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Emergence of superconductivity in the canonical heavy-electron metal YbRh2Si2

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Science  29 Jan 2016:
Vol. 351, Issue 6272, pp. 485-488
DOI: 10.1126/science.aaa9733

Going to extremes to find superconductivity

Quantum phase transitions (QPTs) occur at zero temperature when parameters such as magnetic field or pressure are varied. In heavy fermion compounds, superconductivity often accompanies QPTs, a seeming exception being the material YbRh2Si2, which undergoes a magnetic QPT. Schuberth et al. performed magnetic and calorimetric measurements at extremely low temperatures and magnetic fields and found that it does become superconducting after all. Almost simultaneously with superconductivity, another order appeared that showed signatures of nuclear spin origin.

Science, this issue p. 485

Abstract

The smooth disappearance of antiferromagnetic order in strongly correlated metals commonly furnishes the development of unconventional superconductivity. The canonical heavy-electron compound YbRh2Si2 seems to represent an apparent exception from this quantum critical paradigm in that it is not a superconductor at temperature T ≥ 10 millikelvin (mK). Here we report magnetic and calorimetric measurements on YbRh2Si2, down to temperatures as low as T ≈ 1 mK. The data reveal the development of nuclear antiferromagnetic order slightly above 2 mK and of heavy-electron superconductivity almost concomitantly with this order. Our results demonstrate that superconductivity in the vicinity of quantum criticality is a general phenomenon.

Unconventional (i.e., nonphonon-mediated) superconductivity, which has been attracting much interest since the early 1980s, is often observed at the border of antiferromagnetic (AF) order (1). As exemplified by heavy-electron (or heavy-fermion) metals, the suppression of the AF order opens up a wide parameter regime where the physics is controlled by an underlying quantum critical point (QCP) (2, 3). A central question, then, concerns the interplay between quantum criticality and unconventional superconductivity in strongly correlated electron systems such as heavy-electron metals. In many heavy-electron metals, superconductivity turns out to develop near such a QCP (25). However, the absence of superconductivity in the prototypical quantum critical material YbRh2Si2 (6) has raised the question as to whether the presence of an AF QCP necessarily gives rise to the occurrence of superconductivity. Because YbRh2Si2 exists in the form of high-quality single crystals, we are able to address this issue at very low temperatures without seriously encountering the limitations posed by disorder. YbRh2Si2 exhibits AF order below a Néel temperature TAF = 70 mK. When applied within the basal plane of the tetragonal structure, a small magnetic field of B = 60 mT continuously suppresses the magnetic order and induces a QCP. Along with those of CeRhIn5 (79) and CeCu6–xAux (10, 11), the QCP in YbRh2Si2 has been exclusively demonstrated (12, 13) to be of the unconventional type with Kondo breakdown (1416). Electrical resistivity measurements down to 10 mK have failed to show any indications for superconductivity (6). Because a critical field of 60 mT is likely to destroy heavy-electron superconductivity with a superconducting transition temperature (Tc) of less than 10 mK, a different means of suppressing the antiferromagnetism is needed to eventually reveal any potential superconductivity at its border. We note that the application of pressure does not facilitate realization of a QCP in an AF Yb-based material, as increased pressure will strengthen the magnetic order—contrary to the case of Ce-based systems where magnetism usually becomes weakened by pressure. Compared to Ce, which does not exhibit a nuclear spin, two of the Yb isotopes have finite nuclear spin values [see below and section F of the supplementary materials (17)].

We have carried out magnetic and calorimetric measurements on high-quality YbRh2Si2 single crystals, using a nuclear-demagnetization cryostat with a base temperature of 400 μK (17). Figure 1, A and B, display the temperature dependence of the field-cooled (fc) dc magnetization M(T), measured upon warming at various magnetic fields B ranging from 0.09 to 25 mT, applied within the basal plane of the YbRh2Si2 single crystals. The curves display peaks at 70 mK, which is the well-established Néel temperature for the AF order, as well as additional low-temperature anomalies. There is a second peak in M(T)/B at Embedded Image mK, which indicates an almost-simultaneous onset of a nuclear-dominated AF order (“A phase”) and the Meissner effect (see below). It is visible above 1 mK up to 23 mT and had already been observed previously (18). In addition, there is a shoulder around TB ≈ 10 mK, as defined in Fig. 1C. Below TB, the results of the fc and zero-field-cooled (zfc) measurements become different. This divergence, which is ascribed to superconducting fluctuations [section F of (17)], can be followed as a function of the magnetic field, up to the limit of our setup (B = 0.5 mT) for measurements of the dc magnetization cooled at zero field.

Fig. 1 Temperature dependence of the dc magnetization and ac susceptibility for YbRh2Si2.

(A) Field-cooled (fc) dc magnetization curve of YbRh2Si2 taken at B = 0.09 mT applied within the basal plane. Three main features are clearly visible: the AF phase transition at TAF = 70 mK, a shoulder in magnetization at TB ≈ 10 mK, and a sharp peak at Tc = 2 mK. (B) Series of fc magnetization data taken at fields of 0.10, 1.13, 1.13, 5.01, 7.48, 10.12, 15.01, 20.04, 22.42, and 25.02 mT. (C) Zero-field-cooled (zfc) and fc dc magnetization traces taken at selected small magnetic fields. The traces at 0.028, 0.055, and 0.418 mT were shifted upward for better visibility. For the smallest magnetic field of 0.012 mT, a sharp diamagnetic shielding signal is observed, suggesting a superconducting phase transition. (D) The ac susceptibility was measured using a superconducting quantum interference device magnetometer by modulating a primary coil around the pickup coils. Here we show the in-phase signal χ′ac(T) (at 17 Hz), having compensated the Earth field. The features seen at TAF, TB, and Tc in the dc magnetization curve are also detected by the ac susceptibility. The large negative values of the zfc dc magnetization at B = 0.012 mT (Fig. 1C) and of χ′ac(T) indicate superconducting shielding, whereas the low-temperature peak in the fc dc magnetization (Fig. 1, A to C) signals the onset of the Meissner effect. Measurements in (A) to (D) were performed on samples 1, 2, 3, and 4, respectively.

At Embedded Image mK, the zfc dc M(T)/B (0.012 mT) shows a sharp increase upon warming, starting from negative values (Fig. 1C). This indicates a substantial shielding signal due to superconductivity. Raising the temperature further, the zfc M(T)/B slowly increases until it meets the fc curve at 10 mK. To verify this finding, we carried out measurements of the ac susceptibility, χac, under nearly zero-field conditions [section D of (17)]. Its real part, χ′ac(T), displays an even more pronounced diamagnetic signal (Fig. 1D), larger than what was found for the canonical heavy-electron superconductor CeCu2Si2 (19), again confirming the occurrence of superconducting shielding. In addition, the reduction of the fc magnetization upon cooling below 2 mK reflects flux expulsion from the sample (Meissner effect). The relatively small Meissner volume of ≈3% is most likely due to strong flux pinning [section C of (17)]. As shown in fig. S7, the superconducting phase transition is of first order. This suggests that superconductivity does not coexist on a microscopic basis with AF order, as previously observed for A/S-type CeCu2Si2 [compare with section D of (17)].

In Fig. 2A, the specific heat is displayed as C(T)/T at B = 2.4 and 59.6 mT, respectively. As the electronic specific heat can be completely neglected below T ≈ 10 mK (20), C(T) denotes the nuclear contribution in this low-T regime. The solid lines show the calculated nuclear specific heats at various fields from (20), which include the quadrupolar as well as the Zeeman terms. At zero field, the nuclear specific heat is completely dominated by the nuclear quadrupole states, to which the Zeeman terms due to the nuclear spin states add at B > 0. In Fig. 2B, we display ΔC(T)/T, where ΔC marks the difference between the specific heat measured at the lowest field B = 2.4 mT and the nuclear quadrupole contribution calculated for B = 0 (20). Our ΔC(T)/T results clearly reveal a peak at T ≈ 1.7 mK. Assuming a continuous phase transition, the transition temperature can be obtained by replacing the high-T part of this peak by a sharp jump while keeping the entropy unchanged. This yields a jump height of ~1000 J/KEmbedded Image mol and TA ≈ 2 mK (B = 2.4 mT), almost coinciding with Tc (Fig. 1). Because the effect of the magnetic field on the quadrupole contribution to the nuclear specific heat is of higher order only, we can use the ΔC(T)/T data of Fig. 2B to estimate the nuclear spin entropy (at B = 2.4 mT), SI(T) [section F of (17)]. Embedded Image (where R is the gas constant), the total nuclear spin entropy of YbRh2Si2 for B = 2.4 mT, is reached at T ≈ 10 mK, where ΔC(T) vanishes within the experimental uncertainty (Fig. 2C). Upon cooling to T = TA, SI(T) decreases to ~0.94SI,tot—that is, most of this nuclear spin entropy must be released below the phase transition temperature TA. The entropy of the 103Rh and 29Si spins is temperature independent at T > 1 mK, but the Yb-derived spin entropy SYb(T) decreases by 26% upon cooling from 10 to 2 mK [compare with section F of (17)]. This indicates substantial short-range order, consistent with a second-order (antiferro)magnetic phase transition. We stress that this very large entropy at ultralow temperatures (Fig. 2C) can only be understood if the ordering transition at TA involves the Yb-derived nuclear spins to a substantial degree.

Fig. 2 Nuclear specific heat and entropy of YbRh2Si2.

(A) The temperature dependence of the specific heat C(T) of YbRh2Si2 divided by T is shown for B = 2.4 and 59.6 mT [section E of (17)]. The 2.4-mT data extend down to 1.4 mK. The solid lines denote the calculated nuclear specific heat from (20), which is the sum of the quadrupolar term and the Zeeman term with three selected field-induced Yb-derived ordered magnetic moments: 0.01, 0.05, and 0.15(μYbB). The large errors at temperatures above 10 mK are due to the uncertainty in the subtraction of the addendum [section E of (17)]. (B) ΔC(T)/T was obtained by subtracting the nuclear quadrupolar contribution calculated for B = 0 from the data at B = 2.4 mT. A peak in ΔC(T)/T occurs at ≈1.7 mK. Assuming the transition to be of second order, an equal-area construction yields a nuclear phase transition temperature TA ≈ 2 mK. This coincides with the peak position found in the dc magnetization (compare with Fig. 1)—that is, the superconducting critical temperature Tc at 2.4 mT. The associated jump of ΔC(T)/T is on the order of 1000 J/K2 mol. The error bars reflect the statistical error in the measurements of the specific heat by using a quasi-static heat-pulse technique, as well as the relaxation method. In the latter case, the error bar contains the statistical error in determining both the relaxation time and the heat conductivity of the weak link. In total, two runs have been performed for each field; therefore, four sets of data at the same temperature were used for determining the specific heat. Each data point was weighted by its reciprocal error. At the lowest temperatures, the error associated with the relaxation method is essentially smaller than that of the heat-pulse measurement. (C) From ΔC(T)/T (Fig. 1B), a rough estimate can be made for the nuclear spin entropy SI(T) at B = 2.4 mT (see text). We have normalized SI(T) to SI,tot, the total nuclear spin entropy in YbRh2Si2 at B = 2.4 mT, reached at ~10 mK. By subtracting from SI(T) the contribution of the nuclear Si and Rh spins, which is temperature independent at T ≥ 1 mK, we obtain the corresponding values SYb(T) for the nuclear Yb spins. Measurements were performed on sample 3.

To explore the role of the nuclear spins in the phase diagram [(3, 6) and Fig. 3], we take advantage of the early recognition that hyperfine coupling to nuclear spins can considerably influence the electronic spin properties near a quantum phase transition (21). Furthermore, measurements on PrCu2 and related compounds have demonstrated a large coupling between the electronic and nuclear spins in rare-earth–based intermetallics at temperatures up to 50 mK (22, 23). These considerations raise the possibility of using the presence of nuclear spins to weaken the electronic AF order, thereby enabling the formation of a superconducting state. We have written down a Landau theory of the interplay between the magnetic orders of the electronic and nuclear spins. Consider the electronic AF order, with an order parameter mAF at the AF wave vector QAF, as well as two bilinearly coupled order parameters, mJ and mI, the staggered magnetizations of the electronic and nuclear spins at another finite wave vector Embedded Image. The bilinear coupling arises from the hyperfine coupling between the two order parameters having the same wave vector. The Landau theory will then have the following free-energy functionalEmbedded Image where Embedded Image, Embedded Image, and Embedded Image are, respectively, the normalized order parameters mAF, mJ, and mI; the r terms are quadratic couplings; the u terms as well as ε and η are the intracomponent as well as intercomponent quartic couplings; and λ is the bilinear hyperfine coupling between two normalized order parameters [section G of (17)].

Under suitable conditions (17), this can lead to two stages of phase transitions (Fig. 4). The phase transition at TAF corresponds to the primary AF order setting in at ~70 mK and is not much affected by the nuclear spins. In a suitable parameter range of the Landau theory, the nuclear Embedded Image order dominates over the electronic Embedded Image order and, furthermore, suppresses the primary electronic Embedded Image order. A second transition occurs at Thyb, which represents a hybrid electronic-nuclear spin order. The component that is associated with the nuclear spins generates substantial entropy for the transition, which explains the large nuclear spin entropy that is experimentally observed (Fig. 2C) [section E of (17)]. In addition, the effective g-factor (geff) is on the order of Embedded Image (where gel is the electron g-factor), which is substantially smaller than the bare g-factor for the 4f electrons. This explains the geff < 0.1 observed in our experiment.

Fig. 3 Generic T versus B phase diagram of YbRh2Si2.

This phase diagram (shown on semi-logarithmic scales) is obtained from dc magnetization and ac susceptibility measurements in several magnetic fields. Four samples were measured, and no sample dependence was found. AF indicates the electronic AF order (TAF = 70 mK); PM indicates the paramagnetic state. All data points used to illustrate the AF-PM phase boundary TAF(B) were obtained in the present study. The hatched light-blue area indicates the onset of A-phase fluctuations, which give rise to a reduction of the staggered magnetization and a splitting of the zfc and fc dc magnetization curves (i.e., the beginning of shielding due to superconducting fluctuations) (Fig. 1C). The two data points (gray triangles) determined via field sweeps of the dc magnetization between 3.6 and 6.0 mK (fig. S4) are most likely not related to these A-phase fluctuations. The A + SC phase represents the concurring (dominantly) nuclear AF order and superconductivity, at least at fields below 3 to 4 mT. Only at B = 0 is nearly full shielding observed. The low-temperature limit of our experiment is ~800 μK; therefore, we cannot detect the fc dc M(T) peaks above 23 mT. The two red dashed lines mark the range within which the A-phase boundary line may end. At low fields (B < 4 mT), the transition at ~2 mK is split into two parts (compare with fig. S3). The green circle indicates the superconducting transition temperature seen in the ac susceptibility at B = 0 (compare with Fig. 1D), whereas the yellow circles (partially covered by the green point) result from the shielding signals in the zfc dc magnetization (Fig. 1C). (Inset) These shielding transitions are shown separately on an enlarged scale. The superconducting phase boundary Tc versus B is extremely steep at low fields, with Embedded Image T/K, consistent with results for the canonical heavy-electron superconductor CeCu2Si2 [section A of (17)]. If superconductivity exists at higher fields, Bc2(T) extrapolates to 30 to 60 mT (at T = 0)—that is, close to the critical field of the primary electronic AF phase.

We thus conclude that the A phase forming at Embedded Image mK is an electronic-nuclear hybrid phase dominated by the Yb-derived nuclear spin ordering. We estimate that the small (1 to 2%) 4f electronic component contributes about one-third of the decrease in M(T) below TA [section C of (17)]. As the nuclear phase transition cannot be resolved because of the very small nuclear moment, the major part of this reduction of M(T) (i.e., the other two-thirds) must be due to the Meissner effect [section C of (17)]. A measurement of the fc dc magnetization at very low fields reveals two separated phase transitions close to T = 2 mK: TA and Tc (fig. S3B). Upon increasing the field to ~3 to 4 mT, however, TA and Tc appear to merge within the experimental uncertainty (fig. S3C). As mentioned previously, this peak in the fc dc M(T) curve remains visible (above 1 mK) up to Embedded Image mT (Fig. 1B). By analyzing magnetization data taken between 0.8 and 540 mK at a field of 10.1 mT (fig. S4), we conclude that superconductivity is likely to exist and coincide with the A phase at elevated fields, consistent with the evolution of the M(T) peak as a function of field [section C of (17)].

Figure 4 describes a possible scenario for the two stages of transitions. Below TAF, the Néel order develops. We speculate that the growth of the Néel order parameter mAF is arrested as the temperature is lowered past Thyb, due to the onset of the nuclear spin order. A diminished mAF would place the electronic phase in the regime close to the QCP that underlies the pure electronic system in the absence of any hyperfine coupling. This quantum criticality effectively induced by the nuclear spin order at zero magnetic field would naturally lead to the development of a superconducting state [section I of (17)]. As inferred from the experimental results (Fig. 1C), fluctuations of the A phase are already set in near TB and lead to a substantial reduction of the staggered magnetization and the emergence of superconducting fluctuations well above the A-phase ordering temperature [section I of (17)].

The large initial slope of the superconducting upper critical field Bc2(T) at Embedded Image T/K, extracted from both shielding (Fig. 3, inset) and Meissner measurements (fig. S3C), corresponds to an effective charge-carrier mass of several hundred me (where me is the rest mass of the electron), which implies that the superconducting state is associated with the Yb-derived 4f electrons (heavy-electron superconductivity). Extrapolating the positions of the low-temperature fc M(T) peak to zero temperature, the critical field of the A phase BA = B(TA → 0) is found to be 30 to 60 mT, which corresponds to geff = kBTA(B = 0)/μBBA = 0.03 to 0.06 (where kB is the Boltzmann constant and μB is the Bohr magneton). This value of geff is much smaller than the in-plane electronic g-factor 3.5 (24) but is a factor of 20 to 40 larger than in case of a purely nuclear spin ordering transition. We can understand this geff if the ordered moment is a hybrid of the electronic and nuclear spins with, at most, 2% of the ordered moments being associated with the 4f electron–derived spins.

Fig. 4 Phase transitions at TAF and Thyb.

(A) Sketch of the two phase transitions associated with electronic and nuclear spin orders. (Top line) Without any hyperfine coupling (Ahf), the electronic and nuclear spins are ordered at TAF and TI, respectively. (Bottom line) With a hyperfine coupling, TAF is not affected, but a hybrid nuclear and electronic spin order is induced at Embedded Image. (B) Temperature evolution of the primary electronic spin order parameter (mAF) and the superconducting order parameter Embedded Image. Embedded Image is developed when mAF is suppressed by the formation of hybrid nuclear and electronic spin order directly below Thyb.

The very large entropy near TA ≥ 2 mK is one of the most pronounced features in our observation. An alternative possibility for this entropy is the involvement of a “nuclear Kondo effect”—that is, the formation of a singlet state between the nuclear and conduction electron spins. The resulting superheavy fermions may be assumed to form Cooper pairs and cause a superconducting transition at Tc ≈ 2 mK that would be probed by the magnetic and specific-heat measurements. Though our estimates of the nuclear Kondo temperature and the quasi-particle effective mass reveal discrepancies with this picture [section E of (17)], further theoretical and experimental work is needed to investigate the possible role of the nuclear Kondo effect in generating superconductivity in YbRh2Si2.

It is likely that the coupling of electronic and nuclear spin orders, as well as the concomitant emergence of new physics, is not exclusive to YbRh2Si2 [section H of (17)]. Systematic studies of other heavy-electron antiferromagnets at ultralow temperatures are needed to find out whether a hybrid electronic-nuclear order is a more general phenomenon. In addition, a comparative study would be highly welcome to evaluate whether superconductivity is truly absent in isotopically enriched YbRh2Si2 without Yb-derived nuclear spins, similar to the compound studied in (25).

Our ultralow-temperature measurements on the unconventional quantum critical material YbRh2Si2 reveal heavy-electron superconductivity below Tc = 2 mK. This observation strongly supports the notion that superconductivity near an AF instability is a robust phenomenon.

Supplementary Materials

www.sciencemag.org/content/351/6272/485/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S9

References (2665)

References and Notes

  1. Supplementary materials are available on Science Online.
Acknowledgments: We thank K. Andres, P. Coleman, P. Gegenwart, S. Paschen, and S. Wirth for useful discussions. Part of the work at the Max Planck Institute for Chemical Physics of Solids was supported by the Deutsche Forschungsgemeinschaft Research Unit 960 “Quantum Phase Transitions.” Q.S. was supported by NSF grant DMR-1309531 and Robert A. Welch Foundation grant C-1411. E.S., Q.S., and F.S. thank the Institute of Physics, Chinese Academy of Sciences, Beijing, for hospitality. Q.S. and F.S. acknowledge partial support from the NSF under grant 1066293 and the hospitality of the Aspen Center for Physics. We declare no competing financial interests.
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