  It has been known since decades that the alignment of a beam of high-energy
electrons with particular crystal directions involves a significant increase of
bremsstrahlung radiation emission. This enhancement lies at the conceptual
foundation of innovative positron source schemes for future lepton colliders.
In particular, the so-called hybrid scheme makes use of a heavy-metal radiator
in crystalline form, which is then followed by an amorphous metallic converter
for positron generation from electrons by means of a two-step electromagnetic
process. This work presents the most recent simulation results obtained on the
development of a hybrid positron source for the FCC-$ee$ from the standpoint of
the features of both the crystalline radiator and the amorphous converter.
