  We have performed electronic structure calculations to explore the band-gap
dependence on polytypes for $sp^3$-bonded semiconducting materials, i.e., SiC,
AlN, BN, GaN, Si, and diamond. In this comprehensive study, we have found that
band-gap variation depending on polytypes is common in $sp^3$-bonded
semiconductors; SiC, AlN, and BN exhibit smallest band gaps in $3C$ structure,
whereas diamond does in $2H$ structure. We have also clarified that the
microscopic mechanism of the band-gap variations is attributed to peculiar
electron states $floating$ in internal channel space at the conduction-band
minimum (CBM), and that internal channel length and the electro-static
potential in channel affect the energy level of CBM.
