  We study the effect of magnetism and perpendicular external electric field
strengths on the energy gap of length confined bilayer graphene nanoribbons (or
nanoflakes) as a function of ribbon width and length using a \textit{first
principles} density functional electronic structure method and a semi-local
exchange-correlation approximation. We assume AB (Bernal) bilayer stacking and
consider both armchair and zigzag edges, and for each edge type, we consider
the two edge alignments, namely, $\alpha$ and $\beta$ edge alignment. For the
armchair nanoflakes we identify three distinct classes of bilayer energy gaps,
determined by the number of carbon chains in the width direction ({\it N} =
3{\it p}, 3{\it p}+1 and 3{\it p}+2, {\it p} is an integer), and the gaps
decrease with increasing width except for class 3{\it p}+2 armchair
nanoribbons. Metallic-like behavior seen in armchair bilayer nanoribbons are
found to be absent in armchair nanoflakes. Class 3{\it p}+2 armchair nanoflakes
show significant length dependence. We find that the gaps decrease with the
applied electric fields due to large intrinsic gap of the nanoflake. The
existence of a critical gap with respect to the applied field, therefore, is
not predicted by our calculations. Magnetism between the layers plays a major
role in enhancing the gap values resulting from the geometrical confinement,
hinting at an interplay of magnetism and geometrical confinement in finite size
bilayer graphene.
