P-T-H phase diagram of heavy-electron

At ambient pressure UCd» undergoes a phase transition near 5 K that is believed to arise from antiferromagnetic order within a strongly correlated electron system. Under moderate hydrostatic pressures, two new phase transitions appear below 5 K that are manifested by anomalies in the temperature-dependent electrical resistivity. We present the evolution of these transitions with pressure, temperature, and magnetic field and discuss possible interpretations of their origin given the pressure dependence of the resistivity and magnetic susceptibility.


INTRODUCTION
The observation that superconductivity can appear within a system of strongly correlated electrons having huge effective electronic masses has stimulated the search for additional examples of heavy-electron behavior. ' Such searches have led to the discovery of two uranium-based heavy-electron materials that order magnetically at low temperatures.
Characteristic of the strong electronic correlations in each of these is a very large value of the electronic specific-heat parameter y.
Of the two [UzZn,7 (Ref. 2) and UCd» (Ref. 3)], UCd» has the largest y value, 840 rnJ/mole K, obtained by extrapolating C/T versus T from above the phase transition to T =0 K. C1ear evidence for a phase transition in UCd» comes from a large peak both in the specific-heat and thermal-expansion coefficient at 5.05 K, as well as from a sharp drop in the electrical resistivity at the same temperature.
Both the specific-heat and thermalexpansion anomalies are of a magnitude comparable to those observed in UzZn&7, the ordering in which recently has been established by neutron diffraction to be antiferromagnetic with a much reduced ordered moment and a simple ordered structure.
However, the precise nature of the phase transition in UCd» is considerably less well understood; although, muon spin rotation experiments" indicate antiferromagnetic order. Unlike the sharply defined phase transition in U2Zn, 7, the specific-heat C and thermal-expansion features near 5 Kin UCd&& are characterized by rather broad high-temperature "tails" and a shoulder in C/T versus T near 3.5 K. Because the U-U nearest neighbors are well separated (6.56 A) in UCd & ] it might be expected that the uranium ions carry well-defined 5f-electron moments. The large y value, however, indicates the formation of a narrow band by hybridization.
In analogy with the speculation' that superconductivity in heavy-electron materials is of an unusual type, the nature of magnetic ordering in comparable systems may be similarly atypical in comparison with conventional f-electron magnetic systems.
By viewing the heavy-electron system in UCd» as a Fermi liquid with a large effective mass of the quasiparticles, it is possible from the electronic specific-heat coefficient to estimate a corresponding characteristic degeneracy temperature T* to be on the order of 10 K. Therefore, the phase-transition temperature and T are of comparable magnitudes. Given this observation, one expects the phase transition to be sensitive to slight perturbations of the electronic system, such as can be accomplished by the application of moderate hydrostatic pressures.
With this in mind, we have studied the temperature-dependent electrical resistance and magnetic susceptibility of UCd& i subjected to hydrostatic pressures exceeding 17 kbar and 7 kbar, respectively.

EXPERIMENTAL DETAILS
The electrical resistance was measured using a standard four-lead ac technique from room temperature to typically 1.3 K. Experiments were performed on a small single crystal of UCd» which was grown from excess molten cadmium. X-ray analysis performed on crystals grown at the same time as the one employed in this work confirmed the cubic BaHg» crystal structure and gave a 0 lattice parameter of 9.283 A. There was no evidence from the x-ray study for the presence of second phases.
Hydrostatic pressures were produced in a self-clamping Be-Cu pressure ce11, with a mixture of 1:1isoamyl alcohol and n pentane as the pressure-transmitting medium.
Pressures within the cell were determined at low temperature from the inductively measured superconducting transition of lead. These measurements provide a relative accuracy in pressure determinations of better than +0.5 kbar. Additional details of the pressure cell and measurement procedures have been given elsewhere. A similar, though significantly miniaturized, pressure cell made of binary beryllium-copper was used for susceptibility y measurements in a Faraday magnetometer. The susceptibility of UCd» single crystals from the same batch as the resistivity sample was determined by sub-39 2578 1989 The American Physical Society tracting the susceptibility of the empty cell from that of the cell containing crystals. In order to minimize systematic errors associated with measurements near y=0, the diamagnetism of beryllium-copper was compensated by wrapping Pt foil around the cell body. All measurements were performed in a dc field of 1 T and in the temperature range 2(T &300 K for pressures up to 7.6 kbar. RKSUI,TS In Fig. 1 we show the electrical resistance of UCd&& at four different pressures. The detailed shape of the zeropressure curve agrees with that published previously, including a distinct break in the curve at 5.04 K (see Fig. 1 inset) that signals the phase transition. With increasing pressure the broad resistance maximum centered at T, "=84 K (for P =0) shifts approximately linearly to lower temperatures at a rate of -1.6 K/kbar and becomes more prominent. We also observe in Fig. 1 a rather large, systematic increase in the overall resistance with applied pressure. Near room temperature the resistance increases linearly with pressure at the relative rate (1/Ro)dR/dP=0. 01/kbar. Although we cannot disregard the possibility that the resistance rise might be due to the formation of microcracks, we note that the measurements were performed on a single crystal and that, after an initial pressure increment, there was no detectable hysteresis in the room temperature resistance with repeated pressure cycling.
The occurrence of a phase transition is observed most clearly in plots of the temperature derivative of the resistance dR/BT versus temperature [see Fig. 2(a)]. In the vicinity of the phase transition, this curve appears similar to that of C,]/T versus T measured at ambient pressure.
With increasing pressure [Figs. 2(b) and 2(c)] there is a significant change in the, temperature dependence of R below 4 K that culminates in the appearance of a new phase transition near 3 K at 2.8 kbar. Concurrent with the evolution of the new transition Tz is a gradual shift of the original phase transition T& at temperature T&, to higher temperatures and a diminution of its M /BT signature, i.e. , an apparent decrease in the spin-disorder scattering removed by the phase transition. (In the following we refer to phase transitions T; which are defined by their respective pressureand field-dependent critical temperatures T;, .) T2, falls rapidly toward T=O K [Figs. 2(d) -2(fl] as the pressure is raised above -5 kbar. At the highest pressures [ Fig. 2(i)], a third transition T3 appears whose signature is characterized by a negative-going peak in BR /dT. Both T2 and T3 are also clearly discernible ill plots of R versus T. Specific-heat measurements at ambient pressure show that T&, is depressed somewhat over 1 K by an 11 T magnetic field. %'e also have studied the influence of magnetic fields to 4 T on the transitions Tz and T3 at pressures of 3.8 and 17.3 kbar, respectively. These measurements were performed, with the pressure cell immersed in liquid helium, by fixing the magnetic field and slowly sweeping the temperature through the transition. Starting conditions for each measurement cycle were T=4 K, H=O T.
Both Tz, and T3, were found to be nonlinear functions of H with T2, (3.8 kbar) decreasing by nearly 1.5 K in 4 T and T3, (17.3 kbar) increasing by over 0.5 K at 3 T. The P-T-H phase diagram for UCd» resulting from these studies is shown in Fig. 3.
The dependence of T"on pressure is unusual, increasing initially at a rate of -70 mK/kbar, reaching a plateau between 8 and 13 kbar, and finally increasing again at a rate of about 100 mK/kbar. The initial rate of increase in T"can be compared to that expected on the basis of Ehrenfest's equation appropriate for secondorder phase transitions 8 T& /BP = 3 VT& Ao'/kc& where Aa and Ac are the thermal-expansion and specific-heat changes, respectively, at T"(P =0) and V is the molar volume. From the measurements of Fisk et a/. , we estimate b a =20 X 10 /K and b, cz = 8 J/mole K, which give BT"/'dP =60 mK/kbar, a value in reasonable agreement with that determined directly.
A most striking feature in this work is the observation of two new phase transitions in UCd] & that are separated by only about 14 kbar. The P-T-H diagram suggests a possible inter-relationship between phases T, and Tz as well as between T, and T3. We see in Fig. 3 that Tz disappears near the pressure where T"becomes independent of pressure and T3 appears when T], once again depends on pressure. This correspondence is supported further by the systematics in the resistivity data of Fig. 2. At the same time it is clear from the sign difference in field derivatives of Tz, and T3, that these transitions are quite difFerent. There is some indication from specificheat measurements that perhaps the second transition is beginning to form already at ambient pressure. As mentioned earlier an unexplained shoulder occurs near 3.5 K in a plot of C/T versus T . This temperature agrees well with that obtained by smoothly extrapolating the phase boundary T2, (P) to P =0. Despite attempts to observe an unambiguous signature for the second transition at pressures less than 3 kbar, no evidence for Tz could be  of our pressure measurements. Slopes of these curves give effective moments p, z of 3.60+0.02 and 3.57+0.02 p~/U for P =0.6 and 7.6 kbar, respectively, values consistent with either 5f or 5f configurations at both pressures. The intercepts provide negative paramagnetic Curie temperatures whose magnitudes increase with pressure from 39.4+0.3 to 41.7+0.3 K. Although it is dificult to determine from these measurements the precise pressure dependence of T&"a clearly positive trend is observed, as displayed in the inset of Fig. 4. Because of the limited temperature range over which these measurements could be made and the fact that a 1 T field was applied to the sample, no distinct evidence for phase transition T2 could be detected. However, the systematic change in the temperature dependence of g below T&, might hint of a pressure-induced suppression of T2, . %e also note a progressive broadening of the susceptibility maximum near T" that is consistent with the trends in M IBTshown in Figs. 2(a) -2(e). DrscUsszoÃ s mentioned, muon spin-rotation and relaxation experiments indicate that the phase transition T& is due to antiferromagnetic order. This has not yet been confirmed by neutron scattering but only an upper limit of 1. 5ps/U on the ordered moment has been established. " Such a reduced ordered moment is consistent with neutron measurements on related systems U2Zn, 7 (Ref. 5) and UCu5 (Ref. 12) and with the ratio of the electronic specific-heat contributions above and below the Neel temperature. " That the ordered moment is significantly less than p,& inferred from susceptibility measurements at temperatures higher than T"suggests the presence of Kondo-like interactions leading to partial compensation of the local moment at reduced temperatures. Such interactions, together with Ruderman-Kittel-Kasuya-Yoshida (RKKY) interactions, also appear important for determining the temperature dependence of transport and magnetic properties. ' Many Ceand U-based heavy-electron compounds have a maximum in their electrical resistivity at some moderately low temperature T ", as is also the case in UCd» (Fig. 1). Pressure measurements' show that, unlike UCd», in almost all of these cases dT,"/dP) 0; however, for Yb-based systems with a low-temperature resistivity maximum, d Tm» /dP & 0. ' This behavior also has been attributed to the pressure-dependent competition between intrasite Kondo-like interactions and intersite RKKY-type interactions. '

' Presumably in most
Ceand U-based compounds, pressure ostensibly favors a less-magnetic ground state due to the relatively more rapid increase in Kondo spin compensation than the enhancement of RKKY interactions. The heavy-electron superconductor UBe&3, however, appears to provide' a case in which a more-magnetic-like ground state is favored by pressure even though dT,"/dP &0. On the other hand, in Yb systems, the low volume state is more magnetic and hence generally promoted by applied pressure.
A different example of the behavior described is found in CeA12 in which the resistivity maximum near 100 K moves to lower temperatures with increasing pressure. ' Here, however, T,"has been attributed to Kondo-like scattering off thermally populated crystalline-electric field levels whose splitting from the ground state decreases with pressure. Although the presence of crystal fields has been established in a number of Ce compounds, including CeAlz ' this is not true of U compounds, with the exception of UPd3. ' The 1/y versus T data shown in Fig. 4 would suggest that crystal fields are not well-defined and certainly not manifested clearly in UCd». Even in the absence of crystal-field splittings, we would expect to find an entropy of Rln2 below T&, in UCd» characteristic of the strongly interacting Fermi liquid ground state inferred from the large y value. Our analysis of the specific heat data for UCd» up to Ti, does give an entropy slightly less than Rln2 consistent with this expectation.
It is also not obvious that pressure effects in UCd» and Yb compounds (dT,"/dP &0) can be interpreted similarly. This is made particularly dificult because the two possible 5f configurations have nearly identical efFective

moments. UCd
& & therefore, appears to be somewhat pathological in the sense that it is not straightforwardly analogous to previously studied systems. However"we believe that our observations are generally consistent with the concept of competing intraand intersite interactions. In this perspective, the large electronic specific-heat coefficient y is determined primarily by the intrasite energy scale Tz.
For Tz~Tz, where Tz is a measure of the intersite coupling scale determined by the q-dependent exchange J, the ground state is paramagnetic.
However, when J(q)IT~=1 a magnetic instability occurs and the spinsystem orders. ' Because UCd» has the largest y of any known U-based heavy-electron magnet, this viewpoint suggests that Tz must be small, certainly smaller than Tz. In this regime we expect the magnetic ground state to be favored initially as J is enhanced by pressure, ' pro- however, the data of Fig. 4 indicate d~O~IdP )0, consistent with the expected increase in Tz with pressure. If we assume that spectral weight lost at. q =0 by the application of pressure reappears at some q & 0, then a second magnetic instability [J(q)/TK = 1] could occur. We suggest that this may be the mechanism responsible for the additional phase transition T2. Its extremely strong suppression with pressure, however, is not understood. As mentioned above, the phase diagram presented in Fig. 3 clearly suggests an inter-relationship between phase transitions T, and T2 as well as Ti and T3~A lthough a plausible argument has been given for the origin of T2 and the pressure dependence of T, at low pressures, the source of T3 remains a mystery. Certainly additional experiments, e.g. specific heat, magnetic susceptibility at higher pressures and lower temperatures, as well as neutron scattering under pressure, are required to clarify our understanding of the most interesting I'-T-H behavior of UCd&&. SUMMARY Electrical-resistivity and magnetic-susceptibility measurements on UCd&& as functions of pressure reveal two new phase transitions that are both strongly volume dependent and couple to an applied magnetic field. The pressure dependences of T"and T ", as well as the large electronic specific heat coefficient and reduced ordered moment of UCd», are consistent with the competition between intersite (RKKY-like) and intrasite (Kondo-like) interactions in which at low pressures intersite interactions dominate. We suggest that the phase transition first induced by pressure (T2) arises from a volume-dependent change in the q-dependent susceptibility. At present, no explanation exists for the source of the second pressure-induced transition (T3) except to note that it appears to be coupled to the magnetic transition Ti, which itself may have been modified in a subtle way by pressure sufficiently large to give T3.