Long range order and two-fluid behavior in heavy electron materials

Abstract

The heavy electron Kondo liquid is an emergent state of condensed matter that displays universal behavior independent of material details. Properties of the heavy electron liquid are best probed by NMR Knight shift measurements, which provide a direct measure of the behavior of the heavy electron liquid that emerges below the Kondo lattice coherence temperature as the lattice of local moments hybridizes with the background conduction electrons. Because the transfer of spectral weight between the localized and itinerant electronic degrees of freedom is gradual, the Kondo liquid typically coexists with the local moment component until the material orders at low temperatures. The two-fluid formula captures this behavior in a broad range of materials in the paramagnetic state. In order to investigate two-fluid behavior and the onset and physical origin of different long range ordered ground states in heavy electron materials, we have extended Knight shift measurements to URu₂Si₂, CeIrIn₅, and CeRhIn₅. In CeRhIn₅ we find that the antiferromagnetic order is preceded by a relocalization of the Kondo liquid, providing independent evidence for a local moment origin of antiferromagnetism. In URu₂Si₂ the hidden order is shown to emerge directly from the Kondo liquid and so is not associated with local moment physics. Our results imply that the nature of the ground state is strongly coupled with the hybridization in the Kondo lattice in agreement with phase diagram proposed by Yang and Pines.

Competition between different energy scales gives rise to a rich spectrum of emergent ground states in strongly correlated electron materials. In the heavy fermion compounds, a lattice of nearly localized f electrons interacts with a sea of conduction electrons, and depending on the magnitude of this interaction different types of long-range order may develop at low temperatures (1). The Kondo lattice model strives to capture the essential physics of heavy fermion materials by considering the various magnetic interactions between the conduction electron spins, S_c, and the local moment spins, S_f (2). Different ground states can emerge depending on the relative strengths of the interaction, J, between S_c and S_f and the intersite interaction, J_ff, between the S_f spins (3, 4). Much of the physics of the phase diagram is driven by a quantum critical point, which separates long-range ordered ground states from those in which the local moments have fully hybridized with the conduction electrons to form itinerant states with large effective masses and large Fermi surfaces (5). In several materials quantum critical fluctuations give rise to anomalous non-Fermi liquid behavior in various bulk transport and thermodynamic quantities (6–8).

In recent years, evidence has emerged that in the high temperature disordered phase the electronic degrees of freedom simultaneously exhibit both itinerant and localized behavior (9). Below a temperature T^∗ that marks the onset of hybridization or lattice coherence, several experimental quantities exhibit a temperature dependence that is well described by a fluid of hybridized heavy quasiparticles coexisting with partially screened local moments (3, 10). In the two-fluid picture, the transfer of spectral weight from the local moments to the heavy electrons is described by a quantity f(T) (10). Above a temperature T^∗, the local moments and conduction electrons remain uncoupled. Below T^∗ the local moments gradually dissolve into the hybridized heavy electron fluid as the temperature is lowered. This crossover can be measured directly via nuclear magnetic resonance (NMR) Knight shift experiments (11). The contribution to the Knight shift from the heavy electrons, K_HF, exhibits a universal logarithmic divergence with decreasing temperature below T^∗ in the paramagnetic state. In superconducting CeCoIn₅ this scaling persists down to T_c indicating that the condensate emerges from the heavy electron degrees of freedom (12). Relatively little is known, however, about how this scaling behaves in materials with other types of ordered ground states. Here we report Knight shift data in both URu₂Si₂ and CeRhIn₅. In the former the scaling persists down to the onset of hidden order at T_HO; but in the latter the heavy electron spectral weight reverses course and the local moments relocalize above the Neel temperature, T_N, similar to the antiferromagnet CePt₂In₇ (13). These results imply generally that the heavy electron fluid is either unstable to a Fermi surface instability such as hidden order or superconductivity or collapses to form ordered local moments.

Knight Shift Anomalies

The Zeeman interaction of an isolated nuclear spin in an external magnetic field H₀ is given by , where γ is the gyromagnetic ratio and ℏ is the reduced Planck constant. In this case the nuclear spin resonance frequency is given by the Larmor frequency ω_L = γH₀. In condensed matter systems the nuclear spins typically experience a hyperfine coupling to the electron spins, , where S is the electron spin, A is the hyperfine coupling g is the g-factor and μ_B is the Bohr magneton. The hyperfine coupling is in general a tensor that depends on the details of the bonding and quantum chemistry of the material of interest. S can be the unpaired spin moment of an unfilled shell, the net conduction electron spin of a Fermi sea polarized by a magnetic field, or the spin-orbit coupled moment of a localized electron. S may or may not be located on the same ion as the nuclear moment; the former is referred to as on-site coupling and the latter is transferred coupling. In the paramagnetic state the electron spin is polarized by the external field S = χ·H₀/gμ_B, where χ is the susceptibility. Thus the nuclear spin experiences an effective Hamiltonian:

[1]

where the Knight shift tensor K = A·χ is unit-less. The resonance frequency is given by ω = ω_L(1 + K(θ,ϕ)), where depends on the polar angles θ and ϕ that describe the relative orientation of the field with respect to the principal axes of the Knight shift tensor. For the isotropic case the shift K = Aχ is independent of direction. Strictly speaking the Knight shift was originally defined as the paramagnetic shift from the Pauli susceptibility in metals but in practice refers to any shift K arising from the electronic susceptibility.

The magnetic susceptibility and the Knight shift can be measured independently. One can extract the hyperfine coupling A by plotting the Knight shift versus the susceptibility with temperature as an implicit parameter [such a plot is traditionally known as a Clogston–Jaccarino plot (14)]. If there is a single electronic spin component that gives rise to the magnetic susceptibility, then the Clogston–Jaccarino plot is a straight line with slope A and intercept K₀. K₀ is a temperature independent offset that is usually given by the orbital susceptibility and diamagnetic contributions (15).

Breakdown of Linearity and Failure of Local Pictures.

All known heavy fermion materials exhibit a Knight shift anomaly where the Clogston–Jaccarino plot deviates from linearity below an onset temperature, T^∗ (11). This phenomenon is also visible in a plot comparing K and χ versus temperature, as seen in Fig. 1, where T^∗ marks the temperature below which K no longer tracks χ. This breakdown can arise from the presence of impurity phases, which contribute to the bulk susceptibility but not to the Knight shift. However, Knight shift anomalies occur ubiquitously in single phase highly pure heavy fermion materials and are intrinsic phenomena. Several different theories have been proposed to explain these anomalies. These theories generally fall into two categories: (i) a temperature dependent hyperfine coupling A(T) or (ii) multiple spin degrees of freedom. It has been argued that A acquires a temperature dependence either because of Kondo screening (16) or because of different populations of crystal field levels of the 4f(5f) electrons in these materials (17). A major flaw in such an argument is that the hyperfine coupling is determined by large energy scales involving exchange integrals between different atomic orbitals, and because the Kondo and/or crystal field interactions are several orders of magnitude smaller they cannot give rise to large perturbations in the hyperfine couplings (18). Furthermore, in materials such as the CeMIn₅ system both the Knight shift anomaly and the crystalline electric field interaction are well characterized and are clearly uncorrelated (see Table 1) (19).

Fig. 1.

The Knight shift (solid points) and the bulk susceptibility (solid lines) of the In(1) for field along the c direction in CeMIn₅ for M = Rh ,Ir and Co. T^∗ is the temperature where the two quantities no longer are proportional to one another, and is clearly material dependent. In some cases K appears to exceed χ and in others K decreases.

Table 1.

Crystal field levels and coherence temperatures in the CeMIn₅ materials

Material	T∗ (K)	Δ (K) (from ref. 19)
CeCoIn5	42	100
CeRhIn5	18(1)	80
CeIrIn5	31(5)	78

Two Spin Components.

Because Kondo lattice materials have both localized f electrons and itinerant conduction electrons, a theory that includes different hyperfine couplings to multiple spin degrees of freedom fits better. The two-fluid picture correctly predicts the temperature dependence of the Knight shift anomalies seen experimentally in heavy fermion materials. In the Kondo lattice model the local moment spins, S_f interact with a sea of conduction electron spins, S_c through a Kondo interaction:

[2]

where the sums are over the lattice positions r_i of the localized f electrons. The general solution of the Kondo lattice has not been resolved to date, but the two-fluid description given by Nakatsuji, Fisk, Pines, and Yang provides a phenomenological framework to describe the phase diagram of the Kondo lattice problem (3, 4, 9, 10). This approach offers a natural interpretation of the NMR Knight shift anomaly in terms of two hyperfine couplings to the two different electron spins:

[3]

where A and B are the hyperfine couplings to the conduction electron and local moment spins, respectively (11). The susceptibility is the sum of three contributions χ = χ_cc + 2χ_cf + χ_ff, where with α,β = c,f. In the paramagnetic state and , thus the Knight shift is given by:

[4]

The Knight shift weighs the different correlation functions separately than the total susceptibility, which leads to important consequences if the correlation functions have different temperature dependencies. K ∼ χ if and only if A = B. If A ≠ B, then the Clogston–Jaccarino plot will no longer be linear. By measuring K and χ independently, one can exploit this difference to extract linear combinations of these individual susceptibilities. No other experimental technique can access these susceptibilities.

Free Spin Model.

The general solution for the susceptibilities χ_αβ of the Kondo lattice problem has not been solved to date. It is useful, however, to consider a simplified model. Consider the case of two free spins S_c and S_f that experience a Heisenberg coupling J. The Hamiltonian for this spin system in the presence of an external magnetic field H₀ is given by:

[5]

where μ_B is the Bohr magneton and g_c,f are the g factors of the two different spins. This simplified model is clearly an incorrect physical description of the Kondo lattice problem where the spins are not free but part of a complex many-body problem involving the Fermi sea and various intersite interactions. This model is the simplest case, however, which captures the essence of the Knight shift anomaly and can be solved exactly.

The free energy is given by F = -k_BT ln Z, where Z is the partition function; the susceptibility of the system is given by the second derivative of the free energy with respect to field: . The susceptibility is given as the sum of three contributions:

[6]

[7]

[8]

where x = J/k_BT. The temperature dependence of these three susceptibilities is shown in Fig. 2. In the high temperature limit (T≫J/k_B) χ_cc and χ_ff exhibit Curie behavior and χ_cf → 0 as expected. For low temperature (T ≲ J/k_B) the behavior is modified by the tendency to form a singlet ground state. As shown in Fig. 2 χ_cc, χ_ff and χ_cf change dramatically below T ≈ J/k_B and saturate at low temperatures. The Knight shift, given by Eq. 4, is shown as well in Fig. 2. Clearly K tracks χ for T≫J/k_B but deviates below T ≈ J/k_B, creating an anomaly in the Clogston–Jaccarino plot in Fig. 3. The breakdown of linearity arises because |χ_cf| increases below T ≈ J/k_B and has a different temperature dependence than χ_cc and χ_ff.

Fig. 2.

The components of the magnetic susceptibility for the model spin system. The Knight shift begins to deviate from the bulk susceptibility below a temperature T ≈ J. In this case, we have used , and B/A = 1.5. If g_f = g_c, then χ(T → 0) → 0, as expected for a singlet ground state.

Fig. 3.

Knight shift versus the bulk susceptibility for the model spin system with temperature as an implicit parameter. The Knight shift begins to deviate from the bulk susceptibility below a temperature T ≈ J/k_B. In this case, we have used , and B/A = 1.5.

Two-Fluid Model.

The three individual susceptibilities χ_αβ will exhibit different temperature dependencies in a Kondo lattice than the free spin model considered above. In particular, at high temperatures χ_cc is given by the temperature-independent Pauli susceptibility of the conduction electrons, and χ_ff is given by a Curie–Weiss susceptibility of the local moments. Therefore above a crossover temperature, T^∗, χ_ff dominates and K = K₀ + Bχ. Below T^∗ χ_cf becomes significant and the growth of this component can be measured by the quantity K_HF = K - K₀ - Bχ = (A - B)(χ_cf + χ_cc). The two-fluid model postulates that K_HF is proportional to the susceptibility of the heavy electron fluid; its temperature dependence probes both growth of hybridization and its relative spectral weight (3, 10). Empirically it has been found that in the paramagnetic phase of a broad range of materials, K_HF(T) varies as:

[9]

where is a constant (10). In other words, K_HF exhibits a universal scaling with the quantity T/T^∗, where T^∗ and are material-dependent quantities. is proportional to A–B and can be either positive or negative. T^∗ agrees well with several other experimental measurements of the coherence temperature of the Kondo lattice (10). The Knight shift thus provides a direct probe of the susceptibility of the emergent heavy electron fluid in the paramagnetic state. There are, however, several pieces of missing information regarding the behavior of the heavy electron component that need to be addressed. For example, the Knight shift anomaly is strongly anisotropic in some materials, and it has not been clear whether this is an intrinsic feature of the heavy electron fluid or rather a reflection of the anisotropy of the hyperfine couplings (11). Furthermore, the Yang–Pines scaling works remarkably well for temperatures down to approximately 10% of T^∗, but until recently there has been relatively little information about the evolution of the heavy electron fluid at lower temperature particularly when long-range order develops. We address both of these issues in turn below.

Anisotropy.

One of the striking features of the emergence of the heavy electron component in Kondo lattice materials is the anisotropy of the Knight shift anomaly. For example, in CeCoIn₅ the Knight shift of the In(1) site exhibits a strong anomaly for H₀‖c but no anomaly for H₀⊥c (20). In order to probe this behavior in greater detail, we have measured the full Knight shift tensor in CeIrIn₅. CeIrIn₅ is isostructural to CeCoIn₅ and is a superconductor with T_c = 0.4 K. A single crystal grown in In flux was mounted in a custom-made goniometer probe, and the orientation of the c axis of the crystal was varied from 0 to 90 degrees. ¹¹⁵In has spin , and there are two different crystallographic sites for In in this material; consequently there are several different transitions as seen in Fig. 4. Spectra were obtained as a function of orientation and temperature, and the Knight shift was extracted after correcting for the quadrupolar shift of the resonance (21). Fig. 5 shows K versus θ, where θ is the angle between c and H₀.

Fig. 4.

Spectra of the In(1) and In(2) in CeIrIn₅ at 6 K and constant field of 11.7 T, where θ is the angle between the c axis and the field. There are multiple satellite transitions for each site due to the quadrupolar splitting (37).

Fig. 5.

(A) The Knight shift of the In(1) in CeIrIn₅ versus angle at several different temperatures. Solid points are fits to Eq. 4. (B) K_aa (blue), K_cc (red) and K_ac (gray) versus χ_aa, χ_cc and χ_aa + χ_cc. Solid lines are linear fits as described in the text.

The strong angular dependence can be understood in terms of the tensor nature of the hyperfine coupling. For T > T^∗ the tensor is given by

[10]

in the basis defined by the tetragonal unit cell with unit vectors a, b, and c; and the susceptibility tensor is

[11]

We therefore have

[12]

where θ is the angle between H₀ and the c-axis, , , and . The solid lines in Fig. 5A are fits to this equation, and Fig. 5B shows K_aa, K_cc and K_ac versus χ_aa, χ_cc , and χ_aa + χ_cc, respectively. The fitted values are given in Table 2. The solid lines are fits that yield the hyperfine couplings B_aa, B_cc, B_ac , and the orbital shifts , , and . The fact that B_ac ≈ 0 within error bars implies that the principal axes of the hyperfine tensor coincide with those of the unit cell. On the other hand, implies that the orbital shift tensor is not diagonal in the crystal basis. Rather it is rotated by an angle of approximately 28° from the c direction, suggesting a possible role of directionality between the In 5p orbitals and the Ce 4f orbitals.

Table 2.

Measured hyperfine coupling constants in CeIrIn₅, CeRhIn₅ , and URu₂Si₂

Material	site	Baa (kOe/μB)	Bcc (kOe/μB)	Bac (kOe/μB)	(%)	(%)	(%)
CeIrIn5	In(1)	11.3(1)	13.8(1)	0.4(3)	1.1(2)	0.4(2)	−0.5(2)
CeRhIn5	In(1)	-	21.4(5)	-	-	0.64(5)	-
URu2Si2	Si	-	4.3(1)	-	-	−0.07(1)	-

The emergence of the heavy fermion fluid is evident by computing K_HF(θ,T) = K(θ,T) - K(θ), where K(θ) is given by Eq. 12 and using the fitted parameters. This data in shown in Fig. 6 as well as fits to Eq. 9, which give a value of T^∗ = 31(5) K. From this data, it is clear that T^∗ does not vary as a function of angle, but rather the overall scale of the the quantity decreases with angle until it reaches approximately zero by θ = 90°. The angular dependence clearly indicates that K_HF(θ,T) does not simply vanish for the perpendicular direction but decreases gradually as a function of orientation in a continuous fashion. The constant is anisotropic, but T^∗ is not. In fact, is proportional to A–B and is therefore a tensor quantity itself, which is clearly seen in the angular dependence shown in the inset of Fig. 6. The solid line is a fit to . It is likely that the anisotropy arises primarily because of the hyperfine couplings, but it is not possible to rule out an additional contribution to the anisotropy from the quantity χ_cf + χ_cc. The fact that T^∗ does not depend on orientation suggests that the field H₀ has little effect on the onset of coherence.

Fig. 6.

The heavy electron component of the shift in CeIrIn₅ for the In(1) as a function of angle and temperature. The solid lines are fits to Eq. 9. Inset: The fitted value versus angle. The solid line fit, described in the text, indicates the tensor nature of the hyperfine coupling A.

Superconducting Order

A key question in Kondo lattice materials is how the emergent heavy electron fluid and the remaining local moments interact to give rise to long range ordered states. Several materials exhibit a superconducting ground state, and it has long been assumed that the condensate forms from heavy electron quasiparticles. Recently we showed that this interpretation is supported by direct measurements of the Knight shift in the superconducting state (12). Fig. 7 displays K(T) and K_HF(T) for the In and Co sites in CeCoIn₅. Surprisingly, the total Knight shift K(T) increases for some of the sites below T_c, and decreases for others. The normal expectation is that the spin susceptibility decreases below T_c in a superconductor because the Cooper pairs have zero spin. Application of the two-fluid analysis indicates that the heavy electron component, K_HF(T), decreases as expected for a superconducting condensate. In fact, the data are best fit to a d-wave gap with value Δ(0) = 4.5 k_BT_c, in agreement with specific heat measurements. These results strongly suggest that the superconducting condensate arises as an instability of the heavy electron fluid.

Fig. 7.

(Upper) Knight shift of the In(1), In(2) and Co in CeCoIn₅ in the paramagnetic state for field in the ab plane. For In(2) there are two distinct crystallographic positions depending on the direction of the field. (Lower) The heavy electron susceptibility K_HF in the superconducting state. The solid lines are calculations as discussed in the text. Data are reproduced from ref. 12.

Hidden Order

Another well-known heavy fermion system with long-range order at low temperature is URu₂Si₂, which exhibits a phase transition to a state with an unknown order parameter at T_HO = 17.5 K (22). This unusual phase has been studied for the past two decades and numerous theoretical order parameters have been proposed. Although the nature of the hidden order parameter remains unknown, extensive work has suggested that it has an itinerant nature and involves some type of Fermi surface instability (23). We have measured the Knight shift of the ²⁹Si in URu₂Si₂ in order to investigate the effect of hidden order on K_HF.

Polycrystalline samples of URu₂Si₂ were synthesized by arc melting in a gettered argon atmosphere, and an aligned powder was prepared in an epoxy matrix by curing a mixture of powder and epoxy in an external magnetic field of 9 T. Aligned powder samples are useful to enhance the surface-to-volume ratio and hence the NMR sensitivity. NMR measurements were carried out using a high homogeneity 500 MHz (11.7 T) Oxford Instruments magnet. The ²⁹Si(I = 1/2) (natural abundance 4.6%) spectra were measured by spin echoes, and the signal-to-noise ratio was enhanced by summing several (approximately 100) echoes acquired via a Carr–Purcell–Meiboom–Gill pulse sequence (24). In this field, the hidden order transition is suppressed to 16 K (25). The spin lattice relaxation rate () was measured as a function of the angle between the alignment axis and the magnetic field H₀ in order to properly align the sample, because is a strong function of orientation with a minimum for H₀‖c. Spectra are shown in Fig. 8, and the Knight shift along the c direction, K_c, is shown in Fig. 9. The total magnetic susceptibility, χ_c(T), is shown in Figs. 8 and 9 as a solid line. χ_c exhibited little or no field dependence in the temperature range of interest. The inset of Fig. 9 displays K_c versus χ. There is a clear anomaly that develops around 75 K, in agreement with previous data from Bernal. However, in contrast to previous measurements, we find that below T^∗ the Knight shift turns upward rather than downward (26).

Fig. 8.

Spectra of the ²⁹Si in URu₂Si₂ as a function of temperature. Color intensity (arb units) corresponds to the integral of the NMR spin echo. The solid black line is the bulk susceptibility.

Fig. 9.

The Knight shift of the Si (colored circles) and the susceptibility (solid line) in URu₂Si₂ for field along the c direction. The inset shows K versus χ.

In order to fit the Knight shift data and extract the temperature dependence of K_HF, we have fit both the K_c versus χ data and the K_HF versus T data simultaneously by minimizing the joint reduced . Individually, and provide the best fits to the two datasets, but because K_HF(T) depends on K₀ and B, is an implicit function of these fit parameters as well. The best fit is shown in the inset of Fig. 9. The fit parameters, listed in Table 2 are close to the values originally reported by Kohori (27). K_HF(T) is shown in Fig. 10, and the best fit to Eq. 9 yields T^∗ = 73(5) K. The data clearly indicate that K_HF(T) grows monotonically down to the hidden ordering temperature at 16 K and is well described by the Yang–Pines scaling formula.

Fig. 10.

K_HF(T) versus T for URu₂Si₂ (left) and CeRhIn₅ (right). The solid black line are best fits to Eq. 9. In URu₂Si₂ K_HF grows until the sudden onset of hidden order, whereas in CeRhIn₅ K_HF deviates from the scaling form below 8 K indicating the relocalization of the local moments.

Because the f electrons in URu₂Si₂ are strongly hybridized at low temperatures, the Knight shift and the susceptibility recover a one-component picture below a delocalization temperature T_L (4). This allows us to determine the values of K₀, A and B. The susceptibility of the two components below T^∗ hence can be subtracted and the results are plotted in Fig. 11. The local component also follows the mean-field prediction for a hybridized spin liquid (4), χ_l ∼ f_l/(T + T_l + f_lθ), with f₀ = 1.6, θ = 175 K given by the RKKY coupling and T_l = 70 K by local hybridization.

Fig. 11.

Two-fluid analysis of the magnetic susceptibility in URu₂Si₂ by using the one-component behavior above T^∗ and below T_L. The solid lines are fit to the scaling formula of the Kondo liquid and the mean-field formula of the hybridized spin liquid for T > T_HO and the Yosida function for T < T_HO following the BCS prediction.

Below the hidden order transition temperature K_HF decreases dramatically and saturates at a value ∼K_HF(T_HO)/3 at T ∼ 4 K. This result suggests that, like the superconductivity in CeCoIn₅, the hidden order phase in URu₂Si₂ emerges from the itinerant heavy electron quasiparticles and is not a phenomenon associated with local moment physics (28). The partial suppression of K_HF is also consistent with specific heat measurements which indicate that approximately 40% of the Fermi surface remains ungapped below the hidden order transition (29). Recent theoretical work has suggested the presence of a psuedogap in a range of 5–10 K above the hidden order transition (30). If one neglects the single outlying data point at 16 K, then the data may indicate a subtle change of scaling below approximately 23 K. However, we do not have sufficient precision to make any conclusive statements about the pseudogap.

The Knight shift anomaly K_HF below T_HO also follows the BCS prediction,

[13]

where f(E) is the Fermi distribution function and is the density of states. The gap function Δ(T) has a mean-field temperature dependence,

[14]

where Δ(0) ∼ 4.0 k_BT_HO is the gap amplitude determined by STM experiments. We find a good fit to the Kondo liquid susceptibility below T_HO in Fig. 11.

Antiferromagnetism and Relocalization

In order to investigate the response of K_HF to long-range order we have measured the Knight shift of the ¹¹⁵In(1) site in CeRhIn₅. CeRhIn₅ is an antiferromagnet at ambient pressure with an ordered moment of 0.4 μ_B and T_N = 3.8 K (31). Single crystals were grown in In flux using standard flux growth techniques. The transition of the ¹¹⁵In (I = 9/2) was observed for the In(1) by Hahn echoes, and the alignment of the crystal was confirmed by the splitting of the quadrupolar satellites (32). The Knight shift is shown in Fig. 12. There is a clear anomaly that develops at 17 K, in contrast to a previous report of 12 K (11). In this case the previous data consisted of only six data points over a limited range and did not have the precision to discern the anomaly. By taking more data points at closely spaced intervals, we find K_c increases above the extrapolated value below T^∗, in contrast to that in the CeIrIn₅ and CeCoIn₅ (see Fig. 1). The best fits to the data yield hyperfine couplings that are similar to those in CeIrIn₅ (see Table 2), and we find that T^∗ = 18(1) K for CeRhIn₅.

Fig. 12.

The Knight shift of the In(1) (colored circles) and the susceptibility (solid line) in CeRhIn₅ for field along the c direction. The inset shows K versus χ.

The temperature dependence of the emergent heavy fermion component, K_HF(T) is shown in Fig. 10B for CeRhIn₅. In contrast to the scaling behavior observed in CeIrIn₅, CeCoIn₅ , and URu₂Si₂, K_HF does not follow the Yang–Pines scaling formula down to the ordering temperature, T_N. Rather, K_HF begins to deviate below 9 K, exhibits a maximum at 8 K and then decreases down to T_N. This temperature of the breakdown of scaling corresponds well with the onset of antiferromagnetic correlations observed by inelastic neutron scattering and NMR spin lattice relaxation measurements (33, 34). The physical picture that emerges is that the local moments begin to hybridize below T^∗, but their spectral weight is only gradually transferred to the heavy electron fluid. At 9 K, the partially screened local moments begin to interact with one another and spectral weight is transferred back. This relocalization behavior was first observed in the antiferromagnet CePt₂In₇ (13). The fact that K_HF remains finite at T_N suggests that the ordered local moments still remain partially screened (4).

It is interesting to compare the behavior of the heavy electron Knight shift observed in URu₂Si₂ and CeRhIn₅ with other Kondo lattice materials that undergo long-range order. Fig. 13 shows the scaled K_HF versus T/T^∗ for URu₂Si₂, CeRhIn₅, CeCoIn₅, and the antiferromagnet CePt₂In₇ (T_N = 5.2 K, T^∗ ≈ 40 K) (11, 13). The onset of hybridization at T^∗ is identical for all the compounds, but the low temperature behavior depends critically on the ground state. In both CeCoIn₅ and URu₂Si₂ the spectral weight appears to be transferred continuously to the heavy electron fluid all the way down to the ordering temperature. Similar behavior is present in CeIrIn₅ (T_c = 0.4 K, not shown) down to 2 K. This behavior is surprising, because one might expect that the hidden order and/or superconductivity would emerge from a fully formed heavy electron state. Such a state would be characterized by a temperature independent K_HF, but that is clearly not the case for either material. In fact, the only known cases where K_HF appears to plateau at a constant value are CeSn₃ and Ce₃Bi₄Pt₃, neither of which exhibit long-range order (11).

Fig. 13.

K_HF(T) (normalized) versus T/T^∗ for CeCoIn₅, CePt₂In₇, URu₂Si₂, and CeRhIn₅. Data points in the superconducting and hidden order states are lighter shades. The solid black line is Eq. 9, and the vertical arrows indicate T_N for CePt₂In₇ and CeRhIn₅. Data for the CeCoIn₅ and CePt₂In₇ are reproduced from refs. 12 and 13.

The relocalization behavior in CeRhIn₅ and CePt₂In₇ contrasts starkly with that of the URu₂Si₂ and CeCoIn₅, where K_HF exhibits a peak in the paramagnetic state. In CePt₂In₇ K_HF was determined via ¹⁹⁵Pt NMR in a powder sample, thus the precision was not as high as for a single crystal in which the full susceptibility tensor can be measured (13). The new results in the CeRhIn₅ reported here, however, were indeed acquired in a single crystal and confirm that relocalization behavior appears to be a common feature of antiferromagnetic materials.

Conclusions

We have measured the emergence of the Kondo liquid and investigated its response to long range order in several heavy fermion compounds. In the superconductors CeCoIn₅ and CeIrIn₅ as well as the hidden order compound URu₂Si₂, K_HF rises continuously below T^∗ down to the ordering temperature without saturation. In the antiferromagnets CeRhIn₅ and CePt₂In₇ the Kondo liquid relocalizes as a precursor to long-range order of partially screened local moments. Much of this behavior can be understood in the the framework of hybridization effectiveness introduced in the context of the two-fluid phenomenology (4). In this picture K_HF(T) ∼ f₀, where f₀ is a measure of the collective hybridization that reduces the magnitude of the local f moments, and f₀ = 1 corresponds to complete hybridization. CeCoIn₅, CeIrIn₅, and URu₂Si₂ correspond to materials with f₀ > 1, whereas CeRhIn₅ and CePt₂In₇ have f₀ < 1. Although the results reported here do not directly address issues of non-Fermi liquid behavior and the presence of quantum critical phenomena in Kondo lattice systems, they do have important implications for theory (35, 36). The dual nature of the coexisting local moments and the heavy electron fluid that exists in these materials below T^∗ suggests that the physics cannot be described in terms of a single electronic degree of freedom. Rather, the interactions between the local moments, the heavy electron quasiparticles, and their relative spectral weight should be taken into account.