  Hexagonal SiGe is a promising material for combining electronic and photonic
technologies. In this work, the energetic, structural, elastic and electronic
properties of the hexagonal polytypes (2$H$, 4$H$ and 6$H$) of silicon and
germanium are thoroughly analyzed under equilibrium conditions. For this
purpose, we apply state-of-the-art density functional theory. The phase
diagram, obtained in the framework of a generalized Ising model, shows that the
diamond structure is the most stable under ambient conditions, but hexagonal
modifications are close to the phase boundary, especially for Si. Our
band-structure calculations using the MBJLDA and HSE06 exchange correlation
functionals predict significant changes in electronic states with hexagonality.
While Si crystals are always semiconductors with indirect band gaps, the
hexagonal Ge polytypes have direct band gaps. The branch point energies for Ge
crystals are below the valence band maxima, and therefore the formation of hole
gases on Ge surfaces is favoured. Band alignment based on the branch point
energy leads to type-I heterocrystalline interfaces between Ge polytypes, where
electrons and holes can be trapped in the layer with the higher hexagonality.
In contrast, the energy shift of the indirect conduction band minima of Si
polytypes is rather weak, leading to delocalization of excited electrons at
interfaces, while only holes can localize in the layer with higher
hexagonality.
