#
First Principles Study on the Thermodynamic and Elastic Mechanical Stability of Mg_{2}X (X = Si,Ge) Intermetallics with (anti) Vacancy Point Defects

^{*}

## Abstract

**:**

_{2}X (X = Si, Ge) intermetallics with or without point defects are calculated. The results show that the difference in the atomic radius leads to the instability and distortion of crystal cells with point defects; Mg

_{2}X are easier to form vacancy defects than anti-site defects on the X (X = Si, Ge) lattice site, and form anti-site defects on the Mg lattice site. Generally, the point defect is more likely to appear at the Mg position than at the Si or Ge position. Among the four kinds of point defects, the anti-site defect ${x}_{Mg}$ is the easiest to form. The structure of intermetallics without defects is more stable than that with defects, and the structure of the intermetallics with point defects at the Mg position is more stable than that at the Si/Ge position. The anti-site and vacancy defects will reduce the material’s resistance to volume deformation shear strain, and positive elastic deformation, and increase the mechanical instability of the elastic deformation of the material. Compared with the anti-site point defect, the void point defect can lead to the mechanical instability of the transverse deformation of the material and improve the plasticity of the material. The research in this paper is helpful for the analysis of the mechanical stability of the elastic deformation of Mg

_{2}X (X = Si, Ge) intermetallics under the service condition that it is easy to produce vacancy and anti-site defects.

## 1. Introduction

_{2}X (X = Si, Ge) intermetallics have such advantages as high-temperature resistance, good corrosion resistance, high hardness, high conductivity, high-temperature electric potential and low thermal conductivity, etc.; and have important application prospects in the fields of metal matrix composite materials, structural materials, thermoelectric materials, battery materials, hydrogen storage devices, etc. [1,2,3]. At present, the theoretical research on it mainly focuses on structure stability, electronic structure, mechanical properties, and thermodynamic properties, and so on. According to the first principles and molecular dynamics, the structure, elasticity, and thermodynamic properties of Mg

_{2}Si were predicted in Reference [4,5,6]. The effects of strain, deformation and high pressure on the electronic, optical, and thermoelectric properties of Mg

_{2}Si have been calculated in reference [7,8,9,10]. Hirayama [11,12] theoretically studied the influence of doping on the structural parameters of Mg

_{2}Si. The lattice structure, mechanical properties, and phase transitions Mg

_{2}X (X = Si, Ge) have been studied in Reference [13,14,15,16,17]. Reference [18,19,20,21] studied the electronic structure of doping defects of Mg

_{2}X (X = Si, Ge and Sn) anti-fluorine compounds. References [22,23,24] used the first principle to calculate the interface structure of Mg

_{2}Si and related compounds. Kessair [25] studied the structural characteristics of seven different types of Mg

_{2}Si and found that the thermoelectric potential of cubic CaF

_{2}structure is very high, which may be the preferred material in thermoelectric applications. Vacancy point defects play a decisive role in the physical properties of thermoelectric materials and are used as a new way to improve thermoelectric properties. However, there are few studies on the structures of Mg

_{2}X (X = Si, Ge) with vacancy and anti-site defects.

_{2}X (X = Si, Ge) system with vacancy defect structure are calculated by Density Functional Theory (DFT). The effect of vacancy defect on the structure and properties of Mg

_{2}X (X = Si, Ge) intermetallics is explored from the microscopic point of view.

## 2. Computational Method

_{cut}was set to be 380 eV. The Brillouin zone integral is in the form of 8 × 4 × 8 monkhorst-pack [32], and the super soft pseudopotential of the reciprocal space is selected as the potential function [33].

^{−6}ev/atom, the force on each atom is less than 0.01 eV/Å, the stress deviation is lower than 0.02 GPa, the tolerance deviation is less than 5 × 10

^{−4}Å.; the maximum convergence tolerance of energy and self-consistent field (SCF) convergence threshold are 5.0 × 10

^{−6}eV/atom and 5.0 × 10

^{−7}eV/atom, respectively. Both Mg

_{2}Si and Mg

_{2}Ge belong to the face-centered cubic (FCC) structure of anti-fluorite (CaF

_{2}).

_{16}X

_{8}supercell crystals are shown in Figure 1. There are two vacancy defects V

_{mg}, V

_{x}, two anti-site defects Mg

_{X}, X

_{Mg}in the crystals, and the small red box indicates the location of the point defect.

## 3. Results and Discussion

#### 3.1. Equilibrium Lattice Stability of Intermetallics with or without Point Defects

_{2}X (X = Si, Ge) intermetallic compounds have been calculated and shown in Table 1.

_{16}Si

_{8}and Mg

_{16}Ge

_{8}with point defects are listed in Table 1, and Figure 2 is the comparison of lattice constants of Mg

_{2}X (X = Si, Ge) with point defects. It can be seen from Table 1 that the equilibrium lattice constants calculated in this paper are very close to the experimental values [34,35], which shows that the method adopted is reliable. It also can be seen from Table 1 that when the cell contains vacancy or anti vacancy point defects, lattice distortion occurs in the cell, resulting in cell expansion and a lattice constant increase. At the same time, as shown in Figure 3, the lattice constant of Mg

_{16}Ge

_{8}with vacancy and anti-site defects is larger than that of Mg

_{16}Si

_{8}, which is due to the following relationship between the relative atomic radius: R

_{Si}(0.118 nm) < R

_{Ge}(0.123 nm) < R

_{Mg}(0.160 nm).

#### 3.2. Effects of Vacancy and Anti-Site Points Defects on Thermodynamic Stability

_{16}X

_{8}(X = Si, Ge) intermetallics with point defects is [36]:

_{2}X intermetallic compound, N indicates the number of atoms in the crystal cell, and ${E}_{solid}^{A},{E}_{solid}^{B}$ represent the ground state energy of each atom in the solid cell, ${E}_{solid}^{Mg}=-973.95\mathrm{eV},{E}_{solid}^{Si}=-107.26\mathrm{eV},\text{}\mathrm{and}\text{}{E}_{solid}^{Ge}=-107.29\mathrm{eV}$.

_{2}X (X = Si, Ge) cells with or without point defects are listed in Table 3. The more negative $\Delta {H}_{f}$ of the crystal is, the easier it is to form. That is to say, the absolute ΔH

_{f}of the crystal is larger, it is not easy to be damaged, and it is easier to form.

_{2}X (X = Si, Ge) with point defects is less than that without point defects, which means that the structure with point defects is easy to be destroyed, and it is easy to be in an unstable state before formation, that is to say; it is difficult to form compounds (For example, it can forms a stable structure with point defects at 0 K and 0 GPa, and the formed structure may become unstable under the influence of external factors, which is not contradictory to the relationship between binding energy and crystal stability in the following section of this paper).

_{2}X is easier to form anti-site defects on the Mg lattice site.

_{2}Si intermetallics, the $\Delta {H}_{f}$ (−0.092 eV/atom) of vacancy point defects at Mg position is smaller than that at Si position (−0.084 eV/atom), the absolute value is larger, and it is not easy to be destroyed, so it is easier to form; The formation energy (-0.108 eV/atom) of the anti-site defect at Mg position is smaller than that at Si position (−0.072 eV/atom), the absolute value is larger, it is not easy to be damaged, and it is easier to form, which indicates that the point defect is more likely to appear at the Mg position than at the Si position; The same analysis shows that for Mg

_{2}Ge intermetallics, point defects are more likely to form at the Mg position than at the Ge position. From Table 3, we can also find that this conclusion is also applicable to Mg

_{64}X

_{32}(X = Si, Ge) intermetallics with 2 × 2 × 2 supercells.

_{2}X (X = Si, Ge), respectively; $n{}_{i}$ represent the number of atoms added or removed in Mg

_{2}X (X = Si, Ge) intermetallics, $\mu {}_{i}$ represent the corresponding chemical potential of the atom, i.e.; the defect formation energies of vacancy and anti-site in Mg

_{2}X (X = Si, Ge) are shown in Table 4.

_{2}X (X = Si, Ge), the $\Delta {E}^{vac}$ and $\Delta {E}^{anti}$ at the Si or Ge position is bigger than that at the Mg position, which indicates that the point defect is more likely to appear at Mg position than at Si or Ge position. Among the four kinds of point defects, the defect forming energy of the anti-site defect $\Delta {E}_{Mg}^{anti}$ is the smallest, which indicates that the anti-site defect ${x}_{Mg}$ is the easiest to form in Mg

_{2}X. Compared with other point defects, the $\Delta {E}_{Mg}^{anti}$ is negative, which means it can form spontaneously in 1 × 1 × 2 and 2 × 2 × 2 supercells. The defect formation of vacancy $\Delta {E}^{vac}$ and inversion $\Delta {E}^{anti}$ (except $M{g}_{Si}$) in Mg

_{2}X (X = Si, Ge) can decrease with the increase of supercell size, which shows that only the anti-site defect $M{g}_{Si}$ is easy to form in 1 × 1 × 2 supercells, while other point defects tend to form in 2 × 2 × 2 supercell.

#### 3.3. Effects of Vacancy and Anti-Site Points Defects on Elastic Mechanical Stability

#### 3.3.1. Elastic Constants of Stability Criteria

_{2}X (X = Si, Ge) cells with point defects belong to tetragonal structure, and Mg

_{2}X (X = Si, Ge) cells without point defects belong to cubic structure [40,41].

_{2}X (X = Si, Ge) cells with (without) point defects are listed in Table 5. It can be seen that our calculation results are close to others’ theoretical calculation results [42], indicating that the selection of the parameters is reasonable.

_{2}X (X = Si, Ge) intermetallics can be evaluated. Obviously, the elastic constants ${C}_{ij}$ calculated in Table 3 meet the mechanical stability criteria. Therefore, the Mg

_{2}X (X = Si, Ge) intermetallic compounds with and without vacancy or anti-vacancy point defects are all mechanically stable structures.

#### 3.3.2. Elastic Modulus of Mechanical Stability

_{2}X (X = Si, Ge) intermetallic compounds; it is found that for Mg

_{2}Si type intermetallics, the $B$, $G,$ and $E$ of Mg

_{2}X with Mg vacancy (V

_{Mg}) are higher than those of Mg

_{2}X with X vacancy (V

_{X}). The above results show that the vacancy defects in Mg position, compared with those in the Si position and Ge position, will lead to the stronger resistance to volume deformation, shear strain and elastic deformation of crystal materials, the greater the rigidity, the less deformation, the stronger the brittleness and the worse the plasticity.

_{2}X (X = Si, Ge) with the X

_{Mg}anti-site are higher than those of Mg

_{2}X with the Mg

_{X}anti-site, indicating that when the Si or Ge atom replace the Mg atom to form anti-vacancy defects, compared with Mg atom replacing Si or Ge atom to form anti vacancy defect, the material has a strong resistance to volume deformation, shear strain and elastic deformation, but poor plasticity. Besides, for Mg

_{2}X (X = Si, Ge) with same point defect structure, the $B$, $G,$ and E of 1 × 1 × 2 supercells are smaller than those of 2 × 2 × 2 supercells, which indicates that the Mg

_{2}X (X = Si, Ge) with point defects are easier to form in 2 × 2 × 2 supercells. Therefore, the point defects generated on the Mg position of 2 × 2 × 2 supercells can enhance the mechanical stability of the material, while the plasticity is poor.

**Mg**without point defects is the smallest, and the Poisson’s ratio $\nu $ with defects is larger (greater than 0.18), indicating that the formation of defects makes the transverse deformation of the material larger and the transverse elastic mechanical instability increased. It was also found that the Poisson’s ratio of Mg

_{16}X_{8}_{2}X (X = Si, Ge) with X vacancy is higher than that of Mg

_{2}X with the Mg

_{X}anti-site. Similarly, the Poisson’s ratio of Mg

_{2}X with Mg vacancy was slightly higher than that of Mg

_{2}X with X

_{Mg}anti-site. Thus, the above results show that when there are defects in the Mg, Si, and Ge position, the void point defects increase the instability of transverse deformation more than the anti-void point defects. Note that for Mg

_{2}X (X = Si, Ge) with the same point defect structure (except Mg

_{Si}), the Poisson’s ratio $\nu $ of the 2 × 2 × 2 supercells is smaller than that of the 1 × 1 × 2 supercells, indicating that the point defects in larger supercells can only increase the instability of transverse deformation slightly, compared with 1 × 1 × 2 supercells.

_{2}X (X = Si, Ge) with and without point defects are brittle materials. The maximum G/B value of Mg

_{2}X without point defects indicates that the formation of defects will improve the ductility of materials; the G/B value of vacancy point defects is smaller than that of anti-site defects, indicating that vacancy point defects can improve the plasticity of materials more than anti-site defects. Compared with 2 × 2 × 2 supercells, the G/B of Mg

_{2}X with the same point defect structure (except $M{g}_{Si}$) in 1× 1 × 2 supercells can enhance the plasticity of materials.

#### 3.3.3. Elastic Anisotropy

_{2}X (X = Si, Ge) at 0 GPa with 1 × 1 × 2 supercells and 2 × 2 × 2 supercells, respectively. The near sphere shows that these intermetallic compounds are elastic and anisotropic [48]. Compared with other point defects of Mg

_{2}X, the Mg

_{2}X with X vacancy exhibits a relatively obvious deviation from spherical shape, which indicates that the Mg

_{2}X with X vacancy has a relatively strong anisotropy. The degree of spherical deviation of Mg

_{2}X decreases with the increase of supercell size, implying that Mg

_{2}X with point defects show a relatively weak anisotropy in the 2 × 2 × 2 supercells.

## 4. Conclusions

_{2}X (X = Si, Ge) intermetallic is revealed from the microscopic point of view:

_{2}Ge type intermetallics with (without) point defects is larger than that of Mg

_{2}Si type, and the existence of point defects leads to the instability of crystal cells and the expansion of lattice distortion.

_{f}and E

_{coh}are consistently negative for all Mg

_{2}X (X = Si, Ge) with point defects, implying the thermodynamic stability of Mg

_{2}X (X = Si, Ge) with point defects. The absolute value of $\Delta {H}_{f}$ of Mg

_{2}X (X = Si, Ge) with point defects is smaller than that of the cells without point defects, which indicates that the structure with point defects is relatively difficult to form; intermetallics are easier to form vacancy defects than anti-site on the X (X = Si, Ge) lattice site and form anti-site defects on the Mg lattice site; the ΔH

_{f}of the vacancy or anti-site defects of Mg

_{2}X at the Si or Ge position is bigger than that at the Mg position, which indicates that the point defect is more likely to appear at the Mg position than at the Si or Ge position; the absolute value of the binding energy of the intermetallics without point defects is higher than that of the intermetallics with point defects, which indicates that the energy required for atom bonding is higher and the structure is more stable when there is no defect; the structure of the intermetallics forming point defects at the Mg position is more stable than that at the Si/Ge position.

_{2}X without point defects indicates that the formation of defects will improve the ductility of materials; the $G/B$ value of vacancy point defects is smaller than that of anti-site defects, indicating that vacancy point defects can improve the plasticity of materials more than anti-site defects.

_{2}X without point defects is the smallest, which indicates that the formation of defects makes the transverse deformation of the material larger, and the transverse elastic instability increased. Compared with 2 × 2 × 2 supercells, the G/B of Mg

_{2}X with the same point defect structure (except $M{g}_{Si}$) in 1 × 1 × 2 supercells can enhance the plasticity of materials.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**1 × 1 × 2 supercell crystal structure of Mg

_{2}X (X = Si, Ge) with vacancy point defects or anti-site defects,

**(a)**Mg

_{16}X

_{8},

**(b)**Mg

_{16}X

_{7}(V

_{X}),

**(c)**Mg

_{15}X

_{8}(V

_{Mg}),

**(d)**Mg

_{15}X

_{9}(X

_{Mg}), and

**(e)**Mg

_{17}X

_{7}(Mg

_{X}). The red squares represent the position of the vacancy or anti-site defects.

**Figure 2.**2 × 2 × 2 supercell crystal structure of Mg

_{2}X (X = Si, Ge) with vacancy point defects or anti-site defects,

**(a)**Mg

_{64}X

_{32},

**(b)**Mg

_{63}X

_{32}(V

_{Mg}),

**(c)**Mg

_{64}X

_{31}(V

_{X}),

**(d)**Mg

_{65}X

_{31}(Mg

_{X}) and

**(e)**Mg

_{63}X

_{33}(X

_{Mg}). The red squares represent the position of the vacancy or anti-site defects.

**Figure 4.**Computed 3D Young’s modulus $E$ (GPa) of Mg

_{2}X (X=Si, Ge) Intermetallics in the 1×1×2 supercells.

**Figure 5.**Computed 3D Young’s modulus $E$ (GPa) of Mg

_{2}X (X = Si, Ge) intermetallics in the 2 × 2 × 2 supercells.

**Table 1.**Calculated equilibrium lattice constants of 1 × 1 × 2 Mg

_{2}X (X = Si, Ge) cells with or without two types of point defects.

Intermetallics | a (Å) | b (Å) | c (Å) | V (Å^{3}) |
---|---|---|---|---|

Mg_{16}Si_{8} | 6.351 | 6.351 | 12.702 | 512.338 |

Mg_{16}Si_{7}(V_{Si}) | 6.380 | 6.380 | 12.764 | 519.551 |

Mg_{15}Si_{8}(V_{Mg}) | 6.372 | 6.372 | 12.758 | 518.005 |

Mg_{15}Si_{9}(Si_{Mg}) | 6.369 | 6.369 | 12.962 | 525.793 |

Mg_{17}Si_{7}(Mg_{Si}) | 6.405 | 6.405 | 13.140 | 539.056 |

Mg_{16}Ge_{8} | 6.385 | 6.385 | 12.770 | 520.586 |

Mg_{16}Ge_{7}(V_{Ge}) | 6.434 | 6.434 | 12.793 | 529.584 |

Mg_{15}Ge_{8}(V_{Mg}) | 6.427 | 6.427 | 12.872 | 531.695 |

Mg_{15}Ge_{9}(Ge_{Mg}) | 6.447 | 6.447 | 12.894 | 535.924 |

Mg_{17}Ge_{7}(Mg_{Ge}) | 6.464 | 6.464 | 13.229 | 552.751 |

**Table 2.**Calculated equilibrium lattice constants of 2 × 2 × 2 Mg2X (X = Si, Ge) cells with or without two types of point defects.

Intermetallics | A = b = c (Å) | V (Å^{3}) | Intermetallics | a = b = c (Å) | V (Å^{3}) |
---|---|---|---|---|---|

Mg_{64}Si_{32} | 12.741 | 2068.290 | Mg_{64}Ge_{32} | 12.771 | 2083.054 |

Mg_{64}Si_{31}(V_{Si}) | 12.761 | 2078.015 | Mg_{64}Ge_{31}(V_{Ge}) | 12.870 | 2131.974 |

Mg_{63}Si_{32}(V_{Mg}) | 12.768 | 2081.463 | Mg_{63}Ge_{32}(V_{Mg}) | 12.879 | 2136.209 |

Mg_{63}Si_{33}(Si_{Mg}) | 12.760 | 2077.480 | Mg_{63}Ge_{33}(Ge_{Mg}) | 12.884 | 2138.745 |

Mg_{65}Si_{31}(Mg_{Si}) | 12.808 | 2100.947 | Mg_{65}Ge_{31}(Mg_{Ge}) | 12.918 | 2155.592 |

**Table 3.**Computed formation heat (ΔH

_{f}) and binding energy (${E}_{coh}$) of Mg

_{16}X

_{8}and Mg

_{64}X

_{32}(X=Si, Ge) with or without (anti) vacancy point defects.

1 × 1 × 2 Intermetallics | $\mathbf{\Delta}{\mathit{H}}_{\mathit{f}}\text{}(\mathbf{eV}/\mathbf{atom})$ | ${\mathit{E}}_{\mathit{c}\mathit{o}\mathit{h}}\text{}(\mathbf{eV}/\mathbf{atom})$ | 2 × 2 × 2 Intermetallics | $\mathbf{\Delta}{\mathit{H}}_{\mathit{f}}\text{}(\mathbf{eV}/\mathbf{atom})$ | ${\mathit{E}}_{\mathit{c}\mathit{o}\mathit{h}}\text{}(\mathbf{eV}/\mathbf{atom})$ |
---|---|---|---|---|---|

Mg_{16}Si_{8} | −0.180 | −3.113 | Mg_{64}Si_{32} | −0.170 | -2.949 |

Mg_{16}Si_{7}(V_{Si}) | −0.084 | −2.911 | Mg_{64}Si_{31}(V_{Si}) | −0.148 | -2.899 |

Mg_{15}Si_{8}(V_{Mg}) | −0.092 | −3.078 | Mg_{63}Si_{32}(V_{Mg}) | −0.155 | −2.947 |

Mg_{17}Si_{7}(Mg_{Si}) | −0.072 | −2.852 | Mg_{65}Si_{31}(Mg_{Si}) | −0.142 | −2.880 |

Mg_{15}Si_{9}(Si_{Mg}) | −0.108 | −3.103 | Mg_{63}Si_{33}(Si_{Mg}) | −0.155 | −2.945 |

Mg16Ge8 | −0.281 | −2.953 | Mg_{64}Ge_{32} | −0.259 | −2.761 |

Mg16Ge7(VGe) | −0.167 | −2.756 | Mg_{64}Ge_{31}(V_{Ge}) | −0.236 | −2.716 |

Mg15Ge8(VMg) | −0.197 | −2.910 | Mg_{63}Ge_{32}(V_{Mg}) | −0.248 | −2.761 |

Mg_{17}Ge_{7}(Mg_{Ge}) | −0.155 | −2.708 | Mg_{65}Ge_{31}(Mg_{Ge}) | −0.230 | −2.700 |

Mg_{15}Ge_{9}(Ge_{Mg}) | −0.211 | −3.002 | Mg_{63}Ge_{33}(Ge_{Mg}) | −0.248 | −2.753 |

**Table 4.**Computed $\Delta {E}_{f}{}^{vac}$ and $\Delta {E}_{f}{}^{anti}$ of Mg

_{16}X

_{8}and Mg

_{64}X

_{32}(X = Si, Ge) with vacancy and anti-site point defects.

Intermetallics | $\mathbf{\Delta}{\mathit{E}}_{\mathit{f}}{}^{\mathit{v}\mathit{a}\mathit{c}}\text{}(\mathbf{eV}/\mathbf{atom})$ | $\mathbf{\Delta}{\mathit{E}}_{\mathit{f}}{}^{\mathit{a}\mathit{n}\mathit{t}\mathit{i}}\text{}(\mathbf{eV}/\mathbf{atom})$ | |||
---|---|---|---|---|---|

$\mathbf{\Delta}{\mathit{E}}_{\mathit{M}\mathit{g}}^{\mathit{v}\mathit{a}\mathit{c}}$ | $\mathit{\Delta}{\mathit{E}}_{\mathit{x}}{}^{\mathit{v}\mathit{a}\mathit{c}}$ | $\mathbf{\Delta}{\mathit{E}}_{\mathit{M}\mathit{g}}^{\mathit{v}\mathit{a}\mathit{t}\mathit{i}}$ | $\mathbf{\Delta}{\mathit{E}}_{\mathit{x}}{}^{\mathit{a}\mathit{n}\mathit{t}\mathit{i}}$ | ||

1 × 1 × 2 | Mg_{16}Si_{8} | 3.68 | 7.82 | −2.23 | 6.57 |

Mg_{16}Ge_{8} | 3.45 | 7.24 | −1.68 | 5.90 | |

2 × 2 × 2 | Mg_{64}Si_{32} | 3.09 | 7.67 | −2.55 | 6.62 |

Mg_{64}Ge_{32} | 2.77 | 7.02 | −2.10 | 5.83 |

**Table 5.**Computed elastic constants ${C}_{ij}$ (GPa) of Mg

_{2}X (X = Si, Ge) with or without a point defect (a vacancy or an anti-site defect) in different supercells.

Supercell | Intermetallics | C_{11} | C_{12} | C_{13} | C_{33} | C_{44} | C_{66} |
---|---|---|---|---|---|---|---|

- | Mg_{2}Si | 113.40 | 22.71 | - | - | 45.09 | - |

- | Cal. [42] | 114.07 | 19.56 | - | - | 33.32 | - |

1 × 1 × 2 | Mg_{16}Si_{7}(V_{Si}) | 91.55 | 27.67 | 21.98 | 94.37 | 19.64 | 28.76 |

Mg_{15}Si_{8(}V_{Mg}) | 90.70 | 27.25 | 29.12 | 89.98 | 29.91 | 33.24 | |

Mg_{17}Si_{7}(Mg_{Si}) | 91.83 | 24.34 | 25.26 | 84.48 | 38.07 | 31.50 | |

Mg_{15}Si_{9}(Si_{Mg}) | 100.74 | 27.37 | 27.37 | 97.76 | 29.78 | 30.69 | |

2 × 2 × 2 | Mg_{64}Si_{31}(V_{Si}) | 103.94 | 24.76 | 32.46 | |||

Mg_{63}Si_{32}(V_{Mg}) | 105.98 | 25.23 | 38.96 | ||||

Mg_{63}Si_{33}(Si_{Mg}) | 107.95 | 23.45 | 39.83 | ||||

Mg_{65}Si_{31}(Mg_{Si}) | 102.69 | 25.95 | 36.64 | ||||

- | Mg_{2}Ge | 105.32 | 20.83 | - | - | 42.58 | 105.32 |

- | Cal. [43] | 107.3 | 21.1 | - | - | 41.8 | 107.3 |

1 × 1 × 2 | Mg_{16}Ge_{7}(V_{Ge}) | 86.30 | 24.62 | 19.36 | 89.24 | 22.13 | 86.30 |

Mg_{15}Ge_{8}(V_{Mg}) | 84.38 | 25.08 | 26.89 | 83.96 | 29.24 | 84.38 | |

Mg_{17}Ge_{7}(Mg_{Ge}) | 83.71 | 21.32 | 25.99 | 80.78 | 32.83 | 83.71 | |

Mg_{15}Ge_{9}(Ge_{Mg}) | 94.53 | 25.42 | 24.64 | 92.92 | 30.48 | 94.53 | |

2 × 2 × 2 | Mg_{64}Ge_{31}(V_{Ge}) | 98.69 | 22.22 | 32.46 | |||

Mg_{63}Ge_{32}(V_{Mg}) | 98.05 | 23.14 | 37.08 | ||||

Mg_{63}Ge_{33}(Ge_{Mg}) | 103.14 | 22.97 | 38.42 | ||||

Mg_{65}Ge_{31}(Mg_{Ge}) | 95.57 | 23.53 | 34.59 |

**Table 6.**Lists the calculated bulk modulus, B.; the shear modulus, G.; Young’s modulus, E.; Poisson’s ratio $\nu $ and G/B.

Supercell | Intermetallics | B/GPa | G/GPa | E/GPa | G/B | ʋ |
---|---|---|---|---|---|---|

- | Mg_{2}Si | 52.94 | 45.19 | 105.54 | 0.854 | 0.168 |

- | Cal. [42] | 51.06 | 46.12 | 108.35 | 0.903 | 0.150 |

1 × 1 × 2 | Mg_{16}Si_{7}(V_{Si}) | 46.74 | 27.33 | 68.62 | 0.585 | 0.255 |

Mg_{15}Si_{8(}V_{Mg}) | 49.15 | 31.00 | 76.85 | 0.631 | 0.239 | |

Mg_{17}Si_{7}(Mg_{Si}) | 46.38 | 34.41 | 82.77 | 0.742 | 0.202 | |

Mg_{15}Si_{9}(Si_{Mg}) | 51.49 | 35.53 | 83.61 | 0.752 | 0.239 | |

2 × 2 × 2 | Mg_{64}Si_{31}(V_{Si}) | 51.16 | 35.14 | 85.79 | 0.687 | 0.221 |

Mg_{63}Si_{32}(V_{Mg}) | 52.15 | 39.52 | 94.65 | 0.758 | 0.198 | |

Mg_{65}Si_{31}(Mg_{Si}) | 51.53 | 37.32 | 90.19 | 0.724 | 0.208 | |

Mg_{63}Si_{33}(Si_{Mg}) | 51.62 | 40.78 | 96.84 | 0.790 | 0.187 | |

- | Mg_{2}Ge | 49.00 | 42.45 | 98.81 | 0.866 | 0.164 |

- | Cal. [43] | 49.80 | 42.30 | 98.90 | 0.847 | - |

1 × 1 × 2 | Mg_{16}Ge_{7}(V_{Ge}) | 43.48 | 28.17 | 69.50 | 0.648 | 0.234 |

Mg_{15}Ge_{8}(V_{Mg}) | 45.47 | 29.51 | 72.79 | 0.649 | 0.233 | |

Mg_{17}Ge_{7}(Mg_{Ge}) | 43.88 | 30.16 | 73.62 | 0.688 | 0.220 | |

Mg_{15}Ge_{9}(Ge_{Mg}) | 47.28 | 32.76 | 79.84 | 0.693 | 0.219 | |

2 × 2 × 2 | Mg_{64}Ge_{31}(V_{Ge}) | 47.71 | 34.66 | 83.71 | 0.727 | 0.208 |

Mg_{63}Ge_{32}(V_{Mg}) | 48.11 | 37.23 | 88.78 | 0.774 | 0.193 | |

Mg_{65}Ge_{31}(Mg_{Ge}) | 47.54 | 35.15 | 84.61 | 0.739 | 0.203 | |

Mg_{63}Ge_{33}(Ge_{Mg}) | 49.69 | 39.08 | 92.88 | 0.786 | 0.189 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhao, Y.; Tian, J.; Bai, G.; Zhang, L.; Hou, H.
First Principles Study on the Thermodynamic and Elastic Mechanical Stability of Mg_{2}X (X = Si,Ge) Intermetallics with (anti) Vacancy Point Defects. *Crystals* **2020**, *10*, 234.
https://doi.org/10.3390/cryst10030234

**AMA Style**

Zhao Y, Tian J, Bai G, Zhang L, Hou H.
First Principles Study on the Thermodynamic and Elastic Mechanical Stability of Mg_{2}X (X = Si,Ge) Intermetallics with (anti) Vacancy Point Defects. *Crystals*. 2020; 10(3):234.
https://doi.org/10.3390/cryst10030234

**Chicago/Turabian Style**

Zhao, Yuhong, Jinzhong Tian, Guoning Bai, Leting Zhang, and Hua Hou.
2020. "First Principles Study on the Thermodynamic and Elastic Mechanical Stability of Mg_{2}X (X = Si,Ge) Intermetallics with (anti) Vacancy Point Defects" *Crystals* 10, no. 3: 234.
https://doi.org/10.3390/cryst10030234