**2. Crystal structures and magnetic properties of full-Heusler compounds**

The physical properties of full-Heusler compounds depend on their crystal structures. As shown in **Figure 2**, there are several types of crystal structures with different order degrees [4–6]. The full-Heusler compounds have four interpenetrating fcc sublattices, and each sublattice consists of the *X*, *X'*, *Y* or *Z* atom. The *X*, *X'* and *Y* atoms are transition metals, whereas *Z* is a main group element. In some cases, the *Y* atom is a rare earth element or an alkaline earth metal.

When the *X* and *X'* atoms are of the same element, the chemical composition of the compounds is written as *X*2*YZ*, which generally crystallises in the L21 structure. The prototype of the L21 structure is Cu2MnAl (space group: *Fm*3*m*). The Cu atoms occupy the 8*c* (1/4 1/4 1/4) site, whereas the Mn and Al atoms occupy the 4*b* (1/2 1/ 2 1/2) and 4*a* (0 0 0) sites, respectively. The L21 structure is a highly ordered structure of the full-Heusler compounds. Disorder among the Cu, Mn and/or Al atoms, that is, antisite defects, gives rise to different crystal structures. In a case where the Mn and Al atoms are evenly located at the 4*b* and 4*a* sites, the Cu2MnAl becomes the B2 structure. Its prototype is CsCl (space group: *Pm*3*m*). In a fully

*Magnetic Full-Heusler Compounds for Thermoelectric Applications DOI: http://dx.doi.org/10.5772/intechopen.92867*

#### **Figure 2.**

coefficient, *σ*: electrical conductivity, *κ*: thermal conductivity) of the individual TE

*zT* <sup>¼</sup> *<sup>S</sup>*<sup>2</sup> *σ κ*

(PF), which is a measure of electric power generated using the TE material. To achieve high TE efficiency (standard levels for practical use are *zT* > 1 and

robust compared to that of semiconductors, which is also an advantage.

compounds are introduced. Sections 3 and 4 demonstrate that magnetic full-Heusler compounds are promising for the TE power generation device.

**2. Crystal structures and magnetic properties of full-Heusler**

requirements, a variety of TE materials have been explored, such as chalcogenides, skutterudites, clathrates, silicides, Zintl compounds, half-Heusler compounds and oxides [1–3]. Most of these materials are semiconductors because in general they have high *S* than metals. However, recent theoretical and experimental studies have revealed that metals, in particular, half-metallic full-Heusler compounds have relatively high *S* as well as high *σ*. In addition, their junction with a metal electrode is

In Section 2, the crystal structures and magnetic properties of full-Heusler

The physical properties of full-Heusler compounds depend on their crystal structures. As shown in **Figure 2**, there are several types of crystal structures with different order degrees [4–6]. The full-Heusler compounds have four interpenetrating fcc sublattices, and each sublattice consists of the *X*, *X'*, *Y* or *Z* atom. The *X*, *X'* and *Y* atoms are transition metals, whereas *Z* is a main group element. In some cases, the *Y* atom is a rare earth element or an alkaline earth metal. When the *X* and *X'* atoms are of the same element, the chemical composition of the compounds is written as *X*2*YZ*, which generally crystallises in the L21 structure. The prototype of the L21 structure is Cu2MnAl (space group: *Fm*3*m*). The Cu atoms occupy the 8*c* (1/4 1/4 1/4) site, whereas the Mn and Al atoms occupy the 4*b* (1/2 1/ 2 1/2) and 4*a* (0 0 0) sites, respectively. The L21 structure is a highly ordered structure of the full-Heusler compounds. Disorder among the Cu, Mn and/or Al atoms, that is, antisite defects, gives rise to different crystal structures. In a case where the Mn and Al atoms are evenly located at the 4*b* and 4*a* sites, the Cu2MnAl becomes the B2 structure. Its prototype is CsCl (space group: *Pm*3*m*). In a fully

m), high *S*, high *σ* and low *κ* are required. To meet these

where *T* is the absolute temperature. The product *S*<sup>2</sup>

*Schematic figure of a thermoelectric (TE) power generation device.*

*Magnetic Materials and Magnetic Levitation*

*T*, (2)

*σ* is called the power factor

materials in the device.

**Figure 1.**

PF <sup>&</sup>gt; <sup>2</sup> � <sup>10</sup>�<sup>3</sup> W/K<sup>2</sup>

**compounds**

**66**

*Crystal structures of full-Heusler compounds. The Strukturbericht symbol and a prototype structure are written above and below each crystal structure, respectively.*

disordered phase, all the atoms are randomly distributed in the 8*c*, 4*b* and 4*a* sites, thus resulting in the A2 structure. In such a structure, all the sites are equivalent, which are expressed as a bcc lattice (prototype: W, space group: *Im*3*m*). There are other disordered phases, including the DO3 and B32a structures. The former is caused by the random distribution of the *X*, *X'* and *Y* atoms at the 8*c* and 4*b* sites (prototype: BiF3, space group: *Fm*3*m*). In the B32a structure, the 8*a* (0 0 0) and 8*b* (1/2 1/2 1/2) sites are occupied by the *X*/*Y* and *X'*/*Z* atoms, respectively. The prototype is NaTl (space group: *Fd*3*m*).

When the *X'* and *Y* atoms are of the same element, the chemical composition becomes *XX'*2*Z*, which crystallises in the X (XA or Xa) structure. This structure is called the inverse Heusler phase. The prototype is CuHg2Ti (or AgLi2Sb), and the space group is *F*43 *m*. In the structure, the *X* and *Z* atoms occupy the 4*d* (3/4 3/4 3/4) and 4*a* (0 0 0) sites, respectively, and the *X'* atoms occupy the 4*b* (1/2 1/2 1/2) and 4*c* (1/4 1/4 1/4) sites.

In addition to the above ternary full-Heusler compounds, there are quaternary full-Heusler compounds, *XX'YZ*, which crystallise in the *Y* structure (prototype: LiMgPdSn or LiMgPdSb, space group: *F*43 *m*). The *X*, *X'*, *Y* and *Z* atoms are situated at the 4*d*, 4*b*, 4*c* and 4*a* sites, respectively, occupying one of the fcc sublattices. It should be noted that the inverse Heusler and the quaternary full-Heusler phases are ordered phases, and any disorder among the constituent atoms causes a structural change; the structure changes to the B2, A2, DO3 or B32a structure.

Earlier theoretical studies demonstrated a half-metallic nature in full-Heusler compounds [7, 8]. Since then, many studies have been dedicated to investigate the electronic and magnetic properties of ternary and quaternary full-Heusler compounds. It has been revealed that full-Heusler compounds exhibit a variety of electronic properties; they exhibit the properties of semiconductors [9–18], spingapless semiconductors (SGSs) [19–26], semimetals [27–29], metals [30–34] and half-metals (HMs) [32, 35–78]. Considering the magnetic properties, they have been reported to exhibit nonmagnetism [9–11, 14–18], ferromagnetism [12, 19– 24, 30–33, 36–46, 48–58, 61–66, 68–78], ferrimagnetism [13, 30, 35, 47, 59, 60, 67] and antiferromagnetism [25, 26, 34]. The full-Heusler, inverse Heusler and quaternary Heusler compounds obey the Slater-Pauling rule [79–81]: the total spin

magnetic moment per unit cell scales with the total number of valence electrons in the unit cell.
