First Principle Study on Mg₂X (X = Si, Ge, Sn) Intermetallics by Bi Micro-Alloying

Abstract

Being a positive candidate reinforcement material for laminar composites, the Mg₂X (X = Si, Ge, Sn) based intermetallics have attracted much attention. The elastic properties, anisotropy, and electronic properties of intermetallic compounds with Bi-doped Mg₂X (X = Si, Ge, Sn) are calculated by the first principles method. Results show that the lattice parameters of Mg₂X are smaller than those of Bi-doped Mg₂X. The element Bi preferentially occupies the position of the X (X = Si, Ge, Sn) atom than other positions. Mg₂X (X = Si, Ge, Sn), Mg₆₃X₃₂Bi, Mg₆₄X₃₁Bi, Mg₆₄Ge₃₂Bi, and Mg₆₄Sn₃₂Bi are mechanically stable, while Mg₆₄Si₃₂Bi indicates that it cannot exist stably. The doping of alloying element Bi reduces the shear deformation resistance of the Mg₂X (X = Si, Ge, Sn) alloy. The pure and Bi-doped Mg₂X (X = Si, Ge, Sn) exhibits elastic and anisotropic characteristics. The contribution of the Bi orbitals of Mg₆₃X₃₂Bi, Mg₆₄X₃₁Bi, and Mg₆₃X₃₂Bi are different, resulting in different hybridization effects in three types of Bi-doped Mg₂X.

1. Introduction

Mg alloys have received extensive attention in the past decade due to their optimum strength-to-weight ratio, good corrosion resistance, high-temperature resistance, and pleasant ductility. The application of increased weight in the automotive and aerospace industries is mainly used to reduce weight and improve fuel efficiency [1,2,3,4,5,6,7,8,9]. The Mg₂X (X = Si, Ge, Sn) alloy has a CaF₂-type structure, which has a fairly low density and a reasonably high melting point, hardness, and modulus of elasticity [10,11,12,13]. However, Mg₂X (X = Si, Ge, Sn) alloys have severe room-temperature brittleness [14,15], resulting in a limited application range, and further research is needed.

The addition of alloying elements may improve the mechanical or electronic properties of the material [16,17,18]. After adding trace alloying elements, the low-temperature toughness and high-temperature creep properties of the Mg alloy can be improved by changing the lattice constant and bonding properties [16]. Experimental studies have shown that the addition of Ca in the Mg-Si alloy changes the morphology of the Mg-Si system, and improves the overall performance of the magnesium alloy; adding 0.03 wt.% Bi changes the primary Mg₂Si shape from large to irregular or dendritic to polyhedral [19,20]. Compared with other methods [21,22], the first principles method [23,24] can accurately predict the structure and properties of the phase. Using the first principles method, Zhao Hui et al. [16] showed that the doping of Ca, Sr, and Ba changed Mg₂Si from brittle to ductile; the density of the electronic states shifted, the covalent bond weakened, and the structural stability of the alloy system weakened. Rare earth elements can refine Mg-Si grains, but rare earth elements are expensive [25,26]. M. Ioannou et al. [14] found that Bi is the most stable element in Mg₂Si, compared with other dopants. Based on this, the effect of alloying element Bi on the properties of Mg₂X (X = Si, Ge, Sn) alloys was studied using the first principles method in this paper.

2. Model and Calculation Method

Mg₂X (X = Si, Ge, Sn) belongs to a cubic crystal structure. To guarantee reliable calculated results, we used a 2 × 2 × 2 supercell consisting of 96 atoms to construct the doping structures. In the supercells, Mg or X (X = Si, Ge, Sn) sites can be substituted by a single alloying element Bi, and there are many possible site preferences for Bi. In this paper, three cases were studied, respectively: (1) occupying a position of an Mg atom, (2) occupying a face center position of X (X = Si, Ge, Sn) atoms, and (3) occupying the center position between two neighboring X atoms. The calculation models are shown in Figure 1.

The calculation process was performed using the Cambridge Serial Total Energy Package (CASTEP) [27,28], based on density functional theory (DFT) [29,30]. The electronic exchange association can adopt the GGA-PBE form [31], and the potential function selects the ultra-soft pseudopotential of the reciprocal space. The integral of the Brillouin zone was calculated by the high-symmetric k-point method in the form of Monkhorst-Pack [32], with the k-point grid being 4 × 4 × 4, and the cut-off energy for the plane wave functions was set to 380 eV for Bi-doped Mg₂X. The Broyden–Fletcher–Goldfarb–Shanno (BFGS) [33] algorithm is used to geometrically optimize the unit cell model to obtain the most stable structure. When performing self-consistent iterative SCF calculation, the Pulay density mixing method is used to solve the electron relaxation. The self-consistent convergence condition is: the total energy of the system reaches convergence within 5.0 × 10⁻⁶ eV/atom, the force on each atom is less than 0.01 eV/Å, the stress deviation is less than 0.02 GPa, the tolerance offset is less than 5 × 10⁻⁴ Å, and the SCF convergence accuracy is 5 × 10⁻⁷ eV/atom.

3. Results and Discussion

3.1. Lattice Parameters

After doping, the lattice parameters of Mg₂X (X = Si, Ge, Sn), Mg₆₃X₃₂Bi, Mg₆₄X₃₁Bi, and Mg₆₄X₃₂Bi crystals are in

Table 1. Lattice constant a (Å) and enthalpy of the formation ΔH_f (eV/atom) of Mg₂X, Mg₆₃X₃₂Bi, Mg₆₄X₃₁Bi, and Mg₆₄X₃₂Bi (X = Si, Ge, Sn).

Phase	Lattice Constants a/Å			ΔHf (eV/atom)
	This Work	Cal	Exp
pure Mg2Si	6.371	6.30 [10]	6.35 [34]	–0.170
Mg64Si32	12.741			–0.170
Mg63Si32Bi	12.791	-	-	−0.191
Mg64Si31Bi	12.804	-	-	−0.204
Mg64Si32Bi	12.823	-	-	−0.175
pure Mg2Ge	6.355	6.318 [35]	6.3849 [36]	−0.259
Mg64Ge32	12.710			−0.259
Mg63Ge32Bi	12.906	-	-	−0.279
Mg64Ge31Bi	12.909	-	-	−0.290
Mg64Ge32Bi	12.940	-	-	−0.264
pure Mg2Sn	6.843	6.829 [37]	6.759 [38]	−0.196
Mg64Sn32	13.685			−0.196
Mg63Sn32Bi	13.688	-	-	−0.228
Mg64Sn31Bi	13.670	-	-	−0.237
Mg64Sn32Bi	13.711	-	-	−0.220

. The predicted lattice constants of pure Mg₂X are consistent with other theoretical and experimental values [10,34,35,36,37,38], indicating the reliability of the present computational model. The lattice parameters of Mg₂X are smaller than those of Bi-doped Mg₂X, because the radius of doping element Bi is larger than that of alloying element X and Mg. The enthalpy of formation (ΔH_f) of Mg₂X and Bi-doped Mg₂X is shown in Equation (1):

$$∆H_f=\frac{E_{tot}-N_A{E}_{solid}^A-N_B{E}_{solid}^B-N_C{E}_{solid}^C}{N_A+N_B+N_C}$$

(1)

where E_tot represents the total energy of pure and doped Mg₂X (X = Si, Ge, Sn) phases, $E_{\mathrm{soild}}^{\mathrm{A}}$, $E_{\mathrm{soild}}^{\mathrm{B}}$, and $E_{\mathrm{soild}}^{\mathrm{C}}$ denote the ground state energy of Mg, X, and Bi in the solid cell, N_A, N_B, and N_C are the number of Mg, X, and Bi atoms, respectively.

The calculated ΔH_f of Mg₂X and Bi-doped Mg₂X phases are listed in Table 1. The more negative ΔH_f the crystal is, the easier it is to form. The ΔH_f of Mg₆₄X₃₁Bi is smaller than that of others, which indicates that the element Bi preferentially occupies the position of the X (X = Si, Ge, Sn) atom more than other positions.

3.2. Elastic Properties

The elastic constant is used to describe the ability of a material to resist external force deformation and predict the mechanical properties of a material. The elastic properties are closely related to some important thermodynamic properties (such as the Debye temperature, melting point, and specific heat capacity), so it is necessary to study the elastic properties of the alloy after Bi-doped Mg₂X (X = Si, Ge, Sn) by calculating the elastic constant.

The Bi-doped Mg₂X alloy crystals belong to the cubic crystal type, and have three independent elastic constants [39]: C₁₁, C₁₂, and C₄₄. The stability criterion is: $C_{11}-C_{12}>0$, $C_{11}>0$, $C_{44}>0$, and $C_{11}+2C_{12}>0$ [40]. At the same time, many elastic properties of the crystal can be obtained by the elastic constant C_ij [41,42], for example, bulk modulus B, shear modulus G, Young’s modulus E, Pugh’s index of ductility B/G, Poisson’s ratio ν, and anisotropic Zener ratio A_z. The calculation formula is as follows [43],

$${\rm B}=\frac{C_{11}+2{\mathrm{C}}_{12}}{3}$$

$$G_V=\frac{C_{11}-C_{12}+3{\mathrm{C}}_{44}}{5}$$

$$G_R=\frac{5(C_{11}-C_{12}){\mathrm{C}}_{44}}{3(C_{11}-C_{12})+4{\mathrm{C}}_{44}}$$

$$G=\frac{G_V+G_R}{2}$$

$$E=\frac{9BG}{3B+G}$$

$$\upsilon =\frac{3B-2G}{2(3B+G)}$$

$$A_z=2C_{44}/\left(C_{11}-C_{12}\right)$$

In the current work, elastic constant C_ij, bulk modulus B, shear modulus G, Young’s modulus E, Pugh’s index of ductility B/G, Poisson’s ratio ν, and anisotropic Zener ratio A_z of the Mg₂X, Mg₆₃X₃₂Bi, Mg₆₄X₃₁Bi, and Mg₆₄X₃₂Bi alloys are shown in

Table 2. Elastic constant C_ij (GPa), bulk modulus B (GPa), shear modulus G (GPa), Young’s modulus E (GPa), Pugh’s index of ductility B/G, Poisson’s ratio ν, and anisotropic Zener ratio A_z of Mg₂X (Mg₆₄Si₃₂), Mg₆₃X₃₂Bi, Mg₆₄X₃₁Bi, and Mg₆₄X₃₂Bi alloys.

Phase	C11	C12	C44	B	G	E	G/B	ν	Az
Mg64Si32	111.97	21.55	41.74	51.69	43.10	101.17	0.834	0.174	0.923z
Exp. [44]	126.00	26.00	48.50	59.00	-	-	-	-	-
Cal. [11]	121.20	23.70	49.50	56.20	49.20	113.50	-	-	-
Cal. [45]	118.80	22.27	44.96	-	-	-	-	-	-
Mg63Si32Bi	108.53	24.84	36.21	52.74	38.37	92.64	0.728	0.207	0.865
Mg64Si31Bi	107.64	24.50	36.03	52.21	38.15	92.03	0.731	0.206	0.867
Mg64Si32Bi	37.44	57.65	32.58	-	-	-	-	-	-
Mg64Ge32	103.88	19.65	37.97	49.73	39.57	93.01	0.829	0.175	0.902
Exp. [44]	117.90	23.00	46.50	54.06	-	-	-	-	-
Cal. [11]	118.10	23.60	48.00	55.10	47.70	111.10	-	0.164	-
Cal. [46]	116.10	20.60	44.00	52.50	45.40	105.9	-	0.164	-
Mg63Ge32Bi	101.58	23.26	34.24	49.37	36.13	87.14	0.732	0.206	0.874
Mg64Ge31Bi	101.25	22.21	33.86	48.56	36.02	86.64	0.742	0.203	0.857
Mg64Ge32Bi	52.54	43.72	28.52	46.66	13.91	37.96	0.298	0.364	6.472
Mg64Sn32	68.36	29.39	34.20	40.38	28.11	68.46	0.696	0.217	1.630
Exp. [47]	82.40	20.80	36.60	-	-	-	-	-	-
Cal. [11]	83.71	39.79	21.69	42.36	21.79	74.78	0.51	0.206	-
Cal. [44]	81.10	20.16	34.85	43.73	31.70	-	-	-	-
Mg63Sn32Bi	66.42	27.11	27.42	40.21	24.00	60.04	0.597	0.251	1.395
Mg64Sn31Bi	67.51	25.96	28.63	39.81	25.18	62.38	0.632	0.239	1.378
Mg64Sn32Bi	58.06	29.27	23.25	38.97	19.18	49.41	0.493	0.288	1.615

.

It can be found in Table 2 that Mg₂X (X = Si, Ge, Sn), Mg₆₃X₃₂Bi [48], Mg₆₄X₃₁Bi, Mg₆₄Ge₃₂Bi, and Mg₆₄Sn₃₂Bi satisfy the stability criterion, indicating that these crystals are mechanically stable, while C₁₁ – C₁₂ < 0 of Mg₆₄Si₃₂Bi indicates that the cubic structure cannot exist stably. Thus, it is not an optimal structure.

The bulk modulus B represents the ability of materials to resist deformation under external stress, and the greater the bulk modulus, the stronger the ability to resist deformation [49]. After Bi doping Mg₂Ge and Mg₂Sn: Mg₂X > Mg₆₃X₃₂Bi > Mg₆₄X₃₁Bi > Mg₆₄X₃₂Bi (X = Ge and Sn), but after Bi-doped Mg₂Si, Mg₆₃Si₃₂Bi > Mg₆₄Si₃₁Bi > Mg₂Si, indicating that the ability of Mg₂Si to resist deformation after doping is enhanced. The value of B for Mg₆₃Si₃₂Bi is larger than that for other Bi-doped Mg₂X (X = Si, Ge, Sn) phases, indicating that the Mg₆₃Si₃₂Bi has stronger deformation resistance.

The shear modulus G is used to evaluate the ability of the object to resist shear strain; the greater the value is, and the more obvious the directional bonds between the compounds are, the better the resistance to plastic deformation [50]. The doping of alloying element Bi reduces the shear deformation resistance of the Mg₂X (X = Si, Ge, Sn) alloy. The order of the deformation resistance of Mg₆₃X₃₂Bi, Mg₆₄X₃₁Bi, and Mg₆₄X₃₂Bi is Mg₆₃Si₃₂Bi > Mg₆₃Ge₃₂Bi > Mg₆₃Sn₃₂Bi, Mg₆₄Si₃₁Bi > Mg₆₄Ge₃₁Bi > Mg₆₄Sn₃₁Bi, Mg₆₄Sn₃₂Bi > Mg₆₄Ge₃₂Bi, respectively. This means that the alloy obtained by Bi doping Mg₂Si has better shear resistance than Mg₆₃X₃₂Bi and Mg₆₄X₃₁Bi (X = Ge, Sn).

Young’s modulus E is an important parameter to characterize material stiffness. The smaller the value, the smaller the stiffness, and the better the plasticity of the materials [49]. The doping of the alloying element Bi in Table 2 enhances the plasticity and reduces the stiffness of the Mg₂X (X = Si, Ge, Sn) alloy. The stiffness of Mg₆₃Si₃₂Bi (Mg₆₄Si₃₁Bi) is stronger than that of other Mg₆₃X₃₂Bi (Mg₆₄X₃₁Bi). Poisson’s ratio $\mathsf{\nu }$ refers to the ratio of the absolute value of the transverse positive strain and the axial positive strain when the material is under tension or compression in a single direction. The greater the value, the better the plasticity the corresponding material will have [51]. It can be seen from Table 2 that the doping of the alloying element Bi enhances the plasticity of the Mg₂X (X = Si, Ge, Sn) alloy and corresponds to Young’s modulus calculation result.

According to Pugh, G/B can predict the ductility or brittleness of materials, and the corresponding critical value is 0.57. When G/B > 0.57, materials are brittle, and ductile materials are opposite [52]. The doping of the alloying element Bi causes the Mg₂X (X = Si, Ge, Sn) alloy to be converted from a brittle material to a ductile material, as can be seen in Table 2. In general, the brittleness (extension) of a material can also be measured by C₁₂−C₄₄. If C₁₂–C₄₄ > 0, the material exhibits ductility; on the contrary, it is brittle [16,53]. According to Table 2, it is known that the Mg₂X, Mg₆₃X₃₂Bi, and Mg₆₄X₃₁Bi (X = Si, Ge, Sn) alloys are brittle materials, and Mg₆₄X₃₂Bi (X = Ge, Sn) is a ductile material, which is consistent with the results obtained by G/B.

The elastic anisotropy of the material in the engineering materials shows the possibility of micro-crack in the material, which is closely related to the nanoscale precursor texture of the material, and occupies an important position in the material science [54,55]. Anisotropy, on the other hand, reflects the density distribution of electrons in different directions based on DFT calculations. In different crystal orientations, the density function of electrons is not the same, so it will show a different degree of anisotropy. The material behaves as isotropic when A_z = 1. According to Table 2, it can be seen that the A_z of pure and Bi-doped Mg₂X is not equal to 1, showing the anisotropy of pure and doped Mg₂X. The A_z of Mg₂Si is very close to 1, indicating that the elastic anisotropy of Mg₂Si is relatively small. The anisotropy of Bi-doped Mg₂X (X = Si, Ge) phase is larger than that of Mg₂X, whereas the anisotropy of Bi-doped Mg₂Sn is smaller than that of Mg₂Sn.

Figure 2 plots the 3D Young’s modulus E-surface diagram of pure and Bi-doped Mg₂X (X = Si, Ge, Sn) alloys at 0 GPa. It is clear from the three-dimensional surface that the pure and Bi-doped Mg₂X phases show elastic anisotropy, because their 3D shapes deviate from the spherical shape. Mg₆₄Ge₃₂Bi deviates most from the spherical shape among these phases, indicating that Mg₆₄Ge₃₂Bi shows strong anisotropy. The main effect of impurity doping is mainly to change the charge density distribution, thus affecting the anisotropy. This result is consistent with the calculation results.

3.3. Electronic Properties

Before calculating the electronic structure, we tested the influence of the spin polarization settings on the total energy of the Mg₆₃Si₃₂Bi system, and the results are shown in

Table 3. Total energy of Mg₆₃Si₃₂Bi under different spin polarization settings.

	Non-Spin Polarized	Spin Polarized
Total Energy (eV)	−64958.92186	−64958.92179

. It can be found that the difference between total energy is very small, which means that the effect of spin polarization on the total energy and electronic structure is negligible.

Usually, we study the band gap based on the band structure. We calculated the energy band of Mg₂Si and Mg₂Ge, and the results are shown in Figure 3. It can be seen that there is a band gap of 0.223 eV in Mg₂Si and 0.123 eV in Mg₂Ge, which means that Mg₂Si and Mg₂Ge are both semiconductor materials. However, this cannot be reflected from the DOS diagram. The calculations of Mg2Si in the literature [56,57,58] also has similar results.

As shown in Figure 4, the bonding electrons of the pure and Bi-doped Mg₂X (X = Si, Ge, Sn) alloy compounds are mainly distributed at −10 to 5 eVs. In an energy range from −10 to 0 eV, there is no significant difference in the shape of the total density of states (TDOS) between pure and doped Mg₂X phases. In Figure 4a, the bonding electrons mainly come from the contributions of Mg-3s, Si-3s, Ge-4s, and Sn-5s orbitals. In the −5~0 eV interval, Si-3p, Ge-4p, and Sn-5p orbitals have strong orbital hybridization with Mg-2p and Mg-3s. The energy range from 0~eV to 5~eV is mainly contributed by Mg-2p and Mg-3s orbitals, with small involvement of X states.

As shown in Figure 4b, in the energy range from −10 to −6 eV, the atomic orbital of Mg₆₃X₃₂Bi (X = Si and Sn) alloys are mainly dominated by Bi-6p, Si-3s, Sn-5s, and Mg-3s states, and the TDOS of Mg₆₃Ge₃₂Bi is mainly contributed by Bi-6s, Ge-4s, and Mg-3s states. In the range of −6~0 eV, the Mg₆₃X₃₂Bi (X = Si, Ge, Sn) alloy mainly comes from the interaction between Mg-2p, Mg-3s, Si-3p, Ge-4p, Sn-5p, and Bi-6p, indicating that Mg, X, and Bi have strong bond binding effects in this interval. The Mg-2p and Mg-3s orbitals contribute strongly in the range of 0~5 eV for Mg₆₃X₃₂Bi (X = Si, Ge, Sn).

In Figure 4c, between −6 and 0 eV, the orbital contribution of the Mg₆₄X₃₁Bi alloy is the same as that of the Mg₆₃X₃₂Bi alloy. In the range of −10~−6 eV and 0~2 eV, the contribution of Bi-6s and Bi-6p orbitals of Mg₆₄X₃₁Bi to TDOS has changed compared to Mg₆₃X₃₂Bi. As shown in Figure 4d, the contribution of the Mg and X (S = Ge, Sn) orbitals of Mg₆₄X₃₂Bi to TDOS are similar to the analysis of Mg₆₃X₃₂Bi and Mg₆₄X₃₁Bi. However, the contribution of Bi orbitals of Mg₆₃X₃₂Bi, Mg₆₄X₃₁Bi, and Mg₆₃X₃₂Bi are different, resulting in different hybridization effects in three types of Bi-doped Mg₂X.

4. Conclusions

In this paper, the elastic properties and electronic structures of pure and Bi-doped Mg₂X (X = Si, Ge, Sn) compounds were calculated by the method of plane wave pseudopotential based on density functional theory. The calculation results show that:

(1)

The lattice parameters of Mg₂X are smaller than those of Bi-doped Mg₂X, because the radius of doping element Bi is larger than that of alloying element X and Mg. The ΔH_f of Mg₆₄X₃₁Bi is smaller than that of others, which indicates that the element Bi preferentially occupies the position of the X (X = Si, Ge, Sn) atom than other positions.

(2)

Mg₂X (X = Si, Ge, Sn), Mg₆₃X₃₂Bi, Mg₆₄X₃₁Bi, Mg₆₄Ge₃₂Bi, and Mg₆₄Sn₃₂Bi are mechanically stable, while Mg₆₄Si₃₂Bi indicates that it cannot exist stably. The ability of Mg₂Si to resist deformation after doping is enhanced, and Mg₆₃Si₃₂Bi has stronger deformation resistance. The doping of alloy element Bi makes the Mg₂X (X = Si, Ge, Sn) alloy convert from brittle material to ductile material, and results in plasticity enhancement and stiffness reduction.

(3)

The pure and Bi-doped Mg₂X (X = Si, Ge, Sn) exhibit elastic anisotropic properties. The anisotropy of Bi-doped the Mg₂X (X = Si, Ge) phase is larger than that of Mg₂X, whereas the anisotropy of Bi-doped Mg₂Sn is smaller than that of Mg₂Sn. Mg₆₄Ge₃₂Bi shows strong anisotropy among these phases.

(4)

In an energy range from −10 to 0 eV, there is no significant difference in the shape of TDOS between the pure and doped Mg₂X phases. The contribution of Bi orbitals of Mg₆₃X₃₂Bi, Mg₆₄X₃₁Bi, and Mg₆₃X₃₂Bi are different, resulting in different hybridization effects in three types of Bi-doped Mg₂X.