Orientation of ground-state orbital in CeCoIn 5 and CeRhIn 5
M. Sundermann, A. Amorese, F. Strigari, B. Leedahl, M. W. Haverkort, H. Gretarsson, L. H. Tjeng, M. Moretti Sala, H. Yavş, E. D. Bauer, P. F. S. Rosa, J. D. Thompson, A. Severing
OOrientation of ground-state orbital in CeCoIn and CeRhIn M. Sundermann,
1, 2
A. Amorese,
1, 2
F. Strigari, ∗ B. Leedahl, L. H. Tjeng, M. W. Haverkort, H. Gretarsson,
4, 2
H. Yava¸s, † M. Moretti Sala, ‡ E. D. Bauer, P. F. S. Rosa, J. D. Thompson, and A. Severing
1, 2 Institute of Physics II, University of Cologne, Z¨ulpicher Straße 77, 50937 Cologne, Germany Max Planck Institute for Chemical Physics of Solids, N¨othnitzer Straße 40, 01187 Dresden, Germany Institute for Theoretical Physics, Heidelberg University, Philosophenweg 19, 69120 Heidelberg, Germany PETRA III, Deutsches Elektronen-Synchrotron (DESY), Notkestraße 85, 22607 Hamburg, Germany European Synchrotron Radiation Facility, 71 Avenue des Martyrs, CS40220, F-38043 Grenoble Cedex 9, France Los Alamos National Laboratory, New Mexico 87545, USA (Dated: June 26, 2019)We present core level non-resonant inelastic x-ray scattering (NIXS) data of the heavy fermioncompounds CeCoIn and CeRhIn measured at the Ce N , -edges. The higher than dipole transi-tions in NIXS allow determining the orientation of the Γ crystal-field ground-state orbital withinthe unit cell. The crystal-field parameters of the Ce M In compounds and related substitution phasediagrams have been investigated in great detail in the past; however, whether the ground-state wave-function is the Γ +7 ( x − y ) or Γ − ( xy orientation) remained undetermined. We show that the Γ − doublet with lobes along the (110) direction forms the ground state in CeCoIn and CeRhIn . Acomparison is made to the results of existing DFT+DMFT calculations. I. INTRODUCTION
At high temperature, heavy-fermion materials are de-scribed by decoupled localized f electrons and conduc-tion electron bands. Upon cooling, the localized f elec-trons start to interact with the conduction electrons ( cf -hybridization) and become partially delocalized. The re-sulting entangled fluid consists of heavy quasiparticleswith masses up to three orders of magnitude larger thanthe free electron mass. These quasiparticles may undergomagnetic or superconducting transitions. In the Doniachphase diagram temperature T versus exchange interac-tion J , magnetic order prevails for small J whereas anon-magnetic Kondo singlet state forms for large J . Be-tween these two regimes quantum critical behaviour oc-curs which is often accompanied by a superconductingdome that hides a quantum critical point (QCP). Un-derstanding how these quasiparticles, that have atomic-like as well as itinerant character, give rise to theseground states is a challenging question in condensed-matter physics, and the answer to this question will pro-vide predictive understanding of these quantum states ofmatter. The tetragonal compounds Ce M In ( M = Co, Rh,Ir) are heavy fermion compounds that display differ-ent ground states for different transition metal ions;for M = Co and Ir the ground state is superconducting( T c = 2.3 and 0.4 K) and for M = Rh it is antiferromag-netic ( T N = 3.8 K). High-quality Ce M In crystals canbe grown, making this family suitable for determining theparameter that drives the different ground states. Withinthe above mentioned extended Doniach phase diagram,CeRhIn is on the weak side of hybridization, CeCoIn close to the QCP and CeIrIn is on the side of stronger cf -hybridization, i.e. superconductivity goes along withstronger cf -hybridization. Although there are strong in-dications for localization (Rh) and delocalization (Co,Ir)in, e.g., the size of the Fermi surface, it is not possible to detect the differences in f occupations. They are sosubtle that they are below the detection limit. A light-polarization analysis of soft X-ray ab-sorption spectroscopy (XAS) spectra shows that thecrystal-field wavefunction of the ground state corre-lates with the ground-state properties in the tem-perature - transition metal (substitution) phase diagramof CeCoIn - CeRhIn - CeRh − δ Ir δ In - CeIrIn ; orbitalsmore compressed in the tetragonal ab -plane favor an an-tiferromagnetic ground state as for CeRhIn and the Rhrich compounds with δ ≤ c axis, however, have super-conducting ground states (CeCoIn , CeIrIn and also theIr rich compounds with δ ≥ The obvious conclu-sion is that the more pronounced extension of the groundstate orbitals in the direction of quantization (crystallo-graphic c direction) promotes stronger hybridization inthe z direction and hence superconductivity. This issupported by combined local density approximation plusdynamical mean field theory (LDA+DMFT) calculationsby Shim et al . that find for CeIrIn the strongest hy-bridization with the out-of-plane In(2) ions (see unit cellin Fig. 1 (a)). It was also shown that the suppression ofsuperconductivity in CeCo(In − y Sn y ) by about 3% ofSn is due to a homogeneous increase of hybridization inthe tetragonal ab plane since the Sn ions go preferablyto the In(1) sites. Accordingly, we found that here thehybridization with In(1) ions plays a decisive role; the 4 f ground state orbital extends increasingly in the plane asthe Sn content is increased. Hence, the ground state wavefunction is a very sensi-tive probe for quantifying hybridization. Haule et al. ob-tained a 4 f Weiss field hybridization function for Ce M In based on realistic lattice parameters using density func-tional theory plus dynamical mean field (DFT+DMFT)calculations which they have decomposed into crystal-field components (see Fig. 1 (b)). Here our goal is to ver-ify that the crystal-field components that were extracted a r X i v : . [ c ond - m a t . s t r- e l ] J un in these calculations are in agreement with reality.The tetragonal point symmetry of Ce in Ce M In splitsthe Ce Hund’s rule ground state into three Kramers dou-blets, two Γ doublets Γ + / − = | α | |± / (cid:105) + / − √ − α |∓ / (cid:105) and Γ − / +7 = √ − α |± / (cid:105) − / + | α | |∓ / (cid:105) , and oneΓ = | ∓ / (cid:105) . We write + / − or − / + because the signhas not yet been determined, and this is the scope of thepresent manuscript. Γ as a pure J z state has full rota-tional symmetry around the quantization axis c but themixed states have lobes with fourfold rotational symme-try. The magnitude of α describes the shape and aspectratio of the Γ + / − orbitals whereas the sign in the wave-function determines how the orbitals are oriented withinthe unit cell; with the lobes along [100] (Γ +7 : x − y ) orwith the lobes along [110] (Γ − : xy ).The crystal-field potential of the Ce M In has been de-termined with inelastic neutron scattering (INS) andthe ground state wavefunctions were studied in greaterdetail with linear polarized soft XAS. Hence, thecrystal-field energy splittings, the sequence of states(Γ + / − , Γ − / +7 , Γ ) and also the magnitude of the α -values are known (0.13, 0.38, 0.25 for Co, Rh and Ir).Only the sign of the wavefunction remains unknown be-cause it cannot be determined with any of these dipole-selection-rule based spectroscopies. We, therefore, set upan experiment to determine the sign of the ground-statewavefunction in the Ce M In compounds in order to findout which one of the two scenarios in Fig. 1 (a) applies. II. METHOD
We performed a core level non-resonant inelastic x-ray scattering (NIXS) experiment at the Ce N , -edges(4 d → f ). It has been shown previously that thismethod is able to detect anisotropies with higher thantwofold rotational symmetry. In the following, webriefly recap the principles of NIXS, a photon-in photon-out technique with hard x-rays (E in ≈
10 keV). Becauseof the high incident energies, NIXS is bulk sensitive andallows one to reach large momentum transfers | (cid:126)q | of theorder of 10 ˚A − when measuring in back scattering ge-ometry. At such large momentum transfers, the tran-sition operator in the scattering function S( (cid:126)q , ω ) can nolonger be truncated after the dipole term. As a result,higher order scattering terms contribute to the scatter-ing intensity. For a Ce 4 d → f transition atabout 10 ˚A − , octupole (rank k =3) and triacontadipole( k =5) terms dominate the scattering intensity whereasthe dipole part ( k = 1) is less prominent. Accordingly,the directional dependence of the scattering function ina single crystal experiments follows multipole selectionrules, in analogy to the dipole selection rules in linearlypolarized XAS. Thus single crystal NIXS yields informa-tion not only about the orbital occupation but also thesign of the wavefunction that distinguishes the xy and x − y orientations of a Γ when comparing two direc- (a) (b) (c) In2
In1
In1 Γ Γ Γ Γ Γ M = Co, Rh, Ir Γ Theory Experiment
FIG. 1. (color online) a) Cartoon of unit cell of Ce M In , withorbital shape of Ce in CeCoIn as taken from XAS for thepossibility of a Γ − and Γ +7 Ce ground state orbital. The lightblue triangle emphasizes the In2-In1-In2 triangle. b) Weissfield hybridization function for Ce M In from DFT+DMFTcalculations decomposed into crystal-field components of theCe ion (red orbitals) and the out-of-plane In2 (dark yellowdots) and in-plane In1 (yellow dots) environment, adaptedfrom Ref. 19. c) Crystal-field components of the Ce ions andenvironment of In ions as obtained from the present NIXSexperiment. tions within the xy-plane; here [100] and [110]. III. EXPERIMENT
CeCoIn and CeRhIn single crystals were gown us-ing the standard In-flux technique. CeCoIn crystalsare plate-like with the [001] direction perpendicular tothe plate, whereas CeRhIn crystals are more three-dimensional. A very detailed structural investigation onthe 115 compounds shows that more than 98 % of thecrystal volumes form in the HoCoGa structure. Allsamples were aligned by Laue diffraction before the ex-periment. For each compound two samples were cut, onewith a (100) and a second one with a (110) surface so thatspecular geometry could be realized in the experiment.The experiments were performed at the Max-PlanckNIXS end station P01 at PETRA III/DESY in Ham-burg, Germany. P01 has a vertical scattering geometryand the incident energy was selected with a Si(311) dou- (cid:1) ) "$ + G (cid:1) (cid:5)(cid:13) (cid:1) , "* ! (cid:1) a (cid:9) (cid:16) (cid:7) (cid:6) (cid:10) (cid:14) (cid:17) & (cid:1) (cid:21) (cid:8) (cid:2) (cid:27) (cid:3) (cid:17) (cid:31) (cid:22) ! (cid:19) % (cid:11) . ’ . (cid:16) (cid:15) (cid:6) (cid:12) / (cid:5)(cid:8) D (cid:18) (cid:16) (cid:7) (cid:6) (cid:13) (cid:1) (cid:31) (cid:24)(cid:23) (cid:1) (cid:16) (cid:1) (cid:12) (cid:1) (cid:20) (cid:1) Intensity (rel. units) (cid:1) ’ . . (cid:25) (cid:8) (cid:7) (cid:7) (cid:26) (cid:1) (cid:30) (cid:27) * (cid:27)(cid:1) ’ . . (cid:25) (cid:8) (cid:8) (cid:7) (cid:26) (cid:1) (cid:30) (cid:27) * (cid:27)(cid:1) ) " $ + ) "$ + G (cid:1) (cid:5)(cid:13) (cid:1) , "* ! (cid:1) a (cid:9) (cid:16) (cid:7) (cid:6) (cid:8) (cid:10) (cid:1) (cid:2) (cid:28) (cid:3) (cid:17) (cid:31) (cid:17) & (cid:19) % (cid:11) (cid:1) ) "$ + a (cid:9) (cid:16) (cid:7) (cid:6) (cid:10) (cid:14) (cid:2) (cid:29) (cid:3) (cid:1)(cid:18) % (cid:31) ( - (cid:1) (cid:1) Difference x 10-2 (rel. units) (cid:30) (cid:27) * (cid:27) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) ) " $ + G (cid:1)(cid:5)(cid:13) (cid:1) (cid:1) (cid:1) (cid:1) ) " $ + G (cid:1) (cid:4)(cid:13) (cid:1) ) (cid:29) (cid:27) G (cid:1) (cid:5)(cid:13) (cid:2) (cid:30) (cid:3) ) "$ + a (cid:9) (cid:16) (cid:7) (cid:6) (cid:8) (cid:10) (cid:17) & (cid:1) (cid:21) (cid:8) (cid:1)(cid:18) % (cid:31) ( - (cid:1) (cid:1) FIG. 2. (color online) Non resonant inelastic x-ray scattering (NIXS) data (dots) of CeRhIn5 (a) and CeCoIn5 (b) at theCe N , -edges for the two crystallographic directions (cid:126)q (cid:107) [100] (blue) and (cid:126)q (cid:107) [110] (green) at T = 6 K, plus simulations (lines)for the respective Γ − ground states ( xy orientation) (see text). The bottom panels (c) and (d) show the difference spectraI (cid:126)q (cid:107) [110] - I (cid:126)q (cid:107) [100] (circles) and simulated dichroism for the respective Γ − (orange lines) and Γ +7 (gray lines) crystal-field groundstates, and for a mixed ground state called scaled Γ − (dark red lines), see text. ble monochromator and twelve Si(660) 1 m radius spher-ically bent crystal analyzers were arranged in 3 x 4 ar-ray as shown in Fig. 2 of Ref. 26 so that the fixed fi-nal energy was E final = 9690 eV. The analyzers were po-sitioned at scattering angles of 2 θ ≈ ◦ , 155 ◦ , and160 ◦ which provide an averaged momentum transfer of | (cid:126)q | = 9.6 ± − . The scattered beam was detected bya position sensitive custom-made detector (LAMBDA),based on a Medipix3 chip detector. The elastic line wasconsistently measured and a pixel-wise calibration yieldsinstrumental energy resolutions of ≈ N , edges were measured with the momentum transfer (cid:126)q par-allel to [001] and parallel to [110] ( (cid:126)q (cid:107) [001] and (cid:126)q (cid:107) [110]).We used the full multiplet code Quanty for simulat-ing the NIXS data. A Gaussian broadening of 0.7 eV ac-counts for the instrumental resolution and an additionalLorentzian broadening of 0.4 eV FWHM accounts for life-time effects. The atomic parameters were taken from theCowan code, whereby the Hartree-Fock values of theSlater integrals were reduced to about 60 % for the 4 f -4 f and to about 80 % for the 4 d -4 f Coulomb interactionsto reproduce the energy distribution of the multiplet ex-citations of the Ce N , -edges. This reduction accountsfor configuration interaction processes not included in theHartree-Fock scheme. IV. RESULTS
Figure 2 shows NIXS data (circles) at the Ce N , -edges of CeRhIn (a) and CeCoIn (b) plus simula-tions (lines) for two scattering directions, (cid:126)q (cid:107) [100] (blue)and (cid:126)q (cid:107) [110] (green). The overall shape of the spectralooks very similar and represents the multipole scatter-ing expected for the Ce N , -edges. Figure 2 (c)and (d) show the directional dependencies I (cid:126)q (cid:107) [110] - I (cid:126)q (cid:107) [100] (dichroism), the experimental data as circles and simula-tions for the Γ − and Γ +7 as orange and gray lines, respec- S i m u l a t i o n s f o r G -7 a Difference (rel. units)
E n e r g y t r a n s f e r ( e V )
FIG. 3. (color online) Simulated difference spectra I (cid:126)q (cid:107) [110] -I (cid:126)q (cid:107) [100] for α values of 0 and 0.5, and of 0.38 for CeRhIn and 0.13 for CeCoIn . Co ( α =0.13) y x z 7.0 Γ - Γ Γ
6 top view Γ - Γ Γ
6 top view
Rh ( α =0.36) Γ Γ Γ Ir ( α =0.25) FIG. 4. (color online) Crystal-field splitting of J = 5/2 multi-plet of CeCoIn , CeRhIn and for completeness of CeIrIn asadapted from Ref.s 20–22. For M = Co and Rh the sign of thewavefunction is now determined which has been taken into ac-count when drawing the f charge densities of the respectivestates. tively. The expected dichroisms for a Γ − and Γ +7 groundstate are opposite in sign so that the present experimentprovides an either-or result which makes the interpreta-tion of the data straight forward.For CeRhIn the N , edges in Fig. 2 (a) as well as thedichroism in Fig. 2 (c) are fairly well reproduced by thesimulation with a Γ − ground state (orange line). Here weused the α value of 0.38 as determined with XAS. Thesame simulation with a Γ +7 ground state is clearly in con-tradiction to the observation (gray line). For CeCoIn the agreement between simulated and experimental di-rectional dependence in Fig. 2 (d) is not as good, andwe will discuss below the possible reasons for this. Alsohere the simulation was performed with the correspond-ing α value from XAS, α = 0.13. Most importantly,however, we conclude that the ground state of CeCoIn must be also predominantly of Γ − character because thesize of the scaling between experiment and calculation isclearly positive (0.63 ± and because the Γ +7 is, asfor CeRhIn , in clear contradiction to the observation. V. DISCUSSION
In Figure 3, we compare the directional dependenceI (cid:126)q (cid:107) [110] - I (cid:126)q (cid:107) [100] for several α values. For α = 0 (or 1)the dichroism is zero because in this case the Γ state isa pure J z state and rotational invariant; for α = 0.5 thedichroism is largest. For CeRhIn ( α = 0.38) the mixingfactor α is closer to 0.5 than for CeCoIn ( α = 0.13) sothat the expected dichroism for CeCoIn is smaller thanfor CeRhIn . But, the expected reduction due to the dif-ferent α values still does not account for the stronglyreduced directional effect in CeCoIn . A natural expla-nation could be the stronger cf -hybridization in CeCoIn with respect to CeRhIn : in CeCoIn the coherence tem-perature T ∗ is of the order of 50 K which is compara-ble to the energy splitting of the two lowest crystal-fieldstates (6.8 meV or ≈
75 K) (see Fig. 4) so that the firstexcited crystal-field state will contribute to the groundstate via hybridization. The first excited crystal-field state is the Γ +7 which has the opposite dichroism of thecrystal-field ground state Γ − so that the net dichroismof the hybridized ground state will be reduced. In short,the strongly reduced directional effect in CeCoIn is dueto the presence of strong hybridization. Assuming thefirst excited crystal-field state Γ +7 contributes 19 % to theground state of CeCoIn yields a very good agreement ofmeasured dichrosim and simulation (see dark red line inFig. 2 (d)). However, 19 % of Γ +7 mixed into the groundstate by hybridization must be an overestimation becausethe Γ +7 admixture was not accounted for when describ-ing the linear dichroism in XAS. It turns out that bothdata sets, the directional dependence in NIXS and thelinear dichroism in XAS, can be analyzed consistentlyand are well described with α = 0.10 and 13 % of Γ +7 .Figure 4 summarizes what we know now aboutthe crystal-field splittings of the J = 5/2 multiplet ofCe M In . The splittings and α values are taken frominelastic neutron scattering and XAS as published inRef. 20–22. The present NIXS experiments on CeCoIn and CeRhIn add the missing information that the Γ − dominates the ground state for both compounds, i.e. thelobes of the crystal-field ground state are along the crys-tallographic (110) direction. These Γ − ground state or-bitals extend more in the z -direction than the Γ +7 atabout 6-7 meV so that the scenario as shown in Fig.1 (c)applies to CeCoIn and CeRhIn , whereas the wavefunc-tions projected out by DFT+DMFT calculations haveopposite signs (see Fig.1 (b)).The tips of the lobes of the Γ − ground state wave-functions of CeCoIn and CeRhIn are pointing towardsthe triangle In2-In1-In2 (see Fig. 1 or Fig.1 (c)). It istherefore reasonable to conclude that the impact of thehybridization with the out-of-plane In2 atoms is moreimportant than the hybridization with the in-plane In1atoms. This is in agreement with the results of theCeRh − δ Ir δ In substitution series where the orbitalsthat are more extended along the c -axis tend to hybridizemore strongly. Nevertheless, the present results also showthat hybridization with the In1 atoms is important andthis supports the results of the CeCo(In − y Sn y ) substi-tution series; the Sn atoms go preferably to the In1sites leading to a stronger hybridization in the plane. We would like to note that the revised value of α forCeCoIn that is obtained when taking into account thefirst excited crystal-field state leads to a crystal-fieldground state orbital that is even more extended in the z -direction than for the originally anticipated value. Thesame should apply to CeIrIn when allowing a hybridiza-tion induced contribution of the first excited crystal-fieldstate. Hence, the correlation of stronger hybridizationwith the In2 atoms due to ground state orbitals that aremore extended in z -direction and superconductivity stillholds. A modest increase in the contribution of J z = | ± / (cid:105) to the Γ − -state of CeRhIn will promote overlap of f -orbitals with p -states of In(1) at the expense f -In(2)hybridization. Mixing of Zeeman-split ground and Γ +7 first excited crystal-field levels, in principle, could pro-duce such a modest increase in the J z = | ± / (cid:105) contri-bution. Indeed, recent high-field magnetostriction andnuclear magnetic resonance measurements on CeRhIn are consistent with this possibility that appears to bea significant contributing factor to field-induced Fermi-surface reconstruction in CeRhIn subject to a magneticfield near 30 T.In the limit of strong intra-atomic Coulomb inter-actions, which is typical of strongly correlated metalslike CeCoIn and CeRhIn , the magnetic exchange J is proportional to the square of the matrix element (cid:104) V kf (cid:105) that mixes conduction and f -wavefunctions. Both Kondo and long-range Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions depend on the magnitudeof J that is set by (cid:104) V kf (cid:105) and, consequently, by the f -orbital configuration. These interactions are fundamen-tal for a description of Kondo-lattice systems and theirrelative balance can be tuned by non-thermal control pa-rameters, such as magnetic field and pressure. Modestpressure applied to CeRhIn tunes its antiferromagnetictransition temperature toward zero temperature where adome of unconventional superconductivity emerges witha maximum transition temperature very close to that ofCeCoIn and also changes the Fermi surface from smallto large as in CeCoIn . We do not know if the f -orbitalconfiguration of CeRhIn at these pressures is the sameas that of CeCoIn , but this is an interesting possibilitythat merits study. VI. SUMMARY In f -based materials, the shape of the crystal-field wavefunctions ultimately determines the origin ofanisotropic hybridization in these materials and theirground state. Here, we show that the ground state ofCe M In ( M = Co,Rh) is a Γ − = | α | | ± / (cid:105) − √ − α | ∓ / (cid:105) doublet with lobes Γ − pointing toward the 110 di-rection, i.e., the lobes have xy character. Though carefulDFT+DMFT calculations shed light on these materials,the crystal-field scheme obtained is different from ourexperimental one. Our work settles the question on theorientation of f -orbitals in the ground state of Ce M In and will stimulate theoretical developments that take intoaccount the actual wavefunctions. VII. ACKNOWLEDGMENT
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07) and(0 . ± .
05) for CeRhIn and CeCoIn , respectively. Wecan estimate the scaling between experiment and theoryfrom these values by dividing by the Pearson correlationcoefficients c = 0.72 for CeRhIn and 0.28 for CeCoIn and obtain (0 . ± .
10) for CeRhIn and (0 . ± .
19) for
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