Origin of the magnetic anomaly and tunneling effect of europium on the ferromagnetic ordering in Eu8-xSrxGa16Ge30 (x = 0,4) type-I clathrates.

PHYSICAL REVIEW B 84, 054436 (2011) Origin of the magnetic anomaly and tunneling effect of europium on the ferromagnetic ordering in Eu8-xSrxGa16Ge30 (x=0,4) type-I clathrates Manh-Huong Phan,1Victorino Franco,1,2Anurag Chaturvedi,1Stevce Stefanoski,1George S. Nolas,1,*and Hariharan Srikanth1,† 1Department of Physics, University of South Florida, Tampa, Florida 33620, USA 2Dpto. F´ ısica de la Materia Condensada, ICMSE-CSIC, Universidad de Sevilla, P.O. Box 1065, 41080 Sevilla, Spain (Received 17 September 2010; revised manuscript received 6 April 2011; published 11 August 2011) Systematic dc magnetization studies using the Banerjee criterion, Kouvel-Fisher, and magnetocaloric effect methods provide physical insights into the origin of the magnetic anomaly and the tunneling effect of europium on the ferromagnetic ordering in Eu8Ga16Ge30 type-I clathrates. We show that Eu8Ga16Ge30 undergoes a second-order magnetic transition (SOMT) at TC∼35 K, resulting from the magnetic interaction between the Eu2+ions at the Eu2 sites, followed by a secondary magnetic transition at TL∼10 K (indicated as a magnetic anomaly in previous studies), as a result of the magnetic interaction between the Eu2+ions at the Eu1 and Eu2 sites. The critical exponent β=0.388 is close to that predicted from the three-dimensional Heisenberg model (β=0.365), while the critical exponent γ=0.956 is close to that predicted from the mean-field model (γ=1). The substitution of Sr2+for Eu2+retains the SOMT but largely reduces the transition temperatures (TC∼15 K and TL∼5 K), with the critical exponents β=0.521 and γ=0.917 close to those predicted from the mean-field model (β=0.5 and γ=1). These results point to the important fact that the tunneling of Eu2+between the four equivalent sites in the tetrakaidecahedral cage tends to prevent the occurrence of a long-range ferromagnetic ordering in the type-I clathrate materials. DOI: 10.1103/PhysRevB.84.054436 PACS number(s): 75.30.Sg I. INTRODUCTION In recent years, semiconductors with the clathrate-hydrate crystal structure have been the subject of intense research due to their potential for thermoelectric applications.1–3In these materials “guest” atoms of one or more species reside inside “host” framework polyhedra formed by other species.1 Incorporating rare-earth elements with large magnetic moments as “guest” atoms into an adequate semiconducting host framework has made these materials also interesting for thermomagnetic applications.4–8Among the existing clathrates, the composition Eu8Ga16Ge30 is an example in which the sites for the guest atoms are fully occupied by Eu.4,5The Eu8Ga16Ge30 composition can form in two completely different clathrate structure types.5In the type-I (Pm3n) clathrate crystal structure, Eu resides inside two different polyhedra: two inside dodecahedra at Eu1 (2acrystallographic site) and six inside tetrakaidecahedra at Eu2 (6dcrystallographic site) per unit cell. In the type-VIII (I43m) clathrate structure, eight distorted pentagonal dodecahedra containing 23 vertices surround the Eu2+ions.5The ferromagnetism of the clathrates is attributed to the presence of Eu2+ions with a large saturation moment (7 μB/Eu). The large separation distance between neighboring Eu2+ions (>5.23 ˚ A) excludes the direct exchange interaction between the localized 4fmoments and suggests the long-range Ruderman-Kittel-Kasuya-Yoshida (RKKY) interaction via the charge carriers as the key mechanism for the occurrence of the ferromagnetism in these materials.4–8Since the shortest distance between Eu2+ions in the type-I clathrate (∼5.23 ˚ A) is smaller than that of the type-VIII clathrate (∼5.562 ˚ A), the former possesses a higher Curie temperature (TC∼35 K) compared with that of the latter (TC∼13 K).5,7 Using the standard formulation for the exchange Hamiltonian and the carrier concentrations at TCdetermined from the Hall constants, Paschen et al.5have argued that ferromagnetism should exist for Eu-Eu distances under 6.5 ˚ A for the type-I clathrate and 10 ˚ A for the type-VIII clathrate. Nevertheless, the difference in the Curie temperature between these two clathrates has recently been re-examined and attributed to the different effective masses of the charge carriers.9,10 Our recent studies on the Eu8Ga16Ge30 type-VIII clathrate have revealed the nature of a second-order, paramagnetic to ferromagnetic (PM-FM) transition at TC∼13 K, with long-range ferromagnetic ordering.7However, the case may be different for the Eu8Ga16Ge30 type-I clathrate.4–6,8In this material, the shortest distances for Eu2-Eu2, Eu2-Eu1, and Eu1-Eu1 are 5.23 ˚ A, 5.62 ˚ A, and 10.7 ˚ A, respectively.5The Eu-Eu distance in the [111] direction is 9.27 ˚ A. Since the ferromagnetism only exists within Eu-Eu distances of 6.5 ˚ A,5the magnetic interactions at the Eu2-Eu2 and Eu1-Eu2 distances are expected to contribute to the ferromagnetism of the material. The ferromagnetism is dominated by the magnetic interaction between the Eu2+ions at the Eu2 sites, because the Eu2-Eu2 distance (5.23 ˚ A) is shorter than the Eu1-Eu2 distance (5.62 ˚ A) and more Eu2+ ions are located at the Eu2 sites than at the Eu1 sites per unit cell. However, the magnetic interaction between the Eu2+ions at the Eu1 and Eu2 sites is also expected to be significant at low temperature, which may result in a secondary ferromagnetic ordering. This is likely an important clue in assessing the physical origin of the broad hump observed in ρ(T) and CP(T) at around 10 K (henceforth TL)intheEu 8Ga16Ge30 type-I clathrate, which remained unexplained satisfactorily previously.5,10 Furthermore, it has been noted that Eu2+tunnels between four equivalent sites in the tetrakaidecahedral cages (at the Eu2 sites) in the Eu8Ga16Ge30 type-I clathrate.11 This effect was observed in the ferromagnetic phase below TC∼35 K12,13 and has recently been argued to account for the formation of the modulated ferromagnetic structure in the type-I clathrate.14 However, there are two emerging questions: 054436-1 1098-0121/2011/84(5)/054436(7) ©2011 American Physical Society MANH-HUONG PHAN et al. PHYSICAL REVIEW B 84, 054436 (2011) Why does the jump in Cpat TCsignificantly differ from the value expected for a uniform ferromagnet with S =7/2 from the mean-field theory?4,5,14 What is the universality class that governs the PM-FM transition in the Eu8Ga16Ge30 type-I clathrate? In this context, we believe that a clear understanding of the physical origin of the magnetic anomaly, the nature of the PM-FM transition, and the influence of tunneling of Eu2+ between the four equivalent sites in the tetrakaidecahedral cage on the critical behavior near TCin the type-I clathrate materials is of importance. To address these important and as yet unresolved issues, we have conducted a comprehensive study of the ferromagnetic phase transition and critical exponents in Eu8Ga16Ge30 and Eu4Sr4Ga16Ge30 type-I clathrates. The critical exponents associated with this transition have been determined from dc magnetization data using the Kouvel-Fisher (K-F) and the magnetocaloric effect (MCE) methods. Our results reveal that the magnetic transition is of second-order type in both cases. We show that in Eu8Ga16Ge30 in addition to the PM-FM transition at TC∼35 K, a secondary magnetic transition is present at TL∼10 K. We believe that the transition at TCresults from the magnetic interaction between the Eu2+ ions at the Eu2 sites, while the transition at TLoriginates from the magnetic interaction between the Eu2+ions at the Eu1 (2a) and Eu2 (6d) sites. The finding of the secondary magnetic transition at TL∼10 K can explain the occurrence of the broad hump in ρ(T) and CP(T) at approximately 10 K. Following the usual notation, the critical exponent βdescribes the temperature dependence of the spontaneous magnetization (M0∝(T−TC)β)), the exponent γdescribes that of the zero-field paramagnetic susceptibility χ−1 0∝(T−TC)γ, and n(TC) describes the field dependence of the magnetic entropy change at the critical temperature (SM(T=TC)∝Hn(TC)). For Eu8Ga16Ge30 β=0.388 is close to that predicted from the three-dimensional (3D) Heisenberg model (β=0.365), while γ=0.956 is close to that predicted from the mean-field model (γ=1). n(TC)=0.545 significantly deviates from that predicted by the mean-field model (n(TC)=2/3). The Sr doping largely reduces the transition temperatures (TC∼15 K and TL∼5 K), with the critical exponents β=0.521, γ=0.917, and n(TC)=0.667 close to those predicted by the mean-field model (β=0.5, γ=1 and n(TC)=2/3). These findings suggest the influence of tunneling of Eu2+between the four equivalent sites in the tetrakaidecahedral cage (at the Eu2 sites) on the ferromagnetic correlation in the type-I clathrate and allow us to address why the magnitude of the jump in Cpat TCsignificantly differs from the value expected for a uniform ferromagnet with S=7/2 from the mean-field theory.4,5,14 II. EXPERIMENT Polycrystalline Eu8Ga16Ge30 with the type-I crystal structure was synthesized by reacting the high-purity elements in stoichiometric ratios inside a boron nitride (BN) crucible, which was enclosed in a nitrogen atmosphere, inside a sealed quartz ampoule. The specimen was placed in an induction furnace at 1000 ◦C for 10 minutes followed by a rapid water quench. The Eu4Sr4Ga16Ge30 specimen was prepared by reacting the high-purity elements in a BN crucible inside a nitrogen-filled sealed quartz tube at 950 ◦C 0 1020304050607080 0 2 4 6 8 10 12 369121518 10.95 10.98 11.01 11.04 T C =15 K T C = 35 K μ 0 H = 10 mT M (emu/g) T (K) Eu 8 Ga 16 Ge 30 Eu 4 Sr 4 Ga 16 Ge 30 10.02 10.04 10.06 5 K M (emu/g) T (K) T L =10 K FIG. 1. (Color online) Temperature dependence of magnetization taken at a field of 10 mT for Eu8Ga16Ge30 and Eu4Sr4Ga16Ge30.The inset shows the enlarged M-T curve for Eu8Ga16Ge30. for 3 days, followed by slow cooling to 700 ◦C. After 3 days at 700 ◦C, the product was air quenched.6,15,16 X-ray diffraction (XRD) and energy dispersive spectroscopy (EDS) analyses revealed the presence of the type-I clathrate phase only, with no oxide impurities, and homogenous compositions within the polycrystalline grains.17 Refinement of synchrotron powder diffraction patterns revealed for a stoichiometry of Eu3.47(3)Sr4.53(3)Ga14.48(13)Ge31.52(13) with a 76% preferential europium occupation of the Eu1 site.16 Magnetic and magnetocaloric measurements were conducted using a commercial Physical Property Measurement System from Quantum Design over a temperature range of 2–60 K at applied fields up to3T. III. RESULTS AND DISCUSSION Figure 1shows the temperature dependence of the fieldcooled (FC) magnetization taken at a low applied field of 10 mT for Eu8Ga16Ge30 and Eu4Sr4Ga16Ge30. The PM-FM transitions in both specimens are noted. The Curie temperatures (TC), which are determined from the minimum in dM/dT across the transitions, are 35 K and 15 K for Eu8Ga16Ge30 and Eu4Sr4Ga16Ge30, respectively. A closer examination reveals a sharp increase in the magnetization at ∼10 K for Eu8Ga16Ge30 and at ∼5KforEu 4Sr4Ga16Ge30 (see inset of Fig. 1). This increase in the FC magnetization at TLis a clear indication of a secondary magnetic transition. We believe this transition originates from the magnetic interactions between the Eu2+ions at the Eu1 and Eu2 sites (determined by the Eu1-Eu2 distance). Our previous studies have revealed that Sr2+preferentially replaces Eu2+on the Eu2 sites in Eu4Sr4Ga16Ge30.This substitution increases the Eu-Eu distance to at least ∼10 ˚ A.6 Therefore, a strong decrease in the TC,TL, and saturation magnetization (MS) is expected in the Sr containing specimen. To determine the type of the PM-FM transition in Eu8Ga16Ge30 and Eu4Sr4Ga16Ge30, we have analyzed H/M vs M2curves (which were converted from the isothermal M-H data), using the so-called Banerjee criterion.18 The results are 054436-2 ORIGIN OF THE MAGNETIC ANOMALY AND TUNNELING ... PHYSICAL REVIEW B 84, 054436 (2011) 0 1000 2000 3000 4000 5000 0.00 0.05 0.10 0.15 0.20 0 400 800 1200 0.0 0.1 0.2 0.3 0.4 0.5 (b) 2K Eu8Ga16Ge30 H/M (Oe g/emu) M 2 (emu 2 /g 2 ) 60K (a) Eu4Sr4Ga16Ge30 H/M (Oe g/emu) M 2 (emu 2 /g 2 ) 50K 2K FIG. 2. (Color online) The H/Mvs M2plots at representative temperatures around TCfor Eu8Ga16Ge30 and Eu4Sr4Ga16Ge30. presented in Fig. 2. According to this criterion, the magnetic transition is of second order if all the H/Mvs M2curves have a positive slope.19 On the other hand, if some of the H/Mvs M2curves show a negative slope at some point, the transition is considered to be of first order.19,20 In the present case, the positive slopes of the H/Mvs M2curves of Eu8Ga16Ge30 and Eu4Sr4Ga16Ge30 clearly indicate that these are second-order magnetic transition (SOMT) materials. This result is consistent with the reported λ-shape of the specific heat Cp(T) curve in the Eu8Ga16Ge30 type-I clathrate.4,5 Since the Eu8Ga16Ge30 and Eu4Sr4Ga16Ge30 specimens exhibit SOMT, we have used the K-F method21 to precisely determine the critical exponents near their PM-FM transition temperatures. This method consists of an iterative procedure, which starts by constructing the Arrott-Noakes (A-N) plot (i.e., the plot of M2.5vs (H/M)0.75). From this plot, the values for the spontaneous magnetization M0(T) are computed from the intercepts of various isothermal magnetization vs field curves on the ordinate of the plot (for temperatures below TC). The intercept on the abscissa (for temperatures above TC), allows us to calculate the zero-field paramagnetic susceptibilityχ0(T). 20 25 30 35 40 45 50 0 10 20 30 Y (K) Y(K)=MS(dMS/dT)-1 T(K) 0 5 10 15 20 X (K) X(K)=χ0 -1(d 0 -1/dT)-1 0 500 1000 1500 2000 2500 3000 0 1 2 3 4 5 6 Eu M 1/0.39 (10 4 (emu/g) 1/0.39 ) (H/M) 1/0.96 ((Oe g/emu) 1/0.96 ) 60K 2K FIG. 3. (Color online) Modified Arrott plot isotherms with 2-K temperature interval for Eu8Ga16Ge30. The inset shows the K-F construction for determining the critical exponents and the Curie temperature; the solid lines are fits to Eqs. (1)and(2), respectively. Once the M0(T) and χ0(T)curves have been constructed, two additional parameter data sets, X(T) and Y(T), may be determined: X(T)=χ−1 0dχ−1 0dT−1=(T−TC)/γ (1) Y(T)=M0(dM0/dT )−1=(T−TC)/β (2) The rationale for constructing these quantities is that the spontaneous magnetization should scale with temperature as M0∝(T−TC)βand with zero-field paramagnetic susceptibility as χ−1 0∝(T−TC)γ. Therefore, in the critical region, both X(T) and Y(T) should be linear. The slopes give the values of the critical exponents and the intercepts on the temperature axis correspond to the Curie temperature. The values of the critical exponents are refined by using an iterative method, i.e., once Eqs. (1) and (2) produce the values of the critical exponents, a generalized A-N plot (M1/β vs (H/M)1/γ )is constructed and used to calculate new M0(T) and χ0(T) curves, which are subsequently input into Eqs. (1) and (2), resulting in new values for βand γ. The procedure terminates when the desired convergence of the parameters is achieved. TCis obtained as the intercept on the abscissa of both the Xand Y lines. As an example, Fig. 3shows the A-N plots of Eu8Ga16Ge30 with optimized critical exponents (βand γ) obtained from the K-F method. The inset of Fig. 3shows the K-F plot TABLE I. Table 1. Comparison of the values of the critical exponents of Eu8Ga16Ge30 and Eu4Sr4Ga16Ge30 type-I clathrates with those of theoretical models and of the Eu8Ga16Ge30 type-VIII clathrate from our previous study.7 Material TC(K) β∗γ∗δn(K-F) n(MCE) Ref. Eu8Ga16Ge30 (type-I) 35.2 0.388 0.956 3.474 0.545 0.562 This work Eu4Sr4Ga16Ge30 (type-I) 17.7 0.521 0.917 2.760 0.667 0.675 This work Eu8Ga16Ge30 (type-VIII) 13 0.470 0.964 3.014 — — 7 Mean-field model 0.51.03.0 0.66 36 3D Heisenberg model 0.365 1.336 4.80 36 3D Ising model 0.325 1.241 4.82 36 ∗The errors for calculating βand γare less than 0.05. 054436-3 MANH-HUONG PHAN et al. PHYSICAL REVIEW B 84, 054436 (2011) -2 0 2 4 0.8 1.6 2.4 3.2 -5 0 5 10 15 20 0.3 0.6 0.9 x = 0 MH -1/ δ (emu g -1 Oe -0.29 ) εH -1/( β + γ ) (10 -4 Oe -0.74 ) MH -1/ δ (emu g -1 Oe -0.36 ) εH -1/( β + γ ) (10 -4 Oe -0.70 ) x = 4 FIG. 4. (Color online) Normalized isotherms of Eu8Ga16Ge30 and Eu4Sr4Ga16Ge30 below and above the Curie temperature (TC)using the values of βand γdetermined by the K-F method. for this specimen. The fit parameters of both specimens are summarized in Table I. The best fit yields the values of TC= 35.2 K, β=0.388, and γ=0.956 for Eu8Ga16Ge30 and TC =7.7 K, β=0.521, and γ=0.917 for Eu4Sr4Ga16Ge30. Using the Widom scaling relationship,22 β+γ=βδ,the critical exponent δis determined to be 3.474 and 2.760 for Eu8Ga16Ge30 and Eu4Sr4Ga16Ge30, respectively. This relationship has been tested by plotting M(T=TC)vs Hβ/(β+γ)=H1/δ and checking the linearity of the curve (not shown). The reliability of the obtained exponents and Curie temperatures of Eu8Ga16Ge30 and Eu4Sr4Ga16Ge30 can also be ascertained by checking the scaling of the magnetization curves. It has been shown that for magnetic systems, the scaling equation of state takes the form23 H Mδ=hε M1/β ,(3) where ε=(T−TC)/TCis the reduced temperature, TCis the Curie temperature, h(x) is a scaling function, and βand γ are critical exponents, which characterize the magnetization behavior along the coexistence (H=0, ε<0) and the critical isotherm (ε=0), respectively. Therefore, according to Eq. (3), if the appropriate values for β,γ,and TCare used, the plot of M/H1/δ vs ε/H1/ should correspond to a universal curve onto which all experimental data points collapse. Using the values of β,γ,and TCobtained from the K-F method, the scaled data are plotted in Fig. 4for both specimens. The excellent overlap of the experimental data points clearly indicates that the obtained values of β,γ,and TCfor these specimens are in agreement with the scaling hypothesis. To get additional information about the magnetic ground states and critical behavior near the PM-FM transition in Eu8Ga16Ge30 and Eu4Sr4Ga16Ge30 from a different perspective, we have investigated the MCE of the two specimens and performed MCE-based critical analysis. The magnetic entropy change (SM) was calculated from isothermal magnetization (M-H) curves using the Maxwell relation,7 SM=μ0Hmax 0∂M ∂T H dH, (4) where Mis the magnetization, His the magnetic field, and Tis the temperature. The results of the calculated SMare shown in Figs. 5(a) and 5(b) for Eu8Ga16Ge30 and Eu4Sr4Ga16Ge30, respectively. As evident in Fig. 5, an additional shoulder is present at TL∼10 K for Eu8Ga16Ge30, in addition to the expected peak at TC∼35 K. This feature is largely suppressed for Eu4Sr4Ga16Ge30. The occurrence of the broader peak at 13 K in the SM(T) curves (see Fig. 5(b)) is presumably a combination of the features at TC∼15 K and TL∼5K. A similar result has been reported for the case of a multiple magnetic phase transition system Nd1.25Fe11Ti.24 In a study of MCE in EuO, the minority phase Eu3O4 was also detected at ∼5K. 25 Another clathrate structure type, Eu4Ga8Ge16, has also been reported to undergo antiferromagnetic ordering at ∼8K. 26,27 Although no impurity phases were detected in our specimens, in order to investigate the possibility that TLarises from Eu3O4or other impurity phases, we conducted systematic structural and MCE analyses on three different Eu8Ga16Ge30 specimens. While the XRD patterns reveal only the type-I clathrate phase, MCE data confirmed the presence of TLin all three specimens. This analysis, together with our extensive XRD and EDS analyses, suggests that the presence of TLis an intrinsic property of the Eu8Ga16Ge30 type-I clathrate. Recently, Franco et al.28–32 have shown that for SOMT materials the SM(T) curves measured with different maximum applied fields will collapse onto a universal curve (which is unique for each universality class, regardless of the particular specimen) by normalizing all the SM(T) curves to their respective peak value Spk M(i.e., S=SM(T)/Spk M) and rescaling the temperature axis above and below TC, θ≡θ1=(T−TC)/(Tr−TC),(5) where Tris the reference temperature corresponding to a certain fraction fthat fulfils SM(Tr)/Spk M=f. This choice of fdoes not affect the actual construction of the universal curve, as it implies only a proportionality constant. When the material deviates from ideal behavior (for example a material with multiple magnetic phases), it is necessary to use two well-separated reference temperatures, Tr1and Tr2,to construct the universal curve,28,31 θ≡θ2=−(T−TC)/(Tr1−TC); T⩽TC (T−TC)/(Tr2−TC); T>T C (6) In the present study, the reference temperatures Tr1 and Tr2 have been selected as those corresponding to 0.6Spk M. Figure 6shows the universal curves for Eu8Ga16Ge30 and Eu4Sr4Ga16Ge30 using Eq. (5) for the case of θ1and Eq. (6) for the case of θ2, respectively. As clearly indicated in Fig. 6, the overlap of the different curves is reasonable around TC, however, not in the θ1⩽−2 range (T<TL∼10 K), although the two reference temperature method yields a 054436-4 ORIGIN OF THE MAGNETIC ANOMALY AND TUNNELING ... PHYSICAL REVIEW B 84, 054436 (2011) 0 102030405060 0 2 4 6Eu 8 Ga 16 Ge 30 T L - Δ S M (J/kg K) T (K) T C Δμ 0 H = 0.1 - 3T (a) H 0 1020304050 0 2 3 5 −Δ S M (J/kg K) T (K) Δμ 0 H = 0.1 - 3T Eu 4 Sr 4 Ga 16 Ge 30 T C (b) H FIG. 5. (Color online) Temperature and magnetic field dependencies of magnetic entropy change (−SM)for(a)Eu 8Ga16Ge30 and (b) Eu4Sr4Ga16Ge30. Dashed arrows indicate the direction of the increasing magnetic field. slightly better overlap for temperatures below TC. It has been noted that perfect overlap of the curves is achieved in the entire temperature range for single magnetic phase transition materials28,30 when a single reference temperature is used but not for multiple magnetic phase transition materials,29,34 although the influence of the second phase transition in the close environment of the TCof the transition under this study -6 -4 -2 0 2 4 0.0 0.5 1.0 024 -2 0 2 4 0.0 0.5 1.0 024 Δ S M / Δ S pk M θ 1 x = 0 x = 4 θ 1 x = 0 Δ S M / Δ S pk M θ 2 x = 4 θ 2 FIG. 6. (Color online) The universal curves for Eu8Ga16Ge30 and Eu4Sr4Ga16Ge30 using Eq. (5) in the case of θ1and Eq. (6) in the case of θ2, respectively. 0 102030405060 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 3T Eu 8 Ga 16 Ge 30 n T (K) (a) 1T 0 1020304050 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 3T n T (K) Eu 4 Sr 4 Ga 16 Ge 30 (b) 1T FIG. 7. (Color online) Temperature dependence of the local exponent nfor (a) Eu8Ga16Ge30 and (b) Eu4Sr4Ga16Ge30 at different magnetic fields from 1 to 3 T with a field interval of 0.5 T. can be minimized by using the two reference temperatures. The lack of perfect overlap for Eu8Ga16Ge30 at temperatures well below TCis not a failure of the universal curve model but may be an indication of an additional magnetic transition at TL with different field dependencies for this specimen. The larger differences between both procedures (using θ1or θ2) are seen for the Eu4Sr4Ga16Ge30 due to the closer temperatures TCand TL(∼10 K apart) when compared with Eu8Ga16Ge30 (∼25 K apart). As the contribution to the magnetic entropy change quickly goes to zero when the temperature increases above TL, there is little benefit in the analysis on the Eu8Ga16Ge30 specimen in using θ2. However, the larger convolution of the peaks associated with TCand TLfor the Sr-doped specimen improves the overlap as θ2is used. The need for using two reference temperatures for the Sr-doped specimen is an indication that TLis not completely suppressed by doping. It has also been demonstrated that the field dependence of SMcan be expressed as Spk M∝Hnfor some materials.28,30 The exponent ncan be locally calculated from the logarithmic derivative of the SMwith magnetic field as n(T,H)=dln |SM| dln H,(7) giving n=2 in the paramagnetic range for TTC,n=1for TTC, and n(T=TC)=1+1/δ(1 −1/β)=(1 −α)/ (8) 054436-5 MANH-HUONG PHAN et al. PHYSICAL REVIEW B 84, 054436 (2011) for T=TC.28 It has been reported that ndepends on the magnetic field and temperature of the material.28–31 For the case of single magnetic phase materials, nis field independent at temperature TCor the temperature of the peak entropy change.28,30,33 It evolves with magnetic field at any temperature for the case of multiphase materials.24,34,35 Figures 7(a) and 7(b) show the temperature and magnetic field dependences of ncalculated using Eq. (7)for Eu8Ga16Ge30 and Eu4Sr4Ga16Ge30, respectively. For both specimens, napproaches 2 in the paramagnetic range at T TC.28 As summarized in Table I, the values of nat TC[denoted as n(TC)] calculated from the SM(T,H) data [n(TC)=0.562 and 0.675 for Eu8Ga16Ge30 and Eu4Sr4Ga16Ge30, respectively] are consistent with those calculated from Eq. (8)usingthe values of βand δ(obtained from the K-F method [n(TC)= 0.545 and 0.667 for Eu8Ga16Ge30 and Eu4Sr4Ga16Ge30, respectively]. The agreement between the exponent nobtained from the K-F method and the MCE analysis supports the application of the MCE analysis to determine the critical exponents in the cases when the K-F method cannot be used.31,32 In addition, the anomalous magnetic field dependence of nhas been noted at TL∼10 K for Eu8Ga16Ge30 [see Fig. 7(a)]. The plateau feature below TCin n(T)forEu 4Sr4Ga16Ge30 [see Fig. 7(b)] can be reconciled with the broader nature of the SM(T) peak due to the overlap of the peaks associated with TC and TL[see Fig. 5(b)]. The theoretically predicted relationship between the peak entropy change (Spk M) and the magnetic field, Spk M∝Hn, has been confirmed for both specimens (see Fig. 8). From Table Iwe note that for the Eu8Ga16Ge30 type-I clathrate the critical exponent β=0.388 is close to that predicted from the 3D Heisenberg model (β=0.365) with short-range ferromagnetic correlation, while the critical exponent γ=0.956 is close to that predicted from the mean-field model (γ=1). The exponent of this specimen n(TC)=0.545 significantly deviates from that predicted from the mean-field 0.8 1.2 1.6 .0 -6 -5 -4 -3 -2 Eu 4 Sr 4 Ga 16 Ge 30 Δ S pk M (J kg -1 K -1 ) H 0.55 (T 0.55 ) Eu 8 Ga 16 Ge 30 1.0 1.5 2.0 -5 -4 -3 -2 Δ S pk M (J kg -1 K -1 ) H 0 . 6 7 (T 0 . 6 7 ) 2 FIG. 8. (Color online) Spk Mvs Hnwith n=0.67 and 0.55. The linear Spk M∝Hnrelationship at TCfor Eu8Ga16Ge30 (bottom and left axis) and Eu4Sr4Ga16Ge30 (top and right axis) is shown in the figure. model (n(TC)=2/3). At first glance, one may infer that the magnetic interaction in the Eu8Ga16Ge30 type-I clathrate near TCis of short-range type. However, the ferromagnetism of this material has been proved to be governed by the RKKY mechanism with long-range interaction,4,5,9,10 which thus rules out this hypothesis. In the Eu8Ga16Ge30 type-I clathrate Eu2+tunnels between four equivalent sites in the tetrakaidecahedral cages and the tunneling states correspond to the static positional disorder induced by the split site Eu2.11,12 It has been shown that in addition to the “rattling” motion, or dynamic disorder, of the Eu2+ions, the tunneling of Eu2+ ions plays an important role in producing the glasslike low thermal conductivity in this material.4,5,12 While the thermal conductivity of both Eu8Ga16Ge30 and Sr8Ga16Ge30 was not affected by an applied magnetic field (∼8 T), this magnetic field significantly suppressed the spin-disorder scattering, which consequently increased the electrical conductivity of Eu8Ga16Ge30, leading to the “negative” magnetoresistance effect (the 10% decrease in resistance with an applied field of 8 T) around its Curie temperature (TC∼35 K).4In the present study, we show that the tunneling of Eu2+between four equivalent sites in the tetrakaidecahedral cage may be the reason for the ferromagnetic ordering and MCE in the type-I clathrate materials. It has been noted that the tunneling of Eu2+ions can lead to the coexistence of competing multiple ferromagnetic states or the modulated ferromagnetic structure below TC.2,14 We believe that the formation of the modulated ferromagnetic structure causes the critical exponents βand n(TC)oftheEu 8Ga16Ge30 type-I clathrate to deviate from those predicted by the mean-field theory. This is plausible if one considers the significant difference in the critical exponents between the Eu8Ga16Ge30 type-I clathrate and the Eu8Ga16Ge30 type-VIII clathrate (see Table I). In contrast to the case of the Eu8Ga16Ge30 type-I clathrate, tunneling of Eu2+ ions does not occur in the Eu8Ga16Ge30 type-VIII clathrate, while the critical exponents β=0.470 and γ=0.964 of type-VIII Eu8Ga16Ge30 are close to those predicted from the mean-field model (β=0.5 and γ=1) with long-range ferromagnetic correlations.7This implies that the tunneling of Eu2+ between the four equivalent sites in the tetrakaidecahedral cage tends to prevent the occurrence of a long-range ferromagnetic ordering in the type-I clathrate material. This tunneling could cause the magnitude of the jump in Cpat TCto significantly differ from the value expected for a uniform ferromagnet with S=7/2 from the mean-field theory.4,5,14 From a magnetic cooling application perspective, the refrigerant capacity (RC) is considered as one of the most important factors for assessing the usefulness of a magnetic refrigerant material. The RC depends not only on the magnitude of SMbut also on its temperature dependence (e.g., the full width at half maximum of the SM(T) peak). While the Eu8Ga16Ge30 type-VIII clathrate shows a relatively narrow SM(T) curve resulting in the moderate RC (∼159 J/kg at 3 T),7the tunneling of Eu2+ions in the Eu8Ga16Ge30 type-I clathrate likely leads to the broadened SM(T) curve thus enhancing the RC in this material (∼174 J/kg at 3 T). This finding may provide a route for improving the cooling efficiency in type-I clathrate materials for advanced magnetic refrigeration applications. More importantly, these results provide a better understanding of the fundamental difference 054436-6 ORIGIN OF THE MAGNETIC ANOMALY AND TUNNELING ... PHYSICAL REVIEW B 84, 054436 (2011) in the magnetic and magnetocaloric properties between the type-I and type-VIII clathrates. IV. CONCLUSIONS The nature of the ferromagnetic phase transitions and critical exponents in Eu8Ga16Ge30 and Eu4Sr4Ga16Ge30 has been studied systematically. Eu8Ga16Ge30 undergoes a SOMT at TC∼35 K, as a result of the magnetic interaction between the Eu2+ions at the Eu2 sites, followed by a secondary ferromagnetic transition at TL∼10 K, resulting from the magnetic interaction between the Eu2+ions at the Eu1 and Eu2 sites. Eu4Sr4Ga16Ge30 retains the SOMT but largely reduces the transition temperatures TC∼15 K and TL∼5 K. As shown in Table I, the substitution of Sr2+ for Eu2+in Eu4Sr4Ga16Ge30 increases the critical exponents β=0.521 and n(TC)=0.667 and brings them close to those predicted from the mean-field model ( β=0.5 and n(TC) =2/3). The critical exponents derived from the K-F and MCE methods suggest that the tunneling of Eu2+between the four equivalent sites in the tetrakaidecahedral cage favors the occurrence of a short-range ferromagnetic ordering in Eu8Ga16Ge30. ACKNOWLEDGMENTS This work is supported by the Army Research Office through Grant No. W911NF-08-1-0276. *[email protected] †Corresponding author: [email protected] 1G. S. Nolas, G. A. Slack, and S. B. Schujman, in Semiconductors and Semimetals, edited by T. M. Tritt (Academic Press, New York, 2001), vol. 69, p. 255. 2G. S. Nolas, J. L. Cohn, G. A. Slack, and S. B. Schujman, Appl. Phys. Lett. 73, 178 (1998). 3Y. Sasaki, K. Kishimoto, T. Koyanagi, H. Asada, and K. Akai, J. Appl. Phys. 105, 073702 (2009). 4B. C. Sales, B. C. Chakoumakos, R. Jin, J. R. Thompson, and D. Mandrus, Phys.Rev.B63, 245113 (2001). 5S. Paschen, W. Carrillo-Cabrera, A. Bentien, V. H. Tran, M. Baenitz, Y. Grin, and F. Steglich, Phys.Rev.B64, 214404 (2001). 6G. T. Woods, J. Martin, M. Beekman, R. P. Hermann, F. Grandjean, V. Keppens, O. Leupold, Gary J. Long, and G. S. Nolas, Phys. Rev. B73, 174403 (2006). 7M. H. Phan, G. T. Woods, A. Chaturvedi, S. Stefanoski, G. S. Nolas, and H. Srikanth, Appl. Phys. Lett. 93, 252505 (2008). 8M. H. Phan, V. Franco, A. Chaturvedi, S. Stefanoski, G. T. Woods, G. S. Nolas, and H. Srikanth, J. Appl. Phys. 107, 09A910 (2010). 9V. Pacheco, A. Bentien, W. Carrillo-Cabrera, S. Paschen, F. Steglich, and Y. Grin, Phys.Rev.B71, 165205 (2005). 10A. Bentien, V. Pacheco, S. Paschen, Y. Grin, and F. Steglich, Phys. Rev. B 71, 165206 (2005). 11R. P. Hermann, V. Keppens, P. Bonville, G. S. Nolas, F. Grandjean, G. J. Long, H. M. Christen, B. C. Chakoumakos, B. C. Sales, and D. Mandrus, Phys.Rev.Lett.97, 017401 (2006). 12B. C. Chakoumakos, B. C. Sales, and D. G. Mandrus, J. Alloys Compd. 322, 127 (2001). 13H. Tomono, H. Eguchi, and K. Tsumuraya, J. Phys. Condens. Matter 20, 385209 (2008). 14T. Onimaru, S. Yamamoto, M. A. Avila, K. Suekuni, T. Takabatake, J. Phys. Conf. Ser. 200, 022044 (2010). 15J. L. Cohn, G. S. Nolas, V. Fessatidis, T. H. Metcalf, and G. A. Slack, Phys.Rev.Lett.82, 779 (1999). 16Y. Zang, P. L. Lee, G. S. Nolas, and P. Wilkinson, Appl. Phys. Lett. 80, 2931 (2002). 17G. S. Nolas, T.J.R. Weakley, J. L. Cohn, and R. Sharma, Phys. Rev. B61, 3845 (2000). 18S. K. Banerjee, Phys.Rev.Lett.12, 16 (1964). 19J.Mira,J.Rivas,F.Rivadulla,C.V ´ azquez-V´ azquez, and M. A. Lopez-Quintela, Phys.Rev.B60, 2998 (1999). 20M. H. Phan, S. C. Yu, N. H. Hur, and Y. H. Jeong, J. Appl. Phys. 96, 1154 (2004). 21J. S. Kouvel and M. E. Fisher, Phys. Rev. 136, A1626 (1964). 22B. Widom, J. Chem. Phys. 43, 3898 (1965). 23H. Eugene Stanley, Introduction to Phase Tansitions and Critical Phenomena (Oxford University Press, New York, 1971). 24R. Caballero-Flores, V. Franco, A. Conde, Q. Y. Dong, and H. W. Zhang, J. Magn. Magn. Mater. 322, 804 (2010). 25K. Ahn, A. O. Pecharsky, K. A. Gschneidner, and V. K. Pecharsky, J. Appl. Phys. 97, 063901 (2004). 26M. Christensen, J. D. Bryan, H. Birkedal, G. D. Stucky, B. Lebech, andB.B.Iversen,Phys.Rev.B68, 174428 (2003). 27J. Daniel Bryan, H. Trill, H. Birkedal, M. Christensen, Vojislav I. Srdanov, H. Eckert, Bo B. Iversen, and Galen D. Stucky, Phys. Rev. B68, 174429 (2003). 28V. Franco, J. S. Blazquez, and A. Conde, Appl. Phys. Lett. 89, 222512 (2006). 29V. Franco, A. Conde, V. K. Pecharsky, and K. A. Gschneidner Jr., Europhys. Lett. 79, 47009 (2007). 30V. Franco, A. Conde, J. M. Romero-Enrique, and J. S. Blazquez, J. Phys. Condens. Matter 20, 285207 (2008). 31V. Franco, A. Conde, V. Provenzano, and R. D. Shull, J. Magn. Magn. Mater. 322, 218 (2010). 32V. Franco, A. Conde, D. Sidhaye, B. L. V. Prasad, P. Poddar, S. Srinath, M. H. Phan, and H. Srikanth, J. Appl. Phys. 107, 09A902 (2010). 33V. Franco, A. Conde, M. D. Kuz’min, and J. M. Romero Enrique, J. Appl. Phys. 105, 07A917 (2009). 34V. Franco, R. Cabalero-Flores, A. Conde, Q. Y. Dong, and H. W. Zhang, J. Magn. Magn. Mater. 321, 1115 (2009). 35Q. Y. Dong, H. W. Zhang, J. R. Sun, B. G. Shen, and V. Franco, J. Appl. Phys. 103, 116101 (2008). 36M. Seeger, S. N. Kaul, H. Kronmuller, and R. Reisser, Phys. Rev. B51, 12585 (1995). 054436-7