Investigation of the Itinerant Electron Ferromagnetism of Ni2+xMnGa1−x and Co2VGa Heusler Alloys

Experimental investigations into the field dependence of magnetization and temperature dependences of magnetic susceptibility in Ni2+xMnGa1−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 TC, 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 Ni2+xMnGa1−x, the spin fluctuation parameter in k-space (momentum space, TA) and that in energy space (T0) obtained at TC and 5 K were almost the same. On the contrary, as for Co2VGa, the H vs. M5 dependence was not shown at TC. We obtained TA and T0 by means of an Arrott plot at 5 K. We created a generalized Rhodes–Wohlfarth plot of peff/pS versus TC/T0 for the other ferromagnets. The plot indicated that the relationship between peff/pS and T0/TC followed Takahashi’s theory. We also discussed the spontaneous magnetic moment at the ground state, pS, which was obtained by an Arrott plot at 5 K and the high temperature magnetic moment, pC, at the paramagnetic phase. As for the localized ferromagnet, the pC/pS was 1. As for weak ferromagnets, the pC/pS was larger than 1. In contrast, the pC/pS 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 km 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.


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, M 5 , 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, 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 x Co 1−x Si [11], CoS 2 [12], and Ni [13] followed the relationship provided in Equation (1). The Heusler isotropic ferromagnetic alloy Ni 2+x MnGa 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 (T 0 ). The other approach to obtain T A and T 0 is the analysis of the field dependence of the magnetization by means of an Arrott plot (H/M vs. M 2 ) 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 where g is Lande's g-factor; N 0 is Avogadro's number; and F 1 is the mode-mode coupling term defined as where c is equal to 1/2 and M 0 is the spontaneous magnetization. F 1 is derived from the slope of the Arrott plot (H/M vs. M 2 plot) at low temperatures by Equation (4) where k B is the Boltzmann constant, and ζ is the slope of the Arrott plot. T A and T 0 are obtained by the following relations of 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 T 0 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 Am 2 /kg. Tateiwa et al. evaluated the parameters, T A and T 0 , 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 , T 0 , 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: Equation (7) can be rewritten as: 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+x MnGa 1−x (x = 0.00, 0.02, 0.04) and half-metallic ferromagnets (HMFs) of Co 2 VGa and Co 2 MnGa 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 /T 0 for the other ferromagnets. The plot indicated that the relationship between p eff /p S and T C /T 0 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 .

Materials and Methods
The polycrystalline samples of Ni 2+x MnGa 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 Co 2 VGa 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 χ 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  Table 1. The obtained T A and T 0 by the relations of Equations (5) and (6) are also listed in Table 1. Errors of T A and T 0 were estimated as ±10%, which arose from the error of fitting of the Arrott plot. Within these errors, the T A and T 0 obtained from a low temperature and the values from T C were the same as each other.  (5) and (6) are also listed in Table 1. Errors of TA and T0 were estimated as ± 10%, which arose from the error of fitting of the Arrott plot. Within these errors, the TA and T0 obtained from a low temperature and the values from TC were the same as each other.

Results and Discussion
(a) (b)   [9], which was derived to formulate the magnetic moments ratio, peff/pS, and the critical temperature ratio, TC/T0. Takahashi derived an equation for the relationship between pS, TC, T0 and the effective magnetic moment peff as Equation (7). As for Ni2MnGa, the measured effective moment peff , 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 km was obtained by substituting a peff of 4.75, and pS, TC, and T0 from Table 1 into Equation (8).
In order to investigate the km values of Ni2+xMnGa1−x (x = 0.02, 0.04) and compare them with other ferromagnetic alloy and compounds, we further needed peff values of these alloys. We measured the magnetic susceptibility of these alloys, and peff values were obtained from the Curie constant of the Curie law. Figure 2 shows the inverse magnetic susceptibilities, 1/ = / . The gradient of 1/ vs. T, which is indicated by the dotted lines, is equal to 1/C, where C is a Curie constant.  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, T 0 . The parameters T A (T C ) and T 0 (T C ) were obtained from the M 4 vs. H/M plot at T C [9]. The p eff , T A (5 K) and T 0 (5 K) were the obtained values in this work. Sample T 0 (K) (5 K) T 0 (K) (T C ) [ In a previous study, we analyzed the results of Ni 2 MnGa by means of the generalized Rhodes-Wohlfarth plot (double logarithmic plot of p eff /p S and T C /T 0 ) [9], which was derived to formulate the magnetic moments ratio, p eff /p S , and the critical temperature ratio, T C /T 0 . Takahashi derived an equation for the relationship between p S , T C , T 0 and the effective magnetic moment p eff as Equation (7). As for Ni 2 MnGa, 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 Ni 2 MnGa, a value of 1.61 for k m was obtained by substituting a p eff of 4.75, and p S , T C , and T 0 from Table 1 into Equation (8).
In order to investigate the k m values of Ni 2+x MnGa 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/χ = H/M. The gradient of 1/χ 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: where N is the molecular number per gram. The obtained effective moments peff were 4.72 for Ni2.02MnGa0.98 and 4.68 for Ni2.04MnGa0.96. In Subsection 3.3, we discuss the itinerant electron ferromagnetism by means of these parameters.

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: where ↑ ( ) and ↓ ( ) denote the density of states, DOS, at the Fermi energy, EF, in the majority spin (↑) and minority spin (↓), 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(V1−xMnx)Ga alloys (0 ≤ ≤ 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, ps, and the spin fluctuation parameters, TA and T0, and also measured the magnetic susceptibility to obtain the effective magnetic moment, peff, in the paramagnetic phase. We also obtained TA 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 TC = 337 K. Figure 3b shows the inverse magnetic susceptibility 1/ = / of Co2VGa. The obtained peff was 2.06. The Curie constant C is written as: where N is the molecular number per gram. The obtained effective moments p eff were 4.72 for Ni 2.02 MnGa 0.98 and 4.68 for Ni 2.04 MnGa 0.96 . In Section 3.3, we discuss the itinerant electron ferromagnetism by means of these parameters.

Results of the Magnetic Measurements of Half-Metallic Ferromagnet Co 2 VGa
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. Co 2 VGa is a HMF with a high spin polarization [16]. It has an L2 1 -type cubic crystal structure with a lattice constant a = 0.5782 nm. The spin polarization ratio P is defined as: where N ↑ (E F ) and N ↓ (E F ) denote the density of states, DOS, at the Fermi energy, E F , in the majority spin (↑) and minority spin ( ↓) , 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 P 0 value (P value at T = 0 K) was 75% and the P 0 value of L2 1 -type Co 2 (V 1−x Mn x )Ga alloys (0 ≤ x ≤ 1) was also determined. As for x = 1, Co 2 MnGa, the obtained spin polarization ratio P 0 was 48%. This indicates that Co 2 VGa is a higher polarized HMF. The Curie temperatures of Co 2 VGa and Co 2 MnGa 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 T 0 , and also measured the magnetic susceptibility to obtain the effective magnetic moment, p eff , in the paramagnetic phase. We also obtained T A and T 0 of Co 2 MnGa according to the Takahashi theory by means of the magnetization process at 5 K in Reference [16]. Figure 3a shows the permeability of Co 2 VGa 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/χ = H/M of Co 2 VGa. The obtained p eff was 2.06.    [17]. The magnetization process at TC is expressed as H∝M D with the index D = 4.15 ± 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 TC 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 threedimensional 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       [17]. The magnetization process at TC is expressed as H∝M D with the index D = 4.15 ± 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 TC 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 threedimensional 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   With regard to the Takahashi theory, the magnetization process is expressed as (H/M)∝M 4 by Equation (1) around T C . On the contrary, the H/M was almost proportional to M 3 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∝M D with the index D = 4.15 ± 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 T 0 of Co 2 VGa, we measured the magnetization process of Co 2 VGa at 5 K. Figure 5 shows the Arrott plot (M 2 vs. H/M) of Co 2 VGa. The parameter F 1 was obtained by applying the slope value of the Arrott plot to Equation (4). The parameters T A and T 0 were derived by Equations (5) and (6). The obtained T A and T 0 were 2258 K and 213 K, respectively. fluctuation parameters TA and T0 of Co2VGa, we measured the magnetization process of Co2VGa at 5 K. Figure 5 shows the Arrott plot (M 2 vs H/M) of Co2VGa. The parameter F1 was obtained by applying the slope value of the Arrott plot to Equation (4). The parameters TA and T0 were derived by Equations (5) and (6). The obtained TA and T0 were 2258 K and 213 K, respectively.  Table 2 indicates the Curie temperature TC, the effective magnetic moment peff, the spontaneous magnetization pS, the magnetic moment ratio peff/pS, the spin fluctuation parameters TA and T0, the critical temperature ratio TC/T0, and km, as obtained from Equation (8).  [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.

Analysis According to the Takahashi Theory
The km value was around 1.4. Figure 6 shows the generalized Rhodes-Wohlfarth plot using the parameters in Table 2 [7,26]. The points of Ni2+xMnGa1−x are in accordance with the dotted line as km = 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.  Table 2 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 T 0 , the critical temperature ratio T C /T 0 , and k m , as obtained from Equation (8). Table 2. Basic magnetic parameters and k m as obtained from Equation (8).  [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.

Analysis According to the Takahashi Theory
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+x MnGa 1−x are in accordance with the dotted line as k m = 1.4. It is noteworthy that the HMFs, Co 2 VGa and Co 2 MnGa, 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.

Comparison between the Spontaneous Magnetic Moment at the Ground State, pS, and the Paramagnetic Magnetic Moment, pC, 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 pS, psat, peff, and pC to make the following argument plain. pS is the spontaneous magnetic moment at the ground state (T = 0 K or T << TC). psat is the saturation magnetic moment at the ground state (T = 0 K or T<< TC). peff is the effective magnetic moment in the paramagnetic phase. pC is the magnetic moment in the paramagnetic phase. These four magnetic moments are defined by the unit of . The relation between peff and pC is described as In HMFs, the band for minority spin electrons has a gap at the Fermi level and indicates semimetallic bands. On the other hand, for majority spin electrons, the Fermi level intersects the bands and represents metallic bands. Table 3 represents the magnetic parameters of ferromagnetic Heusler alloys, with the paramagnetic moment pC. The notable point of Table 3 is that the pC/ps of Ni2MnGa and many half-metallic Heusler alloys were smaller than 1. From Equation (9) and (11), the pC is calculated by the Curie constant, C = Nμeff 2 /3kB = Npeff 2 μB 2 /3kB = NpC(pC + 2)μB 2 /3kB. pC refers to the magnetic moment in the paramagnetic phase deduced from the Curie constant C. pc/ps is 1 for the local moment ferromagnetism. For the weak itinerant electron ferromagnetism, the pc/ps is larger than 1 [7].
As for Ni2MnGa, peff was 4.75, as shown in Table 3. Therefore, the pc obtained was 3.85 from Equation (11), and the pC/pS value was 0.980. As a result, the pC/pS was a little smaller than 1. Webster et al. compared the magnetic moment obtained by the saturation magnetization measurement where psat = 4.17 [15]. Then, the psat/ps was 0.92. The magnetization of Ni2MnGa in the magnetic field of 5.0 T at 5 K was 4.10 μB/f.u. Therefore, the psat/ps was 0.96. Regarding the half-metallic Heusler alloys, Co2VGa and Co2MnGa, which are the focus of this article, the psat/ps were 0.70 and 0.80, respectively. The renowned half-metallic Heusler alloys and compounds listed in Table 3 indicate the property of pC/pS < 1. The magnetic properties of the inter-metallic compounds CoMnSb, NiMnSb, PtMnSb,

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 µ B . The relation between p eff and p C is described as 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 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 Ni 2 MnGa 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 eff 2 /3k B = Np eff 2 µ B 2 /3k B = Np C (p C + 2)µ B 2 /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 Ni 2 MnGa, 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 Ni 2 MnGa 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, Co 2 VGa and Co 2 MnGa, 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, Pd 2 MnSn, and Pd 2 MnSb 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 P 0 of Co 2 VGa and Co 2 MnGa were 75% and 48%, respectively [16]. This indicates that Co 2 VGa is a higher polarized HMF. The p C /p S values of Co 2 VGa and Co 2 MnGa 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 Co 2 MnGe experimentally and analyzed the temperature dependence of the spin polarization [29] where the spin polarization of Co 2 MnGe 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 Ni 2 MnGa [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 Co 2 VGa and Co 2 MnGa, which were treated in this article, it is of interest to investigate the temperature dependence of the spin polarization experimentally. 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 eff = p C (p C + 2). * These values were obtained by this work.
In Figure 6, the HMFs of Co 2 VGa and Co 2 MnGa 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.

Conclusions
In this article, experimental investigations and discussions into the field dependence of magnetization and temperature dependences of magnetic susceptibility in Ni 2+x MnGa 1−x (x = 0.00,