Unrenormalized ultrasound attenuation in the heavy-fermion state

Quantitative studies of ultrasound absorption in the heavy-fermion state of UPt3 and UBei3 are reported. The magnitude of the absorption due to electrons in the normal state is not enhanced compared to that of ordinary metals, indicating a cancellation of the mass enhancement by a reduction of the electron-phonon coupling parameter. This implies that the mass enhancement is described by a different Landau Fermi-liquid parameter than in He. The T variation of the normal-state sound velocity at lowest temperatures is consistent with the large electronic specific heat.

The formation of the heavy-fermion state out of what appears at high temperatures to be local magnetic moments embedded regularly in a metallic matrix is the subject of intensive current research. The unconventional properties of the superconducting ground state, into which some of the heavy-fermion systems condense, ' indicates a novel type of superconductivity characterized by large anisotropies of the order parameter ' and, as suggested by Varma" and by Anderson, ' nonsinglet pairing of the electrons. On the other hand, much less is known about the normal state of the heavy fermions.
The enhancement of the effective mass by two orders of magnitude over ordinary metals is accompanied by a corresponding enhancement of the magnetic susceptibility, keeping their ratio of order unity. " The question then arises whether other properties of the heavy ferrnions are similarly enhanced. One of these is the attenuation of sound, a particular type of transport property, which is determined by the electron-phonon coupling strength and the effective mass of the quasiparticles.
Here we present a quantitative study of the ultrasound propagation in UPt3 and UBel3 at low temperatures. The main result is that the magnitude of the ultrasound attenuation is the same as in ordinary metals, and not enhanced by 10 -105 as might be expected from the large effective mass. This implies, as discussed by karma, ' a compensation of the mass enhancement by a reduction of the electronphonon coupling strength and places severe constraints on any theories for the heavy-fermion state. In particular, it is the Landau parameter Fo which characterizes the mass enhancement, and not Fl as in the case of the well-studied Fermi-liquid 'He.
The experiments on UPt3 were performed on a single crystal grown in an ultrahigh-vacuum float-zone apparatus from a previously synthesized ingot. A crystal of 0.7 cm length was cut from a cylindrical sample of 5 cm length and 0.6 cm diameter. The ultrasound transducers (LiNb03) were attached to optically flat opposite surfaces and longitudinal sound was propagated in directions parallel to either the hexagonal c axis or the basal plane. The frequency range extended from -50 to 500 MHz. The experiments on UBei3 were done on single crystals, gro~n from Al flux, and evaporated zinc oxide transducers allowed studies at ultrasound frequencies up to 2 6Hz. The attenuation due to electron-phonon scattering is so weak in UBei3 that it can be measured only at these high frequencies.
Before discussing the experimental results, we recall the physical principles involved in the attenuation of ultrasound by electrons. " Both classical and quantum-mechanical methods have been applied and give identical results for free electrons, but the quantum-mechanical treatment starts from a conceptually more appealing description. The electron's energy in a crystal deformed by a longitudinal elastic wave is described as E(k, 5) =Eo+Ir k /2m '+Ei'75 where m' is the effective mass from the E(k) dispersion, 5 the deformation, and El the deformation potential, a measure of the electron-phonon coupling strength. The result 33 19$6 The American Physical Society UNRENORMAI. IZED ULTRASOUND Arrj:NUATION IN THE. . . for the amplitude attenuation coefficient o. is then readily obtained for the limit of~here the sound wavelength~is smaller than the mean free path i" i.e. , ql, && 1 (q = 2n/X) . The corrections for finite mean free path (q/ ( 1) are done within the classical framework and are found to describe the behavior of real metals very well. %e obtain for a in the limit q/ && 1, Here u is the sound velocity, po the mass density, and v the sound frequency.
In the following we show that (2) indeed describes the results in the normal state of UPt3 and UBe~3. In Fig. 1(a) the attenuation in Upt3 is sho~n as function of the square of the temperature. This is a very useful representation of the data for two reasons. First, it illustrates that a(T) in the normal state (a") has the same T dependence as the electron mean free path /, (T) as deduced from the resistance, shown in Fig. 1 and specific heat support this conclusion. ' " Figure 1 also sho~s how we separate the electronic contribution to the sound attenuation from other contributions, which cause a background attenuation.
Because both a"and a, vary as T2, extrapolation of a, to T = 0 gives the zero for the ordinate, and the total electronic part of a( T -0) is obtained by extrapolating a" to T=0. The uncertainty in a"(0) is very small and irrelevant in the context of this paper. In UBei3, the magnitude of o. " is determined as the difference of o. " just above T, and n, for T 0. No special correction for the temperature dependence of /, (and therefore n") is necessary, because the variation of the resistivity with temperature is much less pronounced in UBei3 than in UPt3.
Here we note that the temperature dependence of n in UBei3 at T && T, is consistent with a T' law, as in Upt3, and sho~s a peak just at T, . This latter feature has been observed for the first time in a superconductor and is the subject of a separate publication. ' In Fig. 2 the frequency dependence of a, (0) is plotted on a double logarithmic scale. The observed f' law again supports the q/((1 description given by (2).
A comparison with other metals can be done in two different ways. Either the microscopic parameters m'Ei are calculated for a given ql or the attenuation is extrapolated (or directly measured) in the q/ &) 1 regime where it is independent of I, . In either case the mean free path has to be estimated from the resistivity, a step which introduces some uncertainty.
%e like to point out, however, that our main conclusion does not rely on the exact value of l, . Given the residual resistivity of the Upt3 sample of -0.5 p, Q cm and following an earlier analysis based on Friedel's maximum scattering argument, we infer -2200 A for I, . '

' At 100
MHz this gives ql = 3.5 x 10 '. The experimental quantity to be compared with other metals is the ql && 1 limiting at- part of the free energy. Whereas this b8 is not dominant in ordinary metals, it can be clearly measurable in heavyfermion systems, and 48 will be proportional to the integral over the specific heat c(T). Given the huge electronic c(T), b8 will vary to first order as T2. This is indeed observed in UPt3, and we ascribe the difference in the prefactor for the two directions to contributions involving the thermal expansion coefficient, which is of different signs in the two directions. ' A quantitative analysis, however, can only be done when all necessary thermodynamical derivatives are measured with respect to the strains corresponding to the elastic deformations associated with the sound propagation in the various directions.
A very similar type of analysis has been successfully applied to CeA13, another heavy-fermion metal,~here the bulk modulus decreases as T' up to -0.4 K. " In conclusion, we note that the attenuation of ultrasound in the heavy-fermion state in UPt3 and UBei3 is of the same order of magnitude as in ordinary metals. The mass enhancement of the quasiparticles by two orders of magnitude is therefore compensated by a reduction of the coupling strength.
These observations demonstrate that the parametrization of the mass enhancement in the heavyfermion state has to involve a different Landau Fermi-liquid parameter than in 'He. According to the physical picture leading to Etl. (2), this nonrenormalization of the ultrasound attenuation implies that the product of the microscopic parameters m' and Ei is of the same magnitude as in ordinary metals, namely, E~m'/m, -5-10 eV (m, is the free electron mass). " Given a mass enhancement of order 10', the attenuation would be expected to be enhanced by -10 over ordinary metals if the coupling constant Ei were not reduced. The experimental observations in UPt3 and UBei3 therefore point to a cancellation of the mass enhancement by a proportional decrease of the electron-phonon coupling strength probed by longitudinal sound. The results of the ultrasound studies are used by Varma" to formulate a phenomenological theory of the heavy™fermion state. The main point is that the effective mass is given by the Landau Fermi liquidparameter F$, and nor by Ff as in 30e. This is also borne out by recent model calculations based on the Gutzwiller approach to the Anderson lattice. ' In addition to the attenuation we have also studied the velocity of sound propagating along the hexagonal axis and in the basal plane for UPt3. In Fig. 3 the variation of the sound velocity u is plotted as a function of the square of the temperature, emphasizing the T2 behavior at lowest temperatures.
In the higher-temperature range the velocity parallel to e decreases further by -1300 ppm before going through a minimum at -18 K. For the sound velocity in the basal plane no minimum up to 20 K is observed, but the change is larger and amounts to -3800 ppm at 22 K. '5 The sound velocity changes at lowest temperatures are consistent with the very large electronic specific heat typical for the heavy-fermion state. The argument is based on the definition of the bulk modulus as the second derivative of the free energy with respect to volume, and can proceed in either a very general way or within a particular model for the electron-lattice coupling. 2 Straightforward thermodynamic derivations give a contribution to the bulk modulus change d 8, due to the electronic contribution to the entropy 'F. Steglich, J. Aarts, C. D. Bredl, %'. Lieke, D. Meschede, %. Franz, andH. Schafer, Phys. Rev. Lett. 43, 1892 (1979).