3.3.3 The Role of Line-Edge-Roughness

In the third step of the design process the effect of edge roughness is introduced. The inset of Fig. 3.22-c shows the influence of edge roughness on the transmission of the ZGNR(20) of length $125~\mathrm{nm}$ . As also described in previous studies [88,19], in the first conduction plateau the effect is negligible. In contrast to ZGNR, ELD-ZGNRs as well as 2ELD-ZGNRs are affected by edge roughness. This is because the bandstructure of these GNRs has undergone a band folding, and therefore, the states in the first conduction plateau have lower wave vectors. As the long range defects can induce only a small momentum transfer, the momentum conservation rule indicates that, in contrast to the ZGNR, the transport of ELD-ZGNRs and 2ELD-ZGNRs will not remain ballistic in the presence of line edge roughness and long range substrate impurities. This is shown in Fig. 3.22-c, where the transmission of a roughened $125~\mathrm{nm}$ long ELD-ZGNR(10,10) channel (solid-blue line) is reduced by $\sim 25\%$ compared to the ballistic value (dashed-black line). Edge roughness degrades the conductivity of holes and electrons by a similar amount, and therefore, the level of asymmetry around the Fermi level is retained.

Figure 3.23: The influence of roughness and positive impurities on the ELD-ZGNR channel. (a) Electronic transmission of ELD-ZGNR(10,10). Rough edges are assumed and the length L is varied. The arrow indicates increasing values of length $L$ . (b) Electronic transmission of ELD-ZGNRs with different widths. The length is assumed to be constant at $250~\mathrm{nm}$ and the arrow indicates the direction of decreasing the ribbon's width. Black-solid and black-dashed lines in (a) and (b): The transmission of the pristine ELD-ZGNR.

Figures 3.23-a and 3.23-b illustrate the influence of roughness in ELD-ZGNR channels on their transmission, for channels of different lengths and widths. In this calculation positive impurities are also included. Figure 3.23-a shows the transmission of edge roughened ELD-ZGNR(10,10) versus energy for the channel lengths $L=250$ , $500$ , and $2000~\mathrm{nm}$ . As the channel length is increased, the transmission drops further compared to the transmission of the ideal channel (black-solid line). This is expected since the channel resistance increases with increasing length. Figure 3.23-b illustrates the effect of the ribbon's width on the transmission of ELD-ZGNRs with rough edges. In this case the length is kept constant at $L=250~\mathrm{nm}$ , and results for three different ribbons with parameters (10,10), (7,7), and (5,5) are shown. As the width of the ribbon is decreased, the effect of line edge roughness scattering on the transmission becomes stronger because the carriers reside on average closer to the edges.

It is worth mentioning that the effect of edge roughness on the transmission is much stronger in AGNR than in ZGNR. Although in the case of some AGNRs a bandgap is naturally present and the asymmetry need not be created with the introduction of line defects and impurities, the conductance is severely degraded by the roughness which renders this type of ribbon not well suited for transport applications [88]. (Note that edge roughness will be needed in order to reduce thermal conductivity as will be shown below.)

Figure 3.24: Transmission at $E=0.2~\mathrm {eV}$ for three different structures versus width. The length is assumed to be constant at $250~\mathrm{nm}$ .

As we mentioned above in Fig. 3.21, the channel which includes two ELDs can shift the majority of the current spectrum in the region between the two ELDs, and thus farther away from the edges. It is therefore expected that the 2ELD-ZGNR will be less affected by edge roughness scattering than the ELD-ZGNR. A comparison of the transmission of these devices with rough edges is shown in Fig. 3.24. The transmission of ELD-ZGNR($n_1$ ,$n_1$ ), and two cases of 2ELD-ZGNR, 2ELD-ZGNR($n_2$ ,4,$n_2$ ) and the 2ELD-ZGNR($n_3$ ,6,$n_3$ ) at $E=0.2~\mathrm {eV}$ versus their width $W$ are compared. The parameters $n_i$ are adjusted such that the three channels have nearly the same width $W$ . The first channel belongs to the category shown in Fig. 3.21-a, the second in the category of Fig. 3.21-b, and the third in the category of Fig. 3.21-c. The third channel as shown in Fig. 3.21 spreads the current spectrum more uniformly in the channel and is expected to be affected the most from edge roughness. All channels have the same length of $L=250~\mathrm{nm}$ . For smaller widths the effect of roughness is strong, and the transmissions of all channels are drastically reduced. Since the 2ELD-ZGNR devices can concentrate the current spectrum around the defect lines as shown in Fig. 3.21-b and 3.21-c, they effectively bring it closer to the edges and the reduction is larger for these devices. For larger widths the transmission of the ribbons approaches its ballistic value, which is 2 for the ELD-ZGNR devices and 3 for the 2ELD-ZGNR devices. The transmission of the 2ELD-ZGNR($n_2$ ,4,$n_2$ ) channels increases faster with increasing channel width, because the current spectrum is located farther from the edges which makes it less susceptible to scattering as the width increases. The transmission of 2ELD-ZGNR($n_3$ ,6,$n_3$ ) channel eventually increases close to the ballistic transmission value as the width increases, but it increases more slowly than that of the 2ELD-ZGNR($n_2$ ,4,$n_2$ ) channel.