
The Seebeck coefficient and power factor are sensitive to the details of the density of states and the asymmetry between electrons and holes [85,86]. The electronic band structures of GALs are calculated using a third nearestneighbor tightbinding method (Sec 2.1.1). By introducing the antidots in the graphene sheet, the zero bandgap graphene can be converted into a narrow bandgap semiconductor [26,27] (see Fig. 3.13). This issue plays an important role in thermoelectric applications. In contrast to pristine graphene, GALs have a beneficial bandgap, so that one can suppress either the electron or the hole current to obtain unipolar conduction. For example, by adjusting the Fermi level near the conduction band the hole current will be negligible. The electronhole asymmetry with respect to the Fermi level depends on the bandgap, on the sharp features of transmission, on the width of the first conduction subband, and on the value of the transmission. At room temperature, the width of the thermal broadening function is about . Therefore, a bandgap around and a first conduction subband width larger than will be ideal for thermoelectric applications.
In RightTri(10,126), there are some localized midgap states, see Fig. 3.13f, as a result of sublatticesymmetry breaking [84,87]. They have a zero group velocity and can not contribute to the carrier transport. Although RightTri(10,126) has the sharpest features in the transmission and its transport bandgap is about , the width of the first conduction subband of RightTri(10,126) is only . As a result, it has a high Seebeck coefficient and a low electrical conductance, see Fig. 3.14. The first conduction subband of a Rect(10,120) has a nonzero group velocity. Therefore, the rectangular GAL is considered as a zero bandgap material and as a result, the Seebeck coefficient will be small which is detrimental to thermoelectric applications. In a Hex(10,120), the first conduction and valence subbands are quasiflat bands due to existence of some edge carbon atoms which have only one nearest neighbor [87]. As shown in Fig. 3.13d, these bands have a small group velocity and have a small contribution to electron transport. As a result, the maximum value of the Seebeck coefficient of Hex(10,120) is not very large and is located close to the bandedge of the second conduction subband. On the other hand, the electrical conductance peaks close to the second subbandedge. Therefore, Hex(10,120) has the third highest power factor among the GALs with different antidot shapes.

On the other hand, the bandgap and the first conduction subband width of Circ(10,108) and IsoTri(10,126) are nearly and , respectively. They also have the highest transmissions. Therefore, as shown in Fig. 3.14, they are the best thermoelectric GALs in terms of the power factor. Because of a sharp feature in the transmission, Circ(10,108) has the highest power factor of the GALs considered. In addition, as shown in Fig. 3.14d the electron contribution to the thermal conductance can be neglected in comparison with the lattice thermal conductance (see Table 3.2).