P-WAVE SUPERCONDUCTIVITY IN UBE13

The specific heat in the superconducting state of UBe~3 shows marked deviations from BCS theory and obeys a T rather than an exponential law at low temperatures. A good description is obtained by the assumption of an Anderson-Brinkman-Morel p-wave superconducting state at all temperatures. The value of the spin-fluctuation parameter deduced is large and consistent with the stability of such a state.

Since the discovery that liquid 3He is a p-wave superfluid at very low temperatures' 4 there has been an added impetus to the search for anisotropic superconducting metals, but there have been no clear candidates among conventional superconductors. However, it has been recognized for some time that certain rare-earth and actinide intermetallic compounds starting with CeA13' are electronic Landau Fermi liquids at low temperatures with very strongly enhanced specific heat and susceptibility analogous to the normal state of liquid 3He. The recent discovery of superconductivity in such materials has led naturally to the speculation that the analogy to 3He may be closer for these materials than for conventional superconductors.
The first system, CeCu2Si2, 6 has had materials difficulties7 9 but the new heavy-electron superconductor' UBet3 has proved free of such problems and also has a higher superconducting transition temperature. In this Letter we wish to demonstrate that the analogy to He is very close for UBet3 by analyzing the specific heat in the normal and superconducting states. The specific heat in the superconducting state at low temperatures shows a very clear deviation from the BCS form but a good description can be obtained by the assumption that the superconducting state is an Anderson-Brinkman-Morel (ABM) p-wave superconductor at all temperatures. Further, the value of the spin-fluctuation parameter deduced is large, larger than that attained in liquid 3He at the solidification curve and consistent with an ABM state at all temperatures in the Brinkman-Serene-Anderson (BSA) theory. 4" We propose from this compar-ision between theory and experiment that UBet3 is the first p wave electronic superconductor and also the first case of an electronic superconducting transition driven by an interaction other than the electron-phonon interaction.
To demonstrate the analogy, mentioned above, between the normal-state properties of the electronic subsystem of UBe~3 and liquid He, we plot the experimental specific heats C~o f both UBe~3 and 3He at low temperatures in the form of C~i Tvs Tin  Fig. 1. The data for 3He were taken from Brewer, Daunt, and Sreedhar'2 and Anderson, Reese, and Wheatley. ts It should be noted that the scales for both C~/ Tand Tare different for the two materials.
The effective Fermi temperatures TF' for UBet3 and 3He are = 8 K and a few tenths of a kelvin, respectively, as estimated from the C~(T)/T data shown in Fig. 1. While 3He is superfluid only below 2.7 mK (=10 TF'), UBet3 is superconducting below 0.9 K ( = 10 t TF' ). The specific heat above and below T, is shown in Fig The fact that C, ( T)~T 3 as T 0 rather than exponential suggests that the gap function goes to zero someplace on the Fermi surface and that we have a p.wave (or possibly higher quantum The magnitude of the gap, (6+6)' 2, now vanishes at the points k"=k"=0, k, = +k". In the weakcoupling limit it is straightforward to evaluate the specific heat of the ABM state and the result is shown also in Fig. 3 (identified as w. c.). The jump in specific heat at T, is smaller than BCS by a factor of -, '. At low temperatures the zeros of the gap give rise to a power law leading to the following results in weak coupling2 4: number) superconducting state. Note that alternative origins of nonexponential behavior in C, (T), such as gapless s-wave pairing due to magnetic impurities or multiphase samples with normal regions, lower the value of 5 C/C"( T, ) and lead to C, ( T) -Tas T 0 in contradiction to experiment.
p-wave superconductivity has been extensively studied in He. Two p-wave superconducting states have been observed there. 2 4 One is the Balian-Werthamer (BW) state in which the magnitude of the gap is constant over the Fermi surface. It has thermodynamic properties identical to an s-wave superconductor.
The second is the ABM state observed near solidification with a gap function c, with 5(0) =1. 65kaT, . A comparison with the experimental values of C~(T) for UBet3 shows that while the overall form of the curve is in better agreement than the BCS form, shown also in Fig. 3, there are substantial quantitative disagreements.
The experimental value of (AC/C")T is much larger, not smaller, than BCS while the measured values of C~( T) as T 0 are smaller than Eq. (2).
Both discrepancies suggest that substantial strongcoupling corrections to the theory are necessary. enhancement region in k space. Thus with a single parameter we can determine the specific heat both near T, and at low temperatures through Eqs. (2) and (3). If we wish to calculate the complete curve of C, (T) we will need to specify the form of the spin susceptibility X(q, co) more precisely and so we have chosen to adopt a simpler scheme with only a single free parameter, 5. We use Eqs.
(2) and (3) to determine the C, (T) for T= T, and T 0 and interpolate between the two regions with a form that gives an entropy C" (T, )  The close analogy in the physical properties between the heavy Fermi liquids UBet3 and He suggests that there should be an underlying similarity in the microscopic descriptions. Anderson and Brinkman4 and also Vollhardtts have successfully interpreted the low-temperature properties of 3He in terms of an almost localized Fermi liquid using the Brinkman-Rice theory'7 for the Hubbard model near the Mott transition. If in UBet3 we assume that the heavy-electron liquid arises because of coherence between the 5f states and that the only role of the Be bands is to determine the size of the U-U hopping-matrix element and the Fermi level is the 5 f band, then it is clear that the Brinkman-Rice theory of a strongly correlated Fermi liquid can be directly applied to the U 5f states. The important requirement is that the number of 5f electrons be very close to an integral number per U and that the on-site Coulomb interactions be strong. These conditions are quite reasonable for UBets. In such a model the very large specific heat arises from the low-lying excitations which arise from the rearrangement of the moments on the U sites with integral occupation, and it is the coupling of the quasiparticles through these excitations which leads to the p-wave superconducting state.
In summary, an examination of the iowtemperature specific heat of UBet3 leads us to identify it as the first metal with an ABM p.wave superconducting state, driven by interactions other than the electron-phonon interaction. There are important strong-coupling corrections and the strongcoupling parameter is larger than that attained in 'He and is consistent with the ABM state being stable at all temperatures. %e suggest that there is a close analogy between the heavy-electron liquid in UBet3 and the nearly localized Fermi liquid in He.