Discrete versus continuous wires on quantum networks

TL;DR

This paper compares the continuum and tight-binding approaches for solving Schrödinger's equation on quantum networks, showing they generally yield different results except at specific energies, with implications for modeling mesoscopic systems.

Contribution

It demonstrates that continuum and tight-binding models of quantum networks are not generally equivalent, highlighting conditions under which they produce similar results.

Findings

The two approaches differ except at special energies.

They coincide only under specific tight-binding parameters.

Differences are illustrated with a T-shaped scatterer example.

Abstract

Mesoscopic systems and large molecules are often modeled by graphs of one-dimensional wires, connected at vertices. In this paper we discuss the solutions of the Schr\"odinger equation on such graphs, which have been named "quantum networks". Such solutions are needed for finding the energy spectrum of single electrons on such finite systems or for finding the transmission of electrons between leads which connect such systems to reservoirs. Specifically, we compare two common approaches. In the "continuum" approach, one solves the one-dimensional Schr\"odinger equation on each continuous wire, and then uses the Neumann-Kirchoff-de Gennes matching conditions at the vertices. Alternatively, one replaces each wire by a finite number of "atoms", and then uses the tight binding model for the solution. Here we show that these approaches cannot generally give the same results, except for special energies. Even in the limit of vanishing lattice constant, the two approaches coincide only if the tight binding parameters obey very special relations. The different consequences of the two approaches are demonstrated via the example of a T-shaped scatterer.