Junctions between a boron nitride nanotube and a boron nitride sheet

For future nanoelectromechanical signalling devices, it is vital to understand how to connect various nanostructures. Since boron nitride nanostructures are believed to be good electronic materials, in this paper we elucidate the classification of defect geometries for combining boron nitride structures. Specifically, we determine possible joining structures between a boron nitride nanotube and a flat sheet of hexagonal boron nitride. Firstly, we determine the appropriate defect configurations on which the tube can be connected, given that the energetically favourable rings for boron nitride structures are rings with an even number of sides. A new formula E = 6+2J relating the number of edges E and the number of joining positions J is established for each defect, and the number of possible distinct defects is related to the so-called necklace and bracelet problems of combinatorial theory. Two least squares approaches, which involve variation in bond length and variation in bond angle, are employed to determine the perpendicular connection of both zigzag and armchair boron nitride nanotubes with a boron nitride sheet. Here, three boron nitride tubes, which are (3, 3), (6, 0) and (9, 0) tubes, are joined with the sheet, and Euler's theorem is used to verify geometrically that the connected structures are sound, and their relationship with the bonded potential energy function approach is discussed. For zigzag tubes (n,0), it is proved that such connections investigated here are possible only for n divisible by 3.