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

Miguel A. Cazalilla, Anı́bal Iucci+2 more|arXiv (Cornell University)|Jul 13, 2012

Graphene research and applicationsMaterials Science19 citations

TL;DR

This paper proposes that local magnetic moments and the Kondo effect in single-atom vacancies in graphene arise due to strong energy-dependent hybridization between a dangling σ orbital and the π-band, driven by vacancy-induced strain and rippling. The hybridization scales as ∼[|ε|ln²(ε/D)]⁻¹ near the Dirac point, enabling Kondo temperatures of 1–100 K, reconciling conflicting experimental observations of magnetic moments and Kondo screening at low temperatures.

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 $σ$ orbital localized at the vacancy is allowed to strongly hybridize with the $π$-band since the scattering with the vacancy turns the hybridization into singular function of the energy ($\sim [|ε| \ln^2 ε/D]^{-1}$, $D\sim$ 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

• To resolve the apparent contradiction between observed magnetic local moments (MLMs) and high-Kondo-temperature (T_K ~ 10–100 K) effects in irradiated graphene.
• To explain how vacancy reconstruction and rippling enable hybridization between σ and π orbitals, which is otherwise symmetry-forbidden in flat graphene.
• To develop a theoretical model that accounts for non-planar effects, strain, temperature, and strong correlations in determining the magnetic and Kondo behavior of vacancies.
• To show that the energy dependence of hybridization drastically alters Kondo physics compared to previous pseudogap models.
• To provide a unified framework for understanding the coexistence of localized moments and Kondo screening in defective graphene.

Proposed method

• Develops a two-level Anderson model incorporating a reconstructed vacancy with a dangling σ orbital (d) and a vacancy-induced π-state (π₀), coupled via energy-dependent hybridization V(ε).
• Models the hybridization as Δ(ε) ∼ [ |ε| ln²(ε/D) ]⁻¹ near the Dirac point, arising from strong scattering at the vacancy, differing from earlier pseudogap forms ∼|ε|ʳ.
• Uses a slave-boson mean-field approach with large-N expansion to treat strong local Coulomb interactions (U) and Hund’s coupling (J_H), enforcing no double occupancy.
• Solves the Kondo problem via self-consistent equations for the slave boson condensate (r₀) and Lagrange multiplier (λ₀), with the Kondo temperature defined as the scale where r₀ → 0.
• Numerically computes the local Green’s function using a tight-binding model with nearest-neighbor (t) and next-nearest-neighbor (t′) hopping to capture the Dirac-like band structure.
• Derives the Kondo temperature condition T_K ln²(T_K/D₀) ∼ J_K from the integral equation involving Δ(ε), linking it to the Kondo coupling J_K.

Experimental results

Research questions

• RQ1 How can high Kondo temperatures (T_K ~ 10–100 K) coexist with stable magnetic local moments (MLMs) in graphene vacancies, given that Kondo screening should quench moments at low T?
• RQ2 What physical mechanism enables hybridization between σ and π orbitals in graphene vacancies, given that such hybridization is symmetry-forbidden in flat, planar graphene?
• RQ3 How does vacancy reconstruction and rippling—induced by strain or temperature—modify the energy dependence of the hybridization and thus the Kondo effect?
• RQ4 What role do local Coulomb interactions (U), Hund’s coupling (J_H), and particle-hole asymmetry play in determining the stability of the magnetic moment and the Kondo temperature?
• RQ5 Can a modified hybridization form Δ(ε) ∼ [ |ε| ln²(ε/D) ]⁻¹ near the Dirac point explain the observed T_K values and weak gate voltage dependence in experiments?

Key findings

• The energy-dependent hybridization Δ(ε) ∼ [ |ε| ln²(ε/D) ]⁻¹, induced by strong scattering at the vacancy, leads to a non-pseudogap Kondo behavior distinct from earlier models.
• This hybridization mechanism allows for Kondo temperatures T_K in the range of 1–100 K, consistent with experimental resistivity fits in irradiated graphene samples.
• The Kondo temperature satisfies the condition T_K ln²(T_K/D₀) ∼ J_K, where J_K is the Kondo coupling, derived from the self-consistent slave-boson solution.
• The model explains the apparent contradiction between Grigorieva and Fuhrer’s experiments: MLM survive at T ≈ 2 K because the Kondo screening is incomplete due to the non-trivial hybridization form.
• Rippling enables sp²–sp³ hybridization, allowing the dangling σ orbital to hybridize with the π-band, which is otherwise forbidden in flat graphene.
• The slave-boson mean-field approach with large-N expansion successfully captures the competition between local moment formation and Kondo screening, with T_K determined by the onset of slave-boson condensation.