## 1 Introduction  
The magnetic tunnel junction (MTJ) usually has a large tunnel magnetoresistance (TMR) value and has become a key component of many advanced magnetic devices, such as the read heads in hard-disk drives [1; 2; 3; 4], magnetoresistive random access memories [5], and the next generation of high-density, nonvolatile memories and logic devices [6; 7; 8; 9; 10] The core component of MTJ is a \"sandwich\" stack usually consisting of two ferromagnetic layers and a seeding layer. The direction of the two magnetizations of the ferromagnetic layers can be switched individually by an external magnetic field. If the magnetizations are in a parallel orientation, the electrons tend to tunnel through the seeding layer more than if they are in the antiparallel orientation. Consequently, such a MTJ can be switched between two states of electrical resistance--one with low and one with very high resistance. In MTJ, the TMR value is the most important parameter, which can be defined by the Julliere formula with the spin polarizations of the ferromagnetic electrodes as follows [11]:  
\[\text{TMR}=2P_{1}P_{2}/(1-P_{1}P_{2}), \tag{1}\]  
where \(P_{1}\) and \(P_{2}\) denote the spin-polarized values of two ferromagnetic layers. The \(P\) value can be defined as follows:  
\[P=(N_{\uparrow}-N_{\downarrow})/(N_{\uparrow}+N_{\downarrow}), \tag{2}\]  
where \(N_{\uparrow}\) and \(N_{\downarrow}\) are the spin-up and spin-down state densities at the Fermi level, respectively. Obviously, the key technologies for MTJ applications aim to increase the spin polarization of ferromagnetic films in MTJ devices. However, achieving such aim is a very complicated task. Atomic disordering [12; 13], surface states [14], interface states [15], and thermal effects [16; 17; 18] are the fatal reasons for depolarizing ferromagnetic films. As a major breakthrough in this field, the fabrication of epitaxial Heusler MTJ can exploit coherent electronic tunneling to produce a large TMR, even at room temperature. In 2004, [PERSON] and co-workers obtained TMR values of up to 220% at room temperature and 300% at low temperatures for CoFeB/MgO/CoFeB MTJ-oriented (100) MgO tunnel barriers and CoFe electrodes [19]. In 2011, a TMR value of about 600% was reported for Co\({}_{2}\)MnAlSi/MgO/CoFe MTJ [20]. The current record for the highest Heusler TMR observed experimentally is held by Co\({}_{2}\)MnSi/MgO/Co\({}_{2}\)MnSi, which produced a TMR ratio of 1995% at 4 K [21]. From the ab initio calculation, a theoretical TMR value of about 10\({}^{6}\) was detected in Co\({}_{2}\)CrSi/Cu\({}_{2}\)CrAl/Co\({}_{2}\)CrSi [22]. Recently, an extremely high TMR value exceeding 25,000% was detected in a binary all-Heusler stack Fe\({}_{3}\)Al/Bi\({}_{3}\)/Fe\({}_{3}\)Al at low bias [23]. Therefore, the MgO-based Heusler MTJ holds great potential for TMR device applications. Heusler alloys are a large family of binary (X\({}_{3}\)Z), ternary (X\({}_{2}\)YZ), and quaternary (X\({}_{1}\)X\({}_{2}\)YZ) compounds with more than 1500 known members and an impressive range of properties [24]. In these compounds, X and Y generally denote a transition metal, while Z denotes an sp-element or a main group element. In general, ternary Heusler alloys have two high-ordered structures, namely, the classic Cu\({}_{2}\)MnAl-type structure and the newly discovered Hg\({}_{2}\)CuTi-type structure. The conventional Cu\({}_{2}\)MnAl-type Heusler alloy X\({}_{2}\)YZ with FM-3M space group consists of four fcc sublattices. In Wyckoff coordinates, X is located at (0, 0, 0) and (0.5, 0.5, 0.5), Y is located at (0.25, 0.25, 0.25), and Z is located at (0.75, 0.75, 0.75). Each X atom has four Y and Z atoms as nearest neighbors, thereby presenting the same X atomic surrounding. Unlike the Cu\({}_{2}\)MnAl-type structure, the Hg\({}_{2}\)CuTi-type Heusler alloy, also known as the inverse Heusler alloy, has composition X\({}_{2}\)YZ bear F-43M space group.