Scaling properties of the magnetic-field-induced specific heat of superconducting UBe13

We report on measurements of the low-temperature speciﬁc heat C p ( T , B ) of the unconventional superconductor UBe 13 in magnetic ﬁelds up to 7 T. At B ’ 2 T, a substantial change in the magnetic-ﬁeld dependence of the temperature derivative of the magnetic-ﬁeld induced contribution to the electronic speciﬁc heat C H ( T , B ), resulting from the ﬂow of supercurrents around the vortices, is observed, suggesting a crossover between two different regions in the superconducting phase diagram of UBe 13 . For ﬁelds B . 2 T, C H ( T , B ) exhibits a scaling behavior with respect to TB 2 1/2 , which provides substantial evidence for the existence of point nodes in the quasiparticle excitation spectrum of the superconductor.

Ever since the discovery of the heavy-electron superconductor UBe 13 , 1 both, experimental as well as theoretical efforts have been made to establish or characterize the nature of superconductivity in this compound. 2,3 Among various other indications, the anomalous behavior of the specific heat at temperatures well below T c (C p ϰT 3 ) 4 or the London penetration depth below T c (⌬ϰT 2 ) 5 have been interpreted as evidence, that superconductivity in this compound is of unconventional nature. Inspired by these power-law temperature dependencies of various properties, it has been proposed that the superconducting state is an axial 4-6 spin-triplet state. However, the definitive identification of the proper symmetry of the superconducting order parameter is still not complete.
The magnetic field induced contribution to the specific heat C p (T,B) of a superconductor with gap-nodes strongly depends on the topology of these nodes, and therefore measurements of the specific heat in external magnetic fields may directly serve as a tool for investigating the topology of the gap nodes. 7 In several recent specific heat studies of the high-T c superconductor YBa 2 Cu 3 O 7 ͑YBCO͒ ͑Refs. 8-11͒ and the heavy-electron superconductor UPt 3 , 12 the magneticfield induced contribution to the specific heat has been investigated and found to match the predictions for a superconductor with line-nodes in the quasiparticle excitation spectrum. In this letter, we demonstrate the existence of a crossover at BϷ2 T between two different regions in the phase diagram of UBe 13 and we further demonstrate that for BϾ2 T, the behavior of C H (T,B) of the heavy-electron superconductor UBe 13 as a function of T and B leads to the conclusion, that its superconducting energy gap exhibits point nodes.
In the mixed state of a type-II superconductor, the superflow circulating around a vortex causes a Doppler shift ⌬E D of the energy scale. For small gap values, i.e., at and close to the gap nodes, this shift causes a nonzero density of electronic states ͑DOES͒ at zero energy which depends on the topology of the gap nodes. In the limit k B TӶ⌬E D and B ӶB c2 , the DOES at zero energy in the case of point nodes, is given by N(0)/N F ϳB/B c2 ln(B c2 /B) for an arbitrary direction of the magnetic field, or N(0)/N F ϳB/B c2 , if the field is exactly parallel to the orientation of the nodes, respectively. For line nodes, N(0)/N F ϳ(B/B c2 ) 1/2 , independent of the direction of the magnetic field. 13 Here N F is the DOES at the Fermi energy E F . This leads, in the limit of TӶT c , to an additional linear-in-T term to C p of the form 7 A more general way to analyze the magnetic-field induced specific heat has recently been discussed by Volovik 14 and by Simon and Lee. 15 They showed that the magnetic-field induced contribution to the specific heat C H (T,B) obeys a scaling behavior, with respect to the scaling parameter x ϳ(T/T c )ͱB c2 /B, of the form C H (T,B)ϳB "(3ϪD)/2… f (x). This scaling may be rewritten as where D denotes the dimension of the nodes, i.e., Dϭ0 for point nodes and Dϭ1 for line nodes. F(x) is a universal scaling function. 14,15 According to Eq. ͑1͒, its asymptotic behavior for xӶ1 is of the form F(x)ϳx DϪ1 . We note that in the case of point nodes an additional term due to the logarithmic term in the DOES, which would invalidate the scaling relation, enters Eq. ͑2͒. Nevertheless, it can be shown that in materials with cubic symmetry, this term may well be small enough to not significantly affect the scaling relation given in Eq. ͑2͒. 16 In order to test these scaling predictions in connection with an unconventional and strongly type-II heavy-electron superconductor, the specific heat C p (T,B) of a small piece of polycrystalline UBe 13 (ϳ50 mg) has been measured in various temperature ranges between 0.08 and 0.38 K and in external magnetic fields between 0 and 7 T, using a thermalrelaxation technique in a dilution refrigerator. Special care was taken for the calibration of the RuO 2 thermometer attached to the sapphire disk serving as the sample platform in external magnetic fields. For this purpose, a calibrated temperature sensor was mounted inside a multilayered superconducting magnetic shield, which was located outside of the core of the magnet solenoid, but was thermally shortcut to the calorimeter. The specific heat of the sample platform alone has been measured in a separate experiment and was only a few ppm of the total measured specific heat at all temperatures and all fields. The absolute accuracy of the calorimeter is better than 3% at the lowest temperatures and further improves with increasing temperature.
This experimental setup allows for C p -measurement scans either parallel to B in the TϪB phase diagram by varying the temperature in a fixed magnetic field or vice versa, thus providing a good sensitivity to thermodynamic features in any direction of the phase diagram.
The sample has been cut from a piece of material used in a previous investigation of the specific heat. 4 The specific heat in zero field and above 0.07 K is well described by C p ϭ␥ 0 Tϩ␤ el T 3 . Earlier measurements of the specific heat of UBe 13 had indicated that C p is sample dependent such that the apparent linear term ␥ 0 T arises from resonant scattering at impurities, imperfections, etc. 6,17,18 The fit parameters ␥ 0 and ␤ el of the zero-field data presented here are consistent with those obtained in earlier measurements. 6 In Fig. 1, we show the low-temperature specific heat as measured for 0.08рTр0.20 K and 0рBр7 T, as ͑a͒ a function of temperature at constant magnetic fields and ͑b͒ at constant temperatures vs field. The slight upturn at low T and high B is due to the nuclear Zeeman contribution of the Be atoms.
For field scans at constant T ͓Fig. 1͑b͔͒, C p (B) shows a broad shoulderlike feature centered around BϷ3 T. In a recent study 19 of the specific heat of UBe 13 in magnetic fields at somewhat higher temperatures, similar features of C p (T,B) at BϷ2 T have been reported and were interpreted as an indication for the occurrence of a second phase in superconducting UBe 13 . In Ref. 20, the results of thermal expansion and specific heat measurements have been combined to show the existence of an additional feature in the phase diagram of UBe 13 . Below, we find further evidence for the existence of an additional feature in the phase diagram of UBe 13 , which might be related to the shoulderlike feature, but which appears at slightly different B.
In this work, we are mainly interested in the magneticfield induced electronic contribution to the specific heat, C H (T,B). The Be nuclei carry a nuclear spin, causing a magnetic field dependent and, especially at high fields and low temperatures, non-negligible contribution to the specific heat, C N (T,B), which is proportional to B 2 T Ϫ2 and is discussed in great detail in Ref. 21. Aiming only at the magnetic-field induced electronic contribution, we consider where C p (T,B) is the measured specific heat, C N (T,B) the evaluated nuclear contribution and C p (T,0) the measured specific heat in zero field. This procedure for obtaining the magnetic-field induced electronic contribution, C H (T,B), to be discussed below does not depend on any fit parameters or unknown background contributions and, therefore, leads to reliable results of C H (T,B). According to Eq. ͑1͒, in the limit of TӶT c and BӶB c2 the magnetic-field induced electronic specific heat should vary linearly with T. Thus, we have plotted C H (T,B)/T vs T measured at constant fields in Fig. 2. The discrepancy between the theoretical predictions and the experimental data is obvious. The model leading to Eq. ͑1͒ is expected to be valid only in the limit where k B TӶ⌬E D , i.e., xϳT/T c ͱ(B c2 /B) Ӷ1. 14 Setting the magnetic field to BϭB c2 /5, which is already at the upper limit of the valid regime, the condition x р 1 5 can only be fulfilled if T/T c р0.09. The critical temperature of the present material is T c ϭ0.91 K. Thus, for the model to be applicable, the temperature has to be below 0.08 K, the lower limit of the temperature range covered in these experiments. This upper limit of the temperature decreases further with decreasing magnetic field and therefore, in this model's context and using these data no conclusions concerning the topology of possible gap nodes can be drawn.

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A more general approach to analyze the topology of possible gap nodes which is not only valid in the limit xӶ1 but for all x, is provided by the universal scaling relation given in Eq. ͑2͒. According to Eq. ͑2͒, C H (T,B)/T 2ϪD B 1/2 should be proportional to a universal function of the parameter x ϳT/T c ͱ(B c2 /B). Although this scaling relation has theoretically only been demonstrated to be valid for TӶT c and BӶB c2 , we have plotted our entire C H (T,B) data set versus yϭT/ͱB, setting Dϭ0 and Dϭ1, respectively. The result for Dϭ0 is shown in Fig. 3. While the open symbols represent the magnetic field scans at constant T, the full symbols correspond to the temperature scans at constant B. We may separate the investigated field regime into a low-field part (BϽB*Ϸ2 T) and a high-field part. Inspecting Fig. 3 it turns out that all the data collapse onto a single common curve, except for those obtained at BϽB*, where the scaling behavior according to Eq. ͑2͒ is not obeyed at all. We note that in Fig. 3 also data for fields up to BϷB c2 /2 are shown, but even for the highest fields applied here, the data reveal the scaling behavior. This scaling with respect to the scaling variable T/ͱB is rather unique and implies the occurrence of nodes in the superconducting energy gap. Since this scaling is appropriate for Dϭ0, we conclude that for BϾB* the energy-gap function of superconducting UBe 13 exhibits point nodes. Hence, we assume that the scaling prediction of Eq. ͑2͒ is more robust than previously expected and holds up to BϽB c2 /2. In the region of the BϪT phase diagram of superconducting UBe 13 where BϽB* and at the lowest temperatures, we cannot make any claims about the topology of possible gap nodes. We note, however, that the broad shoulderlike feature reported in Fig. 1͑b͒ is most probably related to this distinct change in the scaling behavior as may be seen in Fig. 3. In the investigated temperature range, some type of crossover from BϽB* to BϾB* at B*Ϸ2 T is also apparent, if we inspect the temperature derivative of C H (T,B) and display it in a plot of ‫ץ‬C H /‫ץ‬T͉ Tϭconst vs B ͑Fig. 4͒. For BϽB*, the data are rather well described by a power law as indicated by the solid line in Fig. 4, whereas for BϾB*, the behavior is distinctly different. At present, the origin of this crossover at 2 T remains unclear.
As a test, the same procedure, but setting Dϭ1, has been applied to both sets of data. No evidence for a scaling behavior in the whole investigated magnetic field and temperature range has been obtained, as may be seen in the inset of Fig. 3.
In the mixed state of a superconductor, a contribution to the specific heat might also arise from the low lying excitations localized near the core of the vortex. This problem has been discussed for s-wave superconductors by Caroli et al. 22 FIG. 2. Magnetic field induced contribution to the specific heat C H in constant magnetic fields divided by T vs temperature for various fields. FIG. 4. Derivative of the magnetic-field induced specific heat C H with respect to the temperature at Tϭ0.15 K vs magnetic field. The solid line is a power law fit to the data below 2 T, revealing an exponent of 5/3. A remarkable change may be seen at BϷ2 T at which the change of behavior observed in the scaling plot ͑Fig. 3͒.

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and by Bardeen et al. 23 According to that work, the lowest excitation level is expected at E 0 Ϸ⌬ 2 /E F , where ⌬ denotes the energy gap far away from the vortex. It is well established 4,24 and has recently been confirmed 25 that UBe 13 is a strong coupling superconductor, which implies that the energy-gap amplitude is substantially larger than the BCS weak-coupling value of ⌬Ϸ1.76k B T c . The Fermi-energy of UBe 13 , given by E F /k B , is of the order of 10 K. 4 Using these values and the predictions for s-wave superconductors, we find the lowest level of the low-energy excitations in the vortex core to be at significantly higher energies than the typical thermal energies k B T of this experiment. Furthermore, the observed scaling behavior with respect to TB Ϫ1/2 is, as discussed above, rather unique, and cannot be associated with the mentioned vortex-core contribution. 16 In conclusion we note, that for magnetic fields BϾB* Ϸ2 T the magnetic-field induced contribution to the specific heat exhibits a scaling behavior with respect to the scaling parameter T/ͱB which, according to Eq. ͑2͒ for Dϭ0, suggests the existence of pointlike nodes in the superconducting energy gap of UBe 13 . No scaling could be established for the superconducting state of UBe 13 at low temperatures and B ϽB*. A crossover behavior at BϭB* is also observed in the ‫ץ‬C H /‫ץ‬T vs B data. The same data set gives no similarly convincing evidence for the existence of nodes with DϾ0, in any regime of the BϪT phase diagram.
We thank Nippon Steel Corporation for letting us use their magnetic shields, allowing for a very accurate thermometry in magnetic fields. We acknowledge useful discussions with M. Sigrist. Part of this research was supported by the Schweizerische Nationalfonds zur Förderung der wissenschaftlichen Forschung.