Electronic, magnetic and transport properties of graphene ribbons terminated by nanotubes

The atomic structure of graphene edges is important for the determination of the electronic and magnetic properties of graphene, especially for narrow graphene nanoribbons [1–12]. The natural termination of graphene is given either by zigzag (ZZ) or armchair (AC) edges. Theoretical studies [3] found that the minimal energy structure is instead given by a reconstruction of the ZZ edges to form pentagons and heptagon (57) but this structure has been only rarely observed [13, 14], possibly due to a large free energy barrier [15]. A recent theoretical work [16] on the stability of different graphene edge structures has shown that graphene edges can also fold back on themselves and reconstruct as nanotubes, with low formation energy (see the atomic structures in figure 1). Previous theoretical work had considered such a structure among many other possible configurations made by a combination of nanotubes and graphene nanoribbons [17, 18], suggesting that such a structure could be formed in solution.

Zoom In Zoom Out Reset image size

In this paper, we show that, besides protecting the edges from contamination and reconstructions, nanotubes at the edges may lead to magnetism and are not detrimental for the electronic mobility despite the row of sp³ hybridized atoms at the ribbon–tube junction. We study the electronic and magnetic properties of these systems by a combination of DFT and large-scale tight-binding (TB) simulations of transport properties. Our calculations suggest that these systems could be used for a variety of applications that we sketch in figure 2.

Zoom In Zoom Out Reset image size

We consider systems formed by a nanoribbon terminated on both sides by the same AC or ZZ nanotube. We note that a ribbon with AC edges is terminated by ZZ nanotubes and a ribbon with ZZ edges is terminated by AC nanotubes. Nanoribbons terminated by AC nanotubes present interesting magnetic properties. By rolling the ZZ edges of a nanoribbon, two types of AC nanotubes can be formed, as shown in figure 1. If the atoms at the nanoribbon ZZ edge scroll and bind to the same sublattice sites within the nanoribbon, the formed AC nanotube has mirror symmetry with respect to the nanoribbon plane; if the bonding sites belong to opposite sublattice, there is no such kind of symmetry (compare figure 1(b) with (a)). We call these two cases symmetric and asymmetric which correspond to armchair and armchair-like in [16], respectively. The common point of these two cases is that the sublattice symmetry is broken, because all the sp³ hybridized carbon atoms at the junction belong to one sublattice as shown in figure 3. Due to the Lieb theorem [8, 12], this gives the possibility of spin polarization around the junctions. Since the theorem applies to the Hubbard model, accurate calculations for the real system are necessary to investigate this possibility. Besides these symmetry considerations relevant for graphene, a general condition for magnetism that derives from the Stoner criterion is the presence of peaks in the density of states (DOS) at the Fermi energy for non-spin-polarized calculations. As we show in the following, both symmetric and asymmetric ZZ nanoribbons terminated by AC nanotubes present such peaks and therefore are good candidates for magnetism since they satisfy both the Lieb and Stoner requirements.

Zoom In Zoom Out Reset image size

In order to study the magnetic properties, we performed (spin-polarized) DFT calculations by SIESTA [20–22]. We used generalized gradient approximation with the Perdew–Burke–Ernzerhof parameterization [23] and a standard built-in double-ζ polarized [24] basis set to perform geometry relaxation.

In figure 4, we show the non-spin-polarized band structure and DOS of both symmetric and asymmetric ZZ nanoribbons with AC nanotubes at the edges. We see that indeed sharp peaks at the Fermi energy in the DOS are present as a consequence of the flat band located at this energy. This flat band resembles the one due to dangling bonds in ZZ nanoribbons. In [3], it was shown that the 57 reconstruction of the ZZ edges could split this band and shift it away from the Fermi energy. The termination with nanotubes leaves a band at the Fermi energy if no spin polarization is allowed. For the symmetric case, it is easy to visualize the character of this band as a p_z state decaying from the junction both in the nanoribbon and in the nanotube, as shown in figure 4(e). Instead of a change of bond character as for the 57 reconstruction, for our system it is the magnetic polarization that removes this state from the Fermi energy. In figure 5, we show the spin-polarized band structure and DOS, showing the splitting of the flat band at E_F into spin up and spin down levels and the consequent disappearance of the peaks at the Fermi energy in the DOS. We found that, in both the symmetric and asymmetric cases, there is spin polarization near the ribbon–tube junction, i.e. near the sp³ hybridized carbon atoms. Note that the bond distance of these four-fold coordinated atoms is 1.52 Å like in diamond. The spin polarization is mainly located in the nanoribbon for the symmetric case and within the tube for the asymmetric case (see the isosurface plot of the spin density together with its symbolic representation in figures 1(c)–(h)). The up/down spins are distributed over the A/B sublattices, respectively. The label m_max indicates the atom with the highest magnetic moment. A possible explanation for the localization of the spin polarization can be found by realizing that the largest polarization is on the sp² atoms at the junction in between two sp³ atoms (see red atoms in figure 3). The atoms with this particular configuration are either inside the nanotube or inside the nanoribbon and this corresponds to the location of the spin polarization. We are also tempted to conjecture that the localization results from a frustration mechanism. In fact in the symmetric case, where the polarization is located in the nanoribbon, the sp³ atoms break the bipartite symmetry in the ribbon (making magnetization possible) whereas for the asymmetric case, the sp³ atom breaks the bipartite symmetry in the nanotube.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

For the symmetric case, the value of the spin polarization increases with increasing nanoribbon width, and saturates at 1.50μ_B per unit cell when the nanoribbon width is wider than 16 ZZ rows; for the asymmetric case, the spin polarization is always 1.50μ_B per unit cell, irrespective of the nanoribbon width and nanotube radius (see table 1).

For the asymmetric case, the spins are located inside the two nanotubes and therefore the exchange interaction between opposite edges is negligible. For the symmetric case, the spins on the two edges are coupled antiferromagnetically, similarly to hydrogen terminated graphene edges [25, 26]: for the structure shown in figure 1(a), the energy of antiparallel spin configurations (see figure 1(g)) is 22 meV per unit cell lower than for parallel configurations. This sizeable coupling across the ribbon makes the symmetric systems promising as spin valve devices [27]. In figure 2(a), we show a configuration similar to that proposed for dumbbell graphene structures on the basis of the Hubbard Hamiltonian in the mean field approximation [28]. A gate could be used to bring the system from the antiferromagnetically coupled state to the ferromagnetically coupled excited state, favoring spin transport from one nanotube to the other across the ribbon. Moreover, for both the symmetric and asymmetric cases, the magnetic moments along the ribbon–tube junction are qualitatively similar to the case of ZZ edges of nanoribbons. Therefore, high magnetoresistance could be expected, as proposed in [19] for nanoribbons with ZZ edges, by applying magnetic fields of different sign at the ends of the nanoribbon. A sketch of this device for our systems is shown in figure 2(b).

The spin-polarized DOS reveals other features of interest for spintronics related to half-metallic character. In figures 5(a) and (b), we show the spin-polarized DOS for both the symmetric and asymmetric cases, respectively. We see that the symmetric case is metallic for both spins in the whole range of energy. The asymmetric case, instead, is a half-metal near the Fermi energy E_F; namely it is metallic for spin up and insulating for spin down. The half-metallic character of our systems provides opportunities as spin filters without the need of external electric fields [29], magnetic fields [30], ferromagnetic strips [31], impurities [32–34] or defects [35, 36]. Furthermore, there is the opposite half-metallic character at higher energies. Around 0.4 eV, there is insulating character for spin up and metallic character for spin down. As sketched in figure 2(c), a gate along the ribbon could be used to switch between these two half-metallic energy regions and affect selectively the spin transport.

We come now to the case of ZZ nanotubes shown in figures 1(i) and (j). For this case, there is only one type of ribbon–tube junction that preserves sublattice symmetry, implying that there is no magnetization or midgap states [8, 12]. The electronic structure and transport properties, however, strongly depend on the AC ribbon width and on the ZZ tube radius. In TB models, an AC nanoribbon is metallic if the number of AC rows is equal to 3l + 2, where l is a positive integer, and semiconducting otherwise [1, 37]. Furthermore, ZZ nanotubes are metallic for index equal to 3l [38]. In more general models, the properties of AC nanoribbons and ZZ nanotubes may differ from the ones predicted by TB, due to possible self-passivation of the edges for nanoribbons and for the σ–π band mixing for small nanotubes [38]. By using DFT calculations, we found that our joined system becomes a semiconductor with a gap of the order of a few hundreds of meV if both the nanoribbon and the nanotube are semiconducting. The energy gap as a function of geometry is shown in figure 6. The size of the nanotube has to be large enough for the opening of a band gap (figure 6(a)). For the joined system with semiconducting ZZ nanotubes, there is a clear periodicity (three ZZ rows) in the dependence of the energy gap on the nanoribbon width (figure 6(b)). For the studied cases, the value of the gap varies between 30 and 600 meV.

Zoom In Zoom Out Reset image size

Since we want to calculate the transport properties by means of a simpler model, suitable for large samples, we have also calculated the energy structure of our system by π-band TB calculations where we consider only the nearest-neighbor hopping t = 2.7 eV between the carbon atoms [39, 40] for all bonds. The comparison between the gaps calculated by TB and DFT shown in figure 6(a) gives the same periodicity and qualitative agreement. For the case of an AC nanoribbon, with 15 rows terminated by (8,0) ZZ nanotubes, we compare in figure 7(a) the TB and DFT DOS, which again are in qualitative agreement thus supporting the validity of the transport calculations we show next. The electronic transport properties of a semiconducting nanoribbon with or without nanotube terminated edges are obtained by using large-scale TB simulations with about 30 million carbon atoms [39, 40]. In figure 7(b), we show that the electronic mobility parallel to the edges as a function of charge density is quite similar in these two cases. The mobility of the joined system is about 2250 cm² V⁻¹ s⁻¹ at charge density n_e ∼ 10¹³ cm², which is only slightly smaller than that of the AC nanoribbon at the same charge concentration.

Zoom In Zoom Out Reset image size

In summary, we have studied the electronic and magnetic properties of graphene nanoribbons with three types of nanotube terminated edges. The spin magnetization is found to be 1.5μ_B per unit cell in the ground state of both symmetric and asymmetric AC nanotube terminated edges. For symmetric AC nanotube terminated edges, the spin density is located in the ribbon whereas, for the asymmetric case, it is located within the tube. In the ZZ nanotube terminated edges, there is a band gap opening of the order of a few hundreds of meV if the constituent tube and nanoribbon are both semiconducting. The conductivity and mobility in the presence of ZZ nanotube terminated edges are comparable to those of the AC nanoribbon itself.

Our calculations suggest that these systems are not only advantageous because the edges are protected against any kind of chemically induced disorder but also because, by tailoring the ribbon/tube structure, they offer a wealth of possible applications for transport and spintronics.

The support from the Stichting Fundamenteel Onderzoek der Materie (FOM) and the Netherlands National Computing Facilities foundation (NCF) is acknowledged. We acknowledge support from the EU-India FP-7 collaboration under MONAMI and the grant CONSOLIDER CSD2007-00010.