We present evidence for a counter-intuitive behavior of semiconductor mesoscopic networks that is the analog of the Braess paradox encountered in classical networks. A numerical simulation of quantum transport in a two-branch mesoscopic network revea
ls that adding a third branch can paradoxically induce transport inefficiency that manifests itself in a sizable conductance drop of the network. A scanning-probe experiment using a biased tip to modulate the transmission of one branch in the network reveals the occurrence of this paradox by mapping the conductance variation as a function of the tip voltage and position.
We study the relationship between the local density of states (LDOS) and the conductance variation $Delta G$ in scanning-gate-microscopy experiments on mesoscopic structures as a charged tip scans above the sample surface. We present an analytical mo
del showing that in the linear-response regime the conductance shift $Delta G$ is proportional to the Hilbert transform of the LDOS and hence a generalized Kramers-Kronig relation holds between LDOS and $Delta G$. We analyze the physical conditions for the validity of this relationship both for one-dimensional and two-dimensional systems when several channels contribute to the transport. We focus on realistic Aharonov-Bohm rings including a random distribution of impurities and analyze the LDOS-$Delta G$ correspondence by means of exact numerical simulations, when localized states or semi-classical orbits characterize the wavefunction of the system.
Combining Scanning Gate Microscopy (SGM) experiments and simulations, we demonstrate low temperature imaging of electron probability density $|Psi|^{2}(x,y)$ in embedded mesoscopic quantum rings (QRs). The tip-induced conductance modulations share th
e same temperature dependence as the Aharonov-Bohm effect, indicating that they originate from electron wavefunction interferences. Simulations of both $|Psi|^{2}(x,y)$ and SGM conductance maps reproduce the main experimental observations and link fringes in SGM images to $|Psi|^{2}(x,y)$.
