Journal o[ Low Temperature Physics, Iiol, 107, Nos. 5/6, 1997

Energy Gap Symmetry and Thermal Conductivityof YBa2Cu307
Manabu Ikebe, Hiroyuki Fujishiro and Kiminari Nakasato
Faculty of Engineering, Iwate University, Morioka 020, Janan

The characteristic enhancement in the heat conduction below Tc is analyzed
on sintered YBazCu307 from viewpoints of both d-wave and s-wave coupling.
Assuming the existence of a large energy gap Ao (~I.5ABcs), only d-wave
coupling is consistent with experimental observation. It is found that the most
reasonable explana'tion for the enhancement is provided by the weakcoupling phonon conduction model under d-wave energy gap.
PACS number: 74.25.Fy, 74.72.Bk, 74.25.Jb.

1. INTRODUCTION
The symmetry of the Cooper pair coupling in high T c cuprates is one of
the main topics of the current physical interest. The majority of recently
accumulated experimental data seems to support anisotropic dx2.y2 symmetryof the in-plane energy gap A rather than isotropic s-wave symmetry. However,
the results of the tunneling spectroscopy are not so convergent and the con-.
clusion on gap symmetry has not yet been settled. As for the magnitude of A,
many recent reports on the tunneling spectroscopy have suggested the larger
gap values of 2A0/kBTc~'5-'91'2 than the corresponding BCS value
2ABcs/kBTc=3.52.
The thermal conductivity ~: of superconductors has contributions fromboth electrons ( ~ ) and phonons (~phs)" With decreasing temperature, theelectron contribution K=s usually decreases, while the phonon contribution ~:phsincreases. The mechanism which causes these variations is as follows: below T~
an increasing number of electrons condense into the Cooper pairs with zeroentropy, which do not carry heat nor scatter phonons. The number density ofnormal electrons (=quasi-particles) nqr,(T) is determined by the magnitude
and distribution of the energy gap A, and then nqp(T) determines Kes(T)3'4 and467
0022-2291/97/0600-0467512.50/0

9 1997 Plenum Publishing Corporation

468

M. Ikebe, H. F u j i s h i r o and K. Nakasato

Kphs(T)5'6. Thus the gap symmetry, or more precisely, the distribution of thegap A can be discussed by analyzing the temperature dependence of ~:(T) in
the superconducting state. We measured ~:(T) of closely packed sintered
YBa.zCu307 crystals between 15K and 150K and tested the possibility to settle
the gap symmetry in this system. The details of the experimental procedure
have been presented elsewhere7. Merits of the use of a sintered crystal are thatthe electron component K~ is small enough in comparison with the phononcompone~nt ~ph at least in the normal state and that the homogeneity of theoxygen content can be more easily achieved than for single crystals.
2. E X P E R I M E N T A LRESULTS

Figure 1 represents the temperature dependence of the thermal conductivity •(T) and the electrical resistivity p(T) of YBaECU307, which shows asharp superconducting transition at 92K. The 0 vs. T line follows a linear T
dependence in the normal state. By extrapolating the linear T line to the OK
ordinate, the resistivity ratio p(T~)/p(0), which is necessary for the K analyses
in the next section, is estimated to be 9.8.
As can be seen in Fig. 1, K(T) very slightly increases with decreasing temperature down to T~ and then shows the characteristic enhancement below T c.
This type of enhancement directly connected to superconductivity has notbeen observed in the c-direction conductivity ~c of single crystal cuprates.

~,80t . ...
<40""..

Tc=92Ko(m~I/o(O)=ga /_" I-----~.... 1oi0 4 ~,.

I-.

050
Temperature

100(K)

qt"~,150

Fig. 1. p(T) and K(T) vs. T of sintered YBa2Cu307 crystal. By extrapolating p(T) linearly
to OK, the resistivity ratio is estimated to be p(Tc)/p(0)=9.8. The broken line schematically represents hypothetical p(T)~xq'~ curve with P(Tc)/P(0)=9.8 used for Kc estimation(see text).

Energy Gap Symmetry and Thermal Conductivity of YBCO

469

K of polycrystals is an average of the in-plane conductivity Kab and the cdirection K:, which may be given by K=(2/3)Gb+(1/3)K c for the layered
structure. Since many studies have reported Gb>>Kc (Gb,,kr =5), the observed K
in the present study may roughly correspond to (2/3)Kat, of the veryanisotropic system. In the following section, we analyze this enhancement ofK(T) based on the models of s-wave and d-wave energy gap symmetries.

3. DISCUSSION
3.1. Estimation of~:eRecently a variety of experimental results8'9 on high Tr cuprates has
suggested a possibility of drastic suppression of the quasi-particle scattering
rate Zq/~ below T:. Since the electronic thermal conductivity K~ ) in thesuperconducting state is roughly proportional to the product nqp(T)'~qp, theenhancement of K~sbelow T c is possible if the increase of -cqp overcompensates
the decrease of nqp10. We estimate the electron component Kr in thesuperconducting state by use of the formula derived by Kadanoff andMartin4,/ d~ / d Ee2sech2 {21-[e2+([3A)2]1/2} l,otn,
32~:KesKen-oo4zt2e'[{e2+(13A)2}1/2 + at n](1)

where ct is the ratio of the temperature dependent resistance at T e to theresidual impurity resistance, e is the electron energy relative to the chemicalpotential, 13 is the 1/kDT and t is the reduced temperature T,,Tc. The energy gap
A is assumed to be A=A0 for s-wave coupling and A=A0cos2qb for d-wave
coupling with the azimuthal angle d0 in the layer plane. We assumed that theelectrical resistivity 9(T) obeys T" (n-integer) dependence below T~ with theresidual resistivity p(T~)/(a+l) at OK, i.e.,p(T) = p(Tc)(a-~+ a-~-+ltan),

for t~l,(2)where p(T~ is the resistivity value at T:. From the estimated resistivity ratiop(Tc)/p(0) =9.8, a value is determined to be 8.8. Yu et al. 10 proposed n=4 intheir analyses of K(T) data of YBa2Cu307. Since the tunneling spectroscopy
experiments due to Nantoh et al. 2) have reported the maximum energy gap
Ao"5kDT c for YBa.2Cu307, we calculated Eq.(1) for A0=l.5Anc s and n=4 and1. The results are shown for d-wave symmetry in Fig. 2(a) and for s-wave

M. l k e b e , H. F u j i s h i r o and K. N a k a s a t o470s y m m e t r y in Fig. 2(b). As can be seen, no clear enhancement o f ;% occurs
below Tc even for n=4 and d-wave s y m m e t r y because o f the estimated sizable
residual resistivity of the present sample. Thus, if the estimated residual
100v

E(a)

809169d-wave (Ao=1.5Aacs)

E"N0" n60

40/" ' ~ ~ O~tO

0Ic (n=4) / K :20(n=l)Fe-0, i

10080!

Ii50
100
Temperature (K)(b)~aveENE

i0I.--

50(Ao=l.SAecs)_~ 60O"40coO
2OE,-I.--0i0iiI.

50100
Temperature (K)iiiI150Fig. 2. The fitting curves for n v.s~ based on TW-BRT3'5 theory (a) for d-wave symmetryand (b) for s-wave symmetry. With A0=l.5Aacs, the weak coupling model under d-wave
symmetry gives good fitting, while s-wave coupling cannot reproduce tile peak positionof rph" In the present fittings for ICph tile electronic component ~(T) with n=4 is used.
Even if we adopt x:e(T) with n=l, d-wave model again gives a fat better fitting than swave model.

Energy Gap S y m m e t r y and Thermal C o n d u c t i v i t y of YBCO

471

resistivity of this magnitude (p(0)=p(Tc)/9.8) really exists in the sample, theobserved enhancement of • cannot be attributed to the electronic component
Ke 93.2. A n a l y s e s Of Kph
By solving the Boltzman equation on the basis of the relaxation time
approximation, the phonon thermal conductivity is given by 52=3dNR 2 t ' T x3 rOD/I"f X4exor (~2dZ,(3)whereto is the reduced phonon frequency, N(=13) is the number of atoms

composing Y'na2Cu307,R is the gas constant, O D is the Debye temperature, dis the mass density and M is the molar weight. The phonon scattering ratel:phs-1 in the superconducting state is given by the sum of various scattering
centers in the crystal;~phs(x)-l=~b-l+S(Tz)2+p(TX)4+E(TX)g(X,"~y)+UT 3.~exp(-|(4)Here, parameters "%'1, S, P, E and U represent the strength of the phonon
scattering due to grain boundaries, sheet-like faults, point defects, conduction
electrons and other phonons (Umklapp process), respectively, and b is a constant which controls the cut-off frequency of Umklapp process. The functiong(CC, y)="Irphne/~phs e is the phonon scattering ratio due to electrons in thenormal and superconducting state, which depends on the energy gap throughthe parameter y (=A/kBT~J3. As shown in Fig. 2(a), the best fitting curve for
Ao=l.5ABc s was obtained for d-wave coupling with the parameter values
summarized in Table I. If we assume isotropic s-wave coupling with
A0=I.SABcs the peak position of the calculated Kphs shift to a higher
temperature as shown in Fig. 2(b) and agreement with experiment cannot be
achieved irrespective of the choice of the parameter values. It is noteworthy
that the strong coupling model is incompatible with the present observation of~:(T) because then the peak position of K shift toward even higher
temperatures as pointed out by Tewordt and '~tSlkhausen 6.
Finally, we make a rough estimation of the electron-phonon coupling
parameter ~. which is given by 5,2a<t>E71~v

X- - - ,(5)472

M. Ikebe, H. Fujishiro and K. Nakasato

Table I. Fitting parameters for the weak coupling of d-wave symmetry with
Ao=I.5ABr s.
~:b-,(s-,)

S(K-Zs-,)P(K-4sa)E(K-'s -~)U(K-3s-~) OD(K)

5.2x1083.0x106

1.1xl038.5x10 s

3.4x103400

where a is an average lattice constant, <t> the effective hopping matrix
element of electrons and v is the sound velocity. If we take a=4A,<t>=5000K 5 and v=4700m/s (=measured value by us), then ~. is estimated tobe -0.25.
4. S U M M A R Yi) The electronic component Kc of the thermal conductivity does not
explain the observed enhancement below Tc because of the existence of thesizable residual resistivity in the present YBa2CH307sample.
ii) If we assume the existence of a larger maximum energy gap A0---1.5Aacs
as reported by tunneling experiments, only the weak coupling phonon
conduction model under d-wave symmetry is compatible with theexperimental observation. If Ao is of almost the same magnitude as ABcs, thephonon conduction under s-wave coupling may also reproduce theexperimental results.
REFERENCES

1. C. Manabe, M. Oda and M. Ido, Physica C 235-240, 797 (1994).
2. M. Nantoh etal.,J. Appl. Phys. 75,5227 (1994).
3. J. Bardeen, G. Rickayzen and L. Tewordt, Phys. Rev. 113,982 (1959).
4. L.P. Kadanoff and P.C. Martin, Phys. Rev. 124, 670 (1962).
5. L. Tewordt and T. Wolkhausen, Solid State Commun. 70, (1989) 839.
6. L. Tewordt and T. Wolkhausen, Solid State Commupi. 75, (1990) 515.
7. M. Ikebe, H. Fujishiro, T. Naito and K. Noto,J. Phys. Soc. Jpn. 63,3107(1994).
8. D.A.Bonn, P. Dosanjh, R. Liang and W.N. Hardy, Phys. Rev. Lett. 68, 2390
(1992).
9. D.B. Romero et al., Phys. Rev. Lett. 68, 1590 (1992).
10. R.C.Yu, M.B. Salamon, J.P. Lu and W.C.Lee, Phys. Rev. Lett. 69, 1431
(1992).