Liu Futi; Cheng Xiaohong; Zhang Shuhua

[Abstract] The structure of Mg2Si crystal was geometrically optimized using the pseudopotential plane wave method of density functional theory. Based on the optimization, the electronic structure, elastic constants and thermodynamic properties were calculated by first principles. The results show that Mg2Si is an indirect bandgap semiconductor with a band gap of 0.2846eV; The conduction band is mainly composed of 3p, 3s of Mg and 3p electrons of Si; Elastic constants C11=114.39GPa, C12=22.45GPa, C44=42.78GPa; The Debye temperature at zero temperature and zero pressure is 676.4K. The phonon dispersion relationship is determined by using the linear response method, and the relationship between the change of temperature and the thermodynamic functions of Mg2Si such as isovolumetric heat capacity, Debye temperature, enthalpy, free energy and entropy is obtained.% The electronic structure, elastic constants and thermodynamic properties of Mg2Si were calculated based on the optimized structure by using the first-principles pseudo-potential method of density functional theory.The results showed that the indirect band gap of Mg2Si was 0.2846eV; conduction band was constituted mainly by 3p,3s electrons of Mg atoms and 3p electrons of Si atom; elastic constants C11=114.39GPa,C12=22.45GPa,C44=42.78GPa; Debye temperature is 676.4K at zero pressure and zero temperature.The linear response method was applied to determine the phonon dispersion relations,and the relations of thermodynamic functions of heat capacity,Debye temperature,enthalpy,free energy,entropy with temperature were calculated.

Journal of Yibin University

[year (volume), issue] 2012 (000) 006

[Total pages] 4 pages (P39-42)

[Key words] Magnesium silicide; Electronic structure; Elastic constant; Thermodynamic properties

[Author] Liu Futi; Cheng Xiaohong; Zhang Shuhua

[Author's unit] School of Physics and Electronic Engineering, Yibin University, Sichuan 644000, Key Laboratory of Computational Physics, Yibin University, Sichuan 644000; Key Laboratory of Computational Physics, Yibin University, Sichuan Province, 644000; Experiment and Teaching Resource Management Center of Yibin University, Yibin 644000, Sichuan

[Text language] Chinese

[Chinese Library Classification] O48

Metal silicide materials have many excellent thermal, electrical and mechanical properties, among which magnesium silicide (Mg2Si) is the only stable compound of Mg-Si binary system, which has the characteristics of high melting point, high hardness and high elastic modulus. It is a narrow band gap n-type semiconductor material, which is used in optoelectronic devices, electronic devices, energy devices, lasers, semiconductor manufacturing There are important application prospects in the fields of constant temperature control communication [1-5]. Because the raw material resources of alloy elements Mg and Si are very rich, the formation content is large, the price is low, non-toxic and pollution-free, Mg2Si as a new environmental semiconductor material has attracted great attention of researchers. In recent years, many scholars have carried out many studies on various properties of Mg2Si materials, such as lattice dynamics of Mg2Si [6], Energy band structure and dielectric function [7-10], doping and optical properties [11-12], geometric structure, elasticity and thermodynamic properties [13-14] have been studied. These research results are of great significance to the utilization and design of Mg2Si materials, but there are few studies on the thermal capacity, Debye temperature and other thermodynamic properties of Mg2Si. Therefore, this paper uses the pseudopotential plane wave method based on the first principle to study the energy band structure, elastic constants The Debye temperature, heat capacity, free energy, entropy and other thermodynamic functions are calculated theoretically to provide predictions for the experiment and design of Mg2Si materials

The Mg2 Si crystal has an anti-fluorite structure and belongs to a face-centered cubic lattice. The space group is Fm3m, the group number is 225, and the lattice constant is a=0.6391nm [15]. In the crystal structure, each Si atom is located at the (0,0,0) position, and the coordination number is 8, forming a face-centered cubic structure with a side length. The Mg atom is located at the (0.25, 0.25, 0.25) position, and each Mg atom is located at the center of the tetrahedron composed of Si atoms, forming a simple cubic structure with a side length of a/2, The cell structure of Mg2 Si is shown in Figure 1

In this paper, the first-principles pseudo-potential plane wave method based on density functional theory is adopted. The theoretical calculations are all completed by the quantum mechanics module CASTEP package in Material Studios 5.0 software. It is an ab initio quantum mechanics program based on density functional theory, which replaces the ion potential with pseudo-potential and expands the electron wave function with plane wave basis set, The exchange and correlation potential of the interaction between electrons and electrons are corrected by the local density approximation (LDA) or the generalized gradient approximation (GGA), which is currently recognized as a more accurate theoretical method for the calculation of electronic structure. The specific parameters are set as follows. In geometric optimization and electronic structure calculation, the energy cut-off value (Ultrafine) is 380eV, and the energy convergence (Energy tolerance) is 5.0 × 10-6 eV/atom, the maximum force convergence accuracy acting on each atom is 0.01 eV/⊙, the maximum strain convergence is 0.02 GPa, and the maximum displacement convergence is 5.0 × 10-4 ⊙. The Monkhorst-Pack method is used to set k points in the reciprocal space Brillouin, and the density is 4 × four × 4. The pseudopotential selects the super-soft pseudopotential. The valence electrons involved in the calculation of the pseudopotential are Mg: 2p6 3s2 and Si: 3s2 3p2, respectively. The exchange correlation functional between electrons uses the RPBE scheme in the generalized gradient approximation (GGA). In the calculation of phonon scattering, the energy cutoff energy is 900eV, and the convergence accuracy of self-consistent calculation is 1.0 × 10-4 eV/cell, k point selection density is 3 × three × 3. Pseudopotential is the Norm-conserving pseudopotential proposed by Hamann

3.1 Structure optimization

According to the experimental value of Mg2Si lattice parameters, the corresponding crystal structure is established. After geometric optimization, the lattice parameter is a=0.6422 nm, which is very close to the experimental value of 0.6391 nm [15], with an error of 0.5%. It is within the acceptable range of the first-principles calculation. The calculation of the following properties is carried out on the basis of this optimized structure, indicating that the calculation results should have good predictability

3.2 Energy band structure

Through the calculation of the energy band structure of Mg2Si, the energy band structure along the direction of the high symmetry point in the Brillouin region is obtained as shown in Figure 2

It can be seen from Figure 2 that the lowest point of the conduction band is at point X, and the highest point of the valence band is at point G, which is not at the same k point, indicating that Mg2 Si is an indirect bandgap semiconductor. The characteristic energy of the highly symmetric k point in the first Brillouin zone at the bottom of the conduction band Ec and the top of the valence band Ev is shown in Table 1. The minimum value of the energy obtained by the conduction band at point X is 0.2846eV, while the maximum value of the energy obtained by the valence band at point G is 0eV, so the band gap of Mg2 Si is 0.2846eV, It is close to the result of 0.2994eV of Chen Qian et al. [2]

The width of the energy band is very important in the analysis of the energy band. It can be seen from Table 1 that the width of the energy band at the bottom of the conduction band is (2.6761eV-0.2846eV=2.3915eV), and the width of the energy band at the top of the valence band is (-2.5231eV-0eV=-2.5231eV), that is, the energy band at the bottom of the conduction band is narrower than the energy band at the top of the valence band, which means that the effective mass of the electron in the bottom of the conduction band is larger than the effective mass of the hole at the top of the valence band, which means that Mg2Si is a heavy electron, Indirect gap semiconductor with light holes

Figure 3 shows the total density of states of Mg2Si. Through analysis, the energy band of Mg2Si has four regions, of which there are three valence bands: - 42.877 ～ - 42.863 eV is the lowest energy band, and the band width is very small, which is derived from the 3p state electrons of Mg- 8.937 ～ -6.994 eV is a sub-low energy valence band, which is mainly contributed by 3s electrons of Mg and 3s electrons of Si; The remaining energy bands close to the Fermi level correspond to the high energy valence band, with the energy range of -4.580~0 eV. The main contributions of this energy band are the 3p electrons of Mg and the 3p electrons of Si. The 3s electrons of Mg and the 3s electrons of Si make a small contribution to this energy band. The conduction band of Mg2Si is mainly the 3p electrons of Mg, the 3s electrons of Si and the 3p electrons of Si. The contribution of the 3s electrons of Si is relatively small

3.3 Elastic constant

Elasticity is a relatively important research object in the fields of material science, chemistry, physics and geophysics. The equation of state, specific heat capacity, Debye temperature, melting point, etc. of solid matter are all related to elasticity. From the elastic constants, we can obtain important information about the characteristics of crystal anisotropy and the stability of crystal structure. Mg2Si belongs to the face-centered cubic crystal system, and its elastic tensor Cij has three independent components, C11, C12 and C44, After geometric optimization, the elastic constants and bulk elastic moduli of Mg2Si at zero temperature and pressure are calculated as shown in Table 2

According to the isotropic coefficient, the S=21.075 of MgSi can be calculated, indicating that its isotropy is good. The bulk elastic modulus and shear modulus respectively represent the ability of material to resist fracture and plastic deformation, and their ratio can be used as a measure of ductility or brittleness. A high B/G (its critical value is 1.75) value means that the material is malleable, and a low B/G means that the material is fragile. The calculated B/G of Mg2Si=1.206, indicating that Mg2Si is fragile

3.4 Debye temperature

Debye temperature is an important physical quantity of the thermodynamic properties of substances. The Debye temperature of crystals can be calculated by using the elastic constant. The Debye temperature of Mg2Si crystals can be calculated by using the Debye approximation based on the elastic constant calculated previously. According to the Voigt [16] approximation, the shear modulus is based on the Reuss [17] approximation, and the shear modulus is based on the theoretical proof that the polycrystalline modulus is just the arithmetic mean value given by Voigt and Reuss, That is, for cubic crystals, when p=0GPa, the bulk modulus of elasticity is, and then the compression longitudinal wave velocity and shear wave velocity can be obtained from the shear modulus and bulk modulus of elasticity vs=, and the average sound velocity can be obtained. Finally, the Debye temperature can be obtained from the average sound velocity and Debye approximation

118.82 22.27 44.96 54.45 1.07 In the front, ħ Is Planck's constant, k is Boltzmann's constant, NA is Avogadro's constant, n is the number of atoms in the protocell, M is the mass of molecules in the protocell, and V is the volume of the protocell, ρ= M/V is the density

According to the above formula, the Debye temperature of Mg2Si at zero temperature and zero pressure is θ D=676.4K, Debye frequency ω D=8.856 × 1013 Hz, which directly reflect the thermodynamic properties of crystals

3.5 Heat capacity, enthalpy, free energy and entropy function

The linear response method is also used to determine the dispersion relationship of Mg2Si in the first Brillouin region and the density of phonon states, as shown in Figure 4 and Figure 5 respectively. There are two magnesium atoms and one silicon atom in the primary cell of Mg2Si crystal, and there are three acoustic waves and six optical waves in total. It can be seen from the figure that when the wave vector k approaches zero, the frequency of three lattice vibration waves tends to zero, and these three lattice vibration waves are acoustic waves, including two transverse waves, One longitudinal wave and the other six are optical waves. The calculated results show that the frequencies of optical waves of Mg2Si at G point are 7.829 respectively × 1012 Hz、8.336 × 1012 Hz and 9.408 × 1012 Hz.

Under the quasi-harmonic Debye approximation, the thermodynamic properties of Mg2Si are discussed by using the density of phonon states, and the heat capacity formula is used:

The heat capacity and Debye temperature of Mg2Si at a given temperature can be calculated. The isovolumetric heat capacity and Debye temperature in the temperature range of 0~1000K are shown in Fig. 6 and Fig. 7 respectively

At the temperature of 40.5K, the heat capacity is 1.264 J/mol. K, and the Debye temperature is 674.2K, which is very close to the Debye temperature (676.4) at zero temperature and zero pressure calculated by the elastic constant; At 303K, the heat capacity is 62.05 J/mol. K, while the experimental value is 67.9 J/mol. K. At this time, the Debye temperature is 585.5 K. With the increase of temperature, the heat capacity tends to 73.6 J/mol. K, which is very close to the classical limit value of 74.8 J/mol. K, which is consistent with Duron-Petit's law

Enthalpy, free energy and entropy functions all tend to zero when the temperature approaches 0K, which is consistent with the third law of thermodynamics. The specific relationship between them and temperature is shown in Figure 8

The electronic structure and thermodynamic properties of Mg2Si crystal are calculated by using the pseudopotential plane wave method based on density functional theory. The results show that Mg2Si is an indirect bandgap semiconductor with a band gap of 0.2846eV; The conduction band is mainly composed of 3p, 3s of Mg and 3p electrons of Si; Elastic constants C11=114.39GPa, C12=22.45GPa, C44=42.78 GPa; Debye temperature at zero temperature and pressure is 676.4K; The phonon dispersion relation is determined by using the linear response method. The Debye temperature of Mg2Si at 40.5K is 674.2K, and the isovolumetric heat capacity approaches the Dulon-Perty classical limit value of 74.8 J/mol.K with the increase of temperature; The relationship between enthalpy, free energy, entropy and other thermodynamic functions with temperature is consistent with the third law of thermodynamics

[Related literature]

[1] Jiang Hongyi, Zhang Lianmeng. Research status of Mg-Si-based thermoelectric compounds [J]. Introduction to Materials, 2002, 16 (3): 20-22

[2] Chen Qian, Xie Quan, Yan Wanjun, et al. First-principles calculation of electronic structure and optical properties of Mg2Si ［ J ］. Chinese Science Series G, 2008, 38 (7): 825-833

［3］ Song R B，Aizawa T，Sun JQ.SynthesisofMg2 Si1-x Snx solid solutions as thermoelectric materials by bulkmechanical alloying and hot pressingmaterials［J］.Mater Sci Eng B，2007(136)：111-116.

[4] Zang Shujun, Zhou Qi, Ma Qin, et al. Research progress of intermetallic compound Mg2 Si ［ J ］. Today Foundry, 2006, 27 (8): 866-870

[5] Jiang Hongyi, Liu Qiongzhen, Zhang Lianmeng, et al. Quantum chemical calculation of Mg-Si-based thermoelectric materials [J]. Computational Physics, 2004, 21 (5): 439-442

［6］ Tani J I，Kido H.Lattice dynmics of Mg2 Si and Mg2 Ge compounds from first-principles calculations［J］.Intermetallics，2008(16)：418.

［7］ Au-Yang M Y，Cohen M L.Electronic structure and optical propertiesof Mg2 Si，Mg2 Ge，and Mg2 Sn［J］.Phys Rev，1969，178(3)：1358-1364.

［8］ Aymerich F，Mula G.Pseudopotential band structures of Mg2 Si，Mg2 Ge，and Mg2 Sn，and of the solid solution Mg2Ge，Sn［J］.Phys Suatus solid，1970，42(2)：697-704.

［9］ Corkill JL，Cohen M L.Structures and electronic properties of IIA-IV antifluorite compounds［J］.Phys Rev B，1993，48：17138-17144.

［10］Imai Y，Watanabe A.Energetics of alkaline-earth metal silicides calculated using a first-principle pseudoptentialmethod［J］.Intermetallics，2002(10)：333-341.

[11] Min Xinmin, Xing Xueling, Zhu Lei. Study on the electronic structure and thermoelectric properties of Mg2Si and doped series ［ J ］. Functional Materials, 2004 (35): 1154-1159

[12] Chen Qian, Xie Quan, Yang Chuanghua. First-principles calculation of electronic structure and optical properties of doped Mg2Si ［ J ］. Journal of Optics, 2009, 29 (1): 229-235

[13] Peng Hua, Chun Lei, Li Jichao. Theoretical study on the electronic structure and thermoelectric transport properties of Mg2Si [J]. Journal of Physics, 2010, 59 (6): 4123-4129

[14] Liu Nana, Sun Hanying, Liu Hongsheng. First-principles calculation of elastic and thermodynamic properties of Mg2Si [J]. Introduction to Materials, 2009 (23): 278-280

［15］Owen E A，Preston G D.The atomic structure of two intermetallic compounds［J］.Nature(London)，1924(113)：914-914.

［16］VoightW.Lehrbuch der Kristallphysik［M］.Leipzig：Taubner，1996.

［17］Reuss A.Berechung der fliessgrenze von mischkristallen［J］.Z Angew Math Mech，1929(9)：49-58.

［18］Hill R.The elastic behavior of a crystalline aggregate［J］.Proceedings of the Royal Society of London Series A，1952，65(5)：350.

[Abstract] The structure of Mg2Si crystal was geometrically optimized using the pseudopotential plane wave method of density functional theory. Based on the optimization, the electronic structure, elastic constants and thermodynamic properties were calculated by first principles. The results show that Mg2Si is an indirect bandgap semiconductor with a band gap of 0.2846eV; The conduction band is mainly composed of 3p, 3s of Mg and 3p electrons of Si; Elastic constants C11=114.39GPa, C12=22.45GPa, C44=42.78GPa; The Debye temperature at zero temperature and zero pressure is 676.4K. The phonon dispersion relationship is determined by using the linear response method, and the relationship between the change of temperature and the thermodynamic functions of Mg2Si such as isovolumetric heat capacity, Debye temperature, enthalpy, free energy and entropy is obtained.% The electronic structure, elastic constants and thermodynamic properties of Mg2Si were calculated based on the optimized structure by using the first-principles pseudo-potential method of density functional theory.The results showed that the indirect band gap of Mg2Si was 0.2846eV; conduction band was constituted mainly by 3p,3s electrons of Mg atoms and 3p electrons of Si atom; elastic constants C11=114.39GPa,C12=22.45GPa,C44=42.78GPa; Debye temperature is 676.4K at zero pressure and zero temperature.The linear response method was applied to determine the phonon dispersion relations,and the relations of thermodynamic functions of heat capacity,Debye temperature,enthalpy,free energy,entropy with temperature were calculated.

Journal of Yibin University

[year (volume), issue] 2012 (000) 006

[Total pages] 4 pages (P39-42)

[Key words] Magnesium silicide; Electronic structure; Elastic constant; Thermodynamic properties

[Author] Liu Futi; Cheng Xiaohong; Zhang Shuhua

[Author's unit] School of Physics and Electronic Engineering, Yibin University, Sichuan 644000, Key Laboratory of Computational Physics, Yibin University, Sichuan 644000; Key Laboratory of Computational Physics, Yibin University, Sichuan Province, 644000; Experiment and Teaching Resource Management Center of Yibin University, Yibin 644000, Sichuan

[Text language] Chinese

[Chinese Library Classification] O48

Metal silicide materials have many excellent thermal, electrical and mechanical properties, among which magnesium silicide (Mg2Si) is the only stable compound of Mg-Si binary system, which has the characteristics of high melting point, high hardness and high elastic modulus. It is a narrow band gap n-type semiconductor material, which is used in optoelectronic devices, electronic devices, energy devices, lasers, semiconductor manufacturing There are important application prospects in the fields of constant temperature control communication [1-5]. Because the raw material resources of alloy elements Mg and Si are very rich, the formation content is large, the price is low, non-toxic and pollution-free, Mg2Si as a new environmental semiconductor material has attracted great attention of researchers. In recent years, many scholars have carried out many studies on various properties of Mg2Si materials, such as lattice dynamics of Mg2Si [6], Energy band structure and dielectric function [7-10], doping and optical properties [11-12], geometric structure, elasticity and thermodynamic properties [13-14] have been studied. These research results are of great significance to the utilization and design of Mg2Si materials, but there are few studies on the thermal capacity, Debye temperature and other thermodynamic properties of Mg2Si. Therefore, this paper uses the pseudopotential plane wave method based on the first principle to study the energy band structure, elastic constants The Debye temperature, heat capacity, free energy, entropy and other thermodynamic functions are calculated theoretically to provide predictions for the experiment and design of Mg2Si materials

The Mg2 Si crystal has an anti-fluorite structure and belongs to a face-centered cubic lattice. The space group is Fm3m, the group number is 225, and the lattice constant is a=0.6391nm [15]. In the crystal structure, each Si atom is located at the (0,0,0) position, and the coordination number is 8, forming a face-centered cubic structure with a side length. The Mg atom is located at the (0.25, 0.25, 0.25) position, and each Mg atom is located at the center of the tetrahedron composed of Si atoms, forming a simple cubic structure with a side length of a/2, The cell structure of Mg2 Si is shown in Figure 1

In this paper, the first-principles pseudo-potential plane wave method based on density functional theory is adopted. The theoretical calculations are all completed by the quantum mechanics module CASTEP package in Material Studios 5.0 software. It is an ab initio quantum mechanics program based on density functional theory, which replaces the ion potential with pseudo-potential and expands the electron wave function with plane wave basis set, The exchange and correlation potential of the interaction between electrons and electrons are corrected by the local density approximation (LDA) or the generalized gradient approximation (GGA), which is currently recognized as a more accurate theoretical method for the calculation of electronic structure. The specific parameters are set as follows. In geometric optimization and electronic structure calculation, the energy cut-off value (Ultrafine) is 380eV, and the energy convergence (Energy tolerance) is 5.0 × 10-6 eV/atom, the maximum force convergence accuracy acting on each atom is 0.01 eV/⊙, the maximum strain convergence is 0.02 GPa, and the maximum displacement convergence is 5.0 × 10-4 ⊙. The Monkhorst-Pack method is used to set k points in the reciprocal space Brillouin, and the density is 4 × four × 4. The pseudopotential selects the super-soft pseudopotential. The valence electrons involved in the calculation of the pseudopotential are Mg: 2p6 3s2 and Si: 3s2 3p2, respectively. The exchange correlation functional between electrons uses the RPBE scheme in the generalized gradient approximation (GGA). In the calculation of phonon scattering, the energy cutoff energy is 900eV, and the convergence accuracy of self-consistent calculation is 1.0 × 10-4 eV/cell, k point selection density is 3 × three × 3. Pseudopotential is the Norm-conserving pseudopotential proposed by Hamann

3.1 Structure optimization

According to the experimental value of Mg2Si lattice parameters, the corresponding crystal structure is established. After geometric optimization, the lattice parameter is a=0.6422 nm, which is very close to the experimental value of 0.6391 nm [15], with an error of 0.5%. It is within the acceptable range of the first-principles calculation. The calculation of the following properties is carried out on the basis of this optimized structure, indicating that the calculation results should have good predictability

3.2 Energy band structure

Through the calculation of the energy band structure of Mg2Si, the energy band structure along the direction of the high symmetry point in the Brillouin region is obtained as shown in Figure 2

It can be seen from Figure 2 that the lowest point of the conduction band is at point X, and the highest point of the valence band is at point G, which is not at the same k point, indicating that Mg2 Si is an indirect bandgap semiconductor. The characteristic energy of the highly symmetric k point in the first Brillouin zone at the bottom of the conduction band Ec and the top of the valence band Ev is shown in Table 1. The minimum value of the energy obtained by the conduction band at point X is 0.2846eV, while the maximum value of the energy obtained by the valence band at point G is 0eV, so the band gap of Mg2 Si is 0.2846eV, It is close to the result of 0.2994eV of Chen Qian et al. [2]

The width of the energy band is very important in the analysis of the energy band. It can be seen from Table 1 that the width of the energy band at the bottom of the conduction band is (2.6761eV-0.2846eV=2.3915eV), and the width of the energy band at the top of the valence band is (-2.5231eV-0eV=-2.5231eV), that is, the energy band at the bottom of the conduction band is narrower than the energy band at the top of the valence band, which means that the effective mass of the electron in the bottom of the conduction band is larger than the effective mass of the hole at the top of the valence band, which means that Mg2Si is a heavy electron, Indirect gap semiconductor with light holes

Figure 3 shows the total density of states of Mg2Si. Through analysis, the energy band of Mg2Si has four regions, of which there are three valence bands: - 42.877 ～ - 42.863 eV is the lowest energy band, and the band width is very small, which is derived from the 3p state electrons of Mg- 8.937 ～ -6.994 eV is a sub-low energy valence band, which is mainly contributed by 3s electrons of Mg and 3s electrons of Si; The remaining energy bands close to the Fermi level correspond to the high energy valence band, with the energy range of -4.580~0 eV. The main contributions of this energy band are the 3p electrons of Mg and the 3p electrons of Si. The 3s electrons of Mg and the 3s electrons of Si make a small contribution to this energy band. The conduction band of Mg2Si is mainly the 3p electrons of Mg, the 3s electrons of Si and the 3p electrons of Si. The contribution of the 3s electrons of Si is relatively small

3.3 Elastic constant

Elasticity is a relatively important research object in the fields of material science, chemistry, physics and geophysics. The equation of state, specific heat capacity, Debye temperature, melting point, etc. of solid matter are all related to elasticity. From the elastic constants, we can obtain important information about the characteristics of crystal anisotropy and the stability of crystal structure. Mg2Si belongs to the face-centered cubic crystal system, and its elastic tensor Cij has three independent components, C11, C12 and C44, After geometric optimization, the elastic constants and bulk elastic moduli of Mg2Si at zero temperature and pressure are calculated as shown in Table 2

According to the isotropic coefficient, the S=21.075 of MgSi can be calculated, indicating that its isotropy is good. The bulk elastic modulus and shear modulus respectively represent the ability of material to resist fracture and plastic deformation, and their ratio can be used as a measure of ductility or brittleness. A high B/G (its critical value is 1.75) value means that the material is malleable, and a low B/G means that the material is fragile. The calculated B/G of Mg2Si=1.206, indicating that Mg2Si is fragile

3.4 Debye temperature

Debye temperature is an important physical quantity of the thermodynamic properties of substances. The Debye temperature of crystals can be calculated by using the elastic constant. The Debye temperature of Mg2Si crystals can be calculated by using the Debye approximation based on the elastic constant calculated previously. According to the Voigt [16] approximation, the shear modulus is based on the Reuss [17] approximation, and the shear modulus is based on the theoretical proof that the polycrystalline modulus is just the arithmetic mean value given by Voigt and Reuss, That is, for cubic crystals, when p=0GPa, the bulk modulus of elasticity is, and then the compression longitudinal wave velocity and shear wave velocity can be obtained from the shear modulus and bulk modulus of elasticity vs=, and the average sound velocity can be obtained. Finally, the Debye temperature can be obtained from the average sound velocity and Debye approximation

118.82 22.27 44.96 54.45 1.07 In the front, ħ Is Planck's constant, k is Boltzmann's constant, NA is Avogadro's constant, n is the number of atoms in the protocell, M is the mass of molecules in the protocell, and V is the volume of the protocell, ρ= M/V is the density

According to the above formula, the Debye temperature of Mg2Si at zero temperature and zero pressure is θ D=676.4K, Debye frequency ω D=8.856 × 1013 Hz, which directly reflect the thermodynamic properties of crystals

3.5 Heat capacity, enthalpy, free energy and entropy function

The linear response method is also used to determine the dispersion relationship of Mg2Si in the first Brillouin region and the density of phonon states, as shown in Figure 4 and Figure 5 respectively. There are two magnesium atoms and one silicon atom in the primary cell of Mg2Si crystal, and there are three acoustic waves and six optical waves in total. It can be seen from the figure that when the wave vector k approaches zero, the frequency of three lattice vibration waves tends to zero, and these three lattice vibration waves are acoustic waves, including two transverse waves, One longitudinal wave and the other six are optical waves. The calculated results show that the frequencies of optical waves of Mg2Si at G point are 7.829 respectively × 1012 Hz、8.336 × 1012 Hz and 9.408 × 1012 Hz.

Under the quasi-harmonic Debye approximation, the thermodynamic properties of Mg2Si are discussed by using the density of phonon states, and the heat capacity formula is used:

The heat capacity and Debye temperature of Mg2Si at a given temperature can be calculated. The isovolumetric heat capacity and Debye temperature in the temperature range of 0~1000K are shown in Fig. 6 and Fig. 7 respectively

At the temperature of 40.5K, the heat capacity is 1.264 J/mol. K, and the Debye temperature is 674.2K, which is very close to the Debye temperature (676.4) at zero temperature and zero pressure calculated by the elastic constant; At 303K, the heat capacity is 62.05 J/mol. K, while the experimental value is 67.9 J/mol. K. At this time, the Debye temperature is 585.5 K. With the increase of temperature, the heat capacity tends to 73.6 J/mol. K, which is very close to the classical limit value of 74.8 J/mol. K, which is consistent with Duron-Petit's law

Enthalpy, free energy and entropy functions all tend to zero when the temperature approaches 0K, which is consistent with the third law of thermodynamics. The specific relationship between them and temperature is shown in Figure 8

The electronic structure and thermodynamic properties of Mg2Si crystal are calculated by using the pseudopotential plane wave method based on density functional theory. The results show that Mg2Si is an indirect bandgap semiconductor with a band gap of 0.2846eV; The conduction band is mainly composed of 3p, 3s of Mg and 3p electrons of Si; Elastic constants C11=114.39GPa, C12=22.45GPa, C44=42.78 GPa; Debye temperature at zero temperature and pressure is 676.4K; The phonon dispersion relation is determined by using the linear response method. The Debye temperature of Mg2Si at 40.5K is 674.2K, and the isovolumetric heat capacity approaches the Dulon-Perty classical limit value of 74.8 J/mol.K with the increase of temperature; The relationship between enthalpy, free energy, entropy and other thermodynamic functions with temperature is consistent with the third law of thermodynamics

[Related literature]

[1] Jiang Hongyi, Zhang Lianmeng. Research status of Mg-Si-based thermoelectric compounds [J]. Introduction to Materials, 2002, 16 (3): 20-22

[2] Chen Qian, Xie Quan, Yan Wanjun, et al. First-principles calculation of electronic structure and optical properties of Mg2Si ［ J ］. Chinese Science Series G, 2008, 38 (7): 825-833

［3］ Song R B，Aizawa T，Sun JQ.SynthesisofMg2 Si1-x Snx solid solutions as thermoelectric materials by bulkmechanical alloying and hot pressingmaterials［J］.Mater Sci Eng B，2007(136)：111-116.

[4] Zang Shujun, Zhou Qi, Ma Qin, et al. Research progress of intermetallic compound Mg2 Si ［ J ］. Today Foundry, 2006, 27 (8): 866-870

[5] Jiang Hongyi, Liu Qiongzhen, Zhang Lianmeng, et al. Quantum chemical calculation of Mg-Si-based thermoelectric materials [J]. Computational Physics, 2004, 21 (5): 439-442

［6］ Tani J I，Kido H.Lattice dynmics of Mg2 Si and Mg2 Ge compounds from first-principles calculations［J］.Intermetallics，2008(16)：418.

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