Advances in Condensed Matter Physics

Volume 2015 (2015), Article ID 475890, 14 pages

http://dx.doi.org/10.1155/2015/475890

## The Electronic Structure of Short Carbon Nanotubes: The Effects of Correlation

^{1}Laboratoire de Chimie et Physique Quantiques, IRSAMC, Université de Toulouse et CNRS, 118 Route de Narbonne, 31062 Toulouse Cedex, France^{2}Université de Lorraine, Nancy, Théorie-Modélisation-Simulation, SRSMC, Boulevard des Aiguillettes, 31062 Vandœuvre-lès-Nancy, France^{3}CNRS, Théorie-Modélisation-Simulation, SRSMC, Boulevard des Aiguillettes, 31062 Vandœuvre-lès-Nancy, France

Received 20 April 2015; Revised 16 July 2015; Accepted 5 August 2015

Academic Editor: Ashok Chatterjee

Copyright © 2015 Vijay Gopal Chilkuri 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 zig-zag nanotubes of various diameters and lengths. The vertical energy spectra of such nanotubes are presented, as well as their spin multiplicities. The calculations performed using the *tight binding* approach show the existence of quasi-degenerate orbitals located around the Fermi level, thus suggesting the importance of high-quality *ab initio* methods, capable of a correct description of the nondynamical correlation. Such approaches (Complete Active Space SCF and Multireference Perturbation Theory calculations) were used in order to get accurate ground and nearest excited-state energies, along with the corresponding spin multiplicities.

#### 1. Introduction

Besides the two lightest elements, hydrogen and helium, carbon is one of the most widespread elements in the universe and one of the best known ones. Indeed, its three-dimension allotropic forms, diamond and graphite, are well known since the antiquity. For this reason, the recent discovery of new low-dimensional allotropic forms, such as fullerenes, carbon nanotubes, and graphene, came out rather unexpected [1, 2]. In a few years, a completely new branch of science was born, whose scientific and technological impact can hardly be overestimated. Indeed, these findings had, and still have, an enormous importance in the discovery of novel and advanced materials and have been one of the key factors in the development of nanoscience. The peculiar properties of graphene (that concern both the “infinite” ideal sheet and finite nonoislands) raise new challenging theoretical problems, whose understanding and modeling will bring significant insight in our comprehension of the structure of matter. Nanotubes, which can be obtained from a graphene stripe by enrolling it along longitudinal axes, present an even larger spectrum of interesting behaviors. It is clear that a better understanding of these systems would make our knowledge of the general properties of solid-state physics and chemistry much deeper.

The new allotropic forms of carbon share the particularity of being all of low dimensionality. Indeed, the almost spherical fullerene [3] can be considered as a quasi-zero-dimensional (0D) structure, while the graphene one-atom-thick surface is a strictly two-dimensional (2D) structure. Between these two extreme cases are located the carbon nanotubes that can be considered as essentially one-dimensional (1D) materials. The presence of carbon -conjugated lattice, usually resembling a honeycomb hexagonal lattice, is the common feature to all these forms and is at the origin of their striking properties. Indeed, the strength of the carbon-carbon sigma bond and the rigidity of the resulting structure confer graphene and nanotubes a great stability and rigidity and make them appealing to be used as building-block fibers for materials having to resist considerable stress [4]. Another remarkable aspect that should be stressed is the hydrophobicity and the resistance of these materials, a fact that makes them ideal candidates for drug delivery and vectorization [5]. In addition to these* mechanical* properties, which are already extensively exploited nowadays, there are also peculiar and surprising* electronic* properties exhibited by these materials. For instance, graphene can be considered as a zero-gap semiconductor showing an infinite electronic mobility at the Fermi level. Carbon nanotubes [6], on the other hand, which are formally obtained by enrolling a graphene sheet along a helicoidal symmetry axis, show a vast variety of electronic behaviors. They exhibit extremely interesting properties, with a quite complex and rich structure-property correlation pattern. In particular, depending on the way the original graphene sheet is enrolled around the axis, nanotubes can present or not chirality, and their conductivity properties can vary dramatically. So different classes of nanotubes can present a behavior that passes from metallic to low-gap semiconductor, to real semiconductor materials. This peculiar characteristic can efficiently be exploited in the field of opto- and molecular-electronics devices: the potential performance and efficiency of carbon based nanostructures could potentially induce a revolution in that field, comparable to the one produced by silicon based microdevices [1, 7].

Because of the scientific and technological importance of carbon nanotubes, a huge amount of scientific work, both theoretical and experimental, has been devoted to the study of the properties and the electronic structure of these systems. Up to now, however, the great majority of such studies were conducted on infinitely extended systems. Moreover, in many cases, the Hamiltonians used were restricted to semiempirical models,* ab initio* calculations being more often performed at density functional theory level only. However, even if the study of extended systems deserves a considerable importance, both from a theoretical and a practical point of view, finite-size effects occurring in short finite clusters can also reveal new and unexpected properties, which can strongly modify the physical behavior with respect to the infinite system. In particular, it is known that graphene nanoclusters and open carbon nanotubes can present the so-called edge orbitals that are partially filled orbitals whose electron density is mainly concentrated on the border regions of the system. A peculiarity of these orbitals is given by the fact that their energy lies close to the Fermi level; that is, they are the frontier orbitals of the system. Depending on the geometry of the cluster, the edge orbitals show interesting degeneracy patterns. These features will imply the presence of different types of open-shell electronic structures, possibly inducing magnetic behaviors, characterized by a high-spin ground state. Moreover, there is the possibility of a closing of the Fermi level gap, to give rise to a metallic behavior in the limit of large systems.

The presence of an open-shell structure will also induce a strong multiconfigurational nature of the low-lying states of the system (in particular, of those of lower multiplicity). This fact implies that multireference methods are required to properly take into account the effects of the static correlation [8]. Note, as previously evoked, that the magnetic behavior is strongly dependent on the geometry of the cluster and can be rationalized by invoking the Ovchinnicov Rule [9, 10] and Lieb’s theorem [11]. For instance, it has been argued that triangular-shape clusters are ferromagnetic, while hexagonal shapes are expected to be diamagnetic and show a gap at the Fermi level. Note also that the magnitude of the magnetic coupling between the states can be tuned by varying the size of the cluster. All these important characteristics can been exploited by connecting in a network different graphene clusters characterized by different magnetic properties. Let us also cite the fact that the required precise control on the shape and size of the cluster can nowadays be achieved by bottom-up techniques based on the precise deposition of carbon atoms on different surfaces. In fact, such techniques have already allowed to produce very specific graphene clusters, and even graphene antidots [12]. Finally, it is also noteworthy to remind that carbon nanostructures can be used to host and stabilize magnetic linear structures and that the interaction with conjugated carbon structures can also strongly modify the magnetic behavior of the linear aggregate.

In the present contribution, we want to extend our analysis of border effects performed on graphene nanoclusters to the domain of nanotubes. In particular we will concentrate on zig-zag nonchiral nanotubes, and we will consider only open structures that present terminal edges saturated with hydrogens. Indeed, this simple choice allows to keep all the carbon atoms -hybridized and to limit the edge effects. This is often done in theoretical studies even if in most experimental samples there are no terminal hydrogens.

Infinite nanotubes are uniquely characterized (apart from the sign of helicity, if they are chiral) by a pair of nonnegative integer numbers, and (with ), and the corresponding nanotube is denoted as . These two numbers describe how a graphene sheet is enrolled around the helicoidal axes in order to generate the infinite nanotube. As the two extreme cases we find nonchiral nanotubes: the zig-zag nanotubes are of the type , while the corresponding armchair structures are of type . When is different from both 0 and , we have chiral nanotubes. If finite nanotubes are considered, the situation is much more complex, since one must consider not only the length of the nanotube itself, but also the details of its two borders. This is particularly true for chiral nanotubes, whose helicity prevents the possibility of cutting the extremities in such a way to respect the nanotube symmetry along the central axis. Nonchiral nanotubes, however, can be truncated in such a way that the two edges respect the nanotube symmetry. In zig-zag nanotubes (the systems that are the object of this work) the integer number indicates the number of contiguous hexagon units that we find on a circular path around the tube; the value, on the other hand, is redundant, since it is identically equal to 0. Finite-length zig-zag nanotubes are characterized by , as the corresponding infinite systems, and by the* length* of the nanotube. We will indicate by this length, and, in order to avoid confusion with the well-established notation , we indicate by a fragment of a (i.e., zig-zag) nanotube having length .

To study the nature of edge orbitals and their influence on the global electronic structure of the nanostructure, as well as its evolution with the different nanotube structural parameters, we will apply two different computational strategies: a very simple semiempirical Hückel approach, which can treat a large variety of different structures, and highly correlated multireference wave-function-based approaches, able to give quantitative results but whose use is restricted to relatively small systems. The first crude approximation allows us to obtain a qualitative overview of the density of states and in particular to characterize the orbital degeneracies at the Fermi level. Note that this approach will also allow us to consider very large aggregates, that is, to come close to the infinite system limit. The wave-function-based approach, on the other hand, takes care of the static and dynamic correlation and is able to study the low-lying spectrum of the magnetic states, evaluating its evolution with the geometry and the structural parameters. To the best of our knowledge, this is the first contribution applying high level Multireference Perturbation Theory (MRPT) to the study of this kind of systems. A similar approach was already applied to closed polyacenes structures [13] whereas most of previous works on nanotubes [14–22] or nanoribbons [23] principally rely on density functional theory or tight binding results. The main conclusion of studies on There also have been some studies in the solid-state community on the energy spectrum [24] and properties of single walled nanotubes [25] using sophisticated analytic tools. Similar structures as those considered consist in an antiferromagnetic coupling of two electrons localized in the edge orbitals.

This paper is organized as follows. In Section 2, Theoretical Methods, we briefly describe the methods that we employed in this study, and we justify their use. The computational details of the calculations are given in Section 3, in order to permit the reproduction of our calculations in future further studies. In Section 4, we present the nanotube structure and symmetry, along with the particular terminology used in this work. Results at the Hückel and* ab initio* level are presented and discussed in Section 5. Final conclusions are drawn in Section 6, together with plans for future works.

#### 2. Theoretical Methods

In this section, we briefly describe the theoretical methods and the computational approaches used in the present work. The computational details concerning both the* ab initio* and the* tight binding* approaches will be discussed in detail in the next section, in order to facilitate the reproduction of our results.

Carbon nanotubes, as it is the case of most graphene-derived nanostructures, show the presence of edge molecular orbitals (MO) that are neither doubly occupied nor totally empty in the electronic ground state (see, e.g., [12]). The simplest Hamiltonian that is able to give a qualitative description of the electronic wavefunction of -hybridized carbon structures is the Hückel Hamiltonian. It is a tight binding Hamiltonian that was developed in order to describe small polyenes and in general aromatic compounds. At Hückel level, only one atomic orbital per carbon atom is considered (the orbital that is locally orthogonal to the orbitals), and this defines the system. Correspondingly, only one electron per atom is considered. Despite its simplicity, the Hückel Hamiltonian is capable of grasping the essence of the electronic structure of these systems, and their low-lying energy spectrum obtained at this level is remarkably correct. It is worth noticing that the pioneering 1947 paper on graphite (and graphene) written by Wallace describes results obtained with this type of approach [26].

As a general strategy that we employed in previous studies on other carbon nanostructures [13, 27], we performed preliminary tight binding calculations by using the model Hückel Hamiltonian [28], in order to obtain a first approximation of the energy spectrum at one-electron level. This is very important for subsequent calculations at many-electron level by using an* ab initio* Hamiltonian, in order to clearly identify the orbitals that form a quasi-degenerate partly occupied manifold. A proper* ab initio* treatment of this group of quasi-degenerate orbitals requires the introduction of the* nondynamical correlation* [29], and this is most conveniently done by the use of the Complete Active Space Self-Consistent Field (CAS-SCF) formalism [30]. In this approach, the correlation of the electrons distributed among the quasi-degenerate orbitals (the* active* orbitals) is fully considered at a configuration interaction level. At the same time, the shape of* all* the molecular orbitals (active or not) is fully optimized self-consistently. It is known that a CAS-SCF approach is suited for a qualitative description of the low-lying energy spectrum of open-shell systems (magnetic structures, mixed-valence systems). For quantitative results, the introduction of the* dynamical correlation* is also required, since the weights of the Slater determinants belonging to the quasi-degenerate manifold can be strongly modified by the interaction. Among the possible multireference approaches, the Quasi-Degenerate Perturbation Theory is the only one that is capable of treating these relatively large systems. In particular, contracted methods have the advantage of a weak computational dependence on the dimension of the active space and can therefore be used in the present case. In this work, we chose the n-Electron Valence Perturbation Theory at second-order level (NEVPT2), because of its capability of dealing with the intruder-state problem without the need of introduction of arbitrary parameters [31, 32].

In doing a CAS-SCF calculation, the choice of the guess starting orbitals plays a key role in the final result. Indeed, being a nonlinear optimization algorithm, CAS-SCF admits a huge number of different solutions, sometimes not presenting a particular physical interest. It is also frequent, because of this nonlinear character of the formalism, to have instabilities that lead to symmetry-breaking phenomena in the case of strictly degenerate states, as the ones that characterize nanotubes having non-Abelian symmetries. In the present work, we performed High-Spin Restricted Open-Shell Hartree-Fock (ROHF) calculations on the different systems, by choosing as single-occupied orbitals precisely the orbitals that show a quasi-degenerate character at Hückel level. Once the manifold of quasi-degenerate orbitals has been identified, we performed frozen-orbital CAS-SCF (i.e., CAS-CI) calculations on this active space. It is known that, for states having nonionic character, the resulting energy spectrum is very close to the CAS-SCF one, at a considerably lower computational cost.

#### 3. Computational Details

In the next subsections, we describe separately the* tight binding* and* ab initio* approaches.

##### 3.1. Tight Binding

All computations at tight binding level (Hückel calculations in the language of chemists) have been performed by using a specific home-made code[33–35]. The Hückel Hamiltonian has been constructed starting from the connectivity of each carbon atom, by defining, as usual, where is 1 if sites and are connected in the nanostructure skeleton and 0 otherwise. The value of the hopping integral should depend, in principle, on the distance between the two connected atoms and . However, since the C-C bonds have very similar length in all of our structures, we assumed the same values of for all the topologically connected pair of atoms. In our calculations, this parameter was fixed to the arbitrary value . We remind that is sometimes called the hopping integral in the physics literature, where it is usually indicated as . In the section Results and Discussion, we report the energy spectrum of several nanotubes, computed at the Hückel level. This is done as a function of the orbital number. In order to facilitate the comparison between systems having a different number of orbitals, the orbital numbers are normalized in the segment .

##### 3.2.
*Ab Initio*

All the systems have been studied using the minimal STO-3G Gaussian basis [36] for both carbon and hydrogen. Although this basis set is certainly too small to give quantitatively reliable results, it is known to reproduce correctly the qualitative behavior of many organic and inorganic systems. Its reduced size, on the other hand, permits to investigate at a correlated* ab initio* level systems whose size is relatively large. However, in order to explore the effect of a more realistic basis set, a limited number of structures representative of each class of nanotubes have been studied with a larger basis set, the cc-pVDZ correlation consistent basis set of Dunning [37]. These are the structures characterized by and for which a contraction for C and for H were used. This corresponds to a valence double-zeta plus polarization (vdzp) basis set.

The systems’ geometries have been optimized at Restricted Open-Shell Hartree-Fock (ROHF) level for the lowest-energy state. This means the triplet state for the nanotubes characterized by an even value of () and the quintet state for the nanotubes characterized by an odd value of ().

During geometry optimization, the or symmetries have been imposed depending on the nanotubes classes. Thus, all the coordinates have been relaxed allowing the nanotube to lose its cylindrical shape. The only remaining constraint was the order of the rotation axis. Actually, due to restrictions of the Molpro quantum chemistry package [38], optimization has been carried out using the appropriate Abelian subgroups. Even if this may result in symmetry breaking of the electronic wavefunction, it never appeared for the systems considered in this work. For all the nanotubes, it appears that only slight distortions from the regular ideal cylindrical shape have been obtained. Even the terminal hydrogens just slightly point towards the axis of the nanotube. All the detailed geometries are given in the Supplementary Material available online at http://dx.doi.org/10.1155/2015/475890. At these optimized geometries, the lowest states have been studied at Complete Active Space Self-Consistent Field (CAS-SCF) level [39]. Then the effect of dynamical correlation was introduced at Multireference Perturbation Theory (MR-PT) level, using the* partially contracted* version of second-order n-Electrons Valence Perturbation Theory (NEVPT2) formalism [40–42]. In all the CAS-SCF calculations, the active space has been selected by choosing the orbitals that are strictly degenerated or quasi-degenerated, with the Hückel Hamiltonian, at the Fermi level. This means that our active spaces are either CAS(2,2) (for the tubes) or CAS(4,4) (for the tubes). The geometries have been optimized at ROHF level for the high-spin wavefunction. The orbitals obtained thereof have then been kept frozen for the CAS-SCF calculations on the other spin multiplicities. In this way, our calculations are actually of CAS-CI [43, 44] type which in our case give energies close to CAS-SCF ones. The NEVPT2 formalism has then been applied to these CAS-CI wavefunctions in order to recover the dynamical correlation.

Obviously, our geometrical constraints prevent Jahn-Teller distortion which could be expected in the case of degenerate orbitals. In this preliminary work, we decided not to consider this possibility. However, due to the stiffness of the nanotube backbone we believe these distortions to be of small size.

#### 4. Electronic Structure of Carbon Nanostructures

The net of carbon atoms forming a graphene layer is the conceptual starting point to produce most of the carbon -hybridized nanostructures. A crucial feature to rationalize the relation between structure and electronic properties, and, more generally, to study finite-size effects in these structures, is to recognize that the honeycomb graphene skeleton forms a* bipartite lattice* [12], with two compenetrating triangular sublattices and . Each carbon atom belonging to a graphene nanostructure is associated with one of these two sublattices, and one can speak of -type and -type graphene atoms, or centers. A lattice is bipartite if each -type atom has only -type nearest neighbors, and vice versa. So, for instance, zig-zag edges are composed of atoms that all belong to the same sublattices. A necessary condition for a lattice to be bipartite is the absence of odd-number carbon cycles. In order to rationalize the emergence of magnetic properties in graphene nanosystems, one can recall Lieb’s theorem [11]. This theorem, also known as the theorem of itinerant magnetism and usually applied within the framework of the Hubbard one-orbital model, is able to predict the total spin of the ground state in bipartite lattices. In particular, one can see that an imbalance on the number of atom in one sublattice results in a magnetic ground state with spin where and are the numbers of atoms of - and -type, respectively. It is also possible to show that, for a Hückel Hamiltonian, is also the number of eigenvalues equal to zero. It is important to note that the magnetization originates from localized edge-states that give also rise to a high density of states at the Fermi level which in turn can determine a spin polarization instability. Moreover the relation between the unbalanced number of atoms and the ground state spin implies that two centers will be ferromagnetically coupled if they belong to the same sublattice and antiferromagnetically coupled if they do not [45].

We consider now the symmetry of the nanotubes. It is easy to see that these structures fall into two classes, according to the parity of the length . The tubes have symmetry if is odd ( tubes) and symmetry for even ( tubes). In all these cases (with the partial exception of the case , which is however too narrow to form a tube) we are in the presence of non-Abelian symmetry groups. We recall that our* ab initio* calculations have been performed by using Abelian subgroups of the full point group of the system. According to the parity, even or odd of and , we have the following Abelian subgroups for , for , for , and for .

We call* edge carbon atoms* the carbon atoms that are saturated with hydrogens. The role of these edge atoms is crucial in order to understand the finite-size effects in carbon nanotubes. The edge carbon atoms of the tubes of type can support molecular orbitals (MO) of -nature, having alternating sign on any pair of consecutive hexagons. Let us consider the MO combinations that are localized on one of the tube extremities only. At Hückel level, it is straightforward to verify that both these alternating edge combinations are eigenfunctions of the one-electron Hamiltonian corresponding to eigenvalues equal to zero. Therefore, the Hückel energy spectrum presents a pair of degenerate orbitals at the Fermi level for all tubes, regardless of the parity of . The situation is different as far as the* ab initio* Hamiltonian is considered, since these two localized edge combinations are no longer exact molecular orbitals of the Fock Hamiltonian. As a result, at* ab initio* level the exact degeneracy can be lost. In particular, it turns out that while the two edge orbitals are exactly degenerate for tubes, the degeneracy is only approximated in the case of the structures. This point can be easily understood by symmetry considerations. In fact the two localized edge orbitals are interchanged by some of the symmetry operations of the point group of the system ( for and for tubes). For this reason, they give rise to a representation of the molecule symmetry group. In the case of tubes, the representation is irreducible, and it still corresponds to two* exactly degenerate* molecular orbitals of the system (e.g., see Figure 6, where the two degenerate orbitals have symmetry). In the case of the tubes, on the other hand, the two combinations belong to two different irreducible representations of the Abelian symmetry group of the system. The two orbitals are only approximately degenerate (see Figure 7, where the two orbitals have and symmetries, resp.). This argument explains the remarkable difference between the* tight binding* and* ab initio* energy spectra in the Fermi level region for the tubes.

#### 5. Results and Discussion

##### 5.1. Tight Binding

Even if the tight binding approach represents a crude approximation of a chemical system, it can be extremely useful in order to sketch out some general tendencies and to elucidate the behavior of the different classes of compounds. This is particularly true in the present case, since a tight binding approach allows the treatment of very large systems due to its extremely reduced computational costs. It is possible, in this way, to explore the behavior of virtually “infinite” systems.

Short zig-zag nanotubes can be divided into four different classes, characterized by different symmetries and molecular-orbital patterns depending on the* parity* of and . The Hückel calculations for these classes permitted to characterize the different active spaces which were essentially confirmed at the* ab initio* level.

Nanotubes of the type have symmetry, while those of type have symmetry. At the Hückel level, both and have two exactly degenerate orbitals at the Fermi level, hosting two electrons. This is illustrated in Figures 1 and 2, where the energy spectra of and structures are reported. For the sake of comparison, the spectra of long, , or very wide, and , structures are also reported in the figures. None of these spectra shows the presence of an energy gap at the Fermi level, and the wide structures have even an* infinite* density of levels at the Fermi level. Nanotubes of the types and , on the other hand, have four quasi-degenerate levels (two quasi-degenerate pairs of exactly degenerate levels) near the Fermi level, hosting a total of four electrons. This fact is illustrated, for the and structures, in Figures 3 and 4. From these figures it appears that, in the case of very long nanotubes, four degenerate isolated orbitals, hosting a total of four electrons, are located at the Fermi level. As a systematic study of large systems at the Hückel level involves a digression from the present topic, a general analysis of the four classes of the nanotubes will follow.