Magnetic order and crystalline electric field excitations of the quantum critical heavy fermion ferromagnet CeRh_6Ge_4
J. W. Shu, D. T. Adroja, A. D. Hillier, Y. J. Zhang, Y. X. Chen, B. Shen, F. Orlandi, H. C. Walker, Y. Liu, C. Cao, F. Steglich, H. Q. Yuan, M. Smidman
aa r X i v : . [ c ond - m a t . s t r- e l ] F e b Magnetic order and crystalline electric field excitations of the quantum critical heavyfermion ferromagnet CeRh Ge J. W, Shu, D. T. Adroja,
2, 3
A. D. Hillier, Y. J. Zhang, Y. X. Chen, B. Shen, F. Orlandi, H. C. Walker, Y. Liu,
1, 5
C. Cao, F. Steglich,
1, 6
H. Q. Yuan,
1, 5, 7, 8 and M. Smidman
1, 5, ∗ Center for Correlated Matter and Department of Physics, Zhejiang University, Hangzhou 310058, China ISIS Facility, STFC Rutherford Appleton Laboratory,Harwell Oxford, Oxfordshire OX11 0QX, United Kingdom Highly Correlated Matter Research Group, Physics Department,University of Johannesburg, P.O. Box 524, Auckland Park 2006, South Africa Institute for Advanced Materials, Hubei Normal University, Huangshi 435002, China Zhejiang Province Key Laboratory of Quantum Technology and Device,Department of Physics, Zhejiang University, Hangzhou 310058, China Max Planck Institute for Chemical Physics of Solids, Dresden, Germany State Key Laboratory of Silicon Materials, Zhejiang University, Hangzhou 310058, China Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China (Dated: February 26, 2021)CeRh Ge is an unusual example of a stoichiometric heavy fermion ferromagnet, which can becleanly tuned by hydrostatic pressure to a quantum critical point. In order to understand the originof this anomalous behavior, we have characterized the magnetic ordering and crystalline electricfield (CEF) scheme of this system. While magnetic Bragg peaks are not resolved in neutron pow-der diffraction, coherent oscillations are observed in zero-field µ SR below T C , which are consistentwith in-plane ferromagnetic ordering consisting of reduced Ce moments. From analyzing the mag-netic susceptibility and inelastic neutron scattering, we propose a CEF-level scheme which accountsfor the easy-plane magnetocrystalline anisotropy, and suggests that the orbital anisotropy of theground state and low lying excited state doublets is an important factor giving rise to the observedanisotropic hybridization. In heavy fermion materials, the competition betweenmagnetic exchange interactions which couple local mo-ments, and the Kondo interaction between local momentsand the conduction electrons, can frequently be tunedby non-thermal parameters such as pressure, magneticfields, or chemical doping [1]. Consequently, the anti-ferromagnetic (AFM) ordering temperature can often becontinuously suppressed to zero at a quantum criticalpoint (QCP), where there is a breakdown of Fermi liquidbehavior, and the large accumulation of entropy can leadto emergent phases such as unconventional superconduc-tivity [2, 3]. Conversely, ferromagnetic (FM) QCP’s arenot usually found, due to either a first-order disappear-ance of FM order, or the interjection of different groundstates [4]. Theoretically, it was predicted that FM QCP’sare forbidden in clean itinerant FM systems [5, 6], andwhile there have been some reports of their occurrencein some doped materials, including YbNi (P − x As x ) [7],CePd . Rh . [8], URh − x Ru x Ge [9], and Ni − x Rh x [10], in such cases a disorder driven suppression of thefirst-order transition is difficult to exclude.Recently, the heavy fermion ferromagnet CeRh Ge with a Curie temperature T C = 2 . T C , yielding aFM QCP [12, 13]. The QCP is accompanied by a strange metal phase, with a linear temperature dependence ofthe resisitivity and a logarithmic divergence of the spe-cific heat coefficient [12]. To account for this behavior inlight of the previously reported prohibition of FM QCP’sin itinerant systems, it was proposed that CeRh Ge ex-hibits local quantum criticality, where the requisite en-tanglement of the local moments is generated by their xy -anisotropy [12]. Alternatively, it was proposed thatthe pressure-induced first-order transition is avoided bythe soft-modes, which prevent FM quantum criticality,becoming massive due to the antisymmetric spin-orbitcoupling arising from the broken inversion symmetry inthe crystal lattice (space group P ¯6 m
2) [14]. Additionalstudies therefore are necessary to gain insight into theorigin of the FM quantum criticality, and to distinguishbetween the different theoretical scenarios. It is partic-ularly important to both further characterize the natureof the magnetic ordering, and to understand the origin ofthe magnetic anisotropy. Therefore, we performed neu-tron diffraction, muon-spin relaxation ( µ SR), and inelas-tic neutron scattering measurements on polycrystallinesamples of CeRh Ge .Neutron powder diffraction measurements were per-formed on the WISH diffractometer at the ISIS pulsedneutron facility [15, 16], both at 4 K in the paramag-netic state, and well below T C at 0.27 K. The data at FIG. 1. (Color online) (a) Neutron powder diffraction pat-tern of CeRh Ge at 4 K, for one pair of detector banks onthe WISH diffractometer. The results from a Rietveld refine-ment are also displayed. (b) Difference between the patternstaken at 0.27 K, and 4 K, for three pairs of banks. The scat-tering angles for each of the detector banks are displayed inthe panels. (c) Zero-field µ SR spectra of CeRh Ge at twotemperatures below T C , where the solid lines show the resultsfrom fitting with Eq. 1. (d) Temperature dependence of theinternal fields obtained from fitting the µ SR data, where re-sults for the Lorentzian relaxation rate are shown in the inset. a = 7 . c = 3 . d -spacings correspond-ing to non-integer ( hkl ) reflections, in-line with a lack ofan AFM component to the magnetism. For FM order,the magnetic Bragg peaks are situated on the structuralpeaks, and therefore weak magnetic peaks arising fromsmall ordered moments will be more difficult to detect.No additional intensity on any nuclear peaks is consis-tently resolved across multiple pairs of detector banks.As shown in the top panel of Fig 1(b), additional in-tensity is observed on the (002) Bragg peak at 0.27 Kfor one pair of banks, as expected for FM order within-plane moments. However, similar behavior is not re- solved for banks situated at different scattering angles,and therefore this increase might be an artifact arisingfrom imperfect normalization to the monitor.Evidence for magnetic order in CeRh Ge is howeverrevealed by zero-field µ SR measurements, which wereperformed on the MuSR spectrometer at the ISIS facil-ity [17, 18]. As shown in Fig 1(c), coherent oscillationsare observed below T C , demonstrating the occurrence oflong range magnetic order. The data in the magneticallyordered state are best analyzed taking into account threeoscillation frequencies, using A ( t ) = A + X i =1 A i cos( γ µ B i t + φ )e − ( σ i t ) / + A e − λt (1)while in the paramagnetic state only the first and lastterms are utilized. Here A i are the initial asymme-tries corresponding to local fields B i , with Gaussian re-laxation rates σ i , while λ is the Lorentzian relaxationrate of a non-oscillating component, and γ µ is the muon-gyromagnetic ratio. The temperature dependence of thethree values of B i , as well as λ , are displayed in Fig. 1(d). B and B are found to increase with decreasing temper-ature, whereas B reaches a maximum at about 1 K, anddecreases slightly at lower temperatures. λ exhibits apeak around the magnetic transition, as well as a steepincrease below about 0.8 K.The positions of the muon stopping sites were esti-mated via density functional theory calculations with f -electrons as core electrons [19, 20], from the minimumenergy positions for a positive | e | charge. Two crys-tallographically inequivalent sites were identified, s =( , , s = (0 . , . , .
01) which corresponds to a localminimum. The latter corresponds to six equivalent crys-tallographic positions related by three-fold and mirrorsymmetries, and since the three-fold symmetry is brokenby in-plane FM order, this leads to up to three distinctlocal fields associated with the s sites. From low temper-ature magnetization measurements, the ordered in-planemoment is estimated to be around 0 . − . µ B / Ce [12].Estimates for the local magnetic fields at the stoppingsites s and s arising from in-plane FM order were cal-culated using the muesr package [21]. For uniform FMorder with a moment of 0 . µ B / Ce orientated along the a -axis, local fields of 364 and 152 G are calculated for the s sites, and 87 G is calculated for s , in comparison tofitted values for B , B and B of 405, 151, and 56 G, re-spectively. On the other hand, a moment of 0 . µ B / Ceyields 58 G for s , in good agreement with B , but yieldsunderestimates of 235 and 98 G for the s sites. In mag-netic metals, an accurate comparison between the calcu-lated and observed local fields requires accounting for themuon contact hyperfine fields, and similar discrepanciesto calculations have been found for heavy fermion mag-nets [22]. Moreover, a change in the hyperfine fields with ∆ (cid:1) = 5.8 meV ∆ (cid:2) = 22.1 meV ∆ (cid:2) ∆ (cid:1) |±5/2>|±3/2>|±1/2> (cid:1) + s (a) (b) (c) (cid:0) + s (cid:2) (cid:3) FIG. 2. (Color online) (a) Temperature dependence of the inverse magnetic susceptibility of single crystal CeRh Ge for twofield directions [12]. The solid lines show the results from fitting with a CEF model described in the text. (b) CEF levelscheme and wavefunctions obtained from fitting with the CEF model, where the angular distributions of the wave functionsare also displayed. (c) Crystal structure of CeRh Ge , with Ce, Rh, and Ge shown in yellow, grey and purple, respectively.Ferromagnetic order with moments along the a -axis is also illustrated, as well as the proposed muon stopping sites s and s (orange and green spheres), and the orientation of the ground state orbitals from the CEF model. temperature could lead to the non-monotonic behaviorof B , which together with the increase of λ below 0.8 K,may point to the low temperature evolution of the un-derlying correlated state. The differences may also arisefrom uncertainties in the muon stopping sites, the ori-entation of the moments in the basal plane, or a spatialmodulation of the ordered moment, but in the case ofthe latter AFM Bragg peaks would be expected to be ob-served in neutron diffraction. As a result, both neutrondiffraction and ZF- µ SR are consistent with FM order inCeRh Ge , with a small in-plane ordered moment, andindicate the absence of any significant AFM component.In order to determine the splitting of the J = 5 / Ge by crystalline elec-tric fields (CEF), the single crystal magnetic suscepti-bility [12] was analyzed using the Hamiltonian H CF = B O + B O , where B mn and O mn are Stevens CEFparameters and operator equivalents, respectively [23].Note that since the Ce site has hexagonal point sym-metry, there are only two non-zero Stevens parameters B and B , and therefore there is no mixing of differ-ent | m J i states in the atomic wave functions. The re-sults are displayed in Fig. 2(a), with fitted values of B = 1 .
25 meV and B = 0 . λ ab = − . λ c = − . ground state doublet ψ ± GS = | ± i , a low-lying first excited state ψ ± = | ± i separated by ∆ = 5 . ψ ± = | ± i at ∆ = 22 . | ± i ground state is anticipated, since thisis the only eigenstate which corresponds to a non-zeroin-plane moment, with h µ x i = h ψ ∓ | g J ( J + + J − ) | ψ ± i =1 . µ B /Ce. Since h µ x i is much larger than the observedlow temperature moment of around 0 . − . µ B / Ce, thisindicates that there is a reduced ordered moment, eitherdue to Kondo screening processes or significant zero-pointfluctuations [12]. From the large positive value of B , -3 0 3 601020304050 -5 0 5 10 150510152025-10 0 10 20 300510152025 20 40 60 8002468 (a) E i = 12 meV20(cid:176) - 60(cid:176) CeRh Ge
7K LaRh Ge
7K CeRh Ge Ge S ( Q , ) ( m b s r - m e V - f. u . - ) (b) E i = 20 meV20(cid:176) - 60(cid:176) (c) E i = 38 meV 10(cid:176) - 60(cid:176) Energy transfer (meV) (d) E i = 100 meV10(cid:176) - 60(cid:176) FIG. 3. (Color online) Low angle cuts of the inelastic neutronscattering spectra of CeRh Ge and LaRh Ge at 7 K and100 K for incident energies of (a) 12 meV, (b) 20 meV, (c)38 meV, and (d) 100 meV. The integrated angular ranges aredisplayed in the panels. the ab -plane is expected to correspond to the easy direc-tion of magnetization, in-line with the observed high andlow temperature susceptibilities. This suggests that thesingle-ion anisotropy arising from the local environmentof the Ce ions is sufficient to account for the observedeasy-plane anisotropy, in contrast to many Kondo ferro-magnets which order along the hard axis [24].Inelastic neutron scattering measurements were per-formed on powder samples of CeRh Ge and the non-magnetic analog LaRh Ge , using the MERLIN spec-trometer at the ISIS facility [25]. Figure 3 displays lowangle cuts of the data normalized to absolute units attwo temperatures, for four different incident energies E i .No well defined CEF levels can be detected at energytransfers up to 80 meV. On the other hand, over a large E i
12 meV 20 meV 38 meV 100 meV S m ag ( Q , ) ( m b s r - m e V - f. u . - ) Energy transfer (meV) (cid:215)0.1
FIG. 4. (Color online) Magnetic contribution to the inelasticneutron scattering intensity versus energy transfer at 7 K,for four different incident energies. The solid line shows thecalculated inelastic response for the CEF scheme in Fig. 2,where the FWHM of the quasielastic and inelastic peaks are3.3 meV (2 T K ), while the dashed line shows the case wherethe inelastic peak corresponding to the first CEF excitationhas a FWHM of 30 meV. range of energy transfers, the scattering from CeRh Ge is consistently larger than the La-analog indicating thepresence of broad magnetic scattering, extending from atleast the elastic line ( ∼ . E i = 12 meV) upto at least 60 meV. Note that magnetic scattering couldnot be resolved on measurements performed at lower en-ergies on the OSIRIS spectrometer (not displayed) [26],likely due to the weak and broad nature of the mag-netic response. The magnetic scattering for the MER-LIN data was estimated by subtracting the low angleLa-cuts, from the CeRh Ge data, taking into accountthe different neutron scattering cross sections, and theresults are shown in Fig. 4. It can be seen that the mag-netic scattering is strongest at low energies, but with along tail up to high energies. If this broad scatteringwere associated with the ground state doublet, i.e. cor-responding to quasielastic scattering, this would imply avery large Kondo temperature T K on the order of hun-dreds of Kelvin [27]. This is in contrast to the moderatevalue of T K = 19 K deduced from comparing the mag-netic entropy to a spin-1/2 Kondo model [11]. The solidline in Fig. 4 shows the results of calculating the inelasticneutron spectra for the CEF scheme in Fig. 2(b), witha full-width at half maximum (FWHM) for all the exci-tations of 3.3 meV (2 T K ). It can be seen that this failsto account for the broad magnetic scattering, and a welldefined excitation at ∆ would be resolved. Note that noexcitation at ∆ is expected, since the matrix elementsfor the transition from | ± i to | ± i are zero due to theneutron selection rules ∆ m J = ±
1. On the other hand,the dashed line shows the case with a quasielastic FWHMof 2 T K , but a much broader inelastic excitation with aFWHM of 30 meV, and it can be seen that this scenariocan well account for the broad scattering. This suggests that the inelastic neutron scattering results are consis-tent with the CEF scheme deduced from the magneticsusceptibility, but with the low-lying CEF excitation at5.8 meV being significantly broadened due to hybridiza-tion with the conduction electrons.The CEF scheme displayed in Fig. 2(b) can account forthe in-plane orientation of the Ce-moments of CeRh Ge below T C , which was proposed to be vital for generat-ing the necessary entanglement for avoiding a first-ordertransition under pressure, allowing for the occurrence ofa ferromagnetic QCP [12]. The angular distributions ofthe CEF wave functions are also displayed. Notably, boththe ground state and first excited doublet at 5.8 meVprimarily have electron density out of the basal plane,which may explain the strongly anisotropic hybridizationrevealed by angle-resolved photoemission spectroscopy(ARPES), with significantly stronger hybridization alongthe c -axis [28]. Moreover, the low-lying first excited dou-blet appears to hybridize much more strongly with theconduction electrons, which may be a consequence ofgreater overlap with the out-of-plane Rh(2) and Ge(2)atoms, while the ground state charge density is orientatedtowards the neighboring Ce atoms [Fig. 2(c)]. Such a sce-nario with a more strongly hybridized excited CEF levelhas been predicted to give rise to metaorbital transitions[29], which was proposed theoretically for CeCu Si [30],yet has not been observed experimentally [31]. The influ-ence of the first excited state on the low temperature be-havior of CeRh Ge may be inferred from the Kadowaki-Woods ratio corresponding to a ground state degeneracy N = 4, on both sides of the QCP [12]. In fact, the angu-lar distribution of the ground state 4 f orbitals has beenidentified as a key parameter for tuning the hybridizationof the Ce(Co,Ir,Rh)In family of heavy fermion super-conductors [32, 33], where more prolate Γ ground statesare associated with stronger c − f hybridization, likelydue to stronger hybridization with out-of-plane In atoms[34]. Therefore our results suggest that the anisotropichybridization, and by extension anisotropic magnetic ex-change interactions, are not only driven by the quasi-one-dimensional arrangement of Ce-chains, but by theangular distribution of the CEF orbitals arising from the local environment of the Ce atoms.In summary, our neutron diffraction and µ SR mea-surements are consistent with FM order in CeRh Ge ,with a reduced magnetic moment compared to that ex-pected from the CEF ground state. We propose a CEFscheme which can account for the easy-plane anisotropyof CeRh Ge , which was predicted to be crucial for theoccurrence of FM quantum criticality in this system [12].Moreover, the broad magnetic scattering observed in in-elastic neutron scattering suggests the presence of strong c − f hybridization, where the low lying first excited CEFlevel couples more strongly than the ground state. Thiscould potentially reconcile there being significant Kondoscreening processes which reduce the ordered moment,with the conclusion of localized 4 f electrons inferred fromquantum oscillation measurements [19]. These resultssuggest that the anisotropy of the CEF orbitals is animportant factor in the observed anisotropic hybridiza-tion [28], and systems with similarly anisotropic groundstate orbitals could be good candidates for searching foradditional quantum critical ferromagnets. It is of partic-ular interest to examine whether there is correspondinglylarge anisotropy in the magnetic exchange interactions ofCeRh Ge , i.e. quasi-one-dimensional magnetism, whichcould be determined from single crystal inelastic neutronscattering or THz spectroscopy. ACKNOWLEDGMENTS
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