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# Non-Fermi liquid regimes with and without quantum criticality in Ce_{1−x}Yb_{x}CoIn_{5}

Contributed by M. Brian Maple, March 20, 2013 (sent for review September 12, 2012)

## Abstract

One of the greatest challenges to Landau’s Fermi liquid theory—the standard theory of metals—is presented by complex materials with strong electronic correlations. In these materials, non-Fermi liquid transport and thermodynamic properties are often explained by the presence of a continuous quantum phase transition that happens at a quantum critical point (QCP). A QCP can be revealed by applying pressure, magnetic field, or changing the chemical composition. In the heavy-fermion compound CeCoIn_{5}, the QCP is assumed to play a decisive role in defining the microscopic structure of both normal and superconducting states. However, the question of whether a QCP must be present in the material’s phase diagram to induce non-Fermi liquid behavior and trigger superconductivity remains open. Here, we show that the full suppression of the field-induced QCP in CeCoIn_{5} by doping with Yb has surprisingly little impact on both unconventional superconductivity and non-Fermi liquid behavior. This implies that the non-Fermi liquid metallic behavior could be a new state of matter in its own right rather than a consequence of the underlying quantum phase transition.

The heavy-fermion material CeCoIn_{5} is a prototypical system in which strong interactions between conduction and predominantly localized *f* electrons give rise to a number of remarkable physical phenomena (1, 2). Unconventional superconductivity (SC) emerges in CeCoIn_{5} out of a metallic state with non-Fermi liquid (NFL) properties: linear temperature dependence of resistivity below 20 K, logarithmic temperature dependence of the Sommerfeld coefficient, and divergence of low-temperature magnetic susceptibility (3⇓⇓–6). These anomalies disappear beyond a critical value of the magnetic field and the system recovers its Fermi liquid properties. The crossover from NFL to Fermi liquid behavior is thought to be governed by a quantum critical point (QCP), which separates paramagnetic and antiferromagnetic (AFM) phases and is located in the superconducting phase (7, 8). Neutron-scattering studies (9) and more recent measurements of the vortex-core dissipation through current–voltage characteristics (10) provide direct evidence for an AFM QCP in CeCoIn_{5} that could be accessed by tuning the system via magnetic field or pressure.

Nevertheless, a growing number of *f*-electron systems do not conform with this QCP scenario; for example, the NFL behavior and/or SC in some systems occurs in the absence of an obvious QCP (11⇓⇓–14). An intriguing candidate is the alloy Yb-doped CeCoIn_{5} that exhibits an unconventional phase diagram without an apparent QCP, whereas the onset of coherence in the Kondo lattice and the superconducting transition temperature are only weakly dependent on Yb concentration and prevail for doping up to (15). However, the presence of a QCP in the parent CeCoIn_{5} compound and the logarithmic temperature dependence of the normal state Sommerfeld coefficient in lightly doped Ce_{1−x}Yb_{x}CoIn_{5} crystals (16) show that this system is in the vicinity to a QCP. Therefore, we are presented with the remarkable opportunity to elucidate the nature of the NFL behavior and unconventional SC in such a system, to search for possible QCPs, and to determine the specific role the QCP plays in defining the low-temperature properties of this material and, in particular, to determine the degree to which quantum criticality and SC are coupled to each other.

To uncover the field-induced QCP and its evolution on Yb concentration, we study the magnetic field (*H*) and temperature (*T*) dependence of the transverse in-plane magnetoresistivity (MR), , for T and K. Fig. 1*A* and its *Inset* display the field dependence of measured at different temperatures for the and samples, respectively. We note that the data for these samples fall under two groups: (*i*) nonmonotonic field dependence of MR (main panel) with positive MR at low fields and negative quadratic MR at high fields, typical for and (*ii*) negative and quadratic MR over the whole measured field range (*Inset*), typical behavior for the high Yb doping .

Positive MR in heavy-fermion materials at low fields marks the departure from the single-ion Kondo behavior and is determined by the formation of the coherent Kondo lattice state for systems in or close to their Fermi liquid ground state (18⇓⇓⇓⇓–23). The maximum in the MR of a Kondo lattice Fermi liquid at a certain field value is a result of the competition between a *T*-independent residual resistivity contribution that increases with increasing *H*, and a *T*-dependent term that decreases with increasing *H* (23, 24). Thus, in conventional Kondo lattice systems, the peak in the field-dependent MR moves toward lower *H* with increasing *T* because a lower field is required to break the Kondo lattice coherence.

To determine the nature of the positive MR in Ce_{1−x}Yb_{x}CoIn_{5} for , we extract the temperature dependence of the peak in the field-dependent MR and plot these data in Fig. 1*B*. This figure shows that vs. *T* of the samples is nonmonotonic, with a maximum around 20 K, and a linear behavior at low-*T* values. The positive MR measured at K [for which decreases with increasing *T*] could reflect the presence of the coherent Kondo lattice state at low field values as discussed in the previous paragraph. In contrast, the behavior below 20 K is strikingly different from what one would expect for conventional Kondo lattice systems. The increase of with increasing *T* at these lower temperatures had previously been observed in CeCoIn_{5} and has been attributed to field quenching of the AFM spin fluctuations responsible for the NFL behavior (25). Therefore, we conclude that the positive MR measured at K in Ce_{1−x}Yb_{x}CoIn_{5} with reflects the dominant role played by the AFM quantum spin fluctuations.

An important next goal is to identify the associated with these quantum fluctuations. One option is to extrapolate the low temperature linear behavior to K and identify this with . However, because there is a certain error associated with the determination of , we adopted a more accurate procedure to unambiguously determine for different Yb doping. This procedure is discussed in detail in *Materials and Methods*. We show in Fig. 1*C* the values of , obtained using this procedure, as a function of Yb concentration. As expected, the value of of 4.1 T for CeCoIn_{5} (Fig. 1*C*) coincides with the value of determined previously from both resistivity measurements done in the normal state (26) and characteristics measured in the mixed state (10). Therefore, the measurement of along with the analysis used here constitutes an excellent experimental technique to determine the field-induced QCP in the NFL regime. The points for correspond to the case when there is no positive MR, i.e., the maximum in MR shifts to zero field (Fig. 1*A*, *Inset*).

The *Inset* to Fig. 1*C* shows the suppression with doping of (from ref. 15) and for Ce_{1−x}Yb_{x}CoIn_{5} and of the corresponding temperatures (from ref. 17) for Ce_{1−x}La_{x}CoIn_{5}. These doping-dependent and temperature phase diagrams for Ce_{1−x}Yb_{x}CoIn_{5} show that, although SC is robust and survives over the whole Yb doping range, the field-induced QCP is strongly suppressed with Yb doping and disappears for . This implies that SC and quantum criticality are likely to be decoupled in this system, i.e., unconventional SC is not triggered by spin fluctuations.

The experimental technique used to determine also permits the determination of the gyromagnetic factor *g*. There is a significant change in the value of the *g* factor at the QCP. For CeCoIn_{5}, this change is from (*g* factor of free electrons in a metal) just below to just above . Similarly, for the sample, the change is from just below to just above . The experimentally observed changes in the *g* factor reflect the transformations that the electronic system undergoes with the change in the external magnetic field. Therefore, we conclude that for the conduction electrons are only weakly coupled to the local spins (i.e., there is a small amount of admixture between the *f*-electron and the conduction-electron states). However, for , given that the values of 1.3 and 1.0 are only slightly larger then the value of for the crystal field configuration of Ce ions (27), the conduction states are strongly hybridized with the *f* states because the AFM fluctuations between the local moments are suppressed, which is reflected in the reduction in the value of the *g* factor (28). The change in the value of *g* at can be interpreted using the phenomenological theory of “Kondo breakdown” at (28). Within this theory, the changes in the *g* factor are governed by the changes in the size of the Fermi surface: larger values of the *g* factor correspond to a small Fermi surface so that the conduction electrons are effectively decoupled from the localized *f* states. In the opposite limit of smaller *g* values, the Fermi surface is large, reflecting the strong coupling between the conduction and *f* electrons. More importantly, the jump in the size of the Fermi surface at corresponds to the divergence of the quasiparticle’s effective mass.

A very interesting and puzzling behavior of the Ce_{1−x}Yb_{x}CoIn_{5} system is that, even though the QCP disappears for , the system continues to display NFL behavior, as evidenced by the sublinear *T* dependence of its resistivity (15). This means that this NFL behavior at higher Yb doping could be a new state of matter in its own right rather than a consequence of the underlying quantum phase transition. We further investigated the origin of this NFL behavior by studying in more detail the *T* dependence of the resistivity measured in different magnetic fields. The resistivity of all of the Yb-doped single crystals studied follow remarkably well the following expression:for temperatures up to about 15 K (see Fig. 2*A* for the fitting results) and in fields up to 14 T; here , *A*, and *B* are doping- and field-dependent fitting parameters. We show in Fig. 2 *B* and *C* the doping and magnetic field dependences, respectively, of the ratio of the *T*-linear contribution to the total *T*-dependent contribution of the resistivity. These results show that the linear in *T* term in the resistivity is present for low Yb doping in the quantum critical regime ( K). The percentage of the linear in *T* term decreases with increasing *x* and *H* and disappears for and at fields at which the MR is only negative, where only the dependence is present.

It is noteworthy that the average Yb valence saturates to a value of about 2.3 for (29). This result along with the data of Fig. 2 *B* and *C* and the fact that both the superconducting transition and coherence temperatures remain weakly dependent on doping indicate that Yb atoms form a cooperative mixed-valence state that significantly reduces the pair-breaking effects, which could also play an important role in the origin of the NFL behavior at these higher Yb concentrations . This idea is supported by the observed scaling of vs. (see Fig. 3 and the following discussion).

Fig. 3 shows that the normalized Kondo lattice coherence temperature (from this work) scales with the normalized superconducting critical temperature (from ref. 15) over the whole Yb concentration range studied here (black squares) and up to about 10% La doping (red circles, data from ref. 17). The linear scaling function is as follows:whereThe fact that both and are suppressed at the same rate for both types of substitutions, i.e., , is notable. We interpret it as an indication that the onset of the many-body coherence in the Kondo lattice and emergence of SC have the same physical origin in both systems: hybridization between conduction and localized *f*-electron states. In particular, this suggests that Cooper pairing develops primarily on the “heavy” (i.e., large) Fermi surface. In addition, it is intriguing that, although the scaling of these two temperatures for Yb and La conforms with that of the other rare-earth substitutions on the Ce site (30), the nature of the scaling distinguishes the Yb substitution from the other rare-earth substitutions because both and remain much more robust with respect to disorder for Yb, whereas a nonlinear and quite rapid suppression of both temperatures is observed for the other rare-earth substituents, with a suppression of to zero at around % of rare-earth substitution (30) (Fig. 1*C*, *Inset*, and data from refs. 15 and 17). This observation could be explained by noting that Yb atoms are in a mixed-valence state and, therefore, must be correlated. The correlation may arise via local lattice deformations. *Materials and Methods* gives a scenario on how the Yb impurity correlations may slow down the suppression of the coherence temperature.

All of the above observations show that the field-induced QCP governs the normal-state transport and thermodynamic properties. Although the proximity of many heavy-fermion superconductors to a magnetic instability indicates that the exchange of spin-fluctuations between the conduction electrons is a primary mechanism for the Cooper pairing (31⇓–33), our experimental results strongly suggest that an alternative mechanism for Cooper pairing may be at play in Ce_{1−x}Yb_{x}CoIn_{5}: following the full suppression of the QCP, the normal state is fully reconstructed by the ytterbium substitution, whereas the superconducting critical temperature is reduced only by a factor of 2 at . One possible resolution of this conundrum lies in the hypothesis that superconducting pairing is spatially inhomogeneous and must necessarily involve local Ce *f* moments. Specifically, when magnetic ions in a lattice exchange a spin with their metallic environment in two distinct symmetry channels, they become overscreened, forming a condensate of composite pairs. In this picture, both the local moments and the electron pairs are involved in the formation of the superconducting condensate—hence the name composite pairing (34, 35). Therefore, within the framework of composite pairing theory, it is natural to expect that the magnetic QCP will have little effect on the origin of unconventional SC.

To summarize, we have experimentally investigated the evolution of the field-induced QCP in the heavy-fermion alloy Ce_{1−x}Yb_{x}CoIn_{5} upon Yb concentration. In particular, we have shown that the linear temperature dependence of resistivity is governed by the system’s proximity to the QCP, whereas the emergence of SC and Kondo lattice coherence remain weakly dependent on the presence of the QCP. Finally, we proposed a technique to probe the interplay between quantum criticality and SC, which can be used to analyze a variety of strongly correlated electronic materials.

## Materials and Methods

### Experimental Methods.

Single crystals of Ce_{1−x}Yb_{x}CoIn_{5} were grown using an indium self-flux method (36). The crystal structure and composition were determined from X-ray powder diffraction and energy dispersive X-ray techniques. The single crystals have a typical size of mm^{3}, with the *c* axis along the shortest dimension of the crystals. They were etched in concentrated HCl for several hours to remove the indium left on the surface during the growth process and were then rinsed thoroughly in ethanol. Four leads were attached to the single crystal, with . High-quality crystals were chosen to perform in-plane transverse and longitudinal MR measurements, with and , respectively, as a function of temperature (*T*) and applied magnetic field (*H*). In both situations, however, the field is perpendicular to the current to ensure that the Lorentz force is the same.

### Determination of Field-Induced QCP.

Here, we describe the procedure we used to identify the associated with the quantum fluctuations. Based on the data of Fig. 1*A*, we conclude that the MR for low Yb doping has two main contributions: one negative and quadratic in *H*, which we denote (black symbols in Fig. 4*A* at high fields) and the other is positive, denoted (green symbols in Fig. 4*A*), with the latter one obtained by subtracting for all field values the negative quadratic MR from the measured MR. This latter contribution to the MR is isotropic because the green data follow quite well the longitudinal MR (red data in Fig. 4*A*). Also, is linear in *H* at low fields (Fig. 4*B*) and saturates at high fields (Fig. 4*C*).

We define the characteristic fields where is , , , , etc. of the saturation value . We show the *T* dependence of these characteristic fields in Fig. 4*D* by the black, red, green, and blue solid circles, respectively. We then fit the linear low-*T* behavior of these . Fig. 5 is a plot of the slopes *K* (closed symbols), obtained from these linear fits of the different curves with different percentages of (Fig. 4*D*), and the corresponding percentage of (open symbols) vs. the corresponding values of the intercept fields for the and 0.10 samples. We note the sharp increase of the slope *K* at a certain value, which we identify as the quantum critical field for reasons given below.

The origin of the sharp increase in the values of *K* can be interpreted as follows. For a system in the quantum critical regime (i.e., low *H* and *T* data of Fig. 4*D*), the only energy scale is the Boltzmann energy . We compare this energy scale to the quasiparticle Zeeman energy, i.e.,

From Eq. **4**, we see that the slope *K* must be inversely proportional to the gyromagnetic factor *g*. So, the sharp increase in *K* is a result of the sharp decrease in *g*. Previous studies (37) have shown that abrupt changes in the values of the gyromagnetic factor occur at the QCP. Therefore, using this procedure, we are able to unambiguously determine as the value of at which there is the sharp change in the *g* factor.

### Effect of the Correlation Between the Yb Impurity on the Suppression of and .

To see how the impurity correlations may slow down the suppression of the coherence temperature, we consider the characteristic length scale *R* on which the impurity distribution function significantly deviates from unity. The impurity distribution function determines the probability with which one can find one impurity at a certain distance from another. Within the Born approximation, one can show that there will be two contributions to the self-energy of the conduction electrons. One contribution, , corresponds to the scattering of electrons on the same impurity and, upon the averaging over disorder, this contribution is proportional to the concentration of impurities . The second contribution, , describes the scattering of electrons on two different impurities and, therefore, is proportional to . In the presence of impurity correlations, however, becomes proportional to . Thus, if the radius of correlations is large enough, , becomes comparable with the first, linear in , self-energy correction . Consequently, within the large-*N* mean field theory, one can show that impurity correlations may provide the “healing effect”: the rate of suppression of the coherence temperature becomes strongly dependent on the impurity correlation length *R* (38).

## Acknowledgments

This work was supported by National Science Foundation Grants NSF DMR-1006606 and DMR-0844115, Institute for Complex Adaptive Matter Branches Cost Sharing Fund, and Ohio Board of Regents Grant OBR-RIP-220573 at Kent State University, and by Department of Energy Grant DE-FG02-04ER46105 at University of California at San Diego. M.J. gratefully acknowledges financial support by the Alexander von Humboldt Foundation.

## Footnotes

↵

^{1}T.H. and Y.P.S. contributed equally to this work.- ↵
^{2}To whom correspondence may be addressed: mbmaple{at}ucsd.edu or calmasan{at}kent.edu.

Author contributions: T.H., Y.P.S., L.S., M.J., M.D., M.B.M., and C.C.A. designed research; T.H., Y.P.S., L.S., M.J., M.D., M.B.M., and C.C.A. performed research; T.H. and Y.P.S. analyzed data; and T.H., Y.P.S., L.S., M.J., M.D., M.B.M., and C.C.A. wrote the paper.

The authors declare no conflict of interest.

## References

- ↵Coleman P (2007) Heavy fermions: Electrons at the edge of magnetism.
*Handbook of Magnetism and Advanced Magnetic Materials*(Wiley, New York), Vol 1, pp 95–148. - ↵
- ↵
- ↵
- Petrovic C,
- et al.

_{5}at 2.3 K. J Phys Condens Matter 13(17):L337–L342. - ↵
- ↵
- ↵
- ↵
- ↵Kenzelmann M, et al. (2008) Coupled superconducting and magnetic order in CeCoIn
_{5}.*Science*321(5896):1652. - ↵
- ↵
- Matsumoto Y,
- et al.

_{4}. Science 331(6015):316–319. - ↵
- ↵
- Pfleiderer C,
- Böni P,
- Keller T,
- Rössler UK,
- Rosch A

- ↵
- Hardy F,
- et al.

- ↵
- ↵
- Bauer ED,
- et al.

- ↵
- ↵
- ↵
- ↵
- Flouquet J,
- et al.

- ↵
- Rauchschwalbe U,
- Steglich F,
- de Visser A,
- Franse JJM

_{2}Si_{2}and CeAl_{3}. J Magn Magn Mater 63–64:347–348. - ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵

_{1−x}Yb

_{x}CoIn

_{5}

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