Nuclear magnetic resonance in Kondo lattice systems

Transcription

1 Reports on Progress in Physics REPORT ON PROGRESS Nuclear magnetic resonance in Kondo lattice systems To cite this article: Nicholas J Curro Rep. Prog. Phys. Manuscript version: Accepted Manuscript Accepted Manuscript is the version of the article accepted for publication including all changes made as a result of the peer review process, and which may also include the addition to the article by IOP Publishing of a header, an article ID, a cover sheet and/or an Accepted Manuscript watermark, but excluding any other editing, typesetting or other changes made by IOP Publishing and/or its licensors This Accepted Manuscript is IOP Publishing Ltd. During the embargo period (the month period from the publication of the Version of Record of this article), the Accepted Manuscript is fully protected by copyright and cannot be reused or reposted elsewhere. As the Version of Record of this article is going to be / has been published on a subscription basis, this Accepted Manuscript is available for reuse under a CC BY-NC-ND. licence after the month embargo period. After the embargo period, everyone is permitted to use copy and redistribute this article for non-commercial purposes only, provided that they adhere to all the terms of the licence Although reasonable endeavours have been taken to obtain all necessary permissions from third parties to include their copyrighted content within this article, their full citation and copyright line may not be present in this Accepted Manuscript version. Before using any content from this article, please refer to the Version of Record on IOPscience once published for full citation and copyright details, as permissions will likely be required. All third party content is fully copyright protected, unless specifically stated otherwise in the figure caption in the Version of Record. View the article online for updates and enhancements. This content was downloaded from IP address... on // at :

2 Page of CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT ROPP-.R Nuclear Magnetic Resonance in Kondo lattice systems Nicholas J. Curro Department of Physics, University of California, Davis, CA, USA curro@physics.ucdavis.edu October Abstract. Nuclear magnetic resonance has emerged as a vital tool to explore the fundamental physics of Kondo lattice systems. Because nuclear spins experience two different hyperfine couplings to the itinerant conduction electrons and to the local f moments, the Knight shift can probe multiple types of spin correlations that are not accessible via other techniques. The Knight shift provides direct information about the onset of heavy electron coherence and the emergence of the heavy electron fluid. Submitted to: Rep. Prog. Phys.. Introduction In the majority of materials there is at least one atomic site with an isotope with a nuclear spin that can be investigated with nuclear magnetic resonance (NMR). NMR can provide valuable information about the local magnetic and electronic environment of the nucleus through the hyperfine and quadrupolar couplings. Since it is a local probe, NMR can address questions such as phase purity, homogeneity of doped systems, and coexistence of multiples types of order parameters. NMR also probes the density of states at the Fermi level in metallic systems, and therefore is a valuable probe of phenomena in which an energy gap at the Fermi level opens at a phase transition. For example, the temperature dependence of the nuclear spin-lattice relaxation rate can reveal the presence or absence of nodes in the superconducting gap. Furthermore, the behavior of the NMR Knight shift below T c can indicate the symmetry of the spin part of the superconducting condensate wavefunction. As a result, NMR studies traditionally have focused on low temperatures to elucidate the symmetry of the order parameter in nearly all of the known heavy fermion superconductors [,,,,,, ]. In recent years, however, NMR has played an increasing role in measuring the behavior of the normal or paramagnetic state of Kondo lattice systems. In all known such materials, the NMR Knight shift is known to deviate from the bulk magnetic susceptibility below some onset temperature. This behavior is anomalous because

3 CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT ROPP-.R Page of Nuclear Magnetic Resonance in Kondo lattice systems E t Π ka ky a Π Π Π Π Π Π Π Π k x a Figure. (Left) Dispersion relation for a local moment hybridized with itinerant electrons in the absence of hybridization (dashed lines) and for weak finite hybridization (solid lines). The Fermi energy is indicated by the orange line, and the thin vertical lines indicate the Fermi wavevector. (Right) The D Fermi surface for no hybridization (dashed) and finite hybridization (solid). The dotted line indicates the path through k-space shown in the left figure. nominally these two quantities should be proportional to one another. The magnitude of the effect is material-dependent, as is the onset temperature. This anomalous behavior actually probes the dissolution of the local moments in the Kondo lattice and the formation of a coherent heavy fermion state at low temperature. This phenomenon is often described phenomenologically in terms of an Anderson lattice, in which the electronic dispersion is described by a weak hybridization between itinerant conduction electrons and a lattice of localized orbitals, illustrated in Fig.. This model captures the charge degrees of freedom, as seen recently in scanning tunnelling microscopy (STM) measurements []. However, the behavior of the spins has become clearer recently with the understanding of the Knight shift anomaly. At high temperatures, the f-moments behave as local moments coupled by an intersite interaction that may or may not give rise to long-range order at low temperature. However, at lower temperatures the system behaves as if it contains both local moments and heavy electrons simultaneously. The NMR Knight shift probes the correlation function S f S c between the local moments and conduction electron spins that accompanies the emergence of the heavy electron fluid below a crossover temperature, T, the Kondo lattice coherence temperature. Below this temperature the heavy electron fluid coexists with residual local moments over a broad range of temperatures, as described phenomenologically by the two-fluid model [, ]. In this report, we provide a general overview of NMR in condensed matter systems, including a detailed description of the interactions between the nuclear spins and the electronic degrees of freedom. This background is followed by a description of Knight

4 Page of CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT ROPP-.R Nuclear Magnetic Resonance in Kondo lattice systems shift studies in several heavy electron materials, as well as a case study of the heavy fermion material CeRhIn as a function of pressure. Finally, we review recent numerical studies of the relevant spin-spin correlation functions in the periodic Anderson model and show how the Knight shift anomaly is a natural consequence of the hybridization between the itinerant and local electronic degrees of freedom.. Nuclear Magnetic Resonance as a probe of condensed matter.. The hyperfine interaction One of the most important interaction for nuclei in strongly correlated systems is the hyperfine interaction, coupling the nuclear spins to the electron spins: H hyp = AÎ S = AÎzS z + A(Î+S + Î S + ), () where A is the hyperfine constant. This coupling is typically on the order of ev, which is much lower in energy than that of the electronic system. The hyperfine coupling is therefore a negligible perturbation to the electronic degrees of freedom. However, the hyperfine interaction does create a significant perturbation to the system of otherwise free nuclear spins. Consequently this coupling enables the nuclei to probe both the static susceptibility, χ, and the dynamical susceptibility, χ(q,ω), of the electron quasiparticles. The diagonal term AÎzŜz gives rise to a static shift of the resonance frequency. This hyperfine constant is often expressed as A/γ μ B in units of Oe/μ B,and typically varies from - koe/μ B. This unit is often more useful since it gives an estimate of the size of the hyperfine field that the electron spin creates at the nucleus. The Hamiltonian for a nucleus in an external field that experiences a hyperfine interaction is given by: H = ω L Î z + AÎzS z. () For temperatures T A/k B K, the electron and nuclear spins do not develop any coherence and therefore we can replace S z with its thermal averaged value S z = χh, where χ is the magnetic susceptibility of the electronic system. The Hamiltonian can be rewritten then as: H = ω L ( + K)Îz, () where the Knight shift, K, measures the percent shift of the resonance frequency from that of an isolated nucleus (ω = γh ). The Knight shift is given by: K = Aχ / γμ B. In a Fermi liquid, χ is given by the Pauli susceptibility, so K AN() is temperature independent. This scenario works well for simple metals such as Li and Na, yet there are many cases where the Knight shift is more complex. In Pt, for example, there are multiple hyperfine couplings to the d- and sp- bands, and hence several contributions to the total shift []. In many-electron atoms, there is also a core-polarization term, in which the

5 CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT ROPP-.R Page of Nuclear Magnetic Resonance in Kondo lattice systems core s electrons acquire a population difference between the up- and down-spin states. This difference arises because the orthogonal eigenstates of the many-electron atom get mixed by the perturbing influence of the external field. In practice, it is difficult to estimate the contribution of a core-polarization term versus a purely contact term []. As a result, hyperfine couplings are usually taken to be material dependent parameters. Furthermore, the hyperfine coupling is often not isotropic, but rather a tensor quantity such that H hyp = Î A S, () where the matrix A may or may not be diagonal in the crystal basis. In this case, the Knight shift becomes a function of the field direction, and it may be necessary to map out the angular dependence of K in order to discern the components of the tensor. Thus the nuclear spin experiences an effective Hamiltonian: H Z + H hyp = γ Î (I + K) H, () where the Knight shift tensor K = A χ is dimensionless. The resonance frequency is given by ω = ω L ( + K(θ, φ)), where K(θ, φ) =H A χ H /H 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. In the most general case, the hyperfine interaction in the unit cell basis is given by: A aa A ab A ac A = A ab A bb A bc () A ac A bc A cc and the susceptibility tensor is: χ = χ aa χ bb χ cc. () By Neumann s principle, the susceptibility tensor must have the symmetry of the bulk crystal, and is diagonal in the crystal basis. Therefore: where K(θ) =K aa sin θ + K cc cos θ + K ac sin θ cos θ () K aa = A aa χ aa cos φ + A ab (χ aa + χ bb )cosφsin φ + A bb χ bb sin φ () K ac = A ac (χ aa + χ cc )cosφ + A bc (χ bb + χ cc )sinφ () K cc = A cc χ cc. () The diagonal components of the hyperfine coupling tensor can be measured by comparing the Knight shift and the the magnetic susceptibility, which can be measured independently, and plotting the Knight shift versus the susceptibility with temperature as an implicit parameter (such a plot is traditionally known as a Clogston-Jaccarino plot []). If there is a single electronic spin component that gives rise to the magnetic

6 Page of CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT ROPP-.R Nuclear Magnetic Resonance in Kondo lattice systems Intensity Frequency (MHz) In() In() Figure. (Left) Unit cell for CeRhIn. Dashed lines indicate the nearest neighbor transferred hyperfine couplings to the Ce f moments at the In() and In() sites. (Right) Spectra of the In() and In() in CeIrIn at K and constant field of. 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. Reproduced from Ref. []. susceptibility, then the data will form a straight line with slope A and intercept K. K is a temperature independent offset that is usually given by the orbital susceptibility and diamagnetic contributions []. Hyperfine interactions can be both on-site and transferred to other lattice sites. Transferred hyperfine interactions are particularly important for materials with localized electrons. If the nucleus is located at the site of the local moment, then the on-site hyperfine coupling to the local moment can be quite large. In some cases this large coupling leads to a fast relaxation rate, which can potentially be so large that no FID or spin echo can be detected. This is particularly true for nuclei in lanthanide and actinide atoms. Transferred hyperfine interactions to ligand nuclei on atoms that neighbor the local moments can often be much more useful, as demonstrated in the unit cell figure on the left side of Fig.. There are often multiple such transferred couplings, depending on the local symmetry of the nuclear site. In this case, the hyperfine interaction should be written as a sum over multiple sites: H hyp = Î A i S(r i ), () i where the sum is over the neighbors with the local moments. In this case, the net hyperfine field will depend on the long-range spin configuration, and it is useful to consider the q dependence of the hyperfine field. If the spin configuration is given by S(r) = q S(q)eiq r, then the hyperfine field can be written as B hf (q) =A(q) S(q), where: A(q) = r i A i e iq r. () It is important to consider the q-dependence of the hyperfine field when there is long-

7 CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT ROPP-.R Page of Nuclear Magnetic Resonance in Kondo lattice systems range magnetic order at a particular wavevector, Q. In the ordered state, the static hyperfine field alters the resonance condition for the nuclei. In some cases, B hf may even vanish at particular values of q. For on-site interactions, however, there is no q dependence. The Knight shift in the paramagnetic state only probes the q = component. The Knight shift tensor is given by: K =( i A i) χ, and includes all of the couplings. Depending on the particular symmetry of the nuclear site, the transferred couplings A i may be related to one another by reflections or rotations. Therefore, it is often the case that the Knight shift tensor has a higher symmetry than the hyperfine tensor, and the angular dependence in Eq. may be simplified. The q-dependence is also important for considerations of the role of spinfluctuations in relaxing the nuclei. The spin-lattice relaxation rate, T can be written as: T = γ k B T lim A (q) χ (q,ω), () ω ω q where χ (q,ω) is the dynamical spin susceptibility at wavevector q and frequency ω. Near a magnetic phase transition critical fluctuations of the hyperfine field B hf (Q), dominate the relaxation of the nuclei. If the hyperfine coupling happens to vanish at Q, then the nuclei may not be sensitive to the critical fluctuations... The electric field gradient Nuclei with spin I > / can have finite quadrupolar moments, giving rise to an interaction with the surrounding electronic charge distribution. The quadrupolar interaction is given by: H Q = ˆQ αβ V βα, () αβ where Q αβ are the elements of the nuclear quadrupolar tensor, V βα = V () x α x β,andv( r) is the electric potential created by the electrons. The V βα form a second rank tensor in real space known as the Electric Field Gradient, or EFG tensor. The EFG is determined by the electronic system, and Q αβ is a property of the nucleus. Equation can be written entirely in terms of spin operators and parameters of the EFG: H Q = e Qq I(I ) [(Îz Î )+η(îx Îy ) ]. () Here Q is the quadrupolar moment of the nucleus, eq V zz is the largest eigenvalue of the EFG tensor, and η (V xx V yy ) /V zz is the asymmetry parameter of the EFG. Q, like I, is an intrinsic parameter of the nucleus, and typically is on the order of a barn ( cm ). This expression is only valid in the basis which diagonalizes the EFG tensor, so that V xx, V yy and V zz are the principal eigenvalues of the EFG tensor. For arbitrary directions the operators Îα must be rotated using rotation operators. By convention

8 Page of CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT ROPP-.R Nuclear Magnetic Resonance in Kondo lattice systems the direction associated with the largest eigenvalue is notated q and corresponds to V zz > V xx > V yy. Depending on the size of the quadrupolar moment and the EFG, the quadrupolar interaction can be on the same order of magnitude as the Zeeman interaction. In this case, the NMR spectra can be quite complex, with multiple resonances that are strongly orientational dependent. As shown in Fig., the In() and In() spectra in the heavy fermion CeIrIn have multiple resonances that depend on the relative orientation of the applied field and the crystal axes. Since In has spin I =/ and quadrupolar moment Q =. barn, there are nine resonances per site, and there are multiple In sites. The EFG for each site is unique, and therefore each site can be observed independently via NMR. The EFG generally contains two terms, one arising from the distribution of charges in the lattice itself, plus a second term arising from the distribution of charges in the on-site electronic orbitals. The on-site term will vanish for filled shells, but is non-zero for partial occupations. Changes to these occupations can therefore be measured by NMR. For example, the EFG of the In() site in CeRhIn changes under pressure near the quantum critical pressure, P c =. GPa[]. This change may reflect a change in the Fermi surface as the f electrons become itinerant []. Quantitative calculations of the EFG are difficult, however, and require detailed electronic structure computations []. The quadrupolar Hamiltonian is also strongly dependent on the orientation of the magnetic field with respect to the principal axes of the EFG tensor, giving rise to spectra that change dramatically as a function of crystal orientation as shown in Fig.. However, this angular dependence is well-defined and can be used to precisely align the sample in situ.. NMR in Heavy Fermions.. The Knight shift anomaly In metals with local moments such as rare-earth and d-electron systems there are usually two types of hyperfine couplings: an on-site coupling to itinerant conduction electron spins, S c, and a second transferred hyperfine coupling to local moment electron spins, S f []. The total hyperfine interaction becomes: ( H hyp = Î A S c + ) B i S f (r i ), () i where A is an on-site hyperfine tensor interaction to the conduction electron spin, and B is a transferred hyperfine tensor to the f spins. Note that we consider here nuclear spins on the ligand sites, i.e., not on the f atom nucleus. Given the two spin species, S c and S f, there are three different spin susceptibilities: χ cc = S c S c, χ cf = S c S f,andχ ff = S f S f. The full expression for the Knight shift

9 CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT ROPP-.R Page of Nuclear Magnetic Resonance in Kondo lattice systems K c (%) K c (%) K c (%) is given by: T (K) CeRhIn CeIrIn CeCoIn χ c (x - emu/mol) Figure. Knight shifts (data points) and bulk susceptibilities (solid lines) for CeRhIn, CeIrIn and CeCoIn as a function of temperature. The data sets have been offset vertically for clarity. Reproduced from Ref. []. K(T )=Aχ cc (T )+(A + B) χ cf (T )+Bχ ff (T ), () where we have dropped the tensor and summation notation for simplicity. The bulk susceptibility is given by: χ(t )=χ cc (T )+χ cf (T )+χ ff (T ). () Note that if A = B, thenk χ for all temperatures. However, if χ cc (T ), χ cc (T ), and χ cc (T ) have different temperature dependences, then the Knight shift will not be proportional to susceptibility, leading to a Knight shift anomaly at a temperature T. This phenomenon is illustrated in Fig.. Ineachmaterial,χ ff and χ cf have different temperature dependences, thus K and χ stop scaling with one another below a crossover temperature T that depends on the particular material. This temperature corresponds to the coherence temperature of the Kondo lattice below which the local moment and itinerant degrees of freedom become entangled and the heavy fermion fluid begins to emerge [].

10 Page of CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT ROPP-.R Nuclear Magnetic Resonance in Kondo lattice systems K c (%) χ c (x -.. Scaling behavior emu/mol) ΔK (%) T (K) Figure. (Left) Knight shift versus bulk susceptibility for the CeMIn materials. Solid lines are the best fits to the high temperature data. (Right) The difference, ΔK, between the Knight shift and the high temperature fits shown in the left figure. For temperatures T>T, the Knight shift and the susceptibility scale with one another, as seen in Figs. and the left panel of. This linear relationship arises because χ is dominated by the local moments, thus χ χ ff and K Bχ. In fact, χ is generally well described by Curie-Weiss behavior, with effective moments that are expected for the material (i.e., p eff =.μ B for Ce + in the case of the materials shown in Fig. ). The slope of a linear fit to the high temperature K versus χ data yields the hyperfine coupling, B, and the intercept, K, as illustrated in Fig.. It is then straightforward to show that the deviation of the Knight shift from this high temperature behavior is given by: ΔK(T )=K(T) Bχ(T ) K () =(A B)(χ cf + χ cc ). () This quantity, shown in the right panel of Fig., probes that growth of the correlation function between the local moments and the conduction electron spins. In this figure it is clear that both the onset temperature for ΔK and the overall magnitude vary from one material to the other, reflecting the variation of (A B). However, by scaling the temperature axis by the onset temperature, T, and the vertical axis by a materialdependent factor as shown in Fig., the temperature dependence of ΔK(T ) becomes identical for all the materials, despite the fact that the Knight shift anomaly observed in Fig. differs dramatically for each system. Similar scaling behavior has been observed in a wide range of heavy fermion materials []. The scaling behavior of the Knight shift anomaly provides strong support for the two-fluid model, in which the quantity χ cf + χ cc is identified with a heavy electron fluid susceptibility, χ HF []. Empirically it has been found that in a broad range of

11 CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT ROPP-.R Page of Nuclear Magnetic Resonance in Kondo lattice systems ΔK (%) T (K)... Figure. ΔK versus T/T for the CeMIn compounds. The magnitude of ΔK has been normalized for comparison between all data sets. The solid line is given by Eq.. materials, K HF (T )variesas: K HF (T )=K HF ( T/T ) / [ + log(t /T )] () where T and KHF are material dependent constants []. This equation is shown as a solid line in Fig., and indicates that ΔK exhibits universal scaling with the T/T. KHF is proportional to A B and can be either positive or negative depending on the hyperfine couplings. The constant A represents the on-site coupling to the conduction electron spins, and cannot be measured independently. The coherence temperature T agrees well with several other experimental measurements, such as the temperature of the maximum in the resistivity []. Below T in addition to the heavy electron susceptibility, χ HF (T ), there is also a contribution from the local moments, χ LM (T ). Both quantities contribute to the Knight shift with different weights, so the Knight shift is no longer proportional to the total susceptibility. In the two-fluid picture, the relative weight of the local moments is continuously reduced with decreasing temperature whereas the weight of the heavy electron component increases. These two components coexist over a broad range of temperatures starting at T and continuing down in temperature until the onset of either long-range order antiferromagnetism or superconductivity [].

12 Page of CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT ROPP-.R Nuclear Magnetic Resonance in Kondo lattice systems.. Relocalization Figure. The heavy electron susceptibility tends to either grow until the onset of order, or turn around above the ordering temperature in a phenomenon known as relocalization. Reproduced from Ref. []. The Knight shift thus provides a direct probe of the susceptibility of the emergent heavy electron fluid in the paramagnetic state and the two-fluid model provides important context in which to interpret the properties of the NMR Knight shift anomaly across phase diagrams and material families. However, a key question is to what extent the twofluid behavior persists as a function of temperature, field, and pressure. For example, the phenomenological form in Eq. diverges as T, so it is natural to expect that some other behavior should emerge at sufficiently low temperature. Most of the heavy fermion systems that have been measured to date either become superconducting or antiferromagnetic at low temperatures. In superconducting materials, ΔK(T ) is well described by Eq. down to T c, at this point it exhibits a maximum and decreases below, as illustrated in Fig. []. (Similar behavior exists inthecaseofuru Si at the hidden order temperature T HO. K.) This behavior reflects the fact that the superconducting condensate is formed from the itinerant heavy electron degrees of freedom. A different type of behavior has been observed in materials that undergo antiferromagnetic order. In this case, χ HF appears to go through a maximum at a temperature, T above the Neel ordering, and then decrease until T N, as demonstrated in Fig.. This phenomenon has been observed in CePt In (T N =. K,T K and T K) and in CeRhIn (T N =. K,T K,andT K) [,, ]. This breakdown of the two-fluid scaling between T N and T has been termed relocalization;

13 CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT ROPP-.R Page of Nuclear Magnetic Resonance in Kondo lattice systems the interpretation is that the collective hybridization between the local moments and the conduction electrons reverses, shifting spectral weight back to the local moment degrees of freedom prior to long-range order of the local moments themselves. The coexistence of the heavy fermion fluid and the partially screened local moments offers a natural explanation for the coexistence of antiferromagnetism and superconductivity in the heavy fermion materials. In the prototypical Kondo lattice system CeRhIn, superconductivity emerges and coexists with ordered local moments over a range pressure P =.. GPa,asshowninFig.. The heavy fermion fluid begins to form at T through collective hybridization of the local moments, but the local moments do not completely dissolve. The partially screened local moments that remain coexist with the heavy fermion fluid, but can also undergo long-range antiferromagnetic order. The remaining heavy fermion fluid can form a superconducting condensate at lower temperature. However, if the local moments are sufficiently screened such that T N would be lower than T c, then superconducting order in the heavy fermion fluid preempts remaining the degrees of freedom, and long-range antiferromagnetism is quenched. Yang and Pines have developed a comprehensive theory of the evolution of the magnitude of the local moments and their ordering temperature as a function of the hybridization effectiveness []. In this picture the collective hybridization of the local moments and the itinerant electrons becomes complete at T = at a critical pressure. At this point, the local moments have completely dissolved, and T c reaches a maximum. Below the critical pressure, the hybridization effectiveness is small, leading to an incomplete formation of the heavy fermion fluid. Above the critical pressure, the heavy fermion fluid is fully formed and the local moments disappear at a temperature T L (P )increases with pressure... Field dependence Shockley and coworkers have investigated the field dependence of the Knight shift anomaly in CeIrIn. For fields applied along the c-axis, the Knight shift anomaly is essentially field independent for fields up to T []. No bulk susceptibility data was available at these high fields, so rather than comparing K to χ, the authors compared the Knight shifts of the two In sites in the material (see Fig. ). Because the two sites have different hyperfine couplings A i and B i (site i =, ) the Knight shifts K and K have different couplings to χ cc, χ cf and χ ff. Rather than using Eq., in this case the heavy electron susceptibility can be determined by comparing the Knight shifts of the two sites: ΔK(T )=K (T ) (B /B )K (T ) K () =(A A B /B )(χ cf + χ cc ). () This approach is superior because both quantities can be measured simultaneously in the crystal at the same temperature, field and orientation, thus eliminating the possibility of systematic errors when comparing Knight shifts and bulk susceptibilities. On the other hand, the high temperature couplings B and B cannot be measured separately: the

14 Page of CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT ROPP-.R Nuclear Magnetic Resonance in Kondo lattice systems T(K) B cc () ( koe/μ B ). AFM Spin Liquid T*. Relocalization.. P (GPa) Kondo Liquid + Spin Liquid P c. SC. Figure. (Upper) Phase diagram of CeRhIn versus pressure. The blue data points indicate T crossover and the starred data points indicate the relocalization temperature as determined by NMR Knight shift measurements. (Bottom) The transferred hyperfine coupling B cc () of the In() site as a function of pressure. The solid line is a fit as described in the text. Adapted from Ref. []. The data for T N and T c is reproduced from []. slope of the plot of K versus K is given by the ratio B /B, rather than the absolute values of B and B. To determine the absolute value of these couplings, one needs the susceptibility, χ. Shockley and coworkers have also studied the dependence of the Knight shift anomaly as a function of field orientation []. In CeIrIn, the Knight shift of the In() and In() sites is strongly anisotropic. There is a large Knight shift anomaly for the field along the c-direction, as shown in Fig., andδk is easily measurable. However, in the ab-direction the Knight shift anomaly nearly vanishes in this compound, and similar behavior has been observed for other members of the CeMIn family []. A question that emerges, therefore, is whether T could be anisotropic. By measuring the full temperature and angular dependence of the shifts K (T,θ) tok (T,θ) the

15 CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT ROPP-.R Page of Nuclear Magnetic Resonance in Kondo lattice systems Figure. Knight shift of the In() (left) and the In() (right) in CeRhIn as a function of temperature and pressure. Solid lines are guides to the eye. authors were able to extract ΔK(T,θ), where θ is the angle between the c-axis and the applied field. In fact, the hyperfine couplings are such that for field in the ab direction, A B, therefore Knight shift of the In() site nearly tracks χ over the entire temperature range in this orientation. However, for intermediate angles ΔK(T,θ) is non-zero and it is possible to extract the angular dependence of T. The magnitude of angular dependence of ΔK(T,θ) is determined by the hyperfine tensors, but the temperature dependence is determined by T, which appears to exhibit a small angular dependence, reaching a maximum along the Ce-In() bond direction (see Fig. ). This behavior may reflect the anisotropy of the hybridization between the Ce f electrons and the In() p electrons. Dynamical mean-field theory (DMFT) calculations of the electronic structure of CeIrIn indicate that the hybridization is momentum dependent, and that the out of plane In() atoms couple more strongly to the Ce f moments than the in plane In() atoms []. Thus even though NMR is a local probe, the transferred hyperfine coupling to the f moments enables this technique to probe the anisotropy of the hybridization, complementing k-space probes such as angle-resolved photoemission (ARPES) and scanning tunneling microscopy (STM).. Case Study: CeRhIn under pressure The Knight shift anomaly has recently been investigated as a function of pressure in the CeRhIn system []. At ambient pressure, this material is antiferromagnetic below T N =. K, and superconducting below a maximum T c =. K for hydrostatic pressures above P c =. GPa []. At ambient pressure the static susceptibility exhibits Curie- Weiss behavior at high temperatures, and the Knight shift of the In() site tracks χ down to a temperature T K (see Fig. ). This behavior changes dramatically under pressure, as shown in Fig.. The In() Knight shift falls by a factor of two

16 Page of CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT ROPP-.R Nuclear Magnetic Resonance in Kondo lattice systems K c () (%).. K c () (%).. K K Figure. (Left) Experimental data of the Knight shift of the In() versus the Knight shift of the In() site in CeRhIn, adapted from Ref. []. (Right) Simulated Knight shifts of the In() versus the In() shift using the model described in the text. between ambient pressure and GPa, whereas the In() Knight shift remains relatively unchanged, except for a slight increase below T. This behavior cannot be understood as a change in the bulk susceptibility under pressure, otherwise the shifts of both sites would change in the same manner. The data indicate, therefore, that the hyperfine couplings themselves are changing under pressure... Changes to hyperfine coupling Fig. shows the experimental Knight shift of the In() site versus that of the In() site for several different pressures, with temperature implicit. The solid lines are fits to the high temperature data. The slope is given by the ratio of the transferred couplings, B /B, which clearly changes with pressure. Fig. shows how B varies with pressure, assuming that B remains pressure independent. This assumption is reasonable, because if the changes observed in K reflected changes in χ, thenb would have to change in such a manner that K would remain unchanged with pressure. It is more likely, therefore, that the pressure dependence of the ratio reflects that of B. The solid line in Fig. is a fit to B +ΔB P/P,whereB =. ±. koe/μ B,ΔB =. ±. koe/μ B, and P =. ±. GPa. (Note that we report only the single site hyperfine coupling, but K B χ at the In() site due to the four nearest neighbor Ce.) It is curious that the high pressure limit B is close to the transferred hyperfine coupling observed in the isostructural material CeCoIn (B =. koe/μ B ). Both the temperature dependence of K and the coupling B in CeRhIn under pressure are qualitatively similar to those in CeCoIn at ambient pressure, which supports the argument that these materials can be tuned by chemical pressure [].

17 CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT ROPP-.R Page of Nuclear Magnetic Resonance in Kondo lattice systems The hyperfine coupling B plays a role in other NMR observed quantities, such as the internal field in the antiferromagnetic state, H int, and in the spin-lattice relaxation rate, T. Therefore, the pressure dependence of these observed quantities also includes the pressure dependence of the hyperfine coupling. The internal field at the In() site in CeRhIn decreases much faster than the ordered moment measured by neutron scattering [,,, ]. This observation can now be explained by the pressuredependent hyperfine coupling. Furthermore, T has been measured as a function of temperature and pressure in this material, and it decreases with increasing pressure [], suggesting a suppression of spin fluctuations. This contrasts with resistivity measurements that indicate an increase in scattering, as the system approaches a quantum critical point under pressure []. Again, the anomalous behavior of the T can be explained by the evolution of the hyperfine coupling with pressure, so that the spin-lattice relaxation data are consistent withanincreaseinspinfluctuationsnearthe QCP... Coherence and hybridization It is straightforward to determine ΔK(T,P) from the two Knight shifts shown in Fig. and therefore to determine the pressure dependence of T. As shown in Eq., this quantity is proportional to the heavy electron susceptibility. Fitting ΔK(T ) toeq. yields two fitting parameters, T and KHF. T is shown as a function of pressure in Fig.. KHF is proportional to the quantity A A B /B, which is pressure dependent. Therefore, the ratio of the intercept to the slope of a plot of KHF versus B /B (shown in the inset to Fig. ) isgivenbya /A. The best fit yields A /A =. ±., providing a measure of the ratio of the on-site hyperfine couplings in this material. The fact that the data can be well fit by this linear relationship implies that the on-site couplings A and A remain pressure-independent. However, there is no direct method to determine the absolute values of these couplings. Nevertheless, it is reasonable to assume that the on-site couplings do not depend on pressure, whereas the transferred couplings depend on the hybridization between the Ce f orbitals and the In p orbitals. As pressure increases and the atoms move closer to one another, the overlap integrals increase and therefore transferred couplings can change. It is instructive to consider a model system to see how changes to the hyperfine coupling B can contribute to the complex behavior observed in Figs. and. If we use the measured hyperfine couplings to the In() and In() sites, and also the two-fluid expressions to approximate the correlation functions: χ ff ( f(t )) () T + T χ cf f(t ) ( + ln(t /T )) () χ cc, () where f(t )=f ( T/T ) / [], then it is possible to compute the Knight shifts of both sites as a function of temperature and pressure. Fig. shows K versus K

18 Page of CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT ROPP-.R Nuclear Magnetic Resonance in Kondo lattice systems ΔK/(A /A - B /B ) (%) GPa.GPa.GPa.GPa.GPa (%) K HF T/T* B /B. Figure. ΔK versus temperature scaled by T for various pressures in CeRhIn, adapted from Ref. []. INSET: KHF (determined by fitting ΔK to Eq. ) versus the ratio B /B (determined from the high temperature fits in Fig. ). The solid line is a linear fit as described in the text.. computed in this fashion for same pressures as measured in the experiment. The model calculations capture not only the change of slope of the high temperature regime, but also the character of the deviation from linearity at low temperatures. For low pressures, the Knight shift bends back towards the left in the figure, but for increasing pressure the Knight shift bends towards the right below T. This model captures the same trends as the experimental data. Differences from the data are likely due to different susceptibilities χ cc (T )andχ ff (T ) than used in the model. The pressure dependence of T shown in Fig. reveals an increase from K at ambient pressure to nearly K at GPa. This dramatic increase in the coherence temperature reflects an increase in the hybridization between the f moments and the conduction electrons, and is consistent with previous experiments that indirectly probe the coherence temperature under pressure []. As discussed below, the relationship between T and the hybridization can be studied directly using the periodic Anderson model.. NMR in the periodic Anderson model The Knight shift anomaly can be understood in the context of the periodic Anderson model (PAM), in which an itinerant band of conduction electrons hybridizes with a

19 CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT ROPP-.R Page of Nuclear Magnetic Resonance in Kondo lattice systems lattice of local f electrons with strong on-site repulsion. The Hamiltonian for this model is given by: H = t ij,σ(c iσ c jσ + c jσ c iσ ) V (c iσ f iσ + f iσ c iσ ) iσ ( + U f n f i )( n f i ) () i where c iσ (c iσ )andf iσ (f iσ ) are creation(destruction) operators for conduction and local electrons on site i and with spin σ. n c,f iσ are the associated number operators. t is the hopping amplitude between conduction electrons on the near neighbor sites ij of a square lattice, U f the local repulsive interaction in the f orbital and V the hybridization between conduction and localized electrons. The nuclei interact with the electrons via ( Eq. ), where the conduction ( and ) f moment spins are given by: S i c = (c i c i ) σ c i and S f i = (f i f i ) σ f i. This model cannot be solved c i exactly, but various approximations can be made that provide physical insight into the general phenomenon... Exact diagonalization calculations One of the simplest approaches to understanding the model in Eq. is to assume two lattice sites involving two conduction electron spins and two f spins. In this case, the Hilbert space has dimension =, and if we consider only the half-filled states, then this number is reduced to. In this case it is straightforward to diagonalize Eq. exactly. The correlation functions χ cc, χ cf and χ ff can be computed as a function of the parameters t, V, U f and temperature. Fig. shows representative plots for these quantities. The computed susceptibilities capture the principal features of the experiment: χ cf is negligible at high temperature, but grows progressively larger below a temperature that increases with V. Note that in such a finite cluster the model cannot accurately capture the low temperature physics because of the presence of energy gaps. In the thermodynamic limit, these gapss would vanish, altering the low temperature physics []. Nevertheless, this simple model captures much of the relevant features. An even simpler approach is to consider a single lattice site, with one conduction electron coupling to one f-electron. For low temperature, this model can be expressed in terms of a Heisenberg interaction between the spins: Ĥ = JS c S f,wherej =V /U f. This model also captures the relevant features of the Knight shift anomaly, with T J []... Quantum Monte Carlo solutions A much better approach to determine the correlation functions for the Hamiltonian in Eq. is to use quantum Monte Carlo (QMC) methods. This approach enables one to investigate much larger lattices, but is restricted to the half-filled limit. Recently Jiang f i

20 Page of CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT ROPP-.R Nuclear Magnetic Resonance in Kondo lattice systems χ ab / (g a g b μ B ) T/t.. χ cc χ cf χ ff χ total. χ cf / (g c g f μ B ) Figure. (Left) The various correlation functions computed for the half-filled PAM for two lattice sites, using V =., t =., and U f =. (Right) χ cf (T ) for the same system for V =.,.,.,.,.,.,.,.,. and.. and coworkers have computed the correlation functions for several different cases, with lattice sizes up to, for various values of the parameters U f and V,intwoandthree dimensions []. They were able to reproduce the Knight shift anomaly for a given set of hyperfine coupling parameters, A and B. They determined ΔK(T ) as a function of the hybridization, and showed that it was well described by Eq. for.t <T <T. This approach enabled them to extract T as a function of the hybridization, V,shown in Fig., which qualitatively agrees with the phase diagram of CeRhIn under pressure (Fig. ). The hybridization parameter, V, likely increases with pressure as the Ce f orbitals overlap more with the In p orbitals. Similar calculations of the correlation functions that have provided microscopic evidence for two-fluid behavior have also been carried out for the one-dimensional Kondo-Heisenberg model [].. Concluding remarks and future directions The Knight shift anomaly, once considered a non-universal and poorly understood feature to be swept under the rug, has proven to be an important probe of the coherence in heavy fermion systems. Other theories of the Knight shift anomaly have been put forth, including the presence of a Kondo screening cloud that gives rise to a difference between the local and bulk response [], and the effects of crystal field (CEF) splitting of thecemoments[]. The field dependence of the anomaly studied in CeIrIn discussed in section suggests that the CEF contributions are negligible []; however it is not possible to rule out CEF effects in influencing the subtle anisotropy of T observed in this material []. Another theory that has been put forth based on an analysis of the spin lattice relaxation rate is that the hyperfine coupling itself is temperature dependent []. However, to a large extent the hyperfine coupling is determined by the wavefunctions of the core electrons, and therefore these couplings should not change significantly with temperature []. The pressure dependence of the hyperfine coupling in CeRhIn, however, is an unexpected and curious observation. Transferred hyperfine couplings in the presence.. T/t...

21 CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT ROPP-.R Page of Nuclear Magnetic Resonance in Kondo lattice systems T/t T* T N T L.. V/t.. Figure. The temperature - hybridization phase diagram for the model Hamiltonian described by Eq., reproduced from Ref. []. The solid red region corresponds to long-range antiferromagnetic order. The shaded gray region between T and T L corresponds to T/T scaling, and can be described by the two-fluid model. of large spin-orbit couplings have not been well studied at the microscopic level. Both the orbital and spin contributions to the Ce moments contribute to the net hyperfine field at the In() site, but with opposite signs. Thus it is plausible that the hyperfine coupling could decrease with pressure, depending on how the orbital and spin contributions evolve with hybridization. Further studies of the anisotropy of the transferred hyperfine interaction in this material, coupled with microscopic model calculations, may shed important light on the relationship between this coupling and the electronic hybridization. Further studies of the behavior of the spin-lattice relaxation in the PAM can also provide important information about the two-fluid model, especially if coupled with computation of the spectral functions, A ij (k,ω), i, j =(c, f) that can be compared with STM or ARPES data, to understand how the charge and spin degrees evolve simultaneously with pressure and temperature. Furthermore, spin lattice relaxation rate data in CeCoIn has been interpreted in the two-fluid model at a phenomenological level [, ]. However, little is known about how the dynamical susceptibilities, χ cc (q,ω), χ cf (q,ω), and χ ff (q,ω) evolve with temperature and pressure. Acknowledgements We thank D. Pines, R. Scalettar and Y.-f. Yang for stimulating discussions. This work was supported by the National Science Foundation under Grant No. DMR-, and the NNSA under the Stewardship Science Academic Alliances program through U.S. DOE Research Grant No. DE-NA. References [] Tou, H., Kitaoka, Y., Asayama, K., Kimura, N., Ōnuki, Y., Yamamoto, E. & Maezawa, K. Odd- Parity Superconductivity with Parallel Spin Pairing in UPt : Evidence from Pt Knight Shift Study. Phys. Rev. Lett., ().

22 Page of CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT ROPP-.R Nuclear Magnetic Resonance in Kondo lattice systems [] Kuramoto, Y. & Kitaoka, Y. Dynamics of Heavy Electrons (Oxford University Press, Oxford, ). [] Zheng, G.-q., Tanabe, K., Mito, T., Kawasaki, S., Kitaoka, Y., Aoki, D., Haga, Y. & Onuki, Y. Unique Spin Dynamics and Unconventional Superconductivity in the Layered Heavy Fermion Compound CeIrIn : NQR Evidence. Phys. Rev. Lett., (). [] Kawasaki, S., Mito, T., Kawasaki, Y., Zheng, G., Kitaoka, Y., Aoki, D., Haga, Y. & Onuki, Y. Gapless magnetic and quasiparticle excitations due to the coexistence of antiferromagnetism and superconductivity in CeRhIn : a study of In NQR under pressure. Phys. Rev. Lett., (). [] Curro, N., Caldwell, T., Bauer, E., Morales, L., Graf, M., Bang, Y., Balatsky, A., Thompson, J. & Sarrao, J. Unconventional superconductivity in PuCoGa. Nature, (). [] Chudo, H., Sakai, H., Tokunaga, Y., Kambe, S., Aoki, D., Homma, Y., Shiokawa, Y., Haga, Y., Ikeda, S., Matsuda, T. D., Onuki, Y. & Yasuoka, H. Al NMR Evidence for the Strong-Coupling d-wave Superconductivity in NpPd Al. J. Phys. Soc. Jpn., (). [] Sakai, H., Tokunaga, Y., Fujimoto, T., Shinsaku Kambe and, R. E. W., Yasuoka, H., Aoki, D., Homma, Y., Yamamoto, E., Nakamura, A., Shiokawa, Y., Nakajima, K., Arai, Y., Matsuda, T. D., Haga, Y. & Onuki, Y. Anisotropic Superconducting Gap in Transuranium Superconductor PuRhGa : Ga NQR Study on a Single Crystal. J. Phys. Soc. Jpn., (). [] Aynajian, P., da Silva Neto, E. H., Gyenis, A., Baumbach, R. E., Thompson, J. D., Fisk, Z., Bauer, E. D. & Yazdani, A. Visualizing heavy fermions emerging in a quantum critical Kondo lattice. Nature, (). [] Yang, Y.-F., Fisk, Z., Lee, H.-O., Thompson, J. D. & Pines, D. Scaling the Kondo lattice. Nature, (). [] Yang, Y.-f. & Pines, D. Emergent states in heavy-electron materials. Proc. Natl. Acad. Sci., E E (). [] Clogston, A. M. & Jaccarino, V. Susceptibilities and Negative Knight Shifts of Intermetallic Compounds. Phys. Rev., (). [] Walstedt, R. E., Bell, R. F., Schneemeyer, L. F., Waszczak, J. V. & Espinosa, G. P. Diamagnetism in the normal state of YBa Cu O. Phys. Rev. B, (). [] Abragam, A. The Principles of Nuclear Magnetism (Oxford University Press, Oxford, ). [] Shirer, K. R., Shockley, A. C., Dioguardi, A. P., Crocker, J., Lin, C. H., aproberts Warren, N., Nisson, D. M., Klavins, P., Cooley, J. C., Yang, Y.-f. & Curro, N. J. Long range order and two-fluid behavior in heavy electron materials. Proc. Natl. Acad. Sci., E E (). [] Lin, C. H., Shirer, K. R., Crocker, J., Dioguardi, A. P., Lawson, M. M., Bush, B. T., Klavins, P. & Curro, N. J. Evolution of hyperfine parameters across a quantum critical point in CeRhIn. Phys. Rev. B, (). [] Shishido, H., Settai, R., Harima, H. & Ōnuki, Y. A Drastic Change of the Fermi Surface at a Critical Pressure in CeRhIn : dhva Study under Pressure. J.Phys.Soc.Jpn., (). [] Rusz,J.,Oppeneer,P.M.,Curro,N.J.,Urbano,R.R.,Young,B.-L.,Lebègue, S., Pagliuso, P. G., Pham, L. D., Bauer, E. D., Sarrao, J. L. & Fisk, Z. Probing the electronic structure of pure and doped CeMIn (M = Co,Rh,Ir) crystals with nuclear quadrupolar resonance. Phys. Rev. B, (). [] Curro, N., Young, B., Schmalian, J. & Pines, D. Scaling in the emergent behavior of heavy-electron materials. Phys. Rev. B, (). [] Yang, Y.-F. & Pines, D. Universal Behavior in Heavy-Electron Materials. Phys. Rev. Lett., (). [] aproberts Warren, N., Dioguardi, A. P., Shockley, A. C., Lin, C. H., Crocker, J., Klavins, P. & Curro, N. J. Commensurate antiferromagnetism in CePt In, a nearly two-dimensional heavy fermion system. Phys. Rev. B, ().