ELSEVIER Physica B 230-232 (1997) 342-344

Very low-temperature thermal conductivity of UPt3

H. S u d e r o w a,., j . p . B r i s o n b, A.D. H u x l e y a, j . F l o u q u e t a

a D~partement de Recherche Fondamentale sur la Mati~re Condens~e, SPSMS, CEA/Grenoble, 17 rue des Martyrs, 38054 Grenoble Cedex 9, France

bCentre de Recherches sur les Tr~s Basses TempHatures, CNRS, BP 166, 38042 Grenoble-Cedex 9, France

Abstract

We present new measurements of the thermal conductivity of UPt3 down to very low temperatures (16 mK). The implications of these measurements for the order parameter of UPt3 will be discussed in view of new theoretical predictions.

Keywords." Thermal conductivity; UPt3; Superconductivity

UPt3 is today widely believed to be an uncon- ventional superconductor. The existence of several different superconducting phases in the H-P-T phase diagram is experimentally well confirmed, and can only be understood in terms of unconventional super- conductivity, i.e. if additional symmetries (other than gauge symmetry) are broken at the superconducting transition. Much theoretical and experimental work has been done in order to try to find the symmetry of the order parameter (OP) in this compound, but at present this problem is still not settled. Our choice was to map the position and type of the nodes of the superconducting gap (which depend sensitively on the symmetry of the OP) by thermal conductivity measurements. Indeed, earlier work has demonstrated [1-3] that the thermal conductivity x is of electronic origin down to very low temperatures, and thus sensitive to the electronic excitations in the super- conducting state. Compared to the specific heat C, x is a directional probe, thus, a study of x was ex- pected to give much information about the actual gap structure of UPt3. The measurements of Refs. [2, 3] are compatible with the so-called Elg and E2u ir-

* Corresponding author.

reducible representations of the OP. Both lead to a spectrum of thermal excitations with a line node in the basal plane and nodes along the c-axis (UPt3 is hexa- gonal) which vanish linearly (Elg) or quadratically (E2u) to zero with the polar angle 0. Nevertheless, problems in the interpretation related to inelastic scat- tering or Fermi surface effects made it impossible to distinguish between both representations. It became clear that more systematic measurements to much lower temperatures (T,~50mK) were needed. We present here very low-temperature data (T >I 16mK) on high-quality single crystals of UPt3 which give a more stable basis for the correct interpretation of the thermal conductivity. The discussion of our data will be done on the basis of theoretical predictions by Ref. [4]. Our experimental method as well as the samples are described in Ref. [2].

The thermal conductivity in zero field is shown in Fig. 1. No signature of the anomaly observed in the specific heat at 100mK is observed down to 16mK. Thus, this anomaly has to be related with localised excitations which do not contribute to x. The inset shows the anisotropy Kc/K b normalised to 1 at Tc (= 550 inK). The enhanced resolution of our mea- surements permits us to observe a kink at Tc in ro/Xb,

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H~ Suderow et al. I Physica B 230-232 (1997) 342-344 343

a feature not observed in earlier work. In the super- conducting phase, quantitative comparison with the- ory can only be done when inelastic scattering is negligible. At Tc in our sample both inelastic and elas- tic scattering are of equal importance (the resistivity behaves like p = p o + A T 2 with Po ,~AT2). When entering the superconducting state, the inelastic scat- tering rate should diminish more rapidly than in the normal state, as the superconducting gap opens over the Fermi surface. One can then hope, as pointed by the authors of Refs. [5, 6] that a qualitative treatment using the Wiedemann Franz law may be sufficient to account for inelastic scattering. But the importance of Fermi surface effects, shown by Norman et al. [6] greatly complicates the interpretation of the measure- ments at intermediate temperatures. Therefore, we discuss here only our low-temperature results. In- deed, at the lowest temperatures ( T ~ 0 K ) ~: is only sensible to the Fermi surface regions near the nodes of the gap and to impurity-induced effects. The pair- breaking effect of even nonmagnetic impurities leads to a finite density of states at the Fermi level with a corresponding bandwidth 7. Y depends on the scatter- ing phase shift 6 (6 = rt/2 for UPt3 [2]) and on the impurity concentration ni [4]. For 7" ~< 7 Graf et al. [4] predict xj/T = ctj + f l i t 2 giving precise expressions for ~j and flj: - Fo r j = b (heat current in the basal plane) t£ b is the

same for Elg and E2u, as both predict a line node in the basal plane. Moreover, ~b is universal (i.e. dependent of the form of the gap near the node, but independent of hi ) , and fib = ( 7x2k2/6072)~xb.

- For j = c for Elg, ~c oc y and fie = 2.5(flb/%)~c, but for Ezu, ~c (x 0~ b is universal and tic = ( f lb /~b)~c.

As shown in Fig. 2, between 50 and 30mK x/T is proportional to T 2 for the heat current along both axis, but with a zero ( j II b) or a negative ( j II c) zero temperature extrapolation. It is only below 30mK that x/T begins to saturate towards a KIT = const. behaviour, and that we expect the predicted laws to be valid• First, we focus on the thermal conductiv- ity in the basal plane. The fit (Fig. 2) gives ~b = 0.18 m W / K 2 cm (of the same order of magnitude as ~b ~ 1 mW/(K 2 cm) estimated by [4]) and flb/O~b = 4.2 x 103/K 2, therefore 7 ~ 17mK. This low value means that our sample is too pure to observe clearly the properties of the impurity induced band. Assuming that the impurity concentration ni is the same for both

80

60

40

20

' ' ' I ' ' ' I ' ' ' I ' ' ' I ' ' ' 1.01 . . . . . . . .

1 0 % o o o o °

~s'~ ~ ° O ° ° k Oo 1 / . ~ ~.0.99~ T °%t e% 0 . 9 8 Tc

, ... o9, ....... I • e 0 . 4 0 . 5 0 . 6 0.

• e T ( K ) " e g o °

8 "'... j / /c

, I , , , I , , , I , I i ° l °o / 0 01o Oi o

0 0.2 0.4 0.6 0.8 T (K)

Fig. 1. The thermal conductivity of UPt 3 below 1 K for the heat current along the b and along the c-axis. The inset shows the anisotropy rc/Xb normalised to 1 at Tc.

3

2.5

2

1.5

1

0.5

0

C a x i s E a s ~ ' ' " o o o o

• o • • o

c axis E ~ ~ " o 2u • o o o

, , e ,

• jllc

j / l b o

o o

i i i i I i i i i I l l l L I i i i i

0 0.5 1 1.5 2

T 2 (10 .3 K 2)

Fig. 2. The very low temperature thermal conductivity of UPt 3 (between 34 and 16rnK) as a function of T 2, with the fits based on the predictions of Graf et al. The inset shows x/T between 50 and 16mK. Below 30 inK, the properties of the impurity induced band of quasiparticles are observed (see text).

measured crystals, and having fixed % and flb/O~b only • c can be varied as a free parameter in order to repro- duce the measured curve for j [I c [4]. For E2u, the best agreement is obtained with ~c = 0.2 mW/K 2 cm, for Elg with ~c=0.11mW/K2cm. As shown in Fig. 2, Elg reproduces better our data, but only in a small temperature range (16mK~<T~<25mK). Nevertheless, the assumption that 7 is the same for

344 H. Suderow et al./Physica B 230-232 (1997) 342-344

both measured samples may not be justified. Even if both samples come from the same mother crys- tal and had exactly the same heat treatment, differ- ences in the impurity (or defect) concentration can never be excluded. Indeed, the width of the super- conducting transition of our samples measured by resistivity is 17mK for j II b, but only 8mK for j II c. Our data could then be explained within E2u if the impurity concentration ni 0( 72 would be 2.5 times lower for j II c than for j II b [4]. One way of solv- ing this problem is to measure less pure samples, again for both axis and down to the lowest possible temperatures.

One important point not understood at present is the discrepancy between the value ofv estimated from normal phase data (with the impurity scattering rate Fo/Tc ~ 5 x 10 -2 we obtain ~ ~ 50mK; see Ref. [4] for the formulas), which is 3 times larger than the one found from our low-temperature fit. We note also that x was calculated by Norman et al. [6] taking many different possibilities of describing the complicated Fermi surface. The experimentally observed (see in- set Fig. 1 ) temperature independent Kc/l¢ b for T < Tc is only reproduced within one possible description of the Fermi surface (see Ref. [6]) and E2u, although no physical background supports this way of describing the Fermi surface.

Our new very low-temperature measurements pro- vide a first test of the predictions of Graf et al. [4] on the simple regime where x is governed by the impu- rity induced band of quasiparticles. Measurements on less pure crystals in the same temperature range should be a further test of these strong theoretical predictions (universal values of x/T as T ~ 0 K) and may lead to an unambiguous distinction between the Elg and Ezu models.

We would like to thank P. Wrlfle and P.J. Hirsehfeld for very useful discussions.

References

[1] K. Behnia et al., J. Low Temp. Phys. 84 (1991) 261. [2] A.D. Huxley et al., Phys. Lett. A 209 (1995) 365-372 and

references therein; H. Suderow et al., Physica B 223&224 (1996) 47-49.

[3] B. Lussier et al., Phys. Rev. Lett. 73 (1994) 3294; B. Lussier et al., Phys. Rev. B 53 (1996) 5145.

[4] M.J. Graf, S.K. Yip and J.A. Sauls, J. Low Temp. Phys. 102 (1996); M.J. Graf et al., Phys. Rev. B 53 (1996) 15 147.

[5] A. Fledderjohann et al., Solid State Commun. 94 (1995) 163. [6] M.R. Norman and P.J. Hirschfeld, Phys. Rev. B 53 (1996)

5706. [7] Recently Yu.S. Barash et al., JETP Lett. 63 (1996) 296, also

discussed x in UPt3.