  The zero band gap (Eg) graphene becomes narrow Eg semiconductor when graphene
is patterned with periodic array of hexagonal shaped antidots, the resultant is
the hexagonal Graphene Antidot Lattice (hGAL). Based on the number of atomic
chains between antidots, hGALs can be even and odd. The even hGALs (ehGAL) are
narrow Eg semiconductors and odd hGALs (ohGAL) are semi-metals. The Eg opening
up by hGALs is not sufficient to operate a realistic switching transistor. Also
hGAL transistors realized on Si/SiO2 substrate are suffering with low carrier
mobility and ON-OFF current ratio. In order to achieve a sizable Eg with good
mobility, AB Bernal stacked hGALs on hexagonal Boron Nitride (hBN), ABA Bernal
stacked hBN / hGAL / hBN sandwiched structures and AB misaligned hGAL /hBN
structures are reported here for the first time. Using the first principles
method the electronic structure calculations are performed. A sizable Eg of
about 1.04 eV (940+100 meV) is opened when smallest neck width medium radius
ehGAL supported on hBN and about 1.1 eV (940 + 200 meV) is opened when the same
is sandwiched between hBN layers. A band gap on the order of 71 meV is opened
for Bernal stacked ohGAL / hBN and nearly 142 meV opened for hBN / ohGAL /hBN
structures for smallest radius and width of nine atomic chains between
antidots. Unlike a misaligned graphene on hBN, the misaligned ohGAL/hBN
structure shows increased Eg. This study could open up new ways of band gap
engineering for graphene based nanostructures. Keywords: Graphene, graphene
antidots, hexagonal boron nitride, band structure, band gap engineering
