Abstract
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 oxides^{1,2} and a set of Ce and Ubased compounds^{3}. But because their transition temperatures differ by nearly two orders of magnitude, this raises the question of whether a common pairing mechanism applies. PuCoGa_{5} has a transition temperature intermediate between those classes and therefore may bridge these extremes^{4}. Here we report measurements of the nuclear spinlattice relaxation rate and Knight shift in PuCoGa_{5}, 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.
Main
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 T_{c}. On the other hand, Cooper pairs formed by the exchange of antiferromagnetic spin fluctuations possess even parity and nonzero angular momentum^{5}; 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, welldefined electronic excitations or quasiparticles reside at these nodes, and measurements sensitive to the electronic density of states exhibit powerlaw variations with temperature that depend solely on the topology of the gap zeros.
The highT_{c} copper oxides and certain Ce and Ubased compounds, called heavyfermion 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 delectrons in the copper oxides and the felectrons in the heavy fermion materials experience strong onsite Coulomb repulsion that introduces elements of both localized and itinerant behaviour. In the delectron materials these correlations create a Mott insulator in the undoped case and in the felectron systems lead to enhanced effective electron mass. PuCoGa_{5} crystallizes in exactly the same structure as does one of these unconventional, heavyfermion superconductors: CeCoIn_{5}. Their common crystal structure and similarities of magnetic properties derived from nearly localized felectrons suggest that PuCoGa_{5} may also be an unconventional superconductor. However, the T_{c} of PuCoGa_{5} is nearly an order of magnitude higher than that of CeCoIn_{5} (T_{c} = 2.3 K, the highest T_{c} among heavyfermion systems)^{6}, but comparable to that of many conventional superconductors, such as Nb_{3}Sn (T_{c} ≈ 18 K). Without a direct probe of the gap symmetry of PuCoGa_{5} 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 distinction^{7,8,9,10,11,12}. 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, K_{s}, and the nuclear spinlattice relaxation rate, T_{1}^{1}, of the ^{59}Co and ^{69,71}Ga nuclei in the normal and superconducting states of two samples of PuCoGa_{5}. Figure 1a shows a series of ^{71}Ga NMR spectra obtained from Sample A (see Methods) from which the total Knight shift K_{tot} is determined straightforwardly. Similar data were collected from ^{59}Co NMR (not shown). The temperature dependence of these shifts, plotted in Fig. 1c, reveals a pronounced drop in K_{tot} of both ^{71}Ga and ^{59}Co nuclei at T_{c}. In the normal states of metals like PuCoGa_{5}, K_{tot} is the sum of two contributions: K_{tot} = K_{orb} + K_{s}, where K_{orb} is temperatureindependent, and K_{s} = Aχ_{s}, where A is a hyperfine constant, and χ_{s} is the spin susceptibility. The constants A and K_{orb} were determined independently for ^{59}Co and ^{71}Ga nuclei above T_{c}, as shown in Fig. 1b. Quantitatively accounting for both K_{orb} and the demagnetization field in the superconducting state, we obtain the temperature dependence of χ_{s} shown in Fig. 2.
In the superconducting state, χ_{s} probes the spin symmetry of the Cooper pair wavefunction. When two quasiparticles with up and downspin form a Cooper pair with total spin of either S = 0 or S = 1, the wavefunction is either antisymmetric (spinsinglet or more generally evenparity) or symmetric (spintriplet or oddparity) under particle exchange. For singlet pairing, χ_{s} decreases in the superconducting state, but for triplet pairing χ_{s} remains constant (depending on the orientation of the applied magnetic field^{12}). The decrease evident in Fig. 2 clearly establishes PuCoGa_{5} as a spinsinglet 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 PuCoGa_{5} could be either a conventional swave or an unconventional superconductor with L > 0 and even.
Although χ_{s} drops below T_{c}, as expected for spinsinglet 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 PuCoGa_{5} is inevitable owing to lattice defects created by the recoil of uranium atoms during the radioactive decay of the plutonium, for example, ^{239}Pu → ^{235}U + αparticle^{13}. The solid curve in Fig. 2, discussed below, shows that the temperature dependence of χ_{s} is described well by calculations that account for impurity scattering in a dwave superconductor.
Further evidence for nodal superconductivity is provided by spinlattice relaxation rate measurements on Sample B (see Methods). T_{1}^{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, T_{1}^{1} establishes the pairing symmetry. The data are shown in Fig. 3a. Below T_{c}, T_{1}^{1} exhibits a sharp decrease, with roughly T_{1}^{1}∝T^{3} behaviour, followed by T_{1}^{1}∝T at the lowest temperatures. In contrast to expectations for an swave superconductor, the lack of a (Hebel–Slichter coherence) peak in T_{1}^{1} just below T_{c} and the powerlaw dependence of T_{1}^{1} are identical to responses found in the copper oxide superconductors and CeCoIn_{5} (refs 14, 15). Furthermore, the temperature dependence of T_{1}^{1} in the normal state of PuCoGa_{5} 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 fluctuations^{16}. Further, the normalized relaxation rates of PuCoGa_{5}, highT_{c} YBa_{2}Cu_{3}O_{7} and CeCoIn_{5} scale onto a common curve as a function of the dimensionless parameter T/T_{c}. Although we have chosen T_{c} as an easily defined characteristic energy, we also find that, with a spinfluctuation energy of T_{0} ≈ 270 K for PuCoGa_{5} (ref. 17), the T_{c} and T_{0} values of PuCoGa_{5} have the same proportionality observed for several other unconventional superconductors^{18} (see Supplementary Fig. 1). The results of Fig. 3b, consequently, argue that PuCoGa_{5} 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 relaxation^{2} (see Supplementary Fig. 2).
Selfconsistency of our results is provided by calculations of the temperature dependence of T_{1}^{1} below T_{c} for sample B and the Knight shift for sample A within the framework of a selfconsistent Tmatrix approximation (SCTA)^{19,20}, considering the effects of impurity scattering on both s and dwave pairing scenarios. The T_{1}^{1} data are bestfitted (Fig. 3a, solid curve) by a strongcoupling dwave gap function with lines of nodes and the parameters Δ/k_{B}T_{c} = 4 and Γ^{B}/Δ = 0.01, where Γ^{B} is the impurity scattering rate for sample B. This gap value is significantly enhanced relative to the dwave weakcoupling limit Δ/k_{B}T_{c} = 2.14 and very similar to that determined for CeCoIn_{5} (ref. 15). The temperature dependence of χ_{s} (Fig. 2 data) is fitted using the same gap magnitude, but with Γ^{A}/Δ = 0.03 for sample A. The difference in the impurity lifetime broadening ΔΓ = Γ^{A}  Γ^{B} is due to the age difference and hence impurity scattering in samples A and B. From the Abrikosov–Gorkov relationship^{21} ΔT_{c} = (π/4)ΔΓ, this difference (ΔΓ/Δ = 0.02) implies ΔT_{c} = 1.2 K, in agreement with both the difference in T_{c} values of the two samples and timedependent studies of T_{c} suppression (dT_{c}/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 T_{c0} = 19.1 K for pristine, defectfree PuCoGa_{5}, a value roughly half that of recent theoretical predictions^{22}.
The total electronic energy calculations for PuCoGa_{5} and its isostructural neighbour NpCoGa_{5} predict that both should order antiferromagnetically^{23}. As predicted, NpCoGa_{5} is an antiferromagnet below the Néel temperature T_{N} ≈ 47 K (ref. 24), but there is no evidence for longrange magnetic order above 1 K in PuCoGa_{5} (ref. 25). These calculations neglect the role of manyelectron correlations that lead to a nearly magnetic state in Ce and Ubased heavyfermion systems, which, in the absence of such correlations, would order magnetically. The existence of these correlations is predicted for Pubased materials^{26} and is reflected in PuCoGa_{5} through its enhanced electronic specific heat coefficient, γ ≈ 95 mJ mol^{1} K^{2} (ref. 17). As with the copper oxides and heavyfermion systems like CeCoIn_{5}, the Cooper pairing in PuCoGa_{5} is most probably mediated by antiferromagnetic fluctuations arising from proximity to an antiferromagnetic/paramagnetic border.
It thus appears that PuCoGa_{5} establishes continuity in the spectrum of energy scales controlling unconventional superconductivity in other felectron 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.
Methods
Spin lattice relaxation and Knight shift analysis
There are several nuclei in PuCoGa_{5} that can be investigated by NMR: ^{69}Ga (I = 3/2), ^{71}Ga (I = 3/2), and ^{59}Co (I = 7/2). The nuclear hamiltonian is given by Ĥ = γhÎ B_{0}(1 + K_{tot}) + hν_{c}[3Î_{z}^{2}  η(Î_{x}^{2}  Î_{y}^{2})], where h is Planck's constant, γ is the gyromagnetic ratio, B_{0} is the external magnetic field, ν_{c} = eQV_{cc}/20, η = (V_{aa}  V_{bb})/V_{cc}, Q is the quadrupolar moment and the V values are the components of the electric field gradient tensor. In the PuCoGa_{5} 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 ^{59}ν_{Q} = 1.70 MHz, η = 0. There are two different ^{69,71}Ga sites: Ga(1) (1c site) and Ga(2) (4i site). The Ga(1) site has axial symmetry, has four nearestneighbour Pu atoms, and experiences a large electric field gradient: ^{69}ν_{Q} = 28.28 MHz, η = 0. For the Knight shift measurements, both the central line ( I_{z} = 1/2↔ + 1/2) of the Co and the upper satellite of the ^{71}Ga ( I_{z} = +3/2↔ + 1/2) were measured. The resonance frequencies f of these transitions can be written in secondorder perturbation theory (because γ B_{0}≫ν_{Q}) by:
where θ is the angle between B_{0} and the c axis of the crystal. T_{1}^{1} was measured at the zerofield quadrupolar transition ( I_{z} = ± 3/2↔ ± 1/2) of the ^{69}Ga(1) by fitting the time dependence of the magnetization recovery after inversion of the nuclear polarization. Sample A, used to measure K_{s}, 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 T_{c} of sample A was reduced from its asgrown value of 18.5 K and in the measuring field of 8 T was ∼16 K. Sample B, used to measure T_{1}^{1}, consisted of an unoriented powder that had aged two weeks, with T_{c} ≈ 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 2µm pore sizes located along the axis of the coil.
In the superconducting state, there is a third contribution to the total shift: ΔB/B_{0}, 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 (^{59}Co and ^{71}Ga(1)) in PuCoGa_{5}, we determine ΔB ≈ 40 Oe at 4 K (ref. 9). Because of the field orientation in our experiment, we can rule out a spintriplet pairing state with a dvector pointing along the crystallographic c axis, as proposed for the spintriplet superconductors Sr_{2}RuO_{4} and UPt_{3} (ref. 12).
For the theoretical fits we modelled the temperature dependence of the strongcoupling superconducting gap function as ΔT = Δtanh[b(T_{c}/T  1)^{0.5}]. The ratio of the specificheat jump to C_{N} (the normalstate specific heat at T_{c}) constrained the phenomenological parameters Δ and b as follows: ΔC/C_{N} = (bΔ/(πk_{B}T_{c}))^{2}/(2a), with a = 2/3 for a freeelectron gas. Our choice of parameters Δ/k_{B}T_{c} = 4 and b = 1.74 lead to a moderately enhanced coefficient a ≈ 1.1–1.7 for ΔC/C_{N} ≈ 1.43–2.28, consistent with reported values for the Sommerfeld coefficient γ_{s} = C_{N}/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, N_{F} is the density of states at E_{F}, T_{n}^{1} is the normal state relaxation rate at T_{c}, and M(E) is the ‘anomalous’ density of quasiparticle states, which vanishes for dwave superconductors. The ‘dirty’ dwave calculation (assuming a finite density of states from impurities) in Fig. 3a assumes strong impurity scattering in the SCTA with Γ^{B}/Δ = 0.01, Δ/k_{B}T_{c} = 4 and b = 1.74. Note that it is not possible to fit both the suppression of the Hebel–Slichter coherence peak just below T_{c} and the linear lowtemperature behaviour with an isotropic swave pairing model, even when magnetic impurity scattering is included.
The normalstate relaxation rate data shown in Fig. 3b have been normalized to the value at 1.25T_{c} in order to avoid complications associated with the pseudogap effect, which suppresses T_{1}^{1} in the normal state just above T_{c} in the copper oxides^{14}. There is a qualitative difference between the conventional and the dwave superconductors below and above T_{c}. For the dwave superconductors, the relaxation rate is given by a single scaling function f(T/T_{c}) up to at least 3T_{c}.
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Acknowledgements
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.
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Supplementary information
Supplementary Figure S1
This file shows the superconducting transition temperature, T_{c}, versus the characteristic spin fluctuation temperature, T_{0}, of several heavy fermion superconductors, highT_{c} cuprate superconductors and PuCoGa_{5}. This file also contains the Supplementary Figure Legend. (DOC 168 kb)
Supplementary Figure S2
This Supplementary figure shows the normalized spin shift versus T/T_{c} 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 PuCoGa_{5}. Nature 434, 622–625 (2005). https://doi.org/10.1038/nature03428
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