## 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 YbRh_{2}Si_{2}, 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 YbRh_{2}Si_{2} 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 YbRh_{2}Si_{2}, 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 (*2*–*5*). However, the absence of superconductivity in the prototypical quantum critical material YbRh_{2}Si_{2} (*6*) has raised the question as to whether the presence of an AF QCP necessarily gives rise to the occurrence of superconductivity. Because YbRh_{2}Si_{2} 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. YbRh_{2}Si_{2} exhibits AF order below a Néel temperature *T*_{AF} = 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 CeRhIn_{5} (*7*–*9*) and CeCu_{6–}* _{x}*Au

*(*

_{x}*10*,

*11*), the QCP in YbRh

_{2}Si

_{2}has been exclusively demonstrated (

*12*,

*13*) to be of the unconventional type with Kondo breakdown (

*14*–

*16*). 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 (

*T*

_{c}) 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 YbRh_{2}Si_{2} 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 YbRh_{2}Si_{2} 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 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 *T*_{B} ≈ 10 mK, as defined in Fig. 1C. Below *T*_{B}, 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.

At 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, χ_{ac}, 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 CeCu_{2}Si_{2} (*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 CeCu_{2}Si_{2} [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 Δ*C*(*T*)/*T*, where Δ*C* 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 Δ*C*(*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/K mol and *T*_{A} ≈ 2 mK (*B* = 2.4 mT), almost coinciding with *T*_{c} (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 Δ*C*(*T*)/*T* data of Fig. 2B to estimate the nuclear spin entropy (at *B* = 2.4 mT), *S*_{I}(*T*) [section F of (*17*)]. (where *R* is the gas constant), the total nuclear spin entropy of YbRh_{2}Si_{2} for *B* = 2.4 mT, is reached at *T* ≈ 10 mK, where Δ*C*(*T*) vanishes within the experimental uncertainty (Fig. 2C). Upon cooling to *T* = *T*_{A}, *S*_{I}(*T*) decreases to ~0.94*S*_{I,tot}—that is, most of this nuclear spin entropy must be released below the phase transition temperature *T*_{A}. The entropy of the ^{103}Rh and ^{29}Si spins is temperature independent at *T* > 1 mK, but the Yb-derived spin entropy *S*_{Yb}(*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 *T*_{A} involves the Yb-derived nuclear spins to a substantial degree.

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 PrCu_{2} 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 *m*_{AF} at the AF wave vector **Q**_{AF}, as well as two bilinearly coupled order parameters, *m*_{J} and *m*_{I}, the staggered magnetizations of the electronic and nuclear spins at another finite wave vector . 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 functional where , , and are, respectively, the normalized order parameters *m*_{AF}, *m*_{J}, and *m*_{I}; 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 *T*_{AF} 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 order dominates over the electronic order and, furthermore, suppresses the primary electronic order. A second transition occurs at *T*_{hyb}, 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 (*g*_{eff}) is on the order of (where *g*_{el} is the electron *g*-factor), which is substantially smaller than the bare *g*-factor for the 4f electrons. This explains the *g*_{eff} < 0.1 observed in our experiment.

We thus conclude that the A phase forming at 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 *T*_{A} [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: *T*_{A} and *T*_{c} (fig. S3B). Upon increasing the field to ~3 to 4 mT, however, *T*_{A} and *T*_{c} 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 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 *T*_{AF}, the Néel order develops. We speculate that the growth of the Néel order parameter *m*_{AF} is arrested as the temperature is lowered past *T*_{hyb}, due to the onset of the nuclear spin order. A diminished *m*_{AF} 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 *T*_{B} 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 *B*_{c2}(*T*) at T/K, extracted from both shielding (Fig. 3, inset) and Meissner measurements (fig. S3C), corresponds to an effective charge-carrier mass of several hundred *m*_{e} (where *m*_{e} 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 *B*_{A} = *B*(*T*_{A} → 0) is found to be 30 to 60 mT, which corresponds to *g*_{eff} = *k*_{B}*T*_{A}(*B* = 0)/μ_{B}*B*_{A} = 0.03 to 0.06 (where *k*_{B} is the Boltzmann constant and μ_{B} is the Bohr magneton). This value of *g*_{eff} 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 *g*_{eff} 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.

The very large entropy near *T*_{A} ≥ 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 *T*_{c} ≈ 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 YbRh_{2}Si_{2}.

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 YbRh_{2}Si_{2} [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 YbRh_{2}Si_{2} without Yb-derived nuclear spins, similar to the compound studied in (*25*).

Our ultralow-temperature measurements on the unconventional quantum critical material YbRh_{2}Si_{2} reveal heavy-electron superconductivity below *T*_{c} = 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 and Notes

**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.