Nuclear magnetic resonance and heavy-fermion superconductivity in UBe13 and related systems

Abstract We report the use of the 9Be nuclear magnetic resonance to study heavy-fermion superconductivity in the U1−xThxBe13 alloy system, x = 0 and 0.033. The nuclear spin-lattice relaxation rate 1/T1, which yields information on thermal excitations in the superconducting state, is found to vary more slowly the temperature than expected for a conventional BCS superconductor with a nonzero energy gap Δ. This indicates an enhanced density of excitations for low energies E ⪡ Δ. At intermediate temperatures 1/T1T3≃const. for UBe13, which is consistent with highly anisotropic pairing; the specific temperature dependence suggests lines of zeros of Δ on the Fermi surface. The relaxation data do not agree qualitatively with theories of superconducting pair breaking due to paramagnetic impurities. At low temperatures (T 1 T 1 T is approximately constant in UBe13, with a value which decreases with decreasing applied field. This behavior is probably not due to direct relaxation by paramagnetic impurities, for which the temperature dependence would be in the opposite sense. For x= 0.033 1/T1 varies less rapidly than T3. The additional relaxation may be due to filling in the gap by Th-induced pair breaking. The 9Be spectra give no indication of additional broadening due to vortex-lattice inhomogeneity or (for x = 0033) of magnetic ordering at a second transition temperature Tc2 below the superconducting transition (at Tc1). Our results are ompared to NMR studies of CeCu2Si2, and to other experiments in heavy-fermion superconductors.


Introduction
Cerium-and uranium-based intermetallic compounds have recently been discovered [1][2][3] in which very massive (m~ff > 200me) itinerant electrons, or "heavy fermions", are unambiguously involved in the superconducting state. These heavy-fermion superconductors exhibit extremely enhanced normal-state specific heats at low temperatures, with linear specific-heat coefficients y--= C/T hundreds of times greater than in ordinary transition metals. A clear signature of heavyfermion superconductivity is a correspondingly large discontinuity in the specific heat at the superconducting transition temperature T c, which implies that the heavy fermions themselves participate in the superconducting pairing [4]. The possibility has been raised [5] of unconventional Cooper pairing in these materials, with non-BCS orbital and spin symmetries and strong energygap anisotropy.
Furthermore an interesting "double transition" has been observed in the specific heat of the pseudobinary alloy system U1_xThxBe13 for x of the order of a few per cent [6]. Superconductivity appears below the higher transition temperature Tel, and persists below the second transition temperature To2. T~I decreases with Th concentration, but there is an unusual plateau in To~ (x) in the region where the double transition is observed. No evidence for macroscopic phase separation has been found, and the second transition seems to be accompanied by strong anomalies in the ultrasonic sound velocity [7] and thermal expansion [8].
Nuclear magnetic resonance (NMR) techniques have been used to probe superconductivity since before the appearance of the BCS theory [9]. The utility of NMR lies in its sensitivity to the behavior of local microscopic magnetic fields, both static and dynamic, in condensed matter. We review briefly the kinds of information which can be obtained using NMR which are relevant to the study of heavy-fermion superconductors.
Nuclear spin-lattice relaxation, the process by which a nuclear spin population distribution attains thermal equilibrium which the lattice, is due to coupling between nuclear spins and low-lying thermal excitations. In metals the dominant mechanism for nuclear spin-lattice relaxation is often the so-called Korringa mechanism, in which conduction electrons are spin-flip scattered by the nuclear moments. Paramagnetic impurities in a metal can also couple to the nuclei, and impurityspin fluctuations can be an important relaxation time T~ or, equivalently, the relaxation rate R 1/T 1, is the principal quantity of interest.
NMR spectra reflect the distribution of local static fields, and can be used, for example, to map out the field distribution in the mixed-state Abrikosov vortex lattice [9]. Large NMR frequency shifts and line broadening accompany magnetic ordering transitions [11], and NMR can therefore be used to search for such transitions in superconductors. NMR spectra are split by the electrostatic interaction between nuclear quadrupole moments and crystalline electric field gradients. The latter depend on crystal structure, and are correspondingly altered by structural modifications or phase transitions.
Subsequently 63Cu NQR and NMR in CeCu2Si 2 [13,14] and 9Be NMR in UBel3 [15,16] were investigated. The 29Si resonance is also available in CeCuzSi 2. It is unlikely, however, that a t95pt resonance will be easily observable in UPt3, because the estimated relaxation time T~ at low temperatures is prohibitively short.
This paper reports the results of our NMR studies in the superconducting states of the heavyfermion superconductors U1_xTh~Be~3, x = 0 and 0.033. (Data from the normal state will be discussed only as they affect interpretation of superconducting-state results.) Section 2 describes 9Be spin-lattice relaxation in U~ xTh~Be~3, x = 0 and 0.033, and section 3 gives results obtained from 9Be resonance spectra in these materials. We compare our data to other experimental results in these and other HFS systems in section 4, and section 5 summarizes our conclusions. Some of this work has been reported previously [15,16].

Spin-lattice relaxation in U~_xThxBe~3
NMR experiments have been carried out on two specimens of UBe13 (x = 0). Sample no. 1 was ground to a coarse powder (0.5 mm grain size), and sample no. 2 was in the form of several small crystallites (5 mm size). A small-crystallite sample of U~_xThxBe13, x = 0.033, was also investigated. Superconducting transition temperatures T c were measured using an ac inductance technique, and found to be 0.85 K and 0.75 K for UBeI3 sample nos. 1 and 2, respectively, in the applied field B =/%H ~ 1.56T used for most of the experiments. For the x = 0.033 sample Tc(1.56 T) was found to be 0.55 K. Standard pulsed NMR techniques were used to obtain relaxation times and field-swept spectra. Fig. 1 gives the temperature dependence of the 9Be spin-lattice relaxation time T~ in UBe~3 (sample no. 1), in the form 1/T~T vs. tempera-

1~Tin ~ T, I/TIsT in the superconduting state is proportional to Ttn(T)/T1s(T ) =-R~(T)/R,(T),
where R n is the relaxation rate which would be measured in the normal state at temperature T.
Rs/R n is directly available from theory [9], and 1/TI~T is therefore a convenient quantity for comparison. The salient features of these results are as follows: (1) The increase of 1/T~T just below T c usually observed in conventional superconductors [9] is absent in UBel3. (2) The data follow a power law 1/TlsT oc T 2 between approximately 0.2 K and somewhat below T c. This is more easily seen in fig. 2 for the temperature dependence of Rs/R . at low temperatures. The values p = 4 and 2 are found for the axial and polar L = 1 triplet states, respectively. The latter is consistant with the data of fig.  2. This suggests that anisotropic pairing is a candidate for superconductivity in UBe13. Lowlying excitations would then be associated, more generally than in the p-wave triplet model, with lines of zeros of A on the Fermi surface. Fig G 4a gives the temperature dependence of the 11 In 1/TlsT in indium metal, a conventional superconductor, for comparison with the UBe13 data of fig. 1 [18]. It can be seen that a pronounced maximum is observed just below T c in in-dium, but not in UBe13 over the corresponding temperature range. Similarly, Fig. 4b gives the temperature dependence of 1/T1 T3 for indium metal. Downward curvature is observed which, if present, would also have been seen in UBe13 in spite of the poorer accuracy of the latter data ( fig.  2). Indeed, the ~lSln data can be fit very well [18] to the activated temperature dependence expected for a BCS superconductor with only slight anisotropy of the gap parameter [9], and the reasonable value (A)av/kBT c = 1.80 ---0.05 is obtained [18]. This comparison indicates that UBe13 is a highly anisotropic superconductor compared to conventional materials.
The observation 1/TIsT~-const, below 0.2 K in UBe13 (fig. I) suggests a nonzero Ns(E ) for E = 0, i.e. a small residual gaplessness. Other mechanisms exist for this observation, however, in particular pair breaking by paramagnetic spins (impurities or incompletely-compensated Kondo moments.) The consequences for nuclear spin-lattice relaxation of the effect of pair breaking on excitations in a BCS superconductor have been treated in both the weak-coupling [19] and the strong coupling (Kondo) [20] limits. Pair breaking reduces the maximum value of R JR, just below T~, but theory does not yield the required linear density of states Ns(E ) at low energies. Fig. 5 gives the weak-coupling result for RffR, vs T/T c for several values of the pair-breaking parameter a required to suppress superconductivity completely [T~(c~,) =0]. It can be seen that quantitative agreement is lacking: a power law RffR. ~ T is found only for the value a ~ 0.91a, which just suppresses the gap to zero, because in this case N,(E) oc E '/2 [191.
Paramagnetic spins can also relax nuclei directly; indeed, this is more commonly observed than relaxation by superconducting excitations if the concentration of paramagnetic spins is at all large f" . Here Aoo and T~ are the gap parameter and transition temperature in the undoped host superconductor, and a/acT is the normalized pair-breaking parameter. [9]. However, in this case 1/Tls T would be expected to be highly field dependent at low temperatures, due to saturation of the electronic spins by the field. Saturation decreases 1/T 1 with increasing field [7], which is opposite to the observed effect. We conclude that direct paramagnetic relaxation is not likely to be an important contribution to the observed rate. 9Be relaxation data from the x = 0.033 sample are given in fig. 6 in the form 1~TIT 3 vs. T. For 0.2 K< T< To, I/T~T varies less rapidly than T 2, as can be seen from the increase of 1/T1T 3 with decreasing temperature shown in Fig. 6. This result may indeed signal pair breaking due to Th impurities, which would depress Tc, as observed [21], and also broaden the structure in N~(E). The absence of structure in the neighborhood of either Tel or Tc2 should also be noted ( fig.  6).

9Be resonance spectra in the superconducting state
NMR spectra were obtained from traces of integrated spin-echo intensity as a function of swept field. Spectra for UBe13 sample no. 2 are shown in fig. 7. These are typical quadrupole-split spectra for nuclear spin 3/2, with a central (1/2 <--->-1/2) transition and quadrupole satellites. A curious property of these spectra is the absence of large shifts or broadening AH in the superconducting state; broadening (/z0AH=lmT) can be barely resolved at the lowest temperatures in the spectra of fig. 7. Slight broadening was previously found for the x = 0.033 sample [16], where a shift at T= 0.113 K was also observed. No shift is seen in Fig. 7. It is not clear how much of the previously-observed shift was due to field drift, however, since the field was regulated more closely for the x = 0 spectra than for x = 0.033.
A shift and broadening should arise from the inhomogeneous flux expulsion characteristic of the type-II mixed-state vortex lattice [9]. It is not clear why it sets in only for T< 0.15 K in UBel3. The order of magnitude of the broadening, analyzed as described previously [16], leads to an estimated London penetration depth A L > 2000 A and a corresponding Ginzburg-Landau parameter K > 20 for x = 0 as well as 0.033. It has been remarked [16] that the absence of 9Be shift or broadening at the second transition (temperature Tc2 ) for x = 0.033 is evidence for the absence of magnetic ordering below this temperature. For example, Z7AI NMR and NQR in the Kondo compound CeAI 2 [22], which orders antiferromagnetically at T N = 3.8 K, exhibit broadened spectra (/z 0 AH.-~10mT) below T N. NMR is a direct magnetic probe of local magnetic order, and it would be very hard to see how order with an appreciable magnetic moment could occur without causing NMR broadening. A quantitative estimate of this broadening is hard to make, since indirect (RKKY) couplings, etc., are not known in UBel3. A rough estimate of the dipolar coupling is 0.1 T/tz B [16], which leads to an upper limit of ~IO-2/ZB on the moment per U atom if the low-atom if the low-temperature broadening were magnetic in origin.

63Cu NQR and NMR in CeCu 2 Sie
NQR data show an absent [12] or strongly reduced [13] maximum in 1/T1T below To, as in UBe13 , but the temperature dependence of 1/T 1T is much weaker below ~0.6T~ than in UBeI3. In an applied field of 0.572T, however, the 63Cu relaxation rate in CeCu2Si 2 varies with temperature similarly to the 9Be rate in UBe13 [14]. This strong-field dependence in CeCu2Si 2 is opposite to that found in UBe13 , and is consistent with direct relaxation by paramagnetic impurities [14], possibly "remagnetized" Ce ions near defects. A direct mechanism was not favored in the original report [12], because the nuclear magnetization recovery was observed to be exponential [7]. A simple calculation shows, however, that spindiffusion-limited exponential relaxation would be consistent with the data for impurity concentrations =1000 ppm.

Other techniques
The relative ultrasonic attenuation rate as/a n varies as T 2 at low temperatures in both UPt 3 [17] and UBe 13 [23]. This is consistent with the nuclear spin-lattice relaxation results, except that the 1/TiT~const. behavior seen in NMR in UBe13 below ~0.1 K does not seem to be observed in ultrasonic attenuation. NMR relaxation might therefore be sensing a spin-dependent excitation which does not affect ultrasonic attenuation. Such a difference does not seem to arise from the present state of the theory, however.
The absence of strong NMR anomalies at either transition for the x = 0.033 sample makes it unlikely that the recently discovered giant ultrasonic anomalies at Tc2 [7] can be attributed to an antiferromagnetic transition with an appreciable moment per U atom.
Thermal conductivity measurements in UBe13 [24] are also consistent with the picture which seems to emerge, at least at intermediate temperatures, of superconducting pairing with lines of zeros of the gap parameter.

Conclusions
NMR and data from other nonequilibrium techniques exhibit qualitative differences between heavy-fermion and conventional superconductors. The low-lying excitations are much more numerous in the former, and seem to be given by power-law dependences of the density of states on excitation energy. The power laws are consistent with excitation spectra obtained from 3He-like theories of anisotropic pairing, but the experiments do not constitute proof of such pairing. For example, Kondo-lattice theories with singlet pairing are able to obtain lines of zeros of the gap parameter [25].
Surprisingly, 9Be NMR spectra show no rapid onset of vortex inhomogeneity broadening below T c in UBe13. The same absence of anomalies in the NMR spectra are strong evidence against magnetic ordering at To2 in U1_xThxBe13, x = 0.033. No anomaly was observed in the quadrupole splitting, either, but this only puts an upper bound of ~2% on any change of lattice parameter [16]. This would be an enormous structural change; a weaker one, consistent with the NMR spectra, could occur at the lower transition.