Sum Rules and Interlayer Conductivity of High-Tc Cuprates

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Science  01 Jan 1999:
Vol. 283, Issue 5398, pp. 49-52
DOI: 10.1126/science.283.5398.49


Analysis of the interlayer infrared conductivity of cuprate high–transition temperature superconductors reveals an anomalously large energy scale extending up to midinfrared frequencies that can be attributed to formation of the superconducting condensate. This unusual effect is observed in a va- riety of materials, including Tl2Ba2CuO6+ x, La2− xSrxCuO4, and YBa2Cu3O6.6, which show an incoherent interlayer response in the normal state. Midinfrared range condensation was examined in the context of sum rules that can be formulated for the complex conductivity. One possible interpretation of these experiments is in terms of a kinetic energy change associated with the superconducting transition.

Interlayer electron transport in cuprate superconductors has been the focus of both experimental and theoretical efforts (1–11). The radical differences between the in-plane and interlayer behaviors of cuprates are most clearly illustrated in the raw reflectance R(ω) measured with polarized light of frequency ω. In high transition temperature (T c) superconductors, the reflectance of the electric field component parallel to the CuO2planes shows a metallic response, whereas reflectance obtained in the polarization along the interplane c axis direction [R c(ω)] is like that of ionic insulators with characteristic phonon peaks in the far infrared (IR), as shown in Fig. 1 for Tl2Ba2CuO6+ x(Tl2201). Below T c a sharp plasma edge at ω = 37 cm−1 (in Tl2201) emerges out of a nearly “insulating” normal state spectrum because superconducting (SC) currents flow along all crystallographic directions. In other cuprate compounds, this feature appears at 10 to 200 cm−1(2–4, 12–15). A collective (Josephson-like) mode associated with pair tunneling between CuO2 layers implies a plasma edge in the IR reflectance with the frequency position of the minimum in R c(ω) being proportional to the square root of the superfluid density ρs. The magnitude of ρs quantifies the electronic spectral weight of the SC δ function: ρs = 4πn s e 2/m* (16), where n s is the density of SC carriers, m* is their mass, and e is the electron charge. Although development of the plasma edge is expected based on elementary electrodynamics (6), formation of the SC condensate from the incoherent normal state response is an intriguing issue.

Figure 1

Reflectance of Tl2Ba2CuO6+ xmeasured in Ec and Eab polarizations of incident radiation. Thec-axis reflectance is nearly insulating in the normal state but at T < T c ≃ 80 K is dominated by the Josephson-like plasma edge.

We explored the changes in incoherent c-axis conductivity below T c in connection with thec-axis superfluid density. Our analysis uses model-independent arguments based on the oscillator strength sum rule or causality of the electromagnetic response. We found that in several high-T c cuprates, ρs significantly exceeds the spectral weight missing from the real part of the conductivity in a frequency region comparable to the SC gap 2Δ. This discrepancy in spectral weight indicates that a significant fraction of ρs is accumulated from mid-IR frequencies. These results support the hypothesis of a kinetic energy change associated with the superconducting transition (5, 9, 17, 18).

We measured the c-axis response of Tl2201 at the University of California at San Diego and compared our new data with earlier results reported by Timusk's group at McMaster University for lightly underdoped La2− xSrxCuO4(La2l4) with T c = 32 K (12) and underdoped YBa2Cu3O6.6 (Y123) withT c = 59 K (13). Tl2Ba2CuO6+ xis a structurally simple material with just one CuO2 plane per unit cell. Single crystals were grown as described in (7) and exhibited a SC transition atT c = 81 K with ΔT c ≃ 8 to 10 K (from magnetization). Reflectance measurements in the frequency range from 16 to 15,000 cm−1 were performed with an IR interferometer upgraded for spectroscopy of microsamples. Typical dimensions of the Tl2201 crystals were 0.8 × 0.8 × 0.075 mm3. A mosaic of several specimens with similarT c and ΔT c was used for the measurements in the energy interval down to 16 cm−1. The experiment was then repeated in the range from 60 to 9000 cm−1 using the thickest (≃130 μm) single crystal. The difference from the mosaic spectrum did not exceed 6%. The uncertainty of the relative changes in the spectra taken at different temperatures was less than 0.5%; this latter uncertainty is of primary importance for analysis of the spectral weight detailed below.

The c-axis component of the complex conductivity of Tl2201 crystals σ(ω) = σ1(ω) +iσ2(ω) was determined through Kramers-Kronig (KK) transformations of the normal incidence Ec reflectance. Extrapolations to low and high frequencies required for KK integrals do not affect the results in the frequency range where the actual data exist. The real and imaginary parts of the conductivity (Fig. 2) are dominated by a series of phonon peaks (at 85, 153, 366, and 602 cm−1 in Tl2201). The electronic background is relatively weak and atTT c the dc conductivity inferred from σ1(ω → 0) is about 5 Ω−1cm−1. If we ignore phonon peaks and weak features in the electronic background of Tl2201 and La2l4, the conductivity can be described as σ1(ω) ≃ σdc up to at least 200 meV (19). Featureless absorption with the spectral weight being almost uniformly spread over a broad energy scale is indicative of incoherent transport between the CuO2 layers (1). Consistent with this view, σ2(ω) is negative down to the lowest accessible energies, which suggests that the dominant contribution to the conductivity is due to IR-active phonons and possibly high-frequency electronic transitions.

Figure 2

(Left) The real part of the conductivity of Tl2Ba2CuO6+x, La2− xSrxCuO4(12) and YBa2Cu3O6.6 (13). The sharp structure is due to IR-active phonons. (Right) The imaginary part of the conductivity at TT c and at T <<T c. Thin dashed lines show zero parameter fit to the superconducting state data calculated as described in the text. Data for YBa2Cu3O6.6 are reprinted from Physica C, volume 254, C. C. Homes, T. Timusk, D. A. Bonn, R. Liang, W. N. Hardy, “Optical properties along the c-axis of YBa2Cu3O6+x, forx = 0.50 to 0.95. Evolution of the pseudogap,” pp. 265–280, Copyright, 1995, with permission from Elsevier Science.

Spectra of σ2(ω) undergo a qualitative change with the SC transition: below T c in all samples studied one finds a strong positive far-IR contribution. The SC δ function in σ1(ω) implies (by causality of the electromagnetic response) a 1/ω term in σ2(ω) with a prefactor given by the superfluid density: σ2(ω,T) = ρs(T)/4πω. We determined the magnitude of ρs from extrapolation of ωσ2(ω) to zero frequency. This method of extracting ρs from the data does not require model-dependent assumptions. We used this value of ρs to calculate the spectra of σ2(ω, T <T c) from the following expression: σ2(ω, T < Tc ) = ρs(T)/4πω + σ2(ω, T > T c) (Fig. 2 right, thin dashed lines). The spectra calculated in this fashion with no adjustable parameters reproduce the data in all the materials, thus validating the procedure of determining ρs.

The absolute values of the SC plasma frequency ωps= (ρs)1/2 for La2l4 and Y123 are in good agreement with the results reported by others (2,14). Our value for Tl2201, ωps = 130 cm−1, is somewhat greater than the value of 98 cm−1 estimated from a fit of grazing incidence reflectance data to an oscillator model by Tsvetkov et al. (20). The magnitude of thec-axis penetration depth λc =c/(ρs)1/2 ≃ 12 μm from our data is shorter than λc = 17 to 19 μm measured by Moler et al. (7). We believe that the differences may be due to the sensitivity of thec-axis response of Tl2201 to minor variations of oxygen content or to structural defects acting as weak link interconnections between the CuO2 planes. Clearly, these results call for systematic studies of the evolution of ρs with doping.

We now turn to analysis of the changes in the real, dissipative part of the conductivity σ1(ω) belowT c. In all the crystals, the SC state conductivity is suppressed compared with the spectra taken atTT c. None of the studied samples, however, reveal new features in σ1(ω) atT << T c that can be assigned to the SC gap. Notably, σ1(ω) remains finite down to the lowest frequencies, which suggests gapless behavior. We quantify the changes in the electronic background by introducing the effective spectral weight N eff(ω):Embedded Image(1)The magnitude ofN eff(ω) is proportional to the number of carriers participating in the optical absorption below the cutoff frequency ω and has the dimensions of the plasma frequency squared (cm−2). It is instructive to compare Eq. 1 with the oscillator strength sum rule for optical conductivity that relates the integral of σ1(ω) to carrier density n and free electron mass me :Embedded Image(2)One difference between Eqs. 1 and 2 is that, in the latter, integration is performed up to ∞ and this model-independent sum rule insists that the area confined under the whole conductivity spectrum must be equal to a constant. On the contrary, N eff defined according to Eq. 1 over an incomplete frequency range can be a temperature-dependent quantity. In both the La214 and Y123 materials, N eff(ω < 700 cm−1) is strongly suppressed asT is lowered from 300 K down to T c; this effect is usually attributed to a pseudogap leading to transfer of the electronic spectral weight from far-IR to higher energies (3,4, 14, 21).

In conventional superconductors suppression of σ1(ω) atT << T c is connected with the formation of SC condensate. Indeed, σ1(ω) = 0 forT << T c and ω below the energy gap 2Δ, with the “missing area” in the conductivity recovered under the δ function at ω = 0 (Fig. 3 bottom). Then the conservation of the total spectral weight between the normal and SC state (N n and N s, respectively) is expressed in terms of the Ferrel-Glover-Tinkham sum rule (22):Embedded Image(3)where N n = (120/π) ∫0 dωσ1(ω,T > Tc ) andN s = (120/π) ∫0+ dωσ1(ω, T = 0). However, almost every property of the cuprates is anything but conventional. In particular, the absolute values of ρs and [N n(TT c) − N s] show no simple relation analogous to Eq. 3 between the ρsextracted from σ2(ω) and the missing spectral weight determined from σ1(ω) (Fig. 3) (23). It appears that the area confined under the SC δ function is nearly twice [N n(ω) −N s(ω)] when the upper limit of integration in the latter is restricted to 120 to 150 meV.

Figure 3

(Top) Difference between the effective spectral weight at TT c and at T <<T c, normalized by ρs. In all the materials studied, [N n(ω) −N s(ω)]/ρs did not exceed 0.6 (ignoring sharp peaks due to the IR-active phonons). In the case of La2− xSr2CuO4and Tl2Ba2-CuO6+ xwe show [N n(ω) −N s(ω)]/ρs without phonon subtraction (light lines) and with phonons removed as described in (12, 23) (dark lines). Approximate error bars are shown at selected energies. (Bottom left) Calculations for a conventional dirty-limit superconductor with the scattering rate Γ = 20Δ show that about 90% of the ρs is collected from ω < 4Δ—that is, [N n(ω) −N s(ω)]/ρs > 0.9 by ω = 4Δ. This energy scale corresponds to about 0.1 eV in the case of the cuprates if the photoemission result for 2Δ is used (24). (Bottom right) Spectra of σ1(ω) for a conventional dirty superconductor with Γ = 20Δ.

We believe that the inequality ρs > [N n(TT c) − N s] originates from the limited frequency range over which the integration is performed. Nevertheless, the frequency scale in Fig. 3 is quite broad compared with the estimates of 2Δ = 45 meV based on photoemission studies (24) and is sufficient to exhaust the SC sum rule (Eq. 3) in a conventional dirty limit superconductor. In the dirty limit about 60% of ρs originates from ω < 2Δ, and by ω ≃ 6Δ the accumulation of the condensate is 96% complete (Fig. 3 bottom). In cuprates we find that the spectra of [N n(ω) −N s(ω)]/ρs show a very steep slope that saturates at ω ≃ 300 cm−1 (≃ 38 meV), reaching a value of approximately 0.5. The discrepancy between ρs and (N nN s) shows that the spectral weight of thec-axis condensate, at least in several different classes of cuprate superconductors, is collected from frequencies significantly exceeding 120 to 150 meV. The saturation of [N n(ω) −N s(ω)]/ρs at ω ≃ 300 cm−1 suggests that it is unlikely that the remaining portion of ρs will be acquired in the immediate vicinity of the upper-ω limit of our data. It is also unlikely that the high-energy contribution to the SC condensate is confined to some narrow feature located at ω > 150 meV. Our accuracy in this region is sufficient to detect a sharp absorption resonance, which is not found in mid-IR energies. We therefore conclude that about 50% of the spectral weight of the condensate associated with the Josephson-like collective mode of cuprates is distributed over a broad frequency scale starting at 0.15 eV and extending through the mid-IR range (up to ≃ 0.5 eV or even higher energy). In the case of underdoped La214 and Y123 compounds, the presence of the pseudogap seen in the interplane conductivity implies that the mid-IR contribution to ρs is likely even stronger (25). It is interesting that the energy range from which the superconducting condensate is drawn and the energy scale for the pseudogap are comparable.

We emphasize that the ρs > [N nN s] inequality suggesting mid-IR condensation (Fig. 3) is specific only to thec-axis electrodynamics and only to the materials with truly incoherent normal state conductivity. So far, we have been unable to find similar conflict between the values of ρs and [N nN s] in the in-plane response of either Y123, La214, Tl2201, or any other cuprate. Also, the strength of the effect in the c-axis electrodynamics is suppressed as the c-axis response becomes more coherent with increasing doping. Specifically, for a series of YBCO crystals with different oxygen dopings, the discrepancy between ρs and [N nN s] is greatest in the YBa2Cu3O6.53 compound and systematically decreases with increasing doping until the discrepancy vanishes for YBa2Cu3O6.85. In addition, we have observed enhancement of ρs compared with the magnitude of [N nN s] in a series of Ni-doped YBa2Cu3O6.6 crystals and in YBa2Cu4O8 samples. Although there is no sign of the spectral weight discrepancy in optimally doped YBa2Cu3O6.95 crystals withT c ≃ 93 K, other cuprates with high values ofT c including Tl2201 and HgBa2CuO4 (T c ≃ 96 K) (26) do exhibit the ρs > [N nN s] inequality. Considering the gapless nature of σ1(ω) atT << T c in all these materials and the large energy scale of superfluid formation, one can conclude that the magnitude of the SC gap is irrelevant to the interlayer response at least of the cuprates discussed in this work. It remains to be seen whether evidence for mid-IR condensation is found in other cuprates or in other classes of strongly anisotropic noncuprate superconductors including 2D organic materials and NbSe2.

The discrepancy between the magnitude of the superfluid density ρs and the spectral weight missing from the IR part of the conductivity can be interpreted within theoretical approaches that lead to the following form of the conductivity sum rule (9, 10,17, 27)Embedded Image(4)In Eq. 4, W is a cutoff frequency of the order of a bandwidth and α(T) is proportional to an electronic kinetic energy (5, 8–10, 17). In these sum rules, the right-hand side of Eq. 4 is allowed to change with temperature, magnetic field, and other parameters. The validity of the global oscillator strength sum rule (Eq. 2) is assured through proper readjustment of high-energy interband contributions to σ1(ω) at ω > W. We stress that only the sum rule in Eq. 2 is truly model independent. For the case of a superconductor, Eq. 4 can be used to yield the following sum rule (9, 10, 17):Embedded Image Embedded Image(5)where αn and αs are proportional to electronic (kinetic) energies in the normal and SC states. In our experiments, ρs and [N nN s] are obtained independently (28). Therefore, the inequality ρs > [N nN s] (Fig. 3) indicates that kinetic energy change associated with the SC transition may account for the discrepancy in spectral weight.

The change of the electronic kinetic energy at T< T c suggested by our data should be contrasted with the behavior of conventional superconductors where this effect is negligibly small. Moreover, in metallic superconductors ρs − [N nN s] ought to be negative, consistent with the experimental data for lead films (29). At least two models proposed for high-T c superconductors (17,18) predicted the correct sign of the effect but expected it to be dominant in the response of the CuO2 planes. The interlayer tunneling (ILT) theory (5, 8, 9) predicted the ρs > [N nN s] inequality found in the c-axis transport, but the absolute value of ρs in Tl2201 is smaller (7, 20) than is expected within the ILT model (5, 8). Because change in the interlayer kinetic energy has been detected in several classes of high-T csuperconductors, we believe that this unusual effect will be instrumental in narrowing the field of plausible theoretical models of high-T c superconductivity.

  • * Present address: Lucent Technologies, 2000 North Naperville Road, Naperville, IL 60566, USA.


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