Course Hero - We put you ahead of the curve!
You have requested the below document.
- Title: Quijada99prb-ab-bisco
- Type: Notes
- School: UF
- Course: PHZ 6426
- Term: Fall
REVIEW PHYSICAL B VOLUME 60, NUMBER 21 1 DECEMBER 1999-I Anisotropy in the ab-plane optical properties of Bi2Sr2CaCu2O8 single-domain crystals M. A. Quijada* and D. B. Tanner Department of Physics, University of Florida, Gainesville, Florida 32611-8440 R. J. Kelley and M. Onellion Department of Physics, University of Wisconsin, Madison, Wisconsin 53706 H. Berger and G. Margaritondo Institut de Physique Applique, Ecole Polytechnique Federale, CH-1015 Lausanne, Switzerland Received 17 June 1999 The ab-plane optical properties of the high-temperature superconductor Bi2Sr2CaCu2O8 are anisotropic in both the normal and the superconducting state. Consistent with the orthorhombic structure, the principal axes lie along the a and b crystallographic axes, nearly 45 from the Cu-O bond direction. In the normal state, analysis of the temperature-dependent optical conductivity suggests a scattering rate for the free carriers that shows ab anisotropy in both magnitude and temperature dependence. In the superconducting state, the anisotropy in the oscillator strength of the super uid response determined from the far-infrared frequency dependence of 2 ( ) and from a sum-rule analysis leads to a penetration depth D that is larger along the b axis than b a the a axis: ( L L ). S0163-1829 99 01845-7 I. INTRODUCTION There have been many publications about the ab-plane anisotropy of the optical properties of the high-temperature superconductors.1 YBa2Cu3O7 was the focus of much of this work;2 9 its Cu-O chains along the b axis are considered to be the cause of the anisotropy in the optical conductivity as well as in the dc transport10 properties. This assignment follows the conventional approach of separating the response into chain and plane components, with the underlying assumption that the quasi-two-dimensional CuO2 planes are isotropic.5 8 In this paper we describe a study of the ab-plane anisotropy in the optical properties of single-domain crystals of Bi2Sr2CaCu2O8. Unlike YBa2Cu3O7 , there are no Cu-O chains, making it possible to study the anisotropy of the CuO2 planes. One issue is the anisotropy of the order parameter in the superconducting state. A second and equally important issue is the anisotropy of the electronic structure in the normal state. The CuO2 planes in Bi2Sr2CaCu2O8 are separated by double Bi2O2 layers, which are believed to act as a charge reservoir. There is an orthorhombic distortion in the ab plane because of weak superlattice modulation along the b axis, which is attributed to an incommensurate defect structure in the Bi2O2 layers.1 13 Note that the a and b axes in this material are along the Bi-O bonds and nearly 45 from the Cu-O bonds. Based on the very small difference in the a and b bond lengths 0.04 , one would expect an almost isotropic ab-plane conductivity. In the experiments reported here, we measured the re ectance of Bi2Sr2CaCu2O8 single-domain crystals for light polarized along the two principal axes in the ab plane in a frequency range of 80 33 000 cm 1 10 meV 4.1 eV . The re ectance in the far-infrared and mid-infrared regions was measured for temperatures above and below the superconducting transition temperature. The optical conductivity and 0163-1829/99/60 21 /14917 18 /$15.00 PRB 60 related optical constants are obtained from a Kramers-Kronig analysis of the re ectance at each temperature. Analysis of the optical conductivity is carried out using one- and twocomponent models. A discussion of the anisotropy in the superconducting state will be given in the context of these two models. A detailed study of dc transport measurements performed on similar single-domain crystals is reported elsewhere.14,15 II. PREVIOUS WORK ON Bi2Sr2CaCu2O8 The rst single-crystal re ectance spectra of Bi2Sr2CaCu2O8 were reported by Reedyk et al.16 The temperature dependence of samples with T c 85 K was measured in the far infrared and the optical conductivity determined by Kramers-Kronig analysis. Similar spectra were found by other workers.17 29 The re ectance drops steadily throughout the infrared to a minimum around 10 000 cm 1 1.3 eV . Very little temperature dependence is observed above 1000 cm 1. Also visible are a band centered around 16 000 cm 1 2 eV as well as structure with considerable sample-to-sample variation spanning 28 000 32 000 cm 1 3.5 4 eV . The lower band is interpreted as the Cu-O charge transfer band, while the upper one is most likely associated with excitations of the Bi-O layers. As rst shown by Forro et al.,30 the micaceous nature of the bismuthates makes it possible to prepare thin freestanding akes of Bi2Sr2CaCu2O8 that are as little as 1000 thick and to make infrared transmission studies of these akes. Because the samples are free standing, the transmittance may be measured over a wide frequency range without interference from a substrate. Romero et al.31 measured the transmittance T between 80 and 30 000 cm 1 at temperatures between 20 and 300 K. The transmittance is low overall, and increases with increasing frequency. The low-frequency T was rather different above and below the superconducting 14 917 1999 The American Physical Society 14 918 M. A. QUIJADA et al. PRB 60 transition, with a nite intercept for T T c contrasting with T 2 for T T c . At higher frequencies, T increases quasilinearly with out to 2200 cm 1 0.27 eV ; above which it increases more quickly. There is a transmission maximum at 14 000 cm 1 1.8 eV and a second maximum at 25 000 2 cm 1 3.1 eV . The linear increase is different from the behavior that is expected for a simple metal.32 That the ab plane itself is not isotropic was observed by Romero and co-workers.31,33 This anisotropy occurs in spite of the pseudotetragonal crystal structure of this material and the absence of chains. The optical conductivity and other optical constants of Bi2Sr2CaCu2O8 have been estimated by many workers with qualitatively similar results. Reedyk et al.,16 Quijada et al.,24 Puchkov and co-workers,25,27 Basov et al.,26 and Liu and co-workers28,29 reported the results of Kramers-Kronig analysis of re ectance. Romero and co-workers33 35 determined the optical conductivity by Kramers-Kronig analysis of the transmittance. In the normal state the low-frequency conductivity approaches the dc conductivity and falls with increasing frequency. However, 1 above 300 cm 1 the decrease in 1 ( ) is closer to 2 than the behavior expected for free carriers. Furthermore, the T dependence of 1 ( ) at high frequencies is much smaller than at dc or low frequencies. Thus, Bi2Sr2CaCu2O8 displays the non-Drude conductivity that is a common feature of the high-T c superconductors. Below T c , 1 0.15 eV , 1 ( ) has a broad maximum around 1000 cm with some structure in the phonon region. A slight dip in 1 50 meV although this 1 ( ) can be seen around 400 cm antiresonance or notch is not as noticeable as in the YBa2Cu3O7 system. Recently, considerable work has been reported on Bi2Sr2CaCu2O8 samples in the underdoped portion of the phase diagram.25 27,29 These measurements are in the regime where a pseudogap may occur in the normal state. The pseudogap is not evident in the optical conductivity measured in the ab plane. Instead, there is structure in the scattering rate, 1/ ( ,T), as calculated from a memory-function analysis of the optical conductivity. In optimally doped materials, the scattering rate is a nearly linear function of the frequency. In contrast, the underdoped cuprates have a scattering rate that is depressed at frequencies below about 700 cm 1 at temperatures a little above T c . duces samples that are very slightly on the underdoped side of the Bi2Sr2CaCu2O8 phase diagram. Typical crystals are thin platelets with dimensions of a few millimeters in the ab plane. Identi cation of the a and b axes was done by using low-energy electron diffraction techniques. The incommensurate superlattice modulation pattern was seen along the b axis and not in the perpendicular direction a axis , suggesting the samples were single-domain crystals. The alignment of the principal axes in the crystal was con rmed by observing the extinction points when the sample was rotated under a microscope Olympus, model BHM with crossed polarizers. Meissner effect measurements indicate the samples are single phase with the onset of superconductivity around 86 K. This is in good agreement with the onset of superconductivity as determined by using four-probe resistance measurements.14 B. Optical techniques III. EXPERIMENTAL TECHNIQUES AND DATA ANALYSIS A. Crystal growth and characterization The Bi2Sr2CaCu2O8 crystals used in the study were grown by using standard techniques as reported elsewhere.36 In a typical experiment, the starting materials, Bi2O3, SrCO3, CaCO3, and CuO are ground and placed in an alumina crucible. The mixture is then heated to a temperature of 50 70 C above the liquidus temperature and equilibrated for 6 h. The temperature is subsequently lowered to 875 880 C, and after reaching equilibrium for 6 h, the temperature is slowly cooled at 0.5 2 C/h to 820 C, after which the furnace is cooled more quickly. Samples are subsequently annealed in dry oxygen at 600 C for 8 h and later reannealed in argon at 750 C for a period of 12 h. This procedure pro- Normal-incidence re ectance of the samples was measured by using a modi ed Perkin-Elmer 16U grating spectrometer in the near-infrared and ultraviolet regions 2000 33 000 cm 1 . The far-infrared and midinfrared regions were covered using a Bruker IFS-113v Fourier transform spectrometer 80 4000 cm 1 . Linear polarization of the light was achieved by placing a polarizer of the appropriate frequency range in the path of the beam using a gear mechanism that allowed in situ rotation. This setup allowed for very accurate determination of the anisotropy in the re ectance. Low-temperature measurements 10 300 K were done by attaching the sample holder assembly to the tip of a continuous- ow cryostat. A exible transfer line delivered liquid helium from a storage tank to the cryostat. The temperature of the sample was stabilized by using a temperature controller connected to a previously calibrated Si diode sensor and a heating element attached to the tip of the cryostat. The data acquisition process consisted of measuring spectra at each temperature for both the sample and for a reference Al mirror, and then dividing the sample spectrum by the reference spectrum in order to obtain a preliminary re ectance of the sample. After measuring the temperature dependence in this preliminary re ectance for each polarization, the proper normalizing of the re ectance was obtained by taking a nal room-temperature spectrum, coating the sample with a 2000- -thick lm of Al, and remeasuring this coated surface. A properly normalized room-temperature re ectance was then obtained after the re ectance of the uncoated sample was divided by the re ectance of the coated surface and the ratio multiplied by the known re ectance of Al. This result was then used to correct the re ectance data measured at other temperatures by comparing the individual room-temperature spectra taken in the two separate runs. This procedure corrects for any misalignment between the sample and the mirror used as a temporary reference before the sample was coated and, more importantly, it provides a reference surface of the same size and pro le as the actual sample area. The uncertainties in the absolute value of the re ectance reported here are in the order of 1%, while the uncertainty in the relative anisotropy is much smaller, 0.25%. This PRB 60 ANISOTROPY IN THE ab-PLANE OPTICAL . . . 14 919 uncertainty is in good agreement with the reproducibility found from the measurements of three different samples.24 C. Kramers-Kronig analysis We estimated the optical constants by Kramers-Kronig transformation of the re ectance data.37 The low- and highfrequency extrapolations were done in the following way. The conventional low-frequency extrapolation for metals is the so-called Hagen-Rubens relation, R( ) 1 A , where A is a constant determined by the re ectance of the lowest frequency measured in the experiment. For high-T c samples, this procedure is inadequate; it can only be used as a rst approximation. A better procedure extends the lowfrequency data using a Lorentz-Drude model, dominated at the low frequencies by the free-carrier Drude form. Finitefrequency excitations are modeled by Lorentz oscillators. The tted re ectance is then used as an extension below the lowest measured frequency. In the superconducting state, the re ectance is expected to be unity for frequencies close to zero. An empirical formula that represents the way R approaches unity is R 1 B 4 , where B is a constant determined from the lowest frequency measured. However, it is better to use the same Lorentz-Drude model, but with the Drude scattering rate set to zero. At high frequencies, the ab-plane anisotropy in the re ectance is consistent with the anisotropy obtained using ellipsometric technique in the visible and UV ranges.38 We therefore extended the data up to 50 000 cm 1 by appending the results of Kelly et al.38 Above this frequency, the re ectance for higher interband transitions was modeled using the formula R Rf f s FIG. 1. The 300-K re ectance of Bi2Sr2CaCu2O8 for light polarized along the a and b axes. , 1 where R f and f are the re ectance and frequency of the last data point. The exponent s is a number that can take up values between 0 and 4; we used s 2. At very high frequencies ( f ), where the free-electron behavior sets in, the approximation used is R Rf f 4 , 2 with f chosen to be 100 eV and R f adjusted to match smoothly to R( ) from Eq. 1 . We observed some dependence of the results for frequencies close to the highest frequencies on the choice of s and f . For lower frequencies, however, the effects due to the choice of the exponent s and f were insigni cant. IV. THE REFLECTANCE A. Room-temperature spectra Figure 1 displays the room-temperature re ectance of Bi2Sr2CaCu2O8 over a wide frequency range for polarization along the a and b axes. At low frequencies the a axis re ectance is 1 2 % higher than the b axis re ectance. As the frequency increases, the re ectance in both polarizations falls off in a quasilinear fashion. Near the plasmon minimum, we observe that the plasma edge for the polarization parallel to the b axis occurs at a slightly lower frequency than for the a axis. The splitting is about 500 cm 1, making the in-plane anisotropy of Bi2Sr2CaCu2O8 much less pronounced than in YBa2Cu3O7 , where the splitting is about 5500 cm 1. The larger plasma edge along the b direction in has been attributed2,3,5,8 to the presence of YBa2Cu3O7 CuO chains along the b axis. The anisotropy of Bi2Sr2CaCu2O8, which does not have chains, demonstrates that the electronic structure of the CuO2 planes is themselves anisotropic. This anisotropy is of course consistent with the orthorhombic structure of Bi2Sr2CaCu2O8. However, the structural anisotropy of the CuO2 plane is much smaller in Bi2Sr2CaCu2O8 than in YBa2Cu3O7 . Indeed, because the Cu-O bonds are nearly 45 from the a and b axes, there is almost no difference in their lengths. Bi2Sr2CaCu2O8 is thus of particular interest because the orthorhombic distortion of the CuO2 layers appears to be the only reason for the inplane anisotropy in the optical properties. At frequencies above the plasmon minimum the re ectance is substantially higher for E b. Two interband transitions are evident in this region. The rst interband peak is present in both directions, while the second one, centered at 3.8 eV, is more pronounced for the polarization along b and is almost absent along the a direction. The rst peak, at 2.3 eV, is assigned to the charge transfer band of the CuO2 planes of this material, whereas the second one is most likely associated with interband transitions occurring in the Bi2O2 layers. This result is in agreement with ellipsometric measurements of Kelly et al.38 B. Temperature-dependent spectra The temperature dependence of the polarized re ectance in the far-infrared and mid-infrared regions is shown in Fig. 2. Two things should be noticed about these data. First, there 14 920 M. A. QUIJADA et al. PRB 60 FIG. 3. Relative absorptivity (Ra Rb A b A a ) for two different samples in the superconducting state. V. OPTICAL CONSTANTS FIG. 2. Temperature dependence of the a- and b-axis re ectance of Bi2Sr2CaCu2O8. A. Room temperature is an increase in the far-infrared re ectance as the temperature of the sample is lowered, with two shoulderlike features developing in both polarizations as the sample enters the superconducting state. The rst feature is at a frequency between 100 and 200 cm 1; it is followed by a stronger feature 400 450 cm 1 and a weak minimum around 900 at 1 cm . The interpretation of these features will be given when discussing the optical conductivity obtained from the Kramers-Kronig analysis. Second, the re ectance is higher for E a at all frequencies and temperatures. Even when the sample is superconducting, there remains a difference in the re ectance (Ra Rb ) of about 1%. Romero et al.31 found the transmittance of freestanding single crystals of Bi2Sr2CaCu2O8 to be higher for light polarized along the a axis, implying more absorption for the b-axis polarization. The re ectance data in Fig. 2 agree with these results. Figure 3 shows for two different samples the anisotropy in the superconducting-state absorption at 20 K , de ned as Ra Rb A b A a , where A 1 R. In both cases, the lowfrequency absorption anisotropy is about 0.5 1 %. Although the accuracy in absolute re ectance is only 1%, so that we cannot determine whether the a-axis re ectance is indeed unity below 200 cm 1, our accuracy in anisotropy determination is 0.25%. This measure of the uncertainty in the anisotropy determination was obtained in two ways. It represents the statistical variation in repeated measures of a single sample; it also represents the reproducibility found from the measurements of three different samples.24 Therefore, we can say with certainty that the b-axis re ectance is less than 100% down to 150 cm 1 20 meV . The 300-K optical conductivities a and b are shown in Fig. 4. The inset shows the low-energy region on an expanded scale. All of the aspects of the re ectance are also evident here. First, there is a weak anisotropy in the farinfrared conductivity ( a b ) that is in agreement with the dc value for similar samples.14 Second, the low-frequency 2 dependence 1 ( ) falls off much more slowly than the 1 of a simple Drude spectrum. For these samples the slope of 1 0.56 0.02 region is for 1 ( ) between 400 8000 cm both a and b directions. This slope is nearly the same as for the a-axis room-temperature conductivity15 of YBa2Cu3O7 ( 0.53 0.02). Interpretation of this non-Drude behavior of the optical conductivity of the copper-oxide superconductors has been one of the most debated issues related to the optical properties of these materials. The upper panel of Fig. 5 shows the energy-loss function, Im(1/ ). There is a slight difference in the position of the loss function maxima for the two polarizations. In simple metals, the position of this peak gives the screened plasma frequency p and its width is a measure of p/ the scattering rate of the free carriers. However, because of the unusual behavior of the conductivity in the midinfrared, it is dif cult to make the same assignment here, unless some assumptions are made about this midinfrared absorption. If a generalized Drude model39 is used to describe the data, the results lead to a renormalized scattering rate 1/ * and an effective mass enhancement m* for the quasiparticles. The extra absorption is then a consequence of a linear increase of 1/ * as the frequency of the light is increased, and the width of the loss function is the value of 1/ * at the screened plasma frequency. However, if two or more types of carriers contribute, then the broadening in Im(1/ ) is due to the combination of the free Drude-like carriers with other type of bound-carrier excitations in the midinfrared. PRB 60 ANISOTROPY IN THE ab-PLANE OPTICAL . . . 14 921 FIG. 4. Room-temperature conductivity on a wide frequency scale for polarization along the principal axes in the ab plane of Bi2Sr2CaCu2O8. The inset shows the details of the low-frequency region. It has been suggested by Bozovic et al.40 that there is a universal frequency dependence in the loss function of all the copper-oxide superconductors, with Im(1/ ) 2 for small . The low-frequency data for Im(1/ ) are shown on a log-log scale in the inset of Fig. 5. This plot shows that there are actually two regimes to consider in the Bi2Sr2CaCu2O8 samples we measured. For frequencies below 1000 cm 1, the power-law dependence in Im(1/ ) is 0.99 0.02. Only for 1000 cm 1 does the exponent in become on the order of 2. Similar observations were made by Gao et al.41 from measurements on the La2 x Srx CuO4 system. The real part of the dielectric function 1 ( ) is shown in the lower panel of Fig. 5. The zero crossing of 1 ( ), which also is related to the screened plasma frequency p , evidently occurs at a lower energy for E b than for E a. This difference could be due to either a larger bare plasma frequency p for the charge carriers moving along the a direcfor tion or to a larger high-frequency dielectric constant E b. The more pronounced peak in the interband transition 3.8 eV for E b suggests a larger for this centered at direction, making the latter the more likely possibility. Figure 6 shows the results of evaluating the partial sum rule for 1 ( ). This function is given by N eff m mb 2mV cell e2 d 0 , 3 where e and m are the free-electron charge and mass, respectively, m b the effective mass, and V cell the volume occupied by one formula unit of Bi2Sr2CaCu2O8. The curves give the effective number of carriers per formula unit participating in optical transitions below frequency . Although N eff is roughly isotropic at low frequencies, some differences appear at higher energies, particularly in the interband region where transitions in the Bi2O2 layers are thought to occur.38 At a frequency of 12 000 cm 1 1.5 eV , the onset of the charge-transfer band, N eff 0.75. Thus, assuming an effective mass of m b m, there are about 0.37 carriers per CuO2 unit. B. Temperature dependence of the optical conductivity FIG. 5. Upper panel Room-temperature loss function Im(1/ ) on a wide-frequency scale along the principal axes in the ab plane of Bi2Sr2CaCu2O8. Inset: Logarithmic scale to show power-law dependence at low frequencies. Lower panel real part of dielectric function for the ab plane of Bi2Sr2CaCu2O8. The temperature evolution of the optical conductivity is shown in Figs. 7 a and 7 b for E a and E b, respectively. There are several important features to these spectra. First, in the normal state 100 300 K the low-frequency optical conductivity extrapolates reasonably well to the dc conductivity. For example, at 300 K the a-axis conductivity extrapolates to 1 1 cm 1; the b-axis to about 3000 cm 1. about 3400 14 922 M. A. QUIJADA et al. PRB 60 FIG. 6. The results of evaluating the partial sum rule for the room-temperature conductivity of Bi2Sr2CaCu2O8. These values correspond to resistivities of 290 and 330 cm for a and b, respectively. Second, with decreasing temperature, the low-frequency conductivity increases strongly, in agreement with the T-linear resistivity. There is a characteristic narrowing of this far-infrared portion of the spectrum. This narrowing can be seen most easily by noting that the 300-K data are larger than the lower-temperature data above about 500 cm 1 but then crosses through each of the lower temperature curves at progressively lower frequencies, becoming the smallest conductivity near 100 cm 1. Third, at high frequencies, 1 ( ) does not show much temperature variation; all the curves draw together around 3000 cm 1. Below T c , the low-frequency conductivity is considerably reduced, so that the 20-K conductivity is smallest at all frequencies measured. The missing area in the farinfrared conductivity appears as the zero-frequency deltafunction response of the super uid. This aspect is discussed in more detail below. In addition, a minimum develops 400 cm 1 in the superconducting state. This feaaround ture is visible for both polarizations, but is most evident in the b-axis data. The lower re ectance of the b polarization is responsible for the upturn observed in 1 ( ) at the lowest temperature for this polarization. This result may have interesting consequences for the understanding of dynamic properties in these materials. For instance, it indicates the presence of low-lying excitations for carriers moving along the b axis that are not present in the perpendicular direction. It is not clear what the origin of these excitations is. Perhaps, the presence of the superlattice structure in the b axis may introduce an additional source for scattering along this crystallographic direction. However, on account of the periodicity of the superlattice, one should not expect any additional scattering process for this direction. In some studies, structure in the conductivity around 400 cm 1 has been interpreted as evidence of the superconducting energy gap.3 The fact that we observe the feature for the FIG. 7. a Temperature dependence of the a-axis optical conductivity obtained from Kramers-Kronig analysis of the re ectance. b Temperature dependence in the b-axis optical conductivity obtained from Kramers-Kronig analysis of the re ectance. polarization along b, which has re ectance lower than unity, contradicts the conventional s-wave notion of a gap, where for frequencies below 2 no excitations are allowed. We note that photoemission experiments on samples from the same batch as the ones used in this study have shown evidence for the opening of a gap below T c in the energy spectrum around the Fermi level.42,43 The maximum gap parameter obtained from the analysis of the photoemission data has a value of close to 25 meV at least along the a axis . Although twice this energy is close to the minimum position observed in the conductivity data, the minimum in the conductivity cannot be assigned to the superconducting energy gap. Moreover, since similar structures at about this energy are observed in all copper-oxide superconductors, they have PRB 60 ANISOTROPY IN THE ab-PLANE OPTICAL . . . 14 923 also been explained as due to strong bound-carrier interactions with phonon excitations.44 47 VI. APPROACHES TO THE OPTICAL PROPERTIES In this section, we discuss the normal-state optical conductivity in the far-infrared and mid-infrared regions. This is the frequency range below the charge-transfer gap of the insulating phase of the cuprates; thus, it is the region where the dynamics of the doped-in charge carriers may be studied. Two approaches are used, called one-component and two-component. In the rst, there is only a single band or type of carrier, and the unusual frequency dependence in the mid-infrared is attributed to interaction with a spectrum of optically inactive excitations. This interaction causes a frequency dependence to the carrier scattering rate and enhances the low-frequency effective mass. In the two-component approach, two different contributions are assumed: the Drude response of free carriers and a second band of mid-infrared carriers. The properties of the Drude carriers determine the dc conductivity, including its temperature dependence, while the mid-infrared carriers dominate the higher frequency conductivity. A. One-component analyses uid NFL theory of Virosztek and Ruvalds,50 the Luttinger liquid picture of Anderson,51 the proposal that the conductivity is a paraconductivity involving phase separation, advocated by Emery and Kivelson,52 and the nearly antiferromagnetic Fermi liquid picture of Monthoux and Pines.53 In many of these the scattering rate is simply given by the temperature. In the MFL, the imaginary part of the one-particle selfenergy is written as 2 T, , T T, 8 Im Many authors3,7,8,25,26 have analyzed 1 ( ) using a generalized Drude model with a frequency-dependent scattering rate. In this approach, there is only one type of charge carrier. Hence, the dielectric function can be written as39 *2 p 2 i /* , 4 is a constant that includes contributions from inwhere terband transitions, the plasma frequency * (m/m*) p p with m* the effective mass, and p the bare plasma fre4 Ne 2 /m b . The quantity quency, de ned by p 1/ *( ), known as the renormalized relaxation rate, may easily be calculated, 1/ * 4 1 1 T the where is a dimensionless coupling constant. For model predicts a renormalized scattering rate that is linear in temperature, which is in accord with the linear temperature dependence in the resistivity that is observed in nearly all copper-oxide superconductors. As increases a new spectrum of excitations comes into play, causing the scattering rate to grow linearly with frequency up to a cutoff frequency c , which is also included in the model. The physical source of this behavior is the coupling of the charge carriers to a spectrum of charge and/or spin-density uctuations. The strong frequency dependence arises from a Holstein54 process, where an electron can absorb a photon, emit some other excitation, and scatter. Energy conservation requires the photon frequency to be greater than the excitation frequency . Thus the scattering by this process turns on at the onset of the excitation spectrum and becomes stronger with increasing frequency until the maximum excitation frequency is reached. Romero et al.34 analyzed their optical conductivity for Bi2Sr2CaCu2O8 and Bi2Sr2CuO6 to obtain , nding 0.27 for both materials. The MFL equations for the selfenergy agree with the data in several important ways. First, both give a dc resistivity in agreement with experiment. SecT. Third, there ond, Im increases linearly with for is an enhancement of the effective mass at low frequencies by an amount that is larger at lower temperatures. The dielectric function for the MFL can be written as48,49 2 p 2 N 2 pj 1 2 j 2 , 5 2 without knowing p . It is also possible to use a complex relaxation rate, or memory function,39 G 1/ ( ) i ( ) with 1/ ( ) the unrenormalized carrier scattering rate and ( ) the mass ). The dielectric funcenhancement factor. m* m(1 tion is 2 p 2 /2 j i , j 9 1 i/ . 6 In this memory-function approach the relaxation rate may be calculated from 1/ 2 p Im 1 . 7 Models providing a phenomenological justi cation for this approach include the marginal Fermi liquid MFL theory of Varma and co-workers48,49 the nested Fermi liq- where p is the bare plasma frequency for the charge carriers, de ned by 2 4 ne 2 /m b with n the carrier density and p m b the band mass of the carriers. The quantity represents the quasiparticle self-energy. The real part of is related to the effective mass m* of the interacting carriers by49 m*( )/m b 1 2 Re ( /2)/ , whereas the imaginary part is related to the quasiparticle lifetime through 1/ *( ) 2m b Im ( /2)/m*( ). The factors of 2 arise because quasiparticle excitations come in pairs. The second term is a sum of Lorentzian oscillators of strength p j , center frequency j , and width j representing contributions from , is the contriinterband transitions. The remaining term, bution from transitions above the highest measured frequency. Figure 8 shows the result of tting Eq. 9 to our data for the a-axis conductivity. The data are shown as heavy dashed lines whereas the ts are shown as thin solid lines. The con- 14 924 M. A. QUIJADA et al. PRB 60 FIG. 8. Fits to the a-axis conductivity using the marginal Fermi liquid MFL model. The t is the thin solid line; components of the t MFL conductivity, two weak 100-meV bands, a 0.5-eV band, and the 2-eV charge-transfer band are thin dashed lines. nearly as strong as the MFL contribution. Similar results were obtained for the b axis, and the parameters for both principal axes are listed in Table I. Interestingly both the plasma frequency and the coupling constant are larger for the b polarization than for the a polarization. The analysis of Romero and co-workers34 for unpolarized data gave results midway between these. As an alternative to tting, the scattering rate and effective mass functions may be calculated from Eq. 4 once and p are obtained. Figure 9 shows good estimates for m*/m b and 1/ *( ) at room temperature for Bi2Sr2CaCu2O8 along the a and b axes. The values used were 1 1 and and pa 16 200 cm a 4.6; pb 16 240 cm 4.8. In Fig. 9 it is interesting to notice that the mass b enhancement at low frequencies is isotropic in the ab plane, on account of the nearly isotropic values of p . In contrast, Fig. 9 shows an anisotropic renormalized scattering rate that increases linearly with frequency. The linear increase is in agreement with the predicted behavior in the MFL and NFL models. The temperature dependence of the unrenormalized scattering rate is shown Fig. 10. The upper panel shows 1/ at ve temperatures for the a direction and the lower panel shows the same thing for the b direction. We note that these functions are not so linear as 1/ *. The basic behavior, with the low-frequency values increasing with increasing temperature and the high-frequency parts nearly temperature independent, is in accord with the ideas of the MFL and NFL models. B. The pseudogap ductivities of the MFL contribution for T 100 K and the rst four Lorentzian lines are shown as thin dashed lines. The last of these represents the charge-transfer band of the insulating parent compound. The Lorentzian contributions are small for frequencies below the MFL cutoff frequency 1 c 1800 cm ; they contain less than 2% of the low-energy spectral weight. The third band, however, at 3600 cm 1, is Recent one-component analyses of the infrared properties of underdoped Bi2Sr2CaCu2O8 have been used to suggest a pseudogap in the normal state.25 27 The pseudogap is not evident in the optical conductivity measured in the CuO2 planes. Instead, there is structure in the 1/ ( ,T) in the abplane, a depressed scattering rate at low frequencies and at 1 TABLE I. Parameters of MFL model ts frequencies in cm a axis TK MFL 100 150 200 300 TK Lorentzian 1 100 150 200 300 100 150 200 300 All All p c p . b axis c 13 100 13 500 13 800 13 500 pj 0.23 0.24 0.26 0.23 j 1900 1900 1750 1800 j 14 050 14 100 14 300 13 700 pj 0.32 0.30 0.32 0.26 j 1 790 1 790 1 800 1 790 j Lorentzian 2 Lorentzian 3 Charge transfer band 1 570 1 690 1 650 1 570 1 790 1 890 2 160 1 790 12 300 7 700 640 610 650 640 1 000 1 030 1 000 1 020 3 650 17 500 190 290 220 280 350 600 570 430 9200 6800 1 900 1 250 1 760 1 270 750 870 1 330 1 260 12 600 6 400 640 650 660 650 1 000 1 050 1 050 1 050 3 700 17 100 460 300 300 650 120 380 380 550 11 000 5 800 PRB 60 ANISOTROPY IN THE ab-PLANE OPTICAL . . . 14 925 FIG. 9. One-component analysis showing the effective mass enhancement upper panel and the renormalized scattering rate lower panel along the a and b axes. FIG. 10. Temperature dependence of the unrenormalized scattering rate 1/ ( ) for the a upper panel and b lower panel axes. temperatures a little above T c . The scattering rates of Fig. 10 do not show evidence for the pseudogap, suggesting that the materials we have measured are near optimum doping, in agreement with the linear dc resistivity of our crystals. T-linear scattering rate combined with a T-independent plasma frequency is, of course, completely consistent with the dc conductivity. Thus, the dielectric function is assumed to be made up of at least four parts: D MIR interband , 10 C. Two-component analysis Here, we assume that the infrared conductivity is composed of two components. The underlying reason for this analysis is based on two essential observations. First, the optical conductivity 1 ( ) obtained from doping-dependent studies clearly shows the appearance of freelike as well as bound excitations in the midinfrared as doping in the CuO2 planes of the samples is increased.55 60 Second, there is a narrow range of frequencies in the far infrared where 1 ( ) shows a variation with T that is consistent with the T-linear resistivity of optimally doped samples. At higher frequenshows a weaker temperature cies, 1( ) dependence.35,41,61 63 The data shown in Fig. 7 exhibit a behavior that is consistent with this observation. This separation into mid-infrared and free-carrier contributions has been used in several previous studies of Bi2Sr2CaCu2O8. Reedyk et al.16 and Romero et al.34,35 analyzed the optical conductivity of Bi2Sr2CaCu2O8 in this way. The results are similar to the results of Kamaras et al.63 for YBa2Cu3O7 : there is an onset apparent of mid-infrared conductivity around 100 cm 1, structure in the phonon region, and a broad maximum around 1000 cm 1 0.12 eV . The free-carrier component has a nearly T-independent plasma frequency, and a T-linear scattering rate. This where D is associated with the free-carrier or Drude-like part, MIR corresponds to the mid-infrared bound transitions, interband includes higher-frequency interband transitions, and is the limiting high-frequency value. One way of separating individual contributions uses a Drude-Lorentz model. To describe each component requires three parameters: linewidth , plasma frequency p , and center frequency 0 . The model dielectric function is then 2 pD 2 2 pj j 2 j 2 i/ i , j 11 where the second term is the Drude response of the free carriers pD is their plasma frequency and 1/ is their scattering rate , the sum runs over the midinfrared bands, the charge-transfer band the band gap of the insulating parent compound , and higher-energy interband transitions. Each band is characterized by a plasma frequency p j , center frequency j , and width j . The results of tting the re ectance calculated using the dielectric function in Eq. 11 to our data is shown in Fig. 11. The t, employing two oscillators for the midinfrared region, one for the charge-transfer band, and two for higher-energy interband transitions, is good. To t the re ectance below T c we assumed a Drude linewidth of a fraction of a cm 1; i.e., 14 926 M. A. QUIJADA et al. PRB 60 FIG. 11. The a- and b-axis conductivity at 100 K and ts to a two-component model. The t is the thin solid line, and the Drude and midinfrared components are the dashed lines. we collapsed the Drude conductivity to a delta function. The parameters of the t are listed in Table II. The Drude weight is a little larger for the a polarization, as is the strength of the rst 90 meV mid-infrared band. In contrast, the 0.5 eV mid-infrared band has more oscillator strength for the b polarization. As an alternative to least-square tting, a self-consistent approach to separate the Drude-like from the boundlike ex- citations was used by Romero et al..34,35 What follows is a brief description of this method applied to the conductivity curves shown in Fig. 7. In the superconducting state and for T T c , all the free-carrier part is presumed to have collapsed 0. Therefore, in rst approximainto a delta function at tion the total conductivity at the lowest temperature (T 20 K) has a negligible free-carrier contribution; it corresponds to the 1MIR term in Eq. 10 . Hence, the free-carrier 1 TABLE II. Parameters of Drude-Lorentz model ts frequencies in cm a axis TK Drude 20 65 85 100 150 200 300 TK Mid-IR 1 20 65 85 100 150 200 300 20 65 85 100 150 200 300 All pD . b axis 1/ pD 1/ 9 250 9 180 9 200 9 040 9 080 9 100 9 020 pj j 73 128 202 300 423 j 8 890 8 760 8 800 8 750 8 740 8 770 8 700 pj j 100 157 240 327 435 j Mid-IR 2 Charge transfer band 10 800 10 800 10 800 11 000 10 900 10 900 10 600 10 600 10 700 10 500 10 300 10 300 10 300 10 300 7 700 865 759 667 675 631 790 798 4 500 4 320 4 230 4 520 4 430 4 520 4 440 17 500 2480 2510 2430 2410 2360 2330 2280 7930 8030 7710 7740 7660 7850 7480 6800 10 170 10 270 10 150 10 330 10 410 10 360 10 100 12 310 12 220 12 000 11 880 12 120 12 000 12 000 6 400 827 772 765 700 500 793 783 4 060 3 920 3 960 4 120 3 700 4 060 4 120 17 100 2380 2370 2300 2260 2530 2290 2220 8710 8570 8420 8220 9060 8660 8220 5800 PRB 60 ANISOTROPY IN THE ab-PLANE OPTICAL . . . 14 927 FIG. 13. Temperature dependence of the a- and b-axis resistivities obtained from dc transport measurements and extrapolations of the optical conductivity. FIG. 12. Drude part from a two-component analysis of 1 ( ) and ts obtained at each temperature. Top panel, a axis; bottom panel, b axis. part in the normal state can be obtained by subtracting 1MIR from the experimental 1 ( ) at T T c . This rst iteration produces free-carrier conductivity (1) ( 1D ) at individual temperatures above T c . If this conductivity has a Drude line shape, a t of straight line to the curve (1) obtained by plotting 1/ 1D vs 2 will yield a slope and intercept that can be used to get initial guess values for pD and 1/ . Once values for pD and 1/ are obtained, they can be used to calculate a Drude conductivity from t 1D II. The constant plasma frequency means that the oscillator strength and the carrier density do not change with temperature. Figure 13 shows as symbols the Drude resistivity 4 / 2 for both polarizations as a function of temperature. pD The dc resistivity measured for the a and b axes on similar samples is shown as the full and dashed lines. Two things should be noticed about this gure. The rst is that dc and optic both have a linear temperature dependence. Second, there is good agreement between the anisotropy determined from optical and from dc transport measurements. 1 41 2 pD 2 2. 12 Then, a new mid-infrared conductivity 1MIR is generated at t each temperature by subtracting 1D from the total 1 ( ) at each temperature above T c . A self-consistent check of the pD and 1/ obtained at each temperature is done by rst computing an average mid-infrared conductivity 1MIR from the 1MIR obtained as explained above, and using this average as the starting mid-infrared term in a second iteration. We found the parameters converge after performing three or four iterations. 1. Drude component The Drude conductivities 1 ( ) 1MIR obtained from this analysis are shown in Fig. 12 along with ts obtained from the estimates for pD and 1/ at each temperature. The normal-state Drude plasma frequency is nearly 1 temperature independent. We nd a pD 9300 200 cm , b 1 while pD 8900 200 cm . Note that these values are a little larger but within error bars of the parameters in Table FIG. 14. Scattering rate (1/ ) obtained from dc transport and the two-component analysis of the optical conductivity. 14 928 M. A. QUIJADA et al. optic PRB 60 Figure 14 shows a comparison of 1/ the ts with 1/ dc obtained from 1/ 2 pD dc dc , obtained from 4 13 where pD is the average Drude oscillator strength. This gure shows that the temperature variation of 1/ dc and 1/ optic is indeed linear in the normal state as expected. A linear temperature dependence of 1/ in metals is a hightemperature phenomenon, with1,64 / 2 k BT / 0, 14 where is a dimensionless constant that measures the strength of the coupling of the free carriers to whatever excitations are causing the scattering. We nd a 0.35, while b 0.31. These results are in good agreement with previously estimated values for using a two-component analysis in other samples.1 Furthermore, these results suggest the transport properties of the copper-oxide superconductors are in the weak-coupling regime. In the superconducting state, 1/ dc drops to zero on account of dc also going to zero. At the same temperature 1/ optic also exhibits a sudden drop. This sudden drop suggests that the excitations that cause the scattering of the carriers in the normal state are suppressed below T c . The limited number of points and possible uncertainties, especially at the lowest frequencies, prevent us from extracting the temperature dependence of 1/ in the superconducting state. Nonetheless, we do nd that the 1/ optic obtained is larger for the b axis than for the a axis in the superconducting state. This evidence for an additional channel for elastic scattering in the crystallographic b direction is also consistent with the b nal intercept observed in dc from extrapolations to the zero temperature value dc . Sudden drops in the quasiparticle scattering rate as the sample becomes superconducting have been observed in unpolarized infrared measurements of Bi2Sr2CaCu2O8, 35 La2 x Srx CuO4, 41 and YBa2Cu3O7 , 65 and in experiments of femtosecond optical transient and microwave absorption and on measurements on YBa2Cu3O7 , 66 67 BiO 2Sr2Ca2Cu3O10. Similarly, there have been predictions of drop of the quasiparticle scattering rate in the superconducting state within the phenomenology of the marginal Fermi liquid approach.68 This sudden drop in 1/ has been proposed as the reason for the appearance of a coherence peak in 1 ( ) in studies of YBa2Cu3O7 thin lms65,69,70 and in Bi2Sr2CaCu2O8 single crystals35,71 for temperatures just below T c and for frequencies in the microwave region. 2. Mid-infrared absorption FIG. 15. Mid-infrared part of the total conductivity. b-axis absorption extends to lower frequencies 150 200 cm 1 and is higher in the frequency range 150 700 cm 1. In both cases, there is weak structure due to phonons, includ400 and 800 cm 1. Figure 15 shows ing weak minima at that these minima appear in both the normal and superconducting states. Earlier optical studies3,7 interpreted this The mid-infrared conductivity, that which remains after subtracting the Drude term or low-frequency part, is shown in the top and bottom panels of Fig. 15 for the a and b axes, respectively. 1MIR does not have much temperature dependence. Most of the temperature dependence in 1 ( ) comes from the free-carrier contribution. To illustrate the anisotropy, Fig. 16 compares 1MIR for the a and b directions at 100 and 200 K. The onset of absorption appears around 250 cm 1 for the a axis, whereas the FIG. 16. Mid-infrared part of the total conductivity, comparing the a and b axes at two temperatures in the normal state. PRB 60 ANISOTROPY IN THE ab-PLANE OPTICAL . . . VII. SUPERCONDUCTING STATE A. The superconducting condensate 14 929 FIG. 17. Plot of 1 ( ) at T 20 K in the a- and b-axis polarizations along with the c-axis loss function, from Ref. 72. We saw from the two-component analysis that the a-axis scattering rate for the Drude carriers is close to 150 cm 1 at 100 K, and that it is suddenly suppressed in the superconducting state. If this 100 K 1/ is used together with published estimates73,74 for the Fermi velocity v f in the CuO2 planes to compute the mean free path l for quasiparticles 100 . propagating along this direction we obtain l v f This number is considerably larger than the typical ab-plane coherence length reported73,75 for this and other high-T c compounds, 15 . This comparison suggests that , as was rst Bi2Sr2CaCu2O8 is in the clean limit, i.e., l pointed by Kamaras et al.63 for YBa2Cu3O7 . In this limit, absorption associated with the superconducting gap is not observable because in the superconducting state most of the spectral weight moves to the zero-frequency delta function. Thus the only signature of the condensate is its inductive response, seen in the real part of the dielectric function, 1 ( ). The delta function conductivity gives, via the Kramers-Kronig relations,76 2 ps 2, notchlike feature, which in YBa2Cu3O7 is much more evi430 cm 1, as the superconducting endent and occurs at ergy gap. An argument against this interpretation is that the minimum in 1 ( ) is also observed above T c in nearly all samples, making its association with an energy gap very unlikely.31,41,63 An alternative interpretation attributes this structure to electron-phonon interactions.44 47 As discussed by Reedyk and Timusk,46 these interactions are between the c-axis longitudinal optic LO phonons and the ab-plane bound carriers. Supporting evidence for this interpretation is shown in Fig. 17, where the Bi2Sr2CaCu2O8 c-axis energy-loss function72 which has maxima at the LO phonon frequencies are plotted along with a- and b-axis conductivities at 20 K. The coincidence between the loss-function maxima and 1 ( ) minima is quite remarkable. Similar evidence for this single effect has come from measurements on La2CuO4 crystals.60 1 1b 15 where ps is the oscillator strength of the superconducting condensate, de ned as ps 4 n s e 2 /m, with n s the density of super uid carriers. The term 1b is the bound-carrier contribution to 1 ( ). Hence, the condensate contribution ps to 1 ( ) can be determined from a plot of 1 ( ) as a func2 . As shown in Fig. 18, this plot for the a axis tion of gives a straight line whose slope is 2 . From this slope at ps T 20 K, we nd a 9000 200 cm 1 and b 8200 ps ps 200 cm 1. Hence, the superconducting-carrier response is larger for the a-axis direction. An alternative method that obtains the same results is to estimate the missing area under 1 ( ) in the superconducting state. This estimate is done by subtracting the conductivity at the lowest temperature T 20 K in this case from the 2 FIG. 18. 1 vs in the superconducting state for the a direction. 14 930 M. A. QUIJADA et al. PRB 60 conductivity just above T c T 100 K in this case . Then, the s sum rule or density of super uid carriers, N effm/mb , is evaluated by performing the integral s N eff m mb ,100 K 0 ,20 K . 16 s By evaluation of the integral in Eq. 16 , we nd N effm/mb s 0.20 for the a axis and N effm/mb 0.16 for the b axis. 2 2 and Hence, by noticing that (N effm/mb) psmVcell/4 e from the known unit-cell volume of Bi2Sr2CaCu2O8, we nd that a ps s 19 900 N eff 8900 cm 1 in the a-axis polarization, while b 8100 cm 1 for the b ps axis. Both quantities agree with the results from analysis of 1 described above. B. ab-plane anisotropy in the London penetration depth The London penetration length L measures the distance over which an electromagnetic wave is attenuated inside a superconductor. Since this length is a measure of the super uid response in a superconductor, it is also related to the super uid oscillator strength ps by L c/ ps . In anisotropic materials, L is a tensor quantity. Hence, polarized infrared spectroscopic offers a unique opportunity to determine the different components of this tensor. Other techniques, such as magnetic inductance method or muon spin resonance,77,78 only give values of L that are averages of the different components of L . From the values of ps for the a and b axes of Bi2Sr2CaCu2O8, we nd for the London a b length along the a axis L 1800 , while L 1960 . The b a ratio of these two quantities is L / L 1.1. Another way to demonstrate London response of the super uid is to calculate a generalization of the London length via c L FIG. 19. London penetration length as a function of frequency using Eq. 17 . 1 . 1 17 Note that this can also be written in terms of the imaginary part of the conductivity, 2 ( ), as L c/ 4 2 ( ). Figure 19 displays L for the a and b axes. The fact that both curves in Fig. 19 are nearly at in the far infrared, approach0, suggests that the princiing the values given above at pal contribution to 2 ( ) is from the super uid carrier response, which follows 2 ( ) 1/ . There is a de nite anisotropy in the penetration depth. To explain the source of this anisotropy the rst thing that should be considered is whether the anisotropy that is observed in L is due to mass enhancement effects. In the normal state, the ab-plane anisotropy in the dc resistivity, from extrapolations of the optical data and direct dc transport, is b / a 1.25. As discussed previously, the normal-state anisotropy in the Drude plasma frequency derived from a twocomponent analysis of the optical data is found to be smaller b than this anisotropy, i.e., a pD pD 1.04 0.04. Hence, most of the anisotropy in dc is due to a free-carrier relaxation rate that, at T 100 K, is 20% larger for the b axis. This suggests the interactions that are responsible for the relaxation of the free carriers are not isotropic but that the effective masses have a smaller anisotropy. If we compare the super uid plasma frequency to the Drude plasma frequency, we nd for the a axis that a is ps 97% of a essentially identical . In contrast, in the b dipD 1 rection b 8100 200 cm 1 and b ps pD 8900 200 cm , so that the super uid plasma frequency is only about 90% of the Drude plasma frequency. We note further that the zerotemperature extrapolation of dc or 1/ for the b axis appears to be nite, whereas for the a axis, the zerotemperature intercept is very close to zero. That the b-axis plasma frequency is smaller in the superconducting state suggests that the anisotropy is not related to a mass enhancement effect but instead is due to an additional pair-breaking scattering channel in the b or superlattice direction. This additional scattering causes additional absorption at nite frequencies in the superconducting state and a reduced condensate weight. C. Optical conductivity and symmetry of the order parameter The symmetry of the order parameter in the hightemperature superconductors has been addressed by many workers.43,66,79 87 There is a growing consensus for an unconventional d-wave symmetry for this quantity. Angularresolved photoemission spectroscopy ARPES has played a key role in this consensus.43,84,85,88 In one study, Shen et al.85 performed ARPES measurements on Bi2Sr2CaCu2O8 single crystals, nding a condensate peak that is larger and more pronounced along the -X symmetry direction, i.e., from the center of the Brillouin zone to the X point in momentum space. The gap seems to vanish within the experimental , , 45 away from the resolution of 2 meV along maximum-gap direction. Based on the assumption the material has tetragonal rather than orthorhombic symmetry, the authors conclude the symmetry of the order parameter is PRB 60 ANISOTROPY IN THE ab-PLANE OPTICAL . . . 14 931 compatible with d x 2 y 2 symmetry pairing.85 Recent experiments in the temperature dependence of L (T) have found a linear temperature variation that has also been interpreted has evidence of d x 2 y 2 pairing in these materials.66 On the other hand, this interpretation is in contradiction with other ARPES experiments performed by Kelley and co-workers.43 These authors argue against a pure d x 2 y 2 gap symmetry. The authors also point out that other possibilities such as mixing of d-wave with either s- or p-wave symmetries cannot be excluded. In spite of the controversy, one strong conclusion that can be obtained from these results is that the order parameter in the high-T c materials appears to be highly anisotropic. The optical conductivity can say certain things about the symmetry of the electronic structure in the superconducting state. Exactly what can be said depends on several factors: clean or dirty limit, gap symmetry, and underlying crystallographic symmetry. If the material were in the dirty limit, then the s-wave superconductor has a gap 2 in its excitation spectrum, as derived by Mattis and Bardeen.89 The impurity scattering also has the effect of averaging any anisotropy from band-structure effects, so that the gap is reasonably isotropic, even in materials with some electronic anisotropy. 2 . The missing At zero temperature 1s vanishes for 2 appears32 in spectral weight of 1s in the range 0 0. In the case of nite temperatures, the function at thermally exited quasiparticles can give rise to a Drude-like contribution to 1 ( ) with a width in the order of 1/ . A dirty-limit superconductor with a gap function that has nodes on the Fermi surface, such as one with a d-wave or p-wave gap, will have nite contribution to 1 ( ) for all 2 max , even at zero temperature. The reason for this is that it takes only an arbitrarily small energy to break Cooper pairs composed of electrons with momenta close to the nodes of the Fermi surface. Using a self-consistent T-matrix approximation, Hirschfeld et al.90 carried out the calculation of 1s / n for unconventional non-s-wave superconductors, nding a sort of pseudogap at energies below 2 max . There is a narrow Drude-like peak at low frequencies, caused by broken pairs in the nodes of the gap function. In the clean limit, the s-wave superconductor retains its gap in the excitation spectrum but, as discussed above, the oscillator strength of the gap transition is small.63 Moreover, if the scattering is largely electron-electron, the optical threshold is actually at 4 rather than 2 , because if the photon merely breaks a single Cooper pair, the drift momentum is unchanged and hence the electrical current is unaffected. Instead, two pairs must be broken, with subsequent large-angle scattering, for there to be nite 1 ( ). 91 In the clean limit of the d-wave superconductor, with a small scattering phase shift, an expansion of 1s / n in predicts a quadratic frequency dependence for close to zero in the case of a gap function with polar symmetry, whereas a 4 dependence occurs for a gap with axial symmetry. Hence, these results indicate that for a superconductor with anisotropic order parameter, electromagnetic radiation is always 0. However, as in the absorbed for frequencies down to s-wave case, the spectral weight associated with the superconducting-state absorption is found to be small.92 In a polarized measurement, the d-wave superconductor, whether clean or dirty, has an isotropic response. This result can be understood as follows. Suppose the incident electric eld is oriented parallel to the gap nodes, i.e., in the 45 direction. This eld always can be decomposed into components along the a and b directions, which are the directions of the gap minima. Similarly, a eld that is oriented along a direction where the gap is maximum may be constructed from components parallel to the nodes. Because the magnitude of the gap function has fourfold symmetry whereas the ab-plane conductivity tensor has at most twofold dipolar symmetry, the gap anisotropy does not contribute to a conductivity anisotropy. Finally, in an orthorhombic crystal, the dielectric tensor has three different components, with principal axes along the a, b, and c crystallographic directions. Unlike the tetragonal symmetry case, the orthorhombic crystal cannot have a pure d-wave gap, although d s is allowed.93 Our results unambiguously show that the ab-plane optical response of Bi2Sr2CaCu2O8 is anisotropic both above and below T c . As illustrated in Figs. 2 and 3, the a-axis re ectance in the superconducting state reaches almost 100% for frequencies in the far infrared, whereas there is a difference in the re ectance level (Ra Rb ) on the order 1 2 %. A Kramers-Kronig analysis of the re ectance for the two polarizations gives anisotropy in 1 ( ). In the normal state, the far-infrared conductivity is about 20% larger for polarization along the a axis. In the superconducting state, 1 ( ) is a factor of 2 larger for the b axis below 300 cm 1 and down to the lowest frequency measured in the experiment. Finally, the penetration depth is about 10% larger for the a-axis direction. If a one-component analysis with a frequency-dependent 1/ is used to explain these results, the superconducting-state conductivity must then be due to excitations across the superconducting gap. The observed anisotropy in 1 ( ) below T c implies an order parameter consistent with a C 2v rather than a C 4v symmetry. If this interpretation is taken, these data are incompatible with s-wave pairing or with a pure d x 2 y 2 gap symmetry. The results would be compatible with a combination of s and d pairing.43 If the conductivity is decomposed into two components, the observed anisotropy of the ab-plane conductivity could be due to two factors. Above T c the free-carrier damping rate is 20% stronger along b 1/ (100 K) 180 cm 1 for the b axis and 150 cm 1 for the a axis . Above and below T c , the mid-infrared component has a larger contribution along b than along the a axis. This anisotropy is compatible with the observed orthorhombic unit cell in this material. VIII. CONCLUSIONS In conclusion, the anisotropy of the ab-plane of the copper-oxide superconductors has been studied by measuring the polarized re ectance of single-domain crystals of Bi2Sr2CaCu2O8. There is signi cant ab-plane anisotropy in the optical properties of this material. In the normal state, the infrared conductivity is about 20% higher along the a axis than the b axis. A similar ab-plane anisotropy is observed in the normal-state dc resistivity of similar samples. The normal-state ab-plane conductivity exhibits nonDrude behavior, characterized by strong temperature depen- 14 932 M. A. QUIJADA et al. PRB 60 dence in the far infrared and a much weaker temperature variation in the mid-infrared region. If the conductivity is analyzed in the framework of a two-component picture, the low-frequency part can be regarded as Drude-like in nature, with a scattering rate that is linear in temperature. This behavior is consistent with the linear temperature dependence of the dc resistivity. The coupling constant obtained in this analysis is 0.3 0.4 in all samples. The ab-plane anisotropy observed in the normal-state conductivity of these samples persists in the superconducting state as well. The optical conductivity at low frequencies is a factor of 2 larger along the b axis than along the a axis. This suggests that either an anisotropic order parameter with a twofold symmetry (C 2v ) rather than a fourfold symmetry (C 4v ) or that the mid-infrared component of the optical conductivity is anisotropic. Finally, estimates of the London penetration lengths display anisotropy, with b / a 1.1. This result suggests some additional elastic scattering mechanism in the b axis of the crystal is giving rise to some kind of pair-breaking effect that b is causing L to be larger along this direction L 2000 a compared to L 1800 . ACKNOWLEDGMENTS We would like to acknowledge many discussions with P.J. Hirschfeld. Work at Florida was supported by the NSF Solid State Physics through Grant No. DMR9705108. Work at Wisconsin was supported by the DOE. *Present address: NASA/GODDARD, MS 551, Bldg. 5, Rm. C38, 15 16 Greenbelt, MD 20771. 1 D. B. Tanner and T. Timusk, in Physical Properties of High Temperature Superconductors III, edited by D. M. Ginsberg World Scienti c, Singapore, 1992 , p. 363. 2 B. Koch, H. P. Geserich, and T. Wolf, Solid State Commun. 71, 495 1989 . 3 Z. Schlesinger, R. T. Collins, F. Holtzberg, C. Field, S. H. Blanton, U. Welp, G. W. Crabtree, Y. Fang, and J. Z. Liu, Phys. Rev. Lett. 65, 801 1990 . 4 T. Pham, M. W. Lee, H. D. Drew, U. Welp, and Y. Fang, Phys. Rev. B 44, 5377 1991 . 5 H. P. Geserich, B. Koch, M. Durrler, and Th. Wolf, in Electronic Properties of High-T c Superconductors and Related Compounds, edited by H. Kuzmany, M. Mehring, and J. Fink, Springer Series in Solid-State Sciences Vol. 99 SpringerVerlag, Berlin, 1990 , p. 290. 6 J. Schutzmann, B. Gorshunov, K. F. Renk, J. Munzel, A. Zibold, H. P. Geserich, A. Erb, and G. Muller-Vogt, Phys. Rev. B 46, 512 1992 . 7 L. D. Rotter, Z. Schlesinger, R. T. Collins, F. Holtzberg, C. Field, U. W. Welp, G. W. Crabtree, J. Z. Liu, Y. Fang, K. G. Vandervoort, and S. Fleshler, Phys. Rev. Lett. 67, 2741 1991 . 8 S. L. Cooper, A. L. Kotz, M. A. Karlow, M. V. Klein, W. C. Lee, J. Giapintzakis, and D. M. Ginsberg, Phys. Rev. B 45, 2549 1992 . 9 D. N. Basov, R. Liang, D. A. Bonn, W. N. Hardy, B. Dabrowski, M. Quijada, D. B. Tanner, J. P. Rice, D. M. Ginsberg, and T. Timusk, Phys. Rev. Lett. 74, 598 1995 . 10 T. A. Friedmann, M. W. Rabin, J. Giapintzakis, J. P. Rice, and D. M. Ginsberg, Phys. Rev. B 42, 6217 1990 . 11 S. A. Sunshine, T. Siegrist, L. F. Schneemeyer, D. W. Murphy, R. J. Cava, B. Batlogg, R. B. van Dover, R. M. Fleming, S. H. Glarum, S. Nakahara, R. Farrow, J. J. Krajewski, S. M. Zahurak, J. V. Waszczak, J. H. Marshall, P. Marsh, L. W. Rupp, Jr., and W. F. Peck, Phys. Rev. B 38, 893 1988 . 12 T. M. Shaw, S. A. Shivashsnkar, S. J. La Placa, J. J. Cuomo, T. R. McGuire, R. A. Roy, K. H. Kelleher, and D. S. Yee, Phys. Rev. B 37, 9856 1988 . 13 E. A. Hewat, J. J. Capponi, and M. Marezio, Physica C 157, 502 1989 . 14 M. A. Quijada, D. B. Tanner, R. J. Kelley, and M. Onellion, Physica C 235-240, 1123 1994 . M. A. Quijada, Ph.D. thesis, University of Florida, 1994. M. Reedyk, D. A. Bonn, J. D. Garrett, J. E. Greedan, C. V. Stager, T. Timusk, K. Kamaras, and D. B. Tanner, Phys. Rev. B 38, 11 981 1988 . 17 K. Kamaras, S. L. Herr, J. S. Kim, G. R. Stewart, D. B. Tanner, and T. Timusk, in Electronic Properties of High-T c Superconductors and Related Compounds Ref. 5 , p. 260. 18 Yun-Yu Wang, Goufu Feng, and A. L. Ritter, Phys. Rev. B 42, 420 1990 . 19 J. H. Kim, I. Bozovic, J. S. Harris, Jr., W. Y. Lee, C.-B. Eom, and T. H. Geballe unpublished . 20 O. V. Kosogov, M. V. Belousov, V. A. Vasil ev, V. Y. Davydov, K. G. Dyo, A. A. Kopylov, V. D. Petrikov, and V. V. Tret yakov, Pis ma Zh. Eksp. Teor. Fiz. 48, 488 1989 JETP Lett. 48, 530 1989 . 21 J. Huml cek, E. Schmidt, L. Bouanek, M. Garriga, and M. Cardona, Solid State Commun. 73, 127 1990 . 22 A. M. Rao, P. C. Eklund, G. W. Lehman, D. W. Face, G. L. Doll, G. Dresselhaus, and M. S. Dresselhaus, Phys. Rev. B 42, 193 1990 . 23 P. Calvani, M. Capizzi, S. Lupi, P. Maselli, D. Peschiaroli, and H. Katayama-Yoshida, Solid State Commun. 74, 1333 1990 . 24 M. A. Quijada, D. B. Tanner, R. J. Kelley, and M. Onellion, Z. Phys. B: Condens. Matter 94, 255 1994 . 25 A. Puchkov, P. Fournier, D. N. Basov, T. Timusk, A. Kapitulnik, and N. N. Kolesnikov, Phys. Rev. Lett. 77, 3212 1996 . 26 D. N. Basov, R. Liang, B. Dabrowski, D. A. Bonn, W. N. Hardy, and T. Timusk, Phys. Rev. Lett. 77, 4090 1996 . 27 A. V. Puchkov, D. N. Basov, and T. Timusk, J. Phys.: Condens. Matter 8, 10 049 1996 . 28 H. L. Liu, M. A. Quijada, D. B. Tanner, H. Berger, G. Margaritondo, R. J. Kelley, and M. A. Onellion, Eur. Phys. J. B 8, 47 1999 . 29 H. L. Liu, M. A. Quijada, A. Zibold, Y.-D. Yoon, D. B. Tanner, G. Cao, J. E. Crow, H. Berger, G. Margaritondo, L. Forr, BeomHoan O, J. T. Markert, R. J. Kelly, and M. Onellion, J. Phys.: Condens. Matter 11, 239 1999 . 30 L. Forro, G. L. Carr, G. P. Williams, D. Mandrus, and L. Mihaly, Phys. Rev. Lett. 65, 1941 1990 . 31 D. B. Romero, G. L. Carr, D. B. Tanner, L. Forro, D. Mandrus, L. Mihaly, and G. P. Williams, Phys. Rev. B 44, 2818 1991 . 32 R. E. Glover and M. Tinkham, Phys. Rev. 107, 844 1956 ; 108, 1175 1957 . PRB 60 33 ANISOTROPY IN THE ab-PLANE OPTICAL . . . 14 933 D. B. Tanner, D. B. Romero, K. Kamaras, G. L. Carr, L. Forro, ly, and G. P. Williams, in Electronic StrucD. Mandrus, L. Miha ture and Mechanisms for High-Temperature Superconductivity, edited by G. C. Vezolli et al. Plenum, New York, 1991 . 34 D. B. Romero, C. D. Porter, D. B. Tanner, L. Forro, D. Mandrus, L. Mihaly, G. L. Carr, and G. P. Williams, Solid State Commun. 82, 183 1992 . 35 D. B. Romero, C. D. Porter, D. B. Tanner, L. Forro, D. Mandrus, L. Mihaly, G. L. Carr, and G. P. Williams, Phys. Rev. Lett. 68, 1590 1992 . 36 P. D. Han and D. A. Payne, J. Cryst. Growth 104, 201 1990 . 37 Frederick Wooten, Optical Properties of Solids Academic, New York, 1972 . 38 M. K. Kelly, P. Barboux, J.-M. Tarascon, D. E. Aspnes, P. A. Morris, and W. A. Bonner, Physica C 162-164, 1123 1989 . 39 J. W. Allen and J. C. Mikkelsen, Phys. Rev. B 15, 2952 1977 . 40 I. Bozovic, K. Char, S. J. B. Yoo, A. Kapitulnik, M. R. Beasley, T. H. Geballe, Z. Z. Wang, S. Hagen, N. P. Ong, D. E. Aspnes, and M. K. Kelly, Phys. Rev. B 38, 5077 1988 . 41 F. Gao, D. B. Romero, D. B. Tanner, J. Talvacchio, and M. G. Forrester, Phys. Rev. B 47, 1036 1993 . 42 Y. Hwu, L. Lozzi, M. Marsi, S. La Rosa, M. Winokur, P. Davis, M. Onellion, H. Berger, F. Gozzo, F. Levy, and G. Margaritondo, Phys. Rev. Lett. 67, 2573 1991 . 43 R. J. Kelley, Jian Ma, G. Margaritondo, and M. Onellion, Phys. Rev. Lett. 71, 4051 1993 . 44 T. Timusk, C. D. Porter, and D. B. Tanner, Phys. Rev. Lett. 66, 663 1991 . 45 T. Timusk and D. B. Tanner, Physica C 169, 425 1990 . 46 M. Reedyk and T. Timusk, Phys. Rev. Lett. 69, 2705 1992 . 47 M. Reedyk, Ph.D. thesis, McMaster University, 1992. 48 C. M. Varma, P. B. Littlewood, S. Schmitt-Rink, E. Abrahams, and A. E. Ruckenstein, Phys. Rev. Lett. 63, 1996 1989 . 49 P. B. Littlewood and C. M. Varma, J. Appl. Phys. 69, 4979 1991 . 50 A. Virosztek and J. Ruvalds, Phys. Rev. B 42, 4064 1990 . 51 P. W. Anderson, Science 235, 1196 1987 ; in Frontiers and Borderlines in Many Particle Physics, edited by J. R. Schrieffer and R. A. Broglia North-Holland, Amsterdam, 1989 ; Phys. Rev. Lett. 64, 1839 1990 ; Science 256, 1526 1992 . 52 V. J. Emery, S. A. Kivelson, and H. Q. Lin, Phys. Rev. Lett. 64, 475 1990 ; V. J. Emery and S. A. Kivelson, Physica C 209, 597 1993 . 53 P. Monthoux and D. Pines, Phys. Rev. B 49, 4261 1994 . 54 T. Holstein, Phys. Rev. 96, 535 1954 ; Ann. Phys. N.Y. 29, 410 1964 . 55 S. Uchida, T. Ido, H. Takagi, T. Arima, Y. Tokura, and S. Tajima, Phys. Rev. B 43, 7942 1991 . 56 G. A. Thomas, in Proceedings of the Thirty-Ninth Scottish Universities Summer School in Physics, edited by D. P. Tunstall and W. Barford Hilger, Bristol, 1991 , p. 196. 57 S. L. Cooper, D. Reznik, A. Kotz, M. A. Karlow, R. Liu, M. V. Klein, W. C. Lee, J. Giapintzakis, D. M. Ginsberg, B. Veal, and A. P. Paulikas, Phys. Rev. B 47, 8233 1993 . 58 Y. Tokura, H. Takagi, T. Arima, S. Koshihara, T. Ido, S. Ishibashi, and S. Uchida, Physica C 162-164, 1231 1989 . 59 S. Lupi, P. Calvani, M. Capizzi, P. Maselli, W. Sadowski, and E. Walker, Phys. Rev. B 45, 12 470 1992 . 60 M. A. Quijada, D. B. Tanner, F. C. Chou, D. C. Johnston, and S-W. Cheong, Phys. Rev. B 52, 15 485 1995 . 61 G. A. Thomas, M. Capizzi, J. Orenstein, D. H. Rapkine, A. J. Millis, P. Gammel, L. F. Schneemeyer, and J. V. Waszczak, in Proceedings of the International Symposium on Electronic Structure of High T c Superconductors, edited by A. Bianconi Pergamon, Oxford, 1988 , p. 169. 62 J. Orenstein, G. A. Thomas, A. J. Millis, S. L. Cooper, D. H. Rapkine, T. Timusk, L. F. Schneemeyer, and J. V. Waszczak, Phys. Rev. B 42, 6342 1990 . 63 K. Kamaras, S. L. Herr, C. D. Porter, N. Tache, D. B. Tanner, S. Etemad, T. Venkatesan, E. Chase, A. Inam, X. D. Wu, M. S. Hegde, and B. Dutta, Phys. Rev. Lett. 64, 84 1990 . 64 P. B. Allen, T. B. Beaulac, F. S. Khan, W. H. Butler, F. J. Pinski, and J. H. Swihart, Phys. Rev. B 34, 4331 1986 . 65 F. Gao, G. L. Carr, C. D. Porter, D. B. Tanner, G. P. Williams, C. J. Hirschmugl, B. Dutta, X. D. Wu, and S. Etemad, Phys. Rev. B 54, 700 1996 . 66 W. N. Hardy, D. A. Bonn, D. C. Morgan, Ruixing Liang, and Kuan Zhang, Phys. Rev. B 70, 3999 1993 . 67 J. M. Chwalek, C. Uher, J. F. Whitaker, G. A. Mourou, J. Agostinelli, and M. Lelental, Appl. Phys. Lett. 57, 1696 1990 . 68 E. J. Nicol and J. P. Carbotte, Phys. Rev. B 44, 7741 1991 . 69 F. A. Miranda, W. L. Gordon, K. B. Bhasin, V. O. Heinen, and J. D. Warner, J. Appl. Phys. 70, 5450 1991 . 70 M. C. Nuss, P. M. Mankiewich, M. L. O Malley, E. H. Westerwick, and P. B. Littlewood, Phys. Rev. Lett. 66, 3305 1991 . 71 K. Holczer, L. Forro, L. Mihaly, and G. Gruner, Phys. Rev. Lett. 67, 152 1991 . 72 A. Zibold, M. Durrler, A. Gaymann, H. P. Geserich, N. Nucker, V. M. Burlakov, and P. Muller, Physica C 193, 171 1992 . 73 S. A. Wolf and V. Z. Kresin, IEEE Trans. Magn. 27, 852 1991 . 74 P. B. Allen, W. E. Pickett, and H. Krakauer, Phys. Rev. B 36, 3926 1987 . 75 A. Umezawa, G. W. Crabtree, J. Z. Liu, T. J. Moran, S. K. Malik, L. H. Nunez, W. L. Kwok, and C. H. Sowers, Phys. Rev. B 38, 2843 1988 . 76 T. Timusk and D. B. Tanner, in Physical Properties of High Temperature Superconductors I, edited by D. M. Ginsberg World Scienti c, Singapore, 1989 , p. 339. 77 J. R. Cooper, L. Forro, and B. Keszei, Nature London 343, 444 1990 . 78 Y. Tanaka, M. Fukutomi, and T. Asano, Jpn. J. Appl. Phys., Part 2 27, L209 1988 . 79 A. T. Fiory, A. F. Hebard, P. M. Mankiewich, and R. E. Howard, Phys. Rev. Lett. 61, 1419 1988 . 80 D. A. Bonn, R. Liang, T. M. Riseman, D. J. Baar, D. C. Morgan, K. Zhang, P. Dosanjh, T. L. Duty, A. MacFarlane, G. D. Morris, J. H. Brewer, W. N. Hardy, C. Kallin, and A. J. Berlinsky, Phys. Rev. B 47, 11 314 1993 . 81 J. F. Annet, N. Goldenfeld, and S. R. Renn, Phys. Rev. B 43, 2778 1991 . 82 J. F. Annet and N. Goldenfeld, J. Low Temp. Phys. 89, 197 1992 . 83 Z. Ma, R. C. Taber, L. W. Lombardo, A. Kapitulnik, M. R. Beasley, P. Merchant, C. B. Eom, S. Y. Hou, and J. M. Phillips, Phys. Rev. Lett. 71, 781 1993 . 84 D. S. Dessau, Ph.D. dissertation, Stanford University, 1992. 85 Z.-X. Shen, D. S. Dessau, B. O. Wells, D. M. King, W. E. Spicer, A. J. Arko, D. Marshall, L. W. Lombardo, A. Kapitulnik, P. Dickinson, S. Doniach, J. DiCarlo, A. G. Loeser, and C. H. Park, Phys. Rev. Lett. 70, 1553 1993 . 86 D. J. Van Harlingen, Rev. Mod. Phys. 67, 515 1995 . 87 J. P. Kirtley, C. C. Tsuei, M. Rupp, J. Z. Sun, L. S. Yu-Jahnes, A. 14 934 M. A. QUIJADA et al. 91 PRB 60 Gupta, M. B. Ketchen, K. A. Moler, and M. Bhushan, Phys. Rev. Lett. 76, 1336 1996 . 88 D. Hong, J. C. Campuzano, K. Gofron, C. Gu, R. Liu, B. W. Veal, and G. Jennings, Phys. Rev. B 50, 1333 1994 . 89 D. C. Mattis and J. Bardeen, Phys. Rev. 111, 412 1958 . 90 P. J. Hirschfeld, P. Wol e, J. A. Sauls, D. Einzel, and W. O. Putikka, Phys. Rev. B 40, 6695 1989 . J. Orenstein, S. Schmitt-Rink, and A. E. Ruckenstein, in Electronic Properties of High-T c Superconductors and Related Compounds Ref. 5 . 92 S. M. Quinlan, P. J. Hirschfeld, and D. J. Scalapino, Phys. Rev. B 53, 8575 1996 . 93 Q. P. Li, B. E. C. Koltenbah, and Robert Joynt, Phys. Rev. B 48, 437 1993 .
Find millions of documents here - Study Guides, Homework Solutions, Papers, Exam Answer Keys and more.
Course Hero has millions of course related materials that will enable you to learn better, faster and get an A in all your courses.
Below is a small sample set of documents:
Selcuk06prl-TrapModes.pdf
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008
Path: UF >> PHZ >> 6426 Fall, 2008