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**Mesoscopic analog of the Braess paradox in a congested (...)**

**Staff :** Hermann Sellier, Vincent Bayot, Serge Huant

**PhD :** Peng Liu (now at Peking University, China)

The Braess paradox is a counter-intuitive behavior of congested networks where the overall performance can be degraded by the opening of an additional branch that should intuitively relax the congestion. The best example is that of road networks where the opening of a high speed road in a saturated network can paradoxically lead to a longer travelling time for every driver. This paradox is well explained by game theory and its extensions to classical networks (electrical, mechanical) are also well understood.

We have been looking for the existence of a similar paradox in quantum physics. One of our motivation is the possibility to use Scanning Gate Microscopy (SGM) to modulate by gate effect the transmission of a channel in a semiconductor nanostructure, et therefore to open or close a single branch of a mesoscopi network. This way we found numerical and experimental proofs that an analog of the Braess paradox can indeed exist in mesoscopic semiconductor networks.

An example of the simulations that give proofs of a mesoscopic Braess paradox is shown on the figure below. The network is a rectangular corral where electron transport is ballistic and coherent at low temperature. Openings are larger that arms such that the network is congested (some of quantum conduction channels are reflected at the openings). This corral contains a central arm that creates a short-cut for transport between the two openings. Each arm can be closed by a local negative gate voltage corresponding to the SGM tip action (red dot on the figure). The figure shows the current density inside the network at 4 K for a source-drain bias of 1 mV. Closing the central arm (figure c) results in a increase of the current at the openings with respect to the unperturbed situation (figure b), whereas intuition suggests the opposite behavior, i.e. a decrease of the current, as it is the case for lateral arms (figure a). This counter-intuitive effect on the central arm can thus be associated to a new type of Braess paradox for mesoscopic systems of particular geometry.

These results from numerical simulations have been reproduced by preliminary SGM experiments at 4.2 K on a network device patterned in InGaAs/InAlAs heterostructure (electron density 3,5x10^11 /cm² and mobility 100 000 cm²/V/s) with 2 µm mean free path and 1 µm coherence length. A current increase for a negative SGM tip voltage has been observed when the tip is located above the central arm, over a voltage range and with an amplitude larger than universal conductance fluctuations that are also present in such mesoscopic systems (figure below). This work will be continued in order to identify the parameters governing the amplitude of this effect, still with the support of numerical simulations.

**Publications :**

*Transport Inefficiency in Branched-Out Mesoscopic Networks: An Analog of the Braess Paradox*

M.G. Pala, S. Baltazar, P. Liu, H. Sellier, B. Hackens, F. Martins, V. Bayot, X. Wallart, L. Desplanque, and S. Huant

Phys. Rev. Lett. **108**, 076802 (2012)

*A new transport phenomenon in nanostructures: a mesoscopic analog of the Braess paradox encountered in road networks*

M. Pala, H. Sellier, B. Hackens, F. Martins, V. Bayot, and S. Huant

Nanoscale Research Letters **7**, 472 (2012)

**Collaborations :**

IMEP : Marco Pala, Samuel Baltazar

IEMN : Xavier Wallart, Ludovic Desplanque

UCL : Benoit Hackens, Frederico Martins, Vincent Bayot

**Fundings :**

RTRA Fondation Nanosciences

ANR PNANO MICATEC (2008-2010)

- - Very low temperature AFM-STM combined microscopy
- - Mesoscopic analog of the Braess paradox in a congested network
- - Aharonov-Bohm effect and local density of state in quantum rings

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