Observation of magnetic order in the heavy-fermion superconductor UBe13.

We have measured the magnetostriction, L(H), of a single crystal of the heavy-fermion superconductor UBe13, using an all-silicon, high-precision capacitance dilatometer. We find clear evidence for a transition to an antiferromagnetic state at T~-8. 8 K, which is suppressed in a field by dT~/dH-0. 36 K/T. At low temperatures we observe pronounced magnetostrictive oscillations, which we believe are de Haas-van Alphen oscillations due to an unusual aspect of the Fermi surface.

PACS numbers: 74.70.Tx In recent years the intimate relationship between magnetism and superconductivity in the heavy-fermion superconductors (HFSC) has become increasingly clear.
In Upt3, with T, -0.55 K, an antiferromagnetic transition has been observed' in neutron-scattering experiments with TJv-5.0 K. In URu2Si2, with T, -1.2 K, there is a transition to an antiferromagnetic state at T-17.5 K as seen in a number of measurements.
The coupling of the magnetic order parameter to the superconducting order parameter imposes symmetry constraints on the nature of the superconducting state, thought to be paired with angular momentum I~1 in all the HFSC. Such studies have been performed on UPt3 using neutron scattering, and are in progress on URu2Si2 using neutron and magnetic x-ray scattering. Furthermore, it is believed that the pairing mechanism in the HFSC is due to mediation via antiferromagnetic spin fluctuations.
In this context we might expect that the presence of a magnetic transition with T~-10T, is a general feature of superconductivity in this class of materials, and an essential one to understand. The heavy-fermion superconductor UBe~3, with T, -0.8 K, has the most pronounced low-temperature properties of the HFSC, with y-1100 mJ/molK; nevertheless, a magnetic transition above T, has not been reported previously. ' In this Letter we show clear evidence for an antiferromagnetic transition at TJv -8.8 K, which is suppressed in a magnetic field by dTN/dH -0.36 K/T. At low temperatures we see unusual magnetostrictive oscillations, not previously observed in any system, which we believe are related to the de Haas-van Alphen effect and to the magnetism we observe. These may be related to the unusual features observed in both specific-heat"' and resistivity' measurements at T-2 K, neither of which have received a satisfactory explanation. Also of note is that pressure studies' show that the electronic Gruneisen constant y, changes sign at T-8 K. UBei3 is to be distinguished from the other HFSC by its cubic crystal structure (UPt3, hexagonal; URu2Si2, tetragonal). While we find again that Tv -10T, in support of the generality of these as the relevant energy scales, the manifestation of the magnetic properties among the HFSC is quite different. This is in part due to the different crystal structures, and to their accompanying symmetry considerations.
Measurement of the magnetostriction of materials has long been a traditional tool' for the study of weakly magnetic systems, due to the high sensitivity of dilatometry techniques. This technique is especially justified for the heavy-fermion systems, since they not only have strongly enhanced electronic properties due to the mass renormalization, but also have strongly enhanced Gruneisen parameters, ' typically y, =dlny/din V-100, where y is the linear specific-heat coeScient. Thus, because of the strong volume dependence of the thermodynamic properties of the system, the thermal expansion and magnetostriction are more sensitive by y, and y, respectively, than they would otherwise be, and hence more sensitive to magnetic order than the more direct thermodynamic measures (i.e. , C, , M). From a thermodynamic Maxwell relation' V 'aV/aH l, , = -am/a~I T, H, where m =M/V, the magnetization per unit volume.
Assuming that M points along H, m gH, and that the dilatations are isotropic, this is equivalent to L 'dL/dH= -, ' x'y m, where x is the compressibility, and y is the magnetic Gruneisen parameter y =d lnZ/d lnV. Since the compressibility is known, and assuming that y is constant, Eq.
(2) shows that magnetostriction measurements can be simply related to the bulk magnetization of the system.
Using a novel implementation of conventional capacitance dilatometry we have studied the thermal expansion L(T) and magnetostriction L(H) of UBe~3. For magnetostriction measurements it is conventional'" to calibrate a dilatometer using a slug of single-crystal silicon which is expected to have no magnetostriction component. Instead we have constructed the cell itself only from single-crystal silicon to achieve high stability and sensitivity for magnetostrictive measurements. Furthermore, the design is ideal for thermal-expansion measurements, since the length changes at low temperatures are approximately hL/L =2.2X 10 ' T and are thus insignificant below 15 K. No materials other than the sample or the silicon were used, in order to eliminate the need for addendum corrections and to maintain high sensitivity.
With this cell design we achieved a sensitivity of hL/L -3X10 for measurements either as a function of H or T. The sample was grown from an Al melt using standard Czochralski techniques. The sample was approximately cubic, with dimensions -5x5X5 mm . The magnetic field was applied along the [100] direction and the dilatation measured along the perpendicular [100] direction.
The capacitance (Cti-5 pF) was measured between a polished face of the sample (100) and a thin gold film evaporated onto a silicon electrode, using a standard three-terminal capacitance bridge.
The thermal expansion of UBei3 has been previously measured by Ott' at H=O for 0.3 & T & 10 K. Ott, Fisk, and Smith also reported ' no significant magnetostriction up to 15 kG. Our detailed study of the thermal expansion in the superconducting state and up to 20 K is the subject of a later work, ' but is generally consistent with their measurements.
Our measurements up to 8 T show that the magnetostriction is not appreciable in magnitude, but a careful study of it shows significant structure which we now discuss.
In order to directly measure the quantity of interest, dL/dH [see Eq.
(2)], we added a very-low-frequency (f -10 Hz) field modulation of -1 kG to the applied field, while recording the consequent modulations of the length of the sample. The frequency was kept low to eliminate sample heating and to maintain long integration times. The amplitude of the length modulations was measured at a number of temperatures, tracing out a curve, which is equivalent to M(T). This is shown in Fig. 1 for HO=3 and 7 T. The curves have the characteristic shape of an order parameter, being constant at low temperatures and vanishing at 7.65 and 6.33 K, respectively.
We believe that the shape of the dL/dH curves is clear evidence for magnetic ordering and the suppression of the transition temperature in a field indicates that it is antiferromagnetic in nature. However, at a conventional antiferromagnetic transition the magnitude of the magnetic susceptibility decreases below Tz.
The observed increase in y, m may be due to a change in the magnetoelastic coupling (i.e. , y ) associated with the magnetic ordering, or the result of a stronger coupling to one of the sublattice magnetizations, which would naturally increase below T~. We have measured dL/dH vs T for a number of applied fields between 5 and 78.5 kG. The values of (1/L) &dL/dH in the saturated region (T-3 K) are plotted versus H in the inset of Fig. 1. The linear variation clearly supports the assumption made earlier of g being independent of H. From the slope of this line, and a value for the compressibility' of tr=8X 10 '3 cm /erg, we find using Eq. (2) that y g-2.9X10 emu/cm . This value is approximately 16 times higher than the measured o low-temperature value for the Pauli susceptibility go, which confirms the fact that the observed effects are indeed due to a Gruneisen-enhanced susceptibility. Equivalently, we can express this as (I/go)dg/ dP-13 Mbar, which should be measurable with pressure studies provided that dg/dP is constant up to the pressures typically used for measurement (-10kbar). Fig.   2. In the absence of a detailed model for M(T), the transition temperatures T~have been extracted assuming a linear variation of dL/dH near Tv. At low fields the error bars on Ty are high since the signal vanishes as H 0. From Fig. 2 we find that TJv is suppressed by dT~/dH -0 36 K/T. We. conclude that for H =0,

The transition line in the H-T plane is plotted in
While no hysteresis is observed in the curves of dL/dH as a function of temperature, there is significant hysteresis as a function of magnetic field for T & T~. In Fig. 3 we show a number of plots of L(H), taken at constant temperature, and relatively slow sweep rates dH/dt -0.3 mT/s. Figure 3, curve a is for T & Ttv (T=10.0 K) and shows no magnetic hysteresis. The curvature at low fields is most likely an artifact of the dilatometer.  .25 K and only shows a trace for increasing field. It shows sharp jumps in L(H) at specific fields, which appear to fall within an envelope roughly like that of Fig.  3, curve b. Upon repetition such a trace appears qualitatively similar, but the jumps do not occur at the same field each time. Figure 3, curve d is for T=0.825 K, and shows that the frequency of the jumps has increased significantly. They still remain in a well-defined envelope, now reduced in amplitude, but appear more convincingly to be oscillatory, perhaps periodic in I, H With Fig. 3, curve e at T =0. 600 K this trend continues.
The anomaly at low fields is associated with , H( i.0060 K) and is discussed elsewhere.
The oscillatory magnetostriction at low temperatures, shown in Fig. 3, is clearly reminiscent of de Haas-van Alphen oscillations. It is not at all surprising to observe oscillatory magnetostriction in a clean metal at low temperatures due to the de Haas-van Alphen eA'ect as was first noted by Chandresekhar, ' and later measured in bismuth. Magnetostriction measurements are particularly well suited due to their sensitivity, and the possible enhancements discussed earlier. Ho~ever, there are a number of reasons why their appearance would be unlikely in this measurement; notably that the high eA'ective masses implied by the specific heat, and the short mean free path implied by the resistivity should suggest that de Haas-van Alphen oscillations could only be observed at very low temperatures. occur within an envelope similar to the above; curve d, T 0.825 K, rnagnetostrictive oscillations are much more rapid and lower in amplitude; curve e, T 0.600 K, the oscillations continue to decrease in amplitude and increase in rate. The anomaly below 2.6 T for curve e is due to H, 2 for this ternperature.
Furthermore, there are a number of ways in which our observations are not consistent with the conventional expectations for the de Haas-van Alphen eA'ect. First, the oscillatory frequency is given by ni~ya --poS, where is the flux quantum, hc/e, and 5 is an extremal area of the Fermi surface. The area 5 is given by the Fermisurface topology and rarely deviates by more than 10 over the range of temperatures~here measurements are possible. In our measurements co~/g changes drastically with temperature, with co~a~~I/T for 0. For this case, however, the oscillations appear more in the form of first-order jumps governed by nucleation processes.
While our data are clearly inconsistent with the conventional de Haas-van Alphen effect, we nevertheless believe that suitable modifications appropriate to UBe|3 can help to explain our results. It has been shown that the amplitude of the magnetostrictive oscillations can be directly related' to the oscillatory magnetization by hL/L = -, ' try, mB, where y, dlnS/dlnV is a Gruneisen parameter associated with the extremal orbit area S. Thus, there can be a tremendous enhancement over the usual oscillation amplitude if a part of the Fermi surface is particularly volume dependent. We can guess that y,y, -100, but it could easily be higher for one particular part of the Fermi surface. The first-order nature of the jumps can likely be understood in terms of magnetic interaction.
The oscillations are governed not by the applied field, H, but by the internal field 8 H+4 Mtr. Thus if M becomes comparable to the spacing between oscillations the system is in a metastable state during part of the oscillation cycle and undergoes a transition to a lower-energy state after a suitable nucleation process. Ordinarily such nucleation is not impeded and smooth variations are observed, but in our case it appears that the surface barriers are more significant than usual. In support of this argument we note that the jumps are only observed as a function of H and never as a function of T. It may be possible to understand the dependence of the oscillation amplitude on H and T in this context. It is not yet clear whether the observed hysteresis envelopes are a result of the same phenomena that lead to to the oscillations, or due to a separate magnetic effect which constrains the oscillation amplitude.
The strong temperature dependence to the oscillation frequency is difficult to understand. It may, in fact, be a change in the Fermi surface, related to the magnetic transition, such as the formation and opening up of a neck between parts of the Fermi surface. Alternately, in this temperature range the effective-mass renormalization is thought to be rapidly increasing, which may be reflected in the Fermi-surface topology. Finally, the change in oscillation rate may only be an apparent one, due to a change in the nucleation rate of magnetization jumps as a function of temperature.
We have shown that the capacitance dilatometry technique can be implemented to make very sensitive measurements of the magnetization of a system. It is particularly well suited to the study of the heavy-fermion sys-tems, due to their greatly enhanced Gruneisen parameters. We have demonstrated that UBei3 orders into an antiferromagnetic state with T~-8.8 K, which is suppressed in a field with dTtv/dH-0. 36 K/T. Clearly, neutron-scattering experiments are warranted to elucidate the nature of the magnetic ordering. Below -3 K we observe very unusual magnetostrictive oscillations, which we believe can be understood in the context of the de Haas-van Alphen effect with suitable modifications.