Semi-classical study of the Quantum Hall conductivity

TL;DR

This paper analyzes the integer Quantum Hall conductivity using semi-classical methods for electrons in a bi-periodic potential, highlighting the role of tunneling and well configurations in topological conductivity.

Contribution

It provides an analytical and numerical study of how well positions and shapes influence topological quantum Hall conductivity in a bi-periodic potential.

Findings

Hall conductivity arises from tunneling effects.

Specific well configurations are necessary for non-zero topological conductivity.

Analytical results are confirmed by numerical calculations.

Abstract

The semi-classical study of the integer Quantum Hall conductivity is investigated for electrons in a bi-periodic potential $V(x,y)$. The Hall conductivity is due to the tunnelling effect and we concentrate our study to potentials having three wells in a periodic cell. A non-zero topological conductivity requires special conditions for the positions, and shapes of the wells. The results are derived analytically and well confirmed by numerical calculations.