3.2.3 Phononic Structure

We compare the thermal conductance of circular GALs with $L_S=10$ and different radii, including Circ(10,24), Circ(10,108) and Circ(10,258). The phonon density of states (DOS) and phonon transmission of these GALs are shown in Fig. 3.15. As indicated in Table 3.2, by increasing the size of the antidot, the phonon DOS, the phonon transmission, and the thermal conductance are significantly reduced. In Fig. 3.16 two phonon modes of Circ(10,108) at the $\Gamma$ point are shown. Fig. 3.16-a presents a localized phonon mode as a result of introducing antidots, whereas, Fig. 3.16-b shows a propagative mode. By introducing antidots into the graphene sheet, some phonon modes become localized, similar to electrons, and they can not contribute to the thermal conductance.

Figure 3.15: Comparison between (a) phonon density of states and (b) transmission of pristine graphene and circular GALs with different antidot areas.

To investigate the effect of the antidot circumference, we compare GALs with nearly the same area and different shapes, including Circ(10,108), Rect(10,120), Hex(10,120), IsoTri(10,126), and RightTri(10,126). Although the DOS of these GALs have the same order as that of a pristine graphene sheet, the transmissions can be very different. Fig. 3.17-a shows that the phonon transmissions of Circ$(10,108)$ and Rect$(10,120)$ are quite different from that of pristine graphene. However, Circ$(10,108)$ , Rect$(10,120)$ , and Hex$(10,120)$ have nearly the same transmissions, whereas IsoTri$(10,126)$ and Right$(10,126)$ have similar transmissions which are different from the first group, see Fig. 3.17-b.

Figure 3.16: Phonon modes at $\Gamma$ point: (a) represents a localized mode at $E=30 \mathrm{meV}$ and (b) represents a propagating mode at $E=16 \mathrm{meV}$ . The amplitude of the vibrations has been normalized.

The transmissions of Circ$(10,108)$ , Rect$(10,120)$ , and Hex$(10,120)$ are similar because they have similar circumference and thus the same number of boundary carbon atoms. Furthermore, the nearest-neighbor dots in these GALs have nearly the same distance. On the other hand, IsoTri$(10,126)$ and RightTri$(10,126)$ have the same circumference which is different from those of the first group.

Figure 3.17: (a) The comparison between the phonon transmission of a pristine graphene, Circ$(10,108)$ , and Rect$(10,120)$ . (b) The comparison between the phonon transmission of Circ$(10,108)$ , IsoTri$(10,126)$ , and RightTri$(10,126)$ . IsoTri$(10,126)$ and RightTri$(10,126)$ have similar transmission, but generally smaller than that of Circ$(10,108)$ . This can be explained by a larger circumference and a lower distance between the nearest-neighbor antidots of these two GALs.

The thermal conductances of pristine graphene and different GALs are summarized in Table 3.2. Triangular GALs have the smallest thermal conductance, although they have the minimum area of all antidot shapes. This behavior can be explained by considering the fact that triangular antidots have the highest circumference of all antidots with the same area. This indicates that circumference of the antidot has a stronger effect on the thermal conductance rather than its area.

Table 3.2: The comparison of the thermal conductance of pristine graphene and different GALs.

Structure	Thermal conductance [W/K-m]
Pristine Graphene	1.3813
Circ$(10,24)$	0.6948
Circ$(10,108)$	0.3764
Circ$(10,258)$	0.2220
Rect$(10,120)$	0.3378
Hex$(10,120)$	0.3764
IsoTri$(10,126)$	0.2606
RightTri$(10,126)$	0.2509