IEEE TRANSACTIONS ON MAGNETICS, VOL 50, NO 11, NOVEMBER 2014 1200104 Influence of Cr Doping on the Critical Behavior of Amorphous Alloy Ribbons Fe78−xCrxSi4Nb5B12Cu1 T L Phan1 , P Q Thanh2, N Chau2 , C X Huu3 , D.-T Ngo4 , T A Ho1 , T D Thanh1,5, and S C Yu1 Department of Physics, Chungbuk National University, Cheongju 361-763, Korea of Physics, Hanoi University of Science, Vietnam National University, Hanoi, Vietnam Department of Electronics and Communication Engineering, Danang University of Technology, Danang, Vietnam Department of Micro- and Nanotechnology, and Department of Energy and Storage, Technical University of Denmark, Kgs Lyngby 2800, Denmark Institute of Materials Science, Vietnam Academy of Science and Technology, Hanoi, Vietnam Faculty Though many previous works focused on studying Cr-doped Fe–Si–Nb–B–Cu amorphous alloys, magnetic-interaction mechanisms in these materials have not been carefully investigated yet Dealing with these issues, we have prepared the amorphous alloy ribbons Fe78−x Cr x Si4 Nb5 B12 Cu1 with x = 1, 3, and 6, and then studied their magnetic and critical properties Magnetization versus temperature and magnetic-field measurements, MHT, performed on a vibrating sample magnetometer reveal that the Cr-content increase in Fe78−x Cr x Si4 Nb5 B12 Cu1 reduces the T C from 430 K for x = to about 322 K (for x = 6) This indicates the decline of ferromagnetic (FM) exchange interactions between Fe atoms when there is the presence of Cr atoms We have also analyzed the M(H) data at the temperatures in the vicinity of the T C using the modified Arrott plot method and the scaling hypothesis, and determined the values of the critical exponents β = 0.367–0.376 and γ = 1.315–1.338 These values are close to those expected for the 3-D Heisenberg model with β = 0.365 and γ = 1.336, proving the existence of short-range FM order in the amorphous alloy ribbons Index Terms— Critical behavior, Fe-based alloy ribbons, short-range ferromagnetism I I NTRODUCTION I T HAS been known that Fe-based amorphous alloys (either in the bulk or ribbon form) exhibit many noticeable properties, which can give a large saturation magnetization and the soft-magnetic behavior [1] Depending on the application aspects, these properties can be easily modified by changing elemental composition of alloys It has been realized that the presence of Nb, Si, B, Cu, and Cr in Fe-based amorphous alloys improves remarkably the corrosion resistance and the soft-magnetic behavior [1]–[3] Particularly, the formation of the amorphous phase becomes easy when many chemical elements are simultaneously combined in Fe-based alloys because of structural disorders related to the difference in the atomic radii of the elements This also changes the ferromagnetic–paramagnetic (FM–PM) phase-transition temperature (the Curie temperature, TC ) Based on these important features, it has been fabricated and investigated the amorphous alloys for magnetic refrigeration applications using the magnetocaloric effect [4]–[6] For conventional applications in cooling systems, the TC of the amorphous alloys can be controlled in the temperature range of 260–310 K For example, the TC values (>400 K) of amorphous alloys ribbons Fe78 Nb5 Si4 B12 Cu1 and Fe73.5 Nb3 Si13.5 B9 Au1 can be reduced to lower temperatures by substituting partly Fe atoms by Cr ones [4], [7], [8] In reference to our work, there were many previous studies on Cr-doped Fe–Nb–Si–B–Cu-based amorphous alloy ribbons [2], [4], [7], [9] However, magnetic interactions in these Manuscript received March 2, 2014; accepted May 17, 2014 Date of current version November 18, 2014 Corresponding author: S C Yu (e-mail: scyu@chungbuk.ac.kr) Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org Digital Object Identifier 10.1109/TMAG.2014.2325949 materials and their FM–PM phase-transition type have not been carefully studied yet It has been shown that the study of the critical properties based on the Arrott plot method and the scaling hypothesis [10]–[12] are considered as an effect approach to elucidate the mentioned issues In this paper, we present a thorough study on the magnetic properties of amorphous alloy ribbons Fe78−x Crx Si4 Nb5 B12 Cu1 The experimental results reveal that the Cr-content increase reduces remarkably the TC from 430 K (x = 1) to 322 K (x = 6), and leads to the values of the critical exponents shifted toward those for the mean-field theory However, the alloy ribbons still have the nature of short-range FM order because their exponents are more close to those expected for the 3-D Heisenberg model II E XPERIMENTAL D ETAILS Three amorphous ribbon samples of Fe78−x Crx Si4 Nb5 B12 Cu1 (x = 1, 3, and 6) with the widths and thicknesses of 2–6 mm and 20 μm, respectively, were prepared from highpurity metallic elements (including Fe, Si, Nb, Cu, B, and Cr) using the melt-spinning technique The fabrication was carried out in a vacuum chamber at 10−4 torr that pressure was maintained using an Ar gas flow After preparation, X-ray diffraction studies (based on a Bruker D5005 X-ray diffractometer) revealed that all the ribbon products were amorphous without any secondary phases [4] Magnetization versus temperature and magnetic-field measurements, M(H , T ), were performed on a vibrating sample magnetometer III R ESULTS AND D ISCUSSION Fig 1(a) shows temperature dependences of magnetization (M) normalized to the maximum value (Mmax ) for the amorphous alloy ribbons Fe78−x Crx Si4 Nb5 B12 Cu1 under the application of the magnetic field H = 100 Oe For each 0018-9464 © 2014 IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See http://www.ieee.org/publications_standards/publications/rights/index.html for more information 1200104 IEEE TRANSACTIONS ON MAGNETICS, VOL 50, NO 11, NOVEMBER 2014 Fig (a) M(T ) data normalized to the maximum value, M/Mmax and (b) dM/dT versus T curves for the alloy ribbons in the field H = 100 Oe sample, a slight decrease of M with increasing temperature is observed This decrease becomes rapid when the sample is heated above the FM–PM phase transition temperature TC , where magnetic moments become disordered under the impact of the thermal activation energy By plotting the dM/dT versus T curves, see Fig 1(b), the TC values determined from the minima are about 430, 410, and 323 K for the alloy ribbons with x = 1, 3, and 6, respectively A decrease of the TC with increasing Cr-doping content in Fe78−x Crx Si4 Nb5 B12 Cu1 demonstrates the weakening of exchange FM interactions between Fe atoms Such the scenario leads to the reduction of the M and magnetic-entropy change ( Sm ) values [4], [7], [8] To understand more thoughtfully the magnetic properties of the alloy ribbons Fe78−x Crx Si4 Nb5 B12 Cu1 , we have measured the magnetic-field dependence of magnetization at different temperatures around the TC , M(H ) The recorded M(H ) data are shown in Fig Similar to the M(T ) data shown in Fig 1(a), the magnetization is gradually decreased with increasing temperature The nonlinear M(H ) curves in the FM region (as T < TC ) become linear as T > TC , where the alloy ribbons are paramagnetic At a given temperature, the magnetization increases with the applied field H However, no magnetization saturation is achieved This phenomenon was also observed in many FM compounds [7], [13] due to shortrange FM order caused by magnetic inhomogeneity In an attempt to further confirm the presence of short-range FM order in the alloy ribbons, we have used the Arrott plot methods [10], [11], which were developed from the meanfield (MF) theory for a FM system exhibiting a second-order magnetic phase transition and long-range magnetic interactions [14] According to this theory, the free energy G L is expanded in even powers of M as follows: G L = aM2 + bM4 + · · · − HM (1) where a and b are temperature-dependent parameters Minimizing G L gives the following relation: H /M = 2a + 4bM2 (2) Fig M(H ) curves at temperatures around the TC (with temperature increments of K) for the alloy ribbon with (a) x = 1, (b) x = 3, and (c) x = Within the framework of the MF theory, (2) can be generalized to the scaling equation of state [11] (H /M)1/γ = c1 ε + c2 M 1/β (3) where c1 and c2 are also temperature-dependent parameters, and ε = (T − TC )/TC is the reduced temperature Meanwhile, β and γ are the critical exponents associated with the saturation magnetization, Ms , and the inverse initial susceptibility, χ0−1 respectively Equation (3) means that if magnetic interactions in a ferromagnet accurately obey the MF theory (i.e., it has long-range FM order), the M 1/β versus (H /M)1/γ curves with β = 0.5 and γ = 1.0 (the values of the exponents for the MF theory [12]) around the TC are parallel straight lines, as mentioned in the normal Arrott plots [10] However, the M 1/β versus (H /M)1/γ curves shown in Fig 3(a)–(c) for the alloy ribbons not exhibit these features In addition, if plotting the M 1/β versus (H /M)1/γ curves with β = 0.365 and γ = 1.336 (the values of the exponents for the 3-D Heisenberg model [12]), they are parallel straight lines around the TC , see Fig 3(d)–(f) All of these features indicate the absence of longrange FM order and the existence of short-range order in the amorphous alloy ribbons Fe78−x Crx Si4 Nb5 B12 Cu1 It should be noticed that the parallel straight lines of M 1/β versus (H /M)1/γ are just obtained for high enough magnetic fields (H > kOe), where magnetic moments are almost aligned to the field direction Basically, the values of the exponents β and γ characterized for of short-range FM interactions can be determined by using the modified Arrott PHAN et al.: INFLUENCE OF Cr DOPING ON THE CRITICAL BEHAVIOR OF AMORPHOUS ALLOY RIBBONS Fig M 1/β versus (H /M)1/γ curves with (a)–(c) β = 0.5 and γ = 1.0, and (d)–(f) with β = 0.365 and γ = 1.336 for the amorphous alloys with x = 1, 3, and (MA) method, starting from the asymptotic relations [11], [13] Ms (T ) = M0 (−ε)β , ε < χ0−1 (T ) = (h /M0 )εγ , ε > 1200104 Fig Ms (T ) and χ0−1 (T ) data around the TC fitted to (4) and (5), respectively, for the amorphous ribbons with (a) x = 1, (b) x = 3, and (c) x = (4) (5) where M0 and h are the critical amplitudes As mentioned above, because the exponent values of β = 0.365 and γ = 1.336 are more suitable to the descriptions of the MF theory in the plot of M 1/β versus (H /M)1/γ , they are thus used as the trial exponent values for the MA method [11], [13] With these trial values and the MA plots shown in Fig 3(d)–(f), the Ms (T ), and χ0 (T ) data obtained from the linear extrapolation in the high-field region (H > kOe) to the M 1/β and (1/χ0 )1/γ axes are then fitted to (4) and (5), respectively, to achieve better β, γ , and TC values These new values of β, γ , and TC are continuously used for the next MA-plot processes until they converge to the stable values During the best fitting process, the TC values determined from the M(T ) curves would be used as a reference In Fig 4, it shows the MA-plot processes for the Ms (T ) and 1/χ0 (T ) data, and the determined values of β, γ , and TC for the alloy ribbons Fe78−x Crx Si4 Nb5 B12 Cu1 ; here, β = 0.367 ± 0.004, γ = 1.338 ± 0.018, and TC ≈ 430 K for x = 1, β = 0.370 ± 0.004, γ = 1.324 ± 0.013 and TC ≈ 411 K for x = 3, and β = 0.376 ± 0.002, γ = 1.315 ± 0.006 and TC ≈ 322 K for x = With the critical parameters determined, the MA plots (i.e., M 1/β versus (H /M)1/γ curves) for the M(H, T ) data around the TC of the alloy ribbons show a set of parallel straight lines at the fields H > kOe In addition, the M 1/β versus (H /M)1/γ lines at the TC pass through the origin, see Fig These features are in good agreement with the descriptions of the MA-plot method, and reveal the reliability Fig Modified Arrott plots of M 1/β versus (H /M)1γ with the obtained critical exponents for Fe78−x Crx Si4 Nb5 B12 Cu1 with (a) x = 1, (b) x = 3, and (c) x = of the values of the critical parameters Alternatively, checking the reliability of the determined values can be based on the scaling hypothesis, which states that M is a function of ε and H : M(H, ε) = |ε|β f± (H /|ε|β+γ ), where f+ and f − are the analytic functions for T > TC and T < TC , respectively [12] 1200104 IEEE TRANSACTIONS ON MAGNETICS, VOL 50, NO 11, NOVEMBER 2014 lead to the different values of the critical exponents when Crdoping content is varied IV C ONCLUSION We studied the magnetic and critical properties of the amorphous ribbons Fe78−x Crx Si4 Nb5 B12 Cu1 The results revealed that the Cr-doping weakened FM exchange interactions between Fe atoms This reduced the magnetization M and the TC of the alloy ribbons Analyses of the M(H, T ) data around the TC based on the MA-plot method and the scaling hypothesis determined the exponents β = 0.367–0.376 and γ = 1.315–1.338 These values are close to those expected for the 3-D Heisenberg model, and characteristic of the shortrange magnetic interaction type ACKNOWLEDGMENT This work was supported by the Converging Research Center Program through the Ministry of Science, ICT and Future Planning, Korea, under Grant 2013K000405 R EFERENCES Fig Scaling plots of M/|ε|β versus H /|ε|β+γ for the M(H, T ) data around the TC values of Fe78−x Crx Si4 Nb5 B12 Cu1 with (a) x = 1, (b) x = 3, and (c) x = This equation implies that all M(H, T ) data fall onto two universal branches f − and f + in the M/|ε|β versus H /|ε|β+γ curves if the determined values of β, γ , and TC are correct Interestingly, such the conditions completely satisfies with our results if performing the scaling plots of M/|ε|β versus H /|ε|β+γ , as shown in Fig The critical values of β, γ , and TC determined in our work are thus reliable Notably, though the Cr-content increase leads to the values of the critical exponents shifted toward those for the MF theory, the amorphous alloy ribbons still exhibit the nature of short-range FM interactions because their exponents β = 0.367–0.376 and γ = 1.315–1.338 are more close to those expected for the 3-D Heisenberg model We believe that the doping of Cr atoms changes the range of the exchange interaction J (r ), the total magnetic moment, and the spatial dimensionality of Fe-based FM substances These factors directly influences magnetic interactions in the alloy ribbons According to renormalization group theory [15] applied for an exchange-interaction system, the values of the critical exponents depend on the range of the exchange interaction J (r ) = 1/r d+σ , where d and σ are the dimension and the interaction range, respectively The MF exponents are valid for σ < 1/2, while the Heisenberg ones are valid for σ > The exponent values belong to other universality classes if 1/2 < σ < (such as the 3-DIsing model, the tricritical MF theory, the mixture of shortand long-range FM interactions with β and γ values lying in between the MF theory and the 3-D 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Fe atoms This reduced the magnetization M and the TC of the alloy ribbons Analyses of the M(H, T ) data around the TC based on the MA-plot method and the scaling hypothesis determined the exponents... interactions in the alloy ribbons According to renormalization group theory [15] applied for an exchange-interaction system, the values of the critical exponents depend on the range of the exchange... that the doping of Cr atoms changes the range of the exchange interaction J (r ), the total magnetic moment, and the spatial dimensionality of Fe-based FM substances These factors directly influences