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Effects of magnetism and electric field on the energy gap of bilayer graphene nanoflakes

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, $α$ and $β$ 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.