Investigation of the Itinerant Electron Ferromagnetism of Ni_2+xMnGa_1−x and Co₂VGa Heusler Alloys

Abstract

Experimental investigations into the field dependence of magnetization and temperature dependences of magnetic susceptibility in Ni_2+xMnGa_1−x (x = 0.00, 0.02, 0.04) and Co₂VGa Heusler alloy ferromagnets were performed following the spin fluctuation theory of itinerant ferromagnetism, called as “Takahashi theory”. We investigated the magnetic field dependence of magnetization at the Curie temperature T_C, which is the critical temperature of the ferromagnetic–paramagnetic transition, and also at T = 5 K, which concerns the ground state of the ferromagnetic state. The field dependence of the magnetization was analyzed by means of the H vs. M⁵ dependence, and the field dependence of the ground state at 5 K was investigated by means of an Arrott plot (H/M vs. M²) according to the Takahashi theory. As for Ni_2+xMnGa_1−x, the spin fluctuation parameter in k-space (momentum space, T_A) and that in energy space (T₀) obtained at T_C and 5 K were almost the same. On the contrary, as for Co₂VGa, the H vs. M⁵ dependence was not shown at T_C. We obtained T_A and T₀ by means of an Arrott plot at 5 K. We created a generalized Rhodes–Wohlfarth plot of p_eff/p_S versus T_C/T₀ for the other ferromagnets. The plot indicated that the relationship between p_eff/p_S and T₀/T_C followed Takahashi’s theory. We also discussed the spontaneous magnetic moment at the ground state, p_S, which was obtained by an Arrott plot at 5 K and the high temperature magnetic moment, p_C, at the paramagnetic phase. As for the localized ferromagnet, the p_C/p_S was 1. As for weak ferromagnets, the p_C/p_S was larger than 1. In contrast, the p_C/p_S was smaller than 1 by many Heusler alloys. This is a unique property of Heusler ferromagnets. Half-metallic ferromagnets of Co₂VGa and Co₂MnGa were in accordance with the generalized Rhodes–Wohlfarth plot with a k_m around 1.4. The magnetic properties of the itinerant electron of these two alloys appeared in the majority bands and was confirmed by Takahashi’s theory.

1. Introduction

Spin fluctuation theories have been proposed to explain the physical principles of the itinerant electron system [1,2,3,4,5,6,7]. Takahashi proposed the self-consistent renormalization (SCR) theory according to zero-point spin fluctuations, which assimilated both the transverse and longitudinal components of the fluctuations [4,5,6,7]. An outstanding characteristic of this theory is the magnetization at T_C. The theory proposed by Takahashi indicates that the magnetic field dependence, H, is proportional to the magnetization, M⁵, at the Curie temperature, T_C. This property was obtained by the differential calculus of the magnetization of the spin fluctuation free energy [7,8,9].

In this theory, the relation between the magnetic fields H and magnetization M is obtained theoretically by the equation of,

$$H=c(T_{\mathrm{C}})M^5$$

(1)

where c (T_C) is the constant value at T_C (refer to the references for the derivation process of Equation (1) [7,9]). MnSi [10], Fe_xCo_1−xSi [11], CoS₂ [12], and Ni [13] followed the relationship provided in Equation (1). The Heusler isotropic ferromagnetic alloy Ni_2+xMnGa_1−x (x = 0.00, 0.02, 0.04) also followed the relationship mentioned in Equation (1) [8,9,13]. From the spontaneous magnetic moment and magnetization at T_C, we obtained the spin fluctuation parameter in k-space (momentum space, T_A) and in energy space (T₀).

The other approach to obtain T_A and T₀ is the analysis of the field dependence of the magnetization by means of an Arrott plot (H/M vs. M²) at the ground magnetic state, T = 5 K [7,14]. Tateiwa et al. mentioned the derivation method of this approach in detail [14]. The magnetization in the ground state is expressed by the following equation

$$H=\frac{F_1}{N_0^3{\left(g{\mu }_B\right)}^4}\times \left(-M_0^2+M^2\right)M$$

(2)

where $g$ is Lande’s g-factor; N₀ is Avogadro’s number; and F₁ is the mode–mode coupling term defined as

$$F_1=\frac{2T_{\mathrm{A}}^2}{15c{T}_0}$$

(3)

where c is equal to 1/2 and M₀ is the spontaneous magnetization. F₁ is derived from the slope of the Arrott plot (H/M vs. M² plot) at low temperatures by Equation (4)

$$F_1=\frac{N_0^3{\left(2{\mu }_B\right)}^4}{k_{\mathrm{B}}\zeta }$$

(4)

where k_B is the Boltzmann constant, and $\zeta$ is the slope of the Arrott plot. T_A and T₀ are obtained by the following relations of

$${\left(\frac{T_{\mathrm{C}}}{T_0}\right)}^{5/6}=\frac{p_{\mathrm{S}}^2}{5g^2{C}_{4/3}}\times {\left(\frac{15c{F}_1}{2T_{\mathrm{C}}}\right)}^{1/2}$$

(5)

$${\left(\frac{T_{\mathrm{C}}}{T_{\mathrm{A}}}\right)}^{5/6}=\frac{p_{\mathrm{S}}^2}{5g^2{C}_{4/3}}\times {\left(\frac{2T_{\mathrm{C}}}{15c{F}_1}\right)}^{1/2}$$

(6)

where C_4/3 = 1.00608, and p_S is the spontaneous magnetic moment at the ground state (T = 0 K). In the Takahashi theory, it is mentioned that the experimental results of the magnetization measurement can be applied to these equations in units of kOe and emu/g for the magnetic fields H and magnetization M, respectively (p. 66 in Reference [7]). Therefore, we used these units to calculate the T_A and T₀ parameters clearly. Incidentally, the value of the magnetic field H in 10 kOe is equal to the value in T (Tesla), and the value of magnetization M in emu/g is equal to the value in Am²/kg.

Tateiwa et al. evaluated the parameters, T_A and T₀, of actinide 5f electron systems which were analyzed by means of Equations (4)–(6) [14]. Tateiwa et al. also used the units of kOe and emu/g.

The relation between p_S, T_C, T₀, and the effective magnetic moment p_eff in the paramagnetic phase was derived from a formula shown in Equation (3.47) in [7], as follows:

$$\frac{p_{\mathrm{eff}}}{p_{\mathrm{S}}}\approx 1.4\times {\left(\frac{T_0}{T_{\mathrm{C}}}\right)}^{\frac{2}{3}}$$

(7)

Equation (7) can be rewritten as:

$$k_{\mathrm{m}}=\left(\frac{p_{\mathrm{eff}}}{p_{\mathrm{S}}}\right)\times {\left(\frac{T_{\mathrm{C}}}{T_0}\right)}^{\frac{2}{3}}$$

(8)

When k_m is 1.4, Equation (8) is equal to Equation (7).

In this study, experimental investigations into the field dependence of magnetization and temperature dependences of magnetic susceptibility in Ni_2+xMnGa_1−x (x = 0.00, 0.02, 0.04) and half-metallic ferromagnets (HMFs) of Co₂VGa and Co₂MnGa Heusler alloys were performed following the self-consistent renormalization (SCR) spin fluctuation theory of itinerant electron ferromagnetism by Y. Takahashi [7]. We investigated the magnetic field dependence of magnetization at the Curie temperature T_C, which is the critical temperature of the ferromagnetic–paramagnetic transition, and also at T = 5 K, which concerns the ground state of the ferromagnetic phase. We created a generalized Rhodes–Wohlfarth plot of p_eff/p_S versus T_C/T₀ for the other ferromagnets. The plot indicated that the relationship between p_eff/p_S and T_C/T₀ followed Takahashi’s theory. We also discussed the magnetism of Heusler alloys by comparing the spontaneous magnetic moment p_S at the ground state (T = 0 K) and paramagnetic magnetic moment p_C.

2. Materials and Methods

The polycrystalline samples of Ni_2+xMnGa_1−x (x = 0.00, 0.02, 0.04) were prepared by arc melting the constituent elements, nominally, 4N Ni, 3N Mn, and 6N Ga, several times in an Ar atmosphere. Each ingot was melted several times to ensure good homogeneity. The products from the arc melting process were sealed in an evacuated silica tube and solution heat-treatment was applied at 1123 K for 3 days. After these treatments, the sample was quenched in water. The polycrystalline sample of Co₂VGa was fabricated by levitation melting after making a 66.6Co–33.4Ga (at.%) binary alloy by induction furnace melting in order to avoid the reaction of the crucible by the V element. The purity of the starting elements were 99.7% V, 3N Co, and 4N Ga. The obtained ingot was annealed at 1373 K for 3 days and quenched in water.

The magnetization measurements were performed up to 5 T by means of a SQUID magnetometer (Quantum Design Inc., San Diego, USA) at the Institute for Materials Research, Tohoku University. The permeability measurement was performed in AC magnetic fields with a frequency of 73 Hz and maximum field of ± 10 Oe. The AC magnetic fields were measured by a gaussmeter 410 (Lakeshore Cryotronix Inc., Westerville, Ohio, USA). The magnetic susceptibility measurements were performed by means of a vibrating sample magnetometer (VSM, PASCO Co. Ltd, Roseville, CA, USA), which was installed in a water-cooled electromagnet (Tamagawa Seisakusho Co. Ltd., Sendai, Japan) at Ryukoku University. The magnetic susceptibility $\chi$ in the paramagnetic phase was obtained from the temperature dependences of magnetization M, measured at the magnetic fields of H = 0.10 T and the relation of $\chi =M/H$.

3. Results and Discussion

3.1. Results of the Magnetic Measurements of Ni_2+xMnGa_1−x

Figure 1 shows the Arrott plot (M² vs. H/M) of: (a) Ni₂MnGa, and (b) Ni_2.04MnGa_0.96 at T = 5 K. By using the slope value $\zeta$ of the Arrott plot, the parameter F₁ was derived by Equation (4). The spontaneous magnetic moment, p_S; effective moment, p_eff; Curie temperature, T_C; and spin fluctuation parameters T_A and T₀ are listed in

Table 1. The magnetic parameters of Ni_2+xMnGa_1−x (x = 0.00, 0.02, 0.04). The spontaneous magnetic moment, p_S; effective moment, p_eff; Curie temperature, T_C; spin fluctuation parameter in k-space (momentum space) T_A, and that in energy space, T₀. The parameters T_A (T_C) and T₀ (T_C) were obtained from the M⁴ vs. H/M plot at T_C [9]. The p_eff, T_A (5 K) and T₀ (5 K) were the obtained values in this work.

Sample	pS (μB/f. u.)	peff (μB/f. u.)	TC (K)	TA (K) (5 K)	TA (K) (TC) [9]	T0 (K) (5 K)	T0 (K) (TC) [9]
Ni2MnGa	3.93	4.75	375	556	563	254	245
Ni2.02MnGa0.98	3.79	4.72	372	580	566	269	288
Ni2.04MnGa0.96	3.64	4.68	366	583	567	316	345

. The obtained T_A and T₀ by the relations of Equations (5) and (6) are also listed in Table 1. Errors of T_A and T₀ were estimated as $\pm$10%, which arose from the error of fitting of the Arrott plot. Within these errors, the T_A and T₀ obtained from a low temperature and the values from T_C were the same as each other.

In a previous study, we analyzed the results of Ni₂MnGa by means of the generalized Rhodes–Wohlfarth plot (double logarithmic plot of p_eff/p_S and T_C/T₀) [9], which was derived to formulate the magnetic moments ratio, p_eff/p_S, and the critical temperature ratio, T_C/T₀. Takahashi derived an equation for the relationship between p_S, T_C, T₀ and the effective magnetic moment p_eff as Equation (7). As for Ni₂MnGa, the measured effective moment p_eff, which was measured in this work, was 4.75, which was the same value as the result by Webster et al. [15]. For Ni₂MnGa, a value of 1.61 for k_m was obtained by substituting a p_eff of 4.75, and p_S, T_C, and T₀ from Table 1 into Equation (8).

In order to investigate the k_m values of Ni_2+xMnGa_1−x (x = 0.02, 0.04) and compare them with other ferromagnetic alloy and compounds, we further needed p_eff values of these alloys. We measured the magnetic susceptibility of these alloys, and p_eff values were obtained from the Curie constant of the Curie law.

Figure 2 shows the inverse magnetic susceptibilities, $1/\chi =H/M$. The gradient of $1/\chi$ vs. T, which is indicated by the dotted lines, is equal to 1/C, where C is a Curie constant.

The Curie constant C is written as:

$$C=\frac{N{p}_{\mathrm{eff}}^2{\mu }_B^2}{3k_{\mathrm{B}}}$$

(9)

where N is the molecular number per gram. The obtained effective moments p_eff were 4.72 for Ni_2.02MnGa_0.98 and 4.68 for Ni_2.04MnGa_0.96. In Section 3.3, we discuss the itinerant electron ferromagnetism by means of these parameters.

3.2. Results of the Magnetic Measurements of Half-Metallic Ferromagnet Co₂VGa

HMFs are comprised of a metallic band for one spin direction. For the other spin direction, a semiconducting band has an energy gap around Fermi energy. Co₂VGa is a HMF with a high spin polarization [16]. It has an L2₁-type cubic crystal structure with a lattice constant a = 0.5782 nm. The spin polarization ratio P is defined as:

$$P_0(\%)=\left|\frac{N_{\uparrow }\left(E_F\right)-N_{\downarrow }\left(E_F\right)}{N_{\uparrow }\left(E_F\right)+N_{\downarrow }\left(E_F\right)}\right|\times 100$$

(10)

where $N_{\uparrow }\left(E_F\right)$ and $N_{\downarrow }\left(E_F\right)$ denote the density of states, DOS, at the Fermi energy, E_F, in the majority spin ($\uparrow$) and minority spin ($\downarrow )$, respectively. Umetsu et al. calculated the DOS by means of the LTMO method with the atomic spheres approximation (ASA). From the results of this calculation, the P₀ value (P value at T = 0 K) was 75% and the P₀ value of L2₁-type Co₂(V_1−xMn_x)Ga alloys ($0\le x\le 1)$ was also determined. As for x = 1, Co₂MnGa, the obtained spin polarization ratio P₀ was 48%. This indicates that Co₂VGa is a higher polarized HMF. The Curie temperatures of Co₂VGa and Co₂MnGa were 337 K and 695 K, respectively. We measured the magnetic field dependences of the magnetization to obtain the magnetic moments, p_s, and the spin fluctuation parameters, T_A and T₀, and also measured the magnetic susceptibility to obtain the effective magnetic moment, p_eff, in the paramagnetic phase. We also obtained T_A and T₀ of Co₂MnGa according to the Takahashi theory by means of the magnetization process at 5 K in Reference [16].

Figure 3a shows the permeability of Co₂VGa around the Curie temperature. From the differentiation of the permeability for the temperature, denoted as dP/dT, the Curie temperature was obtained as T_C = 337 K. Figure 3b shows the inverse magnetic susceptibility $1/\chi =H/M$ of Co₂VGa. The obtained p_eff was 2.06.

Figure 4 shows the M³ vs. H/M plot and M⁴ vs. H/M plot of Co₂VGa around T_C = 337 K.

With regard to the Takahashi theory, the magnetization process is expressed as (H/M)$\propto$M⁴ by Equation (1) around T_C. On the contrary, the H/M was almost proportional to M³ as shown in Figure 4a. Nishihara et al. also measured the magnetization around T_C [17]. The magnetization process at T_C is expressed as H$\propto$M^D with the index D = 4.15 $\pm 0.05$. Their result was the same as in this study. Nishihara et al. mentioned that the discrepancy between these experimental magnetization results and the Takahashi theory is supposed to arise from the distribution of T_C in the sample because the fourth-order expansion of the magnetic-free energy vanishes at the Curie temperature. In this study, we tried again with other ingots from the former sample used by Nishihara et al. As this experiment reproduced the former experiment, there may be an essential reason. Incidentally, other magnetic models have indicated that the molecular field theory denotes the D value as 3.0, the three-dimensional Heisenberg model denotes the D value as 4.8, and the three dimensional Ising model as 4.82 [18]. None of these matched the analysis in this investigation. In order to obtain the spin fluctuation parameters T_A and T₀ of Co₂VGa, we measured the magnetization process of Co₂VGa at 5 K. Figure 5 shows the Arrott plot (M² vs. H/M) of Co₂VGa. The parameter F₁ was obtained by applying the slope value of the Arrott plot to Equation (4). The parameters T_A and T₀ were derived by Equations (5) and (6). The obtained T_A and T₀ were 2258 K and 213 K, respectively.

3.3. Analysis According to the Takahashi Theory

Table 2. Basic magnetic parameters and k_m as obtained from Equation (8).

	TC (K)	peff (μB)	pS (μB)	peff/pS	TA (K)	T0 (K)	TC/T0	km	Reference 1
Ni2MnGa	375	4.75 *	3.93	1.21	563	245	1.53	1.61	This work *, [9]
Ni2.02MnGa0.98	372	4.72 *	3.79	1.25	566	288	1.29	1.48	This work *, [9]
Ni2.04MnGa0.96	366	4.68 *	3.64	1.28	567	345	1.06	1.34	This work *, [9]
Co2VGa	337	2.06	1.87	1.10	2258	213	1.58	1.50	This work
Co2MnGa	695	4.16	4.09	1.02	1,037	364	1.91	1.57	[16,19]
Ni	623	3.3	0.6	5.5	1.76 × 104	4.83 × 103	0.129	1.41	[13]
MnSi	30	2.25	0.4	5.6	2.18 × 103	155	0.194	1.88	[7,10]
Ni3Al	41.5	1.3	0.075	17.3	3.67 × 104	2.76 × 103	0.015	1.06	[7,20]
Y(Co0.85Al0.15)2	26	2.15	0.138	15.6	7.26× 103	1.41× 103	0.018	1.08	[7,21]
ZrZn2	21.3	1.44	0.12	12	7.4× 103	1390	0.015	0.74	[7,22]
CoS2	120	1.72	0.98	1.76	2.20× 103	294	0.41	0.96	[12,23]
UCoGe	2.4	1.93	0.039	49.5	5.92 × 103	362	0.0065	1.74	[7,24]
UGe2	52.6	3.00	1.41	2.13	442	92.2	0.571	1.61	[14]
NpFe4P12	23	1.55	1.35	1.15	285	16.4	1.40	1.44	[14,25]

¹ Citations in our published paper [9] are incorrect. The correct citations are listed above. We apologize for this mistake. * These values were obtained by this work.

indicates the Curie temperature T_C, the effective magnetic moment p_eff, the spontaneous magnetization p_S, the magnetic moment ratio p_eff/p_S, the spin fluctuation parameters T_A and T₀, the critical temperature ratio T_C/T₀, and k_m, as obtained from Equation (8).

The k_m value was around 1.4. Figure 6 shows the generalized Rhodes–Wohlfarth plot using the parameters in Table 2 [7,26]. The points of Ni_2+xMnGa_1−x are in accordance with the dotted line as k_m = 1.4. It is noteworthy that the HMFs, Co₂VGa and Co₂MnGa, were also in accordance with this line. Originally, the Takahashi theory was applied to weak ferromagnets. It is interesting that this theory can be applied to strongly correlated 5f electron systems as well as Heusler HMFs.

3.4. Comparison between the Spontaneous Magnetic Moment at the Ground State, p_S, and the Paramagnetic Magnetic Moment, p_C, for HMFs

In this subsection, we consider the magnetism of Heusler alloys by comparing the spontaneous magnetic moment at the ground state and paramagnetic magnetic moment.

We rewrote the definitions of p_S, p_sat, p_eff, and p_C to make the following argument plain. p_S is the spontaneous magnetic moment at the ground state (T = 0 K or T << T_C). p_sat is the saturation magnetic moment at the ground state (T = 0 K or T<< T_C). p_eff is the effective magnetic moment in the paramagnetic phase. p_C is the magnetic moment in the paramagnetic phase. These four magnetic moments are defined by the unit of ${\mu }_B.$ The relation between p_eff and p_C is described as

$$p_{\mathrm{eff}}=\sqrt{p_{\mathrm{C}}\left(p_{\mathrm{C}}+2\right)}$$

(11)

In HMFs, the band for minority spin electrons has a gap at the Fermi level and indicates semi-metallic bands. On the other hand, for majority spin electrons, the Fermi level intersects the bands and represents metallic bands.

Table 3. Magnetic parameters of ferromagnetic Heusler alloys. p_C indicates the magnetic moment at the paramagnetic phase. The relation between p_eff and p_C is defined by the equation of $p_{\mathrm{eff}}=\sqrt{p_{\mathrm{C}}\left(p_{\mathrm{C}}+2\right)}$.

Sample	TC (K)	pS (μB/f.u.)	peff (μB/f.u.)	pC (μB/f.u.)	pC/pS	Reference
Ni2MnGa	375	3.93	4.75 *	3.85	0.980	This work *, [9]
Ni2.02MnGa0.98	372	3.79	4.72 *	3.82	1.01	This work *, [9]
Ni2.04MnGa0.96	366	3.64	4.68 *	3.79	1.04	This work *, [9]
Co2VGa	337	1.87	2.06	1.30	0.70	This work
Co2MnSi	1034	5.01	2.86	2.03	0.41	[30]
Co2MnGe	905	4.76	3.70	2.82	0.56	[30]
Co2MnSn	825	5.02	5.29	4.38	0.87	[31]
Co2MnGa	695	4.09	4.16	3.28	0.80	[19]
Co2FeSi	1015	5.42 (300 K)	5.65	4.74	0.875	[32]
Co2FeGa	1089	5.05 (300 K)	4.59	3.69	0.730	[32]
CoMnSb	478	4.2	4.0–4.6	3.1-3.7	0.74–0.88	[27]
NiMnSb	728	4.2	2.9–4.2	2.1-3.3	0.69–0.79	[27]
PtMnSb	572	3.96	4.3–4.9	3.4-4.0	0.86–1.01	[27]
Ni2MnIn	315	4.4	4.69	3.80	0.86	[33,34]
Rh2MnSn	410	4.14	4.83	3.93	0.95	[31]
Pd2MnSn	189	4.23	4.70	3.81	0.90	[28]
Pd2MnSb	255	4.40	4.8	3.9	0.89	[28]

* These values were obtained by this work.

represents the magnetic parameters of ferromagnetic Heusler alloys, with the paramagnetic moment p_C. The notable point of Table 3 is that the p_C/p_s of Ni₂MnGa and many half-metallic Heusler alloys were smaller than 1. From Equation (9) and (11), the p_C is calculated by the Curie constant, C = Nμ_eff²/3k_B = Np_eff²μ_B²/3k_B = Np_C(p_C + 2)μ_B²/3k_B. p_C refers to the magnetic moment in the paramagnetic phase deduced from the Curie constant C. 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 [7].

As for Ni₂MnGa, p_eff was 4.75, as shown in Table 3. Therefore, the p_c obtained was 3.85 from Equation (11), and the p_C/p_S value was 0.980. As a result, the p_C/p_S was a little smaller than 1. Webster et al. compared the magnetic moment obtained by the saturation magnetization measurement where p_sat = 4.17 [15]. Then, the p_sat/p_s was 0.92. The magnetization of Ni₂MnGa in the magnetic field of 5.0 T at 5 K was 4.10 μ_B/f.u. Therefore, the p_sat/p_s was 0.96. Regarding the half-metallic Heusler alloys, Co₂VGa and Co₂MnGa, which are the focus of this article, the p_sat/p_s were 0.70 and 0.80, respectively. The renowned half-metallic Heusler alloys and compounds listed in Table 3 indicate the property of p_C/p_S < 1. The magnetic properties of the inter-metallic compounds CoMnSb, NiMnSb, PtMnSb, Pd₂MnSn, and Pd₂MnSb showed an effective paramagnetic moment above T_C, which was also smaller than the spontaneous and saturation moment of the ground state at T = 0 K [27,28].

As above-mentioned, the spin polarization values P₀ of Co₂VGa and Co₂MnGa were 75% and 48%, respectively [16]. This indicates that Co₂VGa is a higher polarized HMF. The p_C/p_S values of Co₂VGa and Co₂MnGa were 0.70 and 0.80, respectively, as shown in Table 3. The results concerned with these two alloys indicate that the alloy with a larger spin polarization showed a smaller p_C/p_S value.

Dong et al. studied the spin polarization of Co₂MnGe experimentally and analyzed the temperature dependence of the spin polarization [29] where the spin polarization of Co₂MnGe was 27% at 2 K. However, the spin polarization decreased with increasing temperature and vanished at 300 K. It is considered that the magnetic moment decreases at a high temperature with the decrease of the spin polarization. Ott et al. also suggested that this effect could be attributed to a decrease of the conduction electron spin polarization in the paramagnetic phase, which has a higher temperature than T_C [27]. A simple molecular field model, which took into account both local moments and spin-polarized itinerant electrons, explained that p_C/p_S < 1 [27]. They introduced an “Enhanced Temperature-independent Pauli susceptibility”, which comes from the itinerant electron bands intersecting the Fermi level, and explained that the Curie constant was reduced if the interactions between local magnetic moments and holes were antiferromagnetic. Therefore, the reduction in the Curie constant indicates that the magnetic moment p_C at a high temperature in a paramagnetic phase is smaller than that of the spontaneous magnetization p_S as well as the saturation moment p_sat at the ground phase of T = 5 K. Webster et al. pointed out the electronic and spin phases of Ni₂MnGa [15]. In the paramagnetic phase, only the Mn atoms carry a magnetic moment. It is supposed that in the paramagnetic phase a large moment was induced by the electrons around the Mn atom at the Mn site. On the contrary, at the Ni site, the spins fluctuated at high temperature in the paramagnetic phase. Therefore, it is also supposed that p_C at a high temperature in the paramagnetic phase is smaller than that of the p_S. As for Co₂VGa and Co₂MnGa, which were treated in this article, it is of interest to investigate the temperature dependence of the spin polarization experimentally.

In Figure 6, the HMFs of Co₂VGa and Co₂MnGa were in accordance with the generalized Rhodes–Wohlfarth plot with a k_m around 1.4. The majority of the bands of these two alloys intersected the Fermi level [16]. Therefore, the magnetic property of the itinerant electron appeared in the majority bands and was confirmed by Takahashi’s theory.

4. Conclusions

In this article, experimental investigations and discussions into the field dependence of magnetization and temperature dependences of magnetic susceptibility in Ni_2+xMnGa_1−x (x = 0.00, 0.02, 0.04), Co₂VGa, and Co₂MnGa Heusler alloy ferromagnets were performed following the spin fluctuation theory of itinerant electron ferromagnetism by Y. Takahashi.

• As for Ni_2+xMnGa_1−x, the spin fluctuation parameters in k-space (momentum space, T_A) and that in energy space (T₀) obtained at T_C and 5 K were almost the same within $\pm$10% error. This consequently indicates that the spin fluctuation parameters can be obtained from the H/M vs. M⁴ plot at T_C and also from an Arrott plot (H/M vs. M²) at a low temperature of T << T_C.
• As for Co₂VGa, the H vs. M⁵ dependence was not shown at T_C. T_A and T₀ were obtained by means of an Arrott plot at 5 K, which is well below T_C = 337 K;
• In order to obtain a k_m value as defined in Equation 8, the magnetic susceptibility was measured, and the p_eff was obtained by means of Curie law. The k_m of Co₂VGa (1.50) and Co₂MnGa (1.57) were around 1.4, which was proposed in Takahashi’s theory. The generalized Rhodes–Wohlfarth plot of p_eff/p_S versus T_C/T₀ indicated that the relationship between p_eff/p_S and T₀/T_C for the ferromagnets, including Ni_2+xMnGa_1−x and HMFs of Co₂VGa and Co₂MnGa, followed Takahashi’s theory. In HMFs, the band for minority spin electrons has a gap at the Fermi level and indicates semi-metallic bands. On the other hand, for majority spin electrons, the Fermi level intersects the bands and represents metallic bands. The magnetic properties of the itinerant electron of these two HMFs alloys appeared in the majority bands and were confirmed by Takahashi’s theory;
• As for Ni_2+xMnGa_1−x and HFMs, we obtained the spontaneous magnetic moment at the ground state, p_S, by an Arrott plot at 5 K, and the high temperature magnetic moment, p_C, at the paramagnetic phase. The p_C/p_S was smaller than 1 for many Heusler alloys, which is a different property from the localized ferromagnets (p_C/p_S = 1), or, for weak itinerant electron ferromagnets (p_C/p_S > 1). A comparison between Co₂VGa and Co₂MnGa indicates that the alloy with a larger spin polarization showed a smaller p_C/p_S value. Further, an experimental investigation into the temperature dependence of spin polarization is needed to clarify the mechanism of shrinkage of the magnetic moments in the paramagnetic phase at a high temperature.