Two novel SiC phases: structure, mechanical, and transport properties

Two novel phases of SiC are put forward in this paper, in which the crystal structural, mechanical, and electronic properties, as well as effective mass and carrier mobility of SiC in the Pnnm phase (Pnnm-SiC) and Pm phase (Pm-SiC) are researched utilizing first principles calculations. Both of the novel SiC phases are certificated to have good mechanical and dynamic stability. Through analysis of the three-dimensional perspective of Young’s modulus, shear modulus and Poisson’s ratio, visible anisotropies of mechanical properties are found. The band structure calculations predict two wide bandgap semiconductors, that the Pnnm-SiC is an indirect with a bandgap value of 3.12 eV, While the Pm-SiC is a quasi-direct with a bandgap value of 2.64 eV, which indicates the Pm-SiC has a higher application potential in the optoelectronic device area. An extremely large electronic mobility (7200 cm2 V−1s−1) is found in the Pnnm-SiC. Based on the wide band gap, large carrier mobility, good mechanical and dynamic stability, the Pnnm-SiC is a promising material in the field of high performance electronic device in harsh environment.


Introduction
With the continuous progress of the semiconductor material industry, increasing number of researchers are attempting to seek out semiconductors with an enhanced performance analogue to that of IV alloys. SiC is considered as a quite promising technical material owing to its outstanding properties, such as chemical inertness, high hardness and melting temperature, and prominent semiconductor properties [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. The most of polytypes being under ambient conditions are constituted of Si 4 C (SiC 4 ) tetrahedral units with cubic (3C-SiC), hexagonal (2H-, 4H-, and 6H-SiC), or rhombohedral (15R-, 21R-, and 33R-SiC) structures relying on the stacking sequence [2][3][4][5][6][7][8]. The cubic structure is also called beta-SiC (β-SiC), as well as the structures of hexagonal and rhombohedral are all sort out under the general term alpha-SiC (α-SiC) [9][10][11][12]. Daviau et al [13] made a comprehensive review, under different conditions, the phase of SiC changes one polytype to another polytype. They explore high-pressure SiC with experiment and computation, and the effect of measuring pressure on SiC vibration and material properties. In addition, high temperature studies, including thermal expansion and the equation of thermal state of SiC, and the melting behavior at high temperature and pressure have probed. SiC has been observed over 250 polytypes under ambient conditions [14], in order to know the formation of specific polytypes, the transformation conditions, transformation mechanisms, and high temperature between different polytypes much work has been done [15,16].
Over the past decades, many researches have paid much effort on carbon materials or carbon-based materials that emerge hardness and stiffness approach that of diamond. Xing et al [17] put forward C2/m-20 carbon and made a comparison that other monoclinic symmetry was investigated. Moreover, they discovered that C2/m-20 carbon displays the largest elastic modulus, and anisotropy of Young's modulus. In addition, many researchers predicted other carbon allotropes, such as lonsdaleite [18,19], TAL-carbon [20], W-carbon [21], PBCF-graphene [22], C-carbon [23], Z-carbon [24], H-carbon, S-carbon [25], T-carbon [26], and so on. Fan et al [27] came up with four new silicon allotropes, including Amm2, C2/m- 16 confirmed that all structures are mechanically and dynamically stable, and C2/m-20 phase shows the higher anisotropy compared with other structures. Bai et al [28] proposed a novel silicon allotrope which is in the P2/m space group, and discovered that P2/m-Si is brittle. The results also show along different directions the difference of elastic anisotropy is larger than other phases (P222 1 , Fd-3m, C2/m-16, C2/m-20, Amm2). With the continuous deep research of carbon and silicon allotropes by researchers, the development of C-Si alloys has been promoted, and C-Si alloys are fascinating more attention in academic and electronic industry owing to their potential outstanding properties unlike elemental silicon and carbon. Recently, researchers are engaged in finding new kinds of C-Si alloys with fabulous physical properties. Many the three-dimensional silicon carbide materials are found by researchers, Zhang et al [29] put forward two novel phases of P4 2 /mnm symmetry of Si 8 C 4 and Si 4 C 8 and systematically researched their structural, electronic, and elastic properties, and both of them are confirmed to have mechanical and dynamical stability, and the calculations of electronic structure present that Si 8 C 4 and Si 4 C 8 are indirect materials with the band gap values of 0.74 and 0.15 eV. The mechanical, electronic and anisotropic properties of SiC 2 and SiC 4 were investigated by Fan et al [30], and the results indicated that those structures all have anisotropy and that SiC 2 is an indirect material with a band gap value of 2.28 eV, but for SiC 4 is a quasi-direct material with a band gap value of 0.91 eV. Tan, and Zhang et al [31,32] put forward C-Si alloys in the orthorhombic structure of P222 1 space group, and confirmed that all structures have mechanical stability and elastic anisotropy, and the results of band structure display that the put forward structures are all indirect band gap material. Wang et al [33] put forward C-Si alloys (C 16 , C 12 Si 4 , C 8 Si 8 , Si 16 , and C 4 Si 12 ) in C2/m structure, and shown that their alloys are mechanically stable, meanwhile their C-Si alloys are brittle. The results of band structure display that C 16 and Si 16 are indirect material, but for C 12 Si 4 , C 8 Si 8 and C 4 Si 12 are semi-metallic alloys. In addition, theoretical prediction of new SiC allotropes in C2/m-20 structure were proposed by Xu et al [34], and confirmed that all structures have mechanical and dynamic stability. The calculations of the band structures indicate that C 12 Si 8 is a direct bandgap, C 8 Si 12 and C 4 Si 16 are semi-metallic alloys, but for C 16 Si 4 is indirect. It should be noted that dope two indirect band gap semiconductors (C 20 , Si 20 ), and a direct band gap semiconductor (C 12 Si 8 ) is attained.
Based on the previous work, two novel phases of SiC (Pnnm-SiC, and Pm-SiC) are put forward. The structural, mechanical, electronic, anisotropic, and transport properties of SiC in different phases are calculated applying first principles calculations. The Pnnm-SiC is an indirect semiconductor while Pm-SiC is a quasi-direct semiconductor, but all SiC structures have mechanical and dynamic stability. The mechanical anisotropy is analyzed utilizing elastic modulus from the three-dimensional perspectives. What's more, the anisotropy of the carrier mobility is analyzed using the phonon-limited model, and an exceedingly high electron mobility (7200 cm 2 V −1 s −1 ) and hole mobility (170 cm 2 V −1 s −1 ) are found in the Pnnm-SiC.

Computational methods
The theoretical calculations are executed with the Vienna ab initio simulation package (VASP) in view of density functional theory (DFT) [35][36][37]. The projector augmented wave (PAW) method is applied to bewrite the electron-ion interaction, and the Perdew-Burke-Ernzerh (PBE) exchange-correlation functional of the generational gradient approximation (GGA) is taken [38]. In all calculations, k-point samplings with 0.04 Å −1 is adopted to Pnnm-and Pm-SiC, and the value of energy cutoff of the plane wave is set to 500 eV. The geometry optimization parameters are determined using the Broyden-Fletcher-Shenno (BFGS), with the following convergence tolerance: the self-consistent convergence of the energy is´-5 10 7 eV Å −1 , the maximum force on the atom is 0.02 eV Å −1 , and stress is less than 0.02 GPa. The phonon frequencies are computed utilizing linear response theory (DFPT) [39]. DFT-D3(BJ) method is adopted in the structural, elastic properties, and carrier mobility (effective elastic constant) calculations [40,41]. Moreover, on account of the DFT usually underrates the band energy of solid materials, the electronic band structures of Pnnm-and Pm-SiC are calculated applying a hybrid functional (HES06) [42][43][44]. are Pm-SiC, which contains 5and 7-membered rings (5+7) and six-atom hexagonal C-Si rings. After the geometry optimization using PBE, PBE-D3, and LDA method, the calculated lattice parameters of β-SiC, Pnnm-SiC, and Pm-SiC are enumerated in table 1. According to table 1, the computed lattice parameter of β-SiC using PBE-D3 method is in fabulous accordance with experimental results [45] in that its minute difference 0.016 Å between this calculation work and experimental lattice parameters. So the lattice parameter computed by PBE-D3 method is more consistent with previous experiment value relative to PBE method and LDA method.  Table 1. The lattice parameters a, b and c ; α, β, and γ (in°); cell volume; and cell density of β-SiC, Pnnm-SiC, and Pm-SiC, respectively.

Material
Method The enthalpy of SiC in different phases can be utilized as a significant norm to consider whether the semiconductor materials can be synthesized. Enthalpy of formation is the energy absorbed or released by various atoms from simple compounds [46], which expresses the difficulty level of the formation of compounds. The formation enthalpy (ΔH) is computed using the following expression: Where the E total is the entire energy of Pnnm-SiC or Pm-SiC; the n Si is the number of Si atoms in the cell; n C is the number of C atoms in the cell; E solid Si is the energy of a silicon atom in elemental silicon; and E solid C is the energy of a carbon atom in elemental carbon. According to the expression, the formation enthalpies of Pnnm-SiC and Pm-SiC are calculated, and their formation enthalpies are −0.373 eV and −0.261 eV, respectively. It is noteworthy that formation enthalpies of SiC in different phases are negative, it can be seen that both of SiC have relative stability.
The phonon spectrum of SiC in the two different phases are computed to certificate its dynamic stability, and the spectrum of SiC in different phases are displayed in figure 2. From figure 2, the SiC of these different phases all have dynamical stability because there are no imaginary frequencies throughout the entire Brillouin zone for the Pnnm-SiC and Pm-SiC.
On the other hand, the material's mechanical properties determine whether its mechanical stability and it can exist in the nature. The elastic constant and modulus of β-SiC, Pnnm-SiC and Pm-SiC are calculated in this work. There are nine independent elastic constants C ij (C 11 , C 22 , C 33 , C 44 , C 55 , C 66 , C 12 , C 13 , C 23 ) in the orthorhombic symmetry of the space group of Pnnm. But for the monoclinic symmetry of the space group of Pm, besides the nine independent elastic constants are the same as the orthorhombic symmetry, which also contains other four independent elastic constants (C 15  Both of the independent elastic constants of Pnnm-SiC and Pm-SiC accord with the mechanical stability criteria [48,49]. That is to say, both of different phases of SiC have mechanical stability under ambient condition. Moreover, bulk modulus (B) and shear modulus (G) are fracture and plastic deformation resistance. The Voigt-Reuss-Hill approximation [50-52] is taken to compute B and G, and they are defined in the following  Table 2. The elastic constant (in GPa) and the elastic moduli (in GPa) of Pnnm-SiC, Pm-SiC, and β-SiC.

Materials
Method Where B V means the Voigt approximation of B; B R is the Reuss approximation of B; G V is the Voigt approximation of G; and G R is the Reuss approximation of G. The Young's modulus (E) is applied to measure the solid's stiffness, and we find that the higher the value of E is, the stiffer the material is [53]. Meanwhile, the E can be computed by the equation, and the equation is as follows [52]: The ratio of B to G is an identifier that can determine a solid is brittle or not according to the Pugh's theory [47]. Poisson's ratio v is accord with B/G, which indicates the brittle compounds with a small value of v (<0.26), and ductile compounds with a large value of v (>0.26) [54]. v can be attained from the formula, and the formula is as follows: All of the elastic moduli, including B, G, E and v of Pnnm-SiC and Pm-SiC are enumerated in table 2. From the table 2, the elastic constants of β-SiC in this work using PBE-D3 method is in fabulous accordance with the results of previous results [55], which proves our work is accurate and faithful. Besides, on the basis of the above criteria mentioned, both of the elastic constants of Pnnm-SiC and Pm-SiC at ambient condition are positive and meet the mechanical stability standard, demonstrating that both of SiC are mechanically stable. The E can be applied to depict the relevant tensile strain, and according to the above discovery of E we find that Pnnm-SiC is stiffer than Pm-SiC. The B of Pnnm-SiC is slightly larger than that of Pm-SiC, while both of SiC in different phases are slightly smaller than β-SiC. For v of SiC in different phases, as proposed by Pugh [47], β-SiC, Pm-SiC and Pnnm-SiC are brittle materials because they have small values of v (<0.26). Moreover, the B, G and E of Pnnm-SiC and Pm-SiC are possible to reach that of the β-SiC, which indicates both of SiC in different phases have good mechanical stability.

Anisotropic properties
As we all know, the elastic anisotropy is a significant implication in crystallophysics. Moreover, the threedimensional (3D) surface construction is an effective method to depict the elastic anisotropy [56], and the spatial distribution of elastic modulus has been researched. The directional dependence of Young's modulus, the maximum of shear modulus (shear modulus MAX), the minimum of shear modulus (shear modulus MIN), the maximum of Poisson's ratio (Poisson's ratio MAX), and the minimum of Poisson's ratio (Poisson's ratio MIN) for Pnnm-SiC and Pm-SiC at zero pressure are shown in figures 3(a)-(e), and 4(a)-(e), respectively. Moreover, the three-dimensional directional dependence of isotropic material will present a spherical shape; and the higher the spherical shape deviation is, the higher the mechanical anisotropy is [57]. Different from E, the magnitude of v and G in the two directions, including direction a and direction b, and demands three angles to be described [58]. The G is defined as the ratio of shear stress to linear shear strain, and v is defined as the ratio of transverse strain to axial strain [59].   In addition, the carrier effective mass has the greatest influence on the carrier transport properties, such as the lager the effective mass but the smaller carriers mobility, So there is no doubt for us to calculate effective mass. As for the solution of carrier effective mass, which is related to the band dispersion as: ] (i, j=1, 2, 3) is the inverse effective mass tensor. What is more, the band structure effective mass in the certain orientation is defined as the following equation: The electrons effective mass or holes effective mass can be defined as the partial differential at the upper edge of the valence band or at the lower edge of the conduction band. Based on the detailed description of effective mass, we have computed it about Pnnm-SiC, Pm-SiC, and β-SiC as follows. The lattice directions and magnitude of effective mass of SiC in different phases are abstracted and enumerated in table 4. By analyzing the results of table 4, the electronic effective mass and the hole effective mass of β-SiC are in excellent agreement with previous calculation results [60][61][62] because of the minute difference 0.01 between this calculation work and previous calculation work. In addition, we find that a high electron effective mass is 5.638 m 0 and hole effective mass is 0.1872 m 0 in the Pnnm-SiC. Moreover, the electronic effective mass and the hole effective mass of Pm-SiC are all larger than Pnnm-SiC, respectively. The electronic effective mass of Pm-SiC and Pnnm-SiC shows different calculation results in different directions, so both of SiC in different phases have effective mass anisotropy.

Carrier mobility
As for carrier mobility, a phonon-limited scatter model is adopted, in which we ignore ionized impurity scattering and the optical branch scattering, and the reason is that the main mechanism limiting carrier mobility is acoustic phonon scattering in intrinsic semiconductor. A three-dimensional carrier mobility of a solid can be computed as [63]: W i and D -W , i and 1% compression and dilatation have been applied in this calculation work. l 0 is the lattice constant of the transport orientation and Δl is the deformation relative to l 0 ; r and υ s mean the density of crystal and the velocity of acoustic. As for the relationship between r and υ s , which can be defined as ru = C , eff s 2 and C eff is the effective constant of elasticity. The deformation potential constant (in eV), effective elastic constant (in GPa), and carrier mobility (in 10 3 cm 2 V −1 s −1 ) are displayed in table 5. The carrier mobilities m , l m , t1 and m t2 are computed by utilizing the above equation (8), and the temperature (T) is set to 300 K. The carrier mobility of β-SiC is computed to prove the precision of the results of SiC in different phases. According to table 5, the calculated carrier mobilities (m e =710 cm 2 V −1 s −1 , m h =110 cm 2 V −1 s −1 ) of β-SiC are in fabulous accordance with the experimental results (m e =750 cm 2 V −1 s −1 , m h =120 cm 2 V −1 s −1 ) [64][65][66]. A high electron mobility (270 cm 2 V −1 s −1 ) and hole mobility (60 cm 2 V −1 s −1 ) is found in the Pm-SiC, but for the Pnnm-SiC has higher electron mobility (7200 cm 2 V −1 s −1 ) and hole mobility (170 cm 2 V −1 s −1 ) than the Pm-SiC, which indicates the Pnnm-SiC has good transport properties and more potential applications. However, both of SiC in different phases shows different calculation results in different directions, which indicates Pnnm-SiC and Pm-SiC all have carrier mobility anisotropy. In addition, we find that even if the effective mass of β-SiC is smaller than the Pnnm-SiC, the electron mobility of Pnnm-SiC is larger than β-SiC. There are two reasons for the result: on the one hand the Table 4. Direction vector is along the three principal lattice (a, b, c) and magnitude (in m a , m b , and m c ) of the effective mass of β-SiC, Pnnm-SiC and Pm-SiC, respectively.  Table 5. Effective constant of elastic (in GPa), the constant of deformation potential (in eV) and carrier mobility (in 10 3 cm 2 V −1 s −1 ) of β-SiC, Pnnm-SiC, and Pm-SiC.
Carrier type Materials Pnnm-SiC (494 GPa) has larger effective elastic constant than β-SiC (390 GPa), and the other hand the deformation potential constant of β-SiC (15.02 eV) is larger than Pnnm-SiC (8.64 eV), so the Pnnm-SiC has larger electron mobility with a value of 7200 cm 2 V −1 s −1 . Due to its high carrier mobility, the Pnnm-SiC has good transport properties and that is promising materials in the microelectronics area.

Conclusions
The structural, mechanical, and electronic properties, as well as anisotropy, and carrier mobility of SiC in different phases have been investigated using first principle calculations in view of DFT. The phonon spectra and elastic constants calculations illustrate that Pnnm-SiC and Pm-SiC are dynamically and mechanically stable under ambient pressure. What is more, through an analysis of elastic modulus, both of SiC in different phases are mechanically anisotropic, and Pm-SiC and Pnnm-SiC are naturally brittle materials because they have small values of Poisson's ratio v (<0.26). The calculations of band structure predict that Pnnm-SiC is an indirect semiconductor with a value of 3.12 eV, while Pm-SiC is a quasi-direct semiconductor with a value of 2.64 eV, which demonstrates the Pm-SiC has large potential applications in the optoelectronic devices area. The effective mass calculations show that effective mass of Pnnm-SiC is smaller than Pm-SiC, and both of SiC in different phases all have effective mass anisotropy. Finally, carrier mobilities of all different phases of SiC are calculated using a phonon-limited scatter model, and the result shows that a high electronic mobility (7200 cm 2 V −1 s −1 ) and hole mobility (170 cm 2 V −1 s −1 ) in the Pnnm-SiC, and both of SiC in different phases all have carrier mobility anisotropy.