In recent years, Heusler alloys have become a subject of intensive experimental and theoretical investigations. Ferromagnetic shape memory alloys (FSMAs) with the Heusler-type structure (L2₁-type structure), which exhibit both the ferromagnetic and structural transitions, have attracted much attention due to their potential application as smart materials [1,2]. A large magnetic field-induced strain by the rearrangement of twin variants in the martensite phase was observed in the Ni-Mn-Ga FSMAs [3,4]. Moreover, a large magnetocaloric effect (MCE) accompanied by a magnetostructural transition, where both the ferromagnetic and structural transitions occur together, has been expected to be useful for devices [5,6]. Among FSMAs, the stoichiometric Heusler alloy Ni₂MnGa has been the most studied. Ni₂MnGa orders ferromagnetically with a Curie temperature of T_C ≈ 365 K. On cooling below the martensitic transition temperature T_M ≈ 200 K, a superstructure forms, and the ferromagnetic state remains below T_M [7,8]. The spontaneous magnetization M_s just below T_M is larger than the M_s just above T_M for Ni₂MnGa.

Recently, Kataoka et al. [9] and Endo et al. [10] carried out magnetization, permeability and differential scanning calorimetric (DSC) measurements on the FMSAs Ni₂Mn_{1 − x}Cu_xGa (0 ≤ x ≤ 0.40) and Ni₂MnGa_{1 − y}Cu_y (0 ≤ y ≤ 0.25) alloys, respectively. It was found that for Ni₂Mn_{1 − x}Cu_xGa (0 ≤ x ≤ 0.40) alloys the magnetostructural transitions between the paramagnetic austenite phase (Para-A) and the ferromagnetic martensite phase (Ferro-M) occur in the concentration range of 0.23 ≤ x ≤ 0.30 [9]. Similarly, the magnetostructural transitions between the Para-A and Ferro-M were observed in the concentration range of 0.12 ≤ y ≤ 0.14 for Ni₂MnGa_{1 − y}Cu_y (0 ≤ y ≤ 0.25) alloys [10]. Furthermore, the characteristics of the phase diagrams of Ni₂Mn_{1 − x}Cu_xGa (0 ≤ x ≤ 0.40) and Ni₂MnGa_{1 − y}Cu_y (0 ≤ y ≤ 0.25) alloys were found to be very similar to those of Ni_{2 +x}Mn_{1 − x}Ga (0 ≤ x ≤ 0.36) alloys [11]. Kataoka et al. explained the phase diagram of Ni₂Mn_{1 − x}Cu_xGa (0 ≤ x ≤ 0.40) alloys using the Landau-type phenomenological free energy as a function of the martensitic distortion and the magnetization [9]. Their analysis showed that the biquadratic coupling term, together with a higher order term, of the martensitic distortion and the magnetization plays an important role in the interplay between the martensite phase and the ferromagnetic phase. As the result, Kataoka et al. could obtain the satisfactory agreement between the calculated and observed phase diagrams for Ni₂Mn_{1 − x}Cu_xGa (0 ≤ x ≤ 0.40) alloys. Also the phase diagram of Ni_{2 − z}Cu_zMnGa (0 ≤ z ≤ 0.8) alloys was determined from the results of the temperature dependence of the initial permeability [12]. The T_C of Ni_{2 − z}Cu_zMnGa (0 ≤ z ≤ 0.8) alloys increased with the Cu concentration z. While, the T_M decreased abruptly with z.

In this paper, the concentration dependence of the magnetic moment for Ni₂Mn_{1 − x}Cu_xGa (0 ≤ x ≤ 0.40) and Ni₂MnGa_{1 − y}Cu_y (0 ≤ y ≤ 0.25) alloys is examined to gain deeper insight for the magnetic properties of these FSMAs. On the basis of the experimental results, the site occupancy and the magnetic structure of Ni₂Mn_{1 − x}Cu_xGa (0 ≤ x ≤ 0.40) and Ni₂MnGa_{1 − y}Cu_y (0 ≤ y ≤ 0.25) alloys are presented. Furthermore, the site occupancy of Ni_{2 − z}Cu_zMnGa (0 ≤ z ≤ 0.4) alloys is also presented.

The Ni₂Mn_{1 − x}Cu_xGa (0 ≤ x ≤ 0.40) and Ni₂MnGa_{1 − y}Cu_y (0 ≤ y ≤ 0.25) alloys were prepared by repeated arc melting of the constituent elements, namely 99.99% pure Ni, 99.99% pure Mn, 99.99% pure Cu and 99.9999% pure Ga in argon atmosphere. The heat treatments of all reaction products after the arc melting were reported in the references [9,10]. Weight loss between before and after the arc melting is within 0.5%, thus the composition of the specimens is seen to be the same with the nominal ones. By using X-ray powder diffraction measurements all samples were confirmed to be of single phase at room temperature. Magnetization measurements on prepared samples were carried out in magnetic fields up to 50 kOe using a superconducting quantum interference device (SQUID) magnetometer.

Figure 1, Figure 2 show the phase diagrams of Ni₂Mn_{1 − x}Cu_xGa (0 ≤ x ≤ 0.40) and Ni₂MnGa_{1 − y}Cu_y (0 ≤ y ≤ 0.25) alloys reported by Kataoka et al. [9] and Endo et al. [10], respectively. The phase diagrams shown in Figure 1, Figure 2 have characteristics very similar to that [11] of Ni_{2 +x}Mn_{1 − x}Ga (0 ≤ x ≤ 0.36) alloys. As shown in Figure 1, the samples with x ≤ 0.20 of Ni₂Mn_{1 − x}Cu_xGa (0 ≤ x ≤ 0.40) alloys crystallize in the L2₁-type structure at room temperature. However, the details of the crystal structure in the martensite phase for the samples with x ≤ 0.20 are not clear. It was confirmed by the low temperature X-ray powder diffraction measurements that the sample with x = 0.23 for Ni₂Mn_1-xCu_xGa alloy crystallizes in a 14-layered monoclinic (14M) structure (space group: C2/m) well below the martensitic transition temperature [9]. The X-ray powder diffraction pattern of the sample with x = 0.23 at room temperature shows that the cubic phase with the L2₁-type structure and the monoclinic phase with the 14M structure coexist. Similarly, the X-ray powder diffraction patterns at room temperature of the samples with x = 0.25 and 0.27 indicated that the L2₁ phase and the 14M phase coexist although the fraction of the 14M phase increases with increasing the concentration x. The sample with x = 0.35 crystallizes in a tetragonal D0₂₂-like crystal structure with no lattice modulation at room temperature. The crystal structures of the 14M and the D0₂₂-like also appear in the martensite phase of Ni₂MnGa_{1 − y}Cu_y (0 ≤ y ≤ 0.25) alloys depending on the Cu concentration, being similar to that of Ni₂Mn_{1 − x}Cu_xGa (0 ≤ x ≤ 0.40) alloys [10]. The magnetization curves at 5 K for Ni₂Mn_{1 − x}Cu_xGa (0 ≤ x ≤ 0.40) alloys with various concentrations x are shown in Figure 3. All of the magnetization curves are characteristic of ferromagnetism or ferrimagnetism. The magnetization M at 5 K for all samples is saturated in a field of about 20 kOe. The magnetization curves at 5 K for Ni₂MnGa_{1 − y}Cu_y (0 ≤ y ≤ 0.25) alloys with various concentrations y are shown in Figure 4. The spontaneous magnetization at 5 K for Ni₂Mn_{1 − x}Cu_xGa (0 ≤ x ≤ 0.40) and Ni₂MnGa_{1 − y}Cu_y (0 ≤ y ≤ 0.25) alloys was determined by the linear extrapolation to H/M = 0 of the M² versus H/M curves (Arrott plot). The magnetic moments per formula unit, μ_s, at 5 K for Ni₂Mn_{1 − x}Cu_xGa (0 ≤ x ≤ 0.40) and Ni₂MnGa_{1 − y}Cu_y (0 ≤ y ≤ 0.25) alloys were estimated from the values of the spontaneous magnetization and are plotted against concentrations x and y as shown in Figure 5. The concentration dependence of μ_s at 5 K for Ni_{2 − z}Cu_zMnGa (0 ≤ z ≤ 0.40) alloys is also shown in Figure 5 [12], where the values of μ_s for Ni_{2 − z}Cu_zMnGa (0 ≤ z ≤ 0.40) alloys were determined at 4.2 K. The μ_s at 5 K of the stoichiometric Ni₂MnGa is estimated to be about 4 μ_B/f.u. by extrapolations of the μ_s versusx curve for Ni₂Mn_{1 − x}Cu_xGa (0 ≤ x ≤ 0.40) alloys to x = 0 and of the μ_sversusy curve for Ni₂MnGa_{1 − y}Cu_y (0 ≤ y ≤ 0.25) alloys to y = 0. Recently, Ahuja et al. carried out a magnetic Compton scattering study of the near-stoichiometric Heusler alloy Ni_2.03Mn_0.97Ga [13]. For Ni_2.03Mn_0.97Ga, they found the value of μ_s to be 4.01 μ_B/f.u. at 110 K in a field of 2 T. The value of μ_s at 5 K for Ni₂MnGa in the present study is in good agreement with the value reported by Ahuja et al. [13].

Recently, Li et al. investigated theoretically the site preference and elastic properties of Fe-, Co- and Cu-doped Ni₂MnGa alloys by using the first-principles exact muffin-tin orbital method in combination with the coherent-potential approximation [14]. According to the results of the calculation by Li et al. [14], Cu atoms for Ni₂Mn_0.95Cu_0.05Ga occupy the vacant Mn sublattice and the magnetic moments of the Ni, Mn, Cu and Ga atoms, μ_Ni, μ_Mn, μ_Cu and μ_Ga, in Ni₂Mn_0.95Cu_0.05Ga are 0.32 μ_B, 3.37 μ_B, −0.03 μ_B and −0.05 μ_B, respectively. The values of μ_Ni, μ_Mn and μ_Ga calculated by Li et al. [14] are in good agreement with those reported earlier for the stoichiometric Heusler alloy Ni₂MnGa [15,16,17,18,19,20,21,22]. In order to explain the observed concentration dependence of the magnetic moment for Ni₂Mn_{1 − x}Cu_xGa (0 ≤ x ≤ 0.40) alloys we present a simple model in which the following assumptions are made. Cu atoms for Ni₂Mn_{1 − x}Cu_xGa (0 ≤ x ≤ 0.40) alloys always occupy the vacant Mn sublattice. The magnetic moments of the Ni, Mn, Cu and Ga atoms in Ni₂Mn_{1 − x}Cu_xGa (0 ≤ x ≤ 0.40) alloys are collinear and have Li et al.’s values independent of x. The magnetic moments of the Mn atoms are ferromagnetically coupled to the magnetic moments of the Ni atoms. Then, the total magnetic moment per formula unit, μ_s (cal), of Ni₂Mn_{1 − x}Cu_xGa (0 ≤ x ≤ 0.40) alloys is given by μ_s(cal) = 2μ_Ni + (1 − x)μ_Mn + xμ_Cu + μ_Ga. The solid line in Figure 5 is the one calculated by using this equation. As seen in Figure 5, the experimental values are in good agreement with those calculated.

Next, we consider the concentration dependence of μ_s for Ni₂MnGa_{1 − y}Cu_y (0 ≤ y ≤ 0.25) alloys. According to the results of the calculation for the site occupation of Ni₂MnGa_0.95Cu_0.05 by Li et al. [14], Cu atoms always occupy the sublattice of the deficient component; the configuration Ni₂Mn(Ga_0.95Cu_0.05) is the most stable in where the components A and B in (A,B) occupy the same sublattice. In this case, the μ_s at 5 K for Ni₂MnGa_{1 − y}Cu_y (0 ≤ y ≤ 0.25) alloys is almost independent of y under the assumption that Ni₂MnGa_{1 − y}Cu_y (0 ≤ y ≤ 0.25) alloys are collinear ferromagnets and the values of μ_Ni and μ_Mn are independent of x. However, as shown in Figure 5, the experimental μ_s at 5 K for Ni₂MnGa_{1 − y}Cu_y (0 ≤ y ≤ 0.25) alloys decreases steeply with increasing y. We, therefore, suggest a different site-occupation configuration i.e., Ni₂(Mn_{1 − y}Cu_y)(Ga_{1 − y}Mn_y) with the Cu atoms occupying the Mn sublattice, for Ni₂MnGa_{1 − y}Cu_y (0 ≤ y ≤ 0.25) alloys, where some of the Mn atoms move on to the Ga sublattice. Here, we assume that the magnetic moment of the Mn atoms substituted on to the Ga sites in Ni₂(Mn_{1 − y}Cu_y)(Ga_{1 − y}Mn_y) alloys is antiferomagnetically coupled to the magnetic moment of the Mn atoms on the Mn sublattice. The values of the magnetic moments of the Mn atoms on the Mn and Ga sublattices, respectively, are assumed to be 3.37 μ_B and −3.43 μ_B, which remain constant with increasing y from y = 0. The other magnetic moments μ_Ni, μ_Cu and μ_Ga in the Ni₂(Mn_{1 − y}Cu_y)(Ga_{1 − y}Mn_y) alloys are the constant values 0.32 μ_B, −0.03 μ_B and −0.05 μ_B, respectively, as in Ni₂Mn_{1 − x}Cu_xGa (0 ≤ x ≤ 0.40) alloys. A value of −3.43 μ_B was calculated by Li et al. [14] for the magnetic moment of the Mn atoms on the Ga sublattice. The antiferromagnetic coupling between nearest-neighbor Mn atoms in Ni₂(Mn_{1 − y}Cu_y)(Ga_{1 − y}Mn_y) alloys is due to the variation of the exchange interaction that becomes antiferromagnetic for small Mn-Mn interatomic distances. This antiferromagnetic coupling was already proved experimentally and theoretically in many Mn-rich Ni-Mn-Ga Heusler alloys [14,23,24,25,26,27]. Then, the μ_s of Ni₂(Mn_{1 − y}Cu_y)(Ga_{1 − y}Mn_y) alloys is given by μ_s(cal) = 2μ_Ni + (1 − y)μ_MnI + yμ_Cu + (1 − y)μ_Ga + yμ_MnII, where μ_MnI and μ_MnII mean the values of the magnetic moment of the Mn atoms on the Mn sublattice and the Ga sublattice, respectively. The broken line in Figure 5 is the one calculated by using the above equation. As seen in Figure 5, the experimental values are in agreement with those calculated for low y concentrations. For samples with high y concentrations, however, the experimental values of μ_s deviate from the broken line in Figure 5. This may be attributed to any concentration dependence of the μ_MnI, μ_MnII and μ_Ni values or occurrence of disorder of the constituent elements associated with the increase of y.

In the above considerations on Ni₂MnGa_{1 − y}Cu_y (0 ≤ y ≤ 0.25) alloys, we excluded the site-occupation configurations of (Ni_{2 − y}Cu_y)(Mn_{1 − y}Ni_y)(Ga_{1 − y}Mn_y) and (Ni_{2 − y}Mn_y)(Mn_{1 − y}Cu_y)(Ga_{1 − y}Ni_y) by taking into account the theoretical result that the formation energies of the site-occupations in Ni₂Mn(Ga_0.95Cu_0.05) and Ni₂(Mn_0.95Cu_0.05)(Ga_0.95Mn_0.05) are much smaller than those of (Ni_1.95Cu_0.05)(Mn_0.95Ni_0.05)(Ga_0.95Mn_0.05) and (Ni_1.95Mn_0.05)(Mn_0.95Cu_0.05)(Ga_0.95Ni_0.05) [14]. Unfortunately, the calculations in [14] were made for the site occupation of Cu-doped Ni₂MnGa with the L2₁-type structure, instead of the one with the observed martensitic structure at 5 K. Nevertheless, the above satisfactory agreements between the experimental and calculated concentration dependence of the magnetic moment μ_s may rather confirm that the site-occupation configurations in the present analyses exist also in the martensite phase. Lastly, we consider the concentration dependence of μ_s for Ni_{2 − z}Cu_zMnGa (0 ≤ z ≤ 0.40) alloys. As shown in Figure 5, the experimental values of μ_s for Ni_{2 − z}Cu_zMnGa (0 ≤ z ≤ 0.40) alloys are almost independent of concentration z [12]. We assume that Cu atoms for Ni_{2 − z}Cu_zMnGa (0 ≤ z ≤ 0.40) alloys occupy the vacant Ni sublattice according to the results of the calculation by Li et al. [14]. Furthermore, we assume that μ_Ni, μ_Cu, μ_Mn and μ_Ga in Ni_{2 − z}Cu_zMnGa (0 ≤ z ≤ 0.40) alloys, respectively, are the constant values 0.32 μ_B, 0.04 μ_B, 3.37 μ_B and −0.05 μ_B, which are the magnetic moments calculated by Li et al. [14] for Ni_1.95Cu_0.05MnGa. Then, the concentration dependence of μ_s(cal) for FSMAs Ni_{2 − z}Cu_zMnGa (0 ≤ z ≤ 0.40) alloys is given by μ_s(cal) = (2 − z)μ_Ni + zμ_Cu + μ_Mn + μ_Ga, which is shown by the dot dash line in Figure 5. As seen in Figure 5, the experimental values are in good agreement with those calculated.

The magnetization measurements at 5 K of FSMAs Ni₂Mn_{1 − x}Cu_xGa (0 ≤ x ≤ 0.40) and Ni₂MnGa_{1 − y}Cu_y (0 ≤ y ≤ 0.25) alloys have been carried out. The μ_s at 5 K for Ni₂Mn_{1 − x}Cu_xGa (0 ≤ x ≤ 0.40) alloys decreases linearly with increasing concentration x. On the other hand, the μ_s at 5 K for Ni₂MnGa_{1 − y}Cu_y (0 ≤ y ≤ 0.25) alloys decreases steeply with increasing y compared to the μ_s for Ni₂Mn_{1 − x}Cu_xGa (0 ≤ x ≤ 0.40) alloys. To explain the concentration dependence of μ_s for the Cu-modified Ni₂MnGa alloys, we suggested the site-occupation configurations, Ni₂(Mn_{1 − x}Cu_x)Ga for Ni₂Mn_{1 − x}Cu_xGa (0 ≤ x ≤ 0.40), Ni₂(Mn_{1 − y}Cu_y)(Ga_{1 − y}Mn_y) for Ni₂MnGa_{1 − y}Cu_y (0 ≤ y ≤ 0.25) and (Ni_{2 − z}Cu_z)MnGa for Ni_{2 − z}Cu_zMnGa (0 ≤ z ≤ 0.40) alloys. These configurations together with some theoretical values of the magnetic moments of constituent atoms were proved to explain well the concentration dependence of μ_s for Cu-modified Ni₂MnGa.