Research ArticlePHYSICS

Fully gapped superconductivity with no sign change in the prototypical heavy-fermion CeCu2Si2

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Science Advances  23 Jun 2017:
Vol. 3, no. 6, e1601667
DOI: 10.1126/sciadv.1601667


In exotic superconductors, including high-Tc copper oxides, the interactions mediating electron Cooper pairing are widely considered to have a magnetic rather than a conventional electron-phonon origin. Interest in this exotic pairing was initiated by the 1979 discovery of heavy-fermion superconductivity in CeCu2Si2, which exhibits strong antiferromagnetic fluctuations. A hallmark of unconventional pairing by anisotropic repulsive interactions is that the superconducting energy gap changes sign as a function of the electron momentum, often leading to nodes where the gap goes to zero. We report low-temperature specific heat, thermal conductivity, and magnetic penetration depth measurements in CeCu2Si2, demonstrating the absence of gap nodes at any point on the Fermi surface. Moreover, electron irradiation experiments reveal that the superconductivity survives even when the electron mean free path becomes substantially shorter than the superconducting coherence length. This indicates that superconductivity is robust against impurities, implying that there is no sign change in the gap function. These results show that, contrary to long-standing belief, heavy electrons with extremely strong Coulomb repulsions can condense into a fully gapped s-wave superconducting state, which has an on-site attractive pairing interaction.

  • Superconductivity
  • Strongly Correlated Electrons
  • heavy-fermion materials
  • pairing symmetry
  • unconventional superconductors


The discovery of heavy-fermion superconductivity in CeCu2Si2 was an important turning point in the history of superconductivity because it led to the birth of research on non–electron-phonon–mediated pairing (1, 2). Heavy-fermion superconductivity is usually intimately related to magnetism in some form. In particular, superconductivity often occurs in the vicinity of a zero-temperature magnetic instability known as a quantum critical point (QCP) (24). Thus, it is widely believed that in these materials, Cooper pairing is mediated by magnetic fluctuations. The superconducting gap structure is a direct consequence of the mechanism producing the pairing. In phonon-mediated conventional superconductors with a finite on-site pairing amplitude in real space (Fig. 1A), the superconducting gap function Δ(k) is isotropic in momentum space (Fig. 1B). On the other hand, in magnetically mediated unconventional superconductors, the on-site pairing amplitude vanishes because of strong Coulomb repulsion, and superconductivity is caused by a potential that is only attractive for particular displacements between the electrons forming the Cooper pair (Fig. 1C) (5). A net attractive interaction can be realized if the superconducting gap changes sign on the Fermi surface (Fig. 1, D and E). In some materials, such as cuprates (6) and the heavy-fermion CeCoIn5 (7, 8), the sign change of the gap leads to gap functions with nodes along certain momentum directions. However, in certain iron-pnictide superconductors, the gap function has no nodes but may change sign between the well-separated electron and hole Fermi surface pockets (9, 10).

Fig. 1 Pairing interactions and superconducting gap functions.

(A) Pairing interaction in real space for attractive force mediated by electron-phonon interaction. Blue part corresponds to attractive region. Both electrons composing the Cooper pair can occupy the same atom. (B) Isotropic s-wave superconducting state in the momentum space driven by the attractive force shown in (A). The gap function is constant in the entire Brillouin zone. (C) Pairing interaction due to magnetic fluctuations. The red and blue parts correspond to repulsive and attractive regions, respectively. Both electrons cannot occupy the same atom. Superconductivity is caused by the attractive part of the oscillating pairing interaction. (D and E) Examples for the gap structures in momentum space for unconventional superconductors caused by an on-site repulsive force, Embedded Image symmetry (D), and s± symmetry (E). Because of the sign change of the superconducting order parameter, the gap vanishes on the yellow lines. When the Fermi surface crosses these lines, gap nodes appear.

CeCu2Si2 is a prototypical heavy-fermion superconductor near a magnetic instability (1, 11) with a transition temperature Tc ≃ 0.6 K (Fig. 2A) (1). The Fermi surface consists of heavy-electron and light-hole bands (Fig. 2B) (12). Slight variations in stoichiometry lead to “A”-type and “S”-type crystals; the former is antiferromagnetic and the latter is superconducting without magnetic ordering but lying very close to a magnetic QCP (Fig. 2A). The in-plane resistivity above Tc in zero field, which follows a power law ρa = ρa0 + ATϵ, with ϵ = 1.5 (Fig. 2C, inset), along with the heat capacity, which follows Embedded Image in the normal state slightly above the upper critical field, are consistent with non–Fermi liquid behaviors expected for three-dimensional antiferromagnetic quantum critical fluctuations (1315). The magnetic field–induced recovery of Fermi liquid behavior with ϵ = 2 shown in Fig. 2C bears striking resemblance to other heavy-fermion compounds in the vicinity of QCPs, such as CeCoIn5 and YbRh2Si2 (16, 17). A critical slowing down of the magnetic response, revealed by neutron scattering (18) and nuclear quadrupole resonance (NQR) (19) in the normal state, has also been attributed to antiferromagnetic fluctuations near the QCP. These results have led to a wide belief that antiferromagnetic fluctuations are responsible for the pairing interaction in CeCu2Si2. Here, we report a comprehensive study of the gap structure of S-type CeCu2Si2 using several different probes, which together are sensitive to the gap structure on all Fermi surface sheets, and also any possible changing of gap sign between sheets.

Fig. 2 Phase diagrams and electronic structure of CeCu2Si2.

(A) Schematic T-g phase diagram, where g is a nonthermal control parameter, such as pressure, substitution, or Cu deficiency. Red and blue arrows indicate two different types of CeCu2Si2 with antiferromagnetic (A-type) and superconducting (S-type) ground states, respectively. The S-type crystal locates very close to antiferromagnetic (AFM) QCP. (B) Fermi surface colored by the Fermi velocity (in units of 106 m/s) obtained by the local density approximation (LDA) + U calculation (12). Fermi surface consists of separated electron and hole pockets: heavy electron pockets with cylindrical shape around X-point and rather complicated light hole pockets centered at Γ point. (C) H-T phase diagram with color coding of T exponent (ϵ) of the in-plane electrical resistivity, ρ(T) = ρ0 + ATϵ for Hc. Inset shows the T dependence of ρ(T) in zero field and in magnetic fields of 2.5 and 12 T applied along the c axis.


Specific heat

Specific heat C is a bulk probe that measures all thermally induced excitations. Figure 3A and its inset depict the specific heat divided by temperature C/T for a crystal used in the present study. At zero field, C/T exhibits a sharp transition at Tc and tends toward saturation at the lowest temperature. The C/T value at the lowest temperature, 15 mJ/K2 mol, is less than 2% of γN, which indicates a very low number of quasi-particle excitations and that any inclusion of nonsuperconducting A-type material is very small. This value is around half of that previously reported (20) and demonstrates the improved quality of the present samples. The data are well fitted by an exponential T dependence, showing a lack of thermally induced excitations at the lowest temperatures in agreement with the previous studies (20, 21). A linear behavior does not fit our C/T data but if it was forced to, then a fit above 90 mK in Fig. 3A would lead to an unphysical negative intercept at T = 0 K. This is indicative of a fully gapped state with minimal disorder. More precisely, because the specific heat is dominated by the parts of the Fermi surface where the Fermi velocity is low (or mass is large), the C/T data suggest the absence of line nodes in the heavy electron band.

Fig. 3 Temperature dependencies of specific heat and London penetration depth well below the superconducting transition temperature Tc.

(A) Inset shows the specific heat divided by temperature C/T in zero field and in the normal state at μ0H = 2 T for Hab plane. The main panel shows C/T at low temperatures. The gray solid line is an exponential fit of the data, yielding Δ = 0.39 K. (B) Temperature-dependent change in the in-plane penetration depth Δλ in a single crystal of CeCu2Si2. The dashed (solid) line is a fit to a power law (exponential) temperature dependence up to 0.2 K. Inset shows the normalized superfluid density ρs(T) = λ2(0)/λ2(T) as a function of T/Tc, extracted by using a value of λ(0) = 700 nm (section S2). The dashed line is the temperature dependence of ρs(T) in the simple d-wave case.

Penetration depth and lower critical field

By contrast, the magnetic penetration depth measures the surface of the sample (to a depth of a few micrometers) and is dominated by the low-mass, high-velocity parts of the Fermi surface. We find that the in-plane penetration depth λab(T) at low temperatures (TTc) exhibits strong curvature and tends toward becoming T-independent (Fig. 3B), similar to the results for C/T and in contrast to the T linear dependence, expected for clean superconductors with line nodes (22). A fit to power law T dependence Δλ(T) = (λab(T) − λab(0)) ∝ Tn gives a high power n > 3.5 (see section S1 and figs. S1 and S2), which is practically indistinguishable from the exponential dependence expected in fully gapped superconductors. Because λab measures the in-plane superfluid response, our data show that gap nodes, at which quasi-particles with momentum parallel to the ab plane are excited, are absent on the light-hole bands.

For a more detailed analysis of the superconducting gap structure, the absolute value of λab(0) is necessary so that the normalized superfluid density Embedded Image can be calculated. Unfortunately, previous measurements have reported a wide spread of values of λab(0) [120 to 950 nm (23, 24)], which probably reflects differences in sample stoichiometry between studies. We have estimated λab(0) = 700 nm from Hall-probe magnetometry measurements of the lower critical field Hc1 in the same samples, as used for our Δλ(T) study (section S2 and fig. S3). The inset of Fig. 3B shows the T dependence of ρs(T). Near Tc, we find convex curvature in ρs(T), which is a signature frequently observed in multigap superconductors (25). Two-gap behavior has been reported in the recent scanning tunneling spectroscopy (26) and specific heat measurements (20, 21).

Thermal conductivity

Thermal conductivity is a bulk, directional probe of the quasi-particle excitations and, like penetration depth, is dominated by the high-velocity parts of the Fermi surface (27). Figure 4A and its inset show the T dependence of the in-plane thermal conductivity κa/T (with heat current Qa). The thermal conductivity in the normal state at T → 0 K, slightly above the upper critical field for H ∥ c, obeys well the Wiedemann-Franz law, κa/T = L0a (Fig. 4A, dashed line), where L0 is the Lorenz number and ρa is the in-plane resistivity. At the lowest temperatures, κa/T extrapolated to T = 0 K is zero within our experimental resolution and is at least an order of magnitude smaller than that expected for line nodes (section S3 and fig. S4), consistent with the λ(T) results.

Fig. 4 Thermal conductivity of CeCu2Si2 for various directions of thermal current and magnetic field.

(A) Temperature dependence of the in-plane thermal conductivity divided by temperature κa/T in zero field and in magnetic field of μ0H = 2.2 T applied along the c axis. WF refers to κ/T at T → 0 calculated from the Wiedemann-Franz law. (B) Field dependence of κ/T for two different configurations: (i) κa/T (Qa) in Hc and (ii) κc/T (Qc) in Hc. In these configurations, thermal conductivity selectively probes the excited quasi-particles with in-plane momentum. The dashed horizontal lines represent the phonon contribution, κph/T, estimated from the WF law above upper critical field (see the main text). (C) Field dependence of κc/T for configuration (iii), where Qc and Ha. In this case, thermal conductivity selectively probes the excited quasi-particles with out-of-plane momentum. (D) Field-induced enhancement of thermal conductivity Δκ(H) ≡ κ(H) − κ(0) normalized by the normal-state value, Δκ(H)/Δκ(Hc2), for the configurations (i), (ii), and (iii) plotted against the magnetic field normalized by the upper critical fields. Black and green broken lines represent the field dependencies expected for line and point nodes.

Further evidence for the absence of any nodes is provided by magnetic field dependence of κ. In fully gapped superconductors, where all the quasi-particle states are bound to vortex cores, the magnetic field H hardly affects κ except in the vicinity of the upper critical field Hc2. By contrast, in nodal superconductors, heat transport is dominated by the delocalized quasi-particles. In the presence of a supercurrent with velocity vs around the vortices, the energy of a quasi-particle with momentum p is Doppler-shifted relative to the superfluid by E(p) → E(p) − vsp, giving rise to an initial steep increase of Embedded Image for line nodes and κ(H)/TH log H for point nodes. Thermal conductivity selectively probes the quasi-particles with momentum parallel to the thermal current (p · Q ≠ 0) and with momentum perpendicular to the magnetic field (p × H ≠ 0) because Hvs (27). To probe the quasi-particle excitations on the whole Fermi surface, we performed measurements for three different configurations: (i) κa for Hc, (ii) κc for Hc, and (iii) κc for Ha (Fig. 4, B and C). For (i) and (ii), thermal conductivity selectively probes the quasi-particles with in-plane momentum, whereas for (iii), it selectively probes quasi-particles with out-of-plane momentum. For configuration (ii), there is structure at μ0H ~ 1 T, which again indicates the presence of multiple superconducting gaps. The H dependence for configuration (iii) shown in Fig. 4C is similar to configuration (i). Remarkably, in all configurations, magnetic field hardly affects the thermal conduction in the low-field regime (Fig. 4, B and C); the field-induced enhancement, Δκ(H) ≡ κ(H) − κ(0) is less than 1/100 of the normal-state value Δκ(Hc2) even at H/Hc2 ~ 0.15, demonstrating a vanishingly small number of delocalized quasi-particles excited by magnetic field. As shown by the dashed lines in Fig. 4D, Δκ(H)/Δκ(Hc2) is far smaller than that expected for line and point nodes.

Electron irradiation

The above measurements of C(T), Δλ(T), and κ(T, H) demonstrate the absence of any kind of nodes in the gap function on the whole Fermi surface. To further distinguish between the remaining possible gap structures, we have measured the effect of impurity-induced pair breaking on Tc. These measurements are a sensitive test of possible sign changes in the gap function either between different Fermi surface sheets or on a single sheet. Impurity-induced scattering between sign-changing areas of Fermi surface will reduce Tc very rapidly, whereas if there is no sign change, the reduction will be much slower or even zero. To introduce impurity scattering by homogeneous point defects in a controllable way, we used electron irradiation with an incident energy of 2.5 MeV (28), which according to our calculation of electron-scattering cross sections mainly removes Ce atoms. Electronic-structure calculations of CeCu2Si2 (12) show that the bands crossing the Fermi level are mainly composed of a single Ce f-manifold; therefore, removing Ce atoms by electron irradiation will act as a strong point scatterer and induce both intra- and interband impurity scattering with similar amplitude.

Our results show that Tc of CeCu2Si2 is decreased slowly with increasing dose (inset of Fig. 5A). The transition width remains almost unchanged after irradiation, indicating good homogeneity of the point defects. The temperature dependence of resistivity indicates that the primary effect of irradiation is the increase of temperature-independent impurity scattering with dose and that the temperature-dependent inelastic scattering remains unaffected. In- and out-of-plane residual resistivities reach ρa0 ~ 120 microhm·cm and ρc0 ~ 110 microhm·cm for irradiated crystals (inset of Fig. 5A). Using Embedded Image (j = ab or c), we obtain in- and out-of-plane mean free paths, Embedded Image nm and Embedded Image nm, respectively. Here, we used averaged in-plane (out-of-plane) Fermi velocity Embedded Image m/s (Embedded Image m/s) of the light-hole band, λc(0) = λab(0)(ξabc) = 480 nm, where in-plane and out-of-plane coherence lengths are determined by the orbital-limited upper critical fields, ξab = 4.7 nm and ξc = 6.9 nm, respectively (section S3). These mean free paths are obviously shorter than ξab and ξc. For unconventional pairing symmetries, such as d-wave, superconductivity is completely suppressed at Embedded Image. In stark contrast, Tc of CeCu2Si2 is still as high as ~ Tc0/2 even for Embedded Image and Embedded Image. Note that this Embedded Image is the upper limit value because Embedded Image is estimated from the penetration depth and conductivity, both of which are governed by the light bands, whereas ξ is determined by the upper critical field, which is governed by heavy bands. We also note that our electron irradiation data are semiquantitatively consistent with the oxygen irradiation results reported for thin films (29), although the initial Tc = 0.35 K for the pristine film is much lower than that of our single crystals. Thus, these results demonstrate that superconductivity in CeCu2Si2 is robust against impurities. This is also clearly seen by comparison to the d-wave superconductor CeCoIn5 (30), which has comparable effective mass and carrier number. Figure 5A shows the residual resistivity dependence of Tc/Tc0, where Tc0 is the transition temperature with no pair breaking. In CeCoIn5, Tc is suppressed to zero in the sample with ρ0/Tc0 smaller than 10 microhm·cm/K (30), whereas Tc in CeCu2Si2 is still ~ 50% of Tc0 even for the sample with ρ0/Tc0 larger than 150 microhm·cm/K, indicating that the pair-breaking effect in CeCu2Si2 is fundamentally different from that in CeCoIn5.

Fig. 5 Pair-breaking effect of CeCu2Si2.

(A) Suppression of superconducting transition temperature Tc/Tc0 as a function of ρ0/Tc0, which is proportional to the pair-breaking parameter for CeCu2Si2 and Sn-substituted CeCoIn5 (d-wave) (30). Here, Tc0 is the transition temperature with no pair-breaking effect and ρ0 is the residual resistivity. For CeCu2Si2, Tc0 = 0.71 K is used. Inset shows the temperature dependence of resistivity in CeCu2Si2 before and after electron irradiation, which creates point defects. (B) Comparison of impurity effect of CeCu2Si2 with those of other superconductors. Suppression of superconducting transition temperature Tc/Tc0 as a function of dimensionless pair-breaking parameter ℏ︀/τimpkBTc0. The solid line shows the prediction of the Abrikosov-Gor’kov (AG) theory for an isotropic s-wave superconductor with magnetic impurities. We also plot the data for Sn-substituted CeCoIn5 (d-wave) (30), electron-irradiated YBa2Cu3O7-δ (d-wave) (31), electron-irradiated Ba(Fe0.76Ru0.24)2As2 (possibly s± wave) (32), and neutron-irradiated MgB2 (33) and YNi2B2C (35). The value of Tc0 is estimated by extrapolating two initial data points to zero 1/τimp limit. Rather weak pair-breaking effect in Ba(Fe0.76Ru0.24)2As2 has been attributed to a large imbalance between intra- and interband scattering (32). For MgB2 data, we use the value of λab(0) = λc(0) = 100 nm (34). For YNi2B2C data, we use λab(0) = 110 nm (36) and Embedded Image nm (37), where Embedded Image and Embedded Image are upper critical fields parallel and perpendicular to the ab plane.

Comparison to other materials (cuprates and iron pnictides) with sign-changing gaps confirms the much weaker effect of impurities in CeCu2Si2. In Fig. 5B, we plot the impurity-induced Tc reduction in a number of materials [CeCoIn5 (30), YBa2Cu3O7 (31), Ba(Fe0.76Ru0.24)2As2 (32), MgB2 (33, 34), and YNi2B2C (3537)] as a function of dimensionless scattering rate ħ/τimpkBTc0, where τimp is the impurity scattering time estimated from residual resistivity ρ0 and the penetration depth, τimp = μ0λabλc0. Plotting the data in this way removes the effect of difference in carrier density and effective mass between the different materials, and it can be seen that the Tc reduction in CeCu2Si2 is much weaker than the archetypal cuprate YBa2Cu3O7 (31), which has a sign-changing Embedded Image gap function. The iron pnictides present a very unusual system where the k dependence of the scattering is critical to the effect of impurities on Tc. Assuming that the pairing in these materials is caused by interband spin-fluctuation interactions, the gap function is expected to change sign between the electron and hole sheets (s± pairing). Interband impurity scattering will then increase ρ0 and decrease Tc in a similar way to other sign-changing gap materials; however, if the scattering is purely intraband, then this would increase ρ0 but would not decrease Tc (38). It is highly unlikely that this anomalous situation could occur in CeCu2Si2 because the Fermi surface sheets are not well separated and, as described above, Ce vacancies would produce non–k-selective scattering.

The slow but finite reduction in Tc as a function of ρ0 that we see in CeCu2Si2 can be explained qualitatively by the moderate gap anisotropy that we have observed in our C(T) and λ(T) measurements. In cases where there is gap anisotropy, scattering will tend to average out the gap, thus depressing Tc. However, crucially this will be at a much slower rate than for a sign-changing gap, as shown in Fig. 5B by data for the non–sign-changing s-wave superconductors MgB2 (33) and YNi2B2C (35), which are known to have very anisotropic energy gaps. Our observed slower decrease in Tc as a function of impurity scattering in CeCu2Si2 compared to these materials is consistent with our observed moderate anisotropy.


The combination of our measurements and previous results is far more consistent with one possible gap structure. The strong reduction of the spin susceptibility in the superconducting state observed by nuclear magnetic resonance Knight shift indicates spin-singlet pairing (39), which rules out any odd-momentum (p or f) states, including those, such as the Balain-Werthamer state (40), that are fully gapped (41). This is consistent with the observation that Hc2 is Pauli-limited (42). In the present crystal, orbital-limited upper critical fields at T = 0 K calculated from Embedded Image are 10.0 and 14.7 T for Ha and Hc, respectively. These values are much larger than the observed Hc2 of 2.0 T for Ha and 2.3 T for Hc.

Our observation that superconductivity is robust against inter- and intraband impurity scattering rules out any sign-changing gap functions, such as d-wave or the recently proposed sign-changing s± state (12). Both the d-wave and s± states are also highly unlikely because neither could be nodeless in CeCu2Si2 where the electron and hole Fermi surface sheets are not well separated. Finally, unconventional states that combine irreducible representations of the gap function, such as Embedded Image or Embedded Image, can be ruled out because these states would be highly sensitive to impurities and furthermore, because in general, these representations are not degenerate, we would expect to see two distinct superconducting transitions. If there was accidental degeneracy, this would be broken by pressure or doping, but no double transitions are observed under these conditions either (11). This leads us to the surprising conclusion that the pairing in CeCu2Si2 is a fully gapped non–sign-changing s-wave state.

Previously, evidence for line nodes in CeCu2Si2 has been suggested by measurements of the NQR relaxation rate 1/T1 where a T3dependence below Tc was observed (19). However, these results would also be explained by the multigap nature of the superconductivity shown here by our C(T), λ(T), and κ(H) measurements. The absence of the coherence (Hebel-Slichter) peak in 1/T1(T) below Tc may be explained by the quasi-particle damping especially for anisotropic gap and thus does not give conclusive evidence for the sign-changing gap (21). Inelastic neutron scattering shows an enhancement of magnetic spectral weight at about E ~ 2Δ (43), which could be interpreted in terms of a spin resonance expected in superconductors with a sign-changing gap. However, this enhancement is very broad compared with some cuprates (44) and CeCoIn5 (45), therefore is not clearly a resonance peak, which is expected to be sharp in energy. Moreover, recent calculations show that a broad maximum at E ~ 2Δ appears even in superconductors without sign-changing gaps (46). We should add that even in the d-wave CeCoIn5 case, the interpretation of the neutron peak below Tc is still controversial (47, 48). Hence, the NQR and neutron results do not provide conclusive evidence for a sign-changing gap structure and are not necessarily inconsistent with the results here.

At first sight, our finding that CeCu2Si2 has a non–sign-changing s-wave gap function casts doubt on the long-standing belief that it is a magnetically driven superconductor, despite overwhelming evidence that this compound is located near a magnetic QCP. It is unlikely that the conventional electron-phonon interaction could overcome the on-site strong Coulomb repulsive force, which enhances the effective mass to nearly 1000 times the bare electron mass, in this heavy-fermion metal, which does not have high-energy strong-coupled phonons. However, recent dynamic mean field theory calculations have shown that robust s-wave superconductivity driven by local spin fluctuations is found in solutions to the Kondo lattice model, which is commonly used to describe heavy-fermion metals (49). Other recent theoretical work has shown that electron-phonon coupling could be strongly enhanced near a QCP again, stabilizing s-wave superconductivity (50). Our results might therefore support a new type of unconventional superconductivity where the gap function is s-wave but the pairing is nevertheless driven by strong magnetic fluctuations.


S-type single crystals of CeCu2Si2 were grown by the flux method (51). Specific heat was measured by the standard quasi-adiabatic heat pulse method. The temperature dependence of penetration depth λ(T) was measured by using the tunnel diode oscillator technique operating at ~ 14 MHz. Weak ac magnetic field (~ 1 μT) was applied along the c axis, inducing screening currents in the ab plane. Thermal conductivity was measured by the standard steady-state method using one heater and two thermometers, with an applied temperature gradient less than 2% of the sample temperature. The contacts were made by indium solder with contact resistance much less than 0.1 ohm. We examined the effect of superconductivity of indium by applying a small magnetic field and found no discernible difference. We also measured with contacts made of silver paint with contact resistance less than 0.1 ohm and observed identical results. We measured several different crystals and obtained essentially the same results.

Electron irradiation was performed in the electron irradiation facility SIRIUS (Système d’Irradiation pour l’Innovation et les Utilisations Scientifiques) at École Polytechnique. We used electrons with an incident energy of 2.5 MeV for which the energy transfer from the impinging electron to the lattice is above the threshold energy for the formation of vacancy interstitial (Frenkel) pairs that act as point defects (28). To prevent the point-defect clustering, we performed irradiation at 25 K using a H2 recondenser. According to the standard calculations, the penetration range for irradiated electrons with 2.5-MeV energy is as long as ~ 2.75 mm, which is much longer than the thickness of the crystals (typically 50 to 100 μm). This ensures that the point defects created by the irradiation are uniformly distributed throughout the sample thickness. Our simulations also show that for 1 C/cm2 dose, irradiation causes about one to two vacancies per 1000 Ce atoms.


Supplementary material for this article is available at

section S1. Temperature dependence of penetration depth

section S2. Lower critical field

section S3. Zero-field thermal conductivity

fig. S1. Magnetic penetration depth versus temperature for two samples measured.

fig. S2. Parameters obtained for the fits to the Δλ(T) data.

fig. S3. Lower–critical field measurements of CeCu2Si2.

fig. S4. Temperature dependence of thermal conductivity at low temperatures.

References (5254)

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Acknowledgments: We thank A. Chubukov, P. Coleman, P. Hirschfeld, K. Ishida, H. Kontani, H. v. Löhneysen, P. Thalmeier, C. Varma, I. Vekhter, and Y. Yanase for useful discussions. Irradiation experiments were supported by EMIR network, proposal No. 16-0398, 16-9513, and 17-1353. Funding: This work was supported by Grants-in-Aid for Scientific Research (KAKENHI) (nos. 25220710, 15H02106, 15K05158, and 16H04021) and Grants-in-Aid for Scientific Research on Innovative Areas “Topological Materials Science" (no. 15H05852) and “J-Physics” (no. 15H05883) from the Japan Society for the Promotion of Science and by the UK Engineering and Physical Sciences Research Council (grant no. EP/L025736/1 and EP/L015544/1). Author contributions: Y. Matsuda and T. Shibauchi conceived and designed the study. T.Y., Y.T., Y.K., T.O., and D.T. performed the thermal conductivity measurements. T.T., J.A.W., Y. Mizukami, C.P., A.C., and T. Shibauchi performed penetration depth measurements. T.Y., T.T., Y.T., Y. Mizukami, and D.T. performed electrical resistivity measurements. Y.T., S.K., and T. Sakakibara performed specific heat measurements. M.K. and Y. Mizukami electron-irradiated the samples. H.S.J., S.S., and C.G. synthesized the high-quality single-crystalline samples. H.I. performed calculations based on the first-principles calculations. T. Shibauchi, Y.T., A.C., and Y. Matsuda discussed and interpreted the results and prepared the manuscript. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.
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