## 4 Visualizing Electron Flow across the Interface
The inter-edge transmission is favored by the narrow width, so that the electrons belonging to \(C_{e}^{L}\) can more effectively transmit to \(C_{e}^{R}\) and to a lesser extent to \(C_{v}^{R}\). Moving to wider systems, we observe that the edge-to-edge contribution to the current (red dashed lines in Figure 4) weakens in comparison to the valley current (black full lines in Figure 4) as the ribbon width increases. This effect is accompanied by an enhanced transmission of the \(C_{e}^{L}\) channel into the right valley eigenstate \(C_{v}^{R}\). Comparing these results with those of the zGNR-6 and -8 systems (see Figure 4b,c), we now understand how the contribution of the (edge-to-valley) cross-channel tunneling gradually becomes more important than the direct inter-edge transmission as the ribbon gets wider. To further understand the effect of GNR width on transport behavior, we consider the wider ribbon zGNR-40 (Figure 4d) and observe again that the \(C_{v}^{L}\) channel transmits mostly to \(C_{v}^{R}\) (\(\sim 80\%\) at \(E=-0.1\)\(eV\), for instance). However, the mechanism is different for the edge channel, as the inter-edge tunneling (\(C_{e}^{R}\to C_{e}^{L}\sim 2\%\) for \(E=-0.1\)\(eV\)) is suppressed relative to the interchannel (\(C_{e}^{L}\to C_{v}^{R}\sim 10\%\) for \(E=-0.1\)\(eV\)) transmission. Clearly, the widest system does not allow sufficient overlap between the two \(C_{e}^{L}\) and \(C_{e}^{R}\) eigenstates of the leads, so that this reduced inter-edge coupling prevents a strong tunneling from taking place between the opposite edges of the structure. It is not trivial to determine the crossover between edge and valley transport regimes, since valley currents spread over broad spatial ranges, being shared by a number of zigzag strips along the system  
Figure 4: Contributions of the \(C_{v}^{L}\to C_{v}^{R}\) (black full lines), \(C_{v}^{L}\to C_{e}^{R}\), \(C_{e}^{L}\to C_{v}^{R}\) (green dashed double dotted lines) and \(C_{e}^{L}\to C_{e}^{R}\) (red dashed lines) tunneling to the transmission in the heterojunctions for the zGNR-4 (**a**), -6 (**b**), -8 (**c**) and -40 (**d**) in the energy region where there are two conducting channels in the pristine junction (Region (II)). Here, \(C_{v}^{L}\) (\(C_{v}^{R}\)) represents the valley Bloch state from the source (drain) lead, while the edge state is represented by \(C_{e}^{L}\) (\(C_{e}^{R}\)). width. In contrast, edge currents are limited to a narrow portion of the ribbon. In order to gain further insight on this point, we consider spin-up states and plot the atomic position of the largest inter-site current (as a function of the index of the zigzag strip along the ribbon width) for each zGNR unit cell along the junction for the zGNR-4, -6, -8 and -10 cases in Figure 5. Here, we considered the energy values denoted by (a), (c), (d) and (e) in the left side of Figure 2. Note that the largest current is on the edge with a net spin-up polarization for the zGNR-4 case, except for the interface cells. Even though this is not strictly true for the zGNR-6 case, the zigzag strip presenting the largest current is usually the edge one. In other words, despite the fact that the valley states transmit more than the edge ones, the valley current is shared by a number of zigzag strips, so that most current flows along the edges of the system. The largest current is then moved to the valley region for zGNR-8 and wider systems. This shows that the heterospin system has a limiting behavior when moving from the zGNR-6 to the zGNR-8 system. Furthermore, as pointed out before, the \(C_{e}^{L}\to C_{e}^{R}\) transmission for the zGNR-6 system behaves differently compared to wider systems, since it presents two prominent peaks centered around \(-0.35\) eV and \(-0.27\) eV in the energy Range (II), resulting in a dip around \(-0.32\) eV. Analyzing this result carefully, we find that such a double-peak feature is also present in the zGNR-8.