3.2.1 Geometrical Structure

The electrical and optical properties of GALs have been theoretically studied in Refs. [84,26,64]. The results indicate that by introducing regular antidots in a graphene sheet, it is possible to achieve a direct band gap semiconductor from a semi-metallic pristine graphene sheet. Bai and co-workers reported the first field-effect-transistor based on GALs [65]. To investigate the effect of the dot geometry on the thermoelectric properties of GALs, the unit cell of a GAL is described by two parameters $L_S$ and $N$ , where $L_S$ is the side length of the hexagon in terms of the graphene lattice constant ($a=2.46\AA$ ) and $N$ is the number of carbon atoms removed from the pristine supercell. In Fig. 3.12 Circ, Rect, Hex, IsoTri, and RightTri represent a circular, rectangular, hexagonal, iso-triangular, and right-triangular antidot in the hexagonal unit cell, respectively. Fig. 3.12-b shows a circular antidot which is formed by removing $108$ carbon atoms from a cell with $L_S=10$ . It is therefore represented by Circ$(10,108)$ . The number of edge carbon atoms in a unit cell of different GALs is also given in Table 3.1. As shown below, the number of carbon atoms at the boundary plays an important role on the thermal properties of the structure.

Table 3.1: The number of edge carbon atoms in a unit cell of different GALs.

Structure	Number of boundary atoms
Circ$(10,108)$	30
Rect$(10,120)$	32
Hex$(10,120)$	30
IsoTri$(10,126)$	36
RightTri$(10,126)$	38