# Carbon-Isovalent Dopant Pairs in Silicon: A Density Functional Theory Study
## 3 Results  
### Carbon Substitutional-Isovalent Dopant Defect Pairs  
The Ge atoms in the Si lattice are completely soluble and can form Si\({}_{1}\)\(\sim\)Ge\({}_{\text{x}}\) alloys [71]. Conversely, the larger Sn and Pb are not completely soluble over the compositional range but only for a limited range (i.e., for small Sn and Pb compositions) [72]. The incorporation of isovalent dopant atoms is important as they can modify the optical, electronic, and structural properties [71; 72].  
Figure 1 represents the energetically favorable carbon substitutional-isovalent dopant substitutional C\({}_{\text{s}}\)D\({}_{\text{s}}\) (D = Ge, Sn, Pb) defects. These were derived using DFT calculations for all the possible configurations within the supercell. Here, the binding energy (i.e., energy of the defect pair minus the energy of the isolated constituent defects) is the measure of stability. The binding energy of a C\({}_{\text{s}}\)D\({}_{\text{s}}\) defect can be calculated using the following equation:  
\[\text{E}_{\text{b}}(\text{C}_{\text{s}}\text{D}_{\text{s}})=\text{E}_{\text{ C}_{\text{s}}\text{D}_{\text{s}}\text{ supercell}}+\text{E}_{\text{Si}\text{-supercell}}-\text{E}_{\text{Cs}\text{+}\text{Si}\text{-supercell}}- \text{E}_{\text{D}_{\text{s}}\text{-supercell}} \tag{1}\]  
where E\({}_{\text{Cs}\text{D}_{\text{s}}\text{-supercell}}\) is the total energy of a Si supercell containing a C\({}_{\text{s}}\)D\({}_{\text{s}}\) defect, E\({}_{\text{Si}\text{-supercell}}\) is the total energy of a Si supercell, E\({}_{\text{Cs}\text{+}\text{Si}\text{-supercell}}\) is the total energy of the Si supercell containing a C substitutional, and E\({}_{\text{Ds}\text{+}\text{Si}\text{-supercell}}\) is the total energy of the Si supercell containing a D substitutional.  
The electronegativity difference between the C substitutional and the Si lattice atoms is reflected in the shorter Si-C as compared to the Si-Si (or Si-D) bond distances (refer to Figure 1). It should be stressed at this point that the distances and bond angles are given in both Figures 1 and 2 in order to facilitate a comparison, as well as for other scientists to facilitate a comparison with their results in the future and to consider whether they have reproduced the present results or found alternative geometries.  
Figure 1: Schematic representation of the energetically favorable (a) C\({}_{\text{s}}\)Ge\({}_{\text{s}}\), (b) C\({}_{\text{s}}\)Sn\({}_{\text{s}}\) and (c) C\({}_{\text{s}}\)Pb\({}_{\text{s}}\) defects in Si. Important distances and angles in the vicinity of the defects are given. Dimmer atoms represent atoms in the background.  
The C\({}_{\text{s}}\)Ge\({}_{\text{s}}\) pair is only weakly bound (by \(-0.10\) eV) and forms when the C atom is at the second nearest neighbor configuration as compared to Sn (refer to Figure 1). Conversely, in both the C\({}_{\text{s}}\)Sn\({}_{\text{s}}\) and C\({}_{\text{s}}\)Pb\({}_{\text{s}}\) defects, the isovalent atom is at nearest neighbor sites with respect to the C atom (refer to Figure 1). The C\({}_{\text{s}}\)Sn\({}_{\text{s}}\) pair was calculated to be bound by \(-0.42\) eV in good agreement with previous theoretical studies (\(-0.19\) to \(-0.20\) eV, [47; 73]). The C\({}_{\text{s}}\)Pb\({}_{\text{s}}\) pair is bound more (\(-0.56\) eV) as the larger Pb atom takes more advantage of the space provided by the smaller (compared to Si) carbon substitutional atom.