A new transport phenomenon in nanostructures: a mesoscopic analog of the Braess paradox encountered in road networks
© Pala et al.; licensee Springer. 2012
Received: 16 July 2012
Accepted: 2 August 2012
Published: 22 August 2012
The Braess paradox, known for traffic and other classical networks, lies in the fact that adding a new route to a congested network in an attempt to relieve congestion can degrade counterintuitively the overall network performance. Recently, we have extended the concept of the Braess paradox to semiconductor mesoscopic networks, whose transport properties are governed by quantum physics. In this paper, we demonstrate theoretically that, alike in classical systems, congestion plays a key role in the occurrence of a Braess paradox in mesoscopic networks.
KeywordsBraess paradox Mesoscopic physics Congested networks Scanning gate microscopy
Adding a new road to a congested road network can paradoxically lead to a deterioration of the overall traffic situation, i.e., longer trip times for individual road users, or, in reverse, blocking certain streets in a complex road network can surprisingly reduce congestion. This counterintuitive behavior has been known as the Braess paradox[2, 3]. Later extended to networks in classical physics such as electrical or mechanical networks[4, 5], this paradox lies in the fact that adding extra capacity to a congested network can degrade counterintuitively its overall performance.
A key ingredient in the occurrence of classical Braess paradoxes is network congestion. Our previous work was made on a congested mesoscopic network, and it indeed exhibited a marked paradoxical behavior. In this letter, we study numerically in more detail the effect of congestion by simulating three rectangular corrals of different dimensions, i.e., different degrees of congestion. We show that releasing congestion considerably relaxes the paradoxical behavior. Simulations of the spatial distribution of the current density inside the networks for different positions of the local gate help to interpret our predictions in terms of current redistribution inside the network.
The three simulated networks are shown in Figure1a,b,c. The narrowest network in Figure1a is nearly identical to the network simulated in our previous work, 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 < W0 , where W and W0 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 > W0 , as shown in Figure1b, progressively relaxes congestion since all conducting modes injected by the openings can be admitted in the lateral arms. Starting from the network of Figure1b, we will further relax the congestion by increasing the widths L of the horizontal long arms, as shown in Figure1c.
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 EF 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].
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’ 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.
Results and discussion
The key role of congestion in the network
Figure1d,e,f shows the current flowing through the structures depicted in Figure1a,b,c, respectively, as a function of the tip position scanned along the median lines (red lines). Figure1d shows the occurrence of an analog of the classical Braess paradox in a congested mesoscopic network as a distinctive current peak centered at Ytip = 0 nm. When the tip-induced potential closes the central wire connecting the two openings in Figure1a, the current is counterintuitively increased. However, Figure1e,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.
The robustness of the paradox
In this letter, we have studied the geometric conditions of mesoscopic networks for the occurrence of a quantum analog of the Braess paradox, known previously for classical systems only. By analyzing the spatial distribution of current density in different structures, we have shown that congested structures are the most suitable geometries to the occurrence of such a counterintuitive phenomenon. This is reminiscent to what is known for the classical paradoxes, in particular, for the historic road-network Braess paradox.
This work has been supported by the French Agence Nationale de la Recherche (MICATEC project), the FRFC (grant no. 2.4.546.08.F) and FNRS (grant no. 1.5.044.07.F), and by the Belgian Science Policy (Program IAP-6/42). Vincent Bayot acknowledges support from the Grenoble Nanosciences Foundation (Scanning-Gate Nanoelectronics project).
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