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Evidence for strong electron-phonon coupling and polarons in the optical response of La_{2-x}Sr_xCuO_4

The normal state optical response of La_{2-x}Sr_xCuO_4 is found to be consistent with a simple multi-component model, based on free carriers with strong electron-phonon interaction, localized polaronic states near 0.15 eV and a mid-infrared band at 0.5 eV. Normal state reflectance and absorbance of La_{1.83}Sr_{0.17}CuO_4 are investigated and their temperature dependence is explained. Both, the ac and dc response are recovered and the quasi-linear behavior of the optical scattering rate up to 3000- 4000 cm^{-1} is found to be consistent with strong electron-phonon interaction, which also accounts for the value of T_c. Although not strictly applicable in the superconducting state, our simple model accounts for the observed penetration depth and the optical response below T_c can be recovered by introducing a small amount of additional carriers. Our findings suggest that the optical response of La_{2-x}Sr_xCuO_4 could be explained both, in the normal and superconducting state, by a simple multi-fluid model with strong electron-phonon interaction if the gap symmetry and the temperature dependence of the 0.5 eV mid-infrared band are adequately taken into account.


I. INTRODUCTION
Experimental studies in recent years have revealed a variety of anomalous optical properties of high temperature superconductors (HTSC), both in the normal and superconducting state, which are still not fully understood theoretically. The optical conductivity in the midand near-infrared (MIR,NIR) regimes in the metallic phase of these materials is unusually high. The temperature dependence is extremely weak and the superconducting transition can hardly be observed at far-infrared (FIR) frequencies. In addition, powder absorbance measurements 1-3 reveal an unusual temperature dependence of the integrated absorbance in the mid-infrared: An increase in the integrated absorbance upon cooling up to T c is followed by a sharp slop change at T c , and a saturation or even a decrease at lower temperatures. Such behavior can neither be explained by a normal Fermi-liquid approach nor by conventional strong-coupling theory. 4 To explain the optical response of HTSC two main approaches are considered: singleand multi-component models. In single-component models the complex optical conductivity while multi-fluid models 6 generally include a number of Lorentz oscillators, peaked at nonzero frequencies. I.e. localized states which dominate at MIR frequencies are superimposed on a free carrier contribution in multi-component models. Experimental data on the optical conductivity of optimal doped HTSC compounds cannot distinguish between the above approaches since no clear structures are resolved in these data. In the case of underdoped compounds more structure is observed, suggesting that multi-component descriptions are more adequate. 3 A vastly enhanced MIR response of chemically doped parent compounds of HTSC, both n type and p type, is also observed in the isolating state. 7,8 Here the optical response indicates the existence of self-localized carriers: small polarons. The MIR band, which is normally absent in stoichiometric samples, can be interpreted as a composition of overtones of local modes created by the self-trapping of extra charges injected into the lattice. In addition, absorbance measurements of various Sr doped La 2−x Sr x CuO 4 (LSCO) compounds demonstrated that localized states are present near 0.15 eV and have a significant oscillator strength even at high doping levels. 3 These results show that multi-component approaches should be considered in the under-and optimal doped regime, and even in slightly overdoped compounds.
So far multi-component approaches have used a simple Drude term to model the free carriers contribution to the optical response. However, there is considerable evidence for strong electron-phonon coupling in HTSC, and it is now widely accepted that the simple Fermi liquid quasiparticle description is not even applicable to the normal state metallic phase of these materials. Indeed, while the MIR spectra depend only weakly on temperature, the free carrier scattering rate, dominated by the electron-phonon interaction, is strongly temperature dependent and recent studies on YBa 2 Cu 3 O 7−δ (YBCO) 9 and LSCO 3 show that to model this behavior the simple Drude term for the free carriers must be modified to contain complicated frequency and temperature dependencies which yet have to be explained theoretically.
In this paper we will advance the multi-fluid approach by applying strong-coupling theory to determine the free carriers contribution to the optical response. As a model system we chose LSCO which is highly qualified for studying the optical properties of HTSC since it has no Cu-O chains and low c-axis conductivity. Analysis of absorbance data of under-and optimal doped compounds is straightforward using effective medium approach 3,9 , since the c-axis conductivity is non metallic. For slightly overdoped LSCO, discussed here, c-axis conductivity might effect the FIR response but not the MIR and NIR spectra.
We will show that the normal state response of LSCO compounds is well described by a simple multi-component approach based on three main contributions: (a) free carriers response with strong electron-phonon interaction; (b) localized polaronic states near 0.15 eV; (c) mid-infrared band near 0.5 eV. With this model one can account for the optical response and its temperature dependence, the dc resistivity, and the value of the superconducting transition temperature T c based on normal electron-phonon interaction pairing.
Although a variety of simplifying assumptions make our model strictly applicable only in the normal state, it is instructive to apply it to the superconducting state as well. The predicted London penetration depth is in agreement with measured data and the optical response below T c can be accounted for if a small amount of additional charge carriers, "hidden" in the normal state, is introduced. As the most likely source of these carriers we identified the 0.5 eV MIR band. This is in agreement with observed behavior of the total absorbance and would suggest a bipolaronic origin of this band, as proposed by Alexandrov

II. MODEL
We model the optical response within a three liquid model, consisting of two bands of localized charge carriers near 0.15 eV and 0.5 eV, respectively, and a free carrier contribution derived within the classical strong-coupling theory of superconductivity. If we disregard the phonon contribution in the FIR regime, the total dielectric function can be written as Here, ǫ be 1 and ǫ be 2 represent the contributions of the two MIR bands, 0.15 eV and 0.5 eV, respectively, and ǫ E is the free carriers contribution.
In the normal state, with which we are mainly concerned here, the omission of phonons from the dielectric function has no substantial effect. We note however, that their omission magnifies the impact of the superconducting transition on the optical response below T c .
In order to include phonons one should consider their temperature dependent line width, energy shift, and renormalization below T c which are out of the scope of the present paper.
The 0.15 eV band has been experimentally identified to be of polaronic origin 3,12 since its optical activation energy is 3-5 times larger than the thermal one. The origin of the second MIR band, near 0.5 eV, is yet unknown. One possibility is the dissociation of bipolarons which do or do not form a conduction band. Experimentally it was shown that this band is only weakly temperature dependent and that the total absorbance is modified below the superconducting transition. In this work we ignore this temperature dependence since it is hardly observed in reflectance data, and approximate the 0.5 eV band with a temperature independent Lorentzian. Although this simplification is well justified in the normal state, there is evidence that it does not hold in the superconducting state.
The Lorentz dielectric functions for the two MIR bands have the form If we neglect any difference in effective mass between the polaronic and free states, the free carriers plasma frequency and the oscillator strength of the polaronic Lorentzian obey the sum rule and the temperature dependent spectral weight of the 0.15 eV band is given by while the plasma frequency of the free carriers is Here, T 0 denotes a thermal activation energy.
The free carriers contribution to the optical response is calculated by applying con- where µ * is the Coulomb pseudo-potential and ω c is the frequency cutoff. N(Ω) and f (ω) are Bose and Fermi distribution functions. As usual, the impurities are assumed to be non-magnetic here. Consequently, impurity scattering does not appear in the isotropic equations (6a) and (6b).
Using the standard theory of electromagnetic response function, the optical conductivity can subsequently be calculated in the local (London) limit. Therefore, the free carriers contribution to the optical conductivity is given by 13 The functions and include normal impurity scattering effects in Born approximation, γ imp is an average scattering rate, and ω 2 p E denotes the effective plasma frequency of the free carriers in the Cu-O planes. The free carriers contribution to the total dielectric function (2) is then given by The strong-coupling theory is no first principle theory but requires definite assumptions on the Eliashberg function, α 2 F (ω), and needs coupling parameters as an input. Here we started from published phonon density of state 17 but found it incompatible with the optical and the dc data since the scattering rate at low frequencies is too large. Better agreement is achieved when α(ω) is nonuniform and increases at high frequencies. This is also consistent with tunneling data 18,19 where high energy phonons seem to have stronger electron-phonon interaction coefficient. For the sake of simplicity we chose a power law behavior: α 2 (ω) = (ω/ω 0 ) η with η > 0. The phonon spectrum F (ω) is estimated by a set of quadratic Lorentzians and a low frequency cutoff function to account for the low frequency tail which otherwise dominates the electron-phonon interaction constant The resulting Eliashberg function is shown in Fig. 1 and the parameters used to define Table I. The overall coupling coefficient λ was adjusted to give the correct value of T c . To determine the superconducting transition temperature for our choice of parameters we used McMillan's equation. 20 The resulting T c is in good agreement with the transition temperature derived directly from Migdal-Eliashberg theory by calculating the London penetration depth λ L (T). In the superconducting state, for T close to T c , one the transition temperature can be easily determined once σ E (ω) is calculated. Similarly, the impurity scattering rate γ imp is found by the measured dc resistivity, and the strength of localized states is in general agreement with those found in absorbance measurements. 3 We have performed various model calculations using different values for the electronphonon coupling strength λ, ranging from 0.7 − 1.4. These calculations show that it is necessary to assume λ ≈ 1 to achieve reasonable agreement with both, tunneling data and the optical and dc data. In general, we found that the different kinds of data investigated here put severe constraints on possible choices of the fitting parameters and we note that although the number of model parameters is rather large, no significant deviation from the parameters listed below (Table II) Table II. Figure 2 shows the calculated normal state reflectance, together with the measured ones at three different temperatures and an additional calculated curve at 40 K. At low frequencies ( < 1000 cm −1 ) the optical response is dominated by the free carriers and is therefore temperature dependent. At higher frequencies the temperature dependence is largely suppressed since the spectral weights of the two localized states become Another quantity of interest is the dielectric loss function, Im(−1/ǫ(ω)). Experimentally the loss function of HTSC is found to follow a quadratic behavior, Im(−1/ǫ(ω)) ∝ ω 2 , over almost the entire frequency range up to the plasma edge, where it exhibits a (first) peak and subsequently decreases. 23 The height of the peak is temperature dependent, while its position and the curvature of Im(−1/ǫ(ω)) strongly depend on ǫ ∞ . We have calculated the dielectric loss function using a variety of different model parameters, confirming the high sensitivity of the peak position to the choice of ǫ ∞ . For optimally doped (x = 0.15) LSCO the position of the maximum has been reported to be approximately 6500 cm −1 . 23 Figure   3 shows the dielectric loss function for La 1.83 Sr 0.17 CuO 4 at 200 K as calculated within our multi-liquid model using the parameters from Table II. The behavior in the FIR and MIR regimes is quadratic to good approximation and the curve shows a maximum at 6320 cm −1 , close to the experimentally observed value for optimal doping. This confirms our choice of With the parameters obtained from the fit of the reflectance data the absorption coefficient α(ω) = 4 π ω Im(ǫ 1/2 (ω)) can be calculated and compared to measured optical absorbance data. Analysis of absorbance data must take into account grain-size effects and the response of the host material. Therefore, the major disadvantage of this approach is the difficulty in converting qualitative results into quantitative ones. However, it has been shown recently that absorbance measurements at MIR frequencies can be analyzed quantitatively within the framework of the effective medium approach. 9 Figure 4 shows The quasi-linear behavior of 1/τ * up to 3000 − 4000 cm −1 was one of the driving forces for assuming electron-electron interaction (marginal 24 and nested 25 Fermi Liquids for example) rather than electron-phonon one since the phonon spectrum does not exceed 1000 cm −1 . As already mentioned by Shulga et al (Ref. 26) this argument is valid for 1/τ rather than 1/τ * which indeed becomes frequency independent above the phonon spectrum. Therefore the multi-component model with strong electron-phonon interaction supports one of the major properties of HTSC: the quasi-linear response of the optical scattering rate up to energies much higher than the phonon spectrum. It is important to note here that obtaining 1/τ * experimentally is somewhat delicate since this quantity is quite sensitive to the value of ǫ ∞ .
The general behavior of the dc resistivity also support the above picture, namely the co-existence of free and localized charge carriers and the thermal excitations of the later.
Temperature dependent resistivity measurements are done at constant pressure, and the correction to constant volume reveal deviation from quasi-linear behavior where the resistivity data tend to saturate or follow a lower slope at high energies. This change of slope is also noticed in the calculated resistivity, where the larger number of free carriers reduces the slope of the ρ(T) curve, Fig. 7 , where x 0 is the total number of localized carriers per Cu atom. The above ratios yield x 0 = 0.075 in close agreement with the estimated spectral weight from absorbance measurements. 3 Once the parameters of our model are determined as described above, the optical response of LSCO can be calculated for arbitrary temperatures, above and below T c = 37.4 K.
Therefore, we will also apply the model to the superconducting state to gain an insight in how it could and should be altered to be applicable in both, the normal and superconducting state.
The first quantity of interest here is the London penetration depth. According to Eq.
(12), its zero-temperature value, λ L (0), provides a measure for the free carriers plasma frequency ω p E (0), since the zero-frequency limit of the conductivity is entirely due to the free carriers response. Figure 8 shows the normalized inverse penetration depth, λ 2 L (0)/λ 2 L (T), as predicted by our three-liquid model, using the parameters from Table II. At low temperatures the phenomenological relation, is well resembled and we can use this expression to extrapolate λ L (T) to zero temperature.
This yields λ L (0) = 2579Å. Experimentally it is difficult to obtain the absolute value of the penetration depth, especially for thin films. However, it is generally accepted that λ L (0) > 2300Å for optimally doped LSCO [28][29][30] . Using muon spin relaxation rate measurements Aeppli et al found λ L (0) ≈ 2500Å, close to the value predicted by our multi-liquid approach.
It has been observed earlier for various HTSC that the integrated absorbance exhibits an unusual temperature dependence. [1][2][3] In the normal state the integrated absorbance increases upon cooling. At the superconducting transition temperature a sudden slope change occurs, followed by a saturation or even a decrease of the integrated absorbance upon further cooling. Another discrepancy between the measured and calculated optical response can be seen in the reflectance below T c . While the normal state fit curves practically coincide with the experimental reflectance data over a wide range of frequencies (see Fig. 2), in the superconducting state the measured reflectance from Gao et al is slightly underestimated by our model. This could be simply due to the fact that the simplifying assumption of equal effective mass for the free and localized carriers breaks down below T c . Alternatively, the discrepancy can be accounted for by introducing additional free carriers. Figure 11 shows the measured reflectance at 10 K together with the prediction of our model. Here, a slightly modified model, with an additional small Drude term, with ω p D = 2000 cm −1 and γ D = 200 cm −1 , has been included. The close resemblance between the measured curve in Fig. 11 with this modified multi-liquid model indicates the existence of a "hidden" reservoir of charge carriers which has to be taken into account in the superconducting state. There is increasing evidence that the gap symmetry in LSCO is more complicated than simple s-wave. 11 In the case of d-wave pairing, the "hidden" carriers could be explained by the contribution of resonant scattering. Another likely source for these additional carriers is the second MIR band, near 0.5 eV, which was taken to be temperature independent in our simple multi-fluid approach. Indeed, by increasing the free carriers plasma frequency slightly, band. The comparison of our model with the measured reflectance in the superconducting state seems to confirm this picture. In Fig. 12 the reflectance curve at 10 K calculated with the modified free carriers plasma frequency and oscillator strength of the second Lorentzian is shown for frequencies below 2500 cm −1 together with our original model and the measured reflectance. Above 1500 cm −1 the modified spectral weight provides better agreement with the experimental data. Between 100 − 600 cm −1 our model predictions can be seen to deviate from the experimental curve, predicting lower absorbance in this frequency regime. This is due to the simplifying assumption of s-wave pairing for the free carriers. As mentioned above, the gap symmetry in LSCO is probably more complicated and must be taken into account to describe the FIR regime, since its shape leaves long tails which can easily alter the optical response up to 1000 cm −1 . In addition, the omission of phonon modes and the  . ω min = 60 cm −1 , ω α = 1000 cm −1 , A α = 3.8, and η = 1.9, where: α 2 (ω) = A α (ω/ω α ) η ,   Table II (dash-dotted line). In contrast to the good agreement between data and fit in the normal state (compare Fig. 2), the measured reflectance at 10 K is slightly underestimated by our model. Also shown is a modified multi-fluid model (solid line), including an additional Drude term with ω p D = 2000 cm −1 and γ D = 200 cm −1 , which better resembles the measured reflectance.
FIG. 12. Comparison between the measured reflectance at 10 K, the multi-liquid model using the parameters from Table II (dotted line), and the model with weaker MIR response, ω p (be2) = 5748.1 cm −1 and ω p E (0) = 10591 cm −1 (solid line). The discrepancy in the FIR regime is due to various simplifying model assumptions, as explained in the text. Above 1500 cm −1 the measured data is well resembled by the modified model.