## 4 Visualizing Electron Flow across the Interface
Performing this analysis on the heterojunction, we observe that the delocalized states have a large probability to be transmitted across the interface, given the significant overlap between such a state on one side of the interface and the corresponding state on the other side. By the same token, it would seem that the more localized state should have a much smaller transmission probability to the equivalent states on the other side of the interface. However, care must be taken in making such a claim, since these states are no longer stationary states of the interface system, and we expect that cross-state transmission should take place. Such an analysis is best performed within the S-matrix formalism, since it can highlight the important interference effects across the interface, as discussed below. The local current plot in Figure 2a establishes that the left conductance peak in zGNR-4 corresponds mostly to the transfer of the current from one edge to the other, as one of the Bloch states for this energy is highly localized along the edge and benefits from the narrow width of the system to the tunnel. Now, analyzing the plot for the corresponding peak in zGNR-6 (Figure 2b), the edge-to-edge contribution for the current is still present, but with a smaller relative contribution to the current over the ribbon's valley. This is mostly due to the presence of the other delocalized Bloch state at the same energy. The widergeometry also weakens the coupling between opposite edges, compared to narrower ribbons. Note that the peak on the left is suppressed for the zGNR-8 and wider systems as the overlap between the Bloch states localized on opposite edges is significantly reduced (see the left panel of Figure 2). Moving to the peaks in the right-hand side of Region (II), whose corresponding edge Bloch state is slightly less edge-localized than those from the left part of Region (II), we observe that they still produce a dominant inter-edge current in zGNR-6 (Figure 2c), but the relative weight of this contribution in comparison to the valley current decreases as we increase the width of the ribbon (see Figure 2d,e). **Figure 3.** Dispersion relation for the highest occupied band for a zGNR-6 in its AFM state. Regions (I) (highlighted in gray) and (II) are indicated accordingly. The inset shows a comparison between DFT and tight-binding description (TBU) for a broader range of energy and local density of states (LDOS) for selected values of the the \(\mathbf{k}\) vector along the ribbon's length. **5. Eigenchannel Analysis: Scattering of the Electrode's Bloch States**  
A more detailed understanding of the scattering phenomena is obtained by identifying the transmission contribution from the individual states from the leads (Figure 4). Here, we focus on the energy Range (II) corresponding to the discussion on Figure 2. This range is highlighted by the shaded area in the plot from the left-hand side of Figure 3. There are two electronic states for each energy value in this range: one for a \(k\)-vector smaller than \(k_{6}\) (with \(k_{6}\) being the \(k\)-vector of Point 6 in Figure 3) and another one for a \(k\)-vector larger than \(k_{6}\). To simplify the subsequent discussion, we will refer to the first electronic level (\(k<k_{6}\), which is less localized) as the valley state, or simply as channels \(C_{v}^{L}\) and \(C_{v}^{R}\) in the left and right terminals, respectively. The second level (\(k>k_{6}\), which is localized along the edges) will be referred to as the edge state, or simply as channels \(C_{e}^{L}\) and \(C_{e}^{R}\) for the left and right electrodes. First, for zGNR-4 (the narrowest system studied here; see Figure 4a), we observe that the transport is balanced between valley-to-valley (\(C_{v}^{L}\to C_{v}^{R}\); black full line) and edge-to-edge (\(C_{e}^{L}\to C_{e}^{R}\); red dashed line) tunneling, with a minor cross-channel contribution (\(C_{v}^{L}\to C_{e}^{R}\) and \(C_{e}^{L}\to C_{v}^{R}\); green dashed dotted line).