Magnetism trends in doped Ce-Cu intermetallics in the vicinity of quantum criticality: realistic Kondo lattice models based on dynamical mean-field theory
MMagnetism trends in doped Ce-Cu intermetallics in the vicinity of quantum criticality:realistic Kondo lattice models based on dynamical mean-field theory
Munehisa Matsumoto , Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, JAPAN Institute of Materials Structure Science, High Energy AcceleratorResearch Organization, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan (Dated: April 3, 2020)The quantum critical point (QCP) in the archetypical heavy-fermion compound CeCu dopedby Au is described, accounting for the localized 4 f -electron of Ce, using realistic electronicstructure calculations combined with dynamical mean-field theory (DMFT). Magnetism trends inCe(Cu − (cid:15) Au (cid:15) ) (0 < (cid:15) (cid:28)
1) are compared with those in Co-doped CeCu , which resides on thenon-ferromagnetic side of the composition space of one of the earliest rare-earth permanent mag-net compounds, Ce(Co,Cu) . The construction of a realistic Doniach phase diagram shows thatthe system crosses over a magnetic quantum critical point in the Kondo lattice in 0 . < x < . − x Co x ) . Comparison between Au-doped CeCu and Co-doped CeCu reveals that the sweptregion in the vicinity of QCP for the latter thoroughly covers that of the former. The implicationsof these trends on the coercivity of the bulk rare-earth permanent magnets are discussed. PACS numbers: 71.27.+a, 75.50.Ww, 75.10.Lp
I. MOTIVATION
While heavy-fermion (HF) materials and rare earthpermanent magnets (REPM’s) have gone through con-temporary developments since the 1960s , apparentlylittle overlap has been identified between the two classesof materials. One of the obvious reasons for the absenceof mutual interest lies in the difference in the scope ofthe working temperatures: HF materials typically con-cern low-temperature physics of the order of 10K or evenlower while REPM concerns room temperature at 300Kor higher. The other reason is that the interesting re-gions in the magnetic phase diagram sit on the oppositesides, where HF behavior appears around a region wheremagnetism disappears while with REPM the obviousinterest lies in the middle of a ferromagnetic phase. Inretrospect, several common threads in the developmentsfor HF compounds and REPM’s can be seen: one of theearliest REPM’s was Ce(Co,Cu) where Cu was addedto CeCo to implement coercivity, and CeCu was even-tually to be identified as an antiferromagnetic Kondo lat-tice .One of the representative HF compounds is CeCu that was discovered almost at the same time as the cham-pion magnet compound Nd Fe B . While REPM’smake a significant part in the most important materi-als in the upcoming decades for a sustainable solutionof the energy problem with their utility in traction mo-tors of (hybrid) electric vehicles and power generators,HF materials might remain to be mostly of academic in-terests. But we note that a good permanent magnet ismade of a ferromagnetic main-phase and less ferromag-netic sub-phases. For the latter compounds in REPM’s,we discuss possible common physics with HF materials,namely, magnetic quantum criticality where magnetismdisappears and associated scales in space-time fluctua-tions diverge, and propose one of the possible solutions for a practical problem on how to implement coercivity,which measures robustness of the metastable state withmagnetization against externally applied magnetic fields.Even though the mechanism of bulk coercivity on themacroscopic scale in REPM’s is not entirely understood,the overall multiple-scale structure has been clear in thatthe intrinsic properties of materials on the microscopicscale of the scale of O (1) nm is carried over to the macro-scopic scale via mesoscopic scale. Namely, possible sce-narios in coercivity of Nd-Fe-B magnets and Sm-Comagnets have been so far discussed as follows. a. Nd-Fe-B magnets Propagating domain wallsaround a nucleation center of reversed magnetization areblocked before going too far. Infiltrated elemental Ndin the grain-boundary region that is paramagnetic in thetypical operation temperature range of O (100) K neutral-izes inter-granular magnetic couplings among Nd Fe Bgrains . Single-phase Nd Fe B does not show coerciv-ity at room temperature and fabrication of an optimalmicrostructure on the mesoscopic scale, with the infil-trated Nd metals between Nd Fe B grains, seems to becrucial to observe bulk coercivity. b. Sm-Co magnets and Ce analogues
Pinning cen-ters of domain walls are distributed over cell-boundaryphases made of Sm(Co,Cu) which separate hexagonally-shaped cells of Sm (Co,Fe) . The uniformity of the cellboundary phase suggests that the pinning intrinsi-cally happens on the microscopic scale in Sm(Co,Cu) which freezes out the magnetization reversal dynamics.Also for CeCo , addition of Cu has been found to helpthe development of bulk coercivity without much partic-ular feature in the microstructure, suggesting here againan intrinsic origin contributing to the bulk coercivity.Solution of the overall coercivity problem takes out-of-equilibrium statistical physics, multi-scale simulationsinvolving the morphology of the microstructure in theintermetallic materials, electronic correlation in 4 f - a r X i v : . [ c ond - m a t . m t r l - s c i ] A p r T N [ K ] t=(J K -J K,QCP )/J
K,QCP
Kondo-screenedAntiferromagneticCeCu CeCu AuCeCu CeCu CoCeCu Co FIG. 1. (Color online) Realistic Doniach phase diagram forthe target compounds with a rescaled horizontal axis to mea-sure an effective distance to the magnetic quantum criticalpoint for each target compound. It is seen that Co-dopedCeCu moves from the magnetic side towards the Kondo-screened phase crossing QCP, while Au-doped CeCu movesto the opposite direction. Arrows are guide for the eye. electrons, finite-temperature magnetism of Fe-based fer-romagnets, and magnetic anisotropy, each of which byitself makes a subfield for intensive studies. Faced withsuch a seemingly intractable problem, it is important tobuild up fundamental understanding step by step. There-fore, we clarify the magnetism trends around quantumcriticality in Ce-Cu intermetallics, as a part of 4 f -3 d in-termetallics that belong to a common thread betweenHF materials and REPM, in order to pin-point a possi-ble intrinsic contribution to the coercivity, specifically viaexponentially growing length scales in spatial correlationand characteristic time in the dynamics.The magnetization in REPM’s derives from 3 d -electronferromagnetism coming from Fe-group elements and 4 f -electrons in rare-earth elements provide the uni-axialmagnetic anisotropy for the intrinsic origin of coerciv-ity. Sub-phases are preferably free from ferromagnetismto help coercivity e.g. by stopping the propagation of do-main walls. In the practical fabrication of REPM, bothof the main-phase compound and other compounds forsub-phases should come out of a pool of the given set ofingredient elements. Investigations on non-ferromagneticmaterials that appear in the same composition space asthe ferromagnetic material are of crucial importance forcontributing the intrinsic information into the solution ofthe coercivity problem.Thus we investigate the Cu-rich side of the compositionspace in Ce(Co,Cu) and inspect the magnetism trendsaround the HF compound, CeCu . It is found that Codoping into CeCu drives the material toward a mag-netic quantum critical point (QCP), to the extent that3 d -electron ferromagnetism coming from Co does not dominate, which seems to be the case experimentally when the concentration of Co is below 40%. It has alsobeen known that Au-doped CeCu goes into quantumcriticality , a trend which is reproduced in the samesimulation framework. With CeCu as one of the mostrepresentative HF materials, experimental measurementsand theoretical developments have been extensivelydone. Our finding basically reproduces what has alreadybeen agreed on the location of magnetic QCP, but thespirit of our microscopic description may not entirely bethe same as some of the past theoretical works . Ourdescription should be more consistent with even olderworks in the fundamental spirit with the proper in-corporation of realistic energy scales based on electronicstructure calculations. We may fail in catching somesubtlety specific to Au-doped CeCu , but our approachshould be suited rather for general purposes in providingan overview over intrinsic magnetism of f - d intermetallicsto extract the common physics therein.We set up a realistic Kondo lattice model for thesecases and see the followings: 1) CeCu sits very close tothe QCP, 2) Au-induced QCP can also be described onthe basis of a conventional Kondo lattice model as down-folded from realistic electronic structure data featuringlocalized 4 f -electrons, at least concerning the relative lo-cation of QCP, without invoking valence fluctuations orthe specialized Kondo-Heisenberg model to describe localquantum criticality , in contrast to some of those pre-vious developments for Au-doped CeCu , and 3)Co-doping in CeCu drives the material toward the QCPin the opposite direction as Au-doping does in CeCu .The main results are summarized in Fig. 1 where the Au-doped CeCu and Co-doped CeCu are located arounda magnetic QCP following a rescaled realistic Doniachphase diagram .The rest of the paper is organized as follows. In thenext section we describe our methods as specificallyapplied to the target materials: pristine CeCu , CeCu ,and doped cases. In Sec. III magnetism trends in thetarget materials are clarified. In Sec. IV several issuesremaining in the present descriptions and possible impli-cations from HF physics on the intrinsic part of the so-lution of the coercivity problem of REPM are discussed.The final section is devoted to the conclusions and out-look. II. METHODS AND TARGET MATERIALS
We combine ab initio electronic structure calcula-tions on the basis of the full-potential linear muffin-tin orbital method and dynamical mean-field theory(DMFT) for a Kondo lattice model with well-localized4 f -electrons , to construct a Doniach phase diagram adapted for a given target material to identify an effec-tive distance of the material to a magnetic quantum criti-cal point. Electronic structure calculations follow densityfunctional theory (DFT) within the local density ap-proximation (LDA) . Our realistic simulation frame-work can be regarded as a simplified approach inspiredby LDA+DMFT , where electronic structure calcula-tions describing the relatively high-energy scales and asolution of the embedded impurity problem in the low-ermost energy scales are bridged: here a realistic Kondolattice model is downfolded from the electronic struc-ture calculations for Ce-based compounds with well lo-calized 4 f -electrons .More specifically, our computational framework ismade of the following two steps:1. For a given target material, LDA+Hubbard-I isdone to extract hybridization between localized 4 f -electrons and conduction electrons, −(cid:61) ∆( ω ) /π asa function of energy ¯ hω around the Fermi level. Po-sition of the local 4 f -electron level below the Fermilevel is determined as well.2. A realistic Kondo lattice model (KLM) with theKondo coupling J K is defined following the rela-tions : J K = | V | (cid:20) | (cid:15) f | + 1( (cid:15) f + U ff − J Hund ) (cid:21) , (1) | V | ≡ − π (cid:90) D −∞ d ω Tr (cid:61) ∆( ω ) N F , (2)which is a realistic adaptation of Schrieffer-Wolfftransformation to map the Anderson model to Kondo model. Here U ff and J Hund are theCoulomb repulsion energy and an effective Hundcoupling between 4 f electrons, respectively, in(4 f ) configuration and D is an energy cutoff that defines the working energy window for the re-alistic Schrieffer-Wolff transformation. The trace inEq. (2) is taken over all 4 f -orbitals and dividing thetraced hybridization by N F ≡
14 gives the strengthof hybridization per each orbital. Experimental in-formation on the local level splittings is incorpo-rated for the 4 f -electron part. The thus definedKLM is solved within DMFT using the continuous-time quantum Monte Carlo impurity solver . ADoniach phase diagram separating the magneticphase and paramagnetic phase is constructed foreach of the target materials and the magnetic QCPis located.The realistic model parameters that appear inEqs. (1) and (2) are taken on an empirical basisreferring to past works , among which the originof the on-site Coulomb repulsion energy U ff = 5 eVbetween 4 f -electrons can be traced partly back topast electronic structure calculations and analyses ofphotoemission spectroscopy data . Even though onecan argue for material-specific data of U ff , here we aremore concerned with relative trends among the targetmaterials within a realistic model with fixed parametersto get an overview over a group of Ce-based compounds compound a [a.u.], b/a , c/a CeCu a = 15 . b/a = 0 . c/a = 1 . Au a = 15 . b/a = 0 . c/a = 1 . a = 9 . b/a = 1, c/a = 0 . Co (fixed to be the same as CeCu )CeCu Co (fixed to be the same as CeCu )TABLE I. Inputs to LDA+Hubbard-I: the lattice constantsof each target compound. with well localized 4 f -electrons, rather than pursuingpreciseness of each material-specific data point.Below we describe the details of the overall procedureone by one, taking CeCu as a representative case, partlyintroducing the results. A. LDA+Hubbard-I
The overall initial input here is the experimental latticestructure. This is taken from the past experimental liter-ature for pristine CeCu in Ref. 46 and CeCu in Ref. 20,and also for CeCu Au in Ref. 47 together with the par-ticular site preference of the dopant atom, Au. Ourinput lattice constants are summarized in Table I. Wenote that CeCu undergoes a structural phase transitionbetween a high-temperature orthorhombic phase anda low-temperature monoclinic phase , while CeCu Audoes not . In order to compare CeCu and CeCu Auon an equal footing and inspect the relative trends be-tween them and observing that the lattice distortion in-troduced by the structure transition seems to be minor ,we fix the working lattice structure of CeCu to be theorthorhombic phase and proceed to the downfolding tothe realistic Kondo-lattice model. The internal coordi-nates of atoms in CeCu and CeCu Au are shown inTable II. For Co-doped CeCu , various things happenin real experiments starting with the introduction of aferromagnetic conduction band coming from Co and lat-tice shrinkage even before reaching the valence transitionon the Co-rich side. Here in order to simplify the prob-lem and to focus on the magnetism trends concerningthe 4 f -electron QCP, we fix the working lattice to bethat of pristine CeCu and inspect the effects of replace-ments of Cu by Co. Following the site preference of Cofor Cu(3 g ) site as suggested in Ref. 50 for Cu-substitutedYCo , which we also confirm in separate calculations ,we replace Cu by Co in the 3 g sublattice one by one asshown in Table. III for CeCu Co and CeCu Co . Withthis particular set-up, the effects of Co-doping on CeCu has been effectively softened in our calculations. Howeverwe will see that Co-doping on CeCu drives the materialacross the QCP more effectively than Au-doping does forCeCu .LDA+Hubbard-I calculations give the hybridization −(cid:61) ∆( ω ) /π and position of the local 4 f -level, (cid:15) f . The (a)atom Wyckoff internal coordinateCe 4 c (0 . , . , . d (0 . , . , . c (0 . , . , . c (0 . , . , . c (0 . , . , . c (0 . , . , . c (0 . , . , . d (0 . , . , . c (0 . , . , . c (0 . , . , . c (0 . , . , . c (0 . , . , . and (b) CeCu Au. The spatial translation vec-tors are plainly (1 , , , , , , .atom Wyckoff internal coordinateCe 1 a (0 , , c (1 / , / (2 √ , c (1 / , − / (2 √ , g (1 / , −√ / , / a g (1 / , √ / , / b g (1 / , , / a Cu for CeCu and CeCu Co / Co for CeCu Co b Cu for CeCu / Co for CeCu Co and CeCu Co TABLE III. Inputs to LDA+Hubbard-I: internal coordinatesof the atoms in the hexagonal (Space Group No. 191) unit cellof CeCu , CeCu Co, and CeCu Co . Here the spatial transla-tion vectors are taken as ( − / , √ / , / , √ / , , , g ) sublattice. results for (cid:15) f and | V | as defined in Eq. (2) are summa-rized in Table IV. Raw data for −(cid:61) ∆( ω ) /π as traced overall of the 4 f -orbitals is shown in Fig. 2. compound (cid:15) f [eV] | V | CeCu − .
61 0 . Au − .
81 0 . − .
02 0 . Co − .
99 0 . Co − .
72 0 . f -electron level, (cid:15) f , where the offset is taken atthe Fermi level, is shown in the second column for each targetcompound. In the third column, the integrated hybridizationas defined in Eq. (2) is shown. - T rI m ∆ ( ω ) / π ω [eV] CeCu CeCu AuCeCu CeCu CoCeCu Co FIG. 2. (Color online) Calculated hybridization function forthe target compounds within LDA+Hubbard-I.
B. DMFT for the realistic Kondo lattice model
Following Ref. 27, the hybridization function betweenthe localized 4 f -orbital in Ce and conduction electronband defines the material-specific KLM. Here we describethe details of the Kondo impurity problem embedded inthe KLM within DMFT where we use the continuous-time quantum Monte Carlo solver for the Kondo im-purity problem .In the impurity problem embedded in DMFT we in-corporate the realistic crystal-field and spin-orbit levelsplittings in the local 4 f -orbital of Ce. The local 4 f - ∆ ∆ j=7/2multipletj=5/2multipletspin-orbitsplitting crystal-fieldsplitting FIG. 3. Schematic picture for local-level splitting caused byspin-orbit interaction and crystal fields. compound crystal-field splittingsCeCu ∆ = 7 meV, ∆ = 13 meV Ref. 54CeCu ∆ (cid:39) ∆ = 17 meV Ref. 55TABLE V. Input crystal-field splittings following past neu-tron scattering experiments. electron level scheme is shown in Fig. 3. For CeCu andhexagonal CeCu , it is known that the crystal structuresplits the j = 5 / [meV] between the lowest doublet and thesecond-lowest doublet, and ∆ [meV] between the low-est doublet and the third-lowest doublet. Crystal-fieldsplittings have been taken from the past neutron scatter-ing experiments as summarized in Table V. We set thelevel splitting between j = 5 / j = 7 / spin − orbit = 0 . .The input obtained with LDA+Hubbard-I to ourKondo problem is shown in Fig. 2. The Kondo coupling J K via a realistic variant of the Schrieffer-Wolff trans-formation is defined as in Eqs. (1) and (2). There D was the band cutoff that is set to be equal to the Coulombrepulsion U ff = 5 [eV], and J Hund is the effective Hundcoupling in the f multiplet to which the second termof Eq. (1) describes the virtual excitation from the (4 f ) ground state.We sweep J Hund to locate the QCP on a Doniachphase diagram and also to pick up the realistic datapoint at J Hund = 1 [eV]. This particular choice of theHund coupling in the virtually excited state (4 f ) hasbeen motivated by the typical intra-shell direct ex-change coupling of O (1) eV and an overall magnetismtrend in CeM Si (M=Au, Ag, Pd, Rh, Cu, and Ru),CeTIn (T=Co, Rh, and Ir) and pressure-induced quan-tum critical point in CeRhIn as studied in our previousworks, Ref. 27, 28, and 58, respectively. Thus the work-ing computational setup has been applied to elucidate themagnetism trends around QCP as precisely as have beendone for other representative HF compounds. In prac-tice, we define J K at J Hund = 0 as J K , and then sweep amultiplicative factor α = J K /J K , , calculating the tem-perature dependence of staggered magnetic susceptibility χ ( π, T ) for each α . In this way we can see where in theneighborhood of the QCP our target material with α cor-responding to the realistic number, J Hund = 1 eV, resideson the Doniach phase diagram.We calculate the staggered magnetic susceptibility χ ( π ) with the two-particle Green’s function following theformalism developed in Ref. 34 and using a random-dispersion approximation to decouple it into single-particle Green’s functions which would enhance thetransition temperature, in addition to the single-sitemean-field nature in DMFT. The calculated data for1 /χ ( π ) is shown in Fig. 4 for the case of CeCu . Thetemperature dependence of the reciprocal of the stag- / χ ( π ) T [K]CeCu α =1.1 α =1.11 α =1.12 α =1.13 α =1.135 (the realistic point) α =1.14 FIG. 4. (Color online) Calculated temperature depen-dence of the reciprocal of staggered magnetic susceptibilityfor CeCu , the reference compound. T N [ K ] J K [K]CeCu CeCu AuCeCu CeCu CoCeCu Co FIG. 5. (Color online) Realistic Doniach phase diagram forthe target compounds with the bare energy scale of the Kondocouplings. gered magnetic susceptibility 1 /χ ( π ) is observed for each J K = αJ K , and we extrapolate it linearly to the lowtemperature region to see if there is a finite N´eel temper-ature. We identify that the N´eel temperature vanishes inthe parameter range 1 . J K , < J K < . J K , , where J K , is the Kondo coupling at J Hund = 0. The realisticdata point is obtained by plugging in J Hund = 1 [eV] and (cid:15) f = − .
61 [eV] (as can be found in Table IV) toEq. (1) to be J K = 1 . J K , . Thus the data in Fig. 4shows that CeCu is almost right on the magnetic QCPwhere the N´eel temperature disappears.The same procedures are applied to all other targetmaterials. III. RESULTS
Plotting calculated N´eel temperatures with respect to J K = αJ K , , the Doniach phase diagram is constructedfor each target material as shown in Fig. 5.By rescaling the horizontal axis of the Doniach phasediagram as follows, t ≡ ( J K − J K , QCP ) /J K , QCP to inspectthe dimensionless distance to the QCP independently ofthe materials , we end up with the main results asshown in Fig. 1.
A. CeCu vs CeCu Remarkably, CeCu falls almost right on top of mag-netic QCP in Fig. 5. Also it is seen that the energyscales for antiferromagnetic order are on the same scalefor CeCu and CeCu as seen in the vertical-axis scalesfor the calculated N´eel temperatures. This may be rea-sonable considering the similar chemical composition be-tween CeCu and CeCu .Here we note that overestimates of the calculated N´eeltemperature are unavoidable due to the single-site natureof DMFT and approximations involved in the estimationof two-particle Green’s function . Thus the calculatedN´eel temperature for CeCu falling in the range of 20Kshould be compared to the experimental value of 4K only semi-quantitatively. Nevertheless, expecting thatthe same degree of systematic deviations are present inall of the data for the target compounds, we can safelyinspect the relative trends between CeCu and CeCu . B. Magnetic QCP in Au-doped CeCu In Fig. 5 it is seen that doping Au into CeCu onlyslightly shifts the energy scales competing between mag-netic ordering and Kondo screening. Most importantly,Au-doping drives the material towards the antiferromag-netic phase and the magnetic QCP is identified in theregion Ce(Cu − (cid:15) Au (cid:15) ) with (cid:15) (cid:28)
1, which is consistentwith the experimental trends of magnetism . Thishas been achieved with the control of an effective degen-eracy of orbitals incorporating the realistic width of levelsplittings in the localized 4 f -orbital, putting the charac-teristic energy scales in magnetism under good numericalcontrol in the present modeling.While quantitative success for Ce(Cu − (cid:15) Au (cid:15) ) (0 ≤ (cid:15) (cid:28)
1) concerning the location of the magnetic QCPis seen, some qualitative issues may be considered to beon the way to address magnetic quantum criticality, sincethis particular materials family represents the local quan-tum criticality scenario where a sudden breakdownof the Kondo effect is discussed to occur on the basisof a Kondo-Heisenberg model. In our realistic model,the exchange interaction between localized 4 f -electronsnaturally come in as a second-order perturbation pro-cess with respect to the Kondo coupling , which is theRKKY interaction .Even though no special place for another Heisenbergterm is identified in our realistic Kondo lattice model,there should indeed be other terms that are not explic-itly considered: for example, with very well localized 4 f -electrons, there is another indirect exchange coupling that work via the following two-steps: (a) intra-atomicexchange coupling between 4 f -spin and 5 d -spin and (b)inter-atomic hybridization between 5 d -band and other conduction band. Notably, in this channel the couplingbetween 5 d and 4 f is ferromagnetic, which is in princi-ple in competition against the antiferromagnetic Kondocoupling that we mainly consider here.For REPM compounds such as Nd-Fe intermetallics,the latter indirect exchange coupling, which we denote J RT for the convenience of reference as the effective cou-pling between rare-earth elements and transition metals,is dominant because 4 f -electrons are even more well lo-calized than in Ce -based compounds. There the Kondocouplings are not in operation practically, since f - c hy-bridization is weak and Kondo couplings are at too smallenergy scales as compared to other exchange couplings.Now that we bring HF materials and REPM compoundson the same playground, the f - d indirect exchange cou-plings should also have been given more attention eventhough there is at the moment only some restricted pre-scriptions to downfold a realistic number into J RT .This indirect exchange coupling can motivate theHeisenberg term on top of the realistic Kondo latticemodel, even though it is to be noted that the sign ofsuch extra Heisenberg terms is ferromagnetic. This maypave the way to define a realistic version of the Kondo-Heisenberg model . Since J RT ’s can compete againstRKKY at most only on the same order, the presence ofthe J RT terms would not significantly alter the positionof magnetic QCP, which is brought about by the Kondocoupling that competes against RKKY as a function ofexponential of the reciprocal of the coupling constants.This way, it is hoped that there might be a way to recon-cile the local QCP scenario for Au-doped CeCu and thepresent realistic modeling for the magnetic QCP focusingon the characteristic energy scales involving the Kondoeffect.Recent time-resolved measurements and theoreticalanalyses based on DMFT for Ce(Cu,Au) also pro-vides a way to reconcile the local QCP scenario and ex-perimentally detected signals from the possible Kondoquasiparticles on the real-time axis within the non-crossing approximation (NCA) as the impurity solverin DMFT. Since our DMFT results are based on quantumMonte Carlo (QMC) formulated on the imaginary-timeaxis, migrating to the real-time data via analytic continu-ation poses a challenging problem , while the solutionof the quantum many-body problem is numerically exactwith QMC. Thus the location of QCP derived from staticobservables would be better addressed with the presentframework.Still our numerically exact solution is limited tothe imaginary-time direction and effects of the real-space fluctuations are not incorporated in the single-siteDMFT. Recently, theoretical comparison between an ex-act solution of the lattice problem and DMFT has beendone and an artifact of DMFT to overestimate the re-gion of antiferromagnetic phase has been demonstrated.In this respect, the present location of magnetic QCPright below CeCu should also reflect the same artifact:if the spatial fluctuations are properly accounted for, themagnetic phase would shrink and the position of CeCu would shift slightly toward the paramagnetic side. C. QCP to which CeCu is driven by Co-doping Co-doping in CeCu shifts the energy scales morestrongly than seen in Au-doped CeCu . It is seen inFig. 5 that the QCP is driven toward the smaller J K side,reflecting the underlying physics that Kondo-screeningenergy scale is enhanced as Co replaces Cu. The ori-gin of the enhanced Kondo screening is seen in Fig. 2where anomalous peaks below the Fermi level are com-ing in which should come from the almost ferromag-netic conduction band which grows into the ferromag-netism in the Co-rich side of the composition space inCe(Cu,Co) . With 40% of Co, the 4 f -electron QCPis already passed and CeCu Co resides in the Kondo-screened phase. Thus it is found that the magnetic QCPof Ce(Cu − x Co x ) is located in 0 . < x c < .
4. We notethat the crystal structure and crystal-field splitting havebeen fixed to be that of the host material CeCu . Inreality, the QCP may be encountered with smaller Coconcentration.In the present simulations, we have neglected the pos-sible ferromagnetism in the ground state contributed bythe 3 d -electrons in Co. Referring to the past experimentsfor Ce(Cu,Co) described in Ref. 20, the absence of theobserved Curie temperatures for the Cu-rich side with theconcentration of Cu beyond 60% in the low-temperatureregion seems to be consistent with our computationalsetup in the present simulations. Even though other pastwork for an analogous materials family Sm(Co,Cu) does show residual Curie temperature in the Cu-rich re-gion, it should be noted that there is a qualitative dif-ference in the nature of the conduction band of Cu-richmaterials for the Sm and Ce-based families. D. Universal and contrasting trends
Co-doped CeCu and Au-doped CeCu represent thedifferent mechanisms where Co enhances f - d hybridiza-tion with the 3 d -electron magnetic fluctuations in theconduction electrons, while Au rather weakens f - d hy-bridization, being without d -electron magnetic fluctua-tions.The trend in magnetism comes from the relativestrength of exchange coupling between localized 4 f -electron and delocalized conduction electrons. AmongCe-based intermetallic compounds, a general trend in thehybridization ∆ is seen to be like the following J K (Ce-Au) < J K (Ce-Cu) < J K (Ce-Co) (3)as is partly seen in Ref. 75 for other materials familyCe T Si ( T =transition metals) - somewhere in the se-quence of the trend written schematically in Eq. (3), a magnetic quantum critical point between antiferromag-netism located on the relatively left-hand side and para-magnetism located on the relatively right-hand side isencountered within the range where 3 d -electron ferro-magnetism from Co does not dominate. The overallone-way trend from antiferromagnetism on the left-most-hand side to paramagnetism on the right-most-hand sidein Eq. (3) is universal around the magnetic quantum crit-icality, while the contrasting trend between Au-dopedcase and Co-doped case in Ce-Cu intermetallics is seenfrom the position of the Ce-Cu intermetallics concerningthe directions toward which the dopant elements drive.The opposing trends coming from 3 d -metal dopantand 5 d -metal dopant might help in implementing a fine-tuning of the material in a desired proximity to QCP in apossible materials design for REPM’s as discussed belowin Sec. IV B. IV. DISCUSSIONSA. Validity range of the Kondo lattice model
While we have defined the Kondo lattice model refer-ring to the electronic structure of the target materials,the limitations on the validity range of such downfoldingapproach should be kept in mind in assessing the im-plications of the present results. In Sec. III B, we havealready discussed the possible relation of our model tothe Kondo-Heisenberg model that has been extensivelyused in the local QCP scenario for Ce(Cu − (cid:15) Au (cid:15) ) .In a wider context, the spirit of the so-called s - d exchangemodel that was originally introduced by Vonsovskii andZener in the early days of the theory of ferromagnetismis still alive in the indirect exchange coupling J RT . Thishas been dropped in the present modeling for Ce-basedcompounds. Here we have assumed that the localizationof the 4 f -electron in our Ce -based compounds is goodenough to assure the applicability of the Kondo latticemodel: at the same time, it is presumed that our 4 f -electrons in Ce -based f - d intermetallics are not so welllocalized as are the case in Pr or Nd -based f - d inter-metallics. This means that the Kondo coupling comingfrom f - c hybridization would dominate over the J RT ’scoming from the indirect exchange coupling . Such sub-tle interplay between different exchange mechanisms candepend on the material. Since we did not address J RT inthe present studies, the outcome of the possibly compet-ing exchange interactions is not included in the presentscope. Possible subtle aspects coming from the localQCP scenario might reside in this particular leftover re-gion. If one would further opt for an alternative sce-nario , it may be useful to further investigate the effectof these dropped terms, and include them through im-proved algorithms based on better intuition. Althougha completely ab initio description is desirable, to makethe problem tractable we are forced into making someapproximations. Here it is at least postulated that thevalidity of the relative location of magnetic QCP can beassured in the present description because we have putthe most sensitive coupling channel, Kondo physics, un-der good numerical control.A few more discussions on the validity range of theKondo lattice model and possible extensions are in order:
1. Toward more unbiased downfolding
The terms in our low-energy effective models have beendefined targeting at the particular physics, namely, theRKKY interaction and Kondo physics. While this strat-egy has been good enough to address the relative trendsamong the target materials around the magnetic QCP, itmay well have happened that other relevant terms havebeen dropped that do not significantly affect the loca-tion of QCP. In this regard it may be preferred eithera) to downfold from the realistic electronic structure tothe low-energy effective models in a more unbiased way,at least proposing all possible candidate terms and elim-inating some of them only in the final stage accordingto a transparent criterion e.g. referring to the relevantenergy window or b) to work on the observables directlyfrom first principles without downfolding. While b) doesnot look very feasible, a) might pose a feasibly challeng-ing problem with a possible help from machine learning in systematically classifying the candidate terms even forsuch materials with multiple sublattices, multiple orbitalsand relatively large number of orbital degeneracy as im-posed from d -electrons and f -electrons.
2. Effects of valence fluctuations
Valence fluctuations have not been entirely incorpo-rated in the present description of Ce compounds. Otherscenario for Au-doped CeCu that emphasizes the rel-evance of valence fluctuations are recently discussed .We have described at least the magnetism trends aroundthe QCP in CeCu and CeCu Au only with localized 4 f -electrons. Apparently valence fluctuations may not bedominant at least for magnetism. We can restore thecharge degrees of freedom for 4 f -electrons and run ananalogous set of simulations for a realistic Anderson lat-tice model in order to see any qualitative difference comesup on top of the localized 4 f -electron physics. Often thetypical valence states for Ce, Ce or Ce , are not soclearly distinguished: even in the present Kondo latticedescription, (4 f ) state with Ce are virtually involvedin the Kondo coupling and localized 4 f -electrons evencontribute to the Fermi surface . To pick up a few morecases, for actinides or α -Ce, one can either discuss onthe basis of localized f -electrons and define the Kondoscreening energy scale spanning up to 1000K, or convinc-ing arguments can be done also on the basis of delocal-ized 4 f -electrons emphasizing the major roles played byvalence fluctuations. Given that it does not seem quite clear how precisely the relevance or irrelevance of valencefluctuations should be formulated for the description ofmagnetism trends, here we would claim only the rela-tive simplicity of our description for magnetic QCP inCe(Cu − (cid:15) Au (cid:15) ) ( (cid:15) (cid:28) B. Implications on the coercivity of REPM
Observing that the magnetic QCP can be encounteredin the chemical composition space of Ce(Cu,Co) , wenote that slowing down of spin dynamics when the sys-tem crosses over to QCP can be exploited in intrinsicallyblocking the magnetization reversal processes in REPMto help the coercivity. Since coercivity is a macroscopicand off-equilibrium notion, it is still much under develop-ment to formulate a theoretical bridge over the gap be-tween the microscopic equilibrium properties and macro-scopic coercivity. At least with QCP, diverging lengthscales of fluctuations and diverging relaxation times canin principle reach the macroscopically relevant spatialand time scales to help coercivity. Range of the criticalregion on the temperature axis and on the compositionspace would depend on each specific case.In Sm-Co magnets, even though it is clear that the cellboundary phase intrinsically carries the coercivity ,precise characterization of the inter-relation among theintrinsic properties, microstructure, and coercivity hasbeen still under investigation . Since Sm(Cu,Co) can be considered as a hole analogue of Ce(Cu,Co) inthe lowest j = 5 / , with a quest forQCP both for magnetism and possibly also for valencefluctuations, it may help to consider the possible role ofQCP in Sm(Cu,Co) for the intrinsic part of the coerciv-ity mechanism. Considering the electron-hole analogy,possible effect from QCP for Sm(Cu − x (cid:48) Co x (cid:48) ) can beexpected in the concentration range x (cid:48) c (cid:39) − x c (here x c is defined in Sec. III C) which fall in 0 . > x (cid:48) c > .
6. Thismay be compared favorably with the experimentally dis-cussed concentration of Cu in Sm-Co magnets, where upto around 35% of Cu in the cell-boundary phase made ofSm(Co,Cu) , especially in the triple-junction area ,has been correlated with the emergence of good coerciv-ity. V. CONCLUSIONS AND OUTLOOK
Realistic modeling for Au-doped CeCu and Co-dopedCeCu successfully describes the trends in magnetism in-volving QCP on the basis of the localized 4 f -electrons.One of the archetypical HF materials family, CeCu , andits Au-doping-induced QCP can be described within amagnetic mechanism with the terms that can be natu-rally downfolded from the realistic electronic structurein the spirit of the Anderson model , without explicitlyinvoking valence fluctuations or introducing additionalHeisenberg terms. We believe we have just put the char-acteristic energy scales of the target materials aroundQCP under good numerical control in having succeededin addressing the relative trends in magnetism aroundQCP. We do not rule out other subtlety around QCPthat may come from other terms that are not in-cluded in the present simulation framework. As long asthe dominating energy scales are concerned, those otherterms would not significantly alter the magnetism trendaround QCP.Co-doping in CeCu drives the material on a widerscale on the chemical composition axis as compared toAu-doped CeCu . This is caused by magnetic fluctu-ations in the paramagnetic conduction band that is onthe verge of ferromagnetism. For the 4 f -3 d intermetal-lic paramagnets in REPM in general, small changes inthe 3 d -metal concentration can drive the material aroundin the proximity of quantum criticality on the chemicalcomposition space, rendering it easy to encounter criticalregions in a microstructure with an appropriate spatialvariance in the microchemistry.Ce(Co,Cu) represents one of the earliest and mosttypical materials family in REPM . The lattice struc-ture of the materials family RT including Ce(Co,Cu) can be transformed into R T and RT (R=rareearth and T=Fe group elements), and a local structurearound the rare-earth sites in the champion magnet com-pound R Fe B (R=rare earth) resembles RT as de- scribed in Sec. III A of Ref. 14. With our results forCe(Cu,Co) in relation to Ce(Cu,Au) concerning QCP,it has been suggested that potentially various propertiesof derived compounds from the RT archetypical series residing in REPM, especially physics in the crossover toQCP, can be exploited for the possible intrinsic contri-bution to coercivity. ACKNOWLEDGMENTS
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