Coulomb charging energy of vacancy-induced states in graphene

TL;DR

This paper calculates the Coulomb charging energy of vacancy-induced states in graphene, revealing size-dependent scaling and significant edge effects, which are crucial for understanding vacancy magnetism in graphene materials.

Contribution

It provides a detailed computation of the charging energy $U$ in graphene, showing its dependence on system size, edge proximity, and vacancy position, with results differing from previous estimates.

Findings

$U$ scales as $( ext{ln} L)^{-2}$ with system size in bulk graphene.

For realistic sizes, $U$ is on the order of eV, much higher than earlier estimates.

$U$ is highly sensitive to vacancy position near edges, especially in nanoribbons.

Abstract

Vacancies in graphene have been proposed to give rise to $\pi$-like magnetism in carbon materials, a conjecture which has been supported by recent experimental evidence. A key element in this "vacancy magnetism" is the formation of magnetic moments in vacancy-induced electronic states. In this work we compute the charging energy $U$ of a single-vacancy generated localized state for bulk graphene and graphene ribbons. We use a tight-binding model to calculate the dependency of the charging energy $U$ on the amplitudes of the localized wave function on the graphene lattice sites. We show that for bulk graphene $U$ scales with the system size $L$ as $(\ln L)^{-2}$, confirming the predictions in the literature, based on heuristic arguments. In contrast, we find that for realistic system sizes $U$ is of the order of eV, a value that is orders of magnitude higher than the previously reported estimates. Finally, when edges are considered, we show that $U$ is very sensitive to the vacancy position with respect to the graphene flake boundaries. In the case of armchair nanoribbons, we find a strong enhancement of $U$ in certain vacancy positions as compared to the value for vacancies in bulk graphene.