## Abstract

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 Tl_{2}Ba_{2}CuO_{6+}
_{x}, La_{2−}
_{x}Sr_{x}CuO_{4}, and YBa_{2}Cu_{3}O_{6.6}, 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 CuO_{2}planes 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 Tl_{2}Ba_{2}CuO_{6+}
_{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 CuO_{2} 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 ρ_{s}. The magnitude of ρ_{s} quantifies the electronic spectral weight of the SC δ function: ρ_{s} = 4π*n*
_{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.

We explored the changes in incoherent *c*-axis conductivity below *T*
_{c} in connection with the*c*-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, ρ_{s} 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 ρ_{s} 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 La_{2−}
_{x}Sr_{x}CuO_{4}(La2l4) with *T*
_{c} = 32 K (12) and underdoped YBa_{2}Cu_{3}O_{6.6} (Y123) with*T*
_{c} = 59 K (13). Tl_{2}Ba_{2}CuO_{6+}
_{x}is a structurally simple material with just one CuO_{2} plane per unit cell. Single crystals were grown as described in (7) and exhibited a SC transition at*T*
_{c} = 81 K with Δ*T*
_{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 mm^{3}. A mosaic of several specimens with similar*T*
_{c} and Δ*T*
_{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) = σ_{1}(ω) +*i*σ_{2}(ω) was determined through Kramers-Kronig (KK) transformations of the normal incidence *E* ∥*c* 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 at*T* ≃ *T*
_{c} the dc conductivity inferred from σ_{1}(ω → 0) is about 5 Ω^{−1}cm^{−1}. If we ignore phonon peaks and weak features in the electronic background of Tl2201 and La2l4, the conductivity can be described as σ_{1}(ω) ≃ σ_{dc} 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 CuO_{2} layers (1). Consistent with this view, σ_{2}(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.

Spectra of σ_{2}(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 σ_{1}(ω) implies (by causality of the electromagnetic response) a 1/ω term in σ_{2}(ω) with a prefactor given by the superfluid density: σ_{2}(ω,*T*) = ρ_{s}(*T*)/4πω. We determined the magnitude of ρ_{s} from extrapolation of ωσ_{2}(ω) to zero frequency. This method of extracting ρ_{s} from the data does not require model-dependent assumptions. We used this value of ρ_{s} to calculate the spectra of σ_{2}(ω, *T* <*T*
_{c}) from the following expression: σ_{2}(ω, *T *< *T _{c}
*) = ρ

_{s}(

*T*)/4πω + σ

_{2}(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 ρ

_{s}.

The absolute values of the SC plasma frequency ω_{ps}= (ρ_{s})^{1/2} for La2l4 and Y123 are in good agreement with the results reported by others (2,14). Our value for Tl2201, ω_{ps} = 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 the*c*-axis penetration depth λ_{c} =*c*/(ρ_{s})^{1/2} ≃ 12 μm from our data is shorter than λ_{c} = 17 to 19 μm measured by Moler *et al.* (7). We believe that the differences may be due to the sensitivity of the*c*-axis response of Tl2201 to minor variations of oxygen content or to structural defects acting as weak link interconnections between the CuO_{2} planes. Clearly, these results call for systematic studies of the evolution of ρ_{s} with doping.

We now turn to analysis of the changes in the real, dissipative part of the conductivity σ_{1}(ω) below*T*
_{c}. In all the crystals, the SC state conductivity is suppressed compared with the spectra taken at*T* ≃ *T*
_{c}. None of the studied samples, however, reveal new features in σ_{1}(ω) at*T* << *T*
_{c} that can be assigned to the SC gap. Notably, σ_{1}(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):(1)The magnitude of*N*
_{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 σ_{1}(ω) to carrier density *n* and free electron mass *m _{e}
*:(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 as

*T*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 σ_{1}(ω) at*T* << *T*
_{c} is connected with the formation of SC condensate. Indeed, σ_{1}(ω) = 0 for*T* << *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):(3)where *N*
_{n} = (120/π) ∫_{0}
^{∞} *d*ωσ_{1}(ω,*T* > *T _{c}
*) and

*N*

_{s}= (120/π) ∫

_{0+}

^{∞}

*d*ωσ

_{1}(ω,

*T*= 0). However, almost every property of the cuprates is anything but conventional. In particular, the absolute values of ρ

_{s}and [

*N*

_{n}(

*T*≃

*T*

_{c}) −

*N*

_{s}] show no simple relation analogous to Eq. 3 between the ρ

_{s}extracted from σ

_{2}(ω) and the missing spectral weight determined from σ

_{1}(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.

We believe that the inequality ρ_{s} > [*N*
_{n}(*T* ≃*T*
_{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 ρ_{s} 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)]/ρ_{s} show a very steep slope that saturates at ω ≃ 300 cm^{−1} (≃ 38 meV), reaching a value of approximately 0.5. The discrepancy between ρ_{s} and (*N*
_{n} −*N*
_{s}) shows that the spectral weight of the*c*-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)]/ρ_{s} at ω ≃ 300 cm^{−1} suggests that it is unlikely that the remaining portion of ρ_{s} 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 ρ_{s} 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 ρ_{s} > [*N*
_{n} − *N*
_{s}] inequality suggesting mid-IR condensation (Fig. 3) is specific only to the*c*-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 ρ_{s} and [*N*
_{n} − *N*
_{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 ρ_{s} and [*N*
_{n} −*N*
_{s}] is greatest in the YBa_{2}Cu_{3}O_{6.53} compound and systematically decreases with increasing doping until the discrepancy vanishes for YBa_{2}Cu_{3}O_{6.85}. In addition, we have observed enhancement of ρ_{s} compared with the magnitude of [*N*
_{n} −*N*
_{s}] in a series of Ni-doped YBa_{2}Cu_{3}O_{6.6} crystals and in YBa_{2}Cu_{4}O_{8} samples. Although there is no sign of the spectral weight discrepancy in optimally doped YBa_{2}Cu_{3}O_{6.95} crystals with*T*
_{c} ≃ 93 K, other cuprates with high values of*T*
_{c} including Tl2201 and HgBa_{2}CuO_{4} (*T*
_{c} ≃ 96 K) (26) do exhibit the ρ_{s} > [*N*
_{n} − *N*
_{s}] inequality. Considering the gapless nature of σ_{1}(ω) at*T* << *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 NbSe_{2}.

The discrepancy between the magnitude of the superfluid density ρ_{s} 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)(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 σ_{1}(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):
(5)where α_{n} and α_{s} are proportional to electronic (kinetic) energies in the normal and SC states. In our experiments, ρ_{s} and [*N*
_{n} − *N*
_{s}] are obtained independently (28). Therefore, the inequality ρ_{s} > [*N*
_{n}−*N*
_{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 ρ_{s} − [*N*
_{n} −*N*
_{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 CuO_{2} planes. The interlayer tunneling (ILT) theory (5, 8, 9) predicted the ρ_{s} > [*N*
_{n} −*N*
_{s}] inequality found in the *c*-axis transport, but the absolute value of ρ_{s} 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*
_{c}superconductors, 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.