As the first plentiful element on the earth, carbon has kinds of allotropes with a wide range of structures from superhard wide semiconductors to ultra-soft semimetals and even superconductors. The carbon allotropes have been investigated maturely and applied in many fields. The researches show that carbon allotropes own advanced mechanical and electronic properties. Xing et al.^[1] came up with -20 carbon and compared with two other monoclinic symmetry carbon allotropes ( -16 carbon and M-carbon) and found that -20 carbon shows the greatest elastic modulus, anisotropy in Young’s modulus and hardness. Moreover, there are many other superhard carbon allotropes that have been predicted by researchers, such as O-carbon,^[2] W-carbon,^[3] F-carbon,^[4] C-carbon,^[5] lonsdaleite,^[6] Z-carbon,^[7] H-carbon,^[8] ,^[9] S-carbon,^[10] Imma-carbon,^[11] (9, 0)-carbon,^[12] and so on. As the cornerstone of high-tech materials, silicon is widely applied in the production of high-purity semiconductors, high temperature materials, and optical fiber communication materials. Fan et al.^[13] put forwards four novel silicon allotropes ( , -16, -20, and I-4); the mechanical and dynamical stability at ambient condition were proved. In addition, -16, -20, and I-4 phases were indirect band gap semiconductors materials with band gap of 0.56, 0.53, and 1.28 eV, respectively. The phase was a quasi-direct gap phase with a gap of 0.74 eV. The anisotropy properties show that the -20 phase exhibits greater anisotropy in shear elastic anisotropic factors.

C–Si alloys are attracting more and more attention in industry and academia^[14–23] because of their potential standout properties different from elemental carbon and silicon. In the converter steelmaking field, C–Si alloys are a new type of converter alloys, which can replace ferrosilicon, silicon carbide, and carburizer, further reducing the amount of deoxidizing agent. They have a good prospect in the process of converter for the stability, hardness, and mechanical properties which is superior to traditional technology. Recently, physicists are proceeding to find new kinds of C–Si alloys with splendid structural, mechanical, and electronic properties. Especially, direct band gap semiconductors are the direction researchers are interested in because it is easy for the energy transition to occur and they are appropriate for optoelectronic devices. Based on the global particle-swarm optimization algorithm and the density functional theory methods, Zhou et al.^[24] predicted a novel energetically and thermally stable two-dimensional SiC₂ siligraphene (g-SiC₂) which is a semiconductor with a direct band gap of 1.09 eV. From first principles combined with a tight-binding model, Zhao et al.^[25] proposed siligraphenes g-SiC₃ and g-Si₃C, both of which are dynamically stable and the spin–orbital coupling (SOC) band gaps are 0.431 meV (g-SiC₃) and 0.089 meV (g-Si₃C), much larger than those of graphene (0.0008 meV) and planar silicene (0.07 meV). Yan et al.^[26] obtained three stable Si_mC_n graphyne-like structures and studied through the structural relaxation, optimization, electronic structures as well as the thermal stability. Beside two-dimensional structures of silicon carbides, there are many works regarding the three-dimensional silicon carbide materials. Zhang et al.^[27] proposed two new phases of Si₈C₄ and Si₄C₈ with the symmetry and studied their structural, elastic, and electronic properties systematically. Both of them are proved to be mechanically and dynamically stable. The band gaps of Si₈C₄ and Si₄C₈ are 0.74 eV and 0.15 eV, respectively, and both of them are indirect semiconductors. Fan et al.^[28] investigated the structural, mechanical, anisotropic, and electronic properties of SiC₂ and SiC₄. The calculated results show that SiC₂ and SiC₄ are anisotropic materials and that SiC₂ is an indirect band gap semiconductor while SiC₄ is a quasi-direct band gap semiconductor. Tan et al.^[29] proposed C–Si alloys (C₈Si₂, C₆Si₄, C₄Si₆, and C₂Si₈) in the orthorhombic structure of P222₁ space group and proved the mechanical stability and elastic anisotropic. The concluded band structure exhibits that the proposed C–Si alloys are all indirect band gap semiconductors.

Based on the previous works, we propose four new structures of C–Si alloys (C₁₆Si₄, C₁₂Si₈, C₈Si₁₂, and C₄Si₁₆) in structure. In this paper, the lattice parameters and elastic constants are first calculated using first-principles calculations. Then the elastic modulus, Poisson’s ratio, hardness, elastic anisotropy, and Debye temperature are obtained. In addition, the band structures and density of states of C₂₀, Si₂₀, and their alloys are investigated and analyzed.

The structural, mechanical and electronic properties of C₂₀, Si₂₀, and their alloys (C₁₆Si₄, C₁₂Si₈, C₈Si₁₂, and C₄ in structure were studied utilizing Cambridge Serial Total Energy Package (CASTEP)^[30] based on DFT.^[31] The exchange correlation potential was performed with the generalized gradient approximation (GGA) in the form of Perdew–Burke–Ernzerhof (PBE),^[32] PBE for solid (PBEsol),^[33] and the local density approximation (LDA) in the form of Ceperley and Alder data parameterized by Perdew and Zunger (CA-PZ).^{[34, 35]} To describe the core-valence interactions, the ultrasoft pseudo potentials were introduced. The cutoff energy of C₂₀ (C₁₆Si₄, C₁₂Si₈, C₈Si₁₂, and C₄ and was 400 eV and 340 eV, respectively. The valence electron structures of C and Si atoms are 2s²2p² and 3s²3p² and the grid partitions of k-points in the first irreducible Brillouin zone^[36] were set as 7 16 8, 7 16 8, 7 15 7, 6 13 6, 5 13 6, and 5 10 5, respectively. For C₂₀, Si₂₀, and four new C–Si alloys phases, the structures were optimized using Broyden–Fletcher–Goldfarb–Shanno (BFGS)^[37] minimization algorithm. The convergence tolerance was 5 10 eV/atom and the SCF (self-consistent field) tolerance was 5 10 eV/atom. On the atom, the maximum force is 0.01 eV/Å with the maximum stress 0.02 GPa and the maximum displacement Å.

Figure 1 illustrates the crystal structures of C₂₀, Si₂₀, and their alloys (C₁₆Si₄, C₁₂Si₈, C₈Si₁₂, and C₄Si₁₆). They all belong to the space group of in a monoclinic system and every conventional cell includes 20 atoms. By utilizing PBE, PBEsol, and CA-PZ methods, the structures were optimized and the calculated lattice parameters (a, b, c, and β) are listed in Table 1 along with the theoretical value. From Table 1, it is clear that the lattice parameters of C₂₀, Si₂₀ in structure are excellently consistent with other theoretical values.^{[1, 13]} From C₂₀ to C–Si alloys, even to Si₂₀, the lattice parameters (a, b, and c) increase with the number of Si atoms as shown in Fig. 2. Additionally, the curves of the GGA method (PBE and PBEsol) show a little better performance than the one of the LDA method (CA-PZ) in spite of the fact that all the curves in Fig. 2 almost coincide.

Fig. 1.

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	Fig. 1. (color online) Conventional cell crystal structures of C20, Si20, and C–Si alloys (red and blue spheres represent C and Si atoms).

Fig. 2.

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	Fig. 2. (color online) Calculated lattice parameters (a, b, and of C20, Si20, and C–Si alloys by PBE, PBEsol, and CA-PZ methods.

Table 1.

Table 1.

Table 1. Calculated lattice parameters (a, b, c in unit Å and β in unit degree) using PBE, PBEsol, and CA-PZ along with theoretical values. . -20 PBE PBEsol CA-PZ a b c β a b c β a b c β C20 9.006 2.519 8.702 143.44 8.979 2.513 8.675 143.39 8.889 2.490 8.597 143.38 C20 a 9.011 2.519 8.707 143.46 C16Si4 9.865 2.753 9.546 143.49 9.842 2.742 9.523 143.43 9.736 2.712 9.425 143.49 C12Si8 10.441 3.008 10.474 141.59 10.323 2.999 10.439 141.25 10.233 2.957 10.315 141.36 C8Si12 11.873 3.140 10.844 142.05 11.894 3.121 10.809 142.02 11.731 3.079 10.671 142.07 C4Si16 12.355 3.489 11.962 141.37 12.323 3.471 11.923 141.26 12.153 3.421 11.762 141.29 Si20 13.678 3.863 13.232 143.17 13.647 3.857 13.207 143.12 13.443 3.798 13.003 143.11 Si20 b 13.679 3.863 13.238 143.18 a Ref. [1] b Ref. [13].

Table 1. Calculated lattice parameters (a, b, c in unit Å and β in unit degree) using PBE, PBEsol, and CA-PZ along with theoretical values. .

As we all know, the elastic constants of a conventional cell are very significant in researching on the mechanical properties of a crystal, which offer information about the stability and stiffness of the materials. For monoclinic symmetry space group of , there are thirteen independent elastic constants C_ij , , C₃₃, C₄₄, C₅₅, C₆₆, C₁₂, C₁₃, C₁₅, C₂₃, C₂₅, C₃₅, and . The values of elastic constants which were obtained by introducing a small finite strain to the equilibrium structure are listed in Table 2. The elastic constants of C₂₀ and Si₂₀ were consistent with theoretical value. A stable monoclinic structure must satisfy the following mechanical stability criteria:^[38]

Table 2.

Table 2.

Table 2. Calculated elastic constants (in unit GPa) and elastic moduli (in unit GPa) of C20, C16Si4, C12Si8, C8Si12, C4Si16, and Si20. . C11 C22 C33 C44 C55 C66 C12 C13 C15 C23 C25 C35 C46 C20 1109 1103 941 455 401 473 45 111 87 39 17 12 59 C20 a 1105 1249 928 452 398 472 38 108 80 41 16 8 60 C16Si4 672 648 691 271 201 238 64 135 89 –4 20 –9 52 C12Si8 507 457 448 189 173 139 49 69 82 45 47 18 58 C8Si12 363 276 342 139 101 117 76 68 68 1 4 –35 17 C4Si16 255 195 219 75 67 49 38 63 47 –1 8 –3 20 Si20 181 167 142 55 45 51 35 44 46 4 6 0 12 Si20 b 184 167 143 55 52 52 36 46 a Ref. [1] b Ref. [13].

Table 2. Calculated elastic constants (in unit GPa) and elastic moduli (in unit GPa) of C20, C16Si4, C12Si8, C8Si12, C4Si16, and Si20. .

After verification, it can be found that all the elastic constants obey the mechanical stability criteria, which indicates that these six crystals are mechanically stable. To confirm the dynamic stability of their alloys in -20 phase (C₁₆Si₄, C₁₂Si₈, C₈Si₁₂, and C₄Si₁₆), the phonon spectra of C₁₆Si₄, C₁₂Si₈, C₈Si₁₂, and C₄Si₁₆ in -20 phase are displayed in Figs. 3(a), 3(b), 3(c), and 3(d). There is no imaginary frequency along the whole Brillouin zone form Fig. 3, indicating that these alloys are all dynamically stable.

Fig. 3.

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	Fig. 3. The phonon spectra of C–Si alloys.

In order to evaluate the relative stability of C–Si alloys, the phase diagram is calculated based on the regular-solution model.^[39] The formation energy of C–Si alloys is defined as follows:

Fig. 4.

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	Fig. 4. (color online) The formation energy of C–Si alloys.

As we all know, the elastic constants C₁₁, C₂₂, and C₃₃ character the resistance to linear compression in x, y, and z directions,^[40] respectively. The elastic constants C₄₄, C₅₅, and C₆₆ are interpreted by the resistances of the crystal with respect to the shear strain at (100), (010), and (001) planes and are regarded as the important parameters relating to the shear modulus. Figure 5(a) shows the elastic constants C_ii and it is clear that the calculated C₁₁, C₂₂, and C₃₃ are very large among the thirteen elastic constants, which indicates that they are very incompressible under uniaxial stress along x, y, and z axes. It is easily found that C₁₁ is higher than C₃₃ for these six crystals except C₁₆Si₄. Thus, it can be considered that the z axis is more compressible than the x axis for C₂₀, C₁₂Si₈, C₈Si₁₂, C₄Si₁₆, and Si₂₀. In addition, C₁₁ is slightly higher than C₂₂ for all these six crystals indicating that the y axis is a little more compressible than the x axis. In cases of C₄₄, C₅₅, and C₆₆, for all the studied crystals in this work, C₄₄ is larger than C₅₅, so the resistance to the shear strain at the (100) plane is stronger than that of the (010) plane. For C₁₆Si₄, C₁₂Si₈, C₈Si₁₂, C₄Si₁₆, and Si₂₀, the resistance at (100) is larger than that of the (001) plane. As a whole, with the increasing components of Si atoms, the elastic constants C_ii decrease thus the compression property increases.

Fig. 5.

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	Fig. 5. (color online) Elastic constants Cii and elastic moduli (B, G, and of C20, C16Si4, C12Si8, C8Si12, C4Si16, and Si20.

Then using the data of Table 2, the elastic moduli can be calculated. To obtain the bulk modulus B (in unit GPa) and the shear modulus G (in unit GPa), which represent the volumetric elasticity and shearing direction under opposing forces, Voigt^[41] proposed the upper limits and , while Reuss^[42] proposed the lower limits and in the following equations:^[43]

However, the arithmetic average of Voigt and Reuss bounds termed as Voigt–Reuss–Hill (VRH)^[41–43] approximations is the acknowledged best method to calculate the bulk modulus and the shear modulus. From VRH approximations, we have and . The Young modulus E is the ratio of stress to strain and can be obtained by .^[43] Universally, the larger the Young’s modulus E is, the stiffer the material is prone to be. Poisson’s ratio ^[43] is the ratio of the expansion fraction to compression fraction.

Aforementioned elastic moduli, i.e., bulk modulus B (in unit GPa), shear modulus G (in unit GPa), Young’s modulus E (in unit GPa), and Poisson’s ratio v were calculated and shown in Table 3. It is found that the bulk modulus B, the shear modulus G, and the Young modulus E all decrease by 79.72%, 88.31%, and 86.78% with the increase of the Si atoms, meaning that the stiffness decreases. Besides, the slopes of the B, G, and E curves turn smaller and smaller. For the C–Si alloys, the variation tendency of the bulk modulus B and the shear modulus G are almost the same. For the elastic moduli and Poisson’s ratio, the given data of this work are consistent with theoretical values. The ratio of the bulk modulus to the shear modulus ( ) proposed by Pugh^[44] indicates the ductile or brittle feature of a material. If , the material manifests the ductile feature, otherwise, the brittle feature. The crystal with Poisson’s ratio v larger than 0.26 belongs to ductile material. From Table 3, we can see that B/G for C₂₀, Si₂₀ and their alloys (C₁₆Si₄, C₁₂Si₈, C₈Si₁₂ and C₄ is less than 1.75 and Poisson’s ratio v is less than 0.26, it is concluded that they are all brittle crystals.

Table 3.

Table 3.

Table 3. Calculated elastic moduli (in unit GPa) and Poisson’s ratio v of C20, Si20, C16Si4, C12Si8, C8Si12, and C4Si16. . B/GPa G/GPa E/GPa v C20 403 456 0.884 993 0.089 C20 a 412 463 0.890 1010 0.091 C16Si4 287 252 1.139 585 0.160 C12Si8 198 173 1.144 401 0.161 C8Si12 155 120 1.292 286 0.192 C4Si16 106 69 1.536 171 0.232 Si20 82 53 1.547 131 0.232 Si20 a 83 55 1.509 135 0.230 a Ref. [1] b Ref. [13].

Table 3. Calculated elastic moduli (in unit GPa) and Poisson’s ratio v of C20, Si20, C16Si4, C12Si8, C8Si12, and C4Si16. .

The hardness of material can express the elastic and plastic properties as follows:

The elastic anisotropy is another crucial parameter which relates to the probability of inducing micro-cracks in materials. There are many descriptions of anisotropy named the compression anisotropy and the shear anisotropy which represent the percentage of the bulk modulus and shear modulus, respectively, and the universal anisotropy index can be obtained anisotropy as follows:^[45]

From the three anisotropy formulas, it is obvious that , , and are all not less than zero. If , , and are equal to zero, the anisotropy eliminates and the crystal turns to isotropic material. If and equal to 1, then the anisotropy is maximum. From Table 4 and Fig. 6(a), it is easy to be observed that these six studied crystals show weak anisotropy since the values of and are very small. Meanwhile, the column of is a little smaller than the column indicating that the anisotropy is inconspicuous in bulk modulus. As shown in Ref. [13], for silicon allotropes, the value of in structure is larger than that of , -16, and I-4 structures, which indicates that the anisotropy of Si₂₀ in structure is greatest. In our work, the values of of C₄Si₁₆ and C₁₂Si₈ are 0.594 and 0.540, respectively, larger than Si₂₀ (0.241), so C₄Si₁₆ and C₁₂Si₈ own stronger universal anisotropy.

Fig. 6.

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	Fig. 6. (color online) Calculated elastic anisotropy factors (a), shear anisotropy factors (b) of C20, C16Si4, C12Si8, C8Si12, C4Si16, and Si20.

The shear anisotropic factors provide a measure of the degree of anisotropy in the bonding between atoms in different planes. The shear elastic anisotropic factors A₁, , and A₃ can be defined as:^[46]

Table 4.

Table 4.

Table 4. Compression and shear anisotropy percent factors ( and , universal anisotropy index ( and shear anisotropy factors (A1, A2, and . . A1 A2 A3 C20 404.2 401.9 459.8 452.2 0.284 0.831 0.090 0.995 0.857 0.891 C16Si4 287.6 287.0 256.9 246.9 0.113 1.984 0.207 0.994 0.692 0.798 C12Si8 201.3 193.3 181.0 164.6 2.014 4.755 0.540 0.928 0.933 0.643 C8Si12 156.1 153.3 122.6 117.5 0.892 2.122 0.235 0.977 0.839 0.961 C4Si16 107.2 105.3 73.0 65.5 0.889 5.449 0.594 0.866 0.831 0.526 Si20 82.0 81.4 54.5 52.1 0.382 2.279 0.241 0.925 0.825 0.739 Si20 a 83.2 82.6 56.3 54.2 0.347 1.971 0.208 a Ref. [13].

Table 4. Compression and shear anisotropy percent factors ( and , universal anisotropy index ( and shear anisotropy factors (A1, A2, and . .

As a fundamental parameter for evaluating the chemical bonding characteristics and the thermal properties of materials, the Debye temperature _D is generally used to distinguish whether a crystal locates at the high temperature zone or low temperature zone. If the temperature , the energy is satisfying the Dulong-Petit law, otherwise, all the modes of high frequency lose efficacy. Combining the electronic structures and the elastic moduli, the Debye temperature can be expressed by the following equations:^{[47, 48]}

Table 5.

Table 5.

Table 5. Density (ρ in units g/cm3), compressional, shear, and average elastic sound velocities ( , , in units m/s) as well as Debye temperature ( in unit K). . /K C20 3.392 11595 17264 12663 2086 C16Si4 3.278 8765 13783 9637 1451 C12Si8 2.996 7594 11948 8350 1145 C8Si12 2.893 6442 10432 7106 913 C4Si16 2.566 5194 8796 5755 678 Si20 2.226 4894 8286 5422 585 Si20 a 2.225 4972 8382 5506 595 a Ref. [13].

Table 5. Density (ρ in units g/cm3), compressional, shear, and average elastic sound velocities ( , , in units m/s) as well as Debye temperature ( in unit K). .

The electronic band structure describes the law of the electronic motion in a solid and determines the fundamental physical and chemical properties of materials. As is well known, in calculating the band gap, the density functional theory method underestimates the results by 30%–50% compared to the true band gap. The calculated band gaps of C₂₀, Si₂₀, C₁₆Si₄, C₁₂Si₈, C₈Si₁₂, and C₄Si₁₆ are listed in Table 6 by PBE, PBEsol, and CA-PZ methods. The band structure-based PBE are illustrated in Fig. 7 with the dashed line of zero energy representing the Fermi level ( ). The band gap generally refers to the energy difference between the valence band maximum (VBM) and the conduction band minimum (CBM). The minimal-energy state in the conduction band and the maximal-energy state in the valence band are each characterized by a certain crystal momentum (k-vector) in the Brillouin zone. From Fig. 7, it is obvious to obtain that C₂₀, C₁₆Si₄, and Si₂₀ are all indirect band gap semiconductors with band widths of 4.02 eV, 0.33 eV, and 0.53 eV, respectively, while C₁₂Si₈ is a direct band gap semiconductor with band width of 0.17 eV. The remaining C₈Si₁₂ and C₄Si₁₆ are semi-metallic alloys. Notably, a direct band gap semiconductor (C₁₂Si₈) is obtained by doping two indirect band gap semiconductors (C₂₀ and Si₂₀). Meaningfully, C₁₂Si₈ is very appropriate for making optoelectronic devices.

Fig. 7.

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	Fig. 7. (color online) Partial densities of states of C20, Si20, and C–Si alloys.

Table 6.

Table 6.

Table 6. The calculated band gap (in unit eV) of C20, C16Si4, C12Si8, C8Si12, C4Si16, and Si20. . C20 C20 a C16Si4 C12Si8 C8Si12 C4Si16 Si20 Si20 b PBE 4.02 4.04 0.33 0.17 – – 0.53 0.53 PBEsol 3.91 0.13 0.01 – – 0.40 CA-PZ 4.08 0.22 0.04 – – 0.41 a Ref. [1] b Ref. [13].

Table 6. The calculated band gap (in unit eV) of C20, C16Si4, C12Si8, C8Si12, C4Si16, and Si20. .

For illustrating the fundamental features of the chemical bonding in , the partial density of states (PDOS) were calculated as shown in Fig. 8. For C₂₀, C-s states make a main contribution in the energy from −22.5 eV to −12.5 eV while C-p states contribute mainly in energy from −12.5 eV to and 4.02 eV to 10 eV. For C₁₆Si₄, C-s states make a main contribution in energy from −22 eV to −13 eV and C-p states contribute mainly in energy from −13 eV to and 0.33 eV to 5.5 eV. In addition, Si₅-p contributes mainly in energy from 0.33 eV to 5.5 eV. For C₁₂Si₈, C-s states make a main contribution in energy from −18 eV to 11 eV while C-p states contribute mainly in energy from −11 eV to and 0.17 eV to 5.2 eV. It is also found that Si₁-p and Si₅-p contribute mainly in energy from 0.17 eV to 5.2 eV. For C₈Si₁₂, C-s states make a main contribution in energy from −17.5 eV to −10.5 eV while C-p states contribute mainly in energy from −10.5 eV to 3.2 eV. For C₄Si₁₆, C-s states make a main contribution in energy from −16 eV to −8.5 eV while C-p states contribute mainly in energy from −8.5 eV to 3.1 eV. Besides, C4-s contributes mainly in energy from −16 eV to −14 eV. For Si₂₀, Si-s states make a main contribution in energy from −12 eV to −6 eV while Si-p states contribute mainly in energy from −6 eV to and 0.53 eV to 3.35 eV.

Fig. 8.

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	Fig. 8. (color online) Electronic band structures of C20, Si20, and C–Si alloys.

In this work, the structural properties, elastic properties, anisotropies, and electronic properties of C₁₆Si₄, C₁₂Si₈, C₈Si₁₂, and C₄Si₁₆ in structure are investigated. The mechanical stability and the dynamic stability are confirmed by the elastic constants and phonon spectra, respectively. It can be found that the B/G and Poisson’s ratio all decrease with the increment of the Si atoms, which indicates that they are all brittle crystals. Through the analysis of the compression anisotropy, shear anisotropy, and universal anisotropy factors, C–Si alloys perform a smaller anisotropy. By calculating the band structures, the results show that C₁₆Si₄ is indirect band gap semiconductor while C₁₂Si₈ is a direct band gap semiconductor, C₈Si₁₂ and C₄Si₁₆ are semi-metallic alloys. Notably, a direct band gap semiconductor (C₁₂Si₈) is obtained by doping two indirect band gap semiconductors (C₂₀ and Si₂₀).