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- Title: Tanner01ferro-lees99
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AND SUPERFLUID NORMAL-FLUID DENSITY IN THE CUPRATE SUPERCONDUCTORS D.B. Tannera , F. Gaoa , K. Kamar sa , H.L. Liua , M.A. a a , D.B. Romeroa , Y-D. Yoona , A. Zibolda , Quijada H. Bergerb , G. Margaritondob , L. Forr b , R.J. Kellyc , o M. Onellionc , G. Caod , J.E. Crowd , Beom-Hoan Oe , J.T. Markerte , J.P. Ricef , D.M. Ginsbergf , and Th. Wolfg ee of Florida; b Ecole Polytechnique F d ral c University of Wisconsin; d National de Lausanne; High Magnetic Field Laboratory; e University of Texas; f University of Illinois; g Forschungszentrum Karlsruhe. a University Abstract As carriers are introduced into the cuprates (by doping the insulating parent compounds) spectral weight appears in the optical spectrum at photon energies below the charge-transfer gap. This spectral weight increases as the doping level increases. Magnetic penetration depth measurements have shown a good correlation between super uid density and superconducting transition temperature in the underdoped-to-optimally-doped part of the phase diagram. Optical measurements allow independent determination of the total doping-induced spectral weight and the super uid density. These measurements, made on cuprates with transition temperatures from 40 to 110 K, nd that in optimally doped materials only about 20% of the doping-induced spectral weight joins the super uid. The rest remains in nite-frequency, midinfrared absorption. In underdoped materials, the super uid fraction is even smaller. This result implies extremely strong coupling for these superconductors. Keywords: superconductivity; infrared 2 D.B. Tanner et al. INTRODUCTION Essentially every conduction electron participates in the T = 0 super uid of a clean metallic superconductor.1,2 Compelling evidence for this claim is the penetration depth, which, when corrected for nonlocal e ects, gives about the same electron density as the freecarrier optical plasma frequency p , i. e., 2 = 4 ns e2 /cm = ps /c L with ps p . Here, L is the London value for the penetration depth, ns is the density of superconducting electrons and m their e ective mass. The same super uid density ns enters the in nite dc conductivity of the superconductor: the inductive screening of electromagnetic waves is directly related (via the Kramers-Kronig integral) to the spectral weight or oscillator strength of the zerofrequency delta function in the optical conductivity. In what follows, we call the ns extracted from either the penetration depth or the delta-function oscillator strength the super uid density. Strong-coupled metallic systems exhibit a modest reduction in super uid density on account of the Holstein phonon emission process,3 where a charge carrier absorbs a photon of energy , emits an excitation of energy , and scatters, giving rise to absorption. In a clean metal at zero temperature, this process is the dominant cause of infrared absorption, because it the principal mechanism by which the momentum of the charge carriers can be changed. This nite-frequency absorption removes oscillator strength from the delta function, and increases the penetration depth. In metals, the excitation that is emitted is generally a phonon, but in fact it could be any boson that couples linearly to the charge carriers. The interaction leads to a frequency- and temperaturedependent scattering rate, given at high temperatures by 1/ = 2 T , with the dimensionless electron-boson coupling constant. A related consequence of this interaction is a low-energy mass enhancement, which is governed by the same quantity . The e ective mass m becomes m = (1 + )m, where m is the band mass. Because the low-energy penetration depth is proportional to m , the mass enhancement increases the penetration depth and reduces the weight of the delta function. How do these ideas apply to the cuprate superconductors? The cuprate superconductors are generally believed to be in the clean, local limit, and thus to obey London electrodynamics. The reason for this belief is that the three ab-plane length scales are ordered as 3 SUPERFLUID AND NORMAL-FLUID DENSITY . . . A A (1) coherence length ( ab 17 ); (2) mean free path ( ab 100 ), and ), and hence in the clean limit; (3) penetration depth ( ab 1500 A therefore local electrodynamics. Consequently, the cuprates, even with a d-wave gap, should have most of the superconducting-state spectral weight or oscillator strength in the zero-frequency deltafunction. As will be seen in a moment, most of the spectral weight of the doped carriers is not in the delta function; instead, it remains in the broad midinfrared spectrum. The presence of this absorption has been known since the early days of the cuprate materials. Three things motivate a reexamination of the spectral weight at this time: (1) The fraction of the spectral weight in the super uid is small, suggesting an extremely strongly coupled system. (2) The fraction in the super uid is about 20 25% for all optimally-doped systems. (3) In underdoped materials, the super uid fraction is reduced further. OPTICAL TECHNIQUES Normal-incidence, polarized re ectance was measured by using a Bruker IFS-113v Fourier transform spectrometer (80 4000 cm 1 ) in the far-infrared and midinfrared region and a modi ed Perkin-Elmer 16U grating spectrometer in the near-infrared and ultraviolet (2000 33,000 cm 1). We used wire grid polarizers in the far midinfrared and dichroic polarizers in the near infrared ultraviolet. Low-temperature measurements (10 300 K) employed a continuous ow cryostat. The measurement process consisted of obtaining spectra at each temperature for both the sample and for a reference Al mirror. Their ratio gives a preliminary re ectance of the sample. After completing these measurements at each temperature for each polarization, the proper normalizing of the re ectance was obtained by taking a nal room temperature spectrum, coating the sample with 2000 of Al, A and remeasuring this coated surface. The ratio of the spectrum from the uncoated sample to the re ectance of the coated surface was multiplied by the known re ectance of Al to give the most accurate result for the room-temperature re ectance. 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 compensates for any misalignment between the sample and the mirror used as a temporary reference 4 D.B. Tanner et al. before the sample was coated, corrects for interference in the cryostat window, and, most importantly, provides a reference surface of the same size and pro le as the actual sample. The uncertainties in the absolute value of the re ectance are in the order of 1%. This uncertainty is in good agreement with the reproducibility found from the measurements of di erent samples.4 It leads to an uncertainty in the conductivity which varies with frequency, equal to 1%( /(1 ( )2 ). KRAMERS-KRONIG ANALYSIS We estimated the optical constants by Kramers-Kronig transformation of the re ectance data.5 The low- and high-frequency extrapolations were done in the following way. We extended the low-frequency data using a Lorentz-Drude model, dominated at the low frequencies by the free-carrier (Drude) form. Finite-frequency excitations are modeled by Lorentz oscillators. In the superconducting state, the re ectance is expected to be unity for frequencies close to zero, and we used the same Lorentz-Drude model, but with the Drude scattering rate set to zero. We extended the high-frequency end using data from the literature where available, and then using s up to a crossover frequency f and 4 thereafter. The exponent s is a number that can be between 0 and 4; we used s 1. The crossover frequency was chosen to be 1, 000, 000 cm 1 (125 eV) We observed some dependence of the results on the choice of s and f for frequencies close to the highest frequencies. For frequencies below 20,000 cm 1 , however, the e ects of this choice were insigni cant. ( ) ( ) OPTICAL CONDUCTIVITY The optical conductivity at two temperatures for the a-axis of a Bi2 Sr2 CaCu2 O8 single crystal is shown in Fig. 1. In the normal state (100 K), the low-frequency optical conductivity extrapolates reasonably well to the dc conductivity. The temperature dependence4,6 agrees with the T -linear resistivity; and, there is a characteristic narrowing of this far-infrared portion of the spectrum. In contrast, 1 ( ) does not show much temperature variation at high frequencies; the curves draw together around 3000 cm 1 . Below Tc , the low-frequency conductivity is considerably reduced. The missing 5 SUPERFLUID AND NORMAL-FLUID DENSITY . . . Fig. 1. Optical conductivity for the a-axis of Bi2 Sr2 CaCu2 O8 . area in the far-infrared conductivity appears as the zero-frequency delta-function response of the super uid. This aspect is discussed in more detail in the next section. SUM RULE ANALYSIS The doping of insulating cuprates introduces mobile carriers into the CuO2 planes, increasing the low energy spectral weight and decreasing the oscillator strength of the 1.5 2 eV charge-transfer band.7 10 Infrared spectroscopy may be used to estimate the dopinginduced spectral weight, using the partial sum rule for the optical conductivity.5 Ne ) ( m 2mVcell = m e2 1 ( )d 0 (1) where e and m are the free-electron charge and mass respectively, m the e ective mass, and Vcell the volume occupied by one formula unit. For simplicity, we will take m = m in the rest of this discussion and consider Ne ( ) to represent the e ective number of carriers per formula unit participating in optical transitions below frequency . 6 D.B. Tanner et al. Fig. 2. Partial sum rule for the a-axis of Bi2 Sr2 CaCu2 O8 . Figure 2 shows as the upper (solid) curves Ne for the a-axis of a single-domain Bi2 Sr2 CaCu2 O8 crystal at T = 100 K.6 The curve rises, begins to atten out, and then increases slope at the onset of the charge-transfer band. The short dashed line is obtained by subtracting from 1 ( ) the contributions of the charge-transfer and higher-lying bands (obtained by a t of the data to a Drude-Lorentz model) before integration. The value at which the 100 K dashed line saturates is a good estimate of Ne . The optical conductivity may be used to estimate the super uid density, Ns , in two ways. First, one may evaluate Ne ( ) for T < Tc . The data table for 1 ( ) naturally omits the zero-frequency in nite dc conductivity; thus the numerical integral misses the delta-function contribution, and the missing area in Ne ( ) below Tc gives the super uid density. Figure 2 shows (as the long dashed curves) Ne at 20 K for the a-axis of a single-domain Bi2 Sr2 CaCu2 O8 crystal.6 The 20 K data are nearly parallel to the 100 K data; the dotted line is the di erence between them, and is an estimate of Ns . The second method used to estimate the super uid density considers the inductive response of the super uid. One may look at the imaginary part of ( ), at the real part of the dielectric function, 1 ( ), or at a generalized London penetration depth, L ( ). For a 7 SUPERFLUID AND NORMAL-FLUID DENSITY . . . delta-function 1 ( ), one may write L ( ) = c 1 1 ( ) (2) (Note that this can also be written in terms of the imaginary part of the conductivity, 2 ( ), as L = c/ 4 2 ( ). One nds6,11,12 that the function L ( ) is nearly at in the far infrared, and in agreement with SR13 and other measurements made at = 0. From L one may pass directly to the number of super uid electrons, Ns = ns Vcell . Both methods agree relatively well ( 5%), implying that there is no con ict between the missing area in the far-infrared optical conductivity and the super uid density inferred from the penetration depth so far as the ab plane goes. In this the ab plane is quite di erent (and more conventional) than the recent results for light polarized along the c axis.14 In the latter measurement, the farinfrared missing area could only account for about half of the deltafunction area and a transfer of spectral weight from very high energies to the super uid was invoked. As a check of these analyses, one may also make a t of a DrudeLorentz model to the data and get Ne from the sum of the squares of the plasma frequencies of the Drude and those Lorentzians with center frequencies below the charge-transfer gap. Then, with the superconducting response modeled by collapsing the Drude scattering rate to zero, the Drude response gives the super uid density. This modeling produces carrier densities also within about 5% of those obtained by the other methods. DISCUSSION Figure 3 shows the results for single crystals of a number of materials. The left panel shows a Uemura plot,13 displaying Tc as a function of the ab-plane super uid density. (The super uid density is expressed as carriers per copper atom in order to allow for the di ering number of Cu layers and di ering interlayer spacing in the materials studied; however a plot as a function of 3-dimensional carrier density looks very similar.15 ) The typical linear increase of Tc with super uid density is clearly seen. The right panel is similar. It shows Tc as a function of the total doping-induced carrier density, Ne . The linear increase of Tc with 8 D.B. Tanner et al. Fig. 3. Left panel: Tc as a function of the super uid density. Right panel: Tc as a function of the total doping-induced spectral weight. The lines are least square ts to the data for optimally doped crystals. total carrier density is clearly seen. The di erence between the two panels is that the horizontal scale of the right-hand plot is ve times that of the left-hand plot, implying that only about 1/5 of the dopedin carriers join the super uid. This ratio holds quite closely for all optimally-doped materials, from Tc = 40 K La2 CuO4+ to Tc = 110 K Tl2 Ba2 CaCu2 O8 . It even works reasonably well for the b axis of YBa2 Cu3 O7 (represented by the stars in the gure) for which we have assumed 3 coppers per formula unit. For underdoped materials the ratio Ns /Ne becomes smaller and smaller as the doping level decreases from the optimum amount. There are other implications of this result. For example, in the standard model, cuprate infrared conductivity is associated with a strong frequency-dependence to the quasiparticle scattering rate, 1/ ( , T ). This picture was rst presented in the context of the marginal Fermi liquid16 and nested Fermi liquid pictures17 pictures but also occurs in the d-wave theories of the superconductivity.18,19 This picture has a di cult task to account for small value of the super uid spectral weight. Because these materials are in the clean 9 SUPERFLUID AND NORMAL-FLUID DENSITY . . . limit, the only way that the carriers can absorb light is through a process in which the emission of some excitation occurs.20 22 As already mentioned, the oscillator strength in this Holstein sideband and the enhancement of the condensate e ective mass is a measure of the interaction strength between the carriers and the excitation, . Using ps 2 = 4 ns e2 /m with m = (1 + )m, a reduction of the super uid oscillator strength by a factor of 5 implies = 4. This value represents extremely strong coupling; moreover it is not consistent with the value = 0.3 inferred from the temperature or frequency dependence of 1/ ( , T ).6,23,24 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 number DMR-9705108. Research at the EPFL has received support from the Swiss National Science Foundation, at Wisconsin from the DOE, at the National High Magnetic Field Laboratory from NSF Cooperative Agreement No. DMR9527035 and the State of Florida, at Texas from NSF Grant DMR9158089, and at Illinois from NSF Grant DMR-9120000 through the Science and Technology Center for Superconductivity. REFERENCES CITED 1. A.B. Pippard, Proc. Roy. Soc. (London) A216, 547 (1953). 2. M. Tinkham, Introduction to Superconductivity, Second Edition, (McGraw-Hill, New York, 1996). 3. P.B. Allen, Phys. Rev. B 3, 305 (1971). 4. M.A. Quijada, D.B. Tanner, R.J. Kelley and M. Onellion, Z. Phys. B 94, 255 (1994). 5. Frederick Wooten, Optical Properties of Solids (Academic Press, New York, 1972). 6. M.A. Quijada, D.B. Tanner, R.J. Kelley, M. Onellion, and H. Berger, Phys. Rev. B 60, 14000 (1999). 7. S.L. Cooper, G.A. Thomas, J. Orenstein, D.H. Rapkine A.J. Millis, S.-W. Cheong, A.S. Cooper, and Z. Fisk, Phys. Rev. B 41, 11 605 (1990). 8. S. Uchida, T. Ido, H. Takagi, T. Arima, Y. Tokura, and S. Tajima, Phys. Rev. B 43, 7942 (1991). 10 D.B. Tanner et al. 9. M.B.J. Meinders, H. Eskes and G.A. Sawatzky, Phys. Rev. B 48, 3916 (1993). 10. H. Eskes, A.M. Oles, M.B.J. Meinders, and W. Stephan, Phys. Rev. B 50, 17 980 (1994). 11. 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). 12. M.A. Quijada, D.B. Tanner, F.C. Chou, D.C. Johnston, and SW. Cheong, Phys. Rev. B 52, 15 485 (1995). 13. Y.J. Uemura, L.P. Le, G.M. Luke, B.J. Sternlieb, W.D. Wu, J.H. Brewer, T.M. Riseman, C.L. Seaman, M.B. Maple, M. Ishikawa, D.G. Hinks, J.D. Jorgensen, G. Saito, and H. Yamochi, Phys. Rev. Lett. 66, 2665 (1991). 14. D.N. Basov, S.I. Woods, A.S. Katz, E.J. Singley, R.C. Dynes, M. Xu, D.G. Hinks, C.C. Homes, and M. Strongin, Science 283, 49 (1999). 15. 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 , Beomo Hoan O, J.T. Markert, R.J. Kelly, and M. Onellion, J. Phys.: Condens. Matter 11, 239 (1999). 16. C.M. Varma, P.B. Littlewood, S. Schmitt-Rink, E. Abrahams, and A.E. Ruckenstein, Phys. Rev. Lett. 63, 1996 (1989). 17. A. Virosztek and J. Ruvalds, Phys. Rev. B 42, 4064 (1990). 18. P.J. Hirschfeld, W.O. Puttika, and P. W l e, Phys. Rev. Lett. 69, o 1447 (1992). 19. S.M. Quinlan, P.J. Hirschfeld, and D.J. Scalapino, Phys. Rev. B 53, 8775 (1996). 20. T. Holstein, Phys. Rev. 96, 535 (1954). 21. J. Orenstein, S. Schmitt-Rink, and A.E. Ruckenstein, in Electronic Properties of High-Tc Superconductors and Related Compounds, ed. H. Kuzmany, M, Mehrig, J. Fink (Springer-Verlag, Berlin-Heidelberg, 1990). 22. P.B. Littlewood and C.M. Varma, J. Appl. Phys. 69, 4979 (1991). 23. 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). 24. 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).
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