Critical behavior near the ferromagnetic-paramagnetic transformation in the austenite phase of Ni43Mn46Sn8X3 (X = In and Cr) Heusler alloys

Accepted Manuscript Critical behavior near the ferromagnetic-paramagnetic transformation in the austenite phase of Ni43Mn46Sn8X3 (X = In and Cr) Heusler alloys W.Z. Nan, G. Nam, T.D. Thanh, T.S. You, H.G. Piao, L.Q. Pan, S.C. Yu PII: DOI: Reference:

S0304-8853(17)31661-X http://dx.doi.org/10.1016/j.jmmm.2017.07.034 MAGMA 62965

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Journal of Magnetism and Magnetic Materials

Received Date: Accepted Date:

29 May 2017 9 July 2017

Please cite this article as: W.Z. Nan, G. Nam, T.D. Thanh, T.S. You, H.G. Piao, L.Q. Pan, S.C. Yu, Critical behavior near the ferromagnetic-paramagnetic transformation in the austenite phase of Ni43Mn46Sn8X3 (X = In and Cr) Heusler alloys, Journal of Magnetism and Magnetic Materials (2017), doi: http://dx.doi.org/10.1016/j.jmmm.2017.07.034

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Critical behavior near the ferromagnetic-paramagnetic transformation in the austenite phase of Ni43Mn46Sn8X3 (X = In and Cr) Heusler alloys

W.Z. Nan1, G. Nam2, T.D. Thanh1, T.S. You2, H.G. Piao3, L.Q. Pan3, and S.C. Yu1 1

Department of Physics, Chungbuk National University, Cheongju 28644, South Korea

2

Department of Chemistry, Chungbuk National University, Cheongju 28644, South Korea

3

College of Science, China Three Gorges University, Yichang, 443002, China

In this work, we present a detailed study on the magnetic property and critical behavior in the austenitic phase of Ni43Mn46Sn8X3 alloys with X = Cr and In, which were prepared by an arcmelting method in an argon ambience. The M(T) curve of the Cr sample (X = Cr) exhibits a single magnetic phase transition at the Curie temperature of the ferromagnetic (FM) austenitic phase with TAC = 303 K. In contrast, the In sample (X = In) exhibits multiple magnetic phase transitions, including a magnetic phase transition from a FM state to weakly magnetic state at TMC = 165 K of the martensitic phase, a martensitic transition from the weakly magnetic to the FM austenite phase at TM-A = 259 K, and a magnetic phase transition from the FM to paramagnetic (PM) at TAC = 297 K of the austenite phase. Based on the Landau theory and M(H) data measured at different temperatures, we pointed that the FM-PM phase transitions around TAC in both samples were the second-order phase transition. Our results suggest an existence of the long-range FM interactions in the austenite phase. A small deviation from the mean-field theory of the critical exponents has been also observed pointing out an existence of the inhomogeneous magnetism that could be associated with the presence of the anti-FM interactions in these samples. Besides, their effective exponents βeff(ε) and γeff(ε) have been also calculated. Keywords: Critical behavior; Ni-Mn-Sn; Heusler alloys; Mean-field theory; Austenite phase.

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1. Introduction Since the discovery of the ferromagnetic (FM) property in the Cu2MnSn alloy in 1903, Heusler alloys have attracted much attention due to their physical properties and applicability for the magneto-optic and the high-density magnetic recording technologies. In recent years, some interesting physical phenomena in Heusler alloys have been reported, such as magnetoresistance, magnetoelasticity, shape-memory, thermoelectric and magnetocaloric properties. An unusual martensitic transformation from a FM parent phase to a paramagnetic (PM) martensite phase (M phase), which is completely opposite to that in conventional magnetic shape memory alloys, has been observed in the Ni-Mn-Z (Z = In, Sn, Sb) Heusler alloys [1]. Besides, the metamagnetic shape memory effect at room temperature was also found [2], together with many interesting effects derive from the unique transformation of these Heusler alloys [3-8]. Among these, the magnetocaloric effect (MCE) of the Ni-Mn-Z (Z = In, Sn, Sb) Heusler alloys is one of the most interesting properties. It has been widely studied and can be referenced in a large number of publications [5,6,9-14]. It was also shown that an applied magnetic field to the M phase under isothermal conditions realizes magnetic-field-induced martensitic transformation. As a result, a great increase in entropy appears at the transition region, which is opposed to the intuitive expectation of reduction in spin entropy owing to forced spin alignment [2]. This unconventional phenomenon is called the inverse MCE corresponding to positive changes in entropy. Thus, some Heusler alloys exhibit a coexistence of both the MCEs, the conventional and the inverse MCEs corresponding to negative and positive values of magnetic entropy change (ΔS m), respectively [9, 13-15]. These effects are due to the magnetic and structural transformations in materials [5]. Previous reports showed a magnetic inhomogeneous in the M phase that is related to a coexistence of anti-FM and FM interactions. The structures and physical properties of Heusler alloys are thus very sensitive to their compositions [3,10,14,15]. Krenke et al. [3] have found a close relationship between structural and magnetic transformations of the Ni-Mn-Sn systems and their valence electron concentrations per atom ratio (e/a), which is the concentration-weighted sum

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of s, d, and p valence electrons of Ni (3d84s2), Mn (3d54s2), and Sn (5s25p2) atoms. According to Tan et al. [4], the anti-FM interactions in the M phase appear mainly due to the hybridization between the Ni and Mn atoms and obviously, the Ni-Mn distance affect strongly on the strength of this hybridization. On the other hand, Khan et al. [14] show the anti-FM interactions are suppressed when Cr substitution occurs in Ni50Mn37-xCrxSb13 alloys. Several previous reports pointed out that the critical behavior in the vicinity the FM-PM transition in the austenitic phase (A phase) of NiMn-Sn systems is also very sensitive to the composition and the value of e/a. The short-range FM order in the A phase of Ni50Mn37Sn13 will be modified to the long-range FM order by a small change in Sn-content [15] or by a partial replacement of Ni by Ag [13], Cu [16], or Gd [17]. However, detailed investigations into the FM order and the critical behavior have not received much attention yet, and therefore this issue needs the deeper assessment. In this work, we present a detailed investigation into the critical behavior in the vicinity of the FM-PM phase transition of the A phase in the Ni43Mn46Sn8X3 alloys (X = In and Cr, denoted as In and Cr samples, respectively). Our results highlight the fact that Cr substitution leads to suppress the martensitic transformation, contrary to In substitution. For the A phase, the values of the critical exponents (β, γ, and δ) obtained by using the modified Arrott plots (MAP) [18] and the Kouvel-Fisher (K-F) [19] methods are very close to those expected for the mean field theory. The interaction distance decays slower than J (r )  r 4.5 , suggesting an existence of the long-range FM interactions in the samples.

2. Experimental details Two alloy ingot samples of Ni43Mn46Sn8X3 with X = In and Cr were prepared by arcmelting under an Ar atmosphere using Ni, Mn, Sn, In, and Cr metals (99.9 % purity) and then, annealed at 1323 K for 48 h. The compositions of the produces were check by the energy dispersive X-ray (EDX) spectroscopy. Crystal structures of the title compounds were checked at room temperature by a powder X-ray diffractometer using a Cu-Kα radiation source. The magnetization measurements versus temperature as much as magnetic field were performed on a vibrating sample

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magnetometer (VersaLab, Quantum design) with a maximum applied magnetic field of 30 kOe, using a warming mode with a temperature interval of 2 K.

3. Results and discussion Figure 1 show the EDX spectra of the samples recorded at room temperature attached with the scanning electron microscopy (SEM) images. The composition of the samples has been analyzed on the large areas and showed in Table 1. This confirms that the values of the atomic percentage determined based on EDX spectra are quite close to the prescribed ratio, within experimental error. Based on EDX results, we calculated the valence electron concentrations per atom e/a for the samples, thus e/a = 7.95 and 8.02 for X = In and Cr, respectively. These values are very close to those obtained from the nominal compositions (e/a = 7.93 and 8.02 for X = In and Cr, respectively) X-ray diffraction patterns (not show here) obtained at room temperature indicate that the samples are single phase products adopting the cubic L21 structure type (space group Fm3m) belonging to the A phase. There is no feature of the M phase observed suggesting that the M-A phase transition in the samples only occurs at lower room temperature [20]. Here, the valence electrons of In and Cr substitution are 5s25p1 and 3d54s1, respectively, corresponding to the valence electron concentrations per atom e/a = 7.93-8.02. These values are consistent with those reported by Krenke et al. [3] that the structure of Ni-Mn-Sn Heusler alloys at room temperature is cubic L21 if their e/a in the range of 7.723-8.041. Based on X-ray diffraction data, we have also calculated the value of the lattice constant, which is found to be a = 6.001 and 5.917 Å for In and Cr sample, respectively. The a value of the Cr sample is smaller than that of the In sample mainly due to the atomic radius of Cr (1.267 Å) [21] being smaller than that of In one (1.497 Å) [21]. Susceptibility (χ) and its differentiation (dχ/dT) as a function of the temperature measured in the field-cooled mode under a field of 100 Oe for the samples are shown in Fig. 1. Clearly, there are three transitions on χ(T) curves of the In sample in the temperature range from 100 to 340 K,

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including a FM transition of the M phase at TMC = 165 K, a martensitic transformation from M to A phase at TM-A = 259 K, and a FM-PM phase transition of the A phase at TAC = 297 K, which are typically observed in the Heusler alloys [3,9,12,14,15]. In contrast, there only a FM-PM phase transition of the A phase taking place at TAC = 303 K can be observed for Cr sample. This difference can be related to the difference in the valence electrons of In and Cr. It can be seen the valence electrons of In (5s25p1) are very close to those of Sn (5s25p2). The magnetic behaviors of the In sample (Ni43Mn46Sn8In3) are thus quite similar to those reported for Ni43Mn46Sn11 alloy ingot [12]. On the contrary, the valence electrons of Cr (3d54s1) are different far from those of Sn (5s25p2). However, the strength of the hybridization between the Ni and Mn atoms may be reduced as the Cr substitution occurs, which can be contributed to the suppression of the anti-FM interactions in the alloy [14]. A similar behavior was also observed by Ag and Cr substitution [13,14], or increasing Sn concentration up to 18% [3] in Ni-Mn-based Heusler alloys. Although X = In and Cr substitution making the magnetic behavior of Ni43Mn46Sn8X3 alloys are quite different, TAC value in their A phases is not be significantly changed. However, the nature of FM-PM phase transition, and how does the FM interacts in the A phase of these alloys should be clarified. For this reason, we measured the isothermal magnetization M(H) curves at different temperatures around their TAC. Herein, M(H) data for the samples has been corrected by a demagnetization factor (D) that has been determined by a standard procedure from M(Happ) data measurements in the low-field linear response regime at low temperature (H = Happ - DM) [15]. All the isothermal M(H) curves show a progressive decrease of the magnetization values, and the nonlinear M(H) curves become linear as increasing temperature, which is a signature of the FM-PM transition in the vicinity of TAC. Following contents, we shall use TC instead of using TAC to mention the Curie temperature of the A phase in the samples. According to the Landau theory of the phase transitions, the free energy GL is a function of magnetization and temperature. GL(T,M) can be expressed in terms of the order parameter M in the following form neglecting higher order parts [22]

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GL (T , M )  G0 

1 1 A(T ) M 2  B(T ) M 4  HM 2 4

(1)

where, A(T) and B(T) are temperature dependent parameters representing the magnetoelastic coupling and electron condensation energy [23]. With increasing temperature, the coefficient A(T) varies from the negative to the positive sign and the temperature corresponding to the value at A(T) crosses zero is consistent with TC. The nature of a magnetic phase transition can be determined via the sign of B(T). A first-order phase transition corresponds to the negative sign of B(T), while a second-order phase transition (SOPT) is expected for the positive sign of B(T) [24]. By minimizing GL(T,M) as dGL(T,M)/dM = 0, the relation between H/M and M2 can be obtained as below

H / M  A(T )  B(T )M 2

(2)

To understand the nature of the magnetic phase transition in the A phase of Ni43Mn46Sn8X3 alloys, we have plotted M2 vs. H/M curves based on the isothermal M(H) curves measured at different temperatures, see Fig. 2. We can see that M2 vs. H/M curves at high field are quasi linear and nearly parallel. However, at low field region, M2 vs. H/M curves are nonlinear and they are driven towards two opposite directions revealing to the FM and PM states. Using Eq. (2), the coefficients A(T) and B(T) have been determined and plotted in Fig. 3. Clearly with increasing temperature, the coefficient A(T) changes from the negative values to the positive values at 296.5 and 301 K for X = In and Cr, respectively. On the other hand, the coefficient B(T) of these alloys displays the positive sign in the whole temperature. It indicates that the FM-PM phase transition in the A phase of the samples are second order one, and TC values obtained from A(T) (296.5 and 301 K for X = In and Cr, respectively) are quite consistent with those obtained from χ(T) data. Equation (2) also implies that if the magnetic interactions in the samples completely obey the mean-field theory [25], M2 vs. H/M curves in the vicinity of TC should be a series of parallel lines, where the line at T = TC has to pass through the origin. However, M2 vs. H/M curves for both our samples are not absolute parallel to each other. This suggests that the magnetic interactions in the A phase of these alloys do not completely obey the mean-field theory. For this case, the meanfield approximation can be generalized. Thus, the critical behavior around this phase transition is 6

should be investigated. Therefore, we have used the modified Arrott plots (MAP) method to analyze their M(H,T) data based on the Arrott-Noaks equation of state [19], given by  T  TC  1/  ( H / M )1/  a    bM  TC 

(3)

where a and b are temperature dependent parameters. The parameters β and γ are critical exponents corresponding to the spontaneous magnetization (MS) and the initial magnetic susceptibility (χ0), respectively. According to statistical theory, three critical exponents β, γ, and δ obey the Widom scaling relation as following [25]

  1

 

(4)

where critical exponent δ is associated with the critical magnetization isotherm at T = TC. Firstly, we used the MAP using the different models [26], including the mean-field model (β = 0.5 and γ = 1.0), the 3D-Heisenberg model (β = 0.365 and γ = 1.336), the 3D-Ising model (β = 0.325 and γ = 1.241), and the tricritical mean-field model (β = 0.25 and γ = 1.0) to construct the Arrott-Noakes plots M1/β vs. (H/M)1/γ. To choose the best model for the samples, we calculated the relative slope RS = S(T)/S(TC), with S(T) and S(TC) are the slope of M1/β versus (H/M)1/γ at temperatures T and TC in the high field region, respectively. If M1/β vs. (H/M)1/γ plots show a series of absolute parallel lines, the value of RS should be kept to the unit in the whole temperature [27]. Fig. 4 shows RS vs. T data calculated for the samples. We can see that RS value deviates from the unit if using the 3D-Heisenberg model, the 3D-Ising model, and the tricritical mean-field model, but with the mean-field model the value of RS is very close to the unit. Therefore, we can conclude that the mean-field model is the best one to allow the determination the values of the critical exponents (β, γ, and δ) for our samples. According to the scaling hypothesis [25], the critical exponents of the SOPT can be determined from magnetization measurements by using the following power laws MS(T) = M0(-ε)

for ε < 0,

(5)

0-1(T) = (h0/M0)ε

for ε > 0,

(6)

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where ε = (T - TC)/TC is the reduced temperature, M0 and h0/M0 are critical amplitudes. For better critical analysis, the value of |ε| should be limited to about 0.03. Briefly, the fitting procedure can be described as follows. Plotting M1/β vs. (H/M)1/γ by using the initial values β = 0.5 and γ = 1.0, the values of MS(T) and χ0-1(T) are determined from the linear extrapolations of M1/β vs. (H/M)1/γ at high fields to the intercept with the M1/β and (H/M)1/γ axes, respectively. After that, Eqs. (5) and (6) have been used to fit the MS(T) and χ0-1(T) data, respectively, new β and γ values will appear. These steps are repeated until we obtained the stable values of β and γ. Figs. 5(a) and (b) show the MS(T) and 01

(T) data (symbols) at the final step of the MAP method and the fitting curves (solid lines). Thence,

the values of the critical parameters estimated for Ni43Mn46Sn8X3 alloys are β = 0.489 ± 0.016, γ = 0.976 ± 0.011, and TC = 297.2 K for X = In and β = 0.546 ± 0.019, γ = 0.983 ± 0.022, and TC = 301.9 K for X = Cr. The third critical exponent δ has been calculated by using the Widom scaling relation [25] as shown in Eq. (2), δ = 1 + γ/β. δ value is found to be 2.996 and 2.801 for X = In and Cr, respectively. These critical parameters are also given in Table II. Recently, the authors pointed out that the critical behavior of a SOPT can be also investigated by using the Kouvel-Fisher (K-F) method, which can evaluates the values of β and γ that are more accurate because the slope is easily determined [28,29]. According to K-F method [19], Y1(T) and Y2(T) may be derived from Eqs. (5) and (6) Y1(T) = MS(T)[dMS(T)/dT]-1 = (T - TC)/β

(7)

Y2(T) = χ0-1(T)[d χ0-1(T)/dT]-1 = (T - TC)/γ

(8)

 1  Because of MS and χ0-1 should scale with temperature as M S  (T  TC ) and 0  (T  TC ) ,

respectively. Thus both Y1(T) and Y2(T) should be straight lines in the critical region. The slopes of Y1(T) and Y2(T) lines give the values of β and γ, and the intercepts on the temperature axis give the value of TC. Figs. 5(c) and (d) show the K-F plots for both samples together with their critical parameters obtained: β = 0.485 ± 0.013, γ = 0.987 ± 0.017, and TC = 296.8 K for X = In and β = 0.549 ± 0.018, γ = 0.965 ± 0.012, and TC = 301.8 K for X = Cr. The critical exponent δ has been also calculated, δ = 3.035 and 2.758 for X = In and Cr, respectively. These results are also listed in

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Table II, which are very close to those determined by using the MAP method. Besides, δ values deduced from these methods match well, TC value is very close to that obtained from χ(T) and A(T) data, implying that the estimated critical exponents as described above are reliable. Further, the reliability of the critical exponents estimated for the samples can be also checked by using the scaling relation [28,29]. According to the scaling hypothesis [25], the field and temperature dependence of magnetization in the vicinity of TC can be expressed as following M(H, ε) = εβf±(H/ε(β+γ)),

(9)

where f+ and f- are the regular functions for T > TC and T < TC, respectively. In Eq. (9), the renormalized magnetization and field as m ≡ ε-βM(H, ε) and h ≡ Hε-(β+γ), respectively, the scaling relation can be rewritten as [30,31] m = f±(h).

(10)

This implies that for the suitable values of β and γ, the scaled m versus h data will collapse onto two different universal curves corresponding to temperatures below and above TC. In our case, using the values of the critical parameters obtained from K-F method listed in Table II, we performed the m versus h as showed in Fig. 6. Clearly, almost all of the M(H,T) data in the critical region fall onto two independent branches: one for temperatures below TC and another for above TC. It implies that the obtained critical parameters are in good accordance with the scaling hypothesis [31]. Thus, they are thus reasonably accurate and reliable. In Table II, the critical exponents of theoretical models are also listed for comparison [26]. We can see that our results are quite close to those expected for the mean-field model (β = 0.5, γ = 1.0, and δ = 3.0), suggesting the FM coupling in the samples is long-range interaction. However, there is a small deviation from the theoretical values. This could be related to the inhomogeneous magnetism that is due to the coexistence of the FM and the anti-FM interactions in the samples, which were also suggested for Ni-Mn-Sn Heusler alloys [3,15]. However, critical exponents often show a deviation from the values of the theoretical models. This appears as if a magnetic system is governed by various competing coupling and/or disorders [32,33]. Therefore, it is important to

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clarify whether these critical exponents belong to any universality class in the asymptotic region. The exponents in the asymptotic region are universal quantities. They only depend on the system global parameters, like the symmetry, the space dimension, and the dimension of the order parameter [34]. Based on Eqs. (5) and (6), the effective exponents βeff and γeff can be obtained [31,34] eff ( ) 

d [ln M S ( )] d [ln  ]

(11)

 eff ( ) 

d [ln  01 ( )] d [ln  ]

(12)

In the critical regime (T → TC, ε → 0), the effective exponents will be coincide with the asymptotic one (βeff → β, and γeff → γ). The effective exponents βeff(ε) and γeff(ε) for Ni43Mn46Sn8X3 alloys are calculated and plotted in Fig. 7. Clearly, the βeff and γeff varies non-monotonically with ε even in the asymptotic region. It means that βeff and γeff does not match with any theoretical universality classes even in the asymptotic critical region. A similar result has been also observed in several kind materials such as an antiperovskite AlCMn3 [31], partially frustrated amorphous alloys FeMnZr [35], and perovskite manganite Pr0.5Sr0.5MnO3 [36]. In there the non-monotonic changes of βeff and γeff with ε were attributed to magnetic disorders [31,35,36]. These agree with the report by Haug et al. [37]. In a previous report, Mira and coworkers [38] suggested that the critical exponents of a homogeneous magnet should be independent of the microscopic details of the system, which is due to a divergence of the correlation length in the asymptotic critical region. Therefore, the non-monotonic changes of βeff(ε) and γeff(ε) are intrinsic [31,33]. Additionally, Fisher and coworkers [39] pointed out that the universality class of a homogeneous magnet undergoing a SOPT depends on the range of the exchange interactions, J (r )  r ( d  ) , where r is the distance, d is the dimension, and σ is the range of the exchange

interaction. According to the renormalization group analysis for homogeneous magnets with dimensionality d and spin n, the value of σ can be calculated based on the following relation [39]:

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  d  2G   (7n  20)   4 (n  2) 8(n  4)(n  2) 2 1      2   1   2 2 d (n  8) d (n  8)  (n  4)(n  8)   

(13)

where Δσ = σ - ½d and G(½d) = 3 - ¼(½d)2. For three-dimensional materials (d = 3), the exchange integral obeys a relation J (r )  r (3 ) . The mean-field model (β = 0.5, γ = 1.0, and δ = 3.0) is valid for σ ≤ 3/2, J(r) decays slower than r-4.5. Meanwhile the 3D-Heisenberg model (β = 0.365, γ = 1.336, and δ = 4.8) is valid for σ ≥ 2, J(r) decays faster than r-5. In the range 3/2 < σ < 2, the system follows other universality classes. In this work, using the values of γ = 0.965 and 0.987 obtained from the K-F method, we found σ value being in the range of 1.43-1.48 whenever d = 3 and n > 1, which are very close to σ = 3/2. It means that the exchange integral J(r) decays slower than r-4.5 that belongs to the mean-field model. Thus, based on the results obtained, we believe that the long-range FM order exists in both samples. Based on the values of σ obtained above, the remaining critical exponents (ν and α) can be also calculated following the relations [36,40]

 

(14)

  2  d

(15)



where ν is the correlation length critical exponent, and α is the heat capacity critical exponent. Thus the value of ν for the samples is found to be about 0.65-0.69, and α ≈ 0 is obtained for both the samples. This result is consistent with the predicted for the mean-field model (α = 0) [40].

4. Conclusion In conclusion, the magnetic property and the critical behavior around the FM-PM phase transition of the A phase in Ni43Mn46Sn8X3 alloys with X = In and Cr have been investigated. We pointed out that Cr substitution contributed to the suppression the anti-FM interaction, contracting with the case of In. Using the MAP and the K-F methods, the critical parameters in the vicinity of TC of the A phase have been determined. With the obtained critical parameters (β = 0.485 ± 0.013, γ = 0.987 ±

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0.017, and TC = 296.8 K for X = In and β = 0.549 ± 0.018, γ = 0.965 ± 0.012, and TC = 301.8 K for X = Cr), almost all of the isothermal M(H) data at various temperatures were collapsed onto two different universal branches below and above TC confirming the reliability of the obtained results. These results supported that the exchange integral J(r) decays slower than r-4.5. It means that an existence of the long-range FM order in the alloys. However, there is a small deviation of β and γ values from those expected of the mean-field model, which was attributed to the coexistence of the FM and the anti-FM interactions in these alloys. Additionally, the changes non-monotonically of βeff and γeff with ε for the samples have been also observed, which are similar to those proposed for the disordered ferromagnets.

Acknowledgment This work was conducted during the research year of Chungbuk National University 2015, and it was also supported by the National Natural Science Foundation of China (Grant Nos. 11474183 and 51371105).

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14

Table I. Atomic percentage of the chemical species and the valence electron concentrations per atom e/a for Ni43Mn46Sn8X3 alloys deduced from the EDX analyses and their nominal compositions. EDX analysis results

Nominal

Sample Ni (%)

Mn (%)

Sn (%)

X (%)

e/a

Composition

e/a

In

43.42

45.80

7.82

2.96

7.95

Ni43Mn46Sn8In3

7.93

Cr

43.17

45.65

7.99

3.19

8.02

Ni43Mn46Sn8Cr3 8.02

Table II. Critical exponents of Ni43Mn46Sn8X3 (X = In and Cr) alloys compared with those of different theoretical models. Abbreviations: MFT: mean-field theory; 3D-H: 3D-Heisenberg model; 3D-I: 3D-Ising model; TMFT: tricritical mean-field theory; Ref: reference. Model/sample

Method

TC (K)

β

γ

δ

Ref

MFT

Theory

-

0.5

1.0

3.0

[26]

3D-H

Theory

-

0.365

1.336

4.8

[26]

3D-I

Theory

-

0.325

1.241

4.82

[26]

TMFT

Theory

-

0.25

1.0

5.0

[26]

MAP

297.2

0.489 ± 0.016

0.976 ± 0.011

2.996

K-F

296.8

0.485 ± 0.013

0.987 ± 0.017

3.035

MAP

301.9

0.546 ± 0.019

0.983 ± 0.022

2.801

K-F

301.8

0.549 ± 0.018

0.965 ± 0.012

2.758

Ni43Mn46Sn8In3

This work

Ni43Mn46Sn8Cr3

This work

15

(a)

(b)

Fig. 1. EDX spectra attached with the SEM images for (a) In and (b) Cr sample.

0.5

6 (a)

259 K

(b)

X = In

d/dT

4

303 K

0

-2

3

297 K

X = In X = Cr

X = Cr

5

-1

-1

 (10 emu.g .Oe )

7

2 1 0 100

165 K 150

200

250 T (K)

300

350

-0.5 100

150

200

250 T (K)

Fig. 2. (a) χ(T) and (b) dχ/dT(T) measured in the field-cooled mode at H = 100 Oe of Ni43Mn46Sn8X3 alloys with X = In and Cr.

16

300

350

3

280 K

X = Cr

T = 2 K

T = 2 K

1.2 10

3

1 10

3

8 10

2

2

X = In

3

2

2.2 10

3

(b)

276 K

2 2

318 K 6 102 1.1 10

2

2

M (emu /g )

(a)

1.4 10

314 K

3

0

0 10 0

1200 0

400 800 H/M (Oe.g/emu)

M (emu /g )

3.3 10

500 1000 H/M (Oe.g/emu)

4 10

2

2 10

2

0 10 1500

0

Fig. 3. The Arrott plots M2 versus H/M of Ni43Mn46Sn8X3 alloys with X = In (a) and Cr (b) measured at different temperatures around the FM-PM phase transition of the A phase.

(a)

(b)

296.5 K X = In

X = In

1.4

0

-500

3

3

A(T) = 0

3

X = Cr

1.6

X = Cr

3

500

B(T) (Oe.g /emu )

A(T) (Oe.g/emu)

1.8

0.6

0.5 1.2 1

301 K

-1000 270

280

290 300 T (K)

310

320

B(T) (Oe.g /emu )

1000

0.4 270

0.8 280

290

300 T (K)

310

320

Fig. 4. Temperature dependences of the coefficients (a) A(T) and (b) B(T) around the FM-PM phase transition of the A phase.

17

2.5

1.6 Mean field model 3D-Heisenberg model 3D-Ising model Tricritical mean field model

1.5

1.2

X = In

RS

1.4

X = Cr

1

RS

2

Mean field model 3D-Heisenberg model 3D-Ising model Tricritical mean field model

1 T

0.5

T

C

0.8

C

(a)

(b)

0 0.6 275 280 285 290 295 300 305 310 315 280 285 290 295 300 305 310 315 320 T (K) T (K)

Fig. 5. Temperature dependences of RS values obtained around TC for (a) In and (b) Cr samples.

20

175



0

X = In

C

8

X = Cr

110

(b)

10 (a)

4 280

0 280

220

12

-1

-1

20

330

0

350

16

S

C

 = 0.546  = 0.983 T = 301.9 K



(Oe.g/emu)

30

525

M (emu/g)

 = 0.489  = 0.976 T = 297.2 K

S

M (emu/g)

40

(Oe.g/emu)

440

700

288

296 T (K)

304

312

0 290

300 T (K)

310

0

280

290

300

310

 = 0.549  = 0.965 T = 301.8 K

-30

15 10

0

S

C

-20 X = Cr

-1

(d) 0 280

0

T (K)

5

0

-10

-1

-40

 (T)/[d (T)/dT] (K)

(c)

S

5

M (T)/[dM (T)/dT] (K)

X = In

0

S

-10

-1

10

0

C

-20

-1

-30

15

 (T)/[d (T)/dT] (K)

 = 0.485  = 0.987 T = 296.8 K

S

M (T)/[dM (T)/dT] (K)

20 -40

0 290

300 T (K)

310

Fig. 6. (a) and (b) Temperature dependences of the spontaneous magnetization MS(T) and the inverse initial susceptibility 0-1(T) along with the fitting curves based on Eqs. (5) and (6). (c) and (d) The Kouvel-Fisher plots Y1(T) and Y2(T).

18

T
C

T
100

10 T>T 5

 = 0.485  = 0.987 T = 296.8 K

 = 0.549  = 0.965 T = 301.8 K

T>T

C

6

7

10 10 -( h = H|| (Oe)

10

8

10



X = Cr

C

C

10

X = In

m = M||

m = M||

100

C

10

C

5

6

10 10 -( h = H|| (Oe)

(emu/g)

(b)



(emu/g)

(a)

7

Fig. 7. m = M|ε|-β versus h = H|ε|-(β+γ) curves in the log-log scale using the critical parameters obtained from K-F method for (a) In and (b) Cr samples.

10

-1

0.62

10

-3

10

-2

1.04

1.1 1

1 0.58



X = In

0.8

(b)

0.98

X = Cr

0.96

0.56



eff

0.9

0.5

eff

(a)



eff



1.02

0.6

0.55

0.94

0.45

 eff 

0.7

10

-2





0.54



eff

0.4

0.6 10

0.52

-1

10

eff

eff

-2



Fig. 8. Reduced temperature dependences of the effective exponents βeff(ε) and γeff(ε) for Ni43Mn46Sn8X3 alloys (a) X = In, (b) X = Cr.

19

eff

10

0.6





-2

0.92 0.9 -1 10