The unit cell of a zigzag-shaped graphene nanoribbon is described by these three lattice vectors of the honeycomb graphene lattice:
are elementary lattice vectors and
are integers [1, 2].
The crystallographic orientation of the elementary vectors
defines the ribbon edges to be of type zigzag (
) or armchair (
). The apex angle is the angle between vectors
. The edge type and the apex angle are used for classifying the wiggly structures. The snapshots present the four types of zigzag-shaped graphene nanoribbons.
Snapshot 1: zigzag edges with an apex angle of
Snapshot 2: zigzag edges with an apex angle of
Snapshot 3: armchair edges with an apex angle of
Snapshot 4: armchair edges with an apex angle of
The crystallographic orientation of the third elementary vector
determines the width ratio between the two fragments of the zigzag-shaped ribbon. The widths of fragments built on the vectors
are meant to be measured perpendicular to the vectors
. The width ratio is set to unity in this Demonstration, though a more generic case can also be considered .
specify the length and the width of the zigzag-shaped ribbon unit cell. The whole ribbon is obtained by repeated translation of the unit cell by
. In this Demonstration, only a few unit cells are shown for clarity.
A similar approach to the classification and atomic position generation can be applied to other two-dimensional crystalline structures, for example, silicene quantum dots .
The atomic coordinates generated in this Demonstration can be used for calculating physical properties of zigzag-shaped ribbons, for example, their electronic properties [1, 2]. The optical properties of these structures can be calculated as described for zigzag ribbons with straight edges .
Graphene nanoribbons very close in shape to those presented here have been produced by the self-assembling of subtly engineered organic molecular precursors .
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