Symmetry-dependent electronic structure transition in graphether nanoribbons

Since its first fabrication in 2004, graphene has attracted tremendous interest as a typical representative of two-dimensional (2D) materials.^1–4 In order to remedy the zero-bandgap drawback of graphene, control the electronic properties, and enrich the diversity of the monolayer structures, numerous graphene derivatives have been synthesized in recent years. For instance, fully hydrogenated graphene, named graphane,^5,6 would become insulating, while partially hydrogenated graphene exhibits hydrogenation pattern-dependent band structures,^7–11 where a tunable bandgap is realized. Fluorographene, known as one of the thinnest insulators, also exhibits modulatable electronic properties by tuning the stoichiometry.^12–16 Using fluorographene as the passivation layer in field-effect transistors featuring a graphene channel is reported to significantly increase carrier mobility.¹⁷ As a potential precursor material for the mass production of graphene, graphene oxide is one of the most broadly studied graphene derivatives.^3,7,18,19 The outstanding chemical, electronic, and optical properties enable it to be utilized in various fields, such as DNA probes, molecular sieves of water purification, flexible electrodes for rechargeable batteries, ultra-thin flat lenses, hydrogen storage,^20–25 etc.

Recently, Liu et al.²⁶ reported that graphether, a new two-dimensional oxocarbon, could be synthesized by assembling dimethyl ether molecules. It is a semiconductor with a direct bandgap of 0.81 eV and possesses excellent thermodynamic stability, high electron mobility, and greater in-plane stiffness along its armchair direction than that of graphene,^26–28 showing remarkable application potential in nanoelectronic devices. Moreover, due to the progress of experimental technology, the selective growth of graphene, as well as its ribbons, on the substrate has been achieved.²⁹ Patterned graphene derivatives can be fabricated by placing masks or grids over graphene during functionalization.^30,31 Designated attaching atoms could be optionally removed using the probe of a scanning tunneling microscope.³² The above-mentioned experimental approaches achieve precise manipulation in the functionalization of graphene and its derivatives at the atomic level, which lays a solid foundation for their application in transistors and nanodevices.^29,33 These approaches are also expected to be applied to the synthesis of graphether nanoribbons and the selective passivation on their edge atoms.

Inspired by that, we investigate graphether nanoribbons with armchair edges. It is found that symmetry plays an essential role in the band structure properties, which results in a bandgap oscillation with odd–even parity of the nanoribbon. Furthermore, the asymmetry of the electronic structures caused by edge hydrogen passivation will induce a semiconducting–metallic transition. It implies that the electronic structure properties could be modulated by symmetry.

The first-principles calculations are performed in the Atomistix ToolKit (ATK) package^34,35 based on the density functional theory (DFT). The Perdew–Burke–Ernzerhof (PBE)³⁶ functional of the generalized gradient approximation (GGA) is chosen to describe the exchange–correlation interaction. The high basis sets of PseudoDojo pseudopotentials³⁷ and a density mesh cutoff of 150 Ry are adopted. The Monkhorst–Pack k-point sampling is set to 1 × 1 × 15. The calculation supercell is filled with vacuum spaces of more than 15 Å in the aperiodic directions to eliminate the interaction between adjacent images. The atom positions of structures are fully relaxed with a force tolerance of 0.02 eV/Å, unless otherwise specified.

Similar to graphene nanoribbons, graphether nanoribbons can be classified into armchair-edged, zigzag-edged, and chiral ones, according to the edge morphology. Here, we consider armchair-edged graphether nanoribbons (AGENRs), which can be cut off from graphether nanosheets, as shown in the schematic in Fig. 1. The width of an AGENR is defined by the number n of the C atoms in a dimer chain in the transverse direction of the ribbon, as shown in Fig. 1(a). The two edges of an AGENR with odd-numbered n differ from those with even-numbered n, leading to a difference in structure symmetry, which can be identified from the side view of Fig. 1(b). When n is odd, the nanoribbons are symmetrical about the mirror plane σ represented by the yellow dashed line. However, there is obviously no such symmetry for nanoribbons with even-numbered n. It should be noted that the structures shown in Figs. 1(a) and 1(b) have not been optimized.

Considering the stability of optimized structures, the width of AGENRs is set at a minimum of 3. After structural optimization, the most obvious change is that the O atoms at the edge of AGENRs are pushed to the outside, which significantly increases the dihedral angle between the C–O–C plane and the plane of four C atoms in an edge five-membered ring (denoted by θ) from the original 125°, as can be observed in the side views of Figs. 1(b) and 1(d). To be more specific, we list the data of dihedral angles and bond lengths of the relaxed structures in Table I and mark the information of the primitive structure in Fig. 1(a) for comparison. It can be found that the value of θ is slightly larger (about 174°) when the ribbon is very narrow (n = 3, 4) but remains around 171° for other widths. The edge C–O bonds change little regardless of the width. The sequential bond lengths in a transverse C dimer chain of an AGENR show a slight fluctuation in centrally symmetrical distribution. In wide ribbons (n = 9, 10, and11), however, the lengths of the middle C–C bonds are stable at 1.60 Å, which is the value of the primitive structure. This indicates that the structural deformation is mainly located around the edge of the ribbons. The relaxed structure still maintains the same symmetry as the primitive structure, that is, it has a mirror symmetry plane σ when the width n is odd and does not have one when n is even. For convenience, an AGENR with width n is denoted as AGENR-n in what follows.

We next investigate the electronic structure properties of AGENRs. The band structures of AGENR-n are shown in Fig. 2(a), where the locations of the conduction band minimum (CBM) and the valence band maximum (VBM) are indicated by an orange double-headed arrow in each case. The bandgap with increasing n is illustrated in Fig. 2(b), in which the data of the indirect bandgap are marked by pink pentacles. It can be seen that AGENRs are semiconducting regardless of the width, which is similar to armchair-edged graphene nanoribbons.^38–40 It is worth noting that when the nanoribbon is narrow (n ≤ 8), the bandgap type shows an odd–even dependence on the width; that is to say, the nanoribbons with odd-numbered n exhibit indirect bandgaps, while those with even-numbered n exhibit direct bandgaps, and the value of the bandgap also oscillates with the odd–even parity of n. Similar bandgap oscillation with width has also been reported in graphene nanoribbons.⁴⁰ When the width becomes large enough (n > 8), this odd–even dependence disappears, and nanoribbons exhibit direct bandgaps similar to those of the nanosheet, regardless of odd- or even-numbered n. Meanwhile, the oscillation fades, and the bandgap tends to decline monotonically.

It is known that the edge symmetry of the nanoribbon is crucial to their electronic properties.^39–43 When the width n of the nanoribbon increases gradually, it switches between odd and even numbers in turn, which leads to alternating structural symmetries of the nanoribbons, as stated in Sec. III A. The symmetry is believed to have a significant effect on the oscillation of the bandgap. To verify it, we passivate the C atoms on the edges of AGENRs with hydrogen. Three structures with an indirect bandgap, i.e., AGENR-n (n = 3, 5, and7), are passivated at one edge and both edges, and their band structures are shown in Fig. 3. One finds that the three structures maintain indirect bandgaps when both edges are passivated but turn to direct bandgaps when passivated on a single side. Obviously, the structures passivated at both edges still possess mirror symmetry as their structures before passivation, and they maintain indirect bandgaps. However, this symmetry is broken in the structures with unilateral edge passivation, whereupon they exhibit direct bandgaps similar to the other cases without mirror symmetry. It implies that the mirror symmetry tends to trigger an indirect bandgap. Thus, the edge symmetry is essential to a narrow AGENR, which might lead to a direct–indirect transition of the bandgap.

Different symmetries suggest different coupling interactions between the edges. When the ribbon is narrow, the two edges are close to each other, the coupling between them is strong, and the distinction in band structure properties caused by different coupling is obvious, leading to an apparent odd–even oscillation of the bandgap. Nevertheless, with the increase in the width, the distance between the two edges becomes larger, and the edge coupling becomes weaker, which brings less impact on the properties of band structures. One should pay attention to the positions of the VBM and CBM on band structures shown in Fig. 2(a). It can be found that the VBM is always located at the Γ point. For those of a direct bandgap, the CBM is also located at the Γ point. However, the CBM deviates from the Γ point for those of an indirect bandgap. In other words, the location of the CBM is significant as it determines the type of bandgap. Hence, we examine the Bloch states at the CBM of AGENRs with different widths, as shown in Fig. 4. The isosurfaces in all subfigures are set to the same value of 0.2 Å^−3/2. By comparison, we learn that the contribution of edges to CBM decreases gradually as the width n increases. It reveals that the effect of edges on the band properties is indeed weakening. For an AGENR with a width greater than 8, the edge coupling is not strong enough to dominate the bandgap properties. As a result, the system no longer shows an indirect bandgap, and the bandgap oscillation is hidden. Considering a nanoribbon with the maximum limit of width, it will become a 2D nanosheet. The bandgap properties of AGENRs will progressively tend to approach those of the nanosheet with growing width. Since the nanosheet has a direct bandgap of 0.81 eV, AGENRs will retain direct bandgaps, and their bandgaps will decrease gradually.

Further exploration shows that passivating a part of the C atoms at the unilateral edge could induce semiconducting–metallic transition. In a unit cell of the pristine AGENR, there are two C atoms along the edge direction, forming a C dimer. To describe the edge passivation morphology more clearly in what follows, we define four types of passivated C dimers, named A-, B-, C-, and D-types, as shown in Fig. 5(a). In addition, we denote a passivated configuration as the edge type by “@AGENR-n.”

We first investigate the passivation in the minimum unit cell. There are two cases, i.e., A@AGENR-n and C@AGENR-n (D@AGENR-n is equivalent to C@AGENR-n). As examples, Figs. 5(b) and 5(c) illustrate passivated AGENR-5 with A-type and C-type edges, respectively. Band structures show that A@AGENR-5, where both C atoms of the C dimer are passivated, is semiconducting, which is in accordance with the result in section IIIB. However, C@AGENR-5, which has only one edge C atom passivated, is metallic. AGENRs with a width from 3 to 10 are checked, confirming the robustness to the width. It implies that the system exhibits a metallic state when odd-numbered C atoms in the unit cell are passivated but shows a semiconducting state when the number is even. In other words, the conductivity of the system shows odd–even dependence on the number of passivated C atoms in the unit cell.

This finding is corroborated in more complicated passivated structures. We construct a supercell of the AGENR by longitudinally enlarging the unit cell twice, forming two C dimers along the edge direction. Then we passivate a unilateral edge of the supercell selectively. The four C dimer types are combined to represent different passivated edges. In fact, the structures of AC-, CA-, AD-, and DA-terminated AGENR-n with the same n are equivalent in geometry as they are all periodic structures. As for the structures possessing an edge of, for example, AA, there is actually only one longitudinal C dimer in the unit cell, which will not be considered here as they have been discussed above. After removing all the repetitive ones, we obtain four non-equivalent edge types, i.e., AC, BD, AB, and CD. As examples, Figs. 5(d)–5(g) show the structures of AC@AGENR-5, BD@AGENR-5, AB@AGENR-5, and CD@AGENR-5, respectively. Band structure calculations confirm that AC@AGENR-5 and BD@AGENR-5, which own odd-numbered passivated C atoms, are metallic while AB@AGENR-5 and CD@AGENR-5, with even-numbered ones, remain semiconducting, conforming the above-mentioned findings. These results indicate that selective passivation of edge C atoms could induce the transition of AGENRs from semiconducting to metallic, which enables modulation of their conductivity.

In general, electronic structure is critical in determining the conductivity of the system. According to the symmetry of electronic structure, the four types of C dimers can be classified into two categories. In the C-type (or D-type) C dimer group, one C atom is passivated, and another one is unpassivated. The group possesses asymmetrical electronic structure as an unpassivated C atom has one more dangling bond than a passivated one. By contrast, the A-type and the B-type groups exhibit symmetrical electronic structures. Note that each metallic structure in Fig. 5 contains one asymmetrical C dimer group in the unit cell. The structures that do not contain asymmetrical groups remain semiconducting.

To understand the underlying mechanism of the semiconducting–metallic transition, we analyze the density of states (DOS) for the metallic structures in Figs. 5, i.e., C@AGENR-5, AC@AGENR-5, and BD@AGENR-5, as shown in Fig. 6. We also show the contribution of the unpassivated C atom in the asymmetrical C dimer (represented by C_asym) of each structure. It can be found that the C_asym atom contributes the most to the energy states around the Fermi level in each structure, indicating that it plays an important role in the metallicity of the system. The C_asym atom is located in a group with asymmetrical electronic structure, and its dangling bond is unpassivated. Both C atoms in the B-type group of BD@AGENR-5 [Fig. 6(c)] also have unpassivated dangling bonds; nevertheless, they do not make major contributions to the states at the Fermi level. The same is true for the C atoms on the other unpassivated edge. It implies that the configurations possessing asymmetrical electronic structures tend to be metallic. As for the configuration of CD@AGENR-5 [Fig. 5(g)], it has two C_asym atoms in the unit cell but exhibits semiconducting behavior. The reason is that the CD-combined group is symmetrical in the electronic structure as a whole. This further confirms that the asymmetrical electronic structure is the underlying mechanism of the semiconducting–metallic transition.

In summary, we investigate the electronic structure properties of AGENRs based on first-principles calculations. An odd–even oscillation of the bandgap with the width is found when the nanoribbon is narrow, but it fades as the ribbon becomes wider. Further analyses reveal that it can be attributed to the odd–even dependence of the interaction between the two edges, which is derived from the edge symmetry. Moreover, when one edge of an AGENR is partially passivated by hydrogen, the electronic structure asymmetry will lead to semiconducting-metallic transition. Our findings enable the electronic structure modulation of AGENRs and will be useful in the band engineering of related devices and the design of heterojunctions based on 2D nanostructures.

We thank Professor C. S. Liu for helpful discussions. We also acknowledge the support of the Experimental Platform for Basic Science at the Jinling Institute of Technology. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11705097, 11504178, and 11804158), the Natural Science Foundation of Jiangsu Province (Grant No. BK20170895), the funding of the Jiangsu Innovation Program for Graduate Education (Grant Nos. KYCX20_0702 and KYCX20_0792), and the Foundation of New Energy Technology Engineering Laboratory of Jiangsu Province (Grant No. KF0103).

The authors declare no conflict of interest.

The data that support the findings of this study are available from the corresponding author upon reasonable request.