Abstract.: We report the results of measurements of the low-temperature specific heat Cp(T) and the ac susceptibility χac(T) in low applied magnetic fields for a series of samples of Eu1-xCaxB6. The anomalies in Cp(T), together with the results for χac(T) and M(H), confirm the onset of phase transitions to long range magnetic order for x < 0.7 and provide evidence that for x ≥ 0.7, the Eu moments, which are captured in large magnetic clusters with magnetic moments of the order of 260 μB, adopt a spin-glass type ground state. The data set allows to establish the low-temperature [ T,x] phase diagram of this alloy serie

Bianchi, A. D.

Felder, E.

Fisk, Z.

Ott, H. R.

Streule, S.

Weller, M.

Wigger, G. A.

RERO DOC Digital Library

Eur. Phys. J. B 46, 231–235 (2005)DOI: 10.1140/epjb/e2005-00251-y THE EUROPEANPHYSICAL JOURNAL BPercolation limited magnetic order in Eu1−xCaxB6G.A. Wigger1,a, E. Felder1, M. Weller1, S. Streule1, H.R. Ott1, A.D. Bianchi2, and Z. Fisk31 Laboratorium fu¨r Festko¨rperphysik, ETH-Ho¨nggerberg, 8093 Zu¨rich, Switzerland2 National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32306, USA3 Department of Physics, University of California - Davis, CA 95616, USAReceived 27 January 2005 / Received in final form 29 April 2005Published online 11 August 2005 – c© EDP Sciences, Societa` Italiana di Fisica, Springer-Verlag 2005Abstract. We report the results of measurements of the low-temperature specific heat Cp(T ) and the acsusceptibility χac(T ) in low applied magnetic fields for a series of samples of Eu1−xCaxB6. The anomaliesin Cp(T ), together with the results for χac(T ) and M(H), confirm the onset of phase transitions to longrange magnetic order for x < 0.7 and provide evidence that for x ≥ 0.7, the Eu moments, which arecaptured in large magnetic clusters with magnetic moments of the order of 260 µB , adopt a spin-glasstype ground state. The data set allows to establish the low-temperature [T ,x] phase diagram of this alloyseries.PACS. 71.30.+h Metal-insulator transitions and other electronic transitions – 75.20.-g Diamagnetism,paramagnetism, and superparamagnetism1 IntroductionThe phenomenon of magnetic order in solids is most oftenbased on periodic arrays of magnetic moments on regularlattice sites, here denoted as magnetic sites, of the crystalstructure of the respective material. The necessary inter-action between the moments, which more or less deter-mines the transition temperature and the ordered-momentconﬁguration, may be due to diﬀerent causes, not of pri-mary interest here. The present work is devoted to inves-tigating the inﬂuence of the random removal of magneticmoments on the onset of magnetic order at low tempera-tures.A prominent example of magnetic systems is the socalled Heisenberg ferromagnet, where all magnetic sitesare occupied by moments, which are mutually linked bysome interaction. The probability of ﬁnding an unbro-ken path following these links on a random path acrossthe sample is P = 1. This situation is altered, if a num-ber of moments is randomly removed from the magneticsites, breaking all links to the respective nearest neigh-bour moments. The parameter P may also be deﬁned asthe probability of the existence of an inﬁnitely large clus-ter of adjacent moments in an inﬁnitely large array ofsuch moments. In a three-dimensional simple cubic struc-ture, the concentration of moments, below which P = 0, iscthres ≈ 0.31, the so called site percolation limit [1,2]. Forc < cthres it is predicted that the ground state of the solida e-mail: gwigger@standford.eduis no longer characterized by long range magnetic orderand other conﬁgurations of the moments are energeticallymore favourable.The hexaboride compound EuB6 may be regarded asa model system for studying this site percolation prob-lem. It crystallizes with a simple cubic structure. The Euions, each carrying a magnetic moment of 7 µB due to thehalf ﬁlled S = 7/2 4f electron shell [3], occupy the centersof cubes whose corners are decorated with B6 octahedra.Ferromagnetic order sets in at approximately 15 K. Thechemical composition may be changed by, e.g., replacingthe rare-earth element Eu by the alkaline earth elementCa. Because both type of ions adopt a divalent conﬁgura-tion, the substitution is in ﬁrst order expected to be iso-electronic and simply provoking a dilution of magnetic mo-ments. Recently published results of electronic transportmeasurements and their interpretation based on theoreti-cal modelling suggested the inﬂuence of site percolation onthe magnetic and transport properties of Eu1−xCaxB6 [4].In this work we investigated the x-dependence of themagnetic properties of Eu1−xCaxB6 with measurementsof the speciﬁc heat, the magnetic susceptibility and ﬁeldinduced magnetization at low temperatures. The resultsindicate that long range magnetic order among the Eumoments ceases to be established, if x exceeds 0.7 or, inview of site percolation, the concentration of magnetic mo-ments c < cthres. Instead, the ground state seems to adopta spin-glass type conﬁguration of the remaining Eu mo-ments.232 The European Physical Journal BThe samples were all in single-crystalline form andtheir high structural quality was conﬁrmed by correspond-ing characterizations using X-ray and electron-microscopytechniques. For more details, the reader should consult ref-erences [4,8,14].2 Results and discussion2.1 Specific heatWe ﬁrst concentrate on our study of the low-temperaturespeciﬁc heat. The investigated samples were single crys-tals in the form of platelets and the compositions weresuch that x = 0.1, 0.16, 0.37, 0.65 and 0.77, respectively.The speciﬁc heat Cp(T ) was measured in the temperaturerange between 0.34 and 22 K, employing the conventionalrelaxation method. The Cp(T ) data for a crystal of thebinary compound with x = 0 was taken from the workpresented in reference [5]. Since we are mainly interestedin the magnetic part of Cp(T ), we attempted to separatethe diﬀerent contributions. The dominant non magneticterm is due to lattice excitations, Clatt. The contributiondue to conduction electrons can safely be neglected be-cause their concentration is only of the order of 10−3 perunit cell. Earlier successful analyses of speciﬁc-heat andresistivity data of hexaborides were based on the assump-tions that i) the excitations of the Boron network are suﬃ-ciently well accounted for by a Debye-type spectrum withθ = 1160 K in the case of LaB6 [6] and that ii) the cationmotions are well represented by Einstein-type oscillators.Thus the molar speciﬁc heat of the lattice isClatticep (T ) = CDebyep (T ) + CEinsteinp (T ) . (1)The calculations of the two terms is done by usingthe well known integral expressions [23]. For this particu-lar series, the element speciﬁc Einstein temperatures areθEu = 168 K for Eu [8] and θCa = 373 K for Ca [4], re-spectively. The second term on the rhs of equation (1) isthus concentration dependent and a sum of the formCEinsteinp (T ) = (1− x) · CEup (T ) + x · CCap (T ) , (2)where CEup (T ) and CCap (T ) denote the contributions fromthe Eu- and Ca ions, respectively. We assume that theDebye temperature does not vary across the series.The magnetic part Cmag(T ) of the speciﬁc heat is nowobtained by simply subtracting the calculated lattice spe-ciﬁc heat from the measured data. The result is shown inFigure 1. Although shifted to lower temperatures, the twoconsecutive transitions, clearly seen and discussed in pre-vious work for EuB [5,9–12], persist to Ca contents of atleast 16%. Only one transition is observed for x = 0.37 andbeyond, conﬁrming earlier conclusions based on results ofelectronic transport measurements [13,14]. We note thatfrom comparing the shapes of the speciﬁc-heat anomaliesfor x = 0.65 and x = 0.77 alone, the transition from a fer-romagnetically ordered to a spin-glass type ground statecannot be inferred.0 2 4 6 8 10 12 14 16 18 20 220246810121416T (K)Cmag  (J/mole K)x = 0x = 0.10x = 0.16x = 0.37x = 0.65x = 0.77Fig. 1. Magnetic part of the specific heat for x = 0, 0.1, 0.16,0.37, 0.65 and 0.77.0 2 4 6 8 10 12 14 16 18 20 22 240.00.10.20.30.40.50.60.70.80.91.0T (K)Smag/(R*ln(8))  x = 0x = 0.10x = 0.16x = 0.37x = 0.65x = 0.77Fig. 2. Magnetic entropy for x = 0, 0.1, 0.16, 0.37, 0.65 and0.77.The magnetic entropy across these transitions is read-ily obtained fromSmag =∫ T0Cmag(T ′)T ′dT ′ . (3)As may be seen in Figure 2, the full molar magnetic en-tropy of the Eu2+ ions, which amounts to (1−x)Rln(2J+1) with J = 7/2, is recovered at approximately 20 K forall values of x covered in this study.2.2 Magnetic susceptibility and magnetizationThe magnetic phase diagram for x < 0.7 was establishedby Arrott-plot analyses in combination with speciﬁc-heatand susceptibility experiments [4]. In order to probe themagnetic properties of those alloys with x > 0.7 and hencefor moment concentrations c < cthres, we measured the acmagnetic susceptibility χac of Eu1−xCaxB6 between 0.35G.A. Wigger et al.: Percolation limited magnetic order in Eu1−xCaxB6 2330.0 0.5 1.0 1.5 2.0 2.5 3.0 3.52.0.10-72.5.10-73.0.10-73.5.10-74.0.10-74.5.10-7T (K)V/ f  (arbitrary units)22.7 Hz227 Hz2270 Hz9170 Hz22700 Hz0.0 0.5 1.0 1.5 2.0 2.5 3.02.5.10-73.0.10-73.5.10-74.0.10-7 0 Oe10 Oe25 Oe50 Oe100 OeFig. 3. The response voltage divided by the excitation fre-quency V/f for x = 0.77 as a function of temperature for f =22.7, 227, 2270, 9170 and 22700 Hz. The inset shows V/f forf = 227 Hz in various static magnetic fields.and 3.5 K for x = 0.73, 0.77 and 0.84, respectively. Thesamples were placed into an RCH-1000 plastic cell. Inte-grated in the cell body were the primary and secondarycoil systems, each consisting of a pair of coils with 2 × 500and 2 × 2000 windings, respectively. The current ampli-tude in the primary coil was of the order of 1 mA and thefrequencies of the excitation current were set to f = 22.7,227, 2270, 9170 and 22700 Hz respectively. In Figure 3we display the output voltage V of the secondary coils, ameasure for χac, divided by the frequency f , i.e. V/f , ver-sus temperature. The position of the voltage maximum onthe temperature scale, Tmax(f), shifts to higher temper-atures upon enhancing the frequency. The application ofweak static magnetic ﬁelds, of the order of 10 Oe, signif-icantly reduces the amplitude of the maximum. As maybe seen in the inset of Figure 3, at and above 100 Oe,the anomaly is completely quenched. This behavior is dis-tinctly diﬀerent from the features of the ac-susceptibilityresults for x = 0.65 and 0.37, which exhibit a peak atTC but no observable frequency dependence within exper-imental resolution.A frequency dependent maximum in χac(T ) is usuallytaken as evidence for a spin freezing process which leads,upon decreasing the temperature, to a spin-glass conﬁgu-ration, or to the formation of a superparamagnetic phase.Element speciﬁc electron microscopy imaging (EFTEM)revealed [4] that material with x = 0.75 exhibits an inho-mogenous cluster formation with Eu- and Ca rich regions,respectively. This phase separation exceeds the expecta-tions from simply considering the inherent statistical in-homogeneity of the material. The clusters adopt diametersof the order of a few nm. The Eu-rich clusters may be con-sidered as acting like large magnetic molecules with giantmagnetic moments. In view of the response of an isolatedcluster, its dynamics may be characterized by a relaxationtime [15]τ ≡ 1/f = τ0 · eEa/kBTB , (4)-0.9 -0.8 -0.7 -0.6 -0.501234log10((Tmax(f)-Tsg)/Tsg)log(f)0 10 20 30 40 50 6005.0.1041.0.1051.5.1052.0.105B (kOe)M (A/m)Fig. 4. log10(f) is plotted as a function of log10[(Tmax(f) −Tsg)/Tsg] for x = 0.77. In the inset, we display the measuredmagnetization M(H) at 2 K for the same sample as emptycircles and the calculated curve (solid line) as indicated in thetext.where τ0 is the intrinsic relaxation time of the large mo-ment, Ea represents the energy barrier preventing the re-orientation of the moment and TB denotes the so calledblocking temperature. According to literature, however,ﬁtting the available data by employing equation (4) re-sults most often in unrealistic values of these parameters[16]. For real systems, the Vogel-Fulcher relation [17]τ = τ0 · eEa/kB(Tmax−T0) , (5)describing the frequency dependence of the temperaturewhere the magnetic susceptibility exhibits a maximum,seems to be more appropriate. Here, kBT0 is the clusterinteraction energy. This type of behaviour has often beenveriﬁed for typical spin glasses [18]. In our case, the bestﬁts are obtained by setting T0 = 0 K, indicating a pro-gressive freezing of moments which interact over a widerange of energies [19]. However, this procedure leads toτ0 ≈ 10−25 s, obviously unrealistic and orders of magni-tude diﬀerent from approximately 10−13 s, the commonvalue of τ0 for conventional spin glasses [20].In view of the clustering of the Eu moments, a moreadequate description of the dynamics is obtained by in-voking the critical slowing-down process viaf = f0 ·(Tmax(f)− TsgTsg)−zν, (6)where Tsg is the critical temperature for the spin-glassordering, zν is a constant and τ0 = f−10 is the charac-teristic time scale for the spin dynamics. In Figure 4 wedisplay our experimental data for x = 0.77, together witha ﬁt according to equation (6). The resulting parametersare Tsg = 1.79 K, zν = 10.17 and τ−10 = 5.15 × 109 s−1.For conventional spin glasses, experiments and mean-ﬁeldcalculations yield zν ≈ 10 and τ0 ≈ 10−12 s [16]. It is con-ceivable that the signiﬁcant diﬀerence in the magnitude of234 The European Physical Journal Bτ0 is due to the clustering of ionic moments, forming unitswith rather large magnetic moments in our case.It is customary to distinguish between spin-glass andsuperparamagnetic behaviour on the basis of the value ofthe ratio [21]K =∂ln(Tmax)∂ln(f)(7)where Tmax is again the temperature at which, for a givenfrequency f , the maximum of χac is observed.For spin glasses, K ∼ 0.01 and for superparamagnets,K ≥ 0.1. For all our samples with x > 0.7, we obtainK ≈ 0.06. Although this does not allow for a clear ver-dict, we argue that the low-temperature features for thesematerials are those of a spin glass whereby the freezingdoes not involve single spin moments, but the total mo-ments of the mentioned Eu-rich clusters.Additional details of the magnetic response of theseensembles of clusters can be obtained from results of mea-surements of the ﬁeld-induced magnetization. In meanﬁeld approximation, the magnetization of a single clusterincreases with decreasing temperature according tomcl = µsat · B7/2{gµBJ(λEuM + H)/kBT } , (8)where B7/2 is the Brillouin function for J = 7/2, M thesample magnetization, λEu = 5.1 is taken to be identicalto the mean ﬁeld parameter valid for pure EuB6 [8], H isthe external magnetic ﬁeld and the saturation moment isgiven byµsat = N · µEu (9)i.e., by the number of Eu ions in the cluster and the mo-ment of an individual Eu ion. If the clusters would actindependently from each other, the ﬁeld induced magne-tization is expected to vary according toM ∼ mclL(mclHkBT), (10)where L represents the Langevin function. It turned outthat this approximation is not adequate for describing ourM(H) data at low temperatures. The interaction betweenthe clusters can again be taken into account in mean-ﬁeldapproximation, resulting in [22]M(T,H) ∼ mclL[mcl(H + λM(T,H))kBT], (11)where λ captures the internal ﬁeld due to the cluster mag-netization.As usual, equation (11) has to be solved self consis-tently. The corresponding ﬁt to our M(H) data, recordedfor the crystal with x = 0.77 at T = 2 K and displayedin the inset of Figure 4 as open circles, is represented bythe solid line. Demagnetization eﬀects for the experimen-tal data have been considered as described in ref. [23]. Theaverage value for mcl = 260 µB corresponds to a clusterdiameter of approximately 3 nm, which is in good agree-ment with what is observed in the EFTEM images [4].The molecular-ﬁeld constant λ = −1.2 is distinctly smallerthan λEu and implies that the clusters interact predomi-nantly antiferromagnetically.0 10 20 30 40 50 60 70 80 90 100024681012x (%)T (K)paramagneticferromagneticspin-valve phenomena clusterspin glasssemimetal semiconductorFig. 5. Phase diagram for Eu1−xCaxB6. The open circles de-note the ferromagnetic transition obtained from Cmag(T ) andArrot-plot analyses [4]. The filled squares represent the clus-ter spin glass transition temperatures deduced from χac(T, f)data.3 ConclusionIn summary we provide experimental evidence that forx > 0.7 and thus below the site percolation limit, themagnetic ground state of Eu1−xCaxB6 is reached via thefreezing of moments of sizable Eu-rich clusters. This allowsfor a completion of the [x, T ] phase diagram of this hexa-boride series, shown in Figure 5. By increasing x, the ma-terial’s electronic ground state changes from semimetal-lic to a semiconducting, where the electronic propertiesare governed by defect states [24]. We showed previouslythat, at intermediate concentrations, the inhomogeneousdistribution of Eu ions leads to an intrinsic spin-valve fea-tures observed in transport and optical properties [14,25].This experimentally determined phase diagram is in con-ﬂict with the conclusions of a recent theoretical calcula-tion based on a simple Kondo lattice model [26]. It ap-pears that the spontaneous formation of Eu-rich clusters,carrying substantial moments, is requested to be a majoringredient in related future theoretical work.This work has benefited from partial financial support ofthe Schweizerische Nationalfonds zur Fo¨rderung der wis-senschaftlichen Forschung and the US-NSF grant DMR-0203214.References1. S. Kirkpatrick, Solid State Comm. 12, 1279 (1973)2. B.I. Shklovskii, A.L. Efros, Sov. Phys.-Usp. 18, 845 (1975)3. W. Henggeler, H.R. Ott, D.P. Young, Z. Fisk, Solid StateCommun. 108, (1998) 929.4. G.A. Wigger, C. Beeli, E. Felder, H.R. Ott, A.D. Bianchi,Z. Fisk, Phys. Rev. Lett. 93, 147203 (2004)5. L. Degiorgi, E. Felder, H.R. Ott, J.L. Sarrao, Z. Fisk, Phys.Rev. Lett. 79, 5134 (1997)G.A. Wigger et al.: Percolation limited magnetic order in Eu1−xCaxB6 2356. D. Mandrus, B. C. Sales and R. Jin, Phys. Rev. B 64,12302 (2001)7. C. Kittel, Introduction to Solid State Physics (J. Wileyand Sons, New York 1967)8. G.A. Wigger, R. Monnier, H.R. Ott, D.P. Young, Z. Fisk,Phys. Rev B 69, 125118 (2004)9. S. Su¨llow, I. Prasad, M.C. Aronson, S. Bogdanovich, J.L.Sarrao, and Z. Fisk, Phys. Rev. B 62, 11626 (2000)10. J.C. Cooley, M.C. Aronson, J.L. Sarrao, Z. Fisk, Phys.Rev. B 56, 14541 (1997)11. S. Su¨llow, I. Prasad, M.C. Aronson, J.L. Sarrao, Z. Fisk,D. Hristova, A.H. Lacerda, M.F. Hundley, A. Vigliante, D.Gibbs, PRB 57, 5860 (1998)12. M C. Aronson, J.L. Sarrao, Z. Fisk, M. Whitton, B.L.Brandt, Phys. Rev. B 59, 4720 (1999)13. S. Paschen, D. Pushin, M. Schlatter, P. Vonlanthen, H.R.Ott, D.P. Young, Z. Fisk, Phys. Rev. B 61, 4174 (2000)14. G.A. Wigger, C. Wa¨lti, H.R. Ott, A.D. Bianchi, Z. Fisk,Phys. Rev. B 66, 212410 (2002)15. L. Ne´el, Ann. Geophys. 5, 99 (1949)16. J.A. Mydosh, Spin-Glasses: An Experimental Introduction(Taylor and Francis, London, 1993)17. S. Shtrikman, E.P. Wohlfarth, Phys. Lett. A 85, 467(1981)18. F. Holtzberg, T.L. Francavilla, C.Y. Huang, J.L. Tholence,J. Appl. Phys. 53, 2229 (1982)19. H.v. Lo¨hneysen, J.L. Tholence, J. Magn. Magn. Materials13, 136 (1979)20. K. Gunnarson, P. Svedlindh, P. Nordblad, L. Lundgren,H. Aruga, A. Ito, Phys. Rev. Lett. 61, 754 (1988)21. G.V. Bazuev, O.V. Makaroa, G.P. Sheveikin, Russ. J.Inorg. Chem. 28, 1088 (1983)22. L. Wang, J. Ding, Y. Li, H.Z. Kong, Y.P. Feng, X.Z. Wang,J. Phys: Condens. Matter 12, 4253 (2000); or L. Wang, J.Ding, Y. Li, Y.P. Feng, N.X. Phuc, N.H. Dan, J. App.Phys. 89, 8046 (2001)23. C. Kittel, Introduction to Solid State Physics (John Wileyand Sons, New York 1967)24. K. Gianno´, A.V. Sologubenko, H.R. Ott, A.D. Bianchi, Z.Fisk, J. Phys: Cond Matter 14, (2002) 103525. A. Perucchi, G. Caimi, H.R. Ott, L. Degiorgi, A.D.Bianchi, Z. Fisk, Phys. Rev. Lett 92, 067401 (2004)26. V.M. Pereira, J.M.B. Lopes dos Santos, E.V. Castro, A.H.Castro Neto, Phys. Rev. Lett. 93, 147202 (2004)

Percolation limited magnetic order in Eu1-xCaxB6

http://doc.rero.ch/record/322461/files/10051_2005_Article_251.pdf

Percolation limited magnetic order in Eu1-xCaxB6

Abstract

Similar works

Full text

Available Versions

RERO DOC Digital Library