An infinite grid of mesoscopic resistors: investigation and visualization of ballistic conduction

TL;DR

This paper models a square lattice of mesoscopic resistors as waveguides with scattering, analyzing its band structure, conductance, and wave patterns, revealing complex behaviors and potential experimental setups.

Contribution

It introduces a detailed scattering matrix approach for a mesoscopic resistor lattice, linking energy bands to wave pattern complexity and visualizing conductance properties.

Findings

Energy band structure is determined and visualized.

Wave function patterns vary from dull to intricate depending on energy.

Conductance is calculated using the Landauer formula.

Abstract

A square lattice of mesoscopic resistors is considered. Each bond is modeled as a narrow waveguide, while junctions are sources of elastic scattering given by a scattering matrix \mathbf{S}. Symmetry and unitarity constraints are used in a detailed way to simplify all possible matrices \mathbf{S}. Energetic band structure of the system is determined and visualized for exemplary parameters. Conductance between external electrodes attached to two arbitrary nodes is given by the Landauer formula. Thus the transmittance between the electrodes is calculated. The rest of the paper is devoted to studying wave function patterns arising from an electron injected through one electrode into the system. It is observed that depending on its energy both dull and localized or intricate and expansive patterns occur. It is justified mathematically that fitting electron energy into an energy band can be associated with emergence of complex patterns. Finally, possible experimental realizations of the considered model are briefly mentioned.