## 3 Electronic Transmission  
To compute the electronic transport in the heterojunctions, we used the TBU approach to simulate the central channel region made up of 98 zGNR-\(n\) unit cells attached to two zGNR-\(n\) terminals (\(n=4,6,8,10\)). The leads are constrained to maintain the bulk AFM distributions, and the spin configuration in the right terminal is inverted relative to the left one for the case of the heterojunction. In practice, preparing the electrodes in such a magnetic state will require the use of appropriate contacts with a controllable spin state [35]. In Figure 2, we plot the conductance for the occupied states (\(E<E_{F}=0\), where \(E_{F}\) denotes the Fermi energy) for the spin-up states as a function of energy for both the heterojunction (solid black lines) and the perfect system (dashed red lines) for the zGNR-\(n\) junctions. Because the system remains in an AFM overall distribution, the plot is identical for spin-down and is symmetrical relative to \(E=E_{F}\), within the TBU approach (note that while we consider third neighbor interactions, the second neighbor parameter is zero, preserving the electron-hole symmetry [28]). Figure 2: **(Left)** Conductance for a given spin orientation in the AFM homojunction (red dashed lines) and heterojunction (black full lines) for the zGNR-\(n\) (\(n=4,6,8,10\)) systems. The energy ranges (I) and (II) correspond to the dashed and full double arrows, so that they present, respectively, one and two channels in the homojunction (see Figure 3). **(Right)** Local current plots for the peaks corresponding to the (primed) letters in the left panel for the heterojunction (homojunction). In these plots, dark blue corresponds to zero current, while red shows the places with maximal current. We examine the conductance spectra for negative energies in the pristine system and identify two main regions: (I) the energy range characterized by a single conducting channel; and (II) the region for the conductance plateau corresponding to two channels. These regions are indicated by dashed and full double arrows, respectively, in the left-hand side plots in Figure 2. Among the two spin-polarized channels in Region (II), one is strongly localized along one edge [14,16]. The main result from the conductance plot for the heterojunctions in this energy region is that the edge state does have a finite probability to hop from one edge to the opposite one. To better understand the details of the transmission and elucidate the transport process, we performed an eigenchannel analysis, where the electronic transport theory is formulated into the equivalent scattering-matrix (S-matrix) method [36,37]. Here, we express the S-matrix on the basis of the Bloch states of the isolated electrodes, as a natural choice to assess the possibility of transmitting edge states from one edge to the opposite one. It follows from the definitions given above that the S-matrix is a \(2\times 2\) and a \(4\times 4\) matrix in Regions (I) and (II), respectively. For the heterojunction, we observe that the single channel in (I) transmits with a 100% probability, while electronic states impinging at energies located in Region (II) are strongly affected by electronic scattering. We observe that the conductance for energies in the right sector of Region (II) for the heterojunctions has a much lower value compared with the pristine system. This is a quantum mechanical effect that can lead to the construction of logic devices through the turning on and off of the electronic current, as in cross-bar GNRs, where the quantum phase provides a mechanism to control the out of plane electronic currents [38]. In addition, the feature-rich transmission curve has a significant dependence on the system width. For example, the conductance for the zGNR-4 system in (II) shows a local maximum on the left-hand side of the interval, while the zGNR-8 and -10 systems present a peak of conductance on the right-hand side of (II). Interestingly, the zGNR-6 has an intermediate behavior with two clearly distinguishable peaks.