[Paper Review] Quantum Percolation Transition from Graphene to Graphane: Graph Theoretical Approach

Motohiko Ezawa|arXiv (Cornell University)|Apr 14, 2011

Graphene research and applications15 references3 citations

TL;DR

This paper investigates the quantum percolation transition in hydrographene—a partially hydrogenated graphene derivative—using a graph-theoretical approach to a quantum site-percolation model. It reveals a metal-insulator transition at a critical hydrogenation density $ q_c \approx 0.31 $, with hydrographene exhibiting bulk ferromagnetism due to spin polarization in connected clusters, where single-sided hydrogenation is far more efficient for inducing large magnetic moments than double-sided hydrogenation.

ABSTRACT

Graphane is obtained by perfectly hydrogenating graphene. There exists an intermediate material, partially hydrogenated graphene (which we call extit{hydrographene}), interpolating from pure graphene to pure graphane. It has various intriguing electronic and magnetic properties. We investigate a metal-insulator transition, employing a quantum-site percolation model together with a graph theoretical approach. Hydrographene is an exceptional case in which electronic properties cannot be determined solely by the density of states at the Fermi energy. Though there are plenty of zero energy state in wide range of hydrogenation density, most of them are insulating states. We also demonstrate that it shows a bulk ferromagnetic property based on the Lieb theory.

Motivation & Objective

• To understand the electronic and magnetic properties of hydrographene, an intermediate phase between graphene and graphane, which cannot be predicted solely from the density of states at the Fermi energy.
• To investigate how hydrogenation density $ q $ governs the metal-insulator transition in hydrographene using a quantum site-percolation model.
• To explore the emergence of bulk ferromagnetism in hydrographene based on the Lieb theorem and its dependence on hydrogenation type (single- vs. double-sided).
• To establish a mapping between percolation behavior and ferromagnetic transitions via the Kasteleyn-Fortuin correspondence.
• To demonstrate that single-sided hydrogenation is significantly more efficient than double-sided hydrogenation in generating large magnetic moments.

Proposed method

• Modeling hydrographene with a tight-binding Hamiltonian where hydrogenated carbon sites are assigned an infinite on-site potential $ V $, effectively removing $ \pi $-electrons from those sites.
• Applying a quantum site-percolation model on a honeycomb lattice, where hydrogenated sites are removed, leaving a network of carbon atoms connected via $ t $-hopping integrals.
• Using graph theoretical analysis to compute the size distribution of finite clusters and the percolation probability $ P(q) $, identifying the critical hydrogenation density $ q_c $ at which a macroscopic cluster forms.
• Employing the Kasteleyn-Fortuin mapping to relate the percolation problem to the zero-state Potts model, linking the hydrogenation parameter $ q $ to temperature in a ferromagnetic system.
• Calculating magnetization using the Lieb theorem: $ M = \frac{1}{2}|N_A - N_B| $, where $ N_A $ and $ N_B $ are the numbers of A and B sublattice sites in the largest connected cluster.
• Performing Monte Carlo simulations with statistical averaging over 100–500 realizations for system sizes up to 40,000 sites to compute $ S(q) $, $ P(q) $, and $ M(q) $.

Experimental results

Research questions

• RQ1 How does the hydrogenation density $ q $ affect the metal-insulator transition in hydrographene, and what is the critical value $ q_c $?
• RQ2 Why do many zero-energy states in hydrographene not contribute to conductivity, despite their presence in the density of states at the Fermi energy?
• RQ3 What is the origin and magnitude of the bulk ferromagnetic moment in hydrographene, and how does it depend on the hydrogenation pattern (single- vs. double-sided)?
• RQ4 How does the percolation transition in hydrographene map to a ferromagnetic phase transition via the Kasteleyn-Fortuin correspondence?
• RQ5 Why is single-sided hydrogenation more efficient than double-sided hydrogenation in generating large magnetic moments?

Key findings

• The metal-insulator transition occurs at a critical hydrogenation density $ q_c \approx 0.31 $, marked by the divergence of the average cluster size $ S(q) \propto |q - q_c|^{-1} $.
• Hydrographene exhibits bulk ferromagnetism due to spin polarization in the largest connected cluster, with magnetization proportional to the number of sites in that cluster.
• The maximum magnetization per site reaches $ M = 0.17 $ at $ q = 0.44 $ for single-sided hydrogenation, which is about 500 times more efficient than double-sided hydrogenation.
• For double-sided hydrogenation, the maximum magnetization per site is $ M = 0.13 $ at $ q = 0.68 $, indicating a strong dependence on hydrogenation pattern.
• The percolation transition maps to a ferromagnet-paramagnet transition via the Kasteleyn-Fortuin mapping, with $ q $ corresponding to inverse temperature $ T^{-1} $.
• Single-sided hydrogenation leads to a linear increase in magnetization with $ q $, while double-sided hydrogenation shows a non-monotonic behavior, peaking at $ q = 0.68 $.