  We study theoretically two-dimensional single-crystalline sheets of
semiconductors that form a honeycomb lattice with a period below 10 nm. These
systems could combine the usual semiconductor properties with Dirac bands.
Using atomistic tight-binding calculations, we show that both the atomic
lattice and the overall geometry influence the band structure, revealing
materials with unusual electronic properties. In rocksalt Pb chalcogenides, the
expected Dirac-type features are clouded by a complex band structure. However,
in the case of zinc-blende Cd-chalcogenide semiconductors, the honeycomb
nanogeometry leads to rich band structures, including, in the conduction band,
Dirac cones at two distinct energies and nontrivial flat bands and, in the
valence band, topological edge states. These edge states are present in several
electronic gaps opened in the valence band by the spin-orbit coupling and the
quantum confinement in the honeycomb geometry. The lowest Dirac conduction band
has S-orbital character and is equivalent to the pi-pi* band of graphene but
with renormalized couplings. The conduction bands higher in energy have no
counterpart in graphene; they combine a Dirac cone and flat bands because of
their P-orbital character. We show that the width of the Dirac bands varies
between tens and hundreds of meV. These systems emerge as remarkable platforms
for studying complex electronic phases starting from conventional
semiconductors. Recent advancements in colloidal chemistry indicate that these
materials can be synthesized from semiconductor nanocrystals.
