C2-Si: a Novel Silicon Allotrope in Monoclinic Phase

Ⅰ.   INTRODUCTION

Group 14 elements (C, Si, Ge) exist in various forms of stable and metastable allotropes [1-38], some of which have been widely used in semiconductor industry. Silicon plays an important role in semiconductor materials. In fact, silicon is irreplaceable in the modern semiconductor industry. Silicon is a significant photovoltaic material in the solar energy industry. Silicon is the second most abundant element in the Earth's crust, and much experience has been accumulated in the utilization of silicon.

The fact that c-Si (space group: $Fd$-$3m$) is an indirect band gap semiconductor material which limits its application in photoelectric industry. Therefore, it is a significant direction for semiconductor industry to find silicon allotropes with direct band gap. Some silicon allotropes are direct band gap semiconductors, such as $t$P36-Si [19], $h$-Si$_6$ [20], and BC8 (Si-Ⅲ) [22], have been discovered through theoretical research and experimental synthesis. Cai et al. [19] have predicted a new silicon allotrope, named $t$P36-Si, theoretically investigated through first-principle calculations, which has narrow direct band gap which is 0.58 eV. Through X-ray diffraction (XRD) patterns and Raman spectra, it is proven that $t$P36-Si has good optical absorption characteristics and potential application prospects in photovoltaic materials. Guo et al. [20] proposed a new silicon allotrope called $h$-Si$_6$ with three-dimensional silicon structure composed of silicon triangles. $h$-Si$_6$ is a promising material for electronic device to its small effective mass and large carrier mobility. However, most silicon allotropes are still indirect band gap semiconductors [21, 24, 28, 31-36], such as Si$_{96}$ [21], $t$-Si$_{64}$ [24], $C2/m$-16 [31], $P$222$_1$-Si [34], and I926 phase silicon [35, 36]. Fan et al. [24] predicted a new silicon allotrope ($t$-Si$_{64}$) with tetragonal symmetry $I4_1/amd$ space group, they proposed that $t$-Si$_{64}$ is an indirect band gap semiconductor with a lower thermal conductivity, so the Si-Ge alloys in the $I4_1/amd$ phase may be potential thermoelectric materials. Fan el al. [33] studied four silicon allotropes, including one quasi-direct band gap phase ($Amm2$) with a band gap of 0.74 eV and three indirect band gap phases i.e. $C2/m$-16, $C2/m$-20, and $I$-4) with band 0.56, 0.53, and 1.28 eV. Fan et al. [34] studied the structural, dynamic, elastic, anisotropic and electrical properties of $P222_1$-Si silicon by first principles. Electronic structure studies show that $P222_1$-Si silicon is an indirect semiconductor material with band gap of 0.90 eV. Lee et al. [35, 36] proposed the indirect band gap semiconductor I926 phase silicon and other direct band gap and quasi direct band gap silicon allotropes. The optoelectronic industry is an important industry direction, but there are few silicon allotropes with direct band gap, and the silicon germanium alloy constructed with the same family elements can also adjust its band structure to the direct band gap. Incorporation of germanium into the silicon allotropes successfully made indirect band-gap silicon into Si-Ge alloys with a direct band gap [1, 7, 35, 38-40]. Although $C2$-Si proposed in this paper is an indirect band gap semiconductor, SiGe alloy with C2 allotrope constructed by germanium may be able to modulate the energy band to direct band, and the absorption capacity of $C2$-Si in the visible light range is also better than most silicon structures previously reported.

In this work, based on density functional theory (DFT), a novel silicon allotrope, which is called $C2$-Si, is predicted to possess the space group $C2$ by the random strategy combined with group and graph theory (RG$^2$) [3, 39]. $C2$-Si is shown to have an indirect narrow band gap, while the optical absorption ability of $C2$-Si is the highest among the Si allotropes mentioned in this work, so $C2$ phase may be used in photoelectric devices and solar cells. In addition, detailed physical properties, such as structural properties, elasticity, anisotropy and electronic properties are investigated through first-principles calculations.

Ⅱ.   COMPUTATION METHODS

In this work, the Cambridge Serial Total Energy Package (CASTEP) [40] code based on DFT [41, 42] is used for testing the silicon allotrope in the $C2$ structure. The equilibrium crystal structure is obtained by geometric optimization using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) [43] minimization scheme. A plane-wave cut-off energy of 340 eV and less than 2$\pi$$\times$0.025 Å$^{-1}$ Monkhorst-Pack grid for Brillouin zone sampling are used in structural optimization and prediction of properties. The exchange-related potentials are parameterized at the Perdew-Burke Ernzerhof (PBE) level of the generalized gradient approximation (GGA) [44]. The phonon spectrum of $C2$-Si is calculated by density functional perturbation theory (DFPT) [45]. The elastic constants are calculated for the structure optimized by the strain-stress method. The Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional [46] is used for the calculation of the electronic structures of $C2$-Si. For electronic structures within HSE06 hybrid functional, a plane wave cut-off energy of 910 eV is used for $C2$-Si, the Norm-conserving pseudopotentials [47] is adopted for electronic band structures calculations, and the Brillouin zones of $C2$-Si is sampled with 8$\times$2$\times$4 Monkhorst-Pack special $k$-point grids.

Ⅲ.   RESULTS AND DISCUSSION

A   Structural properties

The crystal structure of $C2$-Si in the $C2$ phase is plotted in FIG. 1(a). FIG. 1 (b), (c), and (d) show the crystal structures of $C2$-Si along the $a$-axis, $b$-axis, and $c$-axis, respectively. We use the PBE method to calculate the lattice parameters of c-Si, which are the same as the lattice parameters obtained in experiments, as shown in Table Ⅰ. Thus, the accuracy of the PBE algorithm is proven in this work. The calculated lattice parameters and density of $C2$-Si, c-Si, and $Pm$-$3m$ Si at the PBE level are listed in Table Ⅰ. At 0 GPa, for $C2$-Si, the equilibrium lattice constants calculated at the PBE level are $a$=5.538 Å, $b$=17.408 Å, $c$=10.328 Å, $\beta$=119$^\circ$, and $\rho$=2.133 g/cm$^3$. From Table Ⅰ, the density of $C2$-Si is slightly higher than that of Si$_{64}$ (1.732 g/cm$^3$), while it is slightly lower than that of c-Si (2.329 g/cm$^3$). The crystal structure of $C2$-Si includes four-ring, five-ring, six-ring, seven-ring, eight-ring, nine-ring, ten-ring, and eleven-ring, which are shown in FIG. 2. For the four-ring, it is concluded that the Si$-$Si bonds have two lengths, 2.327 and 2.370 Å, which can prove that the four-ring is symmetric. From FIG. 2, the five-ring, six-ring, seven-ring, and eight-ring have different bond lengths, but their bond lengths are almost equal. Furthermore, the nine-ring and eleven-ring have seven different Si$-$Si bonds, and the ten-ring has eight different Si$-$Si bonds.

Figure  1.  (a) Crystal structure of $C2$-Si, spheres of different colours represent different types of silicon atoms; and crystal structure along (b) $a$-axis, (c) $b$-axis and (d) $c$-axis.

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Table  Ⅰ.  Lattice parameters and density of $C2$-Si, Si$_{64}$ and c-Si.

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Figure  2.  Si rings in the $C2$-Si structure. The bond lengths (Å) are labelled.

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B   Elastic properties

To study the mechanical properties of $C2$-Si, the elastic constants $C_{ij}$, bulk modulus $B_{\rm{H}}$, shear modulus $G_{\rm{H}}$ and Young's modulus $E$ of $C2$-Si, Si$_{64}$, Si$_{96}$, c-Si, Si$_{10}$, Si$_{14}$, $I4/mmm$, and $h$-Si$_6$ are presented in Table Ⅱ. According to the elastic constants ($C_{11}$, $C_{12}$, $C_{13}$, $C_{15}$, $C_{22}$, $C_{23}$, $C_{25}$, $C_{33}$, $C_{35}$, $C_{44}$, $C_{46}$, $C_{55}$, $C_{66}$) given in Table Ⅱ, the mechanical stability of $C2$-Si can be estimated. If the allotrope is a stable monoclinic structure, then the elastic constants of $C2$-Si should satisfy the mechanical stability criteria of monoclinic symmetry. The mechanical stability of $C2$-Si can be estimated based on the following criteria: $C_{11}$$>$0, $C_{22}$$>$0, $C_{33}$$>$0, $C_{44}$$>$0, $C_{55}$$>$0, $C_{66}$$>$0, [$C_{11}$+$C_{22}$+$C_{33}$+2($C_{12}$+$C_{13}$+$C_{23}$)]$>$0, $C_{33}$$C_{55}$$-$${C_{35}}^2$$>$0, $C_{44}$$C_{66}$$-$${C_{46}}^2$$>$0, $C_{22}$+$C_{33}$$-$2$C_{23}$$>$0, $C_{22}$($C_{33}$$C_{55}$$-$${C_{35}}^2$)+2$C_{23}$$C_{25}$$C_{35}$$-$${C_{23}}^2$$C_{55}$$-$${C_{25}}^2$$C_{33}$$>$0, 2[$C_{15}$$C_{25}$($C_{33}$$C_{12}$$-$$C_{13}$$C_{23}$)+$C_{15}$$C_{35}$($C_{22}$$C_{13}$$-$$C_{12}$$C_{23}$) +$C_{25}$$C_{35}$($C_{11}$$C_{23}$$-$$C_{12}$$C_{13}$)]$-$[${C_{15}}^2$($C_{22}$$C_{33}$$-$${C_{23}}^2$)+ ${C_{25}}^2$($C_{11}$$C_{33}$$-$${C_{13}}^2$)+${C_{35}}^2$($C_{11}$$C_{22}$$-$${C_{12}}^2$)]+$g C_{55}$$>$0 [50]. The calculated results show that the elastic constants satisfy the mechanical stability criteria. Therefore, $C2$-Si is mechanically stable under ambient pressure.

Table  Ⅱ.  Elastic constants and elastic moduli of allotropes of silicon.

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The bulk modulus and shear modulus of $C2$-Si are determined using the Voigt-Reuss-Hill approximation:

With $a$, $b$, $c$, $d$, $e$, $f$, $g$, and $\Omega$ in Eq.(3) and Eq.(4) being,

From Table Ⅱ, the bulk modulus $B_{\rm{H}}$ of $C2$-Si is 52 GPa, and the shear modulus $G_{\rm{H}}$ of $C2$-Si is 33 GPa. The bulk modulus $B_{\rm{H}}$ of $C2$-Si is approximately half that of c-Si, and it is slightly less than those of Si$_{10}$, Si$_{14}$ and $h$-Si$_6$, while the bulk modulus $B_{\rm{H}}$ of $C2$-Si is larger than those of Si$_{64}$, Si$_{96}$ and $I4/mmm$ Si. The formula of Young's modulus $E$ and Poisson's ratio $\nu$ are as follows [49-52].:

The ratio of $B_{\rm{H}}/G_{\rm{H}}$ as an indication of ductile versus brittle character was proposed by Pugh [53]. A material is characterized as ductile if the $B_{\rm{H}}/G_{\rm{H}}$ ratio is greater than 1.75; if the $B_{\rm{H}}/G_{\rm{H}}$ ratio is below 1.75, then the material has a brittle character. From Table Ⅱ, it is concluded that Si$_{64}$ (2.29), Si$_{96}$ (1.93), $I4/mmm$ (2.53) and $h$-Si$_6$ (1.76) have the same characteristic of ductilty, whereas $C2$-Si (1.58), c-Si (1.40), Si$_{10}$ (1.20) and Si$_{14}$ (1.22) are characterized as brittle. $I4/mmm$ has the largest ductility and Si$_{10}$ has the largest brittleness in this work.

The phonon spectrum is an important marker to prove the dynamic stability of materials. To confirm the dynamic stability of Si in the $C2$ phase, its phonon spectrum is studied in this work. The phonon spectrum of $C2$-Si is shown in FIG. 3(a). The phonon spectrum shows no imaginary frequency over the entire Brillouin zone, which indicates that $C2$-Si is dynamically stable [57]. The formation energy of $C2$-Si and other allotropes (such as, $h$P30-Si, $h$P36-Si, Si$_{96}$, $t$-Si$_{64}$, $I4/mmm$-Si, Si$_{20}$-T, $Im$-3m Si, and $h$P24-Si) are listed in FIG. 3(b).

Figure  3.  (a) Phonon spectrum of $C2$-Si calculated by DFPT, (b) relative enthalpies of $C2$-Si and other allotropes.

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The formula for calculating the formation energy is as follows:

where $n_1$ is the number of atoms in the new silicon allotrope and $n_2$ is the number of atoms in c-Si. The calculated relative enthalpy of $C2$-Si is 0.345 eV/atom higher than that of c-Si, while the relative enthalpy of $C2$-Si is less than those of $t$-Si$_{64}$ (0.413 eV/atom) and $I4/mmm$-Si (0.364 eV/atom), as shown in FIG. 3(b).

C   Electronic properties

It is well known that the electronic structure determines the most basic physical and chemical properties of materials. However, the DFT method usually underestimates the electronic band structure [21]. Considering this problem, Heyd $et$ $al$. [58] proposed a hybrid functional method that is easier to handle, and the Heyd-Scuseria-Ernzerhof (HSE06) functional was generated. The hybrid functional HSE06 is used in the following form [21, 58]:

where the HF mixing parameter $\mu$ is 0.25 and the screening parameter $\omega$ is 0.207 Å$^{-1}$ [46, 58]. Then, the band structure of $C2$-Si is calculated with the HSE06 functional, as shown in FIG. 4, and the band gap is 0.716 eV. The band gap of c-Si (1.12 eV [25, 59]) is approximately 1.5 times that of $C2$-Si. It is clearly found that the results calculated with the HES06 hybrid functional are much larger than those calculated with GGA-PBE. The grey dotted line in the energy band diagram represents the Fermi level ($E_{\rm{F}}$). The valence band maximum (VBM) is located at the A point, and the conduction band minimum (CBM) is located at the Z point, which indicates that $C2$-Si has indirect semiconductor characteristics, so $C2$-Si is an indirect band gap semiconductor.

Figure  4.  Electronic band structure of $C2$-Si within HES06 hybrid functional.

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D   Elastic anisotropy

If a material is isotropic, its three-dimensional surface constructions of the shear modulus, Young's modulus, and Poisson's ratio appear as spheres, and the measured performance values of the material in different directions are exactly the same, while the degree of deviation from a sphere reflects the strength of anisotropy. In this work, we list four silicon allotropes, the $C2/c$, $C2/m$-16, $C2/m$-20 and $P2_1/m$ phases, because all of them have monoclinic symmetry to compare with $C2$.

As we know, the crystal system is one of the factors that affect the anisotropy of materials. In order to compare all of the five silicon allotropes in detail related to the anisotropy, we use the ratio of the maximum to minimum value ($E_{\max}/E_{\min}$) to measure the elastic anisotropy in different planes, including (001) plane, (010) plane, (011) plane, (100) plane, (101) plane, (110) plane, and (111) plane. The maximum values, minimum values, and $E_{\max}/E_{\min}$ ratios of the Young's modulus for the $C2$, $C2/c$, $C2/m$-16, $C2/m$-20, and $P2_1/m$ silicon allotropes in the seven main planes are given in Table Ⅲ. From Table Ⅲ, the maximum value of $C2$-Si is 107.98 GPa, the minimum value is 64.34 GPa, and the $E_{\max}/E_{\min}$ ratio is 1.68, which is slightly greater than those of the $C2/m$-16 (1.43), $C2/m$-20 (1.52) and $P2_1/m$ (1.60) silicon allotropes, while it is slightly smaller than that of the $C2/c$ (1.80) silicon allotrope.

Table  Ⅲ.  The maximum and minimum Young's modulus, shear modulus and Poisson's ratio for different planes of the $C2$, $C2/c$, $C2/m$-16, $C2/m$-20, and $P2_1/m$ phases along with the ratio $E_{\max}/E_\min$, $G_{\max}/G_\min$, or $\nu_{\max}/\nu_\min$.

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FIG. 5(a) shows, for the structure in the $C2$ phase, the 3D surface of the Young's modulus along the $x$-axis, $y$-axis and $z$-axis largely deviates from a spherical shape, which means that the $C2$ phase is highly anisotropic in Young's modulus. The projections of the Young's modulus onto different planes are demonstrated in FIG. 5(b). Because the minimum and maximum Young's modulus of $C2$-Si appear in the (010) plane and its $E_{\max}/E_{\min}$ ratio is 1.68, $C2$-Si has the greatest mechanical anisotropy in the (010) plane. For the (110) plane, the $E_{\max}/E_{\min}$ ratio is 1.35, showing the smallest mechanical anisotropy of $C2$-Si. The mechanical anisotropy in the Young's modulus of $C2$-Si follows the order of (010) plane$>$(100) plane$>$(011) plane=(101) plane$>$(111) plane$>$(110) plane$>$(001) plane. The greatest $E_{\max}/E_{\min}$ ratio of the $C2/c$ phase is 1.66 in the (101) plane. The largest mechanical anisotropy in the Young's modulus of the $C2/m$-16, $C2/m$-20, and $P2_1/m$ phases appears on the (110) plane, (011) plane, and (001) plane, respectively. In different planes, $C2$-Si has a larger mechanical anisotropy than these silicon allotropes.

Figure  5.  Directional dependence of the Young's modulus for (a) $C2$; 2D representation of the Young's modulus for (b) $C2$-Si; the directional dependence of the Young's modulus for (c) $C2/c$, (d) $C2/m$-16, (e) $C2/m$-20, and (f) $P2_1/m$ phases.

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The maximum surfaces of the shear modulus of the $C2$, $C2/c$, $C2/m$-16, $C2/m$-20, and $P2_1/m$ phases are shown in FIG. 6(a - e), and the minimum surfaces of the shear modulus are shown in FIG. 6(f - j). From FIG. 6, the $C2$, $C2/c$, $C2/m$-16, $C2/m$-20 and $P2_1/m$ silicon allotropes exhibit elastic anisotropy in the 3D surface of the shear modulus. The elastic anisotropy in the shear modulus of different phases is compared based on $G_{\max}/G_{\min}$. The maximum values, minimum values and $G_{\max}/G_{\min}$ ratios of the shear modulus for the $C2$, $C2/c$, $C2/m$-16, $C2/m$-20 and $P2_1/m$ phases in the seven main planes are given in Table Ⅲ. From Table Ⅲ, the maximum and minimum values of shear modulus of $C2$-Si are 45.44 GPa and 26.61 GPa, which appear in the (101) plane at the same time. Therefore, for $C2$-Si, the anisotropy in the (101) plane is greater than that in the other planes. From the overall data of each phase, the largest ratios for the other silicon allotropes are as follows: $C2/c$ (1.65), $C2/m$-16 (1.57), $C2/m$-20 (1.68) and $P2_1/m$ (1.85). It is concluded that the elastic anisotropy of the $C2$ phase is larger than that of the $C2/c$ and $C2/m$-16 phases, less than that of the $P2_1/m$ phase, and equal to that of the $C2/m$-20 phase. In addition, the 2D representations of the maximum and minimum shear modulus of $C2$-Si are illustrated in FIG. 6 (k) and (l). As can be seen from FIG. 6 (k) and (l), the figures of the maximum and minimum values of the shear modulus on each plane are not circular, and each plane of $C2$-Si has different anisotropy. For the (010) plane, the minimum ratio of $G_{\max}/G_{\min}$ is 1.34, which shows the minimum elastic anisotropy in the shear modulus of $C2$-Si. The elastic anisotropy of the $C2$-Si shear modulus follows the order of (101) plane$>$(111) plane$>$(110) plane=(001) plane$>$(011) plane$>$(100) plane$>$(010) plane. It is concluded that the largest elastic anisotropy in the shear modulus of $C2/c$ and $P2_1/m$ is in the (010) plane and (100) plane. For $C2/m$-Si$_{16}$ and $C2/m$-Si$_{20}$, the largest ratios are 1.55 and 1.85, respectively.

Figure  6.  (a) $-$(e) Directional dependence of the maximum shear modulus of the $C2$, $C2/c$, $C2/m$-16, $C2/m$-20 and $P2_1/m$ silicon allotropes; (f)$-$(j) directional dependence of the minimum shear modulus of the $C2$, $C2/c$, $C2/m$-16, $C2/m$-20 and $P2_1/m$ silicon allotropes; (k) 2D representation of the maximum shear modulus of $C2$-Si; (l) 2D representation of the minimum shear modulus of $C2$-Si.

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The 3D surface constructions of the Poisson's ratio of the $C2$, $C2/c$, $C2/m$-16, $C2/m$-20 and $P2_1/m$ silicon allotropes are shown in FIG. 7. The maximum surfaces of the Poisson's ratio of the $C2$, $C2/c$, $C2/m$-16, $C2/m$-20 and $P2_1/m$ phases are shown in FIG. 7(a-e), and the minimum surfaces of the Poisson's ratio are shown in FIG. 7(f-j). From these 3D surface constructions of Poisson's ratio diagrams, it can be seen that the elastic anisotropy of the $C2$ phase is slightly greater than that of the $C2/c$ phase and less than that of the $C2/m$-16, $C2/m$-20, and $P2_1/m$ phases. The 2D representations of the maximum and minimum Poisson's ratio of $C2$-Si are shown in FIG. 7 (k) and (l), which will help us better observe the elastic anisotropy in Poisson's ratio. Here, the magnitude of the elastic anisotropy is calculated by $\nu_{\max}/\nu_{\min}$. The maximum values, minimum values of Poisson's ratio and $\nu_{\max}/\nu_{\min}$ ratios for the $C2$, $C2/c$, $C2/m$-16, $C2/m$-20, and $P2_1/m$ silicon allotropes in the seven main planes are given in Table Ⅲ. From Table Ⅲ, the maximum Poisson's ratio of $C2$-Si is 0.43 and the minimum Poisson's ratio of $C2$-Si is 0.06 in the seven main planes, so the maximum $\nu_{\max}/\nu_{\min}$ value is 7.17 in the (010) plane. The $\nu_{\max}/\nu_{\min}$ value of $C2$-Si is larger than those of the $C2/c$ (2.79), $C2/m$-16 (2.92), $C2/m$-20 (4.00), and $P2_1/m$ (5.13) silicon allotropes, which proves that $C2$-Si has more elastic anisotropy in the Poisson's ratio. $C2$-Si has the greatest Poisson's ratio in the (010) plane and the smallest Poisson's ratio in the (011) plane. The mechanical anisotropy of $C2$-Si in Poisson's ratio follows the order of (010) plane$>$(111) plane=(110) plane$>$(001) plane$>$(101) plane$>$(100) plane$>$(011) plane. From Table Ⅲ, the largest $\nu_{\max}/\nu_{\min}$ ratios of the $C2/c$, $C2/m$-16, $C2/m$-20, and $P2_1/m$ phases are 2.71, 2.64, 3.89, and 4.67 respectively. The largest elastic anisotropy in the Poisson's ratio of the $C2/c$, $C2/m$-16, $C2/m$-20 and $P2_1/m$ silicon allotropes is in the (100) and (110) planes, (110) plane, (001) plane, and (110) plane respectively. $C2$-Si has the largest mechanical anisotropy in Poisson's ratio in the (010) plane.

Figure  7.  (a) - (e) Directional dependence of the maximum Poisson's ratio of the $C2$, $C2/c$, $C2/m$-16, $C2/m$-20 and $P2_1/m$ silicon allotropes; (f) - (j) directional dependence of the minimum Poisson's ratio of the $C2$, $C2/c$, $C2/m$-16, $C2/m$-20 and $P2_1/m$ silicon allotropes; 2D representation of the maximum Poisson's ratio of $C2$-Si (k); 2D representation of the minimum Poisson's ratio of $C2$-Si (l).

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E   Optical properties

Regarding the optical properties, we mainly study the absorption ability of the new silicon allotrope for visible light. The absorption spectra of $C2$-Si in comparison with those of other silicon allotropes are shown in FIG. 8. According to previous studies [20, 60], the absorption ability of c-Si is not as good as that of $h$-Si$_6$, $o$-C12 phase, and $h$P12 phase. We know that the wavelength range from 390 nm to 780 nm is visible light. From FIG. 8, compared with the silicon allotropes in the $o$C12 phase and $h$P12 phase, $h$-Si$_6$ [1, 20], $Cmca$ Si [8, 30], and allotropes in the D239 and D63 phases [35, 36], $C2$-Si has a stronger absorption ability in the ultraviolet range. Zooming in the visible region, one can find that the absorption ability of $C2$-Si is the strongest in the visible region, it is stronger than that of $h$-Si$_6$, $o$C12 phase, and $h$P12 phase; while $o$C12 phase has the weakest absorption ability for visible light, especially in the yellow, orange and red visible light regions. In the near-infrared region with wavelengths over 780 nm, we can still find that $C2$-Si has a superior absorption ability. In the near-infrared region, the absorption capacity of the other silicon allotropes used for comparison in this work is almost negligible. In the visible and near-infrared regions, the absorption capacity of $C2$-Si is larger than that of other phases; that is, $C2$-Si has the largest light absorption range. Although the proposed structure is an indirect band gap, while the absorption ability of $C2$-Si is the largest in visible region and near-infrared region in this work, which include direct band gap, such as D63, D135, D76 silicon allotropes.

Figure  8.  Absorption spectra of $C2$ silicon compared to those of other silicon allotropes.

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Ⅳ.   CONCLUSION

In this work, a new silicon allotrope, $C2$-Si, is theoretically predicted, and its structural properties, elastic properties, and anisotropy are investigated with first-principles calculations. By studying the elastic constants and phonon spectrum, $C2$-Si is found to be mechanically stable and dynamically stable. $C2$-Si is characterized as a brittle material with $B_{\rm{H}}/G_{\rm{H}}$$<$1.75. From the Young's modulus, the mechanical anisotropy of $C2$-Si is slightly greater than that of the $C2/m$-16, $C2/m$-20, and $P2_1/m$ silicon allotropes, while it is slightly smaller than that of $C2/c$. From the shear modulus, the elastic anisotropy of $C2$ is greater than that of $C2/c$, and $C2/m$-16, less than that of $P2_1/m$, and equal to that of $C2/m$-20. $C2$-Si has the largest anisotropy in Poisson's ratio. According to the 2D surface structure diagrams of the Young's modulus, shear modulus and Poisson's ratio of $C2$-Si, the (010) plane has the maximum anisotropy in Young's modulus, the (101) plane has the maximum anisotropy in shear modulus, and the (010) plane has the maximum anisotropy in Poisson's ratio. The band structure of $C2$-Si indicates that it is an indirect and narrow band gap material with the band gap of 0.716 eV within the HSE06 hybrid functional. In addition, the absorption capacity of $C2$-Si is greater than that of other phases in the visible and near-infrared regions.

Ⅴ.   ACKNOWLEDGEMENTS

This work is supported by the National Natural Science Foundation of China (No.61804120 and No.61901162), the China Postdoctoral Science Foundation (No.2019TQ0243 and No.2019M663646), the Key Scientific Research Plan of Education Department of Shaanxi Provincial Government (Key Laboratory Project) (No.20JS066), the Young Talent Fund of University Association for Science and Technology in Shaanxi, China (No.20190110), the National Key Research and Development Program of China (No.2018YFB1502902), and Key Program for International S & T Cooperation Projects of Shaanxi Province (No.2019KWZ-03).