# Accuracy of Classical Conductivity Theory at Atomic Scales for Free Fermions in Disordered Media

## TL;DR

This paper mathematically demonstrates that quantum effects on electric current in disordered media diminish exponentially with size, aligning with experimental observations that classical conductivity models remain valid at larger scales.

## Contribution

It provides a rigorous proof that quantum uncertainty in current density is suppressed in disordered lattice fermions, extending classical conductivity theory to atomic scales.

## Key findings

- Quantum uncertainty diminishes exponentially with volume.
- Classical conductivity theory remains valid beyond a few nanometers.
- Mathematical framework applies to the Anderson model and similar disordered systems.

## Abstract

The growing need for smaller electronic components has recently sparked the interest in the breakdown of the classical conductivity theory near the atomic scale, at which quantum effects should dominate. In 2012, experimental measurements of electric resistance of nanowires in Si doped with phosphorus atoms demonstrate that quantum effects on charge transport almost disappear for nanowires of lengths larger than a few nanometers, even at very low temperature (4.2K). We mathematically prove, for non-interacting lattice fermions with disorder, that quantum uncertainty of microscopic electric current density around their (classical) macroscopic values is suppressed, exponentially fast with respect to the volume of the region of the lattice where an external electric field is applied. This is in accordance with the above experimental observation. Disorder is modeled by a random external potential along with random, complex-valued, hopping amplitudes. The celebrated tight-binding Anderson model is one particular example of the general case considered here. Our mathematical analysis is based on Combes-Thomas estimates, the Akcoglu-Krengel ergodic theorem, and the large deviation formalism, in particular the G\"artner-Ellis theorem.