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Optical properties of doped cuprates and related materials

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Optical properties of doped cuprates and related materials
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Yoon, Young-Duck
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viii, 174 leaves : ill. ; 29 cm.

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Charge transfer ( jstor )
Conductivity ( jstor )
Doping ( jstor )
Electrons ( jstor )
Phonons ( jstor )
Reflectance ( jstor )
Spectral bands ( jstor )
Spectral reflectance ( jstor )
Superconductors ( jstor )
Temperature dependence ( jstor )
Copper oxide superconductors ( lcsh )
Dissertations, Academic -- Physics -- UF
Physics thesis, Ph. D
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non-fiction ( marcgt )

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Thesis:
Thesis (Ph. D.)--University of Florida, 1995.
Bibliography:
Includes bibliographical references (leaves 164-173).
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Typescript.
General Note:
Vita.
Statement of Responsibility:
by Young-Duck Yoon.

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OPTICAL PROPERTIES OF DOPED CUPRATES AND RELATED MATERIALS





By

YOUNG-DUCK YOON


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


1995














ACKNOWLEDGMENTS


I would like to thank my adviser, Professor David B. Tanner, for his advice, patience and encouragement throughout my graduate career. I also thank Professors P.J. Hirschfeld, N. Sullivan, J. Dufty and R. Singh for their interests in serving on my supervisory committee and for reading this dissertation.

Thanks also go to all my past and present colleagues in Tanner's group for their friendship, useful conversations and cooperation. In particular, I would like to thank V. 2eleznf for many enlightening and useful discussions.


ii















TABLE OF CONTENTS



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


ABSTRACT . . . . . . . . . . . . . . .


CHAPTERS


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


II. REVIEW OF PREVIOUS WORK . . . . .


Mid-infrared Bands . . . . . . . .
Sr Doping Dependence of Hole-Doped La2-,SrCuO4 Electron-Doped System . . . . . . .


III. THEORY . . . . . . .


Electronic Structure of Insulating Phases
Cuprates . . . . . . .
BaBiO 3 . . . . . . .
Electronic Models for CuO2 Plane . .
Three Band Hubbard Model . .
One Band Hubbard Model . . .
Spectral Weight Transfer with Doping .
Simple Semiconductor . . . .
Mott-Hubbard System . . . .
Charge Transfer System . . .
Frequency Dependent Conductivity in Sup
Review of Electromagnetic Response in Weak-Coupling Mattis-Bardeen Theory Penetration Depth and Infrared Conduc


~6


~6 ~6 ~7


. . . . . . . . 9


. . . . . . . . 9
. . . . . . . . 9
. . . . . . . . 9
. . . . . . . 11
. . . . . . . 12
. . . . . . . 13
. . . . . . . 17
. . . . . . . 18
. . . . . . . 20
. . . . . . . 21
erconductor . . . . 22 the Normal State . . . 23 . . . . . . . 24
tivity . . . . . 25


iii


. ii vii









Coherence Effect in Superconductor . . . . . . . 26
Strong-Coupled Superconductor . . . . . . . . 27


IV. CRYSTAL STRUCTURE AND SAMPLE CHARACTERISTICS . 30


Crystal Structure and Phase Diagram . . . . . . . 30
La2-,SrzCuO4 . . . . . . . . . . . . 30
Nd2-,CexCuO4 . . . . . . . . . . . . 31
Baj-xKxBiO3 and BaPbi-zBiO3 . . . . . . . . 32
Sample Characteristics . . . . . . . . . . . 34
La2-.SrCuO4 . . . . . . . . . . . . 34
Nd2-xCeCuO4 . . . . . . . . . . . . 34
Bi-O Superconductors . . . . . . . . . . 37


V. EXPERIMENT . . . . . . . . . . . . 38


Background . . . . . . . . . . . . . 38
Dielectric Response Function . . . . . . . . . 38
Optical Reflectance . . . . . . . . . . . 39
Infrared and Optical Technique . . . . . . . . . 41
Fourier Transform Infrared Spectroscopy . . . . . . 41
Optical Spectroscopy . . . . . . . . . . . 43
Instrumentation . . . . . . . . . . . . 43
Bruker Fourier Transform Interferometer . . . . . . 43
Perkin-Elmer Monochromator . . . . . . . . . 45
Michelson Interferometer . . . . . . . . . . 45
dc Resistivity Measurement Apparatus . . . . . . . 46
Data Analysis; Kramers-Kronig Relations . . . . . . . 47
Dielectric Function Models . . . . . . . . . . 49
Two Component Approach . . . . . . . . . 49
One Component Analysis . . . . . . . . . . 50

VI. Ce DOPING DEPENDENCE OF ELECTRON-DOPED Nd2-zCeCuO4 56


Results and Discussion of Insulating Phase . . . . . . 57
Doping Dependence of Optical Spectra . . . . . . . 60
Optical Reflectance . . . . . . . . . . . 60


iv









Optical Conductivity . . . . . . . . . . . 61
Effective Electron Number . . . . . . . . . 63
Loss Function . . . . . . . . . . . . 64
Temperature Dependence of Optical Spectra . . . . . . 66
One Component Approach . . . . . . . . . . 67
Doping Dependence of Low Frequency Spectral Weight . . . . 70
Drude and Mid-infrared Band . . . . . . . . . 70
Transfer of Spectral Weight with Doping . . . . . . 71
Doping Dependence of Charge Transfer Band . . . . . . 73
Summary . . . . . . . . . . . . . . 74


VII. QUASI-PARTICLE EXCITATIONS IN LIGHTLY
HOLE-DOPED La2-,SrzCuO4+ . . . .9


Experimental Results . . . . .
a-b Plane Spectra . . . . . .
c Axis Spectra . . . . . .
Mode Assignment . . . . . .
Hopping Conductivity in Disordered System Optical Excitations of Infrared Bands . .
Summary . . . . . . . .


. . . . . . 9 9
. . . . . . 9 9
. . . . . . 10 1
. . . . . . 10 2
. . . . . . 10 6
. . . . . . 10 9
. . . . . . 111


VIII. INFRARED PROPERTIES OF Bi-O SUPERCONDUCTORS .


Normal State Properties . . . . .
Results for Bai-KBiO3 . . . .
Results for BaPbi..BiO3 . . . .
Comparison of Two Bismuthate Spectra .
Free Carrier Component in BKBO . .
Superconducting State Properties . . .
Superconducting Gap . . . . .
Superconducting Condensate . . .
Discussion of Pairing Mechanism in BKBO
Summary . . . . . . . .


IX. CONCLUSIONS . . . . . . .


. . . . . . 1 3 1
. . . . . . 132
. . . . . . 1 3 2
. . . . . . 1 3 4
. . . . . . 1 3 5
. . . . . . 138
. . . . . . 1 3 8
. . . . . . 14 0
. . . . . . 14 2
. . . . . . 144


156


v


. . 130


98









APPENDIX

OPTICAL AND TRANSPORT PROPERTIES OF LuNi2B2C . . 159 REFERENCES......... ...............................164

BIOGRAPHICAL SKETCH . . . . . . . . . . . 174


vi














Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy





OPTICAL PROPERTIES OF DOPED CUPRATES AND RELATED MATERIALS By

Young-Duck Yoon

August 1995


Chairman: David B. Tanner
Major Department: Physics

The optical properties of cuprates, Nd2-zCeCuO4 and La2-.SrzCuO4, and the related materials, Ba-,K.BiO3 (BKBO) and BaPbi-BiO3 (BPBO), have been extensively investigated by doping-and temperature-dependent reflectance measurement of single crystal samples in the frequency range between 30 cm-1 (4 meV) and 40 000 cm-1 (5 eV). The Nd2-zCe.CuO4 system has been studied at Ce compositions in the range 0 < x < 0.2. La2-xSrxCuO4 has been studied in the spin glass doping regime, (x < 0.04). The two bismuthates have been investigated as superconducting materials with the maximum Tc.

Our results for Nd2-2CezCuO4 show that doping with electrons induces a transfer of spectral weight from the high energy side above the charge transfer excitation band to the low energy side below 1.2 eV, similar to the results observed in hole-doped


vii









La2-.SrCuO4. However, the low frequency spectral weight grows slightly faster than 2x with doping x, as expected for the Mott-Hubbard model.

We find very interesting results at low doping levels in La2-xSrxCuO4. Upon Sr doping the oscillator strength of the phonons is gradually reduced and doping induced modes (Raman modes and carrier-lattice interaction mode) appear in the far-infrared. We also find that the deformation potential by the dynamical tilting of CuO6 octahedra induces a carrier-lattice interaction. The carrier-lattice interaction is characterized by strong infrared active modes and an appearance of the strong Ag Raman modes upon cooling.

Finally, we present the normal and the superconducting properties of Bi-O superconductors. We conclude that the BKBO system is a weak-or moderate-coupling BCS-type superconductor in the dirty limit.


viii













CHAPTER I
INTRODUCTION


Since the discovery of high T, superconductors by Bednorz and Mnller,' extensive efforts have been devoted to identify the nature of the superconducting pairing of an entirely electronic origin in these systems, but the basic mechanism responsible for the superconductivity is not yet known. High T, superconductors are fundamentally different from conventional metallic superconductors. The latter have conventional metallic transport properties above their transition temperatures Tc, and the electronphonon interaction causes the electrons to form pairs, which then condense as bosons in the superconducting state. In contrast, the high T, materials differ from ordinary metal superconductors. They have very high transition temperatures, a linear behavior for their dc resistivities, a strongly temperature dependent Hall coefficient, short coherence lengths, frequency dependence of scattering rate 1/r, etc. The difficulty in understanding these materials stems from their complexity. For example, the large number of atoms in the unit cell and the strong anisotropy of the materials complicate the interpretation of the results.

The undoped parent compounds such as La2CuO4 and Nd2CuO4 are S = 1/2 antiferromagnetic insulators with an optical absorption edge 2 eV. Superconductivity with very high Tc's occurs in the presence of specific chemical doping. Currently, one of the most controversial issues is whether carriers injected in the undoped materials behave like quasiparticles or not. The long range antiferromagnetic order inhibits conduction by creating a spin polarization gap, and therefore the injected carriers which move in the background of spin order nearby need to be reoriented. Thus,


1






2


the hole may be a dressed quasiparticle carrying a reduced antiferromagnetism in its neighborhood. If it is clear that the spin wave excitations will heavily dress the hole, increasing substantially its mass, the dressing of the hole by spin excitations will be a key ingredient in the explanation of the origin of the MIR band which appears near the exchange energy J in the optical conductivity spectrum. Therefore, the optical conductivity can be qualitatively described using models of strongly correlated electrons like t-J model. However, there is little doubt that the properties of high T, materials are dominated by the tendency of the electron correlations, especially, at low doping levels. For example, the electron correlations can be significantly modified by the response of electrons to the lattice. This evidence is observed from local structural distortion which causes non-linear, localized carrier-lattice interaction.

Among the experimental results appearing in the optical conductivity, one of the striking features is a strong doping dependence of spectral weight, i.e., a shift of spectral weight from high to low frequencies. An interesting point is the behavior of the low energy region near the Fermi surface in the case of hole doping and electron doping. Basically, doped holes and electrons show different orbital characteristics in the localized limit of charge transfer materials: holes introduced by doping have O character and so the behavior of spectral weight transfer is expected as that of semiconductors, but doped electrons feel the strong repulsions on Cu 3d sites and will behave as strongly correlated objects, like the Mott-Hubbard (MH) case. However, large hybridization is crucial for high T, materials with strong correlations. This is illustrated by the large value of Cu-C hopping and the large Coulomb repulsion on Cu site. So, each site is not restricted to only one orbital due to large hybridization and instead has a direct mixing of most of the Cu 3d and 0 2p orbitals. As a result, it is proposed that the hole-doped system can be described by a t-J model or a single






3


band Hubbard model, in which the occupation is constrained to at most one electron per orbital. Hence, the low frequency spectral weight (LFSW) is expect to behave as the MH system. However, it is not clear if hole and electron doped systems can both be described by a single band Hubbard model. On the other hand, it is interesting to note that the influence of doping in the antiferromagnetic correlations is nonuniversal between hole and electron materials. For example, single crystal neutron scattering measurements on Nd2-zCe.CuO4 by Thurston et al.2 have shown that 3D antiferromagnetic order persists even with x as high as 0.14, while on La2-xSr.CuO4, a doping of x ~ 0.02 is enough to destroy the long-range order.

The discovery of copper oxide high T, superconducting materials has also generated renewed interest in the Bi-O superconductors, due to many similarities between the bismuthates and the Cu-O cuprates. For example, in spite of a low charge-carrier density (on the order of 1021 cm-3) the T, of the bismuthates is anomalously high; superconductivity occurs near the metal-insulator transition by chemical doping of the insulating BaBiO3. In addition, the high T, conductivity in this system is of great interest because it contains neither a Cu atom nor a two dimensional structural feature which are considered to be crucial for the high T, in the copper oxide superconductors. Thus, understanding this system would undoubtedly facilitate understanding of Cu-O cuprates.

This dissertation describes a detailed study of the optical properties of high T, cuprates and Bi-O superconductors over the infrared to the UV region in the temperature range from 10 K to 300 K.

First, we concentrate on the Ce doping dependence of electron doped Nd2-xCe.,CuO4, which has the simplest structure among high T, materials. The low frequency spectral weight (LFSW) for electron-doped Nd2-.Ce.CuO4 is compared






4


to the results of Uchda et al.3 for hole doped La2-zSrCuO4 and theoretical work of Meinders et al. We find that the far-infrared reflectance has little temperature dependence, indicating the non-Drude behavior of this material. In one component approach, our results illustrate that upon doping the quasiparticle interactions are reduced and hence at high doping levels the imaginary part of quasiparticle self energy,

- Im E, is proportional to w. The optical spectra in the high T, regime of x = 0.15 show a strong Drude band and weak quasiparticle excitations compared to those of neighboring Ce concentration samples.

Second, the low-lying excitations of charge carriers are investigated in the low doping regime for La2- Sr. CuO4+b. The qualitative features of the far-infrared al (w) and the dc transport properties are similar to the behavior of a conventional 2D disordered metal. The resistivity at temperatures below 50 K shows a typical dc variable range hopping behavior. The ai(w) spectrum at 10 K illustrates the photon induced hopping of charge carriers. We also discuss that a narrow band near the antiferromagnetic energy J is likely to have both spin and lattice components. The

1.4 eV band looks like a result of the excitonic effect.

Finally, the optical conductivities of Bi-O superconductors, Bai-.KBiO3 and BaPbi-BiO03, are presented in the superconducting state and in the normal state and compared to a conventional BCS theory. The extrapolated values of the a1(w) spectra at zero frequency for BKBO and BPBO are similar to the results obtained in the dc resistivity measurements, showing metallic and semiconducting behavior, respectively. For BKBO, the electron-phonon coupling constant A 0.6 is estimated. In the superconducting state, the position of the superconducting edge in the reflectance spectra has a strong temperature dependence which is suggestive of the BCS-like energy gap in the dirty limit. From this analysis, a value for the energy gap






5


of 2A = 3.5 0.3 is obtained. The superfluid condensate fractions are determined as 16% at 10 K and 10% at 19 K, and the London penetration depth, AL, is estimated to be 4250 100 A. We also discuss the possibility of pairing mechanism in BKBO.

This dissertation is organized as follows. Chapter II reviews previous optical results on the materials, investigated and issued. In Chapter III, models to describe the behavior of electrons in the CO2 plane and theoretical models for the transfer of spectral weight in the hole doping and electron doping cases are discussed. Some fundamental properties of BCS superconductor are also given. In Chapter IV, the crystal structures and the sample characteristics are presented. Chapter V will describe infrared techniques, experimental apparatus and data analysis. Chapter VI, VII and VIII are devoted to experimental results and discussion. Finally, conclusions are presented in Chapter IX.












CHAPTER II
REVIEW OF PREVIOUS WORK


Mid-infrared Bands

Figure 1 shows optical conductivities in lightly doped YBa2Cu306+y, Nd2CuO4_y, and La2CuO4+y from Thomas et al.5 In Fig. 2, two peaks can be seen in the mid-infrared region. Thomas et al. interprets that the lower energy band is characterized by the exchange energy J and the higher energy band arises from an impurity band near the optical ionization energy of the isolated impurity. Sr Doping Dependence of Hole-Doped La-9_SrCuO4

Uchida et al.3 have measured the reflectance of La2-,SrCuO4 for several doping levels between x = 0 and x = 0.34 at room temperature with large, homogeneously doped, single crystals. They observed in the reflectance spectra that the position of the ~ 0.1 eV plasmon minimum is nearly insensitive to doping due to the contribution of the strong midinfrared band.

The optical conductivity oi(w) is shown in Fig. 3, where the undoped crystal shows a negligible conductivity below 1 eV and a charge transfer gap at about 2 eV. With hole doping, the intensity above the gap is reduced and new features (Drude and midinfrared bands) appear below 1.5 eV i.e., a transfer of spectral weight from above the gap to low energies seems to occur. In the metallic phase, the conductivity at small frequencies decays much more slowly than the Drude-type 1/w2 behavior expected for free carriers.


6






7


250


Nd2CuO4.y
200

E
U








0
0 0.25 0.5 0.75 1.0 1.25
Energy [eV]



Fig. 1. Optical conductivity in the mid-infrared region of YBa2Cu3O6+y
(upper panel), Nd2CuO4-y (center panel), and La2CuO4+y.


Electron-Doped System

As we will discuss later, electron-doped materials, like Nd2-,Ce.CuO4, are structurally very similar to La2- SrCuO4, but doped holes and electrons are introduced in different sites, 0 and Cu sites. It has been found that their optical conductivities are also qualitatively similar for Nd2-..CeCuO46,7 as shown in the bottom of Fig. 3. Other compounds of the same family can be obtained by replacing Nd by Pr, Sm and Gd. The optical properties of Pr2..CeCuO4 have been investigated', and the reported results are very similar to those of Nd2-..CeCuO4.






8




1.5
La2.xSrxCuO4
0.34
0.20 (a)



0.02
0 .06 0.10
0.15
0.5 -'.-.20 .-.02
0

0 1 2 3 4

1.6 1

E 1.2 -\

0.8 "0 (b)


S0.4 '
-. Nd2.xCexCuO4.y

0 1 2 3 4
ho (eV)



Fig. 3. a-b plane optical conductivity of La2-.SrCuO4 (top) and
Nd2-,CeCuO4 (bottom) single crystals for Uchida et al.3













CHAPTER III
THEORY


Electronic Structure of Insulating Phases Cuprates

The CuO2 plane in the insulating cuprates is known as a charge transfer (CT) insulator with a charge transfer energy, A ~ 1.5 2.0 eV, between 0 2p and Cu 3d, depending on structural features such as the in-plane Cu-O distance d and the out-of-plane structural configuration (oxygen coordination number).

The topmost Cu 3d state, a d,2_,2 orbital, is split into upper and lower Hubbard bands by a large on-site Coulomb repulsion U 8 10 eV and, as a result, an occupied 0 2p band is located in between two bands. This band structure is well described by the three band Hubbard model. The three band Hubbard model will be discussed in the next section.

Figure 4 shows a rough scheme of the electronic band structure of a charge transfer insulator. Assuming that the bands do not change with doping (rigid band approximation), then upon hole doping a PES (photoemission) experiment expects that the Fermi energy will be located below the top of the valence band. On the other hand, for an electron doped material the Fermi energy is above the bottom of the conduction band.

BaBiOi

There are two points of view regarding the origin of the CDW instability in insulating phase BaBiO3. One is a Peierls-like scenario, in which Fermi surface nesting


9






10


undlopod
dl Ud dic
dO
X .1


N-1


hole dopin


p-type


N+1O
% %% ctzon doping


1ZL


n-type


Fig. 4. Simple electronic band structure for the charge transfer insulator, for
hole-doped and for electron-doped.
















-u U -


Fig. 5. Schematic representation of the oxygen octahedra. The solid lines
illustrate the symmetric Peierls distortions and the dashed lines illustrate
the undistorted case.






11


and the strong coupling of the conduction band states near EF to bond stretching 0 displacements lead to a commensurate CDW distortion.'0 In another approach, the driving force is the aversion of Bi to the 4+ valence, which leads to a disproportionation into 3+ (6s2) and 5+ (6s') valences on alternate sites."1 In either case one finds a commensurate CDW distortion, in which the 0 octahedra are alternately expanded or contracted as illustrated in Fig. 5. This CDW distortion doubles the unit cell, which splits the half filled metallic band into filled and empty subband, opening a semiconducting gap of ~ 2 eV.




Electronic Models for CuO9 Plane

In this section, a Hamiltonian to describe the behavior of electrons in the high T, materials will be briefly described. Due to the complexity of their structure it is important to make some simplifying assumptions. The very strong square planar Cuo bonds with strong on-site correlations makes it possible to construct a Hamiltonian restricted to electrons moving on the CuO2 plane.

Several models have been introduced for the description of layered strongly correlated systems, as realized in the CuO2 plane. While there is a growing consensus that the high T, materials should be described within the framework of twodimensional (2D) single-band t-Jl2 or three-band Hubbard models13 in the strong coupling limit,14,15 a direct comparison of controlled solutions with experimental data is still lacking. We will discuss these one band and three band Hubbard models in the present section, and the carrier doping effect in these prototype models will be discussed in the following section.






12


Three Band Hubbard Model

First of all, let us consider the bonding of a full Cu-O octahedron (CuO6), that is, the bonding of the 3d orbitals on the Cu ion with the 2p orbitals of the surrounding O ions. There are 17 orbitals in the Cu-O octahedron. Five are from the 3d orbitals of Cu, which are dX2_ 2, dm2, and three dxy types. Also, the four 0 atoms each have three p orbitals which contribute 12 orbitals. However, we here focus on the in-plane bonding and take a more intuitive approach. To do this, consider the two planar 0 atoms with p orbitals that are directed toward the central Cu atom. On the central Cu atom, we only use the d_2_ ,2 orbital, since it is correctly oriented for o- bonding with its neighboring oxygens. It is also the uppermost Cu-d level in the crystal field of the octahedral structure. Thus, only three orbitals (ps, py, and dX2_ ,2) are used. The other 14 orbitals can be taken as nonbonding relative to these orbitals. In addition, the copper ion Cu2+ has a 3d electron configuration which gives the ion spin 1/2. Thus, in the absence of doping, the material is well described by a model of mostly localized spin 1/2 states that give these materials their antiferromagnetic character.

The Hamiltonian in the CuO2 plane can be constructed in the framework of the three orbitals:


H = -tp( Pjdi + dtp + h.c.) tpp (ppj, + h.c.) + Edj~n4+ Eplfn Ud nT + Up InP 1, n,, + Updflfl'(1
S j (,j)


where p, are fermionic operators that destroy holes at the oxygen ions labeled j, while di corresponds to annihilation operators at the copper ions i. (i, J) and (j, J') represent Cu-O and 0-0 neighbors, so that this Hamiltonian contains two hopping terms, tpd and tPP, as well as site energies ei and Coulomb interactions Ui for the two types of






13


sites, i on Cu and j on 0. Upd corresponds to the Coulomb repulsion when two holes occupy adjacent Cu and 0 sites, and may also be very important. It is appropriate to use the hole notation, since there is a one hole per unit cell in the undoped case. Hence, the vacuum state corresponds to the electronic configuration d'Op6. Because Ed < Ep, this hole occupies a d-level, forming the d state. There are two factors that govern the electronic structure. On the other hand, the hybridization tpd is substantial and leads to a large covalent splitting into bonding and antibonding bands, which form the bottom and top of the p-d band complex. Therefore, the bonding orbital is 0-p-like and the antibonding orbital is Cu-d,2_ 2-like. This covalent nature is not restricted to only one orbital per site. There is a direct mixing of most of the Cu 3d and 0 2p states.

On the other hand, the local Coulomb interaction Ud is crucial for the semiconducting properties. In the charge transfer regime (tpd < Ep Ed < Ud),16 the lower Hubbard band is pushed below the 0 level and so three bands are formed as shown in Fig. 6(top). When another hole is added to this unit cell in the charge transfer regime, the new hole will mainly occupy oxygen orbitals due to the on-site Coulomb interaction. The high T, superconducting materials fall into this category (typical parameters are Ep Ed ~ 3 eV, tpd 1.5 eV, tp, ~ 0.65 eV, Ud ~ 10 eV, Up ~ 4 eV, and Ud 1.2 eV).16

One Band Hubbard Model

As originally emphasized by Anderson,12 the essential aspects of the electronic structure of the CuO2 planes may be described by the two dimensional one band Hubbard model. This model is


H = -t (c! c,, + cci,,) + U (ni, )(nd ), (2)
S)a' 07i22






14


3 bands


Ud


o 2p -'- UHB
A


1 band
Uoff

1 n


LHB


UHB


Fig. 6. Simple band structure in the three band (top) and one band (bottom)
Hubbard model.


LHB


I






15


where c! is a fermionic operator that creates an electron at site i of a square lattice with spin o. U is the on-site repulsive interaction, and t is the hopping amplitude. In the limit (t < U < E, Ed), the additional holes sit at Cu sites, and the hybridization may be included by eliminating 0 sites to give an effective Hamiltonian for motion on Cu sites alone. This is obviously a single-band Hubbard model. In a single-band Hubbard model, the conduction band develops a correlation gap of an effective value of the Coulomb repulsion Ueff, and this model yields only two bands, as shown in Fig. 6(bottom).

For large on-site repulsion U, the one band Hubbard-model Hamiltonian can be transformed into the t-J model Hamiltonian. This model describes the antiferromagnetic interaction between two spins on neighboring sites and it allows for a restricted hopping between neighboring sites. Therefore, the Hamiltonian of (2) reduces to a S = Heisenberg model on the square lattice of Cu sites:


H = -t [c,(1 ni-,)(1 n1_,)c,, + h.c.] + J (Si 4ninj), (3)
(ij) ,r (1i)

where Si are spin-1/2 operators at site i of a two dimensional square lattice, and J is the antiferromagnetic coupling between nearest neighbors sites (i]) and is defined as


J = -2 (4)
U

The limit of validity of the t-J model is for J < t or t < U. However, it is often extendable into the regime J ~ t. The hopping term allows the movement of electrons without changing their spin and explicitly excludes double occupancy due to the presence of the projection operators (1 ni_,). The Hamiltonian (3) is just the effective Hamiltonian of the single-band Hubbard model in the large U limit. In this






16


model, the insulating state is created by the formation of spin-density wave. The long range antiferromagnetic order inhibits conduction by creating the spin polarization gap.

When holes are doped on 0 sites, Zhang and Rice17 made a progress in the following argument. The key point is that the hybridization strongly binds a hole on each square of 0 atoms to the central Cu2+ ion to form a local singlet. This singlet then moves through the lattice of Cu2+ ions in a manner which is similar to a hole (or doubly occupied site) in the single-band effective Hamiltonian. This singlet is equivalent to removing one Cu spin 1/2 from the square lattice of Cu spins, and thus the effective model corresponds to spins and holes on the two dimensional square lattice. The 0 ions are no longer explicitly represented in the effective model. Further, two holes feel a strong repulsion against residing on the same square, so that the single-band model is recovered.

It is important to remark that the reduction of the three band model to the tJ model is still controversial. Emery and Reiter14 have argued that the resulting quasiparticles of the three band model have both charge and spin, in contrast to the Cu-0 singlets that form the effective one band t-J model. Their result was based on the study of the exact solution in a ferromagnetic background, and their conclusion was that the t-J model is incomplete as a representation of the low energy physics of the three band Hubbard model. Meinders et al.4 have also shown that the low energy physics in the t-J model behaves as a single-band Hubbard model due to the restriction of double occupancy. However, Wagner et al.18 and Horsch6 have proven that the Zhang-Rice type of singlet17 construction plays a crucial role for the low energy physics in the t-J model. They have suggested that due to an intrinsic strong Kondo exchange coupling between 0-hole and Cu spin the valence band is split into






17


(local) singlet (S) and triplet (T) states. Because the spin singlet states have the lowest energy, the singlet states are located just above the valence bands and act as the lower Hubbard band. Thus, the charge transfer gap of Ueff is formed with the upper Hubbard band. Therefore, the t-J model can produce the low energy spectrum of the three band Hubbard model.


Spectral Weight Transfer with Doping

Insulating CuO2 layered cuprates can be doped with holes or electrons as the charge carriers. A surprising feature with doping is a strong doping dependence of high energy spectral distributions and the redistribution of the spectral weight from high to low energy. Nice examples are the electron-electron loss study19 and 0 is x-ray absorption study20 for the La2-,Sr.Cu04 system. These spectra show a strong decrease with doping x in the intensity of the upper Hubbard band as the lower energy structure develops due to doped-holes in the 0 2p band. Another example is optical absorption experiments,3 where a transfer of spectral weight from a band-gap transition at about 2 eV in insulating La2CuO4 to the low energy scale (< 1 eV) is observed with a strong doping dependence. This redistribution of spectral weight and its doping dependence is due to strong correlation effects and has been observed in several numerical calculations of correlated systems. Naively, doped-carriers may show different orbital characteristics in the case of hole doping and electron doping: holes have 0-2p-like character and electrons have Cu-3d-like character. Thus, we may expect the different doping mechanisms for hole-doped and electron-doped systems. In this section we review the difference between doping mechanisms of a semiconductor, a localized Mott-Hubbard and a CT system and discuss the influence of the hybridization for the Mott-Hubbard and CT system in the framework of Eskes et al.21 and Meinders et al.4






18


eSemflaC4c EFi -- I1S


Moai-Fuhhard





E L








N N.
ITS- PIt


Charge Trunfer










EP





FF


Fig. 7. A schematic drawing of the electron-removal and electron-addition
spectra for semiconductor (left), a Mott-Hubbard system in the localized limit (middle) and a charge transfer system in the localized limit (right).
(a) Undoped (half filling), (b) one-electron doped, and (c) one-hole doped.
The bars just above the figures represent the sites and the dots represent the electrons. The on-site repulsion U and the charge transfer energy A
are also indicated.


Simple Semiconductor

Let us consider a semiconductor with an occupied valence band and an unoccupied conduction band, separated by an energy gap E.. For the undoped semiconductor the total electron removal and addition spectrum is shown in Fig. 7(a)(left). If the total number of sites equals N, then there are 2N occupied states and 2N unoccupied states, separated by E,. If one hole is doped in the semiconductor, the chemical potential will shift into the former occupied band, provided that we can neglect the impurity potential of the dopant.






19


2.50 2.00 1.50


1.00


0.50
0Z .00 .a



1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0
hoic doping clcctron doping


Fig. 8. The integrated low energy spectral weight (LESW) as a function of
doping concentration x for Mott-Hubbard model of Meinders et al.4 The solid line corresponds to the localized limit t = 0. The data points are from the calculations: t = -0.5 eV (lowest) to t = -2 eV steps of 0.5 eV.


The total electron removal spectral weight will be 2N 1 (just the number of electrons in the ground states) and the total electron addition spectral weight will be 2N +1 (total number of holes in the ground states). The electron addition spectrum consists of two parts, a high energy scale (the conduction band) and a low energy scale, which is the unoccupied part of the valence band. Therefore, we can know that the low energy spectral weight equals 1. The same arguments hold for an electrondoped semiconductor. Thus, the low energy spectral weight grows as x with doping x for a hole-doped and electron-doped semiconductor.






20


2.50


2.00


1.50

1.40

0.50




.0 0.8 0.6 0.4 0.2 0.0 2 6
hoic doping clcctron doping


Fig. 9. The integrated LESW as a function of doping concentration x for CT
system of Meinders et al.4 The solid line corresponds to the localized limit tpd = 0. The data points correspond to tpd = 0.5 eV (lowest) to tpd = 2
eV in steps of 0.5 eV.


Mott-Hubbard System

This correlated system is described by the single-band Hubbard Hamiltonian. Figure 7(middle) shows the total photoelectron and inverse photoelectron spectrum at half filling in Mott-Hubbard system. The total electron-removal spectral weight is equal to the number of unoccupied levels. Therefore, each has an intensity equal to N. Upon hole doping there are N 1 singly occupied sites. So the total electronremoval spectral weight will be N 1. For electron addition there are N 1 ways for adding the electron to a site which was already occupied. Therefore, the intensity of the UHB will also be N 1 (not N). We are left with the empty site for which there are two ways of adding an electron (spin up and spin down), both belonging to the LHB. Thus we find N 1 electron removal states near the Fermi-level, two electron






21


addition states near the Fermi level and N 1 electron addition states in the UHB. The same arguments hold for the electron doped case. Thus, a doping concentration x yields a low energy spectral weight 2x and the high energy spectral weight is 1 x. There have been Nx states transferred from high to low energy. However, when the hybridization is taken into account, the low energy spectral weight grows faster than two times the amount of doping as shown in Fig. 8. Charge Transfer System

For the high T, superconductors, an oxygen band is located between the LHB and UHB. These systems are described by the three band Hubbard Hamiltonian. In the localized limit with no hybridization between the oxygen and copper sites (tpd=O), when the electrons are doped in this system, the situation is similar to the MottHubbard case and the spectral weight is transferred from high to low energy. Thus, the low energy spectral weight goes to 2x with doping x. However, upon hole doping the situation is similar to that of the semiconductor without any spectral weight transfer. So, the CT system in the localized limit shows a fundamental asymmetry between hole and electron doping (Fig. 9). That is, electrons will feel the strong repulsions on the d sites, similar to the MH model, and will behave as strongly correlated objects. When the hybridization is taken into account, the low energy spectral weight for the electron-doped CT system behaves more or less the same as found for the Mott-Hubbard system. However, for small hybridization tpd, the low energy spectral weight for the hole-doped CT system behaves as a semiconductor. When the hybridization is increased, the low energy spectral weight for the holedoped CT system rapidly increases and the low energy spectral weight is almost symmetric with respect to hole-electron doping, so the low energy spectral weight is similar to that of the MH system. The high T, superconductors lie in the regime






22


with large hybridization, so the holes in the hole-doped high T, superconductors will behave as strongly correlated particles.


Frequency Dependent Conductivity in Superconductors

Far-infrared conductivity a1(w) is very useful to study particle-hole excitations in the energy range of 4 100 meV for the copper oxide and bismuth oxide superconductors. For example, in the superconducting state, the energy gap 2A of single-particle excitations could be obtained directly from the absorption edge of infrared spectrum. Further information on the nature of quasiparticles and other infrared-active excitations in the superconducting state can be obtained by analyzing the frequency dependence of the absorption spectrum at energies above 2A. Of particular interest are deviations of the measured spectrum or infrared conductivity from the BCS-theory for isotropic superconductors. Mattis and Bardeen22 first calculated the infrared conductivity in the framework of the weak-coupling BCS theory. Deviations might give us some hints on what is basically different in BCS and high T, superconductors.

The Mattis-Bardeen theory does not include the Holstein mechanism,23 where part of the energy of the excited conduction electron is transferred to phonons. This effect is well known for a conventional strong coupling superconductor such as Pb. In this case, one expects characteristic deviations from the Mattis-Bardeen theory. However, little is known about strong coupling corrections to weak-coupling conductivity. The strong-coupling theory of a1(w) which is based on Eliashberg's mode24 was first calculated by Nam.25 Since the early 1960s, the interpretation of energy gap and more detailed measurements of excitation spectra has been based on the Eliashberg theory.24 This theory makes a number of assumptions which may be called into question in the new copper oxide and bismuth oxide superconductors.






23


Review of Electromagnetic Response in the Normal State

The portion of the interaction Hamiltonian of electromagnetic radiation with matter is represented as

H, = +- EA(ri, t).pi, (5) mc

where vector potential A is subject to the gauge condition V A = 0. The fields are as usual the derivatives


1 OA
E= 1.A, and H=VxA. (6)
c &t


We are concerned with the anomalous skin effect ( > 6) only with transverse fields, where the current at a point depends on the electric field not at just the same point but throughout a volume. For metals in the normal state un(w,q) can be calculated in the free-electron approximation by applying Fourier analysis to the well-known Chambers integral expression for the current:


3u R[R.- E(r', t)] RI
J(r,t) = 4r Rj expR/ dr, (7)
47r I R

where a = ne2l/mv0, R=r'-r, t' = t (R/vo), 1 is the mean free path, and vo is the Fermi velocity. Note that the electric field, E, is evaluated at the retarded time, t R/vo. When the field is constant over a mean free path, (7) reduces to Ohm's law, J(0)= aE(0). A familiar result for the complex conductivity aln(w) + io2n(w) in the limit at q= 0 is


nG) Oro I 2n G0WT(8
oin~ 1 = + W2,J2n =+ W272. 8






24


Weak-Coupling Mittis-Bardeen Theory

In the superconducting state, a complex conductivity o,(w, q) may again be defined. In the extreme anomalous limit, q~o > 1 or extreme dirty limit o > 1, Mattis and Bardeen22 showed that the ratio of the superconducting to normal conductivity within weak-coupling BCS theory is


01,3 2 f* IE(E + hw) + A2I[f(E) f(E + hw)] dE
an hW (E2 A2)1/2[(E + hO)2 A2]1/2
1 f-a IE(E + hw) + A21[1_2f(E+hw)] + w (E2 A2)1/2[(E + hw)2 A2]'!2 dE, (9)



a2s 1 jA E(E + hw) + A21[1 2f(E + hw)]dE (10)
an h, (A2 E2)1/2[(E + hw)2 A2]1/2

Eq. (9) is the same as the expression for the ratio of absorption for superconducting to normal metals for case II of BCS theory. Numerical integration is required for T > 0.

Figure 10 shows the Mattis-Bardeen theory for oi,(w)/an and O2,(W)/n as a function of frequency for T = 0. The real part is zero up to hw = 2A and then rises to join the normal state conductivity for hw > 2A.

In the superconducting state for w < 2A, because J(w) = ia2,E(w), the power loss P = (J E) = 0; one can therefore expect a perfect reflector (R = 1) at frequencies below 2A. The imaginary part of o1(w) displays the 11w inductive response for hw < 2A. More simply, this dependence is a consequence of the free-acceleration aspect of the supercurrent response as described by the London equation



E = 8(AJ,)/Ot, A = = A (
ne2 LC (






25


2






0 9




0 I 2 3
2)

Fig. 10. Complex conductivity of superconductors in extreme anomalous (or
extreme dirty limit) at T = 0.


where m, and n., are the mass and density of the superconducting electrons and AL is the London penetration depth. From this relation,

1 nce2
a2 = -- = -. (12)
Aw m.,w

For hw > 2A, a2, falls to zero more rapidly than 11w. Penetration Depth and Infrared Conductivity

The sum-rule argument allows determination of the strength of this supercurrent response from oi,. The oscillator strength sum rule requires that the area under the curve of al(w) have the same value in the superconducting as in the normal state. The missing area A under the integral of a,, appears at w = 0 as AS(w). The amount of conductivity that is transferred from the infrared to the delta function at zero frequency is given by
o
[O'in(w) 019i& Jd = 0. (13)






26


meV
ooo0 2 5V 75 100 2 V P 1

X_ 8000
E
6000 .. f/2&=1.0 /2&=O.10

4000.

2000

0 200 400 600 oo 0 200 400 600 8o 1000
FREQUENCY (cm-)


Fig. 11. The conductivity of a BCS superconductor in the (a) dirty and (b)
clean limits.


The Kramers-Kronig transform of the delta function al (w) is 2A/rw. For comparison with the London equation (11), the penetration depth is related to the missing area by

A = (14)


In the clean limit (1/r < 2A, > 4o), all of the free carrier conductivity collapses into the 6 function, in which case A = re2n/2mb = w,/8, and (13) reduces to the London result. So, the spectral feature is very weak at 2A (Fig. 11(b)). In the dirty limit (1/r > 2A, 1 < 4), the penetration depth tends to be larger than this limiting value and a sharp feature appears at 2A (Fig. 11(a)), and one can write A = re2n,/2mb, where n, < n is the superfluid density. Coherence Effects in Superconductor

At finite temperatures, A(T) < A(O), and also the thermally excited quasiparticles contribute absorption for hw < 2A. This quasi-particle excitation is represented as the distinctive features of the microscopic BCS model of superconductivity,






27


namely a "coherence peak" in the temperature dependence of the conductivity below T, and the logarithmic frequency dependence of o-1(w) near w=O. Coherence effects in superconductors arise because the dynamical properties of the quasiparticle excitations become different from those of normal electron-hole excitations as the gap develops below T. This coherence peak will go to infinity just below T, due to the singularity in the BCS density of states. Thus, as T is lowered below T, the density of excited quasiparticles decreases as these excitations freeze into the condensate, and the properties of the excitations which are present for T > 0 are also modified. There are clearly two fluids, the condensate fraction and the gas of excited quasiparticles. Thus, the condensate response to external electromagnetic fields is described by a 8 function conductivity at w = 0 plus (in the presence of elastic scattering) conductivity with a threshold at w = 2A(T), corresponding to processes in which two quasiparticles are excited from the condensate.

In weak-coupling BCS theory, the energy gap at 0 K is given by



2A = 3.52 kBTc (15)



where kB is Bolzmann's constant and T, is superconducting transition temperature. The gap vanishes at T,, and just below this value, A(T) can be approximated by


A~(T) ;1.74 1 - T ; Tc. (16)
A(O) T,


Strong-Coupled Superconductor

If the electron-phonon coupling is strong (as opposed to weak), then the quasiparticles have a finite lifetime and are damped. This finite lifetime decreases both






28


A(O) and Tc, but T, is decreased more and hence increases the ratio 2A/kBT, above 3.52. The temperature dependence of the superconducting gap is also modified by the damping effect.

A general indicator of strong-coupling in superconductors and hence deviation from weak-coupling BCS is the frequency dependent conductivity in the far-infrared. At low frequencies the electron and its dressing cloud of phonons move together, and one measures fully renormalized conduction electrons. The dressing is affected at higher frequencies, of the order of phonon frequencies, at which an excited conduction can emit phonons, and its renormalization changes. Infrared measurements offer a way to undress the electrons and thus to measure the electron-phonon coupling.

An example of frequency dependent damping is the inelastic scattering of the conduction electrons by phonons in ordinary metals, namely, the Holstein mechanism23 which is an important part of strong-coupling theory. The photon energy is divided between the change in kinetic energy of the electron and the phonon energy. This leads to an enhanced infrared absorption above the threshold energy for creating phonons. The expression for the damping rate is

1 = j a2F(Q)(w O)dQ (17)
r(W) W fo

where a2F(Q) is the Eliashberg function proportional to the phonon density of states F(Q) modified by the inclusion of a factor (1 cosO) to weight large scattering angles 0. The Holstein absorption can be distinguished from the direct absorption by optical phonons because it shifts by 2A in the superconducting state. In addition, the singularities in the superconducting density of states cause the phonon structure to sharpen. As a result, an a2 F(Q) function can be extracted from the optical spectrum.






29


The infrared conductivity a- (w) in strong coupling superconductors is obtained in the framework of Eliashberg's strong-coupling theory.24 This theory incorporates the Holstein mechanism to all orders in the electron-phonon coupling, and is described by an effective scattering potential v, the strength of the electron-phonon interaction by Eliashberg's spectral function a2F(Q), the quasiparticle lifetime due to impurity scattering r, and McMillan's pseudopotential p*. McMillan26 numerically solved the finite temperature Eliashberg theory to find T, for various cases, and the construction from this of an approximate equation relating T, to a small number of simple parameters:

= e 1.04(1 + Aep) (18)
1.45 Aep p*(1 + 0.62Aep)

where E is the Debye temperature and As, is the electron-phonon coupling constant.













CHAPTER IV
CRYSTAL STRUCTURE AND SAMPLE CHARACTERISTICS



Crystal Structure and Phase Diagram


La-.-,Sr,CuO4

The structure of La2-zSrCuO4 shown in Fig. 12(a) is tetragonal and has been known for many years as the K2NiF4 structure. It is also called the "T" structure. In La2-, Sr.CuO4, the Cu-O planes perpendicular to the c axis are mirror planes. Above and below them there are La-O planes. The CuO2 planes are ~ 6.6 A apart, separated by two La-O planes which form the charge reservoir that captures electrons from the conducting planes upon doping. The La-O planes are not flat but corrugated. There are two formula units in the tetragonal unit cell.

Each copper atom in the conducting planes has an oxygen above and below in the c-direction forming an oxygen octahedron. These are the so-called apical 0 atoms or just 0,. However, the distance Cu-0, of ~ 2.4 A is considerably larger than the distance Cu-0 in the planes of ~ 1.8 A. At high temperatures (depending on Sr concentration) there is a transition to an orthorhombic phase (ToIT = 530 K for x = 0), and the copper atoms and the six oxygens surrounding them slightly deviate from their positions.

For x = 0, La2-...SrCu04 is an insulator. Upon doping, La3+ are randomly replaced by Sr2+, and these electrons come from oxygen ions changing their configuration from 02- to 0- (and thus creating one hole in their p shell). Metallic behavior is observed for even small doping concentration, x > 0.04 (Fig. 13). For Sr dopings


30






31


La2CuO4


4


0
I


, 0


e Cu 00

Fig. 12. Crystal structure (a) T


0o

1P001


~-0

I


0


0




0


I



I


Nd2CuO4

*La,Nd phase and (b) T' phase.


between ~ 0.05 and ~ 0.3, a superconducting phase was found at low temperatures. The maximum value (- 40 K) of T, is observed at the optimal doping of x ~ 0.15.


Nd)->..Ce.CuO,

The body centered structure of Nd2CuO4 is shown in Fig. 12(b) and it is called the "T'" structure. It has a close relationship to the T structure of Fig. 12(a). As in the T phase structure, the structure is made of a single Cu02 plane and two Nd-O planes, but the Nd-0 planes are shifted by a/2 in the x-direction, so that the oxygen ions in the Nd-O planes are not on the top of Cu ions. The Nd2CuO4 can be easily electron-doped replacing Nd + by Ce4+.

The phase diagram of this material is shown in Fig. 13 comparing it with holedoped compound. The similarities between the two diagrams are shown, but the


I


I






32


Metallic +-- Insulating Metallic


hi


0.3


300




200



E
1!100


Hole-doped
La2-xSrxCU4y

S


P-type N-typi


AFM
/
SG
SC a


0.2 0.1 0.0 0.1 0.2 0.3
Concentration x in Ln2-xMxCU04.y


0.4


Fig. 13. Phase diagram of Nd2-,Ce CuO4 and La2.-,SrCuO4.


electron-doped system clearly illustrates that superconductivity is a relatively "small" effect compared with antiferromagnetism. Bai-.KBi and BaPb,.Bii-,Oi

BaBiO3 has an almost undistorted ABX3, cubic perovskite structure (Fig. 14). Each Bi atom (B site) is octahedrally coordinated by six 0 atoms. The A site is occupied by Ba or K, while B site is occupied by Bi or Pb. At room temperature, the symmetry of BaPbi-,Bi.03 material changes with doping according to following sequence.27


Orthorhombic Tetragonal Orthorhombic Monoclinic


0 0.90

,* S *j'- - -'
Electron-doped Nd2-xxCeCuO4.y
-4-y



P-type N-type




SC AFM
P0,






33


X





A



Fig. 14. Idealized structure of perovskite ABX3.


Superconductivity exists only in the tetragonal phase and the value of maximum T, is T ~ 13 K for x ~ 0.25. For x > 0.35, the material becomes a semiconductor.

The behavior of Bai-,K.BiO3 is similar to that of the BaPbi-.Bi,03. The superconductivity appears at the boundary of the metal-insulator transition in the cubic phase (z > 0.37) with maximum T, of 30 K and disappears abruptly upon crossing a phase transition to the orthorhombic phase. In spite of the low carrier density (in the order of 1021 cm-3) the value of T, is anomalously high. Also, no magnetism is found in the neighboring compositions. The structures of five phases for 0





34


Sample Characteristics


La,-. Sr,. Cu04

Single crystals of La2-ZSr CuO4 were prepared at Los Alamos Laboratory.30 Sixteen samples of nominal composition La2-.SrZCuO4+6 were grown by conventional solid state reaction at 1050*C using predried La203, SrCO3, and CuO in x increments of 0.002 from x = 0 to 0.04. For each x, the sample was separated into three parts which were treated at 650*C for 5 hours in 1 bar N2 or 1 bar 02, or at 500'C for 72 hours in 230 bar 02, respectively, and then oven cooled. ToIT was measured using a Perkin-Elmer differential scanning calorimeter. Oxygen contents were measured by hydrogen reduction using a Perkin-Elmer thermogravimetric analyzer; the 1 bar N2, 1 bar 02, and 239 bar 02 annealed series showed 8 = 0.00(1), 0.01(1), and 0.03(1), respectively.

The size of all crystals are at least 1.5 mm x 1.5 mm which are suitable for infrared measurements. The surfaces were subsequently etched for 20 min in a solution of 1 % Br in methanol before reflectance measurement. Nd,-.,CeCu04

Nd2--.CeCu04 single crystals were prepared at the University of Texas. All crystals were grown in copper-oxide rich fluxes; normal starting compositions were Nd2-..CexCu4.sOz with various x. The melts were cooled in air from 12600C at 5*C/min. The crystals were mechanically separated from the flux and subsequently reduced in flowing He gas at 910*C for 18 hours. Energy dispersive spectroscopy and electron microprobe (wavelength dependent spectroscopy) analysis on these and many similar crystals have indicated a uniform Ce concentration across the crystal, but absolute concentration determinations are only accurate to Ax = 0.01.






35


1


0


-1


-2


-3


-4


0


5 10 15 20
Temperature (K)


25


30


Fig. 15. Meissner effect measurement on Nd2-.CeCuO4 single crystal.



Table 1. Characteristics of Nd2-.Ce.CuO4 Crystals


x area p (at 300 K) oad (at 300 K)p(T)

(=m2) (Mn-CM) (11-cm)-1
0.0 1 x 1 ~ 10-4
0.11 1 x 1 1 -2 500 ~ 1000 po + AT2

0.14 1 x 1 0.2 0.4 2500 ~ 5000 Po + AT2
0.15 1.2 x 1.2 0.2 5000 po + A T2
0.16 1.2 x 1.2 0.2 5000 p, + A T2

0.19 2 x 2 0.1 ~0.2 5000 ~ 10000 pa + AT2

0.2 2 x 2 0.1 ~0.2 5000 ~ 10000 pa + AT2


E


0

C
0

N


I I It I I I ~ I I I t i l I

- "Nd1 85Ce0.15CuO4a
U
'
















0


36


II I I


I


U


0 4 8
Temperature


12
(K)


1


1


"BaPb1_ Bi,03 SH = 15 OeU

'


II I jI I I--I-


I I I I I I I I


-I-
















-)


0







-2



-3



-4
6 C


-


E
0



0

0
-I

C


10 20 30 40 Temperature (K)


Fig. 16. Meissner effect measurements on BaPbBi...03 single crystal (left)
and Bai...KBiO3 single crystal (right).


Most crystals have good, specular, nearly flux free surfaces, which are suitable for infrared measurements.

Magnetization (Fig. 15) for the superconducting sample shows that although this is a higher T, and has stronger diamagnetism than typically appears in the literature for large crystals, the transition is still somewhat broad, and the field-cooled signal is weak, presumably due to flux pinning. The dc resistivity spectra for all samples roughly exhibit the form of p(T) = po + A T2, quadratic in temperature. The properties of the samples are summarized in Table 1.


0


-1


-2


-3


-4


-5


H =15 Oe -


ml
U
mug
I I I I


-6


uuU-e*m MMMM M 9 O (I 0 00".





Ba-U x0






37


Bi-O Superconductors

The single crystals of Bai_,KBiO3 were grown by a modification of the electrochemical method of Norton.31 The transition to the superconducting phase from dc susceptibility measurement is quite wide with the onset of superconductivity at T = 28 K and the full superconductivity at T = 18 K (Fig. 16), probably associated with the inhomogeniety of the potassium distribution in the crystal. The measured T, (- 22 K) was taken at 50% of transition between 90% and 10% points. Also, a direct measurement of the T, of the same crystal by measuring the temperature at which there is a discontinuity in the tunnel conductance yields T, = 21 K.

For the optical measurement, the sample surface (1.5 mmx 1.5 mm) was mechanically polished using A1203 power of 0.05 pm size. The color was blue after polishing. Figure 16 shows a Meissner effect for BaPb Bi1.203, indicating the onset of superconductivity around 10 K.












CHAPTER V
EXPERIMENT


Background

Dielectric Response Function

The dielectric function e(w, q) describes the response of a crystal to an electromagnetic field. The dielectric function depends sensitively on the electronic band structure of a crystal, and studies of the dielectric function by optical spectroscopy are very useful in the determination of the overall band structure of a crystal. In the infrared, visible, and ultraviolet spectral regions the wavevector q of the radiation is very small compared with the shortest reciprocal lattice vector, and therefore q may usually be taken as zero.

The dielectric constant e of electrostatics is defined in terms of the electric field E and the polarization P, the displacement D: D = eoE + P = eeoE. (19)

The defined e is also known as the relative permittivity. So long as the material is homogeneous, isotropic, linear, and local in its response, the dielectric response may be characterized quite generally by a frequency-dependent complex dielectric function e(w) which we write in terms of its real and imaginary parts as e(w) = e1(w) + -ri(w). (20)

Here, the quantity el (w) is called the real dielectric function whereas o (w) is the frequency dependent conductivity. At zero frequency e (0) becomes the static dielectric constant and a1(O) is the original dc conductivity, cdc.


38






39


Optical Reflectance

The optical measurements that gives an information on the electronic system are measurements of the reflectivity of light at normal incident on single crystals. The reflectance for light impinging onto an ideal solid surface can be derived from the boundary conditions for E and H at the interface. The boundary condition requires


E, + E, = Et. (21)


where the subscripts i, r, and t represent, respectively, the incident, reflected, and transmitted waves at the interface. A similar equation holds for H, but with a change in sign for H7. The magnetic field H is perpendicular to the electric field E and E x H is in the direction of the wave propagation. Thus, we can write


H, H, = Ht. (22)


In the vacuum, E = H, whereas in the medium,


H = N(w)E, (23)


as can be shown by substituting plane-wave expressions of the form exp i(q r -Wt) = expi[(w/c)fi r] into Maxwell's equations. (21), (22), and (23), are easily solved to yield the complex reflectivity coefficient r(w) as the ratio of the reflected electric field E. to the incident electric field Ei:

E, 1 N
r(o) = = 1 + N = p(w)eO(w), (24)


where we have separated the amplitude p(w) and phase 9(w) components of the reflectivity coefficient. By definition the complex refractive index N(w) is related to






40


the refractive index n(w), the extinction coefficient k(w), and the dielectric function e(w) by

N(w) = n(w) + ik(w) = e (25)


One quantity measured in experiments is the reflectance, which is the ratio of the reflected intensity to the incident intensity:

_(1 -n)2 + k2
R = rr* = P2 (1+n)2 + k2 (26)


The measured reflectance R(w) and the phase 9(w) are related to n(w) and k(w) by


v/e9 = r= (1 n) ik (27)
(1 + n) + ik'


and
-2k
tan = 1 2 V (28)


It is difficult to measure the phase O(w) of the reflected wave, but it can be calculated from the measured reflectance R(w) if this is known at all frequencies via the Kramers-Kronig procedure. Then we know both R(w), 9(w), and we can proceed by (27) to obtain n(w) and k(w). We use these in (25) to obtain e(w) = C1(W)+iC2(w), where el(w) and e2(w) are the real and imaginary parts of the dielectric function. The inversion of (25) gives

e1(w) = n2 k2, E2(W) = 2nk. (29)

We will show in data analysis section how to find the phase O(w) as an integral over the reflectance R(w) using Kramers-Kronig relations.






41


Infrared and Optical Technique

Fourier Transform Infrared Spectroscopy

The central component of a Fourier transform infrared spectrometer is a twobeam interferometer, which is a device for splitting a beam of radiation into two paths, the relative lengths of which can be varied. A phase difference is thereby introduced between the two beams and, after they are recombined, the interference effects are observed as a function of the path difference between the two beams in the interferometer. For Fourier transform infrared spectrometry, the most commonly used device is the Michelson interferometer.

The Michelson interferometer, which is depicted schematically in Fig. 17, consists of two plane mirrors, the planes of which are mutually perpendicular. One of the mirrors is stationary and the other can move along an axis perpendicular to its plane. A semi-reflecting film, called the beamsplitter, is held in a plane bisecting the planes of the two mirrors. The beamsplitter divides the beam into two paths, one of which has a fixed pathlength, while the pathlength of the other can be varied by translating moving mirror. When the beams recombine at the beamnsplitter they interfere due to optical path difference. The amplitudes of two coherent waves which at time zero have the same amplitude A(v) at wave number v, but which are separated by a phase difference kS = 2zrv6, can be written as


y1(z) = j A(v)ei2'vzdv, and y2(z) = j A(v)ei2v (Z-0d, (30)


where k is the propagation constant, v is the wave number and 8 is the optical path difference between the two waves. Using the law of superposition, one has


y(z) = y1(z) + y2(z) = j [A(v)(1 + e-i2rvb),i2uz]dv. (31)






42


The complex amplitude of the combined beam reaching the detector is AR(6, v) = A(v)(1 + e-i2r6). (32)

But the intensity B(v, 6) (irradiance or flux density) is

B(v, 6) = A*(6, v)AR(6, v) = A2(v)[1 + cos(2rv6)] = gS(v)(1 + cos2rv6), (33) where S(v) is the power spectrum. The total intensity at the detector is

I(6) = j B(v, 6)dv = j S(v)[1 + cos27rv6]dv. (34)

At zero path difference, the intensity at the detector is 1(0) = S(v)dv. (35)

At zero path difference all of the source intensity is directed to the detector; none returns to the source. At large path differences the intensity at the detector is just half the zero path difference intensity I(oo) = 1 j S(v)dv. (36)

because as 6 -+ oo the cos27rv6 term averages to zero, i.e., it is more rapidly varying with frequency than S(v).

The interferogram is the quantity [I(6) I(oo)]; it is the cosine Fourier transform of the spectrum. For the general case, the final result is obtained: B(v) = [I(6) 1I(0)]e-i2'r6d6. (37)

(37), at a given wave number v, states that if the flux versus optical path I(6) is known as a function of 6, the Fourier transform of [I(b) 1I(0)] yields B(v), the flux density at the wave number v.






43


Optical Spectroscopy

At high frequencies, the Fellgett advantage losses its importance due to the increasing photon noise in the radiation field. For this reason, a grating monochromator is normally used in the near-IR and visible frequency range.

Generally, a grating monochromator is used by applying the rule of diffraction. For a wavelength A,
n1
V=d (38)
2d sin9'

where d is the grating distant. At an angle 9, the first-order component of wavelength A satisfying A = asinO is selected. Meanwhile, any higher order components with wavelengths A, = A/n, or v,, = nv(n = 2,3,...), which could also pass through the slit are absorbed by the filter. The resolution is determined by the slit width and A9, which is the angle of rotation at each step.


Instrumentation

Bruker Fourier Transform Interferometer

To measure the spectrum in the far and mid-infrared (20 4000 cm-1), a Bruker 113V Fourier Transform interferometer is used. Different thickness of Mylar beam splitters, a black polyethylene filter, a bolometer and a Hg arc lamp as detector and source are used for far infrared (20 600 cm-'). A photocell and a globar source are used for mid infrared (450 4000 cm-1). A schematic diagram of the spectrometer is shown in Fig. 18. The sample chamber consists of two identical channels which can be used for either reflectance or transmittance measurements. The entire instrument is evacuated to avoid absorption by water and CO2 present in air.

The principle of this spectrometer is similar to that of a Michelson interferometer. Light from the source is focused onto the beamsplitter and is then divided into two






44


beam; one reflected and one transmitted. Both beams are sent to a two-sided movable mirror which reflects them back to be recombined at the beam splitter site. The recombined beam is sent into the sample chamber and detector. When the two-sided mirror moves at a constant speed v, a path difference 6 = 4vt, where t is the time as measured from the zero path difference. Next, the signal is amplified by a wideband audio preamplifier and then digitalized by a 16-bit analog-to-digital converter. The digitalized data are transferred into the Aspect computer system and are Fourier transformed into a single beam spectrum.


Perkin-Elmer Monochromator

Reflectance spectra from mid-infrared to ultraviolet (UV) frequency region are measured by a model 16U Perkin-Elmer grating monochromator. The basic concept of a grating monochromator involves shining a broadband light source on a grating and selecting a small portion of the resulting diffracted spectrum by letting it pass through an opening known as a slit.

A diagram of the spectrometer is shown in Fig. 19. Three sources-globar (GB), quartz-envelope tungsten lamp (W), and deuterium lamp (D2) are used for different frequency region. The light signal is chopped to give it an AC component which could then be amplified by a lock-in amplifier. Long-pass and bandpass filters eliminate unwanted orders of diffraction. A large spherical mirror images the exit slits of the monochromator onto either a reference mirror or a sample in the case of reflectance measurements. For transmittance measurements, the sample is mounted as close as possible to the focus of the second spherical mirror. The position of the detector is at the focal point of ellipsoidal mirror. Three detectors, a thermocouple (TC), a lead sulfite (PbS) photoconductor, and a silicon photodiode (576) are used to cover






45


the different photon energy regions. Table 2 lists the parameters used to cover each frequency range.

Polarizers could be placed after the exit slit and before the focus of the second spherical mirror if polarized reflectance and transmittance measurements were required. The polarizers used in the far infrared were wire grid polarizers on either calcium fluoride or KRS5 substrates. Dichroic polarizers were used at higher frequencies. (table 2)

The signal from the detector was fed into a standard lock-in amplifier. The lockin is then averaged over a given time interval. The time constant on lock-in could be varied the signal to noise ratio. After having taken a data point, the computer sent a signal to the stepping motor controller to advance to the grating position. This process was repeated until a whole spectrum range was covered. The spectrum was normalized and analyzed through the computer. Michelson Interferometer

A Michelson interferometer is an alternative instrument for measuring the spectrum in the 10 to 800 cm-1 region. In principle, this instrument works in the same way as the Bruker interferometer, but has a better S/N ratio at low frequencies below 100 cm-- due to large size and high power source. A mercury arc lamp is used as a source and the source is chopped to remove all background radiation. The combination of the thickness of a Mylar beam splitter and different filters are used to cover the corresponding frequency range.

The sample and detector are mounted in the cryostat. A doped germanium bolometer operating at 1.2 K is used as a detector. Data acquisition procedure is same as that of Perkin-Elmer grating monochromator.






46


Table 2. Perkin-Elmer Grating Monochromator Parameter


Frequency (cm-1)
801-965 905-1458 1403-1752
1644-2613 2467-4191 4015-5105 4793-7977 3829-5105 4793-7822 7511-10234 9191-13545
12904-20144 17033-24924 22066-28059 25706-37964 36386-45333


Grating" (line/mm)

101 101 101
240 240 590 590 590 590 590
1200 1200 2400 2400 2400 2400


" Note the grating line number


Slit width (micron)
2000 1200 1200 1200 1200 1200 1200 225 75 75 225 225 225 700 700 700


Source Detector


GB GB GB GB GB GB
W

W
W

W
W

W
W

D2 D2 D2


TC TC TC TC TC TC TC PbS PbS PbS PbS PbS 576 576 576 576


per cm should be the same


order of the corresponding measured frequency range in cm-1. dc Resistivity Measurement Apparatus

The experimental arrangement for measuring the resistivity is illustrated in Fig. 20. The measurements were made as a function of temperature from liquid helium temperature (- 4 K) to room temperature (- 300 K) using a lead probe






47


which was thermally anchored to the cold head of a closed-cycle refrigerator (CTI Cryogenics). The probe tip houses four electrodes. The sample can be electrically connected to these electrode with 20 pm diameter gold wire using silver paint. The sample temperature was monitored with a temperature controller (Lake Shore Cryotronics) that was connected to a silicon diode sensor which was attached to the cold head of the cryostat unit.

We measured the resistance, R = V/I, for the configuration of leads schematically shown in Fig. 20, using a standard ac phase-sensitive technique operated at 22 Hz at a current of ~ 700 pA. The results were insensitive to the size of the current. Before measurement, four stride contacts were formed on ab plane by the evaporation of silver plate. An annealing procedure for good Ohmic contact was performed at a temperature of 300 K in flowing 02 for 5 hours. Contact resistance values in the range 10 to 20 1 were obtained by the bonding of Au wires with silver paint. The electrical resistivity values p which is defined as p = RAIL were obtained by measuring the crystal dimensions, where A is the cross sectional area for current flow and L is the length along the voltage drop.

Data Analysis; Kramers-Kronig relations

To obtain the optical conductivity al(w) which is a more fundamental quantity one applies a Kramers-Kronig transform to the reflectance R(w), which yields the phase shift O(w). Formally, the phase-shift integral requires a knowledge of the reflectance at all frequencies. In practice, one obtains the reflectance over as a wide frequency range as possible and then terminates the transform by extrapolating the reflectance to frequencies above and below the range of the available measurements.

Concerning the low frequency extrapolation, we find that the conductivity at frequencies for which there is actual data is not affected significantly by the choice of






48


the low frequency extrapolation. For insulating samples, the reflectance is assumed constant to dc. In the case of metallic samples, a Hagen-Rubens relation, R(w) = 1 AVC, was used. In the superconducting state, we have used the formula R = 1- Bw4, in which R goes to unity smoothly as w approaches zero.

The high frequency extrapolation has significant influence on the results, primarily on the sum rule derived from the optical conductivity. We reduced this effect by merging our data to the reflectance spectra for insulating phase of published papers which extend up to 37 eV (300000 cm-1) for Nd2-.Ce.-CuO4, La2-,SrCuO4,32 and Bi-O superconductors.33 We terminated the transform above 37 eV by using the reflectance varying as 1/w4, which is the free electron asymtotic limit.

The Kramers-Kronig relations enable us to find the real part of the response of a linear passive system if we know the imaginary part of a response at all frequencies, and vice versa. They are central to analysis of optical experiments on solids. Let us consider the response function as a(w) = al(w) + ia2(w). If a(w) has the following properties, a(w) will satisfy the Kramers-Kronig relations:

a1(w) = P 020 2 ds, and a2(W) = -oP j ()ds. (39)

First, a(w) has no singularity, and a(w) -- 0 uniformly as IwI -+ oo. Second, the function a1 (w) is even and a2(w) is odd with respect to real w.

We can apply the Kramers-Kronig relations to reflectivity coefficient r(w) viewed as a response function between the incident and reflected waves in (24). If we apply

(39) to

In r(w) = In R1 + i9(w), (40)

we obtain the phase in terms of the reflectance:

1 |s+wldlnR(s)d
( = In Is+w ds. (41)
2x 7r |o s w| ds






49


According to (41) spectral regions in which the reflectance is constant do not contribute to the integral. Further, spectral region s > w and s < w do not contribute much because the function InI(s + w)/(s w)I is small in these region.

Now, we know R(w), 6(w), and we can use (27) to obtain n(w) and K(w). We use these in (25) to obtain e(w) = 1(w) + ie2(w). In this way we can find every optical constants from the experimental R(w).


Dielectric Function Models

Two Component Aproach

The two component model (Drude and Lorentz) are frequently used to describe the optical properties of materials. The free-carrier component was fit to a Drude model, while the bound carrier interband transition and lattice vibrations were fit by Lorentzian oscillators. The model dielectric function is

2 2
e(w) = e1(w) + 4ri l(w) =- 2 + + (42)
w 2 + iw/r Wj -W2-Wy


where wpD and 1/r are the plasma frequency and relaxation rate of the Drude carriers; Wej, wp, and -yj are the center frequency, strength, and width of the jih Lorentzian contribution; and e,, is the high-energy limiting value of e(w).

In this picture for high T, materials, the free carriers track the temperature dependence of the dc resistivity above T,, while condensing into the superfluid below T. In contrast, the bound carriers have an overdamped scattering rate that exhibits very little temperature dependence.






50


One Component Analysis

Another approach to analyze the non-Drude conductivity is to assume an inelastic scattering of the free carriers in the low frequency with a underlying excitation spectrum. This interaction gives a strong frequency dependence to the scattering rate and an enhanced low frequency effective mass of the free carriers. This approach has been proposed by Anderson34 and applied to heavy Fermion superconductors.35 The one component picture of the optical conductivity can also be described by the "marginal Fermi liquid"31 (MFL) and the "nested Fermi liquid"37,38 (NFL) theories.

According to Varma et al.,36 the quasiparticle self energy E of the marginal Fermi liquid has a imaginary part which qualitatively goes as

-Im EMw ~ rAI (43)
7Aw, w > T
where A is the electron-phonon coupling constant. There is an upturn in the effective mass, with the mass enhancement proportional to A. In the NFL approach of Virosztek and Ruvalds,37,38 the nested Fermi liquid has

-Im E = a max(PT, Iw 1), (44)

where a is a dimensionless coupling constant. This gives a scattering rate that is linear in T at low w and linear in o at high w.

For calculating the frequency dependent scattering rate l/r(w) and effective mass m*(w), the complex dielectric function is described by a generalized Drude model in terms of the complex damping function, also called a memory function, as y = R(w) + iI(w),
2
C(W) = C". 45
O + i7(45) where e. represents interband contribution not involving the charge carriers and op = 4lrne2/m* is the plasma frequency, with n the carrier concentration and m*






51


the effective mass. We can also rewrite (45) in terms of the frequency dependent effective mass m*:


C(W) = f -


w2 pi


w(m*(w)/mb)[w + i/r*(w)],


where 1/r*(w) is the (renormalized) scattering rate. If we compare (45) with (46), we can extract two relations:


m 1(w) 2 w
-1- =1--ReE(-)
Mb WW 2


and


M *W
-1/r*(w) = R(w) = -2ImE(-), mb


(47)


(48)


where R(w) is the "unrenormalized" scattering rate and E is the quasiparticle selfenergy.


(46)





52


Movable mirror Source
B 2


S
Beam x/2
splitter



D0

Detector


Fig. 17. Schematic diagram of Michelson interferometer.







53


g






47.

Itc


a


Source Chamber Near-, mid. or far4R sources Automa~ted Aperture


HII lteromete Chamber e Optical fIfer d Automati beamsplitter changer e Two-sied movable mirro
1 Control interferometer g Reference lanr h Remnote ontrol alignment mirror


---


'U!


It






k


Sample Chamber Sapl focus Reference focus


IV Detector Chamber It Near-. mid-, or far-iR
detectors


Fig. 18. Schematic diagram of IBM Bruker interferometer.


U


)\


\/


Li


r7


i

























vacuum tank


0
C-.

(b


I-1






'-1


C-,

Oq


0

0
0 '*1
0


C-.
0 '-1


o glowbar t~ungsten SOdeuteri















Sor
sample


chop filte Kra


detector


sample rotator






5sanle


01


Lbzft--Ti












Voltage Ik V -0-i


Current J


Four-Point Probe


output
OsclLttor
195


21 KAl






input Lock In v
Amplifier S






196


Fig. 20. Top: Simple arrangement for four probe measurement. Bottom:
Experimental arrangement for the resistivity measurement.


55


Ill


-0 1I 9


Iv


i













CHAPTER VI
CE DOPING DEPENDENCE OF
ELECTRON-DOPED Nd2-..,CeCuO4


In this chapter, we report optical reflectance and conductivity spectra from the far-infrared to UV on the a-b plane of electron-doped Nd2-,Ce.CuO4 for very different Ce concentrations (0 < x < 0.2). This compositional range covers the antiferromagnetic insulator, the high T, superconductor, and non-superconducting "overdoped" metallic samples.

The motivation behind this study is twofold. First, in spite of a lot of theoretical and experimental studies, there is still little understanding of the normal-state excitation spectrum, especially the low-energy-scale physics near the Fermi level, of strongly correlated high T, cuprates. In particular, the differences between hole doping and electron doping in the transfer of spectral weight from high to low frequencies are not well understood. Basically, doped holes and electrons show different orbital characteristics: the holes introduced by doping are mainly in 2p orbitals whereas the doped electrons have 3d orbital character. Therefore, in the three band Hubbard model the motion of holes will depend differently on tpd than motion of the electrons. Eskes et al.21 have shown that in the localized limit (tpd = 0), the transfer of spectral weight with electron doping is similar to the case of the Mott-Hubbard model, whereas the development of spectral weight with hole doping is the same as that of a simple semiconductor. At present, the results for doping dependence in hole-doped systems3,39-46 are somewhat in agreement with one another, but the results for the electron-doped system6-9,47~53 are still controversial.


56






57


A second motivation is due to the nature of the insulating phase of Nd2CuO4, which has the tetragonal T' structure, without the apical 0 atoms of La2-.,.Sr.CuO4 which has the T structure. We expect more simplified electronic structure in the T' phase than in the T phase, allowing us to examine in detail the electronic structure of high T, cuprates.

We first describe the temperature dependent optical reflectance and conductivity for the insulating phase. In the next section, we will present the doping dependent reflectance for the a-b plane of Nd2-,Ce Cu04 and examine a variety of optical functions obtained from a Kramers-Kronig analysis of the reflectance spectra. The optical conductivity al(w) for each doping level is analyzed by the one component and two component models. The doping dependence of the low frequency spectral weight and the high frequency spectral weight are also discussed.

Results and Discussion of Insulating Phase

The room temperature reflectance and conductivity spectra of the undoped compound Nd2Cu04_b are shown in Fig. 21(a) and Fig. 21(b), respectively. The conductivity spectrum exhibits a fundamental absorption edge near 1.5 eV which is attributed to the charge transfer excitations between 0 2p and Cu 3d orbitals on the CuO2 planes. Its energy is lower than in the high T, cuprates with the T and T* phases." (2.0 eV for the T phase La2---SrCu04 and 1.7 eV for the T* phase LaDyCuO4.) In the three-band Hubbard model, this strength is roughly given by 2d/A when A > tpd,55 where tpd represent the nearest neighbor transfer integral. The gap energy A increases as Cu-0 spacing is smaller, because of an effect similar to level repulsion in atoms.

The spectrum is featureless below the gap except for four optical phonons in the far-infrared region, shown in Fig. 22. This reflectance is typical of an insulator. It






58


should be noted that optical transmission spectra in undoped materials show other absorption features in the energy range 0.2 to 1 eV. For example, weak absorption bands near 0.5 eV were first observed in undoped single crystals of single layered T, T*, and T' structures by Perkins et al."6 and multi-layered YBa2Cu307_ by Zibold et al.57 In these studies, they suggest that these bands result from exiton-magnon absorption processes. The spectra in Fig. 21 also show a strong transition around 5 eV, which is observed above 6 eV in optical spectra of the T and T* phases. This peak is located at higher energies in the other structures for the same reason as the larger charge transfer energy.

A group theoretical analysis of the phonon modes in Nd2CuO4 yields 3A2,+4E,.58-60 The A2. modes are observed in the c polarization spectrum and the E. modes corresponds to an atomic motion parallel to the a-b plane. Figure 22 displays the a-b plane reflectance in the far-infrared region as a function of temperature. We clearly observe four strong phonon bands. As the temperature is reduced, the phonon lines become sharper. Since all spectra show an insulating behavior, we extrapolate them to zero frequency assuming asymtotically a constant reflectance. Then we obtain oi(w) and Im(-1/e) by K-K transformations.

The temperature dependence of the a-b plane phonons is shown in Fig. 23. The upper panel shows ai(w), whereas the lower panel shows Im(-1/e). The former determines the TO phonon frequencies, whereas the latter the LO phonon frequencies. Four phonon bands occur at 131, 303, 347 and 508 cm-1 at room temperature. These phonon modes are similar to the case of La2Cu04, but the phonon energies in Nd2CuO4 are lower than in La2CuO4. This difference is primarily due to a result of a larger unit cell dimension (longer bond lengths) in the former material.61 These phonon bands result from four motions: a translational vibration of Nd atom layer







59


against the CuO2 plane (131 cm-1), a Cu-O bending mode from the in-plane Cu-O bond angle modulation (303 cm-1), an out-of-plane 0 translational mode (347 cm-1), and an in-plane Cu-0 stretching mode from the Cu-O bond distance modulation (508 cm-1). The function Im[-1/e(w)] is shown in Fig. 23(b), showing large LO-TO phonon splittings. The temperature dependence of phonon frequencies shows the redshifts with increasing temperature as expected.

The optical conductivity of pure undoped sample should vanish up to 1.5 eV, above which the charge transfer excitations occur. However, the optical conductivity spectrum a1(w) in Fig. 24(b) shows a resonant absorption near 1500 cm-1. This new absorption may be attributed to the deviation from an oxygen stoichiometry of Nd2CuO4 single crystal during oxidation process. This result indicates that our sample is lightly doped with electrons.

In Fig. 24(a), the optical reflectance of the band near 1500 cm-1 is plotted as a function of frequency at several temperatures. Figure 24(b) also shows the temperature dependence of the optical conductivity a1(w) of this peak. This peak is very interesting due to the fact that its energy is close to the antiferromagnetic exchange energy J.62 For this reason, one might expect that the origin of this peak is due to the interaction of doped carriers with magnetic degrees of freedom. Several mechanisms, including self-localized polarons,63-66 photoexcitation of localized holes,6T and magnetic excitations,5 have been proposed to explain this peak.

We have fit this band with the usual Lorentzian. The results for the fitting parameters are shown in Fig. 25. The peak position and peak width shift to higher frequencies by an amount comparable to thermal fluctuation energy kBT as the temperature of sample is increased; that is,


we = WeO + 0.6 kBT, 7 = o + 1.6 kBT,


(49)






60


where kB = 0.695 cm-1/K. The broadening of the line is like the behavior of the free carrier conductivity, which shows a linear temperature dependence of the scattering rate. We find a coupling constant A ~ 0.25 using a formula h/r = 21rAkBT. This is comparable to the coupling constant obtained from the behavior of free carriers in other high T, cuprates. A similar temperature dependence has been observed in the a-b plane conductivity spectrum in lightly doped Nd2CuO4-y single crystal by Thomas et al.5 They suggested that this band is related to a bound charge coupled to the spin and lattice excitations. Unlike the result of Thomas et al., our result shows that the oscillator strength of this band increases with decreasing temperature. We will discuss this band in the next chapter again for lightly hole-doped La2-zSrCuO4 experiments.

The charge transfer band observed near 1.5 eV also has a temperature dependence. Figure 26 shows the reflectance spectra (a) and the optical conductivity spectra (b) calculated from the reflectance spectra using K-K transformations. The reflectance spectrum in Fig. 26(a) at room temperature clearly shows two peaks near 1.36 eV and 1.6 eV. As the temperature decreases, the spectral weight around 1.36 eV at 300 K shifts to the peak near 1.6 eV and the sum rule is satisfied.


Doping Dependence of Optical Spectra

Optical Reflectance

The reflectance spectra for the a-b plane are shown in Fig. 27 between 80 cm-1 (10 meV) and 42 500 cm-1 (5.3 eV) for various Ce concentrations. Other metallic samples with Ce concentrations of x = 0.18 and x = 0.20 were measured, too. But, these spectra are not shown in Fig. 27, because they are very similar to the spectrum of Ce concentration x = 0.19. With doping the spectral weight around 1.5 eV peak is






61


reduced and a reflectance edge rapidly develops below 1 eV. Fig. 27 also shows that the position of the edge shifts to higher frequency with increasing doping and is almost saturated in the metallic regime where 0.14 < x < 0.19. Another notable feature is that the charge transfer band near 1.5 eV moves to higher frequency with increasing dopant concentration x. This behavior is obvious in this system. In addition, there is a systematic change of reflectance between 3 eV and ~ 5 eV with x. A similar behavior has also been observed for hole-doped La2-.SrCuO43 and YBa2Cu307Tsystems.39,44

The magnitude of the reflectance of Nd2-Ce CuO4 at low frequencies is typically larger than the results for hole-doped La2-Sr.CuO4 and YBa2Cu307_6. For example, the magnitude near 600 cm-1 at high doping levels for our results is about ~ 92%, whereas the results for hole-doped La2-xSrxCuO43,39 are ~ 85%.

Among the four E, optical phonons in undoped crystal below 600 cm-1, two infrared active phonons near 301 and 520 cm-1 are visible even in heavily doped crystals. However, two weak phonon bands observed at 131 and 345 cm-1 in the spectrum of undoped crystal are screened out from free carriers in the metallic phase.


Optical Conductivity

The frequency dependent optical conductivities obtained from a KK transformation of the reflectance spectra are shown in Fig. 28 and Fig. 29. We can better observe the influence of doping on spectral response by considering optical conductivity. The a-b plane conductivity of Nd2-...CexCuO4 shows interesting changes with doping. As suggested by the reflectance spectrum in Fig. 27, with doping the conductivity of the charge transfer band above 1.2 eV is systematically reduced, whereas the low frequency spectral weight below ~ 1.2 eV rapidly increases.






62


For a barely metallic sample with x = 0.11, the conductivity below ~ 1.2 eV is composed of two components: a narrow band centered at w = 0 and a midinfrared absorption band centered at 4400 cm-1 (0.55 eV). The narrow band decays much more slowly than the Drude spectrum, which has a w-2 dependence. Upon further doping, this band grows rapidly up to x = 0.14, but grows slowly with dopant concentration x in the metallic phase. On the other hand, upon doping the band near 0.55 eV slightly shifts to lower frequencies and the oscillator strength is a little reduced. However, this peak is not visible as a distinct maximum in the spectra of more highly doped samples due to the mixing with the Drude-like component. Similar qualitative results have also been reported in hole doped La2-..SrCuO4 and YBa2Cu3O7-.6

It is interesting to note that the phonons observed at 301 and 487 cm-1 have about the same intensity with very sharp feature and almost same phonon position at all doping levels, whereas the electronic background increases. This implies that these phonon modes are not screened in the ordinary sense of having their TO-LO splitting decreases to zero.

Significantly, a-1(w) in Fig. 29 shows a dramatic change at frequencies above the 1.2 eV. First, the spectral weight at energies between 1.2 eV and 3 eV systematically decreases with doping. This band has been identified as a charge transfer excitation, in which electrons are transferred from 0 to Cu site. This result should be compared to those obtained in the hole-doped systems, where the charge transfer band shows over a wide energy range between 2 eV and 5 eV due to the contribution of the charge transfer excitations between the apical 0 atoms and Cu atoms. Second, upon doping the spectral weight near the 5 eV peak in the spectrum of undoped crystal is gradually reduced, and another peak which is not observed in the spectrum of






63


undoped crystal is shown near 4.5 eV in the spectrum of x = 0.11. Further, with doping the strength of this peak is reduced and its position shifts to lower frequencies. Third, at high doping levels, a new shoulder near 3.6 eV grows with the decreasing of the strength of 4.5 eV and 5 eV peaks. This seems to transfer the spectral weight of 5 eV peak to energy region between 3 eV and 5 eV with doping. This behavior of doping dependence in high energy region is different from the results obtained from hole-doped systems, where the spectral weight above 1.5 eV systematically decreases with doping.

Effective Electron Number

In order to describe a doping effect quantitatively, we have estimated the effective number of carriers per CuO2 plane. The effective electron number with mass m* = me, Nff(w) is defined according to


2meVeii f'
Nff(w) = 11 fs(w)dw, (50)
0

where e, m, are the free electron charge and mass respectively, and V,11 is the volume of one formula unit. For Nd2-zCexCuO4, we used the unit cell volume, 68 Vel = 187 A3 and the number of Cu atoms per unit cell, Ne, = 2. Neff(w) is the effective number of carriers per formula unit participating in optical transition at frequency below w.

Figure 30 illustrates the effective electron number for the different Nd2-xCexCuO4 samples. In the insulating phase, x = 0, Neff (w) remains nearly zero throughout the optical gap without a mid-infrared band contribution, but increases rapidly above the absorption band of charge transfer excitation. For metallic samples, Nfg (w) exhibits an initial rise due to the Drude band at zero frequency. The following steep rise






64


is the contribution of the mid-infrared bands, which ends around 10 000 cm-1 (1.2 eV). Next, more steep rises come from the contributions of the charge transfer band and high energy interband, respectively. This behavior is very similar to the results obtained in hole-doped cuprates.

The low frequency Neff(w) for metallic samples is plotted in Fig. 31 up to 1000 cm-'. Notably, the integrated spectral weight of superconducting sample of x = 0.15 exhibits a rapid rise at low frequencies below 200 cm-1 due to the strong Drude contribution, and is very strong at frequencies below 800 cm-1 compared with that of slightly overdoped sample of x = 0.16. Finally, two curves for x = 0.15 and x = 0.16 merge near 1000 cm-1. This implies that the strength of the mid-infrared band in x = 0.15 is a weaker than in x = 0.16.

Another important result of our measurements is that Neff(W) at high frequency above 3 eV gradually increases with doping. This is particular in our system. As we will discuss later, this is due to the anomalous strong Drude and mid-infrared bands caused by doping. In order to satisfy sum rule, this result suggests that another high energy band above 4 eV loses spectral weight with electron doping. This is compared to the results of hole-doped La2-..SrCuO4 and YBa2Cu307.6 In these studies, the only spectral weight of the charge transfer region between 1.5 eV and 4 eV is transfered to low frequencies below 1.5 eV, and hence Neff(w) intersects near 3 eV with increasing doping. It is noteworthy that Neff at 3 eV is a factor of 2 larger than that of La2-,Sr.CuO4.

Loss Function

In this section, we describe the energy loss function, Im[1/e(w)]. This function is the probability for energy loss by a charged particle that passes through a solid. It can also be calculated from Im[1/e(w)] = e2(w)/[Eq(w)2 + f2(W)2]. The peak






65


position corresponds to the zeros of e1(w). In a simple Drude model, the maximum of the energy loss function determines the longitudinal plasma frequency of free carriers, corresponding to the zeros of the dielectric function e(wL), and its maximum position shifts to higher frequencies with doping according to Wp = (41rne2/m)1/2. However, the bound carriers in high T, cuprates which contribute a positive dielectric response dielectrically screen the free carrier response, and also lower Wp. The maximum value of Im[1/e(w)] is given approximately by the screened plasma frequency


"4i=ne2
P ec~ m. ect

where ect is the the e (w) value at the charge transfer gap frequency.

Figure 32 shows Im[1/e(w)] with Ce doping as a function of frequency. The result for x = 0 is very small below 1.2 eV except phonon modes in the far-infrared region, and shows a bump near 1.5 eV which is associated with the charge transfer excitation. The spectrum of x = 0.11 shows a featureless continuum near 1000 cm~1 and a broad peak around 7200 cm-1 (0.9 eV). With doping this peak position moves to slightly higher frequencies, where its maximum position corresponds to the appearance of a reflectance edge with doping. For 0.14 < x < 0.2, the peak positions occur near 1.1 eV and are insensitive to Ce doping concentration, inconsistent with the simple Drude model. This indicates that the value of n/m*ect in (51) is insensitive to doping. Figure 32 also shows that the peak position of the superconducting sample with x = 0.15 is observed at higher energy than in slightly overdoped sample with x = 0.16. This may suggest that the superconducting sample has more free carriers or low effective mass of charge carriers. A broad peak width (0.5 eV) in Im[1/e(w)] is due to the anomalous mid-infrared absorption caused by the incoherent motion of free carriers.






66


Our results for Nd2-.Ce, CuO4 axe similar to those of La2-,Sr.CuO4 by Uchida et al.,3 where the zero crossing of el(w) for the metallic samples is pinned near 0.8 eV due to strong mid-infrared absorptions. In contrast, the dielectric response for YBa2Cu307_. obtained by Cooper et al.39 shows almost linear doping dependence of zero crossing of ei (w), exhibiting nearly free carrier behavior.


Temperature Dependence of Optical Spectra

The temperature dependence of the reflectance between 80 cm-1 and 4 000 cm-1 was measured in order to study the applicability of the Drude model. For nonsuperconducting metallic samples, the change of reflectance between 10 K and 300 K is less than 2% in the far-infrared region, as shown in Fig. 33 for metallic samples of x = 0.16 and x = 0.19. However, for superconducting sample of z = 0.15, the reflectance change between these temperatures is about 3.5% near 600 cm-1.

Figure 34(a) shows the temperature dependent reflectance of the superconducting sample, x = 0.15, in the frequency range between 80 cm-1 and 2000 cm-1. As the temperature decreases, the magnitude of the reflectance exhibits a systematic increase. The optical conductivity shows a clear picture of a Drude behavior. Figure 34(b) shows a1(w) obtained after a K-K analysis of reflectance spectra in Fig. 34(a). The al(w) spectra clearly explain the sum rule. As the temperature is reduced, the spectral weight between 500 cm-1 and 2 000 cm-1 is transferred to lower frequencies, corresponding to the narrowing of the Drude band at low temperatures.

We have fit our results with the two component model, a Drude part and several Lorentzian contributions:


2 N 2
)=D + 2 ;j + (52)

j=1 W; W ZL)f






67


where the first term is a Drude oscillator, described by a plasma frequency wpD and a relaxation time r of the free carriers, the second term is a sum of peaks in o1(w), with w2, wj and -y, being the resonant frequency, strength and width of the jth Lorentz oscillator, and the last term is the high frequency limit of e(w).

The Drude components at five temperatures for the superconducting sample, x = 0.15, are depicted in Fig. 35. The Drude component is defined as the conductivity after the average mid-infrared component is subtracted. The temperature dependence of the Drude part satisfies the ordinary Drude behavior. The inset in Fig. 35 shows the temperature dependence of the scattering rate obtained from the fits. The Drude plasma frequency wpD ~ 11200 cm-1, is nearly T-independent, while the temperature dependence of the scattering rate 1/r is consistent with the behavior of the dc resistivity. For example, 1/r is non-linear in T and reduces to half of the 300 K value at 15 K. The dc resistivity value from the four-probe measurement is good agreement with that obtained from simple Drude formula:

2
Pdc = 4( D (53)
42r(1 /r) dc


We emphasize here that the Drude plasma frequency is larger, and 1/r is a little smaller than the values obtained for La2-,SrZCuO4. Also, the value for Wpd is larger than the results obtained by any other experiments for electron-doped system.


One Component Approach

As suggested in previous section, al (w) does not fit the simple Drude formula (a1(w) OC w-2). Especially, a1(w) reveals a strong spectral weight in the mid-infrared region, compared to that at zero frequency. Another approach to analyze this nonDrude conductivity is to assume an inelastic scattering of the free carriers in the






68


low frequency range with a underlying excitation spectrum. The carriers derive a frequency and a temperature-dependent self-energy. The imaginary part goes like ImE ~ max(w, T). This quasiparticle damping has been described in the framework of the"nested Fermi liquid"(NFL)37,38 and the "marginal Fermi liquid"(MFL) models.36 We analyze the non-Drude conductivity of Nd2-.Ce.CuO4 by using a generalized Drude formula with frequency dependent scattering rate.


2
e(w) = -h w[m*(w)/m0[w i/r*(w)]' (54)



where eh is the background dielectric constant associated with the high frequency contribution and the second term represents the effects of frequency dependent damping of carriers. m*/mo represents the effective mass enhancement over the band mass and 1/r*(w) = [l/r(w)](m/m*(w)] the renormalized scattering rate.

Figure 36 shows the m*/mb and 1/r*(w) curves for four samples below 5000 cm-1. We used w, = 20000 22000 cm-1, and e = 5.0 ~ 5.2 in the infrared region for different samples. At low frequencies, the behavior of m*/mb illustrates the coherent motion of carriers, causing the low frequency mass enhancement. This may be due to the interaction of carriers with phonons, or spin and charge excitations of carriers. Our results also suggest that the quasiparticle excitations increase with decreasing doping concentration. This is consistent with other doping dependence results for hole-doped systems. However, the mass enhancement of Nd2-.CeCuO4 is a little bit smaller than those obtained by hole-doped systems. As the frequency is reduced, the effective carrier mass decreases, and approaches to the band mass at high frequency.






69


The renormalized scattering rate 1/r*(w) and the effective mass enhancement m*/mb can be also related to the imaginary part of quasiparticle self energy by M* 1 w
-- = -2 ImE-. (55)
mo r*(w) 2


Figure 37 illustrates the imaginary part of quasiparticle self energy, Im E of Nd2- CeCuO4 crystals below 5000 cm~1. Im E in Fig. 37 is analyzed in several ways.

First, for a barely metallic crystal (x = 0.11), the imaginary part of self-energy deviates from the linearity and reveals a power law between 0.1 eV and 0.6 eV, reflecting the proximity to the phase boundary of the insulator. With doping we see a steady decrease in the quasiparticle interaction and finally, a linear slope of Im E in high doping concentration of x = 0.19.

Second, for superconducting sample (x = 0.15), Im E increases linearly with w below 5 000 cm-1. This is in a good agreement with the predicted behavior in the MFL. According to the MFL theory, the imaginary part of quasiparticle self energy

Im E has the form 'rAw over T < w < we, where w, is the cutoff frequency. We estimated a coupling constant A = 0.15 0.01 from the slope between 500 cm-1 and 2500 cm-1. This estimated value seems to be rather low compared with the results obtained from hole-doped systems. It is also interesting to compare the result for superconducting sample of x = 0.15 with the result for slightly highly doped sample of x = 0.16. The data of x = 0.15 show less quasiparticle excitations than that of x = 0.16. This may suggest that too much quasiparticle interaction causes the reduction of superconductivity in high T, cuprates.

Third, the quadratic temperature dependence of dc resistivity in superconducting sample does not agree with our analysis. We might expect a quadratic dependence






70


in w of scattering rate from dc resistivity. Ordinary Fermi liquid state requires the scattering rate varying as w2. Nevertheless, our result in the high T, regime is consistent with numerous models of the normal state in which strong quasiparticle damping is assumed. Also, our results with doping suggest that the electronic state of very heavily doped CuO2 plane may be acquire the nature of a Fermi liquid.


Doping Dependence of Low Frequency Spectral Weight Drude and Mid-infrared Band

We have emphasized that the spectral weight of the high frequency region above the charge transfer (CT) band is transferred to low frequencies with doping. Such a spectral change indicates that the conduction and valence bands of the CT insulator are reconstructed by doping. In the metallic state, the optical conductivity may be considered as three parts; a free carrier contribution centered at w = 0, mid-infrared bands, and high-energy interband transitions above the charge transfer gap. In order to describe empirically the absorption bands produced by doping, we have fit the o1(w) of each sample to the two component model. We here discuss in detail each band and how its strength changes with Ce doping. The strength of each band j is related to the plasma frequency in the fitting parameters by the relationship


2 47re2 N,
ogj(eV) = ---a (56)
-m* Veii

We estimated w23 (eV2) = 14.88 Ni, using Vceii = 187 A and two Cu atoms per unit cell, where N, is the effective electron number per Cu atom of band j.

For free carrier contribution, we extracted the spectral weight of a Drude oscillator

(ND) in the unit of electron number per Cu atom as a function of Ce concentration x from the sum rule restricted to the Drude conductivity, aD. Figure 38 (circles)






71


represents ND. However, it is difficult to define the Drude part from o (w) in the metallic phase due to the mixing with strong mid-infrared bands. Thus, ND in Fig. 38 represents with large error bars. Figure 38 illustrates that a Drude strength is very low up to Ce concentration of x = 0.11 and is roughly proportional to the dopant concentration x in the metallic phase, as expected for the generation of carrier by adding of electrons in the CuO2 plane. Our results are in good agreement with the phase diagram of Nd2-.,CeCuO4 6970 which shows the insulating phase at a wide Ce compositional range up to z = 0.12. As mentioned earlier in the effective electron number section, the superconducting sample of x = 0.15 has a very strong Drude band. This result is consistent with the theoretical observation71 described by the extended Hubbard model that upon electron doping the Fermi level lies directly in the Van Hove singularity of the upper Hubbard band at a certain doping level. This concentration may be the superconducting sample with x = 0.15.

As shown in the oi(w) of Fig. 28 and Fig. 29, Ce doping in Nd2CuO4 clearly induces the formation of strong mid-infrared bands. Upon doping, these mid-infrared bands continue to grow at high Ce concentrations and tries to merge with the Drude peak. In Fig. 38, we also plot the strength of total mid-infrared bands, NMID, as a function of Ce concentration x (squares). NMID is estimated from several Lorentzian fits to o1(w) in the low-frequency part below the charge transfer gap. NMID also represents with large error bars due to the ambiguity of a mid-infrared band near the Drude part. The strength of the total mid-infrared bands increases rapidly at low doping, but slowly at high doping levels. We stress here that with the Drude band the strength of total mid-infrared bands of Nd2-.Ce.CuO4 is very strong compared to that of hole-doped La2-zSr.CuO4.






72


Transfer of Spectral Weight with Doping

Next, we interpret the low frequency excitation near the Fermi level transferred from the high frequency region as a function of Ce doping x. This is done by computing the effective electron number Neff(w) of the Drude and total mid-infrared bands which corresponds to all electrons that are introduced by doping and comparing with hole-doped La2-..SrCuO4 system of Uchida et al.3

Figure 39 represents the low frequency spectral weight below 1.5 eV of hole-doped La2-..,SrCuO4 of Uchida et al.(left) and the low frequency spectral weight (LFSW) of electron-doped Nd2-_.CeCuO4 for our results (right). The solid lines in Fig. 39 correspond to the localized limit (no p-d hybridization) in the charge transfer system for hole-doping and electron-doping cases. In the localized limit, upon doping the LFSW of electron-doped system is expected to grow similar to the Mott-Hubbard case, where the LFSW goes to 2x with doping x due to the restriction of doubly occupied states of doped carriers, because electrons are doped primarily on Cu sites. For hole-doped system, LFSW grows as x with doping x as semiconductor case, since holes introduced by doping on 0 sites occupy almost free particle levels and scatter weakly off the Cu spins. However, Meinders et al.4 have shown that when the hybridization is large, the LFSW of hole-doped system becomes similar to that of the MH system and the electrons as well as the holes show strongly correlated behavior.

Our results for Nd2-XCe CuO4 show a electron-hole symmetry at low doping levels and a prominent electron-hole asymmetry. The LFSW associated with the CuO2 plane grows faster than 2x with doping x, consistent with the expectation of the MH model, where the lower Hubbard band (LHB) as well as the upper Hubbard band (UHB) loses the spectral weight. The greater LFSW than 2x may result from a large impurity band contributions in T' phase materials and the charge transfer






73


excitations. Especially, a strong spectral weight in the metallic phase around x = 0.15 may reflect the contribution of the charge transfer excitations. This is a spectral weight transferred from the p-like correlated states to the low frequency region. In contrast, the LFSW in La2-,Sr CuO4 goes to 2x at high doping levels and x at high doping levels with doping concentration x.

Doping Dependence of Charge Transfer Band

Figure 40 represents the variation of the charge transfer bands with Ce doping. The charge transfer conductivities are obtained after subtracting high energy interbands, and the Drude and mid-infrared bands. For the charge transfer band in insulating Nd2CuO4, two contributions appear. One (CT1) is a week and narrow band with center frequency near 12900 cm-1 (1.6 eV) and the other band (CT2) is a relatively strong and broad band near 16800 cm-1 (2.08 eV). We can also see two peaks in the spectrum of x = 0.11 near 14 000 cm-1 (1.74 eV) and 16 800 cm-1 (2.08 eV), respectively. At higher doping levels, only one band appears. Figure 40 also shows that the strength of the CT1 and CT2 bands decreases with increasing doping concentration x.

The CT1 band is related to the abrupt decrease of its strength as a result of the decrease of the intensity of UHB upon doping. Upon electron doping the position of CT1 band shifts to higher frequency (from 1.6 eV for x = 0 to 1.74 eV for x = 0.11) and its spectral weight (~ 11 000cm-1) rapidly decreases, finally disappearing for x > 0.14. The spectrum for x = 0.11 in Fig. 40 shows the very weak CT1 band of strength 2 000 cm-1. Figure 41 illustrates the variation of the strength of the CT1 and CT2 bands, NCTI and NCT2, as a function of doping x.

The behavior of the CT2 band with doping is similar to that of the CT1 band. Doping with electrons results in a reduction of the CT2 band and a small shift to






74


higher energies from 2.08 eV for x = 0.11 to 2.29 eV for x = 0.19. However, a transfer of spectral weight only starts after the CT1 band has completely disappeared, as observed in Fig. 41. There is no difference of peak position and strength between the spectra of x = 0 and x = 0.11. The spectral weight is ~ 15000 cm-1 for x = 0.11 and ~ 10300 cm-1 for x = 0.19. Thus, both the CT1 and the CT2 bands seem to due to a transition from the Cu 3d UHB to Zhang-Rice type'7 correlated states.

The squares in Fig. 41 also explain the spectral weight loss of two CT bands upon doping. The spectral weight of two CT bands loses slightly faster than x with doping x. This trend is in good agreement with the behavior of the LFSW with doping x. The amount of the greater spectral weight loss than x is very similar to that of the greater LFSW than 2x, which may be related to p-d charge transfer.

We here have the interesting fact that, when we consider the positions of the CT1 and CT2 bands with doping, the Cu 3d UHB should move to higher energy. From the position differences of the CT1 band between x = 0 and x = 0.11 and the CT2 band between x = 0.11 and x = 0.19 we are led to conclude that the Fermi level should lie ~ 0.35 eV above the bottom of the UHB. This result is consistent with a theoretical estimate72 and the EEL and x-ray absorption spectroscopy.T3 This observation is also compared to the results46,74 observed in La2--.SrCuO4 of 0.7 eV. This narrow energy range induces the strong Drude band and suggests that Nd2-,CeCuO4 of electron-doped system has a large Fermi surface, in good agreement with the angle resolved photoemission experiments. 7'76

Summary

We have examined the change of optical spectra with Ce doping in electrondoped Nd2-,CeCuO4 in the frequency range from the far-infrared to the UV region. We have also made a systematic analysis of the temperature dependence for






75


Nd2-zCeCuO4 at temperatures between 10 K and 300 K. We analyze our data with the one component and two component models. Our results show that the doping mechanism of the electron-doped Nd2-,.CeCuO4 is a little different from that of hole-doped La2- SrCuO4.

The spectrum of the undoped Nd2CuO4 shows a typical insulating characteristic with energy gap of 1.5 eV which is identified to 0 2p-Cu 3d charge transfer excitations. Doping with electrons in insulating Nd2CuO4 induces a shift of spectral weight from the high energy side above the charge transfer excitation band to the low energy side below 1.2 eV. The low energy spectral weight for a barely metallic sample, x = 0.11 is composed of two parts: a narrow Drude-like and mid-infrared parts. Upon further doping the Drude-like band rapidly increases and the mid-infrared band shifts to lower frequency, and hence two parts are hardly separated in the metallic phase.

A weak temperature dependence of the far-infrared reflectance suggests the nonDrude behavior of this material. For example, the change of reflectance between 15 K and 300 K for non-superconducting metallic samples of x = 0.16 and x = 0.19 is less than 2% and for superconducting sample of x = 0.15 the reflectance change between same temperatures is about 3.5% near 600 cm-1. This non-Drude behavior can be analyzed by a frequency dependent scattering rate and a mass enhancement in the one component approach.

In the one component approach, our results show that the mass enhancement at low frequencies is large, and for superconducting sample Im E is linearly proportional to w below 5 000 cm-1, in good agreement with the predicted behavior in the numerical models in which strong quasiparticle damping is assumed. From the slope of Im E a weak coupling constant A 0.15 0.01 is estimated.






76


The low frequency spectral weight (LFSW) with doping is analyzed by the two component model. The Drude strength is very low up to the metal-insulator transition and is roughly proportional to the doping concentration x in the metallic phase. The strength of total mid-infrared bands rapidly increases at low doping but slowly at high doping levels. The LFSW including the Drude and total mid-infrared bands grows faster than 2x with doping x consistent with the MH model. These strong Drude and mid-infrared bands with the result of transport measurements suggests that Nd2-.-xCeCuO4 has a large Fermi surface consistent with photoemission experiments.

The charge transfer (CT) band is also analyzed with the two component model. The CT band in insulating Nd2CuO4 consists of two bands, CT1 and CT2 bands. Upon doping, the CT1 band disappears at high doping levels, while the CT2 band survives even if it partially loses its spectral weight. The two bands correspond to the transition from the Cu 3d UHB to Zhang-Rice type correlated states.






77


Photon Energy (eV)
2 3 4


Nd2Cu04- -- -- 300 K


. \


I".
--


0.1 0.0


2000





1000

b




0


I -


I I I I I I I I I


- I


Nd2CuO4-6
300 K


L




I */ /
I-.---


I
/


0


1000


0 20000 30000 Frequency (cm-')


40000


Fig. 21. (a) Room temperature reflectance spectrum of Nd2CuO4...6 on a-b
plane and (b) a1(w) spectrum after K-K transformation of R in (a).


0


1


Si


1 1 1 1 1 1 1 1


0.4 0.3


5


()

0 CD)
a)
w)


0.2


I--'
I


/
I

/
/


I


I I


I I


-


I


I


I





































200


300
Frequency


Fig. 22. Far-infrared reflectance of Nd2Cu04.6 at several temperatures.


78


1.0 0.8


(D

U


0.6



0.4


* I I


Nd .CuO1


0K


L 4_


ic


1 UUK 200K 300K


400 (cm~)


500


600


'


/



I -


0.2



0.01
100


I I





79


Nd2CuO4_.
TO phonons
.. 10K OOK 200K 300K



-- -


onons
10K
lOOK 200K 300K





-


500 (cml1)


600


700


Fig. 23. Far-infrared (a) a1(w) and (b) Im[-1/e(w)] for Nd2CuO4...6 Peaks
in (a) correspond to TO phonons, in (b) to LO phonons.


B00


b


600


400


200


0
0.8


0.6


0.4


E


LO ph


200


0.2


0.0 100


300 400 Frequency


I I


I


.






80



0.25 .
Nd2Cu04_,5

0.23 -- 10K
100K
.150K c 0.21 200K
2~- ._-_..- 250K
300K C 0.19


0.17


0.15

80


60


C 40 10K


20 --~ 150K
200K 250K


500 1000 1500 2000 2500
Frequency (cm~1)



Fig. 24. (a) Reflectance spectra of Nd2CuO4_6 at several temperatures, and
(b) the real part of the optical conductivity as a function of frequency.






81


2000


0


1500


100


200


Temperature


300


(K)


Fig. 25. The parameters extracted from the Lorentzian fits to the peak near
1500 cm-1 as a function of temperature.


0
0
0
0
0 Center frequency



Band width




-t





Spectral weight


E
C-)


1400


C


1300


E


1450


1200


2125


E
0
C C-






82


Photon Energy (eV)
1.0 1.5


0.21 0 0.19



0.17 0.15


E


800 600


400


200


--





Nd2CuO46
1OOK 200K 300K


a a a i ____


200K 300K


fl


5000


10000
Frequency


15000


(cm~l)


Fig. 26. (a) Temperature dependent-reflectance spectra and (b) optical conductivity spectra of charge transfer band for Nd2CuO4..


2.0


I I I


I






83


Photon Energy (eV)
0.1 1


1000
Frequency


10000 (cm')


Fig. 27. Room temperature reflectance spectra of Nd2-,Ce.CuO4 for various
z on a-b plane.


0.01


*1 I .


I.



0.8 0.6



0.4


CD) (0 rc


0.2 0.0


100


I . Nd2Ce CuO
......... X=.00


x=.15 x=.1 5 x=.1 6 x=.1 9






84


0


2500



2000 1500 1000
b



500



0


0


1


Photon Energy
2 3


10000 20000
Frequency


(eV)
4


30000 (cm~ )


Fig. 28. Room temperature a-b plane optical conductivity spectra of
Nd2-..CeCuO4 with doping x as a logarithmic frequency scale.


I *


Nd2-.Ce.Cu04
.I._.._.._ x=.00
i x=.1 1
...-..... .. x= .1 4
x=.1 5
x=.1 6
x=.19




/ -


5


40000


I I I I I .






85


Energy (eV)
1


6000






4000

E




2000






0


100


1000
Frequency


10000 (cm )


Fig. 29. Room temperature a-b plane al(w) spectra as a function of x (note
the linear frequency scale).


0.01


Photon
0.1


U I I I j I I*


Nd2-xCe CuO4 .x=.00 x=.1 1
.x=.1 4 x=.15 x=.1 6 x=.19




\-






86


0.01


0.6[


0.4 0.2


Photon Energy (eV)
0.1 1


U.0
100 1000 10000
Frequency (cm~1)








Fig. 30. Effective electron number per formula unit for Nd2-zCezCuO4 at
doping levels from 0 to 0.2.


0I
a
E







0
z


Nd2-xCeXCuO4
.. .. .. .. X=.00
x=.1 1
- _ .__.. x=.1 4
x=.15
__..____ x=.16
x=.1 9





-7




~-~/
0, 7

I






87




Photon Energy (meV)
0 20 40 60 80 100 120
0.20

Nd2 -xCeCu04
x=.11 0.16 x=.14
x=.1 5
-...x=.1 6
E x=.1 9
0
0.12



0.08



0.04-0.00
0 200 400 600 800 1000
Frequency (cm~1)








Fig. 31. N*ff per Cu atom of Nd2-...Ce.CuO4 in a frequency range below
1000 cm-.






88


0.5


Photon Energy (eV)
1.0 1.5


2.0


1 0000
Frequency (cm~)


20000


Fig. 32. The energy loss function, Im[-1/e(w)] of Nd2.,.CeCuO4 as a function x.


0.0


0.6


0.4


0-N


0.2 0.0


Nd CeCu04
x=0.00 x=0.1 1
x=0.14 .
x=0.15
x=0.1 6
x=0.1 9


0






89


0.95




(D 0.90 08
0.8


Nd1 84Ce01 6CuO4
15 K
100 K 200 K 300 K


I I


'I


0.95




(D 0.90 0)


0.85


0


~-

Nd 81 Ce019CuO4
15 K
100 K 200 K 300 K


I I


100


200


300
V(cm~)


400


500


600


Fig. 33. Far-infrared reflectance for non-superconducting metallic samples of
Ndi.86Ceo.16CuO4 and Ndi.89Ceo.19CuO4 at several temperatures.


I I


I I I


I I


I I


I


I I


I I


I I






90


1.00 0.95 0.90 0.85


9000 1


7000


b


5000


3000 1000


0


Nd1 .85CeO. 5Cu04
15 K
100 K150 K 200 K 300 K


15 K 100 K 150 K 200 K 300 K




-

S 1 "


500


1000
V(Cm')


1500


2000


Fig. 34. Temperature dependent (a) reflectance and (b) conductivity for superconducting Nd1.85Ceo.15Cu04 as a function of frequency.






91






10000
300


E 250


-200

6000 -- ,I ,,
E 0 100 200 300
Q %Temperature (K)


4000
b Nd 8Ce 15CuO4
......... 15 K
.100 K
2000 -.. 150 K
200 K
.300 K
fit
0'
100 1000
Frequency (cm~1)







Fig. 35. The Drude conductivity obtained by subtracting the mid-infrared
contribution from the total conductivity. The solid line are Drude fits.
Insert shows a Drude scattering rate, 1/7 as a function of temperature.






92


0.
2.21


Photon Energy (eV)
0.2 0.4


0


1.8 -


.0

E

1.4 1.0 2900



2100 1300


A A


0.6


Nd2-xCe CuO4
x=0.1 4-- x=0.15\ x=0.16
x=0.19


'


x=0.1 4


x=0.15
. x=0.16
x=0.1 9

;-


1000


2000 3000
Frequency (cm'1)


4000


5000


Fig. 36. Frequency-dependent mass enhancement (upper panel) and renormalized scattering rate. (lower panel)


500


0


I I


I


I


I


I I




Full Text
I
169
86. J.D. Jorgensen, H.-B. Schttlers, D.G. Hinks, D.W.Capone, I.K. Zhang, M.B.
Brodsky, and D.J.Scalapino, Phys. Rev. Lett. 58, 1024 (1987); P. Bni, J.D.
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Bni, and G. Shirane, Phys. Rev. B 39, 4327 (1989).
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Rev. B 45, 576 (1992); S. Sugai, S. Shamoto, M. Sato, T. Ido, H. Takagi, and
S. Uchida, Solid State Commun. 76, 371 (1990); I. Ohana, M.S. Dresselhaus,
Y.C. Liu, P.J. Picone, D.R. Gabbe, H.P. Jenssen, and G. Dresselhaus, Phys.
Rev. B 39, 2293 (1989); W.H. Weber, C.R. Peters, and E.M. Logotheetis, J.
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Appl. Phys. 26, L495 (1987).
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B 47, 11369 (1993).
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92. N.F. Mott, Metal-Insulator Transition (Taylor and Francis, London, 1990),
Ch 1.
93. A.L. Efros, and M. Poliak, Electron-Electron Interactions in Disordered
Systems (North-Holland, Amsterdam, 1985) Ch 4.
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Rev. B 43, 392 (1991); Phys. Rev. Lett. 63, 2307 (1989).
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Kastner, P.J. Picone, and T. Thio Phys. Rev. B 39, 11563 (1989).
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Rev. B 42, 1045 (1990); Solid State Commun. 76, 365 (1990); Solid State
99.


63
undoped crystal is shown near 4.5 eV in the spectrum of x = 0.11. Further, with
doping the strength of this peale is reduced and its position shifts to lower frequencies.
Third, at high doping levels, a new shoulder near 3.6 eV grows with the decreasing
of the strength of 4.5 eV and 5 eV peaks. This seems to transfer the spectral weight
of 5 eV peale to energy region between 3 eV and 5 eV with doping. This behavior of
doping dependence in high energy region is different from the results obtained from
hole-doped systems, where the spectral weight above 1.5 eV systematically decreases
with doping.
Effective Electron Number
In order to describe a doping effect quantitatively, we have estimated the effective
number of carriers per CuC>2 plane. The effective electron number with mass m* =
me, Neff(u>) is defined according to
LJ
NeffM = 2me^Ce// o\(u)dJ, (50)
7T 6 J
0
where e, me are the free electron charge and mass respectively, and Vceu is the volume
of one formula unit. For Nd2-xCexCu04, we used the unit cell volume,68 Vce¡¡ = 187
3 and the number of Cu atoms per unit cell, Ncu = 2. Neff(u) is the effective
number of carriers per formula unit participating in optical transition at frequency
below id.
Figure 30 illustrates the effective electron number for the different Nd2-xCerCu04
samples. In the insulating phase, x = 0, Neg (u>) remains nearly zero throughout the
optical gap without a mid-infrared band contribution, but increases rapidly above the
absorption band of charge transfer excitation. For metallic samples, Neg (uj) exhibits
an initial rise due to the Drude band at zero frequency. The following steep rise


a(co) (103^'1 cm-1) a(co) (103fl_1 cm-1)
8
Fig. 3. a-b plane optical conductivity of La2-xSrxCu04 (top) and
Nd2-xCexCu04 (bottom) single crystals for Uchida et al.3


Reflectance
116
Fig. 45. Temperature dependence of the c-axis reflectance in
Lai.97Sro.o3Cu04. Inset: high frequency reflectance at room tempera
ture.


7
Fig. 1. Optical conductivity in the mid-infrared region of YBa2Cu306+y
(upper panel), Nd2Cu04_j, (center panel), and La2Cu04+y.
Electron-Doped System
As we will discuss later, electron-doped materials, like Nd2-xCexCu04, axe struc
turally very similar to La2-xSrxCu04, but doped holes and electrons are introduced
in different sites, 0 and Cu sites. It has been found that their optical conductivities
axe also qualitatively similar for Nd2_xCexCu046,7 as shown in the bottom of Fig. 3.
Other compounds of the same family can be obtained by replacing Nd by Pr, Sm
and Gd. The optical properties of Pr2_xCexCu04 have been investigated8,9 and the
reported results are very similar to those of Nd2-xCexCu04.


150
Photon Energy (eV)
0.01 0.1 1
Fig. 63. Comparison of the effective electron number per unit cell,
Bai_xKxBi03 with that of BaPbi_xBix03.


33
Fig. 14. Idealized structure of perovskite ABX3.
Superconductivity exists only in the tetragonal phase and the value of maximum Tc
is T ~ 13 K for x ~ 0.25. For x > 0.35, the material becomes a semiconductor.
The behavior of Bai_xKxBi03 is similar to that of the BaPbi_xBix03. The
superconductivity appears at the boundary of the metal-insulator transition in the
cubic phase (x > 0.37) with maximum Tc of 30 K and disappears abruptly upon
crossing a phase transition to the orthorhombic phase. In spite of the low carrier
density (in the order of 1021 cm-3) the value of Tc is anomalously high. Also, no
magnetism is found in the neighboring compositions. The structures of five phases for
0 < x < 0.5 and temperatures below 473 K have been determined by neutron power
diffraction.28,29 Semiconducting behavior for the monoclinic phase at 0 < x < 0.1 is
explained on the basis of a commensurate charge density wave (CDW). This tendency
suggests unusual electronic interaction, namely strong electron correlation effect, in
this system.


171
116. T.D. Thanh, A. Koma, and S. Tanaka, Appl. Phys. 22, 205 (1980).
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Folkerts and R. N. Shelton, Phys. Rev. B 40, 2662 (1989).
118. S. Tajima, M. Yoshida, N. Koshizuka, H. Sato and S. Uchida, Phys. Rev. B 46,
1232 (1992).
119. S. Pei, N. J. Zaluzec, J. D. Jorgensen, B. Dabrowski, D. Hinks, A. W. Mitchell,
and D. R. Richards, Phys. Rev. B 39, 811 (1989) ; Phys. Rev. B 43, 5511
(1991).
120. B.P. Bonner, R. Reichlin, S. Martin, and H.B. Radousky, Phys. Rev. B 41,
11579 (1990).
121. E. S. Heilman and E. H. Hartford Jr., Phys. Rev. B 47, 11346 (1993).
122. M. Affronte, J. Marcus abd C. Escribe-Filippini, Solid State Commun. 85, 501
(1993); Phys. Rev. B 49, 3502 (1994).
123. H. Sato, T. Ido, S. Uchida, S. Tajima, M. Yoshida, K. Tanabe, K. Tatsuhara
and N. Miura, Phys. Rev. B 48, 6617 (1993).
124. L. F. Mattheiss and D. R. Harmann, Phys. Rev. Lett. 60, 2681 (1988).
125. P. B. Allen, T. P. Beaulac, F. S. Khan, W. H. Butler, F. J. Pinski and J. C.
Swihart, Phys. Rev. B 34, 4331 (1986).
126. M. Shirai, N. Suzuki and K. Motizuki, J. Phys.: Condens. Matter 2, 3553
(1990).
127. W. Jin, M. H. Degani, R. K. Kalia, and P. Vashishta, Phys. Rev. B 45, 5535
(1992).
128. C.-K. Loong, W. Jin, M. H. Degani, R. H. Kalia, P. Vashishta, D. G. Hinks,
D. L. Price, and Y. Zheng, Phys. Rev. B 45, 8052 (1992); Phys. Rev. Lett. 62,
2628 (1989).
129. D.M. Ginsberg and M. Tinkham, Phys. Rev. B 118, 900 (1960).
130. P.J.M. vein Bentum and P. Wyder, Physica B 138, 23 (1986).
131. R.E. Glover, and M. Tinkham, Phys. Rev. B 107, 844 (1956); Phys. Rev.
B 108, 243 (1957).


or(fi cm) Reflectance
121
0 100 200 300 400 500 600 700
y(crrf1)
Fig. 50. Far-infrared (a) reflectance and (b) conductivity cri(u;) for
Lai.97Sro.o3Cu04 at several temperatures.


co O
52
Source
M
i
vmm
Movable mirror
Beam
splitter
I
///
i
%
Detector
Fig. 17. Schematic diagram of Michelson interferometer.


114
Photon Energy (eV)
0.01 0.1 1
Fig. 43. Room temperature u\[u>) spectra, obtained after K-K transformation
of R in Fig. 42.


37
Bi-0 Superconductors
The single crystals of Bai_iKxBi03 were grown by a modification of the elec
trochemical method of Norton.31 The transition to the superconducting phase from
dc susceptibility measurement is quite wide with the onset of superconductivity at
T = 28 K and the full superconductivity at T = 18 K (Fig. 16), probably associated
with the inhomogeniety of the potassium distribution in the crystal. The measured
Tc (~ 22 K) was taken at 50% of transition between 90% and 10% points. Also, a
direct measurement of the Tc of the same crystal by measuring the temperature at
which there is a discontinuity in the tunnel conductance yields Tc = 21 K.
For the optical measurement, the sample surface (1.5 mmxl.5 mm) was mechan
ically polished using AI2O3 power of 0.05 fim size. The color was blue after polishing.
Figure 16 shows a Meissner effect for BaPbxBii_x03, indicating the onset of super
conductivity around 10 K.


13
sites, i on Cu and j on O. Up corresponds to the Coulomb repulsion when two holes
occupy adjacent Cu and 0 sites, and may also be very important. It is appropriate to
use the hole notation, since there is a one hole per unit cell in the undoped case. Hence,
the vacuum state corresponds to the electronic configuration d10p6. Because < tp,
this hole occupies a d-level, forming the d9 state. There are two factors that govern
the electronic structure. On the other hand, the hybridization tpd is substantial and
leads to a large covalent splitting into bonding and antibonding bands, which form the
bottom and top of the p-d band complex. Therefore, the bonding orbital is O-p-like
and the antibonding orbital is Cu-dx2_y2-like. This covalent nature is not restricted
to only one orbital per site. There is a direct mixing of most of the Cu 3d and 0 2p
states.
On the other hand, the local Coulomb interaction Ud is crucial for the semicon
ducting properties. In the charge transfer regime (tp < ep < Ud),16 the lower
Hubbard band is pushed below the 0 level and so three bands are formed as shown
in Fig. 6(top). When another hole is added to this unit cell in the charge transfer
regime, the new hole will mainly occupy oxygen orbitals due to the on-site Coulomb
interaction. The high Tc superconducting materials fall into this category (typical
parameters are ep ed ~ 3 eV, tpd ~ 1.5 eV, tpp ~ 0.65 eV, Ud ~ 10 eV, Up ~ 4 eV,
and Upd ~ 1.2 eV).16
One Band Hubbard Model
As originally emphasized by Anderson,12 the essential aspects of the electronic
structure of the Cu02 planes may be described by the two dimensional one band
Hubbard model. This model is
H = £ (W + 4^>) + c'EKt 5>Ki -1),
(2)


71
represents N¡). However, it is difficult to define the Drude paxt from metallic phase due to the mixing with strong mid-infrared bands. Thus, Np in Fig. 38
represents with large error bars. Figure 38 illustrates that a Drude strength is very
low up to Ce concentration of x = 0.11 and is roughly proportional to the dopant
concentration x in the metallic phase, as expected for the generation of carrier by
adding of electrons in the CuC>2 plane. Our results are in good agreement with the
phase diagram of Nd2-xCezCu0469,70 which shows the insulating phase at a wide Ce
compositional range up to x = 0.12. As mentioned earlier in the effective electron
number section, the superconducting sample of x = 0.15 has a very strong Drude
band. This result is consistent with the theoretical observation71 described by the
extended Hubbard model that upon electron doping the Fermi level lies directly in
the Van Hove singularity of the upper Hubbard band at a certain doping level. This
concentration may be the superconducting sample with x = 0.15.
As shown in the of Fig. 28 and Fig. 29, Ce doping in Nd2Cu04 clearly
induces the formation of strong mid-infrared bands. Upon doping, these mid-infrared
bands continue to grow at high Ce concentrations and tries to merge with the Drude
peak. In Fig. 38, we also plot the strength of total mid-infrared bands, Nmid, as a
function of Ce concentration x (squares). Nmid is estimated from several Lorentzian
fits to oi(w) in the low-frequency part below the charge transfer gap. Nmid also
represents with large error bars due to the ambiguity of a mid-infrared band near the
Drude part. The strength of the total mid-infrared bands increases rapidly at low
doping, but slowly at high doping levels. We stress here that with the Drude band
the strength of total mid-infrared bands of Nd2_ICeICu04 is very strong compared
to that of hole-doped L^-xSrxCuC^.


100
In Fig. 43, we plot the optical conductivity spectra for four samples as a func
tion of frequency as a result of K-K transformations of the reflectance spectra in
Fig. 42. Figure 43 clearly demonstrates that upon doping, the spectral weight above
CT gap near 2 eV decreases and its peak position is shifted to higher frequencies
systematically, consistent with theoretical model and other experiments,81 while the
conductivity below 1.5 eV grows significantly throughout the doping range.
Several TO phonon features axe also identified with peaks in the fax-infrared
region. The tetragonal high temperature phase of La2Cu4 exhibits four in-plane
phonons with Eu symmetry. However, the orthorhombic distortion lowers the sym
metry and activates severed additional phonon modes which account for weak struc
tures around 145 cm-1. Figure 43 also illustrates that with Sr doping the oscillator
strength of phonons decreases due to the electronic screening from free carriers, and
instead doped carrier contribution at u> = 0 and new vibration modes grow in the
fax-infrared region.
Other features that appear in the frequency region below charge transfer gap are
broad mid-infrared bands near 4050 cm-1 (0.5 eV) and 11300 cm-1 (1.4 eV) band
which is not observed in undoped sample and in electron-doped T phase samples
without apical 0 atoms.
Figure 44 shows the plot of ei(u;) as a function of frequency, showing a positive
dielectric response at frequencies less than 100 cm-1 that dielectrically screens the
free-caxrier response. There axe large difference between the static dielectric constant
es ~ 30 (for 3% Sr doped samples) and the high frequency dielectric constant ~ 4.
This large difference seems to be due to the contribution from phonons, and suggests
that La2_zSrzCu04 in low doping regime is very polarizable at low frequencies and
the charge carriers expect to form polarons. Figure 44 in the far-infrared region also


85
Photon Energy (eV)
0.01 0.1 1
Fig. 29. Room temperature a-b plane <7i(u>) spectra as a function of x (note
the linear frequency scale).


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131
cm-1 to 45 000 cm-1 and temperature range from 10 K to 300 K. We subsequently
study the optical conductivity in the normal state and in the superconducting state.
We find a BCS-like gap feature and estimate the electron-phonon coupling constant.
We discuss the infraxed conductivity in the framework of Mattis-Bardeen theory and
finally the the electron pairing mechanism in BKBO.
Samples we have measured are in the K and Pb concentrations with neax max
imum Tc. The sample surface (2 mmx2 mm) of BPBO is very shiny, but that of
BKBO is rough. It is also known that the surface of BKBO is easily degraded in air.
For the optical measurement, the sample surface (1.5 mmxl.5 mm) of BKBO was
mechanically polished using AI2O3 power of 0.05 fim in size. The color was blue after
polishing. To correct the surface roughness, we evaporated an Ag film on the sample
surface, which produces an Ag surface with a roughness comparable to that of the
sample. We again measured the reflectance of an Ag coated sample and obtained
an estimate for the absolute reflectance of BKBO from dividing the sample surface
reflectance by that of an Ag coated sample. The measurements were made for three
different crystals. We have also measured the reflectance before polishing to examine
surface degradation effects due to atmospheric exposure. Our results show that the
degradation of the surface does not affect the fax-infrared spectrum, t.e., the infraxed
gap measurement, because the fax-infraxed light penetrates deeply into the surface.
However, the spectrum in the near-infrared and visible is greatly changed.
Normal State Properties
Results for Bai-TKTBiO^
Figure 59 shows the optical reflectance for Bai_xKxBi03 (BKBO) at several tem
peratures in the frequency range from 30 cm-1 to 40 000 cm-1. As the temperature


127
^ 1200 Center frequency
E
o

E
o
1050
900
2000
E
o

Q.
1750
3200
2950
2700
Band width
0
Spectral weight
i
o
T

0
_L
100 200 300
Temperature (K)
Fig. 56. The center frequency, width, spectral weight extracted from
Lorentzian fits for the 0.15 eV band as a function of temperature.


34
Sample Characteristics
La9_-rSrTCuO/t
Single crystals of La2-xSrxCu04 were prepared at Los Alamos Laboratory.30 Six
teen samples of nominal composition La2_xSrxCu04+ were grown by conventional
solid state reaction at 1050C using predried La203, SrC03, and CuO in x increments
of 0.002 from x = 0 to 0.04. For each x, the sample was separated into three parts
which were treated at 650C for 5 hours in 1 bar N2 or 1 bar O2, or at 500C for 72
hours in 230 bar O2, respectively, and then oven cooled. Tq/j was measured using
a Perkin-Elmer differential scanning calorimeter. Oxygen contents were measured by
hydrogen reduction using a Perkin-Elmer thermogravimetric analyzer; the 1 bar N2,
1 bar O2, and 239 bar O2 annealed series showed 6 = 0.00(1), 0.01(1), and 0.03(1),
respectively.
The size of all crystals are at least 1.5 mm x 1.5 mm which are suitable for infrared
measurements. The surfaces were subsequently etched for 20 min in a solution of 1
% Br in methanol before reflectance measurement.
Nd->_TCeTCu04
Nd2-xCexCu04 single crystals were prepared at the University of Texas. All
crystals were grown in copper-oxide rich fluxes; normal starting compositions were
Nd2-xCexCu4.502 with various x. The melts were cooled in air from 1260C at
5C/min. The crystals were mechanically separated from the flux and subsequently
reduced in flowing He gets at 910 C for 18 hours. Energy dispersive spectroscopy
and electron microprobe (wavelength dependent spectroscopy) analysis on these and
many similar crystals have indicated a uniform Ce concentration across the crystal,
but absolute concentration determinations are only accurate to Ax = 0.01.


OPTICAL PROPERTIES OF DOPED
CUPRATES AND RELATED MATERIALS
By
YOUNG-DUCK YOON
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
1995


Reflectance Reflectance
89
0 100 200 300 400 500 600
y(crrf1)
Fig. 33. Far-infrared reflectance for non-superconducting metallic samples of
Ndi.86Ceo.i6Cu04 and Ndi.8gCeo.i9Cu04 at several temperatures.


Resistivity (10
161
Fig. A-l. Temperature dependence of the electrical resistivity for LUN2B2C.


CHAPTER V
EXPERIMENT
Background
Dielectric Response Function
The dielectric function e(u;, q) describes the response of a crystal to an electro
magnetic field. The dielectric function depends sensitively on the electronic band
structure of a crystal, and studies of the dielectric function by optical spectroscopy
are very useful in the determination of the overall band structure of a crystal. In the
infrared, visible, and ultraviolet spectral regions the wavevector q of the radiation is
very small compared with the shortest reciprocal lattice vector, and therefore q may
usually be taken as zero.
The dielectric constant e of electrostatics is defined in terms of the electric field
E and the polarization P, the displacement D:
D = eoE + P = eeoE. (19)
The defined e is also known as the relative permittivity. So long as the material is
homogeneous, isotropic, linear, and local in its response, the dielectric response may
be characterized quite generally by a frequency-dependent complex dielectric function
e(u;) which we write in terms of its real and imaginary parts as
e(u;) = ci(w) + u)
Here, the quantity ei(u;) is called the real dielectric function whereas c\[u) is the fre
quency dependent conductivity. At zero frequency ei(0) becomes the static dielectric
constant and 38


(/Cu atom)
87
Photon Energy (meV)
0 20 40 60 80 100 120
Fig. 31. Nljj per Cu atom of Nd2_xCexCu04 in a frequency range below
1000 cm-1.


118
b
Fig. 47. Atomic positions in the orthorhombic La2_*SrxCu04 unit cell which
consists of two primitive cells. The hatched circles (a-5) are La atoms,
the filled circles (A and B) Cu atoms, and the open circles (1-8) 0 atoms.


59
against the Cu02 plane (131 cm-1), a Cu-0 bending mode from the in-plane Cu-0
bond angle modulation (303 cm-1), an out-of-plane 0 translational mode (347 cm-1),
and an in-plane Cu-0 stretching mode from the Cu-0 bond distance modulation
(508 cm-1). The function Im[-l/e(u;)] is shown in Fig. 23(b), showing large LO-TO
phonon splittings. The temperature dependence of phonon frequencies shows the
redshifts with increasing temperature as expected.
The optical conductivity of pure undoped sample should vanish up to 1.5 eV,
above which the charge transfer excitations occur. However, the optical conductivity
spectrum new absorption may be attributed to the deviation from an oxygen stoichiometry
of Nd2Cu04 single crystal during oxidation process. This result indicates that our
sample is lightly doped with electrons.
In Fig. 24(a), the optical reflectance of the band neax 1500 cm-1 is plotted as a
function of frequency at several temperatures. Figure 24(b) also shows the temper
ature dependence of the optical conductivity c\(w) of this peale. This peale is very
interesting due to the fact that its energy is close to the antiferromagnetic exchange
energy J.62 For this reason, one might expect that the origin of this peak is due to
the interaction of doped carriers with magnetic degrees of freedom. Several mecha
nisms, including self-localized polarons,63-66 photoexcitation of localized holes,67 and
magnetic excitations,5 have been proposed to explain this peak.
We have fit this band with the usual Lorentzian. The results for the fitting pa
rameters axe shown in Fig. 25. The peale position and peak width shift to higher
frequencies by an amount compaxable to thermal fluctuation energy kgT as the tem
perature of sample is increased; that is,
ue = ujeo -|- 0.6 IcbT, 7 = 7o + 1.6 k^T,
(49)


69
The renormalized scattering rate l/r*(a>) and the effective mass enhancement
m*/mi, can be also related to the imaginaxy paxt of quasipaxticle self energy by
m* 1 u;
TT = 2Im£.
mo r*(w) 2
Figure 37 illustrates the imaginary part of quasiparticle self energy, Im £ of
Nd2-xCeICu04 crystals below 5000 cm-1. Im£ in Fig. 37 is analyzed in several
ways.
First, for a barely metallic crystal (x = 0.11), the imaginary paxt of self-energy
deviates from the lineaxity and reveals a power law between 0.1 eV and 0.6 eV,
reflecting the proximity to the phase boundary of the insulator. With doping we see
a steady decrease in the quasipaxticle interaction and finally, a linear slope of Im £
in high doping concentration of x = 0.19.
Second, for superconducting sample (x = 0.15), Im£ increases linearly with u
below 5 000 cm-1. This is in a good agreement with the predicted behavior in the
MFL. According to the MFL theory, the imaginaxy paxt of quasipaxticle self energy
Im£ has the form ir\u over T < u < uc, where u>c is the cutoff frequency. We
estimated a coupling constant A = 0.15 0.01 from the slope between 500 cm-1 and
2500 cm-1. This estimated value seems to be rather low compared with the results
obtained from hole-doped systems. It is also interesting to compare the result for
superconducting sample of x = 0.15 with the result for slightly highly doped sample
of x = 0.16. The data of x = 0.15 show less quasipaxticle excitations than that
of x = 0.16. This may suggest that too much quasipaxticle interaction causes the
reduction of superconductivity in high Tc cuprates.
Third, the quadratic temperature dependence of dc resistivity in superconducting
sample does not agree with our analysis. We might expect a quadratic dependence


163
Photon Energy (eV)
0.01 0.1 1
Fig. A-3. The optical conductivity a\(u) obtained from the K-K transforma
tion of the reflectance spectrum.


91
Fig. 35. The Drude conductivity obtained by subtracting the mid-infrared
contribution from the total conductivity. The solid line axe Drude fits.
Insert shows a Drude scattering rate, 1/r as a function of temperature.


67
where the first term is a Drade oscillator, described by a plasma frequency up£> and a
relaxation time r of the free carriers, the second term is a sum of peaks in c\[u), with
Uj, upj and 7j being the resonant frequency, strength and width of the jlh Lorentz
oscillator, and the last term is the high frequency limit of e(u;).
The Drude components at five temperatures for the superconducting sample,
x = 0.15, axe depicted in Fig. 35. The Drude component is defined as the con
ductivity after the average mid-infrared component is subtracted. The temperature
dependence of the Drude part satisfies the ordinary Drude behavior. The inset in
Fig. 35 shows the temperature dependence of the scattering rate obtained from the
fits. The Drude plasma frequency upj) 11 200 cm-1, is nearly T-independent, while
the temperature dependence of the scattering rate 1/r is consistent with the behavior
of the dc resistivity. For example, 1/r is non-linear in T and reduces to half of the
300 K value at 15 K. The dc resistivity value from the four-probe measurement is
good agreement with that obtained from simple Drude formula:
_ UPD
Pdc 4tt(1 /r)dc
We emphasize here that the Drude plasma frequency is laxger, and 1/r is a little
smaller than the values obtained for La2_ISrICu04. Also, the value for up is larger
than the results obtained by any other experiments for electron-doped system.
One Component Approach
As suggested in previous section, ct\(u) does not fit the simple Drude formula
(cti(u;) a u;-2). Especially, o\(u) reveals a strong spectral weight in the mid-infrared
region, compared to that at zero frequency. Another approach to analyze this non-
Drude conductivity is to assume an inelastic scattering of the free carriers in the


CHAPTER VI
CE DOPING DEPENDENCE OF
ELECTRON-DOPED Nd2_xCeICu04
In this chapter, we report optical reflectance and conductivity spectra from the
fax-infrared to UV on the a-b plane of electron-doped Nd2-xCexCu04 for very different
Ce concentrations (0 < x < 0.2). This compositional range covers the antiferromag
netic insulator, the high Tc superconductor, and non-superconducting overdoped
metallic samples.
The motivation behind this study is twofold. First, in spite of a lot of theoret
ical and experimental studies, there is still little understanding of the normal-state
excitation spectrum, especially the low-energy-scale physics near the Fermi level, of
strongly correlated high Tc cuprates. In particular, the differences between hole dop
ing and electron doping in the transfer of spectral weight from high to low frequencies
axe not well understood. Basically, doped holes and electrons show different orbital
characteristics: the holes introduced by doping axe mainly in 2p orbitals whereas the
doped electrons have 3d orbital character. Therefore, in the three band Hubbard
model the motion of holes will depend differently on tp than motion of the electrons.
Eskes et a/.21 have shown that in the localized limit (tp = 0), the transfer of spec
tral weight with electron doping is similar to the case of the Mott-Hubbard model,
whereas the development of spectral weight with hole doping is the same as that of
a simple semiconductor. At present, the results for doping dependence in hole-doped
systems3,39-46 axe somewhat in agreement with one another, but the results for the
electron-doped system6-9,47-53 are still controversial.
56


102
Mode Assignment
The crystal structure of La2Cu04 is orthorhombic (D^, Ama, CmCa) at room
temperature and tetragonal (D^, Ii/mmm) above about 515K. The T/O transition
temperature decreases with increasing Sr concentration x. Figure 47 shows the atomic
positions in the orthorhombic unit cell. This structure is defined as a staggered tilting
or rotation around the [110] axis of the CuC>6 octahedra. In the orthorhombic phase
the volume of the primitive cell is doubled with respect to the volume of the tetragonal
phase, so that zone-boundary modes are folded back into the T point, yields 39 optical
modes.83 These modes may be classified into Raman (5Ag + 3B\g + 6f?25 + 4i?3y),
silent (4Au), and infrared active (6B\U + 4i?2u + 7i?3U). There are eleven infrared-
active a-b plane optical phonons and six c-axis polarized phonons. Since the crystals
are twined, we can not separate the i?2u (a axis) from the B$u (b axis) modes.
In Fig. 46, we have shown the c-axis polarized conductivity spectra for single
crystal of Lai.97Sro.o3Cu04 in the temperature range from 10 K to 300 K. We ob
serve four infrared active modes centered at 230, 320, 345 and 510 cm-1 of the six
B\u modes predicted by group theory. A comparison of these frequencies with other
optical studies84 of c-axis spectrum in La2_zSriCu04 in the tetragonal phase indi
cates good agreement with three phonon modes at 230, 345 and 510 cm-1. The 320
cm-1 frequency mode appears to be rendered infrared-active by the orthorhombic
distortion. The two unobserved modes likely have small oscillator strength and high
damping, or at frequencies below 30 cm-1. There are very weak features at 275 and
420 cm-1 in low temperature spectrum that exhibit some temperature dependence.
These modes are the Raman-active Ag modes due to the breaking of the inversion
symmetry by the distortion of Cu06 octahedra.


24
Weak-Coupling Mittis-Bardeen Theory
In the superconducting state, a complex conductivity defined. In the extreme anomalous limit, q£0 > 1 or extreme dirty limit > /, Mattis
and Bardeen22 showed that the ratio of the superconducting to normal conductivity
within weak-coupling BCS theory is
£1£ = i_ /
hu yA
+
-L r
hu JA-h,
1 E(E + hu) + A21 [f{E) f(E + hu)] ,p
(E2 A2)1/2[(E + hu)2 A2]!/2
\E{E + faj) + A2|[l 2f(E + hu)]
hu {E2- A2y/2[{E + hu)2- A2]i/2
dE,
(9)
£2a L /A |E(E + hu) + A2|[l 2f(E + faj)] ,E
crn hu 7a-Au;,-A (A2 E2yl2[(E + hu)2 A2]1/2
Eq. (9) is the same as the expression for the ratio of absorption for superconduct
ing to normal metals for case II of BCS theory. Numerical integration is required for
r>o.
Figure 10 shows the Mattis-Bardeen theory for cr\s{u)¡(yn and cr2s{u)l(Jn as a
function of frequency for T = 0. The real part is zero up to hu = 2A and then rises
to join the normal state conductivity for hu 2A.
In the superconducting state for u < 2A, because J(u;) = <72SE(u;), the power loss
P = (J E) = 0; one can therefore expect a perfect reflector (R = 1) at frequencies
below 2A. The imaginary part of ai(u) displays the l/u inductive response for
hu < 2A. More simply, this dependence is a consequence of the free-acceleration
aspect of the supercurrent response as described by the London equation
E = d(M,)/dt, A = ^ = JaI
(11)


27
namely a coherence peak in the temperature dependence of the conductivity below
Tc and the logarithmic frequency dependence of c\(u) near w=0. Coherence effects
in superconductors arise because the dynamical properties of the quasiparticle ex
citations become different from those of normal electron-hole excitations as the gap
develops below Tc. This coherence peak will go to infinity just below Tc due to the
singularity in the BCS density of states. Thus, as T is lowered below Tc, the density
of excited quasiparticles decreases as these excitations freeze into the condensate, and
the properties of the excitations which are present for T > 0 are also modified. There
are clearly two fluids, the condensate fraction and the gas of excited quasiparticles.
Thus, the condensate response to external electromagnetic fields is described by a
8 function conductivity at u = 0 plus (in the presence of elastic scattering) con
ductivity with a threshold at u = 2A(T), corresponding to processes in which two
quasiparticles are excited from the condensate.
In weak-coupling BCS theory, the energy gap at 0 K is given by
2A =3.52 kBTc
(15)
where kq is Bolzmanns constant and Tc is superconducting transition temperature.
The gap vanishes at Tc, and just below this value, A(T) can be approximated by
A(T)
A(0)
1.74
Tc.
(16)
Strong-Coupled Superconductor
If the electron-phonon coupling is strong (as opposed to weak), then the quasi
particles have a finite lifetime and are damped. This finite lifetime decreases both


26
meV
Fig. 11. The conductivity of a BCS superconductor in the (a) dirty and (b)
clean limits.
The Kramers-Kronig transform of the delta function a\{u) is 2A/xu>. For comparison
with the London equation (11), the penetration depth is related to the missing area
by
(14)
In the clean limit (1/r < 2A, l f0), all of the free carrier conductivity collapses
into the S function, in which case A = 7re2n/2m = u2/8, and (13) reduces to the
London result. So, the spectral feature is very weak at 2A (Fig. 11(b)). In the
dirty limit (1/r > 2A, 1 < f0), the penetration depth tends to be larger than this
limiting value and a sharp feature appears at 2A (Fig. 11(a)), and one can write
A = 7re2n3/2mj, where n, < n is the superfluid density.
Coherence Effects in Superconductor
At finite temperatures, A(T) < A(0), and also the thermally excited quasi-
particles contribute absorption for hu < 2A. This quasi-particle excitation is repre
sented as the distinctive features of the microscopic BCS model of superconductivity,


15
where is a fermionic operator that creates an electron at site i of a square lattice
with spin a. U is the on-site repulsive interaction, and t is the hopping amplitude. In
the limit (t < U < ep e), the additional holes sit at Cu sites, and the hybridization
may be included by eliminating 0 sites to give an effective Hamiltonian for motion
on Cu sites alone. This is obviously a single-band Hubbard model. In a single-band
Hubbard model, the conduction band develops a correlation gap of an effective value
of the Coulomb repulsion Ueff, and this model yields only two bands, as shown in
Fig. 6(bottom).
For large on-site repulsion U, the one band Hubbard-model Hamiltonian can be
transformed into the t-J model Hamiltonian. This model describes the antiferromag
netic interaction between two spins on neighboring sites and it allows for a restricted
hopping between neighboring sites. Therefore, the Hamiltonian of (2) reduces to a
S = ^ Heisenberg model on the square lattice of Cu sites:
H t y [cj.o-(l n- (O'),* (i>
where S, are spin-1/2 operators at site i of a two dimensional square lattice, and J is
the antiferromagnetic coupling between nearest neighbors sites (ij) and is defined as
J =
4?
U '
(4)
The limit of validity of the t-J model is for J extendable into the regime J ~ t. The hopping term allows the movement of electrons
without changing their spin and explicitly excludes double occupancy due to the
presence of the projection operators (1 The Hamiltonian (3) is just the
effective Hamiltonian of the single-band Hubbard model in the large U limit. In this


53
' 7
f. ¡ i 1
y
Vr-
\ ^
~~w
I Souro* Chamber
a Near-, mid-, or lar-IR sources
b Automated Aperture
II Interferometer Chamber
c Optical fitter
d Automatic beamsplitter changer
III Sample Chamber
I Sample focus
J Reference focus
IV Detector Chamber
k Near-, mid-, or far-iR
detectors
a Two-sided movable mirror
f Control interferometer
g Reference laser
h Remote control alignment mirror
Fig. 18. Schematic diagram of IBM Bmker interferometer.


92
Photon Energy (eV)
o.o 0.2 0.4 0.6
Fig. 36. Frequency-dependent mass enhancement (upper panel) and renor
malized scattering rate, (lower panel)


CHAPTER IV
CRYSTAL STRUCTURE AND SAMPLE CHARACTERISTICS
Crystal Structure and Phase Diagram
La*?- t Stt CuOd
The structure of La2_zSrzCu04 shown in Fig. 12(a) is tetragonal and has been
known for many years as the K2NF4 structure. It is also called the T structure. In
La2_zSrzCu04, the Cu-0 planes perpendicular to the c axis are mirror planes. Above
and below them there axe La-0 planes. The Cu02 planes axe ~ 6.6 apaxt, sepaxated
by two La-0 planes which form the charge reservoir that captures electrons from the
conducting planes upon doping. The La-0 planes axe not flat but corrugated. There
axe two formula units in the tetragonal unit cell.
Each copper atom in the conducting planes has an oxygen above and below in the
c-direction forming an oxygen octahedron. These axe the so-called apical 0 atoms
or just Oz. However, the distance Cu-Oz of ~ 2.4 is considerably laxger than
the distance Cu-0 in the planes of ~ 1.8 At high temperatures (depending on
Sr concentration) there is a transition to an orthorhombic phase (Tq/t 530 K for
x = 0), and the copper atoms and the six oxygens surrounding them slightly deviate
from their positions.
For x = 0, La2_zSrzCu04 is an insulator. Upon doping, La3+ axe randomly
replaced by Sr2+, and these electrons come from oxygen ions changing their configu
ration from O2- to 0 (and thus creating one hole in their p shell). Metallic behavior
is observed for even small doping concentration, x > 0.04 (Fig. 13). For Sr dopings
30


5
of 2A = 3.5 0.3 is obtained. The superfluid condensate fractions are determined as
16% at 10 K and 10% at 19 K, and the London penetration depth, Al, is estimated
to be 4250 100 We also discuss the possibility of pairing mechanism in BKBO.
This dissertation is organized as follows. Chapter II reviews previous optical re
sults on the materials, investigated and issued. In Chapter III, models to describe
the behavior of electrons in the CO2 plane and theoretical models for the transfer
of spectral weight in the hole doping and electron doping cases are discussed. Some
fundamental properties of BCS superconductor are also given. In Chapter IV, the
crystal structures and the sample characteristics are presented. Chapter V will de
scribe infrared techniques, experimental apparatus and data analysis. Chapter VI,
VII and VIII are devoted to experimental results and discussion. Finally, conclusions
are presented in Chapter IX.


84
Photon Energy (eV)
0 1 2 3 4 5
Fig. 28. Room temperature a-b plane optical conductivity spectra of
Nd2_zCezCu04 with doping x as a logarithmic frequency scale.


142
The solid lines in Fig. 68 illustrate fitting curves using the standard Mattis-
Bardeen conductivity, based on the dirty limit, weak coupling BCS theory. Fitting
curves axe produced using the Drude formula at 30 K and the Mattis-Bardeen formula
with an energy gap of 54 cm-1, a scattering rate of 300 cm-1 and Tc = 22 K.
Our data axe very well represented by the weak coupling Mattis-Bardeen theory at
low frequencies (uj < 250 cm-1), showing the good agreement of the temperature
dependence below 2A. There is no feature corresponding to strong electron-phonon
coupling in this frequency range. At frequencies between 2A and 150 cm-1 the
measured spectrum at 10 K is less sharply increasing them the calculated spectrum.
Therefore, the present infrared measurements indicate that the BKBO system is a
dirty limit and weak or moderate coupling superconductor with a BCS-like gap and
coupling constant A (V 0.6.
Discussion of Pairing Mechanism in BKBO
So fax, an unsolved question is whether the origin or mechanism of superconduc
tivity in the BKBO system is the usual phonon mechanism or not.145 The BKBO
system does not contain any transition metal element. Hence, a magnetic mechanism
may not be expected for the superconductivity in this compound. In fact, no mag
netic order has been observed in BKBO by muon spin rotation experiments,146 and
the magnetic susceptibility in the normal state in BKBO shows a Pauli paramagnetic
behavior.147 Therefore, we may expect that the superconductivity in BKBO occurs
via the phonon mechanism.
If the weak or moderate coupling mechanism in this high Tc compound is pre
dominantly phonon mediated, then coupling to high frequency phonons is required.
The inelastic neutron scattering and moleculax dynamics simulation studies of Loong
et a/.128 suggested that the oxygen phonon modes soften by ~ 5 to 10 meV with


Optical Conductivity 61
Effective Electron Number 63
Loss Function 64
Temperature Dependence of Optical Spectra 66
One Component Approach 67
Doping Dependence of Low Frequency Spectral Weight 70
Drude and Mid-infrared Band 70
Transfer of Spectral Weight with Doping 71
Doping Dependence of Charge Transfer Band 73
Summary 74
VII.QUASI-PARTICLE EXCITATIONS IN LIGHTLY
HOLE-DOPED La2-zSrxCu04+ 98
Experimental Results 99
a-b Plane Spectra 99
c Axis Spectra 101
Mode Assignment 102
Hopping Conductivity in Disordered System 106
Optical Excitations of Infrared Bands 109
Summary Ill
VIII.INFRARED PROPERTIES OF Bi-0 SUPERCONDUCTORS 130
Normal State Properties 131
Results for Bai_xKxBi03 132
Results for BaPbi_xBix03 132
Comparison of Two Bismuthate Spectra 134
Free Carrier Component in BKBO 135
Superconducting State Properties 138
Superconducting Gap 138
Superconducting Condensate 140
Discussion of Pairing Mechanism in BKBO 142
Summary 144
IX.CONCLUSIONS 156
v


60
where = 0.695 cm-1/K. The broadening of the line is like the behavior of the free
carrier conductivity, which shows a linear temperature dependence of the scattering
rate. We find a coupling constant A ~ 0.25 using a formula fi/r = 2tXk^T. This
is comparable to the coupling constant obtained from the behavior of free carriers
in other high Tc cuprates. A similar temperature dependence has been observed in
the a-b plane conductivity spectrum in lightly doped Nd2Cu04_j, single crystal by
Thomas et al.5 They suggested that this band is related to a bound charge coupled to
the spin and lattice excitations. Unlike the result of Thomas et al., our result shows
that the oscillator strength of this band increases with decreasing temperature. We
will discuss this band in the next chapter again for lightly hole-doped La2_xSrxCu04
experiments.
The charge transfer band observed near 1.5 eV also has a temperature dependence.
Figure 26 shows the reflectance spectra (a) and the optical conductivity spectra (b)
calculated from the reflectance spectra using K-K transformations. The reflectance
spectrum in Fig. 26(a) at room temperature clearly shows two peaks near 1.36 eV
and 1.6 eV. As the temperature decreases, the spectral weight around 1.36 eV at 300
K shifts to the peak near 1.6 eV and the sum rule is satisfied.
Doping Dependence of Optical Spectra
Optical Reflectance
The reflectance spectra for the a-b plane are shown in Fig. 27 between 80 cm-1
(10 meV) and 42 500 cm-1 (5.3 eV) for various Ce concentrations. Other metallic
samples with Ce concentrations of x = 0.18 and x = 0.20 were measured, too. But,
these spectra are not shown in Fig. 27, because they are very similar to the spectrum
of Ce concentration x = 0.19. With doping the spectral weight around 1.5 eV peak is


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164


141
inspection of Fig. 65(b). Thus, the missing area can be estimated from
A = ~ JQ lai(w) ~ (65)
where ups 47rnse2/m is the superconducting plasma frequency. We obtain ujp3 =
3 750 200 cm-1 and 3 000 200 cm-1 at 10 K and 19 K, respectively. The superfluid
condensate fraction is also estimated according to the formula:
na(T) u>l,{T)
n Id
We find fs(T) = 16% and 10% at 10 K and 19 K, respectively, using wp£> = 9400
cm-1 in the normal state.
The area in the 6 function, A, controls the low frequency electromagnetic pene
tration depth. The London penetration depth, X is related to the plasma frequency
in the superconducting state by Xi = 1/2tuP3. We find 4250 100 at 10 K.
This value is somewhat higher than that obtained using microwave methods,142 but
smaller than the results from other infrared measurements.143,144 Since the response
of a superconductor to an electromagnetic field is governed by the imaginary part of
the optical conductivity, can be also determined using the formula:
( (12)
Fig. 67 displays as a function of frequency. For uj < 2A, we expect that X is
independent of uj due to the superfluid response which is cr2 oc l/u, and the zero
frequency extrapolated value is similar to above result. For 2A < u> < 100 cm-1, X
increases with w, corresponding to <72 which falls to zero more rapidly than l/u>.


cr(Q cm) Reflectance
120
0 100 200 300 400 500 600 700
y(cm_1)
Fig. 49. Far-infrared (a) reflectance and (b) conductivity ai(u) for
Lai.99Sro.oiCu04+£ at several temperatures.


152
0 100 200 300 400 500 600
y(cm-1)
Fig. 65. The fax-infrared reflectance (a) and a\[u) (b) at temperatures be
tween 10 K and 300 K.


40
the refractive index n(w), the extinction coefficient k(u), and the dielectric function
e(w) by
N(u) = n(u;) + ik(u) = \J e(u;). (25)
One quantity measured in experiments is the reflectance, which is the ratio of the
reflected intensity to the incident intensity:
R = rr* = p2
(1 n)2 + k2
(l + n)2 + jfc2
(26)
The measured reflectance R(u) and the phase 0(u;) are related to n(u) and k(u) by
VReie = r
(1 n) ik
(1 + n) + ik'
(27)
and
2k
(28)
It is difficult to measure the phase 9(u) of the reflected wave, but it can be
calculated from the measured reflectance R(u>) if this is known at all frequencies via
the Kramers-Kronig procedure. Then we know both R(u>), 9(uj), and we can proceed
by (27) to obtain n(u>) and k(u). We use these in (25) to obtain e(u) = ei(u;)-|-ie2(w),
where ei(u>) and 2(0}) axe the real and imaginary parts of the dielectric function. The
inversion of (25) gives
ei(w) = n2 k2, 2(0;) = 2nk. (29)
We will show in data analysis section how to find the phase 9(u) as an integral over
the reflectance R(w) using Kramers-Kronig relations.


58
should be noted that optical transmission spectra in undoped materials show other
absorption features in the energy range 0.2 to 1 eV. For example, weak absorption
bands near 0.5 eV were first observed in undoped single crystals of single layered T,
T*, and ^ structures by Perkins et al.56 and multi-layered YBa2Cu307_ by Zibold
et al.57 In these studies, they suggest that these bands result from exiton-magnon
absorption processes. The spectra in Fig. 21 also show a strong transition around 5
eV, which is observed above 6 eV in optical spectra of the T and T* phases. This
peak is located at higher energies in the other structures for the same reason as the
larger charge transfer energy.
A group theoretical analysis of the phonon modes in Nd2CuC>4 yields
3A.2U+4.V58-60 The A.2 modes are observed in the c polarization spectrum and
the Eu modes corresponds to an atomic motion parallel to the a-b plane. Figure 22
displays the a-b plane reflectance in the far-infrared region as a function of tempera
ture. We clearly observe four strong phonon bands. As the temperature is reduced,
the phonon lines become sharper. Since all spectra show an insulating behavior, we
extrapolate them to zero frequency assuming asymtotically a constant reflectance.
Then we obtain cr\(u)) and Im(1/e) by K-K transformations.
The temperature dependence of the a-b plane phonons is shown in Fig. 23. The
upper panel shows <7i(u>), whereas the lower panel shows Im( 1/e). The former
determines the TO phonon frequencies, whereas the latter the LO phonon frequencies.
Four phonon bands occur at 131, 303, 347 and 508 cm-1 at room temperature.
These phonon modes are similar to the case of L^CuO-i, but the phonon energies in
Nd2Cu4 are lower than in La2Cu04- This difference is primarily due to a result of
a larger unit cell dimension (longer bond lengths) in the former material.61 These
phonon bands result from four motions: a translational vibration of Nd atom layer


170
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19
Fig. 8. The integrated low energy spectral weight (LESW) as a function of
doping concentration x for Mott-Hubbaxd model of Meinders et al.4 The
solid line corresponds to the localized limit f = 0. The data points axe
from the calculations: t = 0.5 eV (lowest) to t = 2 eV steps of 0.5 eV.
The total electron removal spectral weight will be 2N 1 (just the number of
electrons in the ground states) and the total electron addition spectral weight will be
2N +1 (total number of holes in the ground states). The electron addition spectrum
consists of two parts, a high energy scale (the conduction band) and a low energy
scale, which is the unoccupied part of the valence band. Therefore, we can know that
the low energy spectral weight equals 1. The same arguments hold for an electron-
doped semiconductor. Thus, the low energy spectral weight grows as x with doping
x for a hole-doped and electron-doped semiconductor.


3
band Hubbard model, in which the occupation is constrained to at most one electron
per orbital. Hence, the low frequency spectral weight (LFSW) is expect to behave as
the MH system. However, it is not clear if hole and electron doped systems can both
be described by a single band Hubbard model. On the other hand, it is interesting
to note that the influence of doping in the antiferromagnetic correlations is non-
universal between hole and electron materials. For example, single crystal neutron
scattering measurements on Nd2-ICeICu04 by Thurston et al.2 have shown that 3D
antiferromagnetic order persists even with x as high as 0.14, while on La2_ISrICu04,
a doping of x ~ 0.02 is enough to destroy the long-range order.
The discovery of copper oxide high Tc superconducting materials has also gener
ated renewed interest in the Bi-0 superconductors, due to many similarities between
the bismuthates and the Cu-0 cuprates. For example, in spite of a low charge-carrier
density (on the order of 1021 cm-3) the Tc of the bismuthates is anomalously high; su
perconductivity occurs near the metal-insulator transition by chemical doping of the
insulating BaBiOs. In addition, the high Tc conductivity in this system is of great in
terest because it contains neither a Cu atom nor a two dimensional structural feature
which are considered to be crucial for the high Tc in the copper oxide superconduc
tors. Thus, understanding this system would undoubtedly facilitate understanding of
Cu-0 cuprates.
This dissertation describes a detailed study of the optical properties of high Tc
cuprates and Bi-0 superconductors over the infrared to the UV region in the tem
perature range from 10 K to 300 K.
First, we concentrate on the Ce doping dependence of electron doped
Nd2-iCeICu04, which has the simplest structure among high Tc materials. The low
frequency spectral weight (LFSW) for electron-doped Nd2-ICeICu04 is compared


64
is the contribution of the mid-infrared bands, which ends axound 10 000 cm-1 (1.2
eV). Next, more steep rises come from the contributions of the charge transfer band
and high energy interband, respectively. This behavior is very similar to the results
obtained in hole-doped cuprates.
The low frequency Neff(u>) for metallic samples is plotted in Fig. 31 up to 1 000
cm-1. Notably, the integrated spectral weight of superconducting sample of x = 0.15
exhibits a rapid rise at low frequencies below 200 cm-1 due to the strong Drude
contribution, and is very strong at frequencies below 800 cm-1 compared with that
of slightly overdoped sample of x = 0.16. Finally, two curves for x = 0.15 and
x = 0.16 merge near 1000 cm-1. This implies that the strength of the mid-infrared
band in x = 0.15 is a weaker than in x = 0.16.
Another important result of our measurements is that Neff(u) at high frequency
above 3 eV gradually increases with doping. This is particular in our system. As we
will discuss later, this is due to the anomalous strong Drude and mid-infrared bands
caused by doping. In order to satisfy sum rule, this result suggests that another high
energy band above 4 eV loses spectral weight with electron doping. This is compared
to the results of hole-doped La2_iSrzCu04 and YBa2Cu307_. In these studies,
the only spectral weight of the charge transfer region between 1.5 eV and 4 eV is
transfered to low frequencies below 1.5 eV, and hence Ntf/(u) intersects near 3 eV
with increasing doping. It is noteworthy that Neff at 3 eV is a factor of 2 larger than
that of La2_ISrICu04.
Loss Function
In this section, we describe the energy loss function, Im[l/e(u;)]. This function
is the probability for energy loss by a charged particle that passes through a solid.
It can also be calculated from -Im[l/e(u;)] = e2(u)/[ei(u)2 + e2(w)2]. The peak


165
17. F. C. Zhang and T. M. Rice, Phys. Rev. B 37, 3759 (1988).
18. J. Wagner, W. Hanke, and D.J. Scalapino, Phys. Rev. B 43, 10 517 (1991).
19. H. Romberg, Phys. Rev. B 42, 8786 (1990).
20. C.T. Chen, Phys. Rev. Lett. 66, 104 (1991).
21. H. Eskes, M.B.J. Meinders, and G.A. Sawatzky, Phys. Rev. Lett. 67, 1035
(1991).
22. D.C. Mattis and J. Baxdeen, Phys. Rev. Ill, 412 (1958).
23. T. Holstein, Phys. Rev. 96, 535 (1954).
24. G.M. Eliashberg, Soviet Phys. JEPT 11, 696 (1960).
25. S.B. Nam, Phys. Rev. 156, 470 (1967).
26. W.L. McMilan, Phys. Rev. 167, 331 (1968).
27. J.W. Lynn, High Temperature Superconductivity (Springer-Verger, New
York, 1990), Ch. 4.
28. S. Pei, J. D. Jorgensen, B. Dabrowski, D. G. Hinks, D. R. Richards, A. W.
Mitchell, J. M. Newsam, S. K. Sinha, D. Vaknin and A. J.Jacobson, Phys. Rev.
B 41, 4126 (1990).
29. L.F. Schneemeyer, J.K. Thomas, T. Siegrist, B. Batlogg, L.W. Rupp, and D.W.
Murphy, Nature 335, 421 (1988).
30. J.H. Cho and D.C. Johnston, Phys. Rev. Lett. 71, 2323 (1993); Phys. Rev.
B 5, 3179 (1992); Phys. Rev. Lett. 70, 222 (1993).
31. M.L. Norton, Mat. Res. Bull. 24, 1391 (1989).
32. S. Tajima, S. Uchida, H. Ishii, H. Takagi, S. Tanaka, U. Kawabe, H. Hasegawa,
T. Aita, and T. Ishiba, Mod. Phys. Lett. B 1, 353 (1988).
33. S. Tajima, S. Uchida, A. Masaki, H. Takagi, K. Kitazawa, and S. Tanaka, Phys.
Rev. B 32, 6302 (1985).
34. P.W. Anderson, Mat. Res. Bull. 8, 153 (1973); Science 235, 1196 (1987);
Phys. Rev. Lett. 64, 1839 (1990).
35. B.C. Webb, A.J. Sievers, and T. Mihalisin, Phys. Rev. Lett. 57, 1951 (1986).


10
undoptd
d Ud d10
Ef Ep
p-type n-type
Fig. 4. Simple electronic band structure for the charge transfer insulator, for
hole-doped and for electron-doped.
Fig. 5. Schematic representation of the oxygen octahedra. The solid lines
illustrate the symmetric Peierls distortions and the dashed lines illustrate
the undistorted case.


128
Photon Energy (eV)
0.8 1.2 1.6
Fig. 57. The <7i(u;) spectra near 1.4 eV for La2-xSrxCuC>4+.


45
the different photon energy regions. Table 2 lists the parameters used to cover each
frequency range.
Polarizers could be placed after the exit slit and before the focus of the second
spherical mirror if polarized reflectance and transmittance measurements were re
quired. The polarizers used in the far infrared were wire grid polarizers on either
calcium fluoride or KRS5 substrates. Dichroic polarizers were used at higher frequen
cies. (table 2)
The signed from the detector was fed into a standard lock-in amplifier. The lock-
in is then averaged over a given time interval. The time constant on lock-in could be
varied the signal to noise ratio. After having taken a data point, the computer sent
a signal to the stepping motor controller to advance to the grating position. This
process was repeated until a whole spectrum range was covered. The spectrum was
normalized and analyzed through the computer.
Michelson Interferometer
A Michelson interferometer is an alternative instrument for measuring the spec
trum in the 10 to 800 cm-1 region. In principle, this instrument works in the same
way as the Bruker interferometer, but has a better S/N ratio at low frequencies below
100 cm-1 due to laxge size and high power source. A mercury arc lamp is used as a
source and the source is chopped to remove all background radiation. The combina
tion of the thickness of a Mylar beam splitter and different filters axe used to cover
the corresponding frequency range.
The sample and detector axe mounted in the cryostat. A doped germanium
bolometer operating at 1.2 K is used as a detector. Data acquisition procedure is
same as that of Perkin-Elmer grating monochromator.


144
scattering experiments, and infrared measurements have not been successful. This
can be attributed to the broadening of the phonon linewidth because of the strong
electron-phonon interaction. Also, we do not rule out the possibility of the pairing
mechanism being associated with other electronic excitations.
Summary
In this chapter, we have examined the problems associated with the normal state
and superconducting state properties of Bai-jKxBiOs (BKBO) and BaPbi_xBix03
(BPBO) single crystals. In the normal state, broad bands in the infrared conductivi
ties are observed at about 2500 and 4 000 cm-1 for BPBO and BKBO, respectively,
as shown in the doping dependence experiments. These bands axe associated with
transitions across the charge density wave (CDW) gap. For BKBO, the 1/r associ
ated with the Drude band shows a non-lineaxity with temperature consistent with
the transport measurements, whereas for BPBO the extrapolated values of ) at
zero frequency show a semiconducting behavior. For BKBO the electron-phonon cou
pling constant A ~ 0.6 is estimated from the simple model for the electron-phonon
scattering rate.
In the superconducting state, the positions of the superconducting edge in the fax-
infrared reflectance spectra have a strong temperature dependence which is suggestive
of a BCS-like energy gap. The energy gap of 2A/fcgTc = 3.5 0.3 is obtained from
the oq,,(u>) spectrum at 10 K and is consistent with a weak-or moderate-coupling
limit. Our results show that the BKBO system is a dirty limit superconductor with
a superfluid condensate fraction of 16% at 10 K and 10% at 19 K. The London
penetration depth A is also estimated to be 4 250 100 .
The far-infrared conductivity spectra axe very well represented by the standard
Mattis-Baxdeen conductivity based on a weak coupling BCS theory at frequencies less


48
the low frequency extrapolation. For insulating samples, the reflectance is assumed
constant to dc. In the case of metallic samples, a Hagen-Rubens relation, f?(u) = 1
Ay/u, was used. In the superconducting state, we have used the formula R = 1 i?u>4,
in which R goes to unity smoothly as u approaches zero.
The high frequency extrapolation has significant influence on the results, primarily
on the sum rule derived from the optical conductivity. We reduced this effect by
merging our data to the reflectance spectra for insulating phase of published papers
which extend up to 37 eV (300000 cm-1) for Nd2-zCezCu04, La2_xSrrCu04,32 and
Bi-0 superconductors.33 We terminated the transform above 37 eV by using the
reflectance vaxying as l/w4, which is the free electron asymtotic limit.
The Kramers-Kronig relations enable us to find the real part of the response of a
linear passive system if we know the imaginary part of a response at all frequencies,
and vice versa. They are central to analysis of optical experiments on solids. Let us
consider the response function as a(u;) = ai(u;) + 02(0;). If a(u>) has the following
properties, a(u;) will satisfy the Kramers-Kronig relations:
. 2 n sa2(s) , 2u ai(s) x
ai(w) = -P / xds, and 0*2(0;) = P / -5 K \ds. (39)
7T J0 s1 First, q(w) has no singularity, and <*(u>) v 0 uniformly as |u;| 00. Second, the
function ai(w) is even and a2(w) is odd with respect to real u.
We can apply the Kramers-Kronig relations to reflectivity coefficient r(u;) viewed
as a response function between the incident and reflected waves in (24). If we apply
(39) to
lnr(u;) = lni?5 + i6(u), (40)
we obtain the phase in terms of the reflectance:
1 fc
' |s -f- u\ dIn 7?(s)
in ds.
I s (jj\ ds
(41)


140
energy gap. At T = 10 K, the conductivity spectrum has a minimum at 54 cm-1
and begins to rise up to 120 cm-1 due to photo-excited quasipaxticle absorption. The
difference between cr\n{ strength below 2A (missing area) shifts to the origin to form superconducting con
densate. Theoretically, at T = 0 K, <7ia(u>)=0 up to u=2A. However, our results
show that below 2A the uncertainty in 100% uncertainty for the reflectance spectrum in the superconducting state.
We obtained 2A/kpTc = 3.5 0.3 using 2A = 54 cm-1 and Tc = 22 K, which is
consistent with a weak-or moderate-coupling limit, where 0.3 corresponds to 10%
and 90% value of dc susceptibility for Tc measurement. This value for the energy
gap is in good agreement with the tunneling spectroscopy results of Sharifi et a/.132
for samples from same batch. The observed value for the energy gap is also consis
tent with the results (3.5 ~ 3.8) observed from other tunnelling spectroscopy,133-137
measurements of oxygen isotope effects,138-140 and infrared measurement.141
Superconducting Condensate
We have shown earlier that for the Drude carriers the scattering rate is ~ 300
cm-1 and the mean free path is ~ 168 at 30 K. We can also calculate the Pippard
coherence length ( = hvp/2n2A = 590 using the Fermi velocity vp = 108 cm/s
and 2A = 54 cm-1. The results suggest that BKBO is a dirty limit superconductor,
exhibiting l < £ and 1/r 2A.
In the dirty limit, much of the free carrier conductivity exists at frequencies above
2A and a small part of the Drude strength contributes to the superfluid condensate.
One might expect that from the sum-rule argument the missing area A under the
integral of cti3 appears at u = 0 as A<5(u;). This missing area is easily estimated by


Phonon frequency
122
Temperature (K)
Fig. 51. In-plane phonon frequencies as a function of temperature.


cr(Qcm) Reflectance
149
0 100 200 300 400 500 600
y(cm_1)
Fig. 62. Far-infrared reflectance (a) and conductivity <7i(u;) (b) of
BaPbi_xBir03 at several temperatures.


86
Photon Energy (eV)
0.01 0.1 1
Fig. 30. Effective electron number per formula unit for Nd2-xCexCu04 at
doping levels from 0 to 0.2.


132
decreases from 300 K to 30 K, the infraxed reflectance up to 5000 cm-1 nonlinearly
increases and the reflectance spectra axe exchanged axound the plasma minimum near
14 800 cm-1.
The temperature dependent optical conductivity cri(cj) derived from a Kramers-
Kronig analysis of the reflectance spectra in Fig. 59 are plotted in Fig. 60 up to 30 000
cm-1. The conductivity curves at each temperature show a prominent deviation from
the Drude curve. As the temperature is reduced, the far-infrared conductivity rapidly
increases with increasing temperature while the mid-infrared and high frequency con
ductivities decrease, as expected from the / sum rule. We observe a prominent peak
neax 4 000 cm-1 with a width of about 8 000 cm-1 and a oscillator strength of 20 500
cm-1 at room temperature. This peak is associated with the transitions across the
CDW gap.
Results for BaPbi_TBiTQt
The upper and lower panels in Fig. 61 show the room temperature (a) reflectance
and (b) conductivity for BaPbi_zBiz03 (BPBO). The reflectance and conductivity
spectra of BPBO axe very similar to those of BKBO. The reflectance spectrum shows
a metallic character and has a reflectance minimum axound 12 800 cm-1. Like BKBO,
the conductivity spectrum shows the non-Drude behavior due to a strong CDW band.
However, for BPBO this peak is quite a bit sharper and the gap is narrower than in
BKBO. This is indicative of a high degree of nesting of the band neax the Fermi
surface. On the other hand, the Drude strength is very low, making it difficult to
define the Drude component in the c\{u) spectrum. This trend is likely related to
the formation of Pb-related states within the CDW gap as was observed in the dop
ing dependent experiment of BPBO by Tajima et al.33 They have shown that when
Bi is partially doped with Pb, the CDW band is gradually broadened and shifts to


143
40% K doping of BaBi03 and that the strongest phonon features in superconducting
BKBO occur between 30 and 70 meV (250 and 570 cm-1). The strong features at
these energies axe also observed in the second derivative of the tunneling current and
in the inverted a2F(uj). In addition, Shirai et a/.126 have found that the mode around
60 meV is due to the oxygen stretching vibration towards the nearest neighboring Bi
atoms and the electron-lattice interaction causes remarkable renormalization of the
longitudinal oxygen stretching and breathing mode axound 60 meV. They also find
that for a fixed value of A = 1, Tc ~ 30 K is obtained using the effective Coulomb
repulsion ¡x* ~ 0.1 for a reasonable description of the superconducting properties of
BKBO and a laxge electron-phonon matrix elements from coupling to the high energy
phonons.
Unfortunately, we did not observe the phonon structure nor any sign of the
electron-phonon interaction in this frequency range. Instead, our result in Fig. 65(b)
shows the phonon peale neax 200 cm-1 which is assigned to the bending mode phonon
in the undoped BaBi03. The shape and position of the 200 cm-1 peak do not change
with decreasing temperature, indicating no Holstein mechanism which would axise
from a reasonably strong electron-phonon coupling. However, recently published
infrared measurements143 show that the phonon shape at about 500 cm-1 for the
stretching mode becomes increasingly asymmetric with decreasing temperature. Sim
ilar results have been presented for the breathing mode phonon in Raman exper
iments.148 In my opinion, the breathing mode is not likely to be related to the
pairing mechanism due to the fact that the superconductivity occurs in the cubic
phase 0.37 < x < 0.5 as seen in the phase diagram by Pei et a/.28, but the breath
ing mode only persists to neax the orthorhombic-cubic phase transition. However,
most attempts to find these phonon structures in tunneling spectroscopy, neutron


75
Nd2-iCezCu04 at temperatures between 10 K and 300 K. We analyze our data with
the one component and two component models. Our results show that the doping
mechanism of the electron-doped Nd2-xCexCu04 is a little different from that of
hole-doped La2-xSrxCu04.
The spectrum of the undoped Nd2Cu04 shows a typical insulating characteristic
with energy gap of 1.5 eV which is identified to 0 2p-Cu 3d charge transfer excitations.
Doping with electrons in insulating Nd2Cu04 induces a shift of spectral weight from
the high energy side above the charge transfer excitation band to the low energy side
below 1.2 eV. The low energy spectral weight for a barely metallic sample, x = 0.11
is composed of two paxts: a narrow Drude-like and mid-infrared paxts. Upon further
doping the Drude-like band rapidly increases and the mid-infrared band shifts to
lower frequency, and hence two paxts axe hardly sepaxated in the metallic phase.
A weals temperature dependence of the far-infrared reflectance suggests the non-
Drude behavior of this material. For example, the change of reflectance between 15 K
and 300 K for non-superconducting metallic samples of x = 0.16 and x = 0.19 is less
than 2% and for superconducting sample of x = 0.15 the reflectance change between
same temperatures is about 3.5% near 600 cm-1. This non-Drude behavior can be
analyzed by a frequency dependent scattering rate and a mass enhancement in the
one component approach.
In the one component approach, our results show that the mass enhancement at
low frequencies is large, and for superconducting sample Im S is linearly propor
tional to u below 5 000 cm-1, in good agreement with the predicted behavior in the
numerical models in which strong quasiparticle damping is assumed. From the slope
of ImS a weak coupling constant A ~ 0.15 0.01 is estimated.


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
OPTICAL PROPERTIES OF DOPED
CUPRATES AND RELATED MATERIALS
By
Young-Duck Yoon
August 1995
Chairman: David B. Tanner
Major Department: Physics
The optical properties of cuprates, Nd2-xCe2;Cu04 and La2-xSrICu04, and the
related materials, Bai-jK^BiOs (BKBO) and BaPbi_xBix03 (BPBO), have been
extensively investigated by doping-and temperature-dependent reflectance measure
ment of single crystal samples in the frequency range between 30 cm-1 (4 meV) and
40 000 cm-1 (5 eV). The Nd2-xCexCu04 system has been studied at Ce compositions
in the range 0 < x < 0.2. La2_xSrxCu04 has been studied in the spin glass doping
regime, (x < 0.04). The two bismuthates have been investigated as superconducting
materials with the maximum Tc.
Our results for Nd2-xCezCu04 show that doping with electrons induces a transfer
of spectral weight from the high energy side above the charge transfer excitation band
to the low energy side below 1.2 eV, similar to the results observed in hole-doped
vii


41
Infrared and Optical Technique
Fourier Transform Infrared Spectroscopy
The central component of a Fourier transform infrared spectrometer is a two-
beam interferometer, which is a device for splitting a beam of radiation into two
paths, the relative lengths of which can be varied. A phase difference is thereby
introduced between the two beams and, after they are recombined, the interference
effects are observed as a function of the path difference between the two beams in
the interferometer. For Fourier transform infrared spectrometry, the most commonly
used device is the Michelson interferometer.
The Michelson interferometer, which is depicted schematically in Fig. 17, consists
of two plane mirrors, the planes of which are mutually perpendicular. One of the
mirrors is stationary and the other can move along an axis perpendicular to its plane.
A semi-reflecting film, called the beamsplitter, is held in a plane bisecting the planes
of the two mirrors. The beamsplitter divides the beam into two paths, one of which
has a fixed pathlength, while the pathlength of the other can be varied by translating
moving mirror. When the beams recombine at the beamsplitter they interfere due
to optical path difference. The amplitudes of two coherent waves which at time zero
have the same amplitude A(v) at wave number v, but which cure separated by a phase
difference kS = 2x1/6, cam be written as
yi(z) = r A{vY^vzdv, and y2{z) = f A(u)ei2r^-sUv,
Joo JOO
(30)
where k is the propagation constant, v is the wave number and 6 is the optical path
difference between the two waves. Using the law of superposition, one has
roo
y(z) = yi{z) A y2{z) = [A(i/)(1 +
J OO
(31)


cr(ficm)-1 Reflectance
126
y(cm_1)
Fig. 55. (a) The a-b plane reflectance spectra at temperatures between 10 K
and 300 K for Lai.ggSro.ircCuO^ (b) The real part of the a-b plane con
ductivity <7i(u;) derived from the reflectance spectra R in (a). Inset is the
temperature dependent conductivity of the 0.15 eV band after subtracting
the 0.5 eV band from the data (b).


31
La2Qi04
Nd2Cu04
Cu O O O La, Nd
Fig. 12. Crystal structure (a) T phase and (b) T phase.
between ~ 0.05 and ~ 0.3, a superconducting phase was found at low temperatures.
The maximum value (~ 40 K) of Tc is observed at the optimal doping of x ~ 0.15.
Ndo-^Ce^CuO/t
The body centered structure of Nd2Cu04 is shown in Fig. 12(b) and it is called
the T structure. It has a close relationship to the T structure of Fig. 12(a). As in
the T phase structure, the structure is made of a single Cu02 plane and two Nd-0
planes, but the Nd-0 planes are shifted by a/2 in the x-direction, so that the oxygen
ions in the Nd-0 planes are not on the top of Cu ions. The Nd2Cu4 can be easily
electron-doped replacing Nd3+ by Ce4+.
The phase diagram of this material is shown in Fig. 13 comparing it with hole-
doped compound. The similarities between the two diagrams are shown, but the


125
Fig. 54. Far-infrared conductivity spectra at 10 K for 2 % and 3 % Sr doped
La2_xSrxCu04. Solid lines show fitting curves from Lorentz model.


157
lies directly in the Van Hove singularity of the upper Hubbard band and carriers have
a more itinerant property.
Concerning the second issue, we have first found the evidence for carrier-lattice
interaction by a deformation potential caused by the rotation of Cu06 octahedra.
This is characterized by a strong infrared active modes and an appearance of a strong
Ag Raman modes upon cooling. We have also shown that the qualitative features
of the a\(uj) spectra in the far-infrared and the transport property is similar to the
behavior of a conventional 2D disordered system. At low temperatures below 50 K,
the dc resistivity shows the phenomenon of variable range hopping, where all states
are localized near the Fermi energy and a hole just below the Fermi level jumps to a
state just above it. In addition, the a\(u) spectra at 10 K show a resonant absorption
whose maximum occurs at frequencies between 100 and 130 cm-1, depending on Sr
concentration. This absorption results from the photon-induced hopping of charge
carriers between the localized states. Further, we have observed an absorption band
near 0.15 eV, corresponding to the antiferromagnetic energy J, which also seems to
interact with the lattice, and the peak near 1.4 eV is due to a result of an exitonic
effect. Hence, we conclude that charge dynamics as well as spin dynamics is very
important at low doping levels of the high Tc cuprates.
Finally, in Bi-0 superconductor studies, we have found that the cri(u;) spectra for
both BKBO and BPBO have broad peaks, which are associated with the transitions
across the charge density wave energy gap, are observed in the mid-infrared region.
The extrapolated values of a\ (w) at zero frequency are consistent with the dc resistiv
ity measurements, showing metallic and semiconducting behavior, respectively. The
local CDW order for BKBO seems to be associated with the inhomogeniety of the
potassium concentration near the orthorhombic-cubic phase transition. We have also


66
Our results for Nd2-xCexCu04 axe similar to those of La2-xSrxCu04 by Uchida
et al.,3 where the zero crossing of ei(u>) for the metallic samples is pinned near 0.8
eV due to strong mid-infrared absorptions. In contrast, the dielectric response for
YBa2Cu307_j obtained by Cooper et al.39 shows almost linear doping dependence of
zero crossing of i(c*>), exhibiting neaxly free carrier behavior.
Temperature Dependence of Optical Spectra
The temperature dependence of the reflectance between 80 cm-1 and 4 000 cm-1
was measured in order to study the applicability of the Drude model. For non
superconducting metallic samples, the change of reflectance between 10 K and 300
K is less than 2% in the far-infrared region, as shown in Fig. 33 for metallic samples
of x = 0.16 and x = 0.19. However, for superconducting sample of x = 0.15, the
reflectance change between these temperatures is about 3.5% neax 600 cm-1.
Figure 34(a) shows the temperature dependent reflectance of the superconduct
ing sample, x = 0.15, in the frequency range between 80 cm-1 and 2 000 cm-1.
As the temperature decreases, the magnitude of the reflectance exhibits a system
atic increase. The optical conductivity shows a clear picture of a Drude behavior.
Figure 34(b) shows cri(u>) obtained after a K-K analysis of reflectance spectra in
Fig. 34(a). The reduced, the spectral weight between 500 cm-1 and 2 000 cm-1 is transferred to lower
frequencies, corresponding to the narrowing of the Drude band at low temperatures.
We have fit our results with the two component model, a Drude part and several
Lorentzian contributions:
. ,2 N 2
pD y- upj
iu/r 4^ uj'j u2 i
+ f
luJlj
oo>
e(w) =
(52)


109
In this connection, we would like to note that at u = uimax the separation ru
is approximately equal to 2.5a for 3 % Sr doped sample and 2.7a for 1 % Sr doped
sample. It is of the same order for all other experimental data available.
Optical Excitation of Infrared Bands
In this section, we axe concerned about three infrared absorption bands, which
are considerably different from those of usual doped semiconductors, a narrow band
at 1 200 cm-1 (0.15 eV) and a broad band at 4 050 cm-1 (0.5 eV), and small peale
near 1.4 eV. This strong infrared absorption suggests the importance of electron-
phonon coupling, or other electronic mechanisms in high Tc materials. With further
doping the lower two bands merge with the low frequency free carrier absorption,
and are identified as the mid-infrared band that has been observed in several other
cuprate superconductors. Figure 55 shows the reflectance spectra R(u) (a) and the
conductivity spectra <7\(u) (b) after Kramers-Kronig transformations of R(oj) at sev
eral temperatures for 2% Sr doped La2-xSrxCu04. Two absorption peales are clearly
observed near 0.15 eV and 0.5 eV. (Ti(ui) in Fig. 55(b) also show that the 0.15 eV
band has strong temperature dependence, and the 0.5 eV peak has little temperature
dependence, but the origin of 0.5 eV band is obscure.
First, we are are interested in the 0.15 eV band, because this band appears
(0.12 eV at 10 K and 0.15 eV at 300 K) near the antiferromagnetic exchange en
ergy J calculated from Raman experiments." This band also appears in the t-J
model15,18,72100-102 which is not related with charge excitations. According to the
t-J model, this band is caused by spin fluctuations around the doped hole, and the
energy scale of magnetic interaction is the order of the exchange constant J ~ 0.1
eV. So, one possibility for the origin of this band is the magnetic interaction between
the carriers and the antiferromagnetic spin order.


BIOGRAPHICAL SKETCH
Young-Duck Yoon was born in Seoul, Korea. After completing his undergraduate
course and discharging from military service as a second lieutenant, he then worked
as a researcher at Hyundai Electronic Co., where he took part in the process of
VLSI. During this period, he decided to go abroad for further studies. He started
his graduate studies in physics at the Iowa State University in 1988, where he was
awarded an M.S. degree in Dec. 1990. The topic of his masters thesis was nuclear
magnetic resonance and nuclear quadruple resonance of Cu63,65 on c-axis aligned
YBa2Cu307_$. In 1991, he moved to the University of Florida to pursue a Ph.D.
program in physics and joined with Professor David Tanners group in 1992 to study
optical properties of high Tc superconductors.
174


14
Fig. 6. Simple band structure in the three band (top) and one band (bottom)
Hubbard model.


137
resistivity. The extrapolated value of a\{u) at u = 0 (2 300 (flcm)-1 at 300 K, p =
430 /xil cm) corresponds to one of the lowest values of room-temperature resistivity
which usually ranges between ~ 200 3200 /zficm. Sato et a/.123 demonstrated
that in a sample with a room temperature resistivity lower than 700 [iti cm, the
temperature dependence of the resistivity is metallic with a positive temperature
coefficient and the room temperature resistivity of the semiconducting phase samples
exceeds 700 /fl cm, where the 700 ¡J.CL cm value corresponds to the mean free path
equal to an interatomic spacing of a = 4 A dc resistivity ratio p(300 K)/p(30 K)
of about 2.25 obtained from simple Drude formula,
_ uId
Pdc 4tt(1/t)c
is also consistent with their resistivity ratios p(300 K)/p(30 K) as high as 1.9 ~ 2.6
for the most metallic samples. Thus our data axe essentially in good agreement with
the temperature dependence of the dc resistivity.
Talcing the Fermi velocity to be Uf=108 cm/s from the band calculation124 and
using the relaxation rate (700 cm-1 at 300 K and 300 cm-1 at 30 K), we obtain a
mean free path i vpr ~ 72 at 300 K and 168 at 30 K. The mean free path is
longer than the interatomic spacing a (~ 4 ). The resistivity is expected to saturate
if i < a because the mean free path can no longer be properly defined. The mean
free path for the BKBO sample with a high resistivity value and BPBO samples is
supposed to be close to the lattice constant 4 from the fact that this material shows
semiconductor behavior with a negative temperature coefficient.
We can estimate the electron-phonon coupling constant from the simple model
for the electron-phonon scattering rate, h/r = 2irXk^T.125 This formula applies for T
> 0£>. We obtain a moderate-coupling value for the coupling constant A ~ 0.6 at 300


Reflectance
83
Photon Energy (eV)
0.01 0.1 1
Fig. 27. Room temperature reflectance spectra of Nd2-zCexCu04 for various
x on a-b plane.


73
excitations. Especially, a strong spectral weight in the metallic phase axound x = 0.15
may reflect the contribution of the charge transfer excitations. This is a spectral
weight transferred from the p-like correlated states to the low frequency region. In
contrast, the LFSW in La2_xSrxCu04 goes to 2x at high doping levels and x at high
doping levels with doping concentration x.
Doping Dependence of Charge Transfer Band
Figure 40 represents the variation of the charge transfer bands with Ce dop
ing. The charge transfer conductivities axe obtained after subtracting high energy
interbands, and the Drude and mid-infrared bands. For the charge transfer band in
insulating Nd2Cu04, two contributions appear. One (CT1) is a week and narrow
band with center frequency near 12 900 cm-1 (1.6 eV) and the other band (CT2) is
a relatively strong and broad band near 16 800 cm-1 (2.08 eV). We can also see two
peaks in the spectrum of x = 0.11 near 14 000 cm-1 (1.74 eV) and 16 800 cm-1 (2.08
eV), respectively. At higher doping levels, only one band appears. Figure 40 also
shows that the strength of the CT1 and CT2 bands decreases with increasing doping
concentration x.
The CT1 band is related to the abrupt decrease of its strength as a result of the
decrease of the intensity of UHB upon doping. Upon electron doping the position of
CT1 band shifts to higher frequency (from 1.6 eV for x = 0 to 1.74 eV for x = 0.11)
and its spectral weight (~ 11000cm-1) rapidly decreases, finally disappearing for
x > 0.14. The spectrum for x = 0.11 in Fig. 40 shows the very weak CTl band of
strength ~ 2 000 cm-1. Figure 41 illustrates the variation of the strength of the CTl
and CT2 bands, Ncti and ^CT2i as a function of doping x.
The behavior of the CT2 band with doping is similar to that of the CTl band.
Doping with electrons results in a reduction of the CT2 band and a small shift to


CHAPTER VIII
INFRARED PROPERTIES OF BI-0 SUPERCONDUCTORS
The insulating phase BaBiOs of the bismuthate superconductors, Bai_zKxBi03
(BKBO) and BaPbi-xBijOj (BPBO) exhibits a monoclinical distortion of the per-
ovskite lattice.105,106 Originally attributed to Bi charge ordering (between Bi3+ and
Bi5+), this distortion corresponds to a charge density wave (CDW) instability, open
ing a semiconducting gap at the Fermi level. Doping with K and Pb, on the Ba or
Bi site, respectively, reduces the semiconducting gap and leads to the superconduct
ing state with maximum Tc's of about 31 K and 13 K, respectively. The maximum
Tc occurs neax the composition of the metal-insulator transition. Compaxed to the
conventional BCS and the Cu-0 high Tc superconductors, the bismuthate supercon
ductors have unique properties: (i) Like the layered copper-oxide superconductors,
the bismuthates have a high transition temperatures in spite of a low density of states
at the Fermi level, (ii) Unlike the Cu-0 materials, the insulating phase originates from
the CDW state and is nonmagnetic, (iii) The conduction properties in the normal
state as well as the superconductivity is isotropic, (iv) The maximum Tc is observed
neax the metal-insulator transition.
In order to clarify the mechanism of the superconductivity in the bismuthates,
it is essential to investigate the physical properties in both the superconducting and
normal states. Nevertheless, their normal state and superconducting state properties
so fax axe not well understood.
In this chapter we present an extensive study of the optical properties on BKBO
and BPBO crystals. We first analyze optical reflectance in a frequency range from 30
130


144
scattering experiments, and infrared measurements have not been successful. This
can be attributed to the broadening of the phonon linewidth because of the strong
electron-phonon interaction. Also, we do not rule out the possibility of the pairing
mechanism being associated with other electronic excitations.
Summary
In this chapter, we have examined the problems associated with the normal state
and superconducting state properties of Bai.^KxBiOs (BKBO) and BaPbi_zBix03
(BPBO) single crystals. In the normal state, broad bands in the infrared conductivi
ties are observed at about 2500 and 4 000 cm-1 for BPBO and BKBO, respectively,
as shown in the doping dependence experiments. These bands are associated with
transitions across the charge density wave (CDW) gap. For BKBO, the 1/r associ
ated with the Drude band shows a non-linearity with temperature consistent with
the transport measurements, whereas for BPBO the extrapolated values of cr\(tjj) at
zero frequency show a semiconducting behavior. For BKBO the electron-phonon cou
pling constant A ~ 0.6 is estimated from the simple model for the electron-phonon
scattering rate.
In the superconducting state, the positions of the superconducting edge in the far-
infrared reflectance spectra have a strong temperature dependence which is suggestive
of a BCS-like energy gap. The energy gap of 2A/kgTc = 3.5 0.3 is obtained from
the crla(u;) spectrum at 10 K and is consistent with a weak-or moderate-coupling
limit. Our results show that the BKBO system is a dirty limit superconductor with
a superfluid condensate fraction of 16% at 10 K and 10% at 19 K. The London
penetration depth A is also estimated to be 4 250 100 .
The far-infrared conductivity spectra are very well represented by the standard
Mattis-Bardeen conductivity based on a weak coupling BCS theory at frequencies less


139
temperature dependence which is suggestive of a BCS-like energy gap in the dirty
limit. The spectrum at 10 K consists of a clear peak near 54 cm-1 and a broad dip
axound 100 cm-1, becoming equal to the reflectance at 30 K near about 250 cm-1.
The spectrum at 19 K also has a peale at frequency less than 54 cm-1.
There is no evidence of unity reflectance in the superconducting state that could
be used to identify the energy gap. The reflectance is high at low frequency, but there
is a residual absorption of the order of 7 ~ 8% in the 20~50 cm-1 frequency range.
This is reproducible for three different samples in the same batch. We cant rule out
the existence of the residual absorption in our sample even with the experimental
error of 2%. We expect that a residual absorption is caused by the compositional
inhomogeniety which is partially composed of insulating K concentrations around the
metal-insulator transition. Hence, the residual absorption indicates the normal state
characteristic in the superconducting state. It is unlikely that the residual absorption
is associated with the surface degradation, because the infrared light penetrates deeply
into the sample. This inhomogeneity is also consistent with the persistence of the
CDW band in the mid-infrared region.
The ratio Rs(u)/Rn(u) in Fig. 66 at the peak position is very small (2.5%) com
pared with the other infrared gap measurements, exhibiting a more metallic nature
in the normal state. A shallow broad minimum around 100 cm-1 in Rs
suggests that the BKBO system is a weak or moderate coupling superconductor.
The Kramers-Kronig analysis gives a more detailed picture showing an energy
gap similar to a BCS-like superconductor. Figure 65(b) shows the far-infrared con
ductivity. In Fig. 65(b), the minimum of the conductivity in the superconducting
state moves to higher frequency as the temperature is reduced, indicating the open
ing of a superconducting gap and following the BCS-like model for a superconducting


145
than 250 cm-1. Finally, we conclude that the BKBO system is a dirty limit and weak-
or moderate-coupling BCS-like superconductor and the high Tc may result from an
electron-phonon interaction corresponding to the high energy phonons. Nevertheless
it is still uncleax whether the superconductivity in BKBO can be explained within
the phonon mechanism.


2
the hole may be a dressed quasiparticle carrying a reduced antiferromagnetism in its
neighborhood. If it is clear that the spin wave excitations will heavily dress the hole,
increasing substantially its mass, the dressing of the hole by spin excitations will be
a key ingredient in the explanation of the origin of the MIR band which appears near
the exchange energy J in the optical conductivity spectrum. Therefore, the optical
conductivity can be qualitatively described using models of strongly correlated elec
trons like t-J model. However, there is little doubt that the properties of high Tc
materials are dominated by the tendency of the electron correlations, especially, at
low doping levels. For example, the electron correlations can be significantly modi-
fied by the response of electrons to the lattice. This evidence is observed from local
structural distortion which causes non-linear, localized carrier-lattice interaction.
Among the experimental results appearing in the optical conductivity, one of the
striking features is a strong doping dependence of spectral weight, t.e., a shift of
spectral weight from high to low frequencies. An interesting point is the behavior of
the low energy region near the Fermi surface in the case of hole doping and electron
doping. Basically, doped holes and electrons show different orbital characteristics
in the localized limit of charge transfer materials: holes introduced by doping have
0 character and so the behavior of spectral weight transfer is expected as that of
semiconductors, but doped electrons feel the strong repulsions on Cu 3d sites and will
behave as strongly correlated objects, like the Mott-Hubbard (MH) case. However,
large hybridization is crucial for high Tc materials with strong correlations. This is
illustrated by the large value of Cu-0 hopping and the large Coulomb repulsion on
Cu site. So, each site is not restricted to only one orbital due to large hybridization
and instead has a direct mixing of most of the Cu 3d and 0 2p orbitals. As a result,
it is proposed that the hole-doped system can be described by a t-J model or a single


39
Optical Reflectance
The optical measurements that gives an information on the electronic system are
measurements of the reflectivity of light at normal incident on single crystals. The
reflectance for light impinging onto an ideal solid surface can be derived from the
boundary conditions for E and H at the interface. The boundary condition requires
E{ + Er = Et. (21)
where the subscripts i, r, and t represent, respectively, the incident, reflected, and
transmitted waves at the interface. A similar equation holds for H, but with a change
in sign for Hr. The magnetic field H is perpendicular to the electric field E and Ex
H is in the direction of the wave propagation. Thus, we can write
Hi Hr = Ht. (22)
In the vacuum, E = H, whereas in the medium,
H = N{u)E, (23)
as can be shown by substituting plane-wave expressions of the form exp t(q rut) =
expt[(w/c) r] into Maxwells equations. (21), (22), and (23), are easily solved to
yield the complex reflectivity coefficient r(u;) as the ratio of the reflected electric field
Et to the incident electric field Et:
-<) = | = rTÂ¥ = <24>
where we have separated the amplitude p(u) and phase Q(u) components of the
reflectivity coefficient. By definition the complex refractive index N(u) is related to


76
The low frequency spectral weight (LFSW) with doping is analyzed by the two
component model. The Drude strength is very low up to the metal-insulator transi
tion and is roughly proportional to the doping concentration x in the metallic phase.
The strength of total mid-infrared bands rapidly increases at low doping but slowly
at high doping levels. The LFSW including the Drude and toted mid-infrared bands
grows faster than 2x with doping x consistent with the MH model. These strong
Drude and mid-infrared bands with the result of transport measurements suggests
that Nd2_ICeICu04 has a laxge Fermi surface consistent with photoemission exper
iments.
The charge transfer (CT) band is also analyzed with the two component model.
The CT band in insulating Nd2CuC>4 consists of two bands, CTl and CT2 bands.
Upon doping, the CTl band disappears at high doping levels, while the CT2 band
survives even if it partially loses its spectral weight. The two bands correspond to
the transition from the Cu 3d UHB to Zhang-Rice type correlated states.


167
53. S. H. Wang, Q. Song, B. P. dayman, J. L. Peng, L. Zhang and R. N. Shelton,
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54. Y. Tokura, S. Koshihara, T. Arima, H. Takaki, S. Ishibashi, T. Ido, and S.
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55. S.L. Cooper, G.A. Thomas, A.J. Millis, P.E. Sulewski, J. Orenstein, D.H. Rap-
kine, S.W. Cheong, and P.L. Trevor, Phys. Rev. B 42, 10 785 (1990).
56. J.D. Perkins, J.M. Graybeal, M.A. Kastner, R.J. Birgeneau, J.P. Falck, and
M. Greven, Phys. Rev. Lett. 71, 1621 (1993).
57. A. Zibold, Phys. Rev. Lett., submitted.
58. K. Strobel and R.Geick, Physica C 9, 4223 (1976).
59. Heyen et al., Solid State Commun. 74, 1299 (1990).
60. E. Rampf, U. Schroder, F.W. de Wette, A.D. Kulkarni, and W. Kress Phys.
Rev. B 48, 10 143 (1993).
61. S. Herr and D.B. Tanner, Phys. Rev. B 43, 7847 (1991).
62. S. Sugai, T. Kobayashi, and J. Akimitsu, Phys. Rev. B 40, 2686 (1989).
63. D. Emin, Phys. Rev. B 48, 13691 (1993).
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66. X.X. Bi, P.C. Eklund, Phys. Rev. Lett. 70, 2625 (1993).
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(1993).
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Schneemeyer, and J.V. Waszczak; Phys. Rev. B 45, 2474 (1992).
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16
model, the insulating state is created by the formation of spin-density wave. The long
range antiferromagnetic order inhibits conduction by creating the spin polarization
gap.
When holes are doped on 0 sites, Zhang and Rice17 made a progress in the
following argument. The key point is that the hybridization strongly binds a hole
on each square of 0 atoms to the central Cu2+ ion to form a local singlet. This
singlet then moves through the lattice of Cu2+ ions in a manner which is similar
to a hole (or doubly occupied site) in the single-band effective Hamiltonian. This
singlet is equivalent to removing one Cu spin 1/2 from the square lattice of Cu spins,
and thus the effective model corresponds to spins and holes on the two dimensional
square lattice. The 0 ions are no longer explicitly represented in the effective model.
Further, two holes feel a strong repulsion against residing on the same square, so that
the single-band model is recovered.
It is important to remark that the reduction of the three band model to the t-
J model is still controversial. Emery and Reiter14 have argued that the resulting
quasiparticles of the three band model have both charge and spin, in contrast to the
Cu-0 singlets that form the effective one band t-J model. Their result was based on
the study of the exact solution in a ferromagnetic background, and their conclusion
was that the t-J model is incomplete as a representation of the low energy physics
of the three band Hubbard model. Meinders et air have also shown that the low
energy physics in the t-J model behaves as a single-band Hubbard model due to the
restriction of double occupancy. However, Wagner et a/.18 and Horsch16 have proven
that the Zhang-Rice type of singlet17 construction plays a crucial role for the low
energy physics in the t-J model. They have suggested that due to an intrinsic strong
Kondo exchange coupling between O-hole and Cu spin the valence band is split into


30K
153
Fig. 66. The reflectance in the superconducting state at 10 and 19 K divided
by that at 30 K for Bai_xKxBi03.


La2_ISrICu04. However, the low frequency spectral weight grows slightly faster
than 2x with doping x, as expected for the Mott-Hubbard model.
We find very interesting results at low doping levels in La2-iSrxCu04. Upon
Sr doping the oscillator strength of the phonons is gradually reduced and doping
induced modes (Raman modes and carrier-lattice interaction mode) appear in the
far-infrared. We also find that the deformation potential by the dynamical tilting of
Cu6 octahedra induces a carrier-lattice interaction. The carrier-lattice interaction
is characterized by strong infrared active modes and an appearance of the strong Ag
Raman modes upon cooling.
Finally, we present the normal and the superconducting properties of Bi-0 su
perconductors. We conclude that the BKBO system is a weak-or moderate-coupling
BCS-type superconductor in the dirty limit.
viii


CHAPTER I
INTRODUCTION
Since the discovery of high Tc superconductors by Bednorz and Mller,1 extensive
efforts have been devoted to identify the nature of the superconducting pairing of an
entirely electronic origin in these systems, but the basic mechanism responsible for
the superconductivity is not yet known. High Tc superconductors are fundamentally
different from conventional metallic superconductors. The latter have conventional
metallic transport properties above their transition temperatures Tc, and the electron-
phonon interaction causes the electrons to form pairs, which then condense as bosons
in the superconducting state. In contrast, the high Tc materials differ from ordinary
metal superconductors. They have very high transition temperatures, a linear behav
ior for their dc resistivities, a strongly temperature dependent Hall coefficient, short
coherence lengths, frequency dependence of scattering rate 1/r, etc. The difficulty in
understanding these materials stems from their complexity. For example, the large
number of atoms in the unit cell and the strong anisotropy of the materials complicate
the interpretation of the results.
The undoped parent compounds such as La2Cu04 and Nd2CuC>4 are 5=1/2 an
tiferromagnetic insulators with an optical absorption edge ~ 2 eV. Superconductivity
with very high Tc's occurs in the presence of specific chemical doping. Currently, one
of the most controversial issues is whether carriers injected in the undoped materials
behave like quasiparticles or not. The long range antiferromagnetic order inhibits
conduction by creating a spin polarization gap, and therefore the injected carriers
which move in the background of spin order nearby need to be reoriented. Thus,
1


96
Photon Energy (eV)
10000 15000 20000 25000
Frequency (cm-1)
Fig. 40. o\(u) spectra of the charge transfer band of Nd2_rCezCu04 crystals.


55
Voltage
II
- V-
III
I
(
IV
Current J
Four-Point Probe
21 KA.
Fig. 20. Top: Simple arrangement for four probe measurement. Bottom:
Experimental arrangement for the resistivity measurement.
sample


CHAPTER VII
QUASI-PARTICLE EXCITATIONS IN
LIGHTLY HOLE-DOPED La2-xSrzCu04.
In this chapter, we investigate the quasiparticle excitations in low doping regime
of La2_ISrICu04+5 (0.01 < x,S < 0.04) by temperature dependence reflectance
measurements of single crystals.
The dynamics of dopant-induced charged quasiparticles is currently one of the
central questions in strongly correlated cuprate materials. Due to their connection
with high Tc superconductivity, the main goal is the understanding of the quasiparticle
excitations of doped hole, and the possibility of the superconducting pairing from
an entirely electronic origin in these systems. Properties of quasiparticle in a low
doping regime are much easier to investigate because they can be easily separated
from large electronic contributions. Examples include the hopping of charge carriers
between localized states, the effect of impurity potential for charge carrier localization,
a carrier-phonon interaction and carrier localization by the deformation potential
caused by the tilting of Cu06 octahedra, self-trapped polaron states due to a strong
electron-phonon interaction, and a strong carrier-spin interaction, etc.
Perhaps, the simplest system for the study of these issues is La2-zSrzCu04+.
The parent compound La2Cu04 undergoes a second-order transition from the tetrag
onal K2NF4 structure to an orthorhombically distorted one below Tq/t = 530 K.77
The transition can be described roughly as a staggered tilting, or rotation, of the
oxygen octahedra around the tetrahedral [110] axis, resulting in a \/2 x \/2 doubling
of the unit cell in the a-b plane.78-80 This phase transition folds vibrational modes
at the X point in the bet structure back to the zone center, where they may become
98


CHAPTER IX
CONCLUSIONS
In this dissertation, three major issues of high Tc superconductors have been ex
tensively studied by doping and temperature dependent reflectance measurements.
First, the low-energy-scale physics at frequencies below the charge transfer band and
the doping mechanism in electron-doped Nd2-xCeICu04 have been intensively stud
ied. Second, the low lying excitations near the Fermi level have been investigated
at low doping levels in hole-doped La2_ISrICu04. Finally, the normal state and
superconducting state properties of Bi-0 superconductors have been observed.
For the first issue, we have demonstrated that electron doping induces a transfer
of spectral weight from the high frequency region above the charge transfer excita
tions to the low frequency region near the Fermi level. However, the low frequency
spectral weight grows greater than 2x with doping x as expected in the Mott-Hubbard
model. The Drude component is very strong and narrow compared to that of hole-
doped La2_zSrICu04, and the extrapolated values of cr\{w) at zero frequency are
in good agreement with the results of dc transport measurements. Our results for
the one component approach indicate that upon doping the quasiparticle interaction
substantially decreases and Im E at high level shows a linearity in w below 0.6 eV
consistent with the MFL model, in which strong quasiparticle damping is assumed.
We have also verified that the Fermi level should be stuck on the bottom of the upper
Hubbard band. Furthermore, for superconducting sample of x = 0.15 the Drude band
is stronger and the quasiparticle interactions are less than in neighboring Ce concen
tration samples. This observation suggests that in the high Tc regime the Fermi level
156


36
1
O
-1
-2
-3
-4
O 4 8 12 16 O 10 20 30 40
Temperature (K) Temperature (K)
Fig. 16. Meissner effect measurements on BaPbzBii_z03 single crystal (left)
and Bai_zKzBi03 single crystal (right).
Most crystals have good, specular, nearly flux free surfaces, which are suitable
for infrared measurements.
Magnetization (Fig. 15) for the superconducting sample shows that although this
is a higher Tc and has stronger diamagnetism than typically appears in the litera
ture for large crystals, the transition is still somewhat broad, and the field-cooled
signad is weak, presumably due to flux pinning. The dc resistivity spectra for all
samples roughly exhibit the form of p(T) = p0 + AT2, quadratic in temperature. The
properties of the samples are summarized in Table 1.
3
E

w
I
O
0 -
-2 -
~ -3
o
N
o -4
c
CD
2 -5
P I
D |

BaPb,.xB¡,03
H = 15 Oe.


APPENDIX
OPTICAL AND TRANSPORT PROPERTIES OF LuNi2B2C 159
REFERENCES 164
BIOGRAPHICAL SKETCH 174
vi


158
observed the non-linearity of the 1/r vs. T in BKBO. This result should be compared
to the the linear temperature dependence observed in the resistivity in the high Tc
cuprates. Also, the electron-phonon coupling constant A ~ 0.6 is estimated from the
simple model for the electron-phonon scattering rate. In the superconducting state
for BKBO, we have found that the position of reflectance edge has the temperature
dependence. The energy gap of 2A = 3.5 0.3 has been estimated, consistent with
BCS-type mechanism with moderate or weak coupling. This conventional energy gap
contrasts with the case of the high Tc cuprates, in which a superconducting gap is
not identified in the infrared spectrum. We have estimated the superfluid condensate
fraction as 16% at 10 K and 10% at 19 K, and the London penetration depth to be
(4 200 100) Furthermore, the far-infrared <7i(u;) spectra below 250 cm-1 are well
represented by the standard Mattis-Bardeen conductivity based on a weak coupling
BCS theory. Therefore, we conclude that the BKBO system is a weak or moderate
coupling BCS-type superconductor in the dirty limit. The high Tc about ~ 30 K may
result from the interaction between electrons and high energy phonons. However, the
pairing mechanism for Bi-0 superconductors still remains an open question.


Reflectance
148
Photon Energy (eV)
0 10000 20000 30000
i/(cm_1)
Fig. 61. Room temperature reflectance (a) and cri(u;) (b) obtained from the
K-K transformation of reflectance (a) of BaPbi_xBix03.


-1 A)
79
100 200 300 400 500 600 700
Frequency (cm-1)
Fig. 23. Far-infrared (a) <7i(u;) and (b) Im[-l/e(u;)] for NdoCuO^j. Peaks
in (a) correspond to TO phonons, in (b) to LO phonons.


Resistivity (fi
124
0.10
0.08
E
o
0.06
0.04
0.02
0.00
\
\
*-a1.99^0.01 Cu04+<5
\
i'll I I I I I I I I 1 1 I I 1 I I I I I I I I I I I L.
50 100 150 200 250 300
Temperature (K)
Fig. 53. Temperature dependence of the in-plane resistivity p for
Lai.9gSro.oiCu04+5. Inset shows Inp vs. (1/T)1/2.


Spectral weight (/Cu)
94
Fig. 38. The spectral weight of the Drude and toted mid-infrared as a function
i, estimated from two component model of


105
There axe two types of local polarization due to the orthorhombic distortion.
First, the apical 0 displacements along the c-axis induce changes in the local Cu
charge. Second, the displacement of in-plane 0 atom in the c-direction decreases the
0-0 distance, and therefore changes the electronic states of 0 atom. In each case,
the doped-holes can be easily localized in Cu02 plane and their charge dynamics
are slow enough for the lattice to follow. This slow charge dynamics can induce the
carrier-lattice interaction.
Figure 52 shows the dependence of the oscillator strength of four infrared phonons
and Ag Raman modes at 247 and 278 cm-1 as a function of temperature. The lower
frequency mode is strongly affected by the deformation potential compared with the
high frequency mode as in the case the splitting of mode, showing the enhanced
oscillator strength of phonons upon cooling. This is interpreted as meaning that
the carrier-lattice interaction increases due to the deformation potential caused by
the orthorhombic distortion. The oscillator strength of the Raman active phonons
greatly increases compared to that of the infrared active phonons with decreasing
temperature.
Another mode which is not observed in the Raman and infrared active modes
is a broad peale near 620 cm-1 which is observed at a frequency just below the in
plane stretching mode and shows its broadening with temperature. We have shown
above that the dynamical tilting of the Cu06 octahedra enhances the carrier-lattice
coupling. Also, those vibration which cause a change of bond length in the Cu-0 plane
can have stronger carrier-phonon interaction. Hence, this mode may be associated
with a carrier-phonon interaction. However, presently we can not give a satisfactory
explanation for this mode. Similar modes are observed in lightly doped Nd2Cu4
system90 and a theoretical work.91 They have shown that this mode is induced by


18
Semicowloctc
Mon-Mubhard
Charge Transfer
Fig. 7. A schematic drawing of the electron-removal and electron-addition
spectra for semiconductor (left), a Mott-Hubbard system in the localized
limit (middle) and a charge transfer system in the localized limit (right).
(a) Undoped (half filling), (b) one-electron doped, and (c) one-hole doped.
The bars just above the figures represent the sites and the dots represent
the electrons. The on-site repulsion U and the charge transfer energy A
are also indicated.
Simple Semiconductor
Let us consider a semiconductor with an occupied valence band and an unoccupied
conduction band, separated by an energy gap Eg. For the undoped semiconductor the
total electron removal and addition spectrum is shown in Fig. 7(a)(left). If the total
number of sites equals N, then there are 2N occupied states and 2N unoccupied
states, separated by Eg. If one hole is doped in the semiconductor, the chemical
potential will shift into the former occupied band, provided that we can neglect the
impurity potential of the dopant.


74
higher energies from 2.08 eV for x = 0.11 to 2.29 eV for x = 0.19. However, a transfer
of spectral weight only starts after the CT1 band has completely disappeared, as
observed in Fig. 41. There is no difference of peak position and strength between the
spectra of x = 0 and x = 0.11. The spectral weight is ~ 15 000 cm-1 for x = 0.11
and ~ 10 300 cm-1 for x = 0.19. Thus, both the CT1 and the CT2 bands seem to
due to a transition from the Cu 3d UHB to Zhang-Rice type17 correlated states.
The squares in Fig. 41 also explain the spectral weight loss of two CT bands upon
doping. The spectral weight of two CT bands loses slightly faster than x with doping
x. This trend is in good agreement with the behavior of the LFSW with doping x.
The amount of the greater spectral weight loss than x is very similar to that of the
greater LFSW than 2x, which may be related to p-d charge transfer.
We here have the interesting fact that, when we consider the positions of the CT1
and CT2 bands with doping, the Cu 3d UHB should move to higher energy. From the
position differences of the CT1 band between x = 0 and x = 0.11 and the CT2 band
between x = 0.11 and x = 0.19 we are led to conclude that the Fermi level should lie
~ 0.35 eV above the bottom of the UHB. This result is consistent with a theoretical
estimate72 and the EEL and x-ray absorption spectroscopy.73 This observation is
also compaxed to the results46,74 observed in La2-zSrICu04 of 0.7 eV. This narrow
energy range induces the strong Drude band and suggests that Nd2-xCezCu04 of
electron-doped system has a large Fermi surface, in good agreement with the angle
resolved photoemission experiments.75,76
Summary
We have examined the change of optical spectra with Ce doping in electron-
doped Nd2-iCeICu04 in the frequency range from the fax-infrared to the UV re
gion. We have also made a systematic analysis of the temperature dependence for


(-1/e)
117
i/(cm 1)
Fig. 46. Far-infrared (a) a\(u) and (b) Im(-l/e) along the c-axis for
Lai.9TSro.o3Cu04. Peaks correspond to (a) TO phonons (b) LO phonons.
Inset shows high frequency conductivity.


88
Photon Energy (eV)
o 10000 20000
Frequency (cm-1)
Fig. 32. The energy loss function, Im[l/e(u;)] of Nd2_xCeICu04 as a func
tion x.


133
lower frequencies forming a low frequency tail due to Pb states in the CDW gap.
According to the band structure results,107,108 the undistorted cubic BaPbi_xBix03
alloy possesses a single broad conduction band, which involves (7-antibonding com
binations of Pb-Bi(6s) and 0(2p) states. This suggests that the low energy spectral
weight in BPBO is affected by both band filling and the Pb-related states and is very
complicated.
Figure 62 displays the far-infrared reflectance and conductivity for BPBO. The
reflectance spectra in Fig. 62(a) have small temperature dependence at tempera
tures between 10 K and 300 K. As the temperature decreases, is strongly reduced below ~ 80 cm-1, consistent with the temperature dependence
of the resistivity109 which shows a semiconducting behavior. On the other hand, the
(7i(w) between 80 cm-1 and 500 cm-1 slightly increases with decreasing temperature
to compensate for the decreased oscillator strength below ~ 80 cm-1.
In Fig. 62(b), four optical phonons are cleaxly observed at 102, 171, 222 and
541 cm-1, showing a redshift on heating. Comparing with the phonons110 in the
insulating phase of BaBiOa observed at 97, 137, 230 and 441 cm-1, the frequencies
of three phonons centered at 102, 171 and 541 cm-1 increase with Pb concentration
while the phonon mode at 230 cm-1, which is assigned to a Bi-0 bending mode,
shows the softening of phonon. In contrast, for BKBO only one broad phonon peak
neax 200 cm-1 is observed due to the screening from the free caxriers.
Comparison of Two Bismuthate Spectra
There axe some differences between the two bismuthate spectra. First, the Drude
band in BPBO is weaker and the phonon features axe more distinct than in BKBO.
Second, the CDW band in BPBO is quite a bit sharper and the gap energy is lower
than in BKBO. Third, for BKBO the minimum of <7i(u;) neax 15 000 cm-1 is more


135
While there is originally no CDW band in an ideal cubic perovskite, most infraxed
measurements for BKBO show that the CDW band persists in the metallic phase ad
jacent to the phase transition into the semiconducting phase. The persistence of the
CDW gap is also supported by Raman experiments,117,118 where the breathing mode
phonon at 570 cm-1 exists as a small peak at a composition of x = 0.38 and finally
disappears at a composition of x = 0.45. In addition, Pei et a/.119 using electron
diffraction have reported that BKBO has an incommensurate structural modulation
which is responsible for a partially insulating property in the metallic phase, and
suggested that it may be an incommensurate CDW. However, the persistence of local
CDW order may not be an intrinsic property. The local CDW order may be associ
ated with the inhomogeniety of the potassium distribution at the transition between
the orthorhombic and the cubic phases. A sample prepaxation study and the pressure
dependent experiment120 of optical reflectance support this idea. For example, it is
difficult to prepaxe a clean powder, because the potassium rich phases tend to segre
gate at the grain surface, and visible and infrared reflectance of the superconducting
compound near the phase transition anomalously changes with pressure.
Free Carrier Component in BKBO
We have mentioned earlier that the infraxed conductivity in BKBO may be sep
arated into two parts:
cri(u) = an) + (TiCDW, (62)
where cr\£> is the Drude part and c\cdw corresponds to the CDW band. Thus the
free carrier part in the normal state can be easily obtained by subtracting ct\cdw
from the totcil After the ct\d components at each temperature axe obtained,


51
the effective mass. We can also rewrite (45) in terms of the frequency dependent
effective mass m*\
e(u>) = £oo ~
hJZ
u(m*(uj)/mb)[u) + z/r*(u>)]
(46)
where 1/t*(u>) is the (renormalized) scattering rate. If we compare (45) with (46),
we can extract two relations:
m* i I(lo)
mb u>
2
u
ReS(|)
(47)
and
^l/r = .RM = 2ImE(|), (48)
mb 2
where R(u>) is the unrenormalized scattering rate and E is the quasiparticle self
energy.


Coherence Effect in Superconductor 26
Strong-Coupled Superconductor 27
IV. CRYSTAL STRUCTURE AND SAMPLE CHARACTERISTICS ... 30
Crystal Structure and Phase Diagram 30
La2_xSrxCu04 30
Nd2-xCexCu04 31
Bai_xKxBi03 and BaPbi_xBix03 32
Sample Characteristics 34
La2_xSrxCu04 34
Nd2-xCexCu04 34
Bi-0 Superconductors 37
V. EXPERIMENT 38
Background 38
Dielectric Response Function 38
Optical Reflectance 39
Infrared and Optical Technique 41
Fourier Transform Infraxed Spectroscopy 41
Optical Spectroscopy 43
Instrumentation 43
Bruker Fourier Transform Interferometer 43
Perkin-Elmer Monochromator 45
Michelson Interferometer 45
dc Resistivity Measurement Apparatus 46
Data Analysis; Kramers-Kronig Relations 47
Dielectric Function Models 49
Two Component Approach 49
One Component Analysis 50
VI.Ce DOPING DEPENDENCE OF ELECTRON-DOPED Nd2_xCexCu04 56
Results and Discussion of Insulating Phase 57
Doping Dependence of Optical Spectra 60
Optical Reflectance 60
iv


104
frequency compared to phonon position in the tetragonal phase is associated with the
displacement the apical Oz atoms into lower symmetry sites out of the Cu-0 plane,
as the temperature is lowered. With decreasing temperature, the mean position of
the apical 0* atoms changes significantly moving closer to the Cu-0 planes, which
leads to the different ionic charges of the Oz atoms.
Figure 50 shows that on increasing Sr concentration and decreasing temperature,
new peales appear at 77, 139, 247, 278, 320, 384, 400, 423 and 481 cm-1. The modes
at 77, 139, 320, 384, 400 and 481 cm-1 coincide with the Raman-active B^g + Bzg
modes.87 The B^g and B$g axe active in (a, c) and (a, c) polarization configurations
as shown in Fig. 47. (a, b) denotes that the polarizations of the incident and the
scattered light axe parallel to the a and b axes, respectively. The modes at 247, 278
and 423 cm-1 with strong oscillator strength axe consistent with Ag normal modes.
Here, Ag mode is observed in the (c, c) polaxization which corresponds to A\g mode in
the tetragonal phase. This result shows that the Raman mode becomes to the infrared
active mode. The activity in infrared and Raman is alternative in the crystals with
inversion symmetry such D2h and D^. The appearance of the Raman modes in the
infrared spectrum indicates the breaking of this symmetry. It can be also argued that
as first pointed out by Rice88,89 in the organic materials, linear coupling of charge
carriers to totally symmetry (.A^) phonons can lead to structure in the conductivity
spectrum at the phonon frequencies. In this charged phonon mechanism or electron-
molecular vibration coupling effect, the electron energies depend on the bond lengths,
while at the same time the bond lengths depend on the local charge density. Infrared
radiation at the Ag phonon frequencies can pump charge over long distances, giving
rise to absorption that has electronic oscillator strength and that is polarized in the
a-b plane.


APPENDIX
OPTICAL AND TRANSPORT PROPERTIES OF LuNi2B2C
The superconductivity in the class of quaternary compounds LNi2B2C (L = Y,
Tm, Er, Ho, and Lu) up to 16.6 K was recently discovered by Cava et a/.80 We report
here on resistivity and optical measurements for LuNi2B2C with the highest Tc.
LuNi2B2C has the body centered tetragonal structure (14/mmm) with alternating
LuC and Ni2B2 layers.149 However, band structure calculations150 have proven that
this material is fully three dimensional. In addition, it is known that this material
has a low density of states and a strong electron-phonon coupling constant.151
Figure A-l shows the temperature dependence of the resistivity for LuNi2B2C,
showing a transition Tc of 16.5 K and a typical metallic behavior in the normal state.
It should be note that the slope of the resistivity is linearly proportional to the T
at temperatures above 100 K similar to the case of high Tc cuprates. From the high
temperature slope of the resistivity (T 0£>, where 0£> is the Debye temperature),
we can estimate the value for the electron-phonon coupling constant A using the
formula,152
A = 0.246 (hup)2a, (A-l)
where hup is a plasma frequency and a is the slope of the resistivity dp/dT. Using a of
about 0.3 p cm/K, and {hup)2 =(3.7 eV)2 estimated from the optical measurement,
we get a strong electron-phonon constant A ~ 1.05.
159


28
A(0) and To, but Tc is decreased more and hence increases the ratio 2A/kfiTc above
3.52. The temperature dependence of the superconducting gap is also modified by
the damping effect.
A general indicator of strong-coupling in superconductors and hence deviation
from weak-coupling BCS is the frequency dependent conductivity in the fax-infrared.
At low frequencies the electron and its dressing cloud of phonons move together,
and one measures fully renormalized conduction electrons. The dressing is affected at
higher frequencies, of the order of phonon frequencies, at which an excited conduction
can emit phonons, and its renormalization changes. Infrared measurements offer a
way to undress the electrons and thus to measure the electron-phonon coupling.
An example of frequency dependent damping is the inelastic scattering of the
conduction electrons by phonons in ordinary metals, namely, the Holstein mecha
nism23 which is an important part of strong-coupling theory. The photon energy is
divided between the change in kinetic energy of the electron and the phonon energy.
This leads to an enhanced infrared absorption above the threshold energy for creating
phonons. The expression for the damping rate is
1 o_ r
-A- = / a2F(Sl)(u )d (17)
t(u;) u; J0
where a2F( ft) is the Eliashberg function proportional to the phonon density of states
F(f!) modified by the inclusion of a factor (1 cos#) to weight large scattering an
gles 6. The Holstein absorption can be distinguished from the direct absorption by
optical phonons because it shifts by 2A in the superconducting state. In addition,
the singularities in the superconducting density of states cause the phonon struc
ture to sharpen. As a result, an a2TF($l) function can be extracted from the optical
spectrum.


62
For a baxely metallic sample with x = 0.11, the conductivity below ~ 1.2 eV
is composed of two components: a naxrow band centered at u> = 0 and a mid-
infrared absorption band centered at 4400 cm-1 (0.55 eV). The narrow band decays
much more slowly than the Drude spectrum, which has a u~2 dependence. Upon
further doping, this band grows rapidly up to x = 0.14, but grows slowly with dopant
concentration x in the metallic phase. On the other hand, upon doping the band
near 0.55 eV slightly shifts to lower frequencies and the oscillator strength is a little
reduced. However, this peale is not visible as a distinct maximum in the spectra
of more highly doped samples due to the mixing with the Drude-like component.
Similar qualitative results have also been reported in hole doped La2_ISrxCu04 and
YBa2Cu307_{.
It is interesting to note that the phonons observed at 301 and 487 cm-1 have
about the same intensity with very sharp feature and almost same phonon position
at all doping levels, whereas the electronic background increases. This implies that
these phonon modes are not screened in the ordinary sense of having their TO-LO
splitting decreases to zero.
Significantly, a\(u) in Fig. 29 shows a dramatic change at frequencies above the
1.2 eV. First, the spectral weight at energies between 1.2 eV and 3 eV systematically
decreases with doping. This band has been identified as a charge transfer excitation,
in which electrons are transferred from 0 to Cu site. This result should be compared
to those obtained in the hole-doped systems, where the charge transfer band shows
over a wide energy range between 2 eV and 5 eV due to the contribution of the
charge transfer excitations between the apical 0 atoms and Cu atoms. Second, upon
doping the spectral weight near the 5 eV peak in the spectrum of undoped crystal
is gradually reduced, and another peak which is not observed in the spectrum of


CHAPTER III
THEORY
Electronic Structure of Insulating Phases
Cuprates
The Cu02 plane in the insulating cuprates is known as a charge transfer (CT)
insulator with a charge transfer energy, A ~ 1.5 2.0 eV, between 0 2p and Cu
3d, depending on structural features such as the in-plane Cu-0 distance d and the
out-of-plane structural configuration (oxygen coordination number).
The topmost Cu 3d state, a dxi_yi orbital, is split into upper and lower Hubbard
bands by a large on-site Coulomb repulsion U ~ 8 10 eV and, as a result, an
occupied 0 2p band is located in between two bands. This band structure is well
described by the three band Hubbard model. The three band Hubbard model will be
discussed in the next section.
Figure 4 shows a rough scheme of the electronic band structure of a charge transfer
insulator. Assuming that the bands do not change with doping (rigid band approxi
mation), then upon hole doping a PES (photoemission) experiment expects that the
Fermi energy will be located below the top of the valence band. On the other hand,
for an electron doped material the Fermi energy is above the bottom of the conduction
band.
BaBiOs
There are two points of view regarding the origin of the CDW instability in in
sulating phase BaBiOs. One is a Peierls-like scenario, in which Fermi surface nesting
9


81
o
Temperature (K)
Fig. 25. The parameters extracted from the Lorentzian fits to the peak near
1500 cm-1 as a function of temperature.


29
The infrared conductivity <7i (u;) in strong coupling superconductors is obtained in
the framework of Eliashbergs strong-coupling theory.24 This theory incorporates the
Holstein mechanism to ail orders in the electron-phonon coupling, and is described
by an effective scattering potential v, the strength of the electron-phonon interaction
by Eliashbergs spectral function a2F(ti), the quasipaxticle lifetime due to impurity
scattering r, and McMillans pseudopotential n*. McMillan26 numerically solved the
finite temperature Eliashberg theory to find Tc for vaxious cases, and the construc
tion from this of an approximate equation relating Tc to a small number of simple
parameters:
Tc
Q
1.45
exp[
where 0 is the Debye temperature and
1.04(1 + \tp)
- p*(l + 0.62Aep)
is the electron-phonon coupling constant.


49
According to (41) spectral regions in which the reflectance is constant do not con
tribute to the integral. Further, spectral region s u and s < w do not contribute
much because the function ln|(s + u;)/(s w)| is small in these region.
Now, we know R(u>), 0(uj), and we can use (27) to obtain n(u;) and K{u>). We use
these in (25) to obtain e(u;) = ei(t*;) + 2(^0 la this way we can find every optical
constants from the experimental 72(u;).
Dielectric Function Models
Two Component Approach
The two component model (Drude and Lorentz) are frequently used to describe
the optical properties of materials. The free-carrier component was fit to a Drude
model, while the bound carrier interband transition and lattice vibrations were fit by
Lorentzian oscillators. The model dielectric function is
, 47ri .
e(u;) = ei(w) + o\{u) =
LJ
upD
+ iu/
- + T-
T i "> -
(jj.
VI
u*
1^1j
+ eoo
(42)
where u>pd and 1/r are the plasma frequency and relaxation rate of the Drude carriers;
uej, upj, and 7j are the center frequency, strength, cmd width of the jth Lorentzian
contribution; and is the high-energy limiting value of c(o;).
In this picture for high Tc materials, the free carriers track the temperature de
pendence of the dc resistivity above Tc, while condensing into the superfluid below
Tc. In contrast, the bound carriers have an overdamped scattering rate that exhibits
very little temperature dependence.


Reflectance
80
Frequency (cm 1)
Fig. 24. (a) Reflectance spectra of Nd2Cu04_$ at several temperatures, and
(b) the reed part of the optical conductivity as a function of frequency.


20
hole doping electron doping
Fig. 9. The integrated LESW as a function of doping concentration x for CT
system of Meinders et a/.4 The solid line corresponds to the localized limit
tp = 0. The data points correspond to tp = 0.5 eV (lowest) to tp = 2
eV in steps of 0.5 eV.
Mott-Hubbard System
This correlated system is described by the single-band Hubbard Hamiltonian.
Figure 7(middle) shows the total photoelectron and inverse photoelectron spectrum
at half filling in Mott-Hubbard system. The total electron-removal spectral weight is
equal to the number of unoccupied levels. Therefore, each has an intensity equal to
TV. Upon hole doping there are TV 1 singly occupied sites. So the toted electron-
removal spectral weight will be IV 1. For electron addition there are TV 1 ways for
adding the electron to a site which was already occupied. Therefore, the intensity of
the UHB will also be TV 1 (not TV). We are left with the empty site for which there
are two ways of adding an electron (spin up and spin down), both belonging to the
LHB. Thus we find TV 1 electron removal states near the Fermi-level, two electron


CHAPTER II
REVIEW OF PREVIOUS WORK
Mid-infrared Bands
Figure 1 shows optical conductivities in lightly doped YBa2Cu306+j,,
Nd2Cu04_y, and La2Cu04+¡, from Thomas et al.5 In Fig. 2, two peaks can be seen
in the mid-infrared region. Thomas et al. interprets that the lower energy band is
characterized by the exchange energy J and the higher energy band arises from an
impurity band near the optical ionization energy of the isolated impurity.
Sr Doping Dependence of Hole-Doped La7_TSrTCu04
Uchida et al.3 have measured the reflectance of L^-xSrxCuC^ for several doping
levels between x = 0 and x = 0.34 at room temperature with large, homogeneously
doped, single crystals. They observed in the reflectance spectra that the position of
the ~ 0.1 eV plasmon minimum is nearly insensitive to doping due to the contribution
of the strong midinfrared band.
The optical conductivity cr\(u>) is shown in Fig. 3, where the undoped crystal
shows a negligible conductivity below 1 eV and a charge transfer gap at about 2 eV.
With hole doping, the intensity above the gap is reduced and new features (Drude
and midinfrared bands) appear below 1.5 eV i.e., a transfer of spectral weight from
above the gap to low energies seems to occur. In the metallic phase, the conductivity
at small frequencies decays much more slowly than the Drude-type 1/u2 behavior
expected for free carriers.
6


TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ii
ABSTRACT vii
CHAPTERS
I.INTRODUCTION 1
II.REVIEW OF PREVIOUS WORK 6
Mid-infrared Bands 6
Sr Doping Dependence of Hole-Doped La2-zSrzCu04 6
Electron-Doped System 7
III.THEORY 9
Electronic Structure of Insulating Phases 9
Cuprates 9
BaBiOs 9
Electronic Models for Cu02 Plane 11
Three Band Hubbard Model 12
One Band Hubbard Model 13
Spectral Weight Transfer with Doping 17
Simple Semiconductor 18
Mott-Hubbard System 20
Charge Transfer System 21
Frequency Dependent Conductivity in Superconductor 22
Review of Electromagnetic Response in the Normal State 23
Weak-Coupling Mattis-Bardeen Theory 24
Penetration Depth and Infrared Conductivity 25
iii


103
Figure 48, Fig. 49, and Fig. 50 show the a-b plane phonon spectra in the infrared
region at temperatures between 10K and 300K for La2_zSrICu04+. All samples
that we measured ire in the orthorhombic phase at these temperatures. We see
considerably more than the eleven expected modes. We can classify the phonon
modes in the spectrum as three types: (1) infrared active modes in the orthorhombic
phase, (2) Reiman modes due to the breaking of inversion symmetry, (3) new local
modes caused by the presence of the localized carriers.
In 10 K spectrum for Lai.97Sro.o3CuC>4 in Fig. 50, we observe a strong infrared
features of the seven modes at 107, 154, 168, 188, 352, 366 and 684 cm-1, of the
eleven B2U + Bzu modes. The tetragonal high temperature phase of La2Cu04 exhibits
four Eu symmetry in-plane phonons85 near 135, 164, 360 and 680 cm-1. There are
clear changes in the infrared active modes as a result of the phase transition. For
example, three modes near 135, 164 and 360 cm-1 in the tetragonal phase are splitted
into a B2u + B$u pair in the orthorhombic phase. These modes are assigned to the
translational vibration of the La atoms against the Cu06 octahedra, the bending
vibration of the apical Oz atom against the Cu-0 plane, and the bending vibration
of CuC>2 peme, respectively. This splitting develops upon cooling as a result of the
further rotation of Cu6 octahedra. In this case, in-plane 0 atoms are displaced
in the direction perpendicular to the plane, and thus bring about two different 0-0
distances in Cu-0 plane. The rotation of CuOe octahedra can be also observed with
a soft phonon mode in neutron scattering experiments.86
Figure 51 shows the frequencies of three modes as a function of temperature.
As the phonon mode goes from low to high frequency, the splitting decreases. The
highest mode near 680 cm-1, assigned to the stretching vibrations of the in-plane
Cu-0 bonds, is not split. The splitting of the two apical 0* bending modes to higher


155
Fig. 68. The fax-infrared conductivity in the superconducting state and in
the normal state (30 K). The solid lines axe the conductivity calculated
from Mattis-Bardeen theory.


93
Photon Energy (meV)
0 150 300 450 600
Fig. 37. The imaginary part of self-energy, -ImE with x as a function
frequency.


101
illustrates the coupling of photon field and TO phonons in an ionic crystal, showing
large LO-TO phonon splitting, where LO and TO phonons correspond to the zeros
and the poles of ei(u/), respectively.
c Axis Spectra
The c-axis optical reflectance for Lai.97Sro.o3Cu04 is shown in Fig. 45 at sev
eral temperatures. The room temperature reflectance in the high frequency region is
shown in the inset. The c-axis reflectance has the characteristics of an insulator, show
ing primarily four optical phonons in the far-infrared, almost featureless reflectance
in the high frequency region, and a narrow peak around 10000 cm-1. The optical
conductivity for c-axis of undoped sample is absolutely vanishing up to 1 eV above
which the interband transition start.82 However, our conductivity spectrum in the
inset in Fig. 46 shows a steep rise up to 10 000 cm-1. This rise indicates that upon
doping, some spectral weight is transferred into the low frequency region as in the
case of the in-plane spectrum, but the transferred weight is quite small.
Figure 46 shows the TO and LO phonon spectra at four temperatures in the
far-infrared range obtained from K-K transformation of R(w) in Fig. 45. As the
temperature decreases, the phonon lines become sharper, as expected. The results
show that the c-axis conductivity is dominated by four infrared active modes at 230,
320, 345 and 510 cm-1. In particular, the oscillator strength of the 230 cm-1 phonon
which was assigned to the Cu-0 bond bending mode is very large compared to those
of the other three phonons. The LO-TO splitting is also large, indicating that the
effect of screening due to free carrier is minor for vibration polarized parallel to c-
axis. We also observe very week features at 275 and 420 cm-1 in the low temperature
spectra.


147
Photon Energy (eV)
0.01 0.1 1
Fig. 60. Real paxt of the conductivity, cri(u;), obtained from a Kramers-Kronig
transform of the reflectance in Fig. 59.


4
to the results of Uchda et al.3 for hole doped La2_xSrxCu04 and theoretical work
of Meinders et al,4 We find that the far-infrared reflectance has little temperature
dependence, indicating the non-Drude behavior of this material. In one component
approach, our results illustrate that upon doping the quasiparticle interactions are re
duced and hence at high doping levels the imaginary part of quasiparticle self energy,
ImS, is proportional to u. The optical spectra in the high Tc regime of x = 0.15
show a strong Drude band and weak quasiparticle excitations compared to those of
neighboring Ce concentration samples.
Second, the low-lying excitations of charge carriers are investigated in the low
doping regime for La2_xSrxCu04+. The qualitative features of the far-infrared a\(ui)
and the dc transport properties are similar to the behavior of a conventional 2D
disordered metal. The resistivity at temperatures below 50 K shows a typical dc
variable range hopping behavior. The cr\{u) spectrum at 10 K illustrates the photon
induced hopping of charge carriers. We also discuss that a narrow band near the
antiferromagnetic energy J is likely to have both spin and lattice components. The
1.4 eV band looks like a result of the excitonic effect.
Finally, the optical conductivities of Bi-0 superconductors, Bai_xKxBi03 and
BaPbi_xBix03, are presented in the superconducting state and in the normal state
and compared to a conventional BCS theory. The extrapolated values of the cr\(u;)
spectra at zero frequency for BKBO and BPBO are similar to the results obtained
in the dc resistivity measurements, showing metallic and semiconducting behavior,
respectively. For BKBO, the electron-phonon coupling constant A ~ 0.6 is estimated.
In the superconducting state, the position of the superconducting edge in the re
flectance spectra has a strong temperature dependence which is suggestive of the
BCS-like energy gap in the dirty limit. From this analysis, a value for the energy gap


Reflectance
77
Photon Energy (eV)
0 1 2 3 4 5
Fig. 21. (a) Room temperature reflectance spectrum of Nd2Cu04_ on a-b
plane and (b) c\{u) spectrum after K-K transformation of R in (a).


166
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intzakis, and D.M. Ginsberg, Phys. Rev. B 47, 8233 (1993).
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23
Review of Electromagnetic Response in the Normal State
The portion of the interaction Hamiltonian of electromagnetic radiation with
matter is represented as
Hex = + Y] A(r¡, f)-p (5)
me *r

where vector potential A is subject to the gauge condition V A = 0. The fields are
as usual the derivatives
1 3 A T __
E = and H = V x A.
c at
(6)
We axe concerned with the anomalous skin effect (/ > 6) only with transverse fields,
where the current at a point depends on the electric field not at just the same point
but throughout a volume. For metals in the normal state <7n(u;,q) can be calculated
in the free-electron approximation by applying Fourier analysis to the well-known
Chambers integral expression for the current:
(7)
where a ne2l/mv0, R=r-r, t = t (R/v0), / is the mean free path, and v0 is
the Fermi velocity. Note that the electric field, E, is evaluated at the retaxded time,
t R/v0. When the field is constant over a mean free path, (7) reduces to Ohms
law, J(0)= in the limit at q= 0 is
<7o
1 + u2t2
<^2 n(w)
cr0u>T
1 + u2t2
(8)


cr(n cm) Reflectance
119
0 100 200 300 400 500 600 700
i/(cm_1)
Fig. 48. Fax-infrared (a) reflectance and (b) conductivity ai(u>) for
Lai.98Sro.o2Cu04 at several temperatures.


57
A second motivation is due to the nature of the insulating phase of Nd2Cu4,
which has the tetragonal T structure, without the apical 0 atoms of La2_zSrxCu04
which has the T structure. We expect more simplified electronic structure in the T
phase than in the T phase, allowing us to examine in detail the electronic structure
of high Tc cuprates.
We first describe the temperature dependent optical reflectance and conductivity
for the insulating phase. In the next section, we will present the doping dependent
reflectance for the a-b plane of Nd2-zCezCu04 and examine a variety of optical
functions obtained from a Kramers-Kronig analysis of the reflectance spectra. The
optical conductivity cr\(u) for each doping level is analyzed by the one component
and two component models. The doping dependence of the low frequency spectral
weight and the high frequency spectral weight are also discussed.
Results and Discussion of Insulating Phase
The room temperature reflectance and conductivity spectra of the undoped com
pound Nd2CuC>4_5 axe shown in Fig. 21(a) and Fig. 21(b), respectively. The con
ductivity spectrum exhibits a fundamental absorption edge near 1.5 eV which is
attributed to the charge transfer excitations between 0 2p and Cu 3d orbitals on
the Cu2 planes. Its energy is lower than in the high Tc cuprates with the T and
T* phases.54 (2.0 eV for the T phase La2_xSrxCu04 and 1.7 eV for the T* phase
LaDyCu04.) In the three-band Hubbard model, this strength is roughly given by
tpj/A when A tp,55 where tp represent the nearest neighbor transfer integral.
The gap energy A increases as Cu-0 spacing is smaller, because of an effect similar
to level repulsion in atoms.
The spectrum is featureless below the gap except for four optical phonons in the
far-infrared region, shown in Fig. 22. This reflectance is typical of an insulator. It


Spectral weight (/Cu)
97
Fig. 41. The spectral weight of the CT1 (diamonds) and CT2 (circles) bands,
Ncti and ^£7T2) the spectral weight loss (squares) as a function x.


134
prominent. The results of the sum rule explain the different properties of the two
samples well. Figure 63 illustrates at room temperature for the two samples.
For both samples the initial slow rise of is due to the Drude contribution and
then suddenly increases in the CDW band frequency up to 12 000 cm-1. The
contribution of the next steep rise comes from the CDW band. The BPBO spectrum
shows a weak Drude and strong CDW contribution compared with that of BKBO.
N^ff in Fig. 63 also exhibits a plateau neax 15 000 cm-1. This plateau corresponds
to a prominent minimum in the (cj) spectrum and the plasma minimum in the
reflectance spectrum. This trend is peculiar to BKBO. This result suggests that for
BKBO the low frequency excitations neax the Fermi level axe well sepaxated from
other excitations compaxed to the case of BPBO.
In optical studies of the doping dependence in BPBO,111,112 when Pb is substi
tuted into an active Bi site, the CDW band is gradually broadened and shifts to
lower frequencies. In this picture, the CDW gap persists as a pseudogap even in the
orthorhombic metallic phase for Pb concentrations between 15% and 35%. However,
the case of K doping is somewhat different.113-115 When monovalent K is doped into
the inactive, divalent Ba site, the CDW absorption band energy decreases much faster
than in BPBO and finally disappears in the metallic phase at x = 0.5. However, the
CDW band still persists at the metal-insulator transition composition (x = 0.37),
where there is a phase transition between the cubic and the orthorhombic struc
tures.28 Also, the measurements of the Hall and Seebecks effects113,116 for metallic
BKBO and BPBO show that the carriers axe electrons. Our results for the Drude
strength of both samples are consistent with the above view that BKBO is much
closer to half-filling of the Bi-0 conduction than in BPBO.


138
K. There is a little ambiguity in this estimation due to the non-linearity of 1 /r with
temperature. Nevertheless, the estimation of A seems to be consistent with the gap
measurement and numerical calculations,126-128 where A is suggested to be around 1
in order to explain the conventional electron-phonon mechanism. Thus the normal
state properties may suggest that BKBO is a BCS-like superconductor in which the
electron-phonon interaction plays a significant role.
Superconducting State Properties
Superconducting Gap
In the conventional BCS theory, a bulk superconductor at temperatures below Tc
is a perfect reflector of electromagnetic energy at frequencies below 2A. Above 2A its
behavior is similar to that of a normal metal. In infrared reflectance measurements,
the original inference of the superconducting gap was based on the measurement of
the ratio Rs/Rn of the reflectance in the superconducting state to that in the normal
state. Another case, the superconducting to normal ratio for transmission129,130 shows
a maximum near 2A. According to Glover and Tinkham,131 the superconducting gap
can also be obtained from a\3 in the superconducting state, which has a gap, 2A, at
the threshold for pair excitations. In this section, we examine both the question of
determining a frequency at which the absolute reflectance reaches 100%, and possible
evidence for a BCS size gap in a BKBO crystal.
We have measured the superconducting state reflectance in BKBO. Fig. 65(a)
shows the far-infrared reflectance at various temperatures. This figure illustrates that
in the normal state, BKBO has a very high far-infrared reflectance, characteristic of
free caxriers as expected from the metallic dc resistivity. In the superconducting state
(at 10 K and 19 K), a strong edge appears. The positions of the edge have a strong


115
Photon Energy (eV)
0.01 0.1 1
Fig. 44. Real paxt of the dielectric function f] as a function of u at room
temperature.


112
of phonons is reduced due to the screening by free carriers, and instead doping in
duced modes (Raman active modes, carrier-lattice interaction mode) appear. The
infrared active phonons near 135, 164 and 360 cm-1 observed in the high tempera
ture tetragonal phase of La2Cu04 split into a B^u + B$u pair in the orthorhombic
phase. This splitting develops upon cooling.
The qualitative features of the far-infrared cr\{u) spectra and the transport prop
erty in the low Sr doped La2-xSrzCu04 system is similar to the behavior of a conven
tional 2D disordered metal. In the high temperature region (> 100 K), all impurities
axe ionized and metallic behavior is observed. At lower temperatures (< 100 K),
the freezeout of hole occurs and hence the conductivity results from the thermal
ionization of the shallow impurities. At sufficiently low temperatures, all states axe
localized near the Fermi energy and we expect the phenomenon of variable range
hopping, where a hole just below the Fermi level jumps to a state just above it.
Our experiment suggests another mechanism of hopping, namely, the photon-
induced transitions and their effect on the absorption of electromagnetic radiation
in low doped La2_xSrzCu04 system. In this model, the lowest two energy levels
correspond to a localization of an electron on either one or the other of the donors. We
observe a resonant absorption in the <7i(u;) spectrum at 10 K whose maximum, u)max,
occurs at frequencies between 100 cm-1 and 130 cm-1, depending on Sr concentration.
Finally, we discuss infrared absorption bands observed near 0.15 eV, 0.5 eV and
1.4 eV. The behavior of 0.15 eV band which is observed near the antiferromagnetic
exchange energy J is similar to that of free carriers and this band also seems to have
a lattice component. The origin of 0.5 eV band is still obscure, and the peak near 1.4
eV looks like a result of the excitonic effect.


OPTICAL PROPERTIES OF DOPED
CUPRATES AND RELATED MATERIALS
By
YOUNG-DUCK YOON
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
1995

ACKNOWLEDGMENTS
I would like to thank my adviser, Professor David B. Tanner, for his advice,
patience and encouragement throughout my graduate career. I also thank Professors
P.J. Hirschfeld, N. Sullivan, J. Dufty and R. Singh for their interests in serving on
my supervisory committee and for reading this dissertation.
Thanks also go to all my past and present colleagues in Tanners group for their
friendship, useful conversations and cooperation. In particular, I would like to thank
V. Zelezny for many enlightening and useful discussions.
n

TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ii
ABSTRACT vii
CHAPTERS
I.INTRODUCTION 1
II.REVIEW OF PREVIOUS WORK 6
Mid-infrared Bands 6
Sr Doping Dependence of Hole-Doped La2-zSrzCu04 6
Electron-Doped System 7
III.THEORY 9
Electronic Structure of Insulating Phases 9
Cuprates 9
BaBiOs 9
Electronic Models for Cu02 Plane 11
Three Band Hubbard Model 12
One Band Hubbard Model 13
Spectral Weight Transfer with Doping 17
Simple Semiconductor 18
Mott-Hubbard System 20
Charge Transfer System 21
Frequency Dependent Conductivity in Superconductor 22
Review of Electromagnetic Response in the Normal State 23
Weak-Coupling Mattis-Bardeen Theory 24
Penetration Depth and Infrared Conductivity 25
iii

Coherence Effect in Superconductor 26
Strong-Coupled Superconductor 27
IV. CRYSTAL STRUCTURE AND SAMPLE CHARACTERISTICS ... 30
Crystal Structure and Phase Diagram 30
La2_xSrxCu04 30
Nd2-xCexCu04 31
Bai_xKxBi03 and BaPbi_xBix03 32
Sample Characteristics 34
La2_xSrxCu04 34
Nd2-xCexCu04 34
Bi-0 Superconductors 37
V. EXPERIMENT 38
Background 38
Dielectric Response Function 38
Optical Reflectance 39
Infrared and Optical Technique 41
Fourier Transform Infraxed Spectroscopy 41
Optical Spectroscopy 43
Instrumentation 43
Bruker Fourier Transform Interferometer 43
Perkin-Elmer Monochromator 45
Michelson Interferometer 45
dc Resistivity Measurement Apparatus 46
Data Analysis; Kramers-Kronig Relations 47
Dielectric Function Models 49
Two Component Approach 49
One Component Analysis 50
VI.Ce DOPING DEPENDENCE OF ELECTRON-DOPED Nd2_xCexCu04 56
Results and Discussion of Insulating Phase 57
Doping Dependence of Optical Spectra 60
Optical Reflectance 60
iv

Optical Conductivity 61
Effective Electron Number 63
Loss Function 64
Temperature Dependence of Optical Spectra 66
One Component Approach 67
Doping Dependence of Low Frequency Spectral Weight 70
Drude and Mid-infrared Band 70
Transfer of Spectral Weight with Doping 71
Doping Dependence of Charge Transfer Band 73
Summary 74
VII.QUASI-PARTICLE EXCITATIONS IN LIGHTLY
HOLE-DOPED La2-zSrxCu04+ 98
Experimental Results 99
a-b Plane Spectra 99
c Axis Spectra 101
Mode Assignment 102
Hopping Conductivity in Disordered System 106
Optical Excitations of Infrared Bands 109
Summary Ill
VIII.INFRARED PROPERTIES OF Bi-0 SUPERCONDUCTORS 130
Normal State Properties 131
Results for Bai_xKxBi03 132
Results for BaPbi_xBix03 132
Comparison of Two Bismuthate Spectra 134
Free Carrier Component in BKBO 135
Superconducting State Properties 138
Superconducting Gap 138
Superconducting Condensate 140
Discussion of Pairing Mechanism in BKBO 142
Summary 144
IX.CONCLUSIONS 156
v

APPENDIX
OPTICAL AND TRANSPORT PROPERTIES OF LuNi2B2C 159
REFERENCES 164
BIOGRAPHICAL SKETCH 174
vi

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
OPTICAL PROPERTIES OF DOPED
CUPRATES AND RELATED MATERIALS
By
Young-Duck Yoon
August 1995
Chairman: David B. Tanner
Major Department: Physics
The optical properties of cuprates, Nd2-xCe2;Cu04 and La2-xSrICu04, and the
related materials, Bai-jK^BiOs (BKBO) and BaPbi_xBix03 (BPBO), have been
extensively investigated by doping-and temperature-dependent reflectance measure
ment of single crystal samples in the frequency range between 30 cm-1 (4 meV) and
40 000 cm-1 (5 eV). The Nd2-xCexCu04 system has been studied at Ce compositions
in the range 0 < x < 0.2. La2_xSrxCu04 has been studied in the spin glass doping
regime, (x < 0.04). The two bismuthates have been investigated as superconducting
materials with the maximum Tc.
Our results for Nd2-xCezCu04 show that doping with electrons induces a transfer
of spectral weight from the high energy side above the charge transfer excitation band
to the low energy side below 1.2 eV, similar to the results observed in hole-doped
vii

La2_ISrICu04. However, the low frequency spectral weight grows slightly faster
than 2x with doping x, as expected for the Mott-Hubbard model.
We find very interesting results at low doping levels in La2-iSrxCu04. Upon
Sr doping the oscillator strength of the phonons is gradually reduced and doping
induced modes (Raman modes and carrier-lattice interaction mode) appear in the
far-infrared. We also find that the deformation potential by the dynamical tilting of
Cu6 octahedra induces a carrier-lattice interaction. The carrier-lattice interaction
is characterized by strong infrared active modes and an appearance of the strong Ag
Raman modes upon cooling.
Finally, we present the normal and the superconducting properties of Bi-0 su
perconductors. We conclude that the BKBO system is a weak-or moderate-coupling
BCS-type superconductor in the dirty limit.
viii

CHAPTER I
INTRODUCTION
Since the discovery of high Tc superconductors by Bednorz and Mller,1 extensive
efforts have been devoted to identify the nature of the superconducting pairing of an
entirely electronic origin in these systems, but the basic mechanism responsible for
the superconductivity is not yet known. High Tc superconductors are fundamentally
different from conventional metallic superconductors. The latter have conventional
metallic transport properties above their transition temperatures Tc, and the electron-
phonon interaction causes the electrons to form pairs, which then condense as bosons
in the superconducting state. In contrast, the high Tc materials differ from ordinary
metal superconductors. They have very high transition temperatures, a linear behav
ior for their dc resistivities, a strongly temperature dependent Hall coefficient, short
coherence lengths, frequency dependence of scattering rate 1/r, etc. The difficulty in
understanding these materials stems from their complexity. For example, the large
number of atoms in the unit cell and the strong anisotropy of the materials complicate
the interpretation of the results.
The undoped parent compounds such as La2Cu04 and Nd2CuC>4 are 5=1/2 an
tiferromagnetic insulators with an optical absorption edge ~ 2 eV. Superconductivity
with very high Tc's occurs in the presence of specific chemical doping. Currently, one
of the most controversial issues is whether carriers injected in the undoped materials
behave like quasiparticles or not. The long range antiferromagnetic order inhibits
conduction by creating a spin polarization gap, and therefore the injected carriers
which move in the background of spin order nearby need to be reoriented. Thus,
1

2
the hole may be a dressed quasiparticle carrying a reduced antiferromagnetism in its
neighborhood. If it is clear that the spin wave excitations will heavily dress the hole,
increasing substantially its mass, the dressing of the hole by spin excitations will be
a key ingredient in the explanation of the origin of the MIR band which appears near
the exchange energy J in the optical conductivity spectrum. Therefore, the optical
conductivity can be qualitatively described using models of strongly correlated elec
trons like t-J model. However, there is little doubt that the properties of high Tc
materials are dominated by the tendency of the electron correlations, especially, at
low doping levels. For example, the electron correlations can be significantly modi-
fied by the response of electrons to the lattice. This evidence is observed from local
structural distortion which causes non-linear, localized carrier-lattice interaction.
Among the experimental results appearing in the optical conductivity, one of the
striking features is a strong doping dependence of spectral weight, t.e., a shift of
spectral weight from high to low frequencies. An interesting point is the behavior of
the low energy region near the Fermi surface in the case of hole doping and electron
doping. Basically, doped holes and electrons show different orbital characteristics
in the localized limit of charge transfer materials: holes introduced by doping have
0 character and so the behavior of spectral weight transfer is expected as that of
semiconductors, but doped electrons feel the strong repulsions on Cu 3d sites and will
behave as strongly correlated objects, like the Mott-Hubbard (MH) case. However,
large hybridization is crucial for high Tc materials with strong correlations. This is
illustrated by the large value of Cu-0 hopping and the large Coulomb repulsion on
Cu site. So, each site is not restricted to only one orbital due to large hybridization
and instead has a direct mixing of most of the Cu 3d and 0 2p orbitals. As a result,
it is proposed that the hole-doped system can be described by a t-J model or a single

3
band Hubbard model, in which the occupation is constrained to at most one electron
per orbital. Hence, the low frequency spectral weight (LFSW) is expect to behave as
the MH system. However, it is not clear if hole and electron doped systems can both
be described by a single band Hubbard model. On the other hand, it is interesting
to note that the influence of doping in the antiferromagnetic correlations is non-
universal between hole and electron materials. For example, single crystal neutron
scattering measurements on Nd2-ICeICu04 by Thurston et al.2 have shown that 3D
antiferromagnetic order persists even with x as high as 0.14, while on La2_ISrICu04,
a doping of x ~ 0.02 is enough to destroy the long-range order.
The discovery of copper oxide high Tc superconducting materials has also gener
ated renewed interest in the Bi-0 superconductors, due to many similarities between
the bismuthates and the Cu-0 cuprates. For example, in spite of a low charge-carrier
density (on the order of 1021 cm-3) the Tc of the bismuthates is anomalously high; su
perconductivity occurs near the metal-insulator transition by chemical doping of the
insulating BaBiOs. In addition, the high Tc conductivity in this system is of great in
terest because it contains neither a Cu atom nor a two dimensional structural feature
which are considered to be crucial for the high Tc in the copper oxide superconduc
tors. Thus, understanding this system would undoubtedly facilitate understanding of
Cu-0 cuprates.
This dissertation describes a detailed study of the optical properties of high Tc
cuprates and Bi-0 superconductors over the infrared to the UV region in the tem
perature range from 10 K to 300 K.
First, we concentrate on the Ce doping dependence of electron doped
Nd2-iCeICu04, which has the simplest structure among high Tc materials. The low
frequency spectral weight (LFSW) for electron-doped Nd2-ICeICu04 is compared

4
to the results of Uchda et al.3 for hole doped La2_xSrxCu04 and theoretical work
of Meinders et al,4 We find that the far-infrared reflectance has little temperature
dependence, indicating the non-Drude behavior of this material. In one component
approach, our results illustrate that upon doping the quasiparticle interactions are re
duced and hence at high doping levels the imaginary part of quasiparticle self energy,
ImS, is proportional to u. The optical spectra in the high Tc regime of x = 0.15
show a strong Drude band and weak quasiparticle excitations compared to those of
neighboring Ce concentration samples.
Second, the low-lying excitations of charge carriers are investigated in the low
doping regime for La2_xSrxCu04+. The qualitative features of the far-infrared a\(ui)
and the dc transport properties are similar to the behavior of a conventional 2D
disordered metal. The resistivity at temperatures below 50 K shows a typical dc
variable range hopping behavior. The cr\{u) spectrum at 10 K illustrates the photon
induced hopping of charge carriers. We also discuss that a narrow band near the
antiferromagnetic energy J is likely to have both spin and lattice components. The
1.4 eV band looks like a result of the excitonic effect.
Finally, the optical conductivities of Bi-0 superconductors, Bai_xKxBi03 and
BaPbi_xBix03, are presented in the superconducting state and in the normal state
and compared to a conventional BCS theory. The extrapolated values of the cr\(u;)
spectra at zero frequency for BKBO and BPBO are similar to the results obtained
in the dc resistivity measurements, showing metallic and semiconducting behavior,
respectively. For BKBO, the electron-phonon coupling constant A ~ 0.6 is estimated.
In the superconducting state, the position of the superconducting edge in the re
flectance spectra has a strong temperature dependence which is suggestive of the
BCS-like energy gap in the dirty limit. From this analysis, a value for the energy gap

5
of 2A = 3.5 0.3 is obtained. The superfluid condensate fractions are determined as
16% at 10 K and 10% at 19 K, and the London penetration depth, Al, is estimated
to be 4250 100 We also discuss the possibility of pairing mechanism in BKBO.
This dissertation is organized as follows. Chapter II reviews previous optical re
sults on the materials, investigated and issued. In Chapter III, models to describe
the behavior of electrons in the CO2 plane and theoretical models for the transfer
of spectral weight in the hole doping and electron doping cases are discussed. Some
fundamental properties of BCS superconductor are also given. In Chapter IV, the
crystal structures and the sample characteristics are presented. Chapter V will de
scribe infrared techniques, experimental apparatus and data analysis. Chapter VI,
VII and VIII are devoted to experimental results and discussion. Finally, conclusions
are presented in Chapter IX.

CHAPTER II
REVIEW OF PREVIOUS WORK
Mid-infrared Bands
Figure 1 shows optical conductivities in lightly doped YBa2Cu306+j,,
Nd2Cu04_y, and La2Cu04+¡, from Thomas et al.5 In Fig. 2, two peaks can be seen
in the mid-infrared region. Thomas et al. interprets that the lower energy band is
characterized by the exchange energy J and the higher energy band arises from an
impurity band near the optical ionization energy of the isolated impurity.
Sr Doping Dependence of Hole-Doped La7_TSrTCu04
Uchida et al.3 have measured the reflectance of L^-xSrxCuC^ for several doping
levels between x = 0 and x = 0.34 at room temperature with large, homogeneously
doped, single crystals. They observed in the reflectance spectra that the position of
the ~ 0.1 eV plasmon minimum is nearly insensitive to doping due to the contribution
of the strong midinfrared band.
The optical conductivity cr\(u>) is shown in Fig. 3, where the undoped crystal
shows a negligible conductivity below 1 eV and a charge transfer gap at about 2 eV.
With hole doping, the intensity above the gap is reduced and new features (Drude
and midinfrared bands) appear below 1.5 eV i.e., a transfer of spectral weight from
above the gap to low energies seems to occur. In the metallic phase, the conductivity
at small frequencies decays much more slowly than the Drude-type 1/u2 behavior
expected for free carriers.
6

7
Fig. 1. Optical conductivity in the mid-infrared region of YBa2Cu306+y
(upper panel), Nd2Cu04_j, (center panel), and La2Cu04+y.
Electron-Doped System
As we will discuss later, electron-doped materials, like Nd2-xCexCu04, axe struc
turally very similar to La2-xSrxCu04, but doped holes and electrons are introduced
in different sites, 0 and Cu sites. It has been found that their optical conductivities
axe also qualitatively similar for Nd2_xCexCu046,7 as shown in the bottom of Fig. 3.
Other compounds of the same family can be obtained by replacing Nd by Pr, Sm
and Gd. The optical properties of Pr2_xCexCu04 have been investigated8,9 and the
reported results are very similar to those of Nd2-xCexCu04.

a(co) (103^'1 cm-1) a(co) (103fl_1 cm-1)
8
Fig. 3. a-b plane optical conductivity of La2-xSrxCu04 (top) and
Nd2-xCexCu04 (bottom) single crystals for Uchida et al.3

CHAPTER III
THEORY
Electronic Structure of Insulating Phases
Cuprates
The Cu02 plane in the insulating cuprates is known as a charge transfer (CT)
insulator with a charge transfer energy, A ~ 1.5 2.0 eV, between 0 2p and Cu
3d, depending on structural features such as the in-plane Cu-0 distance d and the
out-of-plane structural configuration (oxygen coordination number).
The topmost Cu 3d state, a dxi_yi orbital, is split into upper and lower Hubbard
bands by a large on-site Coulomb repulsion U ~ 8 10 eV and, as a result, an
occupied 0 2p band is located in between two bands. This band structure is well
described by the three band Hubbard model. The three band Hubbard model will be
discussed in the next section.
Figure 4 shows a rough scheme of the electronic band structure of a charge transfer
insulator. Assuming that the bands do not change with doping (rigid band approxi
mation), then upon hole doping a PES (photoemission) experiment expects that the
Fermi energy will be located below the top of the valence band. On the other hand,
for an electron doped material the Fermi energy is above the bottom of the conduction
band.
BaBiOs
There are two points of view regarding the origin of the CDW instability in in
sulating phase BaBiOs. One is a Peierls-like scenario, in which Fermi surface nesting
9

10
undoptd
d Ud d10
Ef Ep
p-type n-type
Fig. 4. Simple electronic band structure for the charge transfer insulator, for
hole-doped and for electron-doped.
Fig. 5. Schematic representation of the oxygen octahedra. The solid lines
illustrate the symmetric Peierls distortions and the dashed lines illustrate
the undistorted case.

11
and the strong coupling of the conduction band states near Ep to bond stretching 0
displacements lead to a commensurate CDW distortion.10 In another approach, the
driving force is the aversion of Bi to the 4+ valence, which leads to a disproportiona
tion into 3+ (6s2) and 5+ (6s) valences on alternate sites.11 In either case one finds
a commensurate CDW distortion, in which the 0 octahedra are alternately expanded
or contracted as illustrated in Fig. 5. This CDW distortion doubles the unit cell,
which splits the half filled metallic band into filled and empty subband, opening a
semiconducting gap of ~ 2 eV.
Electronic Models for CuO? Plane
In this section, a Hamiltonian to describe the behavior of electrons in the high
Tc materials will be briefly described. Due to the complexity of their structure it is
important to make some simplifying assumptions. The very strong square planar Cu-
0 bonds with strong on-site correlations makes it possible to construct a Hamiltonian
restricted to electrons moving on the Cu02 plane.
Several models have been introduced for the description of layered strongly cor
related systems, as realized in the Cu02 plane. While there is a growing consen
sus that the high Tc materials should be described within the framework of two-
dimensional (2D) single-band t-J12 or three-band Hubbard models13 in the strong
coupling limit,14,15 a direct comparison of controlled solutions with experimental data
is still lacking. We will discuss these one band and three band Hubbard models in
the present section, and the carrier doping effect in these prototype models will be
discussed in the following section.

12
Three Band Hubbard Model
First of all, let us consider the bonding of a full Cu-0 octahedron (CuOe), that is,
the bonding of the 3d orbitals on the Cu ion with the 2p orbitals of the surrounding
0 ions. There are 17 orbitals in the Cu-0 octahedron. Five are from the 3d orbitals
of Cu, which are dx2_y2, dz2, and three dxy types. Also, the four 0 atoms each have
three p orbitals which contribute 12 orbitals. However, we here focus on the in-plane
bonding and take a more intuitive approach. To do this, consider the two planar 0
atoms with p orbitals that are directed toward the central Cu atom. On the central
Cu atom, we only use the dI2_J/2 orbital, since it is correctly oriented for a bonding
with its neighboring oxygens. It is also the uppermost Cu-d level in the crystal field of
the octahedral structure. Thus, only three orbitals (px,py, and dx2^2) are used. The
other 14 orbitals can be taken as nonbonding relative to these orbitals. In addition,
the copper ion Cu2+ has a 3d9 electron configuration which gives the ion spin 1/2.
Thus, in the absence of doping, the material is well described by a model of mostly
localized spin 1/2 states that give these materials their antiferromagnetic character.
The Hamiltonian in the Cu2 plane can be constructed in the framework of the
three orbitals:
H = tvdY^(p)dx + d\pj + h.c.) tpp (PjPj> + h.c.) +
('d> 0',/) i
+udY * 3 (i,j)
(i)
where pj are fermionic operators that destroy holes at the oxygen ions labeled j, while
d, corresponds to annihilation operators at the copper ions i. (i,j) and (j,j) represent
Cu-0 and 0-0 neighbors, so that this Hamiltonian contains two hopping terms, tpd
and tpp, as well as site energies e¡ and Coulomb interactions Ut for the two types of

13
sites, i on Cu and j on O. Up corresponds to the Coulomb repulsion when two holes
occupy adjacent Cu and 0 sites, and may also be very important. It is appropriate to
use the hole notation, since there is a one hole per unit cell in the undoped case. Hence,
the vacuum state corresponds to the electronic configuration d10p6. Because < tp,
this hole occupies a d-level, forming the d9 state. There are two factors that govern
the electronic structure. On the other hand, the hybridization tpd is substantial and
leads to a large covalent splitting into bonding and antibonding bands, which form the
bottom and top of the p-d band complex. Therefore, the bonding orbital is O-p-like
and the antibonding orbital is Cu-dx2_y2-like. This covalent nature is not restricted
to only one orbital per site. There is a direct mixing of most of the Cu 3d and 0 2p
states.
On the other hand, the local Coulomb interaction Ud is crucial for the semicon
ducting properties. In the charge transfer regime (tp < ep < Ud),16 the lower
Hubbard band is pushed below the 0 level and so three bands are formed as shown
in Fig. 6(top). When another hole is added to this unit cell in the charge transfer
regime, the new hole will mainly occupy oxygen orbitals due to the on-site Coulomb
interaction. The high Tc superconducting materials fall into this category (typical
parameters are ep ed ~ 3 eV, tpd ~ 1.5 eV, tpp ~ 0.65 eV, Ud ~ 10 eV, Up ~ 4 eV,
and Upd ~ 1.2 eV).16
One Band Hubbard Model
As originally emphasized by Anderson,12 the essential aspects of the electronic
structure of the Cu02 planes may be described by the two dimensional one band
Hubbard model. This model is
H = £ (W + 4^>) + c'EKt 5>Ki -1),
(2)

14
Fig. 6. Simple band structure in the three band (top) and one band (bottom)
Hubbard model.

15
where is a fermionic operator that creates an electron at site i of a square lattice
with spin a. U is the on-site repulsive interaction, and t is the hopping amplitude. In
the limit (t < U < ep e), the additional holes sit at Cu sites, and the hybridization
may be included by eliminating 0 sites to give an effective Hamiltonian for motion
on Cu sites alone. This is obviously a single-band Hubbard model. In a single-band
Hubbard model, the conduction band develops a correlation gap of an effective value
of the Coulomb repulsion Ueff, and this model yields only two bands, as shown in
Fig. 6(bottom).
For large on-site repulsion U, the one band Hubbard-model Hamiltonian can be
transformed into the t-J model Hamiltonian. This model describes the antiferromag
netic interaction between two spins on neighboring sites and it allows for a restricted
hopping between neighboring sites. Therefore, the Hamiltonian of (2) reduces to a
S = ^ Heisenberg model on the square lattice of Cu sites:
H t y [cj.o-(l n- (O'),* (i>
where S, are spin-1/2 operators at site i of a two dimensional square lattice, and J is
the antiferromagnetic coupling between nearest neighbors sites (ij) and is defined as
J =
4?
U '
(4)
The limit of validity of the t-J model is for J extendable into the regime J ~ t. The hopping term allows the movement of electrons
without changing their spin and explicitly excludes double occupancy due to the
presence of the projection operators (1 The Hamiltonian (3) is just the
effective Hamiltonian of the single-band Hubbard model in the large U limit. In this

16
model, the insulating state is created by the formation of spin-density wave. The long
range antiferromagnetic order inhibits conduction by creating the spin polarization
gap.
When holes are doped on 0 sites, Zhang and Rice17 made a progress in the
following argument. The key point is that the hybridization strongly binds a hole
on each square of 0 atoms to the central Cu2+ ion to form a local singlet. This
singlet then moves through the lattice of Cu2+ ions in a manner which is similar
to a hole (or doubly occupied site) in the single-band effective Hamiltonian. This
singlet is equivalent to removing one Cu spin 1/2 from the square lattice of Cu spins,
and thus the effective model corresponds to spins and holes on the two dimensional
square lattice. The 0 ions are no longer explicitly represented in the effective model.
Further, two holes feel a strong repulsion against residing on the same square, so that
the single-band model is recovered.
It is important to remark that the reduction of the three band model to the t-
J model is still controversial. Emery and Reiter14 have argued that the resulting
quasiparticles of the three band model have both charge and spin, in contrast to the
Cu-0 singlets that form the effective one band t-J model. Their result was based on
the study of the exact solution in a ferromagnetic background, and their conclusion
was that the t-J model is incomplete as a representation of the low energy physics
of the three band Hubbard model. Meinders et air have also shown that the low
energy physics in the t-J model behaves as a single-band Hubbard model due to the
restriction of double occupancy. However, Wagner et a/.18 and Horsch16 have proven
that the Zhang-Rice type of singlet17 construction plays a crucial role for the low
energy physics in the t-J model. They have suggested that due to an intrinsic strong
Kondo exchange coupling between O-hole and Cu spin the valence band is split into

17
(local) singlet (S) and triplet (T) states. Because the spin singlet states have the
lowest energy, the singlet states are located just above the valence bands and act as
the lower Hubbard band. Thus, the charge transfer gap of Ueff is formed with the
upper Hubbard band. Therefore, the t-J model can produce the low energy spectrum
of the three band Hubbard model.
Spectral Weight Transfer with Doping
Insulating CuC>2 layered cuprates can be doped with holes or electrons as the
charge carriers. A surprising feature with doping is a strong doping dependence of
high energy spectral distributions and the redistribution of the spectral weight from
high to low energy. Nice examples are the electron-electron loss study19 and 0 Is
x-ray absorption study20 for the La2-xSrxCu04 system. These spectra show a strong
decrease with doping x in the intensity of the upper Hubbard band as the lower en
ergy structure develops due to doped-holes in the 0 2p band. Another example is
optical absorption experiments,3 where a transfer of spectral weight from a band-gap
transition at about 2 eV in insulating La2Cu04 to the low energy scale (< 1 eV)
is observed with a strong doping dependence. This redistribution of spectral weight
and its doping dependence is due to strong correlation effects and has been observed
in several numerical calculations of correlated systems. Naively, doped-carriers may
show different orbital characteristics in the case of hole doping and electron doping:
holes have 0-2p-like character and electrons have Cu-3d-like character. Thus, we may
expect the different doping mechanisms for hole-doped and electron-doped systems.
In this section we review the difference between doping mechanisms of a semicon
ductor, a localized Mott-Hubbard and a CT system and discuss the influence of the
hybridization for the Mott-Hubbard and CT system in the framework of Eskes et
al.21 and Meinders et al.A

18
Semicowloctc
Mon-Mubhard
Charge Transfer
Fig. 7. A schematic drawing of the electron-removal and electron-addition
spectra for semiconductor (left), a Mott-Hubbard system in the localized
limit (middle) and a charge transfer system in the localized limit (right).
(a) Undoped (half filling), (b) one-electron doped, and (c) one-hole doped.
The bars just above the figures represent the sites and the dots represent
the electrons. The on-site repulsion U and the charge transfer energy A
are also indicated.
Simple Semiconductor
Let us consider a semiconductor with an occupied valence band and an unoccupied
conduction band, separated by an energy gap Eg. For the undoped semiconductor the
total electron removal and addition spectrum is shown in Fig. 7(a)(left). If the total
number of sites equals N, then there are 2N occupied states and 2N unoccupied
states, separated by Eg. If one hole is doped in the semiconductor, the chemical
potential will shift into the former occupied band, provided that we can neglect the
impurity potential of the dopant.

19
Fig. 8. The integrated low energy spectral weight (LESW) as a function of
doping concentration x for Mott-Hubbaxd model of Meinders et al.4 The
solid line corresponds to the localized limit f = 0. The data points axe
from the calculations: t = 0.5 eV (lowest) to t = 2 eV steps of 0.5 eV.
The total electron removal spectral weight will be 2N 1 (just the number of
electrons in the ground states) and the total electron addition spectral weight will be
2N +1 (total number of holes in the ground states). The electron addition spectrum
consists of two parts, a high energy scale (the conduction band) and a low energy
scale, which is the unoccupied part of the valence band. Therefore, we can know that
the low energy spectral weight equals 1. The same arguments hold for an electron-
doped semiconductor. Thus, the low energy spectral weight grows as x with doping
x for a hole-doped and electron-doped semiconductor.

20
hole doping electron doping
Fig. 9. The integrated LESW as a function of doping concentration x for CT
system of Meinders et a/.4 The solid line corresponds to the localized limit
tp = 0. The data points correspond to tp = 0.5 eV (lowest) to tp = 2
eV in steps of 0.5 eV.
Mott-Hubbard System
This correlated system is described by the single-band Hubbard Hamiltonian.
Figure 7(middle) shows the total photoelectron and inverse photoelectron spectrum
at half filling in Mott-Hubbard system. The total electron-removal spectral weight is
equal to the number of unoccupied levels. Therefore, each has an intensity equal to
TV. Upon hole doping there are TV 1 singly occupied sites. So the toted electron-
removal spectral weight will be IV 1. For electron addition there are TV 1 ways for
adding the electron to a site which was already occupied. Therefore, the intensity of
the UHB will also be TV 1 (not TV). We are left with the empty site for which there
are two ways of adding an electron (spin up and spin down), both belonging to the
LHB. Thus we find TV 1 electron removal states near the Fermi-level, two electron

21
addition states near the Fermi level and N 1 electron addition states in the UHB.
The same arguments hold for the electron doped case. Thus, a doping concentration
x yields a low energy spectral weight 2x and the high energy spectral weight is 1 x.
There have been Nx states transferred from high to low energy. However, when the
hybridization is taken into account, the low energy spectral weight grows faster than
two times the amount of doping as shown in Fig. 8.
Charge Transfer System
For the high Tc superconductors, an oxygen band is located between the LHB and
UHB. These systems are described by the three band Hubbard Hamiltonian. In the
localized limit with no hybridization between the oxygen and copper sites (p=0),
when the electrons are doped in this system, the situation is similar to the Mott-
Hubbard case and the spectral weight is transferred from high to low energy. Thus,
the low energy spectral weight goes to 2x with doping x. However, upon hole doping
the situation is similar to that of the semiconductor without any spectral weight
transfer. So, the CT system in the localized limit shows a fundamental asymmetry
between hole and electron doping (Fig. 9). That is, electrons will feel the strong
repulsions on the d sites, similar to the MH model, and will behave as strongly
correlated objects. When the hybridization is taken into account, the low energy
spectral weight for the electron-doped CT system behaves more or less the same as
found for the Mott-Hubbard system. However, for small hybridization tp, the low
energy spectral weight for the hole-doped CT system behaves as a semiconductor.
When the hybridization is increased, the low energy spectral weight for the hole-
doped CT system rapidly increases and the low energy spectral weight is almost
symmetric with respect to hole-electron doping, so the low energy spectral weight is
similar to that of the MH system. The high Tc superconductors lie in the regime

22
with large hybridization, so the holes in the hole-doped high Tc superconductors will
behave as strongly correlated particles.
Frequency Dependent Conductivity in Superconductors
Far-infrared conductivity ai(u;) is very useful to study particle-hole excitations
in the energy range of 4 ~ 100 meV for the copper oxide and bismuth oxide su
perconductors. For example, in the superconducting state, the energy gap 2A of
single-particle excitations could be obtained directly from the absorption edge of
infrared spectrum. Further information on the nature of quasiparticles and other
infrared-active excitations in the superconducting state can be obtained by analyz
ing the frequency dependence of the absorption spectrum at energies above 2A. Of
particular interest are deviations of the measured spectrum or infrared conductivity
from the BCS-theory for isotropic superconductors. Mattis and Bardeen22 first cal
culated the infrared conductivity in the framework of the weak-coupling BCS theory.
Deviations might give us some hints on what is basically different in BCS and high
Tc superconductors.
The Mattis-Bardeen theory does not include the Holstein mechanism,23 where
part of the energy of the excited conduction electron is transferred to phonons. This
effect is well known for a conventional strong coupling superconductor such as Pb.
In this case, one expects characteristic deviations from the Mattis-Bardeen theory.
However, little is known about strong coupling corrections to weak-coupling conduc
tivity. The strong-coupling theory of ai(u>) which is based on Eliashbergs model24
was first calculated by Nam.25 Since the early 1960s, the interpretation of energy gap
and more detailed measurements of excitation spectra has been based on the Eliash-
berg theory.24 This theory makes a number of assumptions which may be called into
question in the new copper oxide and bismuth oxide superconductors.

23
Review of Electromagnetic Response in the Normal State
The portion of the interaction Hamiltonian of electromagnetic radiation with
matter is represented as
Hex = + Y] A(r¡, f)-p (5)
me *r

where vector potential A is subject to the gauge condition V A = 0. The fields are
as usual the derivatives
1 3 A T __
E = and H = V x A.
c at
(6)
We axe concerned with the anomalous skin effect (/ > 6) only with transverse fields,
where the current at a point depends on the electric field not at just the same point
but throughout a volume. For metals in the normal state <7n(u;,q) can be calculated
in the free-electron approximation by applying Fourier analysis to the well-known
Chambers integral expression for the current:
(7)
where a ne2l/mv0, R=r-r, t = t (R/v0), / is the mean free path, and v0 is
the Fermi velocity. Note that the electric field, E, is evaluated at the retaxded time,
t R/v0. When the field is constant over a mean free path, (7) reduces to Ohms
law, J(0)= in the limit at q= 0 is
<7o
1 + u2t2
<^2 n(w)
cr0u>T
1 + u2t2
(8)

24
Weak-Coupling Mittis-Bardeen Theory
In the superconducting state, a complex conductivity defined. In the extreme anomalous limit, q£0 > 1 or extreme dirty limit > /, Mattis
and Bardeen22 showed that the ratio of the superconducting to normal conductivity
within weak-coupling BCS theory is
£1£ = i_ /
hu yA
+
-L r
hu JA-h,
1 E(E + hu) + A21 [f{E) f(E + hu)] ,p
(E2 A2)1/2[(E + hu)2 A2]!/2
\E{E + faj) + A2|[l 2f(E + hu)]
hu {E2- A2y/2[{E + hu)2- A2]i/2
dE,
(9)
£2a L /A |E(E + hu) + A2|[l 2f(E + faj)] ,E
crn hu 7a-Au;,-A (A2 E2yl2[(E + hu)2 A2]1/2
Eq. (9) is the same as the expression for the ratio of absorption for superconduct
ing to normal metals for case II of BCS theory. Numerical integration is required for
r>o.
Figure 10 shows the Mattis-Bardeen theory for cr\s{u)¡(yn and cr2s{u)l(Jn as a
function of frequency for T = 0. The real part is zero up to hu = 2A and then rises
to join the normal state conductivity for hu 2A.
In the superconducting state for u < 2A, because J(u;) = <72SE(u;), the power loss
P = (J E) = 0; one can therefore expect a perfect reflector (R = 1) at frequencies
below 2A. The imaginary part of ai(u) displays the l/u inductive response for
hu < 2A. More simply, this dependence is a consequence of the free-acceleration
aspect of the supercurrent response as described by the London equation
E = d(M,)/dt, A = ^ = JaI
(11)

25
f¡u
2A
Fig. 10. Complex conductivity of superconductors in extreme anomalous (or
extreme dirty limit) at T = 0.
where ma and n3 are the mass and density of the superconducting electrons and
is the London penetration depth. From this relation,
1 nae2
o-2 = = .
Au> m3u>
For > 2A, <72a falls to zero more rapidly than l/u.
Penetration Depth and Infrared Conductivity
The sum-rule argument allows determination of the strength of this supercurrent
response from <7ia. The oscillator strength sum rule requires that the axea under the
curve of The missing axea A under the integral of <7ia appears at u 0 as A8(u). The amount
of conductivity that is transferred from the infrared to the delta function at zero
frequency is given by
[^lnM a\a(tjj)\dui = 0.
(13)

26
meV
Fig. 11. The conductivity of a BCS superconductor in the (a) dirty and (b)
clean limits.
The Kramers-Kronig transform of the delta function a\{u) is 2A/xu>. For comparison
with the London equation (11), the penetration depth is related to the missing area
by
(14)
In the clean limit (1/r < 2A, l f0), all of the free carrier conductivity collapses
into the S function, in which case A = 7re2n/2m = u2/8, and (13) reduces to the
London result. So, the spectral feature is very weak at 2A (Fig. 11(b)). In the
dirty limit (1/r > 2A, 1 < f0), the penetration depth tends to be larger than this
limiting value and a sharp feature appears at 2A (Fig. 11(a)), and one can write
A = 7re2n3/2mj, where n, < n is the superfluid density.
Coherence Effects in Superconductor
At finite temperatures, A(T) < A(0), and also the thermally excited quasi-
particles contribute absorption for hu < 2A. This quasi-particle excitation is repre
sented as the distinctive features of the microscopic BCS model of superconductivity,

27
namely a coherence peak in the temperature dependence of the conductivity below
Tc and the logarithmic frequency dependence of c\(u) near w=0. Coherence effects
in superconductors arise because the dynamical properties of the quasiparticle ex
citations become different from those of normal electron-hole excitations as the gap
develops below Tc. This coherence peak will go to infinity just below Tc due to the
singularity in the BCS density of states. Thus, as T is lowered below Tc, the density
of excited quasiparticles decreases as these excitations freeze into the condensate, and
the properties of the excitations which are present for T > 0 are also modified. There
are clearly two fluids, the condensate fraction and the gas of excited quasiparticles.
Thus, the condensate response to external electromagnetic fields is described by a
8 function conductivity at u = 0 plus (in the presence of elastic scattering) con
ductivity with a threshold at u = 2A(T), corresponding to processes in which two
quasiparticles are excited from the condensate.
In weak-coupling BCS theory, the energy gap at 0 K is given by
2A =3.52 kBTc
(15)
where kq is Bolzmanns constant and Tc is superconducting transition temperature.
The gap vanishes at Tc, and just below this value, A(T) can be approximated by
A(T)
A(0)
1.74
Tc.
(16)
Strong-Coupled Superconductor
If the electron-phonon coupling is strong (as opposed to weak), then the quasi
particles have a finite lifetime and are damped. This finite lifetime decreases both

28
A(0) and To, but Tc is decreased more and hence increases the ratio 2A/kfiTc above
3.52. The temperature dependence of the superconducting gap is also modified by
the damping effect.
A general indicator of strong-coupling in superconductors and hence deviation
from weak-coupling BCS is the frequency dependent conductivity in the fax-infrared.
At low frequencies the electron and its dressing cloud of phonons move together,
and one measures fully renormalized conduction electrons. The dressing is affected at
higher frequencies, of the order of phonon frequencies, at which an excited conduction
can emit phonons, and its renormalization changes. Infrared measurements offer a
way to undress the electrons and thus to measure the electron-phonon coupling.
An example of frequency dependent damping is the inelastic scattering of the
conduction electrons by phonons in ordinary metals, namely, the Holstein mecha
nism23 which is an important part of strong-coupling theory. The photon energy is
divided between the change in kinetic energy of the electron and the phonon energy.
This leads to an enhanced infrared absorption above the threshold energy for creating
phonons. The expression for the damping rate is
1 o_ r
-A- = / a2F(Sl)(u )d (17)
t(u;) u; J0
where a2F( ft) is the Eliashberg function proportional to the phonon density of states
F(f!) modified by the inclusion of a factor (1 cos#) to weight large scattering an
gles 6. The Holstein absorption can be distinguished from the direct absorption by
optical phonons because it shifts by 2A in the superconducting state. In addition,
the singularities in the superconducting density of states cause the phonon struc
ture to sharpen. As a result, an a2TF($l) function can be extracted from the optical
spectrum.

29
The infrared conductivity <7i (u;) in strong coupling superconductors is obtained in
the framework of Eliashbergs strong-coupling theory.24 This theory incorporates the
Holstein mechanism to ail orders in the electron-phonon coupling, and is described
by an effective scattering potential v, the strength of the electron-phonon interaction
by Eliashbergs spectral function a2F(ti), the quasipaxticle lifetime due to impurity
scattering r, and McMillans pseudopotential n*. McMillan26 numerically solved the
finite temperature Eliashberg theory to find Tc for vaxious cases, and the construc
tion from this of an approximate equation relating Tc to a small number of simple
parameters:
Tc
Q
1.45
exp[
where 0 is the Debye temperature and
1.04(1 + \tp)
- p*(l + 0.62Aep)
is the electron-phonon coupling constant.

CHAPTER IV
CRYSTAL STRUCTURE AND SAMPLE CHARACTERISTICS
Crystal Structure and Phase Diagram
La*?- t Stt CuOd
The structure of La2_zSrzCu04 shown in Fig. 12(a) is tetragonal and has been
known for many years as the K2NF4 structure. It is also called the T structure. In
La2_zSrzCu04, the Cu-0 planes perpendicular to the c axis are mirror planes. Above
and below them there axe La-0 planes. The Cu02 planes axe ~ 6.6 apaxt, sepaxated
by two La-0 planes which form the charge reservoir that captures electrons from the
conducting planes upon doping. The La-0 planes axe not flat but corrugated. There
axe two formula units in the tetragonal unit cell.
Each copper atom in the conducting planes has an oxygen above and below in the
c-direction forming an oxygen octahedron. These axe the so-called apical 0 atoms
or just Oz. However, the distance Cu-Oz of ~ 2.4 is considerably laxger than
the distance Cu-0 in the planes of ~ 1.8 At high temperatures (depending on
Sr concentration) there is a transition to an orthorhombic phase (Tq/t 530 K for
x = 0), and the copper atoms and the six oxygens surrounding them slightly deviate
from their positions.
For x = 0, La2_zSrzCu04 is an insulator. Upon doping, La3+ axe randomly
replaced by Sr2+, and these electrons come from oxygen ions changing their configu
ration from O2- to 0 (and thus creating one hole in their p shell). Metallic behavior
is observed for even small doping concentration, x > 0.04 (Fig. 13). For Sr dopings
30

31
La2Qi04
Nd2Cu04
Cu O O O La, Nd
Fig. 12. Crystal structure (a) T phase and (b) T phase.
between ~ 0.05 and ~ 0.3, a superconducting phase was found at low temperatures.
The maximum value (~ 40 K) of Tc is observed at the optimal doping of x ~ 0.15.
Ndo-^Ce^CuO/t
The body centered structure of Nd2Cu04 is shown in Fig. 12(b) and it is called
the T structure. It has a close relationship to the T structure of Fig. 12(a). As in
the T phase structure, the structure is made of a single Cu02 plane and two Nd-0
planes, but the Nd-0 planes are shifted by a/2 in the x-direction, so that the oxygen
ions in the Nd-0 planes are not on the top of Cu ions. The Nd2Cu4 can be easily
electron-doped replacing Nd3+ by Ce4+.
The phase diagram of this material is shown in Fig. 13 comparing it with hole-
doped compound. The similarities between the two diagrams are shown, but the

32
Metallic Insulating Metallic
Fig. 13. Phase diagram of Nd2-zCezCu04 and La2_xSrzCu04.
electron-doped system clearly illustrates that superconductivity is a relatively small
effect compared with antiferromagnetism.
Bai-^K^BiOt and BaPbrBT_.T0-t
BaBiOa has an almost undistorted ABX3, cubic perovskite structure (Fig. 14).
Each Bi atom (B site) is octahedrally coordinated by six 0 atoms. The A site is
occupied by Ba or K, while B site is occupied by Bi or Pb. At room temperature,
the symmetry of BaPbi_zBiz03 material changes with doping according to following
sequence.27
Orthorhombic 0 Tetragonal 0.05 <
Orthorhombic 0.35 Monoclinic 0.90
33
Fig. 14. Idealized structure of perovskite ABX3.
Superconductivity exists only in the tetragonal phase and the value of maximum Tc
is T ~ 13 K for x ~ 0.25. For x > 0.35, the material becomes a semiconductor.
The behavior of Bai_xKxBi03 is similar to that of the BaPbi_xBix03. The
superconductivity appears at the boundary of the metal-insulator transition in the
cubic phase (x > 0.37) with maximum Tc of 30 K and disappears abruptly upon
crossing a phase transition to the orthorhombic phase. In spite of the low carrier
density (in the order of 1021 cm-3) the value of Tc is anomalously high. Also, no
magnetism is found in the neighboring compositions. The structures of five phases for
0 < x < 0.5 and temperatures below 473 K have been determined by neutron power
diffraction.28,29 Semiconducting behavior for the monoclinic phase at 0 < x < 0.1 is
explained on the basis of a commensurate charge density wave (CDW). This tendency
suggests unusual electronic interaction, namely strong electron correlation effect, in
this system.

34
Sample Characteristics
La9_-rSrTCuO/t
Single crystals of La2-xSrxCu04 were prepared at Los Alamos Laboratory.30 Six
teen samples of nominal composition La2_xSrxCu04+ were grown by conventional
solid state reaction at 1050C using predried La203, SrC03, and CuO in x increments
of 0.002 from x = 0 to 0.04. For each x, the sample was separated into three parts
which were treated at 650C for 5 hours in 1 bar N2 or 1 bar O2, or at 500C for 72
hours in 230 bar O2, respectively, and then oven cooled. Tq/j was measured using
a Perkin-Elmer differential scanning calorimeter. Oxygen contents were measured by
hydrogen reduction using a Perkin-Elmer thermogravimetric analyzer; the 1 bar N2,
1 bar O2, and 239 bar O2 annealed series showed 6 = 0.00(1), 0.01(1), and 0.03(1),
respectively.
The size of all crystals are at least 1.5 mm x 1.5 mm which are suitable for infrared
measurements. The surfaces were subsequently etched for 20 min in a solution of 1
% Br in methanol before reflectance measurement.
Nd->_TCeTCu04
Nd2-xCexCu04 single crystals were prepared at the University of Texas. All
crystals were grown in copper-oxide rich fluxes; normal starting compositions were
Nd2-xCexCu4.502 with various x. The melts were cooled in air from 1260C at
5C/min. The crystals were mechanically separated from the flux and subsequently
reduced in flowing He gets at 910 C for 18 hours. Energy dispersive spectroscopy
and electron microprobe (wavelength dependent spectroscopy) analysis on these and
many similar crystals have indicated a uniform Ce concentration across the crystal,
but absolute concentration determinations are only accurate to Ax = 0.01.

35
Fig. 15. Meissner effect measurement on Nd2-xCexCu04 single crystal.
Table 1. Characteristics of Nd2-xCerCu04 Crystals
X
area
(mm2)
p (at 300 K)
(mil-cm)
adc (at 300 K)p(T)
(il-cm)-1
0.0
1 x 1
~ 10"4
0.11
1 x 1
1 ~ 2
500 ~ 1000 Po + AT2
0.14
1 x 1
0.2 ~ 0.4
2500 ~ 5000 po + AT2
0.15
1.2 x 1.2
0.2
5000 po + AT2
0.16
1.2 x 1.2
0.2
5000 po + AT2
0.19
2x2
0.1 ~ 0.2
5000 ~ 10000 po + AT2
0.2
2x2
0.1 ~ 0.2
5000 ~ 10000 po + AT2

36
1
O
-1
-2
-3
-4
O 4 8 12 16 O 10 20 30 40
Temperature (K) Temperature (K)
Fig. 16. Meissner effect measurements on BaPbzBii_z03 single crystal (left)
and Bai_zKzBi03 single crystal (right).
Most crystals have good, specular, nearly flux free surfaces, which are suitable
for infrared measurements.
Magnetization (Fig. 15) for the superconducting sample shows that although this
is a higher Tc and has stronger diamagnetism than typically appears in the litera
ture for large crystals, the transition is still somewhat broad, and the field-cooled
signad is weak, presumably due to flux pinning. The dc resistivity spectra for all
samples roughly exhibit the form of p(T) = p0 + AT2, quadratic in temperature. The
properties of the samples are summarized in Table 1.
3
E

w
I
O
0 -
-2 -
~ -3
o
N
o -4
c
CD
2 -5
P I
D |

BaPb,.xB¡,03
H = 15 Oe.

37
Bi-0 Superconductors
The single crystals of Bai_iKxBi03 were grown by a modification of the elec
trochemical method of Norton.31 The transition to the superconducting phase from
dc susceptibility measurement is quite wide with the onset of superconductivity at
T = 28 K and the full superconductivity at T = 18 K (Fig. 16), probably associated
with the inhomogeniety of the potassium distribution in the crystal. The measured
Tc (~ 22 K) was taken at 50% of transition between 90% and 10% points. Also, a
direct measurement of the Tc of the same crystal by measuring the temperature at
which there is a discontinuity in the tunnel conductance yields Tc = 21 K.
For the optical measurement, the sample surface (1.5 mmxl.5 mm) was mechan
ically polished using AI2O3 power of 0.05 fim size. The color was blue after polishing.
Figure 16 shows a Meissner effect for BaPbxBii_x03, indicating the onset of super
conductivity around 10 K.

CHAPTER V
EXPERIMENT
Background
Dielectric Response Function
The dielectric function e(u;, q) describes the response of a crystal to an electro
magnetic field. The dielectric function depends sensitively on the electronic band
structure of a crystal, and studies of the dielectric function by optical spectroscopy
are very useful in the determination of the overall band structure of a crystal. In the
infrared, visible, and ultraviolet spectral regions the wavevector q of the radiation is
very small compared with the shortest reciprocal lattice vector, and therefore q may
usually be taken as zero.
The dielectric constant e of electrostatics is defined in terms of the electric field
E and the polarization P, the displacement D:
D = eoE + P = eeoE. (19)
The defined e is also known as the relative permittivity. So long as the material is
homogeneous, isotropic, linear, and local in its response, the dielectric response may
be characterized quite generally by a frequency-dependent complex dielectric function
e(u;) which we write in terms of its real and imaginary parts as
e(u;) = ci(w) + u)
Here, the quantity ei(u;) is called the real dielectric function whereas c\[u) is the fre
quency dependent conductivity. At zero frequency ei(0) becomes the static dielectric
constant and 38

39
Optical Reflectance
The optical measurements that gives an information on the electronic system are
measurements of the reflectivity of light at normal incident on single crystals. The
reflectance for light impinging onto an ideal solid surface can be derived from the
boundary conditions for E and H at the interface. The boundary condition requires
E{ + Er = Et. (21)
where the subscripts i, r, and t represent, respectively, the incident, reflected, and
transmitted waves at the interface. A similar equation holds for H, but with a change
in sign for Hr. The magnetic field H is perpendicular to the electric field E and Ex
H is in the direction of the wave propagation. Thus, we can write
Hi Hr = Ht. (22)
In the vacuum, E = H, whereas in the medium,
H = N{u)E, (23)
as can be shown by substituting plane-wave expressions of the form exp t(q rut) =
expt[(w/c) r] into Maxwells equations. (21), (22), and (23), are easily solved to
yield the complex reflectivity coefficient r(u;) as the ratio of the reflected electric field
Et to the incident electric field Et:
-<) = | = rTÂ¥ = <24>
where we have separated the amplitude p(u) and phase Q(u) components of the
reflectivity coefficient. By definition the complex refractive index N(u) is related to

40
the refractive index n(w), the extinction coefficient k(u), and the dielectric function
e(w) by
N(u) = n(u;) + ik(u) = \J e(u;). (25)
One quantity measured in experiments is the reflectance, which is the ratio of the
reflected intensity to the incident intensity:
R = rr* = p2
(1 n)2 + k2
(l + n)2 + jfc2
(26)
The measured reflectance R(u) and the phase 0(u;) are related to n(u) and k(u) by
VReie = r
(1 n) ik
(1 + n) + ik'
(27)
and
2k
(28)
It is difficult to measure the phase 9(u) of the reflected wave, but it can be
calculated from the measured reflectance R(u>) if this is known at all frequencies via
the Kramers-Kronig procedure. Then we know both R(u>), 9(uj), and we can proceed
by (27) to obtain n(u>) and k(u). We use these in (25) to obtain e(u) = ei(u;)-|-ie2(w),
where ei(u>) and 2(0}) axe the real and imaginary parts of the dielectric function. The
inversion of (25) gives
ei(w) = n2 k2, 2(0;) = 2nk. (29)
We will show in data analysis section how to find the phase 9(u) as an integral over
the reflectance R(w) using Kramers-Kronig relations.

41
Infrared and Optical Technique
Fourier Transform Infrared Spectroscopy
The central component of a Fourier transform infrared spectrometer is a two-
beam interferometer, which is a device for splitting a beam of radiation into two
paths, the relative lengths of which can be varied. A phase difference is thereby
introduced between the two beams and, after they are recombined, the interference
effects are observed as a function of the path difference between the two beams in
the interferometer. For Fourier transform infrared spectrometry, the most commonly
used device is the Michelson interferometer.
The Michelson interferometer, which is depicted schematically in Fig. 17, consists
of two plane mirrors, the planes of which are mutually perpendicular. One of the
mirrors is stationary and the other can move along an axis perpendicular to its plane.
A semi-reflecting film, called the beamsplitter, is held in a plane bisecting the planes
of the two mirrors. The beamsplitter divides the beam into two paths, one of which
has a fixed pathlength, while the pathlength of the other can be varied by translating
moving mirror. When the beams recombine at the beamsplitter they interfere due
to optical path difference. The amplitudes of two coherent waves which at time zero
have the same amplitude A(v) at wave number v, but which cure separated by a phase
difference kS = 2x1/6, cam be written as
yi(z) = r A{vY^vzdv, and y2{z) = f A(u)ei2r^-sUv,
Joo JOO
(30)
where k is the propagation constant, v is the wave number and 6 is the optical path
difference between the two waves. Using the law of superposition, one has
roo
y(z) = yi{z) A y2{z) = [A(i/)(1 +
J OO
(31)

42
The complex amplitude of the combined beam reaching the detector is
= A(,)(l + e"2'^). (32)
But the intensity B(i/,8) (irradiance or flux density) is
B(v,8) = A*r(8,i/)Ar(6,i/) = A2(u)[l + cos(27n/<5)] = ^S(i/)(1 + cos2xi/6), (33)
where S(t/) is the power spectrum. The total intensity at the detector is
1(6) = [ B(v,6)di/ = ^ [ 5(i/)[l + cos27ri/i]di/. (34)
Jo 2 Jo
At zero path difference, the intensity at the detector is
7(0) = f S(v)dv. (35)
At zero path difference all of the source intensity is directed to the detector; none
returns to the source. At large path differences the intensity at the detector is just
half the zero path difference intensity
= \fQ S(v)dv- (36)
because as 6 oo the cos2xv8 term averages to zero, t.e., it is more rapidly varying
with frequency than S(v).
The interferogram is the quantity [/() /(oo)]; it is the cosine Fourier transform
of the spectrum. For the general case, the final result is obtained:
B(W = LJiw <37>
(37), at a given wave number v, states that if the flux versus optical path 1(6) is
known as a function of 8, the Fourier transform of [/() ^7(0)] yields B(i/), the flux
density at the wave number u.

43
Optical Spectroscopy
At high frequencies, the Fellgett advantage losses its importance due to the in
creasing photon noise in the radiation field. For this reason, a grating monochromator
is normally used in the near-IR and visible frequency range.
Generally, a grating monochromator is used by applying the rule of diffraction.
For a wavelength A,
n 1
2d sin#
where d is the grating distant. At an angle 9, the first-order component of wavelength
A satisfying A = asin# is selected. Meanwhile, any higher order components with
wavelengths A = A/n, or vn = m/(n = 2,3,...), which could also pass through the
slit axe absorbed by the filter. The resolution is determined by the slit width and A9,
which is the angle of rotation at each step.
Instrumentation
Bruker Fourier Transform Interferometer
To measure the spectrum in the far and mid-infrared (20 ~ 4000 cm-1), a Bruker
113V Fourier Transform interferometer is used. Different thickness of Mylar beam
splitters, a black polyethylene filter, a bolometer and a Hg arc lamp as detector and
source are used for far infrared (20 ~ 600 cm-1). A photocell and a globax source are
used for mid infraxed (450 ~ 4000 cm-1). A schematic diagram of the spectrometer
is shown in Fig. 18. The sample chamber consists of two identical channels which can
be used for either reflectance or transmittance measurements. The entire instrument
is evacuated to avoid absorption by water and CO2 present in air.
The principle of this spectrometer is similar to that of a Michelson interferometer.
Light from the source is focused onto the beamsplitter and is then divided into two

44
beam; one reflected and one transmitted. Both beams are sent to a two-sided movable
mirror which reflects them back to be recombined at the beam splitter site. The
recombined beam is sent into the sample chamber and detector. When the two-sided
mirror moves at a constant speed v, a path difference 8 = 4vt, where t is the time
as measured from the zero path difference. Next, the signal is amplified by a wide
band audio preamplifier and then digitalized by a 16-bit analog-to-digital converter.
The digitalized data axe transferred into the Aspect computer system and axe Fourier
transformed into a single beam spectrum.
Perkin-Elmer Monochromator
Reflectance spectra from mid-infrared to ultraviolet (UV) frequency region are
measured by a model 16U Perkin-Elmer grating monochromator. The basic concept
of a grating monochromator involves shining a broadband light source on a grating
and selecting a small portion of the resulting diffracted spectrum by letting it pass
through an opening known as a slit.
A diagram of the spectrometer is shown in Fig. 19. Three sources-globar (GB),
quartz-envelope tungsten lamp (W), and deuterium lamp (D2) axe used for different
frequency region. The light signal is chopped to give it an AC component which could
then be amplified by a lock-in amplifier. Long-pass and bandpass filters eliminate
unwanted orders of diffraction. A laxge spherical mirror images the exit slits of the
monochromator onto either a reference mirror or a sample in the case of reflectance
measurements. For transmittance measurements, the sample is mounted as close as
possible to the focus of the second spherical mirror. The position of the detector
is at the focal point of ellipsoidal mirror. Three detectors, a thermocouple (TC), a
lead sulfite (PbS) photoconductor, and a silicon photodiode (576) axe used to cover

45
the different photon energy regions. Table 2 lists the parameters used to cover each
frequency range.
Polarizers could be placed after the exit slit and before the focus of the second
spherical mirror if polarized reflectance and transmittance measurements were re
quired. The polarizers used in the far infrared were wire grid polarizers on either
calcium fluoride or KRS5 substrates. Dichroic polarizers were used at higher frequen
cies. (table 2)
The signed from the detector was fed into a standard lock-in amplifier. The lock-
in is then averaged over a given time interval. The time constant on lock-in could be
varied the signal to noise ratio. After having taken a data point, the computer sent
a signal to the stepping motor controller to advance to the grating position. This
process was repeated until a whole spectrum range was covered. The spectrum was
normalized and analyzed through the computer.
Michelson Interferometer
A Michelson interferometer is an alternative instrument for measuring the spec
trum in the 10 to 800 cm-1 region. In principle, this instrument works in the same
way as the Bruker interferometer, but has a better S/N ratio at low frequencies below
100 cm-1 due to laxge size and high power source. A mercury arc lamp is used as a
source and the source is chopped to remove all background radiation. The combina
tion of the thickness of a Mylar beam splitter and different filters axe used to cover
the corresponding frequency range.
The sample and detector axe mounted in the cryostat. A doped germanium
bolometer operating at 1.2 K is used as a detector. Data acquisition procedure is
same as that of Perkin-Elmer grating monochromator.

46
Table 2. Perkin-Elmer Grating Monochromator Parameter
Frequency
(cm-1)
Grating0
(line/mm)
Slit width
(micron)
Source Detector
801-965
101
2000
GB
TC
905-1458
101
1200
GB
TC
1403-1752
101
1200
GB
TC
1644-2613
240
1200
GB
TC
2467-4191
240
1200
GB
TC
4015-5105
590
1200
GB
TC
4793-7977
590
1200
W
TC
3829-5105
590
225
W
PbS
4793-7822
590
75
w
PbS
7511-10234
590
75
w
PbS
9191-13545
1200
225
w
PbS
12904-20144
1200
225
w
PbS
17033-24924
2400
225
w
576
22066-28059
2400
700
d2
576
25706-37964
2400
700
d2
576
36386-45333
2400
700
d2
576
0 Note the grating line number per cm should be the sarnie
order of the corresponding measured frequency range in cm-1.
dc Resistivity Measurement Apparatus
The experimental arrangement for measuring the resistivity is illustrated in
Fig. 20. The measurements were made as a function of temperature from liquid
helium temperature (~ 4 K) to room temperature (~ 300 K) using a lead probe

47
which was thermally anchored to the cold head of a closed-cycle refrigerator (CTI
Cryogenics). The probe tip houses four electrodes. The sample can be electrically
connected to these electrode with 20 pm diameter gold wire using silver paint. The
sample temperature was monitored with a temperature controller (Lake Shore Cry-
otronics) that was connected to a silicon diode sensor which was attached to the cold
head of the cryostat unit.
We measured the resistance, R = V/I, for the configuration of leads schematically
shown in Fig. 20, using a standard ac phase-sensitive technique operated at ~ 22 Hz
at a current of ~ 700 /A. The results were insensitive to the size of the current.
Before measurement, four stride contacts were formed on ab plane by the evaporation
of silver plate. An annealing procedure for good Ohmic contact was performed at a
temperature of ~ 300 K in flowing O2 for ~ 5 hours. Contact resistance values in
the range 10 to 20 fl were obtained by the bonding of Au wires with silver paint.
The electrical resistivity values p which is defined as p = RA/L were obtained by
measuring the crystal dimensions, where A is the cross sectional area for current flow
and L is the length along the voltage drop.
Data Analysis: Kramers-Kronig relations
To obtain the optical conductivity c\{u) which is a more fundamental quantity
one applies a Kramers-Kronig transform to the reflectance R(u>), which yields the
phase shift 0(u). Formally, the phase-shift integral requires a knowledge of the re
flectance at all frequencies. In practice, one obtains the reflectance over as a wide
frequency range as possible and then terminates the transform by extrapolating the
reflectance to frequencies above and below the range of the available measurements.
Concerning the low frequency extrapolation, we find that the conductivity at
frequencies for which there is actual data is not affected significantly by the choice of

48
the low frequency extrapolation. For insulating samples, the reflectance is assumed
constant to dc. In the case of metallic samples, a Hagen-Rubens relation, f?(u) = 1
Ay/u, was used. In the superconducting state, we have used the formula R = 1 i?u>4,
in which R goes to unity smoothly as u approaches zero.
The high frequency extrapolation has significant influence on the results, primarily
on the sum rule derived from the optical conductivity. We reduced this effect by
merging our data to the reflectance spectra for insulating phase of published papers
which extend up to 37 eV (300000 cm-1) for Nd2-zCezCu04, La2_xSrrCu04,32 and
Bi-0 superconductors.33 We terminated the transform above 37 eV by using the
reflectance vaxying as l/w4, which is the free electron asymtotic limit.
The Kramers-Kronig relations enable us to find the real part of the response of a
linear passive system if we know the imaginary part of a response at all frequencies,
and vice versa. They are central to analysis of optical experiments on solids. Let us
consider the response function as a(u;) = ai(u;) + 02(0;). If a(u>) has the following
properties, a(u;) will satisfy the Kramers-Kronig relations:
. 2 n sa2(s) , 2u ai(s) x
ai(w) = -P / xds, and 0*2(0;) = P / -5 K \ds. (39)
7T J0 s1 First, q(w) has no singularity, and <*(u>) v 0 uniformly as |u;| 00. Second, the
function ai(w) is even and a2(w) is odd with respect to real u.
We can apply the Kramers-Kronig relations to reflectivity coefficient r(u;) viewed
as a response function between the incident and reflected waves in (24). If we apply
(39) to
lnr(u;) = lni?5 + i6(u), (40)
we obtain the phase in terms of the reflectance:
1 fc
' |s -f- u\ dIn 7?(s)
in ds.
I s (jj\ ds
(41)

49
According to (41) spectral regions in which the reflectance is constant do not con
tribute to the integral. Further, spectral region s u and s < w do not contribute
much because the function ln|(s + u;)/(s w)| is small in these region.
Now, we know R(u>), 0(uj), and we can use (27) to obtain n(u;) and K{u>). We use
these in (25) to obtain e(u;) = ei(t*;) + 2(^0 la this way we can find every optical
constants from the experimental 72(u;).
Dielectric Function Models
Two Component Approach
The two component model (Drude and Lorentz) are frequently used to describe
the optical properties of materials. The free-carrier component was fit to a Drude
model, while the bound carrier interband transition and lattice vibrations were fit by
Lorentzian oscillators. The model dielectric function is
, 47ri .
e(u;) = ei(w) + o\{u) =
LJ
upD
+ iu/
- + T-
T i "> -
(jj.
VI
u*
1^1j
+ eoo
(42)
where u>pd and 1/r are the plasma frequency and relaxation rate of the Drude carriers;
uej, upj, and 7j are the center frequency, strength, cmd width of the jth Lorentzian
contribution; and is the high-energy limiting value of c(o;).
In this picture for high Tc materials, the free carriers track the temperature de
pendence of the dc resistivity above Tc, while condensing into the superfluid below
Tc. In contrast, the bound carriers have an overdamped scattering rate that exhibits
very little temperature dependence.

50
One Component Analysis
Another approach to analyze the non-Drude conductivity is to assume an inelas
tic scattering of the free carriers in the low frequency with a underlying excitation
spectrum. This interaction gives a strong frequency dependence to the scattering
rate and an enhanced low frequency effective mass of the free carriers. This approach
has been proposed by Anderson34 and applied to heavy Fermion superconductors.35
The one component picture of the optical conductivity can also be described by the
marginal Fermi liquid36 (MFL) and the nested Fermi liquid37,38 (NFL) theories.
According to Varma et al.,36 the quasiparticle self energy S of the marginal Fermi
liquid has a imaginary part which qualitatively goes as
7T2A T, (jJ < T
-ImS(w)
(43)
xXu, u > T
where A is the electron-phonon coupling constant. There is an upturn in the effec
tive mass, with the mass enhancement proportional to A. In the NFL approach of
Virosztek and Ruvalds,37,38 the nested Fermi liquid has
ImS = amax(/?T, | u |),
(44)
where a is a dimensionless coupling constant. This gives a scattering rate that is
linear in T at low u and linear in u at high u>.
For calculating the frequency dependent scattering rate 1 /t(w) and effective mass
m*(u>), the complex dielectric function is described by a generalized Drude model
in terms of the complex damping function, also called a memory function, as 7 =
R(lj) + i/(w),
(w) = Coo -
(45)
u>(l> -f 7)
where e5 represents interband contribution not involving the charge carriers and
up = \J\nnz1 /m* is the plasma frequency, with n the carrier concentration and m*

51
the effective mass. We can also rewrite (45) in terms of the frequency dependent
effective mass m*\
e(u>) = £oo ~
hJZ
u(m*(uj)/mb)[u) + z/r*(u>)]
(46)
where 1/t*(u>) is the (renormalized) scattering rate. If we compare (45) with (46),
we can extract two relations:
m* i I(lo)
mb u>
2
u
ReS(|)
(47)
and
^l/r = .RM = 2ImE(|), (48)
mb 2
where R(u>) is the unrenormalized scattering rate and E is the quasiparticle self
energy.

co O
52
Source
M
i
vmm
Movable mirror
Beam
splitter
I
///
i
%
Detector
Fig. 17. Schematic diagram of Michelson interferometer.

53
' 7
f. ¡ i 1
y
Vr-
\ ^
~~w
I Souro* Chamber
a Near-, mid-, or lar-IR sources
b Automated Aperture
II Interferometer Chamber
c Optical fitter
d Automatic beamsplitter changer
III Sample Chamber
I Sample focus
J Reference focus
IV Detector Chamber
k Near-, mid-, or far-iR
detectors
a Two-sided movable mirror
f Control interferometer
g Reference laser
h Remote control alignment mirror
Fig. 18. Schematic diagram of IBM Bmker interferometer.

54
Fig. 19. Diagram of the Perkin-Elmer grating monochromator.

55
Voltage
II
- V-
III
I
(
IV
Current J
Four-Point Probe
21 KA.
Fig. 20. Top: Simple arrangement for four probe measurement. Bottom:
Experimental arrangement for the resistivity measurement.
sample

CHAPTER VI
CE DOPING DEPENDENCE OF
ELECTRON-DOPED Nd2_xCeICu04
In this chapter, we report optical reflectance and conductivity spectra from the
fax-infrared to UV on the a-b plane of electron-doped Nd2-xCexCu04 for very different
Ce concentrations (0 < x < 0.2). This compositional range covers the antiferromag
netic insulator, the high Tc superconductor, and non-superconducting overdoped
metallic samples.
The motivation behind this study is twofold. First, in spite of a lot of theoret
ical and experimental studies, there is still little understanding of the normal-state
excitation spectrum, especially the low-energy-scale physics near the Fermi level, of
strongly correlated high Tc cuprates. In particular, the differences between hole dop
ing and electron doping in the transfer of spectral weight from high to low frequencies
axe not well understood. Basically, doped holes and electrons show different orbital
characteristics: the holes introduced by doping axe mainly in 2p orbitals whereas the
doped electrons have 3d orbital character. Therefore, in the three band Hubbard
model the motion of holes will depend differently on tp than motion of the electrons.
Eskes et a/.21 have shown that in the localized limit (tp = 0), the transfer of spec
tral weight with electron doping is similar to the case of the Mott-Hubbard model,
whereas the development of spectral weight with hole doping is the same as that of
a simple semiconductor. At present, the results for doping dependence in hole-doped
systems3,39-46 axe somewhat in agreement with one another, but the results for the
electron-doped system6-9,47-53 are still controversial.
56

57
A second motivation is due to the nature of the insulating phase of Nd2Cu4,
which has the tetragonal T structure, without the apical 0 atoms of La2_zSrxCu04
which has the T structure. We expect more simplified electronic structure in the T
phase than in the T phase, allowing us to examine in detail the electronic structure
of high Tc cuprates.
We first describe the temperature dependent optical reflectance and conductivity
for the insulating phase. In the next section, we will present the doping dependent
reflectance for the a-b plane of Nd2-zCezCu04 and examine a variety of optical
functions obtained from a Kramers-Kronig analysis of the reflectance spectra. The
optical conductivity cr\(u) for each doping level is analyzed by the one component
and two component models. The doping dependence of the low frequency spectral
weight and the high frequency spectral weight are also discussed.
Results and Discussion of Insulating Phase
The room temperature reflectance and conductivity spectra of the undoped com
pound Nd2CuC>4_5 axe shown in Fig. 21(a) and Fig. 21(b), respectively. The con
ductivity spectrum exhibits a fundamental absorption edge near 1.5 eV which is
attributed to the charge transfer excitations between 0 2p and Cu 3d orbitals on
the Cu2 planes. Its energy is lower than in the high Tc cuprates with the T and
T* phases.54 (2.0 eV for the T phase La2_xSrxCu04 and 1.7 eV for the T* phase
LaDyCu04.) In the three-band Hubbard model, this strength is roughly given by
tpj/A when A tp,55 where tp represent the nearest neighbor transfer integral.
The gap energy A increases as Cu-0 spacing is smaller, because of an effect similar
to level repulsion in atoms.
The spectrum is featureless below the gap except for four optical phonons in the
far-infrared region, shown in Fig. 22. This reflectance is typical of an insulator. It

58
should be noted that optical transmission spectra in undoped materials show other
absorption features in the energy range 0.2 to 1 eV. For example, weak absorption
bands near 0.5 eV were first observed in undoped single crystals of single layered T,
T*, and ^ structures by Perkins et al.56 and multi-layered YBa2Cu307_ by Zibold
et al.57 In these studies, they suggest that these bands result from exiton-magnon
absorption processes. The spectra in Fig. 21 also show a strong transition around 5
eV, which is observed above 6 eV in optical spectra of the T and T* phases. This
peak is located at higher energies in the other structures for the same reason as the
larger charge transfer energy.
A group theoretical analysis of the phonon modes in Nd2CuC>4 yields
3A.2U+4.V58-60 The A.2 modes are observed in the c polarization spectrum and
the Eu modes corresponds to an atomic motion parallel to the a-b plane. Figure 22
displays the a-b plane reflectance in the far-infrared region as a function of tempera
ture. We clearly observe four strong phonon bands. As the temperature is reduced,
the phonon lines become sharper. Since all spectra show an insulating behavior, we
extrapolate them to zero frequency assuming asymtotically a constant reflectance.
Then we obtain cr\(u)) and Im(1/e) by K-K transformations.
The temperature dependence of the a-b plane phonons is shown in Fig. 23. The
upper panel shows <7i(u>), whereas the lower panel shows Im( 1/e). The former
determines the TO phonon frequencies, whereas the latter the LO phonon frequencies.
Four phonon bands occur at 131, 303, 347 and 508 cm-1 at room temperature.
These phonon modes are similar to the case of L^CuO-i, but the phonon energies in
Nd2Cu4 are lower than in La2Cu04- This difference is primarily due to a result of
a larger unit cell dimension (longer bond lengths) in the former material.61 These
phonon bands result from four motions: a translational vibration of Nd atom layer

59
against the Cu02 plane (131 cm-1), a Cu-0 bending mode from the in-plane Cu-0
bond angle modulation (303 cm-1), an out-of-plane 0 translational mode (347 cm-1),
and an in-plane Cu-0 stretching mode from the Cu-0 bond distance modulation
(508 cm-1). The function Im[-l/e(u;)] is shown in Fig. 23(b), showing large LO-TO
phonon splittings. The temperature dependence of phonon frequencies shows the
redshifts with increasing temperature as expected.
The optical conductivity of pure undoped sample should vanish up to 1.5 eV,
above which the charge transfer excitations occur. However, the optical conductivity
spectrum new absorption may be attributed to the deviation from an oxygen stoichiometry
of Nd2Cu04 single crystal during oxidation process. This result indicates that our
sample is lightly doped with electrons.
In Fig. 24(a), the optical reflectance of the band neax 1500 cm-1 is plotted as a
function of frequency at several temperatures. Figure 24(b) also shows the temper
ature dependence of the optical conductivity c\(w) of this peale. This peale is very
interesting due to the fact that its energy is close to the antiferromagnetic exchange
energy J.62 For this reason, one might expect that the origin of this peak is due to
the interaction of doped carriers with magnetic degrees of freedom. Several mecha
nisms, including self-localized polarons,63-66 photoexcitation of localized holes,67 and
magnetic excitations,5 have been proposed to explain this peak.
We have fit this band with the usual Lorentzian. The results for the fitting pa
rameters axe shown in Fig. 25. The peale position and peak width shift to higher
frequencies by an amount compaxable to thermal fluctuation energy kgT as the tem
perature of sample is increased; that is,
ue = ujeo -|- 0.6 IcbT, 7 = 7o + 1.6 k^T,
(49)

60
where = 0.695 cm-1/K. The broadening of the line is like the behavior of the free
carrier conductivity, which shows a linear temperature dependence of the scattering
rate. We find a coupling constant A ~ 0.25 using a formula fi/r = 2tXk^T. This
is comparable to the coupling constant obtained from the behavior of free carriers
in other high Tc cuprates. A similar temperature dependence has been observed in
the a-b plane conductivity spectrum in lightly doped Nd2Cu04_j, single crystal by
Thomas et al.5 They suggested that this band is related to a bound charge coupled to
the spin and lattice excitations. Unlike the result of Thomas et al., our result shows
that the oscillator strength of this band increases with decreasing temperature. We
will discuss this band in the next chapter again for lightly hole-doped La2_xSrxCu04
experiments.
The charge transfer band observed near 1.5 eV also has a temperature dependence.
Figure 26 shows the reflectance spectra (a) and the optical conductivity spectra (b)
calculated from the reflectance spectra using K-K transformations. The reflectance
spectrum in Fig. 26(a) at room temperature clearly shows two peaks near 1.36 eV
and 1.6 eV. As the temperature decreases, the spectral weight around 1.36 eV at 300
K shifts to the peak near 1.6 eV and the sum rule is satisfied.
Doping Dependence of Optical Spectra
Optical Reflectance
The reflectance spectra for the a-b plane are shown in Fig. 27 between 80 cm-1
(10 meV) and 42 500 cm-1 (5.3 eV) for various Ce concentrations. Other metallic
samples with Ce concentrations of x = 0.18 and x = 0.20 were measured, too. But,
these spectra are not shown in Fig. 27, because they are very similar to the spectrum
of Ce concentration x = 0.19. With doping the spectral weight around 1.5 eV peak is

61
reduced and a reflectance edge rapidly develops below 1 eV. Fig. 27 also shows that
the position of the edge shifts to higher frequency with increasing doping and is almost
saturated in the metallic regime where 0.14 < x < 0.19. Another notable feature is
that the charge transfer band near 1.5 eV moves to higher frequency with increasing
dopant concentration x. This behavior is obvious in this system. In addition, there
is a systematic change of reflectance between ~ 3 eV and ~ 5 eV with x. A similar
behavior has also been observed for hole-doped L^-xSrjCuC^3 and YBa2Cu37_
systems.39,44
The magnitude of the reflectance of Nd2-zCeICu04 at low frequencies is typi
cally larger than the results for hole-doped La2_iSrICu04 and YBa2Cu307_. For
example, the magnitude near 600 cm-1 at high doping levels for our results is about
~ 92%, whereas the results for hole-doped La2-zSrICu043'39 are ~ 85%.
Among the four Eu optical phonons in undoped crystal below 600 cm-1, two
infrared active phonons near 301 and 520 cm-1 are visible even in heavily doped
crystals. However, two weak phonon bands observed at 131 and 345 cm-1 in the
spectrum of undoped crystal are screened out from free carriers in the metallic phase.
Optical Conductivity
The frequency dependent optical conductivities obtained from a KK transforma
tion of the reflectance spectra are shown in Fig. 28 and Fig. 29. We can better observe
the influence of doping on spectral response by considering optical conductivity. The
a-b plane conductivity of Nd2-zCexCu04 shows interesting changes with doping. As
suggested by the reflectance spectrum in Fig. 27, with doping the conductivity of
the charge transfer band above ~ 1.2 eV is systematically reduced, whereas the low
frequency spectral weight below ~ 1.2 eV rapidly increases.

62
For a baxely metallic sample with x = 0.11, the conductivity below ~ 1.2 eV
is composed of two components: a naxrow band centered at u> = 0 and a mid-
infrared absorption band centered at 4400 cm-1 (0.55 eV). The narrow band decays
much more slowly than the Drude spectrum, which has a u~2 dependence. Upon
further doping, this band grows rapidly up to x = 0.14, but grows slowly with dopant
concentration x in the metallic phase. On the other hand, upon doping the band
near 0.55 eV slightly shifts to lower frequencies and the oscillator strength is a little
reduced. However, this peale is not visible as a distinct maximum in the spectra
of more highly doped samples due to the mixing with the Drude-like component.
Similar qualitative results have also been reported in hole doped La2_ISrxCu04 and
YBa2Cu307_{.
It is interesting to note that the phonons observed at 301 and 487 cm-1 have
about the same intensity with very sharp feature and almost same phonon position
at all doping levels, whereas the electronic background increases. This implies that
these phonon modes are not screened in the ordinary sense of having their TO-LO
splitting decreases to zero.
Significantly, a\(u) in Fig. 29 shows a dramatic change at frequencies above the
1.2 eV. First, the spectral weight at energies between 1.2 eV and 3 eV systematically
decreases with doping. This band has been identified as a charge transfer excitation,
in which electrons are transferred from 0 to Cu site. This result should be compared
to those obtained in the hole-doped systems, where the charge transfer band shows
over a wide energy range between 2 eV and 5 eV due to the contribution of the
charge transfer excitations between the apical 0 atoms and Cu atoms. Second, upon
doping the spectral weight near the 5 eV peak in the spectrum of undoped crystal
is gradually reduced, and another peak which is not observed in the spectrum of

63
undoped crystal is shown near 4.5 eV in the spectrum of x = 0.11. Further, with
doping the strength of this peale is reduced and its position shifts to lower frequencies.
Third, at high doping levels, a new shoulder near 3.6 eV grows with the decreasing
of the strength of 4.5 eV and 5 eV peaks. This seems to transfer the spectral weight
of 5 eV peale to energy region between 3 eV and 5 eV with doping. This behavior of
doping dependence in high energy region is different from the results obtained from
hole-doped systems, where the spectral weight above 1.5 eV systematically decreases
with doping.
Effective Electron Number
In order to describe a doping effect quantitatively, we have estimated the effective
number of carriers per CuC>2 plane. The effective electron number with mass m* =
me, Neff(u>) is defined according to
LJ
NeffM = 2me^Ce// o\(u)dJ, (50)
7T 6 J
0
where e, me are the free electron charge and mass respectively, and Vceu is the volume
of one formula unit. For Nd2-xCexCu04, we used the unit cell volume,68 Vce¡¡ = 187
3 and the number of Cu atoms per unit cell, Ncu = 2. Neff(u) is the effective
number of carriers per formula unit participating in optical transition at frequency
below id.
Figure 30 illustrates the effective electron number for the different Nd2-xCerCu04
samples. In the insulating phase, x = 0, Neg (u>) remains nearly zero throughout the
optical gap without a mid-infrared band contribution, but increases rapidly above the
absorption band of charge transfer excitation. For metallic samples, Neg (uj) exhibits
an initial rise due to the Drude band at zero frequency. The following steep rise

64
is the contribution of the mid-infrared bands, which ends axound 10 000 cm-1 (1.2
eV). Next, more steep rises come from the contributions of the charge transfer band
and high energy interband, respectively. This behavior is very similar to the results
obtained in hole-doped cuprates.
The low frequency Neff(u>) for metallic samples is plotted in Fig. 31 up to 1 000
cm-1. Notably, the integrated spectral weight of superconducting sample of x = 0.15
exhibits a rapid rise at low frequencies below 200 cm-1 due to the strong Drude
contribution, and is very strong at frequencies below 800 cm-1 compared with that
of slightly overdoped sample of x = 0.16. Finally, two curves for x = 0.15 and
x = 0.16 merge near 1000 cm-1. This implies that the strength of the mid-infrared
band in x = 0.15 is a weaker than in x = 0.16.
Another important result of our measurements is that Neff(u) at high frequency
above 3 eV gradually increases with doping. This is particular in our system. As we
will discuss later, this is due to the anomalous strong Drude and mid-infrared bands
caused by doping. In order to satisfy sum rule, this result suggests that another high
energy band above 4 eV loses spectral weight with electron doping. This is compared
to the results of hole-doped La2_iSrzCu04 and YBa2Cu307_. In these studies,
the only spectral weight of the charge transfer region between 1.5 eV and 4 eV is
transfered to low frequencies below 1.5 eV, and hence Ntf/(u) intersects near 3 eV
with increasing doping. It is noteworthy that Neff at 3 eV is a factor of 2 larger than
that of La2_ISrICu04.
Loss Function
In this section, we describe the energy loss function, Im[l/e(u;)]. This function
is the probability for energy loss by a charged particle that passes through a solid.
It can also be calculated from -Im[l/e(u;)] = e2(u)/[ei(u)2 + e2(w)2]. The peak

65
position corresponds to the zeros of ei(cj). In a simple Drude model, the maximum of
the energy loss function determines the longitudinal plasma frequency of free carriers,
corresponding to the zeros of the dielectric function e(w£), and its maximum position
shifts to higher frequencies with doping according to up = (4xne2/m)1//2. However,
the bound carriers in high Tc cuprates which contribute a positive dielectric response
dielectrically screen the free carrier response, and also lower up. The maximum value
of Im[l/e(u;)] is given approximately by the screened plasma frequency
Up / 47rne2
uv ^ F= = \ i
y/tct V m ci
where ect is the the ei(u;) value at the charge transfer gap frequency.
Figure 32 shows Im[l/e(w)] with Ce doping as a function of frequency. The
result for x = 0 is very small below 1.2 eV except phonon modes in the far-infrared
region, and shows a bump near 1.5 eV which is associated with the charge transfer
excitation. The spectrum of x = 0.11 shows a featureless continuum near 1000
cm-1 and a broad peak around 7200 cm-1 (0.9 eV). With doping this peak position
moves to slightly higher frequencies, where its maximum position corresponds to the
appearance of a reflectance edge with doping. For 0.14 < x < 0.2, the peale positions
occur near 1.1 eV and are insensitive to Ce doping concentration, inconsistent with
the simple Drude model. This indicates that the value of n/m*ect in (51) is insensitive
to doping. Figure 32 also shows that the peak position of the superconducting sample
with x = 0.15 is observed at higher energy than in slightly overdoped sample with
x = 0.16. This may suggest that the superconducting sample has more free carriers
or low effective mass of charge carriers. A broad peak width (0.5 eV) in Im[l/e(u;)]
is due to the anomalous mid-infrared absorption caused by the incoherent motion of
free carriers.

66
Our results for Nd2-xCexCu04 axe similar to those of La2-xSrxCu04 by Uchida
et al.,3 where the zero crossing of ei(u>) for the metallic samples is pinned near 0.8
eV due to strong mid-infrared absorptions. In contrast, the dielectric response for
YBa2Cu307_j obtained by Cooper et al.39 shows almost linear doping dependence of
zero crossing of i(c*>), exhibiting neaxly free carrier behavior.
Temperature Dependence of Optical Spectra
The temperature dependence of the reflectance between 80 cm-1 and 4 000 cm-1
was measured in order to study the applicability of the Drude model. For non
superconducting metallic samples, the change of reflectance between 10 K and 300
K is less than 2% in the far-infrared region, as shown in Fig. 33 for metallic samples
of x = 0.16 and x = 0.19. However, for superconducting sample of x = 0.15, the
reflectance change between these temperatures is about 3.5% neax 600 cm-1.
Figure 34(a) shows the temperature dependent reflectance of the superconduct
ing sample, x = 0.15, in the frequency range between 80 cm-1 and 2 000 cm-1.
As the temperature decreases, the magnitude of the reflectance exhibits a system
atic increase. The optical conductivity shows a clear picture of a Drude behavior.
Figure 34(b) shows cri(u>) obtained after a K-K analysis of reflectance spectra in
Fig. 34(a). The reduced, the spectral weight between 500 cm-1 and 2 000 cm-1 is transferred to lower
frequencies, corresponding to the narrowing of the Drude band at low temperatures.
We have fit our results with the two component model, a Drude part and several
Lorentzian contributions:
. ,2 N 2
pD y- upj
iu/r 4^ uj'j u2 i
+ f
luJlj
oo>
e(w) =
(52)

67
where the first term is a Drade oscillator, described by a plasma frequency up£> and a
relaxation time r of the free carriers, the second term is a sum of peaks in c\[u), with
Uj, upj and 7j being the resonant frequency, strength and width of the jlh Lorentz
oscillator, and the last term is the high frequency limit of e(u;).
The Drude components at five temperatures for the superconducting sample,
x = 0.15, axe depicted in Fig. 35. The Drude component is defined as the con
ductivity after the average mid-infrared component is subtracted. The temperature
dependence of the Drude part satisfies the ordinary Drude behavior. The inset in
Fig. 35 shows the temperature dependence of the scattering rate obtained from the
fits. The Drude plasma frequency upj) 11 200 cm-1, is nearly T-independent, while
the temperature dependence of the scattering rate 1/r is consistent with the behavior
of the dc resistivity. For example, 1/r is non-linear in T and reduces to half of the
300 K value at 15 K. The dc resistivity value from the four-probe measurement is
good agreement with that obtained from simple Drude formula:
_ UPD
Pdc 4tt(1 /r)dc
We emphasize here that the Drude plasma frequency is laxger, and 1/r is a little
smaller than the values obtained for La2_ISrICu04. Also, the value for up is larger
than the results obtained by any other experiments for electron-doped system.
One Component Approach
As suggested in previous section, ct\(u) does not fit the simple Drude formula
(cti(u;) a u;-2). Especially, o\(u) reveals a strong spectral weight in the mid-infrared
region, compared to that at zero frequency. Another approach to analyze this non-
Drude conductivity is to assume an inelastic scattering of the free carriers in the

68
low frequency range with a underlying excitation spectrum. The carriers derive a
frequency and a temperature-dependent self-energy. The imaginary part goes like
ImS ~ max(u,T). This quasiparticle damping has been described in the frame
work of the nested Fermi liquid (NFL)37,38 and the marginal Fermi liquid (MFL)
models.36 We analyze the non-Drude conductivity of Nd2_xCezCu04 by using a
generalized Drude formula with frequency dependent scattering rate.
c(w) = Ch
ut*
w[m*(w)/m0][w -1- t/r*(w)]
(54)
where e/, is the background dielectric constant associated with the high frequency con
tribution and the second term represents the effects of frequency dependent damping
of carriers, m*/mo represents the effective mass enhancement over the band mass
and 1/t*(u) = (l/r(u;)][m/m*(u;)] the renormalized scattering rate.
Figure 36 shows the m*/mj and 1/t*(u>) curves for four samples below 5000
cm-1. We used utp = 20 000 ~ 2 2000 cm-1, and = 5.0 ~ 5.2 in the infrared
region for different samples. At low frequencies, the behavior of m*/mi illustrates
the coherent motion of carriers, causing the low frequency mass enhancement. This
may be due to the interaction of carriers with phonons, or spin and charge excitations
of carriers. Our results also suggest that the quasiparticle excitations increase with
decreasing doping concentration. This is consistent with other doping dependence
results for hole-doped systems. However, the mass enhancement of Nd2-xCezCu04
is a little bit smaller than those obtained by hole-doped systems. As the frequency
is reduced, the effective carrier mass decreases, and approaches to the band mass at
high frequency.

69
The renormalized scattering rate l/r*(a>) and the effective mass enhancement
m*/mi, can be also related to the imaginaxy paxt of quasipaxticle self energy by
m* 1 u;
TT = 2Im£.
mo r*(w) 2
Figure 37 illustrates the imaginary part of quasiparticle self energy, Im £ of
Nd2-xCeICu04 crystals below 5000 cm-1. Im£ in Fig. 37 is analyzed in several
ways.
First, for a barely metallic crystal (x = 0.11), the imaginary paxt of self-energy
deviates from the lineaxity and reveals a power law between 0.1 eV and 0.6 eV,
reflecting the proximity to the phase boundary of the insulator. With doping we see
a steady decrease in the quasipaxticle interaction and finally, a linear slope of Im £
in high doping concentration of x = 0.19.
Second, for superconducting sample (x = 0.15), Im£ increases linearly with u
below 5 000 cm-1. This is in a good agreement with the predicted behavior in the
MFL. According to the MFL theory, the imaginaxy paxt of quasipaxticle self energy
Im£ has the form ir\u over T < u < uc, where u>c is the cutoff frequency. We
estimated a coupling constant A = 0.15 0.01 from the slope between 500 cm-1 and
2500 cm-1. This estimated value seems to be rather low compared with the results
obtained from hole-doped systems. It is also interesting to compare the result for
superconducting sample of x = 0.15 with the result for slightly highly doped sample
of x = 0.16. The data of x = 0.15 show less quasipaxticle excitations than that
of x = 0.16. This may suggest that too much quasipaxticle interaction causes the
reduction of superconductivity in high Tc cuprates.
Third, the quadratic temperature dependence of dc resistivity in superconducting
sample does not agree with our analysis. We might expect a quadratic dependence

70
in u of scattering rate from dc resistivity. Ordinary Fermi liquid state requires the
scattering rate varying as u1. Nevertheless, our result in the high Tc regime is consis
tent with numerous models of the normal state in which strong quasiparticle damping
is assumed. Also, our results with doping suggest that the electronic state of very
heavily doped CuC>2 plane may be acquire the nature of a Fermi liquid.
Doping Dependence of Low Frequency Spectral Weight
Prude and Mid-infrared Band
We have emphasized that the spectral weight of the high frequency region above
the charge transfer (CT) band is transferred to low frequencies with doping. Such a
spectral change indicates that the conduction and valence bands of the CT insulator
are reconstructed by doping. In the metallic state, the optical conductivity may be
considered as three parts; a free carrier contribution centered at u = 0, mid-infrared
bands, and high-energy interband transitions above the charge transfer gap. In order
to describe empirically the absorption bands produced by doping, we have fit the
of each sample to the two component model. We here discuss in detail each
band and how its strength changes with Ce doping. The strength of each band j is
related to the plasma frequency in the fitting parameters by the relationship
"h(eV2)
47re2 Nj
m* Vct\{
(56)
We estimated (eV2) = 14.88 Nj, using Vceu = 187 and two Cu atoms per unit
cell, where Nj is the effective electron number per Cu atom of band j.
For free carrier contribution, we extracted the spectral weight of a Drude oscillator
(Nq) in the unit of electron number per Cu atom as a function of Ce concentration
x from the sum rule restricted to the Drude conductivity, a¡). Figure 38 (circles)

71
represents N¡). However, it is difficult to define the Drude paxt from metallic phase due to the mixing with strong mid-infrared bands. Thus, Np in Fig. 38
represents with large error bars. Figure 38 illustrates that a Drude strength is very
low up to Ce concentration of x = 0.11 and is roughly proportional to the dopant
concentration x in the metallic phase, as expected for the generation of carrier by
adding of electrons in the CuC>2 plane. Our results are in good agreement with the
phase diagram of Nd2-xCezCu0469,70 which shows the insulating phase at a wide Ce
compositional range up to x = 0.12. As mentioned earlier in the effective electron
number section, the superconducting sample of x = 0.15 has a very strong Drude
band. This result is consistent with the theoretical observation71 described by the
extended Hubbard model that upon electron doping the Fermi level lies directly in
the Van Hove singularity of the upper Hubbard band at a certain doping level. This
concentration may be the superconducting sample with x = 0.15.
As shown in the of Fig. 28 and Fig. 29, Ce doping in Nd2Cu04 clearly
induces the formation of strong mid-infrared bands. Upon doping, these mid-infrared
bands continue to grow at high Ce concentrations and tries to merge with the Drude
peak. In Fig. 38, we also plot the strength of total mid-infrared bands, Nmid, as a
function of Ce concentration x (squares). Nmid is estimated from several Lorentzian
fits to oi(w) in the low-frequency part below the charge transfer gap. Nmid also
represents with large error bars due to the ambiguity of a mid-infrared band near the
Drude part. The strength of the total mid-infrared bands increases rapidly at low
doping, but slowly at high doping levels. We stress here that with the Drude band
the strength of total mid-infrared bands of Nd2_ICeICu04 is very strong compared
to that of hole-doped L^-xSrxCuC^.

72
Transfer of Spectral Weight with Doping
Next, we interpret the low frequency excitation near the Fermi level transferred
from the high frequency region as a function of Ce doping x. This is done by comput
ing the effective electron number Neff(uj) of the Drude and toted mid-infrared bands
which corresponds to all electrons that are introduced by doping and comparing with
hole-doped La2_zSrzCu04 system of Uchida et al.3
Figure 39 represents the low frequency spectral weight below 1.5 eV of hole-doped
La2_zSrzCu04 of Uchida et a/.(left) and the low frequency spectral weight (LFSW)
of electron-doped Nd2_zCezCu04 for our results (right). The solid lines in Fig. 39
correspond to the localized limit (no p-d hybridization) in the charge transfer system
for hole-doping and electron-doping cases. In the localized limit, upon doping the
LFSW of electron-doped system is expected to grow similar to the Mott-Hubbard
case, where the LFSW goes to 2x with doping x due to the restriction of doubly
occupied states of doped carriers, because electrons are doped primarily on Cu sites.
For hole-doped system, LFSW grows as x with doping x as semiconductor case,
since holes introduced by doping on 0 sites occupy almost free particle levels and
scatter weakly off the Cu spins. However, Meinders et a/.4 have shown that when the
hybridization is large, the LFSW of hole-doped system becomes similar to that of the
MH system and the electrons as well as the holes show strongly correlated behavior.
Our results for Nd2_zCezCu04 show a electron-hole symmetry at low doping
levels and a prominent electron-hole asymmetry. The LFSW associated with the
Cu02 plane grows faster than 2x with doping x, consistent with the expectation of
the MH model, where the lower Hubbard band (LHB) as well as the upper Hubbard
band (UHB) loses the spectral weight. The greater LFSW than 2x may result from
a large impurity band contributions in T phase materials and the charge transfer

73
excitations. Especially, a strong spectral weight in the metallic phase axound x = 0.15
may reflect the contribution of the charge transfer excitations. This is a spectral
weight transferred from the p-like correlated states to the low frequency region. In
contrast, the LFSW in La2_xSrxCu04 goes to 2x at high doping levels and x at high
doping levels with doping concentration x.
Doping Dependence of Charge Transfer Band
Figure 40 represents the variation of the charge transfer bands with Ce dop
ing. The charge transfer conductivities axe obtained after subtracting high energy
interbands, and the Drude and mid-infrared bands. For the charge transfer band in
insulating Nd2Cu04, two contributions appear. One (CT1) is a week and narrow
band with center frequency near 12 900 cm-1 (1.6 eV) and the other band (CT2) is
a relatively strong and broad band near 16 800 cm-1 (2.08 eV). We can also see two
peaks in the spectrum of x = 0.11 near 14 000 cm-1 (1.74 eV) and 16 800 cm-1 (2.08
eV), respectively. At higher doping levels, only one band appears. Figure 40 also
shows that the strength of the CT1 and CT2 bands decreases with increasing doping
concentration x.
The CT1 band is related to the abrupt decrease of its strength as a result of the
decrease of the intensity of UHB upon doping. Upon electron doping the position of
CT1 band shifts to higher frequency (from 1.6 eV for x = 0 to 1.74 eV for x = 0.11)
and its spectral weight (~ 11000cm-1) rapidly decreases, finally disappearing for
x > 0.14. The spectrum for x = 0.11 in Fig. 40 shows the very weak CTl band of
strength ~ 2 000 cm-1. Figure 41 illustrates the variation of the strength of the CTl
and CT2 bands, Ncti and ^CT2i as a function of doping x.
The behavior of the CT2 band with doping is similar to that of the CTl band.
Doping with electrons results in a reduction of the CT2 band and a small shift to

74
higher energies from 2.08 eV for x = 0.11 to 2.29 eV for x = 0.19. However, a transfer
of spectral weight only starts after the CT1 band has completely disappeared, as
observed in Fig. 41. There is no difference of peak position and strength between the
spectra of x = 0 and x = 0.11. The spectral weight is ~ 15 000 cm-1 for x = 0.11
and ~ 10 300 cm-1 for x = 0.19. Thus, both the CT1 and the CT2 bands seem to
due to a transition from the Cu 3d UHB to Zhang-Rice type17 correlated states.
The squares in Fig. 41 also explain the spectral weight loss of two CT bands upon
doping. The spectral weight of two CT bands loses slightly faster than x with doping
x. This trend is in good agreement with the behavior of the LFSW with doping x.
The amount of the greater spectral weight loss than x is very similar to that of the
greater LFSW than 2x, which may be related to p-d charge transfer.
We here have the interesting fact that, when we consider the positions of the CT1
and CT2 bands with doping, the Cu 3d UHB should move to higher energy. From the
position differences of the CT1 band between x = 0 and x = 0.11 and the CT2 band
between x = 0.11 and x = 0.19 we are led to conclude that the Fermi level should lie
~ 0.35 eV above the bottom of the UHB. This result is consistent with a theoretical
estimate72 and the EEL and x-ray absorption spectroscopy.73 This observation is
also compaxed to the results46,74 observed in La2-zSrICu04 of 0.7 eV. This narrow
energy range induces the strong Drude band and suggests that Nd2-xCezCu04 of
electron-doped system has a large Fermi surface, in good agreement with the angle
resolved photoemission experiments.75,76
Summary
We have examined the change of optical spectra with Ce doping in electron-
doped Nd2-iCeICu04 in the frequency range from the fax-infrared to the UV re
gion. We have also made a systematic analysis of the temperature dependence for

75
Nd2-iCezCu04 at temperatures between 10 K and 300 K. We analyze our data with
the one component and two component models. Our results show that the doping
mechanism of the electron-doped Nd2-xCexCu04 is a little different from that of
hole-doped La2-xSrxCu04.
The spectrum of the undoped Nd2Cu04 shows a typical insulating characteristic
with energy gap of 1.5 eV which is identified to 0 2p-Cu 3d charge transfer excitations.
Doping with electrons in insulating Nd2Cu04 induces a shift of spectral weight from
the high energy side above the charge transfer excitation band to the low energy side
below 1.2 eV. The low energy spectral weight for a barely metallic sample, x = 0.11
is composed of two paxts: a narrow Drude-like and mid-infrared paxts. Upon further
doping the Drude-like band rapidly increases and the mid-infrared band shifts to
lower frequency, and hence two paxts axe hardly sepaxated in the metallic phase.
A weals temperature dependence of the far-infrared reflectance suggests the non-
Drude behavior of this material. For example, the change of reflectance between 15 K
and 300 K for non-superconducting metallic samples of x = 0.16 and x = 0.19 is less
than 2% and for superconducting sample of x = 0.15 the reflectance change between
same temperatures is about 3.5% near 600 cm-1. This non-Drude behavior can be
analyzed by a frequency dependent scattering rate and a mass enhancement in the
one component approach.
In the one component approach, our results show that the mass enhancement at
low frequencies is large, and for superconducting sample Im S is linearly propor
tional to u below 5 000 cm-1, in good agreement with the predicted behavior in the
numerical models in which strong quasiparticle damping is assumed. From the slope
of ImS a weak coupling constant A ~ 0.15 0.01 is estimated.

76
The low frequency spectral weight (LFSW) with doping is analyzed by the two
component model. The Drude strength is very low up to the metal-insulator transi
tion and is roughly proportional to the doping concentration x in the metallic phase.
The strength of total mid-infrared bands rapidly increases at low doping but slowly
at high doping levels. The LFSW including the Drude and toted mid-infrared bands
grows faster than 2x with doping x consistent with the MH model. These strong
Drude and mid-infrared bands with the result of transport measurements suggests
that Nd2_ICeICu04 has a laxge Fermi surface consistent with photoemission exper
iments.
The charge transfer (CT) band is also analyzed with the two component model.
The CT band in insulating Nd2CuC>4 consists of two bands, CTl and CT2 bands.
Upon doping, the CTl band disappears at high doping levels, while the CT2 band
survives even if it partially loses its spectral weight. The two bands correspond to
the transition from the Cu 3d UHB to Zhang-Rice type correlated states.

Reflectance
77
Photon Energy (eV)
0 1 2 3 4 5
Fig. 21. (a) Room temperature reflectance spectrum of Nd2Cu04_ on a-b
plane and (b) c\{u) spectrum after K-K transformation of R in (a).

Reflectance
78
Frequency (cm 1)
Fig. 22. Far-infrared reflectance of Nd2Cu04_i at several temperatures.

-1 A)
79
100 200 300 400 500 600 700
Frequency (cm-1)
Fig. 23. Far-infrared (a) <7i(u;) and (b) Im[-l/e(u;)] for NdoCuO^j. Peaks
in (a) correspond to TO phonons, in (b) to LO phonons.

Reflectance
80
Frequency (cm 1)
Fig. 24. (a) Reflectance spectra of Nd2Cu04_$ at several temperatures, and
(b) the reed part of the optical conductivity as a function of frequency.

81
o
Temperature (K)
Fig. 25. The parameters extracted from the Lorentzian fits to the peak near
1500 cm-1 as a function of temperature.

Reflectance
82
Photon Energy (eV)
5000 10000 15000
Frequency (cm-1)
Fig. 26. (a) Temperature dependent-reflectance spectra and (b) optical con
ductivity spectra of charge transfer band for Nd2Cu04_.

Reflectance
83
Photon Energy (eV)
0.01 0.1 1
Fig. 27. Room temperature reflectance spectra of Nd2-zCexCu04 for various
x on a-b plane.

84
Photon Energy (eV)
0 1 2 3 4 5
Fig. 28. Room temperature a-b plane optical conductivity spectra of
Nd2_zCezCu04 with doping x as a logarithmic frequency scale.

85
Photon Energy (eV)
0.01 0.1 1
Fig. 29. Room temperature a-b plane <7i(u>) spectra as a function of x (note
the linear frequency scale).

86
Photon Energy (eV)
0.01 0.1 1
Fig. 30. Effective electron number per formula unit for Nd2-xCexCu04 at
doping levels from 0 to 0.2.

(/Cu atom)
87
Photon Energy (meV)
0 20 40 60 80 100 120
Fig. 31. Nljj per Cu atom of Nd2_xCexCu04 in a frequency range below
1000 cm-1.

88
Photon Energy (eV)
o 10000 20000
Frequency (cm-1)
Fig. 32. The energy loss function, Im[l/e(u;)] of Nd2_xCeICu04 as a func
tion x.

Reflectance Reflectance
89
0 100 200 300 400 500 600
y(crrf1)
Fig. 33. Far-infrared reflectance for non-superconducting metallic samples of
Ndi.86Ceo.i6Cu04 and Ndi.8gCeo.i9Cu04 at several temperatures.

cr(ncm) Reflectance
90
0 500 1000 1500 2000
i/(cm~1)
Fig. 34. Temperature dependent (a) reflectance and (b) conductivity for su
perconducting Ndi.85Ceo.i5Cu04 as a function of frequency.

91
Fig. 35. The Drude conductivity obtained by subtracting the mid-infrared
contribution from the total conductivity. The solid line axe Drude fits.
Insert shows a Drude scattering rate, 1/r as a function of temperature.

92
Photon Energy (eV)
o.o 0.2 0.4 0.6
Fig. 36. Frequency-dependent mass enhancement (upper panel) and renor
malized scattering rate, (lower panel)

93
Photon Energy (meV)
0 150 300 450 600
Fig. 37. The imaginary part of self-energy, -ImE with x as a function
frequency.

Spectral weight (/Cu)
94
Fig. 38. The spectral weight of the Drude and toted mid-infrared as a function
i, estimated from two component model of

Low frequency spectral weight (/Cu)
95
Fig. 39. The low frequency spectral weight as a function of x in both
La2_xSrxCu04 (left) and Nd2_xCexCu04 (right). The data from
La2_xSrxCu04 were taken from Ref. 3.

96
Photon Energy (eV)
10000 15000 20000 25000
Frequency (cm-1)
Fig. 40. o\(u) spectra of the charge transfer band of Nd2_rCezCu04 crystals.

Spectral weight (/Cu)
97
Fig. 41. The spectral weight of the CT1 (diamonds) and CT2 (circles) bands,
Ncti and ^£7T2) the spectral weight loss (squares) as a function x.

CHAPTER VII
QUASI-PARTICLE EXCITATIONS IN
LIGHTLY HOLE-DOPED La2-xSrzCu04.
In this chapter, we investigate the quasiparticle excitations in low doping regime
of La2_ISrICu04+5 (0.01 < x,S < 0.04) by temperature dependence reflectance
measurements of single crystals.
The dynamics of dopant-induced charged quasiparticles is currently one of the
central questions in strongly correlated cuprate materials. Due to their connection
with high Tc superconductivity, the main goal is the understanding of the quasiparticle
excitations of doped hole, and the possibility of the superconducting pairing from
an entirely electronic origin in these systems. Properties of quasiparticle in a low
doping regime are much easier to investigate because they can be easily separated
from large electronic contributions. Examples include the hopping of charge carriers
between localized states, the effect of impurity potential for charge carrier localization,
a carrier-phonon interaction and carrier localization by the deformation potential
caused by the tilting of Cu06 octahedra, self-trapped polaron states due to a strong
electron-phonon interaction, and a strong carrier-spin interaction, etc.
Perhaps, the simplest system for the study of these issues is La2-zSrzCu04+.
The parent compound La2Cu04 undergoes a second-order transition from the tetrag
onal K2NF4 structure to an orthorhombically distorted one below Tq/t = 530 K.77
The transition can be described roughly as a staggered tilting, or rotation, of the
oxygen octahedra around the tetrahedral [110] axis, resulting in a \/2 x \/2 doubling
of the unit cell in the a-b plane.78-80 This phase transition folds vibrational modes
at the X point in the bet structure back to the zone center, where they may become
98

99
Raman or infrared active. The dynamical tilting of the Cu-0 plane increases the
dipole potential of Cu-0 plane, and so increases a carrier-lattice interaction with the
increase of doped-holes. Upon Sr doping, Tjv is depressed rapidly from ~ 300 K for
x = 0to~0Kbyx~0.02. The doping also depressed Tq/Ti but at a much lower
rate, such that Tq/j = 0 by x = 0.2.77
We here report new modes induced by doping, and discuss a carrier-lattice in
teraction caused by the tilting of Cu06 octahedra. Further, for 10 K conductivity
spectra, we find resonant absorption peaks which correspond to the photon-induced
hopping of charge carriers. Finally, we discuss the infrared absorption bands near
0.15 eV, 0.5 eV, and 1.4 eV.
Experimental Results
a-b Plane Spectra
Figure 42 shows the reflectance spectra in La2_iSrxCu04+j with (0.01 < x,6 <
0.04) at room temperature in frequency range from 30 cm-1 to 32000 cm-1 (4 eV).
The polarization of the incident light is parallel to a-b plane. The spectrum at fre
quencies below 1 eV exhibits dramatic changes with Sr doping. Among the trends
observed in reflectance with Sr doping are a loss of spectral weight of the charge
transfer (CT) excitations around 2 eV and the development of a plasma edge shift to
higher frequencies with doping, which corresponds to the development of free carrier
band and mid-infrared bands. Figure 42 also shows that all samples we measured
show metallic-like behavior even in the lowest Sr doped sample, i.e., the reflectance is
about 80% near 100 cm-1 and decreases monotonically with the increase of frequency.
In addition, the 1% Sr doped sample shows a larger magnitude of the reflectance than
2% Sr doped sample due to the deviation from the oxygen stoichiometry.

100
In Fig. 43, we plot the optical conductivity spectra for four samples as a func
tion of frequency as a result of K-K transformations of the reflectance spectra in
Fig. 42. Figure 43 clearly demonstrates that upon doping, the spectral weight above
CT gap near 2 eV decreases and its peak position is shifted to higher frequencies
systematically, consistent with theoretical model and other experiments,81 while the
conductivity below 1.5 eV grows significantly throughout the doping range.
Several TO phonon features axe also identified with peaks in the fax-infrared
region. The tetragonal high temperature phase of La2Cu4 exhibits four in-plane
phonons with Eu symmetry. However, the orthorhombic distortion lowers the sym
metry and activates severed additional phonon modes which account for weak struc
tures around 145 cm-1. Figure 43 also illustrates that with Sr doping the oscillator
strength of phonons decreases due to the electronic screening from free carriers, and
instead doped carrier contribution at u> = 0 and new vibration modes grow in the
fax-infrared region.
Other features that appear in the frequency region below charge transfer gap are
broad mid-infrared bands near 4050 cm-1 (0.5 eV) and 11300 cm-1 (1.4 eV) band
which is not observed in undoped sample and in electron-doped T phase samples
without apical 0 atoms.
Figure 44 shows the plot of ei(u;) as a function of frequency, showing a positive
dielectric response at frequencies less than 100 cm-1 that dielectrically screens the
free-caxrier response. There axe large difference between the static dielectric constant
es ~ 30 (for 3% Sr doped samples) and the high frequency dielectric constant ~ 4.
This large difference seems to be due to the contribution from phonons, and suggests
that La2_zSrzCu04 in low doping regime is very polarizable at low frequencies and
the charge carriers expect to form polarons. Figure 44 in the far-infrared region also

101
illustrates the coupling of photon field and TO phonons in an ionic crystal, showing
large LO-TO phonon splitting, where LO and TO phonons correspond to the zeros
and the poles of ei(u/), respectively.
c Axis Spectra
The c-axis optical reflectance for Lai.97Sro.o3Cu04 is shown in Fig. 45 at sev
eral temperatures. The room temperature reflectance in the high frequency region is
shown in the inset. The c-axis reflectance has the characteristics of an insulator, show
ing primarily four optical phonons in the far-infrared, almost featureless reflectance
in the high frequency region, and a narrow peak around 10000 cm-1. The optical
conductivity for c-axis of undoped sample is absolutely vanishing up to 1 eV above
which the interband transition start.82 However, our conductivity spectrum in the
inset in Fig. 46 shows a steep rise up to 10 000 cm-1. This rise indicates that upon
doping, some spectral weight is transferred into the low frequency region as in the
case of the in-plane spectrum, but the transferred weight is quite small.
Figure 46 shows the TO and LO phonon spectra at four temperatures in the
far-infrared range obtained from K-K transformation of R(w) in Fig. 45. As the
temperature decreases, the phonon lines become sharper, as expected. The results
show that the c-axis conductivity is dominated by four infrared active modes at 230,
320, 345 and 510 cm-1. In particular, the oscillator strength of the 230 cm-1 phonon
which was assigned to the Cu-0 bond bending mode is very large compared to those
of the other three phonons. The LO-TO splitting is also large, indicating that the
effect of screening due to free carrier is minor for vibration polarized parallel to c-
axis. We also observe very week features at 275 and 420 cm-1 in the low temperature
spectra.

102
Mode Assignment
The crystal structure of La2Cu04 is orthorhombic (D^, Ama, CmCa) at room
temperature and tetragonal (D^, Ii/mmm) above about 515K. The T/O transition
temperature decreases with increasing Sr concentration x. Figure 47 shows the atomic
positions in the orthorhombic unit cell. This structure is defined as a staggered tilting
or rotation around the [110] axis of the CuC>6 octahedra. In the orthorhombic phase
the volume of the primitive cell is doubled with respect to the volume of the tetragonal
phase, so that zone-boundary modes are folded back into the T point, yields 39 optical
modes.83 These modes may be classified into Raman (5Ag + 3B\g + 6f?25 + 4i?3y),
silent (4Au), and infrared active (6B\U + 4i?2u + 7i?3U). There are eleven infrared-
active a-b plane optical phonons and six c-axis polarized phonons. Since the crystals
are twined, we can not separate the i?2u (a axis) from the B$u (b axis) modes.
In Fig. 46, we have shown the c-axis polarized conductivity spectra for single
crystal of Lai.97Sro.o3Cu04 in the temperature range from 10 K to 300 K. We ob
serve four infrared active modes centered at 230, 320, 345 and 510 cm-1 of the six
B\u modes predicted by group theory. A comparison of these frequencies with other
optical studies84 of c-axis spectrum in La2_zSriCu04 in the tetragonal phase indi
cates good agreement with three phonon modes at 230, 345 and 510 cm-1. The 320
cm-1 frequency mode appears to be rendered infrared-active by the orthorhombic
distortion. The two unobserved modes likely have small oscillator strength and high
damping, or at frequencies below 30 cm-1. There are very weak features at 275 and
420 cm-1 in low temperature spectrum that exhibit some temperature dependence.
These modes are the Raman-active Ag modes due to the breaking of the inversion
symmetry by the distortion of Cu06 octahedra.

103
Figure 48, Fig. 49, and Fig. 50 show the a-b plane phonon spectra in the infrared
region at temperatures between 10K and 300K for La2_zSrICu04+. All samples
that we measured ire in the orthorhombic phase at these temperatures. We see
considerably more than the eleven expected modes. We can classify the phonon
modes in the spectrum as three types: (1) infrared active modes in the orthorhombic
phase, (2) Reiman modes due to the breaking of inversion symmetry, (3) new local
modes caused by the presence of the localized carriers.
In 10 K spectrum for Lai.97Sro.o3CuC>4 in Fig. 50, we observe a strong infrared
features of the seven modes at 107, 154, 168, 188, 352, 366 and 684 cm-1, of the
eleven B2U + Bzu modes. The tetragonal high temperature phase of La2Cu04 exhibits
four Eu symmetry in-plane phonons85 near 135, 164, 360 and 680 cm-1. There are
clear changes in the infrared active modes as a result of the phase transition. For
example, three modes near 135, 164 and 360 cm-1 in the tetragonal phase are splitted
into a B2u + B$u pair in the orthorhombic phase. These modes are assigned to the
translational vibration of the La atoms against the Cu06 octahedra, the bending
vibration of the apical Oz atom against the Cu-0 plane, and the bending vibration
of CuC>2 peme, respectively. This splitting develops upon cooling as a result of the
further rotation of Cu6 octahedra. In this case, in-plane 0 atoms are displaced
in the direction perpendicular to the plane, and thus bring about two different 0-0
distances in Cu-0 plane. The rotation of CuOe octahedra can be also observed with
a soft phonon mode in neutron scattering experiments.86
Figure 51 shows the frequencies of three modes as a function of temperature.
As the phonon mode goes from low to high frequency, the splitting decreases. The
highest mode near 680 cm-1, assigned to the stretching vibrations of the in-plane
Cu-0 bonds, is not split. The splitting of the two apical 0* bending modes to higher

104
frequency compared to phonon position in the tetragonal phase is associated with the
displacement the apical Oz atoms into lower symmetry sites out of the Cu-0 plane,
as the temperature is lowered. With decreasing temperature, the mean position of
the apical 0* atoms changes significantly moving closer to the Cu-0 planes, which
leads to the different ionic charges of the Oz atoms.
Figure 50 shows that on increasing Sr concentration and decreasing temperature,
new peales appear at 77, 139, 247, 278, 320, 384, 400, 423 and 481 cm-1. The modes
at 77, 139, 320, 384, 400 and 481 cm-1 coincide with the Raman-active B^g + Bzg
modes.87 The B^g and B$g axe active in (a, c) and (a, c) polarization configurations
as shown in Fig. 47. (a, b) denotes that the polarizations of the incident and the
scattered light axe parallel to the a and b axes, respectively. The modes at 247, 278
and 423 cm-1 with strong oscillator strength axe consistent with Ag normal modes.
Here, Ag mode is observed in the (c, c) polaxization which corresponds to A\g mode in
the tetragonal phase. This result shows that the Raman mode becomes to the infrared
active mode. The activity in infrared and Raman is alternative in the crystals with
inversion symmetry such D2h and D^. The appearance of the Raman modes in the
infrared spectrum indicates the breaking of this symmetry. It can be also argued that
as first pointed out by Rice88,89 in the organic materials, linear coupling of charge
carriers to totally symmetry (.A^) phonons can lead to structure in the conductivity
spectrum at the phonon frequencies. In this charged phonon mechanism or electron-
molecular vibration coupling effect, the electron energies depend on the bond lengths,
while at the same time the bond lengths depend on the local charge density. Infrared
radiation at the Ag phonon frequencies can pump charge over long distances, giving
rise to absorption that has electronic oscillator strength and that is polarized in the
a-b plane.

105
There axe two types of local polarization due to the orthorhombic distortion.
First, the apical 0 displacements along the c-axis induce changes in the local Cu
charge. Second, the displacement of in-plane 0 atom in the c-direction decreases the
0-0 distance, and therefore changes the electronic states of 0 atom. In each case,
the doped-holes can be easily localized in Cu02 plane and their charge dynamics
are slow enough for the lattice to follow. This slow charge dynamics can induce the
carrier-lattice interaction.
Figure 52 shows the dependence of the oscillator strength of four infrared phonons
and Ag Raman modes at 247 and 278 cm-1 as a function of temperature. The lower
frequency mode is strongly affected by the deformation potential compared with the
high frequency mode as in the case the splitting of mode, showing the enhanced
oscillator strength of phonons upon cooling. This is interpreted as meaning that
the carrier-lattice interaction increases due to the deformation potential caused by
the orthorhombic distortion. The oscillator strength of the Raman active phonons
greatly increases compared to that of the infrared active phonons with decreasing
temperature.
Another mode which is not observed in the Raman and infrared active modes
is a broad peale near 620 cm-1 which is observed at a frequency just below the in
plane stretching mode and shows its broadening with temperature. We have shown
above that the dynamical tilting of the Cu06 octahedra enhances the carrier-lattice
coupling. Also, those vibration which cause a change of bond length in the Cu-0 plane
can have stronger carrier-phonon interaction. Hence, this mode may be associated
with a carrier-phonon interaction. However, presently we can not give a satisfactory
explanation for this mode. Similar modes are observed in lightly doped Nd2Cu4
system90 and a theoretical work.91 They have shown that this mode is induced by

106
doping, and may occur as a result of the carrier-phonon interaction. This mode
is also consistent with the bleaching of phonon modes observed in photo-induced
measurements.64
Hopping Conductivity in Disordered System
A disordered system having electronic states near the Fermi level has localized
states due to strong disorder and small overlap of the wave function. Such systems
axe on the dielectric side of the Anderson transition.92 Lightly doped crystalline
semiconductors and amorphous semiconductors are example of such system. The
electron-electron interaction in such systems determines a large variety of physical
phenomena, especially dc and ac hopping conduction.
Figure 53 shows the resistivity p as a function of temperature for a 1% Sr doped
sample. The resistivity has a minimum at intermediate temperatures, followed by a
low temperature upturn. In the high temperature region (> 100 K), all impurities axe
ionized and metallic behavior is observed due to the overlap of the impurity orbits. At
low temperatures (< 100 K), the freezeout of holes occurs, and hence the conductivity
results from the thermal ionization of the shallow impurities.
At low temperatures below 50 K, p shows the characteristic behavior of disordered
system with strong Coulomb interactions:93
P~ex p(y)1/2. (53)
The inset in Fig. 53 is a plot of lnp vs. (1/T)1/2 at same temperatures, showing
a linear relationship. This is a typical behavior of dc variable range hopping in
localized states near the Fermi surface. At sufficiently low temperatures, under all
circumstances where N(Ep) is finite but states axe localized near the Fermi energy,

107
we expect that the electron jumps from a state below the Fermi level to a nearby
state or distant state. The energy band width 7 neax the Fermi level is determined
by Coulomb interaction 7 ~ e2/ef?, where R is the mean distance between two states.
Also, Chen et al.94 show the power-law frequency dependence of the conductivity in
the microwave frequency region which is characteristic of phonon-assistant tunneling
of electrons between bound states of the dopant atoms.
The behavior of dc resistivity with temperature is very similar to that of free
carrier in the far-infrared 07 (w) spectra. In Fig. 50, at temperatures above 100 K, the
extrapolated values of 07(10) at zero frequency increases with decreasing temperature,
showing a metallic behavior. However, 07(10) below 100 cm-1 at 10 K strongly de
creases and instead 07(10) above 100 cm-1 increases to compensate for the decreased
oscillator strength below 100 cm-1. This indicates that the free carriers at 10 K are
strongly localized.
The far-infrared conductivity spectrum at 10 K in Fig. 54 shows a resonant ab
sorption whose maximum occurs at a frequency between 100 cm-1 and 130 cm-1,
depending on doping concentration. The origin of the resonant absorption is, in the
high frequency region hu > kT, the photon-induced hopping of charge carriers as
a result of carrier transition from one site to another, and thus change its location
with respect to impurity atoms. This process must be distinguished from the usual
phonon-induced hopping mechanism which has been previously studied in Chen et
al.
The first works on the theory of resonant absorption were devoted to the impurity
band of doped semiconductors. Experimentally, Milward and Neuringer95 have first
observed near 30 cm-1 at low temperature in compensated n-type silicon. This value
is lower than those of our results, because the dopant density in n-doped Si is very

108
low (1017 < Nd < 1018 cm 3). The mechanism of the absorption was proposed by
Tanaka and Fan96 and detailed theory was given by Blinowski and Mycielski and
Mott.97 The optical conductivity, <7i(u;), due to resonance absorption in the impurity
band has a maximum at
hu>.
max ~
,
r
un
(57)
where is acceptor concentration, e ie dielectric constant and ru is the average
tunneling distance for pairs of localized states contributing to <7i(u) at frequency ui.
The distance ru is found from the relation
r'u = a \n(2I0/hu>),
(58)
where a is the localized length and I0 is a prefactor of the overlap integral I
I(r) = 70exp(-r/a). (59)
Using (57), 2/0 ~ e2/2ea and a = 8 ~ 5 x 1018 cm-3 from Ref. 98 we obtain
~ 133 cm-1. For the two samples in Fig. 54, the absorption maximum increases
in magnitude and shifts towards higher frequencies with increasing N. This behavior
is in accord with (57). The half-widths of the absorption curves become larger with
increasing Na, and most of this increase in half-width occurs on the high frequency
side of the maximum. The resonant absorption occurs at phonon energies which
axe much smaller than the 35 meV thermal ionization energy of the impurity atoms
estimated from the variation of the dc conductivity and Hall coefficient98 above ~ 50
K. This implies that the resonant absorption occurs at such small photon energies
and low temperatures, where the usual bulk absorption mechanisms axe absent. The
solid lines in Fig. 54 show the curves obtained from Lorentz model.

109
In this connection, we would like to note that at u = uimax the separation ru
is approximately equal to 2.5a for 3 % Sr doped sample and 2.7a for 1 % Sr doped
sample. It is of the same order for all other experimental data available.
Optical Excitation of Infrared Bands
In this section, we axe concerned about three infrared absorption bands, which
are considerably different from those of usual doped semiconductors, a narrow band
at 1 200 cm-1 (0.15 eV) and a broad band at 4 050 cm-1 (0.5 eV), and small peale
near 1.4 eV. This strong infrared absorption suggests the importance of electron-
phonon coupling, or other electronic mechanisms in high Tc materials. With further
doping the lower two bands merge with the low frequency free carrier absorption,
and are identified as the mid-infrared band that has been observed in several other
cuprate superconductors. Figure 55 shows the reflectance spectra R(u) (a) and the
conductivity spectra <7\(u) (b) after Kramers-Kronig transformations of R(oj) at sev
eral temperatures for 2% Sr doped La2-xSrxCu04. Two absorption peales are clearly
observed near 0.15 eV and 0.5 eV. (Ti(ui) in Fig. 55(b) also show that the 0.15 eV
band has strong temperature dependence, and the 0.5 eV peak has little temperature
dependence, but the origin of 0.5 eV band is obscure.
First, we are are interested in the 0.15 eV band, because this band appears
(0.12 eV at 10 K and 0.15 eV at 300 K) near the antiferromagnetic exchange en
ergy J calculated from Raman experiments." This band also appears in the t-J
model15,18,72100-102 which is not related with charge excitations. According to the
t-J model, this band is caused by spin fluctuations around the doped hole, and the
energy scale of magnetic interaction is the order of the exchange constant J ~ 0.1
eV. So, one possibility for the origin of this band is the magnetic interaction between
the carriers and the antiferromagnetic spin order.

no
The mid-infrared conductivity, Fig. 55(b) is composed of two parts,
an(v) (0.15 eV band) and <712(0;) (0.5 eV band). In order to obtain the 0.15 eV
band, first, we have fit the criMID(u) at each temperature using a dielectric function
model for two Lorentz oscillator. The formula is
j=l J
luJ1j
(60)
where u>pj, u>j and 7j correspond to the intensity, center frequency and band width
of each band, respectively. Once values for up2, u>2, and 72 of the 0.5 eV band
axe obtained, they can be used to calculate the Lorentzian spectrum of the 0.5 eV
band, crj^o;). Next, we can obtain the 0.15 eV band after subtracting <7^(u;) from
<7imid{u) in Fig. 55. The inset in Fig. 55 shows the 0.15 eV band.
The upper two panels of Fig. 56 show the temperature dependence of center
frequency u\ and band width 71 for the 0.15 eV. The results shown in Fig. 56 indicate
that with increasing temperature the center frequency increases, and peak position
shift to higher frequency with an amount comparable to the thermal fluctuation
energy k^T of lattice. The behavior of peak position can be described by linearly
varying function:
ue ~ J + ksT, (61)
where kg = 0.695 cm_1/K- This implies that if this band arises from a transition
between states related to the Cu-Cu exchange energy J, the thermodynamic limit
affects the numerical value of peale position J. The behavior of linewidth broadening
with increasing temperature is similar to that of free carriers. The line widths have
a linear temperature dependence, 7(T) ~ 7(0) + 1.5 kgT. A fit of the form h/r =
2it\kBT -f Ti/tq yields a value for the coupling constant A ~ 0.24. However, the
linewidth at each temperature is very broad (260 meV at 300 K).

Ill
The lowest panel in Fig. 56 shows that the oscillator strength for 0.15 eV band
decreases with increasing temperature. This temperature dependence reflects that
other processes may substantially contribute to the this band. The temperature
dependence of the oscillator strength may be described by a polaxonic effect,103 in
which carriers move nonadiabatically with respect to the lattice. In present analysis,
we have also suggested the charge dynamic of doped carriers, namely, the large value
of the static dielectric function, and the carrier-lattice interaction by the deformation
potential etc.Thus, in my opinion, the magnetic polaron which we mentioned above
is also likely have a lattice component, and hence the spin and lattice excitations are
very important at low doping levels of high Tc cuprates.
In Fig. 57, the peak observed near 1.4 eV below the the charge transfer excitation
band is a result of the excitonic effect. This peak is not observed in undoped sample
and in T phase. Suzki et a/.104 have shown that its strength increases with Sr doping
and Uchida et al.have observed this peak at high doping levels. In excitonic model, a
charge transfer excitations from the Cu to the 0 site create the free electrons on Cu
and the free holes on 0 site. An short range attractive interaction (Up) between
them results in the creation of exitons.
Figure 58 shows a fitting curve for the fitting curve is obtained by using the Lorentz model discussed in Chapter V. The
individual contributions of the Lorentzian include the phonon bands, the hopping
conductivity and the mid-infrared bands.
Summary
We have shown that the Cu6 octahedra rotates around [110] axis with decreas
ing temperature, and the deformation potential caused by the tilting of Cu02 plane
enhances the carrier-phonon interaction. As doping proceeds, the oscillator strength

112
of phonons is reduced due to the screening by free carriers, and instead doping in
duced modes (Raman active modes, carrier-lattice interaction mode) appear. The
infrared active phonons near 135, 164 and 360 cm-1 observed in the high tempera
ture tetragonal phase of La2Cu04 split into a B^u + B$u pair in the orthorhombic
phase. This splitting develops upon cooling.
The qualitative features of the far-infrared cr\{u) spectra and the transport prop
erty in the low Sr doped La2-xSrzCu04 system is similar to the behavior of a conven
tional 2D disordered metal. In the high temperature region (> 100 K), all impurities
axe ionized and metallic behavior is observed. At lower temperatures (< 100 K),
the freezeout of hole occurs and hence the conductivity results from the thermal
ionization of the shallow impurities. At sufficiently low temperatures, all states axe
localized near the Fermi energy and we expect the phenomenon of variable range
hopping, where a hole just below the Fermi level jumps to a state just above it.
Our experiment suggests another mechanism of hopping, namely, the photon-
induced transitions and their effect on the absorption of electromagnetic radiation
in low doped La2_xSrzCu04 system. In this model, the lowest two energy levels
correspond to a localization of an electron on either one or the other of the donors. We
observe a resonant absorption in the <7i(u;) spectrum at 10 K whose maximum, u)max,
occurs at frequencies between 100 cm-1 and 130 cm-1, depending on Sr concentration.
Finally, we discuss infrared absorption bands observed near 0.15 eV, 0.5 eV and
1.4 eV. The behavior of 0.15 eV band which is observed near the antiferromagnetic
exchange energy J is similar to that of free carriers and this band also seems to have
a lattice component. The origin of 0.5 eV band is still obscure, and the peak near 1.4
eV looks like a result of the excitonic effect.

Reflectance
113
Photon Energy (eV)
0.01 0.1 1
Fig. 42. Room temperature reflectance R spectra of La2-rSrxCu04+f (0.01 <
x, 6 < 0.04) on a-b plane.

114
Photon Energy (eV)
0.01 0.1 1
Fig. 43. Room temperature u\[u>) spectra, obtained after K-K transformation
of R in Fig. 42.

115
Photon Energy (eV)
0.01 0.1 1
Fig. 44. Real paxt of the dielectric function f] as a function of u at room
temperature.

Reflectance
116
Fig. 45. Temperature dependence of the c-axis reflectance in
Lai.97Sro.o3Cu04. Inset: high frequency reflectance at room tempera
ture.

(-1/e)
117
i/(cm 1)
Fig. 46. Far-infrared (a) a\(u) and (b) Im(-l/e) along the c-axis for
Lai.9TSro.o3Cu04. Peaks correspond to (a) TO phonons (b) LO phonons.
Inset shows high frequency conductivity.

118
b
Fig. 47. Atomic positions in the orthorhombic La2_*SrxCu04 unit cell which
consists of two primitive cells. The hatched circles (a-5) are La atoms,
the filled circles (A and B) Cu atoms, and the open circles (1-8) 0 atoms.

cr(n cm) Reflectance
119
0 100 200 300 400 500 600 700
i/(cm_1)
Fig. 48. Fax-infrared (a) reflectance and (b) conductivity ai(u>) for
Lai.98Sro.o2Cu04 at several temperatures.

cr(Q cm) Reflectance
120
0 100 200 300 400 500 600 700
y(cm_1)
Fig. 49. Far-infrared (a) reflectance and (b) conductivity ai(u) for
Lai.99Sro.oiCu04+£ at several temperatures.

or(fi cm) Reflectance
121
0 100 200 300 400 500 600 700
y(crrf1)
Fig. 50. Far-infrared (a) reflectance and (b) conductivity cri(u;) for
Lai.97Sro.o3Cu04 at several temperatures.

Phonon frequency
122
Temperature (K)
Fig. 51. In-plane phonon frequencies as a function of temperature.

123
13
I
O
x
o
D
-
O
V)
O
Temperature (K)
Fig. 52. Oscillator strength of in-plane phonons and Raman modes at 247
and 278 cm-1 as a function of temperature.

Resistivity (fi
124
0.10
0.08
E
o
0.06
0.04
0.02
0.00
\
\
*-a1.99^0.01 Cu04+<5
\
i'll I I I I I I I I 1 1 I I 1 I I I I I I I I I I I L.
50 100 150 200 250 300
Temperature (K)
Fig. 53. Temperature dependence of the in-plane resistivity p for
Lai.9gSro.oiCu04+5. Inset shows Inp vs. (1/T)1/2.

125
Fig. 54. Far-infrared conductivity spectra at 10 K for 2 % and 3 % Sr doped
La2_xSrxCu04. Solid lines show fitting curves from Lorentz model.

cr(ficm)-1 Reflectance
126
y(cm_1)
Fig. 55. (a) The a-b plane reflectance spectra at temperatures between 10 K
and 300 K for Lai.ggSro.ircCuO^ (b) The real part of the a-b plane con
ductivity <7i(u;) derived from the reflectance spectra R in (a). Inset is the
temperature dependent conductivity of the 0.15 eV band after subtracting
the 0.5 eV band from the data (b).

127
^ 1200 Center frequency
E
o

E
o
1050
900
2000
E
o

Q.
1750
3200
2950
2700
Band width
0
Spectral weight
i
o
T

0
_L
100 200 300
Temperature (K)
Fig. 56. The center frequency, width, spectral weight extracted from
Lorentzian fits for the 0.15 eV band as a function of temperature.

128
Photon Energy (eV)
0.8 1.2 1.6
Fig. 57. The <7i(u;) spectra near 1.4 eV for La2-xSrxCuC>4+.

129
Photon Energy (eV)
0.01 0.1 1
Fig. 58. The Lorentzian fitting curve for the <7i(u;) at 10 K of
La1.97Sro.03 CUO4.

CHAPTER VIII
INFRARED PROPERTIES OF BI-0 SUPERCONDUCTORS
The insulating phase BaBiOs of the bismuthate superconductors, Bai_zKxBi03
(BKBO) and BaPbi-xBijOj (BPBO) exhibits a monoclinical distortion of the per-
ovskite lattice.105,106 Originally attributed to Bi charge ordering (between Bi3+ and
Bi5+), this distortion corresponds to a charge density wave (CDW) instability, open
ing a semiconducting gap at the Fermi level. Doping with K and Pb, on the Ba or
Bi site, respectively, reduces the semiconducting gap and leads to the superconduct
ing state with maximum Tc's of about 31 K and 13 K, respectively. The maximum
Tc occurs neax the composition of the metal-insulator transition. Compaxed to the
conventional BCS and the Cu-0 high Tc superconductors, the bismuthate supercon
ductors have unique properties: (i) Like the layered copper-oxide superconductors,
the bismuthates have a high transition temperatures in spite of a low density of states
at the Fermi level, (ii) Unlike the Cu-0 materials, the insulating phase originates from
the CDW state and is nonmagnetic, (iii) The conduction properties in the normal
state as well as the superconductivity is isotropic, (iv) The maximum Tc is observed
neax the metal-insulator transition.
In order to clarify the mechanism of the superconductivity in the bismuthates,
it is essential to investigate the physical properties in both the superconducting and
normal states. Nevertheless, their normal state and superconducting state properties
so fax axe not well understood.
In this chapter we present an extensive study of the optical properties on BKBO
and BPBO crystals. We first analyze optical reflectance in a frequency range from 30
130

131
cm-1 to 45 000 cm-1 and temperature range from 10 K to 300 K. We subsequently
study the optical conductivity in the normal state and in the superconducting state.
We find a BCS-like gap feature and estimate the electron-phonon coupling constant.
We discuss the infraxed conductivity in the framework of Mattis-Bardeen theory and
finally the the electron pairing mechanism in BKBO.
Samples we have measured are in the K and Pb concentrations with neax max
imum Tc. The sample surface (2 mmx2 mm) of BPBO is very shiny, but that of
BKBO is rough. It is also known that the surface of BKBO is easily degraded in air.
For the optical measurement, the sample surface (1.5 mmxl.5 mm) of BKBO was
mechanically polished using AI2O3 power of 0.05 fim in size. The color was blue after
polishing. To correct the surface roughness, we evaporated an Ag film on the sample
surface, which produces an Ag surface with a roughness comparable to that of the
sample. We again measured the reflectance of an Ag coated sample and obtained
an estimate for the absolute reflectance of BKBO from dividing the sample surface
reflectance by that of an Ag coated sample. The measurements were made for three
different crystals. We have also measured the reflectance before polishing to examine
surface degradation effects due to atmospheric exposure. Our results show that the
degradation of the surface does not affect the fax-infrared spectrum, t.e., the infraxed
gap measurement, because the fax-infraxed light penetrates deeply into the surface.
However, the spectrum in the near-infrared and visible is greatly changed.
Normal State Properties
Results for Bai-TKTBiO^
Figure 59 shows the optical reflectance for Bai_xKxBi03 (BKBO) at several tem
peratures in the frequency range from 30 cm-1 to 40 000 cm-1. As the temperature

132
decreases from 300 K to 30 K, the infraxed reflectance up to 5000 cm-1 nonlinearly
increases and the reflectance spectra axe exchanged axound the plasma minimum near
14 800 cm-1.
The temperature dependent optical conductivity cri(cj) derived from a Kramers-
Kronig analysis of the reflectance spectra in Fig. 59 are plotted in Fig. 60 up to 30 000
cm-1. The conductivity curves at each temperature show a prominent deviation from
the Drude curve. As the temperature is reduced, the far-infrared conductivity rapidly
increases with increasing temperature while the mid-infrared and high frequency con
ductivities decrease, as expected from the / sum rule. We observe a prominent peak
neax 4 000 cm-1 with a width of about 8 000 cm-1 and a oscillator strength of 20 500
cm-1 at room temperature. This peak is associated with the transitions across the
CDW gap.
Results for BaPbi_TBiTQt
The upper and lower panels in Fig. 61 show the room temperature (a) reflectance
and (b) conductivity for BaPbi_zBiz03 (BPBO). The reflectance and conductivity
spectra of BPBO axe very similar to those of BKBO. The reflectance spectrum shows
a metallic character and has a reflectance minimum axound 12 800 cm-1. Like BKBO,
the conductivity spectrum shows the non-Drude behavior due to a strong CDW band.
However, for BPBO this peak is quite a bit sharper and the gap is narrower than in
BKBO. This is indicative of a high degree of nesting of the band neax the Fermi
surface. On the other hand, the Drude strength is very low, making it difficult to
define the Drude component in the c\{u) spectrum. This trend is likely related to
the formation of Pb-related states within the CDW gap as was observed in the dop
ing dependent experiment of BPBO by Tajima et al.33 They have shown that when
Bi is partially doped with Pb, the CDW band is gradually broadened and shifts to

133
lower frequencies forming a low frequency tail due to Pb states in the CDW gap.
According to the band structure results,107,108 the undistorted cubic BaPbi_xBix03
alloy possesses a single broad conduction band, which involves (7-antibonding com
binations of Pb-Bi(6s) and 0(2p) states. This suggests that the low energy spectral
weight in BPBO is affected by both band filling and the Pb-related states and is very
complicated.
Figure 62 displays the far-infrared reflectance and conductivity for BPBO. The
reflectance spectra in Fig. 62(a) have small temperature dependence at tempera
tures between 10 K and 300 K. As the temperature decreases, is strongly reduced below ~ 80 cm-1, consistent with the temperature dependence
of the resistivity109 which shows a semiconducting behavior. On the other hand, the
(7i(w) between 80 cm-1 and 500 cm-1 slightly increases with decreasing temperature
to compensate for the decreased oscillator strength below ~ 80 cm-1.
In Fig. 62(b), four optical phonons are cleaxly observed at 102, 171, 222 and
541 cm-1, showing a redshift on heating. Comparing with the phonons110 in the
insulating phase of BaBiOa observed at 97, 137, 230 and 441 cm-1, the frequencies
of three phonons centered at 102, 171 and 541 cm-1 increase with Pb concentration
while the phonon mode at 230 cm-1, which is assigned to a Bi-0 bending mode,
shows the softening of phonon. In contrast, for BKBO only one broad phonon peak
neax 200 cm-1 is observed due to the screening from the free caxriers.
Comparison of Two Bismuthate Spectra
There axe some differences between the two bismuthate spectra. First, the Drude
band in BPBO is weaker and the phonon features axe more distinct than in BKBO.
Second, the CDW band in BPBO is quite a bit sharper and the gap energy is lower
than in BKBO. Third, for BKBO the minimum of <7i(u;) neax 15 000 cm-1 is more

134
prominent. The results of the sum rule explain the different properties of the two
samples well. Figure 63 illustrates at room temperature for the two samples.
For both samples the initial slow rise of is due to the Drude contribution and
then suddenly increases in the CDW band frequency up to 12 000 cm-1. The
contribution of the next steep rise comes from the CDW band. The BPBO spectrum
shows a weak Drude and strong CDW contribution compared with that of BKBO.
N^ff in Fig. 63 also exhibits a plateau neax 15 000 cm-1. This plateau corresponds
to a prominent minimum in the (cj) spectrum and the plasma minimum in the
reflectance spectrum. This trend is peculiar to BKBO. This result suggests that for
BKBO the low frequency excitations neax the Fermi level axe well sepaxated from
other excitations compaxed to the case of BPBO.
In optical studies of the doping dependence in BPBO,111,112 when Pb is substi
tuted into an active Bi site, the CDW band is gradually broadened and shifts to
lower frequencies. In this picture, the CDW gap persists as a pseudogap even in the
orthorhombic metallic phase for Pb concentrations between 15% and 35%. However,
the case of K doping is somewhat different.113-115 When monovalent K is doped into
the inactive, divalent Ba site, the CDW absorption band energy decreases much faster
than in BPBO and finally disappears in the metallic phase at x = 0.5. However, the
CDW band still persists at the metal-insulator transition composition (x = 0.37),
where there is a phase transition between the cubic and the orthorhombic struc
tures.28 Also, the measurements of the Hall and Seebecks effects113,116 for metallic
BKBO and BPBO show that the carriers axe electrons. Our results for the Drude
strength of both samples are consistent with the above view that BKBO is much
closer to half-filling of the Bi-0 conduction than in BPBO.

135
While there is originally no CDW band in an ideal cubic perovskite, most infraxed
measurements for BKBO show that the CDW band persists in the metallic phase ad
jacent to the phase transition into the semiconducting phase. The persistence of the
CDW gap is also supported by Raman experiments,117,118 where the breathing mode
phonon at 570 cm-1 exists as a small peak at a composition of x = 0.38 and finally
disappears at a composition of x = 0.45. In addition, Pei et a/.119 using electron
diffraction have reported that BKBO has an incommensurate structural modulation
which is responsible for a partially insulating property in the metallic phase, and
suggested that it may be an incommensurate CDW. However, the persistence of local
CDW order may not be an intrinsic property. The local CDW order may be associ
ated with the inhomogeniety of the potassium distribution at the transition between
the orthorhombic and the cubic phases. A sample prepaxation study and the pressure
dependent experiment120 of optical reflectance support this idea. For example, it is
difficult to prepaxe a clean powder, because the potassium rich phases tend to segre
gate at the grain surface, and visible and infrared reflectance of the superconducting
compound near the phase transition anomalously changes with pressure.
Free Carrier Component in BKBO
We have mentioned earlier that the infraxed conductivity in BKBO may be sep
arated into two parts:
cri(u) = an) + (TiCDW, (62)
where cr\£> is the Drude part and c\cdw corresponds to the CDW band. Thus the
free carrier part in the normal state can be easily obtained by subtracting ct\cdw
from the totcil After the ct\d components at each temperature axe obtained,

136
they can be fitted by usual Drude conductivity formula:
1 Idt
&\D 1 o 2
47T 1 +
where upd and r is the plasma frequency and the relaxation rate of the free carriers,
respectively.
Table 3 illustrates the free carrier contribution at several temperatures. The free
carrier contribution at each temperature is in good agreement with the ordinary Drude
behavior. The Drude plasma frequency, upD = 9 500 200 cm-1 obtained from the
above analysis is nearly temperature independent, whereas 1/t in Fig. 64 shows the
non-linearity with temperature. The extrapolated values of cr\{ui) at zero frequency in
Fig. 60 have the same behavior as the temperature dependent 1/t, exhibiting a typical
metallic behavior. This is consistent with the behavior of the normal-state resistivity
of Heilman and Hartford121 on films and Affronte et a/.122 on single crystals. The
non-linearity of the 1/t vs. T is an interesting feature in BKBO in contrast to the
linear temperature dependence observed in the resistivity in high Tc cuprates.
Table 3. Fitting parameters of the Drude conduc
tivity.
T(K)
upd (cm-1)
1/t
300
9700 200
700 20
200
9500 200
450 20
100
9400 200
350 20
30
9400 200
300 20
The normal-state resistivity of films and single crystals of the BKBO system is
interpreted with metallic or semiconductor-like, depending on the room-temperature

137
resistivity. The extrapolated value of a\{u) at u = 0 (2 300 (flcm)-1 at 300 K, p =
430 /xil cm) corresponds to one of the lowest values of room-temperature resistivity
which usually ranges between ~ 200 3200 /zficm. Sato et a/.123 demonstrated
that in a sample with a room temperature resistivity lower than 700 [iti cm, the
temperature dependence of the resistivity is metallic with a positive temperature
coefficient and the room temperature resistivity of the semiconducting phase samples
exceeds 700 /fl cm, where the 700 ¡J.CL cm value corresponds to the mean free path
equal to an interatomic spacing of a = 4 A dc resistivity ratio p(300 K)/p(30 K)
of about 2.25 obtained from simple Drude formula,
_ uId
Pdc 4tt(1/t)c
is also consistent with their resistivity ratios p(300 K)/p(30 K) as high as 1.9 ~ 2.6
for the most metallic samples. Thus our data axe essentially in good agreement with
the temperature dependence of the dc resistivity.
Talcing the Fermi velocity to be Uf=108 cm/s from the band calculation124 and
using the relaxation rate (700 cm-1 at 300 K and 300 cm-1 at 30 K), we obtain a
mean free path i vpr ~ 72 at 300 K and 168 at 30 K. The mean free path is
longer than the interatomic spacing a (~ 4 ). The resistivity is expected to saturate
if i < a because the mean free path can no longer be properly defined. The mean
free path for the BKBO sample with a high resistivity value and BPBO samples is
supposed to be close to the lattice constant 4 from the fact that this material shows
semiconductor behavior with a negative temperature coefficient.
We can estimate the electron-phonon coupling constant from the simple model
for the electron-phonon scattering rate, h/r = 2irXk^T.125 This formula applies for T
> 0£>. We obtain a moderate-coupling value for the coupling constant A ~ 0.6 at 300

138
K. There is a little ambiguity in this estimation due to the non-linearity of 1/r with
temperature. Nevertheless, the estimation of A seems to be consistent with the gap
measurement and numerical calculations,126-128 where A is suggested to be around 1
in order to explain the conventional electron-phonon mechanism. Thus the normal
state properties may suggest that BKBO is a BCS-like superconductor in which the
electron-phonon interaction plays a significant role.
Superconducting State Properties
Superconducting Gap
In the conventional BCS theory, a bulk superconductor at temperatures below Tc
is a perfect reflector of electromagnetic energy at frequencies below 2A. Above 2A its
behavior is similar to that of a normal metal. In infrared reflectance measurements,
the original inference of the superconducting gap was based on the measurement of
the ratio Rs/Rn of the reflectance in the superconducting state to that in the normal
state. Another case, the superconducting to normal ratio for transmission129,130 shows
a maximum near 2A. According to Glover and Tinkham,131 the superconducting gap
can also be obtained from a\3 in the superconducting state, which has a gap, 2A, at
the threshold for pair excitations. In this section, we examine both the question of
determining a frequency at which the absolute reflectance reaches 100%, and possible
evidence for a BCS size gap in a BKBO crystal.
We have measured the superconducting state reflectance in BKBO. Fig. 65(a)
shows the far-infrared reflectance at various temperatures. This figure illustrates that
in the normal state, BKBO has a very high far-infrared reflectance, characteristic of
free carriers as expected from the metallic dc resistivity. In the superconducting state
(at 10 K and 19 K), a strong edge appears. The positions of the edge have a strong

138
K. There is a little ambiguity in this estimation due to the non-linearity of 1 /r with
temperature. Nevertheless, the estimation of A seems to be consistent with the gap
measurement and numerical calculations,126-128 where A is suggested to be around 1
in order to explain the conventional electron-phonon mechanism. Thus the normal
state properties may suggest that BKBO is a BCS-like superconductor in which the
electron-phonon interaction plays a significant role.
Superconducting State Properties
Superconducting Gap
In the conventional BCS theory, a bulk superconductor at temperatures below Tc
is a perfect reflector of electromagnetic energy at frequencies below 2A. Above 2A its
behavior is similar to that of a normal metal. In infrared reflectance measurements,
the original inference of the superconducting gap was based on the measurement of
the ratio Rs/Rn of the reflectance in the superconducting state to that in the normal
state. Another case, the superconducting to normal ratio for transmission129,130 shows
a maximum near 2A. According to Glover and Tinkham,131 the superconducting gap
can also be obtained from a\3 in the superconducting state, which has a gap, 2A, at
the threshold for pair excitations. In this section, we examine both the question of
determining a frequency at which the absolute reflectance reaches 100%, and possible
evidence for a BCS size gap in a BKBO crystal.
We have measured the superconducting state reflectance in BKBO. Fig. 65(a)
shows the far-infrared reflectance at various temperatures. This figure illustrates that
in the normal state, BKBO has a very high far-infrared reflectance, characteristic of
free caxriers as expected from the metallic dc resistivity. In the superconducting state
(at 10 K and 19 K), a strong edge appears. The positions of the edge have a strong

139
temperature dependence which is suggestive of a BCS-like energy gap in the dirty
limit. The spectrum at 10 K consists of a clear peak near 54 cm-1 and a broad dip
axound 100 cm-1, becoming equal to the reflectance at 30 K near about 250 cm-1.
The spectrum at 19 K also has a peale at frequency less than 54 cm-1.
There is no evidence of unity reflectance in the superconducting state that could
be used to identify the energy gap. The reflectance is high at low frequency, but there
is a residual absorption of the order of 7 ~ 8% in the 20~50 cm-1 frequency range.
This is reproducible for three different samples in the same batch. We cant rule out
the existence of the residual absorption in our sample even with the experimental
error of 2%. We expect that a residual absorption is caused by the compositional
inhomogeniety which is partially composed of insulating K concentrations around the
metal-insulator transition. Hence, the residual absorption indicates the normal state
characteristic in the superconducting state. It is unlikely that the residual absorption
is associated with the surface degradation, because the infrared light penetrates deeply
into the sample. This inhomogeneity is also consistent with the persistence of the
CDW band in the mid-infrared region.
The ratio Rs(u)/Rn(u) in Fig. 66 at the peak position is very small (2.5%) com
pared with the other infrared gap measurements, exhibiting a more metallic nature
in the normal state. A shallow broad minimum around 100 cm-1 in Rs
suggests that the BKBO system is a weak or moderate coupling superconductor.
The Kramers-Kronig analysis gives a more detailed picture showing an energy
gap similar to a BCS-like superconductor. Figure 65(b) shows the far-infrared con
ductivity. In Fig. 65(b), the minimum of the conductivity in the superconducting
state moves to higher frequency as the temperature is reduced, indicating the open
ing of a superconducting gap and following the BCS-like model for a superconducting

140
energy gap. At T = 10 K, the conductivity spectrum has a minimum at 54 cm-1
and begins to rise up to 120 cm-1 due to photo-excited quasipaxticle absorption. The
difference between cr\n{ strength below 2A (missing area) shifts to the origin to form superconducting con
densate. Theoretically, at T = 0 K, <7ia(u>)=0 up to u=2A. However, our results
show that below 2A the uncertainty in 100% uncertainty for the reflectance spectrum in the superconducting state.
We obtained 2A/kpTc = 3.5 0.3 using 2A = 54 cm-1 and Tc = 22 K, which is
consistent with a weak-or moderate-coupling limit, where 0.3 corresponds to 10%
and 90% value of dc susceptibility for Tc measurement. This value for the energy
gap is in good agreement with the tunneling spectroscopy results of Sharifi et a/.132
for samples from same batch. The observed value for the energy gap is also consis
tent with the results (3.5 ~ 3.8) observed from other tunnelling spectroscopy,133-137
measurements of oxygen isotope effects,138-140 and infrared measurement.141
Superconducting Condensate
We have shown earlier that for the Drude carriers the scattering rate is ~ 300
cm-1 and the mean free path is ~ 168 at 30 K. We can also calculate the Pippard
coherence length ( = hvp/2n2A = 590 using the Fermi velocity vp = 108 cm/s
and 2A = 54 cm-1. The results suggest that BKBO is a dirty limit superconductor,
exhibiting l < £ and 1/r 2A.
In the dirty limit, much of the free carrier conductivity exists at frequencies above
2A and a small part of the Drude strength contributes to the superfluid condensate.
One might expect that from the sum-rule argument the missing area A under the
integral of cti3 appears at u = 0 as A<5(u;). This missing area is easily estimated by

141
inspection of Fig. 65(b). Thus, the missing area can be estimated from
A = ~ JQ lai(w) ~ (65)
where ups 47rnse2/m is the superconducting plasma frequency. We obtain ujp3 =
3 750 200 cm-1 and 3 000 200 cm-1 at 10 K and 19 K, respectively. The superfluid
condensate fraction is also estimated according to the formula:
na(T) u>l,{T)
n Id
We find fs(T) = 16% and 10% at 10 K and 19 K, respectively, using wp£> = 9400
cm-1 in the normal state.
The area in the 6 function, A, controls the low frequency electromagnetic pene
tration depth. The London penetration depth, X is related to the plasma frequency
in the superconducting state by Xi = 1/2tuP3. We find 4250 100 at 10 K.
This value is somewhat higher than that obtained using microwave methods,142 but
smaller than the results from other infrared measurements.143,144 Since the response
of a superconductor to an electromagnetic field is governed by the imaginary part of
the optical conductivity, can be also determined using the formula:
( (12)
Fig. 67 displays as a function of frequency. For uj < 2A, we expect that X is
independent of uj due to the superfluid response which is cr2 oc l/u, and the zero
frequency extrapolated value is similar to above result. For 2A < u> < 100 cm-1, X
increases with w, corresponding to <72 which falls to zero more rapidly than l/u>.

142
The solid lines in Fig. 68 illustrate fitting curves using the standard Mattis-
Bardeen conductivity, based on the dirty limit, weak coupling BCS theory. Fitting
curves axe produced using the Drude formula at 30 K and the Mattis-Bardeen formula
with an energy gap of 54 cm-1, a scattering rate of 300 cm-1 and Tc = 22 K.
Our data axe very well represented by the weak coupling Mattis-Bardeen theory at
low frequencies (uj < 250 cm-1), showing the good agreement of the temperature
dependence below 2A. There is no feature corresponding to strong electron-phonon
coupling in this frequency range. At frequencies between 2A and 150 cm-1 the
measured spectrum at 10 K is less sharply increasing them the calculated spectrum.
Therefore, the present infrared measurements indicate that the BKBO system is a
dirty limit and weak or moderate coupling superconductor with a BCS-like gap and
coupling constant A (V 0.6.
Discussion of Pairing Mechanism in BKBO
So fax, an unsolved question is whether the origin or mechanism of superconduc
tivity in the BKBO system is the usual phonon mechanism or not.145 The BKBO
system does not contain any transition metal element. Hence, a magnetic mechanism
may not be expected for the superconductivity in this compound. In fact, no mag
netic order has been observed in BKBO by muon spin rotation experiments,146 and
the magnetic susceptibility in the normal state in BKBO shows a Pauli paramagnetic
behavior.147 Therefore, we may expect that the superconductivity in BKBO occurs
via the phonon mechanism.
If the weak or moderate coupling mechanism in this high Tc compound is pre
dominantly phonon mediated, then coupling to high frequency phonons is required.
The inelastic neutron scattering and moleculax dynamics simulation studies of Loong
et a/.128 suggested that the oxygen phonon modes soften by ~ 5 to 10 meV with

143
40% K doping of BaBi03 and that the strongest phonon features in superconducting
BKBO occur between 30 and 70 meV (250 and 570 cm-1). The strong features at
these energies axe also observed in the second derivative of the tunneling current and
in the inverted a2F(uj). In addition, Shirai et a/.126 have found that the mode around
60 meV is due to the oxygen stretching vibration towards the nearest neighboring Bi
atoms and the electron-lattice interaction causes remarkable renormalization of the
longitudinal oxygen stretching and breathing mode axound 60 meV. They also find
that for a fixed value of A = 1, Tc ~ 30 K is obtained using the effective Coulomb
repulsion ¡x* ~ 0.1 for a reasonable description of the superconducting properties of
BKBO and a laxge electron-phonon matrix elements from coupling to the high energy
phonons.
Unfortunately, we did not observe the phonon structure nor any sign of the
electron-phonon interaction in this frequency range. Instead, our result in Fig. 65(b)
shows the phonon peale neax 200 cm-1 which is assigned to the bending mode phonon
in the undoped BaBi03. The shape and position of the 200 cm-1 peak do not change
with decreasing temperature, indicating no Holstein mechanism which would axise
from a reasonably strong electron-phonon coupling. However, recently published
infrared measurements143 show that the phonon shape at about 500 cm-1 for the
stretching mode becomes increasingly asymmetric with decreasing temperature. Sim
ilar results have been presented for the breathing mode phonon in Raman exper
iments.148 In my opinion, the breathing mode is not likely to be related to the
pairing mechanism due to the fact that the superconductivity occurs in the cubic
phase 0.37 < x < 0.5 as seen in the phase diagram by Pei et a/.28, but the breath
ing mode only persists to neax the orthorhombic-cubic phase transition. However,
most attempts to find these phonon structures in tunneling spectroscopy, neutron

144
scattering experiments, and infrared measurements have not been successful. This
can be attributed to the broadening of the phonon linewidth because of the strong
electron-phonon interaction. Also, we do not rule out the possibility of the pairing
mechanism being associated with other electronic excitations.
Summary
In this chapter, we have examined the problems associated with the normal state
and superconducting state properties of Bai.^KxBiOs (BKBO) and BaPbi_zBix03
(BPBO) single crystals. In the normal state, broad bands in the infrared conductivi
ties are observed at about 2500 and 4 000 cm-1 for BPBO and BKBO, respectively,
as shown in the doping dependence experiments. These bands are associated with
transitions across the charge density wave (CDW) gap. For BKBO, the 1/r associ
ated with the Drude band shows a non-linearity with temperature consistent with
the transport measurements, whereas for BPBO the extrapolated values of cr\(tjj) at
zero frequency show a semiconducting behavior. For BKBO the electron-phonon cou
pling constant A ~ 0.6 is estimated from the simple model for the electron-phonon
scattering rate.
In the superconducting state, the positions of the superconducting edge in the far-
infrared reflectance spectra have a strong temperature dependence which is suggestive
of a BCS-like energy gap. The energy gap of 2A/kgTc = 3.5 0.3 is obtained from
the crla(u;) spectrum at 10 K and is consistent with a weak-or moderate-coupling
limit. Our results show that the BKBO system is a dirty limit superconductor with
a superfluid condensate fraction of 16% at 10 K and 10% at 19 K. The London
penetration depth A is also estimated to be 4 250 100 .
The far-infrared conductivity spectra are very well represented by the standard
Mattis-Bardeen conductivity based on a weak coupling BCS theory at frequencies less

144
scattering experiments, and infrared measurements have not been successful. This
can be attributed to the broadening of the phonon linewidth because of the strong
electron-phonon interaction. Also, we do not rule out the possibility of the pairing
mechanism being associated with other electronic excitations.
Summary
In this chapter, we have examined the problems associated with the normal state
and superconducting state properties of Bai-jKxBiOs (BKBO) and BaPbi_xBix03
(BPBO) single crystals. In the normal state, broad bands in the infrared conductivi
ties are observed at about 2500 and 4 000 cm-1 for BPBO and BKBO, respectively,
as shown in the doping dependence experiments. These bands axe associated with
transitions across the charge density wave (CDW) gap. For BKBO, the 1/r associ
ated with the Drude band shows a non-lineaxity with temperature consistent with
the transport measurements, whereas for BPBO the extrapolated values of ) at
zero frequency show a semiconducting behavior. For BKBO the electron-phonon cou
pling constant A ~ 0.6 is estimated from the simple model for the electron-phonon
scattering rate.
In the superconducting state, the positions of the superconducting edge in the fax-
infrared reflectance spectra have a strong temperature dependence which is suggestive
of a BCS-like energy gap. The energy gap of 2A/fcgTc = 3.5 0.3 is obtained from
the oq,,(u>) spectrum at 10 K and is consistent with a weak-or moderate-coupling
limit. Our results show that the BKBO system is a dirty limit superconductor with
a superfluid condensate fraction of 16% at 10 K and 10% at 19 K. The London
penetration depth A is also estimated to be 4 250 100 .
The far-infrared conductivity spectra axe very well represented by the standard
Mattis-Baxdeen conductivity based on a weak coupling BCS theory at frequencies less

145
than 250 cm-1. Finally, we conclude that the BKBO system is a dirty limit and weak-
or moderate-coupling BCS-like superconductor and the high Tc may result from an
electron-phonon interaction corresponding to the high energy phonons. Nevertheless
it is still uncleax whether the superconductivity in BKBO can be explained within
the phonon mechanism.

Reflectance
146
Photon Energy (eV)
0.01 0.1 1
Fig. 59. Normal state reflectance i?(u;) of Bai_xKxBi03 at temperatures
between 30 K and 300 K as a function of frequency.

147
Photon Energy (eV)
0.01 0.1 1
Fig. 60. Real paxt of the conductivity, cri(u;), obtained from a Kramers-Kronig
transform of the reflectance in Fig. 59.

Reflectance
148
Photon Energy (eV)
0 10000 20000 30000
i/(cm_1)
Fig. 61. Room temperature reflectance (a) and cri(u;) (b) obtained from the
K-K transformation of reflectance (a) of BaPbi_xBix03.

cr(Qcm) Reflectance
149
0 100 200 300 400 500 600
y(cm_1)
Fig. 62. Far-infrared reflectance (a) and conductivity <7i(u;) (b) of
BaPbi_xBir03 at several temperatures.

150
Photon Energy (eV)
0.01 0.1 1
Fig. 63. Comparison of the effective electron number per unit cell,
Bai_xKxBi03 with that of BaPbi_xBix03.

151
Temperature (K)
Fig. 64. The Dmde scattering rate, 1/r, as a function of temperature. The
1/r is obtained from the fitting parameters of Table 3.

152
0 100 200 300 400 500 600
y(cm-1)
Fig. 65. The fax-infrared reflectance (a) and a\[u) (b) at temperatures be
tween 10 K and 300 K.

30K
153
Fig. 66. The reflectance in the superconducting state at 10 and 19 K divided
by that at 30 K for Bai_xKxBi03.

154
Fig. 67. The London penetration depth as a function of frequency.

155
Fig. 68. The fax-infrared conductivity in the superconducting state and in
the normal state (30 K). The solid lines axe the conductivity calculated
from Mattis-Bardeen theory.

CHAPTER IX
CONCLUSIONS
In this dissertation, three major issues of high Tc superconductors have been ex
tensively studied by doping and temperature dependent reflectance measurements.
First, the low-energy-scale physics at frequencies below the charge transfer band and
the doping mechanism in electron-doped Nd2-xCeICu04 have been intensively stud
ied. Second, the low lying excitations near the Fermi level have been investigated
at low doping levels in hole-doped La2_ISrICu04. Finally, the normal state and
superconducting state properties of Bi-0 superconductors have been observed.
For the first issue, we have demonstrated that electron doping induces a transfer
of spectral weight from the high frequency region above the charge transfer excita
tions to the low frequency region near the Fermi level. However, the low frequency
spectral weight grows greater than 2x with doping x as expected in the Mott-Hubbard
model. The Drude component is very strong and narrow compared to that of hole-
doped La2_zSrICu04, and the extrapolated values of cr\{w) at zero frequency are
in good agreement with the results of dc transport measurements. Our results for
the one component approach indicate that upon doping the quasiparticle interaction
substantially decreases and Im E at high level shows a linearity in w below 0.6 eV
consistent with the MFL model, in which strong quasiparticle damping is assumed.
We have also verified that the Fermi level should be stuck on the bottom of the upper
Hubbard band. Furthermore, for superconducting sample of x = 0.15 the Drude band
is stronger and the quasiparticle interactions are less than in neighboring Ce concen
tration samples. This observation suggests that in the high Tc regime the Fermi level
156

157
lies directly in the Van Hove singularity of the upper Hubbard band and carriers have
a more itinerant property.
Concerning the second issue, we have first found the evidence for carrier-lattice
interaction by a deformation potential caused by the rotation of Cu06 octahedra.
This is characterized by a strong infrared active modes and an appearance of a strong
Ag Raman modes upon cooling. We have also shown that the qualitative features
of the a\(uj) spectra in the far-infrared and the transport property is similar to the
behavior of a conventional 2D disordered system. At low temperatures below 50 K,
the dc resistivity shows the phenomenon of variable range hopping, where all states
are localized near the Fermi energy and a hole just below the Fermi level jumps to a
state just above it. In addition, the a\(u) spectra at 10 K show a resonant absorption
whose maximum occurs at frequencies between 100 and 130 cm-1, depending on Sr
concentration. This absorption results from the photon-induced hopping of charge
carriers between the localized states. Further, we have observed an absorption band
near 0.15 eV, corresponding to the antiferromagnetic energy J, which also seems to
interact with the lattice, and the peak near 1.4 eV is due to a result of an exitonic
effect. Hence, we conclude that charge dynamics as well as spin dynamics is very
important at low doping levels of the high Tc cuprates.
Finally, in Bi-0 superconductor studies, we have found that the cri(u;) spectra for
both BKBO and BPBO have broad peaks, which are associated with the transitions
across the charge density wave energy gap, are observed in the mid-infrared region.
The extrapolated values of a\ (w) at zero frequency are consistent with the dc resistiv
ity measurements, showing metallic and semiconducting behavior, respectively. The
local CDW order for BKBO seems to be associated with the inhomogeniety of the
potassium concentration near the orthorhombic-cubic phase transition. We have also

158
observed the non-linearity of the 1/r vs. T in BKBO. This result should be compared
to the the linear temperature dependence observed in the resistivity in the high Tc
cuprates. Also, the electron-phonon coupling constant A ~ 0.6 is estimated from the
simple model for the electron-phonon scattering rate. In the superconducting state
for BKBO, we have found that the position of reflectance edge has the temperature
dependence. The energy gap of 2A = 3.5 0.3 has been estimated, consistent with
BCS-type mechanism with moderate or weak coupling. This conventional energy gap
contrasts with the case of the high Tc cuprates, in which a superconducting gap is
not identified in the infrared spectrum. We have estimated the superfluid condensate
fraction as 16% at 10 K and 10% at 19 K, and the London penetration depth to be
(4 200 100) Furthermore, the far-infrared <7i(u;) spectra below 250 cm-1 are well
represented by the standard Mattis-Bardeen conductivity based on a weak coupling
BCS theory. Therefore, we conclude that the BKBO system is a weak or moderate
coupling BCS-type superconductor in the dirty limit. The high Tc about ~ 30 K may
result from the interaction between electrons and high energy phonons. However, the
pairing mechanism for Bi-0 superconductors still remains an open question.

APPENDIX
OPTICAL AND TRANSPORT PROPERTIES OF LuNi2B2C
The superconductivity in the class of quaternary compounds LNi2B2C (L = Y,
Tm, Er, Ho, and Lu) up to 16.6 K was recently discovered by Cava et a/.80 We report
here on resistivity and optical measurements for LuNi2B2C with the highest Tc.
LuNi2B2C has the body centered tetragonal structure (14/mmm) with alternating
LuC and Ni2B2 layers.149 However, band structure calculations150 have proven that
this material is fully three dimensional. In addition, it is known that this material
has a low density of states and a strong electron-phonon coupling constant.151
Figure A-l shows the temperature dependence of the resistivity for LuNi2B2C,
showing a transition Tc of 16.5 K and a typical metallic behavior in the normal state.
It should be note that the slope of the resistivity is linearly proportional to the T
at temperatures above 100 K similar to the case of high Tc cuprates. From the high
temperature slope of the resistivity (T 0£>, where 0£> is the Debye temperature),
we can estimate the value for the electron-phonon coupling constant A using the
formula,152
A = 0.246 (hup)2a, (A-l)
where hup is a plasma frequency and a is the slope of the resistivity dp/dT. Using a of
about 0.3 p cm/K, and {hup)2 =(3.7 eV)2 estimated from the optical measurement,
we get a strong electron-phonon constant A ~ 1.05.
159

160
Figure A-2 shows the room temperature reflectance of LUN2B2C. The reflectance
of this system in the infrared region is higher than that in high Tc cuprates. The spec
trum shows a deviation from the Drude curve due to the bands at energies between
0.7 eV and 3 eV.
It is interesting to examine the optical conductivity cr\{ui). The K-K transfor
mation was used to obtain Figure A-3 shows the room temperature optical
conductivity. We have fit cri(u>) with the two component model. We can separate
o 1 (u) into the Drude component below 0.2 eV and several Lorentz oscillators above
0.5 eV. The solid line illustrates the Drude term with a scattering rate of 1100 cm-1
and an oscillator strength of 30 000 cm-1. The dc conductivity value derived from
the fitting parameters using formula pjc = 4ir(jy~)~ is in good agreement with the
result obtained from dc resistivity measurement.
In conclusion, LUN2B2C has the strong Drude plasma frequency of 30 000 cm-1
and quite large damping factor of 1100 cm-1 compared to the high Tc cuprates, and
a strong electron-photon coupling constant of A ~ 1.05.

Resistivity (10
161
Fig. A-l. Temperature dependence of the electrical resistivity for LUN2B2C.

Reflectance
162
Photon Energy (eV)
0.01 0.1 1
Fig. A-2. Room temperature reflectance spectrum for LUN2B2C.

163
Photon Energy (eV)
0.01 0.1 1
Fig. A-3. The optical conductivity a\(u) obtained from the K-K transforma
tion of the reflectance spectrum.

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BIOGRAPHICAL SKETCH
Young-Duck Yoon was born in Seoul, Korea. After completing his undergraduate
course and discharging from military service as a second lieutenant, he then worked
as a researcher at Hyundai Electronic Co., where he took part in the process of
VLSI. During this period, he decided to go abroad for further studies. He started
his graduate studies in physics at the Iowa State University in 1988, where he was
awarded an M.S. degree in Dec. 1990. The topic of his masters thesis was nuclear
magnetic resonance and nuclear quadruple resonance of Cu63,65 on c-axis aligned
YBa2Cu307_$. In 1991, he moved to the University of Florida to pursue a Ph.D.
program in physics and joined with Professor David Tanners group in 1992 to study
optical properties of high Tc superconductors.
174

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Om
David B. Tanner, Chairman
Professor of Physics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Assistant Professor of Physics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Professor of Physics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Neil S. Sullivan
Professor of Physics

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
I
Rajiv K. Singh
Assitant Professor of Materials Science and
Engineering
This dissertation was submitted to the Graduate Faculty of the Department of
Physics in the College of Liberal Arts and Sciences and to the Graduate School and
was accepted as partial fulfillment of the requirements for the degree of Doctor of
Philosophy.
August 1995
Dean, Graduate School



35
Fig. 15. Meissner effect measurement on Nd2-xCexCu04 single crystal.
Table 1. Characteristics of Nd2-xCerCu04 Crystals
X
area
(mm2)
p (at 300 K)
(mil-cm)
adc (at 300 K)p(T)
(il-cm)-1
0.0
1 x 1
~ 10"4
0.11
1 x 1
1 ~ 2
500 ~ 1000 Po + AT2
0.14
1 x 1
0.2 ~ 0.4
2500 ~ 5000 po + AT2
0.15
1.2 x 1.2
0.2
5000 po + AT2
0.16
1.2 x 1.2
0.2
5000 po + AT2
0.19
2x2
0.1 ~ 0.2
5000 ~ 10000 po + AT2
0.2
2x2
0.1 ~ 0.2
5000 ~ 10000 po + AT2


154
Fig. 67. The London penetration depth as a function of frequency.


Reflectance
78
Frequency (cm 1)
Fig. 22. Far-infrared reflectance of Nd2Cu04_i at several temperatures.


50
One Component Analysis
Another approach to analyze the non-Drude conductivity is to assume an inelas
tic scattering of the free carriers in the low frequency with a underlying excitation
spectrum. This interaction gives a strong frequency dependence to the scattering
rate and an enhanced low frequency effective mass of the free carriers. This approach
has been proposed by Anderson34 and applied to heavy Fermion superconductors.35
The one component picture of the optical conductivity can also be described by the
marginal Fermi liquid36 (MFL) and the nested Fermi liquid37,38 (NFL) theories.
According to Varma et al.,36 the quasiparticle self energy S of the marginal Fermi
liquid has a imaginary part which qualitatively goes as
7T2A T, (jJ < T
-ImS(w)
(43)
xXu, u > T
where A is the electron-phonon coupling constant. There is an upturn in the effec
tive mass, with the mass enhancement proportional to A. In the NFL approach of
Virosztek and Ruvalds,37,38 the nested Fermi liquid has
ImS = amax(/?T, | u |),
(44)
where a is a dimensionless coupling constant. This gives a scattering rate that is
linear in T at low u and linear in u at high u>.
For calculating the frequency dependent scattering rate 1 /t(w) and effective mass
m*(u>), the complex dielectric function is described by a generalized Drude model
in terms of the complex damping function, also called a memory function, as 7 =
R(lj) + i/(w),
(w) = Coo -
(45)
u>(l> -f 7)
where e5 represents interband contribution not involving the charge carriers and
up = \J\nnz1 /m* is the plasma frequency, with n the carrier concentration and m*


22
with large hybridization, so the holes in the hole-doped high Tc superconductors will
behave as strongly correlated particles.
Frequency Dependent Conductivity in Superconductors
Far-infrared conductivity ai(u;) is very useful to study particle-hole excitations
in the energy range of 4 ~ 100 meV for the copper oxide and bismuth oxide su
perconductors. For example, in the superconducting state, the energy gap 2A of
single-particle excitations could be obtained directly from the absorption edge of
infrared spectrum. Further information on the nature of quasiparticles and other
infrared-active excitations in the superconducting state can be obtained by analyz
ing the frequency dependence of the absorption spectrum at energies above 2A. Of
particular interest are deviations of the measured spectrum or infrared conductivity
from the BCS-theory for isotropic superconductors. Mattis and Bardeen22 first cal
culated the infrared conductivity in the framework of the weak-coupling BCS theory.
Deviations might give us some hints on what is basically different in BCS and high
Tc superconductors.
The Mattis-Bardeen theory does not include the Holstein mechanism,23 where
part of the energy of the excited conduction electron is transferred to phonons. This
effect is well known for a conventional strong coupling superconductor such as Pb.
In this case, one expects characteristic deviations from the Mattis-Bardeen theory.
However, little is known about strong coupling corrections to weak-coupling conduc
tivity. The strong-coupling theory of ai(u>) which is based on Eliashbergs model24
was first calculated by Nam.25 Since the early 1960s, the interpretation of energy gap
and more detailed measurements of excitation spectra has been based on the Eliash-
berg theory.24 This theory makes a number of assumptions which may be called into
question in the new copper oxide and bismuth oxide superconductors.


47
which was thermally anchored to the cold head of a closed-cycle refrigerator (CTI
Cryogenics). The probe tip houses four electrodes. The sample can be electrically
connected to these electrode with 20 pm diameter gold wire using silver paint. The
sample temperature was monitored with a temperature controller (Lake Shore Cry-
otronics) that was connected to a silicon diode sensor which was attached to the cold
head of the cryostat unit.
We measured the resistance, R = V/I, for the configuration of leads schematically
shown in Fig. 20, using a standard ac phase-sensitive technique operated at ~ 22 Hz
at a current of ~ 700 /A. The results were insensitive to the size of the current.
Before measurement, four stride contacts were formed on ab plane by the evaporation
of silver plate. An annealing procedure for good Ohmic contact was performed at a
temperature of ~ 300 K in flowing O2 for ~ 5 hours. Contact resistance values in
the range 10 to 20 fl were obtained by the bonding of Au wires with silver paint.
The electrical resistivity values p which is defined as p = RA/L were obtained by
measuring the crystal dimensions, where A is the cross sectional area for current flow
and L is the length along the voltage drop.
Data Analysis: Kramers-Kronig relations
To obtain the optical conductivity c\{u) which is a more fundamental quantity
one applies a Kramers-Kronig transform to the reflectance R(u>), which yields the
phase shift 0(u). Formally, the phase-shift integral requires a knowledge of the re
flectance at all frequencies. In practice, one obtains the reflectance over as a wide
frequency range as possible and then terminates the transform by extrapolating the
reflectance to frequencies above and below the range of the available measurements.
Concerning the low frequency extrapolation, we find that the conductivity at
frequencies for which there is actual data is not affected significantly by the choice of


172
132. F. Sharifi, A. Pargellis, R. C. Djnes, B. Miller, E. S. Heilman, J. Rosamilia,
and E. H. Hartford, Jr., Phys. Rev. B 44, 12521 (1991); Phys. Rev. Lett. 67,
509 (1991); Physica C 185-189, 234 (1991).
133. Q. Huang, J. F. Zasadzinski, N. Tralshawala K. E. Gray, D. G. Hinks, J. L.
Peng, and R. L. Greene, Nature 347, 369 (1990).
134. J. F. Zasadzinski, N. Tralshawala, D. G. Hinks, B. Dabrowski, A. W. Mitchell
and D. R. Richards, Physica C 158, 519 (1989); Physica C 162, 1053 (1989).
135. A. Kussmaul, E. S. Heilman, E. H. Hartford, Jr. ad P. M. Tedrow, Appl. Phys.
Lett. 63, 2824 (1993).
136. H. Sato, H. Takaki, and S. Uchida, Physica C 169, 391 (1990).
137. P. Samuely, N. L. Bobrov, A. G. M. Jansen, P. Wyder, S. N. Barilo and S. V.
Shiryaev, Phys. Rev. B 48, 13904 (1993).
138. S. Kondoh, M. Sera, Y. Anno and M. Sato, Physica C 157, 469 (1989).
139. D. G. Hinks, D. R. Richards, B. Dabrowski, D. T. Marx and A. W. Mitchell,
Nature 335, 419 (1988); Physica C 162-164, 1405 (1989).
140. C. K. Loong, D. G. Hinks and Y. Zheng, Phys. Rev. Lett. 66, 3217 (1991).
141. Z. Schlesinger, R. T. Collins, J. A. Calise, D. G. Hinks, B. Dabrowski, N. E.
Bickers, D. J. Scalapino, Phys. Rev. B 40, 6862 (1989) ; Mrs. Bulletin, June,
(1990).
142. M. S. Pambianchi, S. M. Anlage, E. S. Heilman, E. H. Hartford, Jr., M. Bruns
and S. Y. Lee, Appl. Phys. Lett. 64, 244 (1994).
143. A. V. Puchkov, T. Timusk, W. D. Mosley and R. N. Shelton, Phys. Rev. B,
submitted.
144. F. J. Dunmore, H. D. Drew, E. J. Nicol, E. S. Heilman and E. H. Hartford,
Phys. Rev. B 50, 643 (1994).
145. B. Batlogg, R. J.Cava, L. W. Rupp, Jr., A. M. Mujsce, J. J. Krajewski, J.
P. Remeika, W. F. Peck, Jr., A. S. Cooper and G. P. Espinosa, Phys. Rev.
Lett. 61, 1670 (1988).
146. Y. J. Uemura, B. J. Sternlieb, D. E. Cox and A. W. Sleight, Nature 335, 151
(1988).


107
we expect that the electron jumps from a state below the Fermi level to a nearby
state or distant state. The energy band width 7 neax the Fermi level is determined
by Coulomb interaction 7 ~ e2/ef?, where R is the mean distance between two states.
Also, Chen et al.94 show the power-law frequency dependence of the conductivity in
the microwave frequency region which is characteristic of phonon-assistant tunneling
of electrons between bound states of the dopant atoms.
The behavior of dc resistivity with temperature is very similar to that of free
carrier in the far-infrared 07 (w) spectra. In Fig. 50, at temperatures above 100 K, the
extrapolated values of 07(10) at zero frequency increases with decreasing temperature,
showing a metallic behavior. However, 07(10) below 100 cm-1 at 10 K strongly de
creases and instead 07(10) above 100 cm-1 increases to compensate for the decreased
oscillator strength below 100 cm-1. This indicates that the free carriers at 10 K are
strongly localized.
The far-infrared conductivity spectrum at 10 K in Fig. 54 shows a resonant ab
sorption whose maximum occurs at a frequency between 100 cm-1 and 130 cm-1,
depending on doping concentration. The origin of the resonant absorption is, in the
high frequency region hu > kT, the photon-induced hopping of charge carriers as
a result of carrier transition from one site to another, and thus change its location
with respect to impurity atoms. This process must be distinguished from the usual
phonon-induced hopping mechanism which has been previously studied in Chen et
al.
The first works on the theory of resonant absorption were devoted to the impurity
band of doped semiconductors. Experimentally, Milward and Neuringer95 have first
observed near 30 cm-1 at low temperature in compensated n-type silicon. This value
is lower than those of our results, because the dopant density in n-doped Si is very


99
Raman or infrared active. The dynamical tilting of the Cu-0 plane increases the
dipole potential of Cu-0 plane, and so increases a carrier-lattice interaction with the
increase of doped-holes. Upon Sr doping, Tjv is depressed rapidly from ~ 300 K for
x = 0to~0Kbyx~0.02. The doping also depressed Tq/Ti but at a much lower
rate, such that Tq/j = 0 by x = 0.2.77
We here report new modes induced by doping, and discuss a carrier-lattice in
teraction caused by the tilting of Cu06 octahedra. Further, for 10 K conductivity
spectra, we find resonant absorption peaks which correspond to the photon-induced
hopping of charge carriers. Finally, we discuss the infrared absorption bands near
0.15 eV, 0.5 eV, and 1.4 eV.
Experimental Results
a-b Plane Spectra
Figure 42 shows the reflectance spectra in La2_iSrxCu04+j with (0.01 < x,6 <
0.04) at room temperature in frequency range from 30 cm-1 to 32000 cm-1 (4 eV).
The polarization of the incident light is parallel to a-b plane. The spectrum at fre
quencies below 1 eV exhibits dramatic changes with Sr doping. Among the trends
observed in reflectance with Sr doping are a loss of spectral weight of the charge
transfer (CT) excitations around 2 eV and the development of a plasma edge shift to
higher frequencies with doping, which corresponds to the development of free carrier
band and mid-infrared bands. Figure 42 also shows that all samples we measured
show metallic-like behavior even in the lowest Sr doped sample, i.e., the reflectance is
about 80% near 100 cm-1 and decreases monotonically with the increase of frequency.
In addition, the 1% Sr doped sample shows a larger magnitude of the reflectance than
2% Sr doped sample due to the deviation from the oxygen stoichiometry.


151
Temperature (K)
Fig. 64. The Dmde scattering rate, 1/r, as a function of temperature. The
1/r is obtained from the fitting parameters of Table 3.


Reflectance
113
Photon Energy (eV)
0.01 0.1 1
Fig. 42. Room temperature reflectance R spectra of La2-rSrxCu04+f (0.01 <
x, 6 < 0.04) on a-b plane.


17
(local) singlet (S) and triplet (T) states. Because the spin singlet states have the
lowest energy, the singlet states are located just above the valence bands and act as
the lower Hubbard band. Thus, the charge transfer gap of Ueff is formed with the
upper Hubbard band. Therefore, the t-J model can produce the low energy spectrum
of the three band Hubbard model.
Spectral Weight Transfer with Doping
Insulating CuC>2 layered cuprates can be doped with holes or electrons as the
charge carriers. A surprising feature with doping is a strong doping dependence of
high energy spectral distributions and the redistribution of the spectral weight from
high to low energy. Nice examples are the electron-electron loss study19 and 0 Is
x-ray absorption study20 for the La2-xSrxCu04 system. These spectra show a strong
decrease with doping x in the intensity of the upper Hubbard band as the lower en
ergy structure develops due to doped-holes in the 0 2p band. Another example is
optical absorption experiments,3 where a transfer of spectral weight from a band-gap
transition at about 2 eV in insulating La2Cu04 to the low energy scale (< 1 eV)
is observed with a strong doping dependence. This redistribution of spectral weight
and its doping dependence is due to strong correlation effects and has been observed
in several numerical calculations of correlated systems. Naively, doped-carriers may
show different orbital characteristics in the case of hole doping and electron doping:
holes have 0-2p-like character and electrons have Cu-3d-like character. Thus, we may
expect the different doping mechanisms for hole-doped and electron-doped systems.
In this section we review the difference between doping mechanisms of a semicon
ductor, a localized Mott-Hubbard and a CT system and discuss the influence of the
hybridization for the Mott-Hubbard and CT system in the framework of Eskes et
al.21 and Meinders et al.A


ACKNOWLEDGMENTS
I would like to thank my adviser, Professor David B. Tanner, for his advice,
patience and encouragement throughout my graduate career. I also thank Professors
P.J. Hirschfeld, N. Sullivan, J. Dufty and R. Singh for their interests in serving on
my supervisory committee and for reading this dissertation.
Thanks also go to all my past and present colleagues in Tanners group for their
friendship, useful conversations and cooperation. In particular, I would like to thank
V. Zelezny for many enlightening and useful discussions.
n


Reflectance
82
Photon Energy (eV)
5000 10000 15000
Frequency (cm-1)
Fig. 26. (a) Temperature dependent-reflectance spectra and (b) optical con
ductivity spectra of charge transfer band for Nd2Cu04_.


21
addition states near the Fermi level and N 1 electron addition states in the UHB.
The same arguments hold for the electron doped case. Thus, a doping concentration
x yields a low energy spectral weight 2x and the high energy spectral weight is 1 x.
There have been Nx states transferred from high to low energy. However, when the
hybridization is taken into account, the low energy spectral weight grows faster than
two times the amount of doping as shown in Fig. 8.
Charge Transfer System
For the high Tc superconductors, an oxygen band is located between the LHB and
UHB. These systems are described by the three band Hubbard Hamiltonian. In the
localized limit with no hybridization between the oxygen and copper sites (p=0),
when the electrons are doped in this system, the situation is similar to the Mott-
Hubbard case and the spectral weight is transferred from high to low energy. Thus,
the low energy spectral weight goes to 2x with doping x. However, upon hole doping
the situation is similar to that of the semiconductor without any spectral weight
transfer. So, the CT system in the localized limit shows a fundamental asymmetry
between hole and electron doping (Fig. 9). That is, electrons will feel the strong
repulsions on the d sites, similar to the MH model, and will behave as strongly
correlated objects. When the hybridization is taken into account, the low energy
spectral weight for the electron-doped CT system behaves more or less the same as
found for the Mott-Hubbard system. However, for small hybridization tp, the low
energy spectral weight for the hole-doped CT system behaves as a semiconductor.
When the hybridization is increased, the low energy spectral weight for the hole-
doped CT system rapidly increases and the low energy spectral weight is almost
symmetric with respect to hole-electron doping, so the low energy spectral weight is
similar to that of the MH system. The high Tc superconductors lie in the regime


12
Three Band Hubbard Model
First of all, let us consider the bonding of a full Cu-0 octahedron (CuOe), that is,
the bonding of the 3d orbitals on the Cu ion with the 2p orbitals of the surrounding
0 ions. There are 17 orbitals in the Cu-0 octahedron. Five are from the 3d orbitals
of Cu, which are dx2_y2, dz2, and three dxy types. Also, the four 0 atoms each have
three p orbitals which contribute 12 orbitals. However, we here focus on the in-plane
bonding and take a more intuitive approach. To do this, consider the two planar 0
atoms with p orbitals that are directed toward the central Cu atom. On the central
Cu atom, we only use the dI2_J/2 orbital, since it is correctly oriented for a bonding
with its neighboring oxygens. It is also the uppermost Cu-d level in the crystal field of
the octahedral structure. Thus, only three orbitals (px,py, and dx2^2) are used. The
other 14 orbitals can be taken as nonbonding relative to these orbitals. In addition,
the copper ion Cu2+ has a 3d9 electron configuration which gives the ion spin 1/2.
Thus, in the absence of doping, the material is well described by a model of mostly
localized spin 1/2 states that give these materials their antiferromagnetic character.
The Hamiltonian in the Cu2 plane can be constructed in the framework of the
three orbitals:
H = tvdY^(p)dx + d\pj + h.c.) tpp (PjPj> + h.c.) +
('d> 0',/) i
+udY * 3 (i,j)
(i)
where pj are fermionic operators that destroy holes at the oxygen ions labeled j, while
d, corresponds to annihilation operators at the copper ions i. (i,j) and (j,j) represent
Cu-0 and 0-0 neighbors, so that this Hamiltonian contains two hopping terms, tpd
and tpp, as well as site energies e¡ and Coulomb interactions Ut for the two types of


173
147. S. Uchida, H. Hasegawa, K. Kitazawa and S. Tanaka, Physica C 156, 157
(1988).
148. S. Sugai, Y. Enomoto, and T. Murakami, Solid State Commun. 72, 1193
(1989).
149. T. Siegrist Nature 367, 254 (1994).
150. W.E. Pickett Phys. Rev. Lett. 72, 3702 (1994).
151. J.S. Kim, W.W. Kim, and G.R. Stewart, Phys. Rev. B 50, 3485 (1994).
152. M. Gurvitch, Physica B 126, 276 (1985).


Ill
The lowest panel in Fig. 56 shows that the oscillator strength for 0.15 eV band
decreases with increasing temperature. This temperature dependence reflects that
other processes may substantially contribute to the this band. The temperature
dependence of the oscillator strength may be described by a polaxonic effect,103 in
which carriers move nonadiabatically with respect to the lattice. In present analysis,
we have also suggested the charge dynamic of doped carriers, namely, the large value
of the static dielectric function, and the carrier-lattice interaction by the deformation
potential etc.Thus, in my opinion, the magnetic polaron which we mentioned above
is also likely have a lattice component, and hence the spin and lattice excitations are
very important at low doping levels of high Tc cuprates.
In Fig. 57, the peak observed near 1.4 eV below the the charge transfer excitation
band is a result of the excitonic effect. This peak is not observed in undoped sample
and in T phase. Suzki et a/.104 have shown that its strength increases with Sr doping
and Uchida et al.have observed this peak at high doping levels. In excitonic model, a
charge transfer excitations from the Cu to the 0 site create the free electrons on Cu
and the free holes on 0 site. An short range attractive interaction (Up) between
them results in the creation of exitons.
Figure 58 shows a fitting curve for the fitting curve is obtained by using the Lorentz model discussed in Chapter V. The
individual contributions of the Lorentzian include the phonon bands, the hopping
conductivity and the mid-infrared bands.
Summary
We have shown that the Cu6 octahedra rotates around [110] axis with decreas
ing temperature, and the deformation potential caused by the tilting of Cu02 plane
enhances the carrier-phonon interaction. As doping proceeds, the oscillator strength


Low frequency spectral weight (/Cu)
95
Fig. 39. The low frequency spectral weight as a function of x in both
La2_xSrxCu04 (left) and Nd2_xCexCu04 (right). The data from
La2_xSrxCu04 were taken from Ref. 3.


44
beam; one reflected and one transmitted. Both beams are sent to a two-sided movable
mirror which reflects them back to be recombined at the beam splitter site. The
recombined beam is sent into the sample chamber and detector. When the two-sided
mirror moves at a constant speed v, a path difference 8 = 4vt, where t is the time
as measured from the zero path difference. Next, the signal is amplified by a wide
band audio preamplifier and then digitalized by a 16-bit analog-to-digital converter.
The digitalized data axe transferred into the Aspect computer system and axe Fourier
transformed into a single beam spectrum.
Perkin-Elmer Monochromator
Reflectance spectra from mid-infrared to ultraviolet (UV) frequency region are
measured by a model 16U Perkin-Elmer grating monochromator. The basic concept
of a grating monochromator involves shining a broadband light source on a grating
and selecting a small portion of the resulting diffracted spectrum by letting it pass
through an opening known as a slit.
A diagram of the spectrometer is shown in Fig. 19. Three sources-globar (GB),
quartz-envelope tungsten lamp (W), and deuterium lamp (D2) axe used for different
frequency region. The light signal is chopped to give it an AC component which could
then be amplified by a lock-in amplifier. Long-pass and bandpass filters eliminate
unwanted orders of diffraction. A laxge spherical mirror images the exit slits of the
monochromator onto either a reference mirror or a sample in the case of reflectance
measurements. For transmittance measurements, the sample is mounted as close as
possible to the focus of the second spherical mirror. The position of the detector
is at the focal point of ellipsoidal mirror. Three detectors, a thermocouple (TC), a
lead sulfite (PbS) photoconductor, and a silicon photodiode (576) axe used to cover


Reflectance
146
Photon Energy (eV)
0.01 0.1 1
Fig. 59. Normal state reflectance i?(u;) of Bai_xKxBi03 at temperatures
between 30 K and 300 K as a function of frequency.


123
13
I
O
x
o
D
-
O
V)
O
Temperature (K)
Fig. 52. Oscillator strength of in-plane phonons and Raman modes at 247
and 278 cm-1 as a function of temperature.


32
Metallic Insulating Metallic
Fig. 13. Phase diagram of Nd2-zCezCu04 and La2_xSrzCu04.
electron-doped system clearly illustrates that superconductivity is a relatively small
effect compared with antiferromagnetism.
Bai-^K^BiOt and BaPbrBT_.T0-t
BaBiOa has an almost undistorted ABX3, cubic perovskite structure (Fig. 14).
Each Bi atom (B site) is octahedrally coordinated by six 0 atoms. The A site is
occupied by Ba or K, while B site is occupied by Bi or Pb. At room temperature,
the symmetry of BaPbi_zBiz03 material changes with doping according to following
sequence.27
Orthorhombic 0 Tetragonal 0.05 <
Orthorhombic 0.35 Monoclinic 0.90

65
position corresponds to the zeros of ei(cj). In a simple Drude model, the maximum of
the energy loss function determines the longitudinal plasma frequency of free carriers,
corresponding to the zeros of the dielectric function e(w£), and its maximum position
shifts to higher frequencies with doping according to up = (4xne2/m)1//2. However,
the bound carriers in high Tc cuprates which contribute a positive dielectric response
dielectrically screen the free carrier response, and also lower up. The maximum value
of Im[l/e(u;)] is given approximately by the screened plasma frequency
Up / 47rne2
uv ^ F= = \ i
y/tct V m ci
where ect is the the ei(u;) value at the charge transfer gap frequency.
Figure 32 shows Im[l/e(w)] with Ce doping as a function of frequency. The
result for x = 0 is very small below 1.2 eV except phonon modes in the far-infrared
region, and shows a bump near 1.5 eV which is associated with the charge transfer
excitation. The spectrum of x = 0.11 shows a featureless continuum near 1000
cm-1 and a broad peak around 7200 cm-1 (0.9 eV). With doping this peak position
moves to slightly higher frequencies, where its maximum position corresponds to the
appearance of a reflectance edge with doping. For 0.14 < x < 0.2, the peale positions
occur near 1.1 eV and are insensitive to Ce doping concentration, inconsistent with
the simple Drude model. This indicates that the value of n/m*ect in (51) is insensitive
to doping. Figure 32 also shows that the peak position of the superconducting sample
with x = 0.15 is observed at higher energy than in slightly overdoped sample with
x = 0.16. This may suggest that the superconducting sample has more free carriers
or low effective mass of charge carriers. A broad peak width (0.5 eV) in Im[l/e(u;)]
is due to the anomalous mid-infrared absorption caused by the incoherent motion of
free carriers.


11
and the strong coupling of the conduction band states near Ep to bond stretching 0
displacements lead to a commensurate CDW distortion.10 In another approach, the
driving force is the aversion of Bi to the 4+ valence, which leads to a disproportiona
tion into 3+ (6s2) and 5+ (6s) valences on alternate sites.11 In either case one finds
a commensurate CDW distortion, in which the 0 octahedra are alternately expanded
or contracted as illustrated in Fig. 5. This CDW distortion doubles the unit cell,
which splits the half filled metallic band into filled and empty subband, opening a
semiconducting gap of ~ 2 eV.
Electronic Models for CuO? Plane
In this section, a Hamiltonian to describe the behavior of electrons in the high
Tc materials will be briefly described. Due to the complexity of their structure it is
important to make some simplifying assumptions. The very strong square planar Cu-
0 bonds with strong on-site correlations makes it possible to construct a Hamiltonian
restricted to electrons moving on the Cu02 plane.
Several models have been introduced for the description of layered strongly cor
related systems, as realized in the Cu02 plane. While there is a growing consen
sus that the high Tc materials should be described within the framework of two-
dimensional (2D) single-band t-J12 or three-band Hubbard models13 in the strong
coupling limit,14,15 a direct comparison of controlled solutions with experimental data
is still lacking. We will discuss these one band and three band Hubbard models in
the present section, and the carrier doping effect in these prototype models will be
discussed in the following section.


cr(ncm) Reflectance
90
0 500 1000 1500 2000
i/(cm~1)
Fig. 34. Temperature dependent (a) reflectance and (b) conductivity for su
perconducting Ndi.85Ceo.i5Cu04 as a function of frequency.


42
The complex amplitude of the combined beam reaching the detector is
= A(,)(l + e"2'^). (32)
But the intensity B(i/,8) (irradiance or flux density) is
B(v,8) = A*r(8,i/)Ar(6,i/) = A2(u)[l + cos(27n/<5)] = ^S(i/)(1 + cos2xi/6), (33)
where S(t/) is the power spectrum. The total intensity at the detector is
1(6) = [ B(v,6)di/ = ^ [ 5(i/)[l + cos27ri/i]di/. (34)
Jo 2 Jo
At zero path difference, the intensity at the detector is
7(0) = f S(v)dv. (35)
At zero path difference all of the source intensity is directed to the detector; none
returns to the source. At large path differences the intensity at the detector is just
half the zero path difference intensity
= \fQ S(v)dv- (36)
because as 6 oo the cos2xv8 term averages to zero, t.e., it is more rapidly varying
with frequency than S(v).
The interferogram is the quantity [/() /(oo)]; it is the cosine Fourier transform
of the spectrum. For the general case, the final result is obtained:
B(W = LJiw <37>
(37), at a given wave number v, states that if the flux versus optical path 1(6) is
known as a function of 8, the Fourier transform of [/() ^7(0)] yields B(i/), the flux
density at the wave number u.


61
reduced and a reflectance edge rapidly develops below 1 eV. Fig. 27 also shows that
the position of the edge shifts to higher frequency with increasing doping and is almost
saturated in the metallic regime where 0.14 < x < 0.19. Another notable feature is
that the charge transfer band near 1.5 eV moves to higher frequency with increasing
dopant concentration x. This behavior is obvious in this system. In addition, there
is a systematic change of reflectance between ~ 3 eV and ~ 5 eV with x. A similar
behavior has also been observed for hole-doped L^-xSrjCuC^3 and YBa2Cu37_
systems.39,44
The magnitude of the reflectance of Nd2-zCeICu04 at low frequencies is typi
cally larger than the results for hole-doped La2_iSrICu04 and YBa2Cu307_. For
example, the magnitude near 600 cm-1 at high doping levels for our results is about
~ 92%, whereas the results for hole-doped La2-zSrICu043'39 are ~ 85%.
Among the four Eu optical phonons in undoped crystal below 600 cm-1, two
infrared active phonons near 301 and 520 cm-1 are visible even in heavily doped
crystals. However, two weak phonon bands observed at 131 and 345 cm-1 in the
spectrum of undoped crystal are screened out from free carriers in the metallic phase.
Optical Conductivity
The frequency dependent optical conductivities obtained from a KK transforma
tion of the reflectance spectra are shown in Fig. 28 and Fig. 29. We can better observe
the influence of doping on spectral response by considering optical conductivity. The
a-b plane conductivity of Nd2-zCexCu04 shows interesting changes with doping. As
suggested by the reflectance spectrum in Fig. 27, with doping the conductivity of
the charge transfer band above ~ 1.2 eV is systematically reduced, whereas the low
frequency spectral weight below ~ 1.2 eV rapidly increases.


43
Optical Spectroscopy
At high frequencies, the Fellgett advantage losses its importance due to the in
creasing photon noise in the radiation field. For this reason, a grating monochromator
is normally used in the near-IR and visible frequency range.
Generally, a grating monochromator is used by applying the rule of diffraction.
For a wavelength A,
n 1
2d sin#
where d is the grating distant. At an angle 9, the first-order component of wavelength
A satisfying A = asin# is selected. Meanwhile, any higher order components with
wavelengths A = A/n, or vn = m/(n = 2,3,...), which could also pass through the
slit axe absorbed by the filter. The resolution is determined by the slit width and A9,
which is the angle of rotation at each step.
Instrumentation
Bruker Fourier Transform Interferometer
To measure the spectrum in the far and mid-infrared (20 ~ 4000 cm-1), a Bruker
113V Fourier Transform interferometer is used. Different thickness of Mylar beam
splitters, a black polyethylene filter, a bolometer and a Hg arc lamp as detector and
source are used for far infrared (20 ~ 600 cm-1). A photocell and a globax source are
used for mid infraxed (450 ~ 4000 cm-1). A schematic diagram of the spectrometer
is shown in Fig. 18. The sample chamber consists of two identical channels which can
be used for either reflectance or transmittance measurements. The entire instrument
is evacuated to avoid absorption by water and CO2 present in air.
The principle of this spectrometer is similar to that of a Michelson interferometer.
Light from the source is focused onto the beamsplitter and is then divided into two


70
in u of scattering rate from dc resistivity. Ordinary Fermi liquid state requires the
scattering rate varying as u1. Nevertheless, our result in the high Tc regime is consis
tent with numerous models of the normal state in which strong quasiparticle damping
is assumed. Also, our results with doping suggest that the electronic state of very
heavily doped CuC>2 plane may be acquire the nature of a Fermi liquid.
Doping Dependence of Low Frequency Spectral Weight
Prude and Mid-infrared Band
We have emphasized that the spectral weight of the high frequency region above
the charge transfer (CT) band is transferred to low frequencies with doping. Such a
spectral change indicates that the conduction and valence bands of the CT insulator
are reconstructed by doping. In the metallic state, the optical conductivity may be
considered as three parts; a free carrier contribution centered at u = 0, mid-infrared
bands, and high-energy interband transitions above the charge transfer gap. In order
to describe empirically the absorption bands produced by doping, we have fit the
of each sample to the two component model. We here discuss in detail each
band and how its strength changes with Ce doping. The strength of each band j is
related to the plasma frequency in the fitting parameters by the relationship
"h(eV2)
47re2 Nj
m* Vct\{
(56)
We estimated (eV2) = 14.88 Nj, using Vceu = 187 and two Cu atoms per unit
cell, where Nj is the effective electron number per Cu atom of band j.
For free carrier contribution, we extracted the spectral weight of a Drude oscillator
(Nq) in the unit of electron number per Cu atom as a function of Ce concentration
x from the sum rule restricted to the Drude conductivity, a¡). Figure 38 (circles)


129
Photon Energy (eV)
0.01 0.1 1
Fig. 58. The Lorentzian fitting curve for the <7i(u;) at 10 K of
La1.97Sro.03 CUO4.


106
doping, and may occur as a result of the carrier-phonon interaction. This mode
is also consistent with the bleaching of phonon modes observed in photo-induced
measurements.64
Hopping Conductivity in Disordered System
A disordered system having electronic states near the Fermi level has localized
states due to strong disorder and small overlap of the wave function. Such systems
axe on the dielectric side of the Anderson transition.92 Lightly doped crystalline
semiconductors and amorphous semiconductors are example of such system. The
electron-electron interaction in such systems determines a large variety of physical
phenomena, especially dc and ac hopping conduction.
Figure 53 shows the resistivity p as a function of temperature for a 1% Sr doped
sample. The resistivity has a minimum at intermediate temperatures, followed by a
low temperature upturn. In the high temperature region (> 100 K), all impurities axe
ionized and metallic behavior is observed due to the overlap of the impurity orbits. At
low temperatures (< 100 K), the freezeout of holes occurs, and hence the conductivity
results from the thermal ionization of the shallow impurities.
At low temperatures below 50 K, p shows the characteristic behavior of disordered
system with strong Coulomb interactions:93
P~ex p(y)1/2. (53)
The inset in Fig. 53 is a plot of lnp vs. (1/T)1/2 at same temperatures, showing
a linear relationship. This is a typical behavior of dc variable range hopping in
localized states near the Fermi surface. At sufficiently low temperatures, under all
circumstances where N(Ep) is finite but states axe localized near the Fermi energy,


54
Fig. 19. Diagram of the Perkin-Elmer grating monochromator.


68
low frequency range with a underlying excitation spectrum. The carriers derive a
frequency and a temperature-dependent self-energy. The imaginary part goes like
ImS ~ max(u,T). This quasiparticle damping has been described in the frame
work of the nested Fermi liquid (NFL)37,38 and the marginal Fermi liquid (MFL)
models.36 We analyze the non-Drude conductivity of Nd2_xCezCu04 by using a
generalized Drude formula with frequency dependent scattering rate.
c(w) = Ch
ut*
w[m*(w)/m0][w -1- t/r*(w)]
(54)
where e/, is the background dielectric constant associated with the high frequency con
tribution and the second term represents the effects of frequency dependent damping
of carriers, m*/mo represents the effective mass enhancement over the band mass
and 1/t*(u) = (l/r(u;)][m/m*(u;)] the renormalized scattering rate.
Figure 36 shows the m*/mj and 1/t*(u>) curves for four samples below 5000
cm-1. We used utp = 20 000 ~ 2 2000 cm-1, and = 5.0 ~ 5.2 in the infrared
region for different samples. At low frequencies, the behavior of m*/mi illustrates
the coherent motion of carriers, causing the low frequency mass enhancement. This
may be due to the interaction of carriers with phonons, or spin and charge excitations
of carriers. Our results also suggest that the quasiparticle excitations increase with
decreasing doping concentration. This is consistent with other doping dependence
results for hole-doped systems. However, the mass enhancement of Nd2-xCezCu04
is a little bit smaller than those obtained by hole-doped systems. As the frequency
is reduced, the effective carrier mass decreases, and approaches to the band mass at
high frequency.


136
they can be fitted by usual Drude conductivity formula:
1 Idt
&\D 1 o 2
47T 1 +
where upd and r is the plasma frequency and the relaxation rate of the free carriers,
respectively.
Table 3 illustrates the free carrier contribution at several temperatures. The free
carrier contribution at each temperature is in good agreement with the ordinary Drude
behavior. The Drude plasma frequency, upD = 9 500 200 cm-1 obtained from the
above analysis is nearly temperature independent, whereas 1/t in Fig. 64 shows the
non-linearity with temperature. The extrapolated values of cr\{ui) at zero frequency in
Fig. 60 have the same behavior as the temperature dependent 1/t, exhibiting a typical
metallic behavior. This is consistent with the behavior of the normal-state resistivity
of Heilman and Hartford121 on films and Affronte et a/.122 on single crystals. The
non-linearity of the 1/t vs. T is an interesting feature in BKBO in contrast to the
linear temperature dependence observed in the resistivity in high Tc cuprates.
Table 3. Fitting parameters of the Drude conduc
tivity.
T(K)
upd (cm-1)
1/t
300
9700 200
700 20
200
9500 200
450 20
100
9400 200
350 20
30
9400 200
300 20
The normal-state resistivity of films and single crystals of the BKBO system is
interpreted with metallic or semiconductor-like, depending on the room-temperature


no
The mid-infrared conductivity, Fig. 55(b) is composed of two parts,
an(v) (0.15 eV band) and <712(0;) (0.5 eV band). In order to obtain the 0.15 eV
band, first, we have fit the criMID(u) at each temperature using a dielectric function
model for two Lorentz oscillator. The formula is
j=l J
luJ1j
(60)
where u>pj, u>j and 7j correspond to the intensity, center frequency and band width
of each band, respectively. Once values for up2, u>2, and 72 of the 0.5 eV band
axe obtained, they can be used to calculate the Lorentzian spectrum of the 0.5 eV
band, crj^o;). Next, we can obtain the 0.15 eV band after subtracting <7^(u;) from
<7imid{u) in Fig. 55. The inset in Fig. 55 shows the 0.15 eV band.
The upper two panels of Fig. 56 show the temperature dependence of center
frequency u\ and band width 71 for the 0.15 eV. The results shown in Fig. 56 indicate
that with increasing temperature the center frequency increases, and peak position
shift to higher frequency with an amount comparable to the thermal fluctuation
energy k^T of lattice. The behavior of peak position can be described by linearly
varying function:
ue ~ J + ksT, (61)
where kg = 0.695 cm_1/K- This implies that if this band arises from a transition
between states related to the Cu-Cu exchange energy J, the thermodynamic limit
affects the numerical value of peale position J. The behavior of linewidth broadening
with increasing temperature is similar to that of free carriers. The line widths have
a linear temperature dependence, 7(T) ~ 7(0) + 1.5 kgT. A fit of the form h/r =
2it\kBT -f Ti/tq yields a value for the coupling constant A ~ 0.24. However, the
linewidth at each temperature is very broad (260 meV at 300 K).


25
f¡u
2A
Fig. 10. Complex conductivity of superconductors in extreme anomalous (or
extreme dirty limit) at T = 0.
where ma and n3 are the mass and density of the superconducting electrons and
is the London penetration depth. From this relation,
1 nae2
o-2 = = .
Au> m3u>
For > 2A, <72a falls to zero more rapidly than l/u.
Penetration Depth and Infrared Conductivity
The sum-rule argument allows determination of the strength of this supercurrent
response from <7ia. The oscillator strength sum rule requires that the axea under the
curve of The missing axea A under the integral of <7ia appears at u 0 as A8(u). The amount
of conductivity that is transferred from the infrared to the delta function at zero
frequency is given by
[^lnM a\a(tjj)\dui = 0.
(13)


I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
I
Rajiv K. Singh
Assitant Professor of Materials Science and
Engineering
This dissertation was submitted to the Graduate Faculty of the Department of
Physics in the College of Liberal Arts and Sciences and to the Graduate School and
was accepted as partial fulfillment of the requirements for the degree of Doctor of
Philosophy.
August 1995
Dean, Graduate School


Reflectance
162
Photon Energy (eV)
0.01 0.1 1
Fig. A-2. Room temperature reflectance spectrum for LUN2B2C.


46
Table 2. Perkin-Elmer Grating Monochromator Parameter
Frequency
(cm-1)
Grating0
(line/mm)
Slit width
(micron)
Source Detector
801-965
101
2000
GB
TC
905-1458
101
1200
GB
TC
1403-1752
101
1200
GB
TC
1644-2613
240
1200
GB
TC
2467-4191
240
1200
GB
TC
4015-5105
590
1200
GB
TC
4793-7977
590
1200
W
TC
3829-5105
590
225
W
PbS
4793-7822
590
75
w
PbS
7511-10234
590
75
w
PbS
9191-13545
1200
225
w
PbS
12904-20144
1200
225
w
PbS
17033-24924
2400
225
w
576
22066-28059
2400
700
d2
576
25706-37964
2400
700
d2
576
36386-45333
2400
700
d2
576
0 Note the grating line number per cm should be the sarnie
order of the corresponding measured frequency range in cm-1.
dc Resistivity Measurement Apparatus
The experimental arrangement for measuring the resistivity is illustrated in
Fig. 20. The measurements were made as a function of temperature from liquid
helium temperature (~ 4 K) to room temperature (~ 300 K) using a lead probe


108
low (1017 < Nd < 1018 cm 3). The mechanism of the absorption was proposed by
Tanaka and Fan96 and detailed theory was given by Blinowski and Mycielski and
Mott.97 The optical conductivity, <7i(u;), due to resonance absorption in the impurity
band has a maximum at
hu>.
max ~
,
r
un
(57)
where is acceptor concentration, e ie dielectric constant and ru is the average
tunneling distance for pairs of localized states contributing to <7i(u) at frequency ui.
The distance ru is found from the relation
r'u = a \n(2I0/hu>),
(58)
where a is the localized length and I0 is a prefactor of the overlap integral I
I(r) = 70exp(-r/a). (59)
Using (57), 2/0 ~ e2/2ea and a = 8 ~ 5 x 1018 cm-3 from Ref. 98 we obtain
~ 133 cm-1. For the two samples in Fig. 54, the absorption maximum increases
in magnitude and shifts towards higher frequencies with increasing N. This behavior
is in accord with (57). The half-widths of the absorption curves become larger with
increasing Na, and most of this increase in half-width occurs on the high frequency
side of the maximum. The resonant absorption occurs at phonon energies which
axe much smaller than the 35 meV thermal ionization energy of the impurity atoms
estimated from the variation of the dc conductivity and Hall coefficient98 above ~ 50
K. This implies that the resonant absorption occurs at such small photon energies
and low temperatures, where the usual bulk absorption mechanisms axe absent. The
solid lines in Fig. 54 show the curves obtained from Lorentz model.


I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Om
David B. Tanner, Chairman
Professor of Physics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Assistant Professor of Physics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Professor of Physics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Neil S. Sullivan
Professor of Physics


160
Figure A-2 shows the room temperature reflectance of LUN2B2C. The reflectance
of this system in the infrared region is higher than that in high Tc cuprates. The spec
trum shows a deviation from the Drude curve due to the bands at energies between
0.7 eV and 3 eV.
It is interesting to examine the optical conductivity cr\{ui). The K-K transfor
mation was used to obtain Figure A-3 shows the room temperature optical
conductivity. We have fit cri(u>) with the two component model. We can separate
o 1 (u) into the Drude component below 0.2 eV and several Lorentz oscillators above
0.5 eV. The solid line illustrates the Drude term with a scattering rate of 1100 cm-1
and an oscillator strength of 30 000 cm-1. The dc conductivity value derived from
the fitting parameters using formula pjc = 4ir(jy~)~ is in good agreement with the
result obtained from dc resistivity measurement.
In conclusion, LUN2B2C has the strong Drude plasma frequency of 30 000 cm-1
and quite large damping factor of 1100 cm-1 compared to the high Tc cuprates, and
a strong electron-photon coupling constant of A ~ 1.05.


72
Transfer of Spectral Weight with Doping
Next, we interpret the low frequency excitation near the Fermi level transferred
from the high frequency region as a function of Ce doping x. This is done by comput
ing the effective electron number Neff(uj) of the Drude and toted mid-infrared bands
which corresponds to all electrons that are introduced by doping and comparing with
hole-doped La2_zSrzCu04 system of Uchida et al.3
Figure 39 represents the low frequency spectral weight below 1.5 eV of hole-doped
La2_zSrzCu04 of Uchida et a/.(left) and the low frequency spectral weight (LFSW)
of electron-doped Nd2_zCezCu04 for our results (right). The solid lines in Fig. 39
correspond to the localized limit (no p-d hybridization) in the charge transfer system
for hole-doping and electron-doping cases. In the localized limit, upon doping the
LFSW of electron-doped system is expected to grow similar to the Mott-Hubbard
case, where the LFSW goes to 2x with doping x due to the restriction of doubly
occupied states of doped carriers, because electrons are doped primarily on Cu sites.
For hole-doped system, LFSW grows as x with doping x as semiconductor case,
since holes introduced by doping on 0 sites occupy almost free particle levels and
scatter weakly off the Cu spins. However, Meinders et a/.4 have shown that when the
hybridization is large, the LFSW of hole-doped system becomes similar to that of the
MH system and the electrons as well as the holes show strongly correlated behavior.
Our results for Nd2_zCezCu04 show a electron-hole symmetry at low doping
levels and a prominent electron-hole asymmetry. The LFSW associated with the
Cu02 plane grows faster than 2x with doping x, consistent with the expectation of
the MH model, where the lower Hubbard band (LHB) as well as the upper Hubbard
band (UHB) loses the spectral weight. The greater LFSW than 2x may result from
a large impurity band contributions in T phase materials and the charge transfer


138
K. There is a little ambiguity in this estimation due to the non-linearity of 1/r with
temperature. Nevertheless, the estimation of A seems to be consistent with the gap
measurement and numerical calculations,126-128 where A is suggested to be around 1
in order to explain the conventional electron-phonon mechanism. Thus the normal
state properties may suggest that BKBO is a BCS-like superconductor in which the
electron-phonon interaction plays a significant role.
Superconducting State Properties
Superconducting Gap
In the conventional BCS theory, a bulk superconductor at temperatures below Tc
is a perfect reflector of electromagnetic energy at frequencies below 2A. Above 2A its
behavior is similar to that of a normal metal. In infrared reflectance measurements,
the original inference of the superconducting gap was based on the measurement of
the ratio Rs/Rn of the reflectance in the superconducting state to that in the normal
state. Another case, the superconducting to normal ratio for transmission129,130 shows
a maximum near 2A. According to Glover and Tinkham,131 the superconducting gap
can also be obtained from a\3 in the superconducting state, which has a gap, 2A, at
the threshold for pair excitations. In this section, we examine both the question of
determining a frequency at which the absolute reflectance reaches 100%, and possible
evidence for a BCS size gap in a BKBO crystal.
We have measured the superconducting state reflectance in BKBO. Fig. 65(a)
shows the far-infrared reflectance at various temperatures. This figure illustrates that
in the normal state, BKBO has a very high far-infrared reflectance, characteristic of
free carriers as expected from the metallic dc resistivity. In the superconducting state
(at 10 K and 19 K), a strong edge appears. The positions of the edge have a strong