Abstract
Despite almost 40 years of research, the origin of heavy-fermion superconductivity is still strongly debated. Especially, the pressure-induced enhancement of superconductivity in CeCu2Si2 away from the magnetic breakdown is not sufficiently taken into consideration. As recently reported in CeCu2Si2 and several related compounds, optimal superconductivity occurs at the pressure of a valence crossover, which arises from a virtual critical end point at negative temperature Tcr. In this context, we did a meticulous analysis of a vast set of top-quality high-pressure electrical resistivity data of several Ce-based heavy fermion compounds. The key novelty is the salient correlation between the superconducting transition temperature Tc and the valence instability parameter Tcr, which is in line with theory of enhanced valence fluctuations. Moreover, it is found that, in the pressure region of superconductivity, electrical resistivity is governed by the valence crossover, which most often manifests in scaling behavior. We develop the new idea that the optimum superconducting Tc of a given sample is mainly controlled by the compound’s Tcr and limited by non-magnetic disorder. In this regard, the present study provides compelling evidence for the crucial role of critical valence fluctuations in the formation of Cooper pairs in Ce-based heavy fermion superconductors besides the contribution of spin fluctuations near magnetic quantum critical points, and corroborates a plausible superconducting mechanism in strongly correlated electron systems in general.
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Introduction
Superconductivity (SC) in heavy fermion (HF) systems is most often considered as being mediated by critical spin fluctuations.1,2,3,4 Such a prevailing view is mainly derived from the presence of a magnetic instability regime leading to the collapse of long-range antiferromagnetic (AF) order at a critical pc, concomitant with the emergence of SC. However in a few cases, SC has been ascribed to critical valence fluctuations (CVF) in the pressure region of the highest superconducting transition temperature Tc, in particular for CeCu2Ge2,5,6 CeCu2Si2,7,8 and CeRhIn5.9,10 The main ingredient of this interpretation is the existence in the pressure-temperature (p-T) plane of an underlying first-order valence transition (FOVT), whose critical end point (CEP) occurs at pressure pcr and at slightly negative temperature Tcr (see Fig. 1). With a negative Tcr, only a valence crossover (VCO) regime is accessible at finite temperature and the corresponding crossover line lies close to optimal SC. In the case of the prototype HF superconductor CeCu2Si2, multiple experimental evidence of the VCO and CVF-mediated SC has been reported in.5,7,8,11,12,13,14 For instance, direct, microscopic observation of the VCO and the absence of spin fluctuations close to optimal SC have been reported for CeCu2Si213 and also CeIrIn515,16 via Cu- and In-nuclear quadrupole resonance measurements, respectively.
Selected examples of p-T magnetic and superconducting phase diagrams of Ce-based HF superconductors are represented schematically in Fig. 1. The common feature of all compounds is that SC is optimal at a pressure close to pcr. On the other hand, the magnetic pc can coincide with pcr as in the case of CeRhIn5,17 CePd2Si2 (this work), and CeAu2Si2,18 or be much lower than pcr as in CeCu2Si2.8 The spreading of SC over the pressure axis varies considerably and SC can even emerge deep inside the magnetic phase of CeAu2Si2.18
As a matter of interest, the CVF mechanism shares common aspects with the d-p charge transfer instability in high-Tc cuprates, which has been proposed to be at the origin of marginal Fermi liquid and non-Fermi liquid properties, and the pseudo-gap state.20,21 Moreover, valence fluctuations of Pu ions have been advocated as the source of “high-Tc” in PuCoGa5.22 Thus, we believe that the valence fluctuation physics discussed in this paper is pertinent for a larger community beyond that of HFs.
The microscopic-theoretical basis of the CVF scenario results from the inclusion of the additional term \(H_{U_{{\mathrm{fc}}}} = U_{{\mathrm{f}}c}\mathop {\sum}\nolimits_{i = 1}^N n_i^fn_i^c\) in the periodic Anderson model, where Ufc is the Coulomb repulsion between f and conduction electrons.23,24 The physical origin of the Tc enhancement is the increase in the effective quasiparticle Fermi energy and the constancy of the dimensionless coupling for the Cooper pairing, following a BCS-like expression for Tc. The former factor stems from the VCO from the Kondo to the valence fluctuation region and the latter one is a result of the compensation between the decrease of the quasiparticle density of states and the increase in the pairing interaction, which is mediated by valence fluctuations associated with a sharp VCO.
Five years ago, thanks to an experimental progress25 yielding more accurate electrical resistivity measurements on CeCu2Si2 under pressure up to 7 GPa, we have introduced a method to estimate the temperature Tcr of the CEP.8 Subsequently, the same process was successfully used for CeAu2Si2.18,26,27 In the present paper, this method is applied to all appropriate resistivity data established in Geneva since 1998, including new data notably from CeAg2Si2, CeRhIn5, and CeIrIn5. On the basis of 17 data sets from nine different Ce-based HF compounds, the universal character of the relationship between the superconducting transition temperature Tc and the strength of the valence instability is unveiled. Taking into account the superconducting pair-breaking effect of non-magnetic disorder, quantified by the residual resistivity ρ0, we identify the two main parameters Tcr and ρ0 controlling Tc of a representative part of Ce-based HF superconductors, which is consistent with the CVF theory. Moreover, it is found that, in the VCO regime of the p-T plane, electrical resistivity most often follows scaling behavior, underlining the role of valence fluctuation physics.
Results
Figure 2 displays a 3D plot of the superconducting Tc as function of both the residual resistivity ρ0 and the valence instability parameter Tcr based on published and new results (see Table S1 of the Supplementary Material for details and references). In this paper “Tc” refers to the maximum value of the bulk-superconducting transition temperature versus pressure for a given sample. Evidently in Fig. 2, all compounds except CeCu5Au lie more or less on an empirically drawn inclined surface with a maximum for small ρ0 and Tcr, which suggests that the superconducting Tc of a given sample is mainly controlled by the compound’s Tcr and the sample’s ρ0. Tc seems to culminate at ~2.5 K when Tcr → 0 and ρ0 → 0, i.e., for a quantum CEP and negligible pair breaking effect. However, high ρ0 values or large negative Tcr depress Tc.
We underline that all samples with Tc > 2 K are found to exhibit −15 K < Tcr < 0 K and emphasize the striking relationship between the superconducting Tc, the parameter Tcr of the valence transition CEP, and pair breaking due to non-magnetic impurities (ρ0). We introduce the expression “high-Tc“ to refer to the fact that the compounds with the highest Tc amongst the Ce-based HF superconductors are especially well represented in Fig. 2. Five out of the nine studied compounds have Tc higher than 1.5 K. At the moment, important cases like CeCoIn5,28 CeRh2Si2,29 CePt2In7,30 or non-centrosymmetric CePt3Si31 are lacking for different reasons (see below). Nevertheless, Fig. 2 represents a substantial part of Ce-based HF compounds and gives a unified view on their SC.
Before taking a closer look to the relationships Tc(Tcr) and Tc(ρ0), let us discuss the behavior of electrical resistivity ρ in the VCO regime and summarize the method for extracting Tcr from low-temperature ρ.8 First, in Fig. 3(a) we compare the schematic p-T phase diagrams of CeCu2Si2 and elementary Ce. In Ce, a FOVT occurs at finite temperature due to small Ce–Ce ion spacing and therefore strong Ufc-repulsion between f- and conduction electrons at the same Ce site. The CEP lies at pcr ≈ 1.5 GPa and Tcr ≈ 480 K.32 As a function of pressure, isothermal resistivity of Ce [Fig. 3(b)] exhibits a discontinuous anomaly at the FOVT (T < Tcr).33 In the crossover regime (T > Tcr), isothermal resistivity decreases rapidly but continuously and the resistivity gradient diverges just at the CEP (T → Tcr). In CeCu2Si2, the CEP lies at slightly negative temperature Tcr ≈ −8 K and, in the VCO regime, isothermal ρ decreases more and more rapidly versus pressure as temperature goes down without reaching a first-order discontinuity.8
For a detailed analysis, the p-dependence of ρ* = ρ − ρ0 at several temperatures is derived from ρ(T) of successive pressure runs, as shown for CeAg2Si2 in Fig. 3(c). A preliminary remark is that in all Ce-HF compounds the resistivity ρ* is strongly reduced by 1–2 orders of magnitude, when the system is tuned by pressure through the VCO. Such a reduction, which exceeds that expected for a progressive increase of the c-f hybridization, is attributed to a more or less sudden delocalization of 4f electrons.12 In order to disentangle the intrinsic effect of electron delocalization from that of the temperature-dependent scattering rate, the resistivity has to be normalized. For this purpose we define an initial pressure pin, which signals the onset of the VCO resistivity collapse (see inset of Fig. 3(c) and page 9 of the Supplementary Material for CePd2Si2). Then, the normalized resistivity \(\rho ^{{\mathrm{norm}}} = \frac{{\rho ^ \ast (p) - \rho ^ \ast (p_{{\mathrm{VCO}}})}}{{\rho ^ \ast (p_{{\mathrm{VCO}}})}}\), where pVCO is the pressure of the mid drop of ρ*, is calculated for each temperature.
By way of example, Fig. 3(d) shows ρnorm versus p − pin at 3 K of CeCu2Si2, CeRhIn5, CeAg2Si2, and CePd2Si2. Clearly, it appears that the collapse of ρnorm is always close to optimal SC and a steeper collapse favors higher Tc [see Fig. 3(e)]. With increasing temperature, i.e., increasing distance from the CEP, the pressure scale of the resistivity reduction gets broader and broader and the steepness of the collapse decreases, as shown for CeCu2Si2 in Fig. 3(b). Figure 3(f) displays the temperature dependence of the slope \(\chi = \left| {{\mathrm{d}}\rho ^{{\mathrm{norm}}}/{\mathrm{d}}p} \right|_{p_{{\mathrm{VCO}}}}\) of CeCu2Si2 and CeAg2Si2. χ, which we interpret as valence susceptibility, tends to diverge as χ ∝ (T − Tcr)−1, i.e., a first-order discontinuity would occur in ρ*(p) for T < Tcr. Evidently, a simple fit to χ(T) yields Tcr. The empirical law (T − Tcr)−1 is confirmed by data from several samples of CeCu2Si2,8 CeAu2Si2,18,26,27 and CeRhIn5,17 which are by the way the compounds with highest Tc and least negative Tcr. Note that the plot of χ(T) is limited to a temperature (15 K), which corresponds to a small fraction of the first crystal-field-splitting energy. Such a treatment is repeated on all appropriate data from Ce-based HF compounds (see Supplementary Figs. S4–S13). The extracted Tcr values and other parameters (Tc, ρ0) are summarized in Table S1 of the Supplementary Material.
After identifying Tcr, one can apply the scaling treatment developed in Ref. 8 for CeCu2Si2 within the framework of universal scaling theory of critical phenomena and subsequently applied on data from CeAu2Si218,26,27 and CeRhIn5.17 To this end, a generalized distance h/θ from the CEP is calculated, where h = (p − pVCO)/pVCO and θ = (T − Tcr)/|Tcr|. Then, for a given compound, all ρnorm isotherms in the VCO regime collapse on a single curve ρnorm = f(h/θ) when plotted versus h/θ, as shown in Fig. 3(g). This means that for the generalized distance h/θ from the CEP, the ρnorm isotherms behave in a unique manner, which strongly supports the existence of the valence CEP at (pcr, Tcr). Note that in terms of universal scaling theory of critical phenomena the equation is ρnorm/h1/δ = f(h/θγδ/(δ−1)), with the critical exponents γ and δ (mean-field approach: γMF = 1, δMF = 3). Accordingly to Ref. 8, the critical exponents are fixed as γ = 1 and δ → ∞, which does not correspond to a simple universality class.
The scaling is very robust for 15 data sets from seven different systems and astonishingly the scaling function f(h/θ) is identical for the isovalent systems CeCu2Si2, CeAg2Si2, and CeAu2Si2 [see Fig. 3(g)]. However, f(h/θ) is different for CePd2Si2 and apparently material dependent. For instance f(h/θ) of CeCu5Au (Supplementary Fig. S12) lies in between the two data sets of Fig. 3(g). The scaling fails for two data sets: the ρnorm isotherms of CeCu6 (Supplementary Fig. S11) and CeIrIn5 (Supplementary Fig. S13) do not collapse on a single curve. Measurement errors surely play a role for the CeCu6 sample (see Supplementary Fig. S11) and a change of regime ascribed to the crystal-field effect34 may interfere. In CeIrIn5, low-temperature resistivity properties hint to a pressure-induced change of regime (see Supplementary Fig. S13), which may explain that the isotherms collapse only for h/θ > 0. Though, completely satisfactory explanations are still missing for both.
We now discuss the pair-breaking effect by disorder, which is quantified by the residual resistivity ρ0. Figure 4 shows Tc versus ρ0 of the CeCu2Si2 family and CeRhIn5 from all 26 independent pressure experiments done in Geneva with ρ0 < 50 μΩcm (see Supplementary Fig. S1 for details and references). Not included are still finite Tc values corresponding to very high ρ0, which deviate from the general trend possibly due to alloying or Kondo-hole effects. For instance, in a CeCu2(Si1−xGex)2 alloy, a maximum bulk Tc ~ 0.6 K (ρ = 0 criteria) is reported for ρ0 ~ 70 μΩcm,35 and, in polycrystalline CeCu2Si2, a maximum bulk Tc ~ 0.3 K is given with ρ0 ~ 180 μΩcm, which is higher than the room temperature resistivity.36 The evident decrease of Tc with increasing disorder follows qualitatively well the formula given by the Abrikosov-Gor’kov (AG) theory37 generalized for non-magnetic disorder in a CVF-mediated d-wave superconductor with a critical resistivity \(\rho _0^{{\mathrm{cr}}}=46\,\mu\Omega\mathrm{cm}\).38 Note that every data point in Fig. 4 refers to a set of pressure runs for an experiment on a sample of specific quality as reflected by its ρ0(p = 0) value. Thus in spite of a given data scattering, the systematic dependence of Tc on ρ0 for different samples of different compounds is remarkable.
A similar trend is observed for the CePd2(Si/Ge)2 family, where bulk SC vanishes completely for ρ0 higher than 3 μΩcm (see Supplementary Fig. S2). Tc is already small at ideal sample quality since these compounds are located far from the criticality as signaled by the large negative Tcr ~ 50 K. Therefore, the theory of Okada et al.38 for robustness of Tc versus pair-breaking effect by non-magnetic disorder is not appropriate, and the conventional AG-theory for anisotropic SC can be applied. The latter is valid for the d-wave order parameter predicted by the CVF theory23 and accounts for the rapid decrease of Tc in CePd2Si2.
The critical resistivity \(\rho _0^{{\mathrm{cr}}} \approx 46\,\mu {\mathrm{\Omega cm}}\) of the “high-Tc” HF superconductors is far larger than that expected in the conventional case of weak-coupling SC but is compatible with the generalized AG theory.38 In fact, due to the valence fluctuation renormalization effect of the impurity potential,39 ρ0 is strongly increased at pressures around pcr compared to far lower or higher pressures, which is a hallmark of CVF-mediated HF superconductors (exceptions are CeCu634 and CePd2Si219). The robustness of Tc against impurity scattering is due to the fact that the re-normalized impurity potential is a long-range like bare Coulomb potential.38,39 Indeed, almost all scattering channels with angular momentum \(\ell = 0,1,2 \ldots ,\) i.e., s-, p-, d-wave and so on, are active in the Coulomb-type potential, leading to partial cancellation in the scattering rate among the \(\ell {\mathrm{ - wave}}\) vertex part in the pair susceptibility and the self-energy part in the Green function. This rationalizes the robustness of Tc against the enhanced impurity potential in contrast to the conventional AG-type theory for anisotropic pairing, where essentially an s-wave component of impurity potential is taken into account.40,41
Figure 5 presents the most interesting relationship between Tc and the key parameter Tcr. Tc is maximal for small negative Tcr and decreases as Tcr becomes more negative, which is qualitatively predicted by CVF theory.23,24 About half of the data correspond to samples with almost optimal Tc for the specific compounds thanks to low ρ0 values (see Supplementary Table S1), which underlines the intrinsic character of the Tc-vs-Tcr relationship. Unlike Fig. 4, less data points are presented because of the stringent requirement of accuracy and reliability of resistivity measurement for the scaling analysis, which excludes a part of our results and also those found in literature. In respect of the procedure to deduce Tcr, a prerequisite is the accurate control of the absolute resistivity value as function of pressure (form factor) and temperature and a high-pressure run density. The control of the form factor is far to be an easy task in high-pressure cells and main complication comes from non-hydrostatic conditions in various pressure transmitting medium such as He,42,43 Daphne oil,8,25 or steatite.18,44 Another requirement is the limitation of non-systematic error on pressure and the precise estimation of ρ0. Moreover, due to the presumed 1/(T − Tcr) dependence of χ, the uncertainty on Tcr is magnified for large negative values, while the smallest values are the most accurate. Namely, in the case of CeCu2Si2 with Tcr = −3.7 K, the error is within the symbol size.
Despite reliable small Tcr and moderate ρ0, CeCu6 (no SC) and CeCu5Au (partial SC at 0.11 K) clearly lie below the general trend. From a literature review, small Tcr values should also be expected for some compounds including CeAl2,45 CeAl3,46,47 and CeInCu2.48 However, SC has never been observed in these cases. We have no satisfactory explanation yet for this discrepancy.
The dashed red curve for Tcr → 0 in Fig. 5 is drawn from the theoretical prediction of the 3D model23 in which Tc is paradoxically suppressed just at the critical point of the valence transition, while Tc takes sharp maximum near the VCO line in the Kondo regime. This aspect has also been verified by the density-matrix-renormalization-group calculation for the 1D model,24 which is numerically accurate. Namely, inter-site pairing correlation dominates over spin density wave and charge density wave correlations near the sharp VCO inside the Kondo regime.
Discussion
The systematic behavior of Tc versus ρ0 and Tcr points to a possible maximum Tc ≈ 2.5 K in Ce-based HF superconductors and strongly supports that CVF provide the dominant pairing mechanism. Although, the relation between Tc and Tcr was already inferred in the pioneer work of Onishi and Miyake,23 a quantitative prediction seems almost impossible at the present state of art. Theory also considers that Tcr is influenced by disorder in general, which is less evident in the experimental data (see Supplementary Fig. S3). Naively, one can imagine that disorder induces an additional smearing of the VCO, which depresses Tcr and then Tc.
Let us now comment some generalities about magnetism and SC in HF compounds. A hallmark is the merging close to pcr of the two maxima in ρ(T) [see Fig. 3(c)], which indicates that the rapidly rising Kondo energy starts to exceed the first crystal-field-splitting energy.49 For HF superconductors, this pressure corresponds to optimal SC without exception. Crossing the VCO, the ground state degeneracy of the Ce ion increases from n = 2 to full degeneracy n = 6 of the 4f multiplet,7 and at the pressure of maximal Tc, the energy scale TK is much larger than the magnetic ordering temperature. Moreover, the strength of the f-c hybridization seems to control the position of pc in respect to pcr as shown theoretically.10 In the case of strong hybridization pc and pcr are well separated, but in the case of weak hybridization a hypothetical magnetic QCP would occur at pressure higher than pcr and the VCO drives a first-order collapse of magnetism at pc ~ pcr10,50 in parallel to the traditional competition between the RKKY and Kondo energies. Hence the collapse of magnetism is very abrupt or even first-order-like in CeAu2Si2,18,51 CeAg2Si2,52 and CeRhIn5.53
Although pressure is a clean tuning parameter, clear evidence of a second-order magnetic transition down to zero temperature and a resulting quantum critical point is not well established from pressure investigations on pure lattices. In the particular case of CePd2Si2, results of Refs. 3,43,54,55 support a linear decrease of TN down to zero with p approaching pc. However, a more rapid vanishing of TN appears to correspond to higher superconducting Tc19,56 (see Supplementary Fig. S14). The difficulty of tracking the TN vanishing with resistivity or even heat capacity probes and the unavoidable pressure gradient, due to which the TN decrease appears more progressive, impedes a clear-cut conclusion. Seemingly second-order-like magnetic collapses have only been established for alloy systems, see e.g.35,57 and a lattice58 with relatively high ρ0 value, where disorder likely masks the intrinsic behavior.
The overlap of magnetic order and SC in CeRhIn550,53 and especially in CeAu2Si218 (see Fig. 1) contradicts the longstanding consensus that HF SC emerges in the vicinity of the magnetic border.4 From a global point of view, the CeCu2Si2 family shows quite different magnetic phase diagrams concomitantly with otherwise strikingly similar electric and thermoelectric transport and superconducting properties.18,27,51 For instance, a systematic feature in thermopower precedes the occurrence of SC.27 Consequently, the link between SC and magnetism is primarily a competition, with the possible exception of CeAu2Si2.18 Up to a given delocalization of 4f electrons, magnetism hinders CVF to build up SC. On the other hand, the low-pressure SC pocket in CeCu2Si2 is the best candidate for spin fluctuation mediated SC,59 because the magnetic collapse at pc ≈ 0 and the VCO at pcr ≈ 4.2 GPa are exceptionally well separated. Though, the scenario of single-band nodal-d-wave SC at p = 0 in CeCu2Si2 is now strongly challenged.60,61,62,63
Let us comment on the Kondo-volume-collapse mechanism introduced by Razafimandimby et al.64 To our understanding, it is essentially a phonon-mediated SC mechanism due to enhanced electron-phonon coupling through the Kondo-volume-collapse effect (large Gruneisen parameter). In this regard, it should be different from the valence fluctuation mediated mechanism. According to an almost exact (justified by the Ward identity argument) theoretical discussion based on periodic Anderson model with coupling to phonon by Jichu et al.,65 it seems rather difficult for this mechanism to build up “high-Tc“ SC in Ce-based HF. According to Jichu et al., the enhanced pairing interaction (by Kondo-volume-collapse effect) vanishes at the static limit. Furthermore, it is crucial to note that the valence fluctuation mechanism is not based on density fluctuations but fluctuations of f-c charge transfer with the total charge density (nf + nc) essentially kept constant. Namely, valence fluctuations are rather categorized with orbital fluctuations.
Finally, we comment on CVF in Yb-based HF compounds, which can be approached as electron-hole/inverse-pressure analogues of Ce compounds. Interestingly, the first discovered Yb-based superconductor β-YbAlB466 exhibits normal state properties with unconventional quantum criticality,67 which is naturally explained by the CVF theory.68 Furthermore, common criticality has been observed in some classes of Yb-based periodic crystals and even in the quasicrystal Yb15Au51Al34.69 Search for SC induced by CVF in Yb-based systems as well as the identification of the CEP of the underlying Yb-valence transition on the basis of the method described in this paper is expected to open a new frontier in this field.
For a long time, the spin-fluctuation-mediated mechanism was the mainstream scenario for SC in HF systems. However, the CVF theory has provided a new framework able to account for the high-pressure superconducting phase and several other phenomena in CeCu2Si2. We now have shown that this theory is able to explain salient experimental features in a multitude of systems, corroborating CVF as a plausible Cooper pairing mechanism. Concretely, the present study provides striking evidence that the optimum superconducting Tc in many Ce-based HF superconductors is essentially controlled by the strength of CVF and by non-magnetic disorder. Furthermore, we believe that CVF-induced SC is connected to a much wider part of non-trivial physics in strongly correlated electron systems including high-Tc cuprates.21,70,71 Thus, our work uncovers a new playground for condensed matter physicists.
Methods
The above presented results are based on electrical resistivity data obtained on 17 single crystals from nine different Ce-based HF compounds (see Supplementary Table S1). The four-point electrical resistivity measurements under high-pressure conditions have been carried out in standard helium and dilution cryostats. The high-pressure conditions where obtained using Bridgman pressure cells with tungsten-carbide or diamond anvils and with different pressure transmitting medium. Technical details can be found for each transmitting medium in (helium),7,42 (Daphne oil),8,25 and (steatite).18,44 All relevant information about crystals growth, dimension of sample and pressure cell, and data acquisition can be found in the respective references (see Supplementary Table S1). All relevant information about data treatment can be found in Ref. 8 and in the main text.
Data availability
All relevant data are available from the corresponding author upon request.
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Acknowledgements
We acknowledge H. Wilhelm, S. Raymond, and A. T. Holmes for enabling the analysis of previous data, and M. Lopez, S. Müller, and M. Tran for technical support. D.A. acknowledges for the financial support KAKENHI (15H05882, 15H05884, 15K21732, 16H04006). K.M. acknowledges for the financial support KAKENHI (17K05555). S.W. acknowledges for the financial support KAKENHI (15K05177, 16H01077).
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D.J. conceived the idea and designed the experiment. D.A. and G.L. have grown the single crystal samples. G.S. and Z.R. performed the measurements. G.S. analyzed the data. S.W. and K.M. provided the microscopic-theoretical interpretation. G.S. and D.J. wrote the paper. All authors discussed the results and commented on the manuscript.
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Scheerer, G.W., Ren, Z., Watanabe, S. et al. The dominant role of critical valence fluctuations on high Tc superconductivity in heavy fermions. npj Quant Mater 3, 41 (2018). https://doi.org/10.1038/s41535-018-0111-6
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DOI: https://doi.org/10.1038/s41535-018-0111-6
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