Inter-layer bonds in bilayer graphene with single vacancy defects

If a single vacancy defect occurs in both layers of bilayer graphene, a bond can develop between the two layers under certain circumstances. In this paper we determine if and where a bond between the layers is formed. Using the length of the shortest odd periodic orbit, we can establish the exact position and orientation of the break in each layer. If the configuration is suitable and the planar distance between the breaks is not more than 3 units, then investigating the amplitude for periodic orbits of length two enables us to conclude whether the bond between the layers is created or not.