The Forced Magnetostrictions and Magnetic Properties of Ni2MnX (X = In, Sn) Ferromagnetic Heusler Alloys

Experimental studies into the forced magnetostriction, magnetization, and temperature dependence of permeability in Ni2MnIn and Ni2MnSn ferromagnetic Heusler alloys were performed according to the spin fluctuation theory of itinerant ferromagnetism proposed by Takahashi. We investigated the magnetic field (H) dependence of magnetization (M) at the Curie temperature TC, and at T = 4.2 K, which concerns the ground state of the ferromagnetic state. The M-H result at TC was analyzed by means of the H versus M5 dependence. At 4.2 K, it was investigated by means of an Arrott plot (H/M vs. M2) according to Takahashi’s theory. As for Ni2MnIn and Ni2MnSn, the spin fluctuation parameters in k-space (momentum space, TA) and that in energy space (frequency space, T0) obtained at TC and 4.2 K were almost the same. The average values obtained at TC and 4.2 K were TA = 342 K, T0 = 276 K for Ni2MnIn and TA = 447 K, T0 = 279 K for Ni2MnSn, respectively. The forced magnetostriction at TC was also investigated. The forced linear magnetostriction (ΔL/L) and the forced volume magnetostriction (ΔV/V) were proportional to M4, which followed Takahashi’s theory. We compared the forced volume magnetostriction ΔV/V and mechanical parameter, bulk modulus K. ΔV/V is inversely proportional to K. We also discuss the spin polarization of Ni2MnIn and other magnetic Heusler alloys. The pC/pS of Ni2MnIn was 0.860. This is comparable with that of Co2MnGa, which is a famous half-metallic alloy.


Introduction
Spin fluctuation theories have been proposed to explain the physical properties and the principles of itinerant electron systems [1][2][3][4][5][6][7]. Recently, the spin fluctuation theory of itinerant magnetism, known as Takahashi's theory, was proposed by Takahashi [1][2][3][4]. The self-consistent renormalization (SCR) theory was first proposed by Moriya and Kawabata, taking into account the non-linear mode-mode coupling between spin fluctuation modes [5][6][7]. Concerned about the magnetic field dependence of where M S is spontaneous magnetization in the ground state, p s is the magnetic moment in the ground state (T = 0 K), T A is the spin fluctuation parameter in k-space (momentum space) in units of Kelvin, w A is the molecular weight in units of g, and H is the magnetic field in units of kOe. Takahashi transcribed the spin fluctuation parameter in k-space at temperature T A (K) [2]. The dynamical spin susceptibility, as shown in Equation (3.1) in reference [2], is demonstrated by the double-Lorentzian function of the k-space (parameter: q) and the energy space (frequency ω-space). The Lorentzian function of the k-space is proportional to χ(q = 0, ω = 0). The half-width of this function, ∆q, which indicates a spin fluctuation in k-space, is proportional to the inverse of χ(q = 0, ω = 0). The unit of 1/χ(q = 0, ω = 0) is a dimension of the energy. Finally, ∆q is shown in a dimension of the energy. Therefore, ∆q is proportional to k B T A , where k B is the Boltzmann function and T A is the spin fluctuation parameter, as mentioned above. T A is expressed in the form of T A = Aq 2 B , where q 2 B indicates the effective zone boundary wave vector, and A indicates the non-dimensional parameter, as shown in Equation (3.6) in reference [2]. Another parameter, T 0 , is a spectral distribution Γ qB in the frequency space, which was defined by Γ qB = 2πk B T 0 . In this way, the spin fluctuation parameters in k-space (momentum space), T A , and that in energy space (frequency space), T 0 , were defined. From the spontaneous magnetic moment M S and magnetization at T C , we obtained T A . Investigations into the itinerant magnetism of 3d and 5f electron systems were carried out by means of Equation (1) [1,[8][9][10][11][12][13]. Moreover, this theory has been applied to the ferromagnetic Heusler alloys [11,[14][15][16][17]. The spin fluctuation parameter in energy space T 0 is derived from Equation (3.16) in reference [1]: From Equations (1) and (2), T A and T 0 are obtained. The other method to derive the parameters T A and T 0 is determination from magnetic field dependence of the magnetization in the ground state (T << T C ) [1,13,15].
The magnetization in the ground state is expressed by the following equation: where g indicates the Landé g-factor, N 0 indicates Avogadro's number, and F 1 indicates the mode-mode coupling term of the spin fluctuations written as In Equation (4), c is equal to 1/2 and M 0 is the spontaneous magnetization. Further, F 1 is derived from the slope of the Arrott plot (H/M versus M 2 plot) at low temperatures by Equation (5): where k B indicates the Boltzmann factor, and ζ indicates the slope of the Arrott plot (M 2 versus H/M). Then, T 0 and T A are provided by the following equations, respectively: These equations use units of kOe and emu/g for the magnetic fields H and magnetization M, respectively (p. 66 in reference [1]). The value of the magnetic fields H in 10 kOe is equal to the value in T (Tesla), and the value of magnetization M in emu/g is equivalent to the value in Am 2 /kg.
As for the itinerant ferromagnets, the relation between the effective magnetic moment p eff and the spontaneous magnetic moment p S can be expressed by a generalized Rhodes-Wohlfarth equation (Equation (3.47) in reference [1]): Equation (8) can be rewritten as Therefore, if k m = 1.4, Equation (9) is equal to Equation (8).
The other characteristic property of Takahashi's theory is that the forced volume magnetostriction ∆V/V and the magnetization M at T C can be described as in reference [1]: where ∆V/V can be derived by the following equation: where (∆L/L) // and (∆L/L) ⊥ are the forced linear magnetostriction parallel and perpendicular to an external magnetic field, respectively [18,19].
In this study, we selected Ni 2 MnIn and Ni 2 MnSn alloys. These alloys are ferromagnetic Heusler alloys and do not cause martensitic transformation [20], in contrast to Ni 2 MnGa with a martensitic transformation temperature T M of 195 K [21]. These alloys have L2 1 -type cubic crystal structure. We considered the magnetostriction and magneto-volume effects of these alloys. We measured the forced longitudinal magnetostriction (∆L/L) // and (∆L/L) ⊥ , derived the forced volume magnetostriction ∆V/V as shown by Equation (4), and evaluated the correlation between the magnetization and ∆V/V.

Materials and Methods
Polycrystalline Ni 2 MnIn and Ni 2 MnSn alloys were synthesized from the constituent elements of NI 2 MnIn: Ni (4N), Mn (3N), In (4N); Ni 2 MnSn: Ni (4N), Mn (4N), Sn(5N). The sample of Ni 2 MnIn was prepared by induction melting under an Ar atmosphere. The sample of Ni 2 MnSn was prepared by arc-melting in an Ar atmosphere. The product of Ni 2 MnSn was heated in vacuum at 1123 K for 3 days and then quenched in water. The results of the X-ray diffraction pattern (XRD, Ultima IV, Rigaku Co., Ltd., Akishima, Tokyo, Japan) indicated that these samples were single phase, as shown in Figure 1. The XRD results indicated that the crystal structure is L2 1 cubic, and lattice parameters a were 0.60709 nm and 0.60528 nm for Ni 2 MnIn and Ni 2 MnSn, respectively. A helium-free superconducting magnet at the High Field Laboratory for Superconducting Materials, Institute for Materials Research, Tohoku University, and at the Center for Advanced High Magnetic Field Science, Osaka University was used for the magnetostriction measurements up to 5 T. The magnetization measurement at 4.2 K, which corresponds to the investigation of the magnetic field dependence of the magnetization at the ground state (T << T C ) was performed by means of 30 T pulsed field magnet at the Center for Advanced High Magnetic Field Science, Osaka University. A detailed explanation of the experimental procedure has been given in previous studies [14][15][16][17].  Figure 2 shows the temperature dependence of the permeability P for (a) Ni2MnIn and (b) Ni2MnSn in a zero external magnetic field. The values of dP/dT shown in Figure 2 are the values of the differential of the permeability in the temperature. For Ni2MnIn and Ni2MnSn, the values of TC were obtained from the peak of dP/dT, which were 314 K and 337 K, respectively, using the same approach [14]. Figure 3 for (a) Ni2MnIn and (b) Ni2MnSn shows the plots of M 4 versus H/M at TC. A good linearity can be seen at the origin at TC. The magnetic field dependence of the magnetization indicates that H ∝ M 5 ; therefore, the results agree with Takahashi's theory [1]. In former experimental investigations of Ni2MnGa-type Heusler alloys, such as Ni2+xMnGa1−x (0 ≤ ≤ 0.04) and Ni2Mn1− xCrxGa (0 ≤ ≤ 0.25), Takahashi's theory has also been adapted successfully [11,[14][15][16][17]. The spin fluctuation parameter in k-space, TA, and in energy space, T0, has been calculated from the magnetization process at TC using Equations (3) and (4) by Takahashi's theory [1].     Figure 2 shows the temperature dependence of the permeability P for (a) Ni 2 MnIn and (b) Ni 2 MnSn in a zero external magnetic field. The values of dP/dT shown in Figure 2 are the values of the differential of the permeability in the temperature. For Ni 2 MnIn and Ni 2 MnSn, the values of T C were obtained from the peak of dP/dT, which were 314 K and 337 K, respectively, using the same approach [14].  Figure 2 shows the temperature dependence of the permeability P for (a) Ni2MnIn and (b) Ni2MnSn in a zero external magnetic field. The values of dP/dT shown in Figure 2 are the values of the differential of the permeability in the temperature. For Ni2MnIn and Ni2MnSn, the values of TC were obtained from the peak of dP/dT, which were 314 K and 337 K, respectively, using the same approach [14]. Figure 3 for (a) Ni2MnIn and (b) Ni2MnSn shows the plots of M 4 versus H/M at TC. A good linearity can be seen at the origin at TC. The magnetic field dependence of the magnetization indicates that H ∝ M 5 ; therefore, the results agree with Takahashi's theory [1]. In former experimental investigations of Ni2MnGa-type Heusler alloys, such as Ni2+xMnGa1−x (0 ≤ ≤ 0.04) and Ni2Mn1− xCrxGa (0 ≤ ≤ 0.25), Takahashi's theory has also been adapted successfully [11,[14][15][16][17]. The spin fluctuation parameter in k-space, TA, and in energy space, T0, has been calculated from the magnetization process at TC using Equations (3) and (4)     can be seen at the origin at T C . The magnetic field dependence of the magnetization indicates that H ∝ M 5 ; therefore, the results agree with Takahashi's theory [1]. In former experimental investigations of Ni 2 MnGa-type Heusler alloys, such as Ni 2+x MnGa 1−x (0 ≤ x ≤ 0.04) and Ni 2 Mn 1−x Cr x Ga (0 ≤ x ≤ 0.25), Takahashi's theory has also been adapted successfully [11,[14][15][16][17]. The spin fluctuation parameter in k-space, T A , and in energy space, T 0 , has been calculated from the magnetization process at T C using Equations (3) and (4) by Takahashi's theory [1]. Furthermore, we investigated the magnetization measurement at 4.2 K, which corresponds to the magnetization process that was performed at the ground state (T << TC, T/TC ≈ 1%). Figure 4 plots the magnetic field dependences of the magnetization, M 2 versus H/M, which corresponds to the Arrott plot at 4.2 K for (a) Ni2MnIn and (b) Ni2MnSn [22]. These plots indicated that M 2 was proportional to H/M in high magnetic fields and could be appreciable to Equation (3) of Takahashi's theory [1]. Then, TA and T0 were obtained by means of Equations (3)-(7).

Magnetic Field Dependence of Magnetization
The obtained parameters, TA and T0, are listed in Table 1. These results indicate that Takahashi's theory is applicable to Ni2MnIn and Ni2MnSn alloys. The experimental results followed the relation of (∆ / ) ∝ , which is correct in Equation (10), proposed by Takahashi's theory [1].
(a) (b)  Furthermore, we investigated the magnetization measurement at 4.2 K, which corresponds to the magnetization process that was performed at the ground state (T << T C , T/T C ≈ 1%).  Furthermore, we investigated the magnetization measurement at 4.2 K, which corresponds to the magnetization process that was performed at the ground state (T << TC, T/TC ≈ 1%). Figure 4 plots the magnetic field dependences of the magnetization, M 2 versus H/M, which corresponds to the Arrott plot at 4.2 K for (a) Ni2MnIn and (b) Ni2MnSn [22]. These plots indicated that M 2 was proportional to H/M in high magnetic fields and could be appreciable to Equation (3) of Takahashi's theory [1]. Then, TA and T0 were obtained by means of Equations (3)- (7).
The obtained parameters, TA and T0, are listed in Table 1. These results indicate that Takahashi's theory is applicable to Ni2MnIn and Ni2MnSn alloys. The experimental results followed the relation of (∆ / ) ∝ , which is correct in Equation (10), proposed by Takahashi's theory [1].
(a) (b)  The obtained parameters, T A and T 0 , are listed in Table 1. These results indicate that Takahashi's theory is applicable to Ni 2 MnIn and Ni 2 MnSn alloys. The experimental results followed the relation of (∆V/V) ∝ M 4 , which is correct in Equation (10), proposed by Takahashi's theory [1]. Table 1. Magnetic parameters of Ni 2 MnX (X = Ga, In, Sn). The spontaneous magnetic moment, p S ; effective moment, p eff ; Curie temperature, T C ; spin fluctuation parameter in k-space, T A ; spin fluctuation parameter in energy space, T 0 . The parameter k m was obtained from Equation (9), which was almost the same as k m = 1.4. "This work T C " indicates the values obtained from the magnetization process measurements at T C , and "This work 4.2 K" indicates the values obtained from the magnetization process measurements at 4.2 K. Alloy

Correlation between Magnetization and Forced Magnetostriction
In this subsection, we describe the investigations of forced magnetostrictions for Ni 2 MnIn and Ni 2 MnSn, and the correlation between forced volume magnetostriction and magnetization is discussed. In order to consider the relevance between magnetization and forced magnetostriction, we examined the magnetostriction in the magnetic fields and at T C . Figure 5 shows the external magnetic field dependence of the forced magnetostriction for (a) Ni 2 MnIn and (b) Ni 2 MnSn. The forced volume magnetostriction ∆V/V was derived using Equation (11). For both alloys, the obtained ∆V/V was proportional to the fourth power of the M, (∆V/V) ∝ M 4 , and crossed the origin, (M 4 , ∆V/V) = 0, as indicated by the dotted linearly fitting line. This result is consistent with other Ni 2 MnGa-type Heusler alloys [14,15,17]. Faske et al. conducted an experimental investigation into the magnetization M and magnetostriction ∆V/V of LaFe 11.6 Si 1.4 [12]. They found the relationship between ∆V/V and M as (∆L/L) ∝ M 4 , and crossed the origin, and they suggested that the experimental results of ∆V/V and M were in accordance with Takahashi's theory [1]. As for renowned weak ferromagnet MnSi [8], Takahashi suggested that the relationship between ∆L/L and M is (∆L/L) ∝ M 4 [1]. Not only weak ferromagnet but also L2 1 -type cubic Heusler alloys, and LaFe 11.6 Si 1.4 (NaZn13-type structure), which has a more complex structure, are in accordance with Takahashi's theory. Table 1. Magnetic parameters of Ni2MnX (X = Ga, In, Sn). The spontaneous magnetic moment, pS; effective moment, peff; Curie temperature, TC; spin fluctuation parameter in k-space, TA; spin fluctuation parameter in energy space, T0. The parameter km was obtained from Equation (9), which was almost the same as km = 1.4. "This work TC" indicates the values obtained from the magnetization process measurements at TC, and "This work 4.2 K" indicates the values obtained from the magnetization process measurements at 4.2 K.

Correlation between Magnetization and Forced Magnetostriction
In this subsection, we describe the investigations of forced magnetostrictions for Ni2MnIn and Ni2MnSn, and the correlation between forced volume magnetostriction and magnetization is discussed. In order to consider the relevance between magnetization and forced magnetostriction, we examined the magnetostriction in the magnetic fields and at TC. Figure 5 shows the external magnetic field dependence of the forced magnetostriction for (a) Ni2MnIn and (b) Ni2MnSn. The forced volume magnetostriction ΔV/V was derived using Equation (11). For both alloys, the obtained ΔV/V was proportional to the fourth power of the M, (∆ / ) ∝ , and crossed the origin, (M 4 , ΔV/V) = 0, as indicated by the dotted linearly fitting line. This result is consistent with other Ni2MnGa-type Heusler alloys [14,15,17]. Faske et al. conducted an experimental investigation into the magnetization M and magnetostriction ΔV/V of LaFe11.6Si1.4 [12]. They found the relationship between ΔV/V and M as (∆ / ) ∝ , and crossed the origin, and they suggested that the experimental results of ΔV/V and M were in accordance with Takahashi's theory [1]. As for renowned weak ferromagnet MnSi [8], Takahashi suggested that the relationship between ΔL/L and M is (∆ / ) ∝ [1]. Not only weak ferromagnet but also L21-type cubic Heusler alloys, and LaFe11.6Si1.4 (NaZn13-type structure), which has a more complex structure, are in accordance with Takahashi's theory. In a previous study, we measured the magnetostrictions of Ni2MnGa-type and Heusler alloys at TC and proved that ΔV/V is proportional to the valence electron per atom, e/a [17]. As for Ni2MnGa, In a previous study, we measured the magnetostrictions of Ni 2 MnGa-type and Heusler alloys at T C and proved that ∆V/V is proportional to the valence electron per atom, e/a [17]. As for Ni 2 MnGa, Ni 2 MnIn, and Ni 2 MnSn, the e/a were all the same value as 7.500. Therefore, we compared the forced volume magnetostriction ∆V/V and its mechanical parameter, bulk modulus K [14,15]. The forced volume magnetostriction ∆V/V at 5 T and bulk modulus K are listed in Table 2. The K is inversely proportional to Young's modulus. Therefore, as K becomes smaller, it softens more. The order of ∆V/V at 5 T is Ni 2 MnGa < Ni 2 MnSn < Ni 2 MnIn. The values of M 4 for Ni 2 MnGa and Ni 2 MnIn are comparable. The K of Ni 2 MnIn is smaller than that of Ni 2 MnGa. Therefore, Ni 2 MnIn is softer than that of Ni 2 MnGa. It is conceivable that the strain grows larger for a softer alloy. Then, the ∆V/V of Ni 2 MnIn is larger than that of Ni 2 MnGa. The value of M 4 for Ni 2 MnSn is larger than that of Ni 2 MnGa. Moreover, from the results of K, Ni 2 MnSn is softer than Ni 2 MnGa. Therefore, the ∆V/V of Ni 2 MnSn is larger than that of Ni 2 MnGa.  [25].
The units of M 4 and K are defined by (Am 2 /kg) 4 and Pa, respectively; ∆V and V are measured in m 3 ; K is also defined in N/m 2 . The K∆V is in the dimension of Pa·m 3 = (N/m 2 )·m 3 = Nm = J. Therefore, K·(∆V/V) is in J/m 3 . Here, we defined the parameter E K in J/m 3 . The ∆V/V = E K /K. This equation indicates that the forced volume magnetostriction ∆V/V is inversely proportional to bulk modulus K. The K·(∆V/V) is also listed in Table 2. This is almost the same value. This result also indicates that ∆V/V is inversely proportional to K.

Spin Polarization of Ni 2 MnGa-Type Heusler Alloys
In this subsection, we consider the magnetism of Ni 2 MnGa-type Heusler alloys by comparing the spontaneous magnetic moment at the ground state, p S , and paramagnetic magnetic moment, p C .
The relation between p eff and p C is described as: The p C is obtained from the Curie constant and it is non-dimensional, C = N 0 µ eff 2 /3k B = N 0 p eff 2 µ B 2 /3k B = N 0 p C (p C + 2)µ B 2 /3k B . The p c /p s is 1 for the local-moment ferromagnetism. For the weak itinerant electron ferromagnetism, the p c /p s is larger than 1 [1]. On the contrary, many Heusler alloys have a p c /p s value smaller than 1 [16]. As for the itinerant electron magnets, the minority-spin electrons band has a gap at the Fermi level E F and indicates semi-metallic or insulating bands. On the contrary, the Fermi level intersects the majority-spin electrons band and represents metallic bands. The p c /p s < 1 indicates that the spin polarization occurs, and these alloys can be classified as half-metallic alloys (HMFA). The p S and p C for Ni 2 MnGa-type Heusler alloys are listed in Table 3 [27]. The p C /p S values of Co 2 VGa and Co 2 MnGa were 0.70 and 0.80, respectively. The results for these three alloys indicate that the alloy with a larger spin polarization showed a smaller p C /p S value. The spin polarization of Ni 2 MnSn was obtained by theoretical calculations [25]. The obtained P 0 was about 10%, which indicates that the spin polarization of Ni 2 MnSn is smaller than that of Ni 2 MnIn. Then, the p C /p S of Ni 2 MnSn was almost 1. Even at low temperature, Ni 2 MnIn and Ni 2 MnSn take an L2 1 -type cubic structure. On the contrary, Ni 2 MnGa causes martensitic transformation at T M = 195 K, and below this temperature, 14 M structure was realized [28]. In the martensitic phase, the spin polarization was  [29]. Then, the p sat /p s was 0.92, which is smaller than 1 and deviated from 1 (local moment magnetism). The spin polarization of Ni 2 MnGa affected the deviation of the p sat /p s value. Table 3. Magnetic parameters of ferromagnetic Heusler alloys. p C indicates the magnetic moment at the paramagnetic phase. The relationship between p eff and p C is defined by the equation of p e f f = p C (p C + 2). Takahashi's theory can be applied even to the ferromagnetic Heusler alloy, which has a spin polarization, and further study is needed to clarify the origin of the magnetism and its physical properties. .

Conclusions
In this article, we investigated the itinerant magnetism of Ni 2 MnIn and Ni 2 MnSn alloys. These alloys are ferromagnetic Heusler alloys and do not cause martensitic transformation [20], in contrast to Ni 2 MnGa with a martensitic transformation temperature T M of 195 K [21]. These alloys have an L2 1 -type cubic crystal structure even at low temperature. We considered the magnetostriction and magneto-volume effects of these alloys. We measured the forced longitudinal magnetostriction (∆L/L) // and (∆L/L) ⊥ , and we derived the forced volume magnetostriction ∆V/V. The correlation between the magnetization M and ∆V/V is (∆L/L) ∝ M 4 , and the linear fitting line crossed the origin for both alloys. These results were confirmed by Takahashi's theory [1]. From the magnetization results at T C and 4.2 K, the spin fluctuation parameters were T A in k-space and T 0 in energy space. The obtained k m parameter of the generalized Rhodes-Wohlfarth equation was around 1.4. This result accorded with Takahashi's theory. We considered the results of the examinations and theoretical calculations. We concluded that Takahashi's theory can apply even to the ferromagnetic Heusler alloy, which has a spin polarization. We compared the forced volume magnetostriction ∆V/V and its mechanical parameter, bulk modulus K, and found that ∆V/V is inversely proportional to K.