Observation of a collective mode in superconducting UBe13.

Sound propagation in the heavy-fermion superconductor UBel3 is studied at frequencies from 0. 9 to 2. 4 GHz and temperatures from 0. 01 to 100 K. A peak in the acoustic attenuation is observed below the superconducting transition of 0. 9 K, instead of the rapid drop expected for a BCS super-conductor. Absorption into a collective mode of a nonsinglet, anisotropic superconductor is proposed.

The exotic properties of the heavy-fermion superconductors' (CeCu2Si2, UPt3, and UBe&3) have raised questions concerning the correct description of the su- perconducting ground state of these systems. 2 3 It is a matter of considerable interest to determine whether the pairing interaction is in an L = 1 (or higher) state or whether the L = 0 Bardeen-Cooper-Schrieffer (BCS) state, or some variant, is responsible.Previous experiments with UBet3 and UPt3 have suggested qual- itative deviations from BCS superconductors.
For ex- ample, temperature dependences of the specific heat, thermal conductivity, ultrasonic attenuation, and nu- clear spin-lattice relaxation rate ar unusual in that they seem to obey power laws as T 0.
The present experiments on high-frequency sound propagation in UBet3 show a T temperature-de- pendent absorption coefficient at low temperature.
More significantly, the general features of the absorp- tion near T, are grossly different from the expectations of BCS theory. 9Rather than a rapid decrease below T, reflecting the appearance of the superconducting gap, an enhanced absorption occurs in UBet3 which peaks at or just below T, .' This feature has no precedent in any superconductor.
We interpret the peak as arising from absorption into a collective mode of the super- conducting condensate.It is the analog of similar collective-mode absorptions studied extensively in the triplet superfluid phases of 3He."' 2 These results provide direct evidence for additional degrees of free- dom of the UBe~3 superconducting order parameter and support models based on a nonsinglet ground state.
Sound propagation was studied in a flux-grown sin- gle crystal of UBe~3, 2 & 4 x 1.5 mm, oriented along (001).Longitudinal sound from 0.9 to 2.4 GHz was generated and detected by a sputtered thin film of ZnO.The sound velocity, absorption, and low- frequency ac susceptibility were monitored simultane- ously and continuously as the sample was slowly cooled or heated in a 3He-4He dilution refrigerator.Local regions in the sample could be probed since the 0.5-mm-diam sound beam was much smaller than the crystal dimensions.
Figure 1 shows the attenuation and velocity changes from 0.1 to 1.0 K.The agreement between Av/u and the specific-heat discontinuity AC~confirms that the sound wave is coupled to the heavy-mass fermions responsible for superconductivity.'3  The sound absorption shown in Fig. 1 is anomalous.On the approach to T, from above the absorption starts to increase at the onset of superconductivity as revealed by b, u/v and the susceptibility.The absorp- tion increases to a maximum at the transition midpoint and decreases monotonically as T 0. Experiments were carried out at six frequencies between 0.98 and 2.4 GHz.In all cases the absorption maximum T~o c- curred at the midpoint of the velocity discontinuity.
No systematic shift in T~w ith frequency was detectable to within +10 mK.
The frequency dependence of the absorption relative to the normal state is shown in FIG. 1, Acoustic attenuation and velocity changes in UBei3 below the superconducting transition T, =0.9 K.In contrast to all other superconductors, the attenuation in UBei3 exhibits a peak rather than a sharp decrease below T,.The velocity discontinuity is inversely proportional to the specific-heat jump at T, .The low-frequency ac susceptibility indicates a transition temperture indistinguishable from that given by the velocity.It is nevertheless in agreement with the q/, « 1 limit of the electronic-absorption coefficient'4 (I, is the electron-scattering length extracted from the conductivity) (m Ei)'v shown that m'Et for interacting fermions is unrenor- malized when the energy dependence of the self- energy dominates its momentum dependence.
The existence of an ultrasonic-attenuation peak in a superconductor has not been seen previously.Instead, it is observed that the electronic attenuation decreases below T, in accordance with the predictions of the BCS theory of superconductivity.9'7 In the low-frequency regime, cu « 2A, where b, (T) is the energy gap, quasiparticle scattering of phonons yields ns/nw --2 f(5), (2) where f is the Fermi function, leading to an exponen- tial decrease of n well below T,.
A recent ultrasonic study of the heavy-fermion su- perconductor UPt3 has shown deviations from BCS theory.8 For T & -, ' T"a T attenuation was dis- covered and interpreted in terms of quasiparticle scattering of phonons in an anisotropic gap possessing a line of zeros.No evidence for a peak below T, was found.The attenuation of UBei3 below = -, ' T, is in- consistent with an exponential temperature dependence but is well characterized by a T~d ependence as shown in Fig. 3. Whether the asymptotic T 0 ab- sorption can be attributed solely to quasiparticle scattering, since the additional absorption mechanism responsible for the peak is present, is unclear.Nevertheless, this result is in general agreement with power laws inferred from recent thermal-conductivitys and NMR spin-lattice relaxation measurements. 6 A potential contributor to enhanced attenuation below T, in a superconductor is phonon-induced pair is satisfied.This condition is satisfied only very close to T, ( T, -T ~2 mK) for the acoustic frequencies utilized here.The magnitude of the pair-breaking ab- sorption should be no greater than 10 (a&no), and should drop discontinuously to the BCS value when Eq. ( 3) is no longer satisfied.' This behavior is clearly not observed in Fig. 1.In the presence of an anisotropic gap, pair-breaking contributions should be even less apparent since zeros of the gap allow for Eq.
(3) to be satisfied for some directions in k space, even at T=O.
The most likely candidate for the large acoustic- damping peak below T, is absorption into collective modes.For an anisotropic superconductor, the order parameter will have components corresponding to the additional degrees of freedom associated with oscilla- tions of orbital axes.One can view these modes as ex- cited bound pairs whose excitation energies h Ak lie within the gap.If a particular mode couples to density fluctuations, one can expect absorption of sound when the condition tc0= Q"(T) is satisfied."'9 A recent calculation2 of collective-mode energies in an L =1 polar superfluid has predicted a low-frequency orbital mode which couples to the density.The collective mode occurs at 1.2ho at T= 0 but, more significantly, appears to have an unusually weak temperature dependence near T, .Thus the symmetry and weak temperature-dependent energy of the mode are sug- gestive of our observations near T, .
In He, theory and experiment have shown that the collective-mode frequencies near T, generally scale with the gap, i.e. , Ak= ab, (T), where a is a con- stant.t~T he resonance condition yields a peak occur- ring at temperature T~= [I -(fee/2a'kT, )2] T"with a' -1.The present data do not provide evidence for a frequency dependence of the peak temperature.Two factors may be responsible.First, in contrast to the zero-sound propagation in He, the UBet3 experiments take place in a hydrodynamic, collision-dominated regime.One expects any sharp, resonant structure to disappear.Second, one must take into account the in- homogeneous broadening of the transition and the subsequent averaging over the transition width.Experiments at higher acoustic frequencies should resolve this issue.Inclusion of crystal-field perturba- tions and spin-orbit coupling may be expected to modify collective-mode energies in crystals, 2 3 but such corrections may prove to be minimal in a cubic system such as UBet3.The apparent absence of an at- tenuation peak below T, in hexagonal UPt3 may be in- dicative of the role of crystal-field splittings or aniso- tropic pairing interactions2' in crystals of lower sym- metry.
We have studied the effect of magnetic fields up to 2 T on the absorption.The peak moves to lower tem- perature in general accord with the phase boundary, but, surprisingly, its amplitude increases by approxi- mately 20%.The velocity discontinuity is unchanged, indicating that the transition is still sharp in the rela- tively weak field.In the absence of a theory of the collective-mode energies in a magnetic field, two pos- sibilities for the enhanced absorption seem reasonable.
The first is that preferred alignment of the orbital axes is occurring (H is applied parallel to the sound-wave vector).Additionally, a larger ratio of r0/b, (H) in fin- ite fields should lead to increased absorption.
In conclusion, the appearance of an acoustic- attentuation peak in the superconducting state argues for a new absorption mechanism in heavy-fermion sys- tems.The excitation of a collective mode by sound waves is the most likely explanation for the phenomenon.
We believe that this observation is the most compelling evidence yet offered for non-BCS, anisotropic superconductivity.

Fig. 2 .
The upper data are the attenuation changes at the peak, o. ~-ot&', the lower data represent the attenuation change at about 100 mK, o. 'iv no.Both quantities are consistent with an ~2 law over the frequency range studied and sug- 1985 The American Physical Society 2479 regime.The overall magnitude of the electronic absorption, n&ns, is extremely small.
unrenorrnalized.That is, the prod- uct of the effective mass m' and the deformation po- tential E& possesses a normal metallic value.As dis- cussed elsewhere, ' the sound attenuation in both UBet3 and UPt3 behaves as if m'/m -1.Varma' FIG. 2. Frequency dependence of the attenuation below T,.The open squares represent the attenuation peak amplitude relative to the normal state, o. ~-o.~.The open circles represent the difference between the normal and the superconducting attenuation at the lowest temperature measured ( = 100 mK), nw -no.