Incoherent non-Fermi-liquid scattering in a Kondo lattice

Abstract

One of the most notorious non-Fermi-liquid properties of both archetypal heavy-fermion systems^1,2,3,4 and the high-T_c copper oxide superconductors⁵ is an electrical resistivity that evolves linearly (rather than quadratically) with temperature, T. In the heavy-fermion superconductor CeCoIn₅ (ref. 6), this linear behaviour was one of the first indications of the presence of a zero-temperature instability, or quantum critical point. Here, we report the observation of a unique control parameter of T-linear scattering in CeCoIn₅, found through systematic chemical substitutions of both magnetic and non-magnetic rare-earth, R, ions into the Ce sublattice. We find that the evolution of inelastic scattering in Ce_1−xR_xCoIn₅ is strongly dependent on the f-electron configuration of the R ion, whereas two other key properties—Cooper-pair breaking and Kondo-lattice coherence—are not. Thus, T-linear resistivity in CeCoIn₅ is intimately related to the nature of incoherent scattering centres in the Kondo lattice, which provides insight into the anomalous scattering rate synonymous with quantum criticality⁷.

Main

Although recent theories^4,8,9,10 provide possible routes to an explanation of T-linear resistivity—found in both f-electron systems (for example, Y_1−xU_xPd₃ (ref. 1), CeCu_6−xAu_x (ref. 2), YbRh₂Si₂ (ref. 3) and CeCu₂Si₂ (ref. 4)) and the normal state of the cuprate superconductors⁵—a general interpretation awaits arrival⁷. Several paradoxical features regarding this anomalous scattering rate continue to defy understanding, such as its persistence over decades of energy scales^1,3,5 and down to millikelvin temperatures in three-dimensional materials^1,2,3,4,6, its coexistence with conventional (T²) Hall-angle scattering^11,12 and its inconsistency with one-parameter scaling¹³. Most recently, its observation over three decades of T at the field-tuned quantum critical point (QCP) of CeCoIn₅ has been linked to a violation of the Wiedemann–Franz law¹⁴, an indication that this scattering rate is associated with the failure of Fermi-liquid theory in its most basic form.

Here, we present a rigorous study of the effects of rare-earth substitution on three closely related features of the exotic metal CeCoIn₅: unconventional superconductivity, Kondo-lattice coherence and anomalous charge-carrier scattering. By diluting the Ce lattice within high-quality single-crystal specimens of Ce_1−xR_xCoIn₅ with both non-magnetic (full or empty 4f-shell) and stable-4f-moment substituent ions of varying size and electronic configuration, we are able to inject both ‘Kondo holes’ (isoelectronic ions without magnetic moments) and strongly localized magnetic moments into the coherent Kondo lattice. This has allowed us to probe the spin exchange between the Ce³⁺ localized magnetic moments and the spins of the conduction electrons involved in Cooper pairing, Kondo screening and anomalous transport in a controlled way, revealing a surprising contrast between the response of coherent phenomena and non-Fermi-liquid behaviour to this perturbation.

Figure 1 shows the evolution of both the superconducting transition temperature T_c (identified by the transition in resistivity, ρ) and Kondo-lattice coherence temperature T_coh (identified by the maximum in ρ(T)) for all rare-earth substitutions made in Ce_1−xR_xCoIn₅ through the complete range of concentrations where both features exist. As shown, the salient features are the same for all variants: as a function of residual resistivity (ρ₀∼x—see the Methods section), both T_c and T_coh are suppressed to zero temperature at rates irrespective of the nature of the rare-earth ion, which spans both magnetic (Pr³⁺, Gd³⁺, Dy³⁺, Er³⁺) and non-magnetic (Y³⁺, Yb²⁺, Lu³⁺) f-electron configurations. This highlights the insensitivity of two ‘coherent’ electronic properties of CeCoIn₅, heavy-fermion superconductivity and Kondo-lattice screening, to the magnetic configuration of the substituted rare-earth ions, the implications of each we will consider in turn.

Figure 1: Dependence of superconducting transition temperature, T_c, and Kondo-lattice coherence temperature, T_coh of Ce_1−xR_xCoIn₅ on rare-earth concentration.

Plotted as a function of residual resistivity (ρ₀∼x—see the Methods section), this figure highlights the absence of any effect of the electronic configuration of replacement-ion R on either T_c (filled symbols) or T_coh (open symbols) as they evolve from x=0 (grey triangles). Note the lack of contrast between two particular species that are similar in all respects except f-electron filling: both Y³⁺ (5s²4d¹) and Gd³⁺ (6s²4f⁷5d¹) are isovalent with Ce³⁺ (6s²4f¹5d¹) and have nearly identical metallic radii of 1.801 Å, slightly smaller than that of Ce (1.825 Å) and yielding a similarly small (∼1%) change of the lattice parameters on substitution. The absence of an f-electron shell in Y³⁺ leaves it non-magnetic, whereas the half-filled f-shell of Gd³⁺ has the simplest configuration of the rare earths: a spherically symmetric f-shell with no orbital component (J=S=7/2, L=0) produces a large effective moment μ_eff²=g²J(J+1)=(7.9 μ_B)² with minimal effects from crystalline electric field anisotropy and spin–orbit coupling. The trends in T_c and T_coh are also consistent with those found for the Ce_1−xLa_xCoIn₅ series¹⁵.

The pair-breaking effect in unconventional superconductors arises via both potential (non-magnetic) and spin-flip scattering mechanisms. Potential scattering was shown via La substitution in CeCoIn₅ to follow the Abrikosov–Gor’kov (AG) model for an anisotropic order parameter¹⁵, where it is well known that superconductivity is destroyed once the mean free path, l_mfp, approaches the superconducting coherence length, ξ. Here, we estimate this critical scattering length to be l_cr≃180 Å at the point where T_c→0 (that is, at ρ_cr≃20 μΩ cm, Fig. 1), assuming that the proportionality between l_mfp(x=0)≃1,200 Å (ref. 16) and ρ(x=0) near T_c is independent of doping. This value is roughly twice the in-plane coherence length ξ_a=80 Å (ref. 6) and consistent with previous work¹⁵. Interestingly, the value ρ_cr≃20 μΩ cm coincides with that found in the series CeCoIn_5−xSn_x (ref. 17), where Sn substitution for In preferentially occurs in the Ce–In layers¹⁸. In the absence of any dependence on replacement-ion size, as demonstrated by the contrast in metallic radii of Lu (1.735 Å) and Y (1.801 Å), pair-breaking in CeCoIn₅ thus seems to be dominated by general disorder in the CeIn₃ planes.

The spin-flip interaction imposed on Cooper pairs by magnetic impurities is characterized by a further pair-breaking term , which includes the exchange interaction parameter and the de Gennes factor D_J=(g−1)²J(J+1), with the latter reflecting the classic competition between superconductivity and magnetism¹⁹. The absence of a dependence of ΔT_c on this term in Ce_1−xR_xCoIn₅ is intriguing, but not unprecedented. In UPt₃, the insensitivity of ΔT_c to D_J is attributable to an odd-parity pairing state, where an equal Zeeman shift on parallel spin states renders the spin-flip process ineffective²⁰. In the spin-singlet cuprates, T_c is insensitive to the flavour of the rare-earth ion, R, placed in RBa₂Cu₃O_6−δ (ref. 21) owing to the large physical separation between the R ions and the CuO₂ layers, and hence owing to negligible magnetic interaction. In CeCoIn₅, evidence for even-parity pairing²² also suggests a small value of , given the drastic range of D_J values (from 0.80 for R=Pr to 15.75 for R=Gd, largest in the rare-earth series). However, in contrast to the case of the cuprates, the placement of R ions directly into the active pairing layer¹⁸ of CeCoIn₅ provides the first example of T_c suppression in a spin-singlet superconductor that is truly independent of D_J. Assuming the AG model applies, this places stringent bounds on both the strength of the exchange interaction involved in pair-breaking and the nature of the pairing mechanism itself.

Interestingly, this insensitivity to D_J is mimicked in the suppression of T_coh with rare-earth substitution, as shown in Fig. 1. The temperature T_coh is a characteristic property of the Kondo lattice; associated with the single-ion Kondo temperature, T_K (ref. 23), and hybridization gap²⁴, it signifies the onset of Kondo singlet formation and marks the scale where single-site magnetic scatterers begin to dissolve into a coherent state. Interestingly, in the same way that superconductivity is destroyed when l_mfp→l_cr≈ξ, T_coh also disappears when l_mfp approaches a characteristic coherence length ξ_coh≡ℏv_F/k_BT_coh≃100 Å (using T_coh=50 K and v_F≃6.5×10⁻⁴ m s⁻¹, where k_B and v_F are Boltzmann’s constant and the Fermi velocity, respectively)⁹, again with no dependence on the magnetism of the dopant ion R. Furthermore, note that T_coh→0 near the ∼40% percolation limit for a two-dimensional lattice. Together these support the notion that, regardless of its internal structure, the Ce lattice vacancy, or ‘Kondo hole’, is the dominant contributor to coherence destruction, leading to a universal dilution of the Kondo lattice as expected by the periodic Anderson model²⁵. Thus, both the superconducting electron pair-breaking effect and the suppression of coherent Kondo screening proceed in a manner that is insensitive to the magnetic configuration of the dopant atom, advancing a scenario where spin-independent disorder is the dominant perturbation in both phenomena.

In contrast, the evolution of the non-Fermi-liquid electronic transport in Ce_1−xR_xCoIn₅ shows a striking sensitivity to the dopant atom’s f-moment configuration, with T-linear resistivity persisting only in the presence of strong local-moment exchange. This is introduced in Fig. 2 through a direct comparison of the evolution of ρ(T) as a function of both non-magnetic (Y³⁺) and magnetic (Gd³⁺) Ce-site substitution in Ce_1−xR_xCoIn₅: an increasing Y concentration introduces strong downward curvature in ρ(T) below T_coh (Fig. 2a), whereas T-linear scattering seems to be robust against magnetic Gd substitution (Fig. 2b). We further explore this duality by presenting resistivity data for several characteristic rare-earth substitutions in Fig. 3, fitting ρ(T) for each between T_c and ∼20 K with a simple power law (ρ=ρ₀+A Tⁿ) and plotting Δρ=ρ−ρ₀ versus T to emphasize the exponent n, which appears as the slope on a log–log scale. As shown explicitly in the inset of Fig. 3, n spans a range of sublinear values, with deviations from T-linear being strongest for non-magnetic substitutions.

Figure 2: Comparison of electrical resistivity evolution of Ce_1−xR_xCoIn₅ with both magnetic and non-magnetic rare-earth substitution.

a,b, Resistivity, ρ, plotted for Ce_1−xY_xCoIn₅ (a) and Ce_1−xGd_xCoIn₅ (b) as a function of nominal concentration of rare-earth substitution. Although both the superconducting transition and Kondo coherence temperature (maximum in ρ(T)) are suppressed at the same rate for both substitution series, the temperature dependence of ρ is strongly dependent on the magnetic nature of the substituent ion: Y-doping imposes a strong downward curvature on ρ(T) with increasing concentration, whereas Gd-doping elicits a negligible change in the T-linear resistivity present in pure CeCoIn₅.

Figure 3: Effect of chemical substitution on T-linear resistivity power laws in CeCoIn₅.

The filled symbols represent various rare-earth substitutions in Ce_1−xR_xCoIn₅; the open circles (shifted by ×2 for clarity) represent a single-crystal sample of CeRh_0.15Co_0.85In₅ in its field-induced normal state at 9 T, showing the close connection between sublinear curvature in ρ(T) and the proximity of a spin-density wave instability. The dashed lines are guides showing slopes for various powers of temperature. Inset: Evolution of the temperature power-law exponent n (that is, in Δρ∼Tⁿ) with rare-earth substitution (plotted as residual resistivity ρ₀—see the Methods section), highlighting the isolated behaviour of Gd substitution. Whereas the large effective moment (μ_eff=7.9 μ_B) of Gd³⁺ ions in Ce_1−xR_xCoIn₅ sets it apart from its non-magnetic counterparts, the sublinear exponent observed for Er³⁺ substitution (red square)—with μ_eff=9.6 μ_B—rules out a simple correlation between the moment size and the sublinear power-law exponent, suggesting the importance of the spin configuration of the rare-earth ions and de Gennes factor scaling. The error bars reflect estimates of uncertainty in n on the basis of the temperature range and number of data points used in nonlinear least-squares fits of ρ(T). (The open symbols denote concentrations used in the main figure.)

A sub-T-linear transport scattering rate is highly anomalous, yet not unprecedented. For instance, the resistivity of the strongly correlated f-electron system Sc_1−xU_xPd₃ was indeed observed to follow the form ρ(T)=ρ₀−A Tⁿ with an exponent n≃0.5 (ref. 26), consistent with the n=1/2 expectation of the theoretical multichannel Kondo model for T≪T_K (ref. 9). However, the n<1 curvature in Sc_1−xU_xPd₃ is more likely due to quantum criticality associated with the suppression of spin-glass freezing to T=0 near x_c≃0.3, rather than the multichannel Kondo effect²⁶.

Likewise, the phenomenological trend of n<1 curvature in Ce_1−xR_xCoIn₅ also hints at the proximity of a magnetic instability not unlike that found in CeRhIn₅, where similar sublinear curvature is present in ρ(T) above the antiferromagnetic transition at T_N=3.8 K (ref. 27). In CeRhIn₅, this curvature is proportional to the magnetic entropy, a reflection of the fact that magnetic correlations dominate the transport scattering process²⁷. In CeCoIn₅ the same phenomenon was found to be dependent on the proximity to a field-tuned QCP²⁸. A connection between the two was established via resistivity measurements of the alloy series CeRh_1−yCo_yIn₅, where a crossover to sublinear behaviour in ρ(T) was shown to be intimately related to the antiferromagnetic QCP²⁹. As shown in Fig. 3, ρ(T) of a single-crystal sample of CeRh_1−yCo_yIn₅ with y=0.85 (close to the alloy-tuned QCP) indeed follows an n≃0.5 exponent over almost two decades in T in its field-induced normal state, indicating a strong connection between n<1 scattering and the proximity of a QCP related to the spin-density wave instability in CeRhIn₅.

In stark contrast, Gd substitution in Ce_1−xR_xCoIn₅ fails to disrupt the mechanism of T-linear scattering: as shown in the inset of Fig. 3, the exponent n experiences an almost negligible change, decreasing at a rate at least five times slower than for non-magnetic substitutions. Because the zero-field magnetic entropy in CeCoIn₅ also grows linearly with temperature above T_c (ref. 6), it is suspected that, like CeRhIn₅, magnetic correlations are what shape this anomalous scattering rate. In Ce_1−xGd_xCoIn₅, this must involve a Ruderman–Kittel–Kasuya–Yosida (RKKY)-type exchange, as demonstrated by both a linear increase with x of the effective moment (up to μ_eff=7.0 μ_B at x=1) and long-range antiferromagnetic order (T_N≃32 K at x=1), which is in line with the proportionality between T_N and D_J found in other magnetic RCoIn₅ compounds³⁰.

But what is the underlying property of Gd³⁺ magnetism that is amenable to T-linear scattering? As shown in Fig. 3, the curvature in ρ(T) of a sample doped with 25% Er³⁺—with an even larger moment (μ_eff=9.6 μ_B) than Gd³⁺—surprisingly exhibits a sublinear power law (n≃0.6) much closer to that of the non-magnetic samples. Furthermore, samples doped with Dy³⁺ (μ_eff=10.6 μ_B) exhibit intermediate behaviour, suggesting that the important parameter is not simply moment size itself, but rather involves details of the f-moment configuration. In particular, the wide range spanned by the de Gennes factors of Gd³⁺, Dy³⁺ and Er³⁺ (with D_J values of 15.75, 7.08 and 2.55, respectively) is the only aspect of the magnetic configuration that follows the evolution of n(x) suggested by our data set, with a phenomenological form n≈1+α(D_J−D₀)ρ₀ where D₀≃18 and α is a positive constant. Despite the peculiar position of D_J in the exponent (rather than as a coefficient) of T, its presence highlights the important role of the spin degrees of freedom in the scattering process that gives rise to T-linear resistivity, promoting the notion that the ‘control parameter’ may indeed be the projected spin of the scattering centres.

What remains highly anomalous, and more generic, is that the relatively strong relation between n and D_J must comply with the extremely weak exchange coupling between localized 4f-states and conduction-band states, as demonstrated by the insensitivity of both ΔT_c and ΔT_coh to the magnetic configuration of R. This contrast provides evidence for a separation between the physics of the Kondo lattice and that of the non-Fermi-liquid transport in CeCoIn₅, with the latter necessarily arising from ‘incoherent’ scattering processes. But how can this interaction coexist with the seemingly different long-range interactions that mediate superconductivity and resonant Kondo-lattice screening? One possibility is that the hybridization between f-states and conduction-electron states is incomplete, leaving a fraction of incoherent scatterers that conspire to cause such a dichotomy. Evidence for such two-fluid behaviour does indeed take form in CeCoIn₅, where an ‘incoherent’ fraction of Kondo moments was shown to survive down to T_c (ref. 31). Another scenario is of a more profound nature: recent evidence for (1) a group of conduction electrons that remains unpaired in the T→0 limit³² and (2) a direction-dependent violation of the Wiedemann–Franz law¹⁴ points to a decoupled character of conduction electrons in CeCoIn₅, suggesting that the separation between the mechanisms behind the coherent properties of CeCoIn₅ and its T-linear resistivity is of a very fundamental nature.

Methods

Single-crystal platelets of Ce_1−xR_xCoIn₅ (including R=Y, Pr, Gd, Dy, Er, Yb and Lu) were grown by the self-flux method⁶. Samples for measurements of electrical resistivity were prepared with typical dimensions ∼2×0.5×0.2 mm and measured with an a.c. resistance bridge by applying ∼0.1 mA excitation current, directed parallel to the basal plane of the tetragonal crystal structure. The data in Figs 1 and 2 are plotted as a function of residual resistivity to eliminate the uncertainty in nominal concentration values. However, note that ρ₀∼x to within error as found previously^15,32. The d.c. magnetization was measured using a superconducting quantum interference device magnetometer in a 50 mT field, and analysed using standard Curie–Weiss fits to data between approximately 25 and 300 K to extract effective moments for the magnetic Ce_1−xR_xCoIn₅ series.