# Effects of Single Doping and Composite Doping of Yttrium and Antimony on the Structural, Thermodynamic, Mechanical, and Electronic Properties of Mg2Si by First Principles

Authors: Zhonghao Zhou; Zhen Zhao; Zhi Li
Journal: Journal of Nanomaterials
 (2019)

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## Abstract

The structural, stability, thermodynamic, mechanical, and electronic properties of Y and Sb doped into Mg2Si were studied by first principles. The results show that the stable structures are Mg7Si4Y, Mg8Si4Sb, Mg6Si4YSb, and Mg7Si4YSb. Mg8Si4Sb has the largest ductility among them. Y doping causes strong orbital hybridization between Mg (2p) and Y (4d), while Sb doping causes strong orbital hybridization between Mg (2p), Si (3p), and Sb (5p). However, the thermal conductivity and hardness of Mg2Si will be reduced by Y and Sb doping. The conductivity of Mg8Si4Sb is the best. The anisotropy of the {100}, {010}, and {001} surfaces will increase once doped with Y or Sb. Mg8Si4Sb has the highest degree of anisotropy.


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## Body

## 1. Introduction

The 6xxx series aluminum alloy is widely used in automobile manufacturing because of its excellent qualities, such as reduced weight and good corrosion resistance [1, 2]. Mg2Si is the main strengthening phase [3] of the 6xxx series Al alloy, and it possesses a low density and high melting point and impedes grain boundary sliding [4–9]. However, the application of the 6xxx series Al alloy is limited due to its brittleness. This is due to the Mg2Si phase which is the preferred phase to form coarse Chinese script or dendritic morphology [10–12]. To improve the mechanical properties of Mg2Si, many alloying elements (such as Bi, As, and Sn [13–20]) are doped. Recently, Y and Sb doping into the Mg2Si phase was considered. Zhang and Zhang [21] found that the fine modification effect of Y on the primary Mg2Si phase is better than that of an equal amount of mixed light rare earth, and it can greatly refine the Mg2Si grains and improve the tensile strength and elongation. Emamy et al. [22] found that the addition of Y increases both the hardness and ultimate tensile strength values of the Al-15%Mg2Si cast composite. Jiang et al. [23] systematically investigated the modification of in situ-formed Mg2Si in Mg–Si alloys with yttrium (Y). Ioannou et al. [24] have prepared Sb-doped Mg2Si compounds through ball milling and a solid-state reaction. Hu et al. [25] found that Sb doping can change the type of Chinese script of Mg2Si to a fine polygonal shape and decrease the grain sizes. Tani and Kido [26] studied the formation energy and atomic structure of impurities in Mg2Sn using first-principles plane-wave total energy calculations. The thermoelectric properties of trace Y-doped Mg2Si crystals have been obtained by experiments, and the total energies of Mg7Si4Y, Mg8Si3Y, and Mg8Si4Y crystals have also been calculated [27]. What are the differences between the single doping and composite doping of Y and Sb? What are the micromechanisms of Y and Sb doping into Mg2Si? At present, there is no theoretical research on the brittleness of Mg2Si by Y and Sb codoping as far as we know.In this work, the structural, thermodynamic, and mechanical properties of Y and Sb doped into Mg2Si were studied by first principles. It will provide a theoretical reference for further research on Mg2Si and 6xxx series aluminum alloys.


### 1.1. Computational Models and Methods

Mg2Si has an antifluorite (CaF2) structure [28] (see Figure 1). The atomic coordinates are as follows: Mg (0.25, 0.25, 0.25) and Si (0, 0, 0).Figure 1
The crystal structure of Mg2Si.For Y and Sb single doping, the positions of the Mg (or Si) atoms are replaced by Y or Sb, respectively. Despite that both the atom sizes of Y and Sb are larger than those of Mg and Si atoms, to avoid omitting certain configurations, the interpolation Y or Sb atoms into the interval of Mg and Si was considered (see Figure2).Figure 2
The structures of single doping.
(a)
(b)
(c)
(d)
(e)
(f)For the composite doping of Y and Sb, the Mg and (or) Si atoms are replaced by Y and Sb, respectively. Y or Sb enters into the interstitial space of the Mg2Si cell, and Sb or Y replaces the Mg and (or) Si atoms (see Figure 3).Figure 3
The structures of composite doping.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)The hypothetical structures of Y- and Sb doping or codoping Mg2Si configurations must be optimized. The calculations were performed using the CASTEP package based on the density functional theory (DFT) [29]. The exchange-correlation energy functional was adopted by Perdew, Burke, and Ernzerhof (PBE) from the generalized gradient approximation (GGA) [30]. The plane-wave cutoff energy was set to 340 eV. The Brillouin zone was sampled using a Monkhorst-Pack k-point mesh as follows: 4 × 3 × 4 for (j) Mg8Si2YSb, (l) Mg8Si3YSb, and (m) Mg8Si4YSb and the rest are 4 × 4 × 4. To find the lowest energy structures of Y- and Sb-doped Mg2Si phases, we observe the following convergence thresholds for geometry optimization: total energy convergence tolerance 1.0 × 10−5 eV/atom steps, 0.03 eV/Å for maximum force, 0.05 GPa for maximum stress components, and 1 × 10−3 Å for maximum displacement. After the optimization, the electronic properties are calculated. The density of states (DOS) is computed by means of a scheme developed by Ackland, while the DOS calculation is based on a Mülliken population analysis with the relative contribution of each atom.


## 1.1. Computational Models and Methods

Mg2Si has an antifluorite (CaF2) structure [28] (see Figure 1). The atomic coordinates are as follows: Mg (0.25, 0.25, 0.25) and Si (0, 0, 0).Figure 1
The crystal structure of Mg2Si.For Y and Sb single doping, the positions of the Mg (or Si) atoms are replaced by Y or Sb, respectively. Despite that both the atom sizes of Y and Sb are larger than those of Mg and Si atoms, to avoid omitting certain configurations, the interpolation Y or Sb atoms into the interval of Mg and Si was considered (see Figure2).Figure 2
The structures of single doping.
(a)
(b)
(c)
(d)
(e)
(f)For the composite doping of Y and Sb, the Mg and (or) Si atoms are replaced by Y and Sb, respectively. Y or Sb enters into the interstitial space of the Mg2Si cell, and Sb or Y replaces the Mg and (or) Si atoms (see Figure 3).Figure 3
The structures of composite doping.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)The hypothetical structures of Y- and Sb doping or codoping Mg2Si configurations must be optimized. The calculations were performed using the CASTEP package based on the density functional theory (DFT) [29]. The exchange-correlation energy functional was adopted by Perdew, Burke, and Ernzerhof (PBE) from the generalized gradient approximation (GGA) [30]. The plane-wave cutoff energy was set to 340 eV. The Brillouin zone was sampled using a Monkhorst-Pack k-point mesh as follows: 4 × 3 × 4 for (j) Mg8Si2YSb, (l) Mg8Si3YSb, and (m) Mg8Si4YSb and the rest are 4 × 4 × 4. To find the lowest energy structures of Y- and Sb-doped Mg2Si phases, we observe the following convergence thresholds for geometry optimization: total energy convergence tolerance 1.0 × 10−5 eV/atom steps, 0.03 eV/Å for maximum force, 0.05 GPa for maximum stress components, and 1 × 10−3 Å for maximum displacement. After the optimization, the electronic properties are calculated. The density of states (DOS) is computed by means of a scheme developed by Ackland, while the DOS calculation is based on a Mülliken population analysis with the relative contribution of each atom.


## 2. Results and Discussion

### 2.1. Structures

The optimized lattice parameters of the pure Mg2Si phase and Y- and Sb-doped Mg2Si phases are shown in Table 1. It can be seen that the lattice parameters of Mg2Si are in agreement with the experimental values [31–33]. Y- and Sb-doped Mg2Si phases have larger lattice parameters and larger volume than the pure Mg2Si phase. It is due to the lattice distortion of the Mg2Si phase by Y and (or) Sb doping which possesses a larger atomic radius.Table 1
The optimized lattice parameters for Y and (or) Sb doping of Mg2Si.
Phase
Lattice constants (nm)
Cell volume (nm3)
Ref.
Mg8Si4
a=0.633
V=0.254
a=0.639
V=0.261
[31]
a=0.635
V=0.256
[32]
a=0.634
V=0.255
[33]
(a) Mg7Si4Y
a=0.643
V=0.266
(b) Mg8Si3Y
a=0.637
b=0.687
c=0.687
V=0.301
(c) Mg8Si4Y
a=0.653
V=0.278
(d) Mg7Si4Sb
a=0.640
V=0.263
(e) Mg8Si3Sb
a=0.645
b=0.643
c=0.643
V=0.267
(f) Mg8Si4Sb
a=0.656
V=0.282
(g) Mg6Si4YSb
a=0.659
b=0.629
c=0.659
V=0.273
(h) Mg7Si3YSb
a=0.654
b=0.655
c=0.655
V=0.280
(i) Mg7Si4YSb
a=0.662
V=0.290
(j) Mg8Si2YSb
a=0.607
b=0.833
c=0.607
V=0.307
(k) Mg7Si3YSb
a=0.741
b=0.647
c=0.649
V=0.273
(l) Mg8Si3YSb
a=0.615
b=0.820
c=0.604
V=0.304
(m) Mg8Si4YSb
a=0.662
b=0.787
c=0.662
V=0.345
(n) Mg7Si4YSb
a=0.662
V=0.289
(o) Mg8Si3YSb
a=0.643
b=0.690
c=0.661
V=0.293


### 2.2. Thermodynamic Properties

In order to understand the thermodynamic stability of the doped system, we calculated the formation energy and cohesive energy. The calculated formation energy and cohesive energy of the Mg2Si phase agree well with the calculated values (−25.92 kJ/mol and −271.15 kJ/mol in [34] and −17.7 kJ/mol and −299.27 kJ/mol in [35]). The process of Y and Sb doping of Mg2Si is outlined in Table 2.Table 2
The process of Y and Sb doping of Mg2Si.
Reactants
Products
Mg8Si4
Y
→
(a) Mg7Si4Y
Mg
Mg8Si4
Y
→
(b) Mg8Si3Y
Si
Mg8Si4
Y
→
(c) Mg8Si4Y
Mg8Si4
Sb
→
(d) Mg7Si4Sb
Mg
Mg8Si4
Sb
→
(e) Mg8Si3Sb
Si
Mg8Si4
Sb
→
(f) Mg8Si4Sb
Mg8Si4
Y
Sb
→
(g) Mg6Si4YSb
Mg
Mg
Mg8Si4
Y
Sb
→
(h) Mg7Si3YSb
Mg
Si
Mg8Si4
Y
Sb
→
(i) Mg7Si4YSb
Mg
Mg8Si4
Y
Sb
→
(j) Mg8Si2YSb
Si
Si
Mg8Si4
Y
Sb
→
(k) Mg7Si3YSb
Mg
Si
Mg8Si4
Y
Sb
→
(l) Mg8Si3YSb
Si
Mg8Si4
Y
Sb
→
(m) Mg8Si4YSb
Mg8Si4
Y
Sb
→
(n) Mg7Si4YSb
Mg
Mg8Si4
Y
Sb
→
(o) Mg8Si3YSb
SiThe corresponding formation energies are shown in Figure4. It can be seen that the formation energies of (a) Mg7Si4Y, (c) Mg8Si4Y, (f) Mg8Si4Sb, (g) Mg6Si4YSb, (i) Mg7Si4YSb, (m) Mg8Si4YSb, and (n) Mg7Si4YSb are negative, indicating that only the structures above are stable. The formation energy is successively (m) < (i) < (n) < (c) < (f) < (g) < (a). It means that (m) Mg8Si4YSb is the easiest to form and (a) Mg7Si4Y is the most difficult to form. Whether for single doping or composite doping, the formation energies of Y and Sb entered into the interstitial space < the formation energies of Y and Sb replaced the Mg sites < the formation energies of Y and Sb replaced the Si sites (even if it cannot form stable structures). It confirms that Y or Sb is difficult to interpolate into Mg2Si phases. For the seven stable structures of (a), (c), (f), (g), (i), (m), and (n), the formation energies of composite doping are generally smaller than those of single doping. Only the formation of (g) Mg6Si4YSb is greater than those of (c) Mg8Si4Y and (f) Mg8Si4Sb; this is because Y and Sb in (g) Mg6Si4YSb has replaced all the Mg sites to form substitutional solid solutions.Figure 4
The formation energies of single doping and composite doping.For the stable structures (a), (c), (f), (g), (i), (m), and (n), the calculated cohesive energies are shown in Figure5. It can be seen that the absolute value of (g) Mg6Si4YSb is the largest, indicating that (g) Mg6Si4YSb is more stable than the other six structures and (m) Mg8Si4YSb has the smallest absolute value and the worst stability. Whether by single doping or composite doping, the structures of Y- or Sb-substituted Mg sites are most stable, while the structures of Y or Sb that entered into the interstitial space are most unstable.Figure 5
The cohesive energies of single doping and composite doping.


### 2.3. Mechanical Properties

To satisfy the mechanical stability of the formed structures, the elastic constants of Mg2Si, (a), (c), (f), (g), (i), (m), and (n) are calculated as shown in Table 3. (a) Mg7Si4Y, (c) Mg8Si4Y, and (f) Mg8Si4Sb are still cubic structures, (g) Mg6Si4YSb and (m) Mg8Si4YSb are transformed into tetragonal structures, and (i) Mg7Si4YSb and (n) Mg7Si4YSb are transformed into triclinic structures. The mechanical stability of the phases above can be judged by the criterion for cubic, tetragonal, and triclinic systems in References [36] and [37]. Only Mg2Si, (a) Mg7Si4Y, (f) Mg8Si4Sb, (g) Mg6Si4YSb, and (i) Mg7Si4YSb satisfies the criteria of mechanical stability.Table 3
The elastic constants for Mg2Si, (a) Mg7Si4Y, (c) Mg8Si4Y, (f) Mg8Si4Sb, (g) Mg6Si4YSb, (i) Mg7Si4YSb, (m) Mg8Si4YSb, and (n) Mg7Si4YSb.
Phases
Elastic stiffness constantsCij (GPa)
Mg2Si
C11=131.31
C12=11.19
C44=46.46
(a) Mg7Si4Y
C11=106.00
C12=25.76
C44=29.00
(c) Mg8Si4Y
C11=22.72
C12=60.54
C44=21.17
(f) Mg8Si4Sb
C11=53.73
C12=40.66
C44=16.16
(g) Mg6Si4YSb
C11=99.10
C12=26.56
C13=28.41
C33=99.21
C44=13.71
C66=13.03
(i) Mg7Si4YSb
C11=96.36
C12=23.99
C13=24.26
C14=−1.44
C15=2.91
C16=−1.61
C21=23.99
C22=96.44
C23=24.34
C24=−1.78
C25=2.58
C26=−1.05
C31=24.26
C32=24.34
C33=96.63
C34=−1.17
C35=1.69
C36=−1.56
C41=−1.44
C42=−1.78
C43=−1.17
C44=21.21
C45=−0.91
C46=1.69
C51=2.91
C52=2.58
C53=1.69
C54=−0.91
C55=19.72
C56=−2.92
C61=−1.61
C62=−1.05
C63=−1.56
C64=1.69
C65=−2.92
C66=20.75
(m) Mg8Si4YSb
C11=75.12
C12=20.81
C13=34.21
C33=75.02
C44=−11.73
C66=−9.39
(n) Mg7Si4YSb
C11=52.86
C12=50.91
C13=51.14
C14=−8.13
C15=−3.89
C16=3.88
C21=50.91
C22=53.37
C23=52.35
C24=2.55
C25=8.78
C26=1.21
C31=51.14
C32=52.35
C33=52.71
C34=0.13
C35=−0.49
C36=−8.16
C41=−8.13
C42=2.55
C43=0.13
C44=21.38
C45=−4.73
C46=4.06
C51=−3.89
C52=8.78
C53=−0.49
C54=−4.73
C55=19.08
C56=−3.33
C61=3.88
C62=1.21
C63=−8.16
C64=4.06
C65=−3.33
C66=18.76The calculated bulk modulus, shear modulus, and Young’s modulus (51.23 GPa, 51.50 GPa, and 115.85 GPa) of Mg2Si agree well with the calculated results (57.88 GPa, 48.82 GPa, and 114.32 GPa [38]). Young’s modulus calculated for Mg2Si, (a) Mg7Si4Y, (f) Mg8Si4Sb, (g) Mg6Si4YSb, and (i) Mg7Si4YSb are shown in Figure 6. No matter with what kind of doping, Young’s modulus of pure Mg2Si will decrease significantly. It means that Mg2Si has the highest stiffness. In other words, Y and Sb doping will improve the plasticity of Mg2Si. For single doping, Young’s modulus of doped Sb is lesser than that of doped Y. This shows that the plasticity of (f) Mg8Si4Sb is better than that of (a) Mg7Si4Y, and the stiffness of (a) Mg7Si4Y is larger than that of (f) Mg8Si4Sb. For composite doping, Young’s modulus of (g) Mg6Si4YSb is less than Young’s modulus of (i) Mg7Si4YSb. This shows that the plasticity of (g) Mg6Si4YSb is better than that of (i) Mg7Si4YSb, and the stiffness of (g) Mg6Si4YSb is lesser than that of (i) Mg7Si4YSb.Figure 6
The results of Young’s modulus.The values ofK/G, Poisson’s ratio u, AU, and Debye temperature ΘD for Mg2Si, (a) Mg7Si4Y, (f) Mg8Si4Sb, (g) Mg6Si4YSb, and (i) Mg7Si4YSb are shown in Table 4. Poisson’s ratio u (0.12) and Debye temperature ΘD (600.23 K) calculated for Mg2Si agree with the results (0.161 and 581.6719 K [38]). From the values of K/G, it can be seen that the values of Mg2Si and (a) Mg7Si4Y are less than the threshold value of 1.75, so Mg2Si and (a) Mg7Si4Y are brittle materials, while (f) Mg8Si4Sb, (g) Mg6Si4YSb, and (i) Mg7Si4YSb are ductile materials. Mg2Si has minimal K/G and has the worst brittleness, while (f) Mg8Si4Sb has maximalK/G and has the greatest ductility. From the values of u, it can be seen that (f) Mg8Si4Sb has the best plasticity and Mg2Si has the worst plasticity. The plasticity is successively (f) Mg8Si4Sb > (g) Mg6Si4YSb > (i) Mg7Si4YSb > (a) Mg7Si4Y > Mg2Si. From AU, it can be seen that the five structures are all anisotropic and Mg2Si has minimal value, so the degree of anisotropy is minimal. (g) Mg6Si4YSb has maximal value, and the degree of anisotropy is the highest. The degree of anisotropy of Mg2Si will be raised by doped Y and Sb. From the values of ΘD, it can be seen that the Debye temperature (ΘD) of Mg2Si is the highest, which shows that the thermal conductivity of Mg2Si is the best, the chemical bonds between the cell’s atoms are the strongest, and the hardness is the greatest. (f) Mg8Si4Sb has the lowest Debye temperature (ΘD), which shows that the thermal conductivity of (f) Mg8Si4Sb is the worst, the chemical bonds between the cell’s atoms are the weakest, and the hardness is the least. Furthermore, no matter with what kind of doping, the Debye temperature (ΘD) of Mg2Si will be lowered. It is consistent with the calculated results of Debye temperature (ΘD) [27].Table 4
The values of theK/G, u, AU, and ΘD for Mg2Si, (a) Mg7Si4Y, (f) Mg8Si4Sb, (g) Mg6Si4YSb, and (i) Mg7Si4YSb.
Mg2Si
(a) Mg7Si4Y
(f) Mg8Si4Sb
(g) Mg6Si4YSb
(i) Mg7Si4YSb
K/G
0.99
1.59
3.99
2.53
1.88
u
0.12
0.24
0.38
0.33
0.27
AU
7.93
12.85
104.51
138.31
49.18
ΘD (K)
600.23
445.60
248.77
331.88
346.11The shear anisotropic factorsA1, A2, and A3 are shown in Figure 7. It can be found that the five structures are all anisotropic on the three planes, and the degree of anisotropy on each plane is very similar. (f) Mg8Si4Sb has the highest degree of anisotropy, while Mg2Si has the lowest degree of anisotropy.Figure 7
The results of shear anisotropy.


### 2.4. Electronic Properties

To understand the action mechanism, the total density of states (DOS) and the partial density of states (PDOS) should be studied. Figure8(a) shows the density of states of Mg7Si4Y, between −3 eV and the Fermi level, where there exists a very strong orbital hybridization for Mg (2p) and Y (4d) indicating very strong bonding effects. Figure 8(b) shows the density of states of Mg8Si4Sb, where there is strong orbital hybridization between the atoms of Mg, Si, and Sb. Figure 8(c) shows the density of states of Mg6Si4YSb. The mainly orbital hybridization appears in −6 to 4 eV. The atoms of Mg and Y have strong bonding effects, while the bonding effects of the other atoms are weak. Figure 8(d) shows the density of states of Mg7Si4YSb, where all the four atoms show orbital hybridization. In a word, Y doping causes strong orbital hybridization between Mg (2p) and Y (4d), while Sb doping causes strong orbital hybridization between Mg (2p), Si (3p), and Sb (5p). The Fermi surface shifts towards the high-energy region after Y and Sb doping, resulting in an increase in the density of the electron energy states corresponding to the Fermi level. The values are 2 eV (Mg2Si), 3.05 eV (Mg7Si4Y), 5.4 eV (Mg8Si4Sb), 3.7 eV (Mg6Si4YSb), and 2.9 eV (Mg7Si4YSb) in turn, indicating that doping can improve the electrical conductivity of Mg2Si. This is also consistent with the results which were obtained in Reference [27]. The electrical conductivity of Mg8Si4Sb is the best.Figure 8
The total density of states (DOS) and the partial density of states (PDOS) for five structures.
(a)
(b)
(c)
(d)
(e)


## 2.1. Structures

The optimized lattice parameters of the pure Mg2Si phase and Y- and Sb-doped Mg2Si phases are shown in Table 1. It can be seen that the lattice parameters of Mg2Si are in agreement with the experimental values [31–33]. Y- and Sb-doped Mg2Si phases have larger lattice parameters and larger volume than the pure Mg2Si phase. It is due to the lattice distortion of the Mg2Si phase by Y and (or) Sb doping which possesses a larger atomic radius.Table 1
The optimized lattice parameters for Y and (or) Sb doping of Mg2Si.
Phase
Lattice constants (nm)
Cell volume (nm3)
Ref.
Mg8Si4
a=0.633
V=0.254
a=0.639
V=0.261
[31]
a=0.635
V=0.256
[32]
a=0.634
V=0.255
[33]
(a) Mg7Si4Y
a=0.643
V=0.266
(b) Mg8Si3Y
a=0.637
b=0.687
c=0.687
V=0.301
(c) Mg8Si4Y
a=0.653
V=0.278
(d) Mg7Si4Sb
a=0.640
V=0.263
(e) Mg8Si3Sb
a=0.645
b=0.643
c=0.643
V=0.267
(f) Mg8Si4Sb
a=0.656
V=0.282
(g) Mg6Si4YSb
a=0.659
b=0.629
c=0.659
V=0.273
(h) Mg7Si3YSb
a=0.654
b=0.655
c=0.655
V=0.280
(i) Mg7Si4YSb
a=0.662
V=0.290
(j) Mg8Si2YSb
a=0.607
b=0.833
c=0.607
V=0.307
(k) Mg7Si3YSb
a=0.741
b=0.647
c=0.649
V=0.273
(l) Mg8Si3YSb
a=0.615
b=0.820
c=0.604
V=0.304
(m) Mg8Si4YSb
a=0.662
b=0.787
c=0.662
V=0.345
(n) Mg7Si4YSb
a=0.662
V=0.289
(o) Mg8Si3YSb
a=0.643
b=0.690
c=0.661
V=0.293


## 2.2. Thermodynamic Properties

In order to understand the thermodynamic stability of the doped system, we calculated the formation energy and cohesive energy. The calculated formation energy and cohesive energy of the Mg2Si phase agree well with the calculated values (−25.92 kJ/mol and −271.15 kJ/mol in [34] and −17.7 kJ/mol and −299.27 kJ/mol in [35]). The process of Y and Sb doping of Mg2Si is outlined in Table 2.Table 2
The process of Y and Sb doping of Mg2Si.
Reactants
Products
Mg8Si4
Y
→
(a) Mg7Si4Y
Mg
Mg8Si4
Y
→
(b) Mg8Si3Y
Si
Mg8Si4
Y
→
(c) Mg8Si4Y
Mg8Si4
Sb
→
(d) Mg7Si4Sb
Mg
Mg8Si4
Sb
→
(e) Mg8Si3Sb
Si
Mg8Si4
Sb
→
(f) Mg8Si4Sb
Mg8Si4
Y
Sb
→
(g) Mg6Si4YSb
Mg
Mg
Mg8Si4
Y
Sb
→
(h) Mg7Si3YSb
Mg
Si
Mg8Si4
Y
Sb
→
(i) Mg7Si4YSb
Mg
Mg8Si4
Y
Sb
→
(j) Mg8Si2YSb
Si
Si
Mg8Si4
Y
Sb
→
(k) Mg7Si3YSb
Mg
Si
Mg8Si4
Y
Sb
→
(l) Mg8Si3YSb
Si
Mg8Si4
Y
Sb
→
(m) Mg8Si4YSb
Mg8Si4
Y
Sb
→
(n) Mg7Si4YSb
Mg
Mg8Si4
Y
Sb
→
(o) Mg8Si3YSb
SiThe corresponding formation energies are shown in Figure4. It can be seen that the formation energies of (a) Mg7Si4Y, (c) Mg8Si4Y, (f) Mg8Si4Sb, (g) Mg6Si4YSb, (i) Mg7Si4YSb, (m) Mg8Si4YSb, and (n) Mg7Si4YSb are negative, indicating that only the structures above are stable. The formation energy is successively (m) < (i) < (n) < (c) < (f) < (g) < (a). It means that (m) Mg8Si4YSb is the easiest to form and (a) Mg7Si4Y is the most difficult to form. Whether for single doping or composite doping, the formation energies of Y and Sb entered into the interstitial space < the formation energies of Y and Sb replaced the Mg sites < the formation energies of Y and Sb replaced the Si sites (even if it cannot form stable structures). It confirms that Y or Sb is difficult to interpolate into Mg2Si phases. For the seven stable structures of (a), (c), (f), (g), (i), (m), and (n), the formation energies of composite doping are generally smaller than those of single doping. Only the formation of (g) Mg6Si4YSb is greater than those of (c) Mg8Si4Y and (f) Mg8Si4Sb; this is because Y and Sb in (g) Mg6Si4YSb has replaced all the Mg sites to form substitutional solid solutions.Figure 4
The formation energies of single doping and composite doping.For the stable structures (a), (c), (f), (g), (i), (m), and (n), the calculated cohesive energies are shown in Figure5. It can be seen that the absolute value of (g) Mg6Si4YSb is the largest, indicating that (g) Mg6Si4YSb is more stable than the other six structures and (m) Mg8Si4YSb has the smallest absolute value and the worst stability. Whether by single doping or composite doping, the structures of Y- or Sb-substituted Mg sites are most stable, while the structures of Y or Sb that entered into the interstitial space are most unstable.Figure 5
The cohesive energies of single doping and composite doping.


## 2.3. Mechanical Properties

To satisfy the mechanical stability of the formed structures, the elastic constants of Mg2Si, (a), (c), (f), (g), (i), (m), and (n) are calculated as shown in Table 3. (a) Mg7Si4Y, (c) Mg8Si4Y, and (f) Mg8Si4Sb are still cubic structures, (g) Mg6Si4YSb and (m) Mg8Si4YSb are transformed into tetragonal structures, and (i) Mg7Si4YSb and (n) Mg7Si4YSb are transformed into triclinic structures. The mechanical stability of the phases above can be judged by the criterion for cubic, tetragonal, and triclinic systems in References [36] and [37]. Only Mg2Si, (a) Mg7Si4Y, (f) Mg8Si4Sb, (g) Mg6Si4YSb, and (i) Mg7Si4YSb satisfies the criteria of mechanical stability.Table 3
The elastic constants for Mg2Si, (a) Mg7Si4Y, (c) Mg8Si4Y, (f) Mg8Si4Sb, (g) Mg6Si4YSb, (i) Mg7Si4YSb, (m) Mg8Si4YSb, and (n) Mg7Si4YSb.
Phases
Elastic stiffness constantsCij (GPa)
Mg2Si
C11=131.31
C12=11.19
C44=46.46
(a) Mg7Si4Y
C11=106.00
C12=25.76
C44=29.00
(c) Mg8Si4Y
C11=22.72
C12=60.54
C44=21.17
(f) Mg8Si4Sb
C11=53.73
C12=40.66
C44=16.16
(g) Mg6Si4YSb
C11=99.10
C12=26.56
C13=28.41
C33=99.21
C44=13.71
C66=13.03
(i) Mg7Si4YSb
C11=96.36
C12=23.99
C13=24.26
C14=−1.44
C15=2.91
C16=−1.61
C21=23.99
C22=96.44
C23=24.34
C24=−1.78
C25=2.58
C26=−1.05
C31=24.26
C32=24.34
C33=96.63
C34=−1.17
C35=1.69
C36=−1.56
C41=−1.44
C42=−1.78
C43=−1.17
C44=21.21
C45=−0.91
C46=1.69
C51=2.91
C52=2.58
C53=1.69
C54=−0.91
C55=19.72
C56=−2.92
C61=−1.61
C62=−1.05
C63=−1.56
C64=1.69
C65=−2.92
C66=20.75
(m) Mg8Si4YSb
C11=75.12
C12=20.81
C13=34.21
C33=75.02
C44=−11.73
C66=−9.39
(n) Mg7Si4YSb
C11=52.86
C12=50.91
C13=51.14
C14=−8.13
C15=−3.89
C16=3.88
C21=50.91
C22=53.37
C23=52.35
C24=2.55
C25=8.78
C26=1.21
C31=51.14
C32=52.35
C33=52.71
C34=0.13
C35=−0.49
C36=−8.16
C41=−8.13
C42=2.55
C43=0.13
C44=21.38
C45=−4.73
C46=4.06
C51=−3.89
C52=8.78
C53=−0.49
C54=−4.73
C55=19.08
C56=−3.33
C61=3.88
C62=1.21
C63=−8.16
C64=4.06
C65=−3.33
C66=18.76The calculated bulk modulus, shear modulus, and Young’s modulus (51.23 GPa, 51.50 GPa, and 115.85 GPa) of Mg2Si agree well with the calculated results (57.88 GPa, 48.82 GPa, and 114.32 GPa [38]). Young’s modulus calculated for Mg2Si, (a) Mg7Si4Y, (f) Mg8Si4Sb, (g) Mg6Si4YSb, and (i) Mg7Si4YSb are shown in Figure 6. No matter with what kind of doping, Young’s modulus of pure Mg2Si will decrease significantly. It means that Mg2Si has the highest stiffness. In other words, Y and Sb doping will improve the plasticity of Mg2Si. For single doping, Young’s modulus of doped Sb is lesser than that of doped Y. This shows that the plasticity of (f) Mg8Si4Sb is better than that of (a) Mg7Si4Y, and the stiffness of (a) Mg7Si4Y is larger than that of (f) Mg8Si4Sb. For composite doping, Young’s modulus of (g) Mg6Si4YSb is less than Young’s modulus of (i) Mg7Si4YSb. This shows that the plasticity of (g) Mg6Si4YSb is better than that of (i) Mg7Si4YSb, and the stiffness of (g) Mg6Si4YSb is lesser than that of (i) Mg7Si4YSb.Figure 6
The results of Young’s modulus.The values ofK/G, Poisson’s ratio u, AU, and Debye temperature ΘD for Mg2Si, (a) Mg7Si4Y, (f) Mg8Si4Sb, (g) Mg6Si4YSb, and (i) Mg7Si4YSb are shown in Table 4. Poisson’s ratio u (0.12) and Debye temperature ΘD (600.23 K) calculated for Mg2Si agree with the results (0.161 and 581.6719 K [38]). From the values of K/G, it can be seen that the values of Mg2Si and (a) Mg7Si4Y are less than the threshold value of 1.75, so Mg2Si and (a) Mg7Si4Y are brittle materials, while (f) Mg8Si4Sb, (g) Mg6Si4YSb, and (i) Mg7Si4YSb are ductile materials. Mg2Si has minimal K/G and has the worst brittleness, while (f) Mg8Si4Sb has maximalK/G and has the greatest ductility. From the values of u, it can be seen that (f) Mg8Si4Sb has the best plasticity and Mg2Si has the worst plasticity. The plasticity is successively (f) Mg8Si4Sb > (g) Mg6Si4YSb > (i) Mg7Si4YSb > (a) Mg7Si4Y > Mg2Si. From AU, it can be seen that the five structures are all anisotropic and Mg2Si has minimal value, so the degree of anisotropy is minimal. (g) Mg6Si4YSb has maximal value, and the degree of anisotropy is the highest. The degree of anisotropy of Mg2Si will be raised by doped Y and Sb. From the values of ΘD, it can be seen that the Debye temperature (ΘD) of Mg2Si is the highest, which shows that the thermal conductivity of Mg2Si is the best, the chemical bonds between the cell’s atoms are the strongest, and the hardness is the greatest. (f) Mg8Si4Sb has the lowest Debye temperature (ΘD), which shows that the thermal conductivity of (f) Mg8Si4Sb is the worst, the chemical bonds between the cell’s atoms are the weakest, and the hardness is the least. Furthermore, no matter with what kind of doping, the Debye temperature (ΘD) of Mg2Si will be lowered. It is consistent with the calculated results of Debye temperature (ΘD) [27].Table 4
The values of theK/G, u, AU, and ΘD for Mg2Si, (a) Mg7Si4Y, (f) Mg8Si4Sb, (g) Mg6Si4YSb, and (i) Mg7Si4YSb.
Mg2Si
(a) Mg7Si4Y
(f) Mg8Si4Sb
(g) Mg6Si4YSb
(i) Mg7Si4YSb
K/G
0.99
1.59
3.99
2.53
1.88
u
0.12
0.24
0.38
0.33
0.27
AU
7.93
12.85
104.51
138.31
49.18
ΘD (K)
600.23
445.60
248.77
331.88
346.11The shear anisotropic factorsA1, A2, and A3 are shown in Figure 7. It can be found that the five structures are all anisotropic on the three planes, and the degree of anisotropy on each plane is very similar. (f) Mg8Si4Sb has the highest degree of anisotropy, while Mg2Si has the lowest degree of anisotropy.Figure 7
The results of shear anisotropy.


## 2.4. Electronic Properties

To understand the action mechanism, the total density of states (DOS) and the partial density of states (PDOS) should be studied. Figure8(a) shows the density of states of Mg7Si4Y, between −3 eV and the Fermi level, where there exists a very strong orbital hybridization for Mg (2p) and Y (4d) indicating very strong bonding effects. Figure 8(b) shows the density of states of Mg8Si4Sb, where there is strong orbital hybridization between the atoms of Mg, Si, and Sb. Figure 8(c) shows the density of states of Mg6Si4YSb. The mainly orbital hybridization appears in −6 to 4 eV. The atoms of Mg and Y have strong bonding effects, while the bonding effects of the other atoms are weak. Figure 8(d) shows the density of states of Mg7Si4YSb, where all the four atoms show orbital hybridization. In a word, Y doping causes strong orbital hybridization between Mg (2p) and Y (4d), while Sb doping causes strong orbital hybridization between Mg (2p), Si (3p), and Sb (5p). The Fermi surface shifts towards the high-energy region after Y and Sb doping, resulting in an increase in the density of the electron energy states corresponding to the Fermi level. The values are 2 eV (Mg2Si), 3.05 eV (Mg7Si4Y), 5.4 eV (Mg8Si4Sb), 3.7 eV (Mg6Si4YSb), and 2.9 eV (Mg7Si4YSb) in turn, indicating that doping can improve the electrical conductivity of Mg2Si. This is also consistent with the results which were obtained in Reference [27]. The electrical conductivity of Mg8Si4Sb is the best.Figure 8
The total density of states (DOS) and the partial density of states (PDOS) for five structures.
(a)
(b)
(c)
(d)
(e)


## 3. Conclusions

The structural, stability, thermodynamic, and mechanical properties of Y and Sb doped into Mg2Si are calculated by the GGA-PBE method. The results revealed that the stable structures that can be formed are (a) Mg7Si4Y, (f) Mg8Si4Sb, (g) Mg6Si4YSb, and (i) Mg7Si4YSb. (g) Mg6Si4YSb is the most stable, while (f) Mg8Si4Sb is the least stable. Young’s modulus of Mg2Si will be reduced by Y and Sb doping, the ductility of Mg2Si will be improved, and the brittleness of Mg2Si will be well solved. Among them, the effect of single-doped Sb is better. The Debye temperature (ΘD) of Mg2Si will be lowered, and then the thermal conductivity and hardness of Mg2Si will be reduced by Y and Sb doping. The Debye temperature (ΘD) of (f) Mg8Si4Sb in the five structures is the lowest. Doping with Y and Sb can improve the conductivity of Mg2Si, and Mg8Si4Sb has the best conductivity.



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*Source: 6307163-2019-03-10.xml*