M. Ikebe et al.: Thermal Transport in 90 K- and 60 K-Phase YBa2 Cu3 O7ÿd High-Tc Oxides

413

phys. stat. sol. (b) 209, 413 (1998)
Subject classification: 74.25.Fy; 74.25.Jb; 74.72.Bk; S10.15

Thermal Transport in 90 K- and 60 K-Phase
YBa2Cu3O7ÿd High-Tc Oxides
M. Ikebe (a), H. Fujishiro (a), K. Nakasato1  (a), T. Mikami (a), T. Naito (b),and T. Fukase (b)(a) Department of Materials Science and Technology, Faculty of Engineering,
Iwate University, 4-3-5 Ueda, Morioka 020-8551, Japane-mail address: ikebe@iwate-u.ac.jp; Fax: +81-19-621-6373, Tel.: +81-19-621-6362(b) Institute for Materials Research, Tohoku University, 2-1-1 Katahira,Sendai 980-8577, Japan(Received April 9, 1998; in revised form July 20, 1998)
The thermal diffusivity a and conductivity k of YBa2 Cu3 O7ÿd sintered crystals with oxygen deficiencies d  0:10 (90 K-phase) and d  0:35 (60 K-phase) have been quasi-simultaneously measured between 15 and 170 K. The phonon and electron components of a and k are separated andanalyzed from viewpoints of both the d- and s-wave superconducting energy gap models. It isfound that the enhancement of k below Tc, which is commonly observed for 90 K- and 60 K-phase
YBa2 Cu3 O7ÿd , is consistent only with the phonon origin model under d-wave energy gap anisotropy with the maximum gap 2Dmax =kB Tc  5. The phonon scattering mechanisms are syntheticallyanalyzed for both 90 K- and 60 K-phase YBa2 Cu3 O7ÿd and the oxygen vacancy is found to be oneof the dominant phonon scattering centers in this system.

1. Introduction
The thermal conduction is an important transport phenomenon in solids ranking with theelectrical conduction. From the viewpoint of material physics, the thermal conduction isoften more versatile than the electrical one because the former provides us with valuable
information not only of conductors but also of insulators and even of superconductors below Tc. Thermal conductivity measurement, however, is far less popular than electricalresistivity measurement. It is desirable to accumulate the data of the thermal conductivityof various materials and to establish a standard method to analyze the thermal conductivity.
Recently, we have developed a quasi-simultaneous measuring system of the thermal
diffusivity a and thermal conductivity k under identical experimental set-ups [1, 2]. Because the diffusivity a is given by k=C, i.e., k divided by the specific heat C, we can `in situ'
determine the specific heat by measuring both a and k. To monitor the correct value of
the Debye temperature QD of the sample from the data of the specific heat enhances thereliability of analyses on the phonon thermal conductivity. In this paper we apply the aand k simultaneously measuring method to 90 K- and 60 K-phase YBa2 Cu3 O7ÿd (YBCO)
high-Tc superconductors and develop synthetic and self-consistent analyses for the thermal conductivity and diffusivity.
1 Present address: Shibaura Engineering Works Co. Ltd., 6-25-22 Sagamigaoka, Zama 228,
Japan.

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M. Ikebe, H. Fujishiro, K. Nakasato, T. Mikami, T. Naito, and T. Fukase

In the superconducting state, both the phonon component kph T and electron component ke T of the thermal conductivity are determined intrinsically by the quasi-particle density nqp T. Then nqp T is determined by the magnitude and the distribution of
the energy gap D T. Accordingly, the thermal conductivity can become a useful tool todetermine the gap symmetry, or more precisely, the distribution of D T in a superconductor. We can expect to obtain useful information on the energy gap symmetry fromthe analyses of the thermal conductivity and diffusivity.

2. Experimental
YBa2 Cu3 O7ÿd polycrystals were prepared by a standard solid state reaction method.
The stoichiometric mixtures of Y2 O3, BaCO3 and CuO powders were calcined at
910  C for 24 h in air. They were pulverized, pressed into pellets, sintered at 955  C for
30 h in flowing oxygen and then slowly cooled to room temperature in about 24 h.
These samples were named as 90 K-YBCO. The 60 K-phase (60 K-YBCO) and nonsuperconductive samples were obtained by quenching the 90 K-YBCO into liquid nitrogen from 600 and 900  C, respectively. The amount of the oxygen deficiency d wasdetermined to be d  0:10  0:01 for 90 K-YBCO, 0:35  0:01 for 60 K-YBCO and
0:75  0:01 for non-superconductive YBCO by iodometric titration. The thermal conductivity was measured by a standard continuous heat flow method and the diffusivity
was measured by an arbitrary heating method between 15 and 170 K. A Gifford-McMahon cycle helium refrigerator was used as a cryostat and AuFe(0.07 at%)±chromel thermocouples were used as thermometers. The measurement of the heat transport was
performed automatically and details of the measuring system were described elsewhere[1, 2]. The phonon velocities in the present specimens were measured by a pulse superposition method using  7 MHz longitudinal and transverse ultrasound. The densities ofthe samples were determined by measuring the volume and weight, which were confirmed to be more than 90% of theoretical values. The grain sizes of the specimens
were observed by use of a scanning electron microscope (SEM).

3. Experimental Results
Fig. 1 shows the temperature dependence of the electrical resistivity r T of the present
90 K-, 60 K- and non-superconductive YBCO samples. Both 90 K- and 60 K-YBCO show
sharp superconducting transitions at Tc  92 K and 62 K. The absolute values of r T ofthese samples are pretty small for sintered materials. But in comparison with good single
crystals, the resistivity is large due to sizable residual resistivity r 0. In this sense the present sintered samples may be regarded as `dirty' YBa2 Cu3 O7ÿd . The r T curve of 90 KYBCO follows the linear T law, while r T of 60 K-YBCO below about 200 K shows somewhat steeper reduction than T linear. These characteristics of r T are similar to those of
single crystals [3, 4]. On the other hand, r T of non-superconductive YBCO shows semiconductive characteristics and r T increases with decreasing temperature below 170 K.
Fig. 2 shows the temperature dependence of the thermal conductivity k T of thesame samples as in Fig. 1. With decreasing temperature down to Tc, k T of 90 KYBCO slightly increases and then shows a characteristics enhancement related to theonset of superconductivity. k T of 60 K-YBCO follows a qualitatively similar curve asthat of 90 K-YBCO. The characteristic enhancement below Tc is, however, more con-

Thermal Transport in 90 K- and 60 K-Phase YBa2 Cu3 O7ÿd High-Tc Oxides

415

Fig. 1. r T vs. T for 90 K-, 60 K- and non-superconductive YBCO. The residual resistivity r 0 is obtained by linearly extrapolating r to 0 K. The resistivity ratio r T=r 0 is estimated to be 2.7 for 90 KYBCO and 3.6 for 60 K-YBCO. r T of non-superconductive YBCO is also shown for comparison

spicuous for the 90 K-sample than for the 60 K-sample. k T takes a maximum at
Tmax  45 K for 90 K-YBCO and at Tmax  37 K for 60 K-YBCO. The absolute valuesof k T of 60 K-YBCO are smaller than those of 90 K-YBCO over the entire measured
temperature range. k T of the non-superconductive sample is the smallest and de-Fig. 2. k T vs. T for 90 K-, 60 K- and non-superconductive YBCO. k T takes a maximum at
Tmax  45 K for 90 K-YBCO and at Tmax  37 K for 60 K-YBCO. The solid lines represent thetheoretical curves calculated by use of the parameter values in Table 1. d-wave symmetry is assumed for 90 K- and 60 K-YBCO (see text)

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M. Ikebe, H. Fujishiro, K. Nakasato, T. Mikami, T. Naito, and T. Fukase
Ta b l e 1
Fitting parameters for the weak coupling of d- and s-wave symmetry with
Dmax  D0  1:5DBCS
tÿ1b(108 sÿ1 )s
PEU
QD (K)(106 Kÿ2 sÿ1 ) (103 Kÿ4 sÿ1 ) (108 Kÿ1 sÿ1 ) (103 Kÿ3 sÿ1 )ld-wave90 K-YBCO
60 K-YBCO

4.7
4.9

3.2
3.3

0.23
1.2

12
4.3

3.6
3.4

400
400

0.42
0.16

s-wave90 K-YBCO
60 K-YBCO

4.6
4.8

4.0
5.0

0.7
1.2

4.7
3.5

2.4
2.7

400
400

0.16
0.13

non-super
-YBCO

11

5.6

6.4

0

3.0

400

0

creases monotonically with decreasing temperature. The temperature dependence ofk T of the non-superconductive sample is very small above  40 K.
Fig. 3 shows the thermal diffusivity a T of 90 K- and 60 K-YBCO as a function oftemperature. a T of 90 K-YBCO is larger than that of 60 K-YBCO and increases withdecreasing temperature. It is noticed that a increases more rapidly below Tc than above
Tc . The temperature dependence and the absolute values of a T of 90 K-YBCO areconsistent with other reports [5, 6]. As for 60 K-YBCO, there are no available otherdata on a T to compare as far as we know.
Since we measured both the thermal conductivity k and diffusivity a, the specific heatcan be determined by use of the relation, C  k=a. Fig. 4 shows the estimated specificheat per mole (CM ) together with typical curves of the Debye specific heat. Neglecting

Fig. 3. The thermal diffusivity a vs. T for 90 K- and 60 K-YBCO. The solid lines represent thetheoretical curves calculated by use of the parameter values in Table 1. d-wave symmetry is assumed (see text)

Thermal Transport in 90 K- and 60 K-Phase YBa2 Cu3 O7ÿd High-Tc Oxides

417

Fig. 4. The specific heat C determined from the relation C  k=a. C vs. T curves of both 90 Kand 60 K-YBCO are consistent with Debye specific heat with QD  400 K

the electronic contribution Ce to the total specific heat C in this rather low carrier
density cuprate system and in this rather high temperature range, the Debye temperature QD of each sample is determined to be 400 K for both 90 K- and 60 K-YBCO
from the best fitting Debye curve.
Fig. 5 shows the temperature dependence of the longitudinal sound velocity vLs of thepresent 90 K-, 60 K- and non-superconductive YBCO. The sound velocities of respec-

Fig. 5. Velocity vLs T of longitudinal ultrasound ( 7 MHz) of 90 K-, 60 K- and non-superconductive YBCO

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M. Ikebe, H. Fujishiro, K. Nakasato, T. Mikami, T. Naito, and T. Fukase

tive samples are roughly the same and no systematic dependence on oxygen deficiency
d is noticeable. This result is consistent with the specific heat data in Fig. 4, where nodifference of QD is noted between 90 K- and 60 K-YBCO.

4. Discussion
Similarly to conventional metal and alloy superconductors, both electrons and phonons
contribute to the heat conduction also in high-Tc oxides. With decreasing temperaturein the superconducting state, more and more electrons ( quasi particles) condense into
the Cooper pairs with zero entropy, which do not carry heat nor scatter phonons. Thenthe number density of quasi particles nqp T is a key parameter to determine both theelectron thermal conductivity kes and the phonon thermal conductivity kphs of a superconductor. In the following sections, we make synthetic analyses on the temperaturedependence of kes T and kphs T and try to get information on the energy gap symmetry, which is one of the main topics of current physical interest. In the present analyses
we presume the maximum energy gap Dmax to be larger than the BCS gap DBCS
( 1.76kB Tc ) because the majority of tunneling [7 to 9] and photo-emission experiments[10 to 12] etc. have supported the larger energy gap values of 2Dmax =kB Tc  5 to 9 ofhigh-Tc cuprates.
4.1 Estimation of the electron thermal conductivity kes below T c
In conventional superconductors in which the electron scattering by impurities is dominant, the electron thermal conductivity kes T rapidly decreases relatively to the corresponding normal state component ken T because of rapid reduction of nqp T withdecreasing temperature. For cuprate superconductors, however, a variety of experimental studies have suggested a drastic suppression [12, 13] of the quasi-particle scattering
ÿ1rate tÿ1
qp in comparison to the normal state electron scattering rate t e . Since the electron thermal conductivity is roughly proportional to the product nqp T tqp T, the enhancement of kes below Tc is possible if the increase of the quasi-particle scattering
time tqp T overcompensates the decrease of nqp T [14 to 17]. Here we estimate kes T
in the superconducting state by use of the following formula derived by Kadanoff andMartin [18]:
2p
1


kes
3
1 2
1  at n
2 1=2
2
2
e

dj
de
e
sech

bD

;
1
e
4p2
2
ken
 at n2 1=2002e  bD 
where e is the electron energy relative to the chemical potential, b is 1=kB T and t is the
reduced temperature T=Tc. ken is equal to p2 =3 ne =m kB Tte (we assume the Wiedemann-Franz law), m and ne being the mass and the density of the electron, respectively.
The energy gap D is assumed to be D  D0 for s-wave coupling, and D  Dmax cos 2j ford-wave coupling, j being the azimuthal angle in the two-dimensional circular Fermi surface. We assume that below Tc the electrical resistivity (i.e., quasi-particle resistivity) follows t n temperature dependence with the residual resistivity equal to r Tc = a  1 at 0 K,


1
a
n
r T  r Tc 

t ;
2
a1
a1

Thermal Transport in 90 K- and 60 K-Phase YBa2 Cu3 O7ÿd High-Tc Oxides

419

where r Tc  is the resistivity value at Tc . In eq. (2) the first term corresponds to theresidual resistivity r 0 and the second term stands for the temperature dependent partof the resistivity. The a value in eqs. (1) and (2) is the ratio of the two terms at Tc . For
the electrical resistivity above Tc t > 1, n is set to be unity in eq. (2) to be consistent
with experimental results. From the linear extrapolation of r T lines in Fig. 1, a isestimated to be 1.7 for the 90 K-phase sample and 2.6 for the 60 K-phase sample. Following the results of the tunneling spectroscopy due to Nantoh et al. [7], we take themaximum energy gap value Dmax  D0  1:5DBCS and calculated eq. (1) for various nvalues and for both d- and s-wave symmetries. The calculated curves under d-wave
energy gap are presented in Fig. 6 for 90 K-YBCO and in Fig. 7 for 60 K-YBCO.
According to our calculation, the d-wave model always resulted in a considerablylarger electronic component kes than the s-wave model below Tc. This is reasonablebecause more quasi-particles survive at all temperatures below Tc under anisotropic
d-wave energy gap and carry heat. In Figs. 6 and 7, it is clearly seen that no enhancement in kes can be expected near Tmax (Tmax  45 K for 90 K-YBCO and  37 K for
60 K-YBCO) even if we assume rather drastic t 6 depression of quasi-particle scattering
below Tc and d-wave gap symmetry; kes Tmax  is always smaller than or, at most, almost
equal to ke Tc . This ceiling of kes is caused by the sizable residual resistivity r 0 of our`dirty' YBCO samples. It is plausible that the possible maximum of the quasi-particle
thermal conductivity kes T cannot exceed LT=r 0 (L being the Lorentz number),which is given by the linear line crossing the origin in Figs. 6 and 7. At Tmax , the possible maximum enhancement in kes measured from ke Tc  Dkmax
es  is about 2 mW/cm Kfor 90 K-YBCO and  4 mW/cm K for 60 K-YBCO. Even these values of Dkmax
areesFig. 6. Estimation of electronic thermal conductivity kes for 90 K-YBCO. The energy gap is assumed to have d-wave symmetry, D  1:5DBCS cos 2j. The quasi-particle scattering rate is assumed
to have T n dependence on temperature T. The linear line LT=r 0 (L is the Lorentz number,r 0  1:13 mW cm) represents the possible maximum electronic thermal conductivity. Tmax is thetemperature at which the observed k T takes maximum. Dkmax
is the possible maximum enhancees
ment of kes at Tmax measured from ke Tc 

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M. Ikebe, H. Fujishiro, K. Nakasato, T. Mikami, T. Naito, and T. Fukase

Fig. 7. Estimation of kes for 60 K-YBCO. d-wave energy gap D  1:5DBCS cos 2j is assumed. Theline LT=r 0 r 0  0:14 mW cm) represents the possible maximum electronic thermal conductivity. Tmax is the temperature at which the observed k T takes maximum. Dkmax
is the possibleesmaximum enhancement of kes at Tmax measured from ke Tc 

almost negligibly small compared to the observed enhancement Dkob  30 mW/cm K and 15 mW/cm K for 90 K and 60 K-YBCO. It is also to be noticed that the present estimation of r 0 by the linear extrapolation to 0 K may be the lower limit because r T in thenormal state should be deviate upwards from the T-linear line at low temperatures tocross the 0 K ordinate in the right angle. Thus, k enhancement observed for the present
90 K- and 60 K-YBCO samples cannot be attributed to the electronic component ke .
Although the k enhancement in the present `dirty' YBa2 Cu2 O7ÿd samples cannot be
explained by the electronic component model, it is quite possible that kes also contributes to the k enhancement in `pure' r 0  0 crystals as Yu et al. [14], Krishara et al.
[15] and other research groups have pointed out [16, 17]. Krishara et al. [15] measured
the thermal Hall conductivity and concluded that about a half of the k enhancement
observed in their `pure' 90 K-phase YBa2 Cu3 O7 crystal came from the electronic component. Then the remaining half of the k enhancement in `pure' specimens should be ofphonon origin. The phonon thermal conductivity of the present `dirty' specimens is tobe analyzed in the following section.
4.2 Analyses of the phonon thermal conductivity kph basedon the BRT-TW theory
Assuming the Debye spectrum of phonons, the phonon thermal conductivity is generally given by the following formula [19]:3dn0 Rhv2s i
kph 
2pM



T
QD

3 2p
0

QD =T

dj
0

x4 ex
tph x dx ;ex ÿ 12

3

Thermal Transport in 90 K- and 60 K-Phase YBa2 Cu3 O7ÿd High-Tc Oxides

421

where x is the reduced phonon frequency, n0  13 ÿ d the number of atoms composing YBa2 Cu3 O7ÿd , R the gas constant, d the mass density and M is the molar weight.
Equation (3) contains integral over j to treat the d-wave anisotropy in the superconducting state. The phonon scattering rate tÿ1
ph is assumed to be given by the sum ofcontributions of various scattering centers [19],


QD
2
4
ÿ1
ÿ1
3 2
tph  tb  s Tx  p Tx  E Tx g x; y  UT x exp ÿ:4bTHere, the parameters tÿ1
b , s, p, E and U respresent the phonon scattering strength due
to grain boundaries, sheet-like faults, point defects, conduction electrons and other phonons (umklapp process), respectively, and b  2:1 is a constant which efficiently limitsethe cut-off frequency of the umklapp process. The function g x; y  tephn =Tphs
standsfor the ratio of the phonon scattering times by electrons in the superconducting andnormal states, which was first calculated by Bardeen, Rickayzen and Tewordt (BRT)[20] on the basis of the BCS theory. With decreasing temperature, g x; y promptly
decreases in the superconducting state, depending on the magnitude of the energy gap
D through the parameter y  D=kB T. The decrease of g x; y results in the increase ofkphs in the superconducting state as was observed in many conventional superconductors [21]. We calculated kph on the basis of eqs. (3) and (4) for both the d- and s-wave
energy gap symmetries with Dmax  D0  1:5DBCS . For kes , we assumed t 3 dependence
for r T below Tc, which corresponds to r T limited by the spin fluctuation scattering[17]. The calculated kes was subtracted from experimental k to determine kph. This t 3
assumption does hardly influence on the analyses of kph because of the estimated relatively small electronic component kes . The grain boundary scattering time tb was determined so as to be consistent with the average grain size observed by SEM. The tb
values hardly influence on the results of the kph analyses either, because the boundary
scattering is far weak in comparison to other scattering mechanisms in the temperature
range of the present measurement. For the values of QD we used 400 K determinedfrom the estimated Cph for 90 K- and 60 K-YBCO. The average phonon velocity
hv2s i  f vLs 2  2 vTs 2 g=3 vTs being the velocity of the transverse wave) was calculated
to be 3:5  105 cm/s using the value of vLs in Fig. 5 and the velocity ratio vTs =vLs  0:58
measured for another similarly prepared 90 K phase sample. The best fitting calculation
curves of the 90 K- and 60 K-samples are shown in Fig. 8a and b for d- and s-wave
symmetries, respectively. The parameter values for the fitting curves in Fig. 8a and b aresummarized in Table 1. If we assume the s-wave coupling with D0  1:5DBCS , however,the peak position of calculated kph of 90 K-YBCO appears at a higher temperaturethan that of experiment observations irrespective of the choice of parameter values.
The peak position of kph shifts toward higher temperatures with increasing D0 and thes-wave kph curve in Fig. 8b is the one which shows the kph maximum at the lowest
temperature, keeping reasonable allover fitting. A similar difficulty is noticed also forthe 60 K-sample in s-wave model. Thus, the real best fitting is possible only for thed-wave coupling model. Here it is somewhat puzzling that the strong coupling modelseems to be incompatible with observed k T, because then the peak position of k Tshould again shift toward higher temperatures even for the d-wave model as pointed
out by Tewordt and WoÈlkhausen [22].
According to Tewordt and WoÈlkhausen (TW) [19], the coefficient E which represents
the strength of scattering by electrons is directly proportional to the electron±phonon

422

M. Ikebe, H. Fujishiro, K. Nakasato, T. Mikami, T. Naito, and T. Fukase

Fig. 8. a) Best fitting curves of phonon thermal conductivity kph for 90 K- and 60 K-YBCO. d-wave
energy gap D  1:5DBCS cos 2j is assumed. The values of parameters used for kph calculation aresummarized in Table 1. b) Fitting curves for kph for 90 K- and 60 K-YBCO. s-wave energy gap
D0  1:5DBCS is assumed. The maximum of calculated kph occurs at higher temperatures than thatof experimental observation irrespective of the choice of parameters values

coupling constant l through the relation
l

2chti
E;
pvs

5

where c is the average lattice constant and hti is the effective hopping matrix element
for the two-dimensional electron band. A rough estimation gives l  0:42 for 90 K-

Thermal Transport in 90 K- and 60 K-Phase YBa2 Cu3 O7ÿd High-Tc Oxides423

Ê and hti  5000 K folYBCO and l  0:16 for the 60 K-YBCO with values of c  4 A
lowing TW. The estimated values of l are not so large as to explain such high-Tc values
but the electron±phonon interaction is pretty large in the YBCO system and it is noteworthy that the electron±phonon interaction is larger in 90 K-YBCO than in 60 KYBCO. The contribution of the electron±phonon interaction to superconductivity
should not be neglected, though the main origin for the superconductivity of high-Tc
cuprates must be attributed to some other mechanisms. As we have pointed out inprevious papers [23, 24], the estimated l values are not much affected by substituting
yttrium by other rare-earth elements or by substituting Ba by Sr or Ca atoms.
4.3 Analyses of the thermal diffusivity and the strengthof each phonon scattering mechanism
Since the thermal conductivity is given by the sum of the electron component ke andthe phonon component kph , we also define the respective components of the diffusivity
a as follows:
kph
k
ke
a 
6

 aph  ae :CCCSince we have already estimated ke and kph in the preceding section, thus the separation of aph and ae is straightforward. If we neglect again the electronic contribution to
the specific heat C, aph is given by
QD
 =T

aph 

kph v2sCph
30tph x

QD
 =T
0

x4 ex
dx
ex ÿ 12x4 ex
dx
ex ÿ 12

vs lphv2shtph i :337Equation (7) defines the average phonon relaxation time htph i over the Debye phonon
spectrum. The calculated a T using parameters for d-wave symmetry in Table 1 is presented by solid lines in Fig. 3. The calculated curves reproduce the experimental datapretty well except for the lower temperature region, which supports the self-consistency
and reliability of our present analyses.
Finally, we discuss the strength of each scattering mechanism operative in the present YBa2 Cu3 O7ÿd samples. Figs. 9a and b show the calculated scattering rate htph iÿ1which indicates the contributions of various scattering centers. Each curve in Fig. 9
which represents the strength of respective scattering centers was obtained by calculating ht0ph iÿ1 based on eq. (7) and parameter values in Table 1, assuming that acertain scattering term in eq. (4) was removed, and then by subtracting calculated
ht0ph iÿ1 from the total htph iÿ1 . We see that the point defects pT 4 x4 in eq. (4)) arestrong phonon scatters in a very wide temperature range except for very low temperatures. The point defect scattering is stronger in 60 K-YBCO because of abundant
oxygen vacancies. Electrons also strongly scatter phonons above Tc in 90 K- and
60 K-YBCO but become ineffective at low temperatures in the superconducting state.
Phonon±phonon umklapp scattering becomes important at high temperatures above
100 K. The sheet-like faults (sT 2 x2 due to stacking faults etc.) become efficient scattering centers below about 30 K. The grain boundary scattering is always very small
in the temperature range above 15 K.

424

M. Ikebe, H. Fujishiro, K. Nakasato, T. Mikami, T. Naito, and T. Fukase

Fig. 9. a) Calculated phonon scattering rate htph iÿ1 for 90 K-YBCO and contributions of each scattering mechanism. The contributions of each scattering mechanism is estimated assuming that thevery scattering mechanism is completely removed (see text). Open circles are the experimental
points corresponding to the observed thermal conductivity k T. d-wave symmetry for the energy
gap is assumed and the parameter values of Table 1 are used. b) Calculated phonon scattering rate
htph iÿ1 for 60 K-YBCO and contributions of each scattering mechanism. Open circles are the experimental points corresponding to k T. d-wave symmetry is assumed

5. Summary
We have measured simultaneously the thermal conductivity k and the thermal diffusivity a of YBa2 Cu3 O6:90 (90 K-phase) with Tc  92 K and YBa2 Cu3 O6:65 (60 K-phase)Thermal Transport in 90 K- and 60 K-Phase YBa2 Cu3 O7ÿd High-Tc Oxides

425with Tc  62 K. k and a have been systematically analyzed. Important conclusions are
summarized in the following.
(i) The electronic heat conduction does hardly contribute to the characteristic enhancement of k below Tc in sintered `dirty' 90 K- and 60 K-YBa2 Cu3 O7ÿd because of
relatively large residual resistivity r 0 in the present samples.
(ii) The observed enhancement of k below Tc is almost entirely of phonon origin in
the present specimens.
(iii) Analyses based on the dominant phonon conduction model support the d-wavelike energy gap anisotropy if we assume the maximum energy gap to be equal to or
larger than Dmax  1:5DBCS :(iv) The strong coupling model seems to be incompatible with the behavior of observed k because the effect of the strong coupling is expected to shift peak position of
k to much higher temperatures than experimental results.
(v) Electron±phonon coupling is pretty strong, especially in 90 K-phase YBCO andits contribution to superconductivity should not be neglected.
(vi) Among various scattering mechanisms for phonons, scatterings by electrons andpoint defects are dominant except in the very low temperature region. The phonon
scattering by oxygen vacancies is stronger in YBa2 Cu3 O6:65 (60 K-phase) than in
YBa2 Cu3 O6:90 (90 K-phase).
Acknowledgements The authors wish to thank Dr. M. Kikuchi of Tohoku University for her guidance to the iodometric titration. The contribution to iodometry measurement by Mr. A. Sakamoto of Tohoku University is greatly acknowledged. The
sound velocity measurement and iodometry were performed under the inter-university cooperative research program of the Institute for Materials Research, Tohoku
University.

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