Electronic properties of sculpturenes

We investigate the electronic properties of sculpturenes, formed by sculpting selected shapes from bilayer graphene, boron nitride or graphene–boron nitride hetero-bilayers and allowing the shapes to spontaneously reconstruct. The simplest sculpturenes are periodic nanotubes, containing lines of non-hexagonal rings. More complex sculpturenes formed from shapes with non-trivial topologies, connectivities and materials combinations may also be constructed. Results are presented for the reconstructed geometries, electronic densities of states and current–voltage relations of these new structures.

(BN) and allowing them to reconstruct, new hetero-nanotubes (NTs) can be made, as well as a variety of more complex geometries formed from two or more materials.
In practice, a variety of techniques are available for cutting sculpturenes [3]. For example graphene can be cut using lithographic [4][5][6], chemical [7][8][9][10] and sonochemical [11,12] techniques. In particular, scanning tunnelling microscopy lithography [6] can be used to cut graphene nanoribbons (GNRs) with widths as small as 2.5 nm, with a specified chirality, a specified location and with their ends contacted to (graphene) electrodes and aberrationcorrected transmission electron microscopy can be used to cut holes in graphene and other materials [13,14]. All of these require specific experimental conditions and deliver cuts with different levels of accuracy. The atomic-scale dynamics of these methodologies are largely unknown and therefore in what follows, we circumvent this issue by starting from bi-layers with pre-cut edges and then using density functional theory (DFT) to allow them to reconstruct.
The aim of the present paper is to investigate the electron transport properties of a selection of these novel structures. Initially we shall examine the electronic properties of sculpturenes formed by cutting straight bilayer NRs and allowing them to relax. Depending on the direction of cutting, the resulting sculpturenes can be either perfect NTs, or NTs with lines of non-hexagonal rings. Electronic properties of more complex sculpturenes are also presented, including their densities of states and electron transport properties.

Electronic and geometric properties of all-carbon sculpturenes formed from straight nanoribbons (NRs)
In what follows, all structures are relaxed using the SIESTA implementation of DFT [15] using the Ceperley-Alder exchange correlation functional, with norm-conserving pseudopotentials and double zeta polarized basis sets of pseudo atomic orbitals. (g) A BiAGNR with T 1 T 1 /T 2 T 2 -terminations (96 atoms) where the top AGNR is blue and the bottom AGNR is red. After reconstruction, the edges reconstruction is shown in black. In general, we adopt the notation that a BiGNR with T i T j /T k T l -terminations has a bottom (top) GNR with upper and lower edge terminations T i and T k (T j and T l ). Figure 1 shows examples of pre-cut, straight, bilayer graphene nanoribbons (BiGNRs), which are infinitely periodic in the horizontal direction and of finite width W in the vertical direction, along with the corresponding NTs, which form after allowing them to reconstruct spontaneously. The initial BiGNRs are AB-stacked bilayer zigzag graphene nanoribbons (BiZGNRs), which after relaxation form perfect armchair CNTs with no defects.
The above examples are only a subset of the CNTs that can be obtained by relaxing armchair or zigzag BiGNRs, because the resulting CNTs depend not only on the orientation of the BiGNR, but also on the combination of edges which coalesce during reconstruction. To illustrate how different combinations of edge terminations can affect the resulting CNTs, consider first the zigzag and armchair edge terminations of GNRs, shown in figure 3.  When an AB-stacked bilayer is cut to form BiZGNRs, the bilayer NRs possess two upper and two lower edges associated with each of the stacked monolayer ribbons. Each pair of upper (or lower) edges can be formed from a combination of the edges shown in figure 3. The CNTs shown in figures 4(a)-(g) are the relaxed structures resulting from each of their adjacent BiGNRs. To perform the geometry relaxation, it is necessary to define a supercell in the initial BiGNR. Since the reconstructed sculpturene may have a larger unit cell than the initial  BiGNR, the supercell is chosen to be larger than the unit cell of the BiGNR and sufficiently large to accommodate the periodicity of the reconstructed sculpturene. Although this is not necessary for the sculpturenes of figures 1 and 2, it is clearly necessary for those of figure 4. Figure 4(g) shows that the T 1 T 1 /T 2 T 2 -terminations relax to a perfect (8,0) zigzag CNT. Further terminations such as T 1 T 2 /T 1 T 2 , T 1 T 1 /T 1 T 2 and T 2 T 2 /T 1 T 2 also form a perfect zigzag CNT. This demonstrates that the formation of zigzag CNTs from AGNRs is rather robust. Figure 4(a) shows that T 1 T 1 /T 1 T 1 terminations of BiZGNRs relax to a perfect (6,6) armchair CNT. Figure 4(b) shows that T 1 T 1 /T 2 T 2 terminations produce an armchair CNT with a line of pentagon-heptagon pairs. Figure 4(c) shows that T 2 T 2 /T 2 T 2 terminations produce an armchair CNT with two lines of pentagons-heptagon pairs. Figure 4(d) shows that T 1 T 2 /T 1 T 2 terminations lead to an armchair CNT with two lines of non-hexagonal rings, which contain octagons, horizontal pentagon-pairs and vertical pentagon-pairs. Figure 4(e) shows that T 2 T 2 /T 1 T 2 T 2 terminations lead to two lines of non-hexagonal rings. In this case, the bottom line contains four octagons, one horizontal pentagon-pair and three vertical pentagon-pairs per supercell and the top line contains nine pentagons and heptagons. This tendency for polygons with more than six sides, to be attracted by polygons with less than six sides is a generic topologically-driven feature of networks of three-fold vertices [16,17]. Figure 4(f) shows that T 1 T 1 /T 1 T 2 terminations lead to an armchair CNT with one line of four octagons, one horizontal pentagon-pair and three vertical pentagonpairs per supercell. Such lines of non-hexagonal rings are likely to possess novel spintronic and electronic properties [18][19][20].
We now investigate the electronic properties of the nanotube (NT) sculpturenes shown above, all of which are periodic in the horizontal direction, albeit in some cases with large    This persistence of the energy gap is also found in CNTs containing periodic 'zips' of impurities [21,22]. In all cases, the DOS contains van Hove singularities associated with the periodic nature of these quasi-onedimensional structures.
The For the sculpturene shown in figure 6(b), (which corresponds to the CNT sculpturene of figure 4(d)) the DOS shows many mini energy gaps around the Fermi energy. Similarly, figure 6(d) corresponds to a CNT sculpturene with one line of four octagons, one horizontal pentagon-pair and three vertical pentagon-pairs. The presence of quasi-disordered supercells in these sculpturenes leads to a significant increase in the DOS around the Fermi energy [24]. As shown by the insets of figure 5, the local-density-of-states close to the Fermi energy tends to be concentrated on these lines of defects.
From the number of open channels, it is clear that the conductance of sculpturenes with quasi-disordered unit cells is typically lower than that of conventional conducting NTs, because    (1) confirms that the current is lower than that of a comparable ideal (6,6) NT.

Electronic properties of boron nitride (BN) sculpturenes formed from straight NRs
We now examine the electronic properties of BN sculpturenes. Conventional boron nitride nanotubes (BNNT) are quasi-one-dimensional nanostructures predicted in 1994 [25] and experimentally discovered in 1995. BN is an electrical insulator with a wide band-gap of approximately 5 eV [26][27][28]. In this section we examine the electronic properties of BNNT formed by relaxing bilayer BN NRs. As examples, figure 7 shows three different combinations of edge terminations of zigzag BN NRs.
When an AB-stacked bilayer is cut to form bilayer zigzag BN NRs, the bilayer NRs possess two upper and lower edges associated with each of the stacked monolayer ribbons.   [29] of approximately 1.25 eV (figures 9(a)-(c)), which is similar to that of silicon (1.12 eV) [30]. Furthermore the BNNT sculpturenes in figures 9(b) and (c) possess additional energy gaps in the energy window from −2 to 2 eV.
The number of open channels for each system is shown in the far right sub-figures in figure 9. These show that within the energy window from −2 to 2 eV, there is only one open channel for the sculpturenes in figure 9(a) and one or two open channels for the others. Since the band gap is decreased in the zipped NTs, these can support current at low-bias voltages, which would not be possible in the insulating (6,6) BNNT.

Electronic properties of carbon-BN hetero-sculpturenes formed from straight NRs
In this section, we examine the electronic properties of hetero-NTs obtained by relaxing bilayer ribbons formed from a monolayer of carbon on top of a monolayer of BN. As an example, figure 10 shows the hetero-NT obtained by sculpting a hetero-bilayer from a monolayer of graphene on top of a single layer of BN and allowing reconstruction. The resulting NT consists of a half cylinder of carbon joined to a half cylinder of BN.
To understand the electronic structure of such hetero-sculpturenes, it is useful to examine their properties as a function of the relative widths of the carbon and BN sections of the NT. Such structures are likely to be challenging to realize experimentally, although in-plane junctions between monolayer BN and graphene are known [31] and therefore one can envisage cutting hetero-bilayer ribbons whose upper layer is formed in part from graphene and in part from BN, as shown in figure 11. This figure shows a range of resulting hetero-NTs, starting from (a) a perfect (6,6) BNNT, then containing progressively thicker sections of carbon (b-f) and ending with a perfect (6,6) CNT. In all cases, the hetero-NTs have a boron-carbon and a nitrogen-carbon interface and the overall structure is made of hexagonal rings.
These structures are useful for illustrating the role of the interface between grapheme and BN, because they share features associated with impurities in CNTs [32][33][34][35][36][37], specifically those associated with the doping of CNTs with boron [27,37] and nitrogen [24,34]. For such impurities, a characteristic peak near the Fermi energy is found, with the peak associated with boron below the Fermi energy and the peak associated with nitrogen above the Fermi energy. Similar peaks appear in the DOS presented in figures 11(c)-(f). Beginning with the well-known DOS of the perfect (6,6) CNT ( figure 11(g)), we see that the addition of a one-ring-thick NR of BN to the structure (figure 11(f)) creates a new feature near the Fermi energy. As the thickness of the BN strip increases (figures 11(e)-(c)), the feature persists. Since the feature has two peaks (one above and one below the Fermi energy), it is reasonable to expect the peaks are associated with either the boron-carbon interface or the nitrogen-carbon interface. To demonstrate this, Figure 15. The inset figures show, (a) The initial supercell (top) which contains a sculpted NR region with T 1 T 1 /T 1 T 1 -termination whose ends are connected to an ABstacked BIG, the resulting sculpturene is shown underneath the initial supercell (1224 carbon atoms). (b) The initial supercell (top) which contains a sculpted NR region with T 1 T 1 /T 2 T 2 -termination whose ends are connected to an AB-stacked BIG, the resulting sculpturene is shown underneath the initial supercell (1198 carbon atoms). (c) The initial supercell (top) which contains a sculpted NR region with T 1 T 1 /T 1 T 2 -termination whose ends connected to AB-stacked BIG, the resulting sculpturene is shown underneath the initial supercell (1211 carbon atoms). The black curves represent the zero bais transmission probability of the CNT sculpturenes which are connected to BiG electrodes after relaxation. See movies 2 and 3, available at stacks.iop.org/NJP/16/013060/mmedia. figure 12 shows the local density of states (LDOS) for the structure of figure 11(d), centred on each peak. The orbital distributions shown in figure 12 were obtained by calculating the LDOS in a 0.04 eV of energy window centred on the two peaks shown in figure 11(d). Clearly, the peaks are localized at the two interfaces. Figure 11(b) shows that adding a one-ring-thick graphene NR to a BN NT significantly reduces the gap to approximately 0.8 eV, while the addition of a two-ring-thick graphene NR closes the gap completely. Interestingly, the two-peak feature associated with the boron-carbon and nitrogen-carbon interfaces is robust, being present even in figure 11(b).
The band structures and number of open channels for these hetero-NTs are shown in figure 13. With the obvious exceptions of the perfect BNNT and CNT and the one-ring-thick  (1)) are also similar for all of these hetero-NTs, with the exception of the sculpturenes of figure 11(b).

Electronic properties of sculpturenes with sculpted electrodes
One of the biggest challenges for future nanoelectronics is the design of reliable methods to connect sub-10 nm devices to electrodes. As illustrated in figures 15 and 20, sculpturenes have the potential to overcome this contacting problem. As an example, figure 15 shows that by sculpting a narrow rectangular BiGNR between two wider BiGNRs and allowing them to relax, the resulting sculpturene is a CNT automatically connected to BiGR electrodes. As a second example, figure 20 shows that more complex structures, such as a CNT torus connected to electrodes can also be obtained. The structures of figures 15 and 20 comprise a central scattering region attached to periodic electrodes on the left and right and therefore their transmission coefficients T(E) can be obtained using the non-equilibrium Green's function code SMEAGOL [38,39], which utilizes the DFT-based hamiltonian from SIESTA. In what follows, we investigate the transport properties of such sculpturene-based systems, which are constructed by sculpting a finite AB-stacked BiGNR (with the same terminations shown in figures 4(a)-(f) whose ends are connected to periodic BiGR electrodes.  transmission probability of zipped sculpturenes obtain by sculpting NRs from AB-stacked BiGR with T 1 T 1 /T 2 T 2 and T 1 T 1 /T 1 T 2 terminations respectively and then relaxing the initial structures to form CNT sculpturenes which are automatically connected to AB-stacked BiG electrodes.   The initial supercell (top) which contains a sculpted NR region with T 2 T 2 /T 2 T 2 -termination whose ends connected to ABstacked BIG, the resulting sculpturene is shown underneath the initial supercell (1172 carbon atoms). (b) The initial supercell (top) which contains a sculpted NR region with T 1 T 2 /T 1 T 2 -termination whose ends connected to AB-stacked BIG, the resulting sculpturene is shown underneath the initial supercell (1198 carbon atoms). (c) The initial supercell (top) which contains a sculpted NR region with T 2 T 2 /T 1 T 2 -termination whose ends connected to AB-stacked BIG, the resulting sculpturene is shown underneath the initial supercell (1185 carbon atoms). The black curves represent the zero bais transmission probability of the CNT sculpturenes which are connected to BiG electrodes after relaxation. energy of the CNT sculpturene of figure 15(b) is non-zero along the lines of 5/7 rings, which leads to the relatively-high conductance of this sculpturene system. Figures 18(a), 5, 13(b) and 18(c) show the transmission probability of quasi-disordered sculpturenes (with two lines of non-hexagon rings) which are obtained by sculpting a NR from AB-stacked BiG with T 2 T 2 /T 2 T 2 , T 1 T 2 /T 1 T 2 and T 2 T 2 /T 1 T 2 terminations respectively and allow them to reconstruct.
Like the structures of figures 15(b) and 17, the non-hexagonal carbon rings which appear in the final sculpturenes again strongly influence transport. The left sub-figure of figure 19 shows that at low voltages, the sculpturenes shown in figures 15(a) and (c) can carry the higher currents, while at higher voltages between approximately 0.2 and 1.4 V, the structure shown   figure 15(b) carries the higher current. Similarly the right sub-figure shows the sculpturene system with two lines of 5/7 rings ( figure 18(a)) carries the higher current up to approximately 1 V, whereas the lower current corresponds to the system shown in figure 18(b).
Over a range of energies near the Fermi energy, the value of T(E) shown in figures 15 and 18 never exceeds unity. To illustrate that this feature is shared by other in-situ sculpturenes, For the structure in figure 20(a), there are two open scattering channels in the energy range of from −0.8 to 0.8 eV and therefore in this range, the transmission matrix t is a 2 × 2 matrix of the form: The transmission coefficient T(E) is therefore equal to the sum of the two eigenvalues of the transport matrix τ , given by: Figure 20(b) shows a plot of T(E), while figure 20(c) shows plots of the two eigenvalues. One of the two eigenvalues is dominant (red curve) whereas the second eigenvalue (black curve) is negligible. The presence of a single dominant transmission eigenvalue is also found in of the carbon nanobud-like structure shown in figure 21(a), whose T (E) is shown in figure 21(b). Figure 21(c) again shows that this sculpturene possesses single dominant eigenvalue (red curve) and a negligible second eigenvalue (black curve).
The presence of a single dominant channel, with T(E) never exceeding unity is typical of electron transport through single molecules, where the T(E) exhibit Breit-Wigner and/or Fano resonances when E coincides with molecular energy levels (shifted by the self-energy of the electrodes) [40]. Hence the sculpturenes of figures 15-21 share the advantages of single-molecule devices associated with the presence of transport resonances, but do not suffer from the well-known disadvantages associated with highly-resistive contacts to the electrodes. Like single-molecule devices [41], transport through the above structures is also controlled by geometry, but on a significantly-higher length scale.

Summary
The electronic properties of carbon, BN and BN-carbon-based sculpturenes have been investigated.
For the simplest sculpturenes formed from reconstructed BiGNRs, the resulting NTs depend not only on the chirality of the initial NRs, but also on the combination of edge terminations. Typically the sculpturene NTs possess a unit cell which is longer than the initial BiGNR and in many cases possess lines of non-hexagonal rings, which as shown in figure 6, lead to a reduction or even elimination of energy gaps in the DOS of such structures. For BN sculpturenes, figure 9 shows that the energy gap near the Fermi energy is found to persist, but is significantly reduced for the bulk value. In the case of hetero-NT sculpturenes formed from reconstructed ribbon containing both carbon and BN strips, pronounced peaks in the DOS are found, which arise from carbon-boron and carbon-nitrogen bonds. Sculpturenes connected to sculpted electrodes have also been investigated, including NT-like structures, a torus and a nanobud. In all cases, electron transport was found to be dominated by a single scattering channel, leading to electrical conductance less than or of order the conductance quantum.
The above examples demonstrate that electronic properties such as the band gap and electrical conductance of sculpturenes can be tuned by varying the geometry of the initiallysculpted bilayers, or in the case of hetero-structures, by varying the relative concentrations of the two materials. Furthermore their connectivity to external electrodes can be engineered in a controlled manner, thereby offering a new approach to optimizing contact resistances. In addition to the sculpturenes considered here, one can envisage many other examples, including sculpturenes formed from other 2d materials and combinations thereof. In addition, under realistic conditions, the initially-cut bilayer structures may have disordered edges, which would add to the variety of achievable sculpturenes. On the other hand, a molecular dynamics study of defective CNTs at 1500 K has shown an interesting self-healing property [42], which suggests that, at moderate temperature, the disorder in these NTs may 'heal' into a smaller set of stable structures of the type considered in this paper. For the future it would be of interest to the selfhealing properties of sculpturenes and their atomic-scale dynamics during simultaneous cutting and reconstruction.