Structural and thermodynamic stability of hexagonal-diamond $\text{Si}_{1 - x - y}\,\text{Ge}_{x}\,\text{B}_{y}$ alloys

Pushing dopant concentrations beyond the solubility limit in semiconductors -- a process known as hyperdoping -- has been demonstrated as an effective strategy for inducing superconductivity in cubic-diamond Si and SiGe materials. Additionally, previous studies have reported that several polytypes of Si may exhibit a type-I superconducting state under high pressure. In this work, we employ ground-state Density Functional Theory simulations to investigate the effects of both low and high B doping concentrations on the structural and thermodynamic properties of hexagonal-diamond SiGe alloys, with a systematic comparison to their cubic-diamond counterparts. Our results highlight three key findings: (i) structural analysis confirms that the lattice parameters of SiGeB alloys adhere to a ternary Vegard's law, consistent with observations in cubic-diamond SiGe alloys. However, at high doping concentrations, B incorporation can locally disrupt the hexagonal symmetry, particularly in the presence of B clustering; (ii) dopant formation energy calculations reveal that B is thermodynamically more stable in the hexagonal phase than in the cubic phase across all Ge concentrations, regardless of the doping level; (iii) mixing enthalpy calculations demonstrate that hyperdoped hexagonal-diamond SiGe alloys are thermodynamically stable across the full range of Ge compositions and that their tendency for hyperdoping is more favorable than that of cubic-diamond SiGe alloys. Taken together, these findings indicate that hyperdoping is experimentally viable in hexagonal-diamond SiGe alloys and, in light of previous evidence, position these materials as a promising platform for the exploration of superconductivity in group IV semiconductors.