Electric current in star junctions of molecular wires

Abstract

There is an enormous amount of literature on molecular-size circuits. However, no common regularities have ever been reported regarding the embedding of Y-like or, more generally, starlike molecular junctions into circuits. At the same time, connections of several wires are unavoidable components of any circuit. We show that in star junctions of $\mathcal{N}$ identical molecular wires, which are equally coupled to the feeding leads, the branched current is inversely proportional to ${\mathcal{N}}^{2}$. It is also proved that, independently of molecular structure, the minimal resistance that is associated with the terminal-to-terminal current through such junctions is equal to $h{\mathcal{N}}^{2}/8{e}^{2}$. In words, it is divided by eight von Klitzing resistance times ${\mathcal{N}}^{2}$. These predictions rule the performance of quantum wire junctions and provide useful references for studies of complex multiterminal molecular devices. The current-voltage dependence predicted for the circuits that contain a star junction of molecular wires is compared with I-V characteristics of constructively similar junctions of conventional resistors, where the Ohm law prescribes the proportionality of branched current to $1/\mathcal{N}$. The newly derived basic formulas include the exact solution of Lippman-Schwinger equation for a general model of multiterminal, rigid (coherent) scatterer, and also, a new and general trace formula for multiterminal transmission.