On the minimal conductivity of graphene

TL;DR

This paper investigates how the minimal conductivity of graphene varies depending on the order of limits taken in the calculation, revealing different universal values under different conditions.

Contribution

It demonstrates the sensitivity of graphene's minimal conductivity to the order of limits in the Kubo formula and explores effects of disorder and frequency on its value.

Findings

Minimal conductivity is $4/\pi$ when DC limit is taken first.

Minimal conductivity is $\pi/2$ when energy integration is performed first.

Disorder and high-frequency effects alter the minimal conductivity values.

Abstract

The minimal conductivity of graphene is a quantity measured in the DC limit. It is shown, using the Kubo formula, that the actual value of the minimal conductivity is sensitive to the order in which certain limits are taken. If the DC limit is taken before the integration over energies is performed, the minimal conductivity of graphene is $4/\pi$ (in units of $e^2/h$) and it is $\pi/2$ in the reverse order. The value $\pi$ is obtained if weak disorder is included via a small frequency-dependent selfenergy. In the high-frequency limit the minimal conductivity approaches $\pi/2$ and drops to zero if the frequency exceeds the cut-off energy of the particles.