Lattice thermal conductivity of YBa2Cu3O7−δ

Physica C 192 (1992) 435-442 North-Holland

Lattice thermal conductivity of YBa2Cu307_6 J . L . C o h n and S.A. W o l f

Naval Research Laboratory, Washington, DC 203 75, USA T . A . Vanderah

Naval Weapons Center, China Lake, CA 90124, USA V . S e l v a m a n i c k a m a n d K. Salama

Texas Centerfor Superconductivity at University of Houston. Houston. TX 77004, USA

Received 20 December 1991

We report a systematic study of the ab-plane thermal conductivity (x) on single crystal and liquid-phase processed (LPP) specimens of YBazCu307_6 ( J < 0.16 ) in the temperature range 10 K - T_< 300 K. From measurements of electrical conductivity on the same specimens and application of the Wiedemann-Franz law we estimate the relative contributions to the heat conduction from the carriers and the lattice. The normal-state phonon scattering mechanisms are quantified by calculations which employ the conventional theory of lattice heat conduction by longitudinal acoustic phonons. Differencesin the magnitude and temperature dependence ofx for the LPP and crystal specimens are accounted for by differences in the relative weight of phonondefect, phonon-carrier, and phonon-phonon scattering. For all specimens phonon-defect scattering predominates throughout most of the temperature range.

1. Introduction

The thermal conductivity, x, in the high-Tc superconductors [ 1 ] is unique in providing information on transport mechanisms in both the normal and superconducting states. The dominance of the phonon heat transport over that o f the carriers and the importance o f phonon-carrier scattering m the cuprates are manifested as a sharp upturn and peak in the ab-plane r for T < T ¢ . For YBa2Cu307_a (YBCO), the most widely studied material, there is no consistent "'signature" behavior for the normal state x measure~ in single crystals; increasing, decreasing and near'~y constant x with decreasing temperatm'e have been observed [ 2 - 6 i . For a material where the lattice conducts most o f the heat a ~c~c1/ T behavior, charaOeristic of phonon-phonon umklapp scatteriag, is typical at high-temperatures. The experimental results suggest that there exists in YBCO a delicate, sample-dependent balance among the various phonon scattering mechanisms which

govern the temperature dependence of x. Hagen et al. [2 ] found that the in-plane x of superconducting and insulating YBCO crystals were nearly the same for T> 92 K, and suggested that the thermal resistivities associated with phonon-carrier and phononphonon scattering were comparable. More recent efforts [5-7] to fit the experimental data to simple models for lattice conduction indicate that phonondefect scattering is also an important factor limiting the heat transport in YBCO. In this paper we present a systematic study of the ab-plane thermal conductivity on single crystal and liquid-phase processed YBa2Cu307_6 (d < 0.16 ), We estimate the electronic and lattice heat conductivities in the normal state using the measured thermal and electrical conductivities and applying the Wiedemann-Franz law. The roles of phonon-phonon, phonon-carrier, and phonon-effect scattering in limPin~ the lattice conduction are quantified from calculations using the BRT theory of lattice conduction in superconductors. Phonon-defect scattering is

0921-4534/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reser~,ed.

436

J.L. Cohn et aL / YBCO lattice thermal conductivity

found to predominate at all temperatures. We demonstrate that the height of the conductivity peak correlates with the phonon-defect scattering strength.

2. Experimental Preparation of the liquid-phase processed (LPP) specimens is described elsewhere [ 8 ]. This material is composed of large, twinned crystalline grains (up to 4 mm dimensions in the ab-plane and 10-20 ~tm thick) which are highly oriented along the c-axis during growth. Y2BaCuOs precipitates, occurring as distinct entities within the grains, occupy approximately 20% of the volume. The two LPP specimens studied (8 × 2 × 1 mm 3), designated LPP 1 and LPP2, had sharp resistive transitions ATe<0.5 K, R = O at 92.5 K, and room-temperature resistivities p=280 and 1000 ttf~ cm, respectively Specimen LPPI was annealed ( 150 h at 450°C in ]Latm flowing oxygen) to vary the oxygen content, subsequently remeasured, and is designated LPPIA. LPPIA had p (295 K) =750 l~fl cm. Three YBa2Cu3OT_a crystals (CR) were grown in zirconia crucibles by a self-decanting flux method as described elsewhere [9 ], and subsequently annealed in oxygen at 450-500°C. CRI was a large crystalline composite ( 3 × 1.5×0.3 mm 3) made up of four, oriented individual grains, bound by CuO flux. This crystal had AT¢~IK, R = 0 at 90K, and p (295 K ) ~ 4 mr2 cm. The latter value clearly reflects the presence of the insulating flux inclusions which are common in vet3' large crystals. CR2 and CR3 were high-quality, mm-sized (80 ~tm-thick) single crystals with p ( 2 9 5 K ) = 2 2 0 - 2 5 0 la~em, ATe <0.5 K, and R = 0 at 92 K. We note that a measurement of both x(T) and p(T) on the same specimens has not been previously reported for YBCO and is crucial to our quantitative analysis. A knowledge of the oxygen content is important to an interpretation of thermal conductivity data because the electronic structure of YBCO is highly sensitive to oxygen content and also because oxygen defects may scatter phonons. The oxygen deficiencies of the specimens (except CR3 ~ are estimated from measurements of the~oelcctri,z power, performed simultaneously with the thermal conductivity and reported previously [10]: 6~0.16 ( C R I ) , ~ 0 . 1 2

(LPPI), ~ 0 . 0 8 (CR2), 6,~0.02 (LPPIA), t~0.01 (LPP2). The oxygen deficiencies of the crystals are estimated independently from anomalies in their magnetization hysteresis loops, features which have recently been correlated with oxygen content in a study of magnetization and X-ray diffraction [ 11 ]. The values determined in this way were in excellent agreement with the thermopower estimates. The magnetization data for CR3 indicate a t~ value comparable to that of CR2. The thermal conductivity was measured using a steady-state technique, employing a differential chromel-constantan thermocouple and small resistive heater glued to the specimen with varnish. The temperature difference during measurement, AT, was typically 0.3 K-I.0 K. Linearity in the AT response was confirmed throughout the temperature range by varying the heater power. The background pressure was maintained at < 10-5 Torr during the measurements. Errors due to heat losses through the leads and by radiation are estimated at less than 2% except for T> 200 K where radiation losses become more significant. At room temperature we estimate an error of = 10%.

3. Results and discussion 3.1. Qualitative observations

The thermal conductivity versus temperature is plotted for all specimens in figs. I and 2. Near room temperature x~ 8-11 W/mK, which agrees with previous measurements on single-crystal YBCO [2-6 ]. The most notable feature in the data is the sharp upturn and peak that occur for T< To. This behavior is widely observed in the cuprates [ 1 ], and arises from an increase in the phonon mean free path due to the reduced scattering of phonons by carriers as the latter condense into superconducting pairs. The relative height of the peak and the temperature at which it occurs vary between samples (see table 1 ), reflecting differences in specimen purity (defect scattering). We return to this issue below. The contribution of the carriers to the heat conduction may be calculated using the WiedemannFranz law (WFL), which states that x ~ / a T = L e , where xe ~s the electronic component of the thermal

J.L. Cohn et aL / YBCO lattice thermal conducttvio, 28

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T(K) Fig. 2. Thermal conductivity vs. temperature for YBCO crystals (CR). conductivity, or is the electrical conductivity, and Le is the electronic Lorenz number. Using the ideal Lorenz number L o = 2 . 4 5 × 1 0 -8 Wfl/K-', appropriate in the elastic scattering limit, yields an upper limit for xe and we find for our specimens at 100 K, 0.05_T¢, xe/x
437

phonon scattering from chain-site oxygen defects (those altered during the anneal) does not play a prominent role in limiting the overall heat conductivity for this specimen. However, this is apparently not generally true |or YBCO. Recent measurements [ ! 2 ] of x in an untwinned YBCO single crystal indicate that the relative height of the conductivity maximum is sensitive to oxygen defects, In view of these results we conclude that, for specimen LPPI, phonon scattering by other defects (e.g. twin boundaries, cation disorder) is more prominent than that due to oxygen defects. In fig. 3 we plot the lattice conductivity, x t = x - ~ , versus temperature, where x~ is determined, as above, from the measured electrical resistivity and use of the WFL. Interestingly, consistently higher tq. values with stronger temperature dependences are observed in the LPP specimens. We anticipate three principal sources of phonon scattering: other phonons, free carriers, and defects. Phonon-carrier and phonondefect scattering generally yield a temperature-independent or weakly decreasing xL with decreasing temperature at high temperatures [ 13 ]. We attribute the observed temperature dependence of r r to phonon-pho~,on scattering. It seems likely that the insulating inclusions present in the LPP specimens and CRI are the source of the more strongly /'-dependent ~q in these samples. Indeed there is evidence from data on mixed-phase polycrystals [ 14 ] that the 211 ("green") phase has such a temperature dependence. The lower values of XL and the weaker T dependence observed in CR2 and CR3 suggest that the single crystals have a higher phonon-defect scattering component than do the other samples. Evidently the boundaries between the 211 precipitates and superconducting phase in the LPP material do not degrade the phonon heat flow so that the total thermal conduction is improved. The overall change in ~CLfrom 300 K down to T¢ is, for all specimens, relatively small (_< 50%) in comparison to that expected for a defect-free insuconduction, all of the YBCO materials appear to be in the complicated regime where all three scattering mechanisms (phonons, carriers, and defects) are significant.

438

J.L. Cohn et aL / YBCO lattice thermal conductivity

Table 1 List of parameters for YBCO crvst.:ls (CR) and liquid-phase processed (LPP) specimens• Tin, is the temperature at which ~cattains its maximum value, xm,~. A~, A~,, and App are pho,on scattering parameters used in fitting the lattice thermal conductivity to cq. ( 1). ,8( 100 K) is the relative weight of phonon--defect scattering as compared to all scattering, calculated at T= 100 K using the fitting parameters (see text for details) Sample

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45 55 39 39 45 43 38

1.40 1.14 1.47 1.47 1.52 1.52 1.84

0.01 0.31 0.30 0.18 0.04 0.08 0.20

1.55× 103 0.28X 10~ 2.80 1.4 1.73 6.0 1.73 3.1 0.85 1.4 0.85 1.4 0.23 !2

60 19 40 40 81 81 170

0.74 0.77 0.63 0.69 0.67 0.67 0.52

ing by defects and the influence of crystalline anisotropy on the phonon-carrier scattering. The lattice heat conductivity is written as

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T(K) Fig. 3. Lattice conductivity (XL) vs. temperature (symbols are the same as m fig. 1 ). ~CLwas calculated by subtracting, from the data in figs. 1 and 2, the electronic component as determined from the measured electrical resistivities and application of the WFL. The solid lines (for specimens CR2, CR3, LPPIA) are normal-state curves calculated from eq. ( 1 ) with parameters listed in table !. The dotted line is a fit to the high-temperature data for specimen LPPIA with a phonon-phonon scattering rate ~cxT2 (see text for details).

3.2. Theoretical analysis of the lattice conductivity We now attempt a more quantitative analysis of the phonon scattering mechanisms using the BRT theory [15 ] for lattice heat conduction in a superconductor. This model assumes heat conduction by longitudinal acoustic pho~ons, limited by carrier scattering. Recently Tewordt and W61khausen [ 16] have generalized the BRT theory to include scatter-

Here v=5000 m / s is the sound velocity [17], O = 4 0 0 K is the Debye temperature [18], x = h w / kaT is the reduced phonon frequency, and r(x, ~, T) is the phonon relaxation time. The scattering of phonons by crystal boundaries, point defects, carriers, and other phonons is represented by r - I ( x , ~, T)=v/d+A~a(xT) 4 +A.~'~T( l - ~2) ~/2g(x ' 7") + AopX2T 4 ,

where d is the crystallite thickness. The function g(x, T) is the ratio of the phonon-carrier scattering rates in the normal and superconducting states as defined by BRT, equal to unity for T> Tc and a universal function of 3/kaT for T < T~. The integration over the angtdar variable ( accounts for the c.D,sta!!ine ano isotropy - it is assumed that carriers emit and absorb only the in-plane component of phonon m o m e n t u m . The difficulty in applying eq. ( 1 ) to the measured ~cdata for T < Tc is that we do not know the behavior of the electronic component in the superconducting state. There are two issues of concern regarding ~cc:

439

J.L. Cohnet aL / YBCOlattice thermalconducttvity

the T-dependence of the normal-state ~¢e for T < T¢ and the form for the decrease of xc due to the onset of superconductivity. Theoretically both of these temperature dependences are sensitive to the type of scattering mechanism which limits the electronic thermal conduction. One approach [ 7 ] is to assume that xc decreases from a constant normal-state value (as predicted from the WFL and linear-T electrical resistivity) according to the form predicted theoretically [ 19 ] for weak coupling superconductors. If, however, electron-phonon scattering is important in determining the t.he..,-m,.~!conduction then the applicability o f the WFL is doubtful for T < To. Indeed the normal-state x~ might then be expected to increase with decreasing T toward a maximum, achieved in simple metals [ 13 ] at T~ 0/10. The applicability of weak-coupling theory to describe the decay of ~ce in the superconducting state is also in doubt. A recent analysis [6 ] o f the slope change in x for YBCO close to Tc suggests that the decrease of x~ is too sharp to be accounted for by weak coupling theory. In view of these uncertainties regarding x, we adopt a conservative approach in fitting our data. The scattering parameters Apd, Ape, App are determined selfconsistently by numerically integrating eq. (2) to fit the XL data for T>T¢ (fig. 3) and the x data for T< 20 K, where it is assumed that xe has decreased sufficiently such that X~,XL. Thus we make no assumptions regarding the behavior of r,'~ at T< T¢. Note that the value of the gap is not adjusted in this procedure because the low-temperature lattice conductivity is independent of J. This is illustrated by the calculated curves for samples CR3 (curves labelled a) and LPP2, shown as solid lines in fig. 4. We may investigate the influence of the WFL approximation on the values of the fitting parameters in the following way. Rather than using the ideal temperature-independent value for the Lorenz number L~ = Lo to calculate x~, as we did above, we employ the temperature-dependent L~ appropriate for a pure metal (inelastic scattering limit) [20]. The .,wuttmE; . . . . 1,:_~ h:e is :~uutx zlum K to yield new tcL data, _..t. . . .a~Jtt:t.t . . . a ~-_as shown for specimen CR3 as curve b in fig. 4. The corresponding fits to eq. (1) for this data set are shown as dashed lines. The two XL data sets, derived using upper- and lower-limit expressions for x~, allow us to estimate the uncertainty in the fitting parameters. The parameters for all specimens are listed

in table 1. We now discuss the individual scattering terms in more detail. The appropriate form for the phonon-defect scattering rate is dictated by the type of defect, In general, the scattering o f long-wave phonons is approximated by a power-law in phonon frequency, with a stronger frequency dependence describing smaller defects. Given the complex crystal structure of the cuprates, we anticipate substantial scattering from defects to originate from point-like disorder such as missing atoms (e.g. oxygen defects) and atomic substitutions (e.g. in the cation layers). Thus we have employed, in our phenomenological phonon scattering rate, a point-defect term [ 13] otto 4. We also anticipate scattering from twin boundaries, perhaps characterized by a rate with a lower power of frequency. However, we have found no substantial improvement in the fits to the data by including additional terms, and thus have considered only the point defect term in order to limit the number of fitting parameters. The phonon-phonon scattering is the most difficult to characterize since its temperature and frequency dependence are sensitive to the specific lattice spectrum. At low temperatures the relaxation rate of a single low-energy ( E < kBT) mode of wave vector q via three-phonon scattering has been established by Herring [21 ] as z - t ( q ) x-qST 5-~, where s is an exponent determined by cD,stal symmetry. For longitudinal acoustic modes in a cubic crystal s=2. The layered structure of the cuprates makes the appropriateness of the cubic crystal form doubtful. In addition, the normal state of YBCO does not correspond to "low temperatures" ( T< O) and thus the assumption of dispersionless phonons is also suspect. From our qualitative observations above regarding the temperature dependence ODCL, it is clear that phonon-phonon scattering plays only a minor role in the normal-state scattering; just enough to yield an increasing •L with decreasing temperature. Thus vv~

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440

J.L. Cohn et aL / YBCO lattice thermal conductivit.v

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Fig. 4. The measured X (open symbols) and XL (closed symbols) for two specimens from figs. 1-3. For CR3, the KLdata labelled (a) and (b) were calculated using upper- and lower-limit estimates for K~,respectively. The solid and dashed lines are rL curves, calculated from eq. ( l ) for different values of a scaled BCS superconducting gap (from bottom to top): J/~4acs = 0, 0.7, 1.0, 1.6.

T< 180 K for the crystals (see solid lines in fig. 3 for specimens CR2, CR3, LPP 1A). Though some of the discrepancy between the data and the theory at high temperatures may be attributed to radiation losses, a change in the frequency dependence of the phonon-phonon relaxation rate with increasing T is likely. Theoretically, in the high temperature limit (T>_O) the relaxation rate for three-phonon scattering [22] should vary as x T 2 for all modes. The dotted line in fig. 2 represents a fit to the high temperature data for specimen LPP1A using this form for the phonon-phonon scattering rate, with other scattering terms the same as for the lowertemperature fit (solid line). It is clear that a simple power-law in phonon frequency and temperature is inadequate for describing the phonon-phonon relaxation rate throughout the normal state regime. In addition to the effects of the cffstalline anisotropy and dispersion in the acoustic modes, a more detailed description of the phonon-phonon scattering should also consider the effects of acoustic-optic mode scattering. Given the large phonon density of states at energies close to the "top" of the acoustic spectrum ( = 10 meV) [23], such scattering could play an important role in determining the phonon relaxation rate at high temperatures. Regarding the phonon-carrier scattering, we find that equally good fits can be achieved with the anisotropic (as in eq. ( ! ) ) or isotropic forms. The best

fits for the anisotropic expression yield values for Ape that are consistently ~. 40% larger than when isotropy is assumed. This is relevant if this parameter is to be used to extract a value for the electronphonon coupling constant [ 7,16 ]. In spite of the ambiguities in choosing a specific form for the scattering rates, we find that the relative weights of the various scattering terms determined from the fits are not very sensitive to their specific frequency dependences. The parameters employed in the fitting procedure (table l ) imply that defects are the most prominent scatterers of phonons in all specimens. At T = 100 K we estimate that phonondefect scattering accounts for more than 50% of the total scattering, As mentioned above, the relative height of the peak in x and its position in temperature vary between specimens and presumably reflect the specimen purity. We should expect the relative height of the peak, as given by Xm~/K(To), to correlate with the phonondefect scattering. To check this hypothesis we calculate, using the fitting parameters, the effective thermal resistivities associated with each of the scattering mechanisms, e.g. Wp~=_1--XL/rL(Ap~=0), x~here KL(Apd=O) is the lattice conductivity calculated from eq. ( 1 ) by omitting the defect scattering term. We then define the quantity fl=--Wpd/ ( W ~ + I4%+ Wpp), which is a measure of the relative importance of defect scattering as compared to

,I.L. Cohn et al. / YBCO lattice thermal conductivity

all scattering. The values of,a( 100 K) for all specimens are listed in table 1 and in fig. 5 we plot Xma~/ x(T~) versus r ( 100 K). Also included in this graph are data points for the three single crystals measured and fitted by Peacor et al. [ 7 ]. We have calculated values of ~ using their fitting parameters. The correlation between Xmax/x(Tc) and p evident in fig. 5 provides a self-consistency check on our analysis. Interestingly, Xmax/x(T~) does not correlate with electrical resistivity (see table 1 ). This is quite apparent from the data for CR2 and CR3 which have for p, but very different values of comparable ",,,~ues "-' xmJx(Tc). This clearly indicates that the phonons are sensitive to different defects than are the carriers. Such a "separation" of the carder and lattice subsystems with regard to defect scattering probably has its origin in the layered structure for which the charge is confined to the CuO2 planes. The lattice heat conduction may largely occur via phonons which are sensitive to the disorder in the cation layers. This poses an interesting problem for future investigation.

441

study of the lattice thermal conductivity of YBCO, focusing our analysis on establishing the relative importance of 'he various scattering mechanisms. We find that defects account for more than 50% of the total phonon scattering near To. A larger and more strongly temperature dependent x is found in liquidphase processed YBCO as compared with singlecrystal YBCO. These differences ar attributed to the presence of insulating inclusions in the LPP material. The relative height of the peak in x for T< Tc is shown to correlate with the degree ofphonon-defect scattering as determined from calculations using the simplified model of lattice heat conduction by longitudinal acoustic phonons. It is found that specimens having very similar values of electrical resistivity can differ substantially in their degree of phonon-defect scattering. This indicates that the carriers and acoustic phonons are sensitive to different sources of disorder, a result which we suggest is a characteristic of the cuprate layered structure.

Acknowledgements 4. Conclusions To summarize, we have conducted a systematic 2.0

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We thank S.D. Peacor foi many fruitful discussions and for sharing calculations prior to publication. We also acknowledge valuable suggestions throughout this work from V.Z. Kresin, M.M. Miller, M.S. Osofsky, M.E. Reeves and R.J. Soulen Jr. One of us (JLC) acknowledges fellowship support from the Office of Naval Technology. Preparation of the LPP specimens at Univ. Houston was supported under prime Grant No. MDA972-88-G0002 from the Defense Advanced Projects Agency and the state of Texas.

1.2

References 1 .0

i

I

0.5

•

•

•

•

i

0.6

. . . .

!

. . . .

0.7

|

•

0.8

#'(lOOK) Rg. 5. Relative height of the thermal conductivity peak, normalized to values at To, plotted vs. the calculated parameter p( 100 K ) (defined in the text), which describes the relative weight of phonon-defect scattering to the total scattering. The horizontal line connects the two data points calculated for CR3, corresponding to p a r a m e t e r sets a and b.

[ 1 ] For a recent review o f thermal condtlctivit) in the cuprates, see C. Uher, J. Supercond. 3 (1990) 337. [2] S.J. Hagen, Z.Z. Wang and N.P. Ong, Phys. Rex. B 40 (1989) 9389. [3]T.T.M. Palstra, B. Batlogg, L.F. Schneemeyer and J.V. Waszcak, Phys. Rev. Lett. 64 (1990) 3090. [4] S.D. Peacor, J.L. Cohn and C. Uher, Phys. Rev. B 43 ( 1991 ) 8721.

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J.L. Cohn et al. / YBCO lattice thermal conductivity

[ 5 ] V. Floremiev, A. Inyushkin, A. Taldenkov, O. Melnikov and A. Bykov, in: Progress in High Temperature Superconductivity, Vol. 25, ed. R. Nicolsky (World Scientific, Teaneck, NJ, 1990) p. 462. [6] J.L. Cohn, T.A. Vanderah and S.A. Wolf, Phys. Rev. B 45 Jan. I (1992). [7] S.D. Peacor, R.A. Richardson, F. Nori and C. Uher, Phys. Rev. B44, Nov. 1 (1991). [8] K. Salama, V. Selvamanickam, L. Gao and K. Sun, Appl. Phys. Lett. 54 (1989) 2352; V. Selvamanickam and K. Salama, Appl. Phys. Lett. 57 (1990) 1575. [9] T.A. Vanderah, C.K. Lowe-Ma, D.E. Bliss, M.W. Decker, M.S. Osofsky, E.F. Skelton and M.M. Miller, J. Cryst. Growth, 1991, to be published. [ 10] J.L. Cohn, S.A. Wolf, V. Selvamanickam and K. Salama, Phys. Rev. Lett. 66 (1990) 1098; J.L Cohn et al., Electronic Structure and Mechanisms for High Temperature Superconductivity, eds. J. Ashkenazi et al. (Plenum, New York, 1991 ). [ 111 M.S. Osofsky, J.L. Cohn, E.F. Skelton, M.M. Miller, R.J. Soulen Jr. and S.A. Wolf, Phys. Rev. B 45, March l ( 1992 ), to be published. [ 12] J.L. Cohn, E.F. Skelton, S.A. Wolf and J.Z. Liu, unpublished. [13] R. Berman, Thermal Conduction in Solids (Oxford, New York, 1976).

[ 14] B.A. Merisov, G. Ya. Khadzhai, M.A. Obolenskii and O.A. Gavrenko, Soy. J. Supercond.: Phys., Chem., Eng. (Russ. ed.) 2 (1989) 19; S.E. Buravoi. K.V. Nefebov, V.A. Samoletov, B.A. Tallerczik and E.V. Kharitonov, ibid., 32 ( 1989); see also ref. [ 11. [ 15 ] J. Bardeen, G. Rickayzen and L. Tewordt, Phys. Rev. 113 (1959) 982. [ 16] L. Tewordt and Th. W61khausen, Solid State Commun. 70 (1989) 839; ibid., 75 (1990) 515. [ 17] See e.g., P.B. Allen, Z. Fisk and A. Migliori, in: Physical Properties of High Temperature Superconductors, vol. I, ed. D.M. Ginsberg (World Scientific, Teaneck, NJ, 1989) p. 213. [18]See, e.g., A. Junod, in: Physical Properties of High Temperature Superconductors, vol. 2, ed. D.M. Ginsberg (World Scientific, Teaneck, N J, 1990) p. 13. [ 19] B.T. Geilikman and V.Z. Kresin, Kinetic and Non-SteadyState Effects in Superconductors (Wiley, New York, ! 974); B.T. Geilikman, M.I. Dushenat and V.R. Chechetkin, Soy. Phys. JETP46 (1977) 1213. [20] J.G. Hust and L.L. Sparks, Nat. Bur. Stand. Tech. Note 634 (1973);see also l~f. [ 13], p. 152. [21 ] C. Herring, Phys. Rev. 95 (1954) 954. [22] P.G. Klemens, Solid State Phys. 7 (1958) !. [23]W. Reichardt, D. Ewert, E. Gering, F. Gompf, L. Pintschovius, B. Renker, G. Collin, A.J. Dianoux and H. Mutka, Physica B 156-157 (1989) 897.