Parimagnetism in RCo 2 series (R=Dy, Ho, and Tm)
C. M. Bonilla, J. Herrero-Albillos, A. I. Figueroa, C. Castan-Guerrero, J. Bartolome, I. Calvo-Almazan, D. Schmitz, E. Weschke, L. M. García, F. Bartolome
aa r X i v : . [ c ond - m a t . m t r l - s c i ] F e b Parimagnetism in RCo series (R=Dy, Ho, and Tm) C. M. Bonilla ∗ Instituto de Ciencia de Materiales de Arag´on and Departamento de F´ısica de la Materia Condensada,CSIC - Universidad de Zaragoza, Pedro Cerbuna 12, E-50009 Zaragoza, Spain
J. Herrero-Albillos
Fundaci´on ARAID, Paseo Mar´ıa Agust´ın 36, E-50004 Zaragoza, SpainCentro Universitario de la Defensa, Ctra. de Huesca s/n, E-50090 Zaragoza, Spain andInstituto de Ciencia de Materiales de Arag´on, CSIC - Universidad de Zaragoza, Pedro Cerbuna 12, E-50009 Zaragoza, Spain
A. I. Figueroa, C. Cast´an-Guerrero, and J. Bartolom´e
Instituto de Ciencia de Materiales de Arag´on and Departamento de F´ısica de la Materia Condensada,CSIC - Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain
I. Calvo-Almaz´an
Institute Laue Langevin, 38042, Grenoble, France andInstituto de Ciencia de Materiales de Arag´on and Departamento de F´ısica de la Materia Condensada,CSIC - Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain
D. Schmitz and E. Weschke
Helmholtz-Zentrum Berlin f ¨ u r Materialien und Energie GmbH,Albert-Einstein-Str. 15, 12489 Berlin, Germany L. M. Garc´ıa and F. Bartolom´e
Instituto de Ciencia de Materiales de Arag´on and Departamento de F´ısica de la Materia Condensada,CSIC-Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain (Dated: April 2, 2019)X-ray circular magnetic dichroism (XMCD), longitudinal ( χ ac ) and transverse (TS) ac magneticsusceptibility have been measured in several members of the R Co series ( R = Dy, Ho, and Tm) asa function of temperature and applied magnetic field. We show that parimagnetism is a general be-havior along the R Co ferrimagnetic series ( R being a heavy rare earth ion). XMCD results evidencethe presence of two compensation temperatures, defining two different parimagnetic configurations,which is a fully unexpected result. The inverse χ ac curve exhibits a deviation from Curie-Weissbehavior which is recovered under applied magnetic field. The large excess of polarizability abovethe critical temperature proves the existence of an enhanced effective moment due to the presence ofshort range magnetic correlations, which are also observed in TS measurements. The combination ofTS and XMCD measurements allows to depict new magnetic phase diagrams for the R Co series. Anew scenario allowing to understand the observed phenomenology as a Griffiths phase-like behavioris proposed, where the amorphous R Co represents the undiluted system case. PACS numbers: 81.30.Bx, 71.20Lp, 75.20Hr
I. INTRODUCTION
Among the newly discovered phenomena related toshort-range magnetic correlations, an interesting exam-ple is the parimagnetic phase observed in ErCo . Pari-magnetism consists in the antiparallel alignment of thenet magnetization of each sublattice in the magneticallydisordered, high temperature phase (
T > T c ), of a two-sublattice system under an applied magnetic field, differ-ently to the usual paramagnetic phase where the two sub-lattices would present net magnetizations parallel to theapplied magnetic field. This phenomenon indeed contra-dicts the usual interpretation of the R Co series as simpleexchange-enhanced paramagnets with a rare-earth dom-inated magnetism. The hierarchy of exchange magnetic interactions in R Co is J Co − Co ≫ J R − Co , while J R − R is usually ne-glected. Therefore, the formation of Co short-range mag-netic correlations in the paramagnetic phase is energet-ically favored. Indeed, Co magnetic correlations havebeen previously identified at the origin of some anoma-lies in the transport properties of R Co , but it had beenascribed to critical fluctuations, near the Curie tempera-ture. Nevertheless, previous works on ErCo and HoCo ;including small angle neutron scattering (SANS), lon-gitudinal ( χ ac ) and transverse (TS) magnetic suscep-tibilities and muon spin relaxation spectroscopy ( µ SR), among others, have shown the presence of magnetic cor-relations not only near T c , but also at much higher tem-peratures,. Magnetic correlations at temperatures abovethe ferrimagnetic ordering in ErCo had also been identi-fied in the neutron diffraction pattern and in electri-cal resistivity measurements. Magnetization and trans-port properties show the fingerprint of high tempera-ture magnetic correlations in related compounds, suchas Er(Co x Ni − x ) and Er(Co x Ti − x ) . Other inter-metallic Laves phases do also show magnetic correlationsat high temperatures compared with T C : D´eportes andcoworkers claimed the existence of short-range mag-netic order at four times T C in CeFe , evidencing theimportance of transition metal interactions in the para-magnetic phase of these intermetallic compounds.Previous work in ErCo allowed to identify a Griffiths- like phase within its magnetically disordered phase. AGriffiths phase can be seen as the realization of short-range order, within a diluted magnet of a given concen-tration x , lying between the percolation concentration, x p , and the undiluted “pure” system, x = 1. The Grif-fiths phase appears below a given temperature, T G , abovethe critical ordering temperature of the diluted systemand that of the pure one: T xc < T G < T c . The shortrange order is a kind of low-temperature remnant of themagnetic order of the undiluted system, in the temper-ature range at which the system would order sponta-neously if it were not diluted. This is the scenario in mostof the physical realizations of Griffiths phases claimedup to now, as in germanates , manganites andalloys. The ac-susceptibility curve obtained forErCo shows the usual deviation from a Curie Weiss be-havior below a certain temperature which is, however,recovered under applied magnetic field. Such ex-treme sensitivity to the applied field is characteristic ofa Griffiths phase.
However, the physical origin of aGriffiths - like phase in a pure compound like ErCo isnot easy to understand.From previous experimental work parimag-netism in ErCo is a consequence of short-range correla-tions between Co magnetic moments and it is natural toinvestigate whether this phenomenon is present amongother ferrimagnetic R Co .A systematic X-ray magnetic circular dichroism(XMCD) study together with magnetic susceptibilitymeasurements performed in R Co with R = Dy, Ho andTm, are presented in order to obtain a more complete de-scription of the magnetism above T c in these compounds.Our work indicates that parimagnetism observed in theErCo system is found in the R Co R = Dy, Ho andTm, members of the series too, and the effect of short-range correlations are even stronger than those found inErCo , showing some unexpected features reminiscent ofcompensation points in rare-earth transition metal com-pounds. II. SAMPLE CHARACTERIZATION ANDEXPERIMENTAL DETAILS
Polycrystalline ingots of R Co with R = Dy, Ho andTm were prepared following a standard procedure bymelting together the metallic precursors in an arc fur- nace under Ar atmosphere. An excess of 1% of rareearth has been added to obtain a final sample with thecorrect stoichiometry, taking into account the amount ofrare earth which is evaporated during the melting pro-cess. The samples were annealed under Ar atmosphereat 850 ◦ C for twelve days in order to improve the homo-geneity. X-ray diffraction analysis on powdered sampleswas performed in a Rigaku RTO 500RC diffractometerwith Bragg-Brentano geometry using K α Cu radiationat room temperature. The Rietveld analysis shows verywell crystallized samples, as shown in previous works and no impurities within the 2% of accuracy of powderdiffraction methods.A complete magnetic characterization has been per-formed on the powdered samples. The longitudinal acsusceptibility has been measured under applied fields upto µ H =5 T in a SQUID Quantum Design magnetome-ter from T =4 K to T =400 K. Magnetization ( M ) (Fig. 1)and ac susceptibility χ ac as function of temperature mea-sured for the three ferrimagnetic systems show criticaltemperatures T c = 136 K, 78 K and 4.6 K for DyCo ,HoCo , and TmCo , respectively, which coincide withthose reported in the literature. The presence of short-range magnetic correlationsabove T c has been studied by means of transverse suscep-tibility (TS). These measurements were performed usinga radiofrequency (RF) self-resonant circuit oscillator us-ing CMOS transistors coupled with a LC tank, operat-ing at a frequency of about 12 MHz. The sample andcoil set are designed to fit into a commercial QuantumDesign PPMS probe such that the perturbing RF mag-netic field inside the coil ( ≈ − T) is oriented perpen-dicular to the external dc magnetic field H dc suppliedby the PPMS superconducting magnet, thus setting thetransverse geometry. In a TS measurement at a fixedtemperature, the resonant frequency f is recorded as the H dc field is swept from positive to negative saturation(unipolar scan) and then back to positive (bipolar scan).Since the change in frequency of the circuit, ∆ f , is adirect consequence of the change in inductance as thesample is magnetized, ∆ f is proportional to the varia-tion of the transverse susceptibility of the sample, ∆ χ T ,the magnitude of physical interest. In the present study,the quantity [∆ χ T /χ T ]% = [ χ T ( H dc ) − χ sat T ] /χ sat T × H dc is the one considered,where χ Sat T is the transverse susceptibility at the satu-rating field µ H sat =1 T, within a temperature range 2.5K < T <
300 K. Details on the technique and the circuitcan be found in Ref. 33.The skin depth, δ , of the R Co compounds has beencalculated by the well-known formula δ = p ρ / πfµ where ρ is the resistivity, µ the magnetic permeabilityand f the excitation frequency. The values range be-tween 30 µ m at low temperatures and 180 µ m for T > T c up to room temperature. In order to assure that the RFmagnetic field penetrates the entire grain the sample hasbeen powdered to reach a grain size of average diameter ∼ . µ m, well below δ , to avoid the formation of eddy T(K)
TmCo T(K) m ( B /f. u . ) T(K) FIG. 1. (Color online) Magnetization as function of temperature at µ H =0.25 T (full symbol) and µ H =1 T (open symbol).From left to right for ferrimagnetic systems DyCo , HoCo and TmCo . currents on the surface.X-ray absorption spectroscopy (XAS) and X-ray mag-netic circular dichroism (XMCD) measurements werecarried out at the UE46-PGM1 beamline, at BESSY syn-chrotron facility, Berlin. The measurements were per-formed at the Co L , and the Dy, Ho and Tm M , absorption edges with a polarization rate of 0.9. Thetemperature was varied between 5 K to 350 K under ap-plied fields between µ H =0.25 T and µ H =6 T. Thedetection method used was total electron yield. In orderto prevent the spurious signals due to surface oxidation,the polycrystalline ingots were cleaved in an ultra highvacuum chamber just before starting its exposure to theX-ray beam. III. EXPERIMENTAL RESULTSA. X-ray magnetic circular dichroism
DyCo , HoCo and TmCo have been studied bymeans of XMCD spectroscopy to obtain the element-specific magnetization in these materials. The XMCDsignals show the usual spectral shape, for Co L , and R M , edges, as is mentioned in Refs. 34 and 35 respec-tively. The XMCD signals at the R M edges have in ev-ery case the same sign and (as expected) much reducedintensity with respect to the the R M edge ones, andwill be, therefore, omitted in the majority of this work,for the sake of clarity. Figs. 2, 3 and 4 show XMCDspectra for Co L , edges (left) and Dy, Ho and Tm M edges (right), respectively, at selected temperatures andat an applied magnetic field of µ H =1 T for DyCo andTmCo and µ H =2 T for HoCo .The Co absorption at the L , edges has been normal-ized using the usual 2:1 branching ratio. The normal-ization of the M , , has been performed by setting to 1the maximum of the unpolarized absorption (obtainedby averaging right and left polarized light XAS). The netmagnetic moment of the rare-earth sublattice has been
770 780 790 800 810-0.70.00.71.42.12.83.54.2
T>T c T 345 K228 K188 K140 K T 228 K188 K140 K345 K X M CD ( a r b . un i t s ) E (eV) FIG. 2. (Color online) XMCD spectra at the Co L , (left) andDy M (right) edges in DyCo for selected temperatures at µ H =1 T. Dashed line separates the XMCD spectra measuredabove and below T c . assumed to be proportional to the area under the XMCDsignal at R M edge, in such a way that negative valuesfor the integral calculated from the XMCD spectrum cor-respond to a positive sign of the R net magnetic moment.On the other hand, given A and B , the areas under theCo XMCD curves at the L and L edges respectively,the magnetic moment per Co atom is proportional to − A + 4 B . The spectra at the top of each panel inFig. 2 and 3, show the XMCD signals recorded below T c in DyCo and HoCo , respectively. The sign of thesespectra demonstrates the ferrimagnetic ordering of thesephases, with the larger rare-earth net magnetic momentparallel to the applied field and the Co one, much smaller,antiparallel to it. XMCD could not be measured below T c on TmCo , as T c lies below the lowest temperatureachievable at the experimental setup.The temperature range in which a parimagnetic con- 770 780 790 800 810-1012345 Co L E (eV) X M CD ( a r b . un i t s ) 40 K96 K135 K352 K HoCo T>T c T 352 K135 K96 K40 K E (eV) X M CD ( a r b . un i t s ) T>T c T 770 780 790 800 810-1012345 Co T>T c T 41 K TmCo L E (eV) X M CD ( a r b . un i t s ) Tm 41 K T 188 K, the XMCD signal atthe Co L , changes sign, remaining thereafter unchangedup to the highest studied temperature.Above the temperature at which the net moment of Co M E (eV) X M CD ( a r b . un i t s ) Pari.2 Para 770 780 790 800 8100246810 L HoCo 352 K E (eV) X M CD ( a r b . un i t s ) Pari.2 Para FIG. 5. (Color online) XMCD spectra at the Co L , (left) andHo M , (right) edges in HoCo for selected fields at T =352 K.Dashed line separates the paramagnetic and parimagnetic-2configurations. HoCo T T T c M(T) SQUID m Ho Co Co + m Ho H=1T m ( B /f. u . ) T (K) FIG. 6. (Color online) Magnetic moment in HoCo (left) as afunction of temperature obtained from the XMCD data treat-ment. Open circles ( ◦ ) and open diamonds ( ♦ ) are the netmagnetization per atom for Co and rare-earth respectively,while full diamonds (cid:7) are the sum of the rare-earth and Comagnetization. The continuous line is the SQUID magneti-zation measurement. T c is the long-range magnetic orderingtemperature, T , is the temperature at which the net magne-tization of Co changes from negative to positive and T , is thetemperature at which the net magnetization of the rare-earthsublattice changes from positive to negative magnetization sublattice is zero, T ≈ 220 K, the disor-dered Co atomic magnetic moments are polarized by theapplied magnetic field, creating a positive Co magneticmoment.Analogous results are obtained in HoCo and TmCo (Fig. 3 and Fig. 4). Well above T c , at 96 K and 25 K forHoCo and TmCo respectively, the sign of XMCD signalfor Co L , edge remains unchanged with respect to theferrimagnetic phase, which is the telltale characteristic ofparimagnetism. At a higher temperature, which dependson the magnitude of the applied field and R , the net Comagnetic moment changes from negative to positive. InFigs. 3 and 4 it can be seen that at T =135 K and µ H =2T for HoCo and T =35 K and µ H =1 T for TmCo , theCo net magnetic moment is aligned parallel to the appliedmagnetic field, as expected in paramagnetism.Up to this point, our results in R Co for R = Dy, Ho,and Tm show a behavior that is similar to that previouslyobserved in ErCo . At higher temperatures, an interest-ing feature is found in these compounds. For example,the Dy signal in DyCo is small, (indeed comparable tonoise level) but the minute, positive signal at 1296 eVobservable in the bottom right panel of Fig. 2 suggests areversal on the sign of the Dy XMCD at the M edge for T > 228 K. The Dy sublattice magnetization would be-come opposite to the Co net magnetization and to the ap-plied magnetic field. The experimental results on HoCo and TmCo unambiguously confirm this trend: Ho andTm M XMCD signals change their sign at high tempera-ture, above T = 135 K and T = 35 K, respectively (rightbottom spectra of Fig. 3 and Fig. 4). In view of theseresults, two temperatures at which one of the two sub-lattice magnetization crosses zero well above the criticaltemperature can be defined: T , where the net magneti-zation of Co changes from negative to positive and T ,where the net magnetization of the rare-earth sublatticechanges from positive to negative (see Table I).The separate thermal evolution of Ho and Co net mag-netic moments obtained from XMCD measurements at µ H =1 T is shown in Fig. 6. The signals in the fer-rimagnetic phase have been scaled to the values of Hoand the Co moments, as reported from neutron diffrac-tion experiments. As expected, the total magnetizationmeasured in a SQUID magnetometer coincides with thesum of both magnetization sublattices, i.e., the sum of Hoand twice the Co net magnetization sublattices. Both Hoand Co magnetizations show abrupt jumps at the order-ing transition. T and T are also indicated.One would expect that under a sufficiently high ap-plied magnetic field, the conventional paramagnetic con-figuration should be recovered. To complete our studyof the paramagnetic phase of the R Co series, we mea-sured XMCD for applied fields from µ H =0.25 T up to µ H =6 T, at selected temperatures. Fig. 5 shows theXMCD spectra recorded at the Co L , and Ho M , edges (left and right panel, respectively) measured inHoCo at T =352 K. The experimental results show thata field of µ H =4 T at room temperature is required torecover the polarization signs of conventional paramag-netism for HoCo . The Co XMCD signal is essentiallyunchanged, indicating that Co net magnetization is par-allel to the applied magnetic field at T =352 K, indepen-dently of the orientations of the Ho net magnetic mo-ment. Four configurations, common to the three ferrimag-netic compounds for Co and R net magnetic momentsare identified from the XMCD measurements under ap-plied magnetic fields up to µ H =5 T in a temperaturerange between T =4 K to T =350 K : a) the long-rangeordered antiparallel alignment below T c ; b) the pari-magnetic low-field intermediate-temperature phase, de-noted as parimagnetic-1 , (which is essentially the pari-magnetism observed in ErCo ); c) the expected high-temperature paramagnetic configuration, which almostdisappears at low fields, substituted by d) an unexpected,high-temperature parimagnetic configuration with an in-verted, negative net rare-earth magnetization, denoted as parimagnetic-2 to differentiate it from the low tempera-ture parimagnetic one. B. ac magnetic susceptibility As it has been discussed in previous works, theoccurrence of parimagnetism in ErCo is due to a com-petition of interactions on the Co sublattice. In particu-lar, the Co-Co exchange interaction is responsible for theshort-range order correlations which facilitate the collec-tive reversal of the Co moments at the temperature wherethe net moment of the Co magnetic sublattice crosseszero. The phenomenology found by means of XMCD ex-periments in DyCo , HoCo , and TmCo , is similar tothat observed in ErCo , and therefore short-range corre-lations in the paramagnetic phase of the R Co systemsconsidered in this study, are expected to occur. Indeed,previous SANS measurements have been used recently toprobe the presence of short-range correlations within themagnetically disordered phase of HoCo , yielding a cor-relation length of 7 ˚A in HoCo around T , in coincidencewith our previous result in ErCo . Magnetic ac suscepti-bility measurements will be presented here to get insighton the origin of parimagnetism in those compounds.The inverse of χ ′ ac ( T ), for DyCo (left), HoCo (center)and TmCo (right), at selected applied magnetic fieldsbetween µ H =0 T to µ H =5 T is shown in Fig 7. In astandard paramagnetic system, the inverse of χ ′ ac ( T ) hasa linear dependence with temperature, according to theCurie-Weiss law χ ′ ac ( T ) = CT − θ and C = N µ B k B µ eff (1)where N is the number of dynamic entities, θ is the Curietemperature, and µ eff is the effective magnetic moment. R magnetic moments are two orders of magnitude largerthan the Co one, for T > T c in the R Co , there-fore, µ eff is expected to be entirely due to the rare-earthmoment.Within this approximation, the dashed lines in Fig. 7represent the contribution to the paramagnetic suscepti-bility from R ions by using the S, L and J values pre-dicted by the Hund’s rules, and assuming that only theground state J is populated. The estimated values for Compound T c (K) XMCD TS T (K) T (K) T (K) DyCo 140 210 < T < 228 228 < T < 252 200HoCo 78 88 < T < 112 135 < T < 200 104TmCo < T < 32 38 < T < 41 20TABLE I. Temperatures T (at which the net magnetization of Co changes from negative to positive) and T (at which thenet and magnetization of the rare-earth sublattice changes from positive to negative) estimated from XMCD and T , obtainedfrom TS measurements. DyCo T(K) / a c ’ ( m o l O e / e m u ) . .0 T5 T . Dy =10.63 B HoCo / ’ a c ( m o l O e / e m u ) T(K) Ho =10.60 B . .0 T1 T . . . / a c ’ ( m o l O e / e m u ) TmCo T(K) Tm =7.57 B . FIG. 7. (Color online) Temperature dependence of inverse of ac susceptibility. From left to right: DyCo ; at µ H =0 T, 0.04T, 1 T and 5 T, HoCo ; at µ H =0 T, 0.05 T, 0.3 T and 1T TmCo ; at µ H =0 T, 0.1 T, 1 T and 5 T. Dashed line shows thetheoretical behavior expected for each system in the paramagnetic region according to Curie-Weiss law. µ eff are 10.63 µ B , 10.60 µ B and 7.57 µ B for Dy, Ho andTm ions, respectively. Clear deviations from those valuesand, in general, from a linear dependence with tempera-ture of the χ − ac can be observed for the three compounds,with the presence of one or two minima in the curves for µ H < =1 T. Those strong deviations from the simpleCurie-Weiss curve at lower applied magnetic fields cor-responds to values of µ eff considerably higher than thoseestimated from the rare-earth magnetic moment only.For example, the curve measured at zero applied fieldfor HoCo allows to extract a Curie-Weiss value of µ eff ∼ µ B , about 2.5 µ B higher than the predicted value for R magnetic moment only. The high-temperature XMCDsignal at the Co L , edges shows a non negligible mag-netic moment in Co, that is nevertheless not large enoughto explain the strong µ eff observed. We ascribe the highvalue of µ eff to the formation of volumes of short-lengthcorrelated spins, of about 1.5 nm in diameter near T , asobserved by SANS in ErCo and HoCo .At high applied fields, the ac magnetic susceptibilitydecreases and at µ H =5 T the expected linear trend ofthe inverse susceptibility for R Co with small, uncor-related Co moments is almost recovered. This behav-ior, also observed in magnetically disordered ErCo , hasbeen related with the extreme sensitivity to the appliedfield of χ ac taking place at the onset of a Griffiths-likephase. C. Radio-frequency transverse susceptibility andmagnetic phase diagrams Transverse Susceptibility (TS) technique has been usedto study the anisotropic magnetic properties and mag-netic switching in a variety of systems from multilay-ered thin films to single crystals and nanoparticles. More interesting to our case, a careful analysis of TSprofiles (their shape and magnitude) has also revealedseveral fascinating features, such as phase coexistenceand short-range correlations that are present in dopedmanganites and cobaltites. Indeed, the sensi-tivity of this technique to short-range magnetic cor-relations has been previously demonstrated in R Co compounds , as it provides insight of the spin dy-namics in each temperature region, particularly in theirmagnetically disordered phase. While an unipolar TSscan recorded below T c , on a polycrystalline ferromagnethas a maximum at a field, H M which coincides with theswitching field, H S , which is the field needed to reversethe direction of the net magnetization; an ideal param-agnet shows H M =0 since the magnetic moments aredisordered and uncorrelated, being able to adiabaticallyfollow the RF excitation. In contrast, the TS bipolarscans measured above T c in ErCo showed maxima atnonzero H M in the temperature range where short-rangemagnetic correlations are present. In this work we present TS results performed in DyCo ,HoCo and TmCo . DyCo bipolar scans are shown in FIG. 8. (Color online) TS scans for DyCo , for µ H sat = 1 Tat selected temperatures representative of the TS profile evo-lution in ferri- (panel a), pari-1 (c) and pari-2 (e) phases; TSbipolar scans shown in panels (b) and (d) have been recordedat T c and T , respectively. Fig. 8 for µ H sat = 1 T at selected temperatures: T =100K (panel a), 150 K (b), 180 K (c), 200 K (d) and 250K (e), representative of the TS profile evolution in ferri-(panel a), pari-1 (c) and pari-2 (e) phases. TS bipolarscans shown in panels (b) and (d) have been recorded at T c and T , respectively. Panel (a) shows peaks at theswitching field of ferrimagnetic DyCo . The peaks arebroad, but the switching field can still be properly deter-mined. TS bipolar scans at and above T c are narrowerthan that measured below T c . In particular, H M is al-most zero at T c (panel b), but it is clearly nonzero athigher temperatures. The finite values of H M found at T > T c in DyCo suggest the occurrence of short-rangecorrelations within the paramagnetic region. The short-range order gives rise to an increase in the field neededto reverse the magnetic moments, H M , as they follow theRF excitation, in agreement with previous experimentalworks. A description for a similar study on HoCo can be found in Ref. 8 where the same phenomenologyhas been observed up to temperatures as high as T = 300K.The temperature evolution of H M can be obtainedfrom the analysis of the TS profiles measured at a numberof temperatures, as shown in the top panel of Figs. 9, 10,and 11 for DyCo , HoCo and TmCo respectively. Inthe three studied systems H M = 0 for T < T c due tothe long-range ferrimagnetic alignment between the R and Co magnetization sublattices. Then, the H M pro-file shows a minimum at T c , associated to the collectiveswitching behavior of the rare-earth and Co moments inthe ferrimagnetic ordered state. Above T c , H M neverreaches zero in the measured temperature range. This isan indication of the presence of short-range correlationsin all the R Co compounds, since, as it has been pointedout before, H M = 0 in an ideal paramagnet.The H M curve measured for DyCo shows two min-ima, one at 150 K (at T c ) and a second one at T =200 Kapproximately coincident with T as obtained from theXMCD analysis. The minima in the H M profile matcheswith the maxima observed at the magnetic susceptibility,as it is shown in the middle panel of Fig. 9. At temper-atures above T , the H M curve increases slowly, and nofeature is present at the temperature T due to the in-version of the Dy moments. These observations indicatethat the net magnetic moment in the field direction dueto the Co short-range correlated moments is the domi-nant contribution to the TS susceptibility for T above T c . The R moments, instead, follow adiabatically theCo short-range correlated moments behavior. Magneti-zation measurements at temperatures above T c have beenperformed in DyCo corroborating that the H M valuesdepicted in Fig 9 within the disordered magnetic phasedo not correspond to a switching field of any spuriousferro- or ferrimagnetic phase.The characteristic relaxation time of the fluctuationdue to these short-range correlations can be calculatedfrom Ref. 9 as τ SRC ≃ − s in the temperature rangeof T which is much larger than the experimental time τ exp = 1 /ω exp = 10 − s. Therefore, the maxima in theRF TS at H = 0 may be observed in spite of the param-agnetism shown by DC magnetization. At T > T c , thedrag caused by this slow reaction to the RF excitation re-sults in a displaced profile at each unipolar scan and theTS bipolar scans show two maxima, (at ± H M ). As theCo net moment is reduced near the compensation tem-perature T the τ SRC decreases and therefore the valueof H M decreases. Similar results are found for HoCo and TmCo , but additionally, in the case of HoCo verylarge values of H S can be also seen in the proximities ofthe spin reorientation transition that in this compoundoccurs at T =16 K. The correspondence between the results from the TSand XMCD measurements can be seen more clearly whencomparing the top panel of Figs. 9, 10, and 11 with theircorresponding bottom panels, where new magnetic phasediagrams for DyCo , HoCo , and TmCo are proposed FIG. 9. (Color online) Top: Field at which a maxima isfound in the TS unipolar scans µ H M as function of tem-perature for DyCo . The values of H M obtained from the se-lected temperatures in Fig. 8 are represented in open symbols.Middle: Susceptibility identified from TS measurements with µ H sat = 1 T for DyCo Bottom: New magnetic phase dia-gram proposed for DyCo from XMCD analysis for differentfields and temperatures. Dashed line are a guide to the eye toseparate the different configurations between the Co and Dynet magnetic moments. (cid:3) are selected values of T c from theliterature. Full symbols indicate the field and temperaturevalues at which XMCD measurements have been recorded inDyCo . (cid:4) are used for the ferrimagnetic phase (Co ↓ Dy ↑ ). H , • and N are used in the disordered region, to indicate theparimagnetic-1 (Co ↓ Dy ↑ ), paramagnetic (Co ↑ Dy ↑ ) andparimagnetic-2 configurations (Co ↑ Dy ↓ ) respectively. from our extensive XMCD data. In the magnetic phasediagrams, schemes of the magnetic configurations of Coand R moments are depicted to distinguish the ferrimag-netic, parimagnetic-1, paramagnetic and parimagnetic-2 configurations. The full symbols represent tempera-ture and field values at which XMCD data were acquiredto obtain the phase diagram. Square, inverted trian-gle, circle and triangle corresponds to the ferrimagnetic,parimagnetic-1, paramagnetic and parimagnetic-2 config-urations respectively, as deduced from the Co L , and RM , XMCD spectra. Open squares are selected values FIG. 10. (Color online) Top: Field at which a maxima isfound in the TS unipolar scans µ H M as function of temper-ature for HoCo . The inset shows the spin reorientation atlow temperature in HoCo that can be identified by a jump tovery high H M values at ∼ 30 K. Middle: Susceptibility iden-tified from TS measurements with µ H sat = 1 T for HoCo Bottom: New magnetic phase diagram proposed for HoCo from XMCD analysis for different fields and temperatures.Dashed line are a guide to the eye to separate the differentconfigurations between the Co and Ho net magnetic moments. (cid:3) are selected values of T c from the literature. Full symbolsindicate the field and temperature values at which XMCDmeasurements have been recorded in HoCo . (cid:4) are used forthe ferrimagnetic phase (Co ↓ Ho ↑ ). H , • and N are used inthe disordered region, to indicate the parimagnetic-1 (Co ↓ Ho ↑ ), conventional paramagnetic (Co ↑ Ho ↑ ) and parimagnetic-2 configurations (Co ↑ Ho ↓ ) respectively. of T c obtained from the literature (Ref. 48). The dashedlines are a guide to the eye that suggest a separation be-tween the regions at which the different alignments be-tween the Co and R net magnetic moments take place onthe magnetic phase diagram for T > T c .In general, the H M curve depicts the same regions ob-served in the phase diagrams obtained from XMCD inthe three compounds. However, the temperature T ob-tained from TS measurements, is lower than the temper-ature T obtained from XMCD in DyCo and TmCo . FIG. 11. (Color online) Top: Field at which a maxima isfound in the TS unipolar scans µ H M as function of temper-ature for TmCo . The H M and temperature ranges of themain panel and the inset of Fig. 11 have been chosen to allowthe observation of the ordering transition at T = 4 . H M for TmCo above T c . Middle: Suscep-tibility identified from TS measurements with µ H sat = 1 Tfor TmCo . Bottom: New magnetic phase diagram proposedfor TmCo . Dashed line are guides to the eye to separate thedifferent configurations between the Co and Tm net magneticmoments. (cid:3) are values of T c from magnetization as func-tion of temperature measurements presented in Fig. 1. H arethe temperatures and fields at which the alignment betweenCo and Tm net magnetic moment is antiparallel for temper-atures above T c (parimagnetism-1). • are the temperaturesand fields at which the XMCD analysis indicates a parallelalignment between Co, Tm net magnetic moments and H (conventional paramagnetism). N are the temperatures andfields at which the XMCD analysis indicates that Tm netmagnetic moment is antiparallel to Co net magnetic momentand to the applied field H (parimagnetism-2). We ascribe such a discrepancy to the difference betweenthe processes that are probed by each experiment; whileXMCD is a static technique, TS is sensitive to the mag-netic relaxation processes; therefore the temperature atwhich a given process is observed may vary. The XMCDexperiments reflect R and Co magnetization separately while the TS measurements senses the whole R Co sys-tem. Table I allows to compare the compensation tem-peratures T and T obtained from XMCD at µ H =1 Tand the temperature T obtained from TS experiments. IV. DISCUSSION The onset of parimagnetism can be understood as aresult of the internal and external magnetic fields actingon the Co sublattice. Within the mean field approxima-tion one can consider three contributions to the total fieldacting on one of the Co moments: the external appliedfield, H appl , the molecular field due to the R sublattice, H R − Co , and the molecular field from the Co sublatticeitself, H Co − Co . The direction of the applied magneticfield, H appl defines the positive direction for magnetiza-tion. The magnitude of the molecular field for R and Cosublattices depends on both the total magnetization ofeach sublattice and the molecular field constants: n R − Co ,and n Co − Co , which are proportional to the exchange in-teractions, J R − Co , and J Co − Co , respectively.The key point to understands the R Co magnetismabove T c lies in the fact that the Co-Co interaction( J Co − Co ∼ 10 meV) is much stronger than the Co- R interaction ( J R − Co ∼ The difference of twoorders of magnitude between the interactions would leadto a magnetism dominated by cobalt. However, the elec-tronic structure is such that the Co magnetic moment inthe paramagnetic region is about two orders of magni-tude smaller than the rare-earth magnetic moments ,and therefore, the interaction energies J Co − Co µ Co µ Co and J R − Co µ R µ Co are similar in magnitude. This equi-librium between the two internal fields may be easilydestabilized by small or local variations on the crystalor electronic structure. J Co − Co is so large that a fluc-tuation on the Co magnetic moment, either dynamic orstatic, will locally generate an exchange energy reduc-tion by creating a short-range Co-Co ferromagneticallycorrelated zone. As cited before, the presence of mag-netic short-range correlation have been confirmed by µ SRspectroscopy in ErCo below ∼ 200 K, and short-rangecorrelation lengths of about 8 ˚A below ∼ 150 K have beenevidenced by SANS in ErCo and HoCo .Moreover, the ac magnetic susceptibility in the para-magnetic region of R Co (R=Ho, Dy, or Tm) at zerofield clearly does not follow the expected Curie lineardependence. The actual curve obtained by measuring agiven sample is batch- and history dependent, althoughthe general tendency in polycrystalline samples is the oneshown in Fig. 7: samples have a much larger polariz-ability than expected in the paramagnetic region, withsusceptibilities typically two to five times larger thanexpected. The application of a (rather large) magneticfield is required to remove this extra polarizability and insome cases, µ H =5 T are barely enough to recover theparamagnetic values. A very detailed characterization ofthe samples (x-ray and neutron diffraction, electron mi-0croscopy) do not show a noticeable presence of impuritieswhich may reasonably explain the experimental behavior.However, any local deviation of stoichiometry would gen-erate electronic and structural changes able to stronglycorrelate Co moments.Indeed, density functional theory calculations to bepublished elsewhere show that the formation of smallCo short-range-correlations is favored near defects, va-cancies, grain boundaries or electronic instabilities. Thisresult reminds the strong spin fluctuations which havebeen relevant to understand the paramagnetism and thetransport properties of the Co Laves phases for decades.Any instability that locally enhances a Co magnetic mo-ment generates a very strong internal H Co − Co field, whicheasily may overcome the effect of the applied and H R − Co fields, enhancing short-range correlations around it. Avery interesting novelty in this work is that, in R Co ,the Co short-range correlations not only generate pari-magnetism at low field and temperature but also areable to reverse the (small) net magnetization of the rareearth sub-lattice at higher temperatures under moder-ate applied fields. Therefore, as long as short-rangemagnetic correlations are present; both configurations,parimagnetism-1 and parimagnetism-2, are explained bythe competition between external and molecular fieldsacting on the Co moments.As discussed in the Introduction, a Griffiths-like phasewas identified in ErCo . From the analysis of the longi-tudinal ac susceptibility, we find in DyCo , HoCo , andTmCo the usual enhanced polarizability related withGriffiths phases, while the expected value is recoveredunder applied magnetic field. This sensitivity tothe applied field is also characteristic of the formationof Griffiths phases. The enhanced effective momentvalues found in the magnetically disordered phase canbe ascribed to the formation of spin-correlated volumeswhich are in agreement with previous SANS results , asexpected in a Griffiths phase. Finally, the occurrence ofparimagnetism itself requires that Co sublattice magneti-zation overcomes the lanthanide one, which is only possi-ble if a relatively large number of Co moments ferromag-netically coupled form a larger, although short lived ,cooperative magnetic moment. Therefore, it is plausibleto propose the formation of a Griffiths-like phase above T c in the R Co compounds analyzed in this paper.However, as also mentioned in the Introduction, a Grif-fiths phase occurs in a magnetic diluted system and it isobserved through the formation of short-range magneticcorrelations in the range T c < T < T G , being T G < T ′ c ,the critical temperature of the undiluted system.It is relevant to note that amorphous R Co order mag-netically at T am c well above room temperature (for exam-ple, ErCo orders at T am c ∼ 500 K and T amc > 400 K foramorphous TbCo ) Cobalt in amorphous ErCo has anon negligible magnetic moment, and J Co − Co µ Co µ Co setsthe magnetic order so that T am c ≈ · T cryst c . In contrast,Co in crystalline R Co has a reduced magnetic moment,as a consequence of the electronic structure in the or- dered solid. One may think of the crystalline R Co as asystem in which magnetic Co has been “diluted” by theeffect of the electronic structure. Therefore, R Co couldbe considered as behaving like a magnetically “diluted”system, where the large cobalt moments, present in theamorphous phase, have been removed, indeed substitutedby small, fluctuating magnetic moments (of the order of0 . µ B in ErCo ). However, the Co-Co interaction isvery strong ( J Co − Co ∼ 10 meV) and it comes naturallythat spin fluctuation on Co moments in R Co easily in-duce ferromagnetically correlated volumes. As said be-fore, Co develops a magnetic moment in R Co near anyimperfections or grain boundaries present in the system.These magnetic moments may add to intrinsic, dynam-ical spin fluctuations to generate a background densityof short range order correlations. Those would be at theorigin of the observed Griffiths-like phase behavior, andin some cases may be stable enough to induce parimag-netism, i.e. to reverse the net magnetic moment of therare earth ions at temperatures above room temperature,in a magnetically disordered image of the compensatedstate above T amcomp .In summary, parimagnetism would be a consequence ofa Griffiths-like phase of the undiluted, high-temperaturemagnetically ordered amorphous R Co . The new “di-lution” mechanism at play in this Griffiths phase is notcompositional, but purely magnetic, with origin in a com-bination of electronic-structure and local order aroundCo atoms. Considering the R Co amorphous alloy asthe “pure”, undiluted system sets T ′ c clearly above roomtemperature. Moreover, amorphous ErCo also shows acompensation point at about T amcomp ∼ 230 K : above T amcomp , Co net magnetization is larger than the Er one,whereas the opposite is true below T amcomp . This phe-nomenology closely resembles what has been found in R Co (for R =Dy, Ho, and Tm), giving raise to theparimagnetic-2 configuration. V. CONCLUSIONS Several experimental techniques have shown the exis-tence of parimagnetism (the antiparallel configuration ofthe net magnetization of two sublattices within the para-magnetic phase) among ferrimagnetic systems in the CoLaves phases family. XMCD results in R Co ( R = Dy,Ho, and Tm) show not only the inversion of Co mag-netization sublattice at a compensation temperature T well above T c , as observed in ErCo . Surprisingly, theinversion of R magnetization sublattice at a second com-pensation temperature, T > T , gives rise to a high tem-perature parimagnetic configuration, with the rare earthmoment antiparallel to the Co one and to the field, whichresembles the high temperature state of the magneticallyordered amorphous R Co .These findings depict a new magnetic phase diagramfor DyCo , HoCo and TmCo , with two new parimag-netic configurations above T c and below an applied field1threshold of about 3 − R Co as a consequence of the relativeintensity of both, the exchange interactions present inthe system, and the relative magnitude of the cobalt andrare-earth magnetic moments. As the Co-Co exchangeconstant is two orders of magnitude stronger than theCo- R one, the molecular field of Co sublattice is dom-inant as soon as any fluctuation locally drives the Comagnetic moment higher than its average value, given bythe electronic structure: a local deviation of stoichiom-etry, impurities, vacancies, or even statistical spin fluc-tuations would generate strong correlations between Comoments, originating the formation of short-range cor-related volumes. The strength of the Co-Co interactionwould transmit this correlations along the system. In-deed, the detailed study of TS profiles as function oftemperature has shown a finite value of H M for all tem-peratures and compounds, which is a clear indication ofshort-range correlations. On the other hand χ ac mea-surements confirm a very high effective moment µ eff , cal- culated from the χ − ac ( T ) curves, which is congruent withthe occurrence of Co short-range correlations. The re-covery of the Curie behavior under applied field suggesta Griffiths-like behavior, which has also been observedin ErCo . An interpretation of parimagnetic R Co as aGriffiths phase stands on an electronic structure - baseddilution of the Co moment with respect to the “hightemperature phase” represented by magnetically orderedamorphous R Co . VI. ACKNOWLEDGMENTS The financial support of MAT2011/23791, AragoneseIMANA (partially funded by the European Social Fund)and the European FEDER funds is acknowledged. Theauthors acknowledge HZB for the allocation of syn-chrotron radiation beamtime. C. M. Bonilla acknowl-edge a Spanish MINECO grant and C. Cast´an and A. I.Figueroa acknowledge a JAE-Predoc CSIC grants. ∗ [email protected] J. Herrero-Albillos, F. Bartolom´e, L. M. Garc´ıa, A. Young,T. Funk, J. Campo, and G. J. Cuello., Phys. Rev. B ,094409 (2007). A. M. Stewart, J. Phys. C: Solid State Phys. , 1557(1984). D. Bloch and R. Lemaire, Physical Review B , 2648(1970). J. D´eportes, D. Gignoux, and F. Givord, Phys. Statu Solidi(b) , 29 (1974). J. J. Rhyne, Journal of Magnetism and Magnetic Materials , 88 (1987). A. Castets, D. Gignoux, and B. Hennion, Journal of Mag-netism and Magnetic Materials , 375 (1980). E. Gratz, R. Resel, A. T. Burkov, E. Bauer, A. S.Markosyan, and A. Galatanu, J. Phys.: Condens. Mat-ter , 6687 (2001). C. M. Bonilla, I. Calvo, J. Herrero-Albillos, A. I. Figueroa,C. Cast´an-Guerrero, J. Bartolom´e, J. A. Rodriguez-Velamaz´an, D. Schmitz, E. Weschke, D. Paudyal, V. K.Pecharsky, K. A. G. Jr, F. Bartolom´e, and L. M. Garc´ıa,Journal of Applied Physics , 07E315 (2012). J. Herrero-Albillos, L. M. Garc´ıa, and F. Bartolom´e., Jour-nal of Physics: Condensed Matter , 216004 (2009). A. I. Figueroa, S. Chandra, M. H. Phan, H. Srikanth,C. M. Bonilla, F. B. L. M Garc´ıa and, J. Bartolom´e,and J. Herrero-Albillos, Journal of Applied Physics ,07E118 (2011). C. M. Bonilla, N. Marcano, J. Herrero-Albillos,A. Maisuradze, L. M. Garc´ıa, and F. Bartolom´e., PhysicalReview B , 184425 (2011). A. Pirogov, A. Podlesnyak, T. Strassle, A. Mirmelstein,A. Teplykh, D. Morozov, and A. Yermakov, Applied Ph-ysiscs A: Material Sciences Process , 598 (2002). A. Podlesnyak, T. Strassle, J. Schefer, A. Furrer,A. Mirmelstein, A. Pirogov, P. Markin, and N. Baranov,Physical Review B , 012409 (2002). F. Garc´ıa, M. R. Soares, and A. Y. TAkeuchi., Journal ofMagnetism and Magnetic Materials , 1197 (2001). M. R. Soares, A. Y. Takeuchi, F. Garc´ıa, S. F. da Cunha,and M. E. Massalami, Journal of Magnetism and MagneticMaterials , 473 (1999). Y. ¨Oner and O. Kamer, Journal of Applied Physics ,07E137 (2008). J. D´eportes, , D. Givord, and K. R. A. Ziebeck, Journalof applied physics , 2074 (1981). R. B. Griffiths, Phys. Rev. Lett. , 17 (1969). Z. W. Ouyang, V. K. Pecharsky, K. A. G. Jr., D. L.Schlagel, and T. A. Lograsso, Physical Review B ,094404 (2006). C. Magen, P. A. Algarabel, L. Morell´on, J. P. Ara´ujo,C. Ritter, M. R. Ibarra, A. M. Pereira, and J. B. Sousa,Physical Review Letters , 167201 (2006). A. M. Pereira, L. Morell´on, C. Magen, J. Ventura, P. A.Algarabel, M. R. Ibarra, J. B. Sousa, and J. P. Ara´ujo,Physical Review B , 172406 (2010). N. P´erez, F. Casanova, F. Bartolom´e, L. M. Garc´ıa,A. Labarta, and X. Batlle., Physical Review B , 184411(2011). M. Salamon and S. Chun., Physical Review B , 014411(2003). M. Salamon, P. Lin, and S. Chun., Physical Review Letters , 197203 (2002). W. Jiang, X. Z. Zhou, G. Williams, Y. Mukovskii, andK. Glazyrin, Physical Review B , 092404 (2007). A. H. C. N. G. Castilla and B. A. Jones., Physical ReviewLetters , 35313534 (1998). Y. ¨Oner and M. Guillot, Journal of Magnetism and Mag-netic Materials , 3313 (2012). J. Herrero-Albillos, L. M. Garc´ıa, F. Bartolom´e, A. Young,and T. Funk., Journal of Magnetism and Magnetic Mate-rials , e442 (2007). F. Bartolom´e, J. Herrero-Albillos, L. M. Garc´ıa, A. Young,T. Funk, N. Plugaru, and E. Arenholz., Journal of Mag- netism and Magnetic Materials , 319 (2004). J. Herrero-Albillos, F. Bartolom´e, L. M. Garc´ıa, J. Campo,A. Young, T. Funk, and G. J. Cuello., Journal of Mag-netism and Magnetic Materials , 1645 (2007). M. M´ıˇZek, J. ProkleˇZka, V. Sechovsk´y, D. Turcinkov´a,J. Prchal, A. F. Kusmartseva, K. V. Kamenev, and J. Ka-mar´ad, Journal of Applied Physics , 07E132 (2012). S. Khmelevskyi and P. Mohn., Journal of Physics: Con-densed Matter , 9453 (2000). A. I. Figueroa, J. Bartolom´e, J. M. G. del Pozo, A. Arauzo,E. Guerrero, P. T´ellez, F. Bartolom´e, and L. M. Garc´ıa,J. Magn. Magn. Mat. , 2669 (2012). C. T. Chen, Y. U. Idzerda, H. J. Lin, N. V. Smith,G. Meigs, E. Chaban, G. H. Ho, E. Pellegrin, and F. Sette,Physical Review Letters , 152 (1995). J. B. Goedkoop, Ph. D. thesis (Katolieke Universiteit teNijmegen, 1989). B. T. Thole and G. van der Laan, Europhysics Letters ,1083 (1987). S. Pizzini, L. M. Garc´ıa, A. Fontaine, J. P. Rueff, J. Vogel,R. M. Galera, J. B. Goedkoop, N. B. Brookes, G. Krill,and J. P. Kappler, Journal of Electron Spectroscopy andRelated Phenomena , 165 (1997). D. Bloch and J. Voiron, Solid State Communications ,685 (1973). J. Herrero-Albillos, Ph. D. Tesis Aplicaci´on de nuevas son-das microsc´opicas al estudio del magnetismo de las fasesde Laves RCo (Universidad de Zaragoza, 2007) Chap. 7,pp. 168–174. D. Gignoux, F. Givord, and J. Schweizer, Journal ofPhysics F: Metal Physics , 1823 (1977). D. Gignoux, D. Givord, F. Givord, W. C. Koehler, andR. M. Moon, Phys. Rev. B , 162 (1976). N. A. Frey, S. Srinath, H. Srikanth, M. Varela, S. Penny-cook, G. X. Miao, and A. Gupta, Physical Review B ,024420 (2006). G. T. Woods, P. Poddar, H. Srikanth, and Y. M.Mukovskii, J. Appl. Phys. , 10C104 (2005). P. Poddar, M. B. Morales, N. A. Frey, S. A. Morrison, E. E.Carpenter, and H. Srikanth, Journal of Applied Physics , 063901 (2008). S. Chandra, A. I. Figueroa, B. Ghosh, M. H. Phan,H. Srikanth, and A. K. Raychaudhuri., Journal of AppliedPhysics , 07D720 (2011). C. Leighton, D. D. Stauffer, Q. Huang, Y. Ren, S. El-Khatib, M. A. Torija, J. Wu, J. W. Lynn, L. Wang, N. A.Frey, H. Srikanth, J. E. Davies, K. Liu, and J. F. Mitchell,Physical Review B , 214420 (2009). N. A. Frey Huls, N. S. Bingham, M. H. Phan,H. Srikanth, D. D. Stauffer, and C. Leighton,Phys. Rev. B , 024406 (2011). J. Herrero-Albillos, L. M. Garc´ıa, F. Bartolom´e,F. Casanova, A. Labarta, and X. Batlle, Phys. Rev. B , 134410 (2006). F. G. D. Gignoux and R. Lemaire., Physical Review B ,3878 (1975). C. M. Bonilla and et al., In preparation. T. Nakama, K. Shintani, K. Yagasaki, A. T. Burkov, andY. Uwatoko, Physical Review B , 511D522 (1999). B. Boucher, A. Lienard, J. P. Rebouillat, andJ. Schweizer., Journal of Physics F: Metal Physics , 1421(1979). K. Buschow,