Direct observation of heavy quasiparticles in UPt3 via the dHvA effect

We present the results of an investigation of the de Haas-van Alphen (dHvA) effect in the heavy fermion superconductor UPt3. 
 
Oscillations composed of up to 8 frequency components, corresponding to cyclotron orbits in a plane normal to the a-axis, have been detected in a high purity single crystal and a study of their amplitude as a function of temperature and magnetic field has been performed in the intervals 20–150 mK and 40–115 kG, respectively. From this study we obtain estimates of the cyclotron masses, found to range approximately from 25 to 90 times the bare electron mass, and of the mean free path, found to be in excess of 1000 A. The relationship between these findings and the results of conventional energy band calculations is discussed.


Introduction
The exotic low temperature behavlour of the lntermetaihc compounds called "heavy fermion superconductors", has been attributed to the existence of fermion quaslpartlcles with effective masses of unprecedented magnitude, of order 100 times greater than the bare electron mass [1][2][3][4][5][6] Of fundamental interest is the precise nature of these quaslpartlcles and the origin of the effective interactions which lead to the formatlon of the enigmatic superconducting ground states Although a great deal of indirect evidence for the existence of heavy quasIpartlcles has been collected, no measurement so far has allowed direct probing and unambiguous detection In this paper we report the first direct observation of these particles in a heavy fermlon superconductor, namely in UPt3, via the de Haas-van Alphen (dHvA) technique The dHvA effect consists of an oscillatory variation/~/of the magnetisatlon as a function of the Inverse of the applied magnetic field H In accordance with the traditional theory for a Fermi liquid (see eg refs [7,8]) the frequency and amplitude of M provide a direct measure of the principal properties of quasiparticles near the Fermi level (1 e those quaslparticles which are responsible for the low temperature behavlour) In the simplest case of a paramagnet with one partly filled lsotroplc conduction band, the fundamental component of ~7/ at a temperature T may be expressed as where a and th are quantities which are normally independent of T and H, and hko, m* and lo are, respectively, the momentum, the effective mass, and the effective mean free path of quaslpartlcles near the Fermi level The velocity vo of the quaslpartlcles at the Fermi level is given by m*vo = hko From eq (1) It IS seen that ko is determined from the frequency of the oscillations, m* from the temperature dependence of the (relative) amplitude at fixed H, and In from the field dependence of the (relative) amplitude at fixed T In the more general case of several partly filled anisotropic conduction bands an oscillatory magnetlsation M IS associated with each extremal cross-sectional area M of the Fermi surface lying In a plane normal to H The fundamental component of M parallel to H for a given area M is in general still given by eq (1), where ko, m* and lo are now appropriate averages associated with the orbit around M Specifically, k0 is the average Fermi wavevector defined by M = ~rk~, m* is the cyclotron mass defined by m*= hko/vo, where 1leo is the average of the inverse of the quaslpar-0304-8853/87/$03 50 O Elsevier Science Publishers B V (North-Holland Physics Publishing Division) tlcle velocity on the orbit, and 1/lo is the average of the inverse of the quaslpartlcle mean free path on the orbit Hence for each group of quasipartlcles associated with an extremal area M of the Fermi surface, the corresponding dHvA frequency F = hcM/2"rre and amphtude A wdl yield three orbltally averaged quaslpartlcle parameters a Ferret wavevector/Co, a cyclotron mass m* and a mean free path lo These wdl be the focus of our attention

Details of the experiment
High purity single crystals of UPt3 were prepared in an ultrahigh vacuum by zone-refining a rod of the compound, contamed m a watercooled copper crucible, with radio frequency heating Our purification procedure, previously used for transition metal lntermetalhc compounds [9][10][11], yielded bulk single crystals with residual resistivity ratios (extrapolated to absolute zero from the temperature interval of 1 2 to 10 K) In excess of 400 Low strain samples, 0 5 mm thick and 3 0 mm in diameter, were cut from the purest parts of the ingots by means of spark erosion followed by chemical etching The dHvA experiments were performed in a superconducting magnet-dilution refrigerator system with maximum field of 115 kG and miramum temperature of 20 mK The dHvA magnetisatlon was detected by means of a low frequency (8 to 25 Hz) field modulation method, the mare detatls of which have been described previously [12] Measurement of the sample temperature, which is particularly critical In the determination of the cyclotron masses, was facilitated by immersion of the sample directly m the dilute liquid phase of the mixing chamber, and by means of a thin silver heat link between the sample and a calibrated germanium thermometer The thermometer was embedded m a smtered sdver matrtx located in the mixing chamber and in a zero field region of the magnet Measurements of the cyclotron masses were carried out as a function of the amplitude and frequency of the modulation field The results presented here were obtained by an extrapolation in the hmit of zero amplitude and frequency (1 e zero eddy current heating of the sample by the modulation field) The errors quoted for the cyclotron masses arise entirely from the uncertainty in the sample temperature

Results of the experiment
Our experimental results, for a field direction along the a axis of the hexagonal (SnNI3) structure (parallel to the FK direction in reciprocal space), are presented in figs 1 to 4 and in table 1 Fig 1 shows a typical recording of the dHvA magnetlsatlon and its associated Fourier spectrum at 20mK Five dominant and several weaker frequency components are evident, pointing to the existence of a complex Fermi surface As shown in table 1, the range of frequencies corresponds to a range of average Fermi wavevectors k0 from 0 121 to 0 425 .~-iThe latter value is 67% of the Brlllouin zone wavevector FA (fig 5) and, as we shall see in the next section, is comparable to the largest average wavevector expected from conventional band theory for any closed sheet of the Fermi surface of UPt~ A detailed study of the temperature dependence of the dHvA amphtudes (fig 3) ytelds, from eq (1), the cyclotron masses given m table 2, which range from 25me for the lowest dHvA frequenoes, to 50me for the intermediate fre-Table 1 Summary of dHvA results for a magnetic field apphed along the a-axis (parallel to the FK direction of the reoprocal lattice) F is the dHvA frequency, ko is the average effective wavevector characterising the extremal area associated with F, m* is the cyclotron mass derived from the temperature dependence of the dHvA amphtude, and lots an effectwe mean free path derived from the field dependence of the amplitude An average effective Fermi velooty v. may be defined for each extremal area by hk.= m'v.quencies, and up to 90me for the highest frequency In addition to the average wavevectors ko and masses m* of the orbits, we obtain, from the field dependence of the amplitude (eq (1)), esttmates of the mean free path lo From fig 4 and table 1 we see that in all cases a lower bound for 1o is approximately 1000A, a value comparable to that obtained in many other pure metals which we have investigated We point out that an effective scattering time "to and a scattering (Dingle) temperature To for the quaslpartlcles can be defined by [8] h/~'o = 2"rrkaTo = hvo/lo = h2ko/m*lo (2) Table 2 Comparison of band calculated [17] and experimental values of the average wavevector ko (Fermi surface area M = Irk 2) and of the cyclotron masses The ratio of experimental to calculated masses is the largest thus far reported m any system and is m all cases comparable, within experimental error, to the ratio of the measured and band calculated hnear coefficient of the heat capacity [ It should be stressed that the effective mean free path appearing in the dHvA amplitude reflects the effect of all scattering processes and of phase cancellation due to lnhomogeneltles, and is thus normally much smaller than the corresponding mean free path derived from the electrical resistivity

Discussion of the results
In this section we discuss the relationship of our dHvA results to conventional energy band models based on the local spin density approximation for the exchange correlation potential All calculations of this kind predict very slmdar band models [13][14][15][16][17][18] The extremal cross-sectional areas in planes normal to the a-axis, in the central FALM plane for sheets 1-5 and in a non-central plane for sheet 3' (fig 6), predicted by the Fermi surface model are given in table 2 (in terms of ko), together with the calculated cyclotron masses Also in table 2, along with the calculated ko and m* for each orbit, a possible and tentative identification with the observed results is given for ease of comparison It is seen that the number of extremal areas predicted and the corresponding range of ko agree generally with the dHvA results of table 1 On the other hand, the measured masses are m all cases much larger, typically of the order of 20 times larger, than the calculated masses This The effective mean free path of the carriers, as referred from the field dependence of the dHvA amphtude, is between 1000-2200.A for the various orbits The expertmental Fermt surface areas are not mconststent with recent local denstty band calculations [13][14][15][16][17][18] The observed cyclotron masses are, however, in all cases much greater than predtcted by the local denstty band models, by a factor which is, within the experimental error, of the same order of magnitude as the ratio of the experimental to the band calculated hnear coefficient of the heat capacity We wish to thank P Coleman for many helpful dtscussions and G C F Newcombe for his help with the early measurements We have also benefitted from communlcattons wtth A J Freeman, S M Hayden and M Pepper The high purity uranium used to produce the UPt~ was kmdly provtded by R Hall Thts research was supported by the S E R C of the Umted Kingdom, the US Department of Energy, Dtvtston of Materials Sctence, and the N S E R C of Canada

Fig 1 Fig 2 Fig 3 Fig 4
Fig 1 Typical oscillatory variation of the dHvA magnetlsauon in UPt~ (upper trace) and the corresponding Fourier spectrum (lower trace) at 20inK for an apphed magnetic field H parallel to the a-axis The signal Is detected by a low frequency and low amplitude field modulation technique

Fig 5
Fig 5 The first Brllloum zone of the hexagonal close packed structure , and the results of OguchI and Freeman for the Fermi surface are presented In fig 6 for illustration Their predicted Fermi surface consists of three closed electron surfaces centred on F (1, 2 and 3), closed electron surfaces centered on K (3'), two nested toroidal hole surfaces about FA (4 and 5), and toroldal surfaces about LH (4) linking nelghbourlng zones Because UPt3 is a compensated metal the total volumes enclosed by the electron surfaces and the hole surfaces are equal Surfaces 3 and 4 involve the largest number of carriers and dominate the overall density of states at the Fermi level