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Crystal structures and the electronic properties of silicon-rich silicon carbide materials by first principle calculations

Silicon carbide has been used in a variety of applications including solar cells due to its high stability. The high bandgap of pristine SiC, necessitates nonstoichiometric silicon carbide materials to be considered to tune the band gap for efficient solar light absorptions. In this regards, thermodynamically stable Si-rich SixC1-x materials can be used in solar cell applications without requiring the expensive pure grade silicon or pure grade silicon carbide. In this work, we have used density functional theory (DFT) to examine the stability of various polymorphs of silicon carbide such as 2H–SiC, 4H–SiC, 6H–SiC, 8H–SiC, 10H–SiC, wurtzite, naquite, and diamond structures to produce stable structures of Si-rich SixC1-x. We have systematically replaced the carbon atoms by silicon to lower the band gap and found that the configurations of these excess silicon atoms play a significant role in the stability of Si-rich SixC1-x. Hence, we have investigated different configurations of silicon and carbon atoms in these silicon carbide structures to obtain suitable SixC1-x materials with tailored band gaps. The results indicate that 6H-SixC1-x is thermodynamically the most favorable structure within the scope of this study. In addition, Si substitution for C sites in 6H–SiC enhances the solar absorption, as well as shifts the absorption spectra toward the lower photon energy region. In addition, in the visible range the absorption coefficients are much higher than the pristine SiC.


Introduction
Silicon carbide is a very important semiconductor material [1,2] which has more than 200 poly-types [3] and has great properties which make it an attractive material to be used for applications in extreme environment [4,5,6]. These interesting properties include high strength, high hardness, and low thermal expansion [7,8,9], and it has been used in high-temperature applications because of its high thermal conductivity [10,11,12] which is 3.6 W/cm-K for 3C-SiC, and 4.9 W/cm-K for 4H-SiC and 6H-SiC [13]. 3C-SiC, 2H-SiC, 4H-SiC, and 6H-SiC are the most common phases. SiC can be synthesized by chemical vapor deposition (CVD) [14,15,16,17,18], vapor liquid solid (VLS) [19,20,21,22,23] and physical vapor transport (PVT) [24,25,26,27,28] methods. Silicon carbide has been used in photovoltaic solar cells (PVSC) [29,30,31,32,33,34,35] for decades. However, photovoltaic devices need efficient materials which have high stability and suitable band gap to work efficiently. The band gap of SiC ranges from 2.3 eV to 3.3 eV [36,37,38,39,40]. This is higher than what is suitable for efficient solar energy absorptions. Efficient solar conversion requires a bandgap of less than 1.5 eV. Hence, it is necessary to have tunable band gaps in SiC materials. Previously, it has also been shown from the first principle based atomistic simulations that Si-rich SiC nano-clusters can be very stable and can have tunable energy gaps [41,42,43]. In the bulk limit, the non-stoichiometric SiC may provide such options [44] as well, so we have studied Si-rich SiC with periodic boundary conditions in this work.
It has been shown that increasing carbon atoms in amorphous silicon carbide will increase the optical band gap, and the fraction of carbon affects the optical properties negatively in amorphous silicon carbide [45,46]. Also, the result shows that the optical band gap changes as the silicon concentration increases. A non-stoichiometric Si-rich Si x C 1-x film can increase the percentage of the conversion efficiency of a p-n junction photovoltaic solar cell by up to 3% than that of crystalline silicon within the visible wavelengths (400-600 nm) which is due to enhanced optical absorption. Si-rich Si x C 1-x film can also be used to engineer the visible and near-infrared absorbance and can be enhanced [44,45,47]. These results indicate that the absorption coefficient correlates with the ratio of silicon-rich silicon carbide [44,45]. The experimental result also shows that due to the increasing number of silicon atoms in Si-rich Si x C 1-x , the optical bandgap of the Si-rich Si x C 1-x has been reduced from 2.05 eV to 1.49 eV [44]. The silicon and carbon ratio of the Si-rich Si x C 1-x films can be characterized by utilizing X-ray photoelectron spectroscopy and Mg Kα radiation at 1253.6 eV, and the transmittance and reflectance spectra can be used to analyze the absorption coefficient of Si-rich Si x C 1-x [44]. It has been shown before that the optical absorption capability of non-stoichiometric silicon-rich silicon carbide film is better with reduced optical bandgap than crystalline silicon (c-Si) [44,47], hence non-stoichiometric Si-rich Si x C 1-x can be used as an alternative to silicon in photovoltaic applications.
The silicon-rich silicon carbide films have been synthesized by using low temperature and low power plasma enhanced CVD (PECVD) method in a system which is rich by the inorganic compound silane [44,45]. Moreover, silicon-rich silicon carbide has existed as a nonstoichiometric matrix which corresponds to Si fabrication technology to keep the self-assembled Si-quantum dots (QDs) [48].
Si-rich SiC:Si 0.74 C 0.26 film has an optical bandgap around 1.45 eV which provides a large absorption coefficient of 3.8 Â 10 5 cm À1 in the visible wavelength region. In the limit of all C atoms replaced by Si atoms, the band gap should approach to pure Si gap which is 1.1 eV [47]. However, we would like to avoid the indirect band gap problem which a shortcoming of pristine Si solar cell devices. Due to the tunability of the band gap, this material nowadays is being used in optoelectronic devices such as light-emitting diodes as well [48].

Computational methods
The calculations of this work were done within the framework of density functional theory (DFT) as in Vienna ab initio simulation package (VASP) [49,50]. We have utilized the Projector augmented-wave method (PAW) [51,52] and generalized gradient approximation [53], namely the Perdew-Burke-Ernzerhof (PBE) as an exchange-correlation functional [54]. We relaxed all the structures before the total energies' calculation. Then, we calculated the formation enthalpies to examine the stability of these structures. The formation enthalpy was calculated using the following formula where ΔH f is the formation enthalpy, and EðSi x C y Þ is the total energy of the unit cell of a SiC system, EðSiÞ is the total energy of the stable bulk diamond phase of Si per atom, and EðCÞ is the total energy of stable bulk Graphite phase per C atom. x and y represent the total number of Si and C atoms in the unit cell doped or removed, respectively. We included Van der Waals forces in our calculation to determine the total energy of bulk Graphite [55,56].
We have utilized Materials project [57], and American mineralogist crystal structure database [58] to collect initial structure parameters of 2H, 4H, 6H, diamond, FeSi and wurtzite structures of silicon carbides. VESTA (visualization for electronic and structural analysis) was used throughout the work to visualize all the crystal structures [59]. Tetrahedron method was used to calculate the density of states (DOS) with different denser K-point meshes taking the size of the supercell into account. The kinetic energy cut-off for the plane-wave basis set was set to 400 eV, and all the ions were relaxed until Hellman-Feynman force was equal to or less than 0.01 eV/Å. Note, Gaussian smearing was used when the tetrahedron method was not applicable with some large hexagonal structures.
Due to the prospective applications of Si rich Si x C 1-x materials in photovoltaics, it is important to study its optical absorption under visible light spectrum. Hence the frequency dependent complex dielectric function ϵ(ω) ¼ ϵ₁(ω) þ iϵ₂(ω) is calculated in the independent particle picture using VASP. Here ϵ₁ and ϵ₂ are the real and imaginary parts respectively. ω is the frequency of the incident photon. The expression for ϵ₂(ω) is as follow [60]: In this expression, e and m are the charge and mass of an electron respectively. M is the dipole matrix. i and j are initial and final states respectively. f i is the fermi distribution for the i-th state with wave function vector k. The real part of the dielectric function can be found by the Kramers-Kronig transformation as Hence, the optical absorption coefficient can be found from the following formula:

Pristine silicon carbide
At first, we have examined different structures of pristine silicon carbide such as 2H, 4H, 6H, diamond, wurtzite, and FeSi structures of silicon carbides. We utilized the wurtzite structure of zinc sulfide [58] by replacing all zinc atoms with silicon and sulfide atoms with carbon. Naquite structure of FeSi [61] is used where iron atoms are replaced by silicon, and the silicon atoms which are originally in FeSi structure are replaced by carbon atoms. In such unrelaxed naquite-structure of SiC, a Si atom is bonded with seven equivalent C atoms whereas a fully relaxed pure structure transforms into a diamond structure. For the defect studies in naquite structure, we doped Si atoms at C sites in the original unrelaxed form and then relaxed systems without any constraint to check if similar naquite-diamond phase transformation takes place. Hence both naquite and diamond structures are considered for defect studies. Note that a Si atom possesses a tetrahedral coordination with C atoms in a diamond structure. Table 1 shows the lattice parameters of their unit cells after full relaxation. Our optimized lattice parameters are in excellent agreement with the available experimental results [62]. Since GGA overestimates lattice parameters, our calculated results are slightly larger than the experimental results, but in all the cases these are less than 1%.
The calculated formation enthalpies of SiC structures are presented in Table 2. It shows that the diamond and FeSi structures have the same formation enthalpies per atom and are the lowest which indicates that they are the most stable structures, with 6H being slightly less stable by 0.001eV per atom. Given the small energy difference between 6H and cubic SiC structures, at least computationally at the GGA, it is apparent that wurtzite structure has the third lowest formation enthalpy and the 4H being the lowest stable phase for pristine SiC.

Silicon-rich silicon carbide structures
The main aim of this paper is to study different structures of siliconrich silicon carbide materials and their stability along with electronic properties. The chemical formula of silicon-rich silicon carbide can be defined as nSi C :SiC which states n number of carbon atoms are replaced by n silicon atoms in an otherwise pristine silicon carbide phase. Our Table 1 GGA optimized lattice parameters of pure SiC in 2H, 4H, 6H, diamond, FeSi, and wurtzite structures in their unit cells.

Structure
Lattice parameters work mainly focuses on taking off the carbon atoms and substitute them with silicon to get structures having more silicon than the carbon atoms.
We have examined all the pristine SiC phases which we have mentioned earlier, namely 2H, 4H, 6H, FeSi, wurtzite and diamond structures, and found that the diamond and 6H structures are among the favorable structures of SiC in this respect. Later we will discuss 8H-and 10H-SiC structures as well. In this section, we doped silicon for carbon sites for the diamond and 6H structures only to check the structural properties after full relaxation. We considered (2 Â 2 Â 2) supercells where the pristine diamond structure has 32 silicon and 32 carbon atoms, and the pristine 6H structure has 48 silicon and 48 carbon atoms. First, we made siliconrich silicon carbides by replacing one carbon with one silicon atom, Si C :SiC, where n ¼ 1. Fig. 1 shows the relaxed structures of diamond and 6H structures of silicon-rich silicon carbides. Then, in the second step we replaced two carbon atoms by two silicon atoms in the structures, 2Si C :SiC. We considered different configurations for replacing carbon atoms. For this, we considered situations such as the substituted Si atoms are near to and far from each other (Fig. 2) to examine which structure would be the favorable one.
We have found that the relaxed volume is smaller when the extra silicon atoms are near to each other, and larger when they are far from each other in most of the cases. We continued the same procedure up to eight carbon atoms and replaced them by silicon atoms in the SiC structures in different configurations. It is found that the volume increases progressively as the number of Si atoms are increased in the SiC structures. Note that the volume of the relaxed cells increases faster in the cubic (diamond) structures, which is 16.91% than that in the hexagonal (6H) structures, 9.32% while one to eight Si substitution takes place for C sites. It has also been found that the contribution from covalent bonding in hexagonal structures is higher than that in cubic structures, which will be presented in detail later, and the smaller compact structure of hexagonal structure is a result of that.
To see whether the substitution of extra Si atoms preserves the structural integrity, Table 3 shows the lattice parameters of 2 Â 2 Â 2 supercells of diamond and 6H structures, respectively. It can be seen that in Si C :SiC of diamond structure, the lattice parameters a, b, and c are equal, and the angels are also equal which mean that the symmetry did not break through the modeling processes. We have found the symmetry is preserved for 6H structures as well. However, it can also be argued that in diamond structures of nSi C :SiC (n ¼ 2 to 8), the lattice parameters a, b, and c are almost equal and angles' deviation from 90 is very small.
Similarly, for 6H nSi C :SiC (n ¼ 2 to 8), the lattice parameter a and b are almost equal and angles' deviation from 90 and 120 is not much. These imply that the overall structural integrity considered for Si-rich SiC are preserved here.

Formation enthalpies of silicon-rich silicon carbides
After all the structures are fully relaxed, we have calculated the formation enthalpies per atom for nSi C :SiC (n ¼ 1 to 8) in diamond, FeSi, wurtzite and 6H structures, which are presented in Table 4. It is found that the 6H structure of 1Si C :SiC has the lowest formation enthalpy which is 0.012 eV per atom lower than that of the diamond structure. When two carbon atoms were replaced by silicon atoms (2Si C :SiC), we examined two different situations for diamond and 6H structures only. In the first case doped Si atoms were placed near to each other and in the second   case they were far from each other. The calculated formation enthalpies are less when the extra two silicon are near to each other than when they are far. It is about 0.003 eV less for the 6H structure, and 0.001 eV less for the diamond structure as in Table 5. The reason that formation enthalpies per atom decrease less for the diamond structure is that the Si-Si interaction is less in the diamond structure than that in the 6H structure.
Also, for 3Si C :SiC, we made different arrangements of the extra three silicon atoms in SiC diamond and 6H structures: when they are close to each other, far from each other and two of them are close to each other and the third one is far from the other two. The calculated the formation enthalpies for all three scenarios can be found in Table 6. The results show that when the extra silicon atoms are close to each other, it gives lower energies than when they are relatively far from each other. The formation enthalpy of the 6H structure is -0.068 eV per atom, which is lower than that of the diamond structure. Similarly, we considered different configurations of distributing the extra silicon atoms in 4Si C :SiC  Table 3 The lattice parameters of GGA-relaxed diamond and 6H structures for 2 Â 2 Â 2 supercells for nSi C :SiC (n ¼ 1 to 8).

Structure
Diamond 6H Lattice parameters Lattice parameters   to see the validity of our expectation about choosing the extra silicon atoms to be close to each other. We examined two different configurations of 6H structure and found the most favorable one formed when all the four doped Si atoms were close to each other ( Table 7). The formation enthalpy per atom for the lowest energy structure is -0.057 eV in this case. However, for diamond structure, despite the fact that we have considered numerous configurations (Here, only two cases are presented in Table 7), the formation enthalpies remained positive, which means that 4Si C :SiC diamond structures are no longer thermodynamically favorable.
Repeating the above procedure of replacing carbon by silicon atoms we continued up to eight Si atoms. In all these cases, we have found that 6H structure is the most favorable one. The diamond structure as mentioned in the preceding paragraph from 4 substitutional Si atoms formation enthalpies become all positive. Similar trend was shown for FeSi-type structure. However, for Wurtzite structure, after 5 th substitutional Si atom, formation enthalpies became all positive. Only the 6H SiC structure has shown the thermodynamic stability up to 8 substitutional Si atoms in SiC (Table 4).
It can be seen from Fig. 3 that up to 8Si C :SiC, the 6H structure has the lowest formation enthalpies per atom compared to all other structures, and the only structure with all negative formation enthalpies. The trend shows that out of all Si-rich structures 6H structures are thermodynamically stable. However, it should be noted here that in the case where all carbon atoms are replaced by silicon atoms which is the Si bulk limit, the diamond structure is the ground state structure. The figure also shows that the formation enthalpy has similar trend for diamond, wurtzite, and FeSi structures for up to 3 extra Si atoms, while 6H structure takes a different path. Table 4 summarizes the formation enthalpies of this Si-rich nSi C :SiC structures for 6H, wurtzite, FeSi and diamond structures. We did not consider 2H and 4H in our calculations because it gives positive formation enthalpies even for a single C replaced by a Si atom. Also, we have noticed that cubic structures, namely diamond, and FeSi structures started to give positive formation enthalpy when four carbon atoms are replaced by silicon atoms. It means those structures were no longer thermodynamically stable. We also tried several arrangements of extra silicon atoms to examine it further. However, it was not the case with 6H structures which gave negative formation energies up to eight carbon atoms replaced by silicon atoms. We noticed that the carbon atoms which were replaced by silicon atoms should be near to each other either at the same layer or same plane to get the lowest energy structure for a given number of extra Si atoms.
At this point, we carried out calculations for higher order hexagonal structures such as 8H and 10H structures for both pristine and silicon-rich silicon carbides. In 8H and 10H structures, α ¼ β ¼ 90 and γ ¼ 120 .
After full relaxation, we found that for 8H, the lattice parameters for the pristine structure are a ¼ b ¼  [63] and found that their stability are the same for pure phases and it is -0.1645 eV per atom, which is also comparable to 6H. However, for silicon-rich silicon carbide (1Si C :SiC), we find that 10H structure tends to give the lowest formation enthalpy which equals to -0.042 eV per atom. The energetics are reported in Table 8. Out of all the possible arrangements that we have considered in this work, our investigation suggests that substituted silicon atoms like to stay in the same layer and distribute symmetrically along a and b. However, these want to stay as far as possible along c.
Stability of these structures can also be assessed from another point of view. Electrostatic energy is the potential energy which comes from Coulomb forces due to the ionic nature of the crystal structures and can be found by carrying out Ewald summation over the positive and negative ions as in [64]. We calculated the electrostatic energies for both pure and silicon-rich silicon carbide structures of this work. It is well-known that SiC has both covalent and ionic bonding in the structure due to the electronegativity difference between Si and C atoms. In pure relaxed form, since diamond and FeSi-SiC structures have exactly same geometry, their electrostatic energies are same. Hence, electrostatic energy for pure diamond structure is reported only in Fig. 4 (a). The figure shows that pure diamond SiC has the lowest electrostatic energy. Now, due to extra Si substitution, there should be an overall reduction of electrostatic energy. Fig. 4 (b) shows the calculated electrostatic energies for Table 6 Formation enthalpies of 6H and diamond structures of 3Si C :SiC. For both cases three Si atoms are doped at C sites in 2 Â 2 Â 2 supercells of SiC.    diamond, FeSi, Wurtzite and 6H SiC structures as C atoms are replaced by Si atoms. The reduction of electrostatic energy is the highest in the 6H structure which is about 65.63% for one to eight Si substitution whereas it is only 7.62% in the diamond structures. In fact, both the hexagonal structure, wurtzite and 6H showed similar electrostatic energy reduction than the cubic structures, diamond and FeSi. This analysis suggests that the reason for the stability of 6H nSi C :SiC structure is the stronger covalent bonding enhances in its environment rather than the ionic bonding.

The density of states (DOS)
For brevity, we have presented four density of states plots in this section to show how Si doping at C sites introduces new states near the Fermi level. Fig. 5 shows the total and partial density of states (DOS) plot for pristine 6H-SiC. The top of the valence band, valence band maximum (VBM), has the main contribution from C-p orbitals along some Si-p. As expected, the top part of the valence band, namely -4 eV-0 eV shows strong Si-C hybridization. At the VBM the s/p hybridization is suppressed     due to the part ionic nature of the Si-C bonding. Fig. 6 shows the total and partial DOS plot for 6H-1Si C :SiC. The substitutional Si-p band is now seen on top of the valence band. However, this is not a very localized defect band; rather the width is about 1.43eV. Fig. 7 shows the total and partial DOS plot for diamond 1Si C :SiC structure. Unlike 6H-1Si C :SiC, no dominant Si-p level is seen on top of VBM, rather very localized C-p bands at VBM is found. The nature of CBM in all of these plots did not change much. Now it is important to discuss the band gap trend. Table 9 shows the band gap of different nSi C :SiC structures. For pristine SiC band gap of 6H and diamond structures are 2.034 eV and 1.391 eV, respectively. Reported experimental band gap for these two are 3.02 eV and 2.20 eV, respectively [65]. As is expected the calculated DFT band gaps are under-estimated compared to the experimental band gaps. With one Si substitution band gaps reduced significantly, 0.418 eV and 0.773 eV respectively. For 6H nSi C :SiC, from n ¼ 3 the band gap reduced to zero due to the extra Si s/p hybridized states in the band gap (Fig. 8). At this GGA level of calculations, these bands create metal-like materials. However, for diamond structure, even though it becomes thermodynamically unstable from n ¼ 4 (Table 4), it does have zero band gap at n ¼ 3, probably because of its higher ionic bonding nature compared to the hexagonal structures. Even within hexagonal structures, wurtzite has zero band gap with n ¼ 3, as the ionic contribution is lower than the 6H structure than that of n ¼ 4.

Optical absorption coefficients
In this section, we present the optical absorption properties of pure and silicon rich 6H-Si x C 1-x . Our GGA-DFT calculated optical absorption coefficients are presented in Fig. 9 and are calculated using the formula as given in ref. [60]. Note that the energy gaps here were calculated from DFT, hence are under-estimated as mentioned earlier. The figure shows the first significant peak for pure 6H-SiC is at 3.89 eV. When Si atoms are doped for C sites (6H-Si 1.04 C and Si 1.40 C), the first major peak shifts at lower photon energy range for both cases. It shows that for higher Si concentration, the first major peak moves at 1.14 eV for Si 1.40 C. For a pure system, there are only two major peaks up to 5 eV whereas for a Si-rich system there are multiple peaks for the same photon energy range and there is an overall red shift for Si-rich systems. This indicates that a Si-rich system extends the absorption spectra toward the lower photon  energy region. Such red shift due to higher Si content was also found for silicon-rich Si x C 1-x thin film photovoltaic solar cell [44]. Also, higher Si content enhances the overall photo absorption up to around 5 eV, and most particularly, the absorption increases significantly within the visible spectra as shown in Fig. 9.

Conclusions
We have examined the stability and the electronic properties of various polymorphs of silicon-rich silicon carbide phases with tunable bandgaps which could be suitable for photovoltaic applications. Different structures considered for this work were 2H, 4H, 6H, 8H, 10H, diamond, FeSi, and wurtzite structures of SiC. Within these structures, different numbers of carbon atoms were replaced by silicon systematically and studied within the density functional theory framework. We found that for pristine SiC, 6H and diamond structures have comparable stabilities. However Si-rich SiC favors hexagonal structures. The substituted Si atoms prefer to be near to each other rather than spread out randomly. Also, we have found that it is better to choose the extra silicon atoms to be on the same layers symmetrically distributed along the hexagonal lattice a and b directions to get energetically favorable structures. From electronic structure calculations, it was found that near the Fermi level the most contributions come from Si and C p orbitals. Also, we noticed that s and p orbitals in some structures have strong hybridization in the conduction bands. Within the scope of our study, 6H (or hexagonal) structures show more covalent nature than the diamond (or cubic structures). In addition, our optical absorption calculations show that Sirich 6H-SiC shifts its absorption spectra toward lower photon energy region than its pure phase, and in the visible range the absorption coefficients are very high compared to the pristine SiC.

Declarations
Author contribution statement Noura D. Alkhaldi, Sajib K. Barman: Performed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper. Muhammad N. Huda: Conceived and designed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.