## 1 Introduction  
With the development of nanotechnology and computational materials science, spintronics has developed rapidly in the past 30 years. In order to improve the performance of spin diodes, spin valves, and spin filters, the design of high spin-polarized materials has attracted much attention [1; 2; 3; 4]. For half-metallic ferromagnetic compounds with unique electronic structures, one of the two spin channels is semiconducting and the other is metallic. As promising spintronic candidates, they exhibit a complete spin polarization of carriers near the Fermi level [5; 6; 7; 8; 9; 10; 11; 12]. A spin-gapless semiconductor is another new kind of spintronic material, which has an almost complete spin polarization and good compatibility with the existing semiconductor industry. By shifting the Fermi energy at the finite gate voltage, the spin-polarized transport properties of spin-gapless semiconductors can be tuned, which has great prospects for future spintronic applications [13; 14]. Half metals and spin-gapless semiconductors are special kinds of materials. Because of their unique electronic structures, they have novel spin-dependent electronic properties. The band structure of these materials has only one energy gap for a specific spin direction, which means that the band gap disappears in the opposite spin direction. This phenomenon leads to the application prospects of high carrier spin polarization and spin-controlled electrical and magnetic features. In recent years, in order to develop new technologies involving spintronics, it has been important to search for new materials with these characteristics. Many of these materials are Heusler alloys with a specific crystalline ordering, which is very important for the unique electronic and magnetic properties of these materials. Therefore, it is crucial to study these alloys and compounds with these novel physical properties. Heusler alloys are named for the German mining engineer and chemist [PERSON], who investigated Cu-Mn-Al alloys around the year 1900. Heusler compounds are ternary intermetallic compounds, which have been known since 1903 [15]. These alloys were interesting because some of them were ferromagnetic, even though their constituent atoms were non-ferromagnetic as elements. At that time, Cu\({}_{2}\)MnAl compounds were proven to be ferromagnetic, although none of the three elements were ferromagnetic. The structure of the alloys was later found to be face-centered cubic (fcc) with a four-atom basis consisting of a single formula unit. Thus, members of this Heusler family with a formula unit A\({}_{2}\)BC (known as \"full\" Heuslers) can be viewed as layers consisting of a square lattice of \"A\" atoms alternating with layers of \"B\" and \"C\" atoms, as shown in Figure 1a. A related Heusler family has members with the formula unit ABC, in which half of the \"A\" atoms of the full Heusler are replaced by vacancies, as shown in Figure 1b.These are known as \"half\" Heuslers or semi-Heuslers. A third Heusler family with the formula unit A\({}_{2}\)BC (similar to the \"full\" Heuslers) consists of alloys that can be viewed as \"AB\" layers alternating with \"AC\" layers, as shown in Figure 1c. These are known as \"inverse\" Heuslers. The properties of Heusler compounds can be altered by element substitution, but no single set of properties can characterize the entire Heusler family. Heusler alloys are expected to play an important role in spintronics, magneto-optical reading-recording devices, magnetic tunnel junctions, or tunneling magnetoresistance devices, due to their various multifunctional magnetic properties [16; 17]. In the past years, several inverse binary- and ternary-Heusler alloys with a high Curie temperature, as well as quaternary-Heusler alloys, were theoretically predicated to exhibit spin-gapless semiconducting electronic structures [18; 19]. Moreover, the bulk and thin films of Mn\({}_{2}\)CoAl and Ti\({}_{2}\)MnAl films were experimentally fabricated [20; 21; 22; 23].