[Paper Review] Local Moment Formation and Kondo Eect in Defective Graphene

Miguel A. Cazalilla, Anı́bal Iucci+3 more|arXiv (Cornell University)|Jul 13, 2012

Graphene research and applications17 citations

TL;DR

This paper investigates local moment formation and the Kondo effect at single-atom vacancies in graphene, incorporating vacancy reconstruction, strain, and temperature effects. It reveals that singular energy-dependent hybridization between the $σ$ orbital and $π$-band—scaling as $[|\epsilon| \ln^2 \epsilon/D]^{-1}$—gives rise to multiple impurity phases, determining magnetic moment size and Kondo effect feasibility based on Coulomb interactions, Hund's coupling, doping, and symmetry breaking.

ABSTRACT

We study the local moment formation and the Kondo effect at single-atom vacancies in Graphene. We develop a model accounting for the vacancy reconstruction as well as non-planarity effects induced by strain and/or temperature. Thus, we find that the dangling $\sigma$ orbital localized at the vacancy is allowed to strongly hybridize with the $\pi$-band since the scattering with the vacancy turns the hybridization into singular function of the energy ($\sim [|\epsilon| \ln^2 \epsilon/D]^{-1}$, $D\sim$ the bandwidth). This leads to several new types of impurity phases, which control the magnitude of the vacancy magnetic moment and the possibility of Kondo effect depending on the strength of the local Coulomb interactions, the Hund's rule coupling, the doping level, and the degree of particle-symmetry breaking.

Motivation & Objective

• Understand the formation of local magnetic moments at single-atom vacancies in graphene.
• Account for structural distortions, strain, and temperature effects on vacancy electronic structure.
• Investigate how hybridization between the $σ$ orbital and $π$-band influences magnetic and Kondo behavior.
• Determine the conditions under which the Kondo effect emerges in defective graphene.
• Identify the role of electron correlation, Hund's coupling, doping, and symmetry breaking in controlling impurity phases.

Proposed method

• Develop a theoretical model that includes vacancy reconstruction and non-planar distortions due to strain or thermal effects.
• Model the hybridization between the localized $σ$ orbital and the $π$-band as a singular function of energy: $[|\epsilon| \ln^2 \epsilon/D]^{-1}$, with $D$ as bandwidth.
• Incorporate local Coulomb interactions and Hund's rule coupling to assess their influence on magnetic moment formation.
• Analyze the system's phase diagram under varying doping levels and particle-hole symmetry breaking.
• Use effective Hamiltonian approaches to describe the impurity behavior and derive conditions for Kondo screening.

Experimental results

Research questions

• RQ1 How does vacancy reconstruction and non-planarity affect the hybridization between the $σ$ orbital and $π$-band in graphene?
• RQ2 What determines the magnitude of the local magnetic moment at a single-atom vacancy?
• RQ3 Under what conditions does the Kondo effect emerge at a vacancy in graphene?
• RQ4 How do local Coulomb interactions and Hund's coupling influence the formation of distinct impurity phases?
• RQ5 How does doping and particle-symmetry breaking modulate the possibility of Kondo screening?

Key findings

• The hybridization between the $σ$ orbital and $π$-band exhibits a singular energy dependence proportional to $[|\epsilon| \ln^2 \epsilon/D]^{-1}$, which strongly influences electronic correlations.
• Multiple impurity phases emerge due to the interplay of electron correlations, Hund's coupling, and symmetry breaking, leading to distinct magnetic and Kondo behaviors.
• The strength of local Coulomb interactions and Hund's rule coupling directly control the size of the magnetic moment at the vacancy.
• The Kondo effect is not universal but depends critically on doping level and the degree of particle-hole symmetry breaking.
• Strain and temperature-induced non-planarity significantly alter the hybridization and thus the stability of Kondo and magnetic phases.
• The system exhibits a rich phase diagram where the Kondo effect can be suppressed or enhanced depending on the interplay of interaction parameters and symmetry.