Introduction

In the fields of materials science, increasing interest is now being attracted to graphene, a two-dimensional (2D) sheet composing of carbon atoms arranged in a honeycomb lattice structure1. Because of its compatibility with industry-standard lithographic processing, electron mobility up to 150 times greater than Si and a thermal conductivity of graphene twice of that of diamond2, graphene shows its promise as a future material for the nano- and micro-electronics applications, such as field effect transistors (FETs)3,4, biological membranes5, sensitive platforms6,7 and so on. Among them, as the most successful device concept in electronics, most works on graphene devices so far have been related to FETs8,9. When graphene is used as the channel material, the graphene-based transistors will own an ultrafast speed, making it able to operate in the THz frequency range10. Moreover, the extremely thin channels will possibly allow the transistors to be scaled to shorter channel lengths without encountering the adverse short channel effects, which restrict the performance of existing devices.

To achieve graphene-based electronic devices with high performance, specific bandgaps of graphene are usually required for different electronic applications11,12,13. As instance, since the Ion/Ioff ratios in FETs are largely dependent on Eg of the channel materials on the basis of their proportional relationship with EquationSource math mrow msub miI mrow mtexton mo/ msub miI mrow mtextoff mo∝ msup mtexte mrow msub miE mig mo/ msub mik mtextB miT , graphene materials with sufficiently large bandgaps are demanded for room-temperature FET operation14,15,16, where Eg denotes the bandgap, kB the Boltzmann constant and T the absolute temperature. In general, Eg with only a few tens of meV is too weak for room-temperature operation of FETs14. To have Ion/Ioff ratios as 10–100, Eg of graphene materials around 0.1 eV is required15, while for high Ion/Ioff of 107, it should be large as 0.4 eV15,16. Despite this, however, graphene cannot be directly adopted as semiconductors in electronics because of its zero bandgap [Eg(∞) = 0 eV]17, where ∞ denotes the bulk size in 2D. With such an issue, graphene creates a unique set of challenges for implementation into conventional electronics due to substantial leakage currents in the off state for field-effect devices. To address it, the degeneracy of the conduction and valance bands should be lifted via breaking the equivalence between the A and B sublattices18. Aiming at this purpose, the bandgap openings (BOs) in graphene have been realized by forming graphene quantum dots (GQDs)2,19,20, nanoribbons (GNRs)2,15,21,22,23 and nanoporous graphene sheets (NPGs)24,25.

Upon the investigation on the BOs in GQDs, GNRs or NPGs, different edge natures should be considered. According to its honeycomb lattice, graphene may own ideal armchair (AC) or zigzag (ZZ) edges12. With the present fabrication techniques, however, these perfect edges are difficult to be formed for GQDs and GNRs26,27. Instead, disordered edges are usually observed, which can be regarded as the mixture of AC and ZZ edge shapes2,28. Because of it, the two edge structures of GNRs remain asymmetric2,26,27. Moreover, their edges are kept naked with one dangling bond for each C atom. To stabilize those C atoms, dangling bonds can be chemically saturated with some radicals or groups denoted by R beneficial for the BOs, such as R = H29, F30 or OH31. Such an edge saturation was not performed for GQDs and GNRs in most experiments due to its technical difficulty, while Ritter et al.2 claimed that it can be performed through exposing GQD or GNR flakes to H radicals. Disordered GQDs and GNRs without the edge saturation will be shown as GQD-null and GNR-null, while those saturated with R will be exhibited as GQD-R and GNR-R. As to mesoscopic NPGs24,25, the holes in graphene are usually fabricated by the lithography technique with different geometrical shapes, such as circles or squares (C-NPGs or S-NPGs). Far from the atomistic precision, the internal edges are usually kept disordered and naked, referred to as DO-NPG-null. Owing to these attempts, the BOs were observed in disordered GQDs19,20, GNRs3,15,21,23,32,33,34 and NPGs24,25, while the D-dependence of Eg(D) varies considerably among them, where D denotes the diameter of GQDs or holes in C-NPGs, width of GNRs, or the side length of holes in S-NPGs. On lowering their D, a monotone increase in the openings was observed for disordered and naked GQDs and GNRs3,15,19,20,21,23,32,33,34, while it is more substantial for GQDs. As regards NPGs, a monotone decrease in the BOs has been reported24. In contrast, the BOs are lowered when their edges are saturated2. However, our understanding to the openings in disordered flakes is still in incubation.

Much effort has been devoted to investigating the opening mechanism in graphene. As a powerful technique, the computer simulation approaches, such as the first principles density functional theory (FP-DFT), Dirac equation and the tight-binding (TB) approximations, have been adopted for such studies. It may provide us band structure images, electronic density and other basic information of graphene for analyzing and explaining the electronic performance35. By this means, the opening behavior is usually studied using ideal GQDs and GNRs with regular edge structures13,29,31,36,37 or ideal NPGs with perfect periodic antidot superlattices or holes38,39,40. With these contributions, however, it seems still difficult to illustrate the openings in disordered flakes, since these ideal flakes exhibit much different D-dependences of Eg(D). The BOs in ideal GQDs and GNRs are relevant to edge structures. AC-GQDs have larger Eg(D) than ZZ-GQDs do36. By exploring the electronic properties for a wide class of nanoribbons, GNRs exhibit a rich variety of bandgaps from metals to typical semiconductors13,37. Typically, ZZ-GNRs show a metallic or semiconducting state13,29,31,37, while AC-GNRs present the oscillating BO behaviors from the metallic to semiconducting state with respect to D13,29,37, where the metallic points emerge at the period of 3 with Na = 3p + 2, Na the number of AC chains and p a positive integer. Concerning ideal NPGs38,39,40, the opening behaviors are much complicate, which may be affected by not only the hole diameter but also the hole geometry, lattice geometry, separation between adjacent holes, hole locations and the edge geometry. When D of holes rises, Eg(D) would increase for some antidot lattices39, such as hexagonal or triangular antidot lattices with a hexagon-shaped hole, while for the rectangular, rotated triangular and the honeycomb antidot lattices with a hexagon-shaped hole, only particular hole separations lead to an obvious opening39. The rectangular antidot superlattices perforated with four holes of honeycomb symmetry can exhibits the semiconducting or metallic states, depending on the width ratio of AC chains to ZZ chains38. More impressively, in particular, AC-GNRs with Na = 11 can be changed from the metallic state to the semiconducting state when it is perforated with a honeycomb hole especially at the central position, while it is still metallic if a triangular-shape hole is introduced38.

Besides the above studies, some other theoretical ways have been explored for the BOs in DO-GNRs by considering the impact from the edge disorder31. With the renormalized lateral confinement approach41, Eg(D) of DO-GNRs is decided by the structural disorder at the edges caused by etching or inaccuracies in the width determination, while such a disorder can result in an ″effective transport width″ smaller than the ribbon's nominal width. Alternatively, the BO behaviors can be explained with the Coulomb blockade or charging effects22,26,27, where the BOs are induced not only by confinement but also by Coulomb blockade originated from the edge roughness. However, the BOs in disordered GQDs and NPGs have been less discussed. As shown in Table 1, in addition, some expressions have been presented to plot Eg(D) of disordered GQDs, GNRs or NPGs. In general, these expressions are achieved using the fitting method with those experimentally measured Eg(D) values, while one or two adjustable parameters a and/or b are present there. Since these parameters are case-dependent and vary considerably among different researches, the reported fitting expressions cannot take their essential roles in helping us revealing the opening mechanism either.

Table 1 Equations or expressions reported to depict measured Eg(D) of disordered GQDs, GNRs and NPGs with or without the edge saturation
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Due to the limitation in existing approaches or expressions, some issues on the BOs in disordered graphene flakes have not been addressed. As instances, a theoretical way to describe the difference in Eg(D) induced by the dimension effect is still absent. It also remains unclear how to depict the reverse change in Eg(D) of NPGs relative to those of GQDs and GNRs. As to the role from the edge saturation, on the other hand, its exact mechanism has not been clarified as yet. On account of it, one still wonders how the BOs in saturated flakes will be weakened relative to those naked ones. In practice, the gap opening mechanism of disordered flakes is much complicated, which is relevant to not only the size but the dimension, geometric shape and the edge saturation. However, a systematic theoretical investigation on their roles is still absent, which obstructs our physical insight into the BOs and throws obstacles in our way toward the full-scale application of graphene in electronics. Because of these, a unified theoretical approach should be developed to elucidate the BOs in disordered graphene flakes.

In recent years, with the use of the mesoscopic thermodynamics of materials42,43, a bandgap thermodynamics has been proposed to illustrate the Eg(D) function of 3D III–V and II–VI nanosemiconductors with macroscopic parameters of materials based on the nearly-free-electron approach44,45,46. In light of it, strikingly, it reads,

In equation 1.1, Ec(D)3D is the energy required to break the atoms in the solid into isolated atomic species, which can be obtained experimentally by measuring the latent heat of sublimation at any temperature and extrapolating to zero Kelvin. Inspired by Lindemann′s criterion for solid melting and Mott′s expression for vibrational melting entropy47,48,49,50, utilizing the proportional relationship between Ec(D) and Tm(D), Ec(D)3D/Ec(∞) function for the 3D nanocrystals having free surfaces with broken bonds is given as44,45,

where is a surface physicochemical amount with , σ2 the mean square displacement of thermal vibration at the melting temperature and the subscripts sv and in-3D the surface and interior atoms. Using Lindemann′s criterion for solid melting and Mott′s expression49,50, it was derived that , where Svib(∞) denotes the atomic vibration entropy and R the gas constant. or denotes the critical diameter of crystals where almost all atoms or molecules are located on the surface and a crystalline structure is no longer stable, v/s means the volume/surface ratio, d the dimension of the crystal with d = 0 for nanoparticles, d = 1 for nanowires and d = 2 for thin films. In light of equations 1.1 and 1.2, the bandgap expansion for nanoscaled 3D III–V and II–VI semiconductors induced by the surface or interface effect has been illustrated successfully44,45,46. Stimulated by this success, one may conceive that the aforesaid issues on the BOs in the 2D graphene system may also be addressed in the thermodynamic way. Figure 1 illustrates the schematic plots of (A) GQDs, (B) GNRs, (C) C-NPGs and (E) S-NPGs, while a tripolar flake marked with A1A2B1B2C1C2 in (D) and a cross flake with +A1A2B1B2C1C2D1D2 in (F) denote the nanoscopic functional units of C-NPGs in (C) and S-NPGs in (E), respectively. In view of it, an atomic layer can be observed at the edge of GQDs, GNRs and tripolar or cross flakes, which should be responsible for the BOs in graphene. With this way, the distinct edge physicochemical nature of C atoms, such as coordination imperfection (CI) or chemical bonding, should play an important role in the BOs in graphene with relation to the dimension, geometric shape and the edge saturation. However, no attempts have been made on it so far.

Figure 1
figure 1

Illustration of edge atomic layers (marked in red) for (A) GQDs, (B) GNRs, (C) C-NPGs, (D) tripolar unit of (C), (E) S-NPGs and (F) cross unit of (E).

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In this paper, a general thermodynamic approach will be proposed to illustrate the BO behaviors in disordered graphene flakes. Based on this, significantly, it provides us a new physicochemical insight into the openings in graphene. With the DFT simulation, accordingly, the electronic structure of disordered GNRs will be investigated. Suggestions will be made how to tune Eg(D) of graphene for its application in nanoelectronics.

Results

BOs in disordered and naked GQDs, GNRs and NPGs

In light of equations 2.1 and 2.2 in methods, Eg(D) of naked graphene flakes can be worked out as long as and are known. Since and are decided by the CI of atoms at the surface of 3D nanocrystals or edge of 2D graphene flakes, can be correlated with via the coordination number CN of atoms at the surface Nsv and that at the edge Nedge. According to the inverse proportion relationship between σ2 and CN of atoms, based on the definitions of in Introduction and in Methods, one has ∝ (Nsv/Nin−3D)−1 and ∝ (Nedge/Nin−2D)−1, where Nin−3D and Nin−2D denote the CN in the bulk 3D and 2D crystals. It thus reads = (Nsv/Nin−3D)/(Nedge/Nin−2D). Since Nin−3D = 12 and Nsv = 9 for face-centered cubic crystals and Nin−3D = 8 and Nsv = 6 for body-centered cubic ones47, Nsv/Nin−3D = 3/4. Moreover, Nedge = 2 and Nin−2D = 3 for 2D graphene flakes. On the basis of these, can therefore be elucidated with . Utilizing the published expression cited in Introduction, one has = 9[2Svib(∞)/3R + 1]/8.

As regards of GQDs, GNRs and NPGs, it can be given with equation 2.4. Since l/s = 2(2 − d)/D, of GQDs and GNRs can be given as D/ or , where d = 0 for GQDs and d = 1 for GNRs. As regards NPGs, cannot be decided directly with that for GQDs or GNRs. This is because D here denotes not the size of functional tripolar or cross flake but that of holes in NPGs. Given that the effective size of tripolar or cross flakes is Deff, should be explored using . For the tripolar unit of C-NPGs in Figure 1(D), the edge-C atoms along the concave edges of A1A2, B1B2 and C1C2 own broken bonds, while those along straight lines of A2B1, B2C1 and C2A1 are still embedded in graphene without any CI, where not the latter but the former takes effect in the BOs of NPGs. Hence, the effective free edge l of A1A2B1B2C1C2 comes from three concave sections of A1A2, B1B2 and C1C2 only. Let the distance between the center of any two neighboring circles O1, O2 and O3 be C larger than D, l = A1A2 + B1B2 + C1C2 = 3A1A2 = πD/2 and . Thus, we have . Taking it into the above expression, one has . Concerning the cross unit of S-NPGs in Figure 1(F), in analogy, l/s = 4D/(C2 − D2) and . Inserting these expressions into equation 2.2 instead of , Eg(D) as the function of D can be explored for C- and S-NPGs.

With the above amounts, Figure 2 shows the solid Eg(D) curves as the function of D for (A) GQD-null or GNR-null, (B) C-NPG-null and (C) S-NPG-null plotted with equations 2.1 and 2.2 . For the comparison purpose, the reported reciprocal or exponential fitting expressions are also shown using dashed and/or dotted curves for GNR-null15,41,21,23 and NPG-null24. Available experimental results (symbols) were shown accordingly for verification. In light of Figure 2, the bandgaps are substantially opened for naked GQDs, GNRs and NPGs. As D shrinks, Eg(D) of GQDs and GNRs is expanded exponentially. Moreover, GQDs have larger Eg(D) values than GNRs do. In contrast, much similar D-dependence of Eg(D) has been observed for C-NPGs in (B) and S-NPGs in (C), where Eg(D) shrinks on decreasing D and Eg(D) becomes strong as C declines from 44 nm to 22 nm. Our predicted curves go through experimental results shown in Figure 2(A)–(B), supporting that our theoretical approach can be utilized to elucidate the BOs in graphene regarding the dimension and geometrical shape. As regards the upward or downward deviation of reported Eg(D) values of GNRs from our predictions, it should be attributed to the methods to measure Eg(D). As exampled by GNRs, in particular, its Eg(D) is usually derived from the Ion/Ioff ratios of GNR-based FETs3,15,16,51. Utilizing this means, there will be some possible errors caused by inaccurate D determination due to over-etching underneath the etch mask, uncertainties in the D measurements based on atomic form microscope or the assumption of negligible Schottky barrier for holes in ultra-narrow GNR-based FETs15,41.

Figure 2
figure 2

The solid Eg(D) curves as the function of D using equations 2.1 and 2.2 for (A) DO-GQD-null and DO-GNR-null, (B) DO-C-NPG-null and (C) DO-S-NPG-null at C = 22, 33 and 44 nm, where and are from Results.

For the comparison purpose, the reported reciprocal or exponential fitting expressions from Table 1 are also shown using dashed and/or dotted curves with Eg(D) = a/D15,21, Eg(D) = a/(D − b)23,41 and Eg(D) = a/(DebD)22,23 for GNR-null in (A) and Eg(D) = a/(33 − D)24 in (B). The symbols denote available measured results with 19 and 20 for GQDs and 41, 15, 21, 23, 3, 32, 33 and 34 for GNRs in (A) and ○24 for C-NPGs in (B). For calculation, h = 0.142 nm18,35. Svib(∞) is given with Svib(∞) ≈ Sm(∞)− R44,45,50, where Sm(∞) means the melting entropy using Sm(∞) = Hm(∞)/Tm(∞)44,50 with Hm(∞) and Tm(∞) being the melting enthalpy and melting point, respectively. Svib (∞) or Sm of graphene can be referenced to that from graphite with Hm(∞) = 132 kJ/mol and Tm(∞) = 5500 K58,59.

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BOs in disordered but saturated GQDs and GNRs

Using equations 2.1 and 2.2 , Eg(D) of saturated GQDs and GNRs can be explored if is known, where is the same as that of naked GQDs and GNRs. Since α2D is related to the edge atomic nature, may be resolved via . According to equation 2.3, it reads and . Divided by , it has . Since Ec ∝ σ−2, one gets , where and are the respective atomic cohesive energies of edge-C atoms of naked and saturated GQDs and GNRs. Provided that and have the same D-dependences, the ratio of should be D-independent and thus . With its help, can be given when and are known.

With the above derived expressions, Figure 3 shows the solid Eg(D) curves as the function of D in light of equations 2.1 and 2.2 for DO-GQD-R and DO-GNR-R with (A) R = H, (B) R = F and (C) R = OH. In comparison, the reported reciprocal fitting expression2 to depict the D-dependences of Eg(D) for GQD-H as aforementioned is also given using dashed curves in (A). Available experimental results (symbols) were also shown accordingly. As can be seen in Figure 3, when the edge saturation process is carried out, the BOs are realized, where Eg(D) is enhanced as D decreases. There exists a fair agreement between our curves and experiment results in Figure 3A, which validates our theoretical models.

Figure 3
figure 3

The solid Eg(D) curves as the function of D in light of equations 2.1 and 2.2 for DO-GQD-R and DO-GNR-R with R = H in (A), F in (B) and OH in (C), where and are from Results.

In comparison, the reported reciprocal fitting expression from Table 1 to depict the D-dependences of Eg(D) with Eg(D) = a/Db for DO-GQD-H2 is also given in (A) using dashed curve. The symbols denote the measured results with 2. For naked GQDs or GNRs, where EC = C is the bond energy of C = C. Upon the edge saturation with R, . EC=C and EC−R will be taken as the averaged values from ZZ and AC-GNRs. The respective EC = C values are taken as 4.68 and 5.81 eV for of ZZ and AC-GNRs56. To have , the respective EC = C and EC−R are 4.97 eV and 5.47 eV with R = H, 4.60 eV and 5.22 eV with R = F and 4.44 eV and 5.06 eV with R = OH, where the values of these parameters are given with the simulation method in Supplementary Information. See the caption of Figure 2 for other necessary parameters in calculation.

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Investigation on electronic structure of DO-GNRs

Because of its importance in electronics, in usual, ideal AC or ZZ-GNRs have been adopted to investigate the electronic structure of graphene-based nanostructures13,18,29,37,52. However, seldom attempts have been made to it for DO-GNRs. Hence, the electronic structure of DO-GNRs with the AC or ZZ skeleton along the periodic axis, referred to as DO-GNRs(AC) or DO-GNRs(ZZ), is investigated here. Figure 4 shows the band structures and the corresponding charge density isosurfaces of the lowest unoccupied molecular orbital (LUMO) and the highest occupied molecular orbital (HOMO) at Γ point of DO-GNRs(AC) and DO-GNRs(ZZ) in (a) R = H, (b) R = F and (c) R = OH or of DO-GNRs(ZZ) in (d) R = H, where their width is given at D ≈ 1.2 nm. In light of their band structures, the energy bands for spin-up and spin-down states are degenerate, while the openings are observed substantially for all DO-GNRs. The respective gaps are 0.44, 0.50 and 0.56 eV for DO-GNRs(AC) saturated by R = H, F and OH. Regarding DO-GNRs(ZZ) saturated with R = H, Eg(D) = 0.40 eV, close to that of DO-GNRs(AC) with R = H. In comparison, some differences in HOMO and LUMO can be observed between DO-GNRs and ideal GNRs. In ideal AC-GNRs, the charge densities are concentrated homogenously along the C = C bonds over the whole framework36. In Figure 4(a)–(c) for DO-GNRs(AC), the HOMO and LUMO orbitals distribute mainly along the C = C bonds. However, the densities are inhomogeneous over the whole region, much weak in the inner area instead. Moreover, some electronic states appear at single ZZ-structured edge-C atoms (labeled as 1, 2 and 3). In ideal ZZ-GNRs, on the other hand, the charge densities are centered at single C atoms especially for those along the two ZZ edges36. As for DO-GNRs(ZZ) in (d), however, the LUMO and HOMO charge densities are mainly centered at single C atoms, which seems strong in the interior area. Some charge densities along the C = C bonds have also been observed.

Figure 4
figure 4

Band structure, charge density isosurfaces of LUMO and HOMO at Γ point of DO-GNRs(AC) with (a) R = H, (b) R = F and (c) R = OH and of DO-GNRs(ZZ) with (d) R = H.

The EF is set to zero. Blue and yellow denote the positive and negative wave function contours, respectively and the value of the isosurfaces is 0.02 e/Å3. C, H, F and O atoms are denoted by gray, white, cyan and red balls, respectively.

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Discussion

According to our predictions, the BOs observed for disordered and naked GQDs, GNRs or NPGs in Figure 2 are contributed from the decrease in Ec of C atoms at the edge compared to that in the interior. In a perfect graphene, each C atom has a planar sp2 configuration with three C = C bonds. When graphene flakes are naked, each edge-C atom has one dangling bond and it thus consists of two C = C bonds only, leading to a decrease in Ec of edge-C atoms relative to those in the interior. Consequently, the equivalence between the A and B sublattices is broken, giving rise to the BOs in graphene flakes. As regards the saturated case in Figure 3, the dangling bonds of edge-C atoms are terminated, while each edge-C atom owns two C = C bonds and one C-R bond, different from the case in the interior. On account of it, the equivalence of A and B atoms is broken, resulting in the BOs in GQD-R or GNR-R. This infers that the BOs in saturated GQDs and GNRs should also be ascribed to the bonding change of edge-C atoms. With regard to the difference in Eg(D) among GQDs, GNRs and NPGs, it agrees with their D-dependences of l/s ratio, which rises for GQDs and GNRs but shrinks for NPGs on lowering D (Supplementary Fig. S1). Concerning the large Eg(D) of GQDs relative to GNRs, it should be attributed to the observation that GQDs have larger l/s ratios than GNRs do. It should be noted that the D-dependences of Eg(D) for GNR-null in Figure 2(A), NPG-null in Figure 2(B) and GNR-H in Figure 3(A) can be reflected on decreasing D with those reported fitting expressions. However, these expressions can only be adopted for elucidating the BO behaviors in those graphene flakes numerically, while neither of them can be adopted to address the issues on the essential roles played by the dimension effect, geometric shape and the edge saturation, ascribed to the presence of adjustable parameters a and/or b in them. In Figure 2(A), in particular, several fitting expressions have been reported to elucidate the D-dependence of Eg(D) for GNR-null. However, they vary considerably among different researches and it is difficult for one to see which one is suitable for illustrating the D-dependence of Eg(D) for naked GNRs.

According to the previous studies, a monotonous increase of Eg(D) has been reported for ideal AC and ZZ-GQDs36, which agrees with our predictions on Eg(D) of DO-GQDs. Concerning the difference in the size of Eg(D) between AC-GQDs and ZZ-GQDs36, it should be attributed to their different edge natures53. However, the opening behaviors of disordered GNRs differ to those ideal AC and ZZ-GNRs. The oscillating BO behaviors in AC-GNRs are induced by the full Fermi wavelength effect13. As for ZZ-GNRs, the reported metallicity is caused by the flat bands at the Fermi level EF from the highly localized states formed at the edges18, while the semiconducting state is stemmed from the staggered sublattice potential on the hexagonal lattice induced by the edge magnetization29. One may thus wonder whether the roles from the Fermi wavelength effect, the flat bands at EF or the interedge magnetic interaction observed in AC or ZZ-GNRs would affect the BOs in disordered GNRs or not. Among them, in fact, the roles from the Fermi wavelength effect and the flat bands at EF in DO-GNRs should be negligible. This is reasonable since almost no oscillating BO behaviors or metallic states have been reported for experimentally fabricated DO-GNRs, while the openings in them are observed substantially and rise monotonously on lowering D instead3,15,21,23,32,33,34. As to the role from the interedge magnetic interaction, it was investigated in the present work by performing spin-polarized and non spin-polarized calculations for DO-GNR-R shown in Figure 4(a)–(d). The obtained results show that almost no energy difference can be found for them between these two methods, indicating that all DO-GNRs give the nonmagnetic characteristics. In light of it, the contribution from the interedge magnetic interaction to the BOs in DO-GNRs should also be neglected. Obviously, the negligible roles from these factors should be attributed to the edge disorder. In Figure 4, in addition, the increasing order of Eg(D) for DO-GNR-R at D ≈ 1.2 nm from the simulation method with R = H, F and OH is the same as that from our predictions in Figure 3. However, their sizes are somewhat lower than those from our predictions, which might be related to the investigation approaches. As to the difference in the HOMO and LUMO charge density between DO-GNRs and ideal GNRs, it should also be relevant with the edge disorder in the former.

As mentioned above, in addition, the BOs contributed from the confinement role of D of holes have been reported for some typical ideal antidot superlattices38,39,40, which are in agreement with our predictions on DO-NPGs. In contrast, the metallic state observed in some other superlattices should be induced by the flat bands attributed to some factors, such as the instability of the incomplete benzenoid bonding pattern, large width of the ZZ chains or the presence of the ZZ rims38,39,40. As the metal-semiconductor transition in porous AC-GNRs with Na = 11 regarding the position effect, it is induced by the suppression of the localized states38. However, it seems that the roles from these factors can be neglected in the BOs in DO-NPGs, since they can be much suppressed by the internal edge disorder of holes. As evidence, no metallicity has been observed in experimentally fabricated DO-NPGs24,25,54. Besides it, a more recent investigation on the BOs in ideal AC or ZZ-GNRs perforated with disordered holes claims that the sensitivity of Eg(D) to the lattice symmetry is significantly suppressed in the presence of disorder55.

To see how Eg(D) of saturated DO-GQDs or GNRs will evolve relative to the naked case, Figure 5 shows ΔEg(D) as the function of D with ΔEg(D) = Eg(D)R − Eg(D)null. ΔEg(D) are negative for the edge saturation by H, F and OH, decreasing on the decline of D. This suggests that the BOs are weakened after the edge saturation. In comparison, the D-dependence of ΔEg(D) for GQDs is stronger than that for GNRs where the D-dependence of ΔEg(D) is the strongest for the H saturation and the weakest for the OH saturation. In view of it, further, one can see that Eg(D) of GQDs or GNRs can be lined in sequence with Eg(D)H < Eg(D)F < Eg(D)OH < Eg(D)null. Frankly, such a variation is essentially relevant to the difference in Ec of edge-C atoms for GQDs or GNRs with or without the edge saturation. Relative to naked graphene flakes, Ec of edge-C atoms is increased after the edge saturation56, depending on the radicals or groups. Accordingly, the edge physicochemical amount α2D is suppressed, leading to the change in Eg(D) in an decreasing order with R = null, OH, F and H as observed, Thanks to it, the opening suppression originated from the edge saturation can be understood associated with the edge physicochemical nature.

Figure 5
figure 5

ΔEg(D) as the function of D for disordered GQDs and GNRs between the cases with the saturated and naked edges, where ΔEg(D) = Eg(D)R − Eg(D)null with R = H, F and OH.

The corresponding values of , and for calculation are referenced to those adopted in Figures 2 and 3.

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In light of equations 2.1 and 2.2, moreover, Figure 6 gives a comparison of D necessary to achieve Eg(D) = 0.1 eV (denoted as D0.1eV) and Eg(D) = 0.4 eV (marked with D0.4eV) between naked and saturated GNRs, which are specified by Ec of edge-C atoms along the x-axis. From naked GNRs to saturated GNRs with R = OH, F and H, interestingly, D decreases as Ec rises, while the D0.4eV curve goes below the D0.1eV curve. Relative to the naked case, small D0.1eV and D0.4eV are strictly required for GNR-R, decreasing in the order of R = OH, F and H. Owing to these, one can see that, if a certain Eg(D) value is devised for a graphene flake, the necessary D will be affected by the edge physicochemical nature associated with the saturation.

Figure 6
figure 6

A plot of D necessary to achieve Eg(D) = 0.1 eV and Eg(D) = 0.4 eV for disordered GNRs with or without the edge saturation specified by Ec of edge-C atoms.

The values of , , , and necessary for the calculation are referenced to those adopted in Figures 2 and 3.

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In published papers, special interest has been attracted how to tune the gap of graphene. One important way for it is to control the edge states or nature of graphene flakes via the edge saturation with different chemical radicals or groups31. In ZZ-GNRs, the edge modifications can break the spin degeneracy. This will lead to a semiconducting-metal transition or a half-semiconducting state with the two spin channels having different band gaps, or a spin-polarized half-semiconducting state with the spins in the valence and conduction bands being oppositely polarized. About AC-GNRs, the edge saturation gives electronic states a few eV away from the Fermi level and does not significantly affect their bandgap. Based on it31, hence, it is reasonable that Eg(D) of disordered GQDs or GNRs can be modulated quantitatively by the edge saturation. In view of our predictions, since there exists an obvious discrepancy of Eg(D) between naked and saturated GQDs or GNRs, such a modulation can be realized by controlling the edge coverage upon the saturation, which can be performed through eliminating only a certain percentage of dangling bonds in a uniform way. Given that there exists a linear relationship between α2D and the edge coverage, one has for partially saturated GQDs and GNRs, where xR denotes the percentage of saturated edge-C atoms from 0 (the full edge-naked state) to 1 (the complete edge-saturated state). Inserting the above α2D expression into equation 2.2, Eg(D) as the function of xR at D = 1.6 nm was plotted in Figure 7 using equation 2.1 with R = H in (A), R = F in (B) and R = OH in (C). Eg(D) decreases continuously as xR rises. On increasing xR from 0 to 1, Eg(D) declines from 1.70 eV to 0.93 eV for GQDs and from 0.84 eV to 0.41 eV for GNRs in (A) with R = H, from 1.70 eV to 1.12 eV for GQDs and from 0.84 eV to 0.50 eV for GNRs in (B) with R = F and from 1.70 eV to 1.18 eV for GQDs and from 0.84 eV to 0.53 eV for GNRs in (C) with R = OH. This suggests that Eg(D) of disordered GQDs and GNRs can be adjusted subtly by changing xR, beneficial to the design and engineering of graphene-based nanoelectronic devices.

Figure 7
figure 7

Eg(D) as the function of xR at D = 1.6 nm for disordered GQDs and GNRs with (A) R = H, (B) R = F and (C) R = OH using equations 2.1 and 2.2, where the corresponding values of , , , and are referenced to those adopted in Figures 2 and 3.

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Methods

General consideration on the BOs in disordered graphene flakes

Although the dimensionality of 2D graphene differs from 3D semiconductors, there exists CI for edge-C atoms of graphene. By reason of such a similarity, the coherent energy of edge-C atoms will be decreased definitely. It is natural that the nearly-free-electron approach can also be adopted in predicting the BOs in graphene. In view of this, the decrease in Ec of edge-C atoms will play an essential role in influencing the crystalline field of graphene and the equivalence between the A and B sublattices will resultantly be broken, leading to the BOs in graphene.

According to the tight-binding approach57, electrons are viewed as occupying the standard orbitals of their constituent atoms and then hopping between atoms during conduction. Mathematically, this is represented as a hopping or transfer integral between neighboring atoms, which can be viewed as the physical principle that creates electron bands in crystalline materials, due to overlapping between atomic orbitals. In virtue of it, the BOs in graphene induced by the upward shift of conduction band and downward shift of valence band of the π bonds are associated to the hopping energy Eh between π orbitals with Eh = 2.96 eV57. In light of these, with reference to 3D nanosemiconductors44,45, Eg(D) of graphene flakes is given as,

To obtain Eg(D) with equation 2.1, Ec(D)2D/Ec(∞) function should be explored for 2D flakes, which can be referenced with the 3D case44,45,49,50, where free surfaces of 3D nanocrystals with broken bonds are substituted by free edges of 2D flakes. One thus has,

In equation 2.2, α2D is an edge physicochemical amount, while denotes the critical size of 2D materials. To explore Ec(D)2D, these two amounts should be developed by considering the roles played by the dimension, geometric shape and/or the edge saturation. Referenced by the 3D case44,45,49,50, the physicochemical amount α2D in equation 2.2 for 2D graphene flakes is decided by the edge nature relative to the interior case and one thus has,

where the subscripts edge and in-2D mean the edge and interior atoms of 2D graphene flakes. In principle, α2D of naked graphene flakes marked with is decided by the CI of C edge-atoms. Since the CI of edge-C atoms is eliminated upon the edge saturation, in contrast, a different α2D will be resulted from it for saturated graphene flakes denoted with . On the other hand, is one amount related to the dimensionality or the geometrical shape of graphene flakes, which can be principally given as,

where s/l is the area/edge ratio. Associated to it, the D-dependences of Eg(D) should be relevant to the l/s ratio of graphene flakes.

DFT approach to electronic structure of DO-GNRs

The electronic structure of DO-GNRs was investigated with the DFT approach. DO-GNRs are 1.2 nm wide or so with the AC or ZZ skeleton along the periodic axis, referred to as DO-GNRs(AC) or DO-GNRs(ZZ). Moreover, the two edges of both DO-GNRs(AC) and DO-GNRs(ZZ) are composed of the mixed AC or ZZ structures and kept symmetric. For DO-GNRs(AC), the supercell includes 12 zigzag C = C bonds in the periodic direction, as shown in Figure 4(a)–(c), where the edges are saturated by H, F and OH, respectively. For DO-GNRs(ZZ), the supercell has 12 armchair C = C bonds in the periodic direction [see Figure 4(d)], where the edge is saturated by H atoms.

Spin-polarized DFT calculations are performed using DMOL3 code with GGA-PBE as the exchange correlation function. All electron core treatment and double numeric plus polarization (DNP) basis set are adopted. The Brillouin zone is sampled by 5 × 1 × 1 k-points and the real space global cutoff radius is set to be 5.0 Å for the geometry and electronic calculations. In addition, a uniform vacuum of 15 Å perpendicular to the GNR is employed to ensure no interactions between neighboring cells. These bring out the convergence tolerance of energy of 1 × 10−5 Ha, maximum force of 0.002 Ha Å−1 and maximum displacement of 0.005 Å, respectively.