In the Bardeen–Cooper–Schrieffer theory of superconductivity, electrons form (Cooper) pairs through an interaction mediated by vibrations in the underlying crystal structure. Like lattice vibrations, antiferromagnetic fluctuations can also produce an attractive interaction creating Cooper pairs, though with spin and angular momentum properties different from those of conventional superconductors. Such interactions have been implicated for two disparate classes of materials—the copper oxides1,2 and a set of Ce- and U-based compounds3. But because their transition temperatures differ by nearly two orders of magnitude, this raises the question of whether a common pairing mechanism applies. PuCoGa5 has a transition temperature intermediate between those classes and therefore may bridge these extremes4. Here we report measurements of the nuclear spin-lattice relaxation rate and Knight shift in PuCoGa5, which demonstrate that it is an unconventional superconductor with properties as expected for antiferromagnetically mediated superconductivity. Scaling of the relaxation rates among all of these materials (a feature not exhibited by their Knight shifts) establishes antiferromagnetic fluctuations as a likely mechanism for their unconventional superconductivity and suggests that related classes of exotic superconductors may yet be discovered.
Cooper pairs, which have zero net spin and angular momenta in conventional superconductors, condense into a macroscopic quantum state that is separated energetically from all unpaired electrons by a finite gap Δ. Because of this gap, measurements that probe the electronic density of states near the Fermi energy exhibit a thermally activated temperature dependence below the superconducting transition temperature Tc. On the other hand, Cooper pairs formed by the exchange of antiferromagnetic spin fluctuations possess even parity and non-zero angular momentum5; consequently, the superconducting energy gap is not finite everywhere but vanishes at points or along lines in momentum space. These gap nodes have a profound influence on the properties observed at low temperature. For any finite temperature, well-defined electronic excitations or quasiparticles reside at these nodes, and measurements sensitive to the electronic density of states exhibit power-law variations with temperature that depend solely on the topology of the gap zeros.
The high-Tc copper oxides and certain Ce- and U-based compounds, called heavy-fermion materials, have unconventional, nodal superconducting gaps and support antiferromagnetic fluctuations that lead naturally to these gap structures. These fluctuations common to both classes of materials are a consequence of strong electron–electron correlations: both the d-electrons in the copper oxides and the f-electrons in the heavy fermion materials experience strong on-site Coulomb repulsion that introduces elements of both localized and itinerant behaviour. In the d-electron materials these correlations create a Mott insulator in the undoped case and in the f-electron systems lead to enhanced effective electron mass. PuCoGa5 crystallizes in exactly the same structure as does one of these unconventional, heavy-fermion superconductors: CeCoIn5. Their common crystal structure and similarities of magnetic properties derived from nearly localized f-electrons suggest that PuCoGa5 may also be an unconventional superconductor. However, the Tc of PuCoGa5 is nearly an order of magnitude higher than that of CeCoIn5 (Tc = 2.3 K, the highest Tc among heavy-fermion systems)6, but comparable to that of many conventional superconductors, such as Nb3Sn (Tc ≈ 18 K). Without a direct probe of the gap symmetry of PuCoGa5 it has been impossible to differentiate conclusively between these two scenarios.
Nuclear magnetic resonance (NMR) and nuclear quadrupolar resonance (NQR) are powerful techniques used to make this distinction7,8,9,10,11,12. Both techniques probe the density of quasiparticle excitations, N(E), with excitation energy E above the Fermi energy, and reveal information about the spin state of the Cooper pairs and the pairing symmetry of the gap function in momentum space, Δ(k). We have used NMR and NQR to measure the Knight shift, Ks, and the nuclear spin-lattice relaxation rate, T1-1, of the 59Co and 69,71Ga nuclei in the normal and superconducting states of two samples of PuCoGa5. Figure 1a shows a series of 71Ga NMR spectra obtained from Sample A (see Methods) from which the total Knight shift Ktot is determined straightforwardly. Similar data were collected from 59Co NMR (not shown). The temperature dependence of these shifts, plotted in Fig. 1c, reveals a pronounced drop in Ktot of both 71Ga and 59Co nuclei at Tc. In the normal states of metals like PuCoGa5, Ktot is the sum of two contributions: Ktot = Korb + Ks, where Korb is temperature-independent, and Ks = Aχs, where A is a hyperfine constant, and χs is the spin susceptibility. The constants A and Korb were determined independently for 59Co and 71Ga nuclei above Tc, as shown in Fig. 1b. Quantitatively accounting for both Korb and the demagnetization field in the superconducting state, we obtain the temperature dependence of χs shown in Fig. 2.
In the superconducting state, χs probes the spin symmetry of the Cooper pair wavefunction. When two quasiparticles with up- and down-spin form a Cooper pair with total spin of either S = 0 or S = 1, the wavefunction is either antisymmetric (spin-singlet or more generally even-parity) or symmetric (spin-triplet or odd-parity) under particle exchange. For singlet pairing, χs decreases in the superconducting state, but for triplet pairing χs remains constant (depending on the orientation of the applied magnetic field12). The decrease evident in Fig. 2 clearly establishes PuCoGa5 as a spin-singlet superconductor. To satisfy Fermi statistics, the pair wavefunction must be antisymmetric with respect to particle exchange. Because the spin part of the wavefunction is antisymmetric (singlet), the symmetry of the orbital component of the wavefunction (given by (- 1)L, where the angular momentum L = 0, 1, 2, 3, … or s, p, d, f, …) must be symmetric, so L must be even. Thus, the results of Fig. 2 leave open the possibility that PuCoGa5 could be either a conventional s-wave or an unconventional superconductor with L > 0 and even.
Although χs drops below Tc, as expected for spin-singlet pairing, it should approach zero as T → 0 (neglecting minor vortex core contributions). In an unconventional, nodal superconductor, impurity scattering creates excitations in the superconducting gap nodes, and hence a finite spin susceptibility. Impurity scattering in PuCoGa5 is inevitable owing to lattice defects created by the recoil of uranium atoms during the radioactive decay of the plutonium, for example, 239Pu → 235U + α-particle13. The solid curve in Fig. 2, discussed below, shows that the temperature dependence of χs is described well by calculations that account for impurity scattering in a d-wave superconductor.
Further evidence for nodal superconductivity is provided by spin-lattice relaxation rate measurements on Sample B (see Methods). T1-1 measures the rate at which a nucleus reaches an equilibrium spin temperature; it is dominated by scattering between the conduction electrons and the nuclear spins. In the superconducting state, the condensate cannot relax the nuclei without breaking Cooper pairs, so the nuclei are predominantly relaxed by quasiparticles. Because the symmetry of the orbital part of the Cooper pair wavefunction determines the energy dependence of N(E) below the energy gap Δ, T1-1 establishes the pairing symmetry. The data are shown in Fig. 3a. Below Tc, T1-1 exhibits a sharp decrease, with roughly T1-1∝T3 behaviour, followed by T1-1∝T at the lowest temperatures. In contrast to expectations for an s-wave superconductor, the lack of a (Hebel–Slichter coherence) peak in T1-1 just below Tc and the power-law dependence of T1-1 are identical to responses found in the copper oxide superconductors and CeCoIn5 (refs 14, 15). Furthermore, the temperature dependence of T1-1 in the normal state of PuCoGa5 is qualitatively different than that observed in conventional BCS superconductors (see Fig. 3b), but has been predicted for a relaxation rate dominated by antiferromagnetic spin fluctuations16. Further, the normalized relaxation rates of PuCoGa5, high-Tc YBa2Cu3O7 and CeCoIn5 scale onto a common curve as a function of the dimensionless parameter T/Tc. Although we have chosen Tc as an easily defined characteristic energy, we also find that, with a spin-fluctuation energy of T0 ≈ 270 K for PuCoGa5 (ref. 17), the Tc and T0 values of PuCoGa5 have the same proportionality observed for several other unconventional superconductors18 (see Supplementary Fig. 1). The results of Fig. 3b, consequently, argue that PuCoGa5 is indeed a bridge between extreme cases. A priori, we might expect a similar scaling behaviour among the Knight shifts of these materials; however, the long wavelength (q = 0) response in the normal state of these systems is quite different from the finite q response that dominates the spin lattice relaxation2 (see Supplementary Fig. 2).
Self-consistency of our results is provided by calculations of the temperature dependence of T1-1 below Tc for sample B and the Knight shift for sample A within the framework of a self-consistent T-matrix approximation (SCTA)19,20, considering the effects of impurity scattering on both s- and d-wave pairing scenarios. The T1-1 data are best-fitted (Fig. 3a, solid curve) by a strong-coupling d-wave gap function with lines of nodes and the parameters Δ/kBTc = 4 and ΓB/Δ = 0.01, where ΓB is the impurity scattering rate for sample B. This gap value is significantly enhanced relative to the d-wave weak-coupling limit Δ/kBTc = 2.14 and very similar to that determined for CeCoIn5 (ref. 15). The temperature dependence of χs (Fig. 2 data) is fitted using the same gap magnitude, but with ΓA/Δ = 0.03 for sample A. The difference in the impurity lifetime broadening ΔΓ = ΓA - ΓB is due to the age difference and hence impurity scattering in samples A and B. From the Abrikosov–Gorkov relationship21 ΔTc = (π/4)ΔΓ, this difference (ΔΓ/Δ = 0.02) implies ΔTc = 1.2 K, in agreement with both the difference in Tc values of the two samples and time-dependent studies of Tc suppression (dTc/dt ≈ - 0.24 K per month; Jutier, F. et al., unpublished work). From our fits, we estimate dΓ/dt ≈ 0.25 K per month, which implies that Tc0 = 19.1 K for pristine, defect-free PuCoGa5, a value roughly half that of recent theoretical predictions22.
The total electronic energy calculations for PuCoGa5 and its isostructural neighbour NpCoGa5 predict that both should order antiferromagnetically23. As predicted, NpCoGa5 is an antiferromagnet below the Néel temperature TN ≈ 47 K (ref. 24), but there is no evidence for long-range magnetic order above 1 K in PuCoGa5 (ref. 25). These calculations neglect the role of many-electron correlations that lead to a nearly magnetic state in Ce- and U-based heavy-fermion systems, which, in the absence of such correlations, would order magnetically. The existence of these correlations is predicted for Pu-based materials26 and is reflected in PuCoGa5 through its enhanced electronic specific heat coefficient, γ ≈ 95 mJ mol-1 K-2 (ref. 17). As with the copper oxides and heavy-fermion systems like CeCoIn5, the Cooper pairing in PuCoGa5 is most probably mediated by antiferromagnetic fluctuations arising from proximity to an antiferromagnetic/paramagnetic border.
It thus appears that PuCoGa5 establishes continuity in the spectrum of energy scales controlling unconventional superconductivity in other f-electron systems and in the copper oxides, and that a universal tunable magnetic pairing mechanism may be operative in all materials with functional elements of both localized and itinerant electrons. This leads naturally to the speculation that there may be other materials classes in which antiferromagnetic fluctuations create an exotic form of superconductivity.
Spin lattice relaxation and Knight shift analysis
There are several nuclei in PuCoGa5 that can be investigated by NMR: 69Ga (I = 3/2), 71Ga (I = 3/2), and 59Co (I = 7/2). The nuclear hamiltonian is given by Ĥ = γhÎ B0(1 + Ktot) + hνc[3Îz2 - η(Îx2 - Îy2)], where h is Planck's constant, γ is the gyromagnetic ratio, B0 is the external magnetic field, νc = eQVcc/20, η = (Vaa - Vbb)/Vcc, Q is the quadrupolar moment and the V values are the components of the electric field gradient tensor. In the PuCoGa5 structure (Fig. 2 inset) (space group P4/mmm), the Co (1b site) lies directly below and above each Pu, has axial symmetry, and experiences an electric field gradient given by 59νQ = 1.70 MHz, η = 0. There are two different 69,71Ga sites: Ga(1) (1c site) and Ga(2) (4i site). The Ga(1) site has axial symmetry, has four nearest-neighbour Pu atoms, and experiences a large electric field gradient: 69νQ = 28.28 MHz, η = 0. For the Knight shift measurements, both the central line ( Iz = -1/2↔ + 1/2) of the Co and the upper satellite of the 71Ga ( Iz = +3/2↔ + 1/2) were measured. The resonance frequencies f of these transitions can be written in second-order perturbation theory (because γ B0≫νQ) by:
where θ is the angle between B0 and the c axis of the crystal. T1-1 was measured at the zero-field quadrupolar transition ( Iz = ± 3/2↔ ± 1/2) of the 69Ga(1) by fitting the time dependence of the magnetization recovery after inversion of the nuclear polarization. Sample A, used to measure Ks, was a single crystal oriented with the c axis 74.3° from the applied field (8 T) and was measured six months after its growth. Because of radiation damage, the Tc of sample A was reduced from its as-grown value of 18.5 K and in the measuring field of 8 T was ∼16 K. Sample B, used to measure T1-1, consisted of an unoriented powder that had aged two weeks, with Tc ≈ 18.5 K. Each sample was encapsulated inside an NMR solenoid coil embedded in Stycast epoxy. Thermal contact to the sample was established via gas transfer through stainless steel frits with 2-µm pore sizes located along the axis of the coil.
In the superconducting state, there is a third contribution to the total shift: -ΔB/B0, where ΔB (the demagnetization field) measures the reduction in the internal magnetic induction due to the diamagnetic shielding currents induced by the applied field. By measuring two different sites (59Co and 71Ga(1)) in PuCoGa5, we determine ΔB ≈ 40 Oe at 4 K (ref. 9). Because of the field orientation in our experiment, we can rule out a spin-triplet pairing state with a d-vector pointing along the crystallographic c axis, as proposed for the spin-triplet superconductors Sr2RuO4 and UPt3 (ref. 12).
For the theoretical fits we modelled the temperature dependence of the strong-coupling superconducting gap function as ΔT = Δtanh[b(Tc/T - 1)0.5]. The ratio of the specific-heat jump to CN (the normal-state specific heat at Tc) constrained the phenomenological parameters Δ and b as follows: ΔC/CN = (bΔ/(πkBTc))2/(2a), with a = 2/3 for a free-electron gas. Our choice of parameters Δ/kBTc = 4 and b = 1.74 lead to a moderately enhanced coefficient a ≈ 1.1–1.7 for ΔC/CN ≈ 1.43–2.28, consistent with reported values for the Sommerfeld coefficient γs = CN/T for T → 0 (refs 4, 23).
The spin lattice relaxation and spin susceptibility data were fitted to:
where f(E) is the Fermi–Dirac function, NF is the density of states at EF, Tn-1 is the normal state relaxation rate at Tc, and M(E) is the ‘anomalous’ density of quasiparticle states, which vanishes for d-wave superconductors. The ‘dirty’ d-wave calculation (assuming a finite density of states from impurities) in Fig. 3a assumes strong impurity scattering in the SCTA with ΓB/Δ = 0.01, Δ/kBTc = 4 and b = 1.74. Note that it is not possible to fit both the suppression of the Hebel–Slichter coherence peak just below Tc and the linear low-temperature behaviour with an isotropic s-wave pairing model, even when magnetic impurity scattering is included.
The normal-state relaxation rate data shown in Fig. 3b have been normalized to the value at 1.25Tc in order to avoid complications associated with the pseudogap effect, which suppresses T1-1 in the normal state just above Tc in the copper oxides14. There is a qualitative difference between the conventional and the d-wave superconductors below and above Tc. For the d-wave superconductors, the relaxation rate is given by a single scaling function f(T/Tc) up to at least 3Tc.
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We thank Z. Fisk, D. Pines and C. P. Slichter for discussions. This work was performed at Los Alamos National Laboratory under the auspices of the US Department of Energy Office of Science. Y.B. is supported by KOSEF through CSCMR.
The authors declare that they have no competing financial interests.
This file shows the superconducting transition temperature, Tc, versus the characteristic spin fluctuation temperature, T0, of several heavy fermion superconductors, high-Tc cuprate superconductors and PuCoGa5. This file also contains the Supplementary Figure Legend. (DOC 168 kb)
This Supplementary figure shows the normalized spin shift versus T/Tc in the normal state. This file also contains the Supplementary Figure Legend. (DOC 28 kb)
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Curro, N., Caldwell, T., Bauer, E. et al. Unconventional superconductivity in PuCoGa5. Nature 434, 622–625 (2005). https://doi.org/10.1038/nature03428
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