# Anomalous delocalization of resonant states in graphene \& the vacancy magnetic moment

## TL;DR

This paper investigates the origin of discrepancies in magnetic moment calculations for vacancies in graphene, revealing that anomalous delocalization of resonant states caused by periodic defect arrangements affects magnetic properties, and that hybrid functionals can correct this issue.

## Contribution

The study identifies the fundamental mono-electronic origin of delocalized resonant states in graphene vacancies and demonstrates how hybrid functionals can address the resulting artificial dispersions.

## Key findings

- Resonant states are anomalously delocalized due to periodic defect arrangements.
- Hybrid functionals correct the artificial dispersion of the $ ext{pi}$ midgap bands.
- Discrepancies in magnetic moments are linked to these delocalized resonant states.

## Abstract

Carbon atom vacancies in graphene give rise to a local magnetic moment of $\sigma+\pi$ origin, whose magnitude is yet uncertain and debated. Partial quenching of $\pi$ magnetism has been ubiquitously reported in periodic $first-principles$ calculations, with magnetic moments scattered in the range 1.0 - 2.0 $\mu_{B}$, slowly converging to the lower or the upper end, depending on how the diluted limit is approached. By contrast, (ensemble) density functional theory calculations on cluster models neatly converge to the value of $2$ $\mu_{B}$ when increasing the system size. This stunning discrepancy has sparked an ongoing debate about the role of defect-defect interactions and self-doping, and about the importance of the self-interaction-error in the density-functional-theory description of the vacancy-induced states. Here, we settle this puzzle by showing that the problem has a fundamental, mono-electronic origin which is related to the special (periodic) arrangement of defects that results when using the slab-supercell approach. Specifically, we report the existence of resonant states that are $anomalously$ delocalized over the lattice and that make the $\pi$ midgap band $unphysically$ dispersive, hence prone to self-doping and quenching of the $\pi$ magnetism. Hybrid functionals fix the problem by widening the gap between the spin-resolved $\pi$ midgap bands, without reducing their artificial widths. As a consequence, while reconciling the magnetic moment with expectations, they predict a spin-splitting which is one order of magnitude larger than found in experiments.