Abstract
Superconductivity is due to an attractive interaction between electrons that, below a critical temperature, drives them to form Cooper pairs and to condense into a ground state separated by an energy gap from the unpaired states. In the simplest cases, the pairing is mediated by lattice vibrations and the wavefunction of the pairs is isotropic. Less conventional pairing mechanisms can favour more exotic symmetries of the Cooper pairs. Here, we report on pointcontact spectroscopy measurements in PuCoGa_{5}, a moderate heavyfermion superconductor with a record high critical temperature T_{c}=18.5 K. The results prove that the wavefunction of the paired electrons has a dwave symmetry, with four lobes and nodes, and show that the pairing is likely to be mediated by spin fluctuations. Electronic structure calculations, which take into account the full structure of the forbital multiplets of Pu, provide a hint of the possible origin of these fluctuations.
Introduction
A century on from the discovery of superconductivity, a complete understanding of some of the mechanisms that lead to its manifestation is still missing. Considerable research efforts are currently devoted to elucidating mechanisms by which pairs of electrons can bind together through the mediation of a boson field different than the one associated with the vibrations of a crystal lattice. PuCoGa_{5}, a 5felectron heavyfermion superconductor with a record high critical temperature T_{c}=18.5 K^{1}, is one of the many compounds for which the shortrange, isotropic attraction provided by simple electron–phonon coupling does not appear as an adequate glue for electron pairing. Magnetic, or virtual valence, fluctuations may have an important role in the stabilization of the superconducting ground state in PuCoGa_{5}, but the specific nature of the coupling mechanism remains obscure.
PuCoGa_{5} is a compound with unique properties. The Sommerfeld coefficient γ, as measured by specificheat experiments, is between 77 and 95 mJ mol^{−1} K^{−1} (refs 1,2). The effective mass of electrons is about 1/5 of that measured for the isostructural unconventional superconductor CeCoIn_{5}^{2}, suggesting a smaller degree of electronic correlation. The magnetic properties of PuCoGa_{5} and their possible relationship with the superconducting pairing have been debated, mainly because of conflicting results about the magnitude of the magnetic moment carried by the Pu ions. A Curie–Weiss susceptibility in the normal state was initially observed^{1}, as expected for fluctuating local Pu^{3+} moments. Successive μSR studies showed no static, normalstate electronic magnetism^{3,4}, and polarized neutron diffraction experiments showed a small and temperatureindependent microscopic magnetization dominated by the orbital moment^{5}.
The amplitude of the gap Δ (or at least of the gap ratio 2Δ/k_{B}T_{c}) has been estimated by different techniques. Using nuclear magnetic resonance (NMR) to measure the Knight shift in the superconducting state of a sample with T_{c}=18.5 K, Curro et al.^{6} obtained 2Δ/k_{B}T_{c}=8 (Δ~6.3 meV).
Various bandstructure calculations have been reported^{7,8,9,10,11,12,13,14}, showing that the details of the Fermi surface (FS) and the value of the local magnetic moment in PuCoGa_{5} depend on the approximations used to simplify the description of the electron correlations. In the local density/generalizedgradient approximations (LDA/GGA), the FS is quasitwo dimensional (2D)^{7,8,11,13}, with at least two nearly cylindrical sheets, one holelike around the Γ=(0, 0, 0) point of the reciprocal space (k_{x}, k_{y}, k_{z}), and one electronlike around M=(π, π, 0). FSs with similar features have been observed in Febased superconductors^{15}. Indeed, the coupling mechanism that seems to account for the high T_{c} superconductivity in iron pnictides (based on the nesting between hole and electron FS sheets through a vector associated with a peak in the spin susceptibility) had been proposed earlier for PuCoGa_{5}^{10}. The FS remains qualitatively unchanged with the addition of the CoulombU interaction^{14}.
Several experimental facts^{2,3,6,16,17,18,19} suggest that in PuCoGa_{5} the electrons in the Cooper pairs have a mutual angular momentum =2, corresponding to a superconducting order parameter (OP) with dwave symmetry, that is, the gap in the singleparticle excitation spectrum has nodal lines intersecting the FS. A direct proof of the gap symmetry is, however, still missing. This is an important point, as the symmetry of the OP is closely related to the pairing mechanism. For instance, while isotropic electron–phonon attraction favors the formation of zeroangularmomentum pairs, with a spherically symmetric OP, dwave symmetry is most easily realized if the pairing interaction is repulsive at short range and anisotropic at larger distances, as the one provided by effective spin–spin couplings on the border of antiferromagnetism^{20}.
A magnetic nature of the Cooper pairing mediator in PuCoGa_{5} has been discussed by several authors (a recent review can be found in ref. 21). Flint et al.^{22} considered virtual valence fluctuations of the magnetic Pu configurations, creating Kondo screening channels with different symmetry, and demonstrated that in a lattice of magnetic ions, exchanging spin with conduction electrons in two different channels, a condensate of composite pairs between local moments and electrons is formed. These models must be reconciled with the temperatureindependent susceptibility observed in the normal state^{5}. A phononic mechanism in the framework of the dwave Eliashberg theory has been discussed in ref. 19. Although able to reproduce a number of experimental observations, this model requires an electron–phonon (e–ph) coupling constant λ that is much higher than the value experimentally deduced from the time relaxation of photoinduced quasiparticles^{23} (λ=0.20–0.26).
To determine the amplitude and the symmetry of the superconducting OP, and thus get some insight into the possible coupling mechanism, we performed pointcontact Andreevreflection spectroscopy (PCARS) measurements in PuCoGa_{5} single crystals. PCARS has very often been the key experimental tool to elucidate the unconventional nature of superconductivity in very different materials, from heavy fermions^{24} to ironbased superconductors^{25}.
Based on these PCARS measurements, we prove that the OP of PuCoGa_{5} has a dwave symmetry, consistent with indirect indications from NMR and μSR measurements. In freshly annealed crystals of ^{239}PuCoGa_{5} with bulk , the analysis of different PCAR spectra gives a gap amplitude at T→0 equal to Δ=5.1±0.3 meV, corresponding to a gap ratio 2Δ/k_{B}T_{c}=6.4±0.4, that indicates a strong electron–boson coupling. A similar gap ratio is obtained in ^{242}PuCoGa_{5} with Sb impurities and a reduced T_{c}=14.5 K. In both cases, the temperature dependence of the gap is consistent with the predictions of the Eliashberg theory for strongcoupling superconductivity if spin fluctuations provide the mediating bosons. A characteristic boson energy Ω_{0}=5.3–8.0 meV (within the range of spinfluctuation energies determined by NMR) and a coupling constant λ=2.2–3.7 allow reproducing the lowtemperature gap values. These results show that PuCoGa_{5} is an unconventional, strongcoupling dwave superconductor and indicate spin fluctuations as the probable boson that mediates the electron pairing. Furthermore, we demonstrate that this picture is not in contradiction with the temperatureindependent magnetization. Indeed, electronic structure calculations accounting for the full structure of the 5forbital atomic multiplets and their hybridization with the conduction bands show that the Pu atoms carry nonvanishing average moments as their f shells fluctuate between a singlet and a sextet configuration. These fluctuations are dynamically compensated by fluctuations in the surrounding cloud of conduction electrons such that the compound as a whole remains nonmagnetic.
Results
Pointcontact Andreevreflection measurements
Figure 1 presents the temperature dependence of the raw conductance curves of a point contact whose normalstate resistance is 6.2 Ω. All the curves but the lowesttemperature one are shifted downward for clarity. The top curve is measured at T=1.8 K, which means about T_{c}/10. This ensures that the gap extracted from it is representative of the gap for T→0. The highenergy tails of the conductance curves are temperature independent, as shown in the inset, where the curves measured at T=1.8 K and in the normal state are reported without vertical offset as an example. This demonstrates that the contact is in the perfectly ballistic regime, with no contribution due to Maxwell terms in the contact resistance, that is, there is no diffusion in the contact region and the maximum excess energy with which the electrons are injected into the superconductor is exactly eV. Meeting this condition is essential for energyresolved spectroscopy to be possible. The absence of heating effects in the contact is also witnessed by the coincidence between the Andreev critical temperature (that is, the temperature at which the Andreev signal disappears and the normalstate conductance is recovered, here between 18.0 K and 18.5 K) and the bulk T_{c}=18.1 K determined from resistivity measurements (as shown in Supplementary Fig. S1).
It is worth noticing that the normalstate conductance curves in PuCoGa_{5} do not show the strong asymmetry observed in various heavyfermion compounds like, for instance, CeCoIn_{5}^{26}, UBe_{13}^{27} and URu_{2}Si_{2}^{28,29} (but not, for example, in UPt_{3}^{30}). In some cases, this asymmetry was ascribed to magnetic properties of the normal state. In CeCoIn_{5}, the asymmetry appears below the temperature where a coherent heavyfermion liquid develops^{26}, and has been explained in terms of a Fano resonance involving localized states near the interface and itinerant heavy electrons in the bulk^{31}. The absence of a strong, temperaturedependent asymmetry in PuCoGa_{5} may thus confirm that superconductivity in this compound develops out of an incoherent metallic state^{32}.
In order to compare the experimental curves to the theoretical ones, a normalization is required, that is, a division by the conductance of the same point contact when the superconductor is in the normal state. Owing to the extremely high value of the upper critical field in PuCoGa_{5}, this cannot be achieved by applying a magnetic field to suppress superconductivity at low temperature. However, owing to the moderate T_{c} and the negligible temperature dependence of the normalstate conductance (witnessed by the absence of any change in shape of the tails of the conductance curves at eV>15 meV), we can safely normalize the conductance curves at any by the normalstate conductance curve measured at (or just above) .
The result of this normalization is shown in Fig. 2 (symbols) for two contacts made on different places of freshly broken surfaces of the same sample. The two contacts have the same but different normalstate resistance R_{N}. It is worth noticing that the Andreev signal is very high, while in most heavyfermion compounds it is suppressed to a few percent of the normalstate conductance^{24,31}. This effect has been ascribed to extrinsic causes (for example, elastic scattering in the contact region) or intrinsic ones (for example, an energydependent quasiparticle lifetime or the existence of unpaired light electrons below T_{c} that do not participate in Andreev reflection^{31}). None of these effects seem to have a role in our point contacts.
The curve in Fig. 2a is similar to the best ones observed in cuprates^{33,34,35,36}. Its shape, with a very clear zerobias conductance peak (ZBCP), whose amplitude is greater than 2, is incompatible with a isotropic OP (Fig. 3a, top). This is clear in Fig. 3b that reports theoretical curves calculated within the 2D Blonder–Tinkham–Klapwijk (2DBTK) model for Andreev reflection^{37,38,39,40} at a normal metal/isotropic superconductor contact for different values of the dimensionless barrier parameter Z. All the curves present either a plateau or two maxima symmetric about zero bias, and their amplitude is always <2 even in ideal conditions (T=0, absence of broadening effects). Instead, the experimental curve in Fig. 2a looks very similar to those calculated within the same 2DBTK model in the case of a nodal OP with a change of sign at the FS. In particular, theoretical curves closely resembling the measured one can be obtained if the OP has a dwave symmetry (shown in Fig. 3a, bottom), the current is mainly injected along the nodal direction (that is, at an angle α=π/4 with respect to the lobes of the dwave gap^{38,39}) and , as shown in Fig. 3c. In these conditions, the central peak is ascribed to zeroenergy Andreev bound states, arising from the constructive interference between electronlike and holelike quasiparticles that feel OPs with different signs^{38,39}.
The shape of the conductance curve in Fig. 2b is very different, but can be readily explained within the same scenario of dwave superconductivity if the current is injected along a different direction^{38,39}. Indeed, this curve looks similar to the theoretical Andreevreflection spectra calculated within the 2DBTK model for Z=0.5–0.7 and α=π/8 and shown in Fig. 3d.
Fit of the experimental curves
Solid lines in Fig. 2a,b are the results of a quantitative fit of the experimental spectra by using the aforementioned 2DBTK model. The parameters of the model are Δ (here intended as the maximum amplitude of the gap), the barrier strength Z, the angle α and a spectral broadening parameter Γ that must usually be included in the model when fitting experimental curves^{40}. Here so that it does not add any ambiguity to the determination of the gap amplitude. The finite experimental temperature (1.8 K) has been taken into account. The bestfitting values of the parameters for the curves of Fig. 2a,b are indicated in the legend. The values of the gap (4.85 meV and 5.10 meV, respectively) are very similar and this is certainly the most important result, but also the barrier and broadening parameters are very similar. The only relevant difference, as expected, is in the value of the angle α, as shown pictorially in the insets of Fig. 2.
Figure 4a shows an example of how the normalized conductance curves evolve with increasing temperature. The experimental data (symbols) are compared with the 2DBTK fit (solid lines). The fit is good at all temperatures; note that both Z and α are independent of temperature and were thus kept constant for all the curves. We also kept Γ nearly constant so that the only parameter that varies significantly with temperature is the gap amplitude Δ.
The temperature dependence of the gap as obtained from the fit of three different sets of conductance curves (in three different contacts) is shown in Fig. 4b. It is clear that the gap values are reproducible to a high degree; at low temperature, they range between 4.85 and 5.30 meV. The vertical spread of gap values can be used to evaluate a posteriori the uncertainty on the gap itself; this is obviously much greater than the uncertainty arising from the fit of a single curve, which can be empirically determined as the range of gap values that allow an acceptable (that is, within some confidence limit) fit of the conductance curve, when all the other parameters are varied as well. To be conservative, we can assume the experimental gap at low temperature to be equal to Δ=5.1±0.3 meV, corresponding to a gap ratio 2Δ/k_{B}T_{c}=6.4±0.4, much larger than the value of 4.28 expected in weakcoupling dwave superconductors according to the BardeenCooperSchrieffer (BCS) theory. The spread of gap values is small at low temperature and maximizes at around 14 K. Although the general trend of the gap seems to be compatible with a BCSlike Δ(T) dependence (but of course, with a nonBCS gap ratio), this uncertainty does not favour discussing it in detail.
Analysis of the data within Eliashberg theory
The solid line in Fig. 4b, which is in fairly good agreement with the experimental data, represents the theoretical temperature dependence of the gap calculated within the strongcoupling theory for superconductivity (known as the Eliashberg theory^{41,42}) by assuming mediating bosons with a spinfluctuationlike spectrum^{43}: where Ω_{0} is the energy of the peak and Ω_{max} is a cutoff energy that we chose equal to 4Ω_{0}^{44}. The shape of the spectrum is shown in the inset of Fig. 5a. In addition to the peak energy Ω_{0}, the minimal Eliashberg model contains two other parameters that can be adjusted to reproduce the experimental T_{c} and lowtemperature gap: the electron–boson coupling constant λ and the Coulomb pseudopotential μ^{*} (that describes the effects of the Coulomb repulsion and normally ranges between 0 and 0.2^{42}). The concentration of defects in the material was also taken into account by using a scattering parameter γ_{d}=0.25 meV obtained by fitting the temperature dependence of the local spin susceptibility^{6}.
We first solved the Eliashberg equations in the imaginaryaxis representation (explicitly written in the Supplementary Methods) to determine the pairs of λ and Ω_{0} values that, for some given values of μ^{*} between 0 and 0.2, give the experimental T_{c}=18.5 K. In Fig. 5a these pairs are represented by solid lines in the (Ω_{0}, λ) plane.
Then, we solved the realaxis Eliashberg equations to calculate the lowtemperature gap Δ(T→0) for all sets of Ω_{0}, λ and μ^{*} previously determined. The gap values are shown as a function of Ω_{0} in Fig. 5b. The range of Ω_{0} values that are physically compatible with the PCARS results are finally determined, as shown in Fig. 5b, by intersecting the calculated Δ(T→0) versus Ω_{0} curves with the experimental gap range (grey region). The result is . This interval largely overlaps with the range of spinfluctuation energies (from 4 to 8 meV) determined by NMR measurements^{45}; this clearly indicates that spin fluctuations do possess the right energies to explain the experimental PCARS results. The intersection between the two ranges provides a conservative range for the characteristic boson energy, that is, . The corresponding range of λ values, as shown in Fig. 5, turns out to be . The line in Fig. 4b was obtained by taking Ω_{0}=6.5 meV, μ^{*}=0, and using an electron–boson coupling constant λ=2.37. The slight gap increase at low temperature is a consequence of the strong electron–boson coupling^{46}. Further details of the calculations are given in the Supplementary Methods.
PCARS measurements on crystals with reduced T_{c}
As a test of the reliability and generality of the results discussed so far, we also performed PCARS measurements in ^{242}PuCoGa_{5} crystals featuring a lower bulk , K, due to the presence of Sb impurities (<1%). The point contacts in this case were made by placing a small drop of Ag conductive paste on the side surface of the platelike crystals. Fig. 6a shows the conductance curves of a Ag/^{242}PuCoGa_{5} contact with R_{N}=6.9 Ω measured as a function of temperature. The curves are vertically offset for clarity; as in the cases discussed for the crystals with T_{c}=18.1 K, there is no shift of the tails at increased temperature. The Andreev signal disappears at some temperature between 14.5 K and 14.7 K, that is, .
Figure 6b reports the normalized lowtemperature conductance of the same contact (top, symbols) and that of a different contact with R_{N}=18.8 Ω and (bottom, symbols). The shape of these curves, with a zerobias maximum, confirms that the gap has a dwave symmetry. It is worth noting that these spectra (as well as all those obtained in crystals with reduced T_{c}) show a strongly reduced Andreev signal with respect to the ideal case: the excess conductance at zero bias is <10% of the normalstate conductance. A similar effect is commonly observed in heavyfermion superconductors^{24,31} and has been reported, in particular, for the isostructural compound CeCoIn_{5}^{26,47}. According to ref. 31, this reduction might be explained by assuming that a fraction of the injected current tunnels into a nonsuperconducting band or set of states and does not contribute to Andreev reflection. In our case it seems logical, however, not to ascribe the small signal to intrinsic phenomena but, rather, to the greater amount of disorder and impurities of these samples with respect to the purest crystals (where instead the signal is very high) or to the use of the Ag paste instead of the Au wire as a counterelectrode. In either case, the broadening parameter Γ of the 2DBTK model can be phenomenologically used to account for extrinsic inelastic scattering. A specific model for diffusive metal/dwave superconductor^{48} could be used as well, with no major changes in the conclusions.
Solid lines in Fig. 6b depict two representative fits of the experimental data within the 2DBTK model used so far. The fitting parameters are indicated in the legend of the same figure. As expected, the fit requires a large Γ value, although still smaller than the corresponding Δ. Taken all together, the PCARS measurements in these crystals indicate a lowtemperature gap Δ=3.9±0.3 meV, corresponding to a gap ratio 2Δ/k_{B}T_{c}=6.2±0.4. The latter differs only slightly (about 3%) from that determined in the crystals with the highest T_{c}, supporting the robustness of the picture that emerges from our measurements.
Figure 6c shows an example of the temperature dependence of the gap extracted from the fit (symbols). The line is calculated within the Eliashberg theory, using the same spectral function and the same values of Ω_{0}, λ and μ^{*} as in Fig. 4b. The different T_{c} and the different gap amplitude arise only from the larger value of the quasiparticle scattering rate γ_{d} included in the Eliashberg equations to account for the disorder: while T_{c}=18.5 K was obtained by using γ_{d}=0.25 meV, the reduced T_{c}=14.5 K of this particular contact requires γ_{d}=1.6 meV. This indicates substantial consistency of all the results and generality of the conclusions drawn about the mechanisms of superconductivity in PuCoGa_{5}.
Discussion
A scenario where magnetic fluctuations are responsible for the formation of the Cooper pairs in PuCoGa_{5} must be reconciled with the observed temperatureindependent magnetic susceptibility^{5} that points to vanishing local moments at the Pu sites. A plausible explanation is provided by electronic structure calculations combining the LDA with the exact diagonalization (ED)^{49} of a singleimpurity Anderson model^{50} that is composed of a Pu 5f shell (the impurity) and of those extended states that hybridize with this shell and form the socalled electronic bath. In this approach, the band structure obtained by the relativistic version of the fullpotential linearized augmented plane wave method^{51} is consistently extended to account for the full structure of the forbital atomic multiplets and their hybridization with the conduction bands^{52}. Details on this procedure are given in the Supplementary Methods; the bandstructure and the FS are shown in Supplementary Fig. S2.
The calculated forbital density of states (DOS) is shown in Fig. 7. Below the Fermi energy E_{F}, the DOS exhibits a threepeak structure that is typical for Pu and for a number of its compounds. Our DOS is in reasonably good agreement with the results of the noncrossing approximation reported in ref. 53 and with photoemission experiments performed on single crystals^{54} and on thin film samples^{55}, as shown in Supplementary Fig. S3.
The inset of Fig. 7 shows the valence histogram calculated by projecting the ground state of the Anderson impurity model onto the Pu atomic eigenstates that correspond to an integer 5fshell occupation n_{m}. The plotted probabilities P_{m} determine the 5forbital valence The highest obtained probability is P_{5}=0.63, followed by P_{6}=0.33. In addition, there are small but nonzero probabilities P_{4}=0.03 and P_{7}=0.01. The Pu 5f shell has thus an intermediatevalence nature, being a mixture of a magnetic 5f^{5} sextet and a nonmagnetic 5f^{6} singlet. Similar electronic structures have been suggested for δPu^{56} and several rareearthbased materials. In the latter case, if the intermediatevalence state involves a nonmagnetic atomic configuration, the lowT magnetic susceptibility is temperature independent^{57}. The susceptibility of PuCoGa_{5} is also found to be temperature independent at low temperatures^{5}.
The Pu f shell carries a nonvanishing average magnetic moment as it fluctuates between the singlet and the sextet. At the same time, the ground state as a whole is a singlet characterized by all angular momenta equal to zero (S=L=J=0). The fluctuations of the momenta in the f shell are accompanied by compensating fluctuations in the bath, which can be viewed as a manifestation of the Kondo physics. In analogy to a Kondo singlet state, the magnetic susceptibility is anticipated to behave as , which remains constant for , as observed experimentally^{5}. In addition, our bandstructure calculations suggest an antiferromagnetic instability due to the presence of a FS sheet with a negative second derivative of the Drude plasma energy, as shown in Supplementary Table S1.
This analysis indicates that the dwave superconducting coupling in PuCoGa _{5} can be mediated by spin fluctuations—even though the microscopic magnetization is temperatureindependent—because such fluctuations would involve a timedependent 5f local moment dynamically compensated by a moment formed in the surrounding cloud of conduction electrons.
Methods
Pointcontact Andreevreflection spectroscopy
Pointcontact spectroscopy is a simple but very powerful tool for the investigation of the superconducting OP that, in past decades, has been successfully applied to many families of superconductors, namely cuprates, borocarbides, heavy fermions and the recently discovered ironbased compounds. The technique consists in measuring the differential conductance of a N–S contact between a normal metal (N) and a superconductor (S) whose radius a is smaller than both the electronic mean free path and the coherence length in S. In these conditions, an electron travels through the contact ballistically, that is, without being diffused, so that if a voltage V is kept at the junction's ends, it enters the superconductor with a maximum excess energy eV. When this energy is smaller than the gap in the superconductor, Δ, the electron cannot propagate in S as an electronlike quasiparticle (ELQ), because it finds no available states. Thus, it forms a Cooper pair and a hole is retroreflected in N.
If the OP is isotropic (Fig. 3a, top) and there is no potential barrier at the interface, this results in a doubling of the conductance for V≤Δ/e. If eV>Δ, the conductance decreases again towards the value it would have if the superconductor were in the normal state. This is shown in the top curve of Fig. 3b, calculated for the case of an ideal barrierless junction by using the BTK model^{37} generalized to the 2D case^{39,40}. If a potential barrier is present (that is, the dimensionless barrier parameter Z has a finite value), other phenomena take place that can give rise to a normal reflection of the incoming electron, and also to the transmission of holelike quasiparticles in S. As a result, the conductance presents two maxima at approximately V=±Δ/e and a zerobias minimum, as shown in Fig. 3b.
If the OP is anisotropic, electrons injected along different directions may experience different pairing amplitudes. Figure 3a (bottom) shows for example an OP with wave symmetry, , where θ is the azimuthal angle in the k_{x}, k_{y} plane of the reciprocal space and φ is the inclination angle. In cases like this, the shape of the conductance curves does not depend only on the height of the potential barrier at the interface, but also on the direction of (main) current injection with respect to the kspace axes.
The curves shown in Fig. 3c,d represent the theoretical conductance curves of a normal metal/dwave superconductor calculated by using the 2DBTK model. In this approach, the FS is supposed to be perfectly cylindrical (with its axis parallel to the k_{z} axis), the dependence of the OP on φ is disregarded, and the direction of current injection is simply defined by the angle α between the normal to the interface n and the k_{x} axis. Note, however, that individual electrons approach the N–S junction from any direction, specified by the angle θ_{N} (0≤θ_{N}<π/2) between their wavevectors and n^{39}. In this paper, in view of the mostly 2D shape of the largest FS sheets (and also for simplicity), we have always used this model. In ref. 40 it is seen that this approximation, with respect to a more refined 3D model, generally gives rise to an overestimation of the parameter Z, which is not relevant in our analysis.
If α=0, the normal n is parallel to the k_{x} axis; for any angle of incidence θ_{N} of the incoming electron, ELQ and HLQ transmitted in S with angles +θ_{S} and −θ_{S} with respect to n feel the same OP, in amplitude and sign. However, because of the angular dependence of the gap, the conductance is doubled only at zero bias where it shows a characteristic cusp. If there is no barrier, the same shape is obtained for any value of α: the conductance curve always looks like the top curve in Fig. 3c. If instead a barrier is present, for any α≠0 some values of θ_{N} exist for which HLQ and ELQ feel OPs of opposite sign^{39}. This gives rise to constructive interference between HLQ and ELQ that results in localized zeroenergy states (Andreev bound states). These states manifest themselves in the conductance giving rise to a ZBCP. When α=π/4 the current is injected along the nodal direction, all ELQ and HLQ interfere and the ZBCP is maximum. Examples of calculated (normalized) conductance curves assuming α=π/4 and α=π/8 are shown in Fig. 3c,d for increasing values of Z.
All the curves in Fig. 3 were calculated at T=0 and in ideal conditions (perfectly ballistic conduction, no broadening effects). When the 2DBTK model is used to fit experimental data, however, the calculated conductance at T=0 must be convoluted with the Fermi function^{40}. Moreover, an additional broadening parameter Γ must be often included in the model to account for the finite lifetime of quasiparticles and other extrinsic broadening effects (like inelastic scattering processes occurring near the N–S interface)^{40}.
Growth and characterization of the samples
The ^{239}PuCoGa_{5} crystals were grown by a flux method and characterized by Xray diffraction, electrical resistivity, magnetization and specific heat measurements. Magnetic susceptibility and resistivity measurements were performed in commercial Quantum Design platforms (MPMS 7T SQUID and PPMS9T). The crystals were submitted to a thermal treatment to anneal the selfradiation damage. Supplementary Fig. S1 shows the temperature dependence of the electrical resistivity measured for one of these crystals immediately after a thermal treatment. All the samples exhibited a critical temperature very close to the optimal value T_{c}=18.5 K.
The ^{242}PuCoGa_{5} single crystals were grown from the melt using ^{242}PuSb instead of metallic ^{239}Pu as starting material, leading to traces of Sb (<1%) and to a reduced T_{c}=14.5 K. The use of the ^{242}Pu isotope avoids effects from radiation damage and self heating, especially at low temperature. Magnetization, transport properties and heat capacity confirm their similarity with freshly synthesized single crystals of ^{239}PuCoGa_{5}^{5}.
Fabrication of point contacts
PCARS measurements in ^{239}PuCoGa_{5} were started within 1 day from the thermal treatment. The point contacts were made between a fresh, mirrorlike surface of the crystal (just exposed by breaking the sample) and a thin Au wire (about 10 μm in diameter). The uneven broken surface on which the contact is made prevents a fine control of the direction of current injection with respect to the crystallographic axes. However, as shown above, this does not prevent the unambiguous determination of the amplitude and symmetry of the OP.
Point contacts in ^{242}PuCoGa_{5} with T_{c}=14.5 K were made on the side surface of the platelike crystals so as to inject the current mainly along the ab planes. Unlike in the purest samples, here we used a small spot of Ag conducting paste between the Au wire and the sample to act as the N electrode and also to mechanically stabilize the contacts^{40}.
Additional information
How to cite this article: Daghero, D. et al. Strongcoupling dwave superconductivity in PuCoGa5 probed by point contact spectroscopy. Nat. Commun. 3:786 doi: 10.1038/ncomms1785 (2012).
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Acknowledgements
This work has been performed at the Institute for Transuranium Elements within its 'Actinide User Laboratory' program, with financial support to users provided by the European Commission. The support from Czech Republic Grants GACR P204/10/0330 and GAAV IAA100100912 is thankfully acknowledged.
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Affiliations
Dipartimento di Scienza Applicata e Tecnologia, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy.
 D. Daghero
 , M. Tortello
 & G.A. Ummarino
European Commission, Joint Research Centre, Institute for Transuranium Elements, Postfach 2340, D76125 Karlsruhe, Germany.
 J.C. Griveau
 , E. Colineau
 , R. Eloirdi
 , A.B. Shick
 & R. Caciuffo
Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ18221 Prague, Czech Republic.
 A.B. Shick
 & J. Kolorenc
University of Hamburg, Institute of Theoretical Physics, Jungiusstrasse 9, 20355 Hamburg, Germany.
 A.I. Lichtenstein
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Contributions
R.C., D.D. and M.T. designed the research; D.D., M.T. and J.C.G.performed PCARS experiments; R.E., J.C.G. and E.C. prepared and characterized the samples; G.A.U. performed Eliashberg calculations, A.B.S., J.K. and A.I.L. performed ab initio calculations; D.D., M.T., J.C.G.,E.C., R.C., G.A.U. and A.B.S. analysed and interpreted the data; and R.C., D.D., M.T., A.B.S. and J.K. wrote the paper.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to R. Caciuffo.
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