3.3.2 The Role of Substrate Impurities

The possibility of further enhancing the asymmetry between electron and hole transport near the Fermi level by the introduction of positively charged substrate background impurities is examined. The effect of background impurities is included in the Hamiltonian in a simplified way as an effective negative long range potential energy on the appropriate on-site Hamiltonian elements as described in Ref. [88]. A positive impurity in the substrate will constitute a repulsive potential for holes (a barrier for holes but a well for electrons) and will degrade hole transport more effectively than electron transport. Figure. 3.22-a shows how the transmission of the ELD-ZGNR(10,10) channel (dashed-black line) is affected after the introduction of positive charged impurities in the channel (solid-blue line). Indeed, the transmission of holes below the Fermi level ( $E=0~\mathrm{eV}$ ) is degraded. This effect additionally increases the asymmetry of the propagating bands and improves the Seebeck coefficient. On the other hand, the opposite is observed when negative impurities are introduced in the substrate. Negative impurities are a barrier for electrons and reduce their transmission [94], but do not interfere with the hole subsystem as shown in Fig. 3.22-b. This type of impurities will actually harm the asymmetry and needs to be avoided.

Figure 3.22: The effect of (a) positive substrate impurity, (b) negative substrate impurity, and (c) roughness on the transmission of ELD-ZGNR(10,10) with length of $125~\mathrm{nm}$ . Inset of (c): The transmission of ZGNR(20) in the presence of roughness.