5 K gives
wp, 5800 100 cm-1, from which the London penetration depth can be estimated
to be AL = 1/21rwp = 275 5 nm. This value, which is much less than the film
thickness (820 nm), is comparable to the 250 nm in-plane AL found by muon-spin-
relaxation (pSR) measurements109 for Lal.85Sro.15CuO4 at T = 6 K. We note that
only a fraction f, = w,/lwD ;- 85% of the free carriers condense into superfluid, in
good agreement with the observation that ~15% of the Drude spectral weight has
shifted to MIR region below Tc as outlined above. Further evidence that supports
this argument is obtained from the f-sum rule that will be discussed next.

Sum Rule-Superconducting Condensate

Figure 32 illustrates the spectral weight, Neff (w) m/mb, as defined according to


Nef(w) 2- mV,,2 O(w')dw', (93)


where e, m are the free electron charge and mass, respectively. mb is the averaged
high-frequency optical or band mass, and Vcei is the volume (95 A) of one formula
unit. Note Eq. (93) is also called partial sum rule and is the generalization of Eq. (45).
In this expression, Nff (w) equals to the effective number of carriers per formula unit
participating in optical transition at frequencies below w.68 The normal state Nff (w)
curves at 10,000 cm-1 gives, if mb = m, roughly 0.18 hole per CuO2 layer, which is
a value close to the dopant concentration of our film ( x ~ 0.17) assuming each Sr
atom donates one hole to the CuO2 layer.









In the normal state, the curves exhibit a sharp rise in the far infrared followed by
a broad plateau before another rise beginning near 10,000 cm-1 due to the charge-

transfer transition. As the temperature is lowered, spectral weight transfers to lower

frequency in response to a decreasing relaxation rate. Below Tc, the spectral weight

is reduced as expected due to superconducting condensation. From the difference

between Neff (w) m/mb for the normal and the superconducting states, the plasma

frequency of the superfluid charge carriers [or the missing area in the curve of oa(w)]

can be estimated. This difference gives A(Nff m/mb) = wp,s2mVen,/4re2, from which

we find wp, = 5800 cm-1 at 5 K, in excellent agreement with the value determined

from the real dielectric function as discussed earlier.

One surprising result of our measurements is that the Neff (w) m/mb in the charge
transfer region is larger at T > 300 K than at other temperatures below 250 K, as

shown in the inset of Fig. 32. The mechanism that causes this difference is not

clear at this moment. One speculation is that the structural transition at around
250 K may change the band structure due to the doubling of the unit cell. The

transformation introduces new Brillouin zone planes at which the semiconductor-like

gaps are opened, transferring oscillator strength to higher frequency regions. The

band mass may also change accordingly. This difference disappears, however, above

15 eV, where the Neff (w) m/ma curves come together. 15 eV is the end point of the

interband excitations from the O 2p valence bands to the La 5d/4f conduction bands

above the the Fermi level and the starting point of excitations from the Cu 3d bands

to the La 5d/4f bands.

Figure 33 shows the temperature dependence of the Drude (WpD) and supercon-

ducting (wp,) plasma frequencies. Here WpD is determined from the fit to eal(w) as

described earlier and is consistent with a picture of constant carrier concentration in

the normal state. This magnitude of WpD (~ 0.8 eV) is smaller in comparison with

the values (, 1.2 eV) obtained in YBa2Cu3076_ or BiSrCaCuO crystals, presumably









indicating lower carrier concentration on the CuO2 planes. Below Tc, wps is estimated
from the sum rule, the linear fit to el(w) vs. w-2, and the least-squares fit to the re-
flectance data using a two-fluid model. These three approaches give very close results
in Wp, and we take the average values. Shown in the inset is the superfluid electronic
density fraction f,(T). This superconducting condensate is calculated according to

f,(T) = n,(T)/n = w,(T)/wD with WpD = 6300 cm-1, the normal state value. This
quantity f,(T) is essentially a measure of the strength of the 6 function in al(w, T),
and is related to the T-dependence of the penetration depth AL(T). The solid curves
in Fig. 33 and its inset show the phenomenological behavior predicted by BCS theory
according to

f,(T) rA(T)l2
f,(0) A(0) (94)

where A(T) is the T-dependent BCS order parameter. It gives a nearly constant
A(T) at T < Tc. Near Tc, A(T) drops to zero with a (1 T/Tc)1/2 dependence. The
behavior of f,(T) in our data agrees with this expression and it demonstrates that
the normal carriers condense rapidly into the superfluid below Tc, as expected.

One-Component Approach

An alternative approach to analysis of the optical conductivity is the one-
component model with a frequency dependent mass and scattering rate.40,110-112 In
this approach, the infrared absorption is entirely due to free carriers, in which are di-
vided into "coherent" and "incoherent" parts caused by the interaction of the free car-
riers with some sort of optically inactive excitations (charge or spin fluctuations).105
This approach has been proposed by Anderson113 and applied to heavy-Fermion
superconductors114. The normal Drude component is regarded as the coherent part
centered at w = 0. The incoherent part occurs at frequencies characteristic of the ex-
citations and shifts away from w = 0 due to interactions with the excitations. In this









model, the complex dielectric function is described by a generalized Drude formula:


2
c(w) = Ch W (95)
W[W E(W)P


where eh is the "background" dielectric constant associated with the high frequency
contributions, wp--defined by 4irNe2/mi-is the bare plasma frequency of the free

carriers, and E(w) = El(w) + iE2(w) is the self energy of the carriers.

Because e(w) is causal, El(w) and E2(w) are related by the Kramers-Kronig equa-

tions. It is important to stress that the interband contributions, which can be lumped
into ch, are excluded from wp and E(w). To find E(w), knowledge of wp and Eh is re-

quired. In order to identify the interband components, we fit the experimental al (w)
at frequencies higher than 8000 cm-1 with Lorentz oscillators to parameterize the
interband absorption. By subtracting the contribution due to these interband oscil-

lators from the total conductivity and calculating the area under al(w), we obtain
Wp = 13, 000 cm-1, corresponding to a carrier density of n = 1.8 x 1021 cm-3(M/m)

or 0.17 holes per CuO2 unit if mb = m. As we have found WpD = 6300 100 cm-1

in the two-component analysis, we can also estimate the strength of MIR absorption
or the "incoherent" component as pm = (2 w2D)1/2 ; 11,370 cm-1. eh can be

estimated from the interband oscillators, giving eh ~ 4 in the far infrared region. At
higher frequencies, ch becomes complex and w-dependent.

Mass enhancement m*/mb and self energy E(w)

Once Wp and Ch are determined, the self energy E(w) can be calculated at each
frequency from the experimental e(w) according to Eq. (95). If we rewrite Eq. (95)

as

E(W) = Ch W (96)
W[W + i/T*(a)]









and compare Eq. (96) with Eq. (95), we can extract the renormalized scattering rate

1/T*(w) = -E2(w) mb/m*, and the effective plasma frequency w* = wp(mb/m*)1/2,
where the effective mass enhancement is given by


m*/mb = 1 -- E/w. (97)


Note both the real and imaginary parts of E(w) are negative definite. The computer
routine for the one-component analysis is given in Appendix B. The resulting curves of
m*(w)/mb and E2(w) are shown in Fig. 34. The effective mass m* is greatly enhanced
at low w and m* $ mb at high w, as expected for the MFL and NFL theory.37'38

The behavior of m*(w)/mb and E2(w) as shown can be viewed as arising from a
local Coulomb interaction of carriers with a broad spectrum of other excitations. At
low frequencies, the carriers drag a low-energy excitation cloud along with them, caus-
ing a mass enhancement. As frequency increases, the scattering rate 1/7* increases
when the low-lying states are excited hence a new inelastic scattering occurs. The
carrier mass decreases to approach the band mass as w increases, for the low-lying

excitations can not follow the rapid carrier motion. We can estimate the character-
istic energy range of the low-lying excitations from the frequencies at which m*(w)

and E2(w) change from their low to high frequency behavior. This range appears to
be between 300-1000 cm-1 (0.04-1.2 eV). We note that a pronounced peak near 0.1
eV reported by Uchida et al.29 is not observed in our spectra of m*/mb and E2. The

present values of E2 are comparable with their result for the unnormalized scattering
rate. The mass enhancement here is, however, a factor of 0.15 smaller than their
result. The high value of m* in their data would imply an even stronger coupling be-

tween the free carriers and the low-lying excitations, which is difficult to understand.
Note that the value of m*/mb at low w and low T can also be predicted from the
conductivity sum rule from Fig. 32 or simply from w/o~wD 4.2, which agrees well









with the result in Fig. 34. Writing m*/mb = 1 + A, we find the low-frequency-limit

value of coupling constant A w 3 at low temperatures, suggesting strong interaction

of carriers with a spectrum of other excitations. One major difficulty with this model

is that this large A would give a high Tc, inconsistent with the actually measured Tc

value. To account for this large A, one may speculate that the Tc is suppressed by

other mechanisms. However, Such mechanisms, if any, are not clear at this point.


Effective scattering rate 1/r*(w)

A linear T-dependent scattering rate at w ~ 0 implies it is also linear in w at

higher frequencies. The effective renormalized scattering rate can be obtained by

l/r* = -(mb/m*) E2. This quantity is shown in Fig. 35. The extrapolated w = 0

values of 1/T* are compatible to those obtained above in the two-component fit by

assuming a constant scattering rate. This is not surprising since both the one- and

two-component approaches have described the dc transport behavior well. At higher

frequencies, we observe 1/r* is of order max(T, w) before it saturates. According to

the MFL theory, however, it is not 1/r* but the imaginary part of the quasi-particle

self energy E2 that has the form -E2 = Amax(7rT, w), as long as w < wc 1000

cm-1. Thus E2 would change from constant to linear in w at w > 7rT. At low w, our

results agree with this prediction, and E2 tends to saturate at frequencies above we.

Since A is in principle T-independent, one expects the slope of E2(w) to be constant

at all temperatures in MFL theory. However, our data indicate a gradual decrease of

slope with increasing temperature.

It is difficult to interpret the frequency dependent scattering rate as a consequence

of inelastic scattering due to Holstein effect,107,108 in which a carrier can absorb a

photon of energy iw, emit an excitation (or a phonon) of energy e (e ~ 300 cm-1

in this case), and scatter. First, the large value of A (~ 3) derived from our data of

Fig. 34 suggests a strong coupling between the conduction carriers and the excitation.