Interaction of dopants with the I$_3$-type basal stacking fault in hexagonal-diamond Si

Recently synthesized hexagonal-diamond silicon, germanium, and silicon-germanium nanowires exhibit remarkable optical and electronic properties when compared to cubic-diamond polytypes. Because of the metastability of the hexagonal-diamond phase, I$_3$-type basal stacking faults are frequently observed in these materials. Understanding and modulating the interaction between these extended defects and dopants are essential for advancing the design and performance of these novel semiconductors. In the present study, we employ density functional theory calculations to investigate the interaction of extrinsic dopants (group III, IV, and V elements) with the I$_3$-type basal stacking fault in hexagonal-diamond silicon. Contrary to the behavior observed in cubic-diamond silicon with intrinsic stacking faults, we demonstrate that neutral and negatively charged $p$-type impurities exhibit a marked tendency to occupy lattice sites far from the I$_3$-type basal stacking fault. The interaction of acceptors with the planar defect reduces their energetic stability. However, this effect is much less pronounced for neutral or positively charged $n$-type dopants and isovalent impurities. The thermodynamic energy barrier to segregation for these dopants is small and may even become negative, indicating a tendency to segregate into the fault. Through a detailed analysis of structural modifications, ionization effects, and impurity-level charge density distribution, we show that the origin of this behavior can be attributed to variations in the impurity's steric effects and its wave function character. Finally, all these results are validated by considering the extreme case of an abrupt hexagonal/cubic silicon interface, where acceptor segregation from the cubic to the hexagonal region is demonstrated, confirming the behavior observed for $p$-type dopants near the I$_3$-type defect.