The three simulated networks are shown in Figure 1a,b,c. The narrowest network in Figure 1a is nearly identical to the network simulated in our previous work 6 , apart from slightly larger openings (320 nm instead of 300 nm). Its dimensions are chosen such that the electron flow is congested. Indeed, in a system where electrons can be backscattered solely by the walls defining the structure geometry, a sufficient condition to reach congestion is obtained when the number of conducting modes allowed by internal constrictions is smaller than the number of conducting modes in the external openings, which implies 2 W < W ₀ , where W and W ₀ denote the widths of the lateral arms (both of the same width) and of the external openings (of equal widths too), respectively. In turn, increasing W such that 2 W > W ₀ , as shown in Figure 1b, progressively relaxes congestion since all conducting modes injected by the openings can be admitted in the lateral arms. Starting from the network of Figure 1b, we will further relax the congestion by increasing the widths L of the horizontal long arms, as shown in Figure 1c.

The transport properties of these structures are simulated within an exact numerical approach based on the Keldysh Green’s function formalism. A thermal average is performed around the Fermi energy E _F at the temperature T = 4.2 K. We adopt a mesh size of Δx = Δy = 2.5 nm. The Green’s function of the system is computed in the real space representation that allows us to take into account all possible conducting and evanescent modes. Moreover, in order to reduce the computational time and memory requirements, we exploit a recursive algorithm, which is based on the Dyson equation 6 9 .

In this framework, the current densities along the x-axis (transport direction) and the y-axis (transverse direction) between two adjacent nodes read as follows:

J i , i + 1 ; k , k = − 4 e h ∫ d ω Re H i , i + 1 ; k , k G i + 1 , i ; k , k < ω ,

J i , i ; k , k + 1 = − 4 e h ∫ d ω Re H i , i ; k + 1 , k G i , i ; k + 1 , k < ω ,

where H _i,i';k,k' represents the Hamiltonian discretized on the local basis, and G ^< _i,i';k,k' (ω) is the ‘lesser-than Green’s function’ 9 in the real space representation and energy domain.

The tip-induced potential is simulated by considering a point-like gate voltage of −1 V placed at 100 nm above the 2DEG, which corresponds to a lateral extension of ≈ 400 nm for the tip-induced potential perturbation at the 2DEG level.

Figure 1d,e,f shows the current flowing through the structures depicted in Figure 1a,b,c, respectively, as a function of the tip position scanned along the median lines (red lines). Figure 1d shows the occurrence of an analog of the classical Braess paradox in a congested mesoscopic network as a distinctive current peak centered at Y _tip = 0 nm. When the tip-induced potential closes the central wire connecting the two openings in Figure 1a, the current is counterintuitively increased. However, Figure 1e,f shows that as soon as the condition for congestion is relaxed, allowing a larger number of conducting channels to propagate in the region inside the structure, the paradox disappears, and the total current exhibits a maximum when the tip is placed over the two antidots.

In order to microscopically study this behavior, we have simulated in Figure 2 the spatial distribution of the absolute value of the current |J| inside the three structures for Y _tip = 0 nm and Y _tip = −400 nm. When comparing Figure 2a and Figure 2d for the congested structure, we can notice that the opening of a third central wire connecting the contacts has a twofold effect. The first consequence is to create a direct connection between the source and the drain, which should positively contribute to the total current flowing through the system. The second one is to generate alternative paths that trap electrons in the central region and should promote a longer stay inside the network. We believe that this second effect is the one responsible for the decrease of the total current as long as the third wire is opened. The comparison of Figure 2a and Figure 2d is indeed very instructive, and in particular, the behavior of the current through the right path paradoxically decreases while the depleting tip moves away. This behavior clearly indicates that the current contribution of trapped electrons around the right antidot compensates partially the initial current. This effect is only partly replicated in the networks of Figure 1b,c, whose current redistributions are shown in Figure 2b,e and Figure 2c,f, respectively. In these cases, the reopening of the third wire, obtained by placing the tip over the antidot, induces a number of new internal paths, which are small compared to the large number of semiclassical trajectories already present in the lateral arms. Therefore, the closing of the central path implies only a small current increase in Figure 1e,f around the position Y _tip = 0 nm, which is not sufficient to overcome the current at Y _tip = −400 nm.

Finally, in order to test the robustness of our results, we simulated the non-congested structure of Figure 1c at different Fermi wavelengths (λ _F = 57, 47, and 38 nm). This is shown in Figure 3. The behavior of the three curves is qualitatively very similar: they present two regions of maximum current when the gated tip is placed over the two antidots, allowing the passage of electrons through the central path, but they also show a local increase in current around Y _tip = 0, when the tip closes the central path. This is a signature that the mechanism responsible for the occurrence of the paradox in the congested structure of Figure 1a, even if still present, is not predominant with respect to the direct coupling between the two contacts provided by the third wire.