The dominant role of critical valence fluctuations on high T_c superconductivity in heavy fermions

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 T_cr. 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 T_c and the valence instability parameter T_cr, 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 T_c of a given sample is mainly controlled by the compound’s T_cr 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.

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 p_c, 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 T_c, 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 p_cr and at slightly negative temperature T_cr (see Fig. 1). With a negative T_cr, 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 CeIrIn₅^15,16 via Cu- and In-nuclear quadrupole resonance measurements, respectively.

Fig. 1

Examples of schematic p-T magnetic and superconducting phase diagrams of Ce-based HF superconductors. Symbols stand for representative data points from CeRhIn₅ ¹⁷, CeCu₂Si₂,⁸ CeAu₂Si₂,¹⁸ and CePd₂Si₂:¹⁹ Néel temperature T_N and bulk-superconducting T_c. The graded-colored area represents the valence crossover

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 p_cr. On the other hand, the magnetic p_c can coincide with p_cr as in the case of CeRhIn₅,¹⁷ CePd₂Si₂ (this work), and CeAu₂Si₂,¹⁸ or be much lower than p_cr 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-T_c 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-T_c” 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 U_fc is the Coulomb repulsion between f and conduction electrons.^23,24 The physical origin of the T_c 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 T_c. 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 T_cr 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 T_c 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 T_cr and ρ₀ controlling T_c 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 T_c as function of both the residual resistivity ρ₀ and the valence instability parameter T_cr based on published and new results (see Table S1 of the Supplementary Material for details and references). In this paper “T_c” 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 T_cr, which suggests that the superconducting T_c of a given sample is mainly controlled by the compound’s T_cr and the sample’s ρ₀. T_c seems to culminate at ~2.5 K when T_cr → 0 and ρ₀ → 0, i.e., for a quantum CEP and negligible pair breaking effect. However, high ρ₀ values or large negative T_cr depress T_c.

Fig. 2

Maximal superconducting T_c of Ce-based HF superconductors as a function of the key parameter T_cr and the residual resistivity ρ₀. See text for details. Bold numbers indicate: CeAg₂Si₂ = 1, CeCu₂Ge₂ = 2, CeCu₅Au = 3, and other samples of CeCu₂Si₂ = 4 and CeAu₂Si₂ = 5. The error bars for T_cr represent estimated errors according the scaling analysis (see main text) and the error bars for ρ₀ result from the power-law extrapolation to zero temperature of ρ(T). All compounds lie on or not far from the empirically drawn surface. Blue (violet) data points lie above (below) the surface. The surface is drawn for T_cr ≤−2 K for a reason discussed below and for T_c ≥ 0.28 K, since no reliable information exists for very negative T_cr or very high ρ₀

We underline that all samples with T_c > 2 K are found to exhibit −15 K < T_cr < 0 K and emphasize the striking relationship between the superconducting T_c, the parameter T_cr of the valence transition CEP, and pair breaking due to non-magnetic impurities (ρ₀). We introduce the expression “high-T_c“ to refer to the fact that the compounds with the highest T_c amongst the Ce-based HF superconductors are especially well represented in Fig. 2. Five out of the nine studied compounds have T_c 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 T_c(T_cr) and T_c(ρ₀), let us discuss the behavior of electrical resistivity ρ in the VCO regime and summarize the method for extracting T_cr 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 U_fc-repulsion between f- and conduction electrons at the same Ce site. The CEP lies at p_cr ≈ 1.5 GPa and T_cr ≈ 480 K.³² As a function of pressure, isothermal resistivity of Ce [Fig. 3(b)] exhibits a discontinuous anomaly at the FOVT (T < T_cr).³³ In the crossover regime (T > T_cr), isothermal resistivity decreases rapidly but continuously and the resistivity gradient diverges just at the CEP (T → T_cr). In CeCu₂Si₂, the CEP lies at slightly negative temperature T_cr ≈ −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

Fig. 3

Extracting the negative temperature T_cr of the valence transition critical end point (CEP) from low-T resistivity. a Schematic p-T phase diagrams of elementary Ce³¹ and CeCu₂Si₂.⁸ b Isothermal resistivity ρ versus pressure p of Ce in vicinity of the valence transition CEP³² and normalized resistivity ρ^norm vs p of CeCu₂Si₂ in the VCO regime. c Resistivity ρ − ρ^phonon versus temperature T of CeAg₂Si₂ at selected pressures. Inset: Resistivity isotherms ρ^* = ρ − ρ₀ of CeAg₂Si₂ versus pressure at temperatures from 3 to 15 K. d ρ^norm versus p − p_in of CeCu₂Si₂, CeRhIn₅, CeAg₂Si₂, and CePd₂Si₂ at 3 K. The lines are guides to the eyes. e Schematic diagram of bulk SC for the same compounds (same p-scale as in d). f Slope $\chi = |{\mathrm{d}}\rho ^{{\mathrm{n}}orm}/{\mathrm{d}}p|_{p_{{\mathrm{V}}CO}}$ versus temperature T of CeCu₂Si₂ and CeAg₂Si₂. The red lines represent fits to the data with χ ∝ (T − T_cr)⁻¹. Error bars on χ, shown for representative data points, correspond to the overestimation and underestimation of χ due to a low p-run density. g Normalized resistivity ρ^norm versus the generalized distance h/θ from the CEP of CeCu₂Si₂, CeAu₂Si₂, CeAg₂Si₂, and CePd₂Si₂. c–g See Supplementary Table S1 for references

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 p_in, 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 p_VCO is the pressure of the mid drop of ρ^*, is calculated for each temperature.

By way of example, Fig. 3(d) shows ρ^norm versus p − p_in at 3 K of CeCu₂Si₂, CeRhIn₅, CeAg₂Si₂, and CePd₂Si₂. Clearly, it appears that the collapse of ρ^norm is always close to optimal SC and a steeper collapse favors higher T_c [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 − T_cr)⁻¹, i.e., a first-order discontinuity would occur in ρ^*(p) for T < T_cr. Evidently, a simple fit to χ(T) yields T_cr. The empirical law (T − T_cr)⁻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 T_c and least negative T_cr. 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 T_cr values and other parameters (T_c, ρ₀) are summarized in Table S1 of the Supplementary Material.

After identifying T_cr, 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 CeAu₂Si₂^18,26,27 and CeRhIn₅.¹⁷ To this end, a generalized distance h/θ from the CEP is calculated, where h = (p − p_VCO)/p_VCO and θ = (T − T_cr)/|T_cr|. Then, for a given compound, all ρ^norm isotherms in the VCO regime collapse on a single curve ρ^norm = 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 ρ^norm isotherms behave in a unique manner, which strongly supports the existence of the valence CEP at (p_cr, T_cr). Note that in terms of universal scaling theory of critical phenomena the equation is ρ^norm/h^1/δ = f(h/θ^{γδ/(δ−1)}), with the critical exponents γ and δ (mean-field approach: γ_MF = 1, δ_MF = 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 ρ^norm 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 T_c 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 T_c 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(Si_1−xGe_x)₂ alloy, a maximum bulk T_c ~ 0.6 K (ρ = 0 criteria) is reported for ρ₀ ~ 70 μΩcm,³⁵ and, in polycrystalline CeCu₂Si₂, a maximum bulk T_c ~ 0.3 K is given with ρ₀ ~ 180 μΩcm, which is higher than the room temperature resistivity.³⁶ The evident decrease of T_c 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 T_c on ρ₀ for different samples of different compounds is remarkable.

Fig. 4

Pair-breaking systematics: T_c vs ρ₀ (ρ₀ taken at the pressure of maximal T_c) of the CeCu₂Si₂ family and CeRhIn₅ (published and unpublished data, see Supplementary Fig. S1 for references). The error bars on ρ₀ result from the power-law extrapolation to zero temperature of ρ(T). The solid line represents the generalized Abrikosov-Gor’kov theory with a critical resistivity $\rho _0^{{\mathrm{cr}}} = 46\,\mu {\mathrm{\Omega cm}}$³⁸

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). T_c is already small at ideal sample quality since these compounds are located far from the criticality as signaled by the large negative T_cr ~ 50 K. Therefore, the theory of Okada et al.³⁸ for robustness of T_c 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 T_c in CePd₂Si₂.

The critical resistivity $\rho _0^{{\mathrm{cr}}} \approx 46\,\mu {\mathrm{\Omega cm}}$ of the “high-T_c” 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 p_cr compared to far lower or higher pressures, which is a hallmark of CVF-mediated HF superconductors (exceptions are CeCu₆³⁴ and CePd₂Si₂¹⁹). The robustness of T_c 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 T_c 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 T_c and the key parameter T_cr. T_c is maximal for small negative T_cr and decreases as T_cr becomes more negative, which is qualitatively predicted by CVF theory.^23,24 About half of the data correspond to samples with almost optimal T_c for the specific compounds thanks to low ρ₀ values (see Supplementary Table S1), which underlines the intrinsic character of the T_c-vs-T_cr 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 T_cr, 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 − T_cr) dependence of χ, the uncertainty on T_cr is magnified for large negative values, while the smallest values are the most accurate. Namely, in the case of CeCu₂Si₂ with T_cr = −3.7 K, the error is within the symbol size.

Fig. 5

Building superconductivity up in heavy fermion compounds. The presented T_c-vs-T_cr relation is qualitatively predicted by CVF theory.^23,24 The continuous line is a guide to the eyes. The dashed line represents a qualitative prediction from CVF theory.²³ The error bars on T_cr represent estimated errors according the scaling analysis (see main text). The arrow indicates T_cr of CeCu₆ (no SC)

Despite reliable small T_cr 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 T_cr 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 T_cr → 0 in Fig. 5 is drawn from the theoretical prediction of the 3D model²³ in which T_c is paradoxically suppressed just at the critical point of the valence transition, while T_c 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 T_c versus ρ₀ and T_cr points to a possible maximum T_c ≈ 2.5 K in Ce-based HF superconductors and strongly supports that CVF provide the dominant pairing mechanism. Although, the relation between T_c and T_cr 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 T_cr 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 T_cr and then T_c.

Let us now comment some generalities about magnetism and SC in HF compounds. A hallmark is the merging close to p_cr 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 T_c, the energy scale T_K is much larger than the magnetic ordering temperature. Moreover, the strength of the f-c hybridization seems to control the position of p_c in respect to p_cr as shown theoretically.¹⁰ In the case of strong hybridization p_c and p_cr are well separated, but in the case of weak hybridization a hypothetical magnetic QCP would occur at pressure higher than p_cr and the VCO drives a first-order collapse of magnetism at p_c ~ p_cr^10,50 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 T_N down to zero with p approaching p_c. However, a more rapid vanishing of T_N appears to correspond to higher superconducting T_c^19,56 (see Supplementary Fig. S14). The difficulty of tracking the T_N vanishing with resistivity or even heat capacity probes and the unavoidable pressure gradient, due to which the T_N 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 CeRhIn₅^50,53 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 p_c ≈ 0 and the VCO at p_cr ≈ 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-T_c“ 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 (n_f + n_c) 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 T_c 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-T_c 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.