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TEMPERATURE DEPENDENCE OF INFRARED AND OPTICAL PROPERTIES OF HIGH TEMPERATURE SUPERCONDUCTORS By FENG GAO A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1992 ACKNOWLEDGMENTS It is my great pleasure to thank my advisor, professor David B. Tanner, for giving me the opportunity to study the most exciting new area of solid state physicshigh Tc superconductivityand for his valuable guidance and advice, constant support, patience and encouragement all throughout my graduate work at the University of Florida. I also thank professors N.S. Sullivan, H.A. Van Rinsvelt, J.W Dufty, M.W. Meisel, and J.H. Simmons for serving on my supervisory committee. My recent years of academic and research experience in the "Florida Group" have been interesting, enjoyable, challenging, hardworking and thus the most memorable years in my life. I would like to extend my thanks to all of my colleagues in this group, past and present, for their friendship, useful conversation and cooperation which were essential in developing a pleasant research environment. In addition, I appreciate C.D. Porter's help in computer software and programming. Another special thanks goes to professor D.B Tanner for his great help in using the EZ'EX he wrote, which made it easy for me to type this dissertation. I also want to thank the staff members in the physics department machine shop and the engineers in the condensed matter group for their technical support. I have had many interesting discussions with Drs. S.L. Herr, K. Kamaras, D.B. Romero, V. Zelezny, T. Timusk and S. Etemad. In particular, I am very grateful to Dr. G.L. Carr for his valuable help, suggestions and collaboration during my early work. His kindness and knowledge are admired. I would like to express my acknowledgment to Dr. S. Etemad at the Bell Com munication Research and to Drs. J. Talvacchio and M. Forrester at the Westing house Science and Technology Center for providing good quality highTc films, which were the major topics in this dissertation. Also to be thanked are M. Doss and C. Gallo at 3M company for preparing the polycrystalline samples studied in this work. Other people that deserve thanks are Drs. G.P. Williams and C.J. Hirschmugl at Brookhaven National Laboratory for their help in the farinfrared measurement using the farinfrared beamline at the National Synchrotron Light Source. I also have special thanks to my parents, brother and sisters, my wife, my daughter and son for their love, care, encouragement and understanding during my graduate studies. A final thanks goes to the organizations which supported this research through the U.S. Defense Advanced Research Projects Agency Grant MDA97288J1006, and the National Science Foundation Grant DMR9101676. TABLE OF CONTENTS ACKNOWLEDGMENTS ....................... ABSTRACT . . . . CHAPTERS I. INTRODUCTION ............. II. BRIEF SURVEY OF SUPERCONDUCTIVITY Fundamental Properties ........... Crystal Structure and Phase Diagram . Other Physical Properties . . . .. 1 . . 6 . .. 7 . 9 . . 8 . . 9 III. THEORY . . . Optical Theory . . . Optical Response of the Medium . . Determination of Optical Constants . . Reflectance of thick crystals . . KramersKronig relations . . Combination of reflectance and transmittance of thin films Lorentz and Drude Models . . Sum Rule . . . Superconductivity . . . Perfect Conductor ................... Superconductor .. .. . IV. INFRARED TECHNIQUES . Interferometry ............ Infrared Radiation at Low Frequencies Fourier Transform Spectroscopy . Optical Spectrometers . Bruker Interferometer . Michelson Interferometer . PerkinElmer Monochromator . 13 . 13 . 13 . 16 . 17 . 18 S20 . 24 . 26 . 26 S26 . 27 . 33 . ......... 33 . . 33 . 36 . 38 . 38 . 40 . 40 V. SAMPLE PREPARATION AND CHARACTERISTICS . La2.SSrCuO4 Epitaxial Films . . . YBa2Cu307aO Oriented Films . . . YBa2.SrzCua30_ Polycrystalline Samples . . Reannealing Procedures for YBa2Cu307 . . Meissner Effect Test and Susceptibility . . VI. EXPERIMENTS AND LOW TEMPERATURE TECHNIQUES . Low Temperature Apparatus . . . Reflectance Measurements and UncertaintiesLa2_Sr cCuO4 Films Procedures in KramersKronig Analysis . . Highfrequency extrapolation . . Lowfrequency extrapolation . . . Combination of .A(w) and f(w) MeasurementsYBa2Cu3aO7 Films . Measurement of YBa2aSrCu307_6 Pellets . VII. OPTICAL STUDIES OF La2.Sr.CuO4 FILMS . Results and Discussion . . Infrared Phonons ............... Structural phase transition . . Frequency shift and lifetime . . TwoComponent Approach . . The freecarrier componentwpD and r . The midinfrared absorption . . Holstein effect .. .. . Superconductingtonormal ratios . Extra absorption below T . . Sum RuleSuperconducting Condensate . OneComponent Approach . . Mass enhancement m*/mb and self energy E(w) Effective scattering rate 1/r*(w) . Loss Function .. .. .. . The Superconducting Gap . . Summary . . . . 72 . 76 . 76 . 76 . 77 . 78 . 79 . 81 . 83 84 . 85 . 86 . 87 . 89 . 90 . 92 . 93 . 95 . 97 VIII. FARINFRARED STUDIES OF YBa2Cu3O7_ FILMS Results and Discussion . . FreeStanding Transmittance f (w) . TwoFluid Model Fit .............. Relaxation Rate and Superfluid Condensate . "Coherence Peak" .............. MidInfrared Band and Superconducting Gap . Summary . . . IX. YBa2Cu307_ POLYCRYSTALLINE SAMPLES . Infrared Spectra and Analysis . . Superconducting Condensate . . Phonon Frequency and Linewidth . Effective Medium Approximation . . Anisotropic Medium . . EMA and MGT Approaches ........... Weighted Average ............... Summary . . . X. SUMMARY AND CONCLUSIONS . . Normal State .. .. .. . Superconducting State . . APPENDICES A. DATA FOR OTHER SAMPLES B. COMPUTER PROGRAMS .... REFERENCES ............. BIOGRAPHICAL SKETCH ....... . 115 . 116 . 119 . 120 . 122 . 123 . 124 . 126 . 138 . 138 . 139 . 140 . 142 . 143 . 145 . 147 . 147 . 162 . 162 . 163 . . 165 . . 170 . . 185 . . 194 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy TEMPERATURE DEPENDENCE OF INFRARED AND OPTICAL PROPERTIES OF HIGH TEMPERATURE SUPERCONDUCTORS By Feng Gao December 1992 Chairman: David B. Tanner Major Department: Physics The infrared properties of the newly discovered high temperature super conductors are extremely unusual. We have extensively studied two cuprate families: La2;SrCuO4 and YBa2Cu307_6. The former is the material where high Tc was first discovered; the latter is the first substance with a transition temperature above the liquid nitrogen boiling point. The samples studied were abplaneoriented super conducting thin films deposited on insulating substrates. Optical transmittance and reflectance measurements were made with the films in both the normal and super conducting states. Other superconducting samples in forms of randomly oriented or textured polycrystallines and granular films were also measured. The infrared conductivity of these cuprates in the normal state showed a strong, nearly temperatureindependent broad band in the midinfrared region in addition to a strongly temperaturedependent narrow Drude band in the far infrared. Most of the freecarrier oscillator strength was found to shift into the zerofrequency deltafunction conductivity in the superconducting state as the free carriers condense into Cooper pairs. The CuO2plane London penetration depth AL (~2700 A for La2,SrzCuO4, ~1700 A for YBa2Cu3O76) was estimated from the superfluid density. The low frequency tail of the midinfrared absorption, a direct particlehole excitation, re mained for T < Tc. One striking result observed was the linear temperature dependence of the quasi particle scattering rate above Tc, followed by a rapid drop just below Tc. Another was that the Tdependent conductivity of the YBa2Cu307O films at low frequencies exhibited a peak just below Tc, resembling the "coherence peak" of ordinary super conductors, yet having a different origin. This peak is associated with the dramatic decrease of the scattering rate rather than with the coherence effect. Finally, the superconducting gap absorption was invisible in the infrared spectra, suggesting that these cuprates were "cleanlimit" superconductors. The energy gap for YBazCu3aO7 might be deduced indirectly from the Tdependence of the farinfrared vibrational features of the polycrystalline samples, suggesting 3.0 < 2A/kBTc < 4.2. CHAPTER I INTRODUCTION This dissertation describes a detailed study of the optical propertiesfrom far infrared through ultraviolet frequenciesof the newly discovered copper oxide ma terials which have high superconducting transition temperatures. It concentrates on the investigations on La2zSrzCuO4 (To 30 K) and YBa2Cu307_6 (To 90 K) thin films and polycrystalline samples through measurement and analysis of optical transmittance and/or reflectance as a function of incident photon frequency (w) and sample temperature (T). Much effort in this work has been devoted to study the anomalous nonDrude response in the midinfrared region, the behavior of supercon ducting energy gap, and the role of phonon and lowlying excitations. The discovery of high temperature superconductors (HTSC) by Bednorz and Miiller1 in 1986 and Wu et al.2 in 1987 has stimulated considerable interest in the scientific world. These new materials are interesting because they present an exciting new regime for superconductivity and have the potential of valuable practical applica tions. The most fundamental properties of these highTc cuprates are threefold: high superconducting transition temperature (Tc), short coherence length (t), and large anisotropy. The conventional mechanism of pairing via electronphonon interaction cannot describe this oxide materials simply because Tc is to high. Attempts have been made to determine whether these highTc superconductors are fundamentally different from the conventional superconductors, which have been well described by the BardeenCooperSchrieffer (BCS) theory.3 Infrared spectroscopy, a powerful and successful technique for studying classical superconductors,48 has been widely used to study such fundamental physical proper ties as superconducting gaps, crystal vibrations, electronphonon couplings, lowlying excitations, density of states, and electronic band structure. In the framework of BCS theory, the existence of an energy gap means that a bulk superconductor at T < Tc is a perfect reflector of electromagnetic radiation for photon energy (hw) less than the gap energy (2A). Photons with fiw > 2A can disassociate the Cooper pairs and cause quasiparticle transitions to unoccupied levels above the gap, making the superconductor behave like a normal metal. This is indeed the case for conventional superconductors, as first verified by Glover and Tinkham.4 However, the highTc superconductors show much more complicated behavior. A major difficulty with the observability of the superconducting gap in the infrared spectra is that the reflectance of HTSC can hardly be distinguished from unity at low frequencies within which a gap is expected. Because the transition temperature is comparable to the Debye temperature (Tc Te), the superconducting gap ener gies of HTSC are expected to lie in the frequency range where infrared active optical phonons are present. In return, these phonons (if not well screened by free carri ers) may obscure the observation of the energy gap. It has been reported9 by direct bolometric absorption measurements that these materials have a finite absorption down to the frequencies well below the BCS gap at T < Tc. In both the normal and superconducting states, a nonDrude lowlying excitation spectrum is present in the midinfrared region. Added to the complexity is the anisotropy of these mate rials, which means that for polycrystalline and twinned samples, only an effective response can be measured. Furthermore, thin film measurements are complicated by contributions from the substrates. Although the optical results generally agree among different investigators, the interpretation in many waysparticularly about the infrared determinations of the gap and the origin of the midinfrared bandstill remains controversial and is not clearly understood. One feature common to all cuprate superconductors is the existence of quasitwo dimensional CuO planes, which appear to play a major role in highTo superconduc tivity. Therefore, it is of primary importance to investigate the intrinsic electronic responses of these planes. Most optical studies to date have concentrated on the 90 K transition temperature YBa2Cu307r (YBCO) system, which contains both CuO2 planes and CuO chains. (For reviews, see Refs. 1013.) It has been observed, however, that the quasionedimensional CuO chains in YBCO have a substantial contribution to the optical conductivity,14'15 which has complicated the analysis of this material. In contrast, the La2.SrzCu04 (LSCO) system, which has the simplest crystal struc ture in the cuprate family and contains only single CuO2 layer per formula unit, has been studied in most cases on the sintered polycrystalline samples.1623 Because the LSCO materials are strongly anisotropic, it is difficult to determine the intrinsic na ture of the CuO2 layers from measurements of polycrystalline samples. A few optical measurements, mostly restricted to the compositiondependence studies at room tem perature, on La2zSrzCuO4 single crystals or thin films have been made,2429 and most recently, a systematic temperaturedependent optical study on oriented samples of this material has been reported.30 In no case can the normalstate infrared conductivity be described by a simple Drude model. In many studies,1012,3036 this nonDrude conductivity observed in ceramics, crystals and thin films has been described by a twocomponent approach: a narrow, strongly temperaturedependent Drude absorption centered at the origin and a broad, nearly temperatureindependent midinfrared (MIR) band. The Drude absorption is due to the free carriers which are responsible for the dc transport and which condense into a superfluid below Tc whereas the MIR absorption is due to the bound carriers which have a semiconductorlike gap. This approach has been adopted because it is clear that a single, strongly damped, Drude term, used to model the infrared date in many early measurements, does not work.10'12 An alternative is a singlecomponent approach: all of the infrared absorption is due to one type of carriers, with a strong frequency dependence in the scattering rate and effective mass. This approach also leads to a broad range of optically inactive excitations in the midinfrared region while at low frequencies (including dc) the conductivity goes inversely with the temperature. This approach has been described in the framework of the "marginal Fermi liquid" (MFL) theory of Varma et al.37 and the "nested Fermi liquid" (NFL) theory of Virosztek and Ruvalds.38'39 Attempts have been made 10,15,4046 to assign the superconducting energy gap either to the edge of a rapid drop in the ratio 5,(w)/S,,(w) of the superconducting to normal reflectance or to the absorption onset of the conductivity, ai,(w), in the superconducting state. The values of 2A(O)/kBTc obtained in this way range between 2.5 and 8. There has been a controversy, however, whether these structures are due to the energy gap or are part of the midinfrared absorption.3235,4749 Most of the work to date determines the frequencydependent conductivity through KramersKronig analysis of reflectance measurements.1013,3035,4050 The main advantage of this technique is that a large amount of important information about the material can be extracted easily once the reflectance over a wide frequency range is measured. However, there is a drawback of artificial extrapolations beyond the measured frequency range, which is required by the KramersKronig integral. Furthermore, the optical properties derived from this method is very dependent on the accuracy of the reflectance especially for highly reflecting materials. In contrast, the optical functions derived from transmission measurement are much less sensitive to the errors than from reflectance. In this work, both techniques are employed for data analysis. We will present a method of extracting the optical functions directly from combined measurements of reflectance and transmittance without referring to the KramersKronig analysis. The rest of this dissertation is organized as follows. The second chapter is a re view of the previous research work on superconductivity and of some fundamental properties of HTSC. Chapter III discusses the basic theory about the general optical properties of solids and the phenomenological models in description of superconduc tivity. Chapter IV will describe the infrared technique and experimental apparatus used in this study. Sample preparation and characteristics will be presented in chapter V. Chapter VI explains the procedures of experimental measurement and low temper ature technique along with data collecting and processing. Chapter VII, VIII and IX are devoted to data analysis and result discussion, in which the experimental optical spectra for several highTc samples are presented and discussed in detail. Various theoretical models are used to describe the physical properties of the oxide materials in the normal and superconducting states. Finally, Summary and conclusions are given in chapter X. CHAPTER II BRIEF SURVEY OF SUPERCONDUCTIVITY Since Kamerlingh Onnes' 1911 discovery of superconductivity in mercury cooled to 4.2 K, the observed transition temperatures, Tc, have gradually moved upward but remained strictly a low temperature phenomenon. The most important developments of this area in the recent decades include: the GinzburgLandau theory,51 the Pippard nonlocal electrodynamics,52 and the discovery of isotope effect53,54 in early 1950's; the first satisfactory description of microscopic mechanism (in terms of energy gap and Cooper pairs) by the BCS theory3 in 1957; the prediction of Josephson effect55 in 1962; and, most recently, the discovery of highTc superconductors in late 1980's. The highest record forTo before 1986 was 23.2 K in Nb3Ge found by L. R. Testardi et al.56 in 1973. The extraordinarily high Tc superconductivity era started in 1986 when J. G. Bednorz and K. A. Miiller1 observed that La2,BaCuO4 became super conducting below 35 K. This opened the way for intensive work on high temperature superconductors. Another breakthrough soon arrived by the announcement2 in 1987 of YBa2Cu3O76, with Tc above 90 K capable of becoming superconducting in liquid nitrogen. In 1988, two families of copper oxide compounds with even higher values of Tc, 110 K in Bi2Sr2Ca2Cu30057 and 125 K in Tl2Ba2Ca2Cu3Ol0,58 were found. The discovery of highTc oxides has generated tremendous excitement in public because of the new technological promises of these materials. Superconductivity can now be achieved with a simpler coolantliquid nitrogen. The electronic properties of HTSC can be exploited to make more efficient microelectronic components such as passive microwave devices, logic circuits, computer interconnect boards, and infrared detectors. The fabrication of ultrasensitive sensors and production of Josephson microwave mixers with a wider electromagnetic wave windows become possible with the use of HTSC because of the larger energy gap expected for these materials. Fundamental Properties As stated in the introductory chapter, there are three unusual fundamental prop erties which distinguish these new materials from the conventional superconductors. First, the transition temperature Tc is high. Aside from the promising applications as stated above, high Tc also presents a new challenge to scientific investigators, as many lowlying excitations are present near Tc. These excitations will affect some funda mental properties of the superconductors such as Tc and critical current Je as well as the energy gap 2A, if such excitonic energies are large enough to break Cooper pairs. These highTc materials have low dc conductivity in the normal state due to lower conduction electronic concentration, which results in a longer penetration depth. The high value of Tc has turned out to complicate the behavior of the materials in both the normal state and superconducting state. Second, as a consequence of higher Tc or larger superconducting gap compared to the classical superconductors, HTSC have a shorter coherence length with typical value of ( = hvF/kTc ~ 10 A, which is comparable to the unit cell dimensions. All HTSC are type II superconductors because of ( < A, where A is the electromagnetic penetration depth with a value of the order 1000 A. The shortness of makes the superconductivity sensitive to small scale structures. In turn the fluctuations play a much larger role in highTo materials than in classical superconductors. A small ( also leads to a high value of upper critical magnetic field He2. Thirdly, HTSC show large optical anisotropy. The physical properties such as optical conductivity and other fundamental physical parameters vary in different di rections. The resistivity, for instance, along the caxis (pc) is larger than within the CuOplane (Pab) by a factor of ~ 102. Therefore, high quality crystals and oriented films are essential for experimental studies, because the anisotropy of the layered cop per oxides requires that the samples be measured along different axes of the crystals. Finally, it has been estimated and will be shown in this dissertation that the mean free path I = VFr 100 A, making S<1 It is this condition that places the HTSC in the "clean limit" which is sharply distin guished from most conventional superconductors. The latter ones at low temperatures are usually in the anomalous skin effect limit or dirty limit. Crystal Structure and Phase Diagram The crystal structure and the phase diagrams of La2,SrCuO4 and YBa2Cu307. are shown in Figs. 1 and 2. Note that both figures show the structure in a unit cell which contains two formula units. The parent compounds of both ma terials are antiferromagnetic semiconductors. When doped with holes, they become metallic. For La2_,Sr,CuO4, it has the perovskite K2NiF4 structure and is the body centered tetragonal Bravais lattice (I4/mmm). The typical values of lattice constants are: a b 3.78 A, and c 13.2 A. As the temperature is lowered, the crystal exhibits a second order structural phase transition from tetragonal to orthorhombic phase (Cmca). This transition involves a staggered rotation of the CuO6 octahedra as shown by the arrows in Fig. 1. Upon further cooling the crystal exhibits another transition from metallic to superconducting phase. The optimum Sr doping for su perconductivity lies in the range of 0.1 < x < 0.2. Since La2.SrzCuO4 dose not have chains, one expects that the dynamic conductivity r(w) of an oriented crystal probed to be solely due to the intrinsic response from the CuO2 planes, provided the electric field vector E is parallel to these planes. In contrast with La2SSrzCuO4 crystals, the presence of CuO chains, as shown in Fig. 2, along the baxis in the YBa2Cu3aO7 system complicates the analysis of this materials. A great deal of effort has been devoted to distinguish the role of the quasionedimensional chains from that of the quasitwodimensional planes in recent years. However, most of the YBa2Cu3O7a samples are usually microtwinned, making it difficult to identify the difference. One the other hand, by far YBa2Cu3aOr are the most studied material in highT. family. The parent compound YBa2CuasO is tetragonal while the superconducting YBa2Cu307_ is orthorhombic. Typical values of the lattice constants are: a = 3.82 A, b = 3.88 A, and c = 11.68 A. The transition temperature of this material is very dependent on the oxygen doping concentration as illustrated in Fig. 2. Other Physical Properties A good superconductor is usually a poor electric conductor in the normal state. The reason is that the conventional electron pairing requires a strong electronphonon interaction in order to produce a high transition temperature. It is the same inter action that causes the large electronic scattering rate hence high resistivity in the normal state. The HTSC are quite poor electric conductors above Tc, thus one ex pects a strong electron scattering mechanism in these materials. It is widely believed that the electronphonon interaction plays a minor role in the superconductivity for YBa2Cu307_6. However, a significant isotope shift (a 0.2) due to partial substitution of "80 for 160 in La.s85Sro.15CuO4 has been observed and interpreted as evidence for strong electronphonon coupling.59,60 This implies that phonons may still play an important role, if not a key role, in the pairing mech anism. On the other hand, the observed linear behavior of the dc resistivity for La2aSrzCuO4 up to 1100 K implies a weak electronphonon coupling for the free carriers.61 Therefore the La2ZSrzCu04 system is expected to bridge the classical superconductors and HTSC. In optical studies, a lot of effort has been made in recent years to study the nonDrude response in the midinfrared region and to discover the superconducting energy gap. It has been observed that the MIR absorption is absent in the undoped parent compounds such as La2CuO4 and YBa2Cu306. For La2aSrzCuO4, Uchida et al.29 have reported that the MIR absorption band develops with increasing dopant concentration and then exhibits a saturation in the higher compositional range 0.1 < x < 0.25. Similar effects are observed in doping of ntype Pr2sCezCuO4 by Cooper et al.62 As a consequence of the redistribution of the the O 2p and Cu 3d orbitals upon doping, spectral weight is rapidly transferred from the inplane O 2p + Cu 3d charge transfer (CT) excitations above 2 eV to the freecarrier absorption (Drude band) and the lowenergy excitations (MIR band) below 1.5 eV. Therefore both the Drude and MIR absorptions in HTSC appear to be related to the introduction of holes on the CuO2 layers (or CuO chains) by doping. For La2_Sr,CuO4, the CT gap becomes weaker or fills in and the phonons are obscured as holes are added upon substituting Sr2+ for La3+. In contrast to these changes, the plasma minimum in the reflectance is pinned at 0.9 eV and insensitive to the dopant concentration.22,23,63,64 This unusual behavior is in contradiction with the prediction that the plasma frequency should increase with increasing carrier concentration. In summary, the high temperature superconductors are complex and have many unusual properties that we have been trying to understand. The major issues that challenge to the spectroscopists include: the normal transport properties, the super conducting mechanism and the energy gap, the roles of electronphonon interaction and lowlying excitation. These issues will be addressed in the following chapters. La2 Srx Cu4 Phqse Diagram 0 0.1 0.2 0.3 Sr concentration, x 0.4 Fig. 1. Phase diagram for La2SSrzCuO4 and crystal structure of the par ent compoundLa2Cu04 (Ref. 65). (Note that it has been reported recently that Tc became zero near x = 0.22. See Ref. 66.) I Superconducting Ba2Cu3g? Slab tBa 0 Ba SCu S0 c Lb a CuO '" Chains Planes YBa2306+, 600 500 400 300 200 100 0.0 02 0.4 0.6 0.8 1.0 x Fig. 2. Crystal structure for YBa2Cu3076, and phase diagram (Ref. 65) of YBa2CuaO6+,. AF: antiferromagnet, SC: superconducting. CHAPTER III THEORY Optical Theory The frequency dependent optical conductivity al(w) and dielectric function eI(w) are most directly connected to the absorptive and dispersive nature of a material. In the zero frequency limit, ac(0) becomes the ordinary dc conductivity, and e1(0) is the static dielectric constant. However, neither al(w) nor cl(w) can usually be measured directly. Therefore, the optical properties are usually determined experimentally by measuring the reflectance or transmittance as a function of the energy of the incident light radiation, from which al(w) and el(w) can be derived. The interaction between matter and the applied electromagnetic field is described by Maxwell's equations and the boundary conditions. Optical Response of the Medium In the infrared through ultraviolet region, the wavelength of the light radiation is much larger than the dimensions of the unit cell. The propagation of electromagnetic wave in a medium can be described by a set of four differential equations (known as the macroscopic Maxwell's equations):67 V D = 41pf, V.B=O, xE= 1 OB (1) c at ' Vx1_ ID 47r Sc t c Jf, where E and H are the electric and magnetic fields, D and B the displacement field and magnetic induction, pf and Jf the freecharge and freecurrent densities, respectively. Gausian units are used throughout this dissertation unless otherwise specified. For weak electromagnetic field and in local limit, the response of the medium is linear and can be written by the constitutive relations: D = elE, B = pH, Jf = rlE, (2) where el, i, il are the frequencydependent dielectric function, permeability, and conductivity, respectively, of the medium. For simplicity, we take p =1, the case for most nonmagnetic materials. Thus, B can be replaced by H. If the medium is isotropic and homogeneous, then e1 and a~ are scalar quantities rather than tensors and have no space variation. Assume the fields have the planewave form: {H { exp[i(q x t)], (3) where the vectors Eo, Ho and q (wave vector) are in general complex and independent of space x and time t, then g can be replaced by iw, and V by iq, causing the curl equations in (1) become iq x E = iH, (4) c 4?r w 4?r l w i iq x H = (Jd + p) = i+ i (  =) E, (5) c c c 47r where the first term in Eq. (5) is the displacement current, the second is free (con duction) current, and the third is polarization (bound) current. One can introduce a complex conductivity a = al + ia2 with 02 = w(l el)/47r, or a complex dielectric function e = eC + ie2 with e2 = 4~rCr/w such that .w 47r .w iq x H= i E+ 4 E=iWeE. (6) c c c Finally, Eqs. (4) and (6) are simplified as c (7) q x H= H, qxH= eE. c Equation (7) implies that these three vectors are mutually perpendicular with one other (q I E, H) if e is a scalar, the case for isotropic materials. Such a wave is the well known transverse wave. The solution of Eq. (7) is q2 = (~) c. One can also define a complex refractive index N, yielding the very useful dispersion relationship: W W q = N = (n + i). (8) c c Consider the case of normal incidence and q II x, then Eq. (3) has the form: { E I Eoe ie'N ) Ei (9) Hf Ho; HoJ (9) This solution is an attenuated wave with a skin depth 6 = c/wK or a power absorption coefficient a = 2/6 = 2wI/c; the phase velocity is vp = c/n. In summary, the optical response of a material can be described by various quan tities (called optical "constants") which are not independent and are interrelated by 1 4r N2 Z = 1 + i4 (10) where the complex surface impedance Z = R+iY, with R and Y being the impedance and reactance, has been introduced. Note all the optical "constants" introduced are (in general) frequency dependent. We are particularly interested in the real part of the optical conductivity, o1, because it is directly proportional to the power dissipation of the electromagnetic field per unit volume by the medium: dPdip = Re(J E*) = Re [(7E) E*)] = or E2. (11) dV 2 2 2LJ JI Here J = J1 +Jp = OE, defined by Eq. (5), is the total charge current induced by the electric field E. This indicates that only the inphase conduction current Jf = o7E dissipates power, while the displacement current Jd = i'E and the polarization current Jp = iOzE do not because they are 900 out of phase with E thus the time average of energy flow is zero. Determination of Optical Constants One useful experimental technique to determine the frequency dependence of the optical constants is to measure the fraction of power intensity reflected by, A.(w), or transmitted through, 5(w), by a sample. The incident photons of energy hw inter act with the electrons, ions, spinons ..., causing electronic transitions from occupied states below the Fermi energy (EF) to unoccupied states above EF; or interact with lattice vibrations (phonons), causing polariton excitations. Transmittance measure ments require good quality films with thicknesses (d) being shorter than the electro magnetic penetration depths (6), which are not usually feasible. Thus, reflectance measurements are more frequently adopted in optical experiments. Here we will describe the background of the theory and the techniques of extracting the optical constants from .A(w) in a wide frequency range or from a combination of A.(w) and F(w). The details of the optical measurements and the experimental approaches for the optical constants will be given in chapter VI. Reflectance of thick crystals The expression for the reflectance will be rather simple for normal incidence on bulk samples with surface dimensions much greater than the skin depth (d > 6). In this case, both E and H are parallel to the sample surface. In the absence of the idealized surface current, the boundary conditions require that the tangential components of E and H are continuous at the interface: SE +E, = Et, (12) Hi Hr = Ht, where the subscripts i, r, and t denote the incident, reflected, and transmitted fields, respectively, at the interface. Note that, in Eq. (12), Er and Ei are assumed in the same direction. Thus H, is opposite to Hi to maintain the relation q 1 E x H as required by Eq. (7) for a plane wave. The scalar relation between E and H can be simplified as H = NE according to Eqs. (7) and (8). Thus, when a plane wave is propagating across the interface between medium a and medium b, it satisfies Hi = NaEi, Hr = NE,r, (13) Ht = NbEt, where Na and Nb are, respectively, the complex refractive indices in medium a and medium b. From Eqs. (12) and (13), it is straight forward to find the complex ampli tude coefficients of the reflected (r) and transmitted (t) electric field: (14) SEr Na Nb r E, Na + Nb' (14) Ei 2Na tE +r 1. Ei Na + Nb The light is usually incident from vacuum onto a sample surface so that we take Na = 1, and Nb = N = n + in. The power (intensity) reflectance is then given by S(1 n)2 + 2 (15) (1 + n)2 + r2 The measured reflectance A and q, the phase change of the reflected electric field wave, are related to n and K by (1 n)+ic' (16) VfAei = r = i (16) (1 + n) + ir.' and 2K tan = 1 2. (17) Note again that all quantities in the above equation are frequency dependent. KramersKronig relations If A.(w) is measured over a wide frequency range, the phase dispersion O(w) can be evaluated using the KramersKronig relations68 w o In ,,(w) In (w') d = In w'wI din (w') dw'. 2r Jo w' + w dw' The second expression, obtained from the first by integrating by parts, indicates that the faraway spectral regions (w' < w and w' > w) and the regions in which A(w') is flat (dY'/dw' 0) have very small contributions to the integral. After A(w) and O(w) are determined, one can invert Eq. (16) to obtain n(w) = 1 (w, (19) 1 + (u(w) 2/A() cos O(w) ) = 2(w sin O(w) 1 + a(w) 2 /})cos O(w) Finally, other optical constants, such as e(w), a(w), skin depth 6, absorption coefficient a, and electronic loss function Im(1/e).., can be obtained from Eq. (10), i.e., e 2 2 d1 = n2 K"2, Q2 = 2niK, a1 = Wf2/4r, (21) a2 = w(1 e1)/47r, 6 = CIWK , a = 2w/Kc. Other commonly used KramersKronig equations are () = dw', (22) r J0 :12 W2 2w fl w 1(w') 1 d' e2() = 'I' (23) 7 o W012 02 where 9 stands for principle part of the integral. These equations relate a dispersive process to an absorptive process due to the requirement of causality for linear response functions. In other words, the real and imaginary parts of a linear response function (such as Inr = In A + i4, N = n + ix, E = C1 + ie, r = oa + ia2, etc.) are not independent with each other. Rather, they are rigorously related by the Kramers Kronig dispersion relations. Equation (18) shows that 0 at a single frequency must be determined from A. at all frequencies (and vise versa). In practice, .(w) can be measured only in a limited discrete frequency range wl < w < w2. Therefore, reasonable extrapolations beyond the region of the experimental data must be made. This procedure will be discussed later in detail (see p. 69) for our La2.Sr,CuO4 thin film data. A drawback of the KramersKronig technique is obviously the requirement of a large frequency range measurements, which are not always possible. Errors will be introduced by the artificial extrapolations. Furthermore, if the sample is thin, the single bounce assumption will be no longer valid, hence, the determination of the optical constants become difficult. The situation is even more complicated for a thin sample (film) deposited on a substrate. In such case, it is possible to extract a1 and a2 (without the KramersKronig analysis) from combined reflectance and transmittance measurements over any finite frequency range. This technique will be presented below. Combination of reflectance and transmittance of thin films For a film of thickness d < A, the wavelength of the farinfrared radiation, and d < AL, 6, the penetration depth, the film may be treated as a sheet of conductor of complex admittance yi+iy2. In this case, the transmission through, ff and reflection from, Af, a film on a supporting substrate with index n can be approximated as 4n (yl + n + 1)2 +y ' 2(24) and S(y+ 1)2 +y (y + n (25) (yl + n + 1)2 22 These singlelayer equations are generalizations of expressions given by Glover and Tinkham.69 The dimensionless complex admittance of the thin film is related to the conductivity a by y = Zoad or yl + iy2 = Zo(ol + ia2)d. (26) Here, d is the film thickness and Zo = 377 fl = 4r/c (esu) is the impedance of free space. 21 The exact expressions for the composite Fresnel coefficients of transmission and reflection in normal incidence can be derived easily. A thin film on a substrate can be considered as sandwiched by two media, the air and substrate. Consider a three medium system with complex refractive indices of N1, N2 and N3, respectively, where Nj = nj + ixi (j = 1, 2, 3), the transmission coefficient (from medium 1 through medium 2 and into medium 3) is tf= t2t2ei6 1 + r23r2ei26 + (r23r21ei26)2 + .] t42t23eiS (27) 1 r23r21ei26 ' where rij = (Ni Nj)/(Ni + Nj) and tij = 2Ni/(Ni + Nj) as already derived in Eq. (14). The complex phase depth 6 of medium 2 with thickness d is defined by 27rN2d w 6b N2d. (28) A c Similarly, the reflection coefficient is rf = r12 + lt2r23t21ei26 [1 + r2r23e6 + (r21 r23ei2 +2 .] t12r23t21ei26 = r12 + 1 r2r23e(29) r12 + r23ei26 1 r23r21ei26 ' Note the identity t12t21 rl2r21 1 has been used in deriving the last expression in Eq. (29). The power transmittance and reflectance are finally obtained: S= 3 Itf12 and f = rf12. (30) The subscript f has been chosen for Eqs. (27)(30) to consist with the notations of Eqs. (24) and (25). The denominators in Eqs. (27) and (29) account for the multiple internal reflections. One can recover the approximations of Eqs. (24) and (25) easily from the rigorous expressions of Eqs. (27)(30). This can be done as follows: first, take medium 1 as vacuum (N1 = 1), medium 2 as a metal film with thickness d and refractive index N2, and medium 3 as a weakly absorbing semiinfinite slab with index N3; then, substitute the following approximations into Eqs. (27)(30), IN2z > Ni = 1, IN21 > IN31 n3 = n, (K3 < n3), ei26 1 + i2 (31) 4ir iN26b ad = y. c Here we have applied the long wavelength (or low frequency) limit and assumed that film is thin enough such that 6 < 1. The the calculation is straight forward and is left as an exercise to the interested readers. In reality, the substrate has a finite thickness and it is a four medium problem with medium 4 being air. For a nearly opaque metal film, the overall reflectance of film plus substrate in this 4medium system is A (32) Equation (24) gives the transmittance across the film into the substrate. This quantity is related to the measured transmittance 3 (across the film and substrate into the air) by (1 ,)eax 3 = , (33) 1 AS'e2azff where x is the thickness and a the absorption coefficient of the substrate (for example, MgO). The other terms in Eq. (33) are the substrateincident internal reflection of the film, S(yz n + 1)2 + y2 (yl n )2+y (34) (yj + n + 1)2 + y2 and the single bounce reflection of the substrate, (1 )2 + 2 1 n\2 S= (l++.a nj (35) The approximation in Eq. (35) holds when c a/2w < n, the case of weakly absorbing medium. Equation (33) assumes a thick or wedged substrate, so that there is no coherence among multiple internal reflections within the substrate. The index n and the absorption coefficient a of a substrate can be obtained by measuring the overall transmittance sub(w) and reflectance Asub(w) of the substrate. In general, for normal incidence, the formulas for ffsb(w) and .,sub(w) of an absorbing substrate of parallel faces with thickness x are given by70 [(1 A,)2 + 4, sin2 ]ea (36 ,ub) = (1 ,eax)2 + 4Aeaz sin2(4 + () ' (1 eax)2 + 4ea sin2 (1 Aeax)2 + 4Aeaz sin2(o + ) Here ~, and 0 are defined by Eq. (35) and Eq. (17), respectively, and f = nyx. The expression of these two equations incorporates interference effects due the substrate. [Equations (36) and (37) can also be derived from the expressions of Eqs. (27) (30), taking medium 2 as the substrate and medium 3 the air.] In a low resolution measurement (for example, see p. 166), the periodic interference fringes are averaged out. The averages can be found by integrating Eqs. (36) and (37) over df, to be (1 (1 eax Ysub(w) (1 s)2e (38) 1 2e2a38) s 1 + (1 2,s)e2as 1 e2a s. (39) Therefore, n and a can be solved by inverting Eqs. (38) and (39). After measuring 7 and A at each frequency, we can determine yi and y2, hence oil and o~ by inverting Eqs. (24), (25) and Eqs. (32)(34). In this procedure, we can iterate for a selfconsistent 5', using the values of a and n measured for the substrate. This technique has been applied in data analysis of our YBa2Cu307_s films. The computer programming routines used for this computation are presented in Appendix B. Lorentz and Drude Models Two classical models (Lorentz and Drude) are frequently used to describe the optical properties of materials. The Lorentz model can be employed for either bound carrier interband transitions or lattice vibrations whereas the Drude model is appli cable to freecarrier intraband transitions. Thus we can model the dielectric function by a sum of three terms: 6 = CDrude + ELorentz + Coo (40) Here coo is the contribution from the high frequency absorption beyond the measured range. The Lorentz dielectric function CL can be derived by assuming that the electrons are bound to their cores by harmonic forces and are subject to viscous damping forces which represent the energy loss mechanism. CL is then given by 2 W (41) L=c E 2 W2  i[jW where wj, 7j, and Wpj are the resonant frequency, plasma frequency and damping constant, respectively, of the jth Lorentzian. The plasma frequencydefined by w~i = 47rNie2/m with Ni and m* being the number density and effective mass of the bound carriers may also be written as wp = VS'wj such that Si represents the contribution of the jth oscillator to the static dielectric constant. From the quantum mechanical point of view, hwj is the transition or gap energy between the initial and excited atomic states, 7y the inverse lifetime of the excited carriers or the energy width due to energy uncertainties in the initial and final excited states. The oscillator strength wpj is related to the transition probability. The Drude model describes the optical response of free carriers in good metals. It is just a particular case of the Lorentz oscillator with the resonance frequency equal to zero (no restoring force for "free" carriers): 2 WPD (42) w(w + i/r) where the Drude plasma frequency is defined by W2D = 4rNe2/m* with N being the charge concentration (do not be confused with the index of refraction) and m* the effective mass of the free carriers. The viscous damping mechanismdescribed by a relaxation time ris associated with collisions between electrons (or holes) and impurities or lattice vibrational phonons in metals. The real part of the Drude conductivity is 1 WpD 7 .C 1D = = 1 2 a (43) 47F 1 + W272 1 + L272 ~ with the zero frequency limiting value p2D r Ne2r To = N= (44) 47 m* being the ordinary dc conductivity. Since the reflectance can be calculated using Eqs. (15) and (21), therefore, as an alternative to the KK analysis, we can in return fit the experimental reflectance data with a combined DrudeLorentz model of Eq. (40) to extract the optical parameters. This procedure turns out to be very successful as will be discussed in our data analysis later. Sum Rule One of the most important sum rules is called the fsum rule. It states that the area under the conductivity al (w) is conserved: Jo0 w r Ne2 Oldw = 8 2m (45) Jo 8 2 m Here m and e are the bare mass and electric charge of a free electron. This sum rule means that the area, or oscillator strength, is independent of factors such as the sample temperature, the scattering rate, phase transition, etc. The sum rule has an important impact on a superconductor, in which an energy gap develops below the transition temperature Tc; the spectral weight at w < 2A shifts into the origin, causing an infinite dc conductivity. Superconductivity Perfect Conductor A perfect conductor has no scattering, namely 1/7 = 0. This is the case for ideal metals with perfect translationally invariant periodic lattice described by Bloch's theorem. The complex conductivity in this case can be obtained by iw w2[ i1 (w)= lim () + i(46) r*oo 4r(w + i/r) 4 rw (46 or i = 6(w), a 02 (47) 4 4rw(47) For simplicity, here wp is used instead of WpD to represent the Drude plasma frequency. Note Eq. (47) satisfies the sum rule required by Eq. (45), considering b(w) is an even function thus fo'S 6(w) dw = 1/2. Equation (46), or (47), implies that a perfect conductor has an infinite dc conductivity but al = 0 for w $ 0, and the inductive response (represented by a2 which goes as 1/w) is dominant at low frequencies. The dielectric function e is W2 e2 = Oa = 0, = 1 (w 0) (48) W2 ' thus t i( /2 2 1)1/2, (0' <,< (49) (1 _ W/2)1. (W > Wp) Therefore, because A = [(n 1)2 + K2]/[(n + 1)2 + i2], a bulk perfect conductor is also a perfect reflector (A = 1) for w < wp. Superconductor The optical response of a superconductor is similar to that of a perfect conductor. One major distinction is that a BCS superconductor has an energy gap, 2A, in the excitation spectrum and the electrons are paired when the temperature is lowered below To. In the BCS weakcoupling theory, the energy gap for T < Tc is given by71 1 tanh (2 + A2)1/2 VN(0) J (2 + A2)1/2 d, (50) 0 where N(0) is the density of states at the Fermi level, V is the electronphonon interaction potential, 3 = 1/kBT, and w, is the typical phonon frequency or Debye frequency. The transition temperature is predicted as kBTc = 1.13 hwce1/Nv(O). (51) In the vicinity of Tc, the theory gives A(T) 1.74 A(0) 1( (52) The limiting value at T = 0 is 2 A(0) = 3.5 kBTc. (53) [For a Tc = 90 K superconductor, for example, this would give 2 A(0) = 220 cm1, a range in the far infrared.] Consequently, one expects al = 0 in the range 0 < iw < 2A. This means that photons of energy less than 2A are not ab sorbed because their energies are not sufficient to break Cooper pairs. The incident light is therefore 100% reflected because the impedance mismatch at the interface (n = 0 in the superconducting sample, and n = 1 in vacuum). This property agrees with the Meissner effect that the electromagnetic field is zero in the interior of a bulk superconductor. However, part of the field still does penetrate into the superconductor and is exponentially damped within a length scale called the London penetration depth: AL = 47Ne2 (54) with N, being the superfluid density. If all free carriers condense completely into pairs, then AL = c/Wp. The fraction of the transmitted electric field at the sample surface can be found from Eqs. (14) and (49), t = M , (55) t 2 .2w (55) 1+N wP for w < 2A < Wp. This indicates that the transmitted electric field Et has a phase shift related to, and is much less than, the incident field Ei. Note the transmitted power is zero, namely 5 = nlt 2 = 0, because n = 0 (al = 0, e1 < 0). The induced inductive current (associated with 02) in the superconductor is 900 out of phase with Ei and hence does not dissipate energy. The transmitted magnetic field given by Ht = NEt i Et = 2Hi (56) is much larger than E1 and is in phase with the incident Hi. This is a consequence of the continuity of H at the interface and the 1800 reversal of Er, the reflected wave. At w > 2A (taking h = 1), the photon energy is high enough to disassociate Cooper pairs, causing quasiparticle excitations across the superconducting gap to the unoccupied normal levels. The conductivity thus approaches the normal state value. At finite temperature below Tc, the conductivity al at w < 2A is no longer zero, shown in Fig. 3, due to the existence of thermally excited quasiparticles. The temperature dependence of the lowfrequency conductivity exhibits a peak below Tc (illustrated in Fig. 4) which has been explained by the coherence effect. In BCS theory, the perturbation Hamiltonian can be written as H = EBk'kCk'Ck (57) kk' Here the subscript k represents the quantum state for momentum and spin, Ct and Ck are the quasiparticle creation and annihilation operators, and Bk'k are matrix elements of the perturbation operator. In the normal state, each term in the sum is independent. At T < Tc, however, there exists phase coherence between the wave functions of the occupied states. This interference leads to Bk'k = +B_k'k, where the upper sign for "case I" and the lower sign for "case II" interactions. The Hamiltonian for interaction of electromagnetic radiation with matter is rep resented by a term p A, where p is the momentum of the electrons and A is the vector potential of the external field. Since this term is odd with p (or k), the in teraction obeys the case II process. Tinkham71 has shown that this interference will result a coherence factor for scattering: F(A, E, E) = 1 T, (58) where E is the quasiparticle energy measured from the Fermi level and E' = E + hw. The superconducting to normalstate conductivity can be expressed as 1 00 i= 1 [ F(A, E, E')N,(E)N,(E + fw) [f(E) f(E + hw)] dE (59) 1 wj I(E(E + hw) + A [f(E) f(E + iw)] hw oo (E2 A2)1/2[(E + hw)2 A2)]1/2 where f(E) is the Fermi distribution function and N,(E) is the the superconducting density of states given by IEI N,(E) = N(0) Re (E (60) Here Re stands for real part. Equation (60) indicates that N, = 0 for IEI < A. It diverges near A and approaches the normal state value at IE\ > A. Note the case II coherence factors have been used in Eq. (59). The resulting conductivity calculated from Eq. (59) are shown in Figs. 3 and 4. The lowfrequency upturn in Fig. 3 is due to the coherence factor, and the minimum moves to higher energy as T is lowered, indicating an opening superconducting gap. At T = 0, oal(w) = 0 up to w = 2A; above this frequency, ria(w) begins to rise due to the photoexcited quasiparticle absorption. The difference between ol,,(w) and al,(w) disappears at higher frequencies. The oscillator strength below 2A (or the "missing" area) shifts into the origin to form the superconducting condensate. Figure 4 shows that, at small frequency w, the integral in Eq. (59) gives a peak below Tc because of the divergence of the density of states N,(E) at E A. The peak gradually disappear with increasing frequency. Such peak due to the case II coherence factor has been observed in the nuclear relaxation rate72 and the optical conductivity8 for classical superconductors. 1.5 I I I I MattisBardeen theory 1 I 2Ao = 200 cm1 1/r = 100 cm1 Tc= 90 K 1.0 b S I = OK '. T = 40 K T= 40 K __ T = 60 K ST = 80 K b \ T = 85 K 0.5 \. ... T = 90 K \\ \\ .^.V 0.0 0 100 200 300 400 500 o (cm ) Fig. 3. The conductivity ratio of a superconductor vs. frequency at T < Tc in the framework of MattisBardeen theory. 2.5 . MattisBardeen theory 2Ao = 200 cm1 1/T 00 cm r /T = 100 cm 5 cm 2.0 Tc = 90 K W = 10 cm = 20 cm1 o = 30 cm1 S= 50 cm1 = 100 cm1 1.5 b  /\ b 1.0 / / /' ___ S// / 0.5/ / / / / 1 0.0 I 0 50 100 150 Temperature (K) Fig. 4. Conductivity ratio as a function of temperature at low frequencies, showing a coherence peak below Tc. CHAPTER IV INFRARED TECHNIQUES This chapter describes the principles of Fourier transform interferometry and spectrometers used in this work. The optical response is determined experimentally by measurements of reflectance or transmittance of the sample as a function of a wide range of incident photon energies. This range extends from about 20 cm1 to 40,000 cm1 (2.5 meV5 eV) using variety of optical spectrometers, light sources and detectors. The following conversion factors for energy units are useful: E: 1 meV = 11.6 K = 8.066 cm1 = 0.242 THz f : 1 THz = 4.133 meV = 33.33 cm1 = 48 K w : 1 cm1 = 0.124 meV = 1.44 K =30 GHz T : 1 K = 0.086 meV = 0.695 cm1. Interferometry The spectrometers used to measure the optical spectra of the samples in the farinfrared (20600 cm1) and midinfrared (5003000 cm1) region are a slowscan Michelson interferometer and a fastscan 113 V Bruker Fourier Transform Interferom eter. (A Perkin Elmer 16 U Grating Monochromator, which will be discussed later, is used to collect data at higher frequencies of 100040,000cm1). Infrared Radiation at Low Frequencies Interferometry is widely used for measurements in the farinfrared region primar ily due to the fact of energystarvation for all thermal sources at low frequencies. This fact can be seen from the Plank law for the spectral distribution of blackbody radiation. The power p(w) emitted per unit area of the light source at temperature T and in frequency range between w and w + dw is given by: p(w)dw dw (61) 4r2c2 ehw/kT 1 The radiation spectra are peaked near xw = 2.82 kT (or w w 2 T for w in cm1 and T in kelvin). Figure 5 plots the spectrum of Eq. (61) using logarithmic scales. The intensity is normalized to the peak value at 1000 K. It can be shown that the peak power pm(T) ~ T3 and is given by (T) = 1.42 ( )( ) 1.26 T )3 mW/cm. (62) 2rc h 1000 K Note that the result in Eq. (62) is for per one unit wavenumber (1 cm1) interval. To stress and illustrate the strong temperature dependence, the same spectral distri butions are also plotted in linear scale, shown in Fig. 6. The total radiation power Po emitted from a source of area A can be obtained by integrating Eq. (61) over all frequencies Po = A p(w) dw = aT4A (63) with Vr2 k4 a= 60 2 = 5.67 x 1012 W/cm2 K4 (64) 60 c2%" being the StefanBoltzmann constant. Consider a mercury arc lamp source with A = 3 cm2 at T = 5000 K, the total emitted power is Po = 1.1 x 104 W. To estimate the fraction of radiation power in the farinfrared region, remembering 1 K = 0.7 cm1, one can approximate Eq. (61) in the low frequency limit x hw/kT < 1: kT p(w) 422 (65) This ~ w2 dependence, as seen from the slopes of the curves plotted in Fig. 5, indicates that p(w) decays rapidly with decreasing frequency, which can also be seen in Fig. 6. The emission power up to a frequency w is therefore P(w) = A p(w') dw' = w3A. (66) Jo 12=ir2c2 Of the total radiation power Po, only a fraction 9 = P(w)/Po = Tx (k (67) is emitted below w. For w = 100 cm1 and T = 5000 K, this fraction is r = 1.2 x 106, i.e., only a tiny amount of power, 13 mW out of 11 KW (taking A = 3 cm2), is emitted at w < 100 cm1. Therefore, a dispersion spectrometer such as grating monochromator, which measures a one resolution width at a time, is obviously inefficient in the farinfrared measurement. The situation can be greatly improved if one uses an interferometer, in which entire radiation power at all frequencies is utilized. Consequently, the signaltonoise ratio can be greatly enhanced which is called the Fellgett advantage.73 The details of interferometry have been described in literature,7476 and here only a brief discussion will be given. Fourier Transform Spectroscopy The principle of interferometry is based on the idea of the Michelson interfer ometer as sketched in Fig. 7. Light radiation from an extended source S is divided by a semireflecting beam splitter B (mylar film or thin Ge layer) into two parts of approximately equal intensity. These two beams are reflected by a stationary mirror M1 and a movable mirror M2, and are then recombined to enter the detector D. As M2 moves a distance of x/2 away from its neutral position, a path difference between the two beams, x, is introduced before they are combined, yielding a phase difference 6 = 21rx/A = 2rvx. Here A and v are wavelength and wave number, respectively, of the incident light. Assuming these two beams have an equal amplitude a(v), then the complex amplitude of the combined beam reaching the detector is A() = a(v)(1 + ei2rvx). (68) In the ideal case, a(v) = /S(v2, where S(v) is the spectral intensity of the radiation source (as modulated by losses due to detector absorptivity, transmission of filters, lenses, beamsplitter, windows and samples, and reflection of the mirrors or samples, etc.). From Eq. (68), one can obtain the intensity at the detector as a function of path difference x at frequency v I(x, v) = AA* = 2a2(1 + cos 2rvx) = 1S(v)(1 + cos 2rvx). (69) For a polychromatic source emitting a continuous spectrum from v = 0 to v = oo, Eq. (69) must be integrated to obtain the total intensity which gives 21(x) = S(v) dv+ S(v)cos 27rvxd. (70) I(x) is called the interferogram. The first term in Eq. (70) is constant and is the total intensity, So, emitted from the source. As x+oo, there is no correlation between the two beams, the second term, which is just the Fourier transform of S(v), vanishes because of the rapid oscillation of the cosine function. At x = 0, the interference is constructive for all frequencies hence the detector receives a maximum signal 1(0) called centerburst or "white light" (see the upper panel of Fig. 8). It is straight forward to see from Eq. (70) the relation between these two limits: I(0) = 21(oo) = So, (71) where I(oo) is the average or dc value of the interferogram. One can extend the lower limit in Eq. (70) by noting that S(v) an even function, i.e., S(v) = S(v). By defining D(x) = 4[I(x) I(oo)], which is linearly related to the signal at the detector, one finds D(x) = S(v)e12" dv = FT[S(U)], (72) S(v) = j o D(x)ei2w dx = FT[D(x)], (73) or FT D(z) FT S(V), (74) where FT represents the Fourier transformation and FT' is the inverse FT. There fore, if one knows the interferogram, D(x), for a continuous path difference, the spec tral intensity distribution of the radiation, S(v), can be determined by the Fourier transform of the interferogram. This computation has turned out to be accessible in practical applications with the development of the fast Fourier transformation (FFT) and the availability of modern computers. In practice, however, it is impossible to measure a continuous interferogram over a infinite path difference. Instead, one samples a finite number of discrete points up to some maximum path difference zm and replaces the Fourier integral by Fourier series. The finite maximum path difference approximation introduces side lobes near sharp spectral structures. This problem can be repaired by applying the apodiza tion technique.76 Another problem, caused by the discrete sampling, is the socalled aliasing effect which must be reduced by using some cutoff filters to suppress the high frequency components. Figure 8 illustrates spectra of the real time interfero gram D(x) and its Fourier transformationsingle beam spectrummeasured by the Bruker interferometer. As we can see, the interferogram is not symmetric about its central position, which is caused by the phase error and sampling. Optical Spectrometers Bruker Interferometer Most of the measured reflectance and transmittance spectra in this work is ob tained by using an IBMBruker fastscan Fourier transform interferometer, the prin ciple of which being similar to that of a Michelson interferometer. The frequency range covered is 205000 cm1. As illustrated in Fig. 9, the system is divided into four chamberssource, inter ferometer, sample and detector. A Hg arc lamp is used for far infrared (20700 cm1) and a globar source for mid infrared (4005000 cm1). The sample chamber consists of two identical channels which can be used for either reflectance or transmittance measurements. For reflectance measurement, an optical stage, shown in the top part of Fig. 9, is place into the sample chamber. The entire system is evacuated to avoid H20 and CO2 absorption during measurements. Light from the source is focused onto the beamsplitter and is then divided into two beamone reflected, and one transmitted. Each beam is imaged onto the faces of a movable twosided mirror. These two beams retrace their route back to the beam splitter for recombination. The recombined beam is sent into the sample chamber and then into the detector. When the twosided mirror moves at a constant speed, v, a path difference x = 4vt is made, where t is the time since the mirror is at the zero pathdifference position. Suppose the light is a monochromatic wave of wavenumber vo so that S(v) = So 6(v Vo) + So 6(v + Vo), [the second term is needed to make S(v) an even function,] then Eq. (72) gives D(t) = Docos2rfat, (75) where Do = 2So and fa = 4vvo = (4v/c)fo. This indicates that the optical frequency of the radiation, fo, is reduced by a factor of 4v/c. In other words, the detector sees a signal with an audio frequency fa instead of the much higher optical frequency fo. This signal is amplified by a wideband audio preamplifier and then digitized by a 16bit analogtodigital converter. The digital data are transferred into the Aspect computer system and are Fourier transformed into a singlebeam spectrum, as shown in Fig. 8, after some necessary corrections such as apodization and phase correction.77,78 Bolometric detectors. The fact of weak infrared signals requires not only the use of interferometry techniques, as described earlier in this chapter, but also a detector of high sensitivity. One kind of detector with adequate sensitivity is the Hecooled bolometer. The detectors used in this work are a 4.2 K Si bolometer for FIR, and a 4.2 K Si:B photodetector for MIR. Pyroelectric deuterated triglycine sulfate (DTGS) detectors are also available. The cooled detectors have much better signaltonoise (S/N) ratio as compared with the DTGS. The bolometer system consists of three main parts: detector, liquid helium (LHe) dewar, and preamplifier. Figure 10 illustrates a diagram of the bolometer detector mounting and the LHe dewar (HD3). After the dewar is diffusion pumped to a pressure of ~ 106 torr, it is precooled with liquid nitrogen for about an hour. The precoolant is then removed and the liquid helium is transferred in to the helium can to maintain the detector at 4.2 K. (A temperature as low as 1.2 K can be achieved by reducing the vapor pressure above the liquid helium.) A thermal radiation shield is placed between the helium can and the case to reduce the head load on the cold area. The Si detector is mounted on a cold surface under the helium can. The optical signal is guided by the pipe along the optical axis through a window (poly for FIR, KRS5 for MIR) and optical filters before it finally arrives at the detector. The output electric signal from the detector is amplified and then sent to the A/D converter of the Bruker interferometer. Michelson Interferometer The farinfrared (10600 cm') data are also measured with a slowscan Michelson interferometer. In comparison with the Bruker, it has an even better S/N ratio at low frequencies (particularly below 50 cm1) due to a larger and brighter mercury source. The disadvantage, however, is that it runs much slower. As shown in Fig. 11, the light is interrupted periodically by a rotating chopper in order to allow lockin detection. A beam splitter with various thicknesses of mylar films is used in combination with different optical filters to cover the corresponding frequency range. The detector used is a 4.2 K bolometer as illustrated in Fig. 10. Like the Bruker interferometer, the whole system is evacuated during measurements. PerkinElmer Monochromator Optical Spectra from mid infrared through visible and ultraviolet (UV) at fre quencies of 100040,000 cm1 (0.125eV) are measured using a model 16U Perkin Elmer (PE) monochromator. During measurements, the tank is kept under vacuum to prevent from water absorption, particularly for the mid and nearinfrared re gions. As shown in Fig. 12, three sourcesglobar, quartzenvelope tungsten lamp, and deuterium lampare used to cover this frequency range. A proper source can be selected by turning M2 from outside the vacuum tank. The light is chopped and passes through one or two of a set of bandpass filters. These filters reject the un wanted higher order diffraction from the grating, which occurs at the same angle as the desired firstorder component. This case can be seen from a simple diffraction equation: asin 0 = nA = n/v, where a is the grating constant. At an angle 0, the firstorder component of wavelength A satisfying A = a sin 0 is selected. Meanwhile, any higher order components with wavelengths An = A/n, or v, = nv (n = 2,3...), which could also pass through the slit are absorbed by the filter. Light enters the grating monochromator through an entrance slit and leaves through an exit slit. The dispersed spectrum is scanned across the exit slit as the grating is rotated. The resolution of the monochromator is determined by the slit widths. Increasing the slit widths increases the intensity of the emerging radiation (higher S/N ratio) at cost of lower resolution. Mirror M1 is a reference mirror which can be rotated or replaced by a sample for reflectance measurements. For transmis sion measurements, the sample is mounted in a "sample rotator," as indicated in Fig. 12. The positions of the sample on the rotator and of the detector are the two focal points of an ellipsoidal mirror. Three detectors are used to cover the different photon energy regions: a thermocouple for 0.110.9 eV, a lead sulfur (PbS) photo conductor for 0.52.5 eV, and a silicon photodiode for 2.25 eV. Table 1 lists the parameters used to cover each frequency range. The electric signal from the detector is sent to a lockin amplifier (Ithaco model 393). The output signal from the lockin system is then averaged over a given time interval and converted into digital data by an integrating digital voltmeter (Fluke 8520A). The data are finally transmitted through the IEEE48 Bus and a general purpose interface box to a PDP 1123 computer and recorded on the hard disk for subsequent analysis. Table 1. PekinElmer Grating Monochromator Parameters Frequency Grating Slit width Sourceb Detectorc (cm1) (line/mm) (micron) 801965 101 2000 GB TC 9051458 101 1200 GB TC 14031752 101 1200 GB TC 16442612 240 1200 GB TC 24674191 240 1200 GB TC 40155105 590 1200 GB TC 47937977 590 1200 W TC 38295105 590 225 W PbS 47937822 590 75 W PbS 751110234 590 75 W PbS 919113545 1200 225 W PbS 1290420144 1200 225 W PbS 1703324924 2400 225 W 576 2206628059 2400 700 D2 576 2570637964 2400 700 D2 576 3638645333 2400 700 D2 576 a Note the grating line number per cm should be the same order of the corresponding measured frequency range in cm1. b GB: Globar; W: Tungsten lamp; Da: deuterium lamp. c TC: Thermocouple; PbS: Lead sulfite; 576: Silicon photocell. In the grating monochromator, depicted in Fig. 13, the reflecting grating diffrac tion equation is satisfied: a(sin a + sin/p) = nA, (n = 0, 1, 2...) (76) where a is the grating constant (cm/line), a and / are angles of the incident and diffracted rays, respectively, and n is the order of diffraction. When Eq. (76) is satisfied, the interference is constructive. One can then rewrite Eq. (76) as nA = 2acos 6sin0, (77) where 6 = (a /)/2 and 0 = (a + /)/2. In practice, 6 is fixed (26 = 40) regardless of the grating position because the incident and diffracted light paths are predetermined by the physical geometry, whereas 0 changes as the grating (or its surface normal) is rotated. It can be seen from Eq. (77) or Fig. 13 that at 0 = 0, [i.e., / = a, the incident and diffracted rays are on both sides of and symmetric to the normal of the grating surface N(0)] it will give a zeroorder diffraction (white light) for all frequencies. Therefore, 0 is the rotation angle of the grating surface normal, N(0), with respect to the zeroorder position,N(0), as illustrated in Fig. 13. The first order is the desired one and the higher orders (n > 2) are removed by the proper optical filters as described earlier. Taking n = 1, one gets v = 1/A = Ccsc (78) with C = 1/2a cos 6 being a constant. Equation (78) indicates that the frequency is linearly related to csc 0. As the grating is rotated, a single component at frequency v satisfying Eq. (78) is selected and emerges through the exit slit into the sample chamber. The monochromater is mechanically designed such that the grating, driven by a stepping motor, is moved linearly with csc0, thus the scanning is linear in wavenumber. The rotation angle has been designed in the range 150 < 0 < 600, the optimum quasilinear range in the cosecant function. To find the resolution of the monochromator, one simply needs to take the derivatives of Eq. (78) in its logarithm form: In v = InC Insin 0, (79) du = cot 0 dO, (80) V where dO is the angle subtended by the slit (with a width s) at the collimator with a focal length f = 26.7 cm, i.e., dO = s/f. Equation (80) implies that a larger 0 will give a better resolution. Consider the worst case at the maximum slit opening, s = 2000 pm, Eq. (80) gives 0.4% < Idv/vl < 2.8%, which is adequate since most of the solid materials are lack of sharp features at frequencies above the midinfrared band. A more detailed description of the grating monochromator can be found elsewhere.79 100 1000 Co (cm1 10000 Fig. 5. Blackbody radiation spectra using loglog scales at three temperatures. The power intensity is normalized to the maximum value at T = 1000 K [see Eq. (62) for maximum power]. The slopes are equal to 2 at low frequencies, indicating an w2 dependence. O O O x a E 0 3 H 1 0.1 0.01 0.001 0.0001 0.00001 46 Blackbody radiation spectrum 0 0 0 1.0 X 0.. D 0.5 0 1 x P(w,1000K)/Pmax(1000K) 0 __ 10 x P(c, 300K)/Pmax(1000K) 0 / __ 100 x P(o, 100K)/Pmax(1000K) N .0 I \ Z 0.0 0 1000 2000 3000 4000 cw (cm1) Fig. 6. Normalized blackbody radiation spectra using a linear scale. Note that the relative scale is expanded by factors of 10 for T = 300 K and of 100 for T = 100 K. Source Movable mirror Beam splitter Detector Fig. 7. Schematic of Michelson interferometer. r 48  1 I r Real time spectrum 4Q o L 0) 0 4 L. a)  ... I... I ,1 .II ,I S100 50 0 50 100 Path difference x (Arb. unit) Single beam spectrum C L E 3 0 100 200 300 400 500 w(cm1) Fig. 8. The interferogram D(x) (upper panel), and its Fourier transformation S(v) (lower panel) measured by a fastscanning Bruker interferometer. (Note we have used w instead of v to label the frequency axis for all figures throughout this dissertation for consistency.) 49 S h Hg Iv d ; nb uI ome ChMbe I Ip Chwamr a Near, mir. or trIR sources I Sample locus b Autornadec Aperture I Bl*erence focus I Iateren erCamber IV Detector Chmn S*Optical filler I Near. tor rIR (I Automatic b: rmiptM cOtnger dlteors Twoaide movtle mirror ( Control in1eremn ter I RPernce laser SRewnme control & flnmr nemrror Fig. 9. Schematic diagram of IBMIR/98 BRUKER interferometer. The top part of the figure (enlarged scale) is an optical stage setup for measuring reflectance. DEWAR, MODEL HD3 OUTLINE SKETCH Fig. 10. Bolometer detector. The dimensions are in inches. SHIELD I WORK SURFACE Fig. 11. Michelson interferometer. vacuum tank Fig. 12. Schematic diagram of PerkinElmer monochromator spectrometer. 5samrple Zero order position Rotatable grating entrance slit . Sdiffracted 0' exit slit N(O) 26 N(1) Fig. 13. Schematic of the grating monochromator showing the incident and diffracted rays and the operation of the grating. Note that the grating constant, a, is significantly exaggerated in order to illustrate the path difference given by Eq. (76). CHAPTER V SAMPLE PREPARATION AND CHARACTERISTICS Various highTc superconducting samples have been investigated in this work. This dissertation will concentrate on the La2aSrCuO4 and YBa2Cu3O76 oriented thin films as well as YBa2Cu3O7.. polycrystalline samples. Other samples such as YBa22SrzCu307_ ceramics, YBa2Cu408 textured pellets, YBa2CuaO36 granular films and ultrathin YBa2Cu3aOr_ film (96 A) were also measured but in less detail. La9_,Sr,CuO4 Epitaxial Films Three La2.SrCuO4 thin films were prepared at Westinghouse Research Center in Pittsburgh using offaxis dc magnetron sputtering technique.80 Two of them are deposited on LaAlO3 substrates with dimensions of 6 mm x 6 mm x 270 nm, and the third is grown on the (100) face of a SrTiO3 substrate with dimensions of 10 mm x 10 mm x 820 nm. Both kinds of substrates have a perovskite structure which makes a good lattice match with the films. Figure 14 illustrates the diffractometer position 20 and xray counts measured at Westinghouse, showing that the films are highly abplane oriented. In addition to the caxis texture, the films are epitaxial. In other words, the [100] and [010] directions which lie in the plane of the films are parallel to the [100] and [010] directions in the substrates. The properties of the samples are summarized in Table 2. The CuO2 plane dc resistivity of a La2.Sr.CuO4 film, shown in Fig. 15, exhibits a sharp superconducting transition near 30 K. Above approximately 100 K, the re sistivity for all films is roughly of the form of p(T) = po + aT, linear in temperature (a = 1.2 ~ 1.5 fl cm/K), with a nearly zero extrapolated intercept. Deviations from this behavior are evident in the plateau below ~ 100 K. The inset of Fig. 15 shows Table 2. La2sSrCuO4 Thin Films Characteristics. Sample # Thickness Area x Tc ATc Substrate (nm) (mm2) (K) (K) 1, 2 270 6.3 x 6.3 0.15 27 1.5 LaAlO3 3 820 10 x 10 0.17 31 1.5 SrTiO3 an expanded view near Tc for p(T) and the inductive transition measured by the change of inductance of a coil placed against the film. The composition of the films is x = 1.51.7, near the optimum values for superconducting La2.SrzCuO4 films. The resistivity is consistent with the published reports of good quality La2SSrzCuO4 films.28'29 Details of sample preparation and dc transport properties can be found elsewhere.80 YBagCu2O7 Oriented Films The farinfrared spectra (both reflectance and transmittance) of three YBa2Cu307_. thin films have been measured. The samples were prepared by the research group at Bell Communication Research (Bellcore). The films were de posited on 1mmthick MgO substrates by pulsedlaser ablation from a stoichiomet ric target.sl Other commonly used substrates for highT7 superconducting films are LaAlOa, SrTiOa (as used in our La2,SrCu04 films), LaGaOs, and yttriastabilized zirconia (YSZ). These substrates are usually good for epitaxial growth and for re flectance measurements of thin films. However, they are not suitable for thin film transmission studies as all of them are opaque in much of the farinfrared region. In contrast, MgO is reasonably transparent up to 330 cm1 at low temperatures and is also a good substrate for oriented growth. These properties make it the best choice as a substrate for farinfrared transmission studies of highTc films. Film thicknesses for two films (480 and 1560 A) were measured by Rutherford backscattering (RBS) and step profilometry. These two techniques agreed to within 100 A. Thickness for the third film (1800 A) is estimated from growth conditions. Table 3 lists the parameters of the YBa2Cu30O7_ samples. Table 3. Characteristics of YBa2Cu3O7_ films. Thickness Area Tc ATc ode (at 300 K) Substrate (A) (mm2) (K) (K) (f1 cm1) 1mmthick 1800 5 x 5 89 2 2000 MgO 1560 5 x 5 90 1 2500 MgO 480 5 x 5 83 3 1600 MgO Figure 16 illustrates the temperature dependence of the dc resistivity, Pdc, for a YBa2Cu3O7_ film measured by fourprobe technique. In comparison with La2.SrCuO4, the YBa2Cu3O7a films have higher Tc's (~90 K) and show a T linear resistivity over most of the temperature region above T,. One striking feature is that, however, the magnitudes of p(T) are close for all highTo films as represented by Fig. 15 and Fig. 16. YBa9.,Sr,CuROT Polycrystalline Samples The samples were first prepared at 3M center.82 The starting compounds were Y203, BaCO3 and CuO. The powders were combined to make a homogenous mixture. After a set of lengthy firings, grindings and jetmillings, the compound is pressed (with carbonwax) into pellets before the final firing and annealing in a slow steady oxygen flow. The strontium concentration of these YBa2.SraCu307_6 samples was in the range 0 < x < 1.3. Trarascon et al. have observed that the upper limit of Sr doping is at x 1.4, beyond which the materials are multiphase.83 The transition temperature of this compound decreases gradually from 93 K (x = 0) to 80 K (x f 1.4).83 Typical parameters of the our original samples are listed in Table 4. The table gives the Sr composition, the dc resistivity at room temperature, the mass density and unit cell volume. It appears that the sample density is significantly lower than the theoretical prediction, indicating porous behavior. The detailed information about the preparation procedures of these ceramic samples can be found in Ref. 82. We noted that the initial measurements of these samples showed strong vibra tional features in the farinfrared region84 and there were differences but no systematic variations with x in the reflectance and conductivity spectra. This result was most likely due to oxygen deficiency or a mixed phase in the samples as indicated by a large number of vibrational phonons in the infrared spectra. The CuO plane phonons, if not well screened, would show up as a result of reduced carrier concentration. The 02 deficiency was further confirmed by placing the samples in a 0.3 Tesla magnetic field, with the result that most of the samples did not show a magnetic levitation when they were cooled by liquid nitrogen, implying that they were not superconducting. We also found there were green grains on some sample surfaces. The presence of this "green phase" (2112) indicated that the samples were in mixed phase. It was clear that 02 had to be added into these samples to recover their superconducting phase for further study. Table 4. Typical parameters of YBa2rSrCCuaOTr polycrys talline samples z p (300 K) Density Density b Volume (mi cm) (g/cm3) (g/cm3) (A3) 0 10.0 5.741 6.347 173.55 0.13 18.9 4.992 6.332 172.97 0.26 10.8 6.013 6.292 172.38 0.39 1.7 4.819 6.252 171.79 0.52 2.5 4.921 6.211 171.20 0.65 3.4 4.216 6.169 170.62 0.78 5.1 3.498 6.127 170.04 0.91 15.8 5.420 6.084 169.46 1.04 21.0 5.460 6.048 168.90 1.17 2.5 4.191 5.995 168.30 a Measured mass density. b Theoretical density calculated from the atomic mass and the volume in one unit cell. b Unit cell volume (from Ref. 83). Reannealing Procedures for YBa9CutO7.. As the Sr doping in YBa2,ZSr,Cu307_6 hardly affected the transition tempera ture, we turned our attention to the fully oxygenated ceramics and their temperature dependence, which appeared more interesting. Therefore, we tried to reanneal two YBa2Cu3076_ (a = 0, 6 > 0.5) samples in order to increase their oxygen content in two step procedures. First, the greenblack pellets were ground into powder using a freezer mill at liquid nitrogen (LN2). The power was then placed in a platinum foil and an alumina crucible to fire at 920 OC for 12 hours in order to release any water content. (Care must be taken that the sample should not be overheated as the melting point is around 1000 OC). After having been cooled gradually to room temperature, the product was removed from the furnace. Second, this dark and brittle bulk product was reground and pressed into pellets with a pressure of 105 psi. These pellets, blackish with very smooth surfaces, were returned to the oven for a second retiring. Having been sintered at 920 OC for a 12 hour period, the pellets were annealed in a 1 atm 02 steady flow to start the oxygen doping process with a gradual temperature decrease. The timetable of this firing is illustrated in Fig. 17. Meissner Effect Test and Susceptibility. After the reannealing procedures, we immediately tested the samples in the mag netic field. As the samples were cooled to 77 K (LN2) in the field, we found both pellets levitated. The levitation lasted for more than 25 sec after the LN2 was removed from the samples, indicating that they had returned to good quality superconductors. This test, known as the Meissner Effect, showed that the magnetic flux originally present was expelled from the interior of the bulk samples. The magnetic susceptibility mea surements also showed that the samples had an onset of diamagnetism at Tc ~ 92 K as illustrated in Fig. 18, suggesting that the YBa2Cu307_ pellets had been greatly improved and become almost fully oxygenated (6 < 0.1). It turned out later that they had a very stable transition temperature and showed very little degradation over time. The improvement can also be seen in Fig. 19 which shows the reflectance {(w) and conductivity al(w) of one YBa2Cu307_ sample at room temperature, before 60 and after the reannealing treatment. In contrast to the initial measurement (dashed curves), the postreannealing spectra (solid curves) had higher overall levels in both ((w) and a,(w) and just displayed five pronounced peaks which had been clearly identified as caxis phonons. Phonons confined in the CuO2 planes are screened by the conduction carriers thus only those phonons oscillating along the caxis are visible. 61 820nm La2_xSrxCu04 film 008 SrTIO3 300 006 0 0010 0.1 ratio, 004 200 Au 0 O 0.01 > 002 X 0.00 1 I I I I I . 10 20 30 40 50 60 70 80 28 Fig. 14. Xray diffraction pattern measured at Westinghouse for a La2,SrCu04 film used in this work. The film was grown on a SrTiO3 substrate, and the growth orientation can be seen in this figure. 600 . I. .. . 200 . 0 p (pi cm) Inductance (orb. unit) 500  100 S400  E 0 C 300 25 30 35 T (K) 200 100o 820 nm La2xSrxCuO4 thin film 0 50 100 150 200 250 300 Temperature (K) Fig. 15. Resistivity in the abplane, as a function of temperature, for a La2.SrzCu04 thin film (x ~ 0.17) used in this study. The inset fig ure shows an expanded view of the region near Tc for the same sample and compares the resistive transition to the inductive transition. 600 . . 200 . 500 100 400 0 " 300 85 90 95 100 T (K) 200 100 156nm YBa2CuO3 f 0 50 100 150 200 250 Temperature (K) 300 Fig. 16. Measured dc resistivity in the CuO plane for a 156nm thick YBa2Cu3076 film. It demonstrates a sharp superconducting transition near 90 K. The inset illustrates an expanded view around T,. 64 Sintering and reannealing for YBa2Cu3076 polycrystallines 100ooo 02 flow begins 0 1 0) L 00 _ 500 E I 0 I I 0 5 10 15 20 25 30 35 Time (hours) Fig. 17. Reanneal schedule for YBa2Cu3076 samples used in this study. E CD 0 E r >1 4 an CU (I) 10 12 100 Temperature 130 (K) Fig. 18. Measured ac magnetic susceptibility of a YBa2Cu3076 pellet after reannealing. The lower branch is the "zerofieldcooled" curve (ZFC), and the upper one is "fieldcooled" (FC) which characterizes the true Meissner effect. 2 4 6 8 I I I I I YBo2Cu3O7 ceramic simple /AA A A A A A A A A  A A p I I I 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 400 300 1 200  I' 100 ,A"v: /), 0 I I 0 100 200 300 400 CO (cm1) 500 600 700 Fig. 19. Comparison of the IR spectra of a YBa2CuaOT30 pellet before and after reannealing treatment. The beforereannealing sample shows rich phonons, an indication of mixed phase. E b CHAPTER VI EXPERIMENTS AND LOW TEMPERATURE TECHNIQUES This chapter describes the experimental techniques used to perform the optical measurements on various samples over wide ranges of frequencies and temperatures. Measurements of other physical properties and the experimental apparatus are also discussed. Low Temperature Apparatus The cooling system consists of three major parts: Hansen HighTran refrigerator cryostatt), transfer line and helium supply dewar. The sample temperature can be varied from 4 K to 300 K by a controlled operation of liquid helium transfer. Figure 20 illustrates the flow diagram of the experimental setup. The sample holder is attached to the cryotip end of the refrigerator. An optical spectroscopy vacuum shroud is used to isolate the cold tip from the outside environment. Optical windows can be installed on the vacuum shroud to allow the reflection and transmission measurements. The sample temperature is sensed by a calibrated silicon diode thermometer (Si410A) buried into the cold finger. The accuracy of the diode is 1 K. The sample can be warmed by adding electrical heat to the tip heater and the temperature is controlled automatically and monitored by a temperature controller (Hansen & Associates 8000). A thermal radiation shield is attached to the second cold stage to protect the sample and to absorb the 300 K black body radiation from the vacuum shroud, hence the heat load near the cold tip can be reduced. All these steps are necessary in order to minimize the systematic error in temperature recording. Before the helium flow is started, the cryostat is evacuated to a pressure of 104 torr or less in the vacuum shroud. By pressurizing the He dewar, the liquid helium is transferred from the dewar through the transfer line to the cryostat. The flow rate can be regulated by two flow meters with hoses and shut off valves which control the tip flow and shield gas flow. Reflectance Measurements and UncertaintiesLa~ SrCuO4 Films The reflectance measurements were performed using two spectrometers with a variety of light sources, beamsplitters and detectors for different overlapping frequency ranges. The angle of incidence of the incident light was about 110 from the surface normal, so that the electric field of the infrared radiation was dominantly parallel to the abplane. The reflectance was calibrated with a reference mirror of 2000 A thick aluminum evaporated on an optically polished glass substrate. The sample and Al mirror reference were mounted on a heliumcooled cold tip, along with a silicon thermometer and a resistance heater, to allow the temperature to be varied from 5 K to 350 K. The sample and reference could be exchanged by rotating the cryostat. As the overall scale of the reflectance is very crucial to the analysis of HTSC, we carefully tested the stability and measured the absolute reflectance at each temper ature. Thermal contraction of the sample holder and position variation between the sample and reference were also taken into account. In order to study the temper ature dependence of the midinfrared band and the plasma edge, we measured the reflectance at each temperature up to 4000 cm1 (0.5 eV), and at selected temper atures up to 40,000 cm1 (5 eV). The coincidence of spectra in each of the overlap frequency range was usually within 0.5%. As the film thickness (820 nm) was much greater than the penetration/skin depth (~ 250 nm), features attributable to the SrTiOa substrate effect were not detected. Because the sample surface was extremely smooth and shiny, specular reflection was assumed and there was no need to coat the sample with a metal film to correct for diffuse scattering losses. Also, the large sample area (1x1 cm2) enabled us to obtain a high signaltonoise ratio, making it unnecessary to smooth the data for analysis. The experimental uncertainty in our reflectance measurements is estimated to be 0.5%. This error arises mainly from the difficulty in establishing precise optical alignment as the reference and the sample are interchanged, and partly from the slight temperature dependence of the Al reflectance at low frequencies. This small uncertainly in A.(w), however, will cause a larger propagated error at low frequencies in the optical conductivity a(w) generated by the KramersKronig transformation. Procedures in KramersKronig Analysis After obtaining satisfactory results for a wide range of reflectance spectra .A(w), we have confidence in using the KramersKronig (KK) transform to determine the real part of the optical conductivity al (w), a more fundamental quantity than S(w) in description of particlehole excitations of a material by absorption of photons of energy hw. In principle, the KK integral requires a knowledge of .(w) at all frequencies68 as described on p. 18 of chapter III. Thus reasonable and careful extrapolations of the reflectance beyond the measured range must be made. Highfrequency extrapolation The highfrequency extrapolation usually has a significant influence on the results, primarily on the sum rule derived from the optical conductivity. This effect has been reduced by merging our data to the reflectance spectra of Tajima et al.,2 which extend up to 37 eV (300,000 cm'). We find their spectra are in excellent agreement (within 5% in relative difference; 0.8% in absolute reflectance) with our high frequency data at room temperature. After careful measurements, however, we observe a significant decrease in the overall level of A(w) at frequencies above the plasma edge (~ 7000 cm1) as the temperature is lowered below 250 K. This decrease persists up to 40,000 cm1, the upper limit of our experimental data, the reflectance at 250 K being about 80% of that at room temperature in this frequency region. However, as the temperature is further decreased below 250 K, aside from the steepening of the plasma edge, there is very little temperature dependence down to 5 K in this high frequency region as shown in Fig. 21. We have carefully repeated the measurements several times and found this behavior reproducible in both the cooling and warming processes. At the same time, we have observed no change at all temperatures in the signal level reflected from the Al reference which has been mounted near the sample. In addition, the reflectance remains unchanged as the sample is heated up from 300 K to 350 K. These tests have convinced us that the extraneous influence such as thermal expansion/contraction of the sample holder or condensation of water on the sample surface can be ruled out. We therefore have readjusted the highfrequency roomtemperature reflectivity given by Tajima et al.27 with a relative factor of 5% increase in the range of 5 ~ 8 eV, but no change above this range, before appending it to our data for temperatures below 250 K. After doing so, we have assumed A(w) ~ w4, a freeelectron asymptotic behavior, above 37 eV. These changes preserve the sum rule at 20 eV. Lowfrequency extrapolation The lowfrequency extrapolation is equally important. We find that using the HagenRubens relation, A(w) = 1 Ax/J, for the normal state leads to a slightly depressed conductivity near the low frequency end, followed by a sharp rise towards zero frequency. This distortion may affect the estimate of the dc conductivity and also of the sum rule, from which we want to find the superconducting condensate by calculating the missing area below Tc. Since the HagenRubens relation, a good approximation for ordinary metals, appears to be inappropriate for the HTSC because of the presence of phonons and of lowfrequency (midinfrared) absorption processes, First, we make a leastsquare fit to the optical conductivity, al(w), derived from the initial KK transform of A(w). In this procedure, we use a twocomponent dielectric function (Drude plus midinfrared and phonon oscillators): W2 N 2 (W) +D i/r+E 2 i + +oo, (81) + W / ( ,,,2 jWT7j j=1 where the first term is a Drude oscillator described by a plasma frequency WpD and a relaxation time r of the free carriers; the second term is a sum of oscillators for mid infrared and phonon absorptions with wi, Wpi, and 7y being the resonant frequency, strength, and width of the jth Lorentz oscillator; and the last term coo is the high frequency limit of f(w). This last parameter is found from a fit to ,(w). Using the fit parameters, we recalculate the low frequency reflectance for the nor mal state. Then, after extending the experimental A(w) with calculated reflectance, a second KK transform is made. The results of this "second" ol(w) give a more reason able low frequency behavior. In the superconducting state, we have used the formula A = 1 Bw4, as the way that A goes to unity. For temperatures well below Tc, the low frequency reflectance is nearly constant, with some noise fluctuations around unity. We have set A = 1 in this region for the KK transformation. As mentioned earlier, the experimental uncertainty in .(w) is about A. = 0.5%. As A(w) + 1 at low w and low T, the KK transform will give propagated error in o (w)primarily coming from the propagated error in the real index of refraction n(w)roughly equal to A'I 1 A (82) al 1A (82) namely, the percentage uncertainty in oa is about 1/(1 ~) times higher than that in A. We will address this issue later. Combination of A(w) and f(w) MeasurementsYBa9CuiOT_ Films The farinfrared transmittance F(w) and reflectance A(w) measurements for the oriented YBa2Cu307O films deposited on MgO were made at temperatures from 6 K to 300 K, concentrated around Tc (~ 90 K). The light was incident nearly normal (~ 100 for reflectance and ~ 00 for transmission) to the sample surfaces. In other words, the E vector is polarized on the CuO2 plane of the YBa2Cu3076 films. Reflectance measurement is similar to that discussed above for the La2.SrxCuO4 films. For transmission measurements, the reference was a blank opening. The films, with surface dimensions of 5 x 5 mm2, were circularly masked to reduce their areas to about 4 mm diameter. Transmittance spectra from 20 to 100 cm1 for the 480 and 1560A films, and 20375 cm1 for the 1800A film, were measured using the far infrared beamline at the National Synchrotron Light Source (NSLS).85 Transmittance over 50375 cm1 for the 480 and 1560A films and reflectance over 20375 cm1 for all three films were measured using a Bruker Fouriertransform interferometer. Sample and reference spectra were measured three times each to estimate the random noise. In order to deal with the effect due to the underlying substrates, we have also measured a 1 mmthick bare MgO at each temperature where film data were taken. Measurement of YBa9._SrCu.OTO Pellets Optical reflectance measurements were made with a slowscan home made Michel son spectrometer and a fastscan Bruker interferometer for the far and mid infrared regions and with a grating monochromator for higher frequencies up to ultraviolet region. Initial measurements were made on all pellets (x = 01.3) at room temper ature. After the reannealing treatments, a fully oxygenated sample with x = 0 was chosen and carefully measured at temperatures between 7 K and 300 K. In order to study the role of lattice vibrations in the superconducting transition, a number of measurements were made around T,. Room temperature data above 4000 cm1 were used in analysis for all temperatures. This approximation was justified by the fact of only little Tdependence throughout the midinfrared range. Surface correction. One important difference between ceramic samples and smooth thin films is that diffuse scattering from the granular surface will cause a rapid decrease in reflectance with increasing frequency, particularly when the wave length is comparable to the grain sizes. To compensate for the scattering losses, all pellets were coated with a thin aluminum layer after the initial optical measurement was finished, and a second measurement was carried out on the coated samples at room temperature to estimate the losses due to nonspecular reflection. Two factors must be considered in making the coating. First, it is important that the coating layer be thin enough so that the microstructure of the sample surface remains unaltered. Second, the layer must be thicker than the penetration depth of the coating material. In our case, the Al coating was ~ 2000 A thick, smaller than the sample grain size (~15 pm) but greater than the Al penetration depth (~200 A). The coating was made by using an ion milling (microetch) equipment. The final corrected reflectance was obtained by evaluating the ratio of the initial reflectance (no coating) to the reflectance of the coated sample, then multiplying the ratio by the aluminum reflectance from the literature.86 After the reflectance spectra .9(w) over a wide frequency range were measured, the optical conductivity o(w) was determined by performing the KramersKronig transformation, with reasonable extrapolations similar to the case of our La2,SrzCu04 films described above. Fig. 20. Schematic of low temperature apparatus. *cDEWAR PRESSURIZATION RjOW PRODUCT FLOW u HELIUM EXHAUST c SHROUO VACUUM REGULATOR Photon 0 1 0.6 I S 820n 0.5 0.4  Energy (eV) 0.3 0.2 0.1 0.0 I 0 10000 20000 w (cm1) 30000 Fig. 21. Temperature dependence of the reflectance in the interband region. There is a remarkable change in A((w) between 300 K and 250 K but no appreciable change above or below this temperature range. CHAPTER VII OPTICAL STUDIES OF La2,SrzCuO4 FILMS In this chapter, we present the inplane spectra of reflectance 5(w, T) and con ductivity a(w,T) of high quality La2zSr,CuO4 films over a wide frequency range of 3040,000 cm' (4 meV5 eV) and for temperatures between 5 K and 350 K. We make an extensive optical study on the infrared dynamics of the films. Sample preparation and the characteristic transport properties have been de scribed in chapter V. The parameters of the samples have been summarized in Table 2 (see p. 55), and the dc resistivity has been illustrated in Fig. 15. Thinner films (270 nm thickness) were measured but transparent enough that some features of the substrate could be seen in the reflectance spectra. Consequently, the work described here will focus on an especially thick film with thickness (820 nm) greater than the electromagnetic penetration depth (d > 6) to avoid the substrate complications. Details of optical measurement techniques and the uncertainties in the KramersKronig (KK) analysis have been discussed in chapter VI. Here we will present the spectra of reflectance and other optical functions derived from the KK analysis. Details of the infrared phonons and optical conductivity a(w) in the normal and superconducting state are discussed. Comparisons of the normal state data to both two and onecomponent descriptions of the optical dielectric function are also made. Results and Discussion Infrared Phonons Figure 21 has showed the measured abplane reflectance A(w,T) of a La2,SrzCuO4 thin film on linear scale over most of the measured range. The details of the low frequency behavior are presented in Fig. 22 at several temperatures. The inset, which shows data plotted on a logarithmic frequency scale for the entire mea sured frequency range at three typical temperatures, illustrates the strongly damped plasma edge around 0.8 eV (6000 cm') and the interband features around the visible region. As we can see from Fig. 22, M(w, T) increases over a broad frequency range with decreasing temperature, as expected. A few infraredactive phonons in the ab plane are visible. These phonons are more obvious in the spectrum than in the case of YBa2Cu307_6.1012,3436 This indicates that La2_SrCu04 crystals have a lower free carrier concentration and a higher vibrational oscillator strength. The phonon parameters can also be extracted from oa(w), the real part of the optical conductiv ity, shown in Fig. 23. Of the seven IRactive phonon modes (3A2, + 4E.) expected at the r point for the bodycentered tetragonal D1 I4/mmm symmetry, three distinct abplane E. modes are observed at 126, 359, and 681 cm1 for T = 300 K. These eigenenergies are close to those previously reported by Collins et al.,26 132, 358, and 667 cm1, from a roomtemperature reflectance study of a La2zSrCuO4 single crystal. These three phonons have been assigned as external, bending and stretching modes, respectively.87,88 More details regarding the phonon mode assignment have been reported in Ref. 89. Structural phase transition We note that the lowest phonon mode at w = 126 cm1, corresponding to an inplane translational vibration of the La atoms against the CuO6 octahedron unit, broadens and splits into two distinct modes as T decreases below 250 K. The split ting begins at the tetragonaltoorthorhombic structural transition which involves a staggered rotation of CuO6 octahedra. At 200 K, the degeneracy of the two modes is lifted but their energies are so close that they can barely be resolved. The splitting develops upon further cooling as depicted in Fig. 24. This splitting is probably asso ciated with the folding back of the zoneboundary mode to the zone center because of the unit cell doubling due to orthorhombic distortion (D18 Cmca symmetry). Similar results in neutron scattering measurements have been reported and associated with a soft phonon mode.90 For comparison, the inset in Fig. 24 shows the results of Keane et al.91 for the inplane lattice constants of a Lal.85Sro.15Cu04 sample as a function of temperature. The structural distortion is evident in their data at T < 200 K. Frequency shift and lifetime We also observe that the CuO stretching mode at 681 cm1 hardens by 13 cm1 as the sample cools off from 300 to 100 K, as expected for thermal contraction. It stops shifting, however, upon further cooling. In contrast, the frequency of the Cu O bending mode at 359 cm1 remains constant at all temperatures yet exhibits a discernible splitting at low T. We thus conclude that the stretching mode is much more sensitive to the CuO bond length than the bending mode. Tajima et al.87 have recently found a similar result when they measured the room temperature phonon frequencies of different cuprates with different lattice constants a but almost the same reduced mass by substituting the La atom by other rare earth elements. A similar effect has also been observed in the T'RE2CuO4 system by Herr et al.92 In our case the absence of further hardening at lower temperatures is probably due to the fact that the real part of the phonon selfenergy Eph = A + if has three contributions: A(T) = A(o)(T) + A(1)(T) + A(2)(T) (83) where A() accounts for thermal expansion, A(1) and A(2) for the cubic and quartic anharmonic terms in the lattice potential, respectively. These contributions are gen erally of the same order of magnitude but may have different signs. Thus A(o) may be compensated by the sum of A(1) + A(2) at low temperatures. Another possibility is the saturation of the Tdependence of all three contributions below 100 K. Such an effect has been found in silver and thalliumhalides.93 Indeed, Tranquada et al.94 and Keane et al.91 have observed that the interatomic distances of La2,SraCu04 saturate below 100 K. It has been reported95'96 that the two lowerlying IR active phonons at 149 and 190 cm' for YBa2Cu307 ceramic samples narrow dramatically but have no softening upon entering into superconducting state. In contrast, the phonons above 275 cm1 exhibit opposite behavior (i.e., little change in width but apparently softening below Tc). The anomalous dramatic narrowing in phonon widths for YBa2Cu30aO7 has been attributed to the disappearance of interaction between electrons and phonons with energies less than the superconducting gap when the electrons condense into Cooper pairs below Tc.10 The phonon lifetime will increase as a result of decreased probability of colliding with quasiparticles, because the number of quasiparticle ex citations decreases rapidly below Tc. This issue will be addressed in more details in chapter IX, where polycrystalline samples are discussed. In any event, here we do not see a dramatic Tdependence in the observed abplane phonons for La2,SrzCuO4, perhaps because the lowest phonon mode at 126 cm1 is far above the BCS gap energy, which is ~ 80 cm1 for a Tc = 31 K sample. TwoComponent Approach Returning to the conductivity spectra as shown in Fig. 23, we note that the nor mal state al(w, T) at the low frequency limit is nearly equal to the dc conductivity and exhibits a Drude response. A remarkable depression can be seen at 30 K, just below Tc, for w < 150 cm1, indicating the shift of spectral weight into the origin due to the superconducting condensation. The inductive current represented by the imaginary part of the complex conductivity, o2, is dominant at w < 100 cm1 and it diverges as w+0 for T < Tc, as shown in Fig. 25. Above Tc, a2 changes slope at low frequencies and heads for the origin, and the maximum moves to higher fre quency and decreases rapidly with increasing temperature, as expected. On the other hand, at w > 300 cm1, the normalstate ol(w) in Fig. 23 decays much more slowly than the free carrier o2dependence as one expects in a Drude model. Addition ally, al(w) has much weaker temperature dependence at high frequencies than at low frequencies. This "nonDrude" behavior, which is universal for all copper ox ide superconductors,1013,3136,40,42 can be described in a twocomponent picture, in which a narrow (with a width of order kBT) and strongly Tdependent free carrier (Drude) absorption peaked at w = 0 combines with a broad boundcarrier (MIR) absorption centered at higher frequencies. According to this picture, the cuprates are viewed as consisting of two type of carriers: free carriers which track the dc conduc tivity above Tc and which condense to superconducting pairs below Tc, and bound carriers which are responsible for the broad MIR excitation. The dielectric function is made up of four parts: C(w) = CD + EMIR + Cphonon + foo (84) where ED is the free carrier or normal Drude intraband contribution; eMIR is the boundcarrier contribution; Ephonon is the phonon contributions, a sum of harmonic oscillators; and coo is the high frequency contribution. To decompose the total conductivity into two components, we can assume that the conductivity at 5 K, al(w, 5K), is a good first approximation of a0MIR, namely 1MIR == 1C(w, 5K), for the Drude part is presumed to have collapsed to a 6(w) function (the optical spectra are dominated by the inductive response). Thus the Drude conductivity at higher temperatures can be initially estimated by subtracting l (w, 5K) from the experimental a1(w, T), namely aD) == 0 .(1) Here the superscripts denote the number of iterations. Since 1 WPD' D (85) 4?r 1 + w272 ' we can determine WpD and 1/r from a linear fit to 1/1 s. w2. Once WpD and 1/T are determined from the slope and the intercept of this straight line, we can again estimate the midinfrared conductivity from the difference between a calculated Drude conductivity and the measured conductivity, namely a2MIR = o1 1D, where clD is calculated according to Eq. (85). By averaging (2) at temperatures above Tc, we find the average (o2flR) \ MIR [or l(w, 5K)], but there are noticeable differences. Therefore we repeat the above procedure with (2)i replacing 1MIRi 1MIR 1MIR' and find convergence after a few iterations. The freecarrier componentw and r Figure 26 illustrates the comparison between the free carrier contribution, al (alMIR), and the calculated Drude conductivity. This figure shows that the conductivity is in good agreement with the ordinary Drude behavior after the MIR component is subtracted. The Drude plasma frequency, WpD = 6300 100 cm1, obtained from the above analysis is essentially Tindependent, whereas 1/r is linear in T. Writing A/r = 22rAkBT, we obtain a weakcoupling value for the coupling constant, A = 0.25. This small value of A is consistent with the observed absence of saturation up to 1100 K for the dc resistivity.61 Taking the Fermi velocity in the basal plane to be VF = 2.2 x 107 cm/s, as calculated by Allen et al.97 for Lal.85Sro.15Cu04, and using our relaxation rate we can estimate the mean free path to be 100 K l = VFr (110 A) (86) At T = 1000 K, I ~ 11 A, which is still longer than the interatomic spacing a (here taken to be 3.8 A, the inplane lattice constant). The resistivity is expected to saturate if I < a, because the mean free path can no longer be properly defined in this region.98 On the other hand, at temperature close to Tc, the mean free path I [e.g., 150K 220 A according to Eq. (86)] is much longer than the coherence length ( (~ 10 A). It is this case that places the HTSC in the "clean limit", which in turn gives a significant impact on the observability of the superconducting gap. Figure 27 depicts the temperature dependence of 1/r in comparison with (1/r)d, calculated from the measured fourprobe dc resistivity Pdc and the value of WpD found above, 2 (1/r)dc = bD (87) As seen in Fig. 27, (1/r)dc or Pdc decreases quasilinearly from room temperature followed by a plateau and then a sudden drop as the temperature approaches Tc whereas the farinfrared scattering rate shows a quasilinear T variation followed by a fasterthanlinear drop (1/7 ~ T2) below Tc. This is evident when the same data are plotted on a loglog scale, as shown in the inset of Fig. 27. The excellent agreement in both the slopes and overall levels between the dc transport and infrared measurements strengthens our confidence in the determination of the normal state plasma frequency wpD and scattering rate 1/r. The sudden drop in 1/7 just below Tc is interesting and has received considerable attention recently. Such observations on quasiparticle damping have been reported previously for laserablated YBa2Cua307 films99'36 and a freestanding Bi2Sr2CaCu208 crystal.100'101 Similar behavior has also been found for YBa2Cu307_ and Bi2Sr2Ca2Cu3010 in femtosecond optical transient absorption experiments.102 This result may suggest that the excitation that scatters the free carriers is also strongly suppressed below Tc, or forms its own gap, as the free carriers condense. Another interpretation is that the number of unoccupied states available near the Fermi levels decreases rapidly as a result of the depression of the density of quasiparticle states near EF as the gap opens, causing a dramatic decrease in the probability of quasiparticle elastic scattering. Nicol et al.103 have recently calculated the quasiparticle scattering rate and found such a fast drop within the phenomenological Marginal Fermi Liquid model. However, on account of the large error bars at low frequencies (below 100 cm') and the limited number of temperatures below Tc (31 K) in our data, we are unable to observe a "coherence" peak in al(T), as has been calculated by Nicol et al.103 and found in YBa2Cu3aO7 by Nuss et al.,104 and in Bi2Sr2CaCu2Os by Romero et al.101 This "coherence peak" has also been observed in our YBa2Cu3aOs thin films and will be discussed in detail in the next chapter. The midinfrared absorption Figure 28 presents the MIR conductivity in the normal and superconducting states. This quantity is obtained by subtracting the calculated free carrier contri bution (shown in Fig. 26 as solid lines) from the total conductivity. Some features that are common at all temperatures include: an onset near 80 cm1, a maximum around 250 cm1, a notchlike structure at 400 cm1, and a broad peak around 800 cm1. As we can see, the MIR conductivity taIMIR(w, T) has a relatively weak tem perature dependence. There do appear to be three distinct temperature regimes: > 250 K, T,200 K, and below Tc. In each, there is a noticeable conductivity increase in the region of 1501500 cm1 with decreased temperature. The enhancement is more obvious for T < Tc and will be discussed below. According to the data in Fig. 28, the "twogap" structure of an onset near 80 cm1 (3.7 kBTc) and a notch around 400 cm1 (18 kBTc) is present both below and above Tc. This structure is shown more clearly in Fig. 29, where we plot the average of the curves above and below T,. Thus we cannot associated either feature with the superconducting gap, since that presumably would not appear above Tc. Furthermore, there is no shift in any feature in the superconducting state as would be expected for a Holstein sideband associated with condensate. Such features have also been observed35'100'105 in YBa2Cu307O_ and Bi2Sr2CaCu208 films. The structure at 400 cm1 (50 meV), which appears common to the cuprate superconductors, has been explained as due to strong bound carrier/phonon coupling.48 It can not be accepted as a superconducting gap simply because its magnitude is too large. The value of the lowerenergy onset usually varies for different samples. The presence of this structure above Tc and the lack of evidence of an energy shift with varying temperature below Tc make it difficult to associate it with the BCS gap. Holstein effect Lee et al.106 have calculated the dynamic conductivity in the framework of strong coupling theory, including the Holstein mechanism.107,108 They obtain a twogap structure in the superconducting state. The first onset is presumed to be the super conducting gap, while the "second gap" is interpreted as the consequence of inelastic scattering with phonons due to the Holstein effect. To estimate this effect, we have calculated the conductivity according to the Holstein theory for our film and find that the enhancement of the MIR strength below Tc may not be accounted for by the inelastic scattering contribution. In the Holstein model, the scattering rate at low temperature can be obtained by108 1/r(w) = F(2)(w 0) dil, (88) where a2F(Q) is the Eliashberg function or electronphonon spectral density. The parameters used in our calculation were: Wp = 6300 cm1 (from the two component model fit outlined above), A(w=0) = 0.25, and the average boson frequency flo = 75 cm1. In general, the coupling parameter is given by108 (w) = 2 a2F() In W n Wd2 2 d (89) with a zero frequency limiting value A = A(w ) = 2 f a ) d (90) For simplicity, we have assumed the Eliashberg function has the form (in an Eien stein model) a2F(0) = Ab(fS 1o), where A = 1AI o according to Eq. (90). The 2 A0acrigt Tq 9) h calculated result is illustrated as the dashdotted curve in Fig. 29. The size of the the Holstein side band could be enlarged to match the measured MIR spectral weight by increasing A and wp, but this would be in disagreement with the values determined experimentally. Superconductingtonormal ratios Another unconventional behavior is seen in the superconducting to normalstate conductivity ratio shown in Fig. 30. Ratios of conductivity have been used frequently in the past to suggest superconducting gap structure.41,42 In Fig. 30, we compare ol, and "aln" at the same temperature. We note that if al, and ol, are compared at different temperatures, the result is totally different as shown in the inset, resembling a BCSlike behavior as seen in Fig. 3 on p. 31. To estimate lan(w, T) below Tc, we presume that the "normal state" WpD and 1/7 below Tc follow the "normal" behavior, i.e., WpD remains a constant (6300 cm1) and 1/7 follows the linear extrapolation of the relaxation rate above Tc. Then ain below Tc can be calculated as the sum of the calculated Drude component and the averaged MIR conductivity (alMIR)n, namely measured al, T > Tc ln= 1 W DT (91) 4 1 + w22 + (alMIR)n. < Tc As we can see in Fig. 30, the ratio ai,/lal exhibits a sharp edge near 100 cm1 and has a peak around 180 cm1. The peak is suppressed but does not shift as T approaches Tc from below. al, "overshoots" al, up to 1000 cm1 and then gradually joins the normal state conductivity at higher frequencies. This surprising result can be attributed first to the observed enhancement of the midinfrared conductivity in the superconducting state, and second to the observed fasterthanlinear decrease in the quasiparticle scattering rate as demonstrated in Fig. 27. Extra absorption below T. We turn to the differences between the MIR conductivity above Tc and the below Tc conductivity. The enhancement is evident in the raw data of Fig. 23, in which we can see the conductivity at 5 K is higher than that at 50 K, above Tc, for w Z 360 cm1. By calculating the difference between the averaged midinfrared conductivity in the superconducting state, (alMIR),, and the one in the normal state, (UIMIR),, we find an extra absorption below Tc in the MIR region which counts for roughly 15% of the Drude oscillator strength. This difference is shown in Fig. 29. (Note that the actual fraction may be smaller for the reason of large error bars in o1 at low w below Tc, as will be discussed below; thus the difference, (aiMIR)s (aIMIR)n, may be exaggerated at low frequencies.) This anomalous behavior suggests the existence of another type of excitation visible in the superconducting state, with the normal Drude carriers not completely condensing into the superfluid below Tc. However, this argument can not be taken as rigorous, since our approach of extracting the Drude component has neglected the wdependence of the electronic scattering rate, though it may be weak as suggested by the small value of coupling constant A ~ 0.25. To confirm our observation of the extra absorption below Tc in the MIR con ductivity obtained by the twocomponent analysis, we use two other independent methods to estimate the oscillator strength of the superconducting condensate: the dielectric function and the fsum rule. According to the clean limit picture, when 2A > 1/7 the Drude oscillator strength will condense into a w = 0 delta function for T < Tc. Thus the real part of the dielectric function at low frequencies is w2 fe(w) = elb W (92) where wp, is the superconducting plasma frequency defined as ws, = 47rne2/ma with n, being the density of superfluid carriers; and elb is the bound carrier contribution to e (w), i.e., the lowfrequency sum of all finite frequency absorption. In principle, qb is wdependent. It is constant only at frequencies well below the lowest boundcarrier resonant frequency. Figure 31 shows the plot of el(w) [obtained from KK transform of A(w)] as a function of w2. The data fall on a straight line, as predicted by Eq. (92), in the low frequency range. The slope obtained from the linear regression fit at T = 5 K gives wp, 5800 100 cm1, from which the London penetration depth can be estimated to be AL = 1/21rwp = 275 5 nm. This value, which is much less than the film thickness (820 nm), is comparable to the 250 nm inplane AL found by muonspin relaxation (pSR) measurements109 for Lal.85Sro.15CuO4 at T = 6 K. We note that only a fraction f, = w,/lwD ; 85% of the free carriers condense into superfluid, in good agreement with the observation that ~15% of the Drude spectral weight has shifted to MIR region below Tc as outlined above. Further evidence that supports this argument is obtained from the fsum rule that will be discussed next. Sum RuleSuperconducting Condensate Figure 32 illustrates the spectral weight, Neff (w) m/mb, as defined according to Nef(w) 2 mV,,2 O(w')dw', (93) where e, m are the free electron charge and mass, respectively. mb is the averaged highfrequency optical or band mass, and Vcei is the volume (95 A) of one formula unit. Note Eq. (93) is also called partial sum rule and is the generalization of Eq. (45). In this expression, Nff (w) equals to the effective number of carriers per formula unit participating in optical transition at frequencies below w.68 The normal state Nff (w) curves at 10,000 cm1 gives, if mb = m, roughly 0.18 hole per CuO2 layer, which is a value close to the dopant concentration of our film ( x ~ 0.17) assuming each Sr atom donates one hole to the CuO2 layer. In the normal state, the curves exhibit a sharp rise in the far infrared followed by a broad plateau before another rise beginning near 10,000 cm1 due to the charge transfer transition. As the temperature is lowered, spectral weight transfers to lower frequency in response to a decreasing relaxation rate. Below Tc, the spectral weight is reduced as expected due to superconducting condensation. From the difference between Neff (w) m/mb for the normal and the superconducting states, the plasma frequency of the superfluid charge carriers [or the missing area in the curve of oa(w)] can be estimated. This difference gives A(Nff m/mb) = wp,s2mVen,/4re2, from which we find wp, = 5800 cm1 at 5 K, in excellent agreement with the value determined from the real dielectric function as discussed earlier. One surprising result of our measurements is that the Neff (w) m/mb in the charge transfer region is larger at T > 300 K than at other temperatures below 250 K, as shown in the inset of Fig. 32. The mechanism that causes this difference is not clear at this moment. One speculation is that the structural transition at around 250 K may change the band structure due to the doubling of the unit cell. The transformation introduces new Brillouin zone planes at which the semiconductorlike gaps are opened, transferring oscillator strength to higher frequency regions. The band mass may also change accordingly. This difference disappears, however, above 15 eV, where the Neff (w) m/ma curves come together. 15 eV is the end point of the interband excitations from the O 2p valence bands to the La 5d/4f conduction bands above the the Fermi level and the starting point of excitations from the Cu 3d bands to the La 5d/4f bands. Figure 33 shows the temperature dependence of the Drude (WpD) and supercon ducting (wp,) plasma frequencies. Here WpD is determined from the fit to eal(w) as described earlier and is consistent with a picture of constant carrier concentration in the normal state. This magnitude of WpD (~ 0.8 eV) is smaller in comparison with the values (, 1.2 eV) obtained in YBa2Cu3076_ or BiSrCaCuO crystals, presumably indicating lower carrier concentration on the CuO2 planes. Below Tc, wps is estimated from the sum rule, the linear fit to el(w) vs. w2, and the leastsquares fit to the re flectance data using a twofluid model. These three approaches give very close results in Wp, and we take the average values. Shown in the inset is the superfluid electronic density fraction f,(T). This superconducting condensate is calculated according to f,(T) = n,(T)/n = w,(T)/wD with WpD = 6300 cm1, the normal state value. This quantity f,(T) is essentially a measure of the strength of the 6 function in al(w, T), and is related to the Tdependence of the penetration depth AL(T). The solid curves in Fig. 33 and its inset show the phenomenological behavior predicted by BCS theory according to f,(T) rA(T)l2 f,(0) A(0) (94) where A(T) is the Tdependent BCS order parameter. It gives a nearly constant A(T) at T < Tc. Near Tc, A(T) drops to zero with a (1 T/Tc)1/2 dependence. The behavior of f,(T) in our data agrees with this expression and it demonstrates that the normal carriers condense rapidly into the superfluid below Tc, as expected. OneComponent Approach An alternative approach to analysis of the optical conductivity is the one component model with a frequency dependent mass and scattering rate.40,110112 In this approach, the infrared absorption is entirely due to free carriers, in which are di vided into "coherent" and "incoherent" parts caused by the interaction of the free car riers with some sort of optically inactive excitations (charge or spin fluctuations).105 This approach has been proposed by Anderson113 and applied to heavyFermion superconductors114. The normal Drude component is regarded as the coherent part centered at w = 0. The incoherent part occurs at frequencies characteristic of the ex citations and shifts away from w = 0 due to interactions with the excitations. In this model, the complex dielectric function is described by a generalized Drude formula: 2 c(w) = Ch W (95) W[W E(W)P where eh is the "background" dielectric constant associated with the high frequency contributions, wpdefined by 4irNe2/miis the bare plasma frequency of the free carriers, and E(w) = El(w) + iE2(w) is the self energy of the carriers. Because e(w) is causal, El(w) and E2(w) are related by the KramersKronig equa tions. It is important to stress that the interband contributions, which can be lumped into ch, are excluded from wp and E(w). To find E(w), knowledge of wp and Eh is re quired. In order to identify the interband components, we fit the experimental al (w) at frequencies higher than 8000 cm1 with Lorentz oscillators to parameterize the interband absorption. By subtracting the contribution due to these interband oscil lators from the total conductivity and calculating the area under al(w), we obtain Wp = 13, 000 cm1, corresponding to a carrier density of n = 1.8 x 1021 cm3(M/m) or 0.17 holes per CuO2 unit if mb = m. As we have found WpD = 6300 100 cm1 in the twocomponent analysis, we can also estimate the strength of MIR absorption or the "incoherent" component as pm = (2 w2D)1/2 ; 11,370 cm1. eh can be estimated from the interband oscillators, giving eh ~ 4 in the far infrared region. At higher frequencies, ch becomes complex and wdependent. Mass enhancement m*/mb and self energy E(w) Once Wp and Ch are determined, the self energy E(w) can be calculated at each frequency from the experimental e(w) according to Eq. (95). If we rewrite Eq. (95) as E(W) = Ch W (96) W[W + i/T*(a)] and compare Eq. (96) with Eq. (95), we can extract the renormalized scattering rate 1/T*(w) = E2(w) mb/m*, and the effective plasma frequency w* = wp(mb/m*)1/2, where the effective mass enhancement is given by m*/mb = 1  E/w. (97) Note both the real and imaginary parts of E(w) are negative definite. The computer routine for the onecomponent analysis is given in Appendix B. The resulting curves of m*(w)/mb and E2(w) are shown in Fig. 34. The effective mass m* is greatly enhanced at low w and m* $ mb at high w, as expected for the MFL and NFL theory.37'38 The behavior of m*(w)/mb and E2(w) as shown can be viewed as arising from a local Coulomb interaction of carriers with a broad spectrum of other excitations. At low frequencies, the carriers drag a lowenergy excitation cloud along with them, caus ing a mass enhancement. As frequency increases, the scattering rate 1/7* increases when the lowlying states are excited hence a new inelastic scattering occurs. The carrier mass decreases to approach the band mass as w increases, for the lowlying excitations can not follow the rapid carrier motion. We can estimate the character istic energy range of the lowlying excitations from the frequencies at which m*(w) and E2(w) change from their low to high frequency behavior. This range appears to be between 3001000 cm1 (0.041.2 eV). We note that a pronounced peak near 0.1 eV reported by Uchida et al.29 is not observed in our spectra of m*/mb and E2. The present values of E2 are comparable with their result for the unnormalized scattering rate. The mass enhancement here is, however, a factor of 0.15 smaller than their result. The high value of m* in their data would imply an even stronger coupling be tween the free carriers and the lowlying excitations, which is difficult to understand. Note that the value of m*/mb at low w and low T can also be predicted from the conductivity sum rule from Fig. 32 or simply from w/o~wD 4.2, which agrees well with the result in Fig. 34. Writing m*/mb = 1 + A, we find the lowfrequencylimit value of coupling constant A w 3 at low temperatures, suggesting strong interaction of carriers with a spectrum of other excitations. One major difficulty with this model is that this large A would give a high Tc, inconsistent with the actually measured Tc value. To account for this large A, one may speculate that the Tc is suppressed by other mechanisms. However, Such mechanisms, if any, are not clear at this point. Effective scattering rate 1/r*(w) A linear Tdependent scattering rate at w ~ 0 implies it is also linear in w at higher frequencies. The effective renormalized scattering rate can be obtained by l/r* = (mb/m*) E2. This quantity is shown in Fig. 35. The extrapolated w = 0 values of 1/T* are compatible to those obtained above in the twocomponent fit by assuming a constant scattering rate. This is not surprising since both the one and twocomponent approaches have described the dc transport behavior well. At higher frequencies, we observe 1/r* is of order max(T, w) before it saturates. According to the MFL theory, however, it is not 1/r* but the imaginary part of the quasiparticle self energy E2 that has the form E2 = Amax(7rT, w), as long as w < wc 1000 cm1. Thus E2 would change from constant to linear in w at w > 7rT. At low w, our results agree with this prediction, and E2 tends to saturate at frequencies above we. Since A is in principle Tindependent, one expects the slope of E2(w) to be constant at all temperatures in MFL theory. However, our data indicate a gradual decrease of slope with increasing temperature. It is difficult to interpret the frequency dependent scattering rate as a consequence of inelastic scattering due to Holstein effect,107,108 in which a carrier can absorb a photon of energy iw, emit an excitation (or a phonon) of energy e (e ~ 300 cm1 in this case), and scatter. First, the large value of A (~ 3) derived from our data of Fig. 34 suggests a strong coupling between the conduction carriers and the excitation. 