Magnetic Properties

decrease of thec/a-ratio and a change to a(3¯2)₂-structure. Finally, a plane shift-ing can rearrange the stackshift-ing sequence and period to a nanotwinned(5¯2)₂ 7M adaptive structure [82].

2.2. Magnetic Properties

Stoichiometric Ni2MnGa undergoes a phase transition from a paramagnetic to a ferromagnetic order at a Curie temperature ofT_C = 376K, which only slightly changes for off-stoichiometric compounds [12]. The magnetic moment is around4.17µ_B. Numerous experimental and theoretical studies have shown, that it is largely confined to the Mn sites (µ_Mn ≈3.4−3.7µ_B) and only a small amount is associated with Ni (µ_Ni ≈ 0.16− 0.4µ_B) and Ga (µ_Ga ≈ −0.04−

−0.13µ_B) atoms [12, 86, 87]. The origin of the ferromagnetic order in Ni-Mn-Ga alloys and their relevant magnetic properties are discussed in the following sections.

2.2.1. Origin of Magnetism

In general, the ferromagnetic ordering is a consequence of the magnetic inter-action, mainly between Mn atoms [87] The interaction is strongly dependent on the Mn-Mn distance, which in the case of an austenitic cubic unit cell of Ni2MnGa is≈4.12Å (see Fig. 2.1) [88]. This dependency was demonstrated by Kanomataet al., who have shown that the pressure derivative of the Curie temperaturedT_C/dpis positive for a series of Ni2MnZ alloys⁶(see Fig. 2.9) [88].

This behavior demonstrates an increasing ferromagnetic exchange interaction for decreasing Mn-Mn distance.

For a stoichiometrically ordered Ni2MnGa alloy Mn atoms are not next neigh-bors in the crystallographic unit cell and are largely separated (≈4Å). Due to this large distance it is very unlikely that a direct exchange interaction is responsible for the magnetic interaction between Mn atoms [87]. An exchange interaction mediated by Ni and Ga atoms between the Mn atoms is a more

6 (Z=Al, Ga, In, Sn and Sb)

a) b)

Figure 2.9.: (a) The dependance of the Curie temperature TC and marten-sitic transformation temperature Tt on hydrostatic pressure for Ni2MnGa.

(b)Experimental Curie temperatures as a function of the Mn-Mn distance for L21- and C1b-type alloys in which the main carriers of the magnetic moment are the Mn atoms. The upward arrows attached to the marks express the rise of the Curie temperature with increasing pressure (Images from [88]).

likely scenario. Magnetic interactions mediated by conduction electrons (like Nidand Gapstates for the case of Ni2MnGa), characterized by an effective ex-change parameter having an oscillatory behavior, have long been known as the Rudermann-Kittel-Kasuya-Yosida (RKKY) interaction [87]. The RKKY interac-tion is caused by the polarizainterac-tion of free electrons, which in turn arises from the presence of a magnetic impurity (Mn atom). When the polarization reaches other impurities it results in a magnetic interaction between them. Treating the localized Mndelectrons as a periodic array of ’magnetic impurities’ and the Ni dand Gapelectrons as mediating conduction electrons, a magnetic interaction of RKKY type can be expected in Ni2MnGa [87]. However, as pointed out by Himmetogluet al., the situation in an alloy is more complicated. Especially the conduction electrons that mediate the magnetic interactions are not free and therefore the strength of the interaction depends on the nontrivial topology of the Fermi surface, which is dominated by Ni states in both spin channels (see

2.2. Magnetic Properties section 2.3) [87, 89].

The saturation magnetization of austenite is lower than that of martensite [12, 14, 22]. This can be attributed to changing interatomic distances in the distorted unit cell accompanied by changing magnetic interactions between the Mn atoms. Also the redistribution of electronic charges, due to the struc-tural transition, affects the absolute magnitude of atomic moments [22]. Since the main contribution to the magnetic moment is given by the Mn atoms, the variation of Mn content in off-stoichiometric alloys has a high impact on the sat-uration magnetization. Consequently it decreases with decreasing Mn content.

Somehow counterintuitive is the behavior for Mn-rich alloys. The saturation magnetization decreases with increasing Mn content for alloys with a Mn con-tent>25%. This is ascribed to an antiferromagnetic ordering of the excess Mn atoms with respect to the original Mn atoms [90]. This interaction occurs since the extra Mn atoms residing on the Ni- or Ga sites of the unit cell are nearest neighbors to the original Mn atoms instead of being separated by Ni and Ga atoms (see Fig. 2.1). As a consequence the saturation magnetization has its maximum for a stoichiometric composition with a Mn content of25%.

The study of magnetic interactions in Ni-Mn-Ga and related Heusler alloys is important not only for the knowledge of the magnetic properties of the sys-tem. Recent ab initio calculations have demonstrated that competing ferro- and antiferromagnetic interactions govern the complex magnetic behavior and ex-change parameters of Mn-Ni and Mn-Ga interactions determine the magnetic energy as a function of the tetragonalityc/a[82, 87]. Due to a strong magne-toelastic interaction and a pronounced correlation between the magnetic and electronic properties in these materials a precise account of magnetism and magnetic interactions is essential to predict the relative stabilities of different phases.

2.2.2. Anisotropy

The existence of the MSM effect, i.e. the reorientation of the martensitic twin variants, is based on a high magnetocrystalline anisotropy of the low-symmetry martensitic phase. The magnetocrystalline anisotropy of the cubic austenite is

very low of order of10³Jm⁻³ [91]. After the transformation to martensite, the magnetocrystalline anisotropy increases by two orders of magnitude. It can be determined by measuring the field-dependent magnetization in different crystallographic directions of a martensitic bulk sample which is in a single variant state. The anisotropy constant can be determined as the area between these curves. Fig. 2.10 displays the magnetization curves measured for 5M, 7M and NM martensitic phases [20]. These measurements indicate a strong

Figure 2.10.: Magnetization curves for 5M(a), 7M(b)and NM(c)martensite of Ni-Mn-Ga. The measurements were performed at room temperature for differ-ent crystallographic directions of single crystals being in a single variant state.

(Image from [20]).

anisotropy for the 5M phase with the magnetic easy axis oriented along the short c-axis. In the case of orthorhombic 7M martensite, the short c-axis is the magnetic easy axis, the crystallographic a-axis is the hard axis and the b-axis is the magnetic mid-hard b-axis. For the tetragonal NM martensite there exists an easy a- plane and a hard c-axis, which is the longest crystallographic parameter of the NM unit cell. The calculated anisotropy constants for the three martensitic phases are summarized in Table 2.2.

Temperature behavior of the magnetic anisotropy has an important effect on the magnetic shape memory effect. The temperature dependance of an anisotropy constant Ku for the case of an uniaxial anisotropy can be conve-niently described by the relation:

K_u(T)

M_S(T)³ = K_u(0)

M_S(0)³ ∝const., (2.6)

2.2. Magnetic Properties

Type of 5M 7M NM

martensite Tetragonal Orthorhombic Tetragonal

K1 (10⁵Jm⁻³) 1.65 1.7 -2.3

K2 (10⁵Jm⁻³) <0.07 0.9 0.55

Table 2.2.: Anisotropy parameters for the different martensitic phases of Ni-Mn-Ga obtained at room temperature (Values from [91]).

where M_S(T) is the saturation magnetization at temperature T [22]. There-fore with decreasing temperatureK_u saturates and rapidly decreases when approaching the Curie temperatureT_C. This behavior is shown in Fig. 2.11 for the 5M and 7M martensites. This characteristic of the magnetic anisotropy can

a) b)

Figure 2.11.: Temperature dependance of the saturation magnetization MSand the anisotropy constants K1and K2of five-layered tetragonal 5M(a)and seven-layered orthorhombic 7M(b)martensite. Dotted vertical lines mark the phase transition temperatures. (Image from [91]).

have a large impact on high-temperature applications of the MSM effect. For MSM alloys which have martensitic structural transformation temperature in the vicinity or higher than the Curie temperature, the rapid decrease ofK_ucan drastically lower the temperature limit of the MSM effect.

The microscopic origin of the magnetocrystalline anisotropy was studied within the density-functional theory and by means of x-ray circular dichroism

[92, 93]. Calculated orbital moment anisotropies have shown that the magnetic anisotropy energy results from Ni and Mn atoms, with80% of the anisotropy energy coming from Ni atoms. This results is remarkable, because Ni atoms contribute only about10% to the magnetic moment in Ni2MnGa, as outlined in the beginning of this section [92]. Experimental studies have confirmed the importance of Ni atoms for the magnetocrystalline anisotropy [93]. However, Klaeret al.argue for a simple model of magnetic anisotropy in Ni2MnGa based on the anisotropy of atomic orbitals in a tetragonally distorted system in com-bination with spin-orbit coupling. Their results illustrate that the bulk mag-netocrystalline anisotropy in Ni2MnGa is caused by a reallocation of electron states with3d_z² symmetry located predominantly at the Ni atom [93].

2.3. Electronic Properties and instability of