Academic Journal of Science and Technology ISSN: 2771-3032 | Vol. 10, No. 1, 2024 181 First‐principles Calculations of The Elastic Properties, Electronic Structure and Optical Properties of Hexagonal Al4SiC4 Zongye Li*, Ailiang Yin School of Electronic Information, Southwest Minzu University, Chengdu, China *Corresponding author: ZongYe Li Abstract: The crystal structure, elastic constants, electronic structure and optical properties of Al4SiC4 Have been systematically investigated using the first principles calculation method based on density functional theory. The calculated lattice constants and elastic constants are consistent with the experimental and calculated values, and it shows that the hexagonal Al4SiC4 crystal structure is stable. The calculated bulk, shear, Young’s modulus and Poisson’s ratio of hexagonal Al4SiC4 are in accordance with the values in the literature. The calculated band gap of the Al4SiC4 is 1.076 eV. The changes of optical response functions of hexagonal Al4SiC4 in the (100) and (001) directions with photon energy are calculated, including complex dielectric function, complex refractive in-dex, absorption spectrum and reflection spectrum. In the (100) and (001) directions, the static dielectric con-stants are 7.74 and 8.96,respectively,and the refractive indexes are 2.78 and 2.99,respectively. The calcu-lated results can provide theoretical basis for related applications. Keywords: Al4SiC4; Electronic structure; Optical properties; First-principles. 1. Introduction Al4SiC4 ceramic is characterized by its low density (3.03 g /cm3), high melting point (> 2700℃), high chemical stability, high strength, low thermal expansion coefficient and excellent oxidation and hydration resistance. It has become a promising high-temperature structural material and high- performance refractory material to be developed, and has gradually attracted people's attention in the fields of aerospace structural ceramics and high-temperature electronic semiconductors [1][6].Since Barczak et al.[7] first discovered Al4SiC4, several research groups [2][5][8][10] have focused on the synthesis and preparation of Al4SiC4 (such as solid-state reaction sintering method, chemical vapor deposition).Experimental study on the properties and applications of volume-solid phase sintering method, hot pressing sintering method, high temperature self-spreading method, penetration method, mechanical alloying method, etc.[3][11][20] .Inoue et al. [1] used a mixture of aluminum, silicon and carbon and a mixture of kaolin, aluminum and carbon as raw materials to synthesize Al4SiC4 powder by two-step method, and then conducted pulse current sintering to synthesize massive Al4SiC4. Yamamoto et al. [2] prepared Al4SiC4 powder with aluminum, silicon and carbon black as raw materials. Huang et al. [3] studied the high temperature oxidation resistance and high temperature mechanical properties of Al4SiC4 prepared by reactive hot pressing sintering, and obtained that the bending strength increased with the increase of temperature within the test temperature range of 1000 ~ 1300℃.Wen et Al [4] used Al, graphite powder and polycarbosilane (PCS) as raw materials to prepare Al4SiC4 bulk ceramics by reactive hot pressing method, and studied the mechanical properties, thermal properties and oxidation behavior of Al4SiC4 ceramics.Solozhenko et al. [14] synthesized Al4SiC4 polycrystalline samples by reacting aluminum with graphite and silicon, and fitted the experimental data of pressure and volume (p-V) with Birch equation of state to obtain a volume modulus of 180.2 GPa.In recent years, theoretical studies on Al4SiC4 have also been reported, but not many [21][27], mainly focusing on the calculation of material properties based on first principles, such as lattice constants, elastic properties, band structure, optical properties and thermodynamic properties[21],[22],[24],[28] .Li et al. [24] calculated the phonon spectrum, chemical bond, thermal and thermodynamic properties of Al4SiC4 using density functional theory, and obtained that the linear thermal expansion coefficient of Al4SiC4 in the direction [001] was slightly higher In the direction [100].Liao et al. [25] have studied the bonding properties, elastic stiffness and ideal strength of Al4SiC4, and the results show that the mechanical properties of Al4SiC4 are closely related to the crystal structure, and the volume modulus is 179 GPa, the shear modulus is 140 GPa, and the lattice parameter is a = 3.222 A. c = 21.352 A.So far, there are few theoretical studies on the elastic properties, electronic structure and optical properties of Al4SiC4 including complex dielectric function, complex refractive index, absorption spectrum and reflection spectrum.Therefore, in this paper, the elastic properties, electronic structure and optical properties of Al4SiC4 are calculated using the mode conservation pseudopotential based on density functional theory, and the relationship between them is analyzed theoretically, which provides a theoretical basis for the practical application of Al4SiC4. 2. Computational Methods This paper adopts the first principles calculation method based on density functional theory, and all the calculations are completed on the Cambridge Sequential To tal Energy Package(CASTEP)[29] module.In the calculation process, Perdew-Burke-Ernzerhof (PBE) method on Gener-al gradient approximation (GGA) is selected for exchange correlation terms[30] . The interaction potential between ions and valence electrons is described by the mode conservation 182 pseudopotential.The valence electrons involved in the calculation are Al: 3s23p1, Si: 3s23p2, C: 2s22p2.The Broyden- Fletcher-Goldfarb-Shanno (BFGS) algorithm was used to optimize the structure of the cell model, and the truncation energy was set to 750 eV. The self-consistent cyclic convergence accuracy is 5.0 × 10-7eV/atom, the stress error between atoms is less than 0.02 GPa, the maximum displacement convergence error between atoms is less than 5.0 × 10-4A, and the total energy convergence standard of the system is 5.0 × 10-6eV/atom.The k point grid in Brillouin zone is 5 × 5 × 2. 3. Result and Discussion 3.1. Lattice parameter Al4SiC4 is a hexagonal system with space group P63mc [1 1], group number 186, lattice constant a = b = 3.2771 Å, c = 21.676 Å, V = 201.60 Å, α = β = 90°, γ = 120°.The crystal structure of Al4SiC4 is shown in Figure 1, where the purple is Al atom, the light gray is C atom, and the yellow is Si atom. Through the geometric structure optimization of Al4SiC4 cell, the lattice constant after optimization is obtained, as shown in Table 1. Compared with the experimental value [11], the difference between a and c is -0.63% and 0.31%, indicating that the model and calculation method adopted are credible. Table 1. Calculated lattice parameters a, c ( in Å) , and volume V ( in Å3 ) with the literature results for comparison Figure 1. Crystal structures of Al4SiC4 . The purple, gray and yellow spheres represent the Al, C and Si ions, respectively. 3.2. Elastic property For the hexagonal crystal phase Al4SiC4, there are five independent constants, respectively C11, C12, C13, C33, C44, C66= (C11-C12) /2, and the calculation results are shown in Table 2.As can be seen from Table 2, C33 > C11 > C12 > C44 > C13 > 0. According to the elastic stability criterion of hexagonal crystal structure [31] : C11-| C12| > 0, C44 > 0, (C11+ C12) C33-2 (C13)2 > 0, it can be seen that the calculated elastic constant of Al4SiC4 satisfies the above stability conditions, so the crystal structure of Al4SiC4 is stable. Table 2. Calculated secend order elastic constants ( Cij in GPa) of Al4SiC4 Based on the calculated elastic constant Cij, the volume modulus and shear modulus can be obtained according to the Reuss-Voigt-Hill [32] model: Where Sij is the elastic flexibility tensor, Poisson's ratio and Young's modulus can then be calculated: Where X = Reuss, Voigt or Hill. The calculation results are shown in Table 3.The calculated bulk moduli of Reuss, Voigt and Hill are 186.0, 179.7, 18.2 GPa, respectively, which are consistent with the calculated values (179 GPa [25], 171.9 GPa [26]) and experimental value (182.0 GPA [14]) agree well, indicating that the calculation method is reasonable.The shear modulus is 134.5,129.3, 131.9 GPa, which is close to the value in the literature (140 GPa [25]).Bulk modulus B and shear modulus G represent the ability of materials to resist compressive deformation and shear strain, respectively. The ratio of shear modulus and bulk modulus G /B is widely used to characterize the ductility and brittleness of materials.According to an empirical criterion proposed by Pugh [33], when G /B > 0.57, the material is brittle, and when 183 G /B < 0.57, the material is ductile.In this paper, the G /B value of Al4SiC4 calculated by GGA-PBE method is 0.73, which is greater than the critical value of 0.57, indicating that Al4SiC4 is brittle, which is consistent with the predicted results of literature [26]. Table 3. Calculated bulk modulus BX( X = V, Ｒ, H) ( in GPa) , shear modulus GX( in GPa) , Young’s modulus EX( in GPa) , Poisson’s Ｒatio νXwith the literature results for comparison 3.3. Electronic property Band structure is an important basis for material analysis.The band structure of Al4SiC4 is shown in Figure 2.The dotted line in the figure represents the Fermi level. It can be seen from the figure that the conduction band base of Al4SiC4 is located at the high symmetric point M in the first Brillouin region, and the valence band top is located at the high symmetric point G in the first Brillouin region, so Al4SiC4 has an indirect band gap, which is consistent with the results reported by Forster et al. [21].The energy gap Eg = 1.076eV of Al4SiC4 is consistent with the calculated value (1.05eV [22], 1.12eV [26]), but lower than the experimental value (2-2.5eV[5]).Due to the deficiency of the calculation method in solving the excited state energy, the obtained band gap will be smaller than the experimental value, so the calculated value is smaller than the experimental value, but it does not affect the analysis of the band structure. Figure 2. The band structure of Al4SiC4 Figure 3 shows the total state density and fractal density of Al4SiC4. As can be seen from Figure 3, it can be roughly divided into three regions: -15 eV ~ 10 eV, -8.8 eV ~ 0 eV, and 0 eV ~ 19.1 eV.In the low cost zone (-15 eV - 10 eV), it is mainly composed of Si-2s and C-2s electronic states, and the electron orbitals of the two atoms are hybridized, indicating that there is a strong covalent bond between the two atoms, so that the system has a more stable structure. The high valence zone (-8.8EV-0 eV) is mainly composed of SI-2P and C-2P electron states, and the density of states near the Fermi plane is mainly composed of p electron contributions of Al, Si and C atoms, and s electron contributions are less. The conduction band is mainly composed of Al-2p and Si-2p electron states. Figure 3. The total density of states and partial density of states of Al4SiC4 3.4. Optical property 3.4.1. Complex dielectric function The macroscopic optical response of a solid can usually be described by the complex dielectric function ε(ω) = ε1(ω) + iε2(ω), where ε1 and ε2 are the real and imaginary parts of the complex dielectric function, respectively, and ω is the frequency of the photon.The calculated complex dielectric function curves of hexagonal crystal phase Al4SiC4 in the polarization direction of incident light along the a axis (100) and the c axis (001) are shown in Figure 4. As can be seen from Figure 4, the static dielectric constant of Al4SiC4 in the direction of (100) and (001), that is, the value of zero frequency, is 7.74 and 8.96, respectively. In the direction of (100), in the low energy region, both the real and imaginary parts of the dielectric function increase with the increase of photon energy. The maximum value of the real part of the dielectric function at 4.38eV is 14, and the maximum value of the imaginary part at 5.86eV is 13.8, which is consistent with the results reported in the literature (the real part is 4.38eV [22], the imaginary part is 5.82eV [22]).Subsequently, the virtual part of the dielectric function decreases and approaches 0 after about 20 eV. The real part of the dielectric function decreases first and then rises, reaching the lowest value -4.12 at 9.07eV. The real part of the dielectric function increases slowly within the range of about 9 eV to 184 35 eV.In the direction of (001), the real part of the dielectric function reaches a maximum of 15.7 at 3.71 eV, and the imaginary part reaches a maximum of 15.3 at 5.32 eV. Figure 4. The complex dielectric functions of Al4SiC4 3.4.2. Complex refractive index According to the relationship between dielectric function and complex refractive index: ε1= n2-k2, ε2= 2nk, (n is the refractive index, k is the extinction coefficient), the refractive index n and extinction coefficient k of the hexagonal crystal phase Al4SiC4 can be calculated along the polarization direction of the incident light along the a axis (100) and the c axis (001) with the photon energy. As shown in Figure 5. As can be seen from Figure 5, the refractive index of Al4SiC4 at zero frequency in the direction of (100) and (001) is 2.78 and 2.99, respectively, which is consistent with the results reported in the literature (2.73 [22]).With the increase of photon energy, the refractive index first increases, then decreases and then increases.The refractive index along the direction (100) reaches a maximum value of 3.85 at 4.63 eV and a minimum value of 0.164 at 20.62 eV. The extinction coefficient starts to increase significantly at 2.02 eV, reaches a maximum value of 2.61 at 8.2 eV, and then decreases.The refractive index along the direction (001) reaches a maximum value of 4.06 at 3.94 eV and a minimum value of 0.159 at 19.58 eV.The extinction coefficient starts to increase significantly at 2.04 eV, reaches the first peak at 2.46 eV at 5.95 eV and then declines, then continues to rise to 8.43 eV, reaches the second peak at 2.50 eV, and then slowly decreases. Figure 5. The complex refractive indexes of Al4SiC4 3.4.3. Absorption coefficient Fig.6 shows the absorption coefficients of the hexagonal crystal phase Al4SiC4 along the polarization direction of the incident light along the a axis (100) and the c axis (001) with the photon energy.It can be seen that in the direction of (100) and (001), with the increase of photon energy, the absorption coefficient of Al4SiC4 first increases and then decreases, and the absorption coefficient is close to 0 after (0-2.5 eV) and 30 eV, respectively.The absorption coefficient reaches a peak value of 3.58 × 105/ cm at 8.98 eV in the direction of (100), and 3.62 × 105/ cm at 9.46 eV in the direction of (001). Figure 6. The absorption coefficients of Al4SiC4 3.4.4. Reflectance Based on the relationship between reflectivity and rereflectivity [34]: ,n is refractive index, k is extinction coefficient.The reflectance curve of Al4SiC4 along the a axis (100) and c axis (001) of the incident light polarization direction along with the photon energy can be obtained, and the calculation results are shown in Fig.7.As can be seen from Fig.7, the reflectance of Al4SiC4 in the direction of (100) and (001) generally increases with the increase of photon energy, at first shows a wave rise and then a rapid decline, and reaches a maximum value of 0.642 and 0.628 at 18.0 eV and 17.8 eV, respectively. When the photon energy is greater than 22 eV, the reflectance in both directions is almost the same.In the visible light energy range of 1.6eV ~ 3.2eV, the reflectance of (100) direction is between 24% ~ 29%, and the reflectance of (001) direction is between 27% ~ 35%, but the latter is slightly larger than the former. 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