ANISOTROPY IN THE INFRARED, OPTICAL
AND TRANSPORT PROPERTIES OF HIGH
MANUEL ALBERTO QUIJADA
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
It is with great
pleasure that I thank my advisor, Professor
for his advice, patience and encouragement throughout my graduate career here at
University of Florida.
I feel fortunate to be part of his research group.
thank Professors J. Graybe
Hooper, N. Sullivan, and J.H.
mons for their interests in serving on my supervisory committee and for reading this
Thanks also go to all my past and present colleagues in Tanner's group for their
friendship, useful conversations and cooperation.
In particular, I would like to thank
C.D. Porter for his assistance with computer software. I am also indebted to Drs. G.L.
Carr, D.B. Romero, and V. Zelezny for many enlightening and useful discussions.
I would like to express my gratitude to Drs. J.P. Rice,
D.M. Ginsberg, M. Kelley,
Chou and D.C. Johnston for providing good quality single crystals
that were essential to the completion of this work.
The technical support of the staff members in the physics department machine
shop and engineers in the cryogenic group is appreciated greatly.
like to take this opportunity to thank my wife,
Zunilda, and my
daughter, Melissa, for their support and understanding during the countless nights
they stayed alone while I was working in the laboratory.
Finally, I also thank my parents for giving me their support throughout my aca-
T7 _._. _- -1 ...-.. P I rT
rn .-.. ...f-1_ nL -rv r1'7lr .._. ...
TABLE OF CONTENTS
. a a a a a a a a a a a11
* S S a a a a a a aa a a a a a a S 1
REVIEW OF PREVIOUS
a S S S 0S5
Crystal Structure of Copper-Oxide Materials
La2 CuO4+. . . . .
YBa2 Cu3 07_ *
Bi2Sr2 CaCu2s . . . .
Review of Optical Properties Copper-Oxide Materials
Midinfrared Absorption in the CuO2
Anisotropy in the ab Plane
R"'V -,O a 0 0 5 0 S
Models for Carriers in the CuO2 Planes: Normal State
Three-Band Hubbard Model
t J M odel . S . .S S. ..a a. .
M odels for 1 .(. . . . . . .
T n r Z-\ an 1 1) I 4, -
a S S S S 0 5
* a a S S 0 0 5 0 *
* S S S S S S S S SS8
Y1I II LIVI
Superconducting State Models
Symmetry of the Order Parameter .
Evidence for Proposed Pairing States
Determination of Gap by Optical Spectroscopy
Fourier Transform Infrared Spectroscopy
S S S S S 4 5 4 4 4
S S S S 4 4 4 4 4 8 S 4 4
The Perkin-Elmer Monochromator .
Sample Mounting and Low Temperature Measurements
Normalization Procedure of the Reflectance .
Data Analysis of the Spectra: The Kramers-Kronig Tra
High-Frequency and Low-Frequency Extrapolations
Sample Preparation Techniques
4 4 4 5 S 86
Bi2Sr2 CaCu2 08 Single-Domain Crystals
La2Cu04+6 Single Crystal . . .
OPTICAL STUDY OF La2Cu04+, SINGLE CRYSTAL
c-Axis Reflectance of La2CuO4+ . . .
Room Temperature Spectra
Low Temperature c-Axis Re
Assignment c-Axis Phonons
Effective Charge . .
ab-Plane Reflectance .
Assignment ab-Plane Phonons
* 4 5 4 5 4 4 4 4 5 4 5 5 5
* S S S S S S S S S S 4 4 5
* S S S 4 S 5 4 5 5 5 5 S
* S S S S S S S S S S 4 S S 4
Low Temperature ab-Plane Reflectance
fl _. i1 TI f .l. r- z. 1 /ftit._I-._
* S S 4 5 4 4 I
* S 4 5 4 5
* S S 4 S 5
S S S 9 S SS S S C 5 59
Comparison of ab-Plane Reflectance:
c and q c
S S S 5 9
S S C 9 C S S S C S S S S S 59
ANISOTROPY IN THE AB-PLANE OPTICAL PROPERTIES
OF YBa2Cu30 7_
Room Temperature Spectra
Temperature Dependent Reflectance
Effect of the Chains
S.w S S 103
S S S S S S S S S S S S S S S SS S S S 10?7
ab-Plane Anisotropy in the London Penetration Depth
ANISOTROPY IN THE AB-PLANE OPTICAL PROPERTIES
C S S S C S C S S S S S S S S S S 111
Results of the Optical Reflectance
Room Temperature Spectra
Temperature Dependent Spectra
Discussion of Optical Constants
Temperature Dependent Optical Conductivity
* S S S S S C S S 113
S S S S S C S S S S 5 116
* S S S 125
S S S S S SS S S S S S S 12'7
tion ............. 134
iensate .. ...... 139
ab-Plane Anisotropy in the London Penetration Depth 141
Optical Conductivity and Symmetry of the Order Parameter 143
RESISTIVITY TENSOR OF Bi2Sr2CaCu20s
S S S S S S S S S S S C 9 S 5 5 C C S 14/7
Sample Preparation and Measurement Method
for Anisotropic Materials
U S C S S S S C C 5 5 5 152
Closer Look to the Transition Temperature
Results and Discussion
S C S. . *161
S C S S S S S S C 5 S C P 5 163
Review of Flux-Flow Resistance and Kosterlit
S C S C P C S C S C S C C 168
S S S C P S S S S C S C C S 5 5 5 170
TUDY OF BEDT-TTF(C104)2
S S 5 5 S S S S 5 S 173
S S 5 5 S C S C S C 5 5 186
S S S C S S P 5 5 5 5 5 5 S S S S S S S 5 5 200
p p p p p p p pp p p p p p 2141
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
ANISOTROPY IN THE INFRARED
AND TRANSPORT PROPERTIES OF HIGH
Manuel Alberto Quijada
David B. Tanner
Major Department: Physics
properties of the high-temperature superconductors are extremely
We have extensively studied superconducting high-quality single crystals
of YBa2Cu3O7- Bi2Sr2CaCu20
, and La2CuO4+6.
All these materials have CuO2
planes as the entities responsible for the metallic
behavior and superconductivity.
optical reflectance measurements were taken
both above and
superconducting state on a wide frequency range.
All these materials display interesting anisotropy in their optical properties. In
particular, optical investigations of the oxygen-doped La2CuO4+6 reveal the out-of-
plane (c axis) spectrum of this material is typical of an insulator with the optical
conductivity dominated by optical phonons.
In contrast, the ab-plane optical spec-
response is measured on the face of the crystal that has
the c axis parallel to the
propagation vector of the light.
the midinfrared in
the far infrared.
is also anisotropy in
anisotropy can be mostly attributed to the presence of CuO chains along the b axis.
One striking result is that in spite of the fact that Bi2Sr2CaCu20s does not have
the CuO chains,
we observed anisotropy between the a and b axes infrared conduc-
tivity of this compound as well.
The presence of this
conducting state suggests two possibilities.
superconducting order parameter.
A second explanati
anisotropy even in the super-
bility could be an anisotropic
on is that the overall conduc-
tivity is composed of a
simple Drude term combined with a more broad midinfrared
The observed higher absorption in the low-frequency region along the
by an anisotropic second midinfrared component in
The discovery of superconductivity in the copper oxides by Bednorz and Miiller1
in 1986 has revolutionized the field of condensed matter physics.
The importance of
this remarkable discovery can not be overstated.
On the one hand
, it offers promising
technological applications for materials that lose their resistance to the flow of electri-
cal current above liquid nitrogen temperatures.
On the other hand, many experiments
have provided ample evidence of the exciting new phenomena present in these mate-
Early measurements were designed to learn if the superconducting properties
of these materials could be explained in the context of the Bardeen-Cooper-Schrieffer
) theory2 for conventional superconductors.
Some of those initial results sup-
ed a BCS-like theory.
Among these, flux quantization
and the AC Josephson
effect4 show that the elementary charge in the superconducting state is
e. In addition, photoemission5-7
experiments suggest the presence
of a superconducting energy
the same time,
been an accumu-
nation of evidence for an
unconventional nature of the high-Tc materials.
the most important results that have emerged are high superconducting transition
linear dc resistivity in the normal state,12,13
and extremely small
Perhaps the second most striking property in these materials,
beside their high-Tc value, is the anisotropy in their physical properties.16-21
As soon as these materials were discovered, there began an intense effort to study
their optical properties.22-24
Soon, it was realized that the strong anisotropy that
means has provided some important results but at the same time has raised some
s well known, superconductivity in
these materials is
associated with the quasi-two dimensional CuO2 planes. Most optical studies related
to the anisotropy in these materials have concentrated in the anisotropy between the
directions perpendicular to (c axis) and parallel to the CuO2 planes. Research of the
anisotropy within the CuO2
(ab) planes has been studied to a lesser degree.
of the orthorhombic distortion that exists in these planes there are two important
anisotropy of the
d electronic structure in the normal state and (
) what if any is
the anisotropy of the superconducting order parameter?
ince the energy gap plays a central role in the BCS theory,
to observing this gap
by optical methods.30'31
One of the ad-
vantage of the optical methods compared
to, for example,
tunneling is that direct
electrical contact to the sample surface is not necessary.
This is especially important
since crystals and films may have dead layers near the surface that make it nearly
impossible for current to tunnel between an electrode and the superconductor.
optical experiment, by contrast, the probing radiation can penetrate a few thousand
A into the sample so the presence of dead layers becomes less of an issue.
which has been used with great success in the past
to study energy gaps
conventional superconductors, has also given valuable information in solids about lat-
tice vibrations, electron-phonon coupling, low-lying excitations, and electronic band
In the context of the BCS theory, the presence of a gap means that for
photon energies less than 2A, the bulk properties of the superconductor at T
show only an inductive part, with the real or absorptive part being zero. So, in order
quasiparticles in the sample.
At photon energies well above the gap,
behaves as if it was in the normal state.
This thesis is concerned with the subject of anisotropy in the optical properties
of the high-Tc materials.
The materials investigated are single-domain
C T = 9O K),
2Sr2CaCu2s8 (T7 =
85 K), and an oxygen-dop
gle crystal of La2CuO4+8 (Tc
There are two important issues that will be
considered. The first one is the anisotropy in the optical properties of the ab plane
vs. the c axis. A large surface area containing the c axis in the La2CuO4+s sam-
ple allowed the study of the c-axis polarization as well as the ab-plane response in
also allowed us to examine the ab-plane response in this material
when the propagation vector (q) of the light is parallel and perpendicular to the CuO2
The second issue is the anisotropy in the optical response along the two prin-
cipal axes in the ab plane, and how this is related to the crystal structure anisotropy
in single-domain crystals of Bi2Sr2CaCu20s and YBa2Cu307 _.
One important ob-
servation is the presence of a larger absorption in the superconducting-state optical
conductivity along the b axis below the frequency where a superconducting energy gap
should be observed.
Explanation for this larger absorption in the framework of the
models that have been proposed will be discussed in addition to its
tion with the symmetry of the superconducting order parameter in the copper-oxide
of this thesis is
presents a review of
experimental works that have been done related to the crystal structure and opti-
properties of the high-Tc superconductors.
brief theoretical background for
be presented in
to anisotropy between the ab-plane and c-axis optical measurements of La2CuO4+6
crystal are presented in Chapter V. The presentation of results and discussion related
to the measurements on
YBa2Cu3076- and Bi2Sr2CaCu2 O8
with emphasis of the anisotropy within the ab plane, is done in
VIII is for presentation of results regarding dc-transport
measurements of the resistivity tensor of Bi2Sr2CaCu20s samples.
IX contains concluding remarks.
This chapter will be
OF PREVIOUS EXPERIMENTAL WORK
devoted to a survey of previous experimental works related
to the physical properties of the copper-oxide superconductors.
two major sections.
It will be divided in
The first one is related to investigations of the crystal structures
of the three kind of materials studied in this study.
Information about the lattice
parameters that are relevant to an optical experiment will be
major section is a brief review of relevant previous work of the optical properties in
the copper-oxide superconductors.
Crystal Structure of Copper-Oxide Materials
Good knowledge of the crystallographic structure in single-crystal materials is
essential to the understanding of the optical properties.
case of the copper-oxide materials.
This is certainly true in the
These materials are considered by many as a good
example of a 2-dimensional system.
The reason for this point of view
s that the char
occur mainly in
copper atoms that are strongly bonded to oxygen atoms with an interatomic distance
of 1.9 A. This distance hardly changes from structure to structure. Above and below
there are other
layers of atoms that
are believed to provide the
carriers necessary for conductance in these planes, as well as to provide overall charge
The identity of these atoms depends on the system under study.
the tetragonal K2NiF4 structure was soon corroborated by Takagi
research indicated the nearly stoichiometric La2 CuO4 is an antiferromagnetic insula-
tor that can be hole doped by partially substituting the lanthanum site with some
of the alkaline earths Ca, Sr, or
Ba to produce superconducting materials with Tc
in the 30-40 K range.
, it was also realized that superconducting samples in
this system can be obtained by producing samples where the oxygen stoichiometry is
higher than four.33'34
Figure 1 shows the crystal structure of the undoped L
compound. In this nearly stoichiometric material, the oxidation
state of the individ-
ual species is La3+
, and 02 in order to have charge neutrality.
state of copper leaves this
atom with one unpaired electron in the d-shell; therefore,
the net spin at the copper site is 1/2.
Crystal structure of La2CuO4 material (
Each copper atom is coordinated
After Ref. 33).
to four oxygen atoms in
In addition, there are two more distant apical oxygens above and below the copper
those in the plane.
The Cu ion and its octahedral oxygen configuration would imply
a tetragonal I4/mmm space group in
However, neutron scattering
data for this compound indicate that
below a characteristic temperature there is a
lowering of the symmetry due to a slight tilt of the apical oxygens from their high
This tilting produces
a tetragonal to orthorhombic transformation
at a temperature that is close to 500 oC for the nearly stoichiometric material and
decreases with increasing doping, reaching
As mentioned above, superconductivity
calating additional oxygen atoms in the structure.
provide hole carriers in the Cu02 planes. The first
K at a Sr concentration of around
in this material can be achieved by inter-
This excess oxygen is thought to
report on the structure of oxygen-
enriched samples was done by Chaillout
They reported neutron diffraction
results on a sample that had
3% extra oxygen.
doping level the material will phase separate into a superconducting (oxygen-rich)
phase and a near
stoichiometric region, just as happens in La2NiO4+6.36
Based in a
two-phase model, the conclusion was reached that the superconducting phase has
group from the observation of peaks in the neutron scattering data that
are forbidden in Fmmm symmetry.
They created a model where one apical oxygen
is replaced by two oxygen sites that form a peroxide with a
short 0-0 bond distance
of 1.64 A.
Most recently, Radaelli et al.37 pointed out the need to perform such experiments
on samples with oxygen concentration well beyond the phase separation regime.
reported neutron diffraction data on three electrochemically-oxidized
samples: two ce-
ramics (6 = 0.08,
with Tc = 32 K, and S = 0.12,
with Tc = 42 K) and one single
After a carefully analysis of the neutron scattering data,
they concluded the basic
crystal structure of all samples corresponds to the Fmmm symmetry.
This was ob-
trained from the absence of small diffraction peaks that would be allowed in a Bmab
They argued the Bmab symmetry would be
by the presence
of interstitial oxygen in the
Following up on
ried out a refinement of the
structural parameters for the single-crystal data in the
In the analysis,
they relaxed the apical oxygen po-
sition slightly from the high-symmetry points and introdu
site located nearby and adjacent to a LaO layer (see Fig. 2
ced an additional oxygen
). The lattice parameters
obtained from analysis of the neutron scattering data on the single crystal at
are a b
= 5.34 A and c= 13.22 A.
Model for the crystal structure of La2CuO4+6 (
After Ref. 33).
Fig. 3. Phase diagram for YBa2Cu307_6 as a function oxygen concentration.
AF: antiferromagnet, SC: superconductor (after Ref. 38).
system, the parent compound YBa2Cu306 crystallizes in tetragonal form at high tem-
but converts by oxygen ordering to an orthorhombic form (YBa2Cu307_6)
on cooling down at room temperatures.
There is also a tetragonal to orthorhombic
transition driven by the oxygen content in the material at xc = 0.3,
which also deter-
mines the Tc in the sample as shown in the phase diagram of Fig.
the essential features of the crystal structure of the orthorhombi
been gathered based on x-ray and neutron diffraction data. In t
c phase that have
his material, each
copper atom in the plane is linked to four oxygens atoms at about 1.94 A, and a fifth
weakly bonded apical oxygen at 2.3 A as in the 2-1-4 compounds.
of the oxygen atoms in
the planes gives a perovskite-like structure with
There are two CuO2
planes per unit cell.
Unlike the tetragonal
the orthorhombic phase contains an additional CuO containing layer that
consists of a 1-dimensional network of CuO chains along the b
axis in the unit cell.
contributes 0.25 hole per copper for each of the two planes).
There is a difference
between the a- and b-axis
dimensions in the order of
with the b axis being the
larger on account of the CuO chains present along this direction.
The unit cell has
lattice dimensions of
c = 11.65 A. Effects of the chains
the superconducting properties of the material have been difficult to clarify because
the as-grown samples are usually twinned in the ab plane with alternating strip-like
domains of a- and b-axis oriented material.
Crystal structure of YBa2Cu3O7-6 material (after Ref. 39).
Special growing t
techniques 0 in recent years have pro
ded large enough single-
possible the study
More discussion about this will be given later when discussing sample preparation
The discovery of 20
r9 C U 0 r 2
up the race
u u v
aI' oi Il
Crystallographic structure of Bi2Sr2 CaCu2 08 sample (after Ref.
The structure of this material was quickly identified
as Bi2Sr2CaCu2Os or
The Bi-based compounds form a series of layered material where the CuO2
sheets of Bi202.
formula is (BiO)
where n is the number of consecutively stacked
It is found that Tc increases with the number ni, reaching a maximum
value of 110
when n = 3 and decreasing again as n increases further.
the ideal crystal structure for the case when
this structure the
to be separated
by sheets of Bi202
crystal lattice parameters of the orthorhombic unit cell are a = 5.41 A,b = 5.44 A,
and c = 30.78 A. In site of the relatively simple arrangement of the many atoms, a
is not well
also contributes to
to the Bi202 disorder,
the CuO2 sheets appear to be free of defects as determined
from x-ray and neutron scattering experiments.44
the oxygen content
varies for any fixed cation composition, affecting the transition temperature in the
In general, Tc increases as oxygen is removed.
It is still unclear however
how doping of the CuO2 planes occurs in this structure.
It is generally assumed that
doping of the CuO2
planes in this material comes from
oxygen in the B
and neutron scattering46
experiments have shown the existence of
a superlattice modulation or distortion that resides mostly on the Bi202 layers.
is incommensurate13'47'48 with the b-axis unit cell parameter given above (period
There have been many attempt
s to explain this superlattice by modeling defects
that are observed in the structure of this compound.41'48-51
In one of those models,
the superstructure is claimed48 to be the result of the addition of one in ten oxygen
atoms in the Bi202
layers, that would also cause a displacement of the surrounding
cation of this or any model must await further work.
Review of Optical Properties Copper-Oxide Materials
This section is be devoted to a review of the work related to the status of the
infrared response in the copper-oxide superconductors.
properties in these material
Investigation of the optical
s has been very vigorous since their discovery in 1986.23'24
In spite of this, understanding of the infrared properties in these materials remain a
As mentioned before
the feature common to these materials is
Since the optical
properties of these materials are very
different for directions parallel and perpendicular to the CuO2 planes and even show
substantial anisotropy within these planes, the availability of ab-plane oriented films
(c axis) mainly on La2-. Sra Cu04 single crystals.
The second section will touch the
subject of the midinfrared absorption
off as soon as a few
introduced in the CuO2 planes of these materials. In the third section, the subject of
the anisotropy within the ab plane will be reviewed in the context of the work done
with single-domain crystals of YBa2Cu307_- and B
As a result of the layered nature of the materials,
the in-plane and out-of-plane
optical properties of the copper-oxide superconductors show a remarkable anisotropy.
became evident as soon as the first polarized studies were reported on single
crystals of these compounds.2
In one of the earlier reports,54
of insulating La2CuO4 and of
8% Sr-doped samples were compared for polarizations
parallel and perpendicular to the CuO2 planes.
The results showed that the c-axis
was virtually unchanged suggesting that doping
affect the c-axis response in the system.
The spectra were typical of an insulating
material with phonon-like structures due to infrared-active modes (A2u symmetry).
In contrast, the refle
chance parallel to the ab planes showed substantial differences. In
the undoped sample, the reflectance is dominated by the presence of infrared allowed
the reflectance approaching a constant value at low
frequencies indicative of insulating
planes as well.
other hand, the doped sample showed a rising reflectivity at low frequencies with the
phonons visibility greatly reduced due to screening by the free carriers in the material.
This rising reflectivity implies a metallic conductivity
n the CuO2 planes of this
material, although at this concentration the material is not yet superconducting.
became clear from this study that the anisotropy between the ab-plane and the c-axis
A more systematic study of the c-axis and ab-plane optical response as function
of Sr doping in La2-_.SrzCu04
was done by Uchida et al.28
Let us first review the
Figure 6 displays the
reflectivity for three representative doping levels:
suits confirm28 the previous observation about the unchanged character of the
reflectance between the insulating and superconducting phase.
The electronic back-
ground appears to be very weak from the almost flat slope of the reflectance in the
The spectrum is dominated by two major optical phonons.
Only in the
overdoped nonsuperconducting samples do there appear to be
qualitative changes in
the reflectance. In this case, the spectrum appears to have a free carrier component,
although the highest phonon is not completely screened out.
This is consistent with
an observed metallic behavior in the resistivity as a function of temperature in the c
axis at this concentration.
In this context, the authors regard the overdoped samples
as an example of anisotropic 3-dimensional metals.
Photon Energy (eV)
I I C I4
This semiconducting-like behavior for the c-axis conductivity has also been doc-
umented in other cuprate such as
YBa2Cu3O_6 and Bi2Sr2CaCu20s.25'29,57
case YBa2Cu3 07-_ the results show some variation for various oxygen concentrations.
It appears that samples with the highest oxygen content show a nearly metallic be-
havior along the c axis,25'29 signaling that coupling between the CuO2 planes of this
compound is somewhat stronger.
The temperature dependence of the c-axis optical
properties has also been investigated in superconducting samples of La2-zSrzCu04
The results of the reflectance measured at
three doping levels are shown in Fig.
the far-infrared reflectance
a constant value at
low frequencies for the lowest
while it shows a slight negative slope for higher doping indicative of some
weak dc transport.
Tc, a sharp edge develops in the reflectance that signals
the onset of superconductivity.
In the analysis, the authors find that this edge does
not scale with Tc and it is not due to a superconducting gap excitation, but rather to a
plasma-edge-like feature associated with the superconducting-state carriers. A direct
consequence of this is the appearance of coherent transport of charge across the CuO2
planes below Tc. Similar results have also been obtained in the superconducting state
c-axis response of YBagCu3O -6 single crystals.
Midinfrared Absorption in the CuO9 Planes
The optical spectrum of undoped parent compounds of the copper-oxide supercon-
ductors such as La2Cu04 and Nd2Cu04 shows that these materials are charge transfer
insulators, with a band gap energy in the range 1.7-2.0 eV (14,000-16,000 cm-1), de-
pending on the material. It is generally accepted that this energy corresponds to tran-
sitions in the CuO2 planes between occupied 0 2p levels and the lowest unoccupied
100 150 200 250
Frequency ( cm' )
7. Temperature dependence in c-axis reflectance of La2- SrzCuO4
function of doping (after Ref. 58).
few carriers are introduced in these planes, there is an appearance
midinfrared24 that are peaked in an energy range from 0.1 eV to 0.5 eV (
the material under study.
excitations are thought
the result of photon assisted transition of bound holes (electrons) from their
ground state to
excited states and the continuum.
Due to the peak energy of one
of these midinfrared modes being approximately equal to the antiferromagnetic
J (~ 0.12eV), this band has been attributable to a magnetic origin,
the hopping of a charge from site to site involving the flipping of the nearby
spins.62 In other works, these
excitations have been relat
ed to the polaronic binding
energy of an impurity to the lattice.63'64
conductivity in La2-xSr3Cu04 samples as a function of Sr concentration by Uchida
Doping of the CuO2
planes in this system can be achieved by partially re-
placing the lanthanum sites La3+ by strontium Sr2+
Depending on the doping level,
the properties of the samples change from insulating (x
x < 0.20)
to nonsuperconducting metallic (x
The insulating com-
pound La2Cu04 shows the charge transfer (CT) band at
V (16,000 cm
-1) and no
absorption below this ener
For the lightly doped (x = 0.02) sample, there is clear
of the maximum that develops in the midinfrared conductivity at 0.5
followed by a reduction of the spectral weight above the CT band.
This transfer of
spectral weight from energies above the CT band is regarded as evidence of the itin-
erant nature of the states near the Fermi surface of these materials.
appears to shift to lower energies as a function of doping.
at 0.14 eV (1100 cm
For x = 0.10, it shows up
1); while for x = 0.15, it appears as a shoulder at a even lower
Other materials such as
and Nd2CuO4 show similar evolution
of the infrared conductivity as doping proceeds in these systems.24
Photoinduced absorption, a technique that induces doping in the insulating ma-
trials by shining light onto them to excite photo carriers,
has also shown the pres-
ence of this midinfrared absorption
in several samples.
La2-zSr1CuO4 and Nd2Cu04 by Kim et al.66
The samples studied are
, and in La2Cu04,
Tl2Ba2Cal -Gd Cu208 by Foster et al.67
The optimally-doped samples do not show a discernible maximum in the midin-
But even in this case, the conductivity (rl(w)) decays more slowly than the
typical Drude-type dependence w
Evidence for this non-Drude response in crl(w)
has been provided for nearly all the copper-oxide-based superconductors.23'24'65'68-70
1 2 3 4
as a function
La2-z Sr, CuO4(after Ref. 28).
up in optimally-doped samples.
In the first place, there is a region (800-4000 cm
where this absorption shows very little temperature dependence.
This happens at the
same time the de conductivity in the material changes by a factor of three between
300 and 100 K. Clearly, a Drude formula with a single relaxation rate for the charge
carriers will not account for such a behavior.
there is a definite temper-
ature dependence in the low frequency conductivity that is in good agreement with
the measured dc value.65168,69,71'72
The two most commonly mentioned models for interpretation of this midinfrared
absorption have taken two rather divergent approaches.
In one approach,
only one type of carrier which is responsible for both the dc transport and the con-
scattering rate and an effective mass enhancement at low frequencies.
are expected to form a superconducting condensate below Tc.
The second approach is called the two-component model.
infrared conductivity in the cuprate is the result of two types
first type of carriers,
In this approach,
of charge carriers.
which are considered to be Drude-like in nature, are responsible
for the dc conductivity in the normal state and form the superfluid density below Tc.
The second component, which is formed by bound carriers, has a semiconducting-like
In this model, the Drude component is expected to have a scattering rate
independent in frequency and linear in temperature.
More discussion about
this will be done in Chapter III. The following subsection will address the issue of the
anisotropy of the midinfrared absorption of single-domain crystals of YBa2Cu3O7_6
Anisotropy in the ab Plane
s one of the most studied high-Tc systems.
At the same time,
most of the optical studies
in this material have been done on samples that show
twinning in the ab plane.
Therefore, these measurements only show an average
of the ab-plane optical properties.
Since the presence of the CuO chains along the b
axis is likely to provide substantial conductivity in the midinfrared, polarized optical
measurements of single-domain crystals have become highly desirable.
recent developments40'73 in making large enough twin-free samples have made possible
optical studies along the two principal axes in the ab plane of this material.26'31'74'75
The first room temperature measurements done on a wide frequency scale were per-
for linearly polarized light parallel and perpendicular to the chain direction.
frequencies, the reflectance for Ra and Rb are nearly equal. At higher frequencies, Ra
falls off faster than Rb, reaching the plasmon minimum at a lower frequency.
plasma edge minimum, which signals the zero crossing of e1(w), occurs at 1 eV for the
while it is at 1.5 eV for the b direction.
This shows a splitting of the plasma
edge minimum observed in twinned samples.65
At approximately 3 eV
for higher ener
A Kramers-Kronig analysis of this reflectance reveals
a spectral weight in the midinfrared that is roughly a factor of
larger along the b
et al. interpret the a-axis conductivity
as intrinsic to the Cu02
To obtain the chain conductivity, they subtract the a-axis conductivity from the the
total b-axis conductivity.
In this analysis, the chain conductivity is then modeled by
a broad Drude-like peak with strongly damped carriers.
As pointed out by Koch et
al. the additive nature of the conductivities is violated at higher frequencies (wc
YB a2!Cu O 6+,
as a function of oxygen doping
The doping levels they studied
were x ~
1 (Tc = 90 K)
0.6 (Tc = 66 K), and x 0.1 (insulating).
trum of the insulating (x
_ 0.1) phase shows the charge transfer band at w ~ 2.0
This energy has been associated with a charge transfer across the O 2p and the
Cu 3d levels.28'76'77
Upon doping, the strength ofthis charge transfer transition di-
The reduction is accompany by an enhancement of the conductivity in the
midinfrared in both the a and b axes.
Moreover, there is a lifting in the degeneracy
of the peak at 4.1
that is present in the tetragonal insulating phase.
, Ra = Rb and
Schlesinger et al.31 reported the temperature dependence in the a- and b-axis re-
flectance of mechanically detwinned single-domain crystals (Tc = 90 K). The normal
state data show qualitatively similar results as described above for samples of similar
Tc. In the superconducting state, the authors reported what appears to be 100% re-
flectivity (within 0.5% uncertainty) for the a-axis polarization below an energy of 500
On the other hand, the b-axis reflectance is 2-3% lower in the same energy
This apparent threshold in the a-axis reflectance is interpreted as the BCS
superconducting energy gap (2A
8kBgTc) in the CuO2 planes.
The extra absorp-
tion observed in the b direction is argued as resulting from the stronger midinfrared
absorption due to the presence of the chain excitations in this direction.
dependence of this
threshold in the a-axis reflectance has also been investigated by
The samples investigated are oxygen reduced single-domain crystals
of YBa2Cu307_- with Tc of 56,
and 93 K.
The results show the structure at 500
does not shift neither with doping nor with a change in temperature for all
three doping levels studied.
The claim that this energy corresponds to a superconducting energy gap is now
considered rather questionable. Most recent direct absorption (bolometric) measure-
show a finite an fairly large conductivity below this energy
for polarization along the a-axis direction in 90-K single-domain crystals.
that the accuracy in these measurements is rather high compared to typical uncer-
tainties in reflectance experiments (0.02% vs. 0.5%) makes the results for the a-axis
conductivity shown in Fig. 9 more representative of the real
their short coherence length (
the high-Tc superconductors are considered
In all the analyses of the conductivity, 68'71'72
the mean free path
YBa2Cu307 _6 in the a direction.
The weak peaks are calculated phonon
No evidence of a
superconducting gap is seen in the spec-
trum (after Ref. 30).
pointed out by Kamaras et al.6S
Since the width of the free carrier band is smaller
that of the superconducting gap,
observation of this gap
by optical methods
because most of the spectral weight of the free carrier part has
condensed into a 6-function at zero frequency, leaving a negligible amount of weight
for transition across the gap.
Notice this argument agrees with a superconducting
large compared to the scattering rate of the Drude component.
have been attempts79
to produce dirty enough samples, but still superconducting,
that would make (
so the gap might be observable using opt
far, these experiments have not been successful.
.1~ LI J
of the unpolarized results. Figure 10 displays results of the average reflectance in the
ab plane of this system on a wide frequency range for three samples with different
doping levels.80 Similar results have been reported by other groups.81-83 The low fre-
quency reflectance is characterized by the absence of strong phonon lines. At higher
frequencies, the reflectance falls off in a quasilinear fashion reaching the plasma edge
minimum at around 10,000 cm
The results at high frequency are char-
acterized by a couple of interband transitions.
The first peak,
which is centered at
-1 (2 eV), is attributed to the charge transfer band between the occupied
O 2p levels and the lowest unoccupied Cu 3d orbitals in the CuO2
interband peak, which appears at 30,000 cm
eV), is found to show some sam-
pie to sample variation and has been interpreted as an interband transition occurring
mostly in the Bi20O
As mentioned previously,
this material has no chains.
In spite of this,
there is an orthorhombic distortion of the ab plane resulting from
an incommensurate superlattice modulation presents along the b axis of the material.
There have been only a few studies reporting on the anisotropy of the ab-plane optical
properties of this system.571
85,86 The first of those reports, by Kelly et al.,
of ellipsometric measurements that showed a strong anisotropy in the near-infrared
region and higher frequencies.
the peak at 30,000 cm
(3.8 eV) is
found to be sharper and stronger along the modulation direction.
The temperature dependence in the far-infrared optical conductivity has been ob-
tained from a Kramers-Kronig analysis of the unpolarized reflectance84 and transmit-
studies of free standing single crystals.
The micaceous nature of the Bi-based
material has made possible the preparation of very thin flakes (1000 A) that could be
used for transmittance studies without having to worry about substrates.
Room temperature reflectance of B
2Sr2CaCu20g samples with dif-
ferent oxygen doping (after Ref.
At the same time, the temperature dependence of oa(w) is less pronounce
in the midinfrared.
This is consistent with the non-Drude behavior observed in the
of the cuprate.
the superconducting state,
there is a broad
eV) followed by some
weak phonon structures and
a notch-like minimum at w~-400 cm
sotropy in the midinfrared region were re]
transmittance studies showing
ported by Romero et al.86 The
s quite substantial in spite of the fact the difference in the a- and
b-axis dimensions of the pseudotetragonal unit cell is only 3%.
The transmittance in
the midinfrared is found to be lower for the b direction (more absorbing) than the a
Models for Carriers in the CuO9 Planes:
Superconductivity in the copper-oxide materials arises through
carriers by hole or electron doping the nearly square CuO2 planes. I
properties of these doped planes are very unusual, and so far, the'
fully accounted for.
the addition of
y have not been
The property that is most frequently mentioned as indicative of
unusual normal-state behavior is the linear temperature dependence of the resistivity
This behavior is not the case in, for example a Fermi liquid description, where
T2 is what is expected due to electron-electron scattering.
Other unusual prop-
erties are a temperature dependent Hall coefficient, proximity of superconductivity
to a magnetic phase,
and a very short coherence length.
It is widely believed that
understanding of the normal-state properties will eventually provide knowledge of
the pairing mechanism in the cuprate.
Most theoretical models for the normal-state
properties start with the so-called three-band Hubbard model.
Next sections discuss
the underlying issues of this and other models and the implications for the description
of electron dynamics in the 2-dimensional CuO2 planes.
Three-Band Hubbard Model
The unusual properties of the copper-oxide superconductors prompted the need
to construct a Hamiltonian that properly describes the motion of carriers in the CuO2
planes of these materials. People working in the field soon realized that electron corre-
system leaves one hole (or one electron) per unit cell.
If one neglects interactions,
this material would be expected to be metallic with a half-filled conduction band. In
reality, this material is an antiferromagnetic insulator. Band structure calculations88
showed the available states for this hole could be in either one of the Cu 3d or O 2p
(2p 2py) orbitals.
As mentioned before, each Cu atom in the structure is surrounded
by an octahedron of
six O atoms.
(This number changes for different materials:
YBa2Cu3O7_6 and 4 for Nd2CuO4.)
configuration removes the degeneracy
between the 3d orbitals of the Cu atom.
mainly a 3dz2_y2 character.
It turns out the highest energy level has
Therefore, the hole would reside mostly on the Cu site,
giving this atom a net spin of 1/2.
the material can
a model with localized spin-1/2 states.
would seem to explain both the insula
An antiparallel arrangement of these spins
ting and antiferromagnetic properties of the
as indeed is
A low-lying optical excitation
transfer of one hole from the O 2p level to the upper Hubbard band (Cu 3d).
gap for thi
s excitation is denoted by
A in Fig.
This has been corroborate
optical studies of La2CuO4 and other insulating parent compounds,
where a charge
transfer excitation has been observed in the optical conductivity that is peaked at
around 1.7-2.0 eV.
Then, the next question to ask is how to construct a Hamiltonian that includes
the motion of additional holes introduced by doping?
An answer to this has been
provided in a
dimensional tight-binding model by Emery
and Varma et
al.91 The basic feature of this model is the introduction of an hybridization parameter
between the Cu 3d and
0 2p orbitals.
Other parameters that are included
account for all possible interactions are site energies Ed and Ep,
Coulomb energies Ud
0 2p ---UHB
11. Energetic position of the three bands in the three-band Hubbard
Hamiltonian can be written explicitly
= tpdp(di + h.c.)
p(p' + h'.c.) + edZ
The first term is the hybridization or hopping between nearest neighbors on Cu and 0
The pi are Fermnionic operators that destroy holes at the 0 site labeled j
the di correspond to annihilation operators at the Cu site i. Also, (i,j) refers to pairs
of nearest neighbors on i (Cu) and j (0) sites. A term for direct 0-0 hopping
included for completeness.
Notice also that only near-neighbor interactions are taken
Tntera.ctions a.t larrper ditfiane..r a.r th onrht to he screened bv a. finite
additional holes produced by doping will go mostly into the O
There have been band structure calculations9" and most recently cluster
calculations94 that have placed estimates for the paramet
ers in Eq.
2 3.6 eV, Ud
=8 -11 eV
, Up 4 eV; the remaining terms (tpd,
tpp) are all in the range
It is clear from the value of Ud the appropriate limit
for the physics of the high temperature superconductors is the intermediate to the
strong coupling limit.
Simplified picture of the three-band Hubbard model where the Cu
3d and 0 2p levels are hybridized to form the lower and upper Hubbard
bands (LHB and UHB respectively).
t J Model
that the correct effective theory might be reduced to a one-band Hubbard model. In
the Hamiltonian is defined as
where the dt are fermionic operators that create holes at site
i with spin
hybridization of the Cu 3dz2_,y2
orbitals is denoted
by the parameter t.
The parameter U is the on-site Coulomb repulsion.
Figure 12 shows a schematic in
this case where the three bands of the three-band Hubbard model are combined into
two bands labeled as the lower Hubbard band (LHB) and the upper Hubbard band
the previous model
into the so called t -
was first derived from
the Hubbard model
canonical transformations by Hirsch95 and Gros et al.96 In the context of the high-Tc
problem, the model was introduced by Anderson94 and derived by Zhang and Rice97
by canonical transformation from the three-band Hubbard model.
is a one-band Hubbard model, where the state of the doped hole is only represented
by the spin of the Cu site
on which it resides,
i.e., spin up or down if there is no
hole, or the absence of spin if there is one hole at any particular site.
Out of possible
triplet or singlet mixing states, the latter was found to have the lowest energy for the
hybridization of the Cu and doped-hole wavefunctions.97
H = J (S,.Sj -inj)
The Hamiltonian is
S+ [,cC +h.c],
where J is the antiferromagnetic coupling between nearest neighbors (ij) similar to
1t-_ -- ...__ TT 1 TT *..
ml C 1. i T 1 P .
where the limit of validity is for
The Si are spin-1/2 operators
and ct create electrons of spin a on site i. Hence, electrons move in a 2-dimensional
lattice with hopping amplitude constrained such that there are not doubly occupied
lattice with hopping amplitude t, constrained such that there are not doubly occupied
There have been some controversies in whether the three-band Hubbard and
the reduced t -
J models will lead to the same low-energy physics on a temperature
e in the order of Tc.
There have been some cluster calculations on a small number
of atoms that have addressed this issue.
In one of those studies.92
a cluster of the
form Cu5 016 is used with the full three-band Hubbard Hamiltonian and parameters
determined a priory.
number of spi
For the stoichiometric case, it is found the eigenvalues of the
to the corresponding ones of the Heisenberg Hamiltonian for a finite
is taken as giving some support to the model, although more
work is clearly needed in the area.
Models for al (w)
The one-band Hubbard and t -
J models have been the starting points in many
calculations for studying the dynamics of carriers moving in the Cu02 planes.
include the response of those carriers under the influence of an electromagnetic field.
Results pertaining to the anomalous midinfrared absorption observed in al(w) for
nearly all copper-oxide superconductors were discussed in Chapter II.
Here, there will
be a summary of the theoretical studies related to obtaining oa(w) from numerical
solutions to the one-band Hubbard and the t J models outlined above.
It is generally accepted the one-band Hubbard model is one of the simplest models
which may contain the essential features of the CuO2 planes.
The problem that exists
This is the reason why numerical solutions of finite cluster calculations have been
found useful to obtain approximate solutions to the problem.98s-100
The numerical method commonly used to determine the ground state of the clus-
ters is the Lan
This technique consists
in providing an initial guess
for the ground state
o0) of the system.
The next step is to apply the chosen Hamil-
tonian to this ground state
to obtain a second state
states corresponding to the Hilbert space of the cluster
under study can be constructed in this way to give a matrix of coefficients an and bn
that are defined by
These definitions assure the states that form the basis of the Hilbert space are or-
to each other.
The matrix obtained from these coefficients can be later
diagonalized using standard methods.
The conductivity tensor in linear response theory is obtained from the relationship
between the current density operator, j z(q, w)
and the electric field vector, E,(q,w),
in the limit of q ~ 0.
Hence, it can be
where azx the absorptive part of the optical conductivity at zero temperature might
be written as
-- ---xo) ,
the Hamiltonian whose energy is E0, w is the frequency, and e is a small
number that moves the poles of the Green's
function into the complex plane. In the
, the current operator j, in the x direction at zero momentum can be
z = itZ (c ,ca+x,a
As discussed previously, the one-band Hubbard Hamiltonian contains three charac-
teristic energies that are expected to give interesting optical excitations; the hopping
, the on-site repulsion term U, and the exchange interaction J, given by Eq.
strong coupling limit.
Dagotto et al.100 reported numerical solutions for the
Hubbard model on a
x 4 cluster,
a hole concentration in
x < 0.375.
At half filling,
the results show accumulation of weight in
acr(w) above an energy which is close to 6t. If previous estimates of t are taken,92 this
energy can then be correlated with the corresponding charge transfer gap of excita-
observed in the optical conductivity of the insulating compounds around
0 eV. If such correlation is made, the weight in al(w) is the result of charge
excitations from the lower to
the upper one-band Hubbard model.
Figure 13 also
shows the results at dopin
levels away from half filling.
It is evident in the
the redistribution of spectral weight from this charge transfer gap to lower energies as
function of dopi:
2 holes in the 4
In particular, the doping level of x = 0.125, which corresponds to
x 4 cluster, shows two major features that occur below the CT band.
weight that occurs half way between zero and the CT band has been associated by
Dagotto et al. and others98-100 as the midinfrared band that shows up in the optical
conductivity spectrum of the cuprate.
The results also show that,
doping, the Drude peak grows considerably,
while the midinfrared band only shows
a modest increase.
Dagotto et al.100 pointed out the result
s shown in Fig. 13 would
correspond to the intermediate coupling regime, i.e., U
~ 8t. For large coupling, a
s between the CT excitation and the midinfrared band, whereas for small
values of U/t both
excitations merge, making it difficult to separate them.100
ilar calculations in the context of the t -
Horsch99 for different values of J. The res
ones shown in Fig.
C.-X. Chen and H.-E
J model have been done by Stephan and
ults show a qualitative agreement with the
The Drude and the midinfrared peaks are clearly evident in
similar results in the near-half filling case were also obtained by
. Shilttler from solution of the one-band Hubbard model in the
strong coupling limit.102
From the previous discussion, it is quite clear the numerical results obtained from
the one-band Hubbard and t -
J models favor the approach followed by many re-
searchers regarding the interpretation of the optical conductivity al(w) obtained in
experiments involving the optical properties of the copper-oxide superconductors. In
the infrared conductivity in these materials is considered
to be the
combination of a Drude-like free-carrier component at w = 0,
with a strongly tem-
perature dependent scattering rate, combined with much broader bound excitations
at higher frequencies.
the free carriers track the temperature de-
pendence of the dc resistivity above Tc,
while condensing into the superfluid below
5 10 15
Fig. 13. Optical conductivity obtained from solution of the one-band Hubbard
The extra component in the infrared has also been argued to be the result of direct
transitions from valence band states close to the Fermi level into empty states of the
In the absence of a clear physical origin for these
excitations, the natural choice has been to model those absorptions by Drude-Lorentz
oscillators. Hence, the dielectric function e(w) is fitted to an equation of the form
w2 + iwl/r
j=1 i wry1
, the first
the Drude component describe
frequency upD and scattering rate 1/r; the second term is a sum of midinfrared and
oscillators with wyj,
being the resonant frequency,
strength, and the width of the jth Lorentz oscillator respectively.
The last term,
is the high frequency limit of
which includes higher interband transitions.
An alternative approach that has been proposed to explain the anomalous non-
Drude behavior in the infrared conductivity of the copper-oxide materials is to model
al(w) using a generalized Drude model with a frequency dependent scattering rate
In this model
, there is only one type of charge carriers.
dielectric function can be written as105
w [W- C(W)I
where Wp is the bare plasma frequency for the charge carriers in the far and midin-
defined by 4rNe2/mb, and oo is a constant that includes contributions from
the Kramers-Kronig relations,
they must obey causality.
this causality condition, E(w) is taken to be complex, i.e., E(w) =
Hence, the model requires the introduction of a modified functional
form for both the effective mass and the scattering rate.
and imaginary part of E(w), we arrive at
m*(w) c1 (=)
By decomposing the real
1/r*(w) = -E2(w) m
where now 1/7*(w) is called the renormalizedd" scattering rate, mb is the band mass,
and m* is the frequency dependent effect
Two models that provide a phe-
nomenological justification for this approach are the "marginal Fermi liquid" (MFL)
theory of Varma et al.106'107
the MFL model
and the "nested Fermi liquid" (NFL) theory of Virosztek
For example, the imaginary part of the one-particle self-energy in
is written as
where A is a dimensionless coupling constant.
Hence, for w < T the model predicts
a renormalized scattering rate that is linear in temperature,
which is expected from
the linear temperature dependence in the resistivity that is observed in nearly all
As w increases, reaching a magnitude of order of T
or higher, a new spectrum of excitations arises.
This causes 1/r*(w) to grow linearly
with frequency up to a cutoff frequency wc that is introduced in the model.
effective mass enhancement at low frequencies.
This enhancement is expected to be
to the coupling constant A.
the model requires the presence
of an energy gap that opens up at the Fermi surface as the material enters in the
superconducting state. This gap should show up in the spectrum of a i(w). As it will
be discussed later, observation of this gap by optical means is still an open question.
When data obtained by different groups72'75'87
using different high-Tc materials
are analyzed in the context of this model, there seems to be a qualitative agreement
with the predictions of the model.
In first place, the dc resistivity obtained from the
model agrees with the experiment.
-Im YE(w) is nearly constant for cw
linearly with cw for cw
there is an
enhancement at low frequencies that is larger at low temperatures.
One important argument against the model is that the cutoff frequency deter-
mined by the agreement between the data and
the model is rather low,
i. e.,wc is
This is in conflict with the high
cutoff frequency suggested
In addition, the coupling constant
determined for samples with lower Tc is actually larger when compared with the value
obtained with higher Tc samples.75 The problem arises because the Tc is supposed to
be determined by
Superconducting State Models
Experimental evidence for the unconventional nature of the normal-state
ties in the copper-oxide superconductors has been established without a doubt.
the same time, there is no unambiguous evidence that the properties in the super-
conducting state show anomalous behavior as well.
It was established early on that
pairing of electrons (holes) was indeed present in the superconducting state of these
questions that one can ask about the nature of this pairing.
The first one is what
is the force that media
es the attractive interactions?
In ordinary superconductors,
the pairing is mediated by phonons interacting with free carriers in the material.2
The second question is whether the pairing is accompanied by an energy gap that
opens up on the Fermi surface as happens in conventional superconductors.110 Evi-
dence against the conventional nature of this pairing could then be, for example, the
presence of nodes or states within this gap.
While answer to the pairing mechanism must await the development of a success-
ful theory to explain superconductivity in the high-Tc materials, experiments could,
in principle, provide answer to the presence of a gap.
In fact, many experimental
techniques, such as tunneling, infrared spectroscopy, photoemission and penetration
depth measurements, have tried to demonstrate the existence of the superconducting
gap and its
Unfortunately, there have been contradictions in some of the
The following section will review the many ideas, both theoretical and
perimental, that have been discussed related to the possible symmetry of the pairing
in the superconducting state of the copper-oxide materials.
Symmetry of the Order Parameter
Superconductivity in general is regarded as evidence of a broken symmetry.
transformation involves a change that allows the description of the system, in the
modynamic limit, by a macroscopic wavefunction and a phase. In group-theoretical
calculations, the symmetry of the superconducting state corresponds to one of the
irreducible representations of the total symmetry group of the normal state.111'112
The possible broken symmetries considered are spin and point group rotations as well
as global gauge symmetries.
the high-Tc materials, such as
YBa2 Cu3O -
of this distortion may lead to different conclusions regarding the possible symmetry
classifications of the order parameter in the superconducting state. In addition,
in BCS theory superconductivity results from pairing of electrons, the coupling of the
electron spins could lead to singlet (
0) or triplet (
1) states for the spin part
of the pairing wavefunctions.
In the following discussion spin-orbit coupling is not considered for brevity.
the assumed crystal symmetry in the normal state will be the one of the lattice,"13
although others have been proposed in the literature.114
For example, if a tetrago-
nal point group symmetry is considered for the crystal, the possible superconducting
the irreducible representations of
SO(3) is the group rotation in spin space and D4h is the point
group symmetry of the lattice.
In a singlet state with tetragonal point group sym-
and 2-dimensional irreducible representations corresponding to
the D4h group in the system.
Based on experimental grounds, the two most quoted
possibilities for the high-Tc are the Alg and Big.
The first one
, the Alg, is the only
singlet state with a gap function that is nonzero everywhere on the Fermi surface.
This symmetry corresponds to the so-called s-wave symmetry in the BCS theory for
The gap function has the form
up to a
function with the symmetry of the lattice.
The second possibility that has arisen in
some theoretical models,115 as well as from some recent experimental results5 116 that
will be discussed later, corresponds to the B1g point group symmetry.
station gives a gap function that can represented with the d 2_v2 orbital and that is
On a spherical Fermi surface, this gap
function would have nodes at 450
angles with respect to the lobe maxima.
point group and has s-wave symmetry.
The rests have the Big, B2g, and B3g
having single-state gap symmetries of the following d-
wave like orbitals:
dzy, cdz, and dyz respectively.
It should be pointed out that the
will not be realized in this case.
There are many other possibilities that include triplet state spin wave functions
in both orthorhombic and tetragonal point group symmetries.
The p-wave states, for
example, are important in superfluid 3He.
The interested reader is referred
literature for more details.113
Evidence for Proposed Pairing States
In the context of the t J model, there have been numerical results on finite clus-
ters that suggest an attractive channel for binding of two holes in an antiferromagnetic
background with dz2_y2
The results show the average distance
the two holes decreases as the ratio J/t increases.
this provides evidence that the strong coupling limit is a necessary condition for the
binding of holes
n the t-
J model and that it eventually leads to phase separation in
the system, zi.e.,
the material separates in hole-rich and hole-poor regions.115"119-121
Dagotto et al.'5 extended the calculations to include higher doping in a 4
as a function of the ratio J/t.
x 4 cluster
The numerical results also suggest in this case a signal
for superconductivity that is stronger in the dz2 y2 c
pairing correlation function shows a maximum at J/t
For this channel,
Of course, these results
are not enough evidence to prove a condensate in the bulk of the system. Finite size
effect studies should be carried out before a definite conclusion could be reached. On
the other hand, finite
cluster calculations in the three- and one-band Hubbard mod-
els have not given evidence for binding of holes in the d,_ y2 channel, although the
Let us turn our attention to the experimental data.
The electromagnetic London
penetration length AL(T) is considered one of the basic lengths in superconductivity.
its temperature dependence can give information about
e in a superconductor.
Earlier ab-plane penetration depth measurements on
YBa2Cu3O7-s single crystals and films by Harshman et al.124'125 and Kruisin-Elbaum
et al.126 suggested the gap was nodeless, i.e., the low temperature behavior of AL(T)
thought to be exponential
as in an ordinary BCS superconductor with isotropic
Most recently, a reanalysis of these and more recent kinetic inductance data
have yielded a quadratic variation of
AL(T) from its
Most available results are best fitted129 by using the empirical
L (T) =
t = T/Tc.
Likewise, in more recent penetration depth experiments on single
crystals of YBa2Cu307_6
a stable microwave cavity,
very low temperature dependence of AL(T) is linear rather than quadratic in T.116
The results have
an order parameter that
d 2_y2 symmetry, since this symmetry would introduce a linear power-law variation
in AAL(T) of the form
where Amaz is the maximum gap value over the Fermi surface.
Morover, the tern-
perature dependence of the penetration depth in Bi2Sr2CaCu20s single crystals and
YBa2Cu307_ films have been done by Ma et al.130 In this study, it is of found that
oc T2 for T
Thus far, there has been little progress in
the understanding of the observed differences.
The problem that exists is that any
attempts to model the effect of impurities that could change the linear T
to T2 in less than ideal samples.131"132
It is shown in one of those calculations that
strong resonant scattering could account for the differences that are observed in the
temperature dependence of AAL (T) in films and single crystals that have an order pa-
rameter with a dz2_y2 symmetry.1m
Perhaps, other more subtle mechanisms such as
the CuO chains in YBa2Cu307_6 and the superlattice modulation in Bi2Sr2CaCu208
could have some effect in the electrodynamics response of these materials.
as the one
Shen et al.
performed angular resolved photoemission spectroscopy
(ARPES) measurements and found a condensate peak that is larger and more pro-
nounced along the symmetry
, i.e., from center of the Brillouin zone to
the x-direction in momentum space, and seems to vanish (within the experimental
resolution of 10 meV) 450
away from the previous direction (P
Based on this
observation, they conclude the symmetry of the order parameter is compatible with
_y2 symmetry pairing.?
3 first is that the gap i
This conclusion is strongly dependent on two assumptions.
o in the direction where it is undetectable in the ex-
The second is the point group symmetry of the crystal is tetragonal rather
than orthorhombic, i.e., a rotation of 900
or 1800 in the plane would give the same
magnitude of the gap with just a change of the phase.
it is not clear whether these checks were made. More
From the results presented,
lover, such data do not seem
incompatible with similar measurements done by Kelley et al.7 where another singlet
d-wave symmetry (dzz) is proposed as an allowed possibility.
This is concluded from
of the condensate peak observed in photoemission experiments
as function of photon polarization and photoelectron collection directions.
Finally, the authors concluded that the point group symmetries of both the normal
and superconducting states of this compound show a D2h instead of a D4h character.
Other probes of the pairing state in the high-Tc, such as tunneling, do not reveal
clear evidence of a gap,
but rather show substantial density of states in the range
where a gap might be expected.8'9
Determination of Gap by Optical Spectroscopy
As is well known, a superconductor is a good reflector of light for energies below
is the optical energy gap for quasiparticle excitations in the supercon-
This means that in a reflectance experiment a signature of the gap
will be when R = 1 for photon energies less than 2A.
At higher energies, since there
is enough energy to break a Cooper pair, the material will start absorbing light and
the reflectance will show deviation from unity.
A Kramers-Kronig analysis will then
show an optical conductivity Oai(w) that is zero for w =
and finite for w
This has indeed been confirmed in the experiment, where
predicted by the BC
is very close to the value
theory for most conventional superconductors.110
There have been attempts in many optical studies to associate features in the su-
reflectivity of the high-Tc materials with superconducting energy
the quantity 2A/kbTc reported in the literature have
for two reasons.
However, these claims must be taken with certain caution
In the first place, there is a stringent requirement that R must be
unity below the threshold for quasiparticle excitations.
With the typic
AR in the order 0.5%, it is not possible to rule out any small but finite absorption
when the reflectance is very close to 100%.
important in view of the fact that
such uncertainties in R will introduce errors in acr(w) in the order of
It is clear that
the S/N ratio in cl(w) diminishes as
R approaches one.
there have been reports of direct absorption measurements30
that show a nonzero
value for the a-axis absorptivity of single domain crystals of YBa2Cu307_6 down to
-1 or so.
Such energy falls well below the range of values of 2A/kbTc reported
in the literature.
To reconcile these results
, it has been proposed the extra absorp-
tion observed in al(w)
below Tc is part of a second component in the midinfrared
To explain the absence of a gap feature, it has
argued that since the high-Tc materials are in the clean limit,68 because of the short
coherence length and long mean free path, the rising reflectance in the far infrared is
not due to a superconducting gap but rather to a scattering rate that approaches zero
as the sample enters in the superconducting state.71
a Drude-term conductivity that condenses into a d
This picture is consistent with
elta function at zero frequency.
This leaves very little spectral weight for transitions across the gap that might still
be present in the superconducting state.
Furthermore, it has been argued that some
of these features might be associated with interaction involving longitudinal c-axis
phonons with the ab-plane bound carriers.135'136
An apparent justification for this is
due to the fact that some of these features are still present above Tc.
This issue will
be addressed in Chapter V with results obtained on a La2Cu04+6 single crystal.
the measurement of the
transmittance or the
reflectance of a sample as a function
of the incident light frequency.
is done over a
very wide frequency range,
the far infrared and
the ultraviolet (UV) regions of the optical spectrum, it is necessary to use different
combinations of spectrometers, light sources, and detectors.
the experimental techniques used in this work.
This chapter describes
The first section in the chapter includes
discussion of the Fourier transform spectroscopy technique that was used to cover the
spectral regions in the far and midinfrared.
There is also a discussion on the Perkin-
Elmer monochromator that was used to cover from the midinfrared up to the UV.
Descriptions of the kind of detectors, polari
zation control, sample mountain
analysis procedures will also be presented.
A final section will briefly describe the
preparation techniques of the high-Tc materials that were studied in this work.
Fourier Transform Infrared Spectroscopv
The far infrared
one of the less accessible spectral regions.
The reason for this
is the reduced available power from radiant sources at those low frequencies. For any
given source, the total blackbody power spectrum is given by
PO = aT4 A,
available from zero up to a frequency w can be estimated by using the Rayleigh-Jeans
p(w) = B Aw2
where kB and c are the Boltzmann's constant and speed of light respectively.
the ratio of the emitted to the total power up to frequency
w is given by
P (w) 5 hw 3
Po B4 k T
-1 and T
= 5000 K the fraction is T7 =
, i.e., if the total
power of the source is 1 W, only 1/ will be emitted for frequencies below 100 cm
power of the source is 1 W, only 1 pW will be emitted for frequencies below 100 cm
This energy deficiency
was overcome by the development of Fourier transform spec-
The principle of operation can be understood in terms of the Michelson
interferometer shown in Fig. 14.
j Fixed Mirror
Light coming from a source falls onto a partially transmitting beamsplitter.
half of the radiation will be transmitted to the movable mirror M2.
The other half is reflected onto the fixed mirror labeled M1.
and they recombine again at the beamsplitter.
Both beams are reflected
When M1 and M2 are equidistant with
respect to the beam-splitter position, the sum of the two beams will have maximum
M2 moves away from this maximum position, the two beams will be out
of phase by an amount 0
where v is the frequency of the incident light in
units of cm
-1 and S is the distance as measured away from the position of maximum
intensity or zero path difference.
If the two beams have equal amplitudes a(v),
sum of the amplitudes reaching the detector can be written
A() = a(v)(1 +e e6).
The intensity of the radiation reaching the detector
can then be
as a function of path difference
I(S, v) = AA*
S(u) is the spectral density of the source. For a source having frequencies from
v = 0to v
, the total integrated intensity, or the interferogram I(S),
obtain by integration of Eq. 21.
The result is
S(u) cos 2ruvSdv,
The first term in Eq.
is a constant equal to the total output intensity of the source
The spectrum itself can be found by computing the inverse Fourier transform of
the integral in Eq.
The importance of this is that
the information about
the spectrum is being observed continuously.
This advantage was first recognized by
1(8) is recorded over a finite range of path differences and with lim-
The discreet nature of I(6) changes the Fourier integral into a Fourier
series. In addition, the maximum finite path difference 6m introduces side-lobes near
sharp features of S(v).
This problem can be minimized by the method of apodiza-
The sampling interval of I(6) determines the maximum cutoff frequency
This will introduce aliasing for frequencies higher than (Vmax).
problem is sol
ved by introducing proper optical filters that will attenuate those higher
The instrument used to cover the far and midinfrared (30-4000 cm
-1) is a Bruker
fast-scan Fourier transform spectrometer.
The principle of operation is very
similar to the Michelson interferometer.
A schematic diagram of the spectrometer
is shown in Fig.
The instrument comes equipped with two
a Hg and
lamps for the far
nfrared and midinfrared respectively.
There are also de-
tectors for each of those spectral regions.
the interferometer area,
light from the source
is focused to a beamsplitter that sends the transmitted beam
to one mirror and the reflected light to other mirror facing the first one. Both beams
are sent to a two-sided movable mirror which reflects both beams back to be recom-
bined at the beam-splitter site.
The two interfering beams are then directed to the
the reflectance chamber, that allows reflectivity measurements with near-normal in-
The final destination of the light is the detector chamber.
The whole area of
the spectrometer is evacuated to avoid absorptions by water and CO2 present in the
air. Because the two-side mirror moves with constant speed v, the path difference 8
between the transmitted and reflected beams at the beamsplitter, is changed accord-
ing to the relation S = 4vt,
where t is the time as measured from the moment S
Hence, this moving mirror produces a modulation of the infrared signal,
in the form
D(t) = Do cos(2 rfat),
where D(t) is the signal as received by the detector and the infrared frequency Vo is
turned into an audio frequency in the formula fa = 4vvo.
The next stage is amplifica-
tion and digitizing of this signal before is sent to an Aspect computer for apodization,
phase correction, and finally application of the Fourier transform to the obtained in-
terferogram to finally get the spectrum.
Table 1 shows the parameters used in the
As was mentioned before, the intensity of the source blackbody spectrum becomes
rather weak in the far infrared.
This can be partially overcome by using detectors
that are sensitive enough to give an acceptable signal to noise ratio.
that most infrared detectors have is that the sensitivity is limited by the background
For example, if the detector is operated at room temperature,
the peak of
the blackbody spectrum of this background is centered at around 1000 cm-1
in the middle of the infrared.
This can be overcome by operating the detector at
Bruker FTIR Operating Parameters
Range Beam Split. Opt. Filt. Source Pol. Detect.
cm-1 Material Material Material
35 90 Mylar Black PE Hg arc 1 bolometer
80 400 Mylar Black PE Hg arc 1 bolometer
100 600 Mylar Black PE Hg arc 1 bolometer
450 4000 Germanium on KBr none Globar 2 DTGS, photocell
= polyethylene. Polarizer 1
= wire grid on oriented polyethylene; polarizer 2 = wire
grid on AgBr.
is shown in Fig.
The detector element is made out of Si.
by bolometric means,
i.e., by changing its temperature as the infrared light strikes
This change in temperature is amplified and recorded as a voltage signal,
is then digitized and sent to the Aspect computer.
The Perkin-Elmer Monochromator
At higher frequencies,
the Fellgett advantage losses its importance due to the
availability of brighter sources and more sensitive detectors.
grating spectrometer is an excellent choice to cover frequenci
For these reason
es in the near infrared
and up to the UV
The instrument used i
s a modified Perkin-
The diagram showing the details of the instrument is
Depending on the frequency of interest, there are three sources to chose
a globar for the midinfrared,
a tungsten lamp for the near infrared, and a
* but sflnmmf C-sR
* tlaI filWtr
4 Auterathc b-ur)ator etwigr
* TMan icbenuviw cgwq
* Cn-ad nores tear
I cwitis m.a.n.lw
3 Aslern. har
* Rene cate mflmul thurm
-a~ ~~~~ -
rr, Y m
---slits glo rb
formula 2d sin0 = nA,
where n is the nth order of the diffracted light, A is the wave
length, 0 is the angle of incidence, and d is the spacing between the grating lines.
angle of incidence is changed by rotating the grating with a
allows access to
exits the monochromator and it is focused onto the position labeled R in Fig.
The sample and reference are placed there for reflectance measurements.
of transmittance, the sample and reference (in this case an empty, or "blank,"
In the case
or transmitted light
a detector that
to a lock-in amplifier for
amplification of the signal.
The output of the lock-in is fed
to a digital voltimeter that is remotely controlled
by a PC computer that also controls the step-motor controllers of the grating and
The collection of the data is done through the computer by recording a single
beam spectrum (signal
vs. frequency) for the reference and the sample sequentially
and taking the ratio of these two spectra (S
transmission of the sample.
s/Sr) to obtain the true reflectance or
During normal operation, the spectrometer chamber is
evacuated down to 150 mTorr or so to prevent any absorption due to water or C02
present in the air. For more details about the operation of this machine the interested
reader is referred to Reference 142.
Table 2 lists the parameters used to cover each
The need to resolve the dielectric tensor along the principal axes of the single
crystals used in this work required us to polarize the electric field of the light. Since the
radiation generated at the source is randomly polarized, the polarization of the light
was accomplished by inserting a polarizer in the path of the beam. Tt
Perkin-Elmer Grating Monochromator Parameters
Frequency Grating a Slit width Sourceb Detector C
(cm-1) (line/mm) (micron)
a Note the grating line number per cm should be the same
order of the corresponding measured frequency range in cm1.
c TC: Thermocot
W: Tungsten lamp;
iple; PbS: Lead sulfi
D2: Deuterium lamp.
te; 576: Silicon photo-
the midinfrared spectral range (300-4000 cm
-1) a silver bromide substrate is used,
the substrate is polyethylene.
plastic polarizers were used in
the near infrared,
the desired polarization of the light
easily accomplished by mounting the polarizer in the path of the beam using a gear
mechanism that also allowed rotation from the outside without breaking the vacuum
in the spectrometers.
This in-situ adjustment of the polarizers greatly reduces the
uncertainty in the relative anisotropy of the reflectance (better than -0.25%).
specially important when the relative anisotropy in the optical response of the crystal
is not very large.
Diagram of frame used to mount samples.
Sample Mounting and Low Temperature Measurements
Due to the small size of the samples used in this study, special care had to be
advantage of the whole area, the mounting of the sample was done in the following
way. A copper frame was machined with a small hole at the center (c 2 mm diameter).
One side of the frame (the front side) had a conical surface as illustrated in Fig. 18.
This conical surface was necessary to scatter away any light that may hit the copper
frame when inserted in the beam path inside the spectrometer.
The next step was
to solder two very fine copper wires across the center of the hole on the back side
of the frame.
Then, the sample was carefully mounted on the two wires and glued
there by a good thermal conductor (apiezon grease).
A piece of Al-coated Si with
approximately the same area as the sample was mounted on another frame in the
same way as the sample in order to be used as a reference.
Mounting of the Bi2Sr2CaCu20s
and La2 CuO4+s samples did not require the
the use of wires due to the larger size of these samples.
Instead, each sample was
the back side of a frame, as shown in Fig.
smaller diameter than
the size of the sample.
with a hole of slightly
A frame with same hole-size as the
frame of the sample was used to mount a Al-coated piece of glass to be used as a
Finally, frames containing the sample and
the Al-coated reference were
out in opposite sides of a sample holder assembly that
positioned inside the spectrometer for reflectance measurements.
of the polarization dependence in
the reflectance, samples were mounted so
that the principal axes of the measured face could be studied by setting the polarizer
horizontal or vertical with respect to the spectrometer bench.
Low temperature measurements were possible by attaching the
sembly to the tip of a Hansen and Associates High-Tran cryostat.
A flexible transfer
line was used to flow liquid helium from a
storage tank to the cryostat.
element attached to the tip of the cryostat.
In this set-up, the temperature of the
sample could be lowered by increasing the flow of helium and increased by applying a
current to the heater element. During measurements, the sample holder and cryostat
units were placed inside a shroud equipped with optical windows in the spectrometer
The pressure inside of this shroud was kept below
Torr to prevent
the formation of ice on the cryostat or the surface of the sample.
Since sample and
mirror were on opposite sides of the sample holder, measurements of the sample and
mirror spectra were possible by simply rotating the cryostat assembly by 1800
final step was to take the ratio of these two spectra to obtain the reflectance of the
Normalization Procedure of the Reflectance
After measuring the temperature dependence in
the reflectance of the sample,
the final normalizing of the reflectance was obtained by taking a final room temper-
ature spectrum, coating the sample with a film 2000 A thick of Al, and remeasuring
this coated surface.
A properly normalized room-temperature reflectance was then
obtained after the reflectance of the uncoated sample was divided by the reflectance
of the coated surface and the ratio multiplied by the known reflectance of Al.
result was then used to correct the reflectance data measured at other temperatures
by comparing the two room-temperature spectra taken in the two separate runs.
procedure corrected for any misalignment between the sample and the mirror used
as a temporary reference before the sample was coated and more importantly, it pro-
vided a reference surface of the same size as the actual sample area.
In cases where
the sample surface had some roughness, the procedure also corrected for losses due
to a nonspecular sample reflectance.
Data Analysis of the Spectra:
The Kramers-Kronig Transformations
The power reflectance measured in the experiment is related to the amplitude
reflectivity which contains information about the optical absorption in the sample.
For normal incidence, the amplitude reflectivity is given by
- 1) +i
(n + 1) + is
where n and n are the real and imaginary parts in the complex refractive index of the
sample under consideration.
Information about these quantities can be obtained by
noting that the power reflectance R(w) is
related to the amplitude reflectance r(w)
in the following way:
r(w) = p(w) exp i(w),
) is the phase shift in the light upon being reflected from the sample and
) is related to the power reflectance R(w) by
R(w) = p2 ().
Since the amplitude reflectance and the phase shift are the real and imaginary parts of
a response function respectively, they can be related by means of the Kramers-Kronig
the phase shift can be obtained from the Kramers-
w /3: In R(w')
In principle, knowledge of the phase shift is only possible if the power reflectance is
known over an infinite range of frequencies.
R(ow) is only measured over a
w, and for the regions where R(w) is flat, there are negligible contributions
to the integral in Eq. 28.
This implies that for the frequencies where R(w) is not
available, it is possible to make extrapolations that would not affect very much the
results for the range for which R(w) is known.
The kind of extrapolations that can
be made depend on the type of material under consideration, as it will be discussed
in the next section.
High-Frequency and Low-Frequency Extrapolations
For metals and insulators the high frequency reflectance is usually
by interband transitions of the inner core electrons to excited
Only at very
high frequencies (above 100,000 cm
-1) the free-electron behavior becomes important.
In the absence of any published data that can be append to the existing data,
reflectance in the interband region is usually modeled using to the formula
>= Rf LA)
where Rf and wf are the reflectance and frequency of the last data point measured
in the experiment.
The exponent s is a number that can take up values between 0
At very high frequencies (wf,),
where the free-electron behavior sets in, the
approximation used is
R(w) = Rf (
It is still expected some dependence on the results for frequencies close to the last
frequency measured on account of the choice of s and Wff.
(for metals this happen
for frequencies above the plasma edge minimum.) At low frequencies, the scheme for
on the properties of the material under study. In the situations
relation, R(w) = 1 A ~J, where A is a constant determined by the reflectance of the
lowest frequency measured in the experiment.
For high-Tc samples, it is found this
procedure results inadequate and it can only be used as a first approximation. A more
appropriate procedure is to fit the reflectance using a Lorentz-Drude model where the
free-carrier response is assumed to have the the typical Drude form.
frequency excitations are modeled by Lorentz oscillators.
The fitted reflectance is
used as an extension below the lowest frequency that was measured in the experiment
before the Kramers-Kronig analysis is finally performed on the data.
the superconducting state, it is expected that
unity for frequencies close to zero.
the reflectance will approach
An empirical formula that has been found to rep-
resent the way R approaches unity is R =
where B is a constant determined
from the lowest frequency measured.
It should be pointed out that typical uncer-
in R(w) are in the order AR
Hence, the propagated error in, for
example, the optical conductivity al (w) obtained from the Kramers-Kronig analysis
of the reflectance is roughly
A01 1 AR
al 1 -R R
It is clear the RHS of Eq.
31 diverges, or the signal to noise ratio is very small as R
approaches unity. Implications of this in the high-Tc materials will be discussed later.
Once the proper extrapolations are made and
the phase shift is obtained,
optical constants of the material are easily obtained by means of the formulae that
The frequency dependent refractive index n and extinction coefficient n(w)
2 R(w) sin O(w)
1 + R(w))
These relationships can be re-written in terms of the dielectric function
where the real (e1(
w)) and imaginary (E2(w)) parts are obtained from
E2(w) = 2nn.
Other important relations are the real and imaginary parts of the optical conductivity
a(w), the skin depth 6(w), and the absorption coefficient c(w):
All these equations reflect the fact that the absorptive (real) and inductive (imagi-
nary) parts of a process are all related to each other due to the causality requirement
in the Kramers-Kronig relations.
This section will provide a brief discussion of the sample-preparation techniques
El= N2 = (, i.)2
Untwinned crystals of YBa2Cu307_- that were used in this study were prepared
at the University of Illinois in
The samples were grown using a standard Cu-O flux growth procedure.145
advantage of this method is that micro twinning develops in the crystals together with
the transition from the tetragonal phase at high temperatures to the orthorhombic
phase at room temperatures.
This twinning happens because of randomly oriented
mechanical stress present during the slow-cooling segment from temperatures around
850 OC.146 In order to avoid development of twinning the slow-cooling process was
interrupted by pulling
the sample out of the furnace.
Evidence for the quenching in
the tetragonal phase was obtained by looking isotropic extinctions on the ab plane
of the samples under a microscope with crossed polarizers in reflectance mode.147
The tetragonal crystals were then oxygenated during a post-growth procedure that
converted them to the orthorhombic phase having sharp superconducting transitions
K.40 Since the crystals were not subjected to any mechanical stress in the
process of converting them from tetragonal
there was a reduced
possibility of creating dislocations or defects that otherwise may affect the intrinsic
in the material.
Typical crystals with single-domain regions of 1 x
were obtained every third or fourth successful attempt.
The dimension along the
was approximately 25 pm. Determination of the a and b
in all the crystals
was done by the sample grower before the samples were sent to this author.
as determined b
y cooling the sample in a
field of 10 G,
around 90 K. Fig. 19 shows the Meissner fraction measured for a
crystal grown with this technique for the applied field parallel to the a and b
(Meissner effect) is larger when the field is applied along b axis than when it is applied
along the a axis.
This indicates that flux pinning is larger when the field is along the
86 88 90 92
YBa2 Cu3 O7- 6(After Ref.
A qualitative analyst
of the data suggests
that the criti
cal current density (Jc) is
larger perpendicular to the chain direction than parallel to it (Jca
the ab-plane anisotropy in the dc resistivity,16 performed on crystals from the
batch have given some interesting results regarding the effect of the chains in the dc
These results, which are shown in Fig.
indicate the ab-plane
anisotropy ratio in the dc resistivity is around
with the lower resistivity being
the b axis.
This may suggest,
to a first approximation,
chains provide an additional electronic channel for conductivity along this direction,
assuming that the chain conductivity can be additive to condu
activity of the Cu02
100 125 150 175 200 225 250
20. Resistivity anisotropy in single-domain
(After Ref. 16).
crystals of YBa2Cu3O7_s
, there is indication of anisotropy in the Raman-active phonon lines for
the two polarizations in the ab plane.
Bi Sr CaCu 0s Single-Domain Crystals
The Bi2Sr2CaCu20O crystals used in the study were grown by R. Kelley and M.
Onellion at the University of Wisconsin in Madison,
The technique used is
a flux method with slow cooling in a temperature gradient.152 In a typical experiment,
the starting materials, Bi203, SrCO3, CaC03, and CuO are ground and placed in an
The mixture is then heated to a temperature of 50-70 OC above
the liquidus temperature and equilibrated for 6 hrs.
The temperature is subsequently
75-880 OC, and after reaching
equilibrium for 6 hrs, the temperature is
slowly cooled at 0.5-2 C/h to 820 OC where the experiment is terminated.
are subsequently annealed in dry oxygen at 600 OC for
h an later reannealed in
Argon at 750 OC for a period of 12 hrs.
low electron energy diffraction (LEED) techniques.
The incomemsurate superlattice
modulation pattern was seen along the b axis and not in the perpendicular direction
(a axis), suggesting the samples were single-domain crystals.
The alignment of the
principal axes in the crystal was confirmed by observing the extinction points when
e sample was rotated
under a microscope (Olympus, model BHM) with
Meissner effect measurements were performed on one of the samples to
determine the superconducting state transition temperature.
shown in Fi
21, reveal the onset of superconductivity is around 86 K. Moreover,
the onset to the superconducting state, as determine by using four-probe resistance
is around 91
with a transition
Results of the
resistivity tensor along the a, b, and
on these crystals are reported in Chapter
Meissner effect measurements on a Bi2Sr2CaCu2Os single crystal.
The orientation of the magnetic field is H i c axis.
La9Cu04+A Single Crystal
on an oxygen-doped
a2CuO4+5 single crystal prepared at the University of Iowa by F.C.
Chou and D.C.
A stoichiometric single crystal of La2(
The as-grown crystal was insulating.
was prepared using a self-flux
The oxidation procedure was carried
out using a electrochemical cell151 with the La2Cu04 sample as one of the working
electrodes. A platinum wire working
as the negative electrode was attached to one side
of the crystal using silver paint and the contact was fully covered with silicone rubber.
The set-up of the electrochemical cell was La2CuO4/ 1N NaOH / Pt.
of the cell
done by applying an anodic current of 10 pA to the La2Cu04 crystal
for a period of two months. In order to optimized the oxidation current, an constant
electrical potential of 0.6 V (versus a Ag-AgC1 reference electrode ) was maintained
during the charging
using a potentiostat.
The exact oxygen content of this
particular sample after the oxidation was complete
ed is not known.
on the gained weight could not be used because small pieces of the crystal were lost
during handling. Meissner effect data, which are shown in Fig.
indicate the onset
of superconductivity is at around 40 K. A comparison of this with the onset of other
samples of known oxygen concentration suggests the excess oxygen should be around
The sample that was provide to us contained two optically-smooth faces.
On face contained the ab plane and the c axis, so with the help of a polarizer, it was
possible to study the anisotropy between these two directions.
The other face that
studied only contained the twinned ab-plane direction.
I Cu I
, I I i I I' 1
r 1- I
I11I I It .J
*I I I T1 II
I f 1 1
L!I I I
Meissner effect measurements on L
a9CuOAlS single crystal (After
r r I r-r
1 r-r 1 1
I.I~ t I
OPTICAL STUDY OF La2CuO4+6 SINGLE CRYSTAL
In this chapter, the optical properties of the ab plane and the c axis of supercon-
ducting La2CuO4+4 are described. The ma
state by electrochemical insertion of oxygen.
trial is transformed from its insulating
The sample was prepared at Iowa State
University by F.C. Chou and D. Johnston in collaboration with S-W
AT&T Bell Laboratories.
Although the exact oxygen content is not known in this
sample, it is estimated to be 6 0.11 from comparison of the observed Tc of 40 K
with other samples of known concentration.152
Optical absorption studies of the stoichiometric parent compound La2CuO4 have
revealed this is a charge transfer insulator having marked anisotropy in both phonons
and electronic features for polarization of the light parallel and perpendicular to the
CuO2 planes in the system.52153'691153-155
Most of the optical investigations of super-
conducting materials derived from this parent compound have been on the Sr-doped
A review of some of these studies
was done in Chapter
III. In this system, a maximum Tc in the order of 35 K is obtained by substituting
15% of the La atoms by Sr. In addition, superconductivity is also obtained when ad-
ditional oxygen is intercalated in the crystal structure of La2 CuO4. Normally, oxygen
intercalation is obtained by annealing the sample at high temperatures (~ 500 oC)
in an oxygen-rich environment.33,34
In spite of its success, there are two major draw-
backs in producing superconducting samples using this technique. In first place, there
is the requirement of extremely high oxygen partial pressures (
kbar) in order to
uniformly oxidized samples could be one of the reasons why reports on
investigation of oxygen-doped samples of this material have been limited to lightly
doped nonsuperconducting samples.64,69'154
Recent developments in electrochemical
techniques have made possible the synthesis of uniformly oxidized samples with rel-
atively high oxygen content (6
~ 0.12) and transition temperatures near 40 K.152'158
Details of this technique were given in Chapter IV (p.
Sketch of the sample with the two faces that were used to measured
the reflectance on the LasCuO4+e single crystal.
Results of reflectance studies of this oxygen-doped sample were obtained from
two faces of the crystal.
One of the faces
(the face labeled I in Fi
both the c-axis and either the a- or b-axis
direction (on account of twinning, we were
unable to distinguish).
I allowed us,
the use of linearly polarized light,
to probe the optical response of the
as well as the ab-plane response.
Face I I
E E aab
below the superconducting transition temperature.
As in the undoped material, the
spectrum along the c direction is mainly dominated by optical phonons and no evi-
dence of metallic component is found in the optical conductivity.
At the same time,
the ab-plane response shows metallic reflectance in the far infrared and a plasma edge
around w ~ 7500 cm
The second face that was measured,
the face II in Fig.
, provided an aver-
age of the ab-plane response.
Optical reflectance was measured in a frequency range
that extended from the far-infrared and near-ultraviolet spectral regions (80-38,000
Temperature dependence measurements, above and below Tc,
were also car-
ried out in the same range of frequencies. A Kramers-Kronig analysis of the ab-plane
reflectance reveals the unusual non-Drude behavior in the midinfrared conductivity
al (w) that is typical of the copper-oxide superconductors. For frequencies in the near
infrared and above, a rather unusual temperature dependence was observed in the op-
tical reflectance of the sample. In addition to the expected sharpening of the plasma
, the reflectance became gradually lower as
the temperature of
the sample was lowered.
This result may suggest a temperature dependence in some
high-frequency interband transitions.
Moreover, the reflectance displays some structures in the far-infrared region that
are not present when the ab-plane reflectance is measured on face I of the crystal. A
comparison of the sum rule in both cases reveals that the differences in both spectra
are not likely due to different oxygen compositions in both surfaces of the sample.
, the differences are most likely due to electron-phonon interactions that are
enhanced when the wavevector of the incident light is parallel to the c axis in the
Reflectance of La2 CuO4+6 single crystal for light polarized along the
ane and the c axis (T = 300 K).
Reflectance of La9 CuOal
Room Temperature Spectra
24 shows the room
temperature reflectance for
light linearly polarized
parallel and perpendicular to the c axis of the sample.
The results show a dramatic
anisotropy for the in-plane and out-of-plane optical properties, just as it has been
r-doped superconducting samples derived from the stoichiometric parent
frequencies, the spectrum is almost featureless, showing only a broad electronic-like
feature whose maximum is around w
The spectrum is almost un-
changed with respect to the c-axis spectrum of the undoped material. However,
is a weak structure that appears just below the peak of the phonon mode at 512 cm .
This structure, which is not present in the stoichiometric material,52'54 is resolved as
ond phonon mode as the temperature of the sample is lowered. More discussion
about this will be given later.
Kramers-Kronig analysis of the reflectance shown in Fig. 24.
completely screened out by the free carriers in the CuO2
planes are also visible in
the far infrared.
The real part of the conductivity a<7(w),
obtained from a Kramers-
Kronig analysis of the reflectance, is shown in Fig. 25.
the electronic-like features are more easily seen. The I
Here, the phonon modes and
results presented here indicate
gen doping in the lanthanum cuprate only affects the electronic excitations
related to the copper-oxide planes in the system.
Similar conclusions regarding the
two dimensionality of the electronic properties in the copper-oxide superconductors
have also been drawn from measurements on YBa2Cu307_. and Bi2Sr2CaCu2Os ma-
Low Temperature c-Axis Reflectance
Figure 26 displays the reflectance in the
nfrared region along the
as a function of temperature. As the sample is cooled down, the phonon lines become
sharper, as expected.
The structure that appears in the room temperature spectrum
just below 512 cm
1 is more clearly resolved as a phonon mode at 492 cm
not present in the c-axis spectrum of either undoped or Sr-doped samples.52'54
At the low-frequency end, we notice all spectra above Tc approach a constant value
for the reflectance.
This is indicative of semiconducting behavior.
On the other hand,
the inset in Fig. 26 shows the data at the lowest temperature show a downward trend
towards low frequencies that is not present in the data above Tc.
This trend is most
likely correlated with the appearance of a plasmon minimum in the superconducting-
state reflectance as
first observed in the c-axis spectrum of La2
Tamasaku et al.58
In this study, it was found the minimum forms part of a reflectance
edge that is related to the formation of a superfluid condensate that provides coherent
400 600 800
Temperature dependence in
c-axis reflectance of La2 Cu04+
of this condensate to the dielectric function can be written
where wp, represents the oscillator strength of the superconducting condensate and
iO+ is the scattering rate that tends to zero as the the mean free path becomes infinite
the reflectance edge associated with this condensate moves toward lower frequencies
as the doping level in the sample is decreased.
In order to fit this edge in our data,
we used a condensate oscillator strength with wps
value with the results presented by Tamasaku et al.
indicates that the effective oxygen doping is
~ 85 cm
A comparison of this
for one of their Sr-doped samples
S 0.11), and that the reflectance edge
should appear around w ~-
-1 in the present sample.
Due to the small sample
, the lowest frequency measured in the present experiment was around 33 cm
This would explain why the edge was not observed in the present experiment.
Assignment c-Axis Phonons
Let us turn our attention to a quantitative analysis of the phonon modes
in Fig. 26.
In order to perform a Kramers-Kronig analysis,
frequency end was kept constant,
as is customary for insul
the reflectance at the low-
ators. On the other hand,
the positive slope of R(w) below
a low-frequency extension. The
Tc at low frequencies, required us to use Eq. 38 as
procedure followed for high-frequency extrapolation
n Chapter IV.
The results of the Kramers-Kronig analysis performed
on the data are shown in Fig. 27.
panel of this figure shows the optical
while the bottom panel shows Im(-1/e) both at several temperatures.
by a total
infrared-active modes centered at 230, 340, 492,
and 512 cm
A comparison of these
frequencies with other optical studies52'54'69 of the c-axis spectrum in undoped and Sr-
doped samples of La2CuO4 indicates good agreement with the first two phonon modes.
However, in those studies only one mode is observed at ~ 501 cm
analyses16'162 indicate that modes of ionic displacements (q = 0) along the c axis
the nearly tetragonal
structure of La2CuO4+6
will have the
cal conductivity (upper panel) and loss function (bottom panel)
along the c axis of La2 Cu04.+ at several temperatures.
The peaks in these
The eigenvector for the latter mode at 491 cm
-1 involves in-phase vibrations
of the apical oxygens above and below with respect to the four oxygen atoms in the
Based on this,
the present data suggest that
the presence of two modes at
-1 and 512 cm
-1 is most likely associated with the incorporation of additional
oxygen atoms in
These additional oxygens,
which are located in or
between the LaO
layers, may provide two slightly different force constants between
the apical oxygens and the Cu02 layers.
A quantitative analysis of the intensity and linewidth for each phonon mode as
a function of temperature can be done by modeling the reflectance using a dielectric
function model consisting of four Lorentz oscillators plus a core dielectric constant
Eoo to account for contributions at higher frequency.
The formula is
where each term in the sum corresponds to an optical phonon
with Wpj, cyj, and 7j
being the intensity,
center frequency and
damping of each mode respectively.
shows the temperature dependence of
obtained from a fit
to the reflectance
temperature using Eq. 39.
The bottom panel shows the
corresponding oscillator strength (wpj) for the jth phonon mode.
The results shown
indicate that most of the temperature dependence in
the result of a reduction in 7y
sample is decreased.
(increase in the lifetime) as the temperature of the
This is what should be expected, since at low temperatures, the
thermal motion of the atoms in the structure will be
This diminishes the
chances of scattering among the atomic vibrations in the crystal.
The results show
Temperature dependence of the phonon
parameters showing the
linewidth y: (inner na.nel a.nd t h nncilla.tnr strepn h r :. (h nttnm na.neli
not show any significant temperature variation in the measured range; only the mode
at 340 cm
s some systematic hardening as the temperature is lowered.
center frequency for this mode goes from 340 cm
-1 at room temperature to 347 cm-1
The intensities (wpj) in each phonon line are related to the effective charge that
is carried by each
considered is that each
To establish this relationship, the first thing that should be
phonon mode splits into transverse (TO) and longitudinal
ionic vibrations in
TO frequencies involve transverse
vibrations of the atoms,
are obtained directly from
the peak position in
, the real part of the optical conductivity ai (w).
On the other hand, the
LO frequencies involve long range changes in the dipole moment along the direction
of propagation in the crystal.
Information about the center frequencies of these LO
oscillations can be obtained, in principle, from the peak positions in the loss function
Im(-1/e) as shown in the bottom panel of Fig. 27
. A direct determination of the LO-
possible by noting that this splitting is related to the oscillator
the phonon mode in
the context of the Lyddane-Sachs-Teller relation.
The formula is written as
= j 2 2 o)1
pj(WLO, W TOJ)00.
Table 3 displays the
a Lorentz fit
to the room
reflectance along with the TO and the LO frequencies derived from Eq. 40.
Parameters of a Lorentz fit for the mea-
Oscillator# wTOj WLOj wipj y
(cm-1) (cm-1) (cm-1) (cm-1)
1 230 498 1052 20
2 342 343 65 15
3 492 502 218 24
4 512 515 174 23
Eoo == 5.3
has been applied successfully to other systems with ionic character.162
[we LOj TOj
where j is the sum over all LO-TO splitting,
V is the volume of the unit cell and
k is the sum over all ions with mass mk and effective charge
must obey charge neutrality:
'k. Since the crystal
In general, Eq. 41 can not be solved unless
the number of unknown parameters Zk is less than or equal to two.
In the case of
La2CuO4+s, since oxygen is much lighter compared to
the other atoms,
hand side of Eq. 41 will be dominated, in first approximation, by the term related to
Thus by neglecting all but the oxygen contributions, the result of
will yield the effective charge of oxygen averaged over all sites.
oxygen in the order of
The slightly higher value obtained here (14%) is most
likely due to the insertion of additional oxygen in the structure.
These results differ
from the nominal effective charge of two expected for the oxygen in the structure and
they indicate the high degree of covalency of the bonds in the structure.
Result of the reflectance for polarization of the electric field parallel to the CuO2
is shown in Fig. 24. As mentioned above, there is a marked contrast for polarization
of the light parallel and perpendicular to CuO2 planes.
The ab-plane spectrum
behavior with optical phonon modes at low frequencies that are not
completely screened out
by the free carriers.
At higher frequencies,
we observe the
plasma edge minimum at
This energy is not much different than the
one observed in Sr-doped
For frequencies above the
plasmon minimum, we see the usual charge transfer (CT) peak at
followed by higher energy interband transitions.
Assignment ab-Plane Phonons
In view of the fact that the crystal
with only a weak orthorhombic distortion
structure in La2CuO4 is almost tetragonal,
n that occurs at low temperatures, all phonon
lines can be classified under the D17 point group symmetry.
Therefore, the irreducible
representation of the vibrations that involve in-plane atomic displacements will
respond to the Eu symmetry. Hence, we should expect four infrared-active modes in
the in-plane spectrum.
In our oxygen-doped sample, we observe a total of six major
phonon-like features at 80, 140,
230, 355, 484,
and 680 cm
and assignment of the ab-plane phonon modes of La2 CuO4 indicate that only infrared-
in the present sample suggests that the assumption of a tetragonal symmetry is only
good as a first approximation.
Nonetheless, the close correspondence of three mea-
sured frequencies in our sample with the assigned modes in La2CuO4
conclude they indeed correspond to the Ez symmetry.
allows us to
The low-frequency mode at
-1 corresponds to bending vibrations of the out-of-plane atoms (apex oxygens)
against the Cu02 planes.154
The remaining two modes at 355 cm
-1 and 680 cm-1
are related to bending and stretching vibrations of the Cu-O bonds respectively.
This leaves us with the question for the assignment of the three remaining modes.
Various Raman measurements consistently show Raman-active modes at ~
and ~ 445 cm
-1 in the undoped material.160'163-165 Moreover, a mode at ~ 90 cm-
been seen in Sr-doped samples
using inelastic neutron scattering166
Based on a tetragonal structure for the unit cell,
indicate that the mode at
-1 is double degen-
erate with even vibrations that correspond to the Eg and Aig symmetries.
, the frequency at -445 is related to the A19 irreducible representation.
The assignment of the mode at ~ 90 cm
-1 is regarded as belonging to the Eg sym-
Hence, if a correlation is made between the additional infrared modes in
this oxygen-doped sample and the Raman-active modes in undoped samples, it can be
argued that doping in the material lowers the crystal symmetry making even (gerade)
vibrations in the unit cell become infrared actives.
One explanation for this could be
that the Alg mode may couple to electronic excited states of Eu symmetry making
the former infrared allowed modes. Similar observations have been made by Shimada
et al.69 where modes at
, ~230, and ~ 460 cm
-1 have been observed to grow with
Sr doping in La2z
SrzCuO4 samples. A second explanation for the mode at 483 cm
Measured temperature dependence in the ab-plane reflectance (face
II) of La2CuO48+ sample.
The reflectance at three selected tem-
peratures in the near infrared and the visible.
see later, is enhanced by interactions of the electronic background with the c-axis
LO phonon modes in the sample.
Low Temperature ab-Plane Reflectance
Figure 29 shows the temperature dependence in the ab-plane infrared reflectance
measured on face II of the sample.
the far infrared,
we observe an increase
crystal lattice is also more clear at
low temperatures where weak splitting of the
phonon modes at
and 680 cm
modes at ~ 182 cm
1 and ~ 296 cm
1 are more easily resolved at low temperatures.
These infrared-active modes are visible in the spectrum since they are not completely
screened by the electronic background.
The inset in Fi
near-infrared and visible.
9 shows the reflectance at three selected temperatures in the
We observe a marked temperature dependence of the re-
there is a sharpening and steepening of the
plasma edge minimum at
as the temperature of the sample is reduced.
At the same time, the reflectance is reduced (sample becomes more transparent) for
the visible and above.
A similar temperature dependence has
observed in the ab-plane reflectance of La2-zSraCuO4 thin film72 and lightly oxygen-
doped La2CuO4+ single crystals.64
In contrast to the result of the Sr-doped thin
where a rather abrupt change
s observed in the reflectance at 250
K and no
change below this temperature,72
the temperature dependence here is more gradual.
There are two things that could be said about this temperature dependence.
first place, the fact that the reflectance is decreasing at lower temperatures implies
there should be a strong temperature dependence in the opposite direction in some
interband transition at a higher frequency.
This must be the case in order to satisfy the
ature dependence could be related to the structural phase transition (tetragonal to
orthorhombic) that occurs at low temperatures in the lanthanum-based superconduc-
Such transition is known to be produced by a small staggered tilt of the apical
oxygens in the Cu06 octahedron.
There are two effects associated with this.
rnC *h pm ic a r+bnnreltnn' S +b' nn- rail n +bn n,4r nitnh nrY nb!0ao
rp1,P ccrrnn ;~ 2C
0.6 0.9 1.2
Temperature dependence of sum rule on a wide frequency range to
show transfer of oscillator strength from low to high frequencies.
Results of ab-Plane Optical Constants
This section is devoted to present results of the temperature dependence in the
ab-plane optical properties of La2 CuO4+6 obtained from a Kramers-Kronig analysis
of the reflectance.
The usual requirement of the Kramers-Kronig integrals to extend
Drude model and using the fitted results to extent the reflectance below the lowest
frequency measured in
the experiment as explained in
high-frequency extrapolation of the room temperature results was done by merging
which only extend up to 38,000 cm
with published results on
Sr-doped samples by Tajima et al.,
168 which extend up to 40 eV. The range beyond
40 eV was extended with a power law
which is the free-electron behavior
Since there is a temperature dependence in the reflectance all the way up to
, as the temperature of the sample is reduced, the scaling factor used in
appending the data from Tajimas et al. to our low-temperature results was rescaled
upward in the range 10-15 eV and this was joined with the unchanged data above 15
eV in order to make the sum rule results equal in the range of frequencies above 15
eV. The results of the sum rule, shown in Fig. 30, suggest that what this procedure
does is to transfer some of the spectral weight that is missing in the range where we
measured a lower reflectance to higher frequencies.
This might suggest some strong
temperature dependence that is occurring in some interband electronic transition at
much higher than
procedure did not make a
gnificant effect on the results of the
Kramers-Kronig analysis below 10,000 cm
-1 (1.2 eV).
Here the imaginary part of -1/e is plotted vs. frequency at several temper-
In ordinary metals, the peak position of this
function gives the longitudinal
for plasma excitations of the charge carriers.
this peak is related to the lifetime of these excitations.
As will be discussed later,
absorption in the midinfrared electronic background is the broad width (- 0.4 eV)
that is observed in the present results and in nearly all copper-oxide superconduc-
Estimates for the screened plasma w,
eM Wp /
can also be deduced
from the position of this peak.
5000 10000 15000
Temperature dependence in the loss function.
the present sample
This frequency is nearly the same as
in other Sr-doped samples with similar doping level.28'72
The intensity of this peak
while the position
does not show much
Fig. 32. Real part of the optical conductivity al(w) at several temperatures
in the far and midinfrared.
doped samples, becomes more resolved and the center position shifts slightly towards
lower energies at lower temperatures.
ab-Plane Optical Conductivity
The results of the temperature dependence in the
) are shown in Fig.
These results were obtained from face II of the sample
As mentioned before, any structure that may be present in the reflectance is more
easily resolved in the spectrum of al(w).
We observe that,
as the temperature of the
later. Moreover, the electronic far-infrared background has a strong temperature de-
pendence with a steep enhancement as the sample is cooled down.
This is followed
by a more weaker temperature dependence at frequencies in the midinfrared.
is a reduction in the midinfrared conductivity when cooling the
temperature down to 200 K
sample from room
with not further appreciable change below this temper-
This is connected with the temperature dependence of the reflectance in the
near infrared and visible that is observed in the inset of Fig. 29.
this could be
As mentioned earlier,
due to the phase transition from tetragonal to orthorhombic that occurs
in the material at lower temperatures.
Similar results were reported on a Sr-doped
film by Gao e
above 300 K.
Gao et al. found no further temperature dependence was present
Similar check was not done in the present sample since oxygen could be
driven out of the structure at temperatures above 300 K.
discussed in Chapter IV, there are two ways to analyze the optical condu
the high-Tc superconductors.
The two approaches are the one-
Since the present data strongly suggest
the presence of two or
more components to Ocrl(c),
we performed an analysis based on a decomposition of the
conductivity in two parts, a Drude or free-carrier part and a midinfrared contribution.
Discussion of the results in this analysis is done in next section.
The results of
after subtracting the phonons
free-carrier contribution at zero frequency, are shown in Fig.
33 at several temper-
The scattering rate of the free-carrier or Drude contribution shows a linear
temperature dependence as shown in Fig. 34, v
perature independent with a magnitude of wpD)
vhile the oscillator strength
~ 5800 + 100 cm
A fit of the
500 1000 1500 2000
Midinfrared conductivity of La2 CuO4$+ at several temperatures ob-
trained after subtracting the Drude-like contribution from the data shown
in Fig. 32.
overall oscillator strength of the midinfrared conductivity appears to decrease
temperature of the sample is decreased from room temperature down to 200 K. A
less noticeable change is seen below this temperature.
This temperature dependence
the decrease that is seen in the near-infrared reflectance shown
in the inset of Fig. 29.
Hence, it appears that some spectral weight is being trans-
ferred to higher frequencies,
which is also affecting ao(w
) in the midinfrared.
We also observe that a bump-like structure,
which at 300
K appears at
about 200 K
, diminishes in intensity with a reduction in temperature.
, a second and sharper feature reappears at a lower energy,
growing again in intensity as the temperature of the sample is further reduced. Simi-
lar midinfrared-like modes in the optical conductivity of both lightly doped64'169 and
superconducting samples have also
been reported in
present results underscore the generality of these features in the optical properties of
the copper-oxide-based materials.
100 150 200
The scattering rate of the Drude-like part from a two-component fit to
the optical conductivity shown in Fig. 35 (Notice the linear temperature