From defect to effect: controlling electronic transport in chevron graphene nanoribbons

While bottom-up synthesis allows for precise control over the properties of graphene nanoribbons (GNRs), the use of certain precursor molecules can result in edge defects, such as missing benzene rings that resemble a ‘bite’. We investigate the adverse effect of the ‘bite’ defects on the electronic transport properties in three chevron-type GNRs and discover that the extent of scattering is governed by the different defect positions. Applying the concepts learned in single GNRs, we engineer defects in two nanostructures to construct prototypical components for nanoelectronics. First, we design a switch, consisting of three laterally fused fluorenyl-chevron GNRs, and place a pair of ‘bite’ defects to effectively allow the switching between four binary states corresponding to distinct current pathways. Second, we show that conscientious placement of a ‘bite’ defect pair can increase conductance between two leads in a triple chevron GNR junction. Overall, we outline how the incorporation of ‘bite’ defects affects transport properties in chevron-type nanostructures and provide a guide on how to design nanoelectronic components.


Introduction
Low-dimensional materials such as graphene and derived nanostructures have been attracting interest as potential building blocks of future devices since the first separation and characterization of graphitic films by Novoselov and Geim [1]. Boasting a wide variety of exciting physical properties such as high mechanical stability and structural flexibility due to the sp 2 hybridization [2], massless chiral Dirac fermions [3], and ballistic transport over micrometer length-scale [1], makes graphene appealing candidate for beyond-the-Moore's era nanoelectronics. However, the lack of band gap [1] severely hinders the utilization of graphene in devices, where high on-off current ratios are expected.
One of the most effective methods for opening a band gap in graphene has been the geometric confinement [4][5][6] of charge carriers in one-dimensional graphene nanoribbons (GNRs). This can been achieved by successful 'bottom-up' synthesis of GNRs from rationally shaped precursor molecules [7] that give unprecedented atomic-scale control over width [8], crystallographic orientation [9] and edge termination [10] of the resulting GNR. Moreover, the emergence of novel physical properties such as magnetism [11] or topology [12,13] can now be engineered. As a consequence, GNRs have attracted interest in a multitude of areas like spintronics [14], gas sensing [15], optical devices [16] or nanocircuitry [17], however, integration of GNRs in devices is typically hampered by significant Schottky barriers at the contacts [18] and structural defects.
We have previously shown the detrimental effect on transport properties of similar defects in zigzag GNR [11] and armchair GNR (AGNR) nanoribbons [19]. Here, we continue our investigation of 'bite' defects by considering a new class of GNRs with a chevron-edge type that also exhibit missing benzene rings in the resulting synthesized structure [7]. Similar to 9-AGNRs [19], the defect emerges due to the employed precursor molecule (6,11-dibromo-1,2,3,4-tetraphenyl-triphenylene) that hosts multiple phenyl rings which can cleave due to the steric-hindrance effects during the cyclization step. Interestingly, recent progress has been made in modifying the precursor molecule in order to obtain derivatives of the chevron GNRs (cGNRs)-either laterally extended cGNRs (ecGNRs) [20][21][22] or fluorenyl-cGNRs (fcGNRs) [23] that can be coupled into a two-dimensional superlattice geometry and exhibit emergent interface-localized electronic states. Furthermore, Mutlu et al [24] have used such laterally coupled fcGNRs to fabricate short-channel field-effect transistors with on-off ratios exceeding 10 4 and noted that structural perfection plays a remarkably important role in the charge transport. Nonetheless, due to the atomic structure of the base precursor molecule, 'bite' defects are still expected in the newly synthesized nanostructures and thus cause a concern over the practicality of employing chevron-based GNRs in electronic devices.
To expedite the process of attaining optimal devices, we first explore experimentally observed defects in three chevron-type GNRs and investigate the effects on the electronic transport. Next, we use this knowledge to place defects in complex heterostructures in order to obtain desirable electronic transport characteristics. Overall, our work reveals that defects can be used to selectively control the electronic transport properties, which is instrumental to the realization of novel carbon-based electronic devices.

Methods
We employ a tight-binding (TB) model with one p z orbital per atom and up to third nearest-neighbour hopping integrals. The TB Hamiltonian is expressed as: where ϵ i is the on-site energy at lattice position i, t ij is the hopping integral between the atoms (i, j), and c † i (c i ) creates (annihilates) an electron at lattice site i. We adopt the TB model proposed and benchmarked by Hancock et al [25]. The on-site energies are expressed as ϵ = 0 eV and the hopping terms as t 1 = 2.70 eV, t 2 = 0.20 eV, and t 3 = 0.18 eV for first-, second-and third-nearest neighbour couplings respectively.
We then use non-equilibrium Green's function method to obtain electronic transport properties: where η adds infinitesimally small imaginary character to energy E, I is the identity matrix and Σ L(R) is the self-energy containing information about the left (right) semi-infinite lead. We obtain the self-energies using Dyson equation: where H 0 is the Hamiltonian of the lead's unit cell, while H 1 is the coupling between the cells. The self-energies are used to calculate broadening function: Next, by taking the trace of the following product we can obtain transmission: Finally, we can use the transmission coefficient T to express conductance G in terms of conductance quantum G 0 via the Landauer formula [26]: All our TB calculations are performed using the Kwant package [27].

Results and discussion
In figure 1 we show three experimentally obtainable GNRs, which all share a similar precursor molecule and thus also a common shape. Chevron-edge GNR (figure 1(a)) was first synthesized by Cai et al [7] from tetraphenyl-triphenylene monomers and displays a pure armchair edge structure. Slightly modifying the tetraphenyl-triphenylene precursor molecule by an extra atom yields a fcGNR ( figure 1(b)) that has been shown to host an in-gap edge state and thus makes the ribbon truly metallic. Interestingly, Jacobse and colleagues showed that the metallic fcGNR can cross-link laterally to create nanoporous semiconducting graphene with emergent interface electronic states [23]. Finally, more significant lateral extension of the cGNR can be obtained by attaching another benzene ring to the precursor molecule and the resulting extended-chevron GNR (figure 1(c)) has been shown to exhibit a lower band gap [22] in accordance with width-dependent band gaps in AGNRs. Unusually, all of these GNRs retain glide symmetry that exhibits the peculiar band structure with doubly degenerate bands at X high-symmetry point [28]. We also notice that the low energy bands show only minor dispersion and do not cross each other, hence leading to discrete bands that in turn offers exciting opportunities for selective electronic transport engineering. Moreover, multiple different phenyl-group attachments in chevron-based GNR precursor molecules further extend the tunability over the positioning of the 'bite' defects. In figure 1 we present in red some of the possible cleavage points in the precursor molecules and the resulting final structure that arises from selective introduction of a defect.
As a starting point into engineering defects to control the transport properties, we have to establish the effects of a single point-like defect in an otherwise pristine system. In figure 2(a) we show the unit cell of the cGNR and mark eight possible defect positions, however only two of these are not symmetrically equivalent. We designate these positions as 'Side' (1,4,5,8) and 'Center' (2,3,6,7) defects and plot the corresponding conductance plots in figure 2(b). We focus on the transport in valence and conduction bands and compare the defective conductance curves (G d ) with the pristine cGNR (G p ) by a descriptor τ [19]: which estimates the preserved conductance in an energy interval spanning from E 0 to E 0 + δE. In this case δE is set to extend over the bandwidth. For example, if a defect completely suppresses the conductance in both the valence and conduction bands then τ = 0, while if a defect does not have any influence on the conductance in both bands, then τ = 1. We can notice that the center defect cause a larger disruption, with only τ = 0.39 conductance preserved, while side defect induces significantly smaller scattering with τ = 0.66 conductance preserved in the two bands. We can explain these results by looking at the local density of states (figure 2(c)) at the valence band (E = −0.28t 1 ), where the largest electron density is concentrated on the central atoms. Hence, center defects interact more heavily with the wavefunction associated with the band as opposed to the side defects, where only minuscule electron density is located. Similar observations can be also made for the conductance band due to the approximate electron-hole symmetry. Furthermore, we also plot the spatial probability current maps in figure 2(d) and show how the current pathways are affected by the introduced defects and confirm our observations based on the local density of states (LDOS).
Next, we display the conductance plots for a pair of defects in figure 2(e) and establish that a pair of two center defects have the largest suppression of conductance only retaining τ = 0.18 of the original conductance. Moreover, in the one-band region, e.g. −0.3t 1 < E < 0.26t 1 , this defect pair can completely suppress the conductance. It can be noticed that the pair defects follow the same trends observed with an individual defect as the lowest scattering originating from two side defects (τ = 0.43), while a middle ground is reached when a pair of side and center defects is considered (τ = 0.33). Overall, we show that 'bite' defects have a detrimental effect on conductance close to band edges and caution in the design of the precursor should be exercised, yet we also find that strategic placement of defects can allow a degree of control over the magnitude of conductance.
Similar principles of defect impact also extend to ecGNR, where another defect position is accessible due to the additional benzene ring in the precursor molecule (Sfigure 1(a)). We observed that the extended defect has the smallest impact on the conductance properties (τ = 0.77), followed by side defect (τ = 0.55), while center defect is still the most adverse (τ = 0.36), which agrees very well with our conclusions about bulk LDOS being concentrated in the middle of the GNR (Sfigures 1(b) and (d)). The similarities of the defect response stem from the fact that the LDOS is akin for both systems, being more localized on the central atoms. Although the missing benzene ring in position 9 leads to the smallest transport disruption, the combination of two such defects represent a width-modulated ecGNR-cGNR-ecGNR heterostructure that not only decreases the transport in the two bands by about a third, but also induces an anti-resonance around E = ±0.3t 1 as seen in Sfigure 1(e). Interestingly, no such anti-resonances were observed in the valence or conduction bands of defective cGNRs.
On the other hand, electronic structure and hence electronic transport of fcGNRs differ widely from both cGNRs and ecGNRs. The out-most atom in the five-membered ring gives rise to a in-gap metallic band and opens additional possibilities of engineering the transport properties. We show the possible defect positions in figure 3(a) and the conductance profiles in figure 3(b). Note that τ is defined only for the metallic band now. One can notice that only center defects are possible due to the nature of the precursor molecule and also observe the detrimental effect on the metallic band, where only τ = 0.15 conductance is preserved. The LDOS and current plots at E = 0 seen in figures 3(c) and (d) show that the metallic band arises due to the five-membered ring and the current pathways are strongly confined to the band edges. Interestingly, we see that adding just a single defect at position 3, leads to both current pathways disrupted, not just the top one. Whereas the LDOS plot displays a peculiar localization pathway-the electron density is localized either on the top or the bottom side and hence prompts to negligible overlap between the sides, thus minimizing conductance. Adding a second defect can result in two drastically different outcomes-either further suppressed conductance or inducing resonant tunneling at E = 0. First, adding a second defect at position 2 induces negligible changes in both LDOS and current maps as compared to the single defect, while exhibiting a very low τ = 0.01 value. Second, we can recover perfect conductance G = G 0 in our metallic band if we place the other defect in position 6. The resulting resonance peak corresponds to the localized state observed in figure 2(c) that acts as a quantum dot and facilitates resonant transport. Additionally, we can observe that the band at E ≈ 0.5t 1 qualitatively responds to the defects in the same way as cGNR and ecGNR, hence giving us an opportunity to selectively affect the electronic transport in different bands. With the knowledge gained from placing 'bite' defects in the three chevron-type GNRs, we now turn our attention to more complex structures.
As a first step, we take two fcGNRs and fuse them laterally ( figure 4(a)) to obtain a nanoporous GNR. Interestingly, we see in figure 4(b) the emergence of another in-gap state at E = 0.23t 1 that arises due to the hybridization between the five-membered rings [23]. By exploring the probability current maps in figures 4(d) and (e) we can assign these states to either inner (E = 0) or outer (E = 0.23t 1 ) five-membered rings. Whereas the previously equivalent defect positions can be split in two groups-defects 2 and 7 can be classified as the 'outer' , while defects 3 and 6 can be considered as 'inner' positions. We show in figure 4(b) the effect on conductance, when both of these defects are introduced. It is clear that now the defect positions will play a different role on both in-gap bands and, for example, the 'outer' defect (2 or 7) has a small influence on the metallic band (τ = 0.77), whereas conductance is drastically affected (τ = 0.17) at E = 0.23t 1 . This is a direct consequence of the different LDOS localization patterns and current pathways. Therefore, careful introduction of defects can lead to a selective closure of particular transport channels, while not affecting others.
We continue our investigation by fusing an additional fcGNR to our previous nanoporous GNR and extend it laterally as seen in figures 5(a) and (b). The resulting band structure and conductance plots in figures 5(c) and (d) tell us that there are still two in-gap bands at E = 0 and E ≈ 0.25t 1 with maximum conductance of G = 2G 0 and G = G 0 (except for a very narrow energy range), respectively. Whereas the band at E ≈ 0.45t 1 has the maximum conductance of G = 3G 0 . Although we already observed the spatially resolved current pathways in double fcGNR corresponding to different bands, the effect is more prevalent in triple fcGNR. We show in Sfigure 2 that the metallic band is once again localized only on the out-most atoms similar to fcGNR and double fcGNR, while the E ≈ 0.25t 1 band arises from the fusion of the inner five-membered rings is localized only on the middle ribbon. Finally, at E ≈ 0.45t 1 we have three current pathways equally distributed over the three fused ribbons.
Resolving the spatial current map already gives us a degree of control over the transport properties. For example, in a device consisting of the triple fcGNR, we can vary the gate voltage (V G ) to target a specific band and hence control in which part of the fcGNR the current flows. Furthermore, using the previous results of defect impact on transport properties, we propose a more complex switch that can access four different binary states-(0,0),(1,0),(0,1) and (1,1), where 0 and 1 correspond to the magnitude of conductance G 0 delivered to two regions of the triple fcGNR. We show in figures 5(e)-(g) the three different states associated with a specific gate voltage V G and indicate the state with 'light bulbs' , where the yellow color demonstrates that the pathway is on (1) and the white color indicates that it is off (0). It is straightforward to see that the fourth state (0,0) can be accessed at any energy that is not coinciding with a band, for example, at E = 0.1t 1 .
Strikingly, we see that after the introduction of the 2 + 7 defect pair in the bottom ribbon, we can completely close down one current pathway and decrease the conductance by exactly one conductance quantum G 0 ( figure 5(d)). Although, we already observed some degree of selectivity over different bands by introducing a defect in particular positions in narrower chevron-based GNRs, the laterally fused triple fcGNR shows an unprecedented possibility to carefully control the transport properties. We attribute this characteristic to the distinct separation of different bands and the band's localized nature, where the wavefunction is confined to a particular fcGNR that makes up the extended structure. Interestingly, the strong confinement then allows us to selectively alter only a particular spatial channel, without affecting the transport in other parts of the ribbon.
After showing that chevron-based GNRs can be used as building blocks for potential all-carbon devices and defect engineering gives a selective control over the electronic transport properties, we move on with investigating junctions composed from multiple GNRs. One of the first experimentally synthesized junctions  by Cai et al [7] was a symmetric chevron-based triple junction ( figure 6(a)). First, we show that such junction can be used as an electron beam splitter-the conductance of valence and conduction bands measured from the left lead to the upper(lower) lead exhibits low scattering with τ = 0.38 and hence overall transmission of τ = 0.77 to both leads (figure 6(c)). The low scattering stems from the fact that design of the central scattering region retains the symmetry and shape of the individual leads as shown by Chen et al [29]. Next, we notice that careful engineering of the 'bite' defects can increase the conductance in a selected direction. Figure 6(b) shows a pair of two 'bite' defects placed in the upper lead and the resulting current map at E = 0.36t 1 that shows an increased transmission to the right lead. We calculate that the ratio of the retained conductance is increased from τ = 0.38 to τ = 0.47, while also minimizing the conductance to the upper lead down to τ = 0.04. We propose that adding together multiple such triple-junctions with or without engineered 'bite' defects can be used to selectively control and split the incoming current and effectively act as interconnects between components of an electric circuit.

Conclusions
In summary, we investigated the electronic transport properties of three chevron-type graphene nanorribons in presence of 'bite' defects. Our calculations show the detrimental effect of the defects on the conductance, however, we also notice that different defect positions lead to diverse response in particular energy bands, thus giving a degree of control over the transport properties. Furthermore, it can be seen that the spatial electron density localization and current pathways play a major role in predicting the effects of the defect and therefore this information can be used to selectively affect a distinct conductance channel.
Next, we applied the lessons learned about individual GNRs to construct more complex nanostructures and selectively engineer the electronic transport properties with the help of intentionally introduced 'bite' defects. For example, we design a switch consisting of three laterally fused fluorenyl-cGNRs and place a pair of defects to effectively allow the switching between 4 binary states corresponding to different current pathways. With varying gate voltage we can achieve current flow in one, two or none of the connections, while not introducing any additional electron scattering and maintaining electronic transport of exactly one conductance quantum per connection. Interestingly, experimental realization of such structure could be achieved in near future, as innovative designs of precursor molecules have already been utilized to synthesize GNRs with 'bite'-like edges [30].
Finally, we show how an experimentally synthesized triple cGNR junction can act as an electron beam splitter and preserve 77% of the pristine cGNR's conductance in valence and conduction bands. In addition, deliberate placement of a defect pair can increase conductance between two of the leads from 38% to 47% of the maximum, hence turning the defect into a positive effect. We propose that using a triple junction as a building-block and placing defects strategically offers an excellent control over the current flow and thus can potentially be applied as interconnects in all-graphene nanocircuits. Further tunability of such system could be improved by applying a gate potential [31] to create a switch or transistor-like system. Overall, we establish design rules of defect incorporation in cGNR structures to control electron transport.

Data availability statement
The data cannot be made publicly available upon publication because they are not available in a format that is sufficiently accessible or reusable by other researchers. The data that support the findings of this study are available upon reasonable request from the authors.