# Carbon-Isovalent Dopant Pairs in Silicon: A Density Functional Theory Study
## 1 Introduction
Central to this co-doping of Si with Ge and C, (\(\rm r_{Ge}>\rm r_{Si}\), although \(\rm r_{C}<\rm r_{Si}\)) is a complementary strategy to enhance device operation through controlling defect properties. It has been determined that the simultaneous Ge and C co-doping improves radiation hardness of the Cz-Si substrate [12; 41]. Moreover, Ge and C co-doping has been found to suppress the large-size voids in Si which generate leakage currents that deteriorate the efficiency of electronic devices [10; 40]. Sn is also an important isovalent dopant impurity in Si. Sn, being larger than the Si atom, attracts vacancies thus reducing the formation of VO and VO\({}_{2}\) defects in Si [42; 43; 44]. This tendency to attract vacancies promotes Sn as a promising dopant for radiation hardness in Si [45]. Additionally, the large Sn atoms are used to compensate strain layers in B-doped Si, which is the basic substrate material for power devices [46]. Since Sn attracts vacancies and C attracts self-interstitials, it has been proposed that Sn and C co-doping in Si could further enhance the radiation hardening potential of the material [42; 47], which is valuable for application with solar cells and detectors operating in radiation environments [48]. Furthermore, since the introduction of C, which is a small atom compared to Si, contracts the lattice when introduced at substitutional sites, as opposed to the case of Sn which is larger than Si and expands the lattice, the compensation for each other's strains could effectively enhance their solubility [49]. This is important for the fabrication of carbon-implanted light-emitted diodes [46; 49]. This also raises an interest for any possible interaction between Sn and C and the formation of Sn-C pairs. Thus, Sn and C co-doping was proposed as a tool to assess the degradation of Si-based devices [42]. Lead has an even larger covalent radius than Sn and the consequence is that there is an introduction of significant strains in the lattice that can affect the equilibrium concentrations and defect processes involving intrinsic defects [50; 51]. Regarding radiation defects, Pb doping can affect the production evolution and the reactions of various \(\rm V_{n}O\) and \(\rm VO_{m}\) defects in Si, indicating a practical interest for defect engineering [52; 53; 54; 55]. Notably, Pb in Si does not introduce any levels in the forbidden gap in comparison with Sn [52]. Additionally, when Pb and C are simultaneously in the lattice, the compressive strains introduced by Pb can be compensated by the tensile strains introduced by C. Technically, co-doping with C and Pb has been used to stabilize Pb at substitutional sites and suppress its precipitation [52; 56]. It is therefore of interest to investigate the interaction between Pb and C for the formation of carbon-lead pairs [14]. Among the Ge-C, Sn-C, and Pb-C pairs, it is worth noting that only Sn-C has been studied more thoroughly, both experimentally and theoretically [47; 57; 58; 59]. A number of IR lines at 873, 1025.3, and 6875 cm\({}^{-1}\) were attributed [47] to the (C\({}_{1}\)Sn\({}_{8}\))\({}^{2\text{N}}\) with C\({}_{\text{i}}\) at the second neighboring position, with respect to the Sn. Additional lines at 888.9, 985.3 and 6915 cm\({}^{-1}\) were also attributed to a less stable C\({}_{1}\)Sn\({}_{8}\))\({}^{2\text{N}}\) configuration [58]. Regarding the Ge-C and Pb-C pairs, their presence were proposed theoretically and was indirectly concluded experimentally from variations in the concentrations of the O and C-related defects [14; 60; 61]. Isovalent doping in Si is a valuable technique used to optimize devices. In particular, Ge, Sn, and Pb, when present in the Si lattice, affect the formation, the concentration, thermal stability, and reactions of the oxygen-related and C-related defects. This is important for the radiation hardness of the material, especially when Si-based devices such as detectors are operated in radiation environments.