## 3 Results and Discussion  
Heusler alloys crystallize in a highly-ordered cubic structure, and have a stoichiometric composition of X\({}_{2}\)YZ, where X and Y are transition-metal elements, and \(Z\) is a main group element. Generally, the Heusler structure can be considered as four interpenetrating f.c.c lattices along the space diagonal, in which the transition metal atoms occupy the A (0, 0, 0), B (0.25, 0.25, 0.25) and C (0.5, 0.5, 0.5) Wyckoff positions, respectively. The main group element occupies the D (0.75, 0.75, 0.75) position. The tetragonal Heusler alloys can be considered as tetragonal distortions of the cubic phase along the z direction, and the c/a ratio can be used to quantify the amount of tetragonal distortion [9; 31; 32]. For tetragonal Mn\({}_{3}\)Z (Z = Al, Ga, In Tl, Si, Ge, Sn, Pb) alloys, the Mn(A), Mn(B), Mn(C) and \(Z\) atoms occupy the (0, 0, 0), (0.25, 0.25, 0.25), (0.5, 0.5, 0.5) and (0.75, 0.75) Wyckoff positions, respectively. As a typical example, we first present the results of Mn\({}_{3}\)Ga alloy. Figure 1 shows the total energy as a function of c/a (\(\Delta\)E\({}_{\text{total}}\)-c/a curve) for Mn\({}_{3}\)Ga alloy. The total energy of the cubic phase is set as the zero point. The lattice constant of the cubic phase is obtained by minimizing the total energy and is 5.66 A. The unit cell volume is the same as that of cubic phase, and is fixed when the tetragonal distortion is considered. From Figure 1, it can be seen that there are two local energy minima on the \(\Delta\)E\({}_{\text{total}}\)-c/a curve, i.e., a shallow one is at c/a = 1.35 and a deeper one at c/a = 1. The latter is energetically favorable. Between the two energy minima, there is an energy barrier at c/a = 1.15. The lower and upper insets show the corresponding crystal structures, band structures, and densities of states (DOS) for the cubic (c/a = 1) and distorted (c/a = 1.35) cases. From the band structures, we can see that the cubic phase (c/a = 1) of Mn\({}_{3}\)Ga is close to the half-metal, with a high degree of spin polarization. But in the tetragonal phase (c/a = 1.35), the band structure is completely different from the cubic one, and it can also be seen that the spin polarization declines rapidly at the Fermi level from the density of states patterns. This is mainly due to the fact that cubic symmetry is reduced after the tetragonal distortion. Comparing the work reported by [PERSON] et al. [31] with that by [PERSON] et al. [33], it can be noted that the difference of volume can have a large impact on the \(\Delta\)E\({}_{\text{total}}\)-c/a curves. Therefore, we performed a series of investigations on the tetragonal distortion with different volumes to further understand the relation between the volume and the \(\Delta\)E\({}_{\text{total}}\)-c/a curves for Mn\({}_{3}\)Ga alloy. In Figure 2a, we show the \(\Delta\)E\({}_{\text{total}}\)-c/a curves of Mn\({}_{3}\)Ga alloy with different volumes (v = 17.0 nm\({}^{3}\), 18.0 nm\({}^{3}\), 19.0 nm\({}^{3}\), 20.0 nm\({}^{3}\), 21.0 nm\({}^{3}\), 22.0 nm\({}^{3}\)), which correspond to the different lattice constants in the cubic phase. It can be seen that the shape of the \(\Delta\)E\({}_{\text{total}}\)-c/a curves varies with the change of the volume. There are two local energy minima in the \(\Delta\)E\({}_{\text{total}}\)-c/a curves for all the Mn\({}_{3}\)Ga alloys with different volumes. For V = 17.0 nm\({}^{3}\) and 18.0 nm\({}^{3}\), the total energy of cubic phase is lower than that of the tetragonal phase, which indicates that the cubic phase is more stable than the tetragonal phase. As the volume expands to higher level than 19.0 nm\({}^{3}\), the total energy of the tetragonal phase becomes lower than the cubic phase. At same time, the energy barrier from the cubic phase to the tetragonal phase gradually decreases with the increasing volume, and finally, disappears at V = 21.0 nm\({}^{3}\).