Constraints on Possible Mechanisms for

This paper discusses a phenomenological model used to describe various properties of a dx2−y2 superconductor in its temperature as well as frequency dependence, namely, the London penetration depth, the optical conductivity, the microwave conductivity, and the electronic thermal conductivity. We assume the CuO 2 planes to be the dominant feature for superconductivity and develop a 2D-formalism in which inelastic scattering is modelled explicitly by a spectral density which describes a fluctuation spectrum responsible for the superconducting transition and also for the large inelastic scattering observed in the normal state above the critical temperature Tc . The feedback effect of superconductivity on the spectral density is modelled by a temperature dependent low frequency cutoff. Theoretical results are compared with the experimental data and the fact that such a model allows a consistent description of a variety of phenomena is then used to formulate constraints on possible mechanisms of superconductivity in oxides.


Introduction
Since the discovery of high-T c superconductivity in oxide materials by Bednorz and M uller 1] huge e orts have been made to nd a theoretical description of the pairing mechanism which leads to critical temperatures of the order of 100 K. Nevertheless, it is only recently that a consistent set of experiments performed on high quality twinned and untwinned single crystals of optimally doped YBa 2 Cu 3 O 6:95 (YBCO) seem to have resolved the symmetry of the superconducting order parameter to be predominantly of the d x 2 ?y 2 symmetry with nodes crossing the Fermi surface.Most important is the observation of the linear temperature dependence of the London penetration depth at low temperatures 2] which was discussed earlier by Annett et al. 3].Such a linear dependence at low temperatures has also been reported for the spin susceptibility 4].More evidence was supplied by the discovery of the existence of the so-called unitary limit in the electronic thermal conductivity of YBCO single crystals doped with Zn 5], and the c-axis Josephson tunnelling experiments where the conventional superconductor (Pb) is deposited across a single twin boundary 6].This experiment also o ers a direct evidence for a subdominant s-symmetric contribution to the order parameter which is typical of orthorhombic systems 7].
Modi cations in the low-temperature linear dependence of the penetration depth brought about by the impurity scattering 8] are also naturally understood from theoretical models with the order parameter having d-wave symmetry 9,10].Such models, however, tend to predict slopes for the penetration depth near the critical temperature which are not as steep as those observed.This is true even if inelastic scattering is incorporated in the calculations through the Eliashberg-type formalism which represents a rst approximate attempt of including self-energy e ects 11].
Another set of experiments which can be used to put constraints on possible mechanisms of high T c superconductivity are microwave conductivity measurements as a function of temperature in pure single crystals of optimally doped YBCO 12,13] which have revealed the existence of a very large peak around 40 K, whose size and position depend somewhat on the microwave frequency used.A similar peak can be found in the electronic thermal conductivity, though at much lower temperatures 14].This peak in the microwave conductivity has been widely interpreted to be due to the rapid reduction in the inelastic scattering below T c and is generally referred to as the collapse of the low-temperature inelastic scattering rate.One possible way to describe this experimental result is the introduction of a temperature dependent inelastic scattering time which can be modelled from the spin uctuation theory 15].Schachinger et al. 16{18] proposed quite a di erent explanation which not only allowed a satisfying analysis of the microwave conductivity peak but also described the temperature dependence of the London penetration depth and of the electronic thermal conductivity consistently.Their phenomenological model is based rst of all on the observation that a mechanism responsible for superconductivity in oxides, which also leads to d-wave superconductivity, is most likely to be electronic in origin.In such a model the collapse of the inelastic scattering rate which causes a large peak in the microwave conductivity is explained by a gap which opens up in the uctuation spectrum responsible for superconductivity 19{21].Such an e ect is generic to all electronic mechanisms where the uctuation spectrum causing superconductivity belongs to the superconducting quasiparticle system itself and becomes gapped as superconductivity sets in.
The phenomenological model consists in the application of a temperature dependent low frequency cuto to the uctuation spectrum to give the correct temperature dependence of the London penetration depth of clean, optimally doped YBCO single crystals.(It is interesting to note that the temperature dependence of the low frequency cuto follows quite closely the temperature dependence of the superconducting gap.)This model is then used (within the framework of the Eliashberg-type formalism adjusted to allow for d-wave superconductivity) to calculate optical conductivity, microwave conductivity, and electronic thermal conductivity of clean superconductors and superconducting systems which also contain moderate concentrations of impurities.
It is the purpose of this paper to review the recent results and to expand the application of the model to the calculation of the optical conductivity of a d x 2 ?y 2 superconductor in its normal as well as superconducting state.Section 2 speci es the basic Eliashberg-like equations and the formulae used to calculate the London penetration depth, optical conductivity, optical re ectivity, and electronic thermal conductivity.In Section 3 the results are discussed and, nally, Section 4 presents our conclusions.

Formalism
The simplest description of a d-wave superconductor is obtained within the BCS formalism assuming a separable model for the pairing interaction.In such a model the pairing potential depends on the product cos(2 ) cos(2 0 ) where and 0 are the directions of the initial and nal momenta on a two-dimensional circular Fermi surface.To include the dynamics of the uctuations that are exchanged in the pairing it is necessary to go beyond the BCS and consider self-energy corrections.We restrict ourselves to the Eliashberg-type formalism which was discussed by Schachinger and Carbotte 22] and o ers at least a rst order approximation to the full self-energy corrections.Such a formalism includes a Bose-exchange spectral density I 2 F( ) which enters both the gap channel for the pairing energy ~ (i! n ) and the channel for the renormalized Matsubara frequencies !(i! n ).This last quantity exists in the normal state and carries information on the s-wave part of the interaction.From symmetry considerations, the gap channel involves the d-wave part of the interaction.In principle, these two projections of the full boson-exchange interaction need not involve the same weighting of the Bose energies.In the absence of detailed information on the mechanism we will assume, for simplicity, that a single I 2 F( ) can, nevertheless, be employed as a rst approximation but with a di erent weight g in the gap channel as compared to the renormalization channel.The two nonlinear self-energy equations for !(i! n ) and ~ (i! n ), with i! n = i T(2n + 1); n = 0; 1; 2; : : :, and temperature T will then have the following form in the imaginary axis notation 22]: and Here h i denotes the average over the angle , ?+ = n I =(N(0) 2 ) with n I being the concentration of isotropically scattering impurities, N(0) is the normal state quasiparticle density of states at the Fermi energy, c = cot 0 , and 0 is the T-matrix phase shift.For very large values of c (c ! 1) we are in the Born scattering limit and for c = 0 in the so-called unitary (resonant) scattering limit.In the weak scattering (Born) limit the impurity term in equation ( 1) reduces to t + (i! n ) with c absorbed into t + = n I N(0)jV (k F )j 2 , where V (k F ) is the impurity scattering potential evaluated at the Fermi momentum k F .The London penetration depth L (T) at any temperature T < T c follows from the solution of equations ( 1) and ( 2) and is given, within a numerical constant, by 10]: The optical conductivity ( ) at any temperature T and photon frequency is given by 23,24]: ( ) = i e 2 N(0)v ; (7) where the star symbolizes a complex conjugate, e is a charge on the electron, and v F is the Fermi velocity.Furthermore, E( ; ) = q !2 ( ) ? ~ 2 ( ; ); N( ; ) = !( ) E( ; ) ; P( ; ) = ~ ( ; ) E( ; ) : (10) In the normal state equation (7)  Once the real ( 1 ( )) and imaginary ( 2 ( )) parts of the conductivity are known, the optical re ectivity R( ) can easily be calculated using R( ) = 1 ?p "( ) 1 + p "( ) 2 ; (14) with the dielectric function "( ) de ned as "( ) = " 0 + 4 i ( ); (15) where " 0 describes the response of quasiparticles in completely lled bands.
Using the above notation we nd for the electronic thermal conductivity 25]: : (16) where indices 1 and 2 refer to the real and imaginary parts, respectively.equations (7) to (16) require the renormalized gap function ~ ( ) and the renormalized frequencies !( ) on the real axis.These can be found by the analytical continuation of the results of equations ( 1) and ( 2) from the imaginary axis to the real one employing the method developed by Marsiglio et al. 26].
Most of the parameters of the model have now been speci ed except for g, the dto s-anisotropy of the exchange potential and for the form of the boson-exchange spectral density I 2 F( ) which describes the uctuation spectrum.Many choices could be made for this latter quantity.If we knew the actual mechanism which leads to pairing in oxides, there would be no choice at all as we would know the microscopic origin.In the absence of such information we adopt a very simple form which can be regarded as one guided by the nearly antiferromagnetic Fermi liquid model 27,28].We use up to to some convenient high frequency cuto for the numerical work (400 meV here).The frequency !sf sets the energy scale of the uctuation spectrum and is not arbitrary because our numerical work has to re ect the observation that above the critical temperature the inelastic scattering rate is of the order of several times T c in value.This requires !sf = 30 meV and I 2 is adjusted to give a clean limit critical temperature T c0 = 100 K in solving the linearized equations ( 1) and ( 2) 16].This results in the strong coupling parameter T c0 =! log = 0:31 where !log is de ned in the usual way 29] !log = exp 2 Z d I 2 F( ) ln ; (18) and represents the average boson energy in the system.The only parameter left is g and it has already been shown that the results do not depend qualitatively on the choice of g 16] and we set g = 0:8 to be de nite.

Optical conductivity, normal state
It is the aim of this short subsection to verify the parameters introduced previously to make the uctuation spectrum (17) de nite, namely, !sf , and the high frequency cuto , by comparing the theoretically obtained normal state conductivity, equation (11), with the experiment.Figure 1 presents the results of such a comparison.What is shown here is the real part of the optical conductivity 1 ( ) ( gure 1a) and the inverse of the in-plane optical scattering time as a function of frequency .The data points are for a clean, twinned, optimally doped YBCO single crystal (T c = 93:2 K, solid squares), a similar sample with 0.75% Ni substituted at Cu-sites (T c = 91 K, solid triangles), and a sample with 1.4% Ni substitution (T c = 89 K, solid down-triangles).The normal state conductivity data have been obtained at a sample temperature of 100 K in all the cases.Theoretical results are presented for the clean sample (solid line), the sample with 0.75% Ni (dashed line), and the sample with 1.4% Ni (dotted line).The Ni content was simulated by the value for t + necessary to decrease the clean sample's critical temperature to the required value.The arrows indicate the data points used to scale theory to experiment.This scaling was necessary as the theoretical results are on an arbitrary scale because of the factor ! p =4 which was left out in evaluating equation (11).Theoretically such a scaling should not be necessary for the inverse scattering time because, according to equation (19), the theoretical values are free of this material parameter.Nevertheless, Homes et al. 30] point out in their paper that they were using a value of !p = 1:6 eV which is somewhat ambiguous because it depends on the frequency cuto used in the evaluation of the experimental data.The agreement between theory and experiment is excellent for the clean and the 0.7% Ni sample over the whole frequency range covered by the experiment.We note some deviations for the 1.4% Ni sample at low frequencies and above 1200 wave numbers.But it is quite obvious that this data set escapes the general trend established by the other two samples.
We conclude that the simple model uctuation spectrum (17) with !sf = 30 meV and a high frequency cuto of 400 meV allow an excellent description of the frequency dependence of the normal state optical conductivity and thus, establishes a valid basis for further investigations into the superconducting state.

In-plane London penetration depth
Our results for the temperature dependence of the in-plane London penetration depth given on evaluation of equation ( 6) which requires only the solutions of the self-energy equations ( 1) and ( 2) on the imaginary axis are displayed in gure 2. What is presented is the inverse square of the normalized in-plane penetration depth L (0)= L (t)] 2 as a function of the reduced temperature t = T=T c .The experimental data by Bonn et al. 31] are indicated by solid squares.It is obvious from the gure that these data cannot be described by an s-wave superconductor (dotted line).On the other hand, the low temperature results for a d x 2 ?y 2 super- conductor (solid line) seem to agree rather well with the experiment in the region 0 6 t 6 0:2 but for t > 0:2 very pronounced deviations are noted.In particular, as in the previous work 11], the slope of the penetration depth near T c is not so steep as compared with the experiment.In order to develop a theoretical model able to remove this discrepancy between theoretical predictions and the experiment we recall the result well known from functional derivative methods and applied to conventional anisotropic s-wave superconductors, namely, that very low frequency phonons have the same e ect as static impurities and reduce T c , i.e., they are pair breaking 32].Similar considerations apply to a d-wave superconductor in which case it has been shown that the functional derivative of T c with the uctuation spectral density is negative at low frequencies 33].If, at low temperatures, such low-energy excitations are removed because of the feedback e ect superconductivity has on the uctuation spectrum, one would expect that the superconducting gap itself will be larger than it would otherwise be for the associated value of T c .With the increase of temperature the amount of pair breaking increases, because the low frequency part of the uctuation spectrum is restored as the superconducting gap closes up.This should a ect the temperature dependence of the penetration depth.In fact, the experimental data of Bonn et al. 31] can be used to model the temperature dependence of such a low frequency cuto to make the theoretical results follow quantitatively the experiment.If we apply a low frequency cuto !c = 2:1T c0 at t = 0 to the uctuation spectrum and model the temperature dependence of !c closely to that of the superconducting gap, we achieve, after minor adjustments, optimal agreement between theory and experiment (dashed line, gure 2).This uctuation spectrum, which will also be used in all further calculations, is presented in gure 3 where the temperature dependence of the low frequency cuto is emphasized in the insert which shows the low frequency part of the uctuation spectrum on an extended frequency scale for various reduced temperatures.
Obviously, what pushes up L (0)= L (t)] 2 at intermediate temperatures and, correspondingly, increases the slope near T c , is the pair breaking e ect associated with the introduction of lower frequency uctuations as T increases towards T c .

Microwave conductivity
In discussing the microwave conductivity we concentrate on the case = 34:8 GHz ' 0:144 meV studied by Bonn et al. 12] using a clean, twinned, and optimally doped YBCO single crystal.It has already been pointed out by Schachinger et al. 16,17] that the peak in the microwave conductivity of a d x 2 ?y 2 supercon- ductor falls too low in temperature and is not large enough to agree with the experiment if equations ( 1) and ( 2) are solved in a clean limit employing an unmodi ed uctuation spectrum (17) before equation ( 15) is solved to calculate optical conductivity.But applying a low frequency cuto to (17) according to gure 3 moves the microwave peak towards higher temperatures and increases its magnitude considerably bringing the theoretical predictions into better but still not satisfying agreement with the experiment.
Adding impurity scattering in the Born approximation a ects the size and width of the microwave peak with the position of the peak in temperature remaining relatively unchanged.If, instead, the impurity scattering is treated in the unitary limit, the attenuation of the microwave peak is much more pronounced and shifts to higher temperatures 17].While in the best untwinned single crystal samples of YBCO the residual scattering is believed to be rather small, we have to assume for twinned crystals some residual scattering which can be modelled by adding some Born limit impurity scattering to achieve the best possible agreement between theory and experiment.Figure 4 shows the result of such a tting process in which the impurity parameter t + = 0:822 meV has been chosen which reduces the clean limit critical temperature by 5 K to T c = 95 K. (The solid squares in gure 4 correspond to the experimental data reported by Bonn et al. 12] and the solid curve gives our best theoretical result.)The scale in our theoretical data is arbitrary and was tted to agree with the data at the temperature indicated by the arrow.For comparison we also show in gure 4 the theoretical result one would get if the system's critical temperature were lowered to 95 K by adding the resonant impurity scattering (the dotted line).While for the Born scattering excellent agreement is obtained, the unitary impurity scattering results certainly provide an unacceptable description of the data.This establishes the theoretical equivalent of a clean and twinned YBCO single crystal and completely de nes the phenomenological model by the temperature dependence of the low-frequency cuto in the uctuation spectrum responsible for superconductivity and by a certain amount of the Born impurity scattering to compensate for the residual scattering in a clean, twinned sample.If this model is to be used to de ne additional constraints on possible theoretical explanations of high T c superconductivity, predictions of this model are to be compared with other properties of clean and twinned YBCO single crystals which do not need additional theoretical parameters and do not depend linearly on the properties investigated so far.

Electronic thermal conductivity
A candidate, for which experimental data are readily available, is the electronic thermal conductivity.While the normal state electronic thermal conductivity is linearly related to the d.c.conductivity ( = 0) via the Wiedemann-Franz law, there is certainly no linear relation between the microwave conductivity ( 6 = 0) and the electronic thermal conductivity, as a close inspection of equations ( 7) or ( 15) and ( 16) reveals immediately.It has already been pointed out in the introduction that the in-plane electronic thermal conductivity, ab;e (T), of clean, twinned YBCO single crystals develops a very pronounced low temperature peak in its temperature dependence.This feature can now be used to check on the consistency of the model developed so far.Schachinger and Carbotte 18] demonstrated in an extensive study that ab;e (T) does not show in the clean limit of equations ( 1) and (2) a very pronounced peak around t = 0:15 if the unmodi ed uctuation spectrum (17) is applied.This peak is then enhanced by at least one order of magnitude if the temperature dependent low frequency cuto of the uctuation spectrum is included.No signi cant shift of the peak in temperature occurs.Adding impurities attenuates the peak and it becomes shifted towards higher temperatures.In this, again, the Born impurity scattering is less e ective than resonant scattering impurities.
Without the introduction of any new parameters, the theoretical results of calculations within the phenomenological model are scaled to meet the in-plane electronic thermal conductivity measured by Matsukawa et al. 14], as shown in gure 5 (the solid line).(This scaling is still necessary because in order to set the scale theoretically the value N(0)v 2 F of equation ( 16) is needed but it is not known.Thus, a t to one data point, indicated by the arrow in gure 5, is essential.)Without the low frequency cuto the dashed curve is obtained which shows no agreement with the experiment at all.Thus, the in-plane electronic thermal conductivity is an equally sensitive probe of the feedback e ect which superconductivity has on the uctuation spectrum.

Microwave conductivity in systems with impurities
After the phenomenological model has been de ned and veri ed against the inplane electronic thermal conductivity, it is certainly of some interest to test it even further using experimental results reported by Bonn et al. 8] for the microwave conductivity of twinned YBCO single crystals which have been doped with small concentrations of Zn and Ni.In particular, one sample with 0.3% Zn substitution at the Cu-sites showed a critical temperature of 89.5 K (about 4 K down from the clean sample's T c ) and the other sample with 0.71% Ni substitution had T c of 90.47 K (about 3 K down).This alone establishes Zn as a more powerful dopant.The experiments also revealed that both types of impurities had about the same e ect on the peak in the microwave conductivity (a substantial reduction to less than half the height of the clean sample and only a little shift towards higher temperatures).On the other hand, the in-plane penetration depth followed a T 2 law in its low temperature dependence in the Zn doped sample (typical of the resonant impurity scattering) while the Ni doped sample developed a linear low temperature dependence which is typical of the Born scattering 3,10].This is the reason why Ni impurities have been regarded to be of the Born type.
The experimental situation can be simulated theoretically by either increasing t + beyond its clean sample value of 0.822 meV until the observed decrease in T c by about 3.5 K on the average is achieved or by adding nonzero values of ?+ to describe additional resonant scattering impurities.It was pointed out by Schachinger and Carbotte 17] that only resonant scattering impurities have the observed effect on the peak of the microwave conductivity.As Ni does not show the required T 2 law in the low temperature variation of the in-plane penetration depth, these authors concluded that Ni impurities must be at least of intermediate scattering strength and setting c = 0:5 in equation (1) results in excellent agreement between the experimental and theoretical microwave conductivity data even for Ni doped YBCO.This is demonstrated in gure 6a.Schachinger and Carbotte 17] were also able to prove that the T 2 to T law crossover in the low temperature variation of the in-plane penetration depth occurs in the region 0:3 < c < 0:4 and this explains why the theoretical predictions for the low temperature variation of the in-plane penetration depth agree so well with the experiment, as it is shown in gure 6b.All this proves that a phenomenological model which describes the feedback the superconductivity has on a uctuation spectrum belonging to the same system of quasiparticles which condense is capable of a consistent description of the temperature dependence of various properties of optimally doped samples of YBCO with and without additional doping with Zn or Ni impurities.In the next subsection we would like to investigate how the frequency dependence of the optical conductivity is a ected by the model assumptions and how this agrees with the experiment.

Optical conductivity, superconducting state
In gure 7a we compare the optical re ectivity R( ) and the real part of the optical conductivity 1 ( ) with the data reported by Wang et al. 34] for an untwinned, optimally doped, clean YBCO single crystal (T = 8 K) and with the data measured by Homes et al. 30] for a similar sample (T = 12 K).These authors also report data for a twinned, optimally doped, clean YBCO single crystal.Theoretical results are presented for the theoretical equivalent of the clean sample without (the dotted line) and with (the solid line) a low frequency cuto !c = 2:1T c0 at zero temperature.We also used " 0 = 3:5, !p = 1:6 eV, and T = 9:5 K. Obviously, the theoretical data for a system with the low frequency cuto in the uctuation spectrum are in better agreement with the experiment.It is also interesting to note that the best agreement can be found for the Ejja data of untwinned samples (the solid triangles and the solid down-triangles).This is not surprising as the light polarized along the a-axis probes just the CuO 2 planes and the theory presented here is developed to describe superconductivity in two-dimensional copper-oxygen planes.The worst agreement between theory and experiment is found for the twinned sample (the solid squares).This shows that our theoretical equivalent of the twinned clean sample is not su cient to describe completely the in uence the background scattering has on the optical conductivity at higher frequencies.
A similar comparison can be found in gure 7b for the real part of the optical conductivity 1 ( ) as a function of frequency .The theoretical results for the system with the low frequency cuto in the uctuation spectrum have been scaled to t the data point indicated by the arrow.Using a slightly di erent scaling would have provided an almost equally good agreement with the data reported by Homes et al. (the solid down-triangles).The same scaling was then applied to the system with the full uctuation spectrum.The main feature of these results is that the low frequency cuto in the uctuation spectrum suppresses the real part of the optical conductivity to higher frequencies, thus indicating a bigger gap than one would nd without such a low frequency cuto .This is in very good agreement with the Ejja data for the untwinned sample in the region of 200 < < 1000 wave numbers which is quite remarkable.Again, no agreement at all is found for the twinned sample developing a frequency dependence of 1 ( ) which would correspond to a much higher impurity content as it is required by the critical temperature.

Conclusion
Experimental evidence is very much in favour of a superconducting order parameter which is predominantly of a d-wave symmetry.In orthorhombic systems a subdominant component of a s-wave symmetry can be expected.A mechanism which is responsible for the superconductivity in oxides and which also results in a d-wave symmetry of the superconducting order parameter is most likely to be electronic in origin (no phonons).
In this paper a phenomenological model developed by Schachinger et al. 16{18] has been discussed and proved to be able to describe consistently the temperature dependence of the in-plane penetration depth, the temperature dependence of the microwave conductivity and of the electronic thermal conductivity of clean, twinned and optimally doped YBCO single crystals.Moreover, calculations performed within the framework of this model revealed that the in uence of impurities of various kinds on the temperature dependence of the microwave conductivity and of the in-plane penetration depth can also be explained consistently.Finally, it was possible to demonstrate in this paper that the model is also capable to predict a frequency dependence of the low temperature optical conductivity which is in very good agreement with the experimental data found for the light polarized along the a-axis of untwinned, clean, optimally doped YBCO single crystals.
These results will now be used to put additional constraints on possible mecha-nisms of superconductivity in oxides: it is not only most likely that the mechanism is electronic in origin leading to a d-wave superconductivity.Moreover, it seems necessary for the uctuation spectrum which causes superconductivity to belong to the superconducting quasiparticle system because it becomes gapped as the superconductivity sets in.This has been demonstrated quite clearly by the phenomenological model discussed here.

Figure 1 .
Figure 1.a) The real part of the normal state optical conductivity 1 ( ) and b) the inverse optical scattering time 1= ab ( ) as a function of frequency .The data points represent experimental data reported by Homes et al. 30] for a clean, twinned, optimally doped YBCO single crystal (solid squares), for a similar sample with 0.75% Ni substitution (solid triangles), and for a sample with 1.4% Ni (solid down-triangles).Theoretical results obtained from equation (11) are shown for the clean sample (solid line), the sample with 0.75% Ni (dashed line), and for the sample with 1.4% Ni (dotted line).The arrows indicate the data points used to scale theory to experiment.The temperature is 100 K.

Figure 2 .
Figure 2. The inverse square of the normalized in-plane London penetration depth L (0)= L (t)] 2 as a function of the reduced temperature t = T=T c .The solid curve is for a d x 2 ?y 2 superconductor and without a low frequency cuto in the uctuation spectrum.The dashed curve is for a low frequency cuto !c = 2:1T c in the uctuation spectrum.The solid squares indicate the data by Bonn et al. 31].Finally, the dotted line gives the results one would get for a classical s-wave superconductor like Pb.

1 (Figure 4 .
Figure 4.The real part of the microwave conductivity, 1 , for the frequency = 34:6 GHz ' 0:144 meV as a function of the reduced temperature T=T c .The solid line represents the theoretical results for a model system having T c of 95 K as a result of the additional Born impurity scattering.The dotted line corresponds to a model system with the additional resonant impurity scattering having the same T c as the former system.In both systems a low frequency cuto of !c = 2:1T c at zero temperature has been applied to the uctuation spectrum I 2 F ( ).The solid squares correspond to experimental data reported by Bonn et al. 8] for a clean, twinned, optimally doped YBCO single crystal.The arrow indicates the data point which has been used to scale theory to experiment.

Figure 5 .
Figure5.The in-plane electronic thermal conductivity ab;e (T ) in units of mW/cm K as a function of the reduced temperature T=T c .The solid curve is the result of model calculations using a temperature dependent low frequency cuto in the uctuation spectrum.The solid squares are the data of Matsukawa et al. 14] and seem to agree well with theory.The scale of the theoretical ab;e (T ) was adjusted to t the point indicated by the arrow.The dotted line is a theoretical calculation without a low frequency cuto on the uctuation spectrum.It disagrees strongly with the data.

1 ( 5 BonnFigure 6 .
Figure 6.a) The real part of the microwave conductivity, 1 , for the frequency of = 34:8 GHz = 0:144 meV; b) the di erence in the London penetration depth L (T ) ?L (1:3K) as a function of the reduced temperature T=T c .The solid line represents the theoretical results for a model system the critical temperature of which has been lowered from 95 K to 91.5 K by the resonant impurity scattering (c = 0).The dashed line corresponds to a model system of the same T c = 91:5 K but with the intermediate strength impurity scattering (c = 0:5).In both systems a low frequency cuto !c = 2:1T c0 at zero temperature had been applied to the uctuation spectrum.The solid squares represent the experimental data for a YBCO sample with 0.31% Zn substitution, while the open triangles describe the experimental data for a YBCO sample with Ni substitution 8].The arrows indicate the data points which have been used to scale theory to experiment.

Figure 7 .
Figure 7. a) The re ectivity R( ) and b) the real part of the optical conductivity 1 ( ) as a function of frequency .Theoretical data are presented for the theoretical equivalent of a clean sample with (solid lines) and without (dotted lines) a low frequency cuto in the uctuation spectrum.The temperature is 9.5 K. Included are the experimental data reported by Wang et al. 34] for a clean, untwinned, optimally doped YBCO single crystal.The sample temperature is 8 K, the solid triangles correspond to the light polarized along the a axis (Ejja) while the open circles are for Ejjb.The other data are from the experiments by Homes et al. 30] and are for a clean, untwinned, optimally doped YBCO single crystal and a sample temperature 12 K.The solid down-triangles correspond to Ejja and the open squares to Ejjb.Finally, data for a clean, twinned, optimally doped YBCO single crystal are also included (solid squares).