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Physica B 219&220 (1996) 80 82

Lattice thermal conductivity of Nb-based alloysuperconductors and phonon scattering by electrons
M. Ikebe a'*, T. Naito a, H. Fujishiro a, K. Noto a, N. Kobayashi b, K. M o r i c~Faculty of Engineering, Iwate University, 4-3-5, Ueda, Morioka 020, JapanbInstitute .for Materials Research, Tohoku University, Sendai 980, Japan
CFaculty of Engineering, Toyama University, Toyama 930, Japan

Abstract
Quantitative analyses for the thermal conductivity in the superconducting state of Nbl-xTax and N b l - x M o x alloy
systems were performed applying the revised Bardeen, Rickayzen and Tewordt theory. The correlation between Tc andthe strength of the electron-phonon interaction was clearly demonstrated. Addition of Mo to Nb was found to reduce
promptly the electron-phonon interaction, resulting in the reduction of To.

1. Introduction
One useful method to evaluate phonon electron interaction in superconductors is to analyze the temperature
variation of the thermal conductivity in the superconducting state. This method is especially suitable for alloysuperconductors, in which the electron transport is limited by impurity scattering and a considerable part of theheat flux is due to phonons. In such a case, the classicaltheory of Bardeen, Rickayzen and Tewordt (BRT) [1] isknown to explain well the temperature variation of thethermal conductivity in the superconducting state, i.e.,the quenching of the electron component due to theformation of the Cooper pairs and the enhancement of
the phonon component due to the quenching of phonon
scattering by electrons. Recently, Tewordt and W61khausen (TW) [2] generalized the BRT theory by taking
account of other phonon scattering centers as well. We
apply the TW procedure to typical classical alloy systems, Nbx xTa~ [3, 4] and N b l - x M o x [5] and clarify* Corresponding author.

how the electron-phonon interaction manifests itself inthe thermal conductivity below Tc.

2. Experimental
N b - T a and Nb Mo alloys were prepared by melting
the constituent metals (with 99.9% purity) in an Ar arc
furnace about ten times to promote homogeneity. The
resultant alloy buttons were finally remelted in an electron beam furnace under a vacuum of 10 -6 mmHg. Thethermal conductivity was measured by a usual continuous heat flow method in He 4 (1.2-4.2K) and He 3
(0.5-1.2 K) cryostats. Carbon resistors were used as thermometers. In order to measure the normal state conductivity, magnetic fields were applied perpendicular to theheat flow direction.

3. Results and discussion
Fig. 1 shows, as a typical example, the thermal conductivity of Nb80Ta2o as a function of the temperature

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SSDI 0 9 2 1 - 4 5 2 6 ( 9 5 ) 0 0 6 5 6 - 7

M. Ikebe et al./Physica B 219&220 (1996) 80 82
40OO°oE~°e20•o°Xo•o °°°©O.o_ o~°,oo00",,''

81
I'''' ;I-''.~._~._~~= 30:' --E ~ 20\''I''''

without
electron scattering
Nba0Ta20./!

Nb40Ta60

e-=oolo_//.#'/

Ix.

e•

t

0,,.t",,,~ ---~,..

2T8i~,¢ ' ~ ",,0I,,,.4-~'v

1~ ~ I

2,,,3,I,,,0,4Temperature

0'

123

Temperature(K)

Fig. 1. The thermal conductivity of NbsoTa20 versus T [3].
Solid and open circles denote the data in the superconducting
and normal states, respectively. Dashed and solid curves indicate the electron and phonon component in the superconducting state.

4(K)Fig. 2. Phonon thermal conductivity of Nbl xTaxin the superconducting state and TW fitting lines.

30'I'I'I.>
for both the superconducting and normal states. Theelectron component of the heat flux in the superconducting state ~,'e~relative to that in the normal state ~C~nwas
calculated by the following BRT formula:+ 3'2(1 + lY)-t}/2F(O),f~0(1 +;z+y)dz

andy(1)A(T)=: kBT(2)

The energy gap A(T) was assumed to obey the standard
BCS theory. By subtracting the electronic contribution
following these equations, the obtained p h o n o n thermal
conductivity in the superconducting state Kgs is given inFig. 2 for Nb~ _~Ta~ and in Fig. 3, for Nb~ xMo~, respectively. Then, Kg, was analyzed using the following
TW formulation:

k,(kB~3T3 r(')~"T x4e x~"g~- 2rc2v \h-~]J0i eT~--1) rph~dx,(3)where OD is the Debye temperature, v the sound velocity
and x ( = ho)/kBT ) is the reduced p h o n o n frequency. Thep h o n o n relaxation time in the superconducting state 'rphs
is expressed asTph~ =~ / b 1 _f_ ~ / d 1 +e-

20~g,%,,/t,'e,, = {2F( - y) + 25, ln(1 + / -~')

F(y) ="OT p 1 -I- T e 1= rb L + DTx + pT4x 4 -I- EgTx.

(4)Here, ~'b is the p h o n o n relaxation time due to boundaries
and D, p, and E refer to the strength of the p h o n o nscattering due to dislocations, point defects and elec-ie-,
oe'O
e'D.

10I Nb75,M02
00

1

2

Temperature

34(K)Fig. 3. Phonon thermal conductivity of Nb~ ~Mo~ in thesuperconducting state and TW fitting lines.

trons, respectively. The p h o n o n scattering ratio (4 byelectrons in the superconducting and normal state wasalso given by BRT. In this analysis, p was assumed to be
zero because the point defect scattering is not effective insuch a low-temperature region below 4 K. The best fitting curves are shown in Figs. 2 and 3. The parameters
used and determined in the fitting are summarized inTable 1. In this table, we notice that the values of E, thestrength of the p h o n o n scattering by electrons ( = quasiparticles in the superconducting state), is larger for samples with higher T~. Comparing NbsoMo2o with
Nb80Ta2o, we also notice much reduced E value ofNb~oMo2o accompanied with the remarkable reductionof T~. In order to show the temperature dependence of82

M. Ikebe et al./Physica B 219&220 (1996) 80-82

Table 1
Characteristic parameters of the Nb~_~Ta~ and Nbl_~Mo~ alloys. For comparison, those of YBa2CuaO7 (90 K phase) andYBa2Cu30~_6 (60 K phase) were also presented [7]Parameters

Nb80Ta20

Nb40Ta60

Nb20Taso

NbsoM020

Nb75M025

YBa2Cu3007(90 K phase)

YBa2Cu307 -~(60 K phase)zff I (s 1)
E(K-I s -1 )D(K-1s -1)
p(K-4s -1)
s(K-2s l)aOD(K)To(K)
v(m/s)

2.8x10 7
1.1 x 101°
1.8xl0700250
7.854238

7.9x10 6
4.3 x 1098.6x10600250
5.523794

5.1x10 6
3.2 x l09
5.3x10600250
4.823572

7.6x10 6
2.6 x 1098.8x10600
264 [6]4.28
4912

2.3x10 6
2.0 x 1091.0x10700266 [6]3.415025

4.8x10 8
2.6 x 1090
1.1 x 103
1.0X 107
300923330

6.1x10 8
2.0 x 1090
6.2x 102
8.0X 106
350623530

a s is the strength of the phonon scattering due to sheet-like faults (Zs-1 = s T :x2).

0.4~)

IJ~~

I'''I',~

NbaoTa2o''~,ched, both the dislocation scattering and boundary scattering are effective on the comparative intensity.
In our previous paper [7], the E values of the oxide
superconductors
YBa2Cu307(90 Kphase)
andYBa2Cu3OT-6 (60 K phase) were estimated following thesame procedure as the present analyses. The E values ofY B C O were about 2 x 10 9 ( K - t S- 1) and these values arerather comparable with the present alloy systems. In viewof the small carrier concentration in the oxide system, thee l e c t r o n - p h o n o n interaction is unexpectedly strong andcannot be neglected, at least, in Yttrium-based oxide
superconductors.

To /Nb4oTaeo
N baoTaso__ _ NbaoMO2oo 0.3

I-. . . . .

/

_/

0.2
re•:

0.1
............

op00,

,

I

2_ , = - ~ . ~ . ~ ,,4Temperature

I,,

6,

I

8,

10(K)
R e f e r e n c e sFig. 4. Calculated average phonon scattering rate (rph)-1 versus T. In this figure, (zph)-1 can be regarded as entirely due toelectron scattering.

the p h o n o n scattering more clearly, the calculated
phonon scattering rate (Zph) -1 versus T is plotted inFig. 4. The calculation of (Zph) was performed as anaverage on the Debye phonon spectrum using the parameters in Table 1. O n the scale of the ordinate in Fig. 4,(Zph) -1 can be regarded as entirely due to electronscattering ((Zph) ~- (Zp~)). At He 3 temperatures(0.5-1.2 K) where electron scattering is completely quen-[1] J. Bardeen, G. Rickayzen and L. Tewordt, Phys. Rev. 113(1959) 982.
[2] L Tewordt and T. W61khausen, Solid State Commun. 70(1989) 839.
[3] N. Kobayashi, K. Noto, M. Ikebe and Y. Muto, J. Low
Temp. Phys. 17 (1974) 575.
[4] M. Ikebe, N. Kobayashi and Y. Muto, J. Phys. Soc. Japan37 (1974) 278.
[5] M. lkebe, N. Kobayashi and Y. Muto, Phys. Lett. 47
A (1974) 227.
[6] K. Mori, unpublished.
[7] H. Fujishiro, M. Ikebe, T. Naito and K. Noto, Physica
C 235-240 (1994) 825.