Fully gapped d-wave superconductivity in CeCu2Si2.

The nature of the pairing symmetry of the first heavy fermion superconductor CeCu2Si2 has recently become the subject of controversy. While CeCu2Si2 was generally believed to be a d-wave superconductor, recent low-temperature specific heat measurements showed evidence for fully gapped superconductivity, contrary to the nodal behavior inferred from earlier results. Here, we report London penetration depth measurements, which also reveal fully gapped behavior at very low temperatures. To explain these seemingly conflicting results, we propose a fully gapped [Formula: see text] band-mixing pairing state for CeCu2Si2, which yields very good fits to both the superfluid density and specific heat, as well as accounting for a sign change of the superconducting order parameter, as previously concluded from inelastic neutron scattering results.

The structure of the superconducting order parameter has been frequently studied, due to its close relationship with the underlying pairing mechanism. While the conventional electron–phonon pairing mechanism typically leads to -wave states with fully opened gaps and a constant sign over the Fermi surface (1), unconventional superconductors with different pairing mechanisms often form states with a sign-changing order parameter (2, 3). For instance, cuprate and many Ce-based heavy fermion superconductors are generally believed to be -wave superconductors, with nodal lines in the energy gap on the Fermi surface (4–6). On the other hand, in the high-temperature iron-based superconductors, an state has been proposed, with a change of sign of the gap function between disconnected Fermi surface pockets, but the energy gap remains nodeless (7). In this context, the surprising recent discovery (8–10) of evidence for fully gapped superconductivity in the first heavy fermion superconductor CeCu2Si2 (11) requires further attention.

Superconductivity in CeCu2Si2 occurs in close proximity to magnetism. Samples with either superconducting ( type), antiferromagnetic ( type), or competing phases ( type) are obtained via slight tuning of the composition within the homogeneity range (12). High-pressure measurements of CeCu2 (Si1−xGex)2 reveal two distinct superconducting domes, one centered around an antiferromagnetic (AFM) instability at ambient/low pressure and another near a valence instability at high pressure (13). The close proximity of superconductivity to an AFM instability suggests that in CeCu2Si2 it is driven by the corresponding quantum criticality. Inelastic neutron scattering (INS) measurements clearly indicate that the Cooper pairing is associated with a damped propagating paramagnon mode at the incommensurate ordering wavevector Q of the spin-density wave (SDW) order nearby in the phase diagram (14). The large intensity of the low-energy spin excitation spectrum at Q, which reveals a spin gap in the superconducting state, as well as a pronounced peak well inside the superconducting gap (14, 15), implies a sign change of the pairing function between the two regions of the Fermi surface spanned by Q (16, 17). The absence of a coherence peak and the temperature dependence of the spin-lattice relaxation rate [] in Cu nuclear quadrupole resonance (NQR) measured above 100 mK further suggested an unconventional superconducting order parameter with line nodes in the gap structure (15, 18). Angle resolved resistivity measurements at 40 mK indicate a 4-fold modulation of the upper critical field , as expected for a -wave gap with symmetry (19), while a sign change spanning Q is compatible with pairing symmetry (16). Therefore, CeCu2Si2 behaves as an even-parity -wave superconductor, whose gap structure has yet to be determined.

However, a recent specific heat investigation reported exponential behavior of at very low temperatures, suggesting fully gapped superconductivity in CeCu2Si2 (8). Following this work, scenarios of multiband superconductivity with a strong Pauli paramagnetic effect, loop-nodal superconductivity, and pairing with no sign change were proposed (9, 10, 20, 21). Furthermore, scanning tunneling spectroscopy down to 20 mK also hints at a multigap order parameter (22). Indeed electronic structure calculations reveal that multiple bands cross the Fermi level (8, 23), and renormalized band structure calculations show that the dominant heavy band (with ) leads to Fermi surface sheets mainly consisting of warped cylinders along the axis (23). The aforementioned discrepancies between the pairing symmetries deduced from different measurements show that the superconducting order parameter of CeCu2Si2 is poorly understood. A particular puzzle is how to reconcile the fully gapped behavior with the previous evidence for a sign-changing order parameter and nodal superconductivity. Here we probe the superconducting gap symmetry by measuring the temperature dependence of the London penetration depth and propose a scenario of a fully gapped band-mixing pairing state, which reconciles all of the seemingly contradictory results.

Results

Resistivity and Specific Heat.

The samples were characterized using resistivity and specific heat measurements, as shown for the -type sample in Fig. 1. The residual resistivity of the -type sample in the normal state just above is cm, and a superconducting transition is observed, onsetting around 0.65 K and reaching zero resistivity at about 0.6 K. The transition width of K is in line with recent reports (19). The specific heat also shows a superconducting transition with 0.64 K, similar to previous results (8). The -type sample (Fig. 1) displays a superconducting transition, onsetting around 0.62 K, with a lower residual resistivity of cm. The specific heat shows both an AFM transition at 0.7 K and a superconducting transition at 0.53 K.

Fig. 1.

Specific heat as and resistivity of (A) -type and (B) -type CeCu2Si2.

Temperature Dependence of the Penetration Depth.

Measurements of the change of the London penetration depth for the -type sample are displayed in Fig. 2. As shown in the Inset, a sharp superconducting transition is clearly observed, with an onset at around 0.62 K. To probe the superconducting gap structure, we analyzed the behavior of at low temperatures, and the results are shown in the main panel. The data were fitted with the exponential temperature dependence for a fully open gap, , where is the gap magnitude at zero temperature and the constant allows for some variation in the extrapolated zero temperature value. The fitting was performed up to 0.12 K (), and as shown by the solid line in Fig. 2, the model can account for the data with a gap of . A similar gap value of is obtained from a corresponding fit for the -type sample, as displayed in the main panel of Fig. 2. The small gap values in both cases means that only saturates at very low temperatures. The results indicate similar superconducting properties of the - and -type samples and are consistent with the fully gapped superconductivity reported for an -type single crystal in ref. 8.

Fig. 2.

The change in London penetration depth at low temperature for an (A) -type and (B) -type sample of CeCu2Si2. The solid lines show fits to a fully gapped model described in the text, while the dashed lines show fits to a power law temperature dependence of . The data across the whole temperature range of the superconducting states are displayed in the Inset of A. The Inset of B shows when the data are fitted with up to a temperature .

The penetration depth of the - and -type samples could also be described by a power law dependence (), with and 2.43, respectively, when fitting from the base temperature to 0.12 K. For line nodes in the superconducting gap in the presence of impurity scattering, may show quadratic behavior at low temperatures, which crosses over to linear behavior at an elevated temperature (24). To check how the exponent evolves with temperature, we also fitted with the power law expression from the base temperature up to a range of temperatures , and the dependence of on is shown in Fig. 2 , Inset. It can clearly be seen that for both samples, increases with decreasing , with . This indicates that the true low-temperature behavior is not a dependence, as expected for a dirty nodal superconductor, but increases as expected for superconductivity exhibiting a full gap. Therefore, both the specific heat and data are consistent with fully gapped superconductivity at very low temperatures.

Analysis of the Superfluid Density.

The superfluid density was calculated using and is displayed for both - and -type samples in Fig. 3, where Å (25). The superfluid density was fitted following the method of ref. 26, for a gap integrated over a cylindrical Fermi surface. The superfluid density data were fitted with an isotropic -wave model with a gap (27) as well as a -wave model with line nodes (, = azimuthal angle). As shown in Fig. 3, a single-band isotropic -wave gap cannot account for the data of the -type sample, in contrast to the low-temperature data discussed above (Fig. 2). The single-band -wave model shows reasonable agreement above , but the agreement is poor at low temperatures, since for this model is linear but the data are not. The agreement with the -wave model at higher temperatures is consistent with the previously reported evidence for -wave superconductivity. The data were also fitted using a two-gap -wave model (27), and this gives reasonable agreement. The fitted gap values are and , with a fraction for the larger gap of . Both gap values are slightly larger than the ones obtained for the two-gap model in ref. 8. Similarly, neither the - nor -wave single-band models could describe the superfluid density of the -type sample at low temperatures (Fig. 3), but the data could be fitted using a two-gap model with , , and .

Fig. 3.

Superfluid density of CeCu2Si2 fitted with various models. Fits for the -type sample are shown for (A) two fully open gaps, as well as -, and -wave models denoted by solid, dotted, and dashed lines, respectively, and (B) a band-mixing pairing model. Fits for the -type sample are shown for (C) two fully open gaps, as well as -, and -wave models denoted by solid, dotted, and dashed lines, respectively, and (D) a band-mixing pairing model.

On the other hand, it is difficult to reconcile an -wave model with the evidence for a sign-changing gap function, as concluded from the INS response, where a sharp spin resonance forms at the edge of a spin gap well inside the superconducting gap (14). Moreover, the incommensurate ordering wave vector of the nearby SDW (Q) is identical to the nesting wave vector spanning the flat parallel parts of the warped cylinders (28). This shows that there is a sign change of the pair wave function inside the dominating heavy-fermion band, which is incompatible with a nodeless pairing state (21). On the other hand, if there is no sign change of the gap function across the Fermi surface, , the coherence factor in the spin susceptibility is vanishingly small (29, 30). Consequently, in this case, the spin spectrum will not have a sharp peak, although there may be a broad enhancement of the spectral weight above (31). However, when there is a change of sign of the gap function between regions of the Fermi surface connected by , there is an enhanced coherence factor since . This gives rise to a sharp peak in below , leading to the conclusion that there must be a sign-changing order parameter in CeCu2Si2 (16, 17). By a similar argument, the lack of a coherence peak in NQR measurements also strongly disfavors superconductivity without a sign reversal (15, 18, 32). Furthermore, given the strong Coulomb repulsion in Ce-based heavy fermion superconductors, the order parameter must be anisotropic with a sign change, without running into the issue of a large (33). In other words, the -electrons have a Coulomb repulsion that is much larger than their effective Fermi energy, and they must avoid each other, thereby excluding any sign-preserving pairing function. In such strongly correlated superconductors, even anisotropic and sign-changing pairing states can be robust against disorder (33). Indeed, potential scattering [due to a site exchange between Cu and Si of less than within the homogeneity range (34)], which enhances the residual resistivity by a factor of 4, apparently has an almost negligible influence on (cf., the resistivity results on the and samples in Fig. 1). Also, recent experiments on electron-irradiated samples revealed only a minor change of (9). At the same time, just like in the cuprates, the effect of substitutional disorder on is known to be site- and size-dependent (35). For CeCu2Si2, the superconducting was found to be extremely sensitive to nonmagnetic substitutions on the Cu site: For example, Rh, Pd, and Mn substitution for Cu at a level of 1 fully suppresses superconductivity (35), which is impossible to account for in the scenario of an -wave state without a sign change of the order parameter. Further studies are needed to develop a detailed understanding of all these observations.

In the present work, we consider a pairing function that, by analogy with an pairing state (36), has an effective gap as a result of intraband pairing with symmetry and interband pairing with symmetry; our pairing function preserves both the fully gapped nature and order parameter sign change along Q on a single nested Fermi surface. The pairing state was introduced in the context of the iron-based superconductors () (36), as part of the studies about orbital-selective superconducting pairing (37–40). There, the pairing function has the form (“”), as a product of an -wave form factor and a Pauli matrix in the orbital subspace. For that case, the interorbital mixing in the dispersion part of the Hamiltonian ensures that in the band basis the pairing is equivalent to a superposition of intra- and interband components with and form factors, respectively. The resulting quasiparticle spectrum acquires a nonvanishing contribution, as the two components are added in quadrature, ensuring a full gap on the whole Fermi surface with a sign change of the intraband component of the gap function. It is also shown how this pairing channel can be stabilized within a self-consistent five-orbital model, with a full gap and a resonance in the spin-excitation spectrum. Similar to the case considered in ref. 36, Q of CeCu2Si2 will connect two parts of the Fermi surface with a sign change in the intraband component of the gap function (see Fig. 4), thereby generating an enhanced spin spectral weight just above a threshold energy (14), inside the superconducting gap (see below and ref. 15).

Fig. 4.

An illustration of the warped parts of the cylindrical Fermi surfaces (red) in CeCu2Si2 at particular values of , corresponding to the nesting portions of the 3D Fermi surface, as well as additional smaller pockets (blue) projected onto the wavevector plane (23). The component of the antiferromagnetic wavevector Q projected into the same wavevector plane connects the parts of the heavy Fermi surface with a sign change in the intraband pairing component. The corresponding Fermi surface and nesting wavevector in the 3D space are those displayed in figure 3b of ref. 28.

In the following, we apply a simplified model for the gap structure to CeCu2Si2, given by summing contributions from the two -wave states in quadrature, with , where is the gap temperature dependence from Bardeen–Cooper–Schrieffer theory (1), which we used previously. In general, a band-mixing pairing can introduce corrections to the gap given above, due to the nondegeneracy of the bands throughout the Brillouin zone, which would then lead to an extra parameter, the band splitting. In the following, we will show that the data can be well fit by the simple function without this extra parameter. Although the and states each have two line nodes, the nodes of the two states are offset by in the plane, and as a result, the gap function is nodeless everywhere on a cylindrical Fermi surface. It should also be noted that for this model the same is calculated upon exchanging and . The superfluid density of the -type sample was fitted using this model, and the results are shown in Fig. 3. It can be seen that such a model can also fit the data well, with gap parameters of and . In Fig. 3, for the -type sample is equally well fitted, with similar parameters of and . The values of the larger gap agree almost perfectly with the gap value obtained from Cu-NQR measurements at higher temperatures (15). Furthermore, this model only uses two fitting parameters, while the two-band -wave model of ref. 8 needs three.

Analysis of the Temperature Dependence of the Specific Heat.

We also reanalyzed the specific heat data digitized from ref. 8 using the band-mixing pairing model. As shown in Fig. 5, the data can also be well described using this model, with fitted parameters and . The value of the small gap is similar to that obtained from the superfluid density fit, while the large gap is smaller in comparison. It should be noted that the calculated superfluid density requires an estimation of , where is a calibration constant for the tunnel diode oscillator (TDO) method, and the small differences in the gap values from the fits may arise due to uncertainties in this value.

Fig. 5.

Specific heat of -type CeCu2Si2 digitized from ref. 8. The solid line shows a fit to the band-mixing pairing model.

Discussion and Summary

Both the superfluid density and specific heat results are highly consistent with a model of the band-mixing pairing state, which most importantly also explains the sign change of the superconducting order parameter. Although the fully gapped nature of the pairing state means that the density of states is zero at low energies, is nearly linear above the small gap, much like for pairing states with line nodes. This is also consistent with the literature results showing -wave superconductivity (15, 18), which were not obtained at low enough temperatures to observe clear evidence for fully gapped behavior. The lack of a coherence peak below in the Cu-NQR measurements (15, 18, 32) can also not be accounted for by a two-gap -wave model but is readily taken into account by the anisotropic state, which changes sign along across the Fermi surface. We note that the effective gap corresponding to a band-mixing pairing is formally identical to one obtained from a pairing and the good fits to and the specific heat are also consistent with this pairing. In the state, time-reversal symmetry would be broken and although no clear experimental indication of a time-reversal symmetry-breaking superconducting state was found from SR measurements of -type samples (41), this requires further study. By contrast, a band-mixing pairing is invariant under time reversal, while generating the expected sign change.

From a theoretical perspective, unconventional superconductors in the presence of strong correlations are generally expected to be robust against disorder (33). This has been demonstrated in models for strongly correlated superconductivity driven by short-range spin-exchange interactions (42, 43). The pairing state (36) arises in a similar fashion and is also expected to be robust against disorder. Because the electrons in heavy fermion systems undoubtedly have strong correlations, the band-mixing pairing state proposed for CeCu2Si2 should be similarly robust to disorder, except for atomic substitutions (35, 44).

In conclusion, we have studied the change of penetration depth and normalized superfluid density of the heavy fermion superconductor CeCu2Si2 (both - and -type samples). The behavior of at very low temperatures agrees with fully gapped superconductivity, as concluded from specific heat measurements (8). We demonstrate that a nodeless band-mixing pairing state can account for the temperature dependence of both the superfluid density and specific heat. This state has the necessary sign change of the superconducting order parameter along on the heavy Fermi surface deduced from INS (14) and is consistent with the lack of a coherence peak in . The model also explains the consistency of -wave superconductivity at higher temperatures, previously reported from measurements (15, 18, 32). We therefore propose this band-mixing pairing state to be the superconducting order parameter of CeCu2Si2. Given that this is also a strong candidate pairing state for FeSe-based superconductors (36), such a pairing model may well be applicable to a wider range of fully gapped unconventional superconductors, including the case of a single CuO2 layer (45).

Materials and Methods

CeCu2Si2 single crystals were synthesized by a modified Bridgman technique using a self-flux method (46). The temperature dependence of the London penetration depth shift was measured down to about 40 mK using a TDO-based technique (47), where , is the resonant frequency of the TDO coil, and is a calibration constant (48).