Advances in Condensed Matter Physics

Volume 2018, Article ID 9097045, 14 pages

https://doi.org/10.1155/2018/9097045

## The Electronic Structure of Graphene Nanoislands: A CAS-SCF and NEVPT2 Study

^{1}Université de Toulouse, INSA, UPS, CNRS, LPCNO (IRSAMC), 135 avenue de Rangueil, 31077 Toulouse, France^{2}Laboratoire de Chimie et Physique Quantiques, IRSAMC, Université de Toulouse et CNRS, 118 route de Narbonne, 31062 Toulouse Cedex, France^{3}Equipe de Chimie et Biochimie Théoriques, SRSMC, Université de Lorraine et CNRS, Bp 70236 boulevard des Aguilettes, 54506 Vandoeuvre-lès-Nancy Cedex, France

Correspondence should be addressed to Thierry Leininger; rf.eslt-spu.cmasri@regniniel.yrreiht

Received 12 December 2017; Accepted 28 February 2018; Published 26 April 2018

Academic Editor: Sergio E. Ulloa

Copyright © 2018 Lucy Cusinato et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents a* tight binding* and ab initio study of finite graphene nanostructures. The attention is focused on three types of regular convex polygons: triangles, rhombuses, and hexagons, which are the most simple high-symmetry convex structures that can be ideally cut out of a graphene layer. Three different behaviors are evidenced for these three classes of compounds: closed-shells for hexagons; low-spin open-shells for rhombuses; high-spin open-shells for triangles.

#### 1. Introduction

Extended layers of conjugated carbon atoms disposed in a 2-dimensional (2D) honeycomb lattice are the constituent of the common graphite and were regarded as a sort of intellectual curiosity constituting the model of low dimensional materials. Indeed the attitude toward such systems dramatically changed at the beginning of this century with the works of Novoselov et al. [1, 2]. As reported in their seminal paper they were able to produce and characterize a new allotropic form of carbon, constituted by the 2-dimensional sheet that is graphene. Soon after this work, an impressive number of research papers appeared dealing with the structural and electronic properties of graphene. Indeed, because of the robustness of the graphene skeleton, this allotropic carbon form happens to be one of the strongest materials ever produced; consequently its use as reinforcer in novel high-performance composite materials becomes straightforward. Notably its remarkable electron transport properties and the fact that it is a zero-gap semiconductor are making it one of the materials of choice for future applications in molecular electronic devices [3]. Technical applications of graphene can be related to extremely diverse technological areas and, for instance, can be related to single molecule detection, to field effects transistors of even to quantum information processing. From a more fundamental science point of view it is remarkable that graphene allowed the predictions of quantum electrodynamic to be tested in a solid state-system because of its unusual linear electron excitations. Remarkably enough, electronic excitation close to the Fermi level strongly reminds us of the one exhibited by massless Dirac fermions. The 2D structure of graphene can also be exploited as a support to depose atoms, molecules, or aggregates; in some cases it has been shown that the interaction with graphene was able to significantly change their properties. These few and simple considerations certainly can justify the extremely high interest paid nowadays to graphene chemistry and physics. But unusual and remarkable properties are not only the domain of extended and formally infinite graphene sheets. Indeed, finite nanostructures can exhibit novel and fascinating properties due to the confining effects and the finite size. First of all graphene nanostructures, ribbons or island, show an accumulation of electronic density at the border, giving raise to the so-called edge states [4]. The presence of such states can strongly modify the physical and chemical behavior of such materials and has been demonstrated by different research groups, both theoretically and computationally. Moreover a very strong correlation between the geometry of the graphene nanostructure and the properties of their edge-state exists, and the global shape of the island indeed can modulate its electronic properties and structure in a very impressive way. For instance, the very strong difference in the properties of armchair and zig-zag edges structures has been the subject of quite a number of interesting publications. It is useful to cite here that the perspective of achieving a high control on the precise shape of graphene nanostructures and also of the type and nature of their edges not only is a theoretical fascinating suggestion but is becoming a sounding reality ready to be exploited. Indeed in the early years of the, yet still recent, graphene era, nanoscale systems were built by a top-down approach basically consisting in stripping out graphite sheets. Obviously such a procedure offered very limited, or even inexistent, control on the shape of the product. Today, on the contrary, we may assist to the emergence of much more sounding so-called bottom-up techniques in which nanoislands are built by precise deposition on metallic surfaces. These techniques generally allow a very good control on the shape of the resulting product that can be built with atomically defined edges. Quite recently, as suggested by Fürst et al. [5], the possibility of creating periodically perforated graphene structures (the so-called graphene antidot lattice) has been evoked to obtain even much more enhanced and diverse electronic and magnetic properties. Indeed complex graphene structures, having hexagonal, rhomboidal, or triangular shape, can exhibit extremely complex, yet predictable, electronic, and magnetic properties that make them ideal candidates, to rationally design organic giant nanomagnets, also exploiting the possibility offered by the different connectivity of different graphene units [6, 7]. These kinds of structure have recently attracted both experimental and theoretical consideration. For instance, Lin and coworkers [8] proposed a new route of synthesis of hexagonal and triangular (and also rectangular) graphene nanoflakes from carbon nanotubes and characterized them using photoluminescence. Pelloni and Lazzeretti [9] looked at the density response to an external magnetic field of triangular, rhombus, hexagonal, and rectangular structures using the polygonal current model [10]. Jamaati and Mehri [11] studied the influence of the edge length on the conductance of rhombuses nanoflakes. Mansilla Wettstein and coworkers [12] used a DFTB approach to simulate absorption spectra of triangular and hexagonal graphene nanostructures and evidenced the impact of the shape of the edifice. Many of these interesting properties belonging to finite graphene nanostructure arise from the presence and nature of their edge states [13]. Indeed the electronic density accumulation to the borders gives raise to a high density of electronic states close to the Fermi level. This in turn induces a high multireference character that may results in the presence of a very rich low energy electronic spectrum. Different states of different multiplicities can indeed happen to be the ground states, while even the lowest multiplicity ones can be described as open-shell states, with the electron close to the Fermi level being unpaired, and coupled ferromagnetically or antiferromagnetically to the others. The spin multiplicity of the actual ground state and hence the ferromagnetic or antiferromagnetic nature of the nanoisland will be dictated by the shape of the nanostructure. An easy way to formalize their magnetic properties is to recognize that the graphene lattices can be decomposed in two sublattices; therefore one will have to deal with and carbon atoms. Note that centers will only have atoms as first neighbors and vice versa. This implies that, for instance, zig-zag edges are all composed of the same type of atoms, while armchairs edges alternate between and atoms. Subsequently one should recall Lieb’s theorem [14] that states that the spin multiplicity of a given structure will be given by the balance between the number of atoms ( or ) belonging to the two sublattices Note that in case one uses the simple tight binding approach or single-reference treatment with a minimal basis set, the difference will also be equal to the numbers of degenerate eigenvalues lying at the Fermi level, that is, whose energy is exactly zero. Note also that Lieb’s theorem implies that two-carbon center will be ferromagnetically coupled if they belong to the same lattice and antiferromagnetically coupled if they belong to the opposite. Finally it has to be noted that magnetization arising from edge states in case of large islands and because of the high number of states close to the Fermi level may give rise to spin instability. Recently some of us have shown that coherently with Lieb’s theorem triangular structures have high-spin ground states and are ferromagnetic, while hexagonal and rhomboidal ones are open-shell low-spin structures of very high multireference character. Usually graphene nanostructures are studied at semiempirical level, or when at ab initio level using DFT methodologies, also because of the size and computational cost required to achieve large cluster sizes. On the contrary the presence of high-correlated structures giving rise to a very rich and subtle low energy spectrum, with an impressive magnetic behavior, would call for a wavefunction based multireference treatment. In this contribution triangular, T_*n*, hexagonal, H_*n*, and rhomboidal, R_*n*, structures, where is the number of hexagons per side, will be studied at ab initio level using multireference perturbation theory formalism, a level of theory that represents the best balance between accuracy and computational cost in treating both static and dynamic correlation. On the other hand the energy band structure will be reported also at semiempirical (Hückel) level. This will allow reaching a very high cluster size and, therefore, achieving the limit of infinite structures. The different character of the density of states close to the Fermi level will be considered also to easily rationalize the emergence of magnetic properties. In particular the presence of a gap at Fermi level, whose value strongly diminishes with the system size, will be underlined in the case of low-spin systems, while a continuous band structure is shown for the other structure. Moreover the evolution of the magnetic coupling with the island shape and size will also be particularly taken into account. In Figures 1, 2, and 3, the largest structures considered in this work, for each of the three different types (i.e., H_6, R_7, and T_7), are illustrated. and sublattices are highlighted in different colors. It appears that the hexagons (H) structures have , to which a singlet ground state corresponds. Also the rhombus (R) structures are associated with a zero value of , but this happens by compensation between the two different equilateral-triangle halves of the rhombus. This explains the singlet open-shell character of the corresponding ground states. Finally, the triangles (T) have a nonzero value of and a high-spin open-shell ground state.