Kondo screening and beyond: an x-ray absorption and dichroism study of CePt 5 /Pt(111)
KKondo screening and beyond: an x-ray absorption and dichroism study ofCePt /Pt(111) C. Praetorius and K. Fauth
1, 2, ∗ Physikalisches Institut, Universit¨at W¨urzburg, Am Hubland, 97074 W¨urzburg, Germany Wilhelm Conrad R¨ontgen-Center for Complex Material Systems (RCCM),Universit¨at W¨urzburg, Am Hubland, 97074 W¨urzburg, Germany (Dated: October 27, 2018)We use x-ray absorption spectroscopy as well as its linear and circular magnetic dichroisms tocharacterize relevant interactions and energy scales in the surface intermetallic CePt /Pt(111). Theexperiments provide insight into crystal field splitting, effective paramagnetic moments, their Kondoscreening and mutual interactions and thus into many aspects which typically determine the lowtemperature behavior of correlated rare earth compounds. Exploiting the tuneability of Ce valencethrough the thickness dependent epitaxial strain at the CePt /Pt(111) interface, we are able tosystematically investigate the impact of hybridization strength on these interactions. ConsiderableKondo screening is indeed observed at all CePt thicknesses, and found to be strongest in case ofstrongest hybridization. While the magnetic response is commensurate with an impurity Kondoscale of T K > ∼ K for specimen temperatures
T > ∼
30 K, this is no longer the case at lowertemperature. Its detailed study by XMCD at one specific thickness of CePt reveals an anomalyof the susceptibility at T ∗ ≈
25 K instead, which we tentatively associate with the onset of latticecoherence. At lowest temperature we observe paramagnetic saturation with a small Ce 4 f saturationmagnetization. Within the framework of itinerant 4 f electrons, saturation is due to a field inducedLifshitz transition involving a very heavy band with correspondingly small degeneracy temperatureof T F ≈ T > ∼ PACS numbers: 71.27.+a,75.30.Mb,75.70.-i,78.70.Dm
I. INTRODUCTION
The richness and complexity of physical behavior en-countered in Ce intermetallics derives from the interac-tion of localized and itinerant electronic degrees of free-dom, i. e. the finite hybridization of Ce 4 f states withthe band structure of the periodic solid. The microscopicdetails of the interactions give rise to a rich phenomenol-ogy of physical properties and a variety of ground statesincluding magnetic order, superconductivity and param-agnetic heavy fermion liquids . This variability arisesfrom the occurrence of competing effective interactionswith small associated energy scales. Accordingly, tuningthe interactions by nonthermal control parameters suchas hydrostatic or chemical pressure may result in quan-tum critical points and unconventional behavior in theirvicinity .Identifying and characterizing the relevant energyscales thus constitutes an essential part of understandingthe low temperature behavior and of establishing corre-lations such as e. g. between local hybridization strengthon the one hand and macroscopic properties on the other.In this respect, advanced methods of surface science havedemonstrated tremendous potential and novel insight inrecent years, notably owing to their resolving capabilitiesin real or reciprocal space in combination with great spec-tral resolution. Associating findings from surface sen-sitive experiments to bulk properties of the respective materials may represent a nontrivial task, since relevantinteractions are frequently altered in the vicinity of thesurface .Technical limitations in applying nonthermal controlparameters as well as the unavailability of classical ther-modynamic methods or inelastic neutron scattering re-strict the possibilities of systematically studying theproperties and phase diagrams of systems at surfaces inan analogous manner to bulk materials. In the presentwork, we overcome some of these limitations by exploit-ing the fact that epitaxial strain at an interface may serveas a parameter controlling the strength of hybrizidationbetween Ce 4 f states and those of the metallic bands .X-ray circular magnetic dichroism (XMCD) is then be-ing used as an element and orbital specific probe of theanisotropic Ce 4 f paramagnetic response in these ultra-thin specimens including its temperature dependence. Inthis way, we systematically tune the many body interac-tions via Ce 4 f hybridization and study its relevance forvarious electronic and magnetic properties such as crystalfield splitting and magnetic Kondo screening.Among the ordered binary bulk intermetallic phases ofCe and Pt, CePt is the one richest in Pt . It crys-tallizes in the hexagonal CaCu structure (space groupP6/mmm, No. 191). The local point group at the Cesites is D h . Previous work has established that alloy-ing Ce into the surface of Pt(111) results in surface in-termetallics which adopt the same atomic lattice ,except for the surface termination where a dense Pt a r X i v : . [ c ond - m a t . s t r- e l ] N ov FIG. 1. ( left ) Hexagonal atomic environment of a Ce atom(large red sphere) in CePt . The lower layer exhibits thekagome hole characteristic of the bulk lattice. The top layeris shown with an additional Pt atom at this position, repre-senting the surface termination . ( right ) Atomic arrange-ment of an idealized CePt /Pt(111) specimen at t nom = 6 u.c.(structure C’ in Refs. 19 and 24, which is the majority phaseat this thickness). The periodic brightness modulation of thesurface atoms corresponds to the superstructure corrugationobserved in scanning tunneling microscopy . atomic layer is formed by occupying the kagome hole po-sitions with extra Pt atoms, as shown in Fig. 1. Carefulpreparation results in well defined CePt thickness at thespecimen surface, which we refer to as the nominal thick-ness t nom in multiples of the CePt unit cell (u.c.) alongthe hexagonal axis. The intermetallic thickness amountsto approx. 0 .
44 nm per u.c. of CePt .Early measurements of the bulk magnetic response inpolycristalline CePt were interpreted within a crystalfield scheme involving a fairly large overall Ce 4 f levelsplitting of about 76 meV . Later work concludedon antiferromagnetic ordering at T N = 1 K based onlow temperature susceptibility and specific heat mea-surements. Overall, the thermodynamic data gave noevidence for particular importance of Kondo or heavyfermion physics at the time. A resistivity minimum at T < ∼
10 K, found lateron by Sagmeister et al. mighthint at a Kondo scale of that order, the marked resis-tivity decrease below T = 2 K was linked to magneticordering rather than coherent band formation, however.Matters seem different with the CePt /Pt(111) sur-face intermetallics. Temperature effects in angle resolvedphotoemission were first observed by Andrews et al. and lateron identified as the tail of a Kondo resonanceby Garnier et al. in specimens which we identify asCePt with t nom = 4 . . . . A re-investigation by Klein et al. demonstrated the persistence of the Kondo resonance to T = 66 K and signatures of incipient coherence at T ≈ thus appears to be a Kondo latticematerial with T ∗ < T K , the opposite scenario comparedto a class of heavy fermion materials for which a phe-nomenological two-fluid picture has been proposed in thelast years .Hints at the relevance of Kondo physics inCePt /Pt(111) were also detected recently by x-ray ab-sorption (XA) and magnetic circular dichroism (XMCD) experiments . The temperature dependence of the Cevalence measured by XA hinted at a Kondo scale in ex-cess of 10 K, lending support to the T ∗ < T K scenario.CePt /Pt(111) also displays a remarkable dependence ofthe Ce valence as a function of intermetallic thickness, re-sulting in an interesting tunability of the electronic andmagnetic properties. The CePt thickness may thus beused as a non-thermal control parameter for the interac-tions in this material, while the underlying atomic struc-ture is essentially unchanged .In our preliminary analysis of a restricted XMCDdataset , we were also able to show that a consider-able degree of magnetic Kondo screening must be presentin these CePt thin film specimens. Here, we extendover those results by examining the x-ray linear dichro-ism (XLD) and the anisotropic Ce 4 f magnetic responsedetected by XMCD over a larger temperature range.These experiments yield valuable information on crys-tal field splitting , Kondo screening and magneticcoupling in the specimens and we shall discuss in de-tail how we obtain these quantities from our experimentaldata. Moreover, we report on a low temperature anomalyin the (inverse) susceptibility, which we tentatively inter-pret as an independent signature of a coherence scaleof T ∗ ≈
25 K. X-ray absorption experiments may thusserve as a powerful means to identify the various interac-tion scales which all contribute to the complexity in thebehavior of Kondo lattice materials.
II. METHODS
CePt /Pt(111) specimens were produced by follow-ing the procedures described in our previous work .Clean Pt(111) was prepared by repeated cycles of 1 keVAr + ion sputtering and annealing to 1170 K. Cerium(99.9% purity) was evaporated onto this surface nearambient temperature and interdiffusion was activated bysubsequent annealing to approx. 970 K for 5 to 10 min.This procedure results in well-ordered CePt intermetal-lic phases the thickness of which is predetermined by thequantity of Ce deposited . They are terminated a bya single dense Pt(111) atomic layer, giving rise to theremarkable inertness of these surfaces .Soft x-ray Ce M , XAS and XMCD experiments werecarried out at the PM 3 bending magnet beam line forcircular polarization of BESSY II at Helmholtz CenterBerlin (HZB). Absorption spectra were acquired in thetotal electron yield mode (TEY) using circular polarizedradiation (polarization: ≈ .
93) within a custom XMCDend station ( ± θ X = 60 ◦ with respect to the surface normalprobe the polarization-averaged, “isotropic” spectrum .Additional datasets were acquired at the SOLEILSR facility using the local CroMag end station at theDEIMOS beam line . Measurements were taken withbetter energy resolution, a degree of circular polarizationnear 100% and with the main goal of reaching lower spec-imen temperatures. The same experimental geometrieswere chosen to warrant a maximum of comparability ofthe XMCD data. Based on polarization alone, this choiceentails the linear dichroism between normal and obliqueincidence data to be enhanced by 12 .
5% with respect toto PM3 under otherwise identical settings.The TEY escape depth was previously found to be ofthe order of 1-1 . . In combination with the moder-ate concentration of Ce in the CePt specimens, this cre-ates a situation in which TEY saturation effects maysafely be neglected.In our analyses we make use of simulated absorptionspectra, obtained from full atomic multiplet calculationsas implemented in the Quanty package . The param-eters of the calculations are similar to those employedin our preceding analysis of CeAg x . ff (df ) Slater In-tegrals were reduced to 60% (80%) of their respectiveHartree-Fock values and energy dependent core excita-tion lifetimes were introduced to achieve best agreementwith the experimental line shapes.
III. RESULTS AND DISCUSSIONA. general observations and discussion framework
Figure 2(a) displays a selection of Ce M , spectrafrom CePt specimens with different thicknesses along-side with data acquired on a similar CeAg x sample .All datasets were obtained at oblique incidence (PM3beamline) and hence represent the isotropic spectra .In contrast to the CeAg x XA spectrum, a high energyshoulder appears in the CePt data at ¯ hω ≈
906 eV. Itspresence is indicative of finite hybridization between thelocalized Ce 4 f states and the metallic band structure and hence an admixture of states with n f = 0 character.Analyzing the spectral intensity of this shoulder revealsa non-monotonic dependence of the Ce valence on thethickness of the intermetallic film . Interestingly, thereis a systematic concommitant variation in the line shapeof the main f → d f type excitation spectrum, mosteasily visible as the variation of relative spectral weight offeature A in Fig. 2(a). The greatest similarity with theCeAg x spectrum is given in case of weakest hybridiza-tion, i.e. for t nom = 10 . reveals that the main contribution to itsvariation is an increasingly asymmetric response of theindividual resonances as the hybridization is increased.This is illustrated in Fig. 2(b) by a comparison of theisotropic spectra of weakest and strongest hybridization,i.e. at t nom = 10 . t nom = 2 u.c., respectively.Along with each experimental M spectrum a simula-tion is being displayed. The simulation for t nom = 10 . t nom = 2 u.c.The pronounced asymmetric spectral response causes the(apparent) reduction of the spectral weight of feature Avs. feature B, at the same time the falling edge of the M absorption is significantly broadened. The strength ofasymmetry in the spectral response correlates directlywith the f spectral weight . We therefore assumethat it is a manifestation of an increasingly efficient cou-pling to low energy excitations in the metallic bands withwhich the 4 f states hybridize and as such related to theedge singularity problem (see e.g. Refs. 48–51 and ref-erences therein). Its study for quasi-atomic multipletscoupled to a metal is as a challenging problem . Ourhypothesis therefore is currently under detailed theoret-ical scrutiny .The analysis in Refs. 46 and 47 suggests in particu-lar that hybridization induced ground state admixture ofstates with j = 7 / , may safely be neglected in the present case.Likewise, the neglect of double occupancy of the 4 f or-bital as a result of strong on-site Coulomb repulsion is agood approximation for CePt .Nevertheless, the redistribution of spectral weight to-wards higher excitation energies not only affects the spec-tral appearance of the XA spectra but also modifies thespectral shape of XLD. This is shown in Fig. 2(c), wherewe plot simulated XLD spectra with the same parame-ters as in Fig. 2(b) along with their isotropic XA counter-parts. While the peaks of both XA and XLD spectra arereduced in intensity, we note that the ratios between themagnitudes of XLD and XA are affected in different waysfor the main spectral features. In particular, while theloss of peak intensity of feature B in XA is partly com-pensated for by transfer of spectral weight from featureA, the same spectral weight transfer additionally reducesthe XLD of feature B owing to the sign change in XLDbetween features A and B. Overall, the observation of anasymmetric spectral response leads us to expect a sys-tematic decrease of the relative XLD amplitude as thehybridization is increased.Experimentally, we shall find this expectation con-firmed (see below), but the magnitude of XLD is evensuppressed well beyond the level to be expected by themodified f XA line shape. Similar findings have been re-ported before and have been suggested to result from hy-bridization viz. the Kondo interaction . Within the non-crossing approximation (NCA) to the impurity problemthis may qualitatively be rationalized as follows. We rep-resent the many body state as a superposition of stateswith f and f character | Ψ (cid:105) = c (cid:12)(cid:12) f (cid:11) + c (cid:12)(cid:12) f (cid:11) = c ( T ) (cid:12)(cid:12) f (cid:11) + (cid:88) m j c m j ( B, T ) | m j (cid:105) . (1)In the absence of excited state mixing, the squaredcoefficients c and c directly correspond to the relativespectral weights of the f and f related fractions of the FIG. 2. (a) Isotropic Ce M , XA spectra as a function ofCePt thickness along with a similar spectrum acquired onCeAg x . Most prominent peaks are labeled as A,B, and C.Peak A reduces to a mere shoulder, when the high energyshoulder at 906 eV is strongest ( t nom ≈ spectra (symbols) and analysis of their lineshape. For t nom = 10 . spectrum computed using Quanty . For t nom = 2 u.c., the simulated spectrum has been convolutedwith the asymmetric response function shown in the inset.(c) calculated effect of the asymmetric response on XA andXLD line shapes and magnitudes. Note that relative to theXA strength, the XLD is most strongly reduced at feature B. XA spectra . In a second step we rewrite the f partas a superposition of the six | m j (cid:105) states of the Ce 4 fj = 5 / have proposed a simplified NCA scheme whichis particularly suited to treat CF splitting and meanfield coupling at the same level as the many body Kondophysics. The characteristic failure of the standard NCA FIG. 3. (a) Symbols: selection of normal incidence spectrafor t nom = 10 . T ), extracted fromthe experimental spectra according to eq. (A.1). Solid line:fit to Υ( T ) using eq. (A.3). at low temperature is circumvented in this approach byomission of the divergent term in the spectral functionof the | f (cid:105) state. The application of this scheme doesnot yield a satisfactory description of our experimentalresults though: imposing a Kondo scale of order 10 K inaccordance with the observed Ce 4 f occupation n f ( T ) leads to a much stronger supression of the low temper-ature magnetic response than we observe. Conversely,assuming a small effective Kondo scale ( T K < ∼ T ∗ ) inthe NCA, one computes much too large a susceptibilityat higher temperatures compared with the experimentaldata. We therefore reckon that it is the occurrence of twodistinct energy scales which prevents the applicability ofthe NCA scheme in case of CePt /Pt(111). B. XLD analysis
In our analysis below, we shall therefore use a con-ventional crystal field approach and take the many bodyaspects of the problem into account in a qualitative, phe-nomenological way. In extension of our previous work we represent the possibility of non-thermal CF state oc-cupation by introducing an ‘isotropic fraction’ of weight w i in the normal incidence (NI) spectra, where for sim-plicity we assume this fraction to be temperature inde-pendent and the same in the three Kramers doublets.The model equation for the temperature dependence ofNI spectra ( f part only) thus reads as follows: I NI ( T ) = 1 Z (cid:48) (cid:16) w i I ISO + w i ( I NI | / (cid:105) + p I NI | / (cid:105) + p I NI | / (cid:105) ) (cid:17) . (2)Here, w i = (1 − w i ), and p , = exp( − ∆ , /k B T ) arethe Boltzmann weights representing the thermal excita-tion probabilities according to the CF splittings ∆ = E / − E / and ∆ = E / − E / . The standard par-tition function Z being given by Z = 1 + p + p , Z (cid:48) is constructed such as to take the isotropic fraction intoaccount, i.e. Z (cid:48) = w i + w i Z .Figure 3(a) displays a selection of NI spectra from theCePt specimen with smallest hybridization ( t nom = 10 . = 1 . ± . = 27 ± w i ≈ .
23. We thus find a similar energeticordering of the CF states as in CeAg x while the totalCF splitting (i.e. ∆ ) is larger by about one order ofmagnitude in CePt .While the data presented in Fig. 3(a) demonstrate thatthe fits do adequately capture the essential thermal evo-lution of the XA spectra at normal incidence, the factthat the simulated spectra do not perfectly match theexperimental peak positons and line shape gives rise to anon-negligible residual error and thus a fairly shallow op-timum with associated parameter uncertainties that arerelatively large. The situation further aggravates as theCePt layer thickness is reduced owing to both the re-duction Ce M , TEY signal above background and theincreasing hybridization which reduces the magnitude ofXLD as discussed above. We have therefore sought analternative means to evaluate the CF splitting from thetemperature dependent XA data in a way that does notdepend on an accurate simulation of the XA line shape.A simple yet robust measure of the magnitude of XLDcan indeed be found. It essentially consists of a rela-tion involving ratios of the peak amplitudes of featuresB and C in the normal and oblique incidence spectra, re-spectively (see Appendix for details). The resulting XLDparameter Υ for t nom = 10 . = 0 . ± . = 29 ± > | ± / (cid:105) character must be the one of highest energy. Thesmaller energy scale ∆ is responsible for the increase ofΥ at low temperature in Fig. 3(b). For specimens withsmaller t nom such a low temperature variation of Υ couldnot unambiguously be determined. We are therefore ledto conclude that ∆ assumes such small values that itstemperature effect essentially slips out of the temperaturerange accessible in our XLD experiments. We shall findthis idea to be confirmed by the analysis of our XMCDdata, the basics of which we discuss next. FIG. 4. (a) XA and XMCD datasets for t nom = 4 u.c. atnormal ( θ X = 0) and oblique ( θ X = 60 ◦ ) incidence at T = 20K and with applied field of µ H = ± . C. anisotropic paramagnetic response
The element and orbital specific measurement of thetemperature dependent magnetic response provides di-rect access to the Ce 4 f magnetic moments as well asindependent information on the CF splittings . Fig-ure 4(a) displays normal and oblique incidence XA andXMCD data obtained for t nom = 4 u.c. at a temperatureof T = 20 K. While the small yet finite XLD is dis-cernible in the XA spectra, the XMCD datasets reveala pronounced anisotropy in the paramagnetic response.Just as Υ > | ± / (cid:105) character from XLD, it follows from the strongin-plane single ion anisotropy that | ± / (cid:105) possess largestatistical weight at T = 20 K and thus either constitutethe ground state or are energetically adjacent to it.More insight is obtained from a quantitative evaluationof the (anisotropic) inverse susceptibility and its temper-ature dependence, which is shown in Fig. 4(b). Suscepti-bilities are obtained from XMCD by evaluating the mag-netic Ce 4 f polarization on the basis of the well-knownsum rule for the orbital magnetic moment . The directapplication of the spin moment being prohibitive in caseof Ce we proceed as in our previous work andderive the total moment by assuming the atomic relation m S = − m L / ≈ , XMCD data , since it constitutes only aminor correction to the effects reported below.The experimental data of Fig. 4(b) contain a num-ber of characteristics to be captured by a modelling ap-proach. These include the reduced magnitude of the ef-fective paramagnetic moment compared to the free ionvalue as well as its pronounced anisotropy. In addition,the normal incidence data feature a strong kink at T ≈
50K which results from CF splitting. Finally, we note theoccurrence of a finite, positive and anisotropic paramag-netic Curie-Weiss temperature Θ p .The solid lines in Figure 4(b) were calculated assuminga hexagonal CF, including intersite magnetic couplingat the mean field level . As mentioned above, Kondoscreening is being accounted for in a phenomenologicalway as follows. χ || = g µ B k B T Z (cid:0) C + 9 C p + 25 C p (cid:1) (3) χ ⊥ = g µ B k B T Z · (cid:18) C (cid:18) k B T ∆ (cid:19) + (4)+ C (cid:18) k B T ∆ − ∆ − k B T ∆ (cid:19) p − C k B T ∆ − ∆ p (cid:19) The susceptibility at an arbitrary angle θ with respectto the hexagonal axis is then obtained via χ θ = cos θχ − || − λ + sin θχ − ⊥ − λ , (5)where the interaction between Ce sites is represented bythe mean field coupling constant λ .Quite evidently, our approach is capable of quantita-tively reproducing the experimental susceptibility data.It is in fact the approach with the smallest number of freeparameters which allowed us to model the results acrossthe entire range of CePt thicknesses studied, and wherethese parameter values vary in a sensible way, largely inaccordance with the respective XLD results . All themany body physics is contained in the magnitudes of the C i factors, introduced such as to directly reflect the re-duction of the magnetic moments in the | ± i/ (cid:105) states.In accordance with the discussion above, a small valuefor ∆ ≈ . χ || along the hexagonal axis is in-creased in comparison with the one obtained for a pure | ± / (cid:105) Kramers doublet: the unscreened effective mo-ment at θ X = 0 increases from m / = √ gµ B ≈ . µ B to m / = √ gµ B ≈ . µ B . Consequently, the mo-ment reduction factors C , ≈ / χ || , a value of ∆ = 15 . t nom = 10 . indeed exhibits a systematic dependence onthe intermetallic thickness. The Curie-Weiss behavior isaccounted for by a coupling constant with λ > p follows naturally from the anisotropicsusceptibility.Since Θ p > t nom = 4 u.c. While magnetic order is not observed downto T ≈ T ∗ ≈ K , as we shalldiscuss in section III E below. For now, we conclude our discussion of the magneticanalysis with reference to recent susceptibility calcula-tions within the thermodynamic Bethe Ansatz for animpurity degeneracy of N = 4. This comparison is sen-sible only in a temperature range in which the thermaloccupation of the |± / (cid:105) doublet may safely be neglected.This is the case for T ≈
30 K , where the slope in χ − || vs. T corresponds well to the moment reduction deter-mined by C , . On the one hand, we find from Ref. 60that in the limit of ∆ (cid:28) T K a moment reduction ofthis order is expected at T /T K ≈ . K.On the other, looking at the calculated susceptibility, wefind that χ || should nearly have reached a temperatureindependent value in this temperature range, which isstrongly at variance with our experimental findings. Wethus arrive at a similar conclusion as with respect to theNCA scheme above: while some useful connections withthe solutions to the impurity Kondo problem can be es-tablished, the occurrence of a separate low-energy scaleimpedes a more thorough analysis on their basis. D. thickness dependence of XLD and XMCD
Experiments as described above were carried out forCePt /Pt(111) specimens of various thicknesses in at-tempt to elucidate the impact of hybridization strengthon the observed behaviors. Figures 5 and 6 summarizeour findings concerning both XLD and XMCD in termsof the model equations (A.3) and (5) across the range ofCePt thicknesses studied. Fig. 5 displays the experimen-tal data, i.e. Υ( T ) (where available) and the anisotropicparamagnetic response determined at θ X = 0 ◦ and θ X =60 ◦ .All experimental datasets were subjected to simulta-neous modelling of Υ, χ || and χ ◦ . Except for the caseof t nom = 10 . could be determined fromthe data, we have adopted ∆ = 0 . t nom = 3 . fixed, we are left with a totalof six further parameters to be determined. Besides ∆ ,which is shared by the equations for XLD and XMCD,we determine the strength of XLD reduction ( γ ), whichcomprises both effects induced by hybridization, i.e. thealtered XA line shape and the mixing of | m j (cid:105) weights.The remaining parameters apply to the magnetic dataonly and consist of the C i moment reduction factors andthe mean field coupling λ . For the three specimens rep-resented in the bottom row of Fig. 5 the values of ∆ and C were supplied by hand such to be in accordance withthe adjacent specimens, since the amount of experimentaldata does not warrant their independent determination.These cases are represented by open symbols in Fig. 6below.But for χ − ◦ at t nom = 1 . t nom = 1 . FIG. 5. Overview over XLD and XMCD results obtainedfor CePt specimens of various thicknesses in the range 1u.c. ≤ t nom ≤
11 u.c. Experimental datapoints for the XLDparameter Υ are shown along with the fits according toeq. (A.3), those for the inverse susceptibility with fits accord-ing to eq. (5). Where applicable, identical parameters wereused in both fits. The resulting fit parameters are given inFig. 6. For the sake of clarity, ordinate scales for Υ differbetween the first and third row of panels. Note that particu-larly in the latter the high temperature limiting value Υ = 1is strongly suppressed. trends of the parameter values when plotted vs. inter-metallic thickness.A synopsis of these parameter evolutions is providedin Figure 6. It reveals a number of systematic varia-tions. There is an obvious transition in the magnitudeof the CF splitting ∆ . For the thicker, more weaklyhybridized films ∆ takes on values of ∆ ≈ . . . .∆ determines both the rate of approach of the XLD pa-rameter Υ towards unity towards high temperature aswell as the position of the marked kink in χ − at normalincidence. Also, the general trend of the paramagnetic(single ion) anisotropy becoming less anisotropic as t nom FIG. 6. Thickness dependence of the various parameters de-termined from least squares fitting of the experimental XLDand XMCD data in Fig. 5. (a) CF excitation energy ∆ (∆ = 0 . t nom = 10 . γ of the XLD magnitude.(c) Moment reduction factors C i as indicators of magneticKondo screening. Kondo screening is strongest between 2u.c. < ∼ t nom < ∼ .(d) Mean field molecular field constant λ . is reduced is compatible with a reduction in ∆ (giventhat ∆ is already small). It is not evident to unequiv-ocally identify the cause of this transition. On the onehand it seems unlikely that the variations in lattice pa-rameter play a major role, since the structural changes(see Ref. 19) are small for t nom > ∼ is essentially taking place. On the other, whilethe region of small ∆ coincides with the occurrence ofstrong hybridization, there is no obvious further correla-tion with the nonmonotonous variation in hybridizationstrength vs. t nom which occurs in this range of intermetal-lic thickness.The latter has a direct bearing on the magnitude ofthe magnetic response, however, as is manifest from thebehavior of the C i factors representing the moment re-ductions in Fig. 6(c). Kondo screening is thus strongestwhere hybridization is strongest. We note that thescreening factors C and C assume nearly identical val-ues in the fits for all specimens. We take this as furtherevidence that indeed a quasi quartet CF ground state isformed.Another remarkable result of our experiments is thestrong reduction of XLD towards small t nom , i. e. whenhybridization is strong. This reduction is given by theparameter γ and plotted in Fig. 6(b). The overall XLDreduction notably is much stronger than what would beexpected from the asymmetric XA line shape broaden-ing and thus is obviously dominated by nonthermal | m j (cid:105) mixing as discussed above in section III A.With respect to Fig. 6(d) we note that a small butfinite mean field coupling constant ( λ ) is consistently FIG. 7. (a) Low temperature Ce 4 f XMCD magnetizationcurves measured at oblique incidence for a CePt thicknessof t nom = 4 u.c. (b) Detailed temperature dependence of theinverse Ce 4 f susceptibility at NI, revealing a departure fromthe high temperature Curie-Weiss behavior near T ∗ ≈
25 K.Datasets produced at BESSY and SOLEIL, respectively, yieldvery good agreement in the overlapping temperature range. found for all specimens. It appears, therefore, thatthe dominant magnetic correlations in the local momentregime of CePt /Pt(111) are ferromagnetic in nature,while bulk CePt orders antiferromagnetically at very lowtemperature .The models implemented in eqns. (A.3) and (5) thusprovide a good basis for a systematic analysis of thetrends generated by varying the strength of Ce 4 f hy-bridization by choice of intermetallic thickness. Nev-ertheless, quantitative parameter values resulting fromour fits should in principle be taken with some caution.Kondo screening for example is inherently temperaturedependent while the parameters of our model equationsare not. The fitting procedure will thus produce parame-ter values which best emulate this thermal behavior. It isgratifying therefore to note that most recent experimentsemploying electronic Raman scattering lend strong sup-port to our present conclusions with respect to the CFlevel structure . E. low temperature behavior
Our consistent finding of a mean field coupling con-stant ( λ >
0) indicates the possibility of a ferromagnet-ically ordered ground state at temperatures below, say, T = 5 . . .
10 K. We have therefore tested this possibilityby a more detailed study of the low temperature mag-netic response at SOLEIL for a specimen with t nom = 4u.c. Figure 7(a) displays a selection of XMCD magneti-zation curves for the lowest temperatures, measured at θ X = 60 ◦ , for which Θ p ≈ . χ − || vs. temperature as in Fig. 7(b),such a deviation is indeed observed around T ≈
25 K. Datasets acquired at BESSY and SOLEIL, respectively,agree very well and the anomaly at 20 . . .
25 K is in factalready present in the BESSY dataset.The kink in χ − || ( T ) may be seen as separating twodistinct Curie Weiss regimes with Θ p > T > ∼
25 Kand Θ p ≈ T < ∼
20 K. From extensive simulations werule out that the observed behavior could be obtained byassuming a more refined (e.g. spatially inhomogeneous)crystal field scheme Also, the absence of any peculiarityin the XLD data at a temperature scale of ≈
25 K speaksagainst a CF related effect.Instead, we notice that a departure from Curie Weissbehavior may hint at emerging lattice coherence. Thiswas e. g. also suggested in case of CePb which exhibitsa magnetic anomaly very much reminiscent of the oneobserved here . A coherence temperature of the or-der of T ∗ = 25 K in the 4 u.c. CePt intermetallic islargely in line with the photoemission results by Klein etal. (Ref. 32). Obviously, the then expected crossover tothe low temperature scaling regime with temperature in-dependent Pauli susceptibility is not yet fully undergoneat T = 2 K, indicating a small degeneracy temperature T (cid:28) T ∗ – not infrequent in heavy electron systems .Calculations within the Kondo lattice model (KLM)indicate a considerable robustness of the heavy fermionbands against temperature and magnetic fields . Mo-tivated by this observation, we consider the implicationsof the experimental M ( H ) behavior at lowest experimen-tal temperature in Fig. 7(a) in the framework of heavy,itinerant Ce 4 f states. Such an attempt also seemsworthwhile since analyzing the measured magnetizationcurves in terms of local moment magnetization functionsresults in physically inconsistent parameters.One characteristic experimental feature is that themagnetization curve visibly approaches some satura-tion behavior with a Ce 4 f saturation magnetization of0 . . . . . µ B per Ce atom. This value amounts to only afraction of the expected saturation moment of 1 . µ B peratom in the local moment picture, given the hexagonalCF scheme determined above and θ X = 60 ◦ .With itinerant 4 f electrons, the rationale for observ-ing paramagnetic saturation is different from the case oflocal moments: it is expected to occur as a consequenceof a Lifshitz transition induced by the applied magneticfield, i. e. when the Zeeman splitting shifts the chemi-cal potential into the hybridization gap of the majoritystates . Put differently, at very low temperature thescale on which magnetic saturation is observed is given byequating Zeeman and Fermi energies of the heavy band.The magnitude of saturation magnetization then dependsessentially on the fraction of the Brillouin zone coveredby the heavy band and may indeed be small.From this perspective, the Ce 4 f magnetization ap-proaching magnetic saturation at applied fields of theorder of µ H = 6 T implies a heavy band with a Fermienergy around 0 . T F ≈ f contribution to the Pauli susceptibility will only be ob-tained at temperatures well below the range accessibleto our experiments. The KLM calculations reported inRef. 66, when evaluated for the temperature dependent4 f susceptibility, appear to lend support to such an inter-pretation of our findings. These caclulations do cover aparameter range down to T < ∼ T F (cid:28) T ∗ and a small mo-ment Curie Weiss like magnetic response is indeed foundon this temperature scale .These considerations leave the question untouchedwhether additionally some kind of ‘two liquid’ scenariomight apply in analogy to those cases with T K < T ∗ forwhich this phenomenology was introduced . Fur-ther work shall be required to more firmly establish thevalidity of the heavy fermion scenario to account for theCe 4 f magnetic response in CePt /Pt(111) and is cur-rently in progress.We finally emphasize that no changes of XA line shapeand 4 f occupation occur in the vicinity of the tempera-ture of the magnetic anomaly . This finding illustratesthat the main role of the ‘delocalization process’ at T ∗ consists of establishing phase coherence between the Cesites, the ‘local physics’ remaining essentially unaltered. IV. SUMMARY AND CONCLUSIONS
In conclusion, we have presented a detailed investiga-tion of the spectral and magnetic response as detectedby x-ray absorption and dichroism at the Ce M , edgesof CePt /Pt(111) ordered surface intermetallics. Com-bining pieces of evidence from different spectroscopicmodes and geometries we were able to gather relevantinformation on the interactions and associated energyscales in this material. The general picture emergingfrom our study is that for T > ∼
30 K we are essen-tially concerned with the “impurity regime” featuringsubstantially Kondo-screened local moments, subjectedto a hexagonal crystal field and weak ferromagnetic cor-relations. The CF ground state is essentially a quasi-quartet of the | ± / (cid:105) and | ± / (cid:105) states. The | ± / (cid:105) states are split off by ≈
15 meV in the range of smallintermetallic thickness, whereas the splitting increases to > ∼
25 meV at larger t nom .The tunability of hybridization by epitaxial strain pro-vides us with a non-thermal control parameter in a sur-face science experiment. The Ce 4 f paramagnetic mo-ment clearly depends on hybridization strength and weobserve the strongest moment reduction in the case ofstrongest hybridization. It is maybe an interesting ob-servation that strong hybridization coincides with smalloverall CF splitting. A similar correlation appears tohold in case of CeAg x , albeit on a much smaller energyscale .While the magnitude of Kondo screening is compati-ble with an impurity Kondo scale of order 10 K as pre-viously determined from the temperature dependent Ce valence , the magnetic response at lowest temperatureis not. Instead, for specimens with t nom = 4 u.c. wefind an anomaly in the magnetic response at T ∗ = 25K which we discuss as potentially signalling the onset oflattice coherence. A coherence temperature of this or-der is well in line with previous experimental evidence.Adopting this view, we may understand the occurrenceof paramagnetic saturation with a small saturation mo-ment as to emerge from a field induced Lifshitz transi-tion. The observation of a saturation field of the orderof 6 T is then indicative of a very narrow 4 f band withcorrespondingly small degeneracy temperature. Such asmall inherent energy scale readily accounts for the factthat a temperature independent 4 f contribution to thePauli susceptibility is not observed within the tempera-ture range of our experiment ( T > ∼ f degrees offreedom which allows one to gain insight in several smallenergy scales coexisting in heavy fermion materials. Wehope that our work will stimulate interest in carryingover the methodology to other heavy fermion materials.In particular, we anticipate that the element and orbitalspecificity of XLD and XMCD should provide profoundinsight into the physics behind metamagnetic transitionsin those heavy fermion materials where the required mag-netic field is accessible with current synchrotron radiationinstrumentation. ACKNOWLEDGMENTS
Acknowledgement for assistance on the occasion of var-ious synchrotron radiation beam times is owed to M. Zin-ner, H. Kießling, B. Muenzing, P. Sprau, and S. Br¨uckas well as to the beamline staff, T. Kachel (HZB), F.Choueikani and P. Ohresser (SOLEIL) for their support.We also thank P. Hansmann, M. W. Haverkort, F. F. As-saad, M. Bercx, H. Schwab and F. Reinert for most help-ful and stimulating discussions. This work received fi-nancial support by the Deutsche Forschungsgemeinschaftwithin FOR1162 (TP 7). Access to synchrotron radia-tion was also partially granted by HZB managed fundsand the European Community’s Seventh Framework Pro-gramme (FP7/2007-2013) under the CALIPSO project(Grant Agreement No. 226716). Generous allocation ofbeam time at the synchrotron radiation facilities as wellas their general support is gratefully acknowledged.
Appendix: Modeling the XLD parameter Υ The reduction of both the TEY signal and the rela-tive magnitude of XLD with decreasing CePt thickness0 FIG. 8. (a-c) calculated normal incidence and isotropic spec-trum for each of the m j states. Arrows indicate the intensitiesutilized for determining Υ (eq. (A.1)). (d) ternary diagramrepresenting the dependence of Υ on the fractional occupa-tions w ( m j ). Colored symbols represent the trajectories of Υfollowed for ∆ = ± = 30 meV. Colors representtemperature and range from red ( T = 900 K) to blue, T = 7K). (e) temperature dependence Υ( T ) for the same ∆ , . prompted a search for an alternative, simple and robustmeasure of XLD, suitable for a determination of the CFsplittings ∆ , . It turns out that by using the relativepeak heights of the spectral features B and C at the M and M edges of NI spectra, a suitable parameter Υ canbe obtained. Its sensitivity for the magnitude of XLDrelates to the fact that the XLD of features B and Cpossess opposite sign for all m j . The relation to XLD isobtained by relating this peak ratio to the one obtainedat oblique incidence (i.e. the isotropic spectrum in caseof the data taken at BESSY II). We define Υ asΥ( T ) = I NI B I NI C · I ISO C I ISO B = I NI B I ISO B · I ISO C I NI C . (A.1)In practice, Υ is most readily evaluated by directcomparison of the peak intensities between normal andoblique incidence spectra at the spectral positions of fea-tures B and C, as indicated in the regrouped, final ex-pression of eq. (A.1).At a given temperature T , the magnitude of Υ willdepend on the occupation of the CF levels, each con-tributing in an individual way. As an illustration, weplot the NI spectra for each of the CF states along withthe isotropic spectrum in panels (a)-(c) of Fig. 8, cal-culated here without taking asymmetric broadening intoaccount. The relevant intensities at features B and C areindicated by arrows. For each of the | ± i/ (cid:105) doublets, we determine the de-viation of the ratios I NIi, B /I ISOi, B and I NIi, C /I ISOi, C from unity A i,B = I NIi, B I ISOi, B − , A i,C = I NIi, C I ISOi, C − T )in these quantities as followsΥ( T ) = 1 + γ (cid:18) A ,B + p A ,B + p A ,B ) /Z A ,C + p A ,C + p A ,C ) /Z − (cid:19) , (A.3)with p , and Z as defined in section III B. In this ex-pression, γ represents an overall reduction of the XLDmagnitude, which comprises both its reduction due tothe asymmetric spectral response and the hybridizationinduced mixing of | m j (cid:105) states. The temperature depen-dence Υ( T ) is then encoded in the Boltzmann weights p , ( T ).To more accurately represent the fact that XLD re-duction due to asymmetric spectral response is differ-ent for features B and C (see Fig. 2(c) and correspond-ing text) one can either determine the quantities definedin eq. (A.2) from calculated spectra taking the spectralasymmetry into account or by introducing separate XLDreduction factors γ B and γ C .Υ( T ) = 1 + γ B ( A ,B + p A ,B + p A ,B ) /Z γ C ( A ,C + p A ,C + p A ,C ) /Z (A.4)Likewise, one might wish to represent the possibilitythat since ∆ > ∼ T K the XLD could actually be lessstrongly reduced for the | ± / (cid:105) states. Adding such de-tails to the model, however, does not significantly alterthe fit results concerning the magnitudes of ∆ , , whichwe are primarily interested in here. The Υ( T ) calcula-tions represented in Fig. 3(b) and Fig. 5 were thus allcomputed according to eq. (A.3). The overall XLD re-duction factor γ resulting from the fits is reported inFig. 6(b).Figure 8(d) contains a ternary nomogram which rep-resents the behavior of Υ as a function of the statisticalweights of the | m j (cid:105) states. The center of the trianglecorresponds to the high temperature limit in which all | m j (cid:105) states possess the same weight and where thereforeΥ = 1. Lines represent initial state compositions of equalΥ. It is readily seen that Υ is primarily sensitive to thedegree of admixture of | / (cid:105) character and hence wellsuited to determine ∆ . Υ > specimens, signifies that | ± / (cid:105) is the CF state of highest energy.Panel (e) of Fig. 8 displays Υ( T ), as calculated usingeq. (A.3) with γ = 1 (or, equivalently, eq. (A.4) with γ B = γ C = 1), setting ∆ = 30meV and ∆ = ± and thus similar in both cases, while thelow temperature trends in Υ( T ) are determined by thesign of ∆ .1 ∗ [email protected] A. C. Hewson,
The Kondo Problem to Heavy Fermions (Cambridge University Press, 1993). N. Grewe and F. Steglich, in
Handbook on the Physicsand Chemistry of Rare Earths , Vol. 14, edited by K. A.Gschneider, Jr. and L. Eyring (Elsevier, Amsterdam, 1991)p. 343. H. v. L¨ohneysen, A. Rosch, M. Vojta, and P. W¨olfle, Rev.Mod. Phys. , 1015 (2007). P. Gegenwart, Q. Si, and F. Steglich, Nat. Phys. , 286(2008). Y.-F. Yang, Z. Fisk, H.-O. Lee, J. D. Thompson, andD. Pines, Nature , 611 (2008). F. Steglich and S. Wirth, Reports on Progress in Physics , 084502 (2016). Y.-F. Yang, Reports on Progress in Physics , 074501(2016). M. Klein, A. Nuber, F. Reinert, J. Kroha, O. Stockert, andH. v. L¨ohneysen, Phys. Rev. Lett. , 266404 (2008). Q. Si and F. Steglich, Science , 1161 (2010). Y. Zhong, Y.-F. Wang, Y.-Q. Wang, and H.-G. Luo, Phys.Rev. B , 035128 (2013). C. M. Varma, Reports on Progress in Physics , 082501(2016). S. Ernst, S. Kirchner, C. Krellner, C. Geibel, G. Zwicknagl,F. Steglich, and S. Wirth, Nature (2011), 10.1038/na-ture10148. M. H. Hamidian, A. R. Schmidt, I. A. Firmo, M. P. Allan,P. Bradley, J. D. Garrett, T. J. Williams, G. M. Luke,Y. Dubi, A. V. Balatsky, and J. C. Davis, Proceedings ofthe National Academy of Sciences of the United States ofAmerica , 18233 (2011). Y. Iwamoto, M. Nakazawa, A. Kotani, and J. C. Parlebas,J. Phys.: Condens. Matter , 1149 (1995). C. Dallera, M. Grioni, A. Shukla, G. Vanko, and J. L.Sarrao, J. Synchrotron Rad. , 242 (2002). M. G¨uttler, K. Kummer, S. Patil, M. H¨oppner, A. Han-naske, S. Danzenb¨acher, M. Shi, M. Radovic, E. Rienks,C. Laubschat, C. Geibel, and D. V. Vyalikh, Phys. Rev.B , 195138 (2014). M. Mulazzi, K. Shimada, J. Jiang, H. Iwasawa, andF. Reinert, Phys. Rev. B , 205134 (2014). S. Patil, A. Generalov, M. G¨uttler, P. Kushwaha,A. Chikina, K. Kummer, T. C. R¨odel, A. F. Santander-Syro, N. Caroca-Canales, C. Geibel, S. Danzenb¨acher,Y. Kucherenko, C. Laubschat, J. W. Allen, and D. V.Vyalikh, Nature Communications , 11029 (2016). C. Praetorius, M. Zinner, A. K¨ohl, H. Kießling, S. Br¨uck,B. Muenzing, M. Kamp, T. Kachel, F. Choueikani,P. Ohresser, F. Wilhelm, A. Rogalev, and K. Fauth, Phys.Rev. B , 045116 (2015). B. Predel, in
The Landolt-B¨ornstein Database , Vol. 5c,edited by O. Madelung (SpringerMaterials, 1993). A. Janghorban, M. Lomello-Tafin, J. M. Moreau, andP. Galez, Intermetallics , 2208 (2010). C. J. Baddeley, A. W. Stephenson, C. Hardacre,M. Tikhov, and R. M. Lambert, Phys. Rev. B , 12589(1997). J. M. Essen, C. Becker, and K. Wandelt, e-J. Surf. Sci.Nanotech. , 421 (2009). J. Kemmer, C. Praetorius, A. Kr¨onlein, P.-J. Hsu, K. Fauth, and M. Bode, Phys. Rev. B , 195401 (2014). P. Tereshchuk, M. J. Piotrowski, and J. L. F. Da Silva,RCS Adv. , 521 (2015). C. Praetorius, M. Zinner, G. Held, and K. Fauth, Phys.Rev. B , 195427 (2015). H. Lueken, M. Meier, G. Klessen, W. Bronger, andJ. Fleischhauer, J. Less-Comm. Met. , P35 (1979). A. Schr¨oder, R. Vandenberg, H. von L¨ohneysen, W. Paul,and H. Lueken, Solid State Commun. , 99 (1988). E. Sagmeister, E. Bauer, E. Gratz, H. Michor, andG. Hilscher, Physica B , 148 (1997). A. B. Andrews, J. J. Joyce, A. J. Arko, J. D. Thompson,J. Tang, J. M. Lawrence, and J. C. Hemminger, Phys.Rev. B , 3277 (1995). M. Garnier, D. Purdie, K. Breuer, M. Hengsberger, andY. Baer, Phys. Rev. B , 11399 (1997). M. Klein, A. Nuber, H. Schwab, C. Albers, N. Tobita,M. Higashiguchi, J. Jiang, S. Fukuda, K. Tanaka, K. Shi-mada, M. Mulazzi, F. F. Assaad, and F. Reinert, Phys.Rev. Lett. , 186407 (2011). K. R. Shirer, A. C. Shockley, A. P. Dioguardi, J. Crocker,C. H. Lin, N. apRoberts Warren, D. M. Nisson, P. Klavins,J. C. Cooley, Y.-F. Yang, and N. J. Curro, Proceedingsof the National Academy of Sciences Y.-F. Yang and D. Pines, Proceedings of the Na-tional Academy of Sciences M. Jiang, N. J. Curro, and R. T. Scalettar, Phys. Rev. B , 241109 (2014). P. Castrucci, F. Yubero, F. C. Vicentin, J. Vogel, andM. Sacchi, Phys. Rev. B , 14035 (1995). P. Hansmann, A. Severing, Z. Hu, M. W. Haverkort, C. F.Chang, S. Klein, A. Tanaka, H. H. Hsieh, H. J. Lin, C. T.Chen, B. Fak, P. Lejay, and L. H. Tjeng, Phys. Rev. Lett. , 066405 (2008). T. Willers, Z. Hu, N. Hollmann, P. O. Koerner, J. Gegner,T. Burnus, H. Fujiwara, A. Tanaka, D. Schmitz, H. H.Hsieh, H.-J. Lin, C. T. Chen, E. D. Bauer, J. L. Sarrao,E. Goremychkin, M. Koza, L. H. Tjeng, and A. Severing,Phys. Rev. B , 195114 (2010). T. Willers, J. C. Cezar, N. B. Brookes, Z. Hu, F. Strigari,P. Koerner, N. Hollmann, D. Schmitz, A. Bianchi, Z. Fisk,A. Tanaka, L. H. Tjeng, and A. Severing, Phys. Rev. Lett. , 236402 (2011). C. Praetorius, M. Zinner, P. Hansmann, M. W. Haverkort,and K. Fauth, Phys. Rev. B , 165107 (2016). P. Ohresser, E. Otero, F. Choueikani, K. Chen,S. Stanescu, F. Deschamps, T. Moreno, F. Polack, B. La-garde, J. P. Daguerre, F. Marteau, F. Scheurer, L. Joly,J. P. Kappler, B. Muller, O. Bunau, and P. Sainctavit,Rev. Sci. Instrum. , 013106 (2014). R. Nakajima, J. St¨ohr, and Y. U. Idzerda, Phys. Rev. B , 6421 (1999). M. W. Haverkort, M. Zwierzycki, and O. K. Andersen,Phys. Rev. B , 165113 (2012). M. W. Haverkort et al. O. Gunnarsson and K. Sch¨onhammer, in
Handbook on thePhysics and Chemistry of Rare Earths , Vol. 10, edited byK. A. Gschneider, Jr. and L. Eyring (Elsevier, Amsterdam,1987) Chap. 64, p. 103. C. Praetorius,
Ce M , XAS and XMCD as Local Probesfor Kondo and Heavy Fermion Materials , Ph.D. thesis,Univ. W¨urzburg (2015). C. Praetorius, P. Hansmann, M. Haverkort, and K. Fauth,“Coupling of core hole and itinerant excitations in rareearth intermetallics,” unpublished results. V. Mauchamp, M. Jaouen, and P. Schattschneider, Phys.Rev. B , 235106 (2009). O. ˇSipr, J. Min´ar, A. Scherz, H. Wende, and H. Ebert,Phys. Rev. B , 115102 (2011). J. Vinson, J. J. Rehr, J. J. Kas, and E. L. Shirley, Phys.Rev. B , 115106 (2011). M. W. Haverkort, G. Sangiovanni, P. Hansmann,A. Toschi, Y. Lu, and S. Macke, Europhys. Lett. ,57004 (2014). A. Delobbe, M. Finazzi, B. Buschinger, O. Trovarelli,C. Geibel, J. P. Kappler, and G. Krill, Physica B , 1144 (1999). N. E. Bickers, D. L. Cox, and J. W. Wilkins, Phys. Rev.B , 2036 (1987). T. Jo and A. Kotani, J. Magn. Magn. Mater , 394 (1987). G. Zwicknagl, V. Zevin, and P. Fulde, Zeitschrift FurPhysik B-condensed Matter , 365 (1990). B. T. Thole, P. Carra, F. Sette, and G. van der Laan,Phys. Rev. Lett. , 1943 (1992). J. P. Schill´e, F. Bertran, M. Finazzi, C. Brouder, J. P.Kappler, and G. Krill, Phys. Rev. B , 2985 (1994). Y. Teramura, A. Tanaka, B. T. Thole, and T. Jo, J. Phys.Soc. Jpn. , 3056 (1996). T. Jo, J. Electron Spectrosc. Relat. Phenom. , 73 (1997). H. Desgranges, Physica B , 93 (2015). B. Halbig et al. , unpublished results. The lowest temperature data point of the BESSY appearsto disagree with the data gathered at SOLEIL. We ascribethis difference mostly to the construction details of the re-spective cryostats and positioning of temperature sensorsin particular. For the SOLEIL dataset, toward low temper-ature, sensor reading is a lower limit to the actual sampletemperature, the deviation being estimated to ∆ T ≈ . T cryo = 2 K. In the cryostat used at BESSY, the valueread from the detector is rather an upper limit to the spec-imen temperature, some deviation arising for T < ∼
20 K. D. D¨urkop, E. Braun, B. Politt, H. Schmidt, B. Roden,and D. Wohlleben, Zeitschrift f¨ur Physik B CondensedMatter , 55 (1986). P. Fazekas,
Electron Correlation and Magnetism (WorldScientific, 1999). K. S. D. Beach and F. F. Assaad, Phys. Rev. B , 205123(2008). M. Bercx and F. F. Assaad, Phys. Rev. B , 075108(2012). M. Bercx, Private communication.68