boron-nitride nanomaterials via ab-initio calculations and elasticity theory

Carbon 99 (2016) 523e532

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Electronic and pseudomagnetic properties of hybrid carbon/boronnitride nanomaterials via ab-initio calculations and elasticity theory Farzaneh Shayeganfar a, Rouzbeh Shahsavari b, c, d, * Engineering Physics Department, Polytechnique Montr
eal, Montr
eal, Qu
ebec, Canada Department of Civil and Environmental Engineering, Rice University, Houston, TX 77005, USA c Department of Material Science and NanoEngineering, Rice University, Houston, TX 77005, USA d Smalley Institute for Nanoscale Science and Technology, USA a

b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 16 August 2015 Received in revised form 14 December 2015 Accepted 15 December 2015 Available online 23 December 2015

Low dimensional materials such as Boron Nitride nanotube (BNNT) and graphene are attractive for demonstrating several fundamental physical properties and development of novel technologies in nano/micro-scale devices. Although various 3D carbon-based architectures are reported via covalent connection of carbon allotropes, the introduction of analogous 3D hybrid carbon/BN allotropes and determining their exquisite junction-induced properties remain elusive. Here, we focus on mono- and double-layer hybrid graphene/BNNT and graphene/carbon nanotubes (CNT) architectures and explore their diverse junction configuration-induced electronic and pseudomagnetic properties via ab-initio calculations and elasticity theory. By introducing heptagonal and octagonal rings in the junctions, we find that the mismatch between the defected graphene and the BNNT/CNT diameters creates a bond strain at the junction, thus inducing a gauge field and pseudomagnetic field, which decay exponentially along the radial distance of the junction. Furthermore, our analysis of the band structures and density of states in hybrid double-layer architectures demonstrate that there exists a flat band and band dispersion near the Fermi level of graphene/CNT junctions, a feature not present in the graphene/BNNT junction due to the intrinsic wide band gap of BNNT. Finally, our size-effect study shows that while the band gap energy of heptagonal graphene/CNT junction and octagonal graphene/ BNNT junctions is sensitive to the nanotube length, this is not the case for octagonal graphene/CNT junctions due to the less perturbation of the electronic states of the valence bond (VB) in the octagonal graphene/CNT junctions. Together, these findings have important implications on science-based engineering of numerous hybrid carbon and boron nitride allotropes while significantly broadening the spectrum of strategies for fabricating new hybrid nanomaterials through covalent connection of dissimilar low-dimensional materials. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction Over the past few years, several efforts have been made to fabricate hybrid nanomaterials with modified or novel photonic properties that typically are more attractive than their individual constituents [1,2]. In the context of hybrid, low dimensional materials, a key focus has been on creating carbon-based structures by covalently connecting 0D (zero-dimensional), 1D and/or 2D parent

* Corresponding author. Department of Civil and Environmental Engineering, Rice University, Houston, TX 77005, USA. E-mail address: [email protected] (R. Shahsavari). http://dx.doi.org/10.1016/j.carbon.2015.12.050 0008-6223/© 2015 Elsevier Ltd. All rights reserved.

structures. Examples include carbon nanopeapod [3], carbon nanobuds [4], periodic graphene nanobuds [5], pillared graphene [6], and so on. The hybrid forms of BN are mainly fabricated in the group of ternary BCN materials by doping B and N in carbon systems or vice versa [7]. Examples comprise BCN-based nanotubes [8,9], BCN-based nanostructures [10] and pillared boron nitride [11]. Other classes of hybrid BCN materials are obtained via direct deposition of h-BN on graphene [12], or direct growth of graphene on h-BN [13], or artificially stacked Graphene-BN van der Waals solids [14]. Both theoretical and experimental studies suggest that hybrid nanomaterials can leverage the best aspects of their constituents and/or render new functionalities depending on the structural

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integrity and interfacial chemical bonding of the constituents [6,7]. As an example, it is known that symmetry breaking in graphene layers leads to complex states and emergence of energy gap which convert graphene layers to semiconductor [15]. Whereas the inversion symmetry breaking has now been created with many strategies such as external electric field [16e20], molecular adsorption [21e23], and applying of strain in bilayer graphene [24], creating novel hybrid architectures composed of low dimensional C and BN materials could provide de novo opportunities to symmetry breaking in emergent hybrid states. Recent successes in connecting 1D and 2D low dimensional materials via advanced experiments such as electron beam welding [9] and chemical vapor deposition techniques [10,12], calls for a more predictive, fundamental design of junctions to get a deep insight on the hybrid physical interactions. This, in turn, can impact several novel applications of such hybrid materials in areas such as nanoelectronics [6,7,25,26], detection of biosystems such as nanopore based DNA [13], and energy storage [6] and 3D thermal and mechanical properties [6,27,28]. For instance, pillared graphene, made of covalently connected graphene and carbon nanotubes (CNT), exhibits an excellent balance of thermal conductance and electronic properties in 3D [7,8]. In a recent work from Feng et al., they proposed a novel device composed of CNT and bilayer graphene hybrids [29]. Using TEM experiments, they shows that single wall CNT can create a bridge between bilayer graphenes via electron beam irradiation at high
temperature (>600 C) electron beam irradiation of few layer graphene [29]. Such nanotube-graphene hybrids by lithographic cutting method and self-folding of graphene process, form a metalsemiconductor-metal junction for field effect transistors (FET) device without extra element as electrode, since the graphene itself acts as an electrode [30,31]. Such works open the pathway to design several double layer graphenes connected to BNNT and CNT, as FET devices, since graphene nanopore converts the metallic (6,6) CNT to a semiconductor by inducing partial sp3 into the sp2 of nanotube and opening band gap energy [32]. In this context, junctions play a key role in controlling phonon scattering and band gap between tubes and sheets [33,34]. While there are several reports on hybrid carbon-based allotropes, similar studies on the effect of junctions on hybrid materials made of 2D graphene and 1D BN nanotubes are essentially unexplored. Li et al. [26], reported hetero-epitaxial growth of hexagonal BN by graphene edges. While this study was focused on only 1D covalent boundaries, it inspired us to design new configuration of 2D interfaces with BN nanotube (BNNT) connected to graphene. In particular, given the high bandgap of BN nanotubes, which act as an insulator compared to graphene [35], it will be interesting to explore the covalent junctions between such hetero-structures. From a computational design standpoint, Matsumoto et al. [6], investigated the electronic states of heptagon junctions for three different end geometry structures by tight binding (TB) model. The electronic properties of pillared graphene have been studied by Gonzalez et al. [36] using the continuum theory (Dirac-Fermion fields) and a tight-binding model. More recently, Novaes and his coworkers focused on electronic transport between graphene layers and CNT [7]. In fact, charge carrier mobility differs along the normal direction to ridges for the rippling and buckling of the graphene layer [37]. Topological insulators in IVeVI compounds including multivalley massless Dirac fermions show interface superconductivity at bands and helical snake states by strain engineering [38]. Theoretical and experimental studies have demonstrated that corrugated graphene creates effective gauge field on its low energy charge carriers, which lead to Landau quantization [39]. Strain engineering of graphene as highly flexible nanoscale material [3,41]

unveil that strain field could tune electronic structure of graphene [42] and lead to enormous pseudomagnetic fields [43], providing new bias for study of devices at high magnetic field regimes [31,42,43]. Strain engineering of graphene layers can also tune the energy gaps and cause quantum hall effect [44e46]. The strain induced near the junction affects the electronic structure and creates intravalley pseudo-gauge field, which yields magnetic field on electrons. Meanwhile, bond deformation due to strain and bending at the connecting boundary will induce additional strains because of mismatch between the lattice constant of graphene and the nanotube diameter [36,47e49]. The objective of this work is three-fold: first, we will investigate the effect of junction configuration on structural stability of monoand double-layer graphene covalently connected to CNT and BNNT via density functional theory (DFT) calculations. Second, we will explore the role of junction-induced strains to create pseudomagnetic fields in pillared graphene and hybrid pillared graphene/ BNNT via the tight binding approach and compare the results with elasticity theory of triaxial stresses [44,50]. Finally, we will develop a quantitative analytic approach for strain-induced pseudomagnetic field in corrugated graphene nanopores. 2. Results and discussion Tuning electronic band gap energy between valence and conduction bands in graphene layers manifests great potential for several technological application such as switching the conduction off to controllable level of energy gap [15]. Here, we present DFT results of tuning electronic properties of hybrid carbon network materials by designing multi-dimensional arrays composed of 1D CNT or BNNT arrays covalently connected to 2D single and bilayer graphene. To create junctions, we brought (6,6) CNT and (6,6) BNNT nanotubes to two different defected graphenes, (supporting information section. I) resulting in two different junction configurations (Fig. 1aef): one symmetric junction containing six heptagonal rings, which only occurs in the case of CNT, and one asymmetric junction containing 3 octagonal rings, which occur both in the case of CNT and BNNT. Note that both junction configurations satisfy the Eulers theorem of polygons [51], i.e. the number of faces, F, vertices, V, Edges, E, and Genus, G, obey F þ V ¼ E þ 2  2G [3]. For instance, heptagon graphene/CNT junction is a closed surface of genus 2 (G ¼ 2). Thus, CNT and graphene must share a total bond surplus of 12 (6 each) [11]. The heptagon graphene/CNT satisfies this criterion by 6 heptagons, each having þ1 bond surplus. We organized our study to investigate the electronic band structure properties and strain-induced pseudomagnetic fields on mono- and double-layer defected graphenes connected to CNT and BNNT. To calculate and compare the band structure and pseudomagnetic field of our hybrid graphene/CNT and graphene/BNNT, we briefly present the DFT results in terms of dipole moment, pz, the net Mulliken charge on nanotube, cohesion energy, Ecoh, energy gap, Eg, and binding energy per atom, Eb/atom, for finite and infinite CN and BNNT. In view of Table 1, the following observations deserve attention: (i) BNNT junction is the most stable hybrid structure, followed by octagonal and heptagonal CNT junction, (ii) the net charge on nanotube (NT) is more important for the most stable junction and dipole moment, and (iii) finite and infinite BNNT are energetically more favorable than analogous CNT. Combining the first two observations, the most stable junctions are those in which the deformation of the boundary atoms and charge transfer are more important. Such structural deformation is consistent with pp repulsion minimization because of high Nitrogen electronegativity [52]. From Table 1, it is evident that the cohesion energy and electron charge transfer are different for the two kinds of junctions considered in this study. Within the carbon system alone, the

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Fig. 1. Schematics, density of states and band structures of different defected graphene covalently connected to CNT and BNNT. (a) defected graphene as a host for a heptagonal junction to CNT., (bec) defected graphenes as hosts for octagonal CNT/BNNT junctions. (d) heptagonal graphene/CNT junction, (e) octagonal graphene/CNT junction, and (f) octagonal graphene/ BNNT junction. (gei) represents the comparison of total density of states (DOS) for isolated graphene and different nanotube junctions. (jel) Band structures of graphene covalently interconnected to CNT and BNNT, corresponding to panel (def), respectively. The arrows show band gap energy opening in the Fermi level as 67 meV in heptagonal graphene/CNT (j), 178 meV in octagonal graphene/CNT (k), and 159 meV in the octagonal graphene/BNNT junctions (l). (A colour version of this figure can be viewed online.)

Table 1 Electric dipole moment (pz), net charge transfer, cohesion adsorption energy (Ecoh) and energy gap (Egap) for heptagonal and octagonal junction. Binding Energy per atom Eb/atom for finite and infinite CNT and BNNT. GE þ NT pz(Debye) HeptagonCNT 1.43 OctagonCNT 2.04 OctagonBNNT 2.96 NT Binding Energy (Eb/atom) Finite (eV) CNT BNNT

1.89 5.83

NT net charge (jej)

Ecoh (eV)

Egap (meV)

0.35 0.52 0.75

2.72 2.88 3.71

67 178 159

Infinite (eV) 4.18 7.62

octagonal junction is more stable than the heptagonal junction. The calculated higher cohesion energies for the octagonal junction in CNT are associated with a quite strong pp stacking interactions, which could improve the orbital overlap and the orbital mixing

between C atoms of the graphene and CNT states in the junction boundaries to introduce a band gap in graphene þ CNT junction through a local electronic perturbation. Our DFT results showed a significant charge transfer from graphene to nanotubes, confirming the covalent bonding between the two. In agreement with this statement, Fig. 1g shows the total density of states (DOS) of heptagonal graphene/CNT junction, with respect to the DOS of pristine graphene and isolated nanotube. Besides Fermi energy shift, the DOS of defected graphene is significantly perturbed by the presence of CNT. The states of pillared graphene with heptagons is shifted 0.54 eV towards lower Fermi energy, which indicates p-doping of the defected graphene moiety. This is consistent with values in Table 1 showing a charge transfer from graphene to nanotube. The DOS of pillared graphene with octagonal junctions exhibits similar behavior with larger shift in Fermi level (EF) (Fig. 1h), corresponding to more perturbations and stronger interactions in this junction configuration. However,

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in the case of BNNT junction (Fig. 1i), the perturbation of states decreases due to the presence of Nitrogen atoms with high electronegativity, minimizing the pp repulsion. Fig. 1jel shows the band structure of heptagonal and octagonal CNT and BNNT junctions, where the Fermi level (EF) is shifted into valence bond, consistent with the p-type doping and the emergence of a band gap close to the Dirac cone, neutrality point. Upon connection of graphene and CNT, the calculated band gap (Eg) is 67 meV for the heptagonal junction while this gap opening increases to 178 meV for the octagonal junction. In the case of octagonal graphene/BNNT junction, the calculated band gap opening is 159 meV. One plausible explanation for gap opening takes into account that a nanotube array on graphene breaks the symmetry of the p-states near Fermi energy. As a result, the four fold degeneracy of p-bands of pristine graphene is decreased and lifted (see Fig. 1jel). The presence of interactions and charge transferring in pillared arrays can be revealed by plotting the charge density (Fig. 2aec). Overlapping and mixing of states cause high charge density and accumulation (red color). In order to investigate the effect of strain on the electronic properties, we use the optimized structural positions obtained from our ab initio calculation as an input to Eq. S(6) (supporting information see section. III) to find the LDOS maps, which could be directly accessed by scanning tunneling microscopy STM experiments. The LDOS maps at Fermi energy are shown in Fig. 3a, b. Integrating the LDOS along the graphene width results in the curve plotted in Fig. 3c, d. Several interesting effects can be observed such as the appearance of peaks in the LDOS, which are more pronounced for octagonal interface. We attributed this featureto more significant ripples in octagonal molecular junctions than heptagonal ones. An exponential fit of the form ex/l

demonstrate the decay pattern in both sides of junction gaps. These results suggest that different junctions of pillared graphene can have a significant strain-induced effect on neighboring regions, affecting electronic transportation. 2.1. Strain-induced pseudo magnetic field From a theoretical point of view, external stresses on graphene can change the neighboring carbon distances and create ripples on graphene layer [45,46]. In our case, the origin of external stresses and deformations are bond bending at the junction via cohesion energy between graphene and nanotubes, leading to covalently connected junctions. Strain-induced bond deformations and the mismatch between the nanotube diameter and lattice constant of graphene layer create intervalley gauge field which acts on K valley electrons. The electronic properties of graphene are described by a P y y tight-binding Hamiltonian, H ¼ i;j;s  tðrij Þcis cjs þ h:c: where cis (cjs) creates (destroys) an electron at site i (j), the sum is for nearest neighbors [46,49]. The gauge field potential can be described using the empirical relation, tðrij Þ ¼ t0 ebðjdrij j=a0 1Þ [50], and the tight-binding Hamiltonian, which includes strain in terms of modified hopping parameter t(rij) between delocalized p electron cloud orbitals [50]. In above, tðrij Þ ¼ t0 ebðjdrij j=a0 1Þ ; drij is a function of the atomic position vector, which connects the i and j neighboring atoms in the strained structure; t0 ¼ 2.7eV, a0 ¼ 1.42Å is the CC bond length and ! b ¼ 3.37 is a constant [50]. In the Hamiltonian, the Fermi surface ( k ! e! ! point) in the reciprocal space is replaced by k  Z A where A is the fictitious vector potential leading to a corresponding pseudo! ! ! magnetic field ( B ¼ 1=evF V  A ), where e and vF~106m/s represent charge unit and Fermi velocity in graphene, respectively [50].

Fig. 2. Charge density for mono and bilayer graphene connected to CNT and BNNT. (aec) Charge density for mono-layer graphene connected to CNT in the form of heptagonal (a) and octagonal junction (b). (c) represents octagonal graphene/BNNT junction. (def) Charge density for bi-layer graphene connected to CNT in the form of heptagonal (a) and octagonal junction (b). (c) represents octagonal graphene/BNNT junction. Overlapping and mixing of states in (aef) cause high charge density and accumulation (red color). (A colour version of this figure can be viewed online.)

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Fig. 3. LDOS at Fermi energy in graphene with heptagonal (a) and octagonal (b) junction, evaluated in a plane x,y. (c) and (d) present the integrated LDOS along the width of graphene. (A colour version of this figure can be viewed online.)

By rewriting the TB Hamiltonian with new hopping parameters (as a function of position), the induced gauge field becomes,

Ax þ iAy ¼

X ! d ij

! dtðrÞij ð r Þei

!!

k : d rij

uab ¼ ; B ¼ vy Ax  vx Ay

 2bZ  uxx  uyy ; 2uxy ; 3a0 e

! (3)

(1)

where dt(rij) ¼ t(rij)t0 is the difference between the hopping terms in the strained and the original lattice, and B is a pseudomagnetic field perpendicular to the xy plane. Another approach to analyze the effect of strain on the electronic structure of the system created near the boundary junction is the elasticity theory to model curved surfaces with the use of generalized Dirac equation [36,47]. Rewriting the Dirac Hamiltonian with modified hopping parameters allows to compute the gauge field via [44,48,50],

A¼

1 vua vub v2 h þ þ 2 vxb vxa vxa xb

(2)

where b (~ 3) is a constant. Then, the pseudomagnetic field (perpendicular to the plane) could be calculated [46] by defining the atomic displacements, i.e. u ¼ r'iri, and the strain tensor uab including out of plane displacements,

where h(x,y) is the displacement along the z direction. Fig. 4aec shows the plot of Gauge field A calculated by Equation Eq. (1) for three structural deformation obtained from DFT results. Interestingly, the vector potential patterns are not constant. In all of the three junction types, there are deformed lines around the boundary region of the junction and this deformation for octagonal graphene/ CNT is significantly stronger than octagonal graphene/BNNT. These figures support our finding in Table 1, which indicated that the most stable configuration is graphene/BNNT junction. Therefore, this latter junction configuration induces less corrugation and ripples on the defected graphene, as compared to other two systems. As a result, different pseudomagnetic field pro les can be existed. Furthermore, by computing the pseudomagnetic field B and its density pro le (Fig. 4def), one can clearly realize that the variation of the magnetic field of octagonal graphene/BNNT around the junction boundary is less than those of other two junction configurations comprising CNT, in good agreement with our previous observation. For a more accurate analysis, Fig. 4g shows the pseudomagnetic field along the radial distance, r (grey line at Fig. 4e). It is interesting to note that the density of pseudomagnetic field B

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Fig. 4. Gauge field and pseudo magnetic field for graphene/CNT and graphene/BNNT junctions. (a-c)Vector plot of the gauge field A for heptagonal graphene/CNT junction (a), octagonal graphene/CNT junction (b) and octagonal graphene/BNNT junction (c). (def) density plots of induced pseudo magnetic field B, corresponding to the junctions in (a) to (c), respectively. (geh) Evolution of pseudo magnetic field versus radial distance r (g) and versus the change in the lattice constant of defect, strain parameter, Da0/a0 (h), which our computational results in good agreement with experimental reported (black line) in Refs. [43,44]. The radial distance is shown by a grey line in panel (e). (A colour version of this figure can be viewed online.)

exhibit four-fold (heptagon) and three-fold (octagon) symmetry in all three junction types. Compared to BNNT, CNT induces a pseudomagnetic field B with higher density at center and boundary regions of the junctions. In the case of BNNT, the induced magnetic field varies smoothly in the central region of the defected graphene. This strain decays as a function of r1, where r is the radial distance of defected boundary to bulk of graphene as shown in Fig. 4g (variation of B versus r). By using dimensional analysis, one can show that the strain near the junction is at the order of Da0/a0, where Da0 is the change in lattice constant of the defect, or at the order of DR0/R0, where DR0 denotes the change in the radius of the nanotube [36]. Therefore, as a measure of strain, we computed and plotted the evolution of pseudomagnetic field with respect to Da0/a0 (Fig. 4h), which is in good agreement with experimental results [43,44]. This finding suggests that the elastic strain induces changes on the electronic structure near the junction. To this end, we modeled and derived a quantitative analytical definition of a strain-induced pseudomagnetic field in corrugated graphene (supporting information section. IV), which supports our results for the induced pseudomagnetic field of pillared graphene.

2.2. Double graphene layers covalently connected to nanotubes In view of typical 3D periodic structure of hybrid nanostructures, in this section we designed a new set of structures with double graphene sheets connected by CNT and BNNT with different junction configurations and different tube lengths. First, we discuss the heptagonal junctions with different nanotube lengths between the bilayer graphene. Fig. 5bec demonstrates that the band dispersion is significantly different for short and long nanotubes. In the long nanotube, the flat band is originated from the localized states in nanotube array, which supports our finding of DOS of graphene layers attached to nanotube (Fig. 5d). This is also in agreement with earlier report [36] indicating that the band gap energy decreases with the existence of flat bands (from 425 meV in short nanotube to 311 meV for long ones). The total DOS of short and long CNT in Fig. 5d allows one to confirm the magnitude of the mixing of electronic states in these two structures. Remarkably, there is a peak in conduction band (CB) for the long CNT, suggesting the existence of local states for the long nanotube array. Fig. 5b also demonstrates that the electronic band structure of bilayer graphene includes a Mexican hat dispersion between

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Fig. 5. Electronic band structures and total density of stets for bi-layer graphene systems connected to CNT and BNNT with various length scales. (aed) bi-layer graphene with heptagonal junctions to CNT. The magnitudes of band gap opening for (b) and (c) are 425 and 311 meV, respectively. (eeh) bi-layer graphene with octagonal junctions to CNT. The magnitude of band gap openings are 143 (e), 150 (f) and 155 (g) meV. (iel) bi-layer graphene with octagonal junctions to BNNT. The magnitude of band gap openings are 281 (i), 377 (j) and 377 (k) meV. (d,h,l) show the comparison of DOS versus tube lengths and the pristine structures. (A colour version of this figure can be viewed online.)

conduction and valence band [53]. Using tight-binding approximation, the built-in electric field between graphene layers (Ebi) can be computed by the equation between DEK (energy gap at K point) and d (distance between graphene layers), Ebi zDEd K [22,54,55]. In the case of graphene/CNT with short CNT, the internal (built-in) electric field is calculated to be 0.85 V/nm for the heptagonal junction, and Ebi ¼ 0.31 V/nm for the octagonal junction. In the case of graphene/BNNT with short BNNT, this internal electric field is calculated as 0.56 V/nm, which breaks the inversion symmetry and

create potential difference between top and bottom graphene layers, hence the energy gap. To obtain further insights into the nature of the junctions with heptagonal and octagonal rings, we calculated the electronic band structure for several tube lengths with octagonal junctions in graphene/CNT and graphene/BNNT (Fig. 5eel). Interestingly, the band gap energy does not significantly change in the octagonal graphene/CNT array, but somehow there is a low band dispersion (almost flat) in the case of longer nanotube. Similarly, for the

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graphene/BNNT junction, the energy gap (Fig. 5iek) and band dispersion seem to be constant, owing to the intrinsic widebandgap [35] of BNNT, which exhibit insulating properties. Our results for CNT corroborates the recent finding of electronic transport between connected nanotubes and defected graphene [8]. With carful inspection of graphene/CNT junctions (Fig. 5bec and Fig. 5eeg), we notice that the existence of the flat band and band dispersions near the Fermi level originates from the local states imposed by the long nanotube arrays (Fig. 5c, g). However, in the case of graphene/BNNT junction, there is no flat band (Fig. 5k); only some dispersions in the electronic band structure, which is consistent with the DOS of graphene layers attached to BNNT (Fig. 5l). As Fig. 5l reveals, although the states of graphene layers are perturbed by connecting to BNNT, there is no peak or localized state in the total DOS such as those in the graphene/CNT systems. The increasing band gap energy versus the tube length for the graphene/BNNT system can be attributed to the decreased band dispersion in this structure, as compared to the graphene/CNT junction. Fig. 2def shows the calculated LDOS for different CNT and BNNT junctions (the cases with short tube length), demonstrating that the density of electronic states in the graphene/BNNT junction is quite distinct from those of the graphene/CNT junction. This suggests that the charge transfer between graphene layers and nanotubes is a major parameter responsible for the band dispersion of hybrid graphene/BNNT and graphene/CNT nanostructures. 3. Conclusion This work examined the influence of junction configuration on electronic properties of graphene layers connected to BNNT and CNT, as promising hybrid nanostructures and materials for various nanoelectronic applications. Via ab initio density functional theory calculations, we studied three junction types, i.e.heptagonal graphene/CNT, and octagonal graphene/CNT and graphene/BNNT, and showed that in all cases the covalent connection of graphene to CNT and BNNT induces a bond deformation at the junction boundaries. This mismatch between the lattice constants of defected graphene and nanotube diameter creates a local elastic strain (gauge field), which induces changes in the local electronic structure, thereby creating a pseudomagnetic field near the junction. By performing tight-binding calculations and using the elasticity theory, we found that while the induced pseudomagnetic field decreases exponentially with radial distance from the junction boundary, it increases linearly with Da0/a0, the stress parameter. Furthermore, we extended our analysis to hybrid structures with double layers of graphene sandwiching CNT and BNNT with different length scales. Our results pointed out to two main junction classifications: one with flat bandgaps near the Fermi level and one without. More precisely, longer CNT arrays connected to graphene layers exhibited flat bands, in contrast to the case of BNNT, regardless of the BN tube length. Moreover, we found that while the band gap energy for heptagonal graphene/CNT and octagonal graphene/BNNT junctions are sensitive to the nanotube length, this is not the case in the octagonal graphene/CNT junction because the geometry of octagons creates less perturbation in the electronic states of the VB in the latter junctions, thus reducing the accumulation of localized electronic states at the boundary. Together, such information and findings provide de novo concepts, physical insights and design variables that can have a significant impact on nanoengineering multidimensional arrays of hybrid graphene, CNT and BNNT networks, leading to novel 3D electro- and magnetomutable hybrid nanomaterials. Broadly, our findings lay the foundation for discovering numerous tunable, hybrid 3D C/BN architectures by fusing low

dimensional C and/or BN building blocks including 0D fullerene, 1D nanotubes and 2D nanosheets. A rich set of hybrid carbon nanostructures have been developed and studied over the past years. However, similar advances for mixing C allotropes with BN counterparts have thus far remained elusive. To the best of our knowledge, this paper for the first time introduces 3D graphene/BNNT covalent junctions and explores their diverse electro- and magneto-mutable properties. Such hybrid building blocks and architectures can complement graphene-based nanoelectronics and can also open up a plethora of opportunities to explore several fascinating nanomaterials with tailorable in-plane and out-ofplane opto-electro-thermo-mechanical properties, such as nextgeneration 3D semiconductors with adjustable bandgap [33], fabricating 3D fillers for layered materials and stacked heterostructures [56e61], and creating 3D thermal transport devices and rectifiers with implications in nanoscale calorimeters and microelectronic processes [62]. 3.1. Computational methods All the theoretical calculations were performed based on the density functional theory (DFT) as implemented in the SIESTA package [63] that uses periodic supercell method and localized basis sets. We use the local-density approximation (LDA) to express the exchange-correlation energy. The sampling of the Brillouin zone includes a fine 18181 Monkhorst-Pack k-points grid for geometry optimization to produce an accurate band structure. The geometry optimization was pursued until the convergence criterion was less than 105 eV for total energy and less than 0.01 eV/Å for forces. Despite the weakness of DFT method in describing dispersive forces [64e67] for which the adsorption energies are typically underestimated with LDA and generalized-gradient approximation (GGA) functional, LDA appears more successful in providing reasonable geometries for p-stacked systems [52,68e70]. In contrast, LDA is well-known to overestimate adsorption energies when interactions are stronger than p-dispersion [71]. In fact, DFTLDA can qualitatively and quantitatively give an accurate description of the electronic structure of graphitic and carbon nanotube structures where the band structures are dominated by the chemistry of sp2 orbitals [70,72]. Concerning the role of dispersion force, we compare the dispersion band map and cohesion adsorption energy based on dispersion-corrected function (PBE þ Dispersion calculations in VASP package). The results are given in Table S1 (see section. II of supporting information). As well described elsewhere [73], LDA provides electronic structures for graphene that are similar to Green's function (GW) technique (One-Particle Green Function (G) with Screened Coulomb Interaction (W)) except for a small energy shift. Thus, we are confident that our conclusions are not affected by our computational approach. One of the most important characteristics for the existence and stability of the covalently interconnected graphene-nanotube structures is cohesion energy. The cohesion energy (Ecoh) per each covalent bond is defined using:

Ecoh ¼

EðGE þ NTÞ  ½EðGEÞ þ EðNTÞ n

where E(GE þ NT) is the total energy for the optimized covalently interconnected graphene-nanotube structure, E(GE) is the total energy of an optimized defected graphene, E(NT) is the total energy of an optimized NT (CNT and BNNT), and n is the number of covalent bond in the junction. Finally, DFT calculations were used to determine the stability and binding properties of finite and infinite

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CN and BNNT. We studied the armchair (6,6) nanotubes, which are connected to the graphene layer to create 3D homo- and heteronanostructures. The binding energy per atom of BNNT and CNT reported, is defined by,

Eb ¼

EðBNn ; Cn Þ  ½nEðBN; CÞ n

where E(BNn,Cn) is the total energy for optimized BN or C cluster, E(BN,C) is the total energy for the relaxed single BN or C, and n is the number of atoms in the unit cell. If the value of Eb is negative, it means that the cluster formation is exothermic and therefore stable. Author contributions R.S. and F.S. designed the research; F.S and R.S performed the computational research and analyzed the data; and R.S. and F.S. wrote the paper. Competing financial interest The authors declare no competing financial interest. Acknowledgments RS acknowledges the financial support from Rice University. The computational resources of this work were provided by Calcul Quebec and Compute Canada. Appendix A. Supplementary data Supplementary data related to this article can be found at http:// dx.doi.org/10.1016/j.carbon.2015.12.050. References [1] V. Sazonova, Y. Yaish, H. Ustunel, D. Roundy, T.A. Arias, P.L. McEuen, Tunable carbon nanotube electromechanical oscillator, Nature 431 (2004) 284e287. [2] Q. Cao, H.S. Kim, N. Pimparkar, J.P. Kulkarni, C. Wang, M. Shim, K. Roy, M.A. Alam, J.A. Rogers, Medium-scale carbon nanotube thin-film integrated circuits on flexible plastic substrates, Nature 454 (2008) 495e500. [3] J. Lee, H. Kim, S.J. Kahng, G. Kim, Y.W. Son, J. Ihm, H. Kato, Z.W. Wang, T. Okazaki, H. Shinohara, et al., Bandgap modulation of carbon nanotubes by encapsulated metallofullerenes, Nature 415 (2002) 1005e1008. [4] A.G. Nasibulin, P.V. Pikhitsa, H. Jiang, D.P. Brown, A.V. Krasheninnikov, A.S. Anisimov, P. Queipo, A. Moisala, D. Gonzalez, G. Lientschnig, et al., A novel hybrid carbon material, Nat. Nanotechnol. 2 (2007) 156161. [5] X. Wu, X.C. Zeng, Periodic graphene nanobuds, Nano Lett. 9 (2009) 250e256. [6] G.K. Dimitrakakis, E. Tylianakis, G.E. Froudakis, Pillared graphene: a new 3-D network nanostructure for enhancing hydrogen storage, Nano Lett. 8 (2008) 3166e3170. [7] V. Varshney, S.S. Patnaik, A.K. Roy, G. Froudakis, B.L. Farmer, Modeling of thermal transport in pillared-graphene architectures, ACS Nano 4 (2010) 1153e1161. [8] F.D. Novaes, R. Rurali, P. Ordejn, Electronic transport between graphene layers covalently connected by carbon canotubes, ACS Nano 4 (2010) 7596e7602. [9] M. Terrones, F. Banhart, N. Grobert, J.-C. Charlier, H. Terrones, P.M. Ajayan, Molecular junctions by joining single-walled carbon nanotubes, Phys. Rev. Lett. 89 (2002) 7. [10] D. Kondo, S. Sato, Y. Awano, Self-Organization of nobel carbon composite structure: graphene multi-layers combined perpendicularly with aligned carbon, Nanotub. Appl. Phys. Express 1 (2008) 074003. [11] N. Sakhavand, R. Shahsavari, Synergistic behavior of tubes, junctions and sheets imparts mechano-mutable functionality in 3D porous multifunctional boron nitride nanostructure, J. Phys. Chem. C 18 (2014) 22730e22738. [12] F. Du, D. Yu, L. Dai, S. Ganguli, V. Varshney, A.K. Roy, Preparation of tunable 3D pillared carbon nanotube graphene networks for high-performance capacitance, Chem. Mater. 23 (2011) 4810e4816. [13] A. Barati Farimani, K. Min, N.R. Aluru, DNA base detection using a single-layer MoS2, ACS Nano 8 (2014) 7914e7922. [14] A.A. Kane, T. Sheps, E.T. Branigan, V.A. Apkarian, M.H. Cheng, J.C. Hemminger, S.R. Hunt, P.G. Collins, Graphite electrical contacts to metallic single-walled carbon nanotubes using Pt electrodes, Nano Lett. 9 (2009) 3586e3591.

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