# Electronic inhomogeneity in a Kondo lattice

^{a}Los Alamos National Laboratory, Los Alamos, NM 87545;^{b}Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China;^{c}Department of Physics and Astronomy, University of California, Irvine, CA 92697;^{d}National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32306;^{e}Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan; and^{f}Department de Physique, Universite de Montreal, Montreal, QC, Canada H3C 3J7

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Contributed by Zachary Fisk, March 15, 2011 (sent for review December 29, 2010)

## Abstract

Inhomogeneous electronic states resulting from entangled spin, charge, and lattice degrees of freedom are hallmarks of strongly correlated electron materials; such behavior has been observed in many classes of *d*-electron materials, including the high-*T*_{c} copper-oxide superconductors, manganites, and most recently the iron–pnictide superconductors. The complexity generated by competing phases in these materials constitutes a considerable theoretical challenge—one that still defies a complete description. Here, we report a manifestation of electronic inhomogeneity in a strongly correlated *f*-electron system, using CeCoIn_{5} as an example. A thermodynamic analysis of its superconductivity, combined with nuclear quadrupole resonance measurements, shows that nonmagnetic impurities (Y, La, Yb, Th, Hg, and Sn) locally suppress unconventional superconductivity, generating an inhomogeneous electronic “Swiss cheese” due to disrupted periodicity of the Kondo lattice. Our analysis may be generalized to include related systems, suggesting that electronic inhomogeneity should be considered broadly in Kondo lattice materials.

Electronic inhomogeneity is commonplace in materials in which strong correlations among electrons produce electronic states that compete with one another on multiple length scales (1). One early indication of such heterogeneity came from studies of the high-*T*_{c} cuprate superconductors in which nonmagnetic Zn impurities were introduced into the CuO_{2} planes of YBa_{2}Cu_{3}O_{6+x} (YBCO) and La_{2-x}Sr_{x}CuO_{4} (LSCO)^{2}; the anomalous suppression of the superfluid density of the superconducting condensate was explained within a Swiss cheese model comprised of normal regions around the impurity that healed over a (short) coherence length of order 20 Å within a superconducting matrix (2), later verified by scanning tunneling spectroscopy (3). Not only is superconductivity locally suppressed in the Swiss cheese regions, but new electronic states emerge, such as impurity resonances and other exotic forms of electronic inhomogeneity (e.g., “stripe” and “checkerboard” phases) observed in cuprates and also in other *d*-electron materials (e.g., manganites) (1, 4). In contrast, electronic inhomogeneity has rarely been considered in the prototypical correlated system: *f*-electron materials (5) in which itinerant heavy quasiparticles emerge at low temperature due to a periodic lattice of Kondo ions. In this work, we investigate the underlying electronic structure of the Kondo lattice compound CeCoIn_{5} whose heavy quasiparticles pair to create a *d*-wave superconducting state below 2.3 K (6). As will be discussed, the superconductivity itself serves as a mirror that reflects the presence of electronic inhomogeneity. A thermodynamic analysis of high-purity single crystals of CeCoIn_{5}, doped with different impurities (Y^{3+}, La^{3+}, Yb^{2+}, Th^{4+}, Hg, and Sn), reveals that lattice sites filled by these impurities create “Kondo holes” (7, 8) that produce a nonsuperconducting component within the superconducting state, very much like the Swiss cheese model of the cuprates (2). Our results not only provide strong evidence for an inhomogeneous electronic ground state in this *f*-electron heavy fermion superconductor, they uncover fundamental properties of the Kondo lattice itself.

## Results and Discussion

Substitutions for Ce (or In) in CeCoIn_{5} by nonmagnetic elements R(or Hg, Sn) rapidly suppress *T*_{c}, with *T*_{c} → 0 K typically in the range of 10–15% substitution for Ce (In). Fig. 1 shows that, concomitant with the depression of *T*_{c}, there is a systematic increase in the value of *C*/*T* (*T* → 0 K) ≡ *γ*_{0} that is a measure of a nonsuperconducting electronic contribution to specific heat in the superconducting state. In a magnetic field of *H* = 5 T (*H*∥*c* axis), the normal state Sommerfeld coefficient *γ*_{N} follows a logarithmic temperature dependence, indicating proximity to a quantum critical point (9) for all dopants. An extrapolation of the infield *C*/*T* data to *T* = 0 K, such that the extrapolation conserves entropy between the normal and superconducting states at *T*_{c}, yields *γ*_{N} > 1.2 J/mol Ce K^{2} for all concentrations. We make the ansatz that there is an additional normal component to *C*/*T* below *T*_{c} given by *γ*_{0}/*γ*_{N} and compare this normal component to the reduction of the superconducting condensation energy (properly normalized relative to the condensation energy of pure CeCoIn_{5}), where . As shown in Fig. 2*A*, the doping-induced normal state fraction comes *precisely* at the expense of the superconducting state fraction as evidenced by a common linear variation of *R*_{γ} = *γ*_{0}/*γ*_{N} vs. 1 - *R*_{U}, for all substituents (Y^{3+}, La^{3+}, Th^{4+}, Yb^{2+}, Hg, and Sn—see Fig. 2*B* and Fig. S1), regardless of valence or size of the impurity atom. This unexpected result provides compelling evidence for electronic inhomogeneity in an *f*-electron Kondo lattice. Furthermore, the linear dependence of *γ*_{0}/*γ*_{N} on impurity concentration (Fig. 2*A*, *Inset*) does not follow the expectation for creating electronic states in superconducting nodes through disorder in a “dirty” *d*-wave scenario in the strong scattering (unitary) limit (for which *γ*_{0}/*γ*_{N} ∼ *x*^{1/2}) or in the weak scattering (Born) limit (Fig. 3*A*), implying that the impurities suppress the superconducting energy gap through the creation of intragap states, much like Zn impurities in YBCO and Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} (Fig. 3*B*) (4, 10). In this analysis, we have used the simple Bardeen–Cooper–Schrieffer expression for the condensation energy , where *N*(0) is the density of states at the Fermi level, to allow a comparison of the different dopants substituted into the heavy fermion superconductors. More complete calculations of *U*_{SC} for unitary scatterers is plotted as *γ*_{0}/*γ*_{N} vs. 1 - *U*_{SC}(Γ)/*U*_{SC}(0) in Fig. S2, where Γ is the impurity scattering rate. These calculations do not reproduce the universal linear relation of *R*_{γ} vs. 1 - *R*_{U} (Fig. 2*B*), furthering a scenario of electronic heterogeneity in which the dopants locally suppress superconductivity.

Our thermodynamic analysis of impurities introduced into CeCoIn_{5} further implies that the electronic inhomogeneity arises from disruption of the coherent Kondo lattice by Kondo holes. We estimate the characteristic energy scale of these Kondo holes through a simple binary alloy model, consistent with the creation of Swiss cheese holes, in which the specific heat is composed of a superconducting and normal component: [1]Because *C*_{tot} ∼ ln(*T*^{∗}/*T*) remains virtually unchanged with a Kondo lattice coherence temperature *T*^{∗} ∼ 40 K up to approximately 40% La in CeCoIn_{5} (11), the large contribution to electronic specific heat from these Kondo holes (*γ*_{0} ∼ 9.5 J/mol La K^{2} for *x* = 0.1; see Fig. 1) indicates that their effective mass is huge or, equivalently, that their characteristic energy scale is small, *T*_{KH} = *πR*/6*γ*_{0} ∼ 0.3 K for an effective “spin-1/2” La impurity, where *R* is the gas constant (12); strong scattering from these massive Kondo holes leads to the loss of quantum oscillations (13), even for < 1% La impurities in CeCoIn_{5}. Breaking the translational invariance of the Kondo lattice locally suppresses the superconducting gap significantly as seen in the strong reduction of the specific heat jump Δ*C* at *T*_{c} (Fig. 3*A*) of doped CeCoIn_{5}, analogous to the strong temperature-dependent pair-breaking effects when Ce Kondo impurities, characterized by *T*_{K} ∼ 0.1 K are introduced into the 3.3 K *s*-wave superconductor LaAl_{2} (14); indeed, the suppression of Δ*C* in these two systems is very similar (Fig. 3*A*). Substitutions on the In site lead to either weaker suppression (Sn) of the gap, or stronger suppression (Cd, Hg) possibly due to additional spin-flip pair-breaking effects caused by the local nucleation of magnetism near the Cd or Hg sites (15). Further support for the local suppression of superconductivity around the Kondo holes is provided by analysis of the effective size of an impurity bound state in a *d*-wave superconductor (4), given by , where *ξ*_{0} = 4.9 nm is the superconducting coherence length for the Ce_{0.9}La_{0.1}CoIn_{5} sample, determined from the initial slope of the upper critical field *dH*_{c2}/*dT*_{c} (16). The ratio of the energy of the impurity state (or resonance) to the superconducting gap Δ_{0} is *ε*_{0} = {[1 - (*T*_{KH}/0.3Δ_{0})^{2}]/[1 + (*T*_{KH}/0.3Δ_{0})^{2}]} following ref. 17, where the strong-coupling value Δ_{0} = 2.25*T*_{c} was used (6). From this formula, we find that *R*_{imp} = 5.8 nm is comparable to *ξ*_{0}, using *T*_{KH} = 0.3 K for Ce_{0.9}La_{0.1}CoIn_{5}, consistent with local suppression of superconductivity near the La impurities. [Similar impurity length scales *R*_{imp} ∼ *ξ*_{0} are obtained for other La concentrations *x* = 0.02 and 0.05, which have nearly identical Kondo hole energy scales *T*_{KH} ∼ 0.2–0.3 K (12) and values of *dH*_{c2}/*dT*_{c} (16).] Recent scanning-tunneling spectroscopy on Th impurities in URu_{2}Si_{2} (18) reveals a strong local change of the density of states in this Kondo lattice, demonstrating that Kondo holes significantly affect the normal state as well (19). Our results further strengthen the connection between the heavy fermion superconductors and the cuprates, as the suppression of Δ*C* of Zn-doped YBCO is similar to that of Ce_{1-x}*R*_{x}CoIn_{5} (Fig. 3*A*), and also to the iron–pnictide superconductors (20, 21), in which electronic inhomogeneity has been observed recently (22).

^{115}In nuclear quadrupole resonance (NQR) measurements further characterize the doping distribution of Ce_{1-x}La_{x}CoIn_{5} and provide insight into the nature of the resulting electronic state. Fig. 4 shows the NQR signal for the 4*ν*_{Q} quadrupolar (± 9/2↔ ± 7/2) transition for the in-plane In(1), as well as the temperature dependence of the spin-lattice relaxation rates () for *x* = 0, 0.1, and 1. The NQR peaks are relatively sharp in the pure compound (Fig. 4*B*) with the LaCoIn_{5} frequency (*ν*_{Q} ∼ 8.01 MHz) smaller than that of CeCoIn_{5} (*ν*_{Q} ∼ 8.17 MHz), in good agreement with previous reports (23, 24). The NQR spectrum in the normal state at *T* = 3 K is significantly broadened for the *x* = 0.1 sample, with the main peak (labeled A) virtually at the same frequency as in CeCoIn_{5} and with two adjacent peaks (labeled B and C) resolved. There is no additional broadening or shift in the spectra as the sample becomes superconducting below *T*_{c} = 0.9 K, confirming that the heterogeneous electronic state below *T*_{c} has its origin in the normal state, as a result of doping. The lack of any intensity at the frequency corresponding to pure LaCoIn_{5} and the similar temperature dependence of the spin-lattice relaxation rates of the three peaks—all with essentially the same onset *T*_{c}—rule out chemical segregation.

This thermodynamic analysis extends to other heavy fermion superconductors, as presented in Fig. 2*B*, and such electronic inhomogeneity may provide a framework for resolving several outstanding issues. In the 18.5 K superconductor PuCoGa_{5}, the radioactive decay of Pu-239 produces defects and/or dislocations, mimicking the Swiss cheese hole in Ce_{1-x}*R*_{x}CoIn_{5}. Analysis of the specific heat data of a “fresh” (approximately 2-wk old) and “aged” (approximately 3-mo old) PuCoGa_{5} sample (Figs. S3 and S4), reveals that the induced normal state fraction is comparable to the superconducting state fraction as radiation damage accumulates, in agreement with self-consistent *T*-matrix calculations describing the rate of suppression of *T*_{c} (20). Furthermore, the observed anomalous reduction in superfluid density (25) with time in this strong-coupling *d*-wave superconductor is similar to the reduction observed in the Zn-doped YBCO and LSCO cuprates (2). Likewise, nonmagnetic Y and Th impurities (26) introduced into the exotic (odd-parity, ref. 27) superconductor UPt_{3} fall into this class of rather exceptional dopants. Finally, Tanatar et al. (28) have presented evidence for a normal component arising from an expansion of superconducting gap nodes on the Fermi surface in Ce_{1-x}La_{x}CoIn_{5} (*x* < 0.15), a result interpreted within an extreme multiband model in which electrons with a small effective mass remain unpaired on small 3D Fermi surface pockets (for a different viewpoint, see ref. 29). In contrast to this scenario, a consistent picture emerges in which impurities (*i*) create an inhomogeneous electronic state within the superconducting condensate of CeCoIn_{5} (Fig. 3*B*) and (*ii*) destroy the coherent Kondo lattice near the two-dimensional percolation limit, approximately 40% for the *impurities* (not the magnetic Ce ions), corresponding to a universal scattering rate given by a resistivity of *ρ*_{0} ∼ 35 *μ*Ωcm (30). These novel states of matter within the Swiss cheese regions, which do far more than just suppress the superconducting gap, are ubiquitous in cuprates (4), but our study of this particular state reveals the nature of the underlying Kondo lattice and emphasizes the delicate interplay between unconventional superconductivity and the periodic array of Kondo ions from which it originates.

Our study of CeCoIn_{5} and other heavy fermion superconductors highlights that superconductivity itself provides a previously unappreciated window on electronic inhomogeneity in Kondo lattice materials in the form of Kondo holes. Though some theoretical ideas have been put forth to investigate the disruption of the Kondo lattice by nonmagnetic impurities (7, 8), this problem remains virtually unexplored, aside from a few experimental studies (19, 31). Further investigations of Kondo holes in these *f*-electron Kondo lattices and superconductors, including the application of local probes such as scanning-tunneling spectroscopy, provide an opportunity to unravel their complexity; indeed, electronic inhomogeneity in these materials may well prove to be the norm rather than the exception.

## Materials and Methods

Single crystals of Ce_{1-x}*R*_{x}Co(In_{1-y}*M*_{y})_{5} (*R* = Y^{3+}, La^{3+}, Gd^{3+}, Yb^{2+}, Th^{4+}; *M* = Sn, Cd, Hg) were grown from In flux, whereas single crystals of PuCoGa_{5} were grown from Ga flux. Specific heat measurements were carried out in a Quantum Design Physical Properties Measurement System from 0.4 to 20 K (or from 5 to 25 K for PuCoGa_{5}), or in a ^{3}He/^{4}He dilution refrigerator from 50 mK to 3 K, in magnetic fields up to 9 T. The concentrations of the impurities were determined from energy-dispersive X-ray spectroscopy. The ^{115}In NQR measurements were performed using a phase coherent pulsed NMR/NQR spectrometer. Several crystals with similar *T*_{c} previously investigated by specific heat measurements were gently crushed into powder to improve the signal probed by the rf measurement. The frequency-swept ^{115}In NQR spectra (*I* = 9/2; *γ*/2*π* = 9.3295 MHz/T) were obtained using an auto-tuning probe in a ^{3}He cryostat. The spectra were obtained by stepwise summing the Fourier transform of the spin-echo signal. The values of the spin-lattice relaxation time *T*_{1} were obtained by fits of the recovery of the nuclear magnetization *M*(*t*) after a saturation pulse. The self-consistent *T*-matrix calculations of the specific heat jump are described in detail in ref. 32.

In our thermodynamic analysis, estimates of the normal state electronic specific heat coefficient *γ*_{N} were obtained by linear extrapolation to *T* = 0 K (a determination of *γ*_{N} obtained by fits of the data to the model of Moriya and Takimoto, ref. 33, for critical fermions yields similar values within 5–10%) and requiring entropy balance at *T*_{c}. The superconducting fraction of the heavy electrons is calculated by , where *T*_{c} is determined by equal entropy construction above and below *T*_{c}, *U*_{sc} is the superconducting condensation energy determined from the integration of the entropy difference between the normal (high field) and superconducting (zero field) states up to *T*_{c}, and *E*_{c} is a constant described below. If the high field data were not available for all doping levels, we used the normal state value for *γ*_{N} of the pure compound as an approximation for the doped compounds; the insensitivity of *C*/*T* just above *T*_{c} for all concentrations (Fig. 1) and the entropy balance at *T*_{c} indicate this approximation is reasonable. In a few cases (*x* = 0.05 La, 0.012 Sn, 0.03 Yb), the entropy balance was not satisfied in all available datasets (see Fig. 1); therefore, we added a small linear term (approximately 3–5% of *γ*_{N}) to correct the entropy balance and obtain two different values of *U*_{sc} before and after the correction for a better relative comparison within each doping series. We take the average and their difference gives the error bars for *R*_{U} of these samples in Fig. 2.

In our analysis of Ce_{1-x}*R*_{x}Co(In_{1-y}*M*_{y})_{5}, we have taken approximately *E*_{c} = *U*_{sc}(0)/*T*_{c}(0)^{2} from the pure compound CeCoIn_{5}. This analysis, together with the experimental fact *R*_{γ} + *R*_{U} = 1 in Fig. 2, indicates that the pure compound of CeCoIn_{5} has a negligible amount of impurities (i.e., comparing *γ*_{0} = 0.04 J/mol·Ce K^{2} to *γ*_{N} = 2 J/mol·Ce K^{2}). However, in some heavy fermion compounds such as U_{1-x}*M*_{x}Pt_{3} and PuCoGa_{5}, even the pure compounds have a significant number of defects or a large intrinsic *γ*_{0}. In this case, it is necessary to define a parameter *E*_{c}, which corrects the normalization by for additional disorder and/or systematic errors (see Table S1), to best fit all the doped data of a given system. For example, in PuCoGa_{5}, *R*_{γ} ∼ 0.2 even in the fresh sample, suggesting a large amount of defects caused by radiation damage, which is consistent with theoretical calculations that indicate *T*_{c} = 19.1 K in an undamaged material and account for the decrease in *T*_{c} with time (20, 25). In the case of UPt_{3}, the pure material has a different condensation energy from the doped compounds (Fig. 2*B*), reflecting the double superconducting transition and also the sensitivity of this exotic odd-parity superconductor to impurities.

## Acknowledgments

Z.F. thanks Ilya Vekhter, Piers Coleman, and Lev Gor’kov for useful discussions and the hospitality of the Aspen Center for Physics. E.D.B. and J.D.T. thank H. Yasuoka for helpful discussions. This work was performed at Los Alamos National Laboratory under the auspices of the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering. Work at University of California, Irvine was performed under the National Science Foundation (NSF) Grant NSF-DMR-0801253. Work at the National High Magnetic Field Laboratory was performed under the auspices of the NSF (DMR-0654118) and the State of Florida.

## Footnotes

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^{1}To whom correspondence may be addressed. E-mail: edbauer{at}lanl.gov, yifeng{at}iphy.ac.cn, or zfisk{at}uci.edu.

Author contributions: J.D.T. and Z.F. designed research; E.D.B., C.C., R.R.U., C.F.M., H.S., F.R., M.J.G., A.V.B., R.M., A.D.B., A.P.R., and P.L.K. performed research; E.D.B., C.C., and A.D.B. prepared samples; E.D.B., Y.-f.Y., C.C., R.R.U., C.F.M., H.S., F.R., R.M., A.D.B., and A.P.R. analyzed data; and E.D.B. and Y.-f.Y. wrote the paper.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1103965108/-/DCSupplemental.

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