  The natural and true band profiles at heterojunctions formed by hexagonal
Si$_x$Ge$_{1-x}$ alloys are investigated by a variety of methods: density
functional theory for atomic geometries, approximate quasiparticle treatments
for electronic structures, different band edge alignment procedures, and
construction of various hexagonal unit cells to model alloys and
heterojunctions. We demonstrate that the natural band offsets are rather
unaffected by the choice to align the vacuum level or the branch point energy,
as well as by the use of a hybrid or the Tran-Blaha functional. At interfaces
between Ge-rich alloys we observe a type-I heterocharacter with direct band
gaps, while Si-rich junctions are type-I but with an indirect band gap. The
true band lineups at pseudomorphically grown heterostructures are strongly
influenced by the generated biaxial strain of opposite sign in the two adjacent
alloys. Our calculations show that the type-I character of the interface is
reduced by strain. To prepare alloy heterojunctions suitable for active
optoelectronic applications, we discuss how to decrease the compressive biaxial
strain at Ge-rich alloys.
