Magnetic States of the Zigzag Edge of a Graphene Nanoribbon

Abstract

Using a simple structural model and the multicenter Anderson Hamiltonian, Green’s functions are obtained for the atoms of the zigzag edge of an epitaxial graphene nanoribbon. The electronic structure of the free nanoribbon is discussed in detail. Specifically, expressions for the band spectrum and density of states are found and estimates of the occupation numbers and magnetic moments are given. For a nanoribbon strongly bonded to a metal substrate, the criteria for the appearance of magnetic moments are determined. As it is shown for both free and epitaxial nanoribbons, the probability of the appearance of magnetic moments and their magnitude for zigzag edge atoms that have two nearest neighbors is higher than for atoms with three nearest neighbors.

APPENDIX

To find the occupation numbers, it is necessary to calculate the following integrals:

$${{J}_{ - }} = \int\limits_{{{\omega }_{1}}(\pi /2a)}^{{{\omega }_{1}}(0)} {\frac{{\omega d\omega }}{{{{R}_{ - }}(\omega )}}} ,\quad {{S}_{ - }} = \int\limits_{{{\omega }_{1}}(\pi /2a)}^{{{\omega }_{1}}(0)} {\frac{{d\omega }}{{{{R}_{ - }}(\omega )}},} $$

$${{P}_{ - }} = \int\limits_{{{\omega }_{1}}(\pi /2a)}^{{{\omega }_{1}}(0)} {\frac{{d\omega }}{{\omega {{R}_{ - }}(\omega )}};} $$

$${{J}_{ + }} = \int\limits_{{{\omega }_{2}}(\pi /2)}^{{{\omega }_{2}}(0)} {\frac{{\omega d\omega }}{{{{R}_{ + }}(\omega )}}} ,\quad {{S}_{ + }} = \int\limits_{{{\omega }_{2}}(\pi /2)}^{{{\omega }_{2}}(0)} {\frac{{d\omega }}{{{{R}_{ + }}(\omega )}},} $$

$${{P}_{ + }} = \int\limits_{{{\omega }_{2}}(\pi /2)}^{{{\omega }_{2}}(0)} {\frac{{\omega d\omega }}{{\omega {{R}_{ + }}(\omega )}},} $$

where ω₁(0) = t($\sqrt {17}$ + 1)/2, ω₂(0) = t($\sqrt {17}$ – 1)/2, ω₁(π/2a) = t, and ω₂(π/2a) = Δ. Using the handbook [26] (formulas 3.147–3.149), we obtain the following expressions for the J_–, S_–, and P_– integrals:

$${{J}_{ - }} = 2[(1 + \eta )\Pi ( - \eta ,\eta ) - \eta K(\eta )],$$

$${{S}_{ - }} = (2{\text{/}}{{\omega }_{1}}(0))K(\eta ),$$

$${{P}_{ - }} = (1{\text{/}}2{{t}^{2}})[(1 + \eta )\Pi (1,\eta ) - K(\eta )],$$

$${{J}_{ + }} = 2[(1 + \eta )\Pi (\varphi , - \eta ,\eta ) - F(\varphi ,\eta )],$$

$${{S}_{ + }} = (2{\text{/}}{{\omega }_{1}}(0))F(\varphi ,\eta ),$$

$${{P}_{ + }} = (1{\text{/}}2{{t}^{2}})[(1 + \eta )\Pi (\varphi ,1,\eta ) - \eta F(\varphi ,\eta )],$$

where K(η) and Π(n, η) are the complete elliptic integrals of the first and third kind [26, 27], F(φ, η) and Π(φ, n, η) are the incomplete elliptic integrals of the first and third kind [26, 27],

$$\eta = (\sqrt {17} - 1){\text{/}}(\sqrt {17} + 1),$$

and

$$\varphi = \arcsin (\sqrt {[1 - \Delta {\text{/}}{{\omega }_{2}}(0)]{\text{/}}[1 + \Delta {\text{/}}{{\omega }_{1}}(0)]} ).$$

For what follows, we need to expand the J₊, S₊, and P₊ integrals over φ = π/2 – β, where β ≈ $\sqrt {\Delta {\text{/}}t} {\text{/}}2$$\ll$ 1. Based on the definition of the F(φ, η) and Π(φ, n, η) integrals and assuming

$$F(\varphi ,\eta ) = K(\eta ) - \kappa (\beta ,\eta ),$$

$$\kappa (\beta ,\eta ) = \int\limits_{1 - {{\beta }^{2}}/2}^1 {\frac{{dx}}{{\sqrt {(1 - {{x}^{2}})(1 - {{\eta }^{2}}{{x}^{2}})} }},} $$

$$\Pi (\varphi ,n,\eta ) = \Pi (n,\eta ) - \pi (\beta ,n,\eta ),$$

$$\pi (\beta ,n,\eta ) = \int\limits_{1 - {{\beta }^{2}}/2}^1 {\frac{{dx}}{{(1 + n{{x}^{2}})\sqrt {(1 - {{x}^{2}})(1 - {{\eta }^{2}}{{x}^{2}})} }},} $$

we get

$$\kappa (\beta ,\eta ) \approx \sqrt {\Delta {\text{/}}t} {\text{/}}4\sqrt {1 - {{\eta }^{2}}} ,$$

$$\pi (\beta ,n,\eta ) \approx \sqrt {\Delta {\text{/}}t} {\text{/}}4(1 + n)\sqrt {1 - {{\eta }^{2}}} .$$

Thus, preserving only terms with ~$\sqrt {\Delta {\text{/}}t}$ and taking into account relation Π(–η, η) = [π + 2(1 + η)K(η)]/4(1 + η) [27], we come to formulas (18).