  Lonsdaleite germanium has a direct band gap, but it is not an efficient light
emitter due to the vanishing oscillator strength of electronic transitions at
the fundamental gap. Transitions involving the second lowest conduction band
are instead at least three orders of magnitude stronger. The inversion of the
two lowest conduction bands would therefore make hexagonal germanium ideal for
optoelectronic applications. In this work, we investigate the possibility to
achieve this band inversion by applying strain. To this end we perform ab
initio calculations of the electronic band structure and optical properties of
strained hexagonal germanium, using density functional theory with the modified
Becke-Johnson exchange-correlation functional and including spin-orbit
interaction. We consider hydrostatic pressure, uniaxial strain along the
hexagonal c axis, as well as biaxial strain in planes perpendicular to and
containing the hexagonal c axis to simulate the effect of a substrate. We find
that the conduction-band inversion, and therefore the transition from a
pseudo-direct to a direct band gap, is attainable for moderate tensile uniaxial
strain parallel to the lonsdaleite c axis.
