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# Emergence of superconductivity in heavy-electron materials

Contributed by David Pines, November 19, 2014 (sent for review November 3, 2014)

## Significance

Although the pairing glue for the unconventional superconductivity found in heavy-electron materials has been identified as quantum critical spin fluctuations associated with their proximity to antiferromagnetic order, until now we have lacked a simple expression for their superconducting transition temperature, *T*_{c}, that explains why *T*_{c} changes with pressure, or varies from one material to another. The experiment-based expression proposed here parameterizes the effective frequency-dependent quasiparticle interactions in terms of their unusual normal-state properties; it provides a quantitative explanation of the measured pressure-induced variation in *T*_{c} in the “hydrogen atoms” of unconventional superconductivity, CeCoIn_{5} and CeRhIn_{5}, predicts a similar pressure variation for other heavy-electron quantum critical superconductors, and quantifies their variations in *T*_{c} with a single parameter.

## Abstract

Although the pairing glue for the attractive quasiparticle interaction responsible for unconventional superconductivity in heavy-electron materials has been identified as the spin fluctuations that arise from their proximity to a magnetic quantum critical point, there has been no model to describe their superconducting transition at temperature *T*_{c} that is comparable to that found by Bardeen, Cooper, and Schrieffer (BCS) for conventional superconductors, where phonons provide the pairing glue. Here we propose such a model: a phenomenological BCS-like expression for *T*_{c} in heavy-electron materials that is based on a simple model for the effective range and strength of the spin-fluctuation-induced quasiparticle interaction and reflects the unusual properties of the heavy-electron normal state from which superconductivity emerges. We show that it provides a quantitative understanding of the pressure-induced variation of *T*_{c} in the “hydrogen atoms” of unconventional superconductivity, CeCoIn_{5} and CeRhIn_{5}, predicts scaling behavior and a dome-like structure for *T*_{c} in all heavy-electron quantum critical superconductors, provides unexpected connections between members of this family, and quantifies their variations in *T*_{c} with a single parameter.

Because the unconventional superconductivity found in many heavy-electron materials arises at the border of antiferromagnetic long-range order, it is natural to consider the possibility that its physical origin is its proximity to a quantum critical point that marks a transition from localized to itinerant behavior, and that the associated magnetic quantum critical spin fluctuations provide the pairing glue (1⇓⇓⇓–5), in contrast to conventional superconductors, where phonons provide the pairing glue (6). However, developing a simple physical picture for the behavior of such quantum critical superconductors, including a Bardeen, Cooper, and Schrieffer (BCS)-like expression for their superconducting transition temperature (*T*_{c}), has proven difficult. In part, this is because of the unusual normal state from which superconductivity emerges (7⇓⇓⇓⇓–12), and in part it stems from the difficulty in finding a simple model for an effective frequency-dependent attractive quasiparticle interaction that closely resembles that proposed earlier for the cuprates (13⇓⇓⇓–17).

In finding a way to characterize heavy-electron quantum critical superconductivity it is helpful to begin by recalling the principal features of its remarkably similar emergence in two of the best-studied materials, CeCoIn_{5} and CeRhIn_{5} (18⇓⇓⇓⇓–23). As may be seen in Fig. 1, there are three distinct regions of emergent heavy-electron superconductivity in their pressure–temperature phase diagrams that are defined by a line marking the delocalization cross-over temperature, *T*_{L}, at which the collective hybridization of the local moments becomes complete and the Néel temperature, *T*_{N}, that marks the onset of long-range antiferromagnetic order of the hybridized local moments.

Region I: *T*_{c} ≤ *T*_{L}. Superconductivity emerges from a fully formed heavy-electron state. The general increase in *T*_{c} seen with decreasing pressure is cut off by a competing state, quasiparticle localization, so *T*_{c} reaches its maximum value at the pressure, *p*_{L}, at which the superconducting and localization transition lines intersect.

Region II: *T*_{c} > *T*_{L} and *T*_{N}. Superconductivity emerges from a partially formed heavy-electron state whose ability to superconduct is reduced by the partially hybridized local moments with which it coexists. The region includes the quantum critical point (QCP) at *T* = 0 that marks a zero temperature transition from a state with partially localized ordered behavior to one that is fully itinerant; this QCP is the origin of the quantum critical spin fluctuations that provide the pairing glue in all three regions (2).

Region III: *T*_{c} ≤ *T*_{N}. Partially hybridized local moments are present in sufficient number to become antiferromagnetically ordered at the Néel temperature *T*_{N} despite the presence of coexisting remnant heavy electrons that become superconducting at lower temperatures.

The dominance of superconductivity around the QCP supports the idea that the coupling of quantum critical spin fluctuations to the heavy-electron quasiparticles plays a central role, with the resulting induced attractive quasiparticle interaction being maximally effective near it. Importantly, there is direct experimental evidence that these quantum critical fluctuations provide the superconducting glue: Curro et al. (4) find that the spin-lattice relaxation rate, 1/*T*_{1}, to which these give rise, scales with *T*_{c} at the pressure at which *T*_{c} is maximum, whereas a recent detailed investigation of that scaling (5) explains how this comes about. First, at this “optimal” pressure, *T*_{c} scales with the coherence temperature, *T**, that marks the initial emergence of heavy-electron behavior and is determined by the nearest-neighbor exchange interaction between the *f*-electron local moments (8); second, at this optimal pressure, 1/*T*_{1} scales with *T**, a scaling behavior that is a unique signature of its origin in quantum critical spin fluctuations.

In this paper we use these important scaling results to develop a simple BCS-like phenomenological expression for the superconducting transition temperature, show that it explains the variation of *T*_{c} with pressure for both CeCoIn_{5} and CeRhIn_{5}, and offer a detailed prediction for a similar dome-like structure in other quantum critical heavy-electron superconductors.

## Phenomenological BCS-Like Model for Heavy-Electron Superconductivity

For phonon-induced superconductivity, BCS found a simple expression for *T*_{c} that depended on three quantities (6): the quasiparticle density of states; the average strength, *V*, of the phonon-induced attractive interaction between quasiparticles; and the average energy range over which it is attractive. Our proposed phenomenological heavy-electron quantum critical magnetic expression involves magnetic analogs of these quantities, all of which can be determined from experiment: *N*_{F}(*p*,*T*_{c}), the heavy-electron density of states at *T*_{c}; an effective attraction, *V*(*p*) *= ηk*_{B}*T**(*p*), where *T**(*p*) is the pressure-dependent interaction between local moments, *k*_{B} is the Boltzmann constant, and *η* is a parameter that measures the relative effectiveness of spin fluctuations in bringing about superconductivity for a given material; and, consistent with the above scaling results, a range of energies over which the quantum critical spin-fluctuation-induced interaction will be attractive that is proportional to *T**_{m}, the coherence temperature at the pressure *p*_{L}, at which *T*_{c} is maximum. It takes the form*κ*(*p*) = *N*_{F}(*p*,*T*_{c})*k*_{B}*T**(*p*) and, as discussed below, used experiment to determine the prefactor 0.14.

It is important to note that because experiment shows that the heavy-electron specific heat varies inversely as *T** and grows logarithmically as the temperature is lowered (10, 12), *C*/*T* ∼1/*T**ln(*T**/*T*), the density of states, *N*_{F}(*p*,*T*_{c}), will exhibit a similar dependence on *T**(*p*). Because experiment shows that *T**(*p*) varies monotonically with increasing pressure (Fig. 2*B*, *Inset*), without a countervailing *T**(*p*) dependence in the strength of spin-fluctuation-induced interaction the dimensionless pairing strength would vary monotonically and Eq. **1** could never lead to the dome structure of *T*_{c} seen experimentally.

Eq. **1** may be rewritten as*T*_{c}/*T**_{m}) against 1/*κ*(*p*) therefore provides a test of our BCS-like expression for *T*_{c}. As discussed in *Methods*, in the absence of systematic specific heat measurements *κ*(*p*) may be determined from experiment by using a two-fluid analysis (7⇓⇓⇓⇓–12) to obtain the heavy-electron density of states, *N*_{F}; the resulting values of *κ*(*p*) for CeCoIn_{5} and CeRhIn_{5} are given in Fig. 2*A*. When used to test the validity of Eq. **2**, we find, as can be seen in Fig. 2*B*, that the two materials fall on the same line, a scaling result that provides strong evidence for the validity of our BCS-like equation, whereas the common intercept tells us that 0.14*T**_{m} is the best choice for the range of the spin-fluctuation-induced attraction for the two compounds.

Our model enables us to predict the maximum effectiveness of the spin-fluctuation-induced interaction for a given material; it is given by*κ*(*p*) is the only pressure-dependent quantity in Eq. **1**, our predicted ratio, *T*_{c}(*p*)/

As may be seen in Fig. 2 *C* and *D*, when we use the values of *κ*(*p*) shown in Fig. 2*A* as input and determine *λ*_{max} to be 1.23 for CeCoIn_{5} and 0.62 for CeRhIn_{5} from experiments at *p*_{L}, Eq. **4** provides a remarkably good quantitative explanation of the dome-like structure observed as the pressure is varied in CeCoIn_{5} and CeRhIn_{5} (18, 19). We note that both *κ*(*p*) and *T*_{c}(*p*) are peaked at *p*_{L}, the pressure at which the delocalization line, *T*_{L}, intersects with *T*_{c}. Our model successfully explains the decrease in *T*_{c} above this pressure as being brought about by the reduction in the heavy-electron density of states produced by the increase in *T*_{L}; below this pressure, the decrease in *T*_{c} arises from the reduction in the heavy-electron density of states brought about by the partial localization of the heavy electrons.

## Discussion

Encouraged by the above results, we next apply our approach to the emergence of superconductivity in other heavy-electron materials (24⇓⇓⇓⇓⇓–30) for which *T** has been measured and Curro scaling has been established or seems likely to apply. Our results are given in Table 1, where the characteristic dimensionless coupling strength, *κ*(*p*_{L}), has been calculated using the two-fluid expression for *N*_{F}, and the effectiveness parameter, *η* = *λ*_{max}/*κ*(*p*_{L}), is obtained at the measured (or assumed) optimal pressure, *p*_{L}. We call attention to a striking similarity in the values of *λ*_{max} shown in Table 1: UPt_{3} seems to be a sister element to CeCoIn_{5} and PuCoIn_{5}, even though their superconducting transition and coherence temperatures differ by a factor of five and their superconducting states possess different symmetries. The large value of *η* found for CeRhIn_{5} suggests that in this material the effective interaction, *V*, could be as large as 3*T**, and the fact that *η* > 1 for many materials suggests that the effective attractive interaction is generally somewhat greater than *T**.

Importantly, because there is only a modest variation in *κ*(*p*_{L}) as one goes from one material to another, most of the measured variation in [*T*_{c}/*T**]_{max} is likely due to variations in the impedance match between the spin-fluctuation spectrum and the heavy-electron Fermi surface that we have parametrized by *η*. These variations can be explained by changes in effective dimensionality and crystal structure. As Monthoux and Lonzarich emphasized in their seminal papers (16, 17), near two dimensionality and a tetragonal crystal structure are most favorable to superconductivity; their presence in CeRhIn_{5} at 2.4 GPa and PuCoGa_{5} could explain the relatively large values of *η* seen for these materials, whereas their absence in CeIn_{3} would explain its low value of *η* and its very low *T*_{c}/*T**.

CeRhIn_{5} at 2.4 GPa and PuCoGa_{5} demonstrate how very effective spin fluctuations can be in bringing about superconductivity; their *T*_{c} is an appreciable fraction of the effective heavy-electron Fermi energy, *k*_{B}*T*_{c}/*E*_{F} = 2*k*_{B}*T*_{c}*N*_{F}(*T*_{c})/3, being 0.016 and 0.013, respectively, fractions large compared with those seen in the cuprates and very large compared with those found for conventional superconductors. Our model for heavy-electron superconductivity leads to the prediction that the maximal value of the ratio *k*_{B}*T*_{c}*/E*_{F} is ∼0.03, about twice the above values.

As a first step toward understanding the microscopic origin of Eq. **1**, we can ask whether it is consistent with the anticipated results of a microscopic strong coupling calculation of quantum critical spin-fluctuation-induced superconductivity for heavy-electron materials that takes full account of an experimentally determined frequency-dependent interaction. We find (*Methods*) in the case of CeCoIn_{5}, where neutron-scattering experiments yield direct information on the quantum critical spin fluctuation spectrum, that the range of the effective attractive interaction found in microscopic strong coupling calculations is remarkably close to what we propose phenomenologically, and complete consistency is obtained provided the coupling of quasiparticles to the spin fluctuations scales with *T**. This is but a first step, but we hope that this consistency will encourage the development of a complete microscopic derivation of our simple phenomenological BCS-like equation for *T*_{c} in which quantum critical spin-fluctuation superconductivity can be characterized by a range, ∼0.14*T**_{m}, and a pressure-dependent strength, *ηT**, both of which can be determined directly from experiment.

Another interesting question for future study is whether our phenomenological approach to quantum critical spin-fluctuation-induced superconductivity in heavy-electron materials can be extended to the cuprates and any other unconventional superconductors in which scaling behavior for the spin-lattice relaxation rate with *T*_{c} has been seen at or near optimal doping levels.

## Methods

### Determining the Characteristic Dimensionless Pairing Strength, *κ*(*p*), from Experiment.

In the Fermi liquid regime (region I in Fig. 1), where the density of states can be derived from the specific heat measurements and the coherence temperature, *T**(*p*), can be estimated from the resistivity, the dimensionless pairing strength, *κ*(*p*), can be directly determined from experiment, so that our proposed BCS-like Eq. **4** involves no free parameters and could be verified without any further assumptions. However, because the relevant experimental information on the pressure dependence of the specific heat is not yet generally available, to test the applicability of Eq. **4** to heavy-electron materials under pressure we have used the two-fluid model to determine the pressure dependence of the density of states. This procedure has earlier been shown to yield correct specific heat results for a number of heavy-electron compounds (7).

### The Delocalization Line, Néel Temperature, and Heavy-Electron Density of States.

In the two-fluid model, the three regions in Fig. 1 are determined by the hybridization parameter (7),

which quantifies the fraction of *f*-electrons that become itinerant. The pressure dependence of the hybridization effectiveness, *f*_{0}(*p*), discussed below, can be determined from magnetic experiments (cf. ref. 11). For *f*_{0} > 1, the cross-over line of complete delocalization temperatures, *T*_{L}, in the heavy-electron phase diagram is obtained by setting *f*(*T*_{L}) = 1, so that

For *f*_{0} < 1, a fraction of residual local moments always remains and becomes antiferromagnetically ordered at low temperatures. The two-fluid model predicts that the Néel temperature, *T*_{N}, is given by:

where the frustration parameter, *η*_{N}, is independent of pressure and found to be 0.14 for CeCoIn_{5} and 0.32 for CeRhIn_{5} in Fig. 2 *C* and *D*.

The heavy-electron density of states in Eq. **1** is obtained in the two-fluid model by assuming that *N*_{F}(*T*) follows the heavy-electron specific heat, according to *C*_{HE}/*T* is determined by requiring the entropy at *T**, **4**. The logarithmic growth is cut off by complete delocalization at *T*_{L} in region I, superconductivity at *T*_{c} in region II, and long-range magnetic order at *T*_{N}, or its precursor at *T*_{0}, the temperature at which heavy electrons begin to relocalize before the emergence of hybridized local moment order at *T*_{N} (7, 31), in region III so that the heavy-electron density of states at the superconducting transition at *T*_{c} is

where *T*_{x}(*p*) = *T*_{L}(*p*) in region I, *T*_{c}(*p*) in region II, and *T*_{0/N}(*p*) in region III (Fig. 2 *C* and *D*). Importantly, we see that because *N*_{F} varies inversely with *T**, the characteristic dimensionless pairing strength, *κ*(*p*) = *k*_{B}*T**(*p*)*N*_{F}(*p*,*T*_{c}), depends comparatively weakly on *T*_{x}/*T** in all three regions:*f*(*p*_{L},*κ*(*p*) at *p*_{L}:

### Deducing Other Key Parameters from Experiment.

The pressure dependence of the coherence temperature, *T**(*p*), may be obtained from resistivity measurements (7, 19), and, in the case of Cd-doped CeCoIn_{5}, from Knight shift experiments (22). The results are shown in Fig. 2*B*, *Inset*.

To determine *f*_{0}(*p*), we first note *f*_{0}(*p*_{QC}) = 1 and use experiment to determine *f*_{0} at ambient pressure; for other pressures, we assume that *f*_{0}(*p*) scales linearly with *T**(*p*) (cf. ref. 11) and obtain

where *f*_{0}(0) is the hybridization parameter at ambient pressure and *T**(0) and *T**_{QC} are the coherence temperatures at ambient pressure and the QCP, respectively.

For CeRhIn_{5}, one has *p*_{QC} ∼2.25 GPa and *T**_{QC} ∼33 K (18); an analysis of its magnetic properties yields *T**(0) ∼17 K and *f*_{0}(0) ∼0.65 at ambient pressure. For CeCoIn_{5}, a scaling analysis of the resistivity (20) suggests *p*_{QC} ∼1.1 GPa and *T**_{QC} ∼82 K, whereas an analysis of the temperature–magnetic field phase diagram yields *f*_{0}(0) ∼0.87 and *T**(0) ∼56 K at ambient pressure, a result that yields an excellent fit to the variation of the QCP with pressure (11). For Cd doping, we assume that 5% Cd doping has similar effect on *f*_{0} as a negative pressure of −0.7 GPa, as is suggested by experiment (21). The effect of Cd doping is, however, different from pressurization because *T** is doping-independent, as is seen in the NMR experiment (22). For both materials, our choice of *f*_{0}(*p*) leads to a unique prediction of *T*_{L}(*p*) that can be verified experimentally.

The cutoff temperatures, *T*_{x}(*p*), for the growth in the heavy-electron state density in region III are determined from the Knight shift and/or Hall measurements (7). For CeCo(In_{1−x}Cd_{x})_{5}, experiment shows that *T*_{x} is roughly given by *T*_{N}; for CeRhIn_{5}, experiment shows that *T*_{x} = *T*_{0}∼2*T*_{N} at ambient pressure and decreases to *T*_{N} at *p*_{N} ∼1.8 GPa (7). In this region, a further experimental test of our choice of parameters is provided by the Néel temperature that can be calculated using Eq. **7**.

On combining and inserting these experimental parameters into Eq. **9** we obtain the dimensionless pairing strength, *κ*(*p*), in Fig. 2*A* and the results for *T*_{N} and *T*_{c} shown in Fig. 2 *C* and *D* that are in remarkably good agreement with experiment.

### Prediction of a Dome-Like Structure for *T*_{c}.

Our prediction of a dome-like structure for *T*_{c} versus pressure for any heavy-electron superconductor is based on the behavior of the solutions of Eq. **4** for the three distinct regions of emergent superconductivity.

Region I: *f*_{0} > 1 and *T*_{c} < *T*_{L}. The growth of *N*_{F}(*T*) is cut off at the delocalization temperature, *T*_{L}, below which *f*(*T*) = 1 and Eq. **4** only depends on *f*_{0},

*T*_{c} is maximum at the pressure at which *T*_{c} = *T*_{L}; it decreases at higher pressures because the density of states decreases, being cut off at higher values of *T*_{L} by the increase in *f*_{0}.

Region II: *f*_{0} ∼1 and *T*_{c} > *T*_{L} and *T*_{N}. Because the growth of *N*_{F}(*T*) extends to *T*_{c}, Eq. **4** takes the form

and has to be solved self-consistently. Most heavy-electron quantum critical superconductors fall in this region, where the logarithmically nearly divergent density of states acts to enhance the effective interaction by a factor, [1 + ln(*T**/*T*_{c})], that can vary between 7.0 and 3.8 as one goes from *T*_{c}/*T** = 0.0025 to 0.062.

Region III: *f*_{0} < 1 and *T*_{c} < *T*_{N}. The growth in *N*_{F} is cut off at *T*_{0} so that

With increasing pressure, *f*_{0} increases and *T*_{N} and *T*_{0} decrease, so that *T*_{c} increases and becomes greater than *T*_{N} before one reaches the quantum critical pressure.

### A Consistency Check with Microscopic Strong Coupling Calculations.

It is reasonable to assume that the pairing interaction for heavy-electron superconductivity is given by an expression identical to that used to explain quantum critical cuprate superconductivity (2),

where *g* is the quasiparticle-spin fluctuation coupling strength and *χ*(**q**,*ω*), the dynamic susceptibility, follows the quantum critical form expected from its proximity to an antiferromagnetic state (13):

with a peak at the ordering wave vector, **Q**, of magnitude *ξ* is the antiferromagnetic correlation length, *a* is the lattice constant, and *χ*_{0} is the uniform spin susceptibility, and a temperature-dependent spin fluctuation energy, *ω*_{SF}. Because the measured ratios of the energy gap to *T*_{c} for heavy-electron materials are typically large compared with the weak coupling result, 1.75, any attempt to seek consistency between our proposed phenomenological expression for *T*_{c} and microscopic calculations should begin with the strong coupling numerical results (14, 15) required to take account of the frequency dependence of the interaction, Eq. **13**. Although these have yet to be carried out for heavy-electron materials, it is to be expected that these will yield a BCS-like expression in the strong coupling limit that is analogous to that found for the cuprates, namely,

where *λ*_{1} and *λ*_{2} are constants of order unity.

The microscopic result, Eq. **15**, will be consistent with our phenomenological expression, Eq. **1**, if, first, the proposed microscopic prefactor, *λ*_{1}*ω*_{SF}(*ξ*/*a*)^{2}, is identical to 0.14*T**_{m}, the effective range over which we have proposed that the quantum critical spin fluctuation induced interaction will be attractive, and second, if the coupling, *g*, of the heavy-electron quasiparticles to the spin fluctuations scales with *T**, the nearest-neighbor local moment interaction (8). This last connection is plausible because through collective hybridization the heavy-electron quasiparticles are born coupled by an interaction similar to that of the local moments from which they emerge. Importantly, experimental information on the microscopic prefactor is available for CeCoIn_{5}, where neutron scattering measurements of the spin fluctuation spectrum near *T*_{c} at ambient pressure (23) yield *ω*_{SF} = 0.3 ± 0.15 meV and *ξ* = 9.6 ± 1.0 Å (about twice the in-plane lattice constant *a* = 4.60 Å). One then has *ω*_{SF}(*ξ*/*a*)^{2} = 1.3 meV ∼15.1 K, in remarkably close agreement with our phenomenological result, 0.14*T**_{m} = 12.9 K. The two expressions agree if we take *λ*_{1} = 0.85 in Eq. **15** and assume that neutron scattering at the quantum critical pressure will yield results for this product that are similar to those found at ambient pressure. Future calculations and experiments on other materials can test our prediction that the microscopic prefactor will always be ∼0.14*T**_{m}.

## Acknowledgments

We thank G. Lonzarich for his critical reading and helpful comments on an earlier draft of this manuscript, Z. Fisk for his critical reading and helpful remarks about the framing of this manuscript, and the Aspen Center for Physics (National Science Foundation Grant PHYS-1066293) and the Santa Fe Institute for their hospitality during its writing. Y.-f.Y. thanks the Simons Foundation for its support and this work is supported by National Natural Science Foundation of China Grant11174339 and Strategic Priority Research Program (B) of the Chinese Academy of Sciences Grant XDB07020200.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: yifeng{at}iphy.ac.cn or david.pines{at}gmail.com.

Author contributions: Y.-f.Y. and D.P. designed research; Y.-f.Y. and D.P. performed research; Y.-f.Y. analyzed data; and Y.-f.Y. and D.P. wrote the paper.

The authors declare no conflict of interest.

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