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## Material Information- Title:
- Optical properties of doped cuprates and related materials
- Creator:
- Yoon, Young-Duck
- Publication Date:
- 1995
- Language:
- English
- Physical Description:
- viii, 174 leaves : ill. ; 29 cm.
## Subjects- Subjects / Keywords:
- 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 - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (Ph. D.)--University of Florida, 1995.
- Bibliography:
- Includes bibliographical references (leaves 164-173).
- General Note:
- 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 ,* 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 |

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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. Axe, G. Shirane, R.J. Birgeneau, D.R. Gabbe, H.P.Jenssen, M.A. Kastner, and C.J. Peters, Phys. Rev. B 38,185 (1988); T.R. Thurston, R.J. Birgeneau, D.R. Gabbe, H.P. Jenssen, M.A. Kastner, P.J. Picone, N.W. Preyer, J.D. Axe, P. Bni, and G. Shirane, Phys. Rev. B 39, 4327 (1989). 87. S. Sugai, Phys. Rev. B 39, 4306 (1989);Physica C 185-189, 76 (1991 )]Phys. 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. Opt. Soc. Am. B 6, 455 (1989); S. Sugai, M. Sato, and S. Hosoya, Jpn. J. Appl. Phys. 26, L495 (1987). 88. M.J. Rice, Phys. Rev. Lett. 37, 36 (1976). 89. M.J. Rice, L. Pietronero, and P. Bruesch, Solid State Commun. 21, 757 (1977). 90. G.A. Thomas, D.H. Rapkine, S.-W. Cheong, and L.F. Schneemeyer, Phys. Rev. B 47, 11369 (1993). 91. K. Yonemitsu, A.R. Bishop, and J. Lorenzana, Phys. Rev. Lett. 69, 965 (1992). 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. 94. C.Y. Chen, R.J. Birgeneau, M.A. Kastner, N.W. Preyer, and T. Thio, Phys. Rev. B 43, 392 (1991); Phys. Rev. Lett. 63, 2307 (1989). 95. R.C. Milwaxd and L.J. Neuringer Phys. Rev. Lett. 15, 664 (1965). 96. S. Tanaka and H.Y. Fan, Phys. Rev. 132, 1516 (1963). 97. J. Blinowski and J. Mycielski, Phys. Rev. 136, A266 (1964); Phys. Rev. 140, A1024 (1965); N.F. Mott Phil. Mag. 22, 7 (1970). 98. N.W. Preyer, R.J. Birgeneau, C.Y. Chen, D.R. Gabbe, H.P. Jenssen, M.A. Kastner, P.J. Picone, and T. Thio Phys. Rev. B 39, 11563 (1989). S. Sugai, S. Shamoto, and M. Sato, Phys. Rev. B 38, 6436 (1988); Phys. 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 VceÂ¡Â¡ = 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). 117. K. F. McCarty, H. B. Radousky, D. G. Hinks, Y. Zheng, A. W. Mitchell, T. J. 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 = Â£ (W + 4^>) + c'EKt 5>Ki -1), (2) 71 represents NÂ¡). However, it is difficult to define the Drude paxt from 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). xml version 1.0 encoding UTF-8 REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID E6GNY5JGF_6H3MBA INGEST_TIME 2015-03-31T18:28:11Z PACKAGE AA00029823_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES 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) + 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 (/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 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 = 2ImÂ£. mo r*(w) 2 Figure 37 illustrates the imaginary part of quasiparticle self energy, Im Â£ of Nd2-xCeICu04 crystals below 5000 cm-1. ImÂ£ 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 Â£ in high doping concentration of x = 0.19. Second, for superconducting sample (x = 0.15), ImÂ£ 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 ImÂ£ 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 upÂ£> 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 and Bardeen22 showed that the ratio of the superconducting to normal conductivity within weak-coupling BCS theory is Â£1Â£ = 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) Â£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)Â¡(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- 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 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. Â¡ 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 REFERENCES 1. J.G. Bednorz and K.A. Muller, Z. Phys. B 64, 189 (1986). 2. T.R. Thurston, M. Matsuda, K. Yamada, Y. Endoh, R.J. Birgeneau, P.M. Gehring, and G. Shirane, Phys. Rev. Lett. 65, 263 (1990). 3. S. Uchida, T. Ido, H. Takagi, T. Arima, Y. Tokura, and S. Tajima, Phys. Rev. B 43, 7942 (1991). 4. M. B. J. Meinders, H. Eskes and G. A. Sawatzky, Phys. Rev. B 48, 3916 (1993). 5. G.A. Thomas, D.H. Rapkine, S.L. Cooper, S.W. Cheong, and A.S. Cooper, Phys. Rev. Lett. 20, 2906 (1991). 6. S. Uchida, Mod. Phys. Lett. B4, 513 (1990); S. Uchida, and T. Ido, JJAP Series 7 Mechanisms of Superconductivity, 133 (1992). 7. T. Arima, K. Kikuchi, M. Rasura, S. Koshihara, Y. Tokura, T. Ido, and S. Uchida, Phys. Rev. B 44, 917 (1991). 8. S.L. Cooper, G.A. Thomas, J. Orenstein, D.H. Rapkine A.J. Millis, S.W. Cheong, A.S. Cooper, and Z. Fisk, Phys. Rev. B 41, 11605 (1990). 9. T. Arima, Y. Tokura, S. Uchida, Phys. Rev. B 48, 6597 (1993). 10. E. Jurczek and T.M. Rice, Europhys. Lett. 1, 225 (1986). 11. C.M. Vaxma, Phys. Rev. Lett. 61, 2713 (1988). 12. P.W. Anderson, Science 235, 1196 (1987). 13. C.M. Varma, S. Schmitt-Rink, and E. Abrahams, Solid State Commun. 62, 681 (1987); V. Emery, Phys. Rev. Lett. 58, 2794 (1987). 14. V. Emery and G. Reiter, Phys. Rev. B 38, 4547 (1988); Phys. Rev. B 38, 11938 (1988). 15. C.X. Chen, and H.B. Schtter, Phys. Rev. B 41, 8702 (1990); Phys. Rev. B 43, 3771 (1991). 16. P. Horsch, Helvetica Physica Acta 63, 345 (1990). 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 wpÂ£> = 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+Â£ at several temperatures. 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 Commun. 75, 795 (1990); K.B. Lyons, P.A. Eleury, J.P. Remeika, A.S. Cooper, a nd T.J. Negran, Phys. Rev. B 37, 2353 (1988). 100. A. Moreo and E. Dagotto, Phys. Rev. B 42, 4786 (1990).; Phys. Rev. B 41, 9049 (1990). 101. I. Sega, and P. Prelovsck, Phys. Rev. B 42, 892 (1990). 102. D. Poilblanc, and H.J. Schulz, Phys. Rev. B 46, 6435 (1992).; Phys. Rev. B 47, 3268 (1993). 103. D. Mihailovic, C.M. Foster, K. Voss, and A. J. Heeger, Phys. Rev. B 42, 7989 (1990). 104. M. Suzuki, Phys. Rev. B 39, 2312 (1989). 105. D.T. Mark, P.G. Radaelli, J.D. Jorgensen, R.L. Hitterman, D.G. Hinks and S. Pei, Phys. Rev. B 46, 1144 (1992). 106. C. Chaillout, A. Santoro, J.P. Remeika, and M. Maxezio, Solid State Com mun. 65, 1363 (1988). 107. L.F. Mattheiss, and D.R. Hamann, Phys. Rev. B 28, 4227 (1983) ; Phys. Rev. B 26, 2686 (1982). 108. D.A. Papaconstantopoulos, A. Pasturel, and J.P. Juluen, Phys. Rev. B 40, 8844 (1989). 109. H. Takagi, M. Naito, S. Uchida, K. Kitazawa, S. Tanaka and A. Katsui, Solid State Commun. 55, 1019 (1985). 110. S. Uchida, S. Tajima, A. Masaki, S. Sugai, and S. Tanaka, J. Phys. Soc. Japan 11, 4395 (1985). 111. S. Tajima, S. Uchida, A. Masaki, H. Kitazawa, and S. Tanaka, Phys. Rev. B 35, 696 (1987). 112. K. Kitazawa, S. Uchida, and S. Tanaka, Physica B 135, 505 (1985). 113. H. Sato, S. Tajima, H. Takagi and S. Uchida, Nature 338, 241 (1989). 114. S. H. Blanton, R. T. Collins, K. H. Kelleher, L. D. Rotter, Z. Schlesinger, D. G. Hinks and Y. Zheng, Phys. Rev. B 47, 996 (1993). 115. M. A. Kaxlow, S. L. Cooper, A. L. Kotz, M. V. Klein, P. D. Han and D. A. Payne, Phys. Rev. B 48, 6499 (1993). 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 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 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{ densate. Theoretically, at T = 0 K, <7ia(u>)=0 up to u=2A. However, our results show that below 2A the uncertainty in 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 < Â£ 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 Â¡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 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 Â¡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 > 0Â£>. 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, Phys. Rev. Lett. 64, 1067 (1990). 54. Y. Tokura, S. Koshihara, T. Arima, H. Takaki, S. Ishibashi, T. Ido, and S. Uchida, Phys. Rev. B 41, 11657 (1990). 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). 64. Y.H. Kim, Phys. Rev. Lett. 67, 2227 (1991); Phys. Rev. B 36, 7252 (1987). 65. C.M. Foster, Solid State Commun. 71, 945 (1989). 66. X.X. Bi, P.C. Eklund, Phys. Rev. Lett. 70, 2625 (1993). 67. J.P. Falck, A. Levy, M.A. Kastner, and R.J. Birgeneau, Phys. Rev. B 48, 4043 (1993). 68. G.A. Thomas, D.H. Rapkine, S.L. Cooper, S.W. Cheong, A.S. Cooper, L.F. Schneemeyer, and J.V. Waszczak; Phys. Rev. B 45, 2474 (1992). 69. S. Uchida, Mod. Phys. Lett. B4, 513 (1990). 70. H. Takagi, S. Uchida, and Y. Tokura, Phys. Rev. Lett. 62, 1197 (1989). 71. P. Unger, and P. Fulde, Phys. Rev. B 51, 9245 (1995). 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,.xBÂ¡,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, 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\Â£> 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>) = Â£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 0Â£>, where 0Â£> 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+Â¡, 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 36. C.M. Vaxma, P.B. Littlewood, E. Abrahams, and A. Ruckenstein Phys. Rev. Lett. 63, 1996 (1989) P.B. Littlewood, and C.M. Varma, J. Appl. Phys. 69, 4979 (1991). 37. A. Virosztek, and J. Ruvalds, Phys. Rev. B 42, 4064 (1990). 38. J. Ruvalds, and A.V. Rosztek, Phys. Rev. B 43, 5498 (1991). 39. S.L. Cooper, D. Reznik, M.A. Kaxlow, R. Liu M.V. Klein, W.C. Lee, J. Giap- intzakis, and D.M. Ginsberg, Phys. Rev. B 47, 8233 (1993). 40. M. Shimada, M. Shimizu, J. Tanaka, I. Tanaka, and H, Kojima, Physica C 193, 277 (1992). 41. S. Etemad, D.E. Aspnes, M.K. Kelly, R. Thompson, J.-M. Tarascn, and G. W. Hull, Phys. Rev. B 37, 3396 (1988). 42. M. Suzuki, Phys. Rev. B 39, 2312 (1989). 43. G.L. Doll, J.T. Nicholls, M.S. Dresselhaus, A.M. Rao, J.M. Zhang, G.W. Lehman, P.C. Eklund, G. Dresselhaus, and A.J. Strauss Phys. Rev. B 38, 8850 (1988). 44. J. Orenstein, G.A. Thomas, A.J. Millis, S.L. Copper D.H. Rapkine, T. Timusk, D.F. Schneemeyer, and J.V. Waszczak, Phys. Rev. B 42, 6342 (1990). 45. J. Bouvier, N. Bontemps, M. Gabay, M. Nanot, and F. Queyroux, Phys. Rev. B 45, 8065 (1992). 46. B. Bucher, J. Kaxpinski, and P. Wachter, Phys. Rev. B 45, 3026 (1992). 47. K. Hirochi, S. Hayashi, H. Adachi, T. Mitsuyu, T. Hirao, K. Setsune, and K. Wasa, Physica C 160, 273 (1989). 48. E.V. Abel, V.S. Bagaev, D.N. Basov, O.V. Dolgov, A.F. Plotnikov, A.G. Poiaxkov, and W. Sadovsky, Solid State Commun. 79, 931 (1991). 49. J.G. Zhang, X.X. Bi, E. Mcrae, and P.C. Eklund, Phys. Rev. B 43, 5389 (1991). 50. S. Lupi, P. Calvani, M. Capizzi, P. Maselli, W. Sadowski and E. Walker, Phys. Rev. B 45, 12470 (1992). 51. L. Degiorgi, S. Rusieckim and Wachter, Physica C 161, 239 (1989). 52. Y. Watanabe, Z. Wang, S. A. Lyon, N. P. Ong, D. C. Tsui, J. M. Tarascn and E. Wang, Solid State Commun. 74, 757 (1990). 168 72. W.H. Stephan, W. Linden, and P. Horsch, Phys. Rev. B 39, 2924 (1989).: Phys. Rev. B 42, 8736 (1990). 73. M. Alexander, H. Romberg, N. Nucker, P. Adelman, J. Fink, J. T. Makkert, M. P. Maple, S. Uchida, H. Takaki, Y. Tokura and D. W. Murphy, Phys. Rev. B 43, 333 (1991). 74. T. Toyama and SA. Maekawa, J. Phys. Soc. Japan 60, 53 (1991). 75. R.O. Anderson, R. Classen, J.W. Allen, C.G. Olson, C. Janowitz, and L.Z. Liu, Phys. Rev. Lett. 70, 3163 (1993). 76. D.M. King, Z.X. Shen, D.S. Dessau, B.O. Wells, and W.E. Spicer, Phys. Rev. Lett. 70, 3159 (1993). 77. D.C. Johncton, J. Magn. Magn. Matr. 100, 218 (1991). 78. P. Bni, J.D. Axe, G. Shirane, R.J. Birgeneau, D.R. Gabbe, H.P. Jenssen, M.A. Kastner, C.J. Peters, P.J. Picone, and T.R. Thurston, Phys. Rev. B 38, 185 (1985). 79. J.D. Jorgensen, H.-B. Schttler, D.G. Hinks, D.W. Capone, H.K. Zhang, M.B. Brodsky, and D.J. Scalapino, Phys. Rev. Lett. 58, 1024 (1987). 80. R.J. Cava, A. Santoro, D.W. Johnson, Jr., and W.W. Rhodes, Phys. Rev. B 35, 6716 (1987). 81. W. Stephan and P. Horsch, Phys. Rev. B 42, 8736 (1990).. 82. K. Tamassaku, T. Ito, H. Takagi, and S. Uchida, Phys. Rev. Lett. 72, 3088 (1994). 83. R. Geick and K. Strobel, J. Phys. C 10, 4221 (1977). 84. R.T. Collins, Z. Schlesinger, G.V. Chandrashekhar, and M.W. Shafer, Phys. Rev. B 39, 2251 (1989); F.E. Bates and J.E. Eldridge, Solid State Commun. 72, 187 (1989); P.C. Eklund, A.M. Rao, G.W. Lehman, G.L. Doll, M.S. Dressel- haus, P.J. Picone, D.R. Gabbe, H.P. Jenssen, and G. Dresslhaus, J. Opt. Soc. Am. B 6, 389 (1988); M. Shimada, M. Shimizu, J. Tanaka, I. Tanaka, H. Kojima, Physica C 193, 277 (1992); A.V. Bazhenov, T.N. Fursova, V.B. Tim ofeev, A.S. Cooper, J.P. Remeika, Z. Fisk, Phys. Rev. B 40, 4413 (1989). 85. M. Mostoller, J. Zhang, A.M. Rao, and P.C. Eklund, Phys. Rev. B 41, 6488 (1990). 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(rÂ¡, 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)= <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 ^Â£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+Â¡, 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 (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 eÂ¡ 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 = Â£ (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- 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 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(rÂ¡, 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)= <7o 1 + u2t2 <^2 n(w) cr0u>T 1 + u2t2 (8) 24 Weak-Coupling Mittis-Bardeen Theory In the superconducting state, a complex conductivity and Bardeen22 showed that the ratio of the superconducting to normal conductivity within weak-coupling BCS theory is Â£1Â£ = 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) Â£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)Â¡(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 fÂ¡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 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 Orthorhombic 0.35 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,.xBÂ¡,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) + 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 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 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>) = Â£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. Â¡ 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 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 VceÂ¡Â¡ = 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(wÂ£), 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 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 upÂ£> 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 = 2ImÂ£. mo r*(w) 2 Figure 37 illustrates the imaginary part of quasiparticle self energy, Im Â£ of Nd2-xCeICu04 crystals below 5000 cm-1. ImÂ£ 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 Â£ in high doping concentration of x = 0.19. Second, for superconducting sample (x = 0.15), ImÂ£ 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 ImÂ£ 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, aÂ¡). Figure 38 (circles) 71 represents NÂ¡). However, it is difficult to define the Drude paxt from 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 ^Â£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 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+Â£ 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, 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\Â£> 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 Â¡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 > 0Â£>. 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{ densate. Theoretically, at T = 0 K, <7ia(u>)=0 up to u=2A. However, our results show that below 2A the uncertainty in 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 < Â£ 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 wpÂ£> = 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 Â¡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 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. 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 0Â£>, where 0Â£> 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. REFERENCES 1. J.G. Bednorz and K.A. Muller, Z. Phys. B 64, 189 (1986). 2. T.R. Thurston, M. Matsuda, K. Yamada, Y. Endoh, R.J. Birgeneau, P.M. Gehring, and G. Shirane, Phys. Rev. Lett. 65, 263 (1990). 3. S. Uchida, T. Ido, H. Takagi, T. Arima, Y. Tokura, and S. Tajima, Phys. Rev. B 43, 7942 (1991). 4. M. B. J. Meinders, H. Eskes and G. A. Sawatzky, Phys. Rev. B 48, 3916 (1993). 5. G.A. Thomas, D.H. Rapkine, S.L. Cooper, S.W. Cheong, and A.S. Cooper, Phys. Rev. Lett. 20, 2906 (1991). 6. S. Uchida, Mod. Phys. Lett. B4, 513 (1990); S. Uchida, and T. Ido, JJAP Series 7 Mechanisms of Superconductivity, 133 (1992). 7. T. Arima, K. Kikuchi, M. Rasura, S. Koshihara, Y. Tokura, T. Ido, and S. Uchida, Phys. Rev. B 44, 917 (1991). 8. S.L. Cooper, G.A. Thomas, J. Orenstein, D.H. Rapkine A.J. Millis, S.W. Cheong, A.S. Cooper, and Z. Fisk, Phys. Rev. B 41, 11605 (1990). 9. T. Arima, Y. Tokura, S. Uchida, Phys. Rev. B 48, 6597 (1993). 10. E. Jurczek and T.M. Rice, Europhys. Lett. 1, 225 (1986). 11. C.M. Vaxma, Phys. Rev. Lett. 61, 2713 (1988). 12. P.W. Anderson, Science 235, 1196 (1987). 13. C.M. Varma, S. Schmitt-Rink, and E. Abrahams, Solid State Commun. 62, 681 (1987); V. Emery, Phys. Rev. Lett. 58, 2794 (1987). 14. V. Emery and G. Reiter, Phys. Rev. B 38, 4547 (1988); Phys. Rev. B 38, 11938 (1988). 15. C.X. Chen, and H.B. Schtter, Phys. Rev. B 41, 8702 (1990); Phys. Rev. B 43, 3771 (1991). 16. P. Horsch, Helvetica Physica Acta 63, 345 (1990). 164 165 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). 166 36. C.M. Vaxma, P.B. Littlewood, E. Abrahams, and A. Ruckenstein Phys. Rev. Lett. 63, 1996 (1989) P.B. Littlewood, and C.M. Varma, J. Appl. Phys. 69, 4979 (1991). 37. A. Virosztek, and J. Ruvalds, Phys. Rev. B 42, 4064 (1990). 38. J. Ruvalds, and A.V. Rosztek, Phys. Rev. B 43, 5498 (1991). 39. S.L. Cooper, D. Reznik, M.A. Kaxlow, R. Liu M.V. Klein, W.C. Lee, J. Giap- intzakis, and D.M. Ginsberg, Phys. Rev. B 47, 8233 (1993). 40. M. Shimada, M. Shimizu, J. Tanaka, I. Tanaka, and H, Kojima, Physica C 193, 277 (1992). 41. S. Etemad, D.E. Aspnes, M.K. Kelly, R. Thompson, J.-M. Tarascn, and G. W. Hull, Phys. Rev. B 37, 3396 (1988). 42. M. Suzuki, Phys. Rev. B 39, 2312 (1989). 43. G.L. Doll, J.T. Nicholls, M.S. Dresselhaus, A.M. Rao, J.M. Zhang, G.W. Lehman, P.C. Eklund, G. Dresselhaus, and A.J. Strauss Phys. Rev. B 38, 8850 (1988). 44. J. Orenstein, G.A. Thomas, A.J. Millis, S.L. Copper D.H. Rapkine, T. Timusk, D.F. Schneemeyer, and J.V. Waszczak, Phys. Rev. B 42, 6342 (1990). 45. J. Bouvier, N. Bontemps, M. Gabay, M. Nanot, and F. Queyroux, Phys. Rev. B 45, 8065 (1992). 46. B. Bucher, J. Kaxpinski, and P. Wachter, Phys. Rev. B 45, 3026 (1992). 47. K. Hirochi, S. Hayashi, H. Adachi, T. Mitsuyu, T. Hirao, K. Setsune, and K. Wasa, Physica C 160, 273 (1989). 48. E.V. Abel, V.S. Bagaev, D.N. Basov, O.V. Dolgov, A.F. Plotnikov, A.G. Poiaxkov, and W. Sadovsky, Solid State Commun. 79, 931 (1991). 49. J.G. Zhang, X.X. Bi, E. Mcrae, and P.C. Eklund, Phys. Rev. B 43, 5389 (1991). 50. S. Lupi, P. Calvani, M. Capizzi, P. Maselli, W. Sadowski and E. Walker, Phys. Rev. B 45, 12470 (1992). 51. L. Degiorgi, S. Rusieckim and Wachter, Physica C 161, 239 (1989). 52. Y. Watanabe, Z. Wang, S. A. Lyon, N. P. Ong, D. C. Tsui, J. M. Tarascn and E. Wang, Solid State Commun. 74, 757 (1990). 167 53. S. H. Wang, Q. Song, B. P. dayman, J. L. Peng, L. Zhang and R. N. Shelton, Phys. Rev. Lett. 64, 1067 (1990). 54. Y. Tokura, S. Koshihara, T. Arima, H. Takaki, S. Ishibashi, T. Ido, and S. Uchida, Phys. Rev. B 41, 11657 (1990). 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). 64. Y.H. Kim, Phys. Rev. Lett. 67, 2227 (1991); Phys. Rev. B 36, 7252 (1987). 65. C.M. Foster, Solid State Commun. 71, 945 (1989). 66. X.X. Bi, P.C. Eklund, Phys. Rev. Lett. 70, 2625 (1993). 67. J.P. Falck, A. Levy, M.A. Kastner, and R.J. Birgeneau, Phys. Rev. B 48, 4043 (1993). 68. G.A. Thomas, D.H. Rapkine, S.L. Cooper, S.W. Cheong, A.S. Cooper, L.F. Schneemeyer, and J.V. Waszczak; Phys. Rev. B 45, 2474 (1992). 69. S. Uchida, Mod. Phys. Lett. B4, 513 (1990). 70. H. Takagi, S. Uchida, and Y. Tokura, Phys. Rev. Lett. 62, 1197 (1989). 71. P. Unger, and P. Fulde, Phys. Rev. B 51, 9245 (1995). 168 72. W.H. Stephan, W. Linden, and P. Horsch, Phys. Rev. B 39, 2924 (1989).: Phys. Rev. B 42, 8736 (1990). 73. M. Alexander, H. Romberg, N. Nucker, P. Adelman, J. Fink, J. T. Makkert, M. P. Maple, S. Uchida, H. Takaki, Y. Tokura and D. W. Murphy, Phys. Rev. B 43, 333 (1991). 74. T. Toyama and SA. Maekawa, J. Phys. Soc. Japan 60, 53 (1991). 75. R.O. Anderson, R. Classen, J.W. Allen, C.G. Olson, C. Janowitz, and L.Z. Liu, Phys. Rev. Lett. 70, 3163 (1993). 76. D.M. King, Z.X. Shen, D.S. Dessau, B.O. Wells, and W.E. Spicer, Phys. Rev. Lett. 70, 3159 (1993). 77. D.C. Johncton, J. Magn. Magn. Matr. 100, 218 (1991). 78. P. Bni, J.D. Axe, G. Shirane, R.J. Birgeneau, D.R. Gabbe, H.P. Jenssen, M.A. Kastner, C.J. Peters, P.J. Picone, and T.R. Thurston, Phys. Rev. B 38, 185 (1985). 79. J.D. Jorgensen, H.-B. Schttler, D.G. Hinks, D.W. Capone, H.K. Zhang, M.B. Brodsky, and D.J. Scalapino, Phys. Rev. Lett. 58, 1024 (1987). 80. R.J. Cava, A. Santoro, D.W. Johnson, Jr., and W.W. Rhodes, Phys. Rev. B 35, 6716 (1987). 81. W. Stephan and P. Horsch, Phys. Rev. B 42, 8736 (1990).. 82. K. Tamassaku, T. Ito, H. Takagi, and S. Uchida, Phys. Rev. Lett. 72, 3088 (1994). 83. R. Geick and K. Strobel, J. Phys. C 10, 4221 (1977). 84. R.T. Collins, Z. Schlesinger, G.V. Chandrashekhar, and M.W. Shafer, Phys. Rev. B 39, 2251 (1989); F.E. Bates and J.E. Eldridge, Solid State Commun. 72, 187 (1989); P.C. Eklund, A.M. Rao, G.W. Lehman, G.L. Doll, M.S. Dressel- haus, P.J. Picone, D.R. Gabbe, H.P. Jenssen, and G. Dresslhaus, J. Opt. Soc. Am. B 6, 389 (1988); M. Shimada, M. Shimizu, J. Tanaka, I. Tanaka, H. Kojima, Physica C 193, 277 (1992); A.V. Bazhenov, T.N. Fursova, V.B. Tim ofeev, A.S. Cooper, J.P. Remeika, Z. Fisk, Phys. Rev. B 40, 4413 (1989). 85. M. Mostoller, J. Zhang, A.M. Rao, and P.C. Eklund, Phys. Rev. B 41, 6488 (1990). 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. Axe, G. Shirane, R.J. Birgeneau, D.R. Gabbe, H.P.Jenssen, M.A. Kastner, and C.J. Peters, Phys. Rev. B 38,185 (1988); T.R. Thurston, R.J. Birgeneau, D.R. Gabbe, H.P. Jenssen, M.A. Kastner, P.J. Picone, N.W. Preyer, J.D. Axe, P. Bni, and G. Shirane, Phys. Rev. B 39, 4327 (1989). 87. S. Sugai, Phys. Rev. B 39, 4306 (1989);Physica C 185-189, 76 (1991 )]Phys. 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. Opt. Soc. Am. B 6, 455 (1989); S. Sugai, M. Sato, and S. Hosoya, Jpn. J. Appl. Phys. 26, L495 (1987). 88. M.J. Rice, Phys. Rev. Lett. 37, 36 (1976). 89. M.J. Rice, L. Pietronero, and P. Bruesch, Solid State Commun. 21, 757 (1977). 90. G.A. Thomas, D.H. Rapkine, S.-W. Cheong, and L.F. Schneemeyer, Phys. Rev. B 47, 11369 (1993). 91. K. Yonemitsu, A.R. Bishop, and J. Lorenzana, Phys. Rev. Lett. 69, 965 (1992). 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. 94. C.Y. Chen, R.J. Birgeneau, M.A. Kastner, N.W. Preyer, and T. Thio, Phys. Rev. B 43, 392 (1991); Phys. Rev. Lett. 63, 2307 (1989). 95. R.C. Milwaxd and L.J. Neuringer Phys. Rev. Lett. 15, 664 (1965). 96. S. Tanaka and H.Y. Fan, Phys. Rev. 132, 1516 (1963). 97. J. Blinowski and J. Mycielski, Phys. Rev. 136, A266 (1964); Phys. Rev. 140, A1024 (1965); N.F. Mott Phil. Mag. 22, 7 (1970). 98. N.W. Preyer, R.J. Birgeneau, C.Y. Chen, D.R. Gabbe, H.P. Jenssen, M.A. Kastner, P.J. Picone, and T. Thio Phys. Rev. B 39, 11563 (1989). S. Sugai, S. Shamoto, and M. Sato, Phys. Rev. B 38, 6436 (1988); Phys. Rev. B 42, 1045 (1990); Solid State Commun. 76, 365 (1990); Solid State 99. 170 Commun. 75, 795 (1990); K.B. Lyons, P.A. Eleury, J.P. Remeika, A.S. Cooper, a nd T.J. Negran, Phys. Rev. B 37, 2353 (1988). 100. A. Moreo and E. Dagotto, Phys. Rev. B 42, 4786 (1990).; Phys. Rev. B 41, 9049 (1990). 101. I. Sega, and P. Prelovsck, Phys. Rev. B 42, 892 (1990). 102. D. Poilblanc, and H.J. Schulz, Phys. Rev. B 46, 6435 (1992).; Phys. Rev. B 47, 3268 (1993). 103. D. Mihailovic, C.M. Foster, K. Voss, and A. J. Heeger, Phys. Rev. B 42, 7989 (1990). 104. M. Suzuki, Phys. Rev. B 39, 2312 (1989). 105. D.T. Mark, P.G. Radaelli, J.D. Jorgensen, R.L. Hitterman, D.G. Hinks and S. Pei, Phys. Rev. B 46, 1144 (1992). 106. C. Chaillout, A. Santoro, J.P. Remeika, and M. Maxezio, Solid State Com mun. 65, 1363 (1988). 107. L.F. Mattheiss, and D.R. Hamann, Phys. Rev. B 28, 4227 (1983) ; Phys. Rev. B 26, 2686 (1982). 108. D.A. Papaconstantopoulos, A. Pasturel, and J.P. Juluen, Phys. Rev. B 40, 8844 (1989). 109. H. Takagi, M. Naito, S. Uchida, K. Kitazawa, S. Tanaka and A. Katsui, Solid State Commun. 55, 1019 (1985). 110. S. Uchida, S. Tajima, A. Masaki, S. Sugai, and S. Tanaka, J. Phys. Soc. Japan 11, 4395 (1985). 111. S. Tajima, S. Uchida, A. Masaki, H. Kitazawa, and S. Tanaka, Phys. Rev. B 35, 696 (1987). 112. K. Kitazawa, S. Uchida, and S. Tanaka, Physica B 135, 505 (1985). 113. H. Sato, S. Tajima, H. Takagi and S. Uchida, Nature 338, 241 (1989). 114. S. H. Blanton, R. T. Collins, K. H. Kelleher, L. D. Rotter, Z. Schlesinger, D. G. Hinks and Y. Zheng, Phys. Rev. B 47, 996 (1993). 115. M. A. Kaxlow, S. L. Cooper, A. L. Kotz, M. V. Klein, P. D. Han and D. A. Payne, Phys. Rev. B 48, 6499 (1993). 171 116. T.D. Thanh, A. Koma, and S. Tanaka, Appl. Phys. 22, 205 (1980). 117. K. F. McCarty, H. B. Radousky, D. G. Hinks, Y. Zheng, A. W. Mitchell, T. J. 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). 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). 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). 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 (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 eÂ¡ 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 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 Orthorhombic 0.35 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(wÂ£), 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, aÂ¡). 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 fÂ¡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 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 |