Long range order and two-fluid behaviorin heavy electron materialsKent R. Shirera, Abigail C. Shockleya, Adam P. Dioguardia, John Crockera, Ching H. Lina, Nicholas apRoberts-Warrena,David M. Nissona, Peter Klavinsa, Jason C. Cooleyb, Yi-feng Yangc, and Nicholas J. Curroa,1

aDepartment of Physics, University of California, Davis, CA 95616; bLos Alamos National Laboratory, Los Alamos, NM 87545; and cBeijing NationalLaboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China

Edited by Malcolm R. Beasley, Stanford University, Stanford, CA, and approved August 31, 2012 (received for review June 12, 2012)

The heavy electron Kondo liquid is an emergent state of condensedmatter that displays universal behavior independent of materialdetails. Properties of the heavy electron liquid are best probed byNMR Knight shift measurements, which provide a direct measureof the behavior of the heavy electron liquid that emerges belowthe Kondo lattice coherence temperature as the lattice of localmoments hybridizes with the background conduction electrons.Because the transfer of spectral weight between the localized anditinerant electronic degrees of freedom is gradual, the Kondo liquidtypically coexists with the local moment component until the ma-terial orders at low temperatures. The two-fluid formula capturesthis behavior in a broad range of materials in the paramagneticstate. In order to investigate two-fluid behavior and the onset andphysical origin of different long range ordered ground states inheavy electron materials, we have extended Knight shift measure-ments to URu2Si2, CeIrIn5, and CeRhIn5. In CeRhIn5 we find thatthe antiferromagnetic order is preceded by a relocalization of theKondo liquid, providing independent evidence for a local momentorigin of antiferromagnetism. In URu2Si2 the hidden order is shownto emerge directly from the Kondo liquid and so is not associatedwith local moment physics. Our results imply that the nature ofthe ground state is strongly coupled with the hybridization in theKondo lattice in agreement with phase diagram proposed by Yangand Pines.

nuclear magnetic resonance ∣ heavy fermion ∣ hyperfine couplings

Competition between different energy scales gives rise to a richspectrum of emergent ground states in strongly correlated

electron materials. In the heavy fermion compounds, a lattice ofnearly localized f electrons interacts with a sea of conductionelectrons, and depending on the magnitude of this interaction dif-ferent types of long-range order may develop at low temperatures(1). The Kondo lattice model strives to capture the essential phy-sics of heavy fermion materials by considering the various mag-netic interactions between the conduction electron spins, Sc, andthe local moment spins, Sf (2). Different ground states canemerge depending on the relative strengths of the interaction, J,between Sc and Sf and the intersite interaction, Jf f , between theSf spins (3, 4). Much of the physics of the phase diagram is drivenby a quantum critical point, which separates long-range orderedground states from those in which the local moments have fullyhybridized with the conduction electrons to form itinerant stateswith large effective masses and large Fermi surfaces (5). In sev-eral materials quantum critical fluctuations give rise to anoma-lous non-Fermi liquid behavior in various bulk transport andthermodynamic quantities (6–8).

In recent years, evidence has emerged that in the high tem-perature disordered phase the electronic degrees of freedomsimultaneously exhibit both itinerant and localized behavior (9).Below a temperature T � that marks the onset of hybridization orlattice coherence, several experimental quantities exhibit a tem-perature dependence that is well described by a fluid of hybri-dized heavy quasiparticles coexisting with partially screenedlocal moments (3, 10). In the two-fluid picture, the transfer of

spectral weight from the local moments to the heavy electronsis described by a quantity f ðTÞ (10). Above a temperature T �, thelocal moments and conduction electrons remain uncoupled. Be-low T � the local moments gradually dissolve into the hybridizedheavy electron fluid as the temperature is lowered. This crossovercan be measured directly via nuclear magnetic resonance (NMR)Knight shift experiments (11). The contribution to the Knightshift from the heavy electrons, KHF , exhibits a universal logarith-mic divergence with decreasing temperature below T � in theparamagnetic state. In superconducting CeCoIn5 this scaling per-sists down to Tc indicating that the condensate emerges from theheavy electron degrees of freedom (12). Relatively little is known,however, about how this scaling behaves in materials with othertypes of ordered ground states. Here we report Knight shift datain both URu2Si2 and CeRhIn5. In the former the scaling persistsdown to the onset of hidden order at THO; but in the latter theheavy electron spectral weight reverses course and the localmoments relocalize above the Neel temperature, TN , similar tothe antiferromagnet CePt2In7 (13). These results imply generallythat the heavy electron fluid is either unstable to a Fermi surfaceinstability such as hidden order or superconductivity or collapsesto form ordered local moments.

Knight Shift AnomaliesThe Zeeman interaction of an isolated nuclear spin I in an exter-nal magnetic field H0 is given by HZ ¼ γℏI · H0, where γ is thegyromagnetic ratio and ℏ is the reduced Planck constant. In thiscase the nuclear spin resonance frequency is given by the Larmorfrequency ωL ¼ γH0. In condensed matter systems the nuclearspins typically experience a hyperfine coupling to the electronspins, Hhyp ¼ γℏgμB I · A · S, where S is the electron spin, A isthe hyperfine coupling g is the g-factor and μB is the Bohr mag-neton. The hyperfine coupling is in general a tensor that dependson the details of the bonding and quantum chemistry of the ma-terial of interest. S can be the unpaired spin moment of an un-filled shell, the net conduction electron spin of a Fermi seapolarized by a magnetic field, or the spin-orbit coupled momentof a localized electron. S may or may not be located on the sameion as the nuclear moment; the former is referred to as on-sitecoupling and the latter is transferred coupling. In the paramag-netic state the electron spin is polarized by the external fieldS ¼ χ · H0∕gμB, where χ is the susceptibility. Thus the nuclearspin experiences an effective Hamiltonian:

HZ þHhyp ¼ γℏI · ð1þKÞ · H0; [1]

Author contributions: P.K. and N.J.C. designed research; K.R.S., A.C.S., A.P.D., J.C., C.H.L.,N.a.-W., D.M.N., and J.C.C. performed research; K.R.S., A.C.S., Y.-f.Y., and N.J.C. analyzeddata; and N.J.C. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

See Commentary on page 18241.1To whom correspondence should be addressed. E-mail: curro@physics.ucdavis.edu.

See Author Summary on page 18249 (volume 109, number 45).

www.pnas.org/cgi/doi/10.1073/pnas.1209609109 PNAS ∣ Published online September 24, 2012 ∣ E3067–E3073

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where the Knight shift tensor K ¼ A · χ is unit-less. The reso-nance frequency is given by ω ¼ ωLð1þKðθ; ϕÞÞ, whereKðθ; ϕÞ ¼ H0 · A · χ · H0∕H 2

0 depends on the polar angles θand ϕ that describe the relative orientation of the field with re-spect to the principal axes of the Knight shift tensor. For the iso-tropic case the shift K ¼ Aχ is independent of direction. Strictlyspeaking the Knight shift was originally defined as the paramag-netic shift from the Pauli susceptibility in metals but in practicerefers to any shift K arising from the electronic susceptibility.

The magnetic susceptibility and the Knight shift can be mea-sured independently. One can extract the hyperfine couplingA byplotting the Knight shift versus the susceptibility with tempera-ture as an implicit parameter [such a plot is traditionally knownas a Clogston–Jaccarino plot (14)]. If there is a single electronicspin component that gives rise to the magnetic susceptibility, thenthe Clogston–Jaccarino plot is a straight line with slope A andintercept K0. K0 is a temperature independent offset that isusually given by the orbital susceptibility and diamagnetic contri-butions (15).

Breakdown of Linearity and Failure of Local Pictures.All known heavyfermion materials exhibit a Knight shift anomaly where the Clog-ston–Jaccarino plot deviates from linearity below an onset tem-perature, T � (11). This phenomenon is also visible in a plotcomparing K and χ versus temperature, as seen in Fig. 1, whereT � marks the temperature below whichK no longer tracks χ. Thisbreakdown can arise from the presence of impurity phases, whichcontribute to the bulk susceptibility but not to the Knight shift.However, Knight shift anomalies occur ubiquitously in singlephase highly pure heavy fermion materials and are intrinsic phe-nomena. Several different theories have been proposed to explainthese anomalies. These theories generally fall into two categories:(i) a temperature dependent hyperfine couplingAðTÞ or (ii) mul-tiple spin degrees of freedom. It has been argued that A acquiresa temperature dependence either because of Kondo screening(16) or because of different populations of crystal field levels ofthe 4f(5f) electrons in these materials (17). A major flaw in suchan argument is that the hyperfine coupling is determined by largeenergy scales involving exchange integrals between differentatomic orbitals, and because the Kondo and/or crystal field inter-actions are several orders of magnitude smaller they cannot giverise to large perturbations in the hyperfine couplings (18). Further-more, in materials such as the CeMIn5 system both the Knightshift anomaly and the crystalline electric field interaction are wellcharacterized and are clearly uncorrelated (see Table 1) (19).

Two Spin Components. Because Kondo lattice materials have bothlocalized f electrons and itinerant conduction electrons, a theorythat includes different hyperfine couplings to multiple spindegrees of freedom fits better. The two-fluid picture correctlypredicts the temperature dependence of the Knight shift anoma-lies seen experimentally in heavy fermion materials. In the Kondolattice model the local moment spins, Sf interact with a sea ofconduction electron spins, Sc through a Kondo interaction:

HK ¼ J∑i

Sf ðriÞ · Sc þ∑ij

Jf f ðrijÞSf ðriÞ · Sf ðrjÞ [2]

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

Hhyp ¼ γℏgμBI · ðASc þ BSf Þ [3]

where A and B are the hyperfine couplings to the conductionelectron and local moment spins, respectively (11). The suscept-ibility is the sum of three contributions χ ¼ χcc þ 2χcf þ χf f ,where χαβ ¼ hSαSβi with α; β ¼ c; f . In the paramagnetic statehSci ¼ ðχcc þ χcf ÞH0 and hSf i ¼ ðχf f þ χcf ÞH0, thus the Knightshift is given by:

K ¼ K0 þAχcc þ ðAþ BÞχcf þ Bχf f : [4]

The Knight shift weighs the different correlation functions sepa-rately than the total susceptibility, which leads to important con-sequences if the correlation functions have different temperaturedependencies. K ∼ χ if and only if A ¼ B. If A ≠ B, then theClogston–Jaccarino plot will no longer be linear. By measuringK and χ independently, one can exploit this difference to extractlinear combinations of these individual susceptibilities. No otherexperimental technique can access these susceptibilities.

Free Spin Model. The general solution for the susceptibilities χαβof the Kondo lattice problem has not been solved to date. It isuseful, however, to consider a simplified model. Consider thecase of two free S ¼ 1

2spins Sc and Sf that experience a Heisen-

berg coupling J. The Hamiltonian for this spin system in the pre-sence of an external magnetic field H0 is given by:

H ¼ gcμBSc · H0 þ gfμBSf · H0 þ JSc · Sf ; [5]

where μB is the Bohr magneton and gc;f are the g factors of thetwo different spins. This simplified model is clearly an incorrectphysical description of the Kondo lattice problem where the spinsare not free but part of a complex many-body problem involvingthe Fermi sea and various intersite interactions. This model is thesimplest case, however, which captures the essence of the Knightshift anomaly and can be solved exactly.

The free energy is given by F ¼ −kBT lnZ, where Z is thepartition function; the susceptibility of the system is given bythe second derivative of the free energy with respect to field:

Table 1. Crystal field levels and coherence temperatures in theCeMIn5 materials

Material T � (K) Δ (K) (from ref. 19)

CeCoIn5 42 100CeRhIn5 18(1) 80CeIrIn5 31(5) 78

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 ¼ Rh ,Ir and Co.T � is the temperature where the two quantities no longer are proportional toone another, and is clearly material dependent. In some cases K appears toexceed χ and in others K decreases.

E3068 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1209609109 Shirer et al.

χ ¼ ∂ 2F∂H 2 jH→0. The susceptibility is given as the sum of three

contributions:

χccðxÞ ¼�g2cμ

2B

4kBT

�2ðex þ x − 1Þð3þ exÞx [6]

χf f ðxÞ ¼� g2f μ

2B

4kBT

�2ðex þ x − 1Þð3þ exÞx [7]

χcf ðxÞ ¼ −�gcgfμ

2B

4kBJ

�2ðex − x − 1Þ

3þ ex[8]

where x ¼ J∕kBT. The temperature dependence of these threesusceptibilities is shown in Fig. 2. In the high temperature limit(T ≫ J∕kB) χcc and χf f exhibit Curie behavior and χcf → 0 asexpected. For low temperature (T ≲ J∕kB) the behavior is mod-ified by the tendency to form a singlet ground state. As shown inFig. 2 χcc, χf f and χcf change dramatically below T ≈ J∕kB andsaturate at low temperatures. The Knight shift, given by Eq. 4, isshown as well in Fig. 2. Clearly K tracks χ for T ≫ J∕kB butdeviates below T ≈ J∕kB, creating an anomaly in the Clogston–Jaccarino plot in Fig. 3. The breakdown of linearity arises becausejχcf j increases below T ≈ J∕kB and has a different temperaturedependence than χcc and χf f .

Two-Fluid Model. The three individual susceptibilities χαβ willexhibit different temperature dependencies in a Kondo latticethan the free spin model considered above. In particular, at hightemperatures χcc is given by the temperature-independent Paulisusceptibility of the conduction electrons, and χf f is given by aCurie–Weiss susceptibility of the local moments. Therefore abovea crossover temperature, T �, χf f dominates and K ¼ K0 þ Bχ.Below T � χcf becomes significant and the growth of this compo-nent can be measured by the quantity KHF ¼ K −K0 − Bχ ¼ðA − BÞðχcf þ χccÞ. The two-fluid model postulates that KHF isproportional to the susceptibility of the heavy electron fluid;its temperature dependence probes both growth of hybridizationand its relative spectral weight (3, 10). Empirically it has beenfound that in the paramagnetic phase of a broad range of mate-rials, KHFðTÞ varies as:

KHFðTÞ ¼ K 0HFð1 − T∕T �Þ3∕2½1þ logðT �∕TÞ� [9]

where K 0HF is a constant (10). In other words, KHF exhibits a

universal scaling with the quantity T∕T �, where T � and K 0HF

are material-dependent quantities. K 0HF is proportional to

A–B and can be either positive or negative. T � agrees well withseveral other experimental measurements of the coherence tem-perature of the Kondo lattice (10). The Knight shift thus providesa direct probe of the susceptibility of the emergent heavy electronfluid in the paramagnetic state. There are, however, severalpieces of missing information regarding the behavior of the heavyelectron component that need to be addressed. For example, theKnight shift anomaly is strongly anisotropic in some materials,and it has not been clear whether this is an intrinsic feature ofthe heavy electron fluid or rather a reflection of the anisotropyof the hyperfine couplings (11). Furthermore, the Yang–Pinesscaling works remarkably well for temperatures down to approxi-mately 10% of T �, but until recently there has been relativelylittle information about the evolution of the heavy electron fluidat lower temperature particularly when long-range order devel-ops. We address both of these issues in turn below.

Anisotropy. One of the striking features of the emergence ofthe heavy electron component in Kondo lattice materials is theanisotropy of the Knight shift anomaly. For example, in CeCoIn5the Knight shift of the In(1) site exhibits a strong anomaly forH0‖c but no anomaly for H0⊥c (20). In order to probe thisbehavior in greater detail, we have measured the full Knight shift

1.0

0.5

0.0

-0.5

-1.0

χ/µ B

2 g c2

3.02.01.00.0T/J

0.15

0.10

0.05

0.00

-0.05

-0.10

K (arb units)χ

χcc

χff

χcf

K

Fig. 2. The components of the magnetic susceptibility for the model spinsystem. The Knight shift begins to deviate from the bulk susceptibility belowa temperature T ≈ J. In this case, we have used gf∕gc ¼

ffiffiffi2

p, and B∕A ¼ 1.5. If

gf ¼ gc , then χðT → 0Þ → 0, as expected for a singlet ground state.

0.10

0.08

0.06

0.04

0.02

0.00

K (

arb

units

)

0.60.40.20.0

χ/µB2gc

2

Fig. 3. Knight shift versus the bulk susceptibility for the model spin systemwith temperature as an implicit parameter. The Knight shift begins to deviatefrom the bulk susceptibility below a temperature T ≈ J∕kB. In this case, wehave used gf∕gc ¼

ffiffiffi2

p, and B∕A ¼ 1.5.

Inte

nsity

13012011010090Frequency (MHz)

In(1) In(2)

90°

82°

55°

44°

28°

Fig. 4. Spectra of the In(1) and In(2) in CeIrIn5 at 6 K and constant field of11.7 T, where θ is the angle between the c axis and the field. There are multi-ple satellite transitions for each site due to the quadrupolar splitting (37).

Shirer et al. PNAS ∣ Published online September 24, 2012 ∣ E3069

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tensor in CeIrIn5. CeIrIn5 is isostructural to CeCoIn5 and is asuperconductor with Tc ¼ 0.4 K. A single crystal grown in Influx was mounted in a custom-made goniometer probe, andthe orientation of the c axis of the crystal was varied from 0to 90 degrees. 115In has spin I ¼ 9

2, and there are two different

crystallographic sites for In in this material; consequently thereare several different transitions as seen in Fig. 4. Spectra wereobtained as a function of orientation and temperature, and theKnight shift was extracted after correcting for the quadrupolarshift of the resonance (21). Fig. 5 shows K versus θ, where θis the angle between c and H0.

The strong angular dependence can be understood in terms ofthe tensor nature of the hyperfine coupling. For T > T � the ten-sor is given by

B ¼Baa Bab Bac

Bab Bbb Bbc

Bac Bbc Bcc

0@

1A [10]

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

χ ¼χaa 0 0

0 χaa 0

0 0 χcc

0@

1A: [11]

We therefore have

KðθÞ ¼ Kaa sin2 θþKcc cos2 θþKac sin θ cos θ [12]

where θ is the angle between H0 and the c-axis, Kaa ¼ K 0aa þ

Baaχaa, Kcc ¼ K 0cc þ Bccχcc, and Kac ¼ K 0

ac þ Bacðχaa þ χccÞ.The solid lines in Fig. 5A are fits to this equation, and Fig. 5Bshows Kaa, Kcc and Kac versus χaa, χcc , and χaa þ χcc, respec-tively. The fitted values are given in Table 2. The solid linesare fits that yield the hyperfine couplings Baa, Bcc, Bac , andthe orbital shifts K 0

aa, K 0cc , and K 0

ac. The fact that Bac ≈ 0 withinerror bars implies that the principal axes of the hyperfine tensorcoincide with those of the unit cell. On the other hand, K 0

ac ≠ 0implies that the orbital shift tensor is not diagonal in the crystalbasis. Rather it is rotated by an angle of approximately 28° fromthe c direction, suggesting a possible role of directionality be-tween the In 5p orbitals and the Ce 4f orbitals.

The emergence of the heavy fermion fluid is evident by com-puting KHFðθ; TÞ ¼ Kðθ; TÞ − KðθÞ, where KðθÞ is given byEq. 12 and using the fitted parameters. This data in shown inFig. 6 as well as fits to Eq. 9, which give a value of T � ¼31ð5Þ 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 quantityK 0

HF decreases with angle until it reaches approximately zeroby θ ¼ 90°. The angular dependence clearly indicates thatKHFðθ; TÞ does not simply vanish for the perpendicular directionbut decreases gradually as a function of orientation in a contin-uous fashion. The constant K 0

HF is anisotropic, but T � is not.In fact, K0

HF is proportional to A–B and is therefore a tensorquantity itself, which is clearly seen in the angular dependenceshown in the inset of Fig. 6. The solid line is a fit to K 0

HF ¼K 0

HF;c cos2 θþK 0

HF;a sin2 θ. It is likely that the anisotropy arises

primarily because of the hyperfine couplings, but it is not possibleto rule out an additional contribution to the anisotropy from thequantity χcf þ χcc. The fact that T � does not depend on orienta-tion suggests that the field H0 has little effect on the onset ofcoherence.

Superconducting OrderA key question in Kondo lattice materials is how the emergentheavy electron fluid and the remaining local moments interactto give rise to long range ordered states. Several materials exhibita superconducting ground state, and it has long been assumedthat the condensate forms from heavy electron quasiparticles. Re-cently we showed that this interpretation is supported by directmeasurements of the Knight shift in the superconducting state(12). Fig. 7 displays KðTÞ and KHFðTÞ for the In and Co sitesin CeCoIn5. Surprisingly, the total Knight shift KðTÞ increasesfor some of the sites below Tc, and decreases for others. The nor-mal expectation is that the spin susceptibility decreases below Tcin a superconductor because the Cooper pairs have zero spin.Application of the two-fluid analysis indicates that the heavy elec-tron component, KHFðTÞ, decreases as expected for a supercon-ducting condensate. In fact, the data are best fit to a d-wave gapwith value Δð0Þ ¼ 4.5 kBTc, in agreement with specific heat mea-surements. These results strongly suggest that the superconduct-ing condensate arises as an instability of the heavy electron fluid.

Hidden OrderAnother well-known heavy fermion system with long-range orderat low temperature is URu2Si2, which exhibits a phase transitionto a state with an unknown order parameter at THO ¼ 17.5 K(22). This unusual phase has been studied for the past two dec-ades and numerous theoretical order parameters have beenproposed. Although the nature of the hidden order parameterremains unknown, extensive work has suggested that it has anitinerant nature and involves some type of Fermi surface instabil-ity (23).We have measured the Knight shift of the 29Si in URu2Si2in order to investigate the effect of hidden order on KHF .

Polycrystalline samples of URu2Si2 were synthesized by arcmelting in a gettered argon atmosphere, and an aligned powderwas prepared in an epoxy matrix by curing a mixture of powderand epoxy in an external magnetic field of 9 T. Aligned powdersamples are useful to enhance the surface-to-volume ratio andhence the NMR sensitivity. NMRmeasurements were carried outusing a high homogeneity 500 MHz (11.7 T) Oxford Instrumentsmagnet. The 29SiðI ¼ 1∕2Þ (natural abundance 4.6%) spectrawere measured by spin echoes, and the signal-to-noise ratio wasenhanced by summing several (approximately 100) echoesacquired 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 (T −1

1 ) was measured as afunction of the angle between the alignment axis and the mag-

Table 2. Measured hyperfine coupling constants in CeIrIn5, CeRhIn5 , and URu2Si2

Material site Baa ðkOe∕μBÞ Bcc ðkOe∕μBÞ Bac ðkOe∕μBÞ K0aa (%) K0

cc (%) K0ac (%)

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) -

4

2

0

-2

K (

%)

2520151050

χ (x10-3

emu/mol)

4.0

3.0

2.0K

(%

)

806040200

θ (deg)

A B

Fig. 5. (A) The Knight shift of the In(1) in CeIrIn5 versus angle at severaldifferent temperatures. Solid points are fits to Eq. 4. (B) Kaa (blue), Kcc

(red) and Kac (gray) versus χaa, χcc and χaa þ χcc . Solid lines are linear fits asdescribed in the text.

E3070 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1209609109 Shirer et al.

netic field H0 in order to properly align the sample, because T −11

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 di-rection, Kc, is shown in Fig. 9. The total magnetic susceptibility,χcðTÞ, is shown in Figs. 8 and 9 as a solid line. χc exhibited little orno field dependence in the temperature range of interest. Theinset of Fig. 9 displays Kc versus χ. There is a clear anomaly thatdevelops around 75 K, in agreement with previous data fromBernal. However, in contrast to previous measurements, we findthat below T � the Knight shift turns upward rather than down-ward (26).

In order to fit the Knight shift data and extract the temperaturedependence of KHF , we have fit both the Kc versus χ data andthe KHF versus T data simultaneously by minimizing the jointreduced χ2ðK0; B; K 0

HF; T�Þ ¼ χ2

1 ðK0; BÞ þ χ22 ðK 0

HF; T�Þ. Indi-

vidually, χ21 and χ2

2 provide the best fits to the two datasets,but becauseKHFðTÞ depends onK0 and B, χ2

2 is an implicit func-tion of these fit parameters as well. The best fit is shown in theinset of Fig. 9. The fit parameters, listed in Table 2 are close to the

values originally reported by Kohori (27). KHFðTÞ is shownin Fig. 10, and the best fit to Eq. 9 yields T � ¼ 73ð5Þ K. The dataclearly indicate that KHFðTÞ grows monotonically down to thehidden ordering temperature at 16 K and is well described bythe Yang–Pines scaling formula.

Because the f electrons in URu2Si2 are strongly hybridized atlow temperatures, the Knight shift and the susceptibility recover aone-component picture below a delocalization temperature TL(4). This allows us to determine the values of K0, A and B. Thesusceptibility of the two components below T � hence can be sub-tracted and the results are plotted in Fig. 11. The local compo-nent also follows the mean-field prediction for a hybridized spinliquid (4), χl ∼ f l∕ðT þ Tl þ f lθÞ, with f 0 ¼ 1.6, θ ¼ 175 K givenby the RKKY coupling and Tl ¼ 70 K by local hybridization.

Below the hidden order transition temperature KHF decreasesdramatically and saturates at a value ∼KHFðTHOÞ∕3 at T ∼ 4 K.This result suggests that, like the superconductivity in CeCoIn5,the hidden order phase in URu2Si2 emerges from the itinerantheavy electron quasiparticles and is not a phenomenon associatedwith local moment physics (28). The partial suppression ofKHF isalso consistent with specific heat measurements which indicate

-2

-1

0

1

2.01.51.00.50.0T (K)

1.2

1.0

0.8

0.6

2.0

1.5

1.0

0.5

In(1)ab

In(2)perp

Coab

4

2

0

-2

KH

F (

%)

2.01.51.00.50.0T (K)

In(2)perp

Coab

115 K

(2) p

erp

(%

)

115 K

(1) a

b (

%)

59K

(1) a

b (

%)

Fig. 7. (Upper) Knight shift of the In(1), In(2) and Co in CeCoIn5 in the para-magnetic state for field in the ab plane. For In(2) there are two distinct crys-tallographic positions depending on the direction of the field. (Lower) Theheavy electron susceptibility KHF in the superconducting state. The solid linesare calculations as discussed in the text. Data are reproduced from ref. 12.

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

KH

F(θ

,T)

(%)

5 6 7 8 910

2 3 4 5

T (K)

0° 28° 44° 55° 82° 90°

-0.5-0.4-0.3-0.2-0.10.0

K0 H

F

(%)

806040200θ (deg)

Fig. 6. The heavy electron component of the shift in CeIrIn5 for the In(1) as afunction of angle and temperature. The solid lines are fits to Eq. 9. Inset: Thefitted value K 0

HF versus angle. The solid line fit, described in the text, indicatesthe tensor nature of the hyperfine coupling A.

100

80

60

40

20

0

Tem

pera

ture

(K

)

1.00.80.60.40.2Knight Shift (%)

14121086420χ( x10

-3emu/mole)

Fig. 8. Spectra of the 29Si in URu2Si2 as a function of temperature. Colorintensity (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 (solidline) in URu2Si2 for field along the c direction. The inset shows K versus χ.

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that approximately 40% of the Fermi surface remains ungappedbelow the hidden order transition (29). Recent theoretical workhas suggested the presence of a psuedogap in a range of 5–10 Kabove the hidden order transition (30). If one neglects the singleoutlying data point at 16 K, then the data may indicate a subtlechange of scaling below approximately 23 K. However, we do nothave sufficient precision to make any conclusive statements aboutthe pseudogap.

The Knight shift anomaly KHF below THO also follows theBCS prediction,

KanomðTÞ −Kanomð0Þ ∼Z

dE�−∂f ðEÞ∂E

�NðEÞ; [13]

where f ðEÞ is the Fermi distribution function and NðEÞ ¼jEj∕

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 − ΔðTÞ2

pis the density of states. The gap function

ΔðTÞ has a mean-field temperature dependence,

ΔðTÞ ¼ Δð0Þ tanh½2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðTHO∕T − 1

p�; [14]

where Δð0Þ ∼ 4.0 kBTHO is the gap amplitude determined bySTM experiments. We find a good fit to the Kondo liquid suscept-ibility below THO in Fig. 11.

Antiferromagnetism and RelocalizationIn order to investigate the response of KHF to long-range orderwe have measured the Knight shift of the 115Inð1Þ site inCeRhIn5. CeRhIn5 is an antiferromagnet at ambient pressurewith an ordered moment of 0.4μB and TN ¼ 3.8 K (31). Singlecrystals were grown in In flux using standard flux growth techni-ques. The 3

2↔ 1

2transition of the 115In (I ¼ 9∕2) was observed for

the In(1) by Hahn echoes, and the alignment of the crystal wasconfirmed by the splitting of the quadrupolar satellites (32). TheKnight shift is shown in Fig. 12. There is a clear anomaly thatdevelops 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 theanomaly. By taking more data points at closely spaced intervals,we find Kc increases above the extrapolated value below T �, incontrast to that in the CeIrIn5 and CeCoIn5 (see Fig. 1). The bestfits to the data yield hyperfine couplings that are similar to thosein CeIrIn5 (see Table 2), and we find that T � ¼ 18ð1Þ K forCeRhIn5.

The temperature dependence of the emergent heavy fermioncomponent, KHFðTÞ is shown in Fig. 10B for CeRhIn5. In con-trast to the scaling behavior observed in CeIrIn5, CeCoIn5 , andURu2Si2, KHF does not follow the Yang–Pines scaling formuladown to the ordering temperature, TN . Rather, KHF begins todeviate below 9 K, exhibits a maximum at 8 K and then decreasesdown to TN . This temperature of the breakdown of scaling cor-responds well with the onset of antiferromagnetic correlationsobserved by inelastic neutron scattering and NMR spin latticerelaxation measurements (33, 34). The physical picture thatemerges is that the local moments begin to hybridize below T �,but their spectral weight is only gradually transferred to the heavyelectron fluid. At 9 K, the partially screened local moments beginto interact with one another and spectral weight is transferredback. This relocalization behavior was first observed in the anti-ferromagnet CePt2In7 (13). The fact that KHF remains finite atTN suggests that the ordered local moments still remain partiallyscreened (4).

Fig. 10. KHFðTÞ versus T for URu2Si2 (left) and CeRhIn5 (right). The solidblack line are best fits to Eq. 9. In URu2Si2 KHF grows until the sudden onsetof hidden order, whereas in CeRhIn5 KHF deviates from the scaling formbelow 8 K indicating the relocalization of the local moments.

25

20

15

10

5

0

χc (x10

-3 emu/m

ol)

403020100Temperature (K)

8

6

4

2

0

Kc(

1) (

%) 9.0

8.0

7.0

6.0

Kc

(%)

24222018

χc (x10-3

emu/mol)

Fig. 12. TheKnight shift of the In(1) (colored circles) and the susceptibility (solidline) in CeRhIn5 for field along the c direction. The inset shows K versus χ.

0.6

0.4

0.2

0.0

K (

%)

10x10-3

86420χ (emu/mol U)

10x10-3

8

6

4

2

0

χ (e

mu/

mol

U)

100806040200T (K)

χ χ lχ h

Fig. 11. Two-fluid analysis of the magnetic susceptibility in URu2Si2 by usingthe one-component behavior above T � and below TL. The solid lines are fitto the scaling formula of the Kondo liquid and the mean-field formula ofthe hybridized spin liquid for T > THO and the Yosida function for T < THO

following the BCS prediction.

Fig. 13. KHF ðTÞ (normalized) versus T∕T � for CeCoIn5, CePt2In7, URu2Si2, andCeRhIn5. Data points in the superconducting and hidden order states arelighter shades. The solid black line is Eq. 9, and the vertical arrows indicateTN for CePt2In7 and CeRhIn5. Data for the CeCoIn5 and CePt2In7 are repro-duced from refs. 12 and 13.

E3072 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1209609109 Shirer et al.

It is interesting to compare the behavior of the heavy electronKnight shift observed in URu2Si2 and CeRhIn5 with other Kondolattice materials that undergo long-range order. Fig. 13 shows thescaled KHF versus T∕T � for URu2Si2, CeRhIn5, CeCoIn5, andthe antiferromagnet CePt2In7 (TN ¼ 5.2 K, T � ≈ 40 K) (11, 13).The onset of hybridization at T � is identical for all the com-pounds, but the low temperature behavior depends criticallyon the ground state. In both CeCoIn5 and URu2Si2 the spectralweight appears to be transferred continuously to the heavy elec-tron fluid all the way down to the ordering temperature. Similarbehavior is present in CeIrIn5 (Tc ¼ 0.4 K, not shown) down to2 K. This behavior is surprising, because one might expect thatthe hidden order and/or superconductivity would emerge froma fully formed heavy electron state. Such a state would be char-acterized by a temperature independent KHF , but that is clearlynot the case for either material. In fact, the only known caseswhere KHF appears to plateau at a constant value are CeSn3 andCe3Bi4Pt3, neither of which exhibit long-range order (11).

The relocalization behavior in CeRhIn5 and CePt2In7 con-trasts starkly with that of the URu2Si2 and CeCoIn5, where KHFexhibits a peak in the paramagnetic state. In CePt2In7 KHF wasdetermined via 195Pt NMR in a powder sample, thus the preci-sion was not as high as for a single crystal in which the fullsusceptibility tensor can be measured (13). The new results in theCeRhIn5 reported here, however, were indeed acquired in a sin-gle crystal and confirm that relocalization behavior appears to bea common feature of antiferromagnetic materials.

ConclusionsWe have measured the emergence of the Kondo liquid and inves-tigated its response to long range order in several heavy fermioncompounds. In the superconductors CeCoIn5 and CeIrIn5 as well

as the hidden order compound URu2Si2, KHF rises continuouslybelow T � down to the ordering temperature without saturation.In the antiferromagnets CeRhIn5 and CePt2In7 the Kondo liquidrelocalizes as a precursor to long-range order of partiallyscreened local moments. Much of this behavior can be under-stood in the the framework of hybridization effectiveness intro-duced in the context of the two-fluid phenomenology (4). In thispicture KHFðTÞ ∼ f 0, where f 0 is a measure of the collective hy-bridization that reduces the magnitude of the local f moments,and f 0 ¼ 1 corresponds to complete hybridization. CeCoIn5,CeIrIn5, and URu2Si2 correspond to materials with f 0 > 1,whereas CeRhIn5 and CePt2In7 have f 0 < 1. Although the re-sults reported here do not directly address issues of non-Fermiliquid behavior and the presence of quantum critical phenomenain Kondo lattice systems, they do have important implications fortheory (35, 36). The dual nature of the coexisting local momentsand the heavy electron fluid that exists in these materials belowT � suggests that the physics cannot be described in terms of asingle electronic degree of freedom. Rather, the interactionsbetween the local moments, the heavy electron quasiparticles,and their relative spectral weight should be taken into account.

ACKNOWLEDGMENTS.We thank S. Balatsky, P. Coleman, M. Graf, D. Pines, andJ. Thompson for stimulating discussions. Work at UC Davis on the URu2Si2material was supported by UCOP-TR01; the work on CePt2In7 was supportedby the National Nuclear Security Administration under the StewardshipScience Academic Alliances program through DOE Research Grant numberDOE DE-FG52-09NA29464; the work on CeIrIn5 was supported by theNational Science Foundation under Grant DMR-1005393. Work in Chinawas supported by the Chinese Academy of Sciences and NSF-China (GrantNo. 11174339). Los Alamos National Laboratory is operated by Los AlamosNational Security, LLC, for the National Nuclear Security Administration ofthe US Department of Energy under Contract DE-AC52-06NA25396.

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Shirer et al. PNAS ∣ Published online September 24, 2012 ∣ E3073

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