# Carbon-Isovalent Dopant Pairs in Silicon: A Density Functional Theory Study
## 1 Introduction  
Silicon (Si) is the key semiconductor for integrated circuits, transistors, sensors, detectors, solar cells, as well as for applications related to nuclear medicine, space industry, telecommunications, and quantum computing [1; 2; 3; 4; 5]. An important issue in the performance of the devices is the presence of defects which affect their electrical, optical, and mechanical properties [6; 7; 8; 9]. Obviously, it is significant to understand the behavior of the defects in order to control the performance of the devices, either by suppressing their detrimental effects or by enhancing their beneficial influence in the material. Thus, by proper defect engineering, we can enhance the efficiency of the material for certain applications. The main isovalent impurities in silicon, that is C, Ge, Sn, and Pb, play an important role as their presence in the lattice influence the material's properties. Ge, Sn, and Pb occupy substitutional sites in the Si lattice, and since they are electrically inactive, they do not affect the electrical behavior of the material. Additionally, due to the difference in the covalent radius (r\({}_{\text{C}}\) = 0.77 A, r\({}_{\text{Ge}}\) = 1.22 A, r\({}_{\text{Sn}}\) = 1.41 A, r\({}_{\text{Pb}}\) = 1.44 A) compared to that of Si, (r\({}_{\text{Si}}\) = 1.17 A), they give rise to local elastic strains which can affect the equilibrium between the concentrations of vacancies and self-interstitials, and the interactions between them as well as with other impurities present. Indeed, Ge, Sn, and Pb, which are bigger than Si and therefore introduce compressive strains in the lattice, tend to associate with vacancies; however, C, which is smaller than Si, introduces tensile strains in the lattice and tend to associate with self-interstitials. Notably, isovalent dopants in Si are widely used for defect engineering in Czochralski silicon (Cz-Si) [10; 11; 12; 13; 14]. Aside from oxygen, carbon is an important impurity in Si unintentionally added during material growth [1; 9; 15; 16]. In previous studies, the interactions of C with self-interstitials and various impurities have been reported to mainly be with O and other C atoms, leading to families of (C-O)-related (for instance, \(\rm C_{i}O_{i}\), \(\rm C_{i}O_{i}(Si_{n})_{n}\)[9; 16; 17; 18; 19; 20] and C-related defects (for instance, \(\rm C_{i}C_{s}\), \(\rm C_{i}C_{s}(Si_{n})_{n}\)) [9; 16; 21; 22; 23; 24]. Notably, carbon affects oxygen precipitation [25; 26] in Si. Also, carbon co-doping in doped Si suppresses the formation of boron-oxygen complexes, which is very important for the performance of crystalline Si solar cells [27]. Ge, when introduced into the Si lattice, affects the properties of Si in many ways. It has been found that Ge atoms, by locking dislocations, can enhance the mechanical strength of Si, which improves the production yield of the material [28; 29]. It suppresses thermal donors, which stabilizes the electrical properties of Si [30]. It enhances oxygen precipitation and improves the internal gettering capability of metallic contamination which is very important for device fabrication [31; 32]. It suppresses void defects and thus offers high quality polish\(c\)d surface for epi-layer growth in Si [33]. It suppresses the formation of large-sized crystals originated particles (COPs), thereby improving the quality of the surface of the material for epitaxial growth [34]. Ge also suppresses B-O defects that lead to light-induced degradation (LID) of carrier lifetime, which is an important issue for the solar cells industry [35]. Importantly, Ge affects the production and evolution of O-related and C-related irradiation induced defects in Si [36; 37; 38]. Ge-doping is a valuable technique to control the formation and the thermal stability of O-related and C-related defects in Si. Interestingly, co-doping has proven to be a promising strategy for effectively tuning the electrical, mechanical, and optical properties of semiconductors, particularly Si. [10; 36; 39; 40].